Anisotropic reflection oscillation in periodic multilayer structures of parity-time symmetry

Xue-Feng Zhu; Yu-Gui Peng; De-Gang Zhao

doi:10.1364/OE.22.018401

1. Introduction

Wave phenomena under the framework of non-Hermitian Hamiltonians are often encountered in optical systems with gain and/or losses. In the past few years, a wide class of optical systems that respects parity-time (PT) symmetry was intensely studied with several exotic phenomena predicted and observed, such as loss induced transparency [1], power oscillations and nonreciprocity of light propagation [2], coherent perfect lasing and absorption [3–5], reconfigurable Talbot effect [6], on-chip optical isolation [7–10], and anisotropic transmission resonances [11], etc. Among the miscellaneous PT symmetric systems, the sinusoidal PT symmetric complex crystals with balanced gain/loss modulations show rather unusual scattering and transportation properties [12–16], viz. unidirectional invisibility, and the eigenvalues of Hamiltonian are completely real in the PT symmetric phase despite the non-Hermiticity [17]. Intuitively, the periodic PT symmetric structure modulated in the form of δexp(ik·r) can be regarded as a complex grating providing a unidirectional wave vector k to the incident waves, where the diffraction mode can be selectively excited by satisfying the phase matching condition in one direction but not the other. Recently, Longhi [18] has demonstrated that unidirectional invisibility in sinusoidal PT symmetric complex crystals will break down in the case that the number of unit-cells surpasses a threshold by studying the exact analytical expressions of coupled mode theory, showing the importance of boundary effect in describing the Bragg scattering and coupling of counter-propagating waves in the sinusoidal PT symmetric system of a finite length.

In this paper, we investigate the scattering and transportation properties of periodic multilayer structures with the rectangular PT symmetric modulation based on exact analytical expressions for transfer matrix method. The results show that periodic multilayer structure with rectangular PT symmetric modulation exhibits anisotropic reflection oscillation as crystal length or the number of unit-cells is increasing. In particular, at the lengths corresponding to the minima in the reflection/transmission oscillations, the complex photonic crystal is bidirectionally reflectionless, however with remarkably different field distributions inside the crystal when probed from opposite sides. At the minima, the system is operating in the PT symmetric phase, where all minima are de facto mapped to the central points of PT symmetric phase. Around the minima while still in the PT symmetric phase, the crystal is unidirectionally reflectionless due to the anisotropic reflection oscillations. In the PT broken phase, the transmission through the crystal is larger than unitary. With a dielectric layer inserted between the neighboring unit-cells, more complicated reflection/transmission oscillation patterns can be observed. The exploration of optical characteristics in periodic multilayer structures with the rectangular PT symmetric modulation shows rich physics and new functionalities, such as anisotropic reflection oscillations, where ingenious optical devices can be designed to react differently depending on the direction of wave propagation.

2. Schematic of the PT symmetric complex crystals

In optics, PT symmetry requires the complex refractive index distributed in the form of n(z) = n*(-z) with respect to z = 0 [2]. In this paper, we consider an optical periodic structure with a PT symmetric refractive index distribution (see the schematic diagram in Fig. 1). For the loss regions, the refractive index distributions are n₁(z) = n₀ + Δn + Δni for mL≤z≤(m + 1/4)L and n₂(z) = n₀-Δn + Δni for (m + 1/4)L≤z≤(m + 1/2)L. For the gain regions, the refractive index distributions are n₃(z) = n₀-Δn-Δni for (m + 1/2)L≤z≤(m + 3/4)L and n₄(z) = n₀ + Δn-Δni for (m + 3/4)L≤z≤(m + 1)L. Here m = 1, 2, …, N, with N being the number of unit-cells in the PT symmetric structure, n₀ the refractive index of the uniform background medium, and Δn the amplitude of both real and imaginary modulations. At the Bragg condition, L = 4L_j = λ/(2n₀), with λ the wavelength of light in vacuum and j = 1, 2, 3, 4.

Fig. 1 Schematic diagram of PT symmetric complex crystals. Both real part (blue solid line) and imaginary part (red dashed line) of refractive index have rectangular modulations of equivalent amplitudes, where the modulations differ in phase by π/2.

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3. Theory and formulation

For describing the scattering of plane waves (wave number: k₀ = 2πn₀/λ) impinging on one unit-cell from the left, we can use a unimodular characteristic matrix as expressed into

M_{j} (L_{j}) = [\begin{matrix} m_{11} & m_{12} \\ m_{21} & m_{22} \end{matrix}] = [\begin{matrix} \cos (k_{0} n_{j} L_{j}) & - \frac{i}{n_{j}} \sin (k_{0} n_{j} L_{j}) \\ - i n_{j} \sin (k_{0} n_{j} L_{j}) & \cos (k_{0} n_{j} L_{j}) \end{matrix}],

For the whole periodic medium with N unit-cells, we then have M(NL) = (M(L))^N = [M₁₁, M₁₂; M₂₁, M₂₂]. From the theory of matrices, the Nth power of a unimodular matrix M(L) is [19]

{[M (L)]}^{N} = [\begin{matrix} m_{11} u_{N - 1} (a) - u_{N - 2} (a) & m_{12} u_{N - 1} (a) \\ m_{21} u_{N - 1} (a) & m_{22} u_{N - 1} (a) - u_{N - 2} (a) \end{matrix}]

where a = (m₁₁ + m₂₂)/2 and u_N are the Chebyshev Polynomials of the second kind:

u_{N} (a) = \frac{\sin ((N + 1) \cos^{- 1} a)}{\sqrt{1 - a^{2}}} .

As shown in Fig. 2, it is interesting to find out that a≈–cos(π/(11/9(n₀/Δn)²)), for 0≤Δn/n₀≤0.15. Therefore, under the condition that Δn/n₀<<1, we have

u_{N} (a) \approx \frac{\sin ((N + 1) (1 - \frac{1}{P}) π)}{\frac{π}{P}},

where P = 11/9(n₀/Δn)². In the end, the elements of the characteristic matrix for the whole periodic structure can be written into

Fig. 2 Plots of a = (m₁₁ + m₂₂)/2 (blue solid line) and –cos(π/(11/9(n₀/Δn)²)) (red dashed line) versus Δn/n₀, where 0≤Δn/n₀≤0.15.

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\begin{array}{l} M_{11} \approx m_{11} \frac{\sin (N (P - 1) \frac{π}{P})}{\frac{π}{P}} - \frac{\sin ((N - 1) (P - 1) \frac{π}{P})}{\frac{π}{P}}, \\ M_{12} \approx m_{12} \frac{\sin (N (P - 1) \frac{π}{P})}{\frac{π}{P}}, M_{21} \approx m_{21} \frac{\sin (N (P - 1) \frac{π}{P})}{\frac{π}{P}}, \\ M_{22} \approx m_{22} \frac{\sin (N (P - 1) \frac{π}{P})}{\frac{π}{P}} - \frac{\sin ((N - 1) (P - 1) \frac{π}{P})}{\frac{π}{P}} . \end{array}

For the PT symmetric structure, the left-side reflection coefficient and the transmission coefficient are respectively [19]

\begin{array}{l} r_{L} = \frac{(M_{11} + M_{12} n_{0}) n_{0} - (M_{21} + M_{22} n_{0})}{(M_{11} + M_{12} n_{0}) n_{0} + (M_{21} + M_{22} n_{0})}, \\ t = \frac{2 n_{0}}{(M_{11} + M_{12} n_{0}) n_{0} + (M_{21} + M_{22} n_{0})} . \end{array}

From Eqs. (1) to (6), it is obvious that the reflection and transmission curves should oscillate as the number of unit-cells N is increasing, with the periodicity being exactly P. Let us set the constant P to be an integer. From Eq. (5), we have M₁₁(M₂₂) ≈1 and M₁₂(M₂₁)≈0, when the number of unit-cells N = ξP (ξ being an integer). According to Eq. (6), the left-side reflection coefficient and the transmission coefficient at N = ξP are r_L≈0 and t≈1. For the right-side incidence, the unimodular characteristic matrix for each unit-cell can be obtained by simply exchanging the subscripts $1 \leftrightarrow 4$ and $2 \leftrightarrow 3$ , namely, M₄M₃M₂M_{1 $\leftrightarrow$}M₁M₂M₃M₄. By investigating the analytical expressions of a and m_ij(i,j = 1,2), one can deduce that a, m₁₂, and m₂₁ are invariant, while m₁₁ and m₂₂ take their complex conjugates, after exchanging the subscripts for the reversed incidence. Therefore, indicated by Eqs. (5) and (6), the left-side and right-side reflections are not necessarily equal to each other due to the reason that

m_{11} \neq m_{11}^{*}, m_{22} \neq m_{22}^{*},

in the non-Hermitian system, except for the special points (N = ξP) where the coefficients of m_ij approach nearly zero, viz. u_N–₁(a)≈0. In that case, the right-side reflection coefficient r_R is also close to zero, where the transmission is unitary for both the left-side and right-side incidences in terms of Lorentz reciprocity. As a result, at the periodical points (N = ξP), the PT symmetric medium exhibits bidirectionally reflectionless with unitary transmission, so-called bidirectional transparency, where the anisotropic reflection oscillations for opposite incident directions simultaneously get to the minima of zero. The scattering matrix for this one dimensional (1D) two-port system can be expressed as S = [r_L t; t r_R], where the eigenvalues are

λ_{1, 2} = \frac{r_{L} + r_{R} \pm i \sqrt{{(2 i t)}^{2} - {(r_{L} - r_{R})}^{2}}}{2}, with | λ_{1} | | λ_{2} | = 1.

4. Numerical calculation results

On the basis of analytical transfer matrix method, we will discuss in this section the scattering and transportation properties of the rectangular PT symmetric crystal as shown in Fig. 1. In our case, we have used n₀ = 3, Δn = 0.2, λ = 1200 nm, and L = λ/(2n₀) = 200 nm. It needs to be mentioned that the imaginary part of refractive index is in the range of −0.016<κ<0 at λ = 1550 nm for currently available semiconductor based optical gain media [20]. Here we assume optical gain coefficient (|κ| = Δn = 0.2) to be a big number in order to avoid a very large finite element model, since in this section we will first study the bidirectional transparency of the PT symmetric scattering system with the number of unit-cells N = ξP, and the periodicity P of reflection/transmission oscillation is rapidly decreased as Δn increases. To be specific, P = 11/9(n₀/Δn)² = 275, as predicted by the Eq. (4) with n₀ = 3, Δn = 0.2. We have calculated left-side reflectance, right-side reflectance, and transmittance of the PT symmetric scattering system with the number of unit-cells N increasing from 1 to 800, by using transfer matrix method in optics without applying any approximation [15]. From Fig. 3, the oscillations are distinctively observed in reflection and transmission curves with the periodicity being exactly 275, well agreeable with the prediction from the Eq. (4).

Fig. 3 (a) Left-side reflectance, (b) right-side reflectance, and (c) transmittance in a rectangular PT symmetric crystal for n₀ = 3, Δn = 0.2 and L = λ/(2n₀). The number of unit-cells N is increasing from 1 to 800.

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The numerical calculations also show that the PT symmetric scattering system is bidirectionally transparent at the periodical points N = ξP (marked by the arrows in Fig. 3), and in the vicinity of those periodical points, the system exhibits unidirectional transparency due to the anisotropic reflection oscillations. Physically, the asymmetric amplitudes of left-side and right-side reflectances are caused by the unidirectional wave vector (|k| = 2(2πn₀/λ)) provided by the complex grating, while oscillations are results of the discontinuity at the interfaces between the PT symmetric medium and the uniform background medium. In particular, for the periods of 1≤N≤800, the left-side reflectance R_L<7.117 × 10⁻⁵ (Fig. 3(a)), the right-side reflectance R_R<834 (Fig. 3(b)), and the transmittance 1<T<1.245 (Fig. 3(c)). It needs to be mentioned that T→1 can be obtained from the generalized energy conservation relation [11]

\sqrt{R_{L} R_{R}} = | T - 1 |,

when

R_{L} R_{R} \to 0

. Therefore the PT symmetric scattering system that is either bidirectionally reflectionless or unidirectionally reflectionless has unitary transmission according to Eq. (9).

Due to the odd symmetry of imaginary modulation in material parameters of PT symmetric medium, the field distribution should be very different when probed from opposite directions. We have mapped out the distributions of out-of-plane component of electric field (E_┴) by using a finite element solver (COMSOL Multiphysics), where the incident light is transverse electric polarized (TE waves) and the number of unit-cells in the rectangular PT symmetric crystal is N = 275. In the numerical simulation, the PT symmetric medium is reflectionless no matter in which direction the light is propagating as shown in Fig. 4(a). However, the PT symmetric system still renders directional responses according to the field distribution (|E_┴|) inside the system. In Fig. 4(b), the E_┴ field is highly localized in the PT symmetric system for the right-side incidence, and the intensities (|E_┴|²) of localized fields are enhanced by up to three orders. For left-side incidence in Fig. 4(a), no field localization is observed. In both cases, the field distributions are symmetric with respect to the central line of the scattering system, respectively. Considering the PT symmetry of the scattering system, the E_┴ field is equally distributed in the loss and gain regions, showing that the energy produced by the gain is completely absorbed by the loss in the mirrored position, thus resulting in a unitary transmission.

Fig. 4 The amplitude of electric field E_┴ distribution for the light propagating (a) from the left and (b) from the right through the rectangular PT symmetric crystal with the number of unit-cells N = 275.

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In optics, the change of transmission and reflection from (R_L + R_R)/2−T<1 to (R_L + R_R)/2−T>1 can be related to the transition from PT symmetric phase to PT broken phase [11]. From Eq. (8), if the quantity (R_L + R_R)/2−T<1, the eigenvalues of scattering matrix are unimodular (|λ₁| = |λ₂| = 1) with their absolute values degenerated, which is in the so-called PT symmetric phase. However, if (R_L + R_R)/2−T>1, the eigenvalues of scattering matrix are non-unimodular (|λ₁| = 1/|λ₂|<1), signifying the PT broken phase. The absolute of eigenvalues of scattering matrix is plotted in Fig. 5(a), where the number of unit-cells N increases from 1 to 800. From Fig. 5(a), there exists a series of transitions corresponding to entering and leaving the PT broken phase at the periodical points N = ξP ± 6 (N>0). Comparing Fig. 5(a) with Fig. 3, we find that the minima/maxima in the reflection/transmission oscillation are actually mapped to the central points of PT symmetric/broken phases, respectively. When the scattering system is in PT symmetric phase (ξP−6≤N≤ξP + 6 and N>0), the left-side reflectance R_L<2.7 × 10⁻⁷, the right-side reflectance R_R<3.2, and the transmittance T≈1. Therefore, in the PT symmetric phase, the scattering medium is either bidirectionally transparent (N = ξP) or unidirectionally transparent ( $N \neq ξ P$ ). While in the PT broken phase, especially at the center of PT broken regions (N≈ξP + P/2), the transmission is much larger than unitary (T_max≈1.244), although the scattering system can still be regarded as unidirectionally reflectionless for R_L≈7.116 × 10⁻⁵ and R_R≈839.9. It is also worth noting that the PT symmetric phase will be rapidly expanded with the broken phase shrinked accordingly as the modulation strength of refractive index (Δn/n₀) decreases.

Fig. 5 The absolute of eigenvalues of scattering matrix for the PT symmetric crystal with the number of unit-cells N increasing (a) from 1 to 800 and (b) from 250 to 300, respectively.

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We have carried out finite element simulation to demonstrate the phenomenon of unidirectional transparency in the PT symmetric medium in Fig. 6, where the number of unit-cells N = 269 as marked by the arrows in Fig. 5(b). From Fig. 6, the interference pattern due to strong Bragg reflection (R_R≈3.177) is clearly visualized for the right incidence, whereas the reflected light is barely observed for the left incidence (R_L≈2.692 × 10⁻⁷). When the light is incoming from the right, the E_┴ field is more localized in the gain regions and the strong unidirectional Bragg reflection comes from the extra energy from the gain after being dissipated in the loss.

Fig. 6 The amplitude of electric field E_┴ distribution for the light propagating (a) from the left and (b) from the right through the rectangular PT symmetric crystal with the number of unit-cells N = 269.

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Unidirectional transparency is not equivalent to unidirectional invisibility, because the observer can still sense the existence of PT symmetric medium if the phase of transmitted light is differed from the phase of light passing through a homogeneous background medium with the same length of scattering system. In Fig. 7, we plot the relation between the phase difference and the number of unit-cells N. As analyzed in Sec. 3, the transmission coefficient t≈1 at N = ξP, showing that the phase is almost unchanged for the light traversing the PT symmetric medium with the number of unit-cells N = ξP. At the Bragg condition, the length of each unit-cell is L = λ/(2n₀), so that the phase is changed by π for the light traversing a uniform background medium with the length of one unit-cell. Therefore, when N = ξP is an odd number, the phase difference is close to π and the system is still visible to some phase sensors, whereas the PT symmetric system can be regarded as bidirectionally invisible (phase difference≈0) if N = ξP is an even number. The results in Fig. 7 clearly demonstrate that the bidirectional invisibility condition is satisfied on condition that N = 0 and N = 550 (1≤N≤800), since the phase difference is approaching zero. In the vicinity of those specific points while still in the PT symmetric phase, such as N = 1,…,6, N = 544,…,549, and N = 551,…,556, the PT symmetric medium renders the property of unidirectional invisibility.

Fig. 7 The difference between the phases of transmitted waves propagating through the PT symmetric crystal and through a homogeneous background medium with the same length of scattering system, with the number of unit-cells N increasing from 1 to 800.

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Another interesting case is to insert a homogeneous dielectric slab between neighboring unit-cells as shown in the inset (one unit-cell) of Fig. 8, where the whole periodic system still respects PT symmetry. The refractive index and length of the dielectric slab are n₀ and L₅ = λη/(8n₀), respectively. In this case, we find out that the variable a in Chebyshev Polynomials of the second kind takes the approximate form of

a \approx - \cos (\frac{π}{\frac{11}{9} {(\frac{n_{0}}{Δ n})}^{2}} + \frac{η π}{4 + 0.003 \sin (\frac{η π}{4})}),

as demonstrated in Fig. 8, where 0≤η≤4 and 0≤Δn/n₀≤0.15. When η>0, more complicated reflection/transmission oscillation patterns will be observed as predicted by substituting Eq. (10) into Eqs. (1) to (6).

Fig. 8 Plots of a = (m₁₁ + m₂₂)/2 (blue solid line) and –cos(π/(11/9(n₀/Δn)²) + ηπ/(4 + 0.003sin(ηπ/4))) (red dashed line) versus η, where L₅ = λη/(8n₀) as shown in the inlet is the length of a dielectric layer inserted between neighboring unit-cells.

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For further demonstration, we have also calculated the left-side reflectance, the right-side reflectance, and the transmittance of two different PT symmetric scattering systems (I and II) with the number of unit-cells N increasing from 1 to 500 by using transfer matrix method in optics without any approximation, where the lengths of the additional dielectric layers (L₅) in the scattering systems I and II are λ/(8n₀) and λ/(4n₀), respectively. The calculated results are shown in Fig. 9. Fourfold oscillation and double oscillation are distinctively observed in reflection and transmission curves with the periodicities P_I,II being approximately 75 and 153 for the scattering systems I and II. The numerical calculations show that the scattering systems I and II are both unidirectionally transparent for 1≤N≤500, except for the special points (N = ξP_I,II) where the scattering systems I and II exhibit bidirectional transparency. To be specific, for the points 1≤N≤500, the left-side reflectance R_L<1.47 × 10⁻⁸ (Fig. 9(a)), the right-side reflectance R_R<0.173 (Fig. 9(b)), and the transmittance T≈1 (Fig. 9(c)) for system I; The left-side reflectance R_L<7.47 × 10⁻⁹ (Fig. 9(d)), the right-side reflectance R_R<0.089 (Fig. 9(e)), and the transmittance T≈1 (Fig. 9(f)) for system II.

Fig. 9 (a) Left-side reflectance, (b) right-side reflectance, and (c) transmittance in a rectangular PT symmetric crystal with a dielectric layer [L₅ = λ/(8n₀)] inserted between neighboring unit-cells (system I). (d) Left-side reflectance, (e) right-side reflectance, and (f) transmittance in a rectangular PT symmetric crystal with a dielectric layer [L₅ = λ/(4n₀)] inserted between neighboring unit-cells (system II). The number of unit-cells N is increasing from 1 to 500.

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In principal, one can study the scattering properties of 2D or 3D PT symmetric photonic crystals by combining the transfer matrix method with the multiple-scattering method, so-called layered multiple-scattering method which has already been developed and applied in the acoustic domain [21, 22]. The unidirectional invisibility in higher dimensions is possible to be observed when the 2D or 3D complex gratings are constructed to provide unidirectional wave vectors to the incident light. In addition, by introducing optical nonlinearity into the PT symmetric crystals, some nonreciprocal responses in transmission can be realized. One of the fascinating devices based on nonlinear PT symmetric crystals is the optical isolator without harmonic generation [9].

5. Conclusions

In conclusion, Bragg scattering in rectangular PT symmetric complex crystals of finite thickness has been theoretically studied. Analytical expressions for reflection and transmission coefficients have been derived based on theory of matrices and fine numerical approximation. The analytical results show that anisotropic reflection oscillation can be obtained in this PT symmetric system as crystal length is increasing. The complex crystal is bidirectionally reflectionless at the lengths corresponding to the minima in the reflection/transmission oscillations. However, the PT symmetric set-up can still hold directional responses due to one-way field localization inside the crystal when probed from opposite sides. We emphasize that the crystal is unidirectionally reflectionless due to the anisotropic reflection oscillation around those minima meanwhile in the PT symmetric phase. In the broken phase, the invisibility breaks down since the transmittance is much larger than unitary. We also show that more complicated reflection/transmission oscillation patterns can be observed by inserting a dielectric layer between the neighboring unit-cells.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under No. 11304105 and the financial support from the Bird Nest Plan of HUST.

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Anisotropic reflection oscillation in periodic multilayer structures of parity-time symmetry

Abstract

1. Introduction

2. Schematic of the PT symmetric complex crystals

3. Theory and formulation

4. Numerical calculation results

5. Conclusions

Acknowledgments

References and links

Cited By

Figures (9)

Equations (10)

Optics Express