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- Photonics Research
- Vol. 6, Issue 4, A1 (2018)

Abstract

Keywords

1. INTRODUCTION

The discovery of parity-time symmetric (PT-symmetric) optics has prompted a surge of research activities for engineering synthetic materials with new properties and functionalities. PT-symmetric systems are non-Hermitian, but can exhibit entirely real spectra as long as they respect the conditions of PT symmetry ^{[1]}. PT-symmetric photonic systems ^{[2–5]} that rely on engineering the imaginary part of the refractive index to create balanced gain and loss regions, have many exotic features, including power oscillations, unidirectional invisibility, coherent perfect absorption, nonreciprocal light propagation, double refraction, and various intriguing nonlinear effects ^{[6–20]}. The research into PT-symmetric structures was primarily focused on the nondispersive materials, whereas the systems with dispersion have only lately attracted growing interest ^{[21,22]}. The non-Hermitian optical structures with dispersion were considered in Ref. ^{[22]}. It was shown that such structures could be PT-symmetric only for a discrete set of real frequencies. For gain and loss layers with identical parameters, the PT-symmetry condition can be fulfilled at the emission frequency.

It is necessary to note that in practice it is difficult to design structures with perfectly balanced gain and loss. The aim of this paper is to explore the fundamental properties of one-dimensional (1D) layered non-Hermitian systems (with or without PT-symmetry) by taking into account the dispersion for gain and loss media. We derive the characteristic frequencies for practical realization of PT-symmetry, and we investigate the role of material dispersion in non-Hermitian scattering systems, where the spatial distributions of gain and loss are not subject to any spatial symmetry requirements. We point out the existence of exceptional points (EPs) associated with degenerate scattering eigenstates.

2. THEORETICAL ANALYSIS

Let us consider the bilayer composed of two slabs of thicknesses,

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Figure 1.Geometry of the problem.

The gain and loss media are described by the Lorentzian permittivity. The dielectric permittivity components for the layer of type ^{[22]}, where

First, let us consider a PT-symmetric optical system. In this case, we assume that two slabs have an identical thickness,

We use positive parameters ^{[22]}. The last equation in Eq. (

The dependence of the absorption coefficient on the frequency is presented in Fig. ^{[23,24]}:

Figure 2.Absorption coefficient

Substitution of Eq. (

From Eq. (

Since the layers are assumed to be homogeneous in the ^{[25]}. The reflection and transmission coefficients are readily obtained by satisfying the continuity conditions for the tangential field components at the stack interfaces. We note that when the gain layer thickness exceeds some critical value, lasing begins to develop, and the characteristics of the system must be described in the nonlinear approximation using a field-dependent dielectric constant ^{[25]}. In this case, the solution of the problem cannot be obtained by the aforementioned Fresnel method. It can be shown that for frequency values near the emission frequency

Wave scattering in the proposed system is modeled using the corresponding optical scattering matrix (

The eigenvalues of the ^{[6]}, 1D PT-symmetric structures can undergo spontaneous symmetry-breaking transitions in terms of the eigenvalues and eigenvectors of the corresponding

3. NUMERICAL RESULTS

The reflectance for both left and right incidence and the transmittance of TM waves incident at angle

Figure 3.(a) Reflectance [

Inspired by the so-called, “loss-induced transparency” phenomena ^{[4]}, we examine now the increase of the transmittance in a loss-dominated system [a system for which

Figure 4.(a) Reflectance [

To further investigate this effect of amplification in a loss-dominated multilayer heterostructure, the transmittance coefficient

Figure 5.(a) Geometry of non-Hermitian periodic stack. (b), (c) Geometries of non-Hermitian random stacks. (d), (e) Transmittance of TM wave through the stacks incident at

As mentioned before, in PT-symmetric systems the eigenvalues of the scattering matrix identify the exceptional points at which the symmetry-breaking transitions occur. To illustrate the effect of the gain/absorption coefficient, the dependence of

Figure 6.Eigenvalues of the

In the general case of a non-Hermitian system, the analysis of the eigenvalue spectrum dependence on the parameter

4. CONCLUSIONS

In summary, the basic scattering properties of dispersive non-Hermitian systems, where the spatial distributions of gain and loss are not subject to any spatial symmetry requirements, are systematically examined. We demonstrate that the proper combination of the parameters of constitutive materials of the system helps to implement the PT symmetry at desirable frequencies of incident waves. It is shown that the dispersive system with nonidentical parameters of materials with gain and loss could be PT-symmetric maximum for two real frequencies. One of our main results is that, for a frequency range close to the emission frequency of the gain layer by changing the parameters of the layers and incident waves and composition of the stack, we can have amplification of a transmitted wave in an on-average lossy non-Hermitian structure. The analysis of the eigenvalue spectrum of non-PT symmetric systems demonstrates the existence of EPs associated with degenerate scattering eigenstates.

References

[7] S. Longhi. PT-symmetric laser absorber**. Phys. Rev. A, 82, 031801(2010)**.

[24] A. E. Siegman**. Lasers(1986)**.

O. V. Shramkova, K. G. Makris, D. N. Christodoulides, G. P. Tsironis. Dispersive non-Hermitian optical heterostructures[J]. Photonics Research, 2018, 6(4): A1

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