Beam quality of radial Gaussian Schell-model array beams

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Abstract

Taking the Rayleigh range zR and the M2-factor as the characteristic parameters of beam quality, the beam quality of radial Gaussian Schell-model (GSM) array beams is studied. The analytical expressions for the zR and the M2-factor of radial GSM array beams are derived. It is shown that for the superposition of the cross-spectral density function zR is longer and the M2-factor is lower than that for the superposition of the intensity. For the two types of superposition, zR increases and the M2-factor decreases with increase in beam coherence parameter, and both zR and the M2-factor increase with increase in inverse radial fill-factor. For the superposition of the cross-spectral density function, zR increases and the M2-factor decreases with increase in beam number, while for the superposition of the intensity both the zR and M2-factor are independent of the beam number.

Introduction

Laser beam arrays have attracted much attention because of their wide spread applications in high-power systems, inertial confinement fusion and high-energy weapons. The propagation of laser beam arrays in free space or in paraxial optical systems has been studied [1], [2], [3]. Recently, much work has been carried out concerning the spreading of laser beam arrays propagating through atmospheric turbulence [4], [5], [6], [7]. In addition, the scintillations of laser beam arrays have also been investigated [8].

The Rayleigh range is used in the theory of lasers to characterize the distance over which a beam may be considered effectively non-spreading [9]. The Rayleigh ranges of a single fully and partially coherent beam have been studied [9], [10], [11], [12]. It has been shown that a single partially coherent beam will have a Rayleigh range that is shorter than that of a single fully coherent beam with the same intensity distribution in the beam waist [12]. However, until now the Rayleigh range of partially coherent array beams has not been studied. Furthermore, some interesting questions have not been examined: What about the relationship between the Rayleigh range of a partially coherent array beam and that of a fully coherent array beam? What about the relation between the Rayleigh range of a partially coherent array beam and that of a single partially coherent beam?

The beam propagation factor, namely, the M2-factor, is a useful parameter for characterizing the beam quality of different laser beams [13]. The factor M2 is invariant when a real laser beam passes through a nonaberrated paraxial optical beam train or focusing system. The value of M2 is 1 for any arbitrary beam profile, with the limit of M2≡1 occurring only for a single Gaussian beam [13]. The lower the factor M2 is, the better the beam quality [14]. Using the intensity moment method, the analytical expression for the M2-factor of rectangular Gaussian array beams has been derived for the coherent and incoherent combinations [15]. However, the analytical expression for the M2-factor of radial Gaussian array beams has been derived only for the case of the incoherent combination [16], [17]. In practice, a partially coherent laser source is often encountered. Therefore, it is very important to study the beam quality of radial partially coherent array beams. In this paper, taking the Rayleigh range and the M2-factor as the characteristic parameters of beam quality, the beam quality of radial Gaussian Schell-model (GSM) array beams are studied in detail. The analytical expressions for the Rayleigh range and the M2-factor of radial GSM array beams are derived using the quadratic free-space propagation equation of the mean-squared beam width, where the superposition of the cross-spectral density function and the superposition of the intensity are considered. In the coherent limit, the radial Gaussian array beam can be treated as a special case of the radial GSM array beam. In addition, the comparisons of the Rayleigh range and the M2-factor between the radial GSM array beam and the radial Gaussian array beam, between the radial GSM array beam and a single GSM beam centered at the origin are given.

Section snippets

Superposition of the cross-spectral density function

We assume that a radial array beam consists of N equal elements, which are GSM beams and located symmetrically on a ring with radius r0, and separation angle θ0=2π/N, as shown in Fig. 1. The cross-spectral density function of the element that, in the rectangular coordinate system, is centered at the position (r0, 0, 0) is expressed asW0(x1,x2,y1,y2,z=0)=exp[-(x1-r0)2+y12+(x2-r0)2+y22w02]exp[-(x1-x2)2+(y1-y2)22σ02],where w0 and σ0, respectively, are the waist width and the coherence

Numerical calculation results and analysis

The Rayleigh range zR and the M2-factor versus the radial GSM array beam parameters (i.e., the beam number N, the beam coherence parameter α and the inverse radial fill-factor r0) are given in Fig. 2, Fig. 3, Fig. 4, Fig. 5(a) and (b), respectively, where λ=1.06 μm and w0=1 mm are kept fixed. The solid curves represent zR and M2-factor for the superposition of the cross-spectral density function, the dashed curves denote those for the superposition of the intensity. From Fig. 2, Fig. 3, Fig. 4,

Concluding remarks

It is known that the larger the Rayleigh range means the longer non-spreading distance. On the other hand, the M2 value gives a measure of “how many times diffraction limited” the real beam is in each transverse direction [14]. In this paper, the Rayleigh range zR and the M2-factor have been taken as the characteristic parameters of beam quality, the beam quality of radial GSM array beams have been studied in detail. The analytical expressions for the zR and the M2-factor of radial GSM array

Acknowledgment

The authors are very thankful to the reviewers for valuable comments. This work was supported by the National Natural Science Foundation of China under grant 60778048.

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