Abstract

Hermite-cosine-Gaussian (HcosG) laser beams are studied. The source plane intensity of the HcosG beam is introduced and its dependence on the source parameters is examined. By application of the Fresnel diffraction integral, the average receiver intensity of HcosG beam is formulated for the case of propagation in turbulent atmosphere. The average receiver intensity is seen to reduce appropriately to various special cases. When traveling in turbulence, the HcosG beam initially experiences the merging of neighboring beam lobes, and then a TEM-type cosh-Gaussian beam is formed, temporarily leading to a plain cosh-Gaussian beam. Eventually a pure Gaussian beam results. The numerical evaluation of the normalized beam size along the propagation axis at selected mode indices indicates that relative spreading of higher-order HcosG beam modes is less than that of the lower-order counterparts. Consequently, it is possible at some propagation distances to capture more power by using higher-mode-indexed HcosG beams.

© 2005 Optical Society of America

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References

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  1. L. W. Casperson, A. A. Tovar, “Hermite-sinusoidal-Gaussian beams in complex optical systems,” J. Opt. Soc. Am. A 15, 954–961 (1998).
    [CrossRef]
  2. D. M. Zhao, W. C. Zhang, S. M. Wang, H. J. Liu, Q. H. Zhu, X. F. Wei, “Propagation of one-dimensional off-axial Hermite-cosine-Gaussian beams through an apertured paraxial ABCD optical system,” Optik (Stuttgart) 114, 49–51 (2003).
    [CrossRef]
  3. D. M. Zhao, H. D. Mao, W. C. Zhang, 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]
  4. N. R. Zhou, G. H. Zeng, “Propagation properties of Hermite-cosine-Gaussian beams through a paraxial optical ABCD system with hard-edge aperture,” Opt. Commun. 232, 49–59 (2004).
    [CrossRef]
  5. D. Zhao, H. Mao, D. Sun, “Approximate analytical expression for the kurtosis parameter of off-axial Hermite-cosine-Gaussian beams propagating through apertured and misaligned ABCD optical systems,” Optik (Stuttgart) 114, 535–538 (2003).
    [CrossRef]
  6. Z. R. Mei, D. M. Zhao, D. Sun, J. G. Gu, “The M-2 factor and kurtosis parameter of the off-axial Hermite-cosh-Gaussian beams,” Optik (Stuttgart) 115, 89–93 (2004).
    [CrossRef]
  7. X. Q. Wang, B. D. Lu, “The M-2 factor of Hermite-cosh-Gaussian beams,” J. Mod. Opt. 48, 2097–2103 (2001).
    [CrossRef]
  8. S. R. Luo, B. Lu, “Propagation of the kurtosis parameter of Hermite-cosh-Gaussian beams,” Optik (Stuttgart) 113, 329–332 (2002).
    [CrossRef]
  9. D. M. Zhao, H. D. Mao, H. J. Liu, “Propagation of off-axial Hermite-cosh-Gaussian laser beams,” J. Opt. A Pure Appl. Opt. 6, 77–83 (2004).
    [CrossRef]
  10. Y. L. Qiu, H. Guo, X. Z. Chen, H. J. Kong, “Propagation properties of an elegant Hermite-cosh-Gaussian beam through a finite aperture,” J. Opt. A Pure Appl. Opt. 6, 210–215 (2004).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  13. T. Shirai, A. Dogariu, E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 20, 1094–1102 (2003).
    [CrossRef]
  14. H. T. Eyyuboğlu, Y. Baykal, “Average intensity and spreading of cosh-Gaussian laser beams in the turbulent atmosphere,” Appl. Opt. 44, 976–983 (2005).
    [CrossRef] [PubMed]
  15. H. T. Eyyuboğlu, Y. Baykal, “Reciprocity of cos-Gaussian and cosh-Gaussian laser beams in turbulent atmosphere,” Opt. Express 12, 4659–4674 (2004).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  20. H. T. Eyyuboğlu, Y. E. Yenice, Y. Baykal, “Higher order annular laser beam propagation in free space,” submitted to Opt. Eng. (Bellingham).

2005 (2)

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

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

2004 (7)

Y. Baykal, “Correlation and structure functions of Hermite-sinusoidal-Gaussian laser beams in a turbulent atmosphere,” J. Opt. Soc. Am. A 21, 1290–1299 (2004).
[CrossRef]

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

N. R. Zhou, G. H. Zeng, “Propagation properties of Hermite-cosine-Gaussian beams through a paraxial optical ABCD system with hard-edge aperture,” Opt. Commun. 232, 49–59 (2004).
[CrossRef]

Z. R. Mei, D. M. Zhao, D. Sun, J. G. Gu, “The M-2 factor and kurtosis parameter of the off-axial Hermite-cosh-Gaussian beams,” Optik (Stuttgart) 115, 89–93 (2004).
[CrossRef]

D. M. Zhao, H. D. Mao, H. J. Liu, “Propagation of off-axial Hermite-cosh-Gaussian laser beams,” J. Opt. A Pure Appl. Opt. 6, 77–83 (2004).
[CrossRef]

Y. L. Qiu, H. Guo, X. Z. Chen, H. J. Kong, “Propagation properties of an elegant Hermite-cosh-Gaussian beam through a finite aperture,” J. Opt. A Pure Appl. Opt. 6, 210–215 (2004).
[CrossRef]

Z. R. Mei, D. M. Zhao, D. Sun, J. D. Gu, “The M-2 factor and kurtosis parameter of the off-axial Hermite-cosh-Gaussian beams,” Optik (Stuttgart) 115, 89–93 (2004).
[CrossRef]

2003 (4)

D. Zhao, H. Mao, D. Sun, “Approximate analytical expression for the kurtosis parameter of off-axial Hermite-cosine-Gaussian beams propagating through apertured and misaligned ABCD optical systems,” Optik (Stuttgart) 114, 535–538 (2003).
[CrossRef]

D. M. Zhao, W. C. Zhang, S. M. Wang, H. J. Liu, Q. H. Zhu, X. F. Wei, “Propagation of one-dimensional off-axial Hermite-cosine-Gaussian beams through an apertured paraxial ABCD optical system,” Optik (Stuttgart) 114, 49–51 (2003).
[CrossRef]

D. M. Zhao, H. D. Mao, W. C. Zhang, 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]

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

2002 (1)

S. R. Luo, B. Lu, “Propagation of the kurtosis parameter of Hermite-cosh-Gaussian beams,” Optik (Stuttgart) 113, 329–332 (2002).
[CrossRef]

2001 (2)

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

X. Q. Wang, B. D. Lu, “The M-2 factor of Hermite-cosh-Gaussian beams,” J. Mod. Opt. 48, 2097–2103 (2001).
[CrossRef]

1998 (1)

1979 (1)

Baykal, Y.

Casperson, L. W.

Chen, X. Z.

Y. L. Qiu, H. Guo, X. Z. Chen, H. J. Kong, “Propagation properties of an elegant Hermite-cosh-Gaussian beam through a finite aperture,” J. Opt. A Pure Appl. Opt. 6, 210–215 (2004).
[CrossRef]

Dogariu, A.

Eyyuboglu, H. T.

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

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

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

H. T. Eyyuboğlu, Y. E. Yenice, Y. Baykal, “Higher order annular laser beam propagation in free space,” submitted to Opt. Eng. (Bellingham).

Gradysteyn, I. S.

I. S. Gradysteyn, I. M. Ryzhik, Tables of Integrals, Series and Products, 6th ed. (Academic, 2000), pp. 360.

Gu, J. D.

Z. R. Mei, D. M. Zhao, D. Sun, J. D. Gu, “The M-2 factor and kurtosis parameter of the off-axial Hermite-cosh-Gaussian beams,” Optik (Stuttgart) 115, 89–93 (2004).
[CrossRef]

Gu, J. G.

Z. R. Mei, D. M. Zhao, D. Sun, J. G. Gu, “The M-2 factor and kurtosis parameter of the off-axial Hermite-cosh-Gaussian beams,” Optik (Stuttgart) 115, 89–93 (2004).
[CrossRef]

Guo, H.

Y. L. Qiu, H. Guo, X. Z. Chen, H. J. Kong, “Propagation properties of an elegant Hermite-cosh-Gaussian beam through a finite aperture,” J. Opt. A Pure Appl. Opt. 6, 210–215 (2004).
[CrossRef]

Kong, H. J.

Y. L. Qiu, H. Guo, X. Z. Chen, H. J. Kong, “Propagation properties of an elegant Hermite-cosh-Gaussian beam through a finite aperture,” J. Opt. A Pure Appl. Opt. 6, 210–215 (2004).
[CrossRef]

Liu, H. J.

D. M. Zhao, H. D. Mao, H. J. Liu, “Propagation of off-axial Hermite-cosh-Gaussian laser beams,” J. Opt. A Pure Appl. Opt. 6, 77–83 (2004).
[CrossRef]

D. M. Zhao, W. C. Zhang, S. M. Wang, H. J. Liu, Q. H. Zhu, X. F. Wei, “Propagation of one-dimensional off-axial Hermite-cosine-Gaussian beams through an apertured paraxial ABCD optical system,” Optik (Stuttgart) 114, 49–51 (2003).
[CrossRef]

Lu, B.

S. R. Luo, B. Lu, “Propagation of the kurtosis parameter of Hermite-cosh-Gaussian beams,” Optik (Stuttgart) 113, 329–332 (2002).
[CrossRef]

Lu, B. D.

X. Q. Wang, B. D. Lu, “The M-2 factor of Hermite-cosh-Gaussian beams,” J. Mod. Opt. 48, 2097–2103 (2001).
[CrossRef]

Luo, S. R.

S. R. Luo, B. Lu, “Propagation of the kurtosis parameter of Hermite-cosh-Gaussian beams,” Optik (Stuttgart) 113, 329–332 (2002).
[CrossRef]

Mao, H.

D. Zhao, H. Mao, D. Sun, “Approximate analytical expression for the kurtosis parameter of off-axial Hermite-cosine-Gaussian beams propagating through apertured and misaligned ABCD optical systems,” Optik (Stuttgart) 114, 535–538 (2003).
[CrossRef]

Mao, H. D.

D. M. Zhao, H. D. Mao, H. J. Liu, “Propagation of off-axial Hermite-cosh-Gaussian laser beams,” J. Opt. A Pure Appl. Opt. 6, 77–83 (2004).
[CrossRef]

D. M. Zhao, H. D. Mao, W. C. Zhang, 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]

Mei, Z. R.

Z. R. Mei, D. M. Zhao, D. Sun, J. G. Gu, “The M-2 factor and kurtosis parameter of the off-axial Hermite-cosh-Gaussian beams,” Optik (Stuttgart) 115, 89–93 (2004).
[CrossRef]

Z. R. Mei, D. M. Zhao, D. Sun, J. D. Gu, “The M-2 factor and kurtosis parameter of the off-axial Hermite-cosh-Gaussian beams,” Optik (Stuttgart) 115, 89–93 (2004).
[CrossRef]

Piper, J. A.

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

Plonus, M. A.

Qiu, Y. L.

Y. L. Qiu, H. Guo, X. Z. Chen, H. J. Kong, “Propagation properties of an elegant Hermite-cosh-Gaussian beam through a finite aperture,” J. Opt. A Pure Appl. Opt. 6, 210–215 (2004).
[CrossRef]

Ryzhik, I. M.

I. S. Gradysteyn, I. M. Ryzhik, Tables of Integrals, Series and Products, 6th ed. (Academic, 2000), pp. 360.

Saghafi, S.

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

Sheppard, C. J. R.

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

Shirai, T.

Sun, D.

Z. R. Mei, D. M. Zhao, D. Sun, J. G. Gu, “The M-2 factor and kurtosis parameter of the off-axial Hermite-cosh-Gaussian beams,” Optik (Stuttgart) 115, 89–93 (2004).
[CrossRef]

Z. R. Mei, D. M. Zhao, D. Sun, J. D. Gu, “The M-2 factor and kurtosis parameter of the off-axial Hermite-cosh-Gaussian beams,” Optik (Stuttgart) 115, 89–93 (2004).
[CrossRef]

D. Zhao, H. Mao, D. Sun, “Approximate analytical expression for the kurtosis parameter of off-axial Hermite-cosine-Gaussian beams propagating through apertured and misaligned ABCD optical systems,” Optik (Stuttgart) 114, 535–538 (2003).
[CrossRef]

Tovar, A. A.

Wang, S.

D. M. Zhao, H. D. Mao, W. C. Zhang, 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]

Wang, S. C. H.

Wang, S. M.

D. M. Zhao, W. C. Zhang, S. M. Wang, H. J. Liu, Q. H. Zhu, X. F. Wei, “Propagation of one-dimensional off-axial Hermite-cosine-Gaussian beams through an apertured paraxial ABCD optical system,” Optik (Stuttgart) 114, 49–51 (2003).
[CrossRef]

Wang, X. Q.

X. Q. Wang, B. D. Lu, “The M-2 factor of Hermite-cosh-Gaussian beams,” J. Mod. Opt. 48, 2097–2103 (2001).
[CrossRef]

Wei, X. F.

D. M. Zhao, W. C. Zhang, S. M. Wang, H. J. Liu, Q. H. Zhu, X. F. Wei, “Propagation of one-dimensional off-axial Hermite-cosine-Gaussian beams through an apertured paraxial ABCD optical system,” Optik (Stuttgart) 114, 49–51 (2003).
[CrossRef]

Wolf, E.

Yenice, Y. E.

H. T. Eyyuboğlu, Y. E. Yenice, Y. Baykal, “Higher order annular laser beam propagation in free space,” submitted to Opt. Eng. (Bellingham).

Zeng, G. H.

N. R. Zhou, G. H. Zeng, “Propagation properties of Hermite-cosine-Gaussian beams through a paraxial optical ABCD system with hard-edge aperture,” Opt. Commun. 232, 49–59 (2004).
[CrossRef]

Zhang, W. C.

D. M. Zhao, H. D. Mao, W. C. Zhang, 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]

D. M. Zhao, W. C. Zhang, S. M. Wang, H. J. Liu, Q. H. Zhu, X. F. Wei, “Propagation of one-dimensional off-axial Hermite-cosine-Gaussian beams through an apertured paraxial ABCD optical system,” Optik (Stuttgart) 114, 49–51 (2003).
[CrossRef]

Zhao, D.

D. Zhao, H. Mao, D. Sun, “Approximate analytical expression for the kurtosis parameter of off-axial Hermite-cosine-Gaussian beams propagating through apertured and misaligned ABCD optical systems,” Optik (Stuttgart) 114, 535–538 (2003).
[CrossRef]

Zhao, D. M.

D. M. Zhao, H. D. Mao, H. J. Liu, “Propagation of off-axial Hermite-cosh-Gaussian laser beams,” J. Opt. A Pure Appl. Opt. 6, 77–83 (2004).
[CrossRef]

Z. R. Mei, D. M. Zhao, D. Sun, J. G. Gu, “The M-2 factor and kurtosis parameter of the off-axial Hermite-cosh-Gaussian beams,” Optik (Stuttgart) 115, 89–93 (2004).
[CrossRef]

Z. R. Mei, D. M. Zhao, D. Sun, J. D. Gu, “The M-2 factor and kurtosis parameter of the off-axial Hermite-cosh-Gaussian beams,” Optik (Stuttgart) 115, 89–93 (2004).
[CrossRef]

D. M. Zhao, W. C. Zhang, S. M. Wang, H. J. Liu, Q. H. Zhu, X. F. Wei, “Propagation of one-dimensional off-axial Hermite-cosine-Gaussian beams through an apertured paraxial ABCD optical system,” Optik (Stuttgart) 114, 49–51 (2003).
[CrossRef]

D. M. Zhao, H. D. Mao, W. C. Zhang, 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]

Zhou, N. R.

N. R. Zhou, G. H. Zeng, “Propagation properties of Hermite-cosine-Gaussian beams through a paraxial optical ABCD system with hard-edge aperture,” Opt. Commun. 232, 49–59 (2004).
[CrossRef]

Zhu, Q. H.

D. M. Zhao, W. C. Zhang, S. M. Wang, H. J. Liu, Q. H. Zhu, X. F. Wei, “Propagation of one-dimensional off-axial Hermite-cosine-Gaussian beams through an apertured paraxial ABCD optical system,” Optik (Stuttgart) 114, 49–51 (2003).
[CrossRef]

Appl. Opt. (1)

J. Mod. Opt. (1)

X. Q. Wang, B. D. Lu, “The M-2 factor of Hermite-cosh-Gaussian beams,” J. Mod. Opt. 48, 2097–2103 (2001).
[CrossRef]

J. Opt. A Pure Appl. Opt. (2)

D. M. Zhao, H. D. Mao, H. J. Liu, “Propagation of off-axial Hermite-cosh-Gaussian laser beams,” J. Opt. A Pure Appl. Opt. 6, 77–83 (2004).
[CrossRef]

Y. L. Qiu, H. Guo, X. Z. Chen, H. J. Kong, “Propagation properties of an elegant Hermite-cosh-Gaussian beam through a finite aperture,” J. Opt. A Pure Appl. Opt. 6, 210–215 (2004).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Opt. Commun. (4)

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

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

D. M. Zhao, H. D. Mao, W. C. Zhang, 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]

N. R. Zhou, G. H. Zeng, “Propagation properties of Hermite-cosine-Gaussian beams through a paraxial optical ABCD system with hard-edge aperture,” Opt. Commun. 232, 49–59 (2004).
[CrossRef]

Opt. Express (1)

Optik (Stuttgart) (5)

D. Zhao, H. Mao, D. Sun, “Approximate analytical expression for the kurtosis parameter of off-axial Hermite-cosine-Gaussian beams propagating through apertured and misaligned ABCD optical systems,” Optik (Stuttgart) 114, 535–538 (2003).
[CrossRef]

Z. R. Mei, D. M. Zhao, D. Sun, J. G. Gu, “The M-2 factor and kurtosis parameter of the off-axial Hermite-cosh-Gaussian beams,” Optik (Stuttgart) 115, 89–93 (2004).
[CrossRef]

Z. R. Mei, D. M. Zhao, D. Sun, J. D. Gu, “The M-2 factor and kurtosis parameter of the off-axial Hermite-cosh-Gaussian beams,” Optik (Stuttgart) 115, 89–93 (2004).
[CrossRef]

S. R. Luo, B. Lu, “Propagation of the kurtosis parameter of Hermite-cosh-Gaussian beams,” Optik (Stuttgart) 113, 329–332 (2002).
[CrossRef]

D. M. Zhao, W. C. Zhang, S. M. Wang, H. J. Liu, Q. H. Zhu, X. F. Wei, “Propagation of one-dimensional off-axial Hermite-cosine-Gaussian beams through an apertured paraxial ABCD optical system,” Optik (Stuttgart) 114, 49–51 (2003).
[CrossRef]

Other (2)

H. T. Eyyuboğlu, Y. E. Yenice, Y. Baykal, “Higher order annular laser beam propagation in free space,” submitted to Opt. Eng. (Bellingham).

I. S. Gradysteyn, I. M. Ryzhik, Tables of Integrals, Series and Products, 6th ed. (Academic, 2000), pp. 360.

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Figures (10)

Fig. 1
Fig. 1

Contour plots of two HcosG source beams, with V x r = 0 m 1 , V y r = 0 m 1 and V x r = 50 m 1 , V y r = 50 m 1 .

Fig. 2
Fig. 2

Contour plots of two HcosG source beams, with V x r = 50 m 1 , V y r = 50 m 1 and V x r = 50 m 1 , V y r = 10 m 1 .

Fig. 3
Fig. 3

3D illustrations of two HcosG source beams, with n = 0 , m = 0 and n = 2 , m = 0 .

Fig. 4
Fig. 4

Contour plots of two HcosG source beams, with α s x = 5 cm , α s y = 5 cm and α s x = 5 cm , α s y = 2.5 cm .

Fig. 5
Fig. 5

Contour plots of two HcosG source beams, with a x = 20.0 m 1 , a y = 20.0 m 1 and a x = 14.1 m 1 , a y = 14.1 m 1 .

Fig. 6
Fig. 6

Contour plots of two HcosG source beams, with b x = 0.0 , b y = 0.0 and b x = 2.0 , b y = 0.0 .

Fig. 7
Fig. 7

3D illustration of HcosG source and receiver beams at propagation lengths of L = 0 , 0.4 , 2 , 20 km . Intensity distribution of HcosG beam (a), (b) before propagation and (c), (d) after propagation.

Fig. 8
Fig. 8

Contour plots of a HcosG source beam and a HcosG receiver beam at L = 2 km .

Fig. 9
Fig. 9

(a) Normalized beam size variation of a HcosG beam along the p x axis with different mode indices versus propagation length. (b) Normalized beam size variation of a HcosG beam along the p y axis with different mode indices versus propagation length.

Fig. 10
Fig. 10

Variation of received power of a HcosG beam for an aperture size of α r = ( α s x 2 + α s y 2 ) 1 2 2 versus propagation length at different mode indices.

Equations (14)

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

u s ( s ) = u s ( s x , s y ) = 0.5 A c H n ( a x s x + b x ) H m ( a y s y + b y ) exp [ 0.5 ( s x 2 α s x 2 + s y 2 α s y 2 ) ] { exp [ i ( V x s x + V y s y ) ] + exp [ i ( Y x s x + Y y s y ) ] } ,
I s ( s ) = I s ( s x , s y ) = H n 2 ( a x s x + b x ) H m 2 ( a y s y + b y ) exp [ ( s x 2 α s x 2 + s y 2 α s y 2 ) ] cos 2 [ ( V x r s x + V y r s y ) ] .
u r ( p , z = L , t ) = k exp ( i k L ) ( 2 i π L ) d 2 s u s ( s ) exp [ i k ( p s ) 2 ( 2 L ) + ψ ( s , p ) 2 i π f t ] ,
I r ( p , z = L ) = k 2 ( 2 π L ) 2 d 2 s 1 d 2 s 2 u s ( s 1 ) u s * ( s 2 ) exp { i k [ ( p s 1 ) 2 ( p s 2 ) 2 ] ( 2 L ) } exp [ ψ ( s 1 , p ) + ψ * ( s 2 , p ) ] .
exp [ ψ ( s 1 , p ) + ψ * ( s 2 , p ) ] = exp [ 0.5 D ψ ( s 1 s 2 ) ] = exp [ ρ 0 2 ( s 1 s 2 ) 2 ] ,
I r ( p , z = L ) = b 2 ρ 0 4 ( D s x D s y ) 1 2 exp { 4 ρ 0 4 b 2 [ p x 2 ( α s x 2 D s x ) + p y 2 ( α s y 2 D s y ) ] } [ exp { 2 ρ 0 2 [ V x r 2 ( ρ 0 2 a s x + 1 ) D s x + V y r 2 ( ρ 0 2 a s y + 1 ) D s y ] } { exp [ 8 i ρ 0 4 b 2 ( V x r p x D s x + V y r p y D s y ) ] S x 1 S y 1 + exp [ 8 i ρ 0 4 b 2 ( V x r p x D s x + V y r p y D s y ) ] S x 2 S y 2 } + exp { ρ 0 4 [ V x r 2 ( α s x 2 D s x ) + V y r 2 ( α s y 2 D s y ) ] } ( exp { 4 ρ 0 4 b [ V x r p x ( α s x 2 D s x ) + V y r p y ( α s y 2 D s y ) ] } S x 3 S y 3 + exp { 4 ρ 0 4 b [ V x r p x ( α s x 2 D s x ) + V y r p y ( α s y 2 D s y ) ] } S x 4 S y 4 ) ] ,
b = k 2 L , a s x = [ ( 0.5 α s x 2 ) + ( 1 ρ 0 2 ) ] , a s y = [ ( 0.5 α s y 2 ) + ( 1 ρ 0 2 ) ] , D s x = 4 [ ρ 0 4 ( a s x 2 + b 2 ) 1 ] , and D s y = 4 [ ρ 0 4 ( a s y 2 + b 2 ) 1 ] ,
S x 1 = l x 1 = 0 [ n 2 ] l n x 1 = 0 n 2 l x 1 k x 1 = 0 [ I n x 1 2 ] l n x 11 = 0 l n x 1 2 k x 1 l x 2 = 0 [ n 2 ] l n x 2 = 0 n 2 l x 2 k x 2 = 0 [ ( l n x 11 + l n x 2 ) 2 ] ( 1 ) l x 1 + l x 2 2 2 n l x 1 l x 2 l n x 1 l x n 2 i l n x 1 + l n x 2 2 k x 1 2 k x 2 T l x 1 T l x 2 ( n 2 l x 1 ) ( n 2 l x 2 ) ( n 2 l x 1 l n x 1 ) ( l n x 1 2 k x 1 l n x 11 ) ( n 2 l x 2 l n x 2 ) l n x 1 ! ( l n x 1 2 k x 1 ) ! k x 1 ! ( l n x 11 + l n x 2 ) ! ( l n x 11 + l n x 2 2 k x 2 ) ! k x 2 ! × ( a x ) l n x 1 + l n x 2 ( b x ) 2 n 2 l x 1 2 l x 2 l n x 1 l n x 2 ( ρ 0 2 ) l n x 2 ( 0.25 D s x ) k x 2 l n x 11 l n x 2 ( a s x i b ) l n x 1 + k x 1 + k x 2 ( V x r 2 b p x ) l n x 1 2 k x 1 l n x 11 [ ρ 0 2 ( V x r + 2 b p x ) ( a s x i b ) + ( V x r 2 b p x ) ] l n x 11 + l n x 2 2 k x 2 .
T l x j = 1 × 3 × …. ( 2 l x j 1 ) for l x j 0 , j = 1 , 2 ;
( B 1 B 2 ) = B 1 ! [ ( B 1 B 2 ) ! B 2 ! ] ,
I s N ( s x , s y ) = I s ( s x , s y ) Max [ ( I s ( s x , s y ) ) ] ,
I r N ( p x , p y , z = L ) = I r ( p x , p y , z = L ) Max [ ( I s ( s x , s y ) ) ] .
α p x N ( z ) = { [ 2 p x 2 I r ( p , z ) d p x I r ( p , z ) d p x ] [ 2 s x 2 I s ( s ) d s x I s ( s ) d s x ] } 1 2 .
P r N = 2 π 0 α r r I r ( p , z ) d r [ 2 π 0 r I s ( s ) d r ] ,

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