Optothermal transport behavior in whispering gallery mode optical cavities

Since the first demonstration of liquid microdroplet cavities, whispering gallery mode resonators have emerged as a unique platform for studying fundamental physics phenomena and for developing integrated photonics technologies.¹ One reason for their ubiquitous appeal is their high quality factors (Q), which result in large circulating intensities and extremely narrow linewidths.² While the high circulating power enables the development of many innovative devices, it also gives rise to several non-linear optical effects^3,4 which can de-stabilize the resonant wavelength of the device. For example, optothermal or photothermal behavior, in which coupled optical energy is converted to heat, is commonly observed in high Q devices.^5,6 This dynamic conversion process causes transmission peak broadening and destabilizes the resonant wavelength, degrading the overall device performance. Therefore, a rigorous understanding of the physics which governs the optothermal behavior of optical cavities is critical.

To develop a model for the optothermal behavior in cavities, it is necessary to consider the optical and thermal components, which give rise to this non-linear phenomenon serially, as they are distinct yet coupled problems. Both aspects have temporal elements (photon decay time and thermal transport time) and will be dependent on the device geometry and its material properties. However, previous research has primarily focused on the optical physics element, simplifying the heat transport components.⁷ Specifically, the role of the device geometry in heat transport, which determines the thermal time constant of the system, was overlooked as the majority of experimental work was focused on microsphere resonant cavities.⁸ Therefore, this complex and dynamic process has not been fully explored.

In the present work, we develop a comprehensive, generalizable optothermal theory for whispering gallery mode microcavities, which includes both the optical and thermal components. Our approach relies on a combination of analytical theory and serially coupled optical and thermal finite element method (FEM) simulations. By using FEM, we are able to accurately incorporate geometry and material-specific parameters, capturing their role in the optothermal behavior of these devices. To verify this theoretical construct, the temperature change induced by circulating power is simulated and experimentally verified for the specific case of the toroidal resonant cavity (Fig. 1(a)). This device provides a rigorous verification of our FEM approach, because it includes multiple materials (silica and silicon) as well as a complex geometry with multiple thermal cooling pathways.

Starting with classic resonator physics theory, the resonant wavelength is dependent on the refractive index and the device diameter. Because all materials experience temperature-dependent refractive index and volume changes, the thermally induced resonant wavelength change (Δλ) can be described by⁹ 

where n_eff is the effective refractive index, dn/dT is the effective thermo-optic coefficient, ΔT is the temperature change, λ is the resonant wavelength, and ε is the expansion of the material. From expression (1), it is clear that the index and the device radius can be tuned thermally, inducing a resonant wavelength shift. However, in most resonant cavities, the dn/dT is substantially larger than ε; therefore, the first term dominates the optothermal effect. As such, all subsequent analysis will assume that the second term (λεΔT) is zero.

While any source of heat can change the effective refractive index of the mode, one particularly interesting case is when the optical mode behaves as the heat source. This unique scenario occurs in devices with particularly long photon lifetimes or high quality factors (Q) where the optical loss of the material is converted to heat.¹⁰ In order to predict the optothermal shift, it is first necessary to calculate the circulating power (P_circ) as well as the dissipated power (P_d) in the cavity. Based on these values and the cavity geometry, the temperature change can be modeled in COMSOL Multiphysics. The first two calculations only assume a high Q whispering gallery mode cavity and are therefore generalizable to any whispering gallery mode cavity. However, the optical field distribution and thermal transport simulations are specific to the cavity geometry, material system, and wavelength. Assuming steady state operation, the circulating power inside the resonator is¹¹ 

where Q_o is the intrinsic quality factor, n_eff is the effective refractive index, R_meff is the effective mode radius, and K is the ratio of the cavity loss due to coupling to the intrinsic cavity loss. This ratio is a measure of the power which is coupled into the cavity with respect to the loss of the cavity. Therefore, by measuring the total transmission (T), we can calculate K from T = [(1 − K)/(1 + K)]² and use K to calculate the circulating power.

The dissipated or lost power (P_d) in the cavity is described by the loss coefficient (σ_o), the circulating energy (|a|²), and the round-trip time in the cavity (t_r). Specifically

It is important to emphasize that the term |a|² here is the energy circulating in the cavity and is related to the power through the round-trip circulation time. The loss coefficient (σ_o) is not specific to an individual loss mechanism, but captures all of the loss mechanisms present in the system such as material loss, surface scattering, and radiation loss.² However, only the material loss behaves as a heat source. Therefore, to isolate this loss mechanism, the previous generalized loss equation is written in terms of the absorptive loss time constant (τ_abs), creating an expression for the power lost to material absorption (P_abs)

However, the conversion of optical energy to thermal energy is not completely efficient. To capture the thermal energy conversion inefficiency, an effective power (P_eff) is defined as P_eff = γP_d, where γ is an experimentally determined scaling constant. Specifically, after calculating the theoretical lost power, it is imported into COMSOL Multiphysics, and the thermal shift is determined. By comparing the simulated thermal shift and experimental shift, the effective power and γ are calculated. The effective power describes the amount of power in the optical mode confined in the device which is able to change the temperature of the device. Therefore, for each point, P_eff is calculated from P_d and imported into COMSOL Multiphysics.

However, P_eff is not a single or constant value. Because the optical mode profile depends on the operating wavelength and the device geometry, the thermal profile or heat source used in the simulations should mirror this dependence. Therefore, in the present work, the optical mode profiles are simulated for a series of device geometries and subsequently imported as the heat source profiles for the heat transport modeling. The transient nature of the heating is incorporated into the model by applying the heat source as a single pulse of a finite amplitude and length. The duration of the pulse is sufficiently long to ensure that the device reaches a thermal steady state, and the intensity is the input lost power. As such, the model is able to accurately capture all of the relevant physical parameters of the optothermal coupling. By combining the simulation results and Eq. (1), the wavelength shift for any coupled power can be determined.

Using this systematic approach, we are able to explore the effect of the geometrical parameters on the optothermal threshold and the thermal time constant of the structure. These two parameters provide insight into two very different aspects of this system. The threshold is critical in determining the time averaged maximum power which can be used before this effect first appears. In contrast, the thermal constant, or thermal response of the system, represents the time that the structure requires to return to the initial temperature after a single pulse of thermal energy. Assuming that light is applied in a pulse train, the thermal time constant sets the upper limit on how fast the pulses can be applied before the device reaches thermal runaway. Therefore, these two values govern the maximum pulse frequency and intensity that a device can support, which is a critical parameter in applications like switching and filtering.

To study the role of geometry on these parameters, the specific case of the toroidal cavity is studied. Specifically, the oxide thickness and overhang lengths are varied from 1.0 to 2.0 μm and 5.0 to 12.0 μm, respectively (Fig. 1(b)). In both sets of simulations, the device major (R) and minor radii are held constant at 60 μm and 7 μm.

Figs. 2(a) and 2(b) show representative results from a matched pair of optical and thermal FEM simulations, and Figs. 2(c) and 2(d) show the compiled results for all of the simulations. From the modeling results, it is evident that by increasing the overhanging length or decreasing the oxide thickness, the thermally induced shift increases. Additionally, there is a clear dependence on the wavelength. This change is the result of the decrease in optical mode volume which forms a more localized and intense heat source.

To analyze the thermal time constant (τ_th), the thermal decay from a single heat pulse to the toroid is modeled.¹² As can be observed in Figs. 2(e) and 2(f), by increasing either the overhang length or the membrane thickness, the thermal time constant of the device can be tuned. By combining the modeling results, it is clear that the optothermal threshold is proportional to the oxide thickness and the overhang length.

To experimentally explore the dependence of the optothermal shift on the oxide thickness, a series of silica toroidal cavity devices are fabricated using the classic combination of photolithography, thermal oxide and silicon etching, and carbon dioxide laser reflow.¹² To control the oxide thickness, a dual-oxide etch is performed for a sub-set of the devices.¹² 

One method for quantifying the optothermal behavior is to measure the threshold for the onset of the optothermal response. This characterization experiment is performed by coupling light from a tunable, narrow-linewidth laser into the cavity using a tapered optical fiber waveguide and measuring the resonant wavelength. At the onset of the optothermal response, the resonant wavelength will shift. Tapered fiber waveguides are ideal for this measurement as they allow precise control over the amount of optical power, which is coupled into the device as well as providing nearly perfect phase matching.¹³ The waveguides are aligned using side- and top-view machine vision systems, and the output power from the fiber is monitored and recorded on a LabView-controlled high speed digitizer/oscilloscope card. Resonance spectra are recorded over a range of coupling conditions, and the effect of the thermal build-up on the resonant wavelength is determined.¹² 

The optothermal threshold is quantified in terms of the power coupled into the cavity, which is determined from the transmission spectra. Specifically, the optothermal threshold is the % coupled power = T_on-resonance/T_{off-resonance}, where T_on-resonance is the on-resonance signal level and T_{off-resonance} is the off-resonance signal level. The threshold coupling ratio or the point at which the device first exhibits optothermal behavior is determined.¹² In addition, the ratio of the net heating power to the lost power is defined as the absorbed power ratio. This ratio accounts for the non-absorbed power as well as the conduction into the structure and the fact that the temperature distribution is approximated by the effective temperature at the hottest part of the mode.¹² 

In the first series of experiments, the dependence of the optothermal threshold on the wavelength is characterized by measuring the optothermal behavior at 765 nm, 1330 nm, and 1550 nm. Fig. 3(a) is a representative spectrum at 1550 nm showing a resonance which is thermally broadened. Fig. 3(b) contains the threshold data for all three wavelengths, and the results are also summarized in Table I. To enable direct comparison among the three devices across wavelengths and discussion of the optothermal threshold, the wavelength shift is normalized by input power and Q. Two fits are shown for each data set in Fig. 3(b): (1) a linear fit to the experimental data above the threshold (dashed line) and (2) the FEM modeling results (dotted line). These results provide insight into several aspects of this system.

First, there is a clear wavelength dependence, with the threshold increasing as the wavelength decreases. This dependence indicates that the τ is increasing as the wavelength increases. This behavior mostly likely arises from the interaction of the optical field with the monolayer of water on the surface of the cavity.^2,6,14 Because the optical absorption coefficient of water increases from 765 nm to 1550 nm, the threshold is reduced. Second, the relative optothermal efficiencies can be determined from Fig. 3(b). Specifically, the ability of the optical mode to induce linewidth broadening is improved at higher wavelengths. Therefore, a higher Q device operating at lower wavelength could have similar broadening characteristics to a lower Q device operating at a higher wavelength.

In the second series of experiments, the dependence of the optothermal threshold on the oxide membrane thickness is measured at 1550 nm. However, each device had a unique Q, mode volume, and overhang length. Therefore, to accurately compare the thresholds of the five different devices, these parameters are normalized from the optothermal threshold coupling ratio, creating the scaled optothermal threshold (Fig. 4). As can be observed, there is a clear linear dependence on the oxide thickness, which is in agreement with the modeling (Fig. 2(c)). Additionally, these results indirectly verify the predicted dependence of τ on the oxide thickness (Fig. 2(e)).

In conclusion, we have developed a coupled model for the optothermal behavior of whispering gallery mode optical cavities and verified it experimentally using ultra-high-Q toroidal resonant cavities. By including the temporal and spatial distribution of the optical field in the thermal model, we are able to predict the optothermal behavior and distortion of the resonant linewidth. The toroidal cavity is a particularly interesting case as it includes multiple materials with different heat transfer coefficients and a complex geometry with multiple interfaces for heat transfer. The experiments demonstrated that the theoretical model is able to accurately capture the dependence of both the threshold and the thermal time constant on the device geometry. The ability to precisely tune the optothermal broadening could be used to design optothermally controlled switches and lasers,¹⁵ laser locking,¹⁶ and bio-sensors.¹⁷ 

The authors acknowledge the Office of Naval Research (N00014-11-1-0910) for financial support.