OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 25 — Dec. 10, 2007
  • pp: 16328–16341
« Show journal navigation

Rytov theory for Helmholtz-Gauss beams in turbulent atmosphere

Rodrigo J. Noriega-Manez and Julio C. Gutiérrez-Vega  »View Author Affiliations


Optics Express, Vol. 15, Issue 25, pp. 16328-16341 (2007)
http://dx.doi.org/10.1364/OE.15.016328


View Full Text Article

Acrobat PDF (608 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

The Rytov theory for the propagation of Helmholtz-Gauss (HzG) beams in turbulent atmosphere is presented. We derive expressions for the first and second-order normalized Born approximations, the second-order moments, and the transverse intensity pattern of the HzG beams at any arbitrary propagation distance. The general formulation is applied to study the propagation of several special cases of the HzG beams, in particular, the Bessel-Gauss and Mathieu-Gauss beams and their pure nondiffracting counterparts, the Bessel and Mathieu beams. For numerical purposes, we assume the standard Kolmogorov distribution to model the power spectrum of the atmospheric fluctuations.

© 2007 Optical Society of America

1. Introduction

Atmospheric propagation of laser radiation is currently a very active area of research, with potential applications such as optical communications, imaging, and remote sensing [1

1. L. Andrews and R. Phillips, Laser beam propagation through random media (SPIE Press, 1998).

]. When the optical/IR wave propagates through the atmosphere, both the amplitude and phase of the electric field experience random fluctuations caused by random changes in the index of refraction. Although classical treatments of optical beam propagation in atmosphere are based primarily on the lowest order Gaussian beam [1

1. L. Andrews and R. Phillips, Laser beam propagation through random media (SPIE Press, 1998).

], there is recent evidence that other types of beams might provide better performance. In view of this, several classes of laser beams in turbulent atmosphere have been investigated, such as the higher-order and elliptical Gaussian beams [2

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]

, 3

3. H. T. Eyyuboglu, S. Altay, and Y. Baykal, “Propagation characteristics of higher-order annular Gaussian beams in atmospheric turbulence,” Opt. Commun. 264, 25–34 (2006). [CrossRef]

, 4

4. Y. Cai and S. He, “Average intensity and spreading of an elliptical Gaussian beam propagating in a turbulent atmosphere,” Opt. Lett. 31, 568–570 (2002). [CrossRef]

, 5

5. Y. Cai and D. Ge, “Analytical formula for a decentered elliptical Gaussian beam propagationg in a turbulent atmosphere,” Opt. Commun. 271, 509–516 (2007). [CrossRef]

], cosh-Gaussian beams [6

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

, 7

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

], Hermite-cosh-Gaussian laser beams [8

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

, 9

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

], partially coherent twisted anisotropic Gaussian Schell-model beams [10

10. 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]

], higher order Bessel-Gaussian beams [11

11. H. T. Eyyuboğlu, “Propagation of higher order Bessel-Gaussian beams in turbulence,” Appl. Phys. B , 88, 259–265 (2007). [CrossRef]

], and dark hollow beams [12

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

]. Of particular interest are the vortex beams whose carried orbital angular momentum provides additional resistance to turbulence [13

13. Z. Bouchal, “Resistance of nondiffracting vortex beam against amplitude and phase perturbations,” Opt. Commun. 210, 155–164 (2002). [CrossRef]

], and also the nondiffracting beams [14

14. Z. Bouchal, “Nondiffracting optical beams: physical properties, experiments, and applications,” Czech. J. Phys. 53, 537–578 (2003). [CrossRef]

, 15

15. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo,, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25, 1493–1495 (2000). [CrossRef]

, 16

16. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. 29, 44–46 (2004). [CrossRef] [PubMed]

], which have been shown to be resistant to amplitude and phase perturbations [17

17. Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstructionof a distorted nondiffracting beam,” Opt. Commun. 151, 207–211 (1998). [CrossRef]

, 18

18. T. Aruga, S. W. Li, S. Yoshikado, M. Takabe, and R. Li, “Nondiffracting narrow light beam with small atmospheric turbulence-induced propagation,” Appl. Opt. 38, 3152–3156 (1999). [CrossRef]

, 19

19. G. Gbur and O. Korotkova, “Angular spectrum representation for propagation of arbitrary coherent and partially coherent beams through atmospheric turbulence,” J. Opt. Soc. Am. A 24, 745–752 (2007). [CrossRef]

, 20

20. O. Korotkova and G. Gbur, “Propagation of beams with any spectral, coherence and polarization properties in turbulent atmosphere,” Proc. of SPIE 6457, 64570J1–64570J12 (2007).

, 21

21. O. Korotkova and G. Gbur, “Angular spectrum representation for propagation of random electromagnetic beams in a turbulent atmosphere,” J. Opt. Soc. Am. A 24, 2728–2736 (2007). [CrossRef]

]. The comparison of theoretical models with general computer simulators for optical/IR beams propagating in turbulent atmosphere is also subject of current study [22

22. R. Frehlich, “Simulation of laser propagation in a turbulent atmosphere,” Appl. Opt. 30, 393–397 (2000). [CrossRef]

, 23

23. C. Arpali, C. Yazicioglu, H. Eyyuboğlu, S. Arpali, and Y. Baykal, “Simulator for general-type beam propagation in turbulent atmosphere,” Opt. Express 14, 8918–8928 (2006). [CrossRef] [PubMed]

, 24

24. D.C. Cowan, J. Recolons, L.C. Andrews, and C.Y. Young, Atmospheric Propagation III, Proc. SPIE 6215, 62150 B-1- 62150B-10 (2006).

].

Recently, the free-space propagation of Helmholtz-Gauss (HzG) beams was theoretically [25

25. J. C. Gutiérrez-Vega and M. A. Bandres, “Helmholtz-Gauss waves,” J. Opt. Soc. Am. A 22, 289–298 (2005). [CrossRef]

] and experimentally studied [26

26. C. López-Mariscal, M. A. Bandrés, and J. C. Gutiérrez-Vega, “Observation of the experimental propagation properties of Helmholtz-Gauss beams,” Opt. Eng. 45, 068001 (2006). [CrossRef]

]. The term HzG beam refers to a paraxial wave whose disturbance at the plane z=0 is given by the transverse field of an arbitrary nondiffracting beam (i.e. a solution of the two-dimensional Helmholtz equation) modulated by a Gaussian transmittance. The model of the HzG beam describes in a more realistic way the propagation of ideal nondiffracting beams because HzG beams carry finite power, retain the nondiffracting propagation properties within a finite propagation distance, and can be realized experimentally to a very good approximation [25

25. J. C. Gutiérrez-Vega and M. A. Bandres, “Helmholtz-Gauss waves,” J. Opt. Soc. Am. A 22, 289–298 (2005). [CrossRef]

]. The propagation of optical HzG beams has been studied in absorbing and gain media [27

27. M. Guizar-Sicairos and J. C. Gutiérrez-Vega, “Propagation of Helmholtz-Gauss beams in absorbing and gain media,” J. Opt. Soc. Am. A , 23, 1994–2001 (2006). [CrossRef]

], and through complex paraxial ABCD systems in the scalar [28

28. M. Guizar-Sicairos and J. C. Gutiérrez-Vega, “Generalized Helmholtz-Gauss beams and its transformation by paraxial optical systems,” Opt. Lett. 31, 2912–2914 (2006). [CrossRef] [PubMed]

] and vector domains [29

29. M. A. Bandres and J. C. Gutiérrez-Vega, “Vector Helmholtz-Gauss and vector Laplace-Gauss beams,” Opt. Lett. 30, 2155–2157 (2005). [CrossRef] [PubMed]

, 30

30. Raul I. Hernandez-Aranda, J. C. Gutiérrez-Vega, Manuel Guizar-Sicairos, and Miguel A. Bandres, “Propagation of generalized vector Helmholtz-Gauss beams through paraxial optical systems,” Opt. Express 14, 8974–8988 (2006). [CrossRef] [PubMed]

]. Some meaningful special cases of the HzG beams are the known cosine-Gauss and cosh-Gauss beams [7

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

], the Bessel-Gauss (BG) beams [31

31. F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987). [CrossRef]

], the Mathieu-Gauss (MG) beams [32

32. J. C. Gutiérrez-Vega and M. A. Bandres, “On the normalization of the Mathieu-Gauss optical beams,” J. Opt. Soc. Am. A , 24, 215–220 (2007). [CrossRef]

], and the parabolic-Gauss beams [25

25. J. C. Gutiérrez-Vega and M. A. Bandres, “Helmholtz-Gauss waves,” J. Opt. Soc. Am. A 22, 289–298 (2005). [CrossRef]

].

In this paper we investigate the propagation of HzG beams in turbulent atmosphere within the regime of weak fluctuations. Since a HzG beam is formed as a suitable superposition of tilted plane-wave-Gaussian beams [25

25. J. C. Gutiérrez-Vega and M. A. Bandres, “Helmholtz-Gauss waves,” J. Opt. Soc. Am. A 22, 289–298 (2005). [CrossRef]

, 26

26. C. López-Mariscal, M. A. Bandrés, and J. C. Gutiérrez-Vega, “Observation of the experimental propagation properties of Helmholtz-Gauss beams,” Opt. Eng. 45, 068001 (2006). [CrossRef]

, 27

27. M. Guizar-Sicairos and J. C. Gutiérrez-Vega, “Propagation of Helmholtz-Gauss beams in absorbing and gain media,” J. Opt. Soc. Am. A , 23, 1994–2001 (2006). [CrossRef]

], we first determine the first and second Born approximations for a single constituent tilted plane-wave-Gaussian beam. Then, we employ the Rytov theory to find the statistics of a pair of tilted plane-wave-Gaussian beams propagating in the atmosphere for a given model of its power spectrum. Finally, we sum up the contributions from all pairs of tilted plane-wave-Gaussian beams to determine the second-order statistical parameters and the transverse intensity pattern of the HzG beams in atmosphere.

By developing a general Rytov theory for a HzG beam in turbulent atmosphere, we easily obtain as limiting cases many simpler results for the ideal nondiffracting beams and spherical-Gaussian wave models. In particular, we apply our theoretical results in the study of the propagation of Bessel-Gauss and Mathieu-Gauss beams and compare its evolutions with respect the propagations of their corresponding pure nondiffracting beams, i.e. the Bessel and Mathieu beams.

2. Second-order Rytov theory

Consider a scalar monochromatic optical/IR wave U(R) with time dependence exp(-iωt) traveling in the z direction (unit vector zẑ) through turbulent atmosphere. Let us express the three-dimensional position vector R and the three-dimensional wave vector K as

R=(r,z),r=(x,y)=(rcosθ,rsinθ),
K=(k,kz),k=(kx,ky)=(kcosφ,ksinφ),
(1)

where r and k denote the positions at the transverse planes of the configuration and the spatial-frequency spaces, respectively.

For a turbulent medium with a random index of refraction n(R), the propagation of the field U(R) is governed by the stochastic Helmholtz equation

[2x2+2y2+2z2+K2n2(R)]U(R)=0,
(2)

where K=|K|=ω/c is the wave number in vacuum. The time variations in the index of refraction have been suppressed, since propagation is much faster than the time needed for it to change significantly. For weak atmospheric fluctuations the mean value of n(R) is approximately unity, then n 2(R) can be approximated as

n2(R)=[1+n1(R)]21+2n1(R),n1(R)1,
(3)

where n 1(R) is a small random quantity with mean value zero.

In the Rytov method [1

1. L. Andrews and R. Phillips, Laser beam propagation through random media (SPIE Press, 1998).

], the solution of Eq. (2) at the plane z=L is assumed to take the form

U(r,L)=U0(r,L)+U1(r,L)+U2(r,L)+,
(4a)
=U0(r,L)exp[Ψ1(r,L)+Ψ2(r,L)+],
(4b)

where U 0(r,L) is the unperturbed field, U d(r,L) and U 2(r,L) are the first and second-order Born corrections, and Ψ1(r,L) and Ψ2(r,L) denote the first and second-order complex phase Rytov perturbations, respectively. It is generally assumed that |U 2(R)|≪|U 1(R)|≪|U 0(R)|.

Given the unperturbed solution U 0(r,L), the perturbations Um=1,2,…(r,L) can be sequentially obtained using the recursive relation [1

1. L. Andrews and R. Phillips, Laser beam propagation through random media (SPIE Press, 1998).

]

Um(r,L)=K22π0Ldzd2sexp[iK(Lz)+iKsr22(Lz)]n1(s,z)LzUm1(s,z),
(5)

where r=(x,y) and s=(sx,sy) are the transverse position vectors in the observation and integration planes, respectively, and d2s=dsxdsy is the differential surface element in the integration plane.

U¯m(r,L)Um(r,L)U0(r,L),m=1,2,3,,
(6)

according to

ψ1=ln[1+U¯1(r,L)]U¯1(r,L),U¯1(r,L)1,
(7a)
Ψ2=U¯2(r,L)12U¯12(r,L).
(7b)

For the purpose of calculating the ensemble average of a large number of propagations, it is convenient to represent the index-of-refraction fluctuations n 1(s,z) in the form of a two-dimensional Riemann-Stieltjes integral [1

1. L. Andrews and R. Phillips, Laser beam propagation through random media (SPIE Press, 1998).

]

n1(s,z)=dν(κ,z)exp(iκ·s),
(8)

where dν(κ,z) in the random differential spectrum of the refractive-index fluctuations, and

κ=(κx,κy),κ=κ,
(9)

is the transverse spatial frequency wave vector associated to n 1. Equation (8) implies that the optical wave is expected to be homogeneous and isotropic in the transverse plane but not necessarily in the direction of propagation.

3. Born approximations for a constituent tilted plane-wave-Gaussian beam

3.1. Construction of Helmholtz-Gauss beams

Undistorted HzG beams propagating in vacuum are formed as a suitable superposition of tilted plane-wave-Gaussian beams of the form [25

25. J. C. Gutiérrez-Vega and M. A. Bandres, “Helmholtz-Gauss waves,” J. Opt. Soc. Am. A 22, 289–298 (2005). [CrossRef]

]

u0(R)=exp(ik02z2Kμ)exp(iKz)μexp(r2μw02)exp(ikφ·rμ),
(10)

where

μ=μ(z)=1+izzR,
(11)

zR=Kw 2 0/2 is the Rayleigh distance, w 0 is the waist size of the Gaussian envelope at z=0, and k 0 is the characteristic transverse wave number of the beam u 0(R).

The tilted plane-wave-Gaussian beam u 0(R) propagates along the direction of the wave vector Kφ=(kφ,K2k02), where the transverse wave vector

kφ=(k0cosφ,k0sinφ),k0=kφ,
(12)

forms an angle φ with the x axis and has a constant transverse amplitude k 0.

The HzG beam is constructed with the conical superposition of fields u 0(R) whose mean propagation axes K φ lie on the surface of a cone with vertex semi-angle arctan(k0/K) about the z axis, and whose amplitudes are modulated angularly by the angular spectrum A(φ), that is

U0(R)=A(φ)u0(R)dφ.
(13)

Basic information on the propagation of the HzG beams in turbulent media can be obtained by analyzing the statistical properties of the single tilted plane-wave-Gaussian beam Eq. (10). We begin by determining its first and second normalized Born approximations.

3.2. First-order Born approximation

Replacement of Eq. (10) into Eqs. (6) and (5) yields

u¯1(r,L)=K22πu0(r,L)0Ldzd2sexp[iKsr22(Lz)]
×[1μexp(s2μw02)exp(iσzμ)exp(ikφ·sμ)]  [1Lzdν(κ,z)exp(iκ·s)],
(14)

where σ≡k20/2k.

By expanding |s-r|2=s 2-2s·r+r 2 and rearranging terms, Eq. (14) can be rewritten as

u¯1(r,L)=K22πu0(r,L)0Ldνdν(κ,z)exp(iσzμ)(Lz)μexp[iKr22(Lz)]
×d2sexp(as2)exp(iq·s),
(15)

where the short notations

aa(z,L)LizRμw02(Lz),
(16)
qq(κ,z,kφ,r,L)κKrLz+kφμ,q=q,
(17)

have been adopted for convenience. The d2 s integral in Eq. (15) can be evaluated in closed-form using polar coordinates [33

33. M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions (Dover, 1964).

, 34

34. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 2000) 6th ed.

], we have

d2sexp(as2)exp(iq·s)=πaexp(q24a).
(18)

Replacing this result into Eq. (15), the first-order normalized Born approximation for the single tilted plane-wave-Gaussian beam reads as

u¯1(r,L)=K22u0(r,L)0Ldzdν(κ,z)[1aμ(Lz)]
×exp(iσzμ)exp[iKr22(Lz)]exp(q24a),
(19)

where dν(κ,z) is the random spectrum of the refractive-index fluctuations [Eq. (8)]. Because 〈n 1(s,z)〉=0 by definition, from Eq. (8) it follows that 〈dν(κ,z)=0〉 and therefore the ensemble average of the first-order Born approximation is 〈ū1(r,L)〉=0.

3.3. Second-order Born approximation

Once obtained the expression for the first-order normalized Born approximation [Eq. (19)], the second-order normalized Born approximation can be determined using the recursive relation Eq. (5), we have

u¯2(r,L)=K4w024πu0(r,L)0Ldz0zdzd2sdν(κ,z)dν(κ,z)
×exp[iKr22(Lz)]exp(iσzμ)(Lz)(zizR)exp[iK(Lz)s22(Lz)(zz)]exp[i(κKrLz)·s]exp(q24a),
(20)

where a =a(z,z), q=q(κ,z′,kφ,s,z), and µ(z).

By following the same procedure as in Sect. 3.2 to evaluate in closed-form the d2s integral, we get the second-order normalized Born approximation for the single tilted plane-wave-Gaussian beam

u¯2(r,L)=K4w024u0(r,L)0Ldz0zdzdν(κ,z)dν(κ,z)
×exp[iKr22(L2)]exp(iσzμ)(Lz)(zizR)bexp{[zR(zz)2K(zizR)](κ2μ+k02μ'+2κ·kφ)}exp(p24b),
(21)

where

bb(z)K(zR+iL)2(Lz)(zizR),
(22)
pp(κ,r,L,z,z,κ,kφ)κkrLz+(zizRzizR)(κ+kφμ),p2=p2.
(23)

have been defined for brevity. Since 〈n 2 1(s,z)〉≠0, it follows that 〈dν(κ,z)dν(κ,z)〉=0 and therefore for the second-order Born correction 〈ū2(r,L)〉=0.

Once determined the first and second-order Born approximations, then the complex phase perturbations Ψ1(r,L) and Ψ2(r,L) can be readily determined using Eqs. (7), and the propagated field with Eqs. (4).

4. Second-order statistical moments and averaged intensity

The averaged intensity of the fieldU at the point (r,L) is given by 〈I(r,L)〉=〈U*(r,L)U(r,L)〉, where U* denotes the complex conjugate of U. Using the spectral representation of the HzG beam [Eq. (13)], we write

I(r,L)=U*(r,L)U(r,L)=ππdφππdχA*(φ)A(χ)uφ*(r,L)uχ(r,L),
(24)

where the subscripts (φ,χ) refer to two tilted plane-wave-Gaussian beams [Eq. (10)] propagating in the positive z direction, and whose transverse wave vectors k φ and k χ make angles φ and χ with the x axis, respectively.

Keeping terms to only second order in the Rytov approximation [Eq. (4b)], we may write the cross-spectral density function 〈u*φ(r,L)u χ(r,L)〉 within the integral in Eq. (24) as

uφ*(r,L)uχ(r,L)=u0,φ*(r,L)u0,χ(r,L)
×exp[Ψ1,φ*(r,L)+Ψ2,φ*(r,L)+Ψ1,χ(r,L)+Ψ2,χ(r,L)].
(25)

The averaged exponential function is now simplified using the method of cumulants, namely, if t is a random variable, then the average of its exponential function can be approximated to second-order as [1

1. L. Andrews and R. Phillips, Laser beam propagation through random media (SPIE Press, 1998).

, p. 126]

exp(t)exp[t+12(t2t2)],
(26)

which is an exact relation only if t is a Gaussian random variable.

By applying Eq. (26) to Eq. (25) and noting that for the first-order perturbations 〈Ψ*1,φ〉=〈Ψ1,χ〉=0, we obtain

uφ*(r,L)uχ(r,L)=u0,φ*(r,L)u0,χ(r,L)exp[Eφ(1)*(r,L)+Eχ(1)(r,L)+Eφ,χ(2)(r,L)],
(27)

where

Ej(1)(r,L)Ψ2,j*(r,L)+Ψ1,j*2(r,L)=u¯2,j*(r,L),j={φ,χ},
(28)

is the first second-order statistical moment of the Rytov approximation for the component j={φ,χ}, and

Eφ,χ(2)(r,L)=Ψ1,φ*(r,L)Ψ1,χ(r,L)u¯1,φ*(r,L)u¯1,χ(r,L),
(29)

is the second second-order statistical moment for the combination of both waves.

We show in Appendix A that in the case when the atmosphere is isotropic (azimuthally symmetric), the second-order statistical moments are given by

Ej(1)(r,L)=2π2K20Ldη0κΦn(κ)dκ,j={φ,χ},
(30)
Eφ,χ(2)(r,L)=4π2K20Ldη0dκΦn(κ)κexp[zR(Lη)2κ2K(L2+zR2)]J0(iQκ),
(31)

where J 0 is the zeroth-order Bessel function, Φn(k) is the power spectrum of atmospheric fluctuations which models the atmospheric turbulence, and

Q2zR(Lη)rL2+zR212(Lη)w02kφ,χ,Q=Q,
(32)
kφ,χkφL+izR+kχLizR.
(33)

Equations (30) and (31) give the first and second second-order statistical moments of the Rytov approximation for two tilted plane-wave-Gaussian beams of the form (10) propagating in turbulent atmosphere for a given random spectrum Φn(κ) of the refractive-index fluctuations.

It is noteworthy that the first second-order moment Eq. (30) is exactly the same as that obtained by Andrews and Phillips for the fundamental Gaussian beam [1

1. L. Andrews and R. Phillips, Laser beam propagation through random media (SPIE Press, 1998).

, Eq. 5.54], and that obtained by Gbur and O. Korotkova for two tilted ideal plane waves [19

19. G. Gbur and O. Korotkova, “Angular spectrum representation for propagation of arbitrary coherent and partially coherent beams through atmospheric turbulence,” J. Opt. Soc. Am. A 24, 745–752 (2007). [CrossRef]

, Eq. 3.7]. Equations (28) and (30) reveal that, up to second-order approximation, the average value of the normalized second-order Born approximation of the HzG beams is independent of the observation point r within the beam.

On the other hand, it can be shown that the second second-order moment Eq. (31) reduces to the corresponding expressions derived (a) for a single, normally incident Gaussian beam [1

1. L. Andrews and R. Phillips, Laser beam propagation through random media (SPIE Press, 1998).

, Sect. 5.5.3] in the limit φ→0, χ→0, and k 0→0, and (b) for two tilted ideal plane waves [19

19. G. Gbur and O. Korotkova, “Angular spectrum representation for propagation of arbitrary coherent and partially coherent beams through atmospheric turbulence,” J. Opt. Soc. Am. A 24, 745–752 (2007). [CrossRef]

, Eq. 3.8] in the limit of infinite Gaussian waist size, i.e. w 0→∞.

Once the second-order statistical moments have been calculated for a given medium, the intensity at the plane z=L of a HzG beam of any type may be evaluated by replacing the cross-spectral density function Eq. (27) into the superposition integral Eq. (24), namely

I(r,L)=ππdφππdχA*(φ)A(χ)u0,φ*(r,L)u0,χ(r,L)
×exp{2Eφ(1)(r,L)+Eφ,χ(2)(r,L)}.
(34)

Equation (34) is the main result of this paper. It permits an arbitrary HzG beam to be propagated through turbulent atmosphere whose small refractive-index fluctuations are described by the random spectrum Φn(κ).

Fig. 1. Plot of the ensemble average of the on-axis intensity for a pure Gaussian beam, a zeroth-order pure nondiffracting Bessel beam, and a zeroth-order Bessel-Gauss beam, each propagated both in free space and a turbulent atmosphere

5. Numerical results for circularly and non-circularly symmetric beams

In this section, we present the results of propagating several known special classes of HzG beams employing the theory developed in the last two sections. For numerical purposes we assume in all cases the usual Kolmogorov spectrum given by [1

1. L. Andrews and R. Phillips, Laser beam propagation through random media (SPIE Press, 1998).

, Sect. 3.3.1],

Φn(κ)=T(γ)Cn2κγ2,
(35)

where

T(γ)=Γ(γ+1)4π2sin[π2(γ1)],γ=53,
(36)

and C 2 n=10-14 m-2/3 is the refractive-index structure parameter which is assumed constant for horizontal propagation.

5.1. Circular symmetric beams: Bessel-Gauss beams

The Bessel-Gauss beams of mth-order at the source plane are written as [25

25. J. C. Gutiérrez-Vega and M. A. Bandres, “Helmholtz-Gauss waves,” J. Opt. Soc. Am. A 22, 289–298 (2005). [CrossRef]

, 31

31. F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987). [CrossRef]

]

BGm(r,θ)=Jm(k0r)exp(imθ)exp(r2w02),
(37)

where Jm is the mth-order Bessel function and w 0 defines the waist size of the Gaussian envelope. The Bessel-Gauss beams tend to pure nondiffracting Bessel beams in the limit when w 0→0, and, on the other side, tend to the fundamental Gaussian beam when kt→0. The Bessel-Gauss beams are constructed with a superposition of tilted plane-wave-Gaussian beams of the form (10) with an angular spectrum in Eq. (13) given by A(φ)=exp(imφ).

To determine the averaged intensity for the Bessel-Gauss beams propagating in turbulent media, we first compute numerically the cross-spectral density function 〈u*φ(r,L)uχ(r,L)〉 in Eq. (27) and later evaluate Eq. (24) using the angular spectrum A(φ)=exp(imφ).

Figure 1 shows the on-axis intensity for a zeroth-order Bessel-Gauss beam, and their corresponding fundamental Gaussian beam and zeroth-order pure nondiffracting Bessel beam. For comparison purposes, each beam is propagated both in free space and in turbulent atmosphere, from the source plane z=0 to the plane z=z max=w0k/k 0. The physical meaning of the maximum propagation distance zmax is important to understand its practical relevance and the longitudinal ranges considered in the analysis. In principle, ideal unapertured nondiffracting beams propagate without distortion through an infinite distance, however, from the theory of the nondiffracting beams [14

14. Z. Bouchal, “Nondiffracting optical beams: physical properties, experiments, and applications,” Czech. J. Phys. 53, 537–578 (2003). [CrossRef]

] it is known that practical apertured nondiffracting beams propagate keeping its nondiffracting feature up to a distance given by ak/k 0, where a is the radius of the circular aperture, and k and k 0 are the wave number and the transverse wave number of the nondiffracting beam. This distance delimits the typical conical region where significant interference of the constituent plane waves of the nondiffracting beam occurs. Beyond this distance, the transverse pattern of the beam acquires an annular shape whose diameter increases with propagation distance and, thus, strictly speaking, it cannot be more considered as a nondiffracting beam. In our case, the Helmholtz-Gauss beams are Gaussian apertured nondiffracting beams, where the Gaussian width is taken as the mean radius of an equivalent aperture [25

25. J. C. Gutiérrez-Vega and M. A. Bandres, “Helmholtz-Gauss waves,” J. Opt. Soc. Am. A 22, 289–298 (2005). [CrossRef]

]. In this way, zmax=w0k/k 0 delimits the region where significant interference of the constituent tilted plane-wave-Gaussian beams occurs. For optical frequencies, z max can reach values about 20 or 30 m.

Fig. 2. Plot of the ensemble average of the intensity versus the radial distance from the origin, for the zeroth-order nondiffracting Bessel beam and zeroth-order Bessel- Gauss beam. The intensity profiles correspond to several propagation distances L=0,zmax/4,zmax/2,3zmax/4

The averaged transverse intensity distribution for Bessel-Gauss beams and pure nondiffracting Bessel beams traveling through a turbulent medium are depicted in Fig. 2 for the planes L={0,1/4,1/2,3/4}zmax in the range r/w 0∈[0,2

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]

]. The comparison of the radial dependencies at L=(3/4)zmax of the Bessel-Gauss beams with respect to other circularly symmetrical beam is included in Fig. 3 for both turbulent and free-space propagation. The propagations were conducted using the tilted Gaussian wave spectra for HzG beams, a plane wave spectrum for nondiffracting beams, and the corresponding spectrum for the Gaussian beam [1

1. L. Andrews and R. Phillips, Laser beam propagation through random media (SPIE Press, 1998).

].

It is known that a m-th order Bessel-Gauss beam of the form (37) carries an orbital angular momentum per photon given by mh̄. To investigate the possible additional resistance to turbulence exhibited by the vortex beams, in Fig. 4 we show the ensemble average of the intensity for the third-order Bessel-Gauss beam. Comparing with the propagations for the lowest J 0 Bessel–Gauss beams reported in Figs. 13, it is clear from Fig. 4 that the presence of the orbital angular momentum provides additional resistance to turbulence. These conclusions are in agreement with those reported by H. T. Eyyuboğlu in Ref. [11

11. H. T. Eyyuboğlu, “Propagation of higher order Bessel-Gaussian beams in turbulence,” Appl. Phys. B , 88, 259–265 (2007). [CrossRef]

] who determined the average intensity of the Bessel-Gauss beams using the extended HuygensFresnel principle and evaluating it numerically to a level of double integral [1

1. L. Andrews and R. Phillips, Laser beam propagation through random media (SPIE Press, 1998).

]. In particular, he found that all higher order Bessel-Gauss beams will have a well-like shape and their profiles will further be governed by order and relative magnitudes of width parameter and Gaussian source sizes.

Fig. 3. Plot of the ensemble average of the intensity versus the radial distance from the origin, for the zeroth-order nondiffracting Bessel beam, zeroth-order Bessel-Gauss beam, and a pure Gaussian beam. The propagation distance for all of them was L=3zmax/4, under the same turbulence conditions.
Fig. 4. Plot of the ensemble average of the intensity versus the radial distance from the origin, for the third-order Bessel-Gauss beam, for several propagation distances L=0,zmax/4,zmax/2,3zmax/4.

5.2. Non-circular symmetric beams: Helical Mathieu-Gauss beams

Mathieu beams constitute a complete and orthogonal family of nondiffracting optical beams that are solutions of the wave equation in elliptic coordinates [15

15. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo,, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25, 1493–1495 (2000). [CrossRef]

]. These beams are fundamental in the sense that any optical field can be expanded in terms of Mathieu beams with appropriate weight factors and spatial frequencies [35

35. A. Chafiq, Z. Hricha, and A. Belafhal, “Paraxial propagation of Mathieu beams through an apertured ABCD optical system,” Opt. Commun. 253, 223–230 (2005). [CrossRef]

]. Mathieu beams have been applied for example to photonic lattices [36

36. Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, “Shaping soliton properties in Mathieu lattices,” Opt. Lett. 31, 238–240 (2006). [CrossRef] [PubMed]

], transfer of angular momentum using optical tweezers [37

37. S. Chávez-Cerda, M.J. Padgett, I. Allison, G.H.C. New, Julio C. Gutiérrez-Vega, A.T. O’Neil, I. MacVicar, and J. Courtial, “Holographic generation and orbital angular momentum of high-order Mathieu beams,” J. Opt. B: Quantum Semiclass. Opt. 4, S52–S57, (2002). [CrossRef]

, 38

38. C. López-Mariscal, J. C. Gutiérrez-Vega, G. Milne, and K. Dholakia, “Orbital angular momentum transfer in helical Mathieu beams,” Opt. Express , 14, 4182–4187 (2006). [CrossRef] [PubMed]

], and localized X-waves [39

39. C. A. Dartora and H. E. Hernández-Figueroa, “Properties of a localized Mathieu pulse,” J. Opt. Soc. Am. A 21, 662–667 (2004). [CrossRef]

]. Mathieu-Gauss beams are a subclass of the Helmholtz-Gauss beams, namely they are Mathieu beams apodized by a Gaussian transmittance [25

25. J. C. Gutiérrez-Vega and M. A. Bandres, “Helmholtz-Gauss waves,” J. Opt. Soc. Am. A 22, 289–298 (2005). [CrossRef]

, 32

32. J. C. Gutiérrez-Vega and M. A. Bandres, “On the normalization of the Mathieu-Gauss optical beams,” J. Opt. Soc. Am. A , 24, 215–220 (2007). [CrossRef]

] that carry a finite power and can be generated experimentally to a very good approximation [26

26. C. López-Mariscal, M. A. Bandrés, and J. C. Gutiérrez-Vega, “Observation of the experimental propagation properties of Helmholtz-Gauss beams,” Opt. Eng. 45, 068001 (2006). [CrossRef]

]

Fig. 5. Plot of the ensemble average of the intensity versus the radial distance from the origin, for the second-order Mathieu-Gauss beam along the positive x and y axes, for several propagation distances L=0,zmax/4,zmax/2,3zmax/4.

The transverse field at the source plane z=0 of the mth-order helical Mathieu-Gauss beams is given by [25

25. J. C. Gutiérrez-Vega and M. A. Bandres, “Helmholtz-Gauss waves,” J. Opt. Soc. Am. A 22, 289–298 (2005). [CrossRef]

]

MGmm±(r)=CmJem(ξ,ε)cem(η,ε)±iSmJom(ξ,ε)sem(η,ε),
(38)

x=hcoshξcosη,y=hsinhξsinη,
(39)

where h=2√ε/k 0 is the semifocal distance of the elliptic coordinate system. The transverse distribution of the helical Mathieu beams [Eq. (38)] is characterized by a set of confocal elliptic rings whose eccentricity is determined by the parameter ε. The special case when ε=0 corresponds to the known mth-order Bessel beams Jm (k0r) exp(±imθ). Unlike Bessel-Gauss waves, the Mathieu-Gauss beams are not rotationally symmetric with respect to the propagation axis.

The propagation of Mathieu-Gauss beams in turbulent atmosphere is determined by computing numerically the cross-spectral density function 〈u*φ(r,L)(r,L)〉 in Eq. (27) and later evaluating Eq. (24) using the Mathieu angular spectrum A±m(φ)=cem(φ,εisem(φ,ε).

The intensity distributions along the x and y axes at transverse planes L={0,1/4,1/2,3/4}zmax are plotted in Fig. 5 for a second-order Mathieu-Gauss beam and a transverse range x/w 0∈[0,2

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]

]. Note that the vertical scales are different in the subplots. For comparison purposes we also included the analytical plots of the Mathieu-Gauss beams in free space. The plots in Fig. 5 reveal that the turbulence affects practically in the same way both orthogonal axes of the elliptical rings of the Mathieu beam, thus, at least for low values of the ellipticity parameter, the Mathieu beam does not exhibit an evident anisotropic perturbation as propagates in a random medium with small perturbations. Similar to the Bessel-Gauss beams, the helical Mathieu-Gauss beams of higher-order exhibit more resistance to small perturbation than its purely even or odd counterparts [25

25. J. C. Gutiérrez-Vega and M. A. Bandres, “Helmholtz-Gauss waves,” J. Opt. Soc. Am. A 22, 289–298 (2005). [CrossRef]

].

6. Conclusions

The Rytov theory is a useful tool to study the propagation of HzG beams through turbulent atmosphere within the regime of weak fluctuations of the refractive-index. In the course of obtaining the second order statistical moments and averaged intensity of the HzG beams, we also determined their first and second order normalized Born approximations. These analytical expression can be easily and accurately evaluated numerically for the given spectrum density of the refractive-index variations. The results for the HzG beams reported in this paper generalize those previously presented for a Gaussian beam [1

1. L. Andrews and R. Phillips, Laser beam propagation through random media (SPIE Press, 1998).

], and also for the nondiffracting beams [19

19. G. Gbur and O. Korotkova, “Angular spectrum representation for propagation of arbitrary coherent and partially coherent beams through atmospheric turbulence,” J. Opt. Soc. Am. A 24, 745–752 (2007). [CrossRef]

].

A. Appendix: Derivation of the second-order statistical moments

In this appendix we derive the expressions for E (1) j (r,L) [Eq. (30)] and E (2) φ,χ (r,L) [Eq. (31)].

A.1. Derivation of E(2)φ,χ (r,L) [Eq. (31)]

By replacing the first-order Born approximation Eq. (19) into (29) we obtain

Eφ,χ(2)(r,L)=K2zR2(L2+zR2)u0,φ*(r,L)u0,χ(r,L)0Ldz0Ldzdν*(κ,z)dν(κ,z)
×exp[iσ(zμ*zμ)]exp[iKr22(1Lz1Lz)]exp{14(qφ2a(z,L))*+qχ2a(z,L)]}.
(A-1)

Following the classical treatment [1

1. L. Andrews and R. Phillips, Laser beam propagation through random media (SPIE Press, 1998).

, Sect. 5.5.3], the ensemble average 〈dν*(κ,z)dν(κ, z)〉 is written in the form

dν*(κ,z)dν(κ,z)=Fn(κ,zz)δ(κκ)d2κd2κ,
(A-2)

where δ is the Dirac delta function and Fn(κ, |z-z |)≡Fn(κxy, |z-z|) is the two-dimensional spectral density of the index of refraction. This spectral density has appreciable value when the difference |ξ|=|z-z | is close to zero and relates to the refractive-index spatial power spectrum in the following way

Φn(κ)=12πdξFn(κx,κy,ξ),
(A-3)

where Φn(κ)≡Φn(κxy) is the power spectrum of atmospheric fluctuations.

The change of variables η=12(z+z) and ξ=z-z allows us to rewrite the dz and dz integrals as

0Ldz0Ldz=0Ldη2ηL2ηLdξ.
(A-4)

Since Fn(κ, |ξ|) has appreciable values for very small values of |ξ|, we can extend the limits of the dξ integral from -∞ to ∞ without significant error. In addition, we may write zz η, which finally yields

Eφ,χ(2)(r,L)=2πK2zR2(L2+zR2)u0,φ*(r,L)u0,χ(r,L)exp[zR(2σL2+Kr2)L2+zR2]exp(zRr·kφχ)
Ldηd2κΦn(κ)exp[w02(Lη)2κ22(L2+zR2)]exp(Q·κ),
(A-5)

where

kφχkφL+izR+kχLizR,
(A-6)
Q2zR(Lη)rL2+zR212(Lη)w02kφχ.
(A-7)

For an isotropic azimuthally symmetric medium, the d2 κ integral in Eq. (A-5) can be reduced to a single radial integral using polar coordinates. After some algebraic manipulation, we obtain

Eφ,χ(2)(r,L)=4π2K20Ldη0dκΦn(κ)κexp[zR(Lη)2κ2K(L2+zR2)]J0(iQκ),
(A-8)

where J 0 is the zeroth-order Bessel function.

A.2. Derivation of E(1)φ (r,L) [Eq. (30)].

Replacing Eq.(21) into Eq. (28) yields

Eφ(1)(r,L)=K4w024u0,φ(r,L)0Ldz0zdzdν(κ,z)dν(κ,z)1b(Lz)(zizR)
×exp[iKr22(Lz)]exp[iσzμ]exp[(zR(zz)2K(zizR))(κ2μ+k02μ+2κ·kφ)]exp(pφ24b)
(A-9)

Since n 1(s,z) is a real function, we know that

dν(κ,z)dν(κ,z)=dν(κ,z)dν*(κ,z)=Fn(κ,zz)δ(κ+κ)d2κd2κ.
(A-10)

Making the change of variables η=12(z+z) and ξ=z-z , the approximation zz ≅η, and recalling that Fn(κ, |ξ|) has appreciable values for very small values of |ξ| and depend on |ξ|, we can note that

0Ldz0zdzFn(κ,zz)=120LdηdξFn(κ,ξ).
(A-11)

Using Eqs.(A-2), (A-3), (A-10), and (A-11), the expression for E (1) φ (r,L) [Eq.(A-9)] reduces to

Eφ(1)(r,L)=πK20LdηΦn(κ)d2κ.
(A-12)

When the medium is isotropic (azimuthally symmetric) this double integral reduces to the single radial shown in Eq. (30).

Acknowledgments

The authors acknowledge the financial support from Consejo Nacional de Ciencia y Tecnología of México (grant 42808), and from Tecnológico de Monterrey (grant CAT007).

References and links

1.

L. Andrews and R. Phillips, Laser beam propagation through random media (SPIE Press, 1998).

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]

3.

H. T. Eyyuboglu, S. Altay, and Y. Baykal, “Propagation characteristics of higher-order annular Gaussian beams in atmospheric turbulence,” Opt. Commun. 264, 25–34 (2006). [CrossRef]

4.

Y. Cai and S. He, “Average intensity and spreading of an elliptical Gaussian beam propagating in a turbulent atmosphere,” Opt. Lett. 31, 568–570 (2002). [CrossRef]

5.

Y. Cai and D. Ge, “Analytical formula for a decentered elliptical Gaussian beam propagationg in a turbulent atmosphere,” Opt. Commun. 271, 509–516 (2007). [CrossRef]

6.

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

7.

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

8.

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

9.

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

10.

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]

11.

H. T. Eyyuboğlu, “Propagation of higher order Bessel-Gaussian beams in turbulence,” Appl. Phys. B , 88, 259–265 (2007). [CrossRef]

12.

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

13.

Z. Bouchal, “Resistance of nondiffracting vortex beam against amplitude and phase perturbations,” Opt. Commun. 210, 155–164 (2002). [CrossRef]

14.

Z. Bouchal, “Nondiffracting optical beams: physical properties, experiments, and applications,” Czech. J. Phys. 53, 537–578 (2003). [CrossRef]

15.

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo,, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25, 1493–1495 (2000). [CrossRef]

16.

M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. 29, 44–46 (2004). [CrossRef] [PubMed]

17.

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstructionof a distorted nondiffracting beam,” Opt. Commun. 151, 207–211 (1998). [CrossRef]

18.

T. Aruga, S. W. Li, S. Yoshikado, M. Takabe, and R. Li, “Nondiffracting narrow light beam with small atmospheric turbulence-induced propagation,” Appl. Opt. 38, 3152–3156 (1999). [CrossRef]

19.

G. Gbur and O. Korotkova, “Angular spectrum representation for propagation of arbitrary coherent and partially coherent beams through atmospheric turbulence,” J. Opt. Soc. Am. A 24, 745–752 (2007). [CrossRef]

20.

O. Korotkova and G. Gbur, “Propagation of beams with any spectral, coherence and polarization properties in turbulent atmosphere,” Proc. of SPIE 6457, 64570J1–64570J12 (2007).

21.

O. Korotkova and G. Gbur, “Angular spectrum representation for propagation of random electromagnetic beams in a turbulent atmosphere,” J. Opt. Soc. Am. A 24, 2728–2736 (2007). [CrossRef]

22.

R. Frehlich, “Simulation of laser propagation in a turbulent atmosphere,” Appl. Opt. 30, 393–397 (2000). [CrossRef]

23.

C. Arpali, C. Yazicioglu, H. Eyyuboğlu, S. Arpali, and Y. Baykal, “Simulator for general-type beam propagation in turbulent atmosphere,” Opt. Express 14, 8918–8928 (2006). [CrossRef] [PubMed]

24.

D.C. Cowan, J. Recolons, L.C. Andrews, and C.Y. Young, Atmospheric Propagation III, Proc. SPIE 6215, 62150 B-1- 62150B-10 (2006).

25.

J. C. Gutiérrez-Vega and M. A. Bandres, “Helmholtz-Gauss waves,” J. Opt. Soc. Am. A 22, 289–298 (2005). [CrossRef]

26.

C. López-Mariscal, M. A. Bandrés, and J. C. Gutiérrez-Vega, “Observation of the experimental propagation properties of Helmholtz-Gauss beams,” Opt. Eng. 45, 068001 (2006). [CrossRef]

27.

M. Guizar-Sicairos and J. C. Gutiérrez-Vega, “Propagation of Helmholtz-Gauss beams in absorbing and gain media,” J. Opt. Soc. Am. A , 23, 1994–2001 (2006). [CrossRef]

28.

M. Guizar-Sicairos and J. C. Gutiérrez-Vega, “Generalized Helmholtz-Gauss beams and its transformation by paraxial optical systems,” Opt. Lett. 31, 2912–2914 (2006). [CrossRef] [PubMed]

29.

M. A. Bandres and J. C. Gutiérrez-Vega, “Vector Helmholtz-Gauss and vector Laplace-Gauss beams,” Opt. Lett. 30, 2155–2157 (2005). [CrossRef] [PubMed]

30.

Raul I. Hernandez-Aranda, J. C. Gutiérrez-Vega, Manuel Guizar-Sicairos, and Miguel A. Bandres, “Propagation of generalized vector Helmholtz-Gauss beams through paraxial optical systems,” Opt. Express 14, 8974–8988 (2006). [CrossRef] [PubMed]

31.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987). [CrossRef]

32.

J. C. Gutiérrez-Vega and M. A. Bandres, “On the normalization of the Mathieu-Gauss optical beams,” J. Opt. Soc. Am. A , 24, 215–220 (2007). [CrossRef]

33.

M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions (Dover, 1964).

34.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 2000) 6th ed.

35.

A. Chafiq, Z. Hricha, and A. Belafhal, “Paraxial propagation of Mathieu beams through an apertured ABCD optical system,” Opt. Commun. 253, 223–230 (2005). [CrossRef]

36.

Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, “Shaping soliton properties in Mathieu lattices,” Opt. Lett. 31, 238–240 (2006). [CrossRef] [PubMed]

37.

S. Chávez-Cerda, M.J. Padgett, I. Allison, G.H.C. New, Julio C. Gutiérrez-Vega, A.T. O’Neil, I. MacVicar, and J. Courtial, “Holographic generation and orbital angular momentum of high-order Mathieu beams,” J. Opt. B: Quantum Semiclass. Opt. 4, S52–S57, (2002). [CrossRef]

38.

C. López-Mariscal, J. C. Gutiérrez-Vega, G. Milne, and K. Dholakia, “Orbital angular momentum transfer in helical Mathieu beams,” Opt. Express , 14, 4182–4187 (2006). [CrossRef] [PubMed]

39.

C. A. Dartora and H. E. Hernández-Figueroa, “Properties of a localized Mathieu pulse,” J. Opt. Soc. Am. A 21, 662–667 (2004). [CrossRef]

OCIS Codes
(010.1300) Atmospheric and oceanic optics : Atmospheric propagation
(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence
(010.3310) Atmospheric and oceanic optics : Laser beam transmission

ToC Category:
Atmospheric and oceanic optics

History
Original Manuscript: September 27, 2007
Revised Manuscript: November 8, 2007
Manuscript Accepted: November 20, 2007
Published: November 26, 2007

Citation
Rodrigo J. Noriega-Manez and Julio C. Gutiérrez-Vega, "Rytov theory for Helmholtz-Gauss beams in turbulent atmosphere," Opt. Express 15, 16328-16341 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-25-16328


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. L. Andrews and R. Phillips, Laser beam propagation through random media (SPIE Press, 1998).
  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]
  3. H. T. Eyyuboglu, S. Altay, and Y. Baykal, "Propagation characteristics of higher-order annular Gaussian beams in atmospheric turbulence,"Opt. Commun. 264, 25-34 (2006). [CrossRef]
  4. Y. Cai and S. He, "Average intensity and spreading of an elliptical Gaussian beam propagating in a turbulent atmosphere,"Opt. Lett. 31, 568-570 (2002). [CrossRef]
  5. Y. Cai and D. Ge, "Analytical formula for a decentered elliptical Gaussian beam propagationg in a turbulent atmosphere,"Opt. Commun. 271, 509-516 (2007). [CrossRef]
  6. H. T. Eyyuboglu and Y. Baykal, "Reciprocity of cos-Gaussian and cosh-Gaussian laser beams in turbulent atmosphere," Opt. Express 12, 4659-4674 (2004). [CrossRef] [PubMed]
  7. H. T. Eyyuboglu and Y. Baykal, "Average intensity and spreading of cosh-Gaussian beams in the turbulent atmosphere," Appl. Opt. 44, 976-983 (2005). [CrossRef] [PubMed]
  8. H. T. Eyyuboglu, "Propagation of Hermite-cosh-Gaussian laser beams in turbulent atmosphere,"Opt. Commun. 245, 37-47 (2005). [CrossRef]
  9. H. T. Eyyuboglu, "Hermite-cosine-Gaussian laser beam and its propagation characteristics in turbulent atmosphere," J. Opt. Soc. Am. A 22, 1527-1535 (2005). [CrossRef]
  10. 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]
  11. H. T. Eyyuboglu, "Propagation of higher order Bessel-Gaussian beams in turbulence," Appl. Phys. B,  88, 259-265 (2007). [CrossRef]
  12. Y. Cai and S. He, "Propagation of various dark hollow beams in a turbulent atmosphere," Opt. Express 14, 1353-1367 (2006). [CrossRef] [PubMed]
  13. Z. Bouchal, "Resistance of nondiffracting vortex beam against amplitude and phase perturbations," Opt. Commun. 210, 155-164 (2002). [CrossRef]
  14. Z. Bouchal, "Nondiffracting optical beams: physical properties, experiments, and applications," Czech. J. Phys. 53, 537-578 (2003). [CrossRef]
  15. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, "Alternative formulation for invariant optical fields: Mathieu beams," Opt. Lett. 25, 1493-1495 (2000). [CrossRef]
  16. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, "Parabolic nondiffracting optical wave fields," Opt. Lett. 29, 44-46 (2004). [CrossRef] [PubMed]
  17. Z. Bouchal, J. Wagner, and M. Chlup, "Self-reconstructionof a distorted nondiffracting beam," Opt. Commun. 151, 207-211 (1998). [CrossRef]
  18. T. Aruga, S. W. Li, S. Yoshikado, M. Takabe, and R. Li, "Nondiffracting narrow light beam with small atmospheric turbulence-induced propagation," Appl. Opt. 38, 3152-3156 (1999). [CrossRef]
  19. G. Gbur and O. Korotkova, "Angular spectrum representation for propagation of arbitrary coherent and partially coherent beams through atmospheric turbulence," J. Opt. Soc. Am. A 24, 745-752 (2007). [CrossRef]
  20. O. Korotkova and G. Gbur, "Propagation of beams with any spectral, coherence and polarization properties in turbulent atmosphere," Proc. of SPIE 6457, 64570J1-64570J12 (2007).
  21. O. Korotkova and G. Gbur, "Angular spectrum representation for propagation of random electromagnetic beams in a turbulent atmosphere," J. Opt. Soc. Am. A 24, 2728-2736 (2007). [CrossRef]
  22. R. Frehlich, "Simulation of laser propagation in a turbulent atmosphere,"Appl. Opt. 30, 393-397 (2000). [CrossRef]
  23. C. Arpali, C. Yazicioglu, H. Eyyuboglu, S. Arpali, and Y. Baykal, "Simulator for general-type beam propagation in turbulent atmosphere," Opt. Express 14, 8918-8928 (2006). [CrossRef] [PubMed]
  24. D.C. Cowan, J. Recolons, L.C. Andrews, C.Y. Young, Atmospheric Propagation III, Proc. SPIE 6215, 62150 B-1- 62150B-10 (2006).
  25. J. C. Gutiérrez-Vega and M. A. Bandres, "Helmholtz-Gauss waves," J. Opt. Soc. Am. A 22, 289-298 (2005). [CrossRef]
  26. C. López-Mariscal, M. A. Bandrés, and J. C. Gutiérrez-Vega, "Observation of the experimental propagation properties of Helmholtz-Gauss beams," Opt. Eng. 45, 068001 (2006). [CrossRef]
  27. M. Guizar-Sicairos and J. C. Gutiérrez-Vega, "Propagation of Helmholtz-Gauss beams in absorbing and gain media," J. Opt. Soc. Am. A,  23, 1994-2001 (2006). [CrossRef]
  28. M. Guizar-Sicairos and J. C. Gutiérrez-Vega, "Generalized Helmholtz-Gauss beams and its transformation by paraxial optical systems," Opt. Lett. 31, 2912-2914 (2006). [CrossRef] [PubMed]
  29. M. A. Bandres and J. C. Gutiérrez-Vega, "Vector Helmholtz-Gauss and vector Laplace-Gauss beams," Opt. Lett. 30, 2155-2157 (2005). [CrossRef] [PubMed]
  30. Raul I. Hernandez-Aranda, J. C . Gutiérrez-Vega, Manuel Guizar-Sicairos, and Miguel A. Bandres, "Propagation of generalized vector Helmholtz-Gauss beams through paraxial optical systems," Opt. Express 14, 8974-8988 (2006). [CrossRef] [PubMed]
  31. F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987). [CrossRef]
  32. J. C. Gutiérrez-Vega and M. A. Bandres, "On the normalization of the Mathieu-Gauss optical beams," J. Opt. Soc. Am. A,  24, 215-220 (2007). [CrossRef]
  33. M. Abramowitz, and I.A. Stegun, Handbook of Mathematical Functions (Dover, 1964).
  34. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 2000) 6th ed.
  35. A. Chafiq, Z. Hricha, A. Belafhal, "Paraxial propagation of Mathieu beams through an apertured ABCD optical system," Opt. Commun. 253,223-230 (2005). [CrossRef]
  36. Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, "Shaping soliton properties in Mathieu lattices," Opt. Lett. 31, 238-240 (2006). [CrossRef] [PubMed]
  37. S. Chávez-Cerda, M.J. Padgett, I. Allison, G.H.C. New, JulioC. Gutiérrez-Vega, A.T. O’Neil, I. MacVicar, and J. Courtial, "Holographic generation and orbital angular momentum of high-order Mathieu beams," J. Opt. B: Quantum Semiclass. Opt. 4, S52-S57, (2002). [CrossRef]
  38. C. López-Mariscal, J. C. Gutiérrez-Vega, G. Milne and K. Dholakia, "Orbital angular momentum transfer in helical Mathieu beams," Opt. Express,  14, 4182-4187 (2006). [CrossRef] [PubMed]
  39. C. A. Dartora and H. E. Hernández-Figueroa, "Properties of a localized Mathieu pulse," J. Opt. Soc. Am. A 21, 662-667 (2004). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited