1. Introduction

Optical microresonators of different geometries (spherical cavities [1], disc cavities [2], toroidal cavities [3,4], or photonic crystal cavities [5]) have been attracting a great deal of scientific and technical interest for more than a decade due to their extremely high Q -factors and small mode volumes. In combination with efficient pumping schemes [6] microresonators provide a very convenient tool for studying a diverse set of phenomena: cavity quantum electrodynamic effects [7,8], Raman scattering [9], lasing [10], parametric oscillation [11], biosensing [12] etc. Among a variety of nonlinear effects in microspheres, the thermal ones have the lowest thresholds and can be observed at powers as low as tens of microwatts. It is well known, however, that the temperature based effects are incomparably slower than the nonlinearity of Kerr type, and are, therefore, unfavorable for many applications.

In this paper the difference in response times of the thermal and Kerr nonlinearities is used to separate their contributions to the nonlinear optical properties of microspheres. To this end, nonlinear properties of the transmission spectra of a coupled taper-microsphere system have been investigated. The spectra have been obtained using the scanning technique when the excitation wavelength of incident light is swept through the resonances of the microspheres. Depending on the speed of the wavelength scanning, the Kerr and thermal effects contribute differently to the total nonlinear response. The properties of the system are determined by several characteristic times: resonance building time τ_mode (related to the homogeneous linewidth of the resonance), sweeping time τ_scan (time it takes to scan through a single resonance), thermal relaxation time τ_th between mode volume and the rest of the microsphere and thermal relaxation time τ_envir between microsphere and environment (air in the presented here experiments). For the modes which are of interest the τ_mode is always orders of magnitude shorter than the thermal relaxation time, τ_th. The regime of very fast sweeping (τ_scan≪τ_mode) is of little interest because the scanning completes before the resonance fully builds up so that the mode in the resonator does not have time to develop. For a slow enough sweeping (τ_scan≫τ_mode) the power absorbed due to inevitable passive optical losses heats the microsphere and causes the nonlinear response, which is due to temperature dependence of the refractive index and thermal expansion of the microsphere. The characteristic times of these processes are τ_th and τ_envir respectively, while the part of the nonlinear response due to the Kerr effect is instantaneous. This allows us to manipulate the partial contribution from the thermal and Kerr based nonlinearities.

2. Experiment

An experimental setup used to study the optical response of microresonators in our work is shown in Fig. 1.

 

Fig. 1. Experimental setup used for microsphere resonator measurements.

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A light from the Tunable Laser Source (TLS, Agilent 81600/81640A) is amplified by an EDFA (High Wave Optical Technology), which boosts the signal up to 10 mW. The amplified light passes through a tapered fiber with the minimum waist diameter of about 4 µm, produced with a specially customized splicing and tapering machine (Vytran LDS-1250). The microsphere resonator is positioned close to the waist of the taper in order to excite the whispering gallery modes due to the overlap of the evanescent fields in the taper and the microsphere. Subsequently the light passing through the taper is detected by a receiver with an electrical bandwidth of 125 MHz (Terahertz Technologies Inc.). The electrical signal is visualized by an oscilloscope (Agilent DSO 6104A) triggered by the external function generator (Agilent 33250A), which is used to drive sweeping of the TLS. This setup allows one to change the wavelength in both directions with a variable sweeping speed, a wavelength span, and an optical power. Taking into account that the optical power does not have significant wavelength dependence within the sweeping region, one can take the oscilloscope traces as a direct representation of the transmission spectra.

In the experiments with sweeping wavelength it is necessary to realize that the transmission spectra visualized on an oscilloscope screen do not represent regular homogeneously broadened resonant lines. The distinction is due to the fact, that during the sweep the nonlinearity affects differently the wavelengths within the bandwidth of the resonance. Only in the linear regime the transmission spectrum corresponds to the homogeneously broadened resonance line. When the nonlinearity becomes significant (at higher input powers) it affects the line shape stronger in the vicinity of the resonant wavelength. At high power the visualized transmission dips appears to be broadened compared to the low input powers (homogeneously broadened linewidth).

The cause of this apparent broadening is the dependence of the resonant wavelength on the absorbed power. Due to the thermo optic effect, the refractive index increases and the transmission resonance shifts into the longer wavelength region. Since this process happens simultaneously with the sweeping, the resonant dip becomes trapped by the sweeping wavelength resulting in the broadened image of the transmission spectrum. In order to avoid misinterpretations, here and below the term “broadening” will be used in connection with the experimentally obtained data (visualized on an oscilloscope). While Kerr nonlinearity should only lead to the “broadening” of the observed resonance transmission dips, it has been found, that the thermal nonlinearity, in addition to the “broadening”, results in a distortion of resonance shapes of oscillatory or chaotic type [13].

3. Theoretical modeling and estimations

Inevitable intra-cavity absorption causes an increase of temperature inside the resonator which leads to a deviation of the refractive index and a change of geometrical sizes of the microcavities. Mutual dynamics of the processes of heating and dissipation determines the mentioned above resonance broadening and small scale instabilities, as well as a shift of the entire resonance towards longer wavelengths, which will be discussed below in this work. The temperature distribution on T(r⃗,t), determined by the heating and dissipation processes, is described by a standard thermal diffusion equation:

where k _th is the thermal conductivity, ρ is the material density, C͂ is the specific heat capacity per mass of the resonator medium, α is the linear absorption coefficient, and E is the electric field coupled into the microsphere. The thermal diffusion coefficient k _th/ρC͂ describes the heat dissipation process into the environment, which can be subdivided into two stages with different characteristic time scales: temperature equalization of the mode volume with the entire microresonator (characteristic time τ_th of the order of hundreds of microseconds), and equalization of the microresonator as a whole with the environment of the resonator (characteristic time τ_envir of the order of tens of milliseconds). Usually, in broad wavelength sweeping operation many eigen modes are excited at different wavelengths during one scan. The sweeping time through a single resonance can be shorter or longer than τ_th depending on the wavelength sweeping speed. However, the time between subsequent sweeps through the same resonance is usually much longer than τ_th, and can be shorter or longer than τ_envir depending on the wavelength sweeping speed and sweeping range. This means that between two subsequent sweeps through the same resonance the local heat in the volume, occupied by a particular mode, spreads over the entire microresonator. On the other hand, the resonator is heated by all modes excited during a single sweep and does not have enough time to come in thermal equilibrium with the surrounding between two subsequent sweeps. As a result, after several sweeps the microresonator is heated up to some quasi-equilibrium temperature and then temperature dynamics, taking place inside the resonator, occurs around this quasi-equilibrium level. This quasi-equilibrium temperature level depends on the particular experimental conditions and results in a shift of all resonances to the longer wavelength region as mentioned above. This effect together with a detailed investigation of the influence of the sweeping speed on the observed data, have not been considered comprehensively before and are topics of this contribution.

A theoretical model for the electric field and thermal dynamics in a microsphere adapted in this paper is based on a well known Mie theory [14] combined with approximated treatment of the temperature dynamics developed in [15]. Assuming, that in the vicinity of any given resonance only one whispering gallery mode can be taken into account, the equations describing coupled field and temperature dynamics can be presented in the following form:

Here C_lm (t) and A_lm (t) are amplitudes describing the time dynamics in the expansion of internal and exciting fields in terms of vector spherical harmonics with orbital and azimuthal numbers l,m [14], Δ=ω₀-ω_l+iδ_l is the detuning between the excitation frequency ω₀ and the mode frequency ω_l, where δ_l is the imaginary part of the mode frequency, describing radiative losses of the mode. ${\gamma }_{\mathrm{el},\mathrm{lm}}=\frac{{\omega }_l}{{\epsilon }_0{n}^4{V}_{\mathrm{lm}}}{\chi }^{\left(3\right)}({-\omega }_l;{\omega }_l,{-\omega }_l,{\omega }_l)$ is the Kerr nonlinear coefficient containing the third order susceptibility χ ⁽³⁾ which is the first non-vanishing nonlinear term of the polarization expansion. $V_{\mathrm{lm}}=\frac{{(\int {\mid {\overrightarrow{M}}_{\mathrm{lm}}(\overrightarrow{r},{\omega }_l)\mid }^{2}dV)}^2}{\int {\mid {\overrightarrow{M}}_{\mathrm{lm}}(\overrightarrow{r},{\omega }_l)\mid }^{4}dV}$ is the mode volume and $n=\frac{n_1}{n_0}$ is the relative refractive index, where n ₁ and n ₀ are the refractive indexes of the microresonator material and the environment, M⃗_lm (r⃗) is the vector spherical harmonic (TE-polarization) describing spatial distribution for a given mode. The thermal nonlinear coefficient is described by ${\gamma }_{\mathrm{th},\mathrm{lm}}=\frac{{\omega }_0}{n}\left(\frac{\partial n}{\partial T}\right)$ and p_l is the coefficient describing coupling between the taper and the microsphere, which was used as a fitting parameter. $Q\left(t\right)=\frac{\int \left(T(\overrightarrow{r},t)-T_0\right){\mid {\overrightarrow{M}}_{\mathrm{lm}}(\overrightarrow{r},{\omega }_l)\mid }^{2}dV}{\int {\mid {\overrightarrow{M}}_{\mathrm{lm}}(\overrightarrow{r},{\omega }_l)\mid }^{2}dV}$ is the mean temperature difference between actual microsphere temperature and quasi-equilibrium one weighted by the intensity distribution of the mode. Coefficient $B=\frac{4c\alpha }{{\rho \tilde{C}}_n{V}_{\mathrm{lm}}}$ describes temperature change due to absorption of the internal electromagnetic field, where c is the velocity of light. The thermal relaxation time of the mode volume to the rest of the resonator material, reciprocal to δ_th is:

with b as the half of the thickness of the mode in radial direction. The system (2) describes processes around the quasi-equilibrium temperature level T ₀ and does not describe relaxation of the microresonator as a whole (the difference between the quasi-equilibrium temperature T ₀ and environmental temperature or resonator temperature before the scan starts is responsible for the shift of the whole resonance pattern). Hence, the time scale where the system (2) remains valid is τt≪τ_envir (which is of the order of tens of milliseconds).

The next important time scale is the mode relaxation time τ_mode, which is related to the Q- factor as:

where λ_l and δλ_l are the resonance wavelength (optical frequency) and homogeneous resonance linewidth, respectively - see Fig. 2.

 

Fig. 2. High-Q resonance transmission dip measured in a microsphere with a diameter of 410 µm and a coupling power of 7.4 µW.

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The respective Q -factor for the resonance in Fig. 2 (which is typically observed in the experiments) is 7.4*10⁷ and the respective photon life time (or mode relaxation time) is found to be τ_mode≈61 ns according to Eq. (4). It is worth noting that the coefficient δ_l in the first equation of the system (2) does not determine (but, of course, contributes to) the observed total Q -factor, because this coefficient describes only radiative losses of the microsphere resonator (passive losses, scattering at surface, and material inhomogenities have not been taken into account), while the observed Q -factor is determined by all types of optical losses (see, for example, [6]).

The observed transmission spectra are determined by relations between different characteristic times introduced above. The sweeping speeds used in our experiments vary between 0.5–40 nm/s which corresponds to variations of τ_scan between 40 µs to 0.5 µs for the resonance shown in Fig. 2. In this case even for the fastest sweeping speed τ_scan is about 10 times greater than the mode relaxation time. It should be noted that a possibility to discriminate the temperature effects by increasing the sweeping speed (and/or decreasing the homogeneous linewidth) is not obvious. Indeed, when τ_scan becomes smaller than τ_mode the thermal effects are suppressed, but the mode may not have enough time to develop. This discussion aimed at emphasizing the fact that experimental conditions have to be chosen carefully in order to avoid artificial suppression of the effects which one intends to study. Different characteristic time scales are summarized in Fig. 3.

 

Fig. 3. Different characteristic time scales for the high-Q resonance shown in Fig. 2.

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Taking into account the fact that scanning through a resonance time is always much longer, than the respective relaxation time of the resonance, the field time derivative in the first equation in (2) can be set to zero. The external electric field amplitude ude A_lm remained constant during a sweep and hence does not depend on time. In contrast, frequency mismatch changed during the sweeping, which means that Δ(t) is actually time dependent value. Thus the master system (2) for the field amplitude and temperature dynamics becomes:

Analysis of Eq. (5) requires different approaches depending on the relationships between characteristic times shown in Fig. 3 (All times in Fig. 3 have been calculated for the resonance shown in Fig. 2, with the thermal relaxation time being estimated using Eq. (3)). We first consider slow sweeping regime τ_scan>τ_th, in which local thermodynamic quasi-equilibrium is established during the sweep. In this case temperature becomes time independent and the time derivative in the second of Eq. (5) can be eliminated. In this regime the mean temperature difference Q is directly proportional to the mode field intensity resulting in the additional nonlinear term in the first equation of system (5) which effectively mimics the Kerr nonlinearity:

This regime was previously studied in [16] where two steady state solutions (stable warm and stable cold equilibriums) were found numerically and observed experimentally. In this operation mode it is possible to compare directly the contributions from the optical Kerr and thermal effects. First of all one can note that both effects have the same sign and can only enhance each other. However, estimation based on the known constants for silica glasses gives in our case γ_el,lm≈1.42*10¹¹ MHz/J while ${\gamma }_{\mathrm{th},\mathrm{lm}}\frac{B}{{\delta }_{\mathrm{th}}}\approx 2.2*{10}^{12}$ MHz/J and one can see that the thermal nonlinearity is one order of magnitude higher than the magnitude of the Kerr effect in this operation mode. For the sweeping speed used in our experiments the sweeping time through a resonance is always shorter than the thermal relaxation time (but nevertheless longer than the mode relaxation time) τ_mode<τ_scan<τ_th (see Fig. 3), therefore the quasi Kerr approximation (7) does not describe the situation studied in this paper.

It is also interesting to consider the opposite limiting case τ_scan≪τ_th. Since physically this means that thermal relaxation does not occur during the sweep, we can describe this situation by neglecting the relaxation term in the second of Eq. 5 and describe temperature dynamics as:

Substituting this expression in the equation for the electric field amplitude we obtain the following equation describing the field dynamics inside the resonator:

In order to compare roughly the relative contributions of Kerr and thermal effects in this regime, one can evaluate the integral in Eq.(7) as:

where $\overline{{\mid C_{\mathrm{lm}}\mid }^2}$ is some averaged value of the field amplitude during the scan, and again compare the coefficients γ_el,lm and γ_th,lm Bτ_scan. It turns out that the scanning time for which both effects are comparable is about 100 ns, which for resonances with Q -factors of 10⁷ (a typical value in our experiments) is of the same order of magnitude as the mode relaxation time. In order to separate both effects and to investigate Kerr nonlinearity alone it is, therefore, necessary to employ shorter scanning times (higher sweeping speeds). But in this case the respective modes do not have enough time to build up as discussed above. This situation cannot be improved by using resonances with lower Q (and shorter mode relaxation times, respectively) because the intensity inside the resonator will also decrease making the nonlinear effects too weak for reliable detection.

In recent experimental work [20] a possibility to separate Kerr and thermo optical effects by placing a microsphere in the superfluid helium was discussed. The conclusion in that paper was based on the fact that the thermo-optical coefficient becomes negative for T<20K while the experimentally observed transmission spectra have a form typical for the positive thermo optical coefficient. However, according to the presented above discussion, absence of detailed information about temperature dynamics inside the microsphere and the sweeping speed make an unambiguous interpretation of the results of [20] difficult.

Since sweeping time in our experiments is only moderately smaller than the relaxation time (τ_scan<τ_th but not≪τ_th) a quantitative description of our experimental results should rely on the full system of Eq. (5) which can be conveniently rewritten in terms of experimentally measured field intensity, I, where I=|C_lm(t)|²:

with coefficients $u_1=\frac{2\left(\Delta \omega \left(t\right)+{\gamma }_{\mathrm{th},\mathrm{lm}}Q\left(t\right)\right)}{{\gamma }_{\mathrm{el},\mathrm{lm}}}$ , $u_2=\frac{{\left(\Delta \omega \left(t\right)+{\gamma }_{\mathrm{th},\mathrm{lm}}Q\left(t\right)\right)}^2+{\delta }_l^2}{{\gamma }_{\mathrm{el},\mathrm{lm}}^2}$ and $u_3=\frac{{\mid p_1{A}_{\mathrm{lm}}\mid }^2}{{{\gamma }_{\mathrm{el},\mathrm{lm}}}^2}$ where Δω=ω₀-ω_l. Numerical solution of this system was used in what follows for detailed comparison with experimental results.

4. Comparison with the experimental data

In our experiments we measured transmission spectra of the coupled taper-microsphere system with the different sweeping speeds and input powers. The focus of our experiments in the regime of τ_mode<τ_scan<τ_th is to study in details the dependence of the transmission spectrum on the sweeping speed and coupling power. The system (10) has been used to describe the observed experimental data. In the performed experiments the eigen resonance bandwidth, described by δ_l, has not been measured independently. The parameter δ_l takes into account the radiation losses only, and doesn’t coincide with the reciprocal resonance bandwidth even at the low input powers. In the performed calculations the parameter δ_l has been set to 3.6 MHz which looks reasonable and coincides with the typically reported data. Other parameters - S=|p_l A_lm|², δ_th, γ_th,lm, γ_el,lm and B - have been adjusted in order to find best possible correspondence with the experimental data.

It is clear that the parameter S=|p_l A_lm|² (power coupling to resonator) influences the depth of the resonances and so could be adjusted relatively easy. It has been done for two coupling powers - 0.07 mW and 0.34 mW - and the found values for the parameter S have been kept unchanged for the rest of fitting procedure. The other parameters - δ_th, γ_th,lm, γ_el,lm and B - have been fitted based on some particular reference data, and then have been kept unchanged. For such a reference the resonance curve obtained at the low power in taper P=0.07 mW and sweeping speed of 1 nm/s has been chosen. The key criteria for the fit were the FWHM bandwidth of the resonances and the gradient of the resonance shape to the longer and smaller wavelength direction. The set of parameters giving the best fit has been identified for two cases: with and without effect of the Kerr nonlinearity, which, being at least one order of magnitude lower than the thermal one, supposedly should not affect the resonance shape. Due to structure of the numerical codes, the case “without Kerr effect” has been calculated with γ_el,lm (MHz/J)=10^-6 instead of just γ_el,lm (MHz/J)=0, which guaranties zero influence of the Kerr effect on final results (for particular values of the used in the calculations parameters see two respective tables below in the text).

Found parameters were used to simulate other resonance curves. The resonances obtained with the same coupling power of 0.07 mW under different sweeping speed have been compared with the results of numerical simulations (see Fig. 4, left column), and the same comparison has been made for the higher power of 0.34 mW (see Fig. 4, right column). Experimental data are presented by black points, solid red lines show theoretically calculated curves based on Eq. (10) with both thermal and Kerr effect taken into account and dashed blue line represent the results of calculations with omitted Kerr term.

 

Fig. 4. Transmission resonance shapes for two coupling power at the different sweeping speeds. Right column - 0.34 mW, left column - 0.07 mW. Experimental data are depicted by black points, red solid line - simulation with Kerr and thermal effects, blue dashed line - simulation without Kerr effect (thermal nonlinearity only).

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In agreement with the previously published results, the visualized transmission dips become significantly broadened as the input power increases.

It is seen, that for the low power agreement between measured and calculated values is relatively good (Fig. 4, left column). Also, difference between theoretical curves, obtained with and without Kerr effect is not so significant. Less clear is the fact that the deviation (between theory and experiment, and between theoretical curves with/without Kerr) does not look systematic: for the sweeping 5 and 2 nm/s experimental curves exhibit less broadening in comparison with the theoretical ones, while for 0.5 nm/s experiment shows more pronounced effect. The same conclusion can be made from comparison of the theoretical curves with each other, namely with and without Kerr nonlinearity. For the sweeping 2 nm/s theory gives surprisingly less broadening with the Kerr effect, while for the 0.5 nm/s the result agrees with intuitive expectation and shows more broadening with both Kerr and thermal effects.

For the higher coupling power of 0.34 mW the correspondence becomes even worse. First, theoretical curves without Kerr effect follow experimental ones better for the sweeping 40, 20, and 10 nm/s, while numerical results with the Kerr effect show significant deviation especially around their dips. At 5 nm/s and power 0.34 mW the experiment is better described by the curve with Kerr effect, while for 2 nm/s better correspondence takes place without the curve including Kerr effect. For slower sweepings neither curves give good correspondence. As for the deviation between theoretical curves with and without Kerr effect, the curves without Kerr effect give symmetric shape up to sweepings of 10 nm/s, while the curves with thermal nonlinearities exhibit visible dip shift into the left side (see Fig. 4 for respective sweepings). For 5 nm/s the total effect becomes surprisingly less pronounced in comparison with the effect without Kerr - note, that the same counter intuitive result has been obtained for the low power at slower sweeping of 2 nm/s. Coming back to the high power operation (right column), for the slower sweeping of 2, 1, and 0.5 nm/s the results again follow intuitive expectation and the total effect becomes bigger in comparison with the theory without Kerr nonlinearity.

From the presented above discussion several preliminary conclusions follow, namely:

The most unexpected result, however, is not in a quantitative discrepancy of the theoretical and experimental data, but in the values for parameters given by the fitting procedure. Recall, that the parameters S=|p_l A_lm|², δ_th, γ_th,lm, γ_el,lm and B have been adjusted in order to find best possible correspondence with the particular experimental data, and then have been kept for all other comparisons. The fitted values for the parameters are summarized in Table 1.

The parameter S=|p_l A_lm|² turned out to be not exactly proportional to the coupling power that can be seen from the comparison between relations of powers (0.34 mW/0.07 mW) and values for S (0.00415/0.0007). Both relations, according to the definition of S, should give the same values. Nevertheless, discrepancy in 3,8% can be probably explained by some coupling efficiency dependence on the coupling power. For the thermal constants the fitted values correspond pretty much to ones known from other publications. As for the Kerr coefficient, which should be of the order of γ_el,lm≈1.42*10¹¹ MHz/J, the discrepancy in 7 orders of magnitude can not be explained by just fitting errors.

Table 1. Parameters for best fitting.

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It turned out to be impossible to fit the experimental data with the γ_el,lm≈1.42*10¹¹ MHz/J, whatever values of other parameters have been chosen. It is worth noting again, that for relatively small effects (when the observed resonance shape changes are small in comparison with the initial one) both theoretical options (with and without Kerr effect) give results with approximately the same level of accuracy - see Fig. 4 left column (low coupling power) and Fig. 4 right column for sweepings from 40 down to 10 nm/s. Based on these observations it can be concluded, that:

The resonance broadening data (Fig. 4) have been summarized as a function of sweeping speed for the resonance at λ_resonance=1550.660 nm for two levels of coupling powers. The data are presented in Fig. 5, where the resonance bandwidth has been measured on the level of half maximum transmission power outside the resonance, with the respective results of numerical simulations using system of equations (10). Both, the experiment and the theory, show significant dependence of the resonance width on the sweeping speed up to the values of about 10 nm/s. It means that only those results, which have been obtained with the same sweeping speed, can be compared. While for the lower pump power the theoretical model reproduces the experimental data relatively well, for the higher pump power deviations of the theoretical results from the experiment becomes very strong. Possible reasons for the failure of the model based on Eq. (10) to describe effects due to strong non-linearity have been discussed above. For the sweeping speed values higher than 20 nm/s the effect of the nonlinear broadening virtually disappears. This can be explained by the small pump powers used and the fact that according to Eq. 7 and Eq. 9 there is not enough time for the temperature to be increased in the mode volume.

 

Fig. 5. Dependence of the bandwidth of a high-Q resonance (λ_resonance=1550.6600 nm) on sweeping speed for two values of coupling power: black/red solid lines (modeling with/without Kerr effect) with squares (experimental results) - 0.34 mW; black/red dashed lines (modeling with/without Kerr effect) with triangles (experimental results) - 0.07 mW.

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The dependence of the resonance bandwidth on pump power (both experiment and theory) is shown in Fig. 6 for the resonance at wavelength 1550.6λ_resonance=541 nm. It is seen in Fig. 6 that for relatively small powers, up to about 0.2 mW, the theoretical curve coincides with the experimental points (squares in Fig. 6). For higher pump powers, significant discrepancies between the theory and the experiments again appear with a pronounced jump at about 1.0 mW. One can notice that the theoretical modeling with both Kerr effect and the thermal non-linearity included, produces results, which for higher power are somewhat closer to experimental values than the results of modeling without the Kerr effect.

 

Fig. 6. Dependence of nonlinear broadening of a high-Q resonance (λ_resonance=1550.6541 nm) on coupling power (squares & dashed black line - experiments, solid red line - numerical modeling with Kerr effect, dashed blue line - numerical modeling without Kerr effect).

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At the same time, at lower powers both curves in Fig. 6 are much closer to each other and the experiment. It gives us an additional argument in the favor of the conclusion that in the framework of the theoretical model based on Eq. (10) extraction of the effects due to Kerr effect is problematic.

When plotting theoretical curves in Fig. 6 we did not take into account the effect of the power dependence of the resonator-waveguide coupling parameter, S. This effect, which was uncovered during the fitting procedure, would of course improve somewhat the agreement between theory and experiment, but would not remove the qualitative discrepancy caused by fundamental limitations of the theoretical model used. It should be noted also that the results of simulations shown in Fig. 6 have been obtained with slightly different values of the parameters (listed below in Table 2) compared to results presented in Fig. 4 and Fig. 5. This adjustment was required to reflect the fact that the experimental results shown in Fig.6 were obtained for a slightly different resonance than those shown in previous figures.

Table 2. Parameters for best fitting in Fig. 6.

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Another pronounced effect typical for this kind of experiments, which consists in appearance of small oscillations on the gradually increasing background of the right-hand part of the resonance curve, is shown in Fig. 7. These oscillations were explained in [15] as an Andronov-Hopf bifurcation [17]. This effect has also been observed in our experiments - see Fig. 7. This effect has not been investigated in details here and is supposed to be a topic of separate publication.

 

Fig. 7. Visualized transmission dips broadening of the high-Q resonance shown in Fig. 2 (λ_resonance=1551.22579 nm) for two different input powers. Small oscillations attributed to the Andronov-Hopf bifurcation are clearly seen on the right side of the solid black curve.

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The last effect which is considered in this paper is the shift of the resonance peak pattern as a whole. After multiple scanning through the all resonances which are situated in the whole scanned wavelength region the temperature of the microsphere becomes higher and is finally equalized with the environmental temperature until the scanning stops. It causes in turn the changes of refractive index and the diameter of the microsphere. On the other side, the dimensionless mode frequency (or size parameter) $n{x}_l=\frac{2\pi nR}{{\lambda }_l}$ (λ_l - resonance wavelength, n - refractive index, R - microsphere radius) is fixed for a given resonance to the first approximation. From this fact, one can easily derive a shift of the resonance peak as a function of a change in temperature:

Here ΔT=T-T ₀ is the difference between the temperatures at high and low coupling powers, $\frac{\partial n}{\partial T}\approx 1.2\times {10}^{-5}{K}^{-1}$ [18] is the temperature coefficient of the refractive index and ${\alpha }_V\approx \frac{1}{3V}{\left(\frac{\partial V}{\partial T}\right)}_{T_0}=0.55*{10}^{-6}{K}^{-1}$ [19] is the thermal expansion coefficient of the glass.

In experiments with a coupling power of 1.6 mW the measured resonance wavelength shift was found to be about 82 pm, which corresponds to the temperature change of about 6 K. This value is in agreement with results of [16] where nearly same experimental conditions were used. In Fig. 8 the observed wavelength shift as a function of coupling power is presented. The tunable laser used in the tests had a wavelength positioning accuracy of about 15 pm, which is indicated in Fig. 8 by a dashed line parallel to the x-axis.

Each experimental point has been measured with the same accuracy, which is presented by the error indicators.

 

Fig. 8. Nonlinear shift of a high-Q resonance (λ_resonance=1550.6541 nm) depending on coupling power (solid black line with full rombs - sweeping speed 5 nm/s, dashed red line with empty rombs - 1 nm/s). Dashed line parallel to the x-axis indicates the wavelength positioning accuracy of the laser.

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It is seen, that a definitive resonance wavelength shift can be reliably recognized starting from a coupling power of about 0.7 mW. In addition, in the same figure we show a second set of measured data for a sweeping speed of 1 nm/s. It is clear that for the slower speed the microresonator temperature increases by higher values and that the wavelength shift becomes nearly twice for 1 nm/s (in comparison with one for 5 nm/s) for a coupling power of 1.6 mW.

In order to create a comprehensive model of the temperature dynamics during the sweeps it is necessary to take into account all resonances belonging to the chosen wavelength span, and calculate the energy flow into and out of the microsphere during the sweeps and between sweeps. Even though such a detailed calculation of the temperature dynamics has not been performed, the estimations using (11) are sufficient for the explanation of the observed effects.

5. Conclusion

In conclusion, nonlinear broadening of the microsphere resonances and nonlinear resonance shift has been investigated experimentally and theoretically. It has been demonstrated, that the wavelength sweeping speed significantly influences the observed results. Depending on the scanning time through a resonance, quasi-stationary (τ<_mode<τ_th<τ_scan) and transition (τ_mode<τ_scan<τ_th) operation modes can be realized. Theoretically, the field dynamics in the resonator can be described in these cases by (6) and (5), respectively. According to the performed comparison between theory and experiments, influences of Kerr and thermal effects can not be straightforwardly separated. Our estimations and experiments show, that the hope to suppress the thermal nonlinearity by increasing sweeping speed can not be easily realized due to insufficient time for developing high-Q modes of the resonator. The meaningful interpretation of the experimental results is only possible when it is clear in which of the two sweeping operation modes the experimental data are collected.