A class of random source for circular optical frame is generalized to electromagnetic domain. Analytical formulas for the propagation of the electromagnetic source for circular frames combinations through atmospheric turbulence are derived. As two examples, the statistic characteristics of a single circular frame and two nested frames are comparatively studied in free space and in non-Kolmogorov’s atmospheric turbulence. The evolutions of the degree of polarization and the degree of coherence of such circular frames exhibit unique features. The impacts, arising from the refractive-index structure constant, the fractal constant of the atmospheric spectrum and the upper index in the source degree of coherence, on the statistical characteristics are analyzed in detail.
© 2015 Optical Society of America
It is well known in statistic optics that the structure of the source degree of coherence determines the intensity distribution of its far field, i.e., the famous reciprocity relation known to be of Fourier-type law . This law has been used in developing various new model for source correlation function including Bessel correlated source [2, 3 ], Laguarre-Gaussian  and cosine-Gaussian  that lead to ring-shaped far-fields; two types of sinc Schell-model correlation being capable of producing tunable flat and dark hollow profile in the far field ; non-uniformly correlated source leading to a laterally shifted intensity maxima on propagation ; Multi-Gaussian Schell-model correlation generating shape-invariant flat-topped circular or rectangular intensity distribution in the far field [8, 9 ]. On these basis, recently, an analysis model whose source degree of coherence is the Fourier transform of two multi-Gaussian functions’ difference is established , which essentially consistent with the propositions that the difference between two cross-spectral density is a genuine cross-spectral density . The novel model allows the radiation of circular/elliptical and square/rectangular frames in the far zone of its source, and the linear composition of the degree of coherence lead to the form of the nested frames. Generally, for these augmented models for random sources, the extension from scalar source to full electromagnetic domain is implemented, such as the electromagnetic cosine-Gaussian Schell-model beam , the electromagnetic non-uniformly correlated beams , the electromagnetic sinc Schell-model beam , and electromagnetic Rectangular Gaussian Schell model beam  and so on. The statistic properties of such beams including the spectral density, the spectral degree of coherence and polarization properties of these electromagnetic beams are shown to exhibit unique features. Therefore, it is worthwhile to investigate the statistic properties of electromagnetic random source for optical frame.
Since the unified theory of coherence and polarization presented by Wolf, in which it is demonstrated that the changes of the statistic properties of stochastic electromagnetic beams propagating in deterministic or random media can be determined by the cross-spectral density matrix [16, 17 ], some interesting researches have been performed to explore the statistic properties of such beams passing through turbulence atmosphere [18–23 ]. Due to the vectorial nature, the electromagnetic beams are characterized not only by the state of the coherence but also by their polarization properties. It is shown that correlation-induced changes in the degree of polarization in free space and turbulence atmosphere are different [17, 18 ], and the behaviors of the degree of polarization of an isotropic beam and anisotropic electromagnetic Schell-model beams changes differently with the propagation distance in the homogeneous and isotropic turbulence atmosphere . In addition, it is discovered that the degree of polarization and coherence of the source can affect the level of the scintillation index when the beams propagate in turbulence atmosphere and the unpolarized stochastic beams can reduce the scintillation index compared with polarized ones in atmospheric optical communications . Therefore, the studies of the propagation characteristics of stochastic electromagnetic beam, especially the polarization property, are of importance in a turbulent atmosphere.
In this paper, the scalar model proposed in  is used as a building for developing electromagnetic source for circular optical frame. The main purpose is to explore the statistic properties for such electromagnetic source propagating in atmosphere turbulence. Unlike the dark hollow far-field produced by elegant Laguarre-Gaussian and dark hollow Gaussian Schell beam [25, 26 ], the thickness of a single circular frame constructed as a difference of two multi-Gaussian distribution can be adjusted by the upper index of the multi-Gaussian function. Thus, the flat-topped circular frame can be formed for larger value of upper index. In addition, for several overlapped frames, the robustness of outer frames can be controlled by the corresponding weights in linear combination of the degrees of coherence, which prevails over the partially coherent stand Laguarre-Gaussian . In view of the unique characteristic of the circular frames and few reports about the polarization properties of the beams producing dark hollow far-field apart from the work [27, 28 ], it will be of interest to explore the statistic properties of electromagnetic source for circular frame. After the scalar source is extended to its electromagnetic domain, the statistics properties including spectral density, the degree of coherence and the degree of polarization for a single frame and two overlapped frames are comparatively analyzed in free-space and in non-Kolmogorov’s atmospheric turbulence with different fractal constant of the atmospheric spectrum and refractive-index structure constant .
2. Electromagnetic source for circular optical frame
The second-order correlation properties of a stationary electromagnetic source at two points and can be described in terms of the electric cross-spectral density (CSD) matrix , whose element are given by [1, 16 ]
According to the requirement for non-negative definiteness for the elements of the CSD matrix , needs to satisfy the following integral representationEqs. (3) and (4) into Eq. (2), the elements of the CSD matrix can be expressed10]30] it follows that
Furthermore, according to the required inequalities for the nonnegative definiteness 
3. Electromagnetic source for a single circular frame propagating in non-Kolmogorov’s atmospheric turbulence
Recall that generalized paraxial Huygens-Fresnel principle for the components of the CSD matrix that characterizes any electromagnetic beam propagating in free space or in any random linear medium at two points and in the transverse plane of the half-space has form [20–22 ]21].Eq. (18) the integral in Eq. (17) results in the expression
4. Generalization to the circular frame combinations
In  it was pointed out that the linear superposition of the degree of coherence may lead to optical fields with the nested intensity combination of individual frames. In this case, the degree of coherence of the electromagnetic source for several frames is set to be a linear combination of those in Eq. (7)
Due to the effect of linear superposition of the degree of coherence, the initial element of the CSD matrix of the source becomes the linear combination of the components defined in Eq. (5),
The propagation law, for the components in Eq. (24), is the same with Eq. (16), so the specific process of derivation is omitted. It can be found that the elements of the CSD matrix of the source for the circular frame combinations propagating in turbulent atmosphere at the distance z are expressedEqs. (20) and (21) express. In the following section, utilizing the Eqs. (25)-(27) , we can directly study the changes in the spectral density, the degree of polarization and the degree of coherence of the electromagnetic source for circular frame combinations propagating through free space and turbulent atmosphere. The spectral density and the degree of polarization at the point and the degree of coherence at a pairs of points and are given by the formulas [1, 16 ]
5. Numerical examples
We will now illustrate the behaviors of the spectral density, degree of polarization and degree of coherence of the electromagnetic source for a single circular frame and two superposed circular frames () through the free space and atmospheric turbulence in greater detail. A comparative analysis between the cases of and is made. Considering that quite a few source parameters are needed to be set, here, we mainly analyze the statistics properties of the source with uncorrelated field components, i.e. with zero off-diagonal elements of the CSD matrix. The source and the medium parameters are set as follows: , , , , , , , ,,,,,,, , and .
Figure 1 shows the evolution of the spectral density in the axial direction (left) and the transverse distribution of the spectral density at the distance (middle and right) in the turbulence atmosphere. One can see that with the transmission distance increasing Gaussian profile of the source field gradually self-splits into far fields with a single circular frame for and two overlapped frames for . It is shown that the dark center of the single frame or the inner one of two nested frames is gradually filled as the values of increase, i.e. the atmospheric turbulence starts to dominate. But the shape of the outer frame is slightly affected by the turbulent, the property being agree with the situation that the inner square frame resembles a circle while the shape of the outer one remains the same at certain distances .
Figure 2 exhibits the transverse distribution of the spectral density of the circular optical frame at the propagation distance for different values of parameters , and . Figure 2(a) shows that at larger distances the circular flat-topped frame can be formed in free space () for larger values of , but converts to be the circular frame in the atmosphere. By comparing Figs. 2(a) and 2(d) with Figs. 2(b) and 2(e), it can be found that the effect of the non-Kolmogorov turbulence ( and ) with local refractive-index fluctuation on the frame is much stronger than that of the Kolmogorov turbulence (). Especially, the single circular frame is completely reshaped to be Gaussian-like when , which is shown to be an important value for other beams [14, 21 ]. Figures 2(c) and 2(f) show that the circular frame is less susceptible to the effect of atmospheric turbulence for larger values of , being similar with that of rectangular multi-Gaussian Schell model beam in atmospheric turbulent .
The evolutions of on-axial degree of polarization of the circular frames in free space and atmospheric turbulence for difference values of parameters are demonstrated in Fig. 3 . It is found that the degree of polarization of the circular frame in free space increases to a certain value with the propagation distance increasing and keeps invariant, but returns to its initial value in the source plane after propagating a long distance in turbulent atmosphere. This behavior is in agreement with the cases of partially coherent electromagnetic beam and the electromagnetic sinc Schell model beam [14, 18–20 ]. It is observed that the degree of polarization experiences a different change of fluctuation, which essentially is attributed to the unique coherence property of the source. Figure 3(b) and 3(e) show that the fluctuation is suppressed and the humps disappear in the intermediate propagation range when , and the degree of polarization returns to its initial value at the relatively shorter propagation distance. The degree of polarization of two overlapped circular frames is characterized by more humps during propagation, and it finally resumes the initial value at a much longer distance compared with the case of a single circular frame.
Figure 4 shows the transverse distribution of the degree of polarization at for different values of parameters , and . It can be found that for different values of such parameters the change of the degree of polarization of a single circular frame versus the radial almost follows the same trend: first drop then rise gradually towards a saturated value of unity. Apart from some special points where the degree of polarization is independent on the choice of parameter and , the nearer the axial the off-axial point is, the more obvious the change of the degree of polarization is. Like the case of the distribution of spectral density, the behavior of the degree of polarization is less resistant to the affluence of non-Kolmogorov turbulence as Figs. (b) and (e) shows: when , the central prominent part disappears. In addition, for the two superposed frames, the unpolarized off-axial points exist, which are related to the values of the parameters ,and .
It is discovered that the linear combination of the source degree of coherence has no effect on the whole changing trend for the degree of coherence of the electromagnetic source propagating in free space and in the atmospheric turbulence. Therefore, we will only consider the evolutions of the degree of coherence of a single frame for different values of parameters , and . Figure 5 demonstrates the change of the modulus of the degree of coherence at two points (0, 0, z) and ( z) along the z-axis. It can be found that the degree of coherence at the two field points in free space tends to unity in the far-zone, on the contrary, this quantity tends to zero at a certain distance in turbulent atmosphere. This phenomenon implies that the degree of coherence is also destroyed by the atmospheric turbulence, and the stronger turbulence causes the degree of coherence to tend to zero at shorter distances from the source. In addition, the degree of coherence takes on a fluctuation and attains minimum value (zero) at a special distance both in free space and in the atmosphere, which is well understood based on the fact that the element of the CSD matrix of the circular frame during propagation can be expressed to be the difference of two other ones as Eq. (20) shows.
Figure 6 displays the transverse degree of coherence for the single circular frame in the plane . It is shown that the profile of the degree of coherence of the single circular frame in free space is analogy to the intensity distribution of circular dark hollow beam atin free space, i.e. these is a small bright ring around the brightest center . This phenomenon can be elucidated by one of the reciprocity that the far-zone spectral degree of coherence is proportional to the spatial Fourier transform of the spectral intensity distribution across the source . Meanwhile, it is found that the form of the spatial Fourier transform of the spectral intensity distribution across the source determined by Eq. (6) is similar to the expression of the electric field of the dark hollow beam . However, in the atmosphere, the degree of coherence profile converts to be Gaussian shape, and the width of the Gaussian profile become smaller for a larger structure constant and is smallest when the slope . Figure 6(c) shows that the degree of coherence is almost independent on the values of the summation index , which can be explained by the well-known reciprocity relation involving the Fourier-transform pairs between source and far-zone.
In this article, we generalize the scalar source for circular optical frame to electromagnetic domain. The analytical formula for the electromagnetic source for circular frames combination on propagation in turbulent atmosphere is derived and specially used to explore the evolution of the statistics properties of a single circular frame and two superimposed ones. By numerical calculation, it is of interest to find that the degree of polarization experiences a change of oscillation, in particular that of the two nested frames is characterized by more humps. There is off-axis point and on-axis points where the degree of polarization and the degree of coherence is zero separately. In addition, it is illustrated that the characteristic parameters of the media and the source, including the refractive-index structure constant , the fractal constant of the atmospheric spectrum and the upper index in the source degree of coherence, have differently affected the spectral density, the degree of polarization and the degree of coherence of a single circular frame and two nested ones. For sufficiently large values of and for , the influence of the atmosphere on such statistics properties is the strongest: the single flat-topped circular frame and the inner one of the two superimposed frames are shaped to be Gaussian-like, the humps characterizing the degree of polarization are suppressed and the degree of polarization returns to its initial value at the source plane at a earlier propagation distance, and the degree of coherence also tends to zero at a shorter distances from the source. However, for larger values of, such statistic characteristics are less susceptible to the effect of the atmospheric turbulence.
The results presented here indicate that the novel electromagnetic source for circular optical frames carries potential for practical application involving remote sensing, material surface processing, especially the polarization properties can be utilized to influence the behavior of performance parameter in an atmosphere optical communications link.
This work was supported by the National Natural Science Foundation of China (NSFC) (11274273 and 11474253) and the Fundamental Research Funds for the Central Universities (2015FZA3002).
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