1. Introduction

Since the unified theory of coherence and polarization was formulated [1,2], many issues of statistical theory of stochastic electromagnetic beams have been solved. In particular, the changes in the spectral density, the spectral degree of coherence, and the spectral degree of polarization on propagation can be investigated with the help of the $2\times 2$ cross-spectral density matrix (CSDM) of the stochastic electromagnetic beams [3–8]. Besides, the state of polarization, characterized by the shape and the orientation of the polarization ellipse of the polarized part of the field can also be determined by the CSDM [9,10]. Statistical properties of light beams and their manipulation have always received considerable interest for their applications in a broad area of optics [11–14]. A number of methods have been used to modulate the statistical properties of the random electromagnetic beams, such as random phase screen [15] and slit aperture [16].

The correlation function characterized by the peculiar specularity property, not encountered in the most commonly used source models, was found in a space-time modulated field [17]. This property permits an interesting application to the optical processing in which a hologram of a spatially noncoherent object is accomplished by using a wavefront-folding interferometer (WFI) [18]. The extension of specularity to the correlation function in the space-frequency domain, i.e., cross-spectral density function, was realized shortly [19]. Since then it has received little attention apart from the study on specular and anti-specular partially coherent vector solitons [20]. Recently, the specular and antispecular fields were experimentally implemented by launching a Gaussian Schell-model beam into a WFI [21]. The impacts of turbulent media on specular or anti-specular property of this class of beams were also investigated [22]. Besides, the potential applications of specularity and anti-specularity to free-space optical communication and micro-particle trapping have been discussed [23,24]. However, to the best of our knowledge, the researches on the specularity or anti-specularity are all within the framework of the scalar case, and the specular or anti-specular stochastic electromagnetic fields have not been studied in the literature.

In this paper, we consider the field generated by passing a stochastic electromagnetic beam through a WFI. The specularity and anti-specularity are discussed in the vector case. Take electromagnetic Gaussian Schell-model (EGSM) beam as an example, we investigate the spectral density, the spectral degree of coherence, the spectral degree of polarization and the state of polarization of the polarized part of the field during propagation. In particular, we demonstrate the influences on the polarization properties of the phase difference introduced by the interferometer and the phase retardation produced by the compensated right-angle prism.

2. Transformation of random electromagnetic beams by a wavefront-folding interferometer

Figure 1(a) exhibits the wavefront-folding interferometer [25], which is a Michelson-type interferometer mainly considered for the measurement of the spatial correlation properties of partially coherent fields. The key elements of this device are the two perpendicularly oriented right-angle prisms ${\text{PR}}_x$ and ${\text{PR}}_y$ in the arms, which respectively retroreflect the incident field in the $x$ and $y$ direction. When these prisms are slightly tilted with respect to the optical axis, spatial interference fringes can be seen in the observation plane. However, we assume the device perfectly aligned.

 

Fig. 1 Wavefront-folding interferometer (a) and retroreflection by a compensated right-angle prism (b). S is the source, BS is a non-polarizing beam splitter, PR_x and PR_y are right-angle prisms, WP is a wave plate. The set of coordinate axes in (b) is used to describe the state of polarization of a light beam passing through a compensated right-angle prism.

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Suppose that an electromagnetic wave with the electric field vector $E_0\left(x,y\right)$ is incident on the interferometer. To describe the electromagnetic vibrations conveniently, we use a moving set of Cartesian axes $Ox$, $Oy$, $Oz$ with $Ox$ parallel to the beam splitter and $Oz$ along the direction of propagation of the beam [see Fig. 1(a)]. $E_0\left(x,y\right)$ can be expressed as the sum of two mutually orthogonal components, $E_{0x}\left(x,y\right)$ and $E_{0y}\left(x,y\right)$, which are respectively along the axes $Ox$ and $Oy$. The vibration at the input of the interferometer gives rise to two vibrations at the output. Let $E_1\left(x,y\right)$ and $E_2\left(x,y\right)$ be the electric field vectors at the output of the interferometer, which are from the waves reflected by prisms ${\text{PR}}_x$ and ${\text{PR}}_y$, respectively. The components of the electric field vector $E_1\left(x,y\right)$ are related to the components of the electric field vector $E_0\left(x,y\right)$, by the matrix form

By denoting $\theta$ the phase difference between the two arms of the interferometer, the total electric field vector at the output can be expressed in the form

Consider $E_0\left(x,y\right)$ as a single realization of a stochastic electromagnetic field. We assume that the random fluctuations are statistically stationary, at least in the wide sense. The second-order correlation properties of such a field may be characterized by a $2\times 2$ correlation matrix, i.e., the cross-spectral density matrix

Table 1. Expressions for the coefficients $D_{ij},E_{ij},F_{ij}$ and $G_{ij}$

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It is noted from Eqs. (6), (7) and (9) that the total output field depends on the phase retardation $\phi$ produced by a right-angle prism. For incidence at an angle ${45}^{\circ }$ and the prism of index $n=1.5$, the total phase retardation due to two total internal reflections inside a prism is $\phi \approx {74}^{\circ }$. To study the influence of the phase retardation $\phi$ on the statistical properties of the stochastic electromagnetic beams, we will adopt several values of $\phi$. This can be achieved by cementing a phase plate to the entrance face of each right-angle prism [26], whose principal axes are parallel and perpendicular to the roof ridge. Therefore, the total phase retardation $\phi =2\left({\phi }_1+{{\phi }^{\prime }}_1\right)$ with ${\phi }_1$ and ${{\phi }^{\prime }}_1$ being the phase retardation introduced by the total reflection once and by the wave plate once, respectively. In practice, one can use a zero-order wave plate, which consists of two plates cemented together with the fast axis of the one coinciding with the slow axis of the other, to construct a compensated prism.

One can find that Eq. (9) is analogous to Eq. (3) in [21], which is used to study the specularity and antispecularity of scalar fields. To study the specular and antispecular electromagnetic fields, we will consider two special cases $\phi =2m\pi$ and $\phi =\left(2m+1\right)\pi$ ($m$ is an integer).

Case 1. $\phi =2m\pi$. Under this condition Eq. (9) becomes

Case 2. $\phi =\left(2m+1\right)\pi$. In this particular case Eq. (9) becomes

Comparing these two cases, one immediately finds that the specular case and anti-specular case rely on both the phase retardation $\phi$ and the phase difference $\theta$. The two special cases reveal that the output field of the interferometer is specular if one of $\phi$ and $\theta$ is odd times of $\pi$while the other is even times of $\pi$. For anti-specular field, the condition that $\phi$ and $\theta$ are both odd or even times of $\pi$ is satisfied.

It is worth mentioning that Case 2 is in agreement with the specular or antispecular case of the scalar field [21].

3. The statistical properties of random electromagnetic beams propagating in free space

Let us consider the field at the output of the interferometer as a secondary source, which propagates close to the $z$ axis from the output plane of the interferometer ($z=0$) into the half-space $z>0$ in free space. By employing the propagation law applicable to a paraxial ABCD system [2,27], one may obtain the expression for the elements of the CSDM at any two points $\left({\rho }_1,z\right)$ and $\left({\rho }_2,z\right)$ in any transverse plane $z=\text{const}>0$ from the elements of the CSDM at the output plane of the interferometer, viz.,

The spectral density $S\left(\rho ,z;\omega \right)$ at a point $\left(\rho ,z\right)$, at frequency $\omega$, is given by the expression (Ref [28], Sec. 9.2)

Any statistically stationary light beam may be, at each point, expressed as the sum of contributions from a completely polarized and from a completely unpolarized beam. The ratio of the intensity of the polarized portion to the total intensity is called the degree of polarization, which is given by the expression [1]

4. An example: propagation of a modulated electromagnetic Gaussian Schell-model beam in free space

Suppose an EGSM beam incident on the interferometer, whose elements of the CSDM are given by (Ref [28], Sec. 9.4.2)

Substitute Eq. (23) into Eq. (9), one can express the elements of the CSDM at the output of the interferometer by the formula

Table 2. Expressions for the coefficients $H_{ij}$ and $K_{ij}$

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Using Eqs. (15) and (29), the elements of the CSDM of the field in any transverse plane $z>0$ can be determined

Using Eqs. (16)-(22) and (30), one can investigate the statistical properties (i.e., the spectral density, the spectral degree of coherence, the spectral degree of polarization and the state of polarization of the polarized portion) of the modulated EGSM beams upon propagation in free space.

5. The spectral density, the spectral degree of coherence and the polarization properties of a modulated EGSM beam on propagation

In this section, we will demonstrate the behavior of the statistical properties of the modulated EGSM beams during propagation in free space. For convenience, we assume that the split ratio of the non-polarizing beam splitter for the light at a certain wavelength (for instance, $632.8nm$) is 50:50. Consequently, it is reasonable to get $r_x=r_y=t_x=t_y$, and then $C_{ij}=C$. In such a case the expression (30) can be used to describe the evolution of the statistical properties of the incident EGSM beam without passing through the WFI when $\phi =0$ and $\theta =\pi /2$, i.e., the modulated EGSM beam reduces to the normal EGSM beam. In particular we set ${\sigma }_x={\sigma }_y={\sigma }_0$, implying that the polarization (including the degree of polarization, the orientation angle and the degree of ellipticity) of the incident EGSM beam is uniform across the source. Besides, we suppose that ${\delta }_{xx}={\delta }_{yy}={\delta }_0$.

Substituting Eq. (30) into Eq. (16) yields

 

Fig. 2 Intensity distributions of the modulated EGSM beams with $\phi =\pi$ at $z=1000\text{m}$. $S_{\text{G}\max }(z)=2C\left(A_x^2+A_y^2\right)/{\Delta }_{xx}$ represents the maximum value of the normal Gaussian beam. (a) $\theta =0$; (b) $\theta =\pi$. We choose ${\delta }_0\text{=}1\text{mm}$ in (c). The other parameters of the incident beam are chosen as follows: $\lambda \text{=}632.8nm$, $A_x=2$, $A_y=1$, $B_{xy}=0.2\exp \left(i\pi /3\right)$, ${\sigma }_0=1cm$.

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Fig. 3 Changes with $\rho /{\sigma }_{\text{P0}}$ of the spectral degree of polarization, the orientation angle, the degree of ellipticity and the spectral density for several selected values of $\phi$ and $\theta$ in the output plane of the WFI. ${\sigma }_{\text{P0}}=1/\sqrt{1/{\sigma }_0^2+4/{\delta }_{\text{m}}^2}$ with ${\delta }_{\text{m}}=\max \left\{{\delta }_{ij}\right\}$ represents the modulation width by $\theta$ and $\phi$ in the output plane of the interferometer. Rows 1-3 respectively correspond to $\phi =\pi /6,\pi /4,\pi /2$. The values of $\theta$ are given beside the curves. The parameters are the same as Fig. 2(c), ${\delta }_{xy}=2\text{mm}$. The polarization of the incident EGSM beam is uniform with $P=0.621,\alpha \text{=3}{\text{.797}}^{\circ }$ and $\epsilon =0.113$.

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On substituting from Eq. (29) into Eqs. (18)-(22), the expressions can readily be derived for all the polarization properties of the beam at points located in the output plane of the WFI. The spectral degree of polarization is given by the expression

Substituting from Eq. (30) into Eqs. (18)-(22), the expressions can be obtained for all the polarization properties of the beam at points in any transverse plane. It can readily be found that the above-mentioned cases (i) and (ii) still hold during propagation. In particular, we will discuss the degree of polarization, the orientation angle and the degree of ellipticity along the axis $z$(i.e., $\rho =0$), which are given by the formulae

Figure 4 shows the evolution of on-axis polarization properties for several selected $\theta$ when $\phi =\pi /4$, in which curve $\theta =\pi /4$ corresponds to case (ii). Figure 5 demonstrates the on-axis polarization properties for different $\phi$ and $\theta$ in the far-zone, in which curve $\phi =\pi$ corresponds to case (iii).

 

Fig. 4 Changes with the propagation distance $z$ of on-axis spectral degree of polarization, orientation angle and degree of ellipticity for several selected values of $\theta$ when $\phi =\pi /4$. The values of $\theta$ are given beside the curves. The parameters are the same as Fig. 2(c), ${\delta }_{xy}=2\text{mm}$.

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Fig. 5 Changes with $\theta /\pi$ of on-axis spectral degree of polarization, orientation angle and degree of ellipticity for some selected values of $\phi$ in the far-zone. Black solid curve, $\phi =\pi /6$; red short dashed curve, $\phi =\pi /4$; blue short dotted curve, $\phi =\pi /2$; green short dashed-dotted curve, $\phi =\pi$. The parameters are the same as Fig. 2(c), ${\delta }_{xy}=2\text{mm}$.

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In Fig. 6, we illustrate the spectral degree of coherence as a function of the phase difference $\theta$ at several propagation distances. It is evident that the coherence between two symmetrical points can be controlled ranging from complete incoherence to complete coherence. Curves $\phi =0$ and $\phi =\pi$ shows that complete coherence is achieved when $\theta =l\pi$, i.e., the specular case or the anti-specular case. This result is identical to that of the scalar case (Ref [21], Fig. 5). As the propagation distance increases, the range of complete coherence with $\theta$ increases.

 

Fig. 6 Changes with $\theta /\pi$ of the spectral degree of coherence between two symmetrical points ${\rho }_1\text{=}\left(-\rho /2,0\right)$ and ${\rho }_2\text{=}\left(\rho /2,0\right)$ with $\rho =1mm$. (a) $z=0$; (b) $z=200\text{m}$; (c) $z=1000\text{m}$. Black solid curve, $\phi =0$; red short dashed curve, $\phi =\pi /4$; blue short dotted curve, $\phi =\pi /2$; green short dashed-dotted curve, $\phi =\pi$.

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6. Conclusions

In conclusion, we have considered the field generated by a WFI which is illuminated by a stochastic electromagnetic beam. The concepts of specularity and anti-specularity are introduced to the vector case, which are discussed in two special cases. Take EGSM beam as an example, we have studied the spectral density, the spectral degree of coherence, the spectral degree of polarization as well as the state of polarization of the polarized portion of the field on propagation. We have found that the polarization properties including the degree of polarization, the orientation angle and the degree of ellipticity, in general, can be adjusted by varying the phase retardation produced by the compensated right-angle prism and the phase difference introduced by the interferometer. In particular, the on-axis polarization properties will tend to constants in the far-zone, determined by the phase retardation and the phase difference. A special situation has also been discussed under which the polarization properties are independent of the phase difference. Numerical results have shown that complete polarization and linear polarization can be realized from a partially polarized incident field with the polarized part being elliptical polarization. Therefore, the interferometer can be used as a kind of polarization modulator. The results obtained in this paper may find applications in free-space optical communication.