Defect Mode Frequency Research Articles

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.

Low Threshold DFB Lasing at the Edge and Defect Modes in Chiral Liquid Crystals

A brief survey of the recent experimental and theoretical results on the low threshold distributed feedback (DFB) lasing in chiral liquid crystals (CLC) as well as new original theoretical results in the field are presented. The presentation is concentrated on the lasing thresholds at frequencies corresponding to the stop band edge lasing modes and defect modes in CLC. It is demonstrated that the analytic approach reproduces all features of the DFB lasing in CLC obtained by the traditional numerical approach and, more over, allows to reveal some qualitative effects escaped from the researchers employing the numerical methods. Namely, the effect of anomalously strong absorption at the defect mode frequency, a direct analogue of the corresponding effect at the stop band edge mode frequency, is predicted in the framework of the analytic approach. It is proposed also to reduce the lasing threshold by adjusting the pumping wave to the conditions of the anomalously strong absorption effect at the defect mode frequency. The developed approach helps to clarify the physics of lasing at the stop band edge and defect modes in CLC and manifests an agreement with the corresponding results of the previous investigations obtained by a numerical approach.

Low Threshold DFB Lasing in Chiral Liquid Crystals

A brief survey of the recent experimental and theoretical results on the low threshold distributed feedback (DFB) lasing in chiral liquid crystals (CLC) as well as new original theoretical results in the field are presented. The presentation is concentrated on the lasing thresholds at frequencies corresponding to the stop band edge lasing modes and defect modes in CLC. It is demonstrated that the analytic approach reproduces all features of the DFB lasing in CLC obtained by the traditional numerical approach and, more over, allows to reveal some qualitative effects escaped from the researchers employing the numerical methods. Namely, the effect of anomalously strong absorption at the defect mode frequency, a direct analogue of the corresponding effect at the stop band edge mode frequency, is predicted in the framework of the analytic approach. It is proposed also to reduce the lasing threshold by adjusting the pumping wave to the conditions of the anomalously strong absorption effect at the defect mode frequency. The developed approach helps to clarify the physics of lasing at the stop band edge and defect modes in CLC and manifests an agreement with the corresponding results of the previous investigations obtained by a numerical approach.

Impurity effects on the band structure of one-dimensional photonic crystals: experiment and theory

We study the effects of single impurities on the transmission in microwave realizations of the photonic Kronig–Penney model, consisting of arrays of Teflon pieces alternating with air spacings in a microwave guide. As only the first propagating mode is considered, the system is essentially one-dimensional (1D) obeying the Helmholtz equation. We derive analytical closed form expressions from which the band structure, frequency of defect modes and band profiles can be determined. These agree very well with experimental data for all types of single defects considered (e.g. interstitial and substitutional) and show that our experimental set-up serves to explore some of the phenomena occurring in more sophisticated experiments. Conversely, based on the understanding provided by our formulae, information about the unknown impurity can be determined by simply observing certain features in the experimental data for the transmission. Further, our results are directly applicable to the closely related quantum 1D Kronig–Penney model.

Optical bistability in one-dimensional magnetic photonic crystal with two defect layers

One-dimensional magnetic photonic crystal with two magnetic defect layers is theoretically studied. The dependence of defect modes inside photonic band gap on the distance between defect layers is investigated. It is shown that the cubic nonlinear polarization leads to the effect of bistability on the frequencies of defect modes.

Impact of a stripe waveguide to coupled defect modes of photonic crystals

Coupled defect modes of photonic crystals can be characterized by the transmission spectrum shape that is closely linked to the stripe waveguide connecting the defects. Unlike the direct coupling cases where the phase shift of lightwave depends only on the defect mode frequency, we found that for the waveguide coupling cases the waveguide length and the dispersion relation must be also considered. By changing the waveguide length, the spectrum can exhibit periodically three-peak appearance, two-peak appearance, and flat top appearance. At frequencies close to the defect mode, the results of the coupled mode theory are in good agreement with the simulations using the finite-difference time-domain technique, verifying the validity of the theoretical model even in the complicated cases.

The defect modes in one-dimensional photonic crystal with multiple defects

A one-dimensional photonic crystal containing multilayered half-wave defects is designed and the transmission spectrum is studied by using transfer matrix theory. Under the first-order approximation of Taylor expansion, the analytic expression for the defect mode frequency is derived, from which the coupling factor is deduced. Our results are fitted with the experimental results or numerical simulation, and can be used to describe the characteristics of the defect modes. The result is helpful for designing a multiple channel filter.

Defect mode in periodic and quasiperiodic one-dimensional photonic structures

The properties of one-dimensional periodic and quasiperiodic photonic crystals with a defect layer have been investigated. Transfer matrix method (TMM) has been used throughout this study. For periodic photonic crystals, results demonstrate the independence of the defect mode frequency on the defect layer while the defect mode transmission coefficient varies with the position of the defect layer position. On the other hand, defect mode frequency is not that responsive to the index of refraction of the defect layer. The quality factor of the defect mode has been studied as a function of the defect layer position as well as its thickness. For quasiperiodic photonic crystals, the frequency of the defect mode is very sensitive to the defect layer position as well as its thickness. An enhancement of the quality factor of the defect mode has been observed. This study may be valuable in designing optical devices and it may also provide a more accurate method to measure the index of refraction.

Intrinsic Surface and Bulk Defect Modes in Quasi-Periodic Photonic Crystals

By analyzing resonances in the local density of states (LDOS) for a fivefold 2-D quasi-periodic photonic crystal (QPC), we reveal the presence of a well-defined number of defect modes in the spectral region of a low LDOS, which are intrinsic to this class of photonic structures. We may distinguish between surface modes, depending on the truncation of the finite-sized structure, and bulk modes. All defects can be traced to a configuration of closely spaced cylinders. Due to mutual coupling, their collective eigenresonance is strongly shifted into the spectral region of a low LDOS, and they appear as defect modes, which are comparable to acceptor modes in periodic photonic crystals. Control over the geometrical parameters of such resonators allows for a simple adjustment of the defect mode frequencies in QPCs.

Fast Calculation of Cavity-Mode Characteristics of Photonic Crystal Cavities

A fast approach based on effective index perturbation method is proposed to evaluate the intrinsic characteristics of photonic-crystal-slab-based microcavity with two-dimensional finite-difference time-domain (2-D FDTD) technique. For two widely used single defect structures, less than 2% computational error was obtained in calculating the defect mode frequencies. Accurate prediction of cavity modal properties and resonant peak frequencies is feasible based on 2-D FDTD simulation by adjusting the effective index to match the dielectric band edge for donor-like defect mode. The correlation between the modified effective index and the cavity (lasing) mode with the highest quality factor Q offers an efficient tool in the design of defect mode based photonic crystal microcavities

Third-harmonic generation in a one-dimensional photonic-crystal-based amorphous nanocavity

We report a large enhancement of third-harmonic generation in the one-dimensional dielectric photonic-crystal-based amorphous nanocavity. The enhancement took place at the frequency of defect mode and was a pure χ(3) nonlinear process. A great increase of third-harmonic intensity up to approximately five orders of magnitude was demonstrated, which was attributed to field localization in the range of nanocavity.

Band gap characteristics of compound structure composed of lattices with different periodic constants

Our numerical simulation results show that if a compound structure is composed of a lattice with a larger periodic constant and a lattice with a small periodic constant, the lattice with larger periodic constant will play the role of defects. The frequencies of defect modes are dependent on the rod radius (r2) of square lattice with a larger periodic constant, and the relationship of the defect modes and r2 is almost independent of a2. The defect mode is dependent on incidence angle when a22. Furthermore, by optimizing the size of the rods (r2) of the larger periodic lattice, single defect mode, multiple defect modes with equal or different frequency spacing, including single polarization or TE and TM overlap defect mode, can be gotten respectively. The transmissivity of the defect modes are inversely proportional to a2.

Derivation and characterization of dispersion of defect modes in photonic band gap from stacks of positive and negative index materials

An expression is derived to predict the incident angle dependence of the frequency of defect modes inside the zero- n ¯ gap of one-dimensional photonic crystals stacking with positive and negative index materials. From this expression, the dependence of the dispersion type of the defect mode on both the refractive index and the wave impedance of the component layers of the photonic crystal are given. For situations with a given value of permeability, it is found that, as the adjacent layers of the defect are low refractive index materials, only positive or negative type of angular dispersion exists; as the adjacent layers of the defect are high refractive index materials, all three types of angular dispersion, including the near-zero dispersion, appear with the refractive index of the defect changes. The dispersion relation diagrams of such photonic crystals are presented, the simulated results agree with the theoretical prediction.

Thermally tunable filter for terahertz range based on a one-dimensional photonic crystal with a defect

A high-quality smart filter for terahertz range with relative tunability reaching 20% has been demonstrated. The filter is based on a narrow transmission band, which originates from a defect mode that appears due to insertion of a single crystal of KTaO3 into otherwise periodic one-dimensional photonic crystal. Frequencies of defect modes are controlled by the refractive index of the defect: their high tunability is achieved by the strong temperature dependence of the dielectric properties of KTaO3. The low losses of KTaO3 lead to a high peak transmission of the filter. Influence of the defect losses on the properties of the filter is also discussed.

Tolerance of Photonic Crystal Impurity Bands to Disorder ofDefects in Coupled Cavity Waveguides

We investigate the tolerance of photonic crystal impurity bands to thedisorder of defects in one-dimensional coupled cavity waveguides.Although impurity bands formed by defect modes close to the air bandare quasiflat in the absence of disorder, they are easily deterioratedwhen disorders are introduced into defects. In contrast, impurity bandscreated by defect modes near the dielectric band are lesssensitive to disorder in the defect size. It is found that thesensitivity of defect mode frequency to defect size and the qualityfactor of defect modes are two crucial factors in determining thetolerance of impurity bands to the disorder of defects.

Defect modes caused by twinning in one-dimensional photonic crystals

Propagation of electromagnetic waves in a one-dimensional photonic crystal with a twin-defect—a periodicity break where one half of the photonic structure is a mirror image of the other one—is studied using a transfer-matrix method. This work is done in the general framework of photonic structures composed of isotropic materials exhibiting both dielectric and magnetic properties. Both polarizations of electromagnetic waves impinging at oblique incidence on the structure are considered. We derive analytical expressions for the frequency of defect modes and for the enhancement of the electromagnetic field inside the defect. In particular, we discuss possibilities of tuning of defect levels for a photonic crystal structure with a two-layer elementary cell.

Local density of states analysis of surface wave modes on truncated photonic crystal surfaces with nonlinear material

The local density of states and response to an incident plane wave of a finite sized photonic crystal (PC) with nonlinear material (NLM) is analyzed. Of particular interest is the excitation of surface wave modes at the truncated surface of the PC, which is collocated with the NLM material. We compute the 2D Green function of the PC with linear material and then include the Kerr NLM in a self-consistent manner. The 2D PC consists of a square array of circular rods where one row of the rods is semi-circular in order to move the surface wave defect mode frequency into the band gap. Since the surface modes are resonant at the interface, the NLM should experience at least an order of magnitude increase in field intensity. This is a possible means of increasing the efficiency of the PC as a frequency conversion device.

Magnetic Materials in Photonic Crystals

We studied magnetic materials in one-dimensional photonic crystals considering both dielectric constant and magnetic permeability, as an example. The analytical expressions are found not only for the edge frequencies of omnidirectional reflection band's (ORB's) but also for the equiomnidirectional contours of the ORB's. The ORB's can always exist in a periodic array of two alternating layers with different wave impedance although both layers have the same index of refraction. The wider ORB can be easily achieved by the use of magnetic permeability. The dielectric constant and the magnetic permeability have the similar effect on the frequency of defect modes.

Dispersive optical bistability in one-dimensional doped photonic band gap structures

We introduce a Kerr-type nonlinear defect layer in the center of one-dimensional photonic band gap structure. For linear parts, we use transfer matrix technique, and for the nonlinear layer, we solve the Maxwell equation numerically. When incident light intensity varies, a typical S-shape curve of the transmitted light intensity is obtained. In this case, an optical bistability is produced by dynamic shifting of the defect mode frequency, not dynamic shifting of the band edge, which is different from the case of periodic nonlinear superlattices. It is also found that when the linear defect mode frequency moves from the center of the gap to the edge of the gap, the threshold intensity needed for the bistability increases rapidly.

Defect Studies in a One-Dimensional Photonic Band Gap Structure

We calculate defect modes in a finite one-dimensional quarter-wavelength stack by the transfer matrix technique, and find that the defect mode frequency depends on the refractive index as well as on the thickness of the defect layer and the transmission coefficient depends on its position. Then, by using the extended-beam-propagation method, we investigate the short pulse transmission. It is found that the amplitude of the electromagnetic field in the defect layer is much larger than that of the incident light, which can be used to enhance nonlinear effects if the defect layer is replaced by a nonlinear material.