Bragg Reflection Band Research Articles

Optical FBG filter with minimum group delay for WDM systems

The causality principle is taken into account to couple amplitude and phase of the reflection coefficient in a symmetric grating for the optimization purpose. The distortion functional for group delay minimization is proposed. The minimization of mean square group delay within the Bragg reflection band is carried out at fixed integral reflectivity. The optimal reflection coefficient with minimum phase distortion is found analytically. The described approach is shown to give the close to rectangular FBG filter with low group delay variation over the reflection band.

Polarization-selective color-filter Fresnel lens in polymer-stabilized cholesteric liquid crystals

This study demonstrated a color-filter Fresnel lens in polymer-stabilized cholesteric liquid crystals (PSCLCs). The liquid crystals in the odd zones were in a planar cholesteric texture, the Bragg reflection band of which was located in the infrared region. The even zones had a focal conic texture. The linearly polarized white light passing through the planar texture underwent optical activity. The light with the polarization perpendicular to the transmission axis of an analyzer behind the PSCLC Fresnel lens was eliminated. Changing the orientation of the transmission axis of the analyzer can select the color of the focusing image.

Photonic effect in a hyper-swollen lyotropic lamellar phase

We studied the photonic effect in a hyper-swollen lyotropic lamellar phase. This system is composed of only a small percentage of amphiphilic molecules and more than 98% of water by weight; however, it shows clear Bragg reflection because of the periodicity in the order of the visible wavelength. By mixing a laser dye into the lamellar phase, we confirmed the emission enhancement near the higher energy side of the Bragg reflection band, although the effect is weak. We also discuss whether such a structure can exhibit the photonic effect using Berreman’s 4 × 4 matrix method.

Temperature-Dependent Behaviours of Blue Phase I Observed for a Bent-Core Molecular System

Temperature-dependent behaviours, such as the Bragg reflection band and the EO performance, of the liquid-crystalline BP I observed for a nematogenic achiral bent-core molecule doped with a small amount of chiral additive was investigated. Our experimental results indicate that the physical behaviours of our sample were highly sensitive to temperature changes.

Transmission and reflection phase gratings formed in azo-dye-doped chiral nematic liquid crystals

Transmission and reflection gratings were simultaneously formed in azo-dye-doped chiral nematic liquid crystals (N∗LCs) with planar alignment. The formation process is based on a phototuning of the Bragg reflection band of the N∗LC. The helical pitch of the photoreactive N∗LC was spatially controlled with intensity variation of interference light. The resultant periodic structure showed both transmissive and reflective diffractions due to the spatially modulated light intensities. The observed dependence of diffraction efficiencies on the polarization states of the probe beam was well explained by considering a spatial modulation of the helical pitch.

Influence of statistical errors in refractive index on fibre Bragg grating reflection

Averaged coupled wave equations are derived for a fibre Bragg grating with Gaussian noise added to the refractive index. The corrections to reflection and transmission coefficients are expressed in terms of a Green matrix of the unperturbed grating. The explicit formula for the Green matrix is derived for the uniform grating. The average reflectivity is calculated analytically. The noise is shown to increase reflectivity outside the Bragg reflection band and to smooth out the oscillations in the spectrum.

FBG filter with minimal dispersion

An optimal filter for super-fast fiber-optic communication lines should cause minimal phase distortions. The simplest case of Bragg grating with a symmetric profile is considered. A distortion functional is presented, which is the mean square group delay over the Bragg reflection band at a fixed integral reflectivity. The functional is minimized analytically in the simplest limiting case. Numerical calculations confirm that the optimal reflection spectrum is realizable.

Chiral Azobenzene Nematics Phototunable with a Green Laser Beam

Phototuning of Bragg reflection bandgap from green to blue and from red to green wavelengths was induced with a green (λ = 532 nm) laser beam in a cholesteric liquid crystal based on highly sensitive azobenzene nematic BEAM 1005 (BEAM Co.) and UV transparent non-photoisomerizable optically active dopant ZLI-6248 (R-2011) (Merck Ltd.). Optical switching between reflected and transmitted beams, as well as nonlinear transmission and reflection due to Bragg reflection band shift are studied in detail.

Reflection Band Control of Inverse Opal Film with Photoresponse Properties of Push-Pull Type Azobenzene LC Polymers

DcAz2Mc and DR1Mc molecules having push-pull type of azobenzene groups are synthesized, and their photo-orientational behaviors are investigated in the polymer system. In order to understand the relationship between a molecular structure and its physical properties, electronic structure calculation is computationally carried out. Regarding to their photo response properties, the copolymers ofpoly(DcAz2Mc-co-M6PBMe) andpoly(DR1Mc-co-M6PBMe) are infiltrated into inverse opal films, and Bragg reflection shifts are observed under photostimuli. When the linearly polarized light is irradiated, Bragg reflection bands are shifted to the longer wavelength region as reflective index increases.

NONLINEAR TRANSMISSION OF PHOTOSENSITIVE CHOLESTERIC LIQUID CRYSTALS DUE TO SPECTRAL BANDWIDTH AUTO-TUNING OR RESTORATION

Cholesteric liquid crystals (CLCs) which possess a periodicity in the visible portion of the spectra, exhibit selective reflection of circularly polarized light. The ability to modulate this color through a variety of means has been explored, including work which incorporated azobenzene LCs. Two types of systems have recently been explored utilizing wavelength-specific cis-trans isomerization processes, which enable unprecedented photosensitivity. The first system exhibits large blue or red-shifted changes in reflection wavelength upon visible irradiation. The second system exploits the metastable, long-lived photoinduced isotropic state, whose return to the reflective Grandjean texture can be induced by wavelength-specific radiation. We demonstrate nonlinear transmission from both types of systems, starting with submicrowatt power levels and spanning over four orders of magnitude dynamic range. The power dependence and temporal evolution of this effect (10–100 ms) is documented here for red or green laser wavelengths. The effect for the former case is due to bandgap auto-tuning, when the laser beam is tuning the CLC Bragg reflection band to its own wavelength. For the latter case, autonomous, optical feedback due to bandgap restoration is the cause of the nonlinear transmission properties.

Polarization inhibition of the stop-band in distributed Bragg reflectors

Bragg reflection spectra of high-contrast silicon-based 1D photonic crystals were measured for oblique incidence of light. The angular behaviour of the stop-band width has been investigated in the vicinity of the critical angle of incidence (an analogue of the Brewster angle) at which the stop-band for TM-mode disappears. The stop-band inhibition is experimentally observed against the background of pronounced Fabry–Perot interference due to the finite thickness of a photonic crystal slab. It is shown that the width of equidistant fringes arising from the interference limits the minimal measured Bragg reflection band width. The conditions for stop-band to collapse are considered theoretically and shown to be in good agreement with the experimental data presented.

Improved red color with cholesteric liquid crystals in Bragg reflection mode

The helical pitch of the cholesteric liquid crystal (CLC) can be adjusted to reflect the colors red, green, and blue. Additive mixing of these colors in displays results in multicolor images and it is easy to use pure primary blue and green colors, but the red color is in general very unsaturated. We show by simulations that this poor red color performance is due to reflection sidebands on the smaller wavelength side of the normal red Bragg reflection band. We discuss five approaches to improve the red color performance, namely, two types of spectral filtering (dyes or filters), a very low birefringence CLC, a gradient in the birefringence of the CLC, and the use of a bluish reflector. The two methods of spectral filtering are also experimentally tested.

Behavior of the cholesteric liquid-crystal fabry-perot cavity in the bragg reflection band

In this Letter, we investigate the behavior of the cholesteric liquid-crystal Fabry-Perot cavity in the spectral range of the Bragg reflection band. We show that in this region the 2x2-Jones-matrix calculation gives a false predication. However, by the use of the 4x4-matrix method, interesting features are revealed in the resonance spectra. The resonance spectra split when lambda approximately &nmacr;P. Our experiments confirm this phenomenon. The experimental results show that the splitting occurs in a steplike fashion. This can be explained by the quantization of the pitch that occurs due to the fixed boundary conditions.

Optical Bragg reflection from the TGBA phase

We have studied theoretically the reflection band properties of the twist grain boundary A phase (TGBA). A 4 x 4 matrix approach for the electromagnetic wave propagation is used. At normal incidence, simple analytical expressions are found, while at oblique incidence numerical methods have to be used since no analytical solution is available. The chart of stability of the propagating waves and the reflection spectra are given and discussed. No evidence of the incommensurability properties of the TGBA phases can be found optically. However, the analysis of the output polarization within the Bragg reflection band, on thick enough samples, allows us to determine the physical parameters characterizing the TGBA phase. In particular conditions, we can get the rotation Delta Theta between two homogeneous slabs and the total optical phase phi of one slab. When Delta Theta and phi go to zero, the optical properties of the TGBA phase become identical to those of the cholesteric phase. The essential difference appears when Delta Theta and phi are increased; an infinite number of reflection bands forms each Bragg reflection order, due to the periodicity of the TGBA parameters.

Nonlinear Optical Second Harmonic Generation in Ferroelectric SmC* at Noncollinear Geometry

Second harmonic generation in helically twisted SmC* liquid crystal is analysed for oblique light propagation with respect to the helical axis. Numerical results for the harmonic wave close to the Bragg reflection band show strong resonant enhacement of the second harmonic light intensity.

Second harmonic generation in a photonic crystal

Phase matched second harmonic generation is observed experimentally in a centrosymmetric crystalline lattice of dielectric spheres of optical dimensions. The inversion symmetry is broken locally at the surface of each sphere in such a way that the scattered second harmonic light interferes constructively leading to a nonvanishing macroscopic field. Phase matching of the fundamental and second harmonic waves in such periodic lattice is observed to be naturally provided by the bending of the photon dispersion curve at the edge of the Bragg reflection band of a given set of lattice planes.

A single-wavelength DFB structure with a synchronized gain profile

A distributed-feedback structure having a predominant gain at an edge wavelength of the Bragg reflection band is proposed and analytical results are presented. The structure has both refractive-index and gain-loss profiles with the same period. Since the gains per period at two edge wavelengths of this structure is substantially different, the selective amplification at the single wavelength is less sensitive to both fabrication accuracy and matching conditions than other single-wavelength schemes which utilize refractive-index profile alone.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Many-wave approximations for light propagation in cholesteric liquid crystals.

The eigensolutions of the Maxwell equations for light propagation in a perfect cholesteric liquid crystal are considered. They are Bloch waves having, in principle, infinitely many Fourier components. The relative amplitudes of these components are studied as a function of the angular frequency \ensuremath{\omega} and of the incidence angle ${\ensuremath{\theta}}_{i}$, in order to determine the conditions under which the actual eigensolutions can be suitably approximated by a superposition of only a few plane waves. When these conditions are satisfied, the wave vectors and the amplitudes of the components are given by simple analytical expressions. The main results are the following: (a) Far from the Bragg reflection bands, only one or two plane-wave components need be retained. Regions of the (\ensuremath{\omega},${\ensuremath{\theta}}_{i}$) plane are found, in which a very good approximation is represented by a linearly polarized wave (\ensuremath{\pi}- and \ensuremath{\sigma}- polarized-wave approximation) or by a nearly circular wave (quasinormal approximation). This allows selecting the optical conditions for which practically a single plane wave is excited in a cholesteric sample, a fact which is very important when dielectric inhomogeneities due to thermal fluctuations or defects and Brownian motions or fluorescence of guest molecules are studied. (b) Simple dispersion relations are obtained, which give the correct values of both real and imaginary parts of the wave vectors near the first-order reflection band. (c) It is shown that for the higher-order reflection bands, reasonably simple analytical expressions can be obtained, which give good qualitative results but which are not useful for a quantitative analysis. However, an exact dispersion relation simpler than the ones given in the literature is found. In all these cases, each approximation is discussed in detail. The approximate expressions found are in general simpler and more accurate than the ones previously given for similar cases. Their limits of validity for any practical purposes are found. The ensemble of the regions of validity of given approximations covers the largest portion of the (\ensuremath{\omega},${\ensuremath{\theta}}_{i}$) plane.

Angular Dependence of Selective Reflection from the Chiral Smectic C Phase

Abstract Transmission spectra were measured with obliquely incident light for a chiral smectic C phase of (+)-4-n-hexyloxyphenyl 4−(2″-methyl-butyl)biphenyl 4′-carboxylate. Extra reflection bands were observed at about twice the wavelengths of the “normal” Bragg reflection bands, which had been previously predicted by the computor calculation by Berreman.

An Experimental Study of the Anomalous Transmission (Borrmann Effect) in Absorbing Cholesteric Liquid Crystals

Abstract It has been shown previously that in analogy with the Borrmann effect in X-ray diffraction from crystals there is an anomalous increase in the transmitted intensity near the Bragg reflection band in an absorbing cholesteric liquid crystal. In this communication detailed experimental studies of this effect carried out on thin films of mixtures of cholesteryl nonanoate and p-azoxyanisole of different concentrations are reported. Numerical calculations based on the dynamical theory of reflection are also presented. The theoretical curves of circular dichroism versus wavelength are in qualitative agreement with the experimentally observed features.