Abstract

We propose a scheme for the generation of continuous-wave terahertz (THz) radiation. The scheme requires a medium in which three discrete states in a Λ configuration can be selected, with the THz frequency transition being between the two lower metastable states. The propagation of three-frequency continuous-wave electromagnetic (EM) radiation through a Λ medium is considered. Under resonant excitation, the medium absorption can be strongly reduced owing to quantum interference of transitions, whereas the nonlinear susceptibility is enhanced. This leads to efficient energy transfer among the EM waves, providing the possibility of THz generation. We demonstrate that the photon conversion efficiency is approaching unity in this technique.

© 2000 Optical Society of America

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References

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  1. Special issue on terahertz electromagnetic pulse generation, physics, and applications, D. R. Dykaar and S. L. Chuang, eds., J. Opt. Soc. Am. B 11(12) (1994).
  2. F. Strumia, “A proposal for a new absolute frequency standard, using a Mg or Ca atomic beam,” Metrologia 8, 85–90 (1972).
    [CrossRef]
  3. A. Godone and C. Novero, “The magnesium frequency standard,” Metrologia 30, 163–181 (1993).
    [CrossRef]
  4. S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50, 36–42 (1997).
    [CrossRef]
  5. S. E. Harris, J. E. Feld, and A. Imamoglu, “Nonlinear optical processes using electromagnetically induced transparency,” Phys. Rev. Lett. 64, 1107–1110 (1990).
    [CrossRef] [PubMed]
  6. G. Z. Zhang, D. W. Tokaryk, B. P. Stoicheff, and K. Hakuta, “Nonlinear generation of extreme-ultraviolet radiation in atomic hydrogen using electromagnetically induced transparency,” Phys. Rev. A 56, 813–819 (1997); D. W. Tokaryk, G. Z. Zhang, and B. P. Stoicheff, “Nonlinear optical generation in a hydrogen discharge,” Phys. Rev. A 59, 3116–3119 (1999).
    [CrossRef]
  7. S. Babin, U. Hinze, E. Tiemann, and B. Wellegehausen, “Continuous resonant four-wave mixing in double-Λ level configurations of Na2,” Opt. Lett. 21, 1186–1188 (1996); A. Apolonskii, S. Baluschev, U. Hinze, E. Tiemann, and B. Wellegehausen, “Continuous frequency up-conversion in double-Λ scheme of Na2,” Appl. Phys. B 64, 435–442 (1997).
    [CrossRef] [PubMed]
  8. B. S. Ham, M. S. Shahriar, and P. R. Hemmer, “Enhanced nondegenerate four-wave mixing owing to electromagnetically induced transparency in a spectral hole-burning crystal,” Opt. Lett. 22, 1138–1140 (1997); “Enhancement of four-wave mixing and line narrowing by use of quantum coherence in an optically dense double-Λ solid,” Opt. Lett. 24, 86–88 (1999).
    [CrossRef] [PubMed]
  9. M. Jain, H. Xia, G. Y. Yin, A. J. Merriam, and S. E. Harris, “Efficient nonlinear frequency conversion with maximal atomic coherence,” Phys. Rev. Lett. 77, 4326–4329 (1996).
    [CrossRef] [PubMed]
  10. A. J. Merriam, S. J. Sharpe, H. Xia, D. Manuszak, G. Y. Yin, and S. E. Harris, “Efficient gas-phase generation of coherent vacuum ultraviolet radiation,” Opt. Lett. 24, 625–627 (1999).
    [CrossRef]
  11. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge U. Press, Cambridge, UK, 1997).
  12. D. V. Kosachiov, “Resonant Λ medium as a converter of the laser radiation frequency,” Kvant. Elektron. (Moscow) 22, 1123–1128 (1995) [Quantum Electron. 25, 1089–1094 (1995)].
    [CrossRef]
  13. E. A. Korsunsky and D. V. Kosachiov, “Phase-dependent nonlinear optics with double-Λ atoms,” Phys. Rev. A 60, 4996–5009 (1999).
    [CrossRef]
  14. E. Arimondo, “Coherent population trapping in laser spectroscopy,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1996), Vol. 35, pp. 257–354.
  15. A. Godone, F. Levy, and J. Vanier, “Coherent microwave emission in cesium under coherent population trapping,” Phys. Rev. A 59, R12–R15 (1999); J. Vanier, A. Godone, and F. Levy, “Coherent population trapping in cesium: dark lines and coherent microwave emission,” Phys. Rev. A 58, 2345–2358 (1998).
    [CrossRef]
  16. R. W. Boyd, Nonlinear Optics (Academic, San Diego, Calif., 1992).
  17. D. V. Kosachiov, B. G. Matisov, and Yu. V. Rozhdestvensky, “Coherent phenomena in multilevel systems with closed interaction contour,” J. Phys. B 25, 2473–2488 (1992).
    [CrossRef]
  18. D. V. Kosachiov and E. A. Korsunsky, “Efficient microwave-induced optical frequency conversion,” Eur. Phys. J. D. (to be published).
  19. S. J. Buckle, S. M. Barnett, P. L. Knight, M. A. Lauder, and D. T. Pegg, “Atomic interferometers: phase-dependence in multilevel atomic transitions,” Opt. Acta 33, 1129–1140 (1986).
    [CrossRef]
  20. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
    [CrossRef]
  21. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965).
  22. S. Brandt, A. Nagel, R. Wynands, and D. Meschede, “Buffer-gas-induced linewidth reduction of coherent dark resonances to below 50 Hz,” Phys. Rev. A 56, R1063–R1066 (1997).
    [CrossRef]
  23. J. H. Xu and G. Alzetta, “High buffer gas pressure perturbation of coherent population trapping in sodium vapors,” Phys. Lett. A 248, 80–85 (1998).
    [CrossRef]

1999

1998

J. H. Xu and G. Alzetta, “High buffer gas pressure perturbation of coherent population trapping in sodium vapors,” Phys. Lett. A 248, 80–85 (1998).
[CrossRef]

1997

S. Brandt, A. Nagel, R. Wynands, and D. Meschede, “Buffer-gas-induced linewidth reduction of coherent dark resonances to below 50 Hz,” Phys. Rev. A 56, R1063–R1066 (1997).
[CrossRef]

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50, 36–42 (1997).
[CrossRef]

1996

M. Jain, H. Xia, G. Y. Yin, A. J. Merriam, and S. E. Harris, “Efficient nonlinear frequency conversion with maximal atomic coherence,” Phys. Rev. Lett. 77, 4326–4329 (1996).
[CrossRef] [PubMed]

1995

D. V. Kosachiov, “Resonant Λ medium as a converter of the laser radiation frequency,” Kvant. Elektron. (Moscow) 22, 1123–1128 (1995) [Quantum Electron. 25, 1089–1094 (1995)].
[CrossRef]

1993

A. Godone and C. Novero, “The magnesium frequency standard,” Metrologia 30, 163–181 (1993).
[CrossRef]

1992

D. V. Kosachiov, B. G. Matisov, and Yu. V. Rozhdestvensky, “Coherent phenomena in multilevel systems with closed interaction contour,” J. Phys. B 25, 2473–2488 (1992).
[CrossRef]

1990

S. E. Harris, J. E. Feld, and A. Imamoglu, “Nonlinear optical processes using electromagnetically induced transparency,” Phys. Rev. Lett. 64, 1107–1110 (1990).
[CrossRef] [PubMed]

1986

S. J. Buckle, S. M. Barnett, P. L. Knight, M. A. Lauder, and D. T. Pegg, “Atomic interferometers: phase-dependence in multilevel atomic transitions,” Opt. Acta 33, 1129–1140 (1986).
[CrossRef]

1972

F. Strumia, “A proposal for a new absolute frequency standard, using a Mg or Ca atomic beam,” Metrologia 8, 85–90 (1972).
[CrossRef]

1962

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Alzetta, G.

J. H. Xu and G. Alzetta, “High buffer gas pressure perturbation of coherent population trapping in sodium vapors,” Phys. Lett. A 248, 80–85 (1998).
[CrossRef]

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Barnett, S. M.

S. J. Buckle, S. M. Barnett, P. L. Knight, M. A. Lauder, and D. T. Pegg, “Atomic interferometers: phase-dependence in multilevel atomic transitions,” Opt. Acta 33, 1129–1140 (1986).
[CrossRef]

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Brandt, S.

S. Brandt, A. Nagel, R. Wynands, and D. Meschede, “Buffer-gas-induced linewidth reduction of coherent dark resonances to below 50 Hz,” Phys. Rev. A 56, R1063–R1066 (1997).
[CrossRef]

Buckle, S. J.

S. J. Buckle, S. M. Barnett, P. L. Knight, M. A. Lauder, and D. T. Pegg, “Atomic interferometers: phase-dependence in multilevel atomic transitions,” Opt. Acta 33, 1129–1140 (1986).
[CrossRef]

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Feld, J. E.

S. E. Harris, J. E. Feld, and A. Imamoglu, “Nonlinear optical processes using electromagnetically induced transparency,” Phys. Rev. Lett. 64, 1107–1110 (1990).
[CrossRef] [PubMed]

Godone, A.

A. Godone and C. Novero, “The magnesium frequency standard,” Metrologia 30, 163–181 (1993).
[CrossRef]

Harris, S. E.

A. J. Merriam, S. J. Sharpe, H. Xia, D. Manuszak, G. Y. Yin, and S. E. Harris, “Efficient gas-phase generation of coherent vacuum ultraviolet radiation,” Opt. Lett. 24, 625–627 (1999).
[CrossRef]

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50, 36–42 (1997).
[CrossRef]

M. Jain, H. Xia, G. Y. Yin, A. J. Merriam, and S. E. Harris, “Efficient nonlinear frequency conversion with maximal atomic coherence,” Phys. Rev. Lett. 77, 4326–4329 (1996).
[CrossRef] [PubMed]

S. E. Harris, J. E. Feld, and A. Imamoglu, “Nonlinear optical processes using electromagnetically induced transparency,” Phys. Rev. Lett. 64, 1107–1110 (1990).
[CrossRef] [PubMed]

Imamoglu, A.

S. E. Harris, J. E. Feld, and A. Imamoglu, “Nonlinear optical processes using electromagnetically induced transparency,” Phys. Rev. Lett. 64, 1107–1110 (1990).
[CrossRef] [PubMed]

Jain, M.

M. Jain, H. Xia, G. Y. Yin, A. J. Merriam, and S. E. Harris, “Efficient nonlinear frequency conversion with maximal atomic coherence,” Phys. Rev. Lett. 77, 4326–4329 (1996).
[CrossRef] [PubMed]

Knight, P. L.

S. J. Buckle, S. M. Barnett, P. L. Knight, M. A. Lauder, and D. T. Pegg, “Atomic interferometers: phase-dependence in multilevel atomic transitions,” Opt. Acta 33, 1129–1140 (1986).
[CrossRef]

Korsunsky, E. A.

E. A. Korsunsky and D. V. Kosachiov, “Phase-dependent nonlinear optics with double-Λ atoms,” Phys. Rev. A 60, 4996–5009 (1999).
[CrossRef]

Kosachiov, D. V.

E. A. Korsunsky and D. V. Kosachiov, “Phase-dependent nonlinear optics with double-Λ atoms,” Phys. Rev. A 60, 4996–5009 (1999).
[CrossRef]

D. V. Kosachiov, “Resonant Λ medium as a converter of the laser radiation frequency,” Kvant. Elektron. (Moscow) 22, 1123–1128 (1995) [Quantum Electron. 25, 1089–1094 (1995)].
[CrossRef]

D. V. Kosachiov, B. G. Matisov, and Yu. V. Rozhdestvensky, “Coherent phenomena in multilevel systems with closed interaction contour,” J. Phys. B 25, 2473–2488 (1992).
[CrossRef]

Lauder, M. A.

S. J. Buckle, S. M. Barnett, P. L. Knight, M. A. Lauder, and D. T. Pegg, “Atomic interferometers: phase-dependence in multilevel atomic transitions,” Opt. Acta 33, 1129–1140 (1986).
[CrossRef]

Manuszak, D.

Matisov, B. G.

D. V. Kosachiov, B. G. Matisov, and Yu. V. Rozhdestvensky, “Coherent phenomena in multilevel systems with closed interaction contour,” J. Phys. B 25, 2473–2488 (1992).
[CrossRef]

Merriam, A. J.

A. J. Merriam, S. J. Sharpe, H. Xia, D. Manuszak, G. Y. Yin, and S. E. Harris, “Efficient gas-phase generation of coherent vacuum ultraviolet radiation,” Opt. Lett. 24, 625–627 (1999).
[CrossRef]

M. Jain, H. Xia, G. Y. Yin, A. J. Merriam, and S. E. Harris, “Efficient nonlinear frequency conversion with maximal atomic coherence,” Phys. Rev. Lett. 77, 4326–4329 (1996).
[CrossRef] [PubMed]

Meschede, D.

S. Brandt, A. Nagel, R. Wynands, and D. Meschede, “Buffer-gas-induced linewidth reduction of coherent dark resonances to below 50 Hz,” Phys. Rev. A 56, R1063–R1066 (1997).
[CrossRef]

Nagel, A.

S. Brandt, A. Nagel, R. Wynands, and D. Meschede, “Buffer-gas-induced linewidth reduction of coherent dark resonances to below 50 Hz,” Phys. Rev. A 56, R1063–R1066 (1997).
[CrossRef]

Novero, C.

A. Godone and C. Novero, “The magnesium frequency standard,” Metrologia 30, 163–181 (1993).
[CrossRef]

Pegg, D. T.

S. J. Buckle, S. M. Barnett, P. L. Knight, M. A. Lauder, and D. T. Pegg, “Atomic interferometers: phase-dependence in multilevel atomic transitions,” Opt. Acta 33, 1129–1140 (1986).
[CrossRef]

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Rozhdestvensky, Yu. V.

D. V. Kosachiov, B. G. Matisov, and Yu. V. Rozhdestvensky, “Coherent phenomena in multilevel systems with closed interaction contour,” J. Phys. B 25, 2473–2488 (1992).
[CrossRef]

Sharpe, S. J.

Strumia, F.

F. Strumia, “A proposal for a new absolute frequency standard, using a Mg or Ca atomic beam,” Metrologia 8, 85–90 (1972).
[CrossRef]

Wynands, R.

S. Brandt, A. Nagel, R. Wynands, and D. Meschede, “Buffer-gas-induced linewidth reduction of coherent dark resonances to below 50 Hz,” Phys. Rev. A 56, R1063–R1066 (1997).
[CrossRef]

Xia, H.

A. J. Merriam, S. J. Sharpe, H. Xia, D. Manuszak, G. Y. Yin, and S. E. Harris, “Efficient gas-phase generation of coherent vacuum ultraviolet radiation,” Opt. Lett. 24, 625–627 (1999).
[CrossRef]

M. Jain, H. Xia, G. Y. Yin, A. J. Merriam, and S. E. Harris, “Efficient nonlinear frequency conversion with maximal atomic coherence,” Phys. Rev. Lett. 77, 4326–4329 (1996).
[CrossRef] [PubMed]

Xu, J. H.

J. H. Xu and G. Alzetta, “High buffer gas pressure perturbation of coherent population trapping in sodium vapors,” Phys. Lett. A 248, 80–85 (1998).
[CrossRef]

Yin, G. Y.

A. J. Merriam, S. J. Sharpe, H. Xia, D. Manuszak, G. Y. Yin, and S. E. Harris, “Efficient gas-phase generation of coherent vacuum ultraviolet radiation,” Opt. Lett. 24, 625–627 (1999).
[CrossRef]

M. Jain, H. Xia, G. Y. Yin, A. J. Merriam, and S. E. Harris, “Efficient nonlinear frequency conversion with maximal atomic coherence,” Phys. Rev. Lett. 77, 4326–4329 (1996).
[CrossRef] [PubMed]

J. Phys. B

D. V. Kosachiov, B. G. Matisov, and Yu. V. Rozhdestvensky, “Coherent phenomena in multilevel systems with closed interaction contour,” J. Phys. B 25, 2473–2488 (1992).
[CrossRef]

Metrologia

F. Strumia, “A proposal for a new absolute frequency standard, using a Mg or Ca atomic beam,” Metrologia 8, 85–90 (1972).
[CrossRef]

A. Godone and C. Novero, “The magnesium frequency standard,” Metrologia 30, 163–181 (1993).
[CrossRef]

Opt. Acta

S. J. Buckle, S. M. Barnett, P. L. Knight, M. A. Lauder, and D. T. Pegg, “Atomic interferometers: phase-dependence in multilevel atomic transitions,” Opt. Acta 33, 1129–1140 (1986).
[CrossRef]

Opt. Lett.

Phys. Lett. A

J. H. Xu and G. Alzetta, “High buffer gas pressure perturbation of coherent population trapping in sodium vapors,” Phys. Lett. A 248, 80–85 (1998).
[CrossRef]

Phys. Rev.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Phys. Rev. A

S. Brandt, A. Nagel, R. Wynands, and D. Meschede, “Buffer-gas-induced linewidth reduction of coherent dark resonances to below 50 Hz,” Phys. Rev. A 56, R1063–R1066 (1997).
[CrossRef]

E. A. Korsunsky and D. V. Kosachiov, “Phase-dependent nonlinear optics with double-Λ atoms,” Phys. Rev. A 60, 4996–5009 (1999).
[CrossRef]

Phys. Rev. Lett.

M. Jain, H. Xia, G. Y. Yin, A. J. Merriam, and S. E. Harris, “Efficient nonlinear frequency conversion with maximal atomic coherence,” Phys. Rev. Lett. 77, 4326–4329 (1996).
[CrossRef] [PubMed]

S. E. Harris, J. E. Feld, and A. Imamoglu, “Nonlinear optical processes using electromagnetically induced transparency,” Phys. Rev. Lett. 64, 1107–1110 (1990).
[CrossRef] [PubMed]

Phys. Today

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50, 36–42 (1997).
[CrossRef]

Quantum Electron.

D. V. Kosachiov, “Resonant Λ medium as a converter of the laser radiation frequency,” Kvant. Elektron. (Moscow) 22, 1123–1128 (1995) [Quantum Electron. 25, 1089–1094 (1995)].
[CrossRef]

Other

Special issue on terahertz electromagnetic pulse generation, physics, and applications, D. R. Dykaar and S. L. Chuang, eds., J. Opt. Soc. Am. B 11(12) (1994).

D. V. Kosachiov and E. A. Korsunsky, “Efficient microwave-induced optical frequency conversion,” Eur. Phys. J. D. (to be published).

E. Arimondo, “Coherent population trapping in laser spectroscopy,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1996), Vol. 35, pp. 257–354.

A. Godone, F. Levy, and J. Vanier, “Coherent microwave emission in cesium under coherent population trapping,” Phys. Rev. A 59, R12–R15 (1999); J. Vanier, A. Godone, and F. Levy, “Coherent population trapping in cesium: dark lines and coherent microwave emission,” Phys. Rev. A 58, 2345–2358 (1998).
[CrossRef]

R. W. Boyd, Nonlinear Optics (Academic, San Diego, Calif., 1992).

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge U. Press, Cambridge, UK, 1997).

G. Z. Zhang, D. W. Tokaryk, B. P. Stoicheff, and K. Hakuta, “Nonlinear generation of extreme-ultraviolet radiation in atomic hydrogen using electromagnetically induced transparency,” Phys. Rev. A 56, 813–819 (1997); D. W. Tokaryk, G. Z. Zhang, and B. P. Stoicheff, “Nonlinear optical generation in a hydrogen discharge,” Phys. Rev. A 59, 3116–3119 (1999).
[CrossRef]

S. Babin, U. Hinze, E. Tiemann, and B. Wellegehausen, “Continuous resonant four-wave mixing in double-Λ level configurations of Na2,” Opt. Lett. 21, 1186–1188 (1996); A. Apolonskii, S. Baluschev, U. Hinze, E. Tiemann, and B. Wellegehausen, “Continuous frequency up-conversion in double-Λ scheme of Na2,” Appl. Phys. B 64, 435–442 (1997).
[CrossRef] [PubMed]

B. S. Ham, M. S. Shahriar, and P. R. Hemmer, “Enhanced nondegenerate four-wave mixing owing to electromagnetically induced transparency in a spectral hole-burning crystal,” Opt. Lett. 22, 1138–1140 (1997); “Enhancement of four-wave mixing and line narrowing by use of quantum coherence in an optically dense double-Λ solid,” Opt. Lett. 24, 86–88 (1999).
[CrossRef] [PubMed]

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965).

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Figures (4)

Fig. 1
Fig. 1

Closed Λ system with two metastable states |1〉 and |2〉. ω31 and ω32 are the optical frequencies, ωT is the THz-range frequency.

Fig. 2
Fig. 2

Spatial variations of (a) the optical and (b) the THz field intensities [in terms of input intensity I0I31(ζ=0)] and of (c) the relative phase Φ in a vapor of  24Mg atoms interacting with radiation in a closed Λ configuration of levels 3 3P13 3P24 3S1. For this system, the relaxation rates are γ31=3.46×107 s-1, γ32=1.66γ31, γ21=2.6×10-14γ31, and the wavelengths are λ31=517.27 nm, λ31=518.36 nm. Other parameters are atom velocity vz=0; Γ=0; detunings Δ31=Δ32=0; and Rabi frequencies of input fields g31(τ=0)=10γ31, g32(τ=0)=0.1γ31, and g12(τ=0)=0. The dotted curve in (b) is a calculation for u202=0.55×10-4, performed with formula (18).

Fig. 3
Fig. 3

Spatial variations of (a) the optical and (b) the THz field intensities and of (c) the relative phase Φ in a vapor of  24Mg atoms for vapor temperature T=860 K; Γ=10-4γ31; detunings Δ31=Δ32=0; and Rabi frequencies of input fields g31(τ=0)=60γ31, g32(τ=0)=20γ31, and g12(τ=0)=0. Other parameters are the same as in Fig. 2.

Fig. 4
Fig. 4

Spatial variations of the THz radiation intensity in a vapor of  24Mg atoms for Γ=2×10-3γ31 and Rabi frequencies of the input fields (a) g31(τ=0)=60γ31, g32(τ=0)=20γ31 and (b) g31(τ=0)=300γ31, g32(τ=0)=100γ31. Other parameters are the same as in Fig. 3. Note the different length scales in (a) and (b).

Equations (38)

Equations on this page are rendered with MathJax. Learn more.

E(z, t)=m=1,2e3mE3m(z, t)×12 exp{-i[ω3mt-k3mz+φ3m(z, t)]}+c.c.,
H(z, t)=eTH(z, t) 12 exp{-i[ωTt-kTz+φT(z, t)]}+c.c.,
E3mz+1c E3mt=-N 4πd3mω3mc Im(σ˜3m),
φ3mz+1c φ3mt=-N 4πd3mω3mc 1E3m Re(σ˜3m),
Hz+1c Ht=-N 4πμωTc Im(σ˜21),
φTz+1c φTt=-N 4πμωTc 1H Re(σ˜21),
|NC=g32/g311+g322/g312|1-exp(χ32-χ31) 11+g322/g312|2,
Δ32-Δ31=0.
ω31-ω32-ωT=0.
Im(σ31)=gTg32g02 sin Φ,
Im(σ32)=-gTg31g02 sin Φ,
Im(σ21)=-g31g32g02 sin Φ,
Re(σ31)=-gTg32(g322-g312)g04 cos Φ,
Re(σ32)=gTg31(g322-g312)g04 cos Φ,
Re(σ21)=-g31g32g02 cos Φ,
Φ=(χ31-χ32)-χ12.
dE31dz=-πN2c d31d32μg02ω31E32H sin Φ,
dE32dz=πN2c d31d32μg02ω32E31H sin Φ,
dHdz=πN2c d31d32μg02ωTE31E32 sin Φ.
dIdz=dI31dz+dI32dz+dITdz=N42 d31d32μg02(ω32+ωT-ω31)E31E32H sin Φ=0,
ddz I31ω31=-ddz I32ω32=-ddz ITωT,
um=8πcI ω3mω0-1/2E3m(m=1,2),
uT=8πcI ωTω0-1/2H,
ζ=4πNc d31d32μg02 8πc Iω01/2ωTω31ω32 z.
du1dζ=-u2uT sin Φ,
du2dζ=u1uT sin Φ,
duTdζ=u1u2 sin Φ,
dΦdζ=uT2(u22-u12)-u12u22u1u2uT cos Φ.
cos Φ(ζ)=0,
u12(ζ)=u102 sn2[(ζ+ζ0); u10],
u22(ζ)=1-u102 sn2[(ζ+ζ0); u10],
uT2(ζ)=u102{1-sn2[(ζ+ζ0); u10]},
K(u10)=01 dy[(1-y2)(1-u102y2)]1/2,
ζmax=K(u10).
K(u10)=12 ln16u202.
IT(ζ)=I31(ζ=0) ωTω31(1-cd2[ζ; u10]).
ζ=μd31 ωTω31 γ31d318πcI-1/2τ.
g02Γγ+(kTvp)2g02(k31vp)2.

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