Effect of optical aberration on Gaussian laser speckle

Optical aberration effects up to the second moment of Gaussian laser speckle are theoretically investigated for both partially and fully developed speckle. In the development, a plane-wave illuminated diffuser generates a phase-perturbed random field in the object plane that creates speckle in the image plane. Theoretical derivations show that image field statistics are generally non-circular Gaussian due to aberrations. Speckle statistics are not affected by odd-functional aberrations, such as coma, and dependency of aberrations is asymptotically ignorable for very weak or strong diffusers. Furthermore, Gaussian speckle contrast as a functional of optical aberrations exhibits a stationary point for the aberration free condition, where apparently contrast does not achieve a local maximum. Calculations of speckle contrast for several aberration conditions are also presented. ©2009 Optical Society of America OCIS codes: (030.6140) Speckle; (030.6600) Statistical optics; (110.6150) Speckle imaging; (290.5880) Scattering, rough surfaces References and Links 1. E. M. Gullikson, “Scattering from normal incidence EUV optics,” Proc. SPIE 3331, 72-80 (1998). 2. N. A .Beaudry and T. D.Milster, “Scattering and Coherence in EUVL,” Proc. SPIE 3331, 537-543 (1998). 3. N. A.Beaudry and T. D.Milster, “Effects of mask roughness and condenser scattering in EUVL systems,” Proc. SPIE 3676, 653-662 (1999). 4. P. P. Naulleau, “Relevance of mask-roughness-induced printed line-edge roughness in recent and future extreme-ultraviolet lithography tests,” Appl. Opt. 43, 4025-4032 (2004). 5. J. C. Dainty et al. Laser speckle and related phenomena (Springer, 1984). 6. J. W. Goodman, Speckle phenomena in optics, theory and applications (Robert, 2007) 7. J. W. Goodman, “Dependence of image speckle contrast on surface roughness,” Opt. Commun. 14, 324-327 (1975). 8. K.l A. Stetson, “The vulnerability of speckle photography to lens aberrations,” J. Opt. Soc. Am. 67, 15871590 (1977). 9. K. A. Stetson, “Problem of defocusing in speckle photography, its connection to hologram interferometry, and its solutions,” J. Opt. Soc. Am. 66, 1267-1271 (1976). 10. R. D. Bahuguna, K. K. Gupta, and K. Singh, “Study of laser speckles in the presence of spherical aberration,” J. Opt. Soc. Am. 69, 877-882 (1979). 11. R. D. Bahuguna, K. K. Gupta, and K. Singh, “Speckle patterns of weak diffusers: effect of spherical aberration,” Appl. Opt. 19, 1874-1878 (1980). 12. R. N. Singh, A.K. Singhal, “Formation of laser speckles under extra-axial aberrations,” Opt. Quantum Electron. 12, 519-524 (1980). 13. A. Kumar, and K. Singh, “Elongated laser speckles in imaging of a rough object with slit shaped illuminated region: Effect of off-axis aberrations,” Optik 96, 115-119 (1994). 14. P. K. Murphy, J. P. Allebach, and N. C. Gallagher, “Effect of optical aberrations on laser speckle,” J. Opt. Soc. Am. A 3, 215-222 (1986). 15. J. W. Goodman, Introduction to Fourier optics (McGrwa-Hill, 1996). 16. H. H. Barrett and K. J. Myers, Foundations of image science (Wiley Series, 2004). 17. J. Krim, I. Heyvaert, C. Van Haesendonck, and Y. Bruynseraede, “Scanning tunneling microscopy observation of self-affine fractal roughness in ion-bombarded film surfaces,” Phys. Rev. Lett. 70, 57-60 (1993). 18. G. Palasantzas and J. Krim, “Effect of the form of the height-height correlation function on diffuse x-ray scattering from a self-affine surface,” Phys. Rev. B 48, 2873-2877 (1993). #105104 $15.00 USD Received 9 Dec 2008; revised 28 Jan 2009; accepted 6 Feb 2009; published 17 Feb 2009 (C) 2009 OSA 2 March 2009 / Vol. 17, No. 5 / OPTICS EXPRESS 3084 19. G. Palasantzas, “Roughness spectrum and surface width of self-affine fractal surfaces via the K-correlation model,” Phys. Rev. B 48, 14472-14478 (1993). 20. I. R. Reed, “On a moment theorem for complex Gaussian processes,” Trans. Inform. Theory IT-8, 194-195 (1962). 21. F. D. Neeser and J. L. Massey, “Proper complex random processes with applications to information theory,” IEEE Trans. Inform. Theory 39, 1293-1302 (1993). 22. Bochner and Salomon, Lectures on Fourier integrals (Princeton University, 1959).


Introduction
Characteristics of laser speckle are of great interest, since coherent systems contain speckle.For a coherent optical imaging system, speckle is frequently considered as noise deteriorating image quality.For example, in an extreme ultraviolet lithography (EUV) system, the mirror surface roughness generates significant speckle noise on the image plane due to the very short wavelength relative to the surface polish [1][2][3][4].This speckle effect is a function of the illumination's spatial coherence, where noise increases as the coherence increases.In a different application, speckle can be used as an effective tool to measure surface properties [5].For this case, the random field from a diffuser is a reference beam.Correlation between the reference beam and the random field reflected or transmitted from a test surface gives useful information.Both test and reference random fields are affected by optical system characteristics, like aberrations.Therefore, in order to extract precise information about the test surface, variation of speckle caused by the optical system should be understood.In this paper, effects of optical aberrations on Gaussian speckle in a coherent imaging system are investigated.
We define real and imaginary random electric fields on the image plane as R g and I g , respectively, which are generated from real and imaginary random object fields R f and I f .When there are many independent scatters at the object field contributing to the field at an image point, or equivalently, the correlation length of the object field is much smaller than the effective extent of the coherent point spread function (PSF) in object space, R g and I g obey Gaussian statistics, which follows from the central limit theorem.The resulting variation of image irradiance is known as Gaussian speckle [5,10].If R g and I g are governed by circular Gaussian statistics, where approximately the Airy disk diameter for fully developed speckle [5].When a coherent constant background field R g is not ignorable, the speckle is partially developed, and speckle irradiance statistics are described by a Rician distribution.This situation occurs for surface roughness 2 λ ≤ ∼ , when a coherent beam is reflected or transmitted from a rough object.However, fields R g and I g are not always described by circular Gaussian statistics.For example, Goodman has theoretically shown that variances of real and imaginary fields are different for some optical conditions, so R g and I g can exhibit non-circular Gaussian statistics [7].In this paper, statistics of the image field without aberrations are assumed Gaussian.It is also assumed that statistics of the real and imaginary portions at the image point are Gaussian after the addition of aberrations, but not necessarily circular.This assumption is justified, because optical aberrations only increase dimensions of the coherent PSF.Therefore, a larger number of object scatter points contribute to each point in the image.
For a strongly diffused (surface roughness 2 λ ≫ ) object field exhibiting circular Gaussian statistics and generating fully developed speckle, it is known that statistical characteristics of speckle are nearly independent of optical aberrations [5,6].However, Stetson observed that lens aberrations affect speckle photography [8,9].Bahuguna et al. demonstrated that the effect of spherical aberration to speckle from a strong diffuser is ignorable, and only speckle generated from a weak diffuser is dependent on spherical aberration [10,11].However, his analysis is confined to heuristic explanations using geometrical ray techniques without theoretical descriptions.With a similar geometrical ray method, speckle pattern variation according to off-axis aberrations and illuminated diffuser size has been discussed [12,13].According to the author's knowledge, Murphy et al. theoretically described aberration effects on Gaussian laser speckle for the first time [14].They calculated power spectral density of speckle and generated analytic expressions for speckle contrast by Fourier transforming the power spectral density.Although this work is mathematically splendid, it is difficult to capture physical insights from it about how aberration affects speckle statistics.
In this paper, an alternative theoretical approach is developed for investigating the relationship between aberrations and Gaussian speckle (both partially and fully developed).The object field is phase-perturbed after being transmitted or reflected from a rough surface.The illuminating plane wave is completely coherent, and the illuminated area of the random object is much larger than coherent PSF extent.Although the method starts with assumption of a linear shift invariant system, the result can be used to evaluate second order statistics of speckle for off-axis aberrations.In Section 2, theoretical developments are presented.Based on the theoretical results derived in Section 2, mathematical investigations showing speckle characteristics according to optical aberrations are discussed in Section 3 with associated calculation results.Sections 2 and 3 are summarized in Section 4. Appendixes A and B describe mathematical developments for means and correlations between object fields and image fields, respectively.Appendix C contains a derivation that shows speckle contrast as a functional of an aberration function has a saddle point at an aberration free point.Figure 1 shows a conceptual layout for object-generated laser speckle in imaging systems.A coherent field illuminates a diffuser located in the object plane.Transmitted light from the diffuser is the random object field that generates image speckle irradiance.Under the assumption that the optical system is linear and shift invariant with transverse magnification T m , the coherent image field can be described by [15] ( ) ( )

Theoretical development
where and

( )
W ′ ξ are the entrance pupil scaled to the exit pupil coordinates and aberration function in units of wavelength, respectively.The exit pupil is a curved surface with a radius r ′ centered on axis in image space, as shown in Fig. 1.If r ′ is very large, the pupil transmittance function is effectively planar.
In order to evaluate statistical characteristics of ( ) where Here, ( ) Κ ∆x in units of wavelength squared is the covariance of ( ) o l x , which is a stationary random process.Equations ( 5) and (6)  Κ ∆x can be modeled by a self-affine fractal surface [17][18][19] ( ) ( ) where 2 σ , cor L and H are variance of ( ) o l x , correlation length and Hurst exponent, respectively.Note that 0 1 H < < .
From Eq.( 2), irradiance at observation point i x on the image plane is described by where diff C is a diffraction-related constant.A physical interpretation of Eq. ( 8) is that speckle irradiances at observation points i x are formed from only local areas of the random object field bounded by two coherent PSFs.The local contributing areas for different image points are indicated as solid circles in Fig. 1.Statistics inside all local contributing areas are the same, due to the assumption of stationary.Therefore, statistics of the image field over the entire image plane from these local contributing areas are also the same if the optical system is linear and shift invariant.Usually, off-axis aberrations such as astigmatism and distortion are image-coordinate dependent, and they destroy the linear shift invariance of a system.If the local contributing area is an isoplanatic patch with constant aberration coefficients, further development assuming a locally linear and shift invariant system is justified.Therefore, Eqs.(2), (8) and further theoretical developments are valid for calculating speckle characteristics with off-axis aberrations by considering the appropriate isoplanatic coherent transfer function.
Under the assumption of a stationary random image field, covariance of speckle irradiance is defined as which is the fourth moment of the random image field ( ) i g x and x x x .Application of the Gaussian moment theorem [20,21] to Eq. ( 9) yields the covariance expressed by second moments of real and imaginary parts of ( ) i g x , where with image field correlation R ∆x and mean fields R c and I c .Subscripts R and I indicate real and imaginary fields, respectively.For example, Equation (9) shows that covariance with 0 Eq. ( 10), speckle contrast s C within an isoplanatic patch on the image plane is For circular Gaussian field statistics with 0 , Eq. ( 11) reduces to 1 s C = , which is the well known speckle contrast for circular Gaussian speckle.Additionally, if 0 R c ≠ , which indicates a coherent background field for partially developed speckle, Eq. (11) shows that s C is decreased, and irradiance on the image plane is more uniform.Notice that the denominator of Eq. ( 11) is a mean irradiance, which must be real and positive.However, Eq. ( 11) contains an imaginary part, so is a required condition for further theoretical development.
In order to evaluate ( ) ( ) g x , consider the random object field of Eq. ( 3) and the coherent PSF of Eq. ( 4) as separated real and imaginary parts, where and for a symmetric pupil, ( For convenience, the aberration function is separated into odd ( o W ) and even ( e W ) parts.
Statistical means and correlations of ( ) ( ) g x are derived after considering Eqs ( 12) and ( 13), where and Fourier transform variables are indicated by a subscript on F .Detailed mathematical descriptions are shown in Appendix B. Notice that real and imaginary field correlations are composed of an aberration independent part ( ind χ ) and an aberration dependent part ( ab χ ), as shown in Eqs. ( 14) and ( 15), respectively.Also, cross correlation between real and imaginary image fields is For Eqs. ( 15) and ( 16), it is assumed that the pupil is a symmetric function as before.Also, it is clear that ( ) ( ) The odd aberration function is canceled during the mathematical procedure without further assumption, which means that second order statistics of laser speckle are not affected by odd aberrations.This result is consistent with Murphy's work, where he and his coworkers showed that laser speckle contrast is essentially independent of coma and distortion from simulation [14].From Eqs. ( 14), ( 15) and ( 16), it is observed that speckle mean independent of aberrations, because the aberration dependent part ab χ is canceled.However, as shown in Eq. ( 10), speckle covariance (fourth moment of the random image field) ( ) Κ ∆x contains the sum of squares of ( ) R ∆x , which makes the speckle correlation and contrast depend on optical aberrations.

Speckle characteristics
Further theoretical investigation for speckle characteristics affected by optical aberrations is discussed in this section by analyzing Eqs.( 14), ( 15) and (16).Substitution of Eq. ( 14) into Eq.( 10) reduces speckle covariance to where aberration independent and dependent parts, ( ) ( ) and respectively.Notice that the numerator of speckle contrast in Eq. ( 11) is reduced to Eq. ( 17) if 0 i ∆ = x .Therefore, theoretical investigation for the aberration effect on Gaussian speckle can be simplified to the examination of Eq. ( 19).
It is possible to determine the asymptotic behavior of the dependence speckle covariance and contrast on aberrations for very small or large variances (i.e.very small or large ( ) of a rough surface.If the variance of the surface height is very small ( ( ) in Eq. ( 6) approaches unity.Therefore, ( ) ( ) R ∆x in Eqs. ( 15) and ( 16) are approximated as R ∆x in Eq. ( 19) approach very small values.Therefore, the effect of optical aberrations on second order statistics of Gaussian laser speckle is asymptotically ignorable when the height variance of the rough object becomes very small or very large.
It is conceptually obvious that a piston aberration doesn't affect speckle.This dependence can be theoretically verified from the derived mathematical results in section 2. Speckle dependency on piston aberration is investigated with a partial derivative of Eq. ( 19).The partial derivative is calculated using the chain rule for partial derivatives for each component ( ) ( ) R ∆x .It is not difficult to show that the partial derivative is zero with the following conditions: which follow from Eqs. ( 15) and (16).A value of zero for the partial derivative indicates that speckle covariance and contrast are not affected by a piston aberration.This result indicates that speckle covariance in Eq. ( 17) can be further simplified by assuming zero piston aberration, which indicates 0 I c = from Eq. ( 14).Goodman theoretically verified that image field statistics from a phase-perturbed object field transmitted by a rough surface with a Gaussian height distribution of moderate roughness are generally non-circular Gaussian [7].Theoretical results in Eqs. ( 14) and ( 15) are consistent with Goodman's result, where ( ) ( ) for an aberration free case.He intuitively anticipated that optical aberrations change the image field statistics into circular Gaussian.With optical aberrations, ( ) 14) is generally smaller than ( ) is decreased and the image field statistics approach circular Gaussian.However, aberrations generally cause non-zero cross second moment between real and imaginary image fields ( ) , as shown in Eq.( 16).(Notice ( ) ξ is a zero function, or equivalently, a system is aberration-free.)Therefore, optical aberrations produce non-circular Gaussian image field statistics, unlike Goodman's prediction.Since . This condition is equivalent to aberration-free because it is proved that speckle contrast is independent of a piston term.Furthermore, examining definiteness for the quadratic form of ( ) Even though it is mathematically shown that the Gaussian speckle contrast passes through a saddle point when an optical system is aberration-free, speckle contrast calculations performed with Seidel aberrations show that minimum speckle contrasts are observed when aberration is zero.Figure 2 shows calculation results for speckle contrast s C and ( ) 16) for zero piston aberration.The covariance of the rough surface is calculated from Eq. ( 7) with σ = 5 λ , cor L = 20nm, and H = 0.5, where the wavelength is 13.5nm.For the short wavelength regime such as extreme ultra violet (EUV), surface scattering is significant and effects of scattering are important.For this reason, 13.5nm wavelength is chosen for calculations.However, the same calculation can be applied for visible wavelengths.The exit pupil is a square function with an aperture of 30 by 30mm.T m and r′ in Fig. 1 are -0.375 and Furthermore, calculations show that speckle contrasts are independent of a piston term.ξ in exit pupil space such that ( ) Fig. 2(a) and 2(c), respectively.Speckle contrast is relatively small with these aberrations.Figure 3 (a) is for -0.175λ and 0.4λ of defocus and spherical aberrations, respectively.Figure 3(b) is for the normalized image field at (0.23, 1), where 0.5λ spherical, -0.1λ field curvature and -0.2λ astigmatism.As shown in Fig. 3, the combined aberration values are nearly zero over almost the entire exit pupil area, except edge parts where values of 16) are relatively small compared to the value of the central part.Aberrations minimizing speckle contrast are determined by the condition of ( ) An important observation is that this combination of aberrations is not the combination that minimizes RMS spot size or wave aberration variance.For a different rough surface with different statistics, ( ) ξ is changed, and the aberration minimizing speckle contrast is also changed.This result indicates that these aberration combinations are effectively equivalent to an aberration-free condition for the mechanism of generating speckle, which confirms theoretical developments in the previous section.

Summary
Optical aberration effects on Gaussian laser speckle are theoretically investigated.By dividing real and imaginary parts of the random image field, the second moment of speckle on the image plane is derived as a function of aberrations.The random field on the object plane is assumed a stationary random process.The theory is developed for a linear and shift invariant optical system, but it is valid for off-axis aberrations by considering an isoplanatic approximation.Speckle correlation and contrast are not dependent on odd-functional aberrations, such as coma and distortion.Furthermore, if height variance of a rough surface is very large or small, aberration effects on speckle are asymptotically ignorable.The theory shows that speckle contrast with a zero aberration condition is a saddle point as a function of optical aberrations.Calculations performed with Seidel aberrations show that speckle contrasts are minimized with the zero aberration condition, which supports the theoretical result.According to calculation results, it can be stated that optical aberrations appear to always increase speckle contrast.The most interesting result is that optical aberrations generally cause the cross second moment between real and imaginary image fields to be nonzero, which means non-circular Gaussian speckle statistics are manifested by optical aberrations.

Appendix A: Mathematical derivation for first and second moments of a random object field
The characteristic functional for the Gaussian height distribution ( ) where ( ) r , real and imaginary parts of ( ) and ( ) ( ) ( ) respectively.From the definition of a characteristic functional in Eq. (A1), we can describe From the second line in Eq. (A1), Eq. ( A4) is further developed , ., .
with Eq. ( 6), respectively.It is easy to derive the other results in Eq. ( 5) using a similar mathematical procedure.

Appendix B: Mathematical derivation for first and second moments of a random image field
The real and imaginary coherent PSFs in Eq. ( 13) can be rewritten as where real and imaginary effective coherent transfer functions ( ) ξ and ( ) with odd and even aberration functions ( ) e W ′ ξ , respectively.From Eqs. ( 12)and (B1), real mean field in the image plane is ( ) For the second step, the characteristic of a delta function is used.By applying the similar procedure as Eq.(B3) to ( ) By substituting R g of Eq. ( , where it is used real and imaginary object random fields are uncorrelated.Likewise, correlation between imaginary fields and cross correlation between real and imaginary fields are , respectively.Subscripts R and I in correlation functions indicate real and imaginary image fields, respectively.In order to avoid complexity, only the first term of ( ) B4) is further mathematically developed.Considering Eq. (B2), the first term of Eq. (B4) is From the assumption of a stationary random object field, correlation functions of real and imaginary object fields can be expressed as R ∆x , respectively.By changing integration variable 2 x from the assumption of a stationary random object field, Eq. (B7) is Applying a similar procedure for the second term of Eq. (B4), ( ) Likewise, Eqs.(B5) and (B6) are ( ) respectively.Substituting Eq. ( 5), Eqs.(B9), (B10) and (B11) can be rewritten, ( ) respectively.All correlation functions are composed of products of two effective coherent transfer functions with variable ′ ξ and ′ −ξ .Therefore, odd functional aberrations are canceled by substituting effective transfer functions in Eq. (B2).Substituting Eq. (B2) to each correlation functions and using trigonometric identities, Eqs.(B12) and (B13) are simplified to Eqs. ( 14), (15) and (16).The cross correlation between imaginary and real random image fields which is the same as Eq. ( 16) after substituting effective transfer functions in Eq. (B2).

Appendix C: Mathematical derivation for that speckle contrast shows a saddle point at an aberration free condition
Replacing ( ) 15) and ( 16) with ( ) ξ , the first derivative of ( ) Rayleigh statistics, which is known as fully developed speckle.It is well known that speckle contrast 1 averaged speckle grain size is

Fig. 1 .
Fig. 1.Conceptual layout for generating speckle in the image plane.A plane wave illuminates rough surface located in the object plane.The solid circles indicate contribution areas of the random object field for generating speckle at image points.
the phase-perturbed random field by the rough object, the aberrated coherent PSF, and ( ) o l x is optical path length distribution of the rough object in units of wavelength that exhibits Gaussian statistics.The variables T m and transverse magnification and the spatial frequency vector, respectively, composed of parameters described in Fig. 1. ( ) T T m ′ ξ and are derived.It is assumed that the object optical path length ( ) o l x is spatially stationary.Mathematically, stationary indicates that the mean is constant and the correlation (or covariance) is a function of only coordinate difference [16].Real and imaginary means and correlations of the complex random field ( ) o f x are derived in the Appendix A. Results are .(15), which indicates the difference between ( )

Κ
in Eq. (19) is the only aberration dependent part of speckle contrast, the partial derivative of is useful for the investigation of general dependence of speckle contrast on optical aberrations.evaluated at a particular function are a function and an operator, respectively[16].It is shown in Appendix C that first partial derivative of function, implying ( )

ΚFig. 2 .
Fig. 2. Speckle contrasts (a) and cross second moments (b) between real and imaginary fields at fixed observation point with spherical and defocus aberrations from -0.5λ to 0.5λ.Speckle contrast (c) and cross second moment (d) at normalized image fields from 0 to 1 with 0.5λ spherical, -0.1λ field curvature and -0.2λ astigmatism.All aberrations are of third order.Dotted lines in (b) and (d) indicate the combination of aberrations for a zero cross second moment.Speckle contrasts in (a) and (c) along these lines show relatively minimum values.

#
105104 -$15.00USD Received 9 Dec 2008; revised 28 Jan 2009; 6 Feb 2009; published 17 Feb 2009 (C) 2009 OSA 450mm, respectively.The correlation length is 20nm, which is much smaller than the effective width of coherent PSF.Figures 2(a) and 2(b) are s C and spherical and defocus aberrations from -0.5λ to 0.5λ.The white dotted line in Fig. 2(b) indicates the combination of two aberrations where same dotted line is drawn in Fig. 2(a), where relatively better speckle contrast values are observed along this line.Figure 2(a) shows that speckle contrast for no aberration is a minimum point.Figures 2for the image plane ranged from 0 1 in a normalized image field spherical, field curvature and astigmatism as 0.5λ, -0.1λ and -0.2λ, respectively.For this condition, there is no aberration-free point on the entire image plane.However, similar to Figs.(a) and (b), speckle contrasts show minimum values along the dotted line, where

Fig. 3 . 1 .
Fig. 3. Aberrations when the cross second moments between real and imaginary fields are almost zero.Coordinates are a spatial frequency XP r λ ′ = ξ x [um] -1 .Parameters in (a) and (b) are the same as Figs.2-(b) and (d), respectively.Defocus and spherical aberrations for (a) are -0.175λ and 0.4λ, respectively.Normalized field locations Hx and Hy for (b) are 0.23 and 1, respectively.Units are wavelength of 13.5nm.

Figures 3 (
Figures 3(a) and 3(b) are aberrations (A6) and (A10) are reduced C1)where constant terms are omitted.The function in Eq. (C1) is identically zero if and only if ( ) e W ξ is a constant, because ( ) J ξ cannot be a zero function.Therefore, speckle contrast is at a stationary point (maximum or minimum or saddle point) when ( )

1 F
terms are omitted.Notice that ( ) , Q ′ ′′ ξ ξ is a matrix.The quadratic form corresponding to Eq. (C2) is $15.00 USD Received 9 Dec 2008; revised 28 Jan 2009; accepted 6 Feb 2009; published 17 Feb 2009 (C) 2009 OSA 2 March 2009 / Vol.17, No. 5 / OPTICS EXPRESS 3099 ξ ξ .p and u are arbitrary and unit functions, respectively.This quadratic form, after substituting a constant ( ) of this quantity for arbitrary p determines definiteness of the quadratic form.For example, if Eq. (C4) is neither positive nor negative definite, then the constant be estimated that Eq. (C4) is always less and equal to zero only if J is a non-negative function.Since ( ) ( ) 's theorem can be applied to determine whether ( ) J ′ ξ is a non-negative function or not [22].Bochner's theorem says that ( ) J ′ ξ is nonnegative only if its Fourier transform is positive definite.Positive definite functions must satisfy Eq. (6), which is not true for H >0. This indicates ( )J ′ξ is not a non-negative function by Bochner's theorem.Therefore, speckle contrast of no aberration is not a maximum value.If the arbitrary function p is assumed as a binary function, Eq. (p is further constrained so that non-zero portions of p are confined to positive portions of J , which means ( ),0 J p > .According to the width of non-zero portions of p , ( ) , J p is either larger or smaller than ( ) , J u , which indicates that a constant ( ) e W ξ is a saddle point of speckle contrast as a function of the even optical aberration function ( ) e W ξ .
show that the random field transmitted by the rough surface is stationary when ( ) as shown in Appendix B. Notice that ab