General eigenvalue equations for optical planar waveguides witharbitrarily graded-index profiles

Accurate eigenvalue equations for planar waveguides with arbitrarily graded-index profile are derived and expressed in closed forms. A combination of the modified Airy functions and the Wenzel-Kramers-Brillouin (WKB) solutions are employed as field solutions, which turn out to represent almost exact field profiles. The use of new trial solutions enables us to calculate phase shifts at turning points very precisely, allowing us almost exact eigenvalues. It is demonstrated that the results obtained by the proposed method are in excellent agreement with those by the finite element method, achieving significant improvement over the conventional WKB method     

求解渐变折射率分布平面光波导模式的数值方法

本文用多种数值方法（积分插值法、有限元法及幂法）对具有渐变折射率分布的平面光波导率征方程进行了理论计算。这些方法不仅适用于任意的折射率分布，而且能够达到较高的计算精度。作为一个例子，作者对高斯分布波导进行了具体计算。计算结果表明，Arun Kumar等人用的微扰法是较好的方法。此外，本文作者通过迭代求出了更精确边界条件下的本征模。     

Guided wave characteristics of metal-clad graded-index planar optical waveguides: Analytical approach

The complex eigenvalue equations of metal-clad graded-index waveguides are generally solved by numerical methods. The role of waveguide and material parameters in determining mode dispersion and attenuation becomes difficult to elucidate. We describe a perturbative method of solution which is suited to the class of waveguide in which the guided field is described analytically. Mode dispersion and attenuation are explicitly obtained. Applications to three commonly used waveguide models demonstrate the accuracy of our approach in comparison to numerically obtained results.     

A refined WKB method for symmetric planar waveguides withtruncated-index profiles and graded-index profiles

An accurate method is proposed, based on a modified version of the WKB method in quantum mechanics, for evaluating the propagation constants and field distribution of symmetric planar waveguides with a truncated-index and graded-index profile. Simple formulas are given for calculating the dispersion characteristics including those for the truncated parabolic and exponential profile, and results are much more accurate than those of the ordinary WKB method, especially in the near cutoff regions. This method also can be extended to other more complicated cases     

Local field analysis of bent graded-index planar waveguides

A local field analysis is proposed for bent planar waveguides with arbitrary refractive index profiles. Exact vector wave equations that include the gradient index, or polarization correction, term are derived for both transverse electric and transverse magnetic modes from Maxwell's equations in a local bent coordinate system. The approximate local field and correction to the propagation constants are obtained by perturbation analysis. As an example of the method, an infinitely extended parabolic index profile is studied     

Impedance boundary method of moments with extended boundaryconditions

An improved impedance boundary method of moments is proposed to accurately and efficiently determine the propagation characteristics of multilayer planar optical waveguides whose refractive index profiles contain large step index changes or a graded-index region combined with step index changes. Extended impedance boundary conditions are used in the impedance boundary method of moments to replace regions in the waveguide structure which have refractive index discontinuities. Two methods for determining the extended boundary conditions include the standard transfer matrix method and a procedure utilizing a cascaded transmission line representation with successive calculation of input impedances. Simulation results show significantly improved convergence rates in the determination of propagation constants     

A MATRIX METHOD FOR SOLVING PLANAR DIELECTRIC OPTICAL WAVEGUIDES

A matrix method used in multilayer stack of dielectric films is applied-to planar dielectric optical waveguides. A simple and applicable method for obtaining characteristic equation is presented.     

An iterative method for optical reconstruction of graded indexprofiles in planar dielectric waveguides

A nondestructive technique for the reconstruction of refractive index profiles in planar waveguides is presented and analyzed. The approach is based on the integral scattering equations, which permit one to relate the refractive index of an inhomogeneous layer to the reflected field intensity at different incidence angles. From this formulation, an iterative algorithm is developed, such as at each iteration step the problem is formulated as the minimization of a functional representing the error between the measurements and the model data. The recovered profile is then used to improve the validity of the approximation in performing the next step. In this approach, the unknown index profile is represented as the sum of a finite series of basis functions avoiding to select a priori the particular functional form (e.g., Gaussian function, complementary error function, etc). The practical effectiveness of this approach is demonstrated by numerically simulating the measurements for different planar waveguides. The influence of measurement uncertainty and noise on the stability of the technique is also evaluated     

Optoelectrochemical transduction on planar optical waveguides

Planar optical waveguides overlaid with electrochemically modulated sensing films constitute a powerful new class of chemical transducer. The effects on the modal absorption spectrum of depositing a highly absorbing electrochromic film on an optical waveguide are modeled. Theoretical and experimental results are given for such a device consisting of a lutetium biphthalocyanine film deposited on an indium tin oxide-coated potassium ion-exchanged glass waveguide, for application as a chlorine sensor     

A vectorial boundary element method analysis of integrated optical waveguides

In this paper, a new vectorial boundary element method is introduced and applied to the modal analysis of dielectric waveguides with piecewise homogeneous refractive indexes. The procedure, which is free of spurious modes, determines the full field distribution from the longitudinal fields at the refractive index boundaries. Singular kernels are evaluated through series solutions while the electric field discontinuity at corners is accommodated through either a grid refinement technique or a semianalytic approach. Our formalism generates propagation constants and modal field distributions for several representative refractive index profiles with far higher accuracy than standard finite-difference or finite-element procedures.     

Refractive-index profiling of graded-index planar waveguides fromeffective indexes measured for both mode types and at differentwavelengths

It is well known that, under the WKB approximation, the refractive-index profile of a graded-index planar waveguide can be calculated from the corresponding effective-index function, which can be found approximately by least-squares fitting of a set of measured effective indexes. In the previous work, effective indexes measured for a particular type of modes (the TE or TM modes) at a single wavelength are used. This requires that the waveguide supports at least three modes of the same type. In this paper, techniques of combining the effective indexes measured for both mode types and at different wavelengths are proposed and demonstrated with examples. With these techniques, single-mode and two-mode waveguides can be profiled     

Simple and accurate loss measurement technique for semiconductor optical waveguides

Loss figures for GaAs/GaAlAs heterostructure optical waveguides have been derived from Fabry-Perot cavity resonances. These correlate with the mean guide transmissions within ±0.5 dB for 75% of the guides measured, for losses up to 12 dB per pass.     

Thermal strain analysis in optical planar waveguides

For the first time, we derive an analytical form of the thermal-induced strain equation for the waveguide core layer in an optical planar waveguide, which is based on elastic interaction theory in multilayered isotropic structures off the edges by thin-film approximation. This derived equation addresses how material and structural characteristics affect the magnitude and the distribution of the strain. By this equation, the strain-induced birefringence can be reduced or eliminated by proper waveguide design.     

Low-loss bends in planar optical ridge waveguides

Bending losses lower than 0.7 dB for 90° bend sections with radii of curvature as small as 75 μm were measured on silicon-based Al₂O₃ ridge waveguides with SiO₂ cladding layers at a wavelength of 632.8 nm. These values, which are close to the calculated values, are the lowest thus far reported     

Impedance boundary conditions for finite planar and curved frequency selective surfaces

We consider the time-harmonic electromagnetic scattering problem from a finite planar or curved, infinitesimally thin, frequency selective surface (FSS), the periodic unit cells of which are constituted, exclusively, by electric conductors and free-space. In order to avoid the meshing of these cells, the problem is solved by employing an integral equation formulation in conjunction with approximate impedance boundary conditions (IBC) prescribed on the sheet that models the FSS. The impedance in the IBC is derived from the exact reflection coefficient calculated, for the fundamental Floquet mode, on the infinite planar FSS illuminated by a plane-wave at a given incidence. When the FSS is curved, and/or the direction of the incident wave is unknown, higher order IBCs are proposed that are valid in a large angular range and can be implemented in a standard method of moments formulation. Also, a simple technique is presented that allows to reproduce the radiating Floquet modes in the scattered field even though those are not accounted for in these IBCs. Their numerical efficiencies are evaluated for a curved strip grating translationally invariant along one direction. Finally, we present an alternative approach where the impedance is approximated by its truncated Fourier series, that considerably enhances the accuracy of the results at the cost, however, of a denser mesh of the sheet.     

Simplified numerical analysis of optical fibres and planar waveguides

A convenient and efficient numerical technique is presented for determining modal properties of an optical fibre or planar waveguide. A simple recurrence relation, suitable for programming on only a pocket calculator, leads to parameters such as group velocity and cutoff wavelengths. The method is particularly useful for analysing single mode fibres, and an example of current interest is given.     

Properties of four-layer planar optical waveguides near cutoff

Mode properties of four-layer planar optical waveguides near cutoff are considered. The four-layer structures considered are applicable to waveguiding layers formed on cladding layers above either silicon or gallium arsenide substrates. Perturbation approaches are shown to be accurate except near mode cutoff where a numerical solution of the four-layer equations is required. Numerical calculations of waveguide attenuation due to substrate coupling for thermally nitrided silicon dioxide and for gallium aluminum arsenide waveguides are presented for a variety of layer thicknesses, layer material compositions, and wavelengths. Comparison to some experimental data is included.     

Pseudospectral beam-propagation method for optical waveguides

We propose a wide-angle pseudospectral beam-propagation method using the Pade approximant operator for strongly guiding and longitudinally varying optical waveguides. We validate the numerical results of our method by comparison against results obtained from a robust and widely used finite-element beam-propagation method.     

Prism-coupling of light into narrow planar optical waveguides

The efficiency and selectivity of the prism-coupler for exciting single modes in narrow planar optical waveguides is investigated both theoretically and experimentally. A previously published calculation of the coupling efficiency between a plane input wave with finite cross section and a number of waveguide modes, as a function of both the incidence angle and the skew angle, is extended to include the effects of diffraction, which are shown to play a dominant role for small focal widths of the input beam. The theoretical predictions are compared with experimental results obtained at 633 nm on an embedded ridge-type waveguide structure. A number of mechanisms reducing coupling selectivity and stability are discussed, and some practical rules are given for obtaining good stability and selectivity     

Numerical solution of silicon-clad diffused planar optical waveguides

Computer modelling studies indicate that silicon-clad diffused planar optical waveguides exhibit a damped periodic oscillation in their attenuation and phase constant curves and that the attenuation is reduced significantly by reducing the maximum index change.