On forward and backward modes in 1D periodic bounded structures

Received 11 April 2024; revised 13 August 2024; accepted 25 August 2024. Date of publication 7 October 2024; date of current version 9 October 2024. Digital Object Identifier 10.1109/JMW.2024.3451056 On Forward and Backward Modes in 1D Periodic Bounded Structures OSKAR ZETTERSTROM (Graduate Student Member, IEEE), RAÚL RODRIGUEZ-BERRAL , FRANCISCO MESA (Fellow, IEEE), AND OSCAR QUEVEDO-TERUEL (Fellow, IEEE) (Regular Paper) Division of Electromagnetic Engineering and Fusion Science, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden Department of Applied Physics 1, ETS Ing. Informática, Universidad de Sevilla, 41012 Sevilla, Spain CORRESPONDING AUTHOR: Oskar Zetterstrom (e-mail: [email protected]). The work of Francisco Mesa and Raúl Rodríguez-Berral was supported by Project PID2020-116739GB-I00 and Project MICIU/AEI/10.13039/501100011033. This article has supplementary downloadable material available at https://doi.org/10.1109/JMW.2024.3451056, provided by the authors. ABSTRACT We discuss the issue of identifying the forward/backward nature of modes in bounded onedimensional periodic structures. This identification is based on the possibility of adequately and uniquely defining the phase velocity in these types of structure. We propose a general definition of phase velocity for one-dimensional scalar waves and show that, according to that general definition, the voltage and current waves in nonhomogeneous lossless transmission lines with positive per-unit-length capacitance and inductance are necessarily forward waves. We analyze in detail the particular case of periodic transmission lines and question the conclusions about the forward/backward nature of their modal solutions that are traditionally drawn from the inspection of the Brillouin diagrams. Numerical results for the case of corrugated parallel-plate waveguides support the idea that all modes can be considered forward-like as long as a periodic transmission line model remains a sensible and reliable description of the problem. In more general scenarios, we show that an appropriate definition of the phase velocity can still be found for electromagnetic waves with at least one linearly polarized field and that they are also necessarily forward waves if they propagate through media with positive εand μparameters. Finally, we relate our discussion to the effective refractive index of periodic structures, highlighting that although its definition is not valid for a general periodic structure, it can be useful in many practical cases. INDEX TERMS Dispersion diagram, forward and backward modes, periodic structures. I. INTRODUCTION The propagation of electromagnetic waves can be conveniently controlled by structures composed of engineered scatterers. Such structures can be used to perform various conventional tasks, such as filtering [1],[2] and guiding [3] signals, but also more advanced tasks, such as analog computing [4] and controlling the radiation characteristics in leaky-wave antennas [5]. The analysis of these structures is vastly simplified if the scatterers are arranged periodically or, in other words, the structure is invariant after a translation with an integer multiple of the periodicity of the scatterers. Many interesting properties of the periodic structure can then be obtained by studying a unit cell of the structure with the appropriate boundary conditions using a Floquet analysis [6], [7],[8],[9]. The analysis of periodic structures typically begins with an examination of their dispersion diagram. This diagram establishes a relationship between frequency and wavenumber [6],[7],[8],[9]. It offers valuable insight into various characteristics of the periodic structure, such as pass- /stopband frequencies, fast-/slow-mode behavior, and the impact of losses, stopbands, or radiation on wave attenuation. However, although the dispersion diagram allows for a simple representation of the dispersion properties of the periodic structure, some important characteristics of the modes may be obscured by its simplicity and can therefore be © 2024 The Authors. This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ 690 VOLUME 4, NO. 4, OCTOBER 2024 misunderstood. One particular concept that can be misconstrued is the forward/backward nature of the modes, which corresponds to the phase and group velocities having the same/opposite direction. This can result in a questionable understanding of how devices operate and can complicate their design based on this concept. Moreover, it can also pose a conceptual difficulty in the choice of modes to be included in the spectral expansion of the field excited by a single source in the periodic structure [10],[11],[12]. This article deals with the above-mentioned issue of the forward/backward nature of the modes of one-dimensional (1D) periodic bounded structures. We expand on the research presented in [13] and further develop the discussion provided in that work. Although the usual expression of the phase velocity derived for sinusoidally shaped waves is not applicable to the modes of periodic structures, we propose a general expression of the phase velocity for scalar one-dimensional waves. According to this general definition, we show that the phase velocities of the voltage and current waves in nonuniform transmission lines are aligned with the direction of power flow as long as the per-unit-length (p.u.l.) capacitance and inductance parameters are positive. When applied to the particular case of periodic transmission lines with positive parameters, this implies that all their modes can be considered forward waves. This interpretation can be extended to the modes of 1D periodic waveguides as long as they can be described using a periodic transmission line model. In more general situations in which the multidimensional and/or vector nature of the electromagnetic waves need to be taken into account, the definition of phase velocity should be generalized accordingly. Although a completely general definition of phase velocity does not seem to be possible [7],[8],[14],itispossibleto find an unambiguous and conceptually valid definition for electromagnetic waves in which at least one of the fields is linearly polarized. The results presented in this paper are intended to highlight the challenges in characterizing modes in periodic structures. The discussions are illustrated with several examples in ascending order of complexity. Specifically, the paper is organized as follows. In Section II, we examine a possible general definition of the phase velocity of 1D scalar waves in lossless structures. Section III discusses the phase velocity of nonuniform transmission lines. In Section IV, we derive a specific expression of the phase velocity for the modes in periodic structures and study several application cases. Specifically, we first carried out a quasi-analytical study of a 1D photonic crystal, and later the forward/backwardness of the modes in periodically corrugated parallel plate waveguides was studied with the help of a commercial simulator. In Section Vwe discuss the effective refractive index in bounded periodic structures. Lastly, in Section VI we present the conclusions of the work. All the derivations, numerical results and discussions in the main body of this work deal with ideal lossless scenarios. In Appendix Awe present the generalization of the key theoretical expressions to include material losses, and demonstrate that moderate losses are not expected to have a significant impact on our results and conclusions. II. PHASE VELOCITY OF A GENERAL ONE-DIMENSIONAL SCALAR WAVE First we consider the problem of the definition of phase velocity in a lossless but otherwise general 1D scalar wave scenario (zis taken as the spatial variable). Under the assumption of a time-harmonic regime, any 1D scalar mechanical/electromagnetic wave magnitude can be written as A(z,t)=A0(z) cos(ωt−ϕ(z)) (1) where A0(z) is the position dependent amplitude of the wave and φ(z,t)=ωt−ϕ(z) is its phase. We will refer to ϕ(z)as the phase function of the wave. In phasor form, we have the following: A(z)=A0(z)e −jϕ(z)with A(z,t)=A(z)e jωt.(2) The existence of spatially harmonic traveling wave solutions is a well-known fact when dealing with a homogeneous wave problem. These solutions have a constant amplitude and a linear phase function ϕ(z)=βz+ϕ0, where βrepresents the phase constant. For these waves, which we will refer to as sinusoidal waves, it is customary to define the phase velocity as vp=ω β.(3) Conceptually, the phase velocity can be defined as the velocity of a point with constant phase. In this way, the phase velocity characterizes the movement of the monochromatic wave pattern. On the other hand, the group velocity can be defined as the velocity at which the envelope of a narrowband pulse propagates through the medium [7],[9], and characterizes the power flow (energy transport) associated with the wave [7],[8]. For a sinusoidal wave, the group velocity is given by vg=dω dβ.(4) When the wave pattern moves in the direction of the power flow, vpand vghave the same sign and the wave is said to be forward. On the contrary, if vpand vghave opposite signs, then the wave pattern progresses in the direction opposite to the power flow and the wave is said to be backward. Now, to our understanding, there is no reason why the usual conceptual definition of phase velocity mentioned above for simple sinusoidal waves cannot also apply to 1D waves with the general form given in (1). Indeed, if an imaginary observer moves with the phase velocity of the wave, the phase value is constant at the observer position if d dt(ωt−ϕ[z(t)])=ω−dϕ dz dz dt=0(5) and therefore the phase velocity vp=dz/dt(which, in general, will depend of the position z) is given by vp(z)=ωdϕ dz−1 .(6) VOLUME 4, NO. 4, OCTOBER 2024 691 ZETTERSTROM ET AL.: ON FORWARD AND BACKWARD MODES IN 1D PERIODIC BOUNDED STRUCTURES FIGURE 1. Lossless nonuniform transmission line with per-unit-length capacitance and inductance parameters C(z)andL(z). The direction (positive/negative) of the phase velocity at a particular point will vary depending on the specific problem being studied. However, if the phase velocity is directed along (or opposite to) the power flow at every point, then the wave can be unequivocally classified as forward (or backward) using the same criterion commonly applied to sinusoidal waves. III. NONUNIFORM TRANSMISSION LINE In the particular context under consideration, the general principles discussed in the preceding section can now be applied to examine the nonuniform transmission line problem depicted in Fig. 1. The line is assumed to be lossless and with p.u.l. inductance and capacitance parameters L(z) and C(z). The general expressions for the voltage and current along the line are the following: V(z,t)=V(z)ejωtV(z)=V0(z)e −jϕV(z)(7) I(z,t)=I(z)ejωtI(z)=I0(z)e −jϕI(z)(8) where the amplitudes V0(z) and I0(z), as well as the phases ϕV(z) and ϕI(z), are real functions by definition. The phasors V(z) and I(z) satisfy the time-harmonic telegrapher’s equations; that is, dV dz=−jωL(z)I(z)(9) dI dz=−jωC(z)V(z).(10) Introducing (7) in (9), the current can be written as I=j ωL dV dz=1 ωLV0 dϕV dz+jdV0 dze−jϕV(11) where the z-dependence of the different quantities is omitted, as also done in the remainder of this section. Taking into account that the time-average power that flows in the positive z-direction is given by P=1 2VI∗(12) substituting (7) and (11) into (12) leads to P=V2 0 2ωL dϕV dz.(13) Using now (6) to identify the phase velocity of the voltage wave, vV p,(13) can finally be rewritten as P=V2 0 2L 1 vV p .(14) Therefore, in the case of nonuniform transmission lines with positive p.u.l. inductance parameter, it is found that both vV pand Phave the same sign, resulting in a forward voltage wave. A similar procedure applied to the current wave gives the following expression for the time-average power: P=I2 0 2C 1 vI p .(15) Similarly, the current wave is necessarily forward for nonuniform lines that have a positive p.u.l. capacitance parameter. Equating (14) and (15) provides the following ratio between the phase velocities of current and voltage waves: vI p vV p=Z0 I0 V02 (16) where Z0=√L/Cis the (z-dependent) characteristic impedance of the line. Thus, the phase velocities of the voltage and current waves will have the same value only at points where their amplitudes satisfy the following condition: V0(z)=Z0(z)I0(z).(17) However, regardless of this fact, both waves are forward for transmission lines with positive parameters. The generalization of (14) and (15) to a lossy scenario can be found in Appendix A, where we discuss that moderate losses are expected to represent only a small correction to the relation between power flow and phase velocities. IV. PERIODIC TRANSMISSION LINE A periodic transmission line (PTL) can formally be considered a particular case of a nonuniform transmission line with parameters that vary periodically along the line. The modal solutions of a PTL are Floquet modes (also called Bloch waves), which can be written in phasor form as a harmonic phase factor modulated by a periodic function [6]; namely, A(z)=P(z)e −jkzzP(z+p)=P(z) (18) where P(z) can be complex in general, A(z) stands for either the voltage or current, pis the spatial period and kzis the modal wavenumber. Since P(z) is periodic, we can expand it into a Fourier series as P(z)= n Pne−jknzkn=2πn p(19) and therefore A(z)= n Pne−j(kz+kn)z= n Pne−jkznz.(20) An immediate observation [6] is that the modal wavenumber kzin (18) is not uniquely determined, since any two values 692 VOLUME 4, NO. 4, OCTOBER 2024 FIGURE 2. Example dispersion diagram: (a) phase constant, and (b) attenuation constant. kzthat differ in kmfor any mare equally valid, corresponding to periodic functions P(z) that differ in a phase factor e−jkmz. This is a consequence of the fact that, as (20) explicitly shows, the modal solutions are not sinusoidal waves, but rather a linear combination of an infinite number of sinusoidal waves (the so-called spatial harmonics Pne−jkznz)[7],[8]. In Fig. 2, we show a typical dispersion diagram of a lossless 1D periodic structure. We have decomposed the wavenumber as kz=β−jαwith βand αbeing the phase and attenuation constants. In this example, we observe distinct passbands for frequencies with α=0 and stopbands for frequencies where α>0. The presence of harmonics in the modal solutions manifests itself as periodic repetitions in the phase constant [see Fig. 2(a)]. All information about the phase constant can be obtained from any harmonic repetition, and often only the first (n=0) is represented. The first repetition is called the first Brillouin zone. Additionally, for reciprocal structures, the representation of the dispersion properties is further restricted to positive phase values. Although this reduced dispersion diagram, usually called a Brillouin diagram, is a convenient compact representation that can provide very useful information about the pass/stopband structure, it could potentially result in questionable conclusions regarding the forward/backwardness of the modal solutions. In fact, the second passband in the example shown in Fig. 2, which extends approximately from 20 to 30 GHz, is commonly regarded as a backward mode (terminology that was also used in our earlier research [15]). This interpretation is based on the fact that the phase constant value is positive within the first Brillouin zone, whereas the slope of the dispersion curve is negative. Since the slope of the dispersion curve is the same for every spatial harmonic, it is clear that the sign of the slope of the dispersion curve indicates the direction of energy transport along the line [7],[8]. However, the sign of the phase constant for the n=0 harmonic cannot be directly associated with the direction of the phase velocity of the modal solution. This is because the modal solution consists of an infinite number of harmonics with different phase constant values, some positive and some negative. Therefore, the “conventional” expression of the phase velocity given by (3), valid for sinusoidal waves, is not applicable here. To clarify this issue, we next apply the general definition (6) to a Floquet mode to obtain a specific expression for the phase velocity of the modal solutions of a PTL. Within a passband, the Floquet mode in (18) can be written as A(z)=|P(z)|ejξ(z)e−jkzz=|P(z)|e−j[βz−ξ(z)] (21) where ξ(z) is the argument of the complex function P(z). Therefore, the phase function of the wave is ϕ(z)=βz−ξ(z) (22) and, using (6), its phase velocity is given by vp(z)=ωβ−dξ dz−1 .(23) Note that this expression is independent of the choice of β.Ifβis replaced by β+kn, then the periodic function P(z) must be replaced by P(z)e jknz, and therefore ξ(z) becomes ξ(z)+knz. It should be noted that the contribution of the term dξ/dzto (23) is what makes the phase velocity negative for passbands with a negative slope of β(ω), thus resulting in forward modes. A. 1D PHOTONIC CRYSTAL As a simple quasi-analytic application example, we consider the periodic stack of two different dielectric slabs shown in Fig. 3(a), usually known as a 1D photonic crystal. This periodic structure can be rigorously modeled by the PTL in Fig. 3(b). The wavenumber of the modal solutions of this PTL can be obtained from [7],[9] βp=arccosA(ω)+D(ω) 2(24) VOLUME 4, NO. 4, OCTOBER 2024 693 ZETTERSTROM ET AL.: ON FORWARD AND BACKWARD MODES IN 1D PERIODIC BOUNDED STRUCTURES FIGURE 3. (a) A 1D photonic crystal consisting of a periodic stack of two dielectric slabs with relative permittivities ε1and ε2, and thicknesses d1 and d2. (b) Unit cell of the PTL that models the periodic structure. where Aand Dare entries of the transmission (ABCD)matrix of the unit cell, which can be calculated as [T]=AB CD =[T]1[T]2(25) where [T]i=cos (k0√εidi)j η0 √εisin (k0√εidi) j√εi η0sin (k0√εidi) cos (k0√εidi)(26) is the transmission matrix of each length of line, with η0= √μ0/ε0and k0=ω/cbeing the impedance and wavenumber of the free space, respectively. From the value of βand the voltage-to-current ratio (Bloch impedance) at z=0, given by V(0) I(0) =B ejβp−A(27) we can obtain the modal voltage and current waves at any position along the line using standard transmission line theory [9]. Fig. 4(a) shows the Brillouin diagram for the case with ε1=1, ε2=2, d1=d2=p/2, p=10 mm, where we can observe the usual pattern of alternating passbands and stopbands. Fig. 4(b) shows the power flow as a function of frequency, obtained from (12). Power flows in the positive z-direction in the first passband and in the negative z-direction in the second passband, as expected according to the slope of the wavenumbers of the harmonics as functions of frequency. Therefore, the first and second passband modes would typically be classified as forward and backward modes, respectively. It is important to mention that there is no discontinuity in power at the onset of the third passband near FIGURE 4. (a) Brillouin diagram for the PTL in Fig. 3(b) with ε1=1, ε2=2, d1=d2=p/2, and p=10 mm. (b) Power flow along the positive z direction for the modes in (a) with V(0) =1V. 25 GHz. This could seem to be the case because it increases rapidly with frequency at the beginning of the third passband, and after peaking at approximately 25.7 GHz, it gradually decreases at higher frequencies. However, from our discussions in the preceding sections, we expect every modal solution to be forward for a PTL with positive constitutive parameters. In other words, the sign of the phase velocity (23) should match the sign of P=(VI∗)/2 for every modal solution. Thus, Fig. 5show the amplitude, phase, and phase velocity of the voltage and current waves associated with the modal solutions in the middle of the first and second passbands (at βp/π =0.5), respectively. The amplitude remains relatively constant for the modal solutions in the first passband, whereas in the second passband the amplitude exhibits a more pronounced variation along the line. Regarding the phase, Fig. 5(c) shows that it increases monotonically along the line for the first passband. In contrast, in the second passband, the phase of both voltage and current waves is a decreasing function of z[see Fig. 5(d)]. This observation already indicates that the phase velocities of both the voltage and current waves are positive at all points along the line within the first passband. Conversely, within the second passband, the phase velocities of both waves are negative. This is confirmed by the phase velocity results in Fig. 5(e) and (f), which also show that the phase velocities differ for the voltage and current waves, as expected. Furthermore, there is a discontinuity for the current wave at z=p/2, as also expected from (15) due to the abrupt change 694 VOLUME 4, NO. 4, OCTOBER 2024 FIGURE 5. (a), (b) amplitudes; (c), (d) phase functions; (e), (f) phase velocities and (g), (h) normalized magnitude of the Fourier series expansion coefficients [Pncoefficients in (20)] of the voltage and current in the first (left column) and second (right column) passbands at βp/π =0.5. VOLUME 4, NO. 4, OCTOBER 2024 695 ZETTERSTROM ET AL.: ON FORWARD AND BACKWARD MODES IN 1D PERIODIC BOUNDED STRUCTURES FIGURE 6. (a) Periodically corrugated PPW. (b) Its unit cell. (c) Circuit model of the unit cell. (d) Dispersion diagram of the corrugated PPW with dimensions p=12 mm, h=4.8 mm, d=1.9 mm, and l=0.19 mm. in the p.u.l. capacitance. As for the voltage wave, according to (14) its phase velocity is continuous since the p.u.l. inductance has a constant value given by the free space permeability. Apart from these observations, it is clear that the phase velocities of both the voltage and current waves align with the direction of the power flow at all points in both passbands. Consequently, if we were to classify these modal solutions as either forward or backward, we would propose that both should be considered forward waves. FIGURE 7. Simulated normalized (a) electric field and (b) Poynting vector for the second mode, sampled at βp/π =2/3. Both the phase and power progression are heading toward the left (negative zdirection). We can gain further perspective on the above discussion by examining the harmonic composition of the modal solutions. The phase function in Fig. 5(c) shows that the total phase variation of the modal voltage and current across the unit cell is π/2. Taking into account that these results are obtained at the center of the first passband (i.e., βp/π =0.5), they indicate that the periodic function P(z)in(18) does not contribute to the global phase shift across the unit cell in this case. In other words, the total variation of the phase term ξ(z) in (22) is zero, suggesting that the harmonic n=0 is dominant in this case. On the other hand, Fig. 5(d) shows that the total phase shift is −3π/2 at the center of the second passband, which means that the phase term ξ(z)varies2πacross the unit 696 VOLUME 4, NO. 4, OCTOBER 2024 FIGURE 8. (a) and (c): Normalized magnitude, and (b) and (d): phase of the periodic modulation of the voltage wave phasor. The results are obtained at (a), (b) βp/π =1/3 and (c), (d) βp/π =2/3. cell (it should be noted that the total variation of ξ(z)inthe unit cell must always be an integer multiple of 2πto satisfy the periodicity condition of the modal fields). These results suggest that the n=−1 harmonic is the one that dominates in the second passband. These ideas are confirmed by Fig. 5(g) and (h), which show the relative importance of the harmonics in the voltage and current waves in the center of the first and second passbands, respectively. The importance of harmonics is compared in terms of the magnitude of the Fourier coefficients, |Pn|, where each |Pn|is normalized to the square root of the sum of squares of all |Pn|. For the first passband, Fig. 5(g) clearly shows that n=0 is by far the most important harmonic. For this reason, the “conventional” phase velocity (3) for harmonic waves, with βin the first Brillouin zone, gives the correct interpretation of the forward modes within this passband. However, when applied to the second passband, this reasoning leads to the misinterpretation of the mode as backward, the reason being that the n=−1 harmonic is now dominant, with the n=0 harmonic only of minor significance, as we can clearly observe in Fig. 5(h). Consequently, the conventional interpretation correctly explains the forward-like character of the mode if the sign of the dominant n=−1 harmonic wavenumber (that is, β−2π/p<0) is considered. B. 1D PERIODICALLY CORRUGATED PPW After our previous example of a quasi-analytical problem that can be rigorously described by a PTL model, we utilize a commercial simulator to examine a parallel-plate waveguide (PPW) periodically loaded with corrugations along the propagation direction, as depicted in Fig. 6(a). The structure is uniform in the direction transversal to the propagation. The unit cell of the structure is shown in Fig. 6(b) with the following dimensions (in mm): p=12, h=4.8, d=1.9, and l=0.19. This unit cell is analyzed using the Eigenmode Solver of CST [16]. We also analyze the structure using a circuit model to test the validity of the voltage-wave approximation. The unit cell circuit model is shown in Fig. 6(c) and the susceptances Baand Bbare obtained following [17], with Y0=W/(η0h) being the characteristic admittance of the unloaded PPW ( Wis the transverse width of the PPW, bounded by vertical magnetic walls in the simulator). The dispersion diagram of the structure under study is shown in Fig. 6(d), where the dashed lines represent CST solutions and the solid lines represent circuit modeling. The circuit model dispersion curve is found using (24),withthe ABCD parameters defined in Appendix B. We observe a good agreement between the results of CST and the circuit model, indicating that the voltage-wave approximation is accurate for this configuration. The dispersion diagrams of the 1D photonic crystal and the 1D corrugated PPW show a strong resemblance, and again the conventional definition of phase velocity (3) would indicate that the first mode is forward and the second mode is backward. However, since the circuit model is a reliable representation of the problem, according to the discussions in the preceding sections we can argue that both modes are indeed forward. As a first visual confirmation, consider the CST results in Fig. 7. In order to illustrate the phase progression, Fig. 7(a) provides three time samples of the E-field (an animation of the progression of the E-field is provided in the supplementary material online), while Fig. 7(b) shows the time-averaged Poynting vector. It can be observed in the three E-field snapshots that the wave pattern progresses from right to left, thus indicating that the phase velocity is indeed aligned with the power flow along the negative zdirection. In Fig. 8, we show the magnitude and ξ(z) phase term of the voltage wave in the circuit model for both modes at two different values of the wavenumber. We also show results of the voltage waves calculated from the electric-field values provided by the eigenmode solver as V(z)=h 0 Ey(z,y)dy.(28) Good agreement is observed between both sets of data. Note that here, unlike the plots for the photonic crystal in the previous section, only ξ(z) is plotted and not the complete phase ϕ(z)from(22). Again, similar to the results obtained for the 1D photonic crystal, the ξ-term is approximately zero for the first mode and varies 2πover the unit cell for the second mode. Fig. 9(a) and (b) show the phase velocity of these voltage waves, obtained using (23). It is clear that the phase velocity is positive for mode 1 and negative for mode 2; i.e., the phase velocities of the two modes are aligned with their group velocities at any point, which supports the characterization of both modes as forward. Finally, the relative importance of the harmonics extracted from the CST fields is illustrated in Fig. 9(c) and (d). The most important harmonic for the first mode in the corrugated PPW is the n=0 harmonic, VOLUME 4, NO. 4, OCTOBER 2024 697 ZETTERSTROM ET AL.: ON FORWARD AND BACKWARD MODES IN 1D PERIODIC BOUNDED STRUCTURES FIGURE 9. (a)–(b): Normalized phase velocity using (23). (c)–(d): Relative importance of the different harmonics in the modal fields. The results are sampled in (a), (c) βp/π =1/3and(b,d)βp/π =2/3. and the application of (3) yields the correct interpretation of the forward-like nature of the mode. This would not be the case if the dispersion diagram were analyzed outside the first Brillouin zone. For mode 2, the application of (3) to phase constants in the first Brillouin zone misinterprets the nature of the mode, the reason being that the most relevant harmonic is the n=−1 harmonic. Again, (3) would yield the correct interpretation within the n=−1 Brillouin zone. C. 1D SHORT-PERIOD CORRUGATED PPW The dispersion diagram for the same 1D corrugated PPW studied in Section IV-B but with a shorter period (p=5 mm) is shown in Fig. 10(a). The behavior of the first mode in Fig. 10(a) closely resembles that of the first mode in the structure studied previously shown in Fig. 6. However, the behavior of the second mode in Fig. 10(a) is more complex. The circuit model is no longer suitable for accurately representing this mode since its frequency range exceeds the applicable range for the given configuration [17]. The reason for this can be understood from the results in Fig. 10(b) and (c), which present a time sample of the electric field and the Poynting vector for mode 2 obtained at βp/π =5/6. From the field plot, it is evident that the field is no longer mainly transverse, as a significant longitudinal component is present even in the PPW sections. In other words, we have more than one relevant electric field component, and by studying them separately we may find that they can progress at different speeds and/or in different directions. This is illustrated in Fig. 10(d), which shows the phase progression rate of the Eyand Ezcomponents averaged in the transverse cross section of the waveguide. Since the circuit model is not applicable, only the CST results are presented. Note that near the corrugation, the calculation of the phase progression of the z-component fails due to the electric-field singularity condition at the corrugation corners and the fast variation of the electric field in this region. The phase progression is calculated using (23), but here we do not use the term “phase velocity” to stress that these results should be taken with caution and cannot be used to characterize the forward/backwardness of the mode. In fact, we can observe that the phases of the two components progress in opposite directions. Furthermore, the direction of the time-averaged Poynting vector in Fig. 10(c) varies from point to point, even showing vortexes, which means that power does not flow consistently in one direction throughout the transverse cross section. In other words, this is inherently a 2D wave problem that cannot be satisfactorily described using a 1D model. In the online supplementary material, we provide animations of the instantaneous electric field that show in more detail the complex behavior of the modes. It is then evident that, to describe the forward/backwardness of the modes in more general scenarios like the one presented, it is essential to extend the concept of phase velocity to encompass multidimensional waves. This extension enables the examination of its correlation with the local power flow and also requires consideration of the vector properties of the electromagnetic field. We discuss this generalization in Section IV-D below. For completeness, Fig. 10(e) presents the relevance of the different harmonics to the electric field components averaged over the waveguide cross section (calculated from the CST results). Here, we observe that the two field components also have several harmonics that are relevant. In this structure, the phase progression of the different components is in line with what would be expected based on the most relevant harmonic. Specifically, the n=−1 harmonic is the most relevant for the field component Ey, and the phase of this component progresses in the negative z-direction. In contrast, the harmonic n=0 is the most relevant for the field component Ez, and its phase advances in the positive z-direction. However, in situations where multiple significant harmonics are present, such alignment of the phase progression with the importance of the harmonics may not always occur as anticipated. This is illustrated in Appendix Cthrough the examination of a glide-symmetric 1D periodic corrugated PPW. D. MORE GENERAL WAVES Similar to 1D scalar waves, any two- or three-dimensional scalar wave can be written as A(r,t)=A0(r) cos(ωt−ϕ(r)) (29) or, in phasor form, A(r)=A0(r)e −jϕ(r)(30) in which the amplitude and phase functions, A0and ϕrespectively, are again real functions of position. In this case, the progression of the wave pattern with time can be visualized as the movement of its associated constant-phase surfaces ϕ(r)=constant (cophasal surfaces or phase fronts). This movement can be characterized by a phase velocity vector that 698 VOLUME 4, NO. 4, OCTOBER 2024 [12] A. L. Fructos, S. Campione, F. Capolino, and F. Mesa, “Characterization of complex plasmonic modes in two-dimensional periodic arrays of metal nanospheres,” J. Opt. Soc. Amer. B, vol. 28, pp. 1446–1458, 2011. [13] O. Zetterstrom, F. Mesa, R. Rodriguez-Berral, and O. 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