Abstract

The propagation of partially coherent Hermite-cosh-Gaussian (H-ChG) beams through atmospheric turbulence is studied in detail. The analytical expression for the angular spread of partially coherent H-ChG beams in turbulence is derived. It is shown that the angular spread of partially coherent H-ChG beams with smaller spatial correlation length σ0, smaller waist width w 0, smaller beam parameter Ω0, and larger beam orders m, n is less affected by turbulence than that of partially coherent H-ChG beams with larger σ0, w 0, Ω0, and smaller m, n. Under a certain condition partially coherent H-ChG beams may generate the same angular spread as a fully coherent Gaussian beam in free space and also in atmospheric turbulence. The angular spread of partially coherent Hermite-Gaussian (H-G), cosh-Gaussian (ChG), Gaussian Schell-model (GSM) beams, and fully coherent H-ChG, H-G, ChG, Gaussian beams is studied and treated as special cases of partially coherent H-ChG beams. The results are interpreted physically.

© 2008 Optical Society of America

1. Introduction

In 1978 Collett and Wolf predicted that partially coherent beams, like Gaussian Schell-model (GSM) beams, may have the same directionality as a fully coherent laser beam [1, 2]. It means that full spatial coherence in free space is not a necessary condition for highly directional light beams. The theoretical prediction was confirmed by the experiments [3, 4]. In 2003 Wolf and his collaborators showed that under a certain condition there exist the equivalent GSM beams in turbulence which may generate the same angular spread as a fully coherent laser beam [5]. Very recently, we have found that, besides the equivalent GSM beams, there also exist the equivalent partially coherent Hermite-Gaussian (H-G) beams which may have the same directionality as a fully coherent laser beam in free space and also in atmospheric turbulence [6].

On the other hand, Hermite-sinusoidal-Gaussian (H-SG) beams are solutions of the paraxial wave equation [7]. H-SG beams cover a broad range of beams such as Hermite sinh-(or cosh-) Gaussian (H-ShG or H-ChG), and Hermite sin- (or cos-) Gaussian (H-SiG or H-CoG) beams. The production and propagation of H-SG beams in free space were studied in Ref. [8]. The propagation of laser beams in atmospheric turbulence is a topic that has been of considerable theoretical and practical interest for a long time. The propagation of H-G and Laguerre-Gaussian (L-G) beams, CoG and ChG beams, H-ChG beams, and H-SiG and H-ShG beams etc. through the turbulent atmosphere were investigated by Youngs, Eyyuboğlu, Cai and their coworkers in Refs. [9–15]. However, the studies are limited to the spatially fully coherent case. The complex degree of coherence and degree of polarization for partially coherent general beams in atmospheric turbulence, and the influence of turbulence on the propagation of partially coherent ChG, CoG and twisted anisotropic GSM beams was recently studied in Refs. [16–19]. It was shown both theoretically and experimentally that partially coherent beams are less affected by turbulence than the fully coherent laser beams [18, 20, 21].

The aim of this paper is to study the angular spread of partially coherent H-ChG beams propagating through atmospheric turbulence. In comparison with previous work our study is more general. The main results obtained in this paper are illustrated by numerical examples and interpreted physically.

2. Angular spread of partially coherent Hermite-cosh-Gaussian beams in turbulence

The field distribution of H-ChG beams at the source plane z=0 in the Cartesian coordinate system reads as [7, 8]

U(ρ,z=0)=Hm(2w0ρx)Hn(2w0ρy)exp(ρx2+ρy2w02)cosh(Ω0ρx+Ω0ρy),

where ρ≡(ρx, ρy) is the two-dimensional position vector at the source plane z=0, Hm(·) and Hn(·) denote the mth and nth order Hermite polynomials, w 0 is the waist width of the Gaussian part, Ω0 is the beam parameter associated with the cosh part.

The fully coherent beam can be extended to the partially coherent one by introducing a Gaussian term of the spectral degree of coherence [22]. The cross spectral density function of partially coherent H-ChG beams at the plane z=0 is expressed as [7, 8, 22]

W(ρ1,ρ2,z=0)=Hm(2w0ρ1x)Hn(2w0ρ1y)exp(ρ1x2+ρ1y2w02)
×cosh(Ω0ρ1x+Ω0ρ1y)exp[(ρ1xρ2x)22σ02]
×Hm(2w0ρ2x)Hn(2w0ρ2y)exp(ρ2x2+ρ2y2w02)
×cosh(Ω0ρ2x+Ω0ρ2y)exp[(ρ1yρ2y)22σ02],

where σ0 is the spatial correlation length of the source at the plane z=0.

Using the extended Huygens-Fresnel principle, the cross spectral density function of partially coherent H-ChG beams propagating through atmospheric turbulence is expressed as [23]

W(ρ1,ρ2,z)=(k2πz)2d2ρ1d2ρ2W(ρ1,ρ2,z=0)
×exp{ik2z[(ρ1ρ1)2(ρ2ρ2)2]}exp[ψ(ρ1,ρ1)+ψ*(ρ2,ρ2)]m,

where ρ′≡(ρx, ρy) is the two-dimensional position vector at the plane z, k is the wave number related to the wave length λ by k=2π/λ, ψ(ρ, ρ′) represents the random part of the complex phase of a spherical wave due to the turbulence, <·> m denotes the average over the ensemble of the turbulent medium statistics, and [24–27]

exp[ψ(ρ1,ρ1)+ψ*(ρ2,ρ2)]m
exp{1ρ02[(ρ1ρ2)2+(ρ1ρ2)·(ρ1ρ2)+(ρ1ρ2)2]},

with

ρ0=(0.545Cn2k2z)35,

where ρ 0 is the spatial coherence radius of a spherical wave propagating in turbulence, C 2 n is the refraction index structure constant which describes how strong the turbulence is [23]. It is noted that a quadratic approximation of Rytov’s phase structure function was used in Eq. (4) to obtain an analytical result. This approximation has been shown to be a good approximation in practice [24, 28].

To obtain the analytical result, the new variables of integration are introduced as

u=ρ2+ρ12,v=ρ2ρ1.

By letting ρ′ 1=ρ′ 2=ρ′ in Eq. (3), and making use of Eqs. (2), (4) and (6), the intensity of partially coherent H-ChG beams in turbulence at the z plane is given by

I(ρ,z)=W(ρ,ρ,z)
 =(k4πz)2d2ud2vexp(2u2w02)exp(v2ε2)exp(ikzu·v)exp(ikzρ·v)
 ×Hm[2w0(uxvx2)]Hm[2w0(ux+vx2)]Hn[2w0(uyvy2)]Hn[2w0(uy+vy2)]
×{exp[2Ω0(ux+uy)]+exp[Ω0(vx+vy)]
+exp[Ω0(vx+vy)]+exp[2Ω0(ux+uy)]}

where

1ε2=12w02+12σ02+1ρ02.

The normalized rms beam width is defined as [5]

w(z)=ρ2I(ρ,z)d2ρI(ρ,z)d2ρ.

By using some integral transform techniques (see Appendix A), we obtain the normalized rms beam width of partially coherent H-ChG beams in turbulence. The final result is arranged as

w(z)=P+Qk2z2+4(0.545Cn2k13)65z165,

where

P=R1R0,
Q=R2R0,
R0=exp(w02Ω02)Lm0(w02Ω02)Ln0(w02Ω02)+1,
R1=w02{exp(w02Ω02)(1+w02Ω022Lm0(w02Ω02)Ln0(w02Ω02)+1+2w02Ω022
×[Ln0(w02Ω02)Lm11(w02Ω02)+Lm0(w02Ω02)Ln11(w02Ω02)]+w02Ω02
×[Ln0(w02Ω02)Lm22(w02Ω02)+Lm0(w02Ω02)Ln22(w02Ω02)])+m+n+12},
R2=exp(w02Ω02){2(1w02+1σ02)Lm0(w02Ω02)Ln0(w02Ω02)+2w02[Lm11(w02Ω02)
×Ln0(w02Ω02)+Lm0(w02Ω02)Ln11(w02Ω02)]}+2(1w02+1σ02Ω02)+2(m+n)w02

where Lαj(●) (j=0, 1, 2…) denotes the generalized Laguerre polynomial with indices α, j, P and Q are independent of propagation distance z. The first two terms on the right-hand side of Eq. (10) denote the spread of the beam width in free space due to diffraction, where the first term P is independent of the propagation distance z, whereas the second term Qk2z2 increases with z 2, and the third term 4(0.545Cn2k13)65z165 indicates how the turbulence affects the beam spread, and increases with z 16/5.

From Eq. (10) the angular spread of partially coherent H-ChG beams propagating through atmospheric turbulence turns out to be

θsp(z)=w(z)z|z=Qk2+4(0.545Cn2k13)65z65.

Equations (10) and (16) are the main analytical results obtained in this paper. The first term on the right-hand side in Eq. (16) represents the angular spread of partially coherent H-ChG beams in free space which is dependent on the spatial correlation length σ0 and beam parameters w 0, Ω0, m, n, but independent of the propagation distance. The second term describes the effect of turbulence on the angular spread, which increases with C 2 n and propagation distance z, but does not depend on σ0, w 0, Ω0 and m, n. It means that the turbulence plays a dominant role in the angular spread as the propagation distance becomes large enough.

 

Fig. 1. Angular spread θ sp of a partially coherent H-ChG beam versus the spatial correlation length σ0. The calculation parameters are z=10km, m=n=1, w 0=3cm, Ω0=100m-1.

Download Full Size | PPT Slide | PDF

 

Fig. 2. Angular spread θ sp of a partially coherent H-ChG beam versus the waist width w 0. The calculation parameters are z=10km, m=n=1, σ0=1.732cm, Ω0=100m-1, C 2 n=10-14m-2/3.

Download Full Size | PPT Slide | PDF

 

Fig. 3. Angular spread θ sp of a partially coherent H-ChG beam versus the beam parameter Ω0. The calculation parameters are z=10km, m=n=1, w 0=3cm, σ0=4cm, C 2 n=10-15m-2/3.

Download Full Size | PPT Slide | PDF

 

Fig. 4. Angular spread θ sp of a partially coherent H-ChG beam versus the beam order m, n. The calculation parameters are z=10km, w 0=1cm, σ0=2 cm, Ω0=300m-1, C 2 n=10-14m-2/3.

Download Full Size | PPT Slide | PDF

Numerical examples are given by using Eq. (16) to show the influence of turbulence on the angular spread of partially coherent H-ChG beams, where λ=1.06µm is kept fixed. The angular spread θ sp versus the spatial correlation length σ0 for different values of C 2 n is shown in Fig. 1. From Fig. 1 it can be seen that θ sp increases with decreasing σ0 and increasing C 2 n. In addition, the difference between θ sp in turbulence and θ sp in free space is smaller for partially coherent H-ChG beams with smaller value of σ0 than for those with larger value of σ0. For example, for C 2 n=10-14m-2/3, σ0=1 and 6cm, we have θ sp|turb/θ sp|free=1.59 and 3.43, where θ sp|turb and θ sp|free are the angular spread in turbulence and in free space respectively. Figure 2 gives the angular spread θ sp versus the waist width w 0. Figure 2 indicates that θ sp decreases with increasing w 0. Furthermore, the larger the waist width w 0 is, the larger the difference between θ sp in turbulence and θ sp in free space exhibits. For instance, for w 0=1 and 4cm we have θ sp|turb/θ sp|free=1.32 and 2.30, respectively. The angular spread θ sp versus the parameter Ω0 is depicted in Fig. 3, which shows that θ sp decreases with increasing Ω0, and the difference between θ sp in turbulence and θ sp in free space increases with increasing Ω0, e.g., for Ω0=20 and 120m-1 we have θ sp|turb/θ sp|free=1.15 and 1.26 respectively. Figure 4 gives the angular spread θ sp versus the beam order m, n. Figure 4 indicates that θ sp increases with increasing m, n, and the difference between θ sp in turbulence and θ sp in free space decreases with increasing m, n, e.g., for m=n=1 and 15 we have θ sp|turb/θ sp|free=1.48 and 1.26, respectively. Therefore, the smaller the spatial correlation length σ0, waist width w 0, beam parameter Ω0, and the larger the beam orders m, n are, the less partially coherent H-ChG beams are affected by turbulence.

Generally, the angular spread of beams in turbulence is affected by two mechanisms. One is the free-space diffraction, and the other is the atmospheric turbulence [20]. Physically, the existence of any substantial original angular spread reduces the effect of atmospheric turbulence. Figures 1–4 indicate that in free space θ sp decreases with increasing σ0, w 0, Ω0, and decreasing m, n. Namely, the smaller σ0, w 0, Ω0, and larger m, n mean the larger substantial original angular spread. This is the physical reason why the angular spread of partially coherent H-ChG beams with smaller σ0, w 0, Ω0, and larger m, n is less affected by turbulence than that of partially coherent H-ChG beams with larger σ0, w 0, Ω0, and smaller m, n.

3. Equivalent beams

Partially coherent H-ChG beams represent a more general type of beams. There are some partially coherent beams and fully coherent beams which can be treated as their special cases.

(I) For Ω0=0, Eq. (16) reduces to

θsp(z)=Q1k2+4(0.545Cn2k13)65z65,

where

Q1=2(m+n+1w02+1σ02).

Equation (17) is the angular spread of partially coherent H-G beams in turbulence, which is in agreement with Eq. (29) in Ref. [6].

(II) For m=n=0, Eq. (16) reduces to the angular spread of partially coherent ChG beams in turbulence, i.e.,

θsp(z)=Q2k2+4(0.545Cn2k13)65z65,

where

Q2=2[(1w02+1σ02)Ω021+exp(w02Ω02)].

(III) For Ω0=0 and m=n=0, Eq. (16) reduces to

θsp(z)=Q3k2+4(0.545Cn2k13)65z65,

where

Q3=2(1w02+1σ02).

Equation (21) is the angular spread of GSM beams propagating through atmospheric turbulence. It is noted that Z 6/5 occurs in the second term of Eq. (21) because a quadratic approximation of the Rytov’s phase structure function is used in this paper, otherwise z appears as Eq. (14) of Ref. [5].

(IV) For σ0→∞, Eq. (17) reduces to

θsp(z)=Q4k2+4(0.545Cn2k13)65z65,

where

Q4=exp(w02Ω02)[2w02Lm0(w02Ω02)Ln0(w02Ω02)+2w02[Lm11(w02Ω02)Ln0(w02Ω02)
+Lm0(w02Ω02)Ln11(w02Ω02)]+2(1w02Ω02)+2(m+n)w02.

Equation (23) is the the angular spread of fully coherent H-ChG beams in turbulence.

(V) For Ω0=0 and σ0→∞, Eq. (16) reduces to the angular spread of fully coherent H-G beams in turbulence, which is given by

θsp(z)=Q5k2+4(0.545Cn2k13)65z65,

where

Q5=2(m+n+1w02).

(VI) For m=n=0 and σ0→∞, Eq. (16) reduces to

θsp(z)=Q6k2+4(0.545Cn2k13)65z65,

where

Q6=2[(1w02Ω021+exp(w02Ω02)].

Equation (27) is the angular spread of fully coherent ChG beams.

(VII) For Ω0=0, m=n=0 and σ0→∞, Eq. (16) reduces to the angular spread of fully coherent Gaussian beams, i.e.,

θsp(z)=Q7k2+4(.0545Cn2k13)65z65,

where

Q7=2w02.

In comparison of Eqs. (16), (17), (19), (21) and (29), we conclude that four partially coherent beams, i.e., partially coherent H-ChG, H-G, ChG and GSM beams will have the same angular spread as a fully coherent Gaussian beam if the condition

Q=Q1=Q2=Q3=Q7

is fulfilled, provided that the wavelength λ is kept fixed. Such four partially coherent beams are called the equivalent partially coherent H-ChG, H-G, ChG and GSM beams, respectively [1, 2].

 

Fig. 5. Normalized rms width w(z) of the four equivalent partially coherent beams and the corresponding fully coherent Gaussian beam versus propagation distance z in free space and in turbulence. a: the corresponding fully coherent Gaussian beam; b: the equivalent GSM beam; c: the equivalent partially coherent H-G beam: d: the equivalent partially coherent H-ChG beam; e: the equivalent partially coherent ChG beam. The calculation parameters are listed in Table 1, and the other parameters are λ=1.06µm, C 2 n=10-14m-2/3.

Download Full Size | PPT Slide | PDF

Tables Icon

Table 1. Beam parameters relating to Fig. 5.

Figure 5 gives the normalized rms width w(z) of the four equivalent partially coherent beams mentioned above and the corresponding fully coherent Gaussian beam propagating both in free space and in atmospheric turbulence. The calculation parameters are listed in Tab. 1. As can be expected, under the condition (31) the four equivalent partially coherent beams exhibit the same directionality as the corresponding fully coherent Gaussian beam in free space and also in turbulence.

Except for the equivalent partially coherent beams, there also exist equivalent fully coherent beams. From a comparison of Eqs. (23), (25), (27) and (29), we see that, three fully coherent beams, i.e., fully coherent H-ChG, H-G and ChG beams will have the same angular spread as a fully coherent Gaussian beam if the condition

Q4=Q5=Q6=Q7

is satisfied, provided that the wavelength λ is kept fixed. Such three fully coherent beams are referred to as the equivalent fully coherent H-ChG, H-G and ChG beams, respectively [1, 2].

 

Fig. 6. Normalized rms width w(z) of the three fully coherent beams and the corresponding fully coherent Gaussian beam versus propagation distance z. a: the corresponding fully coherent Gaussian beam; b: the equivalent fully coherent ChG beam; c: the equivalent fully coherent H-ChG beam; d: the equivalent fully coherent H-G beams. The calculation parameters are listed in Table 2, and the other parameters are λ=1.06µm, C 2 n=10-14m-2/3.

Download Full Size | PPT Slide | PDF

Tables Icon

Table 2. Beam parameters relating to Fig. 6.

Figure 6 gives the normalized rms width w(z) of the three equivalent fully coherent beams mentioned above and the corresponding fully coherent Gaussian beam propagating both in free space and in atmospheric turbulence. The calculation parameters are compiled in Tab. 2. As can be seen, equivalent fully coherent H-ChG, H-G and ChG beams have also the same directionality as the corresponding fully coherent Gaussian beam both in free space and in turbulence, if the condition (32) is satisfied.

The results can be physically explained as follows. In fact, partially coherent H-ChG beams are characterized by four parameters (i.e., σ0, w 0, Ω0, m, n), and partially coherent H-G, ChG and GSM beams are characterized by (σ0, w 0, m, n), (σ0, w 0, Ω0) and (σ0, w 0), respectively. From Figs. 1–4 it follows that for partially coherent H-ChG beams the smaller σ0 means the worse spatial coherence, which result in a larger angular spread. The smaller the waist width w 0 and the beam parameter Ω0 are, the larger the free-space diffraction spread is; while the smaller the beam orders m, n are, the smaller the free-space diffraction spread is. In addition, Eq. (16) indicates that the angular spread due to turbulence is independent of σ0, w 0, Ω0 and m, n. Therefore, both in free space and in turbulence there are two competing mechanisms for partially coherent beams regarding to the directionality, i.e., the spatial coherence and diffraction, which can be balanced by a suitable choice of beam parameters σ0, w 0, Ω0 and m, n, whereas for fully coherent beams discussed above, the angular spread resulting from diffraction can be compensated by appropriately adjusting beam parameters w 0, Ω0 and m, n to achieve the same directionality as a fully coherent Gaussian beam in free space and also in turbulence.

4. Concluding remarks

In this paper, the propagation of partially coherent H-ChG beams in turbulence has been studied in detail. The analytical expression for the angular spread of partially coherent H-ChG beams propagating through atmospheric turbulence has been derived by using the extended Huygens-Fresnel principle, quadratic approximation of Rytov’s phase structure function and integral transform techniques. In comparison with previous work our result is more general, because the angular spread of partially coherent H-G, ChG and GSM and fully coherent H-ChG, H-G, ChG, Gaussian beams can be treated as special cases of Eq. (16). It has been shown that the angular spread of partially coherent H-ChG beams with smaller σ0, w 0, Ω0, and larger m, n is less affected by turbulence than that of partially coherent H-ChG beams with larger σ0, w 0, Ω0, and smaller m, n. Partially coherent H-ChG, H-G, ChG and GSM beams may have the same angular spread as a fully coherent Gaussian beam in free space and also in atmospheric turbulence if the condition (31) is satisfied. Under the condition (32) there exist the equivalent fully coherent H-ChG, H-G and ChG beams which may have the same directionality as a fully coherent Gaussian beam both in free space and in atmospheric turbulence. The physical interpretation has been given to show the validity of our results.

Appendix A: Derivation of Eq. (10)

Equation (9) can be written as

w(z)=FF0,

where

F0=I(ρ,z)d2ρ,
F0=ρ2I(ρ,z)d2ρ.

On substituting Eq. (2) into Eq. (A2) and letting ρ 1=ρ 2=ρ, we obtain

F0=W(0)(ρ,ρ,z=0)d2ρ
=122m+n1m!n!w02π[exp(w02Ω02)Lm0(w02Ω02)Ln0(w02Ω02)+1].

In the derivation of Eq. (A4) the law of conservation of energy was used.

Equation (A3) can be rewritten as

F=F1+F2+F3+F4,

where

F1=ρ2I1(ρ,z)d2ρ,
F2=ρ2I2(ρ,z)d2ρ,
F3=ρ2I3(ρ,z)d2ρ,
F4=ρ2I4(ρ,z)d2ρ,

with

I1(ρ,z)=14(k2πz)2d2ud2v
×Hm[2w0(uxvx2)]Hn[2w0(uy-vy2)]Hm[2w0(ux+vx2)]Hn[2w0(uy+vy2)]
 ×exp(2u2w02)exp(v2ε2)exp(ikzu·v)exp(ikzρ·v)exp[2Ω0(ux+uy)],
I2(ρ,z)=14(k2πz)2d2ud2v
×Hm[2w0(uxvx2)]Hn[2w0(uy-vy2)]Hm[2w0(ux+vx2)]Hn[2w0(uy+vy2)]
×exp(2u2w02)exp(v2ε2)exp(ikzu·v)exp(ikzρ·v)exp[Ω0(vx+vy)].

I 3=(ρ′, z) and I 4=(ρ′, z) can be obtained if Ω0 is replaced by -Ω0 in Eqs. (A10) and (A11). Making use of the integral formula

x2exp(i2πxs)dx=1(2π)2δ(s),

the substitution from Eq. (A10) into Eq. (A6) yields

F1=F11+F12,

where

F11=14(zk)2d2ud2v
×Hm[2w0(uxvx2)]Hn[2w0(uyvy2)]Hm[2w0(ux+vx2)]Hn[2w0(uy+vy2)]
×exp(2u2w02)exp(v2ε2)exp(ikzu·v)exp[2Ω0(ux+uy)]δ(vx)δ(vy),

and

F12=14(zk)2d2ud2v
×Hm[2w0(uxvx2)]Hn[2w0(uyvy2)]Hm[2w0(ux+vx2)]Hn[2w0(uy+vy2)]
×exp(2u2w02)exp(v2ε2)exp(ikzu·v)exp[2Ω0(ux+uy)]δ(vx)δ(vy),

with δ denoting the Dirac delta function, and δ″ being its second derivative.

By virtue of the integral formulae

f(x)δ(x)dx=f(0),

and

exp[(xy)2]Hm(x)Hn(x)dx=2nπm!ynmLnnm(2y2),

the integration of Eq. (A14) with respect to vy and uy yields

F11=14(zk)22nn!πw02exp(w02Ω022)Ln0(w02Ω02)dvxdux
×Hm[2w0(uxvx2)]Hm[2w0(ux+vx2)]exp(2ux2w02)
×exp(vx2ε2)exp(ikzuxvx)exp(2Ω0ux)δ(vx).

Recalling the integral formulae

exp(x2)Hm(x+y)Hn(x+z)dx=2nπm!ynmznmLnnm(2yz),

and

f(x)δ(x)dx=f(0),

where f is an arbitrary function and f″ is its second derivative of f, and performing the integration of Eq. (A18) with respect to ux and vx, we obtain

F11=14(zk)22m+n1m!n!πw02exp(w02Ω02)Ln0(w02Ω02)
×[(k2w024z22ε2)Lm0(w02Ω02)]k2w04Ω024z2Lm0(w02Ω02)k2w04Ω02z2Lm22(w02Ω02)
(2w02+k2w022z0)Lm11(w02Ω02)k2w04Ω02z2Lm11(w02Ω02)].

F 12 can be obtained if m and n are replaced by n and m in Eq. (A21), respectively. Thus, F 1 in Eq. (A13) (i.e., Eq. (A6)) can be obtained.

Similarly, F 2 can be written as

F2=F21+F22,

where

F21=14(zk)22m+n1m!n!πw02[(k2w024z22ε2)+Ω02m(2w02+k2w022z2)].

F 22 can be obtained if m and n are replaced by n and m in Eqs. (A23), respectively.

In addition, F 3 and F 4 can be obtained if Ω0 is replaced by -Ω0 in Eqs. (A13) and (A22) respectively.

Finally, from Eqs. (A1), (A4), (A5), (A13) and (A22) it turns out that

w(z)=P+Qk2z2+4(0.545Cn2k13)65z165.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under grant No. 60778048.

References and links

1. E. Collett and E. Wolf, “Is complete spatial coherence necessary for the generation of highly directional light beams?” Opt. Lett. 2, 27–29 (1978). [CrossRef]   [PubMed]  

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

3. P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256–260 (1979). [CrossRef]  

4. J. D. Farina, L. M. Narducci, and E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980). [CrossRef]  

5. T. Shirai, A. Dogariu, and E. Wolf, “Directionality of Gaussian Schell-model beams propagating in atmospheric turbulence,” Opt. Lett. 28, 610–612 (2003). [CrossRef]   [PubMed]  

6. X. Ji, X. Chen, and B. Lü, “Spreading and directionality of partially coherent Hermite-Gaussian beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 25, 21–28 (2008). [CrossRef]  

7. L. W. Casperson and A. A. Tovar, “Hermite-sinusoidal-Gaussian beams in complex optical systems,” J. Opt. Soc. Am. A 15, 954–961 (1998). [CrossRef]  

8. A. A. Tovar and L. W. Casperson, “Production and propagation of Hermite-sinusoidal-Gaussian laser beams,” J. Opt. Soc. Am. A 15, 2425–2432 (1998). [CrossRef]  

9. C. Y. Young, Y. V. Gilchrest, and B. R. Macon. “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. 41, 1097–1103 (2002). [CrossRef]  

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

11. H. T. Eyyuboğlu, “Propagation of Hermite-cosh-Gaussian laser beams in turbulent atmosphere,” Opt. Commun. 245, 37–47 (2005). [CrossRef]  

12. H. T. Eyyuboğlu and Y. Baykal, “Average intensity and spreading of cosh-Gaussian laser beams in the turbulent atmosphere,” Appl. Opt. 44, 976–983 (2005). [CrossRef]   [PubMed]  

13. H. T. Eyyuboğlu and Y. Baykal. “Hermite-sine-Gaussian and Hermite-sinh-Gaussian laser beams in turbulent atmosphere,” J. Opt. Soc. Am. A 22, 2709–2718 (2005). [CrossRef]  

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

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

16. H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Complex degree of coherence for partially coherent general beams in atmospheric turbulence,” J. Opt. Soc. Am. A 24, 2891–2900 (2007). [CrossRef]  

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

18. H. T. Eyyuboğlu and Y. Baykal, “Transmittance of partially coherent cosh-Gaussian, cos-Gaussian and annular beams in turbulence,” Opt. Commun. 278, 17–22 (2007). [CrossRef]  

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

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

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

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

23. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 1998).

24. S. Wang, C. Ouyang, and M. A. Plonus, “Optical beam propagation for a partially coherent source in the turbulent atmosphere,” J. Opt. Soc. Am. 69, 1297–1304 (1979). [CrossRef]  

25. H. T. Yura, “Mutual coherence function of a finite cross section optical beam propagating in a turbulent medium,” Appl. Opt. 11, 1399–1406 (1972). [CrossRef]   [PubMed]  

26. V. A. Banakh and V. L. Mironov, “Phase approximation of the Huygens-Kirhhoff method in problems of laser beam propagation in the turbulent atmosphere,” Opt. Lett. 1, 172–174 (1977). [CrossRef]   [PubMed]  

27. V. A. Banakh and V. L. Mironov, “Phase approximation of the Huygens-Kirhhoff method in problems of spase-limited optical-beam propagation in the turbulent atmosphere,” Opt. Lett. 4, 259–261 (1979). [CrossRef]   [PubMed]  

28. C. Leader, “Atmospheric propagation of partially coherent radiation,” J. Opt. Soc. Am. 68, 175–178 (1978). [CrossRef]  

References

  • View by:
  • |
  • |
  • |

  1. E. Collett and E. Wolf, “Is complete spatial coherence necessary for the generation of highly directional light beams?” Opt. Lett. 2, 27–29 (1978).
    [Crossref] [PubMed]
  2. E. Wolf and E. Collett, “Partially coherent sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978).
    [Crossref]
  3. P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256–260 (1979).
    [Crossref]
  4. J. D. Farina, L. M. Narducci, and E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
    [Crossref]
  5. T. Shirai, A. Dogariu, and E. Wolf, “Directionality of Gaussian Schell-model beams propagating in atmospheric turbulence,” Opt. Lett. 28, 610–612 (2003).
    [Crossref] [PubMed]
  6. X. Ji, X. Chen, and B. Lü, “Spreading and directionality of partially coherent Hermite-Gaussian beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 25, 21–28 (2008).
    [Crossref]
  7. L. W. Casperson and A. A. Tovar, “Hermite-sinusoidal-Gaussian beams in complex optical systems,” J. Opt. Soc. Am. A 15, 954–961 (1998).
    [Crossref]
  8. A. A. Tovar and L. W. Casperson, “Production and propagation of Hermite-sinusoidal-Gaussian laser beams,” J. Opt. Soc. Am. A 15, 2425–2432 (1998).
    [Crossref]
  9. C. Y. Young, Y. V. Gilchrest, and B. R. Macon. “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. 41, 1097–1103 (2002).
    [Crossref]
  10. H. T. Eyyuboğlu and Y. Baykal. “Analysis of reciprocity of cos-Gaussian and cosh-Gaussian laser beams in a turbulent atmosphere,” Opt. Express 12, 4659–4674 (2004).
    [Crossref] [PubMed]
  11. H. T. Eyyuboğlu, “Propagation of Hermite-cosh-Gaussian laser beams in turbulent atmosphere,” Opt. Commun. 245, 37–47 (2005).
    [Crossref]
  12. H. T. Eyyuboğlu and Y. Baykal, “Average intensity and spreading of cosh-Gaussian laser beams in the turbulent atmosphere,” Appl. Opt. 44, 976–983 (2005).
    [Crossref] [PubMed]
  13. H. T. Eyyuboğlu and Y. Baykal. “Hermite-sine-Gaussian and Hermite-sinh-Gaussian laser beams in turbulent atmosphere,” J. Opt. Soc. Am. A 22, 2709–2718 (2005).
    [Crossref]
  14. Y. Cai and S. He. “Average intensity and spreading of an elliptical Gaussian beam propagating in a turbulent atmosphere,” Opt. Lett. 31, 568–570 (2006).
    [Crossref] [PubMed]
  15. Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express,  14, 1353–1367 (2006).
    [Crossref] [PubMed]
  16. H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Complex degree of coherence for partially coherent general beams in atmospheric turbulence,” J. Opt. Soc. Am. A 24, 2891–2900 (2007).
    [Crossref]
  17. H. T. Eyyubolu, Y. Baykal, and Y. Cai, “Degree of polarization for partially coherent general beams in turbulent atmosphere,” Appl. Phys. B: Lasers Opt. 89, 91–97 (2007).
    [Crossref]
  18. H. T. Eyyuboğlu and Y. Baykal, “Transmittance of partially coherent cosh-Gaussian, cos-Gaussian and annular beams in turbulence,” Opt. Commun. 278, 17–22 (2007).
    [Crossref]
  19. Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
    [Crossref]
  20. G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592–1598 (2002).
    [Crossref]
  21. A. Dogariu and S. Amarande, “Propagation of partially coherent beams: turbulence-induced degradation,” Opt. Lett. 28, 10–12 (2003).
    [Crossref] [PubMed]
  22. M. Zahid and M. S. Zubairy, “Directionality of partially coherent Bessel-Gauss beams,” Opt. Commun. 70, 361–364 (1989).
    [Crossref]
  23. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 1998).
  24. S. Wang, C. Ouyang, and M. A. Plonus, “Optical beam propagation for a partially coherent source in the turbulent atmosphere,” J. Opt. Soc. Am. 69, 1297–1304 (1979).
    [Crossref]
  25. H. T. Yura, “Mutual coherence function of a finite cross section optical beam propagating in a turbulent medium,” Appl. Opt. 11, 1399–1406 (1972).
    [Crossref] [PubMed]
  26. V. A. Banakh and V. L. Mironov, “Phase approximation of the Huygens-Kirhhoff method in problems of laser beam propagation in the turbulent atmosphere,” Opt. Lett. 1, 172–174 (1977).
    [Crossref] [PubMed]
  27. V. A. Banakh and V. L. Mironov, “Phase approximation of the Huygens-Kirhhoff method in problems of spase-limited optical-beam propagation in the turbulent atmosphere,” Opt. Lett. 4, 259–261 (1979).
    [Crossref] [PubMed]
  28. C. Leader, “Atmospheric propagation of partially coherent radiation,” J. Opt. Soc. Am. 68, 175–178 (1978).
    [Crossref]

2008 (1)

2007 (3)

H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Complex degree of coherence for partially coherent general beams in atmospheric turbulence,” J. Opt. Soc. Am. A 24, 2891–2900 (2007).
[Crossref]

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

H. T. Eyyuboğlu and Y. Baykal, “Transmittance of partially coherent cosh-Gaussian, cos-Gaussian and annular beams in turbulence,” Opt. Commun. 278, 17–22 (2007).
[Crossref]

2006 (3)

2005 (3)

2004 (1)

2003 (2)

2002 (2)

C. Y. Young, Y. V. Gilchrest, and B. R. Macon. “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. 41, 1097–1103 (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]

1998 (2)

1989 (1)

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

1980 (1)

J. D. Farina, L. M. Narducci, and E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
[Crossref]

1979 (3)

1978 (3)

1977 (1)

1972 (1)

Amarande, S.

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 1998).

Banakh, V. A.

Baykal, Y.

Cai, Y.

Casperson, L. W.

Chen, X.

Collett, E.

J. D. Farina, L. M. Narducci, and E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
[Crossref]

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

E. Collett and E. Wolf, “Is complete spatial coherence necessary for the generation of highly directional light beams?” Opt. Lett. 2, 27–29 (1978).
[Crossref] [PubMed]

De Santis, P.

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256–260 (1979).
[Crossref]

Dogariu, A.

Eyyuboglu, H. T.

Eyyubolu, H. T.

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

Farina, J. D.

J. D. Farina, L. M. Narducci, and E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
[Crossref]

Gbur, G.

Gilchrest, Y. V.

C. Y. Young, Y. V. Gilchrest, and B. R. Macon. “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. 41, 1097–1103 (2002).
[Crossref]

Gori, F.

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256–260 (1979).
[Crossref]

Guattari, G.

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256–260 (1979).
[Crossref]

He, S.

Ji, X.

Leader, C.

Lü, B.

Macon, B. R.

C. Y. Young, Y. V. Gilchrest, and B. R. Macon. “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. 41, 1097–1103 (2002).
[Crossref]

Mironov, V. L.

Narducci, L. M.

J. D. Farina, L. M. Narducci, and E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
[Crossref]

Ouyang, C.

Palma, C.

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256–260 (1979).
[Crossref]

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 1998).

Plonus, M. A.

Shirai, T.

Tovar, A. A.

Wang, S.

Wolf, E.

Young, C. Y.

C. Y. Young, Y. V. Gilchrest, and B. R. Macon. “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. 41, 1097–1103 (2002).
[Crossref]

Yura, H. T.

Zahid, M.

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

Zubairy, M. S.

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

Appl. Opt. (2)

Appl. Phys. B: Lasers Opt. (1)

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

Appl. Phys. Lett. (1)

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

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (6)

Opt. Commun. (6)

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

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256–260 (1979).
[Crossref]

J. D. Farina, L. M. Narducci, and E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
[Crossref]

H. T. Eyyuboğlu, “Propagation of Hermite-cosh-Gaussian laser beams in turbulent atmosphere,” Opt. Commun. 245, 37–47 (2005).
[Crossref]

H. T. Eyyuboğlu and Y. Baykal, “Transmittance of partially coherent cosh-Gaussian, cos-Gaussian and annular beams in turbulence,” Opt. Commun. 278, 17–22 (2007).
[Crossref]

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

Opt. Eng. (1)

C. Y. Young, Y. V. Gilchrest, and B. R. Macon. “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. 41, 1097–1103 (2002).
[Crossref]

Opt. Express (2)

Opt. Lett. (6)

Other (1)

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 1998).

Cited By

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

Alert me when this article is cited.


Figures (6)

Fig. 1.
Fig. 1.

Angular spread θ sp of a partially coherent H-ChG beam versus the spatial correlation length σ0. The calculation parameters are z=10km, m=n=1, w 0=3cm, Ω0=100m-1.

Fig. 2.
Fig. 2.

Angular spread θ sp of a partially coherent H-ChG beam versus the waist width w 0. The calculation parameters are z=10km, m=n=1, σ0=1.732cm, Ω0=100m-1, C 2 n =10-14m-2/3.

Fig. 3.
Fig. 3.

Angular spread θ sp of a partially coherent H-ChG beam versus the beam parameter Ω0. The calculation parameters are z=10km, m=n=1, w 0=3cm, σ0=4cm, C 2 n =10-15m-2/3.

Fig. 4.
Fig. 4.

Angular spread θ sp of a partially coherent H-ChG beam versus the beam order m, n. The calculation parameters are z=10km, w 0=1cm, σ0=2 cm, Ω0=300m-1, C 2 n =10-14m-2/3.

Fig. 5.
Fig. 5.

Normalized rms width w(z) of the four equivalent partially coherent beams and the corresponding fully coherent Gaussian beam versus propagation distance z in free space and in turbulence. a: the corresponding fully coherent Gaussian beam; b: the equivalent GSM beam; c: the equivalent partially coherent H-G beam: d: the equivalent partially coherent H-ChG beam; e: the equivalent partially coherent ChG beam. The calculation parameters are listed in Table 1, and the other parameters are λ=1.06µm, C 2 n =10-14m-2/3.

Fig. 6.
Fig. 6.

Normalized rms width w(z) of the three fully coherent beams and the corresponding fully coherent Gaussian beam versus propagation distance z. a: the corresponding fully coherent Gaussian beam; b: the equivalent fully coherent ChG beam; c: the equivalent fully coherent H-ChG beam; d: the equivalent fully coherent H-G beams. The calculation parameters are listed in Table 2, and the other parameters are λ=1.06µm, C 2 n =10-14m-2/3.

Tables (2)

Tables Icon

Table 1. Beam parameters relating to Fig. 5.

Tables Icon

Table 2. Beam parameters relating to Fig. 6.

Equations (82)

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

U ( ρ , z = 0 ) = H m ( 2 w 0 ρ x ) H n ( 2 w 0 ρ y ) exp ( ρ x 2 + ρ y 2 w 0 2 ) cosh ( Ω 0 ρ x + Ω 0 ρ y ) ,
W ( ρ 1 , ρ 2 , z = 0 ) = H m ( 2 w 0 ρ 1 x ) H n ( 2 w 0 ρ 1 y ) exp ( ρ 1 x 2 + ρ 1 y 2 w 0 2 )
× cosh ( Ω 0 ρ 1 x + Ω 0 ρ 1 y ) exp [ ( ρ 1 x ρ 2 x ) 2 2 σ 0 2 ]
× H m ( 2 w 0 ρ 2 x ) H n ( 2 w 0 ρ 2 y ) exp ( ρ 2 x 2 + ρ 2 y 2 w 0 2 )
× cosh ( Ω 0 ρ 2 x + Ω 0 ρ 2 y ) exp [ ( ρ 1 y ρ 2 y ) 2 2 σ 0 2 ] ,
W ( ρ 1 , ρ 2 , z ) = ( k 2 π z ) 2 d 2 ρ 1 d 2 ρ 2 W ( ρ 1 , ρ 2 , z = 0 )
× exp { i k 2 z [ ( ρ 1 ρ 1 ) 2 ( ρ 2 ρ 2 ) 2 ] } exp [ ψ ( ρ 1 , ρ 1 ) + ψ * ( ρ 2 , ρ 2 ) ] m ,
exp [ ψ ( ρ 1 , ρ 1 ) + ψ * ( ρ 2 , ρ 2 ) ] m
exp { 1 ρ 0 2 [ ( ρ 1 ρ 2 ) 2 + ( ρ 1 ρ 2 ) · ( ρ 1 ρ 2 ) + ( ρ 1 ρ 2 ) 2 ] } ,
ρ 0 = ( 0.545 C n 2 k 2 z ) 3 5 ,
u = ρ 2 + ρ 1 2 , v = ρ 2 ρ 1 .
I ( ρ , z ) = W ( ρ , ρ , z )
  = ( k 4 π z ) 2 d 2 u d 2 v exp ( 2 u 2 w 0 2 ) exp ( v 2 ε 2 ) exp ( i k z u · v ) exp ( i k z ρ · v )
  × H m [ 2 w 0 ( u x v x 2 ) ] H m [ 2 w 0 ( u x + v x 2 ) ] H n [ 2 w 0 ( u y v y 2 ) ] H n [ 2 w 0 ( u y + v y 2 ) ]
× { exp [ 2 Ω 0 ( u x + u y ) ] + exp [ Ω 0 ( v x + v y ) ]
+ exp [ Ω 0 ( v x + v y ) ] + exp [ 2 Ω 0 ( u x + u y ) ] }
1 ε 2 = 1 2 w 0 2 + 1 2 σ 0 2 + 1 ρ 0 2 .
w ( z ) = ρ 2 I ( ρ , z ) d 2 ρ I ( ρ , z ) d 2 ρ .
w ( z ) = P + Q k 2 z 2 + 4 ( 0.545 C n 2 k 1 3 ) 6 5 z 16 5 ,
P = R 1 R 0 ,
Q = R 2 R 0 ,
R 0 = exp ( w 0 2 Ω 0 2 ) L m 0 ( w 0 2 Ω 0 2 ) L n 0 ( w 0 2 Ω 0 2 ) + 1 ,
R 1 = w 0 2 { exp ( w 0 2 Ω 0 2 ) ( 1 + w 0 2 Ω 0 2 2 L m 0 ( w 0 2 Ω 0 2 ) L n 0 ( w 0 2 Ω 0 2 ) + 1 + 2 w 0 2 Ω 0 2 2
× [ L n 0 ( w 0 2 Ω 0 2 ) L m 1 1 ( w 0 2 Ω 0 2 ) + L m 0 ( w 0 2 Ω 0 2 ) L n 1 1 ( w 0 2 Ω 0 2 ) ] + w 0 2 Ω 0 2
× [ L n 0 ( w 0 2 Ω 0 2 ) L m 2 2 ( w 0 2 Ω 0 2 ) + L m 0 ( w 0 2 Ω 0 2 ) L n 2 2 ( w 0 2 Ω 0 2 ) ] ) + m + n + 1 2 } ,
R 2 = exp ( w 0 2 Ω 0 2 ) { 2 ( 1 w 0 2 + 1 σ 0 2 ) L m 0 ( w 0 2 Ω 0 2 ) L n 0 ( w 0 2 Ω 0 2 ) + 2 w 0 2 [ L m 1 1 ( w 0 2 Ω 0 2 )
× L n 0 ( w 0 2 Ω 0 2 ) + L m 0 ( w 0 2 Ω 0 2 ) L n 1 1 ( w 0 2 Ω 0 2 ) ] } + 2 ( 1 w 0 2 + 1 σ 0 2 Ω 0 2 ) + 2 ( m + n ) w 0 2
θ sp ( z ) = w ( z ) z | z = Q k 2 + 4 ( 0.545 C n 2 k 1 3 ) 6 5 z 6 5 .
θ sp ( z ) = Q 1 k 2 + 4 ( 0.545 C n 2 k 1 3 ) 6 5 z 6 5 ,
Q 1 = 2 ( m + n + 1 w 0 2 + 1 σ 0 2 ) .
θ sp ( z ) = Q 2 k 2 + 4 ( 0.545 C n 2 k 1 3 ) 6 5 z 6 5 ,
Q 2 = 2 [ ( 1 w 0 2 + 1 σ 0 2 ) Ω 0 2 1 + exp ( w 0 2 Ω 0 2 ) ] .
θ sp ( z ) = Q 3 k 2 + 4 ( 0.545 C n 2 k 1 3 ) 6 5 z 6 5 ,
Q 3 = 2 ( 1 w 0 2 + 1 σ 0 2 ) .
θ sp ( z ) = Q 4 k 2 + 4 ( 0.545 C n 2 k 1 3 ) 6 5 z 6 5 ,
Q 4 = exp ( w 0 2 Ω 0 2 ) [ 2 w 0 2 L m 0 ( w 0 2 Ω 0 2 ) L n 0 ( w 0 2 Ω 0 2 ) + 2 w 0 2 [ L m 1 1 ( w 0 2 Ω 0 2 ) L n 0 ( w 0 2 Ω 0 2 )
+ L m 0 ( w 0 2 Ω 0 2 ) L n 1 1 ( w 0 2 Ω 0 2 ) ] + 2 ( 1 w 0 2 Ω 0 2 ) + 2 ( m + n ) w 0 2 .
θ sp ( z ) = Q 5 k 2 + 4 ( 0.545 C n 2 k 1 3 ) 6 5 z 6 5 ,
Q 5 = 2 ( m + n + 1 w 0 2 ) .
θ sp ( z ) = Q 6 k 2 + 4 ( 0.545 C n 2 k 1 3 ) 6 5 z 6 5 ,
Q 6 = 2 [ ( 1 w 0 2 Ω 0 2 1 + exp ( w 0 2 Ω 0 2 ) ] .
θ sp ( z ) = Q 7 k 2 + 4 ( . 0545 C n 2 k 1 3 ) 6 5 z 6 5 ,
Q 7 = 2 w 0 2 .
Q = Q 1 = Q 2 = Q 3 = Q 7
Q 4 = Q 5 = Q 6 = Q 7
w ( z ) = F F 0 ,
F 0 = I ( ρ , z ) d 2 ρ ,
F 0 = ρ 2 I ( ρ , z ) d 2 ρ .
F 0 = W ( 0 ) ( ρ , ρ , z = 0 ) d 2 ρ
= 1 2 2 m + n 1 m ! n ! w 0 2 π [ exp ( w 0 2 Ω 0 2 ) L m 0 ( w 0 2 Ω 0 2 ) L n 0 ( w 0 2 Ω 0 2 ) + 1 ] .
F = F 1 + F 2 + F 3 + F 4 ,
F 1 = ρ 2 I 1 ( ρ , z ) d 2 ρ ,
F 2 = ρ 2 I 2 ( ρ , z ) d 2 ρ ,
F 3 = ρ 2 I 3 ( ρ , z ) d 2 ρ ,
F 4 = ρ 2 I 4 ( ρ , z ) d 2 ρ ,
I 1 ( ρ , z ) = 1 4 ( k 2 π z ) 2 d 2 u d 2 v
× H m [ 2 w 0 ( u x v x 2 ) ] H n [ 2 w 0 ( u y - v y 2 ) ] H m [ 2 w 0 ( u x + v x 2 ) ] H n [ 2 w 0 ( u y + v y 2 ) ]
  × exp ( 2 u 2 w 0 2 ) exp ( v 2 ε 2 ) exp ( i k z u · v ) exp ( i k z ρ · v ) exp [ 2 Ω 0 ( u x + u y ) ] ,
I 2 ( ρ , z ) = 1 4 ( k 2 π z ) 2 d 2 u d 2 v
× H m [ 2 w 0 ( u x v x 2 ) ] H n [ 2 w 0 ( u y - v y 2 ) ] H m [ 2 w 0 ( u x + v x 2 ) ] H n [ 2 w 0 ( u y + v y 2 ) ]
× exp ( 2 u 2 w 0 2 ) exp ( v 2 ε 2 ) exp ( i k z u · v ) exp ( i k z ρ · v ) exp [ Ω 0 ( v x + v y ) ] .
x 2 exp ( i 2 π x s ) d x = 1 ( 2 π ) 2 δ ( s ) ,
F 1 = F 11 + F 12 ,
F 11 = 1 4 ( z k ) 2 d 2 u d 2 v
× H m [ 2 w 0 ( u x v x 2 ) ] H n [ 2 w 0 ( u y v y 2 ) ] H m [ 2 w 0 ( u x + v x 2 ) ] H n [ 2 w 0 ( u y + v y 2 ) ]
× exp ( 2 u 2 w 0 2 ) exp ( v 2 ε 2 ) exp ( i k z u · v ) exp [ 2 Ω 0 ( u x + u y ) ] δ ( v x ) δ ( v y ) ,
F 12 = 1 4 ( z k ) 2 d 2 u d 2 v
× H m [ 2 w 0 ( u x v x 2 ) ] H n [ 2 w 0 ( u y v y 2 ) ] H m [ 2 w 0 ( u x + v x 2 ) ] H n [ 2 w 0 ( u y + v y 2 ) ]
× exp ( 2 u 2 w 0 2 ) exp ( v 2 ε 2 ) exp ( i k z u · v ) exp [ 2 Ω 0 ( u x + u y ) ] δ ( v x ) δ ( v y ) ,
f ( x ) δ ( x ) d x = f ( 0 ) ,
exp [ ( x y ) 2 ] H m ( x ) H n ( x ) d x = 2 n π m ! y n m L n n m ( 2 y 2 ) ,
F 11 = 1 4 ( z k ) 2 2 n n ! π w 0 2 exp ( w 0 2 Ω 0 2 2 ) L n 0 ( w 0 2 Ω 0 2 ) d v x d u x
× H m [ 2 w 0 ( u x v x 2 ) ] H m [ 2 w 0 ( u x + v x 2 ) ] exp ( 2 u x 2 w 0 2 )
× exp ( v x 2 ε 2 ) exp ( i k z u x v x ) exp ( 2 Ω 0 u x ) δ ( v x ) .
exp ( x 2 ) H m ( x + y ) H n ( x + z ) d x = 2 n π m ! y n m z n m L n n m ( 2 y z ) ,
f ( x ) δ ( x ) d x = f ( 0 ) ,
F 11 = 1 4 ( z k ) 2 2 m + n 1 m ! n ! π w 0 2 exp ( w 0 2 Ω 0 2 ) L n 0 ( w 0 2 Ω 0 2 )
× [ ( k 2 w 0 2 4 z 2 2 ε 2 ) L m 0 ( w 0 2 Ω 0 2 ) ] k 2 w 0 4 Ω 0 2 4 z 2 L m 0 ( w 0 2 Ω 0 2 ) k 2 w 0 4 Ω 0 2 z 2 L m 2 2 ( w 0 2 Ω 0 2 )
( 2 w 0 2 + k 2 w 0 2 2 z 0 ) L m 1 1 ( w 0 2 Ω 0 2 ) k 2 w 0 4 Ω 0 2 z 2 L m 1 1 ( w 0 2 Ω 0 2 ) ] .
F 2 = F 21 + F 22 ,
F 21 = 1 4 ( z k ) 2 2 m + n 1 m ! n ! π w 0 2 [ ( k 2 w 0 2 4 z 2 2 ε 2 ) + Ω 0 2 m ( 2 w 0 2 + k 2 w 0 2 2 z 2 ) ] .
w ( z ) = P + Q k 2 z 2 + 4 ( 0.545 C n 2 k 1 3 ) 6 5 z 16 5 .

Metrics