Abstract

We present a theory for far-off resonance mode-locked Raman lasers in H2 with high finesse cavity enhancement. The theoretical derivation for the mode-locked Raman laser is based on a time-dependent continuous-wave (cw) Raman theory. Numerically calculated results, including the Stokes threshold and intracavity fields’s amplitude and phase evolution are discussed in three different regimes depending on the relations between the coherence dephasing rate γ31 and the repetition rate Ω of the mode-locked pump laser. The threshold results from the mode-locked pump cases are compared with the cw, single-mode pump field case.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |

  1. 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]
  2. P. A. Roos, J. K. Brasseur, and J. L. Carlsten, "Diode-pumped non-resonant continuous-wave Raman laser in H2 with resonant optical feedback stabilization," Opt. Lett. 24, 1130-1132 (1999).
    [CrossRef]
  3. L. S. Meng, K. S. Repasky, P. A. Roos, 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]
  4. J. Rifkin, M. L. Bernt, D. Macpherson, and J. L. Carlsten, "Gain enhancement in z XeCl-pumped Raman amplifier," J. Opt. Soc. Am. B 5, 1607-1612 (1988).
    [CrossRef]
  5. M. Sargent, M. O. Scully, and W. E. Lamb, "Semiclasscial laser theory," in Laser Physics (Addison-Wesley, 1974), pp. 96-114.
  6. R. W. Boyd, "Nonlinear optics in the two-level approximation," in Nonlinear Optics, 2nd ed. (Academic, 2003), pp. 261-307.
    [CrossRef]
  7. J. V. Moloney, J. S. Uppal, and R. G. Harrison, "Origin of chaotic relaxation oscillations in an optically pumped molecular laser," Phys. Rev. Lett. 59, 2868-2871 (1997).
    [CrossRef]
  8. J. K. Brasseur, "Construction and noise studies of a continuous wave Raman laser," Ph.D. dissertation (Montana State University, 1998) pp. 19-31.
  9. L. Meng, "Continuous-wave Raman laser in H2: semiclassical theory and diode-pumping experiments," Ph.D. thesis (Montana State University, 2002), pp. 17-22.
  10. E. W. Washburn, International Critical Tables of Numerical Data, Physics, Chemistry and Technology (Edward Wight, 1881-1934), Vol. 711 pp.
  11. American Institute of Physics Handbook2nd ed. (McGraw-Hill, 1963), pp. 6-95.
  12. At 10 atm, λp(1)=800 nm, n1=1.00127005738702; 100 modes away λp(100)=800.18 nm, n100=1.00127005036724. 100×[c/(2n1l)−c/(2n100l)]=0.59 kHz.
  13. K. S. Repasky, J. K. Brasseur, L. Meng, and J. L. Carlsten, "Performance and design of an off-resonant continuous-wave Raman laser," J. Opt. Soc. Am. B 15, 1667-1673 (1998).
    [CrossRef]
  14. W. K. Bischel and M. J. Dyer, "Wavelength dependence of the absolute Raman gain coefficient for the Q(1) transition in H2," J. Opt. Soc. Am. B 3, 677-682 (1986).
    [CrossRef]
  15. W. K. Bischel and M. J. Dyer, "Temperature dependence of the Raman lindewidth and line shift for the Q(1) and Q(0) transitions in normal and para-H2," Phys. Rev. A 33, 3113-3123 (1986).
    [CrossRef] [PubMed]

2000 (1)

1999 (2)

1998 (1)

1997 (1)

J. V. Moloney, J. S. Uppal, and R. G. Harrison, "Origin of chaotic relaxation oscillations in an optically pumped molecular laser," Phys. Rev. Lett. 59, 2868-2871 (1997).
[CrossRef]

1988 (1)

1986 (2)

W. K. Bischel and M. J. Dyer, "Temperature dependence of the Raman lindewidth and line shift for the Q(1) and Q(0) transitions in normal and para-H2," Phys. Rev. A 33, 3113-3123 (1986).
[CrossRef] [PubMed]

W. K. Bischel and M. J. Dyer, "Wavelength dependence of the absolute Raman gain coefficient for the Q(1) transition in H2," J. Opt. Soc. Am. B 3, 677-682 (1986).
[CrossRef]

J. Opt. Soc. Am. B (4)

Opt. Lett. (2)

Phys. Rev. A (1)

W. K. Bischel and M. J. Dyer, "Temperature dependence of the Raman lindewidth and line shift for the Q(1) and Q(0) transitions in normal and para-H2," Phys. Rev. A 33, 3113-3123 (1986).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

J. V. Moloney, J. S. Uppal, and R. G. Harrison, "Origin of chaotic relaxation oscillations in an optically pumped molecular laser," Phys. Rev. Lett. 59, 2868-2871 (1997).
[CrossRef]

Other (7)

J. K. Brasseur, "Construction and noise studies of a continuous wave Raman laser," Ph.D. dissertation (Montana State University, 1998) pp. 19-31.

L. Meng, "Continuous-wave Raman laser in H2: semiclassical theory and diode-pumping experiments," Ph.D. thesis (Montana State University, 2002), pp. 17-22.

E. W. Washburn, International Critical Tables of Numerical Data, Physics, Chemistry and Technology (Edward Wight, 1881-1934), Vol. 711 pp.

American Institute of Physics Handbook2nd ed. (McGraw-Hill, 1963), pp. 6-95.

At 10 atm, λp(1)=800 nm, n1=1.00127005738702; 100 modes away λp(100)=800.18 nm, n100=1.00127005036724. 100×[c/(2n1l)−c/(2n100l)]=0.59 kHz.

M. Sargent, M. O. Scully, and W. E. Lamb, "Semiclasscial laser theory," in Laser Physics (Addison-Wesley, 1974), pp. 96-114.

R. W. Boyd, "Nonlinear optics in the two-level approximation," in Nonlinear Optics, 2nd ed. (Academic, 2003), pp. 261-307.
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

Mode-locked laser interaction with hydrogen.

Fig. 2
Fig. 2

Three different regimes in the relation between γ 31 and Ω. (a) Low pressure ( γ 31 Ω ) , (b) high pressure ( γ 31 M Ω ) , and (c) medium pressure, two cases: γ 31 Ω and γ 31 M Ω .

Fig. 3
Fig. 3

Intracavity Stokes intensity versus outside input pump intensity. In this simulation γ 31 = 10 MHz and Ω = 844 MHz . The three curves are totally overlapped, which means in the region where γ 31 Ω , there is no gain enhancement for the mode-locked case.

Fig. 4
Fig. 4

Intracavity fields’s magnitude and phase evolution with three equal in-phase pump mode. Since the pump modes are in-phase, we set ( θ p 1 ) initial = ( θ p 2 ) initial = ( θ p 3 ) initial = 0 and they keep the same all the time as seen in the figure. The initial Stokes phase are random, we set ( θ s 1 ) initial = 0.3757 π , ( θ s 2 ) initial = 0.9813 π , and ( θ s 3 ) initial = 0.8186 π . Once the system reaches steady-state, the final phase of Stokes 0.746 π is the vectorial summation of all the Stokes initial phases, mode-locked Stokes formed. At steady-state, with E p i n = 3000 , the total average Stokes intensity is I s = E s 1 2 + E s 2 2 + E s 3 2 = 5.78 × 10 9 .

Fig. 5
Fig. 5

Intracavity fields’s amplitude and phase evolution with three unequal in-phase pump modes. Since the three pump mode are not equal in magnitude, the final Stokes phase are not the vectorial summation of all the Stokes initial phases, but still reach the same phase. At steady-state, with E p i n = 3000 , the total average Stokes intensity ( I s = E s 1 2 + E s 2 2 + E s 3 2 = 5.78 × 10 9 ) is same as the previous case.

Fig. 6
Fig. 6

Intracavity Stokes field magnitudes versus outside input pump field. In this simulation γ 31 = 100 GHz and Ω = 844 MHz . It is obvious that the more modes, the less threshold.

Fig. 7
Fig. 7

Intracavity fields’ amplitude and phase evolution with three in-phase pump modes. Since the pump modes are in-phase, we set ( θ p 1 ) initial = ( θ p 2 ) initial = ( θ p 3 ) initial = 0 . The pump phases do not evolve from their initial value. In this example we set the initial random Stokes phases to ( θ s 1 ) initial = 0.8894 π , ( θ s 2 ) initial = 0.7691 π , and ( θ s 3 ) initial = 0.4161 π . The Stokes’s phases evolve from their initial value to a final value that is the vectorial summation of the initial Stokes phases because of the same amplitude in input pump fields. At steady-state, with E p i n = 3000 , the total average Stokes intensity is ( I s = E s 1 2 + E s 2 2 + E s 3 2 = 1.561 × 10 10 ) .

Fig. 8
Fig. 8

Intracavity Stokes field versus outside input pump field. In this simulation γ 31 = 1 GHz and Ω = 844 MHz . The gain enhancement in this plot is not as big as shown in Fig. 6.

Fig. 9
Fig. 9

Intracavity fields’s magnitude and phase evolution with three in-phase pump modes. Since pump modes are in-phase, we set ( θ p 1 ) initial = ( θ p 2 ) initial = ( θ p 3 ) initial = 0 and they evolve slightly away from their initial values. Here we set the initial random Stokes phases ( θ s 1 ) initial = 0.03 π , ( θ s 2 ) initial = 0.506 π , and ( θ s 3 ) initial = 0.8902 π . When the system reaches steady-state, the Stokes fields are not quite in-phase due to the extra phase terms introduced from off-resonance pieces of coherence, so the peak power and Raman gain are somehow degraded.

Equations (35)

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

E ̃ p = 1 2 [ E p 1 ( t ) e i ω p 1 t + E p 2 ( t ) e i ω p 2 t ] + c.c. ,
E ̃ s = 1 2 [ E s 1 ( t ) e i ω s 1 t + E s 2 ( t ) e i ω s 2 t ] + c.c. ,
E ̇ p 1 = L p 1 E p 1 + i ω p 1 ε 0 μ 12 ρ 21 e i ω p 1 t + K ( E p i n ( 1 ) , t ) ,
E ̇ p 2 = L p 2 E p 2 + i ω p 2 ε 0 μ 12 ρ 21 e i ω p 1 t e i Ω t + K ( E p i n ( 2 ) , t ) ,
E ̇ s 1 = L s 1 E s 1 + i ω s 1 ε 0 μ 23 ρ 23 e i ω s 1 t ,
E ̇ s 2 = L s 2 E s 2 + i ω s 2 ε 0 μ 23 ρ 23 e i ω s 1 t e i Ω t ,
ρ ̇ 21 = ( i ω 21 + 1 T 2 ( 21 ) ) ρ 21 i μ 21 E ̃ p ( ρ 22 ρ 11 ) i μ 23 E ̃ s ρ 31 ,
ρ ̇ 23 = ( i ω 23 + 1 T 2 ( 23 ) ) ρ 23 i μ 23 E ̃ s ( ρ 22 ρ 33 ) i μ 21 E ̃ p ρ 31 * ,
ρ ̇ 31 = ( i ω 31 + 1 T 2 ( 31 ) ) ρ 31 i μ 23 E ̃ s * ρ 21 i μ 21 E ̃ p ρ 23 * ,
D ̇ 21 = 1 T 1 ( 21 ) ( D 21 + 1 ) 4 Im ( μ 21 E ̃ p ρ 21 * ) 2 Im ( μ 23 E ̃ s ρ 23 * ) ,
D ̇ 23 = 1 T 1 ( 21 ) D 23 2 Im ( μ 21 E ̃ p ρ 21 * ) 4 Im ( μ 23 E ̃ s ρ 23 * ) ,
σ ̇ 31 = [ i ( ω p 1 ω s 1 ω 31 ) 1 T 2 ( 31 ) ] σ 31 + i μ 21 μ 23 2 ω 21 [ ( E ̃ s * E ̃ p e i ( ω p 1 ω s 1 ) t ) ] + ( i μ 12 μ 21 2 ω 23 E ̃ p E ̃ p * i μ 32 μ 23 2 ω 21 E ̃ s E ̃ s * ) σ 31 .
σ ̇ 31 = ( i Δ γ 31 ) σ 31 + i g 31 ( E s 1 * E p 1 + E s 2 * E p 2 + E s 1 * E p 2 e i Ω t + E s 2 * E p 1 e i Ω t ) ,
σ 31 = i g 31 [ ( E s 1 * E p 1 + E s 2 * E p 2 ) ( 1 γ 31 ) + ( E s 1 * E p 2 ) ( e i Ω t γ 31 i ω R F ) + ( E s 2 * E p 1 ) ( e i Ω t γ 31 + i ω R F ) ] .
ρ 31 = σ 31 e i ( ω p 1 ω s 1 ) t = σ 31 e i ω 31 t ,
ρ 31 = i g 31 [ ( E s 1 * E p 1 + E s 2 * E p 2 ) ( e i ω 31 t γ 31 ) + ( E s 1 * E p 2 ) ( e i ( ω 31 + Ω ) t γ 31 i ω R F ) + ( E s 2 * E p 1 ) ( e i ( ω 31 Ω ) t γ 31 + i ω R F ) ] .
E ̇ p 1 = L p 1 E p 1 i g p 1 ρ 31 e i ω 31 t ( E p 1 + E p 1 e i Ω t ) + k ( t , E p i n ( 1 ) ) ,
E ̇ p 2 = L p 2 E p 2 i g p 2 ρ 31 e i ω 31 t ( E p 1 e i Ω t + E p 2 ) + k ( t , E p i n ( 2 ) ) ,
E ̇ s 1 = L s 1 E s 1 i g s 1 ( ρ 31 e i ω 31 t ) * ( E p 1 + E p 2 e i Ω t ) ,
E ̇ s 2 = L s 2 E s 2 i g s 2 ( ρ 31 e i ω 31 t ) * ( E p 1 e i Ω t + E p 2 ) ,
E ̇ p 1 = L p 1 E p 1 g p 1 g 31 [ E s 1 ( E p 1 E s 1 * + E p 2 E s 2 * ) γ 31 + E s 2 ( E p 1 E s 2 * ) γ 31 + i Ω ] + K ( E p i n 1 , t ) ,
E ̇ p 2 = L p 2 E p 2 g p 2 g 31 [ E s 2 ( E p 1 E s 1 * + E p 2 E s 2 * ) γ 31 + E s 1 ( E p 2 E s 1 * ) γ 31 i Ω ] + K ( E p i n 2 , t ) ,
E ̇ s 1 = L s 1 E s 1 + g s 1 g 31 [ E p 1 ( E p 1 * E s 1 + E p 2 * E s 2 ) γ 31 + E p 2 ( E p 2 * E s 1 ) γ 31 + i Ω ] ,
E ̇ s 2 = L s 2 E s 2 + g s 2 g 31 [ E p 2 ( E p 1 * E s 1 + E p 2 * E s 2 ) γ 31 + E p 1 ( E p 1 * E s 2 ) γ 31 i Ω ] .
σ 31 = i g 31 β = M + 1 M 1 e i β Ω t α = 1 M E s α * E p ( α β ) γ 31 + i β Ω ,
E ̇ p n = L p n E p n g p n g 31 β = 1 M E s β α = 1 M E s α * E p [ α ( β n ) ] γ 31 + i ( β n ) Ω + K ( E p i n ( n ) , t ) ,
E ̇ s n = L s n E s n + g s n g 31 β = 1 M E p β α = 1 M ( E s [ α ( β n ) ] * E p α ) * γ 31 + i ( β n ) Ω ,
σ 31 = i g 31 α = 1 M E s α * E p α γ 31 ,
E ̇ p n = L p n E p n g p n g 31 E s n α = 1 M E s α * E p α γ 31 + K ( E p i n ( n ) , t ) ,
E ̇ s n = L s n E s n + g s n g 31 E p n α = 1 M ( E s α * E p α ) * γ 31 .
σ 31 = i g 31 β = M + 1 M 1 e i β Ω t α = 1 M E s α * E p ( α β ) γ 31 ,
E ̇ p n = L p n E p n + K ( E p i n ( n ) , t ) g p n g 31 β = 1 M E s β α = 1 M E s α * E p [ α ( β n ) ] γ 31 ,
E ̇ s n = L s n E s n + g s n g 31 β = 1 M E p β α = 1 M ( E s [ α ( β n ) ] * E p α ) * γ 31 .
σ 31 = e t γ 0 d x [ t e z γ 0 d x i ( A + B 1 e i ω 1 z + B 2 e i ω 1 z + C 1 e i ω 2 z + C 2 e i ω 2 z ) d z + C ] = i e γ 0 t [ t e γ 0 z ( A + B 1 e i ω 1 z + B 2 e i ω 1 z + C 1 e i ω 2 z + C 2 e i ω 2 z ) d z + C ] = i e γ 0 t [ A e γ 0 t γ 0 + B 1 e ( γ 0 + ω 1 ) t ( γ 0 + ω 1 ) + B 2 e ( γ 0 ω 1 ) t ( γ 0 ω 1 ) + C 1 e ( γ 0 + ω 2 ) t ( γ 0 + ω 2 ) + C 2 e ( γ 0 ω 2 ) t ( γ 0 ω 2 ) ] .
σ 31 = i e γ 0 t [ A e γ 0 t γ 0 + B 1 e ( γ 0 + ω 1 ) t ( γ 0 + ω 1 ) + B 2 e ( γ 0 ω 1 ) t ( γ 0 ω 1 ) ] .

Metrics