## Rytov theory for Helmholtz-Gauss beams in turbulent atmosphere

Optics Express, Vol. 15, Issue 25, pp. 16328-16341 (2007)

http://dx.doi.org/10.1364/OE.15.016328

Acrobat PDF (608 KB)

### 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

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]

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]

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]

*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]

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]

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]

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]

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

**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]

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]

## 2. Second-order Rytov theory

*U*(

**R**) with time dependence exp(-

*iωt*) traveling in the

*z*direction (unit vector

**z**ẑ) through turbulent atmosphere. Let us express the three-dimensional position vector

**R**and the three-dimensional wave vector

**K**as

**r**and

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

*n*(

**R**), the propagation of the field

*U*(

**R**) is governed by the stochastic Helmholtz equation

*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

*n*

^{1}(

**R**) is a small random quantity with mean value zero.

*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 Ψ

_{1}(

**r**,

*L*) and Ψ

_{2}(

**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**)|.

*U*

_{0}(

**r**,

*L*), the perturbations

*U*=1,2,…(

_{m}**r**,

*L*) can be sequentially obtained using the recursive relation [1]

**r**=(

*x,y*) and s=(

*s*) are the transverse position vectors in the observation and integration planes, respectively, and d2s=dsxdsy is the differential surface element in the integration plane.

_{x},s_{y}*n*

_{1}(

**s**,

*z*) in the form of a two-dimensional Riemann-Stieltjes integral [1]

*ν*(

**κ**,

*z*) in the random differential spectrum of the refractive-index fluctuations, and

*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

**22**, 289–298 (2005). [CrossRef]

*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**).

*u*

_{0}(

**R**) propagates along the direction of the wave vector

*φ*with the

*x*axis and has a constant transverse amplitude

*k*

_{0}.

*u*

_{0}(

**R**) whose mean propagation axes

**K**

_{φ}lie on the surface of a cone with vertex semi-angle arctan(

*k*) about the

_{0}/K*z*axis, and whose amplitudes are modulated angularly by the angular spectrum

*A*(

*φ*), that is

### 3.2. First-order Born approximation

^{2}

**s**integral in Eq. (15) can be evaluated in closed-form using polar coordinates [33, 34], we have

### 3.3. Second-order Born approximation

*a*

^{′}=

*a*(

*z*),

^{′},z*q*=

^{′}*q*(

*κ*,

^{′},z′**k**φ,s,

*z*), and

*µ*(

^{′}=µ*z*).

^{′}^{2}s integral, we get the second-order normalized Born approximation for the single tilted plane-wave-Gaussian beam

*n*

^{2}

_{1}(

**s**,

*z*)〉≠0, it follows that 〈d

*ν*(

**κ**,

*z*)d

*ν*(

*κ*)〉=0 and therefore for the second-order Born correction 〈

^{′},z^{′}*u*̄

_{2}(

**r**,

*L*)〉=0.

_{1}(

**r**,

*L*) and Ψ

_{2}(

**r**,

*L*) can be readily determined using Eqs. (7), and the propagated field with Eqs. (4).

## 4. Second-order statistical moments and averaged intensity

*U*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

*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.

*u**

_{φ}(

**r**,

*L*)

*u*

_{χ}(

**r**,

*L*)〉 within the integral in Eq. (24) as

*t*is a random variable, then the average of its exponential function can be approximated to second-order as [1, p. 126]

_{1,φ}〉=〈Ψ

_{1,χ}〉=0, we obtain

*first*second-order statistical moment of the Rytov approximation for the component

*j*={

*φ,χ*}, and

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

*J*

_{0}is the zeroth-order Bessel function, Φ

_{n}(

*k*) is the power spectrum of atmospheric fluctuations which models the atmospheric turbulence, and

_{n}(

**κ**) of the refractive-index fluctuations.

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]

**r**within the beam.

*φ*→0,

*χ*→0, and

*k*

_{0}→0, and (b) for two tilted ideal plane waves [19

**24**, 745–752 (2007). [CrossRef]

*w*

_{0}→∞.

*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

_{n}(

**κ**).

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

*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

*m*th-order at the source plane are written as [25

**22**, 289–298 (2005). [CrossRef]

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

*J*is the

_{m}*m*th-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

*k*→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

_{t}*A*(

*φ*)=exp(

*imφ*).

*u**

_{φ}(

**r**,

*L*)

*u*(

_{χ}**r**,

*L*)〉 in Eq. (27) and later evaluate Eq. (24) using the angular spectrum

*A*(

*φ*)=exp(

*imφ*).

*z*=0 to the plane

*z=z*

_{max}=

*w*

_{0}k/k_{0}. The physical meaning of the maximum propagation distance

*z*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

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

*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

**22**, 289–298 (2005). [CrossRef]

*z*=

_{max}*w*

_{0}k/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.

*L*={0,1/4,1/2,3/4}z

_{max}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]

*L*=(3/4)z

_{max}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].

*m*-th order Bessel-Gauss beam of the form (37) carries an orbital angular momentum per photon given by

*mh*̄. 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. 1–3, 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]

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

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]

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]

**22**, 289–298 (2005). [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]

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]

*z*=0 of the

*m*th-order helical Mathieu-Gauss beams is given by [25

**22**, 289–298 (2005). [CrossRef]

*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

*m*th-order Bessel beams

*J*(

_{m}*k*) exp(±

_{0}r*imθ*). Unlike Bessel-Gauss waves, the Mathieu-Gauss beams are not rotationally symmetric with respect to the propagation axis.

*u**

_{φ}(

**r**,

*L*)

*uχ*(

**r**,

*L*)〉 in Eq. (27) and later evaluating Eq. (24) using the Mathieu angular spectrum

*A*±

_{m}(φ)=ce

_{m}(φ,

*ε*)±

*i*se

_{m}(φ,

*ε*).

*x*and

*y*axes at transverse planes

*L*={0,1/4,1/2,3/4}

*z*are plotted in Fig. 5 for a second-order Mathieu-Gauss beam and a transverse range

_{max}*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]

**22**, 289–298 (2005). [CrossRef]

## 6. Conclusions

**24**, 745–752 (2007). [CrossRef]

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

*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)]

*ν**(

**κ**,

*z*)d

*ν*(

*κ*)〉 is written in the form

^{′}, z^{′}*δ*is the Dirac delta function and

*F*(

_{n}**κ**, |

*z-z*

^{′}|)≡

*F*(

_{n}*κ*, |

_{x},κ_{y}*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

_{n}(

**κ**)≡Φ

_{n}(

*κ*) is the power spectrum of atmospheric fluctuations.

_{x},κ_{y}*z-z*

^{′}allows us to rewrite the d

*z*and d

*z*

^{′}integrals as

*F*(

_{n}**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

*z*≅

*z*

^{′}≅

*η*, which finally yields

^{2}

**κ**integral in Eq. (A-5) can be reduced to a single radial integral using polar coordinates. After some algebraic manipulation, we obtain

*J*

_{0}is the zeroth-order Bessel function.

### A.2. Derivation of *E*^{(1)}_{φ} (**r**,*L*) [Eq. (30)].

*n*

_{1}(

**s**,z) is a real function, we know that

*z-z*

^{′}, the approximation

*z*≅

*z*

^{′}≅η, and recalling that

*F*(

_{n}**κ**, |ξ|) has appreciable values for very small values of |ξ| and depend on |ξ|, we can note that

*E*

^{(1)}

_{φ}(

**r**,

*L*) [Eq.(A-9)] reduces to

## Acknowledgments

## References and links

1. | L. Andrews and R. Phillips, |

2. | C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence-induced beam spreading of higher-order mode optical waves,” Opt. Eng. |

3. | H. T. Eyyuboglu, S. Altay, and Y. Baykal, “Propagation characteristics of higher-order annular Gaussian beams in atmospheric turbulence,” Opt. Commun. |

4. | Y. Cai and S. He, “Average intensity and spreading of an elliptical Gaussian beam propagating in a turbulent atmosphere,” Opt. Lett. |

5. | Y. Cai and D. Ge, “Analytical formula for a decentered elliptical Gaussian beam propagationg in a turbulent atmosphere,” Opt. Commun. |

6. | H. T. Eyyuboğlu and Y. Baykal, “Reciprocity of cos-Gaussian and cosh-Gaussian laser beams in turbulent atmosphere,” Opt. Express |

7. | H. T. Eyyuboğlu and Y. Baykal, “Average intensity and spreading of cosh-Gaussian beams in the turbulent atmosphere,” Appl. Opt. |

8. | H. T. Eyyuboğlu, “Propagation of Hermite-cosh-Gaussian laser beams in turbulent atmosphere,” Opt. Commun. |

9. | H. T. Eyyuboğlu, “Hermite-cosine-Gaussian laser beam and its propagation characteristics in turbulent atmosphere,” J. Opt. Soc. Am. A |

10. | Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. |

11. | H. T. Eyyuboğlu, “Propagation of higher order Bessel-Gaussian beams in turbulence,” Appl. Phys. B , |

12. | Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express |

13. | Z. Bouchal, “Resistance of nondiffracting vortex beam against amplitude and phase perturbations,” Opt. Commun. |

14. | Z. Bouchal, “Nondiffracting optical beams: physical properties, experiments, and applications,” Czech. J. Phys. |

15. | J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo,, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. |

16. | M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. |

17. | Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstructionof a distorted nondiffracting beam,” Opt. Commun. |

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. |

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 |

20. | O. Korotkova and G. Gbur, “Propagation of beams with any spectral, coherence and polarization properties in turbulent atmosphere,” Proc. of SPIE |

21. | O. Korotkova and G. Gbur, “Angular spectrum representation for propagation of random electromagnetic beams in a turbulent atmosphere,” J. Opt. Soc. Am. A |

22. | R. Frehlich, “Simulation of laser propagation in a turbulent atmosphere,” Appl. Opt. |

23. | C. Arpali, C. Yazicioglu, H. Eyyuboğlu, S. Arpali, and Y. Baykal, “Simulator for general-type beam propagation in turbulent atmosphere,” Opt. Express |

24. | D.C. Cowan, J. Recolons, L.C. Andrews, and C.Y. Young, Atmospheric Propagation III, Proc. SPIE |

25. | J. C. Gutiérrez-Vega and M. A. Bandres, “Helmholtz-Gauss waves,” J. Opt. Soc. Am. A |

26. | C. López-Mariscal, M. A. Bandrés, and J. C. Gutiérrez-Vega, “Observation of the experimental propagation properties of Helmholtz-Gauss beams,” Opt. Eng. |

27. | M. Guizar-Sicairos and J. C. Gutiérrez-Vega, “Propagation of Helmholtz-Gauss beams in absorbing and gain media,” J. Opt. Soc. Am. A , |

28. | M. Guizar-Sicairos and J. C. Gutiérrez-Vega, “Generalized Helmholtz-Gauss beams and its transformation by paraxial optical systems,” Opt. Lett. |

29. | M. A. Bandres and J. C. Gutiérrez-Vega, “Vector Helmholtz-Gauss and vector Laplace-Gauss beams,” Opt. Lett. |

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 |

31. | F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. |

32. | J. C. Gutiérrez-Vega and M. A. Bandres, “On the normalization of the Mathieu-Gauss optical beams,” J. Opt. Soc. Am. A , |

33. | M. Abramowitz and I.A. Stegun, |

34. | I. S. Gradshteyn and I. M. Ryzhik, |

35. | A. Chafiq, Z. Hricha, and A. Belafhal, “Paraxial propagation of Mathieu beams through an apertured ABCD optical system,” Opt. Commun. |

36. | Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, “Shaping soliton properties in Mathieu lattices,” Opt. Lett. |

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. |

38. | C. López-Mariscal, J. C. Gutiérrez-Vega, G. Milne, and K. Dholakia, “Orbital angular momentum transfer in helical Mathieu beams,” Opt. Express , |

39. | C. A. Dartora and H. E. Hernández-Figueroa, “Properties of a localized Mathieu pulse,” J. Opt. Soc. Am. A |

**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

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