## Abstract

We report the experimental observation and theoretical analysis of thermo-optic effects in high-Q on-chip lithium-niobate (LN) microdisks. We find that the resonance transmission dip of a LN microdisk was broadened or compressed when the wavelength of the input laser was tuned to the shorter or the longer wavelengths at a wavelength sweeping speed of 4.8 pm/s. When the laser wavelength was shifted with a fast rate (4.8 nm/s), the tendencies of the change in the shape of the transmission dip reverse completely. An oscillatory behavior in the transmission spectra was also observed when the laser wavelength was slowly shifted to the shorter wavelengths. The experimental results were successfully explained by using a theoretical mode considering for a fast thermo-optic effect of LN crystal and a slow heat dissipation process from the LN microdisk to the substrate and surroundings that leads to the reduction of the resonance wavelength through the deformation of the LN microdisk.

© 2016 Optical Society of America

## 1. Introduction

Whispering gallery mode (WGM) micro-resonators with high quality (*Q*) factor can confine light in a small mode volume (*V*) for a long period of time by continuous total internal reflection off the cavity surface. Benefiting from the high ratio of *Q*/*V*, light energy density within the resonator can be dramatically enhanced, e.g., light intensity higher than 1 GW/cm^{2} can be produced in a WGM resonator with a diameter of 50 μm and a quality factor of 1 × 10^{8} by a pump with 1 mW power [1]. WGM micro-resonators with the ability to boost light-matter interaction provide a versatile platform for a variety of fundamental studies and practical applications such as opto-mechanics [2, 3], low-threshold nonlinear optics [4–7], ultrahigh-sensitive sensing [8–11], frequency comb [12–14] and so on.

Significant thermo-optic effect often arises in WGM micro-resonators due to the variation of temperature of the resonator and its surroundings owning to the absorption of the intense light within the resonator and the heat conductivity [15]. For special thermo-optic process that has particular effective thermo-optic coefficient and response time, different transmission spectra, such as linewidth broadening [16], linewidth compression [16] and thermo-optic oscillation [17–23], can be obtained by controlling the wavelength sweeping speed, sweeping direction and the power of the tunable laser [24]. Thermo-optic effects of micro-resonators have been used widely as tools to control transmission spectra or resonance wavelengths in various kinds of experiments. Thermal induced change in effective refractive index of a mode has been used as a resonance frequency tuning method in microcavities [25]. Thermal linewidth broadening has been used for mode locking [26] or dynamic control of optical path length [27]. Highly sensitive thermal sensors were realized by compensating the thermal effect of silica by coating a layer of PDMS [28].

Thermal induced oscillations have been reported in varieties of micro-resonators, such as silica microspheres [21], silicon microdisk [22], silica microspheres coated with PDMS [20] and silicon nitride microdisk [19], with two nonlinear mechanisms with effective thermo-optic effects of opposite signs playing roles in the resonators [29]. It was also reported in millimeter sized WGM resonators that made of crystals, such as BaF_{2} [18] and CaF_{2} [23]. Recently on-chip micro-resonators made from lithium niobate (LN) [30–34] have attracted much attention for combining the features of WGM resonators (high quality factor, broad resonance window, tunable coupling) and those of LN crystal itself (high second nonlinearity, electro-optic and acousto-optic tunability). It has the potential to be used for tunable optical filters, optical modulators/switches, and nonlinear optical wavelength convertors. Thermo-optic broadening and oscillation in transmission spectra of on-chip LN micro-resonators was reported very recently [35], however, there are no detailed studies published until now.

In this paper, we report the systematic study of thermo-optic effects in on-chip LN microdisk resonators, especially the self-modulated transmission at high input power and low sweeping speed of laser wavelength. To the best of our knowledge, we first demonstrate the completely different thermo-optic responses of LN microdisk resonators in the fast and slow wavelength sweeping situations. A theoretical model, that takes a fast thermo-optic effect of LN crystal and a slow heat transportation process from the LN microdisk to the substrate and environment into account, was established to simulate the transmission spectra of the LN microdisk resonators. The theoretical results are consistent with the experimental results.

## 2. Samples and experimental setup

The microdisk resonators were fabricated on a LNOI (lithium niobate on insulator) wafer by using UV-lithography, reactive ion etching, and hydrogen fluoride etching [33]. Figures 1(a) and 1(b) are the optical microscope and scanning electron microscope (SEM) photographs showing the top view of a fabricated microdisk resonator. The radius of the cavity is about 40 μm and its pillar formed during the hydrogen fluoride etching process is about 20 μm in radius. Figure 1(c) is the enlarged view of the smooth edge of the microdisk shown in Fig. 1(b). The smooth surface of the cavity helps us to suppress the scattering loss and thus improve the quality factor of the resonator. The quality factor of the fabricated microdisk resonator used in our experiment was measured to be 3.1 × 10^{5}, which is quite high for an on-chip LN micro-resonator.

The schematic diagram of the experimental setup to study thermo-optic effects of LN microdisk resonators is shown in Fig. 2. Light from a tunable laser in 1550 nm band, whose wavelength is periodically scanned towards the longer and the shorter wavelengths via a sawtooth voltage generated by an arbitrary waveform generator, is split into two branches by using a 90/10 coupler. The 10% branch is directly connected to a power meter to monitor the input power of the microdisk resonator. The 90% branch is evanescently coupled to the microdisk resonator via a tapered fiber thermally pulled from a single mode fiber. The resonator is placed on an XYZ piezo-stage to accurately control the distance and thus the coupling efficiency between the microdisk and the tapered fiber under the scrutiny of two microscopes with long working distance objective (not shown here). The transmission of the tapered fiber is monitored by an oscilloscope connected with a photodiode that is employed to measure the transmission intensity of the tapered fiber. When the wavelength of the input tunable laser is scanned at a constant rate, the transmission spectra of the LN microdisk is shown directly on the oscilloscope.

## 3. Experimental results and discussions

By using the aforementioned experimental setup, we measured a series of transmission spectra for the LN microdisk resonator at various pump power and wavelength scanning rates. Figure 3 shows transmission spectra obtained when the sweeping speed of the wavelength of the tunable laser was 4.8 pm/s. The transmission spectra achieved at different pump power are presented in different colors. When the pump power is low, such as 50.9 μW, the resonance transmission dip has an approximate ideal Lorentzian shape; when the pump power is high, such as 541.3 μW, the shape of the measured transmission dip is significantly deformed, the transmission dip is broadened (compressed) in linewidth when the wavelength of the tunable laser shifts toward the shorter (longer) wavelengths as seen from the blue curves in Figs. 3(a) and 3(c), respectively. The tendency of the change in the shape of the transmission dip tells us that a nonlinear mechanism with a negative thermo-optic coefficient, which make the resonance wavelength of the LN microdisk resonator shift toward the shorter wavelengths when more light is coupled into and absorbed by the LN microdisk resonator, determines the thermo-optic effect of the LN microdisk when the wavelength sweeping rate is as low as 4.8 pm/s.

Comparing transmission spectra shown in different colors in either Fig. 3(a) or Fig. 3(c), one can see that the transmission dip was broadened or compressed more significantly when the pump power was increased. Significant thermo-optic oscillation was observed when the pump power was further improved to the order of 1 mW while keeping the wavelength sweeping speed 4.8 pm/s unchanged. Figure 4(a) shows the typical results obtained when the input power was 2.4 mW. As seen from Fig. 4(a), in some wavelength range, the transmission of the LN microdisk goes up and down periodically, the whole oscillation lasts for about 2.5 s. The oscillation cycles change from sparse to dense and then to sparse again as the input wavelength decreases. The duration of a single oscillation ranges from 20 ms to 80 ms as seen from Figs. 4(c), 4(e) and 4(g), which are, respectively, the magnifications of the green-, blue-, and red-square highlighted parts in Fig. 4(a). The duty cycle, that is defined as the ratio of the time duration with high transmission to the oscillation period, decreases with the reduction of the laser wavelength.

It is well known that the oscillation in the transmission spectrum of a WGM resonator is usually a consequence of the competition between nonlinear mechanisms with effective thermo-optic coefficients of opposite signs [29]. So there should be another nonlinear mechanism with a positive effective thermo-optic coefficient that makes the resonance wavelength shift to the longer wavelengths when more light was coupled into the resonator. To find the evidence for the existence of the nonlinear mechanism with a positive effective thermo-optic coefficient, we measured the transmission spectra while tuning the laser wavelength at various sweeping rates. We noticed that, when the wavelength sweeping rate is much higher than 1 pm/s, the variation tendency of the transmission spectra reversed. As an example, the results obtained with 4.8 nm/s wavelength sweeping rate and 1.66 mW pump power are shown in Fig. 5, where the broadening (compression) of the transmission dip was achieved as the laser wavelength was changed toward the longer (shorter) wavelengths, which is entirely different compared with the results shown in Fig. 3. It is worth mentioning that no oscillation was observed when the wavelength sweeping speed of the laser is 4.8 nm/s even tens of mW pump was employed.

## 4. Theoretical analysis

We attribute the change of transmission dip for the fast and slow scan situations to the competition between a fast thermo-optic effect and a slow heat dissipation process with opposite efficient thermo-optic coefficients. The fast thermo-optic effect with a positive effective thermo-optic coefficient is simply associated with the thermo-optic effect of LN crystal itself; the slow heat dissipation process with a negative thermal optical response functions through the deformation of the wafer due to the heat dissipation from the resonator to the substrate and surroundings [19]. The deformation of the wafer causes the reduction of the size of the LN microdisk and thus induces a blue shift of the resonance wavelength.

Accordingly, a theoretical model was proposed to explain the measured experimental results. The resonance wavelength of the resonator *λ*_{T}, which are impacted by the temperatures of the LN microdisk, the substrate and the environment, are given by [19]

*λ*

_{0}is resonance wavelength of the LN microdisk measured under very low pump power in under-coupling regime, that is called cold cavity resonance wavelength;

*T*

_{1},

*T*

_{2}and

*T*

_{0}are the temperature of the disk, the substrate and the surroundings, respectively;

*n*

_{eff}is the effective refractive index of the interested WGM. Considering for the actual distribution of WGMs in LN microdisk resonators and the anisotropy of LN crystal, the effective refractive index of the WGM can be expressed as

*n*

_{eff}=

*η*

_{e}

*n*

_{e}+

*η*

_{o}

*n*

_{o}, where

*n*and

_{o}*n*is the refractive index of ordinary and extraordinary light, while

_{e}*η*

_{e}and

*η*

_{o}= 1-

*η*

_{e}represent the energy fractions of the extraordinary and ordinary light, respectively. Similarly, the effective thermo-optic coefficient

*dn*

_{eff}/

*dT*

_{1}=

*η*

_{e}

*dn*

_{e}/

*dT*

_{1}+

*η*

_{o}

*dn*

_{o}/

*dT*

_{1}with

*dn*

_{e}/

*dT*

_{1}and

*dn*

_{o}/

*dT*

_{1}being the thermo-optic coefficients related to the extraordinary and ordinary components of the interested WGM, respectively.

The values of *η*_{e} and *η*_{o} could be theoretically calculated by performing a finite-element-method (FEM) simulation, which is a full-vectorial model and is able to give the distribution of each electric field component of a WGM. The calculated results of *η*_{e} for the first to the fourth order quasi-TM (quasi-TE) modes from the FEM simulation are 0.992, 0.975, 0.969, and 0.965 (0.022, 0.042, 0.058, and 0.072), respectively; the corresponding ∂*λ*_{0}/∂*T*_{1} = (*dn*_{eff}/*dT*_{1}) (*λ*_{0}/*n*_{eff}) are 23.0, 22.6, 22.5, and 22.4 pm/K (0.5, 1.0, 1.3, and 1.7 pm/K), respectively. As an example, the distributions in the cross section of the radial, azimuthal and axial components of the electric field for the first order quasi-TM mode in a LN microdisk are shown in Figs. 6(a)-6(c), respectively. By comparing these figures, it is evidential that the radial electric field is much stronger than the corresponding azimuthal and axial components for which reason the mode is called quasi-TM mode.

According to the experimental results shown in Fig. 5(a) measured when the wavelength sweeping speed was 4.8 nm/s, the amount of resonance wavelength shift is about 0.02 nm. The temperature of the LN microdisk need to increase by a few Kelvin correspondingly for quasi-TM and tens of Kelvin for quasi-TE modes, respectively, to fulfill such a wavelength shift. Referring to literatures [19, 20], the reasonable increment of temperature in an on-chip microdisk should be less than a few Kelvin in general. Quasi-TM modes agree the actual situation better. In experiments, it is difficult to determine the exact mode number of the WGM that we are interested. For simplicity, we choose first order quasi-TM mode as our interested WGM to analyze the experimental results. This is a very reasonable assumption because the value of the only effective parameter ∂*λ*_{0}/∂*T*_{1} in our theory changes less than 3% between the quasi-TM modes of different orders.

Due to the conservation of energy, the evolution of the temperatures of the LN microdisk *T*_{1} and the substrate *T*_{2} with respect to time *t* satisfy the following equations [16, 19]:

*P*

_{c}= |

*E*

_{c}|

^{2}/

*τ*

_{r}is intracavity power with

*τ*

_{r}= 2

*πRn*

_{eff}/

*c*being the round trip time of light in the resonator,

*R*the radius of the LN microdisk and

*c*the speed of light in vacuum, respectively.

*Γ*

_{abs}=

*Q*/

*Q*

_{abs}represents the thermal absorption coefficient, which is determined by the material absorption, with

*Q*and

*Q*

_{abs}indicating the total and the absorption-determined quality factor, respectively.

*Cp*

_{1}=

*c*

_{LN}

*ρV*is the heat capacity of the disk, where

*c*

_{LN}is the heat capacity of LN crystal,

*ρ*is the density of LN crystal and

*V*=

*πR*is the volume of the disk and

^{2}h*h*is the thickness of the microdisk resonator. The second term on the right side of Eq. (2) corresponds to the heat dissipation from the cavity to the substrate.

*K*

_{1}is the thermal conductivity between the cavity and the substrate. Equation (3) states that the temperature of the substrate is influenced by the heat transportation from the microdisk to the substrate (1st term on the right hand) and the heat dissipation to the environment (2nd term on the right hand).

*K*

_{2}and

*Cp*

_{2}are the heat capacity of the whole structure and the thermal conductivity between the whole structure and the surrounding environment.

*K*

_{1},

*K*

_{2}, and

*Cp*

_{2}were set as the adjustable parameters to fit the experimental results.

According to the coupled mode theory [20, 36], the dynamics of the electric field E_{c} inside the resonator obeys

*ω*-

*ω*

_{T}is the frequency detuning between the angular frequency

*ω*of the tunable laser and the resonance frequency

*ω*

_{T}of the resonator, which is impacted by the temperatures of the LN microdisk and the substrate.

*κ*

_{0}=

*ω*

_{T}/

*Q*

_{0}is the intrinsic loss rate with

*Q*

_{0}being the intrinsic quality factor.

*κ*

_{e}=

*ω*

_{T}/

*Q*

_{e}is the loss rate due to the coupling of light from the WGM resonator to the tapered fiber with

*Q*

_{e}representing the coupling quality factor.

*E*

_{in}= (

*P*

_{in}

*τ*

_{r})

^{1/2}is the electric field of the input light with

*P*

_{in}as the input power.

*κ*= (

*κ*

_{e}

*τ*

_{r})

^{1/2}denotes the coupling coefficient between the microdisk and the tapered fiber. The output electric field

*E*

_{out}passing through the tapered fiber can be expressed asThe normalized transmission

*T*= |

*E*

_{out}|

^{2}/|

*E*

_{in}|

^{2}can be obtained by solving Eqs. (1)-(5).

We calculated the transmission spectra of the tapered-fiber-coupled LN microdisk resonator with various pump power, wavelength sweeping speed and directions. The parameters used in the numerical calculation are listed in Table 1. The thermo-optic coefficients of both extraordinary and ordinary light can be regarded as invariant for the reason that the change in temperature of the LN microdisk is only on the level of a few Kelvin [38]. Figures 3(b) and 3(d) are the numerical calculation results showing the broadening and compression of the transmission dip corresponding to Figs. 3 (a) and 3(c) measured when the laser wavelengths were shifted with a sweeping speed of 4.8 pm/s to the shorter and the longer wavelengths, respectively. In this case, the wavelength sweeping speed is relatively low, thus the system including the LN microdisk and the substrate have enough time to respond. Therefore, the heat dissipation induced blue shift of the resonance wavelength dominates the thermo-optic response of the device make the transmission dip broaden (compress) as the wavelength of the laser is scanned to the shorter (longer) wavelengths. As a comparison, Fig. 5(b) is the simulation results achieved while setting the wavelength sweeping speed as 4.8 nm/s. When the wavelength of the tunable laser was shifted with a speed beyond the response rate of heat dissipation process, the thermo-optic properties of the LN crystal begin to dominate the thermal response of the system. As the effective thermo-optic coefficient for room-temperature LN crystal is positive, the transmission dip is broadened (compressed) when the laser wavelength is tuned to the longer (shorter) wavelengths, which are in dark contrast to the slow sweeping case shown in Fig. 3.

Under some experimental conditions, the positive thermo-optic effect of the LN crystal itself and the negative effective thermo-optic effect related to the heat dissipation process compete and dominate the resonance wavelength of the LN resonator alternately making the transmission of the resonator increase and decrease in turn as shown in Fig. 4(a). In simulation, we also realized such oscillatory phenomena in transmission spectra, the results similar to the experimental transmission curves of Fig. 4(a) are shown in Fig. 4(b). Though the details of the theoretical calculated oscillation are somehow different from those of the experimental results, the variation tendencies of the oscillation period and duty cycle are consistent. The oscillation cycles change from sparse to dense and then to sparse again as the input wavelength decreases. The duty cycle decreases as the laser wavelength decrease. All the simulation results were obtained by the same parameters listed in Table 1, which manifests that the theoretical model is applicable for on-chip LN microdisk resonators.

It is worthy of mentioning that, due to the two thermal processes are coupled with each other, it is difficult to balance them and thus to fix the resonance wavelength of the resonator during the whole period of time in which the wavelength of the input light sweeps. Additionally, the geometric size of the lithium niobate resonator including the radii and thicknesses of the lithium niobate disk and the silica pillar influence the thermo-optical response. Theoretically, the greater the volume of the lithium niobate disk, the smaller the response time *τ*_{1} = *Cp*_{1}/*K*_{1} of the fast thermo-optic process, which induces more cycles and shorter period of oscillation. Additionally, thermal oscillation phenomena are more difficult to be excited in a lithium niobate resonator with a greater radius.

## 5. Conclusions

In conclusion, thermo-optic oscillation of the transmission spectra of LN microdisk resonators was experimentally observed when the laser scan speed is low and the input power is high. We attribute such a behavior to the competition between the relatively fast thermo-optic effect of LN crystal and the deformation of the microdisk due to the heat dissipation from the microdisk to the substrate and the surroundings. The two nonlinear thermo-optic effects have not only different response rates, but also opposite effective thermo-optic coefficients. These two thermo-optic mechanisms were separated and experimentally confirmed by measuring transmission spectra of LN microdisks by tuning the wavelength of the laser with high and low sweeping speed, respectively. Simulated results from the theoretical model considering for these two opposite thermo-optic effects agree well with the experimental results. This work provides a comprehensive understanding of the thermo-optic effects in LN microdisk resonators, which is helpful for tuning and stabilizing the resonance frequencies of the whispering gallery modes in nonlinear optical experiments. The thermo-optic oscillation behavior may also be used in the temperature sensing or as a light modulator.

## Funding

973 program (2013CB328702); National Natural Science Foundation of China (NSFC) (11374165, 11174153, and 61475077); 111 Project (B07013); PCSIRT (IRT_13R29).

## Acknowledgments

J. Wang and B. Zhu contributed equally to this work. B. Zhu and F. Bo thanks Christophe Baker for his kind help on the simulation of the transmission spectra. F. Bo, F. Gao, G. Zhang, and J. Xu are also at the Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan, Shanxi 030006, China.

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