We present optical properties of crescent-shaped dielectric nano-rods that comprise a square lattice periodic structure named as crescent-shaped photonic crystals (CPC). The circular symmetry of individual cells of periodic dielectric structures is broken by replacing each unit cell with a reduced symmetry crescent shaped structure. The created configuration is assumed to be formed by the intersection of circular dielectric and air rods. The degree of freedom to manipulate the light propagation arises due to the rotational sensitivity of the CPC. The interesting dispersion property of designed CPC occurs due to the anisotropic nature of the iso-frequency contours that yield tilted self-collimated wave guiding. Furthermore, this feature allows focusing, routing, splitting and deflecting light beams along certain routes which are independent of the lattice symmetry directions of regular PCs. The propagation direction of light can be tuned by means of the opening angle of the crescent shape. Finally, the property of being all-dielectric structure ensures the absence of optical absorption losses that are reminiscent of employed metallic nano-particles.
© 2012 OSA
The research in the field of photonic crystals (PC) was emerged in 1987 . Since that time, highly symmetric periodic dielectric structures with a large refractive index contrast have been heavily investigated with an ultimate aim of achieving complete photonic band gap (PBG) that may appear in the dispersion diagram [1,2]. The PC acts as a mirror reflecting the entire incident light wave whose wavelength falls inside the PBG region. The band gap features of pure periodic 3D and 2D PCs were soon demonstrated [3,4]. Perturbing the periodicity of the PC may host artificially created optical modes that are surrounded by the upper and lower boundaries of PBG. Waveguides, sharp corners and cavities have become the ingredients of photonics research [5–7]. Meanwhile, it has been realized that the unperturbed structure also possesses rich spectral characteristics such as self-collimation, negative refraction, super-prism and super-lens [8–11]. All these listed peculiar dispersion properties may not need structural defects. The two cases (unperturbed periodicity vs. broken periodicity) comprise various device applications frequently demanded for photonic devices.
Considering all of the previously investigated common PC configurations, we can conclude that PCs are highly symmetric structures and do possess fixed structural patterns. The two mostly explored and utilized PCs are square- and hexagonal- (also known as triangular) lattice photonic structures. The ingredient element of PC is usually circularly shaped unit cell although there are other types of unit cell shapes such as rectangular, elliptical or annular ones [12–16]. The ultimate aim of these studies is to achieve complete PBG for all polarizations (TE and TM) and design polarization insensitive optical devices [17–19]. In a rather different perspective, the re-oriented unit cells of the PCs may give rise to the implementation of graded index (GRIN) mediums [20,21]. The implementations of GRIN via periodic structures provide great flexibilities in terms of designing different index gradient and photonic integrated circuit components such as couplers, lenses and super-bending device [22–27].
In the present work, we propose a novel type of PC structure named as crescent-shaped photonic crystals (CPC). To the best of our knowledge, this structure has not been studied as a periodic dielectric structure, yet. In this study, the designed CPC enables us to arbitrarily route light beams by exploiting the engineered dispersion diagrams. There is no need to infiltrate any type of anisotropic material and the approach does not possess asymmetric PC patterns. In the CPC structure, the geometrical adjustments are implemented at the level of unit cells not that of structural lattice arrangements. This brings extensive parametric tunabilities in realization of ultra-compact photonic integrated devices. Moreover, although CPCs are formed by isotropic materials, designed structure exhibits anisotropic optical properties similar to optical birefringence. The other unique feature of the CPC structure is due to the fact that the operating frequency of the structure can be easily shifted to any spectral region due to the scalability of the Maxwell’s equations and availability of different lossless dielectric materials.
There have been various mechanisms that may induce optical anisotropy for light propagation in PCs. The anisotropy introduced into the periodic medium can be either in terms of selecting specific materials (dielectric parameter) or structural configuration (unit cell’s shape or type) [12,14,15,17,28–35]. The optical properties of the former approaches can be dynamically tuned by an external applied electric field. The infiltrations of liquid crystals in 2D PCs involving anisotropic media were studied for tuning their photonic band structures [31,32]. The lower symmetry periodic structures have been investigated for different applications. The rectangular lattice PC was used in the study of angular super-prism effect in Ref. 36 and broad angle self-collimation characteristic was explored in Ref. 37. PCs made of parallelogram lattice structure were investigated for light focusing device that utilize self-collimation phenomenon . The self-collimated waveguide bends with different angles have also been implemented . Special attention should be given while joining the rotated blocks of parallelogram lattice PC because the junction planes with complex geometries may be induced . Similarly, the interface at the bend region should be carefully handled for the self-collimated waveguide bends . As a result, these approaches may provide limited capabilities for beam deflecting and routing applications. However, the proposed structure in the present work enables us to easily integrate different blocks made of square-lattice crescent PC.
The effects of symmetry reduction in PC were heavily explored with a goal of obtaining larger band gaps. We should emphasize that lower symmetry structures with complex configurations such as crescent-shape have not been investigated for the dispersion contours engineering and light manipulation applications. Instead of altering lattice type or introducing material anisotropy into periodic medium, we preferred to modify the circular shape of dielectric cylinders. The engineering of the iso-frequency contours (IFCs) can be performed at a level of unit cell and composite structures can be realized in such a way that the interfaces are free from complex geometries. It is possible to use other complex shape unit elements such as modified version of the crescent shape, U or V shapes instead of crescent one. However, it is expected that the degree of rotation of IFCs and focusing power may become different in each case. That aspect of the interpretation needs additional work which is kept for a future study.
The paper is organized as follows: In Section 2, we explain the geometrical details of the proposed CPC model and its dispersion characteristics. In Section 3, we analyze of the designed CPC structure using finite-difference-time-domain (FDTD) method and confirm that the expected results calculated by plane wave expansion (PWE) method agree well with the simulation results in the time-domain. The discussions of the findings and future directions are mentioned in Section 4. The conclusions will be listed in Section 5.
2. 2D crescent-shaped photonic crystals and dispersion analysis
In this work, we purposely break the circular (rotational, four-fold) symmetry of the unit cell by replacing it with a crescent shaped structure. The expectation is to enhance light manipulation capability inside the photonic structure without depending on artificially introduced structural defects. The geometrical shape of the individual cell provides the construction of complex photonic structures that may yield distinct spectral features as we show in the present work. It is versatile to tune the focal point locations and deflection angle of a beam via rotationally manipulating the structure. We show that the photon manipulation (propagation direction and focusing point) is greatly tailored due to the anisotropic nature of the IFCs. Introducing certain amount of rotational degree to each individual cells yields shifting of focal points along both x- and y-axes. It is worth noting that while rotational symmetry is lifted, we keep the translational symmetry intact. The beam flows along the direction which is dictated by the IFCs according to the following relation ,
The orientation of the CPCs strongly influences the direction of light propagation inside the medium. Figure 1(a) shows the geometry of corresponding unit cell for two-dimensional (2D) CPCs. When it is spatially distributed in a square-lattice pattern, Fig. 1(b) appears as the schematic of the structure. The combination of two circular rods (one is made of dielectric and the other is air) in an overlapped form gives rise to a crescent shape. The regarding opto-geometric parameters describing the structure are denoted in the same figure as well. The refractive index of the CPCs is taken to beand the radii of the dielectric/air rods are denoted byand. Their values are, where a is the lattice constant. The related unit cell filling factor, is defined by the formula In the case of the value of the filling factor becomes 0.1722. The opening angle of the crescent shape is defined byand is altered by rotating the composite cell in clock-wise (CW) and counter-clock-wise (CCW) directions as shown in Fig. 1(a). The center to center distance of each circle is represented by D and this parameter is set to The dimensions of the complete structure is denoted by (Li)x(Wi).
PWE method is performed in order to extract the dispersion characteristics of the CPC structure . In our case, the photonic band structure calculations are traced along the Brillouin zone edges starting at the Г point as can be seen in Fig. 1(c).
In the spatial beam routing applications, we may not need any type of structural defects. In such a case, the shapes of the IFCs become crucial. Traditional PCs composed of cylindrical rods or holes provide symmetric IFCs with respect to x- and y-directions. On the other hand, lifting the symmetry of the predefined structure by radially shifting the location of inner air-rod brings anisotropic shape to the IFCs. Hence, the light propagation direction can be arranged by solely controlling crescent open-angle The first band of the IFCs is isotropic due to validity of the effective medium theory . The anisotropy occurs with respect tofor the second and higher bands. For these higher order bands, there are three basic spectral characteristics. They are self-collimation, super-prism and focusing properties. The capability of the adjusting self-collimation direction (in this case it occurs not only along x- or y-directions but also along a certain angle) and the focal point of the light beam are the additional benefits of low-symmetry unit cell implementation. As a result, there is no need to alter the structure orientation or the incidence angle to adjust the focusing location.
Figure 2 is a collection of dispersion curves when traces values from 0° to 90° in CCW direction. Frequency domain approach is employed while calculating IFCs. As we change the crescent open-angle different spectral regions appear in the dispersion plots. Figures 2(a)-2(c) correspond to different frequency contours chosen at fixed operating frequencies for each region: (1) = (0°, 5°, 100, 20°, 30°), (2) = (40°, 50°, 60°), and (3) = (70°, 80°, 90°). The three different frequencies are selected based on their IFC shapes so that strong focusing behavior is promoted. It can be clearly observed from Figs. 2(a)-2(c) that CPC has tilted IFCs for the second band and the tilt amount can be regulated by only adjusting the crescent open-angle. It can be deduced from these figures that the orientation of the tilting amount directly depends on the steering of CPC opening. As the direction of light propagation is perpendicular to IFCs, the shift of focal point tracks an opposite path with respect to the crescent opening angle. These curves are selected to be representative cases of anisotropic IFCs that provide manipulation of focal point. In the next part of the paper, we present time-domain outcomes of the numerical studies.
3. FDTD analysis of the Crescent shape PC
The computational analysis of this section is based on time domain methods by employing two-dimensional FDTD . In order to eliminate the back reflections coming from the ends of the finite computational window, the boundaries are surrounded by the perfectly matched layers . We launched a source with a Gaussian distribution in the time-domain. For the numerical studies, transverse magnetic (TM) guided mode is used and the concerned non-zero electric and magnetic field components are Ez, Hx, and Hy. Then, the operational frequencies are chosen according to the different regimes of anisotropic IFCs as demonstrated in Figs. 2(a)-2(c). When the crescent open angle varies from 0° to 90°, the variation of IFCs is presented in Media 1.
The steady state electric field (Ez) intensity distributions of the CPC structures for the TM mode are shown in Figs. 3(a) -3(c). The parameter denotes the angle between the optical axis and focal point. The location of the focus is represented by F. We noticed that the value can be changed by altering. To exemplify, the crescent open angle is in Fig. 3(a) and focal point occurs at above the optical axis. On the other hand, is in Fig. 3(b) and then becomes negative. The crescent open-angle parameter can be set as an input control parameter that is scanned between to. When the crescent open-angle is at 0°, the focal point location in y-direction is not changed and centered at the optical axis. An oscillation occurs in the structure and a strong focusing is observed at the end face of CPC (point F1). The position of focal point is close to end face of CPC, as shown in Fig. 3(c). Due to interference of the side lobes, there occurs another secondary focal point which is represented by F2. Three important remarks can be inferred from Fig. 3. First, the closeness of the focal point to CPC’s end face is an indication of strong curvature (focusing power) due to special form of IFC. Second, the degree of anisotropy determines the amount of focal shift along y-direction, i.e. the values of Finally, the output angledepends on the input angle in a rather different manner. The functional dependency between the two parameters can be summarized in three sections as follows: first case is, the second case is and finally the third case is This dependency is summarized in Fig. 3(d). The maximum shift of focal point occurs when If one desires to obtain focal point residing on the optical axis, then
In Fig. 4 , the different operational frequency regions are displayed by different colors. The center frequencies of the input pulse for each region are set to and respectively. We can see that variance of with respect toresembles a sinusoidal pattern. The maximum lateral shift of focal point occurs at for a selected operating frequency. It can be seen from the figure that initially increases quickly and then starts to decrease slowly as we increase When we consider the employed discretization process in FDTD small discrepancy occurs while reading the locations of focus points. With a finer spatial resolution, odd-symmetric version of the graph can be obtained.
In addition to manipulating the focusing location, there is also a self-collimated behavior of the CPC. Similar to the previous results, we analyzed the IFCs of TM mode for the second band as shown in Fig. 5(a) . The black dashed arrows in Fig. 5(a) represent the directions of the group velocities and blue arrows represent the wave vectors. The asymmetric characteristic of the CPC has direct impaction on IFCs. As a result, the calculated IFCs for the second band are deformed from a square shape to a tilted square as shown in Fig. 5(a). The steady state e-field is extracted by using 2D FDTD method to observe the tilted self-collimation properties of the CPCs whenand the results are presented in Figs. 5(b) and 5(c). To show this effect along tilted direction, the source wave is allowed to propagate along and directions, in Figs. 5(b) and 5(c), respectively. The flat contours can be used to laterally confine light in the CPC structure. In fact, for a range of incident wave-vectors, the propagation will be normal to the IFC. The flat portion of the contour allows input source to propagate inside periodic CPC without diffraction.
4. Discussions and selected applications
In the current work, we propose a novel type of photonic structure, called “CPC”, and by means of this structure, we are able to design miniaturized optical medium that control both the propagation direction and focusing behavior of the electromagnetic fields. The great capability of CPC to adjust the focusing and deflection of light beams is due to lowering the symmetry of the proposed structure. The two fundamental light manipulation schemes were investigated: focusing and self-collimation effects. In addition to these features, one can implement beam splitters, routers and deflectors as well. The design methodologies are briefly depicted in Figs. 6(a) and 6(b). This can be achieved by advisedly combining differently-positioned CPCs with various crescent open-angles. For instance, suppose that the upper half of the structure lying above the optical axis has negative value forand the lower part has a positive value for as shown in Fig. 6(a). Then, the composite structure can act as a beam splitter. Half of the beam can be directed upwards and the other part is molded in the reverse direction. On the other hand, if the value is adjusted as gradually varying along the propagation direction (sweeping from 0° to 90°), then beam routers can be implemented, which is schematically demonstrated in Fig. 6(b). The details of these proposals are kept the outside of the current study. However, we present an example that shows a two-step tilted light collimation process. To achieve this, a composite version of the structure can be obtained by cascading two pieces of CPC as shown in Fig. 7 . While the first part has negative the second part may have positive The consequence of this combination yields self-collimated beam propagation having both positive and negative tilt angles. The source is placed at the left-side of the structure (the position is indicated by an arrow). The central part of the light beam follows the path that is highlighted with white arrows. When light travels inside the first part of the composite CPC, it bends upward. The second part of the structure routes the light wave downward. Due to the equal values of for both sections, the incidence and reflectance angles of the beam at the interface are equal to each other. One of the observations that can be deduced from Fig. 7 is that e-field concentrates strongly at the sharp edges of each crescent shaped cells.
The idea of splitting input power equally into two branches can be achieved by the help of the lower symmetry of CPC. The numerical investigation of Fig. 6(a) was performed and the result is shown in Fig. 8(a) . The source is placed in the middle of the structure at the left side. The normalized operating frequency is selected to be The light is divided into two self-collimated branches as can be observed from the plot. The amount of spatial separation between the two lobes at the end of the structure can be adjusted by means of CPC’s length. The transverse e-field profile is represented in Fig. 8(b). Almost identical peaks show the successful splitting of light beam by using the designed CPC. By adjusting the location of input source, light splitting with variable intensity ratio can be achieved as well. In addition to that, splitting angle can be controlled by altering the opening angle of the crescent shape cells. The media file in Fig. 8(a) designates the propagation of the input light throughout the splitting structure.
One of the interesting properties of the CPCs is that although the material of the structure is itself isotropic, the formed structure may exhibit anisotropic characteristics due to its asymmetric shape of IFCs. For normalized frequencies above 0.40, the anisotropy ratio defined asis higher than 1.50 and . This implies that CPCs can display different optical properties for different propagation directions of the same polarized light wave and can be approximated as an anisotropic media. Usually, the anisotropic feature of materials belongs to certain type of crystals that the nonlinear optics applications heavily use them . The proposed structure may offer an alternative way to create similar optical effect (form birefringence) that can be realized by structural manipulation of pure transparent periodic dielectric materials. The response of the structure should be investigated for both polarizations. The scope of the present work is intended not to cover this property of the CPC.
Thanks to the recent development in the fields of applied physics and photonics, the difficulties on the fabrication of complex shaped PCs can be surmounted [46–49]. Thus, it is expected that the fabrication of theoretically designed CPCs can be realized by the state-of-the-art fabrication methods featured in semiconductor devices. E-beam lithography, focused ion beam lithography and atomic layer deposition technique can be among the choices. Besides, one can always introduce symmetry reduced configurations that may demand less difficulty for fabrication steps but still show similar effects for light beams.
Converting the normalized values such as lattice constant, structure dimensions etc., in terms of measurable physical quantities gives us following results. When we tune frequency at 1550 nm (for the normalized frequency), then the lattice constant a and the radius are and respectively. The structure dimension becomes as. The focal point maximum shifting distance in the y-direction is equal toThe focal lengths for = 30, 0, and −30 are andrespectively. Even though we outline the findings of square lattice dielectric crescent shapes in air background similar behavior can be obtained by utilizing the complementary structure (i.e., air crescent shapes in dielectric background) patterned either by triangular or square lattice type.
In summary, the reduced symmetry of photonic crystals by introducing asymmetric unit cells in terms of crescent shape instead of circular ones improves the light manipulation capability via the appearance of anisotropic iso-frequency contours in the dispersion diagram. The optical characteristics of the structure were numerically investigated by means of finite-difference time-domain and plane wave expansion methods. The crescent-shaped photonic crystal demonstrates a high degree of control over the light propagation behavior in terms of focusing and self collimation of light beams. The routes of light beam can be tuned by altering the opening angle of the crescent shape. Engineering the placement of each crescent-shape cells may offer a platform for implementing various photonic functions including beam splitters and combiners, deflectors and routers without deploying any defects inside the periodic dielectric structure.
The authors are grateful for the partial financial support from the National Science Council of Turkey, TUBITAK under Grant Number: 110T306. H.K. acknowledges partial support from the Turkish Academy of Sciences Distinguished Young Scientist Award (TUBA-GEBIP).
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