## Abstract

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

## 1. Introduction

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 [1]. 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 [5].

A number of partially coherent model sources [6] 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 [7] 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 [44]. 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 [47]. 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 $z=0$ and radiating into half-space $z>0$. Assuming that the fluctuations are statistically stationary, at least in the wide sense, the second-order correlation properties of the source at points ${\rho}_{1}$ and ${\rho}_{2}$ and angular frequency $\omega $ may be characterized by a $2\times 2$ CSD matrix [28]

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 $N=1$ 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. ${W}_{\alpha \beta}^{(0)}({{\rho}^{\prime}}_{1},{{\rho}^{\prime}}_{2})={W}_{\beta \alpha}^{(0)}{}^{\ast}({{\rho}^{\prime}}_{2},{{\rho}^{\prime}}_{1})$, and non-negative definiteness constraint must hold [6]. The former condition holds if

We readily find from Eq. (9) that inequality (4) is always satisfied, and on substituting from Eq. (9) into Eq. (5) results in inequality

Since $\mathrm{cosh}(x)=[exp(x)+exp(-x)]/2$ is a monotonically increasing function in positive half axis of variable $x$ and $\mathrm{exp}(-{x}^{2})$ 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 ${\delta}_{xy}$ for different values of $N$, $R$, $\left|{B}_{xy}\right|$, ${\delta}_{xx}$ and ${\delta}_{yy}$ are shown in Table 1. One can see that the minimum values of ${\delta}_{xy}$ are only related to the values of ${\delta}_{xx}$ and ${\delta}_{yy}$, and independent of the values of $N$, $R$ and $\left|{B}_{xy}\right|$. This is due to the fact that the dependence of function $\mathrm{exp}(-{x}^{2})$ on its argument is quadratic and that it is linear for $\mathrm{cosh}(x)$. On considering inequality (12) for the limiting case $v\to \infty $, the following inequality is found:

which is in agreement with the results of numerical calculations. However, the maximum values of ${\delta}_{xy}$ are related to all of the source parameters. Table 1 illustrates that the maximum values of ${\delta}_{xy}$ decrease with the increase of the value of $N$, $R$ and $\left|{B}_{xy}\right|$ as well as the decrease of the range between ${\delta}_{xx}$ and ${\delta}_{yy}$.## 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 $r=rs$ ($s$ is a unit vector in its direction) is given by the expression [6]

On substituting from Eq. (8) first into Eq. (15) and then into Eq. (14), one find that

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 $z>0$ close to the positive $z$ direction. According to the Huygens-Fresnel principle, the elements of the CSD matrix at two positions ${r}_{1}=({\rho}_{1},z)$and ${r}_{2}=({\rho}_{2},z)$ in any transverse plane are related to those in the source plane as

On substituting from Eq. (8) into Eq. (19) and calculating the integral we obtain the formula

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 [6]

the spectral degree of coherenceIn 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 [52].

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 [28]

## 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 ${A}_{x}={A}_{y}=1,$$\sigma =0.5mm,$$\lambda =632.8nm,$ unless otherwise specified in the figures and captions.

Figure 1 illustrates typical evolution of the spectral density of the EM CRSM beam with $N=6$, $R=1mm$, ${\delta}_{xx}=0.4mm$ and ${\delta}_{yy}=0.45mm$ in the transverse beam cross-sections at several distances $z$ 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 $N,$${\delta}_{xx},$${\delta}_{yy}$ and $R$ at the plane $z=3m.$As is seen from the figure, the value and parity of parameter $N$ determine the number of rings and the intensity of the center point. When $N$ is an even number, the distribution has a dark center and $N/2$ bright ring, and it has a bright center and $(N-1)/2$ bright ring for the case of an odd $N$. Note that for the case ${\delta}_{xx}={\delta}_{yy}$ the width and height of each ring are uniform, this is due to the fact that the spectral densities’ distributions of $x$ and $y$ components are consistent. While for ${\delta}_{xx}\ne {\delta}_{yy}$, the structures of intensity pattern in $x$ and $y$ 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 $z=3m$ from the source. The choice of the source parameters correspond to the second line of Table 1, for ${\delta}_{xy}$ 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 $N,$$R$and ${\delta}_{xy},$as shown in Figs. 4(b)-4(d). The degree of polarization of the far field on the *z*-axis ($\rho =0$) increases with the increase of $R$ and ${\delta}_{xy},$but it remains the same equal for all even or odd $N.$

Figure 5 shows the behavior of the degree of coherence of the EM CRSM beams as a function of the separation distance ${\rho}_{d}=|{\rho}_{1}-{\rho}_{2}|$ where the two points are chosen at locations symmetric with respect to the optical axis, i.e. ${\rho}_{1}=-{\rho}_{2}=\rho /2$. We can see from Fig. 5(a) and 5(b) that the degree of coherence of an EM CRSM beam with $N>1$ 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 $N$ and that with two symmetrical side lobes for even $N$. Figure 5(c) illustrates that the oscillation of the degree of coherence in the source field diminishes with the decreasing value of parameter $R$ 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 $N$ and ${\delta}_{xx}$ when $z=3m$, as shown in Figs. 5(b) and 5(d), which imply that the dependence on $N$ and ${\delta}_{xx}$ gradually disappears with growing distance.

## 5. Summary

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 [31] 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.

## Acknowledgments

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).

## References and links

**1. **L. Amico, A. Osterloh, and F. Cataliotti, “Quantum many particle systems in ring-shaped optical lattices,” Phys. Rev. Lett. **95**(6), 063201 (2005). [CrossRef] [PubMed]

**2. **G. Ruffato, M. Massari, and F. Romanato, “Generation of high-order Laguerre-Gaussian modes by means of spiral phase plates,” Opt. Lett. **39**(17), 5094–5097 (2014). [CrossRef] [PubMed]

**3. **J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. **177**(1-6), 297–301 (2000). [CrossRef]

**4. **Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Rotary solitons in Bessel optical lattices,” Phys. Rev. Lett. **93**(9), 093904 (2004). [CrossRef] [PubMed]

**5. **Z. Mei and O. Korotkova, “Alternating series of cross-spectral densities,” Opt. Lett. **40**(11), 2473–2476 (2015). [CrossRef] [PubMed]

**6. **L. Mandel and E. Wolf, *Optical Coherence and Quantum Optics* (Cambridge University, 1995).

**7. **F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. **32**(24), 3531–3533 (2007). [CrossRef] [PubMed]

**8. **H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. **36**(20), 4104–4106 (2011). [CrossRef] [PubMed]

**9. **Z. Tong and O. Korotkova, “Nonuniformly correlated light beams in uniformly correlated media,” Opt. Lett. **37**(15), 3240–3242 (2012). [CrossRef] [PubMed]

**10. **H. Lajunen and T. Saastamoinen, “Non-uniformly correlated partially coherent pulses,” Opt. Express **21**(1), 190–195 (2013). [CrossRef] [PubMed]

**11. **S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. **37**(14), 2970–2972 (2012). [CrossRef] [PubMed]

**12. **O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A **29**(10), 2159–2164 (2012). [CrossRef] [PubMed]

**13. **O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. **39**(1), 64–67 (2014). [CrossRef] [PubMed]

**14. **Z. Mei, “Two types of sinc Schell-model beams and their propagation characteristics,” Opt. Lett. **39**(14), 4188–4191 (2014). [CrossRef] [PubMed]

**15. **Z. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. **38**(14), 2578–2580 (2013). [CrossRef] [PubMed]

**16. **C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. **39**(4), 769–772 (2014). [CrossRef] [PubMed]

**17. **Z. Mei, “Light sources generating self-splitting beams and their propagation in non-Kolmogorov turbulence,” Opt. Express **22**(11), 13029–13040 (2014). [CrossRef] [PubMed]

**18. **C. Ding, O. Korotkova, Y. Zhang, and L. Pan, “Sinc Schell-model pulses,” Opt. Commun. **339**, 115–122 (2015). [CrossRef]

**19. **L. Ma and S. A. Ponomarenko, “Optical coherence gratings and lattices,” Opt. Lett. **39**(23), 6656–6659 (2014). [CrossRef] [PubMed]

**20. **Z. Mei, D. Zhao, O. Korotkova, and Y. Mao, “Gaussian Schell-model arrays,” Opt. Lett. **40**(23), 5662–5665 (2015). [CrossRef] [PubMed]

**21. **L. Ma and S. A. Ponomarenko, “Free-space propagation of optical coherence lattices and periodicity reciprocity,” Opt. Express **23**(2), 1848–1856 (2015). [CrossRef] [PubMed]

**22. **M. Santarsiero, G. Piquero, J. C. G. de Sande, and F. Gori, “Difference of cross-spectral densities,” Opt. Lett. **39**(7), 1713–1716 (2014). [CrossRef] [PubMed]

**23. **Z. Mei, O. Korotkova, and Y. Mao, “Products of Schell-model cross-spectral densities,” Opt. Lett. **39**(24), 6879–6882 (2014). [CrossRef] [PubMed]

**24. **O. Korotkova and Z. Mei, “Convolution of degrees of coherence,” Opt. Lett. **40**(13), 3073–3076 (2015). [CrossRef] [PubMed]

**25. **Z. Mei, O. Korotkova, and Y. Mao, “Powers of the degree of coherence,” Opt. Express **23**(7), 8519–8531 (2015). [CrossRef] [PubMed]

**26. **F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. **23**(4), 241–243 (1998). [CrossRef] [PubMed]

**27. **E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A **32**(5-6), 263–267 (2003). [CrossRef]

**28. **E. Wolf, *Introduction to the Theories of Coherence and Polarization of Light* (Cambridge University, 2007).

**29. **S. R. Seshadri, “Partially coherent Gaussian Schell-model electromagnetic beams,” J. Opt. Soc. Am. A **16**(6), 1373–1380 (1999). [CrossRef]

**30. **F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. **3**(1), 1–9 (2001). [CrossRef]

**31. **T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. **7**(5), 232–237 (2005). [CrossRef]

**32. **O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. **246**(1-3), 35–43 (2005). [CrossRef]

**33. **X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express **15**(25), 16909–16915 (2007). [CrossRef] [PubMed]

**34. **F. Gori, M. Santarsiero, and R. Borghi, “Modal expansion for J_{0}-correlated electromagnetic sources,” Opt. Lett. **33**(16), 1857–1859 (2008). [CrossRef] [PubMed]

**35. **D. Zhao and X. Du, “Polarization modulation of stochastic electromagnetic beams on propagation through the turbulent atmosphere,” Opt. Express **17**(6), 4257–4262 (2009). [CrossRef] [PubMed]

**36. **X. Ji, E. Zhang, and B. Lü, “Changes in the spectrum and polarization of polychromatic partially coherent electromagnetic beams in the turbulent atomosphere,” Opt. Commun. **275**(2), 292–300 (2007). [CrossRef]

**37. **Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express **16**(11), 7665–7673 (2008). [CrossRef] [PubMed]

**38. **J. Pu and O. Korotkova, “Propagation of the degree of cross-polarization of a stochastic electromagnetic beam through the turbulent atmosphere,” Opt. Commun. **282**(9), 1691–1698 (2009). [CrossRef]

**39. **E. Shchepakina and O. Korotkova, “Second-order statistics of stochastic electromagnetic beams propagating through non-Kolmogorov turbulence,” Opt. Express **18**(10), 10650–10658 (2010). [CrossRef] [PubMed]

**40. **G. Zhou and X. Chu, “Average intensity and spreading of a Lorentz-Gauss beam in turbulent atmosphere,” Opt. Express **18**(2), 726–731 (2010). [CrossRef] [PubMed]

**41. **X. Chu, C. Qiao, X. Feng, and R. Chen, “Propagation of Gaussian-Schell beam in turbulent atmosphere of three-layer altitude model,” Appl. Opt. **50**(21), 3871–3878 (2011). [CrossRef] [PubMed]

**42. **F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. **100**(5), 051108 (2012). [CrossRef]

**43. **M. Yao, I. Toselli, and O. Korotkova, “Propagation of electromagnetic stochastic beams in anisotropic turbulence,” Opt. Express **22**(26), 31608–31619 (2014). [CrossRef] [PubMed]

**44. **O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett. **29**(11), 1173–1175 (2004). [CrossRef] [PubMed]

**45. **H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. **249**(4-6), 379–385 (2005). [CrossRef]

**46. **F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A **25**(5), 1016–1021 (2008). [CrossRef] [PubMed]

**47. **F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A: Pure Appl. Opt. **11**(8), 085706 (2009). [CrossRef]

**48. **Z. Tong and O. Korotkova, “Electromagnetic nonuniformly correlated beams,” J. Opt. Soc. Am. A **29**(10), 2154–2158 (2012). [CrossRef] [PubMed]

**49. **Z. Mei, O. Korotkova, and E. Shchepakina, “Electromagnetic multi-Gaussian Schell-model beams,” J. Opt. **15**(2), 025705 (2013). [CrossRef]

**50. **Z. Mei and O. Korotkova, “Electromagnetic cosine-Gaussian Schell-model beams in free space and atmospheric turbulence,” Opt. Express **21**(22), 27246–27259 (2013). [CrossRef] [PubMed]

**51. **O. Korotkova and Z. Mei, “Random electromagnetic model beams with correlations described by two families of functions,” Opt. Lett. **40**(23), 5534–5537 (2015). [CrossRef] [PubMed]

**52. **J. Tervo, T. Setälä, and A. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express **11**(10), 1137–1143 (2003). [CrossRef] [PubMed]