Photonic Crystal Modelling using Finite Element Analysis

159 Pages·2002·2.038 MB·English

by  Hiett B.

University of Southampton

Photonic Crystal Modelling using
Finite Element Analysis

Ben Hiett

A thesis submitted for the degree of Doctor of Philosophy

Faculty of Engineering and Applied Science
School of Engineering Sciences

December 2002
UNIVERSITY OF SOUTHAMPTON

ABSTRACT

FACULTY OF ENGINEERING AND APPLIED SCIENCES
SCHOOL OF ENGINEERING SCIENCES

Doctor of Philosophy

Photonic Crystal Modelling using Finite Element Analysis

By Ben Hiett

This thesis presents an efficient finite element method for computing spectra of photonic band
gap materials. These periodic dielectric crystals exhibit a photonic band gap analogous to the
electronic band gap present in semiconductors. Photons in the frequency range of the band gap
are completely excluded so that atoms within such materials are unable to spontaneously absorb
and re-emit light in this region. Photonic band gap devices offer enormous potential in the
development of highly efficient narrow band lasers, integrated optical computing and high-
speed optical communication networks, particularly in the production of purely optical circuits
for dense wavelength division multiplexing.

Computational modelling of photonic band gap devices has traditionally been approached using
plane wave expansion techniques. These have the disadvantages of being expensive in terms of
computation and memory. By contrast, the finite element method is considerably more efficient
since the eigensystem matrices are very sparse and the discontinuous dielectric constant is
handled in real space. We have developed finite element software and used it to compute the
band structure for a variety of common photonic crystal structures along with more novel
structures such as the 12-fold symmetric quasicrystal.

We compare our results with those obtained from other sources, including plane wave
expansion techniques, finite difference methods and experimental data. The performance of the
algorithm is analysed in terms of memory and computational cost confirming the O(n) problem
scaling. Photonic band gap device optimisation is performed via multi-dimensional
minimisation algorithms and the analysis of a canonical set of lattice arrangements. High-
performance grid-enabled compute resources were utilised due to the computationally intensive
nature of the process. This research has ascertained those crystal structures that produce the
largest, most robust photonic band gaps with the best being the simple triangular lattice with a
high filling fraction.

II
Acknowledgements

I would like to thank my supervisor, Professor Simon Cox, for his constructive input and
support. I would also like to thank a number of other people who have contributed to this work
including Dr. J. M. Generowicz, Dr. K. S. Thomas, Mr. M. Molinari and Mr. D. H. Beckett. I
am also grateful for the assistance of Dr. M. E. Zoorob and Dr. M. D. B. Charlton from the
microelectronics group. I would also like to express my gratitude towards EPSRC and the
Department of Electronics and Computer Science, University of Southampton, for funding this
work.
III
Authors Declaration

The work in this report is partly the result of collaborative studies. Experimental data was
provided by Dr. M. D. B. Charlton and the FDTD Transmission figures were provided by Dr.
M. E. Zoorob. The Subspace Iterative Eigenvalue Solver was written by Mr. D. H. Beckett. The
FEM code was written as a joint effort between myself, Prof. S. J. Cox, Dr. J. M. Generowicz,
Dr. K. S. Thomas and Mr. M. Molinari.

IV
Contents

ABSTRACT II
ACKNOWLEDGEMENTS III
AUTHOR’S DECLARATION IV
1 INTRODUCTION.................................................................................................1
1.1 PHOTONIC CRYSTALS – AN OVERVIEW..................................................................................1
1.2 PHYSICS OVERVIEW................................................................................................................4
1.3 FINITE ELEMENT OVERVIEW...................................................................................................6
1.4 CONTRIBUTIONS......................................................................................................................8
1.5 OUTLINE OF THESIS...............................................................................................................10
2 THE FINITE ELEMENT METHOD.....................................................................12
2.1 DOMAIN DISCRETISATION.....................................................................................................12
2.2 INTERPOLATION FUNCTIONS.................................................................................................13
2.3 GOVERNING MATHEMATICAL EQUATIONS...........................................................................16
2.4 TWO-DIMENSIONAL SCALAR CASE.......................................................................................17
2.5 THREE-DIMENSIONAL VECTOR CASE...................................................................................19
2.6 CURL-CONFORMING ELEMENTS...........................................................................................21
2.7 MAXWELL VECTOR WAVE EQUATION..................................................................................24
2.8 APPROXIMATION OF THE DIFFERENTIAL EQUATION.............................................................25
2.9 ASSEMBLY OF THE SYSTEM OF EQUATIONS..........................................................................27
2.10 RECIPROCAL LATTICE SPACE AND THE BRILLOUIN ZONE....................................................29
2.11 SOLUTION OF THE SYSTEM OF EQUATIONS...........................................................................33
2.12 DATA VISUALISATION...........................................................................................................34
2.13 COMMENTS ON THE FEM......................................................................................................37
3 RESULTS..........................................................................................................41
V
3.1 COMPARISON WITH ANALYTICAL RESULTS..........................................................................41
3.2 COMPARISON WITH NUMERICAL RESULTS...........................................................................42
3.3 COMPARISON WITH EXPERIMENTAL RESULTS......................................................................49
3.4 BAND GAP MAPS...................................................................................................................50
3.5 THREE DIMENSIONAL FE ANALYSIS.....................................................................................52
3.6 FREE-SPACE PROPAGATION AT THE GAMMA POINT.............................................................53
3.7 FABRICATION CONSTRAINTS.................................................................................................59
4 PERFORMANCE...............................................................................................64
4.1 ACCURACY AND CONVERGENCE...........................................................................................64
4.2 OPTIMISING 2D ALGORITHM PERFORMANCE.......................................................................65
4.3 OBJECT-ORIENTED SOFTWARE APPROACH...........................................................................69
4.4 CLASSES AND INTERFACES....................................................................................................70
4.5 COMPUTING RESOURCES.......................................................................................................72
4.6 IRIDIS: THE BEOWULF CLUSTER...........................................................................................72
4.7 IRIDIS: THE WINDOWS CONDOR CLUSTER...........................................................................73
4.8 DGL: THE BEOWULF CLUSTER.............................................................................................73
5 12-FOLD SYMMETRIC PHOTONIC QUASICRYSTALS..................................75
5.1 QUASICRYSTAL CONFIGURATIONS.......................................................................................75
5.2 RESULTS................................................................................................................................76
5.3 INTRODUCTION OF CRYSTAL DEFECTS.................................................................................82
5.4 CONCLUSIONS.......................................................................................................................83
6 PHOTONIC CRYSTAL DESIGN OPTIMISATION.............................................84
6.1 AUTOMATED MESH GENERATION.........................................................................................84
6.2 EIGENVALUE ANALYSIS........................................................................................................87
6.3 FUNCTION MINIMISATION: THE DOWNHILL SIMPLEX METHOD...........................................87
6.4 PBG DEVICE OPTIMISATION: BRUTE FORCE APPROACH.....................................................90
VI
6.5 PBG DEVICE OPTIMISATION: CANONICAL FORMS APPROACH............................................94
6.6 OPTIMISATION FOR A HOMOGENEOUS FILLING FRACTION.................................................101
6.7 CONCLUSION.......................................................................................................................102
7 CONCLUSION.................................................................................................104
7.1 SUMMARY...........................................................................................................................104
7.2 FUTURE WORK....................................................................................................................112
REFERENCES......................................................................................................113
APPENDIX A: GALERKIN’S METHOD..............................................................119
APPENDIX B: SIMPLEX ELEMENTS................................................................121
B.1 SIMPLEX COORDINATES......................................................................................................121
B.2 2-DIMENSIONAL CARTESIAN-SIMPLEX CONVERSION.........................................................122
B.3 3-DIMENSIONAL CARTESIAN-SIMPLEX CONVERSION.........................................................123
B.4 DIFFERENTIATION IN SIMPLEX COORDINATES....................................................................124
B.5 INTEGRATION IN SIMPLEX COORDINATES...........................................................................125
APPENDIX C: 2D ELEMENTAL MATRICES.....................................................128
C.1 DERIVATION OF (cid:242) a a dW ..............................................................................................128
W m n
C.2 DERIVATION OF (cid:242) (cid:209) a .(cid:209) a dW ......................................................................................129
W m n
C.3 DERIVATION OF (cid:242) a (cid:209) a dW ..........................................................................................131
W m n
APPENDIX D: ANALYTICAL SOLUTION IN FREE-SPACE.............................134
APPENDIX E: THE LEVI-CIVITA TENSOR.......................................................135
APPENDIX F: 3D ELEMENTAL MATRICES .....................................................136
F.1 DERIVATION OF (cid:242) ((cid:209) · N )(cid:215) ((cid:209) · N )dW .......................................................................137
W i j
F.2 DERIVATION OF (cid:242) N (cid:215) N dW ..........................................................................................138
W i j
VII
( ) ( )
F.3 DERIVATION OF (cid:242) N · (cid:209) · N - N · (cid:209) · N dW .....................................................138
W i j j i
F.4 DERIVATION OF (cid:242) (k(cid:215) N )(k(cid:215) N )dW ..............................................................................139
W i j
APPENDIX G: FEMLIB CLASS DIAGRAMS.....................................................141
VIII
List of Tables

TABLE 2-1 EDGE TO VERTEX MAPPING FOR A TETRAHEDRAL ELEMENT.....................................22
TABLE 2-2 LOCAL TO GLOBAL NODE MAP...................................................................................28
TABLE 3-1 THE VALUES OF l (RECIPROCAL LATTICE VECTOR MAGNITUDE SQUARED) AND THE
NUMBER OF TIMES, N, THAT VALUE OCCURS.......................................................................54
TABLE 3-2 THE VALUES OF l (RECIPROCAL LATTICE VECTOR MAGNITUDE SQUARED) AND THE
NUMBER OF TIMES, N, THAT VALUE OCCURS.......................................................................56
TABLE 3-3 THE PROBABILITY OF A TRIANGULAR LATTICE PHOTONIC CRYSTAL EXHIBITING A
PHOTONIC BAND GAP FOR A GIVEN RELATIVE ERROR IN THE ROD RADIUS.........................62
TABLE 6-1 BAND-GAP OPTIMISATION RESULTS FOR THE TOP TWENTY MESHES.........................91
TABLE 6-2 OPTIMISATION DATA FOR THE TOP-TEN CANONICAL FORMS.....................................97
TABLE 6-3 THE SET OF RANDOMLY GENERATED MESHES THAT EXHIBIT A PHOTONIC BAND GAP
GROUPED INTO THOSE OF EQUAL FILLING FRACTION........................................................101
TABLE B-1 3-DIMENSIONAL SIMPLEX COORDINATE COEFFICIENTS..........................................123
IX
List of Figures

FIGURE 1-1 PHOTO OF A PHOTONIC CRYSTAL – COURTESY OF MARTIN CHARLTON ET AL
(SOUTHAMPTON MICROELECTRONICS RESEARCH GROUP)....................................................2
FIGURE 1-2 COMPUTATIONAL MODELLING OF PHOTONIC CRYSTALS...........................................2
FIGURE 1-3 BREAKDOWN OF THE FINITE ELEMENT METHOD........................................................7
FIGURE 2-1 PERIODIC DOMAIN DISCRETISATION EXAMPLES IN TWO DIMENSIONS (A) AND THREE
DIMENSIONS (B)...................................................................................................................12
FIGURE 2-2 UNIT CELL (A); PERIODIC TILING (B), SHOWING 9 UNIT-CELLS TILED TOGETHER...13
FIGURE 2-3 TRIANGULAR ELEMENT WITH CUBIC INTERPOLATION.............................................14
FIGURE 2-4 TETRAHEDRAL ELEMENT WITH QUADRATIC INTERPOLATION..................................15
FIGURE 2-5 LINEAR AND QUADRATIC INTERPOLATION FUNCTIONS............................................16
FIGURE 2-6 A CURL-CONFORMING VECTOR INTERPOLATION FUNCTION....................................21
FIGURE 2-7 EDGE AND VERTEX DEFINITION FOR A TETRAHEDRAL ELEMENT.............................23
FIGURE 2-8 GLOBAL AND LOCAL NODE NUMBERING..................................................................28
FIGURE 2-9 LOCAL TO GLOBAL MATRIX ELEMENTS....................................................................29
FIGURE 2-10 A TWO-DIMENSIONAL CRYSTAL LATTICE (A) AND THE CORRESPONDING UNIT CELL
(B). THE LATTICE VECTORS, A AND A ARE HIGHLIGHTED WITH ARROWS.........................29
1 2
FIGURE 2-11 LATTICE VECTORS IN REAL SPACE (A) AND RECIPROCAL LATTICE SPACE (B). THE
DIRECTION AND MAGNITUDE OF THE RECIPROCAL LATTICE VECTORS ARE GOVERNED BY
EQUATIONS (2.50) AND (2.51) RESPECTIVELY.....................................................................30
FIGURE 2-12 BRILLOUIN ZONE CONSTRUCTION FOR A TRIANGULAR LATTICE (A) DIMENSIONS OF
THE BRILLOUIN ZONE IN TERMS OF THE REAL-SPACE LATTICE PITCH LENGTH (B). THE 1ST
BRILLOUIN ZONE IS HIGHLIGHTED IN GREY WITH THE IRREDUCIBLE BRILLOUIN ZONE
CROSS-HATCHED..................................................................................................................31
FIGURE 2-13 A SOBOL SEQUENCE; A QUASI-RANDOM DISTRIBUTION OF POINTS (A) AND THE
LINKING OF ADJACENT POINTS HIGHLIGHTING THEIR CLOSE PROXIMITY (B)......................33
FIGURE 2-14 EXAMPLE OF A BAND DIAGRAM FOR A PHOTONIC CRYSTAL. THE FREQUENCY
RANGE THAT NONE OF THE BANDS INTERSECT IS HIGHLIGHTED, THIS REPRESENTS A
PHOTONIC BAND GAP...........................................................................................................35
FIGURE 2-15 EXAMPLE OF A DENSITY OF STATES DIAGRAM FOR A PHOTONIC CRYSTAL. THE
FREQUENCY RANGE FOR WHICH THE DENSITY OF STATES DROPS TO ZERO REPRESENTS A
PHOTONIC BAND GAP...........................................................................................................36
X