The calculation of the field distribution is very accurate because of the relation of the MoL to the Discrete Fourier Transformation (DFT) [5].

Transcription

1 CHAPTER 1 THE METHOD OF LINES 1.1 INTRODUCTION The Method of Lines (MoL) is one of the most efficient numerical methods for solving the partial differential equations with which we describe physical phenomena. It has been applied to various problems in theoretical physics [1]. The MoL was developed by mathematicians and the basic principles are described in mathematical books, e.g. in [2,3]. The idea was first applied by the German mathematician Erich Rothe [4] in 1930 to equations of the parabolic type. But it is quite clear that the MoL can be used much more widely. A review from the mathematical point of view is given by O.A. Liskovets [1]. Books which describe the principle of the MoL for use with electromagnetic fields are e.g. [5 8]. The method of lines has certain similarities with the Mode Matching Technique (MMT) and with the Finite Difference Method (FDM). It differs from the latter in the fact that, for a given system of partial differential equations, all but one of the independent variables are discretised to obtain a system of ordinary differential equations. This semi-analytical procedure saves a lot of computing time, because the solution in one coordinate direction is obtained analytically. The method of lines has outstanding properties. It has been shown that: The convergence behaviour is monotonic [5]. Hence extrapolation is possible and gives accurate results. This is because the MoL has stationary behaviour. The calculation of the field distribution is very accurate because of the relation of the MoL to the Discrete Fourier Transformation (DFT) [5]. The method of lines does not yield spurious modes, which are known e.g. in the finite element method. The relative convergence phenomenon [9] (pp. 603ff), as it sometimes occurs in the Mode Matching Technique as a consequence of the Fourier series truncations, is not observed. COPYRIGHTED MATERIAL This book describes based on the author s own and his co-workers work how the MoL can be used for the analysis of a wide range of electromagnetic field problems, e.g. of planar and quasi-planar waveguide structures for integrated microwave and optic circuits, rectangular waveguide circuits, circular and conformal antennas, fibre structures and many other problems. Analysis of Electromagnetic Fields and Waves c 2008 Research Studies Press Ltd R. Pregla

2 2 Analysis of Electromagnetic Fields and Waves This class of waveguide structures can be analysed accurately and nevertheless in an easy way. However, in spite of the size of this book, there are other problems that can be solved with the MoL which we do not address. We would at least mention some of them here. The modelling of VCSELs (vertical cavity surface emitting lasers) including the electrical and thermal model was described in [10 12]. Particularly, the solution of the heat equation with the method of lines was shown in these papers. The analysis of electrostatic problems was presented in [13]. Non-linear material was taken into account, e.g. in [14]. A BPM (beam propagation method) based on the MoL was given in [7,15]. As will be shown in detail later, we deal in this book with generalised transmission line equations (GTL). These GTL expressions can also be used to develop finite difference BPMs as shown in [16 18]. Now, all these problems deal in the frequency domain (we took the steady state in the electrostatic resp. temperature problems). However, the MoL has also been applied to problems in the time domain [19 23]. Let us begin with the quasi-planar waveguide with constant cross-sections, as shown in Fig It has the following features: The number of substrate layers S i may be arbitrary, the metallisations M i can be included in various planes, the boundaries B i consist of electric or magnetic walls, and the lower and/or upper boundaries B 3 and B 4 respectively may be extended to infinity. The waveguide cross-section in Fig. 1.1 includes common microstrips, microslots and finlines. B 4 B 1 M Si 1 M 2 M i B 2 S 2 S 1 B 3 MMPL1240 Fig. 1.1 Cross-section of quasi-planar waveguide B i = boundaries, M i = metallisations, S i = substrate layers Optimum convergence is always assured if the strip edges are located at the right place between the discretisation lines [24]. It should be noted, however, that the convergence of the propagation constant, the characteristic

3 the method of lines 3 impedance or the resonant frequency does not critically depend on the socalled edge parameters (if the discretisation lines are not positioned on edges). Therefore, the problem of convergence on the whole is not critical. In monolithic integrated circuits the metallisation thickness cannot be neglected compared to the conductor width or slot width. Thus a consideration of the finite metallisation thickness is also presented in this book. Circuits in integrated optics are basically realised in the same way as in integrated microwave techniques. Instead of the metallisation, dielectric strips with higher dielectric constant are used to bound the light (Fig. 1.2). ε 0 ε g ε 0 ε g ε s ε s ε b ε b OIWP1140 Fig. 1.2 Cross-sections of dielectric waveguides (models for waveguides in the integrated optics) We will describe how inhomogeneous layers with abrupt transitions in the dielectric constant can be analysed with the MoL. All material parameters can be complex, i.e. losses can be considered. Furthermore, general anisotropic material can also be included. Complex planar circuits consist of concatenations of various waveguide sections of different cross-sections. Some simple examples are shown in Fig (a) (b) (c) MMPL1250 Fig. 1.3 Outlines of longitudinally inhomogeneous waveguides: Examples for (a) periodic structures (b) discontinuities (c) resonators The MoL, as a special FDM, enables analytic calculation in a specific direction. In this direction, the structures to be analysed can consist of many layers or concatenated sections. To assure numerical stability, we will show that the analysis should be performed by impedance and/or admittance transformation. The algorithm that we use can be understood as generalised transmission line theory. Therefore, as in transmission line theory we start at the end of the

4 4 Analysis of Electromagnetic Fields and Waves device and transform (analytically) the load impedance/admittance through the sections and between the sections to the input port of the input waveguide. For cases where analytic expressions cannot be given, we have developed algorithms with finite differences. Instead of impedances/admittances we could also use scattering parameters (particularly the reflection coefficient), as shown e.g. in [25]. As is known from transmission line theory (and used in the Smith chart), impedances/admittances and reflection coefficients can be transformed into each other. Therefore, the procedures are equivalent, though different formulas are applied. If we know the input impedance and the source wave we can go in the opposite direction and calculate the field in each waveguide section. In the case of eigenmode calculations, we transform the impedance/admittance from the upper and from the lower side of the cross-section to the matching plane, where the eigenvalue problem will be formulated. The relations of the discretised tangential fields at the two boundaries of a layer or at the two ports of a section will be written in the form of open circuited or z-matrix parameters, or of short circuited or y-matrix parameters, as known in scalar form from circuit theory. From these relations, the required impedance/admittance transformation formulas are obtained. The formulas for the impedance/admittance transformation at the (e.g. metallised) interfaces between two layers or concatenation of two different sections have to be developed separately. Here we must take into account the fact that the transverse electric and magnetic field components must be continuous. The discretisation lines in the MoL can (at least formally) be understood as real lines in multi-conductor transmission line theory. Therefore, the fundamental equations for analysis can be given as generalised transmission line (GTL) equations. These GTL equations are obtained from the Maxwell s equations and can be written in general orthogonal coordinate systems and for general material tensors. This book is organised as follows. In this first chapter we show the principles of the discretisation and of the analytic solution. In the second chapter we show the analysis of planar structures. We derive the GTL equations, show the solutions and describe particularly the impedance/admittance transformation to obtain stable results. In the following chapters we deal with extensions or special structures. In Chapter 3 we describe rectangular waveguides and how they can be modelled. For an efficient analysis, it is best to use suitable coordinates. Therefore, we introduce cylindrical coordinates in Chapter 4. Of particular interest are periodic structures, which are e.g. used in filters. To study such devices, Floquet s theorem is introduced, as we will show in Chapter 5. Planar structures whose analysis requires an extension of the algorithms presented in the second chapter are presented in Chapter 6. Even more complex devices are given in the seventh chapter, which deals with crossed discretisation lines. Finally, the most general case is described in Chapter 8. Here we introduce arbitrary orthogonal coordinate systems and an

5 the method of lines 5 arbitrary anisotropy. As special cases, we describe spherical coordinates and elliptical ones in Chapter 8 as well. All chapters give substantial numerical results to show that the algorithms could be used successfully to examine a large variety of devices. The book finishes with several appendices, where some basic algorithms and important properties of the MoL are described. As mentioned before, in this book we show the MoL as a special transmission line algorithm and want to give a uniform representation. There are of course many papers by other authors that deal with the MoL. Therefore, the reference lists are by no means complete and we want to apologise for being unable to include all of them. 1.2 MOL: FUNDAMENTALS OF DISCRETISATION Qualitative description In this chapter we will describe the basic principles of electromagnetic field analysis with the method of lines. We will use a relatively simple waveguide structure to demonstrate how we determine the eigenmodes. For most of the structures that are used in integrated microwave techniques or integrated optics, the electromagnetic fields in waveguide structures can be computed from two independent field components or two potentials. The exception is materials with full anisotropy, for which we need four field components. Fig. 1.4 shows a simple waveguide structure, with discretisation lines in the z-direction. In most of the cases described in this book, we are going to use the transverse field components (i.e. x and y) for the analysis. In the following chapters we will develop generalised transmission line equations for this purpose. In this section, however, we will remain general and show that the longitudinal components (z-components) could also be used. H electric wall 0 z =H H h... H = N HN +1 0 magnetic wall plane M E... = 0 E 0 1 h... E N = E N +1 x MMPL2111 Fig. 1.4 Planar waveguide cross-section (only substrate layer and metal strip) with discretisation lines lines for E z, E y, Hx ---linesfor H z, Hy, E x We start with suitable components of the electric and/or magnetic fields. Throughout the book we assume a time dependence according to exp (jωt) of

6 6 Analysis of Electromagnetic Fields and Waves the fields. This time dependence will be omitted. From Maxwell s equations we find that in homogeneous sections all components of the electric and magnetic field must fulfil the Helmholtz equation 2 F x F y F z 2 + ε rf = LF =0 F = E x,y,z,h x,y,z (1.1) in each separate homogeneous layer with the individual permittivity ε r.forf we have to substitute the suitable component. We normalise the coordinates u = x, y, z with the free space wave number k 0 = ω µ 0 ε 0 according to u = k 0 u, and the magnetic field components H u by the wave impedance η 0 = µ0 /ε 0 according to H u = η 0 H u. Moreover, the components tangential to the boundary of [E] (i.e. E z and E y )and[h] (i.e. H z and H y ) must fulfil the following boundary conditions (BCs): electric wall: E t =0 (DC); magnetic wall: Ht =0 (DC); H t n =0 E t n =0 (NC) (NC) (1.2) For the components normal to the boundary (here the x-components) we have: electric wall: Hn =0 (DC); magnetic wall: E n =0 (DC); E n n =0 (NC) H n =0 n (NC) (1.3) The abbreviations DC and NC stand for Dirichlet and Neumann condition, respectively. The terms on the right ( / n) stand for the derivatives in the direction normal to the wall. Here we see the independence of the two components in their dual boundary conditions. One approach to solving the partial differential eq. (1.1) is to approximate the functions f in a suitable way. This is done in the Mode Matching Technique (MMT) or in the Method of Moments (MoM) in all its variations. Another possibility is to approximate the differential operator L. In the Finite Difference Method the derivatives are substituted by the difference quotient. In the Method of Lines the differential operator is partially substituted by differences, but only as far as is necessary, namely to convert the partial differential equation into a system of ordinary ones. For the determination of the eigenmodes in a waveguide propagating in y-direction with the propagator exp( jk y y) (Fig. 1.4) we only have to discretise in one direction. Considering the structure in Fig. 1.4, it becomes clear that the discretisation should be done in the direction parallel to the interfaces of the layers (i.e. in x-direction). The individual layers are homogeneous in y- and, in this chapter, also in x-direction. Generally, the layers may be inhomogeneous in x-direction. This will be described later.

7 the method of lines 7 The discretisation of the operator L has the consequence that the fields are considered on straight lines, which are perpendicular to the interfaces of the layers. Due to symmetry, only half of the cross-section has to be considered. In this case a symmetry (i.e. magnetic or electric) wall has to be inserted in the middle of the structure. Fig. 1.4 shows that two separate line systems are used, e.g. for E z and H z. There are several reasons for this. First of all, the lateral boundary conditions are immediately fulfilled if the lines are in the right position with respect to the boundaries. In order to fulfil the Dirichlet condition, it is best to put a line on the lateral boundary and to set the corresponding field component to zero. In the subsequent calculation it is not necessary to carry along this component. The Neumann condition is easily satisfied by including the boundary between two lines and enforcing the condition that the components on these two lines are equal to each other. The shifting of the two line systems has more advantages: it allows an optimal edge positioning it reduces the discretisation error it results in an easy quantitative description. These advantages will be individually illustrated at convenient points Quantitative description of the discretisation Let the number of E z and H z lines in the cross section of Fig. 1.4 be equal to N. The field components E z and H z on these lines are combined to column vectors E z and H z respectively. E zi is the component of the discretised electric field vector on the ith line. Of course, it is a function of y. In the analysis, we will usually also need other field components. We can e.g. determine E x and H x from the z-components as: )[ ] (ε 2 r + 2 Ex z 2 = x z H x jε r y j y 2 x z [ Ez H z ] (1.4) As previously described, we discretise eqs. (1.1) and (1.4) in x-direction with finite differences. Eq. (1.4) shows that the derivative of E z with respect to x is needed on an H z line and that of H z is needed on an E z line. This request can be fulfilled by using two shifted discretisation line systems, as shown in Fig We will show that the approximation has second-order accuracy for this position. Therefore, the derivative obtained by the finite difference is determined exactly in the middle of the lines. For the vector of the discretised

8 8 Analysis of Electromagnetic Fields and Waves field components, we may write in this case: h E z x D ee z = D t h E z D e = h 1 D e (1.5) h H z x D hh z = D t e H z D h = h 1 D h (1.6) Note: the subscripts e and h correspond to the tangential field components, which are here the z- andy-components (see Fig. 1.4). For other structures, however, differing components might be the tangential ones. Now, the difference operator for the component E z with the boundaries in Fig. 1.4 has the following form: h E z x [ ] E 0 E E N EN } {{ 1 1 } De (1.7) E 0 and E N+1 are positioned outside the computational window. With E 0 = 0 at the electric wall on the left side (Dirichlet boundary condition) in Fig. 1.4, the left upper 1 is omitted. Due to the magnetic wall on the right side (Neumann BC), the last row (below the horizontal line) has to be withdrawn. The construction of D h is analogous, with the conditions exchanged, i.e. Neumann (Dirichlet) BC on the left (right) side. So, we finally obtain: D e = D h =... (1.8) From these examples it is obvious how difference operators for Dirichlet Dirichlet or Neumann Neumann BCs are constructed. The eqs. (1.5) and (1.6) make clear that e.g. the difference operator D e (that is defined for the representation of E z in eq. (1.5)) can also be used to approximate the derivative of H z. This can also be seen in eq. (1.8) and is a further consequence of shifting the line systems. In the difference operator D, the lateral boundary conditions are included. Moreover, the second derivatives

9 the method of lines 9 can also be represented by means of the difference operator D. Thereis: h 2 2 E z x 2 = ( ) Ez h2 De t x x D ee z = D h Dh t E z = P e E z (1.9) h 2 2 H z x 2 = h 2 ( ) Hz D t x x hd h H z = D e DeH t z = P h H z (1.10) Thus the difference operator for the second derivative is obtained as a product of the difference operators for the first derivatives. This can be seen from the chain form of the second derivatives, keeping in mind that the boundary conditions for the outer derivative are dual to those for the inner derivative. The difference operator P canbewritteninthefollowingway: p l P = (1.11) p r Therein we have p l,r = 2 for the Dirichlet condition on the left resp. right wall and p l,r = 1 for the Neumann condition. In our example we chose P e = P DN and P h = P ND. In the examples so far, only Dirichlet or Neumann boundary conditions have been used. However, with these BCs it is not possible to model radiation. For this we have to introduce absorbing boundary conditions (ABCs), and sometimes periodic boundary conditions (e.g. when examining photonic crystals). All these further BCs will be described in detail in Appendix B. Now we substitute eq. (1.9) or (1.10) in eq. (1.1). Further, we assume a wave propagating in y-direction according to exp( jk y y). Then we obtain the following ordinary differential equation system: d 2 dy 2 F z +((ε r ε re )I P)F z = 0 P = h 2 P = D t D (1.12) F z is either E z or H z and P is the corresponding difference matrix. I is the unit matrix in this equation and in the following. We introduced the effective permittivity ε re according to ε re = ky 2/k2 0. In eq. (1.12) the fields on three lines are coupled with each other because of the tridiagonal structure of P. To decouple these equations, we make a transformation to principal axes. With E z = T e E z Hz = T h H z (1.13) we require that T t e P et e = λ 2 e T t h P ht h = λ 2 h (1.14)

10 10 Analysis of Electromagnetic Fields and Waves where λ 2 e,h is a diagonal matrix. λ2 e,h is the eigenvalue matrix and T e,h the eigenvector matrix of P e,h. In the example above (with the chosen BCs) we have λ 2 e = λ 2 h. The calculation of these matrices is given in Appendix A for various boundary conditions. As P e,h is a symmetric matrix, T is orthogonal for a suitable normalisation of the eigenvectors: T 1 e,h = Tt e,h (1.15) With the abbreviation ε d = ε r ε re we get from eq. (1.12) considering eqs. (1.13) and (1.14): (( ) d 2 dy 2 + ε d )I λ 2 F z = 0 λ 2 = h 2 λ 2 (1.16) By introducing Γ 2 y = λ2 ɛ d ɛ d = ε d I (1.17) we obtain the general solution for F z or in an other form F z =cosh(γ y y)a + sinh(γ y y)b (1.18) F z = e Γyy F f + e Γyy F b (1.19) In most cases the components and their derivatives are only needed on the interfaces of the layers. Therefore, we can also give the solution for an arbitrary layer with thickness d (see Fig. 1.5) in the following form: where (d = k 0 d) d dy [ Fz y=y1 F z y=y2 ] [ ][ ] γ α Fz (y = 1 ) α γ F z (y 2 ) (1.20) α = Γ y (sinh(γ y d)) 1 γ = Γ y (tanh(γ y d)) 1 (1.21) y y y 1 2 B d A MMPL1260 Fig. 1.5 Notation of the interfaces of a layer for the calculation of the field components and their derivatives

11 the method of lines Numerical example As a first simple example, we determine the cut-offwavelengthofa rectangular waveguide of width a. In Chapter 3 we will deal with these waveguides in detail. For the H n0 (or TE n0 ) modes, we obtain this wavelength from eq. (1.17) by considering Γy 2 = 0 and determining the wavelength for which the condition ε re =0holds.From and eqs. (A.4) (A.7) we obtain: λ 2 a 2π = ε r =1 h = N +1 λ cn λ cn a = π nπ (N +1)sin 2(N +1) The results for the first two modes (n =1andn = 2) are shown in Fig As is well known, the exact (normalised) values are 2 (H 10 )and1(h 20 ), respectively. As can be seen, both curves approach the analytic value with an increasing number of discretisation lines N for E z. It should also be stated that the value on the very right could be obtained with only one of these lines. The eigenvectors give the exact values of the eigenmodes in the discretisation points. Normalised cut-off wavelength /N H 10 H 20 Fig. 1.6 Normalised cut-off wavelength of the H 10 (or TE 10) andh 20 (or TE 20) mode in a rectangular waveguide

12 12 Analysis of Electromagnetic Fields and Waves References [1] O. A. Liskovets, The Method of Lines, Review, Differential nye Uravneniya, vol. 1, no. 12, pp , [2]W.E.Schiesser,The Numerical Method of Lines Integration of Partial Differential Equations, Academic Press, San Diego, USA, [3] S. G. Michlin and C. Smolizki, Approximate Methods for Solution of Differential and Integral Equations, Teubner, Leipzig, Germany, [4] E. Rothe, Zweidimensionale parabolische Randwertaufgaben als Grenzfall eindimensionaler Randwertaufgaben, Math. Anna., vol. 102, pp , [5] R. Pregla and W. Pascher, The Method of Lines, in Numerical Techniques for Microwave and Millimeter Wave Passive Structures, T. Itoh (Ed.), pp J. Wiley Publ., New York, USA, [6] M.N.O.Sadiku,Numerical Techniques in Electromagnetics, CRCPress, Boca Raton, London, New York, Washington D.C., 2 edition, [7] R. Pregla, MoL-BPM Method of Lines Based Beam Propagation Method, in Methods for Modeling and Simulation of Guided-Wave Optoelectronic Devices (PIER 11), W. P. Huang (Ed.), Progress in Electromagnetic Research, pp EMW Publishing, Cambridge, Massachusetts, USA, [8] R. Pregla and S. F. Helfert, The Method of Lines for the Analysis of Photonic Bandgap Structures, in Electromagnetic Theory and Applications for Photonic Crystals, Kiyotoshi Yasumoto (Ed.), pp CRC Press, Boca Raton Fl, London, [9] T. Itoh (Ed.), Numerical Techniques for Microwave and Millimeter Wave Passive Structures, J. Wiley Publ., New York, USA, [10] E. Ahlers, S. Helfert, and R. Pregla, Accurate Analysis of Vertical Cavity Surface Emitting Laser Diodes (in German), in Deutsche Nationale U.R.S.I Konf., Kleinheubach, Oct. 1995, vol. 39, pp [11] E. Ahlers, S. F. Helfert and R. Pregla, Modelling of VCSELs by the Method of Lines, in OSA Integr. Photo. Resear. Tech. Dig., Boston, USA, 1996, vol. 6, pp [12] O. Conradi, S. Helfert and R. Pregla, Comprehensive Modeling of Vertical-Cavity Laser-Diodes by the Method of Lines, IEEE J. Quantum Electron., vol. 37, pp , 2001.

13 the method of lines 13 [13] R. Pregla, The Method of Lines for the Unified Analysis of Microstrip and Dielectric Waveguides, Electromagnetics, vol. 15, no. 5, pp , [14] M. Bertolotti, M. Masciulli and C. Sibilia, MoL Numerical Analysis of Nonlinear Planar Waveguide, J. Lightwave Technol., vol. 12, pp , [15] J. Gerdes and R. Pregla, Beam-Propagation Algorithm Based on the Method of Lines, J. Opt. Soc. Am. B, vol. 8, no. 2, pp , [16] R. Pregla, Novel FD-BPM for Optical Waveguide Structures with Isotropic or Anisotropic Material, in European Conference on Integrated Optics and Technical Exhibit, Torino, Italy, Apr. 1999, pp [17] S. F. Helfert and R. Pregla, A Finite Difference Beam Propagation Algorithm Based on Generalized Transmission Line Equations, Opt. Quantum Electron., vol. 32, pp , 2000, special issue on Optical Waveguide Theory and Numerical Modelling. [18] G. Guekos (Ed.), Photonic Devices for Telecommunications, Springer- Verlag, Berlin, Heidelberg, [19] S. Nam, S. El Ghazaly, H. Ling and T. Itoh, Time-Domain Method of Lines, Electron. Lett., vol. 24, no. 2, pp , [20] S. Nam, S. El-Ghazaly, H. Ling and T. Itoh, Time-Domain Method of Lines Applied to a Partially Filled Waveguide, in IEEE MTT-S Int. Symp. Dig., 1988, vol. 2, pp [21] S. Nam, H. Ling and T. Itoh, Time-Domain Method of Lines Applied to the Uniform Microstrip Line and its Step Discontinuity, in IEEE MTT-S Int. Symp. Dig., 1989, vol. 3, pp [22] S. Nam, H. Ling and T. Itoh, Time-Domain Method of Lines Applied to Planar Guided Wave Structures, IEEE Trans. Microwave Theory Tech., vol. 37, no. 5, pp , [23] S. Nam, H. Ling and T. Itoh, Characterization of Uniform Microstrip Line and its Discontinuities Using the Time-Domain Method of Lines, IEEE Trans. Microwave Theory Tech., vol. 37, no. 12, pp , [24] U. Schulz, On the Edge Condition with the Method of Lines in Planar Waveguides, AEÜ, vol. 34, pp , [25] S. F. Helfert and R. Pregla, The Method of Lines: A Versatile Tool for the Analysis of Waveguide Structures, Electromagnetics, vol. 22, pp , 2002, Invited paper for the special issue on Optical Wave Propagation in Guiding Structures.

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