## 1. Introduction

Over several decades, studies on the propagation of light through highly forward scattering media has attracted great attention due to their potential applications to underwater wireless communication, imaging and remote sensing (see, e.g., [

1,

2,

3,

4,

5,

6,

7,

8,

9,

10,

11,

12,

13,

14]).

The significant attenuation and distortion of optical signals in sea water due to absorption and multiple scattering severely constrain the maximum operating range of laser-based systems. This places rather stringent requirements on a possibility to predict the optical channel characteristics, depending on actual sea water parameters. Of prime interest is information on the temporal stretching, spatial and angular broadening of a pulsed beam at the source-receiver distances up to several tens of the photon mean free paths in the medium.

Currently, numerical methods [

3,

4,

5,

6,

9,

11] (regarding earlier results see [

15,

16,

17,

18,

19]) based primarily on Monte Carlo simulation are applied to computations of the light field distribution in sea water from non-stationary (pulsed, modulated) sources. These methods enable taking into account essentially all factors inherent in actual experimental conditions (the angular profile of the sea water phase function, the mutual source-receiver orientation, limited aperture and field-of-view of the receiver etc.). However, the results obtained in such a way, as a rule, are related to a particular set of parameters and can not be transfered directly to other cases. In this connection, there remains interest in finding analytical solutions to the radiative transfer equation that would be based on adequate approximations and satisfactory reproduce the light field distribution in sea water for actual values of the optical parameters.

In propagation of a short pulse through the highly forward scattering medium (

$1-\langle cos\gamma \rangle \ll 1$, where

$\langle cos\gamma \rangle $ is the average cosine of the single-scattering angle [

2]) such as sea water, early arrival photons move along weakly curved trajectories and, as a consequence, exhibit relatively narrow angular distribution and small delay,

$\Delta =ct-z\ll z$ (

$ct$ is the photon path length,

c is the speed of light in the medium,

z is the source-receiver distance). These photons are the least depolarized. Just this component of light field is most important to underwater optical communication and imaging [

2,

3,

6,

11].

For the early arrival component of light field, the small-angle approximation can be applied to the non-stationary radiative transfer equation. Within this approximation, many studies of multiple scattering of a pulsed beam were carried out (see, e.g., [

2,

18,

20,

21,

22,

23,

24,

25,

26,

27,

28,

29,

30,

31,

32,

33]). Most of them [

20,

23,

24,

25,

26,

27,

29,

31,

32,

33] were based on the small-angle version of the Fokker-Planck approximation (or the diffusion approximation in the angular domain) for the collision integral in the radiative transfer equation. The results obtained within this exactly solvable model correlate only qualitatively with experimental data on the structure of a pulsed beam in sea water. This model based on the assumption that the mean square

$\langle {\gamma}^{2}\rangle $ of the single-scattering angle exists within the small-angle phase function representation (i.e., that the phase function wings falls off rapidly [

28,

29,

30]) can not describe adequately many of the pulsed beam characteristics (e.g., the distance dependence of the power attenuation, the temporal, angular and spatial broadening etc.). As analysis shows, the reason is that the phase function of sea water decreases rather slowly with increasing the scattering angle

$\gamma $ (roughly as

$1/{\gamma}^{3}$ [

7,

9,

15,

28]). This feature should be allowed for in solving the radiative transfer equation for sea water.

The aforesaid refers equally to the description of the polarization state of multiple-scattered light in sea water as depolarization results from scattering through relatively large angles [

34,

35,

36]. The results derived within the small-angle diffusion approximation (see, e.g., [

31,

32,

37,

38,

39]) can not be applied directly to sea water and other highly forward scattering media, such as aqueous suspension of polystyrene particles for which the depolarization of pulses has been studied both experimentally [

40,

41,

42] and numerically using Monte Carlo [

17,

41] and DISORT [

19] codes. Information on the degree of light polarization in the transmitted pulse is important for applying the polarization difference technique (see, e.g., [

42,

43]). When measuring the intensity difference between cross-polarized components, the pulse tail (i.e., the contribution from delaying photons) is cut off, which can be used in optical communication and imaging [

42,

43].

In what follows the distribution of light field and its polarization state in a

$\delta $-pulse propagating through sea water are studied. The basic mode approximation in the vector radiative transfer equation (VRTE) which we have applied previously to a stationary case (see, e.g., [

35,

36,

44,

45]) is generalized to calculating the temporal profiles of the intensity and the degree of polarization in the pulse. Our approach are based on a solution of the eigenvalue problem for the basic modes. The results of both analytical, within the small-angle approximation, and numerical calculations are presented. When performing the analytical calculations, we rely on the small-angle version of the Reynolds-McCormick phase function [

46] and the Rayleigh approximation for the scattering matrix. Actual data on the sea water phase function [

47,

48] and the Voss-Fry scattering matrix [

49] are used in our numerical calculations. The power-law parametrization is shown to be valid for the eigenvalues appearing in the expressions for the Laplace transform of the basic modes with respect to time. This enable expressing the distribution of light field in the pulse and the degree of polarization in a self-similar form. Our results are in agreement with data of experiment [

7] and Monte Carlo simulations [

17].

## 3. Model of Single Scattering of Light in Sea Water

In sea water, single scattering of light by suspended particles is directed mainly forward, which is reflected in the angular profile of the phase function. Deflection of light by turbulent perturbations occurs through extremely small angles and does not affect the distribution of light scattered by suspended particles. In the radiative transfer theory, the effect of turbulence is not taken into consideration. Analysis of experimental data and numerical calculations (see, e.g., [

1,

7,

9,

15,

47,

48,

59]) shows that the phase functions of sea water and other media with large inhomogeneities (e.g., aqueous suspension of polystyrene particles which is used to simulate scattering properties of sea water) fall off with increasing angle

$\gamma $ by a power law

${a}_{1}\left(\gamma \right)\sim {({\gamma}_{0}/\gamma )}^{\alpha}$, where the exponent

$\alpha $ varies between

$2.5$ and

$3.5$, and typical values of the angle

${\gamma}_{0}$ are in the range

${1}^{\circ}$–

${5}^{\circ}$ [

1,

7,

9,

15,

47,

48,

59]. A number of examples of

${a}_{1}\left(\gamma \right)$ is illustrated in

Figure 2.

For theoretical modeling the angular profile of single scattering in sea water, the Henyey-Greenstein phase function

and its modification proposed be Reynolds and McCormick [

46]

are also widely used [

7,

9,

15,

21,

50] (e.g., in Monte Carlo simulations [

15], the phase function of sea water was fitted by Equation (

18) with

$\alpha =2.7$ and

$g=0.96$).

The scattering matrix elements of sea water can be inferred from data of measurements [

49] and calculations (see, e.g., [

61] and references therein). According to [

49,

62], the values of these elements for sea water are somewhat different from their Rayleigh values (see Equation (

10)). The angular dependence of the elements

${a}_{+}^{(2,3)}$ and

${a}_{4}$ appearing in Equations (

12) and (

13) is shown in

Figure 2. The data for sea water presented in

Figure 3 can be approximated by the relations

where

The phase function

${a}_{1}\left(\gamma \right)$ is conveniently expanded in a series of Legendre polynomials

The expansion coefficients

${a}_{1}\left(l\right)$ are used in solving the radiative transfer equation (see

Appendix A and, e.g., [

50]). The scattering matrix element

${a}_{4}\left(\gamma \right)$ appearing in Equation (

13) can be expanded similarly. A somewhat different representation is valid for the scattering matrix element

${a}_{+}^{(2,3)}\left(\gamma \right)$. The quantity

${a}_{+}^{(2,3)}(\mathbf{\Omega}{\mathbf{\Omega}}^{\prime})exp\left(2\mathrm{i}{\chi}_{+}\right)$ entering into Equation (

12) is expanded in a series of the generalized spherical functions [

51]

For the highly forward scattering medium such as sea water, the expansion coefficients

${a}_{1}\left(l\right)$,

${a}_{+}^{(2,3)}\left(l\right)$ and

${a}_{4}\left(l\right)$ differ from each other only at relatively small values of

l. This is illustrated in

Figure 4 where the results of numerical calculations of these coefficients for sea water and its imitations are presented.

## 5. Results of Numerical Calculations

The results obtained above in

Section 3 and

Section 4 can be validated using the numerical solution of the eigenvalue problem (see

Appendix A) without resort to the small-angle approximation. In what follows it is assumed that all eigenvalues and eigenfunctions for the basic modes

I,

W and

V correspond to the number

$n=0$.

First we outline the numerical results for

${\epsilon}_{I}\left(p\right)$. The

p-dependence of

${\epsilon}_{I}\left(p\right)$ at

$p>{\sigma}_{tr}$ can be approximated by a power law,

The values of

${c}_{\nu}$ and

$\nu $ for sea water (Kopelevich’s model [

47,

48]) and its imitation are given in

Table 1. The calculations of

${\epsilon}_{I}\left(p\right)$ were carried out with Equation (

A4) (see

Appendix A).

The eigenvalues

${\epsilon}_{I}$,

${\epsilon}_{W}$ and

${\epsilon}_{V}$, and the differences

$\delta {\epsilon}_{W}={\epsilon}_{W}-{\epsilon}_{I}$,

$\delta {\epsilon}_{V}={\epsilon}_{V}-{\epsilon}_{I}$ as functions of

$p/{\sigma}_{tr}$ are shown in

Figure 6. For sea water, the results of measurements by Voss and Fry [

49] were used for calculating the scattering matrix elements

${a}_{+}^{(2,3)}$ and

${a}_{4}$. For the Henyey-Greenstein phase function, the calculations were carried out within the Rayleigh approximation (see Equation (

10)). The calculations of

${a}_{+}^{(2,3)}$ and

${a}_{4}$ based on the Mie theory [

60] underlie the results obtained for aqueous suspension of polystyrene spheres.

The angular dependence of the eigenfunctions

${\Phi}_{I}$,

${\Phi}_{W}$ and

${\Phi}_{V}$ is illustrated in

Figure 7 for the reference case of the Henyey-Greenstein phase function. The difference between the eigenfunctions is small in the forward hemisphere, suggesting that the angular profile is unaffected by the initial polarization of light or, in other words, the decay of the initial polarization is virtually angle independent.

The contribution from the geometrical depolarization to

$\delta {\epsilon}_{W}$ is shown in

Figure 8. This contribution is the main for aqueous suspension of relatively large polystyrene spheres (

$d=0.993\phantom{\rule{4pt}{0ex}}\mathsf{\mu}$m,

$\lambda =633$ nm), and for the reference case involving the combination of the Henyey-Greenstein phase function with the Rayleigh scattering matrix and is comparatively modest for sea water. The features of the Voss-Fry scattering matrix [

49] that are inherent in scattering by non-spherical particles make the dynamical depolarization principal for sea water. As

p increases,

$\delta {\epsilon}_{W}$ and

$\delta {\epsilon}_{V}$ decreases according to a power law,

The numerical values of

${D}_{{\nu}_{1,2}}$ and

${\nu}_{1,2}$ are presented in

Table 2. For the latter two lines of

Table 2 the values of

${\nu}_{1}$ and

${\nu}_{2}$ coincide to the corresponding values of

$3-\alpha /2$, correlating with Equations (

35) and (

36).

## 6. Discussion

The results obtained above give an insight into the distribution of light field in the pulse and enable calculating semi-analytically its main characteristics, including the polarization state, in sea water.

For the intensity of radiation in the pulse, the small-angle approximation can be applied in combination with an adequate parametrization of the sea water phase function. A peculiarity of light scattering by sea water is that the wings of the phase function at

$\gamma >{\gamma}_{0}$ (

${\gamma}_{0}$ is of the order of a few degrees) fall off rather slowly, according to a power law. Therefore, the parametrization such as the Henyey-Greenstein function or the Reynolds-McCormick one grasps the key feature of light scattering by sea water in the forward hemisphere. This is confirmed by the data presented in

Figure 2 and

Figure 4. Additionally, to validate the power-law approximation (

28) in application to a realistic situation, we compare the results of our calculations with experiment [

7]. Substituting Equation (

28) with the same parameters as in Monte Carlo simulation [

7,

9] (i.e.,

$\alpha =3$,

${\gamma}_{0}=0.07$) to the small-angle solution of the transfer equation (see, e.g., [

21]), we obtain the attenuation curve which agrees with the experimental data (see

Figure 9). The relative contribution of multiple-scattered radiation to the received signal increases with the optical distance, resulting in the decrease of the attenuation rate as compared to Beer’s law.

Using the power-law approximation for the phase function, we can represent the distribution of light field in the pulse in the self-similar form as a function of dimensionless variables (see Equations (

31) and (

32)). These variables are certain combinations of the “excess” path

$\Delta $, the transport scattering coefficient

${\sigma}_{tr}$, the source-receiver distance

z and the exponent

$\alpha $ appearing in the parametrization of the phase function. Correspondingly, the main characteristics of the light field distribution (the effective time spread, dispersion in angles and etc.) are expressed in terms of these variables.

The important result for the variance of the angular distribution of light field,

${\langle \theta \rangle}_{z,t}$ (see Equation (

30)), enables controlling the validity of the small-angle approximation at given

z and

t.

From the results obtained it follows that the peak in the temporal sweep of the pulse is observed at small values of the “excess” path,

$\Delta \ll z$, or, for a narrow beam,

$\Delta -{\rho}^{2}/\left(2z\right)\ll z$ (

$\rho $ is the transverse displacement from the beam axis, see

Appendix B). This is illustrated in

Figure 10 where the distribution of light field in the forward direction and the integral-over-angle distribution are shown. The latter correlates well with the Monte Carlo simulation data [

11] (see Figures 3 and 4 in [

11]).

The calculations of light field with the use of the numerical results outlined in

Section 5 are performed in the following way. First, the integral over

p in Equation (

15) is evaluated by the stationary phase (or saddle-point) method, and, as before, only the contribution from the minimum eigenvalue

${\epsilon}_{I}={\epsilon}_{I}^{(n=0)}$ is taken into account. The saddle-point

${p}_{0}$ is sought from the equation

The value of

${p}_{0}$ can be found using the power-law approximation for

${\epsilon}_{I}\left(p\right)$ (see Equation (

40)) or, directly, from the numerical values

${\epsilon}_{I}\left(p\right)$ (see

Figure 6). In the latter case, the results obtained are valid even beyond the small-angle approximation. Further the integral in Equation (

15) is calculated routinely (all

p-dependent factor appearing in Equation (

15) are taken at the saddle-point

${p}_{0}$).

Our approach to calculating the light field distribution in the pulse differs from the analytical solutions proposed for the same problem previously (see, e.g., [

18,

20,

22,

23,

24,

25,

26,

27,

29,

33]) in that the specific angular profile of the sea water phase function is allowed for in our calculations.

The basic modes of linear and circular polarization, and the corresponding degree of polarization are calculated similarly to the light field distribution. The degree of polarization can be written as

where

$\delta {\epsilon}_{W,V}={\epsilon}_{W,V}-{\epsilon}_{I}$ and

${p}_{0}$ is the root of Equation (

42). The numerical values of

$\delta {\epsilon}_{W}$ and

$\delta {\epsilon}_{V}$ as functions of

p are given in

Figure 6. Their analytical parametrization is described by Equations (

35) and (

36) or (

41).

The applicability of the expression for the degree of polarization in the form (

43) is not limited by the small-angle approximation.

The results of comparison of our calculations with data of Monte Carlo simulations for aqueous suspension of polystyrene spheres [

17] are shown in

Figure 11. The numerical simulation [

17] was carried out for the linearly polarized pulse propagating through a plane slab of different transport optical thickness

${\sigma}_{tr}z$. Depolarization ratio

$(1-{P}_{L})/(1+{P}_{L})$ was presented in [

17] as a function of the normalized “excess path”

$\Delta /z$. When going from

$\Delta /z$ to novel variable

${\sigma}_{tr}z{(\Delta /z)}^{3-\alpha /2}$ (see Equation (

39)) we obtain virtually universal pattern of the simulation data (see

Figure 11b,d). The agreement between our calculations and the data [

17] shows possibility of application of the power-law parametrization (

41) to realistic cases.

As noted in

Section 2.2 (see also [

36,

45]), two mechanisms underlie the depolarization of light in scattering media. For sea water, as follows from the numerical results presented in

Figure 6 and

Figure 8, the dynamical mechanism of depolarization proves to be dominant. This is due to specific feature of the Voss-Fry scattering matrix at relatively small scattering angles

$\gamma $. From the Voss-Fry measurements (see

Figure 3 and also the approximation (

19) and the parametrization proposed in [

62]) it follows that, for small

$\gamma $, both

$({a}_{+}^{(2,3)}-{a}_{1})/{a}_{1}$ and

$({a}_{4}-{a}_{1})/{a}_{1}$ are proportional to

${\gamma}^{2}$. In the Rayleigh approximation (see Equation (

10)) and in the case of aqueous suspension of polystyrene spheres, contrastingly, these quantities are proportional to

${\gamma}^{4}$. This distinction can be explained by the contribution of large non-spherical particles to the scattering matrix of sea water. For this reason the effect of circular polarization memory that is typical for aqueous suspension of polystyrene spheres (see, e.g., [

36,

45,

65,

66,

67]) can not be observed in propagation of light through sea water. The linearly and circularly polarized beams are depolarized equally (see

Figure 6).

It should be noted that the results presented above can also be applied to a steady-state beam of light. To do this, the expressions for the basic modes should be integrated over the “excess” path

$\Delta $. This is reduced to putting

$p={\sigma}_{a}$ in the corresponding formulas (e.g., in Equations (

15), (

16) or (

43)).