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Abstract
Coherent perfect absorption-lasing (CPAL) and collimation have been intensively studied for normal and small angle wave incidence. Here, we report a two-dimensional non-Hermitian photonic crystal for broad-angle CPAL and super-collimation. The synergy of a nonsymmorphic glide symmetry of the lattice, gain-loss modulation and an optimization of unit cell induces a parity-time phase transition in the band structure along the Brillouin zone boundary. The transition points, i.e., the exceptional points, form a slab-like contour, with nearly zero dispersion in both real and imaginary parts of the band structure. Such dispersionless band structure significantly enhances the range of incident angle for CPAL and collimation.
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1. Introduction
In quantum mechanics, the Hamiltonian in a closed system is assumed to be Hermitian, which ensures real energy eigenvalues [1]. An open system, i.e., a system with gain or dissipation, is described by non-Hermitian Hamiltonian. Non-Hermitian systems exhibit properties that are beyond the paradigm of Hermitian physics [1–10]. In general, a non-Hermitian Hamiltonian possesses complex eigenvalues. However, if the system is PT-symmetric [11], which is invariant under the combined action of parity and time reversal operations, it enables real eigenvalues [12–25]. As a unique platform for light manipulation, PT-symmetric non-Hermitian systems offer versatile applications such as sensing [12–15], non-Hermitian doping [18], super-collimation [19], unidirectional lasing [21,24], unidirectional perfect absorption [23], and coherent perfect absorber (CPA) [25].
One key feature of a PT-symmetric system is exceptional points (EPs), which are singularities in the parameter space of the system. Two or more eigenvalues, and their corresponding eigenvectors of the system, coalesce at EPs [19–22,26,27]. Take a $2 \times 2$ Hamiltonian for example:
To design metamaterials with desired properties, extensive experiments, parameter scanning and trial-and-error are usually required, which are computationally expensive. With advancements in computational science, new intelligent methods have been developed to optimize the design, from heuristic approaches [35,36] to more sophisticated models; using Gaussian processes [37] and artificial neural networks [38–41]. Heuristic algorithms, however, have a slow convergence speed, while artificial neural networks require a large database of training data. In recent years, Gauss-Bayesian (GB) methods have been employed for designing quantum cascade lasers [41], optical metamaterials [42], mantle cloaks [43], and acoustic structures for different functionalities [44,45]. Such approaches construct a stochastic model for the objective function that needs to be optimized, and this statistical inference can help to reduce iterations dramatically for final optimal design. It utilizes an adaptive framework that efficiently samples the design space to find a global optimum by defining the Gaussian kernel and acquisition function as appropriate, without the need for a large database.
In this work, we propose a design of non-Hermitian photonic crystal (PC) to achieve the functionalities of CPAL and super-collimation in 2D systems for a wide range of incident angle. We combine the glide symmetry [46,47] and non-Hermitian perturbations [48] in a PC to get a rectangular shape of EPs along the Brillouin zone boundaries. We use an optimization algorithm with GB model, which efficiently produces dispersionless real and imaginary parts of eigenfrequencies along one Brillouin zone boundary. The CPAL and collimation are achieved with a broad range of incident angle because of the dispersionless band structure.
2. Create the dispersionless band structure in the 2D non-Hermitian PC
We begin with a 2D PC as illustrated in Fig. 1(a). The PC has a unit cell comprising two rectangular silicone blocks with side length ${l_x}$ and ${l_y}$ embedded in air. We consider non-Hermitian potentials by adding gain and loss, with relative permittivity ${\varepsilon _{g,l}} = 12.5 \mp 1i$, in blocks marked by red (gain) and blue (loss), respectively. The distance between the centers of the two blocks in the horizontal direction is fixed at $0.5a$, where a is the lattice constant in the x direction. The lattice constant in the y direction is ${a_y}$. Along the y direction, the distance from the center of blocks to the center of unit cell is ${r_y}$. We rotate the red and blue blocks clockwise and counterclockwise, respectively, by an angle $\alpha $. Such a PC has a glide symmetry ${G_x} = \{ {M_y}|0.5a\hat{x}\} :({x,y} )\to ({x + 0.5a, - y} )$, which transforms one block to the other through a reflection over the x axis followed by a translation of $0.5a$ along x axis. Along high symmetry lines of the first Brillouin zone, the real and imaginary parts of the band structure for transverse electric (TE) polarization are calculated by the electromagnetic wave module in COMSOL Multiphysics and plotted in Figs. 1(b) and 1(c), respectively. Both results are plotted in dimensionless frequency $2\pi c/a$, where c is the wave speed in air. The parameters of the unit cell are chosen as ${a_y} = a$, ${l_x} = {l_y} = 0.29a$, $\alpha = 30^\circ $ and ${r_y} = 0.25a$. Because of glide symmetries in the PC, two bands below the first bandgap are doubly degenerate at Brillouin zone boundaries [46,47]. Additional non-Hermitian perturbations deform the double degeneracy and spawn a narrow slab-like contour of EPs along the Brillouin zone boundaries [48].
Fig. 1. (a) The unit cell of a 2D PC composed of two rectangular blocks (red with ${\varepsilon _g} = 12.5 - i$ and blue with ${\varepsilon _l} = 12.5 + i$) in air (green). (b) Real part and (c) imaginary part of band structure for TE polarization (with the electric field parallel to the blocks) Brillouin zone. The parameters in the unit cell are set as ${a_y} = a$, ${l_x} = {l_y} = 0.29a$, $\alpha = 30^\circ $ and ${r_y} = 0.25a$, respectively.
To achieve unique phenomena in non-Hermitian systems in high dimensions, the band structure should meet some special conditions. For example, in a non-Hermitian system, CPAL may exist at a particular frequency and with certain incident angle. To make CPAL effective for broad incident angles, the band structure of the PC should be dispersionless along one of the Brillouin zone boundaries. Specifically, we aim to flatten the band along the $\mathrm{{\rm X}{\rm M}}$ boundary in the band structure, which requires careful tuning of the design parameters in the unit cell. We develop a GB model to solve the inverse problem of optimizing the PC structure over the five parameters ${a_y},\; {l_x},\; {l_y},\alpha ,\; {r_y}$ for the desired dispersion properties [49]. The objective function that needs to be minimized is defined as $f(\omega ) = \sum\nolimits_{i = X}^M {({\omega _i} - \mu )^2}$. The developed GB model exploits the statistical approach of the Gaussian process to approximate the objective function $f(x )$ and to determine the next sample x to be evaluated in the functional space. This Gaussian process produces a posterior distribution $Q(f )$, with mean $\mu ({x;\theta } )$ and the covariance kernel function $k({x,x^{\prime};\theta } )$ which is then updated using acquisition function after the objective function is evaluated [44]. A well-selected Gaussian kernel and acquisition function allow the GB model to sample the design space effectively to find the global optimum. In our case, we use the kernel function $k({x,x^{\prime}:\theta } )= \sigma _f^2\left( {1 + \frac{{\sqrt 5 r}}{{{\sigma_l}}} + \frac{{\sqrt 5 {r^2}}}{{3\sigma_l^2}}} \right)exp\left( { - \frac{{\sqrt 5 r}}{{{\sigma_l}}}} \right)$ where $r = \sqrt {{{\left( {x - x'} \right)}^T}\left( {x - x'} \right)}$ is the Euclidian distance between x and $x\prime $ in fuction space, $\sigma _f^2$ and ${\sigma _l}$ are standard deviation and length scale of the random distribution, respectively. The expected improvement is used as the acquisition function, i.e., $EI\left( {x,Q} \right) = {E_Q}[{\rm{max}}(0,\;{\mu _Q}\left( {{x_b}} \right) - f\left( x \right)]$, to determine the optimal position in function space for the lowest value of the posterior mean ${\mu _Q}\left( {{x_b}} \right)$ during the optimization process. The optimization strategy is schematically shown in Fig. 2(a). The optimization result shows a convergence behavior with rapid decrease of the objection function after 40 iterations [see Fig. 2(b)]. The efficiency of algorithm comes from the fact that it uses a Gaussian process to approximate the objective function and an acquisition function to direct samples toward minima without computing gradients. Since the optimization process is based on a statistical surrogate model, suboptimal values may appear in some iterations (such as 1-10, 11-14, etc.) and it uses only the optimal value of the objection function compared to previous iteration until it achieves the desired performance.
Fig. 2. (a) Schematics of GB optimization process to find the optimal parameters of the dispersion-less PC structure. (b) The value of objective function at each iteration of the GB algorithm for optimizing the design parameters.
After the optimization, we get a flat band along the boundary with the parameters of the unit cell ${a_y} = 2a,\,{l_x} = {l_y} = 0.45a,\,\alpha = 0^\circ $ and ${r_y} = 0$. The non-Hermitian PC becomes PT-symmetric. Figures 3(a) and 3(b) show the real and imaginary parts of band structures along high symmetry lines of the first Brillouin zone, respectively. Figures 3(c) and 3(d) are the real and imaginary parts of band structures near the XM boundary in ${k_x} - {k_y}$ plane. The PT-symmetric non-Hermitian periodicity in the unit cell results in two trajectories of EPs at ${k_x} = 0.97\left( {\pi /a} \right)$ and ${k_x} = 1.03\left( {\pi /a} \right)$, divide the Brillouin zone into two parts. In the PT-broken phase with $\left| {\left| {{k_x}} \right| - \left( {\pi /a} \right)} \right| \lt 0.03\left( {\pi /a} \right)$, the real parts of eigenvalues are dispersionless and degenerate; in the PT-exact phase with $\left| {\left| {{k_x}} \right| - \left( {\pi /a} \right)} \right| \gt 0.03\left( {\pi /a} \right)$, the imaginary parts of eigenvalues are dispersionless and degenerate. On the boundary, i.e., $|{|{{k_x}} |- ({\pi /a} )} |= 0.03({\pi /a} )$, both real and imaginary parts of band structure are nearly dispersionless.
Fig. 3. Band structure after the optimization of unit cell. The parameters of the unit cell are set as ${a_y} = 2a$, ${l_x} = {l_y} = 0.45a$, $\alpha = 0^\circ $ and ${r_y} = 0$, respectively. (a) Real part and (b) imaginary part of the band structure along high symmetry lines of the first Brillouin zone. The inset in (a) is the optimized unit cell. (c) Real part and (d) imaginary part of the dispersion surfaces near the Brillouin zone boundary ${k_x} = \pi /a$ in the ${k_x} - {k_y}$ plane.
3. Broad-angle CPAL and super-collimation in the 2D non-Hermitian PC
In the PT-broken phase, a PC with PT-symmetric non-Hermitian periodicity can behave both as a laser oscillator and as a CPA, which is the so-called “CPAL” [28,31,50]. Such a CPAL solution can be predicted by a scattering matrix of a PC slab:
where t is the transmission coefficient, ${r_l}$ and ${r_g}$ are the reflection coefficients from the loss side and gain side, respectively. The eigenvalues ${\lambda _1}$, ${\lambda _2}$ of S should be uni-modular $({|{{\lambda_1}} |= |{{\lambda_2}} |= 1} )$ in the PT-exact phase and inverse $({|{{\lambda_1}} |= 1/|{{\lambda_2}} |} )$ in the PT-broken phase [31]. Specially, the zeros and the poles of the eigenvalues of the scattering matrix correspond to the occurrence of CPA and lasing, respectively. Most of the previous discussions of the scattering matrix and its eigenvalues are focused on the case of normal incidence [20,28,29,31,50], leaving broad-angle CPAL less studied, mainly due the complexity in the design of the unit cell. Figures 4(a) and 4(b) show the eigenvalues of the scattering matrix of a PC slab with 12 layers of unit cell with varying frequency and incident angles. At $\omega = 0.212({2\pi c/a} )$, ${\lambda _1}({{\lambda_2}} )$ exhibits zeros (poles) for all incident angles, indicating a broad-angle CPAL as a direct result of the dispersionless PT-broken phase from the optimization algorithm. In Figs 4(c)–4(j), we plot the electric field distribution of two counter-propagating coherent plane waves at frequency $\omega = 0.212({2\pi c/a} )$ incident onto the left and the right boundaries of the PC slab. The incident angles are set as $0^\circ $ in Figs. 4(c) and 4(d), $30^\circ $ in Figs. 4(e) and 4(f), $45^\circ $ in Fig. 4(g) and 4(h), and $60^\circ $ in Figs. 4(i) and 4(j), respectively. When the phase difference between the incident waves is $0.5\pi ({1.5\pi } )$, the electric field in the PC concentrates in the loss (gain) block, and the PC behaves as a CPA (laser). These results clearly demonstrate that our PC can behave as a broad-angle CPAL.Fig. 4. (a) The smaller and (b) the larger eigenvalues of scattering matrix with the varying incident angle and frequency near the slab of EPs. (c-j) The electric field distributions when two plane waves with frequency $\omega = 0.212({2\pi c/a} )$ and a phase difference (c,e,g,i) $\mathrm{\Delta }\varphi = 0.5\pi $ and (d,f,h,j) $\mathrm{\Delta }\varphi = 1.5\pi $ incident from opposite sides of the PC with 12 layers of unit cells. The incident angle is (c,d) $0^\circ $, (e,f) 3$0^\circ $, (g,h) $45^\circ $, and (i,j) $60^\circ $, respectively. I1, I2 and O1, O2 represent incident waves 1, 2 and outgoing waves 1, 2, respectively.
To understand the mechanism of light propagation in the PC, we plot the equal-frequency contour near the PT-broken phase of the band structure in Fig. 5(a). The dashed vertical lines represent the equal frequency curve at $\omega = 0.212({2\pi c/a} )$. The group velocity is therefore along the horizontal direction, indicating that the electromagnetic wave propagates only along the horizontal direction in the PC. Figure 5(b) is the electric field distribution when a point source at frequency $\omega = 0.212({2\pi c/a} )$ is placed in the center of a PC consisting of $16 \times 10$ unit cells. Figure 5(c) is the electric field distribution when the point source is replaced by a Gaussian beam obliquely incident from the left. The incident angle is 30°. Consistent with the equal-frequency contour, electromagnetic wave propagates in the horizontal direction in the PC in both cases. The standing wave shown on the left side implies that there is a reflection from the structure. This is due to the incident wave is a Gaussian beam coming from one side, rather than two counter-propagating plane waves coming from both sides.This super-collimation [19] effect is a consequence of dispersionless band structure along ${k_y}$ over the entire Brillouin zone, and thus can collimate waves with all incident angles, while the previously reported similar effect in Hermitian PC and photonic-band-gap based waveguides relies on flat band in a certain range of the Brillouin zone [51], limiting the range of incident angles for collimation. The simulation indicates that such a PC can be used to confine light from spreading, minimize power loss during light propagation, and combine beams toward a unique direction.
Fig. 5. All-angle super-collimation at frequency $\omega = 0.212({2\pi c/a} )$. (a) Equal-frequency contour of the second band near PT-broken phase. The dashed line corresponds to the frequency $\omega = 0.212({2\pi c/a} )$, the white arrows represent the direction of the group velocity. (b) Electric field distributions when a point source is placed at the middle of a PC with $16 \times 10$ unit cells. (c) Electric field distributions when a Gaussian beam with an incident angle $30^\circ $ shines the PC.
4. Summary
In conclusion, we demonstrate a 2D PC with slab-like contour of EPs in its band structure. The contour of EPs is induced by the combination of glide symmetry and PT-symmetry, and optimized by machine learning algorithms with GB model. The optimization algorithm in this work is efficiently, and does not need a large dataset. In the PT-broken phase of band structure, the eigenfrequencies are dispersionless along ${k_y}$ in both real and imaginary parts. Such a unique property gives rise to intriguing functionalities. At the frequency of EPs, the PC behaves as CPA or Laser when two counterpropagating coherent plane waves with different phase change incident to opposite boundaries. Different from some previous works, such a CPAL works for all incident angles, which extends the CPAL solution from a point to a line in 2D parameter space. Moreover, the PC enables applications such as super-collimation. Our work paves the way for the combination of machine learning with PCs, which has the advantages of low cost, flexibility, and high efficiency.
Funding
King Abdullah University of Science and Technology (AI initiative, BAS/1/1626-01-01, OSR-CRG2020-4374).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but maybe obtained from the authors upon reasonable request.
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