Wave propagation, bi-directional reflectionless, and coherent perfect absorption-lasing in finite periodic PT-symmetric photonic systems

2.1 Transfer matrix of finite periodic PT-symmetric systems

The transfer matrix of a unit cell is denoted as M ₁. Since the unit cell has a PT-symmetry, it obeys M 1 * = M 1 − 1 , [25]. Then, we require that finite periodic structures composed of PT-symmetric unit cells are embedded in homogeneous environments. According to Lorentz reciprocity theorem, we have D e t [ M 1 N ] = 1 . Here N denotes total number of unit cells. Since D e t [ M 1 N ] = 1 is valid for any N, it would lead to Det[M ₁] = 1. Combined with M 1 * = M 1 − 1 and Det[M ₁] = 1, we can obtain M 1 N − 1 = M 1 N * . The detailed analysis is placed in Appendix A. This result ensures that any N-cell structures composed by PT-symmetric unit cells still have a PT-symmetry invariant.

The transfer matrix for the unit cell having PT-symmetry can be parametrized by

here α ₁ and β ₁ are reals and ϕ ₁ is transmission phase of the unit cell bound by 0,2 π , Refs [30, 40]. The exact description for α ₁, β ₁, and ϕ ₁ would depend on system configurations. Due to Lorentz reciprocity theorem established in the unit cell, i.e., Det[M ₁] = 1, there has a constraint for α ₁ and β ₁, i.e., α ₁ β ₁ ≤ 1 [30, 40].

Next, for our PT-symmetric systems composed by N PT-symmetric unit cells, with the help of Chebyshev identity, the resultant transfer matrix reads

where t _N is transmission coefficient, r _R,N is right reflection coefficient, r _L,N is left reflection coefficient, U N = sin [ ( N + 1 ) Φ ] sin Φ , and Φ is Bloch phase [36, 37, 41]. The Bloch phase is related to the eigenvalues (λ ₁ and λ ₂) of M ₁,

The condition of Im[Φ] = 0 enables incident waves to propagate through assembly systems.

2.2 3D generalized parametric space for general PT-symmetric unit cells enabling wave propagation

Equation (3) indicates that the Bloch phase depends on the PT phase and transmission phase of the unit cells, i.e., α ₁, β ₁, and ϕ ₁. Accordingly, with PT-symmetry, Lorentz reciprocity theorem, and real Bloch phases (Im[Φ] = 0), we provide a 3D parametric space in terms of α ₁, β ₁, and ϕ ₁, to display allowable parametrization supporting wave propagation in finite periodic PT-symmetric structures, marked by light-orange colors in Figure 1(a).

Figure 1:

With consideration of PT-symmetry, Lorentz reciprocity theorem, and real Bloch phase, we depict the 3D parametric space in terms of parametrization α ₁, β ₁, and ϕ ₁ for any PT-symmetric unit cells supporting wave propagation in (a). More specifically, by taking transmission phases of unit cells by ϕ ₁ = 0, 1, 4, 6, we provide the corresponding 2D parametric space by α ₁ and β ₁ marked by light-orange colors in (b). The dark-blue regions denote the accessible parametrization for general PT-symmetry systems, while it possesses complex Bloch phase, resulting in forming evanescent waves in periodic systems. Here red-solid, green-dashed, and blue-dashed lines denote as CPAL, exceptional point and exceptional point, respectively. (c) Schematic of finite periodic PT-symmetric photonics made of a PT-symmetric unit cell with simple gain (G)–loss (L) heterostructure. Here the gain slabs are depicted by red colors, while the loss are depicted by blue colors. The refractive index of the gain (loss) slab is denoted as n _g n l = n g * . E 1 + and E 1 − ( E 3 + and E 3 − ) are complex amplitudes of electromagnetic plane waves in the left lead (right lead) toward right and left propagation, respectively. We illustrate the corresponding parametrization from two sets of unit cells in the parametric space marked by block dots as shown in the left panel of (d)–(f), while the corresponding transmittances T _N with N are shown in the right panel of (d)–(f), respectively. With commercial software COMSOL, we plot the instantaneous electric field distribution and time-averaged Poynting vectors (pink arrows) for the corresponding systems composed by 5 unit cells in (e)–(g). In the case of (d)–(e), the parameters for the unit cell are n _l = 1.82 + 0.38i, l ₁ = 0.0159λ ₀ and l ₂ = 0.237λ ₀, while in the case of (f)–(g) the parameters are n _l = 1.92 + 0.134i, l ₁ = 0.048λ ₀ and l ₂ = 0.442λ ₀. Here λ ₀ is operating wavelength.

More specifically, we plot 2D parametric spaces by taking ϕ ₁ = 0, 1, 4, 6, where the real Bloch phases are depicted by light-orange colors in Figure 1(b). For a comparison, we also depict the accessible parametrization for general PT-symmetric systems marked by dark-blue regions. We can see that not every PT accessible regions can support wave propagation in finite periodic systems. In the PT broken symmetry phase and exceptional point, it requires 0 < α ₁ β ₁ ≤ 1 and α ₁ β ₁ = 0 by parametrization, respectively. These conditions would lead to have real Bloch phase, since it follows 0 ≤ 1 − α 1 β 1 ≤ 1 . Thus, any PT-symmetric unit cells operated at the PT broken symmetry phase or exceptional point would ensure wave propagation in any N-cell systems. We also note that in this case, the transmission phase of the unit cell can be arbitrary. On the other hand, we can observe that some regions of PT symmetry phase have complex Bloch phase, as indicated by dark-blue colors in Figure 1(b), where depend on ϕ ₁. Since the symmetry phase corresponds to α ₁ β ₁ < 0 by parametrization, the factor of Eq. (3) would be subject to 1 − α 1 β 1 > 1 by some values of α ₁ β ₁, leading to the emergence of complex Bloch phase.

4.1 Propagating wave

By taking parametrization (α ₁ = 0.1 and β ₁ = 0.174) and geometry (l ₁ = 0.0159λ ₀ and l ₂ = 0.237λ ₀) into Eqs. (4) and (5), we can obtain the corresponding complex permittivities for the loss slab of n _l = 1.82 + 0.38i and the transmission phase of ϕ ₁ = 3.4. This design of the unit cell correspond to a real Bloch phase and broken symmetry phase, whose its parametrization (α ₁ = 0.1 and β ₁ = 0.174) is marked by a black dot in the left panel of Figure 1(d). We calculate the resultant transmittance T _N = |t _n |² with N and plot the instantaneous electric field distribution and time-averaged Poynting vectors for the system with 5-cells in the right panel of Figure 1(d) and (e), respectively, enabling the wave propagation. Next, we choose another parametrization by (α ₁ = −0.76 and β ₁ = 0.78) and geometry parameters by (l ₁ = 0.048λ ₀ and l ₂ = 0.442λ ₀). By Eqs. (4) and (5), we obtain n _l = 1.92 + 0.134i and the transmission phase of ϕ ₁ = 0.536. In this design, the unit cell is operated at PT symmetry phase and has a complex Bloch phase, whose its parametrization is marked by a block dot in the left panel of Figure 1(f). As expected, wave propagation in the periodic system constituted by such unit cell would form an evanescent wave as shown in the right panel of Figure 1(f), also supported by the instantaneous electric field and time-averaged Poynting vectors in Figure 1(g).

4.2 Bi-directional reflectionless

Any PT-symmetric systems operated at an exceptional point would accompany by unidirectional or bidirectional reflectionless, as well as unity transmittance [30, 37, 40, 43]. Since the transmission phase of PT-symmetric system operated at an expectational point can be arbitrary, the exotic scattering phenomenon can not be regarded as rigorous invisibility [43]. We note that the delay time, defined as the reflection coefficient with respect to operating frequency, behaves like a delta function at an exceptional point, due to an absence of reflected wave [44]. This leads to one-way (unidirectional transparency) or two-way (bi-directional transparency) light-trapping resonance in PT-symmetric systems, which the wave would be trapped for a long time until complete absorption by lossy materials. Moreover, the bifurcation of scattering eigenvectors near the exceptional point exhibits the potential applications for designs of enhanced photonic sensors [8, 45]. Recent studies further unveil that the emergence of the exceptional point is not a signature of PT-symmetric systems, while it is associated with non-Hermitian systems [2, 45, 46].

Any PT-symmetric unit cells at an exceptional point follow ( M 1 ) 12 = 0 or ( M 1 ) 21 = 0 or ( M 1 ) 12 = ( M 1 ) 21 = 0 , corresponding to α ₁ = 0 or β ₁ = 0 or α ₁ = β ₁ = 0 in parameterization. With these conditions, we can see that the wave scattering of systems made of the unit cells would also operate at an exceptional point. However, another condition to have an exceptional point in finite periodic systems, in the absence of the unit cell operated at the exceptional point, is

already proposed by Refs. [37, 38]. We note that this condition would exhibit the bidirectional reflectionless, because it results in M 1 N 12 = 0 and M 1 N 21 = 0 (r _R,N = 0 and r _L,N = 0) simultaneously.

To demonstrate this result, we numerically solve sin[NΦ] = 0 with N = 5 and depict the solutions in the parametric phase marked by the black lines in Figure 2(a). The solutions correspond to the symmetry phase and broken symmetry phase, consistent with transmittance relation [47]. Now, we choose parametrization (α ₁, β ₁) = (0.12, 0.3) for our design, which meets the solution of Eq. (6) with N = 5, marked by a yellow dot in Figure 2(a). Then, taking geometry parameters (l ₁ = 0.03λ ₀ and l ₂ = 0.36λ ₀) into Eqs. (4) and (5), we obtain n _l = 1.33 + 0.64i and the transmission phase of ϕ ₁ = 5.03. Here the unit cell is at broken symmetry phase and has real Bloch phase. In Figure 2(b), we calculate the corresponding transmittance T _N , left reflectance R _L,N = |r _L,N |², and right reflectance R _R,N = |r _R,N |² with N. We can see that at N = 5, the assembly system reflects T _N = 1 and R _L,N = R _R,N = 0, also supported by the plot of instantaneous electric field distribution and Poynting vectors in Figure 2(c).

Figure 2:

In (a), we depict the numerical solutions of Eq. (6) with N = 5 by black lines in the parametric space with ϕ ₁ = 5.03. With the design of n _l = 1.33 + 0.64i, l ₁ = 0.03λ ₀, and l ₂ = 0.36λ ₀, we mark the corresponding parametrization by a yellow dot. In this design, the system constituted of the 5-cells exhibits bi-directional reflectionless, as in (b). With commercial software COMSOL, we plot the corresponding instantaneous electric field distribution and time-averaged Poynting vectors (pink arrows) in (c). In (d), we design another unit cell operated at CPAL by using n ₁ = 1.34 + 1.09i, l ₁ = 0.07λ ₀, and l ₂ = 0.67λ ₀, whose the parametrization is marked by a black dot in the parametric space. The transmission phase is ϕ ₁ = 2.52. In (e), we calculate T _N , R _R,N , and R _L,N with N. In (f), we depict the numerical solutions of Eq. (8) with N = 7 in the parametric space marked by blue lines. We design another unit cell by using n ₁ = 3.17 + 3.14i, l ₁ = 0.016λ ₀, and l ₂ = 0.32λ ₀, that the parametrization is depicted by a brown dot. In (g), as expected, when at N = 7, the wave scattering of the system exhibits CPAL.

4.3 Coherent perfect absorption-lasing (CPAL)

Any PT-symmetric systems operated at CPAL have zero and infinity (pole) eigenvalues of the scattering matrix, corresponding to coherent perfect absorption and lasing in wave scattering [15, 25]. The CPAL is a unique feature and discrete point of the PT broken symmetry phase. To have CPAL occurred at N-cell systems, it requires M 1 N 11 = M 1 N 22 = 0 and I m M 1 N 12 I m M 1 N 21 = 1 . Accordingly, there are two solutions. (i) the unit cell exhibits CPAL and the corresponding cell number is odd. By parametrization, we read

To our knowledge, this result is not found in other works.

(ii) The unit cell is not at CPAL, while its transmission phase is null and the Bloch phase should meet cos[NΦ] = 0,

The detailed derivation is placed at Appendix C.

To verify the case (i), with Eqs. (4) and (7), we design the unit cell by n ₁ = 1.34 + 1.09i, l ₁ = 0.07λ ₀, and l ₂ = 0.67λ ₀, marked by a black dot in Figure 2(d), with CPAL property. The corresponding transmission phase for this unit cell is ϕ ₁ = 2.52. In Figure 2(e), we calculate T _N , R _R,N , and R _L,N , with N. When N is odd, the system exhibits CPAL.

To verify the case (ii), we solve cos[NΦ] = 0 with N = 7 illustrated by the blue lines, corresponding to CPAL occurred at assembly 7-cell systems in Figure 2(f). We can observe that these lines are found at broken symmetry phases, consistent with transmittance relation, [47]. Then, taking the parametrization (α ₁, β ₁) = (0.014, 3.43λ ₀) obtained from the numerical solution of Eq. (8) and geometry parameters (l ₁ = 0.016λ ₀ and l ₂ = 0.32λ ₀) into Eq. (4), we obtain n ₁ = 3.17 + 3.14i. We mark this parametrization in the parametric space by a brown dot in Figure 2(f). The corresponding transmission phase of this unit cell is exactly null, ϕ ₁ = 0. Although this unit cell is not operated at CPAL, at N = 7, the system turns to behave CPAL, as shown in Figure 2(g).

Figure 3:

Schematic of a unit cell made of simple gain (G)-gap-loss (L) heterostructure. Here the refractive indexes for the gain and loss are n _g and n l = n g * , respectively, and the thickness of these slabs is denoted as l ₁. In between gain and loss, it is an air gap with a distance of l ₂ between adjacent gain and loss.

Besides the design of the unit cells by simple gain–loss configuration, the inverse calculation demonstrated here can be extended into more complicated designs of unit cells. Last but not least, we emphasize that the use of the parametric space here is not only directly from the consideration of PT-symmetric transfer matrix and Lorentz reciprocity theorem, but also from extended consideration of real Bloch phase. Therefore it could provide the comprehensive relation for PT phase and Bloch phase of general unit cells to construct exotic PT-symmetric wave phenomena in finite periodic systems.