We consider the field generated by a wavefront-folding interferometer which is illuminated by a stochastic electromagnetic beam. The specular property and anti-specular property are discussed in the vector case. Take electromagnetic Gaussian Schell-model beam as an example, we investigate 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. Results show that the polarization properties including the degree of polarization, the orientation angle and the degree of ellipse can be adjusted by the phase difference of the interferometer and the phase retardation introduced by the prism. The results may be applied in free-space optical communication.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
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 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  and slit aperture .
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 . 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) . The extension of specularity to the correlation function in the space-frequency domain, i.e., cross-spectral density function, was realized shortly . Since then it has received little attention apart from the study on specular and anti-specular partially coherent vector solitons . Recently, the specular and antispecular fields were experimentally implemented by launching a Gaussian Schell-model beam into a WFI . The impacts of turbulent media on specular or anti-specular property of this class of beams were also investigated . 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 , 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 and in the arms, which respectively retroreflect the incident field in the and 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.
Suppose that an electromagnetic wave with the electric field vector is incident on the interferometer. To describe the electromagnetic vibrations conveniently, we use a moving set of Cartesian axes , , with parallel to the beam splitter and along the direction of propagation of the beam [see Fig. 1(a)]. can be expressed as the sum of two mutually orthogonal components, and , which are respectively along the axes and . The vibration at the input of the interferometer gives rise to two vibrations at the output. Let and be the electric field vectors at the output of the interferometer, which are from the waves reflected by prisms and , respectively. The components of the electric field vector are related to the components of the electric field vector , by the matrix form26], withFig. 1(b)]. Using Eq. (3), one readily finds that
By denoting the phase difference between the two arms of the interferometer, the total electric field vector at the output can be expressed in the formEqs. (1) and (2) into Eq. (5), and on using Eq. (4), one can express the components of the total electric fieldEq. (6), one immediately finds that the component of the output field is related to the component and irrelevant to the component of the input field. It is similar for the component of the output field. This is caused by the orientations of the prisms and , whose roof ridges are along the direction and direction, respectively.
Consider 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 correlation matrix, i.e., the cross-spectral density matrixEqs. (6) and (7) into Eq. (8) one then obtainsTable 1.
It is noted from Eqs. (6), (7) and (9) that the total output field depends on the phase retardation produced by a right-angle prism. For incidence at an angle and the prism of index , the total phase retardation due to two total internal reflections inside a prism is . To study the influence of the phase retardation on the statistical properties of the stochastic electromagnetic beams, we will adopt several values of . This can be achieved by cementing a phase plate to the entrance face of each right-angle prism , whose principal axes are parallel and perpendicular to the roof ridge. Therefore, the total phase retardation with and 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 , 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 and ( is an integer).
Case 1. . Under this condition Eq. (9) becomesEq. (10) one readily finds thatEq. (11), whatever the random electromagnetic field at the input is, the CSDM of the field at the output is anti-specular when and specular when .
Case 2. . In this particular case Eq. (9) becomesEq. (12) one can find thatEquation (14) indicates that the CSDM of the field at the output of the interferometer is specular when and anti-specular when .
Comparing these two cases, one immediately finds that the specular case and anti-specular case rely on both the phase retardation and the phase difference . The two special cases reveal that the output field of the interferometer is specular if one of and is odd times of while the other is even times of . For anti-specular field, the condition that and are both odd or even times of is satisfied.
It is worth mentioning that Case 2 is in agreement with the specular or antispecular case of the scalar field .
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 axis from the output plane of the interferometer () into the half-space 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 and in any transverse plane from the elements of the CSDM at the output plane of the interferometer, viz.,
The spectral density at a point , at frequency , is given by the expression (Ref , Sec. 9.2)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 9]9]
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 , Sec. 9.4.2)29,30]30]31].Table 2. It should be noticed that Eq. (29) will reduce to Eq. (23) except for the coefficients when and .
Using Eqs. (15) and (29), the elements of the CSDM of the field in any transverse plane 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, ) is 50:50. Consequently, it is reasonable to get , and then . 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 and , i.e., the modulated EGSM beam reduces to the normal EGSM beam. In particular we set , 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 .Eq. (34), one can find that a central peak or dip of width is observed on a Gaussian background with the beam width during propagation, as shown in Fig. 2(c). The width of the central peak or dip could be adjusted by varying the coherence length , which will increase as increases [see Figs. 2(a) and 2(b)]. Equation (34) also reveals that the central distribution (peak or dip) is determined by the phase retardation and the phase difference , which is displayed in Figs. 3(d1), 3(d2) and 3(d3).
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 expressionEqs. (35)-(37), it is evident that the input EGSM beam with uniform polarization (when and ) across the source will become non-uniform at the output of the WFI in general (see Fig. 3). It is worth mentioning that and when (i.e., ), meaning that the polarized part of the field at certain point is linearly polarized along the direction. Two special cases of this situation are considered: (i) if both and are odd times of then the polarized part of the field will be linearly polarized along the direction at all points across the output plane of the WFI [for example, in Figs. 3(b3) and 3(c3)]; (ii) if then the field will be completely polarized with linear polarization along the direction at point [for example, in Figs. 3(a1)-3(c1) and in Figs. 3(a2)-3(c2)].
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 (i.e., ), which are given by the formulae
Figure 4 shows the evolution of on-axis polarization properties for several selected when , in which curve corresponds to case (ii). Figure 5 demonstrates the on-axis polarization properties for different and in the far-zone, in which curve corresponds to case (iii).
In Fig. 6, we illustrate the spectral degree of coherence as a function of the phase difference 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 and shows that complete coherence is achieved when , i.e., the specular case or the anti-specular case. This result is identical to that of the scalar case (Ref , Fig. 5). As the propagation distance increases, the range of complete coherence with increases.
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.
National Natural Science Foundation of China (NSFC) (11474253 and 11274273); Fundamental Research Funds for the Central Universities (2017FZA3005).
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