Two-dimensional Triangular Photonic Crystal Research Articles

Tunable wave localization at the Dirac frequency in a metallic photonic crystal cavity

In this study, the two-dimensional (2D) triangular lattice metallic photonic crystals (PCs) in visible and infrared bands have been utilized to achieve light confinement at the Dirac frequency. Distinct from the traditional bandgap or total internal reflection cavity modes, the unique photonic localization mechanism leads to an unusual algebraic decay of state and a unique frequency located beyond any bandgaps. This investigation delves into the band structure analysis of 2D metallic PCs, specifically focusing on their distinctive features, such as photonic bandgaps and Dirac cones. The plane wave expansion (PWE) method, enhanced with a linearization technique, is employed for band structure calculations, considering both the frequency-dependent dielectric properties and the intrinsic lossy nature of metallic materials described by the Drude model. The study provides a comprehensive derivation of the PWE equations for metallic PCs and investigates their band characteristics under both TM and TE polarizations. Focusing on TM modes in triangular lattice metallic PCs, it reveals zero density of states (DOS) at K points of the Brillouin corner and the existence of Dirac cones with linearly dispersion and linearly vanishing DOS. The study extends to exploring localized modes at Dirac frequencies, employing a relativistic quantum mechanics approach analogous to graphene's charge carriers. Theoretical predictions are corroborated by numerical simulations, and the potential for tunable Dirac localized modes is highlighted. This research not only deepens the understanding of Dirac properties in graphene-like systems but also lays the groundwork for further exploration of the practical quasi-2D devices, which will provide assistance in the integration of micro- and nano- devices, especially in applications requiring long-range coupling, given the critical importance of optical cavities in contemporary optical technologies.

Topological boundary states of two-dimensional restricted isosceles triangular photonic crystals.

We propose an all-media photonic crystal (PC) composed of isosceles triangle dielectric cylinders that realizes the topological phase transition by simply rotating the isosceles triangular dielectric cylinders. Additionally, the topological phase transition is closely linked with the size parameters and rotation angle of the isosceles triangle. The topological boundary states with lossless transmission are constructed on the interface of two different topological structures, and the optical quantum spin Hall effect is simulated. Further, we verified that the boundary state is unidirectional and immune to disorder, cavity, and sharp bend defects. By rotating the angle of the triangle to control the transmission path of the pseudo-spin state, we realize diverse transport pathways of light, such as the "straight line" shape, "Z" shape, "U" shape, and "Y" shape. This topological system shows a higher degree of freedom, which can promote the research on topological boundary states and the development of topological insulators in practical applications.

Dependence on hydrostatic pressure in two-dimensional triangular photonic crystals

The plane wave expansion method is used to calculate photonic band structures for transverse electric (TE) polarization, in a two-dimensional (2D) photonic crystal with a hexagonal lattice composed air holes with triangular cross sections embedded in a semiconductor (GaAs) background. This study focuses on the dependence of the GaAs dielectric constant on pressure and temperature. The air holes are isosceles triangles with fixed areas. We found that the width of the photonic band gap increases, when the internal angle of the triangle and the angle of rotation in the unit cell are increased at constant hydrostatic pressure values. The width of the photonic band gap remains unchanged when the hydrostatic pressure is increased at fixed internal and rotation angle values. In addition, when considering an H1 defect in the crystal, using the supercell technique, we find that the number of defective modes within the photonic band gap increases with the triangle rotation in the unit cell.

Polariton condensation in photonic crystals with high molecular orientation

We study Frenkel exciton-polariton Bose–Einstein condensation in a two-dimensional defect-free triangular photonic crystal with an organic semiconductor active medium containing bound excitons with dipole moments oriented perpendicular to the layers. We find photonic Bloch modes of the structure and consider their strong coupling regime with the excitonic component. Using the Gross–Pitaevskii equation for exciton polaritons and the Boltzmann equation for the external exciton reservoir, we demonstrate the formation of condensate at the points in reciprocal space where photon group velocity equals zero. Further, we demonstrate condensation at non-zero momentum states for transverse magnetic-polarized photons in the case of a system with incoherent pumping, and show that the condensation threshold varies for different points in the reciprocal space, controlled by detuning.

Zero phase delay induced by wavefront modulation in photonic crystals

A new mechanism for the generation of efficient zero phase delay of electromagnetic wave propagation based on wavefront modulation is investigated in this paper. Both numerical simulations and experiments have demonstrated the zero phase delay behaviors of wave propagation in a two-dimensional triangular photonic crystal. The zero phase delay propagation, independent of the propagating distance, is attributed to an invariable wavefront modulation in a photonic crystal in the direction of wave propagation where the phase velocity is perpendicular to the group velocity with parallel wavefronts (or phasefronts) extending along the direction of the energy flow. This effect can be extended to the three-dimensional cases or other artificially engineered materials and may open a new route for obtaining perfect zero phase delay propagation for an electromagnetic wave instead of using zero-index or zero-averaged-index materials and may have significant potential in many applications.

Effective refractive index of the photonic crystal deduced from the oscillation model of the membrane

The oscillation model of the circular membrane is used to calculate the effective refractive index of the two-dimensional triangular photonic crystal at normal incidence within the second photonic band. Negatively effective refractive indices deduced from this model match those calculated by equifrequency surfaces very well. The result reveals that the field distribution has relation with the effective refractive index at certain frequency regions. Besides, the field distribution described by the Bessel function is more compact than the Fourier series expansion.

Dual-negative refraction in photonic crystals with hexagonal lattices

We present a dual-negative refraction effect based on the overlapping bands in a two-dimensional triangular photonic crystal formed by holographic lithography. Under certain conditions, one incident plane wave launched into this photonic crystal can be dispersed into two negative refracted waves at the same frequency and the same perpendicular polarization state with different phase velocities and group velocities. We find that this dual-negative refraction behavior can be easily manipulated by adjusting the incident angle, the frequency of incident wave and the filling ratio of the PhC. This special effect can be applied to realize wave-front division and optical interference in optical holography. Based on this effect, a double focusing imaging phenomenon has been achieved by the PhC slab. These unique features may show great impacts on both fundamental physics and optical device applications.

Order to disorder optical phase transition in random photonic crystals

Transition process of optical modal properties induced by the introduction of randomness into random photonic crystals is investigated by a computational method. We analyze an impulse response of two-dimensional triangular photonic crystals, in which the positions of the air holes are slightly deviated to random directions. It is shown that the appropriate degree of random departure from a perfect crystal state gives rise to multiple scattering of low group velocity band-edge modes and supports their strong Anderson localization. The achieved confinement efficiency of light exceeds the one obtained in the perfect photonic crystal state.

Focusing analysis of a complex photonic crystal slab with negative refraction

The electromagnetic wave propagation in a two-dimensional (2D) complex triangular photonic crystal (PC) consisting of dielectric cylinders of different radii in air is investigated. The plane wave expansion (PWE) and finite element (FE) methods are used to study the optical and focusing properties of the complex PC. Interestingly, in contrast to a triangular PC consisting of dielectric cylinders of the same radius, the complex PC can generate a focusing effect with an effective refractive index of −1. Furthermore, a design is proposed using antireflection structures to enhance the transmission efficiency of light at the interfaces between the air and the PC slab. Numerical simulations show that appropriately adding antireflection structures to the surface of the slab can greatly improve the focusing resolution.

Imaging properties of a two-dimensional photonic crystal with rectangular air holes embedded in a silicon slab

Abstract We investigate the superlensing effect of a two-dimensional (2D) triangular photonic crystal (PC) slab consisting of rectangular air holes in silicon. An analysis of equal-frequency contours (EFCs) shows that geometric parameters of the PC element shape can greatly change the shape of EFCs, thus change the effective phase index of the PC structure. Moreover, an effective negative refractive index n eff = −1 can be obtained at a normalized working frequency of 0.30 × 2 π c / a by adjusting the width of the rectangular air holes. Imaging capacity of this PC slab involving both the power intensity and full width at half-maximum of image is studied by the finite-difference time-domain (FDTD) method. In order to achieve a high-quality image, an appropriate surface termination of the PC slab is chosen. Coupled-mode theory analysis shows that the optimized surface termination can excite two kinds of surface modes which can couple with the Bloch wave in the PC. With the help of these surface modes, both the intensity of image and the super resolution capacity for the PC slab lens can be improved greatly.

Design and analysis of a wideband photonic crystal waveguide with low group-velocity and low dispersion

The slow light propagation in a line waveguide in the two-dimensional triangular photonic crystal has been numerically studied, based on which a wideband photonic crystal waveguide with low group-velocity and low dispersion is proposed. The numerical simulation analysis shows that it is possible to maximize the group index and minimize the group-velocity dispersion in wide bandwidth by increasing the radius of the basic air hole and changing the position of the first two rows of air holes in photonic crystal waveguides. Such a photonic crystal waveguide exhibits low group velocity and low group-velocity dispersion over a broad wavelength range. A larger group index-bandwidth product is achieved in this type of waveguide structure. The numerically computed results present the normalized bandwidth as 0.32%, 0.48% and 0.642% corresponding to the group index of 85, 58 and 45, respectively.

Improvement of transmission properties through two-bend resonance by holographic design for a two-dimemsional photonic crystal waveguide

We have investigated the transmission properties of a photonic crystal waveguide (PCW) formed by holographic lithography for the first time with a two-dimensional (2D) triangular holographic photonic crystal (PhC) including a line defect with two 60 masculine bends. Calculations have shown that for this PCW high transmission (>90%) through sharp corners can be obtained in a wide frequency range from 0.298 to 0.310 (omega alpha/2pi c) with the relative band gap of 4% when the dielectric contrast is 7.6:1. As far as we know, this result should be the widest frequency range with high transmission (>90%) in the waveguide of similar 2D triangular PhCs ever reported. We have also found that the specific holographic designs of PhC have strong influence on the resonance between the two waveguide bends, and thus this fact can be used as an effective means to improve the transmission property of 2D holographic PCW. In addition to the simplicity and low cost of holographic fabrication of PhCs, these features may reveal the possibly better guiding ability of holographic PCW than the conventional waveguide and the promising potential of the former in the application of photonic integrated circuits.

Effect of Oxidation, Etching, and Thin-Film Deposition on Silicon Photonic Crystals

The effects of oxidation, partial oxide etching, and thin-film deposition on the photonic bandgap of two-dimensional triangular photonic crystals are studied. Two structures, air cylinders in silicon background and silicon cylinders in air background, are analyzed by the plane wave expansion method. Fine tuning of the bandgap is demonstrated by varying the thickness of the interfacial layers of grown by thermal oxidation and produced by atomic layer deposition (ALD). In the air cylinder structure the absolute bandgap can be tuned from 0 to during oxidation by varying the oxide thickness from 0 to , where is the lattice constant. During etching it can be tuned from 0 to . In a silicon cylinder structure the transverse magnetic bandgap can be tuned from 0 to during oxidation and from 0 to during etching. Tuning of the defect frequency of a single defect formed by selective oxidation is demonstrated, and the dependence of the defect frequency on the thickness of the oxide is investigated. Results indicate that the effect of complementary metal oxide semiconductor compatible processes such as thermal oxidation and ALD on the photonic bandgap is significant.

Defect modes created by an elliptic defect in two-dimensional triangular photonic crystals

This paper presents the polarization properties of defect modes created by an elliptic defect in a two-dimensional photonic crystal composed of circular cylinders with a triangular lattice. It has been observed that adjustment of the defects’ symmetry results in the splitting of the dipole mode degeneracy, as predicted by group theory, and the magnitude of the frequency splitting increases as the aspect ratio of the elliptic defect increases. We also study the orientation effect of the elliptic defect on the defect modes. The results show that the frequency of the defect modes is independent of the orientation of the defect rods and the field distribution rotates by the same angle as the defects.

Polarization and thickness dependent guiding in the photonic crystal slab waveguide

We investigated the impact of the thickness of the twodimensional triangular photonic crystal (PC) on modal propagation along a PC slab waveguide. A single line defect optical waveguide in photonic crystal slab was designed by three-dimensional finite difference time domain method, plane wave expansion and effective index methods. The thickness of the PC slab waveguide was optimized to provide modal propagation for both TE-like and TM-like polarizations within the normalized frequency band of a/lambda=0.26-0.268.

Photonic band gap of two-dimensional triangular photonic crystals with broken structural and rotational symmetries

Three deformed and two rotational structures are constructed to study the effect of structural and rotational symmetries on the gap widths of E- and H-polarization bands in photonic crystals. The band structures and field patterns of a triangular lattice of hollow tellurium (Te) rods are calculated using the plane-wave method. The H-polarization band gaps are strongly affected by the interaction between the fields of the rods as the rods are deformed and affected by the reduction in the rotational symmetry as whole rods are rotated. Only the shapes of the rods affect the E-polarization band gaps as the rods are either deformed or rotated. Moreover, H-polarization modes determine the absolute photonic band gap (PBG) width as the rods are rotated, whereas E-polarization modes determine the absolute PBG width as the rods are deformed.

Effect of Photonic Crystal Band Gap on Rotation of Hollow Oval Te Rods in Triangular Lattice

The band structures and field patterns of two-dimensional (2D) triangular photonic crystals with hollow tellurium (Te) rods are calculated using the plane-wave method. Inner-rod and whole-rod rotations are considered to investigate how the rotational and structural symmetries affect E- and H-polarization modes. It is indicated that the H-polarization modes are strongly affected by the shape and interaction between the fields of the rods. In addition, the effect of rotational symmetry on H-polarization modes must be considered when whole rods are rotated. The E-polarization modes for both types of rotations are strongly related to the shape of the rods. Moreover, H-polarization modes determine the absolute photonic band gap (PBG) width when whole rods are rotated, while E-polarization modes determine the absolute PBG width when inner rods are rotated.

Design of Polarization Beam Splitter in Two-DimensionalTriangular Photonic Crystals

A design of a polarization beam splitter (PBS) in two-dimensionaltriangular photonic crystals is proposed and numerically demonstrated,for the first time to our knowledge. The principle of the model isbased on the polarization dependence of the photonic band gaps inphotonic crystals; the polarization extinction ratios fortransverse-magnetic and transverse-electric modes are 11.5 dB and46.7 dB at the centre of its operating frequency range, respectively. Whenthe central wavelength is tuned at 1550 nm, the possible operationwavelength range of this structure can be as large as 28.5 nm, whichalmost covers the whole C band in modern optical communication systems.Furthermore, due to its small size and only one kind of material beinginvolved, the PBS structure may have practical applications in theintegrated optics field.

In-plane and out-of-plane band-gap properties of a two-dimensional triangular polymer-based void channel photonic crystal

The in-plane and out-of-plane band-gap properties of two-dimensional triangular void channel photonic crystals fabricated by femtosecond laser drilling in a solid polymer material were characterized for transverse electric (TE) and transverse magnetic (TM) polarization illumination. For a 24 layer structure stacked in the Γ–M direction, the fundamental stop gap resulted in the suppression of infrared transmission of as much as 96% for TE- and 85% for TM-polarized incident light. The midgap wavelength for the TM polarization was longer by 2.5% than that for the TE polarization. Increasing the angle of incidence for both the in-plane and out-of-plane cases shifted the stop gap to short wavelengths for both TE and TM polarizations. The experimental results allowed for the estimation of the cross section of void channels and the effective refractive index of the polymer after the fabrication.

Enhancement of visible second-harmonic generation in epitaxial GaN-based two-dimensional photonic crystal structures

Second-harmonic (SH) radiation generated in reflection is measured from the surface of a two-dimensional triangular photonic crystal in a GaN layer. A very large SH enhancement is observed when the incident radiation is resonant with a leaky photonic mode. The potential of second-harmonic generation as a tool for photonic band mapping is also envisaged. The extended transparency window of III-nitride wide band gap semiconductors coupled with large nonlinearities is an appealing feature pointing toward light control and manipulation in photonic structures.