Abstract

We study the effects of two-photon absorption on the self-phase modulation (SPM) process in silicon waveguides while including both free-carrier absorption and free-carrier dispersion. An analytical solution is provided in the case in which the density of generated carriers is relatively low; it is useful for estimating spectral bandwidth of pulses at low repetition rates. The free-carrier effects are studied numerically with emphasis on their role on the nonlinear phase shift and spectral broadening. We also consider how the repetition rate of a pulse train affects the SPM process.

© 2007 Optical Society of America

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References

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  2. L. Pavesi and D. J. Lockwood, Silicon Photonics (Springer, 2004).
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  4. X. Chen, N. C. Panoiu, and R. M. Osgood, IEEE J. Quantum Electron. 42, 160 (2006).
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  5. H. Rong, Y. Kuo, A. Liu, M. Paniccia, and O. Cohen, Opt. Express 14, 1182 (2006).
    [CrossRef] [PubMed]
  6. L. Yin, Q. Lin, and G. P. Agrawal, Opt. Lett. 31, 1295 (2006).
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  7. G. W. Rieger, K. S. Virk, and J. F. Young, Appl. Phys. Lett. 84, 900 (2004).
    [CrossRef]
  8. O. Boyraz, P. Koonath, V. Raghunathan, and B. Jalali, Opt. Express 12, 4094 (2004).
    [CrossRef] [PubMed]
  9. I. Hsieh, X. Chen, J. I. Dadap, N. C. Panoiu, and R. M. Osgood, Opt. Express 14, 12380 (2006).
    [CrossRef] [PubMed]
  10. R. Dekker, A. Driessen, T. Wahlbrink, J. Niehusmann, and M. Först, Opt. Express 14, 8336 (2006).
    [CrossRef] [PubMed]
  11. H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, Appl. Phys. Lett. 80, 416 (2002).
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  12. Q. Xu and M. Lipson, Opt. Lett. 81, 341 (2006).
    [CrossRef]
  13. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).

2006 (6)

2005 (1)

2004 (2)

G. W. Rieger, K. S. Virk, and J. F. Young, Appl. Phys. Lett. 84, 900 (2004).
[CrossRef]

O. Boyraz, P. Koonath, V. Raghunathan, and B. Jalali, Opt. Express 12, 4094 (2004).
[CrossRef] [PubMed]

2003 (1)

2002 (1)

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, Appl. Phys. Lett. 80, 416 (2002).
[CrossRef]

Agrawal, G. P.

L. Yin, Q. Lin, and G. P. Agrawal, Opt. Lett. 31, 1295 (2006).
[CrossRef] [PubMed]

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).

Asghari, M.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, Appl. Phys. Lett. 80, 416 (2002).
[CrossRef]

Boyraz, O.

Chen, X.

I. Hsieh, X. Chen, J. I. Dadap, N. C. Panoiu, and R. M. Osgood, Opt. Express 14, 12380 (2006).
[CrossRef] [PubMed]

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

Claps, R.

Cohen, O.

Dadap, J. I.

Day, I. E.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, Appl. Phys. Lett. 80, 416 (2002).
[CrossRef]

Dekker, R.

Dimitropoulos, D.

Drake, J.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, Appl. Phys. Lett. 80, 416 (2002).
[CrossRef]

Driessen, A.

Först, M.

Han, Y.

Harpin, A.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, Appl. Phys. Lett. 80, 416 (2002).
[CrossRef]

Hsieh, I.

Jalali, B.

Koonath, P.

Kuo, Y.

Liang, T. K.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, Appl. Phys. Lett. 80, 416 (2002).
[CrossRef]

Lin, Q.

Lipson, M.

Q. Xu and M. Lipson, Opt. Lett. 81, 341 (2006).
[CrossRef]

M. Lipson, J. Lightwave Technol. 23, 4222 (2005).
[CrossRef]

Liu, A.

Lockwood, D. J.

L. Pavesi and D. J. Lockwood, Silicon Photonics (Springer, 2004).

Niehusmann, J.

Osgood, R. M.

I. Hsieh, X. Chen, J. I. Dadap, N. C. Panoiu, and R. M. Osgood, Opt. Express 14, 12380 (2006).
[CrossRef] [PubMed]

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

Paniccia, M.

Panoiu, N. C.

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

I. Hsieh, X. Chen, J. I. Dadap, N. C. Panoiu, and R. M. Osgood, Opt. Express 14, 12380 (2006).
[CrossRef] [PubMed]

Pavesi, L.

L. Pavesi and D. J. Lockwood, Silicon Photonics (Springer, 2004).

Raghumathan, V.

Raghunathan, V.

Rieger, G. W.

G. W. Rieger, K. S. Virk, and J. F. Young, Appl. Phys. Lett. 84, 900 (2004).
[CrossRef]

Roberts, S. W.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, Appl. Phys. Lett. 80, 416 (2002).
[CrossRef]

Rong, H.

Tsang, H. K.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, Appl. Phys. Lett. 80, 416 (2002).
[CrossRef]

Virk, K. S.

G. W. Rieger, K. S. Virk, and J. F. Young, Appl. Phys. Lett. 84, 900 (2004).
[CrossRef]

Wahlbrink, T.

Wong, C. S.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, Appl. Phys. Lett. 80, 416 (2002).
[CrossRef]

Xu, Q.

Q. Xu and M. Lipson, Opt. Lett. 81, 341 (2006).
[CrossRef]

Yin, L.

Young, J. F.

G. W. Rieger, K. S. Virk, and J. F. Young, Appl. Phys. Lett. 84, 900 (2004).
[CrossRef]

Appl. Phys. Lett. (2)

G. W. Rieger, K. S. Virk, and J. F. Young, Appl. Phys. Lett. 84, 900 (2004).
[CrossRef]

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, Appl. Phys. Lett. 80, 416 (2002).
[CrossRef]

IEEE J. Quantum Electron. (1)

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

J. Lightwave Technol. (1)

Opt. Express (5)

Opt. Lett. (2)

Other (2)

L. Pavesi and D. J. Lockwood, Silicon Photonics (Springer, 2004).

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).

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Figures (3)

Fig. 1
Fig. 1

Reduction in nonlinear phase shift at the pulse center because of TPA inside a silicon waveguide for ϕ max = π , 2 π , and 3 π . Inset, ϕ 0 as a function of ϕ max using r = 0.1 ; dashed curve, r = 0 case.

Fig. 2
Fig. 2

SPM-broadened pulse spectra (solid curves) at the end of a 2-cm-long SOI waveguide at three input intensities such that ϕ max = 1.5 π , 7.5 π , and 15.5 π . Dashed curves include TPA but neglect FCA and FCD effects; dotted curves neglect TPA as well. The nonlinear phase profiles in the three situations are shown in the last plot for ϕ max = 15.5 π .

Fig. 3
Fig. 3

Pulse spectra at the same three intensity levels as in Fig. 2 except that all effects are included and three repetition rates are considered. The nonlinear phase shift at three different repetition rates is shown in part (d) at I 0 = 12.5 GW cm 2 .

Equations (6)

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E z + i β 2 2 2 E t 2 = i k 0 n 2 ( 1 + i r ) E 2 E σ 2 ( 1 + i μ ) N c E α l 2 E ,
N c ( z , t ) t = β TPA 2 h ν 0 E ( z , t ) 4 N c ( z , t ) τ c ,
E z = i k 0 n 2 ( 1 + i r ) E 2 E α l E 2 .
I ( L , t ) = I ( 0 , t ) exp ( α l L ) 1 + 2 k 0 n 2 r I ( 0 , t ) L eff ,
ϕ ( L , t ) = ( 2 r ) 1 ln [ 1 + 2 k 0 n 2 r I ( 0 , t ) L eff ] ,
N c ( t ) β TPA I 0 2 T 0 2 h ν 0 π 8 [ 1 + erf ( 2 t T 0 ) ] .

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