Appl. Phys. 4, 201--212 (1974) @ by Springer-Verlag 1974 Applied Physics Space and Wavelength Dependence of Speckle Intensity* Nicholas George and Atul Jain California Institute of Technology, Pasadena, Cal. 91109, USA Received 21 February 1974/Accepted 15May 1974 Abstract. A unified analysis is presented for the spatial and the spectral sensitivity of speckle (the rapid spatial variations which occur in an image when illumination of narrow spectral width is used) in a space-invariant linear system. In prior work considering speckle size, others have shown that its spatial variation is functionally dependent primarily on the autocorrelation function of the system's impulse response, but effects of varying the wavelength were largely ignored. In the present paper we treat the general problem in which a diffuse object, illuminated by a collimated, monochromatic beam, is imaged by a system whose amplitude impulse response is z(x, rl), where x and q are space and normalized (temporal) frequency coordinates, respectively. An expression is derived for the multi- dimensional autocorrelation function Ru(Ax, rh, rl2) of the intensity u(x, rl) in the image plane. Functionally, it depends upon a convolution of the system autocorrelation function Rz(Ax, rla, rl2) with the characteristic function of the distribution function for heights, which is used to model the input object's surface. Examples are presented; and it is shown that one can infer valuable information about the variation of heights for points on the surface of the input diffuse object, which are separated by much less than the classical resolution limit. Index Heading: Coherence The word speckle is now used to describe not only the sparkling, granular and highly contrasted appearance of visible laser light reflected from a wall or other diffuse surfaces; but its usage has been generalized to include analogous spatial interference effects which occur with scattering from diffuse objects for all types of wave-motion phenomena. Speckle con- stitutes a basic noise phenomenon limiting the working resolution of coherent systems in acoustics, microwaves, infrared and visible optics, and un- doubtedly, its manifestations in electron-particle waves and ultra-violet will shortly be recognized. Generally, any wave with a narrow angular spectrum * Research supported in part by the Air Force Office of Scientific Research. which is scattered from a diffuse object will exhibit speckle if (i) the spectral width is below a certain value and (ii) the object has textural roughness finer than, but on the order of, the size of a resolution cell or more grossly on the order of a wavelength. One of the earliest studies on the statistics of speckle was that by Lord Rayleigh who derived the density function for the light scattered by a rough surface [1]. Extensive treatments appeared on this topic in the intervening years, as is evident in the following quotation taken from M. yon Laue's paper on this subject published in 1914 [2]: "The theme of our investigation is an old one; it is treated in many papers and in every optics textbook." Nevertheless, he was the first to describe an experimental observa-
Space and Wavelength Dependence of Speckle Intensity* Nicholas George and Atul Jain
California Institute of Technology, Pasadena, Cal. 91109, USA
Received 21 February 1974/Accepted 15May 1974
Abstract. A unified analysis is presented for the spatial and the spectral sensitivity of speckle (the rapid spatial variations which occur in an image when illumination of narrow spectral width is used) in a space-invariant linear system. In prior work considering speckle size, others have shown that its spatial variation is functionally dependent primarily on the autocorrelation function of the system's impulse response, but effects of varying the wavelength were largely ignored. In the present paper we treat the general problem in which a diffuse object, illuminated by a collimated, monochromatic beam, is imaged by a system whose amplitude impulse response is z(x, rl), where x and q are space and normalized (temporal) frequency coordinates, respectively. An expression is derived for the multi- dimensional autocorrelation function Ru(Ax, rh, rl2) of the intensity u(x, rl) in the image plane. Functionally, it depends upon a convolution of the system autocorrelation function Rz(Ax, rla, rl2) with the characteristic function of the distribution function for heights, which is used to model the input object's surface. Examples are presented; and it is shown that one can infer valuable information about the variation of heights for points on the surface of the input diffuse object, which are separated by much less than the classical resolution limit.
Index Heading: Coherence
The word speckle is now used to describe not only the sparkling, granular and highly contrasted appearance of visible laser light reflected from a wall or other diffuse surfaces; but its usage has been generalized to include analogous spatial interference effects which occur with scattering from diffuse objects for all types of wave-motion phenomena. Speckle con- stitutes a basic noise phenomenon limiting the working resolution of coherent systems in acoustics, microwaves, infrared and visible optics, and un- doubtedly, its manifestations in electron-particle waves and ultra-violet will shortly be recognized. Generally, any wave with a narrow angular spectrum * Research supported in part by the Air Force Office of Scientific Research.
which is scattered from a diffuse object will exhibit speckle if (i) the spectral width is below a certain value and (ii) the object has textural roughness finer than, but on the order of, the size of a resolution cell or more grossly on the order of a wavelength. One of the earliest studies on the statistics of speckle was that by Lord Rayleigh who derived the density function for the light scattered by a rough surface [1]. Extensive treatments appeared on this topic in the intervening years, as is evident in the following quotation taken from M. yon Laue's paper on this subject published in 1914 [2]: "The theme of our investigation is an old one; it is treated in many papers and in every optics textbook." Nevertheless, he was the first to describe an experimental observa-
202 N. George and A. Jain
tion of speckle, together with an adequate theory. Lacking a laser source, he used a prism to band-limit the light (4200/~-4300 ~); this, he scattered using a diffuser made by spreading lykopodium powder on an optically flat glass substrate. These experiments have been recently reinterpreted with today's refined understanding of speckle [3]. Speckle was more or less rediscovered with the first operation of the 6328 ~ line of the helium-neon laser. In the intervening years, several important con- tributions have been made to the statistical theory of speckle. However, this research has been con- centrated on establishing the spatial dependence of speckle at a fixed wavelength, with relatively little thought being given to its frequency dependence. Comprehensive references to the literature are found in the following papers [4-6] and the bibli- ography by Singh [7]. Our central purpose in this paper is to present a theory for the wavelength dependence of speckle. We analyze an imaged diffuse object illuminated by a succession of monochromatic tones, and we derive an expression for the correlation function of intensity in the output of a linear, space-invariant imaging system. In this context, it is pertinent to give a selected bibliography, as follows. The wavelength dependence of speckle in optics was discussed first by Goodman [8]i Langmuir stressed the similarity
which occurs in the ratio of a typical object's rough- ness to wavelength for radar sea-clutter and visible optical speckle [9]; and Elbaum et al. studied the frequency dependence of speckle with experiments in the 10.6 gm band [10]. Both a theory for speckle reduction using multiple tones of monochromatic illumination and experiments with tunable, con- tinuous wave laser sources and band-limited arc sources have been published by the present authors [11-14]. In the new formulation, herein, both the spectral and spatial aspects of speckle are combined by deriving a multi-dimensional correlation function, R,(Ax, t~ 1, t/z), of the output intensity u(x, t/), where x, t/are position and normalized frequency variables and A x, A t/ are arbitrary increments in these variables, respectively. We apply the linear system theory approach used by Burckhardt El5], Lowenthal and Arsenault [16], and Dainty [17] in their analyses of the spatial statistics of speckle at fixed wavelength. Starting with a phase type of diffuser, characterized by the random height h(x) at position x, which is imaged by a linear, space-invariant system with an
impulse response denoted by z(x, t/), we show that the spectral and spatial correlation function for the output intensity is given by
The bracket { } indicates an expectation over the random process; R= is the autocorrelation function of the system's impulse response; F is the Fourier trans- form or characteristic function of the probability dis- tribution function for heights f (h(x + A x), h(x)); and r(Ax) is the correlation coefficient between the heights h(x + Ax) and h(x). We further examine this general result for several special cases. Two different aspects of speckle make it interesting to understand both the spectral and spatial variations of speckle. One aspect of this interest occurs in holography, electron-microscopy, etc., where multi- tone averaging of several images is a practical means of reducing its deleterous effect on working resolution. And a second facet of this interest is the possibility of obtaining partial resolution within a resolution cell, as sized by classical diffraction theory.
1. Theory of Wavelength Diversity
1.1 Space-Invariant Linear System
In the analysis of speckle a simple imaging system, such as that shown in Fig. 1 consisting of an input object I, a lens of focal length F and aperture diameter
I 5
27r n3y
"q= c
n3= nl/cos e I - no/COS eo
Fig. 1. Imaging of a diffuser I by a lens in the (v, w) plane on to an output plane II. Idealized impulse responses are given in (9), [46), and (60)
Space and Wavelength Dependence of Speckle Intensity 203
~ SQUARE LAWI
Fig. 2. Linear system generalization of Fig. l. The input amplitude is g(x, tl) and the output amplitude and intensity are e(x, 0) and u(x, 0), respectively. The space coordinate is x and the normalized frequency variable ~1 is defined by (15)
D, and an output image plane II can be generalized using linear system theory. This is useful since then the treatment of a wide variety of diffuse objects and imaging optics, even to include the important case of electron-optics, becomes simply a set of examples readily calculated from the general equations. In particular the spatial properties of speckle have been analyzed using this approach [15- /7] . Herein, to include the effect of varying the frequency, we define both our input object g(x, r/) and our system response function z(x, q), with a space variable x and a nor- malized frequency variable ~. Then, from this, we derive a correlation function for speckle intensity in the output as a function of Ax, ~ , and ~Iz, thereby ex- pressing both its spectral and spatial dependence. The notation for the generalized space-invariant optical system is defined, with reference to Fig. 2, as follows: The scalar component of input electric field is denoted by g(x, t/); the impulse response is z(x, ~/); and the output electric field amplitude e(x, ~) and the intensity u(x, tl) are related by
u(x, ~) = e(x, ~) e*(x, 'I) , (l)
where the asterisk denotes complex coNugation. Let us state precisely what we mean by the spectral variation of a speckle pattern. We consider only monochromatic illumination, say, at the normalized frequency q~, and we record a speckle pattern u(x, ~71). Then, the frequency is shifted to t/a and we obtain u(x, ~2). The spectral dependence of speckle is studied by calculation of .the correlation function for u(x, tll) and u(x, tlz). Our analysis is limited to the study of the cross-correlation function between these monochromatic speckle patterns. Further, in the analysis herein, the assumption of a space-invariant transfer function leads to the stationarity in the output random process e(x, q); but inherently, this assumption limits the choice of input-output planes, e.g., the back-focal or "Fourier transform" plane and the far-zone region are not included. Another example of a case which is not directly covered by our analysis occurs when the illumination
is from a band-limited source of light extending from 5000 ~ to 5050 A. In particular, if in recording these speckle patterns, very short time averages are used, then "optical-mix" terms are important; and our results are not applicable. However, if quasi-infinite time averages are used, times >> 2Z/(cd2) which is about 10-13sec for the example chosen, then "optical-mix" terms average out; and in this case, goed qualitative understanding of the speckle pat- terns is obtained by a simple superposition of the results using our theory for the monochromatic tones. Let R,(Ax, th,~2 ) represent the two dimensional autocorrelation of the intensity u(x, q). As a defini- tion, we write
where the brackets { } denote the expected value and A t /= t/; - t/l, Explicitly, this expectation includes the ensemble average over the random process 9 (hence e and u). Substitution of (1) into (2) gives the correla- tion function in terms of the electric field:
It should be noted that in (2) and (3), with the notation R,(Ax,~h, q2), we have tacitly assumed a process which is stationary in x, but not in the (temporal) frequency variable ~/. This point is established by the following reasoning: Assume that the input random process g(x, r/) is stationary in x. The output field e(x, tl) for a linear, space-invariant system with impulse response z(x, q) is given by the convolution of z with g, i.e., e = z , g where the asterisk denotes convolution with the spatial coordinate x and not the temporal frequency coordinate t/. From this and the stationarity of g, it is easily shown that the correlation functions, such as {e(xz, q2) e*(xl, t/2)} and {e(x2, t/2) �9 e*(xl, ~h)}, are functions o fAx = x2 - Xl and neither xa nor x2 alone. Then, with the additional assump- tion of gaussian statistics for the output field e(x, q), as in [15-17], higher-order even moments can be expressed in terms of sums of products of these second-order moments. Thus, the correlation func- tion of intensity R, in (3) will also be a function of the difference x 2 - x l and the frequencies t h and q2 alone�9 In our analysis the temporal frequencies r h and ~/2 are not random variables; and so in general, R, is also a function of both r h and t/2 and not simply of their difference.
204 N. George and A. Jain
Now, we use the above mentioned general result from statistics in Reed [ 18] to expand the expectation value of the product of four normal, stationary, random variables in (3) as follows:
R.(Ax, 171, t/z) = {e(x, t/i) e*(x, t/O}
�9 {e(x + Ax, t h + At/) e*(x + Ax, t/~ + At/)}
+ (e(x, t/~) e*(x + Ax, t/1 + At/)} (4)
�9 {e*(x , t/i) e(x + Ax, t/i + At/ )} .
The second member to the right of the equal sign in (4) is equivalent to [Re(A x, t/i, t/2)] 2, where R~(A x, t/1,112) is given by
By (1), (4), and (5), the autocorrelation for the output intensity is expressible as the sum of the autocorrela- tion function of the electric field and the product of the average intensity at two different frequencies and spatial positions
Ru(Ax, t/i ,/12) ~- {u(x,/71)} {u(x + A x , t/1 -~ At/)} (6)
+ IRe(AX, t/1, t/~)[ ~ �9
Let the Fourier transform with respect to x be indicated by capital letters; thus, as an example for the electric field, we have the following transform pair
E(ox, t/) = ~ e(x, t/) e x p ( - ioJ~x) dx ,
(7)
e(x, t/) = ~ - _ ~ E(ox, 0 exp(ko~x) de%.
Since by definition, the spectral intensity of the electric field, S,(c% t/i, t/2), is the Fourier transform of its autocorrelation, (5) gives
S~(co~,th, t/2)= ~ Re(Ax, t/1,t/2) e x p ( - i c o J x ) d A x , --o0
s~(~ , t/~, t/~)= {E(~ox, t/~) ~*(~o~, t/O}. (8)
Now since the electric field at the output is simply the convolution of the input field 9(x, t/) with the impulse response function of the system, z(x, t/); that is
e(x, 0 = o(x, t/) �9 z(x, t/) , (9)
where the asterisk denotes convolution in the variable x. Combining (8) and the Fourier transform
of (9), one obtains
Se(O~, t/1, ~12) = {O(o~, t/2) G *(o . , t/1) Z ( c % t/2)
�9 Z * ( ~ , t / i ) } �9 (lO)
In a general case where both g and z are random processes, but independent, the expectation in (i0) can be rewritten as the product of the spectral density of the input field with that for the impulse response, i.e.,.
In the case for which z(x, t/) is deterministic, {ZZ*} = ZZ* ; and, of course, (1 l) is still applicable. Taking the inverse Fourier transform of (11), we find that the correlation function for the output electric field is given by the convolution of the correlation function of the input electric field R~(dx, t/1, t/2) with the correlation of the impulse response function R~(Ax, t/1 , t/2), i.e.,
This will be recognized as a generalization of a standard result from linear system theory [19]. Now a basic form of the correlation function of the output intensity can be found by substitution of (12) into (6); it relates R~ to the correlation functions of the independent random processes for the input electric field g(x, 17) and the impulse response z(x, t/), as follows
R . ( A x , t/l, t/2) = {u(x, t/i)} {u(x + Ax , t/1 + At/)}
1.2 Autocorrelation Function for a Phase-Type of Diffuser
In this section we derive an expression for the auto- correlation function of the input field, R 9 in (13), using a pure phase model for the diffuser input. A plane-polarized field incident on the diffuser will exit essentially plane-polarized; hence, a single component of tangential electric field is adequate in the analysis at least for near-normal angles [20, 21]. In typical transmission configurations [12, 14], this depolarized intensity measured in the image plane is at least four orders of magnitude smaller than that for the polarized term. Let the input electric field
Space and Wavelength Dependence of Speckle Intensity 205
9(x, rl) be given by
9(x, tl)= exp[ - i t lh (x ) ] , (14)
where h(x) can be viewed as the height of a diffuser as shown in Fig. 1; and t/ is the normalized radian frequency variable, given in this instance, by (7) in [-12]
2~y_( n, no ) (15) = -cos01 cos0 o '
The optical frequency is v; c = 3 x l08 m/s; n I and n o are the indices of refraction for the bulk material of the diffuser and air, respectively; and 01 and 0o are the respective angles of incidence, i.e., nl sin01 = n o sin0o. The random process for the diffuser, h(x), can be defined by the assumption of a probability density function f(h(x), h(x + A x)). Letting h 1 = h(x) and h2 = h(x + A x), we write the two dimensional charac- teristic function F(rh, t/s ) as the Fourier transform of the joint density function, i.e.,
We note, however, that the characteristic function in (16) is also a function of the correlation coefficient r(Ax) between h(x) and h(x+Ax), and since this coefficient is of importance in our later calculations we use the notation F(r/t, t/z; r(Ax)) to indicate this dependence explicitly. To evaluate the autocorrela- tion function, R(Ax, th, th), we substitute (14) into (17) and obtain the integral
Ro(Ax, th,~12)= ~ ~ exp[ithh(x)] - o o - o o
�9 exp [ - i(t h + A r/) h(x + A x)] f(h(x), h(x + A x)) (18)
�9 dh(x) dh(x + Ax).
By (16), we can express (18) for the autocorrelation of the input electric field in terms of the joint charac- teristic function, as follows
Ro(Ax, th, tlz ) = F ( - rl 1, t h + Arl; r(Ax)) (19)
or equivalently,
R~(A x, th, qz)= F ( - t / t , q2 ; r(A x)) .
From (19) and (13), we find the following general expression for the autocorrelation of the speckle intensity in the output
R u ( A X , / 7 1 , ~]2) = {b/(x,/~1)} {u (x -~ A x , rll -~- A/~)} (20) o o
and for a deterministic optical system, it reduces to
oo
Rz(AX, rh, rl2)= .f z(x + Ax, rlz)Z*(X,t/Odx. (22) - o o
Thus, in general, the autocorrelation of the output intensity is the sum of two terms, the first term being the product of the average speckle intensity at coordinates (x, I/l) and the average speckle intensity at (x + A x, t h + A i/); while the second term is the ab- solute square of the two dimensional characteristic function, F ( - t h, tla + Atl; r(Ax)), convolved with the autocorrelation of the impulse response function for the optical system.
2. E x a m p l e for J o i n t l y N o r m a l D i f f u s e r H e i g h t s
As a first example of the wavelength diversity of speckle, consider the calculation of (20) when the diffuser heights, h(x) and h(x+Ax), are jointly normal random variables�9 Then, their joint density function f(hl, h2) is given, as in [22], by
in which h a = h(x); h 2 = h(x + Ax); r is the correlation coefficient where It[ < l ; zero means are assumed for hl and h2; and a common variance 0 "2 iS used, i.e., a 2 = @2) = @2).
For completeness, two well-known properties of (23) are included�9 The marginal density of ht, p(ht), is
206 N. George and A. Jain
readily found from (23)
p(hO= ~ f (h l , h2) dh2, -co
(24) 1 (
p(hl)= ~ exp - 2 a 2 ] .
Second, in the calculation of the expected value of h2, given h,, i.e., the conditional expectation {h21hl}, it is found that
{h2[hl} = rh 1 . (25)
Insofar as physically modeling a diffuser, (25) establishes the rationale for the assertion that r is functionally dependent on A x, the spatial separation in the plane of the diffuser between h(x + A x) and h(x), as shown in Fig. 1. In the analysis of speckle patterns for partially coherent light, Asakura et al. [5] have used the following exponential
r(A x) = exp - , (26)
where ~ is the correlation length for the random heights of the diffuser. The dependence of the auto- correlation functions, Rg(A x, t/1, t/1) in our notation, is shown versus A x with ct as a parameter in [Ref. 5, Fig. 2]. Briefly, for high correlation at small separa- tions and little or no correlation at large distances, one requires that r--* 1 as A x--. 0 and r-~ 0 as A x ~ oo. For other diffuser models, any normalized correla- tion function, having properties as
~ h(x + Ax) h(x) dx/o z, is a possibility [20, 23-25]. - c o In this context, a convenient function for purposes of physical interpretation is the triangle function, A, as follows
In (28), the allowed range for the correlation coef- ficient is Lr[ < 1. Simple notational changes in (28) to bring it into accord with (19) give the spatial correlation function for the input electric field at frequencies t/1, t/1 + A t/
The frequency dependent, spatial correlation func- tion of the output intensity u(x,t/) is found by combining (20) and (29); the result for the phase type diffuser input is
+ ~ e x p { - �89 2 + 2(1 - r) t/l(t/, + At/)]} (30) - c o
�9 Rz(Ax - Y, t/1, t/i + At/) d 7 2
in which r(?) in the integrand is given by (26) or (27)�9 In (30), the expected values for intensity can also be expressed in terms of the basic correlation functions. By (5) and (12), we write
{U(X, /71)} = Ro(Ax, t / l , / 1 1 ) * Rz(AX, t / l , t / 1 ) [ a x = o �9 ( 3 1 )
Now, we write the convolution explicitly; and thereafter, setting A x = 0 yields the result
{u(x, t/l)} = 7 e x p { - azt/211 - r (7) ] } _ ~ (32)
�9 R z ( - 7, t/l, t/i) dT.
An analogous expression holds for
{u(x + Ax, t/1 + At/)},
where t/1 in (32) is replaced by t/2 = t/1 + At/.
, --A~- > l
or equivalently,
Substitution of (23) into (16) and integration give the following result for the joint characteristic function of f (h i , h2)
3. Speckle Sensi t iv i ty in an Aehromatie Optical S y s t e m
Consider a linear, space-invariant optical system that is independent of frequency. For visible optical speckle, this is a reasonable assumption in many cases, e.g., see the discussion of theory and ex- periments in 1-12]. For z(x, t/) independent of frequency, use the notation za(x ) for the achromatic impulse response�9 The basic result in (13) is simplified to yield
Space and Wavelength Dependence of Speckle Intensity 207
where by (21), we have defined the system correlation function R~,(Ax) as follows
R~,(A x) = {z,(x + A x) z*(x)}. (34)
If in addition we consider the pure phase diffuser input of Section 1.2, then by (20) and (34), the cor- relation function for the output intensity is given by R~,(Ax, ti1, t/2)= {u(x, t/i)} {u(x+ Ax, t/t + At/)}
I f ~ - ' ) d Y 2 (35) + F ( - , h , t/~ + Lit/; r(~,)) R ~ ( 3 x
(Pure phase diffuser).
Finally, if we also assume the Gaussian joint density function (23) for f(hl , h2) and its corresponding characteristic function (28), then by (30), the frequency dependent correlation function for the output inten- sity can be written as
(Pure phase diffi~ser Gaussian distribution of heights).
In the evaluation and interpretation of (36), a very important point to notice is that the quantity a2rh(t/1 + A r/) is on the order of (~)~ when the diffuser roughness is on the order of a wavelength. Thus, the expontial term in the integrand of (36) is negligibly small for portions of the integration over 7 at which the correlation coefficient r(7) approaches zero.
4. The White-Noise Diffuser: Spectral and Spatial Sensitivity
Now, consider the limiting case of a white noise diffuser, i.e., one for which there are spatial cor- relations only over a small dimension ~o which is much less than a resolution cell. Using either (35) or (36), one can show that the autocorrelation func- tion of the output intensity depends on the product of the correlation function for the system's impulse response and the characteristic function for the distribution of diffuser heights. Moreover, this fol- lowing case gives a comparison of (36) to the ac- cepted result in the literature. Since in the case of a white noise diffuser r(y)= 0 unless 7 is zero to within a small wavelength-sized
interval, + %, (36) can be integrated to yield
e . ( 3 x , rh, r/2)~- {u(x, t/l)} {u(x + z~x, t/1 + ~t)} (37)
+ (2~o) 2 e x p [ - (aA i/)2] IR~a(A x)[ 2 ,
for the pure phase diffuser, if azt/~ >> 1. A more careful treatment of the correlation length, later, leads to a comparable result in (64) and (67). Two subcases are of special interest. First for At/= 0, we have the correlation function for speckle inten- sity at fixed wavelength, i.e., by (37), the spatial auto- correlation is given by
"~ 2 still assuming ~-t/t >> 1; and R~ is given by (22) or (34)�9 This is the well-known result derived in the works of Enloe and others which are referenced and described in the recent careful experiments of McKechnie [27], who has found close agreement with the theory. Secondly, when A x=O, one is considering the wavelength variation of the intensity at a fixed posi- tion in the output plane. In this case using (32) to rewrite the expectations for the intensities u(x, t/O and u(x, t h + A t/), one finds the spectrally dependent correlation function to be given by
R,(0, t/i, t/2) = [1 + e x p [ - (~yA t/)2]] (2%) 2 I e . = ( 0 ) l 2 ,
(39)
still assuming a2t/2 ~> 1. By (34), the autocorrelation of the system impulse response for an achromatic system is a constant at Ax--O, i.e., R~(0) -- constant. Thus the autocorrelation function depends only on the term exp [ - ( ~ A ~t)2]. Using the definition in (15) for t/and considering the speckle to be decorrelated when the exponential has decreased to l/e, we find that the frequency interval, A v, which is required to decorrelate the speckle is given by
Av 1 - (40)
e 2zoo- cos0t cos00
We note that (40) agrees with the earlier result given by (25) and (28) in [12], where it was derived based on a different argument. In deriving (25) of [12], it should be noted that Gaussian statistics were not assumed�9 Hence, the frequency interval which is required to decorrelate the speckle patterns is found
208
to be inversely proportional to the standard deviation of the heights on the diffuser surface. Finally, this result has been confirmed by our earlier experiments, e.g., see [14].
5. T he Pupi l Funct ion and Frequency Divers i ty
The general form of the correlation function for the output intensities u(x, 171) and u(x + Ax, t h + A17) is given by (13). This is particularized in (20) for a pure phase type of diffuser, (14). In this section, we calculate R,(Ax, 171,712) for a simple imaging system in order to illustrate or interpret both (13) and (20). Consider a single lens of focal length F and rectangu- lar aperture L~ by Ly. We wish to develop an idealized transfer function with only one spatial variable, i.e. z(x, 17), from this consideration. The diffuser is placed in plane I, (x, y), at s' from the lens, as shown in Fig. l ; and the output electric field e and intensity u=ee* occur at plane II,(x2,y2). Cartesian co- ordinates (v, w) will be used to denote distance in the lens plane. The linear system transfer function h(x2, x; Y2, Y) derived by the usual Fresnel-zone ap- proximation of Sommerfeld's formula is summarized from [26] in our notation
e(x2, Y2;171 = ~ ~ h(x2, x;y2, y) g(x, y ldxdy (41) - oo - o o
The aperture function A and the phase ~b are defined by
A(v,w)=rect(V t r e c t ( W l (43) \Lx} ~L,/'
q5 = 2roy Is + s ' )+ x2+ y2 x~ + y21 (44) - c 2s - ~ - - + 2~-s--- "
The peaked nature of h is used to approximate qS, as follows
q~ ~ (s + s') + 2s + 1 . (45)
If one neglects this slowly varying phase term, then (42) is space invariant and (41) is a convolution�9
N. George and A. Jain
Thus, we define the one-dimensional transfer func- tion z(x 2 - x s/s', 17) proportional to h(x2, x; O, 0):
(46) �9 exp + + v ,
c
where v)~ = c; and the normalized frequency 17 is given by (15); and 1/s + 1/s'= 1/F. Substituting (46) into (22) and neglecting the slight frequency variation give the following result for the correlation function of the impulse response, i.e., Rz(AX2, 171,172) is given by
sinl zcLx~ ( Ax2 ]]
Rz(Ax2 ' r11,112)_ CSLx k c \ s /j (47)
where 7= mean of vl and v2, Ax2 =sAxl/s', and Lx is the length of the lens aperture. First, we consider the frequency dependent term in Ru, i.e., ]R o �9 R~[ 2 by (13), in the case of perfect imaging, Lx---, or. In this case, we write
~ 172) Rza(A x2 - 72) d?2 2. (48) IR a * Rzl 2 = Rg(7~, 171,
By (47), taking the limit as L~ ~ oo gives the Dirac delta function, i.e.,
lim Rza(Ax2 - ~2)=6(Ax2 - 72). (49) g x ~ 00
Thus, by (48) and (49) and integrating, we obtain
[Rg * R=[ z = IRa(Ax2, 17t, 172)12. (50)
Analogous reduced forms can be written for the pure phase diffuser by substitution of (49) into (35) and (36). Second, and of practical interest, consider this con- volution when the aperture Lx is finite. Then, for the pure phase diffuser, described by the joint charac- teristic function F, (47) and (35) give
t:o
IR~ �9 R~I ~ = j ~ F(-171,171 + ~17; r(~))
(51l
where the sub-2 is a reminder that integration is in the output plane and r(72) in the function F must be
Space and Wavelength Dependence of Speckle Intensity 209
scaled properly. The usual practice is followed in defining sinc(t)
sin ~t sinc(t) - (52)
7~t and
~. = cs/~L~.
Assuming further the normal distribution of heights, one can write the following result for the phase type of diffuser input by (36) and (47)
IR o �9 R~[ 2 = exp[ - (aA tO e]
�9 7 exp{-~ [t/a +At/] [1-r (72)]} (53) - o o
2
We evaluate (53) by substituting the triangle function given in (27) for the correlation length r(Ax), sepa- rating the integral from - co to co into the four inter- vals, i.e. [ - c o , - ~ ] , [ - ~ , 0], [0, c~], [~, oe]. Using [-Ref. 28, Eq. (8.230.1)] to evaluate the integrals for the intervals I - c o , -c~] and [c~, col, [Ref. 28, Eq. (3.944.1)] to evaluate the integrals for the inter- vals [ - c~, 0] and [0, ~], and after a lengthy computa- tion, we find that
{R o * R~I 2 = (~Lx) 4 e x p [ - (o-A t/) 23
. . / A x + ~ ] + ( 2 ) [ e x p ( - ~ ) ( 7 [ O ' ( O + t ) ~ ) ]
- @, (0 -~ ~ ] j ' " ( A x + ~ ~1
(54)
+ e x p ( ~Ax2)(y[O'(-1)(O+i)(~-~-)]o~
+ ( ,,(o A x 2 ,[o ,- ]
where
fl = o2rh [t/j. + At/], 0 = - - ~z
(55)
and
si(x)= - 7 sint dt (56) t
and 7(a, b) is the incomplete gamma function. From the series representation for ?[a, b] as in [Ref. 28, Eq. (8.354.1)], i.e.
7 [ a , b ] = ~ ( -1 ) ' b in+" ,,=o rn!(a+m) (57)
we can rewrite (54) as follows
]Rg �9 Rz] 2 = (~Lx) 4 e x p [ - (aA t/)a]
[ . [ A X 2 - o~ __ si(Ax2 + c~ . )] + ~ [(o + 0" - (o - 0"]
m = l (58)
- ( e x p ( f l - ~ ) ( - 1)m [( A x~+ ~ )m - (--AA~!) m ]
+ e x p ( flax2-){( Ax2- The above calculation is repeated for the case of a Gaussian transmission function. In this idealization, the characteristic length of the aperture, L~, is used to control the "width" of the apodization function. As we shall show below, this is a convenience for theoretical purposes; and for electron-bearn optics it is physically appropriate as well. For the one-dimensional lens of focal length F and apodization width L,, the transmission function is
~--j[ ircV2~exp[-( 2vl2]. (59) T(v)=exP l \ Lx ] ]
From [Ref. 12, Eq. (3)], again dropping phase terms as a theoretical idealization, a one dimensional trans- fer function corresponding to (59) can be defined as follows
[ (x2_ x)2] exp -
z,o x z - x ~ = l//~ w , (60)
where M=s/s'; and an achromatic w is computed for the average wavelength 2 from the following relationship
27~s w = - - (61)
TCLx "
210 N. George and A. Jain
The auto-correlation function R~(Ax ) is computed by integration of (22) substituting (60); and the result is
R:,(Axa) = ] f ~ w ' (62)
where A x2 is an arbitrary interval in the output plane. Consider (36) for a pure phase diffuser with the Gaussian distribution of heights; and assume the triangular correlation function for r, (27). Writing only the second member of (36), substituting (62), and omitting the square yield the following equation for Ro(Ax 2, t/t, t/t + At/) �9 Rza(Ax2):
R o * Rz~ = V~-~w
�9 ~ exp{- f i l l - r ( A x 2 - 7)] - ~ (~-~12~ dr. 21wJJ
(63)
As in (55), fl=o'2t/l(t/t-l-At/). Integration of (63) using (27), and some algebra give the following lengthy, but simple answer for R o �9 R~,:
Rg * Rz, = exp[ - �89 0)23 (St + S 2 + Sa), (64)
where
S t =
+ exp ~ ~w-~2 )
2
zlx2- �9 err - [~w / - erf
f l W 2 A x 2 - - - - - 0 :
O~
~2-W
$2 =
flax2 f12w2 exp + - T - - + 7U- 2 ]
Ire 2
flW 2 Axa + +~
O~
f l W 2
- e r f ~ ~ - w /J
exp( - fi) $ 3 - 2
. [2_ (erf(Axa + c~ 1 - /Ax2 -c~ ] ~ w - j - e r I 1 - ~ - ) ) ] '
In (64), the error function is defined, as in [Ref. 28, Eq. (8.250.1)], by
2 i e-'2 dt (65) eft(x) = ~ / 7 0
No approximations have been used in the integration of (63); hence there is no restriction implicit in the size of A x 2 relative to c~. It should be noted that the correlation l eng th , for the diffuser has been scaled to the output plane. First, consider the terms S 1, $2, S 3 in (64) when Ax2 = 0. Using the identity erf(+ x) = - e f t ( - x), one can readily show that
$1 + $2 + $3 = exp(+ f12W2
I e r f ( - ~ - + - ~ 7 ) - er f ( - -~-)] (66)
f +exp( - f i ) l - e r f ~ .
When the roughness • ~ 2, then fl ~ 10 and the $3 term in (66) is negligible. In order to present an explicit calculation for the spectral and spatial dependence of the speckle, the general form in (13) is normalized and rewritten using (31) to yield
where Rg �9 R= is given by (64). Also for computational purposes we find it convenient to express St + $2 + $3 in (64) in terms of the w-function, [Ref. 29, Eq. (7.1.3)-1. This regroups the large exponent in $1, $2, i.e., exp(+ fi2w2/2c~2) and leads to a convenient asymptotic form for large arguments [-29]. We omit the details since we are presenting the computer plots shown in Fig. 3. Three-dimensional plots of R', are shown versus wavelength and A x 2 for rms diffuser heights a of l, 5, 25, 12516m in Figs. 3a-d, respectively. The
Space and Wavelength Dependence of Speckle Intensity
(a) R~/z~• ,x 2) (b) R~
~ ~
211
(c) R; (d) R~
~
Fig. 3 a -d . Normalized correlation function for the speckle intensity in the output plane (II in Fig. 1) of a space-invariant imaging system: R~, versus spatiaI offset A x2 and wavelength 22 from (67). The vertical scale peaks to R' u = 2 at A x2 = 0, 22 = 5500 A, and the contour intervals are 100 A in wavelength and 5 gm in Ax2. The wavelength decorretation interval is seen to decrease sharply as the rms diffuser height a increases: a) I gin, b) 5 gin, c) 25 ~tm, d) 125/~m. The spatial variation in R', is seen to be insensitive to a as predicted by (69). Fixed values of diffuser correlation length ~ = 2.5 ~tm, resolution cell size w = 10 gin, and 21 = 5500 ,& have been used
resolution-cell size and the correlation interval of the diffuser heights are given fixed values, i.e., w = 101am and c~=2.5 gin, respectively. A contour interval of 5 gm is used for A x2, and an interval of 100 ,~ is used for the wavelength scale. The wave- length corresponding to 7/1 is 5500 A, and the wave- lengths labeled are for ~/z = ~/2z, ie., in (15) the index- angle factor is taken to be 1/2. The spatial dependence of R', is relatively insensitive to the diffuser roughness ~, as is seen in Fig. 3. It is largely dependent on the autocorrelation function of the system's impulse response R.. A simple con- sideration either of (63) or (64) shows that this spatial dependence in R',- 1 goes as exp [-(A Xz/W)23. On the other hand, the factor exp[-(aAq) 2] in R o dominates the wavelength dependence. Thus, the
cross-correlation function for the input electric field, established by the statistical properties of the diffuser, controls the wavelength diversity of the speckle. In physical terms a decreasing value of R'u either with 22 or A x2 can be interpreted as a lowered cor- relation between the corresponding speckle patterns. For example, if a speckle pattern is recorded on film and then autocorrelated, this correlation function will drop sharply within a spatial interval which defines the average speckle size [27]. Furthermore, if speckle patterns recorded at 21 and 2 2 a r e cross- correlated, then the interval 2~-21 at which the value of R'u - 1 drops to 0.25 defines the wavelength spacing for decorrelation. Measurements of this decorrelation interval which are quantitatively in
212 N. George and A. Jain
good agreement with this theory have been reported previously [12, 14, 30]. Thus, with the Gaussian apodization in (59), the average speckle size d, is given by
dl,.~ 2w
and by (61), this is approximately
d L ~ -s (69)
Finally, the basic frequency decorrelation interval, in (40), is still valid for this case of small correlation interval ~.
Conclusions
A theory has been described which permits the calculation of the spatial and spectral autocorrelation function R,(Ax, t h, th) of the speckle intensity in the image plane of a linear, space-invariant optical system. Thus, (13) gives R, in terms of the convolution of the autocorrelation of the impulse response func- tion with that for the input electric field. In (20), R, is expressed in terms of the joint characteristic function of the heights of the scatterers on the input diffuser. The assumption of a Gaussian distribution of heights leads to the results in (36). Two separate examples, a rectangular aperture and a Gaussian aperture, are analyzed in Section 5. And we show that our results reduce, in the one-dimensional limit, to the established spatial autocorrelation function in the literature. In summary we note that the speckle size is strongly dependent on the aperture of the optical system, while its wavelength dependence is influenced mainly by the distribution function for the heights on the diffuser surface. Thus, measurements of the frequency diversity of speckle in an output plane should permit one to establish certain detailed characteristics of a diffuse object.
Acknowledgement. The authors are pleased to acknowledge helpful discussions with A.C.Livanos, R.Lipes, J.C.Dainty, and Gareth Parry.
References
t. Lord Rayleigh: Phil. Mag. X, 73 (1880) 2. M.v.Laue: Sitzber. Kgl. Preuss. Akad. Wiss. 47, 1144 (1914),
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