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Finite element modeling methods for photonics. (English) Zbl 1279.78002

London: Artech House (ISBN 978-1-60807-531-7/hbk). xv, 247 p. (2013).
The book consists of a Preface, six chapters divided into sections and partly subsections, three appendices, an index, and a short section with the title ‘About the authors’ on 247 pages. References are given at the end of the first five chapters to help the reader to further explore the subject. The content is illustrated by a number of figures.
In the preface, the authors define loosely photonics as a branch of science that ‘refer to a vast array of components, devices, and technologies that in some way involve manipulation of light’. The design and development of photonic devices require numerical methods to reduce time and costs. The book is devoted to the powerful finite element method (FEM) applied to photonics studying electromagnetic fields in space and time. Mathematical concepts are explained in simple steps along with physical ideas, such that also readers without an advanced mathematical background can benefit from this well-written monograph. The presentation enables the user to develop own computer codes. But the reader has also the possibility to download and modify codes from an accompanying web page.
Chapter 1, the introduction, starts with Maxwell’s equations, consisting of Faraday’s law, Ampere’s law and Gauss’s laws, that describe the evolution of optical or electromagnetic fields. The solutions of Maxwell’s equation have to satisfy the constitutive relations for the medium and the boundary conditions. The constitutive relations are presented with the hint that photonic devices commonly consist of different media and thus several interfaces have to be considered between the media. The interfaces conditions are formulated. The boundary conditions are classified as natural and as forced, the last are divided into Dirichlet and Neumann boundary conditions. Because boundaries at infinity occur in photonic applications truncations of the domain, infinity FEM, and special absorbing boundary conditions have to be taken into account.
For many practical problems one need only solutions for single frequencies. Thus, the authors formulate also the time-harmonic Maxwell equations. The Maxwell system of two first-order equations can be combined to a single second-order equation for the electric or magnetic field intensity, the so-called vector wave equations. The vector wave equations are here considered in the time-harmonic form. The assumption of homogeneous media or simplifications for special inhomogeneous media lead to the corresponding scalar wave equations.
The next topics are the so-called modal solutions of the wave equations defined for structures with a refractive index with the property to be homogeneous in one propagation direction and only varying in the transverse directions. A number of analytical, semi-analytical, and numerical methods for optical waveguides are listed including bibliographical references to perform a modal analysis.
Structures whose refractive index varies along the propagation direction require other solution techniques. The authors focus on beam propagation methods (BPM) based on solutions of a reformulation of the wave equation. Finite element BPMs (FE-BPM) in the scalar and vector form are the subject of Chapter 3 taking into account paraxial (Fresnel approximation) and wide-angle BPMs.
The last two sections of the introduction are devoted to questions of which method should to be chosen for a practical problem and of why finite element methods are favored.
It is not the aim of the introduction to give a comprehensive mathematical/physical insight into optics and electromagnetic theory, the main subject of the monograph are the numerical solution methods for problems of the branch photonics. Nevertheless, the representation could be sometimes improved. For instance, the deduction of the time-harmonic wave equations (1.25) and (1.26) is related to the time-depended Maxwell’s equations instead of the also given time-harmonic version, and because the electric current density \(\vec J\) is present in Maxwell’s equations and not present in the wave equations, it should be mentioned that this derivation is reduced to a medium where \(\vec J = 0\).
In Chapter 2, the authors start with the basic concept of the finite element method. The region is divided into a set of small, non-overlapping subdomains, the finite elements, over which the unknown functions of the differential equations are approximated by interpolation functions, generally polynomials. The interpolation polynomial takes the values of the unknown function at the element nodes and can be used to compute the values at any point inside the element. The sum of all elements, the assembly or the finite element mesh of the domain, represents the whole region. The FEM is set up by two models. The variational approach and the weighted residuals method are element-wise applied resulting in a set of algebraic relations among the unknown parameters. The resulting equations for each element are assembled into generally large-scale sparse global equations which have to be solved by excellent algorithms.
The authors point out the advantages of finite elements, such as: The interpolation polynomial can be used to compute the values at any point inside the element. The size and shape of the elements can be adapted to the geometry of the real-world problems. The solutions are continuous across shared boundaries of neighboring elements. The accuracy of the results can be controlled by the number, size, and shape of the elements and by the approximation order of the interpolation functions. Especially, smaller elements can be chosen in parts of the whole domain where the unknown function rapidly varies in contrast to other parts of the region.
The variational approach and the weighted residuals method are presented in two brief subsections.
The variational method is based on the minimization of a functional in terms of the unknown functions of the differential equations. The minimization of this expression yields the Euler-Lagrange equation which has to correspond to the physical problem, i. e. with the governing differential equations and boundary conditions. The determination of the stationary value is generally easier than the direct solution of the Euler-Lagrange equation. Approximating the unknown function in the established variational principle by a set of basis functions with properly chosen coefficients yields the Rayleigh-Ritz method. The original Rayleigh-Ritz method minimizes the functional over the entire region with the disadvantage that one has to find appropriate approximation functions which satisfy the corresponding boundary conditions. These difficulties are overcome by the FEM-Rayleigh-Ritz approach, in which a global expression built on the contributions of every element is minimized. A drawback of the variational approach consists in that a variational formulation does not exist or is not known for every differential equation. The method is especially well suited for tasks in which a single quantity is to determine, such as the resonant frequency of a structure.
After all this, the authors write the wave equation as a generalized eigenvalue equation and establish the corresponding functional to be minimized. Expanding the fields of the functional in a set of basis functions with unknown coefficients and requiring that the derivatives of the expression with respect to the coefficients vanishes, results in a matrix equation for a generalized symmetric eigenvalue problem giving the guided modes in optical structures. For details of the derivation of the variational expression it is referred to the literature.
The weighted residuals method is introduced considering a deterministic problem \(L u = v\), where \(L\) is a linear operator, \(u\) is the unknown function, and \(v\) stands for a source. Looking for a solution the unknown function \(u\) is expanded in a complete set of known basis functions \(u_i\) with unknown coefficients \(b_i\). The corresponding error residual will be only zero if the expansion represents the exact solution, in all other cases, weight functions \(w_i\) have to be introduced, in such a way that the residual is orthogonal to each of the \(w_i\). In the case of photonic problems here the magnetic field \(\vec H\) is chosen as the unknown function \(u\) and the required integration over the corresponding inner products is formulated. Choosing the weight and the basic functions to be identical results in a special variant, the Galerkin method, that is used for the FE-BPM in Chapter 3. If the operator \(L\) contains differential terms like \(\frac{\partial^2}{\partial x^2}\), double differentiable weight functions have to be used. That is the strong form of the weighted residuals method. For the FE-BPM in Chapter 3 the weak form of the Galerkin method is used. The integral is simplified by integrating by parts in this case. The Galerkin method can be used in all cases in which a variational approach is not possible. If an variational principle exists both methods can be applied and the user has to decide either way taking into account special properties of his problem.
Based on the presented theory the FEM formulation for the scalar and the vector wave equations are described.
In the scalar case it is sufficient to determine the quasi-transverse electric (TE) or the quasi-transverse magnetic field (TM) modes. In contrast to planar waveguides this modes are not purely TE or TM modes in optical waveguides. The FE solution for the quasi TE modes of the latter is derived.
If longitudinal and transverse components exist in a wave guide the vector formulation of the wave equation has to be applied. There are mentioned FEM vector formulations for the longitudinal electromagnetic field components, for three kinds of transverse field components, and for three hybrid components. Represented is the according to the authors most advantageous vector \(\vec H\)-field formulation including all three components of the magnetic field. It is outlined that nonphysical solutions, so-called spurious modes, can occur because the divergence condition is not satisfied automatically, an disadvantage of the nodal finite element formulation in electromagnetics. The authors point to methods for detecting these modes. A penalty procedure for the elimination of spurious modes is described.
An overview about the implementation steps of the FEM is represented by a flowchart. The first steps consist in the meshing and the choice of the shape or interpolation functions. Considered are elements with rectilinear shapes, iso-parametric elements with curved edges, infinite elements, and vector or edge elements.
The authors prefer polynomial expressions for the shape functions that are complete in each element. The 2D meshing with first-order triangular elements is exhibited in great detail including the relationship between the shape functions, the nodal values, and the unknown function. Area coordinates are introduced and Lagrange polynomials in terms of the area coordinates are used for the construction of higher-order shape functions.
Iso-parametric elements are needed for the meshing of real-world problems with curved geometry. The coordinate transformations between the local Cartesian coordinate system and global curvilinear coordinates are treated.
Field varieties up to infinity have to be taken into account for open photonic devices. Such problems can be handled dividing the domain into interior and exterior regions where the fields extend up to infinity. The meshing of the interior region can be done by the standard FEM. For the exterior region infinite elements are applied with shape functions that decay exponentially in a proper way.
The basic functions of edge elements (also known as Nedelec-type elements in the electromagnetics community, reviewer’s remark) are vectors. These elements assure a spurious-free approximation of electromagnetic problems. These and some other advantages of edge elements are described. The simple triangular vector element, the Whitney element, is presented in terms of area coordinates. For details it is referred to the monograph by J. Jin [The finite element method in electromagnetics. New York, NY: Wiley (2002; Zbl 1001.78001)]. According to the authors the edge elements have not found yet a broad interest in the photonics community in contrast to the electromagnetics groups. Thus, the vector elements are not a subject of the reviewed book.
After the insofar as described basic concepts of the FEM the element and global matrices that correspond with the chosen basis functions are derived firstly in a general form. Then the mass and stiffness matrices are evaluated for the first-order and the second-order triangular elements in detail. The assembly of the global matrices is demonstrated for a simple grid consisting of three first-order triangles.
The solution of the resulting linear generalized eigenvalue equation or system of linear equations is not a part of this monograph but, the reader will find some hints to possible matrix kinds (dense, sparse, banded, small, large-scale) and corresponding adequate algorithms including references to publications. Well-known software packages are not quoted.
The next topic consists in the use and implementation of boundary conditions. The truncating of domains, the reducing of the degree of freedom by exploiting the symmetry of structures, electric and magnetic wall conditions, adsorbing boundary conditions, perfectly matched layer boundary conditions (PML), and periodic boundary conditions are treated. The PML is considered in the complex stretching form.
The last four sections of this chapter are devoted to FEM applications of real-world photonic structures, to the FEM analysis of bent waveguides, accuracy problems, and the use of convenient computer systems. It is assumed that the reader will have some insight into the physical background of the applications.
Analyzed are a Si nanowire with a rectangular cross section, a cylindrical shaped photonic crystal fiber, two 1D plasmonic waveguides, and a photonic crystal waveguide as an example for periodic boundaries, all based on special publications by the first author et al.
The refractive index \(n(x,y,z)\) of bent waveguides depends on the \(z\)-direction, i. e., 2D FEM cannot be used. Furthermore, the modes can be lossy and leaky. The solution of Maxwell’s equations in a local bent coordinate system, perturbation analysis, and an equivalent straight waveguide approach with PML structure are discussed.
Approximating the continuous differential equation leads to a discretization error. Interpolating the unknown function gives an interpolation error caused by the interpolation of the function and by calculation of the gradient. The geometric elements are to be constructed with moderate angles. The accuracy of the algebraic system depends on the condition number. All these errors, the convergence of the FEM, and the adaptive mesh refinement are briefly outlined. Applying the FEM to an accuracy benchmark example, a simple rib waveguide, some error dependencies are demonstrated using commercial software (COMSOL 4.3a).
Chapter 3 is devoted to the FE-BPM which is derived on the basis of the already known vector wave equation assuming now that the refractive index \(n = n(x,y,z)\) rather than \(n = n(x,y)\) because some practical devices have such index distribution. Furthermore, PML boundary conditions in form of the above mentioned complex coordinate stretching are involved. Applying the zero divergence equation results in solutions without spurious modes. In addition, the deduction of the full vector formulation of the BPM is simplified assuming that the refractive index variation in the chosen propagation direction (say \(z\)) is sufficiently small such that terms containing the first derivative of \(n^{-2}\) with respect to \(z\) can be neglected. The equations are further simplified separating the field as a product of a fast varying phase term and a slowly varying envelope part. These equations describing the evolution of the envelope of the field in \(+z\)-direction are than treated using the Galerkin method with first-order triangular elements. The BPM is a marching algorithm which propagates the field in steps of \(\Delta z\). Assuming that the PML parameters and the refractive index are constant in each element the equations in terms of the global matrices are derived. These equations can be used to handle propagation of waves with wide angles with respect to the propagation direction. Additionally, special paraxial FE-BPMs are sketched and an implementation of the BPM using a finite element approach in \(z\)-direction is discussed.
The composed global matrices contain also parts which describe the polarization dependence and the coupling between different polarization components. Neglecting these parts of the global matrices, what can be done for applications with a weak coupling between the polarization components, results in a semi-vector FE-BPM consisting of two decoupled equations in the magnetic field intensity terms \(H_x\) and \(H_y\) which can be further simplified to the so-called scalar FE-BPM for problems without polarization dependence, all this to reduce the considerable computational resources needed for the full vector formulation.
The algorithm are demonstrated by examples, such as a 3dB power splitter and a semiconductor optical amplifier. Other topics are the bi-directional BPM and the imaginary axis BPM.
Algorithms for photonic devices whose fields vary rapidly in time are the subject of Chapter 4. Starting from the time-dependent vector wave equation for the electric field intensity \(\vec E\), a simpler scalar wave equation in two dimensions is derived mainly based on engineering arguments with respect to practical photonic devices. Only the evolution of the slowly varying envelope is taken into account. The high frequency carrier wave is neglected. Perfectly matched boundary conditions are involved. Applying the Galerkin approach the corresponding matrix equation with the mass and stiffness matrices is formulated. Employing the Padé approximation to handle the second-order derivative with respect to time results in a wide-band equation. Explicit and implicit schemes in the implementation of the finite-element time domain BPM method (FETD BPM) are compared. The unconditional stable implicit Crank-Nicholson method allows larger step sizes than the explicit procedures. A further simplification is reached by matrix lumping. Illustrated is the explicit FETD method for the simulation of the pulse propagation in optical grating.
The optical properties of photonic devices can be influenced by other physical effects, such as thermal, strain/stress, acoustic, and electric field impacts. Chapter 5 is devoted to these coupled problems. Finite element methods are described for photonic devices in which these forces become important. Treated are for instance the thermal modeling of a VCSEL, the stress analysis of highly birefringent optical fibers, the interaction between acoustic waves and the light propagation in optical Silica waveguides, and the electro-optic model of a Lithium Niobate modulator. Another topic of this chapter are nonlinear devices in which the refractive index of the material depends on the optical beam intensity. Several degrees of nonlinearity are described, especially the third-order Kerr nonlinearity. For example, the model of a nonlinear directional coupler requires the solution of a coupled nonlinear Schrödinger equation for the analysis of the soliton dynamics. The FEM based on the Galerkin approach is presented.
In the last chapter, the authors consider the state of the art and the future directions in FE-based methods for photonics. The good properties of the method are listed. The quantum nature of matter, the light-matter interaction, and parameters that follow a random distribution have to be taken into account in prospective effective implementations. Last but not least, the software needs to be more user-friendly because the user is often not a expert in numerical methods.
The subject of the three appendices are perturbation formulae for the scalar and vector FEM and Green’s theorem.

MSC:

78-02 Research exposition (monographs, survey articles) pertaining to optics and electromagnetic theory
78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory
78A25 Electromagnetic theory (general)
78A60 Lasers, masers, optical bistability, nonlinear optics
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65Z05 Applications to the sciences

Citations:

Zbl 1001.78001

Software:

COMSOL
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