Abstract
Paritytime (PT) symmetric structures present the unidirectional invisibility at the spontaneous PTsymmetry breaking point. In this paper, we propose a PTsymmetric circuit consisting of a resistor and a microwave tunnel diode (TD) which represent the attenuation and amplification, respectively. Based on the scattering matrix method, the circuit can exhibit an ideal unidirectional performance at the spontaneous PTsymmetry breaking point by tuning the transmission lines between the lumped elements. Additionally, the resistance of the reactance component can alter the bandwidth of the unidirectional invisibility flexibly. Furthermore, the electromagnetic simulation for the proposed circuit validates the unidirectional invisibility and the synchronization with the input energy well. Our work not only provides an unidirectional invisible circuit based on PTsymmetry, but also proposes a potential solution for the extremely selective filter or cloaking applications.
Introduction
PTsymmetry presents an interesting performance that the Hamiltonian has the entirely realenergy spectrum below the phase transition point, which can be extended to quantum theory^{1,2,3,4}. Recently, the PTsymmetric theory in quantum mechanics is introduced to optical field^{5}. The classical systems consisting of amplifier^{6}, photon^{7} or periodic nanostructure^{8,9,10} lead to the achievement of the PTsymmetry in the optical frequencies, which has open up a new perspective towards achieving the optical waveguide^{6}, power oscillation^{8,11}, lossinduced transparency^{12}, nonreciprocal Bloch oscillations^{13}, laser absorber^{14}, unidirectional invisibility^{15,16,17}, and various extraordinary nonlinear effects^{18,19,20}. Furthermore, the PTsymmetry can be used in the optical device, e. g. the opticallocking component^{21} and promises applications in electric and acoustic field additionally^{22,23}.
In optics, the striking performance of the PTsymmetry is the unidirectional invisibility at the spontaneous PTsymmetry breaking point^{15,16,17}. The optical systems present reflectionless around the Bragg resonance at the one side, and enhanced reflectivity from the other side contrarily^{15}. Furthermore, when the lossy and active sheets are inserted in front and behind the almost fully reflective sheets, the incident waves are replicated behind the PTsymmetric structure in synchronization with the input signal^{24}. Based on the effectivemapping image between the quantum and the classical mechanism, the circuit and the microwave system can be devoted to the PTsymmetry devices in theory and experiment^{25,26}. In the electric system, the lumped resistor and amplifier are analogous to the attenuation and amplification in the complex potential of Hamiltonian operator.
In this paper, we formulate the PTsymmetric circuit consisting a resistor and a microwave TD with the negative impedance which represent the attenuation and amplification. Based on the scattering matrix method, the circuit presents extremely weak reflection at one side and enhanced reflection from the other side respectively. Additionally, the bandwidth of the unidirectional invisibility can be tuned by the resistance of the reactance element. Furthermore, the electromagnetic simulation validates the unidirectional invisibility and the “teleport” performance in the proposed circuit. Our work provides the PTsymmetry in electric circuit and opens up the possibilities to construct the extremely selective filter and electriccloaking applications.
Results
The layout of the PTsymmetric circuit is shown in Fig. 1. The added capacitors C_{a} = 1 μF with package 0805 are placed on the both sides of the circuit for DCblocking. In addition, the strip line L_{1} = 4 mm is the platform of the external interface. The capacitor C_{x} = 4 pF with package 0603 is parallel in the middle of the main thread, and the distance between C_{x} and C_{a} is L_{2} = λ/4 = 27.3 mm. The resistor R = 50 Ω with package 0603 is laid parallel in the input, and the tunnel diode TD261A with the negative resistor R_{d} = −50 Ω is parallel in the output. The width of the strip line is W = 4.7 mm for the characteristic impedance Z_{0} = 50 Ω, and the via connects the electric components to the background of the circuit board, as shown in Fig. 1(b). The DCsource system of the TD261A is implemented at the output of the main thread. The length and the width of the serpentine line is L = L_{3} + L_{4} + 4L_{5} + 3L_{6} = λ/2 = 55.6 mm and 0.4 mm for blocking the AC signal on the main thread and flowing the DC energy effectively, as shown in Fig. 1(c). Under the perfect reflection of the reactance component, the attractive performance of the proposed circuit presents replicating the input signal in synchronization behind the structure^{24}. More specifically, at the spontaneous PTsymmetry breaking point, the reflection from one end of the circuit is diminished while it is enhanced from the other, and the transmission coefficient is nearly unitary which presents the unidirectional invisibility, as shown in Fig. 1(d).
Theory
We formulate the PTsymmetric circuit consisting of the lumped elements and the microstrip line, as shown in Fig. 2. The negative resistor is implemented by the TD which possesses the relation U_{d} = −IR_{d}^{27}, and the loss performance is realized by the lumped resistor R. Here we regulate the voltage U_{d} across the TD to realize the balancedresistance relation R_{d} = R for PTsymmetric distribution. The two parallel resistors are separated by two transmission lines of which the electric lengths are l_{1} = kd_{1}, l_{2} = kd_{2} and the characteristic impedance is Z_{0}, in which k is the wave number and d_{1,2} is the physical lengths of the transmission lines. Furthermore, the resistance of reactance component which consists the capacitor C or the inductor L is X = 1/ωC or X = ωL between the two transmission lines. Here we note r and x which represent the normalized resistances r = R/Z_{0} and x = X/Z_{0}. In order to eliminate the reflection at port 1, the input impedance is consistent with the characteristic impedance of the transmission lines that is R = Z_{0} and r = 1. The scattering matrix of the electric circuit can be calculated by the transmissionmatrix approach (Supplementary Note 1):
It is straightforward to verify that the S matrix fulfills the PTsymmetry formation PTS(ω^{*})PT = S^{−1}(ω)^{28}. When the lengths of the transmission lines satisfy l_{1}, l_{2} = π/2 + Nπ where N is positive integer, the scattering matrix is:
Additionally, the eigenvalues of the Smatrix at this condition are calculated as:
In Eq. 4, the reflection from the input S_{11} is zero and the transmissions S_{12} and S_{21} are unitary independent of x. When the resistance of the reactance component is far less than the characteristic impedance Z_{0} that represents the normalized relation x ≪ 1, the eigenvalues of the scattering matrix are pure imaginary and the reflection from the output S_{22} ≫ 2 presents the highly enhanced reflection than unitary. Furthermore, the unidirectional invisibility is shown in the circuit at this condition accordingly. When the resistance of the reactance component is equal to Z_{0} that satisfies the normalized relation x = 1, the magnitude of the reflection from the output is and the eigenvalues of the scattering matrix are at the spontaneous PTsymmetry breaking point. When the resistance of the reactance component is greater than Z_{0} that presents normalized relation x ≫ 1, the eigenvalues of the scattering matrix changes to complex from the pure imaginary and the magnitude of reflection which leads to the electric circuit system presenting poorly unidirectional performance. Therefore, the resistance of the reactance component can tune the unidirectional states of the proposed circuit, and the normalized relation x = 1 is corresponding to the spontaneous PTsymmetry breaking point which is the watershed of the unidirectional invisibility.
A condition of special interest is given by l_{1} = l_{2} → π/2, for which we obtain the approximation relations that are , , and the amplitude of scattering matrix can be calculated as:
When the electric lengths of the two transmission lines are l_{1} = l_{2} which satisfy the phase relation π − 2l_{1} = 2x for fixed positive value x and d_{1}, we can get the special frequency from the electric lengths expression l_{1} = l_{2} = kd_{1} = 2πfd_{1}/v.
Here the propagation velocity v is determined by the physical condition of transmission line, which is implemented by microstripline formation. For the smaller positive value x that is x ≪ 1, the specific frequency f_{0} is slightly lower than the transparent frequency which is corresponding to the l_{1} = l_{2} = π/2. When the value of normalized quantities of reactance component x is negative, the specific frequency f_{0} is slightly higher than the transparent frequency respectively. In Eq. (6, 8), it is remarkable that the reflection amplitudes of input is fixed unitary at various specific frequencies, and the reflection amplitudes of output is far larger than unitary due to the numerator and the denominator presenting approximation constant and minimal value close to zero which is inversely proportional to the value of x. The amplitudes of transmission and expressed as at the special frequency f_{0} are larger than unitary because the electric length l_{1} → π/2, and present inversely proportion to the value of x due to the phase relation π − 2l_{1} = 2x. Additionally, when the value of x is smaller, the special frequency f_{0} closes to the transparent frequency corresponding to the phase relation l_{1} = l_{2} = π/2. Based on the above theory, we format the electric circuit consisting of the lumped elements and ideal transmission lines in which the the propagation velocity is speed of light in vacuum v = 3 × 10^{8} m/s, and the frequency is normalized by the the transparent frequency. With the various normalized quantities of reactance component x = 0.05, 0.03, −0.03, −0.05, the specialnormalized frequencies are 0.9682, 0.9809, 1.0191, 1.0318 which distribute symmetrically around the normalizedtransparent frequency f = 1. Furthermore, the reflection amplitudes of input at the special frequencies are constant unitary, as shown in Fig. 3(a). The reflection amplitudes of port 2 at the special frequencies 0.9682 and 0.9809 are 1630 and 4447.4, which presents that the reflection amplitude of output are inversely proportional to the value of x, and the reflection at the special frequencies 1.0191 and 1.0318 are mirror symmetry with the lower frequencies, as shown in Fig. 3(c). The scatteringtransmission amplitudes are 40 and 66.7 at the special frequencies 0.9682 and 0.9809, which presents the transmission amplitude are inversely proportional to the value of x, and the transmission performance at the special frequencies 1.0191 and 1.0318 are mirror symmetry with the lower frequencies, as shown in Fig. 3(b).
Electromagnetic simulation
Based on the above theory, we tune the scattering performance of the PTsymmetric circuit by the mcrostrip lines between the electric elements, which are constructed based on the 1.524mmthick Rogers5880 substrate whose dielectric constant is ε_{r} = 2.2. From the equivalently replacing the air and dielectric regions by a homogeneous medium, the effective dielectric constant of the microstrip line is given approximately by
where W and d represent the microstripline width and the substrate height. Additionally, the W/d ratio can be found as
where and .
For a given characteristic impedance Z_{0} = 50 Ω, d = 1.524 mm and the substrate dielectric constant ε_{r} = 2.2, we can get the microstripline width W = 4.7 mm. From the phase velocity and propagation constant , the length of microstrip line is d_{1} = d_{2} = 27.3 mm corresponding to the phase relation l_{1} = l_{2} = π/2^{29}.
The negative differential resistance (NDR) at microwave frequencies is offered by the quantum tunnelling semiconductor devices, which is realized by General Electric’s TD261A^{30,31}. The NDR can be equivalently considered as a negative resistance −R_{d}, acting as a current source linearly controlled by an applied voltage which is shown as in Fig. 4(a). Furthermore, the equivalent circuit of the TD261A is provided in Fig. 4(b). In the NDR region, TD261A is composed of a negative resistance −R_{d} and the parasitic components R_{p}, L_{p} and C_{p} representing the device package. Although the highfrequency limit of TD261A can be up to 26 GHz, to minimize the impact of the parasitic parameters, the working frequency of the PTsymmetric circuit is chosen to be around 2 GHz. By the bias voltage of 0.38 V, the TD261A parameters are: R_{p} = 7 Ω, L_{p} = 1.5 nH, C_{p} = 0.65 pF and −R_{d} = −50 Ω. Furthermore, we add the capacitors C_{a} = 1μF at the front and behind the main thread of the circuit and validate that C_{a} has little effect on the AC state of the circuit (Supplementary Note 3).
Here the layout simulation is carried out using the commericial electromagnetic software package (CST Microwave Studio). The Sparameter comparison of the theory and the layout simulation with the reactance component C_{x} = 4 pF is shown in Fig. 5. The amplitude of S_{11} representing the reflection from the input is about 0.1 at 2 GHz due to the influence of the equivalent parasitic capacitance and inductor of the inputresistance package, as shown in Fig. 5(a). The amplitude of S_{12}/S_{21} is about 0.9 at 2 GHz presenting a nonideal transmitted state because of the loss of the dielectric substrate and the copper. The parameter S_{22} is about 1.9 which presents a strong reflection and thus a welldefined unidirectional performance, as shown in Fig. 5(c).
Discussion
In conclusion, we formulate the PTsymmetric circuit consisting of the losscomponent resistor and the amplified TD261A. Additionally, we insert the lumped capacitor as the fully reflective element parallel between the two transmission lines. Layout simulations show that the circuit exhibits a unidirectional invisibility. Based on the scattering matrix theory, we illustrate that the circuit presents an extremely weak reflection from input and a stronger reflection than unitary from output respectively. Besides, the proposedparity PTsymmetric circuit opens up an electric unidirectional aspect and provides an extremely selective filter and electriccloaking applications.
Additional Information
How to cite this article: Lv, B. et al. Unidirectional invisibility induced by paritytime symmetric circuit. Sci. Rep. 7, 40575; doi: 10.1038/srep40575 (2017).
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Author information
Affiliations
Microwave and Electromagnetic Laboratory, Harbin Institute of Technology, No. 92, Xidazhi Street, Nangang District, Harbin City, Heilongjiang Province, China
 Bo Lv
 , Jiahui Fu
 , Qun Wu
 , Wan Chen
 , Zhefei Wang
 , Zhiming Liang
 , Ao Li
 & Ruyu Ma
School of Electronic Engineering, Xidian University, Xi’an, 710071, China
 Bian Wu
College of Information Science and Electronic Engineering, Zhejiang University, Hangzhou 310027, China
 Rujiang Li
Propagation Group, Wireless Technologies Branch, Communications Research Centre Canada, Government of Canada, 3701 Carling Ave., Box 11490, Station H, Ottawa, Ontario K2H 8S2, Canada
 Qingsheng Zeng
 & Lei Gao
Harbin medical university, No. 157, Baojian Street, Nangang District, Harbin City, Heilongjiang Province, China
 Xinhua Yin
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Contributions
Bo Lv proposed the main method and theory of the manuscript. Jiahui Fu validated the theory of the manuscript. Bo Lv and Bian Wu designed the circuit in the manuscript. Bo Lv and Rujiang Li wrote the main manuscript text. Qingsheng zeng, Xinhua Yin and Qun Wu reviewed the manuscript. Zhiming Liang and Ao Li simulated the circuit. Lei Gao, Wan Chen, Zhefei Wang and Ruyu Ma prepared Figs 1–5.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to Jiahui Fu.
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