Abstract

Analytical formula is derived for the propagation factor (known asM2-factor) of a stochastic electromagnetic Gaussian Schell-model (EGSM) beam in free space and in turbulent atmosphere. In free space, the M2-factor of an EGSM beam is mainly determined by its initial degree of polarization, r.m.s. widths of the spectral densities and correlation coefficients, and its value remains invariant on propagation. In turbulent atmosphere, the M2-factor of an EGSM beam is also determined by the parameters of the turbulent atmosphere, and its value increases on propagation. The relative M2-factor of an EGSM beam with lower correlation factors, larger r.m.s. widths of the spectral densities and longer wavelength is less affected by the atmospheric turbulence. Under suitable conditions, an EGSM beam is less affected by the atmospheric turbulence than a scalar GSM beam (i.e. fully polarized GSM beam). Our results will be useful in long-distance free-space optical communications.

©2010 Optical Society of America

1. Introduction

In 1994 it was found that the degree of polarization of a stochastic electromagnetic beam may change on propagation in vacuum, and such changes depend on the coherence properties of the source of the beam [1]. Electromagnetic Gaussian Schell-model (EGSM) beam [24] was introduced as an extension of scalar GSM beam [58]. Due to its importance in the theories of coherence and polarization of light, numerous theoretical and experimental papers relating to stochastic electromagnetic beams have been published in the past several years [924].

Over the past several decades, many works have been carried out concerning the propagation of various laser beams through the turbulent atmosphere due to their wide applications, e.g. in free-space optical communication, laser radar, atmospheric imaging systems and remote sensing, and it has been found that the behavior of a laser beam in a turbulent atmosphere is closely related to its initial beam profile, coherence and polarization properties [2546]. Behavior of the statistical properties including the averaging intensity, coherence, degree of polarization, state of polarization and scintillation index of an EGSM beam, propagating in turbulent atmosphere has been studied in details [4048]. It was found that the EGSM beams may have reduced levels of scintillations compared to the scalar GSM beams [47], which is useful for free-space optical communications and laser radar systems (LIDARs) [48]. To our knowledge no results have been reported up until now on the propagation factor of EGSM beams passing through the turbulent atmosphere.

The propagation factor (also known as the M2-factor) proposed by Siegman is a particularly useful property of an optical laser beam, and plays an important role in the characterization of beam behavior on propagation. Several methods have been developed to obtain the propagation factors of the laser beams in free space [4951]. The definition of propagation factor in terms of second-order moments of the Wigner distribution function has been adopted widely for characterizing laser beams [5254]. Recently, this method was extended for the analysis of the propagation factor of a laser beam traveling in the turbulent atmosphere [5557]. The purpose of this paper is to investigate, based on this recently extended method, the M2-factor of an EGSM beam propagating in the turbulent atmosphere, by deriving the explicit expression for the propagation factor of an EGSM beam for both free space propagation and atmospheric propagation, and exploring them comparatively.

2. Theory

The second-order statistical properties of an EGSM beam can be characterized by the 2×2 cross-spectral density matrixW(ρ1',ρ2',0) specified at any two points with position vectors ρ1' and ρ2'in the source plane with elements [24]

Wαβ(ρ1',ρ2';0)=AαAβBαβexp[ρ1'24σa2ρ2'24σβ2(ρ1'ρ2')22δαβ2],(α=x,y;β=x,y)
where σα is the r.m.s width of the spectral density along α direction; 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, Bxy is the complex correlation coefficient between the x and y components of the electric field; parametersAα, Bαβ=|Bαβ|exp(iϕαβ)=Bβα*, σα and δαβ are independent of position and, in our analysis, of frequency. The nine real parametersAx, Ay, σx, σy, |Bxy|, ϕxy, δxx, δyy and δxy entering the general model are shown to satisfy several intrinsic constraints and obey some simplifying assumptions (e.g. the phase difference between the x- an y-components of the field is removable, i.e. ϕαβ=0) [9,14,18]. The trace of the cross-spectral density matrix of an EGSM beam is expressed as [24,9,10]

Wtr(ρ1',ρ2';0)=TrW(ρ1',ρ2',0)=Wxx(ρ1',ρ2';0)+Wyy(ρ1',ρ2';0).

Within the validity of the paraxial approximation, the propagation of the trace of the cross-spectral density matrix of an EGSM beam in the turbulent atmosphere can be studied with the help of the following generalized Huygens-Fresnel integral [26,27]

Wtr(ρ,ρd;z)=(k2πz)2Wtr(ρ',ρd';0)×exp[ikz(ρρ')(ρdρd')H(ρd,ρd';z)]d2ρ'd2ρd',
where k=2π/λ is the wave number withλ being the wavelength. In Eq. (3) we have used the following sum and difference vector notation
ρ'=(ρ1'+ρ2')2,ρd'=ρ1'ρ2',ρ=(ρ1+ρ2)2,ρd=ρ1ρ2,
where ρ1,ρ2are two arbitrary points in the receiver plane, perpendicular to the direction of propagation of the beam and Wtr(ρ',ρd';0)=Wtr(ρ1',ρ2';0)=Wtr(ρ'+ρd'2,ρ'ρd'2;0).The term H(ρd,ρd',z) in Eq. (3) is the contribution from the atmospheric turbulence expressed as
H(ρd,ρd';z)=4π2k2z01dξ0[1J0(κ|ρd'ξ+(1ξ)ρd|)]Φn(κ)κdκ,
where J0 is the Bessel function of zero order, Φn represents the one-dimensional power spectrum of the index-of-refraction fluctuations [27].

Equation (3) can be expressed in the following alternative form [5557]

Wtr(ρ,ρd;z)=(12π)2Wtr(ρ'',ρd+zkκd;0)×exp[iρκd+iρ''κdH(ρd,ρd+zkκd;z)]d2ρ''d2κd,
where κd(κdxκdy) is the position vector in spatial-frequency domain. We can express Wtr(ρ'',ρd+zkκd;0) of an EGSM beam as

Wtr(ρ'',ρd+zkκd;0)=Ax2exp[14σx2(ρ"+ρd+zkκd2)214σx2(ρ"ρd+zkκd2)2(ρd+zkκd)22δxx2]+Ay2exp[14σy2(ρ"+ρd+zkκd2)214σy2(ρ"ρd+zkκd2)2(ρd+zkκd)22δxx2].

The Wigner distribution function of an EGSM beam in turbulent atmosphere can be expressed in terms of the trace of the cross-spectral density matrix by the formula [55]

htr(ρ,θ;z)=(k2π)2Wtr(ρ,ρd;z)exp(ikθρd)d2ρd,
where θ(θx,θy) denotes an angle which the vector of interest makes with the z-direction, kθxand kθy are the wave vector components along the x-axis and y-axis, respectively.

Substituting from Eqs. (6) and (7) into Eq. (8) and applying following integral formula,

exp(s2x2±qx)dx=πsexp(q24s2),(s>0),
we obtain (after tedious integration) the expression
htr(ρ,θ,z)=hxx(ρ,θ,z)+hyy(ρ,θ,z)=Ax2σx2k28π3exp[axxκd22zkbxxρdκdiρκdbxxρd2ikθρdH(ρd,ρd+zkκd,z)]d2κdd2ρd+Ay2σy2k28π3exp[ayyκd22zkbyyρdκdiρκdbyyρd2ikθρdH(ρd,ρd+zkκd,z)]d2κdd2ρd,
with aαα=(18σα2+12δαα2)z2k2+σα22, bαα=18σa2+12δaa2, (α=x,y).

Based on the second-order moments of the Wigner distribution function, the M2-factor of an EGSM beam can be defined as follows [4957]

M2(z)=k(ρ2θ2ρθ2)1/2=k[(x2+y2)(θx2+θy2)(xθx+yθy)2]1/2,
where

<xn1yn2θxm1θym2>=1Pxn1yn2θxm1θym2htr(ρ,θ,z)d2ρd2θ,
P=htr(ρ,θ,z)d2ρd2θ.

Substituting Eq. (10) into Eqs. (12) and (13), we obtain (after tedious integration) the formulas for the moments

P=2π(Ax2σx2+Ay2σy2),
ρ2=2πAx2σx2P(4axx+43π2Tz3)+2πAy2σy2P(4ayy+43π2Tz3),
θ2=2πAx2σx2P(4k2bxx+4π2Tz)+2πAy2σy2P(4k2byy+4π2Tz),
ρθ=2πAx2σx2P(4zk2bxx+2π2z2T)+2πAy2σy2P(4zk2byy+2π2z2T).
with

T=0Φn(κ)κ3dκ.

After substituting from Eqs. (14)-(17) into Eq. (11) we obtain the following expression for the M2-factor of an EGSM beam travelling in turbulent atmosphere

M2(z)=k(ρ2θ2ρθ2)1/2=k{[2πAx2σx2P(4axx+43π2Tz3)+2πAy2σy2P(4ayy+43π2Tz3)]×[2πAx2σx2P(4k2bxx+4π2Tz)+2πAy2σy2P(4k2byy+4π2Tz)][2πAx2σx2P(4zk2bxx+2π2z2T)+2πAy2σy2P(4zk2byy+2π2z2T)]2}1/2.

Under the condition of Φn(κ)=0, Eq. (19) reduces to following expression for the free space M2-factor of an EGSM beam

M2(z)=k(ρ2θ2ρθ2)1/2={(Ax2σx4+Ay2σy4)[4Ax2δyy2σx2+δxx2(Ax2δyy2+Ay2δyy2+4Ay2σy2)]δxx2δyy2(Ax2σx2+Ay2σy2)2}1/2.
One finds from Eq. (20) that the M2-factor of an EGSM beam in free space is independent of z, so its value remains invariant on propagation, while being closely determined by parameters Aα,σαandδαα. Under the condition of Ax=0or Ay=0, Eq. (20) reduces to the following expression for the M2-factor of a scalar GSM beam (i.e., fully polarized GSM beam)
M2(z)=(1+4σα2δαα2)1/2,(α=x,y)
Equation (21) agrees well with the result reported in [51]. Under the condition of δαα=, the right side of Eq. (21) reduces to unity, coinciding with the M2-factor of a coherent scalar Gaussian beam propagating in free space [49].

3. Numerical examples

In this section we study the M2-factor of an EGSM beam in free space and in turbulent atmosphere numerically. In the following examples, we choose the Tatarskii spectrum for the spectral density of the index-of-refraction fluctuations, which is expressed as [27]

Φn(κ)=0.033Cn2κ11/3exp(κ2κm2),
where Cn2 is the structure constant of the turbulent atmosphere, κm=5.92/l0 with l0 being the inner scale of the turbulence. Substituting from Eq. (22) into Eq. (18), we obtain the formula
T=0Φn(κ)κ3dκ=0.1661Cn2l01/3.
Substituting Eq. (23) into Eq. (19), we now can calculate theM2-factor of an EGSM beam numerically.

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(ρ1',ρ2';0)=(Wxx(ρ1',ρ2';0)00Wyy(ρ1',ρ2';0)).
The degree of polarization of the initial source beam at pointρ'can be expressed as follows [4048]
P0(ρ';0)=14DetW(ρ',ρ';0)[TrW(ρ',ρ';0)]2.
Under the condition of Wxx(ρ',ρ';0)=0 or Wyy(ρ',ρ';0)=0, the EGSM beam reduces to a scalar GSM beam with P0(ρ';0)=1.

In the following numerical examples, we set σx=σy=0.02m and Ax=1 unless stated otherwise. In this case, the polarization properties are uniform across the source plane with P0(r;ω)=|Ax2Ay2Ax2+Ay2|. Figure 1 shows the dependence of the degree of polarization in the source plane on Ay. It is clear from Fig. 1 that the degree of polarization in the source plane varies as the value ofAychanges, any nonzero P0 can be achieved either forAy<Ax or for Ay>Ax.

 

Fig. 1 Dependence of the degree of polarization at source plane onAywithAx=1

Download Full Size | PPT Slide | PDF

First we study the properties of the M2-factor of an EGSM beam in free space. We calculate in Fig. 2 the dependence of the M2-factor of an EGSM beam on its degree of polarization in the source plane (z = 0) for two different cases with δxx=0.004m,δyy=0.01m. One finds from Fig. 2 that the M2-factor of an EGSM beam is closely determined by its degree of polarization at z = 0. Its value increases as the degree of polarization increases for the case of Ay<Ax or decreases with increase of the degree of polarization for the case of Ay>Ax. This is caused by the fact that the contribution of the element Wxx(ρ1',ρ2';0) to the M2-factor dominates that of the element Wyy(ρ1',ρ2';0) for the case of Ay<Ax, and the contribution of the element Wyy(ρ1',ρ2';0)plays a dominant role otherwise.

 

Fig. 2 Dependence of the M2-factor of an EGSM beam in free space on its degree of polarization at source plane for two different cases (a) Ay<Ax, (b) Ay>Ax

Download Full Size | PPT Slide | PDF

Figure 3 shows the dependence of the M2-factor of an EGSM beam in free space on the r.m.s width (σx) of the spectral density along x direction with Ay<Ax, P0=0.6, σy=0.5σx, δxx=0.004m, δyy=0.01m. From Fig. 3 it is clearly seen that the value of the M2-factor of an EGSM beam in free space increases as its r.m.s. width of the spectral density increases. Figure 4 shows the dependence of the M2-factor of an EGSM beam in free space on the r.m.s width (δxx) of auto-correlation functions of the x component of the field with Ay<Ax, P0=0.6, δyy=0.5δxx, σx=0.025m,σy=0.01m. One finds from Fig. 4 that the value of the M2-factor of an EGSM beam decreases as the correlation factors δxx and δyydecrease, due to the fact that the spectral degree of coherence W(ρ1',ρ2';0)=TrW(ρ1',ρ2';0)/TrW(ρ1',ρ1';0)TrW(ρ2',ρ2';0) decreases with the decrease of the correlation factors.

 

Fig. 3 Dependence of the M2-factor of an EGSM beam in free space on the r.m.s width (σx) of the spectral density along x direction

Download Full Size | PPT Slide | PDF

 

Fig. 4 Dependence of the M2-factor of an EGSM beam in free space on the r.m.s width (δxx) of auto-correlation functions of the x component of the field

Download Full Size | PPT Slide | PDF

In what follows we study the properties of the normalizedM2-factor of an EGSM beam on propagation in turbulent atmosphere. It is clear from Eq. (19) that the M2-factor of an EGSM beam is now determined by both the parameters of the beam and of the turbulence together. We calculate in Fig. 5 the normalized M2-factor of an EGSM beam on propagation in turbulent atmosphere for different values of the initial degree of polarization with δxx=0.004m, δyy=0.01m, λ=632.8nm, Cn2=1014m2/3, l0=0.01m. As seen from Fig. 5, unlike its propagation-invariant properties in free-space, the normalized M2-factor increases on propagation in turbulent atmosphere, which means the beam quality of an EGSM beam degrades. It is clear from Fig. 5(a) that the value of the normalized M2-factor increases more rapidly as the initial degree of polarization P0 decreases for the case of Ay<Ax, which means that the beam quality of a scalar GSM beam is less affected by the atmospheric turbulence than that of an EGSM beam. From Fig. 5(b) we conclude that for the case of Ay>Ax, the beam quality of an EGSM beam with low initial degree of polarization is less affected by the atmospheric turbulence than that of an EGSM beam with large degree of polarization or that of a scalar GSM beam.

 

Fig. 5 Normalized M2-factor of an EGSM beam on propagation in turbulent atmosphere for different values of the initial degree of polarization

Download Full Size | PPT Slide | PDF

Figure 6 shows the normalized M2-factor of an EGSM beam on propagation in turbulent atmosphere for different initial r.m.s. correlations δxx and δyywithAy<Ax, P0=0.6, λ=632.8nm, Cn2=1014m2/3, l0=0.01m. One finds from Fig. 6 that the value of the normalized M2-factor increases slower on propagation as the initial correlation factors decreases, implying that an EGSM beam with low initial spectral degree of coherence is less affected by the atmospheric turbulence. Figure 7 shows the normalized M2-factor of an EGSM beam on propagation in turbulent atmosphere for different initial r.m.s. widths σxand σyof the spectral densities with Ay<Ax, P0=0.6,δxx=0.004m, δyy=0.01m, λ=632.8nm, Cn2=1014m2/3, l0=0.01m. Figure 8 shows the normalized M2-factor of an EGSM beam on propagation in turbulent atmosphere for different values of wavelengthλ with Ay<Ax, P0=0.6,σx=σy=0.02m, δxx=0.004m, δyy=0.01m, Cn2=1014m2/3, l0=0.01m. The results in Fig. 7 and Fig. 8 lead to the conclusion that the EGSM beam with larger r.m.s. width of the spectral densities and longer wavelength λ is less affected by the atmospheric turbulence.

 

Fig. 6 Normalized M2-factor of an EGSM beam on propagation in turbulent atmosphere for different initial correlation factors δxx and δyy

Download Full Size | PPT Slide | PDF

 

Fig. 7 Normalized M2-factor of an EGSM beam on propagation in turbulent atmosphere for different initial r.m.s widths σxand σyof the spectral densities

Download Full Size | PPT Slide | PDF

 

Fig. 8 Normalized M2-factor of an EGSM beam on propagation in turbulent atmosphere for different values of wavelengthλ

Download Full Size | PPT Slide | PDF

The parameters of turbulence also affect the evolution properties of the normalized M2-factor in turbulent atmosphere. We calculate in Fig. 9 the normalized M2-factor of an EGSM beam on propagation using different values of the structure constant (Cn2) of the turbulent atmosphere with Ay<Ax, P0=0.6, δxx=0.004m, δyy=0.01m, λ=632.8nmand l0=0.01m. Figure 10 shows the normalized M2-factor of an EGSM beam on propagation in turbulent atmosphere for different values of inner scale of the turbulence (l0) with Ay<Ax, P0=0.6, δxx=0.004m, δyy=0.01m, λ=632.8nmand Cn2=1014m2/3. As seen in Fig. 9 and Fig. 10, the normalized M2-factor increases more rapidly asCn2 increases or l0 decreases.

 

Fig. 9 Normalized M2-factor of an EGSM beam on propagation using different values of the structure constant (Cn2) of the turbulent atmosphere

Download Full Size | PPT Slide | PDF

 

Fig. 10 Normalized M2-factor of an EGSM beam on propagation in turbulent atmosphere for different values of inner scale of the turbulence (l0)

Download Full Size | PPT Slide | PDF

4. Conclusion

We have derived the analytical formula for the M2-factor of an EGSM beam valid for both free space and atmospheric propagation by means of the extended Huygens-Fresnel integral and the second-order moments of the Wigner distribution function. Our numerical results show that the initial degree of polarization, the r.m.s. width of the spectral densities and the r.m.s. correlation coefficients together determine the M2-factor of an EGSM beam in free space, whose value is independent of the propagation distance. In turbulent atmosphere, the parameters of the turbulence also affect the M2-factor of this class of beam on propagation. Furthermore, the EGSM beam with lower degree of polarization, lower correlation factors, larger r.m.s. widths of the spectral densities and longer wavelength is less affected by the atmospheric turbulence under suitable conditions. Our results may find applications in long-distance free-space optical communications.

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 and the Huo Ying Dong Education Foundation of China under Grant No. 121009. O. Korotkova's research is funded by the US AFOSR (grant FA 95500810102) and US ONR (grant N0018909P1903).

References and links

1. D. F. V. James, “Changes of polarization of light beams on propagation in free space,” J. Opt. Soc. Am. A 11(5), 1641–1643 (1994). [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. G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1-3), 9–16 (2002). [CrossRef]  

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

5. E. Wolf and E. Collett, “Partially coherent sources which produce same far-field intensity distribution as a laser,”Opt. Commun . 25(3), 293–296 (1978). [CrossRef]  

6. F. Gori, “Collet-Wolf sources and multimode lasers,” Opt. Commun. 34(3), 301–305 (1980). [CrossRef]  

7. A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982). [CrossRef]  

8. F. Wang and Y. Cai, “Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics,” J. Opt. Soc. Am. A 24(7), 1937–1944 (2007). [CrossRef]  

9. E. Wolf, Introduction to the theory of coherence and polarization of light (Cambridge U. Press, 2007).

10. F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23(4), 241–243 (1998). [CrossRef]  

11. G. P. Agrawal and E. Wolf, “Propagation-induced polarization changes in partially coherent optical beams,” J. Opt. Soc. Am. A 17(11), 2019–2023 (2000). [CrossRef]  

12. Y. Cai, D. Ge, and Q. Lin, “Fractional Fourier transform for partially coherent and partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 5(5), 453–459 (2003). [CrossRef]  

13. 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]  

14. F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 25(5), 1016–1021 (2008). [CrossRef]  

15. B. Kanseri and H. C. Kandpal, “Experimental determination of electric cross-spectral density matrix and generalized Stokes parameters for a laser beam,” Opt. Lett. 33(20), 2410–2412 (2008). [CrossRef]   [PubMed]  

16. H. Wang, X. Wang, A. Zeng, and K. Yang, “Effects of coherence on anisotropic electromagnetic Gaussian-Schell model beams on propagation,” Opt. Lett. 32(15), 2215–2217 (2007). [CrossRef]   [PubMed]  

17. D. Ge, Y. Cai, and Q. Lin, “Propagation of partially polarized Gaussian Schell-model beams through aligned and misaligned optical systems,” Chin. Phys. 14(1), 128–132 (2005). [CrossRef]  

18. 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]  

19. 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]  

20. O. Korotkova, M. Yao, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “The state of polarization of a stochastic electromagnetic beam in an optical resonator,” J. Opt. Soc. Am. A 25(11), 2710–2720 (2008). [CrossRef]  

21. Z. Tong, O. Korotkova, Y. Cai, H. T. Eyyuboglu, and Y. Baykal, “Correlation properties of random electromagnetic beams in laser resonators,” Appl. Phys. B 97(4), 849–857 (2009). [CrossRef]  

22. Y. Cai and O. Korotkova, “Twist phase-induced polarization changes in electromagnetic Gaussian Schell-model beams,” Appl. Phys. B 96(2-3), 499–507 (2009). [CrossRef]  

23. 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]  

24. S. Zhu and Y. Cai, “Spectral shift of a twisted electromagnetic Gaussian Schell-model beam focused by a thin lens,” Appl. Phys. B 99(1-2), 317–323 (2010). [CrossRef]  

25. A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 2, (Academic Press, New York, 1978).

26. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, Washington, 2001).

27. L. C. Andrews, and R. L. Phillips, Laser beam propagation in the turbulent atmosphere, 2nd edition, SPIE press, Bellington, 2005.

28. J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A 19(9), 1794–1802 (2002). [CrossRef]  

29. H. T. Eyyuboğlu and Y. Baykal, “Analysis of reciprocity of cos-Gaussian and cosh- Gaussian laser beams in a turbulent atmosphere,” Opt. Express 12(20), 4659–4674 (2004). [CrossRef]   [PubMed]  

30. H. T. Eyyuboğlu, C. Arpali, and Y. K. Baykal, “Flat topped beams and their characteristics in turbulent media,” Opt. Express 14(10), 4196–4207 (2006). [CrossRef]   [PubMed]  

31. Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006). [CrossRef]  

32. R. J. Noriega-Manez and J. C. Gutiérrez-Vega, “Rytov theory for Helmholtz-Gauss beams in turbulent atmosphere,” Opt. Express 15(25), 16328–16341 (2007). [CrossRef]   [PubMed]  

33. M. Alavinejad, B. Ghafary, and F. D. Kashani, “Analysis of the propagation of flat-topped beam with various beam orders through turbulent atmosphere,” Opt. Lasers Eng. 46(1), 1–5 (2008). [CrossRef]  

34. Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14(4), 1353–1367 (2006). [CrossRef]   [PubMed]  

35. Y. Cai and S. He, “Average intensity and spreading of an elliptical gaussian beam propagating in a turbulent atmosphere,” Opt. Lett. 31(5), 568–570 (2006). [CrossRef]   [PubMed]  

36. Y. Cai, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation index of elliptical Gaussian beam in turbulent atmosphere,” Opt. Lett. 32(16), 2405–2407 (2007). [CrossRef]   [PubMed]  

37. H. Lin and J. Pu, “Propagation properties of partially coherent radially polarized beam in a turbulent atmosphere,” J. Mod. Opt. 56(11), 1296–1303 (2009). [CrossRef]  

38. Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express 16(11), 7665–7673 (2008). [CrossRef]   [PubMed]  

39. Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, and Y. Baykal, “Average intensity and spreading of an elegant Hermite-Gaussian beam in turbulent atmosphere,” Opt. Express 17(13), 11130–11139 (2009). [CrossRef]   [PubMed]  

40. O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233(4-6), 225–230 (2004). [CrossRef]  

41. H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. 52(11), 1611–1618 (2005). [CrossRef]  

42. O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere,” Waves Random Complex Media 15(3), 353–364 (2005). [CrossRef]  

43. Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagnetic beams in a turbulent atmosphere,” J. Opt. A, Pure Appl. Opt. 9(12), 1123–1130 (2007). [CrossRef]  

44. H. Wang, X. Wang, A. Zeng, and W. Lu, “Changes in the coherence of quasi-monochromatic electromagnetic Gaussian Schell-model beams propagating through turbulent atmosphere,” Opt. Commun. 276(2), 218–221 (2007). [CrossRef]  

45. H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Degree of polarization for partially coherent general beams in turbulent atmosphere,” Appl. Phys. B 89(1), 91–97 (2007). [CrossRef]  

46. 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]  

47. O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281(9), 2342–2348 (2008).

48. O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for radar systems operating through turbulent atmosphere,” Appl. Phys. B 94(4), 681–690 (2009). [CrossRef]  

49. A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE 1224, 2C14 (1990).

50. 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]  

51. 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]  

52. 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]  

53. X. Ji, T. Zhang, and X. Jia, “Beam propagation factor of partially coherent Hermite–Gaussian array beams,” J. Opt. A, Pure Appl. Opt. 11(10), 105705 (2009). [CrossRef]  

54. G. Zhou, “Generalzied M2 factors of truncated partially coherent Lorentz and Lorentz-Gauss beams,” J. Opt. A, Pure Appl. Opt. 12(1), 015701 (2010).

55. 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]  

56. Y. Dan and B. Zhang, “Second moments of partially coherent beams in atmospheric turbulence,” Opt. Lett. 34(5), 563–565 (2009). [CrossRef]   [PubMed]  

57. 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]  

References

  • View by:
  • |
  • |
  • |

  1. D. F. V. James, “Changes of polarization of light beams on propagation in free space,” J. Opt. Soc. Am. A 11(5), 1641–1643 (1994).
    [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. G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1-3), 9–16 (2002).
    [Crossref]
  4. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003).
    [Crossref]
  5. E. Wolf and E. Collett, “Partially coherent sources which produce same far-field intensity distribution as a laser,”Opt. Commun.  25(3), 293–296 (1978).
    [Crossref]
  6. F. Gori, “Collet-Wolf sources and multimode lasers,” Opt. Commun. 34(3), 301–305 (1980).
    [Crossref]
  7. A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982).
    [Crossref]
  8. F. Wang and Y. Cai, “Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics,” J. Opt. Soc. Am. A 24(7), 1937–1944 (2007).
    [Crossref]
  9. E. Wolf, Introduction to the theory of coherence and polarization of light (Cambridge U. Press, 2007).
  10. F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23(4), 241–243 (1998).
    [Crossref]
  11. G. P. Agrawal and E. Wolf, “Propagation-induced polarization changes in partially coherent optical beams,” J. Opt. Soc. Am. A 17(11), 2019–2023 (2000).
    [Crossref]
  12. Y. Cai, D. Ge, and Q. Lin, “Fractional Fourier transform for partially coherent and partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 5(5), 453–459 (2003).
    [Crossref]
  13. 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]
  14. F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 25(5), 1016–1021 (2008).
    [Crossref]
  15. B. Kanseri and H. C. Kandpal, “Experimental determination of electric cross-spectral density matrix and generalized Stokes parameters for a laser beam,” Opt. Lett. 33(20), 2410–2412 (2008).
    [Crossref] [PubMed]
  16. H. Wang, X. Wang, A. Zeng, and K. Yang, “Effects of coherence on anisotropic electromagnetic Gaussian-Schell model beams on propagation,” Opt. Lett. 32(15), 2215–2217 (2007).
    [Crossref] [PubMed]
  17. D. Ge, Y. Cai, and Q. Lin, “Propagation of partially polarized Gaussian Schell-model beams through aligned and misaligned optical systems,” Chin. Phys. 14(1), 128–132 (2005).
    [Crossref]
  18. 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]
  19. 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]
  20. O. Korotkova, M. Yao, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “The state of polarization of a stochastic electromagnetic beam in an optical resonator,” J. Opt. Soc. Am. A 25(11), 2710–2720 (2008).
    [Crossref]
  21. Z. Tong, O. Korotkova, Y. Cai, H. T. Eyyuboglu, and Y. Baykal, “Correlation properties of random electromagnetic beams in laser resonators,” Appl. Phys. B 97(4), 849–857 (2009).
    [Crossref]
  22. Y. Cai and O. Korotkova, “Twist phase-induced polarization changes in electromagnetic Gaussian Schell-model beams,” Appl. Phys. B 96(2-3), 499–507 (2009).
    [Crossref]
  23. 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]
  24. S. Zhu and Y. Cai, “Spectral shift of a twisted electromagnetic Gaussian Schell-model beam focused by a thin lens,” Appl. Phys. B 99(1-2), 317–323 (2010).
    [Crossref]
  25. A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 2, (Academic Press, New York, 1978).
  26. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, Washington, 2001).
  27. L. C. Andrews, and R. L. Phillips, Laser beam propagation in the turbulent atmosphere, 2nd edition, SPIE press, Bellington, 2005.
  28. J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A 19(9), 1794–1802 (2002).
    [Crossref]
  29. H. T. Eyyuboğlu and Y. Baykal, “Analysis of reciprocity of cos-Gaussian and cosh- Gaussian laser beams in a turbulent atmosphere,” Opt. Express 12(20), 4659–4674 (2004).
    [Crossref] [PubMed]
  30. H. T. Eyyuboğlu, C. Arpali, and Y. K. Baykal, “Flat topped beams and their characteristics in turbulent media,” Opt. Express 14(10), 4196–4207 (2006).
    [Crossref] [PubMed]
  31. Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006).
    [Crossref]
  32. R. J. Noriega-Manez and J. C. Gutiérrez-Vega, “Rytov theory for Helmholtz-Gauss beams in turbulent atmosphere,” Opt. Express 15(25), 16328–16341 (2007).
    [Crossref] [PubMed]
  33. M. Alavinejad, B. Ghafary, and F. D. Kashani, “Analysis of the propagation of flat-topped beam with various beam orders through turbulent atmosphere,” Opt. Lasers Eng. 46(1), 1–5 (2008).
    [Crossref]
  34. Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14(4), 1353–1367 (2006).
    [Crossref] [PubMed]
  35. Y. Cai and S. He, “Average intensity and spreading of an elliptical gaussian beam propagating in a turbulent atmosphere,” Opt. Lett. 31(5), 568–570 (2006).
    [Crossref] [PubMed]
  36. Y. Cai, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation index of elliptical Gaussian beam in turbulent atmosphere,” Opt. Lett. 32(16), 2405–2407 (2007).
    [Crossref] [PubMed]
  37. H. Lin and J. Pu, “Propagation properties of partially coherent radially polarized beam in a turbulent atmosphere,” J. Mod. Opt. 56(11), 1296–1303 (2009).
    [Crossref]
  38. Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express 16(11), 7665–7673 (2008).
    [Crossref] [PubMed]
  39. Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, and Y. Baykal, “Average intensity and spreading of an elegant Hermite-Gaussian beam in turbulent atmosphere,” Opt. Express 17(13), 11130–11139 (2009).
    [Crossref] [PubMed]
  40. O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233(4-6), 225–230 (2004).
    [Crossref]
  41. H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. 52(11), 1611–1618 (2005).
    [Crossref]
  42. O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere,” Waves Random Complex Media 15(3), 353–364 (2005).
    [Crossref]
  43. Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagnetic beams in a turbulent atmosphere,” J. Opt. A, Pure Appl. Opt. 9(12), 1123–1130 (2007).
    [Crossref]
  44. H. Wang, X. Wang, A. Zeng, and W. Lu, “Changes in the coherence of quasi-monochromatic electromagnetic Gaussian Schell-model beams propagating through turbulent atmosphere,” Opt. Commun. 276(2), 218–221 (2007).
    [Crossref]
  45. H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Degree of polarization for partially coherent general beams in turbulent atmosphere,” Appl. Phys. B 89(1), 91–97 (2007).
    [Crossref]
  46. 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]
  47. O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281(9), 2342–2348 (2008).
  48. O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for radar systems operating through turbulent atmosphere,” Appl. Phys. B 94(4), 681–690 (2009).
    [Crossref]
  49. A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE 1224, 2C14 (1990).
  50. 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]
  51. 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]
  52. 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]
  53. X. Ji, T. Zhang, and X. Jia, “Beam propagation factor of partially coherent Hermite–Gaussian array beams,” J. Opt. A, Pure Appl. Opt. 11(10), 105705 (2009).
    [Crossref]
  54. G. Zhou, “Generalzied M2 factors of truncated partially coherent Lorentz and Lorentz-Gauss beams,” J. Opt. A, Pure Appl. Opt. 12(1), 015701 (2010).
  55. 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]
  56. Y. Dan and B. Zhang, “Second moments of partially coherent beams in atmospheric turbulence,” Opt. Lett. 34(5), 563–565 (2009).
    [Crossref] [PubMed]
  57. 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]

2010 (2)

S. Zhu and Y. Cai, “Spectral shift of a twisted electromagnetic Gaussian Schell-model beam focused by a thin lens,” Appl. Phys. B 99(1-2), 317–323 (2010).
[Crossref]

G. Zhou, “Generalzied M2 factors of truncated partially coherent Lorentz and Lorentz-Gauss beams,” J. Opt. A, Pure Appl. Opt. 12(1), 015701 (2010).

2009 (9)

Y. Dan and B. Zhang, “Second moments of partially coherent beams in atmospheric turbulence,” Opt. Lett. 34(5), 563–565 (2009).
[Crossref] [PubMed]

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]

X. Ji, T. Zhang, and X. Jia, “Beam propagation factor of partially coherent Hermite–Gaussian array beams,” J. Opt. A, Pure Appl. Opt. 11(10), 105705 (2009).
[Crossref]

O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for radar systems operating through turbulent atmosphere,” Appl. Phys. B 94(4), 681–690 (2009).
[Crossref]

Z. Tong, O. Korotkova, Y. Cai, H. T. Eyyuboglu, and Y. Baykal, “Correlation properties of random electromagnetic beams in laser resonators,” Appl. Phys. B 97(4), 849–857 (2009).
[Crossref]

Y. Cai and O. Korotkova, “Twist phase-induced polarization changes in electromagnetic Gaussian Schell-model beams,” Appl. Phys. B 96(2-3), 499–507 (2009).
[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]

H. Lin and J. Pu, “Propagation properties of partially coherent radially polarized beam in a turbulent atmosphere,” J. Mod. Opt. 56(11), 1296–1303 (2009).
[Crossref]

Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, and Y. Baykal, “Average intensity and spreading of an elegant Hermite-Gaussian beam in turbulent atmosphere,” Opt. Express 17(13), 11130–11139 (2009).
[Crossref] [PubMed]

2008 (9)

Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express 16(11), 7665–7673 (2008).
[Crossref] [PubMed]

M. Alavinejad, B. Ghafary, and F. D. Kashani, “Analysis of the propagation of flat-topped beam with various beam orders through turbulent atmosphere,” Opt. Lasers Eng. 46(1), 1–5 (2008).
[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, “The state of polarization of a stochastic electromagnetic beam in an optical resonator,” J. Opt. Soc. Am. A 25(11), 2710–2720 (2008).
[Crossref]

F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 25(5), 1016–1021 (2008).
[Crossref]

B. Kanseri and H. C. Kandpal, “Experimental determination of electric cross-spectral density matrix and generalized Stokes parameters for a laser beam,” Opt. Lett. 33(20), 2410–2412 (2008).
[Crossref] [PubMed]

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]

O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281(9), 2342–2348 (2008).

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]

2007 (7)

Y. Cai, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation index of elliptical Gaussian beam in turbulent atmosphere,” Opt. Lett. 32(16), 2405–2407 (2007).
[Crossref] [PubMed]

Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagnetic beams in a turbulent atmosphere,” J. Opt. A, Pure Appl. Opt. 9(12), 1123–1130 (2007).
[Crossref]

H. Wang, X. Wang, A. Zeng, and W. Lu, “Changes in the coherence of quasi-monochromatic electromagnetic Gaussian Schell-model beams propagating through turbulent atmosphere,” Opt. Commun. 276(2), 218–221 (2007).
[Crossref]

H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Degree of polarization for partially coherent general beams in turbulent atmosphere,” Appl. Phys. B 89(1), 91–97 (2007).
[Crossref]

H. Wang, X. Wang, A. Zeng, and K. Yang, “Effects of coherence on anisotropic electromagnetic Gaussian-Schell model beams on propagation,” Opt. Lett. 32(15), 2215–2217 (2007).
[Crossref] [PubMed]

F. Wang and Y. Cai, “Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics,” J. Opt. Soc. Am. A 24(7), 1937–1944 (2007).
[Crossref]

R. J. Noriega-Manez and J. C. Gutiérrez-Vega, “Rytov theory for Helmholtz-Gauss beams in turbulent atmosphere,” Opt. Express 15(25), 16328–16341 (2007).
[Crossref] [PubMed]

2006 (4)

2005 (4)

D. Ge, Y. Cai, and Q. Lin, “Propagation of partially polarized Gaussian Schell-model beams through aligned and misaligned optical systems,” Chin. Phys. 14(1), 128–132 (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]

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. 52(11), 1611–1618 (2005).
[Crossref]

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere,” Waves Random Complex Media 15(3), 353–364 (2005).
[Crossref]

2004 (3)

2003 (2)

Y. Cai, D. Ge, and Q. Lin, “Fractional Fourier transform for partially coherent and partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 5(5), 453–459 (2003).
[Crossref]

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

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]

2000 (1)

1999 (1)

1998 (1)

1994 (1)

1993 (1)

1982 (1)

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982).
[Crossref]

1980 (1)

F. Gori, “Collet-Wolf sources and multimode lasers,” Opt. Commun. 34(3), 301–305 (1980).
[Crossref]

1978 (1)

E. Wolf and E. Collett, “Partially coherent sources which produce same far-field intensity distribution as a laser,”Opt. Commun.  25(3), 293–296 (1978).
[Crossref]

Agrawal, G. P.

Alavinejad, M.

M. Alavinejad, B. Ghafary, and F. D. Kashani, “Analysis of the propagation of flat-topped beam with various beam orders through turbulent atmosphere,” Opt. Lasers Eng. 46(1), 1–5 (2008).
[Crossref]

Arpali, C.

Baykal, Y.

Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, and Y. Baykal, “Average intensity and spreading of an elegant Hermite-Gaussian beam in turbulent atmosphere,” Opt. Express 17(13), 11130–11139 (2009).
[Crossref] [PubMed]

Z. Tong, O. Korotkova, Y. Cai, H. T. Eyyuboglu, 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]

O. Korotkova, M. Yao, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “The state of polarization of a stochastic electromagnetic beam in an optical resonator,” J. Opt. Soc. Am. A 25(11), 2710–2720 (2008).
[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]

Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express 16(11), 7665–7673 (2008).
[Crossref] [PubMed]

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]

H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Degree of polarization for partially coherent general beams in turbulent atmosphere,” Appl. Phys. B 89(1), 91–97 (2007).
[Crossref]

Y. Cai, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation index of elliptical Gaussian beam in turbulent atmosphere,” Opt. Lett. 32(16), 2405–2407 (2007).
[Crossref] [PubMed]

H. T. Eyyuboğlu and Y. Baykal, “Analysis of reciprocity of cos-Gaussian and cosh- Gaussian laser beams in a turbulent atmosphere,” Opt. Express 12(20), 4659–4674 (2004).
[Crossref] [PubMed]

Baykal, Y. K.

Borghi, R.

F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 25(5), 1016–1021 (2008).
[Crossref]

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1-3), 9–16 (2002).
[Crossref]

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]

Cai, Y.

S. Zhu and Y. Cai, “Spectral shift of a twisted electromagnetic Gaussian Schell-model beam focused by a thin lens,” Appl. Phys. B 99(1-2), 317–323 (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]

Y. Cai and O. Korotkova, “Twist phase-induced polarization changes in electromagnetic Gaussian Schell-model beams,” Appl. Phys. B 96(2-3), 499–507 (2009).
[Crossref]

Z. Tong, O. Korotkova, Y. Cai, H. T. Eyyuboglu, and Y. Baykal, “Correlation properties of random electromagnetic beams in laser resonators,” Appl. Phys. B 97(4), 849–857 (2009).
[Crossref]

O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for radar systems operating through turbulent atmosphere,” Appl. Phys. B 94(4), 681–690 (2009).
[Crossref]

Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, and Y. Baykal, “Average intensity and spreading of an elegant Hermite-Gaussian beam in turbulent atmosphere,” Opt. Express 17(13), 11130–11139 (2009).
[Crossref] [PubMed]

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]

Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express 16(11), 7665–7673 (2008).
[Crossref] [PubMed]

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]

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, “The state of polarization of a stochastic electromagnetic beam in an optical resonator,” J. Opt. Soc. Am. A 25(11), 2710–2720 (2008).
[Crossref]

F. Wang and Y. Cai, “Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics,” J. Opt. Soc. Am. A 24(7), 1937–1944 (2007).
[Crossref]

H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Degree of polarization for partially coherent general beams in turbulent atmosphere,” Appl. Phys. B 89(1), 91–97 (2007).
[Crossref]

Y. Cai, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation index of elliptical Gaussian beam in turbulent atmosphere,” Opt. Lett. 32(16), 2405–2407 (2007).
[Crossref] [PubMed]

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006).
[Crossref]

Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14(4), 1353–1367 (2006).
[Crossref] [PubMed]

Y. Cai and S. He, “Average intensity and spreading of an elliptical gaussian beam propagating in a turbulent atmosphere,” Opt. Lett. 31(5), 568–570 (2006).
[Crossref] [PubMed]

D. Ge, Y. Cai, and Q. Lin, “Propagation of partially polarized Gaussian Schell-model beams through aligned and misaligned optical systems,” Chin. Phys. 14(1), 128–132 (2005).
[Crossref]

Y. Cai, D. Ge, and Q. Lin, “Fractional Fourier transform for partially coherent and partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 5(5), 453–459 (2003).
[Crossref]

Chen, Y.

Chen, Z.

Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagnetic beams in a turbulent atmosphere,” J. Opt. A, Pure Appl. Opt. 9(12), 1123–1130 (2007).
[Crossref]

Chu, X.

Cincotti, G.

Collett, E.

E. Wolf and E. Collett, “Partially coherent sources which produce same far-field intensity distribution as a laser,”Opt. Commun.  25(3), 293–296 (1978).
[Crossref]

Dan, Y.

Davidson, F. M.

Dogariu, A.

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere,” Waves Random Complex Media 15(3), 353–364 (2005).
[Crossref]

Eyyuboglu, H. T.

Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, and Y. Baykal, “Average intensity and spreading of an elegant Hermite-Gaussian beam in turbulent atmosphere,” Opt. Express 17(13), 11130–11139 (2009).
[Crossref] [PubMed]

Z. Tong, O. Korotkova, Y. Cai, H. T. Eyyuboglu, 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]

O. Korotkova, M. Yao, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “The state of polarization of a stochastic electromagnetic beam in an optical resonator,” J. Opt. Soc. Am. A 25(11), 2710–2720 (2008).
[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]

Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express 16(11), 7665–7673 (2008).
[Crossref] [PubMed]

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]

H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Degree of polarization for partially coherent general beams in turbulent atmosphere,” Appl. Phys. B 89(1), 91–97 (2007).
[Crossref]

Y. Cai, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation index of elliptical Gaussian beam in turbulent atmosphere,” Opt. Lett. 32(16), 2405–2407 (2007).
[Crossref] [PubMed]

H. T. Eyyuboğlu, C. Arpali, and Y. K. Baykal, “Flat topped beams and their characteristics in turbulent media,” Opt. Express 14(10), 4196–4207 (2006).
[Crossref] [PubMed]

H. T. Eyyuboğlu and Y. Baykal, “Analysis of reciprocity of cos-Gaussian and cosh- Gaussian laser beams in a turbulent atmosphere,” Opt. Express 12(20), 4659–4674 (2004).
[Crossref] [PubMed]

Friberg, A. T.

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982).
[Crossref]

Ge, D.

D. Ge, Y. Cai, and Q. Lin, “Propagation of partially polarized Gaussian Schell-model beams through aligned and misaligned optical systems,” Chin. Phys. 14(1), 128–132 (2005).
[Crossref]

Y. Cai, D. Ge, and Q. Lin, “Fractional Fourier transform for partially coherent and partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 5(5), 453–459 (2003).
[Crossref]

Ghafary, B.

M. Alavinejad, B. Ghafary, and F. D. Kashani, “Analysis of the propagation of flat-topped beam with various beam orders through turbulent atmosphere,” Opt. Lasers Eng. 46(1), 1–5 (2008).
[Crossref]

Gori, F.

F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 25(5), 1016–1021 (2008).
[Crossref]

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1-3), 9–16 (2002).
[Crossref]

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, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23(4), 241–243 (1998).
[Crossref]

F. Gori, “Collet-Wolf sources and multimode lasers,” Opt. Commun. 34(3), 301–305 (1980).
[Crossref]

Gutiérrez-Vega, J. C.

He, S.

James, D. F. V.

Ji, X.

X. Ji, T. Zhang, and X. Jia, “Beam propagation factor of partially coherent Hermite–Gaussian array beams,” J. Opt. A, Pure Appl. Opt. 11(10), 105705 (2009).
[Crossref]

Jia, X.

X. Ji, T. Zhang, and X. Jia, “Beam propagation factor of partially coherent Hermite–Gaussian array beams,” J. Opt. A, Pure Appl. Opt. 11(10), 105705 (2009).
[Crossref]

Kandpal, H. C.

Kanseri, B.

Kashani, F. D.

M. Alavinejad, B. Ghafary, and F. D. Kashani, “Analysis of the propagation of flat-topped beam with various beam orders through turbulent atmosphere,” Opt. Lasers Eng. 46(1), 1–5 (2008).
[Crossref]

Korotkova, O.

Z. Tong, O. Korotkova, Y. Cai, H. T. Eyyuboglu, and Y. Baykal, “Correlation properties of random electromagnetic beams in laser resonators,” Appl. Phys. B 97(4), 849–857 (2009).
[Crossref]

O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for radar systems operating through turbulent atmosphere,” Appl. Phys. B 94(4), 681–690 (2009).
[Crossref]

Y. Cai and O. Korotkova, “Twist phase-induced polarization changes in electromagnetic Gaussian Schell-model beams,” Appl. Phys. B 96(2-3), 499–507 (2009).
[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]

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, “The 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(9), 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]

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere,” Waves Random Complex Media 15(3), 353–364 (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]

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]

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233(4-6), 225–230 (2004).
[Crossref]

Li, Q.

Lin, H.

H. Lin and J. Pu, “Propagation properties of partially coherent radially polarized beam in a turbulent atmosphere,” J. Mod. Opt. 56(11), 1296–1303 (2009).
[Crossref]

Lin, Q.

Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express 16(11), 7665–7673 (2008).
[Crossref] [PubMed]

D. Ge, Y. Cai, and Q. Lin, “Propagation of partially polarized Gaussian Schell-model beams through aligned and misaligned optical systems,” Chin. Phys. 14(1), 128–132 (2005).
[Crossref]

Y. Cai, D. Ge, and Q. Lin, “Fractional Fourier transform for partially coherent and partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 5(5), 453–459 (2003).
[Crossref]

Lu, W.

H. Wang, X. Wang, A. Zeng, and W. Lu, “Changes in the coherence of quasi-monochromatic electromagnetic Gaussian Schell-model beams propagating through turbulent atmosphere,” Opt. Commun. 276(2), 218–221 (2007).
[Crossref]

Martinez-Herrero, R.

Mejias, P. M.

Mondello, A.

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1-3), 9–16 (2002).
[Crossref]

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]

Noriega-Manez, R. J.

Piquero, G.

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1-3), 9–16 (2002).
[Crossref]

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]

Ponomarenko, S. A.

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. 52(11), 1611–1618 (2005).
[Crossref]

Pu, J.

H. Lin and J. Pu, “Propagation properties of partially coherent radially polarized beam in a turbulent atmosphere,” J. Mod. Opt. 56(11), 1296–1303 (2009).
[Crossref]

Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagnetic beams in a turbulent atmosphere,” J. Opt. A, Pure Appl. Opt. 9(12), 1123–1130 (2007).
[Crossref]

Qu, J.

Ramírez-Sánchez, V.

Ricklin, J. C.

Romanini, P.

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1-3), 9–16 (2002).
[Crossref]

Roychowdhury, H.

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. 52(11), 1611–1618 (2005).
[Crossref]

Salem, M.

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere,” Waves Random Complex Media 15(3), 353–364 (2005).
[Crossref]

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233(4-6), 225–230 (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]

Santarsiero, M.

F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 25(5), 1016–1021 (2008).
[Crossref]

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1-3), 9–16 (2002).
[Crossref]

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]

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]

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]

Sudol, R. J.

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982).
[Crossref]

Tong, Z.

Z. Tong, O. Korotkova, Y. Cai, H. T. Eyyuboglu, and Y. Baykal, “Correlation properties of random electromagnetic beams in laser resonators,” Appl. Phys. B 97(4), 849–857 (2009).
[Crossref]

Vahimaa, P.

Wang, F.

Wang, H.

H. Wang, X. Wang, A. Zeng, and K. Yang, “Effects of coherence on anisotropic electromagnetic Gaussian-Schell model beams on propagation,” Opt. Lett. 32(15), 2215–2217 (2007).
[Crossref] [PubMed]

H. Wang, X. Wang, A. Zeng, and W. Lu, “Changes in the coherence of quasi-monochromatic electromagnetic Gaussian Schell-model beams propagating through turbulent atmosphere,” Opt. Commun. 276(2), 218–221 (2007).
[Crossref]

Wang, X.

H. Wang, X. Wang, A. Zeng, and W. Lu, “Changes in the coherence of quasi-monochromatic electromagnetic Gaussian Schell-model beams propagating through turbulent atmosphere,” Opt. Commun. 276(2), 218–221 (2007).
[Crossref]

H. Wang, X. Wang, A. Zeng, and K. Yang, “Effects of coherence on anisotropic electromagnetic Gaussian-Schell model beams on propagation,” Opt. Lett. 32(15), 2215–2217 (2007).
[Crossref] [PubMed]

Watson, E.

O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for radar systems operating through turbulent atmosphere,” Appl. Phys. B 94(4), 681–690 (2009).
[Crossref]

Wolf, E.

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. 52(11), 1611–1618 (2005).
[Crossref]

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere,” Waves Random Complex Media 15(3), 353–364 (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]

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]

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233(4-6), 225–230 (2004).
[Crossref]

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

G. P. Agrawal and E. Wolf, “Propagation-induced polarization changes in partially coherent optical beams,” J. Opt. Soc. Am. A 17(11), 2019–2023 (2000).
[Crossref]

E. Wolf and E. Collett, “Partially coherent sources which produce same far-field intensity distribution as a laser,”Opt. Commun.  25(3), 293–296 (1978).
[Crossref]

Yang, K.

Yao, M.

Yuan, Y.

Zeng, A.

H. Wang, X. Wang, A. Zeng, and W. Lu, “Changes in the coherence of quasi-monochromatic electromagnetic Gaussian Schell-model beams propagating through turbulent atmosphere,” Opt. Commun. 276(2), 218–221 (2007).
[Crossref]

H. Wang, X. Wang, A. Zeng, and K. Yang, “Effects of coherence on anisotropic electromagnetic Gaussian-Schell model beams on propagation,” Opt. Lett. 32(15), 2215–2217 (2007).
[Crossref] [PubMed]

Zhang, B.

Zhang, T.

X. Ji, T. Zhang, and X. Jia, “Beam propagation factor of partially coherent Hermite–Gaussian array beams,” J. Opt. A, Pure Appl. Opt. 11(10), 105705 (2009).
[Crossref]

Zhao, C.

Zhou, G.

G. Zhou, “Generalzied M2 factors of truncated partially coherent Lorentz and Lorentz-Gauss beams,” J. Opt. A, Pure Appl. Opt. 12(1), 015701 (2010).

Zhu, S.

S. Zhu and Y. Cai, “Spectral shift of a twisted electromagnetic Gaussian Schell-model beam focused by a thin lens,” Appl. Phys. B 99(1-2), 317–323 (2010).
[Crossref]

Appl. Phys. B (5)

S. Zhu and Y. Cai, “Spectral shift of a twisted electromagnetic Gaussian Schell-model beam focused by a thin lens,” Appl. Phys. B 99(1-2), 317–323 (2010).
[Crossref]

Z. Tong, O. Korotkova, Y. Cai, H. T. Eyyuboglu, and Y. Baykal, “Correlation properties of random electromagnetic beams in laser resonators,” Appl. Phys. B 97(4), 849–857 (2009).
[Crossref]

Y. Cai and O. Korotkova, “Twist phase-induced polarization changes in electromagnetic Gaussian Schell-model beams,” Appl. Phys. B 96(2-3), 499–507 (2009).
[Crossref]

H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Degree of polarization for partially coherent general beams in turbulent atmosphere,” Appl. Phys. B 89(1), 91–97 (2007).
[Crossref]

O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for radar systems operating through turbulent atmosphere,” Appl. Phys. B 94(4), 681–690 (2009).
[Crossref]

Appl. Phys. Lett. (1)

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006).
[Crossref]

Chin. Phys. (1)

D. Ge, Y. Cai, and Q. Lin, “Propagation of partially polarized Gaussian Schell-model beams through aligned and misaligned optical systems,” Chin. Phys. 14(1), 128–132 (2005).
[Crossref]

J. Mod. Opt. (2)

H. Lin and J. Pu, “Propagation properties of partially coherent radially polarized beam in a turbulent atmosphere,” J. Mod. Opt. 56(11), 1296–1303 (2009).
[Crossref]

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. 52(11), 1611–1618 (2005).
[Crossref]

J. Opt. A, Pure Appl. Opt. (6)

Y. Cai, D. Ge, and Q. Lin, “Fractional Fourier transform for partially coherent and partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 5(5), 453–459 (2003).
[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]

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]

Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagnetic beams in a turbulent atmosphere,” J. Opt. A, Pure Appl. Opt. 9(12), 1123–1130 (2007).
[Crossref]

X. Ji, T. Zhang, and X. Jia, “Beam propagation factor of partially coherent Hermite–Gaussian array beams,” J. Opt. A, Pure Appl. Opt. 11(10), 105705 (2009).
[Crossref]

G. Zhou, “Generalzied M2 factors of truncated partially coherent Lorentz and Lorentz-Gauss beams,” J. Opt. A, Pure Appl. Opt. 12(1), 015701 (2010).

J. Opt. Soc. Am. A (8)

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]

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]

F. Wang and Y. Cai, “Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics,” J. Opt. Soc. Am. A 24(7), 1937–1944 (2007).
[Crossref]

D. F. V. James, “Changes of polarization of light beams on propagation in free space,” J. Opt. Soc. Am. A 11(5), 1641–1643 (1994).
[Crossref]

F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 25(5), 1016–1021 (2008).
[Crossref]

G. P. Agrawal and E. Wolf, “Propagation-induced polarization changes in partially coherent optical beams,” J. Opt. Soc. Am. A 17(11), 2019–2023 (2000).
[Crossref]

O. Korotkova, M. Yao, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “The state of polarization of a stochastic electromagnetic beam in an optical resonator,” J. Opt. Soc. Am. A 25(11), 2710–2720 (2008).
[Crossref]

J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A 19(9), 1794–1802 (2002).
[Crossref]

Opt. Commun (1)

E. Wolf and E. Collett, “Partially coherent sources which produce same far-field intensity distribution as a laser,”Opt. Commun.  25(3), 293–296 (1978).
[Crossref]

Opt. Commun. (6)

F. Gori, “Collet-Wolf sources and multimode lasers,” Opt. Commun. 34(3), 301–305 (1980).
[Crossref]

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982).
[Crossref]

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1-3), 9–16 (2002).
[Crossref]

H. Wang, X. Wang, A. Zeng, and W. Lu, “Changes in the coherence of quasi-monochromatic electromagnetic Gaussian Schell-model beams propagating through turbulent atmosphere,” Opt. Commun. 276(2), 218–221 (2007).
[Crossref]

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233(4-6), 225–230 (2004).
[Crossref]

O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281(9), 2342–2348 (2008).

Opt. Express (10)

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]

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]

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]

H. T. Eyyuboğlu and Y. Baykal, “Analysis of reciprocity of cos-Gaussian and cosh- Gaussian laser beams in a turbulent atmosphere,” Opt. Express 12(20), 4659–4674 (2004).
[Crossref] [PubMed]

H. T. Eyyuboğlu, C. Arpali, and Y. K. Baykal, “Flat topped beams and their characteristics in turbulent media,” Opt. Express 14(10), 4196–4207 (2006).
[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]

Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express 16(11), 7665–7673 (2008).
[Crossref] [PubMed]

Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, and Y. Baykal, “Average intensity and spreading of an elegant Hermite-Gaussian beam in turbulent atmosphere,” Opt. Express 17(13), 11130–11139 (2009).
[Crossref] [PubMed]

R. J. Noriega-Manez and J. C. Gutiérrez-Vega, “Rytov theory for Helmholtz-Gauss beams in turbulent atmosphere,” Opt. Express 15(25), 16328–16341 (2007).
[Crossref] [PubMed]

Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14(4), 1353–1367 (2006).
[Crossref] [PubMed]

Opt. Lasers Eng. (1)

M. Alavinejad, B. Ghafary, and F. D. Kashani, “Analysis of the propagation of flat-topped beam with various beam orders through turbulent atmosphere,” Opt. Lasers Eng. 46(1), 1–5 (2008).
[Crossref]

Opt. Lett. (9)

Y. Cai and S. He, “Average intensity and spreading of an elliptical gaussian beam propagating in a turbulent atmosphere,” Opt. Lett. 31(5), 568–570 (2006).
[Crossref] [PubMed]

Y. Cai, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation index of elliptical Gaussian beam in turbulent atmosphere,” Opt. Lett. 32(16), 2405–2407 (2007).
[Crossref] [PubMed]

F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23(4), 241–243 (1998).
[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]

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]

B. Kanseri and H. C. Kandpal, “Experimental determination of electric cross-spectral density matrix and generalized Stokes parameters for a laser beam,” Opt. Lett. 33(20), 2410–2412 (2008).
[Crossref] [PubMed]

H. Wang, X. Wang, A. Zeng, and K. Yang, “Effects of coherence on anisotropic electromagnetic Gaussian-Schell model beams on propagation,” Opt. Lett. 32(15), 2215–2217 (2007).
[Crossref] [PubMed]

Y. Dan and B. Zhang, “Second moments of partially coherent beams in atmospheric turbulence,” Opt. Lett. 34(5), 563–565 (2009).
[Crossref] [PubMed]

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]

Phys. Lett. A (1)

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

Waves Random Complex Media (1)

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere,” Waves Random Complex Media 15(3), 353–364 (2005).
[Crossref]

Other (5)

A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 2, (Academic Press, New York, 1978).

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, Washington, 2001).

L. C. Andrews, and R. L. Phillips, Laser beam propagation in the turbulent atmosphere, 2nd edition, SPIE press, Bellington, 2005.

E. Wolf, Introduction to the theory of coherence and polarization of light (Cambridge U. Press, 2007).

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE 1224, 2C14 (1990).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1 Dependence of the degree of polarization at source plane on A y with A x = 1
Fig. 2
Fig. 2 Dependence of the M 2 -factor of an EGSM beam in free space on its degree of polarization at source plane for two different cases (a) A y < A x , (b) A y > A x
Fig. 3
Fig. 3 Dependence of the M 2 -factor of an EGSM beam in free space on the r.m.s width ( σ x ) of the spectral density along x direction
Fig. 4
Fig. 4 Dependence of the M 2 -factor of an EGSM beam in free space on the r.m.s width ( δ x x ) of auto-correlation functions of the x component of the field
Fig. 5
Fig. 5 Normalized M 2 -factor of an EGSM beam on propagation in turbulent atmosphere for different values of the initial degree of polarization
Fig. 6
Fig. 6 Normalized M 2 -factor of an EGSM beam on propagation in turbulent atmosphere for different initial correlation factors δ x x and δ y y
Fig. 7
Fig. 7 Normalized M 2 -factor of an EGSM beam on propagation in turbulent atmosphere for different initial r.m.s widths σ x and σ y of the spectral densities
Fig. 8
Fig. 8 Normalized M 2 -factor of an EGSM beam on propagation in turbulent atmosphere for different values of wavelengthλ
Fig. 9
Fig. 9 Normalized M 2 -factor of an EGSM beam on propagation using different values of the structure constant ( C n 2 ) of the turbulent atmosphere
Fig. 10
Fig. 10 Normalized M 2 -factor of an EGSM beam on propagation in turbulent atmosphere for different values of inner scale of the turbulence ( l 0 )

Equations (25)

Equations on this page are rendered with MathJax. Learn more.

W α β ( ρ 1 ' , ρ 2 ' ; 0 ) = A α A β B α β exp [ ρ 1 ' 2 4 σ a 2 ρ 2 ' 2 4 σ β 2 ( ρ 1 ' ρ 2 ' ) 2 2 δ α β 2 ] , ( α = x , y ; β = x , y )
W tr ( ρ 1 ' , ρ 2 ' ; 0 ) = Tr W ( ρ 1 ' , ρ 2 ' , 0 ) = W x x ( ρ 1 ' , ρ 2 ' ; 0 ) + W y y ( ρ 1 ' , ρ 2 ' ; 0 ) .
W tr ( ρ , ρ d ; z ) = ( k 2 π z ) 2 W tr ( ρ ' , ρ d ' ; 0 ) × exp [ i k z ( ρ ρ ' ) ( ρ d ρ d ' ) H ( ρ d , ρ d ' ; z ) ] d 2 ρ ' d 2 ρ d ' ,
ρ ' = ( ρ 1 ' + ρ 2 ' ) 2 , ρ d ' = ρ 1 ' ρ 2 ' , ρ = ( ρ 1 + ρ 2 ) 2 , ρ d = ρ 1 ρ 2 ,
H ( ρ d , ρ d ' ; z ) = 4 π 2 k 2 z 0 1 d ξ 0 [ 1 J 0 ( κ | ρ d ' ξ + ( 1 ξ ) ρ d | ) ] Φ n ( κ ) κ d κ ,
W tr ( ρ , ρ d ; z ) = ( 1 2 π ) 2 W tr ( ρ ' ' , ρ d + z k κ d ; 0 ) × exp [ i ρ κ d + i ρ ' ' κ d H ( ρ d , ρ d + z k κ d ; z ) ] d 2 ρ ' ' d 2 κ d ,
W tr ( ρ ' ' , ρ d + z k κ d ; 0 ) = A x 2 exp [ 1 4 σ x 2 ( ρ " + ρ d + z k κ d 2 ) 2 1 4 σ x 2 ( ρ " ρ d + z k κ d 2 ) 2 ( ρ d + z k κ d ) 2 2 δ x x 2 ] + A y 2 exp [ 1 4 σ y 2 ( ρ " + ρ d + z k κ d 2 ) 2 1 4 σ y 2 ( ρ " ρ d + z k κ d 2 ) 2 ( ρ d + z k κ d ) 2 2 δ x x 2 ] .
h tr ( ρ , θ ; z ) = ( k 2 π ) 2 W tr ( ρ , ρ d ; z ) exp ( i k θ ρ d ) d 2 ρ d ,
exp ( s 2 x 2 ± q x ) d x = π s exp ( q 2 4 s 2 ) , ( s > 0 ) ,
h tr ( ρ , θ , z ) = h x x ( ρ , θ , z ) + h y y ( ρ , θ , z ) = A x 2 σ x 2 k 2 8 π 3 exp [ a x x κ d 2 2 z k b x x ρ d κ d i ρ κ d b x x ρ d 2 i k θ ρ d H ( ρ d , ρ d + z k κ d , z ) ] d 2 κ d d 2 ρ d + A y 2 σ y 2 k 2 8 π 3 exp [ a y y κ d 2 2 z k b y y ρ d κ d i ρ κ d b y y ρ d 2 i k θ ρ d H ( ρ d , ρ d + z k κ d , z ) ] d 2 κ d d 2 ρ d ,
M 2 ( z ) = k ( ρ 2 θ 2 ρ θ 2 ) 1 / 2 = k [ ( x 2 + y 2 ) ( θ x 2 + θ y 2 ) ( x θ x + y θ y ) 2 ] 1 / 2 ,
< x n 1 y n 2 θ x m 1 θ y m 2 > = 1 P x n 1 y n 2 θ x m 1 θ y m 2 h tr ( ρ , θ , z ) d 2 ρ d 2 θ ,
P = h tr ( ρ , θ , z ) d 2 ρ d 2 θ .
P = 2 π ( A x 2 σ x 2 + A y 2 σ y 2 ) ,
ρ 2 = 2 π A x 2 σ x 2 P ( 4 a x x + 4 3 π 2 T z 3 ) + 2 π A y 2 σ y 2 P ( 4 a y y + 4 3 π 2 T z 3 ) ,
θ 2 = 2 π A x 2 σ x 2 P ( 4 k 2 b x x + 4 π 2 T z ) + 2 π A y 2 σ y 2 P ( 4 k 2 b y y + 4 π 2 T z ) ,
ρ θ = 2 π A x 2 σ x 2 P ( 4 z k 2 b x x + 2 π 2 z 2 T ) + 2 π A y 2 σ y 2 P ( 4 z k 2 b y y + 2 π 2 z 2 T ) .
T = 0 Φ n ( κ ) κ 3 d κ .
M 2 ( z ) = k ( ρ 2 θ 2 ρ θ 2 ) 1 / 2 = k { [ 2 π A x 2 σ x 2 P ( 4 a x x + 4 3 π 2 T z 3 ) + 2 π A y 2 σ y 2 P ( 4 a y y + 4 3 π 2 T z 3 ) ] × [ 2 π A x 2 σ x 2 P ( 4 k 2 b x x + 4 π 2 T z ) + 2 π A y 2 σ y 2 P ( 4 k 2 b y y + 4 π 2 T z ) ] [ 2 π A x 2 σ x 2 P ( 4 z k 2 b x x + 2 π 2 z 2 T ) + 2 π A y 2 σ y 2 P ( 4 z k 2 b y y + 2 π 2 z 2 T ) ] 2 } 1 / 2 .
M 2 ( z ) = k ( ρ 2 θ 2 ρ θ 2 ) 1 / 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 )
Φ n ( κ ) = 0.033 C n 2 κ 11 / 3 exp ( κ 2 κ m 2 ) ,
T = 0 Φ n ( κ ) κ 3 d κ = 0.1661 C n 2 l 0 1 / 3 .
W ( ρ 1 ' , ρ 2 ' ; 0 ) = ( W x x ( ρ 1 ' , ρ 2 ' ; 0 ) 0 0 W y y ( ρ 1 ' , ρ 2 ' ; 0 ) ) .
P 0 ( ρ ' ; 0 ) = 1 4 D e t W ( ρ ' , ρ ' ; 0 ) [ T r W ( ρ ' , ρ ' ; 0 ) ] 2 .

Metrics