Abstract

The analytical propagation formulae of Hermite-Gaussian (H-G) beams propagating in Kerr media are derived by using the variational approach. The analytical formulae of the self-focusing critical power, the Rayleigh range and the beam quality factor of H-G beams propagating in Kerr media are also derived. It is demonstrated that the ABCD law is valid if a new complex parameter is introduced, which presents a simple method to study the propagation of H-G beams through an optical system in Kerr media. It is shown that, as the beam order increases, the self-focusing critical power of H-G beams increase and H-G beams are less affected by the Kerr nonlinearity. Finally, it is found that the focus point of H-G beams never appears in self-focusing media, and the focal length is not suitable for characterizing the self-focusing effect of H-G beams.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  8. H. Wang, X. L. Ji, H. Zhang, X. Q. Li, and Y. Deng, “Propagation formulae and characteristics of partially coherent laser beams in nonlinear media,” Opt. Lett. 44(4), 743–746 (2019).
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2019 (2)

H. Wang, X. L. Ji, H. Zhang, X. Q. Li, and Y. Deng, “Propagation formulae and characteristics of partially coherent laser beams in nonlinear media,” Opt. Lett. 44(4), 743–746 (2019).
[Crossref] [PubMed]

L. Wang, X. L. Ji, Y. Deng, X. Q. Li, T. Wang, X. L. Fan, and H. Yu, “Self-focusing effect on the characteristics of Airy beams,” Opt. Commun. 441, 190–194 (2019).
[Crossref]

2017 (1)

X. Wang, Q. Wang, J. Yang, and J. Mao, “Scalar and vector Hermite–Gaussian soliton in strong nonlocal media with exponential-decay response,” Opt. Commun. 402, 20–25 (2017).
[Crossref]

2015 (1)

P. Sharma, “Self-focusing of Hermite-Gaussian laser beam with relativistic nonlinearity,” AIP Conf. Proc. 1670(1), 030029 (2015).
[Crossref]

2012 (1)

N. Kant, M. A. Wani, and A. Kumar, “Self-focusing of Hermite-Gaussian laser beam in plasma under plasma density ramp,” Opt. Commun. 285(21–22), 4483–4487 (2012).
[Crossref]

2009 (2)

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]

M. V. Takale, S. T. Navare, S. D. Patil, V. J. Fulari, and M. B. Dongare, “Self-focusing and defocusing of TEM0P Hermite-Gaussian laser beams in collisionless plasma,” Opt. Commun. 282(15), 3157–3162 (2009).
[Crossref]

2007 (1)

2006 (1)

D. Udayakumar, A. J. Kiran, A. V. Adhikari, K. Chandrasekharan, G. Umesh, and H. D. Shashikala, “Third-order nonlinear optical studies of newly synthesized polyoxadiazoles containing 3,4-dialkoxythiophenes,” Chem. Phys. 331(1), 125–130 (2006).
[Crossref]

2003 (1)

Z. Liu, W. Zang, J. Tian, W. Zhou, C. Zhang, and G. Zhang, “Analysis of Z-scan of thick media with high-order nonlinearity by variational approach,” Opt. Commun. 219(1–6), 411–419 (2003).
[Crossref]

2000 (1)

T. Singh, N. S. Saini, and S. S. Kaul, “Dynamics of self-focusing and self-phase modulation of elliptics Gaussian laser beam in a kerr-medium,” Pramana-J. Phys. 55(3), 423–431 (2000).

1993 (1)

1992 (1)

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24(9), 1027–1049 (1992).
[Crossref]

1991 (1)

E. M. Vogel, M. J. Weber, and D. M. Krol, “Nonlinear optical phenomena in glass,” Phys. Chem. Glasses 32(6), 231–254 (1991).

1990 (2)

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–15 (1990).
[Crossref]

F. Cornolti, M. Lucchesi, and B. Zambon, “Elliptic Gaussian beam self focusing in nonlinear media,” Opt. Commun. 75(2), 129–135 (1990).
[Crossref]

1975 (1)

J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975).
[Crossref]

1965 (2)

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

V. I. Talanov, “Self focusing of wave beams in nonlinear media,” JETP Lett. 2(5), 138–140 (1965).

1964 (1)

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13(15), 479–482 (1964).
[Crossref]

Adhikari, A. V.

D. Udayakumar, A. J. Kiran, A. V. Adhikari, K. Chandrasekharan, G. Umesh, and H. D. Shashikala, “Third-order nonlinear optical studies of newly synthesized polyoxadiazoles containing 3,4-dialkoxythiophenes,” Chem. Phys. 331(1), 125–130 (2006).
[Crossref]

Alda, J.

Bernabeu, E.

Cao, Y.

Y. Cao and C. Huang, “Variational investigation of dipole solitons in nonlocal thermal media,” in Proceedings of 2011 International Conference on Electronics and Optoelectronics (IEEE, 2011), 4, pp. 190–192.

Chandrasekharan, K.

D. Udayakumar, A. J. Kiran, A. V. Adhikari, K. Chandrasekharan, G. Umesh, and H. D. Shashikala, “Third-order nonlinear optical studies of newly synthesized polyoxadiazoles containing 3,4-dialkoxythiophenes,” Chem. Phys. 331(1), 125–130 (2006).
[Crossref]

Chiao, R. Y.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13(15), 479–482 (1964).
[Crossref]

Cornolti, F.

F. Cornolti, M. Lucchesi, and B. Zambon, “Elliptic Gaussian beam self focusing in nonlinear media,” Opt. Commun. 75(2), 129–135 (1990).
[Crossref]

Deng, D.

Deng, Y.

H. Wang, X. L. Ji, H. Zhang, X. Q. Li, and Y. Deng, “Propagation formulae and characteristics of partially coherent laser beams in nonlinear media,” Opt. Lett. 44(4), 743–746 (2019).
[Crossref] [PubMed]

L. Wang, X. L. Ji, Y. Deng, X. Q. Li, T. Wang, X. L. Fan, and H. Yu, “Self-focusing effect on the characteristics of Airy beams,” Opt. Commun. 441, 190–194 (2019).
[Crossref]

Dongare, M. B.

M. V. Takale, S. T. Navare, S. D. Patil, V. J. Fulari, and M. B. Dongare, “Self-focusing and defocusing of TEM0P Hermite-Gaussian laser beams in collisionless plasma,” Opt. Commun. 282(15), 3157–3162 (2009).
[Crossref]

Fan, X. L.

L. Wang, X. L. Ji, Y. Deng, X. Q. Li, T. Wang, X. L. Fan, and H. Yu, “Self-focusing effect on the characteristics of Airy beams,” Opt. Commun. 441, 190–194 (2019).
[Crossref]

Fedoruk, M. P.

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]

Fulari, V. J.

M. V. Takale, S. T. Navare, S. D. Patil, V. J. Fulari, and M. B. Dongare, “Self-focusing and defocusing of TEM0P Hermite-Gaussian laser beams in collisionless plasma,” Opt. Commun. 282(15), 3157–3162 (2009).
[Crossref]

Garmire, E.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13(15), 479–482 (1964).
[Crossref]

Guo, Q.

Huang, C.

Y. Cao and C. Huang, “Variational investigation of dipole solitons in nonlocal thermal media,” in Proceedings of 2011 International Conference on Electronics and Optoelectronics (IEEE, 2011), 4, pp. 190–192.

Ji, X. L.

H. Wang, X. L. Ji, H. Zhang, X. Q. Li, and Y. Deng, “Propagation formulae and characteristics of partially coherent laser beams in nonlinear media,” Opt. Lett. 44(4), 743–746 (2019).
[Crossref] [PubMed]

L. Wang, X. L. Ji, Y. Deng, X. Q. Li, T. Wang, X. L. Fan, and H. Yu, “Self-focusing effect on the characteristics of Airy beams,” Opt. Commun. 441, 190–194 (2019).
[Crossref]

Kant, N.

N. Kant, M. A. Wani, and A. Kumar, “Self-focusing of Hermite-Gaussian laser beam in plasma under plasma density ramp,” Opt. Commun. 285(21–22), 4483–4487 (2012).
[Crossref]

Kaul, S. S.

T. Singh, N. S. Saini, and S. S. Kaul, “Dynamics of self-focusing and self-phase modulation of elliptics Gaussian laser beam in a kerr-medium,” Pramana-J. Phys. 55(3), 423–431 (2000).

Kelley, P. L.

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

Kiran, A. J.

D. Udayakumar, A. J. Kiran, A. V. Adhikari, K. Chandrasekharan, G. Umesh, and H. D. Shashikala, “Third-order nonlinear optical studies of newly synthesized polyoxadiazoles containing 3,4-dialkoxythiophenes,” Chem. Phys. 331(1), 125–130 (2006).
[Crossref]

Krol, D. M.

E. M. Vogel, M. J. Weber, and D. M. Krol, “Nonlinear optical phenomena in glass,” Phys. Chem. Glasses 32(6), 231–254 (1991).

Kumar, A.

N. Kant, M. A. Wani, and A. Kumar, “Self-focusing of Hermite-Gaussian laser beam in plasma under plasma density ramp,” Opt. Commun. 285(21–22), 4483–4487 (2012).
[Crossref]

Lan, S.

Li, X. Q.

H. Wang, X. L. Ji, H. Zhang, X. Q. Li, and Y. Deng, “Propagation formulae and characteristics of partially coherent laser beams in nonlinear media,” Opt. Lett. 44(4), 743–746 (2019).
[Crossref] [PubMed]

L. Wang, X. L. Ji, Y. Deng, X. Q. Li, T. Wang, X. L. Fan, and H. Yu, “Self-focusing effect on the characteristics of Airy beams,” Opt. Commun. 441, 190–194 (2019).
[Crossref]

Liu, Z.

Z. Liu, W. Zang, J. Tian, W. Zhou, C. Zhang, and G. Zhang, “Analysis of Z-scan of thick media with high-order nonlinearity by variational approach,” Opt. Commun. 219(1–6), 411–419 (2003).
[Crossref]

Lucchesi, M.

F. Cornolti, M. Lucchesi, and B. Zambon, “Elliptic Gaussian beam self focusing in nonlinear media,” Opt. Commun. 75(2), 129–135 (1990).
[Crossref]

Mao, J.

X. Wang, Q. Wang, J. Yang, and J. Mao, “Scalar and vector Hermite–Gaussian soliton in strong nonlocal media with exponential-decay response,” Opt. Commun. 402, 20–25 (2017).
[Crossref]

Marburger, J. H.

J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975).
[Crossref]

Navare, S. T.

M. V. Takale, S. T. Navare, S. D. Patil, V. J. Fulari, and M. B. Dongare, “Self-focusing and defocusing of TEM0P Hermite-Gaussian laser beams in collisionless plasma,” Opt. Commun. 282(15), 3157–3162 (2009).
[Crossref]

Patil, S. D.

M. V. Takale, S. T. Navare, S. D. Patil, V. J. Fulari, and M. B. Dongare, “Self-focusing and defocusing of TEM0P Hermite-Gaussian laser beams in collisionless plasma,” Opt. Commun. 282(15), 3157–3162 (2009).
[Crossref]

Porras, M. A.

Rubenchik, A. M.

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]

Saini, N. S.

T. Singh, N. S. Saini, and S. S. Kaul, “Dynamics of self-focusing and self-phase modulation of elliptics Gaussian laser beam in a kerr-medium,” Pramana-J. Phys. 55(3), 423–431 (2000).

Sharma, P.

P. Sharma, “Self-focusing of Hermite-Gaussian laser beam with relativistic nonlinearity,” AIP Conf. Proc. 1670(1), 030029 (2015).
[Crossref]

Shashikala, H. D.

D. Udayakumar, A. J. Kiran, A. V. Adhikari, K. Chandrasekharan, G. Umesh, and H. D. Shashikala, “Third-order nonlinear optical studies of newly synthesized polyoxadiazoles containing 3,4-dialkoxythiophenes,” Chem. Phys. 331(1), 125–130 (2006).
[Crossref]

Siegman, A. E.

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–15 (1990).
[Crossref]

Singh, T.

T. Singh, N. S. Saini, and S. S. Kaul, “Dynamics of self-focusing and self-phase modulation of elliptics Gaussian laser beam in a kerr-medium,” Pramana-J. Phys. 55(3), 423–431 (2000).

Takale, M. V.

M. V. Takale, S. T. Navare, S. D. Patil, V. J. Fulari, and M. B. Dongare, “Self-focusing and defocusing of TEM0P Hermite-Gaussian laser beams in collisionless plasma,” Opt. Commun. 282(15), 3157–3162 (2009).
[Crossref]

Talanov, V. I.

V. I. Talanov, “Self focusing of wave beams in nonlinear media,” JETP Lett. 2(5), 138–140 (1965).

Tian, J.

Z. Liu, W. Zang, J. Tian, W. Zhou, C. Zhang, and G. Zhang, “Analysis of Z-scan of thick media with high-order nonlinearity by variational approach,” Opt. Commun. 219(1–6), 411–419 (2003).
[Crossref]

Townes, C. H.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13(15), 479–482 (1964).
[Crossref]

Turitsyn, S. K.

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]

Udayakumar, D.

D. Udayakumar, A. J. Kiran, A. V. Adhikari, K. Chandrasekharan, G. Umesh, and H. D. Shashikala, “Third-order nonlinear optical studies of newly synthesized polyoxadiazoles containing 3,4-dialkoxythiophenes,” Chem. Phys. 331(1), 125–130 (2006).
[Crossref]

Umesh, G.

D. Udayakumar, A. J. Kiran, A. V. Adhikari, K. Chandrasekharan, G. Umesh, and H. D. Shashikala, “Third-order nonlinear optical studies of newly synthesized polyoxadiazoles containing 3,4-dialkoxythiophenes,” Chem. Phys. 331(1), 125–130 (2006).
[Crossref]

Vogel, E. M.

E. M. Vogel, M. J. Weber, and D. M. Krol, “Nonlinear optical phenomena in glass,” Phys. Chem. Glasses 32(6), 231–254 (1991).

Wang, H.

Wang, L.

L. Wang, X. L. Ji, Y. Deng, X. Q. Li, T. Wang, X. L. Fan, and H. Yu, “Self-focusing effect on the characteristics of Airy beams,” Opt. Commun. 441, 190–194 (2019).
[Crossref]

Wang, Q.

X. Wang, Q. Wang, J. Yang, and J. Mao, “Scalar and vector Hermite–Gaussian soliton in strong nonlocal media with exponential-decay response,” Opt. Commun. 402, 20–25 (2017).
[Crossref]

Wang, T.

L. Wang, X. L. Ji, Y. Deng, X. Q. Li, T. Wang, X. L. Fan, and H. Yu, “Self-focusing effect on the characteristics of Airy beams,” Opt. Commun. 441, 190–194 (2019).
[Crossref]

Wang, X.

X. Wang, Q. Wang, J. Yang, and J. Mao, “Scalar and vector Hermite–Gaussian soliton in strong nonlocal media with exponential-decay response,” Opt. Commun. 402, 20–25 (2017).
[Crossref]

Wani, M. A.

N. Kant, M. A. Wani, and A. Kumar, “Self-focusing of Hermite-Gaussian laser beam in plasma under plasma density ramp,” Opt. Commun. 285(21–22), 4483–4487 (2012).
[Crossref]

Weber, H.

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24(9), 1027–1049 (1992).
[Crossref]

Weber, M. J.

E. M. Vogel, M. J. Weber, and D. M. Krol, “Nonlinear optical phenomena in glass,” Phys. Chem. Glasses 32(6), 231–254 (1991).

Yang, J.

X. Wang, Q. Wang, J. Yang, and J. Mao, “Scalar and vector Hermite–Gaussian soliton in strong nonlocal media with exponential-decay response,” Opt. Commun. 402, 20–25 (2017).
[Crossref]

Yu, H.

L. Wang, X. L. Ji, Y. Deng, X. Q. Li, T. Wang, X. L. Fan, and H. Yu, “Self-focusing effect on the characteristics of Airy beams,” Opt. Commun. 441, 190–194 (2019).
[Crossref]

Zambon, B.

F. Cornolti, M. Lucchesi, and B. Zambon, “Elliptic Gaussian beam self focusing in nonlinear media,” Opt. Commun. 75(2), 129–135 (1990).
[Crossref]

Zang, W.

Z. Liu, W. Zang, J. Tian, W. Zhou, C. Zhang, and G. Zhang, “Analysis of Z-scan of thick media with high-order nonlinearity by variational approach,” Opt. Commun. 219(1–6), 411–419 (2003).
[Crossref]

Zhang, C.

Z. Liu, W. Zang, J. Tian, W. Zhou, C. Zhang, and G. Zhang, “Analysis of Z-scan of thick media with high-order nonlinearity by variational approach,” Opt. Commun. 219(1–6), 411–419 (2003).
[Crossref]

Zhang, G.

Z. Liu, W. Zang, J. Tian, W. Zhou, C. Zhang, and G. Zhang, “Analysis of Z-scan of thick media with high-order nonlinearity by variational approach,” Opt. Commun. 219(1–6), 411–419 (2003).
[Crossref]

Zhang, H.

Zhao, X.

Zhou, W.

Z. Liu, W. Zang, J. Tian, W. Zhou, C. Zhang, and G. Zhang, “Analysis of Z-scan of thick media with high-order nonlinearity by variational approach,” Opt. Commun. 219(1–6), 411–419 (2003).
[Crossref]

AIP Conf. Proc. (1)

P. Sharma, “Self-focusing of Hermite-Gaussian laser beam with relativistic nonlinearity,” AIP Conf. Proc. 1670(1), 030029 (2015).
[Crossref]

Appl. Opt. (1)

Chem. Phys. (1)

D. Udayakumar, A. J. Kiran, A. V. Adhikari, K. Chandrasekharan, G. Umesh, and H. D. Shashikala, “Third-order nonlinear optical studies of newly synthesized polyoxadiazoles containing 3,4-dialkoxythiophenes,” Chem. Phys. 331(1), 125–130 (2006).
[Crossref]

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

JETP Lett. (1)

V. I. Talanov, “Self focusing of wave beams in nonlinear media,” JETP Lett. 2(5), 138–140 (1965).

Opt. Commun. (6)

F. Cornolti, M. Lucchesi, and B. Zambon, “Elliptic Gaussian beam self focusing in nonlinear media,” Opt. Commun. 75(2), 129–135 (1990).
[Crossref]

X. Wang, Q. Wang, J. Yang, and J. Mao, “Scalar and vector Hermite–Gaussian soliton in strong nonlocal media with exponential-decay response,” Opt. Commun. 402, 20–25 (2017).
[Crossref]

Z. Liu, W. Zang, J. Tian, W. Zhou, C. Zhang, and G. Zhang, “Analysis of Z-scan of thick media with high-order nonlinearity by variational approach,” Opt. Commun. 219(1–6), 411–419 (2003).
[Crossref]

N. Kant, M. A. Wani, and A. Kumar, “Self-focusing of Hermite-Gaussian laser beam in plasma under plasma density ramp,” Opt. Commun. 285(21–22), 4483–4487 (2012).
[Crossref]

M. V. Takale, S. T. Navare, S. D. Patil, V. J. Fulari, and M. B. Dongare, “Self-focusing and defocusing of TEM0P Hermite-Gaussian laser beams in collisionless plasma,” Opt. Commun. 282(15), 3157–3162 (2009).
[Crossref]

L. Wang, X. L. Ji, Y. Deng, X. Q. Li, T. Wang, X. L. Fan, and H. Yu, “Self-focusing effect on the characteristics of Airy beams,” Opt. Commun. 441, 190–194 (2019).
[Crossref]

Opt. Lett. (1)

Opt. Quantum Electron. (1)

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24(9), 1027–1049 (1992).
[Crossref]

Phys. Chem. Glasses (1)

E. M. Vogel, M. J. Weber, and D. M. Krol, “Nonlinear optical phenomena in glass,” Phys. Chem. Glasses 32(6), 231–254 (1991).

Phys. Rev. Lett. (3)

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]

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13(15), 479–482 (1964).
[Crossref]

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

Pramana-J. Phys. (1)

T. Singh, N. S. Saini, and S. S. Kaul, “Dynamics of self-focusing and self-phase modulation of elliptics Gaussian laser beam in a kerr-medium,” Pramana-J. Phys. 55(3), 423–431 (2000).

Proc. SPIE (1)

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[Crossref]

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[Crossref]

Other (4)

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Y. Cao and C. Huang, “Variational investigation of dipole solitons in nonlocal thermal media,” in Proceedings of 2011 International Conference on Electronics and Optoelectronics (IEEE, 2011), 4, pp. 190–192.

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A. E. Siegman, Lasers (University Science Books, 1986).

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Figures (10)

Fig. 1
Fig. 1 Beam width with wm(z) in a self-focusing medium versus the propagation distance z, P0 = 37.3 kW. Solid symbols: by Eq. (12); Hollow symbols: by numerical simulation.
Fig. 2
Fig. 2 Self-focusing critical power Pcr versus the beam order m.
Fig. 3
Fig. 3 Relative beam width wm/wm-lin versus the propagation distance z. (a) in a self-focusing medium, P0 = 37.3 kW; (b) in a self-defocusing medium, P0 = 19.3 kW.
Fig. 4
Fig. 4 Curvature radius Rm versus the propagation distance z. (a) in a self-focusing medium, P0 = 37.3 kW; (b) in a self-defocusing medium, P0 = 19.3 kW.
Fig. 5
Fig. 5 Rayleigh range ZR versus the beam order m. (a) in a self-focusing medium, P0 = 3.357 kW; (b) in a self-defocusing medium, P0 = 1.737 kW.
Fig. 6
Fig. 6 Relative Rayleigh range ZR/ZR-lin versus the beam order m. (a) in a self-focusing medium, P0 = 3.357 kW; (b) in a self-defocusing medium, P0 = 1.737 kW.
Fig. 7
Fig. 7 M2-factor versus the beam power P0 for different beam order m. (a) in a self-focusing medium; (b) in a self-defocusing medium.
Fig. 8
Fig. 8 Relative M2-factor versus the beam power P0 for different beam order m. (a) in a self-focusing medium; (b) in a self-defocusing medium.
Fig. 9
Fig. 9 Transverse power flow in a self-focusing medium, P0 = 37.3 kW.
Fig. 10
Fig. 10 Transverse power flow in a self-defocusing medium, P0 = 19.3 kW.

Equations (25)

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1 2k 2 E+i E z +k n 2 | E | 2 n 0 E=0,
E= A mn w 0 w x (z) w y (z) H m ( 2 x w x (z) ) H n ( 2 y w y (z) )exp( x 2 w x 2 (z) y 2 w y 2 (z) ) ×exp[ ik( x 2 2 R mx (z) + y 2 2 R ny (z) φ mn (z) ) ],
L= i 2 ( E E * z E * E z )+ 1 2k ( | E x | 2 + | E y | 2 ) k 2 n 2 n 0 | E | 4 ,
δ Ldxdy dz=0.
L r = + + L dxdy,
L r = 2 m+n m!n! A mn 2 w 0 2 π(m+1/2 )[ k w x 2 (z) 8 R mx 2 (z) d R mx (z) dz + 1 2 w m 2 (z) ( 1 k + k w x 2 (z) 4 R mx 2 (z) ) ] + 2 m+n m!n! A mn 2 w 0 2 π(n+1/2 )[ k w y 2 (z) 8 R ny 2 (z) d R ny (z) dz + 1 2 w n 2 (z) ( 1 k + k w y 2 (z) 4 R ny 2 (z) ) ] 2 m+n m!n! A mn 2 w 0 2 π k 2 d φ mn (z) dz k n 2 A mn 4 w 0 4 8 n 0 w x (z) w y (z) g(m)g(n)π,
δ L r δS(z) = z L r ( S(z)/ z ) L r S(z) =0,
d w x (z) dz = w x (z) R mx (z) ,
k 2 w x 4 (z) R mx 2 (z) ( 1+ d R mx (z) dz ) 2 k + k n 2 P 0 g(m)g(n) 2 n 0 π ( 2 m+n m!n! ) 2 1 ( m+1/2 ) w x (z) w y (z) =0,
d φ mn dz = 2 w 0 2 ( m+n+1 k 2 n 2 P 0 g(m)g(n) 2 n 0 π(m+n+1) ( 2 m+n m!n! ) 2 ) 4 z 2 ( 1 k 2 n 2 P 0 g(m)g(n) 2 n 0 π(m+n+1) ( 2 m+n m!n! ) 2 )+ k 2 w 0 4 ( 1+ z R 0 ) 2 .
d 2 w(z) d z 2 = 4 k 2 w 3 (z) [ 1 k 2 n 2 P 0 g 2 (m) 2 n 0 π(2m+1) ( 2 m m! ) 4 ].
w m (z)= (2m+1) 1/2 [ 4 z 2 η/ ( k 2 w 0 2 ) + w 0 2 (1+z/ R 0 ) 2 ] 1/2 ,
R m (z)= k 2 w 0 4 (1+z/ R 0 ) 2 +4 z 2 η k 2 w 0 4 (1+z/ R 0 )(1/ R 0 )+4zη ,
φ m (z)={ 2m+η k η arctan{ [ 4η+ k 2 w 0 4 / R 0 2 ]z+ k 2 w 0 4 / R 0 2k w 0 2 η } ( 0<η<1 ) 2m+η 2k η ln| [ 4η+ k 2 w 0 4 / R 0 2 ]z+ k 2 w 0 4 / R 0 2k w 0 2 η [ 4η+ k 2 w 0 4 / R 0 2 ]z+ k 2 w 0 4 / R 0 +2k w 0 2 η | ( η<0 ) ,
E( x,y, z j+1 )=exp( i 4 k 0 Δz 2 )exp(is)exp( i 4 k 0 Δz 2 )E(x,y, z j ),
Z R = k w 0 2 2 η 1/2 ,
w m 2 (z) w m 2 (z=0) =1+ z 2 Z R 2 , R m (z)=z+ Z R 2 z ,
θ m = lim z w m (z)/z =2 [ ( 2m+1 )η ] 1/2 / (k w 0 ) ,
M m 2 = w m θ m w 0 θ 0 =(2m+1) η 1/2 .
w m2 = w m1 , 1 R m2 = 1 f + 1 R m1 ,
w m2 2 = w m1 2 ( A+B/ R m1 ) 2 + 4 B 2 (2m+1) 2 η/ ( k 2 w m1 2 ) ,
R m2 = k 2 w m1 4 ( A+B/ R m1 ) 2 / ( 2m+1 ) 2 +4 B 2 η k 2 w m1 4 ( A+B/ R m1 )( C+D/ R m1 )/ ( 2m+1 ) 2 +4BDη ,
1 q m = 1 R m i 2 M m 2 k w m 2 .
1 q m2 = C+D/ q m1 A+B/ q m1 .
S = ω 0 4 μ 0 n 2 [ i( E E * E * E )+2 k 0 n | E | 2 z ^ ],

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