Modes In One-dimensional Photonic Crystal Research Articles

Wannier-function methods for topological modes in one-dimensional photonic crystals

In this work, we use Wannier functions to analyze topological phase transitions in one dimensional photonic crystals. We first review the construction of exponentially localized Wannier functions in one dimension, and show how to numerically construct them for photonic systems. We then apply these tools to study a photonic analog of the Su-Schrieffer-Heeger model. We use photonic Wannier functions to construct a quantitatively accurate approximate model for the topological phase transition, and compute the localization of topological defect states. Finally, we discuss the implications of our work for the study of band representations for photonic crystals.

Physical factors affecting the propagation length of Bloch surface modes in one-dimensional photonic crystals

In this work we present a detailed study of the propagation length of Bloch modes along the surface between a bulk medium and a one-dimensional truncated photonic crystal and how this quantity is affected by different parameters like the number of periods composing the photonic crystal, absorption of materials, the leak of energy due to the intrinsic structure of the system, and the roughness of involved surfaces that is inherently present depending on the manufacturing method and the materials themselves. To analyze the influence of different factors affecting the propagation length, we use essentially three different methods: the characteristic matrix for systems composed of smooth surfaces, along with the numerical fitting of a Lorentz curve around the resonance; the boundary integral method based on the Green function for two-dimensional systems that include random rough surfaces; and the numerical analysis of the energy propagation along the surface when a Gaussian beam impinges on it. We have found that the propagation length can vary from a few nanometers to some millimeters depending on the different mentioned factors, along with the relative position of the modes within the bandgaps. Bloch modes located at the center of bandgaps propagate longer distances than those residing near the edges. Absorption and roughness provoke a drastic decreasing of the propagation length as they increase at small rates. Finally, the number of periods composing the structure also produces an increase of the propagation length reaching a maximum and then dropping almost suddenly to zero when it grows at a constant pace.

Topological edge modes in one-dimensional photonic crystals containing metal

Topological phases of matter has been developing rapidly in recent decades due to their unique topological edge states. In this paper, we analyze the surface modes of a one-dimensional periodic metal-vacuum multilayer structure and find that it can be seen as the optical analogy of the Su-Schrieffer-Heeger (SSH) model. There are two symmetric and antisymmetric edge modes, whose fields are mainly concentrated on the two outermost interfaces. By introducing off-diagonal perturbation to some layers, we find that the edge modes are topologically protected, that is, have good robustness. However, the surface modes in the one-dimensional structure are not completely consistent with the SSH model, especially in the number of extended modes and the existence conditions of edge modes. As some extended modes are missing in our model, it can improve the coupling efficiency between atom and edge modes. Our results not only provide a new platform for the study of robust topological edge modes, but also have potential applications in information transmission, power transfer, and so on.

Controlling the band structures and electromagnetic density of modes in one-dimensional photonic crystals with Lamb wave

We have modeled and investigated band structures and the electromagnetic density of modes in one-dimensional photonic crystal whose refractive index is modulated by the excitation of the fundamental symmetric mode of Lamb wave using the Wigner phase time approach and transfer matrix method. The Rayleigh–Lamb dispersion curves are plotted to select the proper thickness of each sub-layer of considered photonic crystal structure which is comprised of silica (SiO2) and rutile (TiO2) layers as sub-layers of unit cell. The propagation of Lamb wave into the considered structures produced phononic band gaps in gigahertz frequency for acoustic waves. Also, the propagation of Lamb wave into the considered structures causes compression in SiO2 and dilation in TiO2 or vice versa, which can further increase or decrease the refractive index of both layers. Thus, there can be three combinations of change in refractive indices: case 1 (both layers have an unperturbed refractive index), case 2 (maximum change of the refractive index in SiO2 and minimum change in the refractive layer of TiO2), and case 3 (minimum change of the refractive index in SiO2 and maximum change in the refractive layer of TiO2). The band gap structures of considered structures are plotted for both polarized electromagnetic waves.

Controlling the electromagnetic density of modes in one-dimensional photonic crystal with defect using acoustic wave

We have theoretically analyzed the effect of acoustic parameters on the electromagnetic density of modes (DOM) in a one-dimensional photonic crystal featuring a defect mode. The whole structure is modulated by a longitudinal acoustic wave. The transfer matrix method and Wigner time approach are used to calculate the transmittance and electromagnetic DOM, respectively. The results show that the electromagnetic DOM is dynamically controlled by the acoustic amplitude. A sharp peak in DOM is found at the defect site whose magnitude decreases with the increase in acoustic amplitude and shows periodic swing with fractional change in time period of acoustic wave. Furthermore, the result of polarization of electromagnetic wave is also discussed and it is found that the location of the defect mode can be modulated in a large frequency rage without changing the structure of the photonic crystal.

Electromagnetic density of modes in one-dimensional photonic crystals containing dispersive metamaterials

The electromagnetic density of modes (DOM) in a finite one-dimensional photonic crystal containing dispersive metamaterials is computed using Wigner’s time approach. The expression of DOM is derived using the transfer matrix method. Different structural parameters, such as relative thickness, incident angle, and total number of unit cells are varied and their effects are investigated. It is observed that relative thickness, angle of incidence, and total number of unit cells all play an important role in determining the properties of DOM depending on the frequency regime under consideration.

Elastic dependence of defect modes in one-dimensional photonic crystals with a cholesteric elastomer slab

We studied the transmission spectra in a one-dimensional dielectric multilayer photonic structure containing a cholesteric liquid crystal elastomer layer as a defect. For circularly polarized incident electromagnetic waves, we analyzed the optical defect modes induced in the band gap spectrum as a function of the incident angle and the axial strain applied along the same axis as the periodic medium. The physical parameters of the structure were chosen in such a way the photonic band gap of the cholesteric elastomer lies inside that of the multilayer. We found that, in addition to the defect modes associated with the thickness of the defect layer and the anisotropy of the elastic polymer, two new defect modes appear at both band edges of the cholesteric structure, whose amplitudes and spectral positions can be elastically tuned. Particularly, we showed that, at normal incidence, the defect modes shift toward the long-wavelength region with the strain; whereas, for constant elongation, such defects move toward larger frequencies with the incidence angle.

Wannier functions and the calculation of localized modes in one-dimensional photonic crystals

The localized modes of a layered photonic crystal with a substitutional defect are calculated by expanding the magnetic field over a truncated set of Wannier functions. Good agreement is found with the results obtained by the transfer-matrix technique. It is shown that frequencies converge very fast (slowly) as the number of unit cells (photonic bands) in the truncated set is increased.

Electrically Controlled Reflection Band and Tunable Defect Modes in One-Dimensional Photonic Crystal by Using Potassium Titanyl Phosphate (KTP) Crystal

Electrically Controlled Reflection Band and Tunable Defect Modes in One-Dimensional Photonic Crystal by Using Potassium Titanyl Phosphate (KTP) Crystal

Tunable photonic defect modes in one-dimensional photonic crystals containing exponentially and linearly graded index defect

We present an analytical study of the defect modes in one-dimensional (1-D) photonic crystals with defect layer of exponentially and linearly graded index materials. The relative refractive index in the defect layer varies continuously along the direction perpendicular to the surface of layer. The transmission spectra demonstrate that there exist two defect modes inside the photonic band gap for the quarter-wave layers stacking photonic crystal structures with a defect layer of graded index material. The grading profiles of the defect layer affect the frequency and intensity of photonic defect modes. The intensity, frequency and number of defect modes have been changed with the thickness and gradation parameter of graded index defect layer. Moreover, results show that different incident angle in TE and TM polarizations can also be changed the frequency and intensity of the defect modes remarkably. This defect mode splitting and tuning phenomenon in 1-D photonic crystals containing defects of graded index materials can be used to design filters, sensors, and other photonic devices.

Dispersionless gaps and cavity modes in photonic crystals containing hyperbolic metamaterials

We theoretically study dispersionless gaps and cavity modes in one-dimensional photonic crystals composed of hyperbolic metamaterials and dielectric. Bragg gaps in conventional all-dielectric photonic crystals are always dispersive because propagating phases in two kinds of dielectrics decrease with incident angle. Here, based on phase variation compensation between a hyperbolic metamaterial layer and an isotropic dielectric layer, the dispersion of the gap can be offset and thus a dispersionless gap can be realized. Moreover, the dispersionless property of such gap has a wide parameter space. The dispersionless gap can be used to realize a dispersionless cavity mode. The dispersionless gaps and cavity modes will possess significant applications for all-angle reflectors, high-$Q$ filters excited with finite-sized sources, and nonlinear wave mixing processes.

Transmission properties of defect modes in one-dimensional photonic crystals containing gradient refractive index defects

The gradient refractive index (GRIN) defect layer is introduced into one-dimensional photonic crystal (1D PC) to develop the optical filters needed. The transmission properties of 1D PC with GRIN defect layer were investigated based on the finite difference time domain (FDTD) method. Defect mode was obtained according to the GRIN distribution functions. With a GRIN structure in the defect layer, simulation results show that the position and width of the photonic band gap can be effectively modulated by varying the thickness and GRIN distribution functions of defect layer and the parameters of symmetrical layers. The new study provides a certain reference for photonic band gap engineering and designing the photonic-based devices.

Kossel diffraction and photonic modes in one-dimensional photonic crystal

Kossel diffraction under standing-wave excitation in a one-dimensional photonic crystal is investigated. It is shown that by combining the reciprocity theorem, the Fermi golden rule and the concept of density of photonic modes, it is possible to predict the behavior of the Kossel diffraction in such a system.

Convergence of the supercell method for computation of defect modes in one-dimensional photonic crystals

In this paper, we analyze the spectrum induced by the supercell method for studying locally defected one-dimensional photonic crystals. By using the propagation matrix method, we show that the continuous spectrum of the periodic structure of the supercell converges to that of the defected photonic crystal. Also, it can be shown that frequencies of localized defected modes in the bandgap are included in a narrow interval (whose length diminishes exponentially with increasing size of the supercell) of the continuous spectrum of the supercell method.

Properties of defect modes in one-dimensional photonic crystals that contain single negative materials

We examine the properties of the defect modes arising in the zero-effective phase (zero-φ ) gap when a dielectric defect layer is introduced in a one-dimensional photonic crystal containing single negative materials. We show that the defect modes inside the zero-φ gap can be as sensitive to the incidence angle as those inside the Bragg gap. In addition, adjusting the properties of the dielectric defect layer can control the frequencies and the number of defect modes. We present a brief design of a polarization-independent spatial filter based on those defects. The proposed spatial filter can work at dual carrier frequencies with low-pass and wide-pass characteristics. Our results can help improve the performance of microwave devices independent of the source wave polarization.

Properties of single and multiple defect modes in one-dimensional photonic crystals containing left-handed metamaterials

Wave propagation is studied in structures consisting of alternate left- and right-handed layers. Bragg gap and zero-n gap appear in different frequency regions of the structure. The periodicity of the structure is broken by simply reversing the order of the layers in one half of the structure, resulting in defect modes located inside the zero-n gap and Bragg gap. These modes can be made very narrow by adding more layers in the structure. The defect mode located inside the zero-n gap is sensitive to the symmetry of the structure and insensitive to the angle ofincidence of the incoming radiation. Multiple modes are also generated inside the gaps by repeating the structural pattern. Thus, a simple structure can be used for single and multiple modes that are important for different applications.

Band gaps and nonlinear defect modes in one-dimensional photonic crystals with anisotropic single-negative metamaterials

We propose a one-dimensional photonic crystal (1DPC) with multilayered periodic structures containing dispersive anisotropic single-negative (ASNG) (permittivity- or permeability-negative) metamaterials. The influences of the ratio of d A/ d B and their scale on ASNG gaps are discussed. Moreover, the effect of a nonlinear defect layer introduced into the proposed structure is also analyzed, and the results show that the defect mode has two peaks of electrical fields in the defect layer and its bistability property is also different from a defect mode in PCs with isotropic SNG gaps.

Properties of defect modes in one-dimensional photonic crystals containing two nonlinear defects

Transmission properties of one-dimensional photonic crystal with two nonlinear defects were investigated based on the nonlinear transfer matrix method. From the inspection of the optical bistabilities around different defect modes, it was found that the threshold of bistability for the bonding mode is always lower than that for the anti-bonding mode. When intensity of the incident wave increases bistability could happen at any wavelength larger than the resonance wavelength of the linear anti-bonding mode in the gap. The threshold of bistability increases initially with the wavelength, then decreases sharply in reaching the resonance wavelength of the linear bonding mode. The coupling behavior of the two defects was also discussed, and it was found that the threshold of bistability increases with the distance between the two defects. A flat spectrum similar to that formed in the photonic crystal molecules was obtained under certain incident intensity.

Electromagnetic surface modes in one-dimensional photonic crystals with dispersive metamaterials

We study the electromagnetic surface modes supported by the interface between semi-infinite one-dimensional photonic crystals composed of dispersive metamaterials and a homogeneous medium in the presence of the cap layer. A new type of surface mode is found in such structure, and these surface modes can exhibit strong resonance at the first period of the photonic crystal. These resonant surface modes originate from the material dispersion. Another type of surface mode with structural resonance can appear in the thick cap layer. The influences of physical parameters of the structure on the surface modes have also been investigated.

Characterization of defect modes in one-dimensional photonic crystals: An analytic approach

We introduce an analytical model to investigate the localized defect modes associated with a defect cell inserted in a one-dimensional photonic crystal for both transverse electric (TE) and transverse magnetic (TM) polarizations, at an arbitrary angle of incidence. The defect cell, which is made of two dielectric constituents, leads to the appearance of several localized defect modes within the photonic band gaps. We develop an analytical approach based on the both transfer matrix and Green's function methods to determine the number and frequency of the defect modes that can be controlled easily by varying the parameter values of the constituent layers of the photonic crystal. The method demonstrates that the electric field amplitude depends on the defect cell position in the photonic crystal while the defect mode frequency is independent of it. This method is useful for photonic band gap engineering and designing the photonic-based devices.