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
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 [1–4]. 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 . 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 gR and gI, respectively, which are generated from real and imaginary random object fields fR and fI. 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, gR and gI obey Gaussian statistics, which follows from the central limit theorem. The resulting variation of image irradiance is known as Gaussian speckle [5,10]. If gR and gI are governed by circular Gaussian statistics, where
speckle irradiance obeys Rayleigh statistics, which is known as fully developed speckle. It is well known that speckle contrast Cs = σI/〈I〉 = 1, and averaged speckle grain size is approximately the Airy disk diameter for fully developed speckle . When a coherent constant background field 〈gR〉 is not ignorable, the speckle is partially developed, and speckle irradiance statistics are described by a Rician distribution. This situation occurs for surface roughness ≤ ~ λ/2, when a coherent beam is reflected or transmitted from a rough object. However, fields gR and gI 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 gR and gI can exhibit non-circular Gaussian statistics . 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 ≫ λ/2) 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 . 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.
2. Theoretical development
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 mT, the coherent image field can be described by 
Here, f(x o) is the phase-perturbed random field by the rough object, hcoh(x i)is the aberrated coherent PSF, and l(x o) is optical path length distribution of the rough object in units of wavelength that exhibits Gaussian statistics. The variables mT and ξ′ = x XP/λr′ are transverse magnification and the spatial frequency vector, respectively, composed of parameters described in Fig. 1. T(mTξ′) 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 g(x i) in Eq. (2), statistical means and correlations for real fR(x o) and imaginary fI(x o) fields are derived. It is assumed that the object optical path length l(x o)is spatially stationary. Mathematically, stationary indicates that the mean is constant and the correlation (or covariance) is a function of only coordinate difference . Real and imaginary means and correlations of the complex random field f(x o) are derived in the Appendix A. Results are
Here, Kl (Δx o) in units of wavelength squared is the covariance of l(x o), which is a stationary random process. Equations (5) and (6) show that the random field transmitted by the rough surface is stationary when l(x o) is stationary. Notice that f(x o) does not obey circular Gaussian statistics, due to the difference between RoRR(Δx o) and RoII(Δx o), even though there is no correlation between real and imaginary fields. It can be shown that, if l(x o)exhibits very large variance, asymptotically RoRR(Δx o) ≃ RoII(Δx o) and mR ~ 0, so f(x o) becomes circular Gaussian with zero mean. Without losing generality, covariance Kl(Δx o) can be modeled by a self-affine fractal surface [17–19]
where σ 2, Lcor and H are variance of l(x o), correlation length and Hurst exponent, respectively. Note that 0 < H < 1.
From Eq.(2), irradiance at observation point x i on the image plane is described by
where Cdiff is a diffraction-related constant. A physical interpretation of Eq. (8) is that speckle irradiances at observation points x i 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 g(x i) and Δx i = x′i - x i. Application of the Gaussian moment theorem [20,21] to Eq. (9) yields the covariance expressed by second moments of real and imaginary parts of g(x i), where
with image field correlation Ri(Δx i) and mean fields cR and cI. Subscripts R and I indicate real and imaginary fields, respectively. For example, RiRI(Δx i) = 〈gR(Δx i + x i)gI(x i)〉. Equation (9) shows that covariance with Δx i = 0 is irradiance variance σI. Therefore, from Eq. (10), speckle contrast Cs within an isoplanatic patch on the image plane is
For circular Gaussian field statistics with cI = cR = 0, RiRI = RiIR = 0 and RiRR = RiII, Eq. (11) reduces to Cs = 1, which is the well known speckle contrast for circular Gaussian speckle. Additionally, if cR ≠ 0 , which indicates a coherent background field for partially developed speckle, Eq. (11) shows that Cs 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 RiIR = RiRI is a required condition for further theoretical development.
and for a symmetric pupil,
For convenience, the aberration function is separated into odd (Wo) and even (We) parts. Statistical means and correlations of gR(x i) and gI(x i) are derived after considering Eqs (12) and (13), where
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 (χind) and an aberration dependent part (χab), 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 RiRI(Δx i) = RiIR(Δx i), as shown in Appendix B. Notice that χab is dependent on We(ε′), which can change for different isoplanatic areas if a significant field-dependent aberration is present.
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 . From Eqs. (14), (15) and (16), it is observed that speckle mean 〈gg *〉 = RiRR + RiII is independent of aberrations, because the aberration dependent part χab is canceled. However, as shown in Eq. (10), speckle covariance (fourth moment of the random image field) Ks(Δx i) contains the sum of squares of RiRR(Δx i) and RiII(Δx i), which makes the speckle correlation and contrast depend on optical aberrations.
3. 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, Kinds (Δx i) and Kabs(Δx i) are
respectively. Notice that the numerator of speckle contrast in Eq. (11) is reduced to Eq. (17) if Δx i = 0. 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 Ku (0)) of a rough surface. If the variance of the surface height is very small (Ku (0) ≪ 1), Rβ(Δx o) in Eq. (6) approaches unity. Therefore, χab(Δx i) and RiRI(Δx i) in Eqs. (15) and (16) are approximated as
Substitution of Eq. (20) into Eq. (19) indicates that Kads(Δx i) is independent of optical aberrations. For a very large variance (Ku (0)≫1), it can be shown that Rβ (Δx i) ~ 0 from Eq. (6), which implies that both χab(Δx i) and RiRI(Δx i) 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 χab(Δx i) and RiRI(Δx i). 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 cI = 0 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 . Theoretical results in Eqs. (14) and (15) are consistent with Goodman’s result, where RiRR (0) ≠ RiII(0) for an aberration free case. He intuitively anticipated that optical aberrations change the image field statistics into circular Gaussian. With optical aberrations, χab(0) in Eq. (14) is generally smaller than χab(0) with no aberration due to the cos[4πWe(ε′) term in Eq. (15), which indicates the difference between RiRR (0) and RiII(0) 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 RiRI(0), as shown in Eq.(16). (Notice RiRI(0) is always zero when We (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 Kabs(0) in Eq. (19) is the only aberration dependent part of speckle contrast, the partial derivative of Kabs(0) with respect to the aberration function We(ξ′) is useful for the investigation of general dependence of speckle contrast on optical aberrations. Kabs(0) is a functional for the input function of We(ξ′), which means that Kabs(0) is mapping from the vector space of functions We(ξ′) to a scalar space. Therefore, the first and second partial derivatives of Kabs(0) respect to We(ξ′) evaluated at a particular function are a function and an operator, respectively . It is shown in Appendix C that first partial derivative of Kabs (0) indicates that speckle contrast is at a stationary point (maximum or minimum or saddle point) when We (ξ′) a constant function, implying We (0). 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 Kabs (0) determined by the second partial derivative operator indicates that Kabs (0) cannot be achieved a local maximum value at an aberration free condition, which means speckle contrast as a function of We(ξ′) shows a saddle point at aberration free. Notice that We(ξ′) is not a single variable, so We (0) is a point in an infinite dimensional space. For this reason, Kabs(0) is a saddle point at We(0).
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 Cs and RiRI (0) of Eq. (16) for zero piston aberration. The covariance of the rough surface is calculated from Eq. (7) with σ = λ/5, Lcor = 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. mT and r′ in Fig. 1 are -0.375 and 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 Cs and RiRI (0), respectively, for the variation of 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 RiRI (0) = 0. The 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 2(c) and 2(d) are Cs(x i) and RiRI (0), respectively, for the image plane ranged from 0 to 1 in a normalized image field with 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 RiRI (0) is zero. Calculations show that speckle contrast is relatively smaller for We (ξ′) = 0 and the non-zero function We (ξ′) that makes RiRI (0) = 0. Furthermore, calculations show that speckle contrasts are independent of a piston term.
Figures 3(a) and 3(b) are aberrations We (ξ′) in exit pupil space such that RiRI(0) = 0 for 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 -0.2X 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 in Eq. (16) are relatively small compared to the value of the central part. Aberrations minimizing speckle contrast are determined by the condition of RiRI (0) = 0. 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, J(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.
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 l(r) is described as,
where . For a new random process f(r) = exp[i2πl(r)], f(r), real and imaginary parts of f(r) are
respectively. From the definition of a characteristic functional in Eq. (A1), we can describe
Therefore, real and imaginary mean fields are
From Eq. (A2), correlation for a real random object field RoRR (r,r′) = 〈fR(r)f * R(r′)〉 is
Equation (A8) is further developed
Similar procedures are applied to other terms in Eq. (A7). Therefore, RoRR (r, r′) is
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 TR (ξ′) and T 1(ξ′) are defined
For the second step, the characteristic of a delta function is used. By applying the similar procedure as Eq. (B3) to gI(x⃗i), imaginary mean field By substituting gR of Eq. (12) to RiRR(x′i,x i) = 〈gR(x′i)gR(x i)〉, correlation between real image fields is
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 RiRR(x′i, x ′) in Eq. (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 RoRR (Δx o) and RoII(Δx o), respectively. By changing integration variable x o2 = x o1 + Δx o from the assumption of a stationary random object field, Eq. (B7) is
Applying a similar procedure for the second term of Eq. (B4), RiRR (Δx i) is 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 RiIR (Δx i) is
Appendix C: Mathematical derivation for that speckle contrast shows a saddle point at an aberration free condition
where constant terms are omitted. The function in Eq. (C1) is identically zero if and only if We (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 We (ξ) is a constant. The second partial derivative of Kabs (0) is evaluated at We (ξ) and is a linear operator with kernel function given by
Again, constant terms are omitted. Notice that Q(ξ′,ξ″)is a matrix. The quadratic form corresponding to Eq. (C2) is
where . P and u are arbitrary and unit functions, respectively. This quadratic form, after substituting a constant We (ξ), reduces to
and the sign 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 We(ξ⃗) is a saddle point for speckle contrast. For Cauchy-Schwartz inequality
with substituting q = J 1/2 and r = J 1/2 p, it can be estimated that Eq. (C4) is always less and equal to zero only if J is a non-negative function. Since the Fourier transform of , Bochner’s theorem can be applied to determine whether J(ξ;′) is a non-negative function or not . Bochner’s theorem says that J(ξ′) is non-negative only if its Fourier transform is positive definite. Positive definite functions must satisfy |Rβ (x)| ≤ Rβ (0) in 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. (C4) is simplified to
The binary function p is further constrained so that non-zero portions of p are confined to positive portions of J , which means (J,p) < 0. 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 We (ξ) is a saddle point of speckle contrast as a function of the even optical aberration function We (ξ).
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