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Liquid crystal substrates have been shown to provide the means to develop low-cost, reconfigurable, adaptive and tuneable microwave devices for mobile and wireless communication systems. In order to take maximum advantage of the possibilities that these materials offer and to design LC-based devices appropriately, techniques for the characterisation of the liquid crystal dielectric properties are needed. Similarly, appropriate modelling methods are required to simulate accurately the switching behaviour of the liquid crystal and the characteristics of the wave propagation through the devices, taking full consideration of the point-by-point variation of the material tensor permittivity.

Research on reducing material absorption in Terahertz (THz) waveguides has lead to development of guiding structures with transmission losses as low as 1 dB/m. Among waveguides that exhibit low loss at THz frequencies are the dielectric-lined hollow cylindrical metallic waveguides. Loss reduction in this waveguide is attributed to an ideal profile of the dominant hybrid HE11 mode. This mode profile also results in relatively low dispersion and very high coupling efficiency. In this contribution we overview properties of dielectric-lined hollow cylindrical metallic waveguides for THz waves, their design principles and the fabrication process. The impact of the mode profile on losses and dispersion at THz frequencies is confirmed experimentally by THz near-field imaging and THz time-domain spectroscopy and numerically by the finite element method.

A finite-element beam propagation method is applied to the optical analysis of small pixel liquid crystal (LC) (micro)displays. It is shown that the near-field profile is affected in these devices by diffraction, even for the ideal case of homogeneous pixels. The influence of the pixel size on the diffraction profile is shown and a fundamental contrast limit is demonstrated. The importance of LC materials with high birefringence is confirmed for small pixel LC devices as these improve the contrast for a fixed pixel size. Simulation of the pixel walk-off phenomenon in gray pixels due to anisotropy in the longitudinal direction illustrates the importance of including the full dielectric tensor in optical modeling of small pixel devices.

A new technique for the measurement of dielectric constants of nematic liquid-crystal (LC) materials at millimeter-wave frequencies is presented. The proposed method utilizes an electric field to align the LC molecules, therefore offering safety and enormous size reduction over the method using the magnetic field to achieve LC directors' orientation. The measurement device is planar and consists of a rectangular patch resonator with the LC cell directly beneath it. Two preconditioned surface preparations, corresponding to transverse and longitudinal rubbing of the patch surface and ground plane in contact with LC, are investigated. Using the same measurement device with the two preconditioned surfaces, the measurements of dielectric constants and dielectric constant anisotropy of a nematic LC commonly known as E7 are conducted, and the results are found to agree to within 1% while confirming earlier published data.

It has been recently experimentally demonstrated that propagation and polarization properties of the photonic liquid crystal fibers can be effectively tuned with the electric field. In particular, effective electric tuning of the phase birefringence has been obtained in the photonic liquid crystal fibers based on the high index glasses. Accurate numerical simulations of the impact of electric field on the guiding properties of the photonic liquid crystal fibers require complex methods, in which all important physical properties of the liquid crystal are taken into account (optical anisotropy, molecular orientation and relatively high losses). In this paper we present two different numerical approaches based on the finite element method. First one utilizes the simplified assumption of the "collective tilt" of molecules, and gives rough estimation of the phase birefringence tuning range. The second approach is much more rigorous, since electrically induced reorientation of the liquid crystal is calculated with a Q-tensor method giving the values of the inhomogeneous permittivity tensor. The value of the electrically induced birefringence calculated with the second method is in a good agreement with the experimental results.

Liquid crystals are attractive materials for use in millimeter-wave band communications devices due to their large birefringence and reconfigurability. However, characterizing these materials is challenging at these frequencies. This work describes the modeling tools that have been developed for the accurate analysis of the liquid crystal orientation and of the wave propagation through the inhomogeneous and anisotropic liquid crystal layer. It also shows how the modeling has been used in conjunction with experimental measurement to characterize liquid crystal materials over a wide frequency range.

The accurate analysis of optical waveguides is an important issue when designing devices for optical communication. Waveguides combined with liquid crystals have great potential because they allow waveguide tuning over a wide range using low voltages. In this paper, we present calculations that combine an advanced algorithm for calculating liquid crystal behavior and a finite-element mode solver that is able to incorporate the full anisotropy of the materials. Calculation examples demonstrate the validity of our program.

A new transflective nematic liquid crystal display is presented. Unlike other transflective cells, almost the entire device area is potentially available for either transmission of backlighting or reflection of ambient illumination, providing efficient use of available light for the desired application. The single cell-gap device does not require patterned alignment layers or a patterned retarder but employs planar and interdigitated electrodes. The number of electrical connections required makes the cell most suitable for low information content displays. Simulations are used to make comparisons with existing dual-gap and single-gap patterned alignment designs.

The operation of a post-aligned bistable nematic (PABN) liquid crystal device is simulated using a dynamic finite-element implementation of the Landau-de Gennes theory in 3D. Two topologically distinct stable director profiles of higher and lower tilt angles are identified. The switching process between these two states is simulated while monitoring the variation in total free energy of the liquid crystal material. Maxima in the free energy, acting as energy barriers separating the two stable states, are shown to coincide with the formation of -1/2 defect lines in the director field.

The anisotropic anchoring effect of a treated solid surface on a nematic liquid crystal is described in the Landau-de Gennes theory using a power expansion on the tensor-order parameter and two mutually orthogonal unit vectors. The expression has three degrees of freedom, allowing for independent assignment of polar and azimuthal anchoring strengths and a preferred value of the surface-order parameter. It is shown that in the limit for a uniaxial constant-order parameter, the expression simplifies to the anisotropic generalization of the Rapini-Papoular anchoring energy density proposed by Zhao et al. Experimentally measurable values with a physical meaning in the Oseen-Frank theory can be scaled and assigned to the scalar coefficients of the tensor-order- parameter expansion. Results of numerical experiments comparing the anchoring according to the study of Zhao et al. in the Oseen-Frank theory and the power expansion in the Landau-de Gennes theory are presented and shown to agree well.

The coupled interaction between the reorientation of a liquid crystal and the migration of ionic impurities in the presence of an applied electric field is investigated for two device types commonly used in displays, the planar and the in-plane switching (IPS) cell. Simulated optical measurements and transient currents are compared to experimental results in order to discover means to alleviate the problems associated with ion migration. IPS cells have a lower electrode surface area than planar cells, and are therefore more sensitive to the movement of a given ion concentration. As a means to reduce this sensitivity, the effect of altering the electrode width and gap is studied

A novel type of liquid crystal device is described, based on a four-electrode unit, arranged in a hexagonal array. Full three-dimensional simulations were performed using a finite elements algorithm demonstrating a 2 pi rotation of the director in the plane parallel to the substrate surface. Applications for the device are situated in the field of multistable wave plates, spatial light modulators and electrically controllable anchoring.

A programmable LCOS reflective spatial light modulator using an array of liquid crystal pixels used as a phase modulating device is studied. Modelling has been performed to determine the pattern of electrode voltages required to achieve a blazed grating with a linear phase ramp and an abrupt phase transition. As two-dimensional steering of incoming wavefronts is required, oblique gratings are also considered and the effect of the angle of the pixel grid with respect to the phase front of the grating is also considered. Here we study this effect using two possible configurations of the director pre-twist.

We present a new formulation for the dynamic modeling of twisted nematic liquid crystal display cells based on a constant charge assumption instead of the commonly used "fixed voltage" model. This formulation is based on a tensor representation of the director field, which is solved using a combination of finite-element and finite-difference methods. It allows the accurate modeling of liquid crystal pixel cells in thin-film-transistor-driven active matrix displays, extensively used in flat panel computer and television displays.

A computer modelling method for the three-dimensional dynamic analysis of liquid crystal display cells is presented. The method is based on a variational approach to the Oseen-Frank free energy formulation considering three elastic constants and uses a vectorial representation of the director field. A simpler approximate form that uses only two elastic constants but is faster to operate is obtained as a by-product of the full three-constant model implementation. The modelling uses finite elements on a mesh of tetrahedral elements for the calculation of directors and electric potentials while finite differences in time are used in the time stepping process. Comparisons are made with results from a 2D dynamic program using a tensor model giving good agreement.

Liquid Crystals have large electro-optic coefficients which make them suitable materials for a number of elements of 3D displays. This paper considers some of the issues relating to these elements, including the polarization modulation and phase modulation that can be obtained with liquid crystal elements as well as the resolution that can be achieved using liquid crystal layers. In particular the use of ferroelectric liquid crystal fast shutters will be discussed.

Induced bulk orientation of nematic liquid crystal in contact with micron-scale patterned surfaces is investigated using the Landau-de Gennes theory by means of three-dimensional simulations. The effect of the size and spacing of square cross-sectional well and post patterns is investigated and shown to influence the orientation of the liquid crystal bulk, far removed from the surface. Additionally, the effective anchoring strength of the induced alignment is estimated using a modified version of the torque balance method. Both azimuthal and zenithal multistability are shown to exist within unique ranges of feature sizes.

A compact microstrip reflection-type phase shifter controlled by liquid crystal is used to realize a phased array of slot antennas. This array has a beam steering capability in one dimension for 60 GHz applications. In this array each phase shifter provides a differential phase shift of 188.5° at the operating frequency of 59.5 GHz. The final design comprises of a 4 × 1 slot array capable of scanning up to 70° in the E-plane as the bias voltage on the liquid crystal changes from zero to saturation voltage.

This paper introduces a new numerical method to calculate the group delay, chromatic dispersion and dispersion slope of weakly-guiding optical fibers with arbitrary radial refractive index profiles. It is based on the analytic differentiation of the propagation coefficient up to the third order. The simulation results are compared to experimental data, with those calculated by other approaches and exact data where possible. Due to the analytical differentiation of the matrix equation, the method is more accurate compared to other approaches, it is also much faster than numerical differentiation as allows avoiding repeated solution of the eigenvalue problem to calculate the derivatives of the propagation coefficient. The precision of the method is limited only by the approximation errors of the mode solver. The Galerkin method with Laguerre-Gauss basis functions is used to determine the propagation coefficients of weakly-guiding structures. The new method enables fiber manufacturers to rapidly design dispersion characteristics of graded index, step index, single- and multiple-clad fibers, as well as few-mode and bend insensitive fibers.

Finite element analysis, employing the H-field vector and scalar formulations, has been used with the aid of perturbation techniques to determine the modal gain and loss characteristics of optical waveguides and semiconductor laser structures. The accuracy and the applicability limits of the perturbation method are examined, and compared to the more accurate but more computer intensive transverse magnetic field formulation for an embedded-channel waveguide. Further, the method is applied to a rib-waveguide laser structure, where the imaginary part of the refractive index of the InGaAsP active layer is seen to vary according to the carrier concentration profile. Finally, the accuracy of a widely used but simpler method for determining the modal gain for QW and MQW laser structures, in terms of the optical mode confinement, is determined.

A patch antenna array with beam steering capability in one dimension for 60 GHz applications is presented. In this array the phase shifters are liquid crystal based meander lines, each providing a maximum differential phase shift of 38°. The array elements (the patch antennas) are coupled to the microstrip feed line through coupling gaps. These gaps are adjusted in order to uniformly excite the antenna patches. The final design comprises of a 4 × 1 patch array, which is able to scan up to 9° in the E-plane as the bias on the liquid crystal changes from zero to saturation voltage.

A simple and natural approach to the single-layer homogeneous absorber problem is developed and used to calculate previously unavailable universal design curves and equations for different orders of solutions. It is shown that only certain combinations of the electric and magnetic loss angles are useful to achieve total absorption. General equations for the bandwidth and for the minimum absorption beyond resonance are found and these parameters are graphically represented in the loss-angle plane. The graphical aids developed allow some optimum types of absorbers to be identified and analyzed. Approximate solutions of the resonance equation are also included.

This paper presents the computer modelling and modal analysis of a complicated multi-quantum-well waveguide structure with active gain by using the modified vector finite element formulation. With the modified formulation, solving the resultant eigensystem by a subspace iteration sparse solver is made easier and faster. From the computing results, some interesting phenomena about the relations of field profiles to material parameters and waveguide structures are revealed and discussed.

A two-dimensional finite-element code is presented for the steady-state analysis of liquid crystal structures in nonuniform electric fields. It is based on a free-energy formulation which includes all three elastic constants and can deal with pure nematic, twisted nematic, and cholesteric liquid crystal materials. The enhanced capabilities of our code allow the design of composite structures made of both dielectric and liquid crystal materials and with arbitrary configuration of electrodes. Unwanted effects, such as the formation of disclination lines in the director orientation, can be accurately predicted. The method has been applied for analyzing pure nematic liquid crystal microlenses with variable focal length and twisted nematic liquid crystal cells for display applications. Numerical results show that a careful choice of the device structures can avoid the formation of defects and improve their performance.

An accurate finite element method is described for the analysis of leaky optical waveguides with arbitrary cross-section and inhomogeneous anisotropic dielectrics including loss or gain. The method leads to the iterative solution of a nonlinear matrix eigenvalue problem without compromising the sparsity of the resultant matrices which depend only on the mesh topology. >

Two forms of implementing the Crank-Nicolson method for the case of nonlinear optical propagation are compared with the faster split operator method. Stability and convergence of the methods are also analyzed. It is shown that in general the split operator method does not conserve power to the same degree as the Crank-Nicolson method does and it fails to converge in the case of strong nonlinearity. >

This paper presents applications of an efficient vector finite element formulation developed by the authors earlier for the analysis of integrated optical waveguides. It also presents for the first time the vector finite element solution of integrated semiconductor waveguides with active regions. This approach can treat a wide range of waveguide problems with arbitrarily-shaped cross-section and regions of inhomogeneity, anisotropy, and loss or gain. This method totally eliminates spurious solutions and gives solutions directly for the complex propagation constant. Unbounded optical waveguide structures are treated by the implementation of finite and infinite elements. The computer program developed is efficient and can be used to compute propagation constants and field distribution of complicated, realistic waveguide problems. >

This paper presents a comparison of two- finite-element programs developed for accurate cross-sectional analysis of arbitrary optical and microwave waveguiding structures. Both techniques were developed in response to the growing demand from scientists and engineers for application-specific expert systems, and employ advanced interactive pre- and post-processing facilities. They show good agreement, strongly complement one another, and could form the core of a whole library of similar "intelligent" software for computer-assisted analysis and design, with a wide range of possible applications within both supercomputer and workstation environments. >

The eigenmodes in electromagnetic cavities, loaded with arbitrarily shaped dielectric materials, are computed by the edge element method. The computation shows that the well known ‘spurious modes’ no longer appear. Formulae are given and verified so that the number of zero eigenvalues, which come from ∇ × H = 0, may be accurately predicted. Formulae for the order and density of the global matrices are given for a rectangular cavity regularly divided into bricks and tetrahedra, allowing an associated comparison between the contrasting edge and nodal element formulations. Comparison of the computed results with available theoretical and previously published data show the edge element approach to be a robust, accurate and effective method. Copyright © 1994 John Wiley & Sons, Ltd

The transmission efficiency of optical waves through a guided-wave structure, incorporating a sharp corner, is investigated by using rigorous numerical approaches based on the finite-element method. To show the versatility of the proposed numerical approaches, the modal power transmission coefficient of two rib waveguides is calculated, and the results are compared with those obtained by using other approaches.

A strategy for adaptive mesh generation is proposed. The method consists of the use of a suitably defined density function, which can either be defined by the user or be calculated from a previous approximate solution, to guide the generation of a new mesh. This new mesh is built starting from a minimal number of triangular elements, which are then, in several sweeps, repeatedly refined according to the density function. The Delaunay algorithm is used in each stage to keep the shape of the triangles as equilateral as possible. >

A finite-element analysis of complex modes in a shielded image guide is presented. The method of analysis, free from spurious solutions, is based on a novel variational expression using only two field components. Results are given for the behavior of complex modes in the presence of varying frequency, permittivity, and losses. It is shown how the presence of losses in the dielectric insert causes the separation of the complex modes, which cease to be conjugates of each other. >

Lossy dielectric waveguide problems are analyzed by applying an efficient finite-element approach which can deal with waveguides of arbitrary cross-section and transverse inhomogeneity, anisotropy, and significant loss. This method gives solutions directly for the complex propagation constant without spurious solutions and retains its efficiency, which is based on (1) the use of the most economical field representation (in terms of only the transverse components of the magnetic field), achieved without sacrificing the sparsity of the resultant matrices, and (2) the use of a highly efficient solver for large, sparse, nonsymmetric, and complex generalized eigenvalue problems. Examples of lossy waveguide problems are studied, and the results are compared with those obtained by other workers. >

An efficient finite element method is presented for the full wave analysis of dielectric waveguides. This method has four major features: (1) the ability to treat a wide range of dielectric waveguide problems with arbitrarily shaped cross section, inhomogeneity, transverse-anisotropy, and significant loss (or gain); (2) total elimination of spurious solutions; (3) direct solution for the (complex) propagation constant at a specified frequency; and (4) the use of only two components of the magnetic field, thus maximizing the numerical efficiency of solution. The resultant matrix eigenvalue problem is of canonical form and is solved with an efficient method, specially developed for this purpose, taking full advantage of the sparsity of the matrices. Numerical results are shown for a variety of microwave and optical waveguides including anisotropy and losses. These examples also include closed and open structures. The computational results agree very well with analytical and previously published results. >

Due to their large birefringence and moderately low loss, liquid crystals (LCs) are a promising dielectric media for development of a variety of reconfigurable millimeter-wave devices. In order to optimize the design of tunable millimeter-wave devices, accurate values of the dielectric and elastic constants, as well as the loss tangents of LCs, are needed. However, characterization of LCs at millimeter-wave frequencies is a very challenging and demanding task. In this work, a transmission line method is used for the broadband characterization of nematic LCs in the frequency range of 30-60 GHz. For this purpose, a unique LC cell is proposed and using this, five different nematic LCs, including E7, K15, E44, E63, and MDA-00-3506, are measured and the values of their electrical and mechanical parameters are extracted. The extraction of these parameters from the measurements involves an optimization using two finite-element computer programs recently developed by the authors for the prediction of the local alignments of LC molecules and the wave propagation within the test cell. The highest values of the dielectric birefringence and the highest values of the loss tangents are recorded for E44 and MDA-00-3506. The loss tangent for all the LCs shows a general downward trend as the frequency increases, which is a useful characteristic in the development of reconfigurable millimeter-wave devices.

For the first time, a compact 60 GHz band phase shifter incorporating liquid crystal (LC) is presented. Based on the Reflection Type Phase Shifter (RTPS) structure, it comprises a 3-dB coupler and tunable reflective loads formed on a nematic LC substrate. The LC used to form the substrate is known as E7. Biasing of the LC is achieved using an external electric field, allowing for the control of the reactance of the reflective loads. To facilitate on-wafer measurements and the application of the electric field to the LC, a transition from the microstrip line to the CPW is used. Preliminary measurement results for this RTPS show a differential phase shift of over 130 degrees, a figure of merit of 11.4° dB and return losses better than −10dB at 61 GHz.

As an alternative to the classical beam propagation method (BPM), a variational method is presented to solve the TE and TM Helmholtz equations in the paraxial approximation for the propagation of polarized beams through optical waveguides. Using the method of local potentials, the paraxial wave equations are first converted into equivalent z-transient variational principles. These functionals are minimized using a combination of the Rayleigh-Ritz finite-element procedure and a Crank-Nicholson-like finite-difference scheme. Solutions for anisotropic materials are obtained by applying standard Galerkin finite-element and finite-difference methods to a variational formulation derived from the coupled TE/TM paraxial Helmholtz equations. >

A spurious-mode-free variational formulation for the finite-element analysis of anisotropic, inhomogeneous dielectric waveguides is derived and demonstrated with examples. Apart from avoiding the occurrence of spurious modes, this formulation has numerous advantages, including the ability to treat problems including significant amount of loss, the direct solution for the propagation constant at a given frequency, and the use of the most efficient representation of the problem, needing only two vector components. This is achieved without losing the sparsity of the resultant (canonical) matrix eigenvalue problem, which depends on the topology of the mesh used. >

The authors review the application of the finite element method to analysis of waveguide problems. They discuss the significance of different variational formulations, the modeling of the infinite domain of open-boundary waveguides, techniques to avoid spurious solutions, and matrix solution techniques. They briefly refer to the application of these techniques to waveguides containing nonlinear materials and to three-dimensional problems. >

This paper describes how design rules are established for photolithographically manufactured acrylate polymer optical multimode waveguide components by optical experimental measurements made on the manufactured waveguide component. The loss of individual waveguide components, such as straight sections, 90deg bends, crossings, tapers and tapered bends must be known so that the combined loss of a cascade of such elements can be found to determine whether the interconnectionpsilas optical power budget is sufficient to achieve a good bit error rate. However, the loss depends on several factors: the materials: the polymer used for the core and for the cladding, the fabrication technique: e.g. the photolithographic procedure and the precise temperature baking regime used, and the measurement technique: the optical source lateral size and angular divergence and precise position relative to the entrance of the waveguide, the output detector lateral size, its angular acceptance angle (if any) and its precise position relative to the exit of the waveguide. The experiments reported on photolithographically manufactured acrylate polymer multimode waveguide were performed at room temperature. A new technique for measure the transmitted power at waveguide crossings is reported for the first time.

This paper reports numerous experimental measurements and modelling result of the loss caused by various amounts of translation and rotation misalignment in the optical backplane system. Combinations of VCSEL as the light source and waveguides of various widths have been input in the system. The source has been moved away from the centre of a channel waveguide in the horizontal and vertical direction gradually to find out the translation and rotation tolerance. Offsets between VCSEL-waveguide and waveguide-receiver have been studied to find the translation and rotation tolerance.

An overview of the pound1.3 million EPSRC and company matched funded Innovative electronics Manufacturing Research Centre (IeMRC) Flagship project between 3 UK universities and 10 companies entitled ldquointegrated optical and electronic interconnect PCB manufacturingrdquo. The project aims to develop of optical waveguide design rules, layout software, fabrication methods compatible with commercial production, characterisation techniques and optical connector design to provide a supply chain for Polymer Multimode Optical Waveguide Printed Circuit Boards (OPCB) for 10 Gb/s board-to-board interconnections.

This work presents a methodology for the use of finite differences in two-dimensional eigenproblems using a mesh of triangles produced for the finite dement method. Expressions are given for the first and second derivatives at a grid point in terms of nodal values at neighboring points. The method has been tested analysing a waveguide resonator including a dielectric rod.

FD-BPM modeling of multimode polymer bent tapered waveguides gives lower insertion loss, reduced tolerance to lateral misalignment of the optical source but a better angular tolerance than straight tapers with the same input/output aperture ratio.

Liquid Crystal Displays require controlled alignment of the liquid crystal molecular director at its confining surfaces. These surfaces may be coated glass or in the case of 'Liquid Crystal On Silicon' (LCOS) technology, a silicon backplane. In the case of Vertically Aligned Nematic (VAN) cells an initially vertical orientation is used and from this the director may tilt in any direction. Some means is required to bias the tilt in a consistent direction to avoid the occurrence of differently oriented domains. For VAN cells one tilt bias method is oblique deposition of silicon oxide. An alternative method which eliminates concerns over consistency of deposition angle over a large substrate area is the use of surface relief structures to induce tilt bias. This is attractive for LCOS devices. Liquid crystal modelling tools [1] have been used to simulate the effects of rectangular and triangular shaped 'bumps' and 'dips' protruding from and extending into the LC's enclosing surfaces respectively. The director orientation and optical transmission of the LC pixels biased in this way are examined as a function of time during the switching cycle and spatially across the pixel to show that the combination provides controllable tilt bias.

The presence of ion impurities in a liquid crystal cell has a detrimental effect on the performance of the cell in terms of switching times and optical response. Here we present simulated and experimental results showing the effect of these impurities at different states of switching.

An accurate analysis is provided of planar optical waveguides that includes significant loss, while retaining the ability to handle all other arbitrary features. The criteria adopted for selection and/or derivation of the actual variational formulation are outlined. A variational expression is given that has been derived by explicitly including the divergence equation. It is currently being tested against a number of basic benchmake examples. >

A computational study of the propagation stability of trains of bright solitons in monomode birefringent fibres is presented for the first time. Several decisive operational conditions are established. These may be used to optimise the propagation distance and bit rate capacity of an optical-soliton-based transmission system.

This paper summarizes and illustrates some of the current forms of using the finite element method in the analysis of microwave and optical structures. Modal and propagation methods are reviewed. An accurate and efficient method for the vectorial modal analysis of waveguides which also permits the treatment of open waveguides including leakage is described and illustrated with examples of complex modes in microwaves and a leaky ARROW structure in optics. The use of finite elements in propagation methods is also briefly described and illustrated.