A class of electromagnetic sources with multi-cosine Gaussian Schell-model correlation function is introduced. The realizability conditions for such sources and the beam conditions for the beams generated by them are established. Analytical formulas for the cross-spectral density matrix of such beams propagating in free space are derived and used to examine their statistical properties by numerical simulations. The results demonstrate that such beams possess invariant far-field concentric rings-like intensity patterns, uniform polarization state on all rings, and generally different degree of polarization values on different rings. By changing the summation index and other source parameters, the number of rings along with the behavior of various beam characteristics can be tuned at will.
© 2016 Optical Society of America
Modulation and characterization of the spatial profiles of laser beams to meet the needs and improve the performance of various optical systems is currently an important research topic. There are many applications where a light field with an intensity distribution forming concentric rings is desirable, particularly in realizing generic quantum systems of many interacting particles . Several deterministic model beams with such patterns have been developed, such as Laguerre-Gaussian laser modes [1,2] and Bessel optical lattices [3,4]. The average concentric rings-shaped field can also be realized in the far field by modulating the coherence state of the source field, through the alternating series operation of a sinc Schell-model source .
A number of partially coherent model sources  have been recently introduced and realized for tuning the properties of the emitted radiation in the intermediate and far zones from the source plane. Superposition rule  for random sources has precipitated the discovery of new sources radiating beam-like fields with prescribed intensity patterns: self-focusing fields [8–10], flat-top fields [11–14], ring-shaped fields or four beams fields [14–18], lattice-like fields [19–21], as well as other derived models obtained by certain legitimate operations on the source correlation functions [22–25]. These results also demonstrated the potential of the source field coherent modulation for beam shaping applications. The characteristics of such models are usually obtained by use of a scalar formulation that ignores the polarization properties of the optical field. Previously, the states of coherence and polarization of light fields were studied separately, the state of polarization was usually assumed to be invariant as the light beams propagate in free space. Actually, the states of coherence and polarization of a light field had been proven to be intimately related and can be treated in a unified way by the use of the beam coherence-polarization matrix in the space-time domain or the cross-spectral density (CSD) matrix in the space-frequency domain [26–28]. All statistical properties of the electromagnetic stochastic beam-like fields propagating in free space and linear media, including spectral, coherence and polarization states, can be rigorously determined from the CSD matrix in the source plane [29–43]. However, not every such matrix represents a source that generates a beam-like field, it rather must satisfy additional restrictions for the source to generate a beam, namely, the so-called beam conditions . In addition, for an electromagnetic stochastic beam, the choice of the mathematical structure of the correlation matrix is limited by the constraint of non-negative definiteness, also known as the realizability condition [45,46]. Recently, a generalized superposition rule ensuring that the CSD matrix satisfies the constraint of non-negative definiteness has been presented in . It played a significant role in recent studies of electromagnetic stochastic beams [48–51].
In this paper, we will introduce a new class of stochastic electromagnetic sources with properly modeled degree of coherence and any intensity distribution in the source plane for radiating a concentric rings-like intensity pattern on the basis of the superposition rule for the CSD matrices. The paper is organized as follows: the new electromagnetic sources for concentric rings-shaped field are introduced and their realizability conditions are derived in section 2; section 3 gives the beam conditions and the propagation analytical formulas of the CSD matrix elements for the beams generated by the new class of sources; section 4 illustrates the evolution of the statistical properties of beams, including the spectral density, and the states of coherence and polarization; section 5 summarizes the obtained results.
2. Electromagnetic concentric rings Schell-model source and its realizability conditions
We begin by considering a planar, secondary, fluctuating electromagnetic source, located in the plane and radiating into half-space . Assuming that the fluctuations are statistically stationary, at least in the wide sense, the second-order correlation properties of the source at points and and angular frequency may be characterized by a CSD matrix 47]47]:Eq. (3) leads to the formEq. (9) into Eq. (8) and setting the Gaussian profile with the r.m.s. source width for the function , one finds the explicit form of the CSD matrix elements:
Equation (10) represents a new family of the Schell-model sources that may be called Electromagnetic concentric rings Schell-model (EM CRSM) source. Clearly, for the EM CRSM sources correspond to the classic Electromagnetic Gaussian Schell-model sources.
We will now determine the restriction conditions for the source parameters guaranteeing that the mathematical model (10) describes a physically realizable field. For an electromagnetic Schell-model source, the choice of the mathematical form of the CSD matrix must be quasi-Hermitian, i.e. , and non-negative definiteness constraint must hold . The former condition holds if
Since is a monotonically increasing function in positive half axis of variable and is a monotonically decreasing function, their product is not a monotonic function. Therefore, it is hard to obtain a simple and unified analytical formula for the choice of source parameters from the complicated inequality (12), but numerical solutions can be readily found. Several examples of the effective ranges of for different values of , , , and are shown in Table 1. One can see that the minimum values of are only related to the values of and , and independent of the values of , and . This is due to the fact that the dependence of function on its argument is quadratic and that it is linear for . On considering inequality (12) for the limiting case , the following inequality is found:Table 1 illustrates that the maximum values of decrease with the increase of the value of , and as well as the decrease of the range between and .
3. The beam conditions and the propagation laws for the beams generated by the new class of sources
We will now derive the conditions that the CSD matrix must satisfy for the new source to generate an electromagnetic beam, and examine the behavior of its main set of statistical properties, including spectral, coherence and polarization states, for such beams at any intermediate distance from the source.
The far-field spectral density at a point specified by a position ( is a unit vector in its direction) is given by the expression Eq. (16) must be negligible except for directions within a narrow solid angle about the axis. Then can be approximated by unity, while for any values of , so we can see from Eq. (16) that this will be the case if
These conditions are the same as that for the electromagnetic Gaussian Schell-model beams and cosine-Gaussian Schell-model beams [6,50].
Suppose that the source in Eq. (8) generates a beam propagating into half-space close to the positive direction. According to the Huygens-Fresnel principle, the elements of the CSD matrix at two positions and in any transverse plane are related to those in the source plane as
From the components of the cross-spectral density matrix (20), the second-order statistical properties for such beams can be determined. Among the statistics of interest we will consider, in what follows, the spectral density 
In Eqs. (23)-(25) Det and Tr stand for the determinant and the trace of the matrix. Note that the degree of coherence and degree of polarization have different definitions for non-paraxial electromagnetic fields .
In general, the optical field is elliptically polarized, the state of polarization of the polarized portion of the beam may be described in terms of two parameters of the polarization ellipse: the orientation angle 
4. Numerical examples
Next, we will now apply the analytical propagation formula (20) to determine the evolution behavior of statistics Eqs. (23)-(29) in the case of the typical EM CRSM beams on propagating in free space by a set of numerical examples. Without loss of generality, the values of the model source are chosen to be unless otherwise specified in the figures and captions.
Figure 1 illustrates typical evolution of the spectral density of the EM CRSM beam with , , and in the transverse beam cross-sections at several distances from the source plane. One clearly sees that the transverse distribution of the beam’s spectral density from a Gaussian profile given in source plane gradually evolves into a flat-top profile with the increase of propagating distance, and eventually turns into a stable concentric rings profile in the far field, each ring has different width and height. In order to demonstrate the dependence of the spectral density behavior on the source parameters we plot in Fig. 2 the transverse cross-sections of the spectral density of the EM CRSM beams with different values of parameters and at the plane As is seen from the figure, the value and parity of parameter determine the number of rings and the intensity of the center point. When is an even number, the distribution has a dark center and bright ring, and it has a bright center and bright ring for the case of an odd . Note that for the case the width and height of each ring are uniform, this is due to the fact that the spectral densities’ distributions of and components are consistent. While for , the structures of intensity pattern in and directions are different, the superposition of the intensities in two directions leading to each ring is non-uniform. In addition, we also note that the flat-topped intensity pattern can be achieved by adjusting the source parameter, such as shown in Fig. 2(d).
Figure 3 shows the typical distributions of the state of polarization of an EM CRSM beam at the distance from the source. The choice of the source parameters correspond to the second line of Table 1, for attaining its minimum value. One clearly sees that all polarization properties of each ring are uniform. Different rings have different degrees of polarization, but have the same orientation angle and degree of ellipticity.
Figure 4 illustrates the typical distributions of the degree of polarization for different values of propagation distance and sets of source parameters. As can be seen from Fig. 4(a), the uniform polarization distribution of the source field starts to involve lateral oscillations as the propagation distance increases. The fluctuation is enhanced with the increase of the values of parameters and as shown in Figs. 4(b)-4(d). The degree of polarization of the far field on the z-axis () increases with the increase of and but it remains the same equal for all even or odd
Figure 5 shows the behavior of the degree of coherence of the EM CRSM beams as a function of the separation distance where the two points are chosen at locations symmetric with respect to the optical axis, i.e. . We can see from Fig. 5(a) and 5(b) that the degree of coherence of an EM CRSM beam with starts as a modulated by multi cosine function Gaussian distribution, with growing distance, the fluctuations of the degree of coherence profiles gradually weaken and the curve becomes a smooth Gaussian profile for odd and that with two symmetrical side lobes for even . Figure 5(c) illustrates that the oscillation of the degree of coherence in the source field diminishes with the decreasing value of parameter and the profile resembles the multi-Gaussian Schell-model sources, so they have similar far-field intensity distribution with the flat-topped profiles, as shown in Fig. 2(d). Furthermore, we find that the curves are coincident for different and when , as shown in Figs. 5(b) and 5(d), which imply that the dependence on and gradually disappears with growing distance.
In this article, a novel class of stochastic electromagnetic sources has been introduced, in which the correlation is prescribed with the help of the multi-cosine Gaussian Schell-model coherence functions. The realizability conditions for such source and the beam conditions for the beams generated by the new class of sources have been established. The propagation analytical formulas of the CSD matrix elements for such beams have been derived and used to explore the evolution of all the major the second-order characteristics for such beams. It is shown that such sources with Gaussian intensity distribution in the source fields can produce far fields having concentric rings profiles around the axis and retain them for unlimited distances. The polarization state on each ring is uniform, while the degree of polarization can vary from ring to ring in adjustable manner. The summation index N and other source parameters provide a convenient tool for adjusting the number of rings and intensity profiles, along with the behavior of various beam characteristics.
Electromagnetic Schell-model beams can be produced with the help of the interferometric technique involving two spatial light modulators described in  with a suitable choice of the window function for the phase correlation. For the EM CRSM source the window function must take the form of the multi-cosine Gaussian function. The results of this work are of particular importance for some applications in which a field with concentric rings-like intensity patterns must be generated by a source with arbitrary source intensity distribution. Such applications may include optical communications, optical particle manipulation and material processing.
Z. Mei’s research is supported by the National Natural Science Foundation of China (NSFC) (11247004) and Zhejiang Provincial Natural Science Foundation of China (LY16F050007). O. Korotkova’s research is supported by US AFOSR (FA9550-12-1-0449) and US ONR (N00014-15-1-2350).
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