M<sup>2</sup>-factor of a stochastic electromagnetic beam in a Gaussian cavity

Shijun Zhu; Yangjian Cai

doi:10.1364/OE.18.027567

1. Introduction

The stochastic EGSM beam (also called vectorial GSM beam) was introduced theoretically as the natural extension of the scalar GSM beam by Wolf and Gori et al. [1–3] owing to its importance in theories of coherence and polarization of light. It was found that the EGSM beams may have reduced levels of intensity fluctuations compared to the scalar GSM beams (i.e. fully polarized GSM beams) [4], which makes them attractive for free-space optical communications and active laser radar systems [5]. Ghost imaging with an EGSM beam was examined in [6], and it was revealed that one may reduce/enhance the ghost image and its visibility by choosing suitable source polarization, which is useful in novel optical imaging. The radiation force of EGSM beams on a Rayleigh dielectric sphere is explored in [7], and it was found that we can increase trapping ranges by choosing suitable source polarization and coherence, which makes them useful in optical trapping. Thus, it is of practical importance to study the vectorial character of stochastic electromagnetic beams, and control the beam properties [8].

The theory of beam propagation in laser resonators has quite a long history [9–13]. Fox and Li first described the structure of modes of the monochromatic fields in the resonator [10]. Wolf, Agarwal, and Gori generalized the Fox-Li theory to light fields with any state of coherence [14–16]. Palma and associates then studied the behavior of the coherence and the spectral properties of scalar partially coherent beams in a Gaussian cavity [17–20]. Recently, the theory of beam propagation in laser resonators was extended to stochastic electromagnetic fields [21, 22], and the behavior of the degree of polarization, the state of polarization, the degree of cross-polarization and the degree of coherence of an EGSM beam in a Gaussian cavity was explored in detail [23–26]. It is found that we can modulate or control the spectral, coherence and polarization properties of a stochastic beam by a Gaussian cavity by choosing suitable cavity parameters and the parameters of the source beam. To our knowledge no results have been reported up until now on the behavior of the $M^{2}$ -factor of an EGSM beam in a Gaussian cavity. In fact, little attention was paid to the $M^{2}$ -factor of an EGSM beam in free space or in turbulent atmosphere [27].

The propagation factor (also known as the $M^{2}$ -factor) proposed by Siegman [28] is a particularly useful property of an optical laser beam, and plays an important role in the characterization of beam behavior on propagation both in cavity and outside cavity [29, 30]. The $M^{2}$ -factor, which is based on variances, is not free from flaws or limitations [31, 32]. In [33, 34], Martinez-Herrero et al. developed the generalized second moments of hard-edge diffracted coherent laser beam to calculate its $M^{2}$ -factor. The definition of $M^{2}$ -factor was then extended to the partially coherent beams [35, 36], and the $M^{2}$ -factor of a partially coherent beam with or without truncation in free space was studied in [35–38]. Up to now, the properties of the $M^{2}$ -factor of various coherent and partially coherent beams outside the cavity have been investigated widely [39–47]. More recently, more and more attention is being paid to the $M^{2}$ -factor of laser beams in turbulent atmosphere due to their important application in free-space optical communications [27, 48–51]. In this paper, our aim is to investigate the $M^{2}$ -factor of an EGSM beam in a Gaussian cavity. Analytical formula for the $M^{2}$ -factor of an EGSM beam on propagation in a Gaussian cavity is derived by use of a tensor method, which is convenient for treating the propagation of scalar and electromagnetic partially coherent beams [5, 23, 52]. Some numerical examples are given. Our results may find uses in applications relating to construction and optimization of laser cavities.

2. $M^{2}$ -factor of an EGSM beam in a Gaussian cavity

For the convenience of analysis, we consider a stochastic electromagnetic beam which propagates close to the z-axis in a Gaussian cavity. Such a cavity consists of two spherical mirrors, with radius of curvature R, and mirror spot size ε. The distance between the centers of the mirrors is L. Propagation of a beam in the cavity is equivalent to its passage through a sequence of identical thin spherical lenses with focal length $f = R / 2$ , combined with the amplitude filters with a Gaussian transmission function (see Fig. 1 ) [12, 13, 20]. As usual, the resonators are classified as stable ( $0 \leq g < 1$ ) or unstable ( $g \geq 1$ ) due to the value of the stability parameter $g = 1 - L / R$ [12, 13, 20]. Note there are different stability conditions for different resonators. For a Gaussian cavity without gain medium and pump beam, the stability condition is developed in [12, 13, 20] and used in this paper. For a Gaussian cavity with gain medium and pump, the stability condition is defined in a different way, the pump beam and waist parameters should be taken into consideration as shown in [53, 54].

Fig. 1 Schematic diagram of a Gaussian cavity and its equivalent (unfolded) version
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We assume that the beam entering the cavity is generated by an EGSM source [1–3]. Such a beam can be characterized by the cross-spectral density matrix evaluated at points $(r_{1}, r_{2}) \equiv \tilde{r}$ of the tensor form [5, 23]

W_{α β}^{} (\tilde{r}) = A_{α} A_{β} B_{α β} \exp [- \frac{i k}{2} {\tilde{r}}^{T} M_{0 α β}^{- 1} \tilde{r}], (α = x, y; β = x, y),

where $r_{1}, r_{2}$ in Eq. (1) represents two transverse position vectors in the plane of mirror $M_{1}$ , $k = 2 π / λ$ is the wave number withλ being the wavelength of light, $A_{α}$ is the square root of the spectral density of electric field component $E_{α}$ , $B_{α β} = | B_{α β} | \exp (i ϕ)$ is the correlation coefficient between the $E_{x}$ and $E_{y}$ field components, T stands for vector transposition and $M_{0 α β}^{- 1}$ is the $4 \times 4$ matrix of the form $M_{0 α β}^{- 1} = (\begin{matrix} \frac{1}{i k} (\frac{1}{2 σ_{a}^{2}} + \frac{1}{δ_{α β}^{2}}) I & \frac{i}{k δ_{α β}^{2}} I \\ \frac{i}{k δ_{α β}^{2}} I & \frac{1}{i k} (\frac{1}{2 σ_{β}^{2}} + \frac{1}{δ_{α β}^{2}}) I \end{matrix}),$ where Iis the $2 \times 2$ identity matrix, $σ_{α}$ is the r.m.s width of the spectral density alongαdirection, $δ_{x x}$ , $δ_{y y}$ and $δ_{x y}$ are the r.m.s widths of auto-correlation functions of the x component of the field, of the y component of the field and of the mutual correlation function of x and y field components, respectively. In Eqs. (1) and (2) as well as in all the formulas below the explicit dependence of the parameters $σ_{x}$ , $σ_{y}$ , $δ_{x x}$ , $δ_{y y}$ and $δ_{x y}$ on the frequency was omitted for simplicity, although these parameters depend on the frequency in the most general case [1–3, 8].
With the help of the ABCD matrix approach, after passing N times between the mirrors of the cavity, the cross-spectral density matrix of the beam at points with transverse position vectors $(ρ_{1}, ρ_{2}) \equiv \tilde{ρ}$ is given by the expression [23]
$W_{α β}^{} (\tilde{ρ}) = A_{α} A_{β} B_{α β} {[Det (\bar{A} + \bar{B} M_{0 α β}^{- 1})]}^{- 1 / 2} \exp [- \frac{i k}{2} {\tilde{ρ}}^{T} M_{1 α β}^{- 1} \tilde{ρ}], (α = x, y; β = x, y),$ where $M_{1 α β}^{- 1} = (\bar{C} + \bar{D} M_{0 α β}^{- 1}) {(\bar{A} + \bar{B} M_{0 α β}^{- 1})}^{- 1},$ and $\bar{A}, \bar{B}, \bar{C} a n d \bar{D}$ are $4 \times 4$ matrices of the form $\bar{A} = (\begin{matrix} A & 0 I \\ 0 I & A^{*} \end{matrix}), \bar{B} = (\begin{matrix} B & 0 I \\ 0 I & - B^{*} \end{matrix}), \bar{C} = (\begin{matrix} C & 0 I \\ 0 I & - C^{*} \end{matrix}), \bar{D} = (\begin{matrix} D & 0 I \\ 0 I & D^{*} \end{matrix}),$ “∗” denoting Hermitian operator. For the resonator shown in Fig. 1, the matrices A, B, C and D take the form [23] $(\begin{matrix} A & B \\ C & D \end{matrix}) = {(\begin{matrix} A_{1} & B_{1} \\ C_{1} & D_{1} \end{matrix})}^{N}, (\begin{matrix} A_{1} & B_{1} \\ C_{1} & D_{1} \end{matrix}) = (\begin{matrix} I & L I \\ (- \frac{2}{R} - i \frac{λ}{π ε^{2}}) I & (1 - \frac{2 L}{R} - i \frac{λ L}{π ε^{2}}) I \end{matrix}),$ where R is the radius of curvature, ε is the mirror spot size of the cavity, L is the distance between the centers of the mirrors and N is the number of passages between the mirrors. The matrix with elements having subscript “1” describes single pass between the two mirrors. When N is an odd number, $ρ_{1}, ρ_{2}$ in Eq. (3) represents two transverse position vectors in the plane of mirror $M_{2}$ . When N is an even number, $ρ_{1}, ρ_{2}$ represents two transverse position vectors in the plane of mirror $M_{1}$ .
The trace of the cross-spectral density matrix of an EGSM beam after passing N times is expressed as [1–8]
$W_{tr} (\tilde{ρ}) = Tr \overset{\leftrightarrow}{W} (\tilde{ρ}) = W_{x x} (\tilde{ρ}) + W_{y y} (\tilde{ρ}) .$
Now we study the $M^{2}$ -factor of an EGSM beam on propagation in a Gaussian cavity. According to [35, 36], the $M^{2}$ -factor of a partially coherent beam is defined as
$M_{x}^{2} = 4 π Δ x Δ p_{x}, M_{y}^{2} = 4 π Δ y Δ p_{y},$ where $M_{x}^{2}$ and $M_{y}^{2}$ are the beam propagation factors in the x and y directions, respectively, and
$Δ x = \sqrt{\frac{1}{P} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} {(ρ_{x} - \bar{ρ_{x}})}^{2} W (ρ_{x}, ρ_{y}, ρ_{x}, ρ_{y}) d ρ_{x} d ρ_{y}},$ $Δ y = \sqrt{\frac{1}{P} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} {(ρ_{y} - \bar{ρ_{y}})}^{2} W (ρ_{x}, ρ_{y}, ρ_{x}, ρ_{y}) d ρ_{x} d ρ_{y}},$ $Δ p_{x} = \sqrt{\frac{1}{P} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} {(p_{x} - \bar{p_{x}})}^{2} \tilde{W} (p_{x}, p_{y}, - p_{x}, - p_{y}) d p_{x} d p_{y}},$ $Δ p_{y} = \sqrt{\frac{1}{P} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} {(p_{y} - \bar{p_{y}})}^{2} \tilde{W} (p_{x}, p_{y}, - p_{x}, - p_{y}) d p_{x} d p_{y}},$
The normalization factor P is given by
$P = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} W (ρ_{x}, ρ_{y}, ρ_{x}, ρ_{y}) d ρ_{x} d ρ_{y} = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \tilde{W} (p_{x}, p_{y}, - p_{x}, - p_{y}) d p_{x} d p_{y},$ where $\tilde{W}$ is the Fourier transform of W. Note that Eqs. (9) and (10) refer to the waist plane. In our case, as shown in Fig. 1, the spherical mirror $M_{1}$ or $M_{2}$ can be approximately expanded as a combination of thin lens and Gaussian amplitude filter [12,13,20]. Each time the beam passes through the mirror $M_{1}$ or $M_{2}$ , the Gaussian amplitude filter transforms the beam profile into a Gaussian beam profile, thus we can approximately regard that the waist plane of the beam is located in the plane of mirror $M_{1}$ or $M_{2}$ . To calculate the $M^{2}$ -factor of an EGSM beam, we should replace Win Eqs. (9) and (10) with $W_{tr}$ given by Eq. (7).
By expressing $M_{1 α β}^{- 1}$ in Eq. (4) in the following alternative form
$M_{1 α β}^{- 1} = (\begin{matrix} Q_{α β 11} & Q_{α β 12} & Q_{α β 13} & Q_{α β 14} \\ Q_{α β 21} & Q_{α β 22} & Q_{α β 23} & Q_{α β 24} \\ Q_{α β 31} & Q_{α β 32} & Q_{α β 33} & Q_{α β 34} \\ Q_{α β 41} & Q_{α β 42} & Q_{α β 43} & Q_{α β 44} \end{matrix}),$ the Eq. (7) can be expressed (after some operation) in the following alternative form $\begin{array}{l} W_{tr} (ρ_{x 1}, ρ_{y 1}, ρ_{x 2}, ρ_{y 2}) \\ = G_{x x} \exp {- \frac{i ω}{2 c} [ρ_{x}_{1}^{2} Q_{x x 11} + ρ_{x}_{2}^{2} Q_{x x 33} + ρ_{y}_{1}^{2} Q_{x x 22} + ρ_{y}_{2}^{2} Q_{x x 44} + (Q_{x x 13} + Q_{x x 31}) ρ_{x 1} ρ_{x 2} + (Q_{x x 24} + Q_{x x 42}) ρ_{y 1} ρ_{y 2}]} \\ + G_{y y} \exp {- \frac{i ω}{2 c} [ρ_{x}_{1}^{2} Q_{y y 11} + ρ_{x}_{2}^{2} Q_{y y 33} + ρ_{y}_{1}^{2} Q_{y y 22} + ρ_{y}_{2}^{2} Q_{y y 44} + (Q_{y y 13} + Q_{y y 31}) ρ_{x 1} ρ_{x 2} + (Q_{y y 24} + Q_{y y 42}) ρ_{y}_{1} ρ_{y}_{2}]}, \end{array}$ where
$G_{x x} = A_{x} {[Det (\bar{A} + \bar{B} M_{0 x x}^{- 1})]}^{- 1 / 2}, G_{y y} = A_{y} {[Det (\bar{A} + \bar{B} M_{0 y y}^{- 1})]}^{- 1 / 2} .$
Taking the Fourier transform of $W_{tr} (ρ_{x}_{1}, ρ_{y}_{1}, ρ_{x}_{2}, ρ_{y}_{2})$ , we obtain
$\begin{array}{l} {\overset{\leftrightarrow}{W}}_{tr} (p_{x}, p_{y}, - p_{x}, - p_{y}, z) = \frac{16 π^{2} G_{x x}}{k^{2} [{(Q_{x x 13} + Q_{x x 31})}^{2} - 4 Q_{x x 11} Q_{x x 33}]} \exp [- 8 π^{2} i \frac{(Q_{x x 33} + Q_{x x 11} + Q_{x x 13} + Q_{x x 31})}{k [{(Q_{x x 13} + Q_{x x 31})}^{2} - 4 Q_{x x 11} Q_{x x 33}]} (p_{x}^{2} + p_{y}^{2})] \\ + \frac{16 π^{2} G_{y y}}{k^{2} [{(Q_{y y 13} + Q_{y y 31})}^{2} - 4 Q_{y y 11} Q_{y y 33}]} \exp [- 8 π^{2} i \frac{(Q_{y y 33} + Q_{y y 11} + Q_{y y 13} + Q_{y y 31})}{k [{(Q_{y y 13} + Q_{y y 31})}^{2} - 4 Q_{y y 11} Q_{y y 33}]} (p_{x}^{2} + p_{y}^{2})] . \end{array}$
Substituting Eqs. (15) and (17) into Eqs. (9)-(13), we obtain (after tedious integration and operation)
$Δ x^{2} = Δ y^{2} = - \frac{2 π}{k^{2} P} [\frac{G_{x x}}{{(Q_{x x 11} + Q_{x x 13} + Q_{x x 31} + Q_{x x 33})}^{2}} + \frac{G_{y y}}{{(Q_{y y 11} + Q_{y y 13} + Q_{y y 31} + Q_{y y 33})}^{2}}],$ $Δ p_{y}^{2} = Δ p_{x}^{2} = \frac{- 1}{8 π P} [\frac{G_{x x} [{(Q_{x x 13} + Q_{x x 31})}^{2} - 4 Q_{x x 11} Q_{x x 33}]}{{(Q_{x x 11} + Q_{x x 13} + Q_{x x 31} + Q_{x x 33})}^{2}} + \frac{G_{y y} [{(Q_{y y 13} + Q_{y y 31})}^{2} - 4 Q_{y y 11} Q_{y y 33}]}{{(Q_{y y 11} + Q_{y y 13} + Q_{y y 31} + Q_{y y 33})}^{2}}],$ $P = \frac{2 π}{i k} [\frac{G_{x x}}{(Q_{x x 11} + Q_{x x 13} + Q_{x x 31} + Q_{x x 33})} + \frac{G_{y y}}{(Q_{y y 11} + Q_{y y 13} + Q_{y y 31} + Q_{y y 33})}] .$
Then substituting Eqs. (18)-(20) into Eq. (8), we obtain the following expression for the $M^{2}$ -factor of an EGSM beam on propagation in a Gaussian cavity
$M_{x}^{2} = M_{y}^{2} = M^{2} = \frac{Q_{N x x} Q_{N y y}}{G_{x x} Q_{N y y} + G_{y y} Q_{N x x}} \sqrt{(\frac{G_{x x}}{Q_{N x x}^{2}} + \frac{G_{y y}}{Q_{N y y}^{2}}) (\frac{G_{x x} Q_{M x x}}{Q_{N x x}^{2}} + \frac{G_{y y} Q_{M y y}}{Q_{N y y}^{2}})},$ where $M_{x}^{2} = M_{y}^{2} = M^{2} = \sqrt{\frac{4 Q_{α α 11} Q_{α α 33} - {(Q_{α α 13} + Q_{α α 31})}^{2}}{{(Q_{α α 11} + Q_{α α 13} + Q_{α α 31} + Q_{α α 33})}^{2}}}, (α = x, y)$ $Q_{M α β} = 4 Q_{α β 11} Q_{α β 33} - {(Q_{α β 13} + Q_{α β 31})}^{2}, Q_{N α β} = Q_{α β 11} + Q_{α β 13} + Q_{α β 31} + Q_{α β 33}, (α = x, y; β = x, y)$ . Under the condition of $A_{x} = 0$ or $A_{y} = 0$ , Eq. (21) reduces to the following expression for the the $M^{2}$ -factor of a scalar GSM beam (i.e., fully polarized GSM beam)
Equation (21) is the main analytical result of the present paper. It provides a convenient way for studying the behavior of the $M^{2}$ -factor of an EGSM beam in a Gaussian cavity. In the above derivations, we have used following integral formulas
$\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \exp [a x^{2} + b y^{2}] d x d y = \frac{π}{\sqrt{a b}}, \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} x^{2} \exp [a x^{2} + b y^{2} + d x y] d x d y = \frac{4 π b}{{(4 a b - d^{2})}^{3 / 2}} .$
For the case of N = 0, Eq. (21) reduces to the following expression for the $M^{2}$ -factor of an EGSM beam in free space
$M^{2} = {\frac{(A_{x}^{2} σ_{x}^{4} + A_{y}^{2} σ_{y}^{4}) [4 A_{x}^{2} δ_{y y}^{2} σ_{x}^{2} + δ_{x x}^{2} (A_{x}^{2} δ_{y y}^{2} + A_{y}^{2} δ_{y y}^{2} + 4 A_{y}^{2} σ_{y}^{2})]}{δ_{x x}^{2} δ_{y y}^{2} {(A_{x}^{2} σ_{x}^{2} + A_{y}^{2} σ_{y}^{2})}^{2}}}^{1 / 2},$ and Eq. (22) reduces to the following expression for the $M^{2}$ -factor of a scalar GSM beam in free space $M^{2} (z) = {(1 + \frac{4 σ_{α}^{2}}{δ_{α α}^{2}})}^{1 / 2}, (α = x, y)$ Equations (24) and (25) agree well with those reported in [27] and [36].
Now we study the evolution properties of the $M^{2}$ -factor of an EGSM beam in a Gaussian cavity. For the convenience of analysis, we only consider the EGSM beam that is generated by an EGSM source whose cross-spectral density matrix is diagonal, i.e. of the form
$\overset{\leftrightarrow}{W} (r_{1}, r_{2}) = (\begin{matrix} W_{x x} (r_{1}, r_{2}) & 0 \\ 0 & W_{y y} (r_{1}, r_{2}) \end{matrix}) .$
The degree of polarization of the initial source beam at point ris expressed as follows [1]
$P_{0} (r) = \sqrt{1 - \frac{4 Det \overset{\leftrightarrow}{W} (r, r)}{{[Tr \overset{\leftrightarrow}{W} (r, r)]}^{2}}} .$
In the following text, we set $σ_{x} = σ_{y} = 2 m m$ , $A_{y} = 1$ , $λ = 632.8 n m$ , $L = 300 m m$ unless stated otherwise. In this case, the polarization properties are uniform across the source plane with $P_{0} (r) = | \frac{A_{x}^{2} - A_{y}^{2}}{A_{x}^{2} + A_{y}^{2}} | .$ As shown in [55], the EGSM beam can be generated through combination of two orthogonal polarized partially coherent beams. $A_{x}$ and $A_{y}$ in fact represent the amplitudes of x and y components of the electric field. We can use spatial light modulator to alter the value of $A_{x}$ or $A_{y}$ . In general $A_{x}$ doesn’t equal to $A_{y}$ , and the degree of polarization of the source EGSM beam in our example are determined by $A_{x}$ or $A_{y}$ together. The degree of polarization in the source plane varies as the value of $A_{y}$ changes for fixed $A_{x}$ , any nonzero $P_{0}$ can be achieved either for $A_{x} > A_{y}$ or for $A_{x} < A_{y}$ (see Fig. 1 of Ref [27].).
We calculate in Fig. 2 the $M^{2}$ -factor of an EGSM beam in a Gaussian cavity versus N for different values of the cavity parameter g and the source correlation coefficients $δ_{x x}$ and $δ_{y y}$ with $A_{x} = 0.707$ and $ε = 1.5 m m$ . One finds from Fig. 2 that the $M^{2}$ -factor of an EGSM beam in a Gaussian cavity decreases on propagation and its value approaches different values for different values of cavity parameter g. In stable cavities ( $0 \leq g < 1$ ), the $M^{2}$ -factor exhibits decrease with oscillations. In unstable cavities ( $g \geq 1$ ), the decrease is monotonic. We can explain this phenomenon by the fact that the behavior of the $M^{2}$ -factor of an EGSM beam on propagation depends closely on the behavior of its spectral degree of coherence on propagation, and the $M^{2}$ -factor decreases as the spectral degree of coherence increases. In unstable cavities, the spectral degree of coherence of an EGSM beam grows monotonically, just like in the case of free-space propagation, while in stable cavities the spectral degree of coherence grows with oscillations due to periodic focusing and free-space diffraction [25], thus leading to the interesting behavior of the $M^{2}$ -factor of an EGSM beam in a Gaussian cavity. Furthermore, one finds from Fig. 2 that the behavior of the $M^{2}$ -factor of an EGSM beam is also closely related with the source correlation coefficients $δ_{x x}$ and $δ_{y y}$ (i.e., source spectral degree of coherence), particularly for small values of N. When N is large enough, the $M^{2}$ -factor of an EGSM beam with different values of $δ_{x x}$ and $δ_{y y}$ approaches to the same constant value. Note that here we have used the definition of coherence introduced by Wolf [1,8] to discuss the effect of coherence on the behavior of $M^{2}$ -factor in the cavity. There are other definitions of degree of coherence, especially the one introduced by Friberg et al. [56]. Since we only consider the EGSM beam having a diagonal density matrix, we can come to the same conclusions using any one of both definitions.

Fig. 2 $M^{2}$ -factor of an EGSM beam in a Gaussian cavity versus N for different values of the cavity parameter g and the source correlation coefficients $δ_{x x}$ and $δ_{y y}$
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Figure 3 illustrates the $M^{2}$ -factor of an EGSM beam in a Gaussian cavity versus N for different values of the cavity parameter g and the degree of polarization $P_{0}$ of the source beam with $δ_{x x} = 0.15 m m$ , $δ_{y y} = 0.05 m m$ and $ε = 1.5 m m$ . One can see from Fig. 3 that the evolution properties of the $M^{2}$ -factor of an EGSM beam is closely related with the degree of polarization $P_{0}$ of the source beam, particularly for small value of N both in stable and unstable resonators. We can explain this by the fact that the $M^{2}$ -factor of an EGSM beam is calculated from the trace of the cross-special density matrix (i.e., $W_{x x} (\tilde{ρ}) + W_{y y} (\tilde{ρ})$ ). The evolution properties of the elements $W_{x x} (\tilde{ρ})$ and $W_{y y} (\tilde{ρ})$ are closely determined by the correlation coefficients $δ_{x x}$ and $δ_{y y}$ , respectively. The initial degree of polarization calculated by Eq. (27) determines which element of cross-special density matrix contributes a dominant role to the $M^{2}$ -factor, thus leading the dependence of the behavior of the $M^{2}$ -factor of an EGSM beam on the degree of polarization shown in Fig. 3. For small value of N, the $M^{2}$ -factor of an EGSM beam increases with the decrease of $P_{0}$ when $A_{x} > A_{y}$ , while the $M^{2}$ -factor increases with the increase of $P_{0}$ when $A_{x} < A_{y}$ . This phenomenon is caused by the fact that the contribution of the element $W_{x x} (ρ_{1}, ρ_{2})$ to the $M^{2}$ -factor dominates that of the element $W_{y y} (ρ_{1}, ρ_{2})$ for the case of $A_{x} > A_{y}$ , and the contribution of the element $W_{y y} (ρ_{1}, ρ_{2})$ plays a dominant role otherwise. When N is large enough, the influence of polarization on the $M^{2}$ -factor is negligible.

Fig. 3 $M^{2}$ -factor of an EGSM beam in a Gaussian cavity versus N for different values of the cavity parameter g and the degree of polarization $P_{0}$ of the source beam
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Figure 4 explores the $M^{2}$ -factor of an EGSM beam in a Gaussian cavity versus N for different values of the mirror spot size of the cavity ε with $A_{x} = 0.707$ , g = 1, $δ_{x x} = 0.15 m m$ and $δ_{y y} = 0.05 m m$ . As shown in Fig. 4, the behavior of the $M^{2}$ -factor on propagation depends closely on the cavity parameter ε. The $M^{2}$ -factor of an EGSM beam decreases more quickly as the cavity parameterε decreases, while it approaches to the same constant value for different values of ε when N is large enough. This phenomenon can be explained by the fact that as the value of the mirror size decreases, the edge diffraction caused by the Gaussian aperture increases, and the degree of coherence of the EGSM beam increases more rapidly on propagation [25], thus leading to the dependence of the $M^{2}$ -factor on the mirror size as shown in Fig. 4. From above discussions, one comes to the conclusion that, in general, the $M^{2}$ -factor of an EGSM beam approaches to a constant value which is larger than 1 when the number of passage N is enough large as shown in Figs. 2-4, which is caused by the fact that the spectral degree of coherence of the EGSM beam approaches to 1 in the case of large N meaning the EGSM beam becomes a coherent vectorial beam (see Refs [23]- [26].). The properties such as the beam width and degree of coherence of the coherent vectorial beam depend closely on the cavity parameters, thus the constant value of the $M^{2}$ -factor in the case of large N is closely determined by the cavity parameters.

Fig. 4 $M^{2}$ -factor of an EGSM beam in a Gaussian cavity versus N for different values of the mirror spot size of the cavity ε
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3. Thermal lens effect induced changes of the M²-factor of an EGSM beam in a Gaussian cavity
Thermal lensing or thermal blooming occurs as energy absorbed from a laser beam produces local heating of an absorbing medium about the beam axis. The formation of a lens in the medium is due to nonuniform heating by the laser beam when the medium is placed inside the cavity of an operating laser. In 1964, Gordon et al. first discovered the thermal lens effect [57]. Since then, a lot of theoretical and experimental research has been carried out on the thermal lens effect [58–61]. In solid lasers, thermal lens medium exists very commonly. Thermal lens effect has been used widely as a method for measuring the low absorption of the media [62,63]. In this section, we study the thermal lens effect induced changes of the M ²-factor of an EGSM beam in a Gaussian cavity.
Figure 5(a) shows a Gaussian cavity containing thermal lens medium, which consists of two spherical mirrors with radius of curvature R and mirror spot size ε, and a section of thermal lens medium with length $l_{2}$ . The distance between the thermal lens medium and the spherical mirror is $l_{1}$ and the distance between the centers of the mirrors is $2 l_{1} + l_{2}$ . Figure 5(b) shows the equivalent version of Fig. 5 (a). In the following text, we set $2 l_{1} + l_{2} = 300 m m$ . The approximate expression for the ABCD matrix of a thermal lens medium is expressed as [60,64]
$(\begin{matrix} A_{t} & B_{t} \\ C_{t} & D_{t} \end{matrix}) = (\begin{matrix} \cos (β l) I & \sin (β l) / (β n_{0}) I \\ - β n_{0} \sin (β l) I & \cos (β l) I \end{matrix}) = (\begin{matrix} (1 + γ l^{2}) I & l / n_{0} I \\ 2 γ n_{0} l I & (1 + γ l^{2}) I \end{matrix}),$ where $β, γ$ are the medium coefficients and they satisfy the relation $β = \sqrt{- 2 γ}$ , $n_{0}$ is the constant refractive index, and l is the length of the medium. By applying the ABCD-matrix approach for the Gaussian aperture and the thermal lens medium, we find that after the EGSM beam travels between the two mirrors for N times, A, B, C, and D for the equivalent optical system become $(\begin{matrix} A & B \\ C & D \end{matrix}) = {(\begin{matrix} A_{1} & B_{1} \\ C_{1} & D_{1} \end{matrix})}^{N},$ with $A_{1} = (1 + 2 n_{0} γ l_{1} l_{2} + γ l_{2}^{2}) I,$ $B_{1} = [\frac{l_{2}}{n_{0}} + 2 n_{0} γ l_{1}^{2} l_{2} + 2 l_{1} (1 + γ l_{2}^{2})] I,$ $C_{1} = [2 n_{0} γ (1 + (- \frac{2}{R} - \frac{i λ}{π ε_{}^{2}}) l_{1}) l_{2} + (- \frac{2}{R} - \frac{i λ}{π ε_{}^{2}}) (1 + γ l_{2}^{2})] I,$ $D_{1} = [1 + \frac{(- \frac{2}{R} - \frac{i λ}{π ε_{}^{2}}) l_{2}}{n_{0}} + 2 n_{0} γ l_{1}^{2} l_{2} (- \frac{2}{R} - \frac{i λ}{π ε_{}^{2}}) + γ l_{2}^{2} + - \frac{4 l_{1}}{R} - \frac{2 i λ l_{1}}{π ε_{}^{2}} + 2 n_{0} γ l_{2} l_{1} + l_{1} (- \frac{4}{R} - \frac{2 i λ}{π ε_{}^{2}}) γ l_{2}^{2}] I .$ Applying Eqs. (3)-(5), (21) and (29)-(33), we can calculate the behavior of the M ²-factor of an EGSM beam in a Gaussian cavity containing thermal lens medium.

Fig. 5 Schematic diagram of a Gaussian cavity containing thermal lens medium and its equivalent (unfolded) version
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We calculate in Fig. 6 the $M^{2}$ -factor of an EGSM beam in a Gaussian cavity for different values of the length of the thermal lens medium and the source correlation coefficients $δ_{x x}$ and $δ_{y y}$ with $γ = 5 \times 10^{- 6}$ , $A_{x} = 0.707$ , g = 1 and $ε = 0.8 m m$ . Figure 7 shows the $M^{2}$ -factor of an EGSM beam in a Gaussian cavity for different values of the thermal lens medium coefficient γ and the source correlation coefficients $δ_{x x}$ and $δ_{y y}$ with $l_{2} = 100 m m$ , $A_{x} = 0.707$ , g = 1 and $ε = 0.8 m m$ . As shown in Figs. 6 and 7, the evolution properties of the $M^{2}$ -factor of an EGSM beam in a Gaussian cavity are affected by thermal lens medium, especially for large value of N. The $M^{2}$ -factor approaches to different constant values for different values of the length of the thermal lens medium or the thermal lens medium coefficientγ, and the constant value increases with the increase of the length of the thermal lens medium or the thermal lens medium coefficientγ. The behavior of the $M^{2}$ -factor in the cavity containing the thermal lens medium is also influenced by the source correlation coefficients $δ_{x x}$ and $δ_{y y}$ . When N is small, the $M^{2}$ -factor increases with the decrease of the correlation coefficients $δ_{x x}$ and $δ_{y y}$ for fixed parameters ( $l_{2}$ and γ) of the thermal lens medium. When N is large, the $M^{2}$ -factor approaches to a constant and is independent of the correlation coefficients $δ_{x x}$ and $δ_{y y}$ for fixed parameters ( $l_{2}$ and γ) of the thermal lens medium. This is caused by the fact that the EGSM beam also becomes a coherent vectorial beam when the Gaussian cavity contains the thermal lens medium as shown in [26].

Fig. 6 $M^{2}$ -factor of an EGSM beam in a Gaussian cavity for different values of the length of the thermal lens medium and the source correlation coefficients $δ_{x x}$ and $δ_{y y}$
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Fig. 7 $M^{2}$ -factor of an EGSM beam in a Gaussian cavity for different values of the thermal lens medium coefficientγ and the source correlation coefficients $δ_{x x}$ and $δ_{y y}$
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Figure 8 shows the $M^{2}$ -factor of an EGSM beam in a Gaussian cavity for different values of the length of the thermal lens medium and the degree of polarization $P_{0}$ of the source beam with $A_{x} > A_{y}$ , $δ_{x x} = 0.15 m m$ , $δ_{y y} = 0.05 m m$ , $γ = 5 \times 10^{- 6}$ , g = 1 and $ε = 0.8 m m$ . Figure 9 shows the $M^{2}$ -factor of an EGSM beam in a Gaussian cavity for different values of the thermal lens medium coefficient γ and the degree of polarization $P_{0}$ of the source beam with $A_{x} > A_{y}$ , $δ_{x x} = 0.15 m m$ , $δ_{y y} = 0.05 m m$ , $l_{2} = 100 m m$ , g = 1 and $ε = 0.8 m m$ . One finds from Figs. 8 and 9 that the behavior of the $M^{2}$ -factor of an EGSM beam in the cavity containing the thermal lens medium is also affected by the degree of polarization $P_{0}$ of the source beam. For the case of $A_{x} > A_{y}$ , the $M^{2}$ -factor increases with the decrease of $P_{0}$ for fixed parameters ( $l_{2}$ and γ) of the thermal lens medium when N is small. When N is large, the $M^{2}$ -factor approaches to a constant and is independent of $P_{0}$ for fixed parameters ( $l_{2}$ and γ) of the thermal lens medium.

Fig. 8 $M^{2}$ -factor of an EGSM beam in a Gaussian cavity for different values of the length of the thermal lens medium and the degree of polarization $P_{0}$ of the source beam
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Fig. 9 $M^{2}$ -factor of an EGSM beam in a Gaussian cavity for different values of the thermal lens medium coefficient γ and the degree of polarization $P_{0}$ of the source beam
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Figure 10 shows the $M^{2}$ -factor of an EGSM beam in a Gaussian cavity for different values of the length of the thermal lens medium and the mirror spot size of the cavity ε with $δ_{x x} = 0.15 m m$ , $δ_{y y} = 0.05 m m$ , $γ = 5 \times 10^{- 6}$ , $A_{x} = 0.707$ and g = 1. Figure 11 shows the $M^{2}$ -factor of an EGSM beam in a Gaussian cavity for different values of the thermal lens medium coefficientγand the mirror spot size of the cavity ε with $δ_{x x} = 0.15 m m$ , $δ_{y y} = 0.05 m m$ , $l_{2} = 100 m m$ , $A_{x} = 0.707$ and g = 1. One finds from Figs. 10 and 11 that the behavior of the $M^{2}$ -factor in the cavity containing the thermal lens medium is also closely related with the mirror spot size ε of the cavity. When the mirror spot size ε is small (see Fig. 10(a) and Fig. 11(a)), the difference between the $M^{2}$ -factor in an empty cavity and that in a cavity containing thermal lens medium is very small, which means the influence of the thermal lens medium is negligible in this case. When the mirror spot size ε is large (see Fig. 10 (b), (c), Fig. 11 (b), (c)), the difference between the $M^{2}$ -factor in an empty cavity and that in a cavity containing the thermal lens medium is very large, which means the influence of the thermal lens medium is very significant. We explain this phenomenon by the fact that in the cavity as shown in Fig. 5, both the Gaussian amplitude filter and the thermal lens medium alter the $M^{2}$ -factor of the EGSM beam. When the mirror spot size ε of the cavity is small, the influence of the Gaussian amplitude filter on the $M^{2}$ -factor play a dominant role due to serious diffraction, and when the mirror spot size ε of the cavity is large, the diffraction effect of the Gaussian amplitude filter is small and the influence of the thermal lens medium play a dominant role.

Fig. 10 $M^{2}$ -factor of an EGSM beam in a Gaussian cavity for different values of the length of the thermal lens medium and the mirror spot size of the cavity ε
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Fig. 11 $M^{2}$ -factor of an EGSM beam in a Gaussian cavity for different values of the thermal lens medium coefficientγ and the mirror spot size of the cavity ε
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4. Conclusion
We have investigated the evolution properties of the $M^{2}$ -factor of a stochastic EGSM beam in a Gaussian cavity with the help of a tensor method. Our numerical results show that the behavior of the $M^{2}$ -factor of an EGSM beam in a Gaussian cavity is closely determined by the cavity parameters, the degree of polarization $P_{0}$ and the correlation coefficients of the source beam. In stable cavities, the $M^{2}$ -factor exhibits decrease with oscillations, and it exhibits monotonic decrease in unstable cavities. When the number of passages is large enough, the $M^{2}$ -factor of an EGSM beam approaches to a constant value. We have also studied the evolution properties of an EGSM beam in a Gaussian cavity containing the thermal lens medium. It is found that the thermal lens effect affects the behavior of the M ²-factor of an EGSM beam, and the thermal lens effect induced changes of the M ²-factor is also closely related with the cavity parameters and parameters (i.e., degree of polarization and correlation coefficients) of the source beam. We can control the beam properties of the EGSM beam by choosing suitable values of the cavity parameters and the source beam parameters. Our results will be useful in many applications, such as free-space optical communications, laser radar system, optical trapping and optical imaging, where it has been shown that the EGSM beam has advantage over scalar GSM beam and coherent Gaussian beam [4–8].
Acknowledgments
Yangjian Cai acknowledges the support by the National Natural Science Foundation of China under Grant No. 10904102, the Foundation for the Author of National Excellent Doctoral Dissertation of PR China under Grant No. 200928, the Natural Science of Jiangsu Province under Grant No. BK2009114, the Huo Ying Dong Education Foundation of China under Grant No. 121009 and the Key Project of Chinese Ministry of Education under Grant No. 210081.
References and links
1. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003). [CrossRef]
2. 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]
3. 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]
4. O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281, 2342–2348 (2008).
5. Y. Cai, O. Korotkova, H. T. Eyyuboğlu, and Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express 16(20), 15834–15846 (2008). [CrossRef] [PubMed]
6. Z. Tong, Y. Cai, and O. Korotkova, “Ghost imaging with electromagnetic stochastic beams,” Opt. Commun. 283(20), 3838–3845 (2010). [CrossRef]
7. C. Zhao, Y. Cai, and O. Korotkova, “Radiation force of scalar and electromagnetic twisted Gaussian Schell-model beams,” Opt. Express 17(24), 21472–21487 (2009). [CrossRef] [PubMed]
8. E. Wolf, Introduction to the theory of coherence and polarization of light (Cambridge U. Press, 2007).
9. N. Hodgson, and H. Weber, Optical resonators-Fundamentals, advanced concepts and applications (London Springer-Verlag, 1997).
10. A. G. Fox and T. Li, “Resonate modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
11. G. D. Boyd and J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961).
12. H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5(10), 1550–1567 (1966). [CrossRef] [PubMed]
13. L. W. Casperson and S. D. Lunnam, “Gaussian modes in high loss laser resonators,” Appl. Opt. 14(5), 1193–1199 (1975). [CrossRef] [PubMed]
14. E. Wolf, “Spatial coherence of resonant modes in a maser interferometer,” Phys. Lett. 3(4), 166–168 (1963). [CrossRef]
15. E. Wolf and G. S. Agarwal, “Coherence theory of laser resonator modes,” J. Opt. Soc. Am. A 1(5), 541–546 (1984). [CrossRef]
16. F. Gori, “Propagation of the mutual intensity through a periodic structure,” Atti Fond. Giorgio Ronchi 35, 434–447 (1980).
17. G. Cincotti, P. DeSantis, G. Guattari, and C. Palma, “Propagation of partially coherent beams in a periodic sequence of lenses and Gaussian apertures,” Pure Appl. Opt. 3(4), 561–571 (1994). [CrossRef]
18. A. Mascello, M. R. Perrone, and C. Palma, “Coherence evolution of laser beams in cavities with variable-reflectivity mirrors,” J. Opt. Soc. Am. A 14(8), 1890–1901 (1997). [CrossRef]
19. P. De Santis, A. Mascello, C. Palma, and M. R. Perrone, “Coherence growth of laser radiation in Gaussian cavities,” IEEE J. Quantum Electron. 32(5), 802–812 (1996). [CrossRef]
20. C. Palma, G. Cardone, and G. Cincotti, “Spectral changes in Gaussian-cavity lasers,” IEEE J. Quantum Electron. 34(7), 1082–1088 (1998). [CrossRef]
21. E. Wolf, “Coherence and polarization properties of electromagnetic laser modes,” Opt. Commun. 265(1), 60–62 (2006). [CrossRef]
22. T. Saastamoinen, J. Turunen, J. Tervo, T. Setälä, and A. T. Friberg, “Electromagnetic coherence theory of laser resonator modes,” J. Opt. Soc. Am. A 22(1), 103–108 (2005). [CrossRef]
23. M. Yao, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “Evolution of the degree of polarization of an electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Lett. 33(19), 2266–2268 (2008). [CrossRef] [PubMed]
24. O. Korotkova, M. Yao, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “State of polarization of a stochastic electromagnetic beam in an optical resonator,” J. Opt. Soc. Am. A 25(11), 2710–2720 (2008). [CrossRef]
25. Z. Tong, O. Korotkova, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Correlation properties of random electromagnetic beams in laser resonators,” Appl. Phys. B 97(4), 849–857 (2009). [CrossRef]
26. S. Zhu, F. Zhou, Y. Cai, and L. Zhang, “Thermal lens effect induced changes of polarization, coherence and spectrum of a stochastic electromagnetic beam in a Gaussian cavity,” Appl. Phys. B DOI: .
27. S. Zhu, Y. Cai, and O. Korotkova, “Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Express 18(12), 12587–12598 (2010). [CrossRef] [PubMed]
28. A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE 1224, 2C14 (1990).
29. A. E. Siegman and S. W. Townsend, “Output beam propagation and beam quality from a multimode stable-cavity laser,” IEEE J. Quantum Electron. 29(4), 1212–1217 (1993). [CrossRef]
30. J. J. Chang, “Time-resolved beam-quality characterization of copper-vapor lasers with unstable resonators,” Appl. Opt. 33(12), 2255–2265 (1994). [CrossRef] [PubMed]
31. J. Hilgevoord, “The standard deviation is not an adequate measure of quantum uncertainty,” Am. J. Phys. 70(10), 983 (2002). [CrossRef]
32. G. N. Lawrence, “Proposed international standard for laser-beam quality falls short,” Laser Focus World 109–114 (1994).
33. R. Martinez-Herrero and P. M. Mejias, “Second-order spatial characterization of hard-edge diffracted beams,” Opt. Lett. 18(19), 1669–1671 (1993). [CrossRef] [PubMed]
34. R. Martínez-Herrero, P. M. Mejías, and M. Arias, “Parametric characterization of coherent, lowest-order Gaussian beams propagating through hard-edged apertures,” Opt. Lett. 20(2), 124–126 (1995). [CrossRef] [PubMed]
35. F. Gori, M. Santarsiero, and A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82(3-4), 197–203 (1991). [CrossRef]
36. M. Santarsiero, F. Gori, R. Borghi, G. Cincotti, and P. Vahimaa, “Spreading properties of beams radiated by partially coherent Schell-model sources,” J. Opt. Soc. Am. A 16(1), 106–112 (1999). [CrossRef]
37. B. Zhang, X. Chu, and Q. Li, “Generalized beam-propagation factor of partially coherent beams propagating through hard-edged apertures,” J. Opt. Soc. Am. A 19(7), 1370–1375 (2002). [CrossRef]
38. X. Chu, B. Zhang, and Q. Wen, “Generalized M2 factor of a partially coherent beam propagating through a circular hard-edged aperture,” Appl. Opt. 42(21), 4280–4284 (2003). [CrossRef] [PubMed]
39. P. A. Belanger, “Beam quality factor of the LP01 mode of the step-index fiber,” Opt. Eng. 32(9), 2107–2109 (1993). [CrossRef]
40. S. Saghafi and C. J. R. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Opt. Commun. 153(4-6), 207–210 (1998). [CrossRef]
41. B. Lü, B. Zhang, and H. Ma, “Beam-propagation factor and mode-coherence coefficients of hyperbolic-cosine Gaussian beams,” Opt. Lett. 24(10), 640–642 (1999). [CrossRef]
42. G. Zhou, “Generalized beam propagation factors of truncated partially coherent cosine-Gaussian and cosh-Gaussian beams,” Opt. Laser Technol. 42(3), 489–496 (2010). [CrossRef]
43. D. Deng, “Generalized M²-factor of hollow Gaussian beams through a hard-edge circular aperture,” Phys. Lett. A 341(1-4), 352–356 (2005). [CrossRef]
44. D. Deng, X. Fu, C. Wei, J. Shao, and Z. Fan, “Far-field intensity distribution and M² factor of hollow Gaussian beams,” Appl. Opt. 44(33), 7187–7190 (2005). [CrossRef] [PubMed]
45. S. Yin, B. Zhang, and Y. Dan, “Effects of the deformation of reflection volume Bragg gratings on the M²-factor of super-Gaussian laser beams,” Opt. Commun. 283(7), 1418–1423 (2010). [CrossRef]
46. D. Deng, H. Guo, X. Chen, and H. Kong, “Characteristics of coherent and incoherent off-axis elegant Hermite-Gaussian beam combinations,” J. Opt. A, Pure Appl. Opt. 5(5), 489–494 (2003). [CrossRef]
47. D. Deng, “Propagation properties of off-axis cosh Gaussian beam combinations through a first-order optical system,” Phys. Lett. A 333(5-6), 485–494 (2004). [CrossRef]
48. Y. Dan and B. Zhang, “Beam propagation factor of partially coherent flat-topped beams in a turbulent atmosphere,” Opt. Express 16(20), 15563–15575 (2008). [CrossRef] [PubMed]
49. Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “M2-factor of coherent and partially coherent dark hollow beams propagating in turbulent atmosphere,” Opt. Express 17(20), 17344–17356 (2009). [CrossRef] [PubMed]
50. F. Wang and Y. Cai, “Second-order statistics of a twisted Gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 18(24), 24661–24672 (2010). [CrossRef] [PubMed]
51. Y. Yuan, Y. Cai, C. Zhao, H. T. Eyyuboğlu, and Y. Baykal, “Propagation factors of laser array beams in turbulent atmosphere,” J. Mod. Opt. 57(8), 621–631 (2010). [CrossRef]
52. Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27(4), 216–218 (2002). [CrossRef]
53. L. Mandel, and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995)
54. J. F. Urchueguía, G. J. de Valcarcel, and E. Roldan, “Laser instabilities in a Gaussian cavity mode with Gaussian pump profile,” J. Opt. Soc. Am. B 15(5), 1512–1520 (1998). [CrossRef]
55. 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]
56. J. Tervo, T. Setala, and A. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11(10), 1137–1143 (2003). [CrossRef] [PubMed]
57. J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36(1), 3–8 (1965). [CrossRef]
58. S. J. Sheldon, L. V. Knight, and J. M. Thorne, “Laser-induced thermal lens effect: a new theoretical model,” Appl. Opt. 21(9), 1663–1669 (1982). [CrossRef] [PubMed]
59. C. A. Carter and J. M. Harris, “Comparison of models describing the thermal lens effect,” Appl. Opt. 23(3), 476–481 (1984). [CrossRef] [PubMed]
60. T. Xu and S. Wang, “Propagation of Ince-Gaussian beams in a thermal lens medium,” Opt. Commun. 265(1), 1–5 (2006). [CrossRef]
61. R. Weber, B. Neuenschwander, and H. P. Weber, “Thermal effects in solid-state laser materials,” Opt. Mater. 11(2-3), 245–254 (1999). [CrossRef]
62. J. R. Whinnery, “Laser measurement of optical absorption in liquids,” Acc. Chem. Res. 7(7), 225–231 (1974). [CrossRef]
63. C. Hu and J. R. Whinnery, “New thermooptical measurement method and a comparison with other methods,” Appl. Opt. 12(1), 72–79 (1973). [CrossRef] [PubMed]
64. S. Wang, “Matrix methods in treating decentred optical systems,” Opt. Quantum Electron. 17(1), 1–14 (1985). [CrossRef]

M ²-factor of a stochastic electromagnetic beam in a Gaussian cavity

Abstract

1. Introduction

2. $M^{2}$ -factor of an EGSM beam in a Gaussian cavity

3. Thermal lens effect induced changes of the M²-factor of an EGSM beam in a Gaussian cavity

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (11)

Equations (33)

Optics Express

Abstract

1. Introduction

2. M2-factor of an EGSM beam in a Gaussian cavity

3. Thermal lens effect induced changes of the M2-factor of an EGSM beam in a Gaussian cavity

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (11)

Equations (33)

Optics Express

2. $M^{2}$ -factor of an EGSM beam in a Gaussian cavity

3. Thermal lens effect induced changes of the M²-factor of an EGSM beam in a Gaussian cavity