Speckled speckled speckle

Speckle is the spatial fluctuation of irradiance seen when coherent light is reflected from a rough surface. It is due to light reflected from the surface's many nooks and crannies accumulating vastly-discrepant time delays, spanning much more than an optical period, en route to an observation point. Although speckle with continuous-wave (cw) illumination is well understood, the emerging interest in non-line-of-sight (NLoS) imaging using coherent light has created the need to understand the higher-order speckle that results from multiple rough-surface reflections, viz., speckled speckle and speckled speckled speckle. Moreover, the recent introduction of phasor-field ($\mathcal{P}$-field) NLoS imaging---which relies on amplitude-modulated coherent illumination---requires pushing beyond cw scenarios for speckle and higher-order speckle. In this paper, we take first steps in addressing the foregoing needs using a three-diffuser transmissive geometry that is a proxy for three-bounce NLoS imaging. In the small-diffusers limit, we show that the irradiance variances of cw and modulated $n$th-order speckle coincide and are $(2^n-1)$-times those of ordinary (first-order) speckle. The more important case for NLoS imaging, however, involves extended reflectors. For our transmissive geometry with extended diffusers, we treat third-order cw speckle and first-order modulated speckle. Our results there imply that speckle is unlikely to impede successful operation of coherent-illumination cw imagers, and they suggest that the same might be true for $\mathcal{P}$-field imagers.


I. INTRODUCTION
When continuous-wave (cw) laser light that has been diffusely reflected by a rough surface illuminates an observation plane some distance away, a speckle pattern is visible in the measured irradiance.This speckle is due to wave-optical interference between reflections from independent wavelength-scale surface patches that vary in height by many optical wavelengths.This phenomenon, which we call ordinary (first-order) speckle, is well studied [1].
The irradiance produced by diffuse reflection is exponentially distributed at any observation point, and thus has a variance equal to its squared mean.The irradiance's covariance function for diffuse reflection from an extended surface is also well understood, as are limited properties of the second-order speckle seen when the speckle pattern from a first diffuse reflection is reflected from a second rough surface and observed, as speckled speckle, on a new observation plane.However, with the growing interest in non-line-of-sight (NLoS) imaging, colloquially referred to as "seeing around corners", there is now an unfulfilled need to understand higher-order speckle effects, e.g., speckled speckled speckle, such as results from laser light sequentially reflecting off three rough surfaces.Moreover, the advent of phasor-field (P-field) NLoS imagers [2][3][4]-which rely on amplitude-modulated coherent illuminationdictates that the preceding speckle questions be addressed for amplitude-modulated as well as for cw illumination.
In this paper, we take first steps in addressing the preceding issues using a paraxial, scalar-wave, transmissive geometry-which is a proxy for three-bounce NLoS imaging-that we have employed in our earlier treatments of P-field imaging with quasimonochromatic coherent illumination [3,5].After some preliminaries, which allow us to obtain the complete statistics of cw and modulated speckled speckled speckle in the small-diffusers limit, we begin our analyses in earnest with third-order speckle for cw illumination of extended diffusers.There, although small-diffuser speckled speckled speckle is seven times stronger than ordinary (first-order) speckle-i.e., its irradiance variance is seven times its squared mean-we find that cw speckled speckled speckle is highly mitigated by the geometry of the problem.In particular, our closed-form expression for the irradiance covariance of cw third-order speckle proves that the geometry of typical NLoS imaging scenarios reduces that third-order speckle to the ordinary cw speckle produced by the final diffuser.Furthermore, the speckle fluctuations that remain will be suppressed in power collection over any reason-able detector area.To quantify the impact of those residual power fluctuations, we evaluate the signal-to-noise ratio (SNR) for direct detection and show that power fluctuations act as an excess noise-above the fundamental shot-noise limit-that sets a maximum attainable SNR.For typical parameter values, we find this saturation SNR to be quite generous, and thus conclude that the impact of third-order speckle is unlikely to be significant in NLoS imaging with cw coherent illumination.
Next, we move on to the speckle produced by the modulated illumination of extended diffusers, as used in P-field imaging.Here, our analysis is limited to first-order speckle.
We establish an upper bound on the zero-frequency component of the speckle when the initial illumination is space-time factorable, finding that, at worst, such speckle is of ordinary strength.This result is seemingly at odds with Teichman's analysis [4] for factorable, single-frequency modulation, which finds the modulation-frequency-component speckle to be stronger than ordinary speckle.To resolve the apparent discrepancy, we analyze a singlefrequency-modulation limiting case in our P-field framework and show that it recovers both Teichman's result for the modulation-frequency speckle and our upper bound for the zerofrequency speckle.Then, using realistic parameter values for NLoS imaging scenarios, we conclude that the speckle enhancement effect reported by Teichman is likely to be minimal.
Moving further, we analyze the first-order-speckle size for the modulated case and find it to be quite small, suggesting that speckle may be greatly suppressed in power collection over a detector of realistic size.Also, we find this first-order, modulated-speckle size to be comparable to that of the cw case's first-order speckle.If similar correspondences exist-for both speckle strength and speckle size-between modulated and cw speckle from extended diffusers in their second-order and third-order cases, then the adverse effects of speckled speckled speckle on P-field NLoS imaging may be inconsequential.

II. PRELIMINARIES
As developed in our earlier analysis of P-field imaging [3,5,6], we use paraxial, scalarwave optics in a transmissive geometry that serves as a proxy for a typical reflective, threebounce NLoS geometry.The light at each plane is characterized by its baseband complexfield envelope E z (ρ z , t), which modulates an optical carrier of frequency ω 0 to produce a W 1/2 /m-units optical field Re[E z (ρ z , t)e −iω 0 t ], where ρ z is the 2D transverse spatial coor-dinate in the plane indicated by z.For cw speckle E z (ρ z , t) = E z (ρ z ) will have no time dependence, whereas for modulated speckle E z (ρ z , t) will have bandwidth ∆ω ω 0 .In both cases, I z (ρ z , t) ≡ |E z (ρ z , t)| 2 will be the short-time-average (STA) irradiance at the z-plane [7].
The geometry for our third-order speckle analysis is depicted in Fig. 1.Coherent spacetime factorable illumination, E 0 (ρ 0 , t) = E 0 (ρ 0 )S(t) with and S(t) = 1 for cw illumination, is incident at plane 0, which contains a diffuser with thickness profile h 0 (ρ 0 ).In the standard NLoS imaging configuration, this diffuser represents the visible wall at which reflection into the hidden space occurs.Plane 1 contains a second diffuser, whose thickness profile is h 1 (ρ 1 ) and whose size is modeled by a Gaussian pupil with e −1 -field-attenuation-diameter d 1 .This diffuser represents a finite-sized, planar, diffuse target in the hidden scene, where, for simplicity, we have ignored any albedo variations across the target.Plane 2 contains a final diffuser, whose thickness profile is h 2 (ρ 2 ) and whose finite size is modeled by a Gaussian pupil with e −1 -field-attenuation-diameter d 2 .In the NLoS scenario, it represents the visible wall where light reflects back to the imager.That imager's entrance pupil lies in plane 3. Note that plane 2's finite pupil enables us to obtain convergent paraxial-regime results for the variance of third-order cw speckle.A finite pupil at plane 0 is not needed for that purpose, because the initial illumination is self-limited to within that wall's boundaries.The distances between planes 0, 1, 2, and 3 are all L, a choice made for convenience rather than necessity.

A. Basic principles
All the analysis to follow rests on four basic principles: Fresnel diffraction for monochromatic light; the van Cittert-Zernike theorem for propagating the mutual coherence function (MCF) of spatially-incoherent light; the central limit theorem for sums of large numbers of independent random variables; and the law of iterated expectation.
To see how these principles come into play in the cw case, we start with how the initial illumination, E 0 (ρ 0 ) from Eq. ( 1), first passes through the plane-0 diffuser to become E 0 (ρ 0 ) and then diffracts over an L-m-long free-space path to become the illumination, E 1 (ρ 1 ), at  < l a t e x i t s h a 1 _ b a s e 6 4 = " 3 + R + q q b + v Z y q 1 s R 0 e I w 4 X + R + i p I   < l a t e x i t s h a 1 _ b a s e 6 4 = " W z q f G j e 3 X p y  plane 1.We have that where c is light speed and we have ignored the diffuser's refractive index.The essence of P-field imaging [2] is that the diffusers are rough at the optical wavelength, λ 0 = 2πc/ω 0 , but smooth at the modulation wavelength, ∆λ = 2πc/∆ω [8].Our previous work [3,5,6] enforced this behavior by taking the {h n (ρ n ) : n = 0, 1, 2} to be statistically-independent [9], identically-distributed, zero-mean Gaussian random processes with standard deviation σ h satisfying λ 0 σ h ∆λ and a homogeneous, isotropic, covariance function K hh (|ρ|) with correlation length ρ h satisfying ρ h ∼ λ 0 .For the cw case, these statistics imply that E 0 (ρ 0 ) 0 ≈ 0, E 0 (ρ 0 )E 0 (ρ 0 ) 0 ≈ 0, and where • 0 denotes ensemble averaging over h 0 (ρ 0 ).
The final principle our analysis will need-iterated expectation-comes in at this point.
Conditioned on knowledge of |E 1 (ρ 1 )| 2 , the procedure we have just employed can be used to show that E 2 (ρ 2 ) is a zero-mean, complex-valued, Gaussian random process that is completely characterized by where we have used for the field emerging from the diffuser at plane 1 and • 1 denotes ensemble averaging over that diffuser.Iterated expectation now gives us for the unconditional (ensemble averaged over the plane-0 and plane-1 diffusers) MCF of E 2 (ρ 2 ), hence we get I 2 ≡ I 2 (ρ 2 ) 0,1 = πd 2 1 I 1 /8L 2 for the unconditional, diffuseraveraged STA irradiance at plane 2. By now it should be clear that we can pursue a similar argument to that just completed and show that I 3 ≡ I 3 (ρ 3 ) 0,1,2 = πd 2 2 I 2 /8L 2 .We can obtain a further result for cw speckled speckle in the small-diffuser limit, wherein 1.There we have that Consequently, the unconditional probability density function (pdf) for the small-diffuser limit, i.e., the pdf for cw speckled speckle in that regime, is [1] p where K 0 (•) is the zeroth-order modified Bessel function of the second kind and u(•) is the unit-step function.
The results in this subsection can be found in Goodman's monograph [1].We have reviewed them for two reasons.First, in Sec.II B, they will let us analyze both cw and modulated speckled speckle speckle in the small-diffusers limit.Second, in Secs.III and IV, respectively, they will be generalized to treat cw third-order speckle from extended diffusers and modulated first-order speckle from an extended diffuser.
B. Third-order speckle in the small-diffusers regime Guided by the small-diffuser result for cw second-order speckle, let us consider cw thirdorder speckle when (d 0 d 1 /4λ 0 L) 2  1 and (d 1 d 2 /4λ 0 L) 2 1, i.e., when the Fresnel-number products for propagation between planes 0 and 1 and between planes 1 and 2 are both very small.The work from the previous subsection immediately shows us that so that, conditioned on knowledge of |E 2 (0)| 2 , we have that E 3 (ρ 3 ) is a zero-mean, complexvalued, Gaussian random process that is completely characterized by its conditional MCF, The unconditional pdf for I 3 (ρ 3 ) = |E 3 (ρ 3 )| 2 in the small-diffusers limit, i.e., the pdf for cw speckled speckled speckle in that regime, is therefore Recourse to integral tables yields where G 3,0 0,3 (•|0, 0, 0) is a Meijer G-function [11].In Fig. 2 we have plotted the pdfs of the normalized irradiances Ĩn ≡ I n (0)/ I n for n = 1, 2, 3, which show the increasing randomness that occurs in progressing from speckle to speckled speckle to speckled speckled speckle.Indeed, more pdf iterations and properties of the Meijer G-function can be used to show that the normalized variance of nth-order speckle in the small-diffusers regime obeys [6] NVar 2. Logarithmic plots of the pdfs for Ĩn ≡ I n (0)/ I n , the normalized nth-order speckle in the small-diffusers limit.
Interestingly, this subsection's results for high-order cw speckle in the small-diffusers limit have immediate translations into corresponding results for high-order modulated speckle in that regime.In particular, with our assumption of quasimonochromatic, space-time factorable, modulated illumination, the small-diffuser assumption (d where E 1 (0) is the cw-illumination complex envelope whose squared magnitude appears in Eq. ( 8).Similarly, because we have just shown that E 1 (ρ 1 , t) for quasimonochromatic, space-time factorable, initial illumination is itself quasimonochromatic and space-time factorable when (d 0 d 1 /4λ 0 L) 2  1, we have that the additional small-diffuser assumption where E 2 (0) is the cw-illumination complex envelope whose squared magnitude appears in Eq. (10).So, except for their having time-delayed temporal modulations and |S(t − 3L/c)| 2 , the behaviors of second-order and third-order modulated speckle in the small-diffusers regime are identical to what we found for their cw counterparts.

III. THIRD-ORDER CW SPECKLE FROM EXTENDED DIFFUSERS
First-order speckle has long been an issue for line-of-sight laser radars.For a roughsurfaced target, the single-pulse, single-pixel SNR of a heterodyne-detection laser radar asymptotes to a saturation SNR of 1-set by first-order speckle-with increasing targetreturn strength [12].Direct-detection laser radars are largely immune to first-order speckle because each pixel is configured to contain sufficient speckles to average out their individual fluctuations without unduly compromising spatial resolution.NLoS laser imagers could potentially suffer third-order speckle's seven-fold increased fluctuation strength that prevails in the small-reflectors regime.If unabated, this increase would result in a saturation SNR of 1/7.Whether or not such will be the case requires understanding the statistics of thirdorder speckle from extended targets, which is the case of interest for NLoS laser imagers.
That task, for the cw case, is this section's mission.We begin by relating direct detection's saturation SNR to third-order speckle's irradiance statistics.
Suppose plane 0 in Fig. 1 is illuminated with cw light from Eq. ( 1) and that a directdetection system integrates the optical power transmitted through a diameter-D circular pupil in plane 3 over the time interval 0 ≤ t ≤ T .We will neglect technical noises, e.g., thermal noise, and normalize the detector's output to represent the number of detected photons, N , in that time interval.By the conditional Poissonian nature of photon-counting statistics for randomized laser light [13], we have that the resulting SNR is where η is the detector's quantum efficiency, ω 0 is the photon energy, and is the detected power.The first term in the SNR's denominator is due to shot noise-the fundamental noise of semiclassical photodetection [13], which is always present-and the second term in that denominator is the excess noise associated with randomness in the detector's illumination.We can rewrite Eq. ( 19) as where the saturation SNR, is the maximum achievable SNR, and it is only approached (from below) as N → ∞.From Sec.II A we have that All that remains, before we can evaluate SNR sat , is to find plane 3's normalized irradiance covariance, where we are anticipating its being spatially homogeneous, as indeed will turn out to be the case.We will find this normalized irradiance covariance in the next subsection, using Gaussian moment factoring and iterated expectation.Along the way we will get the normalized covariances for I 1 (ρ 1 ) and I 2 (ρ 2 ), whose behaviors aid our understanding of how extended diffusers mitigate high-order speckle.
A. Irradiance covariance of cw third-order speckle The normalized covariance of plane 1's irradiance is easily obtained.We know that E 1 (ρ 1 ) is a zero-mean, complex-valued, Gaussian random process that is completely characterized by the MCF from Eq. ( 5).Gaussian moment factoring gives us which is spatially homogeneous, as presumed earlier.Plane 1's normalized irradiance covariance is then found to be where Proceeding now toward obtaining plane 2's irradiance covariance, we start from E 2 (ρ 2 )'s being-conditioned on knowledge of E 1 (ρ 1 )-a zero-mean, complex-valued, Gaussian random process that is completely characterized by its conditional MCF from Eq. ( 6).Gaussian moment factoring now gives us I 2 (ρ 2 )'s conditional correlation function, Using Fresnel propagation, these terms expand to give Now, using the law of iterated expectation and taking advantage of the linearity of expectation, averaging over the first-diffuser's statistics yields for I 2 (ρ 2 )'s unconditional correlation function.Using this result we get which again is spatially homogeneous.This normalized covariance has interesting behavior with an interesting interpretation.When Ω 01 1, it reduces to which resembles the normalized covariance for first-order speckle, cf.
which follows from Eq. ( 27).This is not an accidental coincidence.When Ω 01 1, the speckle size in I 1 (ρ 1 ) is much smaller than d 1 .Moreover, I 2 (ρ 2 ) is conditionally exponential, given I 1 (ρ 1 ), with conditional mean By the law of large numbers, we have that d because of speckle averaging over the plane-1 pupil.Furthermore, this means we can take E 2 (ρ 2 ) ≡ E 2 (ρ 2 )e iω 0 h 2 (ρ 2 )/c to be a zero-mean, complex-valued, Gaussian random process insofar as calculating the statistics of E 3 (ρ 3 ) is concerned, i.e., Ω 01 1 has totally suppressed the speckle generated in propagation from plane 0 to plane 1 insofar as evaluating the speckle incurred in propagating from plane 1 to plane 2.
Putting aside, for now, the Ω 01 1 condition and its consequences, it should be clear that even without that condition, we can proceed with an iterated-expectation procedure to find NCovar I 3 (ρ 3 − ρ3 ).For the sake of brevity, we will omit the details and just give the final answer: where When Ω 01 1 and Ω 12 1, the preceding normalized covariance becomes which is the normalized covariance for first-order speckle produced by propagation from plane 2 to plane 3. Indeed, when Ω 01 1 and Ω 12 1, the law of large numbers implies , so what would have been speckled speckled speckle in I 3 (ρ 3 ) reduces to the first-order speckle for propagation from plane 2 to plane 3 [14].small enough to make cw third-order speckle reduce to first-order speckle.

B. Saturation signal-to-noise ratio
The saturation SNR is related to the normalized covariance I 3 (ρ 3 ) as follows, Switching to sum and difference coordinates, ρ + = (ρ 3 + ρ3 ) /2 and ρ − = ρ 3 − ρ3 , the ρ + integration yields where is the two-circle overlap function.An exact evaluation of Eq. ( 38) is tedious but results in where It should be noted, however, that speckle averaging at the detector is not always desirable.This is especially true for Willomitzer et al.'s [15] synthetic-wavelength-holography approach to NLoS imaging.It is a variant of P-field imaging that: (1) uses sequential, cw illumination at two optical frequencies; (2) heterodyne detects each frequency's E 3 (ρ 3 ) at high spatial resolution, using a detector array, to obtain its speckle pattern; and (3) forms a P-field image of the hidden-space's target plane by processing the two speckle patterns.Spatial integration over multiple speckles at each detector element degrades the speckle-pattern measurements and is thus undesirable [5].

IV. MODULATED FIRST-ORDER SPECKLE FROM AN EXTENDED DIF-FUSER
Our reason for studying modulated speckle is its potential relevance to P-field NLoS imaging.With the exception of Teichman's work [4], prior theoretical treatments of P-field imaging have ignored the possible ill-effects of high-order speckle on such systems.The P field, as defined in [3], is the temporal Fourier transform of the diffuser-averaged STA irradiance, where E z (ρ z , ω) ≡ dt E z (ρ z , t)e iωt and • denotes ensemble averaging over all relevant diffusers.To analyze high-order speckle's impact on estimating P z (ρ z , ω − ) from experimental data, we introduce whose ensemble average equals P z (ρ z , ω − ).Unfortunately, despite our having found quantitative-indeed favorable-results for the cw third-order speckle from extended diffusers, analysis of the modulated speckle for extended diffusers is far more challenging.
Accordingly, we will limit ourselves to the first-order case, i.e., characterizing the P-field fluctuations at plane 1, where the hidden target would be located.Our goals will be the same as those we set for cw speckle: determining P1 (ρ 1 , ω − )'s speckle strength and speckle size.
Equation (47) does not lend itself to further evaluation, but it does allow us to establish an upper bound on the variance of P1 (ρ 1 , 0), which shows that the modulated first-order speckle at zero frequency from an extended diffuser is never stronger than ordinary (cw first-order) speckle.
The preceding bound is seemingly at odds with Teichman's analysis [4] for factorable, single-frequency modulation, which finds the modulation-frequency-component speckle to be stronger than ordinary speckle.It turns out, however, that this apparent discrepancy is Equations ( 56) and ( 58) indicate that the modulation-frequency speckle is as strong as ordinary speckle on axis and is stronger off axis, in complete agreement with Teichman's analysis.However, we can see that in the worst case the increase is only a factor of e (d 0 Ω/2cL) 2 |ρ 1 | 2 < e (2πd 0 /Λ) 2 ≈ 1.08, (59) for |ρ 1 | < 2L, Λ = 5 cm, and d 0 = 2.2 mm.So, Teichman's result for first-order modulated speckle from an extended diffuser is qualitatively correct, but in the paraxial regime that speckle has approximately ordinary strength.It must be emphasized, however, that our analysis for modulated speckle from extended diffusers does not extend beyond the firstorder case.

B. Speckle size
As seen in the cw case for third-order speckle from extended diffusers, P1 (ρ 1 , ω − )'s variance for the modulated speckle from an extended diffuser is not the sole determinant of whether those speckle fluctuations will severely limit estimating P 1 (ρ 1 , ω − ) from experimental data.The speckle covariance is equally important, if not more so.Deriving modulated first-order speckle's irradiance covariance, however, is more difficult than obtaining its cw counterpart, as we shall soon see.To start, by using E 0 (ρ 0 , t)'s being space-time factorable, 8 j Y k d p G a U j n O a W I 1 c S L H g R b L / B p e x 5 / h C c Q r v 4 I X 7 D S 7 J G G W 2 p 7 6 + 8 5 3 v n O c 2 M 9 i l s t e 7 + f C 4 l L n 1 u 0 7 y 3 d X 7 t 1 / 8 H B 1 b f 3 R S Z 4 W g s I x T e N U n P k k h 5 h x O J Z M x n C W C S C J H 8 O p P 9 m 3 + O k n E D l L + Q c 5 y 2 C Y k J C z M a N E m q 3 R 2 p N o 5 G x 6 f h o H + S w x P 5 6 I 0 p F y 9 I v R W r e 3 3 S s X b g d 5 x I f Y M A I Y e 0 e u 8 k x G I c A a V Z 6 E q V S e 6 R p Y y G P C w 8 K K l g J a Z V p r b 5 o b W T O 8 l Y 1 D I n E M c m 6 L w 2 e a J g n h g f I i r s 0 X k Y p r 3 U C i p I K S F r R X I X s t Z L 9 C 9 l t I v 0 L 6 L c S t E L e F T F k F e S Z q g l 9 A k g u 4 j G 2 j r j 0 r b A g 4 L a Q N x w z M 4 1 l P 7 T N + Y D P 7 H 1 U 1 R m b a 9 w I S h i C a d Q x 5 z r 1 i N g i m 0 n W G / d u k H F y I X N W 7 q d z B p Z y 6 0 r u B 7 M 5 l 3 R u 9 u Z W Y W / N m x y Q h F C T G C R D 7 R N W z g k A P n K H y n l U O A o 2 7 j s b e l k 3 t T 7 O U m 9 e U m W w o J 8 3 M t c A 4 E T P M 6 j o A u l K A p j H G L i B z s O V 1 8 c 6 u 1 5 e X Q z s 4 e b n t v N r e e b / T 3 d 2 r L o 5 l 9 B Q 9 R 5 v I Q W / Q L j p E R + g Y U f Q V f U P n 6 H v n v P O j 8 6 v z e 0 5 d X K h y H q P a 6 v z 5 B y W D 2 S U = < / l a t e x i t > h 2 (⇢ 2 ) 7 x w n 9 v O E F b L T + b m w u N S 6 d f v O 8 t 2 V e / c f P H y 0 u v b 4 p M h K E c B x k C W Z O P N p A Q n j c C y Z T O A s F 0 B T P 4 F T f 7 x v 8 d N P I A q W 8 Q 9 y m s M g p R F n I x Z Q a b a G q 0 / j 4 f Y G 8 b M k L K a p + S E i z o Z q W 7 8 c r r Y 7 W 5 3 Z w s 2 g 6 4 I 2 c u t o u L b 0

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FIG. 1 .
FIG. 1. Geometry for third-order speckle analysis.Thin blue rectangles represent idealized, thin diffusers.The black frames in front of the diffusers in planes 1 and 2 represent Gaussian pupils that capture the essence of the target and visible-wall sizes, respectively.The dashed line represents the detection plane.
t e x i t s h a 1 _ b a s e 6 4 = " B 7

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e F e I Q E w k r 3 Z e o S n P k v L 5 J O o + 6 c 1 B s 3 d q 3 Z A F N U w A E 4 B M f A A a e g C a 5 A C 7 Q B B h l 4 A M / g x b g 3 n o x X 4 2 0 6 u m T M d q r g D 4 z 3 H 7 n Z m G c = < / l a t e x i t > n = b a s e 6 4 = " W v B H O c i z + X y T A 4 1 n x 2 y N c g C 4 0 R M = " > A A A B 6 n i c b Z D L S s N A F I Z P 6 q 3G W 9 W l m 8 E i u C p J X O i m W H D j s q K 9 Q B v K Z D p p h 0 4 m Y W Y i l N B H c O N C E Z f 6 D r 6 H G / F t n L R d a O s P A x / / f w 5 z z g k S z p R 2 n G + r s L K 6 t r 5 R 3 L S 3 t n d 2 9 0 r 7 B 0 0 V p 5 L Q B o l 5 L N s B V p Q z Q R u a a U 7 b i a Q 4 C j h t B a O r P G / d U 6 l Y L O 7 0 O K F + h A e C h Y x g b a x b U f V 6 p b J T c a Z C y + D O o X z 5 Y V e T t y + 7 3 i t 9 d v s x S S M q N O F Y q Y 7 r J N r P s N S M c D q x u 6 m i C S Y j P K A d g w J H V P n Z d N Q J O j F O H 4 W x N E 9 o N H V / d 2 Q 4 U m o c B a Y y w n q o F r P c / C / r p D q 8 8 D M m k l R T Q W Y f h S l H O k b 5 3 q j P J C W a j w 1 g I p m Z F Z E h l p h o c x 3 b H M F d X H k Z m l 7 F P a t 4 N 0 6 5 5 s F M R T i C Y z g F F 8 6 h B t d Q h w Y Q G M A D P M G z x a 1 H 6 8 V 6 n Z U W r H n P I f y R 9 f 4 D E K O Q l g = = < / l a t e x i t > n = b a s e 6 4 = " F N d F 0 J H d a F g J Z o Q H s h G c I 3 p h v Q E = " > A A A B 6 n i c b Z D L S g M x F I b P e K 3 j r e r S T b A I r s p M u 9 B N s e D G Z U V 7 g X Y o m T T T h i a Z I c k I Z e g j u H G h i E t 9 B 9 / D j f g 2 p p e F t v 4 Q + P j / c 8 g 5 J 0 w 4 0 8 b z v p 2 V 1 b X 1 j c 3 c l r u 9 s 7 u 3 n z 8 4 b O g 4 V Y T W S c x j 1 Q q x p p x J W j f M c N p K F M U i 5 L Q Z D q 8 m e f O e K s 1 i e W d G C Q 0 E 7 k s W M Y K N t W 5 l p d z N F 7 y i N x V a B n 8 O h c s P t 5 K 8 f b m 1 b v 6 z 0 4 t J K q g 0 h G O t 2 7 6 X m C D D y j D C 6 d j t p J o m m A x x n 7 Y t S i y o D r L p q G N 0 a p 0 e i m J l n z R o 6 v 7 u y L D Q e i R C W y m w G e j F b G L + l 7 V T E 1 0 E G Z N J a q g k s 4 + i l C M T o 8 n e q M c U J Y a P L G C i m J 0 V k Q F W m B h 7 H d c e w V 9 c e R k a p a J f L p Z u v E K 1 B D P l 4 B h O 4 A x 8 O I c q X E M N 6 k C g D w / w B M 8 O d x 6 d F + d 1 V r r i z H u O 4 I + c 9 x 8 S J 5 C X < / l a t e x i t > n = " 7 u O I u 8 d i l n 3 x + h 2 p o B e e d Y K m j + 0 = " > A A A B 6 n i c b Z D L S g M x F I b P 1 F s d b 1 W X b o J F c F V m 6 k I 3 x Y I b l x X t B d q h Z N J M G 5 r J h C Q j l K G P 4 M a F I i 7 1 H X w P N + L b m F 4 W 2 v p D 4 O P / z y H n n F B y p o 3 n f T u 5 l d W 1 9 Y 3 8 p r u 1 v b O 7 V 9 g / a O g k V Y T W S c I T 1 Q q x p p w J W j f M c N q S i u I 4 5 L Q Z D q 8 m e f O e K s 0 S c W d G k g Y x 7 g s W M Y K N t W 5 F x e 8 W i l 7 J m w o t g z + H 4 u W H W 5 F v X 2 6 t W / j s 9 B K S x l Q Y wr H W b d + T J s i w M o x w O n Y 7 q a Y S k y H u 0 7 Z F g W O q g 2 w 6 6 h i d W K e H o k T Z J w y a u r 8 7 M h x r P Y p D W x l j M 9 C L 2 c T 8 L 2 u n J r o I M i Z k a q g g s 4 + i l C O T o M n e q M c U J Y a P L G C i m J 0 V k Q F W m B h 7 H d c e w V 9 c e R k a 5 Z J / V i r f e M V q G W b K w x E c w y n 4 c A 5 V u I Y a 1 I F A H x 7 g C Z 4 d 7 j w 6 L 8 7 r r D T n z H s O 4 Y + c 9 x 8 P H 5 C V < / l a t e x i t > log 10 [p Ĩn ( Ĩ)] < l a t e x i t s h a 1 _ b a s e 6 4 = " y D B Z E S / E 3 f e r U C n D Y c Y t 4 b F L S 8 c = " > A A A C G H i c b V D L S s N A F J 2 o 1 V p f U Z c i B I t Q F 9 a k L n Q l B T e 6 q 2 A f k I Q w m U z a o Z N J m J m I J e Q z 3 P g r b r p Q x G 1 3 / o 2 T t g t t P T B w 5 p x 7 u f c e P 6 F E S N P 8 1 l Z W 1 0 r r G + X N y t b 2 z u 6 e v n / Q E X H K E W 6 j m M a 8 5 0 O B K W G 4 L Y m k u J d w D C O f 4 q 4 / v C 3 8 7 h P m g s T s U Y 4 S 7 E a w z 0 Taking values close to what we might expect in practice-λ 0 = 532 nm optical wavelength, L = 1 m to 10 m scene depth and standoff, d 0 = 1 mm to 1 cm spot size, d 1 = 3 cm to 2 m target size, and d 2 = 1 m to 10 m wall size-we find that 20 ≤ Ω 01 ≤ 10 9 and 2 × 10 9 ≤ Ω 12 ≤ 10 15 .The least favorable attenuation factor in Eq. (35), 1/(1 + Ω 01 ) ≈ 0.05, is already with BI n being the nth-order modified Bessel function of the first kind.Taking reasonable parameter values, viz., λ 0 = 532 nm, L = 1 m, d 0 = 1 mm, d 1 = 10 cm, d 2 = 1 m, and D = 2 cm, we find that SNR sat ≈ 4700.So, cw speckled speckled speckle has minuscule impact in this case.