\section{\label{sec:l1}Introduction}

Periodically shaped microstrip transmission lines (MSTL) are used in microwave technology as a so-called stepped-impedance filters. Those structures can be considered as one-dimensional photonic crystals (PC) operating in GHz frequency range \cite{Guo08,Belozorov14,Belozorov13}. It is assured by the linear scaling of Maxwell equations\cite{Joan08} which allows  to transfer the properties of photonics system to microwave systems \cite{Guo08}. In PC, the surfaces/boundaries are the defects which break the periodicity of the structure. The modes localized on defects \cite{Rodriguez15} (including surfaces \cite{Klos07,Garanovich08}, interfaces \cite{Guo08,Belozorov14,Belozorov13,Vinogradov06,Tarapov12, Zheng18}) can be observed at selected frequencies in the stop bands of PC, if the appropriate bounary conditions are fulfilled \cite{Belozorov13,Alu03}. The surface states are characterized by complex propagation constant (complex wave vector) and therefore they decay exponentially inside the periodic structure being localized on the surfaces.

The  surface states can appear for different kind of periodic structures (electronic superlattices \cite{Klos03} or magnonic crystals \cite{Rychly17}) where the wave excitations of different nature are observed. The existence of surface states depends both on the bulk parameters (sizes and shape of the units cell) and  the surface parameters (describing how the periodic structure is terminated). By changing the bulk parameters we can design the band structure supporting the existence of surface states. The modification of the surface parameters can tune the frequency of surfaces states within the stop band\cite{Zak85}.

One of the simplest geometry in which the  surface states, in the meaning described above, can be observed is the transission line in the form of periodic waveguide. Such system, confined in two dimensions allows the wave to propagate in one direction. The real transmission line is finite structure, terminated in two points – surfaces – being the endings of the first and last cell in the periodic stucture. By adjusting the geometry of the first and last cell one can change the frequency of surface states within the stop band. In the absence of surface states, such geometrical changes can induce this kind of states close to the edge of stop band and then shift them gradually towards the center of stop band.

For the periodic waveguide, symmetric with respect to the flipping of both endings, the surface states appear in pairs, at very close frequencies. Each of this mode occupies both surfaces at once. For one mode from this pair, the temporal oscillations on both surfaces take place in-phase, for the other one – out-of-phase. Excitations on both surfaces are correlated to each other due to the leaked tails penetrating the interior of the periodic structure. The rate of exponential decay of those tails is defined by imaginary part of the propagation constants. The strong localization corresponds to the large value of this parameter, and the surface state is observed at the center of stop band. On contrary, the surface modes are weaker localized, if their frequencies are close to the edges of stop band, where the imaginary part of propagation constant tends to zero. The pair of surfaces modes of the frequencies close to the center of stop band are well separated in frequency domain from the bulk modes (which can be beneficial for the utilization), but such modes can be almost degenerate (which can make their detection difficult).

The properties of surface states described above are general and refer also to the microwave devices under our investigation.  In this paper we investigated experimentally and numerically the conditions of appearance of surface state in MSTL and using the arguments known in photonics, we explain the properties of surface states observed in MSTL. The consideretion based on band therory of PC will be supplemented by the analysis of lumped element model of stop band filter.

The proposed system is interesting, because of potential applications of periodic MSTL, where the localized modes are isolated in frequency domain from the bulk modes grouped in transmission bands. One can consider such MSTL as a bandpass filter for frequency-division multiplexing which allows to introduce the reference pilot signal as a surface state \cite{Tong04}.

\section{\label{sec:l2}The system}

The investigated system has a form of periodic MSTL (see Fig. 1(a,b)) composed of $n$=5 cells (see Fig. 1(c,d)). Each internal (bulk) cell (Fig. 1(c)) consists the section of wider microstrip ($w$ = 4 mm) of the length $l$ = 10~mm linked to the neighboring cells by thinner sections (1.2 mm in width). The bulk cell length $d$ is equal to 16 mm. The width $w_0$ and length $l_0$ of the wider segment in surface cells and  total length of surface cell $d_0$ (Fig. 1(d)) can by different. For experimentally realized structure, the values of surface parameters: $w_0$,  $l_0$ and $d_0$ are equal to 9 mm,  1.5 mm  and 11.5 mm respectively. We studied also experimentally the system with unaltered surface cells, where $w_0 = w$, $l_0 = l$, and $d_0 = d$,  for reference. The microstrip structure is fabricated on the Neltec substrate with dielectric constant $ε$ = 2.2+i0.0012 and thickness $t$ = 0.381 mm. The microstrip etched at one side of substrate. For numerical studies we changed the surface parameters $w_0$ and $d_0$ continuously to induce surface states and to tune their frequencies. We used the simplified lumped element model (Fig. 1(e)) to explain the appearance of stop bands and surface states. However, validity of this model was already shown \cite{Zhou13}.

\begin{figure}
\begin{center}
 \includegraphics[width=10 cm]{microstrip_f1.png
\end{center}
\caption{\label{fig:epsart}Periodic microstrip under investigation. (a) The photograph of experimental setup presenting the microstrip placed on the substrate. The bottom surface of the substrate is metalized and play the role of the ground plane.   The input and output of the microstrip are connected to Vector Network Analyzer with standard 50 $\Omega$ ports to measure $S_{21}$ spectrum. (b) The geometry of the system. The line is composed of (c) $n$ = 5 bulk cells and (d) two surface cells. The bulk and surface cells differ in the size: $d$, $d_0$, and in the geometrical parameters: ($l$,$w$),  ($l_0$ ,$w_0$), describing the corrugation of the microstrip. (c) The simplified, one-dimensional model of periodic microstrip where the bulk and surface cell are represented by the symmetric two-port networks. Each network is composed of two phase-shifters (resulting from the acquiring of the phase at the distance $d/2$ or $d_0/2$ each) and shunt susceptance (resulting from the presence capacitance $C$ or $C_0$, in each cell of the system).}
\end{figure}

\section{\label{sec:l3}The model}

We used Vector Network Analyzer PNA-L5230A to measure the transmission spectra ($S_{21}$). The $S_{21}$ spectra was also extracted numerically. For numerical calculation of the transmission spectra and field distribution we used MEEP  software which is based on classical frequency domain calculation technique. To interpret the experimental and numerical outcomes, we introduced the model where the microstrip was replaced by equivalent lumped element system. We considered ladder multi-stop band filter being the cascade of symmetric two-terminal networks representing each cell of the  MSTL (see Fig.1(e)). Each network is composed of shout capacitance and two phase shifters: one at input and the other one at output. The shout capacitance can be related to area of one cell of microstrip over the ground plate whereas the phase shift  to its length. Each  network can be then described by two parameters: the shunt capacitance $C$ ($C_0$ for first and last network) and the length $d$ ($d_0$ for the first and the last network). The shunt capacitance can be related to the geometry of the cell and dielectric constant of the substrate. The shunt capacitors formed by successive cells of microstrip are partially open and microwave propagate partially in air above MSTL. We introduced, in lumped-element model, the effective dielectric constant $\varepsilon_{\rm eff}$, lower then dielectric constant of the substrate: $\varepsilon_{\rm{eff}} < \varepsilon$. The other parameter, which has to be specified for lumped-elements model, is the characteristic impedance ZC of MSTL. The impedance ZC allows to relate electric and magnetic field in real MSTL to voltage an current in ladder filter.

The formal relation between output and input voltages and currents for elementary network (representing cell of MSTL) can be written with the aid of ABCD matrix \cite{Pozar11}:
\begin{equation}
    \left(\begin{array}{c}V_{n}\\I_{n}\end{array}\right)=
    \underbrace{\left(\begin{array}{cc}A&B\\C&D\end{array}\right)}_{\bm{M}}
    \left(\begin{array}{c}V_{n+1}\\I_{n+1}\end{array}\right).
\end{equation}
For our system the ABCD matrix $\bm{M}$ has a form \cite{Pozar11}:
\begin{equation}
    \left(\begin{array}{cc}\cos \Omega-p\Omega\sin(\Omega)&j\left(p\left(\cos \Omega -1\right)+\sin\Omega\right)\\
    j\left(p\left(\cos \Omega +1\right)+\sin\Omega\right)&\cos \Omega-p\Omega\sin(\Omega)
    \end{array}\right),\label{eq:M}
\end{equation}
where $\Omega= k_0d $ is  dimensionless angular frequency. The parameter $k_0  = \omega/ (c/\sqrt{\varepsilon_{\rm eff}}) i$s the wave number in uniform space characterized by dielectric constant $\varepsilon_{\rm eff}$. The symbol $p$ is defined as: $p = q (c/\sqrt{\varepsilon_{\rm eff}}(1/(2d))$ were $q = Z_C C$.

In infinite sequence of the networks, the voltages and currents are related to each other by propagation constant $k_B$:
\begin{equation}
    \begin{array}{cc}
       V_n=e^{ik_Bd}V_{n+1},  & I_n=e^{ik_Bd}I_{n+1},
    \end{array}
\end{equation}
The propagation constant $k_B$ can be considered as Bloch wave number, known in physics for description of wave excitation in periodic systems \cite{Ashcroft76}. The wave number $k_B$ is real in the pass band and complex in the stop bands. Eqs. 1-3 allow to derive the dispersion relation which links the dimensionless frequency $\Omega$ (or wave number in uniform space: $k_0$) to dimensionless Bloch wave number $\theta_B = k_B d$.

The lumped-elements model contains two parameters: $\varepsilon_{\rm eff}$ and $p$ which are related to the geometrical and material parameters of real MSTL. In order to refer the lumped-elements model to real structure, we tuned the values of $\varepsilon_{\rm eff}$ and $p$ to get the same positions of stop bands as these found from  numerical studies (see Fig.2 and Fig.3a,b). Additionally, we agreed very well the width of the second gap with numerical data because this gap lay in the middle of frequency range accessible in the experiment (8-18 GHz).  The experimental studies of surface state will by limited to this gap (Fig.3a).

Then, having the values of $\varepsilon_{\rm eff}$ and $p$ fixed we proceeded to look for surface states. The frequencies of surface and defect states are in the ranges of stop bands. In stop bands the right hand side of Eq. 4 is complex and the Bloch wave number $k_B$ takes complex values:$\pm i \rm{Im}(k_B)$ or $\pi/d \pm i \rm{Im}(k_B)$ as well. \cite{Ashcroft76} This means that the propagating waves (described by real wave number $k_B$) are not the solutions in the stop bands and the excitation can be localized on defects (surfaces) with exponential decay of the amplitude: $e^{\pm\gamma}$ where the rate $\gamma=Im(k_B)d$ is related to imaginary part of propagation constant $k_B$ (see Eq. 3).

The resultant ABCD matrix for the whole structure can be calculated as a product of ABCD matrices for successive networks (\ref{eq:M}). This matrix can be then converted to the corresponding scattering matrix which allows to obtain the transmission spectrum by taking its $S_{21}$ element \cite{Pozar11}.

\section{Results}
We fabricated the MSTL composed of less number of cells ($n=5$) because we wanted to observe the splitting of the peak of surface states (as it is predicted in the inset in Fig.2(b)). Additionally, we selected the sizes of the terminating cells ($w_0$=9 mm,  $l_0$=1.5 mm, $d_0$=11.5 mm) to get smaller width of individual peaks forming the doublet of surface modes (we show this effect later in Fig.5). This changes (in the number of cells $n$ and the values of surface parameters: $w_0, l_0, d_0$) do not affect the width and the position of the stop bands.

The experimental results were used  to validate the numerical outcomes. For the parameters $q$ = 24 pF$\Omega$ and $\varepsilon_{\rm eff}$ = 1.6 (which gives $p$ = 0.178 for $d$ = 16 mm) of lumped elements model we have got the best fit to numerical outcomes (see Fig.2), according to criteria described above (position of stop bands and the width of the second stop band).

\begin{figure}
\begin{center}
 \includegraphics[width=10 cm]{microstrip_f2.png
\end{center}
\caption{\label{fig:f2} (a) Transmission $S_{21}$ spectrum for the microstip composed of 8 cells ($l$=10 mm, $w$=4 mm, $d=16$ mm) with two terminating cells of different size ($l_0=3$ mm, $w_0=4$ mm, $d_0=d$) - numerical computation were done  with the aid of MEEP. The change of sizes of terminal cells induces the surface states visible as a peak in the second stop band. (b) Transmission spectrum for microstip composed of $n$=10 identical cells does not contain the surface modes - calculations done with the aid of lumped element model (the parameters of the model were adjusted to fit the position and width of second stop band determined both in numerical calculations (a) and in experiment (Fig.2(a,b)). The surface state appear in the second band (see inset for $n$=10) after equivalent perturbation of terminating networks (see Fig.1e). The shortening the microstrip, by reducing the number of cells $n$, results in the splitting of the peak of surface states in to two peaks. }
\end{figure}

Let’s discuss now the  MSTL where two surface states can be observed in the second stop band. In Fig. 3(a) we presented the measured (solid lines) and numerically simulated (dashed lines) transmission spectra for finite MSTL. We considered MSTL composed of $n$ = 5 identical cells (Fig. 1(d)) and MSTL finished by additional cells of different geometry (Fig. 1(b,c)). The transmission spectrum is limited to experimental frequency range 8-18 GHz, covering the second stop band. For the structure containing  $n=5$ identical cells (see black sketch in Fig.3(b)) we do not find the surface states - in transmission spectra  (black lines in Fig.3(a)), the peaks of surface states are absent. We can see (the spectra drawn by red lines in Fig. 3(a)),  that the presence of terminating cells of altered shape leads to the appearance of surface modes. Visible as a sharp doublet of peaks within the stop band. The spatial profiles of out-of-plane components of electric field for those modes are presented in Fig. 3(c,d) at selected time. The modes are localized at the surfaces (terminations) of MSTL and the envelopes of their amplitude decay exponentially inside the structure. The surface modes occupy both surfaces, because the whole structure is symmetric with respect to flipping both endings and there is no mechanism which can preferably select one end for localization. Due to the mirror symmetry one of the surface modes is symmetric (the electric field oscillate in phase on both ends) – see Fig. 3.(c), and the other one – antisymmetric (the electric field oscillates in out-of-phase on both ends) – see Fig. 3(d). For sufficiently large number of cells in the chain the surface modes practically do not penetrate the deep interior of the structure. The field dynamics at one termination of MSTL is then independent on the dynamics on opposite termination. It means that symmetric and antisymmetric surface states will be degenerated in frequency. Such degeneracy makes the experimental detection of surface states difficult. When the MSTL will be driven at one terminal (surface) by the signal of the frequency equal to the frequency of two degenerate surface states, then it will populate both surface states simultaneously. The field barely leaking trough the long structure will be unable to develop its amplitude at opposite terminal (surface) due to destructive interference of symmetric and antisymmetric mode. In our studies we use however MSTL of relatively small number of cells ($n$ = 5). It ensures the reasonable compromise. We are still able to observe the band structure with the distinctive stop bands but the symmetric and antisymmetric surface states found in the second stop band will have still distinguishable frequencies (about 100 MHz separtion). It allows to transmit the signal in-phase (using symmetric surface state) or with phase flipped (for antisymmetric surface state) between two terminals (surfaces) of MSTL. The signal will be well separated in frequency domain from the modes transmitted in bands below and above stop band.

\begin{figure}
\begin{center}
 \includegraphics[width=10 cm]{microstrip_f3.png
\end{center}
\caption{\label{fig:f3} The appearance of the surface states in the second stop band of MSTL. (a) Transmission spectrum $S_{21}$ for MSTL composed of  bulk cells $n$ = 5 in the absence of surface cells (black lines), compared to $S_{21}$ spectrum for the same structure completed by surface cells (red lines) – see Fig. 1(a,b). The experimental and numerical data are plotted using solid and dashed lines, respectively. (b) The $S_{21}$ spectrum for lumped element model equivalent to the MSTLs considered in (a) – black (red) dotted line and red dotted line corresponds to the system  without (with) surface networks – see Fig. 1(e). For the surface networks we took $d_0$ = 0.72$d$. and $q_0 = q S_0/S = 0.53q$, where $S_0$ and  $S$ are the areas of the microstrip within the surface cell and bulk cell. The spatial profiles of  (c) symmetric surface state (S) and (d) antisymmetric surface state (A) calculated numerically, corresponding to the doubled of peaks visible in (a).}
\end{figure}

The effect of geometrical changes of surface cells is also reflected in  lumped-element model by the variation of two(surface) parameters: $C_0$ (or $q_0$) and $d_0$ – see Fig. 1(e). In the studies presented in Fig. 3(b) we read out the length $d_0 = 11.5$ mm = 0.72 $d$ (directly form the considered geometry of surface cells in MSTL) and we took $p_0 = 0.53 p$ (assuming that the ratio of area of surface cell and bulk cell: $S_0/S$ can by related to appropriate ratio of capacitances: $S_0/S \approx C_0/C \approx p_0/p$). For above values of parameters we found the transmission peak located in the center of the third stop band – see Fig. 3(d). By careful inspection we can notice that the peak is doubled. When the number of bulk networks, separating two defected (surface) networks, is lowered then the splitting of doublet is enhanced – see inset in Fig. 2(b).

We showed that surface states appear in considered system as a result of geometrical changes of the terminal parts of periodic MSTL. We are going to investigate now how to tune the frequencies of surface states within the stop band by adjustment of surface parameters describing the geometry of surface cells of MSTL. We changed  length of surface cells $d_0$ and the width of the wider section in surface cell $w_0$ (the length of this section l0 was kept unaltered). We performed the numerical studies where the values of cell $w_0$ and $l_0$ were changed gradually. The computations were supplemented by the outcomes obtained from lumped element model, where the change of the parameter $w_0$ was simulated by the appropriate change of the capacitance $C_0$ (or parameter $q_0$).

In Fig. 4 we presented the dependencies of the surface states frequencies on surface parameters: $d_0$ and $w_0$ ($C_0$) in the second stop band of considered MSTL. We used both numerical simulations (Fig. 4(a,b)) and lumped-element model (Fig. 4(c,d)) to investigate those dependencies. Both methods give similar results which confirms the applicability of considered lumped element-model for investigation of surface states in periodic MSTL.

By changing the length of surface cell we can shift the doublet of surface states in the whole range of stop band. The lumped-element model predicts that for small values of $d_0$ an additional doublet of surface states appears at the bottom stop band. However, such states were not found with the aid of numerical simulations in the investigated range of $d_0$. The doublets tend to split in the vicinity of edges of stop band. This can be explained when we notice that the imaginary part of propagation constant (Bloch wave number) is decreasing in the close to the edges of stop bands (see Fig. 2(b)). Due to the fact, that the modes are less localized,
(see Eq. 3), the excitation on both terminals of the MSTL can be more correlated to each other. This leads to stronger splitting of the modes which oscillate in-phase (Fig. 3(b)) and out-of-phase (Fig. 3(c)) on both surface.

\begin{figure}
\begin{center}
\includegraphics[width=10 cm]{microstrip_f4.png
\end{center}
\caption{\label{fig:f4}
Tuning the frequency of the surface states by structural changes of the surface cells in the range of the second stop band – for MSTL presented in Fig. 1(a,b).  Both for numerical studies (a) and for lumped element model (c) we studied the impact of the length of surface cell $d_0$ on the frequencies of surface states, where we kept the width of  central section of the waveguide $w_0$ and the capacitance of surface network $C_0$ unaltered: $w_0  = 9 mm$ (equal to the experimental values $w_{0,ex}$), $C_0 = 0.53 C$ (being the counterpart of the experimental value). For fixed value of $d_0$ (the same as in the experiment $d_0 = 0.73 d$) we showed that the frequencies of surface states can be also tuned by adjusting the value of $w_0$ (in numerical studies – (b)) and $C_0$ (in the studies based on lumped-element model).}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=10 cm]{microstrip_f5.png
\end{center}
\caption{\label{fig:f5}
Presence of surface state doublet in transmission spectrum – numerical studies. (a) Doublet appears only in MSTL width tow identically perturbed surface cells (red curves). When considered MSTL is terminated at one side by surfaces cell of different geometry and at the other side by the cell identical to the bulk cells we observe the single surface state in the spectrum. (b) By changing the surface parameter $w_0$ (width of the central segment in the surface cells) we can shift the frequency of surface states within the stop gap (see Fig. 4). The splitting of  the frequencies (in the vicinity of the edge of stop band) can be accompanied with the broadening of the width of individual peaks in the doublet which can led to merging of two peaks into one (see also Fig.4(b)).}
\end{figure}

The frequency of surface states can be also tuned in considered system, by the change of the width of the central segment in the surface cells $w_0$ (Fig. 4(b)). The change of the parameter $w_0$ affects the capacitance of surface cells. Therefore, we changed the capacitance $C_0$ (and parameter $q_0$) in lumped element model proportionally to the altered area of surface cells in MSTL (Fig. 4(d)). Such changes of $w_0$ (or $C_0$) do not lead, however to the noticeable increase of splitting between symmetric and antisymmetric surface modes when their frequencies are pushed away from the center of stop band. It can be explained by checking how the width of  peaks of surface states in transmission spectra changes in dependence on $w_0$ (see numerical outcomes in Fig. 5(b)). We can notice that when the doublet of peaks is shifted toward the edges of stop band, then the width of peaks increases faster than their separation. As a result peaks forming doublet merge into one wide peak.

For the MSTL terminated asymmetrically, i.e. when the first and the last cell of the structure are not identical, each surface state is localized at one termination only and is manifested in the transmission spectrum as a single peak. The origin of this peak is however different, than the origin of the wide peak resulting from merging to peak doublet described above.  In Fig. 5(a) we compared the peak doublet composed of symmetric and antisymmetric surface states (for the structure presented in Fig. 1(a, b) – see also the red sketch in Fig. 5(a)) and the peak of single surface state localized on one surface only (for the structure form Fig. 1(b) with one surface cell omitted – see also the green sketch in Fig. 5(a)). We can notice than the width of the peaks of single surface state is in general much narrower that the width of the merged doublet of symmetric and antisymmetric surface states.

\section{Conclusions}
We investigated experimentally and numerically the surface state in the microwave transmission spectra of the periodic microstrip. We showed, that the frequencies of the surface states can be shifted inside the stop bands by changing the dimensions of elements forming the termination of microstipe. For appropriately adjusted values of these geometrical parameters, the peaks of  surface states are split into symmetric and antisymmetric surface states, for which the electromagnetic field oscillates on both terminations of the microstrip in-phase or out-of-phase-phase, respectively. These surface modes, which occupy both terminations (surfaces) of periodic microstrip, can be potentially used to transmit the reference signal of selective frequency trough the stop band being well separated from the signals transmitted in the bands.

\section*{Acknowledgments}
This work was supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 644348 (MagIC) and the Polish Ministry of Science
and Higher Education resources for science in 2017–2019
granted for the realization of an international co-financed
project (W28/H2020/2017). J.W.K. would like to acknowledge the support of the Foundation of Alfried Krupp Kolleg Greifswald.

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