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# Class Note for OPTI 596C at UA 2

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Date Created: 02/06/15

Optical Imaging and Aberrations EXP Virendra N Mahajan Adjunct Professor The Aerospace Corporation College of Optical Sciences El Segundo California 90245 University of Arizona 310 3361783 virendranmahajanaeroorg Lecture 2 Optical Transfer Function OTF 21 Lecture 2 Summary Optical Transfer Function OTF Isoplanatic Image and its Spectrum OTF as FT of PSF or Autocorrelation of Pupil Function Physical Significance of OTF OTF Properties OTF Value at Zero Wavelength Geometrical OTF Comparison of Diffraction and Geometrical OTFs Asymptotic Behavior of PSF PSF Centroid Hopkins Ratio 22 Image of an lsoplanatic Incoherent Object Object of radiance B070 Gaussian image of a point object located at 70 is located at 7g MFO where M is the transverse magnification of the image Gaussian image Mtg nisenziMztigM 035 Pupil function defocused but shift invariant P7pzl P7pexpi ij r13 147 Zl Zg Pointspread function 2 1 W m 6X 157 a 2m a a a JPrpzl exp x R p drp Diffraction image is convolution of Gaussian image and PSF M7195 Jlg7gPSF7i 7gzid7g 156c 23 Optical Transfer Function Decompose the object and its image into sinusoidal spatialfrequency components gt B03 2 JBVOCXp 27139170 7dr0 1M i iVi XP 2 Wi 700 where 32 is the spectral component of object at spatial frequency 170 given by 3a iBoexp2mo W70 and 3a is the spectral component of image at spatial frequency vi vOM Image spectrum Fourier transform of image MW J1i 7iexp2 i 7i39 7007 Jd exp27ci17l 7i Jlgg PSF7 7g d7g 24 idrglgrgexp2mr rgiiPSFh rgexp2nViri rggtidt igVl1Vl 170 Spatialfrequency spectrum 3a of the diffraction image of an isoplanatic incoherent object is equal to the product of the spectrum igm of its Gaussian image which in turn is equal to a scaled spectrum of the object and the OTF 1a of the imaging system Theorem 6 Spectrum of Gaussian image Substituting for lg from 135 igwi i Igrgexp27ci17rquotgdrg SenZ3BM 7i T1 SiertZg1 70 Optical transfer function OTF is Fourier transform of PSF 25 lPSF1 eXp27cii7i7id7i 1 51 V II 1 2 a 4 a a 2mg 4 a midi exp2mvlrii JPrpeXp x Rrp rijdrp WWI 4711 P715Jd exp 39n WWWi 6X P7 jdr P7p 871 7p Mm Pp Mm d7p 2 lPP7p lR id all1 7pl drp 173bgt OTF is the normalized autocorrelation of the pupil function Theorem 7 Region of integration is the overlap area of two pupils one centered at 71 0 and the other at 71 XRi7i OTF is zero for those frequencies Vi for which overlap is zero No need to calculate the PSF first to determine the OTF 26 Physical Significance of OTF 801m 8 T5 1i70 T 301 mi U gtx0 IO1m IOU m U I gtX 1 V T IOU mltl i 1IVI ilt H 0 Xi Consider the image of a sinusoidal object of spatial frequency V0 radiance B0 modulation or contrast m and an arbitrary phase constant p B70 B01 mcos27ci70 70 p Gaussian image I go 2 IO1mcos2m7i39 7i P where IO nS nz M2BO is the average irradiance 27 Image modulation and phase are the same as those of the object Image spatial frequency Vi is of course different from the object frequency V0 by the image magnification M Diffraction Image IN 11Senz3 IB70PSFE M70d70 156a object Substitute B070 2 B01 mReeXp i27ci70 70 115 10M2 I PSFFL M70 170 mRe exp i27ci7i 3 p x PSF7L M70 exp2m39i7l 7i M70d70 1 a a a 0M2 W Re exp 127c vi I p1 10 1 mRe exp tom 7 ltPTW 28 Substitute for the complex OTF in terms of its modulus and phase 1071 lt lexpi 1 l 196 1103 10 1 m Ti7l cos27ci7l 7i p Pvl 197 Diffraction image is also sinusoidal with a frequency 171 However its modulation is different by a factor lavVi called the modulation transfer function MTF and its phase is different by PVl called the phase transfer function PTF Theorem 9 We will see shortly that Tl7lS 1 Reduction in contrast and changes in the phase of the sinusoidal spatial frequency components contribute to image degradation This is true even if the system is aberration free in which case the OTF is real or the PTF is zero The MTF is zero above a certain cutoff frequency corresponding to zero overlap of the two correlating pupils Hence these high frequencies if present in the object will be absent from the Image 29 OTF Properties co JPSF exp2 iVi w JPPFP WgtdpJiP7pgti2d 1 OTF at zero spatial frequency is unity ie image contrast is zero for a zero object contrast 10 2 1 2 OTF is complex symmetric or Hermitian Theorem 10a 1a a vl ReTi7liImquotci7l Re part is even and imaginary part is odd R6107 RCT i7iand1m1i7 Im c i7i 3 MTF at any frequency is S 1 Theorem 10 b M PiiiPP7p wwfpi 6X 3 P 1J PFp 2d7p P7p 1va 2 210 12 6117p 1 1106 4 MTF at any spatial frequency is S corresponding aberrationfree MTF Theorem 100 5 OTF Slope at the Origin Vi vicos sin 517 dim 2 P6361 JPxp yp Pxp XRQ yp 7 Rndxp dyp Axp yp eXpiCIxp yp Pupil function Pxp yp xp yp A2 xp yp Pupil irradiance 211 08409631 The region of integration is the overlap area of two pupils one centered at xp yp O O and the other at xp yp XR n at Me J39J P x 8Pxpyp ax y dx dy 3 21120 p P P axp P P 3 21120 2 x R AexpiCI iA eXp iCIgt dxp dyp Exp Exp 3 E n 2 0 U gt a i E3 E3 Nli k cum M31 a N QJQJ mk e m 3 m Q Q NM J 3 O where BEBQ represents the contribution due to the dependence of the limits of integration on the spatial frequency gm 8R6quot Me al 32 dx dy 1122 znzo 2Pex axp p p 21120 Thus slope of the real part of the OTF at the origin is independent of aberration It does depend on the amplitude variations at EXP t determines the asymptotic behavior of PSF 81m Me 2 ad 271 aW A dx dy I dx dy 8 21120 Pex axp p p Pex axp p p Slope of the imaginary part determines the centroid of the PSF lrampmlz Ret m21m mlz 2quot ZRCT ZImTaImT ag ag ag 213 Ml twl ag n0 ag n0 Thus the slope of the MTF at the origin is equal that of its real part and therefore it is independent of aberration Theorem 11 If the amplitude across the pupil is uniform ie if Axp yp 2A0 then for an ExP of area Sex 2 Pex A0 Sex IO Sex PR ta El 8 n0 8 n0 rm new i 8 21120 Sex axp p p o Similarly it can be shown that the slope aRezv avv l part of OTF 1vi 4 in polar coordinates with respect to the radial frequency v evaluated at the origin is independent of the aberration 0 of the real 214 Diffraction OTF in the Limit of Zero Wavelength dim P6361 Pxp yp Pxp Meg yp 41de dyp 136361 Axp ypAxp Mega yp 7 RnCXPiQXpa ypdxp dyp Aberration difference function QXp yp xp yp CIxp 7 R yp kRn xp 7 R yp 7 Rn xpyp7tR 81ypnaqxp yp07 2 p 31 x y 31 x y Qxpayp ME 3 pgtn ij p0x2 2nRgaWxP ypnaWxp WWW Exp ayp 215 1i 117La0 P 11xpypexp27ciR 3 W dxp dyp xp yp wan 1136b Hence the diffraction OTF approaches the geometrical optics OTF as 7 a 0 Theorem 12 216 Geometrical OTF Ray aberration xiyl representing the location of a ray in the image plane with respect to the Gaussian image point is related to the wave aberration Wxpyp see Eq 311 of Part I aWxp yp aWxp yp Exp ayp Maw R Geometrical PSF of an aberrated system can be obtained by noting that an element of area dSp dxpdyp centered at a point xp yp in the ExP plane is mapped into an element of area dSi dxidyl centered at the point xiyl in the image plane according to 30 amp dsp 217 If Ipxpyp is the irradiance or the density of rays at the point xpyp in the EXP plane then the irradiance or the density of rays Igxy at a point xiyl in the image plane is given by Igxlyl dSi Ipxpyp dSp Hence the geometrical PSF not the Gaussian image is given by see Eq 410 of Part I 30 3xp yp ts Fourier transform yields the corresponding namely the geometrical OTF 74 1351 IgxiaYiCXP2 i xi nyidxidyi 1AM 2 xp yp 1 a P X1JJ1xpayp CXp27El xinyldxidyi 106361 xpyp exp27ci xlnyldxp dyp 1138b 218 Note that it is properly normalized to unity at the origin and it is Hermitian Tg00 1 Unity contrast at zero spatial frequency Tg n T a n Hermitian Substituting for the ray aberration xiyl in terms of the slope of the aberration function a x d x Tg n PmIJ J xpypexp27ciR WE ypn W2 yin dxp dyp P Thus Ig n can be determined from the pupil irradiance and the slope of the aberration function PSF is not needed to determine the OTF Ig n 1 if W20 This may also be seen from the fact that an aberrationfree point image yiedls unity OTF at all frequencies FT of 57 7g is unity 219 Approximate Expression for Geometrical OTF quotcg n Pg xp yp exp27ci xl 11 dxp dyp 2 P6361 exp27ci xc nyc gtltH xp yp exp27ci xi xC 11yi yCdxp dyp 1141 Expand the exponential for small i n 1 g niexpi Pg 11 2 P6311 6Xp2 i xC 11 961 Yp X 1 2 i xi Xe W ya 279K061 xc Yr 3762 dxp dyp The linear term is zero if xCyC is the PSF centroid Hence 272 P6X Tg n 1 xpyp xl xCnyl yc2dxp dyp 1144a Pg n 27t xcnyc 1144b 220 Geometrical PTF depends on the pupil irradiance only through the centroid of the PSF The PTF for a symmetric aberration is zero as expected since the centroid for such an aberration lies at the origin ahg m Z 4TE2 8 P6X allAg ml ag E Similarly d1g 7n 0 an i0n Slope of the geometrical MTF at the origin is zero H xp yp ax xclly ycx xcdxp dyp 0 1145 02 221 Comparison of Diffraction and Geometrical OTFs Characteristic Td Tg Region of integration Cutoff frequency Aberration free Slope at the origin A dependence Overlap area of two pupils Yes Normalized weighted fractional overlap area Nonzero independent of W Yes since W depends on it Whole pupil No Unity for all frequencies Zero No since ray aberration is independent of A except for any dispersion 222 Asymptotic Behavior of PSF Radially symmetric pupil function P7p 2 PW 71 rpcosepsin6p 0 s eplt Zn 7 ricoselsin9i 0 S 6i lt 275 2 1 27139 PSFfz P d 2 Pexszzj rpexp M quotp 3 quotp 27 1 2m PSFiz P 039 G Gd6 2 Pexx2R2J 11 quotp lexp erpriCOSp 1 p 4 2Pexx2R2 JPrp J0 ZurpigXRrp a rp 2 182 223 Similarly 1 51 V II jPSFfleXp27ci17i 17d7 ZTEJPSFOE 1027Wm12dn 183 and PSFrl 275JTltVLJ0ZTEVLI vidvi 184 PSF and the OTF of a system with a radially symmetric pupil function are also radially symmetric and form a zeroorder Hankel transform pair Theorem 8 224 Asymptotic PSF PSFrl 2 l1vl J027cvlrlvldvl mexJ0mxdx f0 1f 0ifw0 Willis 0 2 m2 222 m4 where NR0 3 fxax x Letm27crl and x vl fvlvi17vi yielding f003 2 a f Vii1 a 120 ViT 2T39a f 0 21quot0 avi dvi 83f 4f iv iv vT 3T M 41 f O 417 0 av av H F Willis A formula for expanding an integral as series Philos Mag 39 455 459 1948 There is a minus sign missing on the righthand side of the formula corresponding to Eq 2 255 2 25 3TIIIO 472133 32745 for large values of 1 1154 Hence for large values of ri the PSF is independent of aberration and varies with I as rf3 Theorem 13 It has been shown for circular pupils that Eq 1154 holds even for rotationally nonsymmetric aberrations B Tatian Asymptotic expansions for correcting truncation error in transfer function calculations J Opt Soc Am 61 1214 1224 1971 226 Asymptotic encircled power r0 2n Pi rs I I 12 901239 dri dei 0 0 re 27 27 2 Fax I I dig gdei I vi 1121 1vi exp 27cirLvi cos6i d 0 re 27 2x 2 Fax I r drl 6114 j 17vi 4vl 1121 I eXp 27cirLvi cos6i d6i 0 lb 275 ZTEPex I rid MM TVia 102mivividvi 0 0 r 275 c Z Pex Id JTVia Vid i J J027criviri dri 0 0 27 2 Fax rC gdd 17vi 4 J127crcvldvl 27 196er Id j RCTVilJ127Eerid1i since Pi is real 0 227 Another formula by Willis meXJ1mxdx 100 1010 fng 1 3 f 0 0 1 2 m 222 m5 27 N Rem04gt Rer0 gt Rer0 gt 3a MM 27 2 Mr O 27 N Pex1147E2rc j Rer0 61 1163 0 Hence up to the rst order the encircled power for large circles is also independent of an aberration Theorem 14 228 Unapodized system AlrplAo OTF 12 of an unapodized aberrationfree system corresponding to a spatial frequency vi is equal to the fractional area of overlap of two pupils whose centers are separated by XRVi Let L be the length of the perimeter of the pupil 1 7vR2Sex I vi cosocds S 1 XRviZ Sex l lcosocds for very small values of vi S 229 where ds is a differential length segment along the perimeter of the pupil and 0c is the angle between vi and the normal to the segment The factor of 2 accounts for the fact that the nonoverlap area is counted twice in the integral 27 27 1vi doc 1vi d since docd 0 0 2 27 7 RLvl2S x Inlcosoclda 0 27 ZXRLviSex Differentiating both sides with respect to vi and evaluating in the limit vi a O 2kRL 2ft 1104 d 1167 0 6X Since I 0 is independent of aberration Eq 1167 holds for aberrated but unapodized systems also 230 Substituting into Eq 1163 XRL P r P 1 1168 C Z zrcsex lt gt o Thus the asymptotic behavior of encircled power depends on the ratio of the perimeter length L of the exit pupil and its transmitting area Sex Theorem 15 The effect of an aberration is to increase the value of rC for which Eq 1 168 is valid The larger the aberration the larger the value of rC required for the validity of this equation 231 PSF Centroid Centroid in terms of OTF slope Xian PeQIWXiaYi1ixiayidxidyi 11 Pg MMYiCXP275i XiTm05in a 2 39 39 a myyaxiexpizmcx nmidad 6X 8 2 39 a a znzo g lixi yixidxidyi 2751ltxigt 6X 1 alm 81m 1173 Hence the PSF centroid can be obtained from the slope of the imaginary part of the OTF Theorem 16a It can not be obtained from the MTF unless the OTF is real in which case it lies at the origin 232 If the system is aberration free then T is real and xiyi00 ie the centroid lies at the Gaussian image point regardless of the amplitude variations across EXP Theorem 160 Centroid in Terms of Wavefront Slope Pxp yp Axp ypexpi27 wxp yp dim P6361 Pxp yp Pxp XRQ yp 7 Rndxp dyp 1 81mquot W ag znzo x 8P x y Z gx ImPxpypgxpp dxp dyp 233 xiylgt IpxpypaW dxpdyp 1177 ex Exp ayp Since RaWaxpaWayp represents the ray aberration xiyl the centroid of the diffraction PSF is the same as that of the geometrical PSF Theorem 16b Hence we may trace rays up to the image plane and determine their centroid taking into account their weighting Ipxp yp Centroid in terms of the wavefront perimeter For a uniform pupil Axp ypA0 2 Pex A0 Sex IO Sex R 3W 3W R A M dxpdyp S 3 Wxpypxp ds Exp ayp ex 234 xm Sig wxpypxpypd 1181 636 Surface integral has been converted into a line integral using Stokes theorem where d is an element of arc length vector along the perimeter of the pupil and 92p and yp are unit unit vectors along the x and yp axes Thus the centroid for a uniform pupil can be obtained from the aberration function along the perimeter of the pupil without any knowledge across its interior Theorem 16d For example primary and secondary coma of five waves 57 in a system with a circular pupil yield different PSFs as shown below but the same centroid Primary coma Secondary coma 235 Hopkins Ratio Hopkins ratio Hm is the ratio of the MTFs and ru i of a system at a spatial frequency Vi with and without aberration Q7p W 133 ltDp XRVi Phase aberration difference function mm T 7ilTuvi HAFPA7P kRviexpiQ7pvid7p jA7pA7p kRVid7p KCXPQQM ltexpiQ ltQgt lt1iQ Qgt Q ltQgt2gt l 2 1 on where 0 ltQ2gt Q2 is the variance of Q I 236 o Hence Hopkins ratio for a certain spatial frequency depends on the variance of the phase aberration difference function across the overlap region of two displaced pupils displacement depending on the spatial frequency and not on the type of the aberration Theorem 19a mm TVilTuvi Multiplying both sides by eXpi P7lI WW H17iexpi l 17i I Viexpi P17l lt6XpiQgt exp i Q ltQgt gtexpiltQgt Hence the mean value Q of the phase aberration difference function yields approximately the PTF Theorem 19b An approximate expression for Hopkins ratio for small aberration difference function is 1 Hvl exp 5 6a Strehl ratio depends on the variance of the aberration function across the pupil Hopkins ratio depends on the variance of the aberration difference function across the overlap area of two pupils oThey are image quality criteria in the space and frequency domains respectively 238 Home Work 2 2 1 Show that the PSF is symmetric and the OTF is real for a system with a symmetric pupil function ie show that if P p P7p then PSF 71 mag and 1717 1 2 2 Consider an extended but isoplanatic incoherent image with an irradiance distribution 7 Show that its sharpness defined as 1127d7 is maximum when it is aberrationfree 2 39

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