Abstract

The evolution of non-Gaussian and nonspherical high-power laser beams in cubic nonlinear media is described by means of their mean or gross parameters: width, mean curvature radius, and quality factor. The influence of the beam over its own propagation is contained in a new mean parameter that measures the ability of a beam to build its own waveguide. Beam quality and threshold power for self-focusing are connected. The ABCD and invariance laws for modified complex beam parameter and quality factor allow one to transform in one step the mean beam parameters through a sequence of nonlinear propagations, lenses, mirrors, and nonlinear quadratic graded index.

© 1993 Optical Society of America

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  1. R. Y. Chiao, E. Garmire, C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
    [Crossref]
  2. H. Kogelnik, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
    [Crossref]
  3. J. T. Hunt, P. A. Renard, R. G. Nelson, “Focusing properties of an aberrated laser beam,” Appl. Opt. 15, 1458–1464 (1976).
    [Crossref] [PubMed]
  4. A. Yariv, “The application of the Gaussian beam formalism to optical propagation in nonlinear media,” Opt. Commun. 27, 295–298 (1978).
    [Crossref]
  5. P. A. Bélanger, C. Pare, “Self-focusing of Gaussian beams: an alternate derivation,” Appl. Opt. 22, 1293–1295 (1983).
    [Crossref] [PubMed]
  6. P. A. Bélanger, “Beam propagation and the ABCD ray matrices,” Opt. Lett. 16, 196–198 (1991).
    [Crossref] [PubMed]
  7. M. A. Porras, J. Alda, E. Bernabeu, “Complex beam parameter and ABCD law for non-Gaussian and nonspherical light beams,” Appl. Opt. 31, 6389–6402 (1992).
    [Crossref] [PubMed]
  8. M. T. Loy, Y. R. Shen, “Study of self-focusing and small-scale filaments of light in nonlinear media,” IEEE J. Quantum. Electron. QE-9, 409–422 (1973).
    [Crossref]
  9. Y. R. Shen, Principles of Nonlinear Optics (Wiley, New York, 1984), pp. 286–333.
  10. A. M. Goncharenko, Yu. A. Logvin, A. M. Samson, P. S. Shapovalov, S. I. Turovets, “Ermakov Hamiltonian systems in nonlinear optics of elliptic Gaussian beams,” Phys. Lett. A 160, 138–142 (1991).
    [Crossref]
  11. B. R. Suydam, “Effect of the refractive index nonlinearity on the optical quality of high-power laser beams,” IEEE J. Quantum. Electron. QE-11, 225–230 (1975).
    [Crossref]
  12. A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum. Electron. 27, 1146–1148 (1991).
    [Crossref]
  13. M. J. Baastians, “Propagation laws of second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 82, 173–181 (1989).
  14. S. N. Vlasov, V. A. Petrishchev, V. I. Talanov, “Averaged description of wave beams in linear and nonlinear media,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 14, 1353–1363 (1971).
  15. P. L. Kelley, “Self-focusing of optical beam,” Phys. Rev. Lett. 15, 1005–1008 (1965).
    [Crossref]
  16. L. W. Casperson, “Synthesis of Gaussian beam optical systems,” Appl. Opt. 20, 2243–2249 (1981).
    [Crossref] [PubMed]
  17. C. Paré, P. A. Bélanger, “Beam propagation in a linear or nonlinear lens-like medium using ABCD ray matrices: the method of moments,” Opt. Quantum. Electron. 24, S1051–S1070(1992).
    [Crossref]

1992 (2)

C. Paré, P. A. Bélanger, “Beam propagation in a linear or nonlinear lens-like medium using ABCD ray matrices: the method of moments,” Opt. Quantum. Electron. 24, S1051–S1070(1992).
[Crossref]

M. A. Porras, J. Alda, E. Bernabeu, “Complex beam parameter and ABCD law for non-Gaussian and nonspherical light beams,” Appl. Opt. 31, 6389–6402 (1992).
[Crossref] [PubMed]

1991 (3)

P. A. Bélanger, “Beam propagation and the ABCD ray matrices,” Opt. Lett. 16, 196–198 (1991).
[Crossref] [PubMed]

A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum. Electron. 27, 1146–1148 (1991).
[Crossref]

A. M. Goncharenko, Yu. A. Logvin, A. M. Samson, P. S. Shapovalov, S. I. Turovets, “Ermakov Hamiltonian systems in nonlinear optics of elliptic Gaussian beams,” Phys. Lett. A 160, 138–142 (1991).
[Crossref]

1989 (1)

M. J. Baastians, “Propagation laws of second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 82, 173–181 (1989).

1983 (1)

1981 (1)

1978 (1)

A. Yariv, “The application of the Gaussian beam formalism to optical propagation in nonlinear media,” Opt. Commun. 27, 295–298 (1978).
[Crossref]

1976 (1)

1975 (1)

B. R. Suydam, “Effect of the refractive index nonlinearity on the optical quality of high-power laser beams,” IEEE J. Quantum. Electron. QE-11, 225–230 (1975).
[Crossref]

1973 (1)

M. T. Loy, Y. R. Shen, “Study of self-focusing and small-scale filaments of light in nonlinear media,” IEEE J. Quantum. Electron. QE-9, 409–422 (1973).
[Crossref]

1971 (1)

S. N. Vlasov, V. A. Petrishchev, V. I. Talanov, “Averaged description of wave beams in linear and nonlinear media,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 14, 1353–1363 (1971).

1966 (1)

H. Kogelnik, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[Crossref]

1965 (1)

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

1964 (1)

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

Alda, J.

Baastians, M. J.

M. J. Baastians, “Propagation laws of second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 82, 173–181 (1989).

Bélanger, P. A.

C. Paré, P. A. Bélanger, “Beam propagation in a linear or nonlinear lens-like medium using ABCD ray matrices: the method of moments,” Opt. Quantum. Electron. 24, S1051–S1070(1992).
[Crossref]

P. A. Bélanger, “Beam propagation and the ABCD ray matrices,” Opt. Lett. 16, 196–198 (1991).
[Crossref] [PubMed]

P. A. Bélanger, C. Pare, “Self-focusing of Gaussian beams: an alternate derivation,” Appl. Opt. 22, 1293–1295 (1983).
[Crossref] [PubMed]

Bernabeu, E.

Casperson, L. W.

Chiao, R. Y.

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

Garmire, E.

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

Goncharenko, A. M.

A. M. Goncharenko, Yu. A. Logvin, A. M. Samson, P. S. Shapovalov, S. I. Turovets, “Ermakov Hamiltonian systems in nonlinear optics of elliptic Gaussian beams,” Phys. Lett. A 160, 138–142 (1991).
[Crossref]

Hunt, J. T.

Kelley, P. L.

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

Kogelnik, H.

H. Kogelnik, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[Crossref]

Logvin, Yu. A.

A. M. Goncharenko, Yu. A. Logvin, A. M. Samson, P. S. Shapovalov, S. I. Turovets, “Ermakov Hamiltonian systems in nonlinear optics of elliptic Gaussian beams,” Phys. Lett. A 160, 138–142 (1991).
[Crossref]

Loy, M. T.

M. T. Loy, Y. R. Shen, “Study of self-focusing and small-scale filaments of light in nonlinear media,” IEEE J. Quantum. Electron. QE-9, 409–422 (1973).
[Crossref]

Nelson, R. G.

Pare, C.

Paré, C.

C. Paré, P. A. Bélanger, “Beam propagation in a linear or nonlinear lens-like medium using ABCD ray matrices: the method of moments,” Opt. Quantum. Electron. 24, S1051–S1070(1992).
[Crossref]

Petrishchev, V. A.

S. N. Vlasov, V. A. Petrishchev, V. I. Talanov, “Averaged description of wave beams in linear and nonlinear media,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 14, 1353–1363 (1971).

Porras, M. A.

Renard, P. A.

Samson, A. M.

A. M. Goncharenko, Yu. A. Logvin, A. M. Samson, P. S. Shapovalov, S. I. Turovets, “Ermakov Hamiltonian systems in nonlinear optics of elliptic Gaussian beams,” Phys. Lett. A 160, 138–142 (1991).
[Crossref]

Shapovalov, P. S.

A. M. Goncharenko, Yu. A. Logvin, A. M. Samson, P. S. Shapovalov, S. I. Turovets, “Ermakov Hamiltonian systems in nonlinear optics of elliptic Gaussian beams,” Phys. Lett. A 160, 138–142 (1991).
[Crossref]

Shen, Y. R.

M. T. Loy, Y. R. Shen, “Study of self-focusing and small-scale filaments of light in nonlinear media,” IEEE J. Quantum. Electron. QE-9, 409–422 (1973).
[Crossref]

Y. R. Shen, Principles of Nonlinear Optics (Wiley, New York, 1984), pp. 286–333.

Siegman, A. E.

A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum. Electron. 27, 1146–1148 (1991).
[Crossref]

Suydam, B. R.

B. R. Suydam, “Effect of the refractive index nonlinearity on the optical quality of high-power laser beams,” IEEE J. Quantum. Electron. QE-11, 225–230 (1975).
[Crossref]

Talanov, V. I.

S. N. Vlasov, V. A. Petrishchev, V. I. Talanov, “Averaged description of wave beams in linear and nonlinear media,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 14, 1353–1363 (1971).

Townes, C. H.

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

Turovets, S. I.

A. M. Goncharenko, Yu. A. Logvin, A. M. Samson, P. S. Shapovalov, S. I. Turovets, “Ermakov Hamiltonian systems in nonlinear optics of elliptic Gaussian beams,” Phys. Lett. A 160, 138–142 (1991).
[Crossref]

Vlasov, S. N.

S. N. Vlasov, V. A. Petrishchev, V. I. Talanov, “Averaged description of wave beams in linear and nonlinear media,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 14, 1353–1363 (1971).

Yariv, A.

A. Yariv, “The application of the Gaussian beam formalism to optical propagation in nonlinear media,” Opt. Commun. 27, 295–298 (1978).
[Crossref]

Appl. Opt. (4)

IEEE J. Quantum. Electron. (3)

B. R. Suydam, “Effect of the refractive index nonlinearity on the optical quality of high-power laser beams,” IEEE J. Quantum. Electron. QE-11, 225–230 (1975).
[Crossref]

A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum. Electron. 27, 1146–1148 (1991).
[Crossref]

M. T. Loy, Y. R. Shen, “Study of self-focusing and small-scale filaments of light in nonlinear media,” IEEE J. Quantum. Electron. QE-9, 409–422 (1973).
[Crossref]

Izv. Vyssh. Uchebn. Zaved. Radiofiz. (1)

S. N. Vlasov, V. A. Petrishchev, V. I. Talanov, “Averaged description of wave beams in linear and nonlinear media,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 14, 1353–1363 (1971).

Opt. Commun. (1)

A. Yariv, “The application of the Gaussian beam formalism to optical propagation in nonlinear media,” Opt. Commun. 27, 295–298 (1978).
[Crossref]

Opt. Lett. (1)

Opt. Quantum. Electron. (1)

C. Paré, P. A. Bélanger, “Beam propagation in a linear or nonlinear lens-like medium using ABCD ray matrices: the method of moments,” Opt. Quantum. Electron. 24, S1051–S1070(1992).
[Crossref]

Optik (Stuttgart) (1)

M. J. Baastians, “Propagation laws of second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 82, 173–181 (1989).

Phys. Lett. A (1)

A. M. Goncharenko, Yu. A. Logvin, A. M. Samson, P. S. Shapovalov, S. I. Turovets, “Ermakov Hamiltonian systems in nonlinear optics of elliptic Gaussian beams,” Phys. Lett. A 160, 138–142 (1991).
[Crossref]

Phys. Rev. Lett. (2)

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

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

Proc. IEEE (1)

H. Kogelnik, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[Crossref]

Other (1)

Y. R. Shen, Principles of Nonlinear Optics (Wiley, New York, 1984), pp. 286–333.

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Figures (1)

Fig. 1
Fig. 1

(a) γ factors and critical powers Pc = (M4/γ)P0 for the super-Gaussian beams Ψs = exp(−rs/as,). The unit power is the Gaussian critical power P0 (b) Collapse distance zc for the super-Gaussian beams at several powers; zc, is in meters, the unit power is the Gaussian critical power, the wavelength is 632.8 nm, and the initial width is 1 mm.

Tables (1)

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Table 1 Three Invariant Parameters in a Linear or Nonlinear Medium

Equations (35)

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2 i k Ψ z = Δ T Ψ + k 2 ( β Ψ 2 - α 2 r 2 ) Ψ ,
W 2 = 4 I Ψ ( r ) 2 x 2 d x d y = 4 π I 0 Ψ ( r ) 2 r 3 d r ,
I = 2 π 0 Ψ ( r ) 2 r d r
Θ 2 = λ 2 4 π I 0 ϕ ( ρ ) 2 ρ 3 d ρ = - λ 2 π I 0 r Ψ ( r ) [ Ψ * ( r ) + 1 r Ψ * ( r ) ] d r ,
1 R = i λ I W 2 0 [ Ψ ( r ) Ψ * ( r ) - Ψ ( r ) Ψ * ( r ) ] r 2 d r .
d I d z = 0 ,
d W 2 d z = 2 W 2 R ,
d d z ( W 2 R ) = Θ 2 - β J - α 2 W 2 ,
d d z ( Θ 2 - β J ) = - 2 α 2 W 2 R ,
J = 2 π I 0 Ψ ( r ) 4 r d r .
I 2 = I 1 I ,
W 2 2 = W 1 2 + 2 W 1 2 R 1 z + ( Θ 1 2 - β J 1 ) z 2 ,
W 2 2 R 2 = W 1 2 R 1 + ( Θ 1 2 - β J 1 ) z ,
Θ 2 2 - β J 2 = Θ 1 2 - β J 1 ,
β J = β I π W 2 γ ,
γ = 2 π 2 0 Ψ 0 ( b ) 4 b d b
W 2 2 = W 1 2 + λ 2 π 2 W 1 2 ( M 1 4 - β π λ 2 I γ 1 ) z 2 .
W 2 2 = W 1 2 + λ 2 π 2 W 1 2 ( M 1 4 - P P 0 γ 1 ) z 2 .
P c = M 1 4 γ 1 P 0 .
z c = π W 1 2 / λ [ ( P / P 0 ) γ 1 - M 1 4 ] 1 / 2 .
W 2 2 = W 1 2 ( 1 + z R 1 ) 2 + λ 2 π 2 W 1 2 ( M 1 4 - P P 0 γ 1 ) z 2 ,
W 2 2 R 2 = W 1 2 R 1 ( 1 + z R 1 ) + λ 2 π 2 W 1 2 ( M 1 4 - P P 0 γ 1 ) z .
z c = π W 1 2 / λ ± [ ( P / P 0 ) γ 1 - M 1 4 ] 1 / 2 - ( π W 1 2 / λ R 1 ) ,
- π 2 W 1 4 λ 2 R 1 2 < M 1 4 - γ 1 P / P 0 < 0             { R 1 < 0 as the beam collapses , R 1 > 0 as the beam does not collapse .
M 1 4 - γ 1 P / P 0 < - π 2 W 1 4 λ 2 R 1 2 as the beam collapses .
Θ 2 - β J = λ 2 π 2 W 2 ( M 4 - γ P / P 0 ) + W 2 R 2 .
W 2 = W 1 ,             1 R 2 = - 1 f + 1 R 1 ,
W 2 2 = W 1 2 ( A + B R 1 ) 2 + λ 2 B 2 π 2 W 1 2 ( M 1 4 - γ 1 P P 0 ) ,
W 2 2 R 2 = W 2 2 ( A + B R 1 ) ( C + D R 1 ) + λ 2 B D π 2 W 1 2 ( M 1 4 - λ 1 P P 0 ) ,
M 2 4 - γ 2 P P 0 = M 1 4 - γ 1 P P 0 ,
q 2 = A q 1 + B C q 1 + D ( below threshold ) ,
1 q = 1 R - i λ π W 2 ( M 4 - γ P P 0 ) 1 / 2 .
W 2 2 = W 1 2 [ cos ( α z ) + 1 α sin ( α z ) R 1 ] 2 + λ 2 π 2 W 1 2 ( M 1 4 - γ 1 P P 0 ) 1 α 2 sin 2 ( α z ) ,
W 2 2 R 2 = W 1 2 [ cos ( α z ) + 1 α sin ( α z ) R 1 ] × [ - α sin ( α z ) + cos ( α z ) R 1 ] + λ 2 π 2 W 1 2 ( M 1 4 - γ 1 P P 0 ) 1 α sin ( α z ) cos ( α z ) ,
M 2 4 - γ 2 P P 0 = M 1 4 - γ 1 P P 0 .

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