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 Calitomia 90245 University ot Arizona Sill 336l783 virendranmahajanaeroorg Lecture 16 Random Aberrations imaging Through Kolmogorov Turbulence 161 Lecture 16 Summary Imaging Through Kolmogorov Turbulence Kolmogorov Turbulence Wave Structure Function o Plane Wave o Spherical wave Near Field Far Field Atmospheric Correlation Length TimeAveraged OTF Application With Numerical Results o Circular Pupils Annular Pupils 162 Kolmogorov Turbulence Refractive index N of the turbulent atmosphere at a point 7 in space fluctuates due to fluctuations of its temperature Write N as a sum of its mean value M7 and a uctuating part n 1W M7 M N7gt1 but n7106 ltn7gt0 Structure function n7172 of the refractive index fluctuations representing the mean square value of the difference of refractive indices at two points 71 and 72 WW Mm mm ltn271n272 2n71n72gt Turbulent atmosphere consists of packets of air called eddies each with a characteristic value of its refractive index 163 It is reasonable to assume that the turbulence is locally statistically homogeneous so that the mean square value of the refractive index is the same at every point 2 a 2 s ltn 10gt in 12 Correlation function of refractive index uctuations Rina ltn71n72gt Homogeneity of turbulence also implies that the correlation function depends on the difference 7 72 71 of the position vectors of the two points RA ltn71n71 7 Power spectral density Qn1 of the refractive index fluctuations is given by the 3D Fourier transform of their correlation function Qn1 i Rn7 expamK 7 fr where 1 is a 3D vector representing a spatial frequency in units of cyclesm o Qn1 gives a measure of the relative abundance of eddies with dimensions 161 where K E 164 For a spatial period A of the refractive index fluctuations K 27cA Inverse Fourier transforming Rn j Qn1exp 27ciE7d31lt If the turbulence is statistically isotropic then the correlation function depends only on the distance r 7 between two points R110 ltn0 nrgt For statistically homogeneous and isotropic turbulence the structure function nr may be written in terms of the correlation function Rnr 9M1 2Rn0 Rnr For Kolmogorov turbulence and C5 r0 lt r lt L0 where C3 in units of m 23 is called the refractive index structure parameter E0 and 0 are called the inner and outer scales of turbulence representing the smallest and the largest eddies respectively 165 Typical values of C3 vary from 10 13111 23 for strong turbulence to 10 17111 23 for weak turbulence Values of 0 are on the order of a few millimeters and those of 0 vary from 1 m to 100 m Substituting for Rnr letting 0 a0 and 0 am and noting that d3KK2dK sinGdG d4 0 S 6 S 7 O S q lt 27 27 2E leXp 2niK7sinede 61 475w 0 0 27cm TEKI nr 87EJ Qn1lt1 szK 0 Since fan dex W 3ltalt 1 bx ba1 0 no 05133 if Qn1lt 969x10 3C5K 113 166 o If we divide r by 27 then the numerical coefficient 969 is replaced by 969X10327c23 or 0033 o Similarly if we divide the wavenumber K by 27 as was done by Fried then the coefficient 969 becomes 969gtlt10327c113 or 818 Fluctuations of refractive index of the turbulent atmosphere introduce fluctuations in phase of an optical wave propagating through it Structure function of phase fluctuations introduced by statistically homogeneous and isotropic turbulence may be written gm 2 ltltIgt71gt 071 7gtl2gt 2chltogt Raw where the correlation function Rql of phase fluctuations is given by Revquot E 9071 7 lt OCIgtI gt D Fried Optical resolution through a randomly inhomogeneous medium for very long and very short exposures J Opt Soc Am 56 1372 1379 1966 and Limiting resolution looking down through the atmosphere J Opt Soc Am 56 1380 1384 1966 167 Wave structure function for a spherical wave propagating through Kolmogorov turbulence is given by see Fried39s second reference L ow 2914k2r53 j C5zzL53dz 582 0 where z varies along the atmospheric path of total length L from a value of zero at the source or the object plane to a value of L at the receiver or the image plane Wave structure function can also be written in the form 53 w r 688rr0 583 where L 06 r0 01847x12j C5zzL53dz 584 0 is a characteristic length of Kolmogorov turbulence called its coherence length or diameter We call it Fried s coherence length in honor of his pioneering work in this area 168 r0 varies with the optical wavelength as x If the line of sight makes an angle 6 with the zenith then the path length L increases by sece or r0 decreases by sec 60396 SinceC decreases with altitude which is different for different sites the integral in Eqs 582 and 584 has a higher numerical value when observing a space object from ground looking upwards than when observing a ground object from space looking downwards o Correspondingly r0 is smaller when a satellite is observed from ground compared to when a ground object is observed from a satellite Consequently image degradation is much smaller when a ground object is observed from space than when a space object is observed from ground o In the first case the object is near the region of turbulence and it is observed from far away In the second case the object is away from the region of turbulence but it is observed from nearby 169 The fact that the image quality is superior in the first case is similar to when an object behind a diffuse shower glass is observed One can see some detail in the object when it is in contact with the shower glass However as soon as the object is moved slightly away from the shower glass it appears only as a halo illustrating complete loss of image resolution o This does not however mean that reciprocity of wave propagation does not hold For example if the wavefront errors of a wave from a point source in space propagating downwards are measured on ground and a conjugate correction is introduced in a beam transmitted upwards with a deformable mirror the beam focus in space will be diffraction limited neglecting any measurement or correction error illustrating that the atmosphere introduces the same wavefront errors whether a beam is propagating up or down through it 1610 For plane wave propagation the factor zL53 under the integral in Eqs 582 and 584 reduces to unity which may be seen as follows A plane wave can be thought of as a spherical wave originating at an infinite distance and traveling through a uniform medium for which 03 0 except for the propagation path through the atmosphere The value of zL in the region for which C3 70 is infinitesimally different from unity o Thus for example starlight propagation in groundbased astronomy can be considered as plane wave propagation with zL53 replaced by unity or a spherical wave propagating an infinite distance to reach the earth39s atmosphere with nonzero C3 value only near the end of its path for which zL is negligibly different from unity 1611 Neglecting the variation of C3 for horizontal propagation we obtain 291C3Lk2r53 Plane Wave 1quot W 38291C3Lk2r53 Spherical Wave and 06 168C3Lk2 Plane Wave r0 2 2 06 302CnLk SpherlcaIWave In the near field LltD27L the amplitude variations are negligible and therefore qr Wr In the far field Lgt Dzk echo 12 own For inbetween ranges the multiplying factor varies smoothly between 1 and 12 1612 MCF for Kolmogorov turbulence ltPR JP7Igt exp Wr exp 344rr053 Timeaveraged OTF of the overall system ltIi7Dr0gt TL17CXp WVD TL17ltTaVDr0gt where lt1avDr0gt exp 344vDr053 eXp 344vi7 Rr053 594 is the longexposure MTF reduction factor assosciated with atmospheric tubulence 1a is independent of the pupil diameter D as it should be Since r0 x the exponent in Eq 594 varies as NIB Since exp 344 003 atmospheric turbulence reduces the overall system MTF for a spatial frequency v rOD by a factor of 003 1613 Thus the correlation of complex amplitudes at two points on a wave separated by a distance r0 is 003 Ia represents the mutual irradiance function of the wave Therefore its magnitude describes the degree of spatial coherence of the wave and thus the visibility of fringes formed in a twopinhole experiment and observed in the vicinity of a point that is equidistant from the two pinholes Because of the random nature of atmospheric turbulence the time averaged irradiances at the two pinholes are equal to each other Hence r0 represents a spatial coherence length of the wave so that its degree of coherence corresponding to two points on it separated by r0 is 003 or that the visibility of the fringes formed by the secondary waves from these points is only 003 Thus wave elements at two points separated by a distance r0 are spatially practically incoherent Value of r0 on a mountain site may vary from 5 to 10 cm in the visible region of the spectrum which increases with wavelength as x 1614 Circular Pupils Assume that the turbulencefree system is aberration free with uniform transmittance Timeaveraged PSF of the overall system 1 ltIr Dr0gt 8 I 1712 Dr0gt J027trvvdv where 0 ltTV Dr0gt TLVCXp 344VDr053 TLV cos1v v1 v2l2 O lt i i Timeaveraged Strehl ratio ltSD 0gt ltI0 D 0gt 8JltTV D 0gtVdv 1615 Timeaveraged fractional encircled power 1 ltPrCDr0gt 27 rC I ltquotcvDr0gt J127crcv dv 0 Unnormalized timeaveraged central irradiance aimre ltsltDrogtgt where Sa 2 Tug4 is the coherent area of the atmosphere and TlD 0 2 1 i J 17L v exp 344vDr053 112 d v 599 0 Dr0 Dr02 ltsltDrogtgt 8l For a fixed total power Pa 1 is proportional to the central irradiance ltIi0 D 0gt For example we may have a groundbased laser transmitter whose total power is fixed but whose beam diameter can be selected to change the central irradiance on a target 1616 For small values of DrO the exponential factor is approximately equal to unity Hence the integral reduces to 18 or the Strehl ratio is approximately equal to unity Since Sex depends on D as D2 the central irradiance for a fixed total power Pa increases as D2 as for an aberrationfree system For very large values of DrO the contribution to the integral in Eq 599 comes from values of v small enough that the exponential factor is not vanishingly small Now 110 2 1 Moreover cos 1v 7c 2 for small values of v and for such values it dominates the second term vwll 122 Thus 1L0 1 near the origin Hence for large values of DrOI Dr0 8DIo2 J 6Xp 344vDr053Vdv 0 00 834465 35 l x6516Xp xdx 1 where x 344vDr053 and the integral is the gamma function F65 Thus maximum value of n is unity 1617 Hence for a fixed total power Pm the central irradiance lt1iODr0gt S Pm SaXZRZ regardless of how large Dis Pm SaleR2 is the aberrationfree value for a system with an exit pupil of diameter r0 Since Sa r02 7v2394 the limiting value P XSa7v2R2 of central irradiance varies with wavelength as 14 o At visible wavelengths r0 10 cm Consequently the performance of a groundbased telescope making astronomical observations in the visible region is limited primarily by atmospheric turbulence to a telescope of diameter D 10 cm The purpose of a large groundbased telescope has therefore generally not been better resolution before the advent of adaptive optics but to collect more light so that dim objects may be observed 1618 08 06 39 gtlt Cgt 04 02 gtV Figure 56a Timeaveraged MTF Aberration or turbulencefree MTF which corresponds to DrO O is shown for comparison OTF at any frequency decreases as Dro increases For example when Dro 2 not only is the OTF at any frequency reduced but the cutoff frequency is practically reduced from a value of 1 to 05 Similarly when Dro 5 the cutoff frequency is practically reduced to 02 1619 10 l l l l l l m gt ltSgt O l l 001 l l l l gt Dro Figure 56b Timeaveraged Strehl ratio Strehl ratio decreases to zero monotonically as DrO increases o For a given value of D Strehl ratio decreases rapidly as r0 decreases Even when r0 is as large as D the Strehl ratio is only 0445 1620 0 l l l l l 0 7 7 ltl39 ltl39 7 7 a O 7 Dr02 1 f i i i i i i i i i i i i i i i i i i i C T Dr02ltSgt 01 f i 001 l i l 01 1 10 gt Dro Figure 56c Timeaveraged central irradiance for a fixed total power represented by n o Aberrationfree value of 11 increases as Dr02 as illustrated by the straight line For small values of DrO aberrated value of 11 also varies as Dr02 As D increases 11 increases much more slowly with a negligible increase for DrO gt 1621 Limiting performance As DrO a oo 1 a 1 The two asymptotes of nDr0 intersect at DrO 1 Indeed Fried defined r0 in a way so as to yield this result He called the quantity 1 the normalized resolution In astronomical observations Pm increases as D increases However if the observation is made against a uniform background then the background irradiance in the image also increases with D as D2 o Hence the detectability of a point object is limited by turbulence to a value corresponding to an exit pupil of diameter r0 no matter how large the diameter D of the exit pupil is In the case of a laser transmitter with a fixed value of laser power Pex the central irradiance on a target will again be limited to its aberrationfree value for an exit pupil of diameter r0 no matter how large the transmitter diameter D is 1622 092996 gt lt rgt ltPrcgt gtrrc Figure 57 Timeaveraged irradiance and encircledpower distributions for different values of DrO o As DrO increases the diffraction rings disappear and the PSFs become smoother which may be approximated by Gaussian functions As DrO increases a given fraction of the total power is contained in an increasingly larger circle eg 84 in a circle of radius r6 122 when there is no turbulence but in a circle of radius 19 when DrO 1 1623 Annular Pupils Straight forward extension of equations for circular pupils 8 1 62 1 ltIr eDr0 2 H1 veDr0gt J027c rv vdv 0 8 1 62 ED 0 E ltI0 DI gt ltTveDr0gtvdv 0 1 ltPrceDr0gt 27ch I ltTveDr0gt J127c rcv dv O ltTv6Dr0gt 15W6exp33944VDr053 The irradiance is in units of QXS xeXZR2 so that its central value in the absence of turbulence is unity 1624 Unnormalized central irradiance ltPexgtSex lt1 06Drogt where MED1 0 1 62Dr02lt5 D 0gt For a fixed total power 136x 1 is proportional to the aberrated central irradiance ltIl06Dr0gt Aberrationfree or diffractionlimited value of n varies as 1 62Dr02 For small values of DrO the atmospheric MTF reduction factor eXp 344vDr053 is approximately equal to unity ie the effect of atmospheric tubulence is negligible Accordingly ltSeDr0gt is also approximately equal to unity and the aberrated value of 11 increases with DrO as for the aberrationfree case 1625 However for larger values of DrO it increases slowly with a negligible increase beyond a certain value of DrO depending on the value of e The saturation effects of atmospheric turbulence occur at larger and larger values of DrO as 6 increases Since Tv is approximately equal to unity near the origin irrespective of the value of e neDr0 a 1 as DrO a co as in the case of circular pupils The two asymptotes of neDr0 for a given value of e intersect at the point 1 2 DrO 1 62 Hence for a fixed total power Pm the central irradiance ltIl0eDr0gt S the aberrationfree central irradiance PexSa7v2R2 for a system with an exit pupil of diameter r0 regardless of how large D is equality is obtained as DrO a co Limiting value QXSaXZR2 of the central irradiance is independent of the value of 6 1626 100 086796 10 gtnDr0 e 01 1 1 1 l 1 1 1 1 01 10 100 1000 gtDr0 001 l l Figure 58 Variation of neDr0 with DrO for several values of 6 Its aberrationfree value given by 1 62Dr02 is represented by the straight lines Its aberrated value approaches unity as DrO ace regardless of the value of 6 1627 Limiting performance In astronomical observations Pa TE41 62D2ltIO where 10gtis the timeaveraged irradiance across the exit pupil increases with D as D2 However if the observation is made against a uniform background then the background irradiance in the image also increases as D2 Hence as in the case of a circular pupil the detectability of a point object is limited by turbulence to a value corresponding to an exit pupil of diameterrO no matter how large the diameter D of the actual exit pupil is In the case of a laser transmitter with a fixed value of laser power Pex the central irradiance on a target will again be limited by its aberrationfree value PexSaXZR2 for an exit pupil of diameter r0 1628 To illustrate the difference in Strehl ratios for circular and annular pupils consider AS Dr0 10ltSODr0gt ltSeDr0gt Strehl ratio decreases to zero monotonically as DrO increases irrespective of the value of e o However AS gt 0 for DrO S 3 Le S decreases faster for annular pupils than for circular pupils o The opposite is true for DrO 2 3 Some typical values of the Strehl ratio are given in Table 53 for several values of DrO and 6 1629 12 8 8 8 10 7 A3 075 08 7 3 lt3 Dro 0gt 0 lt A 06 7 7 9 A3 050 9 04 e 7 U i 02 7 A3 025 i 00 02 i i i 0 2 4 6 8 10 gt Dro Figure 59 Variation of AS Dr010ltSODr0gt ltSDr0gt The Strehl ratio for e 0 shown in Figure 56b on a log scale is shown here on a linear scale 1630 Table 53 Timeaveraged Strehl ratio for various values of e and DrO e Dr0 1 2 3 4 5 0 0445 0175 0089 0053 0035 025 0430 0169 0088 0054 0036 050 0391 0160 0090 0058 0040 075 0344 0152 0095 0067 0050 1631 0901 96 gt lt rgt ltPrcgt Figure 510 Timeaveraged PSF and encircled power for several typical values of DrO and e 05 As DrO increases a given fraction of total power is contained in a circle of increasingly larger radius 1632 10 084696 gt ltPDr0 e gt 00 0 gt Dro Figure 511 Timeaveraged encircled power in the Airy disc as a function of DrO for several typical values of e For a given value of DrO the relative loss of power from the Airy disc decreases as 6 increases 1633
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