## Abstract

Comb generation in different mode families via a stimulated Raman scattering (SRS) process is studied using a silica toroid microcavity. The broad gain bandwidth of SRS in silica allows us to excite longitudinal modes at long wavelengths belonging to mode families that are either the same as or different from the pump mode. We found through experiment and numerical analysis, that an SRS comb in a different mode family with a high quality factor (*Q*) is excited when we pump in a low-*Q* mode. No transverse mode interaction occurs when we excite in a high-*Q* mode resulting the generation of a single comb family. We studied the condition of the transverse mode interaction while varying the mode overlap and *Q* of the Raman mode. Our experimental results are in good agreement with the analysis and this enables us to control the generation of one- and two-mode combs.

© 2017 Optical Society of America

## 1. Introduction

Comb generation with microcavities has been intensively researched because it has many attractive applications such as spectroscopy, frequency metrology, and optical communication [1–13]. A high quality factor (*Q*) microcavity enables us to obtain Kerr comb generation, because third-order nonlinearity is proportional to *Q ^{2}/V* at a given input power, where

*V*is the mode volume of the cavity. Four-wave mixing (FWM) is the principal mechanism that contributes to the generation of the Kerr comb, but other effects such as stimulated Raman scattering (SRS) also occur in the same cavity.

The use of SRS is attractive because it allows us to generate long-wavelength light, particularly in the mid-infrared (MIR) wavelength regime, where the wavelengths are useful for sensing applications. Thus, SRS was taken into account and Raman lasing was demonstrated by using high-*Q* cavities made of silica, calcium fluoride, diamond, polymer, and other materials [14–26], including at MIR [21,27,28] wavelengths. Recently, the generation of comb light with the SRS process using a crystalline has even been reported [29,30].

Among various materials, silica is particularly good for demonstrating Raman lasing [14–17], because it has broadband Raman gain that has a full-width half-maximum (FWHM) bandwidth of 260 cm^{−1} with a center Stokes spectrum shift of 450 cm^{−1} [31,32]. The broad bandwidth gain usually makes precise control of the frequency separation of the longitudinal modes of the microcavity unnecessary. It is therefore straightforward to imagine that a different mode family can be simultaneously excited through the SRS process.

In this paper, we study the interaction between different mode families via the SRS process in a silica toroidal microcavity. We study the energy transition via SRS between two different mode families, one with a high *Q* and the other with a low *Q*. The interaction between transverse modes has already been observed in a crystalline cavity [33], but as yet there is no clear understanding of the condition required for transverse mode coupling, particularly with systems that have a broad SRS gain such as a silica microcavity.

The paper is organized as follows. In section 2, we explain analytically the condition of the mode interaction. We discuss the occurrence of transverse mode coupling via SRS by considering the SRS threshold power as a function of the mode overlapping and *Q*s. Section 3 describes our experiment, and we report two different results, one with and one without transverse mode coupling. We show the relationship between *Q* and the mode coupling. Section 4 is a numerical analysis based on the Lugiato-Lefever equation (LLE). We developed a model, where two LLEs are coupled through a Raman scattering term. The obtained results are in good agreement with the experimental result. These results enable us to understand clearly the mechanism of mode interaction through Raman scattering. The paper finishes with a conclusion.

## 2. Threshold power analysis of SRS

We first consider the analytical threshold power of SRS in a silica cavity. The threshold is described as [17]

*λ*and

_{p}*λ*are the respective wavelengths of the pump and a Raman modes,

_{R}*g*is the nonlinear bulk Raman gain coefficient, and

_{R}*Q*,

_{e}^{P}*Q*, and

_{T}^{P}*Q*are the external

_{T}^{R}*Q*factor of the pump mode, the total

*Q*factor of the pump mode, and the total

*Q*factor of the Raman mode, respectively.

*V*

_{eff}is the effective mode volume contributing to this SRS conversion process.

*V*

_{eff}is usually almost identical to the mode volume of the pump mode since SRS occurs in the same mode family. However, we rigorously calculate the effective mode volume to describe the mode interaction accurately, as follows,

*E*and

_{p}*E*are the amplitudes of the electric fields of the pump and Raman modes, respectively. This includes an overlap of the mode profiles of two modes. When the Raman mode is in the same mode family as the pump mode,

_{R}*V*

_{eff}is at its minimum value because the mode overlap is perfect.

We consider three mode families, TE_{00}, TE_{01} and TE_{10}, which are present in a silica toroidal microcavity. The major diameter, the minor diameter and the free spectral range (FSR) are 100 μm, 8 μm and 600 GHz, respectively. Figure 1(a) shows the mode profiles that we used for our analysis. The calculated effective mode areas are *A*_{eff TE00-TE00} = 9.75 μm^{2}, *A*_{eff TE01-TE01} = 12.79 μm^{2}, *A*_{eff TE02-TE02} = 17.75 μm^{2}, *A*_{eff TE00-TE01} = 18.19 μm^{2}, *A*_{eff TE00-TE10} = 21.69 μm^{2}, and *A*_{eff TE01-TE10} = 29.45 μm^{2}. The effective mode areas are given as *V*_{eff} = 2π*r* × *A*_{eff}, where *r* is the radius of a cavity. To compare the ratios of different excited transverse mode families via the SRS process, we define the power ratio *C* as,

*P*

_{th-same}and

*P*

_{th-diff},

*A*

_{eff-same}and

*A*

_{eff-diff},

*Q*

_{T}_{-same}and

*Q*

_{T}_{-diff}are the SRS threshold powers, effective mode areas, and total

*Qs*of the Raman mode. The subscript indicates whether the Raman mode is in the same or a different mode family. When

*C*is higher than 1, the SRS threshold power of a Raman mode in a different mode family is lower than that for one in the same mode family, which means that the SRS to the different mode family will be dominant. Figure 1(b) shows the calculated

*C*as a function of

*Q*

_{T-diff}

*/Q*

_{T-same}for three different mode combinations. The blue, red and black lines show the cases for TE

_{01}(pump)-TE

_{00}(different), TE

_{10}(pump)-TE

_{00}(different), and TE

_{10}(pump)-TE

_{01}(different). Since the

*Q*of a high-order mode is usually lower than that of a lower-order mode, these three cases are sufficient to understand the influence of the mode interaction in SRS. When

*Q*

_{T-diff}

*/Q*

_{T-same}< 1,

*C*is smaller than 1 in all cases because the mode overlapping is not perfect. However,

*C*is larger than 1 in three cases when

*Q*

_{T-diff}

*/Q*

_{T-same}> 2. This suggests that the mode interaction will occur easily when the

*Q*factor of one mode is only double that of the pump mode. Generally, the

*Q*factor of the fundamental mode (TE

_{00}mode) is much higher than that of a high-order mode (i.e. TE

_{01}) in a silica toroidal microcavity.

## 3. Experimental results

To confirm the theory, we conducted experiments with a silica toroidal microcavity. Figure 2(a) shows our experimental setup. A tunable laser diode scans the input laser wavelength and an erbium-doped fiber amplifier amplifies the input power up to 1 W. A tapered fiber with a diameter of about 1 μm is used as an evanescent coupler. The output is measured with a power meter and an optical spectrum analyzer. Figure 2(b) shows a microscope image obtained from the top of our cavity. A typical Raman gain in silica is shown in Fig. 2(c).

First, we pumped one of the modes and observed the spectrum as shown in Fig. 3(a). A comb spectrum ranging from 1400 to 2000 nm was observed. Figure 3(c) is a magnified view of Fig. 3(a), which shows that the SRS occurs in the same mode family as the pump mode. Then, we pumped the cavity in a different mode. The result is shown in Fig. 3(b), where we observe a dual-comb-like spectrum. The magnified view in Fig. 3(b) clearly shows that a different mode family is excited via the SRS process. Please note that the transverse mode is not generated through FWM because of the energy and momentum mismatch. The frequency difference between these two mode families is about 180 GHz.

Next, we measured the *Q*s of the pump and the SRS comb modes. The modes that we measured are indicated in Fig. 3 as H0 and L0 for two different pump modes and L1, L2, H1, and H2 as two different sets of mode families. We performed a conventional transmittance spectrum measurement using a tunable wavelength sweep laser, and obtained *Q*s of 1.1 × 10^{7} for the 1548.96 nm mode (H0 mode) and 3.1 × 10^{6} for the 1543.08 nm mode (L0 mode), as shown in Figs. 4(a) and (b), respectively. Figures 5(a) and 5(b) are the transmittance spectra for the H1 and H2 modes, which exhibit *Q*s of 1.6 × 10^{7} and 1.9 × 10^{7}, respectively. On the other hand, the *Q*s for the L1 and L2 modes are 5.2 × 10^{6} and 4.7 × 10^{6}, respectively.

From this result, we confirmed that an energy transfer occurs when we pump in a low-*Q* mode, but no transverse mode coupling occurs when we pump the cavity in the highest *Q* mode. Therefore, if we want to suppress the generation of a different longitudinal mode family, we must pump the cavity at the lowest order mode with the highest *Q*. This transverse mode coupling even allows us to find the lowest order mode. And this experimental result is in good agreement with our theoretical understanding that SRS converts energy from a low-*Q* mode to a high-*Q* mode.

Figure 6 confirms our discussion by showing pumping performed at different wavelengths. Figure 6(a) explains the high- and low-*Q* values of the pump modes. The spectrum obtained when we pumped at H0 and L0 are already shown in Figs. 3(a) and 3(b). When we compare Fig. 6(b) with Fig. 3(a), which is the spectrum when we pump at mode (b), we find that they are almost identical, showing only one longitudinal mode family. This indicates that the SRS process occurs in the same mode family as the pump. Indeed, we confirmed that the anti-Stokes light is also in the same mode family. On the other hand, Fig. 6(c), when we pump at mode (c), has the same trend as Fig. 3(b) that shows a twin comb spectrum. It is noted that in Fig. 6(c) that anti-Stokes SRS light is excited at ~1450 nm and it is also in a different mode family from the pump, but in the same mode family as the SRS mode. Thus, the generation of the high-*Q* mode family dominates the generation of the low-*Q* mode in the SRS process. These results indicate that this mode interaction behavior depends solely on the relationship between the *Qs* of the modes used for the pump and the generated SRS light.

## 4. Numerical simulation with coupled LLEs

We developed a numerical model to describe the behavior of a nonlinear cavity in which FWM and SRS occur simultaneously. To obtain a full understanding, we modified the LLE [34,35] and took the nonlinear energy transition via SRS into account [36]. The equations are as follows,

*E*

_{p}and

*E*

_{s}are the electrical fields of the pump and signal (Raman) lights.

*r*,

*t*,

*t*

_{R},

*L*, and

*S*

_{in}are the propagation coordinate (step), (short) time, round-trip time, cavity length, and pump light, respectively.

*α*,

*κ*,

*δ, β*

^{(}

^{k}^{)},

*γ,*and

*Γ*are the intrinsic cavity loss, coupling loss with the waveguide, detuning of the light frequency from the resonance (detuning from the center frequency), cavity dispersion, effective nonlinear coefficients, and effective nonlinear coefficients considering mode overlapping, respectively. The subscripts denote pump and signal lights. Cavity dispersion is calculated with a finite element method as shown Fig. 7.

Equation (4) shows the behavior of a pump mode that couples with a signal mode via cross phase modulation and Raman scattering. The Raman scattering terms include the response from its own intensity, the coupled light intensity, and the interaction between two modes. It is noted that only the pump mode is excited with an external source. Therefore, the signal mode receives energy only through the Raman scattering as described in Eq. (5).

Mode overlapping is considered with effective mode area *A*_{ps} described as

*n*

_{2}is the nonlinear coefficient of a material. When calculating LL equations, we assume that

*Γ*and

_{p}*Γ*have the same value for simplification. The Raman scattering terms, namely the Raman contribution

_{s}*f*

_{R}and the Raman response function

*h*

_{R}, are well-known values where

*f*

_{R}= 0.18 and

*h*

_{R}is described as

*τ*

_{1}= 12.2 fs and

*τ*

_{2}= 32 fs [32]. Although Raman scattering has gain in the orthogonal modes, the efficiency is small [37] and the conversion to orthogonal modes can be neglected. Thus, we consider Raman modes with the same polarization as a pump mode.

To explain the experimental results, we set a pump mode with a *Q* of 5.0 × 10^{6}. Based on the theoretical understanding, the *Q* factor ratio, *Q*_{Raman}/*Q*_{pump}, is used as a parameter. Figure 8(a) shows the calculation results when TE_{01} and TE_{00} are set as the pump and Raman modes, respectively. The vertical axis is the integrated light power of the generated SRS mode. Since each calculation time is tens of thousands of round trip times, the cavity is set in a steady state. When the *Q* factor ratio is 2, the Raman power suddenly increases, which means that the gain overcomes the cavity loss. The value agrees with the theoretical prediction as discussed in section 2 and shown Fig. 8(b). The optical spectrum when the ratio is 3, which corresponds to our experimental values, is shown in Fig. 8(c). The spectrum has the same shape as the experimental result shown in Fig. 3(b). On the other hand, when we pump at a higher mode, we obtain the spectrum shown in Fig. 8(d). The Raman power does not increase, and this is in good agreement with Fig. 3(a).

This calculation confirmed that the origin of the dual comb-like spectrum is the result of mode interaction between the pump and Raman modes via Raman scattering. And the ratio of the *Q* values plays an important role in determining the strength of the mode interaction.

## 5. Summary

In this study, we focused on the transverse mode interaction via stimulated Raman scattering in a silica toroidal microcavity. We measured a twin comb spectrum experimentally and confirmed that the *Q* values of the modes are key parameters as regards allowing the generation of combs in different mode families. The dual comb is only present when we pump in a low-*Q* mode. The experimental results are in good agreement with the theoretical understanding where the critical point for the transverse mode interaction is present at a *Q* ratio larger than two. We developed a numerical model based on LLE considering the mode interaction via Raman scattering and the numerical results showed the same spectrum as the experimental results. The mode interaction we discussed could be also observed in other silica microcavities such as microspheres and microdisks. Our findings contribute to both the understanding of Kerr-Raman dynamics in a silica microcavity and the generation of dual-comb spectra [39–41].

## Funding

This work was supported by the Ministry of Education, Culture, Sports, Science, and Technology (MEXT), Japan, KAKEN #15H05429, and the Photon Frontier Network Program. The first author acknowledges to the Grant-in-Aid by the Program for Leading Graduate School for “Science for Development of Super Mature Society” from the Ministry of Education, Culture, Sport, Science, and Technology in Japan.

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