Class Note for OPTI 596C at UA
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Date Created: 02/06/15
Optical Imaging and Aberrations Virendra N Mahajan Adjunct Protessor The Aerospace Corporation College ot Optical Sciences El Segundo Calitornia 90245 University ot Arizona 3i 0 336l 783 virendranmahajanaeroorg Lecture 9 Imaging With Circular Pupils Polychromatic PSF and OTF Image ota Disc Pinhole Camera TwoPoint Resolution 91 Lecture 9 Summary Polychromatic PSF and OTF Image of a Disc Pinhole Camera TwoPoint Resolution Polychromatic PSF Polychromatic OTF Image of a Disc Pinhole Camera TwoPoint Resolution 92 Polychromatic PSF So far we have discussed monochromatic imaging by working at only one wavelength Now we consider polychromatic imaging For simplicity we assume aberrationfree imaging Monochromatic PSF IrP ex Ir 4x2102 2J1rrr 12 r in units of AF and I in units of Dex 62 or IE gt912 The PSF consists of a bright spot of radius 122 in units of AF surrounded by dark and bright diffraction rings The radius of the bright spot and the location of the diffraction rings depend on the wavelength of the object radiation For an object radiating in a wide spectral band a diffraction pattern is formed at each wavelength with the result that the diffraction rings loose their sharpnessthey smear out 93 Quasimonochromatic PSF QPSF Letp be the image power per unit spectral bandwidth so that p5 is the total image power in a very narrow spectral bandwidth 6 QPSF may be written npok 4x2102 2J1m xF 2 1 0 105 nnAF 4x2102 12 Polychromatic PSF PPSF Assume that an object radiates the system images and a sensor detects the image uniformly as a function of wavelength over a range from A1 to A2 K2 2 TE 2J rm AF 1 PPSF WAA 42J A2 dx l M Spectral bandwidth A A2 A1 Mean wavelength Am 122 A A A x1 x x1 1 mt 2km 2 mt 2km 9 4 Normalizing by the central value of the QPSF for a mean wavelength Am A2 2 3 IrAx x371 2J1rcrl F 1 IMAM 0 ax J miAF 12 dx A Normalized central value 1 A2 M 1 A2 1 1 Axax 104Axx 10Ax mJ 7dr m 2 m2 5AA A 5 K1 K2 1 Ax2xm 1 Ax2xm 1 Normalized wavelength An AAm normalized distance r riAmF irML let 10394 a 104dgtvn AF A An Am 5x 1 AK 4 2km 4x10 2 1 A 2km 95 107 106 103 l l l l l l l l l 0 02 04 06 08 1 12 14 16 18 2 gt Amm Figure 2 58a Aberrationfree central irradiance of the PPSF normalized by the central value of the QPSF as a function of the relative spectral bandwidth AAAm 96 106 104 102 1 Axum 199 n r AM 100 10 10 4 10 6 O n Figure 2 58b Aberrationfree PPSF for various relative spectral bandwidths Akxm The units ofr are AmF Ring structure does not disappear until AkAm 1 ie until the spectral bandwidth equals the mean wavelength 97 Polychromatic OTF Monochromatic OTF m cos lvV1V212 05V51 VVi1 F AFVI39 10 m 08 7 7 06 7 7 p 04 7 7 02 7 7 00 w w w w 00 02 04 06 08 10 98 OTF at two wavelengths 10 m 08 7 i A067 7 gttv 04 7 7 02 7 i 00 00 02 04 06 08 10 gtV The cut off frequency vC 1AF and the OTF for any frequency are higher for a shorter wavelength If A1 lt M then vlc gtv2c and the OTF is higher for A1 than for M 99 Polychromatic OTF l2 l2 1p i J Wu nix J cos 1AFvi Fvi1 2F2v ldx X1 X1 Let x xxl q x2 x1 v vi1x1F d de dx dx AFVi iv xv AA 2 1 x2 61 1 A1 5 o q s 3 because detectors respond to a relatively narrow spectral band q 1 for monochromatic q AZAl 07nm04 um 175 for visible light 910 1 08 s s q 1 06 A 2 3 PD 7 7 i 4 04 s s 8 02 0 l l 0 02 04 06 08 1 av Figure 259 Polychromatic OTF of an aberrationfree system Epv decreases for all values of v as q or spectral bandwidth increases 911 Image of an Incoherent Disc As an example of the image of an isoplanatic incoherent extended object we consider the image of a uniformly radiating or illuminated disc The diffraction image may be obtained by convolving the Gaussian image with the PSF or by inverse Fourier transforming its spatial frequency spectrum which is equal to the product of the spectrum of the Gaussian image and the OTF We want to determine the size of an object that can be treated as a point and thereby define the size of a pinhole that can be treated as a point source Consider an object that is a disc of radius ha and radiance B lying at a distance zo from the entrance pupil of area Sen of an imaging system First we determine the Gaussian image then the diffraction image by the convolution approach and finally by the Fourier transform approach 912 Gaussian image Total power entering the system 2 2 Pen Who SenZ0 3 Total power in the exit pupil and therefore in the image for a transmission factor n Pex quotPen This power is uniformly distributed in the circular Gaussian image of radius kg Mho where M is the image magnification Irradiance of the Gaussian image Ig7g Pexrchgz nsenz M2B Igfor7gshg 913 Diffraction image is convolution of the Gaussian image and the PSF M6 NglrglPSFl lm IngSF 7i yd Fglshg Aberrationfree image 2 a S 2J rm AF PSFltri A261 351 kg 2313 99122 0 0 t t 2 2 2 e a 12 S 339rg 339 Mgquot VgCOS 1quot g 2 2J1nsAF mF rg drg deg ml Ii o Image is radially symmetric as expected since functional dependence on 6 disappears when 6g integration is carried out indeed we may let 6i 0 914 12 Let rrl Fpg rgAFbg hgAF and t sAF r2p 2rpcoseg g 2TB air i 1 l g 4 This is a cumbersome double integration first over 6g and then over pg The integration is simpler in the frequency domain 2J1ltT t 3U 2 pgdpgdeg Central irradiance bg 2 10 gJ M 0 p dpg 1g1 J nbg J12nbg 2280 g Variation of Ii0 with 19g is similar to the variation of the encircled power of the Airy pattern It approaches lg asymptotically 915 Diffraction image in the spatial frequency domain Spectrum of the image is equal to the product of the spectrum of the Gaussian image and the OTF 12 Inverse Fourier transform of the spectrum yields the image Since the Gaussian image is radially symmetric its spectrum is given by its zeroorder Hankel transform kg N 2 h 022 2339ng JO2rtvirgrgdrg IghgViJ127Wihg Pex g 0 Image spectrum 2nvlhg Wi fng39WWi Pex 1Wquot Image 1 fo vexp ZJ39Ei i39I d7 Ighg f1viJ12mihgtmexp marma Hi 916 Very small disc ie for very small values of lag eg a pinhole J12mh g TEVl39l l g 2292 115 14 ft7lexp Zni iFiw i Pm ft7lexp Zni iFiw i or 55 PexPSFFl as expected for a point object Radially symmetric aberration FT reduces to zeroorder Hankel transform 113 PexPSFrl 2713 ftviJ02nvlrivi dvl Encircled power in a circle of radius rC 76 7C PM mg illWW 2 2PexfTVVidvlg 102 vmn d1 ZitrcPexftviJ12 virCdi Pex1 J n rc J12mc for aberrationfree imaging 917 Large disc and radially symmetric aberration Z lghgfJ12 VihgJ02 Vi TidVi Let rrl F and VVl1F 1 1r 2n1gbg fJ12rrvbgJ02rcvrtdv 0 1 110 2nlgbng12 vbgt dv 0 1gp bg 141 J8ltnbg J12rcbg for aberrationfree imaging 918 Power in a circle of radius rC in units of AF 1 6 13rc 2n21gbng12nvbgrvdv g J02mr rdr 0 r 1 2P J12IuvbgJ12 vrctv1vdv Much simpler to calculate Iir or Prc by substituting the value of 1v Power in a circle whose radius is equal to that of the Gaussian image 13bg 2Pex1122mbgrv1vdv 2289 0 g in a circle of radius rC due to a source of radius 19g and the power Pbgrc in a circle of radius 19g due to a source of radius rC Relation between the power Prcb 1 bg2PrCbg Zerg fJ12nvbgJ12nvrctv1vdv rCZPbgrC 2291 0 919 08 i 02 r v 4 Figure 260 Normalized aberrationfree image of an incoherent disc whose Gaussian image has a radius of 19g r and 19g are in units of AF The image of an object whose Gaussian image radius 19g 14 in units of AF is approximately the same as the Airy pattern Hence such an object can be treated as a point o Similarly a pinhole with 19g 14 can be treated as a point source 920 08 06 gtPrc 04 i 02 rc Figure 261a Encircled power of the aberrationfree image of an incoherent disc normalized by the total power Pex rC is in units of AF o Prc for the aberrationfree image of a disc for 19g 14 closely resembles the encircled power of the Airy pattern confirming the size of a pinhole that can be treated as a point source 921 08 06 gt mg 02 0 l l l l l l l 0 05 1 15 2 25 3 35 4 b9 Figure 261b Power in a circle of radius equal to that of the Gaussian image of the disc as a function of its size o Pbg increases monotonically as the disc size increases Pbg for small values of 19g contains only a small fraction of the total power since it is spread by diffraction Pbg for 19g gtgt AF contains a significant fraction of the total power since the effect of diffraction for large objects is relatively small 922 Bu 7 K 087 7 e 14xquot 39 n 7 39 1 7 I 12 39 067 v 1 7 3 39 97 1 g 7 7 l 1 v 1 7 1 7 04 3 v 39 7 39 7 39 c 39 027 yr 7 f V 7 1 r 7 y 7 0 L l l l l l l l l o 05 1 15 2 25 3 35 4 45 5 abg Figure 262a Central irradiance of a defocused image of an incoherent disc Central irradiance approaches unity actually lg asymptotically 923 l0 3d Figure 262b Axial irradiance of a defocused image of an incoherent disc normalized to unity at the center The actual central values normalized by lg in increasing order of 19g are 014 0825 091 and 094 For 19g 14 axial irradiance is approximately the same as for a point source For example it is nearly zero for integral values of Ed in units of A It becomes nonzero for larger discs 924 1 1 be Bad bg 2 08 quot 1 1eauss1ar1 39 gt wmage 12quot 05 a 3 7 b i 047 quot ring 027 7 7 O 1 3 35 4 o 05 1 15 2 25 a as 4 a1 1 1 111111 Ba0 bg 23 08 t s a2 14quotquotm 1 Gaussian wmage 1239 05 i nes 047 1 027 O 1 o 05 1 15 Figure 263 Irradiance distribution of defocused images of an incoherent disc Gaussian image of radius 19g is indicated by the dashed vertical line 925 The image generally resembles the object In particular it is bright in the central region and dim in the outer region As defocus increases the irradiance decreases in the central region and increases in the outer region o If the disc is small and the defocus is large the irradiance at the center may be smaller than that in the outer region as for example for 19g 1 and Ed 1 This behavior is similar to that for a point source and it disappears as the disc size increases o The central irradiance of the image of a coherently illuminated disc can be much lower than that in the surrounding region whether or not the image is defocused unless the disc is very small like a point object Moreover defocus can increase the central irradiance see Figures 276 and 277 on pp 236237 926 Pinhole Camera o A pinhole camera or camera obscura has been the subject of many investigations including those by Petzval and Rayleigh o It is simple a pinhole of radius a on one side of a box of length L and the film on the other distortion free with an in nite depth of eld and a very Wide eld of View Petzval approach for a relationship between a and L 1857 The geometrical image of a distant point object is approximately the same size as the pinhole if the pinhole is large Reducing the pinhole size reduces the image size until diffraction by the pinhole spreads it Optimum image is obtained by minimizing the image spot radius representing the sum of the geometrical and diffraction contributions to it 927 a H Figure 264 Diffraction spot radius based on the cancellation of disturbances from the center and the edge of a pinhole L gtgt a o Diffraction image radius is approximately equal to the distance of a point P from the center of the image at which a disturbance from the center 0 of the pinhole cancels that from its edge A ie when OP AP 0C M2 o Diffraction spot radius rd L6 LOCa xLza 928 Adding the geometrical and diffraction contributions we obtain the image spot radius as r a L2a Differentiating with respect to a and equating the result to zero we obtain the optimum pinhole radius GP 39 r Zap l2 L 2298 Defocus aberration tolerance approach The difference between a pinhole camera and a regular camera is that the former does not use a lens to form the image The lens in a regular camera converts a diverging spherical wave from a point object P0 into a spherical wave converging to an image point P0 in the image plane 929 a l Object plane Pinhole Image plane HHLogli Li 4 b a Imaging by a pinhole camera of radius a b Wavefront incident on the pinhole and emerging wavefront shown shaded required for perfect imaging The pinhole size is extremely exaggerated for clarity of the wavefronts The camera length Li gtgt a 930 Without a lens a spherical wave of radius of curvature L0 diverging from the object P0 is incident on the pinhole and continues as a diverging wave toward the image plane at a distance L and forms a defocused image at PO Defocus wave aberration representing the sum of the sags of two spherical wavefronts passing through the center of the pinhole with their centers of curvature lying at the object and image points is given by ABBC AC or Wrp where rp is the radial distance of a point B in the plane of the pinhole from its center L0 is numerically negative according to our sign convention Image will be practically diffraction limited according to the Rayleigh s quarter wave rule if the peak value of defocus wave aberration is less than or equal to M4 931 where f6 ZaZA is the effective focal length of the pinhole For a distant object L0 gt 00 we obtain Bd a gt a lALiZ Same as the Petzval result The image spot for a point object is approximately the Airy disc with a radius of 061ALia 06142xLi or 061 times the value estimated by Petzval 932 Rayleigh approach 1891 Axial irradiance of a beam at a distance z for a point source of intensity IO diffracted by a circular aperture of radius a m Mr Ede 1 iv a d AV zz BdZ A z R A L L0 2 2 7m 41 2rra L L Pex 0 2 a 02511 l 0 L0 Li L0 i 0 For a given object distance L0 maximum central irradiance at an image distance L is obtained when 610 aLi 0 gt a MLI gt pinhole with one Fresnel zone Rayleigh concluded from his experimental observations that the best image was obtained when aR 0954ALi which is equivalent to the pinhole intercepting only 90 area of the rst Fresnel zone 933 OTF approach Defocus tolerance for Hopkins ratio Hv 2 08 for v S 01 Bd 5 xzov M2 a2 Bd 2 fora distant object l Hence a AL same as Rayleigh39s result Aberrations Since the focal length f6 of the pinhole camera approximately equal to ZaZA depends on the wavelength it suffers from chromatic aberration Similarly since the pinhole appears to be elliptical from an offaxis point object its focal length for an object in the horizontal plane differs from that in a vertical plane Hence it suffers from astigmatism 934 It is free of distortion ie the transverse image magnification M is independent of the field angle M h h LiLO The chief ray from an offaxis point object P at a height h ie an object ray incident through the center of the pinhole reaches the image plane at the image point P at a height h without any deviation as illustrated below P h J P6 p0 T P h I I I I HLo I Li I Object plane Pinhole Image plane Distortionfree imaging o Disadvantage of a pinhole camera is the long exposure it requires due to the small size of the pinhole 935 TwoPoint Resolution A measure of the imaging quality of a system is its ability to resolve closely spaced objects Rayleigh resolution Two point objects of equal intensity are just resolved if the principal maximum of the Airy pattern of one of them falls on the first zero of the other ie if the separation between their Gaussian images is 122AF If the Gaussian images are located at x061AF then the irradiance distribution of the aberrationfree image along the x axis is given by 1x N fgg l r l2j i i323mF where x is in units of AF 936 1 my US D7 n6 ma Alix m as D2 Ell Figure 281 lrradiance distribution along the x axis of the aberrationfree image of two incoherent point objects of equal intensity separated by the Rayleigh resolution of 122kF x is in units of AF The dip at the center has a value of 073 compared to a maximum value of unity at x z 061 937 o Aberrationfree image of a point object Ir 2J1 cr 12 Defocused image of a point object 1 2 IrBd 4 f expinp2J0rrrppdp 0 Defocused image of two incoherent point objects of equal intensity separated by the Rayleigh resolution of 122AF along the line passing through them namely the x axis 2 2 1 IxBd4 gexpinp2J0Itx O61ppdp 1 g expinp2J0rrx O61ppdp 938 087 a 08 12 O6 aux 7 047 02 7 02 Bd12 14 ax ax a b Figure 283 a Defocused image of two incoherent point objects separated by the Rayleigh resolution of 122AF Ed is in units of A and x is in units of AF b Distributions in a are normalized to unity at the principal peaks o Central values in a with increasing defocus are 073 061 and 033 and the principal peaks have a value of 1 084 and 048 located at x i 061 Normalization of principal peaks to unity in b shows that the effect of defocus on the relative dip is small 939 Bd 0 Bd 14 Bd 12 Figure 285 Aberrationfree and defocused images two incoherent point objects separated by the Rayleigh resolution If the two point objects are coherent and in phase they cannot be resolved If they are not in phase then their image appears as if they are of unequal intensities See Figures 284 to 286 Mid Term Test 940
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