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Abstract
The wavevector diagrams or eigenfrequency contours (EFCs) (also called dispersion surfaces) are the best tools to explore the optical properties of photonic crystals (PhCs). Many optical phenomena, such as self-collimation, super-prism, negative refraction, and lensing, have been extensively explored in PhCs based on EFCs. Also, several approaches have been continuingly pursued to modulate the EFCs of PhCs for molding the flow of light. This work presents the modulated wavevector diagrams of PhCs formed by asymmetric non-Moiré (NM) patterns. The NM patterns are contours of trigonometric functions that generate attractive tiles and shapes. Employing such shapes to design a PhC tailors the dispersion of PhCs with stretching, squeezing, and shape-modulated EFCs. Based on the modulated EFCs of the proposed structures, we demonstrate the direction-dependent beam steering phenomenon. The ray tracing, full-wave electromagnetic simulations, far-field patterns, and electric field profiles corroborate the beam steering application of the modulated EFCs. We anticipate that the modulated EFCs of non-Moiré pattern-based PhCs are useful for reconfigurable wave optics and beam steering applications.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Photonic crystal (PhC) composed of dielectric constitutions with the periodic variation of refractive index vividly alters the dispersion relation (the plot of wavevector versus frequency) and gives rise to anomalous dispersion phenomena. Over the past few decades, dispersion relation in PhCs, also known as wavevector diagram or equi-frequency contours (EFCs), has been explored rigorously, and novel optical applications such as slow light [1], self-collimation [2], sub-wavelength imaging [3], and super prism effects [4] were reported under the umbrella of photonic crystalline optics [5,6]. It is interesting to note that it is possible to engineer the EFC’s shape and size for efficient light-molding applications.
Since the optical properties of PhCs are derived from their structural parameters, such as lattice type, geometry, filling fraction, and dielectric permittivity of the rods and background material, many attempts have been made to explore each of these parameters separately to modulate EFCs of PhC. Some of the approaches used to modulate dispersion relation in PhCs are to replace conventional PhCs with Archimedean tilling [7], different geometry patterns oriented at an angle (ellipse and rounded rectangle) [8–10], or by tuning the PhCs using external biases such as electrical, magnetic, thermal, mechanical and biochemical stimuli [11,12].
Out of these approaches, in this work, we attempt to explore structural modifications in PhC atoms by replacing circular rods with anisotropic low symmetric patterns. It may be noticed that usually low symmetric patterns were constructed with the help of an ellipse, triangle, and rounded rectangle, and by introducing an additional rod or by modifying the geometry by changing the position of the circular rod to a certain angle [8–10]. In this work, we alter the dispersion relation with the help of asymmetric patterns obtained from Non-Moiré (NM) tiles. The NM patterns are the contours of trigonometric functions expressed as
In the literature, several geometries were reported for achieving wide bandgap [13,14], anomalous dispersion [15,16], polarization sensitiveness [17,18], reflection modulation, and wave propagation applications [19–23]. Since the emergence of 3-D printing technology can realize any complex photonic structures, the proposed patterns and geometries can be very well employed for wave propagation applications.
To get the EFCs of the proposed NM pattern-based PHCs, Maxwell’s wave equation is solved as an eigenvalue problem subjecting to unit-cell Floquet periodic boundary conditions. The eigenvalue problem is solved using a commercially available EM solver COMSOL RF Module [24]. The constraints and conditions used to solve Maxwell’s wave equation as an eigenvalue problem in COMSOL is similar to the planewave expansion method (PWM) [25] in which (i) real and positive dielectric constant only considered, (ii) material is assumed to be nonmagnetic (µr = 1), (iii) the source free regime is considered, (iv) transversality and orthogonality conditions are imposed such that $\nabla \cdot \vec{A}({\vec{r}} )= 0$ and $\left\langle {{{{\vec{E}}_i}}} \mathrel{|{\vphantom {{{{\vec{E}}_i}} {{{\vec{E}}_j}}}} } {{{{\vec{E}}_j}}} \right\rangle = {\delta _{ij}}$, respectively.
To gain the refraction coupling pictures in the proposed NM patterns-based square lattice PhCs, ray tracing is performed and compared with the conventional circular rod PhC. Full-wave EM calculations are carried out to validate the EFC results, and wave propagation is studied. The aspect of beam steering provided by the modulated EFCs is demonstrated numerically. For practical applications, the observed beam-steering is also verified for 3-D PhC against the out-of-plane radiation losses due to the finite height of the PhC rod.
2. Result and discussions
2.1 Equi-frequency contour (EFC) for asymmetric PhC atoms in square lattice
The square lattice unit cells comprised of the proposed patterns are shown in Fig. 2 with a dielectric permittivity value of 12.96, and the background is considered air. Figure 2(a) shows the regular circular rod of normalized radius 0.252a. Figures 2(b) to 2(d) show different NM patterns (L, N and W) with the same filling fraction (0.199 m2) as that of the circular rod. The asymmetrical rods have been named L, N, and W based on the contour's shape; L stands for the liver, N for an asymmetrical shape, and W for wings, respectively. One can note from Fig. 2(a) that the circular rod is symmetric, but L and N patterns shown in Fig. 2(b) and 2(c) lack symmetry as the geometry is anisotropic. The W pattern in Fig. 2(d) deviates from the symmetry in horizontal and vertical dimensions. Therefore, for patterns shown in Fig. 2(b) to 2(d), one can expect strongly modulated EFC curves compared to EFCs of a conventional circular rod PhC.
Fig. 2. Unit cell of square lattice PhC with (a) Circular rod, (b) L, (c) N, and (d) W patterns. The square lattice first Brillouin zone’s corner Bloch wavevectors are shown in each unit cell.
Since EFC refers to the locus of allowed wavevectors for a given frequency, they provide complete information about the PhC’s dispersion compared to the band structure. To obtain the EFC for the proposed patterns, the eigenvalues (frequencies) are searched for Bloch wavevector ranging from $- \frac{\pi }{a}to\frac{\pi }{a}$ represented for each unit cell shown in Fig. 2. The obtained EFCs are plotted in Fig. 3 for the second band transverse electric (TE) modes. It is found that the anisotropic structure exhibits interesting EFCs for all the proposed NM tiles. It is observed that TE2-band EFCs for a circular dielectric rod comprise three different shapes such as rounded square (square shape with rounded edges), closed hyperbolic-like EFCs, and elliptical EFCs as shown in Fig. 3(a). It is also noted that as frequency increases, the size of the EFCs shrinks and thus indicating the negative band slope (where ${{\partial \omega } / {\partial k}}$ is negative at a given k-point in EFC for the TE2 band).
Fig. 3. EFC for (a) Circle (b) L (c) N and (d) W pattern square lattice PhCs and (e) EFC of all the proposed PhCs at a normalized frequency 0.3986(c/a).
In the case of the proposed NM pattern-based PhCs, EFCs are strongly modified with compression, squeezing, and elongation, as shown in Fig. 3(b)–3(d). The distorted EFCs for anisotropic PhCs consist of flat horizontal bands along $\varGamma X^{\prime}$ direction for the normalized frequency range of 0.3582 (c/a)−0.3689(c/a), 0.3258 (c/a)−0.39470(c/a) and 0.34 (c/a)−0.40 (c/a) for L, N and W PhCs respectively. It is observed that the EFC curves are stretched along $\varGamma X$ direction, but along the $\varGamma X^{\prime}$ direction, the EFCs are squeezed. Overall, EFC curves for proposed patterns are also oriented to a specific angle for circular rod PhC EFCs. The oriented EFCs will result in a drastic change in the refractive/propagative behaviour of EM beams in the proposed PhCs, as the slope of EFCs indicates the refraction direction under homogeneous conditions. For example, as shown in Fig. 3(e), a rounded square curve (normalized frequency 0.3986 (c/a)) at the center of Fig. 3(a) is compared to the central EFC curve of Fig. 3(b)-(d). From Table 1, it is observed that the incident angle corresponding to the corner of the EFC for each pattern varies as $45^\circ $ for the circular rod, 25$^\circ $ for the L-pattern, $3^\circ $ for the N pattern, and 13$^\circ $ for the W-pattern. This aspect suggests the degree of squeezing of EFC (It may be computed as a ratio of the corner angle of a given EFC for each pattern chosen at the common frequency and is listed in Table 1) for each pattern in comparison with the circular rod EFC. According to this computation, it is found that the EFC at 0.3986(c/a) is squeezed 1.8, 15 and 3.46 times for the L, N, and W patterns, respectively. Thus N-pattern shows the highest degree of squeezing among the investigated patterns. The squeezed EFC indicates that only the incident beam with a narrow angular span will couple to the PhC structure. Thus, EFC squeezing is helpful for beam-steering with pencil-like incident beams.
Table 1. List of incident angles corresponding to the corner of the EFC for all four patterns at normalized frequency 0.3986 c/a and a measure of the degree of squeezing of EFC.
The EFC modulation investigated in this work is compared with other reported methods, as in Table 2. It is found that EFC modulated under different conditions shows either squeezing/ elongation/ change in frequency range depending on various parameters like rotating unit cell, working on complex lattices, varying the refractive index ratio, introducing new geometry with the help of basic shapes or by applying external stimuli to the proposed PhCs. From these comparisons, we found that the EFC pattern obtained for the L structure can also be obtained by varying the position of the air hole in the dielectric slab of a unit cell proposed by Erdiven et al. (2018) [9]. However, the asymmetric geometry differs from other reported works, such that the degree of elongation and squeezing is unique to the proposed patterns owing to its structural anisotropy.
Table 2. Comparison of different methods of modulating PhC’s EFCs reported.
2.2 Ray tracing and verification of EFC modulation
The EFC at 0.3986(c/a) corresponding to the circular rod and NM pattern-based PhCs is taken, and ray tracing analysis is performed to investigate the beam steering nature due to the elongated/squeezed EFC. Figure 4(a) shows the ray tracing of rounded square EFC at 0.3986(c/a) for a square lattice-based circular rod PhC (single unit cell is shown in the inset of Fig. 4(a)). It is noticed that for an incident ray at a given angle (θ), the parallel line (the phase matching condition) meets the PhC EFC, as shown in Fig. 4(a). The outward normal drawn at the point of intersection between the phase matching line and the PhC EFC suggests the straight collimated beam. However, the EFC pattern at 0.3986(c/a) for L-shaped pattern-based square lattice PhC is entirely different, and the corresponding ray tracing is presented in Fig. 4(b). The refracted wave in an L-shaped pattern PhC is observed to be tilted to an angle for the exact angle of incidence shown in Fig. 4(a). One can also keep that the EFC curve is flattened, indicating the system's self-collimation effect, but the transmitted e-m wave is bent (steered) to an angle. It is also noticeable that the EFC of a circular atom (Fig. 3(a)) is symmetric along $\varGamma X\; and\; \varGamma X^{\prime}$ directions. Still, for an L -shaped atom (Fig. 3(b)), the slope of the EFC curve along $\varGamma X $ direction is different from $\varGamma X^{\prime}$ direction.
Fig. 4. Ray tracing diagram for (a) rounded square EFC of circular rod-based square lattice PhC (unit cell is given in the inset) and (b) The corresponding modulated EFC of L-pattern based square lattice PhC (unit cell given in the inset) at 0.3986 c/a.
To study this observed collimated beam steering in L-pattern-based PhC, full-wave e-m computations are carried out by impinging TE polarized wave along $\varGamma X\; $ direction for circle and L pattern PhCs system. To develop THz beam steering functional devices, we performed this study at terahertz (THz) frequencies. Figure 5 shows the Ez field map for a TE line source normally incident on a $19 \times 19$ layer PhC. Figure 5(a) shows the self-collimation effect for a circular rod PhC at 1.5 THz, whereas the exact normal incidence in L-pattern-based PhC shows the negatively bent collimated beam at an angle of ${20^\circ }$ as shown in Fig. 5(b). This confirms the ray tracing results given in Fig. 4(b). The bending of an EM wave, even for a typical incidence, should be addressed as a beam steering due to the shape of the EFC rather than as the refracted wave.
Fig. 5. Electric field profile (Ez component) shows (a) self-collimation for circular rod PhC and (b) negatively bent and (c) positively bent collimated beam for L-shaped rod PhC and inverted L-shaped rod PhC at 1.5 THz for a normal incidence TE wave impinged along $\mathrm{\Gamma }X\; $ direction of the PhCs from left side respectively.
The second interesting aspect is that if each of the L-shaped atoms shown in Fig. 5(b) is inverted, the EM beam will be positively bent for a normal incident TE wave, as shown in Fig. 5(c). Since the shape of the EFC for the L-pattern PhC (given in Fig. 4(b)) is a distorted parallelogram, the PhC atom and its inversion result in positive and negative beam steering respectively.
2.3 Beam steering and far-field pattern for a finite height 3-D PhC
The above investigation is restricted to 2-D computations with ideal conditions, in which a PhC rod's height is infinite, and a theoretical line source is used to study wave propagation. However, one must work with a finite-height PhC and a valuable EM source for practical realization. In this demonstration, the beam steering functionalities of the realized NM pattern-based PhCs are verified for a finite-height system. For a finite-height 3-D PhC, rods with the height 2$\lambda $ are chosen, and an EM wave is excited from two parallel plates separated by 100 $\mu m$ . The transmission and far-field calculations are done at THz frequencies. 3-D simulations use the finite-integration-based commercial e-m solver CST Microwave Studio [26]. Figure 6(a)–6(c) shows the beam steering geometries where $7 \times 9$ an array of circular rod PhC (Fig. 6(a)), L-shaped rod PhC (Fig. 6(b)) and an inverted L-shaped rod PhC (Fig. 6(c)) are taken. Perfectly matched boundary layers surround the entire setup, and the hexahedral meshing scheme is adopted for meshing the structure.
Fig. 6. (a)- (c) 3-D configurations of the circular rod PhC, L-shaped rod PhC and inverted L-shaped rod PhC, respectively. (d)-(e) are their respective field profiles, and (g)-(i) are their respective far-field patterns recorded at 1.5 THz, respectively.
It is observed that field profiles (Fig. 6(e), 6(f) and 6(g)) corresponding to 3-D structures agree with the 2-D computations. Significantly the negative (Fig. 6(f)) and positive beam steering (Fig. 6(g)) is evident for the realized L-shaped rod PhC and its inverted configurations, respectively. To probe the beam steering characteristics, far-field results for the circular and L-shaped rod PhC are presented in Fig. 6(h), 6(i) and 6(j), respectively. From the far field calculation, the main lobe direction for the beam steering inside the PhCs is ${90^\circ }$ for the cylindrical rod PhC, ${79^\circ } $ and ${101^\circ }$ for the L-shaped rod PhC and its inverted configurations, respectively, at 1.5 THz. From the far-field plot, one can verify that ${79^\circ }$ belongs to negative beam steering (Fig. 6(i)), whereas ${101^\circ }$ belongs to positive beam steering (Fig. 6(j)). The 3-dB angular bandwidth of all the beams is found to be ${89.9^\circ }$, ${91.1^\circ }$ and ${90.4^\circ }$ for circular rod, L-shaped rod and inverted L-shaped rod PhC, respectively. Compared to the circular rod PhC, the beam-steering in L-shaped rod PhC suffers moderate scattering losses as the main-lobe magnitude computed at 1.5 THz for circular rod, L-shaped rod and inverted L-shaped rod PhCs is found to be 0.2 dBi, −6.1 dBi and −7.3 dBi respectively. Here the dBi refers to the normalized radiation efficiency for the isotropic radiating source. These results affirm that the realized geometries are useful for beam steering applications at THz frequencies.
2.4 Molding the wave path with the proposed NM-based PhCs
This section combines L-shaped and circular rod PhCs to mold the light path with user-designed profiles. Figure 7(a) shows the proposed geometric scheme used for the wave path’s molding. It consists of successive stacking of five PhCs from left to right as follows: (i) circular-rod PhC which supports collimated beam, (ii) an L-shaped rod PhC which supports negative beam steering, (iii) again a circular rod PhC which collimates the EM beam directly, (iv) an inverted L-shaped rod PhC which supports positive beam steering and (v) a final circular rod PhC which collimates the beam to the output port. Figure 7(b) shows the Ez profile for a typical incident TE wave impinged on the geometry from left to right at 1.5 THz, as shown in Fig. 7(a). Such steering capabilities for the proposed PhCs help create delay lines, waveguide channels, and stealth technologies for manipulating the beam paths.
Fig. 7. (a) Geometry used for the proposed beam steering using circular and L-shaped rod PhCs. TE line source is excited from left to right of the geometry, and PML boundary conditions are employed on the top/bottom boundaries. The symmetry line indicates that the left side structure is mirrored on the right side. (b) Ez field pattern at 1.5 THz.
The realized beam steering setup is constructed with 3-D geometry for further practical realization, as shown in Fig. 8(a). Here the EM wave excited between the parallel plates is used as a source. Figures 8(b) and 8(c) show electric field and far-field plots at 1.5 THz, respectively, and they verify the EM wave steering phenomena as observed in 2-D calculations. The main lobe direction, power level, and the 3-dB angular bandwidth for the observed beam steering are −5.9 dBi, $97\; ^\circ \; and\; 95^\circ $, respectively. Hence including the observed losses, the realized beam steering based on the NM tile PhC can be implemented for THz waveband operations.
Fig. 8. (a) 3-D configuration of the proposed beam steering using circular rod and L-shaped rod PhC and (b) and (c) show corresponding electric field and far-field pattern recorded at normal incident TE wave at 1.5 THz.
3. Conclusions
To conclude, the wavevector diagram (EFCs) of the PhCs formed by Non-Moiré patterns are obtained in this work. It is demonstrated that three different unit cells with the same filling fraction show interesting EFC contours with elongation and squeezing aspects. It is numerically evaluated that NM tile-based PhC exhibits asymmetric EFC shape squeezing by 15 times in comparison with the conventional regular circular rod PhC. The L-shaped rod PhC displays different beam steering paths depending on the orientation of the atom, as the shape of the corresponding EFC is found to be a distorted parallelogram. It was demonstrated that normal incident TE light steered as positive and negative bending beams inside the L-shaped and inverted L-shaped PhCs, respectively.
Three-dimensional calculations are carried out with a finite-height PhC at THz waveband, and the far-field patterns are recorded at beam steering frequency. Including the scattering losses, 3-D results agree with the 2-D computations. The choice of positive or negative beam steering is found to be sensitive to the orientation of the NM tile pattern in which the inverted pattern exhibits different beam steering with respect to the original NM tile pattern. This work also demonstrated the user-specific light molding technique using the proposed NM tile-based PhCs. Hence, the investigated dispersion properties of NM tiled-based PhCs will be helpful for the beam steering wave channels, delay lines, communication networks, and stealth technologies for manipulating the wave path.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available but may be obtained from the authors upon reasonable request.
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