Abstract

Employing the small-signal analysis, we study the bidirectional operating characteristics in a Raman ring laser. Bidirectional operation is found not stable in a silicon ring because of the existence of two-photon absorption. Using polar crystals which have different forward and backward Raman gain or introducing external optical injections helps to establish stable bidirectional operation in a Raman ring laser.

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  1. H. Rong, Y. H. Kuo, S. Xu, A. Liu, R. Jones, M. Paniccia, O. Cohen, and O. Raday, “Monolithic integrated Raman silicon laser,” Opt. Express 14(15), 6705–6712 (2006).
    [CrossRef] [PubMed]
  2. F. De Leonardis and V. M. N. Passaro, “Modeling and Performance of a Guided-Wave Optical Angular-Velocity Sensor Based on Raman Effect in SOI,” J. Lightwave Technol. 25(9), 2352–2366 (2007).
    [CrossRef]
  3. W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57(1), 61–104 (1985).
    [CrossRef]
  4. M. Krause, H. Renner, and E. Brinkmeyer, “Analysis of Raman lasing characteristics in silicon-on-insulator waveguides,” Opt. Express 12(23), 5703–5710 (2004).
    [CrossRef] [PubMed]
  5. R. Loudon, “The Raman effect in crystals,” Adv. Phys. 50(7), 813–864 (2001).
    [CrossRef]
  6. T. Saito, K. Suto, J. Nishizawa, and M. Kawasaki, “Spontaneous Raman scattering in [100], [110], and [11-2] directional GaP waveguides,” J. Appl. Phys. 90(4), 1831–1835 (2001).
    [CrossRef]
  7. H. Rong, A. Liu, R. Nicolaescu, M. Paniccia, O. Cohen, and D. Hak, “Raman gain and nonlinear optical absorption measurements in a low-loss silicon waveguide,” Appl. Phys. Lett. 85(12), 2196–2198 (2004).
    [CrossRef]
  8. F. De Leonardis, V. Dimastrodonato, and V. M. N. Passaro, “Modelling of a DBR laser based on Raman effect in a silicon-on-insulator rib waveguide,” Semicond. Sci. Technol. 23(6), 064008 (2008).
    [CrossRef]

2008

F. De Leonardis, V. Dimastrodonato, and V. M. N. Passaro, “Modelling of a DBR laser based on Raman effect in a silicon-on-insulator rib waveguide,” Semicond. Sci. Technol. 23(6), 064008 (2008).
[CrossRef]

2007

2006

2004

M. Krause, H. Renner, and E. Brinkmeyer, “Analysis of Raman lasing characteristics in silicon-on-insulator waveguides,” Opt. Express 12(23), 5703–5710 (2004).
[CrossRef] [PubMed]

H. Rong, A. Liu, R. Nicolaescu, M. Paniccia, O. Cohen, and D. Hak, “Raman gain and nonlinear optical absorption measurements in a low-loss silicon waveguide,” Appl. Phys. Lett. 85(12), 2196–2198 (2004).
[CrossRef]

2001

R. Loudon, “The Raman effect in crystals,” Adv. Phys. 50(7), 813–864 (2001).
[CrossRef]

T. Saito, K. Suto, J. Nishizawa, and M. Kawasaki, “Spontaneous Raman scattering in [100], [110], and [11-2] directional GaP waveguides,” J. Appl. Phys. 90(4), 1831–1835 (2001).
[CrossRef]

1985

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57(1), 61–104 (1985).
[CrossRef]

Brinkmeyer, E.

Chow, W. W.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57(1), 61–104 (1985).
[CrossRef]

Cohen, O.

H. Rong, Y. H. Kuo, S. Xu, A. Liu, R. Jones, M. Paniccia, O. Cohen, and O. Raday, “Monolithic integrated Raman silicon laser,” Opt. Express 14(15), 6705–6712 (2006).
[CrossRef] [PubMed]

H. Rong, A. Liu, R. Nicolaescu, M. Paniccia, O. Cohen, and D. Hak, “Raman gain and nonlinear optical absorption measurements in a low-loss silicon waveguide,” Appl. Phys. Lett. 85(12), 2196–2198 (2004).
[CrossRef]

De Leonardis, F.

F. De Leonardis, V. Dimastrodonato, and V. M. N. Passaro, “Modelling of a DBR laser based on Raman effect in a silicon-on-insulator rib waveguide,” Semicond. Sci. Technol. 23(6), 064008 (2008).
[CrossRef]

F. De Leonardis and V. M. N. Passaro, “Modeling and Performance of a Guided-Wave Optical Angular-Velocity Sensor Based on Raman Effect in SOI,” J. Lightwave Technol. 25(9), 2352–2366 (2007).
[CrossRef]

Dimastrodonato, V.

F. De Leonardis, V. Dimastrodonato, and V. M. N. Passaro, “Modelling of a DBR laser based on Raman effect in a silicon-on-insulator rib waveguide,” Semicond. Sci. Technol. 23(6), 064008 (2008).
[CrossRef]

Gea-Banacloche, J.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57(1), 61–104 (1985).
[CrossRef]

Hak, D.

H. Rong, A. Liu, R. Nicolaescu, M. Paniccia, O. Cohen, and D. Hak, “Raman gain and nonlinear optical absorption measurements in a low-loss silicon waveguide,” Appl. Phys. Lett. 85(12), 2196–2198 (2004).
[CrossRef]

Jones, R.

Kawasaki, M.

T. Saito, K. Suto, J. Nishizawa, and M. Kawasaki, “Spontaneous Raman scattering in [100], [110], and [11-2] directional GaP waveguides,” J. Appl. Phys. 90(4), 1831–1835 (2001).
[CrossRef]

Krause, M.

Kuo, Y. H.

Liu, A.

H. Rong, Y. H. Kuo, S. Xu, A. Liu, R. Jones, M. Paniccia, O. Cohen, and O. Raday, “Monolithic integrated Raman silicon laser,” Opt. Express 14(15), 6705–6712 (2006).
[CrossRef] [PubMed]

H. Rong, A. Liu, R. Nicolaescu, M. Paniccia, O. Cohen, and D. Hak, “Raman gain and nonlinear optical absorption measurements in a low-loss silicon waveguide,” Appl. Phys. Lett. 85(12), 2196–2198 (2004).
[CrossRef]

Loudon, R.

R. Loudon, “The Raman effect in crystals,” Adv. Phys. 50(7), 813–864 (2001).
[CrossRef]

Nicolaescu, R.

H. Rong, A. Liu, R. Nicolaescu, M. Paniccia, O. Cohen, and D. Hak, “Raman gain and nonlinear optical absorption measurements in a low-loss silicon waveguide,” Appl. Phys. Lett. 85(12), 2196–2198 (2004).
[CrossRef]

Nishizawa, J.

T. Saito, K. Suto, J. Nishizawa, and M. Kawasaki, “Spontaneous Raman scattering in [100], [110], and [11-2] directional GaP waveguides,” J. Appl. Phys. 90(4), 1831–1835 (2001).
[CrossRef]

Paniccia, M.

H. Rong, Y. H. Kuo, S. Xu, A. Liu, R. Jones, M. Paniccia, O. Cohen, and O. Raday, “Monolithic integrated Raman silicon laser,” Opt. Express 14(15), 6705–6712 (2006).
[CrossRef] [PubMed]

H. Rong, A. Liu, R. Nicolaescu, M. Paniccia, O. Cohen, and D. Hak, “Raman gain and nonlinear optical absorption measurements in a low-loss silicon waveguide,” Appl. Phys. Lett. 85(12), 2196–2198 (2004).
[CrossRef]

Passaro, V. M. N.

F. De Leonardis, V. Dimastrodonato, and V. M. N. Passaro, “Modelling of a DBR laser based on Raman effect in a silicon-on-insulator rib waveguide,” Semicond. Sci. Technol. 23(6), 064008 (2008).
[CrossRef]

F. De Leonardis and V. M. N. Passaro, “Modeling and Performance of a Guided-Wave Optical Angular-Velocity Sensor Based on Raman Effect in SOI,” J. Lightwave Technol. 25(9), 2352–2366 (2007).
[CrossRef]

Pedrotti, L. M.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57(1), 61–104 (1985).
[CrossRef]

Raday, O.

Renner, H.

Rong, H.

H. Rong, Y. H. Kuo, S. Xu, A. Liu, R. Jones, M. Paniccia, O. Cohen, and O. Raday, “Monolithic integrated Raman silicon laser,” Opt. Express 14(15), 6705–6712 (2006).
[CrossRef] [PubMed]

H. Rong, A. Liu, R. Nicolaescu, M. Paniccia, O. Cohen, and D. Hak, “Raman gain and nonlinear optical absorption measurements in a low-loss silicon waveguide,” Appl. Phys. Lett. 85(12), 2196–2198 (2004).
[CrossRef]

Saito, T.

T. Saito, K. Suto, J. Nishizawa, and M. Kawasaki, “Spontaneous Raman scattering in [100], [110], and [11-2] directional GaP waveguides,” J. Appl. Phys. 90(4), 1831–1835 (2001).
[CrossRef]

Sanders, V. E.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57(1), 61–104 (1985).
[CrossRef]

Schleich, W.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57(1), 61–104 (1985).
[CrossRef]

Scully, M. O.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57(1), 61–104 (1985).
[CrossRef]

Suto, K.

T. Saito, K. Suto, J. Nishizawa, and M. Kawasaki, “Spontaneous Raman scattering in [100], [110], and [11-2] directional GaP waveguides,” J. Appl. Phys. 90(4), 1831–1835 (2001).
[CrossRef]

Xu, S.

Adv. Phys.

R. Loudon, “The Raman effect in crystals,” Adv. Phys. 50(7), 813–864 (2001).
[CrossRef]

Appl. Phys. Lett.

H. Rong, A. Liu, R. Nicolaescu, M. Paniccia, O. Cohen, and D. Hak, “Raman gain and nonlinear optical absorption measurements in a low-loss silicon waveguide,” Appl. Phys. Lett. 85(12), 2196–2198 (2004).
[CrossRef]

J. Appl. Phys.

T. Saito, K. Suto, J. Nishizawa, and M. Kawasaki, “Spontaneous Raman scattering in [100], [110], and [11-2] directional GaP waveguides,” J. Appl. Phys. 90(4), 1831–1835 (2001).
[CrossRef]

J. Lightwave Technol.

Opt. Express

Rev. Mod. Phys.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57(1), 61–104 (1985).
[CrossRef]

Semicond. Sci. Technol.

F. De Leonardis, V. Dimastrodonato, and V. M. N. Passaro, “Modelling of a DBR laser based on Raman effect in a silicon-on-insulator rib waveguide,” Semicond. Sci. Technol. 23(6), 064008 (2008).
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Theory and numerical simulation results illustrating the occurrence of the two different ring laser regimes (unidirectional and bidirectional). The left diagram is the small-signal analysis results from (22). Figures A, B, C, and D are, respectively, the simulation results corresponding to the points A, B, C, and D in the left diagram. The numerical simulation is in the case of symmetric pumped. Values of other parameters, common to these diagrams, are: λ s = 1550nm, λ p = 1646nm, φ ¯ = 6×10−10, τ e f f = 0.1ns, β = 0.6cm2/GW, v g τ p = 2.35cm, v g τ s = 8cm, I s e x t = 0, I p e x t / v g τ p c p = 0.15GW/cm3, and the corresponding I p ( 0 ) = 0.1648GW/cm2.

Fig. 2
Fig. 2

Simulation results ring laser regimes with injections at Stokes wavelength: I s e x t / v g τ s c p = 2kW/cm3 (left) and 3kW/cm3 (right), κ f g R / β = κ b g R / β = 5 and values of other parameters are the same with those in Fig. 1.

Equations (25)

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I ˙ p , ± = ( v g / L ) η p ( I p , ± e x t I p , ± ) I p , ± / τ p c v { v g [ ( λ s / λ p ) g R ( κ f I s , ± + κ b I s , ) + β ( I p , ± + 2 I p , + 2 I s , + + 2 I s , ) + φ ¯ λ p 2 N e f f ] } I p , ± I ˙ s , ± = ( v g / L ) η s ( I s , ± e x t I s , ± ) I s , ± / τ s c v + { v g [ g R ( κ f I p , ± + κ b I p , ) β ( I s , ± + 2 I s , + 2 I p , + + 2 I p , ) φ ¯ λ s 2 N e f f ] } I s , ±
N e f f = β τ e f f 2 ω p ( I p , + + I p , + I s , + + I s , ) 2
k = λ s / λ p , G = v g g R , B = v g β , T = L / v g , F = φ ¯ λ p 2 B τ e f f / ( 2 ω p ) , τ p , s c p = η p . s / T , 1 / τ p , s = 1 / τ p , s c v + 1 / τ p , s c p
I ˙ p , ± = I p , ± e x t / τ p c p I p , ± / τ p + [ k G ( κ f I s , ± + κ b I s , ) B ( I p , ± + 2 I p , + 2 I s , + + 2 I s , ) F ( I p , + + I p , + I s , + + I s , ) 2 ] I p , ± I ˙ s , ± = I s , ± e x t / τ s c p I s , ± / τ s + [ G ( κ f I p , ± + κ b I p , ) B ( I s , ± + 2 I s , + 2 I p , + + 2 I p , ) k 2 F ( I p , + + I p , + I s , + + I s , ) 2 ] I s , ±
( 1 / τ p 1 / τ s + G W + + F U + ) a + + + 2 V I p , + ( 0 ) a + + ( κ f k G + 2 V ) I p , + ( 0 ) = 0 ( 1 / τ p 1 / τ s + G W + + F U ) a + + 2 V I p , ( 0 ) a + + + ( κ b k G + 2 V ) I p , ( 0 ) = 0 ( 1 / τ p 1 / τ s + G W + F U + ) a + + 2 V I p , + ( 0 ) a + ( κ b k G + 2 V ) I p , + ( 0 ) = 0 ( 1 / τ p 1 / τ s + G W + F U ) a + 2 V I p , ( 0 ) a + + ( κ f k G + 2 V ) I p , ( 0 ) = 0
U ± = ( I p , + ( 0 ) + I p , ( 0 ) ) [ ( 1 k 2 ) ( I p , + ( 0 ) + I p , ( 0 ) ) + 2 I p , ± ( 0 ) ] W ± = κ f I p , ± ( 0 ) + κ b I p , ( 0 ) , V = B + F ( I p , + ( 0 ) + I p , ( 0 ) )
[ 1 / τ p + B ( 2 I p , ( 0 ) + I p , ± ( 0 ) ) + F ( I p , + ( 0 ) + I p , ( 0 ) ) 2 ] I p , ± ( 0 ) = I p , ± e x t / τ p c p a ± + I s , + e x t / τ s c p a ± I s , e x t / τ s c p
I ˙ s , ± = I s , ± e x t / τ s c p + { ρ ± + B I s , ± + [ G ( κ f a ± + + κ b a + ) θ + ] I s , + + [ G ( κ f a ± + κ b a ) θ ] I s , } I s , ±
ρ ± = 1 / τ s + G ( κ f I p , ± ( 0 ) + κ b I p , ( 0 ) ) 2 B ( I p , + ( 0 ) + I p , ( 0 ) ) k 2 F ( I p , + ( 0 ) + I p , ( 0 ) ) 2
θ ± = 2 B ( 1 + a + ± + a ± ) + 2 k 2 F [ ( a + ± + a ± + 1 ) I p , + ( 0 ) + ( a ± + a + ± + 1 ) I p , ( 0 ) ]
I ˙ s , ± = ( ρ α s l f I s , ± α c r s I s , ) I s , ± + I s e x t / τ s c p
α s l f = G [ ( κ f + κ b ) a B + ( κ f 2 + κ b 2 ) a G + ( κ f κ b ) 2 a Δ ] + θ B α c r s = G [ ( κ f + κ b ) a B + 2 κ f κ b a G ( κ f κ b ) 2 a Δ ] + θ
ρ = 1 / τ s + ( κ f + κ b ) G I p ( 0 ) 4 B I p ( 0 ) 4 k 2 F ( I p ( 0 ) ) 2 θ = 2 [ 1 2 a B ( κ f + κ b ) a G ] ( B + 2 k 2 F I p ( 0 ) ) a B = 2 V I p ( 0 ) / ( 1 / τ p + ρ + 6 V I p ( 0 ) ) , a G = k G I p ( 0 ) / ( 1 / τ p + ρ + 6 V I p ( 0 ) ) , a Δ = 2 V k G ( I p ( 0 ) ) 2 / [ ( 1 / τ p + ρ + 6 V I p ( 0 ) ) ( 1 / τ p + ρ + 2 V I p ( 0 ) ) ]
( ρ α s l f I s , ± ( s ) α c r s I s , ( s ) ) I s , ± ( s ) = 0
I s , ± = I s , ± ( s ) + ε ±
d d t ( ε + ε ) = ( α s l f I s , + ( s ) α c r s I s , + ( s ) α c r s I s , ( s ) α s l f I s , ( s ) ) ( ε + ε ) Θ ( ε + ε )
α s l f 2 α c r s 2 > 0 , α s l f > 0
α s l f α c r s = G ( κ f κ b ) 2 ( a G + 2 a Δ ) B = ( κ f κ b ) 2 k G 2 I p ( 0 ) 1 / τ p + ρ + 2 V I p ( 0 ) B
( κ f κ b ) 2 k G 2 I p ( 0 ) B ( 1 / τ p + ρ + 2 V I p ( 0 ) ) > 0
0 < ( κ f + κ b ) G [ 1 / ( τ s I p ( 0 ) ) + 4 B + 4 k 2 F I p ( 0 ) ] < ( κ f κ b ) 2 k G 2 / B [ 1 / ( τ p I p ( 0 ) ) + 2 B + 4 F I p ( 0 ) ]
ρ = 1 / τ s + 2 G I p ( 0 ) 4 B I p ( 0 ) 4 k 2 F ( I p ( 0 ) ) 2 , α c r s = α s l f + B = 2 G ( a B + a G ) + θ
0 = ( 2 α s l f + B ) ( I s ( s ) ) 2 + ρ I s ( s ) + I s e x t / τ s c p
Θ = ( A + α s l f I s ( s ) α c r s I s ( s ) α c r s I s ( s ) A + α s l f I s ( s ) ) , where A = I s e x t / ( I s ( s ) τ s c p )
I s e x t / τ s c p > B ( I s ( s ) ) 2
I s ( s ) > ρ / ( 2 α s l f ) , I s e x t > B ρ 2 τ s c p / ( 4 α s l f 2 )

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