Unidirectional reflectionless phenomenon in periodic ternary layered material

Erchan Yang; Yonghua Lu; Yonghua Lu; Yong Wang; Yanmeng Dai; Pei Wang

doi:10.1364/OE.24.014311

1. Introduction

Reflection and transmission of natural linear optical materials are usually bidirectional and symmetric because of Lorentz reciprocal theorem. In the view of information optics, the symmetric propagation of light in these materials is attributed to the symmetric coupling between forward and backward propagating modes since the Fourier transform of any real optical potential (refractive index) function is always symmetric [1–3]. Unidirectional reflectionless or transmissionless materials are challenging and yet of particular interest for realizing one-way propagation, optical isolation and optical invisibility. However, as early as 1996, L. Poladian had proposed a unidirectional grating with refractive index modulation having a complex function form: $Δ n (z) ~ e^{- g z} [\cos (2 π z / Λ) + i \sin (2 π z / Λ)]$ [4]. The introduction of an additional imaginary refractive index modulation in the grating broke the symmetry of the coupling between forward and backward propagating modes. This complex grating was subsequently applied as unidirectional coupler [1] and waveguide gratings [2]. Recently, parity-time (PT) symmetric material which sprouts in quantum optics [5,6] attracts amount of researchers’ interests for the possibility of realizing unidirectional invisibility [7–9]. The structures possess a complex optical potential satisfying $n^{*} (r) = n (- r)$ , which means the real part of the optical potential is an even function of position while the imaginary part is odd. In this form, strictly balanced gain and loss is required. However, materials with pure loss or gain could also be PT-symmetry-like as long as a reference value can be subtracted from the complex optical potential and balance the gain and loss effectively. For example, the passive (no gain) unidirectional reflectionless waveguide was experimentally demonstrated by L. Feng et al [10]. In fact, the PT-symmetry-like structure and L. Poladian’s unidirectional grating are essentially consistent with each other in terms of the optical potential. And they both indicate that an asymmetric Fourier transform of a complex optical potential results in an asymmetric reflection. However, the modulation of these complex optical potential are trigonometric function, leading to spatially overlapped real and imaginary modulation parts.

To avoid the additional technological challenge from the spatial overlap of real and imaginary optical potential modulations, here, we propose a periodic ternary layered material (PTLM) to realize the unidirectional reflectionless phenomenon. The optical potential of this PTLM is rectangularly modulated by a small periodic real and an imaginary fluctuations. The corresponding real and imaginary modulation media are spatially separated. Besides, the complex refractive index (imaginary modulation) medium can be either loss or gain.

2. Design theory of PTLM

Figure 1 presents the schematic diagram of PTLM and the corresponding refractive index distribution. Each unit of PTML is composed of three layers of isotropic, homogeneous and dispersionless media. The thickness of each layer is $a$ , $b$ , and $(Λ - a - b)$ in order and refractive index is $n_{0} + n_{R}$ , $n_{0} + i n_{I}$ , and $n_{0}$ respectively. $| n_{R} |$ and $| n_{I} |$ represent the real and the imaginary modulation amplitudes and satisfy $| n_{R} |, | n_{I} | < < n_{0}$ . Here $n_{I}$ can be either positive (loss) or negative (gain). The total thickness of PTLM is $L = N Λ$ . $Λ$ is the period and $N$ is the number of unit cells. The exact refractive index distribution of PTLM in Fig. 1 can be written in piecewise function as follows:

n (z) = {\begin{matrix} n_{0} + n_{R} & j Λ < z \leq a + j Λ \\ n_{0} + i n_{I} & a + j Λ < z \leq a + b + j Λ \\ n_{0} & a + b + j Λ < z \leq (j + 1) Λ \end{matrix}, (0 < a, b < Λ, j = 0, 1, 2, \dots, N - 1) .

Considering a plane wave at wavelength

λ

propagating along z axis, the electric field E(z) in PTLM can be expressed as the superposition of forward propagating mode

E_{f} (z)

and backward propagating mode

E_{b} (z)

:

E (z) = E_{f} (z) + E_{b} (z) = ε_{f} (z) e^{i β z} + ε_{b} (z) e^{- i β z}

. Here,

ε_{f} (z)

and

ε_{b} (z)

are the corresponding slowly varying amplitudes.

β = 2 π n_{e f f} / λ

is the propagation constant with

n_{e f f}

representing the mode effective refractive index. Under slowly varying envelop approximation (SVEA),

E (z)

obeys the Helmholtz equation:

(2 i β \frac{\partial}{\partial z} ε_{f} - β^{2} ε_{f}) e^{i β z} + (- 2 i β \frac{\partial}{\partial z} ε_{b} - β^{2} ε_{b}) e^{- i β z} + k^{2} n^{2} (z) E (z) = 0.

Here,

k = 2 π / λ

is the vacuum propagation constant.

n (z)

is the refractive index distribution function presented in Eq. (1). Because of the periodicity,

n^{2} (z)

can be expanded into Fourier series as

n^{2} (z) = \frac{1}{2} \sum_{m = - \infty}^{+ \infty} C_{m} e^{2 i β_{m G} z},

where

β_{m G} = m π / Λ

is the mth-order reciprocal lattice vector of PTLM. And

C_{0} = 2 n_{0}^{2} + 4 n_{0} n_{R} \frac{a}{Λ} + 4 i n_{0} n_{I} \frac{b}{Λ},

\begin{array}{l} C_{m} = \frac{i}{π m} ({2 n_{0} n_{R} [\cos (m \frac{2 π}{Λ} a) - 1] + 2 n_{0} n_{I} [\sin (m \frac{2 π}{Λ} (a + b)) - \sin (m \frac{2 π}{Λ} a)]} \\ + i {- 2 n_{0} n_{R} \sin (m \frac{2 π}{Λ} a) + 2 n_{0} n_{I} [\cos (m \frac{2 π}{Λ} (a + b)) - \cos (m \frac{2 π}{Λ} a)]}) . \end{array}

Here, the second order terms of .. and

n_{I}

are neglected.

Fig. 1 Schematic diagram of the unidirectional reflectionless PTLM and the corresponding refractive index distribution. Deep orange area for the real modulation layers; blue for the imaginary modulation layers and orange for the non-modulation layers. The refractive index is $n_{0} + n_{R}$ , $n_{0} + i n_{I}$ , and $n_{0}$ with thickness is $a$ , $b$ , and $(Λ - a - b)$ respectively. $Λ$ is the period and $N$ is the number of unit cells. The entire structure is laterally infinite.

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For the wave with propagation constant close to the mth-order reciprocal lattice vector, i.e., $(Δ β_{m} = β_{m G} - β) < < β_{G}$ where $β_{G}$ is the first order reciprocal lattice vector, only keep the synchronous propagation terms in Eq. (2) and the coupled mode equations of the two propagating modes are obtained as:

| C_{- m} |^{2} = 0

Here, $| C_{m} |^{2} = 0$ depends on mode loss or gain. $B = \frac{C_{m} / 2}{2 β} k^{2}$ and $D = - \frac{C_{- m} / 2}{2 β} k^{2}$ are mode coupling coefficients which are directly related to the expansion coefficients $C_{\pm m}$ .

Then, the corresponding transfer matrix equation that relates the electric field at the input and output interfaces can be written as:

(\begin{matrix} E_{f} (\frac{L}{2}) \\ E_{b} (\frac{L}{2}) \end{matrix}) = (\begin{matrix} M_{11} & M_{12} \\ M_{21} & M_{22} \end{matrix}) (\begin{matrix} E_{f} (- \frac{L}{2}) \\ E_{b} (- \frac{L}{2}) \end{matrix}),

where

M_{11} = e^{i β_{m G} L} (\cos (L \sqrt{{(A + Δ β)}^{2} + B D}) - \frac{i (A + Δ β)}{\sqrt{{(A + Δ β)}^{2} + B D}} \sin (L \sqrt{{(A + Δ β)}^{2} + B D})),

M_{12} = \frac{i B}{\sqrt{{(A + Δ β)}^{2} + B D}} \sin (L \sqrt{{(A + Δ β)}^{2} + B D}),

M_{21} = \frac{i D}{\sqrt{{(A + Δ β)}^{2} + B D}} \sin (L \sqrt{{(A + Δ β)}^{2} + B D}),

M_{22} = e^{- i β_{m G} L} (\cos (L \sqrt{{(A + Δ β)}^{2} + B D}) + \frac{i (A + Δ β)}{\sqrt{{(A + Δ β)}^{2} + B D}} \sin (L \sqrt{{(A + Δ β)}^{2} + B D})) .

Equation (5) shows that the components of transfer matrix $M$ satisfy $M_{22} = M_{11}^{*}$ and $M_{12 (21)} = M_{12 (21)}^{*}$ , which further illustrates that the PTLM here is PT-symmetry-like [11,12]. The corresponding reflectivity (R) and transmissivity (T) are then derived according to Eq. (5) as:

$\begin{array}{l} R_{L} = {| - \frac{M_{21}}{M_{22}} |}^{2} \propto {| D |}^{2} \propto {| C_{- m} |}^{2}, T_{L} = {| M_{11} - \frac{M_{12} M_{21}}{M_{22}} |}^{2} \\ R_{R} = {| \frac{M_{12}}{M_{22}} |}^{2} \propto {| B |}^{2} \propto {| C_{m} |}^{2}, T_{R} = {| \frac{1}{M_{22}} |}^{2} \end{array}$ (6)

The transmissivities for both left- and right-side incident cases are equal since the determinant of transfer matrix $| M | = M_{11} M_{22} - M_{12} M_{21} = 1$ . However, the reflectivities are generally different because of ${| M_{12} |}^{2} \neq {| M_{21} |}^{2}$ . The most interesting thing is that even unidirectional reflectionless phenomenon could be achieved when the coupling coefficient satisfies the condition: ${| D |}^{2} = 0$ or ${| B |}^{2} = 0$ ( $| C_{- m} |^{2} = 0$ or $| C_{m} |^{2} = 0$ ) which implies a cutoff in the forward-to-backward coupling or vice versa. In fact, this condition is exactly consistent with that of previously investigated PT-symmetric structure where the invisibility occurs at a unidirectional point [7].

3. Conditions for unidirectional reflectionless phenomenon

Applying the unidirectional reflectionless condition $| C_{- m} |^{2} = 0$ or $| C_{m} |^{2} = 0$ into Eq. (3).2), the necessary and sufficient (NS) condition to achieve unidirectional reflectionless phenomenon is obtained as:

{\begin{matrix} a + b = \frac{2 q + 1}{2 m} Λ \begin{matrix}  \end{matrix} a n d \begin{matrix}  \end{matrix} a, b \neq \frac{q}{m} Λ, q = 0, 1, \dots m - 1 \begin{matrix} \begin{matrix}  \end{matrix} & (7.1) \end{matrix} \\ \begin{matrix}  \end{matrix} n_{R} = \pm n_{I} \cot ϕ_{a}, \begin{matrix}  \end{matrix} ϕ_{a} = m \frac{π}{Λ} a \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} & (7.2) \end{matrix} \end{matrix} \end{matrix} \end{matrix}

where the symbols “

+

” and “

-

” in Eq. (7).2) correspond to the cases of

R_{L} = 0

and

R_{R} = 0

respectively.

Equations (7).1) and (7.2) show that the NS condition puts constraint both on the thicknesses of the layers and the refractive index modulation amplitudes. This constraint is graphically illustrated in Fig. 2 by plotting $\log_{10} \frac{R_{L}}{R_{R}}$ versus the normalized thicknesses $a / Λ$ and $b / Λ$ while supposing Eq. (7).2) is satisfied, e.g., $n_{R} = n_{I} \cot ϕ_{a}$ . In Fig. 2, the dark lines in each subplot formed by the points of $\log_{10} \frac{R_{L}}{R_{R}} < - 6$ indicate the left-side reflectionless cases. These lines just locate at $a + b = (2 q + 1) Λ / 2 m$ , which is consistent with Eq. (7).1). As a byproduct, the symmetric reflection cases, $\log_{10} \frac{R_{L}}{R_{R}} = 0$ , are shown in the graphs marked with red lines which locate at $a + b = q Λ / m$ , $a = q Λ / m$ , and $b = q Λ / m$ . The blue lines at $a = (2 q + 1) Λ / 2 m$ are also for symmetric reflection cases since $\log_{10} \frac{R_{L}}{R_{R}} = 0$ . But $n_{R} = 0$ in these cases and structures degenerate to the undesirable dual layered materials. As for the bright spots at points ( $a = (2 q + 1) Λ / 2 m$ , $b = q Λ / m$ ), they are the mathematical singularities of $\frac{R_{L}}{R_{R}} = \frac{0}{0}$ for the consideration of Eq. (7).2). And they belong to the symmetric reflection cases since $R_{L} = R_{R}$ when Eq. (7).2) is not considered.

Fig. 2 Theoretical results about $\log_{10} \frac{R_{L}}{R_{R}}$ as a function of $a / Λ$ and $b / Λ$ under the condition of $n_{R} = n_{I} \cot ϕ_{a}$ . $m = 1 - 4$ are the cases that the propagation constants of incident wave are around 1st-4th order reciprocal lattice vectors of PTLM respectively. The red lines at $a + b = q Λ / m$ , $a = q Λ / m$ , and $b = q Λ / m$ represent the symmetric reflection cases: $\log_{10} \frac{R_{L}}{R_{R}} = 0$ ( $R_{L} = R_{R}$ ). The blue lines are also for $\log_{10} \frac{R_{L}}{R_{R}} = 0$ but do not meet requirement for the degeneration of structures. The two yellow cross markers in subplot of $m = 2$ denote the parameter coordinates of the two structures simulated in Fig. 3.

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To corroborate the theoretical NS condition, two specific unidirectional reflectionless PTLM (loss and gain) are designed according to Eqs. (7).1) and (7.2) at $m = 2$ and are numerically simulated by finite-element method (FEM). The parameters are chosen as: $a = b = Λ / 8$ , $n_{R} = n_{I} = 0.009$ for loss PTLM and $a = b = 3 Λ / 8$ , $n_{R} = - n_{I} = 0.009$ for gain PTLM (the two yellow cross markers in Fig. 2). Other common parameters are set as: $n_{0} = 3.42$ , $Λ = 300 n m$ , and $N = 100$ . And the peak wavelength of the reflected wave in both structures can be theoretically estimated as: $λ = \frac{2 π}{β_{m G}} n_{e f f} = \frac{2 Λ}{m} n_{e f f} = 1026 n m$ . The coresponding theoretical and simulated reflectivities are shown in Fig. 3.

Fig. 3 Theoretical (red curves) and simulated (black curves) reflectivities of both loss and gain unidirectional reflectionless PTLM. (a), (b) for the loss PTLM with $a = b = Λ / 8$ and $n_{R} = n_{I} = 0.009$ ; (c), (d) for the gain PTLM with $a = b = 3 Λ / 8$ and $n_{R} = - n_{I} = 0.009$ . For both structures $n_{0} = 3.42$ , $Λ = 300 n m$ and $N = 100$ . The peak wavelength here is absolutely decided by $Λ$ and $m$ . Other wavelength can be designed as an example as long as proper $Λ$ and $m$ are chosen.

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Figure 3 shows that the simulated values of $R_{L}$ are at least five orders of magnitude smaller than those of $R_{R}$ at the peak wavelength $λ = 1026.5 n m$ . In Figs. 3(a) and 3(c), the simulated $R_{L}$ are drifted from the theoretical ones by the order of $10^{- 5}$ , which may arise from the simulation error and the neglect of the second order terms of $n_{R}$ and $n_{I}$ in deriving the NS condition. Nevertheless, it demonstrates that the unidirectional reflectionless phenomena have been achieved and the simulated results are consistent with the theoretical results both for loss and gain PTLM.

4. Explanation of unidirectional reflectionless phenomenon

The realization of unidirectional reflectionless phenomenon both in loss and gain PTLM indicates that the vanishing of one-side reflection does not originate from the absorption of loss medium as that in coupled waveguide system [13–15]. The underlying physical mechanism can be revealed by rewriting the Fourier expansion coefficients [Eq. (3).2)] as:

C_{m} = F (n_{R}) + G (n_{I}),

C_{- m} = F^{*} (n_{R}) - G^{*} (n_{I}) .

Here

\begin{array}{l} F (n_{R}) = i \frac{2 n_{0} n_{R}}{π m} {[\cos (m \frac{2 π}{Λ} a) - 1] - i \sin (m \frac{2 π}{Λ} a)}, \\ G (n_{I}) = i \frac{2 n_{0} n_{I}}{π m} {[\sin (m \frac{2 π}{Λ} (a + b)) - \sin (m \frac{2 π}{Λ} a)] + i [\cos (m \frac{2 π}{Λ} (a + b)) - \cos (m \frac{2 π}{Λ} a)]} . \end{array}

Equations (8.1) and (8.2) show that $C_{\pm m}$ which directly relate to the mode coupling coefficients can be mathematically separated into two parts. $F (n_{R})$ is provided by the real modulation and $G (n_{I})$ is provided by the imaginary modulation. At unidirectional reflectionless point, the NS condition is satisfied, giving $F (n_{R}) = \pm G (n_{I})$ . For $F (n_{R}) = G (n_{I})$ , Eq. (8).2) implies that equal amplitude and opposite phase between the two waves reflected by the real and the imaginary modulation layers lead left side reflectionless: $C_{- m} = 0$ ( $R_{L} \propto {| C_{- m} |}^{2}$ ), while Eq. (8).1) implies that the two reflected waves are in-phase on the right side and non-zero reflection appears: $C_{m} = 2 F (n_{R})$ . The unidirectional reflectionless phenomenon occurs. Correspondingly, for $F (n_{R}) = - G (n_{I})$ , the right-side reflection disappears ( $C_{m} = 0$ ) while a non-zero left-side reflection appears ( $C_{- m} = 2 F^{*} (n_{R})$ ).

Numerical simulations are performed to verify the physical pictures explained above. Figures 4(a)-4(f) show the propagating light intensity distributions of $λ = 1026.5 n m$ in the yz plane for three different structures: the PTLM, the corresponding real and imaginary modulation binary multilayers (RBM and IBM). Figures 4(a), 4(c) and 4(e) are for left incident cases and Figs. 4(b), 4(d) and 4(f) are for right incident cases. All the structure parameters are the same as those in Figs. 3(a) and 3(b), but $n_{I} = 0$ for RBM and $n_{R} = 0$ for IBM. Figures 4(g) and 4(h) are the intensity distributions along z axis for the three structures and they are plotted with black, red, and blue curves respectively. The interference fringes in Figs. 4(b)-4(f) originate from the interference between the incident and the reflected beams. In Figs. 4(c) and 4(e), the fringe distributions are spatially complementary, which can be clearly seen from Fig. 4(g) (the red and blue curves). This indirectly indicates that the beams reflected by the RBM and IBM carry opposite phases for the reflectionless case (left-side incidence). For the PTLM, of which real and imaginary modulation media are simultaneously contained, the beams reflected by these two media interfere destructively and cancel each other out. As a result, almost no interference fringes are observed in Fig. 4(a). On the contrary, Figs. 4(d) and 4(f) indicate that the two reflected beams are in-phase for right-side incident case. As a result, in the PTLM, constructive interference occurs between them. And they together re-interfere with the incident wave forming the interference fringes in Fig. 4(b). This physical mechanism implied in Eqs. (8).1) and (8.2) is different from that of metamaterial cloak [16–18] and will benefit the exploration of other valuable applications of complex optical potential structures.

Fig. 4 Intensity distribution of $λ = 1026.5 n m$ in the three structures. (a), (b) for the PTLM; (c), (d) for the RBM; (e), (f) for the IBM. And (a), (c), (e), (g) show the left incident cases; (b), (d), (f), (h) show the right incident cases; (g) and (h) are the intensity distributions along z axis in the three structures and they are plotted with black, red, and blue curves respectively. All the plots presented are four-unit-cells length near the two sides of the structures.

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5. Parameter choice guide

Furthermore, according to the NS condition we give a qualitative theoretical guiding scheme about how to choose appropriate thickness for each layer to ensure an expectant side reflectionless when the three elementary media are determined. Taking $m = 2$ for instance, the guide is divided into two cases according to the sign of $\cot ϕ_{a}$ and is presented in Fig. 5 as a form of quadrant diagram. In Fig. (5), the horizontal and vertical coordinates represent the real and the imaginary modulation amplitudes ( $n_{R}$ and $n_{I}$ ) respectively. $n_{R} > 0$ ( $n_{R} < 0$ ) denotes the high (low) refractive index modulation dielectric, and $n_{I} > 0$ ( $n_{I} < 0$ ) denotes the loss (gain) modulation medium. Thus each quadrant corresponds to a group of elementary media. For $\cot ϕ_{a} > 0$ where $0 < a < Λ / 4$ or $Λ / 2 < a < 3 Λ / 4$ , the qualitative relations between the modulation amplitudes and reflection coefficients are presented in Fig. 5(a). Left-side incidence can be reflectionless ( $R_{L} = 0$ ) when the modulation amplitudes are located in the first or third quadrants. And the right-side incidence can be reflectionless ( $R_{R} = 0$ ) for the cases in the second and fourth quadrants. Figure 5(b) is for $\cot ϕ_{a} < 0$ where $Λ / 4 < a < Λ / 2$ . The reflectionless conditions in all the four quadrants are reversed now compared with the cases of $\cot ϕ_{a} > 0$ , i.e. $R_{L} = 0$ in the second and fourth quadrants and $R_{R} = 0$ in the first and third quadrants. For instance, if the two given modulation media respectively satisfy $n_{R} < 0$ and $n_{I} > 0$ , $a$ should be within $Λ / 4 ~ Λ / 2$ to achieve left-side reflectionless phenomenon [Fig. 5(b)], while $0 ~ Λ / 4$ or $Λ / 2 ~ 3 Λ / 4$ to achieve right-side reflectionless phenomenon [Fig. 5(a)]. Actually, to ensure the same side reflectionless, the two value ranges of $a$ can be exchanged only if the spatial distribution of the real and the imaginary modulation media are correspondingly exchanged. Other cases for $m \neq 2$ can be also analyzed in a similar way.

Fig. 5 Quadrant diagrams about the guiding choice of $a$ and $b$ to realize an expectant side reflectionless when the three elementary media are determined. Here $m = 2$ is taken for instance.

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6. Summary

Finally, it is necessary to mention that only unidirectional reflectionless does not result in perfect unidirectional invisibility because the transmittivity does not exactly equal to 1 due to the existence of the loss (or gain) medium. However, this invisibility can be achieved by adding another periodic gain (or loss) layered medium to compensate for the existing loss (or gain). Moreover, our theory is not limited to the structure with rectangular optical potential distribution. It is also applicable to other periodic complex potential with more complicated function forms, such as liner, parabolic, quadratic, etc.

In brief, we have proposed a design scheme of a ternary layered unidirectional reflectionless material. The structure contains only one kind of complex medium (either loss or gain). It is relatively facile for practical fabrication due to its layered characteristic and spatial separation between the real and the imaginary modulation media. The necessary and sufficient parametric requirements for unidirectional reflectionless phenomenon of the PTLM are derived. The underlying mechanism is explained by destructive interference between waves reflected from the real and the imaginary refractive index modulation layers. Clear quadrant diagrams to guide the choice of the parameters ( $n_{R}$ , $n_{I}$ and $a$ ) to realize unidirectional reflectionless phenomenon are given. We believe that our work is intriguing for the researches of unidirectional light propagation and has great potential applications in on-chip integrated optics.

Acknowledgments

This work is supported by National Key Basic Research Program of China (2012CB921900) and National Natural Science Foundation of China (11274293, 61377053). E. Yang also thanks X. Xiong for the modifications on the essay writing.

References and links

1. M. Greenberg and M. Orenstein, “Unidirectional complex grating assisted couplers,” Opt. Express 12(17), 4013–4018 (2004). [CrossRef] [PubMed]

2. M. Kulishov, J. Laniel, N. Bélanger, J. Azaña, and D. Plant, “Nonreciprocal waveguide Bragg gratings,” Opt. Express 13(8), 3068–3078 (2005). [CrossRef] [PubMed]

3. M. Greenberg and M. Orenstein, “Irreversible coupling by use of dissipative optics,” Opt. Lett. 29(5), 451–453 (2004). [CrossRef] [PubMed]

4. L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 54(3), 2963–2975 (1996). [CrossRef] [PubMed]

5. C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998). [CrossRef]

6. C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40(5), 2201–2229 (1999). [CrossRef]

7. Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011). [CrossRef] [PubMed]

8. A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012). [CrossRef] [PubMed]

9. S. Longhi, “Invisibility in PT-symmetric complex crystals,” J. Phys. A Math. Theor. 44(48), 485302 (2011). [CrossRef]

10. L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012). [CrossRef] [PubMed]

11. L. Ge, Y. D. Chong, and A. D. Stone, “Conservation relations and anisotropic transmission resonances in one-dimensional PT -symmetric photonic heterostructures,” Phys. Rev. A 85(2), 023802 (2012). [CrossRef]

12. S. Longhi, “PT-symmetric laser absorber,” Phys. Rev. A 82(3), 031801 (2010). [CrossRef]

13. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010). [CrossRef]

14. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009). [CrossRef] [PubMed]

15. H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82(4), 043803 (2010). [CrossRef]

16. A. Alù, “Mantle cloak: Invisibility induced by a surface,” Phys. Rev. B 80(24), 245115 (2009). [CrossRef]

17. A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(1), 016623 (2005). [CrossRef] [PubMed]

18. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006). [CrossRef] [PubMed]

Unidirectional reflectionless phenomenon in periodic ternary layered material

Abstract

1. Introduction

2. Design theory of PTLM

3. Conditions for unidirectional reflectionless phenomenon

4. Explanation of unidirectional reflectionless phenomenon

5. Parameter choice guide

6. Summary

Acknowledgments

References and links

Cited By

Figures (5)

Equations (15)

Optics Express