Four-port photonic structures with mirror-time reversal symmetries

We investigate the transport characteristics of a four-port gyrotropic photonic structure with mirror-time reversal symmetry. The structure consists of two coupled cavities with balanced amplification and attenuation. The cavities are placed on top of a gyrotropic substrate and are coupled to two bus waveguides. Using detail simulations in the microwave domain we demonstrate a strong non-reciprocal intra-guide port transport and an enhanced inter-guide port transmittance. The non-reciprocal features are dramatically amplified in the gain–loss parameter domain where an exceptional point degeneracy, for the associated isolated set-up, occurs. These results are explained theoretically in terms of an equivalent lumped circuit.

Given the above success of  -symmetric photonic structures it is natural to extend the investigation into the more general family of  -symmetric systems. Of special interest is the case where magneto-optical effects, associated with a vector potential which cannot be gauged away, are involved [14,20,23,24]. In this case the notion of time-reversal symmetry has to be adopted accordingly, i.e. it is not only associated with a simple conjugation i ibut it also involves a change in the direction of the vector potential. The motivation to investigate this type of photonic structure is associated with the possibility to induce giant non-reciprocal effects -a theme of ongoing intense investigation [25][26][27][28]. Traditionally ferrite materials such as yttrium-iron garnet (YIG) are invoked in the design of photonic (both in the microwave and optical) non-reciprocal devices. The degree of non-reciprocity in any particular application is always a trade-off between the strength of the gyromagnetic coupling, losses associated with spin-wave relaxation, and the geometry of the wave-ferrite interaction.
In this paper we analyze the transport characteristics of a four-port photonic structure with anti-linear mirror-time reversal ( ) symmetries in the presence of magneto-optical effects. The structure consists of two  -symmetric cavities on a YIG substrate shown in figure 1. The cavities are coupled to two bus waveguides. We show that the whole structure possesses strong non-reciprocal behavior between left and right ports associated with the same waveguide (intra-guide ports) while it demonstrates an enhanced transmittivity, due to the active elements, between cross-ports of the two different waveguides (inter-guide ports). This behavior is strongly enhanced in the parameter domain where exceptional point (EP) degeneracies of the underlying isolated resonators occurs. The results are numerically demonstrated in the microwave domain and are accompanied by a thorough analytical study of a lumped circuit equivalent.
The organization of the paper is as follows. In the next section 2 we introduce the microwave photonic structure. A numerical analysis of the eigenmodes is reported in section 2.1 while in section 2.2 we report our numerical results for the transport properties of this system. In section 3 we present the lumped circuit analog and theoretically analyze its transport characteristics. Specifically in section 3.1, we present a general introduction to the lumped elements of an  symmetric system and their equivalence with the microwave components. In section 3.2 we present the lumped analog of the four-port microwave photonic structure. First we analyze the eigenmodes of the isolated set-up in section 3.3. In section 3.4 we analyze the properties of the scattering matrix of this system. Finally section 3.5 discuss the scattering properties of this system and demonstrate the universal nature of the non-reciprocal intra-guide transport in such set-ups. Our conclusions are given at the last section 4.

Microwave photonic structure
The microwave photonic structure that we consider, shown in figure 1, is the minimal four-port microstrip configuration capable of demonstrating  symmetric nonreciprocity. It consists of two transmission lines, each connected to a left and right port. A pair of proximity coupled half-wave resonators, refered to as the dimer, bridge the space between the waveguides. Small gaps end couple the dimer to each of the transmission lines, leading to a coupling between the two transmission line buses dominated by the dimer resonator. All of the microsrtip structures lay on the top of a 8.75 mm thick YIG substrate, with relative permittivity  = 15 r [29], and a ground plane below. The resonator dimer and the transmission waveguide traces are modeled in the thin, perfect conducting limit. The length of the microstrip resonators are chosen to be = l 24.5 mm long, supporting half-wavelength resonances of approximately w p = 2 1.24GHz. The distance between the two microstrip resonators is d=20 mm which correspond to a phase delay º = d kd 2.36 rad . Later on in section 3.5 we discuss the role of the phase delay in our construction. The microstrip ends are separated by a 0.5 mm gap from the sides of the two 3 mm wide transmission lines.
The YIG-substrate is divided into two types: (i) active domains in the regions just underneath the two microstrip resonators, and (ii) a passive domain elsewhere (see figure 1). A balanced gain and loss is introduced in the two active regions by adding opposite imaginary parts to the YIG permittivity,  ag =´ 15 1 i r ( ), to each resonator of the dimer. Here, γ is a parameter designating the strength of the balanced gain or loss defined so that each resonator, if isolated, would experience exponential growth or decay with g w w = 2 Im Re | ( ) ( )|. The parameter α is a numerically determined factor of (order unity for the configuration of figure 1) accounting for the precise geometrical configuration of the modified permittivity. Furthermore, the YIG-substrate is magnetized with a DC bias field H 0 along the y-axis, parallel to the resonator microstrips. The magnetic

Eigenmode analysis of the isolated system
We start our analysis with the parametric evolution of the eigenfrequencies of the isolated dimer associated with the photonic set-up of figure 1.
A novel feature of non-Hermitian systems with antilinear symmetries, is the possibility to possess an entirely real spectrum for some values of the gain/loss parameter γ [30]. In this parameter domain, coined exact phase, the eigenmodes are invariant under the antilinear operator. As the gain/loss parameter γ, increases beyond some critical value  g g > , the spectrum becomes partially or completely complex and the system enters the socalled broken phase. In this domain the eigenmodes do not respect the antilinear symmetry-although the system itself remains invariant under  . At  g g = the eigenfrequencies and the associated eigenmodes coalesce and the system experiences an EP degeneracy.
We investigate the  -symmetry phase transition for the set-up of figure 1 with the bus waveguides removed (see insets of figure 3). We refer to this as the isolated configuration. Using the 3D-finite element electromagnetic (FEEM) eigenfrequency simulation package of COMSOL [31] we explore the  -symmetry phase transition for three scenarios associated with the distribution of the gain and loss media (the modified permittivity) beneath the microstrips: (i) the gain and loss are collapsed into a localized patch at the resonator ends, as shown schematically in the inset of figure 3(a); (ii) the gain and loss is uniformly distributed over the left and right sides of the whole YIG-substrate, as shown in figure 3(b); and; (iii) finally, the case originally described in figure 1 where the gain and loss is confined directly beneath the individual resonator striplines is reported in figure 3(c).
The purpose of the analysis of these isolated system scenarios is to demonstrate that the  -symmetric behavior is only weakly dependent on the details of how the gain and loss is actually achieved. While the latter two scenarios could be realized in the optics framework by appropriately distributed gain and loss media, case (i) is more relevant to the microwave domain. Concentrating the gain and loss into a small patch at the anti-nodes of the half-wave electric potential oscillations in the microstrip resonator mimics the action of descrete electronic gain or loss elements such as transistors or resistors [21,32]. In all cases, the geometric parameter α is calibrated to preserve the stated meaning of the gain/loss parameter γ. Figure 3 illustrates the parametric evolution of the isolated eigenfrequencies as a function of the gain/loss parameter. When g = 0, the coupled microstrip resonators support two low order resonant modes: a symmetric (low frequency) and an antisymmetric (higher frequency) modes defined by the parallel and antiparallel directional flow of the surface currents in the srtiplines. For g = 0 the associated eigenfrequencies have a small and similar imaginary value due to a weak radiative loss to the simulation boundaries. As γ increases the real part of the eigenfrequencies of the modes changes (see figures 3(a)-(c)) while the associated imaginary part remains the same. At some critical value of the gain/loss parameter  g g = , the eigenvalues coalesce and a spontaneous  -symmetric phase transition occurs. For  g g > the real part of the eigenfrequencies remains degenerate while their imaginary part bifurcates to two distinct values of opposite sign. The eigenmodes take on a left/right flavor with the gain side exponentially growing in time, and the loss side exponentially decaying in time. The main results in this figure show that the quantitative behavior of the three set-ups (i)-(iii) is the same and does not depend on the details of how gain and loss are introduced into the system. For the remainder of the simulations of the non-isolated system, we will restrict analysis to the case (iii), where gain and loss are confined to be below the individual resonator microstrips.

Transport properties
Next we turn to the analysis of the transport characteristics of the four-port system of figure 1. The structure is simulated with COMSOL's 3D FEEM numerical software. A mesh density with element size of λ/8 within the active domains underneath the two microstrips and λ/4 in the surrounding bulk YIG-substrate was used, and confirmed for accuracy, where λ is the wavelength inside the medium. Quasi-TEM waves are introduced or collected from the device transmission lines via impedance-matched ports, supplied through the COMSOL software, defined between the edge of the microstrip waveguide and ground plane. An example of the frequency dependence of the scattering matrix elements is shown in figure 4 for three values of the gain and loss parameter γ. For intermediate values of γ we observe a strong non-reciprocal effect for intra-guide transmittance while the inter-guide transmittance is enhanced. A density plot of the propagating electromagnetic field for the value of γ where this effect is observed, is reported in figure 5. Below we further analyze this behavior 2 .
Numerical simulations were carried out to quantify the dependence of the scattering coefficients between various ports as a function of increasing gain-loss parameter. We report only the transmitted signals with incident wave entering the structure from port R 1 and/or port L 1 . All other transmitted coefficients (and also reflections to the incident ports) are related with the reported transmittances via symmetry considerations, see etc. An impressive conclusion of our analysis is the fact that for some Figure 3. Parametric phase transition of the eigenmodes with increased γ, obtained using 3D-FEEM eigenmode solver package [31].
The balanced loss and gain are introduced in a localized (a) and distributed manner (b). In (c) we present the results for the configuration of figure 1.
26. 2 We restrict all of the following discussion in the frequency regime near the symmetric mode where the transmission of waves through the system shows a maximum non-reciprocity.
critical value of the gain-loss parameter (γ = 0.13601) a giant isolation of the L 1 port from the signal transmitted from the R 1 port is achieved (i.e.~- T 90 dB R L 1 1 or essentially zero) while at the same time we have a healthy signal that is radiated towards the L 2 port (~- T 0.2 dB R L 1 2 ). The signal transmitted from port L 1 to port R 2 , is - while we have also a healthy signal from L 1 to L 2 ports, T 7 dBs L L 1 2 . We finally note that a non-reciprocal intra-guide behavior can occur for another value of g = 0.15. However the phenomenon here is not as dramatic as previously. Below we proceed with a qualitative explanation of this strong intra-guide nonreciprocal effect, while in sections 3.4 and 3.5 we provide a detail theoretical explanation based on equivalences with a lumped circuit.
The nonreciprocity shown in figure 6 has it's roots in the different propagation constant for circularly polarized waves travelling parallel to a static magnetic field exhibited in an otherwise isotropic magnetic medium. The question here is how can this difference be used to achieve a nonreciprocal transmission, and under which conditions the  -symmetry enhance the isolation? A classic ferrite-based nonreciprocal device needs to (1) create a circular polarization containing magnetic fields rotating perpendicular to the static bias magnetic field, and (2) allow separate components of the chiral wave to travel along paths that interfere upon recombination. The rotating field components couple differently to the spin precession of the magnetized ferrite, and experience a resulting phase difference. The structure of this four port device includes all of these features in its design.
Although there is not any obvious circular polarization, the symmetric and antisymmetric modes of the resonator dimer allow the equivalent to occur. Quasi-TEM waves on a single microstrip have only one polarization defined by the single mode, but a pair of parallel striplines, having two components, can result in two independent magnetic fields. These wave fields, perpendicular to the static field, also include components perpendicular to each other, the necessary condition for the gyromagnetic coupling. The symmetric mode, with  . Density plot of the field intensity for incident waves entering the structure from: (a) port L2 and; (b) port R1. The field distribution when γ = 0.136, for the case where the incident wave enters the structure from port L1 is mirror symmetric to the results of (a). Similarly the field distribution for the case where the incident wave enters the structure from port R2 is mirror symmetric to the results of (b).
parallel stripline currents, is dominated by a mutual substrate magnetic field parallel to the plane of the substrate. The antisymmetric mode, with anti-parallel currents, is dominated by a mutual substrate magnetic field perpendicular to the plane of the substrate. Figure 7 schematically illustrates the application of the right-handrule to the region of the substrate significantly influenced by the current-induced fields of both modes.
The rotating field requirement imposes further conditions on the resonances. The fields of figure 7 need to be excited simultaneously. Normally this would be difficult to achieve since the two modes are split by nongyrotropic coupling. However, the symmetry of the transmission line coupling helps in this respect. Depending on the path delay of the resonator coupling to the transmission lines, resonant mode and the off-resonant mode can have different effective coupling strengths to the wave excitation in the waveguide. For example, the resonant mode could be weakly coupled, and the off-resonant mode could be strongly coupled, resulting in a more balanced excitation of both modes by the same frequency. Another rotating field requirement is that there be a quadrature phase difference between the two modes of figure 7. Without dissipation, symmetry of the structure prevents such a phase delay: the resonances are either in-phase, or out of phase, with the effective drive. However, a resonant quadrature phase relationship will be introduced by any small dissipation, usually naturally occurring, that assures some finite bandwidth while passing through the quadrature condition. The second  non-reciprocity requirement that there is a path-length difference, is available through both the phase delay associated with the spatial separation of the end coupling to the resonator elements, and as an inherent 'dwell time' of a resonator.
It is now clear why the non-reciprocity could be enhanced by the  symmetric behavior. As the gain/loss parameter γ progresses toward the EP,  g , both the frequencies of the modes and their phases coalesce. The novel eigenfrequency and eigenmode behavior present in these systems introduces a new parameter distinctly capable of tuning the device in unusual ways to enable the gyrotropic coupling.
The next section discusses these features in the context of the simpler lumped-element model containing only the essential parameters.

Lumped elements
The configuration simulated in section 2 consisted of the active, gyromagnetically coupled microstrip dimer coupled to the two microstrip transmission lines making up the ports. As simple as this system is, the subtlety of the physical coupling between the microstrip elements and the mode structure of the coupled resonators required the COMSOL analysis for an accurate prediction of the device behavior. A more intuitive approach that naturally leads to a theoretical analysis is the lumped element electronic analog. Here, the resonator pair with its inter-resonator coupling and gain/loss character are replaced by appropriately coupled LC circuits. All spatial symmetry considerations,with the exception of the phase delay associated with the microstrip sections between the resonator ends, can be reduced to a matter of network topology defined through the application of Kirchoffʼs laws to fundamental electronic elements. Physical symmetry is reduced to the circuit network as defined by its node topology with appropriately valued connecting elements. The parity, mirror, and time reversal operations are equivalent to the interchange of labels corresponding to pairwise associated circuit nodes, with proper consideration of the resistive and gyrotropic elements through a sign change.
Resistors change their sign upon time-reversal, where negative resistance represents the simplest conceptual inclusion of amplification into electronics. Dissipative loss turns into gain and Kirchoff's laws can be used without modification subject to a condition described below. Experimentally, negative resistance can easily be implemented with negative impedance converters but can only approximate the ideal due to inherent limitations of the required amplifying components [21].
Analysis of circuits including negative resistance elements, however, requires respecting a subtle condition: any two terminal circuit structure reducing to a pure negative resistance will be undefined unless the structure is placed in parallel with a capacitance. This conclusion results from the divergence of the pole associated with parallel RC p combinations (with < R 0) in the limit of  C 0 p . The pole with the usual positive resistance has a sign that corresponds to exponential decay and therefore has a physically valid limit as  C 0 p . For example, the solution for the standard series LRC circuit, though it appears to have a mathematically appropriate solution for negative R, is non-physical in that realm: the hidden pole corresponds to an exponentially growing solution with a diverging rate as  C 0 p . The parallel LRC configuration with > R 0 is not subject to this oversight. The gyrotropic nature of the biased magnetic substrate beneath the microstrip resonator dimer is included electronically by the gyrator, shown schematically in figure 8 and defined by its conductance matrix, with G 0 being the strength of the gyration Although as an actual device, the ideal gyrator can only be approximately implemented in the microwave regime, we nevertheless include it for its conceptual simplicity in allowing a lumped model for the substrate mediated interaction. The mirror operation applied to this device changes the sign of G 0 .
Thus, for a  -symmetric circuit incorporating these basic elements, it is necessary that (1) all reactive elements either have representation in parity-associated network pairs, or directly connect parity inverted network nodes, (2) all Ohmic elements are paired with opposite sign, and (3) each negative Ohmic element has an associated parallel capacitance, or AC equivalent, as part of the circuit. Valid  -circuits of arbitrary complexity can be built up using these simple rules, though their stability needs to be independently determined. Figure 9(a) shows the  -symmetric lumped element circuit used to simplify the four-port microstrip device simulated in COMSOL. The microstrip resonator pair is replaced by a pair of LC resonators, with the left side experiencing a gain, and the right side experiencing a loss of equivalent magnitude represented as respective negative and positive parallel resistances of magnitude R. The original microstrip resonators experience both capacitive and inductive coupling due to their proximity, and a gyration component arising from the biased magnetic substrate. The lumped element resonator coupling is limited to mutual inductance, which captures the proper mode splitting of the quasi-TEM proximity coupling, and a pure gyration component.

Electronic analog circuit
The resonators are capacitively coupled by C c to points along an ideal TEM transmission line, separated by a length d. The transmission lines with impedance Z 0 are then continued on to the device ports, which in this lumped model, are collapsed to the respective nodes of figure 9. Realistic values for the lumped components can be chosen to match the observed resonances of the COMSOL simulation.
This model, with the resonator section limited to the four parallel nodes of the LC resonator pair, provides for a convenient separation of the complete dimer into a generic conductance matrix, G dimer , as illustrated in figure 9(b). This separation not only simplifies the expression of Kirchhoff's laws for the complete system, but also simplifies the analysis of the isolated dimer described by the solutions of with the entries V g and V l denoting the voltage across the gain side and the loss side of the dimer, as illustrated in figure 9. In the first term of equation (3), ( ) ( ) is the conductance of the individual parallel LC resonators, while the second term gives the negative and positive conductances associated with gain and loss sides, which in general might not be exactly balanced. The third and fourth terms  where the scaled conductance matrix D has been separated into an MT symmetric part with balanced gain/ loss parameter γ and a small unbalanced intrinsic loss g 0 satisfying g g -= - , the scaled frequency is w w w = 0 , and I 2 is the identity matrix.

Eigenmode analysis
First we discuss the normal modes of the isolated balanced dimer by setting g = 0 0 . The structure of that part of equation (4) implies relatively simple forms for the eigenvalues and eigenvectors. Respecting this structure indeed leads to a tractable form for the eigenvalue equation ⎛ with eigenvalues of the antisymmetric and symmetric modes Note that the given forms explicitly show all of the relationships among the analogous EPs and the real and imaginary parts of the frequencies. The exact phase below the symmetry breaking point, g g < < 0 MT , is characterized by two purely real eigenfrequencies, while in the broken phase between this and the upper critical point, g g g < < MT c there is one real frequency with two opposite imaginary parts, and above g c , four pure imaginary frequencies. The opposite imaginary parts impose at least one exponentially growing mode above g MT making the isolated dimer unstable and inappropriate for consideration of steady-state behavior.
The normal modes in the exact phase are characterized by equal magnitudes for the voltage oscillations in the gain and loss sides, which are given by with a phase f a s of the loss side relative to the gain side of ⎛ , where both the magnitudes and phases of the voltage oscillations are identical. The role of the gyration strength g is subtle: it acts in an orthogonal sense to the mutual inductance coupling because of its inherent non-reciprocal coupling of current and voltage, and results in mode phasers initially related by p 2 usually indicative of dissipation. However, in spite of the gyration being characterized by a real conductance, it is actually a non-dissipative element. Figure 10 schematically illustrates the evolution of the left and right voltage mode phasers for pure mutual inductance coupling in (a); pure gyration coupling in (b); and a combination of the two in (c). In all cases, the high and low mode coalesce as γ increases, approaching  g . We note that the gyration is thus imposing a normal mode character to the electronic circuit analogous to the circularly polarized eigenstates of free electromagnetic waves in the gyromagnetic medium of the device substrate. The combination of the gyrotropic coupling with normal reactive coupling allow the simple electronic circuit to embody all of the important features of the more nuanced microstrip dimer. Now we impose the small unbalanced intrinsic loss, g 0 of equation (4), into the system to generically account for losses in any real system. Specifically, up to g O 0 ( ), the eigenfrequencies of the isolated symmetric and antisymmetric modes pick up nearly equal imaginary parts s a s ā The closed system consisting of the nearly  -symmetric couples oscillators has now been modified by the parameter g 0 to include a small unbalanced intrinsic loss common to both the symmetric and antisymmetric modes.

Scattering set-up
The full device circuit can now be considered by expressing the transmission lines, characterized by their impedance Z 0 and phase velocity v p , as capacitively coupled to the circuit defined by the conductance matrix G dimer , illustrated earlier in figure 9(b) and explicitly expressed in equation (4). The scaled coupling capacitances, = c C C c are spaced at a scaled distance = d d v LC p along the respective transmission lines. Kirchoff's laws can now be rewritten in the framework of coupled-mode theory, expressing relations among the port wave components required to match the conductance terminal currents of G dimer  refer to the corresponding components for the lower transmission line (#2 of figure 9). Using the coupled-mode equations (13)- (15), one can get the relations between the outgoing signals and incoming signals, which defines a 4×4 scattering matrix S with block elements This completes the description of the four-port device in terms of its scattering matrix, as parameterized through a lumped-element model of the coupled microstrip resonator pair.

Giant intra-guide non-reciprocity and inter-guide transmittance
The COMSOL simulation presented in section 2.2 indicated that our four-port  -symmetric photonic structure demonstrates strong non-reciprocal intra-guide transport and an enhanced inter-guide transmittance for a value of the gain and loss parameter close to  g g » . However, the complexity of the electromagnetic problem-involving many parameters-was not allowing for a thorough analytical or even numerical investigation of the properties of the set-up. In contrast, the equivalent lumped circuit, is not only very transparent but it also provides a very attractive framework where detailed numerical and analytical investigations can be performed.
First we demonstrate that the giant intra-guide non-reciprocal behavior and inter-guide transmittance at  g g » is a universal phenomenon and, as such, it pertains also to the transport characteristics of the lump circuit. Our investigation of the electronic model focuses on the frequency region near the symmetric-mode resonance, where the intra-guide non-reciprocity NR describing the R 1 to L 1 port isolation, see equation (21), is typically experiencing its maximum value. This can be seen in figures 11(a), (b) where we observe that the nonreciprocity between L 1 and R 1 take valuesÑR 80 dBs while the transmittance between L 1 and other ports at  g g . The latter can be evaluated from the parametric evolution of the eigen-frequencies equation (6) of the corresponding isolated system, see figure 11(c).
In figure 12 we show the dependence of the isolation properties, between intra-guide ports, on the resonator coupling strength to the transmission lines. The inset illustrates that the position of the maximum NR, denoted by g max , is always slightly below the  EP, consistent with the above discussion, with a linear relationship. The main figure shows the slope of this linear relation. Deviating only slightly from unity as the coupling increases, this shows that the coupling strength is not very important.  versus the scaled coupling capacitances c. Here g max is the value of γ for which maximum NR is achieved.
In the inset, the linear relation between g max and g MT is shown for several different coupling capacitances = c 0.1, 0.4, 0.7 as the scaled gyrator strength g varies from 0.1 to 1. Other parameters are z = 1 0 , g = 0.03 0 , and m = 0.03. Figure 13. A 3D plot of nonreciprocity parameter NR versus gain/loss parameter γ and the scaled distance d˜. Other parameters are as in figure 11 i.e. z = 1 0 , g = 0.3, g = 0.03 0 , c = 0.3, and m = 0.03. The value of  g for the corresponding isolated system is  g = 0.3 (see figure 11). Notice also that for smaller values of  g g < we have a weak non-reciprocal behavior, and that the preferred transmittance direction alternates for each domain p p < < + n d n 1 ( ) .