Intensity Evolution of Cosine-Gaussian-Correlated Schell-Model Pulse Scattered by a Medium

Abstract

According to first-order Born approximation, the scattering of a partially coherent pulse with cosine-Gaussian correlation by a medium was studied. On the basis of analytic expression, the changes in intensity evolution of the scattered pulse are discussed. The influences of pulse and medium characteristics on the intensity of the scattered pulse were investigated. The intensities of a Gaussian Schell-model (GSM) pulse and a cosine-Gaussian-correlated Schell-model (CGSM) pulse, both scattered by the same medium, are compared, and their similarities and differences are examined in detail. The effective angular width of the scattered pulse could be modulated by the parameters of the pulse and medium. The obtained results could find potential applications in pulsed beam scattering.

1. Introduction

Light scattering has received continuous attention from researchers because of its potential applications in some areas, like medical detection and ocean remote sensing. Since pioneering research in 1870, many studies have been carried out in this field, where the optical statistical characteristics of a scattered pulse, such as intensity, polarization, and coherence, were extensively studied [1,2,3,4]. The inverse problem, obtaining information on a scatterer through the characteristics of a scattered pulse, was also examined [5,6]. However, most of the above investigations were carried out on the scattering of statistically stationary fields.

The optical pulse is an important part of a broad class of light beams [7]. The basic concepts and representations of partially coherent pulses were established [8,9]. In the discussion of pulsed-beam scattering [10,11,12,13,14,15,16], the initial pulse was fully or partially coherent, and the complex degree of coherence of the majority of these partially coherent pulses is a conventional Gaussian Schell-mode function [17]. In recent years, a large number of partially coherent sources with other types of Schell-mode coherence were proposed, e.g., nonuniform-correlation [18], multi-Gaussian Schell-model [19,20], Laguerre-Gaussian and Hermite-Gaussian Schell-model [21], and sinc-correlation sources [22]. Some experiments related to the realization of these pulses have also been made [23,24].

We studied the scattering of a cosine-Gaussian-correlated Schell-model (CGSM) pulse on a quasihomogeneous medium, and investigated the intensity evolution of the scattered pulse within the accuracy of first-order Born approximation. We pay more attention to examining the intensity variation of the scattered pulse with order-parameter n, duration, the temporal-coherence length of the initial pulse, and the radius and correlation length of the medium.

2. Theory

The temporal mutual coherence function for a CGSM pulse has the following form [19]:

$$\mathit{\Gamma }(t_1,t_2)=\langle E^{\ast }\left(t_1\right)E\left(t_2\right)\rangle ={\mathit{\Gamma }}_0\exp \left[-\frac{t_1^2+t_2^2}{4T_0^2}-\frac{{(t_2-t_1)}^2}{2T_c^2}+i{\omega }_0(t_1-t_2)\right]\cos \left(\frac{n\sqrt{2\pi }(t_2-t_1)}{T_c}\right),$$

(1)

where E(t₁) and E(t₂) are the complex analytic signals of pulse realizations at time t₁ and t₂, respectively; the asterisk is the complex conjugate; Γ₀ is a positive constant; T₀ represents the pulse duration; T_c is the temporal-coherence length denoting the temporal correlation of the pulse; ω₀ describes the carrier frequency of the pulse [25]; n is a constant greater than zero and does not need to be an integer. Obviously, for the case of n = 0, a CGSM pulse is simplified to a GSM pulse.

By performing Fourier transform on Equation (1), the cross-spectral density function for a CGSM pulse is given by [11]

$$W({\omega }_1,{\omega }_2)=W_0\exp \left[-\frac{{({\omega }_1-{\omega }_0)}^2+{({\omega }_2-{\omega }_0)}^2}{2{\mathit{\Omega }}_0^2}\right]\exp \left[-\frac{{({\omega }_1-{\omega }_2)}^2}{2{\mathit{\Omega }}_c^2}\right]\times \exp \left[-\frac{\sqrt{2\pi }nb}{T_c}\right]\cosh \left[b({\omega }_1+{\omega }_2-2{\omega }_0)\right],$$

(2)

where

$$W_0=\frac{\sqrt{2}{T}_0}{2\pi {\mathit{\Omega }}_0^{}}{\mathit{\Gamma }}_0$$

(3)

$${\mathit{\Omega }}_0=\sqrt{\frac{1}{2T_0^2}+\frac{2}{T_c^2}}$$

(4)

$${\mathit{\Omega }}_c=\frac{\sqrt{2}{T}_c}{2T_0}{\mathit{\Omega }}_0$$

(5)

$$b=\frac{n\sqrt{2\pi }}{{\mathit{\Omega }}_0^2{T}_c},$$

(6)

where Ω₀ denotes the spectral width of the CGSM pulse. Ω_c represents the spectral coherence width of the pulse. cosh(x) is a hyperbolic cosine function.

Suppose a CGSM pulsed beam is incident on a scatterer along the direction described by unit vector s₀ (Figure 1). The cross-spectral density of a CGSM pulse at two points with position vectors r₁′ and r₂′ can be written as [26]

$$\begin{array}{c}{W}^{(i)}({\mathbf{r}}_1^{\prime },{\mathbf{r}}_2^{\prime },{\omega }_1,{\omega }_2)=W_0\exp \left[-\frac{{({\omega }_1-{\omega }_0)}^2+{({\omega }_2-{\omega }_0)}^2}{2{\mathit{\Omega }}_0^2}\right]\exp \left[-\frac{{({\omega }_1-{\omega }_2)}^2}{2{\mathit{\Omega }}_c^2}\right]\\ \times \exp \left[-\frac{\sqrt{2\pi }nb}{T_c}\right]\cosh \left[b({\omega }_1+{\omega }_2-2{\omega }_0)\right]\exp \left[i(k_2{s}_0\cdot {\mathbf{r}}_2^{\prime }-k_1{s}_0\cdot {\mathbf{r}}_1^{\prime })\right]\end{array}$$

(7)

In the scattering process, the correlation function of the scattering potential is used to describe the scattering properties of a random medium [27]. In [10], it was assumed that the resonance frequency of molecules or atoms in the medium is approximately represented by the carrier frequency of the pulse. Therefore, the correlation function of the scattering potential is expressed as [28,29]

$$C_F({\mathbf{r}}_1^{\prime },{\mathbf{r}}_2^{\prime },\omega )={\langle F^{\ast }({\mathbf{r}}_1^{\prime },\omega )F({\mathbf{r}}_2^{\prime },\omega )\rangle }_m={\langle F^{\ast }({\mathbf{r}}_1^{\prime },{\omega }_0)F({\mathbf{r}}_2^{\prime },{\omega }_0)\rangle }_m=C_F({\mathbf{r}}_1^{\prime },{\mathbf{r}}_2^{\prime },{\omega }_0),$$

(8)

where F(r′, ω) is defined as the scattering potential of the medium, and<·>denotes averaging over the ensemble of a random medium [30].

According to the first-order Born approximation, we obtained the formula for the cross-spectral density function of the scattered pulse [31,32]:

$$\begin{array}{c}{W}^{(s)}(r{\mathbf{s}}_1,r{\mathbf{s}}_2,{\omega }_1,{\omega }_2)=W_0\exp \left[-\frac{{({\omega }_1-{\omega }_0)}^2+{({\omega }_2-{\omega }_0)}^2}{2{\mathit{\Omega }}_0^2}-\frac{{({\omega }_1-{\omega }_2)}^2}{2{\mathit{\Omega }}_c^2}\right]\frac{\exp [ir(k_2-k_1)]}{r^2}\\ \times \exp \left[-\frac{\sqrt{2\pi }nb}{T_c}\right]\cosh \left[b({\omega }_1+{\omega }_2-2{\omega }_0)\right]{\displaystyle \underset{D}{\int }{\displaystyle \underset{D}{\int }{C}_F({\mathbf{r}}_1^{\prime },{\mathbf{r}}_2^{\prime },{\omega }_0)\exp \left[-i\left({\mathbf{K}}_1\cdot {\mathbf{r}}_1^{\prime }+{\mathbf{K}}_2\cdot {\mathbf{r}}_2^{\prime }\right)\right]{\mathrm{d}}^3{r}_1^{\prime }{d}^3{r}_2^{\prime }}},\end{array}$$

(9)

where rs₁ and rs₂ are the position vectors of two observation points (s₁ and s₂ are unit vectors). K₁ = −k₁(s₁ − s₀) and K₂ = k₂(s₂ − s₀) are the momentum-transfer vectors. D is the domain that the medium occupies.

A quasihomogeneous medium was considered in this paper of which the correlation function of the scattering potential is given by [1,29,33]

$$C_F({\mathbf{r}}_1^{\prime },{\mathbf{r}}_2^{\prime },{\omega }_0)=C_0\exp \left[-\frac{{\left|{\mathbf{r}}_1^{\prime }+{\mathbf{r}}_2^{\prime }\right|}^2}{8{\sigma }_R^2}\right]\exp \left[-\frac{{\left|{\mathbf{r}}_1^{\prime }-{\mathbf{r}}_2^{\prime }\right|}^2}{2{\sigma }_r^2}\right],$$

(10)

where C₀ is a positive constant, σ_R is the effective radius of the medium, and σ_r is the correlation length of the media, which must satisfy inequality σ_R ≥ σ_r.

When we substitute from Equation (10) into Equation (9), we can obtain the cross-spectral density of a pulse scattered by a medium [34]:

$$\begin{array}{c}{W}^{(s)}(r{\mathrm{s}}_1,r{\mathrm{s}}_2,{\omega }_1,{\omega }_2)=W_0{C}_0{(2\pi {\sigma }_R{\sigma }_r)}^3\exp \left[-\frac{{({\omega }_1-{\omega }_0)}^2+{({\omega }_2-{\omega }_0)}^2}{2{\mathit{\Omega }}_0^2}-\frac{{({\omega }_1-{\omega }_2)}^2}{2{\mathit{\Omega }}_c^2}\right]\exp \left[-\frac{\sqrt{2\pi }nb}{T_c}\right]\\ \times \cosh \left[b({\omega }_1+{\omega }_2-2{\omega }_0)\right]\frac{\exp [ir(k_2-k_1)]}{r^2}\exp \left[-\frac{1}{2}{\sigma }_R^2{\left(K_1+K_2\right)}^2-\frac{1}{8}{\sigma }_r^2{\left(K_1-K_2\right)}^2\right].\end{array}$$

(11)

By using theinverse Fourier transform of Equation (11), the mutual-coherence function of the scattered pulse is expressed by the following formula:

$$\begin{array}{l}\times \exp \left[-\frac{(\frac{2T_0^2}{T_c^2}+a\frac{T_c^2+4T_0^2}{T_c^2{T}_0^2}){(t_1-t_2)}^2}{4(T_0^2+m+a)\left[1+(m-a)\frac{T_c^2+4T_0^2}{T_c^2{T}_0^2}\right]}\right]\exp \left[\frac{i{\omega }_0(t_1-t_2)}{1+(m-a)\frac{T_c^2+4T_0^2}{T_c^2{T}_0^2}}\right]\\ {\Gamma }_{}^{(s)}(r\mathbf{s},r\mathbf{s},t_1,t_2)=\frac{W_0{C}_0{(2\pi {\sigma }_R{\sigma }_r)}^3}{r^2\sqrt{2(T_0^2+m+a)\left[1+(m-a)\frac{T_c^2+4T_0^2}{T_c^2{T}_0^2}\right]}}\exp \left[-\frac{{(t_1-\frac{r}{c})}^2+{(t_2-\frac{r}{c})}^2}{4(T_0^2+m+a)}\right]\\ \times \exp \left[-\frac{2{\omega }_0^2(m-a)-b^2\frac{T_c^2+4T_0^2}{2T_c^2{T}_0^2}}{1+(m-a)\frac{T_c^2+4T_0^2}{T_c^2{T}_0^2}}-\frac{\sqrt{2\pi }nb}{T_c}\right]\cosh \left[\frac{b\left[2{\omega }_0(m-a)-\frac{i}{2}(t_1-t_2)\right]}{\frac{T_c^2{T}_0^2}{T_c^2+4T_0^2}+m-a}\right],\end{array}$$

(12)

where

$$m=\frac{{\sin }^2\frac{\theta }{2}\left(4{\sigma }_R^2+{\sigma }_r^2\right)}{2c^2}$$

(13)

$$a=\frac{{\sin }^2\frac{\theta }{2}\left(4{\sigma }_R^2-{\sigma }_r^2\right)}{2c^2}.$$

(14)

With the help of unified theory [35,36], where spectral density, spectral degree of coherence, and spectral degree of polarization can be treated in the same manner, the intensity of the scattered pulse has the following form.

$$\begin{array}{l}{I}^{(s)}(r\mathrm{s},t)=\frac{W_0{C}_0{(2\pi {\sigma }_R{\sigma }_r)}^3}{r^2\sqrt{2(T_0^2+m+a)\left[1+(m-a)\frac{T_c^2+4T_0^2}{T_c^2{T}_0^2}\right]}}\exp \left[-\frac{{(t-\frac{r}{c})}^2}{2(T_0^2+m+a)}\right]\\ \times \exp \left[-\frac{2{\omega }_0^2(m-a)-b^2\frac{T_c^2+4T_0^2}{2T_c^2{T}_0^2}}{1+(m-a)\frac{T_c^2+4T_0^2}{T_c^2{T}_0^2}}-\frac{\sqrt{2\pi }nb}{T_c}\right]\cosh \left[\frac{2b{\omega }_0(m-a)}{\frac{T_c^2{T}_0^2}{T_c^2+4T_0^2}+m-a}\right]\end{array}$$

(15)

3. Intensity Properties of CGSM Pulse Scattered by a Medium

On the basis of Equation (15), the intensity involution of a CGSM pulse scattered by a random medium can be studied as follows. Figure 2 shows changes in the normalized intensity of the scattered pulse with the scattering angle for four different values of parameter n. In the following numerical calculations, unless specified otherwise, parameters were chosen as: T₀ = T_c = 5 fs, λ₀= 800 nm, σ_R = 10 λ₀, σ_r = λ₀, and t = r/c. Figure 2 shows that the intensity properties of the scattered CGSM pulse were closely related to parameter n. For a GSM pulse, the intensity distribution of the scattered pulse has Gaussian distribution. With an increase of parameter n, the effective angular width of the scattered pulse decreases. Here, the effective angular width of the scattered pulse is defined as the 1/e point of the normalized intensity of the scattered pulse [37].

Figure 3a illustrates the variations of the normalized intensity of the scattered pulse with scattering angle θ, n = 2. In order to display the impact of initial pulse duration T₀ on the intensity of the scattered pulse, Figure 3b gives the changes of the effective angular width of the scattered pulse with initial pulse duration T₀. It is clear from Figure 3a that the effective angular width of the scattered pulse increased when the initial pulse duration T₀ increased. Furthermore, Figure 3b shows that, for small values of parameter n, the effective angular width of the scattered pulse varied rapidly with initial pulse duration T₀; however, it changed slowly for large values of parameter n.

Figure 4 shows (a) the behavior of the normalized intensity of the scattered pulse as a function of scattering angle θ, and (b) the effective angular width of the scattered pulse against temporal-coherence length T_c of the initial pulse. The effective angular width of the scattered pulse increased with increasing temporal-coherence length T_c of the initial pulse. However, the effective angular width of the scattered pulse changed rapidly with temporal-coherence length T_c for large values of parameter n. Especially with an increase of temporal-coherence length T_c of the initial pulse, the effective angular width of the scattered pulse remained nearly unchanged for the case of n = 0, and it converged to a constant 0.241 for the case of n = 0, 2, 6, 10. The reason for this phenomenon is as follows.

When temporal-coherence length T_c of the initial pulse is infinite, the intensity of the scattered pulse takes the following form:

$$I^{(s)}(r\mathbf{s},,t)=\frac{W_0{C}_0{T}_0{(2\pi {\sigma }_R{\sigma }_r)}^3}{r^2\sqrt{2(T_0^2+m+a)\left[T_0^2+(m-a)\right]}}\exp \left[-\frac{{(t-\frac{r}{c})}^2}{2(T_0^2+m+a)}\right]\exp \left[-\frac{2{\omega }_0^2{T}_0^2(m-a)}{T_0^2+(m-a)}\right].,$$

(16)

which is independent of temporal-coherence length T_c of the initial pulse and parameter n. Therefore, the effective angular width of the scattered pulse converged to a constant when temporal-coherence length T_c of the initial pulse increased for different n parameters.

Figure 5 illustrates the influence of effective radius σ_R of the medium on the normalized intensity of the scattered pulse. As shown in Figure 5, the effective angular width of the scattered pulse increased as effective radius σ_R of the medium decreased. For large values of parameter n, effective radius σ_R of the medium had little impact on the effective angular width of the scattered pulse.

Figure 6 displays the effect of correlation length σ_r of the medium on the normalized intensity of the scattered pulse. The effective angular width of the scattered pulse decreased with increasing correlation length σ_r of the medium. In comparison with effective radius σ_R, the correlation length σ_r of the medium resulted in a more rapid change in the effective angular width of the scattered pulse.

4. Conclusions

In summary, we investigated the intensity evolution of a CGSM pulse scattered by a quasihomogeneous medium on the basis of the scattering theory of nonstationary fields. We derived the closed-form formula for the intensity of the scattered pulse in the time domain. We found that the effective angular width of the scattered pulse can be modulated by the pulse and medium parameters. It increased when the pulse parameters increased or the medium parameters decreased. When variations in the correlation length of the medium and the effective radius of the medium were the same, the former resulted in a more rapid change in the effective angular width of the scattered pulse. In addition, the intensity properties of the scattered CGSM pulse were closely related to the n parameter. Variations of the effective angular width of the scattered pulse induced by parameter n were analyzed in detail. For large values of parameter n, the effective angular width of the scattered pulse changed slowly with the initial pulse duration, and sharply with the temporal-coherence length of the pulse. These results might find uses in practical applications of pulsed-beam scattering.