Thin-film lithium niobate has shown promise for scalable applications ranging from single-photon sources to high-bandwidth data communication systems. Realization of the next generation high-performance classical and quantum devices, however, requires much lower optical losses than the current state of the art resonator (*Q*-factor of ∼10 million). Yet the material limitations of ion-sliced thin film lithium niobate have not been explored; therefore, it is unclear how high the quality factor can be achieved in this platform. Here, using our newly developed characterization method, we find out that the material limited quality factor of thin film lithium niobate photonic platform can be improved using post-fabrication annealing and can be as high as *Q* ≈ 1.6 × 10^{8} at telecommunication wavelengths, corresponding to a propagation loss of 0.2 dB/m.

Thin-film lithium niobate (TFLN) platform has enabled a myriad of classical and quantum applications,^{1} many of which crucially rely on low optical loss. For instance, the bandwidth of electro-optic (EO) frequency combs^{2} and the efficiency of microwave-to-optical transducers^{3,4} are proportional to the resonator quality factor *Q* or (loss rate)^{−1}. Currently, the lowest-reported optical loss in ion-sliced TFLN waveguides is ∼3 dB/m,^{5} which compares favorably to many photonic platforms. At the same time, a loss of ∼0.2 dB/m was measured using whispering gallery mode resonators created by polishing bulk congruent LN.^{6} It is currently an open question if TFLN can reach this and ideally even lower, loss rates. For example, it has been speculated that the ion slicing process used to create TFLN from bulk LN^{7} may result in implantation damage that could yield higher optical absorption in TFLN than in polished bulk LN.

Here, we first develop a post-fabrication process based on annealing in an O_{2} atmosphere in order to reduce the material absorption rate *κ*_{abs} of the TFLN platform. The absorption rate *κ*_{abs} contributes to the intrinsic loss rate *κ*_{int} of the resonator, together with the scattering loss rate *κ*_{sca}. To quantify the reduction of material absorption rate in our samples, we adapt the Kerr-calibrated linear response measurements to the TFLN platform. The method exploits the coexistence of the absorption induced photothermal effect and the material Kerr effect, and it has shown great accuracy in other integrated photonics platforms.^{8,9} The calibration of absorption rate requires parameters such as the nonlinear refractive index *n*_{2} as well as the ratio between the photothermal and Kerr-induced cross-phase modulation (XPM) responses (at bandwidths $<$10 MHz) *γ* = *χ*_{therm}/*χ*_{Kerr} of TFLN. These, and other parameters, are determined by performing laser pump–probe measurements on timescales shorter than the response time of deleterious photorefractive (PR) effects in LN.^{10} Using these pump–probe techniques, we determine that the material limited loss in ion-sliced LN is ∼1.5 dB/m and demonstrate an annealing process that reduces this to ∼0.2 dB/m that approaches the limit of the bulk LN.

The micro-rings are fabricated on a 600 nm-thick x-cut LN thin-film bonded to a 4.7 *μ*m-thick layer of thermal oxide on a silicon wafer (NanoLN). Electron-beam lithography followed by physical reactive Ar^{+} ion etching with a target etch-depth of 300 nm yields micro-ring resonators of 140 *μ*m radius (free spectral range of ∼150 GHz) and a waveguide top-width of 2.4 *μ*m [Fig. 1(b)]. We prepare three sets of resonators from this wafer in the same fabrication run: cladded (sample A), annealed (sample B), and annealed–cladded–annealed (sample C) resonators. All resonators are designed to be undercoupled, with a symmetric point coupling between the bus waveguide and the ring resonator. Sample A is fabricated with the process reported in Ref. 5. That is the resonators are cladded with an 800 nm-thick layer of SiO_{2} using plasma-enhanced chemical vapor deposition with a substrate temperature of 300 °C. For samples B and C, the resonators are annealed at atmospheric pressures in O_{2} at 520 °C for two hours. The annealing step is used to improve the crystallinity of TFLN, thereby repairing potential damages^{11} caused by ion slicing.^{12} We clad sample C with an 800 nm-thick layer of SiO_{2} deposited using inductively coupled plasma chemical vapor deposition (ICPCVD) at 80 °C and then re-anneal it under the same conditions. We emphasize the low temperature nature (80 °C) of the ICPCVD process, which we found to be important to maintain the benefits of the annealing step. We measure a mean *Q*_{int} of 1.5, 2.5, and 5.0 × 10^{6} in samples A, B, and C, respectively [Fig. 1(c)]. We note that all resonances from samples B and C achieve low enough intrinsic loss to exhibit visible asymmetric mode splitting due to Rayleigh back-scattering [Fig. 1(c)]. We attribute the lower intrinsic quality factors of the current set of devices (compared to results presented in Ref. 5) to the possible fabrication variations. We note that *Q*-factors as high as 12 × 10^{6} can be measured using the optimized process.

Before the Kerr-calibrated response measurements, we first determine the timescale of PR effects in the micro-rings since the PR-induced resonance frequency change could distort the inferred material response at low modulation frequencies.^{10} To do this, we optically pump a micro-ring then; after extinguishing the pump, we repeatedly measure one of its resonances with a probe, monitoring the time-dependence of the detuning of the resonance [Fig. 2(a)]. The detuning is normalized to the Kerr shift (discussed later) for convenient comparison. For sample A, we observe a blue shift with a time constant of ∼100 s, which is indicative of PR effects. We did not observe PR behavior for samples B and C over time scales of up to a minute [Fig. 2(b)]. Thus, the PR effect can be ignored for measurements at timescales significantly shorter than 100 s (bandwidths ≫ 0.01 Hz).

To calibrate the absorption rate of different devices, we need to evaluate

where *V* is the optical mode volume, *n*_{g} is the group index, *n*_{eff} is the effective index, and *dν*/*dP*_{abs} is the photothermal frequency shift gradient at pump frequency *ν*, mainly determined by the material thermo-optic coefficients,^{13} and is determined from simulation (see Appendix A–Table I). The response ratio *γ* is the DC offset of the measured response function *γ*(*ω*) in the Fourier domain and is obtained through the fitting. The material absorption rate also requires a good knowledge of *n*_{2} from TFLN. We did an auxiliary pump–probe measurement to obtain *n*_{2} = 1.67 × 10^{−19} m^{2} W^{−1} for our TFLN (see Appendix A for details). We did not employ the commonly used thermal triangle technique^{9} to determine *n*_{2} to avoid PR effects.

Sample . | Post-processes . | Q_{int} (10^{6})
. | Q_{abs} (10^{6})
. |
---|---|---|---|

A | PECVD | 1.5 | 21 |

B | Anneal | 2.5 | 97 |

C | Anneal + ICPCVD + anneal | 5.0 | 163 |

Sample . | Post-processes . | Q_{int} (10^{6})
. | Q_{abs} (10^{6})
. |
---|---|---|---|

A | PECVD | 1.5 | 21 |

B | Anneal | 2.5 | 97 |

C | Anneal + ICPCVD + anneal | 5.0 | 163 |

To separate the photothermal and Kerr effects in our samples and to measure *γ*, we exploit the finite response bandwidth of the photothermal effect. The measurement is accomplished by modulating an optical pump and measuring the resultant side-of-fringe modulation of a probe, as induced by the material response of TFLN [see Fig. 3(a)]. The pump modulation frequency is varied to elucidate photothermal and Kerr-induced XPM on the probe, which can be distinguished at low ($<$10 kHz) and high ($>$1 MHz) modulation frequencies, respectively.^{8} The responses of all the devices were measured, and they all yield two plateaus corresponding to either predominantly Kerr induced (*χ*_{Kerr}) or photothermal-induced (*χ*_{therm}) responses. The ratios of the plateaus *γ* are 11.0, 2.5, and 1.5 for samples A, B, and C, respectively, indicating that thermal annealing reduces the magnitude of photothermal response. It should be noted that cascaded sum-frequency generation of the pump with the probe will contribute an additional XPM indistinguishable from the Kerr contribution. Given our system parameters, we expect this change to be ∼10% (see Appendix B for detailed discussion). For all measurements, the wavelength of the pump is ∼1550 nm, while the probe is at wavelengths detuned several free-spectral ranges of the resonator away, both in TE polarization. The pump power is kept low ($<$1 mW) to avoid nonlinearity induced self-feedback.

Finally, using Eq. (1), we calculate *κ*_{abs}/2*π* to be 9.2, 2.0, and 1.2 MHz for samples A, B, and C, respectively. The absorption rates corresponding to each sample are calibrated individually. All the measured values for different samples are listed in Table I in the form of quality factors, with sample C yielding a material-limited quality factor of 163 million (0.2 dB/m), which is among the highest within integrated photonic platforms, see Table II. Our results suggest that the main source of loss in our high-confinement LN waveguides (sample C) is line-edge roughness-induced scattering, which limits the average intrinsic loss rate *κ*_{int}/2*π* of resonances around 1550 nm to 34 MHz [Fig. 1(c)].

Platform . | n
. | χ_{2} (pm/V)
. | n_{2} (10^{−20} m^{2}/W)
. | Q-factor (10^{6})
. |
---|---|---|---|---|

SiO_{2} | 1.45 | 0 | 2.2 | 3900 |

SiN | 2.0 | 0 | 24 | 290 |

Ta_{2}O_{5} | 2.0 | 0 | 62 | 2.4 |

Al_{0.2}Ga_{0.8}As | 3.3 | 119 | 2600 | 2.0 |

TFLN (this work) | 2.2 | 30 | 17 | 163 |

Platform . | n
. | χ_{2} (pm/V)
. | n_{2} (10^{−20} m^{2}/W)
. | Q-factor (10^{6})
. |
---|---|---|---|---|

SiO_{2} | 1.45 | 0 | 2.2 | 3900 |

SiN | 2.0 | 0 | 24 | 290 |

Ta_{2}O_{5} | 2.0 | 0 | 62 | 2.4 |

Al_{0.2}Ga_{0.8}As | 3.3 | 119 | 2600 | 2.0 |

TFLN (this work) | 2.2 | 30 | 17 | 163 |

In conclusion, we demonstrated that post-fabrication annealing and low-temperature oxide cladding can significantly reduce optical absorption in TFLN waveguides. Absorption at telecommunication wavelengths is reduced by removing damage potentially caused by ion implantation and reactive-ion etching even at temperatures in which other chemical bonds (e.g., Si–H and O–H) are still present.

Consequently, annealing reduces the material absorption loss significantly over a broad frequency range.^{11} Our annealing technique yielded the absorption-limited loss on par with the ∼0.2 dB/m measured in bulk LN,^{6} corresponding to a material-limited *Q*-factor of 163 × 10^{6}. We anticipate that, by improving TFLN fabrication strategies, *Q*-factors approaching the material limit can be achieved, reaching the low-loss regime required for transformative quantum and classical technology, e.g., deterministic room-temperature single photon source with periodically poled TFLN micro-ring.^{14}

We acknowledge fruitful discussions with Martin M. Fejer, Linbo Shao, Maodong Gao, and Christian Reimer. Fabrication is performed at the Harvard University Center for Nanoscale Systems (CNS).

This work was supported by the Air Force Office of Scientific Research (Grant No. FA9550-19-1-0376), the Swiss National Science Foundation (SNSF) (Grant Nos. 185870 and 192293), and the Defense Advanced Research Projects Agency (Grant Nos. HR0011-20-C-0137 and HR0011-20-2-0046).

## AUTHOR DECLARATIONS

### Conflict of Interest

L.H., P.K., M.Z., and M.L. are involved in developing TFLN technologies at Hyperlight Corporation.

### Author Contributions

Amirhassan Shams-Ansari and Guanhao Huang contributed equally to this work.

**Amirhassan Shams-Ansari**: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). **Guanhao Huang**: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). **Lingyan He**: Methodology (equal). **Zihan Li**: Data curation (equal); Methodology (equal). **Jeffrey Holzgrafe**: Methodology (supporting). **Marc Jankowski**: Methodology (supporting). **Mikhail Churaev**: Methodology (supporting). **Prashanta Kharel**: Methodology (supporting). **Rebecca Cheng**: Methodology (supporting). **Di Zhu**: Methodology (supporting). **Neil Sinclair**: Methodology (supporting). **Boris Desiatov**: Methodology (supporting). **Mian Zhang**: Methodology (supporting). **Tobias J. Kippenberg**: Conceptualization (equal); Supervision (equal). **Marko Lončar**: Conceptualization (equal); Supervision (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

### APPENDIX A: n_{2} CALIBRATION METHOD

The material absorption rate requires the evaluation of *n*_{2}. We perform measurements on an auxiliary z-cut sample with similar waveguide geometry to determine *n*_{2}. We chose this specific crystal direction to minimize the error of the cascaded *χ*_{2} calculation (see Appendix B) and mitigate the impact of the adiabatic evolution of the modes from TE to TM during propagation in the x-cut sample due to birefringence. We note that even though anisotropy of *χ*^{(3)} nonlinearity was not reported in TFLN, the *n*_{2} might be crystal orientation dependent; therefore, we characterized the value for both the TE and TM modes in the Z-cut sample, which are aligned to the ordinary and extraordinary axes. The measurement results show little difference in *n*_{2} along these two axes.

Due to the presence of the photo refractive (PR) effect, the commonly used thermal triangle technique^{9} that uses high optical power to determine *n*_{2} is not suitable for the thin film lithium niobate (TFLN) platform. Here, we use a pump–probe scheme [setup see Fig. 4(a)], which is similar to the one used in the main text (for calibrating the loss), to determine the value of *n*_{2} in TFLN at low optical powers. The main idea of this measurement is to precisely calibrate the impact of the pump intracavity power density modulation *δρ* on the probe’s cavity frequency modulation *δν* mediated by the Kerr effect. This allows us to evaluate the value of *n*_{2} from

where $x\u0304$ is the mode intensity weighted average of the corresponding physical quantities, retrieved from the simulation. Table III contains the simulated numbers required for the *n*_{2} calibration and the absorption calibration (COMSOL).

TE parameters . | Values . |
---|---|

$ngn\u0304$ | 4.53 |

V | 7.47 × 10^{−16} m^{3} |

TM parameters | Values |

$ngn\u0304$ | 4.14 |

V | 9.11 × 10^{−16} m^{3} |

Abs parameters | Values |

Cladded $Vd\nu \nu dPabs$ | 2.82 × 10^{−18} m^{3}/W |

Uncladded $Vd\nu \nu dPabs$ | 4.05 × 10^{−18} m^{3}/W |

TE parameters . | Values . |
---|---|

$ngn\u0304$ | 4.53 |

V | 7.47 × 10^{−16} m^{3} |

TM parameters | Values |

$ngn\u0304$ | 4.14 |

V | 9.11 × 10^{−16} m^{3} |

Abs parameters | Values |

Cladded $Vd\nu \nu dPabs$ | 2.82 × 10^{−18} m^{3}/W |

Uncladded $Vd\nu \nu dPabs$ | 4.05 × 10^{−18} m^{3}/W |

The intracavity power density modulation *δρ* is determined by the intensity modulation depth *α* and the waveguide circulating power *P*_{WG},

We modulate the pump intensity *I*(*t*) = *I*_{0}[1 + α cos(2*π*Ω_{IM}*t*)] at Ω_{IM} = 10 MHz and determine the modulation-depth *α* using heterodyne measurements [Fig. 4(b) red spectrum], frequency offset by 100 MHz using acousto-optic modulators. The intracavity power density *ρ*(*P*_{WG}) as a function of waveguide circulating power *P*_{WG} depends on many parameters. Apart from cavity coupling rates, the power is also affected by the background etalon formed due to the chip facet reflections. Therefore, the measured cavity transmission trace from the laser scan consists of a sinusoidal background modulation and a Fano-shaped cavity resonance dip. We express the intracavity power density considering all these effects as

with *κ*_{ex} being the cavity external coupling rate, *κ* being the cavity linewidth, Δ being the laser detuning, Δ*ν*_{FP} being the waveguide background etalon fringe periodicity, Δ*θ*_{FP} being the etalon phase offset, *R* being the characteristic etalon reflectivity, and *V* being the mode volume of the pump mode. Here, power *P* is the waveguide circulating power at the quadrature point of the waveguide background etalon fringe. All the parameters used in the power density function is fitted from the pump mode transmission profile shown in Fig. 4(d) (TM mode result), using fitting function,

Laser frequency sidebands at 300 MHz are applied to calibrate the laser detuning. The fitting results are shown in the following table:

Mode | κ_{ex}/2π | κ/2π | R | Δv_{FP}/2π | Δθ_{FP} |

TM | 14.2 MHz | 47.8 MHz | 0.152 | 11.0 GHz | −0.57 rad |

TM | 87.8 MHz | 302 MHz | 0.107 | 9.05 GHz | −1.42 rad |

Mode | κ_{ex}/2π | κ/2π | R | Δv_{FP}/2π | Δθ_{FP} |

TM | 14.2 MHz | 47.8 MHz | 0.152 | 11.0 GHz | −0.57 rad |

TM | 87.8 MHz | 302 MHz | 0.107 | 9.05 GHz | −1.42 rad |

After the intracavity power density function is determined through fitting the cavity transmission trace, we need to calibrate how much Kerr frequency modulation *δν* on the probe cavity is induced from a given waveguide circulating power *P*. To calibrate the cavity frequency modulation depths, we use the method from Ref. 15 by comparing the Kerr frequency modulation signal to a reference phase modulation with known depth *β* [calibrated also using heterodyne measurements, Fig. 4(b) blue spectrum]. The phase modulation $E(t)=E0ei\beta cos(2\pi \Omega PMt)$ is applied to the probe laser at Ω_{PM} = 9 MHz and is visible in Fig. 4(c) right next to the cavity frequency modulation signal at Ω_{IM} = 10 MHz. The reference phase modulation acts as a ruler and allows us to compare and retrieve the cavity frequency modulation depth at different optical powers. To isolate the cavity frequency modulation contributed by the Kerr effect from the one from the thermal effect, we also measured the XPM response at different pump modulation frequencies using a vector network analyzer. We retrieved the fraction of pure Kerr contribution to the total XPM signal Γ(Ω_{IM}) = *χ*_{Kerr}(Ω_{IM})/*χ*_{XPM}(Ω_{IM}) = 0.60 for the TM mode and 1.00 for the TE mode at Ω_{IM} = 10 MHz by fitting the measured response [TM result shown in Fig. 4(e)]. After that, the Kerr induced cavity frequency modulation can be expressed as

where *ξ* = *S*_{XPM}/*S*_{ref} is the power spectral density ratio between the XPM total signal *S*_{XPM} measured on the real-time spectrum analyzer and the reference phase modulation signal *S*_{ref}.

Since we do not have direct access to the on-chip waveguide circulating power and the coupling efficiencies at the chip facets can be different, we mitigate the uncertainties by taking the geometry average of the input power *P*_{in} and output power *P*_{out} of the chip as the waveguide circulating power $P=PinPout$, measured at the etalon quadrature point. We measure the XPM ratio *ξ*_{F} at the given setting and repeat the measurement (measure ratio *ξ*_{R}) after reversing the input and output of the micro-ring, in order to take into account different coupling efficiencies at the chip facets. The spectrum of the TM mode when measuring both *ξ*_{F} and *ξ*_{R} is shown in Fig. 4(c) and we take their geometric average as well $\xi =\xi F\xi R$.

For all measurements, the wavelength of the pump is ∼1550 nm, while the probe is at wavelengths detuned several free-spectral ranges of the resonator away. The on-chip pump power is kept low ($<$1 mW) to avoid nonlinearity induced self-feedback. With all relevant parameters measured/fitted, by inverting Eq. (A1), our measurements allow calibrating *n*_{2} using

and we find a material nonlinear refractive index of *n*_{2} = 1.61 × 10^{−19} m^{2} W^{−1} for the TE mode (aligned to ordinary axis) and *n*_{2} = 1.74 × 10^{−19} m^{2} W^{−1} for the TM mode (aligned to extraordinary axis). For the TE mode in the X-cut samples we characterized in the main text, an averaged value of *n*_{2} = 1.67 × 10^{−19} m^{2} W^{−1} is used.

It should be noted that cascaded sum-frequency generation of the pump with the probe will contribute an additional XPM indistinguishable from the Kerr contribution. Given our system parameters, we expect this change to be ∼10%.

All the physical quantities measured/fitted for the *n*_{2} calibration are presented in Table IV.

Parameters . | TM values . | TE values . |
---|---|---|

Γ(Ω_{IM}) | 0.60 | 1.00 |

ξ | 44.07 | 12.44 |

ξ_{R} | 1.915 | 6.383 |

P_{in} | 680 µW | 2550 µW |

P_{out} | 6.0 µW | 184 µW |

$\rho ([PinPout]1/2)$ | 47.0 J/m^{3} | 91.7 J/m^{3} |

α | 0.1619 | 0.1619 |

β | 0.1245 | 0.1245 |

Parameters . | TM values . | TE values . |
---|---|---|

Γ(Ω_{IM}) | 0.60 | 1.00 |

ξ | 44.07 | 12.44 |

ξ_{R} | 1.915 | 6.383 |

P_{in} | 680 µW | 2550 µW |

P_{out} | 6.0 µW | 184 µW |

$\rho ([PinPout]1/2)$ | 47.0 J/m^{3} | 91.7 J/m^{3} |

α | 0.1619 | 0.1619 |

β | 0.1245 | 0.1245 |

### APPENDIX B: ESTIMATED CONTRIBUTION OF CASCADED SUM-FREQUENCY GENERATION TO XPM

Determining *n*_{2} relies on pump–probe measurements (e.g., the detuning of resonance with modulating the pump) in our experiment. In this case, cascaded sum-frequency generation (SFG) of the bright pump with the dim probe will contribute to an effective XPM term since the shift of the cavity resonances is measured as a function of the pump power rather than the probe power. However, for TFLN, owing to its large *χ*^{(2)}, cascaded second order nonlinearities can contribute an *n*_{2} with nearly equal and opposite signs to the pure electronic *n*_{2} of LN. As a result, values of *n*_{2} inferred from Z-scans in bulk material or the thermal triangle in ring resonators may differ by an order of magnitude from the real value.^{16,17} We may estimate the strength of these contributions for the waveguides under consideration here by finding both the nonlinear coupling and phase-mismatch between the pump, probe, and generated sum-frequency.

The coupled wave equations for three-wave mixing are given (in power normalized units) by

where *κ*_{3}/*ω*_{3} = *κ*_{2}/*ω*_{2} = *κ*_{1}/*ω*_{1}. In the undepleted limit, we may assume that the bright pump is given by *A*_{2}(*z*) = *A*_{2}(0)exp(−*iϕ*_{NL,2}(*z*)), and *A*_{1}(*z*) = *A*_{1}(0)exp(−*iϕ*_{NL,1}(*z*)), where *ϕ*_{NL,1}(*z*) = *δk*_{1}*z* is assumed to be approximately linear in *z*. We assume *δk*_{1(2)} ≪ Δ*k*, in which case the sum frequency is given by

where we have ignored oscillatory exp(−*i*Δ*kz*) terms that do not contribute to the average phase accumulated by *A*_{1} and *A*_{2}. Substituting the approximations *A*_{2}(*z*) = *A*_{2}(0)exp(−*iϕ*_{NL,2}(*z*)) and *A*_{1}(*z*) = *A*_{1}(0)exp(−*iϕ*_{NL,1}(*z*)) into Eqs. (B3b) and (B3a), we find

The cross-phase modulation that occurs during phase-mismatched SFG between the bright pump and dim probe signal is indistinguishable from the XPM typically encountered in *χ*^{(3)} media since both are linear in pump power and have the opposite sign for typical values of Δ*k*. The Δ*k*^{−1} scaling associated with cascaded nonlinearities may render the strength of this effect a strong function of waveguide geometry. We also note here that while Eqs. (B4b) and (B4a) are accurate for TM modes in Z-cut thin films, these expressions need to be corrected for TE modes in X-cut thin films. In this case, *κ*_{3}(*z*) oscillates periodically during propagation around a ring since the *d*_{ijk} tensor associated with the *χ*^{(2)} interaction is not invariant with respect to rotations around the crystalline x axis. We further note that typical waveguide geometries in the LN exhibit avoided crossings between TE and TM modes for propagation angles offset from the crystalline axes. For large radii rings, these avoided crossings cause a TE-polarized mode to undergo adiabatic conversion between TE and TM, thereby contributing more rapid oscillations to both *κ*_{3}(*z*) and Δ*k*(*z*). In this more general case, we may expand Fourier series *κ*_{3}(*z*)exp(*i*Δ*k*(*z*)*z*) and *κ*_{1}(*z*)exp(−*i*Δ*k*(*z*)*z*) and solve Eqs. (B4b) and (B4a) for each Fourier component. The total contribution to the XPM coefficient is given by

where *κ*_{3,m} is the *m*th Fourier component of *κ*_{3}(*z*)exp(*i*(Δ*k*(*z*) − Δ*k*(0))*z*) and Δ*k*_{m} = *k*_{3} − *k*_{2} − *k*_{1} − *m*/*R* is the phase-mismatch associated with each Fourier component. For large radii rings Δ*k*_{m} ≈ *k*_{3} − *k*_{2} − *k*_{1}, the total contribution from each component may be evaluated using Parseval’s theorem. Using this approximation and the waveguide geometry shown in the main text in Fig. 1, we estimate that the contribution of cascaded nonlinearities to the net XPM coefficient is below 10%.