Square pulse effects on polarized radiative transfer in an atmosphere-ocean model

Cun-Hai Wang; Yan-Yan Feng; Yao-Hua Yang; Xun Ben; Xin-Xin Zhang

doi:10.1364/OE.394892

1. Introduction

Applications of short pulses with duration of nanoseconds or even less has attracted significant attention due to the fast development of advanced lasers [1–3]. The short-pulse induced radiative transfer in participating media has been investigated widely [4–10], but most of the previous work only consider the radiation intensity by solving the scalar radiative transfer equation. Guo et al., [11] experimentally studied laser induced radiative transfer with a 60 ps pulse and they found the refractive index has significant influences on the detected transmittance. By numerically solving the transient radiative transfer equation, Mishra et al., [12] investigated the short-pulse induced radiative transfer in a participating media and presented time-resolved solutions of the reflected and transmitted radiation intensity. Lu and Hsu [13,14] developed a reverse Monte Carlo method to simulate time-dependent radiative transfer processes and performed transient radiative simulations in layered media. Yi et al., [15] and Wang et al., [16,17] proposed a modified Monte Carlo method to improve the computational efficiency and studied variable medium properties on the time-resolved reflectance and transmittance.

Light that propagates in a participating medium experiences polarization as it is essentially a transverse electromagnetic wave. The polarization state of the radiated light is described by the Stokes vector containing four components named I, Q, U, and V, and the governing equation of the polarized radiative transfer process is the vector radiative transfer equation (VRTE) [18–21]. Compared to the scalar radiative transfer equation [22–27] which only solves the radiation intensity, the VRTE is much more complex because it simultaneously solves the four Stokes vector components that are coupled by the light propagation events (scattering, reflection, and refraction) [28]. As a result, only a few studies [29–37] take the full polarization effect into account in short-pulse induced radiative transfer problems.

Investigations on the polarized radiative transfer in the atmosphere-ocean system have considerable significance in remote sensing related fields [38–40]. The steady polarized radiative transfer in the coupled atmosphere-ocean model has been well investigated by Zhai et al., [41] and Sommersten et al., [42]. However, short-pulse induced transient polarized radiative transfer in an atmosphere-ocean model has been rarely studied.

In this paper, we numerically solve the short-pulse induced polarized radiative transfer problem in an atmosphere-ocean model via a modified Monte Carlo method (MMC) [37], and analyze the distributions of Stokes vector components varying with time and direction for different locations relative to the atmosphere-ocean interface.

2. Model and method

The schematic of the atmosphere-ocean model is shown in Fig. 1(a). Based on the time-dependent scalar radiative transfer Eq. (4) and the addition of Stokes components [18], the vector radiative transfer equation can be written as

(1)$$\frac{1}{{{c_0}}}\frac{{\partial {\textbf I}(z,{\boldsymbol {\Omega} },t)}}{{\partial t}} + {\boldsymbol {\Omega} } \cdot \nabla {\textbf I}(z,{\boldsymbol {\Omega} },t) + {\boldsymbol{\mathrm{\beta}} {\textbf I}}(z,{\boldsymbol {\Omega} },t) = {\textbf S}(z,{\boldsymbol {\Omega} },t), $$

where c₀ is the light speed in vacuum, z is the axis coordinate, t is the time, I = (I, Q, U, V)^T is the Stokes vector, Ω is the propagation direction as shown in Fig. 1(b), β = diag(β β β β) is the extinction matrix with β denoting the attenuation coefficient, and S is the source term

(2)$${\textbf S}(z,{\boldsymbol {\Omega} },t) = \frac{{{\kappa _s}}}{{4\pi }}\int_{4\pi } {{\textbf Z}({\boldsymbol {\Omega} ^{\prime}} \to {\boldsymbol {\Omega} }){\textbf I}(z,{\boldsymbol {\Omega} ^{\prime}},t)} d{\boldsymbol {\Omega} ^{\prime}},$$

where κ_s is the scattering coefficient and Z is the scattering phase matrix from an incoming direction Ω’ to an outgoing direction Ω.

Fig. 1. (a) Schematic of the atmosphere-ocean model exposed to a short square pulse, (b) the zenith and azimuth angles of a direction Ω.

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In this work, the thickness of the atmosphere and the ocean water is L_a and L_o, respectively. The positive z axis denotes the vertical upward direction as shown in Fig. 1(a), and the zenith angle θ and the azimuth angle φ of the propagation direction Ω are sketched in Fig. 1(b). Initially, a short square pulse with the duration of t_p and Stokes vector of I₀ = (I₀, Q₀, U₀, V₀)^T, which is the only primary source that drives the polarized radiative transfer, is incident at the top of the atmosphere (labeled as A) with a direction of Ω⁰ = (θ₀, φ₀). The top boundary of the atmosphere is totally transparent and the bottom of the ocean water is assumed to be totally absorbing. Then the boundary conditions [4] are written as

(3a)$$I({\kern 1pt} \,z = {L_\textrm{o}} + {L_\textrm{a}},\mu < 0,t) = {I_0}[H(t) - H(t - {t_p})]\delta (\mu - {\mu _0}),$$

(3b)$$I({\kern 1pt} \,z = 0,\mu > 0,t) = 0,$$

where H(t) is the Heaviside step function, and μ = cosθ is the direction cosine of the polar angle.

The atmosphere-water interface is specular. The refractive index of the atmosphere is assume to n_a = 1.0 and the relative refractive index of water to the atmosphere is taken as n = n_o/n_a = 1.338. When a Stokes vector I propagates to the atmosphere-water interface, the reflected and transmitted Stokes components are calculated as R_s·I and T_s·I, respectively. Here R_s and T_s denote the reflection and transmission matrices calculated according to [37]. In the following, the direction cosine μ is used to indicate the polar directions, thus a positive direction cosine indicates the upward direction with θ < 90°, and a negative direction cosine indicates the downward direction with θ > 90°.

The modified Monte Carlo method (MMC) proposed in [37], which significantly improves the computation efficiency via the time shift and superposition principle, is utilized to simulate the polarized radiative transfer processes. In the MMC simulations, the commonly used dimensionless time t* = βc₀t is applied, the duration of the short pulse is divided into M finite time periods Δt_p* = t_p*/M, and only the radiation bundles within the first time period are traced. The initial time of a bundle emitting in the first time period Δt_p* is given by

(4)$$t_0^{\ast } = {R_t}\Delta t_p^\ast ,$$

where R_t is a uniform pseudo random number within (0, 1].

The polarized propagation of the radiation bundles emitted within the first time period are numerically simulated via the standard ray-tracing technique [43,44]. The polarized signals induced by the entire pulse duration, therefore, are obtained by applying the time shift and superposition principle [37]. With the same bundle to time ratio (bundle number/emitting time), the number of simulated radiation bundles in the MMC is reduced by a factor of M compared with the bundle number needed in the traditional Monte Carlo algorithm, and this makes the MMC effective and efficient for simulating short pulse induced polarized radiative transfer problems.

3. Results and discussions

In this section, simulation results of the polarized radiative transfer problem sketched in Fig. 1 are presented and discussed. The intensity of the pulse is I₀ = 1.0 W·m⁻², the Stokes vector is I₀ = (I₀, Q₀, U₀, V₀)^T = (1.0, 0, 0.5, 0.5)^T, and the incident direction is Ω⁰ = (θ₀, φ₀) = (120°, 0°). Both the atmosphere and water are Rayleigh scattering with an attenuation coefficient of β = 1.0 m⁻¹ and a scattering albedo of ω = 1.0. The dimensionless duration of the square pulse is t_p* = 1.0, and the optical thickness of the ocean water keeps constant of τ_o = βL_o = 1.0 for all the following cases. The atmosphere with different optical thicknesses τ_a = βL_a = 0.15, 1.0, and 5.0 are considered to study the effects of the incidence and disappearance of the square pulse on the Stokes vector components. The Stokes vector components at three selected locations, i.e., the location B just above atmosphere-water interface, the location C just below the atmosphere-water interface, and the location D at the ocean bottom as shown in Fig. 1(a), are presented and analyzed. These locations were chosen because of their considerable importance for variable radiation signal related applications.

3.1 Case of τ_a = 0.15

The atmosphere with an optical thickness of τ_a = 0.15 based on [42] is considered in this case. Without any loss of generality, only the Stokes component distributions along the viewing azimuth angle φ = 90° are analyzed in this section, results along the azimuth angle φ = 180° are presented in the Appendix for the consideration of comparisons. The zenithal Stokes vector components at the location B are plotted in Fig. 2 for the dimensionless times t* = 0.2, 0.4, 0.8, 1.2, 2.0, 3.0, 4.0, and 5.0. As the optical thickness of the atmosphere is τ_a = 0.15, it takes at least a time period of t_a* = βc₀t = βc₀(L_a/c₀) = τ_a = 0.15 for the incident photon to reach the interface. It is seen from Fig. 2 that for the case of t* = 0.2, the Stokes components are negligible except for the directions with μ = [−1.0, −0.75], i.e., the downward direction around the negative z-axis. This is because, theoretically, at the dimensionless time t* = 0.2 which is bigger than t_a* when the atmosphere-water interface begins to receive the incident radiation photons, the incident bundle that directly propagates to the interface has a direction cosine of μ < μ(-z) × t_a*/t* = −1.0 × 0.15/0.2 = −0.75. The results presented in Fig. 2 correspond well with the theoretical predications, and this demonstrates the correctness of the numerical results.

Fig. 2. Zenith distributions of the Stokes vector components at the location B in the case of τ_a = 0.15 plotted for azimuth angle φ = 90° and different times.

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From the results in Fig. 2 we also find that the Stokes component signal becomes stronger for a later time within the pulse duration, as more radiation bundles reach the interface and enhances the polarized signals. In the zenith distribution of the Stokes vector component I at t* = 0.4, as shown in Fig. 2(a), a peak arises and the I component changes strongly near this peak. When the time proceeds to t* = 0.8, the distribution line of the I component shows another peak point where there is an obvious mutation. A similar phenomenon occurs in the distribution line of the I component at t* = 1.2. At time points much greater than the pulse duration, such as t* = 3.0, the distribution line of the I component between the two peaks appears to be a cambered platform and the platform lowers at later times, as shown in Fig. 2(a). The distribution line of the Q component varying with the zenith angle, as plotted in Fig. 2(b), has a similar changing trend with that of the I component at the same time. For the U component plotted in Fig. 2(c), only one sharp peak is observed in the zenith distribution line for t* = 0.4, 0.8, and 1.2. At times later than t* = 3.0, the zenith distribution line of the U component also has a cambered platform for the polar directions around μ = 0. From the polar distributions of the V component plotted in Fig. 2(d), it can be found that the V distribution differs significantly from the other three components. At t* = 0.2, the V component decreases from a positive value at μ = −1.0 to a zero value at μ = −0.75. At t* = 0.4, the V component remains at a relatively high level along the directions with μ = [−1.0, −0.5], and decreases to a zero value at μ = 0. It stays near zero for the directions with μ = [0, 0.5], and then decreases to a minimum value at μ = −1.0. The zenith distribution lines of the V component for later times are also plotted in Fig. 2(d) and we find no cambered platform around μ = 0, in contrast to the other Stokes vector components I, Q, and U.

The time-resolved Stokes vector components at location B are plotted in Fig. 3 for zenith angles with μ = −0.95, −0.75, −0.55, −0.35, −0.15 and their symmetric directions, i.e., the zenith angles with μ = 0.15, 0.35, 0.55, 0.75, and 0.95.

Fig. 3. Time-resolved Stokes vector components at the location B in the case of τ_a = 0.15 plotted for azimuth angle φ = 90° and different zenith angles.

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Due to the downward collimated pulse, a number of radiation photons with uniform propagation directions reach the interface at nearly the same time and this leads to an abrupt rise (fall for the U component along μ = −0.95) in the Stokes component distribution lines in the downward direction (negative cosines) and a flattop which persists for t* = 1.0. For the upward directions with μ > 0, the Stokes components are the superposition of the boundary reflected radiation bundles that are incident from the atmosphere, transmitted radiation bundles that are incident from the water, as well as those scattered by the medium above the interface. The interface reflection weakens the similarity of the incident bundles and this leads to the gradual increasing of the I, Q, and U components and a gradual decrease of the V component; this is quite different from the step change observed in the downward directions. For all the Stokes component distribution lines along the selected directions, there is an obvious time point where the components return to zero, corresponding to the disappearance of the short pulse.

Results in Figs. 2 and 3 should be used to analyze the zenithal or time-resolved distributions, but they only present the Stokes vector component distribution for the selected times and polar directions. A more intuitive distribution of the Stokes vector is shown in Fig. 4, where the distribution contours of the Stokes components at location B are shown as functions of time and direction. The data in Figs. 2 and 3 are part of that presented in Fig. 4, but the different plot styles lend themselves to different analyses which makes them both necessary and useful for a broad range of the community. From the results in Fig. 4 we find that, all the Stokes components along the directions with μ < −0.5 show clear step changes corresponding to the beginning and ending of the incident pulse, leading to a striped area with a duration of t* = t_p* = 1.0. While for the directions with μ > −0.5, no such corresponding striped area is observed. For the I, Q, and U components, the strong signals appear around t* = 1.2 and μ = −0.1 in the contours. The signal along the directions near μ = 0 lasts longer in time, compared with the Stokes components along other directions, as shown in Figs. 4(a)–4(c). From Fig. 2(d) it is found that the V component is nonnegative along the directions with μ < 0 while non-positive along the directions with μ > 0. This means the value of V component along the directions adjacent to μ = 0 is quite close to zero. Therefore, the V component along the directions near μ = 0 is marginal as presented in Fig. 4(d), which is quite different from the case for the other Stokes components that have a peak along the directions near μ = 0.

Fig. 4. Stokes vector contour varying with time and zenith angle plotted for the location B in the case of τ_a = 0.15 and azimuth angle φ = 90°.

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The distribution contours of the Stokes vector components just below the atmosphere-water interface (location C in Fig. 1(a)) are presented in Fig. 5. In addition to the step change along the time axis, a discontinuous distribution along the direction axis is also observed. This is due to the fact that the refractive index of the water is larger than that of the atmosphere; when the radiation bundles propagate to the interface from the water, some of them will be totally reflected and the critical direction cosine is μ_c = − [1− (1/n)²]^0.5 = − 0.66. This total reflection leads to the discontinuous abrupt change at μ = − 0.66 along the direction axis, as shown in Fig. 5. Additionally due to the total reflection effect, the striped area induced by the pulse duration can be only observed for the directions with μ < − 0.66, and the peak areas are located around t* = 1.2 and μ = - 0.66 for the Stokes components I, Q, and U, as shown in Figs. 5(a)–5(c). Compared with the distribution contour of the V component just above the interface presented in Fig. 4(b), another area with relative strong signal can be observed around t* = 1.7 and μ = - 0.5 for the Stokes component contour just below the interface, as shown in Fig. 5(d).

Fig. 5. Stokes vector contour varying with time and zenith angle plotted for the location C in the case of τ_a = 0.15 and azimuth angle φ = 90°.

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The Stokes vector component contours at the bottom of the ocean (location D in Fig. 1(a)) are presented in Fig. 6. As the bottom of the ocean is completely absorbing, all the radiation bundles propagating to the bottom with a negative direction cosine will be absorbed without any reflection; this lack of reflection therefore makes the Stokes vector components zero for the directions with μ > 0. No obvious changes corresponding to the incidence and disappearance of the external pulse are found in Fig. 6. This means the incident radiation bundles that have a uniform direction do not have significant influence on the initial appearance of the Stokes components at location D. The Stokes components at the ocean bottom are dominated by the multiple scattering radiation bundles with random directions.

Fig. 6. Stokes vector contour varying with time and zenith angle plotted for the location D in the case of τ_a = 0.15 and azimuth angle φ = 90°.

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3.2 Case of τ_a = 1.0

In this case, the optical thickness of the atmosphere is set as τ_a = 1.0 which is the same as the dimensionless duration of the incident pulse. The distribution contours of the Stokes vector components at location B are presented in Fig. 7. The polarized signals appear from a dimensionless time of t* = 1.0 which exactly corresponds to the optical thickness of the atmosphere in this case. In contract to the case of τ_a = 0.15 where the Stokes component has a step arise along the directions with μ < −0.5, in this case, the time-resolved Stokes components changes gradually from its beginning for all directions, as shown in Fig. 7 where no step change are observable. This is because the incident bundles with a uniform direction are strongly scattered by the atmosphere with large optical thickness, compared with the atmosphere with a small optical thickness, and thus the incident radiation energy reaches the interface gradually. For the same reason, the many scattering bundles still propagates to the interface when the pulse disappears, leading to the gradually disappearing of the time-resolved Stokes vector components. The analysis of Stokes vector components at location B indicates that the time-resolved signals can be used to deduce the incidence time, while not the duration of the external pulse for the atmosphere with an optical thickness of τ_a = 1.0.

Fig. 7. Stokes vector contour varying with time and zenith angle plotted for the location B in the case of τ_a = 1.0 and azimuth angle φ = 90°.

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The distribution contours of the Stokes vector components at location C are shown in Fig. 8. The discontinuous changes along the direction with μ = - 0.66 are still clearly observable. Compared with the case of τ_a = 0.15, the polarized signals for this case are quite weak along the directions with μ > - 0.66 due to the strong attenuation of the atmosphere. The Stokes vector components at location D, as shown in Fig. 9, are only visible for the dimensionless time late than 2.5 which is bigger than τ_a + 1.338τ_o = 2.338, and this confirms the Stokes components at the ocean bottom are dominated by the multiple scattering radiation.

Fig. 8. Stokes vector contour varying with time and zenith angle plotted for the location C in the case of τ_a = 1.0 and azimuth angle φ = 90°.

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Fig. 9. Stokes vector contour varying with time and zenith angle plotted for the location D in the case of τ_a = 1.0 and azimuth angle φ = 90°.

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3.3 Case of τ_a = 5.0

In this case, the optical thickness of the atmosphere is as large as τ_a = 5.0, to study the pulse induced polarized signals after propagating through an optically thick atmosphere.

The Stokes component contours at location B are presented in Fig. 10. It is found that the Stokes components starts from a dimensionless time that is obvious later than t* = 5.0 which is the least time for an incident bundle to reach the interface. This phenomenon indicates that due to the large optical thickness and strong scattering effect of the atmosphere, rare incident bundles can directly propagate to the interface and the polarized signals at location B is dominated by the multiple scattering bundles. As a result, no step arise can be found in the Stokes component contours at location B, not to mention those at locations C and D, as shown in Figs. 11 and 12.

Fig. 10. Stokes vector contour varying with time and zenith angle plotted for the location B in the case of τ_a = 5.0 and azimuth angle φ = 90°.

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Fig. 11. Stokes vector contour varying with time and zenith angle plotted for the location C in the case of τ_a = 5.0 and azimuth angle φ = 90°.

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Fig. 12. Stokes vector contour varying with time and zenith angle plotted for the location D in the case of τ_a = 5.0 and azimuth angle φ = 90°.

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Fig. 13. Stokes vector contour varying with time and zenith angle plotted for the location B in the case of τ_a = 0.15 and azimuth angle φ = 180°.

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Fig. 14. Stokes vector contour varying with time and zenith angle plotted for the location C in the case of τ_a = 0.15 and azimuth angle φ = 180°.

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Fig. 15. Stokes vector contour varying with time and zenith angle plotted for the location D in the case of τ_a = 0.15 and azimuth angle φ = 180°.

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From the Stokes component contours for the case of τ_a = 5.0 we find that the I component can be visible at the long time as t* = 40, while the V component are almost indistinguishable after a time of t* = 12. The Q components is obvious before t* = 30 and the U component are visible at the time earlier than t* = 20. The results in this case indicates that for the atmosphere with an optical thickness of τ_a = 5.0, the uniformly emitted bundles by the collimated pulse have been sufficiently scattered when reaching the atmosphere-ocean interface.

4. Conclusion

The square pulse induced polarized radiative transfer in a simplified atmosphere-ocean model were numerically simulated by a modified Monte Carlo method. Three atmospheres with different optical thicknesses of τ_a = 0.15, 1.0, and 5.0 are considered. The Stokes components varying with time and direction were presented for locations just above and below the atmosphere-water interface as well as for the ocean bottom location. For the case of τ_a = 0.15, a range of zenith angles with a direction cosine smaller than −0.5, where the time-resolved Stokes components correspond well to the incidence and disappearance of the short pulse, is found in the Stokes contour just above the atmosphere-water interface. This range is shortened in the Stokes contours just below the atmosphere-water interface because of the interface refraction effect. No such range can be observed in the Stokes contours at the bottom of the ocean. For the case of τ_a = 1.0, the time-resolved signals just above the interface can be used to deduce the incidence time, while not the duration of the square pulse. For the case of τ_a = 5.0, due to the strong attenuation effect when radiation bundles propagate through the optical thick atmosphere, the collimated incident bundles have been sufficiently scattered when reaching the interface. The detailed distribution of Stokes components can be applied to analysis of the short-pulse induced ultrafast polarized radiation transfer. The simplified model considered in this work is suitable for qualitative studies of some particular polarized radiative transfer problems, such as the polarized pulse transmission from air into the water and vice versa in the close vicinity of the air-water boundary, and in some remote sensing or underwater polarized transmission. It has potential further developments for solving the polarized radiative transfer problems in realistic atmosphere-ocean systems, by applying the exact properties of the seawater and marine atmosphere [45].

Appendix

Stokes vector contours for the case of τ_a = 0.15 plotted for the azimuth angle φ = 180°, Fig. 13 for the location B, Fig. 14 for the location C, and Fig. 15 for the location D, respectively

Funding

National Natural Science Foundation of China (51906014); China Postdoctoral Science Foundation (2018M641196, 2019M651285); Fundamental Research Funds for the Central Universities (FRF-TP-18-072A1).

Disclosures

The authors declare no conflicts of interest related to this article.

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Square pulse effects on polarized radiative transfer in an atmosphere-ocean model

Abstract

1. Introduction

2. Model and method