Abstract

Steady-state bistable lasing and hysteretic mode hops of two adjacent longitudinal modes are measured in a homogeneously broadened cw laser system: a cw rotational Raman laser in H2. As the pump laser’s frequency is cyclically tuned, the mode hops between two adjacent longitudinal Stokes modes show hysteresis. Theoretical modeling of the hysteresis effect indicates that the gain of the subthreshold Stokes mode is suppressed by a Raman-assisted multiwave mixing process. The theoretical calculation agrees well with the experimental measurement.

© 2004 Optical Society of America

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References

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  1. N. B. Abraham, L. A. Lugiato, and L. M. Narducci, “Overview of instabilities in laser systems,” J. Opt. Soc. Am. B 2, 7–14 (1985).
    [CrossRef]
  2. L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, “Mode-mode competition and unstable behavior in a homogeneously broadened ring laser,” Phys. Rev. A 33, 1842–1854 (1986).
    [CrossRef] [PubMed]
  3. M. Harris, R. Loudon, T. J. Shepherd, and J. M. Vaughan, “Mode-hopping hysteresis in a single-frequency laser,” Opt. Commun. 101, 432–441 (1993).
    [CrossRef]
  4. J. K. Brasseur, K. S. Repasky, and J. L. Carlsten, “Continuous-wave Raman laser in H2,” Opt. Lett. 23, 367–369 (1998).
    [CrossRef]
  5. P. A. Roos, J. K. Brasseur, and J. L. Carlsten, “Diode-pumped nonresonant continuous-wave Raman laser in H2 with resonant optical feedback stabilization,” Opt. Lett. 24, 1130–1132 (1999).
    [CrossRef]
  6. L. S. Meng, P. A. Roos, J. K. Brasseur, and J. L. Carlsten, “Widely tunable continuous-wave Raman laser in diatomic hydrogen pumped by an external-cavity diode laser,” Opt. Lett. 25, 472–474 (2000).
    [CrossRef]
  7. P. A. Roos, L. S. Meng, and J. L. Carlsten, “Efficient, tunable, high power cw near-infrared generation through Raman down-conversion of a diode laser in H 2,” in Conference on Lasers and Electro-Optics, Vol. 39 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2000), postdeadline paper CPD24.
  8. L. S. Meng, P. A. Roos, J. K. Brasseur, and J. L. Carlsten, “High-conversion-efficiency, diode-pumped continuous-wave Raman laser,” Opt. Lett. 26, 426–428 (2001).
    [CrossRef]
  9. L. S. Meng, P. A. Roos, J. K. Brasseur, and J. L. Carlsten, “Continuous-wave rotational Raman laser in H2,” Opt. Lett. 27, 1226–1228 (2002).
    [CrossRef]
  10. G. C. Herring, M. J. Dyer, and W. K. Bischel, “Temperature and density dependence of the linewidths and line shifts of the rotational Raman lines in N2 and H2,” Phys. Rev. A 34, 1944–1951 (1986).
    [CrossRef] [PubMed]
  11. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
    [CrossRef]
  12. M. D. Duncan, R. Mahon, J. Reintjes, and L. L. Tankersley, “Parametric Raman gain suppression in D2 and H2,” Opt. Lett. 11, 803–805 (1986).
    [CrossRef] [PubMed]
  13. J. K. Brasseur, P. A. Roos, K. S. Repasky, and J. L. Carlsten, “Characterization of a continuous-wave Raman laser in H2,” J. Opt. Soc. Am. B 16, 1305–1312 (1999).
    [CrossRef]
  14. R. W. Boyd, Nonlinear Optics (Academic, San Diego, Calif., 1992).
  15. A. E. Siegman, Lasers (University Science, Sausalito, Calif., 1986).
  16. S. E. Harris and A. V. Sokolov, “Broadband spectral generation with refractive index control,” Phys. Rev. A 55, R4019–R4022 (1997).
    [CrossRef]
  17. S. E. Harris and A. V. Sokolov, “Subfemtosecond pulse generation by molecular modulation,” Phys. Rev. Lett. 81, 2894–2897 (1998).
    [CrossRef]
  18. Fam Le Kien, J. Q. Liang, M. Katsuragawa, K. Ohtsuki, K. Hakuta, and A. V. Sokolov, “Subfemtosecond pulse generation with molecular coherence control in stimulated Raman scattering,” Phys. Rev. A 60, 1562–1571 (1999).
    [CrossRef]
  19. Other higher-order Stokes and anti-Stokes modes cannot experience cavity enhancement since their wavelengths are out of reflectivity bandwidth of the cavity mirrors.
  20. L. S. Meng, “Continuous-wave Raman laser in H 2 : semiclassical theory and diode-pumping experiments,” Ph.D. dissertation (Montana State University, Bozeman, Mont., 2002).
  21. The coherence dephasing time for high-pressure and room-temperature H 2 gas is of the order of 10−9 s; while for the cw Raman laser, the cavity field buildup time is of the order of 10−6 s or slower.
  22. J. K. Brasseur, P. A. Roos, L. S. Meng, and J. L. Carlsten, “Frequency tuning characteristics of a continuous wave Raman laser in H2,” J. Opt. Soc. Am. B 17, 1229–1232 (2000).
    [CrossRef]
  23. R. Corbalan, J. Cortit, and F. Prati, “Competition and bistability of longitudinal modes in a Raman laser,” Phys. Rev. A 53, 481–485 (1996).
    [CrossRef] [PubMed]
  24. H. Risken and K. Nummedal, “Self-pulsing in lasers,” J. Appl. Phys. 39, 4662–4672 (1968).
    [CrossRef]
  25. R. G. Harrison and Weiping Lu, “Origin of periodic, chaotic, and bistable emission from Raman lasers,” Phys. Rev. Lett. 63, 1372–1375 (1989).
    [CrossRef] [PubMed]

2002 (1)

2001 (1)

2000 (2)

1999 (3)

1998 (2)

S. E. Harris and A. V. Sokolov, “Subfemtosecond pulse generation by molecular modulation,” Phys. Rev. Lett. 81, 2894–2897 (1998).
[CrossRef]

J. K. Brasseur, K. S. Repasky, and J. L. Carlsten, “Continuous-wave Raman laser in H2,” Opt. Lett. 23, 367–369 (1998).
[CrossRef]

1997 (1)

S. E. Harris and A. V. Sokolov, “Broadband spectral generation with refractive index control,” Phys. Rev. A 55, R4019–R4022 (1997).
[CrossRef]

1996 (1)

R. Corbalan, J. Cortit, and F. Prati, “Competition and bistability of longitudinal modes in a Raman laser,” Phys. Rev. A 53, 481–485 (1996).
[CrossRef] [PubMed]

1993 (1)

M. Harris, R. Loudon, T. J. Shepherd, and J. M. Vaughan, “Mode-hopping hysteresis in a single-frequency laser,” Opt. Commun. 101, 432–441 (1993).
[CrossRef]

1989 (1)

R. G. Harrison and Weiping Lu, “Origin of periodic, chaotic, and bistable emission from Raman lasers,” Phys. Rev. Lett. 63, 1372–1375 (1989).
[CrossRef] [PubMed]

1986 (3)

L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, “Mode-mode competition and unstable behavior in a homogeneously broadened ring laser,” Phys. Rev. A 33, 1842–1854 (1986).
[CrossRef] [PubMed]

G. C. Herring, M. J. Dyer, and W. K. Bischel, “Temperature and density dependence of the linewidths and line shifts of the rotational Raman lines in N2 and H2,” Phys. Rev. A 34, 1944–1951 (1986).
[CrossRef] [PubMed]

M. D. Duncan, R. Mahon, J. Reintjes, and L. L. Tankersley, “Parametric Raman gain suppression in D2 and H2,” Opt. Lett. 11, 803–805 (1986).
[CrossRef] [PubMed]

1985 (1)

1983 (1)

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

1968 (1)

H. Risken and K. Nummedal, “Self-pulsing in lasers,” J. Appl. Phys. 39, 4662–4672 (1968).
[CrossRef]

Abraham, N. B.

L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, “Mode-mode competition and unstable behavior in a homogeneously broadened ring laser,” Phys. Rev. A 33, 1842–1854 (1986).
[CrossRef] [PubMed]

N. B. Abraham, L. A. Lugiato, and L. M. Narducci, “Overview of instabilities in laser systems,” J. Opt. Soc. Am. B 2, 7–14 (1985).
[CrossRef]

Bandy, D. K.

L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, “Mode-mode competition and unstable behavior in a homogeneously broadened ring laser,” Phys. Rev. A 33, 1842–1854 (1986).
[CrossRef] [PubMed]

Bischel, W. K.

G. C. Herring, M. J. Dyer, and W. K. Bischel, “Temperature and density dependence of the linewidths and line shifts of the rotational Raman lines in N2 and H2,” Phys. Rev. A 34, 1944–1951 (1986).
[CrossRef] [PubMed]

Brasseur, J. K.

Carlsten, J. L.

Corbalan, R.

R. Corbalan, J. Cortit, and F. Prati, “Competition and bistability of longitudinal modes in a Raman laser,” Phys. Rev. A 53, 481–485 (1996).
[CrossRef] [PubMed]

Cortit, J.

R. Corbalan, J. Cortit, and F. Prati, “Competition and bistability of longitudinal modes in a Raman laser,” Phys. Rev. A 53, 481–485 (1996).
[CrossRef] [PubMed]

Drever, R. W. P.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Duncan, M. D.

Dyer, M. J.

G. C. Herring, M. J. Dyer, and W. K. Bischel, “Temperature and density dependence of the linewidths and line shifts of the rotational Raman lines in N2 and H2,” Phys. Rev. A 34, 1944–1951 (1986).
[CrossRef] [PubMed]

Ford, G. M.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Hakuta, K.

Fam Le Kien, J. Q. Liang, M. Katsuragawa, K. Ohtsuki, K. Hakuta, and A. V. Sokolov, “Subfemtosecond pulse generation with molecular coherence control in stimulated Raman scattering,” Phys. Rev. A 60, 1562–1571 (1999).
[CrossRef]

Hall, J. L.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Harris, M.

M. Harris, R. Loudon, T. J. Shepherd, and J. M. Vaughan, “Mode-hopping hysteresis in a single-frequency laser,” Opt. Commun. 101, 432–441 (1993).
[CrossRef]

Harris, S. E.

S. E. Harris and A. V. Sokolov, “Subfemtosecond pulse generation by molecular modulation,” Phys. Rev. Lett. 81, 2894–2897 (1998).
[CrossRef]

S. E. Harris and A. V. Sokolov, “Broadband spectral generation with refractive index control,” Phys. Rev. A 55, R4019–R4022 (1997).
[CrossRef]

Harrison, R. G.

R. G. Harrison and Weiping Lu, “Origin of periodic, chaotic, and bistable emission from Raman lasers,” Phys. Rev. Lett. 63, 1372–1375 (1989).
[CrossRef] [PubMed]

Herring, G. C.

G. C. Herring, M. J. Dyer, and W. K. Bischel, “Temperature and density dependence of the linewidths and line shifts of the rotational Raman lines in N2 and H2,” Phys. Rev. A 34, 1944–1951 (1986).
[CrossRef] [PubMed]

Hough, J.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Katsuragawa, M.

Fam Le Kien, J. Q. Liang, M. Katsuragawa, K. Ohtsuki, K. Hakuta, and A. V. Sokolov, “Subfemtosecond pulse generation with molecular coherence control in stimulated Raman scattering,” Phys. Rev. A 60, 1562–1571 (1999).
[CrossRef]

Kowalski, F. V.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Le Kien, Fam

Fam Le Kien, J. Q. Liang, M. Katsuragawa, K. Ohtsuki, K. Hakuta, and A. V. Sokolov, “Subfemtosecond pulse generation with molecular coherence control in stimulated Raman scattering,” Phys. Rev. A 60, 1562–1571 (1999).
[CrossRef]

Liang, J. Q.

Fam Le Kien, J. Q. Liang, M. Katsuragawa, K. Ohtsuki, K. Hakuta, and A. V. Sokolov, “Subfemtosecond pulse generation with molecular coherence control in stimulated Raman scattering,” Phys. Rev. A 60, 1562–1571 (1999).
[CrossRef]

Loudon, R.

M. Harris, R. Loudon, T. J. Shepherd, and J. M. Vaughan, “Mode-hopping hysteresis in a single-frequency laser,” Opt. Commun. 101, 432–441 (1993).
[CrossRef]

Lu, Weiping

R. G. Harrison and Weiping Lu, “Origin of periodic, chaotic, and bistable emission from Raman lasers,” Phys. Rev. Lett. 63, 1372–1375 (1989).
[CrossRef] [PubMed]

Lugiato, L. A.

L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, “Mode-mode competition and unstable behavior in a homogeneously broadened ring laser,” Phys. Rev. A 33, 1842–1854 (1986).
[CrossRef] [PubMed]

N. B. Abraham, L. A. Lugiato, and L. M. Narducci, “Overview of instabilities in laser systems,” J. Opt. Soc. Am. B 2, 7–14 (1985).
[CrossRef]

Mahon, R.

Meng, L. S.

Munley, A. J.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Narducci, L. M.

L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, “Mode-mode competition and unstable behavior in a homogeneously broadened ring laser,” Phys. Rev. A 33, 1842–1854 (1986).
[CrossRef] [PubMed]

N. B. Abraham, L. A. Lugiato, and L. M. Narducci, “Overview of instabilities in laser systems,” J. Opt. Soc. Am. B 2, 7–14 (1985).
[CrossRef]

Nummedal, K.

H. Risken and K. Nummedal, “Self-pulsing in lasers,” J. Appl. Phys. 39, 4662–4672 (1968).
[CrossRef]

Ohtsuki, K.

Fam Le Kien, J. Q. Liang, M. Katsuragawa, K. Ohtsuki, K. Hakuta, and A. V. Sokolov, “Subfemtosecond pulse generation with molecular coherence control in stimulated Raman scattering,” Phys. Rev. A 60, 1562–1571 (1999).
[CrossRef]

Prati, F.

R. Corbalan, J. Cortit, and F. Prati, “Competition and bistability of longitudinal modes in a Raman laser,” Phys. Rev. A 53, 481–485 (1996).
[CrossRef] [PubMed]

Reintjes, J.

Repasky, K. S.

Risken, H.

H. Risken and K. Nummedal, “Self-pulsing in lasers,” J. Appl. Phys. 39, 4662–4672 (1968).
[CrossRef]

Roos, P. A.

Shepherd, T. J.

M. Harris, R. Loudon, T. J. Shepherd, and J. M. Vaughan, “Mode-hopping hysteresis in a single-frequency laser,” Opt. Commun. 101, 432–441 (1993).
[CrossRef]

Sokolov, A. V.

Fam Le Kien, J. Q. Liang, M. Katsuragawa, K. Ohtsuki, K. Hakuta, and A. V. Sokolov, “Subfemtosecond pulse generation with molecular coherence control in stimulated Raman scattering,” Phys. Rev. A 60, 1562–1571 (1999).
[CrossRef]

S. E. Harris and A. V. Sokolov, “Subfemtosecond pulse generation by molecular modulation,” Phys. Rev. Lett. 81, 2894–2897 (1998).
[CrossRef]

S. E. Harris and A. V. Sokolov, “Broadband spectral generation with refractive index control,” Phys. Rev. A 55, R4019–R4022 (1997).
[CrossRef]

Tankersley, L. L.

Tredicce, J. R.

L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, “Mode-mode competition and unstable behavior in a homogeneously broadened ring laser,” Phys. Rev. A 33, 1842–1854 (1986).
[CrossRef] [PubMed]

Vaughan, J. M.

M. Harris, R. Loudon, T. J. Shepherd, and J. M. Vaughan, “Mode-hopping hysteresis in a single-frequency laser,” Opt. Commun. 101, 432–441 (1993).
[CrossRef]

Ward, H.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Appl. Phys. B (1)

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

J. Appl. Phys. (1)

H. Risken and K. Nummedal, “Self-pulsing in lasers,” J. Appl. Phys. 39, 4662–4672 (1968).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Commun. (1)

M. Harris, R. Loudon, T. J. Shepherd, and J. M. Vaughan, “Mode-hopping hysteresis in a single-frequency laser,” Opt. Commun. 101, 432–441 (1993).
[CrossRef]

Opt. Lett. (6)

Phys. Rev. A (5)

Fam Le Kien, J. Q. Liang, M. Katsuragawa, K. Ohtsuki, K. Hakuta, and A. V. Sokolov, “Subfemtosecond pulse generation with molecular coherence control in stimulated Raman scattering,” Phys. Rev. A 60, 1562–1571 (1999).
[CrossRef]

G. C. Herring, M. J. Dyer, and W. K. Bischel, “Temperature and density dependence of the linewidths and line shifts of the rotational Raman lines in N2 and H2,” Phys. Rev. A 34, 1944–1951 (1986).
[CrossRef] [PubMed]

L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, “Mode-mode competition and unstable behavior in a homogeneously broadened ring laser,” Phys. Rev. A 33, 1842–1854 (1986).
[CrossRef] [PubMed]

R. Corbalan, J. Cortit, and F. Prati, “Competition and bistability of longitudinal modes in a Raman laser,” Phys. Rev. A 53, 481–485 (1996).
[CrossRef] [PubMed]

S. E. Harris and A. V. Sokolov, “Broadband spectral generation with refractive index control,” Phys. Rev. A 55, R4019–R4022 (1997).
[CrossRef]

Phys. Rev. Lett. (2)

S. E. Harris and A. V. Sokolov, “Subfemtosecond pulse generation by molecular modulation,” Phys. Rev. Lett. 81, 2894–2897 (1998).
[CrossRef]

R. G. Harrison and Weiping Lu, “Origin of periodic, chaotic, and bistable emission from Raman lasers,” Phys. Rev. Lett. 63, 1372–1375 (1989).
[CrossRef] [PubMed]

Other (6)

P. A. Roos, L. S. Meng, and J. L. Carlsten, “Efficient, tunable, high power cw near-infrared generation through Raman down-conversion of a diode laser in H 2,” in Conference on Lasers and Electro-Optics, Vol. 39 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2000), postdeadline paper CPD24.

Other higher-order Stokes and anti-Stokes modes cannot experience cavity enhancement since their wavelengths are out of reflectivity bandwidth of the cavity mirrors.

L. S. Meng, “Continuous-wave Raman laser in H 2 : semiclassical theory and diode-pumping experiments,” Ph.D. dissertation (Montana State University, Bozeman, Mont., 2002).

The coherence dephasing time for high-pressure and room-temperature H 2 gas is of the order of 10−9 s; while for the cw Raman laser, the cavity field buildup time is of the order of 10−6 s or slower.

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

A. E. Siegman, Lasers (University Science, Sausalito, Calif., 1986).

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Figures (7)

Fig. 1
Fig. 1

Energy-level diagram of the far-off-resonance Raman process. The lowest level a is the ground state; the higher level b can be either a molecular vibrational or a rotational state; level j’s are the multiple excited electronic states. The pump laser’s frequency is ωp, and the generated Stokes frequency is ωs. The two fields are coupled by a molecular coherence oscillating at a frequency ωm. δ is the two-photon detuning (δ<0 in the configuration shown).

Fig. 2
Fig. 2

Experimental setup used to study the mode-hopping hysteresis, a rotational cw Raman laser in H2: ECDL, external-cavity diode laser; λ/2 and λ/4, half-wave and quarter-wave plate; PBS, polarization beam splitter; EOM, electro-optic modulator; SM-PM fiber, single-mode, polarization-maintaining fiber; MML, mode-matching lens; HFC, high-finesse cavity.

Fig. 3
Fig. 3

Experimental measurement showing the mode-hopping hysteresis between two adjacent rotational Stokes longitudinal modes in the cw Raman laser. The mode hop occurs at different pump frequencies depending on the tuning direction of the pump. H is the hysteresis amount. Optical frequencies are measured by a Burleigh wavemeter with 10-MHz resolution. The pump frequency is measured relative to 378, 350 GHz, and the Stokes frequency is measured relative to 360, 750 GHz.

Fig. 4
Fig. 4

Transmitted pump power (top) and Stokes power (bottom) as functions of pump-frequency tuning show butterflylike patterns. Power discontinuities indicate mode hops. The pump-frequency tuning is shown relative to 378, 384 GHz. Back and forth tuning of the pump frequency is performed twice.

Fig. 5
Fig. 5

Plot of H as a function of the pumping rate R. Circles are the experimental measurements. Solid curve is the theoretical calculation.

Fig. 6
Fig. 6

Picture showing how the mode-hopping hysteresis occurs. δ(<0) and δ+(>0) are the two-photon detunings of the Stokes modes ωs and ωs+, respectively. Δc=δ+-δ is the longitudinal mode spacing of the two Stokes modes (i.e., cavity FSR). The width of the rotational Raman-gain linewidth is sufficient to cover the two adjacent longitudinal Stokes modes. (a) The “symmetric point,” where the two Stokes modes lie symmetrically away from the line center, |δ|=|δ+|=Δc/2. The subthreshold mode ωs+ loses some gain so it is below threshold. (b) The mode-hop point, where |δ|>|δ+| and the mode ωs+ reaches the threshold since it is closer to the line center and the lost gain is compensated. (c) After the mode hop, the steady-state gain is quickly saturated to a lower level so that the now-lasing mode ωs+ sees equal gain and loss.

Fig. 7
Fig. 7

Theoretical model for mod-hopping hysteresis. (a) The usual Raman process: the pump field at frequency ωp and the lasing-Stokes mode at frequency ωs estabish a strong coherence between the two energy states oscillating at frequency ωm=ωp-ωs. (b) The first FWM process: the subthreshold Stokes ωs+=ωs+Δc beats with the strong pump ωp, producing a weak coherence sideband oscillating at ωp-ωs+=ωm-Δc; then the lasing-Stokes mode is scattered off this coherence sideband to produce a weak pump sideband with frequency ωp-=ωp-Δc. (c) The second FWM process [(a) and (c) together]: the subthreshold Stokes ωs+ is scattered off the strong coherence, producing another new weak field ωp+. (d) A “bonus” new weak field is generated, although it does not contribute to the gain suppression on ωs-: the sideband ωp- is scattered off the strong coherence, producing a new field ωs+. In this plot, we use the thick lines to represent “strong,” and the thin lines mean “weak.”

Equations (86)

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

E˜(r, t)=12 q[Eq(r, t)exp(-iωqt)+c.c.],
P˜(r, t)=12 q[Pq(r, t)exp(-iωqt)+c.c.].
Eq(r, t)=nEq,n(t)uq,n(r).
cavityuq,n(r)uq,m*(r)dxdydz=Vq,nδnm,
Eq(r, t)=Eq(t)uq(r).
E˙q(t)+γcq2-i(ωq-ωcq)Eq(t)
=i ωq20Pq(t)+γep2Eep(t),
Pq(t)=1Vq cavitydxdydzPq(r, t)uq*(r).
ρ˙aa=i(Ωabρba-Ωbaρab)+Γbaρbb,
ρ˙bb=-i(Ωabρba-Ωbaρab)-Γbaρbb,
ρ˙ab=i(Ωaa-Ωbb+δ)ρab+iΩab(ρbb-ρaa)-γabρab,
Ωaa(r, t)=12[ap|Ep(r, t)|2+as|Es(r, t)|2],
Ωbb(r, t)=12[bp|Ep(r, t)|2+bs|Es(r, t)|2],
Ωab(r, t)=ds2Es(r, t)Ep*(r, t),
ap(s)=122 j|μaj|21ωja-ωp(s)+1ωja+ωp(s),
bp(s)=122 j|μbj|21ωjb-ωp(s)+1ωjb+ωp(s),
ds=122 jμajμjb1ωjb-ωs+1ωja+ωs,
Pp(r, t)=2N[apEp(r, t)+dsρba(r, t)Es(r, t)],
Ps(r, t)=2N[asEs(r, t)+dsρab(r, t)Ep(r, t)],
D˙=-2i(Ωabρba-Ωbaρab)-Γba(D-Deq),
ρ˙ab=-[γab-i(Ωaa-Ωbb+δ)]ρab+iΩabD,
D=ΓbaDeq(γab2+δ2)Γba(γab2+δ2)+4|Ωab|2γab,
ρab=iΩabDγab-i(Ωaa-Ωbb+δ).
ρab=iΩabDγab-iδ.
Pp(r, t)=2NapEp(t)up(r)-i Nds2Dγab+iδ|Es(t)us(r)|2Ep(t)up(r),
Ps(r, t)=2NasEs(t)us(r)+i Nds2Dγab-iδ|Ep(t)up(r)|2Es(t)us(r).
|E˙p(t)|+γcp2|Ep(t)|=-ωpωs kpksG(δ)|Es(t)|2|Ep(t)|+γep2|Eep(t)|,
|E˙s(t)|+γcs2|Es(t)|=G(δ)|Ep(t)|2|Es(t)|,
G(δ)=18c20αg(δ) λpλp+λs tan-1(L/b)L/b.
|Ep|ss=γcs2G(δ),
|Es|ss=ωsωp kskp γep|Eep|/|Ep|ss-γcp2G(δ).
|Eep|th(δ)=γcpγep|Ep|ss.
R=|Eep|2|Eep|th2(δ=±Δc/2).
E˜p(r, t)=12{Ep0(r, t)exp(-iωpt)+Ep+(r, t)exp[-i(ωp+Δc)t]+Ep-(r, t)exp[-i(ωp-Δc)t]+c.c.},
E˜s(r, t)=12{Es0(r, t)exp(-iωst)+Es+(r, t)exp[-i(ωs+Δc)t]+Es-(r, t)exp[-i(ωs-Δc)t]+c.c.}.
ρ˜ab(r, t)=ρ˜0(r, t)+ρ˜-(r, t)=ρ0(r, t)exp(iωmt)+ρ-(r, t)exp[i(ωm-Δc)t].
Ep(r, t)=Ep0(r, t)+Ep+(r, t)exp(-iΔct)+Ep-(r, t)exp(iΔct),
Es(r, t)=Es0(r, t)+Es+(r, t)exp(-iΔct)+Es-(r, t)exp(iΔct),
ρab(r, t)=ρ0(r, t)+ρ-(r, t)exp(-iΔct),
E˙s+(t)+γcs+2-i(ωs+-ωcs+)Es+(t)
=i ωs+20 1Vs+ cavitydxdydzPs+(r, t)us+*(r),
E˙p-(t)+γcp-2-i(ωp--ωcp-)Ep-(t)
=i ωp-20 1Vp- cavitydxdydzPp-(r, t)up-*(r),
E˙p+(t)+γcp+2-i(ωp+-ωcp+)Ep+(t)
=i ωp+20 1Vp+ cavitydxdydzPp+(r, t)up+*(r),
E˙s-(t)+γcs-2-i(ωs--ωcs-)Es-(t)
=i ωs-20 1Vs- cavitydxdydzPs-(r, t)us-*(r),
Pp=2Nap[Ep0+Ep+ exp(-iΔct)+Ep- exp(iΔct)]+2Nds[ρ0*+ρ-* exp(iΔct)]×[Es0+Es+ exp(-iΔct)+Es- exp(iΔct)]=Pp0+Pp+ exp(-iΔct)+Pp- exp(iΔct),
Pp0(r, t)=2N[apEp0(r, t)+dsρ0*(r, t)Es0(r, t)],
Pp+(r, t)=2N[apEp+(r, t)+dsρ0*(r, t)Es+(r, t)],
Pp-(r, t)=2N[apEp-(r, t)+dsρ0*(r, t)Es-(r, t)+dsρ-*(r, t)Es0(r, t)].
Ps=2Nas[Es0+Es+ exp(-iΔct)+Es- exp(iΔct)]+2Nds[ρ0+ρ- exp(-iΔct)]×[Ep0+Ep+ exp(-iΔct)+Ep- exp(iΔct)]=Ps0+Ps+ exp(-iΔct)+Ps- exp(iΔct),
Ps0(r, t)=2N[asEs0(r, t)+dsρ0(r, t)Es0(r, t)],
Ps+(r, t)=2N[asEs+(r, t)+dsρ0(r, t)Ep+(r, t)+dsρ-(r, t)Ep0(r, t)],
Ps-(r, t)=2N[asEs-(r, t)+dsρ0(r, t)Ep-(r, t)].
ρ˙ab=-(γab-iδ)ρab+iΩabD.
ρ˙0(r, t)=-(γab-iδ)ρ0(r, t)+i ds2DEs0(r, t)Ep0*(r, t),
ρ˙-(r, t)=-(γab-iδ+)ρ-(r, t)+i ds2D[Es+(r, t)Ep0*(r, t)+Es0(r, t)Ep-*(r, t)],
ρ˙+(r, t)=-(γab-iδ-)ρ+(r, t)+i ds2D[Es-(r, t)Ep0*(r, t)+Es0(r, t)Ep+*(r, t)],
ρ0(r, t)
=i ds2D Es0(r, t)Ep0*(r, t)γab-iδ,
ρ-(r, t)
=i ds2D Es+(r, t)Ep0*(r, t)+Es0(r, t)Ep-*(r, t)γab-iδ+.
E˙s++γcs+2Es+=G(δ+)Ep02Es++12G(δ+)Ep0Es0Ep-+12G(δ)Ep+Es0Ep0,
E˙p-+γcp-2Ep-=-ωp2ωs2 G(δ+)Es02Ep-+12G(δ+)Ep0Es0Es++12G(δ)Ep0Es-Es0,
E˙p++γcp+2Ep+=-ωp2ωs2 12G(δ)Ep0Es+Es0,
E˙s-+γcs-2Es-=12G(δ)Ep-Es0Ep0.
ddt+αs+Es+=κs+Ep-+κs-Ep+,
ddt+αp-Ep-=κp-Es++κp+Es-,
ddt+γcp+2Ep+=κp+Es+,
ddt+γcs-2Es-=κs-Ep-,
αs+=γcs+2-G(δ+)Ep02,
αp-=γcp-2+ωp2ωs2G(δ+)Es02,
κs+=12G(δ+)Ep0Es0,
κp-=-12 ωp2ωs2G(δ+)Ep0Es0,
κp+=-12 ωp2ωs2G(δ)Ep0Es0,
κs-=12G(δ)Ep0Es0.
d2dt2+αs++γcp+2 ddt+αs+ γcp+2-κp+κs-
×d2dt2+αp-+γcs-2 ddt+αp- γcs-2-κp+κs-Es+
=ddt+γcp+2ddt+γcs-2κs+κp-Es+.
Es+=Es+(0)exp(gt),
g2+αs++γcp+2g+αs+ γcp+2-κp+κs-×g2+αp-+γcs-2g+αp- γcs-2-κp+κs-
=g+γcp+2g+γcs-2κs+κp-.
αs+ γcp+2-κp+κs-αp- γcs-2-κp+κs-
-γcp+γcs-4κs+κp-=f(δ, R)=0.
H(R)=2 (|δR|-Δc/2)1-λp/λs 12π=H(R)1-λp/λs 12π,

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