1. Introduction

Wave transmission and reflection by complex crystals, stimulated by experiments in atom optics [1], attract considerable attention over the last two decades. Particular interest is devoted to periodic structures featuring alternating gain and loss domains which are balanced in the sense that the refractive index obeys the symmetry n(x) = n^∗(−x) (along the chosen direction x). Such optical media are referred to as $\mathcal{P}\mathcal{T}$-symmetric due to their analogy with the $\mathcal{P}\mathcal{T}$-symmetric quantum mechanics [2, 3]) and, so far, were considered mainly in two different physical settings. In [4–8] the problem of light incident transversely to the Bragg grating (i.e. to the x−direction in our terminology) was addressed in the paraxial approximation. In such approximation the reflected waves are neglected and mathematically the problem is formulated as initial-value problem for the parabolic equation for the field. It was found the $\mathcal{P}\mathcal{T}$-symmetric crystals are not transparent [7] in the sense that the energy of the input beam is not conserved along propagation. Furthermore, study of the (forward) diffraction patterns produced by the complex crystals, were shown to reveal the physical features of a system in the vicinity of spectral singularity [8] and allows for efficient light control by the diffraction [9]. In a different geometry, where the light propagates along the Bragg grating axis (i.e. along x-axis in our case), one deals with a problem of reflection by a finite layered $\mathcal{P}\mathcal{T}$-symmetric medium. Among the effects obtained in such statement we mention non-reciprocity of light propagation, i.e. sensitivity to the left and right incidence, as predicted in [10, 11] and observed in [12, 13].

Here we address the forward and backward diffraction (transmission and scattering) of a plane wave incident on a $\mathcal{P}\mathcal{T}$-symmetric photonic crystal (PhC) obeying $\mathcal{P}\mathcal{T}$-symmetry in one direction (along x-axis) and preserving parity symmetry in orthogonal direction in the plane of incidence (along z-direction, with properly chosen symmetry axis). Thus we consider the angle of incidence to be of 30° and be close to the angle of the total reflection. This assumption requires that the refractive index of the surrounding medium is nearly twice the average of the refractive index of the layer, what in general may affect the efficiency of the transmitted and diffracted waves. The case of finite angles of incidence, i.e. situation which is “between” the two limits mentioned above (in crystallography it is known as the Laue diffraction). Moreover we are interested in the wave diffraction beyond the paraxial approximation in the geometry where the angle of incidence is close to a resonant one ensuring strong coupling by the grating. Thus, we consider a PhC, which is infinite in the x−direction and has the dielectric permittivity ε_pc(x) = ε_av+ε_r cos(2κx)−iε_i sin(2κx), where ε_av,r,i are positive real constants, 2κ is the constant of the reciprocal lattice, ε_av defining the average permittivity, and ε_r and ε_i characterizing depths of grating and gain-and-loss distribution, respectively. The crystal has the width L in z−direction (as illustrated in Fig. 1) and is homogeneous in y−direction. Clearly, ${\epsilon }_{\text{pc}}(x)={\epsilon }_{\text{pc}}^{*}(-x)$ and thus we are dealing with a structure obeying $\mathcal{P}\mathcal{T}$ symmetry along x-direction ( $\mathcal{P}$ being the parity operator with respect to the x-axis) and space reversal symmetry in z-direction. The PhC is embedded in a medium with higher permittivity ε_ex: ε_ex > ε_av [Fig. 1].

 

Fig. 1 Illustration of waves incident and reflected at a resonant Bragg angle on a periodically modulated layer with the permittivity ε_pc(x) embedded in the medium with the permittivity ε_ex. The parity symmetry axis in z-direction is defined by L/2.

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In the paraxial limit and in the Bragg scattering mentioned above, an important role is attributed to the phenomenon known as spontaneous $\mathcal{P}\mathcal{T}$-symmetry breaking [2]. For the chosen grating the symmetry breaking occurs at ε_r = ε_i [5, 8]. Bearing in mind the relevance of this phenomenon also in our setting, we use the established terminology and refer to the cases ε_i < ε_r and ε_i > ε_r as to the unbroken and broken $\mathcal{P}\mathcal{T}$-symmetric phases.

We consider the transmission and reflection of an incident TE wave: E = (0, E(x, z), 0). Outside the PhC the medium has a constant permittivity ε_ex and the field has the form of a superposition of plane waves with the wavelength λ and the wavevector k (k = 2π/λ). Inside the PhC, i.e. at 0 < z < L, the field is governed by the Helmhotz equation E_xx + E_zz + k²[ε_pc(x)/ε_ex]E = 0. Thus, inside the medium the x-dependences of the eigenmodes is given by the Bloch waves. Thus, an incident plane wave generally speaking excites all Bloch modes modes inside the slab. If however, the x−component of the incident wavevector coincides with a half-vector of the reciprocal lattice of the modulation, i.e. if k = (κ, 0, q) with q being z−component of the incident wavevector, and if the grating is shallow enough, i.e. |ε_r,i| ≪ ε_av, the incident wave resonantly excites only the modes at the boundary of the Brillouin zone of the grating, leaving all other modes negligible. Such modes can be approximated by plane waves and their description can be reduced to the two-waves model [14]. Thus the field inside the slab is described by superposition of modes A_± (z)e^±iκx, where “ + ” and “ – ” stand for waves propagating in positive and negative x−direction and z−dependence of the amplitudes is governed by the system

Here we have defined ϵ_j = (k²/2ε_ex)[ε_r + (−1)^jε_i] for j = 1, 2 and δ² = k²ε_av/ε_ex − κ² which below will be considered positive. Below the waves propagating in the positive (negative) x−direction, are referred to as positive (negative) waves.

We observe that the system (1) is not symmetric with respect to the exchange A₊ ↔ A₋ that reflects broken parity symmetry with respect to x. The $\mathcal{P}\mathcal{T}$-symmetry however is preserved and requires simultaneous change A₊ ↔ A₋ and ϵ₁ ↔ ϵ₂ (the latter corresponding to ϵ_i ↔ −ϵ_i).

The validity of approximation (1) requires all dropped plane waves, i.e. nonresonant harmonics ~ exp(±inκx) with n = 2, 3, … to be negligible compared to the resonant ones, i.e. ~ exp(±iκx). At the same time the terms kept in system (1) should be of the same order [14]. In other words, our consideration is self-consistent if k²ε_av/ε_ex − κ² ~ ϵ_1,2. In what follows we concentrate on the exact resonance where the lattice period coincides with the wavelength λ. This corresponds to κ = π/λ = k/2 and $q=\sqrt{3}k/2$. Thus we require δ² = k²|ε_av/ε_ex − 1/4| ~ |ϵ_r,i| ≪ k². Thus we consider the angle of incidence to be of 30° and be close to the angle of the total reflection (it is to be noticed that the made assumption requires that the refractive index of the surrounding medium is nearly twice the average of the refractive index of the layer, what in general may affect the efficiency of the transmitted and diffracted waves.

Assuming that ε_r ≠ ε_i (i.e. ϵ₁ ≠ 0; the system is not in the symmetry breaking point), the field inside the periodic slab is given by the superposition E_s = A₊e^iκx + A₋e^iκx = E₁ + E₂, where

The solutions (2) and (3) can be described by the column-vectors ${\mathbf{a}}^{l,r}={\left(a_1^{l,r},a_2^{l,r},a_3^{l,r},a_4^{l,r}\right)}^T$ related to each other by the transfer matrix M: a^r = M a^l. which is obtained by requiring the continuity of the field and its z-derivative at the boundaries z = 0 and z = L. Using explicit expressions (3) and (2) by the straightforward algebra we obtain (notice that det M = 1):

Starting with real q₁ and q₂ (notice that q₂ can be pure imaginary also for the unbroken $\mathcal{P}\mathcal{T}$-symmetry) we observe that if at a given k the parameters are chosen to satisfy q₂L = πp₂ and q₁L = π(p₂ + 2p₁) where p_1,2 are positive integers, the transfer matrix becomes diagonal with m_± = 1. In this case the structure is transparent for the incident radiation no excitation of reflected and transmitted negative waves occurs. Meantime not all pairs of integers (p₁, p₂) are physically acceptable, but only ones ensuring the compatibility condition $(2{\delta }^2+\sqrt{{ϵ}_1{ϵ}_2})={\left(1+2p_1/p_2\right)}^2(2{\delta }^2-\sqrt{{ϵ}_1{ϵ}_2})$. Similarly, a total switch between the two propagation angles, corresponding to κ ↔ −κ and ${\tilde{m}}_{\pm }=1$ with all other entries of M zero occurs without any reflection if q₂L = πp₂ and q₁L = π(p₂ + 2p₁ + 1). Some examples are shown Fig. 2(a) below.

 

Fig. 2 Reflection and transmission coefficients (a) the resonance conservative case ε_r = 16δ²/(17k²), ε_i = 0 and δ² = 0.1k² (corresponding to p₁ = 1, p₂ = 3, see the text). In the rest of panels the same ε_r is used while in (b) ε_i = 0.8ε_r, and in (c) ε_i = 1.1ε_r. (d) Split of the transmission resonance at ε_i = 0.24ϵ_r. Black, red, blue, and green line correspond to |t₁₁|², |r₁₂|², |t₁₃|², and |r₁₄|². The cyan lines show the total transmitted and reflected intensity normalized to the intensity of the incident wave $a_1^{(l)}$

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To illustrate these and other effects of the forward and backward diffraction we consider numerically scattering of a single incident $a_1^{(l)}$ wave from the left (i.e. we set $a_3^{(l)}=a_2^{(r)}=a_4^{(r)}=0$), and characterize the scattering by the left transmission $t_{ij}=a_j^{(r)}/a_i^{(l)}$ the left reflection $r_{ij}=a_j^{(l)}/a_i^{(l)}$ coefficients. Clearly, different indexes i and j correspond to the energy pump into the negative (or positive) wave j generated by the incident positive (or negative) wave i (since the system is open, only in the case ε_i = 0 when the total energy is conserved one can speak about energy transfer among the modes).

In Fig. 2 we illustrate typical dependences of the scattering data on the slab width for the conservative case [panel (a)], for nonzero imaginary parts of the dielectric permittivity in the unbroken $\mathcal{P}\mathcal{T}$-symmetric phase [panel (b)], and for the broken $\mathcal{P}\mathcal{T}$-symmetric phase [panel (c)]. In Fig. 2(a) we observe characteristic resonant structure where the most part of the incident energy is reflected back (red dotted line), except the resonances where total transmission occurs shown by narrow black picks separated from each other by the distance determined by the “oscillator frequencies”: L = p₂π/q₂ = (p₂ + 2p₁)π/q₁ ≈ 40.96λ (as predicted above), and the resonances where the incident radiation is split among all four modes (transmitted and reflected, positive and negative, see picks of the blue lines).

For sufficiently strong gain and losses, however still in the unbroken $\mathcal{P}\mathcal{T}$-symmetric phase, weak transmission resonances persist [picks of the dashed black line in panel (b)]. They however are dominated by the enhanced transmitted wave in the negative direction [blue line in Fig. 2(b)]. The first main resonance of the negatively transmitted wave occurs at L = 3π/q₂ ≈ 35.18λ: The smaller resonances preserve their structure similar to the conservative case [cf. blue lines in Figs. 2(a) and 2(b)] although with much larger amplitudes of the negatively transmitted wave as compared to the wave transmitted in the positive x−direction. Above the breaking point [Fig. 2(c)] the resonant structure of the negative wave persists, although these resonances decay when ϵ_i grows. The wave incident in positive direction is reflected without significant absorption or amplification. At large slab widths the reflection totally dominates the diffraction.

By comparing Figs. 2(a) and 2(b) one concludes that there must exist a transition between the dominating transmission in the positive and negative directions, where the maxima of |t₁₁| and |t₁₃| are of the same order (or even equal). This occurs when the ration ε_i/ε_r is about 0.24. Meantime, it is interesting, that there exist no point the maxima of |t₁₁| and |t₁₃| coincide: when they become of the same order the resonance of positive transmission splits in two maxima, while the resonant transmission with negative angle remains a function of the slab width with a single maximum [see the right resonance in Fig. 2(d)]

In Fig. 3 we show characteristic dependences of the scattering data on the amplitude of the gain-loss parameter ε_i. A remarkable feature of the scattering consists in a strong resonance near (slightly below) the symmetry breaking point where the intensities of the transmitted and reflected negative wave are almost equal and are much larger than the intensities of incident and reflected waves in positive x−direction.

 

Fig. 3 Reflection and transmission coefficients vs width the gain/loss parameter for L = 12.3λ (a) and L = 18.1λ (b). Other parameters are δ² = 0.1 and ε_r = 1.6/17 ≈ 0.094. (c) The dependence of the scattering data on the slab width in the exceptional point ε_i = ε_r. Black, red, blue, and green line correspond to |t₁₁|², |r₁₂|², |t₁₃|², and |r₁₄|². In (c) blue and green lines are indistinguishable on the figure scale.

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To understand the resonant amplification of the wave propagating in negative x−direction [Fig. 3] let us perform more detail analysis of the symmetry breaking point where ε_r = ε_i. Then ϵ₁ = 0 and ϵ₂= k²ε_r/ε_ex. This is the exceptional point of the Bragg grating spectrum [5, 7, 8]. Assuming that δ ± 0 (recall that in Fig. 2, δ² > 0) we obtain a general solution of main_s ystto E = c₁ cos(δz) e^iκx + c₂ sin(δz)e^iκx + c₃ [sin(δz) + (ϵ₂z/2δ) cos(δz)] e^−iκx + c₄ {cos(δz) − (ϵ₂/2δ²) [δz sin(δz) + cos(δz)]} e^−iκx, where c_j are dimensionless constants [instead of field_in].

The transfer matrix has the form ${\mathbf{M}}_{EP}=\left(\begin{array}{cc}{ℳ}_1& 0\\ {ℳ}_2& {ℳ}_1\end{array}\right)$, where ${ℳ}_j=\left(\begin{array}{cc}{\mu }_{j1}& {\mu }_{j2}\\ {\mu }_{j2}^{*}& {\mu }_{j1}^{*}\end{array}\right)$ with

The obtained solution (1) features secular growth (~ z) of only reflected and transmitted negative waves (secular growth of the energy of a paraxial beam in the exceptional point, when only one wave transmitted through a $\mathcal{P}\mathcal{T}$-symmetric PhC exists was discussed in [8]). This secular growth leads to strong enhancements of the waves transmitted and reflected with negative angles. The respective reflection and transmission coefficients are computed straightforwardly. We do not show these length formulas, but provide in Fig. 3(c) numerical illustration of the scattering data exactly at the exceptional point ε_i = ε_r. We observe the series of resonance of strongly amplified transmission and reflection (the respective lines are indistinguishable on the scale of the figure) in the negative x-direction. The observed resonances are separated by the lengths L = π/δ (since now the system is characterized by only one “oscillators frequency” δ).

Finally, we mention the analogy of the model considered above, for the unbroken $\mathcal{P}\mathcal{T}$-symmetry, and a system of two coupled mechanical oscillators described by the real Hamiltonian $H=P_{+}^2/(2\eta )+\eta P_{-}^2/2+({\delta }^2\eta /2)A_{+}^2+({\delta }^2/2\eta )A_{-}^2+\sqrt{{ϵ}_1{ϵ}_2}{A}_{+}{A}_{-}$ where P₊ = ηdA₊/dz and P₋ = (1/η)dA₋/dz are the linear momenta, and $\eta =\sqrt{{ϵ}_2/{ϵ}_1}$ and 1/η are the masses of the oscillators. This illustrates the equivalence of the considered $\mathcal{P}\mathcal{T}$-symmetric system to the Hamiltonian one (what in general terms is known [15]). Since H ≥ 0 and the Hamiltonian is z-independent, the field amplitudes |A_±| cannot grow infinitely even when the layer width does. The existence of a sign definite integral, even in a linear a $\mathcal{P}\mathcal{T}$-symmetric system, is an exceptional, rather than a common fact (see e.g. [16]). Thus, the only difference between the conservative propagation [17] and propagation at the unbroken $\mathcal{P}\mathcal{T}$-symmetry consists in the fact that the coupled “oscillators” have the same “masses” in the former case and different “masses” in the $\mathcal{P}\mathcal{T}$-symmetric case. In the vicinity, of the $\mathcal{P}\mathcal{T}$-symmetry breaking the mass of one of the oscillators becomes infinitely small mass while the other mass becomes infinite (since η → ∞).

To conclude, we have described transmission and reflection of an incident TE wave by a $\mathcal{P}\mathcal{T}$-symmetric photonic crystal which is embedded in a medium with a dielectric permittivity higher than the average dielectric permittivity of the crystal. It was found that under condition of transmission resonance in the vicinity of the $\mathcal{P}\mathcal{T}$-symmetry breaking point the slab strongly enhances either transmitted or reflected waves with negative propagation angles. Usually gain is created by doping the propagation medium by active impurities which are pumped by external laser beams. Thus by manipulating such external field one can control the wavelengths at which the resonant effects are observed. In particular, one can design a device based on a given PhC which in the same geometry depending on the control field can transmit, reflect, strongly enhance or redirect (switch) an incident wave.