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We investigated defect states in band gaps of one-dimensional photonic lattices with delicate modulations of gain and loss that respect parity-time-symmetry (PT-symmetry), viz. n(z) = n*(−z). For the sake of generality, we employ not only periodic structures but also quasiperiodic structures, e.g. Fibonacci sequences, to construct aperiodic PT lattices. Differed from lossless systems for which the defect state is related to only one exceptional point (EP) of the S-matrix, we observed the splitting of one EP into a pair after the introduction of judiciously designed gain and loss in those PT systems, where the defect state enters a non-threshold broken symmetry phase bounded by the EP pair. Some interesting properties associated with defect states and EP splitting are demonstrated, such as enhanced spectral localization, double optical phase abrupt change, and wavelength sensitive reversion of unidirectional transparency.
© 2015 Optical Society of America
1. Introduction
Wave dynamics in complex photonic structures with parity-time-symmetry (PT-symmetry) have been one of vital topics in optical physics and laser applications in the past few years [1–13 ]. By studying the fundamentals of PT-symmetry and the associated properties on a platform with delicately balanced gain and loss, we surprisingly observe various unique properties in either linear or nonlinear regimes, such as PT phase transition [2], coherent lasing and perfect absorption [3,4 ], unidirectional invisibility [5–8 ], one-way localization [9], and nonreciprocal optical isolation [10,11 ]. From a physical point of view, the eigenfield distribution plays a dominant role in describing the interaction between light and PT material, and therefore is helpful for us to understand those properties in depth [9]. Tailoring the eigenfield distributions in purpose creates many intriguing physical features. One of the representative examples is the defect state localized at the interface between two photonic structures with overlapping band gaps [14–17 ]. The sufficient condition for a localized state to exist in the interfacial region of two optical materials is generally that their surface impedances are imaginary numbers of opposite signs [18]. Other works show the formation of defect states inside varied cavity-like structures of quasiperiodic crystals [19,20 ]. As a matter of fact, interface states can be regarded as being trapped inside a cavity with zero length, thus giving rise to a more generalized criterion for predicting the existence of defect states in photonic structures [21]. Recently, researchers have found different kinds of defect states in PT lattices with either symmetric or broken PT phases, which are embedded in the continuous spectrum of scattered states [22,23 ].
In this paper, we explore the exceptional point (EP) dynamics associated with defect states in one dimensional (1D) PT systems. According to Kato's book [24], EPs are branch point singularities of the eigenvalues and eigenvectors, which occur when a matrix (here we use the S-matrix for our two-port 1D scattering model [5,7,12 ], see Fig. 1 ) is analytically continued in a related parameter. In Hermitian systems, EP corresponds to the coalescence of eigenvalues but the associated eigenvectors remain linearly independent, while in the non-Hermitian (e.g. PT) systems, EP could correspond to the coalescences of eigenvectors as well. To ensure the generality, we study the scattering properties of periodic and aperiodic PT lattices in Fig. 1 and observe a new class of EP dynamics, viz. EP splitting. The effect is caused by the interplay between PT-symmetry and defect states in those lattices. As we know, the resonance phenomena (Febry-Pérot and Fano resonances, etc.) in lossless systems with an abrupt reflection phase change of π are typically associated with the coalescence of eigenvalues or an EP of the S-matrix. However, a judiciously designed PT modulation of perturbed optical gain and loss will bring about the splitting of one EP into a pair, which fundamentally broadens EP dynamics. In the non-threshold broken symmetry phase bounded by the EP pair, the spectral localization of defect states will be enhanced, while the EP pair permits the possibilities of other interesting features, such as double phase abrupt change and wavelength sensitive reversion of unidirectional transparency. Since high-speed phase and intensity modulators are of great importance in modern optical communication, our findings would provide a unique route towards the next generation of optical networks and sensing devices.
Fig. 1 Schematic of two PT lattices with periodic sequences and Fibonacci sequences, where the complex refractive indices are distributed in n(z) = n*(−z) about z = 0. For the primitive unit-cell layer A, B, C, and D, the thickness is L and the complex refractive indices are n + Δn + 0.01τi, n−Δn−0.01τi, n−Δn + 0.01τi, and n + Δn−0.01τi, respectively. The imaginary part is positive for loss material and negative for gain material.
2. PT lattices composed of periodic sequences
Before studying the PT lattices of periodic sequences shown in Fig. 1, we first consider the photonic crystals (PC) comprising periodic unit-cells (thickness: 2L). The modulation of index of refraction in each unit-cell is real and given by n(z) = n 2, 0<z<L and n(z) = n 1, L<z<2L. In this periodic system, the homogeneous α layer (α = 1,2) of the jth unit-cell has a forward mode aj ( α )exp[ikα(z−j2L)] and a backward mode bj ( α )exp[−ikα(z−j2L)]. Using the transfer matrix method [25], we can relate the coefficients (a 0,b 0) to (aN,bN) through
Here we define an≡an (1), bn≡bn (1), and UN = sin2(N + 1)KL/sin2KL with the Bloch wave number K = arccos[(C 1 + C 4)/2]/2L. The coefficients C 1, C 2, C 3, and C 4 are respectivelywhere kα = 2πnα/λ and λ is the wavelength in vacuum. For the PC of N unit cells, the coefficient of reflection r is derived from Eq. (1) by setting bN = 0where ξ = UN −2/UN −1. In a band gap region (|r|→1), we can find a specific frequency point at which the reflection phase Φ is exactly 0 or π(−π), subject to whether or not considering the half-wave loss in the case of k 1 larger or smaller than k 2 [26]. Therefore, when k 1<k 2, r = −1 at that peculiar point. From Eq. (3), we deduce the relationsBy eliminating ξ, Eq. (4) is further simplified intoFrom Eq. (5), we may predict the frequency at which the reflection phase is exactly π(−π). If we put together two pieces of such PCs in mirror symmetry about z = 0, the sum of reflection phases of the left and right PCs at the interface are 2π(−2π) at that frequency, eventually giving rise to a localized state by satisfying the resonance condition. For another case of k 1>k 2, we can obtainbased on similar derivations mentioned above. Due to the structural symmetry, the field distribution of the localized defect state is also symmetric about z = 0.In the next step, besides the real modulation being symmetric about z = 0, we further apply a weak gain-loss modulation being antisymmetric to z = 0 to realize a PT system [n(z) = n*(−z)]. Here the imaginary modulation strength is only a perturbation compared to the real component. Due to the intrinsic field amplification and attenuation in gain and loss regions, the defect state in PT systems is more localized in the gain part, driving the whole system into a broken symmetry regime. Previous studies [12,27,28 ] show that the scattering PT systems have a pseudounitary conservation relation of RfRb = (T−1)2, where Rf (b)≡|rf ( b )|2 and T≡|t|2 are the forward (backward) reflectances and the transmittance. The eigenvalues and eigenvectors of the scattering matrix S = [t rb; rf t] for such PT systems are given by λ 1,2 = t ± (rfrb)0.5 = t[1 ± i(1/T−1)0.5] [5,12,27 ] and [1 ± (rf/rb)0.5] (rb≠0) or [ ± (rb/rf)0.5 1] (rf≠0), respectively [12]. When the system is in the broken symmetry phase, the eigenvalues are nonunimodular and T>1. We can intuitively infer that the broken symmetry phase is bounded by two EPs at different frequencies where T = 1 and Rf ( b ) = 0, whether or not the broken phase is thresholdless. In the end, we emphasize that the spectral position of the defect state predicted by Eqs. (5) or (6) is not much influenced by the trivial gain-loss parameter, which will be demonstrated later in numerical examples. It also needs to be mentioned that the resonance-based defect state is not necessary to locate in the band gaps when we introduce PT-symmetry [29], since the amplitude and phase of the reflection coefficients for some peculiar PT systems can be independently and arbitrarily tailored at EPs with the transmittance being unitary [8].
As an example, we consider a 1D PT lattice consisting of 7 periodic unit cells (7AB) for the left PC and 7 periodic unit cells (7CD) for the right PC, embedded in a homogeneous background shown in Fig. 1. The parameters are L = 50 nm, n = 3, τ = 1, Δn = 0.2. In our numerical model, we have neglected these material nonlinearity and dispersion as a simplification. Figure 2(a) shows the transmission spectrum of the PT lattice with periodic sequences in Fig. 1, where we observe two defect states associated with sharp transmission peaks in the band gaps as marked by blue stars, respectively locating at 0.08385c/L and 0.2515c/L. In Figs. 2(b) and 2(c), we plot the electric field intensity distributions of those two defect states in the PT system, when the light is incident from the left hand side. It is clear that both defect states are tightly confined at the interface, where the field intensities are exponentially decaying to the sides. As discussed in the previous part, the formation of defect states in the gaps is attributed to local resonances, where the sum of reflection phases of the left and right PCs at the interface are 2π(−2π) at that frequency. Therefore, the spectral positions of defect states can be predicted by Eqs. (5) or (6) at k 1<k 2 or k 1>k 2. To demonstrate this in our example, we calculate the curve of the left hand side of Eq. (5) by taking only real parts of k 1 and k 2, since the PT modulation of gain and loss is a trivial term (~0.01). The spectral positions of defect states in the PT system are predicted by the zero points of the curve and marked by the blue stars at the frequencies of 0.083c/L and 0.249c/L, which is in great consistence with the result of the transmission study in Fig. 2(a).
Fig. 2 (a) The transmission spectrum of the PT lattice with periodic sequences shown in Fig. 1. The refractive indices of A, B, C, and D layers are 3.2 + 0.01i, 2.8−0.01i, 2.8 + 0.01i, and 3.2−0.01i, respectively. The layer thickness L is 50 nm. (b), (c) The electric field intensity (|E(z)|2) distributions of defect states at the frequencies of 0.08385c/L and 0.2515c/L [labeled by blue stars in (a)], when the light is incident from the left hand side. c is the light speed in vacuum. (d) Curve of equation f(ω) = tan[Re(k 1)L]/Re(k 1) + tan[Re(k 2)L]/Re(k 2) versus the frequency ωL/(2πc). The spectral positions of defect states are in excellent agreement with zero points of f(ω) as labeled by blue stars in (d).
In what follows, we will show the outcome of the interplay between PT symmetry and defect states. In Fig. 3 , we calculate the forward and backward reflectances, transmittance, the arithmetic mean of forward and backward reflection phases, as well as the eigenvalues of S-matrix (Rf, Rb, T, (ϕf + ϕb)/2, λ 1 and λ 2) for the lossless system (τ = 0) and the PT system (τ = 1), respectively. As the coalescences of eigenvalues of S-matrix, exceptional points are typically responsible for an abrupt π phase change of reflection coefficients as a function of frequencies and the onset of phase breaking. The abrupt π phase change is related to a complex zero point in reflection coefficients where rf = 0 + i0 and/or rb = 0 + i0.
Fig. 3 (a), (c), (e) The calculated spectra of Rf, Rb, T, (ϕf + ϕb)/2, λ 1 and λ 2, as a function of the normalized frequency ωL/(2πc), when the system involves only real modulation (τ = 0) of index of refraction. (b), (d), (f) The calculated spectra of Rf, Rb, T, (ϕf + ϕb)/2, λ 1 and λ 2, as a function of the normalized frequency ωL/(2πc), when a weak PT modulation (τ = 1) is introduced. Rf and ϕf represents the reflectance and phase of the forward propagating light. Rb and ϕb are the reflectance and phase of the backward propagating light. T is the transmittance. λ 1 and λ 2 are the eigenvalues of the S-matrix.
It is evident that, as shown in Figs. 3(a), 3(c) and 3(e), there exists only one EP at 0.08385c/L in the lossless optical system (τ = 0). And the optical phases of reflected light in both directions experience an abrupt π phase jump at the unitary resonance transmission. Due to the mirror symmetry of real index modulation, the field distribution of the localized defect state at 0.08385c/L is also symmetric about z = 0. In Figs. 3(b), 3(d) and 3(f), the PT modulation of perturbed optical gain and loss (τ = 1) is introduced to form an open (or non-Hermitian) system. Physically, the defect state in PT systems is more localized in the gain part, due to the intrinsic field amplification and attenuation in gain and loss regions. In this case, the whole system will enter a broken symmetry regime with T>1, leading to the splitting of the EP at 0.08385c/L [Figs. 3(a), 3(c) and 3(e)] into a pair at around 0.0837c/L and 0.08399c/L [Figs. 3(b), 3(d) and 3(f)], respectively. From Figs. 3(b) and 3(d), we can clearly observe a wavelength sensitive reversion of unidirectional transparency (Rf≈0.035, Rb = 0, T = 1→Rf = 0, Rb≈0.035, T = 1) that respects the pseudounitary conservation relation. A double phase abrupt change of π/2 at those two EPs is also shown. In Figs. 3(e) and 3(f), the eigenvalue spectra vividly show the process of EP splitting from one into a pair. From the expressions of eigenvalues of S-matrix in PT systems, we find that the eigenvalues are nonunimodular (|λ 1|>1, |λ 2|<1, |λ 1 λ 2| = 1 or |λ 1|<1, |λ 2|>1, |λ 1 λ 2| = 1) in the broken symmetry phase between the EP pair (T>1). In other regions (T<1), the system operates in the symmetric phase with eigenvalues being unimodular (|λ 1| = |λ 2| = 1). Here we point out that the eigenvalues of S-matrix are very different from the eigenvalues of Hamiltonians. For example, for a close (or Hermitian) system, the eigenvalues of S-matrix λ 1,2 = t ± (rfrb)0.5 can be complex as shown in Fig. 3(e) but the eigenvalues of Hermitian Hamiltonians must be real with orthogonal eigenstates [24,27 ]. Therefore, as shown in Ref. 27, the phase transition of S-matrix defined by S = [t rb; rf t] cannot always reflect the PT symmetry breaking in the underlying effective Hamiltonian.
In the end of this section, we will present the “phase diagram” of our system in Fig. 4(a) . The results show that the phase transition (the darken region) is thresholdless as the strength of gain-loss modulation τ increases. The EP at τ = 0 is splitting into a pair when τ>0, and the defect states marked by the green line immediately enter the broken symmetry phase bounded by the EP line pair (blue and red lines). For the EPs at the blue line, the lattice is unidirectionally reflectionless for the forward propagation (rf = 0, rb≠0). However, for the EPs at the red line, the lattice is unidirectionally reflectionless for the backward propagation (rf≠0, rb = 0). We note that the EP at τ = 0 corresponds to a special case of bidirectional transparency (rf = rb = 0). It should be mentioned that for the EP at τ = 0, the lattice is a closed system (or Hermitian) with no loss and gain. The eigenvalues of S-matrix are degenerated at λ 1,2 = t, but the eigenvectors can be arbitrarily chosen and thus independent with each other. For the EPs at τ>0, the lattice turns into an open system (or non-Hermitian), where both eigenvalues and eigenvectors are degenerated. Specifically, the degenerated eigenvectors are [1 0] for the EPs at the blue line and [0 1] for the EPs at the red line. On the other hand, the sharp transmission peak associated with the defect states is more spectrally localized with a higher Q factor (defined by the center frequency over the FWHM of the transmission peak) when the strength of gain-loss modulation increases. For example, Qτ = 5/Qτ = 0≈1.021, inferred from Fig. 4(b). As mentioned previously, the defect state in PT systems is more localized in the gain, due to the intrinsic field amplification and attenuation in gain and loss regions. Therefore, the enhanced spectral localization of the defect states with T>1 is owing to the longer time delay in gain than in loss for the transmitted photons in the broken symmetry phase [28].
Fig. 4 (a) Phase transition and (b) Q factor (normalized by the Q at τ = 0) as the strength of gain-loss modulation increases from zero. The broken symmetry phase is indicated by the darken area, which is thresholdless. Exceptional points (or phase transition points) are located at blue and red lines separating symmetric and broken symmetry phases. Defect states are located at the green line in the broken symmetry phase.
3. PT lattices composed of aperiodic sequences
To demonstrate the generality of EP splitting associated with defect states in PT systems, we also explore the scattering properties of the aperiodic PT lattice, such as Fibonacci sequences, as shown in Fig. 1.
Considered the primitive unit-cell layers A and B. Fibonacci sequences are created in Fig. 5 according to the production rule B→BA, A→B with F 1 = B or Fj = Fj −1|Fj −2 for j≥3 with F 1 = B and F 2 = BA [19]. The aperiodic PT lattice in Fig. 1 is further constructed by combining two different Fibonacci (F 6) sequences together, where the PT symmetry is respected for the whole system.
Fig. 5 Fibonacci sequence produced by the rules B→BA, A→B with F 1 = B or Fj = Fj −1|Fj −2 for j≥3 with F 1 = B and F 2 = BA.
In Fig. 6 , we show the interplay between PT symmetry and defect states for the Fibonacci lattice. We plot the forward and backward reflectances, transmittance, the arithmetic mean of forward and backward reflection phases, and the eigenvalues of S-matrix for both the lossless Fibonacci system (τ = 0) and the PT Fibonacci system (τ = 1). Previous studies [19,20 ] have shown that quasiperiodic systems have a lot of cavity-like structures to trap defect states, which results in the multi-splitting of band gaps. In Figs. 6(a), 6(c), and 6(e), we observe two EPs at 0.0623c/L and 0.073c/L when τ = 0. In this case, the optical phases of reflected light experience an abrupt π phase jump in both directions at the unitary resonance transmission, marked by the arrows in Fig. 6(c). After the introduction of perturbed PT modulation (τ = 1), the two EPs are respectively splitting into pairs (four EPs), as shown in Figs. 6(b), 6(d), and 6(f). Those four EPs are located at around 0.06227c/L, 0.06266c/L, 0.07286c/L, and 0.07314c/L, along with two wavelength sensitive reversions of unidirectional transparency. Specifically, Rf≈0.012, Rb = 0, T = 1 at 0.06227c/L→Rf = 0, Rb≈0.012, T = 1 at 0.06266c/L and Rf≈0.003, Rb = 0, T = 1 at 0.07286c/L→Rf = 0, Rb≈0.003, T = 1 at 0.07314c/L. We also observe the abrupt phase change of π/2 at each of the four EPs, marked by the arrows in Fig. 6(d). Figures 6(e) and 6(f) present the eigenvalue spectra evolution, when the complex modulation strength τ = 0 turns into τ = 1. The coalescences (or EPs) of eigenvalues of the S-matrix in Fig. 6(e) are all splitting into pairs in Fig. 6(f) with loops formed in-between. The loop regions correspond to the broken symmetry phases, where the eigenvalues are nonunimodular and the wave transmission is larger than unitary.
Fig. 6 (a), (c), (e) The calculated spectra of Rf, Rb, T, (ϕf + ϕb)/2, λ 1 and λ 2, as a function of the normalized frequency ωL/(2πc), when the system involves only real modulation (τ = 0) of index of refraction. (b), (d), (f) The calculated spectra of Rf, Rb, T, (ϕf + ϕb)/2, λ 1 and λ 2, as a function of the normalized frequency ωL/(2πc), when a weak PT modulation (τ = 1) is introduced. Rf and ϕf represents the reflectance and phase of the forward propagating light. Rb and ϕb are the reflectance and phase of the backward propagating light. T is the transmittance. λ 1 and λ 2 are the eigenvalues of the S-matrix.
Figure 7 shows the electric field intensity (|E(z)|2) distributions of two defect states in the PT Fibonacci lattice at the frequencies of 0.06246c/L and 0.073c/L [labeled by blue stars in Fig. 6(b)], when the light is incident from the left hand side. In stark contrast to the cases in Figs. 2(b) and 2(c), the defect states in the PT Fibonacci lattice are localized at some cavity-like structures (e.g. BABBAB) after carefully comparing Fig. 7 with Fig. 5, instead of being trapped at the contact interface of two periodic lattices. Despite the different PT lattice structures, EP splitting and the related effects are the same outcome due to the similar underlined physical pictures.
Fig. 7 Electric field intensity (|E(z)|2) distributions of two defect states in the PT Fibonacci lattice at the frequencies of (a) 0.06246c/L and (b) 0.073c/L [labeled by blue stars in Fig. 6(b)], when the light is incident from the left hand side.
4. Conclusions
As a summary, we have shown that defect states can be easily formed in the band gaps of various PT lattices (periodic sequences and Fibonacci sequences). The interplay between PT-symmetry and defect localizations will result in the splitting of one EP into a pair with some unique properties. For example, we find that the defect states exist in a non-threshold broken symmetry phase bounded by the EP pair with an enhanced spectral localization. Some other intriguing effects are numerically demonstrated, such as double phase abrupt change and wavelength sensitive reversion of unidirectional transparency. Our study enriches the EP dynamics and provides a new direction to design sensing devices with an extra freedom. Of interest will be to investigate the interplay between PT-symmetry and defect states in higher dimensions.
Acknowledgments
This work was supported by the National Natural Science Foundation of China under No.11404125, the financial support from the Bird Nest Plan of HUST, and the Fundamental Research Funds for the Central Universities (HUST 2015TS024).
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