Coherent Perfect Absorption Laser Points in One-Dimensional Anti-Parity–Time-Symmetric Photonic Crystals

Abstract

We investigate the coherent perfect absorption laser points (CPA-LPs) in anti-parity–time-symmetric photonic crystals. CPA-LPs, which correspond to the poles of reflection and transmission, can be found in the parameter space composed of gain–loss factor and angular frequency. Discrete exceptional points (EPs) split as the gain–loss factor increases. The CPA-LPs sandwiched between the EPs are proved to be defective modes. The localization of light field and the bulk effect of gain/loss in materials induce a sharp change in phase of the reflection coefficient near the CPA-LPs. Consequently, a large spatial Goos–Hänchen shift, which is proportional to the slope of phase, can be achieved around the CPA-LPs. The study may find great applications in highly sensitive sensors.

1. Introduction

In photonic crystals (PCs), one can manipulate photons in the same way as electrons in semiconductors. By modulating the refractive indices of dielectric materials, it forms periodic distribution in space and obtains special optical properties. PCs, which are microstructures of wavelength-scale, have photonic bandgaps, that is, light with frequency within the bandgaps cannot pass through the PCs [1,2,3,4]. There are natural photonic crystals, such as opals and butterfly wings. What is more, we can achieve specific optical functions by synthesizing PCs. PCs can also realize localization of light field [5,6,7,8], unidirectional transmission [9,10], directional cloaking [11], solitons [12,13,14], and other functions [15,16,17,18,19]. Otherwise, PCs are used to control photon flows, which can realize the miniaturization and integration of photonic devices [20,21,22,23,24].

It is often necessary to modulate the refractive index of material in the study on PCs. In the past, it mainly focused on the modulation of the real part of the refractive index, while in recent years, the regulation of the imaginary part of the refractive index has gradually attracted people’s attention. The imaginary part of the refractive index represents the loss or gain of material. Loss seems to be an unfavorable factor, which causes energy attenuation, so it should be minimized in device design. People usually reduce the energy loss during the propagation of light through the mode matching in appropriate materials and structures, or compensate the loss of devices through gain. According to quantum mechanics, a system is non-Hermitian if there is loss or gain. By using the similarity between the paraxial equation of electromagnetic field propagation under the slowly varying envelope approximation [25] and the Schrödinger equation of quantum mechanics, the concept of non-Hermite can be extended to the regime of optics [26,27]. Relevant studies have contributed to the proposal of non-Hermitian photonics, which has become a hot topic in optics in recent years. In a non-Hermitian optical system, by simultaneously modulating the real part and the imaginary part of the refractive index of the material, single-mode laser [28], loss-induced transparency [29], and optical field localization [30] can be achieved.

The parity–time (PT)-symmetric Hamiltonian can have real spectra, which extends quantum mechanics into a new research field [31,32]. PT-symmetric Hermiticity is a new concept and PT symmetry in electronic systems has been widely concerned and aroused a lot of discussion [33]. PT symmetry is also of great theoretical and applied value in optics and other related regimes [34,35,36,37,38,39,40,41]. The use of PT potentials in optics has been recently reviewed by Longhi [42], which provides a good background for non-Hermitian properties. The pioneering work to use gain and loss in optics to implement PT-symmetric Hamiltonian was Ruschhaupt et al. [43]. The work presented belongs to a set of explorations of non-Hermitian symmetries beyond PT, a vast and promising field that is just beginning to develop. It will surely provide many important results and applications [44]. The gain in the photon system can be obtained by nonlinear two-wave mixing [45], or Ge/Cr doped fiber [46], while the loss can be achieved by acoustic modulator [47]. Thus, optics has the natural advantage of being used to explore all sorts of non-Hermite predictions that used to be verified difficult in quantum mechanics. However, studies on non-Hermitian photonics have shown that loss and gain, satisfied with PT symmetry, provide a new degree of freedom for the system, which can effectively control photons and generate new optical effects, such as unidirectional reflectionless resonance, which can be used to prepare chip-level semiconductor lasers [48].

Anti-PT-symmetric systems, in which the refractive index of material obeys n(x) = −n*(−x), have been proposed and attracted much attention [49,50,51]. Many interesting phenomena are found in anti-PT-symmetric systems, such as flat broadband light transport [50], and chiral mode conversion [52]. Actually, anti-PT symmetry belongs to a more fundamental symmetry called charge-conjugation symmetry [53,54], which consists of two sublattices. The study on asymmetric coupled systems shows that the topological bound states can be generated via a degeneracy induced by charge-conjugation symmetry [53,55]. Moreover, the topological bound modes can be selectively pumped due to their special energy spectrum [54,56], which can be further utilized to enhance the lasing performance with improved single-mode property, higher slope efficiencies, and robustness against perturbation [57,58,59].

In this work, we investigate the coherent perfect absorption laser points (CPA-LPs) (CPA laser states for which such systems can behave simultaneously as a laser oscillator, emitting outgoing coherent waves, and as a CPA (coherent perfect absorber), absorbing incoming coherent waves) [60,61] in the anti-PT-symmetric PCs. We show that CPA-LPs arise in the parameter space composed of gain–loss factor, and angular frequency. The CPA-laser states, which are sandwiched in splitting exceptional points (EPs), are confirmed to be substantially defective modes. We also discuss the crossing and anti-crossing properties of the eigenvalues of scattering matrix around the CPA-LPs. We further explore the phase of reflection coefficient, and demonstrate the singularity of phase around the CPA-LPs. In the end, we illustrate the application of the singular phase near the CPA-LPs, and induce large spatial Goos–Hänchen (GH) shift, which can be utilized for highly sensitive sensing.

2. One-Dimensional Anti-PT-Symmetric PCs

Figure 1 shows the schematic of the one-dimensional anti-PT-symmetric PCs. The dielectrics A and B are arranged alternatively to form the periodic structure comprising periodic unit cells (thickness: 2d). The structure can also be viewed as a Bragg grating, denoted by (AB)^N, where N is defined as the Bragg periodic number. The refraction index in each unit cell is real and given by n(z) = n_a as 0 < z < d and n(z) = n_b as d < z < 2d. Using the transfer matrix method (TMM), we can relate the electromagnetic fields at the two interfaces of a homogeneous medium by [62]

$$\left(\begin{array}{c}{E}_l\\ H_l\end{array}\right)=\left(\begin{array}{cc}\cos {\phi }_l& -\frac{i}{{\eta }_l}\sin {\phi }_l\\ -i{\eta }_l\sin {\phi }_l& \cos {\phi }_l\end{array}\right)\left(\begin{array}{c}{E}_{l+1}\\ H_{l+1}\end{array}\right),$$

(1)

where φ_l = 2πd_l(ε_l − sin²θ)^1/2/λ, d_l is the thickness of the lth layer transmission dielectric, θ is the incident angle, and E_l, H_l, E_l+1, and H_l+1 are the electromagnetic field intensity at two interfaces of homogeneous dielectric, respectively. For transverse electric (TE) waves, η_l = (ε₀/μ₀)^1/2(ε_l − sin²θ)^1/2, and for transverse magnetic (TM) waves, η_l = ε_l (ε₀/μ₀)^1/2/(ε_l − sin²θ)^1/2. The structure’s total transfer matrix is M = ΠM_l = [m₁₁, m₁₂, m₂₁, m₂₂], where M_l is the corresponding transfer matrix of the lth layer dielectric. Then, the transmission and reflection coefficients can be obtained by the transfer matrix elements

$$t=\frac{2{\eta }_1}{m_{11}{\eta }_1+m_{12}{\eta }_1{\eta }_{2N+1}+m_{21}+m_{22}{\eta }_1},$$

(2)

$$r=\frac{m_{11}{\eta }_1+m_{12}{\eta }_1{\eta }_{2N+1}-m_{21}-m_{22}{\eta }_{2N+1}}{m_{11}{\eta }_1+m_{12}{\eta }_1{\eta }_{2N+1}+m_{21}+m_{22}{\eta }_{2N+1}},$$

(3)

where η_2N+1 = η₁ = η₀ is the vacuum resistivity of light at the input and output ports, respectively. The transmittance T is identified as T = tt* = I_t/I_i, where I_t and I_i are the incident and transmitted intensities, respectively.

3. CPA-LPs and EPs Splitting

As an example, we consider a one-dimensional anti-PT-symmetric PC consisting of 20 periodic unit cells, viz. N = 20, embedded in a homogeneous background, such as in air, shown in Figure 1. The parameters are d = 125 μm, n = 1, ∆n = 0.5, θ = 5°. In our numerical model, we have neglected these material nonlinearity and dispersion as a simplification. Now, we calculate the transmission and reflection spectra as the TM wave obliquely impinges upon the anti-PT-symmetric PCs from the left and right sides. Figure 2a plots the transmittance spectrum, where we observe two resonant states associated with sharp transmission peaks T = 3.01 × 10³ and 1.27 × 10² in the curve marked by black stars, respectively locating at ω = 2.3 × 10¹³, and 3.8 × 10¹³ rad/s.

Figure 2b gives the reflectance spectrum for light incident from the left. It shows that there are also two maxima in the curve of R₁. The positions of the peaks are identical in the transmittance and reflectance spectra. This phenomenon demonstrates that two CPA-laser states may exist around the two maxima. The values of the nadirs in R₁ equal nearly to zero, considering the limit of calculation accuracy. The reflection coefficient is represented by r = |r|exp(iφ), where φ is the phase change of reflection beam. The phase of the reflection coefficient varies with the angular frequency as shown in Figure 2c. There are π jumps in phase at the nadirs of R₁. When the reflectivity is zero, the phase uncertainty of the reflection coefficient is caused. It is notable that, around the peaks of the transmittance and reflectance, the reflection coefficient phase changes dramatically with the change of angular frequency. The dramatic change in phase may induce novel physical effects. For light incident from the right, Figure 2d gives the reflectance spectrum T_2, which is not identical with the spectrum R₁. Therefore, the reflection is anisotropic for light incident from the left and right sides. However, the corresponding frequencies of the two maxima in R₂ are identical with the values in R₁. The phase of the reflection coefficient is shown in Figure 2e. The two reflection coefficient phase curves do not seem to coincide superficially. However, these two curves are really the same if we subtract these π jumps in phase at the zero points in the curves of R₁ and R₂.

In the parameter space composed of the gain–loss factor and angular frequency, Figure 3a gives the transmittance of light as light incident from the left or right. We can see that there are some poles in the parameter space such as at CPA-LP₁ (q_LP1 = 1.975, ω_LP1 = 2.3 × 10¹³ rad/s), and so on. The extreme values of the poles approve them to be CPA-laser states, so we call the poles in the parameter space CPA-LPs. The CPA-laser state is a defective mode or resonant state. The output energy of the laser state is provided by the gain in the dielectrics. Figure 3b plots the reflectance of light incident from the left. The maxima of reflectance coincide with the maxima of transmittance in the parameter space, which identifies that the poles of reflection and transmission correspond to the CPA-LPs as well. Otherwise, we find some discrete EPs denoted by EPs₁ in parameter space. The zero point of reflection is confirmed to be EP based on the scattering matrix S = [t, r₁; r₂, t] [63]. The eigenvalues and eigenvectors of S-matrix are given by λ_1,2 = t ± (r₁r₂)^1/2, and (r₁^1/2, ± r₂^1/2), respectively. The two eigenvectors (λ_1,2) coalesce at (r₁r₂)^1/2 = 0 signifying the EPs of the eigenvalues—or, to be more precise, the EPs originate from r₁ or r₂ = 0.

As light impinges upon the structure from the right, Figure 3c provides the reflectance. One can see that there are also some CPA-LPs and EPs₂ in the parameter space. The position of the CPA-LPs in parameter space does not change for light incident from the left or right. But the EPs are located at the left side of the CPA-LPs. To demonstrate the relation between the CPA-LPs and EPs, we give the phase transition shown in Figure 3d as the gain–loss factor increases. The EPs₁ and EPs₂ split, and the interval of splitting EPs increases with the increase of the gain-loss factor. The CPA-LPs, which are proved to be the defective modes, are sandwiched between the EPs₁ and EPs₂.

In the parameter space composed of angular frequency ω and gain–loss factor q, one of the CPA-LP emerge as the gain–loss factor approaches q_LP1 = 1.975. To verify the CPA-LPs in the anti-PT-symmetric PCs, Figure 4a,b depicts the real part Re(λ_1,2) and imaginary part Im(λ_1,2) of the eigenvalues λ_1,2 of S-matrix. It shows that the eigenvalues degenerate at the CPA-LPs as the angular frequency changes. It is the unique property of CPA-LPs usually utilized for confirming CPA-LP in non-Hermitian systems. In the vicinity of the CPA-LP, the curves of the imaginary part are crossing, while the curves of the real part are anti-crossing [52]. The CPA-LPs are connected with the surface mode localized at the gain–loss boundary. Figure 4c gives the electric field intensity distribution of the CPA-LP₁ at (q_LP1 = 1.975, ω_LP1 = 2.3 × 10¹³ rad/s). It demonstrates that most of the light field power is located in the center layers and is strongly restrained at the interfaces of dielectrics. Along the z-axis, the field exponentially decays as it departs from the center.

As light is incident from the left, Figure 5a illustrates the reflection coefficient phase, φ_r1, in parametric space, where the angular frequency and gain–loss gain can be modified in the vicinity of the CPA-LP₁. The phase dislocates around the CPA-LP₁, which is unique for the CPA-LPs in non-Hermitian systems [64]. The singularity in phase indicates that the phase should abruptly change once the parameters undergo a little variation.

Figure 5b plots the coefficient phase, φ_r1, versus the angular frequency for several different gain–loss factors. The sampling gain–loss factors are q = 1.875, 1.975, and 2.075, which distribute around the q_LP1, respectively. There is a hop point in the phase curve as q = 1.875, and the phase difference is −2π. Neglecting the meaningless phase difference at the hop point, the curve is actually continuous with a negative slope. The phase change can approximately reach π in a phase variation range around q_LP1 as the gain–loss factor q < q_LP1. The phase experiences a π abrupt jump at ω_LP1 as q = q_LP1. Comparably, the phase change can nearly reach −π around ω_LP1 with a positive slope as q > q_LP1.

For light incident from the right, as depicted in Figure 5c, there is also a phase dislocation in φ_r2 existing in the vicinity of the CPA-LP₁. Contrary to φ_r1, the position of the phase hop points in the parametric space does not change. Figure 5d plots the reflection coefficient phase φ_r2, which varies with the angular frequency for several different gain–loss factors around the CPA-LP₁. An abrupt phase change also occurs as q = q_LP1. A π phase roughly falls off in a variation range of gain–loss factor about q_LP1 as q < q_LP1, but a π phase climbs as q > q_LP1. The localization of light field and the bulk effect of gain/loss in material may induce the sharp change in phase of the reflection coefficient.

4. Large Spatial GH Shift Around CPA-LPs

GH shift is the lateral shift and angular deviation of the reflected beam, relative to the position predicted by geometrical optics [65]. How to design, and find, appropriate structures or materials to increase GH shift is the key problem to be considered at present. With reference to the lattice structure of solid crystals, people put forward the concept of PCs, which provide a new approach to solve the above problem. At the bandgap edge of the PCs, there is large GH shift [66], and the optical field localization of the defect PCs can also induce large GH shift [62].

The spatial GH shift relates to the phase of the reflection coefficient, and is proportional to the slope of reflection phase in accordance with Δ = –dφ_{r1, r2}/dk_y [65], where k_y = ksinθ and k = λ/2π. For another example, the CPA-LPs are singular points in the phase of the reflection coefficient, which is to say that the phase dislocates at the CPA-LPs. Therefore, a large spatial GH shift may be induced around the CPA-LPs. Figure 6a plots the phase of the reflection coefficient in the parameter space. The focusing region is separated into two parts labeled I and II, respectively. The spatial GH shifts are negative in part I, while it is positive in part II. Except for the CPA-LP₁, the spatial GH shifts approximate to zero along the dotted line. The negative and positive polarities of GH shift convert at the CPA-LP₁, approving the singularity of the GH shift.

Figure 6b plots the spatial GH shift versus angular frequency for the gain–loss factor q = 1.875. One can see that the GH shift is positive, and a peak appears around the CPA-LP₁. Figure 6c gives the GH shift for q = 1.975. It shows that there is also a maximum in the curve, but the peak becomes sharper and narrower when the gain–loss factor is closer to q_LP1. However, as q = 2.075, the spatial GH shift is negative, and there is a valley in the curve shown in Figure 6d. For the three special gain–loss factors, the corresponding maxima of GH shift are 3.4 × 10³, 1.13 × 10⁵, and −3.25 × 10³ times of the incident wavelength around ω_LP1. Here, we show that a large GH shift in a bulk multilayer structure and the imaginary part of refractive index play an important role in realizing CPA-LPs of the system. The spatial GH shift is highly sensitive to the refractive index of dielectric and angular frequency of incident light, as shown in Figure 6b–d around the CPA-LP. However, the applications in sense at the exceptional point appear to be rather challenging, such as the effect of fluctuations and noise [67,68,69].

5. Conclusions

In conclusion, we have studied the CPA-LPs in one-dimensional anti-PT-symmetric PCs. By varying the gain–loss factor and angular frequency, CPA-LPs and discrete EPs can be found in the parameter space. The CPA-LPs which are defective modes are located between the splitting EPs. The light field of the CPA-LPs are constrained at the interfaces of dielectrics, and mainly distributed at the center of structure. The localization of light field and gain/loss in material generate the singularity of reflection coefficient phase around the CPA-LPs, at which point the phase dislocates. Therefore, a large spatial GH shift is induced near the CPA-LPs, and the maximum GH shift approaches the magnitude of 10⁵λ. The study paves the way for development in highly sensitive sensors.