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Lobster eye optics: a theoretical and computational model of wide field of view X-ray imaging optics

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Lobster eye optics: a theoretical and computational model of wide field of view X-ray imaging optics

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Content LOBSTER EYE OPTICS: A THEORETICAL AND COMPUTATIONAL MODEL OF WIDE FIELD OF VIEW X-RAY IMAGING OPTICS by Samuel Barbour A DissertationPresented to the FACULTY OF THE USC VITERBI SCHOOL OF ENGINEERING UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ASTRONAUTICAL ENGINEERING) August 2015 Copyright 2015 Samuel Barbour for Elan Acknowledgements The experience pursuing my PhD has been incredibly rewarding. My growth as an indi- vidual, intellectually, scholastically and personally, has helped to reshape my perspective. Though a doctoral dissertation is the culmination of an individual’s scholastic career, its completiontakes the supportand guidance of a numberof people. Assuch, I wouldliketo properly thank everyone who helped me. First,IwanttothankProfessorErwinforalloftheguidancehehasgivenmethroughout my graduate career. I am grateful for the initial encouragement to transfer into the PhD program. I have learned an incredible amount covering a broad range of topics, due to the adviceonwhichcoursesI shouldpursue. I alsowouldnothavebeenbringmythesistoit’s fruition withouthis help, and advice. Asawhole,Iwanttothanktherestofmydissertationcommittee,ProfessorsGruntman, Tanguay and O’Brien. The dedication and energy put into my qualifying exam helped me find the direction for my thesis. Throughout the years, Professor Gruntman has also given me a lot of support and advice outside of the committee itself. And the work I did inProfessor Tanguay’s courses help lay the foundation for much of the developmentof the deconvolutionalgorithmsand the lobster-eye lens design trades. I would also like to thank Professor Wang, Professor Kunc, Professor Thangavelu, and Professor Hill who helped shape my experience here. Not only did Professor Wang par- ticipate in my qualifying exam, but he provided me with an opportunity to study in Japan. i And I am grateful for the trust Professor Thangavelu, and Professor Hill instilled in me as their TA. The support the rest of the department has given me is also very appreciated. The help Dell, Anna, Marrietta and Norma have given me over the years has ensured that I have remained on track to complete my degree. And John, Ning, Oulang, Pedro, Dayung, Will, Kevin, Doug, Brian and Gupta helped to create a very productive and supportive environment. In my personal life, the person who deservers the most gratitude is Sandee. She has stood by me throughout the entire process and I am not sure I would have been able to have come this far without her support. I am thankful for her patience and understanding in allowing me the time I have needed. Myparentshavealsobeenincrediblysupportive,alwaystakinganinterestandoffering advice when possible. Andmostof all, I am thankfulfor themencouragingme to returnto school, and for offering to help me in any way to ensure that I could complete my degree. My brothersandtheir familieswere helpfulthroughoutmyentire graduate career. Josh and Melissa were extremely supportive whenever I needed help whether it was in my per- sonal or professional life. Jon and Amy also lent support whenever possible, often acting as soundingboards. I am grateful for the help my in-laws Joo, Kyu and Steve were able to give me, always lending a hand with Elan, and willingto giveme the time I needed whenever I fell behind. I appreciate their understanding of the unconventional hours I worked that often required extra care. And finally, my aunts and uncles, and my friends have all shown a lot of support, and were always willing to help whenever I asked. Both my Aunt Nancy and Aunt Janet, who are professors, shared their expectations of their students and gave me advice on what my goals shouldbe. And my friends Jared, Nate and Jon gave me a ton of advice as I beganto learn programming. ii Table ofContents Acknowledgements . . ... .... ... .... ... .... .... ... .... . i ListofFigures. .... ... .... ... .... ... .... .... ... .... . vi ListofTables . .... ... .... ... .... ... .... .... ... .... . x Abstract . ... .... ... .... ... .... ... .... .... ... .... . xii Chapter 1: Introduction . . .... ... .... ... .... .... ... .... . 1 1.1 X-Ray Astronomy . .... ... .... ... .... .... ... .... . 1 1.2 X-rayFocusingOptics . . ... .... ... .... .... ... .... . 3 1.2.1 Reflectivity .... ... .... ... .... .... ... .... . 5 1.2.2 BasicGrazingIncidenceTelescopeDesign . .... ... .... . 7 1.2.3 LimitationsofTraditionalGrazingIncidenceOptics. ... .... . 10 1.2.4 Kirkpatrick-BaezOptics .... ... .... .... ... .... . 11 1.3 LobsterEyeOptics .... ... .... ... .... .... ... .... . 12 1.3.1 SchmidtOptics . . ... .... ... .... .... ... .... . 13 1.3.2 Lobster-EyeOptics ... .... ... .... .... ... .... . 13 1.3.3 BiologyofLobsterEyes .... ... .... .... ... .... . 15 Chapter 2: An Introduction to Lobster-Eye Optics . .... .... ... .... . 18 2.1 StateOfTheArt.. .... ... .... ... .... .... ... .... . 20 2.1.1 FabricationofLobster-EyeLenses . . .... .... ... .... . 24 2.1.2 Focal Surfaces . . . . . . .... ... .... .... ... .... . 29 Chapter3: Lobster-EyesInGreaterDetail .. ... .... .... ... .... . 31 3.1 ComputationalModel... ... .... ... .... .... ... .... . 32 3.1.1 ModificationstotheMainSimulation .... .... ... .... . 34 3.1.2 Supplemental Code . . . .... ... .... .... ... .... . 35 3.2 StudyingFocal Properties in One Dimension . .... .... ... .... . 35 3.3 FocalContributionsofSquareChannels ... .... .... ... .... . 36 3.3.1 The Connection Between δw and the Span of the Channels Contri- butiontotheFocalSpot. .... ... .... .... ... .... . 38 3.3.2 The Diffuse Background of Unfocused Photons . . . . . . .... . 39 iii 3.4 FocalLengthofSquareChannelLobster-EyeLenses .... ... .... . 39 3.5 FocusingEffiencyofSquare-ChannelLobster-EyeLenses . ... .... . 41 Chapter4: MeridionalLobster-EyeLenses . . ... .... .... ... .... . 43 4.1 GeometricDifferencesinMeridionalLenses. .... .... ... .... . 43 4.2 Refining δwforMeridionalChannels . ... .... .... ... .... . 45 4.3 FocalLengthofMeridionalLobster-EyeLenses... .... ... .... . 47 4.4 FocusingEfficencyofMeridionalLobster-EyeLenses.... ... .... . 49 Chapter5: ThePSFofLobster-EyeLenses . . ... .... .... ... .... . 51 5.1 ThePSFUnderIdealConditions .... ... .... .... ... .... . 52 5.1.1 TheTheoreticalOne-DimensionalFocalSpot .... ... .... . 52 5.1.2 Theoretical One DimensionalGeneralized Background . . .... . 55 5.1.3 TheEffectsofPixelSizeonthePSF . .... .... ... .... . 56 5.1.4 TheTwo-DimensionalPSF... ... .... .... ... .... . 59 5.2 TheEffectsofNon-IdealFocusingConditions.... .... ... .... . 61 5.3 OFF-AXISSOURCESANDABERRATION .... .... ... .... . 62 5.3.1 η ii forOff-AxisSources .... ... .... .... ... .... . 63 5.3.2 Effects of Planar Imaging Surfaces . . .... .... ... .... . 64 5.3.3 Redefining the Focal Spot for Non-Ideal f .. .... ... .... . 67 5.3.4 Non-SquareandRotatedChannels .. .... .... ... .... . 68 5.3.5 The Two DimensionalOff-Axis PSF. .... .... ... .... . 69 5.3.6 Offsettingthe Focal Surface . . . . . .... .... ... .... . 74 Chapter6: ImageProcessingforLobster-EyeLenses .... .... ... .... . 78 6.1 MetricsforImageComparison . .... ... .... .... ... .... . 79 6.1.1 ApproximatingSignal to Noise and η eff ... .... ... .... . 81 6.2 TheDeconvolutionOfImagesProducedbyLobster-EyeLenses . .... . 88 6.2.1 GeneralDeconvolutionAlgorithms.. .... .... ... .... . 88 6.2.2 ModifiedLucy-RichardsonDeconvolutionAlgorithm... .... . 90 6.3 Results of the Deconvolutionof Simulated Lobster-Eye Lens Images . . . . 93 6.3.1 DeconvolutionUnderIdealCircumstances.. .... ... .... . 94 6.3.2 Deconvolutionof a Single Source at Varying θ s ... ... .... . 100 6.3.3 Deconvolutionof Two Adjacent Sources . . . .... ... .... . 104 6.3.4 AdvantagesoftheIdealizedLucy-RichardsonAlgorithm . .... . 106 Chapter7: Lobster-EyeLensTradesandDesign .. .... .... ... .... . 108 7.1 TheInfluenceofSourceSelectiononMissionDesign .... ... .... . 108 7.1.1 The Impact of αonSpectralRange . .... .... ... .... . 109 7.2 DefiningLobster-EyeLensCharacteristics. . .... .... ... .... . 115 7.2.1 Setting the Lens Design Trade-Space . .... .... ... .... . 118 7.2.2 Non-ImagingTrades . . .... ... .... .... ... .... . 120 7.3 CaseStudy1:SterileNeutrinoSurvey . ... .... .... ... .... . 122 iv 7.3.1 FirstPass:BaselineSystem... ... .... .... ... .... . 122 7.3.2 SecondPass:ScalingDowntheLensDesign .... ... .... . 124 7.4 CaseStudy2:WideFieldGRBSurvey. ... .... .... ... .... . 128 7.4.1 FirstPass:BaselineSystem... ... .... .... ... .... . 128 7.4.2 Second Pass: Refinement and LimitingFactors . . . . . . .... . 131 Chapter8: ConcludingDiscussion... .... ... .... .... ... .... . 138 8.1 ImplicationsoftheFindings .. .... ... .... .... ... .... . 138 8.1.1 SummaryofFindings.. .... ... .... .... ... .... . 139 8.1.2 DeconvolutionandthePotentialforSub-GARResolution. .... . 140 8.1.3 PotentialforHardX-rayImaging... .... .... ... .... . 140 8.1.4 Flexibilityin Design . . .... ... .... .... ... .... . 141 8.1.5 LimitationsandConstraints... ... .... .... ... .... . 142 8.2 FutureResearch .. .... ... .... ... .... .... ... .... . 143 ReferenceList. .... ... .... ... .... ... .... .... ... .... . 146 v List of Figures Figure1.1: ThecriticalangleforgrazingincidenceX-rays.... ... .... . 4 Figure 1.2: The reflectivity of Ni for X-rays of 1, 3, 6 and 9 keV at 0< θ i <2 ◦ . 5 Figure 1.3: Underwood’sdepictionofWolterType1 and2X-ray telescopes[51] 8 Figure 1.4: Underwood’s depiction of Kirkpatrick-Baez optics [51] . .... . 12 Figure 1.5: Schmidt’s depiction of the foils arranged around cylinders with central axis along the ˆ x and ˆ z axis. The preferred axis of focus is along ˆ y[45]. .. ... .... ... .... .... ... .... . 14 Figure 1.6: Image formation and a structural overview of a lobster’s eye by Land[32] . .... ... .... ... .... .... ... .... . 16 Figure 1.7: Left: Image of the square facets of a lobster’s eye by Land[32], Right: Image of the square facets of a crayfish’s eye by Tokarski and Hafner[48] . . ... .... ... .... .... ... .... . 17 Figure 2.1: (a) Square channels arranged normal to a spherical surface; (b) A two dimensional representation of a lobster eye optic focusing 3 separate collimatedsources. . . . . . .... .... ... .... . 19 Figure 2.2: A two-dimensionalrepresentation of a lobster eye optic imaging a collimated source. . . . .... ... .... .... ... .... . 19 Figure 2.3: Anormalizedthree-dimensionalrepresentationofacollimatedsource imagedbyalobster-eyelensusingalogscale. ... ... .... . 20 Figure 2.4: All Sky Monitordeveloped by Priedhorsky, Peele, and Nugent [43] 22 Figure 2.5: The reflectivity of Si for X-rays of 1, 3, 6 and 9 keV at 0< θ i <2. . 27 Figure 2.6: Construction diagrams of the interlocking foils as proposed by Gertsenshteyn, Forrester et al. [18] . .... .... ... .... . 29 Figure 3.1: Flowchartoutliningtheprocessofthecomputersimulationwritten inC++. .. .... ... .... ... .... .... ... .... . 32 Figure 3.2: A two dimensional representation of 3 channels of a lobster eye lens imaginga collimated source. . . .... .... ... .... . 36 Figure 3.3: A channel singlyreflecting a collimatedsource that is at θ i =0.5α.. 37 Figure 3.4: A channel singlyreflecting a collimatedsource that is at θ i =1.5α.. 38 Figure 3.5: Channel at θ i = α and θ i =−α focusing a collimatedsource. . . . 40 Figure 3.6: Efficiency η ii vs. α cn oflobster-eyelenseswhilevarying(a) α and (b) photon energy. . . . .... ... .... .... ... .... . 42 vi Figure 4.1: a)Metalplatesalignedalongmeridiansofasphere;b)A2-dimensional representation of a meridional lobster eye lens focusing 3 separate collimated sources. . . . .... ... .... .... ... .... . 44 Figure 4.2: a)Non-squarechannelsandb)channeltaperinmeridionallobster- eyelenses. .... ... .... ... .... .... ... .... . 44 Figure 4.3: A two dimensional representation of 3 channels of a meridional lobster eye lens imaginga collimated source. .... ... .... . 46 Figure 4.4: A channel singlyreflecting a collimatedsource that is at θ i =1.5α.. 47 Figure 4.5: Channel walls at θ i =−w b / and θ i =w f / focusinga collimated source. .. .... ... .... ... .... .... ... .... . 48 Figure 4.6: Efficiency η ii vs. α cn of meridional lobster-eye lenses while vary- ing (a) α and (b) photon energy. . . . .... .... ... .... . 50 Figure 5.1: Position within the image plane vs. channel of fully focused pho- tons for a) square channel and b) meridional lobster-eye lenses . . . 53 Figure 5.2: The one-dimensional PSF of the focused photons for a) square channelandb)meridionallenses. .. .... .... ... .... . 54 Figure 5.3: Asimulated6keVsourceimagedbyanidealsquare-channellobster- eye lens with a 2’ GAR and a sensor set for 1 and 3 PPW .... . 58 Figure 5.4: Theenvelopeoffully-focusedphotonsasafunctionoffocallength fora)square-channelandb)meridionallens .... ... .... . 62 Figure 5.5: Focal length variation caused by planar focal surface. . . . .... . 63 Figure 5.6: Off-axis intensity profiles for a) a square channel and b) a merid- ional lens at θ 2w . . ... .... ... .... .... ... .... . 66 Figure 5.7: Approximated off-axis intensity profiles for f set to match a) a square channel and b) a meridionallens near θ 2w . . ... .... . 69 Figure 5.8: The cross correlation of focal spots at varying θ s for a 1’ square channellobstereyelens .... ... .... .... ... .... . 70 Figure 5.9: ThecrosscorrelationofPSF eo atvarying θ s fora1’squarechannel lobstereyelens.. ... .... ... .... .... ... .... . 71 Figure 5.10: ThecrosscorrelationofPSF ⊥ oe atvarying θ s fora1’squarechannel lobstereyelens.. ... .... ... .... .... ... .... . 72 Figure 5.11: The cross correlation of the background at varying θ s for a 1’ squarechannellobstereyelens ... .... .... ... .... . 73 Figure 5.12: ThecrosscorrelationofPSF oe atvarying θ s fora1’squarechannel lobster eye lens with the photon count reduced to ∼8.2×10 4 .. . 74 Figure 5.13: ThecrosscorrelationofPSF ⊥ oe atvarying θ s fora1’squarechannel lobster eye lens with the photon count reduced to ∼8.2×10 4 .. . 74 Figure 5.14: The cross correlation of the background at varying θ s for a 1’ square channel lobster eye lens with the photon count reduced to ∼8.2×10 4 .... ... .... ... .... .... ... .... . 75 Figure 5.15: ThecrosscorrelationoftheEntirePSFatvarying θ s fora1’square channellobstereyelenswiththephotoncountreducedto∼8.2×10 4 75 vii Figure 5.16: The cross correlation of the focal spotat varying θ s for a 1’ square channellobstereyelenswiththefocalplaneoffset . ... .... . 76 Figure 6.1: Spread of fs and bgalongthefocalplane . . .... ... .... . 81 Figure6.2: FlowchartforstandardLucy-Richardsondeconvolution . .... . 90 Figure 6.3: FlowchartforcomponentseparatedspatiallyvariantLucy-Richardson deconvolutionforlobstereyelenses . .... .... ... .... . 91 Figure 6.4: ThemasksforthespatiallyvariantLucy-Richardsondeconvolution algorithm with an image of an off-axis source spanning multiple masks. .. .... ... .... ... .... .... ... .... . 92 Figure 6.5: Idealized1’GARsquare-channellobstereyelensa)imagingacol- limaged source with PPW = 1 and the deconvolved image using Lucy-Richardsonalgorithmsforab)simulatedandc)analyticalPSF 95 Figure 6.6: Idealized1’GARsquare-channellobstereyelensa)imagingacol- limaged source with PPW = 3 and the deconvolved image using Lucy-Richardsonalgorithmsforab)simulatedandc)analyticalPSF 96 Figure 6.7: Idealized 0.5’ GAR square-channel lobster eye lens a) imaging a collimagedsourcewithPPW =1andthedeconvolvedimageusing Lucy-Richardsonalgorithmsforab)simulatedandc)analyticalPSF 98 Figure 6.8: Idealized 0.5’ GAR square-channel lobster eye lens a) imaging a collimagedsourcewithPPW =3andthedeconvolvedimageusing Lucy-Richardsonalgorithmsforab)simulatedandc)analyticalPSF 99 Figure 6.9: FWHM vs. θ s for an imaged by a lens with GAR=1 and PPW =3. 102 Figure 6.10: Deviationofresolvedpeaklocationfrom θ s vs. θ s for acollimated source imaged by a lobster-eye lens with GAR=1 and PPW =3.. 102 Figure 6.11: The intensity profile along a focal arm of a 0.5’ GAR square- channel lobster-eye lens image of a source at θ s =0.75θ 2w decon- volved using the Lucy-Richardson algorithm with the simulatedPSF103 Figure 6.12: The mid-pointvaluesfromthe deconvolvedimageof to2 adjacent sources for PPW =1to4 ... ... .... .... ... .... . 105 Figure 6.13: Deconvolvedimagesof2sourcesseparatedbya)32’andb)33’for a 1’GAR square-channel lobster eye lens with Ni reflecting surfaces.106 Figure 7.1: η ii forhighenergyphotonsimagedbyaNilobster-eyelensdesigned tofocus6keVsources . .... ... .... .... ... .... . 110 Figure 7.2: pre-andpostprocessedhighenergyimagesfora1’GARlenswith PPW =1 . .... ... .... ... .... .... ... .... . 111 Figure 7.3: The initial range for which a lobster-eye lens will focus more effi- ciently .. .... ... .... ... .... .... ... .... . 112 Figure 7.4: a) θ c and b) ∆ θ c vs. photon energy for Ni, Au, Si and GaAs . . . . 113 Figure 7.5: Spectral Range vs. αforNi,Au,andSi ... .... ... .... . 114 Figure 7.6: R v θ i forNi,Au,andSi .... ... .... .... ... .... . 115 viii Figure 7.7: A visualization of how the different lens characteristics are inter- dependent throughR. . .... ... .... .... ... .... . 119 Figure 7.8: Spectral Range vs. αforGaAs ... .... .... ... .... . 125 Figure 7.9: The(a)unprocessedand(b)deconvolvedimagesof3sourcesbased on the descriptionin reference [9] by the lens defined in Table 7.3 . 127 Figure 7.10: Spectral Range vs. αforPt . . ... .... .... ... .... . 129 Figure 7.11: The images of a source modeled after a GRB ((a) unprocessed & (b) deconvolved) and its afterglow ((c) unprocessed & (d) decon- volved) based on the description in references [17, 49] by the lens defined in Table 7.6... .... ... .... .... ... .... . 137 ix List of Tables Table 1.1: X-ray emittingastronomical sources[6, 27, 53].... ... .... . 1 Table 1.2: X-ray imaging telescopes [1, 3, 17, 34, 23, 50].... ... .... . 2 Table 5.1: θ 2w at different GAR for Square-Channel and Meridional Lenses with the same α and R . .... ... .... .... ... .... . 66 Table 5.2: δ f inv in terms of for 0.5’, 1’, and 2’ GAR lenses with PPW = 4 and5. ... .... ... .... ... .... .... ... .... . 76 Table 6.1: PSNRresultsfora)anidealizedNilobster-eyelensimaginga6keV source at varying GAR using TER model and b) idealized Ni, Au and Si lobster-eye lenses imaging a 6keV source with R/ ≈ 47.1 usingbothTERandfullreflectivitymodels . .... ... .... . 84 Table 6.2: η eff results for a) an idealized Ni lobster-eye lens imaging a 6keV source at varying GAR using TER model and b) idealized Ni, Au and Si lobster-eye lenses imaging a 6keV source with R/ ≈ 47.1 usingbothTERandfullreflectivitymodels . .... ... .... . 86 Table 7.1: Lobster-eye lens paramters for a survey imaging the possible ster- ile neutrino decay spectral line at ∼3.57keV with a photon flux on the order of 10 −5 photons/(cm 2 s). The design goals are scaled for collecting 10 3 photons . . .... ... .... .... ... .... . 123 Table 7.2: Lobster-eye lens paramters for a survey imaging the possible ster- ile neutrino decay spectral line at ∼3.57keV with a photon flux on the order of 10 −5 photons/(cm 2 s). The design goals are scaled for collecting 500 photons usingGaAs as the reflecting surface .... . 126 Table 7.3: Potentiallobster-eyelensdimensionsdesignedusingGaAstoasources with a flux ∼10 −5 photons/(cm 2 s) .. .... .... ... .... . 126 Table 7.4: Meridional lobster-eye lens paramters for a GRB study where hard X-rays will be detected using the lobster-eye lens as a coded mask andX-raysinthe10-20keVrangewillbeimaged. Assumes α =15 and t = 40µm using a sensor with the same pixel dimensions as NuSTAR.. . .... ... .... ... .... .... ... .... . 130 x Table 7.5: Meridional lobster-eye lens paramters for a GRB study where hard X-rays will be detected using the lobster-eye lens as a coded mask and X-rays in the 7-14keV range will be imaged using an offset focal plane. Assumes α = 20 and t = 80µm using a sensor with the same pixel dimensionsas NuSTAR. .... .... ... .... . 132 Table 7.6: Potential lobster-eye lens dimensions designed using Pt for GRB imaging and assuming pixelsizes would be reduced to 0.2mm . . . 134 xi Abstract The lobster-eye telescope with square-cross-section channels has been suggested as a pos- siblecandidateforawide-field-of-viewX-rayall-skymonitor. However,duetothedifficult construction, as of yet, no lobster-eye lenses have been deployed. Though prototypes have been constructed, they typically utilizing non-metal reflecting surfaces and have channels that do not meet required width to length ratio necessary for imaging X-rays efficiently. Because of the limitations for construction, the majority of study has been dedicated to fabrication techniques. Additionally, alternative designs with non-square channels have beenproposedtoenableconstructionwithmetals. Onesuchdesign,meridionallobster-eye lenses, is based on aligninginterlocking reflective plates along perpendicular meridians. This work is a computational and theoretical study of lobster-eye lenses with both square-cross-sectionchannelsandnon-squarechannelsarisingfromthemeridionaldesign. The focusing efficiency of each design for sources at varying angles of incidence is com- paredwithpreviouswork. Furthermore,thefocalpropertiesareredefinedtoaccountforthe curvature and physicalstructure of the lenses themselves. In the process,the differences in focallengthandfocalpropertiesforbothsquare-channelandmeridionallobster-eyelenses are identified. From the focal properties, the point spread function is defined for both square-channel and meridional lobster-eye lenses. In one-dimension, the intensity profile of focused pho- tons are derived. For unfocused photons, an approximation for a generalized theoretical xii intensity profile is designed. The two dimensional point spread function was then created from the convolutionof the one-dimensionalintensityprofiles in each dimension. Aberration due to non-ideal imaging surfaces is detailed. The effects of the reduction in focal lengths due to planar imaging surfaces is defined. The point spread function is generalized to account for non-ideal focal lengths. And the limitations to field of view caused by the planar imaging surface are identified. The theoretical point spread function is compared with simulatedcollimated sources. Deconvolution algorithms are designed for idealized and variant point spread func- tions. Weiner filter and Lucy-Richardson algorithms are modified for lobster-eye lenses. The deconvolvedimages of simulatedcollimated sources are compared for on and off axis idealizedlobster-eyelenses. Thepotentialfor bothsinglesourceandtwosourceresolution is compared for varying pixel densities. The full width half max, accuracy of source posi- tion, peak signal to noise ratio, and percentage of overall image intensity contained within the peak signal are used to judge the overall image quality. Design trades are described based on the interconnection of field of view, resolution, aspect ratio and effective area. From the design trades, two case studies are identified for potential X-ray missions utilizing lobster-eye lenses. Several leaps in technology are necessary to scale the potential lenses to dimensions that are similar to prototype and pre- vious proposals. However, assuming the technological development, lobster-eye lens are designed that would be able to accomplishthe missiongoals. xiii CHAPTER 1: INTRODUCTION 1.1 X-RayAstronomy The X-ray sky is vibrant and variable, with astronomical sources such as black holes and stellar coronae emittingwithin the X-ray spectrum [27, 53]. Transient X-ray sources, such as supernovae, active galactic nuclei, and gamma ray bursts are thought to be fairly abun- dant in the X-ray sky [27]. Table 1.1 includes several prominent X-ray sources as well as their key spectral lines: Table 1.1: X-ray emittingastronomicalsources[6, 27, 53] Source X-rayFeatures (keV) Active Galactic Nuclei 0.1-200 keV (Fe K 6.4 keV) Black Holes Fe like He K-α 6.7 keV Gamma Ray Bursts afterglows 0.5-10 keV Type II Supernova Emissionlines 0.1-10 keV X-ray Binaries Keyemission lines between 5-10 keV Solar & Stellar Corona 0.12-10 keV Flares Highlyvariable up to 100 keV CME Soft X-rays (0.12-12 keV) Thoughnon-imagingdevices(whichmeasuresthephotonflux)havebeenusedtostudy the X-ray output of celestial objects, imaging optics (which focuses the X-rays) have the advantage of visually displaying the structures of the X-ray source, and enabling the pos- sibility of spectroscopic imaging [51]. However, due to the high energy, focusing X-rays poses a challenge. Refractive lenses can not be used, and potential reflective materials 1 are limitated in the angles of incidence (θ i ) at which they can reflect X-rays[51]. Deep- field X-ray telescopes have been deployed since the Einstein telescope launched in 1978 [51](Table 1.2 provides a list of X-ray telescopes in orbit). Since then, all X-ray imaging missions utilize telescopes that are designed based on conic sections, similar to traditional reflecting telescopes. Consequently,all current imagingtelescopeshave very narrow fields of view (FOV), typically limitedto ≈1/2 ◦ [21](Table 1.2). Table 1.2: X-ray imaging telescopes [1, 3, 17, 34, 23, 50] Telescope Launch Year FOV Resolution Spectral Range Einstein 1978 75’ 60” 0.5-4keV EXOSAT 1983 120’ 18” 0.05-2keV ROSAT 1990 38’ 2” 0.1-2.5keV YOHKOH 1991 42’ 3” 0.25-4keV ASCA 1993 24’ 180” 0.4-10keV BeppoSAX 1996 37’ 582” 0.1-10keV Chandra 1999 30’ 0.5” 0.8-10keV XMM Newton 1999 30’ 5” 0.1-12keV Swift 2004 23.6’ 2.4” 0.14-10keV Suzaku 2005 18’ 120” 0.3-12keV Hinode 2006 35’ 2” 0.21-2.1keV NuSTAR 2012 6-10’ 18” 3-78.4keV Even within the constrained FOV, X-ray telescopes have played an important role in the study of astronomy, for example, X-ray telescopes helped determine the mechanisms governing the center of our galaxy[3], as well as shaped our understanding of the solar corona and flares [50]. Furthermore, many extra-galactic elements have ions with spectral linesemittedinwithintheX-rayregime[9]. Inearly2014,thediscoveryofanunidentified spectral line has led to speculation of the identification of sterile nuetrinos, a possible dark 2 matter candidate [7, 9]. Though more data is necessary, if the unidentified line is truly the result of dark matter, one of the current mysteries of both astronomy and fundimental physicscould beginto be solved[7]. ExpandingtheFOVwouldnotonlyenablethestudyofextendedsourcesanddiscovery of new sources, but would also aid in the study of transient sources, due to the extended observationtimeassociatedwithawiderFOV[43]. Furthermore,awideFOVcoupledwith increasedsensitivityisnecessaryforthedetectionandstudyofothervariable,transient,and weaker X-raysourcessuchasGammaRay Bursts(GRB, whichare accompaniedbyX-ray transients), X-ray flashes, nearby stellar coronae, type II supernovae (which have X-ray flashes at onset), active galactic nuclei (AGN), and other active galactic objects, most of which have emission spectrum in the 1-10 keV range [28, 38]. With greater sensitivity, not only will fainter and farther sources be studied, but the brighter sources would also be detected with greater accuracy [43]. The range of times scales of interest for transient and variable sources occure is broad. Pulsars, GRB and X-ray flashes have times scales in the milliseconds to minutes, while AGN’s can last over the period of months to years [20]. In order to capture transients, the FOV needs to be broad enough to ensure a high probability that events will be captured. 1.2 X-rayFocusingOptics Because X-rays do not refract or reflect at near normal incidence, traditional optical sys- tems such as lenses or spherical mirrors are ineffective. However,if the angle of incidence betweentheX-rayandthereflectingsurface islargeenough,anX-raywillreflect similarly to total internal reflection. Because the limiting angle at which light will begin to reflect off of the surface (the external critical angle, θ c ) is nearly 90 degrees, θ c is measured from the surface instead of the surface normal 1.1 [51]. As with total internal reflection, θ c is dependent on the reflective properties of the material and the wavelength of the incident 3 Figure 1.1: The critical angle for grazing incidence X-rays. light;higherenergy X-rayswillhaveasmaller θ c . MetalsmakeforexcellentX-rayreflect- ing materials when used at grazing angles because their indices of refraction are less than 1intheX-rayregimeand θ c will approach a degree. An X-ray focusing optical device can be designed around this principle [51]. Such devices are referred to as grazing inci- dence optics because the light reflecting at angles smaller than θ c appear to graze across the materials surface. θ c can be defined from Snell’s law: n i sin(θ i )= n t sin(θ t ) (where i=incident and t=transmitted, and n is the index of refraction) in a similar fasion to total internal reflec- tion. Because the reflecting surface is in contact with air or a vaccuum, n i =1and n t =n material = 1− δ,where δ refers to the absorption properties of the material. Though the index of refraction is actually a complex number, due to approximation, only the real part of the index of refraction is being considered (n and δ are defined in grater detail in Section 1.2.1). Also, as defined by Figure 1.1, θ c =90− θ i , therefore cos(θ c ) can be sub- stitutedforsin(θ i ). Aswithtotalinternalreflection,totalexternalreflectionoccurswhen θ t equals ninety degrees, and sin(θ t ) reduces to one. After substitution,Snell’s law becomes: cos(θ c )=1− δ 4 Because θ c is much less than one, the paraxial assumption can be used with the Taylor Series approximation of cosine: cos(θ)=1− θ 2 2! + θ 4 4! ... . Ignoring the higher order terms (again because θ c <1), Snell’s law becomes 1− θ 2 c 2 =1− δ and θ c can be defined as: θ c = √ 2δ (1.1) 1.2.1 Reflectivity In the total external reflectivity model, the reflectivity of the surface is approximated as 1 for θ i < θ c (perfect specular reflector), and to 0 for θ i > θ c (perfect absorber). In actuality, the reflectivity of a material changes as a function of the angle of incidence, as well as of the energy of the incident X-ray. Figure 1.2 displays the reflectivity of nickel at angles between 0 and 2 ◦ for X-rays at 1, 3, 6 and 9 keV. For higher energy X-rays (> 6keV), the external reflectivity model closely approximates the actual reflectivity. However, at lower photon energy, the total external reflectivity model is nolonger valid. Figure 1.2: The reflectivity of Ni for X-rays of 1, 3, 6 and 9 keV at 0< θ i <2 ◦ . 5 In order to properly determine the reflectivity,especially for metals, the complexindex of refraction (Equation 1.2) must be incorporated into Fresnel’s equation as summarized by Underwood [51]. n=1− δ −iβ (1.2) Where β is defined by: β = µλ 4π . (1.3) µ in Equation 1.3 is the linear absorption coefficient, which is heavily dependent on the energy of incident X-ray. Though the linear absorption coefficient tends to decrease with increasingenergy, there are distinctspikes inabsorptioncorrespondingtoenergies of elec- trons transitioning between energy states in the atoms of the reflecting material. In the X-ray regime for metals such as nickel and gold (typical X-ray reflecting materials), the absorption spikesare particularly noticeable at the L1 shell [25]. δ in Equation 1.2 is defined by: δ = n electrondensity e 2 λ 2 2πmc 2 = r 0 λ 2 N e 2π (1.4) where r 0 is the classic electron radius and Equation 1.4 is used to quantify θ c in terms of wavelength and material properties [51]. With some manipulation, and substituting X = θ θ c and Y = β δ , Fresnel’s equation becomes [51]: R= √ 2X− (X 2 −1) 2 +Y 2 +(X 2 −1) 2 + (X 2 −1) 2 +Y 2 − X 2 −1 √ 2X+ (X 2 −1) 2 +Y 2 +(X 2 −1) 2 + (X 2 −1) 2 +Y 2 −(X 2 −1) (1.5) Therefore, knowingthe material properties of the reflecting surface, the energy of the inci- dent X-ray, and the angle of incidence, the reflectivity of the material can be determined 6 allowing for greater accuracy than assuming the total external reflection model. However, at higher energy X-rays (for example, energies grater than 6keV for Ni [12]), the reflectiv- ity of most materials mimics the first order approximation of the total external reflectivity model. 1.2.2 BasicGrazing Incidence TelescopeDesign Traditional reflecting telescopes (such as Newtonian Telescopes) use parabolic mirrors, which, as shown through geometry, focus collimated light to a single focal point [24]. Similarly,mostcurrentgrazingincidenceopticsuseparabolicmirrorelements. Butinstead of the mirror elements being constructed from portions of the parabola near the vertex, the mirror elements are constructed from thin rings far enough from the vertex such that the angle between collimated incident light parallel to the central axis of the mirror and the surface of the mirror is within θ c . To shorten the overall focal length, a second set of mirrors, thistimehyperbolic,is used to further redirect the reflected X-rays [51], as shown in Figure 1.3. Wolter further refined the telescope’s design by connecting the hyperbolic and parabolic sections according to geometric constraints [51]. To increase the collection area, andthusincreasethecollectedX-rayflux, concentric,ornested,pairsofparaboloidal and hyperboloidal mirrors are used [51]. The FOV is still ultimately limited by geometry to be within θ c . Further variations in individual telescope designs are based on the X- ray source itself. J.H. Underwood separated X-ray astronomy into two categories, solar and non-solar,due tothevastlydifferentrequirmentsnecessary for imagingdistantobjects with low photon fluxes as opposed to the sun[51]. As early as the 1970’s with Skylab, the sun has been imaged in the X-ray spectrum. CapturingthesunrequiresaFOV largeenoughtocovertheentiresolardiskandtheability tohandleextremevariationsinluminositybetweencalmsolar conditionsandsolarstorms. However, ensuring full coverage of the sun limits the available spectrum, in some cases to energies less than 4keV [50]. Non-imaging X-ray optics are typically used for higher 7 Figure 1.3: Underwood’s depiction of Wolter Type 1 and 2 X-ray telescopes [51] energy X-rays [31], though in 2014, the deep-field high-energy X-ray telescope NuSTAR was used to image the sun [14]. Viewing the sun within the X-ray spectrum adds depth to the study of flares, coronal physics and the evaluation of the morphology of the sun’s magnetic fields [50]. The Hinode mission, designed specifically for solar imaging, uti- lizes a variation of Wolter’s design (Wolter-Schwarzchild) in order to obtain a full view of the solar disk. Like the Wolter’s design, the Wolter-Schwarzchild design uses parabolic- hyperbolic mirror elements, but has stricter constraints placed on the relationship between the parabolic and hyperbolic elements. [34]. In order to image the Galactic Center, satellites have utilized X-ray imaging optics as early as the Einsteinmissionin the late 1970’s. Even with its limitations,the Einsteinmis- sionrevealedthatthe majorityof the emissioninthe center of thegalaxywas concentrated in a bright elliptical region [3]. ASCA, in the mid 1990’s, the first hard X-ray telescope (imaging 10 keV X-rays, with a modest spatial resolution of 3 arc minutes) was able to detect thermal emissions with hydrogen-like and helium-like Kα emission lines within a 8 degreeoftheGalacticCenter[3]. DuetotheopticallythinlineofsighttotheGalacticCen- ter in the hard X-ray regime, there have also been observations made using non-imaging instruments[3]. The early successes imaging the Galactic Center were followed by NASA’s Chandra X-ray telescope,whichrepresentsthe stateoftheart ofX-ray grazingincidence telescopes [53]. With a resolution of less than an arc second, Chandra’s four sets of nested parabolic mirrors (followed by nested hyperbolic mirrors) have a field of view that extends to nearly halfadegree[53]. Chandra also contained two sets of spectroscopic gratings designed for differentbandsofincidentX-rays,rangingfromaslowas0.08keVtoashighas8keV.One oftheprimarygoalsofChandrawastohelpdeterminewhetherasupermassiveblackhole was located at the Galactic Center by specifically targeting helium like iron Kα emission (typically at 6.7 keV), and the mission was fairly successful [3]. But despite the successes of Chandra, it still has its limitations, the most glaring of which are signal-to-noise issues and photonstatistics. To improvephoton statics longer viewing timesare used [53]. More recently, the Swift space telescope was also designed to look for helium-like iron Kα using similar filters to those of the Chandra telescope, but with lower resolution and a smaller FOV [10]. However, unlike Chandra, which uses separate mirror elements for the hyperbolic and parabolic sections, the Swift telescope uses concentric Wolter type I mirrors. SwiftalsowasnotdesignedtoimagetheGalacticCenter,butinsteadtheafterglow of Gamma Ray Bursts (GRB)[10]. In order to capture the GRBs and their afterglow, Swift was designed to autonomouslyand rapidly slewin respons to detection of a new discovery with a high energy detecting instrument[17]. In2012theNuclearSpectroscopicTelescopeArray(NuSTAR)launchedwiththeability to focus X-rays between 3 and 79keV, representing an enormous leap in the spectral range of imaging X-ray telescopes [23]. Two nearly identical Wolter-I optics are used with an approximately 10m focal length. Each optic consists of 133 nested shells comprised of glass coated with “depth-graded multilayer structures” designed to increase θ c for higher 9 energy X-rays[23]. Two nearly identical sensors are attached to the optics by a deployed composite mast and an onboard metrology system is used to asses the alignment of the mirror-sensor systems [23]. The construction technique severly impacted the point spread function of the optical system resulting in an angular resolution of 18” despite having a similar focal length to Chandra [23]. And though the spectral range spans 76keV, the effectiveFOViswavelengthdependent,reaching10’at10keVwhilelimitedto6’at68keV [23]. Published in 2014, data from 73 galaxy clusters imaged by XMM-Newton was used in order to search for previously unidentified spectral lines [9]. Relying heavily on post- processingandmodelingofthespectrallinesofknownelements,thepotentialweaksignal was discovered [7, 9]. The spectral line is consistent with a sterile neutrino, a theoretical prticled proposed as a candidate for dark matter [9]. Though the spectral line was also presentintheChandradataofthePerseuscluster,moredataisnecessarytofullyunderstand thenatureofthisdiscovery[9]. However, because the signal was faint and tests the limits of the sensitify of the instrumentation, the discovery comes with a significant amount of uncertainty [9]. 1.2.3 LimitationsofTraditional Grazing Incidence Optics Due to geometry, the FOV of traditional grazing incidence optics are heavily limited. For collimated light entering parallel to the axis of rotation of a telescope to be focused, the angle of incidence between the light and the mirror must be within θ c . This creates a limit on the physical structure of the telescope because any light originating from angles larger than θ c will not be collected. Ultimately,the telescope’s field of view is, in the most idealized case, limited by the critical angle of the highest energy X-rays being imaged. But, even in this simplification, the actual field of view of the telescope is substantially smaller because of the blur from incident light that is not aligned with the axis of rotation 10 (off-axis). To accommodate off-axis incident light, imaging sensors are often offset from the focal point,although this results in a loss in resolution. The dependence of reflectivity on the energy of incident X-rays is also limiting factor on FOV. As described earlier, θ c , which depends on the photon energy, sets the limitations on the geometry of traditional focusing X-ray optics. The end result is a compromise between imaginghigher energy sources or a broader field of view. For deep field missions, where high energy X-rays are more critical, a narrow field of view may be acceptable. For instance, one of the key discoveries early in Chandra’s mission was a Kα emission line around 6.7 keV, present at Sagittarius A (Sgr A, a structure at the center of the Milky Way galaxy), which was a key clue in helping to determine whether or not a black hole exists there [3]. On the other hand, solar missions require a complete view of the coronal disk, and to compensate for restrictions of the bandwidth, separate instruments are sometimes needed,however,non-imaginginstrumentsareoftenutilizedforhigherenergyX-rays[50]. Though, in the X-ray regime, FOV is independent of focal length, for a given imaging sensor, resolution is proportional to the focal length of the telescope (where the term focal length has been used to defined by the distance from the back of the mirror array to the focal plane as apposed to the geometric properties of the mirrors’ curvature). Longer focal lengths have higher resolutions for identical imaging devices. For a telescope to have a high resolution, larger and larger telescopes must be constructed. For instance, in order to obtain the 1 arc second resolution, Chandra is over 10 meters in length, and was the heaviest and largest payload deployed from a space shuttle at the time of launch [3, 53]. 1.2.4 Kirkpatrick-Baez Optics Early variations of imaging X-ray telescopes included an adaptation of the Kirkpatrick- Baez X-ray microscope [51]. Instead of nesting the conic sections, an assembly of plates curved on one side (typically parobolic) is used to direct the incident X-rays to an initial linear focus [2], followed by a second assembly, positioned perpendicularly to the first, 11 in order to overcome the astigmatism of a single mirror[51] (Figure 1.4). The parabolic geometry of the curved side focuses the incident X-rays in the same fashion as the nested parabolic mirrors, however,the parabola only focuses alongthe singular dimensionof cur- vature. But because this design initially depends upon conic geometry, it retains several of the limitations that hamper the more traditional grazing incidence optic designs. For instance, the FOV is stilllimitedby the critical angle of incident X-rays, and there is stilla large deteriorationof spatial resolutionfor off-axissources [2]. Furthermore, the construc- tion of a Kirkpatric-Baez telescope would require extremely precise alignment in order to effectivelyfocus incident X-rays [51]. Figure 1.4: Underwood’s depiction of Kirkpatrick-Baez optics [51] 1.3 LobsterEyeOptics To alleviate the geometric limitation of grazing incidence optics, designs have been pro- posed that utilize an agregation of smaller grazing incidence elements that combine to create an image. In doing so, the θ c is decoupled from directly limiting the FOV. Schmidt proposed an arrangement of parallel mirrors, which reflected a point source to a line focus from grazing angels of incidence (shown in Figure 1.5). When arranged cylindrically, this geometry focuses collimated light to a linear focus. For two dimensional focus, a sec- ond array of reflecting plates can be implemented directly behind the first [45]. Alterna- tively, Angel devised an optical element patterned after the eyes of macruran crustaceans, 12 such as lobsters. Instead of refracting lenses, macruran crustaceans’ eyes contain small tubes with square cross sections that have reflective inner surfaces. These small tubes are thenarrangedacrossahemisphericalsurface,focusingincominglightontoahemispherical retina [2]. 1.3.1 Schmidt Optics Thefirstone-dimensionalfocusingopticutilizinga“venetianblind”typearrayofreflecting plates was flown on September 30, 1963 by the Lockheed group [51]. Schmidt further defined the design of of a two dimensionalfocusing device using stacked sets of reflecting platesarangedalongorthogonalcylinders(Figure1.5)[45]. Foreachsetofreflectingplates focuses, collimated light focuses to a line that is half the distance to the central axis (R/2, where R is the radius of curvature for the cylinder) [45]. Though the focus is not perfect andthe imagesizeisfinite,thistypeoffocusingdevicehasthe potentialofa widerfield of viewthanconic-section-basedopticaldesigns[27]. Becauseofit’sgeometry,andreflective surfaces on both sides of the foils, incident X-rays from sources outside of the critical angles expanding the FOV as compared to more traditional optics. 1.3.2 Lobster-EyeOptics To create a broad field of view and still retain a relatively broad spectrum, Angel devised an optical element patterned after the eyes of macruran crustaceans, such as lobsters and crayfish. Instead of refracting lenses, macruran crustaceans’ eyes contain small tubes with square cross-sections that have reflective inner surfaces. These small tubes are then arranged across a hemispherical surface, focusing incoming light onto a hemispherical retina [2]. Since the outer surface is spherical, there is no preferred axis in the ideal case, which eliminates off axis aberrations seen in the traditional grazing incidence telescopes, and even in the Schmidt design. Furthermore, in theory, there is no limit to field of view, 13 Figure 1.5: Schmidt’s depiction of the foils arranged around cylinders with central axis along the ˆ x and ˆ z axis. The preferred axis of focus is along ˆ y[45]. especially if symmetry throughout the spherical surface can be preserved [2]. However, because current CCD arrays are flat, an artificial preferred axis normal to the sensor, as wellaslimitationstoboththefieldofviewandsphericalaberration,areintroducedintothe system. But even with the limitations created by a flat imaging sensor, the field of view is stillmarkedly larger than for other X-ray optical designs. Though the two-dimensional focusing Schmidt design expands the FOV when com- pared to the more traditional nested conic section grazing incidence telescope designs, it is still limited compared to lobster-eye optics [2]. Furthermore, lobster-eye lenses can be constructedwithfarlessmassthantheSchmidtdesign. Themassincreaseismainlycaused by the necessity for a structure to hold the foils in place, as well as additional structure to accommodate the off set between the two sets of plates. The additional structure has the potential to double the weight of the system as compared to a device based on Angel’s design [21]. Another advantage of the lobster-eye geometry, as apposed to the Schmidt design, is the lack of a preferred axis, or set of axes. The two-dimensionalSchmidt design 14 is based upon two cylinders of revolution, each of which has a central axis, which, when combined, create a single preferred axis for incident X-rays that is orthogonal to the plane containing the linear focus of the two cylindrical sections. For instance, the optical device in Figure 1.5 would have the axis of revolution for the first set of foils along ˆ x, the second set along ˆ z, and the preferred axis would be along ˆ y. 1.3.3 BiologyofLobsterEyes Because of the unique environmental challenges facing crustaceans, the structure and method image formation of their compound eyes are varied and in many cases special- ized [15]. Water restricts the angular divergence, bandwidth and contrast of the light, as well as reduces the overall intensity [15]. And as a result, more light is needed for image formation. Inresponse,severaldifferentfamiliesofcrustaceanshavedevelopedcompound eyes that utilize superposition instead of the apposition type eyes where each inlet has its own photoreceptor, which is the common compound eye found in other arthropods such as insects[15]. In the superposition eye, there is a space between the facets and the photo receptors,allowingmultiplefacetstocontributetotheimage[32]. Macrurancrustaceans,or long bodied decapods such as lobsters, crayfish and shrimp, utilize multilayer film within theireyestocreatereflectivesurfacestofocuslightfrommultiplefacetsintoanimage[32]. Figure 1.6 shows the structure of a lobster’s eye as well as how it utilizes the reflective wallsof its facets in order tobring lightintoa focus (as drawn byLand, one of the original researchers to describe the reflective properties of lobster eyes)[32]. 15 Figure 1.6: Image formation and a structural overviewof a lobster’s eye by Land[32] One of the key features in the reflective compound eyes found in adult macruran crus- taceans is that the facets are square instead of hexagonal (Figure 1.7)[32]. The square geometry aids in the focusing of the eye, though it is more problematic for arranging on the surface of a sphere [32]. Typically, in crustaceans, the length of the facets are twice the width [32]. However, the length can vary due to light conditions, acting as an effec- tive pupil by limiting the number of facets contributing to an image in brighter settings [8]. The facets themselves are composed of a lens that has a slight taper before attaching to a rhabdom in the retina[8]. Along with the length, the rhabdom diameter also changes in brighter conditions, as a film can extend over portions of the rhabdom to shield it from unwanted contributions[8]. Furthermore, in lowlight conditions,the number of rhabdoms can expand from a single photo receptor to as many as four [8]. 16 Figure 1.7: Left: Image of the square facets of a lobster’s eye by Land[32], Right: Image of the square facets of a crayfish’s eye by Tokarski and Hafner[48] 17 CHAPTER 2: AN INTRODUCTION TO LOBSTER-EYE OPTICS Though Angel first proposed a lobster eye telescope in 1979, only prototype Lobster-Eye opticshave been tested, and none have been includedon X-ray missions. One of the major reasonslobster-eyelenseshaveyet tobe manufacturedhasbeen thedifficultyinproducing optics that meet necessary requirements for generating a useful image in a desired spec- tral range. And, though a few fabrication ideas were presented by Angel, none were fully developed or physicallyrealized, instead Angel demonstrated the optical properties in vis- ible light using microscope slides to construct a flat square celled optical device [2]. Since the concept was first conceived, the theory governing the optics has been refined, more sophisticated simulations have been conducted along with experimentation on prototypes, and several fabrication techniques have been attempted, studiedand analyzed. Each channel in a lobster-eye lens is characterized by the ratio of width to length (α = w/)[2]. Photons which are imaged by the lens enter a channel, reflect from one or more channel walls, and propagate toward a spherical imaging surface (Fig. 2.1b). Pho- tons may also pass through the channel without striking a wall, or may be absorbed by a wall. Because thereflectance ofmetalsforX-raysisverylowexceptatglancingincidence, photons are mostly absorbed if they strike a wall at an angle of incidence greater than a critical angle (θ c ), which is wavelength- and material-dependent but is typically a fraction ofadegree[2]. For incident photons to be focused, the photons must reflect from each set of parallel walls an odd numberof times (Fig.2.2). Photonsreflected an evennumber of timesin both dimensions are unfocused and become part of a diffuse background. Photons reflected 18 Figure 2.1: (a) Square channels arranged normal to a spherical surface; (b) A two dimen- sional representation of a lobster eye optic focusing 3 separate collimated sources. Figure 2.2: A two-dimensional representation of a lobster eye optic imaging a collimated source. an odd number of times in only one dimension are linearly focused and imaged along an straight line radiating from the center of the image. The resulting image of a collimated sourceisacrosswith2focalarmsemanatingfromabrightcentralfocalspotwithadiffuse background (Figure 2.3)[2]. 19 Figure 2.3: A normalized three-dimensional representation of a collimated source imaged by a lobster-eye lens using a log scale. 2.1 StateOf TheArt OneofthefocusesofAngel’sinitialpaperwasacomparisonoftheeffectiveareaofLobster Eye optics with conic section based optics. And, though he showed the capabilities of the ideal lobster-eye optic [2], Angel did not delve into the details of the imaging properties. Chapman and Nugent et. al. further refined the idealized imaging capabilities of lobster- lenseswhilealsodefingtheefficiencyatwhichlobster-eyelensesfocuslight[12]. Focusing on local point sources being imaged by arrays of square channels on a flat plate, Chapman and Nugent et. al. were able to define specifically how each channel contributes to the image, as well as showing the inherent width of focus [12]. To better understand how efficientlylobster-eyelensesfocuslight,ChapmanandNugentet. al. definedtheefficiency of fully focused, singly focused and unfocussed photons (η oo ,η oe ,and η ee respectively) theoretically under the TER model, verifying their findings with a ray trace simulation (also utilizing TER model for photon energies greater than 6keV) [12]. They were able to show that the most efficient lobster eyes (when the greatest percentage of imaged photons 20 are fully focused) were designed with α = √ 2θ i ,andthat η oo and η ee assymptote at 0.25 as α decreases [12]. Over the past three decades since the inception of the concept, several simulations and experiments have tested the theory and capabilities of lobster eye optics. Priedhorsky, Peele, and Nugent modeled a proposed ASM, consisting of a glass micro channel plate (MCP) produced optic with the intention of monitoring fainter X-ray sources [43]. The MCP was designed with a 1 arc minute resolution with 25 micro-meter width channels and a 5 meter focal length. Because the production of the MCP will create non-square pore channels away from the center of the lens, the non-square geometry of the channels were taken intoconsiderationin the simulation,as was surface roughnessand other imper- fections such as tilt, rotations and tapering in the channels. Mission realities were also simulated such as flux patterns and radiation effects. To reduce the effects of spherical aberration caused by imaging onto a flat CCD array and deviation from the square chan- nels away from the central channel, the proposed optic was designed with a small angular field of view, but to compensate multipleoptics would be used on a rotating systemresult- inginanaggregated4π steradian field of view (Figure 2.4). In the simulation, a confused background was imaged and successfully resolved [43]. Peele and Nugent compared an experimental flat MCP constructed lobster eye optic with a ray-trace simulation which was similar to their proposed ASM [42]. Both the sim- ulation and experiment used a monochromatic source at 1.5 keV. By comparing the sim- ulation with an experiment they were able to quantify and qualify the causes of several perturbations from the ideal found in the experiment. Specifically they were analyzing the effectsofchannelrotationsaroundthelongaxis,channeltwistsandnonsquareness(which they did not separate), channel tilt, and surface roughness. Channel rotations causes the focal square to become circular and the focal arms to flare. Channel tilt creates blur within the image. Both non-squareness and channel twisting create a broader focus and flaring, where the non-square effect is due to the second reflection not being orthogonal. Surface 21 Figure 2.4: All Sky Monitor developedby Priedhorsky,Peele, and Nugent [43] Roughnessreduces the intensityof the imageas wellas reducing the focus throughdiffuse scatteringoftheincidentradiationcreatingalowintensityhalo. Fortheirdevice,Peeleand Nugent used a flat micro channel plate (MCP) made from lead glass, noting that curvature can be added through thermally slumping the device. The X-ray source was created by a focused laser on a coper plate, creating a point like source with a 150 µm diameter. All experimentationwascarriedoutwithinavacuumchamber[42]. Bycomparingresultsfrom the experiment to the simulation with various changes to the perturbing effects, Peele and Nugent found channel tilt to have a negligibleeffect on the overall image, where as, along with surface roughness, channel twists and rotations have a fairly sizable effect upon the image produced by an MCP based lobster eye optic [42]. Peele also compared optics built using crossed arrays of planar reflectors (or merid- ional lobster-eye lens) and lobster eye optics produced from MCP and photolithography techniques [38]. The meridional lobster-eye is a variation on the Schmidt design utilizing two sets of one dimensional arrays of flat plates, however, unlike the Schmidt design, the two sets were curved into a spherical shape and interconnect inorder to simulataniously 22 focus the incident X-rays in two dimensions. The results showed that lobster-eye lenses manufactured by MCP or photolithographictechniques were not as limitedin their field of view as the meridional lobster-eye lenses, though both had comparable resolutions. How- ever, Peele found that the choice of materials had a greater effect on effective area than the geometry of the optic [38]. Hudvec, Sveda et.al. produced, tested, and compared a Schmidt based optic and a lobster-eyeoptic [27]. The Schmidtbased optic utilizeda foilbased designwhere the foils were made of glass coated with gold, while the lobster eye optic was produced by electro- forming composite materials with square pores in triangular shaped segments which were then slumped into a spherical surface. The electroforming approach was taken with the intention of using electroformed replication. A ray-trace simulation was run with compa- rable lenses modeling the experimentation. The experimentation results showed that they were able to get the Schmidt based design to give comparable performance to the MCP lobster eye optic, though the optic itself was larger and heavier [27]. Martin, Bruton, and Fraser proposed the use of an MCP based lobster eye optic for theESA’s SMART-1lunar mission[37], thoughtheirproposalwasultimatelynotselected. With the mission constraints in mind, they designing the lobster eye optic to have a radius of0.75meters,composedoffourseparateMCPsweretiledintoatelescopeoptic. Thefocal surface was approximated by a flat CCD array. To resolve the image, Fourier Transform methodswereused. Theprototypeperformedasexpected,achievingsubmillimeterspatial resolution[37]. Sources imaged by a Lobster Eye optic will be convolved (distorted through a mathe- matical processes) by the point spread function describing the optic. The deconvolutionof the image depends upon the source being imaged. Using the known point spread function of a point source, Priedhorsky, Peele, and Nugent designed an algorithm for the deconvo- lution of a confused background, consisting of well separated point sources, for use with an ASM lobster eye optic based upon MCP technology [43]. To resolve the background, 23 the algorithm identifies the brightest spot and removes the point spread function of a point source based around the location of that point. The algorithm then moves on to the next brightestpointintheimage,repeatingtheprocessuntilallbrightspotsare resolved. Inthis fashion, a confused background can be resolved, even for points that are hidden beneath the focal arms of nearby brighter sources [43]. It is important to note that the algorithm was designed specifically for an ASM with the assumption that the image being resolved is of several point sources which are all well separated [39], not an extended source. Todemonstratethedeconvolutionofanextendedsource,Peelesimulatedasquarechan- nel MCP imaging a formation in the shape of ”HI” imprinted on the surface of the moon [39]. In order to resolve the image, a large number of photons were required, though increasingcontrasthelpedreduce thedependence onphotonflux. For deconvolution,max- imum likelihood method was compared with maximum entropy method, though no dis- cernible difference was discovered. The maximum likelihood method is an iterative form ofdeconvolutionwherethesolutionsconvergetoamaximumlikelihoodsolution. Wereas, maximum entropy method adheres to the postulate that the probability distribution which bestdescribestheincidentwaveformistheonewiththelargestentropy. ThoughPeele was abletodeconvolvetheimageoftheextendedsource,thesourcespointspreadfunctionwas well known, and the source needed to be well sampled [39]. 2.1.1 Fabricationof Lobster-EyeLenses MCPs have been a major focus of research and experimentation with lobster eye optics due to their light weight, compact structure. To create an MCP, an assembly of fibers with a square inner core and an outer core clad in glass are fused together. The inner cores are then etched away leaving square hollow channels within the outer glass framework. Once formed, the glass can be formed in to a spherical shape by thermally slumping on a mandrel[36],whichhasalreadybeendemonstratedusingabeampipe[37]. BecauseMCPs are typically made from glass, the band width of X-rays that are capable of being imaged 24 by an MCP produced optic is relatively limited. However, coating the optic with a more reflectivematerial,suchasgoldornickel,maybeabletoexpandthebandwidthoftheoptic [37]. The surface roughness of MCPs has also been shown to be difficult to control [38], and channel twists and rotations that arise from the construction of the MCP create some sizable perturbations in the image [42]. Thermally slumping the device into a spherical shell has been shown to exacerbate the imperfections and degrade the structure and shape of the MCP. As alternative to glass, silicone MCP have been shown to be lighter and more stablethanglassMCPs,andcanbecoatedwithareflectivesurfaceforacontinuoussmooth surface [18]. Because MCPs tend to be relatively small, multiple MCP based lobster eye optics can becombinedtocreateanoveralllargerinstrument. Forinstance,Martin,BrutonandFrasier suggested tiling four MCP’s to expand the overall aperture of the optic to 10 square cen- timeters[37]. Similarly,Hudvec,Sveda et.al. had suggestedproducingtriangularlyshaped MCPsectionswhichcouldbethenshapedintoasphericalsurface[27]. Incontrasttomul- tiple MCP sections creating a single optic, Priedhorsky, Peele, and Nugent proposed using multiple lobster eye optics with discrete imaging sensors arranged along several planes of a rotating satellite [43]. Lead glass designed MCP are also limited in the potential bandwidths of study. Although, coating the glass structure of MCPs with a higher reflectivity material is pos- sible, it has not been shown whether that will significantly improve the throughput of the device [41]. As an alternative, investigations into micromachining techniques have been conducted. LIGA (lithography,electroplating, and molding)was used to etch optics out of metalswithhigherX-rayreflectivity. LIGAisamicro-machiningtechniquewhichinvolves exposing a substrate, through a mask, to intense X-ray radiation (because of the need for high energy exposure to get the aspect ratios required by lobster eye optics, the masking techniqueis slightlydifferent thanother LIGA processes). Thisradiationthenetches away 25 thesubstrate. Afterrepeated exposures,whentheprocesshasfinished,thefinalstructureis electroplated with nickel to create the lobster eye optic [41]. Peele, Mancini et.al. intended to use LIGA as a means for improving the structural form of the lobster eye optic and avoid the other issues present in MCPs such as channel tiltandrotations[41]alongwiththeimprovementinreflectivity. However,intheprocessof constructingthe LIGA based lobstereye optic,Peele, Manciniet.al. discovereddifficulties inobtainingthenecessary aspectratioduetoadhesionissuesintheintermediatematerials, andwereforcedtouseloweraspectratioopticstheirinitialexperimentation. Butevenwith the smaller aspect ration, they were able to produce an image. Study of the optical device itself showed squareness of channels to be on the order of MCP devices with channel rotations improved. Channel tilts, however, were still present. Surface roughness was estimatedbasedonthesurface roughnessattheedge ofthedevice,buttheopticperformed better than expected. Though minimal focusing was observed, for LIGA to be a viable lobster eye manufacturing technique, the surface roughness and aspect ratio need to be drastically improved[41]. Peele, Irving et.al. investigated the possibility of expanding the aspect ratio of LIGA produced lobster eye optics [40]. Though ultimately a larger radius optic would be ideal, theinitialexperimentattemptedtomatchanaverageMCPtypelenswitharadiusof0.375m and a resolution of 4 arc minutes. The intent was for the aspect ratio to be expanded from 7:1 to 30:1, a more reasonable aspect ratio for a lobster eye optic. In order to create curva- ture, the substrate can be bent during X-ray exposure avoiding further misalignments and physical deformations that occur when thermally slumping an MCP. However, multiple issues still need to be resolved from the low aspect ratio results before high aspect ratio lobster eye optics can be produced. Tapered channels, he dominantchannel issue resulting from the LIGA production technique, is due to the upper substrate being exposed longer, mask transmission, and diffraction from the radiation and etching process, causing broad- ening of the central focus and focal arms. Additionally,channel orientation misalignments 26 also exist as do imperfections along the lengths of the channels. Furthermore, it is unclear whether a closely packed system can be produced due to the difficulty in circulating the developer. To increase adhesion, graphite substrates were substituted for the silicon wafer substrates. However, even with the graphite substrate, Peele, Irving et.al. were unable to get full adhesion on high aspect ratio substrates [40]. Using a different etching technique, Chen, Kaaret and Kenny were able to etch silicon to form lobster-eye lenses with a higher degree of alignment accuracy than MCPs [13]. They were able to etch channels with α aproximately 86 arcminutes (40:1 : w), however they were unable to reduce the surface roughness of the channels to acceptable levels[13]. Though they did not investigate coating techniques, silicon is a feasible low energy X-ray reflector (Figure 2.5 shows the X-ray reflectivity for silicon). In order to image higher energy X-rays with silicon substrates, either a viable metal coating technique will need to be established or significantly smaller α will need to be achieved. Figure 2.5: The reflectivity of Si for X-rays of 1, 3, 6 and 9 keV at 0< θ i <2. Assembling an array of foils into a lobster-eye design has been proposed as an alterna- tive to MCP and LIGA technology. Though typically assembled foil structures tend to be patternedafteraSchmidtdesigns,aswiththeexperimentscarriedoutbyGorenstein,Whit- becket.al. [21]andHudec,Svedaet.al. [27],alobster-eyestructurecanbeassembled. The 27 use of foils allows for a reduction in surface roughness because each foil can be smoothed separately before assembly [18]. The foils themselves can be constructed using methods that are currently used for construction of X-ray optics such as electro-formed foils [27]. However,unlikeconic based telescopes,bothsidesof a foilina lobstereye opticwillneed to be reflective. This can be accomplished by either using an epoxy to sandwich two one sided foils together or by coating glass foils. In the either case, making stiff reflective flats is necessary for optical foil devices [27]. An assembly utilizing foils enables for a greater diversity of X-ray reflective materials in the construction of a lobster-eye lens. The choise of reflective material can allow for a greater band width than an uncoated MCP or etched Si lens. Also, since the foils are assembled from separate pieces, not only will the surface roughness be reduced, but the corners of the channels will not be rounded, instead orthogonal plates will form sharp corners and the likelihood of channel rotations will be vastly reduced by the geometric precision in construction [18]. Schmidt designs have been assembled with a high level of precision in foil placement [21]. Furthermore, the foils can be designed with the curvature alreadyinplace,avoidingthedegradationtothestructurethatMCPsexperienceduringthe thermal slumpingprocess. In astronomicalsources, the reduction in structural degradation and surface roughness can lead to spherical aberration being the dominantaberration [18]. The assembly of a lobster eye optic from a set of reflective foils can be accomplished by using interlocking plates as suggested by Gertsenshteyn, Forrester et al. [18]. The interlocking foils use slots that dovetail together between orthogonal pieces as shown in Figure 2.6. One set of foils will have slots that extend from the back of the foil, half the lengthofthethicknessofthefoil. Theothersetoffoilshaveslotsthatextendhalfthewidth of thefoilsfrom thefront of thefoil. The foilsare required tofit snugglytogetherat a high tolerance for proper focussing [18]. Some current etching techniques, such as electron cyclotron resonance (ECR) plasma etching, may provide a viable means for etching foil substrates with the necessary high precision[46]. 28 Figure 2.6: Construction diagrams of the interlocking foils as proposed by Gertsenshteyn, Forrester et al. [18] This design, however, is not a pure lobster-eye lens, mimicking the cross meridian or meridional lobster-eye lens described by Peele [38]. Because the channels are non- square,thelenshasadefinitepreferredaxisandthechannel’scrosssectionalshapechanges throughout the lens. However, for a small FOV, the curvature of the lens can be small enough for the channels to approximate square cross sections within desired tollerances [38]. Furthermore, in nature, the facets of lobstersand crayfish are tapered [32]. 2.1.2 FocalSurfaces The vast majority of X-ray optics that have either been proposed or constructed have relied upon flat CCD arrays. However, the ideal focal surface for lobster-eye lenses is spherical [2] and there has yet to be a realized field flattener designed for lobster-eye lenses. In addition to designing a foil based lobster eye optic, Gertsenshteyn, Forrester et al. proposed creating spherical detection surface, using optically tapered light guides (a polly capillary device) that output onto a CCD array [18]. Polly capillary devices transmit an image, therefore, while the spherical aberration will be minimized, there should not be any degradation in the image. The creation of a spherically shaped image surface would 29 be useful in all versions of lobster eye optics, and non-planar focal surfaces in general are useful in all imaging X-ray optics, even conic based telescopes. However, the design has not yet be fabricated or tested and all current lobster-eye lens experiments and proposals rely upon flat imaging surfaces [18, 43]. 30 CHAPTER 3: LOBSTER-EYES IN GREATER DETAIL 1 Although Chapman and Nugent et.al. detailed how flat arrays of square channels focus local point sources [12], they did not extend their study to include a curved array imaging a collimated source. Furthermore, due to the difficulties in manufacturing, the predom- inant area of study have been heavily influenced by perturbations and limitation created by proposed manufacturing processes. To better understand the geometric and theoretical limitationsoflobster-eyelensesandidentifydefiningdesigncharacteristicsamoredetailed theoretical model is needed. In doing so, the upper limits of capabilities can be identified and the progress of fabrication techniques can be assesed against an ideal. The understanding of defining characteristics of lobster-eye lenses will guide lens design based on mission goals. With the ability to expand the FOV without strict geo- metric constraints, the required image resolution needs to be traded against the desired FOV with considerationof the impact of lens desing on image quality. The effects of α on collection area and spectral range will also need be included within the design trades. The importance of the different design characteristics will be based on the target sources, for example, imaging the short (≤1s), GRB’s requires the ability to handle photon energies in the10-150keVrange [17], whilea surveysearchingfor the potentialsterileneutrinosignal will require a high sensitivity in the 3-4keV range[9, 7]. In the GRB case, a wide FOV is paramount for ensuring GRBs will be within view during the event, while the survey may put a greater importance on spectral sensitivityand angular resolutionthan purely FOV. 1 The findingsas well as muchof the materialin thischapterwere publishedin Reference[5] 31 3.1 ComputationalModel A Monte Carlo ray trace simulation, originally designed and written by Professor Erwin, wasusedtomodelasetofsphericallobster-eyelensesimagingacollimatedsource(Figure 3.1 details the flow of data within the simulation). At onset, the program reads a user specified definition file, which outlines some basic parameters for the simulation, such as the location of all optics and sources, input file names for the source and lens parameters, the number of photonsbeing simulatedand desired outputdata. The simulatedexperiment isthen builtbasedon the parameters outlinedin allinputfiles. After initialization,photons are launched individually and followed as they travers into the lobster eye lens. At each Figure 3.1: Flow chart outliningthe process of the computer simulationwritten in C++. 32 potential collision with the structure of the lens, the program determins wether the photon hasbeenabsorbed,unperturbedorreflected. Allreflectionsarecounted,thenewdirectional vector is calculated, and the reflectivity is used to determine the resulting energy of the photon. The programrepeats the procedure, determiningthe nextpotentialcollision(Asin Figure3.1). Afteraphotoniseitherabsorbedorreachestheimagingsurface,anewphoton is created and followed (Figure 3.1). Based on the experiment, different data sets can be output,suchasimagesurfacelocationofeach photon,thefinalphotondirectionalvectorat the focal surface, and photon counts based on number of reflections. Initially, experiments designedtoemulatepreviousstudieswere used toverifythesimulationwasfunctioningas expected. Because of the use of C++, the simulation utilizes object oriented programing. Object orientedprogramingallowsfortheprogramtobedividedintounits(calledclassesinC++), each of whichoutlinean objectas a groupof dataas well asthe rulesthatgovernthatdata. In doing so, classes were created for the lobster eye optics, reflectivity, and sources, as well as some base classes for linear algebra, handling files, and other necessary functions. A more specific class can also be derived from a general class (base class), refining the rules and alteringthe stored data for specific cases. Having base classes for the lobster-eye optics, reflectivity,and sources along with several derived classes allowsfor easy handling ofdifferentconfigurations,aswellasanabilitytoaddnewconstraints,designs,andoptions with minimal modification to existing code. For example, though initially the simulation only allowed for gold reflecting surfaces, the current program contains both a TER and full reflectivity models for multiple materials. Adding a new material is simple, and only involves adding a few additional lines of code including a derived reflectivity class for the material,theidentifierofthatmaterialtothefunctionthatdetermineswhichmaterialshould be used, and a data file containing the necessary material data. Structuring the simulation in this manner is especially useful when debugging newly expanded code, because only 33 the added feature needs to be evaluated when problems arrise. Furthermore, the flexibil- ity inherent in object oriented programming allows for expanding the capabilities into the current program to be fairly seamless,with the bulkof the new capabilitiesbeing writenin new classes and tied into the main simulationwith minimal invasiveness. 3.1.1 Modifications tothe MainSimulation The simulation originally based the Lobster Eye optic on the multi-plate meridian based design,withlimitedoptionsforalternativelobstereyeopticconfigurations. Thelobstereye classwasredesignedsothatthesimulationcannowbespecifiedtoseveraldifferentlobster eyeconfigurations,andmoreeasilyupdatedwithnewalternativelobstereyedesigns. Each lensconfigurationhasitsownsubclass(derivedclass)allowingthelenstobedefinedfairly autonomously. The simulation initially assumed gold reflecting surfaces. However, many simulations and experiments utilize other reflecting materials or approximate the reflectivity using the totalexternalreflectivitymodel[12]. Enablinga greater flexibilitywiththe potentialmate- rials also allows for a greater variety of lenses to compare and new materials to be tested. A new reflectivity class was created that calculates the reflectivity of a material from the equationsoutlinedbyUnderwood[51]fromavailablematerialpropertydataobtainedfrom Reference [25]. Currently, the class is capable of creating reflecting surfaces with Au, Ni, Si, and GaAs. The class can provide either the full reflectivity model or the TER simplifi- cation for all available materials. Severaladditionaloutputswere addedintothesimulationaswell. Forexample,photon counts based on the number of reflections were designed for comparisons with Chapman and Nugent, etal [12] and all photon locations along the focal surface (planar or spherical) can be outputinplace of a simulatedCCD to allowfor multipleconfigurations tobe tested on the same data set. Even the directional vector of all photons can be output in order to enable the inputchannel to be indentified. 34 3.1.2 Supplemental Code A series of python scripts were writen in order to iterate simulations while varying input parameters and compare results. Several sets of scripts were created based on the desired information and design parameters being tested. One set was designed for the comparison of focusing efficiency [12]. A second set was used to vary the focal length while studying the effects on focusing. Additionally, code was written to compare the computational to theoretical models of the focal properties. Deconvolutionclasses were also created as well as the iterative code to test and vary different lens configurations and analyse the resulting images pre and post processing. All results and analysis were compiled and visualized using pythonand python packages includingnumpy,scipy and matplotlib. 3.2 StudyingFocalPropertiesin OneDimension Insquarechannellobstereyelenses,thephotonwillbefocusedalongeither ˆ xor ˆ yindepen- dent of the other axis, and understanding the focal properties of the lens in one dimension can be studied in ˆ x or ˆ y. Once the one dimensional focal properties are understood, the twodimensionalpointspreadfunction(PSF) istheconvolutionofthetwoindependentone dimensionalPSFs. However,thesameisnotentirelytrueinthemeridionallobstereyelens because the channels are non-square. But, restricting the problem in one dimension will giveanapproximationofthefocalpropertiesofmeridionallenses(aswellasSchimdtstyle X-ray optics) and how they differ from square channel lenses. Furthermore, the lobster eye lenses have been restricted to singly reflected photons contributing to the focal spot. The lenses have been shown to have the greatest focusing efficiency (η ii ) for a specific X-ray energy when α = √ 2θ c [12]. Though η ii was defined undertheassumptionofTER,forX-rayenergieswherethereflectivityofthesurfacemate- rial approximates TER, the energy of the reflected photons after multiple reflections will 35 be negligible. Therefore, designing a lens to take andvantage of an ideal η ii restricts the number of reflecting for focused photons along in one dimensionto single reflections. 3.3 FocalContributionsofSquareChannels The same approach that Chapman and Nugent et.al. use to describe how square channels arranged on a flat plate could focus a local point source [12] can be employed to describe the focal properties of lobster-eye lenses imaging a collimated source when curvature is added. Just as with the flate plate, the contribution of every channel to the focal spot is a function of the incident angle of the source and channel [12]. Figure 3.2 shows 3 channels focusing a collimatedsource at 0.5α, α and 1.5α from the central axis that is aligned with the source. The channel that is aligned with α will completelycontribute to the focal spot. Figure 3.2: A two dimensionalrepresentationof 3 channels of a lobster eyelens imaginga collimated source. However,thechannelsthatarenotalignedwith α willonlypartiallycontributetothefocal spot (as in Figure 3.2). Photons that enter a channel that is between 0 and α (or, due to symmetry,between0 and-α) thatare notreflected a singletime,willpassthroughwithout any reflections. Similarly, photons that enter a channel that is between α and 2α that are not reflected a single time, will instead be reflected twice. The portion of the width of 36 the channel through which singly reflected photons enter is δw. Geometrically,though the configuration is a spherical lens imaging a collimated source instead of a flat lens imaging a point source, the definition of δw is identical [12]. For channels that are between 0 and α the entire upper wall of a channel reflects pho- Figure 3.3: A channel singly reflecting a collimated source that is at θ i =0.5α. tons, and therefore δw can be found fairly straightforward through geometry (Figure 3.3): δw=tan(θ i ) (3.1) where δw is measured orthogonally from the top of the channel and θ i is the angle of incidence between the photon and the channel wall (note, θ i is also the angular location of the channel with respect to the source-lens vector). At θ i = 0, δw = 0 as expected [12]. As θ i increases from zero, δw approaches the channel width. At θ i = α, δw = w again matching expectations[12]. Forchannelsthatarebetween α and2α,onlyafractionoftheupperchannelcontributes to the focal spot. A photon entering the channel at δw will reflect off the upper wall at a distance δ from the channel exit, and leave at the bottom edge of channel exit (Figure 3.4). Again, using basic trigonometry, δ can be shown to equal w/tan(θ i ). And because 37 Figure 3.4: A channel singly reflecting a collimated source that is at θ i =1.5α. the distance from the top of the channel is w− δw, which equals (− δ)tan(θ i ), δw can be shownto be: δw=2w−tan(θ i ) (3.2) where δw is measured orthogonally from the bottom of the channel. As θ i increases from α, δw approaches zero once again once again matching the results defined by Chapman and Nugent et.al.[12]. 3.3.1 TheConnectionBetween δwandtheSpanoftheChannelsCon- tribution tothe Focal Spot Because X-ray reflection require angles of incidence much less than 1 degree [2, 51], all of these channels fall within the range where the paraxial approximation is applicable. Furthermore, the photon will reflect at an angle θ r =θ i , the photon will exit the channel at an angle 2θ i from its original direction. If we assume that the focal plane has been arranged perpendicularly to the direction of the source, then δw will be projected onto the focal plane with the span of δw cos(2θ i ). However, due to the paraxial approximation, cos(2θ i ) can be approximated as 1 and the channel will contribute to the focal spot within 38 aspanof δw.From δw, thespanofthecontributionofanychannelcanbedetermined,and the total intensity at a given point within the focal spot is the superposition of all channel contributionsat that point. 3.3.2 TheDiffuseBackground ofUnfocused Photons The photons that reflect an even number of times, remaining unfocused, will result in a checkereddiffusebackground. Thedarkspacesofthebackgroundareduetotheshadowing of the structure of the lens, as well as the portion of the channel which contributes to the focal spot. Anymovementinthe sourcewillresultinvariationsintheshadowingpaternof the background as the local orientation of the structure to source varies. The contribution to the background made by any indiviual channel will span w− δw. Because the channels are rectilinear, photons emerge parallel to the incidnet direction [2], meaning the photons willbeimagedinrelationtothedistancebetweenthechannelandthecentralaxisalongthe source lens vector (off set from it’s original trajectory by the number of reflections). The locationofunfocusedphotonsalongtheimageplane(x f )fromachannelwillbewithinthe range: x f,min ,x f,max =        Rsin(θ i )− w 2 ,Rsin(θ i )+ w 2 − δw if 0≤ θ i < α Rsin(θ i )− 3w 2 + δw,Rsin(θ i )− w 2 if α < θ i ≤2α (3.3) 3.4 FocalLengthofSquareChannelLobster-EyeLenses Angel[2]showedthatlobster-eyelensesfocusinananalogousfashiontosphericalmirrors with a focal length f = R/2. Chapman et al. [12] further refined the treatment of the focusing characteristics of lobster-eye lenses by takingdimensionalityof the channels into consideration. However,they only considered parallel lenses. i.e., lenses with R= ∞ [12]. 39 Figure 3.5: Channel at θ i = α and θ i =−α focusing a collimatedsource. The focal length f can be found from the superpositionof δw from channels located at θ i =±α where δw =w (Figure 3.5). The reflecting surface is offset from the actual radii of curvature by w/2 and the reflected photons are at an angle of incidence of 2θ i from the central axis (defined by the source-lens vector). The points at which the photons cross the central axis are then be closer to the center of curvature of the lens by (w/2)/(2α), which reduces to /4. Therefore a photon reflecting from the outer edge of the upper wall of the lenswillcrossatthecentralaxisat R/2+/4andphotonsreflectingfromtheinneredgeof the upper wall will cross the central axis at (R+)/2+/4, or R/2+3/4, from the front of the lens. The narrowest breadth of the focal spot is the midpoint of where the photons cross the central axis, at R/2+/2 from the front the lens. At this point, the focal spot spans from −w/2to w/2. Measured from the front of the lens along the central axis aligned with the collimated source, the ideal focal length is f = R+ 2 . (3.4) 40 3.5 Focusing Effiency of Square-Channel Lobster-Eye Lenses In their analysis of lobster-eye lenses, Chapman et al. [12] defined the focusing efficiency as η ii =N ii /N imaged (3.5) where the subscript ii refers to the class of imaged photon: fully focused (oo), linearly focused (oe), or unfocused (ee). (Here o denotes an odd number of reflections from a pair of walls, while e denotes an even number.) Though their study concentrated on increasing the efficiency of a square-channel flat-plate lens, the analysis of focusing efficiency can be applied to spherical lenses directly. Theoretically, the results should be independant of radiusofcurvatureyeildingconsistentresultswithChapmanandNugentetal.’sstudy[12]. Chapman et al. used a normalized lens aspect ratio: α cn = θ c /α = θ c (/w) (3.6) allowingfor multiplesource-lens configurations to be compared. Twosetsofsimulationswereconducted. Thefirstconsideredalenswithfixedgeometry whilevaryingthephotonenergyofthesource,asinRef.[2]. Thesecondsetofsimulations considered a fixed-wavelength monochromatic source while varying the aspect ratio α,as in Ref. [12]. In the latter case, the channel width was varied, but the radii of curvature and channel length were fixed. In both cases, the results were in agreement, matching the resultsfound by Chapman et al. with η oo peaking at α cn = √ 2[12] (Figure 3.6 results that arerepresentativeofeachsetofexperiments). As α cn approaches ∞(increasingthenumber of contributing channels), η ii asymptotes at approximately 0.25 for all classes of reflected photons(0.5 for the combined focal arms efficiency - η eo + η oe ). 41 Figure 3.6: Efficiency η ii vs. α cn of lobster-eye lenses while varying (a) α and (b) photon energy. Even though both experiments were in agreement with previous findings, there were some differences that resulted from varying the photon energy in place of α. Chapman and Nugent et al. used the TER model in their theoretical treatment for η ii . In order for the reflectivity of the surface material to be approximated, their simulation used a photon energy of 6keV with Ni reflecting surfaces. For photon energies that are less than 6keV, the approximation no longer holds for Ni and a greater number of channels contribute to the imaged photons. As a result, the η ii appears to assymptote by α cn = 4 corresponding to photon energies of approximately less than 3 keV (Figure 3.6b). θ c also does not vary linearly, which may contribute to the slight variations between Figures 3.6aand 3.6bfor α cn less than 2. 42 CHAPTER 4: MERIDIONAL LOBSTER-EYE LENSES 1 With the increased interest in the use of flat foil plates to create the channel walls as an alternativeto producingsquare channels in a substrate [21, 22, 27, 28, 43], a more detailed study of the focusing mechinism is necessary to further understand the differences when compared to square-channel lenses. Previously, the center of a meridional lobster-eye lens (Fig. 4.1) was considered an approximation of a square-channel lenses, with differences in imaging properties considered negligible [38]. However, even along the central axis channels are not rectilinear, which has been shown to affect image quality [42]. To quan- tify whether meridional lobster-eye lenses are viable alternatives to square channel lenses, studyingthe focalmechanismisnecessary to moreaccurately describe the focal properties and understand any inherent degradation or advantages. Recent papers propose to utilize a Schmidt-style X-ray focusing device in place of lobster-eye lenses (sometimesevenrefering to them as lobster-eye lenses) [20, 27, 28, 47]. Although the focusing in two dimensions is different, when reduced to one dimension the meridional lobster eye lens (along its central meridion) is identical to a one-dimensional Schmidt optic. Though some modifications will be needed to fully realize the two dimen- sional Schimdt optic, much of the work presented here can be appropriated. 4.1 GeometricDifferencesin MeridionalLenses Becausethechannelwallsofmeridionallobstereyelensarealignedalongasphere’smerid- ians, the channels’ cross sections are not square. As such, meridional lobster eye lenses 1 The findingsandmuchof the materialin thischapter were publishedin Reference[5] 43 Figure 4.1: a) Metal plates aligned along meridians of a sphere; b) A 2-dimensionalrepre- sentation of a meridionallobster eye lens focusing 3 separate collimated sources. Figure 4.2: a) Non-square channels and b) channel taper in meridional lobster-eye lenses. have a central optical axis along the radial vector that intersects the point where the two central meridians cross. To a degree that increases with distance from the central axis, the walls of each channel are no longer aligned with ˆ x or ˆ y and the angle between intersecting meridians varies (Fig. 4.2a). Unlike the square channel design, in which the centers of the channels are aligned along the radii of curvature (Fig. 2.1b), the walls of the channels in a meridional design are aligned along the radii of curvature (Fig. 4.1b). As a result, the channels of meridional lensestaperthroughtheirlength(Fig. 4.2b). Thesegeometricdifferencesalterthefocusing and imagingcharacteristics and δw, f and even α all have to be redefined. 44 For meridional lenses, the global locations of the channels are straightforward because the channel walls are aligned along orthogonal sets of meridians [21, 43, 29]. In compar- ison, the global architecture of square-channel lenses is not quite so straightforward. For simplicity,thepresentworktakesthecentersofthesquarechannelstobealsoalignedalong meridians, with the spacing between the channel meridianslarge enough to ensure that the channels would not overlapat the outer edges of the lens. Inmeridionallenses,foraphotonreflectingfrommultiplewalls,eachsuccessivereflec- tionisincreasedbytheanglebetweenthechannelwalls. Asaresult,theunfocusedphotons are no longer parrallel for 2 or more reflections, and focused photons reflecting 3 or more times will no longer emerge from the channel at 2θ i . Therefore, multiple reflections will likely degrade the overall image, and the lens should be designed to avoid multiple reflec- tions being included within the focal spot. 4.2 Refining δwforMeridionalChannels Figure 4.3 shows a collimated source being imaged by 3 meridional channels located at θ i =0.5α, α, and 1.5α. As with the square channel lobster eye lenses, the contribution from each channel varies based on the angle of incidence. However, the taper inherent in meridional lobster eye lenses creates varying widths at the enterence (w f )andexit(w b )of the channel. Because the channel walls are aligned with the the radii of curvature (R)and they have finite thickness (t): w b = w f +t R− R −t (4.1) 45 Figure4.3: Atwodimensionalrepresentationof3channelsofameridionallobstereyelens imaginga collimatedsource. and the contributions of each channel is limitted to w b . Furthermore, the definition of α needs to be modified to account for the varying channel width. Since, the channel width varieslinearly,theaveragechannelwidthisused. Themeridionalaspectratio(α m )isthen: α m = w b +w f 2 (4.2) For channels where the upper wall is between 0 and w b /, δw is once again measured orthogonallyfromtheupperwall. Thecontributionsofthechannelsareidenticaltosquare- channel lobster-eye lens channels with θ i ≤ α. δw will once again be given by Equation 3.1.At θ i =w b /, δw=w b , andthechannel willcontributetothe entireextentofthe focal spot. Unlike the square channel design, there is a range of angles at which the channels will contributeto the full extentof the focal spot,spanningfrom w b / to w f /. For the channel at w b /, the singly reflected photons will all enter the channel within w b of the upper wall ofthechannel,andreflectalongtheentireextentofthewall. Photonsthatenterthechannel between w b and w f of the upper wall will pass through the channel without reflecting at all. Conversely, the photonsthat enter a channel at w f / withinw b of the lower wall of the 46 channel will be singly reflected. Photons that enter the channel between w b and w f of the lower wall will reflect twice before exitingthe channel. For channels that are at angles of incidence between w f / and 2α m , δw will be mea- sured from the lower channel wall, as was the case for square channel lobster eye lens channels where θ i ≥ α. The same methodology for determining δw as in square channel Figure 4.4: A channel singly reflecting a collimated source that is at θ i =1.5α. lenses can be used, substitudingw b and w f where appropriate. Doing so, δw can be found from w f −(− δ)tanθ i (Figure 4.4). Substituting w b /tan(θ i )for δ: δw=w f +w b −tan(θ i ) (4.3) Note that at w f / Equation 4.3 will propperly reduce to δw = w b ,andat2α m , δw = 0, indicatingthat 2α m is the limit of singlyreflected photons. 4.3 FocalLengthofMeridionalLobster-EyeLenses In meridional lenses, any channel within w b / ≤ θ i ≤ w f / will contribute to the entire focal spot if the contributions superimpose exactly. Considering only the focused photons from this range of channels, the outermost photons, between the lens and the ideal focal 47 Figure 4.5: Channel walls at θ i =−w b / and θ i =w f / focusing a collimatedsource. length f (where the channel contributions superimpose) reflect from the outer edge of the channel wall at ±w f /. Conversely, between f and the center of curvature of the lens, the outermost photons reflect from the inner edge of the channel wall at ±w b /. Therefore, to find f, the intersectionof rays from θ i =−w b / and θ i =w f / (Figure 4.5) can be used in the same manner as θ i =±α in square-channel lobster-eye lenses. Inmeridionallobster-eyelenses,thereflectingsurfaceisoffsetby−t/2fromtheradius ofcurvature. Therefore,photonsreflectingfromthechannelsat θ i =w f /and θ i =−w b / intersect between (R+)/2−t/w f and R/2−t/w b (Figure 4.5) from the front of the lens. The ideal meridionalfocal length from the front of the lens is f = R+ 2 − 2 w b +w f (w b +t) (4.4) Compared to square-channel lenses, the focal length of meridional lenses is shorter by ∆ f = 2 w b +w f (w b +t). (4.5) ∆ f is minimizedas approaches R (i.e., as w b approaches zero) andt approaches 0. How- ever, in most cases, will be much less than R,and t will be significant enough to create 48 structural integrity in the lens itself. Therefore, the discrepancy in focal lengths will be apparent. In previous work [29], channel taper was considered to severely degrade the image of lobster-eye lenses. Here, however, channel taper is considered part of the design of the lens. Because the taper is consistent and defined, the placement of f mitigates the the degredationin image quality seen in previous work. 4.4 FocusingEfficencyofMeridionalLobster-EyeLenses Usingthesamemethodologyassquare-channellenses, η ii wasfoundformeridionallenses. All lenses were simulatedwith the same physicalparameters. By forming both meridional and square channel lens architecture with identical angular separation between channels, channel placement would remain relatively consistent regardless of lens type. However, the global geometry toward the edges of the lens differ. Square-channel lenses maintain channel dimensions while the thickness of the channel walls decrease between them. On theotherhand,meridionallensesmaintainchannelwallthicknesswhilethechannelwidths diminish. When imaging off axis sources, the lenses were designed to be large enough to ensure all potentially focused photons were imaged. η ii was found to be identical to the square-channel simultaions (Figure 4.6). Once again, η ii matched the previousresults with a maximumefficiency at α cn = √ 2[12]. 49 Figure 4.6: Efficiency η ii vs. α cn of meridional lobster-eye lenses while varying (a) α and (b) photon energy. 50 CHAPTER 5: THE PSF OF LOBSTER-EYE LENSES 1 The point spread function (PSF) describes the spread of a single point in the image[19, 24]. Because lobster-eye lenses are designed to focus collimated light in astronomical applications, the normalized intensity profile of a collimated source is equivalent to the PSF.Theintensityatagivenpointalongthefocalspot(I x )inonedimensionisasummation of all contributingchannels [12]: I x =I 0 N ∑ i=0 R θ i ,E ph n (5.1) where R(θ i ,E ph ) is the reflectivity of the channel wall at θ i from the photonsource at pho- ton energy E ph , n is the number of reflections, and I 0 is the incident intensity. Though the Equation 5.1 is independent of lens geometry, the determination of the set of channels which contribute to any given point along the focal plane will differ and result in unique intensity profiles for the focal spot of each lens geometry (Figure 5.2). The diffuse back- groundisa checkered patternmatchingtheopen areas of channelswhich donotcontribute to the focal spot. Unlike the focal spot, the diffuse background is not composed of the superpositionof multiplechannels (Equation 5.1 no longer requires a summation). 1 WiththeexceptionofSections5.1.3,5.3.5and5.3.6,thefindingsandmuchofthematerialinthischapter were publishedin Reference[5] 51 5.1 ThePSFUnderIdealConditions Because the ˆ x and ˆ y sets of reflecting surfaces are independent, the two dimensional focus forlobster-eyelensescanbeconstructedfromtheone-dimensionaldefinitionsoftheimage components: the focal spot and background. 5.1.1 TheTheoreticalOne-Dimensional FocalSpot To determine the one dimensional focal spot intensity at any given point within the focal spot (x fs ), the angular range of contributing channels can be determined by finding the channelsforwhich δwextendstox fs [12]. Allchannels(between θ i =0to θ c )contributing to a x fs will fall within an angular range around the θ i = α channel. In square-channel lobster-eye lenses at ideal f (assuming α < 1 and the paraxial approximation holds), the upper bound of a contribution from channels 0< θ i ≤ α is w/2 (Figure 5.1a). δw(θ i ) increases linearly from 0 to w for (Equation 3.1)as θ i approaches α. Setting for x fs as the lower bound of a focal spot contribution, x min fs = w/2−(θ i /α)w (substitutingw/α for). Forchannelsat α ≤ θ i <2α,thelowerboundofthechannelcon- tributionwillbe−w/2(Figure5.1a). Forthesechannels, δw(θ i )decreaseslinearlyfrom w to 0 as θ i approaches 2α (Equation 3.2). Setting for x fs as the upper bound of a focal spot contribution,x max fs =3w/2−(θ i /α)w(Equation3.2). Solvingx min fs andx max fs for θ i ,Equation 5.2 shows the imits[ ¯ θ min , ¯ θ max ] for channels from the source-lens vector in terms of α. ¯ θ min , ¯ θ max = 0.5− x fs w ,1.5− x fs w (5.2) Finding the angular range for channels contributing to a point within the focal spot of meridional lobster-eye lenses can be accomplished in the same manner as with square- channel lenses. Though,The the boundsstillvary linearlyaround the θ i = α m channel, the 52 Figure 5.1: Position within the image plane vs. channel of fully focused photons for a) square channel and b) meridional lobster-eye lenses governing equations depend upon the position within the focal spot because upper bounds of θ I < α m and the lower bounds of θ I > α m channel contributions do not align with the edges of the focal spot. The focal spot itself will span −w ave /2 ≤ x fs ≤ w ave /2(Fig- ure 5.1b), similarly to square-channel lobster-eye lenses. However, contributions from the channel at θ i = 0 will be an infinitesimal point at −t/2 (Figure 5.1b). For chan- nels where 0 ≤ θ i ≤ α m , the upper bound will increase linearly from −t/2to w ave /2 and the lower bound will decrease linearly from −t/2to −w ave /2. Therefore, x min fs (x fs ≥ −t/2)=(θ i /α m )(w ave +t)/2−t/2and x min fs (x fs ≤−t/2)= −(θ i /α m )(w ave −t)/2−t/2 for 0< θ i ≤ α m . Solving for ¯ θ min : ¯ θ min =        x fs −t/2 (t−w f )/2 for x fs <−t/2 x fs −t/2 (t+w o )/2 for x fs ≥−t/2 (5.3) The condtribution for the channel at θ i = 2α m is an infinitesmial point at x(2α m )= (t/2)(1−t/(w b +t)) (Figure 5.1b). Again, approximating the upper and lower bounds of contributions as linear, x max fs (x fs ≤ x(2α m ))=(θ i /α m )(x(2α m )−w ave /2)+w ave −x(2α m ) 53 Figure 5.2: The one-dimensionalPSF of the focused photonsfor a) square channel and b) meridional lenses. and x max fs (x fs ≥x(2α m ))=(θ i /α m )(x(2α m )+w ave /2)−w ave −x(2α m ) for α m ≤ θ i <2α m as in Figure 5.1b. Solving for ¯ θ max : ¯ θ max =        2− x fs −x(2α m ) −w ave /2−x(2α m ) for x fs <x(2α m ) 2− x fs −x(2α m ) w ave /2−x(2α m ) for x fs ≥x(2α m ) (5.4) For bothgeometries,theintensityatan infinitesmalpointwithinthefocal spotisfound by calculating Equation 5.1 for all channels within the given θ min and θ max . Figure 5.2 compares simulated and theoretical one dimensional focal spot for both meridional and square-channel lobster-eye lenses. In both cases the analytical focal spot constructed from Equation 5.1 (using the angular limits found in Equations 5.2, 5.3,and 5.4) were in agree- ment with the simulation. The one-dimensional PSF of the focal spot of meridional and square-channel lenses differ in their profile, even for the idealized case aligned with the central axis. The square- channel focal spot PSF (Figure 5.2a) is nearly constant, slightly peaked in the center due to the reduction of reflectivity at larger angles of incidence. The meridionalfocal spot PSF 54 (Figure 5.2b), on the other hand, is more tapered on the edges before flattening toward the centerofthePSF.Theflatportionofthemeridionalfocalspotmatchest (forthesimulation, t/w ave ≈ 0.4(where w ave =(w f +w b )/2), and as t approaches zero, the meridional focal spot PSF approaches the triangular profile described by Schmidt [45]. The tapering of the meridional PSF concentrates more of the focused photons toward the center of the focal spot. On the other hand, the square-channel PSF is nearly flat, evenly distributing the channel contributions throughout the entire focal spot. Optimization of the lens will require careful consideration of the effects created by the dimensionality of the channels and channel walls. Furthermore, image reconstruction algorithms will have to be designed to incorporate the correct geometry-specific PSF. 5.1.2 TheoreticalOne Dimensional Generalized Background The photons that reflect an even number of times, remaining unfocused, will result in a checkered diffuse background. The dark spaces of the background are due to the shadow- ingofthestructureofthelens,aswellastheportionofthechannelwhichcontributestothe focal spot. Anymovementinthe sourcewillresultinvariationsintheshadowingpaternof the background as the local orientationof the structure to source varies. Though variations can be used to aid in source location similarly to a coded mask [20], for imaging purposes the background can be approximatedby consideringan averaged approach. Since imaging devices rely on quantized sensors, the background can be thought of as a summationof all infinitesmal contributions within the range of the quanta. Assuming a uniform, coherent collimated source, the resulting intensity within a quanta can be approximated as the frac- tion of the area filled by incident radiation. If shadowing is ignored, the intensity for the background of square-channel lobster-eye lenses can be approximated by Equation 5.5 for the channel that corresponds to the specific location along the image plane (where n is the number of reflections). bg(x bg )=I ◦ (w− δw)R n (θ i ,E ph ) (5.5) 55 This approach ignores the shadowing created by the structure of the lens and instead smooths out checkered pattern instead tracing the envelope that approximates the back- ground at any potential source-lens angles (θ s ). Doing so generalizes the approximation which allows for a single theoretic background to be applied throughout an entire image. For meridionallenses, the background can be obtained by substitutingw b for w into Equa- tion 5.5 and using the appropriate equations for δw. The other difference with meridional lensesisthatbecausethewallsarenotparallel,thereflectivitywillnotbethesameforeach reflection. Therefore, in Equation 5.5 the reflectivity is calculated by: R n (θ i ,E ph )= n ∏ j=1 R(θ i +(j−1)θ ch ,E ph ) where θ ch is the angle between the two reflecting surfaces (which is equal to the angle between channels: (w f +t)/R). 5.1.3 The EffectsofPixelSize onthe PSF Water restricts the angular divergence, bandwidth and contrast of the light, as well as reducestheoverallintensity;asaresult,morelightisneededforimageformationinmarine animals[15]. Inresponse,severaldifferentfamiliesofcrustaceans,includinglobsters,have developedcompoundeyesthatutilizesuperposition[15]inwhichspacebetweenthefacets and the photo receptors allow for multiple facets to contribute to an image[32]. For crus- taceans such as lobsters and crayfish, the square cross-sectional facets have a slight taper as the lens that makes up the facet in the cornea tapers before attaching to a rhabdom (light receptor) in the retina [8]. In bright conditions, the effective focal spot will cover an area of approximately a single rhabdomn [8]. As light conditions change, the film that provides the reflecting surface for each facet can vary, and under dark conditions as many as 4 photo receptors can be illuminated by the superimposed light [8]. Furthermore, the photo-receptors themselvescan vary in size using reflective film [8]. 56 In lobster-eye lenses, the central focal spot will be equal to the channel width [2, 12]. However,becausethechannelsandtheimagesurfacearenotphysicallyconnected(asthey are in nature [32]), there is more freedom in designing and choosing a pixel size. With the goal of resolving the image to a higher accuracy than the geometric angular resolution (GAR=w/f [2]),thebalancebetweenhigherfidelityandover-definitionneedstobefound. When the entire focal spot is contained within a pixel, abberations and the structure of the focal spot can be somewhat hidden, and the shadowing of the background is smoothed over (Figure 5.3). As the pixel size decreases, the effects of shadowing becomes more prominant, as do any abberations. In the original image, as PPW (pixels per channel) increases, the width of the focal spot grows to match, maintaining the central focal spot to be equal to w (Figure 5.3). The width of the focal arms also expand to match the PPW in the exact manner of the focal spot. The unfocused dimension of the focal arm, and the unfocused background becomes more defined, with the shadowing from the structure becoming apparent (Figure 5.3). With increased PPW, descrepencies between the analytical and simulated background becomemore pronounced,asdodifferences inthesimulatedbackgroundatvaryingangles of incidence. Because the pixel size has become more refined, the portion of the channel contributing to the background can completely illuminate a single pixel even for θ i > 0, however a single pixel can be completely shaded by the structure of the lens as well. To account for the decreased pixel size, the algorithm for the theoretical background must be modified to represent the potential for completely illuminating an individual pixel. Nor- malization also needs to account for the assumption that all pixels within−θ c ≤ θ i ≤ θ c will be illumated (i.e. ignoring shadowing), and the resulting excess within the integrated intesity. In nature, because the retina is curved and the structure of the facets extends to the recepors, all facets are aligned with a rhabdom (Figure 1.6 [32]). In lobster-eye lenses, the disconect between the channels and pixels as well as the planar nature of the imaging 57 Figure 5.3: A simulated 6keV source imaged by an ideal square-channel lobster-eye lens with a 2’ GAR and a sensor set for 1 and 3 PPW surfaces result in varying alignment between pixels and channels depending on θ s .The channel-pixelaligmnentwillnotaffecttheresultingimagefromfocusedphotonsnoticably because the central peak is builtfrom the superpositionof allchannels within∼±θ c of θ s . On the other hand, the background willbe significanlyaltered bychannel-pixel alginment. And because a greater pixel density results in a greater fidelity of the shadowing in the background,thedifferencesinthebackgroundandfocalarmsatvarying θ i duetoalignment also become more pronounced. 58 5.1.4 The Two-Dimensional PSF In order to recombine the components, the background (bg) and focal spot (fs)mustbe normalized appropriately. Typically, bg and fs would be required to be normalized to the same integral. However,because the background is averaged and ignores shadowing,each pixel is calculated as if it is perfectly aligned with a channel. Consequently, the open area of the lens (O) is effectively 1, while fs was propperly calculated under the assumption of the one-dimensionalO ( √ O = w/(w+t)). Therefore, the integral of the background is larger than it should be. To acount for the excess intensity in the background due to the exclusion of shadowing, the integral of bg is multiplied byO during normalization. After normalization, the idealized two dimensional point spread function (PSF) can be constructed by convolvingthe fs and bg for each dimension: PSF = η oo ·fs x ∗fs y + η oe ·fs x ∗bg y + η eo ·bg x ∗fs y + η ee ·bg x ∗bg y (5.6) Using a known PSF, the resulting blur can be mathematically removed from an image through deconvolution [19]. The focal efficiency (η ii [12]) properly scales each contribu- tion derifed for a specific α and source energy pair. However, because η ii is approximated using the total external reflectivity model[12], η ii can be set to 0.25 for more realistic reflectivity models to account for the excess photons entering at θ i ≥ θ c (the asymptote of efficiency for an infinite number of channels contributing to an image[12]) or for broader spectral ranges where η ii will vary. Note that each component of the point spread function can be separated outof Equation5.6 (i.e. for the focal spot PSF oo = fs x ∗fs y ) allowingfor each component, as well as the entire PSF, to be compared to simulateddata. Using the same source energy and α, lenses with varying GAR were tested with the pixel width initially set to the channel width. For square-channel and meridional lobster- eyelenseswithGAR=1’, normalizedsimulatedandtheoreticPSF oo hadacrosscorrelation 59 ofapproximately1. Thegeneralizedapproximationtothebackground(PSF ee )hasasignif- icantly lower cross correlation, as expected, due to ignoring the shaddowingpresent in the structure of the lens. For PPW = 1, the cross correlation of the background was ≈ 0.77. The cross correlation figures for the focal arms (PSF eo and PSF oe ) were also lower than PSF oo as expected due to the inclusion of the background approximation. However, each arm wasstillcorrelated significantlyhigher thanthe backgroundat≈0.95. When compar- ingtheentirePSF(Equation5.6),thecrosscorrelationwasstillnearlyoneduetothemuch greater photon concentration within the focal spot. In both cases, decreasing the pixel size did not have a noticable effect on PSF oo but it did significanlyreduce the cross correlation of the focal arms and background. Because an increase in pixelsresults in a greater fidelity of the shadowing,the loss in cross correlation was expected. When usingthe same initaldata set butsettingpixelwidthsto w/2(PPW = 2), cross correlation of focal arms droped to≈0.78 and the background to≈0.47. The lobster-eyelenses with GARset to 0.5’ and 2’ showedsimilarresults, with no sig- nificant changes in the focal spot or either focal arms. For the 2’ lens, the background had a slightly higher cross correlation of ≈ 0.84 while the 0.5’ lens had a slightly lower cross correlationof≈0.73forpixelwidthssettow. Aswiththe1’lobster-eyelenses,increasing PPW resulted in a decrease in cross correlation for the focal arms and background. Tominimizethestatisticalinfluenceofphotoncounts,alargeenoughsamplingofpho- tons were simulated such that the variability between identical simulations were minimal (∼ 1.6×10 6 total imaged photons). When the photon count was decreased, the focal spot remained well correlated but the correlation of the background and focal arms decreased both between the simulation and the theoretical model, as well as between individual sim- ulations. In general, when varying the photon count the theoretical model and simula- tion responded in the same fashion. Furthermore, the correlation between the entire PSF decreased to 0.8 when the total number of image photons was ∼8.2×10 4 for the 1’ GAR PPW =5 configuration, less than the∼0.9 cross correlation of the theoretical models. As 60 photon counts decrease, the theoretical model more accurately approximates the PSF than individual simulations. However, the higher photon count allowed for comparison of the approximationagainstas uniformasamplingas possiblefor the simulatedimagetoensure the imagingcapabilities were properly being modeled. 5.2 TheEffectsofNon-IdealFocusingConditions The focal spot width broadens for all imaging surfaces that are not aligned with the ideal focallength. Forplanarfocalsurfaces, thefocallengthwilldeviatefromtheidealthrough- out the imagingplane for sources that are not aligned with the central axis. And just as the ideal focal length differs between square-channel and meridionallenses, the nature of how the focal spot widens also differs. Figure 5.4a shows the envelope of the central focal spot created by fully focused pho- tons at varying focal lengths in the simulated square-channel lens. The theoretical limits of focused photons for channels at θ i = 0, α and 2α were included in the plot as guides. The focal length was varied from f = R/2− to R/2+ from the front of the lens. The results confirm that the ideal focal length (Equation 3.4) as well as show the variability in focal spot width over a relatively small range of focal lengths. The effect is very similarto spherical aberration. For focal lengths that are less than R/2+/4 from the front of the lens, the focal spot splits into 2 separate spots. Starting with the ideal focal length and reducing the focal length, the inner envelope of the focal spot expands to w/2 between R/2+/4and R/2. The outer envelope initially expands along the outer rays from θ i = α, before expanding alongthe θ i =2α rays for focal lengthsless thanR/2. For focal lengthsgreater than f,the outer envelope expands along the θ i =2α rays. Figure 5.4b shows the envelope of the central focal spot created by fully-focused pho- tons at varying focal lengths in the simulated meridional lens. The limiting trajectories of 61 Figure 5.4: The envelope of fully-focused photons as a function of focal length for a) square-channel and b) meridionallens focused photons for channels at θ i = 0, the innermost contribution from w b /, the outer- most contributionfrom w f /,and2α m are plotted as guides. Again the results confirm the ideal focal length givenby Equation4.4 and showthe variabilityinfocal spotwidthover a relativelysmall range of focal lengths. For forfocal lengthsgreater than f, theouterenvelopeofthecentral focalspotinitially expands along the contributions from w b /. Begining where photons reflecting from the inner most edge of the channel at w f / cross the central axis ((R+)/2−t/(4w f ) as showninFigure4.5)thecentralspotwillspit. Theinnerenvelopeofthefocalspotexpands to t/2 between (R+)/2−t/(4w f ) and R/2+/2. For focal lengths greater than R/2+ /2, the outer envelopeexpands along the θ i =2α m rays. For focal lengths less than f,the outer envelope initially expands along the outer rays from w f /, before expanding along θ i =2α m rays. 5.3 OFF-AXISSOURCESANDABERRATION The ideal focal surface of a lobster-eye lens is spherical [2], as is the case in nature [32]. However, the current image sensor technology is planar [18, 43] which creates a preferred 62 Figure 5.5: Focal length variation caused by planar focal surface. axis normal to the focal plane. Sources that are not aligned with the preferred axis will no longer be imaged at an ideal focal length (Figure 5.5) for both meridional and square- channellobster-eyelenses. Evenifmultiplelensesandsensorsareusedtocreatetheoverall FOV, the limitations created by a single planar sensor need to be considered in order to properly identify the design parameters for each lens and sensor element. Meridional lenses, however, have the added effects of non-square channels (Figure 4.2a). All images regardless of θ s will have some contributionsfrom non-square channels. The larger θ s , the greater the number of non-square channels, and the more non-square the channels become. 5.3.1 η ii for Off-AxisSources A set of simulations was used to compare the efficiency for imaging off axis sources in meridional and square channel lobster eye lenses. The angle θ s between the source direc- tion and the central axis of the lens was varied from 0 ◦ to 25 ◦ , first along ˆ x, then along ˆ x = ˆ y. The simulation was limited to 25 ◦ in order to remain well within the region where the square channels would not intersect. Again,meridionalandsquarechannellensesperformedcomparably,withnodeviations in η ii , indicating that neither the variation in global geometry of the lens nor variations in channel geometry affect the efficiency within the measured FOV. 63 The collection efficiency η col was also compared. Because the lenses were designed based on α = √ 2θ c , η col =N imaged /N 2α (5.7) where N 2α is the number of photonsthat reach the portion of lens within2α of the source- lensvector. Inthiscase,themeridionallensisslightlylessefficientthanthesquarechannel lens starting around 10 ◦ for sources that are off axis along ˆ x and ˆ x = ˆ y. Intuitively, this makes sense because the open area fraction of meridional lenses will decrease in compari- son to square channel lenses. Proposed ASMs are typically comprised of multiple lobster-eye lenses [43, 22], each with a much more constrained FOV than those considered here in the evaluation of focal efficiencies. The proposed FOVs of individual lenses fall well within the regime where efficiency will not be affected. 5.3.2 Effectsof PlanarImagingSurfaces Ifthefocalplaneissettobeidealforthepreferred axis,thensourcesat θ s willhaveafocal length that deviates by δ f p (Figure 5.5)andimageat x i along the focal plane regardless of lens configuration: δ f p = f c 1 cosθ s −1 (5.8) where f c = R− f.As θ s increases, the focal length decreases, as measured from the front of the lens. For square-channel lenses imaging to focal surfaces that are closer than f,the focal spotbroadens,eventuallysplittinginto2separate spots(Figure 5.4a). Formeridional lenses imaging to focal surfaces closer than f, the focal spot broadens (Fig. 5.4b). 64 Inordertomaintainasinglefocalspotinthefocusofsquare-channellenses, δ f p ≤/4. This is also the point at which the focal spot doubles in width. The angle at which δ f p is maximized can be found from θ s ≤cos −1 f c δ f p + f c (5.9) The first column in Table 5.1showsexamplesof θ 2w for the 0.5’, 1’ and 2’ GAR test lobster-eye lenses imaging a 6keV Ni source. With a constant α, larger GAR result in larger FOV. For the lobster-eye lens with GAR = 0.5’, θ 2w (which is measured from the centerofthelens)isanorder ofmagnitudelargerthanChandra’sentireFOV.However,the central focal spot of a source at θ s = θ 2w will be expanded to 2w with 2 peaks at ±w/2 from the the center of the elongated focal spot (Figure 5.6a). Therefore, only a portion of the potential2θ 2w FOV may be usable. The focal spot of meridional lenses broadens for focal lengths closer to the front of the lens than f as well. The point at which the focal spot doubles in size is when δ f p ≈ 2(w b +w f) w b − t 2 1− t w b +t , which approaches /4as t →0and w b → w f .The angles at which meridional lobster-eye lenses reach θ 2w is different, and can become sig- nificant for large GAR (over a constant α). The second column of Table 5.1 shows θ 2w for comparitive 0.5’, 1’ and 2’ GAR meridional lobster-eye lenses. When is much smaller than R (resulting from a smaller GAR), there is little difference between θ 2w of the merid- ional and square-channel lobster-eye lenses. But as GAR increased, the difference in θ 2w increased proportionally(doublingas R doubled). 65 Figure 5.6: Off-axis intensity profiles for a) a square channel and b) a meridional lens at θ 2w . Table 5.1: θ 2w at different GAR for Square-Channel and Meridional Lenses with the same α and R GAR Square-Channel Meridional 0.5’ 5.9 ◦ 5.4 ◦ 1’ 8.4 ◦ 7.6 ◦ 2’ 12.1 ◦ 10.9 ◦ For meridional lenses, the focal spot of a source imaged at θ s = θ 2w has a single peak in the center and broadens along shoulders starting at±t/2(Fig. 5.6b). Because the focal arms are a convolution of the one-dimensional focal spot and the background, these effects are present along each of the focal arms. Additionally, the focal length varies along each focal arm, causing variations along the focal arms of the PSF for a givensource at θ s . 66 5.3.3 Redefining theFocalSpot forNon-Ideal f The analytical one-dimensional focal spot was defined specifically for ideal focal lengths. However,becauseofplanarfocalsurfaces,theresultingsystemhasanimagingsurfacewith avaryingfocallength. Furthermore,thefocalsurfaceisnolongernormaltoasourcebeing imaged at θ s = 0. As a result, all components of the PSF will be expanded by 1/cos(θ s ). However, because of the discretization of the focal plane, even for angles θ s ≤10 ◦ will be sufficiently small enough to assume the paraxial approximation(1/cos(θ s )≈1.01). For nearly all channels in lobster-eye lenses, unfocused photonsemerge from channels parallel to θ s (for θ i > α m channels in meridional lenses the phtons emerge at θ s + θ ch ), therefore the variation in focal length does not affect the unfocused background signifi- cantly. On the other hand, varying the focal length affects the focal spot. The one dimen- sional focal spot can be generalized for regions based on the envelope of the focal spot depicted in Figure 5.4. Forsquare-channellobstereyelenses,theangularlimitsforthechannelsthatcontribute to any given point within the focal spot still will vary linearly. However, much like the meridional lobster-eye lens, the upper and lower bounds of the limits will no longer align with the edges of the focal spot. Defining the location of the infinitesimal contributions for θ i =0and2α, the angular limits can be approximated using the same methodology as was used for the idealized meridional lobster-eye lens. Though x(0) will be unchanged regardless of θ s , x(2α) will be offset from −w/2by4αδ f. Within the 0 ≤ δ f ≤ /4 regime,thecontributionsfromchannels0≤ θ i ≤ θ c are stillboundwithinthespreadofthe θ i = α channel and the limits of the θ i = α contributions can be generalized to [−w/2+ 2αδ f,w/2+2αδ f]. Equation 5.10 gives the angles limits for a position along the focal 67 spotforchannelswhere θ s ≤ θ i ≤ θ s +θ c and θ s iswithinthelimitof θ 2w andthefocalspot will have spread by w/2 both above and belowthe central source-lens axis (Figure 5.4). [θ min ,θ max ]        x fs /w−0.5 2(δ f/) , x fs /w−1.5 2(δ f/)−1 for x fs >w/2 x fs /w−0.5 2(δ f/)−1 , x fs /w+0.5 2(δ f/) for x fs ≤w/2 (5.10) For meridionallobster-eyelenses,the one-dimensionalfocal spotcan be generalizd for both Equations 5.3 and 5.4. The equation governing the determination of θ min still varies for positionsalongthefocal spotthatare eitherless thanor greater than−t/2(the position for photons from θ i = 0). Because the position of photons from θ i = 2α (Equation 5.12) changes based on θ s s, the generalization of θ max , x(2α)=(t/2)(1−t/(w f +t))+4α m δ f in order to account for the offset (as was the case for the square-channel geometry). θ min        x fs −t/2 (1/2)(t−w f )+2δ f/) for x fs <−t/2 x fs −t/2 (1/2)(t+w o )+2δ f/) for x fs ≥−t/2 (5.11) θ max        2− x fs −x(2α) 2(δ f/)−w ave /2−x(2α) for x fs <x(2α) 2− x fs −x(2α) w ave /2+2w f (δ f/)−x(2α) for x fs ≥x(2α) (5.12) 5.3.4 Non-Square and RotatedChannels In meridional lenses, because the channel walls are aligned with orthogonal sets of merid- ians, they do not intersect at perfectly right angles throughout the lens. Previous studies have shown that non-square channels show minimal degradation in images so long as the channels remain rectilinear [29], though there is some potential to broaden the focal spot and focal arms [42]. Because the channel walls themselves are not aligned with the local ˆ x and ˆ y,theyhave the appearance of being rotated along their long axis. The apparent rotation increases with 68 Figure 5.7: Approximated off-axis intensity profiles for f set to match a) a square channel and b) a meridionallens near θ 2w . θ s . Previous studies have shown channel rotation to degrade images by causing the central focus to become circular and the focal arms to flare [42]. Because of the similarity in causation, non-squareness and the rotation of channels are notseparately identified. Aberrations fromboth rotationand non-squarenessare presentin the images produced by the meridional lens, however, neither are as drastic as the effects produced by the flat imaging surface at small angles. Though the effects are present at 3 ◦ (the limit previously determined for planar focal surfaces), they are fairly minimal. At larger angles the central focal spot became noticeably broader and circular. Furthermore, the abberations caused by non-squareness and ratations are not apparent for PPW ≤5at angles θ c ≤ θ 2w in the tested lens configurations. 5.3.5 TheTwo DimensionalOff-Axis PSF The PSF and PSF components were compaired throughout a field of view from 0 to 0.75θ 2w . The FOV was limited to 0.75θ 2w to give sufficient room for the focal arms and background to be contained with θ 2w . To ensure the pixel alignment with the PSF was consistent, the data was centered with the lens before being “imaged”. In this manner, the 69 variationsinlensstructureare preserved buttherelativepositionof pixelswithintheentire PSF will be consistent. For the focal spot, both the theoretic focal spot corresponding to the focal length at each angle and the ideal focal spot was compared against the simulated focal spots at each angle. For PSF oo , the simulated and theoretic results were similar, with the variant theoretic algorithmnearlymatchingtheoffaxisresultsthroughouttheFOV.Theidealtheoreticfocal spot and the central simulated PSF had similar cross correlations throughout the FOV as well. For the pixel widths set to w, the deviation from ideal began at ∼ 0.5and0.6θ 2w (Figure5.8). Aspixelwidthdecreased,thedeviationfromidealbeginsclosertothecentral axis. At PPW =5, the deviation began at ∼ 0.4θ 2w (Figure 5.8). Changing the GAR does not affect the relative positonwithin θ 2w where deviationsbecome apparent. Figure5.8: Thecrosscorrelationoffocalspotsatvarying θ s fora1’squarechannellobster eye lens Because θ s was varied along a single dimension, the two focal arms were affected in different ways. For the focused dimension of the focal arm aligned with θ s (PSF eo ), the allignment of the channels does not vary as much as when the focused dimension of the focal arm was perpendicular to θ s (PSF ⊥ oe ). 70 For PSF eo with PPW = 1, the simulated focal arms were fairly consistent throughout theFOV,deviatingfromtheidealataroundthesamepointas PSF oo (∼0.4θ 2w asinFigure 5.9). Ontheotherhand,thetheoreticfocalarmswerenotaswellcorrelatedasthesimulated throughout most of the FOV in general. The variant theretical focal arm had a relatively consistent cross correlation throughout the FOV while the idealized theoretical focal arm begantodeviatefromat aroundthesame pointasithad forPSF oo (Figure 5.9). In boththe simulated and theoretical PSFs, decreasing the pixel size caused the deviations from ideal tobecomeapparentatsmallerangles(Figure5.9). Decreasingthepixelsizealsodecreased the correlation between the simulated and theoretical focal arms as had been the case for the on-axis analysis. Figure 5.9: The cross correlation of PSF eo at varying θ s for a 1’ square channel lobstereye lens For PSF ⊥ oe with PPW =1, the simulatedfocal arms varied throughoutthe FOV (Figure 5.10). As the orientation of the source and lens structure changed, the cross correlation decreased and increased with the alginment. For the lens configuration used to produce Figure 5.10, the channel source missalignment relative to θ s = 0 ◦ were approximately 0, 46, 92, 61, 14, 32, and 76% for each tested angle, matching expectations of the influence of source lens alignment. On the other hand, the theoretic results remained consistent with 71 the results from PSF eo showing no influence based on source lens alignment. Decreasing the pixel size causes the deviations from ideal to become apparent at smaller angles (Fig- ure 5.10). The consistency throughout the FOV for both theoretical focal arms shows the Figure 5.10: The cross correlation of PSF ⊥ oe at varying θ s for a 1’ square channel lobster eye lens advantage of the approximated background. Variations in the source-lens alignment do not impact the accuracy of the approximation as drastically as the variations in shadowing affect the correlation between images at different θ s . The results for PSF ee with PPW = 1 were similar to those of PSF ⊥ oe though the cross correlations were lower in general. The simulated background deviated throughout the FOV based on source-lens alignment (Figure 5.11). The theoretic results were consis- tent throughout the FOV as well, nearly averaging the simulated background. The effects of the structure can be clearly seen in the patern of the cross correlation. As pixel size decreased, thecorrelationdecreasedaswell,andthecrosscorrelationofthesimulateddata more closely matched than that of the theoretical due to the increased prominance of the the effects of shadowing (Figure 5.11). Unlike the focal spot and focal arms, the cross correlationbetweenseparatesimulationsatthesameangledecreasedwithincreasingPPW (Figure 5.11). 72 Figure 5.11: The cross correlation of the background at varying θ s for a 1’ square channel lobster eye lens Thegoalofaveragingthebackgroundwastofindaconsistantapproximatiionthatcould beusedthroughouttheFOV,notnecessarilytoperfectlydefinethebackgroundforasingle source-lens alignement. For small PPW, and large photon counts the approximation holds well. As was the case when comparingthe on-axis 2 dimensionalPSF, the simulatedimaged included a large enough photon sampling to minimize the effects of photon statistics. Reducing the photon count results in a stark difference in the simulated cross correlations. In all cases, the theoretical PSF and PSF componenets had higer cross correlations to the simulated data than individual simulations, both at the same angle, and at varying angles. At low photon counts (∼ 8.2×10 4 total imaged photons), the variations due to structure becomelessapparentasphotonstatisticsbecomeadominantdifferentiationbetweensimu- lationsforthefocalarmsandbackground. Thecrosscorrelationfiguresforbothfocalarms remainedaround0.8withPPW =1, however,asPPW increasedto5, thecrosscorrelation fell to ∼0.2 (Figures 5.12 and 5.13). Even at PPW =1, the cross correlation of the back- ground was ∼0.1, and approached 0 as PPW increased (Figure 5.14. As a result, the fully 73 Figure 5.12: The cross correlation of PSF oe at varying θ s for a 1’ square channel lobster eye lens with the photoncount reduced to∼8.2×10 4 Figure 5.13: The cross correlation of PSF ⊥ oe at varying θ s for a 1’ square channel lobster eye lens with the photoncount reduced to∼8.2×10 4 PSF is not nearly as well correlated for PPW = 5 as had been the case with high photon counts and the theoretical models more closely represent all simulations(Figure 5.15). 5.3.6 OffsettingtheFocalSurface An alternative to defining the PSF for varying θ s is to attempt to extend the span over which the PSF is relativelyinvariant by offsetting the focal surface. Since PSF oo remained 74 Figure 5.14: The cross correlation of the background at varying θ s for a 1’ square channel lobster eye lens with the photon count reduced to∼8.2×10 4 Figure 5.15: The cross correlation of the Entire PSF at varying θ s for a 1’ square channel lobster eye lens with the photon count reduced to∼8.2×10 4 relativelyinvariantcloser tothecentral axis, thefocal surface can be movedawayfromthe front of the lens to a point that matches the increase in the focal spot width at the angle of invariance(θ s = θ inv ). FromEquation5.8, δ f inv = f (1/cos(θ inv )−1). Forsquare-channel lenses, δw inv is then just 2δ f inv α because the focal spot is expanding by 2α at that point (Figure5.4). Becausetheenvelopeexpandsat4α forfocallengthsgreaterthan f,theoffset δ f off = δw inv /(4α)= δ f inv /2. Table 5.2 has the calculating δ f inv for the 0.5’, 1’, and 2’ 75 Figure 5.16: The cross correlation of the focal spot at varying θ s for a 1’ square channel lobster eye lens with the focal plane offset GAR lenses for PPW =4and5where θ inv were found to be 0.5 and 0.4 θ 2w respectively. All values of δ f are givenin terms of. The 0.5’ and 1’ GAR lobster-eye lenses produced the same δ f inv while the 2’ was approximatelyhalf. Table 5.2: δ f inv in terms of for 0.5’, 1’, and 2’ GAR lenses with PPW =4and5 PPW θ inv 0.5’ GAR 1’ GAR 2’ GAR 4 .5θ 2w /16 /16 /8 5 .4θ 2w /25 /25 /12 Simulations verified the relative invariance of the focal spot over and extended FOV. Forthe0.5’,1’and2’GARlens,theinvariantFOVwasextendedto∼0.75θ 2w forPPW=4 (Figure 5.16a shows the results for the 1’ GAR lens). The FOV of the 1’ and 2’ GAR was only extended to between ∼0.5and0.6θ 2w for PPW =5 and negligablyextended for PPW =5 with the 0.5’ GAR lens (Figure 5.16b showsthe results for the 1’ GAR lens). Formeridionallenses, δ f inv canbefoundusingEquation5.8aswell. Howeverbecause the envelope expands at 2w f / for focal lengths less than f and 2w b / for focal lengths greater than f δw inv and δ f off willbedifferent. Instead δw inv = δ f inv 2w f /andthe offset 76 δ f off = δw inv /(2w b /)= δ f inv (w f /w b ). With the meridional lens, deviation from the ideal occures at approximately 0.4θ 2w and 05.θ 2w for PPW =4 and 5 respectively, as was the case with square-channel lenses. Substituting the appropriate values the meridional lens, δ f inv for the 1’ GAR lens is ∼/52. Once again, the focal spot remained invariant at larger angles with the offset focal surface. However, unlike the square-channel lens, for both PPW = 4 and 5, the increase only reached ∼ 0.6θ 2w . With an offset imaging plane, decreasing to PPW =3 does extend the apparent invarience closer to∼0.75θ 2w . 77 CHAPTER 6: IMAGE PROCESSING FOR LOBSTER-EYE LENSES ThetwodimensionalPSFofalobster-eyelenscoversasignificantportionofafocalsurface due to the broad area covered by the background and focal arms. Though the central spot will span approxamately w, only ∼ 25 to 35% of the incident photons are imaged within the central focal spot, depending on η ii . The focal arms and background, containing the rest of the imaged photons, will extend approximately to ±2θ c (R−)+w along the focal plane (Figure 6.1). Processing is necessary in order to properly separate sources and to view the image more clearly by removing the breadth of the focal arms and background. Because the PSF is known, it can be removed mathematically through deconvolution [44]. However,even after processing, artifacts will still remain and the ideal intial signal(where apointsourcewouldrepresentasinglepoint)willneverbefullyresolvedinrealsituations. Furthermore, the size of the individual pixel with respect to w will affect the degree to which an image can be resolved. Each set of experiments is run at varying pixel sizes in order to compare the findings and identify an appropriate pixel sizes with respect to the w. Becausetheimagingexperimentsinvolvedonlyindividualoradjacentsourcesandsen- sor noise was not modeled, the bulk of the simulated images were empty. To allow for a higher effiency for deconvolution which enabled larger PPW than could realistically be expected (resulting in much larger data arrays than would have been possible other- wise),imageswerecroppedbeforedeconvolution. Preliminarytestsshowednodifferences betweendeconvolvedimagesbeforeandaftersimulatedimageswascroppedofemptypix- els. Whenever spatial variance was being included, the algorithm was designed to crop all masks after being constructed to ensure the correct range of focal lengths at the correct 78 location within the final image were being considered. The main simulation was designed to output the pecise coordinates along a focal plane such that the same data set could be processed at varying pixel dimensions. The flexibility of pixel dimensionality allowed for extremely high pixel densities to be tested, however, the resulting imaging sensors where often much larger than existing technology even after cropping the image. For exam- ple some of the experiments resulted in theoretical PSFs (which were just a fraction of the FOV) that covered approximately the same number of pixels as each Chandra CCD (1024×1024 [53]). Furthermore, the pixel dimensions were not always realistic as the PPW increased for an individual w. In real designs, the relative channel to pixel ratio can be obtained by enlarging the channel width compared to actual pixel sizes. Allimagingsystemsusedforevaluatingthedifferentalgorithmswereidealizedinorder to isolate the geometric influences on imaging. Therefore, no surface roughness or image sensor noise was included in the model. The sensor was also modeled in countingmode (a technique which has been used on X-ray telescopes such as Swift [17]). Only the sources were imaged, background was not included. Aside from the two point resolution tests, only isolated sources were included in individual simulations. Furthermore, the sources were all considered collimated, coherent and monochromatic. The lens itself was ideal, with α = √ 2θ c and f settotheidealfocallengthforthegivenlensgeometry(exceptwhen purposefully offset to accomodate a wider FOV as in Section 5.3.6). 6.1 MetricsforImageComparison To measure whether the processing improved a simulated imaged, a set of metrics was chosen for comparison. Assessing the quality of an image is not trivial. Quantitative mea- sures, such as peak signal to noise ratio, do not always match the qualitative experience [52]. However, the nature of the data being accumulated, qualitative distinctions can be influencedbychoicesinvisualizationof thedata,andnotnecessarilyaresultoftheoptical 79 instrumentation. For simplicity, quantitative measures - specifically full-width half-max (FWHM), peak signal to noise ratio (PSNR) and η eff where used to compare the original images of collimated sources with processed images using different deconvolution meth- ods. The determinationof FWHM was straightforward (asumingonly a single peak exists withintheimage),definedasthenumberofpixelsaroundthepeakintesityimagedthatwas greater than 0.5 the maximum. Because the simulation assumes an idealized system with no surface roughness or detector noise, the PSNR and η eff are more accurately described ascomparingthefocalpeak tothebackgroundandfocal arms. Thefocalpeakisnotsolely the focal spot but also includes the background and focal arm contributions to the central pixels. η eff was based on the η oo , however, instead of isolating fully focused photons, the ratio compares the integrated intensity within the central peak to the overall integrated intensity of the entire image. Because of the potential for the background and focal arms to contributetothe central peak if thesource isaligned withat least a portionof a channel, η eff ≥ η oo . η ii is independent of R [12], meaning the relative proportion of photons within each componentremainsconstantregardlessofFOVorGAR.Thespanoverwhichfocusedpho- tons,inthedimensionoffocus,willbeimagedisalsoconstrainedtowregardlessofR[12]. On the other hand, the span over which the unfocused photons will be imaged varies pro- portially to R. In one dimension and assuming a total external reflection model, bg spans approximately ±2θ c (R−)+w (Figure 6.1). As R increases with respect to , a greater number of pixels and a larger overall area will be involved in imaging the same relative proportionof photonsfor the background and focal arms. As a result, as the area the back- ground and focal arms expand with increased R while the same proportion of photons are being imaged, the relative intensityappears decrease with respect to the focal peak. 80 Figure 6.1: Spread of fs and bg along the focal plane 6.1.1 Approximating Signal toNoiseand η eff Peak signal to noise ratio (PSNR) is a common method for determining one aspect of the quality of an image [52]. In the case of lobster-eye lenses, the image peak will contain a linearcombinationofthecentralfocalspotandanycombinationsfromthebackgroundand focal arms thatfallwithinthe area coveredby thefocal peak. Because of thediscretization of the focal surface into pixels, source-channel-pixel alignment will impact the PSNR to some extent, as well as the determination of the width of the focal peak. With that caveat, considering both ideal and worst case conditions will give a reasonable approximation of a range within which to expect the PSNR. The relative size of pixels to channel widths willfurther impact PSNR, the simplestcase is toassume the pixelshave the same widthas the channels resulting in the pixel equaling fs. Under the TER approximation, only chan- nels that fall within θ c of the central source-lens axis will be imaged. Because reflectivity will be either 1 or 0 using the TER model, the ratio of intensities between two points can be approximated by the ratio of the number of photons imaged in the given pixels. And assuming the source is collimated, the flux will be constant throughout the lens, meaning thephotondensitycanbeusedtoapproximatethenumberofphotonsimagedinaspecified 81 pixel. Despite the large number of assumptions,the PSNR can still be approximated fairly closely and the lens deign parameters that affect PSNR can be identified. The flux for a PSF component, PSF ii , can be approximated by η ii N/A ii where N is the total number of imaged photons and A ii is the area of PSF ii prior to discretization (for example A oo = w 2 ). The intensity at a given pixel can then be determined by the area of the imaged packet of photons (A ph ), prior to discretization, multiplied by the flux. Therefore, the intensity of the focal spot for contributionsof fully focused photons will be η oo N because A oo =w 2 =A ph .TheA eo can be determined by multiplyingw by 2 ∑(w−dw) forallchannelswithin0and θ c fromthecentralsource-lensvector. Thenumberofchannels N ch (along a single axis) is given simply by the θ c divided by the angle between channels (θ ch =(w+t)/R). SubstitutingO for(w/(w+t)) 2 , theopenarea ofthechannelwithinthe channel spacing, and √ 2α for θ c , the number of channels within θ c is: N ch = √ 2O R (6.1) The number of channels within any angular limits can be found using the same method- ology. And the total number of channels contributing to the entire PSF would be (2N ch ) 2 (assuming|θ i |≤|θ c | under the TER model). Forchannelsinsquare-channellobster-eyelensesat θ i < α,w−dw=w−θ i ,whilefor channels α < θ i < θ c , w−dw=w−(2w−θ i )=θ i −w. The summation is distributive, expandingto: N α ∑ (w−θ i )+ N ch ∑ (θ i −w)− N α ∑ (θ i −w). WhereN α isthenumberofchannelsbetween0and α,andisequalto √ OR/(orN ch / √ 2). Because the channels are discrete, θ i =iθ ch and the summationbecomes N α ∑ (w−iθ ch )+ N ch ∑ (iθ ch −w)− N α ∑ (iθ ch −w). 82 And in all cases the summationscan be simplified: N i ∑ w=N i w N i ∑ iθ ch =θ ch N i ∑ i=θ ch N i (N i +1)/2. Substitutinginto the summation: N ch ∑ (w−dw)=wN α −θ ch N α (N α +1) 2 +θ ch N ch (N ch +1) 2 −wN ch −θ ch N α (N α +1) 2 +wN α which reduces to: N ch ∑ (w−dw)=wN ch √ 2−1 for the cumulativelengthof the focal arm. The area can thensimplybe foundby multiply- ing by w (the focused dimensionof the focal arm). A eo =2w 2 N ch √ 2−1 (6.2) where the factor of 2 is necessary to account for the focal arm spanning from−θ c to θ c . With fs equaling the pixel width, and assuming the source is alligned with the pixel, the overall intensity of the pixel will be the area of fs times the contributing flux of fully focused photons, linearly focused photons and background. Because the background only hascontributionsfromasinglechannelatanygivenpoint,thecontributionsfromtheback- ground to fs will be negligible. The peak of the focal arms that is outside of the focal spot will be the contributionsfrom the nearest channel. The area of the contributionat from the nthchannel(for θ i < α)fromthecentralchannelwillbeA n eo =w(w−dw)=w(w−nθ ch ). Substitutingfor θ ch ,A n eo =w 2 (1−n/(R √ O))=w 2 (1− √ 2n/N ch ) and the PSNR can be 83 approximatedbythetotalintensityofthecentralpeakdividedbytheintensityatthenearest contributionfrom both focal arms. PSNR= η oo Nw 2 /w 2 +2η eo Nw 2 / 2w 2 N ch √ 2−1 η eo Nw 2 1− √ 2n/N ch / 2w 2 N ch √ 2−1 which reduces to: PSNR= η oo /η eo 2N ch √ 2−1 +2 1− √ 2n/N ch (6.3) The PSNR is dependent solely on N ch (which is composed of R, andO Equation 6.1), and independent of α. To verify, simulationswere run varying R but maintaining α,using both the TER and full reflectivity models for Ni and a 6keV source. The results matched expectations (to allow for missalingment with pixels and ambiguity of peak edge, PSNR was calculated with n=1 and 3) (Table 6.1a). The results using the full reflectivity model Table 6.1: PSNR results for a) an idealized Ni lobster-eye lens imaging a 6keV source at varying GAR using TER model and b) idealized Ni, Au and Si lobster-eye lenses imaging a 6keV source with R/≈47.1 usingboth TER and full reflectivity models (a) GAR R/ Theoretic (n=1 - 3) PPW =1 PPW =2 PPW =3 0.5’ 95.2 114.2-117.7 113.82 95.57 83.15 1’ 47.1 58.4-62.2 61.96 50.95 46.20 2’ 23.1 30.6-35.2 34.61 25.00 21.72 (b) Material GAR Theoretic (n=1 - 3) TER Full Model Ni 1’ 58.4-62.2 61.96 60.52 Si 0.5’ 58.4-62.2 61.97 59.81 Au 1.35’ 58.4-62.2 61.47 48.46 did not vary significantly from the simulations using the TER model because at 6keV the reflectivity of Ni approaches TER. Tests were also conducted maintaining R/ andO but varying α by changing the reflecting material to Si and Au. Again a 6keV source was simulated and both TER and full reflectivity models were used. Because the same w and O were used to define the lens, regardless of reflecting material, both and R would vary 84 proportionally in order to maintain R/, and the GAR for each reflecting material was different: 0.5’and 1.35’for Si andAu respectivelyfor a comparableR/tothe1’ GARNi lobsteryeyelens(Table6.1b). ThoughtheGARwaschanging,thePSNRresultsremained thesameregardlessof thereflecting material(Table6.1b) whenusingtheTERmodel. The reflectivity of Si at 6keV can be effectively modeled using TER, and the PSNR using both TER and full reflectivity models were not significantly different (Table 6.1b). Auonthe other hand can not be modeled as accurately usingthe TERmodel at 6keV, and as a result, the PSNR decreased when usingthe fullreflectivity model(Table 6.1b). As the reflectivity ofthematerialisnolongeraswellrepresentedbytheTERmodel,morechannelsatangles θ i ≥ θ c are able to contribute to the image resulting in a lower focusing efficiency (η oo is no longer maximized). As with PSNR, η eff incorporates the entirety of the focal peak, including the contri- butions of the background and focal arms. However, η eff compares the fraction of the all imaged photons that are imaged within the focal peak (the integrated intensity of the focal peaktotheoverallintegratedintesityoftheentireimage). TheareadefinedbyEquation6.2 willagain be used in the determinationof the percentage of area covered in the focal peak. Thediscretizationofthefocalsurfaceonceagaincreatesambiguityindeterminingthearea of coverage for the focal spot, as does the exact location of the edges of the focal spot as defined withinthe program. Instead of the FWHM, the entire focal spot was determinedto be thewidthat whichthe focal spotreached the noiselevelinherentin theimage (FWNL). Determining the noise level by use of approximating the outer edge may seem recursive, however,initiallysetting the edge by the point at which the signalbeginsto either increase or falls below a predefined “zero” level (i.e. setting a value well bellow the expected focal arm or background signal). Assuming ideal pixel allignment and pixels with width w will result in the central peak containinga focal arm segment of approximately FWNL*w.The portion of the background contained within the focal peak, on the other hand, will span and area of approximately FWNL 2 . η eff can then be approximated by summing the η ii 85 multipliedby the approximate ratio of area contained within the focal peak to the toal area of the focal component: η eff = η oo + η eo FWNL N ch √ 2−1 + η ee   FWNL 2N ch √ 2−1   2 (6.4) Where FWNL has been normalized by w. As with PSNR, η eff is dependent on N ch while independent of α. Varying R/ (and thus N ch ) resulted in the expected changes in η eff , with larger radii of curvature resulting in smaller η eff in agreement Equation 6.4 (Table 6.2a). On the otherhand, the simulations maintaining R/ but varying the reflecting mate- rial (and therefore α), showed no variation in η eff (Table 6.2b), as had been the case with PSNR. Once again the results using the full reflectivity model were not significantly dif- ferent from the TER model for Si and Ni, however, η eff did decrease for Au when the full reflectivitymodelwasused(Table6.2b). AswithPSNR,inordertomaximize η eff ,choos- ingamaterialforwhichthereflectivitycloselyresemblestheTERmodelforthebandwidth of interest is essential. Table 6.2: η eff results for a) an idealized Ni lobster-eye lens imaging a 6keV source at varying GAR using TER model and b) idealized Ni, Au and Si lobster-eye lenses imaging a 6keV source with R/≈47.1 usingboth TER and full reflectivity models (a) GAR R/ Theoretic (FWNL=1 - 4) PPW =1 PPW =2 PPW =3 0.5’ 95.2 0.36-0.38 0.39 0.36 0.36 1’ 47.1 0.36-0.40 0.391 0.385 0.373 2’ 47.1 0.38-0.46 0.47 0.41 0.38 (b) Material GAR Theoretic (FWNL=1 - 4) TER Full Model Ni 1’ 0.36-0.40 0.391 0.393 Si 0.5’ 0.36-0.40 0.391 0.390 Au 1.35’ 0.36-0.40 0.392 0.363 Decreasingthepixelsizereduces bothPSNRand η eff asthepeaksignalisspreadover a larger number of pixels. With the increase in fidelity, the edge of the central peak also 86 becomesmorecleanlydefined. Therefore,theexcessareaofthefocalpeakcontainingonly background or focal arms that may have been contained within the larger pixels may not havebeenincludedasthepixelsizedecreased. Because asR/decreases thepercentageof the overall PSF contianed within a single pixel increases, the relative difference in PSNR and η eff that results from excludinga pixel willincrease as well. Therefore, the change in η eff and PSNR from varying pixel size increases as R/ decreases (Tables 6.1 6.2). PhotoncountsplayasignificantroleinthevariabilityofPSNR.Varyingthepixelcounts between ∼10 5 and 10 7 , the deviation in the simulated images’ PSNR reduced by a factor of ∼ 2 for every increase in order of magnitude. The simulated PSNR also more closely resmebled the theoretical model as the background became more “filled” by larger photon counts. Photon counts were not as significant in determination of η eff because η eff relies on the percentage of total photons contained within the focal peak, instead of the relative intesity of individual pixels (PSNR being the ratio of the pixel with the greatest intensity to the pixel with the greatest intesitynot contained within the focal peak). In general, η eff was far less variable than PSNR, typically consistantto two orders of magnitude. Because η ii isindependentoflensgeometryandN ch willbeapproximatelythesamefor meridional and square-channel lobster-eye lenses, Equations 6.3 and 6.2 can be applied to meridional lenses in order to estimate PSNR and η eff . The 1’ and 2’ GAR Ni lenses were tested in the meridional configuration. The PSNR and η eff were identical to the square channel lenses for PPW = 1. However, the differences in the focal spot and background becomemoreapparentasthePPW increases. Asaresult,thePSNRand η eff didnotmatch the square-channel results for PPW =2 and 3. In the 1’ GAR meridional lens, the PSNR was relatively unchanges, most likely a result of the bulk of the focal spot intesity being centered within the focal spot as apposed to the more evenly distribution of the focal spot intesityof square-channel lobster-eye lenses. Therefore, at increasing PPW for meridional lobster-eye lenses, the pixel-channel allignmentwill affect the way in which the focal spot is distrubuted differently than had been the case with square-channel lenses. On the other 87 hand, η eff for the meridionallens matched the results from the square-channel lenses. For the2’GARmeridionallens,forallPPW bothPSNRand η eff weresignificantlylargerthan anticipated. But theedges ofthe centralpeak were lessdefined inthesimulatedimagesfor the 2’ GAR meridional lens. To accomodate for the potential inclusion of noise within the central peak, the theoretical values were recalculated with n set to 4 and 5. Allowing for a broader central peak resulted in agreement between the theoretical and simulated PSNR and η eff . 6.2 The Deconvolution Of Images Produced by Lobster- EyeLenses Previous work have tested both maximum likelihood as well as maximum entropy decon- volutionmethodsforresolvingimagesproducedbysimulatedlobstereyelenseswithsome success[39]. Though the exact methods were not specified, two common deconvolution methods that are likely candidates are the Wiener Filter and Lucy-Richardson deconvolu- tion algorithm (also known as Richardson-Lucy). For deconvolution of simulated images, both the Weiner Filter and Lucy-Richardson algorithm were utilized. A modifided Lucy- Richardson algorithmwas also appliedto handlespatial variationscaused by the non-ideal focal surface. 6.2.1 GeneralDeconvolution Algorithms The Wiener Filter is a least mean square error approach to deconvolution[19] constructed from the pointspread function (Equation6.5). In Equation 6.5, PSF represents the Fourier Transform of the PSF (psf). Φ n /Φ o represents the noise to signal ratio, or the relative noise in the image[19]. W psf = PSF ∗ |PSF | 2 + Φ n Φ o (6.5) 88 In a noiselesssystemwhere Φ n /Φ o reduces to zero, the Wiener Filtersimplifiesto 1/PSF. But, if thereare anyzeros inthe pointspread function,usingtheinversepointspread func- tionwillresultinsingularities. Therefore, in a noiselesssystem, Φ n /Φ o can be assumedto beseveralordersofmagnitudelessthanthepointspreadfunction,ensuringnosingularities will arise while having a negligible impact on the results [19]. Once the Wiener filter is constructed and image (i) can be resolved taking the by inverse Fourier transform of the product of Fourier transform of the image and the Wiener filter: s=F −1 {W psf ·F{i}} (6.6) where s isthe bestestimateof the source after deconvolution. However,artifactsmay arise in the best estimate of the source because the Wiener filter tends to amplifynoise [19]. Lucy-Richardson deconvolution is an iterative forward convolving method specifically designed to minimize a nonlinear log-likelihood function [44]. Beginning with an initial guess at the input signal (s ◦ ), each step updates the previous results until Equation 6.7 converges (where s g is the current signal estimate) or the maximum number of steps is reached[44]. s g+1 = psf T ∗ i s g ∗psf ·s g (6.7) Typically, a uniform field is used for s ◦ . For computational purposes, Equation 6.7 can be transformedintoamulti-stepalgorithm(Figure6.2)firstconvolvings g withthe psf,which results in a current best estimate image. Next the ratio of the input image to the current best estimate image (i g ) is found and convolved with psf T to form a correction factor (c). Finally, the current best estimate for the source can be multiplied by this correction factor to find s g+1 , the updated best estimate for the source[33]. 89 Figure 6.2: Flow chart for standard Lucy-Richardson deconvolution 6.2.2 Modified Lucy-Richardson Deconvolution Algorithm Previous work has modified the Lucy-Richardson algorithm to incorporate spatial varia- tions in the point spread functions while determining i g and c [33]. In the case of lobster eye lenses, the separated PSF as well as any spatial variations (Figure 6.3) can be incor- porated into the algorithm while determining i g and c. Using the separated components of the PSF reduces artifacts that can arise from preconvolving the PSF (through the pro- cess of Fourier and inverse Fourier transforms). The flexibility afforded by the forward convolving nature of Lucy-Richardson deconvolution is advantageous compared to other methodsbecauseitreadilylendsitselftomodification. Furthermore,becausetheinitialsig- nal will be a point source (assuming collimation), masking s g will not impact the resolved image in the same manner that masking i will if a source is on the boundary between two masks. Though previous studies have shown least mean square error approaches (such as the Wiener Filter) to be effective, the PSF was considered constant throughout the field of view [39]. 90 Figure 6.3: Flow chart for component separated spatially variant Lucy-Richardson decon- volutionfor lobster eye lenses In the modified algorithm,the PSF is broken intoits componentsin order to handle the one dimensional spatial variations in the focal spot. M i is the mask for the focal length coresponding to fs i . The masks are concentric circles about the origin to match the radial variationsin focal length(Figure 6.4). Though the maskswere only calculated to the FOV, theCCD wasoftensimulatedaslarger thantheFOV toensure novignettingwouldoccure, therefore the outermost mask was significantly larger in order to contain the margins. For sources toward the edges of the FOV, the focal arms and background will cover multiple masks (Figure 6.4). Because the background is an approximated average of all θ s ,the source is convolved with the background prior to being masked for the focal arms, and no masks are necessary for the background. Parallelization can be applied to increase the efficiency in calculating i g and c. Currently the algorithm is designed to convolve each PSF component (i.e. i oo in Figure 6.4) simultaniouslythen sum the results before stepping forward(i.e. i ii andc ii inFigure6.3). DuetothenecesitytobreakthePSFintocomponents, the analytical PSF is used within the algorithm. For an analytical idealized PSF algorithm, 91 Figure 6.4: The masks for the spatially variant Lucy-Richardson deconvolution algorithm with an image of an off-axis source spanning multiplemasks. the components can once again be handled separately however, the masks are omited and sumationstofindi ii andc ii arenotnecessary. Intheabsenseofmasks,thePSF components do not need to be pre-convolved, and s g ∗ fs x ∗ fs y can be entirely calculated in Fourier space. However, masks do need to be applied prior to taking the Fourier transform for the spatiallyvariantalgorithm,asaresult,whenconvolvingthefocalarms,theonedimensional background does need to be convolved with s g prior to being masked. Each mask within the sumations used to determine i ii and c ii require Fourier trans- forms. Furthermore, each focal arm requires pre-convolutionbetween the one dimensional background and s g before being masked, which necessitates additionalFourier and inverse Fourier transforms. Because of the number of Fourier transformed necessary within each iteration of the spatially variant Lucy-Richardson algorithm, the computational time and expense for using the modified algorithm increases with each additional mask. Choosing the minimal number of masks necessary can help eliviate computation time and expense, however the number needs to be chosen at a small enough interval that the variation of focal length (and resulting variation in focal spot) within the mask will be negligible for a specifiedpixeldimension. Asaresult,asPPW increases,thenumberofmasksrequiredfor 92 deconvolution increases as well. To set the masks, a defined variation in focal spot width wassetatonsetofdeconvolution,andthepositionofthecorrespondingfocallengthsalong thefocalplanewereusedtodefinetheinnerandouterradiiofeachmask. Forthetestedset of configurations, the masks were set to match changes in focal length that would result in increasingthefocalspotbythepixelwidth. Thebandsforeachmaskbecomeprogressively narroweratgreaterdistancesfromthecenteroftheimagingplane,andwiderfieldsofview will require much greater numbers of masks. And though offsetting the focal plane was designedtoextendtheinvariantportionofthefocalsurface, thespaciallyvariantalgorithm can still be applied. The algorithm was designed such that the deviations in focal length account for non-ideal placement. However, offsetting the focal surface was considered an alternativemeans of extendingtheFOV withthehopesof reducing computationalexpense while still expandingthe usable FOV. 6.3 Results of the Deconvolution of Simulated Lobster- EyeLensImages To assesimprovementsinpost-processedimages,simulatedimageswere compared before and after deconvolution. Individual images were deconvolved using Weiner Filter as well as Lucy-Richardson algorithms (both variant and invariant). For spatially invariant algo- rithms, both idealized analytical and simulated PSFs were used in the deconvolution. FWHM, PSNR, and η eff were compared for the original simulated image as well as all deconvolved images for each individual simulation. In adition to the image quality of the original and deconvolved images, the two-point resolution using the Reighly limit and the accuracy of angular positionof peak signal were tested for the analytical Lucy-Richardson algorithm using both idealized and spatially variant PSFs. The images were kept to a reasonable size in order for processing on a standard laptop CPU, however, as the pixel 93 densityincreased, the upperlimitsof the pocessingpower were quicklyreached evenwhen the simulatedimage was cropped. 6.3.1 Deconvolution Under IdealCircumstances A single collimated source was imaged by lenses of varying focal lengths and reflective materials. Spacially variant and idealized analytical Lucy-Richardson algorithms were tested against a standard Lucy-Richardson algorithm using a simulated ideal PSF (from an on-axis source) and a Weiner Filter utilzing both a simulated and analytical idealized PSF. Initially a 1’ GAR, Ni based lobster-eye lens was used to image a 6keV source with ideal f and α alongthecentral axiswith PPW =1. There were nodifferencesbetweenthe variantandinvariantalgorithms,asthespatiallyvariantalgorithmreducedtoasinglemask for the on-axis source for the tested fidelity. For square-channel lobster-eye lenses, the Lucy-Richardson algorithm produced significantly improvements in PSNR, FWHM, and η eff with the analytical algorithms outperfoming the simulated PSF (Figure 6.5). Though the PSNR and η eff of the imagewas higher after being processed bythe Weiner Filter, the PSNR and η eff were not as high as the Lucy-Richardson algorithm. In fact, the PSNR for the images deconvolved by the Weiner Filter using both the analytical and simulated PSF wereorderofmagnitudelowerthantheLucy-RichardsonalgorithmsusinganalyticalPSFs. η eff for the images deconvolved by the Weiner Filter were ∼0.6 as apposed to ∼0.95 for theimagesdeconvolvedbytheLucy-Richardsonalgorithms. TheFWHMwasidenticalfor all processed and unprocessed images. Increasing PPW resulted in a reduction in the FWHM for the Lucy-Richardson algo- rithms. For example, setting PPW = 3, the original image’s FWHM expands to 3 pixels in each dimension while the processed images using the Lucy-Richardson’s algorithms resolve to a single pixel in each dimension (Figure 6.6). Effectively, the image is resolved to 1/3 the GAR in each dimension. The PSNR remained at the same order of magnitude 94 Figure 6.5: Idealized 1’ GAR square-channel lobster eye lens a) imaging a collimaged source with PPW =1 and the deconvolvedimage using Lucy-Richardson algorithms for a b) simulatedand c) analytical PSF 95 Figure 6.6: Idealized 1’ GAR square-channel lobster eye lens a) imaging a collimaged source with PPW =3 and the deconvolvedimage using Lucy-Richardson algorithms for a b) simulatedand c) analytical PSF 96 while η eff increased for theLucy-RichardsonalgorithmusingaanalyticalPSF. The Lucy- Richardson algorithm using a simulated PSF, however, had a reduction in both PSNR and η eff . With the variability of the background between simulations, the increased fidelity of the background exacerbates the differences between the simulated image and simulated PSF, while the analytical approximation essentially blurs over any differences. Qualita- tively, viewing the images matches the quantitative measures. Not only does Figure 6.6c appear signficantly less noisy than Figure 6.6a and b, but it is also drastically less noisy than Figure 6.5c. The Weiner Filter becomes unusable at larger PPW, as the system noise begins to overtake the signal. Considering the success of the Lucy-Richardson algorithm in comparison, the Weiner Filter was not included in any other experiments. 2’ and 0.5’ GAR lenses were also tested in order to compare how varying R/ affected the ability for deconvolution. As was the case with the unprocessed image, the increased fidelity of the 0.5’ GAR resulted in a greater PSNR. Increasing PPW can result in a res- olution that mimics a smaller GAR (i.e. PPW = 2 at 1’ GAR has a resolvability equal to PPW =1 at 0.5’ GAR), however there is a loss in the observed PSNR in doing so (as had been the case in the unprocessed image). η eff of the 1’ GAR lens with a PPW =2was nearly identical the 0.5’ GAR lens with a PPW =1(∼0.99). Qualitatively, however, the imagesproduced bythe 0.5’GAR lenswitha PPW =1 appear tohave more noise(Figure 6.7) than the 1’ GAR lens witha PPW =2, but the increase in focal spot intensityresulted in the greater PSNR. As before, increasing PPW increased η eff and resulted in a much lessnoisyimage(frombothaqualitativeandquantitativestandpoint6.8)forthe0.5’GAR lens. In the Lucy-Richardson algorithm using analytical PSFs, maintaining the photon flux within the image, (i.e. N ph remained constant over the span of the lens within 2θ c of the central axis) the PSNR increased for smaller GAR. However, η eff remained nearly con- stant. For the 0.5’, 1’, and 2’ GAR lenses, PSNR were 8097, 5691 and 2546 respectively, 97 Figure 6.7: Idealized 0.5’ GAR square-channel lobster eye lens a) imaging a collimaged source with PPW =1 and the deconvolvedimage using Lucy-Richardson algorithms for a b) simulatedand c) analytical PSF 98 Figure 6.8: Idealized 0.5’ GAR square-channel lobster eye lens a) imaging a collimaged source with PPW =3 and the deconvolvedimage using Lucy-Richardson algorithms for a b) simulatedand c) analytical PSF 99 while η eff ≈0.99in each case. In each of these simulation,∼1.5e5 photonswere imaged. However, it is important to note that PSNR is extremely variable, fluxuating by as much as 1000 between individual simulations. And unlike the orginal image, the photon count sigificantly impacts the value of PSNR, with larger photon counts increasing the value of PSNR. For the tested configurations, at higher PPW as the photon counts increased, η eff approached1inthedeconvolvedimages. As η eff approached1,thesignalisnearlyentirely contained in the central peak. Because the simulation did not model noise, the signal that remained in the background after deconvolution approached 0. As a result, the PSNR at timeswerecalculatedtobeapproaching ∞,whichwasanartifactoftheassumptionsandnot a physical result. Though the PSNR value may not represent the image quantitatively, the qualitative experience of the image is well represented by the apparent increase in PSNR. However, η eff was seen as a more accurate quantitative value to represent the signal to noise in the resolved image. DeconvolutionusingtheLucy-RichardsonalgorithmsandWeinerFilterwiththemerid- ional lobster-eye lens also produces significant improvementsin PSNR, FWHM, and η eff compared to the unprocessed image. The same trends for the deconvolution algorithms were present when applied to the meridional geometry. However, the PSNR was slightly smaller in the meridional lens, but, when considering the variability of PSNR, the differ- ence was not consideredsignificant. Increasing PPW didnot affect PSNR (as was the case with the original image) however,once again, as PPW increased, η eff approached 1. 6.3.2 Deconvolution ofa Single Source atVarying θ s Simulatedimagesof collimatedsourcesat varyinganglesfromthe centralsource-lens axis (θ s ) were deconvolvedusingtheLucy-Richardsonalgorithmswithidealand spatiallyvari- ant analytical PSFs as well as a simulated PSF. θ s was varied from 0 to 0.75θ 2w as was the case when comparing the cross correlation of the PSF components. Square-channel 100 0.5’, 1’, and 2’ GAR as well as the 1’ meridional lobster-eye lenses were intially tested with idealized focal lengths. After the intial tests, the focal plane was offset according to the findings in Section 5.3.6. Along with FWHM, PSNR and η eff , the accuracy of the determined angular location of peak signal was compared to the input θ s (for the unpro- cessed image, a simple peak finder was used). Because the discretization of the pixels, the deviationof angular positionis givenin terms of the angle subtended by a single pixel. For the 1’ GAR lens with PPW =1, the PSNR decreased at increasing θ s but the vari- ations in the focal spot were not as apparent. As such, there was no decernable differences between the spatially variant and idealized analytical Lucy-Richardson algorithms until ∼0.75θ 2w . TheFWHManddeviationinsourcepossitionwereidenticalforalltestedalgo- rithms. Increasing PPW resulted in the PSNR having a greater consistency throughout the image deconvolved using the spatially variant Lucy-Richardson algorithm. For PPW =2, images deconvolved by the Lucy-Richardson algorithm using both theoretical PSFs main- tained FWHM=1 throughout most of the FOV. On the other hand, the simulated PSF and the orignal image had FWHM=2 before expanding to 3 at ∼ 0.75θ 2w for PPW = 2. At PPW = 3, the image deconvolved by the idealized analytical PSF no longer maintained a FWHM=1throughouttheentireFOV(Figure6.9). Theidentifiedlocationofthefocalpeak was more accurate for both analytical PSFs than the simulated PSF at PPW = 3 (Figure 6.10). These findings were consistent withthe 0.5’ and 2’ GAR square-channel and the 1’ GAR meridional lobster-eye lenses. However, the source possition was about a pixel off for the image of the source at ∼ 0.75θ c for the 0.5’ square-channel lobster-eye lens with PPW =3 deconvolved using the idealized PSF. In determining the FWHM, some ambiguity arose in instances where the central focal peakconsistedoftwomaxima. Inmostcases,thealgorithmidentifiedthecentralpeakasa singleextenededsource,buttherewere afewinstanceswherethesecondpeakwasconsid- eredseparated. Whenthecenralpeakwasidentifiedasseparated,theapparentFWHMwas calculated from a single peak, resulting in a lower value than was actually being covered. 101 0 1 2 3 4 5 ˆ x −0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 1 2 3 4 5 ˆ y P eak F inder Invariant PSF Spatially V ariant PSF Simulated PSF FWHM (pix el) Source Angle of Incidence ( θ max ) along ˆ x Figure6.9: FWHMvs. θ s foracollimatedsourceimagedbyalobster-eyelenswithGAR=1 and PPW =3 0.0 0.5 1.0 1.5 ˆ x −0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.0 0.5 1.0 1.5 ˆ y P eak F inder Invariant PSF Spatially V ariant PSF Simulated PSF Deviation from Source P osition (pix el) Source Angle of Incidence ( θ max ) along ˆ x Figure 6.10: Deviation of resolved peak location from θ s vs. θ s for a collimated source imaged by a lobster-eye lens with GAR=1 and PPW =3 102 −600 −400 −200 0 200 400 600 0 10 0 10 1 10 2 10 3 10 4 10 5 −500 −400 −300 −200 −100 0 0 50 100 150 200 −10 −5 0 5 10 0 5000 10000 15000 20000 Figure 6.11: The intensityprofile along a focal arm of a 0.5’ GAR square-channel lobster- eye lens image of a source at θ s = 0.75θ 2w deconvolved using the Lucy-Richardson algo- rithm with the simulated PSF Figure 6.11 is an example of two central peaks separating in the deconvolved image of a single source at 0.75θ 2w . The focal planefor the1’ GARsquare-channel was offsetin order to extendthe invari- ant FOV. Setting PPW =3, where differences between the variant and invariant algorithm were apparent, a single source was imaged at varying θ s throughout the image plane. The focal plane was offset by /32 based on the findings from the cross correlation of the focal spot at varying θ s (Section 5.3.6). The FWHM and source location of the invariant algorithm matched the variant through 0.75θ s .thePSNRand η eff remained comparable throughout as well. Offsetting the lens is a viable means of extending the FOV, and is far lesscomputationallyexpensivethan thespatiallyvariantalgorithm. Thoughat larger PPW the cross correlation did not extend far (in Section 5.3.6 PPW =5 only extended the FOV to∼0.6θ 2w ), under more realistic conditionsthe PPW will mostlikely be small enoughto take advantage of this technique. 103 6.3.3 Deconvolution ofTwo Adjacent Sources The Reighly limitwas used todetermine the angular separation needed to properly resolve two adjacent sources [19]. Two sources were simulated at increasing angular separation (θ d ) along the ˆ x focal arm, and the images were deconvolved at increasing PPW.Two sourceswere consideredseparatedand independentiftheminimumbetweentheimagesof thesourceswas≤0.73ofthepeakvalue(theReighlylimit[19]). Thoughtheinputsources areofequalintensity,thedeconvolvedimageisnotnecessarilyasclearlydefined. Whenthe two resolved peaks were of different magnitudes, the smaller peak was used to determine the Reighly limit. To find the angle at which the Reighly limit was met, the simulation was run at 2 extrema, a minimum angle before which the Reighly limit was met, and a maximumangle(initiallyestimatedattwicetheGAR)beyondwhichtheReighlylimitwas first met. Then simulationwas run again at the midpoint of the two extrema. Based on the results, the midpoint would become either the new minima or maxima for the calculation of a new midpoint. Then the new midpoint would be tested and the process repeated. In this manner, the simulationwas able to increase fidelity and more accurately find the point at which the Reighly limitwas first met. Using the Lucy-Richardson algorithm with the ideal PSF, the angle at which the two sources became resolvable initially decreased with increased PPW (Figure 6.13). For the 1’GARsquare-channelandmeridionallobster-eyelenses,withPPW =1,twosourceshad to be separated by approximately 1.7’ in order to be resolved (red data in Figure 6.13). As PPW increased from 2 to 3, the two sources became resolvable at increasingly smaller angles (black and blue data in Figure 6.13 respectively) reaching approximately half the GAR. However, increasing the PPW to 4 (yellow data in Figure 6.13) did not decrease the angle at which the Reighly limit was met (Figure 6.13). Though the PSNR was not calculated,qualitatively,theapparantnoiselevel,evenwiththeaddedsourcedata,matched 104 Figure6.12: Themid-pointvaluesfromthedeconvolvedimageofto2adjacentsourcesfor PPW =1to4 theresultsforasinglesource. TheFWHMand η eff werenotmeasuredbecausethemetric algorithmswere not designed to handle multiplesources. For comparison,a simplepeak finder was used to determinethe locationof the sources on the unprocessed original image. PPW was varied from 1 to 4 as was the case with the deconvolved images. The sources were resolvable at comparable angles for PPW = 1 and2(where theseparationbetween thesourcewaslarger thantheGAR). Atsmallerpixel dimensions θ d remainedlargerthantheGARfortheoriginalimage,thoughitdidappearto assymptoteattheGAR.Usingthedeconvolutionalgorithmimprovedthe2Pointresolution by a factor of 2. The ability to resolve two adjacent off-axis sources was tested for the sources located at θ s . First the two sources were located at 0.5θ 2w (at the edge of the approximate range for an invariant PSF for PPW = 4) and again at 0.75θ 2w (well beyond the range for an invariant PSF for PPW = 4). In both the Lucy-Richardson algorithm using an idealized and simulated PSF for square-channel and meridional lobster-eye lenses, the test sources 105 Figure 6.13: Deconvolved images of 2 sources separated by a) 32’ and b)33’ for a 1’GAR square-channel lobster eye lens with Ni reflecting surfaces. were found to be separated at approximately half the GAR, matching the on-axis results. Offsettingthefocal surfacedidnotalterthecapabilitiesoftheLucy-Richardsonalgorithm. However,at both 0.5 and 0.75θ 2w the spatiallyvariantalgorithmdidnot appear to separate the sources as narrowly as the idealized PSF or the original image. This may be an artifact of the misalignment of f and the tendency for a single source to appear to have split at larger angles. The spatially variant PSF may incorrectly approximate the second nearby sources as the split second peak created by misalignment. 6.3.4 Advantagesof theIdealized Lucy-Richardson Algorithm For single point resolution, when PPW =1or θ s ≤ 0.4θ 2w for larger PPW, the analytical idealized algorithm performs nearly identically to the spacially variant algorithm in terms oflocatingthesourceposition,FWHM,PSNRand η eff . Offsettingthefocalplaneexpands 106 theFOVtonearlythesameextentasthespatiallyvariantalgorithm. Furthermore,theLucy- Richardson algorithm was able to achieve a higher degree of 2 point resolution. Because there is only one set of PSF components to consider, the idealized analytical algorithm is signicantly less computationally intensive and the time required for deconvolution is considerably reduced. The effects of added PSFs is compounded as PPW increases and moremasksareincludedtocapitalizeontheincreasedfidelity. Insituationsthatrequirethe systemtobecapableofautomated-onboardprocessing(forinstancedetectionandresponse to a GRB as with Swift [17]), using the spatially variant algorithm may not be feasible. Coupling the Lucy-Richardson algorithm using the idealized PSF with a PPW of 3 would enable a resolvability of GAR/3 and a 2 point resolution of ∼GAR/2. At the same time, setting PPW to 3 will mostlikely not result in an oversized sensor array. 107 CHAPTER 7: LOBSTER-EYE LENS TRADES AND DESIGN Duetothedifficultyofconstruction,theproposeddesignsforlobster-eyelenseshaveoften been drivenby the limitationsof the manufacturing techniques[13, 22, 28, 38, 40, 42, 43]. Metrologyhasalsobeendesignedaroundtheimperfectionsarisingduringtheconstruction of a prototype [29]. However, understanding the theoretical limitations and the impact of designchoices onthe capabilitiesof a lobster-eyelensis necessary togaugethe approapri- ate goals for enabling technologies and areas of future research. Furthermore, the benefits and limitations of an idealized lobster-eye lens will help to identify appropriate missions that take advantages the unique abilitiesof lobster-eye lenses. 7.1 TheInfluenceofSourceSelectionon MissionDesign X-ray sources vary in the spectral characteristics, intensity, and timescale. The design of any X-ray telescope is driven by the target source’s limiting characteristics[51], lobster- eye lenses will need to be designed with the same considerations. For example, if the target source is a shorter lived transient the FOV will need to be designed large enough to ensure the events will be captured. At the same time, the effective area will need to be designed such that the required number of photons for imaging will be collected within the timescale of the event. On the other hand, imaging relatively faint sources will require a high enough sensitivity to separate the signal from the X-ray background. However, becausefaintsourcesofinterestaretypicallylonglivedcomparedtotransients,thelobster- eye lens may be able to utilize long exposure times (current telescopes have had a total 108 commulativeexposuretimesontheorderof100sofksforsomesources[17]). Furthermore, becauseimagingoportunitiesarenotnecessarilytimedependent,limitingtheFOVtoafew degrees may be acceptable in terms of overall mission goals. Though not necessarily an ASM, a FOV of a few degrees would represent nearly an order of magnitude increase compared to current X-ray telescopes. Similarly, the spectral range and sensitivity will vary based on the sources of interest, driving the selection of the reflecting material and α. 7.1.1 TheImpactof α onSpectral Range Though each α is ideal for a single energy, all situations will require a range of photon energies to be imaged. For example, even though the study using the XMM-Newton data focused on identifying a single spectral line linked to sterile neutrinos, all of the sources included were redshifted between 0.009 and 0.354[7, 9]. As a result, the single spectral lineappeared tobe imagedatdifferingenerigesdependingonthesource beingstudied[9]. For comparison, the data had to be shifted into the same frame of reference to account for the redshift[9]. Because η ii was defined using TER[12], the approximation is more accurate for ener- gies at which the reflectivity of a material will more closely be approximated by TER. At energies below the limit for which TER is an acceptable approximation, the acceptance angle for recorded photons will extend beyond θ c and photons with 3 or more reflections will contribute to the focal spot and focal arms. In this case, η oo will approach 0.25 as more and more channels contribute to the image. Figure 7.3 depicts η ii (α cn ) for a lobster- eye lens with Ni reflecting surfaces while the photon energy was varied for a single lens. The reflectivity of Ni for photon energies less than 6keV do not adhere closely to the the TER model, and as a result η ii deviates from the idealized values. For example at α cn =2, η oo = η ee as would be the case based on the TER model [12] (note, α cn is inversely pro- portional to photon energy). Furthermore, for the Ni lens using the real reflectivity model, 109 η ii quicklyassymptotesasphotonenergydecreases andmoreandmorechannelsat θ i > θ c are included in the image. Ontheotherhand,theeffectiveareaforhigherenergyphotonswillbemoreconstrained than for the ideal energy. For θ c < α, though the unfocused photons will contribute from all channels within α, the fully focused photons will only contribute within θ c .Once θ c is less than the the separation angle between channels, reflected photons that contribute to the image will be heavily dependent upon source-channel alignment. The unfocused pho- tons from channels within α will stillbe imaged because theypass throughthe lobster-eye lens without reflection. As a result, the image will consiste of predominantly the diffuse background. At this point, the lens will act similarly to a coded mask, however, there may still be some contributionsfrom the focal arms. For example, Figure 7.1 depicts η ii versus Figure 7.1: η ii for high energy photons imaged by a Ni lobster-eye lens designed to focus 6keV sources photon energies for a lobster-eye lens consisting of Ni channels idealized for 6keV photon sources as the photon energy was increased from the ideal. Though the nearly no fully focused photons exist within the image by 40keV, the focal arms still have contributions. However, by 100keV, the background will contain ∼90% of the recorded imaged photons 110 (Figure 7.1). As a result, the image will appear to be the diffuse background, and idivid- ual sources are not readily identifiable (Figure 7.2a). Though the lobster-eye lens is no Figure 7.2: A 1’ GARsquare-channel lobstereye lens a) imaginga collimagedhigh enegy sourcewithPPW =1andb)thedeconvolvedimageusingLucy-Richardsonalgorithmsfor a simulatedPSF longer predominantly imaging through superposition in this regime, the modified Lucy- Richardson deconvolution algorithm can be applied. Using the idealized PSF, η ii can be adjustedaccordingtothespecifiedphotonenergyandthenormalizationofthecomponents can be modifiedtoaccount for thedifferingacceptance areas of thefocused and unfocused photons(∼ θ c /α). Individualsourcescanbeidentified,however,theFWHMofthedecon- volved images were approximately 7GAR (Figure 7.2b). Results were similar for 0.5 and 1’ GAR lobster-eye lenses with PPW =1 and 3. For comparison, the BAT instrument on Swift is designed to give source positions within 4’ of accuracy as compared to the Swift X-raytelescope’s18”resolution(or∼1/13 th theresolution)[17]. Usingalobster-eyelens 111 Figure 7.3: The initial range for which a lobster-eye lens will focus more efficiently in place of a coded mask for identification of sources at energies much greater than the energy design energy for α is possible,but imagingwill not be feasible. For a specific photon energy, the ideal α is the first local maxima of η oo (Figure 7.3) [12]. Defining η oo = η ee as the boundary can give an approximate spectral range for energies at which the lens will more efficiently focus the incident X-rays. Assuming the material is chosen such that the entire spectral range can be approximated using TER, for η oo ≥ η ee , α cn willbe between 1 and2 [12] (note, at photonenergies for whichTERis not appropriate, themaximum α cn willbeslightlylarger than 2as inFigure 7.3). Theapproxi- mate spectral range can be found based on the photon energies corresponding to θ min c = α and θ max c =2α. Therateatwhich θ c varieswithrespecttophotonenergy(∆ θ c )dependsuponthemate- rial and the photon energy. Figure 7.4ashows θ c for .8-10keV for Ni, Au, Si and GaAs. Differences in ∆ θ c between materials was more evident at lower photon energies (Figure 7.4a). For all materials, as photon energy decrease, the ∆ θ c increases, narrowing the spec- tralrangeavailableforwhich η oo ≥ η ee . ForexampleforX-raysgreaterthan9keV, ∆ θ c for allfourmaterialswillbelessthan0.5arc minutescomparedwiththe3arcminutevariation 112 Figure 7.4: a) θ c and b) ∆ θ c vs. photonenergy for Ni, Au, Si and GaAs for Au at approximately 3keV (Figure 7.4b). Though a smaller variation in θ c will result in a larger range of energies for which η oo ≥ η ee , θ c itself also decreases as photon energy increases. For example, α will be on the order of 10s of arcminutes for a photon energy of 9keV for all materials. The approximate spectral range for a material such that η oo ≥ η ee (assuming TER is valid for the material in the desired regime) can be found from θ min c and θ max c . Figure 7.5 displays the spectral range for Ni, Au and Si for α between 20 and 100 arcminutes. For perspective, the approximate w to ratios are included for each α.EvenusingAuasthe reflecting material, the η oo ≥ η ee spectral range remains fairly limited, spanning less than 10 keV for α greater than 27’. Choosing between materials with overlapingspectral range capabilities(suchasNiandAu)willrequireconsiderationoftheoverallimpact α willhave on the rest of the lens design. For example, for a specified w and spectral range, choosing a material with a smaller α (i.e. Ni as apposed to Au) will result in a larger and overall structure because α is inversely proportionalto. Designing lobster-eye lenses for hard X-rays (for example > 10keV) or for a broad η oo ≥ η ee spectralrange(forexample>10keV)willrequireeither α tobeanorderofmag- nitude smaller than prototypes which have been produced[27, 42] or new X-ray reflecting 113 Figure 7.5: Spectral Range vs. α for Ni, Au, and Si materials. NuSTAR uses depth graded multilayer reflecting surfaces to extend the spectral range ofthe reflecting surface to78.4keV,Pt/Cfor theinnermirrorsand W/Sifor theouter mirrors[23]. When considering W or Pt independently, the spectral performance was very similar to that of Au. Developing a multi-layer coating may provide a means for extend- ing the spectral range, but no such reflective surface has been tested with with lobster-eye lenses. Because thetheoreticalefficiencyofalobster-eyelensistiedto θ c ,themoreaccurately theTERmodelsthereflectivityofamaterialforagivenenergy,thegreaterthepotentialfor maximizingthe focusing efficiency of a lens. And as photons from channels at θ i > θ c are imaged, higher order reflection will be included in the focus, further degrading the image. For Ni and Au, low energies may not be ideal because of the poor fit of the TER model even (Figure 7.6). 114 Figure 7.6: R v θ i for Ni, Au, and Si 7.2 DefiningLobster-EyeLensCharacteristics Whendefininglobster-eyelenses,Angleidentified α asadefiningcharacteristicofanindi- viduallens[2]. Angelfurther demonstratedhowthe effectivearea for each PSF component can be found from α,O and R and defined the GAR as w/f[2]. However, potential con- straints to the FOV were not discussed. Nor were the relationship between effective area and α with the FOV. Because of the appearance of R, and w in multiple of the equations governinglens design, relationshipsbetween lens parameters can be established. GAR is dependent on w and f.And f is dependent on R and . On the other hand, the FOV, in terms of θ 2w , is dependent only on R and . As a result, the FOV and GAR are interconnected through R and. However,varying α will result in different pairs FOV and GAR. Within 2θ c , under the TER assumption, all photons entering a channel will be imaged. Therefore, the total effective area for a collimated source can be approximated by the total number of channels within 2θ c multiplied by w 2 . And the overall effective area (all PSF components) for a collimated source can be approximated as (2N ch ) 2 . And once again, the ratio of R to also appears in N ch (Equation 6.1). Furtheromore both PSNR and η eff are effected by the photon throughputwhich is ultimatelygoverned by effective area. DefiningR as the ratio of R/,lobster-eye lenses can be defined around dimensionless lens characteristics. The FOV, N ch and GAR can be rewritten in terms of α andR.For 115 square-channel lobster eye lenses, f can be written (R+1)/2. Substituting for f, GAR =2α/(R+1). Normalizing GAR in terms of α (Γ =GAR/α), the normalized geometric angular resolutionis reduced to Γ= 2 R+1 (7.1) Solving forR, Equation7.1 becomes: R = 2 Γ −1 (7.2) The desired FOV of a lobster-eye lens can be defined in terms of θ 2w , for instance, setting the half FOV to be within 0.4θ 2w of the central axis in order to maintain relative invariance of the focal spot throughout the FOV. Therefore, for a given FOV, the corre- sponding θ 2w canbedefined. InEquation5.9,normalizingbyandsubstituting(R−1)/2 for f c and 1/4 for/4, the angular limitswhere θ s ≤ θ 2w becomes: θ s ≤cos −1 R−1 R−0.5 (7.3) Solving Equation 7.3 for θ s = θ 2w : R = 1−0.5cos(θ 2w ) 1−cos(θ 2w ) (7.4) The affective area for a collimated source can be used to ensure that the photon fluence of a given source is large enough for imaging to take place. Because the total number of channels within 2θ c is (2N ch ) 2 , the effective area can be defined from N ch which is dependent on bothR andO (Equation 7.5). N ch = √ 2OR (7.5) 116 Insituationsoflowphotonfluence, or whereeffectivearea isadrivingrequirement,R can be defined from N ch (Equation 7.6). R = N ch √ 2O (7.6) However, because N ch is dependent upon bothR andO, there is flexibility in creating an appropriate effective area. From Equations 7.2, 7.4 and 7.6, by knowing a required FOV, GAR or N ch and the desired spectral range (to determine α), all other lens parameters can be found. Because R, w and can be varied proportionally, setting θ 2w , Γ or N ch only definesR, choosing the final lens dimensions can still be manipulated based on image sensor dimensionality or desired open area. As such, α,R andO can be used to define the non-dimensional design parameters of a lens in order to meet FOV, resolution, spectral and photon fluence capabilities. And from α,R andO, the physicalstructure of the lens can be defined. Though mission requirements may specify pairings which may not be possible by a single lens, multiple lens/sensor pairing can be nested to extend the overall field of view [22, 37, 43]. Similarly, because increasing the PPW reduces the resolution to sub GAR levels,the desired resolutioncan be set as a fraction of the GAR based on the PPW. The equations defined in this section are derived for square-channel lobster-eye lenses. For meridional lobster-eye lenses, a correction factor accounting for the different in focal length arrising from the goemetric influences will need to be included. However, for a first order approximationand initialdesign,usingthesquare channel parameters shouldbe sufficent. After establishing the initial proposed design, the final lens design parameters can be adjusted to account for the reduced f inherint in meridionallobster-eye lenses. 117 7.2.1 Setting the Lens DesignTrade-Space Settingatrade-space foran individualmissionwillrequire balancingoverallmissiongoals with the individual lens capabilities. Designing around non-dimensional characteristics enables flexibility in design such that multiple options can meet the mission requirements. For example, α can be set in the 25-30’ range for Ni or 35-40’ range for Au in order to imaging a source with a strong signal in the 5-7 keV range. Regardless of the material, potential lenses can be designed with the same Γ andR value. Basing w from the size of pixels in an image sensor and scaling to meet the desired PPW, the Ni lens will have a smallerGAR(andresolution),buta larger overallstructuredue totheincreased. Individ- ually, the increased structure may not be prohibitive,but when nesting multiple lenses, the overall structural size for each configuration grows by a factor of N 2 lens . Three basic approaches to lens design can be defined based on setting field of view, resolution or senstivityas the driving requirement (assuming spectral range will always be well defined by the target sources). The difference between each approach rests in howR is define. For example, for a faint source where fluence is the primary consideration, N ch will drive the rest of the design in order to acheive the necessary effective area. Figure 7.7 is a visualization of howR connects FOV, resolution, spectral range and N ch . Each design approach starts at a different edge and works its way throughR to define potential Lobster-Eye lenses. For an FOV centric design, θ 2w can be determined from θ inv for all potential PPW. However,because individuallensescan benestedtoextendtheFOV ofthesystem[22,37, 43], θ inv has to be determined from the fraction of the FOV covered by an individual lens. As a result, several lens designs can be generated based on the number of lenses required to cover the entire mission FOV, each with a specific θ 2w . For each θ 2w ,R is found using Equation 7.4. 118 Figure 7.7: A visualization of how the different lens characteristics are interdependent throughR. Similarly,fortheresolutioncentricdesign,a setofGARbasedonthepixeldimensions and PPW that will meet the required resolution will be determined initially. A set of α for aspecific sourceandpotentialmaterialscan beusedtodeterminea setof Γ. Foreach Γ,R is defined (Equation7.2). For designs driven by the energy of the source, N ch needs to first be determined such that the resulting effective area will be capable of capturing a large enough photon count within an estimated exposure time (τ). Previous studies have assumed that as few as 100 imagedphotonswouldbesufficienttodetectasource[47]. However,basedonsimulations, when only ∼ 100 photons are imaged, η eff ≈ 0.6 for deconvolved images. Increasing the imaged photon count to ∼ 1000 resulted in the deconvolved images with η eff ≈ 0.9. Using 1000 as a minimum photon count, the effective area can be found from the source characteristics and τ.R can then be found from the resulting N ch (Equation 7.6). Because N ch is dependent on the open area fraction as well asR,O can be estimated based on 119 pixel dimensions, PPW and t (based on the scale of prototypes or assumed fabrication technique). In all cases, onceR is known, the rest of a lens’ dimensions can be found. Γ can be set from Equation 7.1 and the potential GAR can be found from α. θ 2w can be found from Equation7.3,andthereforeFOVcanbedefined. N ch canbedeterminedforthegivensetof O or PPW. Assumingwcan beestimatedfromreasonablesensordimensions,thephysical dimensionsof the lens can be found from α andR. To ensure there is no vignetting throughout the desired FOV, the overall lens and sensor dimensions need to be designed with enough of a margin to include the entire PSF. The outermost edge of the lens needs to be an additional 2θ c beyond the desired FOV (to account for ±θ s ). For the imaging surface, because the background extends approximately (R−)θ c from the center of the PSF, the imaging plane needs to span 2((R− f)/tan(θ s )+((R−)θ c )/cos(θ s )) at a minimum. The impact offsetting the focal surface will have on a lens design will depend on how the offset is factored into the design. For a given FOV, if θ 2w is calculated to adjust for the larger θ inv (i.e. for example, θ 2w = θ 2w /0.6inplaceof θ 2w = θ inv /0.4), the GAR will be smaller butR will be larger resulting in a larger physically structure and larger effective area. On the otherhand, if the FOV is calculated for a given θ 2w (R remains the same and the GAR can be adjusted for the slight increase in f). Therefore, if the goal is to increase the resolution or effective area of a given FOV, offsetting the focal plane needs to be factored in at the beginningof lens desing. However,if the goal isto extendthe FOV of a givenlens, the offset needs to be considered afterR has been defined. 7.2.2 Non-ImagingTrades A thorough mission trade study requires the inclusion of all subsystems, however for the purposes of understanding the limitations of lobster-eye lenses, the case studies used as 120 examples will focus solely on imaging requirements. Because support structure, manu- facturing techniques, and specific sensor technology are not being evaluated, mass, power, thermal and computational requirements are not being discussed in detail. For example, sensors (from a capability and dimensionality stand point) will be based on existing tech- nology. As such, power, comunication and theremal considerations will be assumed con- sistentwith existingsystemsand not discussed when evaluatinglens design. Similarly,the current techniques used for thermal regulation of the reflecting surface, such as the baffles employed by Swift [17], will be assumed to be applicable to lobster-eye lenses such that thermal conditionscan be ignored. In general, mass increases with size. Comparitively, the relative masses of lenses themselves can be considered by estimating the increase in material volume, assuming material density is approximately constant (i.e. m 1 /m 2 ≈V 2 /V 1 for example, a uniform substrate). Scaling potential lobster-eye lenses to similar dimensions as previous proto- types will be considered sufficient for sizing considerations. Mass differences between meridional and square-channel lobster-eye lenses exist due to the geometric differences in design. Even within a single channel meridional lenses will contain more material. The plate thickness remains constant in meridional lenses, where as the walls for square- channel lenses will taper thoughout their length (for lenses with the same characteris- tics). For each wall, the excess material will be in the form of a wedge created by cut- ting (w f −w b ) from the inner edge of a meridional channel. The difference in volume of a single channel is then 4/3w 2 ave θ ch /(α)(1+w f /(2w ave )), accounting for all 4 wedges. The overall ratio of the volumes of meridional and square-channel lobster-eye lenses is 4/3θ ch /(α)(1+w f /(2w ave ))=4/3(1+w f /(2w ave ))/( √ OR). FortheNilensesdesigned fora6keVsourcewithO =1/2thatweretested,the0.5’,1’and2’GARmeridionallenses have approximately 3, 6 and 12% additional mass respectively. The difference in mass is inversely proportional toR;asR increases the difference will eventuallybecome negligi- ble. However, this approximate increase in mass is based on the channels aligned with the 121 central axis and neglectsvariationsin channel and wall dimensionsat varying θ i .Atlarger angles, the meridional channels will narrow while the walls between square-channel nar- row. As a result, the increase in mass of the meridional lens is somewhat underestimated, though, for lobser-eye lens with a FOV on the order of degrees (which is a more realistic expectation), the difference shouldbe small enough to ignore. 7.3 CaseStudy1: SterileNeutrinoSurvey TherecentlydiscoveredspectrallinesintheXMM-Newtondataof73galaxyclustersneeds furtherstudytotrulyidentifythecauseofthesignal[9]. However,thesignalwasconsistent withtheoreticalmodelspredictingtheexistenceofsterileneutrinos,aproposeddarkmatter candidate[9]. Consideringthepotentialforbeginingtounraveloneofthebiggestmysteries facingastronomyandfundimentalphysics[7,9],designingamissionspecificallytosearch for, image, and study the sky in the spectral range of the signal would draw interest. The signal was identified at 3.5keV (the chosen galactic clusters all were redshifted between .009and .354)[7,9]. Thesignalitselfwasconsidered relativelyfaint, thoughdistinctfrom the known spectral lines within the 2-4keV range [9]. In the study, the photon flux were typically on the order of 10 −5 to 10 −6 photons/(cm 2 s) range [9] (the sample designs are based assuming the photon count will be closer to 10 −5 photons/(cm 2 s)). Exposure times were listed on the order of 300 ks [9]. Using the data from the XMM-Newton study, a missionutilizinglobster-eye lenses can be designed in order to survey the skyand identify additionalsources. Duetothe needtoidentifythe sourcewithintheX-ray background[9], photon fluence will be the drivingdesign constraint. 7.3.1 FirstPass: BaselineSystem The XMM-Newton’s MOS CCD contains 600×600 40µm pixels within 2.5×2.5 cm [35]. Prototypelobster-eyelenseshavebeenconstructedwithchannelsontheorderofmagnitude 122 matching the XMM-Newton’s MOS CCD (assuming the channels will result in PPW ≥2). Schmidt style 40µm foils have been assembled with 50µm separations [27]. And square channel glass MCPs with w=200µmandt =40µm have also been tested [42]. Basing the design from the XMM-Newton’s MOS CCD, at PPW =2 and 5, the area of a given channel is 6.5×10 −5 and 4×10 −4 cm 2 respectively. Because 1000 photons were found to provide sufficient imaging levels, initial lens designs were constructed in order to image 1000photonswithin τ =300ks. Table 7.1showsthe parameters for potentiallenses imaging 1000 photons with PPW = 2 through 5 for a lobster-eye sterile nuetrino survey using XMM-Newton’s MOS CCD. Table 7.1: Lobster-eye lens paramters for a survey imaging the possible sterile neutrino decay spectral line at ∼3.57keV with a photon flux on the order of 10 −5 photons/(cm 2 s). The design goals are scaled for collecting10 3 photons PPW w (µm) O(t =40µm) N ch R Γ θ 2w (deg) 2 80 0.44 1291 1080.1 0.0018 1.74 3 120 0.56 861 684.9 0.0029 2.19 4 160 0.64 646 500 0.0040 2.56 5 200 0.69 517 393.3 0.0051 2.89 For comparison,R for the 0.5’GAR test lens was 95.2 and N ch was approximately 96, both of which were significantly smaller than the results in Table 7.1. The increasedR will result physically larger lens than was assumed for the test lenses. To find R, f and , α needs to be defined for a given material. To account for redshift, the spectral range needs to span the ∼2-4keV range to ensure the potential neutrino emission line is within the capabilities of the lobster-eye lens (larger redshifts were observed to be too faint to contribute adequetly to the previous study [9]). Though Au and Ni would only require an α of approximately 70’ and 50’ respectively (Figure 7.5) to span the 2-4keV range, both deviate drastically from the TER model for photon energies below 6keV (Figure 7.6aand 123 b) and would produce images that included more than 2 reflections. For X-ray energies below 4keV, the reflectivity of Si can still be modeled by TER fairly well. Though the reflectivity does fall bellow 0.7 as θ i approaches θ c , the reflectivity is approximately 0 for θ i ≥ θ c (Figure 7.6c). Si will require an α ≈25 in order to span the 2-4keV range. With α =25 , the potential GAR would be between 0.05’ and 0.13’ for PPW between 2 and 5. Though the GAR for PPW = 5 is greater than PPW = 2, the deconvolution for single point resolutionwill be accomplished to a higher degree, while two pointresolution would be fairly similar. Therefore, the resolution will be in the range of current X-ray telescopes. However, using Si as the reflecting material, R will be ∼14m regardless of PPW, resulting in f ≈ 7m. A 7m focal length is significantly larger than the prototype and proposed lobstery-eye lens designs which are typicaly on the order of a meter or less [13, 22, 28, 38, 40, 42, 43]. Furthermore, of the existing deployed X-ray telescopes, only Chandra and NuSTAR have focal lengths on the order of 10m[3, 23]). 7.3.2 Second Pass: Scaling Down the Lens Design In order to design for a more manageable size, either τ must be increased, the required photon count decreased, α increased or a combination of all three. When N ch is scaled for τ that is increased by an order of magnitude, R is decreased to ∼4.5m. Similarly, if N ch isrecalculated under the assumptionof a required photoncount loweredto 500,R will be reduced to ∼3m, with f = 1.5m for PPW = 5. Though increasing τ and lowering the required photoncountreduced the physicalscale of the lensby an order of magnitude,it is stillnearly twice the size of most proposed designs. α can be varied slightly with Si as the reflecting material. At α =28 , η oo >0.25 will span approximately 1.8 to 3.7keV, sufficient for imaging potentially redshifted 3.57keV source. The relatively small change in α (3’) reduces the focal length by nearly 20cm. Though using a metal reflective materials (such as Ni and Au) would further reduce the 124 physical dimensions of the lens by a factor of 2, the loss in image quality would be pro- hibitive. Instead, if GaAs is used in place of Si, α = 40 would result in the ∼1.8 and 3.7keV range based on the reflectivity model provided by Underwood [51] (Figure 7.8). Though the use of GaAs for an X-ray reflective surface is not as well establishedas metals suchasNiorAu,experimentationhasshownthefeasibiltyofGaAsas areflectivematerial [16]. And though α is ∼ 40% larger for GaAs compared to Si, the reflectivity at 3keV 20 25 30 35 40 45 50 α (arcmin) 1000 2000 3000 4000 5000 6000 7000 8000 Bandwidth Extrema (eV) GaAs Figure 7.8: Spectral Range vs. α for GaAs does not deviate too drastically from the TER model as had been the case with Ni and Au. Using τ = 3000ks and requiring 500 photons to be collected, f for the GaAs lobster-eye lens would be approximately 1m. Table 7.2 shows the redesigned lobster-eye lens param- eters using GaAs as the reflecting material. Assuming an offset focal plane and designing for an spacially invariant PSF, the FOV for PPW = 3 is the largest in the set of lenses shown in Table 7.2 at 4.9 ◦ . The GAR would be 0.4’, with the potential for 0.13’ single point resolution and 0.2’ two point resolution after deconvolution. Table 7.3 has the full physical design for the lobster-eye lens with PPW = 3. Though most of the parameters listedinTable7.3are feasible,theoveralllenswidthmayrequirenestingsmallersegments based on previous prototype dimensions. However, the potential for a FOV of 4.9 ◦ would 125 Table 7.2: Lobster-eye lens paramters for a survey imaging the possible sterile neutrino decay spectral line at ∼3.57keV with a photon flux on the order of 10 −5 photons/(cm 2 s). Thedesigngoalsarescaledforcollecting500photonsusingGaAsasthereflectingsurface PPW w (µm) O(t =40µm) N ch R Γ θ 2w (deg) 2 80 0.44 361 312.5 0.0064 3.24 3 120 0.56 241 196.4 0.0101 4.09 4 160 0.64 181 142.6 0.0139 4.81 5 200 0.69 145 111.8 0.0177 5.43 Table 7.3: Potential lobster-eye lens dimensions designed using GaAs to a sources with a flux ∼10 −5 photons/(cm 2 s) Parameter Value EnergyRange......... 1.8-3.7keV α .................... 40’ PixelWidth........... 40µm ChannelWidth........ 120µm ChannelLength........ 1.03cm FocalLength.......... 1.02m GAR................. 0.4’ Resolution(single point) 8.2” LensWidth............ 17.4cm FOV.................. 4.9 ◦ representasignificantleapinFOVforX-rayimagingoptics,withoutsacrificingresolution. The 8.2”singlepointresolutionison the order ofmostX-ray telescopesas showninTable 1.2. However, in order to scale the lens dimensions to a more realistic size, τ was assumed to be an order of magnitude longer than the total exposure time in previous missions. Assuming τ on the order of 1000ks may not be a reasonable. Furthermore, the required photon count was set to 500, at which point the imaging capabilities are still feasible with η eff ≈0.87fortheNitestlensesimagingacollimated6keVsource. However,smallerpho- ton counts were shown to result in significantlydegraded images. And because the reflect- ing surfaces were considered perfectly smooth, scattering due to surface roughness will further degrade the image. Furthermore, background noise was not modeled nor included 126 in the determination of image quality. Therefore, more than 500 photons may be required forimagingundermorerealisticconditionsandresolvingsourceswithlowerphotoncounts may not be possible. Designing for larger photon fluence would ensure sources similar to those in the XMM-Newton study would still be within the range of reasonable imaging capabilities. A simulation of 3 randomly positioned collimated sources were imaged by a test lens fitting the parameters in Table 7.3. The sources were based on the energy and photon flux levels given in reference [9], with the energies adjusted to allow for redshift. Each of the 3 sources was assumed to be at a slightly different redshift resulting in 3 different energy spectrums represented in the single image. The simulated image (Figure 7.9a) was then deconvolvedusingthemodifiedLucy-RichardsonalgorithmwiththeidealizedPSF(Figure 7.9b). Due to the low photonflux and 2500×2500 pixelfocal plane, the background in the Figure 7.9: The (a) unprocessed and (b) deconvolved images of 3 sources based on the description in reference [9] by the lens defined in Table 7.3 simulated image is faint, even when emphasising the signal in the image representation of the data (as outlined in Figure 7.9a). However, when the imaged is cropped to a ∼20×20 127 pixel frame surrounding the central peak of each source, the narrowing of the central peak and reduction in background signal is evident (Figure 7.9). 7.4 CaseStudy2: Wide FieldGRBSurvey Gammarayburstsareintenseandshortlived,rangingindurationfrommillisecondstotens ofminutes,yetGRBsare amongthebrightestobjectsintheskyandoftenaccomaniedbya high energy afterglow[17]. In order to capture the afterglow of GRBs, Swift was designed toautonomouslyslewtoanidentifiedsource[17]. SwiftconsistsoftwoX-raycomponents, a high energy detector (BAT) for identifying the intial GRB burst, and the main imaging telescope(XRT)fortheX-rayafterglow. TheBATisacodedmaskwithaCdZnTedetector which has the ability to detect 15-150keV X-rays (with a threshold of 10keV)[17]. The XRT consistsof nestedWolter mirrorswithan effectivearea of 110cm 2 and a FOV=23.6”. TheXRTwasdesignedtoimagetheafterglowofGRBsbetween0.5-10keV[17]. Oncethe BAT detects the high energy output at the onset of GRBs Swift autonomously reorients in order to image the afterglow in soft X-ray within1 minute [17]. 7.4.1 FirstPass: BaselineSystem Because ofthetemporalconstraints,thelobster-eyelensdesignwillcenteraroundexpand- ing the FOV of the imaging system. Furthermore, combining the ability to detect high energy burst from the onset of GRBs while imaging at lower energies would eliminate the lostimagingopportunitiesduringthe1minuteslewtimeusedbySwift. Thoughadifferent implementation, a dual imaging design has been proposed using a hybrid Schmidt style array with a Kirpatric-Biaz telescope [20]. And though the energy resolution would need tobevariableallowingforahigherenergyresolutionforimagingconditons,detectorshave had variable energy resolution depending upon the spectrum in study [4]. 128 Figure 7.10: Spectral Range vs. α for Pt Ideally, the lobster-eye lens would be designed to extending imaging capabilities into the hard X-ray regime, however, based on the materials studied the α required for hard X-rays are too prohibitiveto consider possible for square-channel lobster-eye lenses in the foreseable future (as was discussed in Section 2.1.1). Though a square-channel design is not realistic, a meridional design may be feasible. For platnum, which has been shown by NuSTAR to be a good X-ray reflector in below their k-absorbtion edge at 78keV [23], the spectral range at α =15 spans approximately 10-20keV (Figure 7.10). Higher energies can still be captured using the lobster-eye lens as a quasi-coded aper- ture. Because GRBs are temporallyand spaciallyseparated, individualburstscouldstillbe identified, though at a lower resolution. The BAT uses a 50% open random coded mask to cover a FOV of θ inv 100 ◦ ×60 ◦ or 1.4 sr[17]. However, because of the design, the mask is much larger than the detection area resulting. Compared to the compared to the 2.7m 2 covered by the mask, the detector itself is 1.2×0.6 m with each pixel spanning 4 mm[17]. Operating in a similar spectral range (3-78keV), NuSTAR utilizes a CdZnTe detector as well, however, each pixel is 0.6 mm wide with four 32×32 pixel array detectors [23]. Because a larger w will result in a 129 much larger instrument(especialy for a small α which results in large), the near order of magnitude difference in size makes the NuSTAR configuration more appealing for design- ingalobster-eyelens. However,thepixelsonNuSTARareapproximatelythreetimeslarger than the 200µm channels from the neutrino study. For comparison, Swift’s XRT CCD has approximate pixel widthsof 67 µm (600×602 pixelscovering40×40mm) [17]. The lobster-eye lens design will center around FOV, which is based on θ inv . A single lens can not accomodate the entire 1.4 sr FOV, therefore nested lenses will be necessary. Nesting affords greater flexibilityin designingthe overallFOV and the indiviuallens FOV and resolution. For data management purposes, only the individual lens that is capturing an event would have to be active at any given time. For an individual lens to have a FOV greater than 10 ◦ , θ inv wouldhavetobeatleast5 ◦ requiring θ 2w to be greater than 12.5 ◦ (using θ inv =0.4 θ 2w formeridionallenses). Table7.4showsthelensparametersfordifferent values of θ 2w . Because w is dictatedby the pixelsize, in all casesO willbe .88 and .96 for PPW =1 and 3 respectively,assumingt =40µm. Table 7.4: Meridional lobster-eye lens paramters for a GRB study where hard X-rays will be detected using the lobster-eye lens as a coded mask and X-rays in the 10-20keV range willbeimaged. Assumes α =15 andt =40µmusingasensorwiththesamepixeldimen- sions as NuSTAR. 2θ inv (deg) θ 2w (deg) R Γ GAR for α =15 N ch (PPW=1-3) 10 12.5 21.9 0.09 1.8’ 29-30 15 18.75 9.9 0.18 3.57’ 14-14 20 25 5.84 0.29 5.8’ 8-9 Asaresultofusingan α thatisextremelynarrow(i.e. ontheorderof∼100×w),the resultingR valueswithinthecalculatedrangewillresultinalarge radiusofcurvatureand, therefore, a refined GAR. However, the resulting overall size of the lens will be large and massive with sizeable reflecting plates. Given the pixel size from NuSTAR,for PPW = 1 130 and 3, would be 13.75 and 41.25 cm respectively. To build a lens that meets the values given in Table 7.4, R would be between 80cm (PPW = 1, FOV=20 ◦ ) and 9m (PPW = 3, FOV=10 ◦ ). Furthermore, a viable structural integrity while using t = 40µm may not be a reasonable assumption. Testing the structural integrety of a lenses a varyingO will be essential to the viability of large lenses with extensive open areas. Further comparing the structural properties to existing lenses with nested concentric paraboloids will illuminate theadvantagesanddisadvantagesoflobster-eyelensesfromastructuralstandpoint,aswell as identified the limitto usuableO. 7.4.2 Second Pass: Refinement andLimiting Factors Desinging a meridional lobster-eye lens with R on the scale of meters is significantly larger than most prototypes that have been fabricated or proposed. For example, a pro- posed Schmidt style array was designed with f = 0.5m = 9cm, w = 1.75mm, and t = 210µm[20]. Similarly, typical MCP style ASM’s have been proposed with R < 1m [37, 43]. However, there has been a proposed meridional style lobster-eye lens had an R = 20m, f = 10m (under the assuption that f = R/2), w = 700µm, = 28cm, and t =100µm[22]. In all cases, the proposed ASM have involved nesting individual lobster- eye lenses to create an overall lens [20, 22, 37, 43]. But with the bulk of proposals requiring R and f that are less than a meter, the design will be refined to reducing the overall lens size (with f approaching 1m). In order to scale down the lobster eye lens options, α can be adjusted. Considering Pt with α = 20 shifts the spectral range to approximately 7-14keV (Figure 7.10), overlaping the upper half of Swift’s XRT spectral range [17]. For α =20, the GAR in Table 7.4 will become 1.8’, 3.6’ and 5.8’ for the 10, 15 and 20 degree lenses. Withthegoalofimagingtransients,awideFOVtakesprecidencetoGAR.However,as θ 2w increases, Γ increases more rapidly, as was the case in Table 7.4. Increasing the FOV from 10 to 15 ◦ resulted in a smaller loss in GAR than expanding from 15 to 20 ◦ (1.8’ as 131 compared to 2.2’). Furthermore, due to the transient nature of GRBs, the assumption that the events will be separated temporally and spatially allows for single point resolution to be the primary consideration (instead of two point resolution). OffsettingthefocalplanecanhelpextendtheFOVforagiven θ 2w (andR). Formerid- ional lenses, offsetting the focal plane extended the invariant FOV to 0.6θ 2w . The offset focal plane extends the FOV of the lens from 10 ◦ ,25 ◦ and 20 ◦ to 15 ◦ , 22.5 ◦ and 30 ◦ respcetively. Doubling t will also most likely be necessary for a more structurally sound lens. To fully test the lens structural integrity, FEM models of the lens including the lens substrait, adhesion or attachement methods, and external structure need to be designed. Setting t = 80µm(similartopreviousdesigns[22]),O is0.78and0.92forPPW =1and3respectively. Table 7.5 contains the updated lens parameters. Table 7.5: Meridional lobster-eye lens paramters for a GRB study where hard X-rays will be detected using the lobster-eye lens as a coded mask and X-rays in the 7-14keV range will be imaged using an offset focal plane. Assumes α =20 andt =80µmusingasensor with the same pixel dimensionsas NuSTAR. 2θ inv (deg) θ 2w (deg) R Γ GAR for α =20 N ch (PPW=1-3) 12.5 10.4 30.8 0.06 1.2’ 37-42 15 12.5 21.9 0.09 1.8’ 28-30 22.5 18.75 9.9 0.18 3.57’ 13-14 30 25 5.84 0.29 5.8’ 8-9 Becauset=80µmdidnotdecreaseO toodrasticallyforthecalculatedw,thenumberof channelswithin θ c remainedrelativelyunchanged. Asaresult,thephotonfluenceshouldbe fairlysimilarfor α =15 and20’. However,willbe10.3and30.9cmfor PPW =1and3 respectively,andRwillrangebetween60.1cm(PPW =1,FOV=30 ◦ )and6.76m(PPW =3, FOV=15 ◦ ). Because N ch will only be 8 to 9, the FOV=30 ◦ design would not result in an 132 adequateopenarea. Ontheotherhand,theFOV=15 ◦ designresultsinlensesthatarefartoo large. TheFOV=22.5 ◦ ,PPW =3lenswouldrequiringanRof3.06mand f of1.61minthe offset-meridional configuration, which is still much larger than most previously proposed lobster-eye lens designs [20, 37, 43]. Because the GAR was calculated from the ideal square-channel geometry, the adjusted offset-meridional GAR will be 3.85’ with a single point resolution of 1.28’. The overall lens size required for the offset-meridional lobster- eye lens with a FOV=22.5 ◦ and PPW =3is1.2×1.2m, which is significantly larger than the prototypes which have been developed (in some cases, orders of magnitude larger) [20, 22, 37, 43]. Reducing the PPW to 2, however, results in a design with = 20.6cm, R = 2.04m, f=1.07m, GAR=3.85’, and a single point resolution of 1.93’. The lens width for 22.5 ◦ and PPW =3is83×83cm, still approximately 10 times the size of previous designs [22]. At PPW = 1, the design begins to become more managable with R = 1.02m, f = 0.54m (including the offset), howeverthe GAR and resolutionare 3.85’. Furthermore, the decon- volved images typically contained more noise for images produced when PPW = 1, and twopointresolutionwaslimitted. Thelensdimensionsforthefull22.5 ◦ areapproximately 40×40cm which is still an order of magnitude larger than the previous prototypes (5-8cm [43, 22]). One of the factors driving driving the design of a large lobster-eye lens centers around α needing to be small for a bandwidth spanning either 7-14keV or 10-20keV. Reducing the the energies at which the lobster-eye lens will image, and thus reducing α would help reduce the size of the lens, however, the breadth of the bandwidth would also be reduced. On the other hand, the relatively large pixel size of 0.6mm resulted in large channels (and thus a large overall lens). Decreasing pixel size would also reduce the overall size of the lens even if α remained in the 15’-20’ range. For example, if the pixel size could be scaled down to the 1/3 the size, the channel dimensions at PPW = 3 would match the dimensionsofthechannelsinthedesignutilizingPPW =1withpixelsat0.6mm. However, 133 Table 7.6: Potential lobster-eye lens dimensions designed using Pt for GRB imaging and assumingpixel sizes would be reduced to 0.2mm Parameter Value EnergyRange......... 7-14keV α .................... 20’ PixelWidth........... 0.2mm ChannelWidth........ 0.6mm ChannelLength........ 10.3cm FocalLength.......... 1.615m GAR................. 1.26’ Resolution(single point) 0.42’ LensWidth............ 69cm FOV.................. 12.5 ◦ becauseN ch isapproximately14,theeffectiveareawillbelimitted,andthetotalnumberof collectedphotonswillbefartosmallforimagingpurposes. Swift hasobservedGRB X-ray afterglowwithphotoncountrates onthe order of 0.09photons/(cm 2 s)duringthe first 300s of imaging[49]. Assuming the photon count rate will average 0.09 photons/(cm 2 s) over a 300s exposure, a lobster-eye lens designed around a FOV=22.5 ◦ with w = 0.6mm would resultinapproximately68photonsbeingcaptured. SettingtheFOVto15 ◦ withw=0.6mm andt =80µmwillresultinapproximately323photonsbeingimaged. NarrowingtheFOV to12.5 ◦ ,thew=0.6mmandt =80µm willcaptureapproximately700photons. However, theoveralllenswillhaveR=318cmand f =161cmbefore addingtheoffset. Theoverall lens width necessary to cover 12.5 ◦ would be 69cm. Table 7.6 displays the potential lens parameters for a lobster-eye lensassumingthe pixelsize could be designedto 0.2mm. The scale of this lobster-eye lens is over an order of magnitude larger than any constructed prototype. The feasibility of constructing a lobster-eye lens on this scale would have to be demonstrated, even if tiling smaller segments is employed. Furthermore, an array of 5×5 lobster-eye lenses with the current 12.5 ◦ FOV will have to be nested to cover a 60 ◦ overall FOV (similar to the BAT aboard Swift [17]). Ignoring the structure required to hold each individual lobster-eye lens, the nested array would span at least 3.2m. And though larger, 134 thelobster-eyelenswouldbeonthesameorderofmagnitudeastheoverallsizeoftheBAT aboard Swift [17]. Though considered bright [17, 49], the temporal constraints of GRBs and the effective arearequiredfortheminimalphotonfluencealsodrivethesizeofthelobster-eyelens. Even if the pixel size was further reduced to ∼60µm scale in the soft X-ray regimed, the FOV wouldneed to be restricted to within10 ◦ and the designsthe overallsize of the lobster-eye would remain relatively consistent for a given photon fluence. If longer exposure times of the X-ray afterglow are utilzed, a greater fluence can be achieved for smaller lobster-eye lenses, but some of the temporal features at onset would be lost. A simulation of a randomly positioned source modeled after a GRB was imaged by a test lens fitting the parameters in Table 7.6. The sources were based on the energy and photonfluxlevelsgiveninreferences [17,49], withthespectrumspanningtheentirerange available for the test lens. The test lens imaged 2 separate simulations, a 30 second high energy burst, spanning 20-150keV, followed by a 300 second low energy afterglow span- ning7-14keV.Thesimulatedimages(Figure7.11aand7.11c)werethendeconvolvedusing the modified Lucy-Richardson algorithm with the idealized PSF (Figure 7.11b and Figure 7.11d). The high energy GRB was deconvolved as expected, the background resolved into a large spot which is sufficient to identify the source (once again the image representation of the data overemphasized the signal resulting in a spot that is larger than the FWHM). The afterglow was also resolved, however, the image appears to have more noise than had been the case for monochromaticsources. For a viable lobster-eye lens to be desgined with hard X-ray capabilities, not only will manufacturing techniques have to be capable of creating channels with α in the 20’ range, but large overall structures need to be manufacture. Furthermore, desiging the appropriate overlap between the FOV of adjacent lenses needs to be studied in terms of increasing imaging capabilities. Finally, a detector which can propperly adjust between high energy detectionand imagingat lowerenergies on orbitneedsto be propperlydesignedand tested 135 and hard X-ray sensor technology will have to be designed with pixel dimensions that are 1/3 the size of current technology. 136 Figure 7.11: The images of a source modeled after a GRB ((a) unprocessed & (b) decon- volved) and its afterglow ((c) unprocessed & (d) deconvolved) based on the description in references [17, 49] by the lens defined in Table 7.6 137 CHAPTER 8: CONCLUDING DISCUSSION Since 1979, a few years after the reflective nature of long-bodied crustacean were first described [32], lobster-eye lenses have appeared in the literature as a potential for wide fieldX-ray telescope[2]. Because ofthedifficultyinconstruction,themajorityofresearch has centered on potential fabrication attempts. And though prototypes have been built and tested,thetheoreticallimitationsofthelenseshasbeenlargelyleftoutofthediscussion(as described in Section 2.1). Lobster-eye lenses have continued to draw interest due to the advantage of decoupling the strict dependence of FOV on the θ c . The potential to image over an extended FOV in hard X-ray would be invaluable. However, just as in the Wolter designs, θ c is a limiting factor in the design and spectral range of lobster-eye lenses. To truly take advantage of the potential of lobster-eye lenses, either manufacturing techniques that can enable extremely small α or new materials with different X-ray reflective properties need to be found. 8.1 ImplicationsoftheFindings Thedetailedstudyofthefocusingmechanismoflobster-eyelensesdirectlyledtothedevel- opement of a deconvolution algorithm as well as criteria for building a trade space for mission design. Furthermore, a better understanding of the limitations in the design have provided some pointsof guidence for future design goals and research. 138 8.1.1 Summaryof Findings The focal properties of square-channel and meridional lobster-eye lenses were described in detail. In both cases, R and f were redefined to account for the dimentionality of the lens structure. η ii was also found to be comparable to previous findings of flat lobster-eye lenses (R= ∞) under the assumptionof TER [12]. In one dimension, both the focused and unfocused properties of the PSF were defined. For the unfocused background, an approximation was used in order to average all possi- ble source-channel alignments. For the central focal spot, the one dimensional focus was defined for both meridional and square-channel lobster eye lenses under the assumption thatonly singlyreflected photonswouldbe includedinthe focused portionsof the PSF. To constructatwodimensionalPSF,theonedimensionalPSFsin ˆ xand ˆ ywereconvoledforall 4 components (the central spot, focal arms and background). Comparison with simulated collimated sources verified the analytical approximationof the PSF. Because current imagingdetectors are planar, the effects of non-ideal imagingsurfaces wasstudied. Specifically,theeffectsduetothereductioninfocallengthforoffaxissources while using a focal plane was defined. Both a spatially variant one-dimensional focal spot andtherangeofspatialinvariancewere definedfor meridionalandsquare-channel lobster- eye lenses with varying pixel resolution. Deconvolutionalgorithmswere designed and tested for square-channel and meridional lobster-eye lenses assuming a collimated, monochormatic source. Weiner Filter and Lucy Richardson algorithms using simulated and theoretical PSFs were compared. Sub-GAR resolution was achived for both single point and two point resolution using a modified Lucy-Richardson algorithmwith an idealized theoretical PSF and an offset focal plane. Lobster-eye lens design trades based on the interconnection of FOV, GAR, effective area, and α were developed. Additionally, the relationship between α and the potential spectral range of efficiently focused photon energies was investigated. Non-dimensional 139 design constraints were identified that allowed for mission goals and target source proper- tiestodrivelensdesign. Twopotentiallobster-eyelensmissionswere identifiedand lenses were designed. A square-channel lobster-eye lens designed for search for sterile neutrinos utilizing GaAs as the reflecting material. The mission parameters were based on finding from the XMM-Newton study [7, 9]. Next a meridional lobster-eye lens was designed to detect high energy GRBs then image the X-ray afterglow using Pt reflecting plates. The mission parameters were based on the capabilities of Swift [17]and NuSTAR[23]. In both cases, severaladvancementsin technologywouldbe necessaryin order tofabricate a func- tional lobster-eye lens which could meet missiongoals. 8.1.2 Deconvolution andthe Potentialfor Sub-GAR Resolution Dating back to Angel’s initial study, lobster-eye lenses are typically discussed in terms of the potential for wide FOV with the resolution of the instrument often assumed to be the GAR[2]. However, the Lucy-Richardson deconvolution algorithms was able to achieve sub-GAR resolution, with η eff approaching one under ideal conditions. As such, the res- olution is not directly limited to the channel width. Therefore, lobster-eye lenses can be designed to allow for larger channels (and therefore larger effective areas) without neces- sarilysacrificingtheresolution. Furthermore,thenear one η eff indicatesa lackofartifacts after processing the image, which, under ideal conditions, demonstrates the effectiveness of the algorithm. 8.1.3 PotentialforHardX-ray Imaging θ c is indirectly coupled to GAR, FOV and spectral range through α. In the GRB case study, the limiting factor of the spectral range centered on the appropriate scaling of α based on current materials. If new reflecting materials can be developed and channels can be constructed with α in the sub 15’ to 20’ range, X-ray imaging in the 7-14keV 140 range is a realistic possibility (as was shown in Section 7.4). Though NuSTAR covered a much broader spectral range, FOV was significantly limited and a 10m focal length was required[23]. NuSTAR demonostrated the benefits of expanding imaging capabilities into thehardX-rayregime[23]. Because of the potentialfor imaging X-rays at photonenergies greater than 6keV with FOV on the order of degrees, lobster-eye lenses could play a role in continuing those delevelopements. Furthermore, though the FOV would be larger than conic based telescopes, the resolution can be designed to similar levelsof refinement. 8.1.4 Flexibility inDesign R wasadeterminingfactorforimagingcapabilitiesregardlessofphysicaldimensions. For anyR value that met a mission requirement, there were sets of potential lenses that could befoundtosatisfyotherconstraints. Consideringtightmarginsofmissiondesignforspace system, the flexibility in lobster-eye lens design makes them an intriguing candidate for developement. For space missions,allowing for multiple options that can potentially meet the requirements for source imagingwithin systemlevel budgets is necessary. Though only ideal α were studied in detail, one interesting advantage of lobster-eye lenses (in either geometry) is the repurposing of the lens at X-ray energies high enough such that η ee approaches one. The ability for a single instrument to handle high energy X- raysinaquasi-codedapertureconfigurationwhileimagingatlowerenergieswouldenablea singlelobster-eyelenstohandletheresponsibilitiesofboththeBATandXRTaboardSwift [17]. Ifasensorarrayorsuitecanbedesignedtovaryenergysensitivityandbandwidth,the lenswillfunctionasaduelinstrumentpotentiallyreducingthemass(andtherefore cost)of the mission. Because the image qualitycan be maintained for different sized lobster-eye lenses, rel- ativelysmallX-rayimaginginstrumentscanbedesignedwithimagingcapabilitiesnearing currenttelescopes. Alongwithreducingmissioncost,smallinstrumentationenabablesnon X-ray centric missionsto add X-ray imagingcapabilities by including a lobster eye lens in 141 the instrumentsuite. Thoughthe lobster-eyelens would be a secondary payload, including X-ray imaging on a wider variety of missions would lead to new data collection oportu- nities and potential discoveries. Furthermore, X-ray data would be coupled with the data collected from the other instruments. 8.1.5 LimitationsandConstraints Currentlytherearenomanufacturingtechniqueswhichhavebeendemonstatedtobeableto produce lobster-eye lenses. First, channels have yet to be constructed with α on the order needed for X-ray imaging. Second, typically the prototypes have been constructed from materials that are not ideal for X-ray imaging, especially at higher energies (for example, MCPs were constructed from glass and coating the class provedproblematic[38]). Finally, adding curvature and controlling the roughness of the reflecting surfaces has been prob- lematic [42]. Though the FOV is not strictly defined by θ c , as with conic basic telescopes [51], α is dependent on θ c . In general, as photon energy increases, θ c and α decrease, making the construction of lobster-eye lenses more difficult for higher energy sources. Further- more, the breadth of the spectral range over which η oo ≥ η ee was shown to increase as α decreased. Therefore, not only are smaller α required for imaging higher energy sources, but for broad spectral ranges as well. For example, while the sterile neutrino case study resulted in a lens which covered a spectral range only spanning 2-4keV while the GRB case study resulted in a lens spaning 7-14keV. Though the materials were different, the GRB lens was designed with α 1/2 that of the sterile nuetrino study. Therefore, even though FOV is not directely limitedby photonenergy, α is. Effective area for a single point source is limited by θ c andO.Furthermore,R and N ch will need to be large enough to ensure a viable PSNR and η eff for imaging. For low intensity sources, achieving a large enough effective area will restrict the FOV or result in a very large instruement. An alternative to building a single large lens, nesting can be 142 used to increase the collection area has been proposed[22]. However, testing the nested configuration of parallel lenses will be required in order to propperly assess whether the processing of multiple lobster-eye lenses will be advantagous compared to a single larger lens. 8.2 FutureResearch Because of the challenges of construction, lobster-eyelenses remain a relativelyopen field with a variety of potential research directions. Manufacturing and construction, generaliz- ing the theory to include aberrations such as surface roughness, imagingextendedsources, and broadening the spectral ranges all need to be studiedin order to fully realize a mission ready lobster-eye lens. Enabling technology, such as new etching techniques, need to be researched for poten- tial advantages that can be exploited for the developement of lobster-eye lenses. Previ- ous research has focused on the potential for etching narrow channels and adding coating [13, 36, 40, 41]. However, if the meridional lens geometry is considered as a potential constructionmethod, then the selectivityof the etchingtechnique becomes a key consider- ation. Forexample,theprecisionneededforprecisematingofreflectiveflats(followingthe proposed construction design in Reference [18]) may be achievable with plasma etching. One such technique, electron cyclotron resonance, has been shown to be able to acheive high orders of selectivitybetween the substrate and mask during etching processes [46]. Experimenting with multi-layer coatings (such as the ones used on NuSTAR [23]) may be necessary in order to extend the spectral range of a lobster-eye lens. For α ≥ 20 , individualreflecting materialsare fairlylimitedinthespectralrange (Figure7.5) assuming the bandwidth is limited to photon energies for which η oo ≥ η ee . Creating multi-layer reflectingsurfaceswillexpandthespectralrangeoflobster-eyelensesregardlessofwhether the bandwidthis limittedto η oo ≥ η ee . 143 The effects from relaxing the constraint on η oo may not be as prohibitive as antici- pated, however the degredation in imaging quality for non-ideal α and broad spectrum sources need to be studied in detail. Allowing for higher order reflections to be included in the image would extend the potential spectral range. By not limittingthe spectral range to energies for which the reflectivity can be modeled by TER, a wider range of potential reflectingmaterialwouldbepossible. Forexample,Nimaybeacceptableforsourcesinthe 2-4keV range if higher order reflections are allowed in the image. Furthermore, because a larger number of channels would be included in the imaging, the effective area and col- lection ability for smaller lobster-eye lenses would increase. However, because the focal spotand backgroundwillbroaden, theone dimensionalPSF componentswouldneed tobe reworked to include channels with 3 or more reflections. And though the PSF of the origi- nal image may appear degraded, the theoretical Lucy-Richardson deconvolution algorithm mayremaineffectivewithaproperlydefinedPSF.Furthermore,studieshaveshownthat,in lowlightconditions,crayfishcompensatebyallowingforagreaternumberofchannelsand a broader focal spot[8], essentiallya biologicalprecedent for allowinghigherorder reflec- tions to contribute to imaging. By altering the amount of reflective film lining the lengths oftheirfacets,crayfisheffectivelyincreasingthenumberoffacetscapableoffocusinglight [8]. However,the size of the focal spotwas seen to increase with more photoreceptors [8]. Though the deconvolution of an extended source has been accomplished in a previ- ous study[39], the ability to propperly deconvolve extended sources with the analytical Lucy-Richardson deconvolution algorithm needs to be studied. During the testing for two point resolution, the lack of artifacts remained relatively consistent with the single point experiements. For sources too closely spaced to be individually resolved, the image appeared to be a single long source. Though an extended source has yet to be tested, the results of the two adjacent sources showed promise and the need for continued research and experimentation. If possible, a wider variety of sources and missions are avialable, includingsolar and deep field imaging. 144 Many of the theoretial models and algorithms developed for lobster eye lenses can be extended to Schmidt style lenses as well. The primary differences between the Schmidt array and the meridionallobster-eye lens arise from the offset of the ˆ x and ˆ y sets of reflect- ing plates. Therefore, the Schmidt optic may be able to be modeled as a set of two one dimensionalmeridionallobster-eyelenseswithvaryinglensparameters. Andthedeconvo- lution algorithm can be modified to incorporate the two sets of one dimensional focal spot and background approximations. Thepotentialimagingapplicationsforlobster-eyelensesalsoextendstoterrestrialuses [26]. For example, imaging back scatered X-rays with lobster-eye lenses would create a new non-invasive medical diagonostic tool [26, 30]. Similarly, using lobster-eye lenses to image back scatter X-rays would be advantagous for National Defense [26]. The use of back scattered X-rays enables the detection of objects that are located in difficult to inspect areas [11]. Furthermore, the radiation dose necessary for back scattered X-rays would be lower than traditional X-ray technology (which is advantageous in both medical anddefense applications)[11]. Thoughseveralchallengesstillexistfor theconstructionof lobster-eyelenses, their place in potentialfor growthof the field of X-ray imagingremains valuable for both terestrial and astronomical applications. 145 ReferenceList [1] Nasa website. http://www.nasa.gov. Accessed: 2010. [2] J.R. P. Angel. Lobster eyesas x-ray telescopes. The AstrophysicalJournal,233:364, 1979. [3] F. K. Baganoff, Y. Maeda, M. Morris, M. W. Bautz, W. N. Brandt, W. Cui, J. Doty, E. D. Feigelson, G. P. Garmire, S. H. Pravdo, G. R. Ricker, and L. K. Townsley. Chandra x-ray spectroscopic imaging of sgr a and the central parsec of the galaxy. The AstrophysicsJournal,591:891,2003. [4] D. Band, J. Matteson, L. Ford, B. Schaefer, D. Palmer, B. Teegarden, T. Cline, M. Briggs, W. Paciesas, G. Pendleton, G. Fishman, C. Kouveliotou, C. 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Abstract (if available)

Abstract The lobster‐eye telescope with square‐cross‐section channels has been suggested as a possible candidate for a wide‐field‐of‐view X‐ray all‐sky monitor. However, due to the difficult construction, as of yet, no lobster‐eye lenses have been deployed. Though prototypes have been constructed, they typically utilizing non‐metal reflecting surfaces and have channels that do not meet required width to length ratio necessary for imaging X‐rays efficiently. Because of the limitations for construction, the majority of study has been dedicated to fabrication techniques. Additionally, alternative designs with non‐square channels have been proposed to enable construction with metals. One such design, meridional lobster‐eye lenses, is based on aligning interlocking reflective plates along perpendicular meridians. ❧ This work is a computational and theoretical study of lobster‐eye lenses with both square‐cross‐section channels and non‐square channels arising from the meridional design. The focusing efficiency of each design for sources at varying angles of incidence is compared with previous work. Furthermore, the focal properties are redefined to account for the curvature and physical structure of the lenses themselves. In the process, the differences in focal length and focal properties for both square‐channel and meridional lobster‐eye lenses are identified. ❧ From the focal properties, the point spread function is defined for both square‐channel and meridional lobster‐eye lenses. In one‐dimension, the intensity profile of focused photons are derived. For unfocused photons, an approximation for a generalized theoretical intensity profile is designed. The two dimensional point spread function was then created from the convolution of the one-dimensional intensity profiles in each dimension. ❧ Aberration due to non‐ideal imaging surfaces is detailed. The effects of the reduction in focal lengths due to planar imaging surfaces is defined. The point spread function is generalized to account for non‐ideal focal lengths. And the limitations to field of view caused by the planar imaging surface are identified. The theoretical point spread function is compared with simulated collimated sources. ❧ Deconvolution algorithms are designed for idealized and variant point spread functions. Weiner filter and Lucy-Richardson algorithms are modified for lobster‐eye lenses. The deconvolved images of simulated collimated sources are compared for on and off axis idealized lobster‐eye lenses. The potential for both single source and two source resolution is compared for varying pixel densities. The full width half max, accuracy of source position, peak signal to noise ratio, and percentage of overall image intensity contained within the peak signal are used to judge the overall image quality. ❧ Design trades are described based on the interconnection of field of view, resolution, aspect ratio and effective area. From the design trades, two case studies are identified for potential X‐ray missions utilizing lobster‐eye lenses. Several leaps in technology are necessary to scale the potential lenses to dimensions that are similar to prototype and previous proposals. However, assuming the technological development, lobster‐eye lens are designed that would be able to accomplish the mission goals.

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Creator Barbour, Samuel (author)

Core Title Lobster eye optics: a theoretical and computational model of wide field of view X-ray imaging optics

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Astronautical Engineering

Publication Date 07/06/2015

Defense Date 07/05/2015

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag astronautics,Astronomy,deconvolution,image processing,lobster‐eye lenses,OAI-PMH Harvest,optics,X‐ray

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Erwin, Daniel A. (committee chair), Gruntman, Michael (committee member), O'Brien, John D. (committee member), Tanguay, Armand R. (committee member)

Creator Email sbarbour@usc.edu,snbarbour@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-585098

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Legacy Identifier etd-BarbourSam-3538.pdf

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Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

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