Abstract

Stability in a Raman ring resonator is studied by using the adiabatic approximation. The analysis is based on far-off resonance Raman scattering in hydrogen. A medium-power approximation is employed, which is good for intensities less than 30 MW/cm2. The resulting differential equations retain their standard low-power Raman gain dependance in addition to an intensity-dependent phase. The steady-state intensity input–output behavior, as well as the linear stability analysis, is accomplished analytically without invoking the mean-field approximation. Feedback is applied to the Stokes beam, a gain situation, or to the depleted pump beam. The Stokes frequency is assumed to be perfectly tuned to the atomic and cavity resonances. It is shown that both situations are multistable and that the power-dependent phase largely determines the stability characteristics. Furthermore, we show that the negative slope branches can be stable when feedback is applied to the pump if the output pump intensity is decreasing with increasing input pump intensity.

© 1990 Optical Society of America

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References

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  1. H. J. Carmichael, in Optical Instabilities, R. W. Boyd, M. G. Raymer, L. M. Narducci, eds. (Cambridge U. Press, Cambridge, 1986), pp. 111–128.
  2. R. Bonifacio, L. Lugiato, Lett. Nuovo Cimento 21, 510 (1978).
    [CrossRef]
  3. J. A. Hermann, B. V. Thompson, in Optical Bistability, C. M. Bowden, M. Ciftan, H. R. Robl, eds. (Plenum, New York, 1981), pp. 199–220;J. A. Hermann, Opt. Commun. 44, 62 (1982);J. A. Hermann, J. N. Elgin, P. L. Knight, J. Phys. B 45, 255 (1982);J. A. Hermann, D. F. Walls, Phys. Rev. A 26, 2085 (1982);J. A. Hermann, B. V. Thompson, Opt. Lett. 7, 301 (1982);J. A. Hermann, Opt. Commun. 37, 431 (1981);B. V. Thompson, J. A. Hermann, Phys. Lett. 83A, 376 (1981);J. A. Hermann, J. N. Elgin, Phys. Lett. 86A, 461 (1981).
    [CrossRef] [PubMed]
  4. P. Alsing, P. R. Peterson, D. A. Cardimona, A. Gavrielides, IEEE J. Quantum Electron. QE-23, 557 (1987).
    [CrossRef]
  5. E. Abraham, W. J. Firth, Opt. Acta 30, 1541 (1983).
    [CrossRef]

1987 (1)

P. Alsing, P. R. Peterson, D. A. Cardimona, A. Gavrielides, IEEE J. Quantum Electron. QE-23, 557 (1987).
[CrossRef]

1983 (1)

E. Abraham, W. J. Firth, Opt. Acta 30, 1541 (1983).
[CrossRef]

1978 (1)

R. Bonifacio, L. Lugiato, Lett. Nuovo Cimento 21, 510 (1978).
[CrossRef]

Abraham, E.

E. Abraham, W. J. Firth, Opt. Acta 30, 1541 (1983).
[CrossRef]

Alsing, P.

P. Alsing, P. R. Peterson, D. A. Cardimona, A. Gavrielides, IEEE J. Quantum Electron. QE-23, 557 (1987).
[CrossRef]

Bonifacio, R.

R. Bonifacio, L. Lugiato, Lett. Nuovo Cimento 21, 510 (1978).
[CrossRef]

Cardimona, D. A.

P. Alsing, P. R. Peterson, D. A. Cardimona, A. Gavrielides, IEEE J. Quantum Electron. QE-23, 557 (1987).
[CrossRef]

Carmichael, H. J.

H. J. Carmichael, in Optical Instabilities, R. W. Boyd, M. G. Raymer, L. M. Narducci, eds. (Cambridge U. Press, Cambridge, 1986), pp. 111–128.

Firth, W. J.

E. Abraham, W. J. Firth, Opt. Acta 30, 1541 (1983).
[CrossRef]

Gavrielides, A.

P. Alsing, P. R. Peterson, D. A. Cardimona, A. Gavrielides, IEEE J. Quantum Electron. QE-23, 557 (1987).
[CrossRef]

Hermann, J. A.

J. A. Hermann, B. V. Thompson, in Optical Bistability, C. M. Bowden, M. Ciftan, H. R. Robl, eds. (Plenum, New York, 1981), pp. 199–220;J. A. Hermann, Opt. Commun. 44, 62 (1982);J. A. Hermann, J. N. Elgin, P. L. Knight, J. Phys. B 45, 255 (1982);J. A. Hermann, D. F. Walls, Phys. Rev. A 26, 2085 (1982);J. A. Hermann, B. V. Thompson, Opt. Lett. 7, 301 (1982);J. A. Hermann, Opt. Commun. 37, 431 (1981);B. V. Thompson, J. A. Hermann, Phys. Lett. 83A, 376 (1981);J. A. Hermann, J. N. Elgin, Phys. Lett. 86A, 461 (1981).
[CrossRef] [PubMed]

Lugiato, L.

R. Bonifacio, L. Lugiato, Lett. Nuovo Cimento 21, 510 (1978).
[CrossRef]

Peterson, P. R.

P. Alsing, P. R. Peterson, D. A. Cardimona, A. Gavrielides, IEEE J. Quantum Electron. QE-23, 557 (1987).
[CrossRef]

Thompson, B. V.

J. A. Hermann, B. V. Thompson, in Optical Bistability, C. M. Bowden, M. Ciftan, H. R. Robl, eds. (Plenum, New York, 1981), pp. 199–220;J. A. Hermann, Opt. Commun. 44, 62 (1982);J. A. Hermann, J. N. Elgin, P. L. Knight, J. Phys. B 45, 255 (1982);J. A. Hermann, D. F. Walls, Phys. Rev. A 26, 2085 (1982);J. A. Hermann, B. V. Thompson, Opt. Lett. 7, 301 (1982);J. A. Hermann, Opt. Commun. 37, 431 (1981);B. V. Thompson, J. A. Hermann, Phys. Lett. 83A, 376 (1981);J. A. Hermann, J. N. Elgin, Phys. Lett. 86A, 461 (1981).
[CrossRef] [PubMed]

IEEE J. Quantum Electron. (1)

P. Alsing, P. R. Peterson, D. A. Cardimona, A. Gavrielides, IEEE J. Quantum Electron. QE-23, 557 (1987).
[CrossRef]

Lett. Nuovo Cimento (1)

R. Bonifacio, L. Lugiato, Lett. Nuovo Cimento 21, 510 (1978).
[CrossRef]

Opt. Acta (1)

E. Abraham, W. J. Firth, Opt. Acta 30, 1541 (1983).
[CrossRef]

Other (2)

J. A. Hermann, B. V. Thompson, in Optical Bistability, C. M. Bowden, M. Ciftan, H. R. Robl, eds. (Plenum, New York, 1981), pp. 199–220;J. A. Hermann, Opt. Commun. 44, 62 (1982);J. A. Hermann, J. N. Elgin, P. L. Knight, J. Phys. B 45, 255 (1982);J. A. Hermann, D. F. Walls, Phys. Rev. A 26, 2085 (1982);J. A. Hermann, B. V. Thompson, Opt. Lett. 7, 301 (1982);J. A. Hermann, Opt. Commun. 37, 431 (1981);B. V. Thompson, J. A. Hermann, Phys. Lett. 83A, 376 (1981);J. A. Hermann, J. N. Elgin, Phys. Lett. 86A, 461 (1981).
[CrossRef] [PubMed]

H. J. Carmichael, in Optical Instabilities, R. W. Boyd, M. G. Raymer, L. M. Narducci, eds. (Cambridge U. Press, Cambridge, 1986), pp. 111–128.

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Figures (7)

Fig. 1
Fig. 1

Ring resonator immersed in hydrogen. The incident Stokes and pump fields are Sin and Pin, and the transmitted fields are St and Pt. Two mirrors are 100% reflecting at both wavelengths. The input mirror reflects the Stokes and transmits the pump for feedback on the Stokes.

Fig. 2
Fig. 2

(a) Input Stokes intensity versus output Stokes intensity for a reflectivity of R = 0.01 and an input pump intensity of 10 MW/ cm2. In one curve the dark solid portions are stable, and the negative slope regions are unstable. The light solid curve is the output versus input amplifier equation. The feedback is on the Stokes. (b) Eigenvalues versus output Stokes intensity. Parameters are the. same as for (a).

Fig. 3
Fig. 3

Input Stokes intensity versus output Stokes intensity for a reflectivity of R = 0.01 and an input pump intensity of (a) 5, (b) 10, and (c) 20 MW/cm2. Feedback is on the Stokes.

Fig. 4
Fig. 4

Input Stokes intensity versus output Stokes intensity for a reflectivity of R = 0.25 and an input pump intensity of 5, 10, and 20 MW/cm2. Feedback is on the Stokes.

Fig. 5
Fig. 5

(a) Input pump intensity versus output pump intensity for a reflectivity of R = 0.01 and an input Stokes intensity of 10 MW/ cm2. In one curve the dark solid portions are stable, and the dashed are unstable. The light solid curve is the output versus the input amplifier equation. The feedback is on the pump. (b) Eigenvalues versus output pump intensity. Parameters are the same as for (a).

Fig. 6
Fig. 6

Input pump intensity versus output pump intensity for a reflectivity of R = 0.01 and input Stokes intensities of 5, 10, and 20 MW/cm2. Feedback is on the pump.

Fig. 7
Fig. 7

Input pump intensity versus output pump intensity for a reflectivity of R = 0.25 and input Stokes intensities of 5, 10, and 20 MW/cm2. Feedback is on the pump.

Equations (43)

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d d t Q + Γ Q = i Δ Q + i Ω * W ,
d d t W = i 2 ( Ω Q Ω * Q * ) γ ( W + 1 ) ,
Δ = Δ s + δ p | P | 2 + δ s | S | 2 , Ω = 2 δ p s P * S .
Q s s = γ D ( Δ i Γ ) Ω * ,
W s s = γ D ( Γ 2 + Δ 2 ) ,
D = γ ( Δ 2 + Γ 2 ) + Γ | Ω | 2 .
d d ξ S = i 2 [ n 3 ( ω 3 ) 1 ] k s ( 1 + W ) S i 2 [ n 1 ( ω s ) 1 ] k s ( 1 W ) S i κ Q * P
d d ξ P = i 2 [ n 1 ( ω p ) 1 ] k p ( 1 W ) P i 2 [ n 3 ( ω p ) 1 ] k p ( 1 + W ) P i κ ( k p / k s ) Q S ,
d d ξ S ( z , t ) = i α s | P | 2 | S | 2 S + β s | P | 2 S ,
d d ξ P ( z , t ) = i α p | P | 2 | S | 2 P β p | S | 2 P ,
α s = [ n 3 ( ω s ) n 1 ( ω s ) ] k s 2 δ p s 2 / γ Γ ,
α p = [ n 3 ( ω p ) n 1 ( ω p ) ] k p 2 δ p s 2 / γ Γ ,
β s = 2 κ δ p s / Γ , β p = β s k p / k s .
I s ( z ) = I s 0 ( 1 + ω p I s 0 ω s I p 0 ) exp ( 2 g I z ) 1 + ω p I s 0 ω s I p 0 exp ( 2 g I z ) ,
I p ( z ) = I p 0 ( 1 + ω p I s 0 ω s I p 0 ) 1 + ω p I s 0 ω s I p 0 exp ( 2 g I z ) .
ψ s ( z ) = 2 π α s Γ × 10 13 δ p s κ c [ I s 0 I s ( z ) ] ,
ψ p ( z ) = 2 π α p Γ × 10 13 δ p s κ c [ I s 0 I s ( z ) ] ,
S ( 0 , t ) = T S i + R S ( L , t ) .
S ( z , t ) = S ( z ) + δ S ( z , t )
P ( z , t ) = P ( z ) + δ P ( z , t ) ,
d δ S d ξ = i α s [ S | P | 2 ( S * δ S + S δ S * ) + S | S | 2 ( P * δ P + P δ p * ) + | P | 2 | S | 2 δ S ] + β s [ S ( P * δ P + P δ P * ) + | P | 2 δ S ]
d δ P d ξ = i α p [ P | S | 2 ( P * δ P + P δ P * ) + P | P | 2 ( S * δ S + S δ S * ) + | P | 2 | S | 2 δ P ] β p [ P ( S * δ S + S δ S * ) + | S | 2 δ P ] .
d S + d ξ = S * d δ S d ξ ± S d δ S * d ξ + δ S d S * d z ± δ S * d S d z
d S + d ξ = 2 β s ( | S | 2 P + + | P | 2 S + ) ,
d S d ξ = i α s ( 2 | P | 2 | S | 2 S + + 2 | S | 2 | S | 2 P + ) + 2 β s | P | 2 S ,
d P + d ξ = 2 β p ( | P | 2 S + + | S | 2 P + ) ,
d P d ξ = i α p ( 2 | P | 2 | S | 2 P + + 2 | P | 2 | P | 2 S + ) 2 β p | S | 2 P ,
S + ( ξ ) β s + P + ( ξ ) β p = constant ,
| S | 2 k s + | P | 2 k p = constant = 8 π c ( I s 0 k s + I p 0 k p ) .
S + ( ξ ) = S + ( z ) [ exp ( λ τ ) + exp ( λ * τ ) ]
S ( ξ ) = i S ( z ) [ exp ( λ τ ) + exp ( λ * τ ) ] ,
d S + ( z ) d z = 2 β s ( | S | 2 P + + | P | 2 S + ) ,
d S ( z ) d z = i α s β s | S | 2 d S + d z + 2 β s | P | 2 S ,
d P + ( z ) d z = 2 β p ( | P | 2 S + + | S | 2 P + ) ,
d P ( z ) d z = i α p β p | P | 2 d P + d z 2 β p | S | 2 P ,
S + ( z ) β s + P + ( z ) β p = constant = S + ( 0 ) β s ,
S + ( z ) = f ( z ) S + ( 0 ) ,
f ( z ) = υ exp ( 2 g I z ) [ 1 + υ exp ( 2 g I z ) ] 2 [ 2 g I z + υ exp ( 2 g I z ) + ( 1 + 2 υ ) υ ] ,
δ S = ( S + + S ) / ( 2 S * ) .
S ( z ) = α s β s | S ( z ) | 2 [ 1 f ( z ) ] S + ( 0 ) + | S ( z ) | 2 | S ( 0 ) | 2 S ( 0 ) .
exp ( 2 λ L / c ) 1 R f ( L ) [ I s 0 I s ( L ) ] 1 / 2 { ( I s 0 I s ( L ) + f ) cos [ ψ s ( L ) ] α s β s ( 8 π × 10 13 / c ) I s ( L ) ( 1 f ) sin [ ψ s ( L ) ] } × exp ( λ L / c ) + 1 R f = 0 .
exp ( 2 λ L / c ) 1 R f ( L ) [ I p 0 I p ( L ) ] 1 / 2 { [ I p 0 I p ( L ) + f ] cos [ ψ p ( L ) ] α p β p ( 8 π × 10 13 / c ) I p ( L ) ( 1 f ) sin [ ψ p ( L ) ] } × exp ( λ L / c ) + 1 R f = 0 ,
f ( z ) = υ exp ( 2 g I z ) [ 1 + υ exp ( 2 g I z ) ] 2 { 2 g I z + [ υ exp ( 2 g I z ) ] 1 + ( 2 + υ ) } .

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