A semianalytical method of mode determination for a multilayer planar optical waveguide

The argument principle is one of the basic theorems from the classical complex analysis and it's crucial role in complex analysis is very nicely underlined in the book of A. F. Beardon Complex Analysis – the Argument Principle in Analysis and Topology, John Wiley and Sons, 1979. We shall generalize a result of T. Radó and P.V. Reichelderfer from Continuous Transformations in Analysis, Springer-Verlag, Berlin, Göttingen, Heidelberg, 1955, p. 399, and a recent paper of P. Duren, w. Hengartner and R. S. LaugesenAmer. Math. Monthly 103(5) (1996), 411-415, showing that the argument principle is an entirely topological theorem (because it holds for continuous, open, discrete mappings in the plane) and hence this theorem belongs more to topology than to complex analysis. We also present as an application a generalization of Rouché's theorem.

We demonstrate that the three-dimensional vectorial transmission line matrix (TLM) method is applicable to the analysis of lossy multilayer optical waveguiding structures. Any lossy multilayer waveguide geometry, including sharp discontinuities in the transverse plane, can be treated taking into account the coupling between all optical field components. The complex propagation constants (propagation constants and the attenuation coefficients) for the fundamental TE-like and TM-like modes can be determined. These parameters of the fundamental TM-like mode of a typical lossy multilayer rib dielectric waveguide are obtained as functions of free-space wavelength. Calculation of the electric-field pattern is also included. Numerical comparisons with the argument principle method (for the case of lossy multilayer slab waveguides) and the spectral-index technique (for the case of lossy multilayer rib waveguides) are also included, and it is shown that the application of the TLM method to lossy multilayer optical waveguides is accurate.

COUNTING AND COMPUTING THE EIGENVALUES OF A COMPLEX TRIDIAGONAL MATRIX, LYING IN A GIVEN REGION OF THE COMPLEX PLANE

Eight unique mode configurations may occur in a multilayer planar waveguide.1 A correspondence exists between these modes and zero entries in the composite transfer matrix of the structure.2 The modes of such a structure are obtainable by equating transfer matrix entries to zero and solving for the complex propagation constants of individual mode types. Based on this theory to find the bound mode of waveguides with any number of layers, a numerical technique was developed that minimizes the magnitude of the bound entry in the transfer matrix.3 This downhill method can find minima of an arbitrary function, but it only converges with first-order accuracy and requires four evaluations of the transfer matrix at each iteration. A new method that takes advantage of the mathematical properties of the transfer matrix to achieve second-order convergence with a reduced computational stencil is presented. This method is capable of finding the propagation constants and field distributions of structures containing materials with complex permittivities. This new method offers improved accuracy and efficiency, compared to the first-order downhill method. Results accurate to within 0.01 dB/cm are typically obtained with one-seventh the computation of the downhill method.

In complex analysis, the winding number measures the number of times a path (counter-clockwise) winds around a point, while the Cauchy index can approximate how the path winds. We formalise this approximation in the Isabelle theorem prover, and provide a tactic to evaluate winding numbers through Cauchy indices. By further combining this approximation with the argument principle, we are able to make use of remainder sequences to effectively count the number of complex roots of a polynomial within some domains, such as a rectangular box and a half-plane.

Exact and variational Fourier transform methods for the analysis of generalised planar waveguide structures are described in detail. Results are presented for specific structures of technological importance, namely the rib and channel waveguides and the rib waveguide directional coupler. These results establish the accuracy of Fourier transform methods in calculating critical design parameters such as propagation constant, field profile and coupling length. Moreover, they compare very favourably with purely numerical techniques in terms of speed and required computer resources.

In this work, the finite element method and the FEAST eigensolver are used to explore applications in fiber optics. The present interest is in computing eigenfunctions u and propagation constants β satisfing the Helmholtz equation Δu + k2n2u = β2u. Here, k is the freespace wavenumber and n is a spatially varying coefficient function representing the refractive index of the underlying medium. Such a problem arises when attempting to compute confinement losses in optical fibers that guide laser light. In practice, this requires the computation of functions u referred to as guided modes and leaky modes. For guided modes, the location of the corresponding propagation constants in the complex plane is known in the optics literature, making the FEAST algorithm an ideal candidate to tackle this problem. In practice, one solves the Helmholtz equation by prescribing zero Dirichlet boundary conditions on a bounded, circular domain representing the fiber cross-section. In this work, we compute numerical solutions to this problem using the FEAST algorithm with the Discontinuous Petrov-Galerkin method for the underlying discretization. To compute leaky modes, we must find complex-valued propagation constants β with corresponding outgoing functions u. To compute such quantities, we employ a Perfectly Matched Layer (PML) to force exponential decay of the solutions, rendering the problem computationally tractable on a bounded domain. A frequency-dependent approach is taken for the PML: This transforms the weak formulation of the Helmholtz problem into a polynomial eigenvalue problem, thus motivating our the adaptation of the FEAST algorithm to solve such eigenvalue problems. We verify the results of our algorithm by finding leaky modes and propagation constants of a step-index fiber, where one can compare against known analytic solutions. Our algorithm is then applied to the task of computing confinement losses of a microstructure fiber, where lossy modes are expected. Confinement losses are computed from the imaginary parts of the corresponding propagation constants, and quantify the loss of power as light travels through an optical fiber. Our numerical results show that in practice, there is a large preasymptotic regime prior to convergence of confinement losses, suggesting that confinement losses reported in the literature could be matched in this regime. In addition, we show that our computed confinement losses remain stable as the strength of the decay in the PML region and the PML region's size are varied. Our results also show that computed confinement losses are extremely sensitive to minor perturbations in

The aim of this thesis is to demonstrate that x-ray waveguide optics can be gen- eralized from a single guiding film to an array of planar waveguides, enabling more complex and controllable phenomena of field propagation both in partic- ular in the near-field in the vicinity of objects to be probed by coherent imaging. Two advanced x-ray multilayer waveguides (MWGs) structures, i.e. the waveg- uide array (WGA) and the multi-guide resonant beam couplers (RBCs) are de- signed and discussed. Starting from basic theoretical analysis, the structural model of MWGs is built up. Then the MWGs are studied in detail by numerical simulations based on finite-difference (FD) simulations, fabricated with preci- sion methods for controlled layer thickness, and finally characterized experi- mentally by phase retrieval methods. Chap. 1 introduces the basic theory of x-ray waveguides (see section 1.1), and3 presents geometries and mechanisms of different coupling devices (see section 1.2). FD simulations and phase retrieval methods are discussed to characterize the field propagation in the near-field and far-field (see section 1.3). In the final section 1.4, the fabrication processes of multilayers and MWGs are presented. Thereafter, the x-ray beams exiting from MWGs are characterized by x-ray re- flectivity and far field measurements with focused synchrotron radiation. Chap. 2 presents the concept of WGA, using the WGA structure with Mo/C multilayer at 19.9 keV photon energy. The controlled variation in guiding layer thickness is introduced to achieve the desired phase shifts between the guided output beams. The FD simulations show that multi-beam interference with the desired phase shifts can lead to a quasi-focal spot sized sub-50 nm in free space behind the waveguide. Chap. 3 uses two iterative phase retrieval algorithms to reconstruct the near- field distribution behind tailored WGA and - for comparison - simpler periodic waveguide multilayers (WGM) with N i /C multilayer structure for 13.8 keV pho- ton energy. These are shown to yield distinctly different near-field patterns. Im- portantly, the WGA also exhibits the desired secondary quasi-focal spot outside the structure. Chap. 4 presents the coupling of finite (sub-μm) x-ray beams into RBCs with three guiding layers in the [N i /C ] 3 /N i structure. Using especially resonant mode excitation, more than one reflected beams are generated with different beam offsets along the RBC surface constituting an exceptionally large Goos- Hänchen effect. Possible applications of such devices are beam splitters for co- herent imaging and interferometry. Chap. 5 summarizes the outcomes of this thesis, and discusses future applica- tions and investigations of the WGA and multi-guide RBCs structures.

Multilayer planar waveguides have become increasingly important in modern integrated optic devices. Numerous numerical methods have been devised to address the fundamental issue in multilayer waveguide design and analysis, that is, the determination of the bound modes.

LetD be a bounded domain in the complex plane whose boundary consists of finitely many pairwise disjoint simple closed curves. GivebD the standard orientation, and letA(D) be the algebra of all continuous functions on $$\bar D$$ which are holomorphic onD. We prove that a continuous functionf onbD extends to a function inA(D) if and only if for eachg∈A(D) such thatf+g≠0 onbD, the change of argument off+g alongbD is nonnegative.

Program index to volume 29

Eigenvalue-counting methods for non-proportionally damped systems

In this paper, the topological nature of the generalization of the classical argument principle well-known in multidimensional complex analysis is discussed. The topological approach offered here ensures some topological results on the structure of pole and zero sets of holomorphic maps of bounded domains in complex manifolds. Some connections with integral representations of holomorphic functions are studied and a geometric interpretation of the Martinelli-Bochner complex-valued, differential-form realization is given.

High slope efficiency and low threshold in a diode-pumped epitaxially grown Yb:YAG waveguide laser

In this work, we present a stochastic theoretical analysis of an electromagnetic wave propagating inside a planar slab waveguide filled with a cholesteric material. In this system, the propagation constant of the wave is assumed to be a randomly fluctuating parameter. By means of the van Kampen systematic expansion method, Maxwell’s electromagnetic equations with random coefficients, and appropriate boundary conditions, are obtained. Numerical examples of the effects of the randomness on the propagating constant, on the ratio between magnetic and electric modes, on the field profiles, and on the energy flow are given. The results show that the stochastic waveguide can support multiple Transverse Magnetic (TM) or Transverse Electric (TE) modes with the same cutoff frequency, and furthermore, they support a distribution of electric and magnetic fields with phase shift. Our results are consistent with those of an isotropic waveguide.