## Abstract

A new kind of partially coherent beam with non-conventional correlation function named elliptical Laguerre-Gaussian correlated Schell-model (LGCSM) beam is introduced. Analytical propagation formula for an elliptical LGCSM beam passing through a stigmatic ABCD optical system is derived. The elliptical LGCSM beam exhibits unique features on propagation, e.g., its intensity in the far field (or in the focal plane) displays an elliptical ring-shaped beam profile, being qualitatively different from the circular ring-shaped beam profile of the circular LGCSM beam. Furthermore, we carry out experimental generation of an elliptical LGCSM beam with controllable ellipticity, and measure its focusing properties. Our experimental results are consistent with the theoretical predictions. The elliptical LGCSM beam will be useful in atomic optics.

© 2014 Optical Society of America

## 1. Introduction

Recently, more and more attention is being paid to partially coherent beams with nonconventional correlation functions (i.e., non-Gaussian correlation functions) [1–20]. Gori and collaborators first discussed the sufficient condition for devising the genuine correlation function of a scalar or vector partially coherent beam [1, 2]. Based on their pioneer work, a variety of partially coherent beams with nonconventional correlation functions have been introduced [3–17], and it was found that those beams exhibit unique and interesting propagation properties. Circular Laguerre-Gaussian correlated Schell-model (LGCSM) beam was introduced in [15], and such beam displays a circular ring-shaped beam profile (i.e., dark hollow (DH) beam profile) in the far field (or in the focal plane) although it has a Gaussian beam profile in the source plane. We introduced an experimental setup for generating partially coherent beams with nonconventional correlation functions in [16], and we reported experimental generation of a circular LGCSM beam for the first time. In [17], the statistical properties of a circular LGCSM beam in turbulent atmosphere were investigated, and it was found that such beam has advantage over a Gaussian Schell-model (GSM) beam for reducing the turbulence-induced degradation, which will be useful in free-space optical communications. In [18], we have found both theoretically and experimentally that we can generate a controllable optical cage by focusing a circular LGCSM beam near the focal plane, which will be useful for trapping particles or atoms. A circular LGCSM beam with vortex phase was proposed and generated just recently [19].

On the other hand, it is well-known that ring-shaped beams (also named DH beams) have important applications in free-space optical communications, laser optics, particles trapping, medical sciences, atomic and binary optics [20–32]. Different theoretical models have been proposed to describe various circular DH beams [26–32]. To describe a DH beam of elliptical symmetry (i.e., elliptical DH beam), several theoretical models have been introduced [33–36]. Several methods have been developed to generate an elliptical DH beam with the help of triangular prism or elliptical hollow fiber or Mathieu and Bessel functions [37–39]. In [40], it was found that the elliptical DH beam can be used to control atomic rotation. In [41], it was revealed that an elliptical DH beam displays smaller scintillation than a circular DH beam, a flat-topped beam and a Gaussian beam in turbulent atmosphere. Partially coherent DH beam of circular or elliptical symmetry has also been introduced [42], and experimental generation of a circular partially coherent DH beam with the help of a multimode fiber was reported in [43].

As shown in [33–35], the elliptical DH beam profile of an elliptical DH beam usually disappears in the far field (or in the focal plane). In this paper, we introduce a new kind of partially coherent beam with nonconventional correlation function named elliptical LGCSM beam, which displays elliptical DH beam profile in the far field (or in the focal plane). We derive the analytical propagation formula for an elliptical LGCSM beam passing through a stigmatic ABCD optical system, and study its focusing properties both numerically and experimentally. Some interesting and useful results are found.

## 2. Ellipitcal Laguerre-Gaussian correlated Schell-model beam: theory

In the space-time domain, the statistical properties of a scalar partially coherent beam are characterized by the mutual coherence function [44]. According to [1], the mutual coherence function of a partially coherent beam should satisfy the condition of nonnegative definiteness and can be written in the following form

*H*is an arbitrary kernel, and

*I*is a nonnegative function,${r}_{1}\equiv \left({x}_{1},{y}_{1}\right)$ and ${r}_{2}\equiv \left({x}_{2},{y}_{2}\right)$ are two arbitrary transverse position vectors. Equation (1) can be expressed in the following alternative form [16]

*H*and

*I*. Here

*H*and

*I*denote the response function of the optical path and the intensity of the incoherent source, respectively.

We set *H* and *I* as follows

*H*denotes the response function of the optical path which consists of free space with length

*f*, a thin lens with focal length

*f*and a Gaussian amplitude filter with transmission function

*T*(see Fig. 1 of Ref [16].),

*I*denotes the intensity of an incoherent elliptical DH beam [33] with ${\omega}_{0x}$and ${\omega}_{0y}$being the beam widths along the

*x*and

*y*directions, respectively.

Substituting Eqs. (4)-(6) into Eqs. (2) and (3), we obtain (after integration) the following expression for a partially coherent beam with special degree of coherence

*x*and

*y*directions, respectively, ${L}_{n}^{0}$ denotes the Laguerre polynomial of mode order

*n*and 0. We call the partially coherent beam whose mutual coherence function and degree of coherence are given by Eqs. (7) and (8) as elliptical LGCSM beam. Under the condition of

*n*= 0, elliptical LGCSM beam reduces to elliptical GSM beam [16]. Under the condition of ${\delta}_{0x}={\delta}_{0y}$, elliptical LGCSM beam reduces to circular LGCSM beam [15, 16]. Under the condition of ${\delta}_{0x}={\delta}_{0y}$ and

*n*= 0, elliptical LGCSM beam reduces to circular GSM beam [44]. Figure 1 shows the density plot of the square of the modulus of the degree of coherence of the elliptical LGCSM beam for different values of ${\delta}_{0x}$and ${\delta}_{0y}$with beam order

*n*= 5. One finds from Fig. 1 that the density plot is of elliptical symmetry and the ellipticity is controlled by ${\delta}_{0x}$and ${\delta}_{0y}$, which are controlled by the parameters ${\omega}_{0x}$and ${\omega}_{0y}$of the incoherent elliptical DH beam. Due to the ellipticity symmetry of the degree of coherence, the newly proposed elliptical LGCSM beam exhibits unique and interesting features on propagation as shown below, although its intensity distribution in the source plane has a circular Gaussian beam profile.

Within the validity of the paraxial approximation, the propagation of the mutual coherence function of an elliptical LGCSM beam through a stigmatic ABCD optical system can be studied with the help of the following extended Collins formula [45, 46]

*A*,

*B*,

*C*and

*D*are the elements of a transfer matrix for an optical system, $k=2\pi /\lambda $ is the wavenumber with$\lambda $ being the wavelength.

For the convenience of integration, we introduce the following “sum” and “difference” coordinates

After integration over ${r}_{s}$, Eq. (11) reduces to

Applying the following expansion formulae [47]

Applying the following integral formula

*n*, Eq. (15) becomes

As a numerical example, we study the focusing properties of an elliptical LGCSM beam by applying the derived formula. We assume that an elliptical LGCSM beam is focused by a thin lens with focal length *f* located in the source plane. The output plane is located in the geometrical plane. Then the transfer matrix between the source plane and the output plane reads as

Applying Eqs. (17)-(21), we calculate in Fig. 2 the density plot of the intensity distribution of an elliptical LGCSM beam in the geometrical focal plane for different values of ${\delta}_{0x}$and ${\delta}_{0y}$with beam order *n* = 5, $\lambda =632.8\text{nm},f=400\text{mm},{\sigma}_{0}=1.0\text{mm}$. It is interesting to find from Fig. 2 that we can obtain elliptical DH (i.e., elliptical ring-shaped) beam profile in the focal plane, in other words elliptical DH beam profile can be formed in the far field due to the fact that the beam profile of the far-field intensity is equivalent to that in the focal plane. The ellipticity of the elliptical DH beam profile in the focal plane (or in the far field) is controlled by the parameters ${\delta}_{0x}$and ${\delta}_{0y}$. Under the condition of ${\delta}_{0x}$ = ${\delta}_{0y}$, circular DH (i.e. circular ring-shaped) beam profile is formed as expected (see Fig. 2(c)) [15,16]. Thus, modulating the correlation function of a partially coherent beam provides a novel way for generating an elliptical DH beam in the focal plane, which will be useful in atomic optics.

## 3. Elliptical Laguerre-Gaussian correlated Schell-model beam: experiment

In this section, we report experimental generation of an elliptical LGCSM beam with controllable spatial coherence and ellipticity, and carried out experimental measurement of its intensity in the geometrical focal plane.

Figure 3 shows our experimental setup for generating an elliptical LGCSM beam, measuring the square of the modulus of its degree of coherence and its focused intensity. Part 1 of Fig. 3 shows the experimental setup for generating an elliptical LGCSM beam with controllable parameters ${\delta}_{0x}$and ${\delta}_{0y}$. A beam emitted from a He-Ne laser ($\lambda =632.8\text{nm}$) is reflected by a reflecting mirror and passes through a beam expander, then it goes towards a spatial light modulator (SLM, Holoeye LC2002), which acts as a phase grating designed by the method of computer-generated holograms. To generate an elliptical DH beam whose intensity is given by Eq. (5), the grating pattern of holograms loaded on the SLM is calculated by the interference of a plane wave and the desired elliptical DH beam. The phase gratings for generating elliptical DH beams (*n* = 5) of different values of ${\omega}_{0x}/{\omega}_{0y}$are shown in Fig. 4. When the laser beam illuminates the SLM, diffraction patters appear, and the first-order diffraction pattern can be regarded as an elliptical DH beam and is selected out by a circular aperture. After passing through a thin lens *L*, the generated elliptical DH beam illuminates the RGGD, producing an incoherent elliptical DH beam. Here *L* is used to control the beam spot size on the RGGD through varying the distance between *L* and RGGD. The transmitted beam from the RGGD can be regarded as an incoherent elliptical DH beam if the diameter of the beam spot on the RGGD is larger than the inhomogeneity scale of the RGGD [48], and this condition is satisfied in our experiment. After passing through the thin lens *L*_{1} and the GAF, the incoherent elliptical DH beam becomes an elliptical LGCSM beam.

Part 2 of Fig. 3 shows our experiment setup for measuring the degree of coherence of the generated elliptical LGCSM beam. The generated elliptical LGCSM beam from the GAF passes through the thin lens ${L}_{2}$with focal length${f}_{2}$, and arrives at the charge-coupled device (CCD), which is used to measure the instantaneous intensity. Both distances from GAF to ${L}_{2}$ and from ${L}_{2}$ to CCD are $2{f}_{2}$(i.e., 2*f*-imaging system). Thus, the degree of coherence of the beam in the plane of the CCD is the same as that just behind the GAF. The output signal from the CCD is sent to a personal computer to measure the normalized fourth-order correlation function (FOCF) of the beam which is defined as

*x*and

*y*being pixel spatial coordinates. Here

*m*denotes each realization and ranges from 1 to 2000. Then the square of the modulus of the degree of coherence of the generated elliptical LGCSM beam is obtained as

Part 3 shows our experimental setup for measuring the intensity at the focal plane. The generated elliptical LGSM beam passes through a thin lens ${L}_{3}$ with focal length ${f}_{3}=400\text{mm}$ which is located just behind the GAF, then arrives at the beam profile analyzer (BPA), which is used to measure its intensity at the focal plane. The transfer matrix between the GAF and the BPA is given by

Figure 5 shows our experimental results of the intensity distribution and the corresponding cross line (dotted curve) of the generated elliptical LGCSM beam just behind the GAF. One finds that the generated elliptical LGCSM beam has a Gaussian beam profile in the source plane as expected, and the beam width ${\sigma}_{0}$ is determined by the transmission function of the GAF. Via theoretical fit (solid curve) of the experimental results, we obtain that ${\sigma}_{0}$ is about 1mm in our experiment.

Figure 6 shows our experimental results of the square of the modulus of the degree of coherence and the corresponding cross lines (dotted curves) of the generated elliptical LGCSM beam (*n* = 5) just behind the GAF for different values of coherence widths ${\delta}_{0x}$and ${\delta}_{0y}$. One finds that the distribution of the square of the modulus of the degree of coherence of the beam just behind the GAF indeed exhibits elliptical symmetry when ${\delta}_{0x}\ne {\delta}_{0y}$, and the ellipticity varies as the parameter ${\omega}_{0x}/{\omega}_{0y}$in Fig. 4 varies. Via theoretical fit (solid curve) of the experimental results, the values of ${\delta}_{0x}$and ${\delta}_{0y}$in Fig. 6(a)-(e) are obtained as (a) ${\delta}_{0x}=0.2\text{mm},$${\delta}_{0y}=0.08\text{mm},$(b) ${\delta}_{0x}=0.2\text{mm},$,${\delta}_{0y}=0.16\text{mm}$ (c) ${\delta}_{0x}={\delta}_{0y}=0.2\text{mm},$(d) ${\delta}_{0x}=0.2\text{mm},$ ${\delta}_{0y}=0.24\text{mm}$, (d) ${\delta}_{0x}=0.2\text{mm},{\delta}_{0y}=0.5\text{mm,}$respectively.

Figure 7 shows our experimental results of the intensity distribution and corresponding cross lines (dotted curve) of the generated elliptical LGCSM beam (*n* = 5) in the geometrical focal plane for different values of coherence widths ${\delta}_{0x}$and ${\delta}_{0y}$. For the convenience of comparison, the corresponding theoretical results (solid curves) calculated by Eqs. (17)-(21) are also shown in Fig. 7. One finds that elliptical DH (i.e., elliptical ring-shaped) beam profile indeed is formed in the geometrical focal plane, and the ellipticity is controlled by the values of the parameters ${\delta}_{0x}$and${\delta}_{0y}$, as expected in Fig. 2. Our experimental results agree well with theoretical predictions.

## 4. Summary

We have introduced a kind of partially coherent beam with nonconventional correlation function named elliptical LGCSM beam as a natural extension of recently introduced circular LGCSM beam. We have derived analytical propagation formula for such beam passing through a stigmatic ABCD optical system. Furthermore, we have reported experimental generation of the newly proposed beam and studied its focusing properties both theoretically and experimentally. We have found that the elliptical LGCSM beam exhibits interesting properties, i.e., its intensity in the focal plane (or in the far field) displays an elliptical DH beam profile, which is quite different from that of a circular LGCSM beam. One can control the ellipticity of the elliptical DH beam profile through varying the initial values of the coherence widths ${\delta}_{0x}$and${\delta}_{0y}$. Thus, our methods provides a novel way for generating elliptical DH beam profile in the focal plane (or in the far field) or for beam shaping. Our results will be useful in atomic optics.

## Acknowledgments

This research is supported by the National Natural Science Foundation of China under Grant Nos. 11274005, 11104195 and 11374222, the Huo Ying Dong Education Foundation of China under Grant No. 121009, the Key Project of Chinese Ministry of Education under Grant No. 210081, the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions, the Universities Natural Science Research Project of Jiangsu Province under Grant No. 11KJB140007, the Key Lab Foundation of The Modern Optical Technology of Jiangsu Province, Soochow University, PR China (KJS1301),and the Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

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