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The attenuation of a coherent field by scattering
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15 June 1995
CQs 32
OPTICS COMMUNICATIONS
ErSEVIER
Optics Communications
117 ( 1995) 532-540
Full length article
The attenuation of a coherent field by scattering H.-J. Schnorrenberg Ins&at fiir Medizinische
‘,
M. Hengstebeck,
Optik, Ludwig-Maximilians-Universitiir Received 30 November
K. Schlinkmeier
Miinchen, Barbarastr.
16, 80797 Miinchen,
1994; revised version received 3 February
Germany
1995
Abstract The intensity of a coherent field attenuated by scattering is described by an ordinary differential equation. This differential equation is derived in the same way as Lambert-Beer law of absorption, but the coherent forward scattering is taken into account. The analytical solution of the differential equation depends on two quantities of single scattering, concentration of scatterers and thickness of the medium. Measurements of coherent light transmitted through suspensions of latex microspheres confirm the resulting equation.
1. Introduction The problem of multiple scattering is important for many domains of physics and has been studied extensively. First, problems of multiple scattering have been treated by transport theory, based on the integrodifferential equation of Boltzmann [ l-51. In the treatment by transport theory interference phenomena are neglected. For that reason, multiple scattering was described by wave theory [ 6-91. Thereby the average field is deduced for a finite volume containing randomly distributed scatterers. The integral equation for the average field can in principle be solved directly [ 61. However, an equation of attenuation of intensity of a coherent field by scattering could not yet be derived by this formulation. In this paper multiple scattering is treated by wave theory, too. But instead of considering the wave function for a finite volume, the wave function is derived for an infinitesimale thin layer. For the thin layer the total scattered flux and the coherent scattered flux in 1To whom correspondence should be addressed. Present address: Kellererstr. 23b, 82256 Fiirstenfeldbruck, Germany. 0030-4018/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDIOO30-4018(95)00240-5
forward direction are determined. The difference of total scattered flux and coherent forward scattered flux is considered as a loss of the flux falling onto the thin layer. This treatment yields an ordinary differential equation for the attenuation of intensity of a coherent field. The analytical solution of this differential equation depends on quantities of single scattering, concentration of scatterers and thickness of the medium. Experiments of attenuation of light, propagating in suspensions of latex microspheres, are presented. The results can be well described by the presented formalism.
2. Theory The following treatment of the attenuation of a coherent field by scattering is based on elementary wave theory. With regard to the experiments, the treatment is done for scattering of light: a light beam with cross section A propagates within the scattering medium along the z-axis (Fig. 1) . A thin layer contains IZscattering particles. These particles can be distributed randomly or regularly. The 12vectors ri define the posi-
H.-J. Schnorrenberg
et al. / Optics Communications I1 7 (1995) 533-540
533
The product Z (z. ) g( k) do of incident intensity, differential scattering cross section and element of solid angle substitutes the radiant flux ZO(k)d3 of the single scattered light wave:
d@%(k)= I( Fig. 1. The scattered electromagnetic field seen from a point in the far zone. A light beam with cross section A propagates along the z-axis. A thin layer contains n particles, which scatter the electromagnetic field falling onto the layer. At a point P in the far zone the scattered
flux is calculated.
tions of the particles. The electromagnetic field E( z, t) enters the thin layer and is scattered by the particles. The radiant flux d@s of the scattered light waves on the area d3 at a point P in the far zone is calculated. Assuming, that the scattering particles are identical, the n electromagnetic fields Ei of the scattered light waves at point P can be expressed by
&(Q,k,l)
=&(k)
COS[Q.(k,
-k)
-wt].
(1)
In Eq. ( 1) the wave vector k points from the scattering particles to the point P in the far zone, EO(k) denotes the amplitude of a single scattered lightwave at point P. The electromagnetic field ES (k, t) at the point P is the superposition of the n scattered fields: Es(k,t)
= Eo(k)
-&os,r,.(k, i=l
-k)
-wtl
.
(2)
Averaging in time and squaring of Eq. (2) yields the intensity Is (k) of the scattered light: Is(k)
=Zo(k)
F:kcos[(ri-rj). i=l j=1
jti
n+)P:kcosI(ri-rj)*(kz ;=1
j=l
(k, -k)]
-k)l
1 .
(3) Multiplying Eq. (3) by the element of area d3 at point P, the scattered flux d@s (k) is obtained:
=Z,,(k,t)
= Zs(k, t) d3
j=1 jti
n+kkcos((ri-q)-@l-k)) i=l
1
d3. (4)
dR.
j+i
The scattered radiant flux d@s( k) can be split into two parts d@.$,‘)(k) and d@p) ( k) : d@“‘(k) S
= Z(z)~(k)ndO,
dGc2’(k) S
= Z(z)a(k) COS((ri-rj)
j=l
(6)
.(k,
-k))dO.
(7)
j=1 j+i
Commonly, d@$‘) (k) is called the incoherent scattering term. The incoherent scattering term d@i’) (k) represents the scattered radiant flux as a sum of iz single scattering events. The second term d@$) (k) takes the interference of the scattered light waves into account. Therefore, the term d@ (k) represents the change of the angular distribution of the scattered flux by interference. In the following context this term is called the interference term. First, the total scattered flux of the layer is derived, which yields the differential equation of LambertBeer law. Although the derivation is well-known, it is carried out in detail to demonstrate the method. Later, the coherent scattered flux in forward direction is derived in the same way, which yields an additional differential equation. In order to get the total scattered flux @s, the incoherent scattering term and the interference term are integrated over the unit sphere: Qbs= I d@“‘(k) S J
4?T
d&(k)
1 (5)
j=l i=l
n+~~;cos((r,-r,),(k,k))
+ I dGc2’(k) s J
*
(8)
477
The value of total scattered flux is not affected by interference, because flux cannot be generated or de-
stroyed by interference. Therefore, the integration of the interference term d@i2) over the whole solid angle 4~ yields no flux. So, the total scattered flux is
534
H.-J. Schnorrenberg et al. /Optics Communications 117 (1995) 533-540
given by the integral of the incoherent @$‘) (k) over the unit sphere:
Gs =
scattering term
d@)(k)
s
Dividing this equation by the cross section A of the light beam, the differential dls (z ) for the intensity of a coherent field decreased by the total scattered flux is obtained:
4n
dk(z>
=
Z(z>~(k>ndfi=
I(z)u
s
In Eq. (9) UT denotes the total scattering cross section for one scatterer. If the scatterers are distributed statistically in the medium, rt represents the average number of scattering particles within the volume V. In order to form the differential equation, it is neccessary to replace discrete particles by a homogeneous material. The homogeneous material must have the same scattering characteristics as the particles. So, in the replacement the differential scattering cross section c(k) is not changed. The homogeneous material is represented by the concentration. The concentration c depends on the volume VPof a scattering particle and the number n of the scattering particles within the volume V and is defined by
c vvp:
=
-Z(z)pscdz
(15)
.
If the total scattered flux is considered as a loss of the coherent field , the differential equation (15) would yield Lambert-Beer law. But the coherent forward scattered lightwave is superposed with the unscattered lightwave, and so, the coherent forward scattered flux must not be considered as a loss. As a consequence, the differential equation of the attenuation by scattering consists of two parts: differential equation ( 15), which describes the total scattered flux, and an additional term, which takes coherent scattering in forward direction into account. In Eq. (5) the scattered flux is defined per solid angle in dependence on scattering direction. For the coherent scattered flux in forward direction the scattering direction k and the propagation direction k, of the lightwave falling onto the layer are equal. So the cosine in Eq. (5) becomes one, and the coherent scattered flux in forward direction can be written as d@u =Z(z)~(k)
j‘JkldR i=l j=l
= Z(z)a(k,)n2df2.
@s =Z(z)
$?cv.
(11)
P
The ratio of total scattering cross section and volume of a particle is identical to the extinction coefficient per unit concentration z+: f_&=%.
(12) P
The finite volume V is replaced by the infinitesimale volume Adz : (13)
=Z(z),uscAdz.
The flux falling onto the layer is decreased by scattering. Therefore, the change d& (z ) of the incident flux is given by d&(z)
d
(10)
The number n of scatterers in Eq. (9) is replaced by
d%(z)
d&(z) ~
(9)
4n
V c:=n-?-. V
=
= -d@,(z)
= -Z(z)pscddz.
(14)
(16)
Thereby the forward direction is denoted by the subscript 0, which indicates scattering angle 6 = O. In order to get the finite flux in forward direction of single scattering, Eq. ( 16) has to be integrated over a solid angle 0c around the forward direction. However, it is important to distinguish the scattered flux in forward direction of single and multiple scattering. For a single scattering process the intensity of scattered light depends on the distance from the scatterer 2 . However, the coherent forward scattered light is the superposition of many scattered lightwaves. So the forward scattered light has to be represented by a plane wave. In contrast to the single scattered light wave, the intensity of a plane wave is constant. In order to get a representation of the multiple scattered *That is expressed inversely propoaional
by the so-called to ?).
inverse
square
law (I is
H.-J. Schnorrenberg et al. /Optics
flux in forward direction independent of solid angle, the function 5(a) of a solid angle & is defined by
J
u(k)
da.
(17)
no
The function 5(Q)) is defined in such a way, that for & = 47r the total scattering cross section is obtained. For forward scattering an solid angle L$ around the forward direction k, is chosen. Then the limit for K& + 0 is considered:
Ja(k) da.
(18)
& The limiting distribution:
process
J
by the 15
Ju(k)
4%~0 = lim @l--10 00 no = 4~
can be expressed
a(k)6(k
dG
- k,)dL? = 4m(k,)
.
(19)
4w
To ease the understanding of the quantity au, an ensemble of isotropic scatterers is considered. The single scattered light of each scatterer is seen as an elementary lightwave. Applying the construction of Huygens, in a high dense medium only the coherent transmission in forward direction is observable. That means, due to interference of the scattered lightwaves the total scattered flux propagates only in the forward direction. This is expressed by Eq. (19). So, the quantity ~a represents the scattered flux in forward direction per particle, if the medium is so dense, that only coherent transmission is observable. As this example shows, the quantity ~0 represents both scattered flux in forward direction per particle and the interference of scattered lightwaves. So, in general it makes no sense to interpret ~0 as the forward scattered flux per particle. Using Eq. ( 19) the finite scattered flux in forward direction is obtained from Eq. ( 16) :
Communications 117 (1995) 533-540
scatterers has to be replaced by a tion. In Eq. ( 10) the number n of the volume V was expressed by and the volume VP of a scattering 1 n=---_cV=v,
(20)
In Eq. (20) the scattered flux @Odepends on the square n* of the number of scatterers. For the derivation of the differential equation the square of the number of
v,
continuous distributhe scatterers within the concentration c particle:
J
(21)
cdV.
V
The number 12 of scatterers depends linearly on the volume V. Therefore, the derivation of a differential equation is simple: dn = $cdV.
(22)
P
However, the square of the number n* does not depend linearly on the volume V. Squaring Eq. (22) makes no sense. To get the correct formulation of the differential equation, the quantities n and rr* have to be interpreted statistically. In Eq. (21) n represents the mean value of the number of scatterers within the volume V. Therefore the concentration c has to be considered as a continuous distribution. The factor l/V, normalizes the distribution (0 5 c < 1) . Hence, the quantity n* in Eq. (20) has to be interpreted as the second moment of the distribution: (23) Although the quantities n and n* are interpreted statistically, no assumption is made on the distribution of the particles in the medium. The particles can be distributed regularly or randomly within the volume. For a random medium, n* represents the average of the square of particles in the layer (and not the square of the average number!). Consequently, the continuous form of Eq. (20) is given by @a =Z(z)
WV. v,
Analogous
to Eq. ( 12), the abbreviation
pJ
@o =Z(z)r?q.
1
535
:=
(24) pa is used:
F. P
In Eq. (24) the finite volume V is substituted infinitesimale volume Adz : dGo = Z(z)~ac*ddz.
(25) by the
(26)
536
H.-J. Schnorrenberg
et al. /Optics
Dividing Eq. (26) by the cross section A of the light beam, the differential dZa(z) for the intensity of the forward scattered light is obtained: dZ,,(z) = 9
= Z(z)pac*dz
.
(27)
The complete differential dZ ( z ) for the intensity of a coherent field attenuated by scattering is the sum of the differentials dZs (z ) (Eq. ( 15) ) and dZo( z ) (Eq. (27)): dZ(z)
=dZs(z)
+dZo(z).
(28)
Consequently, the differential equation for the intensity of a coherent field attenuated by scattering is given by dZ(z)
= (-,usc
+ POC*) Z(z) dz .
(29)
Integration of this equation over the path up to the thickness d of the medium yields Z(c,d)
= exp( -pscd
+ ,uac2d)Za.
(30)
The attenuation of the intensity depends both on thickness d of the scattering medium and concentration c. In contrast, the attenuation by absorption described by Lambert-Beer law depends only on the product cd. The attenuation of intensity by scattering can be written in the same form as Larnbert-Beer law. Therefore in Eq. (30) the extinction coefficient ,us is replaced by a function of concentration ,Gs (c) :
Communications
117 (1995) 533-540
attenuation by scattering an implicit assumption was made by using the quantities aa and or for the multiple scattered lightwave in forward direction. These quantities are defined for an incident plane wave. But only in the far zone, the forward scattered lightwave can be considered as a plane wave. That means, the distance between the scatterers must be large enough, so that the far field approximation is valid. Hence, the (average) distance D between the scattering particles with size a must satisfy following condition, wellknown from Fraunhofer diffraction: a<
Thereby A denotes the wavelength of the light. The condition (33) can be written in dependence on concentration c and particle size a. Therefore in the definition ( 10) of concentration the volume per particle vn is approximated by the cube D3 of distance between the particles. Also, the volume of one scatterer VPis substituted by the cube u3 of particle size: V,
:=.s(l*-
This substitution Z(c,d)
EC)
=Ps(l
- zc).
a3
(34)
C=ny=D3*
Eq. (34) is solved with respect to the distance D:
So, using Eq. (35) the condition approximation can be written as
a
(33)
(31)
a lf4 qz.
(33)
for far field
(36)
Therefore a condition for the validity of Eq. (32) for the attenuation by scattering is
yields
=Zoexp[-@s(c)cd]
;&.
(37) fi
=Zaexp[-,s(l
- Ec)cd]
.
(32)
For low concentration Eq. (32) yields Lambert-Beer law, and attenuation by scattering and absorption can be described by the same formula. With increasing concentration, the coherent forward scattered light is not negligible and the attenuation by scattering deviates from Lambert-Beer law. Obviously, Eq. (32) fails, if (,xo/,u~)c becomes greater than one. However, in the derivation of the
According to condition (37)) the maximal concentration, for which Eq. (32) is valid, depends on particle size a. For small particles this maximal concentration is higher than for large particles.
3. Experiments In this part of the paper the attenuation by scattering is described by the extinction. The extinction E( c, d)
H.-J. Schnorrenberg SP
LI
D
et al. /Optics
LZ
-f---f-cf_fFig. 2. Experimental setup. The light transmitted through the sample (S) is analyzed by a polarizer (P). A spatial filter of two lenses (f = 100 mm) with a moveable diaphragm (D = 200 pm) allows to detect transmission in dependenceon the scattering angle. A photomultiplier (PMT) registrates the intensity of the filtered transmission.
is defined as the (natural) logarithm of the ratio of the incident intensity la and the coherent transmitted intensity Z(c, d) : E(c,d)
=ln(l(_C,\ \ 10
=-p.scd+poc2d.
(38)
/
In order to test Eq. (38), the intensity of coherent transmitted light attenuated by a scattering medium is measured. The measurements require a technique to distinguish incoherent multiple scattered light from the coherent transmission. A commonly used method is to reduce the field of view (FOV) of detection [ 10,111. Thereby incoherent scattered radiant flux, which illuminates a wide solid angle, is reduced. In contrast, the coherent transmission has the same divergence as the incident light beam. For a moderate number of scattering particles, the incoherent scattered radiant flux in the FOV is negligible. In this case, the detected signal corresponds to the intensity of coherent transmitted light. However, for high number of scatterers, incoherent multiple scattered light can not be neglected in the FOV of detection, and the signal does not correspond to the coherent transmitted flux [ 12-151. Therefore, for the attenuation measurements another technique is used here, taking polarization characteristics into account. By this method incoherent scattered light is suppressed even for high concentrations. The experimental setup is shown in Fig. 2. The sample is illuminated by a linearly polarized laser beam (HeNe, A = 633 nm, P = 14 mW) . Behind the sample the transmitted light is analyzed with a polarizer. A spatial filter, consisting of two lenses and a diaphragm, is placed behind the sample, too. The diaphragm in the focal plane of the first lens is mounted on a translation stage. This moveable diaphragm allows to measure the transmission as a function of scattering an-
Communications
117 (199.5) 533-540
531
gle. A photomultiplier positioned in the focal plane of the second lens registrates the flux. The four-f-setup provides, that at the photomultiplier the same spot is illuminated for all positions of the diaphragm. Furthermore an opal glass diffuser in front of the window of the photomultiplier provides homogeneous irradiation of the photocathode. Lock-in technique is used to improve signal-to-noise ratio. Uniform polystyrol (latex) spheres manufactured by Dow Diagnostics are used as scatterers. At the used wavelength absorption by polystyrol can be neglected [ 161. The index of refraction of polystyrol is known [ 171. Latex spheres are supplied in suspensions with a concentration of 10%. For measurements with lower concentrations, these suspensions are simply diluted with water. Measurements with higher concentrations than 10% are not carried out. For each diaphragm position the transmitted light is measured twice: In the first measurement, the transmission axis of the polarizer is parallel, in the second one perpendicular to the plane of polarization of the incident lightwave. Typical measurement curves are shown in Fig. 3. For a scattering angle greater than fO.1 degree, only scattered light is detected. For scatterers much smaller than the wavelength, the registrated intensity of scattered light is equal for both polarizer adjustments. That means, within accuracy of measurement the scattered light is nonpolarized. In contrast, with scatterers much larger than the wavelength, the measured intensity of scattered light depends on the adjustment of transmission axis of the polarizer. With the transmission axis parallel to the plane of polarization of incident light, more intensity is measured, than with an orthogonal adjustment. Consequently, light multiple scattered by large particles is partially polarized 3 . For scattering angles smaller than fO.1 degree the signal of the orthogonal polarizer adjustment has the same value as for large scattering angles. In contrast, the curve for the parallel polarizer adjustment shows a peak within the central range. The width of this central peak depends neither on concentration of scattering
3This can be explained by Mie theory: Particles larger than the wavelength scatter light mainly about small angles, which preserves polarization. In contrast, small particles scatter light over a wide angle range. This disorders polarization, and so multiple scattered light is nonpolarized (within accuracy of measurment)
538
H.-J. Schnorrenberg
latex
et al. /Optics
272
Communications
117 (1995) 533-540
nm
latex
0
1.09
F
0 --I I”
-0.5 scattering
1
’
I”’
1
0.0
I I 0.5
angle
-0.5
[o ]
r
0.0
scattering
’
I 0.5
angle
["I
Fig. 3. Measurements of the intensity of light transmitted through latex suspensions in dependence on scattering angle. The curve for the polarizer adjusted perpendicular to the plane of polarization of incident light is constant over the whole registrated angle range (thick curve). In contrast, the curve for the parallel polarizer adjustment shows a central peak (thin curve). The amplitude A of the peak corresponds to the intensity of coherent transmission. The registrated intensity of light, scattered by particles much smaller than the wavelength, is for large scattering angles independent of the polarizer adjustment (left figure).
but corresponds to the divergence of the incident light beam. Obviously, the central peak corresponds to the coherent part of the transmitted light. So, the intensity of coherent transmission can be measured by determining the amplitude of the central peak. For scattering particles much smaller than the wavelength the amplitude of the peak is determined in a more accurate way: Transmission is not measured for a wide angle range, but only for scattering angle zero. Then the difference of the measured signals for both polarizer adjustments yields the amplitude of the peak (see Fig. 3). In this way errors due to the movement of the diaphragm do not occur. With this technique intensity of coherent transmission can be measured even for highly scattering media. The accuracy of measurement is limited by signal-to-noise ratio of detection and errors due to the preparation of the scattering samples. In order to measure the deviation of extinction from Lambert-Beer law, the concentration must be sufficiently high (E$ (38) ) . However, the intensity of coherent transmission decreases with increasing concentration and sample thickness. That means, for high concentrations the signal of the intensity of coherent transmission is only measurable for sufficiently small sample thicknesses. For several sample thicknesses (between 1 mm and 10 mm) the extinction by latex spheres (c$= lOO& 10 nm, 272f3 nm and 1.09 pm) was measured. For high concentrations, the measured intensity of particles
nor on sample
thickness,
Table 1 Measured and calculated extinction coefficients for 100 nm latex and 272 nm latex suspension
Measured Calculated
Latex 100 nm
Latex 272 nm
ps [cm-‘]
m [cm-‘]
ps [cm-‘]
m [cm-‘]
1.2 x lo3 1.1 x lo3
1.8 x lo3 1.9 x IO3
9.5 x 103 9.6 x lo3
4.5 x 104 4.4 x lo4
For the numerical calculations of the scattering coefficient /.~s and the constant ~0, the refraction index of latex (polystyrol 1171) and water were taken as n=1.59 and n=1.33, respectively.
coherent transmission deviates from Lambert-Beer law about some orders of magnitude. However, the dependence of extinction on concentration can be adequately represented by a polynom of second degree. For reasons explained above, the most precise results are obtained with latex spheres smaller than the wavelength
and small
sample
thicknesses.
Measure-
ments with Latex spheres with diameter of 100 nm in a 2 mm cuvette and with a diameter of 272 nm in a 1 mm cuvette are very accurate. The coefficients of least-square-fit of these measurements agree with the scattering coefficient ps and the constant P.O (as defined in Eq. (25)) calculated by Mie theory (Table 1). So, the measurements confirm the derived formalism.
H.-J. Schnorrenberg et al. /Optics Communications 1 I7 (1995) 533-540 100
latex
nm
1j\
..
I”“““‘1 0
IXU
-y\ -20
...
: I”““‘I’I’I”““‘I”
10
5
concentration
272
latex
539
0
1
2
concentration
[vol%]
[vol%]
Fig. 4. The extinction deviates from the Lambert-Beer law (dashed line), but can be adequately represented by a polynom of second degree (solid line). The sample thickness for the measurement with the 100 nm latex spheres was 2 mm, for the 272 nm latex spheres 1 mm.
4. Summary
Averaging this equation particles yields
In this paper an equation for the intensity of a coherent field attenuated by scattering was derived. Thereby the coherent forward scattering was taken into account, which yields a deviation from Lambert-Beer law for high concentrations of scatterers. The resulting equation was confirmed by measurements with light attenuated by suspensions of latex microspheres.
Es(k,
We thank Malin Vonier for the initial tests on the experimental setup.
g
cos[ri
of the
. (k, - k) - wt] .
i=l
(A.21 Therefore the differential scattering cross section a(k) in Eq. (5) has to be replaced by the average differential scattering cross section c+(k) : d@s(k)
Acknowledgements
t) = Eo(k)
over the distribution
= I(z)
a(k) cos[(q
evaluation
-rj).
(k, -k)]
da. I
jiti (A.3)
Appendix A. Distribution
of different scatterers
For the derivation of the attenuation by scattering only identical scatterers were considered. The generalization for a distribution of different scatterers is simple. Instead of Eq. (2) for a distribution of different scatterers the superposition of the scattered electromagnetic field has to be written as
So in Eq. (30) for the attenuation by scattering, only the quantities UT and ~a have to be averaged over the distribution of the different scatterers. For the average value of the quantities ,us and pa the average volume s of a scatterer of the distribution has to be considered, too: (A.4)
(A.3 Es(k,t)
= k@‘(k)
cost ri . (k, - k) - wt] .
i=l (A.1)
Thereby the index i of the electromagnetic fields Ef’ ( k) represents the different scatterers in the layer.
So the equation of the attenuation of intensity distribution of scatterers is given by I( c, d) = exp( -&cd
+ F;cc*d) IO.
by a
(A.61
540
H.-J. Schnorrenberg
et al. /Optics
References
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A. Schuster, Astrophys. J. 21 (1905) 1. L. Silberstein, Phil. Mag. 4 (1927) 1291. J.W. Ryde, Proc. Roy. Sot. 131 (1931) 451. J.W. Ryde and B.S. Cooper, Proc. Roy. Sot. 131 (1931) 464. S.Q. Duntley, J. Opt. Sot. Am. 32 (1942) 61. L.L. Foldy, Phys. Rev. 67 (1945) 107. M. Lax, Rev. Modern Phys. 23 (1951) 287. M. Lax, Phys. Rev. 85 (1952) 621. V. Twersky, J. Opt. Sot. Am. 69 (1979) 1567.
Communications
117 (1995) 533-540
[IO] A. Zshimaru and Y. Kuga, J. Opt. Sot. Am. 72 ( 1982) 1317. [ 111 G. Zaccanti and P Bruscaglioni, J. Mod. Optics 35 (1988) 229. [12] A. Deepak and M.A. Box, Appl. Optics 17 (1978) 2900. [13] A. Deepak and M.A. Box, Appl. Optics 17 (1978) 3169. [ 141 W.G. Tam and A. Zardecki, Appl. Optics 21 (1982) 2405. [ 151 A. Zardecki and W.G. Tam, Appl. Optics 21 (1982) 2413. [ 161 L.B. Bangs, Uniform Latex Particles (Serva Feinbiochemica G.m.b.H. & Co. KG, Heidelberg). [ 171 Landolt-Bornstein, Optische Konstanten, II. Band, 8. Teil, 6. Auflage, eds. K.-H. Hellwege and A.M. Hellwege (Springer, Berlin, 1962) 9 3, pp. 505.
15 June 1995
CQs 32
OPTICS COMMUNICATIONS
ErSEVIER
Optics Communications
117 ( 1995) 532-540
Full length article
The attenuation of a coherent field by scattering H.-J. Schnorrenberg Ins&at fiir Medizinische
‘,
M. Hengstebeck,
Optik, Ludwig-Maximilians-Universitiir Received 30 November
K. Schlinkmeier
Miinchen, Barbarastr.
16, 80797 Miinchen,
1994; revised version received 3 February
Germany
1995
Abstract The intensity of a coherent field attenuated by scattering is described by an ordinary differential equation. This differential equation is derived in the same way as Lambert-Beer law of absorption, but the coherent forward scattering is taken into account. The analytical solution of the differential equation depends on two quantities of single scattering, concentration of scatterers and thickness of the medium. Measurements of coherent light transmitted through suspensions of latex microspheres confirm the resulting equation.
1. Introduction The problem of multiple scattering is important for many domains of physics and has been studied extensively. First, problems of multiple scattering have been treated by transport theory, based on the integrodifferential equation of Boltzmann [ l-51. In the treatment by transport theory interference phenomena are neglected. For that reason, multiple scattering was described by wave theory [ 6-91. Thereby the average field is deduced for a finite volume containing randomly distributed scatterers. The integral equation for the average field can in principle be solved directly [ 61. However, an equation of attenuation of intensity of a coherent field by scattering could not yet be derived by this formulation. In this paper multiple scattering is treated by wave theory, too. But instead of considering the wave function for a finite volume, the wave function is derived for an infinitesimale thin layer. For the thin layer the total scattered flux and the coherent scattered flux in 1To whom correspondence should be addressed. Present address: Kellererstr. 23b, 82256 Fiirstenfeldbruck, Germany. 0030-4018/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDIOO30-4018(95)00240-5
forward direction are determined. The difference of total scattered flux and coherent forward scattered flux is considered as a loss of the flux falling onto the thin layer. This treatment yields an ordinary differential equation for the attenuation of intensity of a coherent field. The analytical solution of this differential equation depends on quantities of single scattering, concentration of scatterers and thickness of the medium. Experiments of attenuation of light, propagating in suspensions of latex microspheres, are presented. The results can be well described by the presented formalism.
2. Theory The following treatment of the attenuation of a coherent field by scattering is based on elementary wave theory. With regard to the experiments, the treatment is done for scattering of light: a light beam with cross section A propagates within the scattering medium along the z-axis (Fig. 1) . A thin layer contains IZscattering particles. These particles can be distributed randomly or regularly. The 12vectors ri define the posi-
H.-J. Schnorrenberg
et al. / Optics Communications I1 7 (1995) 533-540
533
The product Z (z. ) g( k) do of incident intensity, differential scattering cross section and element of solid angle substitutes the radiant flux ZO(k)d3 of the single scattered light wave:
d@%(k)= I( Fig. 1. The scattered electromagnetic field seen from a point in the far zone. A light beam with cross section A propagates along the z-axis. A thin layer contains n particles, which scatter the electromagnetic field falling onto the layer. At a point P in the far zone the scattered
flux is calculated.
tions of the particles. The electromagnetic field E( z, t) enters the thin layer and is scattered by the particles. The radiant flux d@s of the scattered light waves on the area d3 at a point P in the far zone is calculated. Assuming, that the scattering particles are identical, the n electromagnetic fields Ei of the scattered light waves at point P can be expressed by
&(Q,k,l)
=&(k)
COS[Q.(k,
-k)
-wt].
(1)
In Eq. ( 1) the wave vector k points from the scattering particles to the point P in the far zone, EO(k) denotes the amplitude of a single scattered lightwave at point P. The electromagnetic field ES (k, t) at the point P is the superposition of the n scattered fields: Es(k,t)
= Eo(k)
-&os,r,.(k, i=l
-k)
-wtl
.
(2)
Averaging in time and squaring of Eq. (2) yields the intensity Is (k) of the scattered light: Is(k)
=Zo(k)
F:kcos[(ri-rj). i=l j=1
jti
n+)P:kcosI(ri-rj)*(kz ;=1
j=l
(k, -k)]
-k)l
1 .
(3) Multiplying Eq. (3) by the element of area d3 at point P, the scattered flux d@s (k) is obtained:
=Z,,(k,t)
= Zs(k, t) d3
j=1 jti
n+kkcos((ri-q)-@l-k)) i=l
1
d3. (4)
dR.
j+i
The scattered radiant flux d@s( k) can be split into two parts d@.$,‘)(k) and d@p) ( k) : d@“‘(k) S
= Z(z)~(k)ndO,
dGc2’(k) S
= Z(z)a(k) COS((ri-rj)
j=l
(6)
.(k,
-k))dO.
(7)
j=1 j+i
Commonly, d@$‘) (k) is called the incoherent scattering term. The incoherent scattering term d@i’) (k) represents the scattered radiant flux as a sum of iz single scattering events. The second term d@$) (k) takes the interference of the scattered light waves into account. Therefore, the term d@ (k) represents the change of the angular distribution of the scattered flux by interference. In the following context this term is called the interference term. First, the total scattered flux of the layer is derived, which yields the differential equation of LambertBeer law. Although the derivation is well-known, it is carried out in detail to demonstrate the method. Later, the coherent scattered flux in forward direction is derived in the same way, which yields an additional differential equation. In order to get the total scattered flux @s, the incoherent scattering term and the interference term are integrated over the unit sphere: Qbs= I d@“‘(k) S J
4?T
d&(k)
1 (5)
j=l i=l
n+~~;cos((r,-r,),(k,k))
+ I dGc2’(k) s J
*
(8)
477
The value of total scattered flux is not affected by interference, because flux cannot be generated or de-
stroyed by interference. Therefore, the integration of the interference term d@i2) over the whole solid angle 4~ yields no flux. So, the total scattered flux is
534
H.-J. Schnorrenberg et al. /Optics Communications 117 (1995) 533-540
given by the integral of the incoherent @$‘) (k) over the unit sphere:
Gs =
scattering term
d@)(k)
s
Dividing this equation by the cross section A of the light beam, the differential dls (z ) for the intensity of a coherent field decreased by the total scattered flux is obtained:
4n
dk(z>
=
Z(z>~(k>ndfi=
I(z)u
s
In Eq. (9) UT denotes the total scattering cross section for one scatterer. If the scatterers are distributed statistically in the medium, rt represents the average number of scattering particles within the volume V. In order to form the differential equation, it is neccessary to replace discrete particles by a homogeneous material. The homogeneous material must have the same scattering characteristics as the particles. So, in the replacement the differential scattering cross section c(k) is not changed. The homogeneous material is represented by the concentration. The concentration c depends on the volume VPof a scattering particle and the number n of the scattering particles within the volume V and is defined by
c vvp:
=
-Z(z)pscdz
(15)
.
If the total scattered flux is considered as a loss of the coherent field , the differential equation (15) would yield Lambert-Beer law. But the coherent forward scattered lightwave is superposed with the unscattered lightwave, and so, the coherent forward scattered flux must not be considered as a loss. As a consequence, the differential equation of the attenuation by scattering consists of two parts: differential equation ( 15), which describes the total scattered flux, and an additional term, which takes coherent scattering in forward direction into account. In Eq. (5) the scattered flux is defined per solid angle in dependence on scattering direction. For the coherent scattered flux in forward direction the scattering direction k and the propagation direction k, of the lightwave falling onto the layer are equal. So the cosine in Eq. (5) becomes one, and the coherent scattered flux in forward direction can be written as d@u =Z(z)~(k)
j‘JkldR i=l j=l
= Z(z)a(k,)n2df2.
@s =Z(z)
$?cv.
(11)
P
The ratio of total scattering cross section and volume of a particle is identical to the extinction coefficient per unit concentration z+: f_&=%.
(12) P
The finite volume V is replaced by the infinitesimale volume Adz : (13)
=Z(z),uscAdz.
The flux falling onto the layer is decreased by scattering. Therefore, the change d& (z ) of the incident flux is given by d&(z)
d
(10)
The number n of scatterers in Eq. (9) is replaced by
d%(z)
d&(z) ~
(9)
4n
V c:=n-?-. V
=
= -d@,(z)
= -Z(z)pscddz.
(14)
(16)
Thereby the forward direction is denoted by the subscript 0, which indicates scattering angle 6 = O. In order to get the finite flux in forward direction of single scattering, Eq. ( 16) has to be integrated over a solid angle 0c around the forward direction. However, it is important to distinguish the scattered flux in forward direction of single and multiple scattering. For a single scattering process the intensity of scattered light depends on the distance from the scatterer 2 . However, the coherent forward scattered light is the superposition of many scattered lightwaves. So the forward scattered light has to be represented by a plane wave. In contrast to the single scattered light wave, the intensity of a plane wave is constant. In order to get a representation of the multiple scattered *That is expressed inversely propoaional
by the so-called to ?).
inverse
square
law (I is
H.-J. Schnorrenberg et al. /Optics
flux in forward direction independent of solid angle, the function 5(a) of a solid angle & is defined by
J
u(k)
da.
(17)
no
The function 5(Q)) is defined in such a way, that for & = 47r the total scattering cross section is obtained. For forward scattering an solid angle L$ around the forward direction k, is chosen. Then the limit for K& + 0 is considered:
Ja(k) da.
(18)
& The limiting distribution:
process
J
by the 15
Ju(k)
4%~0 = lim @l--10 00 no = 4~
can be expressed
a(k)6(k
dG
- k,)dL? = 4m(k,)
.
(19)
4w
To ease the understanding of the quantity au, an ensemble of isotropic scatterers is considered. The single scattered light of each scatterer is seen as an elementary lightwave. Applying the construction of Huygens, in a high dense medium only the coherent transmission in forward direction is observable. That means, due to interference of the scattered lightwaves the total scattered flux propagates only in the forward direction. This is expressed by Eq. (19). So, the quantity ~a represents the scattered flux in forward direction per particle, if the medium is so dense, that only coherent transmission is observable. As this example shows, the quantity ~0 represents both scattered flux in forward direction per particle and the interference of scattered lightwaves. So, in general it makes no sense to interpret ~0 as the forward scattered flux per particle. Using Eq. ( 19) the finite scattered flux in forward direction is obtained from Eq. ( 16) :
Communications 117 (1995) 533-540
scatterers has to be replaced by a tion. In Eq. ( 10) the number n of the volume V was expressed by and the volume VP of a scattering 1 n=---_cV=v,
(20)
In Eq. (20) the scattered flux @Odepends on the square n* of the number of scatterers. For the derivation of the differential equation the square of the number of
v,
continuous distributhe scatterers within the concentration c particle:
J
(21)
cdV.
V
The number 12 of scatterers depends linearly on the volume V. Therefore, the derivation of a differential equation is simple: dn = $cdV.
(22)
P
However, the square of the number n* does not depend linearly on the volume V. Squaring Eq. (22) makes no sense. To get the correct formulation of the differential equation, the quantities n and rr* have to be interpreted statistically. In Eq. (21) n represents the mean value of the number of scatterers within the volume V. Therefore the concentration c has to be considered as a continuous distribution. The factor l/V, normalizes the distribution (0 5 c < 1) . Hence, the quantity n* in Eq. (20) has to be interpreted as the second moment of the distribution: (23) Although the quantities n and n* are interpreted statistically, no assumption is made on the distribution of the particles in the medium. The particles can be distributed regularly or randomly within the volume. For a random medium, n* represents the average of the square of particles in the layer (and not the square of the average number!). Consequently, the continuous form of Eq. (20) is given by @a =Z(z)
WV. v,
Analogous
to Eq. ( 12), the abbreviation
pJ
@o =Z(z)r?q.
1
535
:=
(24) pa is used:
F. P
In Eq. (24) the finite volume V is substituted infinitesimale volume Adz : dGo = Z(z)~ac*ddz.
(25) by the
(26)
536
H.-J. Schnorrenberg
et al. /Optics
Dividing Eq. (26) by the cross section A of the light beam, the differential dZa(z) for the intensity of the forward scattered light is obtained: dZ,,(z) = 9
= Z(z)pac*dz
.
(27)
The complete differential dZ ( z ) for the intensity of a coherent field attenuated by scattering is the sum of the differentials dZs (z ) (Eq. ( 15) ) and dZo( z ) (Eq. (27)): dZ(z)
=dZs(z)
+dZo(z).
(28)
Consequently, the differential equation for the intensity of a coherent field attenuated by scattering is given by dZ(z)
= (-,usc
+ POC*) Z(z) dz .
(29)
Integration of this equation over the path up to the thickness d of the medium yields Z(c,d)
= exp( -pscd
+ ,uac2d)Za.
(30)
The attenuation of the intensity depends both on thickness d of the scattering medium and concentration c. In contrast, the attenuation by absorption described by Lambert-Beer law depends only on the product cd. The attenuation of intensity by scattering can be written in the same form as Larnbert-Beer law. Therefore in Eq. (30) the extinction coefficient ,us is replaced by a function of concentration ,Gs (c) :
Communications
117 (1995) 533-540
attenuation by scattering an implicit assumption was made by using the quantities aa and or for the multiple scattered lightwave in forward direction. These quantities are defined for an incident plane wave. But only in the far zone, the forward scattered lightwave can be considered as a plane wave. That means, the distance between the scatterers must be large enough, so that the far field approximation is valid. Hence, the (average) distance D between the scattering particles with size a must satisfy following condition, wellknown from Fraunhofer diffraction: a<
Thereby A denotes the wavelength of the light. The condition (33) can be written in dependence on concentration c and particle size a. Therefore in the definition ( 10) of concentration the volume per particle vn is approximated by the cube D3 of distance between the particles. Also, the volume of one scatterer VPis substituted by the cube u3 of particle size: V,
:=.s(l*-
This substitution Z(c,d)
EC)
=Ps(l
- zc).
a3
(34)
C=ny=D3*
Eq. (34) is solved with respect to the distance D:
So, using Eq. (35) the condition approximation can be written as
a
(33)
(31)
a lf4 qz.
(33)
for far field
(36)
Therefore a condition for the validity of Eq. (32) for the attenuation by scattering is
yields
=Zoexp[-@s(c)cd]
;&.
(37) fi
=Zaexp[-,s(l
- Ec)cd]
.
(32)
For low concentration Eq. (32) yields Lambert-Beer law, and attenuation by scattering and absorption can be described by the same formula. With increasing concentration, the coherent forward scattered light is not negligible and the attenuation by scattering deviates from Lambert-Beer law. Obviously, Eq. (32) fails, if (,xo/,u~)c becomes greater than one. However, in the derivation of the
According to condition (37)) the maximal concentration, for which Eq. (32) is valid, depends on particle size a. For small particles this maximal concentration is higher than for large particles.
3. Experiments In this part of the paper the attenuation by scattering is described by the extinction. The extinction E( c, d)
H.-J. Schnorrenberg SP
LI
D
et al. /Optics
LZ
-f---f-cf_fFig. 2. Experimental setup. The light transmitted through the sample (S) is analyzed by a polarizer (P). A spatial filter of two lenses (f = 100 mm) with a moveable diaphragm (D = 200 pm) allows to detect transmission in dependenceon the scattering angle. A photomultiplier (PMT) registrates the intensity of the filtered transmission.
is defined as the (natural) logarithm of the ratio of the incident intensity la and the coherent transmitted intensity Z(c, d) : E(c,d)
=ln(l(_C,\ \ 10
=-p.scd+poc2d.
(38)
/
In order to test Eq. (38), the intensity of coherent transmitted light attenuated by a scattering medium is measured. The measurements require a technique to distinguish incoherent multiple scattered light from the coherent transmission. A commonly used method is to reduce the field of view (FOV) of detection [ 10,111. Thereby incoherent scattered radiant flux, which illuminates a wide solid angle, is reduced. In contrast, the coherent transmission has the same divergence as the incident light beam. For a moderate number of scattering particles, the incoherent scattered radiant flux in the FOV is negligible. In this case, the detected signal corresponds to the intensity of coherent transmitted light. However, for high number of scatterers, incoherent multiple scattered light can not be neglected in the FOV of detection, and the signal does not correspond to the coherent transmitted flux [ 12-151. Therefore, for the attenuation measurements another technique is used here, taking polarization characteristics into account. By this method incoherent scattered light is suppressed even for high concentrations. The experimental setup is shown in Fig. 2. The sample is illuminated by a linearly polarized laser beam (HeNe, A = 633 nm, P = 14 mW) . Behind the sample the transmitted light is analyzed with a polarizer. A spatial filter, consisting of two lenses and a diaphragm, is placed behind the sample, too. The diaphragm in the focal plane of the first lens is mounted on a translation stage. This moveable diaphragm allows to measure the transmission as a function of scattering an-
Communications
117 (199.5) 533-540
531
gle. A photomultiplier positioned in the focal plane of the second lens registrates the flux. The four-f-setup provides, that at the photomultiplier the same spot is illuminated for all positions of the diaphragm. Furthermore an opal glass diffuser in front of the window of the photomultiplier provides homogeneous irradiation of the photocathode. Lock-in technique is used to improve signal-to-noise ratio. Uniform polystyrol (latex) spheres manufactured by Dow Diagnostics are used as scatterers. At the used wavelength absorption by polystyrol can be neglected [ 161. The index of refraction of polystyrol is known [ 171. Latex spheres are supplied in suspensions with a concentration of 10%. For measurements with lower concentrations, these suspensions are simply diluted with water. Measurements with higher concentrations than 10% are not carried out. For each diaphragm position the transmitted light is measured twice: In the first measurement, the transmission axis of the polarizer is parallel, in the second one perpendicular to the plane of polarization of the incident lightwave. Typical measurement curves are shown in Fig. 3. For a scattering angle greater than fO.1 degree, only scattered light is detected. For scatterers much smaller than the wavelength, the registrated intensity of scattered light is equal for both polarizer adjustments. That means, within accuracy of measurement the scattered light is nonpolarized. In contrast, with scatterers much larger than the wavelength, the measured intensity of scattered light depends on the adjustment of transmission axis of the polarizer. With the transmission axis parallel to the plane of polarization of incident light, more intensity is measured, than with an orthogonal adjustment. Consequently, light multiple scattered by large particles is partially polarized 3 . For scattering angles smaller than fO.1 degree the signal of the orthogonal polarizer adjustment has the same value as for large scattering angles. In contrast, the curve for the parallel polarizer adjustment shows a peak within the central range. The width of this central peak depends neither on concentration of scattering
3This can be explained by Mie theory: Particles larger than the wavelength scatter light mainly about small angles, which preserves polarization. In contrast, small particles scatter light over a wide angle range. This disorders polarization, and so multiple scattered light is nonpolarized (within accuracy of measurment)
538
H.-J. Schnorrenberg
latex
et al. /Optics
272
Communications
117 (1995) 533-540
nm
latex
0
1.09
F
0 --I I”
-0.5 scattering
1
’
I”’
1
0.0
I I 0.5
angle
-0.5
[o ]
r
0.0
scattering
’
I 0.5
angle
["I
Fig. 3. Measurements of the intensity of light transmitted through latex suspensions in dependence on scattering angle. The curve for the polarizer adjusted perpendicular to the plane of polarization of incident light is constant over the whole registrated angle range (thick curve). In contrast, the curve for the parallel polarizer adjustment shows a central peak (thin curve). The amplitude A of the peak corresponds to the intensity of coherent transmission. The registrated intensity of light, scattered by particles much smaller than the wavelength, is for large scattering angles independent of the polarizer adjustment (left figure).
but corresponds to the divergence of the incident light beam. Obviously, the central peak corresponds to the coherent part of the transmitted light. So, the intensity of coherent transmission can be measured by determining the amplitude of the central peak. For scattering particles much smaller than the wavelength the amplitude of the peak is determined in a more accurate way: Transmission is not measured for a wide angle range, but only for scattering angle zero. Then the difference of the measured signals for both polarizer adjustments yields the amplitude of the peak (see Fig. 3). In this way errors due to the movement of the diaphragm do not occur. With this technique intensity of coherent transmission can be measured even for highly scattering media. The accuracy of measurement is limited by signal-to-noise ratio of detection and errors due to the preparation of the scattering samples. In order to measure the deviation of extinction from Lambert-Beer law, the concentration must be sufficiently high (E$ (38) ) . However, the intensity of coherent transmission decreases with increasing concentration and sample thickness. That means, for high concentrations the signal of the intensity of coherent transmission is only measurable for sufficiently small sample thicknesses. For several sample thicknesses (between 1 mm and 10 mm) the extinction by latex spheres (c$= lOO& 10 nm, 272f3 nm and 1.09 pm) was measured. For high concentrations, the measured intensity of particles
nor on sample
thickness,
Table 1 Measured and calculated extinction coefficients for 100 nm latex and 272 nm latex suspension
Measured Calculated
Latex 100 nm
Latex 272 nm
ps [cm-‘]
m [cm-‘]
ps [cm-‘]
m [cm-‘]
1.2 x lo3 1.1 x lo3
1.8 x lo3 1.9 x IO3
9.5 x 103 9.6 x lo3
4.5 x 104 4.4 x lo4
For the numerical calculations of the scattering coefficient /.~s and the constant ~0, the refraction index of latex (polystyrol 1171) and water were taken as n=1.59 and n=1.33, respectively.
coherent transmission deviates from Lambert-Beer law about some orders of magnitude. However, the dependence of extinction on concentration can be adequately represented by a polynom of second degree. For reasons explained above, the most precise results are obtained with latex spheres smaller than the wavelength
and small
sample
thicknesses.
Measure-
ments with Latex spheres with diameter of 100 nm in a 2 mm cuvette and with a diameter of 272 nm in a 1 mm cuvette are very accurate. The coefficients of least-square-fit of these measurements agree with the scattering coefficient ps and the constant P.O (as defined in Eq. (25)) calculated by Mie theory (Table 1). So, the measurements confirm the derived formalism.
H.-J. Schnorrenberg et al. /Optics Communications 1 I7 (1995) 533-540 100
latex
nm
1j\
..
I”“““‘1 0
IXU
-y\ -20
...
: I”““‘I’I’I”““‘I”
10
5
concentration
272
latex
539
0
1
2
concentration
[vol%]
[vol%]
Fig. 4. The extinction deviates from the Lambert-Beer law (dashed line), but can be adequately represented by a polynom of second degree (solid line). The sample thickness for the measurement with the 100 nm latex spheres was 2 mm, for the 272 nm latex spheres 1 mm.
4. Summary
Averaging this equation particles yields
In this paper an equation for the intensity of a coherent field attenuated by scattering was derived. Thereby the coherent forward scattering was taken into account, which yields a deviation from Lambert-Beer law for high concentrations of scatterers. The resulting equation was confirmed by measurements with light attenuated by suspensions of latex microspheres.
Es(k,
We thank Malin Vonier for the initial tests on the experimental setup.
g
cos[ri
of the
. (k, - k) - wt] .
i=l
(A.21 Therefore the differential scattering cross section a(k) in Eq. (5) has to be replaced by the average differential scattering cross section c+(k) : d@s(k)
Acknowledgements
t) = Eo(k)
over the distribution
= I(z)
a(k) cos[(q
evaluation
-rj).
(k, -k)]
da. I
jiti (A.3)
Appendix A. Distribution
of different scatterers
For the derivation of the attenuation by scattering only identical scatterers were considered. The generalization for a distribution of different scatterers is simple. Instead of Eq. (2) for a distribution of different scatterers the superposition of the scattered electromagnetic field has to be written as
So in Eq. (30) for the attenuation by scattering, only the quantities UT and ~a have to be averaged over the distribution of the different scatterers. For the average value of the quantities ,us and pa the average volume s of a scatterer of the distribution has to be considered, too: (A.4)
(A.3 Es(k,t)
= k@‘(k)
cost ri . (k, - k) - wt] .
i=l (A.1)
Thereby the index i of the electromagnetic fields Ef’ ( k) represents the different scatterers in the layer.
So the equation of the attenuation of intensity distribution of scatterers is given by I( c, d) = exp( -&cd
+ F;cc*d) IO.
by a
(A.61
540
H.-J. Schnorrenberg
et al. /Optics
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Communications
117 (1995) 533-540
[IO] A. Zshimaru and Y. Kuga, J. Opt. Sot. Am. 72 ( 1982) 1317. [ 111 G. Zaccanti and P Bruscaglioni, J. Mod. Optics 35 (1988) 229. [12] A. Deepak and M.A. Box, Appl. Optics 17 (1978) 2900. [13] A. Deepak and M.A. Box, Appl. Optics 17 (1978) 3169. [ 141 W.G. Tam and A. Zardecki, Appl. Optics 21 (1982) 2405. [ 151 A. Zardecki and W.G. Tam, Appl. Optics 21 (1982) 2413. [ 161 L.B. Bangs, Uniform Latex Particles (Serva Feinbiochemica G.m.b.H. & Co. KG, Heidelberg). [ 171 Landolt-Bornstein, Optische Konstanten, II. Band, 8. Teil, 6. Auflage, eds. K.-H. Hellwege and A.M. Hellwege (Springer, Berlin, 1962) 9 3, pp. 505.