Abstract

The self-focusing effect of annular beams propagating in the atmosphere to assist delivering powerful laser beams from orbit to the ground is studied in detail. It is found the annular beam is compressed more strongly due to the self-focusing effect as the beam obscure ratio increases. On the other hand, the self-focusing effect between annular, flat-topped and Gaussian beams is compared. It is shown that the self-focusing effect on the annular beam is stronger than that on flat-topped and Gaussian beams. However, the threshold critical power values of annular, flat-topped and Gaussian beams should be in sequence from small to large. Furthermore, the expression of the B integral of annular beams propagating from orbit to the ground in the atmosphere is derived, and the fitting equation related to the B integral of annular beams for maximal compression without filamentation is presented.

© 2017 Optical Society of America

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References

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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  6. F. E. S. Vetelino and L. C. Andrews, “Annular Gaussian beams in turbulent media,” Proc. SPIE 5160, 86–97 (2004).
    [Crossref]
  7. Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14(4), 1353–1367 (2006).
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  8. L. Dou, X. Ji, and P. Li, “Propagation of partially coherent annular beams with decentered field in turbulence along a slant path,” Opt. Express 20(8), 8417–8430 (2012).
    [Crossref] [PubMed]
  9. X. Ji, H. Chen, and G. Ji, “Characteristics of annular beams propagating through atmospheric turbulence along a downlink path and an uplink path,” Appl. Phys. B 122(8), 221 (2016).
    [Crossref]
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    [Crossref]
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    [Crossref]
  15. S. Gladysz, J. C. Christou, L. W. Bradford, and L. C. Roberts, “Temporal Variability and Statistics of the Strehl Ratio in Adaptive-Optics,” Publ. Astron. Soc. Pac. 120(872), 1132–1143 (2008).
    [Crossref]
  16. A. E. Siegman, “How to (maybe) measure laser beam quality,” OSA Trends Opt. Phonotic Ser. 17, 184–199 (1998).
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    [Crossref] [PubMed]
  18. S. V. Chekalin and V. P. Kandidov, “From self-focusing light beams to femtosecond laser pulse filamentation,” Phys. Uspekhi 56(2), 123–140 (2013).
    [Crossref]
  19. J. A. Fleck, J. R. Morris, and E. S. Bliss, “Small-scale self-focusing effects in a high-power glass laser amplifier,” IEEE J. Quantum. Electron. 14(5), 353–363 (1978).
    [Crossref]
  20. A. M. Rubenchik, S. K. Turitsyn, and M. P. Fedoruk, “Modulation instability in high power laser amplifiers,” Opt. Express 18(2), 1380–1388 (2010).
    [Crossref] [PubMed]
  21. C. A. Palla, C. Pacheco, and M. E. Carrín, “Production of structured lipids by acidolysis with immobilized Rhizomucor miehei lipases: selection of suitable reaction conditions,” J. Mol. Catal., B Enzym. 76, 106–115 (2012).
    [Crossref]

2016 (2)

H. Deng, X. Ji, X. Li, H. Zhang, X. Wang, and Y. Zhang, “Effect of spatial coherence on laser beam self-focusing from orbit to the ground in the atmosphere,” Opt. Express 24(13), 14429–14437 (2016).
[Crossref] [PubMed]

X. Ji, H. Chen, and G. Ji, “Characteristics of annular beams propagating through atmospheric turbulence along a downlink path and an uplink path,” Appl. Phys. B 122(8), 221 (2016).
[Crossref]

2015 (1)

2013 (1)

S. V. Chekalin and V. P. Kandidov, “From self-focusing light beams to femtosecond laser pulse filamentation,” Phys. Uspekhi 56(2), 123–140 (2013).
[Crossref]

2012 (2)

L. Dou, X. Ji, and P. Li, “Propagation of partially coherent annular beams with decentered field in turbulence along a slant path,” Opt. Express 20(8), 8417–8430 (2012).
[Crossref] [PubMed]

C. A. Palla, C. Pacheco, and M. E. Carrín, “Production of structured lipids by acidolysis with immobilized Rhizomucor miehei lipases: selection of suitable reaction conditions,” J. Mol. Catal., B Enzym. 76, 106–115 (2012).
[Crossref]

2010 (2)

2009 (1)

A. M. Rubenchik, M. P. Fedoruk, and S. K. Turitsyn, “Laser beam self-focusing in the atmosphere,” Phys. Rev. Lett. 102(23), 233902 (2009).
[Crossref] [PubMed]

2008 (1)

S. Gladysz, J. C. Christou, L. W. Bradford, and L. C. Roberts, “Temporal Variability and Statistics of the Strehl Ratio in Adaptive-Optics,” Publ. Astron. Soc. Pac. 120(872), 1132–1143 (2008).
[Crossref]

2006 (1)

2004 (1)

F. E. S. Vetelino and L. C. Andrews, “Annular Gaussian beams in turbulent media,” Proc. SPIE 5160, 86–97 (2004).
[Crossref]

2000 (1)

U. Roth, F. Loewenthal, R. Tommasini, J. E. Balmer, and H. P. Weber, “Compensation of nonlinear self-focusing in high-power lasers,” IEEE J. Quantum Electron. 36(6), 687–691 (2000).
[Crossref]

1998 (1)

A. E. Siegman, “How to (maybe) measure laser beam quality,” OSA Trends Opt. Phonotic Ser. 17, 184–199 (1998).

1987 (1)

1978 (2)

J. T. Hunt, J. A. Glaze, W. W. Simmons, and P. A. Renard, “Suppression of self-focusing through low-pass spatial filtering and relay imaging,” Appl. Opt. 17(13), 2053–2057 (1978).
[Crossref] [PubMed]

J. A. Fleck, J. R. Morris, and E. S. Bliss, “Small-scale self-focusing effects in a high-power glass laser amplifier,” IEEE J. Quantum. Electron. 14(5), 353–363 (1978).
[Crossref]

1974 (1)

S. N. Vlasov, V. A. Petrishev, and V. I. Talanov, “Average description of wave beams in linear and nonlinear media,” Radiophys Quant. El. 14(9), 1062–1070 (1974).
[Crossref]

1973 (1)

R. H. Hardin and F. D. Tappert, “Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equation,” SIAM Rev. Chronicles 15(2), 805–809 (1973).

1965 (1)

P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15(26), 1005–1008 (1965).
[Crossref]

Andrews, L. C.

F. E. S. Vetelino and L. C. Andrews, “Annular Gaussian beams in turbulent media,” Proc. SPIE 5160, 86–97 (2004).
[Crossref]

Balmer, J. E.

U. Roth, F. Loewenthal, R. Tommasini, J. E. Balmer, and H. P. Weber, “Compensation of nonlinear self-focusing in high-power lasers,” IEEE J. Quantum Electron. 36(6), 687–691 (2000).
[Crossref]

Baykal, Y.

Bliss, E. S.

J. A. Fleck, J. R. Morris, and E. S. Bliss, “Small-scale self-focusing effects in a high-power glass laser amplifier,” IEEE J. Quantum. Electron. 14(5), 353–363 (1978).
[Crossref]

Bradford, L. W.

S. Gladysz, J. C. Christou, L. W. Bradford, and L. C. Roberts, “Temporal Variability and Statistics of the Strehl Ratio in Adaptive-Optics,” Publ. Astron. Soc. Pac. 120(872), 1132–1143 (2008).
[Crossref]

Cai, Y.

Carrín, M. E.

C. A. Palla, C. Pacheco, and M. E. Carrín, “Production of structured lipids by acidolysis with immobilized Rhizomucor miehei lipases: selection of suitable reaction conditions,” J. Mol. Catal., B Enzym. 76, 106–115 (2012).
[Crossref]

Chekalin, S. V.

S. V. Chekalin and V. P. Kandidov, “From self-focusing light beams to femtosecond laser pulse filamentation,” Phys. Uspekhi 56(2), 123–140 (2013).
[Crossref]

Chen, H.

X. Ji, H. Chen, and G. Ji, “Characteristics of annular beams propagating through atmospheric turbulence along a downlink path and an uplink path,” Appl. Phys. B 122(8), 221 (2016).
[Crossref]

Christou, J. C.

S. Gladysz, J. C. Christou, L. W. Bradford, and L. C. Roberts, “Temporal Variability and Statistics of the Strehl Ratio in Adaptive-Optics,” Publ. Astron. Soc. Pac. 120(872), 1132–1143 (2008).
[Crossref]

Deng, H.

Dou, L.

Fedoruk, M. P.

A. M. Rubenchik, S. K. Turitsyn, and M. P. Fedoruk, “Modulation instability in high power laser amplifiers,” Opt. Express 18(2), 1380–1388 (2010).
[Crossref] [PubMed]

A. M. Rubenchik, M. P. Fedoruk, and S. K. Turitsyn, “Laser beam self-focusing in the atmosphere,” Phys. Rev. Lett. 102(23), 233902 (2009).
[Crossref] [PubMed]

Fleck, J. A.

J. A. Fleck, J. R. Morris, and E. S. Bliss, “Small-scale self-focusing effects in a high-power glass laser amplifier,” IEEE J. Quantum. Electron. 14(5), 353–363 (1978).
[Crossref]

Gerçekcioglu, H.

Gladysz, S.

S. Gladysz, J. C. Christou, L. W. Bradford, and L. C. Roberts, “Temporal Variability and Statistics of the Strehl Ratio in Adaptive-Optics,” Publ. Astron. Soc. Pac. 120(872), 1132–1143 (2008).
[Crossref]

Glaze, J. A.

Hardin, R. H.

R. H. Hardin and F. D. Tappert, “Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equation,” SIAM Rev. Chronicles 15(2), 805–809 (1973).

He, S.

Hunt, J. T.

Ji, G.

X. Ji, H. Chen, and G. Ji, “Characteristics of annular beams propagating through atmospheric turbulence along a downlink path and an uplink path,” Appl. Phys. B 122(8), 221 (2016).
[Crossref]

Ji, X.

Kandidov, V. P.

S. V. Chekalin and V. P. Kandidov, “From self-focusing light beams to femtosecond laser pulse filamentation,” Phys. Uspekhi 56(2), 123–140 (2013).
[Crossref]

Kelley, P. L.

P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15(26), 1005–1008 (1965).
[Crossref]

Li, P.

Li, X.

Loewenthal, F.

U. Roth, F. Loewenthal, R. Tommasini, J. E. Balmer, and H. P. Weber, “Compensation of nonlinear self-focusing in high-power lasers,” IEEE J. Quantum Electron. 36(6), 687–691 (2000).
[Crossref]

Miller, R. I.

Morris, J. R.

J. A. Fleck, J. R. Morris, and E. S. Bliss, “Small-scale self-focusing effects in a high-power glass laser amplifier,” IEEE J. Quantum. Electron. 14(5), 353–363 (1978).
[Crossref]

Nakiboglu, C.

Pacheco, C.

C. A. Palla, C. Pacheco, and M. E. Carrín, “Production of structured lipids by acidolysis with immobilized Rhizomucor miehei lipases: selection of suitable reaction conditions,” J. Mol. Catal., B Enzym. 76, 106–115 (2012).
[Crossref]

Palla, C. A.

C. A. Palla, C. Pacheco, and M. E. Carrín, “Production of structured lipids by acidolysis with immobilized Rhizomucor miehei lipases: selection of suitable reaction conditions,” J. Mol. Catal., B Enzym. 76, 106–115 (2012).
[Crossref]

Petrishev, V. A.

S. N. Vlasov, V. A. Petrishev, and V. I. Talanov, “Average description of wave beams in linear and nonlinear media,” Radiophys Quant. El. 14(9), 1062–1070 (1974).
[Crossref]

Renard, P. A.

Roberts, L. C.

S. Gladysz, J. C. Christou, L. W. Bradford, and L. C. Roberts, “Temporal Variability and Statistics of the Strehl Ratio in Adaptive-Optics,” Publ. Astron. Soc. Pac. 120(872), 1132–1143 (2008).
[Crossref]

Roberts, T. G.

Roth, U.

U. Roth, F. Loewenthal, R. Tommasini, J. E. Balmer, and H. P. Weber, “Compensation of nonlinear self-focusing in high-power lasers,” IEEE J. Quantum Electron. 36(6), 687–691 (2000).
[Crossref]

Rubenchik, A. M.

A. M. Rubenchik, S. K. Turitsyn, and M. P. Fedoruk, “Modulation instability in high power laser amplifiers,” Opt. Express 18(2), 1380–1388 (2010).
[Crossref] [PubMed]

A. M. Rubenchik, M. P. Fedoruk, and S. K. Turitsyn, “Laser beam self-focusing in the atmosphere,” Phys. Rev. Lett. 102(23), 233902 (2009).
[Crossref] [PubMed]

Siegman, A. E.

A. E. Siegman, “How to (maybe) measure laser beam quality,” OSA Trends Opt. Phonotic Ser. 17, 184–199 (1998).

Simmons, W. W.

Talanov, V. I.

S. N. Vlasov, V. A. Petrishev, and V. I. Talanov, “Average description of wave beams in linear and nonlinear media,” Radiophys Quant. El. 14(9), 1062–1070 (1974).
[Crossref]

Tappert, F. D.

R. H. Hardin and F. D. Tappert, “Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equation,” SIAM Rev. Chronicles 15(2), 805–809 (1973).

Tommasini, R.

U. Roth, F. Loewenthal, R. Tommasini, J. E. Balmer, and H. P. Weber, “Compensation of nonlinear self-focusing in high-power lasers,” IEEE J. Quantum Electron. 36(6), 687–691 (2000).
[Crossref]

Turitsyn, S. K.

A. M. Rubenchik, S. K. Turitsyn, and M. P. Fedoruk, “Modulation instability in high power laser amplifiers,” Opt. Express 18(2), 1380–1388 (2010).
[Crossref] [PubMed]

A. M. Rubenchik, M. P. Fedoruk, and S. K. Turitsyn, “Laser beam self-focusing in the atmosphere,” Phys. Rev. Lett. 102(23), 233902 (2009).
[Crossref] [PubMed]

Vetelino, F. E. S.

F. E. S. Vetelino and L. C. Andrews, “Annular Gaussian beams in turbulent media,” Proc. SPIE 5160, 86–97 (2004).
[Crossref]

Vlasov, S. N.

S. N. Vlasov, V. A. Petrishev, and V. I. Talanov, “Average description of wave beams in linear and nonlinear media,” Radiophys Quant. El. 14(9), 1062–1070 (1974).
[Crossref]

Wang, X.

Weber, H. P.

U. Roth, F. Loewenthal, R. Tommasini, J. E. Balmer, and H. P. Weber, “Compensation of nonlinear self-focusing in high-power lasers,” IEEE J. Quantum Electron. 36(6), 687–691 (2000).
[Crossref]

Zhang, H.

Zhang, Y.

Appl. Opt. (2)

Appl. Phys. B (1)

X. Ji, H. Chen, and G. Ji, “Characteristics of annular beams propagating through atmospheric turbulence along a downlink path and an uplink path,” Appl. Phys. B 122(8), 221 (2016).
[Crossref]

IEEE J. Quantum Electron. (1)

U. Roth, F. Loewenthal, R. Tommasini, J. E. Balmer, and H. P. Weber, “Compensation of nonlinear self-focusing in high-power lasers,” IEEE J. Quantum Electron. 36(6), 687–691 (2000).
[Crossref]

IEEE J. Quantum. Electron. (1)

J. A. Fleck, J. R. Morris, and E. S. Bliss, “Small-scale self-focusing effects in a high-power glass laser amplifier,” IEEE J. Quantum. Electron. 14(5), 353–363 (1978).
[Crossref]

J. Mol. Catal., B Enzym. (1)

C. A. Palla, C. Pacheco, and M. E. Carrín, “Production of structured lipids by acidolysis with immobilized Rhizomucor miehei lipases: selection of suitable reaction conditions,” J. Mol. Catal., B Enzym. 76, 106–115 (2012).
[Crossref]

J. Opt. Soc. Am. A (1)

Opt. Express (4)

Opt. Lett. (1)

OSA Trends Opt. Phonotic Ser. (1)

A. E. Siegman, “How to (maybe) measure laser beam quality,” OSA Trends Opt. Phonotic Ser. 17, 184–199 (1998).

Phys. Rev. Lett. (2)

P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15(26), 1005–1008 (1965).
[Crossref]

A. M. Rubenchik, M. P. Fedoruk, and S. K. Turitsyn, “Laser beam self-focusing in the atmosphere,” Phys. Rev. Lett. 102(23), 233902 (2009).
[Crossref] [PubMed]

Phys. Uspekhi (1)

S. V. Chekalin and V. P. Kandidov, “From self-focusing light beams to femtosecond laser pulse filamentation,” Phys. Uspekhi 56(2), 123–140 (2013).
[Crossref]

Proc. SPIE (1)

F. E. S. Vetelino and L. C. Andrews, “Annular Gaussian beams in turbulent media,” Proc. SPIE 5160, 86–97 (2004).
[Crossref]

Publ. Astron. Soc. Pac. (1)

S. Gladysz, J. C. Christou, L. W. Bradford, and L. C. Roberts, “Temporal Variability and Statistics of the Strehl Ratio in Adaptive-Optics,” Publ. Astron. Soc. Pac. 120(872), 1132–1143 (2008).
[Crossref]

Radiophys Quant. El. (1)

S. N. Vlasov, V. A. Petrishev, and V. I. Talanov, “Average description of wave beams in linear and nonlinear media,” Radiophys Quant. El. 14(9), 1062–1070 (1974).
[Crossref]

SIAM Rev. Chronicles (1)

R. H. Hardin and F. D. Tappert, “Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equation,” SIAM Rev. Chronicles 15(2), 805–809 (1973).

Other (1)

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, 1995), Chap. 2.

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Figures (16)

Fig. 1
Fig. 1 At the initial plane z = F, 2D initial intensity distributions I(x, y = 0, z = F) of annular beams with different values of the obscure ratio ε.
Fig. 2
Fig. 2 At the initial plane z = F, contour lines of 3D intensity of annular beams with different values of the obscure ratio ε.
Fig. 3
Fig. 3 At the initial plane z = F, 2D initial intensity distributions I(x, y = 0, z = F) of annular, flat-topped and Gaussian beams.
Fig. 4
Fig. 4 At the initial plane z = F, contour lines of 3D intensity of annular, flat-topped and Gaussian beams.
Fig. 5
Fig. 5 Intensity distributions I(x, y = 0, z) versus the propagation distance z, P = 50 Pcr, ε = 0.9.
Fig. 6
Fig. 6 2D intensity distributions I(x, y = 0, z) for different values of the obscure ratio ε on the ground, P = 50 Pcr.
Fig. 7
Fig. 7 3D intensity distributions for different values of the obscure ratio ε on the ground in the atmosphere, P = 50 Pcr.
Fig. 8
Fig. 8 For different values of initial power P, (a) maximum intensity Imax, and (b) SR versus obscure ratio ε on the ground.
Fig. 9
Fig. 9 For different values of initial power P, (a) w, and (b) w / wfree versus obscure ratio ε on the ground.
Fig. 10
Fig. 10 For different values of initial power P, (a) w86.5%, and (b) w86.5%/w86.5%free versus obscure ratio ε on the ground.
Fig. 11
Fig. 11 B integral versus obscure ratio ε on the ground.
Fig. 12
Fig. 12 Maximum B integral (Bmax) versus the outer radius w0 and the obscure ratio ε on the ground. (Red dots: numerical simulation results; Curve surface: fitting surface by using Eq. (15)).
Fig. 13
Fig. 13 3D intensity distributions for annular, flat-topped and Gaussian beams on the ground, P = 2 Pcr.
Fig. 14
Fig. 14 Relative intensity distributions I(x, y = 0, z)/I0max for annular, flat-topped and Gaussian beams on the ground, P = 2 Pcr.
Fig. 15
Fig. 15 Beam radius w versus the propagation distance z for annular, flat-topped and Gaussian beams, P = 2 Pcr.
Fig. 16
Fig. 16 Beam radius w on the ground versus P/Pcr for annular, flat-topped and Gaussian beams.

Tables (1)

Tables Icon

Table 1 Values of the coefficients in Eq. (15).

Equations (15)

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2ik A z + 2 A+2 k 2 n 2 n 0 | A | 2 A=0,
n 2 (z)= n 2 (0)exp[(z/h)],
A z =( D ^ + N ^ )A,
A(z=F,r)= A 1 exp[ i C 0 w 0 2 r 2 ] u=1 N (1) u1 N ( N u )[ exp( u r 2 w 0 2 )exp( u r 2 ε w 0 2 ) ],
P= 0 2π dθ 0 | A | 2 rdr=const,
A 1 = P 2π w 0 2 u=1 N v=1 N (1) u+v2 N 2 ( N u )( N v )[ 1+ε 2(u+v) ε 2(u+εv) ε 2(εu+v) ] .
A(z=F,r)= A 2 exp[ (i C 0 ) w 0 2 r 2 ] u=1 N (1) u1 N ( N u )exp( u r 2 w 0 2 ).
A 2 = P 2π w 0 2 u=1 N v=1 N (1) u+v2 N 2 ( N u )( N v )[ 1 2(u+v) ] .
A(z=F,r)= 2P π w 0 2 exp[ (1+i C 0 ) w 0 2 r 2 ].
B=k 0 z 0 I 0 n 2 (z)dz ,
B=k I 0 n 2 (0)h[ 1exp( z 0 h ) ].
A(z=L,r=0)= A 1 ik 2L exp(ikL) u=1 N (1) u1 N ( N u ) ×{ [ u w 0 2 + ik (1L/F) 1 2L ][ u ε w 0 2 + ik (1L/F) 1 2L ] }.
I 0 = A 1 2 ( k 2L ) 2 w 0 4 u=1 N v=1 N (1) u+v2 N 2 ( N u )( N v )[ ( 1ε ) 2 uv ] .
B=k n 2 (0)h[ 1exp( z 0 h ) ] A 1 2 ( k 2L ) 2 w 0 4 u=1 N v=1 N (1) u+v2 N 2 ( N u )( N v )[ ( 1ε ) 2 uv ] .
B max = C+D 01 ε+ F 01 w 0 + F 02 w 0 2 + G 02 ε w 0 1+D 1 ε+ F 1 w 0 + D 2 ε 2 + F 2 w 0 2 + G 2 ε w 0 ,

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