High Accuracy Electrodynamic Calculationsusing Hermite Finite Elements

A MAJOR QUALIFYING PROJECT REPORTSubmitted to the Faculty of

WORCESTER POLYTECHNIC INSTITUTEin partial fulfillment of the requirements for the

DEGREE OF BACHELOR OF SCIENCEin

PHYSICSand in

ELECTRICAL AND COMPUTER ENGINEERINGby

Siddhant Pandey

Submitted: April 26, 2018

Advisor: L. R. Ram-Mohan

This report represents the work of a WPI undergraduate student submitted to the faculty as evidence ofcompletion of a degree requirement. WPI routinely publishes these reports on its website without editorialor peer review. For more information about the projects program at WPI, please seehttp://www.wpi.edu/academics/ugradstudies/project-learning.html

AbstractWe show that the present discretization approaches for the solution of Maxwell’s equations formulti-scale physics have limitations that can be overcome through the use of C(1)-continuousHermite interpolation polynomials. Our approach of calculating fields in a variational for-mulation, using Hermite polynomials in the finite element method (HFEM), yields bet-ter accuracy by several orders of magnitude than comparable applications of the so-callededge-based vector finite element method (VFEM). We note that VFEM-based commercialsoftware packages that are widely used yield “pixellated” solutions, and typically have ill-defined vector solutions at nodes. We reexamine the issue of removing spurious unphysicalzero-frequency solutions. (i) We employ our results to investigate the group theory of fieldsin an empty cubic metallic cavity to explain the level degeneracy that is much larger thanwhat is to be expected from the Oh symmetry of the cube. This is identified as an exam-ple of “accidental degeneracy," and is explained in detail. (ii) We show that the inclusionof a smaller dielectric cube of relative permittivity ε2 within the cubic cavity leads to theremoval of accidental degeneracy so that the eigenfields have the geometrical symmetry ofthe cube, Oh. The behavior of the mode frequencies as the ratio ε2/ε1 is varied is shown.(iii) A further reduction of symmetry is obtained by allowing the dielectric function in theenclosed cube to have a linear dependence on z. This mimics the effect of an electric field ona quantum well. The method delineated here should be particularly effective in obtainingresults for scalar-vector coupled field problems, such as modeling quantum cavity well lasersand plasmonics.

i

AcknowledgementsThis project has been a rewarding journey, and I have many people to express my gratitudeto. Firstly, I would like to thank my research advisor, Prof. L. R. Ram-Mohan, for allthe opportunities he has provied me with, over the past few years. Working with him hastaught me a lot, and has made me a better researcher. He has been very resourceful. I wishhim the best for the future. Secondly, I would like to thank my colleague and collaborator,Sathwik Bharadwaj, for sharing his own research experience with me, and for his advice ona multitude of things. His work on parallel computing laid the foundation for this project.

I would like to thank my collaborators at Michigan State University, Prof. J. D. Albrecht,and his student Marco Santia. Their expertise on the vector finite element method wasuseful for this project. I would also like to thank my dear friend Dominic Chang, for fruitfuldiscussions, and sharing my wonder for science.

I thank the Center for Computational NanoScience (CCNS) at Worcester PolytechnicInstitute, for the computational resources. I would also like to acknowledge the Office of Artsand Sciences for a summer research fellowship, and a generous travel funding to represent myresearch at the APS March meeting 2018. Finally, I am grateful to all the previous membersof my research group, whose work has aided my research.

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Contents

1 Introduction 1

2 The HFEM Formulation 42.1 The Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Origin and Nature of Spurious Solutions . . . . . . . . . . . . . . . . . . . . 62.4 The Penalty Method and the Divergence-free Constraint . . . . . . . . . . . 6

3 Fields in Empty Cubic Cavity 10

4 Accidental Degeneracies in Electromagnetic Cavities 17

5 Fields in Dielectrically Loaded Cubic Cavity 22

6 Concluding Remarks 38

Appendix 40

iii

List of Figures

2.1 Spurious solutions in the empty cubic cavity of unit dimensions, with divergence-to-curl ratio equal to (a) 1783.8, (b) 1481.7, and (c) 380.1. Note the fielddistribution, which is indicative of the presence of sources, even though thecavity is empty. Light yellow regions correspond to amplitude antinodes, andlight blue regions correspond to amplitude nodes. . . . . . . . . . . . . . . . 7

2.2 Explicity imposing the divergence-free condition by performing global matrixrow and column operations. One of the derivative DoF is eliminated in favorof the remaining two. Here X = 1 for the left-hand side matrix, and X ≈ 105

for the right-hand side matrix, in the generalized eigenvalue problem. Thischoice of a large number pushes the eigenvalue of the redundant 1×1 subspaceout of the spectral range of interest. . . . . . . . . . . . . . . . . . . . . . . . 8

3.1 The convergence of errors in eigenvalue of the first mode for the 3rd orderVFEM, 5th order VFEM, cubic Hermite, and quintic Hermite interpolationpolynomials. Using quintic Hermite interpolation polynomials we can reducethe error in the first mode upto 10−9 with just 27 elements and 8232 DoF,with further reduction in error possible with mesh refinement. The total DoFcorresponds to the global matrix dimension. . . . . . . . . . . . . . . . . . . 11

3.2 The convergence of errors in eigenvalue of the higher frequency modes forquintic HFEM only are shown for the case of an empty cubic cavity of unitdimensions. The total DoF corresponds to the global matrix dimension. . . . 12

3.3 The convergence of the errors in eigenvalues of the first mode in COMSOL,HFSS, MFEM, and HFEM. The total DoF corresponds to the global matrixdimension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.4 Triply degenerate modes (a) (0, 1, 1), (b) (1, 0, 1), and (c) (1, 1, 0) for theempty cavity of unit dimensions with eigenvalue k2

0 = 19.7392. Light yellowregions correspond to amplitude antinodes, and light blue regions correspondto amplitude nodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.5 Doubly degenerate mode (1,1,1) for the empty cavity of unit dimensions witheigenvalue k2

0 = 29.6088. Light yellow regions correspond to amplitude antin-odes, and light blue regions correspond to amplitude nodes. . . . . . . . . . . 15

iv

5.1 Schematic of a dielectrically loaded cavity with Oh symmetry. The conduct-ing cavity has dimensions 1 × 1 × 1 mm3; the cubical dielectric loading hasdimensions 0.5× 0.5× 0.5 mm3, with dielectric constant ε2. The permittivityin the rest of cavity is ε1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.2 Symmetry-adapted triplet (0, 1, 1), (1, 0, 1), (1, 1, 0) of the dielectrically loadedcavity with eigenvalue k2

0 = 18.5267. The conducting cavity has dimensions1 × 1 × 1 mm3; the cubical dielectric loading has dimensions 0.5 × 0.5 × 0.5mm3, with dielectric constant ε2/ε1 = 1.2. Light yellow regions correspond toamplitude antinodes, and light blue regions correspond to amplitude nodes. . 24

5.3 Symmetry-adapted doublet (1, 1, 1) of the dielectrically loaded cavity witheigenvalue k2

0 = 28.8995. The conducting cavity has dimensions 1 × 1 × 1mm3; the cubical dielectric loading has dimensions 0.5× 0.5× 0.5 mm3, withdielectric constant ε2/ε1 = 1.2. Light yellow regions correspond to amplitudeantinodes, and light blue regions correspond to amplitude nodes. . . . . . . . 25

5.4 Symmetry-adapted singlets (1, 1, 3), (1, 3, 1), (3, 1, 1) of the dielectrically loadedcavity with eigenvalues k2

0 = 107.4315, 107.5911. The conducting cavity hasdimensions 1 × 1 × 1 mm3; the cubical dielectric loading has dimensions0.5×0.5×0.5 mm3, with dielectric constant ε2/ε1 = 1.2. The states transformlike the 1D representations (a) A1g, and (b) A2g of the group Oh. Light yellowregions correspond to amplitude antinodes, and light blue regions correspondto amplitude nodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.5 First symmetry-adapted doublet (1, 1, 3), (1, 3, 1), (3, 1, 1) of the dielectricallyloaded cavity with eigenvalue k2

0 = 103.7275. The conducting cavity hasdimensions 1 × 1 × 1 mm3; the cubical dielectric loading has dimensions0.5 × 0.5 × 0.5 mm3, with dielectric constant ε2/ε1 = 1.2. Light yellow re-gions correspond to amplitude antinodes, and light blue regions correspondto amplitude nodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.6 Second symmetry-adapted doublet (1, 1, 3), (1, 3, 1), (3, 1, 1) of the dielectri-cally loaded cavity with eigenvalue k2

0 = 107.5967. The conducting cavityhas dimensions 1 × 1 × 1 mm3; the cubical dielectric loading has dimensions0.5 × 0.5 × 0.5 mm3, with dielectric constant ε2/ε1 = 1.2. Light yellow re-gions correspond to amplitude antinodes, and light blue regions correspondto amplitude nodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5.7 Surface currents of the triply degenerate modes (a) (0, 1, 1), (b) (1, 0, 1), and(c) (1, 1, 0) of the dielectrically loaded cavity with eigenvalue k2

0 = 18.5267.The conducting cavity has dimensions 1 × 1 × 1 mm3; the cubical dielectricloading has dimensions 0.5×0.5×0.5 mm3, with dielectric constant ε2/ε1 = 1.2.Light yellow regions correspond to current antinodes, and light blue regionscorrespond to current nodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

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5.8 Surface currents of the doubly degenerate mode (1, 1, 1) of the dielectricallyloaded cavity with eigenvalue k2

0 = 28.8995. The conducting cavity has dimen-sions 1×1×1 mm3; the cubical dielectric loading has dimensions 0.5×0.5×0.5mm3, with dielectric constant ε2/ε1 = 1.2. Light yellow regions correspond tocurrent antinodes, and light blue regions correspond to current nodes. . . . . 30

5.9 Surface currents of the symmetry-adapted singlets (1, 1, 3), (1, 3, 1), (3, 1, 1) ofthe dielectrically loaded cavity with eigenvalues k2

0 = 107.4315, 107.5911. Theconducting cavity has dimensions 1×1×1 mm3; the cubical dielectric loadinghas dimensions 0.5 × 0.5 × 0.5 mm3, with dielectric constant ε2/ε1 = 1.2.Light yellow regions correspond to current antinodes, and light blue regionscorrespond to current nodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.10 Surface currents of the symmetry-adapted doublet (1, 1, 3), (1, 3, 1), (3, 1, 1) ofthe dielectrically loaded cavity with eigenvalue k2

0 = 103.7275. The conduct-ing cavity has dimensions 1 × 1 × 1 mm3; the cubical dielectric loading hasdimensions 0.5×0.5×0.5 mm3, with dielectric constant ε2/ε1 = 1.2. Light yel-low regions correspond to current antinodes, and light blue regions correspondto current nodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.11 Surface currents of the symmetry-adapted doublet (1, 1, 3), (1, 3, 1), (3, 1, 1) ofthe dielectrically loaded cavity with eigenvalue k2

0 = 107.5967. The conduct-ing cavity has dimensions 1 × 1 × 1 mm3; the cubical dielectric loading hasdimensions 0.5×0.5×0.5 mm3, with dielectric constant ε2/ε1 = 1.2. Light yel-low regions correspond to current antinodes, and light blue regions correspondto current nodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.12 Non-degenerate modes (1, 1, 1) of the perturbed loaded cavity belonging tothe irreducible representations (a) A1 with k2

0 = 26.303562, and (b) B1 withk2

0 = 26.309544. Light yellow regions correspond to amplitude antinodes, andlight blue regions correspond to amplitude nodes. . . . . . . . . . . . . . . . 34

5.13 Evolution of eigenvalues as the dielectric constant in the cavity is varied from1.0 to 12.0. Six level degenerate modes can be seen to split into triplets. . . . 35

1 Nodes on a hexahedral element that are interpolated by the cubic Hermitepolynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2 Nodes on a hexahedral element that are interpolated by the quintic Hermitepolynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

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List of Tables

2.1 Enhancement of the divergence-to-curl ratio by imposing the divergence-freecondition at each node, at the matrix level. . . . . . . . . . . . . . . . . . . . 9

3.1 Eigenvalues for the forty lowest-energy modes of the empty cubic cavity, cal-culated numerically using quintic Hermite interpolation polynomials. Theeigenvalues are compared against their analytical values, and the absoluteerror is displayed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.1 Different possible even and odd combinations of eigen-modes and their cor-responding irreducible representations for the symmetry group C4v. Here themodes n,m are non-zero integers. . . . . . . . . . . . . . . . . . . . . . . . . 19

4.2 Different possible even and odd combinations of eigen-modes in an emptycubic cavity, their degeneracy, and corresponding irreducible representationsfor the symmetry group Oh are listed. Here the modesm,n and k are non-zerointegers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5.1 Numerically calculated eigenvalues for the first few lowest frequency modesof the dielectrically loaded cubic cavity with 17576 total DoF using quinticHermite interpolation polynomials. The conducting cavity has dimensions1 × 1 × 1 mm3; the cubical dielectric loading has dimensions 0.5 × 0.5 × 0.5mm3, with dielectric ratios ε2/ε1 = 1.2 and ε2/ε1 = 5.0 in the interior. Theeigenvalues are compared against their values for the modes in the emptycavity of unit dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.2 Lowering of symmetry, and splitting of mode degeneracy in the presence of adielectric inclusion that has a preferential z-axis. . . . . . . . . . . . . . . . . 37

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Chapter 1

Introduction

Accurately modeling electromagnetic fields (EM) has profound implications for problemsranging from antenna design to on-chip signal propagation. Most design cycles require ex-tensive back-fitting to the measurement and further redesign. This iterative process is veryexpensive in time, and in fabrication costs. The development of ‘right first time’ compu-tational schemes for obtaining solutions for EM fields with spatial resolution adequate formulti-scale problems has thus been an area of active investigation for decades.1 A compu-tational scheme that can obtain high spatial resolution without poor computational scalingand discretization errors is necessary for integrating full-wave electromagnetics when small(nanometer sized) active devices interact with metals and dielectrics that are far from ideal.

EM fields in a physical system should satisfy the wave equation along with a divergence-free condition. For example, in the electric field formulation (E-field), we solve equations ofthe form2–4

∇×∇× E = ε µ ω2 E; (1.1)

∇ · εE = 0, (1.2)

where ω is the eigenfrequency. Computationally if we simply solve the wave equationEq.(1.1), we are not guaranteed that the obtained solutions satisfy the divergence-free con-dition given in Eq.(1.2). Solutions with either zero frequency (ω = 0) or with non-zerodivergence (∇· εE 6= 0) obtained while solving Eq.(1.1) are known as the spurious solutions.Such spurious solutions corrupt the desired eigen-spectrum.

In 2D waveguides, we get around this issue by solving for only one of the field components(Hz or Ez), and obtain the other two components using boundary conditions (BCs) andthe divergence-free condition.5 However, such freedom does not exist in three dimensions(3D). In a numerical calculation, zero frequency solutions will have non-zero values due todiscretization, and will occur in between the physical spectrum.

After four decades of investigation the academic literature has focused on vector finite

1

element method (VFEM) for EM simulations. The physical region is discretized in to trian-gles in 2D and tetrahedra in 3D; the representation of fields in each of these finite elementsis in terms of so-called edge-based interpolation polynomials. Due to the nature of vectorelements, field directions are ill-defined at each node, hence making VFEM unsuitable to beemployed in several problems: particle trajectories in electron microscope design, vacuumtubes and accelerators. VFEM is also ineffective in modeling multi-physics problems whichwill involve quantum mechanical and electrodynamic aspects in tandem, such as designingcavity lasers. In such instances, independent numerical treatments are needed to obtain thewavefunctions and EM fields. Further more, Nedelec compliant edge elements6 do not havenormal continuity along element interfaces.7 This discontinuity is unphysical, and leads tothe formation of artificial charges,8 which make the solutions to have non-zero divergence.This problem worsens with increase in the mesh density. Commercial software packages suchas COMSOL,9 HFSS,10 MFEM11 and others first model the null-space (k0 = 0) of the prob-lem, and then iteratively eliminate them from the total spectrum; this is computationallyquite expensive a procedure.

Nodal representation of field components with scalar functions and their derivatives re-sults in an unambiguous field description at shared nodes among adjacent elements, andtreats the field components on a uniform footing.12 The built-in derivative degrees of free-dom (DoF) allow us readily calculate the additional quantities such as surface currents. Thistreatment results in global functions without severe coarsening which will be suitable for an-alyzing interactions with small features, such as those found in high frequency transistorcircuits.

In this article, we propose a method of using scalar Hermite interpolation polynomials(or as we call it “HFEM”), which employ polynomial, first and second derivative DoF ateach node. We consider polynomials with both tangential and normal derivative continuityacross the element. We note that:

1. This approach yields better accuracy, with a smoother representation of fields, thanthose obtained using VFEM. The HFEM scheme yields several orders of higher accu-racy with fewer elements than those needed in the presently prevalent 3D implemen-tations of VFEM.9–11

2. This representation guarantees consistency in the direction of fields at the vertices ofelements, and hence, well suited to model transport problems. Multiscale calculationscan now be done easily, something that is not feasible with VFEM.

3. We impose the divergence free condition through a constant Lagrange multiplier in-troduced into the action integral, and explicitly imposing a zero-divergence conditionat each node through derivative DoF.

4. Spurious solutions are then eliminated by identifying them using their large |∇·E|/|∇×

2

E| ratio. This procedure does not alter or influence the accuracy of the physicalsolutions.

In VFEM implementations the eigenfrequencies of spurious solutions are pushed to zero,either through the Nedelec conditions or through their removal at each iteration. In eithercase this is an expensive numerical procedure. In the literature, the first approach foreliminating spurious solutions with scalar polynomials was the penalty factor method (i. e.,a Lagrange multiplier scheme).15 However, a fixed choice of the penalty factor fails to imposethe zero-divergence condition adequately for all frequencies.13 Furthermore, the penalty termitself introduces an additional set of spurious solutions.14 Within the HFEM framework, wenow have the luxury of explicitly imposing a zero-divergence condition at each node whileusing Hermite interpolation polynomials since we have derivative degrees of freedom.16–18While this does not ensure the removal of the divergence in the interior of the finite element,it reduces it substantially specially as the dimensions of the element are reduced. In thisarticle, we use a constant penalty factor, and impose zero-divergence at all nodes to identifythe spurious solutions.

We calculate the electric fields in a metallic cubic cavity. We recognize the existence oflarger degeneracy of frequencies in the spectrum than predicted by the geometrical symme-try of the system. We attribute this to the presence of accidental degeneracy. We presentthe additional operators that explain the additional degeneracies in the frequency spectrumbeyond the degeneracies of 1, 2 and 3, allowed by the geometric symmetry. Introducing aregion filled with a dielectric material inside the cavity removes this additional degeneracy.The splitting in mode frequencies corresponding to such a reduction in degeneracy is pre-dicted using group representation theory. These results are verified from eigenvalues andfield distributions calculated using HFEM.

In Sec. 2, the HFEM formulation is discussed, along with a synopsis of the origin of spuri-ous solutions in EM problems. This is in contrast with the earlier use of finite elements wherethe use of these additonal constraints haven’t been able to resovle the issues completely. InSec. 3, we use HFEM to solve Maxwell’s equations in an empty cubic cavity with conductingboundaries. We benchmark the present method and quantify its qualities. In Sec. 4, wediscuss group representation theory, used in later sections. The effects of partially filling thewaveguide with a dielectric medium are analyzed in detail in Sec. 5. Group representationtheory is used to determine theoretically the splitting of mode degeneracies when the dielec-tric is included. The magnetic vector fields for various modes are displayed in figures thatyield further insights into the symmetry and structure of the modes. Concluding remarksare given in Sec. 6.

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Chapter 2

The HFEM Formulation

2.1 The ActionWe wish to solve Maxwell’s equations for time-harmonic fields, so that the electric fieldE(r, t) = E(r)e−iωt, and similarly for the magnetic field H. MaxwellâĂŹs equations thenreduce to the 3D wave equation. For the electric field,

∇×[µ−1r ∇× E

]= k2

0 εr E, (2.1)

where εr is the relative permittivity tensor, and µr is the relative permeability tensor. If themedium is isotropic, εr and µr are scalar quantities rather than second-rank tensors. Thisdifferential equation can be converted into an action integral,

A/T =∫VdV

[∇× E∗ · µ−1

r ∇× E− k20 E∗ · εr E

]. (2.2)

In the finite element method, the integral above is discretized over the physical domain.Additionally, to impose Eq.(1.2), we introduce a penalty term λ

∣∣∣∇ · εrE∣∣∣2 in the actionintegral,

A/T =∫VdV

[∇× E∗ · µ−1

r ∇× E− k20 E∗ · εr E + λ

∣∣∣∇ · εrE∣∣∣2 ],where λ is the Lagrange multiplier. The fields are represented by Hermite interpolation poly-nomials on hexahedral elements multiplied by the values of the fields and their derivativesat the vertices (nodes) of each element. The integral can be discretized over the elementsto obtain a matrix equation in terms of the nodal parameters. We invoke the principle ofstationary action, and set the variation with respect to E∗ equal to zero. This yields a gen-eralized eigenvalue problem which is solved to obtain the frequencies and field distributions

4

of the resonating modes. The magnetic fields are readily obtained from the electric fieldsusing Maxwell’s equations.

Traditionally, while using Hermite finite elements, the discontinuity of the field across adielectric interface is employed by patching the corresponding row vectors, to account fordifferent material properties. This is a computationally expensive and slow process. Wetackle this problem by smoothing the step function behaviour of the dielectric constant bya Fermi distribution function in position space.20 Thus, the dielectric function is given by

ε(x, y, z) = ε1 − (ε1 − ε2)1 + exp

(x+ a

2

) (x− a

2

)δ2

−11 + exp(y + b

2

) (y − b

2

)δ2

−1

×

1 + exp(z + c

2

) (z − c

2

)δ2

−1

. (2.3)

Here δ is the smoothing parameter in the Fermi distribution function, which controls thesteepness of the dielectric fall-off between the two regions. By decreasing the parameter δ, wecan mimic a discontinuous dielectric function. The most important benefit of this smoothingis that the properties of the fermi function are the same on either side of the interface atany frequency, so that there are no jump conditions to implement in the calculations. Wenote that the Fermi function smoothing may be argued as a more physical material interface,since at an atomistic level, material diffusion leads to smoother interfaces.

2.2 Boundary ConditionsAs an example, we consider a cubic cavity with perfectly conducting metallic boundaries.We assume that the dielectric regions of the cavity are charge-free and current-free. At thesurface of a perfect electrical conductor (PEC), the electric and magnetic fields satisfy theboundary conditions (BCs)2–4

n× E = 0 and n ·H = 0. (2.4)

These relations give the BCs on the periphery of the cavity. When working with electricfields, the tangential components of the field are set to zero at the boundary, while thenormal components are determined variationally. Defining En as the normal, and Eτ as thetangential field components, the BCs are given by

Eτ = 0, ∂τEτ = 0. (2.5)

5

2.3 Origin and Nature of Spurious SolutionsNumerical solutions of Maxwell’s equations are polluted with non-physical spurious solutions.These have their origin in the wave equation itself. Taking the divergence on both sides ofEq.(2.1) gives

k20∇ · εr E = 0. (2.6)

This condition is satisfied when either k0 = 0 or ∇·εr E = 0. Note that solutions with k0 = 0are the trivial solution E = 0, when ∇ · εr E = 0 on the boundary. The solutions can nowbe classified into two categories:

1. k0 6= 0, and ∇ · εr E = 0,

2. k0 = 0, and ∇ · εr E 6= 0.

Solutions falling in the first category are the physical solutions, while those falling in the sec-ond category are spurious solutions. In theory, these spurious solutions have zero frequency.However, due to discretization, the eigenvalues of the spurious modes are not computedexactly as zero, and have numerical values comparable to those of the physical solutions.Consequently, the spurious eigenvalues cannot be easily separated from the desired eigen-values.21;22 For the time-harmonic problem, spurious solutions have zero curl and a finitedivergence. This manifests numerically as a very large divergence-to-curl ratio, compared tophysically admissible solutions. We use this criterion to separate spurious solutions from thephysical admissible solutions in post-processing. A few examples of the spurious solutionsare shown in Fig. 2.1. Note how the field distribution shows source-like behaviour withinthe cavity, indicative of non-zero divergence.

2.4 The Penalty Method and the Divergence-free Con-straint

The penaly method15 has been proposed to remove spurious solutions in nodal finite elementimplementations. The penalty term pushes most spurious solution eigenfrequencies outsidethe spectral range of interest. However, a fixed Lagrange multiplier does not remove all spu-rious modes. One proposed algorithm is to use a different multiplier for each mode.13 This,however, is an expensive iterative scheme. Nevertheless, the penalty offers an inexpensivemethod for removing the zero-frequency spurious modes, and can be further enhanced.

One key feature of spurious modes is a large divergence-to-curl ratio |∇ · E|/|∇ × E|.13There is then the possibility of identifying and removing any remaining spurious solutions,based on this ratio, during post-processing. However, the divergence-to-curl ratio for spuriousand physically admissible modes can become comparable, as seen in the first and secondcolumns of Table 2.1.

6

Figure 2.1: Spurious solutions in the empty cubic cavity of unit dimensions, with divergence-to-curl ratio equal to (a) 1783.8, (b) 1481.7, and (c) 380.1. Note the field distribution, whichis indicative of the presence of sources, even though the cavity is empty. Light yellow regionscorrespond to amplitude antinodes, and light blue regions correspond to amplitude nodes.

(a) (b)

(c)

7

We resolve this issue by explicitly imposing the divergence-free condition at each node,using the derivative degrees of freedom.16;17;23 At the matrix level, one of the derivativedegrees of freedom in the equation ∇ · εrE is eliminated in favor of the other two. Theprocedure is demonstrated in Fig. 2.2 for the simpler case of a constant εr.

Applying this technique drives the divergence-to-curl ratio of physically admissible solu-tions even lower, and that of spurious solutions higher, as can be seen from the third andfourth columns of Table 2.1.

Additionally, since the divergence condition is applied at each node, the total divergenceof the admissible solutions decreases further with mesh refinement. This is in constrast toVFEM, where the normal discontinuity of edge elements leads to the formation of artificialcharges at element interfaces, thus increasing the total divergence of the solutions; thisproblem worsens with mesh refinement.8

Note that in VFEM, the zero-frequency spurious solutions are separated by filtering outthe null-space of the curl operator from the spectrum using iterative techniques,24;25 or byfinding eigenvalues in the interior of the spectrum. This is necessary, since if the physicalsolution space is not normal to the null space, the physical solutions will be polluted by nullvectors.

Figure 2.2: Explicity imposing the divergence-free condition by performing global matrix rowand column operations. One of the derivative DoF is eliminated in favor of the remainingtwo. Here X = 1 for the left-hand side matrix, and X ≈ 105 for the right-hand side matrix,in the generalized eigenvalue problem. This choice of a large number pushes the eigenvalueof the redundant 1× 1 subspace out of the spectral range of interest.

-

--

-

X

∂xEx ∂yEy ∂zEz ∂xEx ∂yEy ∂zEz

8

Table 2.1: Enhancement of the divergence-to-curl ratio by imposing the divergence-freecondition at each node, at the matrix level.

Penalty factor method only With Divergence-free condition Solutionk2

0 |∇ · E|/|∇ × E| k20 |∇ · E|/|∇ × E|

19.739334159 0.000000000 19.739334159 0.000000000 Physical19.739334159 0.000000000 19.739334159 0.000000000 Physical19.739334159 0.000000000 19.739334159 0.000000000 Physical29.609001232 0.005891138 29.609001232 0.000001933 Physical29.609001232 0.000389894 29.609001232 0.000001933 Physical29.609001232 172.872234679 31.177339513 182.987276711 Spurious49.358373749 0.000000000 49.358373749 0.000000000 Physical49.358373749 0.000000000 49.358373749 0.000000000 Physical49.358373749 0.000000000 49.358373749 0.000000000 Physical49.358373749 0.000000000 49.358373749 0.000000000 Physical49.358373749 0.000000000 49.358373749 0.000000000 Physical49.358373749 0.000000000 49.358373749 0.000000000 Physical59.228040826 0.572767636 59.228040826 0.000063471 Physical59.228040826 0.575823907 59.228040826 0.000064756 Physical59.228040826 0.137630780 59.228040826 0.000071421 Physical59.228040826 0.135054548 59.228040826 0.000071690 Physical59.228040826 0.000065277 59.228040826 0.000000970 Physical59.228040826 0.277005817 59.228040826 0.000134486 Physical59.228040826 1.063357739 62.209018046 152.766245738 Spurious59.228040826 1.083589808 62.209018046 152.766245738 Spurious59.228040826 3.580885859 62.209018046 152.766245738 Spurious

9

Chapter 3

Fields in Empty Cubic Cavity

To demonstrate our method, we first model an empty cubic cavity with conducting bound-aries, with εr = 1 and µr = 1. Consider a cavity of unit dimensions. The frequency of themodes ω can be obtained from the relation k2

0 = ω2/c2, where k20 is an eigenvalue of the

matrix problem. In Figs. 3.4 and 3.5 we show the electric field distributions for the first fivemodes. The calculations are done using HFEM, with 8232 DoF, within a parallel computingenvironment.27–29 We use the Krylov-Schur algorithm as implemented in SLEPc.30

From Table 3.1, it is clear that the eigenvalues of the empty cavity obtained through ourscheme have very small errors, when compared to the analytical values. In Fig. 3.1, we showthe convergence of the calculated frequencies to their analytical values in an empty cube ofunit dimensions for both HFEM and VFEM (obtained using the package MFEM11). As theglobal number of DoF is increased through mesh refinement, the accuracy improves steadily.Quintic HFEM delivers an accuracy of 1 part in 109 with just 8232 DoF. The second curvefrom bottom (in green) obtained using 5th order VFEM shows about 10 times larger errorfor comparable DoF. Even with further mesh refinement, VFEM has an error higher thanour HFEM scheme. HFEM gives a higher accuracy than VFEM, even with half the numberof DoF. This reduction in required number of DoF can lead to improvement in computationtime. Whereas MFEM took about 34 minutes, with 11520 total DoF, to calculate the firsttwenty eigen-modes of the cavity, HFEM took only about 17 minutes with comparable DoF.

We also consider the error in the eigenfrequencies of higher frequency modes in Fig. 3.2.The errors converge at the same rate as the error in the first mode.

In Fig. 5.12 further comparison is made with similar commercially available software, likeCOMSOL9 and HFSS10, for the empty cavity problem. Compared to HFEM, COMSOL andHFSS show poor convergence with mesh refinement. As seen from Fig. 5.12, the error ineigenvalues computed using COMSOL and HFSS seems to converge to a value of 10−4.

10

Figure 3.1: The convergence of errors in eigenvalue of the first mode for the 3rd orderVFEM, 5th order VFEM, cubic Hermite, and quintic Hermite interpolation polynomials.Using quintic Hermite interpolation polynomials we can reduce the error in the first modeupto 10−9 with just 27 elements and 8232 DoF, with further reduction in error possible withmesh refinement. The total DoF corresponds to the global matrix dimension.

10-12

10-10

10-8

10-6

10-4

0

DoF (104)

Rela

tive E

rror

(k

02/k

02)

2 4 6 8

Quintic HFEM

Cubic HFEM

3rd order VFEM

5th order VFEM

11

Figure 3.2: The convergence of errors in eigenvalue of the higher frequency modes for quinticHFEM only are shown for the case of an empty cubic cavity of unit dimensions. The totalDoF corresponds to the global matrix dimension.

10-12

10-10

10-6

10-8

0 2 4 6 8

state (5,1,1)

state (2,2,2)

state (1,2,2)

state (1,1,2)

state (1,1,1)

DoF (104)

Rela

tive E

rror

(k

02/k

02)

12

Figure 3.3: The convergence of the errors in eigenvalues of the first mode in COMSOL,HFSS, MFEM, and HFEM. The total DoF corresponds to the global matrix dimension.

10-7

10-6

10-5

10-4

10-3

10-2

Rela

tive E

rror

(k

02/k

02)

DoF (104)

0 2 4 6 8

COMSOL 3rd Order

HFSS 3rd Order

MFEM 3rd Order

HFEM 3rd Order

13

Figure 3.4: Triply degenerate modes (a) (0, 1, 1), (b) (1, 0, 1), and (c) (1, 1, 0) for the emptycavity of unit dimensions with eigenvalue k2

0 = 19.7392. Light yellow regions correspond toamplitude antinodes, and light blue regions correspond to amplitude nodes.

(a) (b)

(c)

14

Figure 3.5: Doubly degenerate mode (1,1,1) for the empty cavity of unit dimensions witheigenvalue k2

0 = 29.6088. Light yellow regions correspond to amplitude antinodes, and lightblue regions correspond to amplitude nodes.

(a)

(b)

15

Table 3.1: Eigenvalues for the forty lowest-energy modes of the empty cubic cavity, calculatednumerically using quintic Hermite interpolation polynomials. The eigenvalues are comparedagainst their analytical values, and the absolute error is displayed.

Mode Degeneracy Analytical Hermite FEM Error(0, 1, 1) 3 19.7392088021787172 19.7392088021791601 4.4E -13(1, 1, 1) 2 29.6088132032680758 29.6088132032692926 1.2E -12(0, 1, 2) 6 49.3480220054467930 49.3480220078586953 2.4E -9(1, 1, 2) 6 59.2176264065361517 59.2176264089494637 2.4E -9(0, 2, 2) 3 78.9568352087148689 78.9568352135412113 4.8E -9(1, 2, 2) 6 88.8264396098042275 88.8264396146369393 4.8E -9(0, 1, 3) 6 98.6960440108935861 98.6960442784766911 2.6E -7(1, 1, 3) 6 108.5656484119829448 108.5656486795616189 2.6E -7(2, 2, 2) 2 118.4352528130723034 118.4352528203570074 7.2E -9

16

Chapter 4

Accidental Degeneracies inElectromagnetic Cavities

Physical properties arising from the symmetry of the system can be treated efficiently usinggroup representation theory. It has been well appreciated in Quantum mechanics that thedegeneracies in the energy spectrum arise from the symmetry group of the correspondingHamiltonian.31;32 The degeneracy of an eigenvalue is equal to the dimensionality of the corre-sponding irreducible representation of the symmetry group. If we have any other additionaldegeneracy in the spectrum which cannot be explained by the obvious geometrical symmetryof the system, it is labeled as “accidental degeneracy.” In this section we discuss the presenceof such accidental degeneracy and its removal for EM modes in 2D and 3D cavities.

Let G be an infinitesimal transformation for the coordinate system. Given a HamiltonianH, if [G, H] = 0 and G does not explicitly depend on time, then we say that G is a constant ofmotion. Such constants of motion generate symmetries since they transform one eigenstateto another of the same eigenvalue. We expect to find additional constants of motion wheneverwe observe accidental degeneracies as explained below. If the Hamiltonian is separable in acoordinate system, then the separation constants may be considered as constants of motion.33These are just the generators of the additional symmetry operations. Typically, accidentaldegeneracies are then rendered normal by identifying the hidden covering group.

The example of the familiar H-atom best illustrates the symmetry argument. In the H-atom problem, the conservation of the 3 components of the angular momentum provide usthree constants of motion associated with the rotation group O(3). Equivalently, we considerL+, L−, Lz as the set of 3 operators which commute with the Hamiltonian of the H-atom.We know that an eigenstate |E, `,m〉 of the H-atom transforms under the operation of ladderoperators as

L± |E, `,m〉 =√

(`∓m)(`±m+ 1) |E, `,m± 1〉 . (4.1)

Hence angular momentum operators transform degenerate eigenstates of the same ` butof different azimuthal quantum numbers m into one another. For a given ` we find that

17

(2` + 1) states are degenerate. However the eigenstates of different allowed ` are also de-generate here, leading to a text-book example of accidental degeneracy. Fock34 identifiedthe hidden four-dimensional rotational symmetry group O(4) as the true symmetry of theH-atom, which explains these additional degeneracies manifesting as the familiar s, p, d, f, . . .states being degenerate for a given principal quantum number n. We expect to find addi-tional operators (constants of motion) which commute with the Hamiltonian and connecteigenstates of different ` quantum numbers. These operators are just the three componentsof the new conserved vector, the Runge-Lenz vector35 A which transform degenerate eigen-states of different ` quantum numbers,36 analogous to Eq.(4.1). The components of theangular momentum L and the Runge-Lenz vector A generate the symmetry group O(4). Itcan be shown that the H-atom Hamiltonian is separable in parabolic coordinates.37 We notethat even though the components of L and A commute with the Hamiltonian, they will notmutually commute with each other. These components are subject to kinematic constraintsof the Casimir operators for the group O(4). Hence the eigenstates of the Hamiltonian arerepresented by the complete set of commuting operators H,L2, Lz.

Such analysis, based on the symmetries of a physical system, is also feasible for a cubiccavity, in the context of electromagnetic theory. The electric or magnetic fields correspondingto a degenerate eigen-frequency will form a set of vector basis-functions for the irreduciblerepresentations of the symmetry group. Previously, we have derived a coefficient formula torecognize the irreducible representation corresponding to an eigen-energy, and obtain thesymmetry adapted wavefunctions in quantum dots.20 This coefficient formula will be valideven for the electric (magnetic) modes in EM cavities. We define the projection operatorsthat identify the symmetries and group representations of the field distributions.

Let G be a group of order g and Γ(i) be an li-dimensional representation of G. For agroup element R in G, its representation is given by a li × li square matrix Γ(i)(R). Thenthe projection operator31 corresponding to Γ(i) is given by

P(i) = lig

∑R

χ(i) (R) · PR, (4.2)

where χ(i)(R) is the character and PR is the operator corresponding to the element R. Thetransformation of electric fields under the operation PR is defined by

PR · E (r) = R · E(R−1 · r

). (4.3)

The projection operator P(i) projects out the part of the field E that belongs to the represen-tation Γ(i). Let Eini=1 be the set of eigen-fields for the physical system under consideration.Then the coefficient formula20 is given by

c(i)jk =

∫Vd3r E†j ·

(P(i)Ek

). (4.4)

18

If the coefficient is nonzero, then the field Ek has a component in the ith-representation andEj is a partner. Using these coefficients we can construct the symmetry-adapted electric(magnetic) fields which are exclusively in the ith-representation.

As an example, we consider a 2D empty square cavity, of length a, and surrounded withmetal boundaries, having C4v geometrical symmetry. The character table for different pointgroups are given in Refs.31;32. We know that the eigenvalues supported in the cavity aregiven by

k20 =

(n2x + n2

y

) π2

a2 , (4.5)

where nx and ny are non-zero integers. In Table 4.1, we list all symmetry-adapted basis

Table 4.1: Different possible even and odd combinations of eigen-modes and their corre-sponding irreducible representations for the symmetry group C4v. Here the modes n,m arenon-zero integers.

Mode number Irreduciblerepresentation

Basis functions

(2n− 1, 2n− 1) B1 E(2n−1,2n−1)

(2n, 2n) A2 E(2n,2n)

(2m− 1, 2n− 1)

B1 E(2m−1,2n−1) + E(2n−1,2m−1)

A1 E(2m−1,2n−1) − E(2n−1,2m−1)

(2m, 2n)

A2 E(2m,2n) + E(2n,2m)

B2 E(2m,2n) − E(2n,2m)

(2m− 1, 2n) E E(2m−1,2n), E(2n,2m−1)

functions and irreducible representations for different combinations of (nx, ny) modes. The(odd, odd) or (even, even) doublet with nx 6= ny belongs to two distinct irreducible rep-resentations. Hence, the degeneracy of these modes is not a consequence of the symmetrygroup C4v; therefore, we say they exhibit accidentally degeneracy. This is analogous to thesituation in an infinite square quantum well, where the accidental degeneracy occurs forthe eigen-energies due to the separability of the infinite well potential.42 Such an acciden-tal degeneracy is rendered normal, in the usual parlance, by recognizing that an additionaloperator Ω =

(∂2x − ∂2

y

)exists, which connects the basis functions of A1 (A2) and B1 (B2)

19

representations. Hence, the true symmetry of a 2D empty square cavity will be a cov-ering group, which is a semidirect product of the geometrical symmetry group C4v and aone-dimensional compact continuous group generated by the operator Ω =

(∂2x − ∂2

y

).43 We

can remove the accidental degeneracy in a 2D empty square cavity by intriducing a concen-tric square dielectric inclusion. Such accidental degeneracy and its removal occurs even inrectangular cavities.

Next, we consider the case of an empty cubic cavity resonator of length a with conductingboundaries. The eigen-modes supported by the cubic resonator have the eigenvalues

k20 =

(n2

1 + n22 + n2

3

) π2

a2 , (4.6)

and the electric field components are given by

Ex = E0x cos(n1πx

a

)sin

(n2πy

a

)sin

(n3πz

a

);

Ey = E0y sin(n1πx

a

)cos

(n2πy

a

)sin

(n3πz

a

); (4.7)

Ez = E0z sin(n1πx

a

)sin

(n2πy

a

)cos

(n3πz

a

),

where n1, n2, n3 are non-zero integers, and E0x, E0y, E0z are the field amplitudes in eachdirection. Note that there are three kinds of degeneracies present. The first kind is dueto the permutation of mode numbers; the second occurs when two disitinct sets of modenumbers give the same frequency according to Eq.(4.6). The third kind is a consequenceof the divergence-free condition ∇ · E = 0. On substituting Eq.(4.7) in the divergence-freecondition we obtain the constraint n1 E0x + n2 E0y + n3 E0z = 0. Hence, if n1, n2, n3 6= 0 wesee that there are two independent field components, hence for a given mode (n1, n2, n3) wewill have at least 2 degenerate field solutions.26 The third kind is less transparent, occurringwhen the following relation is satisfied:

n21 + n2

2 + n23 = m2

1 +m22 +m2

3, (4.8)

with ni 6= mi, for i = 1, 2, 3.We know that the cubic cavity has the geometrical Oh symmetry. In Table 4.2, we

list all different possible combinations of mode numbers and their corresponding irreduciblerepresentations from the symmetry group Oh. Here, we have accounted for only the firsttwo kinds of degeneracies. For most of the combinations of mode numbers we observeaccidental degeneracy since they belong to two or more distinct irreducible representations.The accidental degeneracy associated with permutation of mode numbers can be renderednormal in the usual parlance, by identifying the existence of two dynamical operators Ω1 =

20

Table 4.2: Different possible even and odd combinations of eigen-modes in an empty cubiccavity, their degeneracy, and corresponding irreducible representations for the symmetrygroup Oh are listed. Here the modes m,n and k are non-zero integers.

Mode number Degeneracy Irreducible representations

(0, 2n− 1, 2n− 1) 3 T1u

(0, 2n− 1, 2m) 6 T1g⊕T2g

(0, 2n, 2n) 3 A2u⊕Eu

(0, 2n− 1, 2m− 1) 6 T1u⊕T2u

(0, 2n, 2m) 6 A1u⊕A2u

⊕ 2Eu

(2n− 1, 2n− 1, 2n− 1) 2 Eg

(2n, 2n, 2n) 2 Eu

(2n− 1, 2n− 1, 2m) 6 T1u⊕T2u

(2n, 2n, 2m− 1) 6 T1g⊕T2g

(2m− 1, 2n− 1, 2n− 1) 6 A1g⊕A2g

⊕ 2Eg

(2m, 2n, 2n) 6 A1u⊕A2u

⊕ 2Eu

(2n− 1, 2m− 1, 2k) 12 2T1u⊕ 2T2u

(2n, 2m, 2k − 1) 12 2T1g⊕ 2T2g

(2m, 2n, 2k) 12 2A1u⊕ 2A2u

⊕ 4Eu

(2m− 1, 2n− 1, 2k − 1) 12 2A1g⊕ 2A2g

⊕ 4Eg

(∂2x − ∂2

y

)and Ω2 =

(2 ∂2

z − ∂2x − ∂2

y

)which connect the degenerate field solutions.44 By

introducing inside the cavity, a concentric cubical dielectric inclusion, the larger symmetrygroup of the cavity is reduced to Oh. As shown in the following section, this removes theaccidental degeneracy.

21

Chapter 5

Fields in Dielectrically Loaded CubicCavity

We consider a cubical conducting cavity of dimensions 1× 1× 1 mm3. This cavity is loadedwith a concentric cubical dielectric inclusion, of dimensions 0.5×0.5×0.5 mm3, and permit-tivity ε2, as shown in Fig. 5.1. The permittivity in the rest of cavity is ε1. The eigenvaluesof the first few modes are tabulated for the dielectric ratios ε2/ε1 = 1.2 and ε2/ε1 = 5.0. Thecalculations are done with 17576 DoF, to accurately model the dielectric function.

In Figs. 5.2–5.6 we show electric field distributions for the first few resonating modesin the loaded cavity with ε2/ε1 = 1.2. As shown in Fig. 5.2, the first three modes in theempty cavity remain degenerate, as they belong to the three dimensional representationT1u of the group Oh. The doublet in Fig. 5.3, corresponding to the mode numbers (1, 1, 1)also remains degenerate; however, the two independent modes are now symmetry-adaptedpartners, and related by a C4 rotation. An instance of the removal of accidental degeneracycan be seen in the (1, 1, 3), (1, 3, 1), (3, 1, 1) modes, which in the empty cubic cavity forma degenerate sextuplet. From Table 4.2, we see that this sextuplet decomposes into fourseparate irreducible representations of the group Oh. Fig. 5.4 shows the singlet modes in theloaded cavity, belonging to the irreducible representations A1g and A2g respectively. Notehow the magnitude of the electric field has complete Oh symmetry, while the vectors inthe A2g mode flip their directions under a C4 rotation. Similarly, Figs. 5.5 and 5.6 showsymmetry-adapted partners, which belong to the two dimensional representation Eg.

In Fig. 5.13, the evolution of mode frequency on varying the dielectric constant ε2 in theinterior is shown for the lowest few modes. We observe level crossings akin to the case ofbound states in a finite quantum well as the well depth is varied.20

As another example, we consider a linear z-dependent perturbation to the dielectricfunction in the interior dielectric block. This perturbation reduces the symmetry groupof the loaded cavity from the group Oh to C4v, resulting in a further reduction in modedegeneracies.. In Table 5.2, we have listed the eigenfrequencies obtained with our method,

22

and classified them into corresponding irreducible representations of the group C4v. InFig. 5.12, we show electric field distributions for the modes (1, 1, 1), which are degeneratein the unperturbed loaded cavity, but split in frequency in the perturbed loaded cavity. Asseen from Fig. 5.12, the electric field magnitudes have complete C4v symmetry; the firstmode transforms like the representation A1 of C4v, while the second mode transforms likethe representation B1.

In Figs. 5.7-5.11, we plot the surface currents on the conducting cavity. These currentsensure that the magnetic field outside the cube is identically zero. Note that the surfacecurrents are symmetry-adapted, and appear in singlets, doublets and triplets. This is inagreement with the Oh symmetry of the system.

Figure 5.1: Schematic of a dielectrically loaded cavity with Oh symmetry. The conductingcavity has dimensions 1 × 1 × 1 mm3; the cubical dielectric loading has dimensions 0.5 ×0.5× 0.5 mm3, with dielectric constant ε2. The permittivity in the rest of cavity is ε1.

23

Figure 5.2: Symmetry-adapted triplet (0, 1, 1), (1, 0, 1), (1, 1, 0) of the dielectrically loadedcavity with eigenvalue k2

0 = 18.5267. The conducting cavity has dimensions 1× 1× 1 mm3;the cubical dielectric loading has dimensions 0.5 × 0.5 × 0.5 mm3, with dielectric constantε2/ε1 = 1.2. Light yellow regions correspond to amplitude antinodes, and light blue regionscorrespond to amplitude nodes.

(a) (b)

(c)

24

Figure 5.3: Symmetry-adapted doublet (1, 1, 1) of the dielectrically loaded cavity with eigen-value k2

0 = 28.8995. The conducting cavity has dimensions 1×1×1 mm3; the cubical dielectricloading has dimensions 0.5×0.5×0.5 mm3, with dielectric constant ε2/ε1 = 1.2. Light yellowregions correspond to amplitude antinodes, and light blue regions correspond to amplitudenodes.

(a)

(b)

25

Figure 5.4: Symmetry-adapted singlets (1, 1, 3), (1, 3, 1), (3, 1, 1) of the dielectrically loadedcavity with eigenvalues k2

0 = 107.4315, 107.5911. The conducting cavity has dimensions1 × 1 × 1 mm3; the cubical dielectric loading has dimensions 0.5 × 0.5 × 0.5 mm3, withdielectric constant ε2/ε1 = 1.2. The states transform like the 1D representations (a) A1g,and (b) A2g of the group Oh. Light yellow regions correspond to amplitude antinodes, andlight blue regions correspond to amplitude nodes.

(a)

(b)

26

Figure 5.5: First symmetry-adapted doublet (1, 1, 3), (1, 3, 1), (3, 1, 1) of the dielectricallyloaded cavity with eigenvalue k2

0 = 103.7275. The conducting cavity has dimensions 1×1×1mm3; the cubical dielectric loading has dimensions 0.5 × 0.5 × 0.5 mm3, with dielectricconstant ε2/ε1 = 1.2. Light yellow regions correspond to amplitude antinodes, and light blueregions correspond to amplitude nodes.

(a)

(b)

27

Figure 5.6: Second symmetry-adapted doublet (1, 1, 3), (1, 3, 1), (3, 1, 1) of the dielectricallyloaded cavity with eigenvalue k2

0 = 107.5967. The conducting cavity has dimensions 1×1×1mm3; the cubical dielectric loading has dimensions 0.5 × 0.5 × 0.5 mm3, with dielectricconstant ε2/ε1 = 1.2. Light yellow regions correspond to amplitude antinodes, and light blueregions correspond to amplitude nodes.

(a)

(b)

28

Figure 5.7: Surface currents of the triply degenerate modes (a) (0, 1, 1), (b) (1, 0, 1), and(c) (1, 1, 0) of the dielectrically loaded cavity with eigenvalue k2

0 = 18.5267. The conductingcavity has dimensions 1×1×1 mm3; the cubical dielectric loading has dimensions 0.5×0.5×0.5 mm3, with dielectric constant ε2/ε1 = 1.2. Light yellow regions correspond to currentantinodes, and light blue regions correspond to current nodes.

(a) (b)

(c)

29

Figure 5.8: Surface currents of the doubly degenerate mode (1, 1, 1) of the dielectricallyloaded cavity with eigenvalue k2

0 = 28.8995. The conducting cavity has dimensions 1× 1× 1mm3; the cubical dielectric loading has dimensions 0.5 × 0.5 × 0.5 mm3, with dielectricconstant ε2/ε1 = 1.2. Light yellow regions correspond to current antinodes, and light blueregions correspond to current nodes.

(a)

(b)

30

Figure 5.9: Surface currents of the symmetry-adapted singlets (1, 1, 3), (1, 3, 1), (3, 1, 1) of thedielectrically loaded cavity with eigenvalues k2

0 = 107.4315, 107.5911. The conducting cavityhas dimensions 1×1×1 mm3; the cubical dielectric loading has dimensions 0.5×0.5×0.5 mm3,with dielectric constant ε2/ε1 = 1.2. Light yellow regions correspond to current antinodes,and light blue regions correspond to current nodes.

(a)

(b)

31

Figure 5.10: Surface currents of the symmetry-adapted doublet (1, 1, 3), (1, 3, 1), (3, 1, 1) ofthe dielectrically loaded cavity with eigenvalue k2

0 = 103.7275. The conducting cavity hasdimensions 1× 1× 1 mm3; the cubical dielectric loading has dimensions 0.5× 0.5× 0.5 mm3,with dielectric constant ε2/ε1 = 1.2. Light yellow regions correspond to current antinodes,and light blue regions correspond to current nodes.

(a)

(b)

32

Figure 5.11: Surface currents of the symmetry-adapted doublet (1, 1, 3), (1, 3, 1), (3, 1, 1) ofthe dielectrically loaded cavity with eigenvalue k2

0 = 107.5967. The conducting cavity hasdimensions 1× 1× 1 mm3; the cubical dielectric loading has dimensions 0.5× 0.5× 0.5 mm3,with dielectric constant ε2/ε1 = 1.2. Light yellow regions correspond to current antinodes,and light blue regions correspond to current nodes.

(a)

(b)

33

Figure 5.12: Non-degenerate modes (1, 1, 1) of the perturbed loaded cavity belonging to theirreducible representations (a) A1 with k2

0 = 26.303562, and (b) B1 with k20 = 26.309544.

Light yellow regions correspond to amplitude antinodes, and light blue regions correspondto amplitude nodes.

(a)

(b)

34

Figure 5.13: Evolution of eigenvalues as the dielectric constant in the cavity is varied from1.0 to 12.0. Six level degenerate modes can be seen to split into triplets.

0

20

40

60

80

2 4 6 8 10 12

Dielectric Ratio

Eig

en

va

lue

s (

k0

2)

011

111

012

112

022

35

Table 5.1: Numerically calculated eigenvalues for the first few lowest frequency modes of thedielectrically loaded cubic cavity with 17576 total DoF using quintic Hermite interpolationpolynomials. The conducting cavity has dimensions 1 × 1 × 1 mm3; the cubical dielectricloading has dimensions 0.5×0.5×0.5 mm3, with dielectric ratios ε2/ε1 = 1.2 and ε2/ε1 = 5.0in the interior. The eigenvalues are compared against their values for the modes in the emptycavity of unit dimensions.

Mode Numbers Irreduciblerepresentation

k20

ε2/ε1 = 5.0 ε2/ε1 = 1.2 ε2/ε1 = 1.0(0, 1, 1), 7.919791186 18.526768246 19.739208802(1, 0, 1), T1u 7.919791186 18.526768246 19.739208802(1, 1, 0). 7.919791186 18.526768246 19.739208802(1, 1, 1) Eg 18.202154429 28.899507681 29.608813203

18.202154429 28.899507681 29.608813203(0, 1, 2), 21.354532231 47.387744521 49.348022007(2, 1, 0), T1g 21.354532231 47.387744521 49.348022007(1, 0, 2), 21.354532231 47.387744521 49.348022007(2, 0, 1), 23.416822776 47.404561358 49.348022007(0, 2, 1), T2g 23.416822776 47.404561358 49.348022007(1, 2, 0). 23.416822776 47.404561358 49.348022007

35.614536102 57.284302371 59.217626408T1u 35.614536102 57.284302371 59.217626408

(1, 1, 2), (1, 2, 1), 35.614536102 57.284302371 59.217626408(2, 1, 1). 38.219214013 58.327254711 59.217626408

T2u 38.219214013 58.327254711 59.21762640838.219214013 58.327254711 59.217626408

(0, 2, 2),Eu

38.305292320 76.979818578 78.956835213(2, 0, 2), 38.305292320 76.979818578 78.956835213(2, 2, 0). A2u 42.790037374 77.001769904 78.956835213

55.137384588 103.727559345 108.565648679

2Eg55.137384588 103.727559345 108.565648679

(1, 1, 3), (1, 3, 1), 77.568304705 107.596740348 108.565648679(3, 1, 1). 77.568304705 107.596740348 108.565648679

A1g 67.549698454 107.431515611 108.565648679A2g 77.460283215 107.591198480 108.565648679

36

Table 5.2: Lowering of symmetry, and splitting of mode degeneracy in the presence of adielectric inclusion that has a preferential z-axis.

Oh C4v k20

T1u

A1 14.840987

E14.841426

14.841426

EgA1 26.303562

B1 26.309544

T1g

A2 39.928882

E39.937310

39.937310

T2g

B2 40.430146

E40.431522

40.431522

T1u

A1 51.226115

E51.202143

51.202143

T2u

B1 54.625215

E54.631936

54.631936

EuA2 68.152735

B2 68.160536

A2u B2 68.993259

37

Chapter 6

Concluding Remarks

We have shown that electromagnetic simulations done with Hermite elements deliver highaccuracy and smoother representation of fields. We have compared our formalism withanalytical results and shown that it is superior to VFEM implementations in commercialsoftware. Fewer finite elements are needed to achieve comparable results for eigenvaluecalculations.

The divergence free constraint for the electromagnetic fields results in spurious solutionsfor the wave equation. Their eigenfrequencies are pushed to zero in the VFEM, eitherthrough Nedelec compliance or through their removal at each iteration. In either case,there is an expensive numerical procedure. In our approach, we imposed the divergence-freecondition by adding a constant penalty term (with a Lagrange multiplier set to unity), andfor derivative degrees of freedom at each node, we have imposed an explicit constraint. Thenon-zero frequency spurious solutions are eliminated by identifying them through their large|∇ ·E|/|∇×E| ratio. This procedure does not alter or influence the accuracy of the physicalsolutions.

Group theoretical classification of eigen-modes in photonic crystals,38 in radio-frequencycavities39;40, and in metamaterials41 were previously presented in the literature. In thispaper we have considered the symmetries of a metallic cubic cavity, with and without adielectric inclusion. We have identified the origin of the high degeneracy of frequencies ina cubic cavity, and have attributed it to accidental degeneracy. We have recognized threefactors contributing to the degeneracy. The operators additional to those of the symmetrygroup Oh have been determined. The computed field distributions have been shown to besymmetry-adapted, as predicted from group theory.

The accidental degeneracy is lifted with the insertion of a concentric cubical dielectric ofsmaller size. The variation of the spectrum as the ratio ε2/ε1 is changed has been explored.We have shown that this leads to a reordering of some of the mode frequencies.

By considering a spatially linearly varying dielectric we can reduce the symmetry of thesystem further from the group Oh to C4V . This is analogous to applying an external electric

38

field in a semiconductor quantum dot, but with flat potential barriers outside the smallercube.

Since this method is based on geometry discretization, we are now free to change theshape of the cavity and still get the high accuracy using HFEM. This method is well suitedto the application for mixed physics, such as for quantum well lasers in electromagneticcavities. This is because we have node based finite elements with scalar shape functions.

Applications to multiscale analysis is now feasible using the presented method. Thisoption is not open to VFEM due to the lack of directionality for fields at shared nodes inthe finite element mesh. Very dense meshes lead to larger regions in which field directionsare ill-defined. For example, it is not possible to predict particle trajectories in electronmicroscope design, vacuum tubes, and particle accelerator design due to the ill-defined fielddirections around nodes.

The approach presented here is expected to show great promise for the simulation ofelectrodyamics, plasmonics, high frequency circuitry, and especially mixed physics problems.

39

AppendixHermite Interpolation Polynomial BasisThe components of the EM fields are represented using basis functions within each hexahedral(brick) element.

f(x, y, z) =18∑i=1

fiφi(x, y, z) (1)

These 3D interpolation polynomials are constructed by taking the product of 1D Hermitepolynomials, each with a different coordinate as the argument.45 The 1D cubic Hermitepolynomials, which interpolate between two nodes on a line at ξ = ±1 are

N1(ξ) = 14(2− 3ξ+ξ3),

N1(ξ) = 14(1− ξ−ξ2+ξ3),

N2(ξ) = 14(2 + 3ξ−ξ3),

N2(ξ) = 14(−1− ξ+ξ2+ξ3). (2)

The product of these 4 polynomials gives 64 polynomials, which interpolate between thenodes on a hexahedron as shown in Fig. 1, with 8 degrees of freedom on each node.

The 1D quintic Hermite polynomials, which interpolate a function over three nodes on aline at ξ = −1, 0,+1 are

N1(ξ) = 14(4ξ2−5ξ3−2ξ4+3ξ5),

N1(ξ) = 14(ξ2−ξ3−ξ4+ξ5),

N2(ξ) = 1−2ξ2+ξ4,

N2(ξ) = ξ−2ξ3+ξ5,

N3(ξ) = 14(4ξ2+5ξ3−2ξ4−3ξ5),

N3(ξ) = 14(−ξ2−ξ3+ξ4+ξ5). (3)

The product of these 6 polynomials gives 216 polynomials, which interpolate the nodes ona hexahedron as shown in Fig. 2, with 8 polynomials on each node. The degrees of freedom

40

at each node on the hexahedron correspond to

f, f ′x, f′y, f

′z, f

′xy, f

′yz, f

′zx, f

′xyz.

Figure 1: Nodes on a hexahedral element that are interpolated by the cubic Hermite poly-nomials.

111 211

112

222

221

122

121

212

Figure 2: Nodes on a hexahedral element that are interpolated by the quintic Hermitepolynomials.

111211 311

113

112

223

221

233

332

333

331

41

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