## Abstract

A class of electromagnetic sources with sinc Schell-model correlations is introduced. The conditions on source parameters guaranteeing that the source generates a physical beam are derived. The evolution behaviors of statistical properties for the electromagnetic stochastic beams generated by this new source on propagating in free space and in atmosphere turbulence are investigated with the help of the weighted superposition method and by numerical simulations. It is demonstrated that the intensity distributions of such beams exhibit unique features on propagating in free space and produce a double-layer flat-top profile of being shape-invariant in the far field. This feature makes this new beam particularly suitable for some special laser processing applications. The influences of the atmosphere turbulence with a non-Kolmogorov power spectrum on statistical properties of the new beams are analyzed in detail.

© 2014 Optical Society of America

## 1. Introduction

Schell-model sources have played a role of utmost importance in optical coherence theory since it was introduced in 1967 [1]. Among the most classic models is the Gaussian Schell-model source which comes with the degree of coherence of Gaussian distribution [2]. Some other augmented models for scalar random sources have been recently introduced, in which the classic Gaussian degree of coherence is modulated by a well-behaved function, such as Multi-Gaussian Schell-model sources [3], Bessel-Gaussian and Laguerre-Gaussian Schell-model sources [4], cosine-Gaussian Schell-models sources [5], etc. Several random scalar sources producing far field with rectangular intensity profiles have also been introduced by modeling the source degree of coherence with the help of two one-dimensional Schell-model sources [6–8]. These scalar sources and beams are specified by a single correlation function, while the electromagnetic random beams are characterized by the $2\times 2$ cross-spectral density (CSD) matrices. It has been shown that the changes in the propagation of the spectral density, the spectral degree of coherence, and the spectral degree of polarization of a random electromagnetic beam can be determined from knowledge of a CSD matrix [9]. This matrix has already been used in studies of the behaviors of random electromagnetic beams propagating through turbulent atmosphere [10,11], an optical resonator [12] and optical focusing systems [13,14], etc. For an electromagnetic stochastic beam, the spatial correlation functions must satisfy additional restrictions, leading to restrictions for source parameters known as the realizability conditions [15]. Roychowdhury and Korotkova had derived realizability conditions for electromagnetic Gaussian Schell-model (EM GSM) sources [16]. Recently, a basic theory on devising genuine CSD matrices for an electromagnetic stochastic beam was exploited and extended [17,18]. Based on this method, several models for scalar random sources have been extended to electromagnetic domain, such as the electromagnetic Multi-Gaussian Schell-model sources [19], electromagnetic non-uniformly correlated sources [20,21], electromagnetic cosine-Gaussian Schell-models sources [22], and so on.

The degree of coherence of all the known Schell-model sources is Gaussian or based on the Gaussian degree of coherence. Recently, we introduced a new random source class with sinc Schell-model (SSM) correlated function [23], whose degree of coherence in the source plane presents sinc function profile rather than the traditional Gaussian distribution. The beams generated by SSM sources have been found to form a stable single layer flat profile in the far filed. In this paper, we use a scalar model proposed in [23] as a building for developing electromagnetic sources with SSM correlations, terming the novel class of beams the electromagnetic sinc Schell-model (EM SSM) beams, in which all the correlations are prescribed with the help of the scalar SSM distributions. As we illustrate by numerical examples, the EM SSM source can produce a double-layer flat-top intensity distribution and remain shape-invariant in the far field. It is not like the known partially coherent flat-top beams produce a single layer flat-top intensity profile and disappear gradually in propagation and becomes Gaussian in the far field. The double-layer flat-top intensity profile may be of use in applications dealing with laser processing.

After extending the scalar SSM sources to the full electromagnetic domain and deriving the realizability conditions, our major task in this study is to explore the behavior of the main statistical properties for EM SSM beams propagating in free space and atmosphere turbulence. Usually, it is difficult to obtain a general analytical formula for the propagated CSD matrix of EM SSM beams. However, the integral structure of the elements of the CSD matrix constructed through the mode superposition retains its form on propagation. The elementary modes change on propagation, while the weight functions remain unchanged [18,20]. Therefore, we can consider separately the propagation of the kernel functions and apply the weighted superposition method to study the statistical properties of EM SSM beams by a set of numerical examples.

## 2. Electromagnetic Sinc Schell-model source

The second-order correlation properties of a statistically stationary electromagnetic source at two points ${{\rho}^{\prime}}_{1}$ and ${{\rho}^{\prime}}_{2}$ will be described by the $2\times 2$ CSD matrix ${\widehat{W}}^{(0)}({{\rho}^{\prime}}_{1},{{\rho}^{\prime}}_{2};\omega )$, whose element ${W}_{\alpha \beta}^{(0)}({{\rho}^{\prime}}_{1},{{\rho}^{\prime}}_{2};\omega )$ are given by [9]

Recall the non-negative definiteness condition for CSD matrices to be physically realizable, it suffices to have an integral of the form [18]

*Electromagnetic Sinc Schell-model*(EM SSM) sources.

The CSD matrix of a physically realizable field must be quasi-Hermitian, i.e. that ${W}_{\alpha \beta}^{(0)}({{\rho}^{\prime}}_{1},{{\rho}^{\prime}}_{2})={W}_{\beta \alpha}^{(0)}({{\rho}^{\prime}}_{2},{{\rho}^{\prime}}_{1})$. It is sufficient that the condition holds if

Furthermore, the non-negative definiteness constraint for the genuine CSD matrices means that, the following inequalities have to be satisfied for any $v$. We readily find from Eq. (6) that inequality (9) is always satisfied, and on substituting from Eq. (6) into inequalities (10) implies that it is satisfied if## 3. Propagation laws for the beams generated by the new class of sources in linear random medium

Suppose that source (7) generates a beam-like field propagating into half-space $z>0$ filled with turbulent atmosphere. According to the extended Huygens-Fresnel integral principle adjusted for propagation in linear random medium, the elements of the CSD matrix at two points $({\rho}_{1},z)$ and $({\rho}_{2},z)$ in the same transverse plane are related to those in the source plane as [24–26]

On inserting Eq. (2) into Eq. (13) we obtain, after interchanging the orders of integrals, the expression

On substituting from Eqs. (3), (4) and (14) into Eq. (15), after long integral calculations, we obtain, for any mode, the formula

To model the atmosphere turbulence, a non-Kolmogorov spectrum is used [26–28]

It follows from above derivation that the propagated kernel of the elements of CSD matrix on propagation due to random medium is only included in the integral term in the right side of Eqs. (17) and (18), thus, for the free space propagation, Eqs. (17) and (18) can be express as

Based on the weighted superposition (15) and the propagating formulae of the modes (17) and (23), the spectral density $S$, the spectral degree of coherence $\mu $ and the spectral degree of polarization $P$ of the electromagnetic stochastic beam in the turbulent atmosphere and free space can be calculated by the expressions [9]

## 4. The statistical properties of EM SSM beams in free space and non-Kolmogorov atmosphere turbulence

We will now illustrate the behavior of the statistical properties, including spectral density, the degree of coherence and the degree of polarization, of typical beams generated by source (7) on propagating in free space and atmosphere turbulence by a set of numerical examples. Without loss of generality, the values of source and medium parameters are chosen to be ${A}_{x}={A}_{y}=1$, $\left|{B}_{xy}\right|=0.2$, ${\sigma}_{0}=10mm$, $k={10}^{7}{m}^{-1}$, ${\delta}_{xx}=1mm$, ${L}_{0}=1m$, ${l}_{0}=1mm$, otherwise specified in the text and captions.

*A. Spectral density S*

*A. Spectral density S*

Figure 1 shows typical evolution of the spectral density *S* of an EM SSM beam with ${\delta}_{yy}=2.5mm$ in the transverse beam cross-sections at several distances $z$ from the source plane on propagating in free space. One clearly sees that the Gaussian profile of source field gradually transforms into a double-layer flat-top distribution with increasing propagation distance. This is because the coherence properties of light field in the source plane are closely related to the propagation characteristics and the transverse intensity distribution of the far field. Two flat-top profiles correspond to the spectral densities’ distributions of $x$ and $y$ components of electric field with different r.m.s. correlation width. In order to demonstrate the dependence of the spectral density behavior on the degree of coherence, we plot in Fig. 2 its evolution in the transverse beam cross-section at the plane $z=1km$ in free space for different ${\delta}_{yy}$. Note that for the case ${\delta}_{yy}={\delta}_{xx}=1mm$ the transverse beam cross-section is a single-layer flat-top profile, this is due to the degree of coherences of $x$ and $y$ components are the same, which results in two directions with the same far-field spectral density distributions. However, for the cases${\delta}_{yy}\ne {\delta}_{xx}$ shown as Figs. 2(b) and 2(c), the far-field spectral density distributions of $x$ and $y$ directions are different, resulting in the appearance of double-layer profile, the lower layer corresponds to the $x$ component and the higher layer corresponds to the $y$ component. This is the same reason that the electromagnetic cosine-Gaussian Schell-model beams possess double-ring intensity profiles.

Figure 3 shows the transverse distribution of the spectral density *S* of an EM SSM beam with ${\delta}_{yy}=2.5mm$ propagating in the non-Kolmogorov turbulence atmosphere at the distance z = 5km for different values of turbulence parameters ${\tilde{C}}_{n}^{2}$ and $\alpha $. The longitudinal orientations indicate the changes in the Kolmogorov turbulence as the values of $\alpha $ increase and the transverse orientations indicate the changes in the Kolmogorov turbulence as the values of ${\tilde{C}}_{n}^{2}$ increase. One can see from Fig. 3 that, for the weakest turbulence the spectral density keeps the double-layer distribution as shown in Fig. 3(c), while for Figs. 3(a) and 3(f) with decreasing $\alpha $and increasing ${\tilde{C}}_{n}^{2}$ the double-layer profile of spectral density is destroyed gradually. Finally, in Fig. 3(d) for substantially strong turbulence the upper layer and lower layer are blended completely and the beams’ intensity resembles Gaussian profile. Figures 3(b) and 3(e) show the spectral density distributions is destroyed the most when $\alpha =3.1$, which just like other beams [22, 26].

*B. Spectral degree of polarization *P

*B. Spectral degree of polarization*P

We now turn our attention to the changes in the spectral degree of polarization of EM SSM beams in free space and non-Kolmogorov atmosphere turbulence. Figure 4 illustrates the typical distributions of the polarization properties for the most general case of partially correlated electric field components, as beam propagates at 1km in free space. Note that for the case ${\delta}_{xx}={\delta}_{yy}={\delta}_{xy}$ the spectral degree of polarization does not change upon propagation. It is the same as the source plane and uniform for all the points at the any propagation plane, which was also shown to be critical for other beams. However, for other cases, unlike the degree of polarization of EM GSM beams to form an inverted Gaussian distribution in the far field, Figs. 3(b) and 3(c) clearly show the formation of an inverted flat distribution with prominent and uniform polarization distribution in the central region. The height of the prominent part increases with the increase of the coherence length.

Figure 5 shows the transverse distribution of the degree of polarization *P* of an EM SSM beam with ${\delta}_{yy}={\delta}_{xy}=2.5mm$ at the propagation distance z = 5km in the turbulence atmosphere for different values of parameters ${\tilde{C}}_{n}^{2}$ and $\alpha $. For the case Fig. 5(c) of the weakest turbulence, the distribution of the degree of polarization is similar to the case of free space. With the strengthening of turbulence, the steepness of edges gradually become decline and the flat central prominent region change into a Gaussian profile, as shown in Figs. 5(a), 5(f), 5(b) and 5(d). For the case Fig. 5(e) of the strongest turbulence, the central projection disappears and the distribution of the degree of polarization present the inverted Gaussian profile.

In Fig. 6 we illustrate the changes in the spectral degree of polarization along the z-axis of the same beam as in Fig. 5 propagating in the atmosphere turbulence for different values of turbulent parameters. For the case of free space, i.e., ${\tilde{C}}_{n}^{2}=0$, the spectral degree of polarization increases to a certain value with increasing propagation distance and remains unchanged. However, in atmosphere turbulence, this quantity returns to its value in the source plane after propagating a certain distance, the shorter distance required for the stronger turbulence. This result is similar to the case of EM GSM beams [29].

*C. Spectral degree of coherence *μ

*C. Spectral degree of coherence*μ

We will now turn to the analysis of the spectral degree of coherence of the EM SSM beam as it travels in the atmosphere turbulence. Figure 7 illustrates the behavior of the degree of coherence $\mu $ of an EM SSM beam with ${\delta}_{yy}=2.5mm$ as a function of ${\rho}_{d}=|{\rho}_{1}-{\rho}_{2}|$ between two points symmetric with respect to the optical axis, i.e. ${\rho}_{1}=-{\rho}_{2}={\rho}_{d}/2$, for different values of atmosphere parameters and propagation distances. We can see from Fig. 7 that there are two effects occurring with the degree of coherence, as either the propagation distance or the strength of turbulence increase. With the propagation distance increases, the fluctuations of the degree of coherence profiles gradually weaken and eventually degenerate into a Gaussian shape, and the width of profile is reduced in the further field. And the width of Gaussian profile also decreases with the strengthening of turbulence effects.

In Fig. 8 we illustrate the changes in the spectral degree of coherence $\mu $ along the *z*-axis of the same beam as in Fig. 7 propagating in the atmosphere turbulent for different values of atmosphere parameters when ${\rho}_{d}=0.5mm$. For the case of ${\tilde{C}}_{n}^{2}=0$, corresponding to the free space, the spectral degree of coherence tends to 1 in the far-zone. On the contrary, this quantity tends to zero after propagating a certain distance in atmosphere turbulence. This result implies the atmosphere turbulence also destroys spectral coherence of EM SSM beams and it is so at smaller distances from the source for the stronger turbulence.

## 5. Concluding remarks

In this article, we have introduced a new class of stochastic electromagnetic sources with Sinc Schell-model correlations properties, and derived the realizability conditions for such sources. And then, we have explored the variation of the three main statistical properties of the EM SSM beams: the spectral density, and the states of coherence and polarization, on propagating in free space and in atmosphere turbulence based on the weighted superposition method and by numerical simulations. It is shown that, for such novel source which can initially have any intensity distribution, as Gaussian profile in our examples, can produce a double-layer flat-top intensity distribution in the far field in free space. The difference of height and diameter of two layer intensity distribution can be controlled by adjusting the coherent properties of the source field. The results also illustrated that the double-layer flat-top profile is preserved for any propagation distances in free space and weak atmosphere turbulence but it is destroyed by the strong atmosphere turbulence. The double-layer flat-top profile gradually disappears and the Gaussian-like distribution is produced when it propagates in the strong turbulent atmosphere over sufficiently long distances. The spectral degree of polarization and the degree of coherence are also destroyed by the turbulence. The distributions in far field eventually transformed into a divergent inverted Gaussian profile for the spectral degree of polarization and a shrink Gaussian profile for the degree of coherence relative to free space. Just like for other beam classes, the influence of the turbulence to statistical properties of EM SSM beams is the most for sufficiently large ${\tilde{C}}_{n}^{2}$ and for $\alpha $ in a region about 3.1. The propagation of light beams in non-Kolmogorov weak turbulence depends on the choice of the length unit [30,31]. We state that the turbulence structure parameter in this paper uses meter as a unit. If a different unit is selected, the plots would be different.

The electromagnetic random beams can be produced with help of the interferometric technique involving two spatial light modulators described in Ref [32]. For the EM SSM the phase correlation function of the modulators should take forms of Sinc functions instead of Gaussian function as suggested in Ref [32]. for generation of EM GSM model beams. Unlike conventional flat light field with a single layer flat-top profile, the EM SSM source can produce a double-layer flat-top intensity distribution and remain shape-invariant in the far field in free space and weak atmosphere turbulence, and can be controlled the structure of two layer by adjusting the coherent properties of the source field. This feature makes this new beam particularly suitable for applications involving some laser processing, such as laser drilling for some stepped complex hole with different aperture and depth.

## Acknowledgments

The research is supported by the National Natural Science Foundation of China (NSFC) (11247004).

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