Abstract

Novel analytical expressions for the cross-spectral density function of a Gaussian Schell-model pulsed (GSMP) beam propagating through atmospheric turbulence are derived. Based on the cross-spectral density function, the average spectral density and the spectral degree of coherence of a GSMP beam in atmospheric turbulence are in turn examined. The dependence of the spectral degree of coherence on the turbulence strength measured by the atmospheric spatial coherence length is calculated numerically and analyzed in depth. The results obtained are useful for applications involving spatially and spectrally partially coherent pulsed beams propagating through atmospheric turbulence.

© 2011 OSA

1. Introduction

Recently, the propagation characteristics of pulsed beams have attracted much attention due to many applications, such as remote sensing, lidar operation and high-speed free-space optical communication (FSOC) links. Berman et al. [1] proposed a design of FSOC links with Gbps data rate based on spectral amplitude encoding of radiation from a Ti:Sapphire pulsed laser. As is well known, the performance of FSOC links is significantly reduced by atmospheric turbulence; controlling the spatial coherence of transmitted beams has been suggested as a means of improving the statistics of the received optical signal [2,3]. Hence, the propagation characteristics of spatially partially coherent pulsed beams through atmospheric turbulence are of great interest in practical FSOC applications transmitting pulsed beams. Traditionally, the Gaussian Schell-model (GSM) beam [4] is used as a mathematical tool for the description of spatially partially coherent beam-like optical fields.

For pulsed beams, which belong to non-stationary optical fields, it is commonly assumed that the exact temporal shape of the pulses in a pulse train is fixed [5], i.e., the spectral components are considered as being completely coherent. However, in practice, this assumption does not generally hold true, and pulsed beams exhibit spectrally partial coherence. Wang et al. [6] investigated the Gaussian Schell-model pulsed (GSMP) beam, which can be used to characterize pulsed optical fields that are both spatially and spectrally partially coherent, and its propagation formula was derived by using the partially coherent light theory. Lajunen et al. [5] derived an analytical formula describing the free-space propagation of the cross-spectral density of a GSMP beam and also presented its coherent-mode representation. An analytical expression for the far-field spectrum of a diffracted GSMP beam was derived by Ding et al. [7], and the spectral shifts and the spectral switches in the far field were numerically calculated. It is noted that none of these studies have taken atmospheric turbulence into account.

The mutual coherence function and the cross-spectral density function [4], which form a Fourier-transform pair, are two important second-order statistical quantities in coherence theory of stationary optical fields, which were also extended to characterize the coherence of a pulsed beam by Ref. [5], and from them, other statistics, such as the average spectral density and the spectral degree of coherence, can be deduced. The need for understanding the propagation behaviors of a GSMP beam in atmospheric turbulence is important for the successful engineering of related applications. Considering a narrow-band, planar source with an arbitrary coherence state radiating into atmospheric turbulence, Fante [8] derived a general expression in the form of a multiple integral for the two-frequency, two-position mutual coherence function of the optical field. The effects of atmospheric turbulence on the spectral switches of a diffracted GSMP beam and on the polarization properties of a stochastic electromagnetic GSMP beam were formulated by Ding et al. [9,10]. Due to the difficulties to obtain a general analytical expression for the cross-spectral density function of spatially and spectrally partially coherent pulsed beams propagating through atmospheric turbulence, Mao et al. [11] assumed the different spectral components of the pulsed beams to be completely uncorrelated and obtained an analytical formula for the cross-spectral density function of a broadband partially coherent flat-topped beam with a certain angular frequency; however, the light beams used in many practical applications are usually not spectrally completely incoherent. To the best of our knowledge, no results have been reported up until now on the analytical solution for the mutual coherence function or the cross-spectral density function of a GSMP beam propagating through atmospheric turbulence; thus, a deeper investigation on this issue is desirable.

The purpose of this paper is to study the second-order propagation characteristics of a GSMP beam in atmospheric turbulence. The propagation of the GSMP beam is considered in the space-frequency domain. Based on the extended Huygens-Fresnel principle and the quadratic approximation of the random phase structure function, we obtain novel analytical expressions for the cross-spectral density function of a GSMP beam propagating through atmospheric turbulence. With the help of the expressions, the average spectral density and the spectral degree of coherence are in turn investigated.

2. Analytical expressions for the cross-spectral density function

Suppose that a GSMP beam is generated in the source plane z = 0 and propagates into the half-space z > 0, nearly parallel to the positive z direction. The mutual coherence function of a GSMP beam in the source plane is given by [5]

Γ(0)(r1,r2,t1,t2)=Γ0exp[R(r1,r2)T(t1,t2)],
where r 1 and r 2 denote two position vectors in the source plane, Γ0 is a constant, and
R(r1,r2)=r12+r22w02+|r1r2|22σ02,
T(t1,t2)=t12+t222T02+(t1t2)22Tc2+iω0(t1t2),
where w 0 is the beam width, σ 0 is the transverse correlation length, rj is the modulus of r j (j = 1, 2), T 0 is the expectation value of the initial pulse width, Tc is the temporal coherence length, ω 0 is the central angular frequency of the pulses. Using the Fourier transform, the cross-spectral density function of a GSMP beam in the source plane is given by [5]
W(0)(r1,r2,ω1,ω2)=W0exp[R(r1,r2)F(ω1,ω2)],
where ωj (j = 1, 2) is the angular frequency, W 0 = Γ0 T 0/(2πΩ0) is a coefficient independent of both the position vectors and the angular frequencies, Ω0 = (1/T 0 2 + 2/Tc 2)1/2 represents the spectral width and
F(ω1,ω2)=(ω1ω0)2+(ω2ω0)22Ω02+(ω1ω2)22Ωc2,
where Ωc = TcΩ0/T 0 describes the spectral coherence width.

Assuming that any source fluctuations are statistically independent of the fluctuations of atmospheric turbulence and using the extended Huygens-Fresnel principle [12] together with the paraxial approximation [12], the cross-spectral density function of a GSMP beam in the half-space z > 0 can be written as

W(ρ1,z1,ρ2,z2,ω1,ω2)=ω1ω24π2z1z2c2exp[i(ω2z2ω1z1)/c]×d2r1d2r2W(0)(r1,r2,ω1,ω2)×exp[iω2|ρ2r2|22cz2iω1|ρ1r1|22cz1]×exp[ψ*(r1,ρ1,z1,ω1)+ψ(r2,ρ2,z2,ω2)]m,
where ρ j (j = 1, 2) is a position vector in the plane z = zj, ψ(r j,ρ j,zj,ωj) represents the random part of the complex phase of a spherical wave with the angular frequency ωj propagating from (r j, 0) to (ρ j, zj) in atmospheric turbulence, the asterisk denotes the complex conjugation, c is the speed of light, and <∙>m denotes the average over the ensemble of atmospheric turbulence.

In most practical applications, we are interested in the statistics of a GSMP beam in an observation plane transverse to the direction of propagation. Hence, we let z 1 = z 2 = L, where L is the distance from the source plane to the observation plane. In the case of homogeneous turbulence, considering the quadratic approximation of the random phase structure function which gives a good approximation of the second-order statistical properties of optical fields in turbulence under many circumstances [13], <exp[ψ*(r 1,ρ 1,L,ω 1) + ψ(r 2,ρ 2,L,ω 2)]>m can be written by [11,1315]

exp[ψ*(r1,ρ1,L,ω1)+ψ(r2,ρ2,L,ω2)]mΓ2exp[|ρ1ρ2|2+(ρ1ρ2)(r1r2)+|r1r2|2ρ02],
where Γ2 = exp(−0.391Cn 2 Lkd 2 L 0 5/3) is related to the angular frequency difference ω 1ω 2 by kd = (ω 1ω 2)/c, L 0 is the outer scale of turbulence, the refractive-index structure constant Cn 2 is a proportionality constant of the refractive-index structure function, ρ 0 = (0.55Cn 2 kc 2 l 0 −1/3 L)−1/2 is the atmospheric spatial coherence length of a spherical wave, l 0 is the inner scale of turbulence, kc = (ω 1 + ω 2)/(2c). Note that Γ2 and ρ 0 are derived based on the von Karman spectrum [15].

Inserting Eq. (7) into Eq. (6), making use of the identity [16]

exp(p2x2±qx)dx=exp(q24p2)πp,         (Re{p2}>0),
and performing the integration, after a long but straightforward calculation, the cross-spectral density function of a GSMP beam is reduced to
W(ρ1,ρ2,L,ω1,ω2)=W0Γ2g1g2Wbexp[F(ω1,ω2)]exp(|ρ1ρ2|2ρ02)×exp{1Wb2[g12g22R(ρ1,ρ2)+X(ρ1,ρ2)w02wc2+Q1(ρ1,ρ2)+Q2(ρ1,ρ2)ρ02]}×exp{iWb2[ag1g2(g1g2)R(ρ1,ρ2)u(g1,g2)Y(ρ1,ρ2)+Q3(ρ1,ρ2)ρ02]}×exp[iη(g1,g2)]exp[i(ω2ω1)L/c],
where

a=1w02+12σ02+1ρ02,
X(ρ1,ρ2)=g12ρ12+g22ρ22w02+|g1ρ1g2ρ2|22σ02,
Y(ρ1,ρ2)=g1ρ12g2ρ22w02wc2,
1wc2=1w02+1σ02+2ρ02,
Wb2=u(g1,g1)u(g2,g2)+(g1g2)24(1σ02+2ρ02)2,
u(g1,g2)=1w02wc2+g1g2,
Q1(ρ1,ρ2)=g1g22(1ρ04+1σ02w02)(ρ12+ρ22){(g1+g2)2w02(12ρ02+1w02)+5g1g22ρ02w02+(g12+g22)[12σ02w0212ρ02(12σ02+1ρ02)]}ρ1ρ2+[2g12g22+1ρ02(14σ02+1w02)g1g212wc2ρ02w04]|ρ1ρ2|2,
Q2(ρ1,ρ2)=g12ρ22+g22ρ128ρ02σ02+[(22w021σ02)18ρ021w04](g12ρ12+g22ρ22)(1ρ02+1w02)|g1ρ2+g2ρ1|24ρ02|g1ρ1+g2ρ2|24ρ04+3X(ρ1,ρ2)w02,
Q3(ρ1,ρ2)=[1wc2(g12g2g1g22g1w04)+g1g24ρ02(g1g21w04)g1g22w02]ρ12+[1wc2(g12g2g1g22+g2w04)+g1g24ρ02(g1g21w04)+g12g2w02]ρ22+[1wc2(2g12g2+2g1g22+g1g2w04)g1g22ρ02(g1g21w04)g1g2(g1g2)w02]ρ1ρ2,
η(g1,g2)=arctan[a(g1g2)/u(g1,g2)],     ρj=|ρj|,     gj=ωj2cL,     (j=1, 2).

Equation (9) shows the important theoretical results obtained by this paper. Note that Cn 2 = 0 corresponds to the absence of turbulence in which Γ2 = 1 and ρ 0 = ∞; under this condition, Eq. (9) readily proves to be consistent with the formulae obtained by Lajunen et al. (see Eqs. (50)−(60) of [5]) except that our expression for η(g 1,g 2) has a negative sign before arctan(∙). Indeed, Eq. (60) of [5] contains a mathematical error of lacking this negative sign.

3. Average spectral density

Setting ρ 1 = ρ 2 and ω 1 = ω 2 in Eq. (9), we obtain the average spectral density of a GSMP beam propagating through atmospheric turbulence as follows:

S(ρ,L,ω)=W0Δ(L,ω)exp[2ρ2w02Δ(L,ω)]exp[(ωω0)2Ω02],
where Δ(L,ω), referred to as the spectral expansion coefficient, is written as

Δ(L,ω)=1+(2cLw0wcω)2=1+4c2L2w02ω2(1w02+1σ02+2ρ02).

Note that Eqs. (10)(11) are identical with Eqs. (61)−(62) of Ref. [5] except for a different definition of wc. In fact, S(ρ,L,ω) is a product of the average spectral density of a GSM beam and the spectrum of a Gaussian pulse. The average spectral density of a GSMP beam decreases as the propagation distance L increases. We can see from Eq. (10) that the reduction in the average spectral density caused by the propagation of a GSMP beam is independent of the spectrally partial coherence of the source. It can be readily found from Eq. (11) that the effect of atmospheric turbulence, associated with ρ 0, on the spectral expansion coefficient resembles the effect of spatially partial coherence of the source, associated with σ 0. In addition, the contribution of ρ 0 and σ 0 toward the spectral expansion coefficient does not couple together as a GSMP beam propagates through atmospheric turbulence; this manifests that atmospheric turbulence is equivalent to spatially partial coherence of the source in enlarging the spectral expansion coefficient.

4. Spectral degree of coherence

The spectral degree of coherence of a GSMP beam, used to quantitatively characterize the strength of the field correlation at a pair of positions with two arbitrary angular frequencies, is defined as the normalized cross-spectral density function and given by

μ(ρ1,ρ2,L,ω1,ω2)=W(ρ1,ρ2,L,ω1,ω2)S(ρ1,L,ω1)S(ρ2,L,ω2).

The substitution of Eqs. (9) and (10) into Eq. (12) leads to

μ(ρ1,ρ2,L,ω1,ω2)=g1g2Γ2Δ(L,ω1)Δ(L,ω2)Wb×exp(|ρ1ρ2|2ρ02)exp[(ω1ω2)22Ωc2]×exp{1Wb2[g12g22R(ρ1,ρ2)+X(ρ1,ρ2)w02wc2+Q1(ρ1,ρ2)+Q2(ρ1,ρ2)ρ02]+V(ρ1,ρ2)w02}×exp{iWb2[ag1g2(g1g2)R(ρ1,ρ2)u(g1,g2)Y(ρ1,ρ2)+ρ02Q3(ρ1,ρ2)]}×exp[iη(g1,g2)]exp[i(ω2ω1)L/c],
where

V(ρ1,ρ2)=ρ12Δ(L,ω1)+ρ22Δ(L,ω2).

Equation (13) shows another important theoretical contribution of this paper. It can be readily found that the terms Γ2 and exp(−|ρ 1ρ 2|2/ρ 0 2) in Eq. (13) are two turbulence-induced attenuation factors associated with the angular frequency separation |ω 1ω 2| and the position separation |ρ 1ρ 2|, respectively, which have an influence on the modulus of the spectral degree of coherence. Furthermore, the argument of the spectral degree of coherence is the same as the cross-spectral density function.

Now, we numerically determine the spectral degree of coherence of a GSMP beam propagating through atmospheric turbulence and analyze its dependence on the turbulence strength measured by the atmospheric spatial coherence length ρ 0. The greater the turbulence strength becomes, the smaller ρ 0 is. We consider a GSMP beam of which the central wavelength λ 0, corresponding to the central angular frequency ω 0, is 1.55μm, and the expectation value of the initial pulse width T 0 is 200fs. Note that the angular frequency ω is related to the wavelength λ by ω = 2πc/λ. If Tc = ∞, the spectral width Ω0 = 5 × 103GHz which corresponds to 40nm, and for the case of Tc = 100fs, the spectral width Ω0 = 1.5 × 104GHz which corresponds to 120.3nm.

Figure 1 illustrates the dependence relationships between |μ(ρ 1,ρ 2,L,ω,ω)| and the wavelength λ with different combinations of ρ 0 and σ 0. Note that the quantity |μ(ρ 1,ρ 2,L,ω,ω)| actually describes the frequency-varying degree of transverse spatial coherence of the beam, and indeed, |μ(ρ 1,ρ 2,L,ω,ω)| does not depend on T 0 and Tc. It can be seen that |μ(ρ 1,ρ 2,L,ω,ω)| ≡ 1 if a spatially completely coherent pulsed beam propagates through free space without atmospheric turbulence, i.e., ρ 0 = ∞ and σ 0 = ∞; this result agrees with what one might expect. For other combinations of ρ 0 and σ 0, |μ(ρ 1,ρ 2,L,ω,ω)| shows a very slow monotonic growth with the increasing wavelength; the maximum values of |μ(ρ 1,ρ 2,L,ω,ω)|λ = 1.61μm − |μ(ρ 1,ρ 2,L,ω,ω)|λ = 1.49μm are 5.9 × 10−3 and 1.1 × 10−2 for Figs. 1(a) and 1(b), respectively. This variation behavior of |μ(ρ 1,ρ 2,L,ω,ω)| with the wavelength is also consistent with the known theory for a GSM beam (see Eqs. (5.6-84) and (5.6-107) of [4]) when ρ 0 = ∞. Based on these numerical results, one can say that in the presence of spatially partial coherence of the source and (or) atmospheric turbulence, the degree of transverse spatial coherence in an observation plane z > 0 is slightly smaller for a spectral component of a GSMP beam with a higher frequency than one with a lower frequency, that is, as opposed to the source plane in which the transverse correlation length for all spectral components is the same, the degree of transverse spatial coherence in the observation plane for a given spectral component of a GSMP beam is dependent on the frequency. Moreover, it can be observed from Fig. 1 that |μ(ρ 1,ρ 2,L,ω,ω)| reduces as the values of ρ 0 and (or) σ 0 decrease with the same wavelength λ.

 

Fig. 1 Dependence relationships between |μ(ρ 1,ρ 2,L,ω,ω)| and the wavelength λ with different combinations of ρ 0 and σ 0.

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Figure 2 shows |μ(ρ,ρ,L,ω 0,ω 1)| as a function of the wavelength separation δλ associated with the two spectral components of a GSMP beam, where the central angular frequency ω 0 = 2πc/λ 0 is fixed and the angular frequency ω 1 = 2πc/(λ 0 + δλ) varies with δλ. It should be noted that the quantity |μ(ρ,ρ,L,ω 0,ω 1)| is a measure of spectral coherence at the position ρ for the frequencies of ω 0 and ω 1. Figure 2(a) corresponds to a spatially and spectrally completely coherent pulsed beam, and Fig. 2(b) represents a spatially and spectrally partially coherent pulsed beam. We can see from Figs. 2(a) and 2(b) that |μ(ρ,ρ,L,ω 0,ω 1)| reduces monotonically as ρ 0 decreases with the same δλ; this variation behavior of |μ(ρ,ρ,L,ω 0,ω 1)| with ρ 0 manifests the fact that atmospheric turbulence induces the decrease of spectral coherence of a pulsed beam, which is just the reason for the temporal pulse broadening. It needs to be pointed out that the curves of |μ(ρ,ρ,L,ω 0,ω 1)| are actually not rigorously symmetric with respect to δλ = 0; this fact proves that |μ(ρ,ρ,L,ω 0,ω 1)| is determined exactly by a given combination of ω 0 and ω 1 instead of the frequency separation |ω 0ω 1|.

 

Fig. 2 |μ(ρ,ρ,L,ω 0,ω 1)| as a function of the wavelength separation δλ with ρ 0 = ∞, 3cm, 2cm and 1cm.

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Figure 3 illustrates |μ(ρ 1,ρ 2,L,ω 0,ω 1)| in terms of the relative atmospheric spatial coherence length ρ 0/[w 0 2∙Δ(L,ω 0)]1/2 which is defined as the ratio of the atmospheric spatial coherence length to the beam width. In Fig. 3(a), the frequency separation |ω 0ω 1| is fixed and the position separation d = |ρ 1ρ 2| is specified as 0cm, 2cm, 4cm and 6cm, respectively. In Fig. 3 (b), the position separation |ρ 1ρ 2| = 2cm is fixed and the frequency separation |ω 0ω 1| is specified by various values of the wavelength separation δλ. We can see from Figs. 3(a) and 3 (b) that |μ(ρ 1,ρ 2,L,ω 0,ω 1)| rises quickly as the relative atmospheric spatial coherence length increases. In Fig. 3(a), for the case of |ρ 1ρ 2| = 0, |μ(ρ 1,ρ 2,L,ω 0,ω 1)| approaches to saturation as the relative atmospheric spatial coherence length increases past 1, and for other cases, with the increase of |ρ 1ρ 2|, the saturation point moves gradually toward the right side. However, in Fig. 3(b), we can find that almost for all values of the wavelength separation δλ, |μ(ρ 1,ρ 2,L,ω 0,ω 1)| approaches to saturation as the relative atmospheric spatial coherence length increases beyond 1.

 

Fig. 3 |μ(ρ 1,ρ 2,L,ω 0,ω 1)| in terms of the relative atmospheric spatial coherence length ρ 0/[w 0 2∙Δ(L,ω 0)]1/2.

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5. Conclusions

In this paper, novel analytical expressions for the cross-spectral density function of a GSMP beam propagating through atmospheric turbulence have been derived by using the extended Huygens-Fresnel principle together with the quadratic approximation of the random phase structure function, which coincide with the expressions obtained previously for a GSMP beam in free space if the absence of atmospheric turbulence is assumed. From the cross-spectral density function, the expressions for the average spectral density and the spectral degree of coherence of a GSMP beam were obtained. It has been shown that atmospheric turbulence is equivalent to spatially partial coherence of the source in enlarging the spectral expansion coefficient; in the presence of spatially partial coherence of the source and (or) atmospheric turbulence, the degree of transverse spatial coherence in an observation plane z > 0 for a given spectral component of a GSMP beam depends on the frequency, and the higher the frequency is, the slightly smaller the degree of transverse spatial coherence becomes; the spectral degree of coherence is determined exactly by a given combination of the two frequencies instead of the corresponding frequency separation; the modulus of the spectral degree of coherence rises quickly with the increasing relative atmospheric spatial coherence length and then approaches to saturation as the relative atmospheric spatial coherence length increases beyond a given value determined by the parameters of the positions and the frequencies.

The research work in this paper provides a better understanding of the second-order statistics of a GSMP beam propagating though atmospheric turbulence in the space-frequency domain and our results are useful for applications involving spatially and spectrally partially coherent pulsed beams. It should be mentioned that our analytical expressions for the cross-spectral density function also provide a basis for numerically studying the propagation characteristics of a GSMP beam through atmospheric turbulence in the space-time domain, for instance, by using the fast-Fourier-transform algorithm.

Acknowledgments

The authors are very grateful to the reviewers for valuable comments. This work was supported by the National Natural Science Foundation of China (NSFC) under grant 61007046 and the Jilin Provincial Development Programs of Science & Technology of China under grant 201101096.

References and links

1. G. P. Berman, A. R. Bishop, B. M. Chernobrod, D. C. Nguyen, and V. N. Gorshkov, “Suppression of intensity fluctuations in free space high-speed optical communication based on spectral encoding of a partially coherent beam,” Opt. Commun. 280(2), 264–270 (2007). [CrossRef]  

2. C. Chen, H. Yang, X. Feng, and H. Wang, “Optimization criterion for initial coherence degree of lasers in free-space optical links through atmospheric turbulence,” Opt. Lett. 34(4), 419–421 (2009). [CrossRef]   [PubMed]  

3. K. Drexler, M. Roggemann, and D. Voelz, “Use of a partially coherent transmitter beam to improve the statistics of received power in a free-space optical communication system: theory and experimental results,” Opt. Eng. 50(2), 025002 (2011). [CrossRef]  

4. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), Chaps. 4 and 5.

5. H. Lajunen, P. Vahimaa, and J. Tervo, “Theory of spatially and spectrally partially coherent pulses,” J. Opt. Soc. Am. A 22(8), 1536–1545 (2005). [CrossRef]   [PubMed]  

6. L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003). [CrossRef]   [PubMed]  

7. C. Ding and B. Lü, “Spectral shifts and spectral switches of diffracted spatially and spectrally partially coherent pulsed beams in the far field,” J. Opt. A, Pure Appl. Opt. 10(9), 095006 (2008). [CrossRef]  

8. R. L. Fante, “Two-position, two-frequency mutual-coherence function in turbulence,” J. Opt. Soc. Am. 71(12), 1446–1451 (1981). [CrossRef]  

9. C. Ding, L. Pan, and B. Lü, “Effect of turbulence on the spectral switches of diffracted spatially and spectrally partially coherent pulsed beams in atmospheric turbulence,” J. Opt. A, Pure Appl. Opt. 11(10), 105404 (2009). [CrossRef]  

10. C. Ding, Z. Zhao, X. Li, L. Pan, and X. Yuan, “Influence of turbulent atmosphere on polarization properties of stochastic electromagnetic pulsed beams,” Chin. Phys. Lett. 28(2), 024214 (2011). [CrossRef]  

11. H. Mao and D. Zhao, “Second-order intensity-moment characteristics for broadband partially coherent flat-topped beams in atmospheric turbulence,” Opt. Express 18(2), 1741–1755 (2010). [CrossRef]   [PubMed]  

12. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 2005), Chaps. 4 and 7.

13. Y. Gu and G. Gbur, “Measurement of atmospheric turbulence strength by vortex beam,” Opt. Commun. 283(7), 1209–1212 (2010). [CrossRef]  

14. G. Wu, B. Luo, S. Yu, A. Dang, and H. Guo, “The propagation of electromagnetic Gaussian-Schell model beams through atmospheric turbulence in a slanted path,” J. Opt. 13(3), 035706 (2011). [CrossRef]  

15. C. Y. Young, “Broadening of ultra-short optical pulses in moderate to strong turbulence,” Proc. SPIE 4821, 74–81 (2002). [CrossRef]  

16. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic Press, 2007), Chap. 3.

References

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  1. G. P. Berman, A. R. Bishop, B. M. Chernobrod, D. C. Nguyen, and V. N. Gorshkov, “Suppression of intensity fluctuations in free space high-speed optical communication based on spectral encoding of a partially coherent beam,” Opt. Commun. 280(2), 264–270 (2007).
    [Crossref]
  2. C. Chen, H. Yang, X. Feng, and H. Wang, “Optimization criterion for initial coherence degree of lasers in free-space optical links through atmospheric turbulence,” Opt. Lett. 34(4), 419–421 (2009).
    [Crossref] [PubMed]
  3. K. Drexler, M. Roggemann, and D. Voelz, “Use of a partially coherent transmitter beam to improve the statistics of received power in a free-space optical communication system: theory and experimental results,” Opt. Eng. 50(2), 025002 (2011).
    [Crossref]
  4. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), Chaps. 4 and 5.
  5. H. Lajunen, P. Vahimaa, and J. Tervo, “Theory of spatially and spectrally partially coherent pulses,” J. Opt. Soc. Am. A 22(8), 1536–1545 (2005).
    [Crossref] [PubMed]
  6. L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003).
    [Crossref] [PubMed]
  7. C. Ding and B. Lü, “Spectral shifts and spectral switches of diffracted spatially and spectrally partially coherent pulsed beams in the far field,” J. Opt. A, Pure Appl. Opt. 10(9), 095006 (2008).
    [Crossref]
  8. R. L. Fante, “Two-position, two-frequency mutual-coherence function in turbulence,” J. Opt. Soc. Am. 71(12), 1446–1451 (1981).
    [Crossref]
  9. C. Ding, L. Pan, and B. Lü, “Effect of turbulence on the spectral switches of diffracted spatially and spectrally partially coherent pulsed beams in atmospheric turbulence,” J. Opt. A, Pure Appl. Opt. 11(10), 105404 (2009).
    [Crossref]
  10. C. Ding, Z. Zhao, X. Li, L. Pan, and X. Yuan, “Influence of turbulent atmosphere on polarization properties of stochastic electromagnetic pulsed beams,” Chin. Phys. Lett. 28(2), 024214 (2011).
    [Crossref]
  11. H. Mao and D. Zhao, “Second-order intensity-moment characteristics for broadband partially coherent flat-topped beams in atmospheric turbulence,” Opt. Express 18(2), 1741–1755 (2010).
    [Crossref] [PubMed]
  12. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 2005), Chaps. 4 and 7.
  13. Y. Gu and G. Gbur, “Measurement of atmospheric turbulence strength by vortex beam,” Opt. Commun. 283(7), 1209–1212 (2010).
    [Crossref]
  14. G. Wu, B. Luo, S. Yu, A. Dang, and H. Guo, “The propagation of electromagnetic Gaussian-Schell model beams through atmospheric turbulence in a slanted path,” J. Opt. 13(3), 035706 (2011).
    [Crossref]
  15. C. Y. Young, “Broadening of ultra-short optical pulses in moderate to strong turbulence,” Proc. SPIE 4821, 74–81 (2002).
    [Crossref]
  16. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic Press, 2007), Chap. 3.

2011 (3)

K. Drexler, M. Roggemann, and D. Voelz, “Use of a partially coherent transmitter beam to improve the statistics of received power in a free-space optical communication system: theory and experimental results,” Opt. Eng. 50(2), 025002 (2011).
[Crossref]

C. Ding, Z. Zhao, X. Li, L. Pan, and X. Yuan, “Influence of turbulent atmosphere on polarization properties of stochastic electromagnetic pulsed beams,” Chin. Phys. Lett. 28(2), 024214 (2011).
[Crossref]

G. Wu, B. Luo, S. Yu, A. Dang, and H. Guo, “The propagation of electromagnetic Gaussian-Schell model beams through atmospheric turbulence in a slanted path,” J. Opt. 13(3), 035706 (2011).
[Crossref]

2010 (2)

2009 (2)

C. Chen, H. Yang, X. Feng, and H. Wang, “Optimization criterion for initial coherence degree of lasers in free-space optical links through atmospheric turbulence,” Opt. Lett. 34(4), 419–421 (2009).
[Crossref] [PubMed]

C. Ding, L. Pan, and B. Lü, “Effect of turbulence on the spectral switches of diffracted spatially and spectrally partially coherent pulsed beams in atmospheric turbulence,” J. Opt. A, Pure Appl. Opt. 11(10), 105404 (2009).
[Crossref]

2008 (1)

C. Ding and B. Lü, “Spectral shifts and spectral switches of diffracted spatially and spectrally partially coherent pulsed beams in the far field,” J. Opt. A, Pure Appl. Opt. 10(9), 095006 (2008).
[Crossref]

2007 (1)

G. P. Berman, A. R. Bishop, B. M. Chernobrod, D. C. Nguyen, and V. N. Gorshkov, “Suppression of intensity fluctuations in free space high-speed optical communication based on spectral encoding of a partially coherent beam,” Opt. Commun. 280(2), 264–270 (2007).
[Crossref]

2005 (1)

2003 (1)

L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003).
[Crossref] [PubMed]

2002 (1)

C. Y. Young, “Broadening of ultra-short optical pulses in moderate to strong turbulence,” Proc. SPIE 4821, 74–81 (2002).
[Crossref]

1981 (1)

Berman, G. P.

G. P. Berman, A. R. Bishop, B. M. Chernobrod, D. C. Nguyen, and V. N. Gorshkov, “Suppression of intensity fluctuations in free space high-speed optical communication based on spectral encoding of a partially coherent beam,” Opt. Commun. 280(2), 264–270 (2007).
[Crossref]

Bishop, A. R.

G. P. Berman, A. R. Bishop, B. M. Chernobrod, D. C. Nguyen, and V. N. Gorshkov, “Suppression of intensity fluctuations in free space high-speed optical communication based on spectral encoding of a partially coherent beam,” Opt. Commun. 280(2), 264–270 (2007).
[Crossref]

Chen, C.

Chen, H.

L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003).
[Crossref] [PubMed]

Chernobrod, B. M.

G. P. Berman, A. R. Bishop, B. M. Chernobrod, D. C. Nguyen, and V. N. Gorshkov, “Suppression of intensity fluctuations in free space high-speed optical communication based on spectral encoding of a partially coherent beam,” Opt. Commun. 280(2), 264–270 (2007).
[Crossref]

Dang, A.

G. Wu, B. Luo, S. Yu, A. Dang, and H. Guo, “The propagation of electromagnetic Gaussian-Schell model beams through atmospheric turbulence in a slanted path,” J. Opt. 13(3), 035706 (2011).
[Crossref]

Ding, C.

C. Ding, Z. Zhao, X. Li, L. Pan, and X. Yuan, “Influence of turbulent atmosphere on polarization properties of stochastic electromagnetic pulsed beams,” Chin. Phys. Lett. 28(2), 024214 (2011).
[Crossref]

C. Ding, L. Pan, and B. Lü, “Effect of turbulence on the spectral switches of diffracted spatially and spectrally partially coherent pulsed beams in atmospheric turbulence,” J. Opt. A, Pure Appl. Opt. 11(10), 105404 (2009).
[Crossref]

C. Ding and B. Lü, “Spectral shifts and spectral switches of diffracted spatially and spectrally partially coherent pulsed beams in the far field,” J. Opt. A, Pure Appl. Opt. 10(9), 095006 (2008).
[Crossref]

Drexler, K.

K. Drexler, M. Roggemann, and D. Voelz, “Use of a partially coherent transmitter beam to improve the statistics of received power in a free-space optical communication system: theory and experimental results,” Opt. Eng. 50(2), 025002 (2011).
[Crossref]

Fante, R. L.

Feng, X.

Gbur, G.

Y. Gu and G. Gbur, “Measurement of atmospheric turbulence strength by vortex beam,” Opt. Commun. 283(7), 1209–1212 (2010).
[Crossref]

Gorshkov, V. N.

G. P. Berman, A. R. Bishop, B. M. Chernobrod, D. C. Nguyen, and V. N. Gorshkov, “Suppression of intensity fluctuations in free space high-speed optical communication based on spectral encoding of a partially coherent beam,” Opt. Commun. 280(2), 264–270 (2007).
[Crossref]

Gu, Y.

Y. Gu and G. Gbur, “Measurement of atmospheric turbulence strength by vortex beam,” Opt. Commun. 283(7), 1209–1212 (2010).
[Crossref]

Guo, H.

G. Wu, B. Luo, S. Yu, A. Dang, and H. Guo, “The propagation of electromagnetic Gaussian-Schell model beams through atmospheric turbulence in a slanted path,” J. Opt. 13(3), 035706 (2011).
[Crossref]

Lajunen, H.

Li, X.

C. Ding, Z. Zhao, X. Li, L. Pan, and X. Yuan, “Influence of turbulent atmosphere on polarization properties of stochastic electromagnetic pulsed beams,” Chin. Phys. Lett. 28(2), 024214 (2011).
[Crossref]

Lin, Q.

L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003).
[Crossref] [PubMed]

Lü, B.

C. Ding, L. Pan, and B. Lü, “Effect of turbulence on the spectral switches of diffracted spatially and spectrally partially coherent pulsed beams in atmospheric turbulence,” J. Opt. A, Pure Appl. Opt. 11(10), 105404 (2009).
[Crossref]

C. Ding and B. Lü, “Spectral shifts and spectral switches of diffracted spatially and spectrally partially coherent pulsed beams in the far field,” J. Opt. A, Pure Appl. Opt. 10(9), 095006 (2008).
[Crossref]

Luo, B.

G. Wu, B. Luo, S. Yu, A. Dang, and H. Guo, “The propagation of electromagnetic Gaussian-Schell model beams through atmospheric turbulence in a slanted path,” J. Opt. 13(3), 035706 (2011).
[Crossref]

Mao, H.

Nguyen, D. C.

G. P. Berman, A. R. Bishop, B. M. Chernobrod, D. C. Nguyen, and V. N. Gorshkov, “Suppression of intensity fluctuations in free space high-speed optical communication based on spectral encoding of a partially coherent beam,” Opt. Commun. 280(2), 264–270 (2007).
[Crossref]

Pan, L.

C. Ding, Z. Zhao, X. Li, L. Pan, and X. Yuan, “Influence of turbulent atmosphere on polarization properties of stochastic electromagnetic pulsed beams,” Chin. Phys. Lett. 28(2), 024214 (2011).
[Crossref]

C. Ding, L. Pan, and B. Lü, “Effect of turbulence on the spectral switches of diffracted spatially and spectrally partially coherent pulsed beams in atmospheric turbulence,” J. Opt. A, Pure Appl. Opt. 11(10), 105404 (2009).
[Crossref]

Roggemann, M.

K. Drexler, M. Roggemann, and D. Voelz, “Use of a partially coherent transmitter beam to improve the statistics of received power in a free-space optical communication system: theory and experimental results,” Opt. Eng. 50(2), 025002 (2011).
[Crossref]

Tervo, J.

Vahimaa, P.

Voelz, D.

K. Drexler, M. Roggemann, and D. Voelz, “Use of a partially coherent transmitter beam to improve the statistics of received power in a free-space optical communication system: theory and experimental results,” Opt. Eng. 50(2), 025002 (2011).
[Crossref]

Wang, H.

Wang, L. G.

L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003).
[Crossref] [PubMed]

Wu, G.

G. Wu, B. Luo, S. Yu, A. Dang, and H. Guo, “The propagation of electromagnetic Gaussian-Schell model beams through atmospheric turbulence in a slanted path,” J. Opt. 13(3), 035706 (2011).
[Crossref]

Yang, H.

Young, C. Y.

C. Y. Young, “Broadening of ultra-short optical pulses in moderate to strong turbulence,” Proc. SPIE 4821, 74–81 (2002).
[Crossref]

Yu, S.

G. Wu, B. Luo, S. Yu, A. Dang, and H. Guo, “The propagation of electromagnetic Gaussian-Schell model beams through atmospheric turbulence in a slanted path,” J. Opt. 13(3), 035706 (2011).
[Crossref]

Yuan, X.

C. Ding, Z. Zhao, X. Li, L. Pan, and X. Yuan, “Influence of turbulent atmosphere on polarization properties of stochastic electromagnetic pulsed beams,” Chin. Phys. Lett. 28(2), 024214 (2011).
[Crossref]

Zhao, D.

Zhao, Z.

C. Ding, Z. Zhao, X. Li, L. Pan, and X. Yuan, “Influence of turbulent atmosphere on polarization properties of stochastic electromagnetic pulsed beams,” Chin. Phys. Lett. 28(2), 024214 (2011).
[Crossref]

Zhu, S. Y.

L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003).
[Crossref] [PubMed]

Chin. Phys. Lett. (1)

C. Ding, Z. Zhao, X. Li, L. Pan, and X. Yuan, “Influence of turbulent atmosphere on polarization properties of stochastic electromagnetic pulsed beams,” Chin. Phys. Lett. 28(2), 024214 (2011).
[Crossref]

J. Opt. (1)

G. Wu, B. Luo, S. Yu, A. Dang, and H. Guo, “The propagation of electromagnetic Gaussian-Schell model beams through atmospheric turbulence in a slanted path,” J. Opt. 13(3), 035706 (2011).
[Crossref]

J. Opt. A, Pure Appl. Opt. (2)

C. Ding, L. Pan, and B. Lü, “Effect of turbulence on the spectral switches of diffracted spatially and spectrally partially coherent pulsed beams in atmospheric turbulence,” J. Opt. A, Pure Appl. Opt. 11(10), 105404 (2009).
[Crossref]

C. Ding and B. Lü, “Spectral shifts and spectral switches of diffracted spatially and spectrally partially coherent pulsed beams in the far field,” J. Opt. A, Pure Appl. Opt. 10(9), 095006 (2008).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Commun. (2)

G. P. Berman, A. R. Bishop, B. M. Chernobrod, D. C. Nguyen, and V. N. Gorshkov, “Suppression of intensity fluctuations in free space high-speed optical communication based on spectral encoding of a partially coherent beam,” Opt. Commun. 280(2), 264–270 (2007).
[Crossref]

Y. Gu and G. Gbur, “Measurement of atmospheric turbulence strength by vortex beam,” Opt. Commun. 283(7), 1209–1212 (2010).
[Crossref]

Opt. Eng. (1)

K. Drexler, M. Roggemann, and D. Voelz, “Use of a partially coherent transmitter beam to improve the statistics of received power in a free-space optical communication system: theory and experimental results,” Opt. Eng. 50(2), 025002 (2011).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003).
[Crossref] [PubMed]

Proc. SPIE (1)

C. Y. Young, “Broadening of ultra-short optical pulses in moderate to strong turbulence,” Proc. SPIE 4821, 74–81 (2002).
[Crossref]

Other (3)

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic Press, 2007), Chap. 3.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 2005), Chaps. 4 and 7.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), Chaps. 4 and 5.

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Figures (3)

Fig. 1
Fig. 1

Dependence relationships between |μ(ρ 1,ρ 2,L,ω,ω)| and the wavelength λ with different combinations of ρ 0 and σ 0.

Fig. 2
Fig. 2

|μ(ρ,ρ,L,ω 0,ω 1)| as a function of the wavelength separation δλ with ρ 0 = ∞, 3cm, 2cm and 1cm.

Fig. 3
Fig. 3

|μ(ρ 1,ρ 2,L,ω 0,ω 1)| in terms of the relative atmospheric spatial coherence length ρ 0/[w 0 2∙Δ(L,ω 0)]1/2.

Equations (24)

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Γ ( 0 ) ( r 1 , r 2 , t 1 , t 2 ) = Γ 0 exp [ R ( r 1 , r 2 ) T ( t 1 , t 2 ) ] ,
R ( r 1 , r 2 ) = r 1 2 + r 2 2 w 0 2 + | r 1 r 2 | 2 2 σ 0 2 ,
T ( t 1 , t 2 ) = t 1 2 + t 2 2 2 T 0 2 + ( t 1 t 2 ) 2 2 T c 2 + i ω 0 ( t 1 t 2 ) ,
W ( 0 ) ( r 1 , r 2 , ω 1 , ω 2 ) = W 0 exp [ R ( r 1 , r 2 ) F ( ω 1 , ω 2 ) ] ,
F ( ω 1 , ω 2 ) = ( ω 1 ω 0 ) 2 + ( ω 2 ω 0 ) 2 2 Ω 0 2 + ( ω 1 ω 2 ) 2 2 Ω c 2 ,
W ( ρ 1 , z 1 , ρ 2 , z 2 , ω 1 , ω 2 ) = ω 1 ω 2 4 π 2 z 1 z 2 c 2 exp [ i ( ω 2 z 2 ω 1 z 1 ) / c ] × d 2 r 1 d 2 r 2 W ( 0 ) ( r 1 , r 2 , ω 1 , ω 2 ) × exp [ i ω 2 | ρ 2 r 2 | 2 2 c z 2 i ω 1 | ρ 1 r 1 | 2 2 c z 1 ] × exp [ ψ * ( r 1 , ρ 1 , z 1 , ω 1 ) + ψ ( r 2 , ρ 2 , z 2 , ω 2 ) ] m ,
exp [ ψ * ( r 1 , ρ 1 , L , ω 1 ) + ψ ( r 2 , ρ 2 , L , ω 2 ) ] m Γ 2 exp [ | ρ 1 ρ 2 | 2 + ( ρ 1 ρ 2 ) ( r 1 r 2 ) + | r 1 r 2 | 2 ρ 0 2 ] ,
exp ( p 2 x 2 ± q x ) d x = exp ( q 2 4 p 2 ) π p ,           ( Re { p 2 } > 0 ) ,
W ( ρ 1 , ρ 2 , L , ω 1 , ω 2 ) = W 0 Γ 2 g 1 g 2 W b exp [ F ( ω 1 , ω 2 ) ] exp ( | ρ 1 ρ 2 | 2 ρ 0 2 ) × exp { 1 W b 2 [ g 1 2 g 2 2 R ( ρ 1 , ρ 2 ) + X ( ρ 1 , ρ 2 ) w 0 2 w c 2 + Q 1 ( ρ 1 , ρ 2 ) + Q 2 ( ρ 1 , ρ 2 ) ρ 0 2 ] } × exp { i W b 2 [ a g 1 g 2 ( g 1 g 2 ) R ( ρ 1 , ρ 2 ) u ( g 1 , g 2 ) Y ( ρ 1 , ρ 2 ) + Q 3 ( ρ 1 , ρ 2 ) ρ 0 2 ] } × exp [ i η ( g 1 , g 2 ) ] exp [ i ( ω 2 ω 1 ) L / c ] ,
a = 1 w 0 2 + 1 2 σ 0 2 + 1 ρ 0 2 ,
X ( ρ 1 , ρ 2 ) = g 1 2 ρ 1 2 + g 2 2 ρ 2 2 w 0 2 + | g 1 ρ 1 g 2 ρ 2 | 2 2 σ 0 2 ,
Y ( ρ 1 , ρ 2 ) = g 1 ρ 1 2 g 2 ρ 2 2 w 0 2 w c 2 ,
1 w c 2 = 1 w 0 2 + 1 σ 0 2 + 2 ρ 0 2 ,
W b 2 = u ( g 1 , g 1 ) u ( g 2 , g 2 ) + ( g 1 g 2 ) 2 4 ( 1 σ 0 2 + 2 ρ 0 2 ) 2 ,
u ( g 1 , g 2 ) = 1 w 0 2 w c 2 + g 1 g 2 ,
Q 1 ( ρ 1 , ρ 2 ) = g 1 g 2 2 ( 1 ρ 0 4 + 1 σ 0 2 w 0 2 ) ( ρ 1 2 + ρ 2 2 ) { ( g 1 + g 2 ) 2 w 0 2 ( 1 2 ρ 0 2 + 1 w 0 2 ) + 5 g 1 g 2 2 ρ 0 2 w 0 2 + ( g 1 2 + g 2 2 ) [ 1 2 σ 0 2 w 0 2 1 2 ρ 0 2 ( 1 2 σ 0 2 + 1 ρ 0 2 ) ] } ρ 1 ρ 2 + [ 2 g 1 2 g 2 2 + 1 ρ 0 2 ( 1 4 σ 0 2 + 1 w 0 2 ) g 1 g 2 1 2 w c 2 ρ 0 2 w 0 4 ] | ρ 1 ρ 2 | 2 ,
Q 2 ( ρ 1 , ρ 2 ) = g 1 2 ρ 2 2 + g 2 2 ρ 1 2 8 ρ 0 2 σ 0 2 + [ ( 22 w 0 2 1 σ 0 2 ) 1 8 ρ 0 2 1 w 0 4 ] ( g 1 2 ρ 1 2 + g 2 2 ρ 2 2 ) ( 1 ρ 0 2 + 1 w 0 2 ) | g 1 ρ 2 + g 2 ρ 1 | 2 4 ρ 0 2 | g 1 ρ 1 + g 2 ρ 2 | 2 4 ρ 0 4 + 3 X ( ρ 1 , ρ 2 ) w 0 2 ,
Q 3 ( ρ 1 , ρ 2 ) = [ 1 w c 2 ( g 1 2 g 2 g 1 g 2 2 g 1 w 0 4 ) + g 1 g 2 4 ρ 0 2 ( g 1 g 2 1 w 0 4 ) g 1 g 2 2 w 0 2 ] ρ 1 2 + [ 1 w c 2 ( g 1 2 g 2 g 1 g 2 2 + g 2 w 0 4 ) + g 1 g 2 4 ρ 0 2 ( g 1 g 2 1 w 0 4 ) + g 1 2 g 2 w 0 2 ] ρ 2 2 + [ 1 w c 2 ( 2 g 1 2 g 2 + 2 g 1 g 2 2 + g 1 g 2 w 0 4 ) g 1 g 2 2 ρ 0 2 ( g 1 g 2 1 w 0 4 ) g 1 g 2 ( g 1 g 2 ) w 0 2 ] ρ 1 ρ 2 ,
η ( g 1 , g 2 ) = arctan [ a ( g 1 g 2 ) / u ( g 1 , g 2 ) ] ,       ρ j = | ρ j | ,       g j = ω j 2 c L ,       ( j = 1 ,  2 ) .
S ( ρ , L , ω ) = W 0 Δ ( L , ω ) exp [ 2 ρ 2 w 0 2 Δ ( L , ω ) ] exp [ ( ω ω 0 ) 2 Ω 0 2 ] ,
Δ ( L , ω ) = 1 + ( 2 c L w 0 w c ω ) 2 = 1 + 4 c 2 L 2 w 0 2 ω 2 ( 1 w 0 2 + 1 σ 0 2 + 2 ρ 0 2 ) .
μ ( ρ 1 , ρ 2 , L , ω 1 , ω 2 ) = W ( ρ 1 , ρ 2 , L , ω 1 , ω 2 ) S ( ρ 1 , L , ω 1 ) S ( ρ 2 , L , ω 2 ) .
μ ( ρ 1 , ρ 2 , L , ω 1 , ω 2 ) = g 1 g 2 Γ 2 Δ ( L , ω 1 ) Δ ( L , ω 2 ) W b × exp ( | ρ 1 ρ 2 | 2 ρ 0 2 ) exp [ ( ω 1 ω 2 ) 2 2 Ω c 2 ] × exp { 1 W b 2 [ g 1 2 g 2 2 R ( ρ 1 , ρ 2 ) + X ( ρ 1 , ρ 2 ) w 0 2 w c 2 + Q 1 ( ρ 1 , ρ 2 ) + Q 2 ( ρ 1 , ρ 2 ) ρ 0 2 ] + V ( ρ 1 , ρ 2 ) w 0 2 } × exp { i W b 2 [ a g 1 g 2 ( g 1 g 2 ) R ( ρ 1 , ρ 2 ) u ( g 1 , g 2 ) Y ( ρ 1 , ρ 2 ) + ρ 0 2 Q 3 ( ρ 1 , ρ 2 ) ] } × exp [ i η ( g 1 , g 2 ) ] exp [ i ( ω 2 ω 1 ) L / c ] ,
V ( ρ 1 , ρ 2 ) = ρ 1 2 Δ ( L , ω 1 ) + ρ 2 2 Δ ( L , ω 2 ) .

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