1. Introduction

Optical microresonators offer an exciting new platform for the efficient generation of new optical frequencies [1–3]. Microresonator optical parametric oscillators based on both second- [4–6] and third-order optical nonlinearities [7–16] have been reported with widely tunable output frequency ranges beyond those of conventional laser sources. In this article, we focus on microresonator parametric oscillators with a third-order Kerr nonlinearity. Here, the high finesse and small mode area of the microresonator can be harnessed to drive efficient continuous-wave (CW) four-wave-mixing (FWM) at milli-Watt power levels [17–19]. The range of optical frequencies attainable from such a device is set by the phasematching of the underlying FWM interaction, with many different phasematching configurations outlined in the literature [7,9,10,14–16,20,21]. One popular method to generate large frequency shift FWM signals is to drive the microresonator at a pump frequency close to its zero-dispersion-wavelength (ZDW) [8,9,12–15,22,23]. In this regime, the contribution of higher-order dispersion enables the generation of FWM sidebands symmetrically detuned from the original pump frequency, at frequency shifts of the order of 10–100 THz. These results represent an impressive range of tunability from a low-power microresonator device. However, the requirement that the dispersion of the resonator remains close to zero at the pump wavelength ultimately limits the range of viable pump wavelengths, and hence the final range of output frequencies.

To circumvent these limitations, the authors of [7] and [16] invoked and demonstrated a multimode FWM process that exploits the additional wavevector mismatch available between different mode families of the resonator. The extra degree of freedom provided by this wavevector mismatch enables phasematching of large-frequency-shift FWM sidebands, even at pump wavelengths that experience strong second-order dispersion. The configuration used in [7], with a pump in one resonator spatial mode driving two parametric sidebands in a second spatial mode, has strong parallels to the multimode FWM processes first reported in single-pass optical fiber experiments [24–27]. Despite these demonstrations, a complete analytic description of this multimode FWM process has yet to be presented.

In this article, we present a full small-signal analysis of the parametric gain of this multimode FWM interaction in a Kerr resonator. This allows us to derive an expression for the frequency shift of the phasematched parametric sidebands that includes the effects of dispersion to all orders, modal mismatch, nonlinear phaseshifts, and resonator detuning. We then harness this multimode phasematching to observe the experimental generation of visible signals from 602 to 750 nm in a MgF$_2$ micro-disk driven at 1064 nm. The detuning between the two parametric sidebands for the largest of these phasematched FWM frequency shifts exceeds 400 THz, corresponding to almost three octaves. To the best of our knowledge, this represents the largest FWM shift reported to date in an optical microresonator, highlighting the ability of this process to generate new frequencies far from the original pump frequency.

2. Theory

We consider a multimode FWM interaction in which a pump in one mode family of the resonator drives two symmetrically detuned sidebands in a second mode family. To model this situation, we introduce the coherently coupled Lugiato-Lefever equations (CCLLE) that describe the full phase-coherent nonlinear interaction between two modes of a Kerr resonator [28]:

Here, $E_{1,2}$ are the slowly-varying intracavity field envelopes of the two mode families, $t$ is the slow time that describes the evolution of the field over successive round trips, while $\tau$ is the fast time that describes the evolution of the field over a single round trip. In addition to the standard nonlinear self- and cross-phase modulation terms, these equations include phase-sensitive multimode FWM terms, that when correctly matched, drive the cross-mode parametric interactions we wish to consider [27]. The frequency dependence of the wavevectors of the two modes $(\beta _{(1)}$ and $\beta _{(2)})$ are modelled via Taylor expansions of arbitrary order, with $\beta _{m(n)}$ defined as the $m$th derivative of the wavevector of the $n$th mode with respect to angular frequency evaluated at the pump frequency $\omega _p$. This allows us to write the wavevector mismatch between the two modes as: $\delta \beta _0=\beta _{0(1)}-\beta _{0(2)}$, and the group delay mismatch as: $\delta \beta _1=\beta _{1(1)}-\beta _{1(2)}$.

A continuous-wave external field $E_{\textrm {in}(1)}$ drives the first mode (intensity coupling coefficient $\theta _1$ with a phase detuning $\delta _1$). The second mode remains undriven, but inherits the phase detuning of the pump via the four-wave mixing term. Consequently, the second mode is subject to an effective detuning, given by $\delta \beta _0 L+\delta _1$. The resonator losses are given by $\alpha _i$, $L$ is the resonator’s roundtrip length, and $t_R$ is the resonator roundtrip time. The nonlinear interaction coefficient is given by $\gamma$ and is dependent on the mode area [12]. However, for simplicity, we assume that $\gamma$ is identical for both fields. The parameter $B$ controls the strength of the inter-mode nonlinear terms and is set by the polarisation and spatial overlap of the two mode families, with $B = 2$ corresponding to co-polarised modes with perfect spatial overlap [27]. For coupling of the pump to the first and second higher-order spatial mode, using parameters considered below, $B\approx 1$ and 0.75, respectively, and continues to diminish for coupling to even higher order modes. We also note that any FWM process involving all three of these spatial modes is forbidden, as $B=0$.

We first present a small-signal analysis of the parametric gain experienced by two weak sidebands in the second mode ($E_2$) when driven by a strong CW pump in the first mode ($E_1$). The weak sidebands are symmetrically detuned from the pump by a frequency $\Omega$. Setting $E_1(t,\tau )=A_1$, where $A_1$ is the steady-state CW solution of Eq. (1), we look for solutions to Eq. (2) of the form:

Assuming $|b_\pm |\ll |A_1|$, yields the eigenvalue equation:

Within this range, we find a maximum value of parametric gain $(\gamma _\textrm {max}=\gamma BLP_1/2)$ that occurs at the phasematched frequency detuning $\Omega _\textrm {pm}$:

As for other symmetric FWM processes [9,12,27], Eq. (6) depends only on the even-orders of dispersion, with odd-orders playing no role in setting the phasematched frequency shift. The key difference between this multimode phasematching condition and the standard scalar result stems from the presence of the wavevector mismatch between the two modes ($\delta \beta _0$). As this parameter is typically large, and can be both positive or negative (depending on whether the pump drives the high- or low-index mode), it enables the phasematching of large-frequency shift sidebands in regions of both strong anomalous and normal dispersion. In addition, for realistic microresonator parameters, the nonlinear and phase detuning terms are typically much smaller than this modal term and can be neglected. Under these conditions, Eq. (6) is found to reduce to the requirement that the linear wavevector-mismatch of the FWM process ($\delta \beta _{12}$) is zero:

As a final approximation, we note that when operating in a resonator with strong second-order dispersion, the influence of higher-order dispersive terms can also be neglected, allowing us to still further simplify the expression for the phasematched frequency shift to:

Examining Eq. (8) reveals that, when the dispersion of the second mode is normal at the pump wavelength $(\beta _{2(2)}>0)$, phasematching requires the modal wavevector mismatch between the two modes to be positive $(\delta \beta _0>0)$. This implies the pump must be coupled to the mode with the higher effective-mode-index of the two modes. The situation is reversed, however, when operating in the anomalous dispersion regime $(\beta _{2(2)}<0)$. Here, phasematching now requires the pump to be coupled to the mode with the lower effective-mode-index.

To illustrate the range of phasematched frequencies possible using multimode phasematching in Kerr microresonators, we consider a $z$-cut MgF$_2$ micro-disk resonator with minor and major radii of 100 and 500 $\mathrm{\mu}$m respectively, resulting in a free-spectral-range (FSR) of 69.5 GHz. These parameters correspond to the experimental device we will present in the next section. A commercial finite-element program (Comsol Multiphysics) is then used to calculate the effective-mode-indices of the three lowest-order modes over a wavelength range from 0.5 to 4 $\mathrm{\mu}$m. Physically, these three modes correspond to the resonator’s fundamental mode (TE01), and the first two lowest-order transverse modes (TE02 and TE03).

We first consider the multimode FWM interaction between the TE01 mode (driven with a CW external pump at 1064 nm) and the TE02 mode. Our simulations verify that the ZDW of these modes is at $\sim$1.5 $\mathrm{\mu}$m, and at the chosen pump wavelength, both modes exhibit strong normal dispersion ($\beta _2\sim 15$ ps$^2$/km); no scalar phasematching [8,9] is possible. In Fig. 1(a), we plot the wavevector mismatch $\delta \beta _{12}(\Omega )$ for this process, with the second-order dispersion of the TE02 mode shown in the inset. From this figure we see that $\delta \beta _{12}$ rises monotonically from a minimum of $-2\delta \beta _0$ at the pump wavelength until it crosses zero at the frequency detuning of 123 THz. This is the phasematched frequency shift at which the modal and dispersive mismatches exactly cancel, yielding phasematched parametric sidebands at wavelengths of 0.74 and 1.89 $\mathrm{\mu}$m. This plot clearly demonstrates that, even when pumping in a region of large positive chromatic dispersion, the large phase mismatch provided by the modal wavevector mismatch can be used to compensate, allowing for the generation of large frequency shift phasematched sidebands.

 

Fig. 1. (a) Wavevector mismatch $\delta \beta _{12}$ of multimode FWM between a pump field in the TE01 mode and two symmetrically-detuned sidebands in the TE02 mode. Inset: the dispersion of the TE02 mode as a function of wavelength. (b) Phasematched sideband wavelengths as a function of pump wavelength, with the pump in the TE01 mode and the sidebands in the TE02 (solid lines) and TE03 mode (dashed lines), respectively.

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Next, we investigate how this phasematching relationship changes as the pump wavelength is varied. In Fig. 1(b), we plot the phasematched sideband wavelengths of two different multimode FWM processes as the pump wavelength is varied from 0.8 to 1.1 $\mathrm{\mu}$m. In the first of these (solid lines), we consider a pump in the TE01 mode and sidebands in the TE02 mode, in the second (dashed lines) we consider the TE01 mode and the TE03 mode. These curves illustrate the key prediction of Eq. (8), with the larger value of wavevector mismatch for the second process resulting in corresponding larger sideband frequency shifts. Interestingly, Fig. 1(b) also shows that, unlike the scalar phasematching close to the ZDW, surprisingly little change in sideband frequency shift is observed as the pump wavelength is varied. This is due to the fact that neither of the key parameters of Eq. (8), $\beta _{2(2)}$ and $\delta \beta _0$, vary rapidly with wavelength.

In addition to parametric gain, parametric oscillation in a microresonator requires a pair of symmetrically detuned resonances in the sideband mode family that lie within the parametric gain bandwidth given by Eq. (5) [9]. Considering typical pump parameters $P_1=100$ mW, $\gamma =1\times 10^{-3}$ m$^{-1}$W$^{-1}$ and $\delta _2=0$, and using the same resonator parameters as in Fig. (1), yields a narrow $\sim 5$ GHz ($\sim ~0.07$ FSR) region of parametric gain around the phasematched shift frequency of 123 THz.

3. Experiment

To observe multimode FWM in a Kerr microresonator, we use a MgF$_2$ micro-disk with a major and minor radius of 500 and 100 $\mathrm{\mu}$m, respectively, matching the parameters of the device modelled in Section 2. The resonator (shown in Fig. 2(a)) was fabricated using single-point diamond turning [12,29,30], and polished with a polycrystalline diamond slurry to achieve a finesse $\sim 5 \times 10^4$. The resonator is driven by an external cavity diode laser with a center wavelength of 1064 nm. A Ytterbium-doped fiber amplifier is then used to boost the laser’s CW power to 100 mW. An optical fiber taper with a waist of 1 $\mathrm{\mu}$m is used to evanescently couple light to and from the resonator. The use of an independent external pump laser, allows the laser frequency to be tuned continuously, and enables us to separately excite each of the multimode parametric processes accessible in this resonator. This is in contrast to the injection-locked pump configuration of Ref. [7], where feedback from the resonator determines the laser frequency. This makes it considerably more different to controllable excite each of the separate multimode parametric processes we report.

 

Fig. 2. (a) Image of the microresonator device. (b) Top-down microscope image of the microresonator showing scattering of red light corresponding to multimode FWM. (c)–(d) Measured spectra of multimode FWM, between different spatial modes. Two different OSAs were used to measure the combined spectrum shown in (c). For the measurement in (d)m the long-wavelength sideband is beyond the range of either OSA.

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Pump modes that produce large frequency shift multimode FWM sidebands can be easily identified via the strong scattering of visible light from the microresonator which is clearly observable by eye (Fig. 2(b)). By scanning the pump laser over one FSR, we observe 4 different pump modes that produce large frequency shift visible sidebands. Figure 2(c) shows the measured spectrum of one of these multimode processes with a short wavelength sideband at 750 nm and a matching long-wavelength sideband at 1.82 $\mathrm{\mu}$m clearly evident. The large spectral separation of the sidebands ($\pm 118$ THz) required the use of two different spectrum analysers to make this measurement (shown as blue and red traces respectively). We note the complete absence of any competing FWM effects which we attribute to the strong normal dispersion of the pump mode at 1064 nm. Tuning the pump laser into a different nonlinear mode yields the spectrum shown in Fig. 2(d), with a sideband measured at 667 nm, equivalent to a frequency separation of 167 THz from the pump. In this case, the corresponding long-wavelength sideband is located in the mid-IR at 2.6 $\mathrm{\mu}$m, beyond the measurement range of our long-wavelength spectrometer. These sidebands correspond to a new FWM process, driven by different coupled spatial modes.

In another nonlinear pump mode, we observe multimode FWM sidebands at 697 nm and 2.24 $\mathrm{\mu}$m (Fig. 3(a)), equivalent to a 150 THz phasematched frequency shift. We note that the detected signal at 2.24 $\mathrm{\mu}$m is significantly weaker than the equivalent sideband shown in Fig. 2(c). We attribute this decrease in signal strength to the increased material absorption in the out-coupling silica fiber, as well as the poor coupling efficiency of the taper at this wavelength. For this mode, we experimentally record the phasematching curve as a function of pump wavelength by stepping the pump across 20 FSRs with a step size matched to one FSR of the pump mode. As shown in Fig. 3(b), these measurements qualitatively reproduce the predictions of the theoretical phasematching curve of Fig. 1(b) with only modest tunability of the visible sideband wavelength observed. This shows that, unlike FWM phasematched close to the ZDW of the pump, only limited tunability of the sideband’s frequency shift is available through small variations in pump wavelength. To achieve a large change in sideband frequency shift requires the use of different mode family pairs with substantially different values of wavevector mismatch ($\delta \beta _0$).

 

Fig. 3. (a) Spectrum showing a 150 THz frequency shift. The feature in the grey band appearing at 2.1 $\mathrm{\mu}$m is an artifact and corresponds to the second-order diffraction of the pump inside the OSA. (b) Measured wavelength dependence of the sideband frequencies. Solid lines correspond to a fit of Eq. (8).

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Finally, we report the observation of the generation of orange light at 602 nm in the same resonator, corresponding to a 216 THz frequency shift. The long-wavelength sideband of this process is located at 4.6 $\mathrm{\mu}$m, and could not be measured by our IR spectrum analyser. Due to the use of a single taper for coupling, and the requirement that the in-coupling remains optimised for the pump at 1064 nm, the out-coupling of the orange sideband is poor, resulting in a low output signal level (see Fig. 4). We note, however, that the intensity of the visible sideband light scattered directly from the resonator is observed by eye to be at least as bright as that of the three other multimode FWM processes reported above (see Fig. 4 inset). The amount of out-coupled visible light at 602 nm could be significantly enhanced through the addition of a second independently optimised fiber taper to act as a drop port. We also note that, to our knowledge, this result represents the largest FWM shift reported to date in an optical microresonator. The long- and short-wavelength sidebands are located at 65 and 497 THz respectively, yielding a frequency gap between these newly generated frequencies of almost three octaves.

 

Fig. 4. Measurement of an orange sideband at 602 nm. (Inset) Photograph of strong orange light scattering, generated through multimode FWM.

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4. Discussion and conclusions

In conclusion, we have presented a theoretical framework for multimode FWM interactions in a Kerr microresonator and used this to derive the full phasematching expression for a multimode FWM process, in which a pump in one mode family drives the formation of a pair of symmetrically detuned parametric sidebands in a second mode family. A detailed analysis of this phasematching expression clearly demonstrates that the generation of large frequency shift parametric sidebands is possible even under conditions of strong second-order dispersion at the pump wavelength, where FWM is typically prohibited. Explicitly, phasematching remains possible with strong normal dispersion at the pump wavelength provided that the pump is coupled to the mode family with the higher effective index ($\delta \beta _0>0$). Conversely, phasematching for strong anomalous dispersion requires the pump to drive the low-index mode ($\delta \beta _0<0$); however, in this second case, competition between scalar and multimode FWM processes is likely to occur. We are able to observe the predicted multimode phasematching in an MgF$_2$ micro-disk resonator, and experimentally we demonstrate four different phasematched FWM processes with frequency shifts ranging from 118 to 216 THz. The presence of strong normal second-order dispersion at the pump wavelength yields pure FWM spectra in each case, with only the pump and two sidebands present in the output spectrum. We believe these results demonstrate the potential of Kerr microresonators to provide a discretely tunable wideband parametric output even when favourable dispersion conditions are not available at the pump wavelength.