## Abstract

With the help of a tensor method, an explicit expression for the ${M}^{2}$-factor of a stochastic electromagnetic Gaussian Schell-model (EGSM) beam in a Gaussian cavity is derived. Evolution properties of the *M*
^{2}-factor of an EGSM beam in a Gaussian cavity are studied numerically in detail. It is found that the behavior of the ${M}^{2}$-factor of an EGSM beam in a Gaussian cavity is determined by the statistical properties of the source beam and the parameters of the cavity. Thermal lens effect induced changes of the *M*
^{2}-factor of an EGSM beam in a Gaussian cavity is also investigated. Our results will be useful in many applications, such as free-space optical communications, laser radar system, optical trapping and optical imaging, where stochastic electromagnetic beams are required.

©2010 Optical Society of America

## 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 ($0\le g<1$) or unstable ($g\ge 1$) 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].**

**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]**

**$${W}_{\alpha \beta}^{}(\tilde{r})={A}_{\alpha}{A}_{\beta}{B}_{\alpha \beta}\mathrm{exp}\left[-\frac{ik}{2}{\tilde{r}}^{T}{M}_{0\alpha \beta}^{-1}\tilde{r}\right],(\alpha =x,y;\beta =x,y),$$**

**where ${r}_{1},{r}_{2}$ in Eq. (1) represents two transverse position vectors in the plane of mirror${\text{M}}_{1}$, $k=2\pi /\lambda $ is the wave number with$${M}_{0\alpha \beta}^{-1}=\left(\begin{array}{cc}\frac{1}{ik}\left(\frac{1}{2{\sigma}_{a}^{2}}+\frac{1}{{\delta}_{\alpha \beta}^{2}}\right)I& \frac{i}{k{\delta}_{\alpha \beta}^{2}}I\\ \frac{i}{k{\delta}_{\alpha \beta}^{2}}I& \frac{1}{ik}\left(\frac{1}{2{\sigma}_{\beta}^{2}}+\frac{1}{{\delta}_{\alpha \beta}^{2}}\right)I\end{array}\right),$$where **

*λ*being the wavelength of light, ${A}_{\alpha}$is the square root of the spectral density of electric field component ${E}_{\alpha}$, ${B}_{\alpha \beta}=\left|{B}_{\alpha \beta}\right|\mathrm{exp}\left(i\varphi \right)$ is the correlation coefficient between the ${E}_{x}$and${E}_{y}$field components,*T*stands for vector transposition and ${M}_{0\alpha \beta}^{-1}$is the $4\times 4$ matrix of the form**I**is the $2\times 2$ identity matrix, ${\sigma}_{\alpha}$ is the r.m.s width of the spectral density along*α*direction, ${\delta}_{xx}$, ${\delta}_{yy}$ and ${\delta}_{xy}$ 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 ${\sigma}_{x}$, ${\sigma}_{y}$, ${\delta}_{xx}$, ${\delta}_{yy}$ and ${\delta}_{xy}$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 $({\rho}_{1},{\rho}_{2})\equiv \tilde{\rho}$ is given by the expression [23]

**,**

*A***,**

*B***and**

*C***take the form [23]**

*D**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, ${\rho}_{1},{\rho}_{2}$ in Eq. (3) represents two transverse position vectors in the plane of mirror${\text{M}}_{2}$. When

*N*is an even number, ${\rho}_{1},{\rho}_{2}$ represents two transverse position vectors in the plane of mirror${\text{M}}_{1}$.

The trace of the cross-spectral density matrix of an EGSM beam after passing *N* times is expressed as [1–8]

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

*x*and

*y*directions, respectively, and

The normalization factor *P* is given by

*W*. Note that Eqs. (9) and (10) refer to the waist plane. In our case, as shown in Fig. 1, the spherical mirror ${\text{M}}_{1}$or ${\text{M}}_{2}$ 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 ${\text{M}}_{1}$or ${\text{M}}_{2}$, 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${\text{M}}_{1}$or ${\text{M}}_{2}$. To calculate the ${M}^{2}$-factor of an EGSM beam, we should replace

*W*in Eqs. (9) and (10) with ${W}_{\text{tr}}$ given by Eq. (7).

By expressing ${M}_{1\alpha \beta}^{-1}$ in Eq. (4) in the following alternative form

Taking the Fourier transform of ${\text{W}}_{\text{tr}}({\rho}_{x}{}_{1},{\rho}_{y}{}_{1},{\rho}_{x}{}_{2},{\rho}_{y}{}_{2})$, we obtain

Substituting Eqs. (15) and (17) into Eqs. (9)-(13), we obtain (after tedious integration and operation)

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

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

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

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

The degree of polarization of the initial source beam at point **r**is expressed as follows [1]

In the following text, we set ${\sigma}_{x}={\sigma}_{y}=2mm$, ${A}_{y}=1$, $\lambda =632.8nm$, $L=300mm$ unless stated otherwise. In this case, the polarization properties are uniform across the source plane with ${P}_{0}(\mathbf{r})=\left|\frac{{A}_{x}^{2}-{A}_{y}^{2}}{{A}_{x}^{2}+{A}_{y}^{2}}\right|.$ 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 ${\delta}_{xx}$ and ${\delta}_{yy}$ with ${A}_{x}=0.707$ and $\epsilon =1.5mm$. 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 ($0\le g<1$), the ${M}^{2}$-factor exhibits decrease with oscillations. In unstable cavities ($g\ge 1$), 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 ${\delta}_{xx}$ and ${\delta}_{yy}$ (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 ${\delta}_{xx}$ and ${\delta}_{yy}$ 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.

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 ${\delta}_{xx}=0.15mm$, ${\delta}_{yy}=0.05mm$ and $\epsilon =1.5mm$. 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., ${W}_{xx}(\tilde{\rho})+{W}_{yy}(\tilde{\rho})$). The evolution properties of the elements ${W}_{xx}(\tilde{\rho})$ and ${W}_{yy}(\tilde{\rho})$ are closely determined by the correlation coefficients ${\delta}_{xx}$ and ${\delta}_{yy}$, 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 ${W}_{xx}({\rho}_{1},{\rho}_{2})$ to the ${M}^{2}$-factor dominates that of the element ${W}_{yy}({\rho}_{1},{\rho}_{2})$ for the case of ${A}_{x}>{A}_{y}$, and the contribution of the element ${W}_{yy}({\rho}_{1},{\rho}_{2})$plays a dominant role otherwise. When *N* is large enough, the influence of polarization on the ${M}^{2}$-factor is negligible.

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, ${\delta}_{xx}=0.15mm$and ${\delta}_{yy}=0.05mm$. 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 parameter*ε* 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.

## 3. Thermal lens effect induced changes of the *M*^{2}-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 $2{l}_{1}+{l}_{2}=300mm$. The approximate expression for the ABCD matrix of a thermal lens medium is expressed as [60,64]

*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

**$${D}_{1}=\left[1+\frac{\left(-\frac{2}{R}-\frac{i\lambda}{\pi {\epsilon}_{}^{2}}\right){l}_{2}}{{n}_{0}}+2{n}_{0}\gamma {l}_{1}^{2}{l}_{2}\left(-\frac{2}{R}-\frac{i\lambda}{\pi {\epsilon}_{}^{2}}\right)+\gamma {l}_{2}^{2}+-\frac{4{l}_{1}}{R}-\frac{2i\lambda {l}_{1}}{\pi {\epsilon}_{}^{2}}+2{n}_{0}\gamma {l}_{2}{l}_{1}+{l}_{1}\left(-\frac{4}{R}-\frac{2i\lambda}{\pi {\epsilon}_{}^{2}}\right)\gamma {l}_{2}^{2}\right]I.$$**

**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.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 ${\delta}_{xx}$ and ${\delta}_{yy}$ with $\gamma =5\times {10}^{-6}$, ${A}_{x}=0.707$, g = 1 and $\epsilon =0.8mm$. 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 ${\delta}_{xx}$ and ${\delta}_{yy}$ with ${l}_{2}=100mm$, ${A}_{x}=0.707$, g = 1 and$\epsilon =0.8mm$. 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 coefficient*γ*, and the constant value increases with the increase of the length of the thermal lens medium or the thermal lens medium coefficient*γ*. The behavior of the ${M}^{2}$-factor in the cavity containing the thermal lens medium is also influenced by the source correlation coefficients ${\delta}_{xx}$ and ${\delta}_{yy}$. When *N* is small, the ${M}^{2}$-factor increases with the decrease of the correlation coefficients ${\delta}_{xx}$ and ${\delta}_{yy}$ 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 ${\delta}_{xx}$ and ${\delta}_{yy}$ 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].

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 ${A}_{x}>{A}_{y}$, ${\delta}_{xx}=0.15mm$, ${\delta}_{yy}=0.05mm$, $\gamma =5\times {10}^{-6}$, g = 1 and $\epsilon =0.8mm$. 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 ${A}_{x}>{A}_{y}$, ${\delta}_{xx}=0.15mm$, ${\delta}_{yy}=0.05mm$, ${l}_{2}=100mm$, g = 1 and$\epsilon =0.8mm$. 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.

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${\delta}_{xx}=0.15mm$, ${\delta}_{yy}=0.05mm$, $\gamma =5\times {10}^{-6}$, ${A}_{x}=0.707$ 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 coefficient*γ*and the mirror spot size of the cavity *ε* with ${\delta}_{xx}=0.15mm$, ${\delta}_{yy}=0.05mm$, ${l}_{2}=100mm$, ${A}_{x}=0.707$ 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.

## 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.

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