Realization of complex conjugate media using non-PT-symmetric photonic crystals

3.1 Realizing the pseudo-Hermitian condition using a Dirac-like cone

The previous section shows that, by setting the average non-Hermiticity τ±=0, the non-Hermitian Hamiltonian will become pseudo-Hermitian, and we can obtain the real spectra. For a PT-symmetric PC, τ±=0 is automatically satisfied as the distributions of loss and gain within the unit cell are equal (see Supplementary Note 2 for details) [30]. However, for a non-PT-symmetric PC, achieving τ±=0 is not an easy task. Now, we will show that, for a PC exhibiting a Dirac-like cone in the Brillouin zone center, we can always achieve the pseudo-Hermitian condition τ±=0.

The eigenmode uk±(0)(r) in the quantity F_Ω,mm′ depends on the structural details (such as the filling ratio r_c and the contrast between ε_A and ε_B) of the PC and can be computed numerically using COMSOL. Substituting the orthonormal relationship Equation (3) into the pseudo-Hermitian criteria τ±=0, we solve a specific value of the loss-gain ratio ℓ_r

for both m=+ and m=−. Equation (12) shows that, to determine the value of ℓ_r, we must tune the system parameters so that the system satisfies F_Ω,++=F_Ω,−−.

The band dispersions near the Dirac-like cone are shown in Figure 1A, and the quantities ε_ΩF_Ω,mm of the three bands are plotted in Figure 1B. We can see that there is an intersection (F_Ω,++=F_Ω,−−) at k_xa/2π=0.019 (marked by the dashed line), and then Equation (12) can be used to determine the gain-to-loss ratio, which is found to be ℓ_r=−0.15235. Physically, the existence of such an intersection is guaranteed by the band inversion in the Dirac-like cone [25]. The condition F_Ω,++=F_Ω,−− can always be achieved near the Dirac-like cone (see Supplementary Note 6 and Supplementary Figure S1 for details); therefore, τ±=0 is realized at k_xa/2π=0.019 as shown in Figure 1C. Note that the quantity F_Ω,mm should be a function of k, but near the Dirac-like cone (except the Γ point), this quantity is almost a constant (as shown in Figure 1B), as the eigenmodes of the same point group are almost unchanged when the change of k is small. We can therefore assume that the values τ_m, and κ are independent of k in the considered Brillouin zone region. As shown in Figure 1C, by tuning ℓ_r=−0.15235, we can obtain τ±≈0 near the Dirac-like cone (except very close to the Γ point, where τ₋≈−τ₊). We can demonstrate that the pseudo-Hermitian condition at the Γ point is τ₋=−τ₊, which is less demanding than other points, as the symmetry at the Γ point is higher than that at other k points (see Supplementary Note 5 for details). Therefore, the pseudo-Hermitian criterion is approximately fulfilled for all the k points near the Dirac-like cone.

Now, we can calculate the band structures of a PC with loss-gain ratio ℓ_r=−0.15235 to verify the validity of the pseudo-Hermitian Hamiltonian shown in Equation (7). We choose a non-Hermitian strength γ=+0.367 and plot the complex band structure along the M−Γ−X direction in Figure 2A and B. The analytical results are shown by solid green lines, and the numerically results are calculated using COMSOL by open circles. The validity of our non-Hermitian Hamiltonian model is demonstrated by the good agreement between the two sets of results. We can see that EPs appear in both Γ−X and Γ−M directions. To show that these EPs form a ring in k space, we plot the three-dimensional complex band structure in Figure 2C and D using Equation (9) and COMSOL as shown by green surfaces (analytical results) and filled dots (numerical results), respectively. The analytical model (9) predicts that k=k_c=0.019(2π/a) form a ring of EPs, which is verified by COMSOL results. The eigenfrequencies inside the ring form complex-conjugate pairs, and outside the ring, we obtain the entirely real spectra.

Figure 2:

Band dispersions of pseudo-Hermitian bands.

The real parts (A and C) and the imaginary parts (B and D) of the complex eigenfrequencies along the M−Γ−X direction and in the 2D Bloch k space. Open circles and solid dots are calculated using COMSOL, whereas green solid lines and green surfaces are calculated using the analytical model (v_g=0.295c and |κ|²=0.00318). The bands outside the ring are entirely real. The parameters of 2D PC are r_c=0.1999a, ε_A=12.5+iγ, ε_B=1+iγℓ_r, γ=+0.367, and ℓ_r=−0.15235.

However, entirely real spectra do not mean loss-gain compensation. To see this, we compare the system with non-Hermiticity γ₊=+0.367 with its complex conjugate-paired system γ₋=−0.367. If we replace γ₊ by γ₋, the eigenfrequencies described by Equation (9) are the same, but the Hamiltonians (7) are different as described by Equation (11). In other words, these two systems (γ₊ and γ₋) possess the same band dispersion, but the eigenfunctions are different (see Supplementary Note 7 for details); hence, they can display different scattering behaviors, as we will show later.

3.2 Real spectra and CCM

The optical properties of an Hermitian PC carrying a Dirac-like cone in the Brillouin zone center can be described using an effective refractive index medium in which the effective permittivity and permeability are simultaneously zero at the Dirac-like point frequency. We have shown in the previous subsection that, by setting the average non-Hermiticity as zero, the spectra are real outside the ring of the EPs. It is interesting to see what type of EM corresponds to the non-PT-symmetric PC with real spectra. In this section, we will establish the EMT of non-Hermitian PC for describing the physics near the Γ point and study the related scattering properties.

We calculate the effective parameters using a boundary field averaging method [31]. Assuming that the wave k=kx^ propagates along the x direction, we define Z=E_z/H_y=−ωμ/k=−k/(ωε) for a plane wave traveling in a homogeneous medium to express the ratio between the electric fields and the magnetic fields. For the inhomogeneous PC system, the average field ratio is defined as

where E^PC and H^PC are obtained from the eigenfields at the incident boundary I (the boundary of the unit cell along y direction). When we calculate effective parameters of the non-Hermitian PC as functions of frequency, the frequencies should take real-valued numbers, as in actual experiments the incident light comes with a real frequency. Therefore, within the PC, the eigenfields E^PC and H^PC are obtained by solving the “complex-valued k(ω) vs. real-valued ω” band structures [32]. In Figure 3A, we plot the “complex-valued k(ω)” band structure of the non-Hermitian 2D PC calculated using COMSOL. We note that the imaginary parts of the bands (red lines) are almost zero, and for real bands, the “complex-valued k(ω)” band structure should agree well with the “complex-valued ω(k) vs. real-valued k” band structure (see Supplementary Figure S7 for details). In Figure 3B, we plot Z^PC defined in Equation (13) for the bands with positive (dashed lines) and negative (solid lines) wave vectors, respectively, in Figure 3A. The effective permittivity and permeability can be calculated as follows:

Figure 3:

Effective parameters obtained from the real spectra as shown in Figure 2.

(A) Complex-valued k(ω)/k₀ band and (B) averaged field ratio Z^PC calculated numerically from COMSOL. Bands with positive and negative wave vectors are represented by dashed and solid lines, respectively. The real (green) and imaginary (red) parts of the effective parameters ε_e and μ_e are plotted in squares and stars, respectively (C and D).

We plot the real and imaginary parts of ε_e(μ_e) by squares (stars) in Figure 3C and D, respectively. Note that the effective parameters obtained from the two bands (k and -k) are the same.

It is known that, for the Hermitian case, the effective parameters of the PC are real and approach zero near the Dirac-like point frequency, indicating that the PC can be treated as a zero refractive index medium [25]. When we introduce loss and gain into the PC, the effective parameters ε_e and μ_e obtain imaginary parts as shown in Figure 3D. However, we note that the effective refractive index n_e²=ε_eμ_e=(k/k₀)² is a real number, where k₀ is the wave vector in air. This automatically indicates that the inhomogeneous PCs behave like the homogeneous CCM whose refractive index is real but whose permittivity and permeability are complex numbers. The real effective refractive index does not necessarily imply a simple loss-gain compensation, as ε_e and μ_e have imaginary parts. Also, it can be proven that εe(γ+)=εe(γ−)* and μe(γ+)=μe(γ−)* (see Supplementary Note 8 for details).

The complex conjugate-paired systems (γ₊ and γ₋) have dramatically different effects on the scattering properties of electromagnetic waves. To demonstrate this, we consider a slab with thickness d formed by the homogeneous EM with the parameters ε_e and μ_e, as shown in Figure 4A. We assume that plane waves propagate along the x direction with an electric field polarized along the z direction and that the EM slab (blue region) is embedded in air (gray region). The electric field in the air can be written as E(x)= a₁exp(ik₀x)+b₁exp(−ik₀x) for x<−d/2 and as E(x)=b₂exp(ik₀x)+a₂exp(−ik₀x) for x>d/2. The amplitudes of incoming-propagating waves (a₁, a₂)^T and outcoming-propagating waves (b₂, b₁)^T are related by a scattering matrix [23].

Figure 4:

Scattering properties of CCM with effective parameters as shown in Figure 3.

Schematics of wave scattering in (A) EM slab and (B) PC slab. The contour plot of the value 1/Max|λ±| in the (ω, d) plane for EM slab with parameters (C) εe(γ+),μe(γ+) and (E) εe(γ−),μe(γ−). Black and red solid lines represent the reflection R and transmission T for PC slab and black and red squares represent EM slab, respectively. The non-Hermiticity strength of PCs is (D) γ₊=+0.367 and (F) γ₋=−0.367. Slab thicknesses are d/a=17 in (B) and d/a=9 in (D).

where r and t represent the transmission and reflection coefficients of the slab for normal incident plane waves. The elements in S are

where k=k₀n_e is the wave vector in the slab. The corresponding eigenvalues of the S matrix are

These eigenvalues can be used to determine the S matrix poles (1/Max|λ±|→0) where lasing occurs. For a slab composed of CCM [Im(n_e)=Im(k/k₀)=0], kd is a real number. For the S matrix to have poles, we require M₂₂=0; therefore, we obtain

To satisfy Equation (19), μ_e must be an imaginary number. Figure 3C and D shows the effective parameters of PCs with non-Hermiticity γ₊=+0.367. We find Re[μe(γ+)]=0 at ωa/2πc=0.5416, and we can solve Equation (19) to obtain the slab thickness d/a=17.4. For PCs with non-Hermiticity γ₋=−0.367, the effective parameters are εe(γ+)=εe(γ−)* and μe(γ+)=μe(γ−)*, and we can solve the slab thickness d/a=8.9.

Figure 4C and E shows the contour plots of 1/Max|λ±| in the (ωa/2πc, d/a) plane for EM slab with (εe(γ+), μe(γ+)) and (εe(γ−), μe(γ−)) respectively. To verify the scattering poles (1/Max|λ±|→0), in Figure 4B, we plot the transmission T=|t|² and reflection R=|r|² of slabs formed by homogeneous EM by squares and PCs by solid lines. The schematic of the one-dimensional scattering problem of slab formed by PCs is plotted in Figure 4B. We set the thickness of the slab d/a=17, as the thickness of the slab formed by PCs must be an integer. The good agreement between the EM and PCs results shows that the effective parameters shown in Figure 3C and D essentially reproduce the physics. We can see that the system indeed has the transmission and reflection singularities at ωa/2πc=0.5416 in Figure 4D. This indicates that we can observe the poles of the S matrix from the transport configuration. Similarly, the pole in Figure 4E at d/a=8.9 and ωa/2πc=0.5416 for (εe(γ−), μe(γ−)) is also verified by the transmission and reflection singularities in Figure 4F, where d/a=9. When we change the sign of γ, the real bands of the PCs remain almost the same, but the non-Hermitian PCs do not have time reversal symmetry. Although the refractive index satisfies ne(γ+)≈ne(γ−), the scattering phenomena change drastically as the effective permeability (or impedance) has imaginary parts, and μe(γ+)≠μe(γ−). This illustrates that the S matrix poles depend not only on the refractive index but also on the permeability as shown in Equation (19). The sum of the transmission and reflection is not equal to unity, although the effective refractive index of the slab is real. A real refractive index does not necessarily imply energy compensation. The focusing effect of our PCs as a lens is also different for these complex paired systems (see Supplementary Note 9 for details).