Abstract

We present approximate analytical expressions describing nonlinear interaction of two optical waves copropagating inside a silicon waveguide in the presence of linear losses, stimulated Raman scattering, and free-carrier absorption. Our approach avoids the undepleted-pump approximation, which we show to be inadequate to describe accurately the Raman-gain process. Based on our calculations, we propose a new generalized definition for the effective length and show that it provides better insight into the impact of nonlinear absorption on Raman amplification in silicon waveguides.

© 2009 Optical Society of America

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  1. B. Jalali, V. Raghunathan, D. Dimitropoulos, and O. Boyraz, IEEE J. Sel. Top. Quantum Electron. 12, 412 (2006).
    [CrossRef]
  2. R. Claps, D. Dimitropoulos, V. Raghunathan, Y. Han, and B. Jalali, Opt. Express 11, 1731 (2003).
    [CrossRef] [PubMed]
  3. A. Liu, H. Rong, M. Paniccia, O. Cohen, and D. Hak, Opt. Express 12, 4261 (2004).
    [CrossRef] [PubMed]
  4. R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, Opt. Express 12, 2774 (2004).
    [CrossRef] [PubMed]
  5. T. K. Liang and H. K. Tsang, Appl. Phys. Lett. 84, 2745 (2004).
    [CrossRef]
  6. T. K. Liang and H. K. Tsang, IEEE J. Sel. Top. Quantum Electron. 10, 1149 (2004).
    [CrossRef]
  7. M. Krause, H. Renner, and E. Brinkmeyer, Opt. Express 12, 5703 (2004).
    [CrossRef] [PubMed]
  8. Q. Lin, O. J. Painter, and G. P. Agrawal, Opt. Express 15, 16604 (2007).
    [CrossRef] [PubMed]
  9. R. Claps, D. Dimitropoulos, and B. Jalali, Electron. Lett. 38, 1352 (2002).
    [CrossRef]
  10. X. Chen, N. C. Panoiu, and R. M. Osgood, IEEE J. Quantum Electron. 42, 160 (2006).
    [CrossRef]
  11. C.Headley and G.P.Agrawal, eds., Raman Amplification in Fiber-Optic Communication Systems, (Academic, 2005).
  12. D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, Appl. Phys. Lett. 86, 071115 (2005).
    [CrossRef]

2007 (1)

2006 (2)

B. Jalali, V. Raghunathan, D. Dimitropoulos, and O. Boyraz, IEEE J. Sel. Top. Quantum Electron. 12, 412 (2006).
[CrossRef]

X. Chen, N. C. Panoiu, and R. M. Osgood, IEEE J. Quantum Electron. 42, 160 (2006).
[CrossRef]

2005 (1)

D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, Appl. Phys. Lett. 86, 071115 (2005).
[CrossRef]

2004 (5)

2003 (1)

2002 (1)

R. Claps, D. Dimitropoulos, and B. Jalali, Electron. Lett. 38, 1352 (2002).
[CrossRef]

Agrawal, G. P.

Boyraz, O.

B. Jalali, V. Raghunathan, D. Dimitropoulos, and O. Boyraz, IEEE J. Sel. Top. Quantum Electron. 12, 412 (2006).
[CrossRef]

Brinkmeyer, E.

Chen, X.

X. Chen, N. C. Panoiu, and R. M. Osgood, IEEE J. Quantum Electron. 42, 160 (2006).
[CrossRef]

Claps, R.

D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, Appl. Phys. Lett. 86, 071115 (2005).
[CrossRef]

R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, Opt. Express 12, 2774 (2004).
[CrossRef] [PubMed]

R. Claps, D. Dimitropoulos, V. Raghunathan, Y. Han, and B. Jalali, Opt. Express 11, 1731 (2003).
[CrossRef] [PubMed]

R. Claps, D. Dimitropoulos, and B. Jalali, Electron. Lett. 38, 1352 (2002).
[CrossRef]

Cohen, O.

Dimitropoulos, D.

B. Jalali, V. Raghunathan, D. Dimitropoulos, and O. Boyraz, IEEE J. Sel. Top. Quantum Electron. 12, 412 (2006).
[CrossRef]

D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, Appl. Phys. Lett. 86, 071115 (2005).
[CrossRef]

R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, Opt. Express 12, 2774 (2004).
[CrossRef] [PubMed]

R. Claps, D. Dimitropoulos, V. Raghunathan, Y. Han, and B. Jalali, Opt. Express 11, 1731 (2003).
[CrossRef] [PubMed]

R. Claps, D. Dimitropoulos, and B. Jalali, Electron. Lett. 38, 1352 (2002).
[CrossRef]

Hak, D.

Han, Y.

Jalali, B.

B. Jalali, V. Raghunathan, D. Dimitropoulos, and O. Boyraz, IEEE J. Sel. Top. Quantum Electron. 12, 412 (2006).
[CrossRef]

D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, Appl. Phys. Lett. 86, 071115 (2005).
[CrossRef]

R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, Opt. Express 12, 2774 (2004).
[CrossRef] [PubMed]

R. Claps, D. Dimitropoulos, V. Raghunathan, Y. Han, and B. Jalali, Opt. Express 11, 1731 (2003).
[CrossRef] [PubMed]

R. Claps, D. Dimitropoulos, and B. Jalali, Electron. Lett. 38, 1352 (2002).
[CrossRef]

Jhaveri, R.

D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, Appl. Phys. Lett. 86, 071115 (2005).
[CrossRef]

Krause, M.

Liang, T. K.

T. K. Liang and H. K. Tsang, IEEE J. Sel. Top. Quantum Electron. 10, 1149 (2004).
[CrossRef]

T. K. Liang and H. K. Tsang, Appl. Phys. Lett. 84, 2745 (2004).
[CrossRef]

Lin, Q.

Liu, A.

Osgood, R. M.

X. Chen, N. C. Panoiu, and R. M. Osgood, IEEE J. Quantum Electron. 42, 160 (2006).
[CrossRef]

Painter, O. J.

Paniccia, M.

Panoiu, N. C.

X. Chen, N. C. Panoiu, and R. M. Osgood, IEEE J. Quantum Electron. 42, 160 (2006).
[CrossRef]

Raghunathan, V.

Renner, H.

Rong, H.

Tsang, H. K.

T. K. Liang and H. K. Tsang, IEEE J. Sel. Top. Quantum Electron. 10, 1149 (2004).
[CrossRef]

T. K. Liang and H. K. Tsang, Appl. Phys. Lett. 84, 2745 (2004).
[CrossRef]

Woo, J. C. S.

D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, Appl. Phys. Lett. 86, 071115 (2005).
[CrossRef]

Appl. Phys. Lett. (2)

T. K. Liang and H. K. Tsang, Appl. Phys. Lett. 84, 2745 (2004).
[CrossRef]

D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, Appl. Phys. Lett. 86, 071115 (2005).
[CrossRef]

Electron. Lett. (1)

R. Claps, D. Dimitropoulos, and B. Jalali, Electron. Lett. 38, 1352 (2002).
[CrossRef]

IEEE J. Quantum Electron. (1)

X. Chen, N. C. Panoiu, and R. M. Osgood, IEEE J. Quantum Electron. 42, 160 (2006).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (2)

T. K. Liang and H. K. Tsang, IEEE J. Sel. Top. Quantum Electron. 10, 1149 (2004).
[CrossRef]

B. Jalali, V. Raghunathan, D. Dimitropoulos, and O. Boyraz, IEEE J. Sel. Top. Quantum Electron. 12, 412 (2006).
[CrossRef]

Opt. Express (5)

Other (1)

C.Headley and G.P.Agrawal, eds., Raman Amplification in Fiber-Optic Communication Systems, (Academic, 2005).

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Figures (2)

Fig. 1
Fig. 1

Effective length, L eff , versus propagation distance for different input intensities I 0 (solid curves). Dashed curves shows the linear-loss limit, L eff ( z ) , realized in the absence of TPA and FCA. Dotted line corresponds to the lossless case. In the calculations the following parameters were used: α = 1 dB cm , β = 0.5 cm GW , τ c = 1 ns , g R = 76 cm GW , and λ s = 1686 nm .

Fig. 2
Fig. 2

Pump and signal intensity evolution for the exact numerical solution of Eq. (1) (solid curves) compared with the corrected (dashed curves) and uncorrected (dotted curves) solutions [Eq. (4)] as well as with the solution that corresponds to the undepleted pump approximation (dashed–dotted curves). In both cases I s 0 = 0.01 I p 0 . For other parameters see the text.

Equations (14)

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d I p d z = α p I p β p I p 2 ζ p s I p I s σ p τ c ( ρ p I p 2 + ρ s I s 2 + ρ p s I p I s ) I p ,
d I s d z = α s I s β s I s 2 ζ s p I s I p σ s τ c ( ρ p I p 2 + ρ s I s 2 + ρ p s I p I s ) I s .
ζ p s = 2 β p s + 4 g R γ R 2 Ω R Ω p s ( Ω R 2 Ω p s 2 ) 2 + 4 γ R 2 Ω p s 2 ,
ζ s p = ω s ω p ( 2 β p s 4 g R γ R 2 Ω R Ω p s ( Ω R 2 Ω p s 2 ) 2 + 4 γ R 2 Ω p s 2 ) ,
d I p d z α I p κ ( I p 2 + 4 I p I s + I s 2 ) I p γ I s I p ,
d I s d z α I s κ ( I p 2 + 4 I p I s + I s 2 ) I s + γ I p I s .
d I p d z α I p κ ( I p + I s ) 2 I p γ I s I p ,
d I s d z α I s κ ( I p + I s ) 2 I s + γ I p I s .
I ( z ) = I 0 exp ( α z ) 1 + I 0 2 ( κ α ) [ 1 exp ( 2 α z ) ] ,
I s ( z ) = I ( z ) 1 + ( I p 0 I s 0 ) exp [ γ I 0 L eff ( z ) ] ,
I p ( z ) = I ( z ) I s ( z ) .
L eff ( z ) = f ( 0 ) f ( z ) α κ I 0 , f ( z ) = tan 1 [ κ α I ( z ) ] .
I p ( z ) = I s ( z ) ( I p 0 I s 0 ) exp [ ζ p s I 0 L eff ( z ) ] ,
I s ( z ) = I 0 exp ( α z ) 1 ( ζ s p ζ p s ) ( I p 0 I s 0 ) exp [ ζ p s I 0 L eff ( z ) ] ,

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