Abstract

The extension of the conventional finite-difference time-domain solution of the full vector Maxwell equations to modeling femtosecond optical-pulse propagation in a nonlinear Kerr medium that exhibits a finite response time is presented. Numerical results are given for nonlinear self-focusing in two space dimensions and time; the technique can be generalized to three space dimensions with adequate computer resources. Comparisons with previously reported and anticipated results are made. Several novel phenomena that are not observed with scalar models of self-focusing and that can be attributed only to the complete solution of the vector Maxwell equations are discussed.

© 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. V. I. Talanov, “Self-focusing of waves in nonlinear media,” JETP Lett. 2, 138–141 (1965) [Zh. Eksp. Teor. Fiz. Pis’ma Red. 2, 218 (1965)].
  3. E. Garmire, R. Y. Chiao, C. H. Townes, “Dynamics of the self-trapping of intense light beams,” Phys. Rev. Lett. 16, 347–349 (1966).
    [CrossRef]
  4. F. Shimizu, “The effect of dispersion on pulse distortion in optical filaments,” IEEE J. Quantum Electron. 8, 851–854 (1972).
    [CrossRef]
  5. J. Reintjes, R. L. Carman, F. Shimizu, “Study of self-focusing and self-phase-modulation in the picosecond-time regime,” Phys. Rev. A. 8, 1486–1503 (1973).
    [CrossRef]
  6. For a comprehensive review of experimental work on self-focusing until the mid 1970’s, see Y. R. Shen, “Self-focusing: experimental,” Prog. Quantum Electron. 4, 1–34 (1975).
    [CrossRef]
  7. J. A. Fleck, P. L. Kelley, “Temporal aspects of the self-focusing of optical beams,” Appl. Phys. Lett. 15, 313–315 (1969).
    [CrossRef]
  8. For a comprehensive review of theoretical work on self-focusing until the mid 1970’s, see J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975).
    [CrossRef]
  9. S. N. Vlasov, “Structure of the field of wave beams with circular polarization near a nonlinear focus in a cubic medium,” Sov. J. Quantum Electron. 17, 1191–1193 (1987).
    [CrossRef]
  10. M. D. Feit, J. A. Fleck, “Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams,” J. Opt. Soc. Am. B 5, 633–640 (1988).
    [CrossRef]
  11. M. Miyagi, S. Nishida, “TM-type soliton in nonlinear self-focusing media,” Proc. IEEE 62, 1284–1285 (1974).
    [CrossRef]
  12. M. Miyagi, S. Nishida, “TM waves in nonlinear self-focusing media,” Radio Sci. 10, 833–838 (1975).
    [CrossRef]
  13. D. Pohl, “Vectorial theory of self-trapped light beams,” Opt. Commun., 2, 305–308 (1970).
    [CrossRef]
  14. K. Hayata, M. Koshiba, “Full vectorial analysis of nonlinear-optical waveguides,” J. Opt. Soc. Am. B 5, 2494–2501 (1988).
    [CrossRef]
  15. D. Pohl, “Self-focusing of TE01and TM01light beams: influence of longitudinal field components,” Phys. Rev. A 5, 1906–1909 (1972).
    [CrossRef]
  16. M. S. Sodha, V. P. Nayyar, V. K. Tripathi, “Asymmetric focusing of a laser beam in TEM01doughnut mode in a nonlinear dielectric,” J. Opt. Soc. Am. 64, 941–943 (1974).
    [CrossRef]
  17. K. Hayata, A. Misawa, M. Koshiba, “Spatial polarization instabilities due to transverse effects in nonlinear guided-wave systems,” J. Opt. Soc. Am. B 7, 1268–1280 (1990).
    [CrossRef]
  18. R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, M. Schneider, “A frequency dependent finite difference time domain formulation for transient propagation in plasmas,” IEEE Trans. Electromagn. Compat. 32, 222 (1990).
    [CrossRef]
  19. C. F. Lee, R. T. Shin, J. A. Kong, “Finite difference method for electromagnetic scattering problems,” in PIER4: Progress in Electromagnetics Research, J. A. Kong, ed. (Elsevier, New York, 1991), Chap. 11.
  20. R. M. Joseph, S. C. Hagness, A. Taflove, “Direct time integration of Maxwell’s equations in linear dispersive media with absorption for scattering and propagation of femtosecond electromagnetic pulses,” Opt. Lett. 16, 1412–1414 (1991).
    [CrossRef] [PubMed]
  21. P. M. Goorjian, A. Taflove, “FD–TD computational modeling of nonlinear electromagnetic phenomena using a nonlinear convolution approach,” presented at the 1991 North American Radio Science Meeting, University of Western Ontario, London, Ontario, Canada; “Direct time integration of Maxwell’s equations in nonlinear dispersive media for propagation and scattering of femtosecond electromagnetic solitons,” Opt. Lett. 17, 180–182 (1992).
  22. L. Lapidus, G. R. Pinder, Numerical Solution of Partial Differential Equations in Science and Engineering(Wiley, New York, 1982), pp. 489–496.
  23. R. W. Ziolkowski, J. Judkins, “Propagation characteristics of ultrawide-bandwidth pulsed Gaussian beams,” J. Opt. Soc. Am. A 9, 2021–2030 (1992).
    [CrossRef]
  24. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Sec. 17.4.

1992

1991

1990

R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, M. Schneider, “A frequency dependent finite difference time domain formulation for transient propagation in plasmas,” IEEE Trans. Electromagn. Compat. 32, 222 (1990).
[CrossRef]

K. Hayata, A. Misawa, M. Koshiba, “Spatial polarization instabilities due to transverse effects in nonlinear guided-wave systems,” J. Opt. Soc. Am. B 7, 1268–1280 (1990).
[CrossRef]

1988

1987

S. N. Vlasov, “Structure of the field of wave beams with circular polarization near a nonlinear focus in a cubic medium,” Sov. J. Quantum Electron. 17, 1191–1193 (1987).
[CrossRef]

1975

M. Miyagi, S. Nishida, “TM waves in nonlinear self-focusing media,” Radio Sci. 10, 833–838 (1975).
[CrossRef]

For a comprehensive review of experimental work on self-focusing until the mid 1970’s, see Y. R. Shen, “Self-focusing: experimental,” Prog. Quantum Electron. 4, 1–34 (1975).
[CrossRef]

For a comprehensive review of theoretical work on self-focusing until the mid 1970’s, see J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975).
[CrossRef]

1974

1973

J. Reintjes, R. L. Carman, F. Shimizu, “Study of self-focusing and self-phase-modulation in the picosecond-time regime,” Phys. Rev. A. 8, 1486–1503 (1973).
[CrossRef]

1972

F. Shimizu, “The effect of dispersion on pulse distortion in optical filaments,” IEEE J. Quantum Electron. 8, 851–854 (1972).
[CrossRef]

D. Pohl, “Self-focusing of TE01and TM01light beams: influence of longitudinal field components,” Phys. Rev. A 5, 1906–1909 (1972).
[CrossRef]

1970

D. Pohl, “Vectorial theory of self-trapped light beams,” Opt. Commun., 2, 305–308 (1970).
[CrossRef]

1969

J. A. Fleck, P. L. Kelley, “Temporal aspects of the self-focusing of optical beams,” Appl. Phys. Lett. 15, 313–315 (1969).
[CrossRef]

1966

E. Garmire, R. Y. Chiao, C. H. Townes, “Dynamics of the self-trapping of intense light beams,” Phys. Rev. Lett. 16, 347–349 (1966).
[CrossRef]

1965

V. I. Talanov, “Self-focusing of waves in nonlinear media,” JETP Lett. 2, 138–141 (1965) [Zh. Eksp. Teor. Fiz. Pis’ma Red. 2, 218 (1965)].

1964

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

Carman, R. L.

J. Reintjes, R. L. Carman, F. Shimizu, “Study of self-focusing and self-phase-modulation in the picosecond-time regime,” Phys. Rev. A. 8, 1486–1503 (1973).
[CrossRef]

Chiao, R. Y.

E. Garmire, R. Y. Chiao, C. H. Townes, “Dynamics of the self-trapping of intense light beams,” Phys. Rev. Lett. 16, 347–349 (1966).
[CrossRef]

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

Feit, M. D.

Fleck, J. A.

M. D. Feit, J. A. Fleck, “Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams,” J. Opt. Soc. Am. B 5, 633–640 (1988).
[CrossRef]

J. A. Fleck, P. L. Kelley, “Temporal aspects of the self-focusing of optical beams,” Appl. Phys. Lett. 15, 313–315 (1969).
[CrossRef]

Garmire, E.

E. Garmire, R. Y. Chiao, C. H. Townes, “Dynamics of the self-trapping of intense light beams,” Phys. Rev. Lett. 16, 347–349 (1966).
[CrossRef]

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

Goorjian, P. M.

P. M. Goorjian, A. Taflove, “FD–TD computational modeling of nonlinear electromagnetic phenomena using a nonlinear convolution approach,” presented at the 1991 North American Radio Science Meeting, University of Western Ontario, London, Ontario, Canada; “Direct time integration of Maxwell’s equations in nonlinear dispersive media for propagation and scattering of femtosecond electromagnetic solitons,” Opt. Lett. 17, 180–182 (1992).

Hagness, S. C.

Hayata, K.

Hunsberger, F. P.

R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, M. Schneider, “A frequency dependent finite difference time domain formulation for transient propagation in plasmas,” IEEE Trans. Electromagn. Compat. 32, 222 (1990).
[CrossRef]

Joseph, R. M.

Judkins, J.

Kelley, P. L.

J. A. Fleck, P. L. Kelley, “Temporal aspects of the self-focusing of optical beams,” Appl. Phys. Lett. 15, 313–315 (1969).
[CrossRef]

Kong, J. A.

C. F. Lee, R. T. Shin, J. A. Kong, “Finite difference method for electromagnetic scattering problems,” in PIER4: Progress in Electromagnetics Research, J. A. Kong, ed. (Elsevier, New York, 1991), Chap. 11.

Koshiba, M.

Kunz, K. S.

R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, M. Schneider, “A frequency dependent finite difference time domain formulation for transient propagation in plasmas,” IEEE Trans. Electromagn. Compat. 32, 222 (1990).
[CrossRef]

Lapidus, L.

L. Lapidus, G. R. Pinder, Numerical Solution of Partial Differential Equations in Science and Engineering(Wiley, New York, 1982), pp. 489–496.

Lee, C. F.

C. F. Lee, R. T. Shin, J. A. Kong, “Finite difference method for electromagnetic scattering problems,” in PIER4: Progress in Electromagnetics Research, J. A. Kong, ed. (Elsevier, New York, 1991), Chap. 11.

Luebbers, R.

R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, M. Schneider, “A frequency dependent finite difference time domain formulation for transient propagation in plasmas,” IEEE Trans. Electromagn. Compat. 32, 222 (1990).
[CrossRef]

Marburger, J. H.

For a comprehensive review of theoretical work on self-focusing until the mid 1970’s, see J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975).
[CrossRef]

Misawa, A.

Miyagi, M.

M. Miyagi, S. Nishida, “TM waves in nonlinear self-focusing media,” Radio Sci. 10, 833–838 (1975).
[CrossRef]

M. Miyagi, S. Nishida, “TM-type soliton in nonlinear self-focusing media,” Proc. IEEE 62, 1284–1285 (1974).
[CrossRef]

Nayyar, V. P.

Nishida, S.

M. Miyagi, S. Nishida, “TM waves in nonlinear self-focusing media,” Radio Sci. 10, 833–838 (1975).
[CrossRef]

M. Miyagi, S. Nishida, “TM-type soliton in nonlinear self-focusing media,” Proc. IEEE 62, 1284–1285 (1974).
[CrossRef]

Pinder, G. R.

L. Lapidus, G. R. Pinder, Numerical Solution of Partial Differential Equations in Science and Engineering(Wiley, New York, 1982), pp. 489–496.

Pohl, D.

D. Pohl, “Self-focusing of TE01and TM01light beams: influence of longitudinal field components,” Phys. Rev. A 5, 1906–1909 (1972).
[CrossRef]

D. Pohl, “Vectorial theory of self-trapped light beams,” Opt. Commun., 2, 305–308 (1970).
[CrossRef]

Reintjes, J.

J. Reintjes, R. L. Carman, F. Shimizu, “Study of self-focusing and self-phase-modulation in the picosecond-time regime,” Phys. Rev. A. 8, 1486–1503 (1973).
[CrossRef]

Schneider, M.

R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, M. Schneider, “A frequency dependent finite difference time domain formulation for transient propagation in plasmas,” IEEE Trans. Electromagn. Compat. 32, 222 (1990).
[CrossRef]

Shen, Y. R.

For a comprehensive review of experimental work on self-focusing until the mid 1970’s, see Y. R. Shen, “Self-focusing: experimental,” Prog. Quantum Electron. 4, 1–34 (1975).
[CrossRef]

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Sec. 17.4.

Shimizu, F.

J. Reintjes, R. L. Carman, F. Shimizu, “Study of self-focusing and self-phase-modulation in the picosecond-time regime,” Phys. Rev. A. 8, 1486–1503 (1973).
[CrossRef]

F. Shimizu, “The effect of dispersion on pulse distortion in optical filaments,” IEEE J. Quantum Electron. 8, 851–854 (1972).
[CrossRef]

Shin, R. T.

C. F. Lee, R. T. Shin, J. A. Kong, “Finite difference method for electromagnetic scattering problems,” in PIER4: Progress in Electromagnetics Research, J. A. Kong, ed. (Elsevier, New York, 1991), Chap. 11.

Sodha, M. S.

Standler, R. B.

R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, M. Schneider, “A frequency dependent finite difference time domain formulation for transient propagation in plasmas,” IEEE Trans. Electromagn. Compat. 32, 222 (1990).
[CrossRef]

Taflove, A.

R. M. Joseph, S. C. Hagness, A. Taflove, “Direct time integration of Maxwell’s equations in linear dispersive media with absorption for scattering and propagation of femtosecond electromagnetic pulses,” Opt. Lett. 16, 1412–1414 (1991).
[CrossRef] [PubMed]

P. M. Goorjian, A. Taflove, “FD–TD computational modeling of nonlinear electromagnetic phenomena using a nonlinear convolution approach,” presented at the 1991 North American Radio Science Meeting, University of Western Ontario, London, Ontario, Canada; “Direct time integration of Maxwell’s equations in nonlinear dispersive media for propagation and scattering of femtosecond electromagnetic solitons,” Opt. Lett. 17, 180–182 (1992).

Talanov, V. I.

V. I. Talanov, “Self-focusing of waves in nonlinear media,” JETP Lett. 2, 138–141 (1965) [Zh. Eksp. Teor. Fiz. Pis’ma Red. 2, 218 (1965)].

Townes, C. H.

E. Garmire, R. Y. Chiao, C. H. Townes, “Dynamics of the self-trapping of intense light beams,” Phys. Rev. Lett. 16, 347–349 (1966).
[CrossRef]

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

Tripathi, V. K.

Vlasov, S. N.

S. N. Vlasov, “Structure of the field of wave beams with circular polarization near a nonlinear focus in a cubic medium,” Sov. J. Quantum Electron. 17, 1191–1193 (1987).
[CrossRef]

Ziolkowski, R. W.

Appl. Phys. Lett.

J. A. Fleck, P. L. Kelley, “Temporal aspects of the self-focusing of optical beams,” Appl. Phys. Lett. 15, 313–315 (1969).
[CrossRef]

IEEE J. Quantum Electron.

F. Shimizu, “The effect of dispersion on pulse distortion in optical filaments,” IEEE J. Quantum Electron. 8, 851–854 (1972).
[CrossRef]

IEEE Trans. Electromagn. Compat.

R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, M. Schneider, “A frequency dependent finite difference time domain formulation for transient propagation in plasmas,” IEEE Trans. Electromagn. Compat. 32, 222 (1990).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

JETP Lett.

V. I. Talanov, “Self-focusing of waves in nonlinear media,” JETP Lett. 2, 138–141 (1965) [Zh. Eksp. Teor. Fiz. Pis’ma Red. 2, 218 (1965)].

Opt. Commun.

D. Pohl, “Vectorial theory of self-trapped light beams,” Opt. Commun., 2, 305–308 (1970).
[CrossRef]

Opt. Lett.

Phys. Rev. A

D. Pohl, “Self-focusing of TE01and TM01light beams: influence of longitudinal field components,” Phys. Rev. A 5, 1906–1909 (1972).
[CrossRef]

Phys. Rev. A.

J. Reintjes, R. L. Carman, F. Shimizu, “Study of self-focusing and self-phase-modulation in the picosecond-time regime,” Phys. Rev. A. 8, 1486–1503 (1973).
[CrossRef]

Phys. Rev. Lett.

E. Garmire, R. Y. Chiao, C. H. Townes, “Dynamics of the self-trapping of intense light beams,” Phys. Rev. Lett. 16, 347–349 (1966).
[CrossRef]

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

Proc. IEEE

M. Miyagi, S. Nishida, “TM-type soliton in nonlinear self-focusing media,” Proc. IEEE 62, 1284–1285 (1974).
[CrossRef]

Prog. Quantum Electron.

For a comprehensive review of theoretical work on self-focusing until the mid 1970’s, see J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975).
[CrossRef]

For a comprehensive review of experimental work on self-focusing until the mid 1970’s, see Y. R. Shen, “Self-focusing: experimental,” Prog. Quantum Electron. 4, 1–34 (1975).
[CrossRef]

Radio Sci.

M. Miyagi, S. Nishida, “TM waves in nonlinear self-focusing media,” Radio Sci. 10, 833–838 (1975).
[CrossRef]

Sov. J. Quantum Electron.

S. N. Vlasov, “Structure of the field of wave beams with circular polarization near a nonlinear focus in a cubic medium,” Sov. J. Quantum Electron. 17, 1191–1193 (1987).
[CrossRef]

Other

C. F. Lee, R. T. Shin, J. A. Kong, “Finite difference method for electromagnetic scattering problems,” in PIER4: Progress in Electromagnetics Research, J. A. Kong, ed. (Elsevier, New York, 1991), Chap. 11.

P. M. Goorjian, A. Taflove, “FD–TD computational modeling of nonlinear electromagnetic phenomena using a nonlinear convolution approach,” presented at the 1991 North American Radio Science Meeting, University of Western Ontario, London, Ontario, Canada; “Direct time integration of Maxwell’s equations in nonlinear dispersive media for propagation and scattering of femtosecond electromagnetic solitons,” Opt. Lett. 17, 180–182 (1992).

L. Lapidus, G. R. Pinder, Numerical Solution of Partial Differential Equations in Science and Engineering(Wiley, New York, 1982), pp. 489–496.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Sec. 17.4.

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Figures (19)

Fig. 1
Fig. 1

1/e profiles of the electric-field intensity I = |Ex|2 + |Ez|2 and the total field energy density integrated along the z axis: ∫ dz1/2eff[|Ex|2 + |Ez|2] + 1/2μ0|Hy|2 versus time are plotted for the T = 2τ case. The original intensity waist of the input pulse was 7.07 μm.

Fig. 2
Fig. 2

Intensity of the transverse electric-field component |Ex|2 along the propagation axis, given for the T = 2τ case at the time t = 103.0 fs when the maximum intensity focus occurs.

Fig. 3
Fig. 3

Power spectrum of the signal |Ex|2 (shown in Fig. 2) along the propagation axis, given for the T = 2τ case at the time t = 103.0 fs when the maximum intensity focus occurs. The power spectrum of the input pulse is given for comparison.

Fig. 4
Fig. 4

Contour plot of |Ex|2 + |Ez|2 for the T = 2τ case at the time t = 103.0 fs when the maximum intensity focus occurs, showing the breakup of the pulse into two lobes, the null appearing in the transition to the negative-resistivity region.

Fig. 5
Fig. 5

Intensity of the longitudinal electric-field component |Ez|2 along the axial slice x = 0.72 μm, given for the T = 2τ case at the time t ~ 103.0 fs when its maximum value occurs.

Fig. 6
Fig. 6

Contour plot of |Ez|2 for the T = 2τ case at the time t ~ 103.0 fs when its maximum value occurs, showing the appearance of its peak off the propagation axis in the region where the transverse field components exhibit a null.

Fig. 7
Fig. 7

Longitudinal power flow Sz along the propagation axis, given for the T = 2τ case at the time t = 103.0 fs, corresponding to that when the maximum intensity of the transverse field component Ex occurs.

Fig. 8
Fig. 8

Total field density 1/2eff[|Ex|2 + |Ez|2] + 1/2μ0|Hy|2 integrated over the spatial band (x, z) [0.0, 1.2 μm] × [0.0, 40.0 μm] and over the spatial band (x, z) [0.0, 2.4 μm] [0.0, 40.0 μm] and then normalized to the areas of those bands, plotted versus time for the T = 2τ case.

Fig. 9
Fig. 9

Contour plots of the intensities |Ex|2 (top row) and |Ez|2 (second row) and plots of the on-axis intensity |Ex|2 (third row) and the power flow density Sz = ExHy (bottom row) are given for the T = 5τ case at times (a) t = 83.3 fs, (b) t = 96.6 fs, and (c) t = 109.9 fs.

Fig. 10
Fig. 10

1/e profiles of the electric-field intensity I = |Ex|2 + |Ez|2 and the total field energy density integrated along the z axis: ∫ dz1/2eff[|Ex|2 + |Ez|2] + 1/2μ0|Hy|2 versus time, plotted for the T = 20τ case. The original intensity waist of the input pulse was 7.07 μm.

Fig. 11
Fig. 11

Intensity of the transverse electric-field component |Ex|2 along the propagation axis, given for the T = 20τ case at the time t = 97.0 fs when the maximum intensity focus occurs.

Fig. 12
Fig. 12

Power spectrum of the signal |Ex|2 (shown in Fig. 11) along the propagation axis, given for the T = 20τ case at the time t = 97.0 fs when the maximum intensity focus occurs. The power spectrum of the input pulse is given for comparison.

Fig. 13
Fig. 13

Intensity of the longitudinal electric-field component |Ex|2 along the axial slice x = 0.24 μm, given for the T = 20τ case at the time t ~ 97.0 fs when its maximum value occurs.

Fig. 14
Fig. 14

Longitudinal power flow Sz along the propagation axis, given for the T = 20τ case at the time t = 97.0 fs, corresponding to the maximum intensity of the transverse field component Ex. The negative power flow from energy reflected from the focal region is apparent.

Fig. 15
Fig. 15

Total field energy density 1/2eff[|Ex|2 + |Ex|2] + 1/2μ0|Hy|2 integrated over the spatial band (x, z) [0.0, 1.2 μm] × [0.0, 40.0 μm] and over the spatial band (x, z) [0.0, 2.4 μm] × [0.0, 40.0 μm] and then normalized by the areas of those bands, plotted versus time for the T = 20τ case.

Fig. 16
Fig. 16

Intensity of the transverse electric-field component |Ex|2, given for the T = 5τ TM case at the time t = 100.0 fs when the maximum-intensity focus occurs.

Fig. 17
Fig. 17

Contour plot of |Ez|2 for the T = 5τ, TM case, given at the time t = 100.0 fs when its maximum value occurs.

Fig. 18
Fig. 18

Intensity of the transverse electric-field component |Ey|2 given for the T = 5τ, TE case at the time t = 100.0 fs when the maximum intensity focus occurs.

Fig. 19
Fig. 19

Contour plot of |Ey|2 for the T = 5τ, TE case, given at the time t = 100.0 fs when its maximum value occurs.

Equations (34)

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t H y = - 1 μ 0 ( z E x - x E z ) ,
t E x = - 1 eff z H y - σ eff eff E x ,
t E z = + 1 eff x H y - σ eff eff E z ,
eff = L + 0 χ NL
σ eff = 0 t χ NL ,
· D L = - · P NL ,
D L = L E ,
P NL = χ NL ( r , t , E 2 ) E .
t H x = + 1 μ 0 z E y ,
t H z = - 1 μ 0 x E y ,
t E y = + 1 eff ( z H x - x H z ) - σ eff eff E y ,
t χ NL + 1 τ χ NL = 1 τ 2 E 2 ,
t χ NL + 1 τ χ NL = 1 τ 2 ( E x 2 + E z 2 ) ,
t χ NL + 1 τ χ NL = 1 τ 2 E y 2 .
n ^ · ( eff t E NL + σ eff E NL ) = n ^ · ( eff t E L + σ eff E L ) .
χ NL = exp [ - ( t / τ ) ] 0 t d t exp [ + ( t / τ ) ] 2 τ E 2 ,
E x = - exp [ - ( σ eff / eff ) t ] 0 t d t exp [ + ( σ eff / eff ) t ] eff H y z ,
E z = + exp [ - ( σ eff / eff ) t ] 0 t d t exp [ + ( σ eff / eff ) t ] eff H y x .
z 2 E - μ 0 eff t 2 E + x 2 E - 2 μ 0 σ eff t E - ( c t 2 χ NL ) E = - ( 1 / 0 ) ( · P NL ) .
z 2 E - μ 0 eff t 2 E + x 2 E - 2 μ 0 σ eff t E - ( c t 2 χ NL ) E = 0.
E x ( r center , t center ) ( 1 / 2 ) { E x [ ( i + 1 / 2 ) Δ x , j Δ z , n Δ t ] + E x [ ( i + 1 / 2 ) Δ x , ( j + 1 ) Δ z , n Δ t ] } ,
( eff / σ eff ) ( r center , t center ) E x ( r center , t center ) ( 1 / 2 ) ( eff / σ e f f ) [ ( i + 1 / 2 ) Δ x , ( j + 1 / 2 ) Δ z , ( n + 1 / 2 ) Δ t ] × { E x [ ( i + 1 / 2 ) Δ x , j Δ z , n Δ t ] + E x [ ( i + 1 / 2 ) Δ x , j Δ z , ( n + 1 ) Δ t ] } .
E x [ ( i + 1 / 2 ) Δ x , j Δ z , ( n + 1 ) Δ t ] = 1 A ( i , j , n ) ( - { H y [ ( i + 1 / 2 ) Δ x , ( j + 1 / 2 ) Δ z , ( n + 1 / 2 ) Δ t ] Δ z - H y [ ( i + 1 / 2 ) Δ x , ( j - 1 / 2 ) Δ z , ( n + 1 / 2 ) Δ t ] Δ z } + B ( i , j , n ) E x [ ( i + 1 / 2 ) Δ x , j Δ z , n Δ t ] ) ,
1 2 [ A ( i , j , n ) + B ( i , j , n ) ] = eff [ ( i + 1 / 2 ) Δ x , ( j + 1 / 2 ) Δ z , ( n + 1 / 2 ) Δ t ] Δ t ,
1 2 [ A ( i , j , n ) - B ( i , j , n ) ] = σ eff [ ( i + 1 / 2 ) Δ x , ( j + 1 / 2 ) Δ z , ( n + 1 / 2 ) Δ t ] 2 .
H y ( x , z = 0 , t ) = H 0 exp [ - ( x / w 0 ) 2 ] G ( t ) ,
G ( t ) = { 1.0 - cos ( ω 0 t ) for 0 t T 0 for t > T ,
G ˜ ( ω ) = exp ( i ω T / 2 ) sin [ ( ω / ω 0 ) π ] ω 1 1 - ( ω / ω 0 ) 2 .
L R = π w 0 2 λ rad = 30.24 μ m ,
E cr = ( n 2 / n 0 ) - 1 / 2 λ rad / ( π w 0 ) = 3.307 × 10 8 ( V / m )
T R = L R c = π w 0 2 c λ rad = 100.8 fs .
θ int = 8.317 ° 0.439 λ rad π w 0 ,
z cr w 0 θ int = 68.89 μ m ,
z f = ( E 0 cr E 0 ) z cr = 12.48 μ m .

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