Abstract

Propagation of an externally focused or defocused Gaussian beam in a cubically nonlinear material is studied analytically and experimentally. The theoretical analysis is applied to determine the sign and magnitude of n2 for a material by means of a single-beam experiment with a finite nonlinear sample within which propagational diffraction cannot be neglected. Experimental results for a solution of chlorophyll in ethanol are reported. Based on available theory, an average n2 can be defined for a nonlinearity of thermal origin, and this value is found to be in good agreement with experimental results. Finally, the theoretical analysis and the experimental results of a double-beam experiment performed to determine the sign and magnitude of n2 are also presented.

© 1991 Optical Society of America

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  1. R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
    [CrossRef]
  2. P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1008 (1965).
    [CrossRef]
  3. J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
    [CrossRef]
  4. S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Self-focusing and self-trapping of intense light beams in a nonlinear medium,” Sov. Phys. JETP 23, 1025–1033 (1966).
  5. K. E. Reickhoff, “Self-induced divergence of cw laser beams in liquids—a new nonlinear effect in the propagation of light,” Appl. Phys. Lett. 9, 87–88 (1966).
    [CrossRef]
  6. A. G. Litvak, “Self-focusing of powerful light beams by thermal effects,” JETP Lett. 4, 230–233 (1966).
  7. S. A. Akhmanov, D. P. Krindach, A. P. Sukhorukov, and R. V. Khokhlov, “Nonlinear defocusing of laser beams,” JETP Lett. 6, 38–42 (1967).
  8. W. R. Callen, B. G. Huth, and R. H. Pantell, “Optical patterns of thermally self-defocused light,” Appl. Phys. Lett. 11, 103–105 (1967).
    [CrossRef]
  9. W. G. Wagner, H. A. Haus, and J. H. Marburger, “Large-scale self-trapping of optical beams in the paraxial ray approximation,” Phys. Rev. 175, 256–266 (1968).
    [CrossRef]
  10. S. A. Akhmanov, D. R. Krindach, A. V. Migulin, A. P. Sukhorukov, and R. V. Khokhlov, “Thermal self-actions of laser beams,” IEEE J. Quantum Electron. QE-5, 568–575 (1968).
    [CrossRef]
  11. F. W. Dabby and J. R. Whinnery, “Thermal self-focusing of laser beams in lead glasses,” Appl. Phys. Lett. 13, 284–286 (1968).
    [CrossRef]
  12. J. R. Whinnery, D. T. Miller, and F. Dabby, “Thermal convection and spherical aberration distortion of laser beams in low-loss liquids,” IEEE J. Quantum Electron. QE-3, 382–383 (1967).
    [CrossRef]
  13. F. W. Dabby, R. W. Boyko, C. V. Shank, and J. R. Whinnery, “Short time-constant thermal self-defocusing of laser beams,” IEEE J. Quantum Electron. QE-5, 516–520 (1969).
    [CrossRef]
  14. A. E. Kaplan, “External self-focusing of light by a nonlinear layer,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 12, 869–874 (1969) [Radiophys. Quantum Electron. 12, 692–696 (1969)].
  15. D. C. Smith, “Thermal defocusing of CO2laser radiation in gases,” IEEE J. Quantum Electron. QE-5, 600–607 (1969).
    [CrossRef]
  16. F. G. Gebhardt and D. C. Smith, “Self-induced thermal distortion in the near field for a laser beam in a moving medium,” IEEE J. Quantum Electron. QE-7, 63–73 (1971).
    [CrossRef]
  17. S. A. Akhmanov, R. V. Khokhlov, and A. P. Sukhorukov, “Self-focusing, self-defocusing and self-modulation of laser beams,” in Laser Handbook, F. T. Arecchi and E. O. Schulz-Dubois, eds. (American Elsevier, New York, 1972), Vol. 2, pp. 1151–1228.
  18. A. Feldman, D. Horowitz, and R. M. Waxler, “Mechanisms for self-focusing in optical glasses,” IEEE J. Quantum Electron. QE-9, 1054–1061 (1973).
    [CrossRef]
  19. C. Hu and J. R. Whinnery, “New thermo-optical measurement method and a comparison with other methods,” Appl. Opt. 12, 72–79 (1973).
    [CrossRef] [PubMed]
  20. J. R. Whinnery, “Laser measurement of optical absorption in liquids,” Acc. Chem. Res. 7, 225–231 (1974).
    [CrossRef]
  21. J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975).
    [CrossRef]
  22. A. Yariv and P. Yeh, “The application of Gaussian beam formalism to optical propagation in nonlinear media,” Opt. Commun. 27, 295–298 (1978).
    [CrossRef]
  23. A. K. Ghatak and K. Thyagarajan, Contemporary Optics (Plenum, New York, 1978).
    [CrossRef]
  24. A. Yariv, Quantum Electronics (Wiley, New York, 1989).
  25. H.-J. Zhang, J.-H. Dai, P.-Y. Wang, and L.-A. Wu, “Self-focusing and self-trapping in new types of Kerr media with large nonlinearities,” Opt. Lett. 14, 695–696 (1989).
    [CrossRef] [PubMed]
  26. M. Sheik-bahae, A. A. Said, and E. W. Van Stryland, “High-sensitivity, single-beam n2measurements,” Opt. Lett. 14, 955–957 (1989).
    [CrossRef] [PubMed]
  27. M. Sheik-bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, “Simple analysis and geometric optimization of a passive optical limiter based on internal self-action,” Proc. Soc. Photo-Opt. Instrum. Eng. 1105, 146–153 (1989).
  28. J. Hermann and P. Chapple, “External self-focusing in a two-lens system: shift and compression of the focal profile,” J. Mod. Opt. (to be published).
  29. H. H. Lin, A. Korpel, D. Mehrl, and D. R. Anderson, “Nonlinear Chinese tea,” Opt. News 15(12), 55 (1989).
    [CrossRef]
  30. M. Sheik-bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
    [CrossRef]
  31. H. A. Haus, Waves and Fields in Optoelectronics (Prentice Hall, Englewood Cliffs, N.J., 1984).
  32. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972).
  33. R. E. Bolz and G. L. Tuve, Tables for Applied Engineering Science (CRC, Boca Raton, Fla., 1973).
  34. P. P. Banerjee and T. C. Poon, Principles of Applied Optics (Irwin, Boston, Mass., 1991).
  35. R. H. Hardin and F. D. Tappert, “Applications of the split-step Fourier method to the numerical solution of nonlinear and variable-coefficient wave equations,” SIAM (Soc. Ind. Appl. Math.) Rev. Chron. 15, 423 (1973).
  36. A. Korpel, K. E. Lonngren, P. P. Banerjee, H. K. Sim, and M. R. Chatterjee, “Split-step-type angular plane-wave spectrum method for the study of self-refractive effects in nonlinear wave propagation,” J. Opt. Soc. Am. B 3, 885–890 (1986).
    [CrossRef]

1990 (1)

M. Sheik-bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

1989 (4)

H.-J. Zhang, J.-H. Dai, P.-Y. Wang, and L.-A. Wu, “Self-focusing and self-trapping in new types of Kerr media with large nonlinearities,” Opt. Lett. 14, 695–696 (1989).
[CrossRef] [PubMed]

M. Sheik-bahae, A. A. Said, and E. W. Van Stryland, “High-sensitivity, single-beam n2measurements,” Opt. Lett. 14, 955–957 (1989).
[CrossRef] [PubMed]

M. Sheik-bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, “Simple analysis and geometric optimization of a passive optical limiter based on internal self-action,” Proc. Soc. Photo-Opt. Instrum. Eng. 1105, 146–153 (1989).

H. H. Lin, A. Korpel, D. Mehrl, and D. R. Anderson, “Nonlinear Chinese tea,” Opt. News 15(12), 55 (1989).
[CrossRef]

1986 (1)

1978 (1)

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

1975 (1)

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

1974 (1)

J. R. Whinnery, “Laser measurement of optical absorption in liquids,” Acc. Chem. Res. 7, 225–231 (1974).
[CrossRef]

1973 (3)

R. H. Hardin and F. D. Tappert, “Applications of the split-step Fourier method to the numerical solution of nonlinear and variable-coefficient wave equations,” SIAM (Soc. Ind. Appl. Math.) Rev. Chron. 15, 423 (1973).

A. Feldman, D. Horowitz, and R. M. Waxler, “Mechanisms for self-focusing in optical glasses,” IEEE J. Quantum Electron. QE-9, 1054–1061 (1973).
[CrossRef]

C. Hu and J. R. Whinnery, “New thermo-optical measurement method and a comparison with other methods,” Appl. Opt. 12, 72–79 (1973).
[CrossRef] [PubMed]

1971 (1)

F. G. Gebhardt and D. C. Smith, “Self-induced thermal distortion in the near field for a laser beam in a moving medium,” IEEE J. Quantum Electron. QE-7, 63–73 (1971).
[CrossRef]

1969 (3)

F. W. Dabby, R. W. Boyko, C. V. Shank, and J. R. Whinnery, “Short time-constant thermal self-defocusing of laser beams,” IEEE J. Quantum Electron. QE-5, 516–520 (1969).
[CrossRef]

A. E. Kaplan, “External self-focusing of light by a nonlinear layer,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 12, 869–874 (1969) [Radiophys. Quantum Electron. 12, 692–696 (1969)].

D. C. Smith, “Thermal defocusing of CO2laser radiation in gases,” IEEE J. Quantum Electron. QE-5, 600–607 (1969).
[CrossRef]

1968 (3)

W. G. Wagner, H. A. Haus, and J. H. Marburger, “Large-scale self-trapping of optical beams in the paraxial ray approximation,” Phys. Rev. 175, 256–266 (1968).
[CrossRef]

S. A. Akhmanov, D. R. Krindach, A. V. Migulin, A. P. Sukhorukov, and R. V. Khokhlov, “Thermal self-actions of laser beams,” IEEE J. Quantum Electron. QE-5, 568–575 (1968).
[CrossRef]

F. W. Dabby and J. R. Whinnery, “Thermal self-focusing of laser beams in lead glasses,” Appl. Phys. Lett. 13, 284–286 (1968).
[CrossRef]

1967 (3)

J. R. Whinnery, D. T. Miller, and F. Dabby, “Thermal convection and spherical aberration distortion of laser beams in low-loss liquids,” IEEE J. Quantum Electron. QE-3, 382–383 (1967).
[CrossRef]

S. A. Akhmanov, D. P. Krindach, A. P. Sukhorukov, and R. V. Khokhlov, “Nonlinear defocusing of laser beams,” JETP Lett. 6, 38–42 (1967).

W. R. Callen, B. G. Huth, and R. H. Pantell, “Optical patterns of thermally self-defocused light,” Appl. Phys. Lett. 11, 103–105 (1967).
[CrossRef]

1966 (3)

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Self-focusing and self-trapping of intense light beams in a nonlinear medium,” Sov. Phys. JETP 23, 1025–1033 (1966).

K. E. Reickhoff, “Self-induced divergence of cw laser beams in liquids—a new nonlinear effect in the propagation of light,” Appl. Phys. Lett. 9, 87–88 (1966).
[CrossRef]

A. G. Litvak, “Self-focusing of powerful light beams by thermal effects,” JETP Lett. 4, 230–233 (1966).

1965 (2)

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

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

1964 (1)

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

Akhmanov, S. A.

S. A. Akhmanov, D. R. Krindach, A. V. Migulin, A. P. Sukhorukov, and R. V. Khokhlov, “Thermal self-actions of laser beams,” IEEE J. Quantum Electron. QE-5, 568–575 (1968).
[CrossRef]

S. A. Akhmanov, D. P. Krindach, A. P. Sukhorukov, and R. V. Khokhlov, “Nonlinear defocusing of laser beams,” JETP Lett. 6, 38–42 (1967).

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Self-focusing and self-trapping of intense light beams in a nonlinear medium,” Sov. Phys. JETP 23, 1025–1033 (1966).

S. A. Akhmanov, R. V. Khokhlov, and A. P. Sukhorukov, “Self-focusing, self-defocusing and self-modulation of laser beams,” in Laser Handbook, F. T. Arecchi and E. O. Schulz-Dubois, eds. (American Elsevier, New York, 1972), Vol. 2, pp. 1151–1228.

Anderson, D. R.

H. H. Lin, A. Korpel, D. Mehrl, and D. R. Anderson, “Nonlinear Chinese tea,” Opt. News 15(12), 55 (1989).
[CrossRef]

Banerjee, P. P.

Bolz, R. E.

R. E. Bolz and G. L. Tuve, Tables for Applied Engineering Science (CRC, Boca Raton, Fla., 1973).

Boyko, R. W.

F. W. Dabby, R. W. Boyko, C. V. Shank, and J. R. Whinnery, “Short time-constant thermal self-defocusing of laser beams,” IEEE J. Quantum Electron. QE-5, 516–520 (1969).
[CrossRef]

Callen, W. R.

W. R. Callen, B. G. Huth, and R. H. Pantell, “Optical patterns of thermally self-defocused light,” Appl. Phys. Lett. 11, 103–105 (1967).
[CrossRef]

Chapple, P.

J. Hermann and P. Chapple, “External self-focusing in a two-lens system: shift and compression of the focal profile,” J. Mod. Opt. (to be published).

Chatterjee, M. R.

Chiao, R. Y.

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

Dabby, F.

J. R. Whinnery, D. T. Miller, and F. Dabby, “Thermal convection and spherical aberration distortion of laser beams in low-loss liquids,” IEEE J. Quantum Electron. QE-3, 382–383 (1967).
[CrossRef]

Dabby, F. W.

F. W. Dabby, R. W. Boyko, C. V. Shank, and J. R. Whinnery, “Short time-constant thermal self-defocusing of laser beams,” IEEE J. Quantum Electron. QE-5, 516–520 (1969).
[CrossRef]

F. W. Dabby and J. R. Whinnery, “Thermal self-focusing of laser beams in lead glasses,” Appl. Phys. Lett. 13, 284–286 (1968).
[CrossRef]

Dai, J.-H.

Feldman, A.

A. Feldman, D. Horowitz, and R. M. Waxler, “Mechanisms for self-focusing in optical glasses,” IEEE J. Quantum Electron. QE-9, 1054–1061 (1973).
[CrossRef]

Garmire, E.

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

Gebhardt, F. G.

F. G. Gebhardt and D. C. Smith, “Self-induced thermal distortion in the near field for a laser beam in a moving medium,” IEEE J. Quantum Electron. QE-7, 63–73 (1971).
[CrossRef]

Ghatak, A. K.

A. K. Ghatak and K. Thyagarajan, Contemporary Optics (Plenum, New York, 1978).
[CrossRef]

Gordon, J. P.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Hagan, D. J.

M. Sheik-bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

M. Sheik-bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, “Simple analysis and geometric optimization of a passive optical limiter based on internal self-action,” Proc. Soc. Photo-Opt. Instrum. Eng. 1105, 146–153 (1989).

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 equations,” SIAM (Soc. Ind. Appl. Math.) Rev. Chron. 15, 423 (1973).

Haus, H. A.

W. G. Wagner, H. A. Haus, and J. H. Marburger, “Large-scale self-trapping of optical beams in the paraxial ray approximation,” Phys. Rev. 175, 256–266 (1968).
[CrossRef]

H. A. Haus, Waves and Fields in Optoelectronics (Prentice Hall, Englewood Cliffs, N.J., 1984).

Hermann, J.

J. Hermann and P. Chapple, “External self-focusing in a two-lens system: shift and compression of the focal profile,” J. Mod. Opt. (to be published).

Horowitz, D.

A. Feldman, D. Horowitz, and R. M. Waxler, “Mechanisms for self-focusing in optical glasses,” IEEE J. Quantum Electron. QE-9, 1054–1061 (1973).
[CrossRef]

Hu, C.

Huth, B. G.

W. R. Callen, B. G. Huth, and R. H. Pantell, “Optical patterns of thermally self-defocused light,” Appl. Phys. Lett. 11, 103–105 (1967).
[CrossRef]

Kaplan, A. E.

A. E. Kaplan, “External self-focusing of light by a nonlinear layer,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 12, 869–874 (1969) [Radiophys. Quantum Electron. 12, 692–696 (1969)].

Kelley, P. L.

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

Khokhlov, R. V.

S. A. Akhmanov, D. R. Krindach, A. V. Migulin, A. P. Sukhorukov, and R. V. Khokhlov, “Thermal self-actions of laser beams,” IEEE J. Quantum Electron. QE-5, 568–575 (1968).
[CrossRef]

S. A. Akhmanov, D. P. Krindach, A. P. Sukhorukov, and R. V. Khokhlov, “Nonlinear defocusing of laser beams,” JETP Lett. 6, 38–42 (1967).

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Self-focusing and self-trapping of intense light beams in a nonlinear medium,” Sov. Phys. JETP 23, 1025–1033 (1966).

S. A. Akhmanov, R. V. Khokhlov, and A. P. Sukhorukov, “Self-focusing, self-defocusing and self-modulation of laser beams,” in Laser Handbook, F. T. Arecchi and E. O. Schulz-Dubois, eds. (American Elsevier, New York, 1972), Vol. 2, pp. 1151–1228.

Korpel, A.

Krindach, D. P.

S. A. Akhmanov, D. P. Krindach, A. P. Sukhorukov, and R. V. Khokhlov, “Nonlinear defocusing of laser beams,” JETP Lett. 6, 38–42 (1967).

Krindach, D. R.

S. A. Akhmanov, D. R. Krindach, A. V. Migulin, A. P. Sukhorukov, and R. V. Khokhlov, “Thermal self-actions of laser beams,” IEEE J. Quantum Electron. QE-5, 568–575 (1968).
[CrossRef]

Leite, R. C. C.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Lin, H. H.

H. H. Lin, A. Korpel, D. Mehrl, and D. R. Anderson, “Nonlinear Chinese tea,” Opt. News 15(12), 55 (1989).
[CrossRef]

Litvak, A. G.

A. G. Litvak, “Self-focusing of powerful light beams by thermal effects,” JETP Lett. 4, 230–233 (1966).

Lonngren, K. E.

Marburger, J. H.

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

W. G. Wagner, H. A. Haus, and J. H. Marburger, “Large-scale self-trapping of optical beams in the paraxial ray approximation,” Phys. Rev. 175, 256–266 (1968).
[CrossRef]

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972).

Mehrl, D.

H. H. Lin, A. Korpel, D. Mehrl, and D. R. Anderson, “Nonlinear Chinese tea,” Opt. News 15(12), 55 (1989).
[CrossRef]

Migulin, A. V.

S. A. Akhmanov, D. R. Krindach, A. V. Migulin, A. P. Sukhorukov, and R. V. Khokhlov, “Thermal self-actions of laser beams,” IEEE J. Quantum Electron. QE-5, 568–575 (1968).
[CrossRef]

Miller, D. T.

J. R. Whinnery, D. T. Miller, and F. Dabby, “Thermal convection and spherical aberration distortion of laser beams in low-loss liquids,” IEEE J. Quantum Electron. QE-3, 382–383 (1967).
[CrossRef]

Moore, R. S.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Pantell, R. H.

W. R. Callen, B. G. Huth, and R. H. Pantell, “Optical patterns of thermally self-defocused light,” Appl. Phys. Lett. 11, 103–105 (1967).
[CrossRef]

Poon, T. C.

P. P. Banerjee and T. C. Poon, Principles of Applied Optics (Irwin, Boston, Mass., 1991).

Porto, S. P. S.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Reickhoff, K. E.

K. E. Reickhoff, “Self-induced divergence of cw laser beams in liquids—a new nonlinear effect in the propagation of light,” Appl. Phys. Lett. 9, 87–88 (1966).
[CrossRef]

Said, A. A.

M. Sheik-bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

M. Sheik-bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, “Simple analysis and geometric optimization of a passive optical limiter based on internal self-action,” Proc. Soc. Photo-Opt. Instrum. Eng. 1105, 146–153 (1989).

M. Sheik-bahae, A. A. Said, and E. W. Van Stryland, “High-sensitivity, single-beam n2measurements,” Opt. Lett. 14, 955–957 (1989).
[CrossRef] [PubMed]

Shank, C. V.

F. W. Dabby, R. W. Boyko, C. V. Shank, and J. R. Whinnery, “Short time-constant thermal self-defocusing of laser beams,” IEEE J. Quantum Electron. QE-5, 516–520 (1969).
[CrossRef]

Sheik-bahae, M.

M. Sheik-bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

M. Sheik-bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, “Simple analysis and geometric optimization of a passive optical limiter based on internal self-action,” Proc. Soc. Photo-Opt. Instrum. Eng. 1105, 146–153 (1989).

M. Sheik-bahae, A. A. Said, and E. W. Van Stryland, “High-sensitivity, single-beam n2measurements,” Opt. Lett. 14, 955–957 (1989).
[CrossRef] [PubMed]

Sim, H. K.

Smith, D. C.

F. G. Gebhardt and D. C. Smith, “Self-induced thermal distortion in the near field for a laser beam in a moving medium,” IEEE J. Quantum Electron. QE-7, 63–73 (1971).
[CrossRef]

D. C. Smith, “Thermal defocusing of CO2laser radiation in gases,” IEEE J. Quantum Electron. QE-5, 600–607 (1969).
[CrossRef]

Soileau, M. J.

M. Sheik-bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, “Simple analysis and geometric optimization of a passive optical limiter based on internal self-action,” Proc. Soc. Photo-Opt. Instrum. Eng. 1105, 146–153 (1989).

Sukhorukov, A. P.

S. A. Akhmanov, D. R. Krindach, A. V. Migulin, A. P. Sukhorukov, and R. V. Khokhlov, “Thermal self-actions of laser beams,” IEEE J. Quantum Electron. QE-5, 568–575 (1968).
[CrossRef]

S. A. Akhmanov, D. P. Krindach, A. P. Sukhorukov, and R. V. Khokhlov, “Nonlinear defocusing of laser beams,” JETP Lett. 6, 38–42 (1967).

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Self-focusing and self-trapping of intense light beams in a nonlinear medium,” Sov. Phys. JETP 23, 1025–1033 (1966).

S. A. Akhmanov, R. V. Khokhlov, and A. P. Sukhorukov, “Self-focusing, self-defocusing and self-modulation of laser beams,” in Laser Handbook, F. T. Arecchi and E. O. Schulz-Dubois, eds. (American Elsevier, New York, 1972), Vol. 2, pp. 1151–1228.

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 equations,” SIAM (Soc. Ind. Appl. Math.) Rev. Chron. 15, 423 (1973).

Thyagarajan, K.

A. K. Ghatak and K. Thyagarajan, Contemporary Optics (Plenum, New York, 1978).
[CrossRef]

Townes, C. H.

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

Tuve, G. L.

R. E. Bolz and G. L. Tuve, Tables for Applied Engineering Science (CRC, Boca Raton, Fla., 1973).

Van Stryland, E. W.

M. Sheik-bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

M. Sheik-bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, “Simple analysis and geometric optimization of a passive optical limiter based on internal self-action,” Proc. Soc. Photo-Opt. Instrum. Eng. 1105, 146–153 (1989).

M. Sheik-bahae, A. A. Said, and E. W. Van Stryland, “High-sensitivity, single-beam n2measurements,” Opt. Lett. 14, 955–957 (1989).
[CrossRef] [PubMed]

Wagner, W. G.

W. G. Wagner, H. A. Haus, and J. H. Marburger, “Large-scale self-trapping of optical beams in the paraxial ray approximation,” Phys. Rev. 175, 256–266 (1968).
[CrossRef]

Wang, P.-Y.

Waxler, R. M.

A. Feldman, D. Horowitz, and R. M. Waxler, “Mechanisms for self-focusing in optical glasses,” IEEE J. Quantum Electron. QE-9, 1054–1061 (1973).
[CrossRef]

Wei, T. H.

M. Sheik-bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

Whinnery, J. R.

J. R. Whinnery, “Laser measurement of optical absorption in liquids,” Acc. Chem. Res. 7, 225–231 (1974).
[CrossRef]

C. Hu and J. R. Whinnery, “New thermo-optical measurement method and a comparison with other methods,” Appl. Opt. 12, 72–79 (1973).
[CrossRef] [PubMed]

F. W. Dabby, R. W. Boyko, C. V. Shank, and J. R. Whinnery, “Short time-constant thermal self-defocusing of laser beams,” IEEE J. Quantum Electron. QE-5, 516–520 (1969).
[CrossRef]

F. W. Dabby and J. R. Whinnery, “Thermal self-focusing of laser beams in lead glasses,” Appl. Phys. Lett. 13, 284–286 (1968).
[CrossRef]

J. R. Whinnery, D. T. Miller, and F. Dabby, “Thermal convection and spherical aberration distortion of laser beams in low-loss liquids,” IEEE J. Quantum Electron. QE-3, 382–383 (1967).
[CrossRef]

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Wu, L.-A.

Yariv, A.

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

A. Yariv, Quantum Electronics (Wiley, New York, 1989).

Yeh, P.

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

Zhang, H.-J.

Acc. Chem. Res. (1)

J. R. Whinnery, “Laser measurement of optical absorption in liquids,” Acc. Chem. Res. 7, 225–231 (1974).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (3)

K. E. Reickhoff, “Self-induced divergence of cw laser beams in liquids—a new nonlinear effect in the propagation of light,” Appl. Phys. Lett. 9, 87–88 (1966).
[CrossRef]

W. R. Callen, B. G. Huth, and R. H. Pantell, “Optical patterns of thermally self-defocused light,” Appl. Phys. Lett. 11, 103–105 (1967).
[CrossRef]

F. W. Dabby and J. R. Whinnery, “Thermal self-focusing of laser beams in lead glasses,” Appl. Phys. Lett. 13, 284–286 (1968).
[CrossRef]

IEEE J. Quantum Electron. (7)

J. R. Whinnery, D. T. Miller, and F. Dabby, “Thermal convection and spherical aberration distortion of laser beams in low-loss liquids,” IEEE J. Quantum Electron. QE-3, 382–383 (1967).
[CrossRef]

F. W. Dabby, R. W. Boyko, C. V. Shank, and J. R. Whinnery, “Short time-constant thermal self-defocusing of laser beams,” IEEE J. Quantum Electron. QE-5, 516–520 (1969).
[CrossRef]

D. C. Smith, “Thermal defocusing of CO2laser radiation in gases,” IEEE J. Quantum Electron. QE-5, 600–607 (1969).
[CrossRef]

F. G. Gebhardt and D. C. Smith, “Self-induced thermal distortion in the near field for a laser beam in a moving medium,” IEEE J. Quantum Electron. QE-7, 63–73 (1971).
[CrossRef]

S. A. Akhmanov, D. R. Krindach, A. V. Migulin, A. P. Sukhorukov, and R. V. Khokhlov, “Thermal self-actions of laser beams,” IEEE J. Quantum Electron. QE-5, 568–575 (1968).
[CrossRef]

A. Feldman, D. Horowitz, and R. M. Waxler, “Mechanisms for self-focusing in optical glasses,” IEEE J. Quantum Electron. QE-9, 1054–1061 (1973).
[CrossRef]

M. Sheik-bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

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

A. E. Kaplan, “External self-focusing of light by a nonlinear layer,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 12, 869–874 (1969) [Radiophys. Quantum Electron. 12, 692–696 (1969)].

J. Appl. Phys. (1)

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

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

JETP Lett. (2)

A. G. Litvak, “Self-focusing of powerful light beams by thermal effects,” JETP Lett. 4, 230–233 (1966).

S. A. Akhmanov, D. P. Krindach, A. P. Sukhorukov, and R. V. Khokhlov, “Nonlinear defocusing of laser beams,” JETP Lett. 6, 38–42 (1967).

Opt. Commun. (1)

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

Opt. Lett. (2)

Opt. News (1)

H. H. Lin, A. Korpel, D. Mehrl, and D. R. Anderson, “Nonlinear Chinese tea,” Opt. News 15(12), 55 (1989).
[CrossRef]

Phys. Rev. (1)

W. G. Wagner, H. A. Haus, and J. H. Marburger, “Large-scale self-trapping of optical beams in the paraxial ray approximation,” Phys. Rev. 175, 256–266 (1968).
[CrossRef]

Phys. Rev. Lett. (2)

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

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

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

M. Sheik-bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, “Simple analysis and geometric optimization of a passive optical limiter based on internal self-action,” Proc. Soc. Photo-Opt. Instrum. Eng. 1105, 146–153 (1989).

Prog. Quantum Electron. (1)

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

SIAM (Soc. Ind. Appl. Math.) Rev. Chron. (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 equations,” SIAM (Soc. Ind. Appl. Math.) Rev. Chron. 15, 423 (1973).

Sov. Phys. JETP (1)

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Self-focusing and self-trapping of intense light beams in a nonlinear medium,” Sov. Phys. JETP 23, 1025–1033 (1966).

Other (8)

S. A. Akhmanov, R. V. Khokhlov, and A. P. Sukhorukov, “Self-focusing, self-defocusing and self-modulation of laser beams,” in Laser Handbook, F. T. Arecchi and E. O. Schulz-Dubois, eds. (American Elsevier, New York, 1972), Vol. 2, pp. 1151–1228.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice Hall, Englewood Cliffs, N.J., 1984).

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972).

R. E. Bolz and G. L. Tuve, Tables for Applied Engineering Science (CRC, Boca Raton, Fla., 1973).

P. P. Banerjee and T. C. Poon, Principles of Applied Optics (Irwin, Boston, Mass., 1991).

A. K. Ghatak and K. Thyagarajan, Contemporary Optics (Plenum, New York, 1978).
[CrossRef]

A. Yariv, Quantum Electronics (Wiley, New York, 1989).

J. Hermann and P. Chapple, “External self-focusing in a two-lens system: shift and compression of the focal profile,” J. Mod. Opt. (to be published).

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Figures (10)

Fig. 1
Fig. 1

(a) Experimental setup for nonlinear refractive-index measurement. (b) Far-field diffractions (i) without the nonlinear sample, (ii) with light focused in front of the sample, (iii) with light focused within the nonlinear sample close to the exit face.

Fig. 2
Fig. 2

Experimental setup for the single-beam experiment. The dotted curves show the path taken by the beam in absence of the medium.

Fig. 3
Fig. 3

Plot of the square of the beamwidth within the nonlinear medium versus the propagation distance within the nonlinear medium with n2 as a parameter (λ0 = 640 nm, f0 = 5 cm, s = 4.5 cm, P = 10 mW, n0 = 1.36, and the laser-beam-intensity diameter is 0.77 mm). Note that the beam waist shifts to the left (right) and that the beamwidth is greater (less) than that in the diffraction-limited case for n2 < 0 (n2 > 0).

Fig. 4
Fig. 4

(a) Plot of the square of the beamwidth at the exit the nonlinear sample versus the lens–sample separation s with n2 as a parameter (λ0 = 640 nm, f0 = 5 cm, zexit = 1 cm, P = 10 mW, n0 = 1.36, and the laser-beam-intensity diameter is 0.77 mm). (b) Plot of the inverse of the radius of curvature of the beam immediately to the left of the exit face of the nonlinear sample with n2 as a parameter (λ0 = 640 nm, f0 = 5 cm, zexit = 1 cm, P = 10 mW, n0 = 1.36, and the laser-beam-intensity diameter is 0.77 mm).

Fig. 5
Fig. 5

Plot of the square of the far-field observation width at a distance of 1 m from the exit face of the sample versus the lens–sample separation distance s with n2 as a parameter. (λ0 = 640 nm, f0 = 5 cm, zexit = 1 cm, P = 10 mW, n0 = 1.36, and the laser-beam-intensity diameter is 0.77 mm).

Fig. 6
Fig. 6

Plot of the square of the far-field observation width at a distance of 1 m from the exit face of the chlorophyll–ethanol sample versus the lens–sample separation distance s0 = 640 nm, f0 = 5 cm, zexit = 1 cm, P = 10 mW, n0 = 1.36, and the laser-beam-intensity diameter is 0.77 mm).

Fig. 7
Fig. 7

Experimental setup for probe-beam measurement of n2.

Fig. 8
Fig. 8

Plot of n2 versus Δ and A2/|ψeL|2, showing the physical as well as the nonphysical solutions.

Fig. 9
Fig. 9

Plot of n2 versus Δ and A2/|ψeL|2, magnified at the first point of intersection (physical solution).

Fig. 10
Fig. 10

Solution of Eq. (5.13) for the experimental average of Δ. Δ = 1.0062, n2 = −3.8 × 10−14 m2/V2.

Equations (48)

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n = n 0 + n 2 ψ e 2 ,
ψ ( x , y , z , t ) = Re [ ψ e ( x , y , z ) exp j ( ω 0 t - n 0 k 0 z ) ] .
d 2 w / d z 2 = λ 0 2 / ( n 0 2 π 2 ) - 8 n 2 η P / ( π n 0 ) w 3 ,
1 / R = ( 1 / w ) d w / d z ,
a 2 ( z ) w 2 ( z ) = ( 2 / π ) η P .
w 0 2 = w 1 2 f 0 2 λ 0 2 / ( π 2 w 1 4 + f 0 2 λ 0 2 )
w 2 ( z ) = w 0 2 { 1 + [ λ 0 ( z - f 0 + s ) / π w 0 2 ] 2 } ,
w 0 2 = w 0 2 [ 1 + λ 0 2 ( f 0 - s ) 2 / π 2 w 0 4 ] ,
( d w / d z ) z = 0 + = - [ ( f 0 - s ) / w 0 ] ( λ 0 2 / n 0 π 2 w 0 2 ) .
w 2 ( z ) = w 0 2 - 2 z d + ( c / w 0 2 ) z 2 ,
c = λ 0 2 / n 0 2 π 2 - b + d 2 , d = λ 0 2 ( f 0 - s ) / n 0 π 2 w 0 2 , b = 8 n 2 η P / π n 0 .
z waist = n 0 ( f 0 - s ) { 1 + [ λ 0 ( f 0 - s ) / π w 0 2 ] 2 } 1 - b π 2 n 0 2 / λ 0 2 + [ λ 0 ( f 0 - s ) / π w 0 2 ] 2 ,
w min 2 = w 0 2 [ 1 + [ λ 0 ( f 0 - s ) / π w 0 2 ] 2 ] 1 + [ ( λ 0 2 / n 0 2 π 2 ) / ( λ 0 2 / n 0 2 π 2 - b ) ] ( λ 0 ( f 0 - s ) / π w 0 2 ) 2 ,
w 2 ( z ) = w 0 2 [ 1 + ( λ 0 2 / n 0 2 - b ) z 2 w 0 4 ] .
w exit 2 = w 0 2 - 2 z exit d + ( c / w 0 2 ) z exit 2 ,
1 / R exit - = [ ( c / w 0 2 ) z exit - d ] / w exit 2 .
1 / R exit + = n 0 / R exit - .
w obs 2 = w exit 2 [ ( 1 + D / R exit + ) 2 + D 2 λ 0 2 / π 2 w exit 4 ] .
a 1 c 2 + a 2 c + a 3 = 0 ,
a 1 = ρ 3 2 + ρ 3 ρ 5 z exit + ρ 6 z exit 2 , a 2 = 2 ρ 2 ρ 3 - ρ 3 w obs 2 - ρ 3 ρ 4 ρ 5 + ρ 2 ρ 5 z exit - 2 ρ 4 ρ 6 z exit , a 3 = ρ 2 2 - ρ 2 w obs 2 - ρ 2 ρ 4 ρ 5 + ρ 4 2 ρ 6 + ρ 1 D 2 , ρ 1 = λ 0 2 / π 2 , s             ρ 2 = w 0 2 - 2 d z exit , ρ 3 = z exit 2 / w 0 2 ,             ρ 4 = d w 0 2 , ρ 5 = 2 D n 0 / w 0 2 ,             ρ 6 = D 2 n 0 2 / w 0 4 .
n = n 0 + ( d n / d T ) Δ T ( r , z ) ,
n = n 0 - 0.12 α P ( d n / d T ) r 2 / π κ w 0 2 ,
n 2 = 0.03 α ( d n / d T ) w 0 2 n 0 / η 0 κ .
φ ( x , y ) = 2 k 0 w ( z 0 ) ( n 0 + n 2 ψ e ( x , y ) 2 ) ,
ψ e ( x , y ) 2 = a 2 ( z 0 ) exp [ - 2 ( x 2 + y 2 ) / w 2 ( z 0 ) ] ,
ψ e L ( x , y , z ) = [ 1 - P ( x , y ) ] a L × exp [ - ( z 2 + y 2 ) / w 1 2 - 2 j k 0 w ( z 0 ) n 0 ] + P ( x , y ) a L exp [ - ( z 2 + y 2 ) / w 1 2 - j φ ( x , y ) ] ,
P ( x , y ) = P ( 0 , y ) = exp [ - y 2 / w 2 ( z 0 ) ] .
ψ e L ( 0 , y , z ) = a L { exp [ - ( z 2 + y 2 ) / w 1 2 ] - exp ( - z 2 / w 1 2 ) exp [ - y 2 / w 2 ( z 0 ) ] + exp ( - z 2 / w 1 2 ) exp [ - y 2 / w 2 ( z 0 ) + j β y 2 / w 2 ( z 0 ) ] × exp [ - j β / w ( z 0 ) ] } ,
β = ( 4 / π ) η P k 0 n 2 .
q z = x + j π w 1 2 / λ 0 ,             q y = x + j π w c 2 ( x = 0 ) / λ 0 ,
w c 2 ( x = 0 ) = w 2 ( z 0 ) / [ 1 - j β / w ( z 0 ) ] .
w 2 ( x ) = w 1 2 [ 1 + ( λ 0 x / π w 1 2 ) 2 ] ,
R ( x ) = x [ 1 + ( π w 1 2 / λ 0 x ) 2 ] ,
ϕ ( x ) = - tan - 1 ( λ 0 x / π w 1 2 ) ,
w c 2 ( x ) = ( λ 0 / Λ ) w 2 ( z 0 ) { 1 + [ Λ X / π w 2 ( z 0 ) ] 2 } ,
R c ( x ) = X { 1 + [ π w 2 ( z 0 ) / Λ X ] 2 } ,
ϕ c ( x ) = { tan - 1 [ β w 2 ( z 0 ) ] - tan - 1 [ Λ X / π w 2 ( z 0 ) ] } / 2 ,
X = x - π β w ( z 0 ) / Λ ,
Λ = λ 0 [ 1 + β 2 / w 2 ( z 0 ) ] .
ψ e L ( x 0 , 0 , 0 ) = a L ( A exp [ - j ϕ 0 ( x 0 ) ] - ( A B ) 1 / 2 × exp { - j [ ϕ 0 ( x 0 ) + ϕ 1 ( x 0 ) ] / 2 } + ( A C ) 1 / 2 × exp { - j [ ϕ 2 ( x 0 ) + ϕ 0 ( x 0 ) / 2 ] } ) ,
A = [ 1 + ( λ 0 x 0 / π w 1 2 ) 2 ] 1 / 2 , B = { 1 + [ λ 0 x 0 / π w 2 ( z 0 ) ] 2 } 1 / 2 , C = ( [ 1 + β 2 / w 2 ( z 0 ) ] / { 1 + [ Λ X / π w 2 ( z 0 ) ] 2 } ) 1 / 2 , ϕ 0 ( x 0 ) = - tan - 1 ( λ 0 x 0 / π w 1 2 ) , ϕ 1 ( x 0 ) = - tan - 1 [ λ 0 x 0 / π w 2 ( z 0 ) ] , ϕ 2 ( x 0 ) = { tan - 1 [ β / w ( z 0 ) ] - tan - 1 [ Λ X / π w 2 ( z 0 ) ] } / 2 + β / w ( z 0 ) .
Δ = A 2 / ψ e L ψ e L * .
Δ q Δ z + q 2 / f ,
ψ e ( x , y , z ) = a ( z ) exp [ - ( x 2 + y 2 ) / w 2 ( z ) ] ,
n [ n 0 + n 2 a 2 ( z ) ] - 2 n 2 a 2 ( z ) ( x 2 + y 2 ) / w 2 ( z ) n 0 - 2 n 2 a 2 ( z ) ( x 2 + y 2 ) / w 2 ( z ) .
f ( z ) = n 0 w 2 ( z ) / 4 n 2 a 2 ( z ) Δ z .
d q / d z = [ 4 n 2 a 2 ( z ) / n 0 w 2 ( z ) ] q 2 + 1.
1 / q = 1 / R - j λ 0 / n 0 π w 2 ,

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