Optics and Spectroscopy, Vol. 96, No. 5, 2004, pp. 804–809. Translated from Optika i Spektroskopiya, Vol. 96, No. 5, 2004, pp. 874–879. Original Russian Text Copyright © 2004 by Gavrilenko, Sorokin, G. Jandieri, V. Jandieri.
PHYSICAL AND QUANTUM OPTICS
Evolution of the Angular Power Spectrum of Scattered Radiation from a Point Source upon Propagation in a Turbid Medium V. G. Gavrilenko*, A. V. Sorokin*, G. V. Jandieri**, and V. G. Jandieri** * Nizhni Novgorod State University, Nizhni Novgorod, 603950 Russia ** Georgian Technical University, Tbilisi, 380075 Georgia Received June 6, 2003
Abstract—The propagation of radiation from a point source through a plane layer of an absorbing medium with smooth random permittivity inhomogeneities is considered for the case in which the source and a receiver are spaced by different distances from the layer boundaries. The angular power spectra of the scattered radiation are calculated by the method of statistical modeling for different values of the layer thickness, positions of the source and the receiver relative to the layer, and absorption in the layer. The results for the moments of the angular power distribution obtained earlier in the small-angle approximation are fully confirmed. The transformation of the angular power spectrum upon variation of the source or receiver position with respect to the layer is analyzed for the first time. © 2004 MAIK “Nauka/Interperiodica”.
INTRODUCTION The absorption of the energy of light waves upon scattering in chaotic media is known to affect the statistical characteristics of the waves considerably [1, 2]. In the case of an asymmetric formulation of the problem, scattering may significantly distort the angular power spectrum of the scattered radiation [3–5]. Practically, it is of interest to consider the case in which a point source and a receiver of radiation are positioned on either side of a layer of a chaotically absorbing medium. This arrangement corresponds, for example, to examination of dense clouds in transmitted light or to sensing of ocean water. In such a formulation of the problem, the angular power spectrum of the received scattered radiation significantly depends not only on the medium properties, but also on the mutual arrangement of the source and the receiver with respect to the layer. This problem was solved earlier in the smallangle approximation using the method of complex geometrical optics [6]. The solution in the small-angle approximation is simple enough and physically illustrative, but the domain of its applicability is restricted by the assumptions involved. The most important among these assumptions is that of a relatively thin layer of an absorbing chaotic medium, in which, although multiple scattering occurs at smooth inhomogeneities, the width of the angular spectrum and the shift of its maximum with respect to the direction to the source remain small. At the same time, the solution of this problem beyond the assumptions mentioned above is of scientific and practical significance. However, analytical methods for such solution are not yet available. Therefore, information about the properties of the scattered radiation can be obtained only experimentally or through numerical
calculations. In this paper, the problem is solved numerically using the method of statistical modeling. This approach provides far more complete information about the statistical characteristics of the scattered radiation. Thus, this paper continues and generalizes investigations started earlier in [6]. FORMULATION OF THE PROBLEM AND METHOD OF SOLUTION Suppose that a point source (Fig. 1) is in a homogeneous nonabsorbing medium with the permittivity ε = ε0 at a distance L1 above a plane layer of a randomly inhomogeneous absorbing medium with the thickness Z. The permittivity of this layer is ε = ε 0 + iε'' + ε 1 ( r ), where ε1(r) is a random zero-mean variable that describes fluctuations of the real part of the permittivity inside the layer and ε'' is the imaginary part of the permittivity, describing wave absorption in the layer. The source has a conical direction pattern, whose boundaries are shown by dashed lines in Fig. 1. The apex angle of the cone is assumed to be so large that, at L1 ≈ L2 ≈ Z, the illuminated area at the top of the layer permits the scattered radiation to propagate along three directions 1–3, indicated by the dashed-and-dotted lines in Fig. 1. A receiver is located in the homogeneous nonabsorbing medium ε = ε0 in the plane xz at a distance L2 from the layer. From here on, ε0 is assumed to be unity without loss of generality. The straight line drawn from the source to the receiver makes an angle z with the axis θ. In what follows, this angle and the apex angle of the light cone are considered to be fixed as the
0030-400X/04/9605-0804$26.00 © 2004 MAIK “Nauka/Interperiodica”
EVOLUTION OF THE ANGULAR POWER SPECTRUM
source and the receiver move with respect to the layer. Let the characteristic size of inhomogeneities in the layer be much larger than the radiation wavelength and ε1 Ⰶ ε0. Therefore, the problem is to find the angular power spectrum of the scattered radiation. As was mentioned above, this problem was solved in [6] in the small-angle approximation using the method of complex geometrical optics. With this method, we can solve the phase transfer equation using the perturbation technique and find the correlation function. The angular power spectrum of the scattered field was determined as a Fourier transform of this correlation function. For the case of strong phase fluctuations, in which the angular spectrum has the Gaussian form, the equations have been found for the statistical moments of the angular spectrum, namely, for the shift of the centroid and for the variance in both coordinate planes. The dependences of the moments on the mutual arrangement of the source and the receiver relative to the layer were obtained by numerical calculations using these equations for the Gauss model of the spectrum of permittivity fluctuations. In addition to the above restriction on the layer thickness, the domain of applicability of this solution is restricted by the assumption about the Gaussian character of both the spectrum of permittivity fluctuations and the angular power spectrum of the scattered radiation. To obtain reliable information about the behavior of the angular power spectrum without these restrictions, it is necessary to use the most general ideas about the processes of wave propagation and scattering in a randomly homogeneous medium. Statistical modeling (the Monte Carlo method) is an approach based on these general ideas. Therefore, it has been selected for solving the problem formulated. A significant disadvantage of statistical modeling is its low rate of convergence, which makes computations very time-consuming. The literature devoted to both theoretical and applied aspects of the Monte Carlo method is voluminous. We employed the so-called weighted modification of the Monte Carlo method. In this version, as the radiation propagates along the model path (ray tube), absorption is taken into account as some parameter that contributes to the effect of this propagation path on the output angular power distribution of the scattered radiation. This version of the Monte Carlo method is characterized by a higher speed as compared to the classic scheme [7]. However, practical experience has shown that implementation of the weighted algorithm in higher-level languages does not enable acceptable productivity (IA-32 Pentium III 733 MHz computer), and only the development of the procedure in an assembly language has helped in solving the problem formulated above. OPTICS AND SPECTROSCOPY
Vol. 96
No. 5
2004
805 x
θ ε = ε0
L1
ε = ε0 + iε'' + ε1 (r) Z 3
2
1
ε = ε0
L2
z Fig. 1. Formulation of the problem (see explanations in the text).
NUMERICAL ANALYSIS OF THE POWER SPECTRUM To get the most reliable idea about the dependence of the angular spectrum of the scattered wave on the arrangement of the source and the receiver, the layer thickness, and absorption in the layer, the spatial spectrum of permittivity fluctuations was simulated as follows: 2 –2.3 2 ------- k 0 , k ∈ 0, ------- k 0 90 90 Φ ( k ) = C –2.3 2 - k , 2k 0 k , k ∈ -----90 0 0, k > 2k , 0 where k is the modulus of the difference between the wave vectors of the singly scattered wave and the initial wave and k0 is the modulus of the wave vector of the electromagnetic radiation in a vacuum. The coefficient C can be determined from the normalization condition +∞
+∞
∫ ∫ ∫ Φ ( k ) dk = 4π ∫ k Φ ( k ) dk 2
–∞
= 〈 ε1 〉 , 2
0
and, for the chosen model of the spectrum of fluctua2 – 0.7 tions, C = 0.045189 〈 ε 1 〉 k 0 . From the theory of single scattering of waves, it is known [8] that the single scattering phase function σ(α, ϕ) is connected with the spatial spectrum of fluctuations as follows: σ ( α, ϕ ) = πk 0 Φ ( k )/2, 4
GAVRILENKO et al.
806
for the scattering and the absorption, respectively. The parameter σs is the integral of the scattering phase function over all possible scattering directions:
I, rel. units 1.0
σs = 0.8
0.6
0.4
2
0.2 1 0
0.4
0.8
1.0 sx
Fig. 2. (1) Single scattering phase function and (2) cross section of the angular power spectrum of the scattered radiation I(sx , sy = 0) in the absence of absorption at σsZ = 40, σsL1 = 210, and σsL2 = 80.
where α and ϕ are the zenithal and azimuthal angles between the directions of propagation of the initial and singly scattered waves. Since the scattered medium is assumed to be statistically isotropic (the spatial spectrum of fluctuations depends on the modulus of the difference between the wave vectors), the scattering phase function is independent of the azimuthal angle ϕ: σ(α, ϕ) = σ(α). In its turn, 1 α k = --- k 0 sin ---, 2 2 consequently, 1 4 α σ ( α ) = πk 0 Φ --- k 0 sin --- /2. 2 2 This scattering phase function describes rather well the processes of single scattering of light in seawater, as well as infrared radiation from water droplets in clouds and in living tissue. In the algorithm of statistical modeling, this dependence is the probability density that the ray turns by the zenithal angle α in a particular scattering event. The azimuthal angle of ray turning in the plane normal to the wave vector in the ray tube before the scattering event is uniformly distributed in the range from 0 to 2π. In all the numerical experiments, θ amounted to 36.89° (sin θ = 0.6), and the absorption was determined by the so-called photon survival probability Λ = σs /(σs + σa) [2], where σs and σa are the extinction coefficients
∫ σ ( α, ϕ ) dο = ( πk ∫ Φ ( k ) dο )/2, 4 0
where dο is the infinitely small elementary solid angle in which the scattering occurs. In Fig. 2, the projection of the unit vector of the wave normal sx = kx /k onto the x axis is plotted as the abscissa, and the dashed curve shows the cross section of the spectrum of the singly scattered radiation I(sx , sy) s y = k y /k = 0 (in fact, the single scattering phase function) in the absence of absorption. The cross section is normalized to its maximum value, and the data are presented for a plane wave incident at the chosen angle of 36.89°. In the radiative transfer theory, the parameter reciprocal to the extinction coefficient for scattering corresponds to the mean length of the rectilinear path of the radiation between two events of scattering from chaotic medium inhomogeneities. Therefore, the parameter σa = k0ε'' has the meaning of the inverse path length at which the amplitude of radiation decreases e (e = 2.71…) times due to absorption in the medium. For comparison with the results of [6], we carried out a series of calculations for the layer thickness σsZ = 20 and the absorption corresponding to the survival probability Λ = 0.5. The scattering of radiation to large angles at such a layer thickness is quite insignificant; therefore, the condition for correct use of the smallangle approximation is fulfilled and comparison with the results of [6] is valid. The simulation showed that, despite the strong qualitative difference between the spectra of permittivity fluctuations obtained in [6] and in this work, the spatial power spectrum obtained from the numerical experiment is nearly Gaussian. The dependences of the shift of its centroid ∆kx and the vari2 2 ances 〈 k x 〉 and 〈 k y 〉 on L1 and L2 in both coordinate planes are qualitatively identical to the analogous results from [6]. The results of numerical simulation for the layer thickness σsZ = 40 are plotted in Figs. 2–7. Note that, in this case, the angular power spectrum is significantly distorted even in the absence of absorption. This can be judged from Fig. 2, where the solid curve shows the cross section of the angular power spectrum at σsL1 = 210 and σsL2 = 80. The right peak corresponds to the radiation having passed, on the average, along path 3 in Fig. 1. This peak is formed owing to two circumstances: (i) The scattering in this direction takes a smaller amount of energy from the mean field because of the shorter path in this layer. (ii) The radiation that propagates, on the average, along path 3 is multiply scattered by chaotic inhomogeneities, and, near the layer bottom, there is some probability that the next scattering event directs the radiation to the observation point. OPTICS AND SPECTROSCOPY
Vol. 96
No. 5
2004
EVOLUTION OF THE ANGULAR POWER SPECTRUM
807
I, rel. units 1.0 3 0.9
1 2
0.8
4
0.2 0.1 –0.8
–0.6
–0.4
–0.2
0
0.2
0.4
0.6
0.8
1.0 sx
Fig. 3. Evolution of the angular power spectrum of the scattered radiation I(sx , sy) integrated over sy in the range from –0.05 to 0.05 with increasing L1 at σsZ = 40, σsL2 = 80, and the photon survival probability Λ = 0.5: σsL1 = (1) 0, (2) 80, (3) 280, and (4) ∞.
The presence of absorption leads to a more considerable variation in the angular spectrum of the received radiation. Figure 3 illustrates the evolution of the spectrum (photon survival probability Λ = 0.5) with increasing height of the source L1. In this figure, the parameter sx is plotted as the abscissa, and the integral radiation power in the range of sy from –0.05 to 0.05 for different L1 is plotted as the ordinate. All the spectra are normalized to their maximum values. The separation of the receiver from the bottom of the layer remains constant, σsL2 = 80. The angular distribution (Fig. 3, curve 4) was obtained from numerical simulation of incidence of a plane wave onto the layer, which corresponds to the limiting case L1 ∞. It can be seen from Fig. 3 that the angular spectrum has two or three peaks differing in amplitude, position, and angular width. Their values and coordinates on the axis of projections are determined by the mutual arrangement of the source and the receiver with respect to the layer. Here we can separate the three most characteristic paths of propagation of the scattered radiation, each leading to formation of a peak (Fig. 1). It is quite obvious that, at a low height of the source, the wave propagation along path 1 is impossible. At a high height of the source, the intensity of the radiation propagating along path 3 is very low because the turning angle needed to reach the receiver upon OPTICS AND SPECTROSCOPY
Vol. 96
No. 5
2004
leaving the layer is large. The angular coordinates of all the peaks are in good agreement with the corresponding directions in Fig. 1. As far as we know, such a character of the angular power spectrum of the scattered radiation has not been noticed earlier. M[sx] 0.8 0.6
1 3
0.4 0.2
0
2 100
200
300 400 L1, in units of σZ
Fig. 4. Position of the centroid M[sx] of the angular distribution I(sx , sy) versus the height of the source L1 at σsZ = 40: σsL2 = (2) 0 and (1, 3) 80; Λ = (1, 2) 0.5 and (3) 1. The dashed line indicates the value of M corresponding to ∞. L1
GAVRILENKO et al.
808 D[sx] 0.15
0.10
1
0.05
2 0
100
200
300 400 L1, in units of σZ
Fig. 5. Variance D[sx] of the angular distribution I(sx , sy) versus the height of the source L1 at σsZ = 40 and σsL2 = 80: Λ = (1) 0.5 and (2) 1. The dashed line indicates the value of ∞. D corresponding to L1
M[sx] 0.8 0.6
2 1
0.4 0.2
occur in a layer of such a thickness. These dependences of the statistical moments of the angular spectrum on the source height above the layer are qualitatively similar to the analogous dependences obtained in [6], and all the effects of transformation of the angular spectrum that were found in [6] (shift of the mean, nonmonotonic increase of the variance, and anomalous broadening of the angular spectrum) are much more pronounced than in the case of small-angle scattering. These effects are caused by the radiation scattered to large angles with respect to the initial direction of propagation. In the present paper, the strongly scattered radiation plays a significant role in formation of the output power spectrum due to the considerable thickness of the layer and the power-type model of the spectrum of permittivity fluctuations. In [6], the contribution of the strongly scattered radiation to the spectrum was insignificant; therefore, the transformation effects were less pronounced there. The numerical experiment has also confirmed the effect predicted in [6], namely, the shift of the centroid in the direction away from the normal in the initial range of L1 values for nonzero L2. It was shown rigorously in [6] that, at L1, L2 Ⰷ Z, the centroid first shifts away from the direction of the normal and then tends toward it, with the shift being absent when L1 = L2. Curve 1 in Fig. 4 illustrates these conclusions, with the only difference that the equality L1 = L2 is approximate, since, in the numerical experiment, L1 and L2 were comparable to the layer thickness Z. The dependences of the variance D[sx] of the angular spectrum on L1 are shown in Fig. 5. By definition, the variance is calculated as a second-order central statistical moment about sx: D[sx]
0
40
80
120 160 L2, in units of σZ
Fig. 6. Position of the centroid M[sx] of the angular distribution I(sx , sy) versus the distance to the receiver L2 at σsZ = 40 and Λ = 0.5: σsL1 = (1) 40 and (2) 0.
The dependence of the centroid position M[sx] on L1 for different values of L2 is shown in Fig. 4. The centroid is calculated as the first-order initial statistical moment of the resulting angular power spectrum I(sx , sy) with respect to the argument sx: 1 1
M [sx] =
1 1
∫ ∫ s I ( s , s ) ds ds / ∫ ∫ I ( s , s ) ds ds . x
–1 –1
x
y
x
y
x
y
x
y
–1 –1
Curve 1 corresponds to the case σsL2 = 80, while curve 2 is for the case L2 = 0 at Λ = 0.5. For comparison, Fig. 4 also shows the analogous dependence in the absence of absorption (curve 3, σsL2 = 80, Λ =1). It is clearly seen from the plots that a deep scattering mode still does not
1 1
=
∫ ∫ (s
1 1
– M [ s x ] ) I ( s x, s y ) ds x ds y / 2
x
–1 –1
∫ ∫ I(s , s ) ds ds . x
y
x
y
–1 –1
The numerical simulation shows that, in the presence of –1 significant absorption at L2 varying from 0 to 160 σ s , the dependences are similar to that shown by curve 1 (σsL2 = 80, Λ = 0.5) in Fig. 5, with only the position and the amplitude of the peak being varied. This is also confirmed by the nonmonotonic character of this dependence found in [6] for this formulation of the problem. The dependence of the variance on L1 in the absence of absorption is shown by curve 2 in Fig. 5 (σsL2 = 80, Λ = 1). As was mentioned above, the angular spectrum in this case has a small secondary peak (Fig. 2), and this favors an increase in the variance in the range of L1 values where this peak exists. However, because of the very long computation time needed to obtain statistically reliable results, we failed to calculate the depen–1 dence of the variance for L1 larger than 400 σ s . The dashed lines in Figs. 4 and 5 show the values of the centroid M[sx] and the variance D[sx] in the limiting OPTICS AND SPECTROSCOPY
Vol. 96
No. 5
2004
EVOLUTION OF THE ANGULAR POWER SPECTRUM D[sx] 0.06 1 0.04
0.02
0
2
40
80
120 160 L2, in units of σZ
Fig. 7. Variance D[sx] of the angular distribution I(sx , sy) versus the distance to the receiver L2 at σsZ = 40 and Λ = 0.5: σsL1 = (1) 40 and (2) 0.
case of incidence of a plane wave (that is, for L1 ∞) onto the layer with the thickness σsZ = 40 at an angle of 36.89° for the photon survival probability Λ = 0.5. These values are obtained from the distribution shown by the bold dotted curve (curve 4) in Fig. 3. A similar pattern is observed in the case when the receiver is removed from the bottom of the layer at a fixed height of the source L1. In this case, as was noted above, the positions of the peaks in the angular spectrum on the projection axis also correspond to the characteristic propagation paths of the scattered radiation shown in Fig. 1. All the features of the behavior of the dependence of the statistical moments M[sx] and D[sx] on the distance to the receiver L2 that were observed in [6], in particular, the nonmonotonic dependence of the centroid shift on L2 and the zero shift at L1 ≈ L2, are also observed and are more pronounced for the thicker layer (Figs. 6, 7). In Figs. 6 and 7, curves 1 correspond to the case in which σsL1 = 40, and curves 2, to the case in which L1 = 0 at Λ = 0.5. CONCLUSIONS Therefore, our calculations allow us to draw some important conclusions. For a relatively thin layer, in which the conditions of the small-angle approximation are met but multiple scattering occurs, the angular spectrum of the scattered radiation is nearly Gaussian, in spite of a significantly non-Gaussian character of the spectrum of fluctuations of the permittivity of the medium. Accordingly, all the geometrical optics predictions from [6] concerning the qualitative dependence of the statistical moments of the angular power spectrum of the scattered radiation on the mutual arrangement of the source and the receiver OPTICS AND SPECTROSCOPY
Vol. 96
No. 5
2004
809
with respect to the layer have been confirmed by the numerical experiment. For a relatively thick layer, the numerical experiment has shown that, in the case of a certain arrangement of the source and the receiver relative to the layer, the spectrum of the received radiation can be strongly distorted even without absorption. For a thick layer with absorption, we have analyzed how the angular spectrum and its statistical moments depend on the mutual arrangement of the source and the receiver with respect to the layer. It has been found that, under these conditions, the angular power spectrum is strongly non-Gaussian and has several peaks, corresponding to certain directions of the wave propagation. Depending on the position of the source or the receiver, the propagation of the radiation along these directions may be virtually impossible, and this determines the number of peaks, their position, and the fraction of the energy of the scattered radiation accumulated in them. In our opinion, this result has been obtained for the first time. The dependences of the statistical moments of the angular spectrum turned out to be qualitatively identical to those from [6], but some effects (shift of the mean and nonmonotonic increase in the variance) proved to be much more pronounced as compared to those in [6]. This is explained by the combined effect of the larger layer thickness and the power-type character of the spectrum of permittivity fluctuations. ACKNOWLEDGMENTS This study was supported in part by the Competition Center for Basic Natural Science (grant no. E00-3.5469) and the Russian Foundation for Basic Research (project no. 00-15-96741). REFERENCES 1. Optics of the Ocean, Vol. 1: Physical Optics (Nauka, Moscow, 1983) [in Russian]. 2. L. S. Dolin and I. M. Levin, Handbook on the Theory of Underwater Vision (Gidrometeoizdat, Leningrad, 1991) [in Russian]. 3. V. G. Gavrilenko and S. S. Petrov, Waves Random Media 2 (4), 273 (1992). 4. A. V. Aistov and V. G. Gavrilenko, Izv. Ross. Akad. Nauk, Fiz. Atmos. Okeana 31, 792 (1995). 5. A. V. Aistov and V. G. Gavrilenko, Opt. Spektrosk. 78, 672 (1995) [Opt. Spectrosc. 78, 605 (1995)]. 6. V. G. Gavrilenko and A. A. Semerikov, Opt. Spektrosk. 85, 819 (1998) [Opt. Spectrosc. 85, 751 (1998)]. 7. S. M. Ermakov and G. A. Mikhaœlov, Statistical Simulation (Nauka, Moscow, 1982) [in Russian]. 8. S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskiœ, Introduction to Statistical Radiophysics (Nauka, Moscow, 1978), Part 2 [in Russian].
Translated by A. Malikova
