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Abstract
In situ tunable photonic filters and memories are important for emerging quantum and classical optics technologies. However, most photonic devices have fixed resonances and bandwidths determined at the time of fabrication. Here we present an in situ tunable optical resonator on thin-film lithium niobate. By leveraging the linear electro-optic effect, we demonstrate widely tunable control over resonator frequency and bandwidth on two different devices. We observe up to ∼50 × tuning in the bandwidth over ∼50 V with linear frequency control of ∼230 MHz/V. We also develop a closed-form model predicting the tuning behavior of the device. This paves the way for rapid phase and amplitude control over light transmitted through our device.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Tunable couplers for electromagnetic waves have been proposed and demonstrated for a variety of applications in many regimes, including microwave systems for cavity dumping, catch-and-release, and photon pulse shaping [1–3], as well as optical systems for spectral compression, pulse storage and shaping, modulation and filtering, and optical quantum gates [4–13]. These applications benefit from high-speed amplitude and phase control over the light in the cavity, which are governed respectively by the cavity bandwidth and resonant frequency. Whereas previous demonstrations of tunable resonators on silicon rely on thermo-optic phase shifters or charge injection schemes [14–16], more recent demonstrations on thin-film lithium niobate (TFLN) leverage the linear electro-optic effect (“Pockel’s” effect) to achieve faster phase modulation with low loss [17,18]. During review of this manuscript, we became aware of similar work from another group [19].
Here, we build on this TFLN platform, demonstrating simultaneous and arbitrary control of both the resonant frequency and bandwidth of optical resonator modes. This dual control is necessary for achieving arbitrary phase and amplitude control over light transmitted through the device. We also derive a closed-form model describing the bandwidth and detuning behavior as a function of the applied voltage. This model is important to calibrate device performance and predict its behavior in pulse shaping applications.
2. Results
2.1 Device geometry and operation
Our device comprises a resonator with an output waveguide that incorporates phase-sensitive feedback to tune the coupling rate. It consists of a TFLN ridge-slab waveguide and racetrack resonator atop a sapphire handle. A similar device to those measured in this work is pictured in Fig. 1(c). The ridge waveguide is $\sim 1.2$ $\mu$m wide, $\sim 300$ nm tall, and rests atop a $\sim 200$ nm-thick slab layer. It is designed for single-mode operation at telecommunications wavelengths. We fabricate our device following the techniques developed in [20]; waveguides are patterned via an HSQ hardmask, exposed with electron-beam lithography (JEOL JBX-6300FS), and etched using argon ion milling. We pattern Ti:Au electrodes via photolithography and lift-off. Lastly, we make on-chip wirebonds to ensure proper polarity between the electrodes. The electrodes and photonic device are positioned in order to align the applied electric field with the TFLN crystal z-axis, thereby taking advantage of the large $d_{33}$ electro-optic coefficient of LN [21].
Fig. 1. Device geometry and operation. (a) Schematic of the device. Light enters from the left and can couple into the racetrack resonator at points “1” and “2.” Red and blue coloring represent two independent paths the light can take, with purple corresponding to the light in both paths overlapping. Gold represents electrodes. The light can accumulate a different phase $\phi _1$ or $\phi _2$ in each path. (b) An alternative, free-space optics schematic of the system. Black lines are mirrors, and grey lines are tunable mirrors. Two cavities are formed by the light oscillating in the broad ring and the light in the small cavity. Interference between the light in both cavities varies the coupling in and out of the smaller cavity. (c) Optical microscope image of a similar device, representative of the devices measured in this paper.
The racetrack resonator is coupled at two points to the feedback waveguide (Fig. 1(a)). Each coupling point acts as a beam splitter. At the first coupling point, the light splits into two paths; one path consists of the bottom half of the racetrack resonator, while the other consists of the feedback waveguide. The electrodes are positioned both across the racetrack resonator and the feedback waveguide, with independent voltage control over the bias applied to each set of electrodes. In each path, the light accumulates a voltage-dependent phase shift,
Here, $i$ refers to either the first (racetrack) or second (waveguide) path, and $\beta _{V,i}(\omega ; V)$ is the voltage- and frequency-dependent optical propagation constant in each path for a given mode. At the second coupling point, the light in both paths recombines with a phase difference $\Delta \phi =\phi _2-\phi _1$. If the light between the two paths is exactly in-phase ($\Delta \phi =0$), the interference is constructive and coupling between the racetrack and the waveguide is enhanced. This appears as a broadening in the mode’s linewidth. If the light is perfectly out of phase ($\Delta \phi =\pm \pi )$, the interference is destructive, and the racetrack is completely decoupled from the waveguide. The voltage thus controls the resonator bandwidth. This coupling can be thought of as a Mach-Zehnder Interferometer (MZI) with an output port looped back to an input to form a resonator.
2.2 Measurement
We measure our device using the apparatus schematically depicted in Fig. 2(a). A laser is split into two arms. One passes through the optical device, whereas the other passes through an off-chip MZI for the purpose of calibrating the wavelength axis of the device spectrum (see Supplement 1). We vary the applied bias voltage across the device electrodes and record the transmission spectrum for each voltage in different bias configurations. Figure 2(b) depicts one such spectrum. In this spectrum, we observe transmission dips corresponding to resonant frequencies of the racetrack. The dip contrast changes with wavelength across the spectrum, corresponding to coupling differences of the modes. In this case, with fixed voltage bias, the change in coupling can be attributed to each mode having a different propagation constant $\beta$, due to changing the mode wavelength, and therefore a different phase difference accumulated in the feedback region of the device. Figure 2(c) presents the evolution of a single mode from this spectrum as the applied voltage is varied for a particular configuration. It can be seen that this mode undergoes a frequency shift, as well as bandwidth tuning from very under-coupled, through critical coupling, to very over-coupled.
Fig. 2. Measurement Apparatus and Optical Transmission. (a) Simplified schematic of our measurement setup. The input laser wavelength is swept across multiple resonances of the device. Two remotely controlled voltage sources are used to simultaneously apply static voltage bias to either the waveguide or racetrack electrodes. (b) An example of a normalized transmission spectrum of the device. In this case, a bias voltage of 22 V is applied to the racetrack electrodes, while a bias of −22 V is applied to the waveguide electrode. The red arrow indicates a particular mode at $\sim 1603.6$ nm, which we track in (c). (c) Overlaid normalized transmission curves for the mode indicated in (b), plotted for every two volts from 22 V to 42 V. The color indicates the magnitude of voltage. The same magnitude but opposite sign is applied to the waveguide electrode versus the racetrack electrodes.
In order to calibrate the performance of our device and to demonstrate independent bandwidth and frequency tunability, we repeat these transmission measurements while varying the voltages on the electrodes. First, we apply a voltage only across the feedback waveguide. For each voltage, we fit a Fano resonance to a single mode [22]. We define “bandwidth,” $\kappa (V)$, as the bandwidth parameter of the Fano lineshape, which corresponds to the 3 dB bandwidth of a symmetric Fano (i.e., full-width half-maximum of a Lorentzian). We plot in Fig. 3 the “detuning,” $\Delta (\omega )$, between the mode at the current voltage bias and its frequency under zero applied voltage, as well as the bandwidth. Both the bandwidth and detuning exhibit near-sinusoidal oscillations on the order of a few GHz, with a bandwidth extinction ratio (defined as the ratio of maximum bandwidth over minimum bandwidth) of $\sim 20$.
Fig. 3. Independent Electrode Bias Calibration. (a) Tuning behavior of the bandwidth (top) and mode detuning (bot) as a function of voltage applied to the waveguide electrode. The inset schematically depicts the device and where the voltage is applied. Blue datapoints represent data with errorbars, and the red line is a fit of our model to the data, weighted by the errorbars. (b) Tuning behavior of the bandwidth (top) and mode detuning (bot) as a function of voltage applied to the racetrack electrode. The inset schematically depicts the device and where the voltage is applied. For this particular device, however, we have an additional racetrack electrode positioned on the upper straight-length of the racetrack. The applied voltage is equivalent across both electrodes. Blue datapoints represent data with errorbars, and the red line is a fit of our model to the data, weighted by the errorbars. Note the large linear detuning component, indicated by the overlaid black dashed line.
Next, we apply a voltage only to the racetrack electrodes. We again observe near-sinusoidal oscillation in both bandwidth and detuning, but we also see large linear detuning proportional to the applied voltage (Fig. 3(b)(bot)). From this linear fit, we infer the DC electro-optic tuning rate, $g_{EO}/2\pi \approx 674$ MHz/V. We also note that the oscillation in bandwidth in this case arises predominantly from the shifting resonant wavelength in the device. That is, as the resonance frequency of a mode is shifted, its propagation constant varies, thereby altering the phase-difference the mode accumulates between the racetrack and feedback waveguide.
Table 1 presents the bandwidth extinction ratios and EO tuning strength observed for two different devices (the device measured in Figs. 2, 3, as well as that presented in Supplement 1). We observe that the maximum achievable bandwidth scales with the strength of the single-pass power coupling at each racetrack-waveguide coupling point (denoted as “$r$”). This coupling strength can be increased by making the fabricated racetrack-to-waveguide gap smaller.
Table 1. Bandwidth extinction ratios and EO tuning strength observed for two different devicesa
2.3 Fitting to a model
In order to fit the tuning curves in Fig. 3, we use scattering matrix theory to derive a full model of the device transmission (see Supplement 1) [23–26]. This transmission is given by:
In Eq. (2), $r_i$ ($t_i$) is the amplitude cross-coupling (transmission) coefficient at each coupling point, $i$. In the case of lossless coupling, $(r_i^2+t_i^2)=1$. The phase $\phi _r(\omega ; V_r)$ corresponds to the phase accumulated in a single half of the racetrack resonator (i.e., the phase accumulated between the coupling points in the racetrack), and $\phi _{w}(\omega ; V_w)$ is the phase accumulated in the feedback waveguide between the coupling points. Each accumulated phase is in general complex and is given by Eq. (1), where $\beta (\omega ; V)$, the complex propagation constant, accounts for both the resonant frequency as well as propagation loss.
The poles of this transfer function correspond to the complex resonances of the system and therefore contain information about both the frequency and bandwidth of each resonance. We solve for the poles by Taylor expanding the denominator close to each resonant frequency, around a frequency $\omega _0$, and set it equal to zero. We make the assumption that higher-order phase derivatives are negligible ($\delta ^n\phi _i/\delta \omega ^n \ll \left (\delta \phi _i/\delta \omega \right )^n$). This loosely follows from the fact that the inverse group velocity has a much greater magnitude than the group velocity dispersion, which leads to the condition that $(\delta ^2\beta /\delta \omega ^2) \ll \left (\delta \beta /\delta \omega \right )^2L$, where $L$ is the length of the waveguide (see section 3C of Supplement 1). We also assume that the voltage-dependent portion of the phase is not frequency-dependent close to $\omega _0$. In this regime, we arrive at the following form for the expansion (see Supplement 1). We omit the argument labels of the phases ($\phi _i(\omega _0;V_i)\to \phi _i$) for clarity:
In the above, we have factored propagation losses, $\gamma _i$ (the complex component of each phase $\phi _i = \phi _i(\omega _0)$) and the amplitude coupling coefficients, $t_i$, $r_i$ into the coefficients $A=r_1r_2\gamma _{w}\gamma _{r}$ and $B = t_1t_2\gamma _{r}^2$. We also define the length of half the racetrack to be $L_r$ and the length of the interferometric waveguide between coupling points to be $L_w$. $v_g$ is the group velocity of the mode, which is taken here to be equivalent in each part of the device. The complex detuning from the expansion frequency, $\omega _0$, is given by $\zeta (\omega ) = \omega -\omega _0$.
We then solve this expression to first order to find the complex pole frequency $\omega _p$, such that the complex detuning, $\zeta (\omega _p)$, from a frequency close to the zero-voltage resonance $\omega _0$, results in $f(\omega _p) = 0$:
In this way, we obtain the complex pole $\omega _p$ with real and imaginary parts representing the detuning from $\omega _0$ and half of the linewidth $\kappa$. The full model, its fit parameters, and additional details on the fitting procedure are presented in Supplement 1. The minimum of the bandwidth fit corresponds to the completely decoupled resonant linewidth ($\kappa _i$), whereas the maximum corresponds to the maximally coupled linewidth.
2.4 Arbitrary bandwidth and frequency control
In order to achieve arbitrary control over bandwidth and frequency of the mode, we can apply bias simultaneously to both the waveguide and racetrack electrodes. We predict the behavior of the device by feeding the fit parameters from Fig. 3 into our model, Eqs. (4), S15-17, and simulating the expected bandwidth and detuning of the mode as functions of the applied racetrack and waveguide voltages. The full list of parameters for the model are presented in Supplement 1. The simulated results are presented in Fig. 4. As expected, we observe a periodic modulation of both the bandwidth and detuning with voltage. For fixed waveguide voltage but varied racetrack voltage (i.e., vertical slices in Fig. 4(a)), we observe a strong linear detuning in the resonant frequency and a modulation in bandwidth. For a fixed racetrack voltage but varied waveguide voltage (i.e., a horizontal slice), we observe complete modulation between maximum and minimum coupling, as well as a modulation in detuning. Therefore, by varying the applied voltages, we can access any bandwidth and frequency within this simulated 2D space. Furthermore, we can compensate for any undesired detuning modulation arising from the waveguide bias with linear frequency shift controlled by the racetrack bias.
Fig. 4. Arbitrary bandwidth and frequency control. (a) Simulations of the expected bandwidth (left, $\kappa$) and detuning (right, $\Delta$) of the $\lambda \sim 1603.6$ nm resonance as a function of applied voltage across the racetrack (y-axis) and waveguide (x-axis). The simulation consists of the predictions from our closed form model, using the results of the fits in Fig. 3. The dashed black line indicates the voltages over which we measure actual data in (b). (b) Comparison of measured data (blue datapoints) and prediction from our model (red line) for a “dual bias” configuration. The stars indicate predictions for the same voltage combinations as those used for the measurement (blue datapoints). We plot the fit bandwidth against detuning of the mode for each applied voltage. In this case, equal magnitude voltage is applied to both the racetrack electrodes and the waveguide electrode, but with opposite polarity. Error bars indicate a 10% change in the fit error of a Fano fit applied to the resonance from which we extract bandwidth and resonant frequency. The detuning is measured relative to the zero-voltage resonant frequency. In many cases, the error bars are so small they are blocked by the plotted datapoint.
We verify this simulated result experimentally by applying voltages corresponding to the dashed diagonal lines in Fig. 4(a). We apply voltages of equal magnitude but opposite polarity to the waveguide and racetrack electrodes. Sweeping this voltage from 0 V to ~160 V, we take repeated transmission measurements and plot the bandwidth versus detuning of the $1603.6$ nm mode for each voltage (Fig. 4(b)). We observe a close match between our measured device behavior and the simulated predictions from Fig. 4(a).
3. Discussion
3.1 Experimental considerations
Overall, we have demonstrated independently tunable control over both the resonant frequencies and bandwidths of modes of an integrated optical resonator. The achievable frequencies are limited only by how much bias voltage can be applied to the racetrack. The required voltage for a desired phase shift can be reduced by making the straight lengths of the racetrack (and therefore the electrodes) much longer, and also by making the FSR of the racetrack shorter (so that one could operate on a different mode to achieve a frequency change). The maximum bandwidth achievable in this device is limited by the amount of power exchange between the racetrack and the waveguide at each coupling point. We could increase coupling in the future by either reducing the waveguide-to-racetrack coupling gap or moving to longer coupling regions, as in directional or pulley couplers.
Furthermore, when fabricating the device, the physical symmetry of the two coupling points is important for ensuring the maximum bandwidth extinction ratio. If one coupler allows for more power transfer than the other, this will appear as a loss channel on the racetrack resonator, increasing the minimum resolvable bandwidth in the system (see Supplement 1). This is similar to the behavior in an MZI in which asymmetric couplers would cause incomplete extinction of the light in each arm.
In the device exhibited in this work, our wavelength calibration is also tricky. The MZI we used at the time of measurement has an FSR of approximately $7.65$ GHz, which is much larger than our minimum resonance bandwidth. By calibrating to the peaks, valleys, and zero-crossings of the MZI, we are able to obtain wavelength references at every $\sim 1.9$ GHz, and we interpolate the wavelengths in between. For our supplemental device 2, we used an MZI with a much narrower FSR ($\sim 325$ MHz) and a much more accurate calibration scheme. Both calibration algorithms work best with a smooth MZI transmisison. Our MZI transmission data features a regular fast ripple, likely coming from some equipment in the laser path. We smooth the MZI transmission data prior to calibration to eliminate the ripple. Furthermore, the device 1 transmission is measured through a preamplifier with an active low-pass filter (this filter is turned off when measuring device 2 in Supplement 1). However, we note that this filtering only serves to increase our minimum-resolvable bandwidth, thereby placing a lower-bound on the bandwidth extinction ratio we measure.
3.2 Data considerations
In Fig. 3, we observe near-sinusoidal tuning behavior in bandwidth and detuning. Previous demonstrations of feedback-coupled waveguides depicted nearly exact sinusoidal tuning behavior [14,15]. We attribute the asymmetry we observe in the oscillation to the large length mismatch between arms of the feedback coupling region. In order to reduce the voltage necessary to achieve a complete phase shift in the waveguide, we choose to make the waveguide very long. In future device iterations, we would make the racetrack straight length similarly long, thereby mitigating the issues of the length mismatch. We investigate this imbalance more thoroughly via simulation in Supplement 1.
We also notice asymmetry in the voltage required for a complete period of bandwidth tuning in Fig. 3(a). That is, the $V_{\pi }$ of our device is greater at low voltages. We attribute this difference to charge carriers in TFLN. Along these lines, we expect the tuning period to shift with optical power, which would alter the rate of free carrier generation. We leave further study of these effects to studies focused on modulators.
Lastly, we note that our modeling requires a first-order approximation for the complex poles in the device to arrive at semi-analytic expressions for the linewidth and detuning. Therefore, it is only valid in a narrow region around each resonant frequency, and outside of this region, the model may predict non-physical behavior. An example of this is shown in Fig. 4(b). For particular applied voltages (e.g., for detuning values around $-40$ GHz and beyond $\sim 100$ GHz), the predicted bandwidth of the mode can be negative. We believe this stems from the local nature of our Taylor expansion. We improve our model’s accuracy by including parameters that capture the predominantly linear frequency shift that results from applying racetrack voltage (see Supplement 1). We account for this frequency shift in the propagation phase at each applied voltage. However, any error in our wavelength calibration or non-linearity in the local frequency shift would lead to inaccuracy in our fit and model. These ideas warrant further theoretical and experimental investigation.
3.3 Conclusion
We have presented an approach to tune the transmission properties of integrated photonic resonators. By leveraging the fast electro-optic effect in TFLN, our approach will also enable optical pulse shaping and photon catch and release experiments important for emerging quantum networks. By developing a closed-form model to calibrate and predict tuning behavior of this device, we pave the way for these experiments and others that require precise models and repeatable operation.
Funding
National Science Foundation (GRFP (DGE-1656518)); National Nanotechnology Coordinating Office (ECCS-2026822); Stanford University (Q-FARM, Bloch Fellowship Program); Natural Sciences and Engineering Research Council of Canada; Air Force Office of Scientific Research (MURI (FA9550-17-1-0002)); Defense Advanced Research Projects Agency (LUMOS (HR0011-20-2-0046), Young Faculty Award (D19AP00040)); U.S. Department of Energy (DE-AC02-76SF00515, Q-NEXT NQI Center); NTT Research.
Acknowledgments
We thank NTT Research for their financial and technical support. We thank the United States government for their support through the Department of Energy Grant No. DE-AC02-76SF00515, the Defense Advanced Research Projects Agency (DARPA) LUMOS program (Grant No. HR0011-20-2-0046), the DARPA Young Faculty Award (YFA, Grant No. D19AP00040), the U.S. Department of Energy (Grant No. DE-AC02-76SF00515) and Q-NEXT NQI Center, and the U.S. Air Force Office of Scientific Research MURI grant (Grant No. FA9550-17-1-0002). JFH and DJD would like to acknowledge support from the NSF GRFP (No. DGE-1656518). VA acknowledges support from the Stanford Q-FARM Bloch Fellowship Program. KM acknowledges support from the Natural Sciences and Engineering Research Council of Canada (NSERC). Part of this work was performed at the Stanford Nano Shared Facilities (SNSF), supported by the National Science Foundation under award ECCS-2026822. Work was performed in part in the nano@Stanford labs, which are supported by the National Science Foundation as part of the National Nanotechnology Coordinated Infrastructure under award ECCS-2026822.
Disclosures
The authors declare no conflicts of interest.
Data availability
All data generated in this study are available from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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