1. Introduction

Zero-Refractive Index Metamaterials (ZRIMs) are one of the newest type of metamaterials that can be made of dielectric Photonic Crystals (PCs) [1,2]. Due to the well-known and relatively simple manufacturing process, Two–Dimensional (2D) PC-based ZRIMs can be the most important type of optical metamaterials (MMs) [3]. In fact, 2D PC-based ZRIMs can be widely used for Photonic Integrated Circuits (PICs) because of their phenomenological properties of zero-refraction effect [4–16], Zitterbewegung effect [4,6,15], Klein tunneling [4–6], Anderson localization [5], and super-collimation [5]. These structures work based on stimulation of the Dirac-points (DPs) in an accidental degeneracy format [4–9,16–26]. DP, or in another definition Dirac-like point (DLP), in these structures is an intersection point of three degenerate optical states in the dispersion diagram of the PC; two quadratic curve with one flat surface [1,2,4–6,8–12,15,16,18,20,22–24,26]. In DP (DLP) due to zero group velocity ($\partial \omega /\partial \kappa =0$) refractive index tends to be zero [1,5,7,12,15,20,25,27]; zero dispersion in just one operational point of ordered pair of ($\kappa$ , $f$) related to just one Eigen-frequency and just one certain direction of Bloch’s wave-vector. In some researches [6,9,12,16,22,25,26], this point explicitly is called DLP because of its location in the center of the Brillouin zone ($\kappa =\Gamma$) and also because of its triple degeneracy and this fact that DPs exhibit themselves in double degenerate high-symmetric points (Brillouin zone corners), not in triple degenerate states [5–11,19–22,27,28]. Dielectric PC-based ZRIMs with their almost zero electromagnetic intrinsic losses (material losses or absorption) can be a good alternative for Metallic MMs (MMMs) with high electromagnetic intrinsic losses in a wide range of frequencies [1]. In addition, DLPs can suppress radiative (leaky) losses [1,3–6,8 10,13,20,25,28]. As reported in [2], DLP in 2D PC-based ZRIMs embedded in a waveguide is an ultra-sensitive point with a huge value of operational sensitivity to the dielectric rods refractive index changes. In fact, when the refractive index of the rods changes, the system will no longer operate at the Dirac-point. The source of these changes in refractive index can be various factors such as ambient temperature and incident wave intensity (Non-linear effects). Some other parameters can also cause distortion in the operation of the ZRIM in terms of getting away from the Dirac-point, such as irradiated wave frequency, PC lattice constant and radius of dielectric rods. We considered a monochrome electromagnetic wave excitation into a temperature-independent ambient or waveguide ($\Delta T=0$), in which the behavior of embedded dielectric PC-based ZRIMs, is almost ideal. By the way, it should be mentioned that, with respect to a simple dielectric phase shifter, the proposed structure, has some other superior factors such as much smaller footprints of micro-fabricated device for same values of phase shift, wider range of possible phase shifts with much smaller variations, etc. And it is worth mentioning that having a simple dielectric slab which shows both zero refraction index behavior and arbitrary phase shift tuning applicability might be an impossible task. In addition, in the case of a simple dielectric slab, basically there is no running from some reflection from the input.

As shown in [2,4–6,9,13–16,20,24,26,28], ZRIMs have been raised in different specific structures and analyzed in various points of view which are led to discover the characteristics and out-coming sequences such as phase shifter [2,13], cloaking [9], electrically tunable electromagnetic switches [14], phoxonic crystals [16,26], fiber guiding [20], all-optical switching near the Dirac-point [28], Zitterbewegung effect [4–6,15] and super-scattering phenomenon [5,24]. Also, in [1,2,4–13,16,17,19–28] different geometrical types of ZRIM lattices such as square lattices [1,2,5,6,11,13,19,22,23,28], rectangular lattices [9], triangular lattices [4,7,12,17,20,24,26], hexagonal lattices [16,21,27], cubic rod lattices [1,10], non-symmetrical lattices [25] and complex 3D lattices [8] are studied.

In this article, a similar structure as illustrated in [2], a 2D square lattice PC-based ZRIM with 30$\mu$m (radius) Si cylindrical rods and lattice constant of 150$\mu$m embedded in a terahertz (THz) waveguide with Perfect Magnetic Conductor (PMC) side walls and vacuum as a background medium, is considered with the same condition; input THz plane-wave with frequency of 1.082 THz, to analyze some of the characteristics building the idea of an interaction length between incident THz plane-wave and ZRIM structure.

It is apparent, every ideal PC, in our case a 2D square lattice, has a unit cell (UC), representing the whole PC structure. But in this case, the structure has limited number of unit cells, and therefore will not act exactly the same as the ideal one. We embedded a 2D PC-based ZRIM lattice structure with number of 5 columns and 5 rows ($5\times 5$) in the designed THz waveguide and illuminated it with a THz plane-wave and compare the resultant wave versus the propagating original THz plane-wave in an empty waveguide. Through this procedure we utilized the idea of obtaining the differential wave between original THz plane-wave and the computed resultant wave, by which some aspects will come to light. Comparing to a simple dielectric slab, one of the differences of a finite PC-based ZRIM embedded in a THz waveguide (proposed in this work) is indeed a parameter introduced here as "interaction length". This is because, the effective beginning and ending boundaries of this medium are not exactly set physically, on the grounds that, it is an inhomogeneous structure. The objective in this paper was to obtain the length in which the structure can be modeled as a homogeneous media. This is illustrated as the distance between the starting and ending points of the effective homogenized ZRIM system.

In section 2, we will show the input interaction lengths, which are placed just before and after and close to a single ZRIM lattice. In the next step, using a small range of distance between two separate cascaded ZRIM lattices with sizes of $8\times 12$ UCs, the almost strictly descending spatial phase shift tuning by increasing the distance between two lattices is investigated (section 3.1). The spatial phase shift procedure is performed through a system including two THz waveguides with and without ZRIM lattices and is applicable for multi-waveguide systems like the system proposed in [29]. The effects of displacement between these lattices on transmission and reflection coefficients are numerically investigated and one more time the total interaction length (summation of left and right side lengths) is obtained (section 3.2). Although our 2D dielectric PC-based embedded ZRIM lattice is not ideal in terms of infinite number of cylindrical Si rods in both directions of lattice vectors (ideal PC), but due to existence of PMC sidewalls as boundary conditions as well as DLP operational condition in the THz waveguide, it will be shown that for the formation of a complete PC-based ZRIM embedded lattice (two ZRIM lattices with just one lattice constant distance), the transmission loss will be negligible. In our proposed waveguide the boundaries are assumed to be PMC, while the THz wave is passing through the whole PC-based ZRIM along the waveguide. Therefore, the propagating wave is already confined completely in the waveguide and cannot pass through the adjacent PMC walls; hence there is no confinement loss as long as the boundaries remain ideal.

Finally we will summarize the results and present them in the conclusion section while charting the future path on how to achieve more accurate results and provide wider possibilities.

2. Singe ZRIM procedure

Different studies of PC-based ZRIM structures in terms of some characteristics such as the electromagnetic field distribution, geometry, and boundary conditions has been scored in the relevant scientific previous works. All these systems have one thing in common which is having at least one of the three degenerated propagated modes namely electric dipole, magnetic dipole, and electric monopole, positioned in the area of the ZRIM UC [1,2,4–7,9,11–13,17,19,20,22–24,26,28]. It is said that an ideal dielectric 2D PC-based ZRIM structures have no electromagnetic losses; no absorption or leaky radiation. Considering the fact, it is resulted that the wave coming out of the ZRIM does almost have the same amplitude as the incident. On the other hand, since the system passes the electromagnetic wave phase freely [2,13], without effectively changing or accumulating the spatial phase, an interaction length can be introduced for the whole dielectric 2D PC-based ZRIM lattice area. In fact, as shown in Fig. 1(a), we are looking for the entering and leaving boundaries of this phase freely passing electromagnetic wave path. We will follow this behavior by computing wave eigenmodes and then investigating the shape of the propagating expected THz plane-wave, Fig. 2(a). Although the considered square 2D PC-UC has a side length of $150\mu$m, lattice constant of $a=150\mu$m, but it is very useful to obtain the interaction length when the whole 2D PC-based ZRIM lattice is embedded inside the considered THz waveguide with PMC sidewalls. For this purpose, we consider an appropriate THz waveguide with PMC side walls and with an embedded dielectric 2D PC-based ZRIM lattice structure with size of $5\times 5$ (Fig. 1). Then we will compute the scattered wave and subtract it from univalent 1-watt powered 1.082 THz main input Transverse Electric (TE) polarized plane-wave; $E=E_z=real(e^{-i \kappa x})$V/m, where $\kappa$ is $x$-directed wave-number. More extended studies just about this operational frequency and results of its deviation with consideration of PC-UC geometric parameters, lattice dispersion diagram, and also relation between them have been conducted in [2]. It should be noted that in this section and also subsequent section, wherever the electromagnetic wave field profile is plotted, a cut-line in $x$-$y$ plane passing through just the middle of the waveguide is investigated; inset of Fig. 2(a). Hence, this cut-line is used as a reference line and resulted electromagnetic waves are plotted across it. This line is chosen only to compare the results and could be any other direction in the waveguide, while the Maxwell’s equations are solved in the whole waveguide. It is also worth noting that all numerical simulations performed in this paper were based on the finite element method (FEM) with maximum mesh size of $5\mu$m.

 

Fig. 1. (a) Schematic of a $5\times 5$ dielectric 2D square PC-based ZRIM lattice with Si rods embedded in a desired air background THz waveguide with PMC side walls. The first and last columns of rods are centered at $2000\mu$m and $2600\mu$m, respectively. The topmost and bottommost rows each are placed $75\mu$m away from the PMC side walls. The left and right side interaction lengths are shown with ${\textit {d}}_{L}$ and $d_{R}$, respectively. (b) Unit cell of the PC lattice with lattice constant of $a=150\mu$m with air ($\varepsilon =1$) as background medium and Si ($\varepsilon =12.5$) rod with radius of $r=30\mu$m ; $r/a=0.2$

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Fig. 2. (a) The original (blue) and scattered (red) THz waves across the waveguide. Top rectangular inset shows the 2D midline as a reference line for investigating the propagation. (b) computed differential THz wave by subtraction of THz waves in (a). Left-top and left-bottom blue insets are magnified views around crucial points for obtaining the left-side and the right-side interaction lengths, respectively.

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In Fig. 1(a), we have defined a distance from the left-side, Interaction Length ${\textit {d}}_{L}$, up to center of the first column rods (positioned in $2000\mu$m; similarly a distance from the right-side, ${\textit {d}}_{R}$, from center of last column of rods ($2600\mu$m).

Fig. 2(b) shows the differential wave as the result of the subtracting scattered and original THz waves introduced Fig. 2(a). It is expected that, this differential wave would be periodic in the coordinates far enough from the ZRIM lattice. But we are focusing on the variations close to the ZRIM lattice in both sides, which experienced deformations, illustrated in magnified insets of Fig. 2(b). According to the wave amplitude characteristics, one can find the crucial point to obtain both side interaction lengths. In our approach, for the left-side interaction length, we pin the last upside differential wave peak, local maxima (30V/m), before the ZRIM lattice area, in this case somewhere around $1700\mu$m, then we can follow up getting closer to the ZRIM lattice area and search for finding this magnitude again; this value reveals itself at $1925\mu$m. To obtain the right-side interaction length, we have almost followed the same procedure, except that, at first we have focused on the minimum points and found the first local minima after the ZRIM lattice area, while knowing that this local minima would have a different value than the following ones. As shown Fig. 2(b), below inset box, this point is located in coordinate of almost 2670$\mu$m. In fact, outside of these two boundaries, the behavior of the propagating wave is like a simple plane-wave in the waveguide, while in between, it scattered and experienced fluctuations. Hence, we could calculate ${\textit {d}}_{L}$ and ${\textit {d}}_{R}$ by the above definition as $\sim 75\mu$m and $70\mu$m, respectively.

If we consider the maximum element size of $5\mu$m for meshes, we can see in the Fig. 2(b) top inset box that the point with amplitude of $30$V/m does not exactly fall to $1925\mu$m and is a little behind it, which means that the left-side interaction length is slightly more than $75\mu$m. If we add this value to right-side interaction length of $70\mu$m, it gives us the value of almost $150\mu$m, which is equal to PC lattice constant $a$.

It is worth mentioning that although it seems both THz waves before the ZRIM lattice area are almost of same phases (Fig. 2(a)), but the reason of non-zero differential wave existence in left side of ZRIM lattice area in Fig. 2(b), is a very small amount of phase mismatch of the scattered wave compared to the original plane-wave, which is not well seen in Fig. 2(a). This is due to a slight non-ideal reflection from the ZRIM back to the input port. As it turns out, the amplitude of this part of differential wave is much smaller than the rest of the differential wave. We have tried to utilize this non-ideality for our purpose and achieve the input interaction length.

3. Two cascaded ZRIMs procedure

3.1 Spatial phase shift versus the displacement between two ZRIMs

Like [2], we consider a THz waveguide having the same proper conditions composed of two separate cascaded 2D PC-based ZRIM lattices with arbitrary sizes of two $8\times 12$ lattices, which is studied in terms of spatial phase difference value compared to a second rectangular THz waveguide with same dimensions, input-output ports, background medium, and boundary conditions including no ZRIM lattices, Fig. 3. The second waveguide is proposed to better associate the subject of spatial phase difference in the case of two waveguides together, and of course as shown in Fig. 3 with dashed-dotted red connecting line, will exactly change in size same as changes in the main waveguide. As shown in Fig. 3, we want to change the distance between two lattices, $d$, from zero (two touching lattices) to 90$\mu$m (corresponding to $150\mu$m center-to-center separated ZRIM lattices, forming a single 16$\times$m12 ZRIM lattice), by a step of $10\mu$m, and compute the spatial phase difference between two THz waves, propagating in both waveguides, in each case. We consider two rectangular domains with fixed lengths of $1000\mu$m before and after the middle ZRIM area.

 

Fig. 3. (a) Schematic of a considered air background main THz waveguide with two embedded cascaded 8$\times$12 ZRIM lattices and a variable boundary-to-boundary distance d of 0 (two touched lattices) to $90\mu$m (a $16\times 12$ lattice) by step of $10\mu$m. Two beginning and ending areas are fixed with lengths of two $1000\mu$m regions. As indicated, left-bottommost Si rod of the left ZRIM lattice is centered at ($x$, $y$)=($1075\mu$m, $75\mu$m). PMC side walls, input, and output ports are adopted just like the Fig. 1(a). (b) Second waveguide without any embedded ZRIM with air as background medium. All boundaries, dimensions, input, and output ports are exactly like the main waveguide (a). Red dashed-dotted line indicate this fact that with sweeping the distance between two ZRIM lattices in the main waveguide the second waveguide length corresponds to the length of the main one. In this waveguide again we have two end fixed $1000\mu$m areas.

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In order to compute the spatial phase difference, we introduce an approach as follow: we fix two points, each inside of these $1000\mu$m domains and far enough away from the ZRIMs, having the same phase (Fig. 3). In the second waveguide, the distance between these points is an integer multiple of the wavelength, while in the first waveguide, the distance might not have the same character according to the effect of ZRIMs.

One can find the spatial phase difference by the value of $\Delta x$ between the locations of these two points from $\Delta \theta =[\Delta x/\lambda -Floor(\Delta x/\lambda )]\times 360^{\circ }$ [2]. This spatial phase difference in a way represents a comparison between the two proposed THz waveguides, with and without ZRIMs.

As shown in Fig. 4, we considered these two same-phase points in locations of the first and the last upper peaks in two complete cycles of the propagating THz wave in the first THz waveguide (3 cycles behind and ahead of the first and second ZRIM lattice, respectively), say $x_{1}$ and $x_{2}$, respectively. There is no need to show the propagating waves in the second waveguide, since $\Delta \theta$ between these two points in the second THz waveguide without any ZRIM is zero.

 

Fig. 4. Propagated THz waves through the shown main THz waveguide in Fig. 2(a) with two embedded cascaded ZRIM lattices for swept distances of $\textit {d}=0$ to $\textit {d}=90\mu$m by step of $10\mu$m. Left and right-top insets are shown the magnified first and last complete peaks around $280\mu$m and $4080\mu$m, respectively.

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Diagram of the spatial phase difference versus displacement between two ZRIM lattices is plotted in Fig. 5. The exact number of computed spatial phase differences $\Delta \theta$ and spatial differences $\Delta x$ for variations of lattice distance, are shown in the top-right inset of Fig. 5. Because at $d=0$ two lattices are touching and forming a connected lattice there is no longer any discussion of separation between two ZRIMs, and new propagated modes will certainly become more complex and require more advanced discussions, which do not fit in this article; although, the propagated THz wave in the first THz waveuide for $d=0$ (two touching lattices) is plotted in Fig. 4.

 

Fig. 5. Spatial phase shift $\Delta \theta$ versus the boundary-to-boundary displacement between two ZRIM lattices of $d=10\mu$m to $d=90\mu$m by the step of $10\mu$m. Standard error of the mean of $SE=3.69^{\circ }$ in half-top-half-bottom for the data is plotted on each data point. Top-right inset box is shown accurate computed spatial phase difference, and also spatial difference in each case.

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We started our calculations from $d=10\mu$m (according to the mesh size of 5$\mu$m) with the phase shift of $\Delta \theta =276^{\circ }$, while the spatial phase descends to its final value in farther distances. The lowest spatial phase difference belongs to the final data point of $d=90\mu$m which is $\Delta \theta =241.844^{\circ }$. In fact, this condition is corresponding to the spatial phase shift of a single 16$\times$12 embedded ZRIM lattice, which was reported in [2] as $242^{\circ }$. Since spatial phase calculations are highly dependent on the meshing size, finer sizes can provide more accurate data. In our data, we have calculated the standard error of the mean of SE=$3.69^{\circ }$ that is shown for the data in a symmetrical shape.

We can also see the displacements of the desired peaks for different distances ($d=0$ to $d=90\mu$m) in the insets of Fig. 4. In the top insets, the location of peaks located in the both same-phase points are moving while the distance between two ZRIMs, $d$, is varied.

In despite of phase shifts introduced in [13] which were obtained with considereation of a THz wave self-comparison before and after the two separate cascaded ZRIMs in a single waveguide, one of the highlights of this subsection is the spatial phase shift tuning in the case of two adjacent THz waveguides with and without embedded ZRIM lattices with PMC side walls. Based on the performed simulations, it can be seen that a limited adjustable range of about 34 degrees phase shift, from $241.844^{\circ }$ to $276^{\circ }$, for considered boundary-to-boundary displacement range of $d=10\mu$m to $d=90\mu$m less than a lattice constant of $a=150\mu$m is achievable. An important feature of the proposed structure, is that one can achieve almost significant phase shifts in small distance variations (below 90$\mu$m).

3.2 Transmission and reflection coefficients versus displacement between two ZRIMs

As shown in Fig. 2(a), with a very good approximation the amplitudes of the original and scattered THz waves after the ZRIM lattice area with size of $5\times 5$ UCs are almost equal, incidating that the transmission coefficient of this 1.082 THz operated system is unity. This also can be seen in Fig. 4, for a boundary-to-boundary lattice displacement of $d=90\mu$m shown with a black curve (a single embedded 16$\times$12 ZRIM lattice), two illustrated peaks in top insets, somewheres around $x_1=280\mu$m (before the first ZRIM lattice) and $x_2=4070\mu$m (after the second ZRIM lattice), are of same value and slightly more than 600V/m. From these two statements it can be concluded that in terms of energy, the original 1.082 THz plane-wave ideally travels through ZRIM lattice (Fig. 1(a)) or lattices (Fig. 3) embedded in the proper THz waveguide, but in this subsection we have computed the transmission ($T=S_{21}=|E_{z(out}) /E_{z(in)}|^{2}$, $f=1.082$ THz) and reflection ($R=1-T$) coefficients of the last subsection with a same THz wageuide containing two embedded separate cascaded $8\times 12$ ZRIM lattices versus the distance, $d$, between two lattices.

Transmission and reflection coefficients versus displacement between two ZRIMs are plotted in the Fig. 6 and as it can be seen, the transmission (reflection) coefficient rises (decreases) pseudo-exponentially. The minimum value of the transmission coefficient of this THz system occurs at the case of two touching ZRIMs with $d=0$, while the reflection coefficient is at it’s maximum value. As we expected, the transmission coefficient reaches to the maximum value of 1 in the case of two $150\mu$m center-to-center separated ($d=90\mu$m) $8\times 12$ ZRIMs or a corresponding single $16\times 12$ lattice, which represents the minimum value of reflection, 0. Although the exact values of transmission and reflection coefficients are displayed on each plotted data point of the diagram, but for better understanding the interaction procedure, the electric field of the propagating THz wave inside the whole THz waveguide is plotted in the left and right inset boxes for the two starting and ending points of $d=0$ and $d=90\mu$m, respectively. Accoridng to the inset color bars, it can be clearly seen that we get the highest reflection and transmission coefficients for the starting and ending points, respectively.

 

Fig. 6. Transmission and reflection coefficients shown as red and blue curves, respectively. Left and right insets are top view of propagated THz wave profiles in the case of $d=0$ and $d=90\mu$m, respectively. Symmetric value color bars with the units of V/m are shown beside the inset boxes.

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As it is perceived from the numerical values illustrated on each data point, for distances smaller than $d=90\mu$m the transmission coefficient is less than unity while at the final point of $150\mu$m ($=d+2\times 30\mu$m), it reaches unity.

4. Conclusion

Near field behavior of a 2D PC-based ZRIM with Si cylindrial rods embedded in a THz desired waveguide with PMC side walls is studied. PC lattice was a 2D square lattice of Si rods with Vaccum as background medium. For the first step we illuminated a $5\times 5$ PC-based ZRIM lattice embedded in a THz waveguide with PMC side walls with a TE polarized THz plane-wave and shown that when we get closer than$~\sim 75\mu$m to the input side of the ZRIM lattice area the THz wave profile will deform considerably and we called it the input intercation length. This process was baesd on the idea of a computed differential wave between the original THz plane-wave and the scattered THz wave. It was shown that a similar coincidence also occurs for the propagation THz wave on the output side of the ZRIM lattice, which is called the output interaction length slightly smaller than $75\mu$m. It is interesting that the summation of these two interaction lengths reaches to a lattice constant of these ZRIMs, which means we can consider an effective distance of wave interaction, almost as $a/2$ in each side of the ZRIM lattice area. Furthermore, we considered two separate cascaded ZRIMs in a THz proper waveguide with PMC side walls to investigate the interaction lengths. Hence, we computed spatial phase shift of the whole system versus the displacement between two ZRIMs.

We have shown that before $90\mu$m boundary-to-boundary lattice distance, moving towards lower distances, the spatial phase shift will change it’s behavior and rises effectively. A range of $34$ degrees with the cost of the transmission reduction has been provided for phase shift tuning between two (multiple) THz propagating waves.

We have also obtained the transmission and reflection coefficients, and shown that if center-to-center distance between two ZRIMs reaches $150\mu$m, the transmission and the reflection coefficients would be $1$ and $0$, respectively. This is while there is a direct relation between the lattice distance and these coefficients; lower the distance, lower the transmission (higher the reflection).

All three approaches of "the differential wave profile", "the spatial phase difference", and "transmission and reflection coefficients", emphasize an effective total interaction length value equals to a lattice constant of $150\mu$m. An important feature of the proposed structure, is that one can achieve almost significant phase shifts in small distance variations (below $90\mu$m).

In future, considering the effects of the modal analysis of the ZRIM lattice embedded in the waveguide in phase shift engineering, or including multiple multipole behavior of the structure would be interesting. Also, investigating the behavior at much smaller displacements, for example two touching ZRIM lattices, could lead us to more finer results.

These systems are based on an ultra-sensitive Dirac-point which could be somehow used as an actuator in 2D PC-based ZRIM lattice sensors, temperature sensor, incident-wave spatial phase sensor, etc. Beside of the applications which are directly based on Dirac-point, field enhancement with multiple ZRIM lattices and modulating the propagating waves are also applicable.