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- Photonics Research
- Vol. 6, Issue 5, A23 (2018)

Abstract

1. INTRODUCTION

High-quality whispering-gallery mode (WGM) optical resonators ^{[1]} have been widely used in various areas, including low-threshold lasers ^{[2,3]}, optomechanics ^{[4]}, cavity quantum electrodynamics ^{[5]}, and optical communications ^{[6,7]}. Particularly, the strong light–matter interaction makes WGM resonators suitable platforms for nanoparticle/biomolecule sensing ^{[8–17]}. With circular geometry, the WGM resonator supports two degenerate modes at the same eigenfrequency with opposite propagating directions, i.e., clockwise (cw) and counterclockwise (ccw) directions. When a nanoparticle enters the mode volume, the interaction of the optical mode with the nanoparticle lifts the degeneracy of the eigenfrequency and therefore leads to mode splitting, which can be used for the detection and measurement of nanoparticles ^{[9]}. For example, WGM microtoroid resonators ^{[18]}, fabricated from pure silica ^{[9]}, (i.e., passive resonators with loss) and from rare-earth-ion doped silica (i.e., active resonators with optical gain) ^{[10,11]}, have been used as highly sensitive sensors to count and size individual nanoparticles with a radius down to a few tens of nanometers based on the mode splitting technique in both air and liquids ^{[9–12]}.

The detection limit for the mode splitting technique is set by the condition that the frequency splitting could be resolved in the transmission spectra ^{[9]}. Two main approaches have been introduced to improve the detection limit for nanoparticle sensing. One approach is to enhance light–matter interaction by reducing the mode volume ^{[9,11]} or increasing the overlap of optical modes with nanoparticles, which subsequently enhances the mode splitting. The second approach is the use of a gain medium to compensate optical losses in optical modes so that their linewidths become narrower. Below the lasing threshold, the gain compensates the losses of optical modes, increases the effective quality factor, and thus improves the resolvability ^{[10,14]}. In the lasing regime, the linewidth as narrow as several Hz could be obtained ^{[19]}, hence largely improving the detection limit ^{[11,14,15]}. In all these methods, the frequency splitting in principle is proportional to the strength of perturbation

Recently, Wiersig reported sensitivity enhancement of nanoparticle detection with a WGM resonator by utilizing exceptional points (EPs) at which both the eigenvalues and the corresponding eigenstates of a non-Hermitian system coalesce ^{[20–22]}. It was shown that the WGM sensor at (second-order) EPs exhibits a frequency splitting proportional to the square root of the perturbation strength, whereas the response of conventional WGM sensors is proportional to the perturbation strength. Therefore, for sufficiently small perturbations, the sensors operating at EPs can exhibit much larger sensitivity than the conventional sensors. This sensitivity enhancement stems from the complex-square-root topology near EPs. Later, Chen ^{[23]}. The EP was obtained by carefully tuning two nanoscatterers within the mode volume ^{[23,24]}. The square-root dependence of the frequency splitting on the perturbation strength as well as the sensitivity enhancement compared with the conventional WGM sensors was shown. In addition, Zhang ^{[25]}.

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Such a complex-square-root topology can also be found in systems respecting parity-time (PT) symmetry ^{[26,27]}, which was originally developed within quantum field theory and has emerged as one of the most exciting concepts in photonics in the past few years ^{[28–37]}. It was shown that the eigenvalues of non-Hermitian Hamiltonians can be entirely real if they satisfy the condition of PT-symmetry, i.e., ^{[26,27]}. With balanced gain and loss, a coupled-resonator system can serve as a good platform to implement PT-symmetry in optics ^{[32,34,37]}. Hodaei ^{[37]}. Moreover, they also exploited the third-order exceptional points at which three eigenvalues and corresponding eigenstates coalesce to achieve even higher sensitivity that is proportional to the cube root of the perturbation strength. PT-symmetric coupled resonators can also be applied to metrology ^{[38]} and optical gyroscope ^{[39]}.

In this paper, we show that a PT-symmetric system, consisting of two directly coupled resonators with balanced gain and loss, offers a promising platform to improve the detection limit for nanoparticle sensing. The PT-symmetric WGM sensor operating at the phase transition points exhibits a square-root dependence of the sensitivity on the perturbation of nanoparticles, and the sensitivity can be much larger than that of a single WGM sensor. The dependence of sensitivity enhancement on the perturbation strength and the gain (or loss) strength is also studied. With the assistance of gain in the system, the resolvability of mode splitting as well as the detection limit for nanoparticle sensing is significantly improved. Furthermore, we extend our study to multiple-nanoparticle detection and show that, in this case, the PT-symmetric WGM sensor also has larger sensitivity, compared with the single WGM sensor.

2. PT-SYMMETRIC WGM NANOPARTICLE SENSOR

The Hamiltonian of a photonic molecule consisting of two coupled resonators (i.e., ^{[32]}

By properly selecting parameters

Figure 1.WGM nanoparticle sensors based on (a) the single passive resonator, (b) the single active resonator, and (c) PT-symmetric coupled resonators with balanced gain and loss. The gray-colored small circle denotes a nanoparticle within the mode volume of the resonator. The spectra illustrate sensing mechanisms of the sensors. The dashed red curve shows the spectrum before the nanoparticle binding event; the solid blue curve corresponds to the spectrum after the nanoparticle binding event. The sensors in (a) and (b) exhibit a frequency splitting proportional to

In the PT-symmetric coupled resonators shown in Fig.

When the coupling strength increases from zero, the system will experience (i) the PT-symmetry broken regime with

Figure 2.Evolution of the (a) real part and (b) imaginary part of eigenfrequencies in PT-symmetric coupled resonators when the coupling strength between two resonators is varied. The PT phase transition point is obtained when the coupling strength

The numerical simulations were performed in COMSOL Multiphysics, where the model consists of two coupled 2D circular microdisk resonators. The two resonators have the same size with radius 5 μm and the same real part of the refractive index, which is 2. Therefore, the two resonators have the same resonance frequency. The optical modes selected for the numerical study have a wavelength of about 1.5 μm. The imaginary parts of the refractive indexes of the resonators are chosen such that one resonator has gain and the other has loss, i.e., the imaginary part of the refractive index of one resonator is positive, whereas that of the other resonator is negative. The values of the imaginary parts vary from

3. SENSITIVITY ENHANCEMENT AT THE PT PHASE TRANSITION POINT

Without loss of generality, we assume that the resonator

The incoming nanoparticle lifts the eigenfrequency degeneracy of the supermodes: two supermodes experience a frequency shift and linewidth change, whereas the other two supermodes are not affected and can serve as reference signals. Evolution of the real part of eigenfrequencies is shown in Fig.

Figure 3.(a) Evolution of the real part of the eigenfrequencies

Next, we study the sensitivity enhancement for the PT-symmetric WGM sensor operating at the phase transition point, i.e.,

Figure 4.(a) Dependence of sensitivity of a PT-symmetric sensor operating at the phase transition point on the perturbation strength

For a single WGM sensor with the same perturbation, the eigenfrequency splitting is

When the nanoparticle is small enough, so that ^{[40]}.

4. IMPROVEMENT OF THE DETECTION LIMIT

The nanoparticle not only lifts the degeneracy of the real part of the eigenfrequencies in the PT-symmetric sensor but also affects the imaginary part of the eigenfrequencies, as shown in Fig.

Figure 5.(a) Evolution of the imaginary part of the eigenfrequencies

For the PT-symmetric WGM sensor in the broken-phase regime, the optical fields of one pair of supermodes grow, whereas the optical fields of the other pair decay, i.e., one pair of supermodes experiences gain, whereas the other pair feels loss ^{[33,34]}. When the system enters the PT-symmetry unbroken regime, the imaginary part of eigenfrequencies coalesces to zero, i.e., both pairs of supermodes will remain neutral. For the PT-symmetric sensor operating at the phase transition point or in its vicinity, the frequency splitting is enhanced due to the square-root topology of the complex energy eigensurface; meanwhile, the linewidth of the supermodes can be narrow due to the balanced gain and loss. As a result, the detection limit can be significantly improved.

5. DETECTION OF MULTIPLE NANOPARTICLES

For the case of detecting multiple nanoparticles, both the perturbation strength and position of each nanoparticle need to be considered. The strength of light scattering from cw mode to ccw mode is determined by the interference of scattered light from cw to ccw mode induced by each nanoparticle and vice versa. For two nanoparticles located at the resonator ^{[21]} where

Figure 6.(a) Illustration of the detection of two nanoparticles in a PT-symmetric WGM nanoparticle sensor. The two nanoparticles are placed within the mode volume of the resonator with gain. (b), (c) Variation in frequency splitting as a function of the angular position of the second nanoparticle when (b) the two nanoparticles are identical with the same perturbation strength, and (c) the two nanoparticles are different and hence have different perturbation strengths. The angular position of the first nanoparticle is fixed and set to be zero. Blue squares are numerical simulation results; red solid curves are theoretical predictions. The results are normalized by the frequency splitting induced by the first nanoparticle on a single resonator sensor.

The off-diagonal element ^{[21,41–43]}. For sufficiently small nanoparticles, ^{[21,41]}, where the nanoparticles are sufficiently large; thus,

Two different cases are considered: (i) two identical nanoparticles with the same perturbation strength, i.e.,

The above discussion can be directly extended to the case of three or more nanoparticles. The total perturbation Hamiltonian of ^{[21]} where ^{[43]}.

Figure 7.(a) Illustration of the detection of multiple nanoparticles in a PT-symmetric WGM nanoparticle sensor. Ten different nanoparticles (numbered gray circles) are randomly deposited within the mode volume of the resonator with gain one by one. (b) Numerical results of frequency splitting variation for 10 nanoparticles deposited on the PT-symmetric WGM sensor (blue lines and squares) and a single WGM sensor (red lines and circles). The dashed vertical lines are used as eye guides. Results for the single WGM sensor are obtained by removing the lossy resonator in the numerical simulation. Results are normalized by the frequency splitting induced by the first nanoparticle deposited on a single WGM sensor with a value of 73 MHz.

The first nanoparticle entering the mode volume of the PT-symmetric WGM sensor will be detected with the maximal enhancement due to the square-root topology and move the system away from the phase transition point. The second and the consecutive nanoparticles will start with a system that is not exactly at the phase transition point but away from it, depending on the amount of the perturbation induced by the nanoparticles. The sensitivity enhancement can still be obtained provided that the total perturbation of the nanoparticles remains small compared with the gain (or loss) strength in the PT-symmetric sensor ^{[20]}.

6. CONCLUSION

In summary, we study nanoparticle detection using PT-symmetric WGM sensors operating at the phase transition points, where significant enhancement of detection sensitivity is predicted for small perturbations. We have confirmed this enhancement and its dependence on the perturbation strength and the gain (loss) strength through both theoretical analysis and numerical simulations. Two mechanisms contribute to the improvement of detection limit using the PT-symmetric WGM sensor: the first is the square-root topology of complex energy eigensurfaces in the parameter space; the second is the narrow linewidth of the system due to the gain. In addition to the study of single nanoparticle detection, we have also derived a theoretical model and performed numerical simulations to study a multi-nanoparticle detection scenario where the nanoparticles fall onto the PT-symmetric sensor one by one. Our results clearly show that the sensing performance of a PT-symmetric sensor operating at the phase transition point surpasses that of conventional configurations.

It is worth noting that single passive or active WGM resonators have already been used for nanoparticle detection ^{[9,11,14,15]}, and PT-symmetric WGM resonators have also been realized in different configurations ^{[32–34]}. Each of the two resonators could be associated with a micro-heater for thermal tuning so that the frequency detuning between the two resonators can be tuned to zero ^{[44]}. The sensitivity enhancement achieved in practice could be affected by the stability of the system, environmental noise, etc. But, overall, the required experimental platforms are readily available to test the capability of PT-symmetric systems for nanoparticle detection.

Acknowledgment

**Acknowledgment**. J. Zhang is supported by the NSFC. X. Fan is supported by the NSF.

Weijian Chen, Jing Zhang, Bo Peng, Şahin Kaya Özdemir, Xudong Fan, Lan Yang. Parity-time-symmetric whispering-gallery mode nanoparticle sensor [Invited][J]. Photonics Research, 2018, 6(5): A23

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