1. Introduction

In recent years, the propagation characteristics of partially coherent electromagnetic beam have attracted much attention, due to the unique advantages in beam regulation comparing with the coherent or incoherent beam. The partially coherent light not only provides a bridge between the classical and quantum theories [1,2], but also has important applications in, such as free space optical communications, remote sensing, optical imaging, and optical trapping [1–8]. In general, by calculating the statistical second moment of cross-spectral density (CSD) function [9], various properties of static random electromagnetic field can be obtained, such as beams widths, divergences, orbital angular momentum [10–13]. The CSD function must be a Hilbert-Schmidt kernel. In addition, such a Hermitian kernel has to be non-negative definite. This property can be employed to design and implement various partially coherence structured beams [14–22]. It also can be used to construct the coherent-mode representation of CSD function [23]. Gaussian Schell-model (GSM) beams is the most fundamental class of partially coherent beams, whose spatial distribution and correlation function have Gauss profiles [1,2]. The coherent-mode representation of GSM beam was obtained by Gori and Starikov et al. [24,25]. This coherent-mode theory is generalized to the partially polarized, partially coherent beams [26,27], followed by the extension of CSD function from scalar to tensor [10,28]. Besides, the realizability condition for partially coherent electromagnetic fields is investigated in detail by Gori et al. [15,16]. More recently, many new kinds of electromagnetic beams with more complex spatial structure and correlation function have been studied [29–34].

On the other hand, the propagation of fully coherent paraxial beam in anisotropy uniaxial crystal exhibits interesting phenomena, such as the vortex generation and the spin-orbital coupling [35,36]. The analytical solution for Gaussian beam is first derived by Ciattoni et al. within angular spectrum representation, which stimulates many works of uniaxial crystal in the past decades [37,38]. However, the properties of partially coherent beam in uniaxial crystal have not been fully studied. Previous works are limited to the case of propagating direction orthogonal to the optical axis. The first work is by Liu et al. [39] in 2009, who deduced the analytical expression for partially coherent Gaussian beam. The idea was then extended to various different structured partially coherent electromagnetic beams [40–43]. When propagating orthogonal to the optical axis, the two Cartesian components of electric field do not couple each other, which makes the diffraction integral solvable [39]. However, the diffraction integral is more complicated when the beam propagates along the optical axis, for the Cartesian (or circular) components do couple. This situation is similar to the nonparaxial beam propagation orthogonal to the optical axis [44]. As far as we know, no solution is reported for the partially coherent beams propagating along the optical axis, which will be the topic of the present paper.

The paper is organized as follows: In section 2, we present the general angular-spectrum theory for a partially coherent paraxial beam propagating along the optical axis of uniaxial crystal. In section 3, we apply the angular-spectrum theory to the electromagnetic Gaussian Schell-model (EGSM) beams, and obtain the analytical propagating solutions. We further show that the analytical solutions for EGSM beams can be expressed as a quasi-coherent-mode representation, whose modes are constructed by the elegant Laguerre-Gaussian (eLG) beams. In section 4, we calculate two physical quantities, the energy density and degree of polarization, and reveal the propagating characters of EGSM beams with different degree of coherence. Finally, we give our conclusions in section 5.

2. Angular spectrum representation of the CSD tensor for a partially coherence beam propagating along the optical axis

Let us begin with the fully coherent beam in uniaxial crystal propagating along the optical axis. We assume that the optical axis of uniaxial crystal is along the $z$ axis and the input electric field is incident on the transverse plane at $z = 0$. The dielectric tensor of uniaxial crystal is given by

The second order statistical properties of partially coherent beams in the space-frequency domain can be described by CSD tensor, which is defined by [2]

3. Analytic solution of EGSM beams

In this section, we apply the angular-spectrum theory derived in Section 2 to the EGSM beams. Assume a uniformly circularly polarized EGSM beam is incident on the input plane, with the following CSD tensor,

Performing the integral in Eq. (15) is not an easy work because of the vortex term ${e^{i2\phi }}$ and correlated term $\exp ( - 2d{{\textbf k}_{1 \bot }} \cdot {{\textbf k}_{2 \bot }})$ in the integrand. Our key step to solve this problem is like following: we firstly decompose the CSD function into a series of vortex modes and then perform the angular integrals one by one. We take one element of CSD function in Eq. (15) as an example,

Utilizing the relation between Laguerre polynomial $L_m^n(x)$ and confluent hypergeometric function $_1{F_1}(a,b,x)$, we can reshape the Eq. (23) into a form of eLG beam, whose expression in isotropic media is given by [46]

Utilizing the normalization condition of eLG beams, it is straightforward to prove that $\int\!\!\!\int {{{[{\mathbf \Psi }_{}^{nm}({{\textbf r}_ \bot },z)]}^\ast } \cdot {\mathbf \Psi }_{}^{nm}({{\textbf r}_ \bot },z)d} {{\textbf r}_ \bot } = 1$, and furthermore the total energy on transverse plane is conserved, i.e., $I(z) = \sum\nolimits_{m = 0}^{ + \infty } {\sum\nolimits_{n ={-} \infty }^{ + \infty } {{\varLambda _{nm}} = {\pi / {(2a)}}\sum\nolimits_{N = 0}^\infty {{{({b / a})}^N}} = 2\pi {\sigma ^2}} }$, with $N = 2m + |n|$. Equation (29) indicates that the CSD function is an incoherent superposition of coherent mode ${\mathbf \Psi }_{}^{nm}$. However, the mode ${\mathbf \Psi }_{}^{nm}$ is not orthogonal, due to the non-orthogonality of eLG beams [48]. Therefore, we refer to Eq. (29) as a quasi-coherent-mode representation for the EGSM beam propagating along the optical axis of uniaxial crystals. Specifically, we depicted in Fig. 1 the distribution of coefficient ${\varLambda _{mn}}$ for different coherence degree of ESGM, which all satisfy the realizability condition [15]. One can see that, with the decrease of coherence degree, the proportion of higher-order coherent modes increases, and thus more terms are needed to calculate the CSD function in Eq. (29).

 

Fig. 1. Coefficient ${\varLambda _{mn}}/{\varLambda _{00}}$ of the EGSM beam for different degree of coherence. The ${\varLambda _{00}}$ is the coefficient with $n = 0$ and $m = 0$. The values of coherence width $\delta$ are choose as (a) $\delta = \infty$, (b) $\delta = 2\sigma$, (c) $\delta = \sigma$, (d) $\delta = 0.5\sigma$.

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Now let us discuss three extreme cases of Eq. (29). (1) The case of fully coherent beam, i.e., the coherent width $\sigma = \infty$. The coefficient ${\varLambda _{mn}}$ dose not equal zero only when $m = 0$ and $n = 0$, as shown in Fig. 1(a), which means that the CSD function just contains one coherent mode, the fundamental Gaussian beam. The elements of CSD tensor are exactly the same as those given by the analytical solutions of Eq. (25) in Ref. [36]. (2) The case of isotropic crystal with ${\eta _o} = {\eta _e}$. In this case, only one element ${W_{ +{+} }}$ is nonzero in the CSD tensor, for $\varPsi _ - ^{nm}$ is zero in Eq. (27). The coherent mode is exactly the right-handed polarized eLG beam ${\mathbf \Psi }_{}^{nm}({{\textbf r}_ \bot },z) = eLG_m^n({{\textbf r}_ \bot },z,{\eta _o}){\hat{{\textbf e}}_{\textbf + }}$, and the CSD element is ${W_{ +{+} }}({{\textbf r}_{1 \bot }},{{\textbf r}_{2 \bot }},z) = \sum\nolimits_{n,m} {{\varLambda _{nm}}{{[eLG_m^n({{\textbf r}_{1 \bot }},z)]}^\ast }eLG_m^n({{\textbf r}_{2 \bot }},z)}$, whose structure is similar to the standard Hermite-Gaussian (HG) coherent-mode representation of GSM beam, proposed in Ref. [24,25]. We have numerically verified that the results of eLG representation with large cut-off are in good agreement with the analytical results in Ref. [1]. (3) The solution of ${W_{ +{+} }}$ in uniaxial crystal. Following the same procedure used by Wolf to derive the analytical expression of CSD function in isotropic medium, we can deduce the analytic expression of ${W_{\textrm{ +{+} }}}({{\textbf r}_{1 \bot }},{{\textbf r}_{2 \bot }},z)$ beginning from Eq. (15). The final result is

4. Evolution of energy density and degree of polarization

In this section, we use the quasi-coherent-mode representation in Eq. (29) to calculate physical quantities. If only consider the statistical properties of one point on the output plane we can replace the arguments of CSD function by ${{\textbf r}_{1 \bot }} = {{\textbf r}_{2 \bot }} = {{\textbf r}_ \bot }$ and rewrite it as spectral density ${\textbf S}({{\textbf r}_ \bot },z) = {\textbf W}({{\textbf r}_ \bot },{{\textbf r}_ \bot },z)$. The energy density of circular components are just the elements of CSD tensor ${S_{ +{+} }}(r,z)$ and ${S_{ -{-} }}(r,z)$, which do not depend on the azimuth angle $\theta$ (see Eq. (21)). The total energy density is the sum of contributions from two components $S(r,z) = {\textbf TrS}({{\textbf r}_ \bot },z)$. All elements of ${\textbf W}({{\textbf r}_{1 \bot }},{{\textbf r}_{2 \bot }},z)$ can be numerically calculated using Eq. (29) with finite cut-off towards the index n and m. Considering the convergent behavior of coefficient ${\varLambda _{nm}}$, we fix $N = 2m + |n|$ and choose a cut-off $N = 100$ for all calculations in this section.

The calculated energy density of right-handed and left-handed polarized components are displayed in Fig. 2 and Fig. 3, respectively, for various degree of coherence. The values of parameters for the EGSM and uniaxial crystal are chosen as follows: refractive index ${n_o} = 1.658$ and ${n_e} = 1.486$, wavelength $\lambda = 0.5\mu m$, width of beam waist $\sigma = 7\mu m$, and five values of coherence width $\delta$ are used ($\infty$, $2\sigma$, $\sigma$, $0.5\sigma$, $0.2\sigma$), which satisfy the realizability condition [15]. From Fig. 2, one can see that, after the beam propagating to a large distance as in Fig. 2(d), the distribution curve of energy density forms a wavy line. Interestingly, the locations of minimums on the wavy lines seems to coincide together, that is, the oscillations in radial direction are not sensitive to the coherence width. In Fig. 3, the energy density for small degree of coherence develops rapidly in the small propagating regime. The reason is that the proportion of high-order coherent modes is high when the degree of coherence is small, as shown in Fig. 1(d). The high-order modes generally transfer quickly at small distance [48,49]. In the regime of large propagating distance, a saturated behavior is observed again.

 

Fig. 2. The square root of energy density $\sqrt {{S_{ +{+} }}(r,z)}$ for the EGSM beam propagating along the optical axis of uniaxial crystal. The values of parameters used in the calculation are ${n_o} = 1.658$, ${n_e} = 1.486$, $\lambda = 0.5\mu m$, and $\sigma = 7\mu m$. The results with different degree of coherence are displayed by different colors; see the legend in panel (a). The propagating distances are chosen as (a) $z = 500\mu m$, (b) $z = 1500\mu m$, (c) $z = 6000\mu m$, (d) $z = 50000\mu m$.

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Fig. 3. The square root of energy density $\sqrt {{S_{ -{-} }}(r,z)}$ for the EGSM beam propagating along the optical axis of uniaxial crystal. All parameters used in the calculation are the same as those in Fig. 1. The propagating distances are chosen as (a) $z = 500\mu m$, (b) $z = 1500\mu m$, (c) $z = 6000\mu m$, (d) $z = 50000\mu m$.

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If integrating the energy density over the whole transverse plane, one can obtain the total energy $I(z)$ at a propagating distance z. By using the Eq. (30), the energy of right-handed component ${I_ + }(z)$ can be straightforwardly worked out

Also, we can calculate the degree of polarization (DOP), defined by [2,28]

 

Fig. 4. The degree of polarization (DOP) for the EGSM beam propagating along the optical axis of uniaxial crystal. All parameters used in the calculation are the same as those in Fig. 1. The propagating distances are chosen as (a) $z = 500\mu m$, (b) $z = 1500\mu m$, (c) $z = 6000\mu m$, (d) $z = 50000\mu m$.

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One can see that, due to the partially coherence, the uniformly polarized EGSM beam on the input plane becomes partially polarized (i.e., $P < 1$) when propagating in the uniaxial crystal. At large propagating distance as in Fig. 4(d), the DOP around the centers of transverse plane is roughly equal to 1, which means the field is fully polarized. This is consistent with the saturated behaviors in Fig. 2(d) and Fig. 3(d).

5. Conclusion

In summary, we studied the statistical properties of partially coherent beam propagating in uniaxial crystal along the optical axis. The CSD tensor is expressed as multiple integrals within the angular-spectrum representation. For the uniformly polarized EGSM beam, the integrals can be solved by using the vortex expansion, and therefore the analytical solution for CSD tensor is obtained, which is in a form of hypergeometric function. We further showed that this analytical solution can be written into a quasi-coherent-mode representation, whose modes are constructed by the eLG functions. The properties of this representation are discussed and several limits of the analytical solution are examined, which exactly recover the results of previous theories. Finally, we calculated the energy density and degree of polarization of EGSM beam. In small propagating distance regime, these quantities evolve rapidly, while a saturated behavior is observed when propagating to large distance. Our work will provide theoretical support for the regulation of partially coherent beams.