Outlook

So far, the ability of modeling 2-D optical devices has been illustrated and discussed.

Additionally, further examples are shown in Appendix C.4. Nevertheless, there are effects that might be of importance characterizing realistic designs which have not been included in the simulation so far. This subsection aims to give a brief outlook of ideas and possible obstacles when including Gaussian Beams and consider planar devices in form of slabs and layered structures in the modeling.

Usually the term Gaussian Beam refers to 3-D wave, in planar optics however, the term refers to wave that is propagating in thexy-plane with a Gaussian like amplitude distribution perpendicular to the propagation direction and in z-direction it matches the assumptions (for the CIM, it is constant). To consider an excitation with such a Gaussian beam in

3.3 Modeling for Optical Applications

0.00.2 0.40.6 0.81.0 1.21.4 1.6

|Ez|(V/m) (a)

CIM FIT

(b)

0 2 4 6 8

Path length (Λ) 0

0.5 1 1.5 2

|Ez|(V/m) (c)

0 2 4 6 8 10

Path length (Λ)

(d)

Figure 3.21: Magnitude of electric field along the path illustrated in Figure 3.19a for the normalized frequencies (a) ^Λ_λ = 0.03, (b) ^Λ_λ = 0.167, (c) ^Λ_λ = 0.37, (d) ^Λ_λ = 1.67 computed with FIT using [171] and CIM with the proposed extensions. Adapted from [69].

CIM, the Gaussian Beam could be expanded into a series of plane waves analogous to the approach in [172].

Planar optical devices are not bounded by conductors inz-direction and thus the fields of the propagating modes extend to infinity. Considering a homogeneous dielectric with respect to the z-direction, the fundamental modes are plane waves in TM and TE polarization.

The TM polarization is modeled directly by the classical CIM. The TE polarized case can be modeled by making use of duality as described in Section 2.9.

Dielectric slab waveguides can be seen as a layered media, where the dielectric is placed between semi-infinite vacuum domains. Like proposed in [111, 138] for the modeling of TSVs, layered media could be included for the modeling of optical devices by considering an effective complex wavenumber obtained with the TRM. However, a problem is expected to occur when enforcing the boundary condition at inclusions to model inhomogeneous substrates with respect to the xy-plane. Because formally, a mode matching has to be conducted at any interface. In the case of a homogeneous dielectric with respect to the z-axis this problem did not occur, as the modes in z-direction remain orthogonal.

When introducing layered materials, the modes are no longer orthogonal and the effect of evanescent fields that are excited needs to be taken into account. Another effect that may cause problems is the fact that propagating modes in asymmetrically layered material in

4.351 mA/m 13.051 mA/m

(a)

0.062 mA/m 15.986 mA/m

(b)

0.008 mA/m 26.500 mA/m

(c)

0.013 mA/m 28.367 mA/m

(d)

Figure 3.22: Magnitude of magnetic field for a plane wave incident on a finite photonic crystal as illustrated in Figure 3.19b computed with the FIT using [158] on the left side and the CIM withK = 5 on the right side. For the normalized frequencies of (a) ^Λ_λ = 0.11, (b)

Λ

λ = 0.563, (c) ^Λ_λ = 1.13, (d) ^Λ_λ = 1.68. (c) is adapted from [69].

infinite space (with respect to the z-direction) become leaky⁷. It is not clear if a complex effective propagation constant sufficiently takes this effect into account.

7Meaning the wave vector does not lie in thexy-plane.

3.4 Summary

02 46 108 1214 16

|Hz|(mA/m) (a)

CIM FIT

(b)

0 2 4 6 8

Path length (Λ) 0

5 10 15 20 25

|Hz|(mA/m) (c)

0 2 4 6 8 10

Path length (Λ)

(d)

Figure 3.23: Magnitude of magnetic field along the path illustrated in Figure 3.19b for the normalized frequencies (a) ^Λ_λ = 0.11, (b) ^Λ_λ = 0.563, (c) ^Λ_λ = 1.13, (d) ^Λ_λ = 1.68 computed with FIT using [171] and CIM with the proposed extensions.

3.4 Summary

After the CIM has been introduced and extended in Chapter 2, this chapter focused on the application of the CIM for the modeling of SI/PI related problems, microwave devices, and 2-D optical devices. In order for that, the idea of physics-based modeling has been introduced and employed throughout the chapter and different modifications and extensions have been proposed to apply the CIM within the scheme of physics-based modeling in the three areas of application.

The term physics-based modeling is understood as the idea to dissect a problem with a complex geometry and complex field distribution into smaller, easy to handle problems where the dominant electromagnetic effect (or field pattern) can be represented in an efficient way. Usually these smaller problems are represented by network parameter blocks of which the parameters are obtained from equivalent circuit models, analytical or semi-analytical models, or full-wave simulations of small size. In general the terms of physics-based modeling has to be seen as more general and also allow for the modeling of more complex systems.

A short review of physics-based modeling of vias, known as PBV model, was given. In the conventional PBV model, the CIM provides the model for the waves propagating inside a single cavity of a multilayer PCB. In an application example, the extensions to the

CIM proposed in Sections 2.3 and 2.4 are applied to model a realistically shaped PCB with a PCPL aimed to reduce SSN by implementing dielectric rods in the substrate. The conventional PBV model in combination with the proposed extension to CIM has proven to be applicable to this problem and provided accurate results. The computational benefit of the shown approach to full-wave simulations is more than one order of magnitude.

Consecutively, the applicability of the conventional PBV model to SIW structures has been studied. The approach has shown to been applicable and a novel PBV model for the efficient modeling of straight SIW in multilayer PCBs has been proposed. The speedup of the conventional PBV model as it is used also for SI/PI problems compared to full-wave simulations is again about one order of magnitude. The proposed novel PBV model models SIWs as transmission lines and, hence, further accelerate the computations by four to tive orders of magnitude. Next, the limits of the conventional PBV model to the application of planar microwave devices when microstrip lines are employed were illustrated.

The third application area which is the modeling of optical devices has been outlined by the modeling of photonic crystals. Here, the excitation with plane waves and the modeling of the TE polarization has been illustrated and validated. For the modeling of photonic crystals, the CIM shows a speedup of several orders of magnitude compared to 3-D full-wave simulations. Ideas for other extensions that become necessary when including more geometric features, like dielectric slabs, have been discussed in an outlook.

This chapter demonstrated the applicability of the CIM to various kinds of application examples and concludes the discussions about the deterministic CIM and physics-based models. In the next chapters, the introduced models will be considered to be stochastic and analyzed by the means of PCE.

Chapter 4

Polynomial Chaos Expansion (PCE) for Uncertainty Quantication

Up to this point, this thesis was about deterministic modeling of planar wave propagation and components where the effect of planar wave propagation is dominant. In this chapter, the stochastic modeling approach of PCE will be introduced and applied. Before applying PCE to physics-based models and the CIM in consecutive chapters, PCE will be used to stochastically model simple expressions and circuits that arise in the context of physics-based modeling. These expressions will be used to apply PCE in a comprehensive way to obtain approximations for stochastic values in an analytical fashion. Furthermore, the consideration of stochastic network blocks, their properties, efficient numerical generation, and connection will be discussed. Some of the approaches proposed and results shown in this section have been published previously in [8,10–12,14].

This chapter is organized as follows: first, an overview of the state-of-the-art regarding stochastic modeling in electromagnetics is given in order to position PCE in this context before PCE itself is introduced. Then, Stochastic Galerkin Method (SGM) is introduced and some properties and aspects of the underlying mechanisms are discussed. Next, PCE in terms of SGM is applied to simple models in an analytical fashion and a method to efficiently obtain applying it is introduced. The final section of this chapter deals with the efficient handling of problems depending on multiple stochastic variables where the underlying model consists of connected or cascaded subsystems depending only on a subset of variables. The proposed approach will be formulated in the form of network parameter blocks, but is generally applicable to various kinds of models. Application examples and further validation of the methods and techniques introduced and discussed in this chapter can be found in Chapter 5.