## Abstract

We analyze the process of cascaded four-wave mixing in a high-*Q* microcavity and show that under conditions of suitable cavity-mode dispersion, broadband frequency combs can be generated. We experimentally demonstrate broadband, cascaded four-wave mixing parametric oscillation in the anomalous group-velocity dispersion regime of a high-*Q* silica microsphere with an overall bandwidth greater than 200 nm.

© 2009 OSA

## 1. Introduction

It is known that the interplay of optical nonlinearity and group-velocity dispersion in a ring cavity can lead to self sustained parametric oscillation [1]. More recently, high quality factor (*Q*) microcavities, such as the silica microsphere [2] and the microtoroid [3], have allowed for ultralow power nonlinear optical interactions in the continuous-wave (CW) regime. Since the strength of *χ*
^{(3)} nonlinear interactions scales with the square of the quality factor [2], the threshold and bandwidth for parametric oscillation, such as that based on four-wave mixing (FWM) [3], are comparable to those in highly nonlinear atomic systems. Tapered-fibers allow for coupling of light into a high-*Q* silica cavity with nearly 100% efficiency and render the cavity a fiber-in, fiber-out source, which is low-cost, compact, operates at room temperature, and does not require any special or high-power laser sources. Applications for these highly nonlinear microcavities range from low-power parametric oscillators [4] and hyperparametric oscillators [5] to monolithic frequency-comb generators [6] that convert a single CW beam to a broadband frequency comb.

In the nearly-degenerate FWM process [7], two pump photons annihilate, creating a signal-idler pair situated symmetrically about the pump frequency. In the cavity geometry, this allows for doubly resonant parametric oscillation, which has been demonstrated in silica microspheres and microtoroids [2,3] with only a few hundreds of microwatts of pump power. One common feature of these experiments is operation at or near the zero-group-velocity-dispersion (GVD) point of the cavity modes. A consequence of working near the zero-GVD point is that only low input powers may be used, resulting in few signal-idler pairs being generated [2] (due to the rapidly incurred phase-mismatch as the input power is increased). More recently, CW cascaded FWM was observed for the first time in a high-*Q* microtoroid [6]. Cascaded FWM in a high-*Q* cavity is characterized by both narrow spectral features due to the high finesse and a large bandwidth, which makes it ideal for frequency metrology and for optical spectroscopy applications.

It is well known in silica fibers and other dispersive materials that operation near the zero-GVD point allows for efficient and broadband FWM [7]. However, increasing the pump power incurs a phase-matching penalty due to the combination of self and cross-phase modulation by the pump wave, which limits high-power operation. This can be easily seen from the equation for the phase mismatch term Δ*k _{NL}*,

*k*,

_{s}*k*, and

_{i}*k*are the wavevectors, for the signal, pump and idler waves, respectively, $\Delta {k}_{NL}={k}_{s}+{k}_{i}-2{k}_{p}$is the wavevector mismatch in the absence of nonlinearity,

_{p}*γ*is the nonlinear coupling coefficient, and

*P*is the power inside the medium. Near the point of zero GVD, the wavevector mismatch is small, and the process is phase matched at low pump power. However, since the expression for the gain [8],is proportional to the pump power when phase-matched, the signal/idler pair experiences little amplification. Increasing the pump power increases the second term within the square root in (2), which leads to a clamping of the gain. On the other hand, the deeper the pump frequency is located in the anomalous-GVD regime, the more negative the wavevector mismatch becomes. Thus, phase matching can now be achieved by increasing the pump power until Δ

_{p}*k*vanishes.

_{NL}In a cavity geometry, an additional boundary condition on the electromagnetic modes arises, which leads to a strict value for the *k*-vectors on resonance, namely $k=2\pi m/L$, where *m* is the cavity mode number, and *L* the effective cavity length. This implies that on resonance $\Delta k=0$, provided the signal/idler are situated symmetrically with respect to their order number *m* about the pump mode, that is, ${\omega}_{s}={\omega}_{m-n},\text{}{\omega}_{p}={\omega}_{m},\text{}{\omega}_{i}={\omega}_{m+n}$, where *n* is the number of orders between the pump and the signal/idler modes. In the strong-pump approximation, the cavity modes are given by ${\omega}_{p}=2\pi mc/({n}_{eff}+{n}_{2}{I}_{p})L$ and ${\omega}_{s/i}=2\pi (m\pm n)c/({n}_{eff}+2{n}_{2}{I}_{p})L,$ where *n _{eff}* is the effective refractive index (including material and cavity/waveguide dispersion),

*n*

_{2}the nonlinear index coefficient, and

*I*the intracavity pump intensity. In order for the four-photon scattering process to occur efficiently at strong pump powers, the conditions $\Delta {k}_{NL}=0,$ and $\text{2}{\omega}_{p}\text{=}{\omega}_{i}\text{+}{\omega}_{s}$ must be satisfied. Due to the one-to-one relationship between wavevector and frequency in a cavity, it can be shown that these two conditions are satisfied if and only if ${\omega}_{p}\text{=}{\omega}_{s}+n\Delta \omega ={\omega}_{i}-n\Delta \omega $, where

_{p}*n*Δω is such that the signal, idler, and pump are resonant with cavity modes and the nonlinear refractive index is included in the calculation of the mode frequencies. Thus, to predict which FWM signal/idler pairs will oscillate first at a particular input pump power, we calculate which signal/idler modes will be ‘symmetrized’ in the presence of the strong pump circulating in the cavity, i.e., which set of cavity modes does the nonlinear refractive index balance the material/cavity dispersion and lead to frequency matching between the signal/idler and cavity resonances. Since

*n*

_{2}is positive in most materials, this can only occur when the pump frequency is located in the anomalous-GVD regime. In addition, since

*n*is an integer, there exists a unique value of the pump power that maximizes the gain for a specific pair of signal/idler modes. For example, for a fused-silica cavity 480 μm in length, an effective mode area of 10 μm

^{2}, and considering only material dispersion (no waveguide/cavity dispersion), a 17-mW input pump power at 1550 nm will lead to oscillation of the 5th order signal/idler modes, which are separated by 18 nm from the pump wavelength. On the other hand, at 96 mW of input power, oscillation will occur at the 21st order modes, which corresponds to a shift of 72 nm from the pump wavelength. At these high input powers, enough power builds up at the signal/idler pairs for efficient re-mixing with the pump, which leads to cascading of the FWM process [9]. In order for the broadband cascaded FWM process to dominate over 1st order FWM (where each signal-idler pair is generated from two pump photons), the pump frequency must be located deep within the anomalous-GVD regime, such that the first signal-idler pair that oscillates is phase matched at the high input powers.

## 2. Numerical modeling

As discussed earlier, as the fields build up in a high-*Q* cavity, the cavity resonances and the phase-matching conditions shift due to the nonlinear refractive index. Thus, the fields associated with the FWM process that are initially resonant may no longer be so as the intensity dynamics in the cavity change. Due to the number of field modes (> 100 for a 100 mW pump) being generated, predicting the steady-state frequencies is problematic, and a coupled-amplitude model is not viable [9]. The model presented here eliminates this hindrance by removing the requirement of knowing the mode frequencies *a priori*. We treat all the field modes together as a single field that is launched into a one-dimensional ring cavity - with an effective refractive index that accounts for the cavity-mode dispersion of a silica microsphere [2], and is back-calculated from an exact solver [10] taking into account all terms of dispersion -and a beam splitter for output coupling. The mode field area used is approximately 10 μm^{2}, which was also extracted from the exact solver [10]. The full field includes both the pump at a single frequency, as well as random phase and amplitude noise (150 dB below the pump) to initiate the FWM process. The full field is allowed to propagate single pass through the cavity via the split-step Fourier method [11,12], where the boundary conditions are applied at the beam splitter. The input field, *E _{in}* (pump + noise) enters the cavity through a partially reflecting beam splitter, of reflectivity

*R*, transmittivity

*T*, that are chosen to satisfy the critical-coupling condition often encountered in tapered-fiber coupling, as well as to match the quality factor of our silica microspheres (2 × 10

^{7}). For the cavity of length

*L*, The intracavity field at position

*z*= 0, iteration

*t*is given by,

In the full-field formalism, enough points in the frequency domain must be present to account for the cavity modes shifting in time, which can be ensured by having a frequency resolution that is less than the cavity linewidth, or in other words, by insuring that each cavity mode is sampled by at least one point in the FFT array. On the other hand, the overall bandwidth required to model cascaded FWM should be enough to account for the tens/hundreds of nanometers of bandwidth that is expected to be generated. For a 250-μm microsphere with a *Q* = 2 × 10^{7}, the inclusion of 20 signal/idler cascaded pairs, which corresponds to our experimental observations, requires one million points in the FFT array, which sets a severe limit on the speed of the split-step method. In order to run a simulation to steady state, less bandwidth needs to be considered to reduce the size of the FFT arrays and the time for the simulation to be completed. This is achieved with a 17-mW input pump and 90-nm overall bandwidth. Upon reaching steady state, the most important parameter to extract is the separation between the individual peaks [13]. While limiting the power and bandwidth prevents us from accurately studying the power/bandwidth relation especially in comparison with experimental observations, the simulation still allows us to determine whether the oscillation comes from first-order spontaneous FWM or from a cascaded process. In a cascaded FWM process, all the signal/idler pairs contribute to the gain at each of the signal/idler modes, or in other words, all the signal/idler pairs are ’locked’ together through a well-defined phase relationship, which leads to an equal separation between the modes in frequency space. On the other hand, if each of the signal/idler peaks arises from a nearly-degenerate interaction with the pump (2*ω _{p}* =

*ω*+

_{s}*ω*), then the individual peaks will no longer be equidistant; the separation between the peaks will be determined solely by the cavity-mode dispersion, which is not flat. In these simulations, the frequency resolution is 13 MHz, which allows for one data point per mode (for

_{i}*Q*= 2 × 10

^{7}, the linewidth is 13 MHz).

The results of our simulations (Fig. 1 ) show that in steady-state the cascaded FWM peaks are separated by 427.27 ± 0.013 GHz. Over the 90 nm of bandwidth in the simulation, the drift in the positions of the peaks is within the resolution of the FFT simulation and was randomly distributed over the bandwidth of the simulation, which indicates that these peaks follow the stricter, cascaded FWM condition on positions, i.e. that they are equidistant in frequency space and form a frequency comb. The 13-MHz error bar, which is a result from the fact that the actual peaks can fall anywhere between two adjacent maxima (due to the under sampling of the modes) could not be improved by attempting to locate the actual position of the maxima via simple curve fitting algorithms. However, since the drift in the cavity FSR exceeds 500 MHz over the 90-nm window, and since first-order FWM oscillation falls on-resonance with cavity modes, this cavity-mode drift should be mapped into the position of the peaks in the first order FWM case, which leads us to conclude that the simulation clearly shows the presence of cascaded FWM oscillation.

In addition, the separation ∆*f* between the FWM peaks is detuned by 109 MHz from the cavity FSR due to pump cross-phase modulation, which also indicates that the dynamics are governed by cascaded FWM rather than by the first-order nearly degenerate FWM process [2,3]. The results of the simulation lead to the conclusion that when phase-matched in the anomalous-GVD regime cascaded FWM dominates over nearly-degenerate FWM and leads to equidistant peaks in frequency space and the generation of a frequency comb. In the time domain, while a frequency comb might be expected to yield a pulsed output, the results of the simulation remain inconclusive to this date, which could be attributed to either the complex phase difference between the modes or to insufficient sampling in our FFT arrays. Investigation of this matter is ongoing.

## 3. Experiment

When operating at the high powers required for cascaded FWM, thermal effects must be considered since the thermo-refractive response of fused silica can result in detrimental effects to the process [14]. Thermal heating in the microsphere due to pump absorption leads to changes in the refractive index [15] that produce frequency shifts in the cavity modes in the range of several GHz for 1 mW of input pump power (Fig. 2 ). In order to maintain input coupling, a PID lockbox is implemented to keep track of the cavity-mode position as the pump power is increased (Fig. 3 ). However due to large thermal shifts of the cavity modes, we find that no more than 10 mW of pump power can be coupled into and locked to the peak of the resonance of the microsphere before the tuning range on the diode laser head (New Focus Velocity 6328) is reached.

An alternative approach to overcome thermal effects at higher coupled pump powers is to use laser pulses with a duration that is large compared to the cavity lifetime (≈20 ns for *Q* = 2 × 10^{7}) but small compared to the thermal response (≈10 μs). This allows for power buildup in the cavity mode, which leads to the desired nonlinear response while minimizing the thermal drift due to the average power. The average power can be reduced by decreasing the duty cycle of the pump or by increasing the period. The easiest way to achieve this pulsed operation is to scan the laser about the desired mode of the microsphere. We increase the scan speed to reduce the dwell time in the mode (and hence the instantaneous heating) and increase the scan range to reduce the ”average power” heating. For a silica microsphere with a *Q* = 2 × 10^{7} and for a frequency scan of 2.5 GHz at a 1-kHz rate (both within the range of the piezoelectric transducer in the laser cavity), the dwell time in the mode is 2 μs, which is shorter than the thermal response time and longer than the optical buildup time such that the average coupled power is negligible.

To demonstrate broadband cascaded FWM in a silica microsphere, we choose a 250-μm sphere since it yields zero-GVD at wavelengths below the zero-GVD point of bulk fused silica (i.e., 1284 nm) [2]. For 1550-nm operation, the cavity modes are located deep within the anomalous-GVD regime, as required for high pump-power operation. The experimental setup for generating cascaded FWM is shown in Fig. 3. The laser is repeatedly scanned about the fundamental whispering-gallery mode over 2.5 GHz at a 1-kHz scan rate (Fig. 3). At 10 mW of pump power, the peaks close to the pump wavelength are phase matched [Fig. 4(a) ]. This is a much higher threshold than the 500 μW reported for the cavity operating at the zero-GVD point [2], which agrees with the analysis that when the cavity is deep within the anomalous-GVD regime, higher pump-powers are required to phase-match the FWM process. At 30 mW of pump power, cascaded FWM occurs, and we observe a bandwidth of 60 nm [Fig. 4(b)], which is less than the bandwidth predicted by the theory, and could be limited by inefficient detection or other losses in the system. We further increase the pump power to 80 mW and achieve a bandwidth of 150 nm [Fig. 4(c)]. At 100 mW of pump power, which is the highest peak power coupled into the microsphere in this experiment, the onset of Raman oscillation is observed [Fig. 4(d)], and the overall generated bandwidth (FWM + Raman) is 250 nm. Such large bandwidths generated from a single-frequency CW pump, are typically not possible in optical fibers/waveguides in the continuous wave regime.

## 4. Conclusion

We have developed a theory of cascaded FWM in a high-*Q* cavity from a basic ring-cavity model that was chosen to mimic the dispersion in a high-*Q* silica microsphere. A full-field simulation based on the ring-cavity model and a split-step Fourier method was developed and implemented, and it confirms that the observed FWM peaks arise from a cascaded FWM process. Experimental realization of broadband cascaded FWM in a silica microsphere is demonstrated resulting in an overall bandwidth > 200 nm.

## Acknowledgments

This work was financed by the Cornell Center for Materials Research under the NSF Grant No. DMR-0520404, AFOSR, and the DARPA Slow-Light programs. The authors would like to thank Reza Salem and Pablo Londero for useful discussions.

## References and links

**1. **M. Nakazawa, K. Suzuki, and H. A. Haus, “Modulational instability oscillation in nonlinear dispersive ring cavity,” Phys. Rev. A **38**(10), 5193–5196 (1988). [CrossRef] [PubMed]

**2. **I. H. Agha, Y. Okawachi, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Four-wave-mixing parametric oscillations in dispersion-compensated high-Q optical microspheres,” Phys. Rev. A **76**(4), 043837 (2007). [CrossRef]

**3. **T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett. **93**(8), 083904 (2004). [CrossRef] [PubMed]

**4. **A. A. Savchenkov, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, “Low threshold optical oscillations in a whispering gallery mode CaF(_{2}) resonator,” Phys. Rev. Lett. **93**(24), 243905 (2004). [CrossRef]

**5. **A. A. Savchenkov, A. B. Matsko, M. Mohageg, D. V. Strekalov, and L. Maleki, “Parametric oscillations in a whispering gallery resonator,” Opt. Lett. **32**(2), 157–159 (2007). [CrossRef]

**6. **P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature **450**(7173), 1214–1217 (2007). [CrossRef] [PubMed]

**7. **G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2001).

**8. **R. Stolen and J. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. **18**(7), 1062–1072 (1982). [CrossRef]

**9. **T. T. Ng, J. L. Blows, J. T. Mok, R. W. McKerracher, and B. J. Eggleton, “Cascaded four-wave mixing in fiber optical parametric amplifiers: Application to residual dispersion monitoring,” J. Lightwave Technol. **23**(2), 818–826 (2005). [CrossRef]

**10. **I. H. Agha, J. E. Sharping, M. A. Foster, and A. L. Gaeta, “Optimal sizes of silica microspheres for linear and nonlinear optical interactions,” Appl. Phys. B **83**(2), 303–309 (2006). [CrossRef]

**11. **R. A. Fisher and W. Bischel, “The role of linear dispersion in plane-wave self-phase modulation,” Appl. Phys. Lett. **23**(12), 661–663 (1973). [CrossRef]

**12. **A. Korpel, K. E. Lonngren, P. P. Banerjee, H. K. Sim, and M. R. Chatterjee, “Split-step-type angular plane-wave spectrum method for the study of self-refractive effects in nonlinear wave propagation,” J. Opt. Soc. Am. B **3**(6), 885–890 (1986). [CrossRef]

**13. **P. Del’Haye, O. Arcizet, A. Schliesser, R. Holzwarth, and T. J. Kippenberg, “Full stabilization of a microresonator-based optical frequency comb,” Phys. Rev. Lett. **101**(5), 053903 (2008). [CrossRef] [PubMed]

**14. **A. E. Fomin, M. L. Gorodetsky, I. S. Grudinin, and V. S. Ilchenko, “Nonstationary nonlinear effects in optical microspheres,” J. Opt. Soc. Am. B **22**(2), 459–465 (2005). [CrossRef]

**15. **T. Carmon, L. Yang, and K. Vahala, “Dynamical thermal behavior and thermal self-stability of microcavities,” Opt. Express **12**(20), 4742–4750 (2004). [CrossRef] [PubMed]