zotero-db/storage/955EIP69/.zotero-ft-cache

Photonic Crystals: From Theory to Practice
by Steven G. Johnson Submitted to the Department of Physics in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy in Physics
at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2001 © Massachusetts Institute of Technology, 2001. All Rights Reserved.
Author ................................................................................................... Department of Physics May 18, 2001
Certiﬁed by ............................................................................................ John D. Joannopoulos
Francis Wright Davis Professor of Physics Thesis Supervisor
Accepted by ........................................................................................... Thomas J. Greytak
Associate Department Head for Education

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Photonic Crystals: From Theory to Practice
by
Steven G. Johnson
Submitted to the Department of Physics on May 18, 2001, in partial fulﬁllment of the requirements for the degree of
Doctor of Philosophy
Abstract
In this thesis, we explore the design, computation, and analysis of photonic crystals, with a special emphasis on structures and devices that make a connection with practically realizable systems.
First, we analyze the properties of photonic-crystal slabs: 2d periodic dielectric structures that have a band gap for propagation in a plane and that use index-guiding to conﬁne light in the third dimension. Such structures are more amenable to fabrication than photonic crystals with full 3d band-gaps, but retain or approximate many of the latter’s desirable properties. We show how traditional band-structure analysis can be adapted to slab systems in the context of several representative structures, and describe the unique features that arise in this framework compared to ordinary photonic crystals. We study the possibility of lossless linear waveguides in such systems, and highlight their differences with both conventional waveguides and waveguides in true photonic crystals. Finally, we consider the creation of high-Q cavities in slabs, for which the lack of a complete gap entails unavoidable radiation losses. Two mechanisms for minimizing such losses are described and demonstrated: mode delocalization and the novel far-ﬁeld multipole cancellation.
Next, we present a 3d periodic dielectric structure with a large, complete photonic bandgap. The structure is distinguished by a sequence of planar layers, identical except for a horizontal offset, and repeating every three layers to form an fcc lattice. The high symmetry of the layers means that complex devices could be formed by modifying only a single layer, and their similarity to common 2d photonic crystals allows the direct application of results and experience from those simpler systems.
Third, we present and demonstrate general criteria for crossing perpendicular waveguides without crosstalk, based on a priori principles of symmetry and resonance.
Finally, we describe a fully-vectorial, 3d algorithm to compute the deﬁnite-frequency eigenstates of Maxwell's equations in arbitrary periodic dielectric structures, including systems with anisotropy (birefringence) or magnetic materials, using preconditioned block-iterative eigensolvers in a planewave basis. Many different numerical techniques are compared and characterized. Our implementation is freely available on the Web.
Thesis Supervisor: John D. Joannopoulos Title: Francis Wright Davis Professor of Physics

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Biographical Note
Steven G. Johnson was born on September 8, 1973, amid the cornﬁelds, or rather the suburbs, of Illinois—growing up in the promised land of Saint Charles after a ﬂeeting golden age of Naperville. His ambitions for the future formed a continuous progression, starting from the goal of being Dopey of the Seven Dwarfs, to the closely-related career of astronaut, thence to mission specialist, and ﬁnally to scientist (after being reassured by his sister Teresa of how “scientiﬁc” he looked in his new glasses). Any photographs of him dressed as a ballerina are utter fabrications. Mrs. Fitz’ chemistry class at Saint Patrick’s grade school (motto: “If it isn’t green, it’s not from God”) was terribly exciting, showing him for the ﬁrst time how to make quantitative predictions of natural events (here, the proportions of different chemicals that could combine to make new compounds, such as explosives). He sensibly shifted his sights to physics, however, after a highschool chemistry text explained that the laws of chemistry actually come from physicists, “who are very clever at ﬁguring out those kinds of things.” His high-school, by the way, consisted of one year making friends for life at Saint Charles High School (motto: “All school buildings should ﬁt comfortably inside the swimming pool”), followed by three years in search of a girlfriend at the Illinois Mathematics and Science Academy (IMSA, a.k.a. “High-Tech High”). He had wanted to go to MIT since his freshman year of high school, based in part on televised robot competitions, which turned out to be a good thing as he seems to have received a life sentence at this institution. Here, surrounded by hackers wearing capes and machines that go “ping,” Steven’s family, friends, and colleagues agree that his lifelong goal has been fulﬁlled: he is Dopey, after all.
Assuming that he’s not dead, Steven can best be reached at his “lifetime” email address of:
stevenj@alum.mit.edu
The following pages contain a less colloquial summary of Steven’s professional activities.
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Education and Awards
Massachusetts Institute of Technology (MIT), 1995–2001: Doctoral program in physics. Recipient of a National Defense Science and Engineering Graduate Fellowship (NDSEG) and an MIT Karl Taylor Compton Fellowship; also offered National Science Foundation (NSF) and Hertz Foundation fellowships.
MIT, 1991–1995: B.S. Physics, B.S. Electrical Engineering and Computer Science, B.S. Mathematics. 5.0/5.0 GPA. Thesis on semi-classical analyses of quantum chaos. Recipient of MIT Orloff Award for scholarship in physics. Member of honor societies: Phi Beta Kappa (liberal arts), Sigma Pi Sigma (physics), Tau Beta Pi (engineering), and Eta Kappa Nu (electrical engineering).
Illinois Mathematics and Science Academy, 1988–1991. Eagle Scout. Voted “biggest I-think-I-know-it-all.”
Co-recipient of the 1999 J. H. Wilkinson Prize for Numerical Software, for work on the FFTW fast Fourier transform library (http://www.fftw.org/) with Dr. Matteo Frigo of MIT. This is awarded every four years at the Intl. Conf. on Industrial and Applied Math (ICIAM) by Argonne National Laboratory, the National Physical Laboratory (UK), and the Numerical Algorithms Group to the software that “best addresses all phases of the preparation of high-quality numerical software.” For the same work, was co-recipient of a 1999 Laurels Award in “Electronics” from Aviation Week & Space Technology for “signiﬁcant contributions to the global ﬁeld of aerospace in 1999.”
Research and Teaching Experience
OmniGuide Communications, Inc. (Cambridge, MA), Sep. 2000–Jan. 2001: lead theory and simulation research group in the investigation and theoretical characterization of a new class of hollow optical ﬁbers. (Part-time work there continues to the present day.)
Femtosecond Technology Research Association (Tsukuba, Japan), Summer 2000: introduced researchers at FESTA to the theoretical study of photonic crystals, and performed calculations to characterize the structures fabricated at their laboratory.
AT&T Bell Laboratories, summers of 1993 and 1994 (MIT VI-A internship program): developed real-time visualization software for high-speed optics, built laser-alignment system for free-space optics, and developed program for 3d capacitance extraction.
MIT Mathematics Dept. and School of Mgmt., spring 1993 (MIT UROP program): developed numerical studies of options contracts and Black-Scholes equations.
Fermi Natl. Accelerator Laboratory, summers/winters 1989–1991, 1995: developed simulation software for liquid Helium cryogenics systems.
MIT Experimental Study Group, 1994–present: taught freshman physics courses to small groups of 1–3 students, for roughly 6–8 semesters up to the present date, in this optional alternative academic program for MIT freshmen.
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Publications
S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Optics Express 8 (3), 173–190 (Focus Issue on Photonic Bandgap Calculations) (2001).
E. Chow, S. Y. Lin, J. R. Wendt, S. G. Johnson, and J. D. Joannopoulos, “Quantitative analysis of bending efﬁciency in photonic-crystal waveguide bends at λ = 1.55µm wavelengths,” Optics Lett. 26, 286–288 (2001).
S. Fan, S. G. Johnson, J. D. Joannopoulos, C. Manolatou, and H. A. Haus, “Waveguide branches in photonic crystals,” J. Opt. Soc. Am. B 18 (2), 162–165 (2001).
S. G. Johnson and J. D. Joannopoulos, “Three-dimensionally periodic dielectric layered structure with omnidirectional photonic band gap,” Appl. Phys. Lett. 77, 3490–3492 (2000).
E. Chow, S. Y. Lin, S. G. Johnson, P. B. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000).
S. Y. Lin, E. Chow, S. G. Johnson, and J. D. Joannopoulos, “Demonstration of highly efﬁcient waveguiding in a photonic crystal slab at the 1.5-µm wavelength,” Opt. Lett. 25, 1297–1299 (2000).
S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
C. Manolatau, S. G. Johnson, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “High-Density integrated optics,” J. Lightwave Tech. 17, no. 9 (Sep. 1999).
S. G. Johnson, C. Manolatou, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Elimination of cross talk in waveguide intersections,” Optics Letters 23, 1855–1857 (1998).
P. R. Villeneuve, S. Fan, S. G. Johnson, and J. D. Joannopoulos, “Three-dimensional photon conﬁnement in photonic crystals of low-dimensional periodicity,” IEE Proc. Optoelec. 145, 384 (1998).
M. Frigo and S. G. Johnson, “FFTW: An adaptive software architecture for the FFT,” Proc. ICASSP 1998, vol. 3, 1381 (1998).
W. M. Soyars and S. G. Johnson, “Simulating the Tevatron liquid Helium satellite refrigerators,” Advances in Cryogenic Engineering 39, 1231–1235 (1994).
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Acknowledgments
I don’t know half of you half as well as I should like; and I like less than half of you half as well as you deserve.
J. R. R. Tolkien, The Fellowship of the Ring
It would be impossible to express my thanks on this page to all those who have supported me, without whose help I could never have come so far. I will attempt, at least, to satisfy the barest demands of decency by saying a few words here.
No gratitude is sufﬁcient to repay the unceasing love of my parents and family, who have stood behind me from my ﬁrst steps through all the moments of skinned knees and shaken conﬁdence. They read me my ﬁrst book, let a young boy wander through a particle accelerator, and never failed to call when my studies overwhelmed me. They gave me the ground I could stand on whenever the path ahead seemed dim. No son (or brother) could ask for better.
I will always be a student, I hope, and I pray that my future studies will be blessed by teachers as gifted and inspiring as the ones I have had. From Mrs. Fitz and Mr. Cordier in grade school who made their classrooms the best place in the world, to my calculus teacher Mr. Choi who urged us to see the beauty that others overlook, to Dr. Barr who brought me poetry and philosophy, to Peter Dourmashkin who showed me that teaching is the highest form of learning, they and many others have opened new worlds before my eyes.
As a teacher, as an advisor, as a colleague, as a friend, John Joannopoulos sets a standard that I will always strive to attain. I have been truly lucky to have learned from one so generous and so wise in the ways of life as well as those of physics. Thanks for putting up with such an obstinate student, John.
Finally, I am grateful to MIT for supporting me with a Karl Taylor Compton fellowship, and to the United States government for a National Defense Science and Engineering Graduate Fellowship.
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Table of Contents
1 Photonic Crystals and Other Obscure Topics ............................................................21 1.1 In the Beginning, There Was Maxwell ............................................................21 1.2 Periodic Surprises ............................................................................................22 1.3 Photonic Crystals .............................................................................................24 1.4 Things to Come................................................................................................29
2 Band Gaps & Maxwell’s Equations: A Whirlwind Tour...........................................35 2.1 Maxwell’s Equations .......................................................................................35 2.2 The Bloch-Floquet Theorem and Electromagnetism.......................................37 2.3 The Origin of the Photonic Band Gap .............................................................39
3 Elimination of Crosstalk in Waveguide Intersections ...............................................43 3.1 Introduction......................................................................................................43 3.2 Eliminating Crosstalk by Symmetry................................................................44 3.3 Intersecting Photonic-Crystal Waveguides......................................................46 3.4 Intersecting Conventional Waveguides ...........................................................49 3.5 Summary ..........................................................................................................50
4 Guided Modes in Photonic-Crystal Slabs ..................................................................53 4.1 Introduction......................................................................................................53 4.2 Computational Method ....................................................................................56 4.3 Photonic-Crystal Slab Band Structures............................................................57 4.4 Effects of Slab Thickness.................................................................................63 4.5 Slabs with Solid Backgrounds (Sandwiches) ..................................................67 4.6 Slabs with Periodic Backgrounds ....................................................................69 4.7 Slabs with Symmetry-Breaking Backgrounds .................................................72 4.8 Summary ..........................................................................................................74
5 Linear Waveguides in Photonic-Crystal Slabs ..........................................................77 5.1 Introduction......................................................................................................77 5.2 Computational Method ....................................................................................83 5.3 Reduced-Index Waveguides ............................................................................84 5.4 Increased-Index Waveguides ...........................................................................90 5.5 Strip Waveguides in Photonic-Crystal Slabs ...................................................93 5.6 Waveguides in Other Directions......................................................................96 5.7 Estimating the Field Energy in the Dielectric..................................................98 5.8 Summary ........................................................................................................102
6 High-Q Cavities in Photonic-Crystal Slabs .............................................................103 6.1 Introduction....................................................................................................103 6.2 Computational Method ..................................................................................105 6.3 Mode Delocalization......................................................................................106 6.4 Multipole Cancellation...................................................................................110 6.5 Summary and Conclusions ............................................................................115
7 Three-dimensionally periodic dielectric layered structure with omnidirectional photonic bandgap .............................................................................................................117
7.1 Introduction....................................................................................................117 7.2 Other Photonic-Crystal Structures .................................................................119
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7.3 Characterizing the New Structure..................................................................122 7.4 Possible Fabrication Methods ........................................................................123 7.5 Summary ........................................................................................................124 8 Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis125 8.1 Introduction....................................................................................................125 8.2 The Maxwell Eigenproblem ..........................................................................128 8.3 Iterative Eigensolvers.....................................................................................138 8.4 Summary ........................................................................................................152 9 Concluding Remarks................................................................................................153 9.1 Looking Back.................................................................................................153 9.2 Looking Forward ...........................................................................................154 Bibliography ...............................................................................................................157
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List of Figures
Figure 1.1: Schematic of a one-dimensional photonic crystal: alternating dielectric layers (represented by blue and yellow), with a band gap in one direction. Incident light is reflected for colors of light in the band gap, but it cannot be used to trap light in 3d—the light inevitably escapes “sideways.” ..........................................................................27
Figure 1.2: Schematic of the photonic-crystal slab chapters, covering first the bulk systems, then waveguides formed by linear defects, then cavities formed by point defects....31
Figure 1.3: A portion of a novel 3d photonic crystal that we propose in Chapter 7. This structure has a large, complete band gap, can be constructed layer-by-layer using traditional lithography, and its behaviour is understandable in terms of much-simpler twodimensional photonic-crystal structures.....................................................................32
Figure 3.1: Abstract diagram of symmetry requirements for waveguide crossing, showing waveguide mode profiles and resonant-cavity mode contours. By symmetry, the solidline (red) modes cannot couple with the dashed-line (blue) modes and vice-versa...44
Figure 3.2: Waveguide intersections in a two-dimensional photonic crystal consisting of a square lattice of dielectric rods in air. ........................................................................46
Figure 3.3: TM transmission spectra for the four intersections shown in Fig. 3.2: (a) throughput from the input port to the output port and (b) crosstalk from the input port to one of the transverse ports. ........................................................................................48
Figure 3.4: Intersection of two-dimensional, high index-contrast waveguides in air, both (a) without holes and (b) with air holes added to create a resonant cavity using a photonic band gap. ....................................................................................................................49
Figure 3.5: TE transmission spectra for the two intersections shown in Fig. 3.4...........50
Figure 4.1: Two-dimensional photonic crystals. (a) Square lattice of dielectric rods in air, with lattice constant a and radius 0.2a. (b) Triangular lattice of air holes in dielectric, with lattice constant a and radius 0.45a. In both case, the dielectric constant of the highindex material is 12. ...................................................................................................54
Figure 4.2: Band diagrams for the photonic crystals from (a) Fig. 4.1(a) and (b) Fig. 4.1(b). The shaded region indicates the frequencies of states introduced when vertical propagation (i.e., perpendicular to the plane of periodicity) is permitted...............................55
Figure 4.3: Photonic-crystal slabs analogous to the two-dimensional structures from Fig. 4.1. (a) Square lattice of rods in air with height 2.0a. (b) Triangular lattice of holes in a dielectric slab with thickness 0.6a. (Other parameters are as in Fig. 4.1.) Both slabs are suspended in air. .........................................................................................................57
Figure 4.4: Projected band diagrams corresponding to the two slabs in Fig. 4.3. Whether states are even or odd with respect to the horizontal mirror plane of the slab is indicated by open or filled circles, respectively. .......................................................................59
Figure 4.5: Lowest-frequency states of the rod and hole structures from Fig. at M and K,
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respectively, depicted in the unit cell. The shading indicates the z components of the electric and magnetic ﬁelds, respectively, which happen to be positive throughough the unit cell for these states. Outlines of the rods/holes are shown in white. (a) Vertical cross-section. (b) Horizontal cross-section. In both cases the cross-sections bisect the holes or rods. ...................61
Figure 4.6: (a) Horizontal cross-section of Ez for the first odd (TM-like) excited state at M in the rod structure from Fig. 4.4(a). (b)Vertical cross-section of the Hz for the ﬁrst TE-like (but odd) excited state at K in the hole structure shown in Fig. 4.4(b). .....................................62
Figure 4.7: Gap size (as a percentage of mid-gap frequency) versus slab thickness for the two slabs from Fig. 4.3...............................................................................................................63
Figure 4.8: Side-view of the slabs from Fig. 4.3 with a uniform, low-index background above and below the slabs. (a) is the rod slab and (b) is the hole slab. The background has a dielectric constant of 2.0 and all other parameters are as in Fig. 4.3...............................................67
Figure 4.9: Projected band diagrams for the structures shown in Fig. 4.8. ....................68
Figure 4.10: Side-view of the slabs shown in Fig. 4.3 with a periodic background formed by “extruding” the structures with a low-index material (dielectric constant 2.0). All other parameters are as in Fig. 4.3. ........................................................................................................69
Figure 4.11: Projected band diagrams for the structures shown in Fig. 4.10. ................70
Figure 4.12: Side-view of the slabs shown in Fig. 4.10 (extruded backgrounds) with the upper background replaced by air. All other parameters are as in Fig. 4.10. This background breaks the mirror symmetry of the two photonic-crystal slabs. ............................................72
Figure 4.13: Projected band diagrams for the structures shown in Fig. 4.12. The bands can no longer be segregated into even and odd modes, and there is no longer a band gap in the guided modes. ........................................................................................................................73
Figure 4.14: A solid substrate may lie below a slab without affecting the band structure by being sufficiently far from the slab. (a), (b), (c), and (d) depict such solid substrates below the symmetric and antisymmetric rod and hole structures from Figs. 10(a), 12(a), 10(b), and 12(b), respectively. ..................................................................................................................74
Figure 5.1: Photonic-crystal slabs. (a) Square lattice of dielectric rods in air with ε of 12.0, lattice constant a, radius 0.2a, and height 2.0a, with low-index (ε of 2.0) rods extending inﬁnitely above and below. (b) Triangular lattice of air holes extending through both a high-index (ε of 12.0) ﬁnite-height dielectric slab and low-index (ε of 2.0) semi-inﬁnite dielectric regions above and below. The holes have lattice constant a and radius 0.3a, while the high-index slab is of thickness 0.5a.............................................................................................................79
Figure 5.2: Band structure for the rod slab from Fig. 5.1(a). Hollow and filled circles represent modes which are even and odd with respect to the horizontal (z = 0) mirror plane bisecting the slab. There is a band gap in the odd (TM-like) modes in the frequency range 0.3362–0.4292 c/ a..................................................................................................................................80
Figure 5.3: Band structure for the hole slab from Fig. 5.1(b). Hollow and filled circles denote even and odd modes as in Fig. 5.1. There is a band gap in the even (TE-like) modes in the frequency range 0.2708–0.3471 c/a. ...........................................................................................81
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Figure 5.4: Linear defects, which give rise to waveguide modes, in the rod and hole photonic-crystal slabs from Fig. 5.1. The low-index material is not shown, but “extrudes” above and below the structures as in Fig. 5.1. (a) Reduced-index waveguides, created by decreasing/increasing the radii of a line of nearest-neighbor rods/holes, respectively. (b) Increased-index waveguides, created by increasing/decreasing the radii of a line of nearest-neighbor rods/holes, respectively. (c) Dielectric-strip waveguide surrounded by the rod slab, from which a line of nearest-neighbor rods has been removed and replaced with the waveguide. ...........................................................................................................82
Figure 5.5: Projected band structure for the reduced-index rod-slab waveguide from Fig. 5.4(a), showing the (odd-symmetry) guided bands for various defect rod radii (compared to the bulk radius of 0.20a). .............................................................................85
Figure 5.6: Field cross-sections for the rod-radius 0.14a guided mode at X (0.3707 c/a), showing the z component of the electric ﬁeld. The contours of the dielectric function are shown in black. (a) Horizontal (xy) cross-section in the plane bisecting the slab (z = 0). (b) Vertical (yz) cross-section in the plane perpendicular to the waveguide and bisecting a row of rods...............................................................................................................86
Figure 5.7: Projected band structure for the reduced-index hole-slab waveguide from Fig. 5.4(a), showing the (even-symmetry) guided bands for various defect hole radii (compared to the bulk radius of 0.30a). .....................................................................87
Figure 5.8: Diagram of how the reciprocal lattice is projected onto Γ–K (vertical dashed line) for a nearest-neighbor line defect in the hole slab. The black dots are the reciprocal lattice points, the outlined hexagon is the ﬁrst Brillouin zone, and the grey triangle within it is the reduced Brillouin zone. K' is the boundary of the ﬁrst Brillouin zone in the projected lattice. .........................................................................................................88
Figure 5.9: Field cross-sections for the hole-radius 0.45a guided mode at K' (0.3026 c/a), showing the z component of the magnetic ﬁeld. The contours of the dielectric function are shown in black. (a) Horizontal (xy) cross-section in the plane bisecting the slab (z = 0). (b) Vertical (yz) cross-section in the plane perpendicular to the waveguide and bisecting a defect hole. ........................................................................................................89
Figure 5.10: Projected band structure for the increased-index rod-slab waveguide from Fig. 5.4(b), showing the (odd-symmetry) guided bands for defect rod radius 0.25a (compared to the bulk radius of 0.20a). .............................................................................91
Figure 5.11: Horizontal (z = 0) Ez field cross-sections for the rod-radius 0.25a guided modes at X. (a) px guided mode (0.3887 c/a). (b) py guided mode (0.4205 c/a). ..................91
Figure 5.12: Projected band structure for the increased-index rod-slab waveguide from Fig. 5.4(b), showing the (odd-symmetry) guided bands for defect rod radius 0.275a.92
Figure 5.13: Projected band structure for the increased-index hole-slab waveguide from Fig. 5.4(b), showing the (even-symmetry) guided bands for defect hole radii 0.15a and 0.25a (compared to the bulk radius of 0.30a). For each radius, there are two or three guided bands: px (solid line), py (dashed line), and dxy (dotted line, triangles). Guided bands for a defect hole radius of 0, in which a row of holes is entirely removed, are
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shown in bold. ............................................................................................................93
Figure 5.14: Horizontal (z = 0) Hz field cross-sections for the hole-radius 0.15a guided modes at K'. (a) px guided mode (0.2874 c/a). (b) py guided mode (0.3148 c/a). (c) dxy guided mode (0.3377 c/a). ...............................................................................................................94
Figure 5.15: Projected band structure for the strip waveguide in a rod slab, from Fig. 5.4(c), showing the (odd-symmetry) guided bands for waveguide width 0.25a. There are two pairs of guided bands: gap-guided (solid lines) and index-guided (dashed lines). The gap-guided bands are split at X into a 0.07% gap, while the index-guided bands are split at X into a 0.8% gap. 95
Figure 5.16: Horizontal (z = 0) Ez field cross-sections for the 0.25a strip-waveguide gap-guided modes at X. (a) Lower gap-guided mode (0.3626 c/a). (b) Upper gap-guided mode (0.3629 c/ a). ...............................................................................................................................97
Figure 5.17: Vertical (y = 0) Ez field cross-sections of a unit cell, parallel to and bisecting the waveguide, for the strip-waveguide gap-guided and index-guided modes at X. (a) Upper gapguided mode (0.3629 c/a). (b) Lower index-guided mode (0.2632 c/a)....................97
Figure 6.1: Photonic-crystal slabs in 2d and 3d, with defects (resonant cavities) formed by increasing the radius of a central rod. (a) 2d slab: a sequence of dielectric rods with lattice constant a, radius 0.2a, and ε of 11.56. (b) 3d slab: a square lattice of dielectric rods with lattice constant a, radius 0.2a, height 2a, and ε of 12. ..........................................................................104
Figure 6.2: Total Q (solid circles) and radiation Q⊥ (hollow circles) plotted vs. ∆ω above the band edge for monopole defects in the 11x11 slab of rods from Fig. 6.1(b). Black points are for a single-rod defect (left inset, black), and red points are for a 5-rod super-defect (upper-right inset, red). ..........................................................................................................................108
Figure 6.3: Monopole defect mode for ε=9 super-defect (Q = 11000 , Q⊥ = 13000 ): electricﬁeld z-component in the plane bisecting the rods....................................................109
Figure 6.4: Q vs. ω for a dipole state in the 2d rod-sequence structure (inset) of Fig. 6.1(a). Both the total Q (black line, ﬁlled circles) and also the Q±m of the ﬁrst few nonzero multipole moments (colored lines, hollow circles) are shown. .....................................................112
Figure 6.5: Ez radiation pattern for 2d dipole states of Fig. 6.4, using a color table that exaggerates small ﬁeld magnitudes, with dielectric boundaries shown in black. (a) Point just before the peak (Q=1773, ω =0.328). (b) Point at the peak (Q=28700, ω =0.309), showing nodal lines from cancellation of the lowest multipole moment. (c) Point just beyond the peak (Q=6624, ω =0.300). 113
Figure 6.6: (a) Q vs. ω for a quadrupole state in the 3d rod-slab structure of Fig. 6.1(b). Ez in the mid-plane at peak Q is shown i n the inset. The central rod is r=0.45a with ε=13; ε of the four r=0.25a neighbors was varied to control ω , and labels the points. The solid line is a Lorentzian curve ﬁtted to the peak (with R2 = 0.9994 ). (b) Electric-ﬁeld energy density for this mode, plotted in a plane 2a above the rods. (i) Point just before the peak (Q=426, ω =0.346). (ii) Point at the peak (Q=1925, ω =0.349), showing nodal planes from cancellation of the lowest multipole moment. (iii) Point just beyond the peak (Q=408, ω =0.352)..........................114
Figure 7.1: Computer rendering of a novel 3d photonic crystal, showing several horizontal periods
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and one vertical period, consisting of an fcc lattice of air holes (radius 0.293a, height 0.93a) in dielectric (21% ﬁll). This structure has a 21% gap for a dielectric constant of 12..............................................................................................................................118
Figure 7.2: Schematic of the crystal in Fig. 7.1. (a) The vertical structure, showing the different material layers as they might be deposited during fabrication. (The layers are given different colors for clarity, but would normally be the same material and thickness.) (b) Plan view of a horizontal cross-section intersecting the “A” cylinders, with the offset locations of cylinders in other layers also labelled. .................................................119
Figure 7.3: (a) Band diagram for the structure in Fig. 7.1, showing frequency in scale-invariant units vs. wavevector along important symmetry directions in the irreducible Brillouin zone. The inset shows the ﬁrst Brillouin zone and its symmetry points. (b) Vertical transmission spectrum for slightly over one period (3 layers plus a capping hole slab, as shown in the inset) of this structure, showing the Γ – L' gap...................................120
Figure 8.1: Eigenvalue convergence as a function of grid resolution (grid points per lattice constant a) for three different methods of determining an effective dielectric tensor at each point: no averaging, simply taking the dielectric constant at each grid point; averaging, the smoothed effective dielectric tensor of Eq. (8.12); and backwards averaging, the same smoothed dielectric but with the averaging methods of the two polarizations reversed.) ..................................................................................................................135
Figure 8.2: Eigensolver convergence for two variants of conjugate gradient, FletcherReeves and Polak-Ribiere, along with preconditioned steepest-descent for comparison. 143
Figure 8.3: The effect of two preconditioning schemes from section 2.4, diagonal and transverse-projection (non-diagonal), on the conjugate-gradient method. ......................145
Figure 8.4: Comparison of the Davidson method with the block conjugate-gradient algorithm of Sec. 8.3.1. We reset the Davidson subspace to the best current eigenvectors every 2, 3, 4, or 5 iterations, with a corresponding in increase in memory usage and computational costs..................................................................................................147
Figure 8.5: Conjugate-gradient convergence of the lowest TM eigenvalue for the “interior” eigensolver of Eq. (8.27), solving for the monopole defect state formed by one vacancy in a 2D square lattice of dielectric rods in air, using three different supercell sizes (3 × 3 , 5 × 5 , and 7 × 7 ). ....................................................................................................148
Figure 8.6: Scaling of the number of conjugate-gradient iterations required for convergence (to a fractional tolerance of 10–7 ) as a function of the spatial resolution (in grid points per lattice constant, with a corresponding planewave cutoff), or the number p of bands computed. .................................................................................................................151
17

18

List of Tables
Table 7.1: Optimal parameters and gaps for various dielectric contrasts 123
19

20

Chapter 1
Photonic Crystals and Other Obscure Topics
1.1 In the Beginning, There Was Maxwell
The thing that hath been, it is that which shall be; and that which is done is that which shall be done: and there is no new thing under the sun.
Ecclesiastes 1:9.
In 1864, James Clerk Maxwell presented his treatise, “A dynamic theory of the electromagnetic ﬁeld,” and revealed the equations underlying all phenomena in electricity and magnetism [1]. This remarkable work laid bare the deep connections between the electricity of Franklin, the magnetic lodestones of the ancient mariners, and the light that had ﬁlled the ﬁrst eyes gazing into the heavens. And yet Maxwell’s equations, encompassing as they are, were only the beginning.
Of course, it is not enough to understand electricity and magnetism—one must also comprehend the stuff upon and within which those forces operate. Maxwell himself knew the behavior of certain kinds of materials, such as dielectrics (of which glass is a famous example), and many other important systems would emerge over the years: from the semiconductors in our computers to the black holes that warp the universe itself along with its light. (On the other hand, one material of Maxwell’s analyses turned out not to exist: the “luminiferous ether” was vanquished by relativity’s assertion that light propagates without a ﬁxed medium.) Indeed, Maxwell’s electromagnetism has ﬁgured prominently in virtually every subsequent advance of physics, from quantum mechanics to relativity to the “standard model” of particle theory. Forgive us our myopia, however, if we allow ourselves to forget those extraordinary discoveries for a moment—what, then, remains to be found in Maxwell’s equations themselves?
21

An equation is only useful to the extent that you can solve it, and even in simple circumstances the solution of Maxwell’s equations is not easy (as any student in an electromagnetism course will unhappily inform you). Nevertheless, the phenomena that can emerge in such basic problems were largely solved and understood by clever Maxwell and his contemporaries. Since then, it often seemed that the only remaining task was the “engineering” challenge of solving and utilizing those equations in ever more realistic and messier systems. Even the glassy ﬁber-optic cables that today span our oceans for telecommunications would surprise Maxwell only in their uses and in their perfection of manufacture, not in their underlying principles. The story of condensed-matter physics, however, has been that even simple, well-known equations can have unexpected consequences, especially in complicated circumstances.
1.2 Periodic Surprises
Milo tried very hard to understand all the things he’d been told, and all the things he’d seen, and, as he spoke, one curious thing still bothered him. “Why is it,” he said quietly, “that quite often even the things which are correct just don’t seem to be right?”
Norton Juster, The Phantom Tollbooth
Imagine shining a light through a large block of glass with a single bubble of air in it. When the light strikes the bubble, some of it will reﬂect and some of it will continue forward at a slightly different angle (be refracted). As you look through the glass, this scattered light allows you to see the bubble, perhaps with an attractive sparkling caused by all of the reﬂections and refractions. Picture now a second bubble in the glass, just like the ﬁrst but at a different place. As before, the light will reﬂect and refract, this time from both bubbles, sparkling in a more intricate pattern than before. All of this is exactly predicted by Maxwell’s equations, although solving for the precise details would probably require
22

the help of a computer. Next, suppose that we ﬁll the glass with millions of bubbles, all identical and arranged in a perfectly periodic lattice extending in all directions. Surely, this is not much different from before, although the sparkles will form a far more complicated mess? (“Ugh,” says the engineer.) In fact, if you are very careful, you may see something quite different: nothing at all. What would Maxwell say?
James would be gratiﬁed to learn that he has not been contradicted—everything is predicted by his equations, as always. The engineer, working out the integrals carefully on her computer, would discover to her amazement that all of those ﬁendish reﬂections and refractions cancel one another before reaching our eyes, and the light passes through the crystal unimpeded. This is no accident, but is a general consequence of the periodicity of the structure—actually, the explanation is closely related to one of the great mysteries of physics from a century ago, in which the cancellation was of electrons rather than of light.
When you ﬂip a wall switch to illuminate the dirty socks on your ﬂoor, an electric current—a ﬂow of tiny, charged particles called electrons—carries your good intentions to the bulb. Unfortunately, this current requires constant power to maintain, as your monthly electric bill attests; otherwise, the electrons will halt their ﬂow after caroming off an atom or two in the wire (generating heat and light as a side effect). Scientists of the 19th century, however, couldn’t leave well-enough alone: they set out to calculate and measure the rate at which electrons bounced off atoms. What they found was terribly strange. You see, the forces between charged particles are enormous—powerful enough to keep an electron hurtling around a Hydrogen nucleus at a speed of over 1000 miles per second in a circle four billionths of an inch across—so, an electron ﬂying through the metal wire, surrounded by charged nuclei every few billionths of an inch (not to mention all the other electrons), shouldn’t get very far. And yet it was found that, in a pure chunk of a good conductor such as copper wire, an electron apparently meanders blithely past 100 or more nuclei before
23

bumping into one! The solution to this puzzle came in two pieces (some assembly required). First, in the 20th century, it was reluctantly realized that electrons are actually waves: rather than hard little bullets whizzing through space, they are smeared out in a ghostly probability envelope covering an entire volume at once—and, unlike bullets, they can interfere and even cancel if two waves are superimposed. Second, since a conductor like copper forms a periodic lattice of nuclei, a crystal, we can apply the lovely* theorem that we alluded to earlier with the bubbles.
This theorem, called alternatively “Bloch’s” [2] or “Floquet’s” [3] theorem depending upon whether you drink beer or wine, says that, in a periodic medium, waves can be found that propagate without scattering, and is equally true whether the waves consist of electrons or of light beams (or sound waves, or…). This is why the electrons can drift so far in the metal—they only bounce off of impurities or imperfections—and why a light beam could travel through our bubbly glass without reﬂecting towards our eyes. And that’s not all: the machinery of the Bloch-Floquet theorem leads us to many other curious phenomena…

1.3 Photonic Crystals

How do you solve a problem like [∇ × 1-ε-∇ × H

=

 

ω-c--

2

H

]?

How do you hold a moonbeam in your hand?

Oscar Hammerstein II, The Sound of Music

By combining Maxwell’s equations with the theorems of solid-state physics, a surprising and simple result has emerged from what seemed like a horrendous problem, that of light bouncing among an inﬁnity of periodic scatterers. But there’s still more to be learned from the humble electron, a behavior of light that Maxwell never guessed although the equa-

* Physicists have strange attractions. (Mathematicians, contrariwise, have “strange attractors.”)

24

tions were right in front of his eyes: just as there are electrical insulators, which keep the currents in the wires where they belong, one can also build an optical insulator, a photonic crystal.
Photonic crystals again arise from the cooperation of periodic scatterers—thus, they are called “crystals” because of their periodicity and “photonic” because they act on light.* They can occur when the period (the separation of the bubbles or scatterers) is on the order of the wavelength of the light.† When that’s true, and if your bubbles are the right shape, it is possible that all of those reﬂections and refractions will cancel not only the light scattered sideways, but the light moving forward as well. Then, because the light has to go somewhere (energy is conserved), it has no choice but to go back—it is forbidden from entering the photonic crystal. This happens no matter what direction the light is coming from, in a certain range of wavelengths called the photonic band gap.‡ (A similar effect, an electronic band gap, is what gives electrical semiconductors their lucrative properties.)
Once you have such a medium, impervious to light, you can manipulate photons in many interesting ways. By carving a tunnel through the material, you have an optical “wire” from which no light can deviate. Even more dramatically, by making a cavity in the center of the crystal, you have an optical “cage” in which a beam of light could be caught and held—although you couldn’t easily admire it, because the very fact that it cannot escape would render it invisible. These kinds of abilities to trap and guide light have many potential applications in optical communications and computing, where one would like to make tiny optical “circuits” to help manage the ever-increasing trafﬁc through the world’s
* Perversely, at about the same time that electrons were reborn as waves, Einstein discovered that the light waves of Maxwell’s equations are “particles” called photons. † Light, like all waves, has a characteristic wavelength: the distance from one wave crest to the next, where each wavelength corresponds to a different color. For example, all of the colors that you can see fall in the range of about 0.4 to 0.6 microns—one micron is 1/1000 millimeters—but the wave crests are too small and move too fast for you to observe directly. ‡ Some people call these systems “photonic band-gap materials” instead of “photonic crystals.”
25

optical communications networks. Other devices, too, are made possible by this increased control over light: from more-efﬁcient lasers and LED light sources, to opening new regimes for operating optical ﬁbers, to cellular phones that don’t pump half of their signal power into your head (as harmless heat, but wasteful).
What’s so new about photonic crystals? Aren’t they only mirrors, just like the one in which we admire ourselves every morning? Metal mirrors, however, besides operating on entirely different principles, have a major disadvantage: some of the light that strikes them is absorbed rather than reﬂected, and this absorption becomes intolerably large as you move towards the short wavelengths (infrared and visible) that are most important for lasers and optical communications.* On the other hand, photonic crystals rely only on weak interactions of the light with the material, and can have low absorption at nearly any wavelength.
Actually, a form of photonic crystal can be found in nature, producing the iridescent colors of abalone shells, butterﬂy wings, and some crystalline minerals. These beautiful materials archetypically comprise alternating microscopic layers of two substances (or two arrangements of the same substance)—they are only periodic in one direction, and thus form a one-dimensional (1d) photonic crystal, depicted in Fig. 1.1. When light shines on the layers, it is reﬂected by the photonic band gap.† Because the natural optical contrast between the layers is weak, however, the band gap is narrow—only a speciﬁc color is reﬂected, resulting in the brilliant hues we so admire. Moreover, as the angle of the light changes, the periodicity in that direction alters and the band gap shifts wavelengths—that is, different colors are reﬂected at different angles, shimmering as we tilt our heads.
* Metal mirrors work just ﬁne for microwave wavelengths, and Bell Labs tried for many years to use metal tubes as microwave communications “pipes.” How these tubes eventually lost the battle against today’s glass ﬁbers makes an interesting story. (For one thing, you couldn’t bend the tube.) † The classical way to understand the 1d band gap is that, if the periodicity is half a wavelength, the cumulative reﬂections are in phase: the reﬂected wave crests line up, and so are additive.
26

incident light

1d photonic crystal

…is reflected
…but light escapes from a cavity
Figure 1.1: Schematic of a one-dimensional photonic crystal: alternating dielectric layers (represented by blue and yellow), with a band gap in one direction. Incident light is reﬂected for colors of light in the band gap, but it cannot be used to trap light in 3d—the light inevitably escapes “sideways.”
Just such a crystalline mineral attracted the interest of Lord Rayleigh in 1887, and he was (it seems) the ﬁrst to formulate a precise theory of these phenomena [4]–[5]. In fact, Rayleigh stumbled onto a form of the Bloch-Floquet theorem, deriving quite a general solution for waves in 1d periodic media;* he even correctly showed that any such medium exhibits a band gap (although he didn’t call it that). Unfortunately, he failed to pursue this result to its full generality—instead, he later abandoned it for a more direct (but less illuminating) method: performing a heroic calculation to sum the waves reﬂected and refracted from each interface [6]. No one suspected that the same magical cancellation could happen for light shining from any direction in a crystal periodic along all three axes. No one, that is, until 100 years later, in 1987, when the ﬁrst papers were published marrying solid-state physics with Maxwell’s equations and suggesting that a 3d crystal could produce an omnidirectional band gap [98]–[99]. “Suggested,” because an actual example
* Rayleigh was guided in his analysis of periodic media by earlier studies of the lunar orbit, in which the moon is periodically tugged by the sun. If the generality of that work had been recognized, we might now call it the “Hill” theorem (since it predates Floquet by six years) [7].
27

(even in theory) was not found until three years later—unlike in 1d, any old periodic structure will not do, and much trial-and-error was required. The ﬁrst concrete design involved levitating dielectric balls [101], so the engineers asked to build the thing were understandably upset—especially since the balls had to be sub-microns in size for it to work at infrared wavelengths. Nevertheless, after several years, more realistic structures were proposed and even fabricated, using technologies such as those by which tiny circuits are etched onto computer chips.
The challenge of realizing the beneﬁts of photonic crystals in practical systems continues to be a major subject for research. The only such devices currently in production are still based on one-dimensional band gaps like the crystals of Lord Rayleigh (often called Bragg mirrors*), and range from dielectric mirrors and reﬂective coatings to DFB lasers.
Before we continue, we must take a moment to apologize: if it has seemed to you that our explanations of photonic crystals have often failed to explain, then we are in complete agreement. A more substantial understanding can only come after one has had the pleasure of swimming through the actual mathematics of Maxwell’s equations and Bloch-Floquet eigensystems, and that would require a work somewhat longer than this chapter. Fortunately, we are able to recommend to the interested reader a marvelous introductory text by an author unrivaled in his eloquence and clarity, and whose signature is coincidentally required on this thesis. That book, cited in Ref. [8], should be accessible to anyone with an undergraduate background in electromagnetism. (A briefer review can be found in the Nature article of Ref. [9].) Alternatively, Chapter 2 of this thesis provides a whirlwind tour for the impatient and the mathematically inclined.
* Named after Sir William Henry Bragg, who in 1915 received the Nobel prize in physics for his work on X-ray crystallography—probing periodic atomic crystals by bouncing X-rays off them.
28

1.4 Things to Come
I’m very well acquainted too with matters mathematical, I understand equations, both the simple and quadratical, About binomial theorem I’m teeming with a lot o’ news— With many cheerful facts about the square of the hypotenuse.
Sir William Schwenk Gilbert, The Pirates of Penzance
Although we cannot take credit for the discoveries described in the previous sections, we will endeavor in the subsequent chapters to contribute some marginal advance to the understanding of photonic crystals and related systems. Chapter 2, however, is simply a mathematical introduction to photonic crystals, while Chapter 9 is our conclusion. The remaining chapters are derived from a series of articles published in various scientiﬁc journals, meaning that they are either pleasantly self-contained or irretrievably abstruse, depending upon one’s perspective. In any case, we shall summarize them here. 1.4.1 Intersecting Optical Pipes
To begin with, we consider an interesting effect that one can achieve with photonic crystals, and more generally with the mathematical and conceptual techniques that they embody. In particular, we show how two optical “wires” can be crossed without interfering, which might be surprising if you think in terms of another wave: sound.
Imagine that you face an intersection of two corridors, deep within the bowels of MIT. On the opposite side of the intersection is one of your fellow graduate students, but you can’t reach her because it’s noon* and the janitors have roped off the crossing to wax the ﬂoor. Instead, you call across to brag that you used the Boston White Pages to pad out your thesis: “Our batty advisor won’t even notice!” Meanwhile, speak of the devil, he is having his own shouted conversation with a colleague across the same intersection, but in the per-
* In Tech time—2AM, Boston time.
29

pendicular direction: “No need to feel guilty—he’s young, and has plenty of time to ﬁnd another trade if we reject his dissertation.” Naturally, neither of you wants the other to overhear his words—is this possible? (No fair talking through tubes, either; both voices must pass through the same space.) It turns out that, in certain cases, the answer is yes.
You can see where this is heading: an almost identical situation shows up with light, and poses a substantial problem in designing optical circuits. In order to weave together a group of components, one needs the ability to cross waveguides (optical “corridors”). Doing so in the naïve way, however, would allow the crossed channels to overhear one another—this is called crosstalk, and will turn your device into a lemon. (With electronic circuits, this difﬁculty is circumvented by passing the wires over or under one another, but that solution is less practical in optics for a variety of reasons.)
In Chapter 3, we show how, by exploiting general principles of symmetry and resonance, it is possible to design an optical crossing where the two waveguides do not “see” one another—light travels across in one direction or the other, through the same space, without leaking into the perpendicular pipe. Although the inspiration and the ideal realization of this concept resides in full photonic crystals, we show how it can be applied to conventional and hybrid optical systems as well. The application to eavesdropping advisors will, perhaps, be the subject of a future publication. 1.4.2 Photonic-Crystal Slabs
Photonic crystals, in their full three-dimensional glory, are hard to make. Imagine baking ten-layer cake, where each layer is a checkerboard pattern of chocolate and vanilla, and the patterns of adjacent layers must be offset by exactly half a checker. Oh, and the cake is only 1/1000 of an inch high. Wouldn’t it be much easier to bake only a single layer? You can roll out the dough as ﬂat as you need, and even use a cookie-cutter to make the pattern. The optical equivalent of this recipe shortcut is a photonic-crystal slab: a
30

Chapter 4: Bulk Slabs

Chapter 6: Slab Cavities

Chapter 5: Slab Waveguides
Figure 1.2: Schematic of the photonic-crystal slab chapters, covering ﬁrst the bulk systems, then waveguides formed by linear defects, then cavities formed by point defects.
dielectric (e.g., Silicon) with only a two-dimensional pattern and a ﬁnite thickness, which can be baked via the well-known lithographic techniques used to manufacture computer chips. Slabs are very attractive to the engineers, but they involve a compromise—what conﬁnes the light in the vertical direction? Actually, there is a conﬁnement mechanism, albeit imperfect: index guiding, sometimes misleadingly called total internal reﬂection.
Index guiding is the principle behind today’s optical ﬁbers; simply put, it is the tendency of light to stay in stronger dielectrics (which have a higher index of refraction) compared to weaker dielectrics such as air. Just as a bump in the road may throw your car into the ditch, however, any irregularity in the medium can cause light to scatter out, and this ease of loss makes life more difﬁcult with a slab than in a full 3d crystal (where the road is surrounded by impenetrable walls). Fortunately, thanks again to the Bloch-Floquet theorem, at least the periodic variation of the slab itself does not cause scattering; if only regularly-spaced bumps did not disturb your driving!
In the next three chapters, as depicted in Fig. 1.2, we explore and explain the unique phenomena that can occur in such hybrid systems, probing both the limitations and the possibilities of photonic-crystal slabs. Chapter 4 lays the foundations by characterizing the
31

Figure 1.3: A portion of a novel 3d photonic crystal that we propose in Chapter 7. This structure has a large, complete band gap, can be constructed layer-by-layer using traditional lithography, and its behaviour is understandable in terms of much-simpler twodimensional photonic-crystal structures. bulk slabs themselves, without defects. Chapter 5 breaks the slab’s periodicity in only one direction by a linear defect, showing how lossless waveguides can be formed. Finally, Chapter 6 breaks the periodicity entirely and attempts to trap light in a cavity. This is not possible, however, due to the lack of a full band gap, and so we propose and investigate various mechanisms to keep the light conﬁned for as long as we can. 1.4.3 A New Photonic Crystal
Despite their difﬁculty of manufacture, tremendous progress has been made in the fabrication of full three-dimensional photonic crystals. Thus far, however, the menu of practical structures has been limited to a mere handful of possibilites, and so the prospect of new designs with unique and desirable properties holds great attraction in the ﬁeld. In Chapter 7, we propose a novel photonic crystal that can be (and, in fact, has been) realized even at microscopic lengthscales. Although it may look complex, such a periodic system can easily be analyzed in its entirety with the help of our friends Bloch and Floquet.
32

Our structure, depicted in Fig. 1.3, has a number of appealing qualities whose combination is unique. First, its photonic band gap covers a wide range and persists even if only weak dielectric materials are employed. Second, it consists of a sequence of patterned layers that can be constructed one at a time via the “cookie-cutter” of modern micro-lithography. Third, each layer has high symmetry (it looks the same from six directions in the plane). Physicists love symmetry for its own sake, but here it has substantial practical advantages as well—it means that complex optical circuits can be constructed by modifying only a single layer of the structure. Finally, each layer resembles a common twodimensional photonic-crystal structure, and this allows us to understand its behavior in terms of those vastly simpler systems (for which a substantial body of literature, intuition, and computation is already available). 1.4.4 So, You Want To Study Photonic Crystals?
Finally, we consider a matter less of physics than of mathematics—given the equations of Maxwell and Bloch-Floquet, it is still a substantial challenge to compute their solutions. Several methods are already known, of course, but in Chapter 8 we attempt to bring together the best developments in numerical analysis, computer science, and physics in order to take a certain class of techniques further towards its ultimate conclusion, and to provide a comprehensive review and comparison of the ideas of ourselves and others. In this way, we hope to furnish a valuable reference for the reader who wishes to engage in these kinds of computations. We offer more than that, however…
In the literature, all too often one sees a proposal for a new computational algorithm, usually with a few sample results, that nevertheless leaves the reader with no way to evaluate or employ the method for herself without investing several months or even years in programming her own implementation on a computer. Moreover, despite the fact that an entire ﬁeld of scientists and engineers runs the same type of computation on similar types
33

of problems, to a large extent each group of scientists develops their own software. This duplication of effort is, it seems to us, extraordinarily wasteful and unnecessary. With that in mind, and in the scientiﬁc spirit of contributing freely to human knowledge, we have made the results of our work available as free software on the Web—see Ref. [122], http://ab-initio.mit.edu/mpb/. Using it, we invite you analyze for yourself the band gaps and the propagation of light in arbitrary three-dimensional photonic crystals.
And like a silver clarion rung The accents of that unknown tongue,
Excelsior! Henry Wadsworth Longfellow, “Excelsior”
34

Chapter 2

Band Gaps & Maxwell’s Equations: A Whirlwind Tour

2.1 Maxwell’s Equations
In this chapter, we very brieﬂy review the properties of Maxwell’s equations and the Bloch-Floquet theorem, and discuss the origin of the band gap in photonic crystals. We assume herein a fair amount of mathematical sophistication; see Ref. [8] for a gentler introduction. Our starting point, naturally, is Maxwell’s equations in cgs units (written in a slightly more modern form than the 20 equations in 20 variables used by Maxwell [1]):

∇ × E = –1-c-∂--∂--H-t-- ,

(2.1)

∇×H

=

4----π-c

J

+

1-c

∂---∂-ε--t-E--

,

(2.2)

∇ ⋅ εE = 4πρ ,

(2.3)

and ∇ ⋅ H = 0 ,

(2.4)

where E and H are the electric and magnetic ﬁelds, J is the free current density, ρ is the

free charge density, and ε is the dielectric function; all of these are potentially functions of

both the position x and the time t . c , of course, is the speed of light in vacuum.

Recall that a dielectric is a material in which tiny dipoles, the bound charges, line up to oppose any applied electric ﬁeld—this process is encapsulated entirely by the dielectric function ε . In particular, we consider here the case of linear dielectrics, for which ε is independent of E —this is reasonable, as nonlinear effects in optical materials are typically so weak that you cannot observe them without enormous ﬁelds or great distances of propagation. We will also assume that ε is independent of frequency (and therefore independent of time as well), which is reasonable for the relatively narrow frequency ranges

35

that will be of interest in photonic crystals. Finally, we assume that we can neglect any absorption of light by the material; in this case, ε is a real number.

Furthermore, in optics we are normally interested only in cases where no free charges or currents are included, so J and ρ are zero. If we then combine Eq. (2.1) and Eq. (2.2), we can eliminate E in favor of H to write:

∇ × 1-ε-∇ × H

=

-∂---2---H--- . ∂ct2

(2.5)

This equation can be compared to Schrödinger’s equation from quantum mechanics, with

a Hermitian (self-adjoint) “Hamiltonian” operator ∇ × 1-ε-∇ × on the left (instead of

Schrödinger’s

– ∇2 +

V)

and

a time derivative

---∂---2---∂ct2

on

the right (instead of

i∂--∂--t ). Here,

1-ε-

plays a role analogous to that of the potential function V . Unlike in quantum mechanics,

we also have the transversality constraint of Eq. (2.4), but this merely means that we oper-

ate in a subspace of the set of all possible eigensolutions; it has no effect on most of the

following discussion.

Next, like in quantum mechanics, we look for time-harmonic states whose time dependence is e–iωt for some (angular) frequency ω —all possible solutions can be expressed via this form, since the equation is linear—and thus Eq. (2.5) becomes:

∇ × 1-ε-∇ × H

=

 

ω-c--

2

H

.

(2.6)

This is a Hermitian eigenproblem over an inﬁnite domain, and generally produces a con-

tinuous spectrum of eigenfrequencies ω . Things become much more interesting, however,

when we consider in the next section the case of a periodic dielectric structure ε .

When we say that the eigen-operator is “Hermitian,” we are referring to the fact, easily proved from the identities of vector calculus, that:

∫ A* ⋅ ∇ × 1-ε-∇ × B

=

∫

 

∇

×

1-ε-

∇

×

A * ⋅

B,

(2.7)

36

where A and B are arbitrary ﬁelds with boundary conditions such that the boundary terms in the above integrals disappear (for example, if they are periodic or have compact support). In other words, the operator acts the same when applied to the left or to the right under inner products (integrals). It’s also worth noting that this operator is positive semideﬁnite. The combination of these two properties implies that the eigenfrequencies ω in Eq. (2.6) are real numbers.
We should note one important property of Maxwell’s equations in general, and of Eq. (2.6) in particular: they are scale-independent. If you multiply all the dimensions of your system by 10, say, the solutions are exactly the same except that the frequencies are divided by 10. This means that we can solve the equations once and then apply the same results to problems at all lengthscales and frequencies. Because of this scale invariance, it is convenient to use dimensionless units for distance and time: we pick some natural lengthscale a in the system (usually the periodicity), and then express all distances as a multiple of a and all angular frequencies ω as a multiple of 2πc ⁄ a . (This is equivalent to writing the frequencies as a ⁄ λ , where λ is the vacuum wavelength.)
2.2 The Bloch-Floquet Theorem and Electromagnetism
The Bloch-Floquet theorem [152] tells us that, for a Hermitian eigenproblem (or actually, a larger class of differential equations) whose operators are periodic functions of position, the solutions can always be chosen of the form eik ⋅ x ⋅ (periodic function) .* A periodic function f ( x) is one such that f ( x + Ri) = f ( x) for any primitive lattice vector Ri of the crystal—there can be up to three such lattice vectors, if the function is periodic in all three dimensions. A corollary of this theorem is that the wavevector k is a conserved quantity, and hence propagating waves with a ﬁxed k cannot scatter.
* This theorem is easily proven by noting that the translation operator of the crystal commutes with the Hamiltonian operator, and then recalling that simultaneous eigenfunctions can be chosen for commuting operators.
37

When applied to our situation, the Bloch-Floquet theorem means that the solution of Eq. (2.6) for a periodic ε can be chosen of the form:

H = ei(k ⋅ x – ωt)Hk ,

(2.8)

where Hk is a periodic function of position. If we substitute Eq. (2.8) into Eq. (2.5), we

ﬁnd that the function Hk satisﬁes the “reduced” Hermitian eigenproblem:

(∇ + ik) × 1-ε-(∇ + ik) × Hk

=

 

ω-c--

2

Hk

.

(2.9)

Because Hk is periodic, we need only consider this eigenproblem over a ﬁnite domain: the

unit cell of the periodicity. There is a general theorem that eigenproblems with a ﬁnite

domain have a discrete set of eigenvalues. Thus, the eigenfrequencies ω are a countable

sequence of continuous functions: ωn(k) (for n = 1, 2, 3, … ). When they are plotted as a function of the wavevector k , these frequency “bands” form the band structure of the

crystal.

Another important consequence of the Bloch-Floquet theorem is that the solutions are

periodic as a function of the wavevector k ; this is easy to see if we examine a one-dimen-

sional example. Suppose that we have a one-dimensional structure with period a

(ε( x + a) = ε( x) ) and consider the eigenstates for the wavevector k + 2π ⁄ a . The func-

tion ei2πx ⁄ a is periodic (with period a), however, so this part of Eq. (2.8) can be absorbed

into the function Hk ! Since the resulting expression then solves the eigenproblem for a

wavevector k , and those solutions are unique, we can therefore conclude that the eigenso-

lutions at k + 2----π-- are the same as those at k . Because k is only unique to within multiples

a

of

-2---π-a

,

then,

it

is

sufﬁcient

to

only

solve

the

eigenproblem

for

k

in

a

ﬁnite

range,

–π-a- …

π-a

,

called the ﬁrst Brillouin zone. Moreover, we usually have additional symmetries that fur-

ther reduce the range of unique wavevectors—for example, time-reversal symmetry tells

us that k

and –k

produce the same results, so in this case we can look at just k

=

0…

π-a

.

38

This symmetry-reduced region is called the irreducible Brillouin zone, and in three dimensions is given by well-known polyhedra [13] (whose vertices have canonical, obscure labels like Γ , X , K , etc.).
Let us make two additional comments before we continue. First, in the special case where the crystal has translational symmetry (a → 0 ), the irreducible Brillouin zone extends to k = ∞ , and is called a dispersion relation; this is commonly used to characterize waveguides. Second, when one is interested in just the extrema of the frequency bands, it is usually sufﬁcient to consider only wavevectors along the edges of the irreducible Brillouin zone (or often merely the vertices)—this allows the band structure to be plotted in an ordinary two-dimensional diagram, many examples of which are found in this thesis.

2.3 The Origin of the Photonic Band Gap
Now, we will attempt to make some general arguments regarding the possibility of a pho-

tonic band gap: a range of ω into which there intrude no propagating states ωn(k) . In order to better understand the gap, we will exploit two general properties of Hermitian

eigenvalue problems. First of all, the eigenfunctions must be orthogonal; that is, their inner

product (integral over the unit cell) is zero:

∫ Hk(m)*

⋅

H

(n k

)

=

0

(2.10)

for any two eigenstates (m ≠ n ) at a given k point. Second, the lowest band solves a vari-

ational problem:

∫ ∫ ω1(k)2 =

min -----(--∇------+-----i--k---)----×-----H----k----2----⁄--ε--c2 .

Hk

Hk 2

(2.11)

(The numerator is just the expectation value of the eigen-operator.) The second band satis-

ﬁes the same variational problem, with the added constraint that it be orthogonal to the

ﬁrst band according to Eq. (2.10), and so on. Note that the curl of the magnetic eigenﬁeld,

39

in the numerator, is simply proportional to the electric ﬁeld from Eq. (2.2). In order to minimize the variational problem of Eq. (2.11), the ﬁeld for the lowest band
should try to satisfy two conditions. First, its curl should not be too big—that is, it should vary slowly, and not contain any fast oscillations. This is analogous, in quantum mechanics, to having a low kinetic energy. Second, the curl—the electric ﬁeld—should be concentrated in regions of high dielectric constant ε ; this is analogous to having a low potential energy. So far, so good, but where’s the band gap? In order to see the gap, we naturally must look at the second band. Like the ﬁrst band, the second band also wants to oscillate slowly and be concentrated in the high dielectric. However, it must also be orthogonal to the ﬁrst band from Eq. (2.10). In order to accomplish this, the second band must be concentrated in the lower dielectric and/or have a sign oscillation inside the high dielectric to make the integral zero—both of these possibilities will incur a large increase in frequency in Eq. (2.11), a band gap!
Actually, we’re not quite there yet. True, there will generally be a large ω difference between the ﬁrst and second bands, a “gap” at each k point. In order to have an omnidirectional band gap, however, the gaps at all k points must overlap in some frequency range. For this reason, a fairly large contrast in ε between the high and low dielectric regions is usually necessary to make the gaps in different directions large enough to intersect.
The situation is even worse than that, however, because the ﬁelds have a direction as well as a sign and a magnitude. In order to be orthogonal to the ﬁrst band according to Eq. (2.10), the second band need only be perpendicular to the ﬁrst, without sacriﬁcing its concentration in the high dielectric! To combat this phenomenon, we need to make use of the boundary conditions on the ﬁelds, from Maxwell’s equations, that apply at dielectric interfaces. In particular, the surface-parallel component of the curl of H is allowed to be discontinuous across a dielectric boundary, while the surface-normal component must be
40

continuous. Now, suppose that our high dielectric consists of thin veins surrounded by lower dielectric. In that case, the numerator of Eq. (2.11) can be “weighted” discontinuously in these veins for electric ﬁelds running parallel to them, but not for electric ﬁelds perpendicular to the veins. This allows us to discriminate between bands with ﬁelds in different directions, and makes a complete band gap possible despite the variety of ﬁeld polarizations. (Even so, complete gaps typically arise between the second and third bands, not between the ﬁrst and second, precisely because of polarization-related degeneracies.)
When these conditions are satisﬁed, and if you are very lucky, your periodic crystal will have a complete photonic band gap in three dimension. We’ve waved our hands quite a bit in this section, partially in the interest of brevity, but partially because there are no rigorous criteria for the existence of a complete three-dimensional gap. In contrast, for one dimensional systems, life is easy—any nonzero periodic dielectric variation will result in a complete band gap (although it may be small)! This has a simple explanation via perturbation theory that, unfortunately, exceeds the scope of this chapter. A very similar argument, however, implies that your best chance for a complete gap in three dimensions resides in crystals with nearly spherical Brillouin zones. The crystal lattice with the B. Z. closest to a sphere is the fcc (face-centered cubic) lattice,* and almost all known photonic crystals in three dimensions fall into this category (including the crystal we introduce in Chapter 7). Furthermore, the same derivation also explains why the gap arises only for wavelengths of about twice the crystal periodicity. We hope, nevertheless, to have armed the reader with sufﬁcient knowledge to brave the remaining chapters.
* There is class of non-periodic structures called quasi-crystals that, in some sense, have morespherical Brillouin zones, at the cost of additional complexity and some other disadvantages. There is evidence that these structures can have substantial band gaps even with low dielectric contrast.
41

42

Chapter 3
Elimination of Crosstalk in Waveguide Intersections*
3.1 Introduction
The ability to intersect waveguides is crucial in constructing integrated optical circuits, due the desire for complex systems involving multiple waveguides. We present a novel theoretical framework for achieving low crosstalk and high throughput in perpendicular intersections, based on symmetry considerations that can be systematically applied to any optical system. The transmission spectra are predictable without detailed calculations and are robust under perturbations that do not break the symmetry. The focus of this design is the elimination of crosstalk; additional considerations are required to maximize throughput when radiation losses are present. We illustrate the theory with two example systems—a two-dimensional photonic crystal and a conventional dielectric-waveguide intersection—and show how an efﬁcient crossing can be produced by assembling wellunderstood elements that are independently tuned. In contrast, previous studies of waveguide intersections [10]–[11] have lacked general principles that could be applied a priori to diverse systems. Moreover, they have typically been concerned with shallowangle crossings for wavelengths many times smaller than the waveguide width. Although perpendicular crossings in such systems exhibit negligible crosstalk, the crosstalk is close to 10% when the waveguide width is on the order of half a wavelength (as seen below). Our design applies even when the waveguide width is small, permitting single-mode waveguides with optimal miniaturization. Simulations of our structures show peak throughputs of nearly unity while crosstalk falls as low as 10–9.
* This chapter is based on: S. G. Johnson, C. Manolatou, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Elimination of cross talk in waveguide intersections,” Optics Lett. 23, 1855– 1857 (1998).
43

Figure 3.1: Abstract diagram of symmetry requirements for waveguide crossing, showing waveguide mode proﬁles and resonant-cavity mode contours. By symmetry, the solid-line (red) modes cannot couple with the dashed-line (blue) modes and vice-versa.
3.2 Eliminating Crosstalk by Symmetry
The fundamental idea is to consider coupling of the four branches, or ports, of the intersection in terms of a resonant cavity at the center. If the resonant modes excited from the input port can be prevented by symmetry from decaying into the transverse ports, then crosstalk is eliminated and the system reduces to the well-known phenomenon of resonant tunneling through a cavity. This situation can be achieved by means of the following conditions (see also Fig. 3.1):
• Each waveguide must have a mirror symmetry plane through its axis and perpendicular to the other waveguide, and have a single guided mode in the frequency range of interest. This mode will be either even or odd with respect to the mirror plane.
• The center of the intersection must be occupied by a resonant cavity that respects the mirror planes of both waveguides.
44

• Two resonant modes must exist in the cavity, each of which is even with respect to one waveguide’s mirror plane and odd with respect to the other. These should be the only resonant modes in the frequency range of interest.
If these requirements are satisﬁed, then each resonant state will couple to modes in just one waveguide and be orthogonal to modes in the other waveguide, as shown in Fig. 3.1. (For simplicity, we depict a lowest-order even waveguide mode.) Therefore, under the approximation that the ports only couple to one another through the resonant cavity, crosstalk will be prohibited. The transmission to the output port is described by resonant tunneling, and one can use coupled-mode theory [12] to show that the throughput spectrum will be a Lorentzian peaked at unity on resonance. The width of the Lorentzian is given by the inverse of the cavity’s quality factor Q, which is proportional to the lifetime of the resonance mode.
Perfect throughput will not be attained, however, due to a combination of two effects: direct coupling and radiation loss. First, the input port can directly couple with the transverse ports, resulting in crosstalk—this can be decreased arbitrarily, but at the expense of the throughput bandwidth. Direct coupling is reduced by making the cavity larger, since this increases the distance between the localized waveguide modes. The drawback to increased cavity size is that it can either create extraneous resonance modes or, as in the case of the photonic crystal cavities below, increase the Q of the resonance. Second, many systems suffer from losses due to radiation or other mechanisms. If these losses do not couple strongly to the transverse waveguide, then their primary effect will be to decrease throughput and increase reﬂection [12].
The required parity of the cavity modes is easy to achieve in practice, due to the fact that a perpendicular intersection of identical waveguides typically has C4v symmetry (the
45

(a) empty

(c) 3x3

(b) 5x5

(d) 1x1

Figure 3.2: Waveguide intersections in a two-dimensional photonic crystal consisting of a square lattice of dielectric rods in air. symmetry group of the square). In this case, any non-accidental degenerate modes (such as many higher-order modes) will necessarily be a pair having the requisite symmetry from Fig. 3.1, as that is the only two-dimensional irreducible representation of C4v [13].
3.3 Intersecting Photonic-Crystal Waveguides
A realization of the intersection design can be made in a photonic crystal [14], for which the band gap in the bulk material prevents any radiation losses. For ease of computation, we use a two-dimensional crystal, consisting of a square lattice of dielectric rods in air. The rods have lattice constant a, radius 0.2a, and dielectric constant 11.56. This structure prohibits propagation of TM light (in-plane magnetic ﬁeld) in the frequency range 0.286 to 0.421 c/a [14]. Removing a row or column of rods produces a waveguide [14]. A resonant cavity is created by increasing the radius of a single rod to 0.33a, leading to doubly-degenerate modes of the requisite symmetry with a frequency of 0.361 c/a (i.e. a wavelength in air of 2.77a) [15].
These elements are then combined into an intersection, and we simulate the collective behavior of several structural variations (Fig. 3.2) using a ﬁnite-difference time-domain simulation [16]. In each case, we send a wide-spectrum TM gaussian pulse to the input
46

port (left). The fractional power transmission is then evaluated as a function of frequency for the output port (right) and one of the transverse ports (top), yielding the throughput and crosstalk, respectively. The resulting transmission spectra are shown in Fig. 3.3. Up to ﬁve percent of each throughput and crosstalk value can be due to reﬂections from the boundary of the simulation, which is why the peak throughput in cases (c) and (d) appears to exceed unity.
As predicted, the throughput for cases (b) and (c) is a Lorentzian centered on the resonance frequency (which varies due to the ﬁnite size of the cavity). At their resonance frequencies, (b) and (c) achieve crosstalks of 5 × 10–10 and 4 × 10–4 , respectively. In case (d), there are no photonic crystal layers separating the central rod from the waveguide, and conﬁnement is due only to index contrast. The cavity conﬁnement is so weak that coupledmode theory is no longer accurate. Even so, case (d) reaches nearly 100% throughput with a crosstalk at the same frequency of 1.8%. In contrast, both the throughput and the crosstalk for the empty intersection lie in the 20–40% range.
Both the crosstalk and the resonance Q derive from the exponential decay of light through the bulk photonic crystal. Because the crosstalk must tunnel through twice as many crystal layers as the resonance decay, the crosstalk should be approximately Q-2. Looking at Fig. 3.3(b), this estimate for the crosstalk is seen to be valid to within about an order of magnitude, corresponding to the Q’s of roughly 1000, 100, and 10 for cases (b), (c), and (d). Here, we do not consider the additional resonance minima apparent in the crosstalk at some frequencies, which are independent of the throughput resonance and are due to a destructive-interference phenomenon that is still under investigation.
47

1
(a)
1x1
0.8

0.6

0.4
empty 0.2

0 1x100
(b)
1x10-2

empty 1x1

3x3 1x10-4

1x10-6

5x5

1x10-8

3x3 5x5

1x10-10

0.32

0.33

0.34

0.35

0.36

0.37

0.38

frequency (c/a)

Figure 3.3: TM transmission spectra for the four intersections shown in Fig. 3.2: (a) throughput from the input port to the output port and (b) crosstalk from the input port to one of the transverse ports.

48

(a) no holes

(b) holes

Figure 3.4: Intersection of two-dimensional, high index-contrast waveguides in air, both (a) without holes and (b) with air holes added to create a resonant cavity using a photonic band gap.
3.4 Intersecting Conventional Waveguides
The same principles of intersection design can be applied to other systems, such as conventional dielectric waveguide crossings. In particular, we considered an intersection of high index-contrast waveguides in air, of width a and dielectric constant 11.56, as shown in Fig. 3.4(a). In Fig. 3.4(b), we have modiﬁed the crossing to include a resonant cavity by introducing a periodic sequence of air holes on each branch of the intersection, creating a band gap in the TE guided modes of the waveguides [17]. The holes have radius 0.2a and period a. A defect center-to-center separation of 2.2a produces a pair of doublydegenerate, p-like modes with the desired symmetry at a frequency of 0.22 c/a (i.e. a wavelength in the dielectric of 1.3a).
As before, we use a two-dimensional simulation to evaluate the throughput and crosstalk of these two structures, this time with TE-polarized light. The results are shown in Fig. 3.5. Not only does the resonant cavity increase the throughput from 75% to over 90%, but the crosstalk is decreased by two orders of magnitude, from 7% to 0.08%. The resonance Q is close to 40 and, as before, the crosstalk can be estimated by Q-2. The main
49

1
(a)
0.8 no holes
0.6
0.4
0.2 holes
0 1
(b)
0.1 no holes
0.01
holes 0.001

0.0001 0.2

0.204

0.208

0.212

0.216

0.22

Figure 3.5: TE transmission spectra for the tfwo interse(ct/io)ns shown in Fig. 3.4.

difference between the behavior of this crossing and the one from Fig. 3.2 is that there are now radiation losses from scattering off the intersection, resulting in lower throughput.

3.5 Summary
We have shown how, by the application of general principles and symmetry considerations, it is possible to systematically construct efﬁcient waveguide intersections in diverse structures. Although our examples are limited to two-dimensional systems, the same method is applicable in three dimensions. Photonic crystal cavities have been used because of the ﬂexibility and tunability they provide, but the theory is independent of the conﬁnement mechanism. Furthermore, several additional reﬁnements are possible. Many

50

degrees of freedom remain in high index-contrast crossings (as in Fig. 3.4), such as the shape of the intersection and the size and number of holes, that could be used to further reduce radiation losses and increase throughput. Another important possibility is to broaden and ﬂatten the resonance peak, which could be accomplished by adding additional resonators in the waveguides adjacent to the intersection [18].
51

52

Chapter 4 Guided Modes in Photonic-Crystal Slabs*
4.1 Introduction
The discovery of photonic crystals, periodic dielectric materials with a photonic band gap, has opened up new methods for controlling light, leading to proposals for many novel devices [9]. Straightforward application of these results to three dimensions requires a structure with a three-dimensional band gap. Fabricating such structures, however, has been a challenge because they tend to have complex three-dimensional connectivity and strict alignment requirements [19]–[22]. Such designs have been the subject of many recent developments [23]–[27]. An alternative system, the photonic-crystal slab, has been proposed that promises easier fabrication using existing techniques [28]–[35]. This is a dielectric structure that has only two-dimensional periodicity and uses index guiding to conﬁne light in the third dimension. Photonic-crystal slabs retain or approximate many of the desirable properties of true photonic crystals, but at the same time are much more easily realized at submicron lengthscales. We present in this chapter a novel band-structure analysis of photonic-crystal slabs, providing a systematic understanding of many important features of these systems.
* This chapter is based on: S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
53

(a)

(b)

Figure 4.1: Two-dimensional photonic crystals. (a) Square lattice of dielectric rods in air, with lattice constant a and radius 0.2a. (b) Triangular lattice of air holes in dielectric, with lattice constant a and radius 0.45a. In both case, the dielectric constant of the high-index material is 12.
In many ways, photonic-crystal slabs are analogous to two-dimensional photonic crystals, such as those depicted in Fig. 4.1, and this analogy aids greatly in the visualization and analysis of slab systems. Two-dimensional calculations, however, cannot be applied directly to three-dimensional slab structures. In particular, the band structure computed for a two-dimensional structure, as shown in Fig. 4.2, applies in three dimensions only to a structure that is inﬁnitely “extruded” in the third dimension. Moreover, these two-dimensional bands correspond only to states that have no wavevector component in the vertical direction (perpendicular to the plane of periodicity). The inclusion of vertical wavevectors produces a continuum of states depicted by the shaded region in Fig. 4.2, destroying the band gap of the two-dimensional structure. The restriction of a slab to ﬁnite height recreates the band gap in the guided modes of a slab, but also forces a new analysis of the system that is fundamentally three-dimensional and distinct from the two-dimensional calculations. New issues such as slab thickness, index contrast with the substrate, and mirror symmetry assume a prominent role in determining the properties of photonic-crystal slabs.

54

(a)

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

E JE

E JJJE
JE

EE JJE J
JE

E E J JE J
E J

E E J JE J
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JE
JE J E J
E J

JE J E J E J E
J

E J J E J
E J E
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JEE
J

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JEE JEE
in JJ

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J J

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E
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TM gap two dimensions JJJJJJJJJJJ

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J

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(b)

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E E
JEE J J

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J J

E E JE
JJ

E E JEE
JJ

E EE JE J

EE JE J

0.4 0.3 0.2 0.1

J

J JE

J JE

J JE

J JE

J JE

J E J

J E J

TE

JJ

E J

E J

gap in

JJE

JE J

JJE

JE J

two dimensions

JE J

JE J

JE J

JJE

JJE

JJE

JE

JE J

JE J

J TM E TE

J E J

J E J

J E J

J JE

J JE

J JE

J JE

J

0 JE

JE

Γ

M

K

Γ

Figure 4.2: Band diagrams for the photonic crystals from (a) Fig. 4.1(a) and (b) Fig. 4.1(b). The shaded region indicates the frequencies of states introduced when vertical propagation (i.e., perpendicular to the plane of periodicity) is permitted.

55

The chapter is structured as follows. We ﬁrst outline the numerical methods that were used in our calculations. We then introduce slab band structures for two characteristic systems, the slab analogues of Fig. 4.1, suspended in air. The selection of an appropriate slab thickness is discussed, and an estimate for the optimal slab thickness is derived using a variational approach. Finally, the effects of different backgrounds, the regions above and below the slab, are examined in the context of the two example structures. Band structures are given for solid, periodic, and symmetry-breaking backgrounds.
4.2 Computational Method
The computation of a slab band structure is performed in two stages. First, the states of a slab are computed in a periodic cell. Then, the light cone is calculated and overlaid as an opaque region on the band diagram. The resulting non-obscured bands are the guided modes of the system, as described in the following section.
The eigenstates of the slab are computed using preconditioned conjugate-gradient minimization of the Rayleigh quotient in a plane-wave basis [36]. Such a computation requires a periodic cell. The slab is already two-dimensionally periodic, and we impose a three-dimensional periodicity by assuming a periodic sequence of slabs separated by a sufﬁcient amount of background region. Because the guided modes are localized within the slab, the addition of periodic slabs at large intervals does not affect their frequencies noticably. The non-guided modes are affected, but since they fall inside the light cone their exact frequencies are inconsequential.
The light cone is a continuous region indicating all possible frequencies of the bulk background. It is sufﬁcient to compute only the lower boundary of the light cone, since all higher frequencies are automatically included. In the case of a uniform background region, the boundary is simply the wavevector divided by the index. For a periodic back-
56

ground, the boundary is the lowest band of the corresponding two-dimensional system, as is discussed in a later section, and is computed using the two-dimensional version of the method in [36]. In all cases, the light cone is depicted with a uniform shading that does not reﬂect the varying density of states in this region.
a
(a)

(b)

a

z y x

Figure 4.3: Photonic-crystal slabs analogous to the two-dimensional structures from Fig. 4.1. (a) Square lattice of rods in air with height 2.0a. (b) Triangular lattice of holes in a dielectric slab with thickness 0.6a. (Other parameters are as in Fig. 4.1.) Both slabs are suspended in air.
4.3 Photonic-Crystal Slab Band Structures
Shown in Fig. 4.3 are the photonic-crystal slab analogues of the two-dimensional structures from Fig. 4.1. These two systems, a square lattice of dielectric rods in air and a triangular lattice of air holes in a solid dielectric slab, embody the two basic topologies of twodimensional crystals, and are characteristic of many possible slab structures. (An air-
57

bridge structure similiar to Fig. 4.3(b) was fabricated in [35].) As seen from Fig. 4.2, in two dimensions the rod structure has a band gap in the TM modes (magnetic ﬁeld in plane) and the hole structure has a band gap in the TE modes (electric ﬁeld in plane). (The hole structure also has a small band gap in the TM modes, but we focus here on the more robust TE gap, which is larger and separates the lowest two TE bands.)
The corresponding projected band structures for the three-dimensional slab structures are shown in Fig. 4.4. These graphs, whose computation was discussed in the previous section, illustrate many features that are common to all photonic crystal band diagrams.
Perhaps the most important feature of the projected band diagram, the element that distinguishes slabs from ordinary photonic crystals, is the light cone, a continuum of states indicated by a shaded region in the plot. The light cone consists of states, or radiation modes, that are extended inﬁnitely in the region outside the slab; we refer to this region as the “background.” Guided modes, which are states localized to the plane of the slab, can only exist in the regions of the band diagram that are outside the light cone. The primary interest in the radiation modes lies in how they interact with and constrain these guided modes.
Any state that lies below the light cone in the band diagram cannot couple with modes in the bulk background. Thus, the discrete bands below the light cone are guided—the states are inﬁnitely extended within the plane of the slab, but decay exponentially into the background region. This conﬁnement is analogous to total internal reﬂection, and is due to the guided modes seeing a higher effective index in the slab than in the background regions (air, in this case). When a guided band reaches the edge of the light cone, it becomes a resonant state: it extends, albeit with low amplitude, inﬁnitely far into the background, and cannot be used to permanently conﬁne light within the slab. We restrict our
58

discussion, and our use of the term “guided modes,” to truly localized states, which grow arbitrarily small as the distance from the slab becomes large

.

0.8
(a)

light cone

0.7

0.6 0.5 0.4 0.3 0.2

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EJ

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band gap

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(b)

0.7 0.6 0.5 0.4 0.3 0.2 0.1

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odd even

EJ

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Figure 4.4: Projected band diagrams corresponding to the two slabs in Fig. 4.3. Whether states are even or odd with respect to the horizontal mirror plane of the slab is indicated by open or ﬁlled circles, respectively.

59

The reason we refer to these systems as “photonic crystal” slabs is that, like their twodimensional brethren, they have a band gap—but not of the traditional sort. A “band gap” in this case is a range of frequencies in which no guided modes exist. It is not a true band gap because there are still radiation modes at those frequencies. Still, the lack of guided modes in the band gap gives rise to many of the same phenomena that occur in two-dimensional crystals, such as the ability to conﬁne light in the plane to waveguides or resonators [32]. (The presence of the radiation modes in the gap has the consequence that resonant cavity modes will eventually decay into the background.) The fact that light in the band gap of the slab is forbidden from propagating in the plane of the slab, and can only radiate into the background, was used in [30] to design an efﬁcient LED.
As in two dimensions, one is able to decompose the guided modes into two non-interacting classes. The lack of translational symmetry in the vertical direction, however, means that the states are not purely TM or TE polarized. Instead, due to the presence of a horizontal symmetry plane bisecting the slab, the guided bands can be classiﬁed according to whether they are even or odd with respect to reﬂections through this plane, and are indicated on the band diagram by hollow or ﬁlled circles, respectively. As shall be seen below, these even and odd states have strong similarities with TE and TM states, respectively, in two dimensions. (In fact, in the mirror plane itself, the even and odd states are purely TE, and TM, respectively.) It is not surprising, then, that the slab of rods has a gap in its odd modes (Fig. 4.4(a)), and the slab of holes has a gap in its even modes (Fig. 4.4(b)).
60

rods (a)

holes

(b)

0.0

field value

1.0

Figure 4.5: Lowest-frequency states of the rod and hole structures from Fig. 4.3 at M and K, respectively, depicted in the unit cell. The shading indicates the z components of the electric and magnetic ﬁelds, respectively, which happen to be positive throughough the unit cell for these states. Outlines of the rods/holes are shown in white. (a) Vertical crosssection. (b) Horizontal cross-section. In both cases the cross-sections bisect the holes or rods.
In Fig. 4.5, z-components of the electric and magnetic ﬁelds are shown for the lowest-
order odd and even guided modes from the rod and hole slabs at the M and K symmetry
points, respectively. It is apparent from these ﬁgures that these states are strongly guided
within the thickness of the slab. Moreover, within the slab they are TM- and TE-like, and
closely resemble the corresponding states in the two-dimensional system.

61

(a) rods

(b) holes

-1.0

field value

1.0

Figure 4.6: (a) Horizontal cross-section of Ez for the ﬁrst odd (TM-like) excited state at M in the rod structure from Fig. 4.4(a). (b)Vertical cross-section of the Hz for the ﬁrst TElike (but odd) excited state at K in the hole structure shown in Fig. 4.4(b).
Higher-order states in the slabs are formed in two ways. First, they can gain additional nodes or other structure in the horizontal plane, corresponding to higher-order states in the two-dimensional system. Second, they can be formed by adding vertical structure such as horizontal nodal planes. In the case of the rods, the lowest higher-order odd state is of the ﬁrst type, and is depicted in Fig. 4.6(a). The second odd mode at the M point in the holes slab, as shown in Fig. 4.6(b), is of the second type—it corresponds to the lowest-order even mode modiﬁed by the addition of a single horizontal nodal plane. Note that adding a nodal plane bisecting the slab transforms a state from odd to even and vice-versa, so even and odd higher-order modes aren’t necessarily TE-like and TM-like, respectively.
Higher-order modes of the second type do not correspond to excitations in the twodimensional system, and are responsible for destroying the gap if the slab becomes too thick, as is discussed in the following section.

62

35 30

EEEEE

J Rods (odd gap)

25

E

20
E
15

E

E Holes (even gap)

E

JJJ

JJ J

J J

10 E

JE

5

J

0

0

0.5

1

1.5

2

2.5

3

Slab Thickness (a)

Figure 4.7: Gap size (as a percentage of mid-gap frequency) versus slab thickness for the two slabs from Fig. 4.3.
4.4 Effects of Slab Thickness
The slab thickness plays an important role in determining whether a photonic-crystal slab has a band gap in its guided modes. Shown in Fig. 4.7 is a graph of the band gap size as a function of slab thickness for the rod and hole slabs from Fig. 4.3. The existence of an optimal thickness for each slab is easily understood. If the slab is too thick, then higherorder modes can be created with little energy cost by adding horizontal nodal planes. Such modes will lie only slightly above the lowest-order mode, preventing any gap. If the slab is too thin, then the slab will provide only a weak perturbation on the background dielectric constant. Guided modes will still exist, but they will hug the edge of the light cone and be only weakly guided; any gap will be miniscule. Below, we propose a method for estimating slab thicknesses that produces large gaps, and explain the sharp distinction seen in Fig. 4.7 between the optimal thicknesses for the rod and hole slabs.
We postulate that the optimal thickness will be on the order of half the two-dimensional gap-bottom wavelength. The justiﬁcation for this is based on the fact that the wave-

63

length is the lengthscale for variations in ﬁeld amplitude at a given frequency. (We use the

gap-bottom frequency instead of, say, mid-gap, because the state at the bottom of the gap

is the basis for both the state at the lower edge of the slab gap and the excited states at

upper edge.) If the slab thickness is a wavelength or more, then there will be little energy

barrier to creating a higher-order state via a nodal plane. If the slab thickness is less than

half a wavelength, on the other hand, then the mode cannot be strongly conﬁned within the

slab.

Only the frequency, rather than the wavelength, of the gap bottom is known, however.

An effective dielectric constant must be determined in order to compute the corresponding

wavelength. We accomplish this by constructing an estimate for the slab gap-bottom state

using the two-dimensional wavefunction, and evaluating the dependence of its frequency

expectation value on the vertical wavelength. The wavefunction can be approximated by:

∫ |H˜ 〉 = H0( x, y)eikza(k)dk .

(4.1)

Here, H0 is the eigenfunction of the two-dimensional system at the lower edge of the gap.

The a(k) amplitudes are assumed to be chosen so that they produce a state strongly local-

ized within the slab. A Fourier basis is used for the z-dependence so that the vertical

wavevector may be an explicit parameter. We have chosen this wavefunction based on our

experience (e.g. in Fig. 4.5) that the lowest-order guided band in the slab system is similar

in appearance to the corresponding two-dimensional state with the addition of vertical

conﬁnement. (We note that the ﬁeld in (1) is not divergenceless, however.)

We now evaluate the energy (i.e., frequency) of this state. Since the state is assumed to

be localized within the slab, the frequency should not be much affected if we evaluate it in

an inﬁnitely thick slab, which can be done exactly. (In the case shown in Fig. 4.5, more

than 90% of the computed wavefunction’s energy was found to lie within the height of the

slab.) In this approximation, the frequency is found to be one of:

64

ωT2M = 〈---H-˜---|--∇-----×-----ε---〈---(--H--˜--x--1--|--,-H--˜---y----〉--)--∇------×-----|-H-˜----〉 = ω02 + c---2-ε---〈T--k-M--2---〉

(4.2)

ωT2E = -〈--E〈˜--E˜-|--∇|--ε---(-×-x---,∇---y---×)---|-E-˜|-E-˜-〉--〉 = ω02 + c---2--ε--〈T--k-E--2---〉 ,

(4.3)

where E represents the electric ﬁeld, and ω0 is the two-dimensional frequency. Equations

(2) and (3) are equivalent only for divergenceless (transverse) ﬁelds. In the case of (1),

which is not transverse, we use (2) for TM ﬁelds and (3) for TE ﬁelds, for simplicity of the

resulting expression. In both cases, the vertical wavelength is related to frequency simply

by an effective dielectric constant that depends upon the polarization of the corresponding

two-dimensional state, and is given by:

εTE = 〈---E〈--E-0---|0-ε-|-E-|--E-0--〉0---〉 ≡ 〈 ε〉 E ∼ εhigh

(4.4)

εT

–1
M

=

-〈--H--〈--0H---|-ε-0--–|--H1---|-0H---〉--0--〉- ≡

〈 ε–1〉 H ∼ εl–o1w .

(4.5)

In the TE case, the dielectric constant is weighted by the electric ﬁeld, which is concen-

trated in high dielectric for the lower bands [9], and thus one should get an ε near to the

high dielectric constant. For the example of air holes in dielectric (Fig. 4.1(b)), the top-

most state of the bottom TE band in the two-dimensional system gives an ε of 5.06 from Eq. (4.4). This value, while greater than the uniform-weight mean ε of 3.92, is still far

from 12 (the high dielectric), as a result of the structure’s dielectric veins being so thin.

(This structure has both a TE and a TM gap, but we focus on the TE gap corresponding to

the even modes.) In Eq. (4.5), the TM case, we average the inverse dielectric constant, a

mean that favors lower values. Moreover, the mean is weighted by the magnetic ﬁeld,

which tends to be less in the high dielectric than the electric ﬁeld. (The magnetic ﬁeld

must loop around the electric ﬁeld in the high dielectric, since the relationship of the mag-

netic and electric ﬁelds is the same as that of a current loop and its magnetic ﬁeld.) There-

65

fore, the TM average dielectric should be closer to the low dielectric constant. In the case

of the dielectric rods in air (Fig. 4.1(a)), Eq. (4.5) for the highest-frequency mode of the bottom TM band in two-dimensions produces an ε of 1.25 (versus the uniform-weight

inverse mean of 1.13).

From the above calculations, then, an estimate for the optimal slab thickness is given

by:

slab thickness h ∼ --------------1---------------- . 2ωgap-bottom ε

(4.6)

Here, the frequency is given in units of c/a and the thickness is in units of a. The effective

dielectric constant ε can be estimated from the averages in (4) or (5), according to

whether a TE or a TM gap is of interest, and can use as a weighting factor either the state

from a two-dimensional calculation or simply a uniform value.

When applied to the rod and hole systems with the effective dielectric constants com-

puted above, Eq. (4.6) predicts optimal thicknesses of 1.6a and 0.7a, respectively, com-

pared to the computed gap maxima at 2.3a and 0.6a from Fig. 4.7. The estimates come

close enough to the optimal values that large gaps are produced (whereas there would be

no gap at all if the two estimates were applied to each others’ systems). Moreover, this

approximation explains how the sharp difference in optimal thickness between the rod and

the hole systems derives from the polarizations of the modes exhibiting the gap.

It should be noted that the size of the gap is not necessarily the only consideration in

selecting the slab thickness. For example, when localizing states in a resonant cavity,

longer decay times can sometimes be achieved by using slightly thicker slabs—this causes

the states to be more localized within the slab, and also pulls the frequencies down to

where the density of radiation modes is lower (since the density of states in vacuum goes as ω2 ). It is often better to tune parameters, such as the slab thickness, based upon the

66

actual phenomenon that is being optimized, rather than indirectly via the size of the band gap. Nevertheless, the approximation in (6) gives a reasonable starting point for subsequent ﬁne-tuning.

(a) rods

(b) holes

Figure 4.8: Side-view of the slabs from Fig. 4.3 with a uniform, low-index background above and below the slabs. (a) is the rod slab and (b) is the hole slab. The background has a dielectric constant of 2.0 and all other parameters are as in Fig. 4.3.
4.5 Slabs with Solid Backgrounds (Sandwiches)
So far, we have focused on the idealized system of slabs suspended in air. We now turn to the case where the background regions above and below the slab are occupied by a uniform dielectric material forming a substrate or “sandwich” as shown in Fig. 4.8. (We focus ﬁrst on having the same dielectric above and below the slab in order to maintain mirror symmetry; the effects of symmetry breaking are considered in a later section.) The dielectric constant of the substrate is 2, but the holes and the space between the rods continue to be occupied by air. The resulting band structures are given in Fig. 4.9. The background index remains lower than the effective index in the slab, and hence index guiding is still able to produce the guided bands (and band gap) evident in Fig. 4.9.

67

(a)

0.5 0.4 0.3 0.2 0.1

EJ EJ EJ EJ EJ

EJ EJ EJ EJ EJ

EJ EJ
EJ EEJ EJ

EJ
EJ
EEJJ EEJJ EJ

E EJJ EEJ
EEJJ EEJJ EJ

EJJ J EEJJ
EEJJ E EEJJJ

E EEEEJJJ
EEJJ E EEJJJ

EEEEJJJ
EEJJ E EEJJ J

J EEEEEEJJJJJ EEJJ J

E EEEEEJJJJJJ J

E EEEEEJJJJJ J

E EEEEEJJJJJ J

EEEEEEJJJJJ J

EEEEEEEEJJJJJ J

EEEEEEJJJJJJ J

EEEEEEEJJJJJ J

EEEEEEJJJJJ E
J

EEEEEEEJJJJ E
J

EEEEEEJJJJ E
J

EEEJJJJ EEEEEEJJJJJJ EE JJ
J E

EEEEEEEJJJJJJJ J

lEEEEEEEEEJJJJJJJJigEEEEEEEEEJJJJJJJJJJhtEEEEEEEEJJJJJJJJJJcEEEEEEEEJJJJJJJJJonEEEEEEEJJJJJJJeEEEEEJJJJ

odd even

EJ

EJ EJ EJ EJ EJ

0 EJ

EJ

Γ

X

M

Γ

(b)

0.5 0.45
0.4 0.35
0.3 0.25
0.2 0.15

J EJ EJ

J EJ EJ

J EJ EJ

J EJ
EEJJ EJ

J EEJ
EEJJ EJ

J EEJ
EEJJ EJ

EEJ
EEJJ EJ

EEJ J EEJJ EJ

EEJ EEJJ J E

EEEEJJ J J E

EEEEJJ J J E

EEEEJJ J J E

EEEEJJ
J J E

EEEEJJ
J J E

EEEJJ
J J E

EEEJJ
J J E

EEEJJ
J J E

EEEJJ
J J E

EEEJJJ
JJ E

EEJJJ
J E

EEJ EEJJ
J J E

EEJ EEJJJ EJ

lEEEEEJJJJJigEEEEEJJJJJhtEEEEEJJJJJcEEEEJJJJJonEEEJJJJeEEJJJ

J EJ EJ

EJ

EJ
0.1
EJ

J odd E even

EJ

0.05 EJ

EJ

0 EJ

EJ

Γ

M

K

Γ

Figure 4.9: Projected band diagrams for the structures shown in Fig. 4.8. While such a substrate will have a ﬁnite thickness in reality, the localization of the
guided modes within the slab means that the substrate can be considered inﬁnite as long as it is sufﬁciently thick. (In this case, a substrate thickness of several wavelengths is sufﬁ-

68

cient for the guided mode amplitude to be negligible beyond the substrate.) Thus, the light cone states are those of an inﬁnite uniform dielectric, whose frequencies are reduced from those in vacuum by a factor of the index. The increased index above and below the slab also has the effect of pulling down the frequencies of the guided modes, allowing them to remain under the now-lowered light cone. In addition, the guided modes are somewhat less localized—for example, 89% of the energy of the lowest band for the hole structure at K is within the height of the slab for the solid background, versus 96% for an air background.

(a) rods

(b) holes

Figure 4.10: Side-view of the slabs shown in Fig. 4.3 with a periodic background formed by “extruding” the structures with a low-index material (dielectric constant 2.0). All other parameters are as in Fig. 4.3.
4.6 Slabs with Periodic Backgrounds
The effective index above and below the slab can be reduced from that of a solid background by using an “extruded,” low-index version of the slab, as shown in Fig. 4.10. For example, in the case of the hole slab, the holes extend through the low-dielectric substrate as well as through the slab. The resulting band structures are shown in Fig. 4.11, and again demonstrate guided modes and a band gap. Such a structure has advantages both in ease of fabrication (both slab and substrate can be etched at the same time) and in conﬁnement of resonant cavity states (since localized states couple less strongly with a lower-index background). The periodicity of the background does not, however, produce useful photonic-
69

crystal slab effects by itself. Even if there were some sort of band gap in the background states, it would lie at higher frequencies than the band gap of the guided modes and would therefore not provide any additional conﬁnement capabilities.

(a)

0.6 0.5 0.4 0.3 0.2 0.1

EJJ EJ EJ
EJ EJ

EJJ EJ
EJ
EJJ EJ

EE J EJ
EJ
EJJ EJ

E J EJ
EEJ
EJJ E J

EE EJJJ
EEJ
EEJJ E J

J E EJJ
EEJ
EEJJ E J

J E EJ J EEJ
EEEJJ
J

J EJ E EEJJ EEEJJ
J

E J EEJ EJJ EE
J

J EEJJJJ EEEEJJ
J

J EEJJJJ EEEEJJ
J

J EEEEEEJJJJJJ E
J

J EEEEEEJJJJJJ E
J

E EEEEEEJJJJJJ E
J

EEEEEEEJJJJJJ E
J

EEEJ EEEJJJJ E
J

EEEJ EEEEJJJJ E
J

EEJJJJ EEJJ E
J

E EJJJ EEJ E
J

J

EJJJ EJ

EEEJJJJ

EE

JJ

J E

EEEEEEJJJJ E
J

lEEEEEEEJJJJJJigEEEEEEEJJJJJJJhtEEEEEEEJJJJJJJcEEEEEJJJJJJJJonEEEEJJJeEEEJJJJ

J

E J

EEJJ E

EJJ

J

odd

EJ

even

EJJ EJ EJ
EJ EJ

0 EJ

EJ

Γ

X

M

Γ

(b)

0.6 0.5 0.4 0.3 0.2 0.1

J J EJ
J E

J J EJ
J E EJ

J J EJ
EEJJ EJ

J
J EJ
EEJJ EJ

EJJ
EEJJ EJ

EJJ J EEJ J E

EJJ J J EJ
J E

EJJ EJJ
J E

EJJ
EEJJ J J E

EEEJJJ J J E

EEEJJJ J J E

EEEJJJJ J J E

EEJJJJ J J E

EEJJJJ J J E

J EEJJ EJ
J J E

J EEJ EJ
J J E

EEEJJJJ
JJ E

EJJ EEJ J
JJ E

EJJ EEJ J
JJ E

EJ EJ J

J EEEJJJ

J

J J

EE

J E

EJJ EEJJ J

light cJonJeJ

EJJ EEJJ

EJJ J J

EJJ

EJJ

J EJ

J EJ

J E

J E

EJ J E

J EEJ EJ

EEJJ

EEJJ

JJ

EJ E

EJ

odd even

EJ

J J EJ
J E

EJ

EJ

0 EJ

EJ

Γ

M

K

Γ

Figure 4.11: Projected band diagrams for the structures shown in Fig. 4.10.

70

The light cone in this system consists of all the states existing in the bulk background,

an inﬁnitely-extruded two-dimensional photonic crystal. This is similar to the shaded

regions from Fig. 4.2, except that the material has a smaller index. That is, the lower edge

of the light cone is simply the lowest band of the two-dimensional system. The fact that

the lowest band of the two-dimensional structure forms a lower bound for the frequencies

of the extruded structure in three dimensions may not be immediately apparent, and so we

consider it below.

Because the dielectric function of the bulk background, ε( x, y) , has translational sym-

metry in the z direction, any background state’s magnetic ﬁeld can be written in the form:

|H〉 = H( x, y)ei(kz – ωt) .

(4.7)

Such a ﬁeld is “manifestly” propagating in the positive-z direction (upwards) for positive k

and

downwards

for

negative

k.

Therefore,

the

group

velocity

d----ω-dk

,

which

can

be

expressed

as the upwards ﬂux divided by the energy density (using the derivative of the eigenvalue

equation), will ordinarily have the same sign as k. It then follows that the minimum fre-

quency occurs at zero k, at which point the solution |H〉 is simply the two-dimensional

eigenstate. This argument holds at each point in the Brillouin zone, and so the light cone is

bounded by the lowest two-dimensional band.

Although the above is true for the present system, there are actually rare cases in

which the group velocity can be opposite to the phase velocity in a structure with transla-

tional symmetry! These “backward waves” have been found in coaxial cables (dielectric

or metallic) and also in anisotropic materials [37]–[38]. When this occurs, the minimum-

frequency states are at a (small) non-zero k—this is the frequency that determines the light

line. To compute the exact light line in general, then, one must solve for the lowest “2d”

band at small positive k and check to make sure that it is not at a lower frequency.

71

(a) rods

(b) holes

Figure 4.12: Side-view of the slabs shown in Fig. 4.10 (extruded backgrounds) with the upper background replaced by air. All other parameters are as in Fig. 4.10. This background breaks the mirror symmetry of the two photonic-crystal slabs.
4.7 Slabs with Symmetry-Breaking Backgrounds
It is also possible to have a background that is not symmetric, for example a substrate below the slab and air above the slab. (Rod and hole slabs with solid substrates have been studied experimentally, e.g., in [39]–[41].) The light cone in this case is the union of the light cones for the backgrounds above and below the slab. Such a structure is depicted in Fig. 4.12, in which the periodic backgrounds from the previous section here lie only below the slabs. The resulting band diagram is given in Fig. 4.13.
The most important consequence of a symmetry-breaking background is that the guided modes can no longer be classiﬁed as even or odd. Thus, there is no longer any band gap in the guided modes, and the photonic crystal properties of the slab are ostensibly lost. If the guided modes are sufﬁciently localized within the slab, however, so that the background is only a small perturbation, the wavefunctions may still be approximated as even or odd and some effects of the band gap will persist.
In order to maintain the distinction between even and odd guided modes, it is only necessary to preserve mirror symmetry where the guided modes have non-negligible amplitude. Thus, a solid substrate can be used below the slab (and not above) with no effects on the band gap as long as the substrate is separated from the slab by a buffer region that is sufﬁciently thick. Shown in Fig. 4.14 are examples of how a solid substrate might be used
72

below the structures from the previous two sections without affecting the band structures signiﬁcantly.

0.6

(a)

0.5 0.4 0.3 0.2 0.1

B B B B B

B B B B BBB

B B
B B B B

B
BB
BB B B BB

B
BB
BB BBB B B

B BB BB BBB B
B

BBB B BB B
B

BBBBBBBB B

BBBBB B
B

BBBB BB
B

BBBB BB B

BBBBB BB B

BB BBBB B B

BB BBB B B

BB BBB B B

BBBBBB B
B

BBBBBB B
B

BB BBB B
B

BB BBB B
B

BB BBB B
B

BB BBB B
B

BBBBBB B
B

light cone

BBBBBB BB B

BB BBBB B B

B BB BB B BB
B

B
BB BB BBBB

B BB B B

B B B

B

B BB

B

BB

B B B B BB

B

B

0B

B

Γ

X

M

Γ

0.6

(b)

0.5 0.4 0.3 0.2

B
B B

B
B BB

B
BB BB

B
BB BB B

B
BB B B B

B
BB B B B

B B BB B B
B

B BB B B
B

B BB B B B
B

B B B B
B

B B BB
B

B B BB
B

B B BB
B

BB B BB
B

BB B BB
B

B B BB
B

B B BB
B

BB B BB B

BB B BB B

BBB B B

BB B B B B

B B B B B
B

light cone

B

BB B

BB B

B

B

B

B

B B BB

B B BB

B

B

B B

B B B

BB BB

B BB

B

B
B B

B

0.1 B

B

B

B

0B

B

Γ

M

K

Γ

Figure 4.13: Projected band diagrams for the structures shown in Fig. 4.12. The bands can no longer be segregated into even and odd modes, and there is no longer a band gap in the guided modes.

73

(a) rods

(c) holes

substrate
(b) rods

substrate
(d) holes

substrate

substrate

Figure 4.14: A solid substrate may lie below a slab without affecting the band structure by being sufﬁciently far from the slab. (a), (b), (c), and (d) depict such solid substrates below the symmetric and antisymmetric rod and hole structures from Figs. 10(a), 12(a), 10(b), and 12(b), respectively.
4.8 Summary
Photonic-crystal slabs are an important system in the practical application of photonic crystals, and the band structure formalism provides a powerful tool in their analysis. Unlike two- and three-dimensional photonic crystals, the band structures for slabs are projected and have the unique feature of a light cone enveloping the states. The presence of the light cone means that a complete band gap is impossible—still, waveguides and resonant cavities are possible [32]. Furthermore, the presence of a band gap in the guided modes is useful in its own right. For example, Fan et al. have proposed using slab band gaps to prevent in-plane losses from LED emission [30]. Perhaps the greatest promise of photonic-crystal slabs, however, is that they will allow exciting results from two-dimensional crystals such as waveguide bends [42] and channel-drop ﬁlters [43] to be imple-

74

mented easily on optical and infrared lengthscales.
75

76

Chapter 5
Linear Waveguides in Photonic-Crystal Slabs*
5.1 Introduction
Photonic crystals, which prohibit the propagation of light for frequencies within a band gap, have enabled exciting new ways to control light and construct integrated optical devices [44]. An important element of optical circuits is a linear waveguide to carry light to and from components, and photonic crystals also provide unique advantages for waveguides. Photonic-crystal waveguides, guided by the band gap of the bulk crystal [44]–[47], can exhibit near-zero reﬂection and loss through sharp bends [48] and when coupled with resonant cavities [49]–[50], due to the gap’s prevention of radiation losses. Previous theoretical studies of photonic-crystal waveguides, however, have been restricted to purely two-dimensional systems and have not examined the effects of vertical conﬁnement. In this chapter, we present a systematic study of waveguide modes in a three-dimensional system, photonic-crystal slabs. Photonic-crystal slabs are two-dimensionally periodic dielectric structures of ﬁnite height that have a band gap for propagation in the plane and use index-conﬁnement in the third dimension; they have been proposed as a more-easily fabricated alternative to true three-dimensionally periodic photonic crystals [45],[51]–[62]. Although their structure and properties strongly resemble those of twodimensional crystals, slab systems require a fundamentally different, three-dimensional analysis [30],[59],[62]. We will demonstrate how such analyses apply to waveguides and explain the novel considerations that arise for line defects in photonic-crystal slabs. We shall show that, merely to produce guided modes (as well as to be single-mode and ingap), the parameters of the defect must be carefully chosen.
* This chapter is based on: S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000).
77

Waveguides must satisfy three criteria in order to achieve optimal performance in many integrated-optical-circuit applications; e.g., for maximum transmission through sharp bends (on the scale of the wavelength) and resonant cavities. First, of course, the waveguide must support true guided modes (as opposed to resonances, as deﬁned later). This also implies that the waveguide structure must be periodic along the direction of propagation, in order to have a well-deﬁned Bloch wavenumber k and to thereby propagate without reﬂections. Second, the waveguide should be single-mode in the frequency range of interest—this condition is necessary in [48] and the increased reﬂections in a multi-mode waveguide are a straightforward consequence of coupled-mode theory [63]. (Extraneous modes may have little effect if coupling to them through the bend or cavity can be made negligible, but this is not generally the case for wavelength-scale structures.) Third, the guided mode should lie within the band gap of a photonic crystal in order to prohibit radiation losses (which both decrease transmission and increase reﬂection [63]); otherwise, some losses are inevitable at a bend or resonator, which breaks translational symmetry.
Conventional linear dielectric waveguides operate by index-conﬁnement (total internal reﬂection, in the short-wavelength limit). Such waveguides can be made to satisfy the ﬁrst two criteria from above (i.e., be guiding and single-mode), and thereby have relatively high transmission through sharp bends and resonators [50],[64],[65]. However, the absence of a photonic band gap means that transmission is always limited by radiation losses. In a purely two-dimensional photonic-crystal linear waveguide, a linear (onedimensionally periodic) defect is introduced into the crystal, creating a localized band that falls within and is guided by the photonic band gap [44]–[47]. Light is therefore prohibited from escaping the waveguide, and all three criteria can be satisﬁed to achieve perfect transmission through bends and resonant cavities [48]–[50]. These results have been
78

experimentally conﬁrmed using structures that are uniform in the third dimension for

many wavelengths, and thus approximate a two-dimensional system [66]; the large aspect

ratios of such geometries and their lack of vertical conﬁnement or a true band gap in the

guided modes limit their practical applications, however. There are several other possibili-

ties for realizing photonic-crystal waveguides in three dimensions. The most ideal struc-

ture, and the only way to totally eliminate radiation losses, would be a three-dimensionally

periodic structure with a complete band gap, in which the photonic crystal completely sur-

rounds the waveguide. Such structures, though, are challenging to construct at submicron

lengthscales.[67]–[70] In this chapter, we instead focus on photonic-crystal slabs.

Waveguides in photonic-crystal slabs are analogous to those in two-dimensional crystals,

except that they use index conﬁnement in the third dimension. Similar waveguides, which

use photonic-crystal conﬁnement in two dimensions and index conﬁnement in the third,

have been subject to experimental studies [71]–[72], as well as theoretical work [73] sub-

sequent to the submission of the article upon which this chapter is based. There is also

another type of photonic-crystal waveguide that we do not consider in this chapter, the

(a) rod slab

(b) hole slab

z y
x
Figure 5.1: Photonic-crystal slabs. (a) Square lattice of dielectric rods in air with ε of 12.0, lattice constant a, radius 0.2a, and height 2.0a, with low-index (ε of 2.0) rods extending inﬁnitely above and below. (b) Triangular lattice of air holes extending through both a high-index (ε of 12.0) ﬁnite-height dielectric slab and low-index (ε of 2.0) semi-inﬁnite dielectric regions above and below. The holes have lattice constant a and radius 0.3a, while the high-index slab is of thickness 0.5a.
79

photonic-crystal ﬁbre [74]–[75]. This is a two-dimensionally periodic system of (in principle) inﬁnite thickness, in which the waveguide mode propagates perpendicular to the plane of periodicity; in contrast, photonic-crystal slab waveguides are of ﬁnite thickness and run parallel to the plane of periodicity. A waveguide operating on principles similar to the ﬁbres’ was also considered in [76].
In this chapter, we analyze linear-defect waveguides in two characteristic photoniccrystal slabs, a square lattice of dielectric rods in air and a triangular lattice of air holes in dielectric, illustrated in Fig. 5.1. Above and below the slabs are semi-inﬁnite “extruded” substrates with the same cross-sections as the slabs, but having a lower dielectric constant. (We use a substrate in order to model a more practical system than a suspended slab, and having the substrate on both sides of the slab preserves mirror symmetry.) The band diagrams of these two structures are given in Fig. 5.2 and Fig. 5.3, respectively; their calculation and interpretation is described in [62]. Each band diagram includes a shaded region (deﬁning the light cone) and discrete (guided) bands lying below the light cone, which are

0.6 0.5 0.4 0.3 0.2

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JE

JJE JE JE JE

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J

E J

JJE JE

JJE

JE

JE

0.1

JE

JE

JE

0 JE

JE

Γ

X

M

Γ

Figure 5.2: Band structure for the rod slab from Fig. 5.1(a). Hollow and ﬁlled circles rep-

resent modes which are even and odd with respect to the horizontal (z = 0) mirror plane

bisecting the slab. There is a band gap in the odd (TM-like) modes in the frequency range

0.3362–0.4292 c/a.

80

0.5

0.45 0.4
0.35 0.3
0.25 0.2
0.15 0.1

EJ
J E

EJ
J E EJ

EJ
JJ EE EJ

even

EJoEJddEEJ

E EJJ

J EJ

EEJJ

EJJ J

EJ
J E

J E

JE

E

E EEEJJJ
J
E

EEJ EJ J
E

EEEJJ

EEEJJ

EEEJJ

EEJ E

0.3471
EJJ EJJ EJJ EJJ

EEEE

EEJ E EJJ
E

E EJJ E EJJ
E

E EJ J E EJJ
E

E EJ J EEJJ

E EJ
J EEJJ

EEJ
J EJJ

E J
J EJ

E EJ J EJ J

EEJJ EJ
J

EEEJJ

0.2708

J

EEEEEEE

MK

Γ

EJ J EEJ
J E

light cone

EEJ J

EJ

EJ

EJ

EJ

EJ
J E

EJJ J

EEJJ

JJ E

J

E EJ E

EJ

0.05 EJ

EJ

0 EJ

EJ

Γ

M

K

Γ

Figure 5.3: Band structure for the hole slab from Fig. 5.1(b). Hollow and ﬁlled circles denote even and odd modes as in Fig. 5.2. There is a band gap in the even (TE-like) modes in the frequency range 0.2708–0.3471 c/a.

conﬁned vertically in the vicinity of the slab by index conﬁnement. (Thus, the slab forms a

planar waveguide, but in this chapter we use the term “waveguide” only to describe linear

waveguides.) The presence of the light cone, with the corresponding constraint that all

guided modes be index-conﬁned in the vertical direction, gives rise to the most signiﬁcant

differences between photonic-crystal slabs and their two-dimensional cousins. The guided

modes are either even or odd with respect to the horizontal mirror symmetry plane of the

slab; this is analogous to TE and TM, respectively, in two dimensions, although here the

modes are not completely polarized except in the symmetry plane. The rod and hole slabs

have band gaps in their odd and even modes, respectively, though these are not complete

gaps due to the light cone. Nevertheless, in the gap no guided modes exist for the corre-

sponding symmetry. The slab band gap has been shown to support resonant cavities in

point defects [59]–[60], and we will show in this chapter that it can also conﬁne

waveguide modes in linear defects. The differing thicknesses of the slabs derive from the

polarizations of the modes exhibiting the band gap, and were chosen to achieve large gaps.

81

(a)
(b)
(c)
z y
x
Figure 5.4: Linear defects, which give rise to waveguide modes, in the rod and hole photonic-crystal slabs from Fig. 5.1. The low-index material is not shown, but “extrudes” above and below the structures as in Fig. 5.1. (a) Reduced-index waveguides, created by decreasing/increasing the radii of a line of nearest-neighbor rods/holes, respectively. (b) Increased-index waveguides, created by increasing/decreasing the radii of a line of nearest-neighbor rods/holes, respectively. (c) Dielectric-strip waveguide surrounded by the rod slab, from which a line of nearest-neighbor rods has been removed and replaced with the waveguide.
In order to create a linear waveguide, we break the periodicity of the slabs in one direction by creating linear defects of various kinds, as shown in Fig. 5.4. The possible perturbations of the slab fall into two main categories: reduced-index waveguides, as in Fig. 5.4(a), for which the amount of high-index material is decreased; and increased-index waveguides, as in Fig. 5.4(b), for which the amount of high-index material is increased. We also consider a third case, shown in Fig. 5.4(c), in which a column of rods is removed
82

and replaced by a conventional dielectric “strip” waveguide. In all cases, we refer to the direction along the waveguide as x, the perpendicular in-plane direction as y, and the vertical direction (out of plane) as z.
The remainder of the chapter begins with a discussion of the computational methods that we employed, followed by detailed analyses of reduced-index, increased-index, and strip waveguides in photonic-crystal slabs. We then show that waveguides along nearestneighbor directions are the most feasible. Finally, by estimating a lower bound on the fraction of the electric-ﬁeld energy in the dielectric, we argue that it is not possible for photonic-crystal slabs to guide modes that reside primarily in air (unlike two- or threedimensional crystals with complete gaps). (Guided modes can still reside, however, in a reduced-index region.)
5.2 Computational Method
The band structures of Fig. 5.2 and Fig. 5.3 were computed as described in [62], using preconditioned conjugate-gradient minimization of the Rayleigh quotient [77] in a threedimensional plane-wave basis [78], with an imposed periodicity at a large interval in the vertical direction. This is a general, non-separable, vectorial solution of the full Maxwell’s equations in a complete three dimensional basis [79]–[80]—the only approximations are the discretization of the system (i.e., the planewave cutoff) and the imposition of a vertical supercell (which has a negligible effect on the localized, exponentially-decaying guided modes). The light cone boundary derives from the lowest two-dimensional band of the low-index extruded region.
Calculation of projected band structures for line defects (linear waveguides) is similar to the above, except that an additional supercell of seven or eight periods is used in the direction perpendicular to the waveguide. (Waveguide modes are sufﬁciently localized so
83

that the adjacent waveguides introduced by the supercell have a negligible effect on the mode frequencies.) The bands from the unperturbed slab are projected onto the Brillouin zone of the line defect, and form the boundaries of the slab band continuum. Both the slab bands and the light cone are depicted with a uniform shading in the projected band structure, despite the varying density of states in these regions. Only modes that fall outside both the light cone and the slab bands are truly guided in the line defect, decaying exponentially away from the waveguide. When the guided bands cross into the continuum regions, they become resonances that extend inﬁnitely away from the waveguide, albeit with low amplitude. We do not consider such resonance modes in this chapter.
5.3 Reduced-Index Waveguides
Photonic-crystal waveguides formed by removing high-index material [44],[47],[48], effectively reducing the index of the waveguide compared to its surroundings, are dramatic examples of the photonic band gap, as they have no analogues in conventional index-guiding. Such waveguides are also possible in photonic-crystal slabs, such as in Fig. 5.4(a) where we have reduced the radius of a line of rods or increased the radius of a line of holes. The effective index is lower in the waveguide than in the surrounding slab, so that modes can only be constrained horizontally by the band gap. On the other hand, the index is higher in the waveguide than the regions above and below the slab, so that modes can be guided vertically by index-conﬁnement. In this section, we will ﬁrst consider the reduced-radius rod-slab waveguide and then the increased-radius hole-slab waveguide.
The dispersion relations for various reduced-radii rod-slab waveguides are shown in Fig. 5.5. There is only one guided band for each radius, so the waveguide is single-mode. The line defect is periodic along its axis, and so the dispersion relation is plotted on the Γ –X reduced Brillouin zone of this lattice. The light cone and the bulk slab bands from Fig. 5.2 are projected onto their Γ –X wavevector component, resulting in the continuous
84

0.44

0.42

light cone

J

J

J r = 0.10a

0.4
0.38
0.36
0.34
0.32 0.3
(← Γ)

J

J

J

J

J

J

J

J

J

J

J

J

J

J

J

slab bands

0.35

0.4

0.45

wavevector k (2π/a)

J r = 0.12a
J r = 0.14a J r = 0.16a J r = 0.18a
0.5
(X)

Figure 5.5: Projected band structure for the reduced-index rod-slab waveguide from Fig. 5.4(a), showing the (odd-symmetry) guided bands for various defect rod radii (compared to the bulk radius of 0.20a).

shaded regions of Fig. 5.5. This band diagram continues symmetrically beyond X, so all of the bands must have zero group velocity (slope) at X in order to be analytic functions of the wavevector. This condition of zero slope at the Brillouin-zone edge also applies to all of the other waveguide dispersion relations that we consider in this chapter. (The zero slope is not apparent in Fig. 5.5 because the bands were only computed at a few points.)

These discrete bands, all of odd (TM-like) symmetry, decay exponentially away from the defect because they fall in neither the light cone (vertically-radiating modes) nor the continuum of odd slab-bands (which propagate freely within the slab), and so cannot couple with those extended modes. This strong conﬁnement is illustrated in Fig. 5.6, which depicts horizontal and vertical cross sections of the electric-ﬁeld z component at X for the radius 0.14a line defect. We call this an s mode because the ﬁeld distribution around a rod resembles an s electron state, in contrast to other waveguide modes in subsequent sections.

By the time the defect radius decreases to 0.08a (versus 0.20a in the bulk), the guided mode has entirely disappeared into the light cone. This stands in stark contrast to a twodimensional lattice of rods, in which a line defect with the rods entirely removed still sup-

85

x→

z→

y →
Figure 5.6: Field cross-sections for the rod-radius 0.14a guided mode at X (0.3707 c/a), showing the z component of the electric ﬁeld. The contours of the dielectric function are shown in black. (a) Horizontal (xy) cross-section in the plane bisecting the slab (z = 0). (b) Vertical (yz) cross-section in the plane perpendicular to the waveguide and bisecting a row of rods. ports a guided mode [44],[47],[48]. This difference is due solely to the slab’s constraint of vertical index conﬁnement—that is, the guided bands must lie outside the light cone. The two-dimensional dispersion relation of the removed-rod waveguide disappears into the upper gap edge at a wavevector of roughly 0.4 (2π)/a [44], so it is not surprising that it falls into the light cone of the slab in this case. Even more generally, however, we do not expect a photonic-crystal slab waveguide to support guided modes that exist primarily in
86

0.36

light cone

0.34

E

E r = 0.48a

E
0.32

E

0.3

E

E

E r = 0.45a

E

0.28

E

E

E

E r = 0.40a

0.26 slab bands

0.24

0.3

0.35

0.4

0.45

0.5

(← Γ)

wavevector k (2π/a)

(K')

Figure 5.7: Projected band structure for the reduced-index hole-slab waveguide from Fig. 5.4(a), showing the (even-symmetry) guided bands for various defect hole radii (compared to the bulk radius of 0.30a).

air—a large fraction of the electric ﬁeld must lie within the dielectric. The basic reason for

this is that the vertical conﬁnement of the mode entails an “energy cost,” and having much

of the ﬁeld within the dielectric is the only way to bring the mode back down below the light cone. We quantify this argument in section VII. In the case of the Fig. 5.5

waveguides, the fraction of the electric-ﬁeld energy inside the high dielectric at X ranges

from 55% for radius 0.10a to 68% for radius 0.18a. (Nevertheless, the mode is still within the reduced-index region and is not index-guided in the plane.)

A reduced-index waveguide in the hole slab is formed by increasing the radius of a line of nearest-neighbor holes. The resulting band diagrams for various radii are shown in Fig. 5.7. Here, only modes with even (TE-like) symmetry are depicted, since that is the

symmetry of the bands exhibiting a gap. As with the rods, there is only a narrow range of radii that supports guided modes (all of which are single-mode). Although the radius 0.40a

band is guided, it is barely in the band gap, which begins at the maximum frequency of the lower slab bands. The projected band structure for line defects retains a wavevector quantum number, and this means that the range of frequencies available for guided modes may

87

m

m'

n

K

M

K'

Γ

Figure 5.8: Diagram of how the reciprocal lattice is projected onto Γ–K (vertical dashed line) for a nearest-neighbor line defect in the hole slab. The black dots are the reciprocal lattice points, the outlined hexagon is the ﬁrst Brillouin zone, and the grey triangle within it is the reduced Brillouin zone. K' is the boundary of the ﬁrst Brillouin zone in the projected lattice. be greater than the range of the band gap, as is the case here at the K' point. Since this quantum number disappears at bends and other places that break translational symmetry, however, it is important to consider only guided modes within the true band gap.
The nearest-neighbor direction corresponds to the Γ –K direction in the slab Brillouin zone, but K is not the edge of the line-defect Brillouin zone. Figure 5.8 depicts how the reciprocal lattice of the slab is projected to form the reciprocal lattice of the line defect. The boundary of the projected Brillouin zone lies halfway between Γ and m' (projected from m)—this is the point K', projected from M (which lies halfway between Γ and m). Use of the correct projection is important for the computation of the light cone and the slab bands continuum for Fig. 5.7.
As before, the waveguide band is s-like and strongly localized both vertically and horizontally, as shown in Fig. 5.9. In this case, we depict the z component of the magnetic ﬁeld because the mode is TE-like. Since the electric ﬁeld loops around the magnetic ﬁeld
88

x→

(a) s guided mode at K' for r = 0.45a holes
y →
(b) same mode, vertical cross-section

z→

y →

–1

Hz

+1

Figure 5.9: Field cross-sections for the hole-radius 0.45a guided mode at K' (0.3026 c/a), showing the z component of the magnetic ﬁeld. The contours of the dielectric function are shown in black. (a) Horizontal (xy) cross-section in the plane bisecting the slab (z = 0). (b) Vertical (yz) cross-section in the plane perpendicular to the waveguide and bisecting a defect hole.

like ﬁelds around a current, it is concentrated in the dielectric just as it was for the rods—

the fraction of the electric-ﬁeld energy inside the high dielectric at K' ranges from 63% for

radius 0.48a to 86% for radius 0.40a. The critical parameter in this case is the thickness of

the narrow veins between the holes. It is only as that thickness approaches zero that the

guided modes rise up far into the gap, while for small changes in defect hole radius there

are no modes at all in the gap. In contrast, when the rod-slab radius is decreased, the elec-

tric ﬁeld is forced directly into the air and the frequency quickly increases even for small

perturbations.

There are many other ways to create reduced-index waveguides. For example, one

could remove a row of rods and then shift the lattices towards one another on either side of

the waveguide. This might, however, be inconvenient for applications in high-density inte-

89

grated optical devices, since lattice dislocations from different components may conﬂict. When a row of rods is removed, one can run into additional problems due to the electric ﬁeld being concentrated in the adjacent rows of remaining rods. This leads to near-degeneracies between states that are even and odd with respect to the y = 0 mirror plane, creating a multi-mode waveguide. Yet another way to create a reduced-index waveguide is to actually decrease the dielectric constant within a linear (one-dimensionally periodic) region, without necessarily changing the geometric structure.
5.4 Increased-Index Waveguides
A waveguide can also be created by a line defect in which high-dielectric material is added, leading to an increased effective index. As shown in Fig. 5.4(b), we examine two types of increased-index waveguides: ﬁrst, a rod slab with a line of increased-radius rods; and second, a hole slab with a line of decreased-radius holes. In general, increased-index defects can introduce two sorts of guided modes. First, there are modes that lie below both the light cone and the slab band continuum—these modes are conventionally index-guided horizontally as well as vertically. They do not satisfy our third waveguide criterion, however: they do not lie in the band gap, being below the lowest slab bands. Thus, we do not consider such guided bands here. The second type of guided mode, which we shall study, is a state that is pulled down into the gap from the upper slab bands. Such a mode is conﬁned horizontally purely by the band gap and index-conﬁned vertically, just like the reduced-index waveguides in the previous section. Because these increased-index modes derive from the upper slab bands, which have a higher density of states and often come in degenerate pairs (at symmetry points), there is a greater tendency here towards multimode waveguides. As we shall see, however, the single-mode criterion can be satisﬁed if the effective index is not increased too greatly.
90

0.44
0.42
0.4
(r = 0.25a)
0.38

light cone

J

J J

J

J py J px

0.36

0.34

slab bands 0.32

0.3

0.35

0.4

0.45

0.5

(← Γ)

wavevector k (2π/a)

(X)

Figure 5.10: Projected band structure for the increased-index rod-slab waveguide from Fig. 5.4(b), showing the (odd-symmetry) guided bands for defect rod radius 0.25a (compared to the bulk radius of 0.20a).

(a) px guided mode at X for r = 0.25a rods

x→

x→

(b) py guided mode at X for r = 0.25a rods
y →
Figure 5.11: Horizontal (z = 0) Ez ﬁeld cross-sections for the rod-radius 0.25a guided modes at X. (a) px guided mode (0.3887 c/a). (b) py guided mode (0.4205 c/a).
Increasing the radii of a line of nearest-neighbor rods results in the band diagram of Fig. 5.10, showing the odd bands for rod radii of 0.25a. This rod radius gives rise to a pair of non-degenerate guided modes, labeled px and py, but because these modes do not overlap in frequency the waveguide remains single-mode. (Note that here, one can identify modes with the same wavevector but opposite group velocities.) Fig. 5.11 shows cross-
91

sections of the electric ﬁeld for the two states at defect radius 0.25a. As can be seen in Fig. 5.11, the ﬁeld around a rod is similar to a p electron state, oriented in the x and y directions for the px and py states, respectively.
When the defect rod radius is increased to 0.275a, three additional states are pulled down into the gap, as shown in Fig. 5.12. In this case, the px and py states cross (which is possible because they have different symmetry with respect to the y = 0 mirror plane), meaning that they are not single-mode over their whole guided range. The states labeled px(2) and py(2) have similar horizontal cross-sections to px and py, but are second-order excitations in their vertical cross section (i.e., their electric ﬁelds have two vertical nodes). The state labeled sH(1) is a ﬁrst-order vertical excitation of the lowest TE-like band (i.e., a single vertical node in the magnetic ﬁeld), which is s-like in its Hz cross-section. (The lowest TE-like band is of even symmetry, but adding a vertical node transforms it to odd symmetry and allows it to be guided by the gap.) The px(2), py(2), and sH(1) states have no analogues in two dimensions.
Figure 5.13 shows the dispersion relation of even modes for the decreased-radius hole waveguides with radii of 0, 0.15a and 0.25a. The waveguides are single-mode at each fre-

0.44

0.42

0.4
(r = 0.2J75a)
0.38

0.36

0.34

J

light cone

XH

H H

H

J J J
J

XH s (1)

H

p (2)

y

H

p (2)

x

J py

J px

slab bands 0.32

0.3

0.35

0.4

0.45

0.5

(← Γ)

wavevector k (2π/a)

(X)

Figure 5.12: Projected band structure for the increased-index rod-slab waveguide from

Fig. 5.4(b), showing the (odd-symmetry) guided bands for defect rod radius 0.275a.

92

0.36

light cone

py (r = 0.25a)

0.34 0.32

E J

E CEH E

CEEH E

CEH E

dxy (r = 0.15a)
dxy (r = 0) px (r = 0.25a)

E py (r = 0.15a)

0.3

E

J E

J E

J py (r = 0) E px (r = 0.15a)

0.28

J

J

J

J px (r = 0)

0.26 slab bands

0.24

0.3

0.35

0.4

0.45

0.5

(← Γ)

wavevector k (2π/a)

(K')

Figure 5.13: Projected band structure for the increased-index hole-slab waveguide from Fig. 5.4(b), showing the (even-symmetry) guided bands for defect hole radii 0.15a and 0.25a (compared to the bulk radius of 0.30a). For each radius, there are two or three guided bands: px (solid line), py (dashed line), and dxy (dotted line, triangles). Guided bands for a defect hole radius of 0, in which a row of holes is entirely removed, are shown in bold.

quency, and there are px and py modes for each radius as for the rod slabs. In addition, the radius 0 and 0.15a waveguide supports a higher-order dxy mode. All three modes for radius 0.15a are depicted in Fig. 5.14.

In both the rod and the hole slabs, increasing the amount of dielectric brings more and more states into the gap, which makes it harder to achieve single-mode waveguides. One can ameliorate that situation by removing a row of holes and then shifting the lattices on either side of the waveguide towards one another (reducing the effective index in the waveguide). Such a waveguide was considered for the hole structure in two dimensions by [46], and similar geometries should be feasible in photonic-crystal slabs.

5.5 Strip Waveguides in Photonic-Crystal Slabs
In this section, we study another possible photonic-crystal slab waveguide, formed by replacing a row of rods in the slab of Fig. 5.1(a) with a conventional dielectric strip waveguide of the same thickness as the slab, as shown in Fig. 5.4(c). As before, there is a

93

x→

x→

x→

(a) px guided mode at K' for r = 0.15a holes
(b) py guided mode at K' for r = 0.15a holes
(c) dxy guided mode at K' for r = 0.15a holes
y →
Figure 5.14: Horizontal (z = 0) Hz ﬁeld cross-sections for the hole-radius 0.15a guided modes at K'. (a) px guided mode (0.2874 c/a). (b) py guided mode (0.3148 c/a). (c) dxy guided mode (0.3377 c/a). low-index substrate with the same cross-section as the slab and waveguide both above and below this structure.
In Fig. 5.15 we show the odd-symmetry band diagram for such a waveguide with a width of 0.25a. In this case, the frequency axis extends down below the slab band continuum so that both the index-guided modes (dashed lines, below the slab bands) and the gap-guided modes (solid lines) are visible. The two gap-guided modes at X are shown in Fig. 5.16—they differ by a 90° phase shift—and the horizontal cross-sections of the index-guided modes are nearly identical to those of the gap-guided modes. The vertical cross-sections of the gap- and index-guided modes differ sharply, however, as shown in
94

0.4
0.35
0.3
0.25
0.2 0.3
(← Γ)

light cone

J
J J sJlab bands
J
J J

gap-guided

J

J

J

J J
J
index-guided

0.35

0.4

0.45

0.5

wavevector k (2π/a)

(X)

Figure 5.15: Projected band structure for the strip waveguide in a rod slab, from Fig. 5.4(c), showing the (odd-symmetry) guided bands for waveguide width 0.25a. There are two pairs of guided bands: gap-guided (solid lines) and index-guided (dashed lines). The gap-guided bands are split at X into a 0.07% gap, while the index-guided bands are split at X into a 0.8% gap.

Fig. 5.17—the index-guided mode is the fundamental mode and the gap-guided mode is a

second-order excitation.

In the absence of the rod slab, both of the modes shown would lie below the light cone and be purely index-guided. Furthermore, since the strip has translational symmetry, the

Brillouin zone that we apply in Fig. 5.15 would be artiﬁcial—the guided bands would simply fold back and forth at the zone boundaries and be degenerate at X. Because this folding is synthetic, the guided modes do not couple with the light cone even when they fold on

top of it. Adding the rod slab to this system has three effects: ﬁrst, the frequencies of the strip are perturbed; second, the degeneracy at X is broken by the true periodicity of the

system; and third, an upper cutoff for the guided modes is introduced by the light cone

(since the band folding is no longer artiﬁcial). (Also, if it were not for the band gap of the rods, the gap-guided modes would only be resonances—they would couple with the slab

and leak slowly away.) Because the guided modes are strongly localized around the dielec-

95

tric strip, however, the effect of the rods is small. The bands shown in Fig. 5.15 have an root-mean-square deviation of less than 1% from the bands of an isolated strip, and the splitting at X is also less than 1%. Because of the true periodicity of the system, the folded band structure must be analytic and the bands thus have zero group velocity at X—due to the weakness of the perturbation, though, the slope drops towards zero only as the wavevector comes within 1% of X. The greater the periodic perturbation, the greater the splitting and the larger the region of low group velocity, leading eventually to the highlynonlinear dispersions of the previous sections.
Despite the similarity in frequency with the modes of an isolated strip, the waveguide here has a signiﬁcant advantage—it supports (single-mode) bands in the photonic band gap, which will inhibit losses in the cases of sharp bends and cavity interactions. The index-guided lower modes in this structure, on the other hand, have no advantage over a conventional waveguide. These modes might otherwise be preferred since they are the fundamental modes of the waveguide; if this is an important consideration, they can be pushed up into the band gap by reducing the waveguide width to 0.10a (at the expense of increased height-to-width aspect ratio).
5.6 Waveguides in Other Directions
In the preceding sections, we have only considered waveguides in the nearest-neighbor directions of the slabs. Another possibility might be waveguides in the other symmetry directions (i.e., along next-nearest neighbors), as this is known to give rise to guided modes in two dimensions [47]. Because of the presence of the light cone, however, such waveguides are not possible in the photonic-crystal slab structures of Fig. 5.1.
The problem is that, for the next-nearest-neighbor directions, the edge of the projected Brillouin zone comes at too small a wavevector, and the resultant folding of the light cone
96

(a) lower gap-guided mode at X

x→

(b) upper gap-guided mode at X

x→

y →
Figure 5.16: Horizontal (z = 0) Ez ﬁeld cross-sections for the 0.25a strip-waveguide gapguided modes at X. (a) Lower gap-guided mode (0.3626 c/a). (b) Upper gap-guided mode (0.3629 c/a).

(a) upper bands (gap-guided)

(b) lower bands (index-guided)
(waveguide top)

z→

(waveguide bottom)

x →

x →

Figure 5.17: Vertical (y = 0) Ez ﬁeld cross-sections of a unit cell, parallel to and bisecting the waveguide, for the strip-waveguide gap-guided and index-guided modes at X. (a)
Upper gap-guided mode (0.3629 c/a). (b) Lower index-guided mode (0.2632 c/a).

97

eliminates the band gap. For example, the next-nearest-neighbor direction in the hole slab corresponds to the M direction. When the reciprocal lattice is projected onto Γ –M, however, one of the reciprocal lattice points (n in Fig. 5.8) falls onto M. Therefore, the Brillouin zone edge is at M/2, at which point the light-cone frequency is only 0.223 c/a. Since this upper cutoff for guided modes is lower than the gap bottom, there is no band gap and photonic-crystal waveguides are not possible. Similarly, in the rod slab (square lattice), the Brillouin-zone boundary for the next-nearest-neighbor direction is also at M/2, corresponding to a light-cone frequency of 0.332 c/a—again below the gap bottom.
If the rod slab were suspended in air (i.e., no substrate), the light cone at M/2 would be at a frequency of 0.354 c/a, while the band-gap bottom rises to only 0.3397 c/a, leaving a slight (4%) band gap. So, in general, next-nearest-neighbor waveguides in photonic-crystal slabs are not impossible, but the available band gap is greatly reduced from the nearestneighbor case.
5.7 Estimating the Field Energy in the Dielectric
In section III, we suggested that much of the ﬁeld energy must lie within the dielectric material, not in the air, in order for a waveguide mode to fall underneath the light cone and be guided. In this section, we justify that claim; moreover, we ﬁnd an analytic relationship between a mode’s localization and the fraction of it inside the high-index material, explaining why some waveguide modes from two-dimensional systems cannot exist in a slab. First, we derive a rigorous lower bound on the fraction of the electric ﬁeld energy inside the dielectric, in terms of the spatial extent of the mode and the degree to which it is transverse. Then, we approximate the transversality and spatial conﬁnement of the mode to compute a numerical value for this lower bound, and compare with the results of our exact calculations.
98

The frequency of a guided mode can be expressed as an expectation value of the Max-

well operator for the electric ﬁeld, evaluated over the unit cell:

ω2 = 〈---E---〈|--E∇---|--ε-×--(--∇x----)--|×-E----〉|-E----〉 ,

(5.1)

where we take c to be unity. Now, we suppose that the dielectric function is piecewise con-

stant, and write the ﬁeld |E〉 as a sum of ﬁelds |E〉i , such that |E〉i equals |E〉 where the

dielectric is εi and is zero elsewhere. Then, we can rewrite Eq. (5.1) as:

∑ ∑ ω2

=

〈---E---|--∇----〈--×E----|∇-E----〉-×-----|-E----〉

i

ε-1--i

 

〈--i-E-〈--E|--ε-|--(ε--x-i-|-)-E--|-E〉---i-〉

=

〈---E---|--∇----〈--×E----|∇-E----〉-×-----|-E----〉

i

-εf---ii .

(5.2)

Here, f i is the fraction of the electric ﬁeld energy that is in the dielectric εi , and must sum

to unity. In order to be guided, the frequency in Eq. (5.2) must be less than the light cone

frequency ωlc at that point in the Brillouin zone. We will write the electric ﬁeld in a Fourier basis, in which case the curl operations become simply the total Fourier wavevector k

squared. Because the electric ﬁeld is not divergenceless, however, the curl also projects out

the longitudinal ﬁeld component |EL〉 , leaving only the transverse component |ET〉 . (|ET〉 is divergenceless but not curl-free, while |EL〉 is curl-free but not divergenceless. |EL〉 is the static electric ﬁeld of the bound charges.) In this case, the combination of Eq. (5.2)

with the guiding constraint becomes:

∑ 〈 k2〉 ET 〈---E〈---ET---||-EE----〉T---〉

i

-εf---ii

<

ωl

2 c

≤

k

2 B

.

(5.3)

Here, kB is the Bloch wavevector in the Brillouin zone of the line defect, and the second

relation derives from the fact that the light cone must lie beneath the light cone of vacuum.

〈 k2〉 ET is the mean-square wavevector of the transverse ﬁeld, which is inversely related to

the mode’s spatial conﬁnement. We will denote the transverse fraction of the ﬁeld,

〈ET|ET〉 ⁄ 〈E|E〉 , by f T . If we have a two-component system (dielectric ε and air), Eq. (5.3) leads to the following rigorous lower bound on the fraction of energy in the

99

dielectric:

f diel > -1----–----k----B---12---⁄-–--(---1f---T-⁄--〈ε---k---2--〉---E---T--)- .

(5.4)

The more localized the ﬁeld is, the greater becomes the mean-square wavevector and the

greater the minimum f diel . In order to evaluate Eq. (5.4) further, we need to estimate both f T and a lower bound
on the mean-square wavevector. We treat the wavevector by breaking it into its Cartesian

components and examining each component separately (dropping the ET subscript):

〈 k2〉 = 〈 kx2〉 + 〈 ky2〉 + 〈 kz2〉 .

(5.5)

Since the ﬁeld is of the Bloch form (for a system periodic in x) and kB is restricted to the

ﬁrst Brillouin zone, it can be shown

〈 kx2〉

is bounded below by

k

2 B

.

Furthermore,

we

will

simply drop the 〈 ky2〉 term, which corresponds to horizontal conﬁnement of the

waveguide mode—we consider the best case for creating a guided mode, when it is very

weakly conﬁned in the plane (e.g., near the gap edge).

〈 k2〉

≥

k

2 B

+

〈 kz2〉

.

(5.6)

Up to this point, we have maintained a strict lower bound on f diel . Now, however, we

will make two approximations. The ﬁrst is based upon the observed mode proﬁle, such as

in Fig. 5.6 and Fig. 5.9. We are only interested in modes that are localized in z, so we

assume that the vertical mode proﬁle is a Gaussian with a standard deviation less than the

height h of the slab, and thus:

〈

kz2〉

≥

--1-h2

.

(5.7)

Second, in the case of the rod slab, we will assume f T to be nearly unity—that is, neglect

the longitudinal component of the ﬁeld. We justify this assumption by noting that the

ﬁelds of the rod modes are very similar to those of the two-dimensional structure [62], in

100