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Extended angular-spectrum modeling (EASM) of light energy transport in scattering media

Meng Yan, Mali Gong, and Jianshe Ma

Open Access Open Access

Abstract

The exact modeling of light transport in scattering media is critical in biological imaging, free-space communication, and phosphor-converted lighting. Angular spectrum is proved to be a fast and effective approach to reconstructing the wavefront dynamics during the propagation in scattering media, however, finding it difficult in acquiring the wavefront and energy change simultaneously. Besides, conventional methods for energy tracing, such as the Monte Carlo method, are inefficient in speed and hard to simulate the wavefront change. Here, we propose an extended angular-spectrum modeling (EASM) approach using tenuous scattering approximate solutions to obtain a time-efficient and accurate method for reconstruction of energy and wavefront dynamics in various scattering media. The generality of our method is numerically simulated and experimentally verified with a set of scattering media with different properties. EASM has a time advantage under the guarantee of calculation accuracy, especially when calculating several thickness changes after the calculation model is established. Furthermore, multi-layered media can also be simulated by EASM with a good precision. The results suggest that EASM performs certain computations more efficiently than the conventional method and thus provides an effective and flexible calculation tool for scattering media.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

For the past decades, extensive attention has been paid to the theory of light transport in scattering media. The light-media interaction has been widely studied in several fields, such as the biological imaging [1–3], scattering imaging [4], optical neural network [5–7], photonic devices [8–10] and phosphor-converted lighting [11,12]. In biological imaging, obtaining the optical field variation inside a scattering medium plays a key role in refocusing inside or through a scattering medium (shown in Fig. 1(a)) [3,13,14]. However, the transport of light in scattering medium is complex, which leads to a rigorous calculation of its process.

 figure: Fig. 1.

Fig. 1. (a) Optical field propagates through a scattering medium. The energy and wavefront of the light change due to scattering effects. (b), Conventional angular-spectrum modeling. A medium is modeled into several scattering layers with intervals. The wavefront of the incident light will change when it passes through a scattering layer, which is considered as a phase transformation. (c), Light energy change happened in a single scattering layer. The light will be absorbed and scattered (forward or backward) by a single scattering layer. (d), Extended angular-spectrum modeling (EASM). The calculation of energy transport in a scattering medium is also added to the angular spectrum model.

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Now, several methods have been proposed to analyze or calculate the optical field variation when light interacts with scattering media. The Monte Carlo method (MCM) [15–17] and the radiative transfer equation (RTE) [18,19] are generally considered applicable for calculating the energy transport of light in a scattering medium. However, MCM requires a large computational amount to achieve convergence and consumes a lot of time, which means it is difficult to achieve real-time computing. RTE is always approximate to the diffusion equation since it is difficult to solve with an analytical solution, which reduces its calculation accuracy compared with MCM [20]. In addition, the optical wavefront inside a scattering medium is hard to be simulated by MCM or RTE. The wavefront variation of light inside the scattering medium is analyzed extensively by the transfer matrix method [21–23] and angular-spectrum modeling [24,25].

The conventional angular-spectrum modeling, proposed by Yang et al. [26] and improved by Cheng et al [27]., is a flexible and fast method to simulate the wavefront variation or point spread function degradation in biological imaging. Yang et al. apply this method to trace the field propagation during the entire optical phase conjugation process in the presence of scattering media. Cheng et al. establish a comprehensive relation between the model parameters of the angular-spectrum modeling to the macroscopic properties of a scattering medium. As shown in Fig. 1(b), a scattering medium is modeled into several scattering layers based on its length and number of layers, and each layer is considered as a phase transformation [25]. The optical field will be multiplied by the scattering phase once it propagates through a scattering layer (phase transformation) until the last one. Even so, the angular-spectrum modeling is unable to simulate energy variation inside a scattering medium, which limits its scalability.

Here, we propose a universal framework based on the angular-spectrum modeling extended by the transport theory with tenuous scattering approximate solutions [18] to simulate the energy and wavefront variation of the optical field inside a scattering medium simultaneously, that is the extended angular-spectrum modeling (EASM). As shown in Fig. 1(c), the forward, backward and absorption energy of the optical field will be calculated by EASM when it propagates through a scattering layer. Figure 1(d) illustrates that the total energy variation (total transmittance, diffuse reflectance and absorption) after passing through a scattering medium can be obtained by the energy variation of each layer and considering the Fresnel reflection of the front and rear surfaces of the medium. EASM reduces the computation time substantially compared with the conventional MCM. Moreover, the optical field variation inside a multi-layered medium is also considered by EASM when the coefficients (scattering and absorption) and the refraction indexes of the media are different. In particular, the energy variation of samples of any thickness can be simulated at the same time when calculating a scattering medium with a maximum thickness, which is not available in any existing method. We demonstrate the universality of this approach by comparing the computation results with that simulated by conventional MCM. In addition, the effectiveness of our method is also verified by experiments with several samples. This suggests that EASM is able to perform certain computations more efficiently than the conventional method, and thus provide a route to an effective and faster simulation method for light-tissue interaction, scattering imaging and laser lighting.

2. Methods

The proposed EASM, based on the angular spectrum model and transport theory with tenuous approximation, is an effective and fast method to simulate the energy change of light transport in a scattering medium. In our method, shown as the flow chart in Fig. 2, optical field propagation in a scattering medium is modeled into several scattering layers, which effect light energy and phase. Every scattering layer is considered a phase transformation for scattering phase calculation and the space between scattering layers is considered a tenuous distribution of scattering under approximate conditions for transport theory calculation. In addition, the propagation of the optical field from one layer to another between scattering layers is calculated by the angular spectrum method with the Fourier transform.

 figure: Fig. 2.

Fig. 2. Flow chart for EASM.

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Next, scattering energy changes, including forward, backward and absorption energy, are obtained by transport theory with the first-order multiple scattering approximate solutions for the tenuous medium. The energy proportion of each plane wave component of the wavefront is required in the transport theory. Therefore, the components of plane waves at various angles used in the transport theory are calculated by the angular spectrum of the optical field. The optical field after propagating a distance from one layer to another is computed by angular spectrum propagation theory with the inverse Fourier transform. The optical field will be superimposed by the phase transformation once it passes through a scattering layer. Ultimately, the energy variation and propagation of the optical field in the scattering medium follow the above principles until all scattering layers are calculated. It is worth noting that the Fresnel reflection of the light and the energy change caused by the multiple propagation of the reflection light inside the medium need also be considered at the boundary when the refractive index is mismatched. See Supplement 1 for detailed description of EASM.

2.1 Propagation of angular spectrum

Light propagation and phase change are calculated by the propagation theory of angular spectrum in EASM. A scattering medium is divided into several scattering layers and the transformation of the optical field is shown in Fig. 3.

 figure: Fig. 3.

Fig. 3. Processes of light field transports between scattering layers.

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We define $E_m^\textrm{1}(x,y)$ and $E_m^\textrm{2}(x,y)$ as the optical field before and after a scattering layer m, respectively. The scattering layer m is considered as a phase transformation with a two-dimensional phase ${\varphi _m}(x,y)$ to simulate the phase change caused by scattering. The optical field $E_m^\textrm{2}(x,y)$ after scattering layer m is expressed as

$$E_m^\textrm{2}(x,y) = E_m^\textrm{1}(x,y) \cdot \exp [i{\varphi _m}(x,y)]\textrm{.}$$
The propagation of light field in space is calculated by the angular spectrum propagation theory. The angular spectrum of $E_m^\textrm{2}(x,y)$ is expressed by $\tilde{E}_m^\textrm{2}(x,y)$. Based on the propagation theory of the angular spectrum [26,28], the optical field after propagating for an inter-layer distance Δl to the next scattering layer m + 1 is given by
$$\tilde{E}_{m + 1}^\textrm{1}({k_x},{k_y}) = \tilde{E}_m^\textrm{2}({k_x},{k_y}) \cdot \exp \left[ {i \cdot n \cdot \Delta l\sqrt {{{(\frac{{2\pi }}{\lambda })}^2} - k_x^2 - k_y^2} } \right],$$
where n is the refractive index of the medium, and λ is the wavelength of the incident light. When the scattering medium with a thickness l is divided into Nmax scattering intervals (Nmax + 1 layers), the inter-layer distance Δl is calculated by
$$\Delta l = \frac{l}{{{N_{\max }}}}.$$
The optical field $E_{m\textrm{ + 1}}^1(x,y)$ in layer (m + 1) can be obtained by the inverse Fourier transform of the spectrum $\tilde{E}_{m\textrm{ + 1}}^1(x,y)$. Moreover, the parameters of a scattering medium that can be measured have an important connection with the parameters of a scattering layer (phase transformation), which has been studied in detail by Cheng et al [27].

2.2 Transport theory with tenuous scattering approximate solution

The equation of radiation transfer is always solved with approximate solutions because it is difficult to be calculated numerically and obtained exact solutions. The scattering layer in EASM can be regarded as a tenuous scattering medium when under certain approximate conditions because the medium is layered and divided virtually. The scattering radiation intensity in a position inside the scattering medium equals to the sum of the intensities scattered by all particles when the total radiation intensity irradiates them. By the first order multiple scattering approximation, the total radiation intensity is equal to the known reduced incident intensity [18].

The radiation intensity $I(r,\hat{s})$ of the optical field $E_{m\textrm{ + 1}}^1(x,y)$ at a position inside the scattering medium includes the reduced incident intensity ${I_{ri}}(r,\hat{s})$ and the diffuse intensity ${I_d}(r,\hat{s})$, and it is given by

$$I(r,\hat{s}) = {I_{ri}}(r,\hat{s}) + {I_d}(r,\hat{s}).$$
Based on the first-order approximate solution of the equation of the radiation transfer, the diffuse intensity ${I_d}(r,\hat{s})$ can be approximately calculated by
$${I_d}(r,\hat{s}) \approx \int_0^s {\exp [{ - (\tau - {\tau_1})} ]} \cdot \left[ {\frac{{\rho {\sigma_t}}}{{4\pi }}\int\limits_{4\pi } {p(\hat{s},\hat{s}^{\prime}) \cdot {I_{ri}}(r,\hat{s}^{\prime})} d\omega^{\prime} + \varepsilon ({r_1},\hat{s})} \right]d{s_1},$$
where ρ is the number of scattering points per unit volume, r is the position vector, $\hat{s}$ is the direction vector, and s is the distance from the light incident point to the scattering point. For the convenience of calculation, the dimensionless optical thickness and the cosine of azimuth are expressed by
$$\tau = \rho {\sigma _t}z = \rho {\sigma _t}s\cos \theta ,$$
$${\mu _0} = \cos {\theta _0},$$
$$\mu = \cos \theta ,$$
where z is the position of the plane for energy calculation and θ0 is incident propagation angle and θ is the scattering angle. For convenience, σt is equal to the sum of the scattering cross section σs and absorption cross section σa, and ρσt is always replaced with extinction coefficient μt. The z-axis is as the reference axis, and the propagation angle represents the angle with the z-axis in this method.

Scattering energy is related to the angle of an incident plane wave. When the incident light is a plane wave with a propagation direction μ0, the reduced incident intensity Iri is

$${I_{ri}}(\tau ,\mu ,\phi ) = {F_0}\exp ( - \tau /{\mu _0})\delta (\hat{\omega } - {\hat{\omega }_0}),$$
where F0 is the initial incident radiation flux, ω and ω0 are the unit vector in direction $(\mu ,\phi )$ and $({\mu _0},{\phi _0})$. As shown in Fig. 4, the forward radiation intensity Id+ contains the intensity between 0 to z in the scattering medium when 0 ≤ θ ≤ π/2 (0 ≤ µ ≤ 1). The forward radiation intensity Id+ can be written as
$$\begin{array}{c} {I_{d + }}(\tau ,\mu ,{\mu _0},\phi ) = \int_0^\tau {\exp \left[ { - \frac{{(\tau - {\tau_1})}}{\mu } - \frac{{{\tau_1}}}{{{\mu_0}}}} \right]} \cdot \frac{{p(\mu ,\phi ;{\mu _0},{\phi _0})}}{{4\pi }}{{F^{\prime}}_0}\frac{{d{\tau _1}}}{\mu }\\ \textrm{ } = \frac{{p(\mu ,\phi ;{\mu _0},{\phi _0})}}{{4\pi }}\frac{{\exp ( - \tau /{\mu _0}) - \exp ( - \tau /\mu )}}{{({\mu _0} - \mu )}}{\mu _0}{{F^{\prime}}_0}, \end{array}$$
where ${F^{\prime}_0}$ is incident radiation flux taking absorption into account, $p(\mu ,\phi ;{\mu _0},{\phi _0})$ is the phase function, which is widely used in scattering calculations, $({\mu _0},{\phi _0})$ and $(\mu ,\phi )$ represent the azimuths of the incident light and scattering light. Similarly, the backward radiation intensity Id- containing the intensity from z to d when π/2 ≤ θ ≤ π (-1 ≤ µ ≤ 0) is expressed as
$$\begin{array}{c} {I_{d - }}(\tau ,\mu ,{\mu _0},\phi ) = \int_\tau ^{{\tau _0}} {\exp \left[ { - \frac{{(\tau - {\tau_1})}}{\mu } - \frac{{{\tau_1}}}{{{\mu_0}}}} \right]} \cdot \frac{{p(\mu ,\phi ;{\mu _0},{\phi _0})}}{{4\pi }}{{F^{\prime}}_0}\frac{{d{\tau _1}}}{{ - \mu }}\\ \textrm{ } = \frac{{p(\mu ,\phi ;{\mu _0},{\phi _0})}}{{4\pi }}\frac{{\exp ( - \tau /{\mu _0}) - \exp [ - {\tau _0}/{\mu _0} + ({\tau _0} - \tau )/\mu ]}}{{({\mu _0} - \mu )}}{\mu _0}{{F^{\prime}}_0}, \end{array}$$
where τ0 is the dimensionless optical thickness of the scattering medium (equals to $\rho {\sigma _t}d$). In particular, when θ = θ0, the forward radiation intensity Id+ need to be calculated by
$${I^{\prime}_{d + }}(\tau ,{\mu _0},\phi ) = \frac{{p(\mu ,\phi ;{\mu _0},{\phi _0})}}{{4\pi }}\frac{{\tau \exp ( - \tau /{\mu _0})}}{{{\mu _0}}}{F^{\prime}_0}.$$

 figure: Fig. 4.

Fig. 4. Schematic diagram of a plane wave irradiating parallel scattering medium under tenuous scattering approximation.

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The phase function $p(\mu ,\phi ;{\mu _0},{\phi _0})$ is used to describe the probability distribution density of the scattering angle for calculation. We find that the probability distribution function for the cosine of the deflection angle cos(γ) commonly used in the scattering calculation of biological tissues is very suitable for the anisotropic scattering calculation in our method [29]. The deflection angle γ equals to the scattering angle θ minus incident propagation angle θ0. It is noted that using the cosine of the scattering angle u instead of deflection angle γ will greatly simplify the calculation, and the approximation arise errors only when the incident propagation deviates greatly from the reference axis (z-axis). Fortunately, in the tenuous scattering medium, the scattering energy mainly propagates in the forward direction and is concentrated near the incident direction when the light is vertically incident on the scattering medium. It is also noted that $(\phi ,{\phi _0})$ can be eliminated by integrating the phase function. Therefore, the phase function can be reduced and expressed by

$$p(\mu ) = \frac{{1 - {g^2}}}{{2{{(1 + {g^2} - 2g\mu )}^{3/2}}}},$$
where g is the anisotropy factor which is 0 to 0.9 for most biological tissues. Although the approximation of phase function makes the scattering calculation feasible, it still need to be improved in strong scattering media. In addition, to describe both scattering and absorption in a scattering medium, the incident radiation flux after absorption F0 is modified to
$${F^{\prime}_0} = \frac{{{\mu _s}}}{{{\mu _s} + {\mu _a}}}{F_0} = \frac{{{\mu _s}}}{{{\mu _t}}}{F_0}$$
where µs is scattering coefficient, µa is absorption coefficient, and µt is extinction coefficient (µt equals to µs plus µa). The forward radiation intensity Id+ and backward radiation intensity Id- are
$${I_{d + }}(\tau ,\mu ,{\mu _0},g) = \frac{1}{{4\pi }}\frac{{1 - {g^2}}}{{2{{(1 + {g^2} - 2g\mu )}^{3/2}}}}\frac{{\exp ( - \tau /{\mu _0}) - \exp ( - \tau /\mu )}}{{({\mu _0} - \mu )}}{\mu _0}\frac{{{\mu _s}}}{{{\mu _s} + {\mu _a}}}{F_0}$$
$$\scalebox{0.95}{$\displaystyle{I_{d - }}(\tau ,{\tau _0},\mu ,{\mu _0},g) = \frac{1}{{4\pi }}\frac{{1 - {g^2}}}{{2{{(1 + {g^2} - 2g\mu )}^{3/2}}}}\frac{{\exp ( - \tau /{\mu _0}) - \exp [ - {\tau _0}/{\mu _0} + ({\tau _0} - \tau )/\mu ]}}{{({\mu _0} - \mu )}}{\mu _0}\frac{{{\mu _s}}}{{{\mu _s} + {\mu _a}}}{F_0}$}$$
When θ equals to θ0, the forward radiation intensity Id+ is specifically expressed as
$${I^{\prime}_{d + }}(\tau ,{\mu _0},g) = \frac{1}{{4\pi }}\frac{{1 - {g^2}}}{{2{{(1 + {g^2} - 2g\mu )}^{3/2}}}}\frac{{\tau \exp ( - \tau /{\mu _0})}}{{{\mu _0}}}\frac{{{\mu _s}}}{{{\mu _s} + {\mu _a}}}{F_0}$$

The approximate solutions of transport theory lay a foundation for energy calculation in EASM. However, it is necessary to further calculate the proportion of each plane wave component in the scattering optical field since this theory is applicable to plane waves in a certain direction u0.

2.3 Calculating energy efficiency in mono-layer medium

Mono-layer medium in this chapter indicates that the physical property of the medium is single, not that the scattering layer is single-layer. The forward radiation flux consists of three parts: 1) Radiation flux calculated by the integral of Id+ in forward solid angle; 2) Radiation flux when θ = θ0, and 3) Reduced radiation flux calculated by the integral of reduced incident intensity Iri. Therefore, the forward radiation flux F + (µ0) is expressed by

$${F_ + }({\mu _0}) = 2\pi \int\limits_0^1 {{I_{d + }}(\tau ,\mu ,{\mu _0},g)\mu d\mu } + 2\pi \int\limits_0^1 {{{I^{\prime}}_{d + }}(\tau ,\mu ,{\mu _0},g)\mu d\mu } + \int {{I_{ri}}(\tau ,{\mu _0},\omega )d\omega } .$$

Similarly, the backward radiation flux is calculated by the integral of Id- in backward. The backward flux F-(µ0) is given by

$${F_ - }({\mu _0}) = 2\pi \int\limits_{ - 1}^0 {{I_{d - }}(\tau ,\mu ,{\mu _0},g)\mu d\mu } .$$

The integral operation of the formulas applied in this method is relatively complex and can be solved by numerical methods. It is worth noting that the flux calculation depends on the incident light angle µ0 when calculating the energy. Therefore, it is necessary to calculate the energy proportion of each plane wave component in different propagation directions in the incident light field. Any complex optical field can be considered to be composed of various plane waves with different spatial frequencies, and each plane wave is a basis. The incident wavefront is decomposed into a linear combination of different plane waves. The angle of the plane wave in (x, y, z) directions (α, φ, θ). The propagation direction uz along the z-axis of a plane wave is expressed as

$$uz = \cos (\theta ).$$

It should be noted that the effective range of the angular is 1-(cosα)2-(cosφ)2 > 0. According to the angular spectrum theory, the proportion (weights) of each component is obtained by the Fourier transform of the optical field $E_m^1(x,y)$ before incident to each scattering layer. The proportion A0(uz) in different directions of the incident optical field $E_m^1(x,y)$ is given by

$${A_0}(uz) = {A_0}(ux,uy) = |{{\cal F}\{ E_m^1(x,y)\} } |,$$
where ${\cal F}\{{\cdot} \}$ represents the operator of the Fourier transform. Then the energy weights wz of the optical field is obtained by
$${w_z}(i) = \frac{{{{\{ {A_0}(uz(i))\} }^2}}}{{\sum\limits_{i = 1}^{nz} {{{\{ {A_0}(uz(i))\} }^2}} }},$$
where nz is the total number of the sampled angles in z direction of the optical field. The angles and energy proportions of the plane waves in different propagation directions in the incident light field can be obtained through these approximate calculations.

Based on the radiation flux calculation, the forward energy efficiency (normalized) ηf of the optical field after passing through a scattering layer m is

$$\eta _f^m = \sum\limits_{i = 1}^{nz} {{w_z}(i)\frac{{{F_ + }({\mu _0}(i))}}{{{F_0}}}} ,$$
where F0 is the initial incident radiation flux and $\eta _f^1$ equals to 1. Similarly, the backward efficiency ηb of the optical field after passing through the scattering layer m can be calculated by
$$\eta _b^m = \sum\limits_{i = 1}^{nz} {{w_z}(i)\frac{{{F_ - }({\mu _0}(i))}}{{{F_0}}}} .$$

Since energy efficiency (transmittance, absorption and reflectance) are normalized, the absorption efficiency $\eta _a^m$ in the scattering layer m is

$$\eta _a^m = 1 - \eta _f^m - \eta _b^m,$$
where $\eta _f^m$ and $\eta _b^m$ are forward and backward efficiency. Then, the forward efficiency ηf of light after passing through all layers of a scattering medium can be obtained by calculating the efficiency of every layer. When the scattering medium has the refractive index matching boundary, the total transmittance (forward) efficiency is expressed as
$${\eta _f} = \prod\limits_{m = 2}^{{N_{\max }} + 1} {\eta _f^m} ,$$
where Nmax is scattering intervals, and Nmax +1 is the number of total layers. Since the first layer is the starting layer, the scattering efficiency is calculated from the second layer to the last one. The reflectance (backward) efficiency ηb is
$${\eta _b} = \sum\limits_{k = 2}^{{N_{\max }}\textrm{ + 1}} {\left\{ {\left[ {\prod\limits_{m = 1}^{k - 1} {\eta_f^m} } \right] \cdot \eta_b^k} \right\}} .$$

Therefore, the normalized absorption efficiency ηa is calculated by

$${\eta _a} = 1 - {\eta _f} - {\eta _b}.$$

The influence of the Fresnel reflection should be considered when EASM is used in a scattering medium with a refractive index mismatching boundary. In our method, the average Fresnel refection equation is used to calculate the energy change that occurred in the interface of a medium with different inside and outside refractive indexes. The Fresnel reflection [30] between the interface with refractive indexes ni and nt is given by $R({\psi _i})$.

Similarly, the energy proportion of the optical field is calculated by

$${w_r}(j) = \frac{{{{\{ {A_1}(uz(j))\} }^2}}}{{\sum\limits_{j = 1}^{nz} {{{\{ {A_1}(uz(j))\} }^2}} }}$$
where A1(uz) is the proportion in different directions of the optical field on the surface of a medium. It is worth noting that half of the space frequency (nz/2) can be used for calculation considering the energy distribution of angular spectrum is approximately symmetrical when improving the calculation speed. The Fresnel reflection Rf in the boundary of a scattering medium can be expressed as ${R_f}$.

The transmittance (forward) efficiency ${\eta ^{\prime}_f}$, reflectance (backward) efficiency ${\eta ^{\prime}_b}$ and absorption efficiency ${\eta ^{\prime}_a}$ in refractive index mismatching boundary are approximately calculated by

$${\eta ^{\prime}_f} = (1 - {R_i}) \cdot {\eta _f} \cdot (1 - {R_f})$$
$${\eta ^{\prime}_a} = (1 - {R_i}) \cdot ({R_f}{\eta _f}{\eta _a} + {R_f}{\eta _f}{\eta _f}{R_f} + {\eta _a})$$
$${\eta ^{\prime}_b} = 1 - {\eta ^{\prime}_f} - {\eta ^{\prime}_a}$$
where Ri is the Fresnel refraction when the incident light irradiating the surface of the scattering medium vertically. It should be noted that we assume that the scattering medium is large enough so that there is no side energy leakage and reflection. This approximation is common in biological tissues and scattering phosphors.

2.4 Calculating energy efficiency in multi-layered medium

EASM can also be used to calculate the change of light energy in multi-layered scattering media with different physical properties (refractive index, scattering and absorption coefficients). The interaction of the light and multi-layered media or tissues is more complex than the single-layer medium. In this chapter, the change of light energy in two media with different properties are introduced. The calculation of more layers of the media is similar to this.

Energy transportation is mainly dependent on the difference of refractive indexes between two kinds of scattering media. When light from one medium to another, the forward efficiency ηf2 in the interface of two media is

$${\eta _{f2}} = {\eta _{f1}}(1 - {R_l})$$
where ηf1 is the energy efficiency before passing through the interface of two kinds of media, and Rl is the Fresnel refraction based on the refractive indexes of two kinds of media. The backward efficiency ${\eta ^{\prime}_{b1}}$ is calculated by
$${\eta ^{\prime}_{b1}} = {\eta _{b1}} + {\eta _{f1}}{R_l}$$
where ηb1 is the backward efficiency in the first medium. The backward efficiency is the sum of the backward efficiency that occurred in the first medium and the energy obtained from the interface by Fresnel reflection.

Due to the Fresnel reflection on the emitting surface, the total forward efficiency ηf should be expressed as

$${\eta _f} = {\eta _{f2}}(1 - {R_{in}})$$
where Rin is the Fresnel reflection on the emitting surface. The total reflectance (backward) efficiency ηb can be approximately expressed as
$${\eta _b} = {\eta _{f2}}{R_{in}} + {\eta ^{\prime}_{b1}} + {\eta _{b2}}$$
where ηb2 is the backscattering caused by the second medium. The energy calculation method in each medium is the same as that in the mono-layer medium. Then, the normalized absorption efficiency in multi-layered medium is calculated by
$${\eta _a} = 1 - {\eta _f} - {\eta _b}$$

It is worth noting that the Fresnel reflection still occurred in the incident surface of the medium, and the total energy calculation needs to consider it. Significantly, the energy variation of more kinds of media can also be calculated according to the example of two-layered media.

3. Results

3.1 Example: EASM simulates a strongly scattering and absorbing medium

Figure 5 illustrates a simulation example of EASM on the light propagating through a strongly scattering and absorbing sample. The main parameters of this sample are the scattering coefficient of 50 cm-1, the absorption coefficient of 50 cm-1, the anisotropy factor of 0.8, refractive index of 1 and the thickness of 200 µm. The scattering sample is divided into 21 scattering layers (20 intervals) with an interval of 10 µm, and each scattering layer has been given a random phase. Compared with the size of the incident light (the diameter of incident light is 20 µm), the size of the medium in the x-y directions can be considered to be nearly infinite.

 figure: Fig. 5.

Fig. 5. The normalized intensity of the optical field propagating through scattering layers No. 2, 6, 11, 16, 21 (the total number of layers is 21). The interval between two scattering layers is 10 µm, and the energy of the incident light is normalized to 1.

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The forward light energy of the optical field propagating through the scattering sample after layers (number m = 2, 6, 11, 16, 21, the incidence layer and other layers are omitted) is shown in Fig. 5. The results show that the forward light energy decreases gradually due to the absorption and backward scattering effects when it passes through each scattering layer. The forward energy, obtained by EASM, decays by about 25% when the light passes through a distance 50 µm. The results show that EASM can calculate the forward energy change of light passing through any length of the scattering medium. We compare the simulation results, including the normalized total transmittance, absorption and diffuse reflectance, with that simulated by MCM. In addition, for the convenience of comparison, EASM and conventional MCM are both compiled in Matlab language. We rewrite the MCM from C language to Matlab language according to Ref. [15]. The MCM applied in this paper are compared with that in the references to obtain the accuracy of MCM (see Supplement 1 for details). To keep a satisfactory precision for simulation, 100000 simulated photon packets are participated in MCM. As shown in the yellow box in Fig. 5, difference of the absorption and reflectance are both within 0.004. The results are indicative of a great match of EASM and MCM in light energy calculation in a scattering and absorbing sample.

It is worth noting that the wavefront simulation and control in the scattering layers by the conventional angular spectrum method have been studied in detail [26,27], and this part will not be repeated here. The calculation accuracy and time advantage will be described in the subsequent section.

3.2 Accuracy verification of EASM with various scattering media

To verify the accuracy of our method in the calculation of the light-medium interaction, we compute the total transmittance and absorption energy of the optical field in several simulated and experimental samples by EASM compared with that obtained by the conventional MCM. The total incident light energy, transmittance, absorption, and reflectance are normalized and the main parameters of the scattering samples are depicted in Table 1.

Tables Icon

Table 1. Parameters for simulated and experimental samples

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The influence of the refractive index and Fresnel reflection must be considered when calculating the energy transport in a medium. The refractive index matching boundary means that the refractive index of a scattering sample is equal to that of the surrounding medium. As shown in Table 1, the types of samples include I. Simulated samples No. 1-3 with refractive index matching boundary, II. Simulated samples No. 4-15 with refractive index mismatching boundary and III. Experimental samples No. 16-18 Ce:YAG glass, single crystal and ceramics, respectively. The parameters in Table 1 are the scattering coefficient µs (cm-1), the absorption coefficient µa (cm-1), the anisotropy factor g, refractive index n, number of intervals Nl (number of layers is Nl +1), the interval between two layers Δl (µm) and total thickness l (µm). The surrounding medium is considered as air with a refractive index of 1.

The Ce:YAG glass (No. 16), Ce:YAG single crystal (No. 17), and Ce:YAG ceramic (No. 18) samples are selected in the experiment due to their strong absorption of blue laser. The samples are obtained from Rayshine Optics & Film materials Co., Ltd and Shenzhen Graduate School of Tsinghua University. The laser diode (LD) module with a central wavelength 445 ± 5 nm under power 200 mW (MDL-III-445L, Changchun New Industries Optoelectronics Tech Inc.) is applied to emit pump blue laser. The anisotropy factor g of the sample is usually between 0.75 and 0.9 [16,31]. The refractive index n is measured by the material supplier. The scattering coefficient and absorption coefficient are measured and calculated by the collimated transmission method with the non-absorption laser and absorption laser of the materials. The detailed measurement and calculation method can refer to [32]. The scattering coefficients of the non-absorption laser and blue laser are usually considered the same to simplify the calculation and measurement when their central wavelengths are similar [16]. The experimental setup of the transmittance measurement and the sample pictures are shown in Fig. S4 in Supplement 1.

Figure 6(a) and (b) illustrate the calculated and measured total transmittance and absorption of the light. Since forward and absorption energy are crucial to the biological imaging and laser lighting, we mainly discuss the calculation of these two parameters. Detailed results are shown in Supplement 1. From the results of the comparison, the difference in the transmittance of the low scattering (µs/µa is 0.11) and medium scattering (µs/µa is 1) simulated samples are quite small, the maximum difference of 0.028 appears in sample No. 4. Similarly, the maximum difference in the absorption is 0.033 in sample No. 4. The results indicate that EASM expresses excellent calculation accuracy with the standard results obtained by the conventional MCM in low and medium scattering materials.

 figure: Fig. 6.

Fig. 6. (a), Total normalized transmittance of light when it passes through the scattering samples. (b), Total normalized absorption by the scattering samples after light passing through. I. Samples with refractive index matching boundary, II. Samples with refractive index mismatching boundary and III. Experimental samples: glass, single crystal and ceramics.

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However, the results of the simulation samples with strongly scattering (µs/µa is 9, samples No. 2, 8, 11 and 14) have a relatively large error in the absolute value of the difference. The main reason for the error is that the transport theory adopts a tenuous approximation in each modeled scattering layer, which suggests that the accuracy of EASM is weaker than that of MCM in strongly scattering media. When the refractive index does not match, more light is reflected back to the sample and reduces the transmittance due to the strong Fresnel reflection and total reflection in the sample surface, making the strong scattering error weakened. However, we still believe that this is only a numerical approximation, which does not mean that it will improve the calculation accuracy of strong scattering when the refractive index is matched. We suggest that EASM should be applied carefully under strong scattering conditions. More divided layers and shorter inter-layer distance are helpful to improve the calculation accuracy at the cost of computing time (see Results 3.3 and Fig. 8).

To further validate the validity of EASM, the total transmittance of the experimental samples (No.16 glass, No.17 single crystal, and No.18 ceramic) are measured by integrating spheres, respectively. The maximum difference of the transmittance between EASM and measurement is 0.014, appearing in sample No. 16. In general, the simulation and experimental results prove its computational adaptability in most scattering samples.

3.3 Time performance and accuracy affected by layer numbers

The relationship between the number of layers and calculation accuracy, and the computing time are also analyzed. EASM and MCM are implemented by Matlab R2017b in the hardware platform based on the Intel Core i7-10710U CPU and 16 GB RAM. MCM takes 100000 photon packets in the simulation to guarantee calculation accuracy [15]. The transmittance, absorption, and time consumption of the simulated sample No. 10 obtained with different scattering layers are shown in Fig. 7(a).

 figure: Fig. 7.

Fig. 7. (a), Total transmittance (blue line) and absorption (black line) of No. 10 sample with several numbers of layers calculated by EASM. The computing time (red line) is represented by the right red axis. The blue and black dashed lines represent the transmittance (0.299) and absorption (0.653) obtained by MCM. (b), No. 17 single crystal sample. The blue and black dashed lines represent the transmittance (0.082) and absorption (0.833) obtained by MCM. (c), Transmittance and absorption of the sample No. 10 are calculated by EASM with thickness of 600 µm, layers 601 and interval 1 µm, represented by black and blue lines, respectively. The transmittance and absorption calculated by MCM are represented by red circle and triangle, respectively. (d), The transmittance and absorption of the sample No. 18 are calculated by EASM with thickness 1200 µm, layers 121 and interval 10 µm, represented by black and blue lines, respectively. The transmittance and absorption calculated by MCM are represented by the red circle and triangle. The experimental transmittance is represented by the yellow fork.

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The calculation results indicate that the transmittance and absorption of EASM are closer to that of MCM with the increasing layers. If the number of layers is small, it will deviate from the approximate conditions, resulting in a decrease in calculation accuracy. When the number of layers is 6, the difference in transmittance obtained by EASM and MCM is 0.007, and the difference in absorption is 0.008. The difference is minimal, within 0.01, when the scattering sample is divided into 11 scattering layers. The results indicate that there is no need to divide too many layers to achieve a good calculation accuracy in medium scattering material. Too many divided layers may cause too large cumulative error. Furthermore, EASM also shows a remarkable time advantage compared with MCM. As shown in Fig. 7(a), when the number of scattering layers is 11, EASM only takes 6.5 s, which is significantly faster than MCM (96 s). More importantly, the time of phase calculation is also included in the EASM computing time, which is a lack in MCM. The more scattering layers are calculated, the slower the calculation speed will be.

As shown in Fig. 7(b), the single crystal sample No. 17 is also calculated by EASM and MCM. As we can see from the results, the number of scattering layers has little effect on the transmittance and absorption calculation. The possible reasons are that the calculation is more consistent with the tenuous condition and the forward cumulative error is smaller when the scattering effect is weak. The maximum difference of the transmittance is within 0.001 when the number of layers is 26. MCM takes 87 s in computing sample No. 17 in time consuming. By comparison, the computing times are 1.3 s, 3.2 s, and 6.5 s for the number of layers by EASM 3, 6 and 11, respectively. It suggests EASM has a faster computing speed compared with the conventional method.

More importantly, the time advantage of EASM is also in the simultaneous simulation of several scattering samples under different thicknesses with uniform properties. As shown in Fig. 7(c), the scattering sample No. 10 is simulated by EASM to obtain the transmittance and absorption within a thickness 600 µm, layers 601 and an interval of 1 µm. The calculated results of any samples with a thickness within the maximum value of 600 µm could be obtained, only taking 542 s. For comparison, five simulated samples with several thicknesses (100 µm, 200 µm, 400 µm, and 600 µm) are calculated by MCM and it takes an average of about 90 s for each one. It means that more than 54000 s will be consumed when calculating 600 samples with different thicknesses.

We also experimentally verify the accuracy while maintaining time advantage by measuring the transmittance of two samples No. 18 with thicknesses of 500 µm and 1160 µm using the integrating sphere. The difference in the transmittance of the measurement and EASM are 0.006 and 0.002 for thicknesses 500 µm and 1160 µm, respectively. 125 s is spent by EASM to calculate the sample with 121 layers and an interval 10 µm. If the same number of samples is calculated by MCM, it will take a lot of computing time (about 90 s for each one). Hence, the time advantage of EASM is excellent compared with the conventional method when ensuring the calculation accuracy.

It can be seen from the analysis that the selection of the scattering layers and inter-layer distance have an impact on the time performance and accuracy of the EASM calculation. As shown in Fig. 8, the normalized absorption efficiency of the samples No. 1, 2 and 3 at different inter-layer distance (total thickness is 200 µm) are calculated by EASM and compared with MCM (see Fig. 6). The red dotted circles represent the values closest to that calculated by MCM.

 figure: Fig. 8.

Fig. 8. (a), Normalized transmittance efficiency and (b) absorption efficiency of the samples No. 1, 2, and 3 calculated by EASM in several inter-layer distances (total thickness is 200 µm). The red dotted circles represent the values closest to that calculated by MCM.

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It can be seen from the results that, the calculation error under the strong scattering is greater than that under the medium and low scattering due to the tenuous scattering approximate solution. The transmittance error of sample 2, 2 µm inter-layer distance and 101 layers, is only 5.3%, which is more accurate than that obtained when the inter-layer distance is 10 µm. More and thinner scattering layers in the calculation of strong scattering media are helpful to meet the approximate conditions. When calculating the medium scattering samples, the result suggests that the sample interval is about 10 µm, and the inter-layer distance can be adjusted appropriately considering the calculation time. In addition, the experiment results show that larger inter-layer distance and fewer layers can be selected under low scattering to improve the calculation speed and maintain the calculation accuracy, which is also consistent with the results in Fig. 7(b). It is worth noting that the parameters of the scattering medium vary greatly. It is necessary to select parameters according to the calculation accuracy and computing time.

3.4 Light transport in multi-layered media

Biological tissue or colorful phosphor usually has a multi-layered structure, so an interaction of multi-layered scattering media and light should also be considered. As shown in Fig. 9, two media with different properties are combined to be calculated by EASM and MCM, respectively.

 figure: Fig. 9.

Fig. 9. (a), Multi-layered media with refractive index matching boundary. (b), Energy variation (transmittance, absorption and reflectance) of light passing through multi-layered media in Fig. 9(a). (c), Multi-layered media with refractive index mismatching boundary. (d), Energy variation of light passing through multi-layered media in Fig. 9(c). (e), Multi-layered media with refractive index mismatching boundary and different refractive indexes. (f), Energy variation of light passing through multi-layered media in Fig. 9(e).

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The parameters of multi-layered samples are shown in Fig. 9(a), (c), and (e). Every multi-layered sample has a thickness of 200 µm and is divided into 21 scattering layers and an interval of 1 µm. When calculating the multi-layered media with refractive index matching boundary, the maximum difference between the two methods is 0.003 appeared in transmittance simulation (Fig. 9(b)). As shown in Fig. 9(d), the energy variation of the scattering media with refractive index mismatch can also be calculated by EASM. The Fresnel reflection on the front and rear surfaces of the sample will occur. In addition, the energy calculation became more complicated when the refractive indexes of two scattering media are even different (Fig. 9(f)). The energy calculation by two methods matched well in this complex multi-layered media. It is noteworthy that the energy variation of more kinds of media can also be calculated according to the two-layered media. The result is indicative of a good accuracy of EASM in calculating light transport inside multi-layered media.

4. Discussion

Our results show that EASM provides an effective and fast tool to analyze and calculate the energy and wavefront variation of the optical field inside a scattering media. By this method, the energy and wavefront can be simulated simultaneously without being separated as in the conventional method. On the premise of ensuring the calculation accuracy, EASM has an excellent time advantage, especially when calculating several media with the same properties under different thicknesses. Furthermore, multi-layered media can also be calculated by EASM with a good precision.

However, there are some caveats to note. First, it has a relatively large error when applied to the calculation of strongly scattering media due to the tenuous scattering approximation of the transport theory. Second, the verification of the calculation accuracy in complex media still needs a large number of simulations and experiments, as this work only covers limited possible cases. Third, excessive division of scattering layers may cause cumulative error. Thus, we expect that EASM plays a unique role in some particular simulation (for example, high-speed and real-time simulation) rather than a replacement for the conventional method.

So far, this method has focused on the application of the biological media in biological imaging and the scattering phosphor in laser lighting. However, EASM is promising for other potential applications as well, such as scattering imaging, optical fiber imaging, and optical neural network. The application of this method in those fields needs further work in the future.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Supplemental document

See Supplement 1 for supporting content.

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Supplementary Material (1)

NameDescription
Supplement 1       detailed description of EASM, detailed data in figures

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (9)
Fig. 1.
Fig. 1. (a) Optical field propagates through a scattering medium. The energy and wavefront of the light change due to scattering effects. (b), Conventional angular-spectrum modeling. A medium is modeled into several scattering layers with intervals. The wavefront of the incident light will change when it passes through a scattering layer, which is considered as a phase transformation. (c), Light energy change happened in a single scattering layer. The light will be absorbed and scattered (forward or backward) by a single scattering layer. (d), Extended angular-spectrum modeling (EASM). The calculation of energy transport in a scattering medium is also added to the angular spectrum model.

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Fig. 2.
Fig. 2. Flow chart for EASM.

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Fig. 3.
Fig. 3. Processes of light field transports between scattering layers.

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Fig. 4.
Fig. 4. Schematic diagram of a plane wave irradiating parallel scattering medium under tenuous scattering approximation.

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Fig. 5.
Fig. 5. The normalized intensity of the optical field propagating through scattering layers No. 2, 6, 11, 16, 21 (the total number of layers is 21). The interval between two scattering layers is 10 µm, and the energy of the incident light is normalized to 1.

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Fig. 6.
Fig. 6. (a), Total normalized transmittance of light when it passes through the scattering samples. (b), Total normalized absorption by the scattering samples after light passing through. I. Samples with refractive index matching boundary, II. Samples with refractive index mismatching boundary and III. Experimental samples: glass, single crystal and ceramics.

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Fig. 7.
Fig. 7. (a), Total transmittance (blue line) and absorption (black line) of No. 10 sample with several numbers of layers calculated by EASM. The computing time (red line) is represented by the right red axis. The blue and black dashed lines represent the transmittance (0.299) and absorption (0.653) obtained by MCM. (b), No. 17 single crystal sample. The blue and black dashed lines represent the transmittance (0.082) and absorption (0.833) obtained by MCM. (c), Transmittance and absorption of the sample No. 10 are calculated by EASM with thickness of 600 µm, layers 601 and interval 1 µm, represented by black and blue lines, respectively. The transmittance and absorption calculated by MCM are represented by red circle and triangle, respectively. (d), The transmittance and absorption of the sample No. 18 are calculated by EASM with thickness 1200 µm, layers 121 and interval 10 µm, represented by black and blue lines, respectively. The transmittance and absorption calculated by MCM are represented by the red circle and triangle. The experimental transmittance is represented by the yellow fork.

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Fig. 8.
Fig. 8. (a), Normalized transmittance efficiency and (b) absorption efficiency of the samples No. 1, 2, and 3 calculated by EASM in several inter-layer distances (total thickness is 200 µm). The red dotted circles represent the values closest to that calculated by MCM.

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Fig. 9.
Fig. 9. (a), Multi-layered media with refractive index matching boundary. (b), Energy variation (transmittance, absorption and reflectance) of light passing through multi-layered media in Fig. 9(a). (c), Multi-layered media with refractive index mismatching boundary. (d), Energy variation of light passing through multi-layered media in Fig. 9(c). (e), Multi-layered media with refractive index mismatching boundary and different refractive indexes. (f), Energy variation of light passing through multi-layered media in Fig. 9(e).

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Tables (1)

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Table 1. Parameters for simulated and experimental samples

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Equations (37)

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$$E_m^\textrm{2}(x,y) = E_m^\textrm{1}(x,y) \cdot \exp [i{\varphi _m}(x,y)]\textrm{.}$$
$$\tilde{E}_{m + 1}^\textrm{1}({k_x},{k_y}) = \tilde{E}_m^\textrm{2}({k_x},{k_y}) \cdot \exp \left[ {i \cdot n \cdot \Delta l\sqrt {{{(\frac{{2\pi }}{\lambda })}^2} - k_x^2 - k_y^2} } \right],$$
$$\Delta l = \frac{l}{{{N_{\max }}}}.$$
$$I(r,\hat{s}) = {I_{ri}}(r,\hat{s}) + {I_d}(r,\hat{s}).$$
$${I_d}(r,\hat{s}) \approx \int_0^s {\exp [{ - (\tau - {\tau_1})} ]} \cdot \left[ {\frac{{\rho {\sigma_t}}}{{4\pi }}\int\limits_{4\pi } {p(\hat{s},\hat{s}^{\prime}) \cdot {I_{ri}}(r,\hat{s}^{\prime})} d\omega^{\prime} + \varepsilon ({r_1},\hat{s})} \right]d{s_1},$$
$$\tau = \rho {\sigma _t}z = \rho {\sigma _t}s\cos \theta ,$$
$${\mu _0} = \cos {\theta _0},$$
$$\mu = \cos \theta ,$$
$${I_{ri}}(\tau ,\mu ,\phi ) = {F_0}\exp ( - \tau /{\mu _0})\delta (\hat{\omega } - {\hat{\omega }_0}),$$
$$\begin{array}{c} {I_{d + }}(\tau ,\mu ,{\mu _0},\phi ) = \int_0^\tau {\exp \left[ { - \frac{{(\tau - {\tau_1})}}{\mu } - \frac{{{\tau_1}}}{{{\mu_0}}}} \right]} \cdot \frac{{p(\mu ,\phi ;{\mu _0},{\phi _0})}}{{4\pi }}{{F^{\prime}}_0}\frac{{d{\tau _1}}}{\mu }\\ \textrm{ } = \frac{{p(\mu ,\phi ;{\mu _0},{\phi _0})}}{{4\pi }}\frac{{\exp ( - \tau /{\mu _0}) - \exp ( - \tau /\mu )}}{{({\mu _0} - \mu )}}{\mu _0}{{F^{\prime}}_0}, \end{array}$$
$$\begin{array}{c} {I_{d - }}(\tau ,\mu ,{\mu _0},\phi ) = \int_\tau ^{{\tau _0}} {\exp \left[ { - \frac{{(\tau - {\tau_1})}}{\mu } - \frac{{{\tau_1}}}{{{\mu_0}}}} \right]} \cdot \frac{{p(\mu ,\phi ;{\mu _0},{\phi _0})}}{{4\pi }}{{F^{\prime}}_0}\frac{{d{\tau _1}}}{{ - \mu }}\\ \textrm{ } = \frac{{p(\mu ,\phi ;{\mu _0},{\phi _0})}}{{4\pi }}\frac{{\exp ( - \tau /{\mu _0}) - \exp [ - {\tau _0}/{\mu _0} + ({\tau _0} - \tau )/\mu ]}}{{({\mu _0} - \mu )}}{\mu _0}{{F^{\prime}}_0}, \end{array}$$
$${I^{\prime}_{d + }}(\tau ,{\mu _0},\phi ) = \frac{{p(\mu ,\phi ;{\mu _0},{\phi _0})}}{{4\pi }}\frac{{\tau \exp ( - \tau /{\mu _0})}}{{{\mu _0}}}{F^{\prime}_0}.$$
$$p(\mu ) = \frac{{1 - {g^2}}}{{2{{(1 + {g^2} - 2g\mu )}^{3/2}}}},$$
$${F^{\prime}_0} = \frac{{{\mu _s}}}{{{\mu _s} + {\mu _a}}}{F_0} = \frac{{{\mu _s}}}{{{\mu _t}}}{F_0}$$
$${I_{d + }}(\tau ,\mu ,{\mu _0},g) = \frac{1}{{4\pi }}\frac{{1 - {g^2}}}{{2{{(1 + {g^2} - 2g\mu )}^{3/2}}}}\frac{{\exp ( - \tau /{\mu _0}) - \exp ( - \tau /\mu )}}{{({\mu _0} - \mu )}}{\mu _0}\frac{{{\mu _s}}}{{{\mu _s} + {\mu _a}}}{F_0}$$
$$\scalebox{0.95}{$\displaystyle{I_{d - }}(\tau ,{\tau _0},\mu ,{\mu _0},g) = \frac{1}{{4\pi }}\frac{{1 - {g^2}}}{{2{{(1 + {g^2} - 2g\mu )}^{3/2}}}}\frac{{\exp ( - \tau /{\mu _0}) - \exp [ - {\tau _0}/{\mu _0} + ({\tau _0} - \tau )/\mu ]}}{{({\mu _0} - \mu )}}{\mu _0}\frac{{{\mu _s}}}{{{\mu _s} + {\mu _a}}}{F_0}$}$$
$${I^{\prime}_{d + }}(\tau ,{\mu _0},g) = \frac{1}{{4\pi }}\frac{{1 - {g^2}}}{{2{{(1 + {g^2} - 2g\mu )}^{3/2}}}}\frac{{\tau \exp ( - \tau /{\mu _0})}}{{{\mu _0}}}\frac{{{\mu _s}}}{{{\mu _s} + {\mu _a}}}{F_0}$$
$${F_ + }({\mu _0}) = 2\pi \int\limits_0^1 {{I_{d + }}(\tau ,\mu ,{\mu _0},g)\mu d\mu } + 2\pi \int\limits_0^1 {{{I^{\prime}}_{d + }}(\tau ,\mu ,{\mu _0},g)\mu d\mu } + \int {{I_{ri}}(\tau ,{\mu _0},\omega )d\omega } .$$
$${F_ - }({\mu _0}) = 2\pi \int\limits_{ - 1}^0 {{I_{d - }}(\tau ,\mu ,{\mu _0},g)\mu d\mu } .$$
$$uz = \cos (\theta ).$$
$${A_0}(uz) = {A_0}(ux,uy) = |{{\cal F}\{ E_m^1(x,y)\} } |,$$
$${w_z}(i) = \frac{{{{\{ {A_0}(uz(i))\} }^2}}}{{\sum\limits_{i = 1}^{nz} {{{\{ {A_0}(uz(i))\} }^2}} }},$$
$$\eta _f^m = \sum\limits_{i = 1}^{nz} {{w_z}(i)\frac{{{F_ + }({\mu _0}(i))}}{{{F_0}}}} ,$$
$$\eta _b^m = \sum\limits_{i = 1}^{nz} {{w_z}(i)\frac{{{F_ - }({\mu _0}(i))}}{{{F_0}}}} .$$
$$\eta _a^m = 1 - \eta _f^m - \eta _b^m,$$
$${\eta _f} = \prod\limits_{m = 2}^{{N_{\max }} + 1} {\eta _f^m} ,$$
$${\eta _b} = \sum\limits_{k = 2}^{{N_{\max }}\textrm{ + 1}} {\left\{ {\left[ {\prod\limits_{m = 1}^{k - 1} {\eta_f^m} } \right] \cdot \eta_b^k} \right\}} .$$
$${\eta _a} = 1 - {\eta _f} - {\eta _b}.$$
$${w_r}(j) = \frac{{{{\{ {A_1}(uz(j))\} }^2}}}{{\sum\limits_{j = 1}^{nz} {{{\{ {A_1}(uz(j))\} }^2}} }}$$
$${\eta ^{\prime}_f} = (1 - {R_i}) \cdot {\eta _f} \cdot (1 - {R_f})$$
$${\eta ^{\prime}_a} = (1 - {R_i}) \cdot ({R_f}{\eta _f}{\eta _a} + {R_f}{\eta _f}{\eta _f}{R_f} + {\eta _a})$$
$${\eta ^{\prime}_b} = 1 - {\eta ^{\prime}_f} - {\eta ^{\prime}_a}$$
$${\eta _{f2}} = {\eta _{f1}}(1 - {R_l})$$
$${\eta ^{\prime}_{b1}} = {\eta _{b1}} + {\eta _{f1}}{R_l}$$
$${\eta _f} = {\eta _{f2}}(1 - {R_{in}})$$
$${\eta _b} = {\eta _{f2}}{R_{in}} + {\eta ^{\prime}_{b1}} + {\eta _{b2}}$$
$${\eta _a} = 1 - {\eta _f} - {\eta _b}$$
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