Abstract

The Wigner distribution function (WDF) has been used to study the beam propagation factor (M 2-factor) for partially coherent flat-topped (PCFT) beams with circular symmetry in a turbulent atmosphere. Based on the extended Huygens–Fresnel principle and the definition of the WDF, an expression for the WDF of PCFT beams in turbulence has been given. By use of the second-order moments of the WDF, the analytical formulas for the root-mean-square (rms) spatial width, the rms angular width, and the M 2-factor of PCFT beams in turbulence have been derived, which can be applied to cases of different spatial power spectra of the refractive index fluctuations. The rms angular width and the M 2-factor of PCFT beams in turbulence have been discussed with numerical examples. It can be shown that the M 2-factor of PCFT beams in turbulence depends on the beam order, degree of global coherence of the source, waist width, wavelength, spatial power spectrum of the refractive index fluctuations, and propagation distance.

©2008 Optical Society of America

1. Introduction

The propagation of light beams in atmospheric turbulence has been extensively studied for a long time [1-3]. In recent years, because of many practical applications such as free-space optical communications, it is especially interesting that what types of light beams and under what conditions the effects of the turbulent atmosphere on the beams are the smallest [4-8]; i.e., variations of the parameters (such as beam width etc.) characterizing beams due to turbulence are the smallest. Preliminary works [4, 5] have shown that the Gaussian-Schell model (GSM) beam with the worse coherence is less affected by turbulence. Consequently, lots of theoretical studies on the spatial and angular spreading of partially coherent beams in atmospheric turbulence have been carried out, indicating that partially coherent beams are less influenced by turbulence than fully coherent beams [6-8]. Subsequently, experiments supporting these findings have been performed [9]. It is well known that the Wigner distribution function (WDF) can characterize partially coherent beams in space and in spatial frequency domain simultaneously [10-13]. Consequently, the beam propagation factor (M 2-factor) for characterizing laser beams can be studied. The M 2-factor of partially coherent beams propagating in free space and in gain or absorbing media has been investigated [14-18]. Very recently, the M 2-factor of fully coherent Gaussian beams in atmospheric turbulence has been studied by numerical approach [19]. However, to our knowledge, there have been no reports on the beam propagation factor of partially coherent beams propagating in atmospheric turbulence by analytical approach.

The main purpose of this paper is to generalize the definition of M 2-factor in terms of second-order moments of the WDF to the case in atmospheric turbulence and to study the propagation law of M 2-factor for partially coherent flat-topped (PCFT) beams with circular symmetry in a turbulent atmosphere. First, by use of the extended Huygens–Fresnel principle and the definition of the WDF, an expression for the WDF of PCFT beams in a turbulent atmosphere has been given. Then, according to method of second-order moments of the WDF, the analytical formulas for the rms spatial width, the rms angular width and the M 2-factor of PCFT beams in turbulence have been derived, which can be applied to cases of different types of spatial power spectra of the refractive index fluctuations. Finally, the rms angular width and the M 2-factor of PCFT beams in turbulence have been discussed with numerical examples.

2. WDF of PCFT beams in a turbulent atmosphere

According to the beam model proposed by Li for a flat-topped (FT) beam [20, 21] and assuming that FT beams of any order contain equal power and under the assumption of the quasi-monochromatic scalar field, the field of an FT beam with circular symmetry can be expressed as [22]

EN(ρ)=m=1Nαmexp[(mpNρ2w02)]

where σ is the radial coordinate, w 0 is the waist width of the Gaussian beam, N is the order of the FT beam,

αm=(1)m+1N!m!(Nm)!

and

pN=2m=1Nm=1Nαmαmm+m.

For the sake of convenience, a constant is omitted in Eq. (1). For the case of N=1, Eqs. (1) and (3) are reduced to the field of a Gaussian beam and p 1=1, respectively.

The fully coherent beam can be extended to the partially coherent beam by introducing a Gaussian coherence function, and this type of partially coherent beam can be produced by the fully coherent beam passing through a random phase plate or a liquid crystal [23, 24]. Consequently, based on the beam model described by Eqs. (1)-(3) for FT beams and the Schell model used for describing the partially coherent light [25, 26], the cross-spectral density function of PCFT beams at the plane of z=0 can be written as [22, 23]

Γ(ρ1,ρ2,0)=<E(ρ1,0)E*(ρ2,0)>m
=m=1Nm=1Nαmαmexp[(mpNρ12w02+mpNρ22w02+ρ1ρ222σ02)],

where the angle brackets with subscript m denote averaging over the field ensemble, ρ 1 and ρ 2 are two vectors defining points on the transverse plane of z=0, w 0 represents the waist width of the Gaussian beam, σ 0 is the correlation length of the source. For the case of N=1, Eq. (4) is reduced to the cross-spectral density function of GSM beams.

Let us consider a PCFT beam propagating in a turbulent atmosphere from the plane z=0 into the half-space z>0. By using the paraxial form of the extended Huygens–Fresnel principle [3], the cross-spectral density function of PCFT beams through the turbulence can be expressed as [2, 3]

W(ρ,ρd,z)=(k2πz)2W(ρ,ρd,0)
×exp{ikz[(ρρ)·(ρdρd)]H(ρd,ρd,z)}d2ρd2ρd,

where z is the propagation distance, k=2π/λ is the wave number, λ is the wavelength, and

W(ρ,ρd,0)=Γ(ρ1,ρ2,0)=Γ(ρ+ρd2,ρρd2,0)

where the central abscissa coordinate systems are chosen, that is,

ρ=(ρ1+ρ2)2, ρd=ρ1ρ2,
ρ=(ρ1+ρ2)2, ρd=ρ1ρ2.

In Eq. (5) the term exp[-H(ρ d, ρ d, z)] represents the effect of the turbulence, and H can be written as

H(ρd,ρd,z)=4π2k2z010[1J0(κρdξ+(1ξ)ρd)]Φn(κ)κdκ,

where J0 is the Bessel function of the first kind and zero order, Φ n is the spatial power spectrum of the refractive index fluctuations of the turbulent atmosphere.

It is well known that the WDF is especially suitable for the treatment of partially coherent beams. The WDF can be expressed in terms of the cross-spectral density W(ρ, ρ d,z) as [11, 12]

h(ρ,θ,z)=(k2π)2W(ρ,ρd,z)exp(ikθ·ρd)d2ρd

where vector θ=(θ x,θ y), k θ x and y is the wave vector component along the x-axis and y-axis, respectively. Hence θ=(θ 2 x+θ 2 y)1/2 represents an angle of propagation (without taking the evanescent waves into account). Integration of function h over the angular variables θ x and θ y gives the beam intensity, and its integral over the spatial variables x and y is proportional to the radiant intensity of the field.

On the basis of inverse Fourier transform of the Dirac delta function and its property of even function (Chap. 5 of Ref. [27]), we obtain

δ(ρρ)=1(2π)2exp[±iκd·(ρρ)]d2κd.

Then, the cross-spectral density of the beams in the source plane (z=0) can be rewritten as

W(ρ,ρd,0)1(2π)2W(ρ,ρd,0)exp[iκd·(ρρ)]d2κdd2ρ.

On substituting from Eq. (12) into Eq. (5) and using Eq. (11) and the property of the Dirac delta function (p. 75 of Ref. [27])

f(x)δ(x)dx=f(0),

we obtain

W(ρ,ρd,z)=1(2π)2W(ρ,ρd+zkκd,0)
×exp[iρ·κd+iκd·ρH(ρd,ρd+zkκd,z)]d2κdd2ρ

where the term W(ρ,ρd+zkκd,0) is given by the expression from Eqs. (4) and (6), i.e.,

W(ρ,ρd+zkκd,0)=m=1Nm=1Nαmαmexp({pNw02(m+m)ρ2
+pNw02(mm)ρ·(ρd+zkκd)
+[pN4w02(m+m)+12σ02]ρd+zkκd2}).

On substituting from Eq. (14) into Eq. (10) and calculating the integral with respect to ρ by using the formula (3.3232) in Ref. [31]

exp(s2x2±qx)dx=πsexp(q24s2), (s>0)

the WDF of PCFT beams through a turbulent atmosphere can be expressed as

h(ρ,θ,z)=w02k216π3pNm=1Nm=1N[αmαmm+m
×exp(aρd2bκd2+cρd·κdiρ·κdikθ·ρdH)d2κdd2ρd],

where κ d=|κ d|=(κ 2 dx+κ 2 dy)1/2, and

H=4π2k2z01dξ0[1J0(κzkκdξ+ρd)]Φn(κ)κdκ,
a=1w02(mmpNm+m+12α2),
b=w024(m+m)pN+z2k2w02(mmpNm+m+12α2)iz2k(mmm+m),
c=2zkw02(mmpNm+m+12α2)+i2(mmm+m),

where

α=σ0w0

denotes the degree of global coherence of the source.

3. The rms spatial and angular spreads of PCFT beams in a turbulent atmosphere

The moments of the order n 1+n 2+m 1+m 2 of WDF h for three-dimensional beams are given by the expression [12, 13]

<xn1yn2θxm1θym2>=1Pxn1yn2θxm1θym2h(ρ,θ,z)d2ρd2θ

where P=∬h(ρ,θ,z)d2 ρd2 θ is the total irradiance of the beams, and P=πw 2 0/2 for the case of PCFT beams.

As far as the second-order field correlations are concerned, a partially coherent beam can be characterized by the WDF. It can be seen from Eq. (23) that the moments at the plane of z=const are determined by the WDF at the same plane. Consequently, using Eqs. (17) and (23), we can obtain the moments of PCFT beams at the plane of z=const in turbulence.

On substituting from Eq. (17) into Eq. (23), and recalling the formulas (p. 82, p. 95, and p. 141 of Ref. [27])

δ(s)=12πexp(isx)dx,
δ(n)(s)=12π(ix)nexp(isx)dx,(n=1,2)
f(x)δ(n)(x)dx=(1)nf(n)(0),(n=1,2)

(where δ (n) is nth derivative of the Dirac delta function, f(x) is an arbitrary function and f (n)(x) is its nth derivative), some second-order moments of PCFT beams in turbulence turn out to be

<ρ2>=<x2>+<y2>
=2w02pN2[m=1Nm=1Nαmαm(m+m)2]+2k2w02[1α2
+4m=1Nm=1Nmmαmαm(m+m)2]z2+43π2Tz3,
<θ2>=<θx2>+<θy2>
=2k2w02[1α2+4m=1Nm=1Nmmαmαm(m+m)2]+4π2Tz,
<ρ·θ>=<xθx>+<yθy>
=2k2w02[1α2+4m=1Nm=1Nmmαmαm(m+m)2]z+2π2Tz2,

whereas the other second-order moments (i.e., <xy>, <θ x θ y>, < y> and < x>) and all the first-order moments turn out to be zero. In Eqs. (27)-(29), the quantity T is given by the expression, i.e.,

T=0Φn(κ)κ3dκ

which denotes the intensity of the turbulent atmosphere.

It should be noted that the quantity T, which can only be determined by the spatial power spectrum of the refractive index fluctuations, is independent of the parameter of the beams.

According to the definition of rms spatial and angular width of light beams [12], the rms spatial and angular width of PCFT beams in turbulence can be expressed as

wN(z)(<ρ<ρ>2>)12=(<ρ2>)12
={2w02pN2[m=1Nm=1Nαmαm(m+m)2]+2k2w02[1α2
+4m=1Nm=1Nmmαmαm(m+m)2]z2+43π2Tz3}12,
θN(z)(<θ<θ>2>)12=(<θ2>)12
={2k2w02[1α2+4m=1Nm=1Nmmαmαm(m+m)2]+4π2Tz}12.

Equation (31), which is same as Eq. (44) of Ref. [22], represents the spatial spreading of PCFT beams in turbulence; and it is also consistent with the formula (30) of Ref. [6] derived by method of the “equivalent source”. Equation (32) represents the angular spreading of PCFT beams in turbulence, and is consistent with the formula (25a) of Ref. [7] derived by method of the “phase-space dynamics of beam”. The angular width given by Eq. (32) can be regarded as the width of the beams in spatial frequency domain. In other words, the angular width θ N(z ) given by Eq. (32) is equal to the divergence of the “source” at the plane of z=z′ in the free space, but the “source” is described by the WDF h(ρ,θ,z ) in turbulence, i.e., Eq. (17).

It can be readily seen from Eq. (32) that the rms angular spread of PCFT beams propagating in turbulence increases with the propagation distance z, whereas it maintains invariance in free space (i.e., for the case of T=0).

For the sake of comparison, the normalized rms angular width, which represents the relative angular spreading of the beams, is defined as

θrN(z)θN(z)θN(0)
={1+2π2k2w02[1α2+4m=1Nm=1Nmmαmαm(m+m)2]1Tz}12.

From Eq. (33), it is clear that the normalized rms angular spreading of PCFT beams in turbulence decreases with decreasing α; i.e., the beams with the worse coherence are less affected by turbulence. For a given propagation distance z, Eq. (33) indicates that the relative angular spreading increases with waist width w 0 and decreasing wavelength λ, whereas based on Eq. (31) the relative spatial spreading w N(z)/w N increases with decreasing w 0 and increasing λ. That is, the influence of atmospheric turbulence on the relative angular spreading is larger for PCFT beams with the larger w 0 and the smaller λ, but the influence on the relative spatial spreading is smaller. Consequently, in these cases neither the normalized rms spatial nor angular spreading can represents comprehensively the effect of turbulence on PCFT beams.

For the case of N=1, Eqs. (31)-(33) are easily simplified to the results of GSM beams, i.e.,

w1(z)=[w022+2k2w02(1+1α2)z2+43π2Tz3]12,
θ1(z)=[2k2w02(1+1α2)+4π2Tz]12,
θr1(z)=[1+2π2k2w02(1α2+1)1Tz]12.

Taking into account the modified von Karman spectrum [3], Eq. (34) is equal to Eq. (13) in [28]. For the case of α→∞, Eqs. (34)-(36) are further reduced to the expressions of fully coherent Gaussian beams in turbulence.

4. Beam propagation M2-factor of PCFT beams in a turbulent atmosphere

For three-dimensional beams, the beam propagation M 2-factor can be defined as [12]

M2(z)=k(<ρ2><θ2><ρ·θ>2)12
=k[(<x2>+<y2>)(<θx2>+<θy2>)(<xθx>+<yθy>)2]12.

It should be noted that in Eq. (37) the M 2-factor of a beam can be defined in terms of second-order moments of the WDF in the same plane, and it can also be calculated from the WDF h at any plane of z=const. Consequently, the definition of M 2-factor, i.e. Eq. (37), is valid for the case of the presence of the turbulence. In general, the value of M 2(z) in turbulence is equal to the one that is measured at the distance z.

On substituting from Eqs. (27)-(29) into Eq. (37), the beam propagation M 2-factor of PCFT beams through a turbulent atmosphere can be expressed as

MN2(z)=[MN4(0)+k2w02ATz+Bw01Tz3+43π4k2T2z4]12

where T is given by Eq. (30), and

MN4(0)=3AB16π4,
A=8π2pN2m=1Nm=1Nαmαm(m+m)2,

and

B=8π23[1α2+4m=1Nm=1Nmmαmαm(m+m)2].

Note that the value of M 2(z ) given by Eq. (38) is equal to the M 2-factor of the “source” at the plane of z=z′ in the free space, but the “source” is described by the WDF h(ρ,θ,z ) described by Eq. (17) in turbulence. Equation (38), which is the main result of this paper, indicates that the M 2-factor of PCFT beams in turbulence increases with the propagation distance z. For the case of T=0 (in free space), Eq. (38) shows that the M 2-factor, which is only determined by the beam order and coherent parameter of the source, maintains constant (cf. Refs. [29, 30]).

For the sake of convenience, the normalized M 2-factor, which represents the relative M 2-factor of the beams, is introduced by the expression

MrN2(z)MN2(z)MN2(0)
=(1+16π4k2w023BTz+16π43w02ATz3+64π8k29ABT2z4)12.

It can be shown from Eqs. (40) and (41) that in Eq. (42), the coefficient A only depends on N, the coefficient B only depends on N and α, and the value of B increases with decreasing α. Further, it can be readily seen from Eq. (42) that the normalized M 2-factor of PCFT beams decreases with decreasing α and increasing λ. Consequently, the effect of atmospheric turbulence on the normalized M 2-factor of PCFT beams for the worse coherence and the longer wavelength is smaller. It can be also seen from Eq. (42) that for a given propagation distance z, there exists a waist width w 0=w 0min corresponding to the minimal value of the normalized M 2-factor for the beams with different waist widths.

For the cases mentioned in section 3, neither the spatial nor angular spreading can denote comprehensively the effect of atmospheric turbulence on PCFT beams at the plane of z=const for different values of w 0 and of λ, but now we can use the normalized M 2-factor for these cases. Namely, in these cases the smaller value of the normalized M 2-factor of a beam indicates that the beam is less affected by turbulence. Therefore, the normalized M2-factor can show more comprehensively the effect of the turbulence on the optical beams than the normalized spatial and angular width.

For the case of N=1, Eqs. (38) and (42) are simplified to the M 2-factor and the normalized M 2-factor of GSM beams, respectively, namely

M12(z)=[(1+1α2)+2π2k2w02Tz
+8π23w02(1+1α2)Tz3+4π4k23T2z4]12,
Mr12(z)=[1+2π2k2w02(1+1α2)1Tz
+8π23w02Tz3+4π4k23(1+1α2)1T2z4]12.

The formula (43) is a generalization of known formula for the M 2-factor of GSM beams in free space [16]. Further, for the case of α→∞, Eqs. (43) and (44) are reduced to the M 2-factor and the normalized M 2-factor of fully coherent Gaussian beams in turbulence respectively.

5. Numerical calculation results and analysis

In this section, the Tatarskii spectrum is used as a model for atmospheric turbulence [3], i.e.,

Φn(κ)=0.033Cn2κ113exp(κ2κm2)

where C 2 n is the structure constant of the refractive index fluctuations of the turbulence and κ m=5.92/l 0 with l 0 being the inner scale turbulence.

On substituting from Eq. (45) into Eq. (30) and by using the formula 3.4621 in Ref. [31], the quantity T can be easily simplified to

T=0.1661Cn2l013

Numerical calculations for the rms angular width and normalized rms angular width of PCFT beams were performed on the basis of Eqs. (32), (33) and (46). Some illustrative examples are compiled in Figs. 1 and 2. The calculation parameters are α=2, λ=850 nm, w 0=0.05 m, C 2 n=10-15 m-2/3.

Figure 1 gives the rms angular and normalized rms angular widths for different values of beam order N versus the propagating distance in atmospheric turbulence, respectively. It can be seen from Fig. 1(a) that differing from the case of free space (absence of the turbulence), the rms angular width in turbulence obviously increases with the propagation distance, and it is larger for the beams with a higher beam order. Figure 1(b) implies that the normalized rms angular width in turbulence is smaller for the beams with a higher beam order; i.e., the relative angular spreading is less affected by atmospheric turbulence for the beams with a higher beam order.

 figure: Fig. 1.

Fig. 1. (a). rms angular width and (b) normalized rms angular width versus propagation distance for different values of beam order N, respectively. The calculation parameters are α=2, λ=850 nm, w 0=0.05 m, C 2 n=10-15 m-2/3.

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Figure 2 shows the variation of the normalized rms angular width for different values of coherent parameter α on the propagating distance in turbulence. It follows from Fig. 2 that the relative angular spreading of PCFT beams in turbulence decreases with decreasing α, meaning that the relative angular spreading for the beams with the worse spatial coherence is less affected by turbulence.

 figure: Fig. 2.

Fig. 2. Normalized rms angular width versus propagation distance for different values of α. The calculation parameters are N=10, λ=850 nm, w 0=0.05 m, C 2 n=10-15 m-2/3.

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Moreover, based on Eqs. (38), (42) and (46), numerical calculations for the M 2-factor and normalized M 2-factor of PCFT beams in turbulence were carried out. Some typical examples are compiled in Figs. 3-6.

Figure 3 shows the M 2-factor and normalized M 2-factor for different values of beam order N versus the propagating distance in atmospheric turbulence, respectively. The calculation parameters are α=2, λ=850 nm, w 0=0.05 m, C 2 n=10-15 m-2/3. It can be shown in Fig. 3(a) that the M 2-factor in turbulence apparently increases with the propagation distance, indicating that the beam quality obviously becomes worse with increasing propagation distance in turbulence. Figure 3(b) implies that the normalized M 2-factor in turbulence is smaller for PCFT beams with a higher beam order, showing the less influence of the turbulence on the relative M 2-factor of the beams with a higher beam order.

 figure: Fig. 3.

Fig. 3. (a). M 2-factor and (b) normalized M 2-factor versus propagation distance for different values of beam order N, respectively. The calculation parameters are α=2, λ=850 nm, w 0=0.05 m, C 2 n=10-15 m-2/3.

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Figure 4 gives the normalized M 2-factor for different values of coherent parameter α versus the propagating distance in atmospheric turbulence. The calculation parameters are N=10, λ=850 nm, w 0=0.05 m, C 2 n=10-15 m-2/3. It can be easily seen that the normalized M 2-factor of PCFT beams in turbulence decreases with decreasing α, showing the less influence of the turbulence on the relative M 2-factor of the beams with the worse spatial coherence.

 figure: Fig. 4.

Fig. 4. Normalized M 2-factor versus propagation distance for different values of α. The calculation parameters are N=10, λ=850 nm, w 0=0.05 m, C 2 n=10-15 m-2/3.

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Figure 5 gives the normalized M 2-factor for different values of beam order N at the plane of z=1 km versus the waist width w 0 of the GSM beam. The calculation parameters are α=2, z=1 km, C 2 n=10-15 m-2/3, λ=850 nm. It can also be seen from Fig. 5 that there exists a minimum of the normalized M 2-factor as the waist width w 0 increases. For example, for the case of N=1 (i.e. GSM beams), the minimal value of the normalized M 2-factor is of about 1.11, whereas for the case of N=10, the minimal value of the normalized M 2-factor is of about 1.10.

 figure: Fig. 5.

Fig. 5. Normalized M 2-factor versus w 0 for different values of beam order N. The calculation parameters are α=2, z=1 km, C 2 n=10-15 m-2/3, λ=850 nm.

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Figure 6 represents the normalized M 2-factor for different values of beam order N at the plane of z=1 km versus the wavelength of PCFT beams. The calculation parameters are α=2, z=1 km, C 2 n=10-15 m-2/3, w 0=0.05 m. It can be clearly seen from Fig. 6 that the normalized M 2-factor decreases with increasing wavelength λ, indicating that the relative M 2-factor of the beams with the longer wavelength is less affected by turbulence.

 figure: Fig. 6.

Fig. 6. Normalized M 2-factor versus wave length λ for different values of beam order N. The calculation parameters are α=2, z=1 km, C 2 n=10-15 m-2/3, w 0=0.05 m.

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6. Conclusions

In the present paper, by use of the extended Huygens–Fresnel principle and definition of the WDF, an expression for the WDF of PCFT beams with circular symmetry in a turbulent atmosphere has been given. The analytical formulas for the rms spatial width, the rms angular width, and the beam propagation M 2-factor of PCFT beams in turbulence have been derived based on the method of second-order moments of the WDF. For the case of beam order N=1, all results obtained in this paper can be simplified to those of GSM beams. Furthermore, the rms angular width and M 2-factor in turbulence have been discussed with numerical examples.

It can be shown that the beam propagation M 2-factor of PCFT beams depends on the beam order, the degree of global coherence of the source, the waist width, the wavelength, and the spatial power spectrum of the refractive index fluctuations of atmospheric turbulence. Moreover, the beam propagation M 2-factor of PCFT beams increases obviously with the propagation distance in turbulence. From the viewpoint of relative variation of the M 2-factor due to the turbulent atmosphere, the effect of the turbulence on PCFT beams is smaller for beams with a higher order, a lower degree of global coherence of the source, and a larger wavelength.

Acknowledgment

This research was supported by the Program for New Century Excellent Talents in University (NCET-05-0784)

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20. Y. Li, “Light beams with flat-topped profiles,” Opt. Lett. 27, 1007–1009 (2002). [CrossRef]  

21. Y. Li, “New expressions for flat-topped light beams,” Opt. Commun. 206, 225–234 (2002). [CrossRef]  

22. Y. Dan, B. Zhang, and P. Pan, “Propagation of partially coherent flat-topped beams through a turbulent atmosphere,” J. Opt. Soc. Am. A 25, 2223–2231 (2008). [CrossRef]  

23. X. Ji, X. Chen, S. Chen, X. Li, and B. Lü, “Influence of atmospheric turbulence on the spatial correlation properties of partially coherent flat-topped beams,” J. Opt. Soc. Am. A 24, 3554–3563 (2007). [CrossRef]  

24. M. Zahid and M. S. Zubairy, “Directionality of partially coherent Bessel-Gauss beams,” Opt. Commun. 70, 361–364 (1989). [CrossRef]  

25. A. Starikov and E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. 72, 923–928 (1982). [CrossRef]  

26. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge,1995).

27. R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York,1986).

28. X. Ji and B. Lü, “Turbulence-induced quality degradation of partially coherent beams,” Opt. Commun. 251, 231–236 (2005). [CrossRef]  

29. G. Wu, H. Guo, and D. Deng, “Paraxial propagation of partially coherent flat-topped beam,” Opt. Commun. 260, 687–690 (2006). [CrossRef]  

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

31. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic, New York,1980).

References

  • View by:

  1. A. Ishimaru, “Theory and application of wave propagation and scattering in random media,” Proc. IEEE 65, 1030–1061 (1977).
    [Crossref]
  2. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York,1978).
  3. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, Bellingham,2005).
    [Crossref]
  4. J. Wu, “Propagation of a Gaussian-Schell beam through turbulent media,” J. Mod. Opt. 37, 671–684 (1990).
    [Crossref]
  5. J. Wu and A. D. Boardman, “Coherence length of a Gaussian-Schell beam and atmospheric turbulence,” J. Mod. Opt. 38, 1355–1363 (1991).
    [Crossref]
  6. G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592–1598 (2002).
    [Crossref]
  7. S. A. Ponomarenko, J. -J. Greffet, and E. Wolf, “The diffusion of partially coherent beams in turbulent media,” Opt. Commun. 208, 1–8 (2002).
    [Crossref]
  8. T. Shirai, A. Dogariu, and E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 20, 1094–1102 (2003).
    [Crossref]
  9. A. Dogariu and S. Amarande, “Propagation of partially coherent beams: turbulence-induced degradation,” Opt. Lett. 28, 10–12 (2003).
    [Crossref] [PubMed]
  10. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968).
    [Crossref]
  11. M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A 3, 1227–1238 (1986).
    [Crossref]
  12. J. Serna, R. Martínez-Herrero, and P. M. Mejías, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
    [Crossref]
  13. R. Martínez-Herrero, G. Piquero, and P. M. Mejías, “On the propagation of the kurtosis parameter of general beams,” Opt. Commun. 115, 225–232 (1995).
    [Crossref]
  14. A. E. Siegman, “New developments in laser resonators,” Proc. SPIE. 1224, 2–14 (1990).
    [Crossref]
  15. F. Gori, M. Santarsiero, and A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82, 197–203 (1991).
    [Crossref]
  16. 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, 106–112 (1999).
    [Crossref]
  17. C. Palma, P. De Santis, G. Cincotti, and G. Guattari, “Propagation and coherence evolution of optical beams in gain media,” J. Mod. Opt. 43, 139–153 (1996).
    [Crossref]
  18. B. zhang, Q. Wen, and X. Guo, “Beam propagation factor of partially coherent beams in gain or absorbing media,” Optik 117, 123–127 (2006).
    [Crossref]
  19. M. H. Mahdieh, “Numerical approach to laser beam propagation through turbulent atmosphere and evaluation of beam quality factor,” Opt. Commun. 281, 3395–3402 (2008).
    [Crossref]
  20. Y. Li, “Light beams with flat-topped profiles,” Opt. Lett. 27, 1007–1009 (2002).
    [Crossref]
  21. Y. Li, “New expressions for flat-topped light beams,” Opt. Commun. 206, 225–234 (2002).
    [Crossref]
  22. Y. Dan, B. Zhang, and P. Pan, “Propagation of partially coherent flat-topped beams through a turbulent atmosphere,” J. Opt. Soc. Am. A 25, 2223–2231 (2008).
    [Crossref]
  23. X. Ji, X. Chen, S. Chen, X. Li, and B. Lü, “Influence of atmospheric turbulence on the spatial correlation properties of partially coherent flat-topped beams,” J. Opt. Soc. Am. A 24, 3554–3563 (2007).
    [Crossref]
  24. M. Zahid and M. S. Zubairy, “Directionality of partially coherent Bessel-Gauss beams,” Opt. Commun. 70, 361–364 (1989).
    [Crossref]
  25. A. Starikov and E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. 72, 923–928 (1982).
    [Crossref]
  26. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge,1995).
  27. R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York,1986).
  28. X. Ji and B. Lü, “Turbulence-induced quality degradation of partially coherent beams,” Opt. Commun. 251, 231–236 (2005).
    [Crossref]
  29. G. Wu, H. Guo, and D. Deng, “Paraxial propagation of partially coherent flat-topped beam,” Opt. Commun. 260, 687–690 (2006).
    [Crossref]
  30. H. T. Eyyuboğlu, Ç. Arpali, and Y. Baykal, “Flat topped beams and their characteristics in turbulent media,” Opt. Express 14, 4196–4207 (2006).
    [Crossref] [PubMed]
  31. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic, New York,1980).

2008 (2)

M. H. Mahdieh, “Numerical approach to laser beam propagation through turbulent atmosphere and evaluation of beam quality factor,” Opt. Commun. 281, 3395–3402 (2008).
[Crossref]

Y. Dan, B. Zhang, and P. Pan, “Propagation of partially coherent flat-topped beams through a turbulent atmosphere,” J. Opt. Soc. Am. A 25, 2223–2231 (2008).
[Crossref]

2007 (1)

X. Ji, X. Chen, S. Chen, X. Li, and B. Lü, “Influence of atmospheric turbulence on the spatial correlation properties of partially coherent flat-topped beams,” J. Opt. Soc. Am. A 24, 3554–3563 (2007).
[Crossref]

2006 (3)

B. zhang, Q. Wen, and X. Guo, “Beam propagation factor of partially coherent beams in gain or absorbing media,” Optik 117, 123–127 (2006).
[Crossref]

G. Wu, H. Guo, and D. Deng, “Paraxial propagation of partially coherent flat-topped beam,” Opt. Commun. 260, 687–690 (2006).
[Crossref]

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

2005 (1)

X. Ji and B. Lü, “Turbulence-induced quality degradation of partially coherent beams,” Opt. Commun. 251, 231–236 (2005).
[Crossref]

2003 (2)

T. Shirai, A. Dogariu, and E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 20, 1094–1102 (2003).
[Crossref]

A. Dogariu and S. Amarande, “Propagation of partially coherent beams: turbulence-induced degradation,” Opt. Lett. 28, 10–12 (2003).
[Crossref] [PubMed]

2002 (4)

G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592–1598 (2002).
[Crossref]

S. A. Ponomarenko, J. -J. Greffet, and E. Wolf, “The diffusion of partially coherent beams in turbulent media,” Opt. Commun. 208, 1–8 (2002).
[Crossref]

Y. Li, “Light beams with flat-topped profiles,” Opt. Lett. 27, 1007–1009 (2002).
[Crossref]

Y. Li, “New expressions for flat-topped light beams,” Opt. Commun. 206, 225–234 (2002).
[Crossref]

1999 (1)

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, 106–112 (1999).
[Crossref]

1996 (1)

C. Palma, P. De Santis, G. Cincotti, and G. Guattari, “Propagation and coherence evolution of optical beams in gain media,” J. Mod. Opt. 43, 139–153 (1996).
[Crossref]

1995 (1)

R. Martínez-Herrero, G. Piquero, and P. M. Mejías, “On the propagation of the kurtosis parameter of general beams,” Opt. Commun. 115, 225–232 (1995).
[Crossref]

1991 (3)

J. Serna, R. Martínez-Herrero, and P. M. Mejías, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
[Crossref]

F. Gori, M. Santarsiero, and A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82, 197–203 (1991).
[Crossref]

J. Wu and A. D. Boardman, “Coherence length of a Gaussian-Schell beam and atmospheric turbulence,” J. Mod. Opt. 38, 1355–1363 (1991).
[Crossref]

1990 (2)

J. Wu, “Propagation of a Gaussian-Schell beam through turbulent media,” J. Mod. Opt. 37, 671–684 (1990).
[Crossref]

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE. 1224, 2–14 (1990).
[Crossref]

1989 (1)

M. Zahid and M. S. Zubairy, “Directionality of partially coherent Bessel-Gauss beams,” Opt. Commun. 70, 361–364 (1989).
[Crossref]

1986 (1)

M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A 3, 1227–1238 (1986).
[Crossref]

1982 (1)

1977 (1)

A. Ishimaru, “Theory and application of wave propagation and scattering in random media,” Proc. IEEE 65, 1030–1061 (1977).
[Crossref]

1968 (1)

Amarande, S.

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, Bellingham,2005).
[Crossref]

Arpali, Ç.

Bastiaans, M. J.

M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A 3, 1227–1238 (1986).
[Crossref]

Baykal, Y.

Boardman, A. D.

J. Wu and A. D. Boardman, “Coherence length of a Gaussian-Schell beam and atmospheric turbulence,” J. Mod. Opt. 38, 1355–1363 (1991).
[Crossref]

Borghi, R.

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, 106–112 (1999).
[Crossref]

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York,1986).

Chen, S.

X. Ji, X. Chen, S. Chen, X. Li, and B. Lü, “Influence of atmospheric turbulence on the spatial correlation properties of partially coherent flat-topped beams,” J. Opt. Soc. Am. A 24, 3554–3563 (2007).
[Crossref]

Chen, X.

X. Ji, X. Chen, S. Chen, X. Li, and B. Lü, “Influence of atmospheric turbulence on the spatial correlation properties of partially coherent flat-topped beams,” J. Opt. Soc. Am. A 24, 3554–3563 (2007).
[Crossref]

Cincotti, G.

M. Santarsiero, F. Gori, R. Borghi, G. Cincotti, and P. Vahimaa, “Spreading properties of beams radiated by partially coherent Schell-model sources,” J. Opt. Soc. Am. A 16, 106–112 (1999).
[Crossref]

C. Palma, P. De Santis, G. Cincotti, and G. Guattari, “Propagation and coherence evolution of optical beams in gain media,” J. Mod. Opt. 43, 139–153 (1996).
[Crossref]

Dan, Y.

Y. Dan, B. Zhang, and P. Pan, “Propagation of partially coherent flat-topped beams through a turbulent atmosphere,” J. Opt. Soc. Am. A 25, 2223–2231 (2008).
[Crossref]

De Santis, P.

C. Palma, P. De Santis, G. Cincotti, and G. Guattari, “Propagation and coherence evolution of optical beams in gain media,” J. Mod. Opt. 43, 139–153 (1996).
[Crossref]

Deng, D.

G. Wu, H. Guo, and D. Deng, “Paraxial propagation of partially coherent flat-topped beam,” Opt. Commun. 260, 687–690 (2006).
[Crossref]

Dogariu, A.

A. Dogariu and S. Amarande, “Propagation of partially coherent beams: turbulence-induced degradation,” Opt. Lett. 28, 10–12 (2003).
[Crossref] [PubMed]

T. Shirai, A. Dogariu, and E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 20, 1094–1102 (2003).
[Crossref]

Eyyuboglu, H. T.

Gbur, G.

G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592–1598 (2002).
[Crossref]

Gori, F.

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, 106–112 (1999).
[Crossref]

F. Gori, M. Santarsiero, and A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82, 197–203 (1991).
[Crossref]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic, New York,1980).

Greffet, J. -J.

S. A. Ponomarenko, J. -J. Greffet, and E. Wolf, “The diffusion of partially coherent beams in turbulent media,” Opt. Commun. 208, 1–8 (2002).
[Crossref]

Guattari, G.

C. Palma, P. De Santis, G. Cincotti, and G. Guattari, “Propagation and coherence evolution of optical beams in gain media,” J. Mod. Opt. 43, 139–153 (1996).
[Crossref]

Guo, H.

G. Wu, H. Guo, and D. Deng, “Paraxial propagation of partially coherent flat-topped beam,” Opt. Commun. 260, 687–690 (2006).
[Crossref]

Guo, X.

B. zhang, Q. Wen, and X. Guo, “Beam propagation factor of partially coherent beams in gain or absorbing media,” Optik 117, 123–127 (2006).
[Crossref]

Ishimaru, A.

A. Ishimaru, “Theory and application of wave propagation and scattering in random media,” Proc. IEEE 65, 1030–1061 (1977).
[Crossref]

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

Ji, X.

X. Ji, X. Chen, S. Chen, X. Li, and B. Lü, “Influence of atmospheric turbulence on the spatial correlation properties of partially coherent flat-topped beams,” J. Opt. Soc. Am. A 24, 3554–3563 (2007).
[Crossref]

X. Ji and B. Lü, “Turbulence-induced quality degradation of partially coherent beams,” Opt. Commun. 251, 231–236 (2005).
[Crossref]

Li, X.

X. Ji, X. Chen, S. Chen, X. Li, and B. Lü, “Influence of atmospheric turbulence on the spatial correlation properties of partially coherent flat-topped beams,” J. Opt. Soc. Am. A 24, 3554–3563 (2007).
[Crossref]

Li, Y.

Y. Li, “Light beams with flat-topped profiles,” Opt. Lett. 27, 1007–1009 (2002).
[Crossref]

Y. Li, “New expressions for flat-topped light beams,” Opt. Commun. 206, 225–234 (2002).
[Crossref]

Lü, B.

X. Ji, X. Chen, S. Chen, X. Li, and B. Lü, “Influence of atmospheric turbulence on the spatial correlation properties of partially coherent flat-topped beams,” J. Opt. Soc. Am. A 24, 3554–3563 (2007).
[Crossref]

X. Ji and B. Lü, “Turbulence-induced quality degradation of partially coherent beams,” Opt. Commun. 251, 231–236 (2005).
[Crossref]

Mahdieh, M. H.

M. H. Mahdieh, “Numerical approach to laser beam propagation through turbulent atmosphere and evaluation of beam quality factor,” Opt. Commun. 281, 3395–3402 (2008).
[Crossref]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge,1995).

Martínez-Herrero, R.

R. Martínez-Herrero, G. Piquero, and P. M. Mejías, “On the propagation of the kurtosis parameter of general beams,” Opt. Commun. 115, 225–232 (1995).
[Crossref]

J. Serna, R. Martínez-Herrero, and P. M. Mejías, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
[Crossref]

Mejías, P. M.

R. Martínez-Herrero, G. Piquero, and P. M. Mejías, “On the propagation of the kurtosis parameter of general beams,” Opt. Commun. 115, 225–232 (1995).
[Crossref]

J. Serna, R. Martínez-Herrero, and P. M. Mejías, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
[Crossref]

Palma, C.

C. Palma, P. De Santis, G. Cincotti, and G. Guattari, “Propagation and coherence evolution of optical beams in gain media,” J. Mod. Opt. 43, 139–153 (1996).
[Crossref]

Pan, P.

Y. Dan, B. Zhang, and P. Pan, “Propagation of partially coherent flat-topped beams through a turbulent atmosphere,” J. Opt. Soc. Am. A 25, 2223–2231 (2008).
[Crossref]

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, Bellingham,2005).
[Crossref]

Piquero, G.

R. Martínez-Herrero, G. Piquero, and P. M. Mejías, “On the propagation of the kurtosis parameter of general beams,” Opt. Commun. 115, 225–232 (1995).
[Crossref]

Ponomarenko, S. A.

S. A. Ponomarenko, J. -J. Greffet, and E. Wolf, “The diffusion of partially coherent beams in turbulent media,” Opt. Commun. 208, 1–8 (2002).
[Crossref]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic, New York,1980).

Santarsiero, M.

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, 106–112 (1999).
[Crossref]

F. Gori, M. Santarsiero, and A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82, 197–203 (1991).
[Crossref]

Serna, J.

J. Serna, R. Martínez-Herrero, and P. M. Mejías, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
[Crossref]

Shirai, T.

T. Shirai, A. Dogariu, and E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 20, 1094–1102 (2003).
[Crossref]

Siegman, A. E.

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE. 1224, 2–14 (1990).
[Crossref]

Sona, A.

F. Gori, M. Santarsiero, and A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82, 197–203 (1991).
[Crossref]

Starikov, A.

Vahimaa, P.

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, 106–112 (1999).
[Crossref]

Walther, A.

Wen, Q.

B. zhang, Q. Wen, and X. Guo, “Beam propagation factor of partially coherent beams in gain or absorbing media,” Optik 117, 123–127 (2006).
[Crossref]

Wolf, E.

T. Shirai, A. Dogariu, and E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 20, 1094–1102 (2003).
[Crossref]

S. A. Ponomarenko, J. -J. Greffet, and E. Wolf, “The diffusion of partially coherent beams in turbulent media,” Opt. Commun. 208, 1–8 (2002).
[Crossref]

G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592–1598 (2002).
[Crossref]

A. Starikov and E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. 72, 923–928 (1982).
[Crossref]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge,1995).

Wu, G.

G. Wu, H. Guo, and D. Deng, “Paraxial propagation of partially coherent flat-topped beam,” Opt. Commun. 260, 687–690 (2006).
[Crossref]

Wu, J.

J. Wu and A. D. Boardman, “Coherence length of a Gaussian-Schell beam and atmospheric turbulence,” J. Mod. Opt. 38, 1355–1363 (1991).
[Crossref]

J. Wu, “Propagation of a Gaussian-Schell beam through turbulent media,” J. Mod. Opt. 37, 671–684 (1990).
[Crossref]

Zahid, M.

M. Zahid and M. S. Zubairy, “Directionality of partially coherent Bessel-Gauss beams,” Opt. Commun. 70, 361–364 (1989).
[Crossref]

Zhang, B.

Y. Dan, B. Zhang, and P. Pan, “Propagation of partially coherent flat-topped beams through a turbulent atmosphere,” J. Opt. Soc. Am. A 25, 2223–2231 (2008).
[Crossref]

B. zhang, Q. Wen, and X. Guo, “Beam propagation factor of partially coherent beams in gain or absorbing media,” Optik 117, 123–127 (2006).
[Crossref]

Zubairy, M. S.

M. Zahid and M. S. Zubairy, “Directionality of partially coherent Bessel-Gauss beams,” Opt. Commun. 70, 361–364 (1989).
[Crossref]

J. Mod. Opt. (3)

J. Wu, “Propagation of a Gaussian-Schell beam through turbulent media,” J. Mod. Opt. 37, 671–684 (1990).
[Crossref]

J. Wu and A. D. Boardman, “Coherence length of a Gaussian-Schell beam and atmospheric turbulence,” J. Mod. Opt. 38, 1355–1363 (1991).
[Crossref]

C. Palma, P. De Santis, G. Cincotti, and G. Guattari, “Propagation and coherence evolution of optical beams in gain media,” J. Mod. Opt. 43, 139–153 (1996).
[Crossref]

J. Opt. Soc. Am. (9)

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, 106–112 (1999).
[Crossref]

A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968).
[Crossref]

M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A 3, 1227–1238 (1986).
[Crossref]

J. Serna, R. Martínez-Herrero, and P. M. Mejías, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
[Crossref]

G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592–1598 (2002).
[Crossref]

T. Shirai, A. Dogariu, and E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 20, 1094–1102 (2003).
[Crossref]

Y. Dan, B. Zhang, and P. Pan, “Propagation of partially coherent flat-topped beams through a turbulent atmosphere,” J. Opt. Soc. Am. A 25, 2223–2231 (2008).
[Crossref]

X. Ji, X. Chen, S. Chen, X. Li, and B. Lü, “Influence of atmospheric turbulence on the spatial correlation properties of partially coherent flat-topped beams,” J. Opt. Soc. Am. A 24, 3554–3563 (2007).
[Crossref]

A. Starikov and E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. 72, 923–928 (1982).
[Crossref]

Opt. Commun. (8)

Y. Li, “New expressions for flat-topped light beams,” Opt. Commun. 206, 225–234 (2002).
[Crossref]

M. Zahid and M. S. Zubairy, “Directionality of partially coherent Bessel-Gauss beams,” Opt. Commun. 70, 361–364 (1989).
[Crossref]

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Figures (6)

Fig. 1.
Fig. 1. (a). rms angular width and (b) normalized rms angular width versus propagation distance for different values of beam order N, respectively. The calculation parameters are α=2, λ=850 nm, w 0=0.05 m, C 2 n=10-15 m-2/3.
Fig. 2.
Fig. 2. Normalized rms angular width versus propagation distance for different values of α. The calculation parameters are N=10, λ=850 nm, w 0=0.05 m, C 2 n=10-15 m-2/3.
Fig. 3.
Fig. 3. (a). M 2-factor and (b) normalized M 2-factor versus propagation distance for different values of beam order N, respectively. The calculation parameters are α=2, λ=850 nm, w 0=0.05 m, C 2 n=10-15 m-2/3.
Fig. 4.
Fig. 4. Normalized M 2-factor versus propagation distance for different values of α. The calculation parameters are N=10, λ=850 nm, w 0=0.05 m, C 2 n=10-15 m-2/3.
Fig. 5.
Fig. 5. Normalized M 2-factor versus w 0 for different values of beam order N. The calculation parameters are α=2, z=1 km, C 2 n=10-15 m-2/3, λ=850 nm.
Fig. 6.
Fig. 6. Normalized M 2-factor versus wave length λ for different values of beam order N. The calculation parameters are α=2, z=1 km, C 2 n=10-15 m-2/3, w 0=0.05 m.

Equations (64)

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E N ( ρ ) = m = 1 N α m exp [ ( m p N ρ 2 w 0 2 ) ]
α m = ( 1 ) m + 1 N ! m ! ( N m ) !
p N = 2 m = 1 N m = 1 N α m α m m + m .
Γ ( ρ 1 , ρ 2 , 0 ) = < E ( ρ 1 , 0 ) E * ( ρ 2 , 0 ) > m
= m = 1 N m = 1 N α m α m exp [ ( m p N ρ 1 2 w 0 2 + m p N ρ 2 2 w 0 2 + ρ 1 ρ 2 2 2 σ 0 2 ) ] ,
W ( ρ , ρ d , z ) = ( k 2 π z ) 2 W ( ρ , ρ d , 0 )
× exp { ik z [ ( ρ ρ ) · ( ρ d ρ d ) ] H ( ρ d , ρ d , z ) } d 2 ρ d 2 ρ d ,
W ( ρ , ρ d , 0 ) = Γ ( ρ 1 , ρ 2 , 0 ) = Γ ( ρ + ρ d 2 , ρ ρ d 2 , 0 )
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 0 [ 1 J 0 ( κ ρ d ξ + ( 1 ξ ) ρ d ) ] Φ n ( κ ) κ d κ ,
h ( ρ , θ , z ) = ( k 2 π ) 2 W ( ρ , ρ d , z ) exp ( ik θ · ρ d ) d 2 ρ d
δ ( ρ ρ ) = 1 ( 2 π ) 2 exp [ ± i κ d · ( ρ ρ ) ] d 2 κ d .
W ( ρ , ρ d , 0 ) 1 ( 2 π ) 2 W ( ρ , ρ d , 0 ) exp [ i κ d · ( ρ ρ ) ] d 2 κ d d 2 ρ .
f ( x ) δ ( x ) d x = f ( 0 ) ,
W ( ρ , ρ d , z ) = 1 ( 2 π ) 2 W ( ρ , ρ d + z k κ d , 0 )
× exp [ i ρ · κ d + i κ d · ρ H ( ρ d , ρ d + z k κ d , z ) ] d 2 κ d d 2 ρ
W ( ρ , ρ d + z k κ d , 0 ) = m = 1 N m = 1 N α m α m exp ( { p N w 0 2 ( m + m ) ρ 2
+ p N w 0 2 ( m m ) ρ · ( ρ d + z k κ d )
+ [ p N 4 w 0 2 ( m + m ) + 1 2 σ 0 2 ] ρ d + z k κ d 2 } ) .
exp ( s 2 x 2 ± q x ) d x = π s exp ( q 2 4 s 2 ) ,   ( s > 0 )
h ( ρ , θ , z ) = w 0 2 k 2 16 π 3 p N m = 1 N m = 1 N [ α m α m m + m
× exp ( a ρ d 2 b κ d 2 + c ρ d · κ d i ρ · κ d ik θ · ρ d H ) d 2 κ d d 2 ρ d ] ,
H = 4 π 2 k 2 z 0 1 d ξ 0 [ 1 J 0 ( κ z k κ d ξ + ρ d ) ] Φ n ( κ ) κ d κ ,
a = 1 w 0 2 ( mm p N m + m + 1 2 α 2 ) ,
b = w 0 2 4 ( m + m ) p N + z 2 k 2 w 0 2 ( mm p N m + m + 1 2 α 2 ) iz 2 k ( m m m + m ) ,
c = 2 z k w 0 2 ( mm p N m + m + 1 2 α 2 ) + i 2 ( m m m + m ) ,
α = σ 0 w 0
< 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 ( ρ , θ , z ) d 2 ρ d 2 θ
δ ( s ) = 1 2 π exp ( isx ) d x ,
δ ( n ) ( s ) = 1 2 π ( ix ) n exp ( isx ) d x , ( n = 1 , 2 )
f ( x ) δ ( n ) ( x ) d x = ( 1 ) n f ( n ) ( 0 ) , ( n = 1 , 2 )
< ρ 2 > = < x 2 > + < y 2 >
= 2 w 0 2 p N 2 [ m = 1 N m = 1 N α m α m ( m + m ) 2 ] + 2 k 2 w 0 2 [ 1 α 2
+ 4 m = 1 N m = 1 N mm α m α m ( m + m ) 2 ] z 2 + 4 3 π 2 T z 3 ,
< θ 2 > = < θ x 2 > + < θ y 2 >
= 2 k 2 w 0 2 [ 1 α 2 + 4 m = 1 N m = 1 N mm α m α m ( m + m ) 2 ] + 4 π 2 Tz ,
< ρ · θ > = < x θ x > + < y θ y >
= 2 k 2 w 0 2 [ 1 α 2 + 4 m = 1 N m = 1 N mm α m α m ( m + m ) 2 ] z + 2 π 2 Tz 2 ,
T = 0 Φ n ( κ ) κ 3 d κ
w N ( z ) ( < ρ < ρ > 2 > ) 1 2 = ( < ρ 2 > ) 1 2
= { 2 w 0 2 p N 2 [ m = 1 N m = 1 N α m α m ( m + m ) 2 ] + 2 k 2 w 0 2 [ 1 α 2
+ 4 m = 1 N m = 1 N mm α m α m ( m + m ) 2 ] z 2 + 4 3 π 2 T z 3 } 1 2 ,
θ N ( z ) ( < θ < θ > 2 > ) 1 2 = ( < θ 2 > ) 1 2
= { 2 k 2 w 0 2 [ 1 α 2 + 4 m = 1 N m = 1 N mm α m α m ( m + m ) 2 ] + 4 π 2 Tz } 1 2 .
θ r N ( z ) θ N ( z ) θ N ( 0 )
= { 1 + 2 π 2 k 2 w 0 2 [ 1 α 2 + 4 m = 1 N m = 1 N mm α m α m ( m + m ) 2 ] 1 Tz } 1 2 .
w 1 ( z ) = [ w 0 2 2 + 2 k 2 w 0 2 ( 1 + 1 α 2 ) z 2 + 4 3 π 2 T z 3 ] 1 2 ,
θ 1 ( z ) = [ 2 k 2 w 0 2 ( 1 + 1 α 2 ) + 4 π 2 T z ] 1 2 ,
θ r 1 ( z ) = [ 1 + 2 π 2 k 2 w 0 2 ( 1 α 2 + 1 ) 1 Tz ] 1 2 .
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 .
M N 2 ( z ) = [ M N 4 ( 0 ) + k 2 w 0 2 ATz + B w 0 1 Tz 3 + 4 3 π 4 k 2 T 2 z 4 ] 1 2
M N 4 ( 0 ) = 3 AB 16 π 4 ,
A = 8 π 2 p N 2 m = 1 N m = 1 N α m α m ( m + m ) 2 ,
B = 8 π 2 3 [ 1 α 2 + 4 m = 1 N m = 1 N mm α m α m ( m + m ) 2 ] .
M r N 2 ( z ) M N 2 ( z ) M N 2 ( 0 )
= ( 1 + 16 π 4 k 2 w 0 2 3 B Tz + 16 π 4 3 w 0 2 A Tz 3 + 64 π 8 k 2 9 AB T 2 z 4 ) 1 2 .
M 1 2 ( z ) = [ ( 1 + 1 α 2 ) + 2 π 2 k 2 w 0 2 T z
+ 8 π 2 3 w 0 2 ( 1 + 1 α 2 ) T z 3 + 4 π 4 k 2 3 T 2 z 4 ] 1 2 ,
M r 1 2 ( z ) = [ 1 + 2 π 2 k 2 w 0 2 ( 1 + 1 α 2 ) 1 T z
+ 8 π 2 3 w 0 2 T z 3 + 4 π 4 k 2 3 ( 1 + 1 α 2 ) 1 T 2 z 4 ] 1 2 .
Φ n ( κ ) = 0 . 033 C n 2 κ 11 3 exp ( κ 2 κ m 2 )
T = 0 . 1661 C n 2 l 0 1 3

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