Abstract

Nowadays it has been accepted that the Kolmogorov model is not the only possible turbulent one in the atmosphere, which has been confirmed by the increasing experimental evidence and some results of theoretical investigation. This has prompted the scientist community to study optical propagation in non-Kolmogorov atmospheric turbulence. In this paper, using the method of effective beam parameters and a non-Kolmogorov power spectrum which has a more general power law instead of standard Kolmogorov power law value 11/3 and a more general amplitude factor instead of constant value 0.033, the fiber coupling efficiency for a Gaussian-beam wave has been derived for a horizontal path in both weak and strong fluctuation regimes. And then the influence of spectral power law variations on the fiber coupling efficiency has been analyzed. It is anticipated that this work is helpful to the investigations of atmospheric turbulence and optical wave propagation in the atmospheric turbulence.

© 2015 Optical Society of America

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References

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2015 (1)

C. Zhai, L. Tan, S. Yu, Y. Cao, and J. Ma, “Fiber coupling efficiency in non-Kolmogorov satellite links,” Opt. Commun. 336, 93–97 (2015).
[Crossref]

2013 (1)

Y. Baykal, “Coherence length in non-Kolmogorov satellite links,” Opt. Commun. 308, 105–108 (2013).
[Crossref]

2012 (1)

2011 (1)

2008 (1)

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88(1), 66–77 (2008).
[Crossref]

2007 (2)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for laser beam propagation through Non-Kolmogorov turbulence,” Proc. SPIE 6457, 026003 (2007).

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through Non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).

2006 (4)

G. Wang, “A new random-phase-screen time series simulation algorithm for dynamically atmospheric turbulence wave-front generator,” Proc. SPIE 6027, 602716 (2006).

M. S. Belen’kii, E. Cuellar, K. A. Hughes, and V. A. Rye, “Experimental study of spatial structure of turbulence at Maui Space Surveillance Site (MSSS),” Proc. SPIE 6304, 63040U (2006).

J. C. Ricklin, S. M. Hammel, F. D. Eaton, and S. L. Lachinova, “Atmospheric channel effects on free-space laser communication,” J. Opt. Fiber. Commun. Rep. 3(2), 111–158 (2006).
[Crossref]

K. Kazaura, K. Omae, T. Suzuki, M. Matsumoto, E. Mutafungwa, T. O. Korhonen, T. Murakami, K. Takahashi, H. Matsumoto, K. Wakamori, and Y. Arimoto, “Enhancing performance of next generation FSO communication systems using soft computing-based predictions,” Opt. Express 14(12), 4958–4968 (2006).
[Crossref] [PubMed]

2005 (1)

2004 (1)

1998 (1)

1997 (1)

M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).

1996 (3)

S. S. Moiseev and O. G. Chkhetiani, “Helical scaling in turbulence,” Sov. Phys. JETP 83(1), 192–198 (1996).

T. Elperin, N. Kleeorin, and I. Rogachevskii, “Isotropic and anisotropic spectra of passive scalar fluctuations in turbulent fluid flow,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 53(4), 3431–3441 (1996).
[Crossref] [PubMed]

G. D. Boreman and C. Dainty, “Zernike expansions for non-Kolmogorov turbulence,” J. Opt. Soc. Am. A 13(3), 517–522 (1996).
[Crossref]

1995 (4)

A. S. Gurvich and M. S. Belen’kii, “Influence of stratospheric turbulence on infrared imaging,” J. Opt. Soc. Am. A 12(11), 2517–2522 (1995).
[Crossref]

R. R. Beland, “Some aspects of propagation through weak isotropic non-Kolmogorov turbulence,” Proc. SPIE 2375, 1111–1126 (1995).

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).

L. C. Andrews, R. L. Phillips, and P. T. Yu, “Optical scintillations and fade statistics for a satellite-communication system,” Appl. Opt. 34(33), 7742–7751 (1995).
[Crossref] [PubMed]

1994 (2)

D. T. Kyrazis, J. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2110, 43–55 (1994).

L. C. Andrews, W. B. Miller, and J. C. Ricklin, “Spatial coherence of a Gaussian-beam wave in weak and strong optical turbulence,” J. Opt. Soc. Am. A 11(5), 1653–1660 (1994).
[Crossref]

1959 (1)

G. K. Batchelor, “Small-scale variation of convected quantities like temperature in turbulent fluid. Part I. General discussion and the case of small conductivity,” J. Fluid Mech. 5(1), 113–133 (1959).
[Crossref]

Andrews, L. C.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for laser beam propagation through Non-Kolmogorov turbulence,” Proc. SPIE 6457, 026003 (2007).

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through Non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).

L. C. Andrews, R. L. Phillips, and P. T. Yu, “Optical scintillations and fade statistics for a satellite-communication system,” Appl. Opt. 34(33), 7742–7751 (1995).
[Crossref] [PubMed]

L. C. Andrews, W. B. Miller, and J. C. Ricklin, “Spatial coherence of a Gaussian-beam wave in weak and strong optical turbulence,” J. Opt. Soc. Am. A 11(5), 1653–1660 (1994).
[Crossref]

Arimoto, Y.

Batchelor, G. K.

G. K. Batchelor, “Small-scale variation of convected quantities like temperature in turbulent fluid. Part I. General discussion and the case of small conductivity,” J. Fluid Mech. 5(1), 113–133 (1959).
[Crossref]

Baykal, Y.

Beland, R. R.

R. R. Beland, “Some aspects of propagation through weak isotropic non-Kolmogorov turbulence,” Proc. SPIE 2375, 1111–1126 (1995).

Belen’kii, M. S.

M. S. Belen’kii, E. Cuellar, K. A. Hughes, and V. A. Rye, “Experimental study of spatial structure of turbulence at Maui Space Surveillance Site (MSSS),” Proc. SPIE 6304, 63040U (2006).

M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).

A. S. Gurvich and M. S. Belen’kii, “Influence of stratospheric turbulence on infrared imaging,” J. Opt. Soc. Am. A 12(11), 2517–2522 (1995).
[Crossref]

Bishop, K. P.

D. T. Kyrazis, J. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2110, 43–55 (1994).

Boreman, G. D.

Brown, J. M.

M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).

Cao, Y.

C. Zhai, L. Tan, S. Yu, Y. Cao, and J. Ma, “Fiber coupling efficiency in non-Kolmogorov satellite links,” Opt. Commun. 336, 93–97 (2015).
[Crossref]

Chen, C.

Chkhetiani, O. G.

S. S. Moiseev and O. G. Chkhetiani, “Helical scaling in turbulence,” Sov. Phys. JETP 83(1), 192–198 (1996).

Cuellar, E.

M. S. Belen’kii, E. Cuellar, K. A. Hughes, and V. A. Rye, “Experimental study of spatial structure of turbulence at Maui Space Surveillance Site (MSSS),” Proc. SPIE 6304, 63040U (2006).

Dainty, C.

Davidson, F. M.

Dikmelik, Y.

Eaton, F. D.

J. C. Ricklin, S. M. Hammel, F. D. Eaton, and S. L. Lachinova, “Atmospheric channel effects on free-space laser communication,” J. Opt. Fiber. Commun. Rep. 3(2), 111–158 (2006).
[Crossref]

Elperin, T.

T. Elperin, N. Kleeorin, and I. Rogachevskii, “Isotropic and anisotropic spectra of passive scalar fluctuations in turbulent fluid flow,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 53(4), 3431–3441 (1996).
[Crossref] [PubMed]

Ferrero, V.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through Non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for laser beam propagation through Non-Kolmogorov turbulence,” Proc. SPIE 6457, 026003 (2007).

Fugate, R. Q.

M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).

Gerçekcioglu, H.

Golbraikh, E.

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88(1), 66–77 (2008).
[Crossref]

E. Golbraikh and N. S. Kopeika, “Behavior of structure function of refraction coefficients in different turbulent fields,” Appl. Opt. 43(33), 6151–6156 (2004).
[Crossref] [PubMed]

Gurvich, A. S.

Hammel, S. M.

J. C. Ricklin, S. M. Hammel, F. D. Eaton, and S. L. Lachinova, “Atmospheric channel effects on free-space laser communication,” J. Opt. Fiber. Commun. Rep. 3(2), 111–158 (2006).
[Crossref]

Hughes, K. A.

M. S. Belen’kii, E. Cuellar, K. A. Hughes, and V. A. Rye, “Experimental study of spatial structure of turbulence at Maui Space Surveillance Site (MSSS),” Proc. SPIE 6304, 63040U (2006).

Karis, S. J.

M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).

Kazaura, K.

Keating, D. B.

D. T. Kyrazis, J. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2110, 43–55 (1994).

Kleeorin, N.

T. Elperin, N. Kleeorin, and I. Rogachevskii, “Isotropic and anisotropic spectra of passive scalar fluctuations in turbulent fluid flow,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 53(4), 3431–3441 (1996).
[Crossref] [PubMed]

Kopeika, N. S.

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88(1), 66–77 (2008).
[Crossref]

E. Golbraikh and N. S. Kopeika, “Behavior of structure function of refraction coefficients in different turbulent fields,” Appl. Opt. 43(33), 6151–6156 (2004).
[Crossref] [PubMed]

Korhonen, T. O.

Kupershmidt, I.

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88(1), 66–77 (2008).
[Crossref]

Kyrazis, D. T.

D. T. Kyrazis, J. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2110, 43–55 (1994).

Lachinova, S. L.

J. C. Ricklin, S. M. Hammel, F. D. Eaton, and S. L. Lachinova, “Atmospheric channel effects on free-space laser communication,” J. Opt. Fiber. Commun. Rep. 3(2), 111–158 (2006).
[Crossref]

Leeb, W. R.

Liu, R.

Lou, Y.

Ma, J.

C. Zhai, L. Tan, S. Yu, Y. Cao, and J. Ma, “Fiber coupling efficiency in non-Kolmogorov satellite links,” Opt. Commun. 336, 93–97 (2015).
[Crossref]

Matsumoto, H.

Matsumoto, M.

Miller, W. B.

Moiseev, S. S.

S. S. Moiseev and O. G. Chkhetiani, “Helical scaling in turbulence,” Sov. Phys. JETP 83(1), 192–198 (1996).

Murakami, T.

Mutafungwa, E.

Omae, K.

Phillips, R. L.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for laser beam propagation through Non-Kolmogorov turbulence,” Proc. SPIE 6457, 026003 (2007).

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through Non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).

L. C. Andrews, R. L. Phillips, and P. T. Yu, “Optical scintillations and fade statistics for a satellite-communication system,” Appl. Opt. 34(33), 7742–7751 (1995).
[Crossref] [PubMed]

Preble, A. J.

D. T. Kyrazis, J. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2110, 43–55 (1994).

Ricklin, J. C.

J. C. Ricklin, S. M. Hammel, F. D. Eaton, and S. L. Lachinova, “Atmospheric channel effects on free-space laser communication,” J. Opt. Fiber. Commun. Rep. 3(2), 111–158 (2006).
[Crossref]

L. C. Andrews, W. B. Miller, and J. C. Ricklin, “Spatial coherence of a Gaussian-beam wave in weak and strong optical turbulence,” J. Opt. Soc. Am. A 11(5), 1653–1660 (1994).
[Crossref]

Rogachevskii, I.

T. Elperin, N. Kleeorin, and I. Rogachevskii, “Isotropic and anisotropic spectra of passive scalar fluctuations in turbulent fluid flow,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 53(4), 3431–3441 (1996).
[Crossref] [PubMed]

Roggemann, M. C.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).

Rye, V. A.

M. S. Belen’kii, E. Cuellar, K. A. Hughes, and V. A. Rye, “Experimental study of spatial structure of turbulence at Maui Space Surveillance Site (MSSS),” Proc. SPIE 6304, 63040U (2006).

Shtemler, Y.

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88(1), 66–77 (2008).
[Crossref]

Stribling, B. E.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).

Suzuki, T.

Takahashi, K.

Tan, L.

C. Zhai, L. Tan, S. Yu, Y. Cao, and J. Ma, “Fiber coupling efficiency in non-Kolmogorov satellite links,” Opt. Commun. 336, 93–97 (2015).
[Crossref]

Tong, S.

Toselli, I.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through Non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for laser beam propagation through Non-Kolmogorov turbulence,” Proc. SPIE 6457, 026003 (2007).

Virtser, A.

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88(1), 66–77 (2008).
[Crossref]

Wakamori, K.

Wang, G.

G. Wang, “A new random-phase-screen time series simulation algorithm for dynamically atmospheric turbulence wave-front generator,” Proc. SPIE 6027, 602716 (2006).

Welsh, B. M.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).

Winzer, P. J.

Wissler, J.

D. T. Kyrazis, J. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2110, 43–55 (1994).

Yang, H.

Yu, P. T.

Yu, S.

C. Zhai, L. Tan, S. Yu, Y. Cao, and J. Ma, “Fiber coupling efficiency in non-Kolmogorov satellite links,” Opt. Commun. 336, 93–97 (2015).
[Crossref]

Zhai, C.

C. Zhai, L. Tan, S. Yu, Y. Cao, and J. Ma, “Fiber coupling efficiency in non-Kolmogorov satellite links,” Opt. Commun. 336, 93–97 (2015).
[Crossref]

Zilberman, A.

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88(1), 66–77 (2008).
[Crossref]

Appl. Opt. (3)

Atmos. Res. (1)

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88(1), 66–77 (2008).
[Crossref]

J. Fluid Mech. (1)

G. K. Batchelor, “Small-scale variation of convected quantities like temperature in turbulent fluid. Part I. General discussion and the case of small conductivity,” J. Fluid Mech. 5(1), 113–133 (1959).
[Crossref]

J. Opt. Fiber. Commun. Rep. (1)

J. C. Ricklin, S. M. Hammel, F. D. Eaton, and S. L. Lachinova, “Atmospheric channel effects on free-space laser communication,” J. Opt. Fiber. Commun. Rep. 3(2), 111–158 (2006).
[Crossref]

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

Opt. Commun. (2)

Y. Baykal, “Coherence length in non-Kolmogorov satellite links,” Opt. Commun. 308, 105–108 (2013).
[Crossref]

C. Zhai, L. Tan, S. Yu, Y. Cao, and J. Ma, “Fiber coupling efficiency in non-Kolmogorov satellite links,” Opt. Commun. 336, 93–97 (2015).
[Crossref]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

T. Elperin, N. Kleeorin, and I. Rogachevskii, “Isotropic and anisotropic spectra of passive scalar fluctuations in turbulent fluid flow,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 53(4), 3431–3441 (1996).
[Crossref] [PubMed]

Proc. SPIE (8)

R. R. Beland, “Some aspects of propagation through weak isotropic non-Kolmogorov turbulence,” Proc. SPIE 2375, 1111–1126 (1995).

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for laser beam propagation through Non-Kolmogorov turbulence,” Proc. SPIE 6457, 026003 (2007).

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through Non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).

M. S. Belen’kii, E. Cuellar, K. A. Hughes, and V. A. Rye, “Experimental study of spatial structure of turbulence at Maui Space Surveillance Site (MSSS),” Proc. SPIE 6304, 63040U (2006).

M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).

D. T. Kyrazis, J. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2110, 43–55 (1994).

G. Wang, “A new random-phase-screen time series simulation algorithm for dynamically atmospheric turbulence wave-front generator,” Proc. SPIE 6027, 602716 (2006).

Sov. Phys. JETP (1)

S. S. Moiseev and O. G. Chkhetiani, “Helical scaling in turbulence,” Sov. Phys. JETP 83(1), 192–198 (1996).

Other (2)

J. A. Buck, Fundamentals of Optical Fibers (Wiley, 1995).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE Optical Engineering, 2005).

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Figures (8)

Fig. 1
Fig. 1 Maximum coupling efficiency for a collimated beam (Θ0 = 1) as a function of Λ0 for various values of α.
Fig. 2
Fig. 2 Optimum design parameter for a collimated beam (Θ0 = 1) as a function of Λ0 for various values of α.
Fig. 3
Fig. 3 Maximum coupling efficiency for a convergent beam (Θ0 = 0.1) as a function of Λ0 for various values of α.
Fig. 4
Fig. 4 Optimum design parameter for a convergent beam (Θ0 = 0.1) as a function of Λ0 for various values of α.
Fig. 5
Fig. 5 Maximum coupling efficiency for a divergent beam (Θ0 = 10) as a function of Λ0 for various values of α.
Fig. 6
Fig. 6 Optimum design parameter for a divergent beam (Θ0 = 10) as a function of Λ0 for various values of α.
Fig. 7
Fig. 7 Optimum design parameter as a function of Λ0 for various values of Θ0, where α = 11/3.
Fig. 8
Fig. 8 Coherence length as a function of Λ0 for various values of Θ0, where α = 11/3.

Equations (36)

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Φ n ( κ,α )=A( α ) C ˜ n 2 exp( κ 2 / κ m 2 ) ( κ 2 + κ 0 2 ) α/2 , 0κ< , 3<α<4,
Φ n ( κ,α )=h( α ) exp( κ 2 / κ m 2 ) ( κ 2 + κ 0 2 ) α/2 , 0κ< , 3<α<4,
h( α )= Γ( α ) ( k/L ) α/211/6 C n 2 8 π 2 Γ( 10.5α ) [ Γ( 0.5α ) ] 2 sin( 0.25πα ) .
η= P c P a = | A U i ( r ) U m ( r )dr | 2 A | U i ( r ) | 2 dr ,
η= 1 P a A Γ i ( r 1 , r 2 ) U m ( r 1 ) U m ( r 2 )d r 1 d r 2 ,
U m ( r )= k W m 2π f exp[ ( k W m 2f ) 2 r 2 ],
Γ i ( r 1 , r 2 )= Γ i 0 ( r 1 , r 2 )exp 4 π 2 k 2 L 0 1 0 κ Φ n ( κ ) × { 1exp( ΛL κ 2 ξ 2 k ) J 0 [ | ( 1 Θ ¯ ξ )p2iΛξr |κ ] }dκdξ ,
Γ i 0 ( r 1 , r 2 )= W 0 2 W 2 exp( 2 r 2 W 2 ρ 2 2 W 2 i k F pr ),
Θ=1+ L F = Θ 0 Θ 0 2 + Λ 0 2 , Λ= 2L k W 2 = Λ 0 Θ 0 2 + Λ 0 2 ,
Γ i ( r 1 , r 2 ,α )= Γ i 0 ( r 1 , r 2 )exp 4 π 2 k 2 L 0 1 0 κ Φ n ( κ,α ) × { 1exp( ΛL κ 2 ξ 2 k ) J 0 [ | ( 1 Θ ¯ ξ )p2iΛξr |κ ] }dκdξ .
Γ i ( ρ,α )= W 0 2 W 2 exp{ 1 4 Λ( k ρ 2 L )4 π 2 k 2 L 0 1 0 κ Φ n ( κ,α ) ×[ 1exp( ΛL κ 2 ξ 2 k ) J 0 ( | 1 Θ ¯ ξ |ρκ ) ]dκdξ } .
Γ i ( ρ,α )= W 0 2 W LT 2 exp [ 1 4 Λ e ( k ρ 2 L )4 π 2 k 2 L 0 1 0 κ Φ n ( κ,α ) dκdξ +4 π 2 k 2 L 0 1 0 κ Φ n ( κ,α ) exp( Λ L e κ 2 ξ 2 k ) J 0 ( | 1 Θ ¯ e ξ |ρκ )dκdξ ],
Θ e =1+ L F LT = Θ2qΛ/3 1+4qΛ/3 , Λ e = 2L k W LT 2 = Λ 1+4qΛ/3 ,
ρ p ( α )= [ 2 3α h( α ) π 2 k 2 L Γ( 1α/2 ) Γ( α/2 ) ] 1/( α2 ) .
U( a;c;z )= 1 Γ( a ) 0 e zt t a1 ( 1+t ) ca1 dt, a>0 , Re( z )>0 ,
J p ( x )= n=0 ( 1 ) n ( x/2 ) 2n+p n!Γ( n+p+1 ) , | x |< ,
Γ i ( ρ,α )= W 0 2 W LT 2 exp [ 1 4 Λ e ( k ρ 2 L )2 π 2 k 2 h( α )L κ 0 2α U( 1;2 α 2 ; κ 0 2 κ m 2 ) +2 π 2 k 2 h( α )L κ 0 2α 0 1 n=0 ( 1 ) n ( | 1 Θ ¯ e ξ |ρ κ 0 /2 ) 2n n! U( n+1;n+2 α 2 ; κ 0 2 κ m 2 + Λ L e ξ 2 κ 0 2 k )dξ ].
l 0 2 L 0 2 + 1 1+4qΛ/3 2 π 2 W 0 2 ξ 2 ( 1+ Θ 0 2 Λ 0 2 ) L 0 2 1,
U( a;c;z ) Γ( 1c ) Γ( 1+ac ) + Γ( c1 ) Γ( a ) z 1c , | z |1 ,
I p ( x )= n=0 ( x/2 ) 2n+p n!Γ( n+p+1 ) , | x |< ,
F 1 1 ( a;c;z )= n=0 ( a ) n ( c ) n z n n! , | z |< ,
Γ i ( ρ,α )= W 0 2 W LT 2 exp{ 1 4 Λ e ( k ρ 2 L )2 π 2 k 2 h( α )L[ Γ( α 2 1 ) Γ( α 2 ) κ 0 2α + Γ( 1 α 2 ) Γ( 1 ) κ m 2α ] +2 π 2 k 2 h( α )L 0 1 ( Λ L e ξ 2 k + 1 κ m 2 ) α 2 1 F 1 1 [ 1 α 2 ;1; ( 1 Θ ¯ e ξ ) 2 ρ 2 κ m 2 4( 1+Λ L e ξ 2 κ m 2 /k ) ]Γ( 1 α 2 )dξ 2 π 2 k 2 h( α )L κ 0 2α 0 1 I 1 α 2 ( | 1 Θ ¯ e ξ |ρ κ 0 )Γ( 1 α 2 ) ( | 1 Θ ¯ e ξ |ρ κ 0 2 ) α 2 1 dξ }.
F 1 1 ( a;c;z ) Γ( c ) Γ( ca ) z a , Re( z )1 ,
I p ( x ) ( x/2 ) p Γ( 1+p ) , p1,2,3, , z 0 + ,
Γ i ( ρ,α )= W 0 2 W LT 2 exp[ 1 4 Λ e ( k ρ 2 L )+M ρ α2 B ], l 0 ρ L 0 ,
B=2 π 2 k 2 h( α )L[ ( Γ( α 2 1 ) Γ( α 2 ) + Γ( 1 α 2 ) Γ( 2 α 2 ) ) κ 0 2α + Γ( 1 α 2 ) Γ( 1 ) κ m 2α ],
M= 2 3α π 2 k 2 h( α )L Γ( 1 α 2 ) a e Γ( α 2 )( α1 ) ,
a e ={ 1 Θ e α1 1 Θ e , Θ e 0, 1+ | Θ e | α1 1 Θ e , Θ e <0.
P a = 0 D 2 0 2π W 0 2 W LT 2 exp ( 2 r 2 W LT 2 )rdθdr= π W 0 2 2 [ 1exp( D 2 2 W LT 2 ) ],
η= 4 W m 2 λ 2 f 2 W LT 2 [ 1exp( D 2 2 W LT 2 ) ] × 0 D 2 0 D 2 0 2π 0 2π exp [ ( π W m λf ) 2 ( r 1 2 + r 2 2 ) 1 4 Λ e ( k ρ 2 L )+M ρ α2 B ] r 1 r 2 d θ 1 d θ 2 d r 1 d r 2 .
ρ 2 = | r 1 r 2 | 2 = r 1 2 + r 2 2 2 r 1 r 2 cos( θ 1 θ 2 ),
I= 0 2π 0 2π exp{ k Λ e 4L [ r 1 2 + r 2 2 2 r 1 r 2 cos( θ 1 θ 2 ) ]+M [ r 1 2 + r 2 2 2 r 1 r 2 cos( θ 1 θ 2 ) ] α 2 1 }d θ 1 d θ 2 .
I=4π 0 π exp{ k Λ e 4L ( r 1 2 + r 2 2 )[ 1 2 r 1 r 2 r 1 2 + r 2 2 cos( θ d ) ]+M ( r 1 2 + r 2 2 ) α 2 1 [ 1 2 r 1 r 2 r 1 2 + r 2 2 cos( θ d ) ] α 2 1 }d θ d .
I=4π 0 π exp{ k Λ e 4L v[ 1ucos( θ d ) ]+M v α 2 1 [ 1ucos( θ d ) ] α 2 1 }d θ d ,
η= 4 β 2 D 2 exp( B ) π W LT 2 [ 1exp( D 2 2 W LT 2 ) ] 0 1 0 1 exp[ β 2 ( x 1 2 + x 2 2 ) ]F( D 2 4 ( x 1 2 + x 2 2 ), 2 x 1 x 2 x 1 2 + x 2 2 ,α,M ) x 1 x 2 d x 1 d x 2 ,
F( v,u,α,M )= 0 π exp{ k Λ e 4L v[ 1ucos( θ ) ]+M v α 2 1 [ 1ucos( θ ) ] α 2 1 }dθ .

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