Optical heating of resonant nanostructures is one of the key issues in modern nanophotonics, being either harmful or desirable effect depending on the applications. Despite a linear regime of light-to-heat conversion being well-studied both for metal and semiconductor resonant systems is generalized as a critical coupling condition, the clear strategy to optimize optical heating upon high-intensity light irradiation is still missing. This work proposes a simple analytical model for such a problem, taking into account material properties changes caused by the heating. It allows us to derive a new general critical coupling condition for the nonlinear case, requiring a counterintuitive initial spectral mismatch between the pumping light frequency and the resonant one. Based on the suggested strategy, we develop an optimized design for efficient nonlinear optical heating, which employs a cylindrical nanoparticle supporting the quasi bound state in the continuum mode (quasi-BIC or so-called ‘super-cavity mode’) excited by the incident azimuthal vector beam. Our approach provides a background for various nonlinear experiments related to optical heating and bistability, where self-action of the intense laser beam can change resonant properties of the irradiated nanostructure.

The strong resonant response of all-dielectric resonant nanosystems in the visible and infrared region along with the diversity of their optical properties opens the way for various applications in nonlinear and laser optics [

From this point of view, the problem of efficient heating of all-dielectric nanostructures requires special optimization, depending on the final application of the nanophotonic design. Based on this approach and employing advanced methods of nanothermometry, the case of linear optical heating of all-dielectric Mie-resonant nanoparticles was successfully described theoretically [

In this work, we firstly develop a simple analytical model for the nonlinear optical heating of a single-mode resonator supporting efficient light-to-heat conversion and bistable regime of operation, and then employ numerical simulations to propose a realistic design based on the super-cavity. The developed formalism and numerical design dealing with doped silicon cylindrical particles (see

A schematic illustration of the main idea: optical heating of the nanoresonator leads to a spectral shift of the mode frequency inducing strong thermooptical nonlinearity. As the result, the bistable state can be achieved allowing for further efficient heating in the nonlinear critical coupling regime.

Optical heating of matter is a rather complex process, which combines free carrier generation, their interaction with light and phonons, and transport of the phonons across the material [_{abs} is the light power absorbed inside the resonator due nonradiative (ohmic) losses, _{eff} is the effective thermal conductivity of surrounding media, and

The problem of efficient heating of nanoresonator is, thus, reduced to maximization of the absorbed power, which has its fundamental limitation [

We consider a subwavelength resonator with resonance at the frequency _{0} having radiative and non-radiative (ohmic) losses rates _{r} and _{nr} correspondingly. The latter are responsible for light absorption inside the resonator and its consequent heating. Indeed, the absorbed power _{abs} is proportional to nonradiative losses _{nr} and the total electric energy _{abs} = 2_{nr}
_{0} is the eigenfrequency of the resonator, _{r} + _{nr} is the total loss rate, _{0} − ^{2}. Thus, the absorbed power _{abs} = 2_{nr}|^{2} also has resonant Lorentz spectral profile and at the resonance reaches its maximal value

In these terms, the critical coupling condition manifests itself in equal radiative and non-radiative losses _{r} = _{nr}. Indeed, once this condition is fulfilled, the absorbed power tends to its maximal value of _{abs} → |^{2}/2.

Due to the thermorefractive effect, both real and imaginary parts of the refractive index of the material may start to depend on temperature resulting in correspondent dispersive and absorptive Kerr-type nonlinearities. One of the main consequences is the appearance of a bistability regime [

Within a single-mode approximation, the thermooptical Kerr nonlinearity can be accounted for in the first order of perturbation theory through the shift of frequency spectral position _{0} → _{0} − ^{2} and varied nonradiative losses _{nr} → _{nr} + ^{2}. Here, we assume that both nonlinear coefficients are positive

Here, we introduce dimensionless parameters for the mode intensity ^{2}/_{0} = _{r}|^{2}/^{3}, the relative nonlinear coefficient _{0}. Interestingly, to analyze the pump intensity _{0} = Δ_{eff}/_{eff} = ^{2}/^{2} over the spectral width of the resonance _{0} one can increase the pump field intensity, decrease the total losses in the system, or increase the nonlinear coefficient making the system more sensitive to heating. Alternatively, the equation can be rewritten in a more compact form

_{0} is shown in _{0} not shown in the plot.

Bistability in a single-mode nonlinear optical resonator.

(a) Dimensionless mode intensity ^{2}/_{0} = _{r}|^{2}/^{3} for different detuning parameters _{h} and width _{h}. (b) The map of the heating efficiency as the function of the detuning and the pump intensity for the upper branch of the solution

Careful analysis of _{0} and detuning _{max} in

Further analysis shows that the bistability region strongly depends on

Such behavior has a clear physical explanation. Large _{0}. However, what is less obvious, that the bistability exists only for values of _{cr} the point _{cr} in the following manner: at large

Finally, it is important to discuss the hysteresis loop parameters such as hysteresis width _{h} and height _{h}, which is the

One can see that the height and width of the hysteresis loop appear to be dependent on the detuning frequency and are mutually dependent. For large frequency detuning values _{h} ∼ ^{1/2} and _{h} ∼ ^{3/2}. Consequently, for small values of detuning near the critical point hysteresis height increases more rapidly than its width. Thus, for potential optical switching applications, it is more prospective to work closer to the critical frequency

Till now, we have discussed the possible states of the resonator with thermally induced Kerr nonlinearity. The maximal possible mode intensity is provided by the nonlinear critical coupling condition based on the straightforward analysis of _{max}. Moreover, it turns out that there are a certain set of parameters _{max} is reached and they are given by the equation

This equation defines the

The critical coupling regime is seen in _{0}. One can see that the maximal value is reached along the black dashed line, which is obtained as the solution of the nonlinear critical coupling equation

The nonlinear critical coupling condition formulated in _{max} will have a for similar to the classical one with slight modification_{r} the maximal absorption will be observed at

The behavior of the nonlinear resonator strongly depends on the values of dispersive and absorptive nonlinear coefficients _{0} is the permittivity of vacuum, _{0} and _{0} are the initial values of complex refractive index at room temperature _{1} = d_{
T=298K} and _{1} = d_{
T=298K} are the linear thermorefracitve coefficients. The imaginary part of the dielectric permittivity _{abs} through the linear coefficient

For a dielectric resonator the frequency shift is governed by the change of the refractive index, thus providing Δ_{0}/_{0} = −Δ_{0} is unperturbed resonator eigenfrequency. Combination of expressions _{0} ≪ 1.

Similarly, we can derive the absorptive nonlinear coefficient

In this section, we implement the developed approach for optimizing the optical heating of a single nanoresonator. In strong contrast to nanoplasmonic designs, we aim at semiconductor materials, which demonstrate strong thermooptical nonlinear coefficient [

Quasi-BIC states are characterized by the destructive interference of radiation in the far-field zone and, hence, efficient localization of electromagnetic energy in the resonator volume [^{imφ
}, where

Linear optimization of optical resonator parameters.

(a) The cylinder eigenmodes positions depending on the dimensionless wavevector _{d} = 4.6 ⋅ 10^{18} cm^{−3}. (c) Comparison of linear (without thermorefraction) and nonlinear (with thermorefraction) regimes of heating the nanoparticle for zero detuning ^{2}.

Next, we have optimized the nonradiative losses to get close to the critical coupling condition. The doping of semiconductors provides an additional degree of freedom for precise nonradiative losses control. We choose silicon for the resonator material since it possesses zero optical losses in the near-IR region [

For chosen geometry of the resonator, we perform rigorous optimization for nonradiative losses by simulating in numerical commercial software COMSOL Multiphysics temperature increase inside of the nanoparticle excited by the azimuthal vector beam [_{e} = 1 fs is the electron momenta relaxation time; plasmonic frequency _{eff} = 0.18_{e} is the effective mass of electrons in the conduction band of c-Si [_{0} is the permittivity of vacuum, and _{d} = 4.6 ⋅ 10^{18} cm^{−3} corresponding to the optimal value of _{r} + _{nr} = 2.4 ⋅ 10^{12} 1/s.

We have performed the simulations on the optical heating of the designed nanoresonators demonstrating a huge temperature increase Δ^{2} in the linear regime when the thermooptical effects are omitted. However, once the nonlinearity of the system is taken into account, heating is significantly suppressed under the resonant excitation (compare linear and nonlinear regimes in

In our design of the nanoresonator tuned for IR-region, the relative absorptive nonlinear coefficient _{1} = d_{
T=298K} = 0. At the same time, the real part of the thermooptical coefficient at the wavelength _{1} = 2 ⋅ 10^{−4} 1/K [^{28} J^{−1} s^{−1} and ^{25} J^{−1} s^{−1} nonlinear coefficients.

With the given material parameters, we perform full-wave simulations coupled with the heat transfer module in commercial software COMSOL Multiphysics. The thermooptical coupling provides the nonlinear response of the simulated system which may initiate computational difficulties once the bistability regime is reached. In this case, the final state of the iterative numerical solution depends on the initial solutions guess whether the solution is located on the upper, lower, or unstable part of the s-curve shown in _{0} is shown in

Numerical simulations of bistability behavior in single optical resonator.

(a) The dependence of the temperature on the pump intensity for different detuning parameters Δ_{0} of the doped silicon _{0} and detuning factor

It is also worth noting that at the point of maximal heating efficiency in the nonlinear regime, its value becomes equal to the heating efficiency in the linear regime (see the linear dash-dotted line in

The main idea of this paper is to reveal the key aspects of optical heating of a semiconductor resonator: (i) once the resonator is heated in the CW regime, the efficiency of heating immediately drops due to thermooptical reconfiguration of the resonator; (ii) one can reach high efficiency only the heating is accompanied with proper spectral detuning of the pumping laser from the ‘cold’ resonance of the mode; (iii) The maximal heating efficiency is reached in the bistability regime which appears under certain conditions on the pumping intensity, thermooptical coefficients, and of the resonant mode.

However, despite the discussed design being aimed at the near-IR range, where the intrinsic losses of silicon are vanishing, one can tune the proposed design closer to the visible spectra and balance the radiative losses with the intrinsic losses of silicon, which occur at around _{1} = 8.5 ⋅ 10^{−6} 1/K [

It is also worth mentioning that the suggested effect of nonlinear critical coupling can be observed not only in the CW regime, but also under the pulsed excitation once the pulse duration is long enough so that the equilibrium temperature is achieved, i.e. for nanosecond laser pulses. In that case, the quasi-CW regime can be considered and chirped laser pulses could be used to achieve the efficient heating of nanoresonators. Alternatively, the heating effects under the short pico- and femto-second pulses excitation requires more complex models based on the analysis of nonequilibrium dynamics of carriers [

Finally, we would like to provide a brief comparison of the proposed nanophotonic design in terms of heating efficiency to existing analogs of nanoscale optical heaters. Indeed, ^{2}), which is the highest value of heating efficiencies for the structures with direct thermal contact with substrates. For a single c-Si nanodisk laser heating on a substrate at magnetic dipole and quadrupole modes the efficiency reaches 150 K/(mW/μm^{2}) [^{2}) [^{2}) for complex semiconductor structures with quasi-infinite c-Si nanorod covered with a-Si film [^{2}) in aqueous media. Nevertheless, it remains questionable of fabrication, experimental feasibility, and temperature detection possibility [^{2}) [

In conclusion, we have developed a new simple theoretical approach to optimization of the resonator optical heating in the nonlinear regime. The proposed design based on the super-cavity mode in doped silicon cylindrical particles has allowed for efficient light-to-heat conversion when the initial spectrum of incident laser is detuned from the initial spectral position of the resonance. Moreover, we have revealed a bistability regime in the optical heating at intensity around 1 mW/μm^{2}. Our results are also helpful for resolving the thermal challenges for all-dielectric resonator-based photonic devices [

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