Exceptional Points in Non-Hermitian Photonic Crystals Incorporated with a Defect

We explored exceptional points (EPs) in one dimensional non-Hermitian photonic crystals incorporated with a defect. The defect was asymmetric with respect to the center. Two EPs could be derived by modulating the normalized frequency and the gain-loss coefficient of defect. The reflection coefficient complex phase changed dramatically around EPs, and the change in complex phase was π at EPs. The electric field of EPs was mainly restricted to the defect, which can induce a giant Goos–Hänchen (GH) shift. Moreover, we found a coherent perfect absorption-laser point (CPA-LP) in the structure. A giant GH shift also existed around the CPA-LP. The study may have found applications in highly sensitive sensors.


Introduction
Non-Hermite, which stems from quantum mechanics [1], has promptly expanded to optics [2][3][4][5], acoustics [6], and electronics [7]. Systems including gain or loss are non-Hermitian [8][9][10][11][12]. There is an energy exchange between non-Hermitian systems and the external environment. Compared with the quantum structures, gain and loss in optics are relatively easy to realize [13][14][15], so non-Hermitian optics has become an intriguing research subject. The eigenvalues of Hamiltonian are real in Hermitian systems, while they are generally complex in non-Hermitian systems. However, as the Hamiltonian in optical systems meets parity-time (PT) symmetry, that is, the refractive indices of materials are satisfying n(r) = n*(−r) [16,17], the eigenvalues can be real. The PT-symmetry will spontaneously break if the gain and loss surpass some critical value, which is defined as the exceptional point (EP) [8,13]. The eigenvalues and eigenvectors of Hamiltonian degenerate in non-Hermitian systems at the EPs, around which many fascinating optical properties may be induced, consist of sharp changes in the complex phase [18], unidirectional invisibility [19,20], and topological boundary states [21].
EPs and coherent perfect absorption-laser points (CPA-LPs) exist in PT-symmetric PCs [46]; however, it is more difficult to make complex dielectrics obey PT symmetry experimentally. Here, we embedded a defect in PCs. The refractive index of the defect is complex and the imaginary part of the refractive index (denoted by a gain-loss coefficient σ) can be tuned. We searched EPs and CPA-LPs in the parameter space composed of the frequency and gain-loss coefficient. We then explored the complex phase changes of reflection and transmission coefficients around EPs and CPA-LPs. Next, we investigated the degeneration of the eigenvalues of Hamiltonian at EPs and gave the distribution of the electric field. Finally, we simulated the Goos-Hänchen (GH) shift of reflected and transmitted beams around EPs and CPA-LPs.

Non-Hermitian Photonic Crystals
We constructed the non-Hermitian PCs, which were composed of two Bragg gratings and a defect, as shown in Figure 1. Two different dielectrics, A and B, arrayed alternatively to form two Bragg gratings. The dielectrics A and B were MgF 2 and ZnS, respectively. The defect containing dielectrics C and D was embedded in the Bragg gratings. The dielectrics C and D could be obtained by doping SiO 2 with high and low impurities of refractive indices, respectively. We denoted the refractive indices of dielectrics by n a , n b , n c , and n d , respectively. The thickness of each layer was set as a quarter of the optical wavelength, viz, L a,b,c,d = λ 0 /4n a,b,c,d , where λ 0 = 100 µm is the midgap wavelength of the defect PCs. The refractive indices of materials were n a = 1.38, n b = 2.35, n c = 1.4 + 0.01σi, and n d = 1.6 + 0.01σi, respectively, where σ is the gain-loss coefficient. The positive gain-loss coefficient represented loss and the negative gain-loss coefficient represented gain. The gain in the defect could be realized by doping Ge/Cr or nonlinear two-wave mixing [47,48], and the loss of material could be achieved by acoustic modulators [49]. The designed PCs could also be abbreviated to (AB) N CD(BA) N , where N is the periodic number of Bragg gratings, and here it shows N = 5. In simulations, the incident wavelength was in the range of terahertz band and the transmission matrix method (TMM) was used to derive the reflection coefficient r and transmission coefficient t [18]. Subsequently, the reflectance of light and transmittance can be denoted by R = rr* and T = tt*, respectively. The reflection coefficient is generally complex and can be expressed as r/t = |r/t|exp(iϕ r/t ), where ϕ r/t is the complex phase of the reflection/transmission coefficient. As a transverse magnetic (TM) polarization wave obliquely impinged upon the PC with the incident angle θ, the lateral GH shift of reflected/transmitted light beam was proportional to the slope of the reflection/transmission coefficient complex phase, with respect to the incident angle, that is D ry/ty = −λdϕ r/t /2πdθ [18], where λ is the incident wavelength.
We set the incident light as a TM wave, and the incident angle was θ = 20 • . When light obliquely impinged upon the PCs from the left, Figure 2a provides the reflectance spectra for three different values of gain-loss coefficient σ, where (ω − ω 0 )/ω gap is the normalized frequency and ω gap = 4ω 0 arcsin|(n b − n a )/(n b + n a )| 2 /π is the photonic bandgap [50]. The center frequency of the bandgap was defined as ω 0 = 2πc/λ 0 . One can see that there was a bandgap structure in the reflectance spectrum for each fixed coefficient and a defect mode located at the middle of the bandgap. The reflectance of defect mode was zero, viz. R 1 = 0 was the coefficient σ = 0. For σ > 0, such as σ = 1, the reflectance 1 > R 1 > 0 and a partial incident light was reflected by the PCs. The reflectance surpassed 1 as the coefficient σ < 0. The loss and gain in the defect can both enhance the reflectance of the defect mode. Figure 2b gives the transmittance varying with the normalized frequency. Light could not transmit the PCs as the frequency of wave was in the range of the bandgap, except for the defect mode. As σ = 1, the incident wave could overall pass through the structure for the defect mode, i.e., T = 1. The transmittance was T < 1 for the defect mode as the coefficient was σ > 1, while the transmittance was T > 1 for the defect mode as the coefficient was σ < 1. The loss or gain in the defect can reduce or enhance the reflectance of the defect mode, respectively.

Exceptional Points and Sharp Change in Phase
Figure 2a demonstrates the reflectivity in the parameter space composed of the normalized frequency (ω − ω 0 )/ω gap and gain-loss coefficient. One can see there is a peak and a valley in the parameter space. The peak value was R 1 = 1.03 × 10 5 and it can be viewed as +∞ in our calculating accuracy. The maximum point was the CPA-LP, which located at [(ω LP − ω 0 )/ω gap = 0.0654, and was an EP labeled by EP 1 . The reflectance of light at EP 1 was R 1 = 1.33 × 10 −6 , which is approximately equal to 0 in our calculating accuracy. The non-Hermitian system can be viewed as a resonant cavity, of which the defect is the cavity body and the Bragg gratings are reflectors. The resonance cavity and gain-loss coefficient of the defect collectively induced the CPA-LP and EP 1 . Next, we demonstrate the characteristics which approve the CPA-LP and EP.
For the light incident from the left, the reflection coefficient can be written as r 1 = |r 1 |exp(iϕ r1 ), where ϕ r1 is the complex phase of reflection coefficient, as shown in Figure 2b. The EP 1 and CPA-LP were two singular points in the complex phase of the reflection coefficient. The complex phase ϕ r1 sharply changes around the EP 1 and CPA-LP. That is, a slight increase in the normalized frequency or in the gain-loss coefficient may lead to a great variation in the complex phase near the EP 1 and CPA-LP. For three specific values σ = −0.893, −0.793, and −0.693, which were near the CPA-LP, Figure 2c provides the complex phase of reflection coefficient varying with the normalized frequency. For the coefficient σ = −0.793, approaching to σ LP , the complex phase change was ±π around the CPA-LP as the frequency increased. For σ = −0.893, lower than σ LP , the slope of complex phase curve was negative, while for σ = −0.693, larger than σ LP , the slope of complex phase curve was positive and there was a meaningless 2π hopping in the complex phase. Figure 2d demonstrates the complex phase cures for three given values, σ = −0.199, −0.099, and 0.001, around the EP 1 . One can see that the complex phase change was also ±π as the coefficient approached σ EP1 , such as σ = −0.099. Different from the situation near the CPA-LP, the slope of complex phase curve was positive as the coefficient σ < σ EP1 , while the slope was negative as σ > σ EP1 around the EP 1 . The common characteristic was that the closer the parameters were to the CPA-LP and EP 1 , the higher the slope of curve was, which may have resulted in a giant GH shift. Figure 2e shows the transmittance light in the parameter space. There was a peak in transmission in the parameter space as the normalized frequency and gain-loss coefficient changed. The maximum point was exactly at the CPA-LP, which approves the properties of the CPA laser state. The maximum transmittance was T LP = 1.34 × 10 5 , which can be viewed as +∞ in our accuracy. The CPA-LP is a boundary state (the power of mode is mainly restricted at the interface of two different materials). The complex phase of transmission coefficient changed dramatically with the parameters around the CPA-LP, as shown in Figure 2f. There was a hopping in the complex phase of transmission coefficient at the CPA-LP and the variation was ±π. The hopping was meaningless ±2π at the other complex phase jump points, where the complex phase could therefore be viewed to be continuous. The resonance of defect cavity and the gain in dielectrics acted together to result in the CPA-LP. The extreme value in transmittance meant that there was complex phase uncertainty of the transmission coefficient. The sharp change in the complex phase of the transmission coefficient can also induce the lateral GH shift of the transmitted light.
As the light incident from the right, the reflection coefficient was denoted by r 2 = |r 2 |exp(iϕ r2 ). Figure 3a gives the reflectance in the parameter space. There were also two extreme points as the normalized frequency and gain-loss coefficient increased. The minimum was labeled by EP 2 and located at [(ω EP1 − ω 0 )/ω gap = 0.0654, σ EP2 = 0.1], while the maximum was the CPA-LP and located at [(ω LP − ω 0 )/ω gap = 0.0654, σ LP = −0.79]. The positive gain-loss coefficient represented loss and negative gain-loss coefficient represented gain, respectively. When light was incident from the left, the position of EP 1 showed that gain in materials was required, and while the light was incident from the right, loss was required to carry out the singularity of EP 2 . These properties indicate that the EPs were dependent on the directions of light incident form the left and right. The dependence on directions for EPs resulted from the asymmetry of the defect and the non-Hermiticity of PCs. As the gain-loss coefficient σ changed, the real and imaginary parts of eigenvalues are depicted in Figure 4a,b, respectively. For the incident and scattered waves, the scattering matrix in PCs can be written as S = [t r 1 ; r 2 t], which is the counterpart of Hamiltonian in quantum systems [51,52]. The eigenvalues of S-matrix were λ 1,2 = t ± (r 1 r 2 ) 1/2 and the eigenvectors were (r 1 1/2 , ±r 2 1/2 ). The eigenvalues coalesce and the eigenvectors degenerated at (r 1 r 2 ) 1/2 = 0, which was defined as EPs of eigenvalues. That is, the EPs could be derived by r 1 or r 2 = 0. We generally searched for EPs in the parameter space on conditions of R 1 or R 2 = 0. The reflection coefficient complex phase ϕ r1 and ϕ r2 were bound to experience a dislocation as R 1 R 2 = 0, which is an unique characteristic of EPs. The zero reflection and complex phase dislocation can be utilized to seek out EPs. When σ < σ EP1 and σ > σ EP2 , the real parts of the eigenvalues split into two branches. When σ EP1 < σ < σ EP2 , the real parts of eigenvalues degenerated. On the whole, the curves of real parts were anti-crossing and the imaginary part curves were crossing around the EPs for the eigenvalues. The imaginary parts of eigenvalues degenerated at EPs as well. These characteristics are often used to confirm EPs.  Figure 5a gives the distribution of the electric field for EP 1 . It demonstrates that the power of electric field was mainly restrained in the defect. The electric field intensity decreased exponentially as the coordinate extended to the left and right from the 0 point. The distribution of electric field further manifested that EP 1 was the defect state (or transmission state). Figure 5b provides the electric field distribution of CPA-LP. The electric field mainly distributed the defect as well. This means that CPA-LP was also a defect state. Two Bragg gratings formed the resonance of cavity, together with some appropriate gain in defect, to result in the CPA laser state. In the parameter space, the complex phase of the reflection coefficient changed sharply as the normalized frequency and gain-loss coefficient increased near EPs and CPA-LP. Figure 6a gives the GH shift of reflected light around EP 1 and CPA-LP. The interested area is divided into three parts by dotted lines. EP 1 and CPA-LP are exactly on the dotted lines. At EP 1 and CPA-LP, one can see that the GH shift was infinite, so EP 1 and CPA-LP were the singularity of the GH shift. Except for EP 1 and CPA-LP, the GH shift on the dotted lines was 0. Otherwise, the GH shift in region I and III was positive, while the GH shift in region II was negative. The GH shift was greater as the parameters increasingly approached EP 1 and CPA-LP. In our accuracy, the positive GH shift could be as high as 10 6 λ, and the negative GH shift could be as high as −10 5 λ. The sharp change in the complex phase of reflection coefficient near EPs and CPA-LP generated the giant GH shift. The singularity of the GH shift at EPs and CPA-LP was also an intriguing characteristic.  Figure 6b illustrates the relationship between the GH shift and normalized frequency for five special gain-loss coefficients σ = −1.5, ±1 and ±0.5. One can see that the GH shift changed with the normalized frequency. There was a peak in each curve as the coefficient σ > σ EP1 or σ < σ LP and a valley existed in each curve as σ LP < σ < σ EP1 . The positive and negative maximum GH shifts were at the normalized frequency (ω − ω 0 )/ω gap = 0.0654. The incident beam was composed of many different frequency components of plane waves. We have showed that the light field penetrated more deeply into the PCs as the parameters became closer to EP 1 and CPA-LP, so the dispersion of plane waves was more serious. For this reason, the reflected beam consisting of scattering waves had a giant lateral shift. Figure 6c gives the GH shift around EP 2 and CPA-LP as light was incident from the right. Similarly, EP 2 and CPA-LP were the singular points of the GH shift. Along the dotted lines, the GH shift was 0. The GH shift in region I and III was positive, while the shift in region II was negative. The maximum positive and negative GH shifts were 10 5 λ and −10 4 λ, respectively, indicating that the GH shift of the reflected light from the right was smaller than that of the reflected light from the left. Therefore, the GH shift of the reflected beam was dependent on the direction incident from the left and right.

Applications and Giant GH Shift
The lateral shift also existed in the transmitted light. Figure 6d shows the GH shift of the transmitted light beam near CPA-LP. The region was divided into two parts by the dotted line. The GH shift in part I was positive, while the GH shift in part II was negative. The positive and negative GH shifts were 2 × 10 4 λ and −6.5 × 10 4 λ, respectively. CPA-LP was the singular point for the GH shift and the GH shift was infinite at CPA-LP. The GH shift along the dotted line was 0. Even though the CPA laser state was not stable enough, the incident light and transmitted light were located at the two sides of the PCs, respectively. Therefore, the spatial discrimination of sensors based on the GH shift of the transmitted beam was better than that of sensors based on the GH shift of the reflected beam.
Based on the GH shift, the PCs can be utilized for highly sensitive sensors in probing the gain or loss factor of materials. Here, we take the normalized frequency (ω − ω 0 )/ω gap = 0.065, for example, to demonstrate the application. To make the phenomenon more obvious, the given normalized frequency was near ω LP and ω EP . Figure 7a shows the lateral GH shift. One can see that the GH shift changed with the gain-loss coefficient of the defect and there were positive and negative GH shifts around the EPs and LP. The positive and negative maxima could reach as high as the magnitude of 10 3 λ. Figure 7b gives the sensitivity coefficient of sensors based on the GH shift in defecting gain or loss coefficient of materials. The maximum sensitivity coefficient (SC) was 6.65 × 10 4 , since the frequency of incident wave was fixed. The SC can be further improved if the chosen frequency is closer to ω LP and ω EP . Different from our one-dimensional PC of the multilayer type, PC fibers can also utilized for highly sensitive sensors for temperature sensing application via a Sagnac interferometer [53] or for detection of high refractive index liquid analytes via surface plasmon resonance [54]. The incident angle was selected as 20 • and the purpose was to take a specific incident angle as an example to demonstrate that EPs and CPA-LP can be induced in the parameter space when a TM wave is obliquely incident into the non-Hermitian PCs. EPs and the CPA-LP can also be derived as the incident angle is chosen as some other values. The positions of EPs and the CPA-LP changed with the incident angle in the parameter space. If the incident wave was transverse electromagnetic (TE) wave instead, EPs and CPA-LP could also obtained, while the lateral shift of the reflected beam was defined as the Imbert-Federov shift in this case [55].

Conclusions
To summarize, we have theoretically studied EPs in one-dimensional non-Hermitian PCs incorporated with a defect. The defect which contains gain and loss is asymmetric with respect to the center. Two EPs have been searched by increasing the normalized frequency and gain-loss coefficient of defect. EPs are directionally dependent on the light incident from the left and right. The complex phases of the reflection and transmission coefficients changed dramatically with the parameters around EPs and there was a phase hopping of ±π at EPs. The electric field of EPs mainly distributed at the center and EPs were confirmed to be defect modes. A giant GH shift of reflected and transmitted light beams was induced around EPs. A CPA-LP was also found in the parameter space and a giant GH shift existed around the CPA-LP as well. EPs and the CPA-LP were singularities of the GH shift. This research provides an option for development in highly sensitive sensors.