Localized modes in optics of photonic liquid crystals with local anisotropy of absorption

The localized optical modes in spiral photonic liquid crystals are theoretically studied for the certainty at the example of chiral liquid crystals (CLCs) for the case of CLC with an anisotropic local absorption. The model adopted here (absence of dielectric interfaces in the structures under investigation) makes it possible to get rid of mixing of polarizations on the surfaces of the CLC layer and of the defect structure and to reduce the corresponding equations to only the equations for light with polarization diffracting in the CLC. The dispersion equations determining connection of the edge mode (EM) and defect mode (DM) frequencies with the CLC layer parameters (anisotropy of local absorption, CLC order parameter) and other parameters of the DMS are obtained. Analytic expressions for the transmission and reflection coefficients of CLC layer and DMS for the case of CLC with an anisotropic local absorption are presented and analyzed. It is shown that the CLC layers with locally anisotropic absorption reduce the EM and DM lifetimes (and increase the lasing threshold) in the way different from the case of CLC with an isotropic local absorption. Due to the Borrmann effect revealing of which is different at the opposite stop-band edges in the case of CLC layers with an anisotropic local absorption the EM life-times for the EM frequencies at the opposite stop-bands edges may be significantly different. The options of experimental observations of the theoretically revealed phenomena are briefly discussed.     

Optical defect modes at an active defect layer in photonic liquid crystals

An analytic approach to the theory of the optical defect modes in photonic liquid crystals in the case of an active defect layer is developed. The analytic study is facilitated by the choice of the problem parameters related to the dielectric properties of the studied structures. The chosen models allow eliminating polarization mixing at the external surfaces of the studied structures. The dispersion equations determining the relation of the defect mode (DM) frequency to the dielectric characteristics of an isotropic, birefringent and absorbing (amplifying) defect layer and its thickness are obtained. Analytic expressions for the transmission and reflection coefficients of the defect mode structure (DMS) (photonic liquid crystal-active defect layer-photonic liquid crystal) are presented and analyzed. The effect of anomalously strong light absorption at the defect mode frequency for an absorbing defect layer is discussed. It is shown that in a distributed feed-back lasing at the DMS with an amplifying defect layer, adjusting the lasing frequency to the DM frequency results in a significant decrease in the lasing threshold and the threshold gain decreases as the defect layer thickness increases. It is found that, generally speaking, the layer birefringence and dielectric jumps at the interfaces of the defect layer and photonic liquid crystal reduce the DM lifetime in comparison with the DMS with an isotropic defect layer without dielectric jumps at the interfaces. Correspondingly, generally speaking, the effect of anomalously strong light absorption at the defect mode frequency and the decrease in the lasing threshold are not so pronounced as in the case of the DMS with an isotropic defect layer without dielectric jumps at the interfaces. The case of a DMS with a low defect layer birefringence and sufficiently large dielectric jumps are studied in detail. The options of effectively influencing the DM parameters by changing the defect layer dielectric properties, and the birefringence in particular, are discussed.     

Defect modes of chiral photonic crystals with an isotropic defect

Specific features of the defect modes of cholesteric liquid crystals (CLCs) with an isotropic defect, as well as their photonic density of states, Q factor, and emission, have been investigated. The effect of the thicknesses of the defect layer and the system as a whole, the position of the defect layer, and the dielectric boundaries on the features of the defect modes have been analyzed. It is shown that when the CLC layer is thin the density of states and emission intensity are maximum for the defect mode, whereas when the CLC layer is thick, these peaks are observed at the edges of the photonic band gap. Similarly, when the gain is low, the density of states and emission intensity are maximum for the defect mode, whereas at high gains these peaks are also observed at the edges of the photonic band gap. The possibilities of low-threshold lasing and obtaining high-Q microcavities have been investigated.     

Optical edge modes in photonic liquid crystals

An analytic theory of localized edge modes in chiral liquid crystals (CLCs) is developed. Equations determining the edge-mode frequencies are found and analytically solved in the case of low decaying modes and are solved numerically for the problem parameter values typical for the experiment. The discrete edge-mode frequencies specified by the integer numbers n are located close to the stop-band edge frequencies outside the band. The expressions for the spatial distribution of the n’s mode field in a CLC layer and for its temporal decay are presented. The possibilities of a reduction of the lasing threshold due to the anomalously strong absorption effect are theoretically investigated for a distributed feedback lasing in CLCs. It is shown that a minimum of the threshold pumping wave intensity may be reached, generally, for the pumping wave propagating at an angle to the helical axis. However, for lucky values of the related parameters, it may be reached for the pumping wave propagating along the helical axis. The lowest threshold pumping wave intensity occurs for the lasing at the first low-frequency band-edge lasing mode and the pumping wave propagating at an angle to the spiral axis corresponding to the first angular absorption maximum of the anomalously strong absorption effect at the high-frequency edge of the stop band. The study is performed in the case of the average dielectric constant of the liquid crystal coinciding with the dielectric constant of the ambient material. Numerical calculations of the distributed feedback lasing threshold at the edge-mode frequencies are performed for typical values of the relevant parameters.     

Zone structure and polarization properties of the stack of a metamaterial-based cholesteric liquid crystal and isotropic medium layers

The optical properties of a stack of metamaterial-based cholesteric liquid crystal (CLC) layers and isotropic medium layers are investigated. The problem is solved by a modification of Ambartsumian’s layer addition method. CLCs with two types of chiral nihility are defined. The peculiarities of the reflection spectra of this system are investigated and it is shown that the reflection spectra of the stacks of CLC layers of these two types differ from each other. Besides, in contrast to the single CLC layer case, these systems have multiple photonic band gaps. There are two types of such gaps: those selective with respect to polarization of the incident light and nonselective ones. It is shown that the system eigenpolarizations mainly coincide with the quasi-orthogonal, quasi-circular polarizations for normally incident light, except the regions of diffraction reflection selective with respect to the polarization of incident light. The influence of the CLC sublayer thick-nesses, the incidence angle, the local dielectric (magnetic) anisotropy of the CLC layers, and the refractive indices and thicknesses of the isotropic media layers on the reflection spectra and other optical characteristics of the system is investigated.     

Dynamical dielectric susceptibility of ferroelectric thin films and multilayers

The thickness dependence of the real and imaginary parts of the dynamical dielectric susceptibility is investigated phenomenologically for a multilayer structure consisting of alternating ferroelectric and paraelectric layers. It is shown that the frequency dependence of the linear dielectric response can be closely approximated by that of a damped harmonic oscillator, with the static susceptibility, relaxation time, and soft-mode frequency depending on the layer thickness and temperature. When the layer thickness and temperature are equal to their critical values corresponding to the onset of a size-driven ferroelectric phase transition, the static susceptibility and the relaxation time become anomalously large and then decrease with further increasing layer thickness. A spectrum of natural polarization oscillations is predicted to exist with thickness-dependent frequencies. This spectrum includes a soft-mode frequency which vanishes at the critical thickness and at the critical temperature. The frequency spectrum lies below the soft-mode frequency of a thick film (in which the gradient of polarization is negligible). The calculations are compared with experimentally measured dispersion of the dielectric response of a PbTiO₃-Pb_0.72La_0.28TiO₃ multilayer structure. The agreement between the theory and experiment is found to be good.     

Optical properties of a stack of layers of a cholesteric liquid crystal and an isotropic medium

Optical properties of a stack consisting of layers of a cholesteric liquid crystal (CLC) and an isotropic medium are investigated. The problem is solved using the modified Ambartsumyan layer-summation method. Particular features of reflectance spectra of this system are studied. It is shown that, in contrast to a single CLC layer, this system exhibits multiple photonic band gaps. There are two types of photonic band gaps: selective and nonselective with respect to polarization of the incident light. It is shown that eigenpolarizations in the system generally coincide with quasi-orthogonal quasi-circular polarizations, except for regions that are selective with respect to polarization of the diffractive reflection. It is shown that, for an even number of layers, the system under consideration is nonreciprocal and can function as an optical diode. The influence of thickness of CLC sublayers, angle of incidence, local dielectric anisotropy of CLC layers, refractive indices and thicknesses of layers of an isotropic media on reflectance spectra and other optical characteristics of the system is investigated.     

Optical properties of a stack of cholesteric liquid crystal and isotropic medium layers

Some new optical properties of a stack consisting of cholesteric liquid crystal (CLC) and isotropic medium layers are studied. The problem is solved by the modified Ambartsumyan method for the summation of layers. Bragg conditions for the photonic band gaps of the proposed system are presented. It is shown that the choice of proper sublayer parameters can be used to control the band structure of the system. In the general case, the effect of full suppression of absorption, which is observed in a finite homogeneous CLC layer, is not detected in the presence of anisotropic absorption in CLC sublayers. It is shown that this effect can be generated in the system under study if certain conditions are imposed on the isotropic sublayer thickness. Under these conditions, the maximum photonic density of states (PDS) increases significantly at the boundaries of the corresponding band. The influence of a change in the CLC sublayer thickness and the system thickness on PDS is investigated.     

Cholesteric liquid crystals with tunable defect inside as polarization azimuth rotation amplifiers and polarization azimuth stabilizers

In the present paper, we investigate the polarization properties of the cholesteric liquid crystals (CLCs) with an isotropic/anisotropic defect inside them, and possibilities of amplification of the polarization plane rotation and stabilization of the light polarization azimuth by these systems are investigated.     

On the quasi-soliton propagation of few-cycle optical pulses in isotropic dielectrics

A new combined approach for constructing approximate soliton-like solutions of nonlinear wave equations is proposed. The approach includes the method of analytic continuation of dispersion parameters to the complex plane and the averaged variational principle of the Ritz-Whitham type. Based on this approach, the solution of the nonlinear equation describing the propagation of an optical pulse in a transparent isotropic dielectric is found. The obtained solution involves both envelope solitons and breather-like pulses with duration down to one period of electromagnetic oscillations.     

Eigenmodes of index-modulated layers with lateral PMLs

Maxwell equations are solved in a layer comprising a finite number of homogeneous isotropic dielectric regions ended by anisotropic perfectly matched layers (PMLs). The boundary-value problem is solved and the dispersion relation inside the PML is derived. The general expression of the eigenvalues equation for an arbitrary number of regions in each layer is obtained, and both polarization modes are considered. The modal functions of a single layer ended by PMLs are found, and their orthogonality relation is derived. The present method is useful to simulate scattering problems from dielectric objects as well as propagation in planar slab waveguides. Its potential to deal with more complex problems such as the scattering from an object with arbitrary cross section in open space using the multilayer modal method is briefly discussed.     

Reflection chromaticity of cholesteric liquid crystals with sandwiched periodical isotropic defect layers

We study one-dimensional photonic crystals made of cholesteric liquid crystals with sandwiched isotropic defect layers. Based on the Berreman Fast 4 × 4 matrix method, the dispersion relation of one-dimensional photonic crystals is calculated and the corresponding reflection chromaticity is obtained. It is found that the color shift could be controlled by adjusting the thickness and refractive index of the isotropic defect layers. Compared with conventional structures, the reflection chromaticity of this structure is insensitive to the incident angle, if the thickness ratio of the cholesteric liquid crystals to that of the isotropic defect layers and the refractive index of periodical isotropic defect layers are properly set. Furthermore, the common forbidden bands for both left and right circular polarizations can be obtained, and we also take the wavelength-dependent refractive indices into consideration and obtain the reflected light chromaticity with the incident angle increasing. The proposed device can be used as a reflective color filter in the display industry.     

Exact analytical expressions for the dispersion relation of one-dimensional chiral photonic crystals

We consider one-dimensional photonic crystals made of alternating layers of two kinds of isotropic chiral media. The two layers are characterized by the dielectric permittivities ε₁ and ε₂, the magnetic permeabilities μ₁ and μ₂, the chirality parameters γ₁ and γ₂ and the thicknesses d₁ and d₂. By diagonalizing an 8 × 8 matrix derived from the two coupled wave equations satisfied by our system, we obtain exact analytical expressions for the dispersion relation for plane electromagnetic waves propagating at an arbitrary angle with respect to the layers. We present a detailed analysis of our dispersion relation and explore the evolution of the photonic band structure as the frequency, the chirality parameter and the incident angle vary, with a special emphasis on the generation of the cross- and co-polarization bandgaps.     

Polarization plane's weak rotation amplifiers and polarization azimuth stabilizers

A simple and effective method to measure an electromagnetic wave polarization plane's weak rotations in various media is proposed. The specific features of the polarization plane's rotation amplification for the light reflection and transmission through the absorbing and amplifying isotropic layers are calculated. The amplification effects on an anisotropic and cholesteric liquid crystal (CLC) layers are also considered. In conclusion the question of probable choice of the amplifier's noise/signal ratio is discussed.     

Distributed feedback laser with optoelectronic tunability in dye-doped cholesteric liquid crystal with coated photoconductive layer

This work investigates, for the first time, an optoelectronically tunable distributed feedback laser that is based on a planar DDCLC cell with a coated photoconductive (PC) layer. Experimental results show that the CLC reflection band and the lasing wavelength of the DDCLC can both be tuned optoelectronically by varying the intensity of one irradiating CW green beam or the magnitude of the applied dc voltage. The tunability of the DDCLC laser depends on the controllability of the optoelectronic properties of the PC layer and, therefore, on the voltage dropping on the CLC layer. Therefore, the CLC pitch can be controlled by exploiting the optoelectronically induced electrohydrodynamic effect which causes the spatially periodic deformation of the CLC structure. In addition, the dependences for other critical lasing parameters, e.g., energy threshold, lasing efficiency, and lasing linewidth, on external controlling signals are also measured and discussed in the current study.     

Coaxial configuration of the dielectric Cherenkov maser

The linearized Lorentz force, continuity equation, and Maxwell's equations are used to calculate the system dispersion relation for a coaxial configuration of the dielectric Cherenkov maser. The system consists of two coaxial conductors lined with dielectric and an annular relativistic electron beam, which propagates between the two liners. The dispersion relation for the beam and dielectric-lined coaxial waveguide structure and the no-beam system that describes the dependence of the generated frequency on the coaxial waveguide parameters are presented. Using the linearized dispersion relation, the growth rate for the beam-TM_0n waveguide mode instability is calculated in the strong-coupling tenuous beam limit     

Optical properties of Wurtzite-type semiconducting crystals near the isotropic point

Summary Using Stahl's coherent wave approach we calculated the linear interband susceptibility near the band gap. The starting point was the ?MGM? formula with a smeared-out dipole densityM(r) and the Green's functionG for the effective-mass Hamiltonian with a screened Coulomb potential. The susceptibility so obtained was used for calculating transmission spectra of CdS platelets for energies near the so-called isotropic point lying far below the nearest excitonic resonance. By fitting experimental spectra we determined components of the residual dielectric tensor for polarization of the incident beam perpendicular and parallel to thec-axis, the mixed-mode dipole moment corresponding toB-exciton, and decay radii characterizing transition dipole densities. The established set of parameters determines the dielectric tensor for a quite wide energetic range.     

Properties of omnidirectional photonic band gap in one-dimensional staggered plasma photonic crystals

In this paper, the properties of the omnidirectional photonic band gap (OBG) realized by one-dimensional (1D) photonic crystals (PCs) with a staggered structure which is composed of plasma and isotropic dielectric layer have been theoretically studied by the transfer matrix method (TMM). From the numerical results, it has been shown that such OBG is insensitive to the incident angle and the polarization of electromagnetic wave (EM wave), and the frequency range and central frequency of OBG can be effectively controlled by adjusting the plasma frequency, the average thickness of plasma layer, the average thickness of dielectric layer and staggered parameters, respectively. The frequency range of OBG can be notably enlarged with increasing the plasma frequency, average thickness of plasma layer, respectively. Moreover, the bandwidth of OBG can be narrowed with increasing the average thickness of dielectric layer. Changing staggered parameters of dielectric and plasma layer means that the OBG can be tuned. It is shown that 1D plasma dielectric photonic crystals (PPCs) with such staggered structure have a superior feature in the enhancement of frequency range of OBG compared with the conventional 1D binary PPCs. This kind of OBG has potential applications in filters, microcavities, and fibers, etc.     

Plasmon modes of a superlattice — Classical vs quantum limits

The collective plasmon excitations of a superlattice are investigated in both the classical and quantum limits. Using a model that is applicable to superlattices whose constituent layers are either semiconductor- semiconductor, semiconductor-metal, or metal-metal, we show that the surface plasmon interface modes of each layer (slab) couple via the long range Coulomb interaction into two bands of plasmons with dispersion along the superlattice axis. Results for plasmon dispersion are presented for the classical limit (de Broglie wavelength less than the layer width) where the response is treated via a solution of Maxwell's equations using the bulk 3-D dielectric constant to describe each intervening layer. These results are compared to the plasmon dispersion in the quantum regime where the wave-vector frequency dependent dielectric constant of the superlattice is calculated taking into account quantization effects (subband structure). The relationship between the modes in both limits is derived.     

An eigenvalue method to study the threshold behaviors of plasmonic nano-lasers

An eigenvalue method is proposed to study the threshold behaviors of plasmonic nano-lasers. The medium gain and dispersion are taken into consideration based on semi-classical laser dynamics, and therefore the lasing threshold, mode pattern, and lasing frequency can be theoretically predicted. The lasing properties of dielectric, plasmonic core, and plasmonic shell nano-lasers are investigated in details. It is found that the lasing thresholds of nano-lasers can be reduced by two orders of magnitude when introducing localized surface plasmon modes.