1. Introduction

Silicon is increasingly being considered as a dominant platform for photonic integrated circuits, which can be integrated with silicon CMOS electronics [1,2]. Subwavelength passive silicon photonic devices such as bends, splitters, and filters have been developed. Recently, with strong light confinement in silicon microstructures such as planar waveguides and microring resonators, nonlinear optical properties in silicon are enhanced and active functionalities in highly integrated silicon devices have been realized, such as all-optical switches with two-photon absorption and the thermal-optic effect [3], and all-silicon Raman amplification and lasing [4–12].

Two-dimensional (2D) photonic crystal (PhC) slabs are widely used for optical applications as they can confine light by the photonic band gap effect in plane with the aid of total internal reflection vertically. Introduction of point and line defects into 2D PhC slabs create nanocavities and PhC waveguides with arbitrary dispersion control. Through k-space design of cavity modes [13], remarkable high-Q/V_m nanocavities [14–17] have been achieved. High-Q silicon microdisks have also been recently reported [18,19]. The strong light confinement and light-matter interactions in these cavities permit fundamental studies and integrated nanophotonics applications [20], such as channel add/drop filters, low-threshold quantum well lasers [21], cavity quantum electrodynamics [22], enhancement of optical nonlinearities [23], slow light in coupled resonator optical waveguides [24], and ultrasmall nonlinear bistable devices [25,26].

Concurrently, recent work on silica-based high-Q microcavities have shown remarkable ultra-low threshold Raman lasing in silica microspheres [27] and silica microdisks [28]. Light generation and amplification through stimulated Raman scattering (SRS) in planar silicon waveguides have been observed recently [4–8], where the bulk Raman gain coefficient g_R is 10³ to 10⁴ times higher in silicon than in silica with a correspondingly narrower gain bandwidth. In particular, Raman lasing using silicon waveguides as the gain medium was recently demonstrated [9–12]. Two-photon absorption induced free-carrier absorption [29] is addressed using pulsed operation or reversed biased p-i-n diodes. These elements can be directly integrated onto CMOS-compatible silicon chips.

To achieve significant amplification and ultimately lasing, the gain medium should be placed in a cavity to achieve low thresholds or minimization of device footprints on a silicon chip: the challenge becomes achieving and designing sufficiently high-Q cavities with small modal volumes V_m . Here we suggest, for the first time, the enhanced stimulated Raman amplification and ultralow threshold Raman lasing possible in high-Q/V_m photonic band gap nanocavities [30]. This permits applications such as wavelength-selectable signal amplification and lasing in monolithic silicon for subwavelength photonic integrated circuits, novel nanocavity-based lasers at new wavelengths, and further facilitates and advances the study of nonlinear phenomena at small lengthscales. As a specific example, we will illustrate designs of a particular L5 photonic band gap nanocavities for enhanced SRS and Raman lasing, in which lasing thresholds on order of tens to hundreds of µW can be achieved.

2. Design concept and theoretical background

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 in our case). The strongest Stokes peak arises from single first-order Raman phonon (three-fold degenerate) at the Brillouin zone center. Coupling between the pump and Stokes waves in SRS can be understood classically with nonlinear polarizations P ⁽³⁾ , where P ⁽³⁾ is ${\chi }_{\mathit{\text{jkmn}}}^{\mathit{\left(}\mathit{3}\mathit{\right)}}$ E _p E*_p E_s, ${\chi }_{\mathit{\text{jkmn}}}^{\mathit{\left(}\mathit{3}\mathit{\right)}}$ the third-order fourth-rank Raman susceptibility, and E_p and E_s the electric fields of the pump and Stokes waves respectively. The dynamics of SRS is governed through a set of time-dependent coupled nonlinear equations [31]. Interaction of the Raman effect for ultrashort pulses, including contribution to the Kerr effect, has been carefully studied [32–33]. Here we employ the slowly varying envelope approximation for the forward and backward propagation pump wave amplitudes $A_p^{\mathit{\pm }}$ and Stokes wave amplitudes $A_S^{\mathit{\pm }}$ . The coupled nonlinear equations have been explored [34–37] and in particular described by Perlin et al. [38] and Krause et al. [39] as:

Here material dispersion is considered negligible compared to the cavity dispersion, and only the 1^st order Stokes is considered. The first two terms on the right-hand side represent SRS and Kerr nonlinearities respectively, with g_p,s the Raman gain coefficients at the pump and Stokes frequencies (reported in silicon ~20–76 cm/GW [39] at the C-band) and γ_p,s the Kerr nonlinear coefficients (=2πn₂/λ _p,s ) at the pump and Stokes frequencies. In the Kerr term, the 2|A_j |² A_k terms represent cross-phase modulation while the |A_j |² A_k terms represent self-phase modulation. v_p,s are the respective group velocities for the pump and Stokes waves. The α-terms account for linear loss, the κ-terms account for forward and backward coupling, and the δβ-terms account for signal detuning from Bragg resonance. β is the two-photon absorption (TPA) coefficient of silicon, $\overline{\phi }$ the efficiency of the free-carrier absorption (FCA) process, and N̄_eff the effective charge-carrier density. A full discussion of this framework for periodic structures can be found in the reference 38. The above coupled equations can be employed for continuous-wave or pulsed operations. The effective $A_{p\mathit{,}S}^{\mathit{\pm }}$ can be calculated inside high-Q/V_m nanocavities through a Fabry-Perot model [40].

Classical field accumulation of both |A_S |² and |A_p |² intensities, as seen in Equations (1) and (2), within a cavity enhances the spontaneous and stimulated Raman scattering process, since the Raman gain coefficient is intensity dependent. For laser oscillation, the gain condition requires the round-trip gain G_R to exceed the round-trip loss α for initiation of oscillation: (G_R -α)>0. The power build up in a cavity has a Q/V_m dependence, while the cavity round-trip loss has a 1/Q dependence from the definition of Q. Equating G_R with α, the lasing threshold P_threshold can thus be described by:

where V_m is the effective modal volume, ξ the modal overlap, Q_p,s the cavity quality factors, n_p,s the refractive indices at the pump and Stokes wavelengths λ_p and λ_s respectively. Both the pump and Stokes in the nanocavity are designed to have the same (e.g. even) symmetry to maximize the overlap ξ between the modes. The character of this threshold is similar to that derived for whispering gallery modes in microspheres and microcavities [27, 28, 41]. Note that in this classical formulation, the Raman gain coefficient gs is still equivalent to the bulk Raman gain efficient g_b , without considerations of possible cavity quantum electrodynamics (QED) enhancements.

The lasing threshold scales with V_m/Q_sQ_p as illustrated in Equation (3). This therefore suggests the motivation for small V_m cavities with high-Q factors. In fact, for microscopic spherical cavities (with ~70 µm radius) where V_m has an approximate quadratic dependence on the lengthscale R, a R² dependence for the Raman lasing threshold has been observed experimentally [27]. For even smaller spherical microdroplets (with ~4 µm radius), a stronger R⁴ dependence on the Raman lasing threshold has been observed experimentally [42]. This additional R² dependence in the microdroplets is attributed to cavity QED enhancements, particularly when the cavity linewidths are significantly smaller than the homogenous linewidth of the scattering process (when Fermi’s golden rule breaks down). Here the gain coefficient gs should be the Raman gain in a cavity gc, which can be approximated by the ratio of the density of states with the cavity compared to that of free space and looking only at the transition rate per mode [43, 44]. Even without considering these cavity QED enhancements and considering only classical local field enhancement, the Raman lasing threshold in silicon photonic band gap nanocavities can be estimated through Equation (3) to be on order tens to hundreds of µW, based on the high-Q/Vm nanocavities to be described in following section.

Analysis of the possible cavity enhancement of the Raman gain coefficient g_c is explored by several groups [27, 42, 45, 46]. For example, Wu et al. [46] and Lin et al. [42] have discussed cavity enhancement, either through a three-level atom-cavity Λ configuration or an estimate directly from the density of states in the cavity respectively. The resulting enhancements have a 1/L² or 1/L dependence [47]. Using the same analysis for our nanocavities, the cavity-enhanced gain coefficient is estimated on order 10² larger than the bulk gain coefficient. An alternative first-order approximation would be to use an effective SRS path length interaction approach, such as described in Lin et al. [42] and Matsko et al. [41], where the Raman gain of the Stokes signal (~exp(g_c I_c L_c )) involves the intensity of pump light I_c stored in the cavity, and an effective interaction length L_c described as: L_c =Qλ/2πn. For example, the high-Q (~4.2×10⁴) subwavelength nanocavity described in the next section has an effective interaction length on order 3 mm, a factor of 1.2×10³ larger than the physical length (~2.5 µm) of the cavity.

The next section focuses on the numerical design of specific photonic band gap nanocavities to support Raman amplification and lasing in monolithic silicon.

3. Design and analysis

We consider here a specific photonic band gap nanocavity with linearly aligned missing air holes. The nanocavity is designed numerically with MIT Photonic Bands (MPB) package [48] and the 3D FDTD method [49]. Using MPB, the photonic band structure and the defect resonant frequencies can be obtained. With 3D FDTD method, the defect frequencies, field profiles and Qs can be calculated. The goal of the design is to tune the frequencies of pump mode (f_pump ) and Stokes mode (f_Stokes ) with spacing 15.6 THz, corresponding to the optical phonon frequency in monolithic silicon. The wavelengths are also tuned to operate around 1550 nm, with high Qs (on order 10,000 or more) for at least the Stokes mode. The numerical design process is as following: (1) fine-tune the cavity geometry; (2) calculate resonant frequencies f_pump and f_Stokes with MPB; (3) calculate the lattice constant a based on the frequencies (f_pump -f_Stokes )(c/a)=15.6THZ and calculate the wavelength λ_pump =a/f_pump , λ_Stokes =a/f_Stokes ; (4) calculate Q_pump and Q_Stokes with 3D FDTD method. The same process can also be used to design nanocavities for anti-Stokes cavity-enhancement, where anti-Stokes generation typically has appreciably lower scattering magnitudes.

The structure investigated is an air-bridge triangular lattice photonic crystal slab with thickness of 0.6a and the radius of air holes is 0.29a, where a is the lattice period. The photonic band gap in this slab for TE-like modes is around 0.25~0.32 [c/a] in frequency. Two even modes supported in L3, L4, L5 cavities with linearly aligned missing air holes are studied, with which the modal overlap will be maximized. For small cavities such as L3 and L4, calculated a and λ are large, which will not match the telecommunication applications (around 1550 nm wavelength). For example, in L3 cavity, S₁ =0.15a, a=685 nm, λ_pump =2266 nm and λ_Stokes =2568 nm. Finally, two even modes in single L5 cavity are used as the pump and Stokes modes for Raman lasing respectively. Figure 1(a) shows the two defect modal frequencies when a Gaussian impulse is launched at the center of a particular L5 cavity.

 

Fig. 1. (a) Resonant frequencies of the pump and Stokes modes within the photonic band gap. (b) Q_pump and Q_Stokes as a function of shift S₁ .

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Table 1. Design summary of photonic crystal L5 nanocavity for Raman lasing in silicon.

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Table 1 gives the design summary of fine-tuning the shift S ₁ of two air-holes at cavity edge. With increasing S₁ , the calculated lattice period a decreases and the resonant wavelength λ also decreases due to the constant optical phonon frequency. The quality factors increase because the electric field profile is close to Gaussian function and has less leakage. Figure 1(b) compares the quality factor of pump mode and Stokes mode for different shift of two air holes S₁ =0~0.15a. Q values are obtained by calculating modal transient energy decay with 3D FDTD method: Q=ω₀U/P=-ω₀U/(dU dt), where U is the stored energy, ω₀ is the resonant frequency, and P=-dU/dt is the power dissipated. For higher Q nanocavities, a filter diagonalization method [50] can be used. Q_pump and Q_Stokes are in the order of 10³ and 10⁴ respectively. In comparison with the recently reported work [14, 25, 26], such Q factors can be achievable experimentally, with careful control of the surface roughness and fabrication steps. Higher-Q nanocavities than the results reported here can be achieved by fine-tuning the shift of additional air-holes at cavity edge such as S₂ and S₃ or using double-heterostructure nanocavities without significantly changing the effective modal volume [17]. The designed wavelength of Stokes mode is around 1550 nm and the L5 cavity with S₁ =0.05a and a=414 nm is considered in the following calculation.

Figure 2 and Fig. 3 show the electric field profile (E_y ) and 2D Fourier transform (FT) spectrum of E_y at the middle of the slab for pump mode and Stokes mode respectively, where the shift S₁ of air holes is 0.05a. Both modes are of even symmetry to maximize the modal overlap. Currently, the modal overlap integral is around 0.5833 between pump mode and Stokes mode due to the small cavity size and the discrete distribution of air holes in PhC slabs. The white circle is the boundary of the light cone. There is a trade-off between the wavelength λ and the quality factor Q. In Fig. 2(b), the FT spectrum of the pump mode contains large components inside the light cone, which indicates large radiative losses and hence lower Q; the FT spectrum of Stokes mode (Fig. 3(b)), on the other hand, contains much smaller components inside the light cone and corresponds to a higher Q. The higher-order mode (the pump mode in our case) has more serious edge effect (cross-sectional profile differing more significantly from an ideal Gaussian), resulting in a lower Q factor. Figure 4 shows a SEM picture of the specific designed L5 nanocavity fabricated at Columbia University.

 

Fig. 2. The electric field profile (E_y ) (a) and 2D FT spectrum (b) of pump mode.

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Fig. 3. The electric field profile (E_y ) (a) and 2D FT spectrum (b) of Stokes mode.

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Fig. 4. SEM picture of the PhC L5 nanocavity for Raman lasing in silicon.

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The estimated modal volume for Stokes mode of the L5 cavity is V_m ~0.12 µm³. With the calculated Q_Stokes at 4.2×10⁴ and Q_pump at 1550, the resulting Raman lasing threshold in the particular L5 nanocavity is conservatively estimated to be on order 344 µW. This is based on parameters g_s =70 cm/GW and ξ=0.5833. With nanocavities with higher Q_s on order 10⁵ [17], the threshold can be further decreased to several to tens of µW. With corresponding silicon microdisk resonators [18, 19], the achievable thresholds are on order several to hundreds of µW depending on specific designs of the microdisks. In comparison with silicabased ultrahigh-Q microspheres and microtoroids, the lower Q in photonic band gap nanocavities is compensated by the 10³ to 10⁴ larger bulk Raman gain in silicon, and the significantly smaller modal volume of the silicon photonic crystal nanocavities. Each of these approaches-the silicon photonic band gap nanocavities, silicon microdisks, or ultra-high Q silica microspheres and microtoroids-supports the push towards achieving ultralow thresholds for Raman lasing on-chip and fundamental nonlinear optics studies.

4. Conclusion

The concept and design of L5 photonic band gap nanocavities in two-dimensional photonic crystal slabs for enhancement of stimulated Raman amplification and lasing in monolithic silicon is presented for the first time. Specifically, we describe the numerical design of a high quality factor and small modal volume nanocavity that supports the required pump-Stokes modal spacing in silicon, with ultralow lasing thresholds. This concept and design supports the push towards on-chip all-optical signal amplification and lasing in monolithic silicon.