OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 8 — Apr. 16, 2007
  • pp: 4763–4780
« Show journal navigation

Coupled-mode theory for stimulated Raman scattering in high-Q/Vm silicon photonic band gap defect cavity lasers

Xiaodong Yang and Chee Wei Wong  »View Author Affiliations


Optics Express, Vol. 15, Issue 8, pp. 4763-4780 (2007)
http://dx.doi.org/10.1364/OE.15.004763


View Full Text Article

Acrobat PDF (822 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

We examine the dynamics of stimulated Raman scattering in designed high-Q/Vm silicon photonic band gap nanocavities through the coupled-mode theory framework towards optically-pumped silicon lasing. The interplay of other χ(3) effects such as two-photon absorption and optical Kerr, related free-carrier dynamics, thermal effects, as well as linear losses such as cavity radiation and linear material absorption are included and investigated numerically. Our results clarify the relative contributions and evolution of the mechanisms, and demonstrate the lasing and shutdown thresholds. Our studies illustrate the conditions for continuous-wave and pulsed highly-efficient Raman frequency conversion for practical realization in monolithic silicon high-Q/Vm photonic band gap defect cavities.

© 2007 Optical Society of America

1. Introduction

Here we employ a coupled-mode theory framework to study the various contributions on Raman scattering and lasing [41

41. H. A. Haus, Waves and Fields in Optoelectronics. (Prentice-Hall, Englewood Cliffs, N.J., 1984).

, 42

42. A. Yariv, Optical Electronics. (Sanders College Publishing, Philadelphia, 1991).

]. Coupled-mode equations are widely used in passive photonic devices such as optical waveguide direction couplers, channel add-drop filters [43

43. C. Manolatou, M.J. Khan, S. Fan, P.R. Villeneuve, H.A. Haus, and J.D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322 (1999). [CrossRef]

, 44

44. H. Ren, C. Jiang, W. Hu, M. Gao, and J. Wang, “Photonic crystal channel drop filter with a wavelength-selective reflection micro-cavity,” Opt. Express 14, 2446–2458 (2006). http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-6-2446. [CrossRef] [PubMed]

] and also in the analysis of optical nonlinearities [33

33. P. E. Barclay, K. Srinivasan, and O. Painter, “Nonlinear response of silicon photonic crystal microresonators excited via an integrated waveguide and fiber taper,” Opt. Express 13, 801–820 (2005). http://www.opticsexpress.org/abstract. cfm?URI=OPEX-13-3-801. [CrossRef] [PubMed]

, 45–47

45. T. J. Johnson, M. Borselli, and O. Painter, “Self-induced optical modulation of the transmission through a high-Q silicon microdisk resonator,” Opt. Express 14, 817–831 (2006). http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-2-817. [CrossRef] [PubMed]

]. The coupled-mode equations for stimulated Raman scattering in silicon-on-insulator waveguides [10

10. A. Liu, H. Rong, M. Paniccia, O. Cohen, and D. Hak, “Net optical gain in a low loss silicon-on-insulator waveguide by stimulated Raman scattering,” Opt. Express 12, 4261–4268 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-18-4261. [CrossRef] [PubMed]

, 48–50

48. D. Dimitropoulos, B. Houshmand, R. Claps, and B. Jalali, “Coupled-mode theory of the Raman effect in silicon-on-insulator waveguides,” Opt. Lett. 28, 1954–1956 (2003). [CrossRef] [PubMed]

], fiber Bragg grating [51

51. V. E. Perlin and H. G. Winful, “Stimulated Raman Scattering in nonlinear periodic structures,” Phys. Rev. A 64, 043804 (2001). [CrossRef]

] and silica microsphere [52

52. B. Min, T. J. Kippenberg, and K. J. Vahala, “Compact, fiber-compatible, cascaded Raman laser,” Opt. Lett. 28, 1507–1509 (2003). [CrossRef] [PubMed]

, 53

53. D. Braunstein, A. M. Khazanov, G. A. Koganov, and R. Shuker, “Lowering of threshold conditions for nonlinear effects in a microsphere,” Phys. Rev. A 53, 3565–3572 (1996). [CrossRef] [PubMed]

] have been studied. Following from the work of Ref [33

33. P. E. Barclay, K. Srinivasan, and O. Painter, “Nonlinear response of silicon photonic crystal microresonators excited via an integrated waveguide and fiber taper,” Opt. Express 13, 801–820 (2005). http://www.opticsexpress.org/abstract. cfm?URI=OPEX-13-3-801. [CrossRef] [PubMed]

, 45

45. T. J. Johnson, M. Borselli, and O. Painter, “Self-induced optical modulation of the transmission through a high-Q silicon microdisk resonator,” Opt. Express 14, 817–831 (2006). http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-2-817. [CrossRef] [PubMed]

, 47

47. T. Uesugi, B. Song, T. Asano, and S. Noda, “Investigation of optical nonlinearities in an ultra-high-Q Si nanocavity in a two-dimensional photonic crystal slab,” Opt. Express 14, 377–386 (2006). http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-1-377. [CrossRef] [PubMed]

], we derive, in this present paper, the coupled-mode equations for stimulated Raman scattering in high-Q/Vm silicon photonic band gap nanocavities. The dynamics of coupling between the pump cavity mode and the Stokes cavity mode is explored in the presence of cavity radiation losses, linear material absorption, two-photon absorption, and free-carrier absorption. The refractive index shift from the Kerr effect, free-carrier dispersion, and thermal dispersion are also considered in the coupled-mode equations. These equations can be numerically integrated to describe the dynamical behavior of the system for the designed L5 photonic band gap nanocavities. Specific examples such as lasing threshold, both pump and Stokes seed in continuous wave (CW) operation, pulsed pump with CW Stokes seed, on-off gain (loss), and the interaction of pump pulse and Stokes pulse are investigated in detail.

2. Design concept and coupled-mode theory

Stimulated Raman scattering is an inelastic two-photon process, where an incident photon interacts with an excited state of the material (the LO and TO phonons of single-crystal silicon). The strongest Stokes peak arises from single first-order Raman phonon (three-fold degenerate) at the Brillouin zone center. We have proposed a photonic band gap cavity with five linearly aligned missing air holes (L5) in an air-bridge triangular lattice photonic crystal slab with thickness of 0.6a and the radius of air holes is 0.29a, where the lattice period a = 420 nm [38

38. X. Yang and C. W. Wong, “Design of photonic band gap nanocavities for stimulated Raman amplification and lasing in monolithic silicon,” Opt. Express 13, 4723–4730 (2005). http://www. opticsinfobase. org/abstract. cfm?URI=oe-13-12-4723. [CrossRef] [PubMed]

]. The designed cavity supports two even modes, pump mode and Stokes mode, with spacing 15.6 THz, corresponding to the optical phonon frequency in monolithic silicon. Figure 1 shows the scanning electron micrograph (SEM) of the designed and fabricated L5 cavity coupled with photonic crystal waveguide. Figure 2(a) and 2(b) show the electric field profile (Ey) at the middle of the slab for pump mode and Stokes mode calculated from 3D FDTD method.

Coupling between pump and Stokes cavity mode in SRS can be understood classically with nonlinear polarizations P (3) NL . The dynamics of SRS is governed through a set of time-dependent coupled nonlinear equations (in MKS) [54

54. Y. R. Shen, The Principles of Nonlinear Optics, Chap. 10 (Wiley, Hoboken, New Jersey, 2003).

],

××Ep=εpε0c22Ept2=1ε0c22PNL(3)(ωP)t2
(1)
××ES=εSε0c22ESt2=1ε0c22PNL(3)(ωS)t2
(2)
Fig. 1. SEM picture of L5 cavity coupled with photonic crystal waveguide.
Fig. 2. The electric field profile (Ey) of pump mode (a) and Stokes mode (b).

where E p and E S are the electric fields of pump and Stokes cavity mode respectively, P (3) NLand P (3) NL (ωS are the third-order nonlinear polarizability, εp and εS are the dielectric constants. The third-order nonlinear polarization,

PNL(3)(ωp)=6ε0χijkl(3)(ωp)ESES*Ep
(3)
PNL(3)(ωS)=6ε0χijkl(3)(ωS)EpEp*ES
(4)

where χ (3) ijkl is the third-order nonlinear electric susceptibility.

Assume the electric fields of pump mode and Stokes mode are,

Eprt=ap(t)Ap(r)Npept
(5)
ESrt=aS(t)AS(r)NSeSt
(6)

where Ap(r) and AS(r) are the spatial part of the modes. ap(t) and aS(t) are slowly varying envelopes of the pumps and Stokes modes respectively. The amplitude is normalized by the spatial part to represent the energy of the mode (in units of Joules)

Ui=ai2=12εi(r)Eirt2d3r,andNi=12εi(r)Ei(r)2d3r,i=p,S
(7)

Then the third-order nonlinear polarization,

PNL(3)(ωp)=6ε0χR(3)(ωp)aS2apAS2ApNSNpept
(8)
PNL(3)(ωS)=6ε0χR(3)(ωS)ap2aSAp2ASNpNSeSt
(9)

χR(3)=χijkl(3)α*βλδ
(10)

α is the unit vector of the induced polarization P(3)NL, β,λ and δ are unit vectors of the interacting fields Ep,s. Next, by substituting the expression of Ep,s and P(3)NL into the coupled wave equations, taking the slowly varying envelope approximation 2at2<<ωat and only the real part of amplitudes, we obtain

dapdtAp=3ωpIm(χR(3)(ωp))εpε0(AS2NSAp)aS2ap
(11)
daSdtAS=3ωSIm(χR(3)(ωS))εSε0(Ap2NpAS)ap2aS
(12)

Then multiply the equations by the operator 12Siεp,S(r)Ap,S*(r)d3r, for which Si only integrates over the silicon region of the photonic crystal cavity. This results in the coupled-mode rate equations, relating the pump and Stokes evolutions without any loss terms currently,

dapdt=(ωpωS)gScaS2ap
(13)
daSdt=gScap2aS
(14)

where the Raman gain coefficient in the photonic band gap nanocavities, gcS [J-1s-1], is,

gSc=6ωSIm[χR(3)(ωS)]ε0np2nS2VR
(15)

VR=np2(r)Ap(r)2d3rnS2(r)AS(r)2d3rSinp2(r)Ap(r)2nS2(r)AS(r)2d3r
(16)

The bulk gain coefficient, gBR [m/W], is

gRB=12ωSIm[χR(3)(ωS)]ε0npnSc2
(17)

so that

gSc=(c22npnSVR)gRB
(18)

Note that in this classical formulation, the Raman gain coefficient in the photonic band gap nanocavities gcS is still equivalent to the bulk Raman gain coefficient gBloss rates into waveguide (in-plane) and intoR, since possible cavity quantum electrodynamics enhancements are not yet considered.

The electric field of input pump wave and input Stokes wave in the waveguide are,

Ei,inrt=Si(t)Si(r)Ni,ineiωit
(19)

where the field amplitude is normalized by Ni,in=12ni,in(r)ε0cSi(r)2d2r(i = p, S), to represent the input power Pi,in=si2=12ni,in(r)ε0cEi,in(r,t)2d2r. Now, considering the in-plane waveguide coupling loss 1/τi,in and the vertical radiation loss 1/τi,v [41

41. H. A. Haus, Waves and Fields in Optoelectronics. (Prentice-Hall, Englewood Cliffs, N.J., 1984).

], the coupled-mode rate equations are

dapdt=12τpap(ωpωS)gScaS2ap+κpsp
(20)
daSdt=12τSaS+gScap2aS+κSsS
(21)

where 1/τi = 1/τi,in + 1/τi,v , and 1/τi,in/v = ωi/Qi,in/v, 1/τi = ωi/Qi, (i = p, S). 1/τi,in and 1/τi,v are the loss rates into waveguide (in-plane) and into freespace (vertical). κi is the coupling coefficient of input pump wave sp(t) or Stokes wave SS(t) coupled to the pump mode ap(t) or the Stokes mode as(t) of the cavity, and κi=1τi,in. The threshold pump power for the stimulated Raman lasing is obtained from Equations (20) and (21),

Pin,th=π2npnSλpλSVRgRB(Qp,inQp2Qs)
(22)

The lasing threshold scales with VR/QPQS as illustrated in Equation (22). This therefore suggests the motivation for small VR cavities with high-Q factors.

Now, considering the total loss rate 1/τi,total total and the shifted resonant frequency Δωi of pump mode and Stokes mode, the coupled-mode rate equations are therefore [45

45. T. J. Johnson, M. Borselli, and O. Painter, “Self-induced optical modulation of the transmission through a high-Q silicon microdisk resonator,” Opt. Express 14, 817–831 (2006). http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-2-817. [CrossRef] [PubMed]

, 47

47. T. Uesugi, B. Song, T. Asano, and S. Noda, “Investigation of optical nonlinearities in an ultra-high-Q Si nanocavity in a two-dimensional photonic crystal slab,” Opt. Express 14, 377–386 (2006). http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-1-377. [CrossRef] [PubMed]

]

dapdt=(12τp,total+iΔωp)ap(ωpωS)gScaS2ap+κpsp
(23)
daSdt=(12τS,total+iΔωS)aS+gScap2aS+κSsS
(24)

This framework has been reported earlier in Johnson et al. [45

45. T. J. Johnson, M. Borselli, and O. Painter, “Self-induced optical modulation of the transmission through a high-Q silicon microdisk resonator,” Opt. Express 14, 817–831 (2006). http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-2-817. [CrossRef] [PubMed]

] and Uesugi et al. [47

47. T. Uesugi, B. Song, T. Asano, and S. Noda, “Investigation of optical nonlinearities in an ultra-high-Q Si nanocavity in a two-dimensional photonic crystal slab,” Opt. Express 14, 377–386 (2006). http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-1-377. [CrossRef] [PubMed]

] for a single frequency in cavities, as well as in Kippenberg et al. [36

36. T. J. Kippenberg, S. M. Spillane, B. Min, and K. J. Vahala, “Theoretical and experimental study of stimulated and cascaded Raman scattering in ultrahigh-Q optical microcavities,” IEEE J. Sel. Top. Quantum Electron. 10, 1219–1228 (2004). [CrossRef]

] for Raman scattering in cascaded microspheres and microtoroids. We further advance these investigations for the pump-Stokes interactions in photonic crystal cavities, as well as studying the lasing thresholds and dynamics under various specific conditions. The total loss rate for each cavity mode is:

1τi,total=1τi,in+1τi,v+1τi,lin+1τi,TPA+1τi,FCA
(25)

1τp,TPA=βSic2np2Vp,TPAap2+βSic2np2Vo,TPA2aS2
(26)
1τS,TPA=βSic2nS2VS,TPAaS2+βSic2nS2Vo,TPA2ap2
(27)

where the first terms represent the TPA due to two pump photons or two Stokes photons. The second terms represent that one pump photon and one Stokes photon are absorbed simultaneously. βSi is the TPA coefficient of bulk silicon. The effective mode volume for TPA, Vi,TPA is

Vi,TPA=(ni2(r)Ai(r)2dr3)2Sini4(r)Ai(r)2dr3
(28)

1τi,FCA=cniαi,FCA=cni(σi,e+σi,h)N(t)
(29)

From the Drude model [56

56. R. A. Soref and B. R. Bennett, “Electrooptical Effects in Silicon,” IEEE J. Quantum Electron. 23, 123–129 (1987). [CrossRef]

], the absorption cross-sections for electrons and holes, σi,e/h, is

σi,eh=e2cniωi2ε0meh*τrelax,eh
(30)

Here e is the electron charge, τrelax,e/h the relaxation time of carriers, and m*e/h the effective mass of carriers. The mode-averaged free carriers density (electron-hole pairs) generated by TPA is N(t), which is governed by the rate equation [45

45. T. J. Johnson, M. Borselli, and O. Painter, “Self-induced optical modulation of the transmission through a high-Q silicon microdisk resonator,” Opt. Express 14, 817–831 (2006). http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-2-817. [CrossRef] [PubMed]

],

dNdt=Nτfc+G
(31)

The mode-averaged generation rate of free-carriers G can be calculated from the mode-averaged TPA loss rate.

G=βSic22ħωpnp2Vp,FCA2ap4+βSic22ħωSnS2VS,FCA2aS4+βSic2ħ(ωp+ωS)np2VSp,FCA22aS2ap2+βSic2ħ(ωp+ωS)nS2VpS,FCA22ap2aS2
(32)

The expressions of effective mode volume for FCA, VFCA, are

Vi,FCA2=(ni2(r)Air)2dr3)3Sini6(r)Ai(r)6dr3
(33)
VSp,FCA2=(np2(r)Ap(r)2dr3)2(nS2(r)AS(r)2dr3)Sinp2(r)Ap(r)4nS2(r)AS(r)2dr3
(34)
VpS,FCA2=(nS2(r)AS(r)2dr3)2(np2(r)Ap(r)2dr3)SinS2(r)AS(r)4np2(r)Ap(r)2dr3
(35)

τfc is the effective free-carrier lifetime accounting for both recombination and diffusion. Time constants of radiative and Auger recombination, as well as from bulk defects and impurities, are assumed to be significantly slower than the free-carrier recombination and diffusion lifetime [55

55. H. W. Tan, H. M. van Driel, S. L. Schweizer, and R. B. Wehrspohn, “Influence of eigenmode characteristics on optical tuning of a two-dimensional silicon photonic crystal,” Phys. Rev. B , 72, 165115 (2005). [CrossRef]

]. We note that while free-carrier lifetime can vary with carrier density and carrier density can vary spatially with intensity in the cavity, an effective lifetime is used here for simplicity. A quiescent carrier density of N0 = 1022 m-3 is used in the initial condition for silicon.

In equations (23) and (24

24. H. Ryu, M. Notomi, G. Kim, and Y. Lee, “High quality-factor whispering-gallery mode in the photonic crystal hexagonal disk cavity,” Opt. Express 12, 1708–1719 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1708. [CrossRef] [PubMed]

), Δωi is the detuning of the resonance frequency of the cavity from the input light frequency due to the Kerr effect, free-carrier dispersion (FCD), and thermal dispersion. Δωi = ωi - ωi, ωi is the shifted resonant frequency of the cavity and ωi is the input light frequency in the waveguide. Under first-order perturbation, the detuning of the resonance frequency can be expressed as [33

33. P. E. Barclay, K. Srinivasan, and O. Painter, “Nonlinear response of silicon photonic crystal microresonators excited via an integrated waveguide and fiber taper,” Opt. Express 13, 801–820 (2005). http://www.opticsexpress.org/abstract. cfm?URI=OPEX-13-3-801. [CrossRef] [PubMed]

]

Δωiωi=Δnini=(Δni,Kerrni+Δni,FCDni+Δni,thni)
(36)

The detuning due to Kerr effect is

Δnp,Kerrnp=cn2np2Vp,Kerrap2+cn2np2Vo,Kerr2aS2
(37)
ΔnS,KerrnS=cn2nS2VS,KerraS2+cn2nS2Vo,Kerr2ap2
(38)

Δni,FCDni=1ni(ζi,e+ζi,h)N(t)
(39)

From the Drude model [56

56. R. A. Soref and B. R. Bennett, “Electrooptical Effects in Silicon,” IEEE J. Quantum Electron. 23, 123–129 (1987). [CrossRef]

], the material parameter with units of volume, ζi,e/h , is

ζi,eh=e22niωi2ε0meh*
(40)
Δni,thni=1nidnidTΔT
(41)

The mode-averaged temperature difference between the photonic crystal cavity and its environment ΔT is governed by [45

45. T. J. Johnson, M. Borselli, and O. Painter, “Self-induced optical modulation of the transmission through a high-Q silicon microdisk resonator,” Opt. Express 14, 817–831 (2006). http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-2-817. [CrossRef] [PubMed]

, 47

47. T. Uesugi, B. Song, T. Asano, and S. Noda, “Investigation of optical nonlinearities in an ultra-high-Q Si nanocavity in a two-dimensional photonic crystal slab,” Opt. Express 14, 377–386 (2006). http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-1-377. [CrossRef] [PubMed]

]

dΔTdt=ΔTτth+PabsρSicp,SiVcavity
(42)

where ρSi, cp,Si and Vcavity are the density of silicon, the constant-pressure specific heat capacity of silicon and the volume of cavity respectively. The temperature decay life-time τth is determined by the thermal resistance R of the air-bridge silicon photonic crystal cavities.

τth=ρSicp,SiVcavityR
(43)

The total absorbed power is given by

Pabs=Pp,abs+PS,abs+PR,abs
(44)
Pi,abs=(1τi,lin+1τi,TPA+1τi,FCA)ai2
(45)

Absorbed power due to Raman scattering generated optical phonon, PR,abs, is

PR,abs=2(ωpωS1)gScap2aS2
(46)

Equations (23–25), (31), (36), and (42) therefore describe the dynamic behavior of SRS in photonic crystal nanocavities, and is numerically integrated in our work to describe the dynamical behavior of pump-Stokes interactions in our L5 photonic crystal cavity system that supports the desired two-mode frequencies at the appropriate LO/TO phonon spacing.

3. Numerical analysis

3.1 Lasing threshold

We now consider the case of lasing threshold. Around the lasing threshold, the Stokes gain equals the losses, and the Stokes mode energy |aS|2 is much smaller than the pump mode energy |ap|2, equation (24) is simplified to

gScapth2=12τS,total
(47)

The loss rate due to TPA of pump mode is

1τS,TPA=βSic2nS2Vo,TPA2apth2
(48)

The free carriers generated by TPA of pump mode is

N=τfcG
(49)
G=βSic22ħωpnp2Vp,FCA2apth4
(50)

Figure 3 shows the threshold pump mode energy |ap|2 th as a function of QS for different free-carrier lifetimes τfc. All the parameters used in calculation are presented in Table 1. By comparing the curve in the absence of TPA and FCA (βSi = 0) and the curve with TPA but without FCA (τfc = 0), it is observed that TPA increases the threshold pump mode energy but the effect of TPA is relatively weak. The effect of TPA-induced FCA is much more dramatic as shown for different free-carrier lifetimes τfc. The lasing threshold increases when τfc is larger. There is a minimum Stokes QS required for lasing, as seen in the solutions plotted in Figure 3. If QS is lower than a critical value for certain τfc, there is no solution numerically and physically this translates to an absence of a lasing threshold regardless of the pump intensity. For increasing τfc, the critical value of QS increases monotonically as can be seen in Figure 3. The solid and dotted curves show the lasing and shutdown thresholds, respectively [49

49. M. Krause, H. Renner, and E. Brinkmeyer, “Analysis of Raman lasing characteristics in silicon-on-insulator waveguides,” Opt. Express 12, 5703–5710 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-23-5703. [CrossRef] [PubMed]

]. The shutdown threshold is the pump power in which the lasing output power returns to zero due to increasing TPA and FCA. For the L5 cavity studied in the present paper, QS = 21,000, the maximum τfc is around 0.175 ns, and the threshold pump mode energy is 29 fJ. For air-bridged silicon photonic band gap nanocavities, τfc= 0.5 ns [47

47. T. Uesugi, B. Song, T. Asano, and S. Noda, “Investigation of optical nonlinearities in an ultra-high-Q Si nanocavity in a two-dimensional photonic crystal slab,” Opt. Express 14, 377–386 (2006). http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-1-377. [CrossRef] [PubMed]

], which is much higher than the maximum τfc. In order to get lasing for this cavity, instead of using CW pump signal, pulse pump signal can be used to reduce the TPA-induced FCA for loss reduction.

We now solve for the input-output characteristics of photonic band gap defect cavity laser by considering equation (24) in steady state.Figure 4 shows the laser input-output characteristics with different free-carrier lifetimes τfc. QS = 30,000 and QS = 60,000 are considered for comparison. Cavities with higher QS have higher output Stokes signals, lower lasing threshold pump mode energies and higher shutdown thresholds. The required corresponding pump power in the input waveguide can also be calculated based on equation (23) in steady state.

Table 1. Parameters used in coupled-mode theory

table-icon
View This Table
Fig. 3. Threshold pump mode energy versus QS of L5 cavity for different values of free-carrier lifetimes τfc, the solid curve and dotted curve show the lasing and shutdown thresholds, respectively.
Fig. 4. Input-output characteristics of photonic band gap defect cavity laser for different values of free-carrier lifetimes τfc, the solid curve and dashed curve correspond to QS = 30,000 and QS = 60,000, respectively.

3.2 Lasing dynamics

We now consider the dynamics of the Raman lasing interactions. Equations (23–25), (31), (36), and (42) are numerically integrated with a variable order Adams-Bashforth-Moulton predictor-corrector method (Matlab ® ode113 solver). All the parameters used in calculation are presented in Table 1.

Figure 5 shows the dynamics of Raman amplification with 60 mW CW pump wave and 10 μW CW Stokes seed signal, free carrier lifetime is 0.5 ns. In the beginning when free carrier density is low, Raman gain is greater than loss and there is amplification. When free carrier density increases, FCA dominates the loss and Stokes signal is suppressed. The temperature difference then increases significantly. The cavity resonance is red shifted and the pump mode energy goes down. From the numerical results, the Kerr effect is predominantly weak. The FCD effect dominates at first when the temperature difference is low, with a resulting blue shift. Eventually thermal effect dominates, with a resulting red shift. Consider the case of a different carrier lifetime at 0.1 ns. Figure 6 shows the calculated results with free carrier lifetime of 0.1 ns. With lower free carrier lifetime, the free carrier density and the temperature difference are lower, so that the net Raman gain is greater than zero. The oscillation of Stokes mode energy near t = 0.5 ns is due to the dispersion induced Stokes resonance frequency shift.

In order to get lasing from this cavity, instead of using CW pump signal, pulse pump signal with pulse width narrower than the free carrier lifetime is used to reduce the TPA-induced FCA, so as to reduce loss and increase net gain. Figure 7 shows the dynamics of Raman amplification with pump pulse of 60 mW peak power, pulse width TFWHM = 50 ps and 10 μW CW Stokes signal. The free carrier density and the temperature difference are significantly reduced, and a strong Stokes pulse is generated by the pump pulse. The resonance frequency shift is also significantly reduced by the pump pulse operation.

Consider now the case of on-off gain and loss in our cavity system, where the probe signal changes between the pump pulse on and off when the probe frequency is on- (off-) resonance with the Stokes frequency [10

10. A. Liu, H. Rong, M. Paniccia, O. Cohen, and D. Hak, “Net optical gain in a low loss silicon-on-insulator waveguide by stimulated Raman scattering,” Opt. Express 12, 4261–4268 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-18-4261. [CrossRef] [PubMed]

]. Pulsed pump beam with 60 mW peak power and TFWHM = 50 ps, and CW probe beam with 1 mW power are used. Figure 8 shows the on-off gain and on-off loss. When the probe is on the Stokes frequency, an increase in the probe signal due to the SRS is observed, and the on-off gain is around 8 dB. When the probe frequency is detuned from the Stokes frequency (no Raman gain), the loss in the probe signal due to the pump pulse generated free carriers is observed, and the on-off loss is around 20 dB. Figure 9 and Figure 10 show the dynamics of the system.

We also consider the interaction of both pump and Stokes pulses with comparable peak power, with numerical results shown in Figure 11. Pump peak power is 60 mW and Stokes peak power is 20 mW. Both pump pulse and Stokes pulse have TFWHM = 50 ps. The Stokes pulse is amplified by the pump pulse. Due to the high free carrier density induced FCD effect and high Qs, there is an observed oscillation in the amplified Stokes pulse.

Fig. 5. Dynamics of Raman amplification with 60 mW CW pump wave and 10 μW CW Stokes seed signal, τfc, is 0.5 ns.
Fig. 6. Dynamics of Raman amplification with 60 mW CW pump wave and 10 μW CW Stokes seed signal, τfc is 0.1 ns.
Fig. 7. Dynamics of Raman amplification with pulse pump of 60 mW peak power, TFWHM = 50 ps and 10 μW CW Stokes signal, τfc, is 0.5 ns.
Fig. 8. Raman on-off gain and on-off loss with pump pulse of 60 mW peak power, TFWHM = 50 ps and 1 mW CW probe signal, τfc is 0.5 ns.
Fig. 9. Dynamics of Raman on-off gain with pump pulse of 60 mW peak power, TFWHM = 50 ps and 1 mW CW probe signal, τfc is 0.5 ns.
Fig. 10. Dynamics of Raman on-off loss with pulse pump of 60 mW peak power, TFWHM = 50 ps and 1 mW CW probe signal, τfc is 0.5 ns.
Fig. 11. Dynamics of Raman interaction of pump pulse with 60 mW peak power and Stokes pulse with 20 mW peak power, TFWHM = 50 ps, τfc is 0.5 ns.

4. Conclusions

In this work we have derived the coupled-mode equations for stimulated Raman scattering in high-Q/Vm silicon photonic band gap nanocavities towards optically-pumped silicon lasing. Both the lasing threshold and the lasing dynamics are numerically studied in the presence of cavity radiation losses, linear material absorption, two-photon absorption, and free-carrier absorption, together with the refractive index shift from the Kerr effect, free-carrier dispersion, and thermal dispersion. With increasing the cavity Q factors and decreasing the free carrier lifetimes, the reduction in the threshold pump energy is solved numerically, considering all mechanisms and realistic conditions. With CW pump operation, the Stokes signal is suppressed due to strong TPA-induced FCA. With pulse pump operation, the TPA-induced FCA is significantly reduced and Stokes net gain increases, which shows that compact Raman amplifiers and lasers based on high-Q/Vm silicon photonic band gap nanocavities are feasible.

Acknowledgments

The authors thank Xiaogang Chen for helpful discussions. This work was partially supported under the National Science Foundation (award ECS-0622069) and the Columbia University Initiatives in Science and Engineering for Nanophotonics. Xiaodong Yang would like to acknowledge the support of an Intel Fellowship.

References and links

1.

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. 88, 041112 (2006). [CrossRef]

2.

T. Asano, B. -S. Song, and S. Noda, “Analysis of the experimental Q factors (~ 1 million) of photonic crystal nanocavities,” Opt. Express 14, 1996–2002 (2006). http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-5-1996. [CrossRef] [PubMed]

3.

T. Baehr-Jones, M. Hochberg, C. Walker, and A. Scherer, “High-Q ring resonators in thin silicon-on-insulator,” Appl. Phys. Lett. 85, 3346 (2004). [CrossRef]

4.

D. K. Sparacin, S. J. Spector, and L. C. Kimerling, “Silicon waveguide sidewall smoothing by wet chemical oxidation,” IEEE J. Lightwave Technol. , 23, 2455 (2005). [CrossRef]

5.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431, 1081–1084 (2004). [CrossRef] [PubMed]

6.

V. R. Almeida and M. Lipson, “Optical bistability on a silicon chip,” Opt. Lett. 29, 2387–2389 (2004). [CrossRef] [PubMed]

7.

R. Claps, D. Dimitropoulos, V. Raghunathan, Y. Han, and B. Jalali, “Observation of stimulated Raman amplification in silicon waveguides,” Opt. Express 11, 1731–1739 (2003). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-15-1731. [CrossRef] [PubMed]

8.

R. L. Espinola, J. I. Dadap, R. M. Osgood Jr., S. J. McNab, and Y. A. Vlasov, “Raman amplification in ultrasmall silicon-on-insulator wire waveguides,” Opt. Express 12, 3713 – 3718 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-16-3713. [CrossRef] [PubMed]

9.

T. K. Liang and H. K. Tsang, “Efficient Raman amplification in silicon-on-insulator waveguides,” Appl. Phys. Lett. 85, 3343–3345 (2004). [CrossRef]

10.

A. Liu, H. Rong, M. Paniccia, O. Cohen, and D. Hak, “Net optical gain in a low loss silicon-on-insulator waveguide by stimulated Raman scattering,” Opt. Express 12, 4261–4268 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-18-4261. [CrossRef] [PubMed]

11.

Q. Xu, V. R. Almeida, and M. Lipson, “Time-resolved study of Raman gain in highly confined silicon-on-insulator waveguides,” Opt. Express 12, 4437 – 4442 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-19-4437. [CrossRef] [PubMed]

12.

O. Boyraz and B. Jalali, “Demonstration of a silicon Raman laser,” Opt. Express 12, 5269–5273 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-21-5269. [CrossRef] [PubMed]

13.

R. Jones, H. Rong, A. Liu, A. W. Fang, M. J. Paniccia, D. Hak, and O. Cohen, “Net continuous wave optical gain in a low loss silicon-on-insulator waveguide by stimulated Raman scattering,” Opt. Express 13, 519–525 (2005). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-2-519. [CrossRef] [PubMed]

14.

H. Rong, A. Liu, R. Jones, O. Cohen, D. Hak, R. Nicolaescu, A. Fang, and M. Paniccia, “An all-silicon Raman laser,” Nature 433, 292–294 (2005). [CrossRef] [PubMed]

15.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433, 725–728 (2005). [CrossRef] [PubMed]

16.

O. Boyraz and B. Jalali, “Demonstration of directly modulated silicon Raman laser,” Opt. Express 13, 796–800 (2005). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-3-796. [CrossRef] [PubMed]

17.

R. Jones, A. Liu, H. Rong, M. Paniccia, O. Cohen, and D. Hak, “Lossless optical modulation in a silicon waveguide using stimulated Raman scattering,” Opt. Express 13, 1716–1723 (2005). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-5-1716. [CrossRef] [PubMed]

18.

E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. 58, .2059–2062 (1987). [CrossRef] [PubMed]

19.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987). [CrossRef] [PubMed]

20.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light. (Princeton, NJ: Princeton University Press, 1995).

21.

K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Opt. Express 10, 670–684 (2002). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-15-670. [PubMed]

22.

D. Englund, I. Fushman, and J. Vuckovic, “General recipe for designing photonic crystal cavities,” Opt. Express 13, 5961–5975 (2005). http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-16-5961. [CrossRef] [PubMed]

23.

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425, 944–947 (2003). [CrossRef] [PubMed]

24.

H. Ryu, M. Notomi, G. Kim, and Y. Lee, “High quality-factor whispering-gallery mode in the photonic crystal hexagonal disk cavity,” Opt. Express 12, 1708–1719 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1708. [CrossRef] [PubMed]

25.

Z. Zhang and M. Qiu, “Small-volume waveguide-section high Q microcavities in 2D photonic crystal slabs,” Opt. Express 12, 3988–3995 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-17-3988. [CrossRef] [PubMed]

26.

B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4, 207–210 (2005). [CrossRef]

27.

B. S. Song, S. Noda, and T. Asano, “Photonic devices based on in-plane hetero photonic crystals,” Science 300, 1537 (2003). [CrossRef] [PubMed]

28.

K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003). [CrossRef] [PubMed]

29.

H. Takano, B. -S. Song, T. Asano, and S. Noda, “Highly efficient multi-channel drop filter in a two-dimensional hetero photonic crystal,” Opt. Express 14, 3491–3496 (2006).http://www. opticsinfobase. org/abstract. cfm?URI=oe-14- 8-3491. [CrossRef] [PubMed]

30.

H. G. Park, S. H. Kim, S. H. Kwon, Y. G. Ju, J. K. Yang, J. H. Baek, S. B. Kim, and Y. H. Lee, “Electrically driven single-cell photonic crystal laser,” Science 305, 1444–1447(2004). [CrossRef] [PubMed]

31.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004). [CrossRef] [PubMed]

32.

M. Soljacic and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. 3, 211–219 (2004). [CrossRef] [PubMed]

33.

P. E. Barclay, K. Srinivasan, and O. Painter, “Nonlinear response of silicon photonic crystal microresonators excited via an integrated waveguide and fiber taper,” Opt. Express 13, 801–820 (2005). http://www.opticsexpress.org/abstract. cfm?URI=OPEX-13-3-801. [CrossRef] [PubMed]

34.

M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi, and T. Tanabe, “Optical bistable switching action of Si high-Q photonic-crystal nanocavities,” Opt. Express 13, 2678–2687 (2005). http://www.opticsexpress.org/abstract. cfm?URI=OPEX-13-7-2678. [CrossRef] [PubMed]

35.

M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature 415, 621–623 (2002). [CrossRef] [PubMed]

36.

T. J. Kippenberg, S. M. Spillane, B. Min, and K. J. Vahala, “Theoretical and experimental study of stimulated and cascaded Raman scattering in ultrahigh-Q optical microcavities,” IEEE J. Sel. Top. Quantum Electron. 10, 1219–1228 (2004). [CrossRef]

37.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, “Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5 μm wavelength,” Appl. Phys. Lett. 80, 416–418 (2002). [CrossRef]

38.

X. Yang and C. W. Wong, “Design of photonic band gap nanocavities for stimulated Raman amplification and lasing in monolithic silicon,” Opt. Express 13, 4723–4730 (2005). http://www. opticsinfobase. org/abstract. cfm?URI=oe-13-12-4723. [CrossRef] [PubMed]

39.

L. Florescu and X. Zhang, “Semiclassical model of stimulated Raman scattering in photonic crystals,” Phys. Rev. E 72, 016611 (2005). [CrossRef]

40.

J. F. McMillan, X. Yang, N. C. Paniou, R M. Osgood, and C. W. Wong, “Enhanced stimulated Raman scattering in slow-light photonic crystal waveguides,” Opt. Lett. 31, 1235–1237 (2006). [CrossRef] [PubMed]

41.

H. A. Haus, Waves and Fields in Optoelectronics. (Prentice-Hall, Englewood Cliffs, N.J., 1984).

42.

A. Yariv, Optical Electronics. (Sanders College Publishing, Philadelphia, 1991).

43.

C. Manolatou, M.J. Khan, S. Fan, P.R. Villeneuve, H.A. Haus, and J.D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322 (1999). [CrossRef]

44.

H. Ren, C. Jiang, W. Hu, M. Gao, and J. Wang, “Photonic crystal channel drop filter with a wavelength-selective reflection micro-cavity,” Opt. Express 14, 2446–2458 (2006). http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-6-2446. [CrossRef] [PubMed]

45.

T. J. Johnson, M. Borselli, and O. Painter, “Self-induced optical modulation of the transmission through a high-Q silicon microdisk resonator,” Opt. Express 14, 817–831 (2006). http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-2-817. [CrossRef] [PubMed]

46.

C. Manolatou and M. Lipson, “All-Optical Silicon Modulators Based on Carrier Injection by Two-Photon Absorption,” IEEE J. Lightwave Technol. 24, 1433–1439 (2006). [CrossRef]

47.

T. Uesugi, B. Song, T. Asano, and S. Noda, “Investigation of optical nonlinearities in an ultra-high-Q Si nanocavity in a two-dimensional photonic crystal slab,” Opt. Express 14, 377–386 (2006). http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-1-377. [CrossRef] [PubMed]

48.

D. Dimitropoulos, B. Houshmand, R. Claps, and B. Jalali, “Coupled-mode theory of the Raman effect in silicon-on-insulator waveguides,” Opt. Lett. 28, 1954–1956 (2003). [CrossRef] [PubMed]

49.

M. Krause, H. Renner, and E. Brinkmeyer, “Analysis of Raman lasing characteristics in silicon-on-insulator waveguides,” Opt. Express 12, 5703–5710 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-23-5703. [CrossRef] [PubMed]

50.

X. Chen, N. C. Panoiu, and R. M. Osgood, “Theory of Raman-mediated pulsed amplification in silicon-wire waveguides,” IEEE J. Quantum Electron. 42, 160–170 (2006). [CrossRef]

51.

V. E. Perlin and H. G. Winful, “Stimulated Raman Scattering in nonlinear periodic structures,” Phys. Rev. A 64, 043804 (2001). [CrossRef]

52.

B. Min, T. J. Kippenberg, and K. J. Vahala, “Compact, fiber-compatible, cascaded Raman laser,” Opt. Lett. 28, 1507–1509 (2003). [CrossRef] [PubMed]

53.

D. Braunstein, A. M. Khazanov, G. A. Koganov, and R. Shuker, “Lowering of threshold conditions for nonlinear effects in a microsphere,” Phys. Rev. A 53, 3565–3572 (1996). [CrossRef] [PubMed]

54.

Y. R. Shen, The Principles of Nonlinear Optics, Chap. 10 (Wiley, Hoboken, New Jersey, 2003).

55.

H. W. Tan, H. M. van Driel, S. L. Schweizer, and R. B. Wehrspohn, “Influence of eigenmode characteristics on optical tuning of a two-dimensional silicon photonic crystal,” Phys. Rev. B , 72, 165115 (2005). [CrossRef]

56.

R. A. Soref and B. R. Bennett, “Electrooptical Effects in Silicon,” IEEE J. Quantum Electron. 23, 123–129 (1987). [CrossRef]

57.

E. Palick, Handbook of optical constants of solids, ed., (Academic Press, Boston, MA, 1985).

58.

M. Dinu, F. Quochi, and H. Garcia, “Third-order nonlinearities in silicon at telecom wavelengths,” Appl. Phys. Lett. 82, 2954–2956 (2003). [CrossRef]

59.

A. Cutolo, M. Iodice, P. Spirito, and L. Zeni, “Silicon Electro-Optic Modulator Based on a Three Terminal Device Integrated in a Low-Loss Single-Mode SOI Waveguide,” IEEE J. Lightwave Technol. 15, 505–518 (1997). [CrossRef]

60.

S. Sze, Physics of semiconductor devices, 2nd ed. (John Wiley and Sons, New York, New York, 1981).

61.

T. Asano, W. Kunishi, M. Nakamura, B. S. Song, and S. Noda, “Dynamic wavelength tuning of channel-drop device in two-dimensional photonic crystal slab,” Electron. Lett. 41, 37–38 (2005). [CrossRef]

62.

G. Cocorullo, F. G. Della Corte, and I. Rendina, “Temperature dependence of the thermo-optic coefficient in crystalline silicon between room temperature and 550 K at the wavelength of 1523 nm,” Appl. Phys. Lett. 74, 3338–3340 (1999). [CrossRef]

OCIS Codes
(140.3550) Lasers and laser optics : Lasers, Raman
(190.4390) Nonlinear optics : Nonlinear optics, integrated optics
(230.5750) Optical devices : Resonators

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: November 1, 2006
Revised Manuscript: December 18, 2006
Manuscript Accepted: December 18, 2006
Published: April 4, 2007

Citation
Xiaodong Yang and Chee Wei Wong, "Coupled-mode theory for stimulated Raman scattering in high-Q/Vm silicon photonic band gap defect cavity lasers," Opt. Express 15, 4763-4780 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-8-4763


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, "Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect," Appl. Phys. Lett. 88, 041112 (2006). [CrossRef]
  2. T. Asano, B. -S. Song, and S. Noda, "Analysis of the experimental Q factors (~ 1 million) of photonic crystal nanocavities," Opt. Express 14, 1996-2002 (2006). [CrossRef] [PubMed]
  3. T. Baehr-Jones, M. Hochberg, C. Walker, and A. Scherer, "High-Q ring resonators in thin silicon-on-insulator," Appl. Phys. Lett. 85, 3346 (2004). [CrossRef]
  4. D. K. Sparacin, S. J. Spector, and L. C. Kimerling, "Silicon waveguide sidewall smoothing by wet chemical oxidation," J. Lightwave Technol. 23, 2455 (2005). [CrossRef]
  5. V. R. Almeida, C. A. Barrios, R. R. Panepucci and M. Lipson, "All-optical control of light on a silicon chip," Nature 431, 1081-1084 (2004). [CrossRef] [PubMed]
  6. V. R. Almeida and M. Lipson, "Optical bistability on a silicon chip," Opt. Lett. 29, 2387-2389 (2004). [CrossRef] [PubMed]
  7. R. Claps, D. Dimitropoulos, V. Raghunathan, Y. Han, and B. Jalali, "Observation of stimulated Raman amplification in silicon waveguides," Opt. Express 11, 1731-1739 (2003). [CrossRef] [PubMed]
  8. R. L. Espinola, J. I. Dadap, R. M. Osgood, Jr., S. J. McNab, and Y. A. Vlasov, "Raman amplification in ultrasmall silicon-on-insulator wire waveguides," Opt. Express 12, 3713 - 3718 (2004). [CrossRef] [PubMed]
  9. T. K. Liang and H. K. Tsang, "Efficient Raman amplification in silicon-on-insulator waveguides," Appl. Phys. Lett. 85, 3343-3345 (2004). [CrossRef]
  10. A. Liu, H. Rong, M. Paniccia, O. Cohen, and D. Hak, "Net optical gain in a low loss silicon-on-insulator waveguide by stimulated Raman scattering," Opt. Express 12, 4261-4268 (2004). [CrossRef] [PubMed]
  11. Q. Xu, V. R. Almeida, and M. Lipson, "Time-resolved study of Raman gain in highly confined silicon-on-insulator waveguides," Opt. Express 12, 4437 - 4442 (2004). [CrossRef] [PubMed]
  12. O. Boyraz and B. Jalali, "Demonstration of a silicon Raman laser," Opt. Express 12, 5269-5273 (2004). [CrossRef] [PubMed]
  13. R. Jones, H. Rong, A. Liu, A. W. Fang, M. J. Paniccia, D. Hak, and O. Cohen, "Net continuous wave optical gain in a low loss silicon-on-insulator waveguide by stimulated Raman scattering," Opt. Express 13, 519-525 (2005). [CrossRef] [PubMed]
  14. H. Rong, A. Liu, R. Jones, O. Cohen, D. Hak, R. Nicolaescu, A. Fang and M. Paniccia, "An all-silicon Raman laser," Nature 433, 292-294 (2005). [CrossRef] [PubMed]
  15. H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang and M. Paniccia, "A continuous-wave Raman silicon laser," Nature 433, 725-728 (2005). [CrossRef] [PubMed]
  16. O. Boyraz and B. Jalali, "Demonstration of directly modulated silicon Raman laser," Opt. Express 13, 796-800 (2005). [CrossRef] [PubMed]
  17. R. Jones, A. Liu, H. Rong, M. Paniccia, O. Cohen, and D. Hak, "Lossless optical modulation in a silicon waveguide using stimulated Raman scattering," Opt. Express 13, 1716-1723 (2005). [CrossRef] [PubMed]
  18. E. Yablonovitch, "Inhibited Spontaneous Emission in Solid-State Physics and Electronics," Phys. Rev. Lett. 58, 2059-2062 (1987). [CrossRef] [PubMed]
  19. S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486-2489 (1987). [CrossRef] [PubMed]
  20. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light. (Princeton, NJ: Princeton University Press, 1995).
  21. K. Srinivasan and O. Painter, "Momentum space design of high-Q photonic crystal optical cavities," Opt. Express 10,670-684 (2002). [PubMed]
  22. D. Englund, I. Fushman, and J. Vučković, "General recipe for designing photonic crystal cavities," Opt. Express 13, 5961-5975 (2005). [CrossRef] [PubMed]
  23. Y.  Akahane, T. Asano, B. S. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature 425, 944-947 (2003). [CrossRef] [PubMed]
  24. H. Ryu, M. Notomi, G. Kim, and Y. Lee, "High quality-factor whispering-gallery mode in the photonic crystal hexagonal disk cavity," Opt. Express 12,1708-1719 (2004). [CrossRef] [PubMed]
  25. Z. Zhang and M. Qiu, "Small-volume waveguide-section high Q microcavities in 2D photonic crystal slabs," Opt. Express 12,3988-3995 (2004). [CrossRef] [PubMed]
  26. B. S. Song, S. Noda, T. Asano, and Y. Akahane, "Ultra-high-Q photonic double-heterostructure nanocavity," Nat. Mater. 4, 207-210 (2005). [CrossRef]
  27. B. S. Song, S. Noda, and T. Asano, "Photonic devices based on in-plane hetero photonic crystals," Science 300, 1537 (2003). [CrossRef] [PubMed]
  28. K. J. Vahala, "Optical microcavities," Nature 424, 839-846 (2003). [CrossRef] [PubMed]
  29. H. Takano, B. -S. Song, T. Asano, and S. Noda, "Highly efficient multi-channel drop filter in a two-dimensional hetero photonic crystal," Opt. Express 14, 3491-3496 (2006). [CrossRef] [PubMed]
  30. H. G. Park, S. H. Kim, S. H. Kwon, Y. G. Ju, J. K. Yang, J. H. Baek, S. B. Kim, and Y. H. Lee, "Electrically driven single-cell photonic crystal laser," Science 305, 1444-1447(2004). [CrossRef] [PubMed]
  31. T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, "Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity," Nature 432, 200-203 (2004). [CrossRef] [PubMed]
  32. M. Soljacic and J. D. Joannopoulos, "Enhancement of nonlinear effects using photonic crystals," Nat. Mater. 3, 211-219 (2004). [CrossRef] [PubMed]
  33. P. E. Barclay, K. Srinivasan, and O. Painter, "Nonlinear response of silicon photonic crystal microresonators excited via an integrated waveguide and fiber taper," Opt. Express 13, 801-820 (2005). [CrossRef] [PubMed]
  34. M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi, and T. Tanabe, "Optical bistable switching action of Si high-Q photonic-crystal nanocavities," Opt. Express 13, 2678-2687 (2005). [CrossRef] [PubMed]
  35. M. Spillane, T. J. Kippenberg, and K. J. Vahala, "Ultralow-threshold Raman laser using a spherical dielectric microcavity," Nature 415, 621-623 (2002). [CrossRef] [PubMed]
  36. T. J. Kippenberg, S. M. Spillane, B. Min, and K. J. Vahala, "Theoretical and experimental study of stimulated and cascaded Raman scattering in ultrahigh-Q optical microcavities," IEEE J. Sel. Top. Quantum Electron. 10, 1219-1228 (2004). [CrossRef]
  37. H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, "Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5 µm wavelength," Appl. Phys. Lett. 80, 416-418 (2002). [CrossRef]
  38. X. Yang and C. W. Wong, "Design of photonic band gap nanocavities for stimulated Raman amplification and lasing in monolithic silicon," Opt. Express 13, 4723-4730 (2005). [CrossRef] [PubMed]
  39. L. Florescu and X. Zhang, "Semiclassical model of stimulated Raman scattering in photonic crystals," Phys. Rev. E 72, 016611 (2005). [CrossRef]
  40. J. F. McMillan, X. Yang, N. C. Paniou, R. M. Osgood, and C. W. Wong, "Enhanced stimulated Raman scattering in slow-light photonic crystal waveguides," Opt. Lett. 31, 1235-1237 (2006). [CrossRef] [PubMed]
  41. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N. J., 1984).
  42. A. Yariv, Optical Electronics (Sanders College Publishing, Philadelphia, 1991).
  43. C.  Manolatou, M. J.  Khan, S.  Fan, P. R.  Villeneuve, H. A.  Haus, and J. D.  Joannopoulos, "Coupling of modes analysis of resonant channel add-drop filters," IEEE J. Quantum Electron.  35, 1322 (1999). [CrossRef]
  44. H. Ren, C. Jiang, W. Hu, M. Gao, and J. Wang, "Photonic crystal channel drop filter with a wavelength-selective reflection micro-cavity," Opt. Express 14, 2446-2458 (2006). [CrossRef] [PubMed]
  45. T. J. Johnson, M. Borselli, and O. Painter, "Self-induced optical modulation of the transmission through a high-Q silicon microdisk resonator," Opt. Express 14, 817-831 (2006). [CrossRef] [PubMed]
  46. C. Manolatou and M. Lipson, "All-Optical Silicon Modulators based on Carrier Injection by Two-Photon Absorption," J. Lightwave Technol. 24, 1433-1439 (2006). [CrossRef]
  47. T. Uesugi, B. Song, T. Asano, and S. Noda, "Investigation of optical nonlinearities in an ultra-high-Q Si nanocavity in a two-dimensional photonic crystal slab," Opt. Express 14, 377-386 (2006). [CrossRef] [PubMed]
  48. D. Dimitropoulos, B. Houshmand, R. Claps, B. Jalali, "Coupled-mode theory of the Raman effect in silicon-on-insulator waveguides," Opt. Lett. 28, 1954-1956 (2003). [CrossRef] [PubMed]
  49. M. Krause, H. Renner, and E. Brinkmeyer, "Analysis of Raman lasing characteristics in silicon-on-insulator waveguides," Opt. Express 12, 5703-5710 (2004). [CrossRef] [PubMed]
  50. X. Chen, N. C. Panoiu, and R. M. Osgood, "Theory of Raman-mediated pulsed amplification in silicon-wire waveguides," IEEE J. Quantum Electron. 42, 160-170 (2006). [CrossRef]
  51. V. E. Perlin and H. G. Winful, "Stimulated Raman Scattering in nonlinear periodic structures," Phys. Rev. A 64, 043804 (2001). [CrossRef]
  52. B. Min, T. J. Kippenberg, and K. J. Vahala, "Compact, fiber-compatible, cascaded Raman laser," Opt. Lett. 28, 1507-1509 (2003). [CrossRef] [PubMed]
  53. D. Braunstein, A. M. Khazanov, G. A. Koganov, and R. Shuker, "Lowering of threshold conditions for nonlinear effects in a microsphere," Phys. Rev. A 53, 3565-3572 (1996). [CrossRef] [PubMed]
  54. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, Hoboken, New Jersey, 2003) Chap. 10.
  55. H. W. Tan, H. M. van Driel, S. L. Schweizer, and R. B. Wehrspohn, "Influence of eigenmode characteristics on optical tuning of a two-dimensional silicon photonic crystal," Phys. Rev. B,  72, 165115 (2005). [CrossRef]
  56. R. A. Soref and B. R. Bennett, "Electrooptical Effects in Silicon," IEEE J. Quantum Electron. 23, 123-129 (1987). [CrossRef]
  57. Handbook of optical constants of solids, E. Palick, ed., (Academic Press, Boston, MA, 1985).
  58. M. Dinu, F. Quochi, and H. Garcia, "Third-order nonlinearities in silicon at telecom wavelengths," Appl. Phys. Lett. 82, 2954-2956 (2003). [CrossRef]
  59. A. Cutolo, M. Iodice, P. Spirito, and L. Zeni, "Silicon Electro-Optic Modulator Based on a Three Terminal Device Integrated in a Low-Loss Single-Mode SOI Waveguide," J. Lightwave Technol. 15, 505-518 (1997). [CrossRef]
  60. S. Sze, Physics of semiconductor devices, 2nd ed. (John Wiley and Sons, New York, New York, 1981).
  61. T. Asano, W. Kunishi, M. Nakamura, B. S. Song, and S. Noda, "Dynamic wavelength tuning of channel-drop device in two-dimensional photonic crystal slab," Electron. Lett. 41, 37-38 (2005). [CrossRef]
  62. G. Cocorullo, F. G. Della Corte, and I. Rendina, "Temperature dependence of the thermo-optic coefficient in crystalline silicon between room temperature and 550 K at the wavelength of 1523 nm," Appl. Phys. Lett. 74, 3338-3340 (1999). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited