## Abstract

The primary bottlenecks in designing and implementing PT-symmetric systems at microwave frequency ranges are noise and instability which can weakly break PT-symmetry resulting in system performance degradation. Practical implementation of such systems and devices require significant level of control and stability and it is crucial to analyze the noise performance of such systems in terms of noise figure and signal-to-noise ratio. We describe and develop a simulation model to calculate noise figure of PT-symmetric system and evaluate the performance degradation. We also discuss application design and circuit configurations that could reduce the noise figure resulting in better performance.

Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

## 1. Introduction

Parity-time (PT) symmetry is a unique space-time reflection symmetry that describes the invariance of a system to the combined parity and time-reversal operators. The parity (P) operator corresponds to the interchange or inversion of spatial coordinates, whereas the time-reversal (T) operator inverts the time such that if it runs backwards. Systems that are invariant under combined parity and time transformation are regarded as PT-symmetric systems. The concept of such systems originated when Bender and Boetcher explained in the field of quantum mechanics that non-Hermitian Hamiltonians can potentially possess a real spectrum if the PT-symmetry condition $V(r) = {V^\ast }(r)$ is satisfied, where *V* describes the potential of quantum system and ${\ast} $ denotes complex conjugation [1–10]. The implementation of PT-symmetry in optics and electronics has been demonstrated by using balanced loss and gain proportions because it can uniquely provide non-Hermitian Hamiltonians with real eigenvalues [11–15]. The degree of non-Hermiticity in a system is described by its non-Hermiticity parameter and a non-Hermitian system can have real eigenvalues if this parameter remains below a specific threshold. Above this threshold the eigenvalues of such system no longer remain real and become complex as the system transitions into a new state. Such threshold points in PT-symmetric systems are regarded as exceptional points which have been studied to provide unique electromagnetic behavior and applications such as non-linear wave propagation [16–18], laser absorbers [19–20] and negative refraction [21].

Another exotic application of PT-symmetry is unidirectional cloaking and reflectionless transmission at exceptional points where an incident wave from input port is fully transmitted through the structure without any perturbation whereas enhanced reflection is observed when the structure is excited from the output port [22–24]. The effect has been demonstrated in optics by using geometric PT-symmetric regions that employ loss and gain in balanced proportions, modelled as even and odd functions of position-dependent real and complex refractive indices [25]. Similarly, PT-symmetry in electronic circuits has been studied to provide reflectionless transmission and cloaking in [26] where loss and gain are modelled as positive and negative resistors connected in pairs with satisfied parity and time reversal conditions. The aforementioned studies have been conducted in loss-free domains where the inherited system-associated losses and fundamental noise sources have been not considered. The presence of such sources can degrade the system performance and hence it is crucial to study the system noise performance prior to any practical implementation.

In this study, we show that an imaginary impedance originating through a capacitor or an inductor can effectively manipulate the reflection and transmission properties of PT-symmetric systems while sustaining combined Parity (P) and Time-reversal (T) symmetry. We show the design of PT-symmetric circuits in series or shunt circuit configuration, to behave as either, a unidirectional cloaking structure or as an attenuator or a switch, based on the selection of electrical length of the transmission line and capacitive or inductive impedances. In order to benchmark the system performance, the dominant noise sources for PT-symmetric systems have been identified and analytical and simulated noise calculation has been performed to estimate the noise figure. In regards to cloaking applications, the effect of an increased noise figure has been visualized by computing radar cross section of a PT-symmetric cloaked target.

## 2. Parity-Time symmetry using lumped elements

#### 2.1 Circuit schematic in series configuration

Consider the two-port network with its schematic shown in Fig. 1 that consists of two series resistors ${R_1}, {R_2}$ connected by lossless transmission line of electrical length *x* with a propagation constant *k*, physical length *d* and a characteristic impedance of ${Z_0}$. In order to imply time-reversal symmetry, the resistors have opposite values such that ${R_1} = r{Z_0}$ and ${R_2} ={-} r{Z_0}$ where *r* is defined as a dimensionless, non-Hermiticity parameter that is normalized with respect to the characteristic impedance ${Z_0}$ of the transmission line. The series element *X* has a complex impedance originating through a capacitor or an inductor.

From the definition of PT-symmetry we can conclude that the network is PT-symmetric because it is invariant under combined parity (P) and time-reversal (T) operators. The conditions are readily satisfied because interchanging the spatial coordinates of the two resistors and applying time-reversal transformation, the network remains invariant. The system possess loss as positive resistor ${R_1}$ acting as a power sink, and gain as a negative resistor ${R_2}$ as a power source or amplification therefore yielding combined parity and time-reversal symmetry [11].

### 2.1.1 Scattering parameters

A microwave network could be described by its scattering parameters or scattering matrix that defines the incident and reflected signals at the specific ports. The point of interest here is to compute these parameters and find the system exceptional point where the eigenvalues transform from real to complex. The scattering matrix formulation is achieved by cascading the transmission or $ABCD$ matrix of each individual element and then computing the scattering parameters by following [27]. The scattering-parameters at an electrical length of $x = n\pi /2$ are given by Eqs. (1) – (3), where $n = 1,3,5\ldots $ and $ \Delta = X{r^2} - X - 2{Z_0}$.

From Eqs. (1) – (3) it is evident that the scattering parameters of the system are dependent on the non-Hermiticity parameter $r$, which itself depends on the resistors ${R_1}, {R_2}$ and the characteristic line impedance $ {Z_0}$. The scattering parameters also depend on the electrical line length*x*and the imaginary impedance

*X*originating through the series reactance connecting the two transmission lines and the resistor pair.

The Eqs. (1) – (3) show the system exceptional point when the two resistors are equal to the transmission line impedance, i.e. at $r = 1$ and for electrical line length $x = n\pi /2$ where $n = 1,3,5\ldots ,$ the reflection coefficient at port-1 vanishes whereas the magnitude of the transmission coefficient remains unity, regardless of the imaginary impedance $ X$. The forward and reverse transmission coefficients are found to be equal i.e. ${S_{21}} = {S_{12}}$. However, if the system is excited from port-2 the reflection coefficient magnitude is obtained by the selection of imaginary impedance *X* by following Eq. (2). The scattering parameters in the complex plane have been shown in Fig. 2 where an inclusive capacitive or inductive impedance in the PT-symmetric circuit can yield a broad range of output reflection coefficient. For instance, at a frequency of 1.5 GHz, the reflection coefficient of unit magnitude is achievable at port-2 through a capacitance or inductance of 4.25 pF or 2.65 nH respectively. Similarly, a capacitance of 2.12 pF or an inductance of 5.3 nH yields an output reflection coefficient of 2.

#### 2.2 Circuit schematic in shunt configuration

The PT symmetric system shown in Fig. 1 can also be described in shunt configuration. Consider the two port network schematic in Fig. 3 where the resistors ${R_1}, {R_2}$ and the reactive element *X* have now been placed in a shunt configuration. The parity and time-reversal invariance still hold and by PT-symmetry postulates, the system is PT symmetric [11].

### 2.2.1 Scattering parameters

The scattering parameters for the shunt configuration can be obtained by following similar formulation as explained earlier. The s-parameters at an electrical length of $x = n\pi /2$ where $n = 1,3,5\ldots ,$ can be calculated by following [23] as

where $ \Delta = 2X{r^2} + {r^2}{Z_0} - {Z_0}$.From Eqs. (4) – (6), we can identify the symmetry breaking or exceptional point at $ r = 1$. The scattering parameters for the shunt circuit configuration are shown in Fig. 4 in a complex plane where the origin of unit circle marks zero reflection and a radius of 1 represents a full reflection. It can be observed that the PT symmetric circuit designed in shunt configuration now necessitates a capacitance of 1.1 pF or an inductance of 10.6 nH to render a unity output reflection magnitude.

Similarly, a capacitance of 2.12 pF or an inductance of 5.3 nH is able to achieve an output reflection magnitude of 2. This concludes that PT-symmetric circuits designed in shunt configurations require lower capacitance to achieve a particular value of reflection at the output. This is favorable in terms of a practical circuit design perspective. For instance a varactor diode in shunt can provide an appropriate variable junction capacitance in a small range at design frequencies hence resulting in a broader variance of reflection magnitude. However, the tradeoff comes at an increased noise figure of the system. Further discussion on the noise performance of PT-symmetric systems has been explored in subsequent section of this article where the noise figure of series and shunt systems have been estimated.

To develop a deeper understanding of the concept, we have performed full-wave simulations for an incident electromagnetic wave on the PT-symmetric system. Figure 5(a) shows the transverse component ${E_z}$ of the electric field where the system is being excited from port-1. The incident plane wave is launched from the loss side of the system and is fully transmitted through the structure with no reflection and a unitary transmission to port-2 is noticed. The black arrows represent the Poynting vector or average power flow in the system where as Fig. 5(b) shows that when the structure is illuminated from the port-2, the impinging wave undergoes enhanced reflection whose magnitude is given by a selection of capacitive or inductive impedance followed from Eq. (5). For instance, a capacitance of 1.1 pF in shunt configuration provides a reflection coefficient of unit magnitude at port-2. Similar results can also be achieved by an appropriate inductance inclusion in the system.

## 3. PT-Symmetry with electrical length parametrization

The PT-symmetric systems explained in Fig. 1 and Fig. 3 in series and shunt domains showed that a capacitive or inductive impedance can yield broad range of output reflection magnitude at an electrical length of $x = n\pi /2$ where *n* is an odd integer multiplier. This is potentially favorable for the design applications of tunable unidirectional cloaks or unidirectional transmission devices where the input port is perfectly matched with zero reflection and unity transmission towards port-2 with an output reflection coefficient defined by specific capacitance or inductance. However, it is noticeable that if an electrical line length is chosen such that $x = n\pi$ where $n = 1,2,3\ldots $ the scattering parameters for the Fig. 1 are now given as

## 4. A solid state PT-symmetric switch

The choice of electrical length is important in such that the transmission and reflection parameters can be defined by an appropriate imaginary impedance by a capacitive or inductive element in either shunt or series configuration. The proposed PT-symmetric circuits can behave like an attenuator or a switch where a signal transmission is needed to be manipulated. For this purpose, the scattering parameters for the circuit given in Fig. 1 are calculated and described by Eqs. (7) – (8) at an electrical length of $ x = n\pi$. This implies that the reflection and transmission parameters are now dependent on the selection of the reactance. A full-wave time-harmonic simulation verifies the potential of the PT-symmetric circuit to behave as switching device. For instance, Fig. 8(a) shows that at a selected capacitance of 0.1 pF the incident wave is fully transmitted to port-2 with zero input reflection and a transmission magnitude of 1, representing an ON state. On the contrary, Fig. 8(b) shows that at a selected capacitance of 20 pF, the system shows full reflection at port-1 and a transmission magnitude of 0, therefore representing an OFF state. Since PT-symmetric systems are invariant under combined parity and time-reversal operations, the following energy conservation relationship is satisfied [28].

## 5. Noise modelling and performance of parity-time symmetric systems

Although PT-symmetric systems with loss and gain can be designed to minimize overall system losses by compensating loss with equivalent gain proportions, noise being a fundamental property of RF systems with loss (passive) and gain (active) components, is unavoidable and hence it is important to benchmark the noise figure to determine the system performance. In realistic domains, PT-symmetric systems employ non-ideal active and passive circuit components that generate various types of noise. Primary noise contributions originate through lossy resistor $\textrm{ + }R$, which is a source of Johnson-Nyquist or thermal noise, and through the gain resistor $ - R$. The negative resistor does not occur as a standalone circuit element and hence it is implemented through operational amplifiers (Op-Amp) and negative impedance converters (NIC) at low frequencies or a resonant tunnel diode (RTD) at microwave frequencies, exhibiting thermal noise, shot noise and flicker noise contributions. The overall system noise figure is given by

The conventional resistor $\textrm{ + }R$ in series and shunt PT-symmetric circuits discussed in this study can be described by noiseless resistors with equivalent voltage noise sources in series or an equivalent noise current source in parallel. Similarly, the negative resistor can be replaced by equivalent active RTD noise model and the overall system noise figure can then be calculated using analytical methods by following [27,29] and verified using a commercial microwave software simulator. The noise figure in terms of s-parameters is given as where*k*is the Boltzmann’s constant,

*T*is the temperature, ${Z_0}$ is the characteristic line impedance and ${\bar{V}_n}$ is the thermal noise voltage due to resistors and is given as Figure 9 shows the calculated noise figure of the PT-symmetric systems in both series and shunt topologies.

For the purpose of brevity, we have considered only the thermal noise sources in our calculations since it is the most dominant noise source present in the system. However, modelling of other noise sources such as shot noise and flicker noise requires further consideration and the noise figure is expected to increase with inclusion of such sources. The aforementioned techniques for series and shunt design configurations have their unique advantages. For instance, with the shunt configuration the noise figure of the PT-symmetric circuit is estimated at 9.47 dB, which is sufficiently high considering the application to cloaking applications. On the contrary, the series design configuration offers a reduced noise figure of 5.05 dB which is much suitable for cloaking, RCS reduction and scattering applications. A comparison of the noise figure levels in PT-symmetric systems designed in shunt or series configurations have been shown in Fig. 9.

#### 5.1 Performance of a unidirectional cloak in the presence of noise

Inclusion of noise in RF system not only degrades the performance but also causes instabilities and can render the system internally or externally noise limited. A study in [29] found that thermal noise inclusions in a PT-symmetric system with loss and gain components results in a weakly broken PT-symmetry hence degrading the performance. Since PT-symmetric systems utilize active and passive circuit elements that generate various noise types due to which the cloaking achieved by such a metasurface cannot be noise free. It can be followed from [26] that a PT-symmetric cloak with balanced loss and gain metasurfaces can potentially achieve perfect invisibility under ideal loss-free and noiseless conditions where a lossy metasurface fully absorbs the incoming signal by the concept of impedance matching and the gain surface emits the signal with similar amplitude and phase rendering the object undetectable. In this section we will discuss the cloaking performance of a PT-symmetric cloak in realistic domains with the inclusion of thermal noise.

Consider Fig. 10 where a target is being illuminated by the incident electromagnetic wave. We assume that the target is in the far-field region where the incoming wave from a radar antenna can be assumed by a plane wave approximation. In the absence of any cloaking metasurface, the target is detected from the scattered field distribution and radar cross section scattering calculation which has been shown in Figs. 12(a) and 12(f). If the target is covered by a PT-symmetric metasurface cloak described in [26] under ideal noise-free conditions, perfect unidirectional-invisibility is observed from lossy side and the target RCS is suppressed by a factor of 99% as shown in Figs. 12(b) and 12(f).

Since a realistic PT-symmetric cloak cannot be noise free, the lossy part of metasurface which is realized through passive resistive elements would be a source of added thermal noise. The noisy surface current density of the loss metasurface can be written as a sum of noiseless surface current density and equivalent noise current density.

Similarly, the gain metasurface $- R$ is realized through RTD which will have active noise and the surface current density of a noisy gain metasurface can be written as where ${J_{n(passive)}}$ and ${J_{n(active)}}$ are the noise current densities of loss and gain metasurface respectively. The equivalent noise current densities are calculated from thermal noise current given by It is to be noted that Eq. (15) provides the root mean square (rms) noise current, and because the thermal noise current is random and exhibits a Gaussian distribution therefore instantaneous peak noise current would be within six times the rms value for more than 99% of the time. The noise current waveform in time-domain with its statistical distribution has been shown in Fig. 11.The noise current density is then found by following Eq. (16).

where*l*is the length of the individual metasurface and ${i_p}$ is the peak noise current given by ${i_p} = 6{i_{rms}}$.

Using results from section 5 and utilizing shunt configuration from Fig. 3, the noise figure of the cloaking metasurface is estimated to be 9.7 dB. However, utilizing a series design configuration from Fig. 1, the noise figure is calculated to be 5.05 dB. This shows that PT-symmetric circuits designed in series configuration offer a reduced noise figure and are much suitable for cloaking applications in comparison with shunt topology. For instance, Figs. 12(c) and 12(d) show the metasurface cloaking performance with series (5.05 dB) and shunt (9.7 dB) noise figure respectively, whereas Fig. 12(f) shows the RCS comparison of a cloaked target in the presence of 5.05 dB and 9.7 dB noise levels. A further increase in noise levels with a high noise figure of 20 dB, the cloaking performance is severely degraded and the RCS is further increased.

The cloaking achieved by such noisy circuit model is imperfect and it can be seen that while a PT-symmetric cloak may be able to reduce the RCS of the target, a perfect invisibility is never achieved. Further, the receiver receive sensitivity is also important because it will ultimately determine the minimum detection threshold. For instance, a receiver sensitivity may be low enough to a certain level where 10 dB of noise figure does not potentially causes detection. On the contrary, a receiver with high receive sensitivity may cause detection at same noise figure.

The noise in PT-symmetric metasurface primarily originates through loss and gain resistors. Thermal noise from the passive lossy resistor is fixed however the gain resistor requires further attention. The noise originating through the gain resistor depends on the implementation technique such as by the means of operational amplifier, resonant tunnel diode (RTD) or a transistor based negative impedance converter. For the purpose of simplicity, we have considered the equivalent RTD noise model and thermal noise to calculate the overall noise figure of the metasurface in series and shunt design configurations. However, with the inclusion of other noise sources such as shot noise or flicker noise and complex NIC models with greater number of active and passive components, the noise figure is expected to rise. The noise analysis of NIC and non-Foster loaded metamaterials was studied in [30] where noise figure of an actively loaded loop was estimated. However, the noise can be reduced by using series circuit topology and much careful design of gain resistor. Further study of noise figure in cloaking applications was performed in [31] where the target radar cross section (RCS) was calculated in the presence of various noise levels.

## 6. Conclusion

We have shown that an imaginary impedance originating through a capacitor or an inductor can effectively manipulate reflection and transmission behavior in PT-symmetric systems while sustaining parity and time-reversal symmetry. Circuit configuration in series and shunt have been studied to provide broad range of reflection and transmission magnitudes by the inclusive capacitive and inductive impedances. Further studies showed that the electrical line length plays a crucial role in PT-symmetric application design. For instance, at an electrical length of $n\pi /2$ where *n* is an odd multiplier, the output reflection coefficient magnitude of the PT-symmetric circuit is defined by the capacitive or inductive impedances. The application can be useful for a signal reflector or unidirectional cloaking purposes where output reflection is required to be maintained. Similarly, at an electrical length of $n\pi$ where *n* is an integer multiplier, the reflection and transmission coefficients of PT-symmetric system can be manipulated through the capacitive or inductive impedances. This is particularly useful for applications that require switching, filtering or attenuation. The aforementioned studies have been verified analytically and the transmission and reflection behavior is validated and visualized by a full-wave circuit simulation model for series and shunt configurations. Further, we have shown that the PT-symmetric systems are prone to high noise figure levels however careful design techniques can reduce the noise figure with some application limitations. Finally, we have performed noise characterization of the cloaking metasurface and discussed the performance of a noisy PT-symmetric cloak and concluded that the noise degrades the cloaking performance and effectively increases the scattering cross section of the cloaked object and in such circumstances where noise figure is sufficiently high, the system may get internally noise limited.

## Funding

Engineering and Physical Sciences Research Council (EP/R035393/1); Institution of Engineering and Technology (AF Harvey Research Prize).

## Acknowledgments

H. Farooq would like to thank School of Electronic Engineering and Computer Science, Queen Mary University of London for the PhD studentship award. Yang Hao would like to acknowledge IET, United Kingdom for the AF Harvey Research Prize.

## Disclosures

The authors declare no conflicts of interests.

## References

**1. **C. Bender and S. Boettcher, “Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry,” Phys. Rev. Lett. **80**(24), 5243–5246 (1998). [CrossRef]

**2. **C. M. Bender, M. V. Berry, and A. Mandilara, “Generalized PT symmetry and real spectra,” J. Phys. A: Math. Gen. **35**(31), L467–L471 (2002). [CrossRef]

**3. **C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. **70**(6), 947–1018 (2007). [CrossRef]

**4. **C. Bender, D. Brody, and H. Jones, “Extension of PT-symmetric quantum mechanics to quantum field theory with cubic interaction,” Phys. Rev. D: Part., Fields, Gravitation, Cosmol. **70**(2), 025001 (2004). [CrossRef]

**5. **C. M. Bender, D. C. Brody, and H. F. Jones, “Complex Extension of Quantum Mechanics,” Phys. Rev. Lett. **89**(27), 270401 (2002). [CrossRef]

**6. **C. M. Bender, D. C. Brody, H. F. Jones, and B. K. Meister, “Faster than Hermitian Quantum Mechanics,” Phys. Rev. Lett. **98**(4), 040403 (2007). [CrossRef]

**7. **A. Mostafazadeh, “Pseudo-Hermiticity versus PT Symmetry: The Necessary Condition for the Reality of the Spectrum of a Non-Hermitian Hamiltonian,” J. Math. Phys. **43**(1), 205–214 (2002). [CrossRef]

**8. **G. Lévai and M. Znojil, “Systematic Search for PT-Symmetric Potentials with Real Energy Spectra,” J. Phys. A: Math. Gen. **33**(40), 7165–7180 (2000). [CrossRef]

**9. **H. F. Jones, “The Energy Spectrum of Complex Periodic Potentials of the Kronig-Penney Type,” Phys. Lett. A **262**(2-3), 242–244 (1999). [CrossRef]

**10. **S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of Branch Points in PT-Symmetric Waveguides,” Phys. Rev. Lett. **101**(8), 080402 (2008). [CrossRef]

**11. **J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with PT symmetries,” Phys. Rev. A **84**(4), 040101 (2011). [CrossRef]

**12. **S. Bittner, B. Dietz, U. Günther, H. Harney, M. Miski-Oglu, A. Richter, and F. Schäfer, “PT Symmetry and Spontaneous Symmetry Breaking in a Microwave Billiard,” Phys. Rev. Lett. **108**(2), 024101 (2012). [CrossRef]

**13. **Y. D. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser absorber modes in optical scattering systems,” Phys. Rev. Lett. **106**(9), 093902 (2011). [CrossRef]

**14. **L. Ge, Y. D. Chong, S. Rotter, H. E. Tureci, and A. D. Stone, “Unconventional modes in lasers with spatially varying gain and loss,” Phys. Rev. A **84**(2), 023820 (2011). [CrossRef]

**15. **A. E. Miroshnichenko, B. A. Malomed, and Y. S. Kivshar, “Nonlinearly PT-symmetric systems: Spontaneous symmetry breaking and transmission resonances,” Phys. Rev. A **84**(1), 012123 (2011). [CrossRef]

**16. **A. Guo, G. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. Siviloglou, and D. Christodoulides, “Observation of PT Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. **103**(), 093902 (2009). [CrossRef]

**17. **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]

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

**19. **Y. D. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser absorber modes in opitcal scattering systems,” Phys. Rev. Lett. **103**, 093902 (2010). [CrossRef]

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

**21. **R. Fleury, D. L. Sounas, and A. Alù, “Negative Refraction and Planar Focusing Based on Parity-Time Symmetric Metasurfaces,” Phys. Rev. Lett. **113**(2), 023903 (2014). [CrossRef]

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

**23. **A. Mostafazadeh, “Invisibility and PT symmetry,” Phys. Rev. A **87**(1), 012103 (2013). [CrossRef]

**24. **X. Zhu, L. Feng, P. Zhang, X. Yin, and X. Zhang, “One-way invisible cloak using parity-time symmetric transformation optics,” Opt. Lett. **38**(15), 2821–2824 (2013). [CrossRef]

**25. **Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. Christodoulides, “Unidirectional Invisibility Induced by PT-Symmetric Periodic Structures,” Phys. Rev. Lett. **106**(21), 213901 (2011). [CrossRef]

**26. **D. L. Sounas, R. Fleury, and A. Alù, “Unidirectional cloaking based on metasurfaces with balanced loss and gain,” Phys. Rev. A **4**(1), 014005 (2015). [CrossRef]

**27. **D. Pozar, * Microwave Engineering*, 4th ed. (John Wiley, 2011).

**28. **C. Shi, M. Dubois, Y. Chen, L. Cheng, H. Ramezani, Y. Wang, and X. Zhang, “Accessing the exceptional points of parity-time symmetric acoustics,” Nat. Commun. **7**(1), 11110 (2016). [CrossRef]

**29. **K. V. Kepesidis, T. J. Milburn, J. Huber, K. G. Makris, S. Rotter, and P. Rabl, “PT-symmetry breaking in the steady state of microscopic gain–loss systems,” New J. Phys. **18**(9), 095003 (2016). [CrossRef]

**30. **Y. Fan, K. Z. Rajab, and Y. Hao, “Noise analysis of broadband active metamaterials with non-Foster loads,” J. Appl. Phys. **113**(23), 233905 (2013). [CrossRef]

**31. **H. Farooq, D. S. Nagarkoti, K. Z. Rajab, and Y. Hao, “Noise Figure of a Unidirectional Cloaking Circuit Based on Parity-Time Symmetry” in EuCap, (2019), (2019).