An enhancement of instruments for solution ofgeneral transmission line equations with method oflines, impedance-/admittance and �eldtransformation in combination with �nitedifferencesWaldemar Spiller  ( walspill@yahoo.de )

FernUniversitat in Hagen https://orcid.org/0000-0001-6365-4958

Research Article

Keywords: Method of lines, generalized transmission line equations, impedance/admittancetransformation, waveguide structures, �nite differences, �nite differences with second order accuracy

Posted Date: October 5th, 2021

DOI: https://doi.org/10.21203/rs.3.rs-901151/v1

License: This work is licensed under a Creative Commons Attribution 4.0 International License.  Read Full License

An enhancement of instruments for solution of general transmission

line equations with method of lines, impedance-/admittance and field

transformation in combination with finite differences

Waldemar Spiller

Received: date / Accepted: date

Abstract The Method of Lines (MoL) in combination with impedance-/admittance and field transformation

(IAFT) is used to analyze electromagnetic waves. The used cases are wave-guiding structures in microwave

technology and optics. The core of the theory is the solution of generalized transmission line equations (GTL). In

the case of complex structures, a combination with finite differences (FD) can be used. The quality of this solution

essentially depends on the effectiveness of the used interpolation of the differences. The individual steps of the

FD are permanently linked to the steps of the fully vectorial impedance-/admittance and/or field transformation,

so standard libraries cannot be used. Two approaches based on the linear and quadratic interpolation were built

into the impedance-/admittance and field transformation in the past. However, the degree of improvement due

to one or another kind of interpolation depends on the concrete behavior of the solution sought. In the case of

complex structures, choosing the appropriate type of interpolation should be an effective aid. In this paper, an

extension of the family of built-in methods is proposed - with the possibility of being able to build any known

numerical method from the class of one-step or multi-step methods into the GTL solution. These can be higher-

order methods, including fast explicit methods, or particularly stable implicit methods. The transmission matrices

for the impedance-/admittance and field transformation serve as the building site. To illustrate the procedure,

some different methods are integrated into the GTL solution. The accuracy of the solutions is tested on selected

complex structures and compared with each other and with existing solutions. It is shown that the optimal choice

of method and the quality of the solution can depend on concrete structures.

Keywords Method of lines, generalized transmission line equations, impedance/admittance transformation,

waveguide structures, finite differences, finite differences with second order accuracy.

Notation and symbols

– Scalar quantities are denoted by oblique letters, e.g. Ey or N

– Vectors are written in bold and straight, e.g. H

– Matrices are written in bold italics, e.g. RE

– With k0 normalized coordinates are written with an overline, e.g. u

– Field values or matrices in the Floquet domain are marked with “ ˜ ”, e.g. H

– Field values marked with a “ ” denote discretized quantities, e.g. H

E electric field strength vector

E electric field strength, scalar

H magnetic field strength vector

H magnetic field strength, scalar

Q, RE,H operator matrices

I adequate unit matrix

N Number of lines, number of discretization points

ω angular frequency

ε0, µ0 vacuum permittivity and magnetic permeability of free space

k0 = ω√µ0ε0 free space wave number

W. Spiller, E-mail: walspill@yahoo.de

2 Waldemar Spiller

η0 =√

µ0/ε0 wave impedance in vacuum

Hu = η0Hu normalized vector of magnetic field strength

u= x,y,z generalized coordinate

u normalized generalized coordinate, u= k0u

εr relative dielectric constant

1 Introduction

1.1 General considerations

The main aim of this paper is to expand the usability of MoL in combination with impedance-/admittance and

field transformation with finite differences (MoL-IAFT-FD). The aspects of accuracy, stability, implementation

effort, and portability1 are important in everyday scientific life. The specialty of the MoL-IAFT-FD is that the

individual steps of the FD are permanent linked to the steps of the fully vectorized impedance-/admittance and

field transformation. Therefore, the standard software libraries can scarcely be used. In addition, the quality of

the MoL-IAFT-FD essentially depends on the effectiveness of the used interpolation of differences. However,

the quality of the one or another kind of interpolation depends on the concrete behavior of the solution sought,

e.g., (Bultheel and Cools, 2010), (Curtiss and Hirschfelder, 1952), comp. (Spiller et al, 2019), i.e., on specific

applications, or, on each specific wave-guiding structure:

– on the spatial configuration and specific values of the material parameters

– on the order of accuracy of the method and its approach: explicitly or implicitly.

The way to increase the usability of MoL-IAFT-FD is to incorporate further methods from numerical mathematics

into the individual steps of the GTL solution. The methods should be interchangeable with as little effort as

possible, and the user can choose to use them for various specific applications. Two procedures for the numerical

solution of ordinary differential equations comes into consideration: one-step and multi-step methods of some

order of accuracy (Bronstein et al, 2005), (Samarski, 1982), (Samarski, 1986), (Hairer et al, 2007), (Bultheel

and Cools, 2010), (Hairer and Wanner, 1996). The one-step method needs only one of the previous values yn to

calculate the function value at the next step yn+1 (n= 1,2,3, ...,N and N is the number of discretized points). The

multistep procedures need several preceding values, e.g., yn and yn−1. The assignment to one or the other method

will be of particular interest to us with regard to accuracy and stability: For certain complex structures such as

Bragg gratings, photonic crystals, etc., the accuracy and stability can be a challenge. For the improvement of

accuracy, — while preserving numerical stability, — the higher-order methods can be used. However, the aspect

of numerical stability limits the choice of methods: The Second Dahlquist Barrier states that the consistency

order of an A-stable multistep method can be at most two (Dahlquist, 1963). But the Second Dahlquist Barrier

does not apply to the one-step methods such as Runge-Kutta. Moreover, the Daniel-Moore conjecture follows,

that the implicit Runge-Kutta methods can be of any order of accuracy (Hairer et al, 2007), (Bultheel and Cools,

2010), (Hairer and Wanner, 1996). Therefore, a wide choice of one-step methods of the higher-order appears to

be useful.

The two interpolations that have been integrated2 into the MoL-IAFT-FD in the past are the linear and the

square (Pregla, 2006-a), (Pregla, 2006-b), (Pregla, 2008). They are the representatives of the one-step and mul-

tistep methods, respectively. In principle, the multi-step methods can be built into the MoL-IAFT-FD the same

way as the one-step methods. To keep the papers a little simpler, let’s turn to the one-step methods.

However, higher-order methods are not necessarily more accurate, - because of stability problems or dis-

cretization errors (Bultheel and Cools, 2010). In some cases, an interpolation of the lower order may be even

advantageous (Spiller et al, 2019). This consideration also speaks in favor of a broader choice, including the

lower orders.

Another numerical aspect can also be relevant for complex structures: the mathematical stiffness of the GTL.

In general, differential equations are mathematically “stiff” if they contain some constructs or parameters that

cause rapid variations in the solutions. Here, too, the example of certain photonic crystals would apply. It is

generally difficult to integrate the “stiff” equations by ordinary numerical methods. Small errors may rapidly

accumulate (Bronstein et al, 2005), (Curtiss and Hirschfelder, 1952), (Hairer and Wanner, 1996). In many cases,

the implicit methods that are more tolerant of the stiffness can be considered.

1 The portability represents the effort when the already implemented solution (or its parts) is to be transferred to another structure2 In this sense, the possible term “integrated in ...” should be understood in the sense of a “built-in”. It should be distinguished from

the understanding in the sense of “integration of differential equations” - that is, a numerical solution of these equations.

An enhancement of instruments for solution of GTL 3

The existing experience in industry and research also speaks in favor of a broader choice of methods: The

standard libraries of many universal mathematical software products have a variable-step, variable-order special

solvers of orders 1–53. However, the use of these solvers in the MoL-IAFT-FD is not easily possible; The steps

of the FD solution are integrated into the context of the impedance-/admittance transformation and field transfor-

mation, (see section 1.4). From these considerations, it follows: An appropriate choice of the appropriate type of

interpolation being MoL-IAFT-FD should be an effective aid. For the purpose of applicability, the MoL-IAFT-FD

set of instruments should have the option of a broader selection of integrated methods. These methods should be

easily interchangeable or as portable as possible for the user.

This paper takes up the possibility of incorporating practically every numerical method from the class of one-

step and multi-step methods (Bronstein et al, 2005), (Samarski, 1982), (Samarski, 1986) into the GTL solution.

This is done in a uniform way by modifying transmission matrices. These can be methods of the higher-orders,

explicit and implicit. In this paper, the one-step methods are built into the GTL solution. The installation of some

additional multistep methods would, however, be possible in the same way.

The paper is organised as follows. The section 1 introduces the topic, presents the relevant background and

shows the two already established context-related methods or types of interpolation. The section 2 represents the

proposed extension. The verification of numerical results can be found in the section 3. The conclusions are listed

in the section 4.

1.2 Method of Lines

The Method of Lines (MoL) is a semi-analytical versatile tool for the solution of partial differential equations

(Helfert and Pregla, 2002). In general, one can differentiate between two numerical approaches to realizing the

solutions. In the first, very common approach, the differential equations are solved directly by temporarily “freez-

ing” the partial derivatives, e.g., (Schiesser, 1991), (Hamdi et al, 2007). In the second approach discussed in this

paper, the solution is more oriented towards the purpose of analyzing electromagnetic waves, e.g., (Helfert and

Pregla, 2002),(Pregla and Helfert, 2002), (Helfert et al, 2003), (Barcz, Helfert and Pregla, 2002), (Pregla and

Pascher, 1989), (Pregla, 2008) etc. It is integrated in the individual steps of the impedance-/admittance and field

transformation.

Various complex structures, e.g., microwave technology and optics, can be modeled with it, e.g., fiber grat-

ings, (Pregla, 2004), photonic crystals, (Barcz, Helfert and Pregla, 2002), effects of heat propagation, (Conradi,

Helfert and Pregla, 2001), and many others (Pregla and Helfert, 2002), (Pregla, 2008). The procedure is as fol-

lows: The calculation area is covered with lines. The image of a structure is divided into homogeneous sections

in the direction of the analytical solution. The numerical analysis generally consists of two parts:

– Solving wave equations in homogeneous sections

– Field matching at ports between the homogeneous sections

The wave equations are mostly derived with the help of generalized transmission line equations (GTL) (Pregla,

1999), (Pregla, 2002). Then the wave equations or GTL are discretized. Various discretization schemes are avail-

able for an efficient analysis (Pregla and Helfert, 2002), (Pregla, 2008), (Greda, 2004). They can be set up in

different coordinates, equidistant and non-equidistant, in 2D or 3D (Pregla and Helfert, 2002), (Greda, 2004),

(Pregla, 2008). For most practical cases, the discretization of the coordinates perpendicular to the direction of

propagation is assumed to be favorable (Vietzorreck, 2001). In the case of 3D, for example, the cross-section

of a structure is discretized and the analytical solution is used in the direction of propagation. All details of the

discretization and boundary conditions are shown in (Pregla, 2008).

1.3 Generalized transmission line equations

The Generalized Transmission Line equations (GTL) describe the relationship between the transverse compo-

nents of the electric and the magnetic field. The starting point for deriving the GTL is Maxwell’s equations, taking

into account the boundary conditions of concrete structures. The detailed representation of the general GTL, cor-

responding wave equations, the tensor of the material parameters and the normalization of the linear masses, as

well as the magnetic field strength is shown in (Pregla, 2006-a), (Pregla, 2006-b), (Pregla, 2008). At this point,

the following relevant aspects are briefly repeated. The GTL have the form:

d

duF = QF (1)

3 A solver is a piece of software and can perform basic operations to solve, e.g., the initial value problem for ordinary differential

equations using a slew of numerical methods.

4 Waldemar Spiller

Es applies to any homogeneous section, e.g., for each step of the FD (section 1.5). The vector F contains the

discretized values of the electric and magnetic fields F =[Et,H

t]t

. The generalized coordinate u = x,y,z is

normalized with the free space wave number k0 = ω√µ0ε0 according to u= k0u. In addition, the magnetic field

components Hu are normalized with the wave impedance η0 =√

µ0/ε0 according to Hu = η0Hu. The matrix Q

is given here in a general form:

Q =−[

SE jRH

jRE SH

](2)

In most cases, the submatrices SE,H are equal to zero, except in the case of anisotropy (Pregla, 2002), (Pregla,

2008). Therefore, the GTL for the electrical and magnetic components have the general form:

d

duH =−jREE

d

duE =−jRHH (3)

The structure and content of the matrices RE and RH see the example in the section 1.4 below.

The GTL solution for the entire structure is carried out in two procedures: First, an impedance-/admittance

transformation takes place and then the field transformation. Each of these procedures is carried out for the

entire structure in the FD steps. The field transformation serves to determine the fields in each homogeneous

section or (which is identical) in each step of the FD. The field transformation is only possible if the impedances

and/or admittances of all sections (or all FD steps) have already been determined during the previously run

impedance-/admittance transformation. However, in terms of the paper, it is important that the finite differences

are interpolated for both the impedance-/admittance transformation and the field transformation.

1.4 Impedance-/admittance transformation

In the case of a structure composed of different homogeneous sections, the tangential field components must be

adapted to the transitions. The impedance-/admittance transformation is one such adaptation over the homoge-

neous sections. The two-port network parameters of the sections can be calculated from the conditions for open

circuit and short circuit set at the output of the structure. This calculation is done step by step, section by section,

in the direction from the output of the structure to the input. As a final result, all two-port network parameters of

all sections and thus also their respective loads are known.

With the known z-parameter of the two-port network zij and the wave impedance of the section Z0 applies to

the input impedance at input (subport) A:

ZA = z11− z12 (z22 +Z0)−1

z21 (4)

The admittances (YA,B,0 and yij) can also be used for the same purpose. The (4) clarifies the basic idea. In many

cases, however, it is advantageous to carry out the impedance-/admittance transformation with the aid of the

transfer matrices or their components (see below). These components can be represented by the coefficients

of the GTL or also by the z-parameters. The following expressions of the step n of an impedance-/admittance

transformation are shown. The input impedance of the following section (n+1), ZB, is carried out in relation to

the input impedance-/admittance of the current section ZA. The Z0 is the characteristic impedance of the current

homogeneous section.

ZA = (ZH +VEZB)(VH +YEZB)−1

(5)

YA = (YE +VHYB)(VE +ZHYB)−1

(6)

The possible relationship (e.g. in 1.6) between elements of the transmission matrix VAB and coefficients of the

GTL is:

VAB =

[VE ZH

YE VH

]=

[(I+RHRE)

−1 (I−RHRE) (I+RHRE)−1

j (2RH)

(I+RERH)−1

j (2RE) (I+RERH)−1 (I−RERH)

](7)

The elements VE, ZH, YE and VH are auxiliary quantities; a recursive formula can be set up from them, which is

not considered further here.

The submatrices of the transmission matrix VAB can also be assembled from the z parameters.

Working with transformed quantities often proves to be advantageous (mode or Floquet domain). The paper

deals with the original domain, but the procedure discussed can also be applied to the transformed domains (e.g.,

mode or Floquet domain, (Pregla, 2008)).

An enhancement of instruments for solution of GTL 5

1.5 Impedance-/admittance transformation and field transformation with finite differences

In the case of structures with a complex distribution of the material parameters, e.g., photonic crystals, the

impedance-/admittance transformation and field transformation can be combined with finite differences, (Helfert

and Pregla, 1996), (Pregla, 2008).

The structure is divided into short sections with subports A and B. For sufficiently small distances ∆u = τ

between the subports, the method of finite differences with a corresponding interpolation of the differences can

be used. In the past, the two methods of interpolation were built into the GTL solution, the linear and the square,

(Pregla, 2006-a), (Pregla, 2006-b), (Pregla, 2008) 2.5.3.

This paper takes up the possibility of expanding the usability of the MoL-IAFT-FD. The approach extends

the two previously built-in solution methods in principle to all one-step and multi-step methods of various orders

of accuracy known in numerical mathematics (Bronstein et al, 2005), (Samarski, 1982), (Samarski, 1986). The

methods are built into the transmission matrices VBA and/or VAB for each section of the structure, or in other

words, for each step of the FD.

1.6 Field transformation through sections

During field transformation, the electric and magnetic fields are determined using the known impedance and

admittances. The field transformation runs section by section in the direction from the input to the output of the

structure. The start value of the field at the input of the structure is specified. For example (as in the paper below),

it can be a Floquet basic mode transformed into the original domain.

This paper focuses on the recursive field transformation using the transmission matrices V in the original

domain. The calculation of the field can be done both “forward”, from the input to the output of the homogeneous

section and “backward”, from the output to the input. The side facing the input of the structure is denoted to

as (Subport) A, and the side facing the output of the structure as (Subport) B. The corresponding transmission

matrices VAB and VBA are set up for each individual section A-B. The field transformation occurs according to

FA = VABFB (8)

FB = VBAFA (9)

The vector F consists the discretized values of the electric and magnetic fields:

FA,B =[Et

A,B,Ht

A,B

]t(10)

The content of the transmission matrices consists of the material parameters and the differential operators with

the corresponding boundary conditions. All details are listed in (Pregla, 2006-a), (Pregla, 2006-b), (Pregla, 2008).

At this point, only an optional example is shown without further details. Some results will be obtained with this

example later (section 3.2).

Example: Defect waveguide in a 2D photonic crystal in Cartesian coordinates .

This example relates to a straight defect waveguide in a 2D photonic crystal with hexagonal aligned circular

dielectric rods in air. The content of the coefficients of the GTL RE and RH corresponds to (Pregla, 2008): The

tensor of the material parameters goes into the diagonal matrices ε. The main diagonal represents the dielectric

permeability in the respective cross-section “W ” of the waveguiding structure on the respective homogeneous

section (see Fig. 1). The lengths of the (approximately) homogeneous sections in the y-direction (not shown)

are equal to the step of the finite differences. The H- and E field components are discretized on two H and E

line systems that are shifted from one another to a discretization distance across the direction of propagation

y. These lines are called “◦” and “•” (see the discretization schemes in Fig. 1, above). The matrices D are

differential operators and contain the corresponding boundary conditions. The analytical solution is carried

out in the y-direction, whereby the TM and the TE polarization can be analyzed.

TEy -Polarization:

Ez, Hx, Hy are calculated on the • lines, which are vectors of the fields of the GTL E = Ez and H = −Hx,

with Hy = jµ−1yyD

•xEz, Dz = 0, Ex = 0:

GTL equations: ddy

(−H

•x

)=−jR

TEy

E E•z and ddy

E•z =−jRTEy

H

(−H

•x

)

R•TEy

E = εzz−D•tx µ−1

yyD•x R

•TEy

H = µxx (11)

6 Waldemar Spiller

Fig. 1 Discretization scheme. The wave propagation and the analytical solution are in the y-direction. The bullets and circles are

assigned to E and H lines, respectively. ”W” denotes the width of the structure or 1D discretized cross-section. The structure is

shown schematically and approximately, e.g., without the defect waveguide. It is also assumed that the lengths of the (approximately)

homogeneous sections in the y-direction are small enough that they cannot be represented on the image.

TMy -Polarization:

Hz, Ex, Ey are calculated on the ◦ lines, which are vectors of the fields of the GTL E = Ex and H = Hz. The

third field component is obtained from Ey =−jε−1yyD

◦xHz, Dz = 0, Ez = 0:

GTL equations: ddy

(H◦z

)=−jR

TMy

E E◦x and ddy

E◦x =−jRTMy

H H◦z

R◦TMy

E = εxx R◦TMy

H = µzz−D◦tx ε−1

yyD◦x

(12)

1.7 The interpolation methods already integrated in the MoL-IAFT-FD

Let us first briefly introduce the already built-in interpolations, the linear and the quadratic (Pregla, 2006-a),

(Pregla, 2006-b), (Pregla, 2008).

1.7.1 The linear interpolation according to (Pregla, 2006-a)

The linear interpolation in (Pregla, 2006-a) corresponds to a more general case of the weighted Euler method

(see 2.4) with the parameters α = 1 and σ = 0.5. The accuracy and stability depend on these parameters, and

thus also the quality of the solution for specific structures. Further combination of the parameters is considered.

At this point, however, the representation according to (Pregla, 2008) follows. Despite the complex content, the

general GTL has the form of a partial differential equation of the 1st order

d

duF = QF ←→

dy

du= f(u,y) (13)

It is assumed that the step of the finite differences ∆u = τ is small enough to guarantee the adequacy of the

numerical solution. It is also assumed that the index n is assigned to the input side of the (approximately) ho-

mogeneous section, and n+ 1 corresponding to the output side. Then the recursive formula for the numerical

solution would be

yn+1 = yn+τ

2(fn+1 +fn) (14)

with n= 1,2,3, ...,N and N is the number of discretized points for the finite differences. It is applied to the GTL

for a homogeneous section:

FB− FA

τ=

1

2

(QABFB + QABFA

)(15)

with QAB = Q(um) and um = 0.5(uA+uB).The indices “A” and “B” denote the input and output of the homogeneous section in the direction of propa-

gation. FA and FB are separated and yield

FA =(

I+τ

2QAB

)−1(I−

τ

2QAB

)FB VAB =

(I+

τ

2QAB

)−1(I−

τ

2QAB

)(16)

An enhancement of instruments for solution of GTL 7

or

FB =(

I−τ

2QAB

)−1(I+

τ

2QAB

)FA VBA =

(I−

τ

2QAB

)−1(I+

τ

2QAB

)(17)

The result of this procedure is completely identical to the results of the interpolation of the 1-order in (Pregla,

2008).

1.7.2 The quadratic interpolation according to (Pregla, 2006-b)

The final expressions are shown here. The further details are listed in (Pregla, 2006-a), (Pregla, 2006-b), (Pregla,

2008). The structure and the use of the transmission matrix VBA is shown, for the calculation of the field FB from

the known field FA. There is a difference between the first step (or the first homogeneous section of the structure

or the first step of the FD) and the subsequent sections.

For F2 = V1F1 applies:

V1 =

[(I−

3

4Q1

)+

1

8Q1

(I−

3

8Q2

)−1(I+

3

4Q2

)]−1[(I+

3

8Q1

)+

1

64Q1

(I−

3

8Q2

)−1

Q2

](18)

and for Fn+1 = VnFn applies:

Vn =

(I−

3

8Qn

)−1(I+

3

4Qn−

1

8QnV

−1

n-1

)n≥ 2 (19)

where:

Qn =∆uQ(umn ) umn = 0.5(un+un+1) u1 ≡ uA (20)

and VBA = Vn.

The composition of the transmission matrix VAB is now shown. It should be noted that here, too, when the

calculation process moves from the output of the structure towards the input, the calculation of the first step

(section) differs from the following steps.

For FN-1 = VNFN applies:

VN =

[I+

3

4QN−

1

8QN

(I+

3

8QN-1

)−1(I−

3

4QN-1

)]−1[I−

3

8QN +

1

64QN

(I+

3

8QN-1

)−1

QN-1

](21)

QN =∆uQ(umN ) QN-1 =∆uQ(umN−1) (22)

and for Fn-1 = VnFn:

Vn =

(I+

3

8Qn

)−1(I−

3

4Qn +

1

8QnV

−1

n+1

)n≤N −1 (23)

Interpreting Fn-1 as FA and Fn as FB from the further sections are calculated using VAB = Vn. For further details,

see e.g., (Pregla, 2008).

2 An expansion of the GTL solutions with further one-step procedures

2.1 General considerations

Based on the considerations in section 1.1, it will focus on the one-step methods in a uniform way.

– The transmission matrices VBA and/or VBA serve as “containers” for various built-in methods and as the “end

product” of our considerations

– An individually assembled transmission matrix is required for each step of the analysis procedure

– The calculation of the field with the aid of transmission matrices is in principle equivalent to the recursive pro-

cedures with the z- or y-parameters in the transformed domains (modes or Floquet mode domains). However,

the use of transmission matrices appears to be the easiest way to incorporate other different methods.

Seven further methods are built in: Both explicit and implicit Euler methods of order 1 of accuracy, Euler method

with weighting as a general case for the Euler explicit/implicit approach, two kinds of the Runge-Kutta methods of

order 2 (RK2-I and RK2-II), classical explicit Runge-Kutta method of order 4 (RK4) and implicit Gauss-Runge-

Kutta method of order 4 (GRK4), (Hoellig, 2011), (Hoellig, 1998). These are tested on simple and “difficult”

structures, and the quality of the results is compared with one another and with the already known solutions (see

section 3).

8 Waldemar Spiller

2.2 Euler method explicit

The relation can be written asyn+1−yn

τ= αfn =⇒ yn+1 = yn+ατfn (Samarski, 1986). This is used to solve

the GTL:

FB = FA +ατ QnFA =(

I+ατ Qn

)FA (24)

It is formally

Qn =∆uQ(umn ) umn = 0.5(un+un+1) u1 ≡ uA (25)

although applies to the entire homogeneous section.

VAB =(

I+ατ Qn

)−1

VBA =(

I+ατ Qn

)(26)

The subscript “AB” shows that the matrices VAB and VBA only applies to the respective section.

2.3 Euler method implicit

The implicit procedureyn+1−yn

τ= αfn+1 =⇒ yn+1 = yn +ατfn+1 is assumed, (Samarski, 1986). α is a

selectable dimensionless parameter. For a better comparison of the methods, α = 1/2 is chosen. This results in

terms of (13) in

FA = FB−ατ QnFB =(

I−ατ Qn

)FB (27)

It all results in

VAB =(

I−ατ Qn

)VBA =

(I−ατ Qn

)−1

(28)

2.4 Euler method with weighting (Euler-W)

The procedure with weighting is a general case (also for the linear Interpolation in (Pregla, 2006-a), see 1.7.1) in

which one can use the parameter σ to vary the properties of the method between explicit and implicit, (Samarski,

1986):

yn+1−yn

τ= α(σfn+1 +(1−σ)fn) =⇒ yn+1 = yn+ατ (σfn+1 +(1−σ)fn) (29)

It is used to solve the GTL:

FB− FA

τ= ασQnFB +α(1−σ) QnFA (30)

It results in

FB =(

I−ατσQn

)−1(I+ατ (1−σ) Qn

)FA (31)

or

VBA =(

I−ατσQn

)−1(I+ατ (1−σ) Qn

)VAB = V

−1

BA (32)

α= 1 and σ = 0.8 is selected for comparability with the other methods.

2.5 Runge–Kutta methods

The possible Runge-Kutta methods of the second and fourth order of accuracy, (Samarski, 1982), (Samarski,

1986), are built-in into all the solution steps of the GTL.

An enhancement of instruments for solution of GTL 9

2.5.1 Second-order methods (RK2)

In general, the calculations of the 2nd order method are performed in two steps. First, the intermediate result ynis found according to the Euler scheme with the step length ατ :

yn = yn+ατf(un,yn) (33)

In the second step yn+1 is found:

yn+1 = yn+ τ(1−σ)f(un,yn)+στf(un+ατ ,yn) (34)

where α > 0 and σ > 0 are the selectable parameters. After excluding yn, the Runge-Kutta scheme of the 2nd

order results inyn+1−yn

τ= (1−σ)f(un,yn)+σf(un+ατ ,yn+ατf(un,yn)) (35)

The order of accuracy depends on the parameters α and τ . Satisfying the condition σα = 1/2 results in a

schemes family (35) of the 2nd order of accuracy (Samarski, 1982), (Samarski, 1986).

2.5.2 The predictor-corrector scheme of the 2nd order, variant I (RK2-I)

The variant α= 1/2, σ = 1 is being considered. This is the well-known scheme “predictor-corrector” (Samarski,

1982), (Samarski, 1986). It can be represented in the form

yn = yn+τ

2f(un,yn), yn+1 = yn+ τf

(un+

τ

2,yn

)(36)

or, after excluding yn,yn+1−yn

τ= f

[un+

τ

2,yn+

τ

2f (un,yn)

](37)

After applying (34) to the parameters of the GTL, the result is

FB− FA

τ= Qn

(FA +

τ

2QnFA

)(38)

FB =[τ Qn

(I+

τ

2Qn

)+ I]

FA (39)

VBA = τ Qn

(I+

τ

2Qn

)+ I VAB = V

−1

BA (40)

2.5.3 The predictor-corrector scheme of the 2nd order, variant II (RK2-II)

α= 1 and σ = 1/2: This scheme can also be interpreted as a “predictor-corrector” (Samarski, 1982), (Samarski,

1986): The Euler scheme with the step τ (predictor)

yn = yn+ τf(un,yn) (41)

is calculated first, then the scheme with half the sum (corrector):

yn+1−yn

τ=

1

2[f (un,yn)+f (un+1,yn)] (42)

After excluding yn:yn+1−yn

τ=

1

2[f (un,yn)+f (un+1,yn+ τf (un,yn))] (43)

After applying (43) to the parameters of the GTL, the result is

FB− FA

τ=

1

2

[QnFA + Qn

(FA + τ QnFA

)](44)

and

FB =[τ

2Qn

(2I+ τ Qn

)+ I]

FA (45)

that results in

VBA =τ

2Qn

(2I+ τ Qn

)+ I VAB = V

−1

BA (46)

10 Waldemar Spiller

2.5.4 The 4th order Runge-Kutta method (RK4)

yn+1−yn

τ=

1

6[K1 +2K2 +2K3 +K4] (47)

where, in according to (Samarski, 1982), (Samarski, 1986),

K1 = f (un,yn)⇒ K1 = QnFA (48)

K2 = f

(un+

τ

2,yn+

τ

2K1

)⇒ K2 = Qn

(FA +

τ

2QnFA

)=(

I+τ

2Qn

)QnFA (49)

K3 = f

(un+

τ

2,yn+

τ

2K2

)⇒ K3 = Qn

[FA +

τ

2Qn

(FA +

τ

2QnFA

)]=[I+

τ

2Qn

(I+

τ

2Qn

)]QnFA (50)

K4 = f (un+ τ ,yn+ τK3)⇒ K4 = Qn

{FA + τ

[Qn

(FA +

τ

2Qn

(FA +

τ

2QnFA

))]}=

={

I+ τ Qn

[I+

τ

2Qn

(I+

τ

2Qn

)]}QnFA (51)

The coefficients are considered without a field FA, where: K1,2,3,4 = Kwf

1,2,3,4FA. It results in

FB = FA +τ

6

(K

wf

1 +2Kwf

2 +2Kwf

3 + Kwf

4

)FA

FB =[I+

τ

6

(K

wf

1 +2Kwf

2 +2Kwf

3 + Kwf

4

)]FA

(52)

VBA = I+τ

6

(K

wf

1 +2Kwf

2 +2Kwf

3 + Kwf

4

), VAB = V

−1

BA (53)

2.5.5 The 4th order Gauss-Runge-Kutta implicit method (GRK4)

The implicit method, according to Gauss-Runge-Kutta, (Seyrich, 2016), (Grothmann, 2012), (Grothmann, 2015),

(Hoellig, 2011), (Hoellig, 1998), (Pulch, 2020) has the order of convergence 4, the order of accuracy 5 and is

suitable for stiff GTL. The predictor is shown first:

yn+1−yn

τ=

K1 +K2

2(54)

where K1 and K2 are given implicitly:

K1 = f

(un+

3−√

3

6τ ,yn+

1

4τK1 +

3−2√

3

12τK2

)(55)

K2 = f

(un+

3+√

3

6τ ,yn+

3+2√

3

12τK1 +

1

4τK2

)(56)

The expressions (55) and (56) therefore require an iterative procedure (corrector steps) for K1 and K2. These

iterations are carried out on each step n of the FD (n = 1,2,3, ...,N). K1(0) = f(un,yn) and K2

(0) = K1 can

serve as starting values (predictor step). The criterion for the end of the iterations depends on the convergence of

the result: The iterations are ended, for example, as soon as the difference between two successive improvements

is within the specified tolerance limits.

Next, this consideration is used to solve the GTL with MoL-IAFT-FD. For the predictor

FB− FA

τ=

K1 + K2

2(57)

K1 and K2 are put together according to (55) and (56):

K1 = Qn

(FA +

1

4τ K1 +

3−2√

3

12τ K2

)(58)

K2 = Qn

(FA +

3+2√

3

12τ K1 +

1

4τ K2

)(59)

An enhancement of instruments for solution of GTL 11

In order to determine the transmission matrices, the field FA is separated from (58) and (59):

K1 = K(w)

1 FA K2 = K(w)

2 FA (60)

FB =[I+

τ

2

(K

(w)

1 + K(w)

2

)]FA (61)

The starting values for the predictor step are

K(w0)

1 = Qn

(FA +

1

4τ QnFA +

3−2√

3

12τ QnFA

)K

(w0)

2 = K(w0)

1 (62)

Thus the transmission matrices are

VBA = I+τ

2

(K

wf

1 + Kwf

2

)VAB = V

−1

BA (63)

As a summary for the GRK4: The procedure for calculating for each section n is as follows:

1. Assembling the matrices RE and RH from the material parameters and the differential operators, see (1) and

(2).

2. Assembling the matrix Qn from the RE and RH. The isotropy of the material parameters in (2) is assumed,

therefore SE,H = 0, i.e., (Pregla, 2002).

3. Setting of the start values K(w0)

1 and K(w0)

2 and an iterative improvement of the values K(w)

1 and K(w)

2

4. Assembling the transmission matrices VBA and/or VAB

3 Verification

3.1 Test procedure

To verify the function of the methods as part of the MoL-IAFT-FD and to recognize some of their properties,

some test structures are used. Dielectric, periodic and infinitely long 2D test structures are used for simplicity

and efficiency. The structures are analyzed by calculating Floquet modes for only one period (Helfert and Pregla,

1998), (Pregla, 2004), (Pregla, 2008). However, the corresponding parameters of the eigenmodes of a period, i.e.,

Floquet modes, as well as transformation rules, must first be determined. They then apply to the beginning of A

and the end of B of each period.

For the purpose of further simplification, a symmetry of the material parameters in the direction of propaga-

tion is considered. Otherwise, the spatial and the longitudinal distribution of the material parameters is selected

empirically so that the quality of the interpolation can be tested on it, e.g., sharp and smooth transitions in the

spatial distribution of the material parameters. The working frequencies were chosen far enough away from the

Bragg frequency.

A full test for all conceivable structures, scaling and requirements is hardly possible: In this case, an applica-

tion of the basic principles and experience of the software test appears helpful, see e.g., (ISTQB, 2021), (Spillner

and Linz, 2019). Therefore, the aim of a rudimentary examination of the function and basic properties of the new

methods appears to be rational.

The testing procedure is as follows. A period of a structure is 2D discretized - in the propagation direction

and across it. The test consists of several identical test calculations. With each next calculation, the number

of discretization points in the propagation direction is increased, i.e., Ni ∈ [20,30,40, ...,1000]. For each i-th

calculation, a reference excitation EA,f that is the same for all calculations is assumed and the resulting energy

transported through the periodic structure S(P)A,B is examined. If the FD calculation is stable, the values of the

energy transport S(P)A,B = f(Ni) converge to a value that is characteristic of the given structure. The course of

this convergence can serve as a quality feature for each interpolation method used. The subscript “f” stand for a

forward propagation part of the entire field.

The Floquet-modal matrices can be calculated from the eigenvalue problem

Lh = S−1E z11hy11hSE = S−1

H y11hz11hSH (64)

very efficiently by using the open- and short circuit matrix parameters of half the periods (Pregla, 2008) (s.

“impedance-/admittance transformation” in section 1.4). The subscript “h” symbolizes half of the period. In the

12 Waldemar Spiller

case of the symmetrical period z22 = z11 and z12 = z21. The relation between Lh and the Floquet modes phase GF

is:

tanh

(1

2GF

)= L

− 12

h = I/√

S−1E z11hy11hSE = I/

√S−1

H y11hz11hSH (65)

The characteristic wave impedance of any structure in the Floquet domain is an adequate unit matrix Z0 = I. It is

transformed into the original domain using the Floquet-modal matrices SE, SH specific to each structure:

Z0 = SEIS−1H (66)

The excitation is simulated by the vector of the forward propagating Floquet modes:

EA,f = [1,0, · · · ,0]t (67)

The first Floquet mode, the basic mode, is used as a test field. An arrangement of the Floquet-modal matrix SE,H

is assumed that the basic mode can be found in the first column. The vector EA,f is also transformed into the

original domain:

EA,f = SEEA,f (68)

Before that, the z-parameters of the analyzed structure are calculated using the impedance-/admittance transfor-

mation. The following applies to the input impedance:

ZA = z11− z12 (z22 +Z0)−1

z21 (69)

The magnetic field distribution at input A is

HA = 2(ZA +Z0)−1

EA,f EA = ZAHA (70)

and the energy flux density (energy transport) from the Poynting vector

S(P)A = Et

AH∗A (71)

The calculation for input A is sufficient because the output should ideally have exactly the same values. How-

ever, the additional calculation (the same values) for output B can be useful for verification purposes. The field

distribution at output B would then be

HB = (z11 +Z0)−1

z12HA EB = Z0HB (72)

and the energy transport at the output of the periodic section is

S(P)B = Et

BH∗B (73)

3.2 Numerical results and discussion

3.2.1 Findings of general characters

First, all methods of interpolation, both the two already established and all-new, deliver comparable results.

Overall, the differences are in the order of 0.1-2.5% of accuracy.

Second, the results are comparable to those in (Pregla, 2006-b) or (Pregla, 2008) (in the range 0.02-0.07% of

the final result of convergence). However, one should consider: In the example in (Pregla, 2006-b) there should be

significantly fewer modes capable of propagation than in the example in PPP (the conditions see e.g., in (Jahns,

2001)).

So the agreement can be rated as very good.

Thirdly, the accuracy of the individual methods for concrete structures or their scaling with regard to the

working wavelength is different. This shows that being able to choose an optimal method for specific conditions

is helpful.

3.2.2 Typical convergence curves

As a typical example, the convergence curves for the old and new methods are shown in Fig. 3.2.2-3.2.2.

An enhancement of instruments for solution of GTL 13

Fig. 2 Convergence curves for a symmetrical period with a longitudinal pulse profile, n = 1.444− 1.4495. The

length of the period is λ = 1550 nm. The pulse width is λ/2. The smoothing is carried out using the

Savitzky-Golay filter method by the range of 20% of the total number of data points.

Fig. 3 Convergence curves for a symmetrical period with a smooth longitudinal course of the refractive index as a

semicircle. Not smoothed. Otherwise, the data are as in the previous picture.

14 Waldemar Spiller

Fig. 4 Convergence curves for a symmetrical period with a smooth longitudinal course of the refractive index as a

semicircle. Not smoothed. Otherwise, the data are as in the previous picture.

Fig. 5 Example of a straight defect waveguide in a 2D photonic crystal with hexagonal aligned circular dielectric

rods (one period).

Fig. 6 Convergence curves for the photonic crystal similar to the image 3.2.2, the number of the rods per width

of the structure is doubled. n= 1.444−1.4495. The length of the period is 1.9016 nm, λ= 1550 nm. The

width of the structure is 68λ/2.

An enhancement of instruments for solution of GTL 15

3.2.3 About the accuracy of the solution for concrete structures

The next aspect is the different suitability of different methods for different use cases. As discussed in section 3, a

complete test was hardly possible. The following test scenarios for periodic dielectric structures were considered:

– The period of the longitudinal variation of the refractive index is significantly smaller than that required for

the Bragg reflection (sub-wavelength range).

– The period of the longitudinal variation of the refractive index is in the range of the Bragg reflection or greater

(with a fixed working wavelength of 1550 nm).

– The variation in the width of the structure.

– The variation of the refractive index contrast

In all of these scenarios, the solution behaved as expected using all methods.

In the sub-wavelength range: All methods were found to be relatively insensitive to spatial variations in the

refractive index (the contrast was constant). A profile-independent ranking of the methods according to their

accuracy became visible. This remained qualitatively the same for different refractive index profiles (step index

profile, cosine, parabole, circle, photonic crystal), i.e., Fig. 3.2.3. Apparently, the wave “sees” an effective index

of refraction rather than its subtle local variations. As expected, increasing the contrast (and thus the effective re-

fractive index) required a finer discretization in order to avoid any numerical instability that might have occurred.

Fig. 7 The convergence curves are similar to the Fig. 3.2.2, but for the length of the period 0.5λ= 1550 nm. The

smoothing is carried out using the Savitzky-Golay filter method by the range of 20% of the total number of

data points.

With larger spatial periods of the refractive index variations: As expected, a noticeable dependence of the

precision of the methods on different profiles of the refractive index and the spatial scaling of the profiles was

found.

Compared to some others, a certain method can be both less precise and more precise - depending on the

profile of the refractive index and its scaling with respect to the working wavelength (Fig. 3.2.2-3.2.2). The

structures such as the defect waveguide in the photonic crystal can be challenging for the methods (Fig. 3.2.2).

As expected, the increase in the width of the structure also made a possibly finer discretization necessary -

because of more and more propagating modes with finer spatial field distribution.

To illustrate that the suitability of one or another method can depend on the specific application conditions,

e.g., the spatial distribution of the material parameters and/or the scaling with regard to the working wavelength,

an example of some selected applications is shown in the table 1.

16 Waldemar Spiller

Table 1 Methods ranking of accuracy vs. length of the period. The width of the structure is 17λ, with the Neumann boundary con-

ditions on both sides. The relative deviations in accuracy are in the range of 0.1-2.5%. The refractive index varies in the range of

1.444−1.4495.

more accurate←→ less accurate

ε: Grating Remark

0.25λ RK4 Euler all others * * of the same accuracy

0.5λ RK4 EulerW Lin/Quad GRK4 RK2 Euler

1λ Euler RK4 EulerW Line GRK4 Quad RK2

2λ EulerW Line Quad GRK4 RK2 RK4 Euler

3λ Euler EulerW Line GRK4 Quad RK2 RK4

4λ EulerW Euler Line Quad RK2 RK4/GRK4 * * big sporadic errors

5λ Line Quad Euler EulerW RK2/GRK4 RK4 instable

ε: Arc ∩ Remark

0.25λ Euler * RK4 * all others * faster convergence

0.5λ RK2 Quad GRK4 Line EulerW RK4 Euler

1λ Euler RK4 EulerW Line GRK4 Quad RK2

2λ EulerW Line GRK4 RK2 Euler Quad RK4

3λ Line Euler EulerW GRK4 Quad RK2 RK4

4λ Euler GRK4 Line Quad EulerW RK2 RK4

5λ Quad Line Euler EulerW/RK2 GRK4 RK4 instable

4 Conclusions

An enhancement of instruments for solution of general transmission line equations with method of lines, impedance-

/admittance and field transformation in combination with finite differences has been proposed. It was shown that

the precision of a method can depend in an individual way on the refraction index profile of the structure used or

on its scaling. Therefore, a broader choice of different methods is a helpful aid when there are increased demands

on the precision or on the stability of the solution.

Acknowledgment

The analysis presented here was based on the excellent theory and earlier works by Univ. Prof. Dr. Reinhold

Pregla. His advice is greatly appreciated.

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