Abstract

The propagation of an elegant Hermite-Gaussian beam (EHGB) in turbulent atmosphere is investigated. Analytical propagation formulae for the average intensity and effective beam size of an EHGB in turbulent atmosphere are derived based on the extended Huygens-Fresnel integral. The corresponding results of a standard Hermite-Gaussian beam (SHGB) in turbulent atmosphere are also derived for the convenience of comparison. The intensity and spreading properties of EHGBs and SHGBs in turbulent atmosphere are studied numerically and comparatively. It is found that the propagation properties of EHGBs and SHGBs are much different from their properties in free space, and the EHGB and SHGB with higher orders are less affected by the turbulence. What’s more, the SHGB spreads more rapidly than the EHGB in turbulent atmosphere under the same conditions. Our results will be useful in long-distance free-space optical communications.

© 2009 OSA

1. Introduction

Standard Hermite-Gaussian beams (SHGBs) are commonly encountered higher-order modes in practice. In the past decades, SHGBs have been extensively stuided both in theoretical and experimental aspects, and have found wide applications [110]. Carter first introduced the definition of effective spot size for a SHGB and stuided its divergence [2]. The diffraction properties of SHGBs by various gratings have been investigated [3,4]. Luk et al. produced the SHG modes in experiment by mulipoles with complex source points [5], and Chen produced the SHG modes in fiber-coupled laser-diode end-pumped lasers [6]. Laabs studied the propagation of SHGB beyond the paraxial approximation [7]. Cai and Lin introduced the elliptical Hermite-Gaussian beam to describe an astigmatic higher-order beam and studied its propagation properties [8,9]. More recently, SHGB was extended to the partially coherent case, and the propagation properties of partially coherent SHGB in free space and through paraxial ABCD optical system were studied [10,11].

It is well known that SHG modes are eigenmodes of the paraxial wave equation. The Gaussian part of the SHG mode has a complex argument, but its Hermite part is purely real. Siegman first introduced new Hermite-Gaussian solutions named elegant Hermite-Gaussian (EHG) modes that satisfy the paraxial wave equation but has a more symmetrical form [12]. The argument of the Hermite part of the EHG mode is complex. The EHG modes are not orthogonal in the usual sense, but they are biorthogonal. Up to now, various methods have been developed to obtain EHGBs [13,14]. Laabs et al. has demonstrated that the three-dimensional SHG modes can be converted to twisted Hermite-Gaussian modes with complex argument [15]. The propagation properties of EHGB and SHGB in free space and through paraxial ABCD optical system have been studied extensively [1618].

The propagation of various laser beams in a turbulent atmosphere has been widely studied in the past due to their important applications in free-space optical communications and remote sensing, and various methods have been proposed to overcome or reduce the turbulence-induced degradation of laser beams [1938]. It has been found that the use of a higher-mode source beam can reduce the turbulence-induced degradation [20,25,26]. Up to now, although the propagation of various standard higher-order beams in turbulent atmosphere has been investigated extensively, to our knowledge, the propagation of elegant higher-order beams in turbulent atmosphere hasn’t been reported. In this paper, our aim is to investigate the propagation of an EHGB in turbulent atmosphere, and to explore the advantage of the EHGB for application in free-space optical communications. Analytical propagation formulae are derived and some numerical examples are given.

2. Formulation

The electric field of an EHGB at z = 0 is expressed as follows [1218]

E(x1,y1,0)=Hm(x1w0)Hn(y1w0)exp(x12w02y12w02), 
where Hm and Hnare the Hermite polynomials of order m or n in x and y directions, respectively, w0 is the beam waist width of the fundamental Gausian mode. Figure 1 shows the normalized intensity distribution I(x1,y1,0)/I(x1,y1,0)max of an EHGB for different values of m with m = n and w0=0.02m at z = 0. The intensity in Fig. 1 and following figures is normalized with respect to its maximum value.

 

Fig. 1 Normalized intensity distribution of an EHGB for different values of m with m = n at z = 0 (a) m = 1, (b) m = 2, (c) m = 3.

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The electric field of a SHGB at z = 0 is expressed as follows [1,2]

E(x1,y1,0)=Hm(2x1w0)Hn(2y1w0)exp(x12w02y12w02). 

One finds from Eqs. (1) and (2) that the main difference between EHGB and SHGB is 2in Hermite polynomials. For m=n=0, both EHGB and SHGB reduce to Gaussian beams. Figure 2 shows the normalized intensity distribution of a SHGB for different values of m with m = n and w0=0.02m at z = 0. It is clear from Fig. 1 and Fig. 2 that the intensity distributions of SHGB and EHGB with the same orders are completely different for m>1.

 

Fig. 2 Normalized intensity distribution of a SHGB for different values of m with m = n and w0=0.02m at z = 0 (a) m = 1, (b) m = 2, (c) m = 3.

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The paraxial propagation of a laser beam in a turbulent atmosphere can be treated with the well-known extended Huygens-Fresnel integral formula, and the average intensity of a laser beam at the output plane is given as follows [2035]

I(px,py,z)=k24π2z2E(x1,y1,0)E*(x2,y2,0)exp[ik2z(x1px)2ik2z(y1py)2]                        ×exp[ik2z(x2px)2+ik2z(y2py)21ρ02(x1x2)21ρ02(y1y2)2]dx1dx2dy1dy2,
whereρ0=(0.545Cn2k2z)3/5is the coherence length (induced by the atmospheric turbulence) of a spherical wave propagating in the turbulent medium with Cn2 being the structure constant [2035], k=2π/λ is the wavenumber and λ is the wavelength of the light, z is the propagation axis. In the derivation of Eq. (3), we have employed Kolmogorov spectrum and quadratic approximation for Rytov’s phase structure function [2035]. The extended Huygens-Fresnel integral formula Eq. (3) has been approved to be reliable and has been used widely (see e.g [20]- [35].).

Substituting Eq. (1) into Eq. (3), after tedious integration, we obtain

<I(px,py,z)>=k24π2z2w02πe1(11e1)m/2exp(k2w02px24e1z2)j1=0[m2](1)j1m!j1!(m2j1)!l1=0[m2](1)l1m!l1!(m2l1)!t1=0m2l1(m2l1t1)                         ×(2)2m2j12l1(g1)t1(h1x)m2l1t1(m+t12j1)!exp(f1x2e2)(f1xe2)m+t12j1πe2q1=0[m+t12j12]1q1!(m+t12j12q1)!                       ×(e24f1x2)q1w02πe1(11e1)n/2exp(k2w02py24e1z2)j'1=0[n2](1)j'1n!j'1!(n2j'1)!l'1=0[n2](1)l'1n!l'1!(n2l'1)!t'1=0n2l'1(n2l'1t'1)(2)2n2j'12l'1                       ×(g1y)t'1(h1y)n2l'1t'1(n+t'12j'1)!exp(f1y2e2)(f1ye2)n+t'12j'1πe2q'1=0[n+t'12j'12]1q'1!(n+t'12j'12q'1)!(e24f1y2)q'1,
where [x] gives the greatest integer less than or equal to x, and

e1=1ikw022z+w02ρ02,  g1=(1e12e1)1/2w02ρ02, h1x=ikw0px2z(1e12e1)1/2, h1y=ikw0py2z(1e12e1)1/2e2=1+ik2zw02+w02ρ02w04e1ρ04,   f1x=ikw0px2zw03ikpx2ze1ρ02,   f1y=ikw0py2zw03ikpy2ze1ρ02.       

In the above derivations, we have used the following formulae [39]

Hn(l)=k=0[n2](1)kn!k!(n2k)!(2l)n2k,                 
exp[(xy)22u]Hn(x)dx=2πu(12u)n/2Hn[y(12u)1/2],  
+xnexp(px2+2qx)dx=n!exp(q2p)(qp)nπpk=0[n2]1k!(n2k)!(p4q2)k.  

Equation (4) is the analytical propagation formula for the average intensity of an EHGB in turbulent atmosphere, and it provides a convenient way for studying the propagation properties of an EHGB in turbulent atmosphere. In the absence of turbulence(ρ0,i.e., Cn2=0), Eq. (4) reduces to the propagation formula for an EHGB in free space.

According to [2], by use of twice the variance of x or y, the effective beam size of an EHGB in x direction at plane z is defined as

Wxz(z)=2x2<I(x,y,z)>dxdy<I(x,y,z)>dxdy.   
Substituting Eq. (4) into Eq. (9), after tedious integration, we obtain the expression for the effective beam size Wxz(z) as follows
Wxz(z)=A1A2,
where
A1=k24π2z2w02πe1(11e1)m/2j1=0[m2](1)j1m!j1!(m2j1)!l1=0[m2](1)l1m!l1!(m2l1)!t1=0m2l1(m2l1t1)       ×(2)2m2j12l1(g1)t1(m+t12j1)!πe2q1=0[m+t12j12]1q1!(m+t12j12q1)!(1e2)m+t12j1       ×(e24)q1(f2)m+t12j12q1(h2)m2l1t1×Γ(2m2j12q12l1+32)(k2w024e1z2f22e2)2m2j12q12l1+3,              
A2=k24π2z2w02πe1(11e1)m/2j1=0[m2](1)j1m!j1!(m2j1)!l1=0[m2](1)l1m!l1!(m2l1)!t1=0m2l1(m2l1t1)      ×(2)2m2j12l1(g1)t1(m+t12j1)!πe2q1=0[m+t12j12]1q1!(m+t12j12q1)!(1e2)m+t12j1     ×(e24)q1(f2)m+t12j12q1(h2)m2l1t1×Γ(2m2j12q12l1+12)2(k2w024e1z2f22e2)2m2j12q12l1+1,                 
with
f2=ikw02zw03ik2ze1ρ02,h2=ikw02z(1e12e1)1/2. 
Using Eq. (10), the spreading properties of an EHGB in turbulent atmosphere can be analysed. Equations (4) and (10) are the main analytical results of present paper.

For the convenience of comparison, we also derived the analytical propagation formula for the average intensity of a SHGB in turbulent atmosphere as follows

<I(px,py,z)>=k24π2z2w022πes1(11es1)m/2exp(k2w02px28es1z2)j1=0[m2](1)j1m!j1!(m2j1)!l1=0[m2](1)l1m!l1!(m2l1)!t1=0m2l1(m2l1t1)                          ×(2)2m2j12l1(gs1)t1(hs1x)m2l1t1(m+t12j1)!exp(f3x2es2)(f3xes2)m+t12j1                          ×πes2q1=0[m+t12j12]1q1!(m+t12j12q1)!(es24f3x2)q1w022πes1(11es1)n/2                          ×exp(k2w02py28es1z2)j'1=0[n2](1)j'1n!j'1!(n2j'1)!l'1=0[n2](1)l'1n!l'1!(n2l'1)!t'1=0n2l'1(n2l'1t'1)(2)2n2j'12l'1(gs1)t'1                          ×(hs1y)n2l'1t'1(n+t'12j'1)!exp(f3y2es2)(f3yes2)n+t'12j'1πes2                          ×q'1=0[n+t'12j'12]1q'1!(n+t'12j'12q'1)!(es24f3y2)q'1
where
es1=12ikw024z+w022ρ02, gs1=(1es12es1)1/2w022ρ02, hs1x=ikw0px22z(1es12es1)1/2, hs1y=ikw0py22z(1es12es1)1/2es2=12+ikw024z+w022ρ02w044es1ρ04,  f3x=ikw0px22zw03ikpx42zes1ρ02, f3y=ikw0py22zw03ikpy42zes1ρ02. 
In a similar way, we obtain the following expression for the effective beam size Wxz(z)of a SHGB in turbulent atmosphere
Wsxz(z)=As1As2, 
where
As1=k24π2z2w022πes1(11es1)m/2j1=0[m2](1)j1m!j1!(m2j1)!l1=0[m2](1)l1m!l1!(m2l1)!t1=0m2l1(m2l1t1)         ×(2)2m2j12l1(gs1)t1(m+t12j1)!πes2q1=0[m+t12j12]1q1!(m+t12j12q1)!         ×(1es2)m+t12j1(es24)q1(f4)m+t12j12q1(hs2)m2l1t1×Γ(2m2j12q12l1+32)(k2w028es1z2f42es2)2m2j12q12l1+3, 
As2=k24π2z2w022πes1(11es1)m/2j1=0[m2](1)j1m!j1!(m2j1)!l1=0[m2](1)l1m!l1!(m2l1)!t1=0m2l1(m2l1t1)       ×(2)2m2j12l1(gs1)t1(m+t12j1)!πes2q1=0[m+t12j12]1q1!(m+t12j12q1)!       ×(1es2)m+t12j1(es24)q1(f4)m+t12j12q1(hs2)m2l1t1×Γ(2m2j12q12l1+12)2(k2w028es1z2f42es2)2m2j12q12l1+1,
with

f4=ikw022zw03ik42zes1ρ02,hs2=ikw022z(1es12es1)1/2.

3. Numerical examples

Now we study the average intensity and spreading of EHGB in turbulent atmosphere numerically by use of the formulae derived in Section 2.

Figure 3 shows the normalized intensity distribution of an EHGB at several different propagation distances in free space with m = n = 2, w0=0.02m, λ=632.8nm and Cn2=0. One finds from Fig. 3 that the transverse pattern of the EHGB becomes a pattern with four-beamlets in the far field of free space. Our numerical results also show that any EHGB with m>1 and n>1 will converse into a laser beam with four-beamlets, and the distances between these beamlets increase by increasing the orders m and n. For a SHGB however, its beam profile remains invariant on propagation in free space, although its beam spot spreads on propagation. Our results agree well with those reported in [1,2,17,18].Figure 4 shows the normalized intensity distribution of an EHGB at several different propagation distances in turbulent atmosphere with m = n = 2, w0=0.02m,λ=632.8nmand Cn2=1014m2/3. Comparing Figs. 3 and 4, it is clear that the propagation properties of the EHGB in turbulent atmosphere are much different from its propagation properties in free space. In turbulent atmosphere, the EHGB finally becomes a circular Gaussian beam spot in the far field under the isotropic influence of the atmosphere turbulence. Figure 5 shows the normalized intensity distribution of an EHGB for different values of m with m = n at z = 3km in turbulent atmosphere. One finds from Fig. 5 that the intensity distribution of the EHGB with larger m and n in turbulent atmosphere is more similar to its far field intensity distribution in free space, which means the EHGB with larger m and n is less affected by the turbulence. Our numerical results (not shown here to save space) also show that the EHGB with larger values of w0 and λ are less affected by turbulence. What’s more, our numerical results (again not shown here to save space) also show that the SHGB also becomes a circular Gaussian beam in the far field of turbulent atmosphere, and the SHGB with larger values of m, n, w0 and λ are less affected by the turbulence, which agree well with those reported in [20].

 

Fig. 3 Normalized intensity distribution of an EHGB with m = n = 2 at several different propagation distances in free space (a) z = 1km, (b) z = 3km, (c) z = 5km, (d) z = 10km.

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Fig. 4 Normalized intensity distribution of an EHGB with m = n = 2 at several different propagation distances in turbulent atmosphere with Cn2=1014m2/3 (a) z = 1km, (b) z = 1.5km, (c) z = 2km, (d) z = 3km.

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Fig. 5 Normalized intensity distribution of an EHGB for different values of m with m = n at z = 3km in turbulent atmosphere with Cn2=1014m2/3 (a) m = 1, (b) m = 3, (c) m = 5, (d) m = 8.

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To learn about the spreading properties of the EHGB in turbulent atmosphere, we calculate in Figs. 6 (a) -6(c) the effective beam sizes of an EHGB (solid lines) and an SHGB (dotted line) with m = n = 2 and λ=632.8nm versus the propagation distance z in turbulent atmosphere for different values of the structure constant. The corresponding results in free space are also plotted in Fig. 6 (d). Note that in Fig. 6 we have set w0=0.02928m for the EHGB and w0=0.02m for the SHGB, so the EHGB and SHGB have the same effective beam size at the source plane (z = 0). One finds from Fig. 6 that the SHGB spreads more rapidly than the EHGB both in free space and in turbulent atmosphere. As the structure constant becomes larger (i.e., atmospheric turbulence becomes stronger), the advantage of the EHGB over the SHGB becomes smaller. So the EHGB has advantage over the SHGB for application in free-space optical communication in weakly turbulent atmosphere. What’s more, our numerical results (not shown here to save space) also show that the EHGB spreads more rapidly in turbulent atmosphere for a larger structure constant, a shorter wavelength, larger beam orders, and a smaller waist size of the initial beam.

 

Fig. 6 Effective beam size of an EHGB (solid lines) and an SHGB (dotted line) with m = n = 2 andλ=632.8nmversus the propagation distance z in turbulent atmosphere for different values of the structure constant (a) Cn2=1013m2/3 (b) Cn2=1014m2/3 (c) Cn2=1015m2/3 (d) Cn2=0 (free space).

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4. Conclusion

We have derived the analytical propagation formulae for the average intensity and effective beam size of an EHGB in turbulent atmosphere based on the extended Huygens-Fresnel integral. The propagation properties of the EHGB and SHGB have been studied numerically and comparatively. Our results show that the propagation properties of the EHGB and SHGB are much different from their properties in free space, and the EHGB and SHGB with higher orders are less affected by turbulence. What’s more, the SHGB spreads more rapidly than the EHGB in turbulent atmosphere under the same conditions (i.e., the same initial effective beam size, orders and wavelength). The propagation properties of an EHGB are closely related to the beam parameters and the structure constant of the atmospheric turbulence. Our results will be useful in long-distance free-space optical communications.

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References

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  1. A. E. Siegman, LASERS, University Science Books, Mill Valley, 1986.
  2. W. H. Carter, “Spot size and divergence for Hermite-Gaussian beams of any order,” Appl. Opt. 19(7), 1027–1029 (1980).
    [CrossRef]
  3. T. Kojima, “Diffraction of Hermite-Gaussian beams from a sinusoidal conducting grating,” J. Opt. Soc. Am. A 7(9), 1740–1744 (1990).
    [CrossRef]
  4. O. Mata-Mendez and F. Chavez-Rivas, “Diffraction of Gaussian and Hermite-Gaussia beams by finite gratings,” J. Opt. Soc. Am. A 18(3), 537–545 (2001).
    [CrossRef]
  5. K. M. Luk and P. K. Yu, “Generation of Hermite-Gaussian beam modes by multipoles with complex source points,” J. Opt. Soc. Am. A 2(11), 1818–1820 (1985).
    [CrossRef]
  6. Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, “Generation of Hermite-Gaussian modes in fiber-coupled laser-diode end-pumped lasers,” IEEE J. Quantum Electron. 33(6), 1025–1031 (1997).
    [CrossRef]
  7. H. Laabs, “Propagation of Hermite-Gaussian-beams beyond the paraxial approximation,” Opt. Commun. 147(1-3), 1–4 (1998).
    [CrossRef]
  8. Y. Cai and Q. Lin, “The elliptical Hermite-Gaussian beam and its propagation through paraxial systems,” Opt. Commun. 207, 139–147 (2002).
  9. Y. Cai and Q. Lin, “Decentered elliptical Hermite–Gaussian beam,” J. Opt. Soc. Am. A 20(6), 1111–1119 (2003).
    [CrossRef]
  10. Y. Qiu, H. Guo, and Z. Chen, “Paraxial propagation of partially coherent Hermite-Gauss beams,” Opt. Commun. 245(1-6), 21–26 (2005).
    [CrossRef]
  11. Y. Cai and C. Chen, “Paraxial propagation of a partially coherent Hermite-Gaussian beam through aligned and misaligned ABCD optical systems,” J. Opt. Soc. Am. A 24(8), 2394–2401 (2007).
    [CrossRef]
  12. A. E. Siegman, “Hermite-Gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. Am. 63(9), 1093–1094 (1973).
    [CrossRef]
  13. S. Y. Shin and L. B. Felsen, “Gaussian beam modes by multipoles with complex source points,” J. Opt. Soc. Am. 67(5), 699–700 (1977).
    [CrossRef]
  14. E. Zauderer, “Complex argument Hermite-Gaussian and Laguerre-Gaussian beams,” J. Opt. Soc. Am. 3(4), 465–469 (1986).
    [CrossRef]
  15. H. Laabs, C. Gao, and H. Weber, “Twisting of three-dimensional Hermite-Gaussian beams,” J. Mod. Opt. 46, 709–719 (1999).
  16. S. Saghafi and C. J. R. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Opt. Commun. 153(4-6), 207–210 (1998).
    [CrossRef]
  17. S. Saghafi, C. J. R. Sheppard, and J. A. Piper, “Characterising elegant and standard Hermite-Gaussian beam modes,” Opt. Commun. 191(3-6), 173–179 (2001).
    [CrossRef]
  18. B. Lu and H. Ma, “A comparative study of elegant and standard Hermite-Gaussian beams,” Opt. Commun. 174(1-4), 99–104 (2000).
    [CrossRef]
  19. Z. I. Feizulin and Y. A. Kravtsov, “Broadening of a laser beam in a turbulent medium,” Radiophys. Quantum Electron. 10(1), 33–35 (1967).
    [CrossRef]
  20. C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence-induced beam spreading of higher-order mode optical waves,” Opt. Eng. 41(5), 1097–1103 (2002).
    [CrossRef]
  21. S. C. H. Wang and M. A. Plonus, “Optical beam propagation for a partially coherent source in the turbulent atmosphere,” J. Opt. Soc. Am. 69(9), 1297–1304 (1979).
    [CrossRef]
  22. X. Chu, “Propagation of a cosh-Gaussian beam through an optical system in turbulent atmosphere,” Opt. Express 15(26), 17613–17618 (2007).
    [CrossRef]
  23. O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere,” Waves Random Complex Media 15(3), 353–364 (2005).
    [CrossRef]
  24. Y. Cai, O. Korotkova, H. T. Eyyuboğlu, and Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express 16(20), 15834–15846 (2008).
    [CrossRef]
  25. Y. Baykal, “Correlation and structure functions of Hermite-sinusoidal-Gaussian laser beams in a turbulent atmosphere,” J. Opt. Soc. Am. A 21(7), 1290–1299 (2004).
    [CrossRef]
  26. H. T. Eyyuboğlu, “Hermite-cosine-Gaussian laser beam and its propagation characteristics in turbulent atmosphere,” J. Opt. Soc. Am. A 22(8), 1527–1535 (2005).
    [CrossRef]
  27. Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006).
    [CrossRef]
  28. Y. Cai, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation index of elliptical Gaussian beam in turbulent atmosphere,” Opt. Lett. 32(16), 2405–2407 (2007).
    [CrossRef] [PubMed]
  29. M. Alavinejad, B. Ghafary, and F. D. Kashani, “Analysis of the propagation of flat-topped beam with various beam orders through turbulent atmosphere,” Opt. Lasers Eng. 46(1), 1–5 (2008).
    [CrossRef]
  30. Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14(4), 1353–1367 (2006).
    [CrossRef] [PubMed]
  31. R. J. Noriega-Manez and J. C. Gutierrez-Vega, “Rytov theory for Helmholtz-Gauss beams in turbulent atmosphere,” Opt. Express 15(25), 16328–16341 (2007).
    [CrossRef]
  32. Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagnetic beams in a turbulent atmosphere,” J. Opt. A, Pure Appl. Opt. 9(12), 1123–1130 (2007).
    [CrossRef]
  33. K. Zhu, G. Zhou, X. Li, X. Zheng, and H. Tang, “Propagation of Bessel-Gaussian beams with optical vortices in turbulent atmosphere,” Opt. Express 16(26), 21315–21320 (2008).
    [CrossRef] [PubMed]
  34. Y. Cai, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation index of elliptical Gaussian beam in turbulent atmosphere,” Opt. Lett. 32(16), 2405–2407 (2007).
    [CrossRef] [PubMed]
  35. X. Ji, X. Chen, and B. Lu, “Spreading and directionality of partially coherent Hermite-Gaussian beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 25(1), 21–28 (2008).
    [CrossRef]
  36. A. Belmonte, A. Comerón, J. A. Rubio, J. Bará, and E. Fernández, “Atmospheric-turbulence-induced power-fade statistics for a multiaperture optical receiver,” Appl. Opt. 36(33), 8632–8638 (1997).
    [CrossRef] [PubMed]
  37. I. I. Kim, H. Hakakha, P. Adhikari, E. J. Korevaar, and A. K. Majumdar, “Scintillation reduction using multiple apertures,” in Free-Space Laser Communication Technologies IX, G. Mecherle, ed., Proc. SPIE 2990, 102–113 (1997).
  38. Y. E. Yenice, and B. G. Evans, “Adaptive beam-size control for ground-to-space laser communications,” in Free-Space Laser Communication Technologies X, G. Mecherle, ed., Proc. SPIE 3266, 221–230 (1998).
  39. A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

2008

2007

2006

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

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

2005

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

H. T. Eyyuboğlu, “Hermite-cosine-Gaussian laser beam and its propagation characteristics in turbulent atmosphere,” J. Opt. Soc. Am. A 22(8), 1527–1535 (2005).
[CrossRef]

Y. Qiu, H. Guo, and Z. Chen, “Paraxial propagation of partially coherent Hermite-Gauss beams,” Opt. Commun. 245(1-6), 21–26 (2005).
[CrossRef]

2004

2003

2002

Y. Cai and Q. Lin, “The elliptical Hermite-Gaussian beam and its propagation through paraxial systems,” Opt. Commun. 207, 139–147 (2002).

C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence-induced beam spreading of higher-order mode optical waves,” Opt. Eng. 41(5), 1097–1103 (2002).
[CrossRef]

2001

O. Mata-Mendez and F. Chavez-Rivas, “Diffraction of Gaussian and Hermite-Gaussia beams by finite gratings,” J. Opt. Soc. Am. A 18(3), 537–545 (2001).
[CrossRef]

S. Saghafi, C. J. R. Sheppard, and J. A. Piper, “Characterising elegant and standard Hermite-Gaussian beam modes,” Opt. Commun. 191(3-6), 173–179 (2001).
[CrossRef]

2000

B. Lu and H. Ma, “A comparative study of elegant and standard Hermite-Gaussian beams,” Opt. Commun. 174(1-4), 99–104 (2000).
[CrossRef]

1999

H. Laabs, C. Gao, and H. Weber, “Twisting of three-dimensional Hermite-Gaussian beams,” J. Mod. Opt. 46, 709–719 (1999).

1998

S. Saghafi and C. J. R. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Opt. Commun. 153(4-6), 207–210 (1998).
[CrossRef]

H. Laabs, “Propagation of Hermite-Gaussian-beams beyond the paraxial approximation,” Opt. Commun. 147(1-3), 1–4 (1998).
[CrossRef]

1997

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, “Generation of Hermite-Gaussian modes in fiber-coupled laser-diode end-pumped lasers,” IEEE J. Quantum Electron. 33(6), 1025–1031 (1997).
[CrossRef]

A. Belmonte, A. Comerón, J. A. Rubio, J. Bará, and E. Fernández, “Atmospheric-turbulence-induced power-fade statistics for a multiaperture optical receiver,” Appl. Opt. 36(33), 8632–8638 (1997).
[CrossRef] [PubMed]

1990

1986

E. Zauderer, “Complex argument Hermite-Gaussian and Laguerre-Gaussian beams,” J. Opt. Soc. Am. 3(4), 465–469 (1986).
[CrossRef]

1985

1980

1979

1977

1973

1967

Z. I. Feizulin and Y. A. Kravtsov, “Broadening of a laser beam in a turbulent medium,” Radiophys. Quantum Electron. 10(1), 33–35 (1967).
[CrossRef]

Alavinejad, M.

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

Bará, J.

Baykal, Y.

Belmonte, A.

Cai, Y.

Carter, W. H.

Chavez-Rivas, F.

Chen, C.

Chen, X.

Chen, Y.

Chen, Y. F.

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, “Generation of Hermite-Gaussian modes in fiber-coupled laser-diode end-pumped lasers,” IEEE J. Quantum Electron. 33(6), 1025–1031 (1997).
[CrossRef]

Chen, Z.

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

Y. Qiu, H. Guo, and Z. Chen, “Paraxial propagation of partially coherent Hermite-Gauss beams,” Opt. Commun. 245(1-6), 21–26 (2005).
[CrossRef]

Chu, X.

Comerón, A.

Dogariu, A.

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

Eyyuboglu, H. T.

Feizulin, Z. I.

Z. I. Feizulin and Y. A. Kravtsov, “Broadening of a laser beam in a turbulent medium,” Radiophys. Quantum Electron. 10(1), 33–35 (1967).
[CrossRef]

Felsen, L. B.

Fernández, E.

Gao, C.

H. Laabs, C. Gao, and H. Weber, “Twisting of three-dimensional Hermite-Gaussian beams,” J. Mod. Opt. 46, 709–719 (1999).

Ghafary, B.

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

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(5), 1097–1103 (2002).
[CrossRef]

Guo, H.

Y. Qiu, H. Guo, and Z. Chen, “Paraxial propagation of partially coherent Hermite-Gauss beams,” Opt. Commun. 245(1-6), 21–26 (2005).
[CrossRef]

Gutierrez-Vega, J. C.

He, S.

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

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

Huang, T. M.

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, “Generation of Hermite-Gaussian modes in fiber-coupled laser-diode end-pumped lasers,” IEEE J. Quantum Electron. 33(6), 1025–1031 (1997).
[CrossRef]

Ji, X.

Kao, C. F.

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, “Generation of Hermite-Gaussian modes in fiber-coupled laser-diode end-pumped lasers,” IEEE J. Quantum Electron. 33(6), 1025–1031 (1997).
[CrossRef]

Kashani, F. D.

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

Kojima, T.

Korotkova, O.

Y. Cai, O. Korotkova, H. T. Eyyuboğlu, and Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express 16(20), 15834–15846 (2008).
[CrossRef]

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

Kravtsov, Y. A.

Z. I. Feizulin and Y. A. Kravtsov, “Broadening of a laser beam in a turbulent medium,” Radiophys. Quantum Electron. 10(1), 33–35 (1967).
[CrossRef]

Laabs, H.

H. Laabs, C. Gao, and H. Weber, “Twisting of three-dimensional Hermite-Gaussian beams,” J. Mod. Opt. 46, 709–719 (1999).

H. Laabs, “Propagation of Hermite-Gaussian-beams beyond the paraxial approximation,” Opt. Commun. 147(1-3), 1–4 (1998).
[CrossRef]

Li, X.

Lin, Q.

Y. Cai and Q. Lin, “Decentered elliptical Hermite–Gaussian beam,” J. Opt. Soc. Am. A 20(6), 1111–1119 (2003).
[CrossRef]

Y. Cai and Q. Lin, “The elliptical Hermite-Gaussian beam and its propagation through paraxial systems,” Opt. Commun. 207, 139–147 (2002).

Lu, B.

Luk, K. M.

Ma, H.

B. Lu and H. Ma, “A comparative study of elegant and standard Hermite-Gaussian beams,” Opt. Commun. 174(1-4), 99–104 (2000).
[CrossRef]

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(5), 1097–1103 (2002).
[CrossRef]

Mata-Mendez, O.

Noriega-Manez, R. J.

Piper, J. A.

S. Saghafi, C. J. R. Sheppard, and J. A. Piper, “Characterising elegant and standard Hermite-Gaussian beam modes,” Opt. Commun. 191(3-6), 173–179 (2001).
[CrossRef]

Plonus, M. A.

Pu, J.

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

Qiu, Y.

Y. Qiu, H. Guo, and Z. Chen, “Paraxial propagation of partially coherent Hermite-Gauss beams,” Opt. Commun. 245(1-6), 21–26 (2005).
[CrossRef]

Rubio, J. A.

Saghafi, S.

S. Saghafi, C. J. R. Sheppard, and J. A. Piper, “Characterising elegant and standard Hermite-Gaussian beam modes,” Opt. Commun. 191(3-6), 173–179 (2001).
[CrossRef]

S. Saghafi and C. J. R. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Opt. Commun. 153(4-6), 207–210 (1998).
[CrossRef]

Salem, M.

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

Sheppard, C. J. R.

S. Saghafi, C. J. R. Sheppard, and J. A. Piper, “Characterising elegant and standard Hermite-Gaussian beam modes,” Opt. Commun. 191(3-6), 173–179 (2001).
[CrossRef]

S. Saghafi and C. J. R. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Opt. Commun. 153(4-6), 207–210 (1998).
[CrossRef]

Shin, S. Y.

Siegman, A. E.

Tang, H.

Wang, C. L.

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, “Generation of Hermite-Gaussian modes in fiber-coupled laser-diode end-pumped lasers,” IEEE J. Quantum Electron. 33(6), 1025–1031 (1997).
[CrossRef]

Wang, S. C.

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, “Generation of Hermite-Gaussian modes in fiber-coupled laser-diode end-pumped lasers,” IEEE J. Quantum Electron. 33(6), 1025–1031 (1997).
[CrossRef]

Wang, S. C. H.

Weber, H.

H. Laabs, C. Gao, and H. Weber, “Twisting of three-dimensional Hermite-Gaussian beams,” J. Mod. Opt. 46, 709–719 (1999).

Wolf, E.

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

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(5), 1097–1103 (2002).
[CrossRef]

Yu, P. K.

Zauderer, E.

E. Zauderer, “Complex argument Hermite-Gaussian and Laguerre-Gaussian beams,” J. Opt. Soc. Am. 3(4), 465–469 (1986).
[CrossRef]

Zheng, X.

Zhou, G.

Zhu, K.

Appl. Opt.

Appl. Phys. Lett.

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

IEEE J. Quantum Electron.

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, “Generation of Hermite-Gaussian modes in fiber-coupled laser-diode end-pumped lasers,” IEEE J. Quantum Electron. 33(6), 1025–1031 (1997).
[CrossRef]

J. Mod. Opt.

H. Laabs, C. Gao, and H. Weber, “Twisting of three-dimensional Hermite-Gaussian beams,” J. Mod. Opt. 46, 709–719 (1999).

J. Opt. A, Pure Appl. Opt.

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

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

Y. Qiu, H. Guo, and Z. Chen, “Paraxial propagation of partially coherent Hermite-Gauss beams,” Opt. Commun. 245(1-6), 21–26 (2005).
[CrossRef]

H. Laabs, “Propagation of Hermite-Gaussian-beams beyond the paraxial approximation,” Opt. Commun. 147(1-3), 1–4 (1998).
[CrossRef]

Y. Cai and Q. Lin, “The elliptical Hermite-Gaussian beam and its propagation through paraxial systems,” Opt. Commun. 207, 139–147 (2002).

S. Saghafi and C. J. R. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Opt. Commun. 153(4-6), 207–210 (1998).
[CrossRef]

S. Saghafi, C. J. R. Sheppard, and J. A. Piper, “Characterising elegant and standard Hermite-Gaussian beam modes,” Opt. Commun. 191(3-6), 173–179 (2001).
[CrossRef]

B. Lu and H. Ma, “A comparative study of elegant and standard Hermite-Gaussian beams,” Opt. Commun. 174(1-4), 99–104 (2000).
[CrossRef]

Opt. Eng.

C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence-induced beam spreading of higher-order mode optical waves,” Opt. Eng. 41(5), 1097–1103 (2002).
[CrossRef]

Opt. Express

Opt. Lasers Eng.

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

Opt. Lett.

Radiophys. Quantum Electron.

Z. I. Feizulin and Y. A. Kravtsov, “Broadening of a laser beam in a turbulent medium,” Radiophys. Quantum Electron. 10(1), 33–35 (1967).
[CrossRef]

Waves Random Complex Media

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

Other

I. I. Kim, H. Hakakha, P. Adhikari, E. J. Korevaar, and A. K. Majumdar, “Scintillation reduction using multiple apertures,” in Free-Space Laser Communication Technologies IX, G. Mecherle, ed., Proc. SPIE 2990, 102–113 (1997).

Y. E. Yenice, and B. G. Evans, “Adaptive beam-size control for ground-to-space laser communications,” in Free-Space Laser Communication Technologies X, G. Mecherle, ed., Proc. SPIE 3266, 221–230 (1998).

A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

A. E. Siegman, LASERS, University Science Books, Mill Valley, 1986.

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Figures (6)

Fig. 1
Fig. 1

Normalized intensity distribution of an EHGB for different values of m with m = n at z = 0 (a) m = 1, (b) m = 2, (c) m = 3.

Fig. 2
Fig. 2

Normalized intensity distribution of a SHGB for different values of m with m = n and w0=0.02m at z = 0 (a) m = 1, (b) m = 2, (c) m = 3.

Fig. 3
Fig. 3

Normalized intensity distribution of an EHGB with m = n = 2 at several different propagation distances in free space (a) z = 1km, (b) z = 3km, (c) z = 5km, (d) z = 10km.

Fig. 4
Fig. 4

Normalized intensity distribution of an EHGB with m = n = 2 at several different propagation distances in turbulent atmosphere with Cn2=1014m2/3 (a) z = 1km, (b) z = 1.5km, (c) z = 2km, (d) z = 3km.

Fig. 5
Fig. 5

Normalized intensity distribution of an EHGB for different values of m with m = n at z = 3km in turbulent atmosphere with Cn2=1014m2/3 (a) m = 1, (b) m = 3, (c) m = 5, (d) m = 8.

Fig. 6
Fig. 6

Effective beam size of an EHGB (solid lines) and an SHGB (dotted line) with m = n = 2 and λ=632.8nmversus the propagation distance z in turbulent atmosphere for different values of the structure constant (a) Cn2=1013m2/3 (b) Cn2=1014m2/3 (c) Cn2=1015m2/3 (d) Cn2=0 (free space).

Equations (19)

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

E(x1,y1,0)=Hm(x1w0)Hn(y1w0)exp(x12w02y12w02), 
E(x1,y1,0)=Hm(2x1w0)Hn(2y1w0)exp(x12w02y12w02). 
I(px,py,z)=k24π2z2E(x1,y1,0)E*(x2,y2,0)exp[ik2z(x1px)2ik2z(y1py)2]                        ×exp[ik2z(x2px)2+ik2z(y2py)21ρ02(x1x2)21ρ02(y1y2)2]dx1dx2dy1dy2,
<I(px,py,z)>=k24π2z2w02πe1(11e1)m/2exp(k2w02px24e1z2)j1=0[m2](1)j1m!j1!(m2j1)!l1=0[m2](1)l1m!l1!(m2l1)!t1=0m2l1(m2l1t1)                         ×(2)2m2j12l1(g1)t1(h1x)m2l1t1(m+t12j1)!exp(f1x2e2)(f1xe2)m+t12j1πe2q1=0[m+t12j12]1q1!(m+t12j12q1)!                       ×(e24f1x2)q1w02πe1(11e1)n/2exp(k2w02py24e1z2)j'1=0[n2](1)j'1n!j'1!(n2j'1)!l'1=0[n2](1)l'1n!l'1!(n2l'1)!t'1=0n2l'1(n2l'1t'1)(2)2n2j'12l'1                       ×(g1y)t'1(h1y)n2l'1t'1(n+t'12j'1)!exp(f1y2e2)(f1ye2)n+t'12j'1πe2q'1=0[n+t'12j'12]1q'1!(n+t'12j'12q'1)!(e24f1y2)q'1,
e1=1ikw022z+w02ρ02,  g1=(1e12e1)1/2w02ρ02, h1x=ikw0px2z(1e12e1)1/2, h1y=ikw0py2z(1e12e1)1/2e2=1+ik2zw02+w02ρ02w04e1ρ04,   f1x=ikw0px2zw03ikpx2ze1ρ02,   f1y=ikw0py2zw03ikpy2ze1ρ02.       
Hn(l)=k=0[n2](1)kn!k!(n2k)!(2l)n2k,                 
exp[(xy)22u]Hn(x)dx=2πu(12u)n/2Hn[y(12u)1/2],  
+xnexp(px2+2qx)dx=n!exp(q2p)(qp)nπpk=0[n2]1k!(n2k)!(p4q2)k.  
Wxz(z)=2x2<I(x,y,z)>dxdy<I(x,y,z)>dxdy.   
Wxz(z)=A1A2,
A1=k24π2z2w02πe1(11e1)m/2j1=0[m2](1)j1m!j1!(m2j1)!l1=0[m2](1)l1m!l1!(m2l1)!t1=0m2l1(m2l1t1)       ×(2)2m2j12l1(g1)t1(m+t12j1)!πe2q1=0[m+t12j12]1q1!(m+t12j12q1)!(1e2)m+t12j1       ×(e24)q1(f2)m+t12j12q1(h2)m2l1t1×Γ(2m2j12q12l1+32)(k2w024e1z2f22e2)2m2j12q12l1+3,              
A2=k24π2z2w02πe1(11e1)m/2j1=0[m2](1)j1m!j1!(m2j1)!l1=0[m2](1)l1m!l1!(m2l1)!t1=0m2l1(m2l1t1)      ×(2)2m2j12l1(g1)t1(m+t12j1)!πe2q1=0[m+t12j12]1q1!(m+t12j12q1)!(1e2)m+t12j1     ×(e24)q1(f2)m+t12j12q1(h2)m2l1t1×Γ(2m2j12q12l1+12)2(k2w024e1z2f22e2)2m2j12q12l1+1,                 
f2=ikw02zw03ik2ze1ρ02,h2=ikw02z(1e12e1)1/2. 
<I(px,py,z)>=k24π2z2w022πes1(11es1)m/2exp(k2w02px28es1z2)j1=0[m2](1)j1m!j1!(m2j1)!l1=0[m2](1)l1m!l1!(m2l1)!t1=0m2l1(m2l1t1)                          ×(2)2m2j12l1(gs1)t1(hs1x)m2l1t1(m+t12j1)!exp(f3x2es2)(f3xes2)m+t12j1                          ×πes2q1=0[m+t12j12]1q1!(m+t12j12q1)!(es24f3x2)q1w022πes1(11es1)n/2                          ×exp(k2w02py28es1z2)j'1=0[n2](1)j'1n!j'1!(n2j'1)!l'1=0[n2](1)l'1n!l'1!(n2l'1)!t'1=0n2l'1(n2l'1t'1)(2)2n2j'12l'1(gs1)t'1                          ×(hs1y)n2l'1t'1(n+t'12j'1)!exp(f3y2es2)(f3yes2)n+t'12j'1πes2                          ×q'1=0[n+t'12j'12]1q'1!(n+t'12j'12q'1)!(es24f3y2)q'1
es1=12ikw024z+w022ρ02, gs1=(1es12es1)1/2w022ρ02, hs1x=ikw0px22z(1es12es1)1/2, hs1y=ikw0py22z(1es12es1)1/2es2=12+ikw024z+w022ρ02w044es1ρ04,  f3x=ikw0px22zw03ikpx42zes1ρ02, f3y=ikw0py22zw03ikpy42zes1ρ02. 
Wsxz(z)=As1As2, 
As1=k24π2z2w022πes1(11es1)m/2j1=0[m2](1)j1m!j1!(m2j1)!l1=0[m2](1)l1m!l1!(m2l1)!t1=0m2l1(m2l1t1)         ×(2)2m2j12l1(gs1)t1(m+t12j1)!πes2q1=0[m+t12j12]1q1!(m+t12j12q1)!         ×(1es2)m+t12j1(es24)q1(f4)m+t12j12q1(hs2)m2l1t1×Γ(2m2j12q12l1+32)(k2w028es1z2f42es2)2m2j12q12l1+3, 
As2=k24π2z2w022πes1(11es1)m/2j1=0[m2](1)j1m!j1!(m2j1)!l1=0[m2](1)l1m!l1!(m2l1)!t1=0m2l1(m2l1t1)       ×(2)2m2j12l1(gs1)t1(m+t12j1)!πes2q1=0[m+t12j12]1q1!(m+t12j12q1)!       ×(1es2)m+t12j1(es24)q1(f4)m+t12j12q1(hs2)m2l1t1×Γ(2m2j12q12l1+12)2(k2w028es1z2f42es2)2m2j12q12l1+1,
f4=ikw022zw03ik42zes1ρ02,hs2=ikw022z(1es12es1)1/2.

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