OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 5 — Feb. 28, 2011
  • pp: 3945–3951
« Show journal navigation

Propagation of a higher-order cosh-Gaussian beam in turbulent atmosphere

Guoquan Zhou  »View Author Affiliations


Optics Express, Vol. 19, Issue 5, pp. 3945-3951 (2011)
http://dx.doi.org/10.1364/OE.19.003945


View Full Text Article

Acrobat PDF (1027 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

The propagation of a higher-order cosh-Gaussian beam through a paraxial and real ABCD optical system in turbulent atmosphere has been investigated. The analytical expressions for the average intensity, the effective beam size, and the kurtosis parameter of a higher-order cosh-Gaussian beam through a paraxial and real ABCD optical system are derived in turbulent atmosphere. The average intensity distribution and the spreading properties of a higher-order cosh-Gaussian in turbulent atmosphere are numerically demonstrated. The influences of the beam parameters and the structure constant of the atmospheric turbulence on the propagation of a higher-order cosh-Gaussian beam in turbulent atmosphere are also examined in detail.

© 2011 OSA

1. Introduction

Hermite-sinusoidal-Gaussian (HSG) beams have been introduced as exact solutions of the paraxial wave equation [1

1. L. W. Casperson and A. A. Tovar, “Hermite-sinusoidal-Gaussian beams in complex optical systems,” J. Opt. Soc. Am. A 15(4), 954–961 (1998). [CrossRef]

,2

2. A. A. Tovar and L. W. Casperson, “Production and propagation of Hermite-sinusoidal-Gaussian laser beams,” J. Opt. Soc. Am. A 15(9), 2425–2432 (1998). [CrossRef]

]. Cosh-Gaussian beams are the special cases of HSG beams. The propagation of cosh-Gaussian beams in free space and through complex optical systems has been studied [1

1. L. W. Casperson and A. A. Tovar, “Hermite-sinusoidal-Gaussian beams in complex optical systems,” J. Opt. Soc. Am. A 15(4), 954–961 (1998). [CrossRef]

4

4. D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236(4-6), 225–235 (2004). [CrossRef]

]. The propagation characteristics of cosh-Gaussian beams in turbulent atmosphere have been also extensively investigated [5

5. H. T. Eyyuboğlu and Y. Baykal, “Analysis of reciprocity of cos-Gaussian and cosh- Gaussian laser beams in a turbulent atmosphere,” Opt. Express 12(20), 4659–4674 (2004). [CrossRef] [PubMed]

9

9. X. Chu, “Propagation of a cosh-Gaussian beam through an optical system in turbulent atmosphere,” Opt. Express 15(26), 17613–17618 (2007). [CrossRef] [PubMed]

]. On the basis of the superposition of laser beams, a group of virtual sources that generates a cosh-Gaussian beam has been presented [10

10. Y. Zhang, Y. Song, Z. Chen, J. Ji, and Z. Shi, “Virtual sources for a cosh-Gaussian beam,” Opt. Lett. 32(3), 292–294 (2007). [CrossRef] [PubMed]

]. By means of the theorem of the vectorial structure and the method of stationary phase, the vectorial structure of cosh-Gaussian beam has been examined in the far-field [11

11. G. Zhou and F. Liu, “Far field structural characteristics of cosh-Gaussian beam,” Opt. Laser Technol. 40(2), 302–308 (2008). [CrossRef]

]. The coherent combination of certain cosh-Gaussian beams results in flattened beams with an axial shadow [12

12. K. Zhu, H. Tang, X. Wang, and T. Liu, “Flattened light beams with an axial shadow generated through superposing cosh-Gaussian beams,” Optik (Stuttg.) 113(5), 222–226 (2002). [CrossRef]

]. The incoherent combination of some cosh-Gaussian beams with different parameters has been shown to be a flattened Gaussian beam [13

13. Q. Tang, Y. Yu, and Q. Hu, “A new method to generate flattened Gaussian beam by incoherent combination of cosh Gaussian beams,” Chin. Opt. Lett. 5, S46–S48 (2007).

]. The higher-order cosh-Gaussian beam, which is defined as a higher-order cosh function multiplied by a Gaussian function, can be produced by coherent superposition of cosh-Gaussian beams or by superposition of decentered Gaussian beams with the same waist width. The higher-order cosh-Gaussian beam is a broader beam model than the cosh-Gaussian beam. By varying the beam parameters, the higher-order cosh-Gaussian beam in the source plane will take on a Gaussian-like distribution, a flattened distribution, and a dark hollow distribution. Therefore, the higher-order cosh-Gaussian beam is an interesting beam model. Based on the second-order moment, the beam propagation factor and the kurtosis parameter of a higher-order cosh-Gaussian beam have been derived [14

14. G. Zhou and J. Zheng, “Beam propagation of a higher-order cosh-Gaussian beam,” Opt. Laser Technol. 41(2), 202–208 (2009). [CrossRef]

]. The fractional Fourier transform has been applied to treat the propagation of higher-order cosh-Gaussian beams, and the properties of a higher-order cosh-Gaussian beam have been demonstrated in the fractional Fourier transform plane [15

15. G. Zhou, “Fractional Fourier transform of a higher-order cosh-Gaussian beam,” J. Mod. Opt. 56(7), 886–892 (2009). [CrossRef]

]. The vectorial structural characteristics of a higher-order cosh-Gaussian beam have been revealed in the far-field reference plane [16

16. J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Far-field vectorial structure of a higher-order cosh-Gaussian beam,” J. Mod. Opt. 57(20), 2039–2047 (2010). [CrossRef]

]. Propagation properties of higher-order cosh-Gaussian beams in uniaxial crystals orthogonal to the optical axis have been examined [17

17. J. Li, Y. Chen, Y. Xin, and S. Xu, “Propagation of higher-order cosh-Gaussian beams in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. D 57(3), 419–425 (2010). [CrossRef]

]. The research in the propagation of laser beams in a turbulent atmosphere is vital to the applications in free-space optical communications and remote sensing. To the best of our knowledge, the propagation of a higher-order cosh-Gaussian beam in turbulent atmosphere has not been reported elsewhere. In the remainder of this paper, therefore, the propagation of a higher-order cosh-Gaussian beam through a paraxial and real ABCD optical system in turbulent atmosphere is investigated. Analytical propagation formulae of the average intensity, the effective beam size, and the kurtosis parameter are derived and illustrated by a numerical example.

2. Propagation of a higher-order cosh-Gaussian beam in turbulent atmosphere

In the Cartesian coordinate system, the z-axis is taken to be the propagation axis. The higher-order cosh-Gaussian beam in the source plane z = 0 takes the form as
E(x0,y0,0)n=E(x0,0)nE(y0,0)n,
(1)
with En(x 0, 0) and En(y 0, 0) given by
E(j0,0)n=coshn(Ωj0)exp(j02w02),n=1,2,3,,
(2)
where j 0 = x 0 or y 0. n is the beam order. Ω is the cosh parameter, and w 0 is the waist width of Gaussian part. If n = 0, Eq. (1) reduces to be the well-known Gaussian beam. If n = 1, Eq. (1) reduces to be a cosh-Gaussian beam. En(j 0, 0) can also be written in the form as
E(j0,0)n=m=0nCmexp[-(j0bm)2/w02],
(3)
with the coefficients Cm and bm given by
Cm=amexp[(mn2)2δ],am=n!2nm!(nm)!,bm=(mn2)δΩ,
(4)
where δ = w 0 2Ω2. Therefore, En(j 0, 0) can be produced by superposition of n + 1 decentered Gaussian beams with the same waist width. Based on the extended Huygens-Fresnel diffraction integral, the higher-order cosh-Gaussian beam propagating through a paraxial and real ABCD optical system in turbulent atmosphere can be obtained by
E(x,y,z)=1iλBexp[ikz+ikD(x2+y2)2B]En(x0,y0,0)×exp{ik2B[A(x02+y02)2(xx0+2yy0)]+ψ(x0,y0,x,y)}dx0dy0,
(5)
where k = 2π/λ with λ being the optical wavelength. ψ(x 0, y 0, x, y) is the solution to the Rytov method that represents the random part of the complex phase. The optical system of concern is orthogonal and is described by the same ABCD matrix in each of the mutually orthogonal planes, x-z and y-z. A, B, C, and D are the transfer matrix elements of the paraxial optical system. Moreover, there is no inherent aperture between the source and the output planes, which denotes that A, B, C, and D are all realvalued. Therefore, the average intensity of the higher-order cosh-Gaussian beam passing through a paraxial and real ABCD optical system in turbulent atmosphere is found to be
<I(x,y,z)>=1λ2B2En(x01,y01,0)En(x02,y02,0)<exp[ψ(x01,y01,x,y)+ψ(x02,y02,x,y)]>×exp{ik2B[A(x012x022+y012y022)2x(x01x02)2y(y01y02)]}dx01dy01dx02dy02,
(6)
where the angle brackets denote the ensemble average over the medium statistics covering the log-amplitude and phase fluctuations due to the turbulent atmosphere. The asterisk means the complex conjugation. The ensemble average term is given by
<exp[ψ(x01,y01,x,y)+ψ(x02,y02,x,y)]>=exp[(x01x02)2+(y01y02)2ρ02],
(7)
where ρ 0 is the spherical-wave lateral coherence radius due to the turbulence of the entire optical system and is defined as
ρ0=B[1.46k2Cn20Lb5/3(z)dz]3/5,
(8)
where Cn 2 is the constant of refraction index structure and describes the turbulence level. b(z) corresponds to the approximate matrix element for a ray propagating backwards through the system. L is the axial distance between the source and the output planes. Substituting Eqs. (1), (3), and (7) into Eq. (6), the average intensity of the higher-order cosh-Gaussian beam in the output plane reads as
<I(x,y,z)>=<I(x,z)><I(y,z)>,
(9)
with <I(x, z) and <I(y, z) given by
<I(j,z)>=πλBα1jα2jm=0nl=0namalexp(β1j24α1j+β2j24α2j),
(10)
where j = x or y (hereafter). The parameters α 1 j, α 2 j, β 1 j, and β 2 j are defined by
α1j=1w02+1ρ02+ikA2B,α2j=1w02+1ρ021α1jρ04ikA2B,β1j=2blw02+ikjB,β2j=2bmw02ikjB+β1jα1jρ02.
(11)
The effective beam size of the higher-order cosh-Gaussian beam in the j-direction of the output plane yields
Wj=[2j2<I(x,y,z)>dxdy<I(x,y,z)>dxdy]1/2=12ξ[m=0nl=0namal(2ξj+ηj2)exp(τ)m=0nl=0namalexp(τ)]1/2,
(12)
with τ given by
τ=ηj24ξj+bl2α1jw04+γ1j24α2j,
(13)
where γ 1 j, γ 2 j, ξj, and ηj are defined by
γ1j=2bmw02+2blα1jw02ρ02,γ2j=ikα1jBρ02ikB,ξj=k24α1jB2γ2j24α2j,ηj=ikblBα1jw02+γ1jγ2j2α2j.
(14)
The kurtosis parameter, which is defined by the fourth-order moment, is employed to describe the flatness degree of the beams and is an important parameter to valuate the beam propagation. The kurtosis parameter of the higher-order cosh-Gaussian beam in the j-direction turns out to be

Kj=[j4<I(x,y,z)>dxdy][<I(x,y,z)>dxdy][j2<I(x,y,z)>dxdy]2=m=0nl=0namalexp(τ)m=0nl=0namal(12ξj2+12ξjηj2+ηj4)exp(τ)[m=0nl=0namal(2ξj+ηj2)exp(τ)]2.
(15)

3. The numerical results and analyses

Now, the average intensity and spreading of a higher-order cosh-Gaussian beam in turbulent atmosphere are calculated by using the formulae derived above. For simplicity, no other optical element is placed in turbulent atmosphere, which denotes that the matrix elements are A = 1, B = z, C = 0, and D = 1. As the x- and y-directions are separable in the formulae derived above, only the x-direction is considered hereafter. Moreover, λ is set to be 0.8μm. Figures 1
Fig. 1 Normalized average intensity distribution in the x-direction of a higher-order cosh-Gaussian beam in the reference plane of z = 1km in turbulent atmosphere. (a) n = 2 and w 0 = 0.02m. (b) w 0 = 0.02m and Ω = 60m−1. (c) n = 2 and Ω = 60m−1.
and 2
Fig. 2 Normalized average intensity distribution in the x-direction of a higher-order cosh-Gaussian beam in the reference plane of z = 2km in turbulent atmosphere. (a) n = 2 and w 0 = 0.02m. (b) w 0 = 0.02m and Ω = 60m−1. (c) n = 2 and Ω = 60m−1.
represent the normalized intensity distributions of higher-order cosh-Gaussian beams with different beam parameters in the reference plane z = 1km and z = 2km in turbulent atmosphere. In Figs. 1 and 2, Cn 2 is set to be 10−14m-2/3. Under different conditions of the beam parameters, the higher-order cosh-Gaussian beam in the source plane can be a Gaussian-like beam, a flattened beam, and a dark hollow beam. The intensity distributions of a flattened beam and a dark hollow beam will change upon propagation. When the propagation distance is large enough, the intensity distributions of a flattened beam and a dark hollow beam will tend to a Gaussian-like distribution. With varying one of the cosh parameter Ω, the beam order n, and the Gaussian waist w 0, the normalized intensity distribution of a higher-order cosh-Gaussian beam propagating in turbulent atmosphere also takes on different distribution, which is shown in Fig. 1. From Figs. 1 and 2, we can conclude the following conclusion. When one of the cosh parameter Ω, the beam order n, and the Gaussian waist w 0 is large, the normalized intensity distribution of a higher-order cosh-Gaussian beam will undergo first the dark hollow distribution, then the fattened distribution, and finally the Gaussian-like distribution with increasing the propagation distance in turbulent atmosphere. To further reveal the spreading properties of a higher-order cosh-Gaussian beam in turbulent atmosphere, the effective beam sizes of higher-order cosh-Gaussian beams versus the propagation distance z in turbulent atmosphere are depicted in Fig. 3
Fig. 3 The effective beam size in the x-direction of a higher-order cosh-Gaussian beam versus the propagation distance z in turbulent atmosphere. (a) n = 2 and w 0 = 0.02m. (b) w 0 = 0.02m and Ω = 60m−1. (c) n = 2 and Ω = 60m−1.
, where Cn 2 = 10−14m-2/3. When the propagation distance in turbulent atmosphere is small, the higher-order cosh-Gaussian beam with the large cosh parameter has the large effective beam size. When the propagation distance in turbulent atmosphere is large enough, the effective beam sizes of higher-order cosh-Gaussian beams with the different cosh parameters are approximately equivalent. The higher-order cosh-Gaussian beam with the large beam order has the large effective beam size. However, the difference between the effective beam sizes of higher-order cosh-Gaussian beams with different beam orders will shrink upon propagation in turbulent atmosphere. The higher-order cosh-Gaussian beam with the smaller Gaussian waist first has the smaller effective beam size and finally has the larger effective beam size with increasing the propagation distance in turbulent atmosphere. The gradients of the curves in Fig. 3 denote that the higher-order cosh-Gaussian beam spreads more rapidly in turbulent atmosphere for the smaller beam parameters. Figure 4
Fig. 4 The kurtosis parameter in the x-direction of a higher-order cosh-Gaussian beam versus the propagation distance z in turbulent atmosphere. (a) n = 2 and w 0 = 0.02m. (b) w 0 = 0.02m and Ω = 60m−1. (c) n = 2 and Ω = 60m−1.
shows the kurtosis parameters of higher-order cosh-Gaussian beams versus the propagation distance z in turbulent atmosphere. When first propagating in turbulent atmosphere, the higher-order cosh-Gaussian beam with the small cosh parameter, or the small beam order, or the small Gaussian waist has the large kurtosis parameter. When the propagation distance in turbulent atmosphere is large enough, the kurtosis parameters of the higher-order cosh-Gaussian beams all tend to 3, which is the value of the kurtosis parameter of the Gaussian distribution. The influence of the structure constant of the atmospheric turbulence on the propagation of a higher-order cosh-Gaussian beam in turbulent atmosphere is shown in Fig. 5
Fig. 5 (a) Normalized average intensity distribution in the x-direction of a higher-order cosh-Gaussian beam in the reference plane of z = 2km. (b) The effective beam size in the x-direction of a higher-order cosh-Gaussian beam versus the propagation distance z. (c) The kurtosis parameter in the x-direction of a higher-order cosh-Gaussian beam versus the propagation distance z. n = 2, Ω = 80m−1, and w 0 = 0.02m.
. We find that the higher-order cosh-Gaussian beam spreads more rapidly in turbulent atmosphere for a larger structure constant.

4. Conclusions

The propagation of a higher-order cosh-Gaussian beam through a paraxial and real ABCD optical system in turbulent atmosphere is investigated. Based on the extended Huygens-Fresnel integral, the analytical expressions of the average intensity, the effective beam size, and the kurtosis parameter of a higher-order cosh-Gaussian beam are derived in turbulent atmosphere. The average intensity distribution and spreading properties of a higher-order cosh-Gaussian are numerically examined. The higher-order cosh-Gaussian beam spreads more rapidly in turbulent atmosphere for the smaller beam parameters and a larger structure constant. This research is useful to the practical applications in free-space optical communications and remote sensing involving the higher-order cosh-Gaussian beam.

Acknowledgements

This work was supported by National Natural Science Foundation of China (No. 10974179) and Zhejiang Provincial Natural Science Foundation of China (No. Y1090073).

References and links

1.

L. W. Casperson and A. A. Tovar, “Hermite-sinusoidal-Gaussian beams in complex optical systems,” J. Opt. Soc. Am. A 15(4), 954–961 (1998). [CrossRef]

2.

A. A. Tovar and L. W. Casperson, “Production and propagation of Hermite-sinusoidal-Gaussian laser beams,” J. Opt. Soc. Am. A 15(9), 2425–2432 (1998). [CrossRef]

3.

D. Zhao, H. Mao, W. Zhang, and S. Wang, “Propagation of off-axial Hermite-cosine-Gaussian beams through an apertured and misaligned ABCD optical system,” Opt. Commun. 224(1-3), 5–12 (2003). [CrossRef]

4.

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236(4-6), 225–235 (2004). [CrossRef]

5.

H. T. Eyyuboğlu and Y. Baykal, “Analysis of reciprocity of cos-Gaussian and cosh- Gaussian laser beams in a turbulent atmosphere,” Opt. Express 12(20), 4659–4674 (2004). [CrossRef] [PubMed]

6.

H. T. Eyyuboğlu and Y. Baykal, “Average intensity and spreading of cosh-Gaussian laser beams in the turbulent atmosphere,” Appl. Opt. 44(6), 976–983 (2005). [CrossRef] [PubMed]

7.

H. T. Eyyuboğlu and Y. Baykal, “Scintillation characteristics of cosh-Gaussian beams,” Appl. Opt. 46(7), 1099–1106 (2007). [CrossRef] [PubMed]

8.

X. Chu, Y. Ni, and G. Zhou, “Propagation of cosh-Gaussian beams diffracted by a circular aperture in turbulent atmosphere,” Appl. Phys. B 87(3), 547–552 (2007). [CrossRef]

9.

X. Chu, “Propagation of a cosh-Gaussian beam through an optical system in turbulent atmosphere,” Opt. Express 15(26), 17613–17618 (2007). [CrossRef] [PubMed]

10.

Y. Zhang, Y. Song, Z. Chen, J. Ji, and Z. Shi, “Virtual sources for a cosh-Gaussian beam,” Opt. Lett. 32(3), 292–294 (2007). [CrossRef] [PubMed]

11.

G. Zhou and F. Liu, “Far field structural characteristics of cosh-Gaussian beam,” Opt. Laser Technol. 40(2), 302–308 (2008). [CrossRef]

12.

K. Zhu, H. Tang, X. Wang, and T. Liu, “Flattened light beams with an axial shadow generated through superposing cosh-Gaussian beams,” Optik (Stuttg.) 113(5), 222–226 (2002). [CrossRef]

13.

Q. Tang, Y. Yu, and Q. Hu, “A new method to generate flattened Gaussian beam by incoherent combination of cosh Gaussian beams,” Chin. Opt. Lett. 5, S46–S48 (2007).

14.

G. Zhou and J. Zheng, “Beam propagation of a higher-order cosh-Gaussian beam,” Opt. Laser Technol. 41(2), 202–208 (2009). [CrossRef]

15.

G. Zhou, “Fractional Fourier transform of a higher-order cosh-Gaussian beam,” J. Mod. Opt. 56(7), 886–892 (2009). [CrossRef]

16.

J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Far-field vectorial structure of a higher-order cosh-Gaussian beam,” J. Mod. Opt. 57(20), 2039–2047 (2010). [CrossRef]

17.

J. Li, Y. Chen, Y. Xin, and S. Xu, “Propagation of higher-order cosh-Gaussian beams in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. D 57(3), 419–425 (2010). [CrossRef]

OCIS Codes
(010.1300) Atmospheric and oceanic optics : Atmospheric propagation
(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence
(080.2730) Geometric optics : Matrix methods in paraxial optics

ToC Category:
Atmospheric and Oceanic Optics

History
Original Manuscript: January 4, 2011
Revised Manuscript: February 6, 2011
Manuscript Accepted: February 7, 2011
Published: February 14, 2011

Citation
Guoquan Zhou, "Propagation of a higher-order cosh-Gaussian beam in turbulent atmosphere," Opt. Express 19, 3945-3951 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-5-3945


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. L. W. Casperson and A. A. Tovar, “Hermite-sinusoidal-Gaussian beams in complex optical systems,” J. Opt. Soc. Am. A 15(4), 954–961 (1998). [CrossRef]
  2. A. A. Tovar and L. W. Casperson, “Production and propagation of Hermite-sinusoidal-Gaussian laser beams,” J. Opt. Soc. Am. A 15(9), 2425–2432 (1998). [CrossRef]
  3. D. Zhao, H. Mao, W. Zhang, and S. Wang, “Propagation of off-axial Hermite-cosine-Gaussian beams through an apertured and misaligned ABCD optical system,” Opt. Commun. 224(1-3), 5–12 (2003). [CrossRef]
  4. D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236(4-6), 225–235 (2004). [CrossRef]
  5. H. T. Eyyuboğlu and Y. Baykal, “Analysis of reciprocity of cos-Gaussian and cosh- Gaussian laser beams in a turbulent atmosphere,” Opt. Express 12(20), 4659–4674 (2004). [CrossRef] [PubMed]
  6. H. T. Eyyuboğlu and Y. Baykal, “Average intensity and spreading of cosh-Gaussian laser beams in the turbulent atmosphere,” Appl. Opt. 44(6), 976–983 (2005). [CrossRef] [PubMed]
  7. H. T. Eyyuboğlu and Y. Baykal, “Scintillation characteristics of cosh-Gaussian beams,” Appl. Opt. 46(7), 1099–1106 (2007). [CrossRef] [PubMed]
  8. X. Chu, Y. Ni, and G. Zhou, “Propagation of cosh-Gaussian beams diffracted by a circular aperture in turbulent atmosphere,” Appl. Phys. B 87(3), 547–552 (2007). [CrossRef]
  9. X. Chu, “Propagation of a cosh-Gaussian beam through an optical system in turbulent atmosphere,” Opt. Express 15(26), 17613–17618 (2007). [CrossRef] [PubMed]
  10. Y. Zhang, Y. Song, Z. Chen, J. Ji, and Z. Shi, “Virtual sources for a cosh-Gaussian beam,” Opt. Lett. 32(3), 292–294 (2007). [CrossRef] [PubMed]
  11. G. Zhou and F. Liu, “Far field structural characteristics of cosh-Gaussian beam,” Opt. Laser Technol. 40(2), 302–308 (2008). [CrossRef]
  12. K. Zhu, H. Tang, X. Wang, and T. Liu, “Flattened light beams with an axial shadow generated through superposing cosh-Gaussian beams,” Optik (Stuttg.) 113(5), 222–226 (2002). [CrossRef]
  13. Q. Tang, Y. Yu, and Q. Hu, “A new method to generate flattened Gaussian beam by incoherent combination of cosh Gaussian beams,” Chin. Opt. Lett. 5, S46–S48 (2007).
  14. G. Zhou and J. Zheng, “Beam propagation of a higher-order cosh-Gaussian beam,” Opt. Laser Technol. 41(2), 202–208 (2009). [CrossRef]
  15. G. Zhou, “Fractional Fourier transform of a higher-order cosh-Gaussian beam,” J. Mod. Opt. 56(7), 886–892 (2009). [CrossRef]
  16. J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Far-field vectorial structure of a higher-order cosh-Gaussian beam,” J. Mod. Opt. 57(20), 2039–2047 (2010). [CrossRef]
  17. J. Li, Y. Chen, Y. Xin, and S. Xu, “Propagation of higher-order cosh-Gaussian beams in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. D 57(3), 419–425 (2010). [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.

Figures

Fig. 1 Fig. 2 Fig. 3
 
Fig. 4 Fig. 5
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited