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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 26 — Dec. 24, 2007
  • pp: 17613–17618
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Propagation of a cosh-Gaussian beam through an optical system in turbulent atmosphere

Xiuxiang Chu  »View Author Affiliations


Optics Express, Vol. 15, Issue 26, pp. 17613-17618 (2007)
http://dx.doi.org/10.1364/OE.15.017613


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Abstract

The propagation of a cosh-Gaussian beam through an arbitrary ABCD optical system in turbulent atmosphere has been investigated. The analytical expressions for the average intensity at any receiver plane are obtained. As an elementary example, the average intensity and its radius at the image plane of a cosh-Gaussian beam through a thin lens are studied. To show the effects of a lens on the average intensity and the intensity radius of the laser beam in turbulent atmosphere, the properties of a collimated cosh-Gaussian beam and a focused cosh-Gaussian beam for direct propagation in turbulent atmosphere are studied and numerically calculated. The average intensity profiles of a cosh-Gaussian beam through a lens can have a shape similar to that of the initial beam for a longer propagation distance than that of a collimated cosh-Gaussian beam for direct propagation. With the increment in the propagation distance, the average intensity radius at the image plane of a cosh-Gaussian beam through a thin lens will be smaller than that at the focal plane of a focused cosh-Gaussian beam for direct propagation. Meanwhile, the intensity distributions at the image plane of a cosh-Gaussian beam through a lens with different w0 and Ω0 are also studied.

© 2007 Optical Society of America

1. Introduction

2. The average intensity of a cosh-Gaussian beam through an optical system in turbulent atmosphere

In the Cartesian coordinate system, a cosh-Gaussian beam propagates along the z-axis, the source plane and the receiver plane are located at (x 0, y 0, z=0) and (x, y, z=L), respectively. The optical field of a cosh-Gaussian beam at the source plane is of the form [19

19. K. Duan and B. Lu, “Propagation of Hermite-Laguerre-Gaussian beams through a paraxial optical ABCD system with rectangular hard-edged aperture,” Opt. Commun. 250, 1–9 (2005). [CrossRef]

]

E0(x0,y0,0)=exp[x02+y02w02ik(x02+y02)2F]cosh(Ω0x0)cosh(Ω0y0),
(1)

where (x 0, y 0) denotes the transverse coordinates of the source plane, (x, y) denotes the transverse coordinates of the receiver plane, L denotes the optical distance along the z-axis, w 0 is the initial waist width of the beam associated with the Gaussian part, k=2π/λ is the wave number andλ is wavelength, Ω0 is the parameter associated with the cosh part, and F is the phase front radius of curvature (F>0 for a convergent beam and F<0 for a divergent beam). Based on Huygens-Fresnel diffraction integral, the fields at receiver plane of a beam through an ABCD optical system in turbulent atmosphere can be expressed as [23

23. H. T. Yura and S. G. Hanson, “Optical beam wave propagation through complex optical system,” J. Opt. Soc. Am. A 4, 1931–1948 (1987). [CrossRef]

]

E(x,y,L)=1iλBexp(ikL)E0(x0,y0,0)
×exp{ik2B[A(x02+y02)+D(x2+y2)2(xx0+yy0)]+ψ(x0,y0,x,y)}dx0dy0,
(2)

where A, B and D are the geometrical ray-matrix elements for the complete optical system between source and receiver planes, Ψ(x 0, y 0, x, y) is the solution to the Rytov method that represents the random part of the complex phase. The average intensity at the receiver plane is

I(x,y,L)=1λ2B2E0(x01,y01,0)E0*(x02,y02,0)
×exp{ik2B[A(x012x022+y012y022)2(xx01xx02+yy01yy02)]}
×exp[ψ(x01,y01,x,y)+ψ(x02,y02,x,y)]dx01dy01dx02dy02,
(3)

where angle brackets indicate the ensemble average over the medium statistics, asterisk denotes the complex conjugation. The last term of Eq. (3) can be denotes as [24

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

]

exp[ψ(x01,y01,x,y)+ψ(x02,y02,x,y)]=exp[Dw(r01r02)2],
(4)

where Dw(r 01r 02) is the wave structure function. If a quadratic approximation for the wave structure function is employed, Dw(r 01r 02) can be expressed as

Dw(r01r02)2=[(x01x02)2+(y01y02)2]ρ02,
(5)

where ρ 0 is the spherical-wave lateral coherence length. If a laser beam passes through an optical system ρ 0 can be expressed as [23

23. H. T. Yura and S. G. Hanson, “Optical beam wave propagation through complex optical system,” J. Opt. Soc. Am. A 4, 1931–1948 (1987). [CrossRef]

]

ρ0=Bσ0=B[1.46k2Cn20Ldzb53(z)]35,
(6)

where C 2 n is the structure constant, b(z)is the corresponding matrix element for a ray whose origin is at the receiver plane and whose endpoint is at a distance z from receiver plane. Substituting Eqs. (1), (4) and (5) into Eq. (3), the average intensity can be obtained as

I(x,y,L)=w024w2exp[2(x2+y2)w2+4B2Ω02k2w2]{cos(4BΩ0kw2x)+exp(4Ω02w02k2σ02w2)cosh[2(BAF)Ω0w02xFw2]}
×{cos(4BΩ0kw2y)+exp(4Ω02w02k2σ02w2)cosh[2(BAF)Ω0w02yFw2]},
(7)

where w is the 1/e 2 average intensity radius of a fundamental Gaussian beam after passing through an ABCD optical system in turbulent atmosphere and can be written as [23

23. H. T. Yura and S. G. Hanson, “Optical beam wave propagation through complex optical system,” J. Opt. Soc. Am. A 4, 1931–1948 (1987). [CrossRef]

]

w2=4B2(k2w02)+w02(ABF)2+8(k2σ02).
(8)

The three terms on the right-hand side of Eq. (8) represent the contributions to the beam spread that are due to diffraction, geometrical magnification and turbulence, respectively. From Eqs. (7) and (8) we can see that the average intensity and the intensity radius depend on only ray-matrix elements A and B.

3. Special cases and numerical calculation

As an elementary example, the case of a single thin lens of focal length f located at the propagation path is considered, as shown in Fig. 1. The source plane and the receiver plane are conjugate image plane, namely, 1/f=1/z 1+1/z 2. If we set z 1>f, it is easy to find that A=m=-z 2/z 1 and B=0, where m denotes the magnification.

Fig. 1. Propagation geometry of image system in turbulent atmosphere.

Therefore, the average intensity at the image plane is

I(x,y,L)=w024w2exp[2(x2+y2)w2][1+exp(4Ω02w02k2σ02w2)cosh(2mΩ0w02xw2)][1+exp(4Ω02w02k2σ02w2)cosh(2mΩ0w02yw2)],
(9)

where

w2=m2w02+8(k2σ02),
(10)
b(z)={z2zz10<zz1Lzz1<zL.
(11)

Substituting Eq. (11) into Eq. (6), it is easy to obtain

σ0=z21(0.5475k2Cn2L)35.
(12)

From Eqs. (8) and (10) we can see that when a beam passes through a lens, the contributions to the beam spread at the image plane that are due to diffraction disappear and the contributions to the beam spread that are due to turbulence become small. Because the cosh-Gaussian beam adopted in this paper is x-y symmetric, the properties of one-dimensional intensity distributions can reveal the properties of a two-dimensional intensity distribution. From Eq. (7) the one-dimensional average intensity of a collimated cosh-Gaussian beam (F→∞) for direct propagation (B=L, A=1) in turbulent atmosphere can be derived as

I(x,L)=w02wexp(2x2w2+2L2Ω02k2w2)[cos(4LΩ0kw2x)+exp(4Ω02w02k2σ02w2)cosh(2Ω0w02xw2)],
(13)

where

w2=w02+4L2(k2w02)+8L2(k2ρ02).
(14)

The one-dimensional average intensity at focal plane (L=F) of a focused cosh-Gaussian beam for direct propagation in turbulent atmosphere can be represented as

I(x,L)=w02wexp(2x2w2+2L2Ω02k2w2)[cos(4LΩ0kw2x)+exp(4Ω02w02k2σ02w2)],
(15)

where

w2=4L2(k2w02)+8L2(k2ρ02).
(16)

The one-dimensional average intensity at image plane of a cosh-Gaussian beam through a lens in turbulent atmosphere is

I(x,L)=w02wexp(2x2w2)[1+exp(4Ω02w02k2σ02w2)cosh(2mΩ0w02xw2)].
(17)

To see the effects of a lens on the intensity distributions, the one-dimensional normalized average intensity profiles of the beam through a lens and for direct propagation with different parameters are plotted in Fig. 2. Normalized average intensity IN(x,L) at receiver plane is defined as

IN(x,L)=I(x,L)Max[I(x0,0)],
(18)

where Max[〈I(x 0,0)〉] denotes the maximum value of the one-dimensional average intensity at the source plane.

Fig. 2. The normalized average intensity profiles of a cosh-Gaussian beam in turbulent atmosphere. (a) At image plane of a lens with w 0=0.2m and Ω0=30m -1; (b) At receiver plane for direct propagation with F→∞, w 0=0.2m and Ω0=30m -1; (c) At focal plane (L=F) for direct propagation with w 0=0.2m and Ω0=30m -1; (d) At image plane of a lens with L=20km.

The parameters are adopted as L=2z 1=2z 2=4f, C 2 n=10-14 m -2/3, and λ=1.06×10-6 m. From Fig. 2(a) we can see that the normalized average intensity profiles at the image plane of a cosh-Gaussian beam through a lens in turbulent atmosphere are the hollow shapes when L=10km, 16km and 20km. With increase in propagation distance the peak values of the normalized average intensity decrease, meanwhile, the central dips become smaller and the beam spot spreads. From Eqs. (9) and (10) we can see that the changes of the average intensity profiles is only due to the effects of turbulence. Figure 2(b) shows that the intensity profile of a collimated cosh-Gaussian is hollow shape at L=10km. But, with increase in propagation distance, the central dip of the hollow shape gradually disappears [see the dashed line in Fig. 2(b)]. With further increase in the propagation distance (L=16km) the central dip disappears completely and the hollow shape evolve into a bell shape [see the dotted line in Fig. 2(b)]. From Eqs. (13) and (14) we can see that the evolution of the average intensity profiles of a collimated cosh-Gaussian is due to the effects of diffraction and turbulence. The average intensity profiles of a focused cosh-Gaussian beam at the focal plane are bell shapes [see Fig. 2(c)]. The peak values decrease with the increase in the propagation distance. Eqs. (15) and (16) denote that the decreases of the peak value at the focal plane with the increase in the propagation distance are due to the diffraction, geometrical magnification and the turbulence. From Figs. 2(a), 2(b) and 2(c) we can see that the hollow shapes of a cosh- Gaussian beam after passing through an image system can be sustained for longer propagation distance. From Eqs. (9) and (10) we can also see that the intensity distributions has a close relation with w 0 and Ω0. Since the intensity distribution at source plane is different with different Ω0, the different intensity distribution at source plane results in the different intensity distributions at the image plane [see the solid line and dotted line in Fig. 2(d)]. Because large initial waist width can reduce the effects of turbulence on intensity distribution, the hollow shapes can be sustained for longer propagation distance with large initial waist width [see the dashed line and dotted line in Fig. 2(d)].

4. Conclusion

The propagation of a cosh-Gaussian beam through an arbitrary ABCD optical system in turbulent atmosphere is studied. To show the effects of an ABCD system on the propagation of a laser beam in turbulent atmosphere, an elementary ABCD system of a single lens is studied. From the studies and the comparisons with the properties of a collimated cosh-Gaussian and a focused cosh-Gaussian beam for direct propagation in turbulent atmosphere, we can see that a cosh-Gaussian beam through a lens can have a shape similar to that of the initial beam for a longer propagation distance in turbulent atmosphere than that of a collimated cosh-Gaussian beam for direct propagation in turbulent atmosphere. With the increment in the propagation distance, the average intensity radius of a cosh-Gaussian beam at the image plane after passing through a thin lens will be smaller than that of a focused cosh-Gaussian beam at the focal plane for direct propagation in turbulent atmosphere. Meanwhile, the intensity distributions at the image plane with different w 0 and Ω0 are also studied. The investigations show that the average intensity distribution and radius of a cosh-Gaussian beam can be improved by using optical elements on its propagation path in turbulent atmosphere.

Acknowledgments

This work is supported by Zhejiang Provincial Education Committee Foundation of China.

References and links

1.

T. Shirai, A. Dogariu, and E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 20, 1094–1102 (2003). [CrossRef]

2.

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]

3.

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]

4.

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]

5.

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

6.

Y. Baykal, “Log-amplitude and phase fluctuations of higher-order annular laser beams in a turbulent medium,” J. Opt. Soc. Am. A 22, 672–679 (2005). [CrossRef]

7.

H. T. Eyyuboğlu, C. Arpali, and Y. Baykal, “Flat topped beams and their characteristics in turbulent media,” Opt. Express. 14, 4196–4207 (2006). [CrossRef] [PubMed]

8.

Y. Baykal and H. T. Eyyuboğlu, “Scintillation index of flat-topped Gaussian beams,” Appl. Opt. 45, 3793–3797 (2006). [CrossRef] [PubMed]

9.

H. T. Eyyuboğlu, Y. Baykal, and E. Sermutlu, “Convergence of general beams into Gaussian intensity profiles after propagation in turbulent atmosphere,” Opt. Commun. 265, 399–405 (2006). [CrossRef]

10.

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

11.

Y. Cai, “Propagation of various flat-topped beams in a turbulent atmosphere,” J. Opt. A: Pure Appl. Opt. 8, 537–545 (2006). [CrossRef]

12.

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

13.

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects, Opt. Eng.45, 076001–12 (2006). [CrossRef]

14.

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, “Beam wander effects on the scintillation index of a focused beam,” Proc. SPIE 5793, 28–37 (2005). [CrossRef]

15.

X. Chu and G. Zhou, “Power coupling of a two-Cassegrain-telescopes system in turbulent atmosphere in a slant path,” Opt. Express 15,7697–7707 (2007). [CrossRef] [PubMed]

16.

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

17.

X. Ji, T. Huang, and Baida Lu “Spreading of partially coherent cosh-Gaussian beams propagating through turbulent atmosphere,” Acta Physica Sinica 55, 978–982 (2006).

18.

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, 5–12 (2003). [CrossRef]

19.

K. Duan and B. Lu, “Propagation of Hermite-Laguerre-Gaussian beams through a paraxial optical ABCD system with rectangular hard-edged aperture,” Opt. Commun. 250, 1–9 (2005). [CrossRef]

20.

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

21.

H. T. Eyyuboğlu and Y. Baykal, “Generalized beams in ABCDGH systems,” Opt. Commun. 272, 22–31 (2007). [CrossRef]

22.

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

23.

H. T. Yura and S. G. Hanson, “Optical beam wave propagation through complex optical system,” J. Opt. Soc. Am. A 4, 1931–1948 (1987). [CrossRef]

24.

S. C. H. Wang and M. A. Plonus, “Optical beam propagation for a partially coherent source in the turbulent atmosphere,” J. Opt. Soc. Am. 69, 1297–1304 (1979). [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: September 25, 2007
Revised Manuscript: November 26, 2007
Manuscript Accepted: December 6, 2007
Published: December 11, 2007

Citation
Xiuxiang Chu, "Propagation of a cosh-Gaussian beam through an optical system in turbulent atmosphere," Opt. Express 15, 17613-17618 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-26-17613


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References

  1. T. Shirai, A. Dogariu, and E. Wolf, "Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence," J. Opt. Soc. Am. A 20, 1094-1102 (2003). [CrossRef]
  2. H. T. Eyyubo?lu and Y. Baykal, "Reciprocity of cos-Gaussian and cosh-Gaussian laser beams in turbulent atmosphere," Opt. Express 12, 4659-4674 (2004). [CrossRef] [PubMed]
  3. H. T. Eyyubo?lu and Y. Baykal, "Average intensity and spreading of cosh-Gaussian beams in the turbulent atmosphere," Appl. Opt. 44, 976-983 (2005). [CrossRef] [PubMed]
  4. H. T. Eyyubo?lu, "Hermite-cosine-Gaussian laser beam and its propagation characteristics in turbulent atmosphere," J. Opt. Soc. Am. A 22, 1527-1535 (2005). [CrossRef]
  5. H. T. Eyyubo?lu, "Propagation of Hermite-cosh-Gaussian laser beams in turbulent atmosphere," Opt. Commun. 245, 37-47 (2005). [CrossRef]
  6. Y. Baykal, "Log-amplitude and phase fluctuations of higher-order annular laser beams in a turbulent medium," J. Opt. Soc. Am. A 22, 672-679 (2005). [CrossRef]
  7. H. T. Eyyubo?lu, C. Arpali, and Y. Baykal, "Flat topped beams and their characteristics in turbulent media," Opt. Express. 14, 4196-4207 (2006). [CrossRef] [PubMed]
  8. Y. Baykal and H. T. Eyyubo?lu, "Scintillation index of flat-topped Gaussian beams," Appl. Opt. 45, 3793-3797 (2006). [CrossRef] [PubMed]
  9. H. T. Eyyubo?lu, Y. Baykal, and E. Sermutlu, "Convergence of general beams into Gaussian intensity profiles after propagation in turbulent atmosphere," Opt. Commun. 265, 399-405 (2006). [CrossRef]
  10. Y. Cai and S. He, "Propagation of various dark hollow beams in turbulent atmosphere," Opt. Express 14,1353-1367 (2006). [CrossRef] [PubMed]
  11. Y. Cai, "Propagation of various flat-topped beams in a turbulent atmosphere," J. Opt. A: Pure Appl. Opt. 8, 537-545 (2006). [CrossRef]
  12. Y. Cai and D. Ge. "Analytical formula for a decentered elliptical Gaussian beam propagating in a turbulent atmosphere," Opt. Commun. 271, 509-516 (2007). [CrossRef]
  13. L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, "Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects, Opt. Eng. 45, 076001-12 (2006). [CrossRef]
  14. L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, "Beam wander effects on the scintillation index of a focused beam," Proc. SPIE 5793, 28-37 (2005). [CrossRef]
  15. X. Chu and G. Zhou, "Power coupling of a two-Cassegrain-telescopes system in turbulent atmosphere in a slant path," Opt. Express 15, 7697-7707 (2007). [CrossRef] [PubMed]
  16. X. Chu, Y. Ni, and G. Zhou, "Propagation of cosh-Gaussian beams diffracted by a circular aperture in turbulent atmosphere," Appl. Phys. B. 87, 547-552 (2007). [CrossRef]
  17. X. Ji, T. Huang, and B. Lu "Spreading of partially coherent cosh-Gaussian beams propagating through turbulent atmosphere," Acta Physica Sinica 55, 978-982 (2006).
  18. 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, 5-12 (2003). [CrossRef]
  19. K. Duan and B. Lu, "Propagation of Hermite-Laguerre-Gaussian beams through a paraxial optical ABCD system with rectangular hard-edged aperture," Opt. Commun. 250, 1-9 (2005). [CrossRef]
  20. A. A. Tovar and L. W. Casperson, "Production and propagation of Hermite-sinusoidal-Gaussian laser beams," J. Opt. Soc. Am. A 15, 2425-2432 (1998). [CrossRef]
  21. H. T. Eyyubo?lu and Y. Baykal, "Generalized beams in ABCDGH systems," Opt. Commun. 272, 22-31 (2007). [CrossRef]
  22. Y. Zhang, Y. Song, Z. Chen, J. Ji, and Z. Shi "Virtual sources for a cosh-Gaussian beam" Opt. Lett. 32, 292-294 (2007). [CrossRef] [PubMed]
  23. H. T. Yura and S. G. Hanson, "Optical beam wave propagation through complex optical system," J. Opt. Soc. Am. A 4, 1931-1948 (1987). [CrossRef]
  24. S. C. H. Wang and M. A. Plonus, "Optical beam propagation for a partially coherent source in the turbulent atmosphere," J. Opt. Soc. Am. 69, 1297-1304 (1979). [CrossRef]

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