Abstract

We analyze theoretically the nonlinear phenomenon of optical bistability inside a ring resonator formed with a silicon-waveguide nanowire and derive an exact parametric relation connecting the output intensity to the input intensity. Our input–output relation accounts for linear losses, the Kerr nonlinearity, two-photon absorption, free-carrier-induced absorption and dispersion, and thermo-optic effects within the resonator. Based on our study, we generalize the standard definition of effective length to allow for all possible losses within a silicon ring resonator. We also present a simplified version of the bistable phenomenon valid for resonators operating in a regime in which losses resulting from two-photon absorption are relatively small. Our analytical results provide clear insight into the physics behind optical bistability and may be useful for designing silicon-based optical memories.

© 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. Q. Lin, O. J. Painter, and G. P. Agrawal, Opt. Express 15, 16604 (2007).
    [CrossRef] [PubMed]
  3. H. J. Eichler, T. Brand, M. Glotz, and B. Smandek, Phys. Status Solidi B 150, 705 (1988).
    [CrossRef]
  4. V. R. Almeida and M. Lipson, Opt. Lett. 29, 2387 (2004).
    [CrossRef] [PubMed]
  5. G. Priem, P. Dumon, W. Bogaerts, D. V. Thourhout, G. Morthier, and R. Baets, Opt. Express 13, 9623 (2005).
    [CrossRef] [PubMed]
  6. Q. Xu and M. Lipson, Opt. Lett. 31, 341 (2006).
    [CrossRef] [PubMed]
  7. Q. Xu and M. Lipson, Opt. Express 15, 924 (2007).
    [CrossRef] [PubMed]
  8. I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, Opt. Express 17, 22124 (2009).
    [CrossRef] [PubMed]
  9. X. Chen, N. C. Panoiu, and R. M. Osgood, Int. J. Quantum Chem. 42, 160 (2006).
  10. I. D. Rukhlenko, M. Premaratne, C. Dissanayake, and G. P. Agrawal, Opt. Lett. 34, 536 (2009).
    [CrossRef] [PubMed]
  11. G. P. Agrawal, Applications of Nonlinear Fiber Optics, 2nd ed. (Academic, 2008).

2009

2007

2006

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, Int. J. Quantum Chem. 42, 160 (2006).

Q. Xu and M. Lipson, Opt. Lett. 31, 341 (2006).
[CrossRef] [PubMed]

2005

2004

1988

H. J. Eichler, T. Brand, M. Glotz, and B. Smandek, Phys. Status Solidi B 150, 705 (1988).
[CrossRef]

Agrawal, G. P.

Almeida, V. R.

Baets, R.

Bogaerts, W.

Boyraz, O.

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

Brand, T.

H. J. Eichler, T. Brand, M. Glotz, and B. Smandek, Phys. Status Solidi B 150, 705 (1988).
[CrossRef]

Chen, X.

X. Chen, N. C. Panoiu, and R. M. Osgood, Int. J. Quantum Chem. 42, 160 (2006).

Dimitropoulos, D.

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

Dissanayake, C.

Dumon, P.

Eichler, H. J.

H. J. Eichler, T. Brand, M. Glotz, and B. Smandek, Phys. Status Solidi B 150, 705 (1988).
[CrossRef]

Glotz, M.

H. J. Eichler, T. Brand, M. Glotz, and B. Smandek, Phys. Status Solidi B 150, 705 (1988).
[CrossRef]

Jalali, B.

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

Lin, Q.

Lipson, M.

Morthier, G.

Osgood, R. M.

X. Chen, N. C. Panoiu, and R. M. Osgood, Int. J. Quantum Chem. 42, 160 (2006).

Painter, O. J.

Panoiu, N. C.

X. Chen, N. C. Panoiu, and R. M. Osgood, Int. J. Quantum Chem. 42, 160 (2006).

Premaratne, M.

Priem, G.

Raghunathan, V.

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

Rukhlenko, I. D.

Smandek, B.

H. J. Eichler, T. Brand, M. Glotz, and B. Smandek, Phys. Status Solidi B 150, 705 (1988).
[CrossRef]

Thourhout, D. V.

Xu, Q.

IEEE J. Sel. Top. Quantum Electron.

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

Int. J. Quantum Chem.

X. Chen, N. C. Panoiu, and R. M. Osgood, Int. J. Quantum Chem. 42, 160 (2006).

Opt. Express

Opt. Lett.

Phys. Status Solidi B

H. J. Eichler, T. Brand, M. Glotz, and B. Smandek, Phys. Status Solidi B 150, 705 (1988).
[CrossRef]

Other

G. P. Agrawal, Applications of Nonlinear Fiber Optics, 2nd ed. (Academic, 2008).

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Figures (2)

Fig. 1
Fig. 1

Schematic of the silicon ring resonator and details of the notation employed.

Fig. 2
Fig. 2

Bistable characteristics of a silicon ring resonator of 5 μ m radius: (a) comparison between approximate (solid curve) and exact (open circles) solutions; (b) bistable curves at three operating wavelengths; (c) electronically assisted optical switching at a fixed input intensity; and (d) the impact of the thermo-optic effect (TOE) on optical bistability. TOE is absent in panels (a)–(c).

Equations (16)

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1 A d A d z = α 2 ( β 2 i γ ) | A | 2 ( ξ r 2 + i ξ i ) | A | 4 ,
ξ r = σ τ β ( 2 ω ) , ξ i = ( μ 2 ) ξ r , μ = 2 k σ n σ r ,
I ( z ) = I 0 exp ( α z ) 1 + I 0 2 ( ξ r α ) [ 1 exp ( 2 α z ) ] ,
ϕ ( z ) = ϕ 0 + γ I 0 L eff ( z ) ξ i ξ r ( ln I 0 I ( z ) α z ) ,
L eff ( z ) = tan 1 [ I 0 ξ r α ] tan 1 [ I ( z ) ξ r α ] I 0 α ξ r .
E 4 = r E 1 + i t E 3 , E 2 = i t E 1 + r E 3 ,
I 4 ( I 0 ) = r 2 I 0 + I ( 2 π R ) 2 r I 0 I ( 2 π R ) cos Δ ϕ 1 r 2 ,
I 1 ( I 0 ) = I 0 I ( 2 π R ) + I 4 ( I 0 ) .
I in ( I 0 ) = I 1 ( I 0 ) exp ( α L 1 ) 1 + I 1 2 ( I 0 ) ( ξ r α ) [ 1 exp ( 2 α L 1 ) ] ,
I tr ( I 0 ) = I 4 ( I 0 ) exp ( α L 2 ) 1 + I 4 2 ( I 0 ) ( ξ r α ) [ 1 exp ( 2 α L 2 ) ] .
I 1 ( I ) = ( α ξ r exp ( 2 α L 1 ) 1 ) 1 2 .
2 α z + β I 0 L eff ( z ) = ln α I 2 ( z ) + β I 1 ( z ) + ξ r α I 0 2 + β I 0 1 + ξ r ,
ϕ ( z ) = ϕ 0 + γ I 0 L eff ( z ) ξ i ξ r [ ln I 0 I ( z ) α z β 2 γ L eff ( z ) ] ,
L eff ( z ) = q β I 0 ln [ q K ( z ) + 1 q K ( z ) 1 q K ( 0 ) 1 q K ( 0 ) + 1 ] ,
I in ( I 0 ) = I [ I 1 ( I 0 ) , L 1 ] , I tr ( I 0 ) = I [ I 4 ( I 0 ) , L 2 ] ,
ξ r I 1 2 ( I ) exp ( 2 α L 1 ) α + β I 1 ( I ) + ξ r I 1 2 ( I ) = ( χ ( I ) 1 χ ( I ) + 1 ) q

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