ABDELSHAFY, OTHMAN, OSHMARIN, AL-MUTAWA, CAPOLINO: EPD IN CTLS UC IRVINE, SEP 2018

Transcription

1 arxiv: [physics.app-ph] 13 SEP 2018 ABDELSHAFY, OTHMAN, OSHMARIN, AL-MUTAWA, CAPOLINO: EPD IN CTLS UC IRVINE, SEP 2018 Exceptional Points of Degeneracy in Perioically- Couple Waveguies an the Interplay of Gain an Raiation Loss: Theoretical an Experimental Demonstration Ahme F. Abelshafy, Mohame A. K. Othman, Dmitry Oshmarin, Ahma Al-Mutawa, an Filippo Capolino Abstract We present a novel paraigm for ispersion engineering in couple transmission lines (CTLs) base on exceptional points of egeneracy (EPDs). We evelop a theory for fourth-orer EPDs consisting of four Floquet-Bloch eigenmoes coalescing into one egenerate eigenmoe. We present unique wave propagation properties associate to the EPD an evelop a figure of merit to assess the practical occurrence of fourth-orer EPDs in CTLs with tolerances an losses. We experimentally verify for the first time the existence of a fourth EPD (the egenerate ban ege), through ispersion an transmission measurements in microstrip-base CTLs at microwave frequencies. In aition, we report that base on experimental observation an the evelope figure of merit, the EPD features are still observable in structures that raiate (leak energy away), even in the presence of fabrication tolerances an issipative losses. We investigate the gain an loss balance regime in CTLs as a mean of recovering an EPD in the presence of raiation an/or issipative losses, without necessarily resorting to Parity-Time (PT)-symmetry regimes. The versatile EPD concept is promising in applications like high intensity an power-efficiency oscillators an spatial power combiners, or low-threshol oscillators an opens new frontiers for boosting the performance of large coherent sources. Inex Terms Degeneracies, Electromagnetic Bangap, Perioic Structures, Multi-transmission lines. E I. INTRODUCTION LECTROMAGNETIC guiing structures or resonators are characterize by their evolution equations in terms of the eigenmoes (eigenvalues an eigenvectors). Among many features of the evolution of these eigenmoes, we explore the points in the parameter space of such system at which two or more eigenmoes coalesce into a single egenerate eigenmoe [1] [4]. We enote these points as exceptional points of egeneracy (EPD), an the egeneracy orer represents the number of coalescing eigenmoes. Perioic guiing structures enable the occurrence of a funamental class of EPDs at the so-calle regular ban ege (RBE) at which staning waves This material is base upon work supporte by the Air Force Office of Scientific Research awar numbers FA an FA , an by the National Science Founation uner awar NSF ECCS The authors are with the Department of Electrical Engineering an Computer Science, University of California, Irvine, CA USA. ( abelsha@uci.eu, mothman@uci.eu, f.capolino@uci.eu). Fig. 1. Example geometries of waveguie microstrip lines on a groune ielectric slab that support a fourth orer EPD, visible in the (k-) ispersion iagram at microwave frequencies. Examples in (a) an (b) represent two cases of coupling, with proximity fiels an with a physical connection, respectively. Results in Sections III an VI are base on a CTL that moels the geometry in (a), whereas results in Sections IV an V are base on the microstrip CTL geometry in (b). In the lossless case, the fourth orer EPD is calle DBE, however such structures exhibit both secon an fourth orer EPDs in the case of gain an loss balance. Fig. 2. Representation of the eigenvectors of the couple waveguies near an EPD, schematically showing that the four-inepenent vectors coalesce into a egenerate eigenvector at the EPD when one system parameter is varie. with zero group velocity are manifeste at the ban ege (i.e., the separation between pass an stop bans). EPDs also occur in Parity-Time (PT)-symmetric structures such as couple waveguies an resonators when the system s refractive inex obeys n(x) = n * ( x) where x is a coorinate in the system [3], [5] [7] an * enotes complex conjugation. EPDs in PT-

2 symmetric structures occur in couple waveguies [1] [4]. We point out that EPDs cannot occur in systems whose evolution is escribe by a Hermitian matrix (see Section II). Thus, the EPD occurs in systems whose the system vector s evolution, in space or time, is escribe by a non-hermitian matrix which can be impose by perioicity or also by having losses an gain in the system [3], [4]. Recently, some of the authors have shown that EPDs of secon orer can occur with asymmetric istributions of gain an loss in uniform CTLs [2]. Furthermore, in [8] secon orer EPDs in uniform CTLs with loss an gain have been investigate also from the bifurcation theory point of view. In this paper, we investigate secon an fourth orer EPDs occurring in perioic guiing structures whose wave ynamics are represente by two perioically-couple TLs. In particular we are intereste in fourth orer EPDs that occur when all four inepenent Floquet-Bloch eigenmoes coalesce in their eigenvalues an eigenvectors an form one single egenerate eigenvector [1], [9], [10] at the ban ege as it will be clear later on in the paper, in absence of losses an gain. When a fourth orer EPD occurs at the ban ege of a lossless structure it is calle egenerate ban ege (DBE). This DBE conition is the basis for a possible enhancement of gain in active evices comprising DBE structures [11], [12]. Although the DBE was first shown in perioic layers of anisotropic materials where two inepenent polarizations are couple throughout the structure [1], [10], other investigations were also carrie out using microstrip lines [20], [21], for filtering an antenna applications [13] [15]. Moreover, couple silicon waveguies were also esigne [22], [23], with potential application in low threshol an high efficiency lasers [11]. However, in connection with experimental stuies, the existence of EPD features was shown experimentally in perioic anisotropic layere meia consisting of ielectric layers fabricate of low-loss microwave ceramic isks [18]; such geometry exhibits the split ban ege [1], [19]. Also, in optical waveguies the giant resonance an giant Q factor scaling associate with DBE were observe [17], [20]. More recently, the authors propose that the DBE can also be manifeste in all-metallic perioically-loae waveguies [21] an experimentally etecte even in the presence of losses an tolerances [22]. The application of such waveguie in high power microwave oscillators base on electron beams emonstrates low starting (threshol) current an better threshol scaling with length compare to conventional backwar wave oscillators [12], [23]. The same concept has been aopte for investigating laer circuit oscillators [24], [25] with low threshol an potential power efficiency. Note that the DBE (i.e. fourth orer EPD) occurs rigorously only in lossless structures as shown in [1], [11], [13], [21], [22] but fourth orer EPDs can also be achieve in gain an loss balance CTLs as will be shown in this paper. This is ifferent from the CTL regime iscusse in [2] since there secon orer EPDs have been consiere in uniform CTLs, while here we eal with perioic CTLs an the investigate EPD regimes are of the fourth orer. It is also worth noting that the gain compensation conition, the so-calle gain an loss balance conition in CTLs investigate in this paper to obtain fourth orer EPDs oes not necessarily mean PT-symmetric systems. Moreover, in the realization of EPDs with the gain an loss balance conition, it is important to observe that losses can actually represent istribute raiation in a perioically opene waveguie structure. This concept paves the way for a new class of raiating array oscillators base on EPDs. Besies the presente general formulation that is applicable to any CTL, we iscuss in etails the two microstrip couple transmission lines shown in Fig. 1. However, the conclusions rawn can be extene to many other geometries or structures since our formalism is general; operating from RF to optical frequencies. The rest of the paper is organize as follows. In Section II we evelop the theory of perioic CTLs an iscuss the characteristics of EPDs in perioic couple waveguies. In Section III we introuce the concept of hyperistance an investigate the effect of losses an coupling on the ispersion iagram as well as the effect of perturbations. In Section IV an V we show the experimental emonstration of EPDs in perioic as well as finite structures at microwaves base on couple microstrip lines. In Section VI we escribe EPDs in CTL, base on gain an loss balance. II. SYSTEM DESCRIPTION OF CTLS We consier a pair of CTLs such that two inepenent moal fiels are able to propagate along each z-irection, so a total of four inepenent moes are allowe if we inclue also the z irection. At an EPD some of these moes (two or four, epening on the case, in a system satisfying reciprocity) will not be inepenent anymore an actually will coalesce. We refer entirely to the formulation in [2] for homogeneous (i.e., uniform) TLs that escribe the fiel evolution using a CTL j t e approach, assuming a time harmonic evolution as. In this section we consier the general case of a perioic CTL that may or may not possess issipative as well as raiation losses an/or gain. A. State vector an wave propagation in CTLs We start by representing the fiel amplitues in the two CTLs by equivalent voltage an current vectors ( z) = V ( z) V ( z) T I ( z) = I ( z) I ( z) T. It is V an convenient to efine the four-imensional state vector V( z) Ψ( z) I( z) that comprises voltages an currents at any coorinate z in the CTL. This technique has been employe in [1], [12], [26], [27] to investigate the moal properties of photonic crystals an perioic waveguies. The system evolution along z is escribe by the first orer ifferential equations [2] (1) ( z) = j ( z) ( z) z Ψ M Ψ (2) where M is a 4 4 CTL system matrix, where M is given by 0 jz( z) M( z) = jy( z) 0 (3)

3 where 0 is the 2 2 zero matrix, an Z an Y are the series impeance an shunt amittance matrices escribing the per unit parameters of the CTL [2], [27], [28]. Note that in [2] we have investigate the uniform CTL case, i.e., the case when M is invariant along z. The properties escribe in this paper are relate to a perioic variation of the system matrix. M( z) The series impeance an shunt amittance matrices escribing the per unit parameters of the CTL are efine as Z = jl + R, an Y = jc + G. The CTL per unit length inuctance L an capacitance C matrices are 2 2 symmetric an positive efinite matrices [26], [29], whereas the per unit length series resistance R an shunt conuctance G matrices are 2 2 symmetric matrices [26], [29] accounting for losses an for small-signal linear gain introuce for by negative resistance or conuctance. Note that an G are positiveefinite matrices if an only if they represent only the losses [26], [29]. Moreover, we recall that the capacitance an conuctance matrices have negative off-iagonal entries (see Ch. 4 in [26]). Cutoff conitions coul be moele by resonant series an shunt reactive elements as was one in [12], [27]. For instance, cut-off series capacitances (inuctances) per unit length can moel cutoff conitions for TM (TE) waves in the waveguie an coul be inclue in the impeance (amittance) matrix (see Ch. 8 in [30]). However, for the sake of simplicity here we ignore cutoff conitions since we analyze microstrip lines in terms of their funamental quasi-tem moes that ieally o not have a cutoff frequency. The homogenous solution of (2), assuming a certain bounary conition at z = z 0, namely Ψ( z 0) = Ψ 0 insie a uniform (i.e., z-invariant) CTL segment, is foun by representing the state vector solution at any arbitrary coorinate via z 1 R Ψ( z ) = T( z, z ) Ψ ( z ) (4) where we efine T (z 1, z 0) as the transfer matrix which translates the state vector between the two points an z 1 Ψ( z). Within a uniform segment the transfer matrix is easily calculate as ( ) T( z, z ) = exp j( z z ) M (5) an the transfer matrix satisfies the group property T( z, z ) = T( z, z ) T ( z, z ) an the symmetry property T( z, z ) T( z, z ) = 1 (6) where 1 is the 4 4 ientity matrix. The previous iscussion ientifies the transfer matrix of a uniform segment of a CTL, an it is use to escribe perioic waveguies mae of a cascae of uniform CTL segments. z 0 B. Evolution of waves in perioically couple CTLs Let us assume a perioic CTL compose of two uniform segments A an B cascae as shown in Figs. 1(a) an 1(b). The transfer matrix of each iniviual CTL segment is given by T T A B T( z +, z ) = e A A, 0 A 0 T( z +, z ) = e 0 B 0 jm jmbb where, an are efine in terms of the per-unit-length impeance an inuctances of the segments A an B, respectively using (3), while an are the lengths of M A M B A segments A an B respectively. The transfer matrix of the unit cell of the CTL shown in Fig. 1(a), is expresse as the prouct of the two transfer matrices of the iniviual segments of the unit cell calculate in (7) as. On the other han, the transfer matrix of a unit cell of the perioic CTL in Fig. 1(b) incorporates an aitional coupling matrix ue to the coupling microstrip. Particularly in Fig. 1(b), segments A an B are uncouple while the coupling between TL1 an TL2 is meiate through another transfer matrix enote by. T B = T T U B A Hence, the unit cell transfer matrix is TU = TB TC T A. The transfer matrix unit cell as (7) T C translates the state vector across a Ψ( z + ) = T Ψ ( z) (8) where is the perio in Figs. 1(a) an 1(b). For an infinitely long stack of CTL unit cells, a perioic homogenous solution for the state vector exists in the form ( ) jkz U Ψ z e. This form enotes a Floquet-Bloch type solution where k is the complex e jkz Floquet-Bloch wavenumber an is referre to as the Floquet-Bloch multiplier. To fin such Floquet-Bloch wavenumbers an the eigenvectors, we write the following eigenvalue equation ( ) n 1 Ψ n( z ) = 0 (9) where are the regular eigenvectors, corresponing to four eigenvalues (or Floquet-Bloch multipliers) given by Ψ n jkn n = e, with n = 1,2,3,4. The eigenvalues can be reaily obtaine as solutions of the characteristic equation et T 1 = 0. We introuce the matrix as a 2 2 ( ) U iagonal matrix, whose iagonal elements are the Floquet- Bloch wavenumbers with positive real values, i.e. k = iag( k, k ). We efine the Brillouin zone in our perioic 1 2 structure from k = 0 to k = /. Therefore k1, k2, k1 + 2 /, an k2 + 2 / are the four moal wavenumbers of the four inepenent Floquet-Bloch moes in the perioic structure, insie the Brillouin zone. In fact, all the Floquet-harmonics whose wavenumbers are kn, p = kn 2 p / with p = 1,2,3, obey symmetry because we consier reciprocal systems (k an k are both solutions). We will also efine Λ as a iagonal matrix whose iagonal elements are the eigenvalues jkn e via k

4 (Note that e jk 2 j e k Λ e 0 jk = jk e 0 is a iagonal matrix with elements e jk 1.) Therefore, it follows that the transfer matrix written as U 1 (10) an T = U ΛU (11) where U is 4 4 matrix that serves as a non-singular similarity transformation that iagonalizes, an is compute using the four regular, normalize eigenvectors of as U = Ψ Ψ Ψ Ψ meaning that the column vectors of U are the regular eigenvectors of is that are linearly inepenent. If (11) is satisfie with a non-singular U we say that is similar to a iagonal matrix. In principle, the matrix is non-hermitian, but it satisfies some other important properties in the absence of gain an loss, where each constitutive CTL segment has a z-evolution matrix M that is J-Hermitian an a Hermitian characteristic matrix ZY, mainly the J- unitarity property, = U 1 0 TU ( z, z ) JT ( z, z ) J (12) where J is efine in eq. (9) in [2]. Importantly, the constitutive matrices an for instance, are iagonalizable in the absence of gain an loss in each segment, an therefore both an have a complete set of eigenvectors. In aition, in the lossless case their eigenvalues lie on the unit circle of the complex eigenvalue plane. The prouct, on the other han, is not necessarily iagonalizable an inee EPDs can manifest in the parameter space of the perioic structure escribe by, an the complex Floquet multiplier, which T A T A T B T B are the eigenvalues of TT B, may not lie on a circle in the complex eigenvalue plane even in lossless structures as we will show in Section III. C. Fourth-orer exceptional points of egeneracy We efine the fourth-orer EPD as a point in the parameter space of the perioic CTL at which the four eigenvectors coalesce into a single egenerate eigenvector. At an EPD the matrix becomes efective, i.e., it cannot be iagonalize. At a fourth orer EPD, the eigenvalue problem (9) oes not provie a complete basis of inepenent eigenvectors: they coalesce into a single one, hence U in (11) becomes singular. Strictly speaking, it means that it is not possible to fin a nonsingular similarity transformation as in (11) that iagonalizes. It means that must be similar to a matrix that contains a non-trivial Joran block. Hence linearly inepenent generalize eigenvectors in such efective system are foun in the generalize Floquet-Bloch form as q g 1 Ψ q( z ) = 0, q = 1,2,3,4 (13) A g z where here Ψ1 ( ) Ψ 1 ( z ) is a regular or orinary eigenvector (same as a generalize eigenvector of rank 1) while Ψ g 2 () z Ψ g 3 () z an Ψ g 4 () z are the generalize eigenvectors of ranks 2, 3, an 4, respectively (see etails of generalize eigenvectors in Ch. 7 in [31]). A fourth orer EPD in our CTL system occurs if an only if the transfer matrix is similar to a Joran canonical matrix given by TU = S Λ S, Λ = (14) g g g g with S = Ψ1 Ψ2 Ψ3 Ψ 4 being compose of one regular eigenvector an three generalize eigenvectors corresponing to a coincient eigenvalue with multiplicity four, with = exp( jk) an Λ in (14) is a 4 4 Joran matrix. This is the highest orer egeneracy that may be obtaine in reciprocal an linear structures with only two couple TLs, since it combines all supporte waves (forwar an backwar, an/or propagating an evanescent). As the system is reciprocal the symmetry must hol. Therefore if we require to have only one egenerate eigenvalue k = exp( jk ) with multiplicity of four, this implies that k must be either k = / or k = 0. Therefore, the fourth orer EPD can occur either at the Brillouin zone ege (k = 0) or at the center (k = π /). Here we have efine the funamental Brillouin zone (BZ) within the range from k = 0 to 2π. The evolution of the four eigenvectors near the fourth orer EPD, varying either frequency or any other parameter in the CTL, is schematically epicte in Fig. 2. We also stress that such an EPD (i.e. 4 th orer EPD) occurs in an entirely passive structure without gain or loss. Later in Section VI, we show that it may also occur in the presence of gain an loss when a particular balance conition is reache. In the following section, we explore examples where these three kins of exceptional points manifest themselves in CTL structures. III. FOURTH ORDER EPD IN PERIODIC CTLS The examples in Fig. 1(a) an (b) illustrate two couple waveguie geometries that can potentially support a DBE. We recall that the DBE occurs in lossless an gainless structures. Therefore, uner this assumption, when the CTL is mae of a perioic structure like Fig. 1, both M A an M B are J- Hermitian, an the characteristic matrices ZY an YZ in are Hermitian. The microstrip geometries in Fig. 1 constitute perioically cascae segments of couple/uncouple microstrip lines that support four Bloch moes (two in each irection). In this section we elaborate on the perioic microstrip TL in Fig. 1(a). The other example in Fig. 1(b) will be utilize in Sections IV an V. In the following, we erive the ispersion relation for the fourth orer EPD as well as introuce a figure of merit which we call hyperistance to assess the quality of such EPD subject to any kin of perturbation, e.g.,,

5 losses, frequency or structural perturbations like a coupling capacitance. We first consier lossless CTLs an at the en of this section we consier also losses in the CTL when iscussing the hyperistance concept. A. Fourth orer egeneracy We first assume the perioic CTL to be lossless (no gain is introuce yet; we will investigate the case with gain an loss in Section VI), hence the matrix M for each segment satisfies the J-Hermiticity property [1] (see also [31, Ch. 6]) 1 1 M = JMJ, M = NMN (15) where J an N are efine in eq. (9) in [2]; an as such each iniviual CTL segment has four moal solutions that have purely real propagation wavenumbers. Inee, we recall that each of the constitutive CTL segments has the characteristic matrices ZY an that are Hermitian an has real eigenvalues as proven in the Appenix of [2]. YZ The perioic CTL has eigenvalues n = exp( jkn), with, where is the Bloch wavenumber, that is obtaine from the solution of (9). In orer to realize the fourth orer EPD at a certain frequency, we impose the Joran block similarity (14) on the transfer matrix of the perioic unit cell. There are several possible unit cell TL parameters (L, C) combinations that make satisfies (14). A set that moels the geometry in Fig. 1(a) is provie as an example with parameters as in Appenix A, leaing to a DBE at 4.03 GHz, which is the case shown in Fig. 3. n =1,2,3,4 k n B. Dispersion perturbation analysis an Puiseux series When a system parameter of the CTL (a geometry or electrical parameter like, or frequency, or because of the introuction of losses) is perturbe by a small parameter, the perturbe eigenvalue is written as a perturbation of the egenerate eigenvalue in the neighborhoo of an EPD eigenvalue e = exp( jke) in terms of a fractional power expansion (also calle Puiseux series [33] [35]) in the perturbation parameter. Since in this section we focus on the fourth orer EPD in absence of losses an gain (i.e., on the DBE), the eigenvalue is = exp( jk) which can be relate to CTL wavenumber k as (the proof is provie in Appenix B) C m 2 2 j( n 1) j( n 1) 4 1/4 2 1/2 kn( ) = k + a1e + a2e + (16) Here an ' s are the fractional series expansion coefficients for the nth eigenmoes, an n = 1,2,3,4 provie the four possible quartic roots near the EPD. The perturbation factor coul be, for example, efine as the normalize etuning of angular frequency from the exact DBE one, i.e. = ( ) /, where is the DBE angular frequency. Note that in (16), we enote 1/4 as the quartic root in the first quarter of the complex plane since we take the four ifferent quartic roots into account using the exponential terms exp( jn ( 1) / 2). Keeping only the lowest orer terms in the fractional power expansion in (16) leas to the approximate form of the ispersion near the DBE 4 ( ) h( k k ), (17) where h is a parameter that efines flatness of the ispersion near. Inee, from (16) an (17), by substituting = ( ) /, it is clear that the flatness parameter is 4 1 h= / a. C. Dispersion relation for fourth orer egeneracy In Fig. 3 we show the ispersion iagram for a lossless CLT with circuit parameters in Appenix A, that moel the geometry in Fig. 1(a), with three ifferent values of the coupling capacitance per-unit-length (note that the coiagonal elements of C are ). In Figs. 3(, e, f) we only plot the moal curves that are relate to moes with purely real wavenumber for the lossless CTL. Figs. 3(a, b, c) show the complex wavenumber evolutions varying frequency. In the following we iscuss the occurrence of three possible EPDs in lossless CTL: the RBE, the DBE an split ban ege (SBE) by varying. We efine with as the value at which the C m -k C m C m, DBE occurs. Losses are consiere at the en of the section an in the last two columns of Fig. 3 (curves with ashe lines). In Fig. 3(e) we plot the ispersion relation of moes with purely real wavenumber. Since there are four solutions, for there are other two other moes that are evanescent (not shown in Fig. 3(e) but shown in Fig. 3(b)) an all four moes coalesce at. For all four moes are evanescent. Note that the perioic structure s ispersion iagram is perioic in ( ) with perio, hence we plot the ispersion in the first Brillouin zone efine as 0 Re k( ) /, an because of reciprocity wavenumbers are symmetric about the Brillouin zone center k = k /. Re k / -k The so-calle frozen moe regime associate with the DBE is relate to the vanishing group velocity an has been iscusse in previous publications (see [1], [10] [12], [21], [23], [27], [36] [38] an references therein). In this section we focus on the characteristic of the fourth orer EPD, the DBE, an what happens near it, an in Section VI the effect of gain an loss on the DBE is analyze. The eterminant of the similarity transformation U of the transfer matrix in (11) is epicte in Figs. 3(g, h, i) varying frequency, for three choices of C m : we observe that U is singular ( et( U ) = 0) only at the exceptional points. Note that there exist two ifferent kins of EPDs: the DBE (fourth orer egeneracy, associate to four coalescing eigenvectors) that is the most interesting case an the main subject of the stuy in this paper, an various RBEs (secon orer egeneracies associate to two coalescing eigenvectors). In Figs. 3 (a, b, c) we show the evolution of complex wavenumbers in the complex Re(k)-Im(k) plane for increasing frequency (the irection of the arrows), for three values of the capacitance perunit-length C m. By varying C m, three ifferent situations are observe in the ispersion iagrams an they are iscusse C m C m

6 Fig. 3. First two columns: Complex k plane showing the trajectory of the wavenumber k for increasing frequency near the fourth orer EPD conition for ifferent values of the couple capacitances in (a), (b), an (c), for a lossless CTL, an the corresponing ispersion iagram showing the evolution from RBE to DBE an to SBE in (), (e), an (f), respectively, when varying the coupling capacitance. Dots in the left panel plots inicate the starting point = 0.95 of the frequency sweep. In (g)-(l) we show the measures of the closeness to the EPD calculate base on the four eigenvectors (g), (h), (i) epicts et( U ) = 0 while (j), (k), (l) epicts Ψ n of the lossless an lossy perioic structure: for ifferent values of C m. Any EPD necessarily has et( U) = 0 an a fourth orer egeneracy has DH ( ) = 0. Mathematically speaking, only the lossless case with C m = C m, shows a fourth orer EPD (i.e., the DBE). Other values of C m lea to secon orer EPDs (i.e., the RBEs). Losses perturb the RBE an the DBE. Soli lines represent results for a lossless CTL, whereas ashe lines represent results for a CTL incluing raiation loss in which EPD is no longer strictly manifeste, but still it can occur in practice, epening on how small is. Results are base on the microstrip geometry in Fig. 1(a) whose parameters are given in the Appenix A. Here e 2 (4.03GHz) an ke k = /. in the following. The proper coupling capacitance that allows the DBE is enote as C m,. For small coupling istribute capacitance (C m< C m,), the CTL exhibits only one lower an one upper RBE at, as typical in perioic single TL at the ban ege. In Fig. 3(a) trajectories of eigenmoe coalesce twice at at two ifferent frequencies (the upper an lower ban k = k k = k eges) esignating two istinct secon orer EPDs. At this secon orer EPD, two eigenmoes coalesce an this is in principle analogous to the secon orer EPD that occurs in the uniform CTL investigate in [2] where a balance gain an loss conition was necessary to evelop an EPD. Therefore, a secon orer EPD can be realize either with a uniform CTL with balance gain an loss as in [2] or simply by using a perioic TL. In the latter case the EPD occurs at the center of the BZ, i.e. at k = /. Increasing C m, such that C m = C m,, we fin the proper conitions for the fourth orer EPD (the DBE) to be manifeste. At the DBE, four Floquet-Bloch eigenmoes coalesce to a single egenerate eigenmoe; as explaine in Fig. 3(b) where four moal trajectories of complex k varying as a function of frequency intersect at a single point k = k. At the DBE frequency the ispersion iagram in Fig. 3(e) is very flat an thereby approximate by the quartic law (17). Note that this conition cannot occur in uniform CTL mae of only two TLs with L an C istribute parameters, as in this paper, since it requires the presence of a bangap that is enowe by perioicity. (Other important cases with two uniform CTLs with ifferent istribute parameters for backwar waves an evanescent waves will be iscusse in the future.) Increasing the coupling such that C m > C m, will lea to altering the DBE to a split ban ege (SBE) [39] [41] that has been sometimes referre to as ouble ban ege (DbBE) [15], [37] where three istinct EPDs are foun in the ispersion iagram in Fig. 3(f), in the shown frequency range. Each EPD is associate with a secon orer egeneracy in the eigenmoes as seen in Fig. 3(c). At each EPD the ispersion iagram is flat an it exhibits a vanishing group velocity. It can be observe that when C m< C m, or C m> C m,, two EPDs are manifeste, all of secon orer, an either all of them on the same sie of the stop ban (Fig. 3(f)) or on both sies of the stop ban (Fig. 3()). These EPDs are separate, yet for C m=c m, these secon orer EPDs coalesce an form a fourth orer EPD (DBE) (Fig. 3(e)). D. Hyperistance in four-imensional complex vector space for ientifying vicinity to a fourth orer EPD In orer to istinguish a fourth orer EPD among others, an in orer to unerstan how far from the EPD a system is, one ought to efine a figure of merit (or hyperistance) to assess the quality of such EPD subject to any kin of perturbation, like losses, frequency etuning, or tolerances of CTL parameters. Here we focus on losses an coupling as the cause of EPD perturbation. Structures, such as that in Fig. 1 are naturally lossy

7 ue to issipative losses (ielectric an ohmic losses) as well as raiation losses that limit the intrinsic quality factors of the constitutive components. Therefore, a perfect egeneracy conition like the DBE corresponing to a lossless structure oes not exist in practice when losses are present but can be met in an approximate way an still retain the main features of the four coalescing eigenvectors. In case the perioic CTL is a raiating array, then raiation losses (i.e., istribute power extraction) are consiere necessary. However, as we will clarify in Section VI, CTLs with (raiation) losses may rigorously exhibit EPDs when a gain is introuce in a balance fashion. Just a few stuies have shown the sensitivity of fiel enhancement in DBE structure to losses [28], [40], [42], an in general any imperfection can be thought as perturbations that affect the eigenvalues (wavenumber) in the way escribe by (16). Moreover, in esigning realistic waveguies, numerical or experimental methos are use an numerical or systematic errors woul require a quantitative measure for observing an exact EPD. Since our state vector in the CTL is four imensional, we evelop the concept of hyperistance between the four eigenvectors of the transfer matrix of one-unit cell to etermine the closeness to an EPD. Various DH ( ) choices coul be mae for its efinition, an here the hyperistance that represents a figure of merit (FOM) is efine as 4 1 Re H = sin ( mn ), cos( mn ) = 6 m= 1, n= 1 Ψm mn D ( Ψm Ψn) Ψn (18) with representing the angle between two vectors an in a four-imensional complex vector space with norms Ψ n Ψ m mn an ( ) = Ψ n m n m n Ψ m. Angles are efine via the inner prouct Ψ Ψ Ψ Ψ, where the agger symbol enotes the complex conjugate transpose operation, an is efine to be always positive. This FOM yiels a hyperistance ieally equal to zero when all four eigenvectors in the CTL system coalesce, i.e., when the CTL system experiences a fourth orer EPD. Mathematically this is escribe by the transfer matrix of the unit cell becoming similar to a 4 4 Joran Block as in (14). Therefore, when losses (or any isorer of the structural parameters seen as perturbation) occur, the propose FOM is not zero. When using numerical methos or measurements (see Sec. IV) we can assume that the EPD is met in practical terms when the FOM measure is less than a very small threshol value, i.e., DH, where is a small number. It is natural to question when such an EPD occurs in practical terms, i.e., how small shall be to claim an EPD is verifie. Furthermore, it is important also to quantify how much losses or perturbations eteriorate the EPD. To better illustrate this concept, we plot in Figs. 3(j), (k) an (l) the hyperistance varying as a function of frequency, for a lossless CTL (enote by soli lines), i.e., for the three values of coupling capacitance C m consiere in Figs. 3(a, b, c). We can now compare the two FOMs introuce: the one associate to vanishing ( ) et U in Fig. 3(g, h, i) an the one associate to vanishing in Fig. 3(j, k, l). The first one vanishes when at least two eigenvectors coalesce, the secon one is the proper one to escribe a fourth orer EPD because it vanishes only if all four eigenvectors coalesce. Inee, we see that et ( U ) 0 at any EPD, whereas DH 0 only at the DBE frequency ( ) an only when Cm = Cm,, i.e., only in Fig. 3(k). = The two FOMs are also observe when losses are introuce in the CLT as istribute series resistance (ashe line in Figs. 3(g)-(l)). Losses are assume to be in both TLs with a quality factor Q TL of 1000, efine as. We also see that the perturbation ue to losses can eteriorate the DBE is now non-vanishing at =, when Cm = Cm,, in Fig. 3(k). This is in agreement with perturbation theory of eigenmoes leaing to (15), implying that a small parameter leas to a significant QTL = L / R change in the eigenvalues since when 1 ; which takes place in the close vicinity of the fourth orer EPD. We again stress that the FOM provies a more quantitative ientification of the fourth orer egeneracy compare to the simpler measure. The eviation in the ieal EPD et ( U) 1/4 conitions ue to losses can be seen in Fig. 3(j, k, l) by comparing ashe lines to soli lines for the ifferent kins of EPDs in Figs. 3(), (e) an (f) respectively. In a raiating array the limitation in the occurrence of such egeneracies is ue raiation loss (it is natural to introuce gain in the structure to balance the losses an potentially recover the exceptional point as we will show in Section VI). However, the question remains whether we can observe the EPD in experiments even in the presence of losses or not. The concept of hyperistance evelope previously is very useful to ecie how near a system is to an EPD an therefore if an EPD occurs in practice. Furthermore, the concept of hyperistance is helpful in etermining if introucing gain in some parts of the system is useful for compensating for losses, i.e., in efining the concept of gain an loss balance as it will be shown in Section VI. IV. EXPERIMENTAL VERIFICATION OF EPD FEATURES IN PERIODIC CTLS In this section we experimentally verify the existence of the EPD in the microstrip example in Fig. 1(b). First, instea of fabricating a long (multi-unit cell) couple microstrip, we experimentally emonstrate the occurrence of EPDs by performing measurement on a single unit cell fabricate as shown in Fig. 4(a). The unit cell is fabricate on a groune substrate (Rogers substrate RO4360G2) with a ielectric constant of 6.15, height of 1.52 mm, an with a loss tangent of The TL appearing on the top of the figure was esigne to have a characteristic impeance of 50 Ohms an all the unit cell imensions are reporte in Fig. 4(a). To confirm the existence of EPDs in the unit cell, we perform scattering (S)- parameter measurements at the four ports of the unit cell using a four-port Rohe & Schwarz Vector Network Analyzer (VNA) ZVA 67 as shown in Fig. 4(b).

8 Fig. 4. (a) A microstrip unit cell of a perioic structure that exhibits a fourth orer EPD. The substrate is Rogers RO4360G2 with a ielectric constant of 6.15 an height of 1.52 mm. (b) The fabricate unit cell uner test with the 4 ports is attache to a VNA to extract the 4-port S parameters versus frequency use to calculate the eigenvalues. Fig. 5. (a) Dispersion relation an (b) hyperistance measurement versus full-wave simulation of the unit cell of a perioic CTL exhibiting a 4 th EPD. The results show that the 4 th orer EPD occurs at 2 ifferent frequencies in the range shown in this plot. Full-wave simulations are performe with Keysight ADS using the metho of moments (MoM). From the S-parameters measurements, we then retrieve the transfer matrix as shown from well-known conversion tables [43], [44], for the range of frequencies shown in Fig. 5. The four wavenumbers are erive an plotte in Fig. 5 by solving (9) for complex k, at any frequency shown in the plot. The ispersion base on measurements shown in Fig. 5(a) is in a goo agreement to the results from full-wave simulations of the S-parameters performe using Keysight Technologies ADS base on the Metho of Moments (MoM). The ispersion shows several frequencies at which the four wavenumbers have values very close to each other enoting the occurrence of the 4 th orer EPD, with some perturbation ue to ohmic, ielectric an raiation losses. Inee losses, fabrication tolerances, effect of connectors an other realistic factors affect ieal EPD conitions. Even though mathematically speaking the EPD is not verifie exactly, in practical terms, the EPD s prominent features can be well-preserve as we iscuss in the following. In orer to assess the existence of the EPD in practical terms, we utilize the hyperistance concept evelope in Section III for the first time. We use the transfer matrix of the unit cell in Fig. 4(a), base on measurements, to calculate the four eigenvectors Ψ n, with n = 1,2,3,4, of the eigenvalue problem in (9). The eigenvectors are use first to calculate the hyperistance in (18) that is suppose to vanish when approaching a 4 th orer EPD. They are also use to buil the similarity transformation matrix U in (10) an hence et(u) that will ten to zero when at least two eigenvectors become epenent. We plot both an et(u) FOMs in Fig. 5(b) base on measurements an compare them to the same FOMs obtaine from full-wave MoM simulations. We can see that at ifferent frequencies, 2.83 GHz, an 3.11 GHz occurring at k = π the simulate an measure show a minimum enoting vicinity of a 4 th orer EPD espite the presence of raiative an issipative loss. This emonstrates the EPD properties of eigenvector coalescence is observe at microwaves. V. FINITE LENGTH CTLS WITH EPDS: GIANT RESONANCE AND RADIATION We investigate the properties of a finite length perioic CTL as in Fig. 6, with a fourth orer EPD. In principle, the coalescence of four eigenmoes into one egenerate eigenmoe at the DBE causes a quartic ispersion relation (17) an leas to a vanishing group velocity in the infinitely-long perioic structure in the absence of losses. The presence of losses an structural variations/perturbations leas to a non-zero group velocity an reners the hyperistance (18) non-vanishing as seen from the experiment in Fig. 5. In aition, even in a lossless structure, waves in a finite-length CTL with EPD o not have zero-group velocity since the peak of transmission in Fig. 7(c) occurs at a frequency close to the DBE one. Nonetheless, the finite-length CTL evelops very interesting resonance features relate to the EPDs as we iscuss in the following. Consier the finite length microstrip CTL mae of a finite number N of unit cells shown in Fig. 6. The microstrip an substrate s parameters are the same as those use in relation to Figs. 4 an 5. We perform full-wave simulations using Ansys HFSS utilizing the finite element metho (FEM) solver to retrieve the scattering parameters. The ispersion iagram of such structure is shown in Fig. 7(a), obtaine by simulating a single unit cell. The 4 th orer EPD is evient at 2.83 GHz. The

9 Fig. 6. A fabricate microstrip 8 cell array with an EPD. The CTL is base on the unit cell escribe in Fig. 4. Fig. 8. (a) Measurements versus simulations of the power loss parameter efine as 1 S 11 2 S 21 2 varying as a function of frequency near the EPD resonance as well as the simulation of the antenna raiation efficiency. (b) Raiation pattern of the EPD array at the resonance frequency taken along the z-axis, calculate via full-wave simulations using Ansys HFSS. Fig. 7. (a) Dispersion relation for the unit cell of the perioic structure in Fig. 6 obtaine using full-wave finite element metho (Ansys HFSS) accounting also for raiation, ohmic an ielectric losses. (b-c) Simulations an measurements of the scattering parameters S 11 (b) an S 21 (c) for an 8- unit-cell array in Fig. 6; where a goo agreement between full-wave simulations an measurements is shown at the resonance frequency associate with the EPD. The frequency of the resonance peak is estimate by (19). 4-port, N-cell, finite-length CTL is terminate with coaxial connectors at both extene sies of TL1, the length of the extene part on each sie equals 50 mm. Whereas TL2 is terminate in short circuits at the beginning of the 1 st unit cell an at the en of the 8 th unit cell, as seen in Fig. 6. We are intereste in the case where the excite fiels insie the CTL create a staning wave because of the Fabry-Pérot cavity (FPC) resonance near the EPD frequency (see etails in [1], [11]). At such conition constructive interference of the four synchronous eigenmoes leas to a sharp transmission resonance [1], [11]. Such Fabry-Pérot resonance closest to the DBE frequency occurs at the angular frequency enote by r,, referre hereafter as DBE resonance, characterize by a peak in the transmission coefficient S 21 an a ip in S 11. The DBE resonance frequency is approximately given by [3],[7] r, h N 4. (19) Therefore r, approaches for large number N of unit cells. Thanks to the DBE egeneracy conition, a large enhancement in the energy store is expecte because we approach the zero-group velocity conitions when r, an causing a giant enhancement in the Q-factor [11], [45]. We show in Fig. 7(b) an (c) FEM full-wave simulations as well as measurement of the magnitue of the scattering parameters S 21 an S 11. Results show goo agreement between simulation an measurement. Results also emonstrate the occurrence of the DBE resonance at 2.78 GHz, that is close to the DBE frequency of 2.83 GHz. The transmission resonance at the DBE has a transmission coefficient S 21 of 3.6 B. It is important to point out that such transmission resonance is affecte by losses; both issipative an raiative. Inee, the exact mathematical DBE conition is not met as seen in Fig. 7(a) an as iscusse in Section IV. Nonetheless, the DBE resonance peak is still observe in Fig. 7. We emphasize that most of the losses incurre by this CTL are in fact raiation losses. The power loss factor, shown in Fig. 8(a), is efine as 1 S 11 2 S 21 2 combining both effects of issipative Ohmic loss as well as raiation. Results show goo agreement between FEM full-wave simulations an measurements. When the finite-length CTL is seen as an antenna, its raiation efficiency (the ratio of raiate power over input power) is shown in Fig. 8(a) obtaine from full-wave simulations preicting an overall raiation efficiency of ~87% at the DBE resonance frequency. We also plot in Fig. 8(b) the raiation pattern obtaine from full-wave simulation to show that the CTL with DBE is raiating mainly en fire. This is because the progressive phase

10 shift of the guie wave between the unit cells is almost π at the DBE resonance. We emphasize that implementations of DBE in the literature are reporte at k= π, however the DBE can also be attaine at k = 0 an this will lea to a mostly broasie raiation as we will use in Section VII to conceive a raiating oscillator. VI. EPDS WITH BALANCED GAIN AND LOSS The RBE an DBE (EPDs of orer 2 an 4, respectively) exist in structures that are lossless, inee the presence of losses woul inhibit the existence of the DBE, i.e., it woul egrae the egree of coalescence of the four eigenmoes. This phenomenon is monitore by the hyperistance parameter that increases with increasing losses. An this is the reason why we have observe a non-ieal DBE ispersion relation in Fig 5. Increasing raiation losses (e.g., for antenna applications) will further egrae the DBE. Here we aim at proviing a mean to recover EPDs eteriorate by losses by strategically incorporating a proper istribution of gain in each unit cell. The perfect conition that allows the existence of an EPD is name perfect loss an gain balance. (For perfect EPD conition we mean the exact mathematical conition escribe in Section II that leas to ). The balancing between gain an loss implies that both the z-evolution matrices an are not J-Hermitian, an that the characteristic matrices an YZ DH 0 in (3) are non-hermitian. This oes not happen for the DBE case, therefore we will not refer to this conition as DBE. This loss an gain balance conition is ifferent from the situation previously stuie in PT-symmetric literature [2] [4], [6], [46] in two aspects: (i) The EPD stuie here is of fourth orer; (ii) the gain compensation conition for a fourth orer EPD in perioic CTLs oes not necessarily mean that the system is PTsymmetric [2], [3], [46], [47], i.e., it oes not mean that one TL has gain an the other has loss of exact symmetry an magnitue. Note that gain compensation of losses may, in various circumstances, not lea to a perfect EPD as will be shown in the stuy case B. However, in many practical cases the exact (i.e., perfect) conition may not even be necessary. Inee, when is sufficiently small we can observe qualitative features relate to the EPD even though the structure oes not possess the precise mathematical EPD conition. Furthermore, we point out that gain/loss compensation in a unit cell oes not necessarily imply absolute instability or selfoscillation in a finite length structure, inee loaing effect must be taken into account in the stability criterion especially near these EPD as one in [23]. The following iscussion is base on the perioic couple transmission line as in Fig. 1(a), where all parameters are given in Appenix A, investigate with the theoretical tools in Section III. We will stuy ifferent configuration for losses an incorporating gain. Case A: Series loss an shunt gain Losses in the CTL are assume to be in the form of a per unit length series resistance in both TLs, representing raiation an/or issipation losses. Here, gain is introuce in the perioic CTL using a negative per-unit-length conuctance G in both TLs (therefore G is positive with units of Siemens/m). Note that here, we assume that such gain is introuce in both M A ZY M B Fig. 9. (a) Plot of the hyperistance (18) to assess the occurrence of a fourth orer EPD in a CTL with losses, as a function of gain G to compensate for losses. The EPD is achieve in practical terms with G = G e = ms/m at which hyperistance has the minimum value = (keeping the angular frequency fixe at, at which the DH ( ) lossless CTLs evelops a DBE). (b) The corresponing ispersion iagram for the lossy CTL (i.e. before introucing gain) where losses are moele as series resistance R n such that all TLs have Q TL = 100. (c) The ispersion iagram after introucing the appropriate gain G = G e to achieve the balance gain an loss conition, showing the typical flatness of a DBE is recovere. Fig. 10. (a) Dispersion iagram for a lossy CTL structure (i.e. before introucing gain) as in Fig. 9(a) but with one orer of magnitue larger losses than in Fig. 9(a), i.e., now all TLs have Q TL = 10. (b) The corresponing ispersion iagram after introucing the gain G e = ms/m associate to minimum D ( ) = 0.2. H TLs, i.e., each TL has a self-negative conuctance G in aition to a series resistance R n associate to raiation an metal losses for the n th TL, with n = 1,2. Losses are escribe using the quality factor QTL, = L / R associate to the per-unit-length n n n TL parameters, an here we assume that all TLs have the same quality QTL, n = QTL for simplicity. Accoringly, here each constitutive CTL segment has a z-evolution matrix M that oesn t satisfy the J-Hermiticity an a characteristic matrix ZY that is non-hermitian. To estimate the value of the neee G in the CTL to recover the fourth orer EPD occurring at more or less the same angular frequency as that woul occur in the lossless structure = a simple proceure is evelope. Note that here for recovering an EPD we mean that the require hyperistance shall be smaller than a specific low threshol, that is arbitrarily preetermine (at a lossless 4 th orer EPD,

11 i.e. at a DBE, this vanishes). Therefore, we fin the value of the optimum istribute gain parameter, namely G e such that the at, where here we assume = 0.1. In Fig. 9 we show the ispersion iagram of a CLT with losses an with compensating gain, as well as the hyperistance plotte in DH = Fig. 9(a) versus gain parameter G at Q TL = 100 =. We observe that for all TLs with, a chosen values G = G e = ms/m, leas to a minimum DH 0.1 which inicates that the fourth orer egeneracy has been achieve in practical terms, i.e., the four eigenvectors are almost parallel. Inee, we Ψ n plot the ispersion relation of the lossy structure (i.e. before introucing gain) in Fig. 9(b). Then we chose the best G, i.e. G = G e, an in Fig. 9(c) we plot the ispersion with such gain an loss balance conition. It is clear that the ispersion in Fig. 9(c) is very similar to the ieal case with no loss an no gain plotte in Fig. 3(e). Aitionally, in Fig. 10 we show that this proceure is useful to recover EPDs also when the CTL is strongly perturbe by losses. Inee in Fig. 10 the CTL an iscussion is the same as for the case in Fig. 9, except that now the TLs have losses that are an orer of magnitue larger than in Fig. 9, i.e. all TLs consiere for the result in Fig. 10 have Q TL = 10. It is obvious from Fig. 10(a) that the ispersion of the moes in the lossy CTL (i.e. before introucing gain) is strongly perturbe from the ieal case, inee the DBE is not visible at all. Then, using the same proceure previously iscusse, one can recover the 4 th orer EPD in practical terms as shown in Fig. 10(b). We recall one more time that this is not guiing structures satisfying PT-symmetric since gain an loss are not isplace in a symmetric fashion. Case B: shunt loss in both TLs, shunt gain in only one TL We analyze another CTL configuration that has losses an gain not satisfying PT-symmetry. Losses in the CTL are assume to be in the form of a per unit length shunt conuctance G L,n in both TLs, representing raiation an/or issipation losses. Shunt per-unit-length losses are analogously escribe by a quality factor QTL,n = Cn / GL, n, an in the following we assume that all TLs have shunt conuctances G L,n such that Q =. Gain is introuce in the perioic CTL using Q TL,n = TL 100 a negative per-unit-length conuctance G in only TL1 (therefore G is positive with units of Siemens/m). Note that this CTL configuration iffers from the one in Case A in: (i) gain is introuce to one TL only, not in both TLs as in the previous case, an (ii) losses are introuce using shunt conuctance not a series resistance per unit length. Therefore, TL1 has a self-negative conuctance G in aition to a G L,1 associate to losses, whereas TL2 has only G L,2 associate to losses. Similar to Case A, each constitutive CTL segment has a characteristic matrix ZY in (3) that is non-hermitian, an the value of the neee gain G in TL1 to recover the fourth orer EPD is etermine with same proceure use in Case A. In Fig. 11(a) we plot DH versus the gain parameter G at = (that is the angular frequency at which the DBE for the ieal lossless Fig. 11. (a) Plot of the hyperistance (18) to assess the presence of a fourth orer EPD in a CTL with losses, as a function of gain G to compensate for losses. The EPD is achieve in practical terms with value of gain parameter G e = ms/m (attache to TL1) at which hyperistance has the minimum value = (i.e., less than the preetermine threshol DH ( ) ). Angular frequency is fixe at at which the lossless CTLs evelops a DBE. (b) The corresponing ispersion iagram for lossy structure (i.e. before introucing gain) where losses are moele as shunt G L,n such that all TLs have same Q TL = 100. (c) The corresponing ispersion iagram after introucing the gain associate to minimum, showing non-ieal EPD. structure is obtaine) an we observe that for G = G e = ms/m we have minimum an that such minimum has DH 0.1 which inicates that the fourth orer has been achieve in practical terms. The ispersion relation of the lossy CTL, i.e., before introucing gain, is plotte in Fig. 11(b), an in Fig. 11(c) we show the ispersion after introucing the gain associate to minimum. It is worth noting that, the gain compensation leas to a non-ieal EPD. However, in many practical cases the exact EPD may not even be necessary. Inee in Fig. 11(c), one can see that the ispersion iagram is very similar to the one of the ieal DBE, therefore we can still observe qualitative features relate to the DBE. Note that, increasing the gain beyon the value of G = G e may lea to self-oscillations in the unit cell an woul rener the structure absolutely unstable. Such technique however can be very useful in realizing novel schemes for low threshol oscillators which we will investigate in the near future, an have been alreay explore in the areas of electron-beam evices [23] an lasers [48]. VII. CONCLUSION We have experimentally emonstrate for first the time the occurrence of a fourth orer EPD (the DBE) in microstrip CTLs at microwave frequencies, first through four-port measurements of a unit cell leaing to the DBE ispersion relation an then through the transmission characteristics of a finite-length CTL. We have also introuce the novel concept of a hyperistance figure of merit to estimate the effect of perturbations like imperfect coupling, presence of losses an any other perturbation that may arise from fabrications or numerical

12 simulation on the EPD. The hyperistance concept is a way to measure the eigenvectors mutual istance in a multiimensional vector space, that shoul ieally vanish at the exact EPD conition. The smaller the hyperistance the closer a system is to the EPD occurrence. Base on the efine figure of merit, our experimental verification has confirme for the first time that the main EPD features of almost parallel eigenvectors can still exist in realistic perioic arrays that inclue raiation an issipative losses. Furthermore, in CTLs that experience significant raiation an issipative losses, we have shown that the so-calle gain an loss balance conition leas to recovering an EPD, up to a level that can be quantifie via the hyperistance concept, even in schemes that o not necessarily imply PT-symmetry. The gain an loss balance conition scheme can be applie in principle for any amount of loss in a CTL, paving the way to use it in the esign of active gri array antennas. The capability of obtaining EPDs in CTLs with large raiation losses when aing gain in a proper manner in each unit cell paves also the way to use this multi-eigenmoe egenerate scheme in the esign of array oscillators an highintensity spatial power combiners. Potential benefits may inclue low oscillation threshol, or even high-intensity raiation with high power efficiency, an spectral purity. Fig. A1. Geometries of the microstrip configurations aopte in Section III, that evelops an EPD. (a), (b) the equivalent passive CTL system of the geometries in Fig. 1(a). ACKNOWLEDGMENT The authors woul like to thank Prof. A. Figotin, UC Irvine, for very useful iscussions. Also, they woul like to thank Ansys, Inc. for proviing HFSS; Keysight for proviing Avance Design Systems (ADS); an Rogers Corporation for proviing the RF laminates. APPENDIX A: NUMERICAL PARAMETERS USED IN SECTIONS III & VI Perioic CTL parameters use in Section III an VI. The perioic CTL use has the following parameters pertaining to the microstrip lines shown in Fig. A1(a) over a groune ielectric slab with a ielectric constant of 2.2 an height of 1.5 mm, which provies a DBE at 4.03 GHz. All microstrips have a with of 1 mm. The couple line segment has length A = 10 mm an separation gap 0.2 mm, while the uncouple segment is mae of an uncouple TL1 with length uncouple TL2 length B2 =10 mm as shown in Fig. A1(b). Note that the use of the bene junction impacts the assumption of having a uniform segment of a TL1 by slightly increasing the self-inuctance an capacitance ue to bening [49]. Therefore, the corresponing equivalent CTL parameters of the microstrip line unit cells are as follows. For the couple section: C 11=C 22= pf/m, L 11=L 22=0.467 μh/m, C 12 = C 21= C m = 27.2 pf/m, L 12= L 21=L m=0.25 μh/m, are the matrix entries of the C an matrices. For the uncouple section: TL1 has L = 0.54 μh/m, an C = pf/m whereas TL2 has L = 0.5 μh/m, an C = 35 pf/m. Most of the results in Section III are for lossless CTLs, losses are consiere at the en of Sec. III an in the L B1 = 14 mm an Fig. A2. (a) Real an (b) imaginary parts of the k-ω ispersion iagram obtaine for microstrip CTL in Fig. A1(a) using two ifferent methos. First, a circuit simulator (blue curves) aopte in Keysight Technologies ADS. Seconly, by solving eq. (9) for the equivalent passive CTL system shown in Fig. A1(b) (re curves), whose circuit parameters (L, C) are state in this Appenix. In both simulations losses have been neglecte. Fig. A3. (a) Real an (b) imaginary parts of ispersion iagram obtaine for microstrip CTL in Fig. A1(a) using MoM full-wave simulation (Keysight Technologies ADS) accounting for raiation, ohmic an ielectric losses. examples in Section VI an in the full-wave simulation results shown in Fig. A3 for the structure in Fig. A1(a). In Fig. A2, we show the ispersion iagram (for both real an imaginary parts of k-ω) using circuit simulator, which is base on preefine moels to each piece of circuit an using those moels the system response is escribe by a system of equations solve using implicit integration methos, aopte in Keysight Technologies ADS (enote by blue curves). Also, we plot the ispersion obtaine using eq. (9) for the passive