The subject of parity-time (PT) symmetry is more alive than ever after the implementation of PT-related notions in optics, owing to the similarity between the paraxial wave equation and the Schrödinger equation.^[1,2] The PT symmetry is a notion initially in quantum mechanics, which describes the non-Hermitian Hamiltonians ( that respect . Since PT-symmetric potentials can be effectively realized in optics through delicately arranging gain/loss regions, tremendous theoretical and experimental studies have been conducted in such complex systems, enriching our understanding of wave dynamics in PT optical potentials.^[3–12] For example, spontaneous PT symmetry breaking was firstly reported in optical coupled systems.^[2] Unidirectional invisibility was proposed and verified at exceptional points in PT lattices.^[6–8,13] PT lasing and absorption were observed in the PT symmetry breaking phase.^{[4, 9, 10]} Those interesting phenomena were demonstrated to be closely related to singularity points in eigenvalue spectra of the system scattering matrix or Hamiltonian. Recently, researchers found out that singularity dynamics in finite PT systems can be much affected by boundary effects. A representative example is that unidirectional invisibility in PT lattices will break down as the number of period increases.^[14,15] Therefore, it shows the very possibility to tailor singularity effects in the confined PT systems through boundaries.

In a previous work,^[15] we focused only on the Bragg condition in PT-symmetric lattices, and found out that the light field is one-way localized in the lattices with certain periods at the Bragg condition. In this paper, we explore the whole spectrum and obtain varied overtones of one-way localized Fabry–Pérot (FP) resonances. As a very unique thing, we observe, for the first time, that overtones in PT cavities can be classified into a blue shifted branch and a red shifted branch with respect to the fundamental frequency. For the blue shifted and the red shifted overtones of the same absolute order, despite very different wavelengths, they have almost the same interference patterns. Since the FP resonances in PT cavities are featured with unidirectional field localizations, distinctive interference patterns emerge only for the light propagating in one direction but not for the reversed direction. Our finding may provide a unique route towards achieving directional responses to photons.

A PT optical structure has a complex refractive index distribution respecting the form .^[2] In this section, we design a grating structure with a period L as shown in Fig. 1. The periodic unit-cell has four same-size sub-sections (length ) with refractive indices being , , , and , respectively. The incident light is TE polarized. The previous work of Ref. [15] has detailedly demonstrated that when and is a positive integer, the PT lattice having unit-cells is always completely reflectionless for light propagating in opposite directions under the Bragg condition , with the light field one-way localized. From the expression , we conclude that a weak index modulation will lead to a large number of unit-cells in the desired PT lattice. In the following, we choose and as an example, where is a big number only for the sake of avoiding a very large finite element model.^[15] We emphasize that can be chosen with other more practical values (<0.002) and the results to be demonstrated in the following will remain the same.

Fig. 1.

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	Fig. 1. (color online) Unidirectional field localization may occur at Fabry–Pérot resonances in a well-designed PT optical resonator.

We start from the transfer matrix method, where the characteristic matrix for one unit-cell is given by^[16]

Fig. 2.

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	Fig. 2. (color online) Calculations of (a) , (b) , (c) , and (d) . Here the solid blue lines and dashed red lines show the real part and imaginary part, respectively. The black dots mark the positions at which and .

The calculations of , , and in Fig. 3(a) further verify that the aforementioned EPs indeed correspond to FP resonances due to unitary transmission as well as bidirectional transparency, except for a special EP as marked by the dashed arrow in the vicinity of 1.2 μm. At such a peculiar point, the system still has a unitary transmission (T = 1), but the reflection is unidirectionally vanished (, ). As inferred later from the FP resonance patterns in Fig. 4(a), we can find that the fundamental resonance (marked by the order n = 0 in Fig. 3(a) and the black dot in Fig. 3(b)) occurs at 1.2 μm with only one antinode in the PT resonator.

Fig. 3.

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	Fig. 3. (color online) Scattering properties of the PT resonator. (a) Calculations of T, , and . (b) Spectral positions of Fabry–Pérot resonances (marked by the numbers in panel (a)]. At the resonances (n = 0, , , , ), the PT resonator is bidirectionally transparent.

Fig. 4.

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Fig. 4. (color online) Field distributions at FP resonances in the PT resonator. (a) Electric field distributions for the light propagating from the left and the right through the PT resonator at FP resonances (n = 0, , , ), where the field is observed to be unidirectionally localized. The blue and red curves of field distributions in panels (b), (c), and (d) present the spatial modal degeneracy for , , and , respectively.

However, in stark contrast to the FP resonances in the lossless optical resonator (e.g., a high-refractive-index layer), the fundamental frequency in the PT resonator is not the lowest one, since the overtones have a blue shifted branch (blue dots) as well as a red shifted branch (red dots), as shown in Fig. 3(b). Here we set the orders of the blue shifted overtones to be n = 1, 2, 3, …, while the orders of the red shifted overtones are n = −1, −2, −3, …. In Fig. 3(b), we find , , , , , , and . The calculations clearly show that the frequency deviations of the same absolute order overtones from the fundamental one ( are the same.

As exemplified in Fig. 4(a), we observe an interesting property that the localization of light only occurs for the backward propagation at the FP resonances in the PT resonator due to the designed anti-symmetric distribution of the imaginary part of the refractive index. In Figs. 4(b)–4(d), we also notice that for the blue shifted and red shifted overtones of the same absolute order (, , ), they share similar interference patterns, even though they have very different wavelengths. Here the blue curves in Figs. 4(b)–4(d) describe the electric field distributions of the blue-shifted overtones at n = 1, 2, 3, while the red curves show the electric field distributions of the red-shifted overtones at , −2, −3. Very different from the FP resonances in the lossless optical resonator, the FP resonances in the designed PT resonator exhibit beat-like electric field distributions, where the fast oscillation is caused by the oscillated PT modulation.

Unidirectional field localization could be useful in many aspects, such as directional sensing,^[19,20] one-way nonlinearity, and non-blind invisibility. In order to evaluate the degree of unidirectional field localization, we calculate the contrast ratio for different orders of FP resonances in Fig. 5, where and denote the maximum intensities for the light propagating from the right and the left through the PT resonator, respectively. The results show that the contrast ratio reaches maximum (η = 629.5) at the fundamental resonance (n = 0), indicating a remarkable intensity difference up to three orders of magnitude. For the overtones of the same absolute order, the contrast ratio of the red shifted overtone is always larger than that of the blue shifted one. For example, at and 1, equals to 178 and 107.8, respectively. This can be explained by the existence of weak field enhancement for the forward propagation at the blue shifted overtones, enabling of the blue shifted overtones to be larger than that of the red shifted ones.

Fig. 5.

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	Fig. 5. (color online) Contrast ratio for different orders of resonances. and are the maximum intensities for the light propagating from the right and the left through the PT resonator at FP resonances (n = 0, , , , …).

Asymmetric field localization still exists when the PT symmetry is breaking. In this case, the system will not be transparent due to reflections and non-unitary transmission (, ). The reported FP resonances with bidirectional transparency, spatial modal degeneracy, and unidirectional field localization are unique in the PT systems. As aforementioned, for the FP resonances in lossless systems, the fundamental frequency should be the lowest one. Also under the restraint of T-reversal symmetry, the FP resonances in lossless systems must have strong field localization for the light propagating in both directions. In open systems not obeying the PT symmetry, the eigenvalues of -matrix at EPs usually have imaginary parts, indicating that such non-Hermitian systems operate in either gained or lossy conditions and cannot be regarded as conservative.

We have studied an unusual phenomenon of one-way localized FP resonances with spatial modal degeneracy. The FP resonances have both blue shifted and red shifted overtones, and the spatial modal degeneracy is observed for the overtones of the same absolute order. We reveal that one-way field localization occurs at the EP, where the PT system is bidirectionally transparent for the outside observer but inside the light field is localized only for the backward propagation.