Abstract

With the help of a tensor method, an explicit expression for the M2-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 M2-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. [13] 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 [913]. 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 [1416]. Palma and associates then studied the behavior of the coherence and the spectral properties of scalar partially coherent beams in a Gaussian cavity [1720]. 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 [2326]. 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 theM2 -factor of an EGSM beam in a Gaussian cavity. In fact, little attention was paid to the M2-factor of an EGSM beam in free space or in turbulent atmosphere [27].

The propagation factor (also known as the M2-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 M2-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 M2-factor. The definition of M2-factor was then extended to the partially coherent beams [35, 36], and the M2-factor of a partially coherent beam with or without truncation in free space was studied in [3538]. Up to now, the properties of the M2-factor of various coherent and partially coherent beams outside the cavity have been investigated widely [3947]. More recently, more and more attention is being paid to the M2-factor of laser beams in turbulent atmosphere due to their important application in free-space optical communications [27, 4851]. In this paper, our aim is to investigate the M2-factor of an EGSM beam in a Gaussian cavity. Analytical formula for the M2-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. M2-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 (0g<1) or unstable (g1) due to the value of the stability parameter g=1L/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].

 figure: Fig. 1

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 [13]. Such a beam can be characterized by the cross-spectral density matrix evaluated at points (r1,r2)r˜ of the tensor form [5, 23]

Wαβ(r˜)=AαAβBαβexp[ik2r˜TM0αβ1r˜],(α=x,y;β=x,y),
where r1,r2 in Eq. (1) represents two transverse position vectors in the plane of mirrorM1, k=2π/λ is the wave number withλ being the wavelength of light, Aαis the square root of the spectral density of electric field component Eα, Bαβ=|Bαβ|exp(iϕ) is the correlation coefficient between the ExandEyfield components, T stands for vector transposition and M0αβ1is the 4×4 matrix of the form
M0αβ1=(1ik(12σa2+1δαβ2)Iikδαβ2Iikδαβ2I1ik(12σβ2+1δαβ2)I),
where Iis the 2×2 identity matrix, σα is the r.m.s width of the spectral density alongαdirection, δxx, δyy and δxy 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 σx, σy, δxx, δyy and δxyon the frequency was omitted for simplicity, although these parameters depend on the frequency in the most general case [13, 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 (ρ1,ρ2)ρ˜ is given by the expression [23]

Wαβ(ρ˜)=AαAβBαβ[Det(A¯+B¯M0αβ1)]1/2exp[ik2ρ˜TM1αβ1ρ˜],(α=x,y;β=x,y),
where
M1αβ1=(C¯+D¯M0αβ1)(A¯+B¯M0αβ1)1,
and A¯,B¯,C¯andD¯ are 4×4 matrices of the form
A¯=(A0I0IA*),B¯=(B0I0IB*),C¯=(C0I0IC*),D¯=(D0I0ID*),
“∗” denoting Hermitian operator. For the resonator shown in Fig. 1, the matrices A, B, C and D take the form [23]
(ABCD)=(A1B1C1D1)N,(A1B1C1D1)=(ILI(2Riλπε2)I(12LRiλLπε2)I),
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, ρ1,ρ2 in Eq. (3) represents two transverse position vectors in the plane of mirrorM2. When N is an even number, ρ1,ρ2 represents two transverse position vectors in the plane of mirrorM1.

The trace of the cross-spectral density matrix of an EGSM beam after passing N times is expressed as [18]

Wtr(ρ˜)=TrW(ρ˜)=Wxx(ρ˜)+Wyy(ρ˜).

Now we study the M2-factor of an EGSM beam on propagation in a Gaussian cavity. According to [35, 36], the M2-factor of a partially coherent beam is defined as

Mx2=4πΔxΔpx,My2=4πΔyΔpy,
whereMx2 and My2 are the beam propagation factors in the x and y directions, respectively, and

Δx=1P(ρxρx¯)2W(ρx,ρy,ρx,ρy)dρxdρy,
Δy=1P(ρyρy¯)2W(ρx,ρy,ρx,ρy)dρxdρy,
Δpx=1P(pxpx¯)2W˜(px,py,px,py)dpxdpy,
Δpy=1P(pypy¯)2W˜(px,py,px,py)dpxdpy,

The normalization factor P is given by

P=W(ρx,ρy,ρx,ρy)dρxdρy=W˜(px,py,px,py)dpxdpy,
where 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 M1or M2 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 M1or M2, 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 mirrorM1or M2. To calculate the M2-factor of an EGSM beam, we should replace Win Eqs. (9) and (10) with Wtr given by Eq. (7).

By expressing M1αβ1 in Eq. (4) in the following alternative form

M1αβ1=(Qαβ11Qαβ12Qαβ13Qαβ14Qαβ21Qαβ22Qαβ23Qαβ24Qαβ31Qαβ32Qαβ33Qαβ34Qαβ41Qαβ42Qαβ43Qαβ44),
the Eq. (7) can be expressed (after some operation) in the following alternative form
Wtr(ρx1,ρy1,ρx2,ρy2)=Gxxexp{iω2c[ρx12Qxx11+ρx22Qxx33+ρy12Qxx22+ρy22Qxx44+(Qxx13+Qxx31)ρx1ρx2+(Qxx24+Qxx42)ρy1ρy2]}+Gyyexp{iω2c[ρx12Qyy11+ρx22Qyy33+ρy12Qyy22+ρy22Qyy44+(Qyy13+Qyy31)ρx1ρx2+(Qyy24+Qyy42)ρy1ρy2]},
where

Gxx=Ax[Det(A¯+B¯M0xx1)]1/2,Gyy=Ay[Det(A¯+B¯M0yy1)]1/2.

Taking the Fourier transform of Wtr(ρx1,ρy1,ρx2,ρy2), we obtain

Wtr(px,py,px,py,z)=16π2Gxxk2[(Qxx13+Qxx31)24Qxx11Qxx33]exp[8π2i(Qxx33+Qxx11+Qxx13+Qxx31)k[(Qxx13+Qxx31)24Qxx11Qxx33](px2+py2)]+16π2Gyyk2[(Qyy13+Qyy31)24Qyy11Qyy33]exp[8π2i(Qyy33+Qyy11+Qyy13+Qyy31)k[(Qyy13+Qyy31)24Qyy11Qyy33](px2+py2)].

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

Δx2=Δy2=2πk2P[Gxx(Qxx11+Qxx13+Qxx31+Qxx33)2+Gyy(Qyy11+Qyy13+Qyy31+Qyy33)2],
Δpy2=Δpx2=18πP[Gxx[(Qxx13+Qxx31)24Qxx11Qxx33](Qxx11+Qxx13+Qxx31+Qxx33)2+Gyy[(Qyy13+Qyy31)24Qyy11Qyy33](Qyy11+Qyy13+Qyy31+Qyy33)2],
P=2πik[Gxx(Qxx11+Qxx13+Qxx31+Qxx33)+Gyy(Qyy11+Qyy13+Qyy31+Qyy33)].

Then substituting Eqs. (18)-(20) into Eq. (8), we obtain the following expression for the M2-factor of an EGSM beam on propagation in a Gaussian cavity

Mx2=My2=M2=QNxxQNyyGxxQNyy+GyyQNxx(GxxQNxx2+GyyQNyy2)(GxxQMxxQNxx2+GyyQMyyQNyy2),
where
Mx2=My2=M2=4Qαα11Qαα33(Qαα13+Qαα31)2(Qαα11+Qαα13+Qαα31+Qαα33)2,(α=x,y)
QMαβ=4Qαβ11Qαβ33(Qαβ13+Qαβ31)2,QNαβ=Qαβ11+Qαβ13+Qαβ31+Qαβ33,(α=x,y;β=x,y). Under the condition of Ax=0 or Ay=0, Eq. (21) reduces to the following expression for the the M2-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 M2-factor of an EGSM beam in a Gaussian cavity. In the above derivations, we have used following integral formulas

exp[ax2+by2]dxdy=πab,x2exp[ax2+by2+dxy]dxdy=4πb(4abd2)3/2.

For the case of N = 0, Eq. (21) reduces to the following expression for the M2-factor of an EGSM beam in free space

M2={(Ax2σx4+Ay2σy4)[4Ax2δyy2σx2+δxx2(Ax2δyy2+Ay2δyy2+4Ay2σy2)]δxx2δyy2(Ax2σx2+Ay2σy2)2}1/2,
and Eq. (22) reduces to the following expression for the M2-factor of a scalar GSM beam in free space
M2(z)=(1+4σα2δαα2)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 M2-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

W(r1,r2)=(Wxx(r1,r2)00Wyy(r1,r2)).

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

P0(r)=14DetW(r,r)[TrW(r,r)]2.

In the following text, we set σx=σy=2mm, Ay=1, λ=632.8nm, L=300mm unless stated otherwise. In this case, the polarization properties are uniform across the source plane with P0(r)=|Ax2Ay2Ax2+Ay2|. As shown in [55], the EGSM beam can be generated through combination of two orthogonal polarized partially coherent beams. Ax and Ay 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 Ax or Ay. In general Ax doesn’t equal toAy, and the degree of polarization of the source EGSM beam in our example are determined by Ax or Ay together. The degree of polarization in the source plane varies as the value ofAychanges for fixed Ax, any nonzero P0 can be achieved either forAx>Ay or for Ax<Ay (see Fig. 1 of Ref [27].).

We calculate in Fig. 2 the M2-factor of an EGSM beam in a Gaussian cavity versus N for different values of the cavity parameter g and the source correlation coefficients δxx and δyy with Ax=0.707 and ε=1.5mm. One finds from Fig. 2 that the M2-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 (0g<1), the M2-factor exhibits decrease with oscillations. In unstable cavities (g1), the decrease is monotonic. We can explain this phenomenon by the fact that the behavior of theM2-factor of an EGSM beam on propagation depends closely on the behavior of its spectral degree of coherence on propagation, and the M2-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 M2-factor of an EGSM beam in a Gaussian cavity. Furthermore, one finds from Fig. 2 that the behavior of the M2-factor of an EGSM beam is also closely related with the source correlation coefficients δxx and δyy (i.e., source spectral degree of coherence), particularly for small values of N. When N is large enough, the M2-factor of an EGSM beam with different values of δxx and δyy 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 M2-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: Fig. 2

Fig. 2 M2-factor of an EGSM beam in a Gaussian cavity versus N for different values of the cavity parameter g and the source correlation coefficients δxx and δyy

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Figure 3 illustrates the M2-factor of an EGSM beam in a Gaussian cavity versus N for different values of the cavity parameter g and the degree of polarization P0 of the source beam with δxx=0.15mm, δyy=0.05mm and ε=1.5mm. One can see from Fig. 3 that the evolution properties of the M2-factor of an EGSM beam is closely related with the degree of polarization P0 of the source beam, particularly for small value of N both in stable and unstable resonators. We can explain this by the fact that theM2-factor of an EGSM beam is calculated from the trace of the cross-special density matrix (i.e., Wxx(ρ˜)+Wyy(ρ˜)). The evolution properties of the elements Wxx(ρ˜) and Wyy(ρ˜) are closely determined by the correlation coefficients δxx and δyy, respectively. The initial degree of polarization calculated by Eq. (27) determines which element of cross-special density matrix contributes a dominant role to the M2-factor, thus leading the dependence of the behavior of theM2-factor of an EGSM beam on the degree of polarization shown in Fig. 3. For small value of N, the M2-factor of an EGSM beam increases with the decrease of P0 whenAx>Ay, while the M2-factor increases with the increase of P0 whenAx<Ay. This phenomenon is caused by the fact that the contribution of the element Wxx(ρ1,ρ2) to the M2-factor dominates that of the element Wyy(ρ1,ρ2) for the case of Ax>Ay, and the contribution of the element Wyy(ρ1,ρ2)plays a dominant role otherwise. When N is large enough, the influence of polarization on the M2-factor is negligible.

 figure: Fig. 3

Fig. 3 M2-factor of an EGSM beam in a Gaussian cavity versus N for different values of the cavity parameter g and the degree of polarization P0 of the source beam

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Figure 4 explores the M2-factor of an EGSM beam in a Gaussian cavity versus N for different values of the mirror spot size of the cavity ε with Ax=0.707, g = 1, δxx=0.15mmand δyy=0.05mm. As shown in Fig. 4, the behavior of the M2-factor on propagation depends closely on the cavity parameter ε. The M2-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 M2-factor on the mirror size as shown in Fig. 4. From above discussions, one comes to the conclusion that, in general, theM2-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 M2-factor in the case of large N is closely determined by the cavity parameters.

 figure: Fig. 4

Fig. 4 M2-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 [5861]. 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 l2. The distance between the thermal lens medium and the spherical mirror is l1 and the distance between the centers of the mirrors is 2l1+l2. Figure 5(b) shows the equivalent version of Fig. 5 (a). In the following text, we set 2l1+l2=300mm. The approximate expression for the ABCD matrix of a thermal lens medium is expressed as [60,64]

(AtBtCtDt)=(cos(βl)Isin(βl)/(βn0)Iβn0sin(βl)Icos(βl)I)=((1+γl2)Il/n0I2γn0lI(1+γl2)I),
whereβ,γare the medium coefficients and they satisfy the relationβ=2γ, n0 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
(ABCD)=(A1B1C1D1)N,
with
A1=(1+2n0γl1l2+γl22)I,
B1=[l2n0+2n0γl12l2+2l1(1+γl22)]I,
C1=[2n0γ(1+(2Riλπε2)l1)l2+(2Riλπε2)(1+γl22)]I,
D1=[1+(2Riλπε2)l2n0+2n0γl12l2(2Riλπε2)+γl22+4l1R2iλl1πε2+2n0γl2l1+l1(4R2iλπε2)γl22]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.

 figure: Fig. 5

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 M2-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 δxx and δyy with γ=5×106, Ax=0.707, g = 1 and ε=0.8mm. Figure 7 shows the M2-factor of an EGSM beam in a Gaussian cavity for different values of the thermal lens medium coefficient γ and the source correlation coefficients δxx and δyy with l2=100mm, Ax=0.707, g = 1 andε=0.8mm. As shown in Figs. 6 and 7, the evolution properties of the M2-factor of an EGSM beam in a Gaussian cavity are affected by thermal lens medium, especially for large value of N. The M2-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 M2-factor in the cavity containing the thermal lens medium is also influenced by the source correlation coefficients δxx and δyy. When N is small, the M2-factor increases with the decrease of the correlation coefficients δxx and δyy for fixed parameters (l2 and γ) of the thermal lens medium. When N is large, the M2-factor approaches to a constant and is independent of the correlation coefficients δxx and δyy for fixed parameters (l2 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: Fig. 6

Fig. 6 M2-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 δxx and δyy

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 figure: Fig. 7

Fig. 7 M2-factor of an EGSM beam in a Gaussian cavity for different values of the thermal lens medium coefficientγ and the source correlation coefficients δxx and δyy

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Figure 8 shows the M2-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 P0 of the source beam with Ax>Ay, δxx=0.15mm, δyy=0.05mm, γ=5×106, g = 1 and ε=0.8mm. Figure 9 shows the M2-factor of an EGSM beam in a Gaussian cavity for different values of the thermal lens medium coefficient γ and the degree of polarization P0 of the source beam with Ax>Ay, δxx=0.15mm, δyy=0.05mm, l2=100mm, g = 1 andε=0.8mm. One finds from Figs. 8 and 9 that the behavior of the M2-factor of an EGSM beam in the cavity containing the thermal lens medium is also affected by the degree of polarization P0 of the source beam. For the case of Ax>Ay, the M2-factor increases with the decrease of P0 for fixed parameters (l2 and γ) of the thermal lens medium when N is small. When N is large, the M2-factor approaches to a constant and is independent of P0 for fixed parameters (l2 and γ) of the thermal lens medium.

 figure: Fig. 8

Fig. 8 M2-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 P0of the source beam

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 figure: Fig. 9

Fig. 9 M2-factor of an EGSM beam in a Gaussian cavity for different values of the thermal lens medium coefficient γ and the degree of polarization P0 of the source beam

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Figure 10 shows the M2-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 withδxx=0.15mm, δyy=0.05mm, γ=5×106, Ax=0.707 and g = 1. Figure 11 shows the M2-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 δxx=0.15mm, δyy=0.05mm, l2=100mm, Ax=0.707 and g = 1. One finds from Figs. 10 and 11 that the behavior of the M2-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 M2-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 M2-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 M2-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 M2-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.

 figure: Fig. 10

Fig. 10 M2-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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 figure: Fig. 11

Fig. 11 M2-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 M2-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 M2-factor of an EGSM beam in a Gaussian cavity is closely determined by the cavity parameters, the degree of polarization P0 and the correlation coefficients of the source beam. In stable cavities, the M2-factor exhibits decrease with oscillations, and it exhibits monotonic decrease in unstable cavities. When the number of passages is large enough, the M2-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 [48].

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]  

References

  • View by:

  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 World109–114 (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]

2010 (6)

Z. Tong, Y. Cai, and O. Korotkova, “Ghost imaging with electromagnetic stochastic beams,” Opt. Commun. 283(20), 3838–3845 (2010).
[Crossref]

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]

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]

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]

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]

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]

2009 (3)

2008 (5)

2006 (2)

T. Xu and S. Wang, “Propagation of Ince-Gaussian beams in a thermal lens medium,” Opt. Commun. 265(1), 1–5 (2006).
[Crossref]

E. Wolf, “Coherence and polarization properties of electromagnetic laser modes,” Opt. Commun. 265(1), 60–62 (2006).
[Crossref]

2005 (4)

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]

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]

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]

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]

2004 (2)

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]

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]

2003 (4)

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003).
[Crossref]

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

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]

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]

2002 (3)

2001 (1)

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]

1999 (3)

1998 (3)

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]

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]

C. Palma, G. Cardone, and G. Cincotti, “Spectral changes in Gaussian-cavity lasers,” IEEE J. Quantum Electron. 34(7), 1082–1088 (1998).
[Crossref]

1997 (1)

1996 (1)

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]

1995 (1)

1994 (3)

G. N. Lawrence, “Proposed international standard for laser-beam quality falls short,” Laser Focus World109–114 (1994).

J. J. Chang, “Time-resolved beam-quality characterization of copper-vapor lasers with unstable resonators,” Appl. Opt. 33(12), 2255–2265 (1994).
[Crossref] [PubMed]

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]

1993 (3)

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]

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]

P. A. Belanger, “Beam quality factor of the LP01 mode of the step-index fiber,” Opt. Eng. 32(9), 2107–2109 (1993).
[Crossref]

1991 (1)

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]

1985 (1)

S. Wang, “Matrix methods in treating decentred optical systems,” Opt. Quantum Electron. 17(1), 1–14 (1985).
[Crossref]

1984 (2)

1982 (1)

1980 (1)

F. Gori, “Propagation of the mutual intensity through a periodic structure,” Atti Fond. Giorgio Ronchi 35, 434–447 (1980).

1975 (1)

1974 (1)

J. R. Whinnery, “Laser measurement of optical absorption in liquids,” Acc. Chem. Res. 7(7), 225–231 (1974).
[Crossref]

1973 (1)

1966 (1)

1965 (1)

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]

1963 (1)

E. Wolf, “Spatial coherence of resonant modes in a maser interferometer,” Phys. Lett. 3(4), 166–168 (1963).
[Crossref]

1961 (2)

A. G. Fox and T. Li, “Resonate modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).

G. D. Boyd and J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961).

Agarwal, G. S.

Arias, M.

Baykal, Y.

Belanger, P. A.

P. A. Belanger, “Beam quality factor of the LP01 mode of the step-index fiber,” Opt. Eng. 32(9), 2107–2109 (1993).
[Crossref]

Borghi, R.

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]

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]

Boyd, G. D.

G. D. Boyd and J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961).

Cai, Y.

Z. Tong, Y. Cai, and O. Korotkova, “Ghost imaging with electromagnetic stochastic beams,” Opt. Commun. 283(20), 3838–3845 (2010).
[Crossref]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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: .

Cardone, G.

C. Palma, G. Cardone, and G. Cincotti, “Spectral changes in Gaussian-cavity lasers,” IEEE J. Quantum Electron. 34(7), 1082–1088 (1998).
[Crossref]

Carter, C. A.

Casperson, L. W.

Chang, J. J.

Chen, X.

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]

Chu, X.

Cincotti, G.

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]

C. Palma, G. Cardone, and G. Cincotti, “Spectral changes in Gaussian-cavity lasers,” IEEE J. Quantum Electron. 34(7), 1082–1088 (1998).
[Crossref]

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]

Dan, Y.

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]

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]

De Santis, P.

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]

de Valcarcel, G. J.

Deng, D.

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]

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]

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]

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]

DeSantis, P.

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]

Eyyuboglu, H. T.

Fan, Z.

Fox, A. G.

A. G. Fox and T. Li, “Resonate modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).

Friberg, A.

Friberg, A. T.

Fu, X.

Gordon, J. P.

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]

G. D. Boyd and J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961).

Gori, F.

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]

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]

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]

F. Gori, “Propagation of the mutual intensity through a periodic structure,” Atti Fond. Giorgio Ronchi 35, 434–447 (1980).

Guattari, G.

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]

Guo, H.

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]

Harris, J. M.

Hilgevoord, J.

J. Hilgevoord, “The standard deviation is not an adequate measure of quantum uncertainty,” Am. J. Phys. 70(10), 983 (2002).
[Crossref]

Hu, C.

Knight, L. V.

Kogelnik, H.

Kong, H.

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]

Korotkova, O.

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]

Z. Tong, Y. Cai, and O. Korotkova, “Ghost imaging with electromagnetic stochastic beams,” Opt. Commun. 283(20), 3838–3845 (2010).
[Crossref]

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]

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]

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]

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]

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]

O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281, 2342–2348 (2008).

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]

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]

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]

Lawrence, G. N.

G. N. Lawrence, “Proposed international standard for laser-beam quality falls short,” Laser Focus World109–114 (1994).

Leite, R. C. C.

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]

Li, Q.

Li, T.

H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5(10), 1550–1567 (1966).
[Crossref] [PubMed]

A. G. Fox and T. Li, “Resonate modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).

Lin, Q.

Lü, B.

Lunnam, S. D.

Ma, H.

Martinez-Herrero, R.

Martínez-Herrero, R.

Mascello, A.

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]

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]

Mejias, P. M.

Mejías, P. M.

Mondello, A.

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]

Moore, R. S.

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]

Neuenschwander, B.

R. Weber, B. Neuenschwander, and H. P. Weber, “Thermal effects in solid-state laser materials,” Opt. Mater. 11(2-3), 245–254 (1999).
[Crossref]

Palma, C.

C. Palma, G. Cardone, and G. Cincotti, “Spectral changes in Gaussian-cavity lasers,” IEEE J. Quantum Electron. 34(7), 1082–1088 (1998).
[Crossref]

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]

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]

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]

Perrone, M. R.

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]

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]

Piquero, G.

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]

Porto, S. P. S.

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]

Qu, J.

Roldan, E.

Saastamoinen, T.

Saghafi, S.

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]

Salem, M.

Santarsiero, M.

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]

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]

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]

Setala, T.

Setälä, T.

Shao, J.

Sheldon, S. J.

Sheppard, C. J. R.

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]

Shirai, T.

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]

Siegman, A. E.

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]

Simon, R.

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]

Sona, A.

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]

Tervo, J.

Thorne, J. M.

Tong, Z.

Z. Tong, Y. Cai, and O. Korotkova, “Ghost imaging with electromagnetic stochastic beams,” Opt. Commun. 283(20), 3838–3845 (2010).
[Crossref]

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]

Townsend, S. W.

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]

Turunen, J.

Urchueguía, J. F.

Vahimaa, P.

Wang, F.

Wang, S.

T. Xu and S. Wang, “Propagation of Ince-Gaussian beams in a thermal lens medium,” Opt. Commun. 265(1), 1–5 (2006).
[Crossref]

S. Wang, “Matrix methods in treating decentred optical systems,” Opt. Quantum Electron. 17(1), 1–14 (1985).
[Crossref]

Weber, H. P.

R. Weber, B. Neuenschwander, and H. P. Weber, “Thermal effects in solid-state laser materials,” Opt. Mater. 11(2-3), 245–254 (1999).
[Crossref]

Weber, R.

R. Weber, B. Neuenschwander, and H. P. Weber, “Thermal effects in solid-state laser materials,” Opt. Mater. 11(2-3), 245–254 (1999).
[Crossref]

Wei, C.

Wen, Q.

Whinnery, J. R.

J. R. Whinnery, “Laser measurement of optical absorption in liquids,” Acc. Chem. Res. 7(7), 225–231 (1974).
[Crossref]

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]

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]

Wolf, E.

E. Wolf, “Coherence and polarization properties of electromagnetic laser modes,” Opt. Commun. 265(1), 60–62 (2006).
[Crossref]

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]

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]

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003).
[Crossref]

E. Wolf and G. S. Agarwal, “Coherence theory of laser resonator modes,” J. Opt. Soc. Am. A 1(5), 541–546 (1984).
[Crossref]

E. Wolf, “Spatial coherence of resonant modes in a maser interferometer,” Phys. Lett. 3(4), 166–168 (1963).
[Crossref]

Xu, T.

T. Xu and S. Wang, “Propagation of Ince-Gaussian beams in a thermal lens medium,” Opt. Commun. 265(1), 1–5 (2006).
[Crossref]

Yao, M.

Yin, S.

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]

Yuan, Y.

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]

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]

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Figures (11)

Fig. 1
Fig. 1 Schematic diagram of a Gaussian cavity and its equivalent (unfolded) version
Fig. 2
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
Fig. 3
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
Fig. 4
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
Fig. 5
Fig. 5 Schematic diagram of a Gaussian cavity containing thermal lens medium and its equivalent (unfolded) version
Fig. 6
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
Fig. 7
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
Fig. 8
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
Fig. 9
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
Fig. 10
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
Fig. 11
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

Equations (33)

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W α β ( r ˜ ) = A α A β B α β exp [ i k 2 r ˜ T M 0 α β 1 r ˜ ] , ( α = x , y ; β = x , y ) ,
M 0 α β 1 = ( 1 i k ( 1 2 σ a 2 + 1 δ α β 2 ) I i k δ α β 2 I i k δ α β 2 I 1 i k ( 1 2 σ β 2 + 1 δ α β 2 ) I ) ,
W α β ( ρ ˜ ) = A α A β B α β [ Det ( A ¯ + B ¯ M 0 α β 1 ) ] 1 / 2 exp [ i k 2 ρ ˜ T M 1 α β 1 ρ ˜ ] , ( α = x , y ; β = x , y ) ,
M 1 α β 1 = ( C ¯ + D ¯ M 0 α β 1 ) ( A ¯ + B ¯ M 0 α β 1 ) 1 ,
A ¯ = ( A 0 I 0 I A * ) , B ¯ = ( B 0 I 0 I B * ) , C ¯ = ( C 0 I 0 I C * ) , D ¯ = ( D 0 I 0 I D * ) ,
( A B C D ) = ( A 1 B 1 C 1 D 1 ) N , ( A 1 B 1 C 1 D 1 ) = ( I L I ( 2 R i λ π ε 2 ) I ( 1 2 L R i λ L π ε 2 ) I ) ,
W tr ( ρ ˜ ) = Tr W ( ρ ˜ ) = W x x ( ρ ˜ ) + W y y ( ρ ˜ ) .
M x 2 = 4 π Δ x Δ p x , M y 2 = 4 π Δ y Δ p y ,
Δ x = 1 P ( ρ x ρ x ¯ ) 2 W ( ρ x , ρ y , ρ x , ρ y ) d ρ x d ρ y ,
Δ y = 1 P ( ρ y ρ y ¯ ) 2 W ( ρ x , ρ y , ρ x , ρ y ) d ρ x d ρ y ,
Δ p x = 1 P ( p x p x ¯ ) 2 W ˜ ( p x , p y , p x , p y ) d p x d p y ,
Δ p y = 1 P ( p y p y ¯ ) 2 W ˜ ( p x , p y , p x , p y ) d p x d p y ,
P = W ( ρ x , ρ y , ρ x , ρ y ) d ρ x d ρ y = W ˜ ( p x , p y , p x , p y ) d p x d p y ,
M 1 α β 1 = ( 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 ) ,
W tr ( ρ x 1 , ρ y 1 , ρ x 2 , ρ y 2 ) = G x x exp { 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 { 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 ] } ,
G x x = A x [ Det ( A ¯ + B ¯ M 0 x x 1 ) ] 1 / 2 , G y y = A y [ Det ( A ¯ + B ¯ M 0 y y 1 ) ] 1 / 2 .
W tr ( p x , p y , p x , p y , z ) = 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 ( 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 ) ] + 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 ( 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 ) ] .
Δ x 2 = Δ y 2 = 2 π k 2 P [ G x x ( Q x x 11 + Q x x 13 + Q x x 31 + Q x x 33 ) 2 + 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 = 1 8 π P [ 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 + 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 = 2 π i k [ G x x ( Q x x 11 + Q x x 13 + Q x x 31 + Q x x 33 ) + G y y ( Q y y 11 + Q y y 13 + Q y y 31 + Q y y 33 ) ] .
M x 2 = M y 2 = M 2 = Q N x x Q N y y G x x Q N y y + G y y Q N x x ( G x x Q N x x 2 + G y y Q N y y 2 ) ( G x x Q M x x Q N x x 2 + G y y Q M y y Q N y y 2 ) ,
M x 2 = M y 2 = M 2 = 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 )
exp [ a x 2 + b y 2 ] d x d y = π a b , x 2 exp [ a x 2 + b y 2 + d x y ] d x d y = 4 π b ( 4 a b d 2 ) 3 / 2 .
M 2 = { ( 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 ,
M 2 ( z ) = ( 1 + 4 σ α 2 δ α α 2 ) 1 / 2 , ( α = x , y )
W ( r 1 , r 2 ) = ( W x x ( r 1 , r 2 ) 0 0 W y y ( r 1 , r 2 ) ) .
P 0 ( r ) = 1 4 Det W ( r , r ) [ Tr W ( r , r ) ] 2 .
( A t B t C t D t ) = ( cos ( β l ) I sin ( β l ) / ( β n 0 ) I β n 0 sin ( β l ) I cos ( β l ) I ) = ( ( 1 + γ l 2 ) I l / n 0 I 2 γ n 0 l I ( 1 + γ l 2 ) I ) ,
( A B C D ) = ( A 1 B 1 C 1 D 1 ) N ,
A 1 = ( 1 + 2 n 0 γ l 1 l 2 + γ l 2 2 ) I ,
B 1 = [ 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 + ( 2 R i λ π ε 2 ) l 1 ) l 2 + ( 2 R i λ π ε 2 ) ( 1 + γ l 2 2 ) ] I ,
D 1 = [ 1 + ( 2 R i λ π ε 2 ) l 2 n 0 + 2 n 0 γ l 1 2 l 2 ( 2 R i λ π ε 2 ) + γ l 2 2 + 4 l 1 R 2 i λ l 1 π ε 2 + 2 n 0 γ l 2 l 1 + l 1 ( 4 R 2 i λ π ε 2 ) γ l 2 2 ] I .

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