In the past decade, we have witnessed the great advances of parity-time (PT) symmetry of non-Hermitian quantum physics[
Chinese Optics Letters, Volume. 19, Issue 7, 073601(2021)
Optical resonance in inhomogeneous parity-time symmetric systems
We show that inhomogeneous waveguides of slowly varied parity-time (PT) symmetry support localized optical resonances. The resonance is closely related to the formation of exceptional points separating exact and broken PT phases. Salient features of this kind of non-Hermitian resonance, including the formation of half-vortex flux and the discrete nature, are discussed. This investigation highlights the unprecedented uniqueness of field dynamics in non-Hermitian systems with many potential adaptive applications.
1. Introduction
In the past decade, we have witnessed the great advances of parity-time (PT) symmetry of non-Hermitian quantum physics[
From the non-Hermitian quantum theory, EPs are shown to satisfy the self-orthogonality condition[
In this Letter, we investigate the field dynamics in inhomogeneous PT symmetric systems. We pay attention to a two-parallel-WG configuration, where the PT phase varies continuously and slowly, and the region of the exact PT phase surrounded by two EPs can form a localized optical resonator. Many features of this localized non-Hermitian resonance are discussed, including its discrete nature and the formation of half-vortex flux inside. The difference of this non-Hermitian resonance from those in Hermitian systems[
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2. Theory
A schematic of two PT symmetric WGs under investigation is shown in Fig. 1. One of the WGs (WG1) has a positive imaginary component in the refractive index
Figure 1.Schematic of a two-WG inhomogeneous PT symmetric system in the x–y plane. When the PT phase changes very slowly, is it feasible to get a resonance in the exact PT region?
To modulate the PT phase in the system, we can refer to the standard non-Hermitian operator of
3. Simulation and Analysis
To investigate the field dynamics in the spatial-varying PT system, we would like to utilize the full-wave numerical simulation software of COMSOL Multiphysics 5.4. The width
The COMSOL simulation shows that the region of exact PT phase can form a resonator. Figure 2 displays the results at two different frequencies of 227 GHz and 215 GHz, respectively. We can see that the field is localized and resonant around the location of minimum distance
Figure 2.Distribution of field amplitude Ez at the frequencies of (a) 227 GHz and (b) 215 GHz, respectively.
To provide more information about the physical mechanism of the localized resonance, we calculate the eigenvalues
Here,
The solutions of Eq. (2) at 227 GHz are shown in Fig. 3(b), together with the distribution of field amplitude
Figure 3.(a) Distribution of field amplitude from the COMSOL simulation at 227 GHz. (b) The calculated real part of kPT in the structure. At the central of the structure, the PT phase is exact, while across EPs, the PT phase is broken.
The physical mechanism of the localized resonance must be attributed to the complex field dynamics of the non-Hermitian system. We believe that the main factor comes from the impedance mismatching between the broken PT phase (complex solution of
The distribution of energy flux
Figure 4.(a) Distribution of energy flux Sx inside the two WGs at 227 GHz. (b) Distribution of the two-dimensional energy flux in the xy plane. Green arrows (with a number of M = 8) at the lower blank space represent the direction of energy flux in the coupled WGs.
It is well known that resonance not only requires mirrors with high reflectivity, but also asks for coherent conditions on the accumulated phase after a round trip. As a result, resonance is usually discrete and can be characterized by a Q factor of
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Figure 5.(a) Maximum field Emax and (b) the effective resonator length Leff versus f.
The order of the discrete resonance can be defined by the number of half-vortices inside. For example, the resonance shown in Fig. 4(b) has an order M of eight. By analyzing the number of half-vortices at these discrete resonances, we find that M is different by a number of two between two adjacent resonances. The results are labeled in Fig. 5(a). Furthermore, from Fig. 2, we can see that the geometric size of the resonance varies with
Although the curve of Fig. 5(a) is very similar to the transmittance of an ordinary Fabry–Perot resonator, the non-Hermitian resonance discussed in this Letter is different from Fabry–Perot resonances because the effective resonator length
We perform COMSOL simulation and prove this feasibility, as shown in Fig. 6. Compared with Fig. 2, we can see that the field distribution shown in Fig. 6 is no longer symmetric with respect to the center of the WGs because the source is placed only at the left side of the PT symmetric region. Nevertheless, Fig. 6 proves unambiguously that the non-Hermitian resonance can be excited by outside excitation. Furthermore, the excited field at 227 GHz is stronger than that at 215 GHz, which is in good agreement with the results of Fig. 2, because the former one corresponds to the discrete resonance of
Figure 6.Source being placed outside the PT symmetric region can still excite the non-Hermitian resonances at (a) 227 GHz and (b) 215 GHz.
4. Discussion
Above we provide adequate evidence on the existence of localized resonance in an inhomogeneous PT system. We also study the scenarios in some other configurations, including two straight WGs with spatially varied gain/loss, and two curved WGs that are symmetric with respect to the
With the above analysis, we can see that the non-Hermitian resonance is different from other kinds of ordinary resonances[
For future experiments, we can firstly fabricate two geometrically identical WGs (such as the cores of optical fibers) with proper gain and loss and then artificially displace them. As for potential applications, it is worth paying attention to the adaptive applications related to the self-determined geometric feature of the resonance. At different operational frequencies, the localized resonances could choose their own spatial regions (including the geometric size) in the inhomogeneous PT system. Various linear and nonlinear applications can be imaged. Future investigation could pay attention to revealing the unsolved issues of the non-Hermitian resonance, e.g., how to model the field dynamics by using the transfer matrix method and how to explain the phenomena in the broken PT region.
Before ending this Letter, we would like to emphasize other reported resonant features of PT symmetric systems. In fact, spontaneous emission at EPs and the associated density of states (DOS) have been discussed by various groups[
5. Conclusion
In summary, paying attention to coupled WGs with slowly varying PT symmetry, we show that localized resonance can be achieved in the region of exact PT phase. We show that the non-Hermitian resonance is discrete in frequency, and the order of resonance can be defined by the number of half-vortices inside that run from the gain WG to the loss WG. The physical mechanism is briefly discussed. This investigation highlights the unprecedented uniqueness of field dynamics in non-Hermitian systems with many potential adaptive applications[
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Linshan Sun, Bo Zhao, Jiaqi Yuan, Yanrong Zhang, Ming Kang, Jing Chen, "Optical resonance in inhomogeneous parity-time symmetric systems," Chin. Opt. Lett. 19, 073601 (2021)
Category: Nanophotonics, Metamaterials, and Plasmonics
Received: Oct. 29, 2020
Accepted: Dec. 19, 2020
Posted: Dec. 21, 2020
Published Online: Mar. 30, 2021
The Author Email: Jing Chen (jchen4@nankai.edu.cn)