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Dependence of image speckle contrast on surface roughness
Volume 14, number 3
OPTICS COMMUNICATIONS
July 1975
DEPENDENCE OF IMAGE SPECKLE CONTRAST ON SURFACE ROUGHNESS* J.W. GOODMAN Department of Electrical Engineering, Stanjbrd University, Stanjord, California 94305, USA Received 15 April 1975
I. Introduction The dependence of speckle contrast on the roughness of transmitting or reflecting surfaces is a subject of considerable current interest [1-3]. Most recently, experimental work of Ohtsubo and Asakura [4[ has shown the curious result that speckle contrast is smaller in the image plane of the rough surface than in other out-of-focus planes. The purpose of this letter is to examine in detail the statistical properties of the diffusely reflected (or transmitted) fields in the image of a rough surface, and by so doing to explain quantitatively the observed dip of speckle contrast in the image plane.
(C n) .I n ,
((; '7")
~
SPUTTER
f/[I OBJECT PLANE
,
ETECTO.
PUPIL
IMAGE
STOP
PLANE
Fig. 1. Experimental geometry for measuring image-plane speckle contrast.
from its mean value and X is the optical wavelength. The variance of the phase is accordingly 2 = (47r/X)2a2,
O0
2. Mathematical model We assume that the experimental geometry is as shown in fig. 1. The rough surface is illuminated by a normally incident plane wave via the beam splitter B.S. Lenses L 1 and L 2 form a two-lens imaging system (z is one focal length), and the image intensity is examined in the output plane. Of particular interest will be the contrast C= oi/q) of the observed speckle, where (/) represents the mean image intensity and oI its standard deviation. The dependence of C on surface roughness is of prime concern. As in previous analyses [2,3], the phase of the light in the (~, 7/) plane immediately to the right of the reflecting surface is taken to be 0 (~, r/) = (47r/X)h(~, r/), where h(~, r/) is the departure of the surface height * Work supported by the Office of Naval Research.
324
(1)"
(x, y)
(2)
where 02 represents the variance of the surface height. Since the image contrast is of primary concern, we can, without loss of generality, set the reflectivity and incident intensity equal to unity, with the result that the reflected field takes the form a(~, r/) = exp [i(47r/X)h(~, 77)].
(3)
Assuming that L 1 and L 2 have equal focal lengths, producing a magnification of unity, the field at image coordinates (x, y) is related to a(~, r~) by the convolution equation
A(x,y) = ~ K(x-~,y-r?)c~(~, r/)d~dr/,
(4)
where K ( ' ) represents an amplitude weighting function in the (~, r/) plane. This weighting function can be expressed in terms of the pupil function k(~', 71')by
Volume 14, number 3
OPTICS COMMUNICATIONS
July 1975 oo
K(~, r/) = (1/Xz)J d
([A(r)] 2) = exp ( - a 0 2 ) f f cK(A~, At/)
k(~', r/')
--oo
'-ld~'~ d 77,, X exp[i(27r/Xz)(~' + 7777)1
(5)
X cosh[a02Ph(A~, At/)] dA~ dAr/, oo
where k has the property that
([A¢i)] 2) = / 1, (~', r/') in the pupil;
k(~' r/')
[ O,
exp(-o2o)ff ~(a~, an)
X sinh [O2oPh(A~, Ar/)] dA~edAr/,
otherwise.
(,4 (r) A (i)) = 0,
(9)
For most pupils of practical interest, sufficient symmetry exists to assure that K(~, 7) is a real-valued function, a property we shall make use of later.
where
3. Statistical properties of the image fields
Ph is the normalized autocorrelation function of the
If we assume that the surface height function h(~, r/) is a stationary, zer:* mean gaussian random process, we can prove two important properties of the image field A(x, y). First, its real part has a non-zero mean value, corresponding to the specular component of light transmitted by or reflected from the rough surface. Second, and of prime interest to us here, the diffuse component of A is not a circular complex random variable, as explained below. Consider first the mean values of the real and imaginary parts of the field A(x, y). Noting that K(~, r/) has been assumed to be real valued, we have oo
(A (r)) = f ~ K(x-~,y-r/)(cos 0(~, r/)) d~dr/ = Xz exp (-½ o2), (,4 (i)) = f f
(7)
oo
~(a~, a,)a=ff
surface heights, while cosh [" ] and sinh [. ] represent hyperbolic cosine and sine functions, respectively. We subtract off the squared mean of A (r) to find variances, and note that ~ ( A ~ , Ar/) is to a good approximation equal to oK(0, 0) over the range of (A~, Ar/) for which the rest of the integrand has value. In addition, we have
(o, o) = f f K2(~, r/) d~dn oo
--//
k"2""' t ~ , r/') d~'dr/' = Sp,
(11)
where Sp represents the area of the pupil. Thus, we obtain for the variances of the real and imaginary parts of the image field 0 2 = S p e x p ( - o 2 ) f f {cosh[o02Ph(A~, Ar/)]--l}dA~dAr/, _oo (12)
02 = Sp
K(x--~,y-r/) (sin 0(~, 7?))d~dr/
K(~, ~)K(~-A~, r/--A~)d~dr/, (10)
exp(-o2o)ff sinh [O2ph(A~, Ar/)] dA~dAr/. oo
=0, where (') indicates a statistical averaging operation, and we have used the facts that 0 is a zero-mean stationary gaussian random process and that oo
. ; ; K(~, r/) d~dr/= Xz k(0, 0) = Xz. As for the second moments, we can show with some straightforward manipulations that
(8)
The results presented in eq. (12), together with the lack of correlation between A (r) and A (i), demonstrate that the diffuse component of A is not a circular complex random variable (i.e., contours of constant probability are not circles in the (A (r), A (i)) plane). It is of some interest to find asymptotic values of 0 2 and 0 2 for o 2 small and large, and hence to determine the departures from circularity in these two limiting cases. For small 0 2 , we have cosh[O2ph(A~e, At/)] ~-- 1, 325
Volume 14, number 3
July 1975
OPTICS ('OMMUNI('ATIONS
sinh[o2Oh(A~, A~)] ~
O2oOh(A~,A~7),
(13)
- 2-2n+ 1 to o ) 0"2= Sprrr2exp(0"2) F 0 ( 2 n + l ) ( 2 n + l )
!
and with the help of the definition So
=f ; ph(A~, Ar~)dA~dA~'/
(19) (14)
for the correlation area of the surface itself, we find
0"~~- S p SO02.
0 r2 2 0 ,
(15)
Thus we see that for very smooth surfaces, the diffuse component of the image field is primarily in quadrature with the specular component of the image field, and the statistics of the diffuse component are highly non-circular. For large 0-2, we use the asymptotic properties cosh(o2Oh) = ~ exp(0-2Oh), (16)
si,&(0-o2ph) : ~ exp(0"2Oh), plus the fact that ½exp(o2oh) >> 1 over most of the range of integration of interest, to write exp(0"2) 0"2 = 0"2 r
i = Sp
X
2
.~" exp[o2ph(Ae~, A'4) ] dA~dAT~.
(17)
Thus in the limit of" very rough surfaces, circularity of the statistics has been restored. Previous analyses of image plane speckle [5] have assumed circularity of the statistics, an assumption which we see was well justified for the strong diffusers then of interest. Note that our above conclusions in the two limiting cases hold regardless of the correlation function Ph of the surface under study. The solutions for o r2 and off can be found for arbitrary 0"2 once a specific form for Ph is adopted. We choose a two-dimensional gaussian form,
Ph(r) = exp[ (r/rc)2],
(18)
which may be more realistic than the forms assumed in previous analyses [2]. Expanding the integrands ot" eq. (12) in a power series and integrating term-byterm, we obtain (0";)2n
0"2: Sprrr2ceXp(_0"2o)n~=l2n(2n), =[chi(0"02) 326
C
1,,0"21,
= shi(o2), where chi(') and shi(') are the hyperbolic cosine and sine integrals, respectively [6], and C is Euler's constant.
4. Speckle contrast Attention is now turned to the contrast of the speckle pattern and its dependence oi1 surface roughness. in order to calculate the contrast, we find it necessary to make some specific assumptions about the first-order statistics of the real and imaginary field components A (r) and A (i). We assume that the weighted region of object space contributing to the image field at ally point consists of at least several independent correlatic.n areas of the rough surface (in practice, 8 or 10 independent correlation areas will suffice). In this case the central limit theorem implies that the field components A (r) and A (i) are approximately gaussian random variables. Using this fact, with a modest amount of algebra we can show that the variance of image intensity is 02 =41sO2+ 2 o 4 + 2o 4 ,
(20)
where I s represents the intensity of the specular component and equals X2e2exp(-o2). In addition, it is straightforward to show that the mean intensity is (/)=Is + °2r + 02.
(21)
Thus the contrast of the speckle pattern must be given by
Ol C=
[41sO2 + 204 + 204] 1/2 =
q)
(22) Is + °'2r + 0"2
2 "~ o 1 Incorporating eqs. (19), the defining N = (Sprrrc/X"z"y as the effective number of correlation areas of the surface contributing to the intensity at any given image point,
Volume 14, number 3
OPTICS COMMUNICATIONS
Q = {4N-1 (chi 02 - C - In 02) + 2N -2 [(chi 02 - C - In a2) 2 + (shi 02)21 }1/2 X [1
+N-l(eiao 2
c-In @l,
_
where ei o 2 is the exponential integral [6]. This function is plotted in fig. 2, using solid lines, for various values of N and as a function of o0/27r. Note that the contrast always saturates at the "rough-surface" value of unity for sufficiently large a 0 . For small a 2, the contrast behaves asymptotically as
e ~--o2/N
(0¢2 "~ 1),
a result which differs considerably from that of Pedersen [2], who assumed circular statistics for A.
July 1975
fields. The conclusion that the statistics are non-circular depends heavily on our assumption that the weighting function is real valued. It is straightforward to show that if we examine the fields in a plane which is some distance away from the image plane, the weighting function becomes complex valued. The effect of a complex-valued K is essentially to mix the two field components studied above, with the result that the statistics of the diffuse component of field becomes very nearly circular. Thus, in passing through the plane of focus, the diffuse fields pass from circular statistics to non-circular statistics and back to circular statistics. If the statistics are circular, the dependance of speckle contrast on o 0 becomes
K(x, y)
K(x, y)
21 +N1--~-Feio2-C-lno212}l/2 2L
" 02 - C - l n o 0 Q = {2~ I elo 5. Discussion
1 +(1/N)[ei o 02 - C We have found the relation between surface roughness and speckle contrast in the image plane using some reasonable assumptions about the statistical properties of the reflecting or transmitting surface. The most interesting result of the analysis is the noncircular statistics of the diffuse component of image
,o
/// /,/oo/ /o /
C
0.5
in o 02 ]
'
which is plotted with dashed lines in fig. 2. These curves agree well with those of Pedersen [2], derived under the assumption of circular statistics but using non-gaussian surface correlation functions. It appears that Pedersen's results are quite accurate for all planes except the image plane. The dip in contrast observed by Ohtsubo and Asakura [4] undoubtedly corresponds to a change from circular statistics to non-circular statistics and back again. The unique character of the image plane arises from the fact that it is in this plane that the image fields follow the object fields most closely. Since the object fields have been assumed to vary only in phase, the image fields come as close to being a speckle-free phase function as the finite pupil will allow.
0
,,,'/,/I ,'L , 7 ,/ /
/
/
/ /" /
References
/,
///"
I
0.2
0.4
06
0.8
o-e/2"rr
Fig. 2. Speckle contrast versus a0/27r for various values of N. Solid curves represent results for non-circular statistics, dotted curves correspond to circular statistics.
[ 1] [2] [3] [4] [5]
H. Fujii and T. Asakura, Opt. Commun. 11 (1974) 35. H.M. Pedersen, Opt. Commun. 12 (1974) 156. H. Fujii and T. Asakura, Opt. Commun. 12 (1974) 32. J. Ohtsubo and T. Asakura, Opt. Commun. 14 (1975) 30. S. Lowenthal and H. Arsenault, J. Opt. Soc. Am. 60
(1970) 1478. [6] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions Dover Publications, New York (1972) p. 232.
327
OPTICS COMMUNICATIONS
July 1975
DEPENDENCE OF IMAGE SPECKLE CONTRAST ON SURFACE ROUGHNESS* J.W. GOODMAN Department of Electrical Engineering, Stanjbrd University, Stanjord, California 94305, USA Received 15 April 1975
I. Introduction The dependence of speckle contrast on the roughness of transmitting or reflecting surfaces is a subject of considerable current interest [1-3]. Most recently, experimental work of Ohtsubo and Asakura [4[ has shown the curious result that speckle contrast is smaller in the image plane of the rough surface than in other out-of-focus planes. The purpose of this letter is to examine in detail the statistical properties of the diffusely reflected (or transmitted) fields in the image of a rough surface, and by so doing to explain quantitatively the observed dip of speckle contrast in the image plane.
(C n) .I n ,
((; '7")
~
SPUTTER
f/[I OBJECT PLANE
,
ETECTO.
PUPIL
IMAGE
STOP
PLANE
Fig. 1. Experimental geometry for measuring image-plane speckle contrast.
from its mean value and X is the optical wavelength. The variance of the phase is accordingly 2 = (47r/X)2a2,
O0
2. Mathematical model We assume that the experimental geometry is as shown in fig. 1. The rough surface is illuminated by a normally incident plane wave via the beam splitter B.S. Lenses L 1 and L 2 form a two-lens imaging system (z is one focal length), and the image intensity is examined in the output plane. Of particular interest will be the contrast C= oi/q) of the observed speckle, where (/) represents the mean image intensity and oI its standard deviation. The dependence of C on surface roughness is of prime concern. As in previous analyses [2,3], the phase of the light in the (~, 7/) plane immediately to the right of the reflecting surface is taken to be 0 (~, r/) = (47r/X)h(~, r/), where h(~, r/) is the departure of the surface height * Work supported by the Office of Naval Research.
324
(1)"
(x, y)
(2)
where 02 represents the variance of the surface height. Since the image contrast is of primary concern, we can, without loss of generality, set the reflectivity and incident intensity equal to unity, with the result that the reflected field takes the form a(~, r/) = exp [i(47r/X)h(~, 77)].
(3)
Assuming that L 1 and L 2 have equal focal lengths, producing a magnification of unity, the field at image coordinates (x, y) is related to a(~, r~) by the convolution equation
A(x,y) = ~ K(x-~,y-r?)c~(~, r/)d~dr/,
(4)
where K ( ' ) represents an amplitude weighting function in the (~, r/) plane. This weighting function can be expressed in terms of the pupil function k(~', 71')by
Volume 14, number 3
OPTICS COMMUNICATIONS
July 1975 oo
K(~, r/) = (1/Xz)J d
([A(r)] 2) = exp ( - a 0 2 ) f f cK(A~, At/)
k(~', r/')
--oo
'-ld~'~ d 77,, X exp[i(27r/Xz)(~' + 7777)1
(5)
X cosh[a02Ph(A~, At/)] dA~ dAr/, oo
where k has the property that
([A¢i)] 2) = / 1, (~', r/') in the pupil;
k(~' r/')
[ O,
exp(-o2o)ff ~(a~, an)
X sinh [O2oPh(A~, Ar/)] dA~edAr/,
otherwise.
(,4 (r) A (i)) = 0,
(9)
For most pupils of practical interest, sufficient symmetry exists to assure that K(~, 7) is a real-valued function, a property we shall make use of later.
where
3. Statistical properties of the image fields
Ph is the normalized autocorrelation function of the
If we assume that the surface height function h(~, r/) is a stationary, zer:* mean gaussian random process, we can prove two important properties of the image field A(x, y). First, its real part has a non-zero mean value, corresponding to the specular component of light transmitted by or reflected from the rough surface. Second, and of prime interest to us here, the diffuse component of A is not a circular complex random variable, as explained below. Consider first the mean values of the real and imaginary parts of the field A(x, y). Noting that K(~, r/) has been assumed to be real valued, we have oo
(A (r)) = f ~ K(x-~,y-r/)(cos 0(~, r/)) d~dr/ = Xz exp (-½ o2), (,4 (i)) = f f
(7)
oo
~(a~, a,)a=ff
surface heights, while cosh [" ] and sinh [. ] represent hyperbolic cosine and sine functions, respectively. We subtract off the squared mean of A (r) to find variances, and note that ~ ( A ~ , Ar/) is to a good approximation equal to oK(0, 0) over the range of (A~, Ar/) for which the rest of the integrand has value. In addition, we have
(o, o) = f f K2(~, r/) d~dn oo
--//
k"2""' t ~ , r/') d~'dr/' = Sp,
(11)
where Sp represents the area of the pupil. Thus, we obtain for the variances of the real and imaginary parts of the image field 0 2 = S p e x p ( - o 2 ) f f {cosh[o02Ph(A~, Ar/)]--l}dA~dAr/, _oo (12)
02 = Sp
K(x--~,y-r/) (sin 0(~, 7?))d~dr/
K(~, ~)K(~-A~, r/--A~)d~dr/, (10)
exp(-o2o)ff sinh [O2ph(A~, Ar/)] dA~dAr/. oo
=0, where (') indicates a statistical averaging operation, and we have used the facts that 0 is a zero-mean stationary gaussian random process and that oo
. ; ; K(~, r/) d~dr/= Xz k(0, 0) = Xz. As for the second moments, we can show with some straightforward manipulations that
(8)
The results presented in eq. (12), together with the lack of correlation between A (r) and A (i), demonstrate that the diffuse component of A is not a circular complex random variable (i.e., contours of constant probability are not circles in the (A (r), A (i)) plane). It is of some interest to find asymptotic values of 0 2 and 0 2 for o 2 small and large, and hence to determine the departures from circularity in these two limiting cases. For small 0 2 , we have cosh[O2ph(A~e, At/)] ~-- 1, 325
Volume 14, number 3
July 1975
OPTICS ('OMMUNI('ATIONS
sinh[o2Oh(A~, A~)] ~
O2oOh(A~,A~7),
(13)
- 2-2n+ 1 to o ) 0"2= Sprrr2exp(0"2) F 0 ( 2 n + l ) ( 2 n + l )
!
and with the help of the definition So
=f ; ph(A~, Ar~)dA~dA~'/
(19) (14)
for the correlation area of the surface itself, we find
0"~~- S p SO02.
0 r2 2 0 ,
(15)
Thus we see that for very smooth surfaces, the diffuse component of the image field is primarily in quadrature with the specular component of the image field, and the statistics of the diffuse component are highly non-circular. For large 0-2, we use the asymptotic properties cosh(o2Oh) = ~ exp(0-2Oh), (16)
si,&(0-o2ph) : ~ exp(0"2Oh), plus the fact that ½exp(o2oh) >> 1 over most of the range of integration of interest, to write exp(0"2) 0"2 = 0"2 r
i = Sp
X
2
.~" exp[o2ph(Ae~, A'4) ] dA~dAT~.
(17)
Thus in the limit of" very rough surfaces, circularity of the statistics has been restored. Previous analyses of image plane speckle [5] have assumed circularity of the statistics, an assumption which we see was well justified for the strong diffusers then of interest. Note that our above conclusions in the two limiting cases hold regardless of the correlation function Ph of the surface under study. The solutions for o r2 and off can be found for arbitrary 0"2 once a specific form for Ph is adopted. We choose a two-dimensional gaussian form,
Ph(r) = exp[ (r/rc)2],
(18)
which may be more realistic than the forms assumed in previous analyses [2]. Expanding the integrands ot" eq. (12) in a power series and integrating term-byterm, we obtain (0";)2n
0"2: Sprrr2ceXp(_0"2o)n~=l2n(2n), =[chi(0"02) 326
C
1,,0"21,
= shi(o2), where chi(') and shi(') are the hyperbolic cosine and sine integrals, respectively [6], and C is Euler's constant.
4. Speckle contrast Attention is now turned to the contrast of the speckle pattern and its dependence oi1 surface roughness. in order to calculate the contrast, we find it necessary to make some specific assumptions about the first-order statistics of the real and imaginary field components A (r) and A (i). We assume that the weighted region of object space contributing to the image field at ally point consists of at least several independent correlatic.n areas of the rough surface (in practice, 8 or 10 independent correlation areas will suffice). In this case the central limit theorem implies that the field components A (r) and A (i) are approximately gaussian random variables. Using this fact, with a modest amount of algebra we can show that the variance of image intensity is 02 =41sO2+ 2 o 4 + 2o 4 ,
(20)
where I s represents the intensity of the specular component and equals X2e2exp(-o2). In addition, it is straightforward to show that the mean intensity is (/)=Is + °2r + 02.
(21)
Thus the contrast of the speckle pattern must be given by
Ol C=
[41sO2 + 204 + 204] 1/2 =
q)
(22) Is + °'2r + 0"2
2 "~ o 1 Incorporating eqs. (19), the defining N = (Sprrrc/X"z"y as the effective number of correlation areas of the surface contributing to the intensity at any given image point,
Volume 14, number 3
OPTICS COMMUNICATIONS
Q = {4N-1 (chi 02 - C - In 02) + 2N -2 [(chi 02 - C - In a2) 2 + (shi 02)21 }1/2 X [1
+N-l(eiao 2
c-In @l,
_
where ei o 2 is the exponential integral [6]. This function is plotted in fig. 2, using solid lines, for various values of N and as a function of o0/27r. Note that the contrast always saturates at the "rough-surface" value of unity for sufficiently large a 0 . For small a 2, the contrast behaves asymptotically as
e ~--o2/N
(0¢2 "~ 1),
a result which differs considerably from that of Pedersen [2], who assumed circular statistics for A.
July 1975
fields. The conclusion that the statistics are non-circular depends heavily on our assumption that the weighting function is real valued. It is straightforward to show that if we examine the fields in a plane which is some distance away from the image plane, the weighting function becomes complex valued. The effect of a complex-valued K is essentially to mix the two field components studied above, with the result that the statistics of the diffuse component of field becomes very nearly circular. Thus, in passing through the plane of focus, the diffuse fields pass from circular statistics to non-circular statistics and back to circular statistics. If the statistics are circular, the dependance of speckle contrast on o 0 becomes
K(x, y)
K(x, y)
21 +N1--~-Feio2-C-lno212}l/2 2L
" 02 - C - l n o 0 Q = {2~ I elo 5. Discussion
1 +(1/N)[ei o 02 - C We have found the relation between surface roughness and speckle contrast in the image plane using some reasonable assumptions about the statistical properties of the reflecting or transmitting surface. The most interesting result of the analysis is the noncircular statistics of the diffuse component of image
,o
/// /,/oo/ /o /
C
0.5
in o 02 ]
'
which is plotted with dashed lines in fig. 2. These curves agree well with those of Pedersen [2], derived under the assumption of circular statistics but using non-gaussian surface correlation functions. It appears that Pedersen's results are quite accurate for all planes except the image plane. The dip in contrast observed by Ohtsubo and Asakura [4] undoubtedly corresponds to a change from circular statistics to non-circular statistics and back again. The unique character of the image plane arises from the fact that it is in this plane that the image fields follow the object fields most closely. Since the object fields have been assumed to vary only in phase, the image fields come as close to being a speckle-free phase function as the finite pupil will allow.
0
,,,'/,/I ,'L , 7 ,/ /
/
/
/ /" /
References
/,
///"
I
0.2
0.4
06
0.8
o-e/2"rr
Fig. 2. Speckle contrast versus a0/27r for various values of N. Solid curves represent results for non-circular statistics, dotted curves correspond to circular statistics.
[ 1] [2] [3] [4] [5]
H. Fujii and T. Asakura, Opt. Commun. 11 (1974) 35. H.M. Pedersen, Opt. Commun. 12 (1974) 156. H. Fujii and T. Asakura, Opt. Commun. 12 (1974) 32. J. Ohtsubo and T. Asakura, Opt. Commun. 14 (1975) 30. S. Lowenthal and H. Arsenault, J. Opt. Soc. Am. 60
(1970) 1478. [6] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions Dover Publications, New York (1972) p. 232.
327