Photonic band-gap structures - · PDF filePhotonic band-gap structures E. Yablonovitch*...

Vol. 10, No. 2/February 1993/J. Opt. Soc. Am. B 283

Photonic band-gap structures

E. Yablonovitch*

Department of Electrical Engineering, University of California, Los Angeles, Los Angeles, California 90024-1594

Received June 17, 1992

The analogy between electromagnetic wave propagation in multidimensionally periodic structures and electron-wave propagation in real crystals has proven to be a fruitful one. Initial efforts were motivated by the prospectof a photonic band gap, a frequency band in three-dimensional dielectric structures in which electromagneticwaves are forbidden irrespective of the propagation direction in space. Today many new ideas and applicationsare being pursued in two and three dimensions and in metallic, dielectric, and acoustic structures. We reviewthe early motivations for this research, which were derived from the need for a photonic band gap in quantumoptics. This need led to a series of experimental and theoretical searches for the elusive photonic band-gapstructures, those three-dimensionally periodic dielectric structures that are to photon waves as semiconductorcrystals are to electron waves. We describe how the photonic semiconductor can be doped, producing tiny elec-tromagnetic cavities. Finally, we summarize some of the anticipated implications of photonic band structurefor quantum electronics and for other areas of physics and electrical engineering.

1. INTRODUCTION

In this paper we pursue the rather appealing analogy'2

between the behavior of electromagnetic waves in artifi-cial, three-dimensionally periodic, dielectric structuresand the rather more familiar behavior of electron waves innatural crystals.

These artificial two- and three-dimensionally periodicstructures we call photonic crystals. The familiar nomen-clature of real crystals is carried over to the electromag-netic case. This means that the concepts of reciprocalspace, Brillouin zones (BZ's), dispersion relations, Blochwave functions, Van Hove singularities, etc. must be ap-plied to photon waves. It then makes sense to speak ofphotonic band structure (PBS) and of a photonic recipro-cal space that has a BZ approximately 1000 times smallerthan the BZ of electrons. Because of the periodicity,photons can develop an effective mass, but this implica-tion is in no way unusual, since it occurs even in one-dimensionally periodic, optically layered structures. Wefrequently leap back and forth between the conventionalmeaning of a familiar concept such as conduction bandand its new meaning in the context of PBS's.

Under favorable circumstances a photonic band gap canopen up, a frequency band in which electromagnetic wavesare forbidden irrespective of propagation direction inspace. Inside a photonic band gap optical modes, sponta-neous emission, and zero-point fluctuations are all absent.Because of its promised utility in controlling the sponta-neous emission of light in quantum optics, the pursuit of aphotonic band gap has been a major motivation for study-ing PBS.

2. MOTIVATION

Spontaneous emission of light is a major natural phenome-non that is of great practical and commercial importance.For example, in semiconductor lasers spontaneous emis-sion is the major sink for threshold current and must besurmounted in order to initiate lasing. In heterojunction

bipolar transistors, which are all-electrical devices, spon-taneous emission nevertheless rears its head. In certainregions of the transistor current-voltage characteristic,spontaneous optical recombination of electrons and holesdetermines the heterojunction-bipolar-transistor currentgain. In solar cells, surprisingly, spontaneous emissionfundamentally determines the maximum available outputvoltage. We shall also see that spontaneous emission de-termines the degree of photon-number-state squeezing, animportant new phenomenon' in the quantum optics ofsemiconductor lasers. Thus the ability to control sponta-neous emission of light is expected to have a major effecton technology.

The easiest way to understand the effect of a photonicband gap on spontaneous emission is to take note of Fermi'sgolden rule. Consider the spontaneous-emission event il-lustrated in Fig. 1. The downward transition rate w be-tween the filled and the empty atomic levels is given by

W = A -V I 2p(E), (1)

where IVI is sometimes called the zero-point Rabi matrixelement and p(E) is the density of final states per unitenergy. In spontaneous emission the density of finalstates is the density of optical modes available to the pho-ton emitted in Fig. 1. If there is no optical mode avail-able, there will be no spontaneous emission.

Before the 1980's spontaneous emission was often re-garded as a natural and inescapable phenomenon, oneover which no control was possible. In spectroscopy itgave rise to the term natural linewidth. However, in1946, an overlooked note by Purcell 4 on nuclear spin-levelsalready indicated that spontaneous emissin could be con-trolled. In the early 1970's interest in this phenomenonwas reawakened by the surface-adsorbed dye moleculefluorescence studies of Drexhage.' Indeed, during themid-1970's Bykov6 proposed that one-dimensional perio-dicity inside a coaxial line could influence spontaneousemission. The modern era of inhibited spontaneous

0740-3224/93/020283-13$05.00 © 1993 Optical Society of America

E. Yablonovitch

284 J. Opt. Soc. Am. B/Vol. 10, No. 2/February 1993

hv

I I\/

Fig. 1. Spontaneous-emission event from a filled upper level toan empty lower level. The density of final states is the availablemode density for photons.

Co

CutoffFrequency

No ElectromagneticModes

kFig. 2. Electromagnetic wave dispersion between a pair of metalplates. The waveguide dispersion for one of the two polariza-tions has a cutoff frequency below which no electromagneticmodes and no spontaneous emission are allowed.

emission dates from the Rydberg-atom experiments ofKleppner. A pair of metal plates acts as a waveguide witha cutoff frequency for one of the two polarizations, asshown in Fig. 2. Rydberg atoms are atoms in high-lyingprincipal quantum-number states, which can sponta-neously emit in the microwave region of wavelengths.Hulet et al.7 shows that Rydberg atoms in a metallicwaveguide could be prevented from undergoing sponta-neous decay. There were no electromagnetic modes avail-able below the waveguide cutoff.

There is a problem with metallic waveguides, however.They do not scale well into optical frequencies. At highfrequencies metals become more and more lossy. Thesedissipative losses allow for virtual modes, even at frequen-cies that would normally be forbidden. Therefore itmakes sense to consider structures made of positive-dielectric-constant materials, such as glasses and insula-tors, rather than metals. These materials can have lowdissipation, even all the way up to optical frequencies.This property is ultimately exemplified by optical fibers,which permit light propagation over many kilometers withnegligible losses. Such positive-dielectric-constant mate-rials can have an almost purely real dielectric responsewith low resistive losses. If these materials are arrayedinto a three-dimensionally periodic dielectric structure, aphotonic band gap should be possible, employing a purelyreal, reactive, dielectric response.

The benefits of such a photonic band gap for direct-gapsemiconductors are illustrated in Fig. 3. On the right-

hand side of Fig. 3 is a plot of the photonic dispersion,(frequency versus wave vector). On the left-hand side ofFig. 3, sharing the same frequency axis, is a plot of theelectronic dispersion, showing conduction and valencebands appropriate to a direct-gap semiconductor. Sinceatomic spacings are 1000 times shorter than optical wave-lengths, the electron wave vector must be divided by 1000to fit on the same graph with the photon wave vectors.The dots in the electron conduction and valence bands aremeant to represent electrons and holes, respectively. Ifan electron were to recombine with a hole, they would pro-duce a photon at the electronic-band-edge energy. As isillustrated in Fig. 3, if a photonic band gap were tostraddle the electronic band edge, then the photon pro-duced by electron-hole recombination would have no placeto go. The spontaneous radiative recombination of elec-trons and holes would be inhibited. As can be imagined,this has far-reaching implications for semiconductor pho-tonic devices.

One of the most important applications of spontaneous-emission inhibition is likely to be the enhancement ofphoton-number-state squeezing, which has been playingan increasing role in quantum optics lately. The form ofsqueezing introduced by Yamamoto3 is particularly ap-pealing in that the active element that produces thesqueezing effect is none other than the common resistor,as is shown in Fig. 4. When an electrical current flows, itgenerally carries the noise associated wih the graininessof the electron charge, called shot noise. The correspond-ing mean-square current fluctuations are

((Ai) 2 ) = 2eiAf, (2)

where i is the average current flow, e is the electroniccharge, and Af is the noise bandwidth. While Eq. (2) ap-plies to many types of random physical process, it is far

ElectronicBand Gap

k

1000

co

k

Electronic .- -.o,. PhotonicDispersion Dispersion

Fig. 3. Right-hand side, the electromagnetic dispersion, with aforbidden gap at the wave vector of the periodicity. Left-handside, the electron wave dispersion typical of a direct-gap semicon-ductor; the dots represent electrons and holes. Since the pho-tonic band gap straddles the electronic band edge, electron-holerecombination into photons is inhibited. The photons have noplace to go.

-

E. Yablonovitch


ilt

<Ai2> _2eidf

Fig. 4. In a good-quality metallic resistor the current flow isquite regular, producing negligible amounts of shot noise.

from universal. Equation (2) requires that the passage ofelectrons in the current flow be a random Poissonian pro-cess. As early as 1954 Van der Ziel,5 in an authoritativebook called Noise, pointed out that good-quality metal-film resistors, when they carry a current, generally exhibitmuch less noise than predicted by Eq. (2). Apparently theflow of electrons in the Fermi sea of a metallic resistorrepresents a highly correlated process. This is far frombeing a random process: the electrons apparently senseone another, producing a level of shot noise far belowEq. (2) (so low as to be difficult to measure and to distin-guish from thermal or Johnson noise). Sub-Poissonianshot noise entails the following: Suppose that the aver-age flow consists of 10 electrons per nanosecond. Forrandom flow, the count in successive nanoseconds could beanywhere from 8 to 12 electrons. With good-qualitymetal-film resistors, the electron count would be 10 foreach and every nanosecond.

Yamamoto put this property to good use by driving ahigh-quantum-efficiency laser diode with such a resistor,as shown in Fig. 5. Suppose that the laser diode quantumefficiency for emission into the cavity mode were 100%.Then for each electron that passes through the resistorthere would be one photon emitted into the laser cavitymode. A correlated stream of photons is produced withproperties that are unprecedented since the initial exposi-tion of Einstein's photoelectric effect. If the photons areused for optical communication, then a receiver woulddetect exactly 10 photoelectrons each nanosecond. If 11photons were detected, then the deviation would be nomere random fluctuation but would represent an inten-tional signal. Thus information in an optical communi-cations signal could be encoded at the level of individualphotons. The name photon-number-state squeezing is as-sociated with the fixed photon number per time interval.Expressed differently, the bit-error rate in optical commu-nication can be diminished by squeezing.

There is a limitation to the squeezing, however. Thequantum efficiency for propagation into the lasing mode isnot 100%. The 47r-sr outside the cavity mode can capturea significant amount of random spontaneous emission. Ifunwanted electromagnetic modes were to capture 50% ofthe excitation, then the maximum noise reduction insqueezing would be only 3 dB. Therefore it is necessaryto minimize the spontaneous recombination of electronsand holes into modes other than the laser mode. If suchrandom spontaneous events were reduced to 1%, permit-ting 99% quantum efficiency into the lasing mode, the cor-responding noise reduction would be 20 dB, which is well

worth fighting for. Thus we see that control of sponta-neous emission is essential for deriving the full benefitfrom photon-number-state squeezing.

We have advocated the study of photonic band structurefor its applications in quantum optics and optical commu-nications. Positive dielectric constants and fully three-dimensional forbidden gaps were emphasized. It is nowclear that the generality of the concept of the artificial,multidimensional band structure allows for other types ofwaves, other materials, and various lower dimensional ge-ometries, limited only by imagination and need. Some ofthese ideas are being presented in other papers in thisJournal of the Optical Society of America B feature onphotonic band gaps.

3. SEARCH FOR THE PHOTONIC BAND GAP

Having decided to create a photonic band gap in threedimensions, we need to settle on a three-dimensionally pe-riodic geometry. For electrons the three-dimensionalcrystal structures come from nature. Several hundredyears of mineralogy and crystallography have classifiedthe naturally occurring three-dimensionally periodic lat-tices. For photonic band gaps we must create an artifi-cial structure by using our imagination.

The face-centered-cubic (fcc) lattice appears to be fa-vored for photonic band gaps and was suggested indepen-dently by John2 and by me' in our initial proposals. Letus consider the fcc BZ as illustrated in Fig. 6. Variousspecial points on the surface of the BZ are marked. Clos-est to the center is the L point, oriented toward the bodydiagonal of the cube. Farthest away is the W point, a ver-tex where four plane waves are degenerate (which willcause problems below). In the cubic directions are thefamiliar X points.

electron - good-quality resistorsflow is i I have little or no

correlated + shot noise

hv

stimulated0

hvFig. 5. High-quantum-efficiency laser diode, which converts thecorrelated flow of electrons from a low-shot-noise resistor intophoton-number-state squeezed light. Random spontaneous emis-sion outside the desired cavity mode limits the attainable noisereduction.

E. Yablonovitch


Fig. 6. Fcc BZ in reciprocal space.

I II II I

The photonic band gap is different from the idea of aone-dimensional stop band as understood in electrical en-gineering. Rather, the photonic band gap should be re-garded as a stop band with a frequency overlap in all47r-sr of space. The earliest antecedent to photonic bandstructure, dating to 1914 and Sir Lawrence Bragg,"0 is thedynamic theory of x-ray diffraction. Nature gives us fcccrystals, and x-rays are bona fide electromagnetic waves.As early as 1914 narrow stop bands were known to openup. Therefore, what was missing?

The refractive-index contrast for x rays is tiny, gener-ally 1 part in 104. The forbidden x-ray stop bands formextremely narrow rings on the facets of the BZ. As theindex contrast is increased, the narrow forbidden ringsopen up, eventually covering an entire facet of a BZ andultimately all directions in reciprocal space. We shall seethat this requires an index contrast -2. The high indexcontrast is the main new feature of PBS's beyond dynamicx ray diffraction. In addition, we shall see that electro-magnetic wave polarization, which is frequently over-looked for x rays, will play a major role in PBS.

In approaching this subject we adopted an empiricalview-point. We decided to make photonic crystals on thescale of microwaves, and then we tested them by usingsophisticated coherent microwave instruments. The testsetup, shown in Fig. 9, is what we call in optics a Mach-

BCC[If ~~~~~~II

L X k

Fig. 7. Forbidden gap (shaded) at the L point, which is centeredat a frequency 14% lower than the X-point forbidden gap.Therefore it is difficult to create a forbidden frequency band thatoverlaps all points along the surface of the BZ.

Consider a plane wave in the X direction. It will sensethe periodicity in the cubic direction, forming a standingwave and opening a forbidden gap as indicated by theshading in Fig. 7. Suppose, on the other hand, that theplane wave is going in the L direction. It will sensethe periodicity along the cubic-body diagonal, and a gapwill form in that direction as well. But the wave vector tothe L point is 14% smaller than the wave vector to the Xpoint. Therefore the gap at L is likely to be centered at a14% smaller frequency than the gap at X. If the two gapsare not wide enough, they will not overlap in frequency.In Fig. 7, as shown, the two gaps barely overlap. This isthe main problem in achieving a photonic band gap. It isdifficult to ensure that a frequency overlap is ensured forall possible directions in reciprocal space.

The lesson from Fig. 7 is that the BZ should most closelyresemble a sphere in order to increase the likelihood of afrequency overlap in all directions of space. Therefore letus look at the two common BZ's in Fig. 8, the fcc BZ andthe body-centered-cubic (bec) BZ. The bcc BZ has pointyvertices, which make it difficult for us to achieve a fre-quency overlap in all directions. Likewise, most othercommon BZ's deviate even further from a spherical shape,Among all the common BZ's the fcc has the least percent-age deviation from a sphere. Therefore, until now allphotonic band gaps in three dimensions have been based9

on the fcc lattice.

Fig. 8. Two common BZ's for bcc and fcc. The fcc case deviatesleast from a sphere, favoring a common overlapping band in alldirections of space.

X-Y RECORDERFig. 9. Homodyne detection system for measuring phase and am-plitude in transmission through the photonic crystal under test.A sweep oscillator feeds a 10-dB splitter. Part of the signal ismodulated (MOD) and then propagated as a plane wave throughthe test crystal. The other part of the signal is used as localoscillator for the mixer (MXR) to measure the amplitude changeand phase shift in the crystal. Between the mixer and the X-Yrecorder is a lock-in amplifier (not shown).

CO

FCC

E. Yablonovitch

IIIIIIII

IIIIIII I


(a)

(b)

Fig. 10. WS real-space unit cell of the fcc lattice, a rhombic do-decahedron. (a) Slightly oversized spherical voids are inscribedinto the unit cell, breaking through the faces as illustrated. Thisis the WS cell, corresponding to the photograph in Plate II. (b)WS cell structure with a photonic band gap. Cylindrical holesare drilled through the top three facets of the rhombic dodecahe-dron and pass through the bottom three facets. The resultingatoms are roughly cylindrical and have a preferred axis in thevertical direction. This WS cell corresponds to the photographin Plate III.

Zehnder interferometer. It is capable of measuring phaseand amplitude in transmission through the microwave-scale photonic crystal. In principle one can determine thefrequency versus the wave-vector dispersion relationsfrom such coherent measurements. We used a powerfulcommercial instrument for this purpose, the HP8510 net-work analyzer. Our approach in the experiments was tomeasure the forbidden gap in all possible internal direc-tions of reciprocal space. Accordingly the photonic crys-tal was rotated and the transmission measurementsrepeated. Because of wave-vector matching along thesurface of the photonic crystal, some internal angles couldnot be reached. To overcome this problem, large micro-wave prisms, made of polymethyl methacrylate, wereplaced on either side of the test crystal in Fig. 9.

Early the question arose: Of what material should thephotonic crystal be made? The larger the refractive-indexcontrast, the easier it would be to find a photonic bandgap. In optics, however, the largest practical index con-trast is that of the common semiconductors Si and GaAs,with a refractive index n = 3.6. If that index were inade-quate, then photonic crystals would probably never fulfillthe goal of being useful in optics. Therefore we decidedto restrict the microwave refractive index to 3.6 and the

microwave dielectric constant to n2= 12. A commercial

microwave material, Emerson & Cumming Sycast 12, wasparticularly suited to the task, since it was machinablewith carbide tool bits. Any photonic band structure thatwas found in this material could simply be scaled down insize and would have identical dispersion relations at opti-cal frequencies and optical wavelengths.

With regard to the geometry of the photonic crystal,there is a universe of possibilities. So far the only re-striction that we have made is the choice of fcc lattices. Itturns out that a crystal with an fcc BZ in reciprocal space,as shown in Fig. 6, is composed of fcc Wigner-Seitz (WS)unit cells in real space, as shown in Fig. 10. The problemof creating an arbitrary fcc dielectric structure reduces tothe problem of filling the fcc WS real-space unit cell withan arbitrary spatial distribution of dielectric material.Real space is then filled by repeated translation and closepacking of the WS unit cells. The decision before us iswhat to put inside the fcc WS cells. There are an infinitenumber of possible fcc lattices, since anything can be putinside the fundamental repeating unit. The problem:What do we put inside the fcc WS unit cell in Fig. 10?

The question provoked strenuous difficulties and falsestarts over a period of several years before finally beingsolved. In the first years of this research we were un-aware of how difficult the search for a photonic band gapwould be. A number of fcc crystal structures were pro-posed, each representing a different choice for filling therhombic dodecahedron fcc WS cells in real space. Forexample, the first suggestion' was to make a three-dimensional checkerboard as in Fig. 11, in which cubeswere inscribed inside the fcc WS real-space cells in Fig. 10.Later the experiments" adopted spherical "atoms" cen-tered inside the fcc WS cell. Plate I is a photograph ofsuch a structure in which the atoms are precision Al203spheres, n - 3.06, each -6 mm in diameter. The spheresare supported by a thermal-compression-molded bluefoam material of dielectric constant near unity. Thereare roughly 8000 atoms in Plate I. This structure wastested at a number of filling ratios, from close packing tohighly dilute. Nevertheless, it always failed to produce aphotonic band gap.

Then we tested the inverse structure, in which sphericalvoids were inscribed inside the fcc WS real space cell.

Actve m 3 1 a rnrn 3ELayer

n2eRR

n

Fig. 11. Fcc crystal, in which the individual WS cells are in-scribed with cubes stacked in a three-dimensional checkerboard.

E. Yablonovitch


8000atoms! A}~~~~~~

Fig. 12. Construction of fcc crystals, consisting of sphericalvoids. Hemispherical holes are drilled on both faces of a dielec-tric sheet. When the sheets are stacked up, the hemispheresmeet, producing a fcc crystal.

These could be easily fabricated by drilling hemispheresonto the opposite faces of a dielectric sheet with a spheri-cal drill bit, as shown in Fig. 12. When the sheets werestacked so that the hemispheres faced one another, theresult was an fcc array of spherical voids inside a dielec-tric block. These blocks were also tested over a widerange of filling ratios by progressively increasing the di-ameter of the hemispheres. These also failed to producea photonic band gap.

The typical failure mode is illustrated in Fig. 13. Asexpected, the conduction band at the L point falls at a lowfrequency, while the valence band at the W point falls at ahigh frequency. The overlap of the bands at L and Wresults in a band structure that is best described assemimetallic.

The empirical search for a photonic band gap led no-where until we tested the structure shown in Plate II.This is the spherical-void structure, with oversized voidsbreaking through the walls of the WS unit cell as shown inFig. 10(a). For the first time the measurements seemedto indicate a photonic band gap, and we published" theband structure shown in Fig. 14. There appeared to be anarrow gap, centered at 15 GHz and forbidden for bothpossible polarizations. Unbeknownst to us, however,Fig. 14 harbored a serious error. Instead of a gap at theW point, the conduction and the valence bands crossed atthat point, allowing the bands to touch. This produced apseudogap with zero density of states but no frequencywidth. The error arose because of the limited size of thecrystal. The construction of crystals with _104 atomsrequired tens of thousands of holes to be drilled. Sucha three-dimensional crystal was still only 12 cubicunits wide, limiting the wave-vector resolution and re-stricting the dynamic range in transmission. Under

these conditions it was experimentally difficult to notice aconduction-valence band degeneracy that occurred at anisolated point in k space, such as the W point.

While we were busy with the empirical search, theoristsbegan serious efforts to calculate the PBS. The mostrapid progress was made not by specialists in electromag-netic theory but by electronic-band-structure (EBS)theorists, who were accustomed to solving Schrodinger'sequation in three-dimensionally periodic potentials. Theearly calculations 2 "15 were unsuccessful, however. Asa short cut the theorists treated the electromagetic fieldas a scalar, much as is done for electron waves inSchrodinger's equation. The scalar wave theory of pho-tonic band structure did not agree well with experiment.For example, it predicted photonic band gaps in thedielectric-sphere structure of Plate I, whereas none wereobserved experimentally. The approximation of Maxwell'sequations as a scalar wave equation was not working.Finally, when the full vector Maxwell equations wereincorporated, theory began to agree with experiment.Leung 6 was probably the first to publish a successful vec-tor wave calculation in PBS, followed by others 7 8 withsubstantially similar results. The theorists agreed wellwith one another, and they agreed well with experiment"except at the high-degeneracy points U and particularlyW What the experiment failed to reveal was the degener-ate crossing of valence and conduction bands at thosepoints.

The unexpected pseudogap in the crystal of Plate IItriggered concern and a search for a way to overcome theproblem. A worried editorial'9 was published in Nature.But even before the editorial appeared, the problem had

50% VOLUME FRACTION fcc AIR SPHERES

X rn1-= 1.6no

PREDOMINANTLY P POLARIZEDFig. 13. Typical semimetallic band structure for a photonic crys-tal with no photonic band gap. An overlap exists between theconduction band at L and the valence band at W

E. Yablonovitch

Vol. 10, No. 2/February 1993/J. Opt. Soc. Am. B

Plate I. Photograph of a three-dimensional fcc crystal consisting Plate II. Photograph of the photonic crystal corresponding toof A12 03 spheres of refractive index 3.06. The dielectric spheres Fig. 10(a), which had only a pseudogap rather than a full photonic

are supported in place by the blue foam material of refractive in- band gap. The spherical voids were closer than close-packed,dex 1.01. These spherical-dielectric-atom structures failed to overlapping and allowing holes to pass through as shown.show a photonic band gap at any volume fraction.

Plate III. Top-view photograph of the nonspherical-atom structure of WS unit cells as shown in Fig. 10(a), constructed by the method ofFig. 15.

E. Yablonovich


X U r X W K

X U L F x w K

Fig. 14. Purported PBS of the spherical-void structure shown inFigs. 10(a) and Plate II. The rightward-sloping lines representpolarization parallel to the X plane, while the leftward-slopinglines represent the orthogonal polarization, which has a partialcomponent out of the X plane. The cross-hatched region is thereported photonic band gap. This figure fails to show the cross-ing of the valence and conduction bands at the W point, whichwas first discovered by theory.' 6

already been solved by the Iowa State group of Hoet al.'8 The degenerate crossing at the W point washighly susceptible to changes in symmetry of the struc-ture. If one lowered the symmetry by filling the WS unitcell, not by a single spherical atom but by two atoms posi-tioned along the (111) direction, as in the diamond struc-ture, then a full photonic band gap opened up. Theirdiscovery of a photonic band gap in the diamond structureis particularly significant, since the diamond geometryseems to be favored by Maxwell's equations. A form ofthe diamond structure" gives the widest photonic bandgaps, requiring the least index contrast, n 1.87.

More generally, one can lower the spherical-void symme-try in Fig. 10(a) by distorting the spheres along the (111)direction, lifting the degeneracy at the W point. The WSunit cell in Fig. 10(b) has great merit for this purpose.Holes are drilled through the top three facets of the rhom-bic dodecahedron and exit through the bottom threefacets. The beauty of the structure in Fig. 10(b) is that astacking of WS unit cells results in straight holes thatpass through the entire crystal. The atoms are odd-shaped, roughly cylindrical voids centered in the WS unitcell with a preferred axis pointing to the top vertex, (111).An operational illustration of the construction that pro-duces an fcc crystal of such WS unit cells is shown inFig. 15.

A slab of material is covered by a mask that contains atriangular array of holes. Three drilling operations areconducted through each hole, 35.26° off normal incidenceand spread out 1200 on the azimuth. The resulting criss-cross of holes below the surface of the slab produces a fullythree-dimensionally periodic fcc structure with the WSunit cells given by Fig. 10(b). The drilling can be done by

a real drill bit for microwave work or by reactive ion etch-ing to create an fcc structure at optical wavelengths. Atop-view photograph of the microwave structure is shownin Plate III.

In spite of the nonspherical atoms in Fig. 10(b), the BZis identical to the standard fcc BZ shown in textbooks.Nevertheless, we have chosen an unusual perspective fromwhich to view the BZ in Fig. 16. Instead of having the fccBZ resting on one of its diamond-shaped facets, as is usu-ally done, we have chosen in Fig. 16 to present it restingon a hexagonal face. Since there is a preferred axis forthe atoms, the distinctive L points centered in the top andbottom hexagons are threefold symmetry axes and arelabeled L3. The L points centered in the other sixhexagons are symmetric only under a 3600 rotationand are labeled L1. It is helpful to know that the U3 andK3 points are equivalent, since they are a reciprocal lat-tice vector apart. Likewise, the U, and K, points areequivalent.

Figure 17 shows the dispersion relations along differentmeridians for our primary experimental sample of nor-malized hole diameter d/a = 0.469 and 78% volume frac-tion removed (where a is the unit cube length). The ovalsrepresent experimental data with s polarization (perpen-dicular to the plane of incidence, parallel to the slab sur-face), while the triangles represent p-polarization data(parallel to the plane of incidence, partially perpendicularto the slab surface). The horizontal abscissa in the lowergraph of Fig. 17, L3-K3-Ll-U3-X-U3-L3, represents a full

Fig. 15. Method for constructing a fcc lattice of the WS cellsshown in Fig. 10(b). A slab of material is covered by a mask thatconsists of a triangular array of holes. Each hole is drilledthrough three times at an angle 35.260 away from normal andspread 1200 on the azimuth. The resulting crisscross of holesbelow the surface of the slab, suggested by the cross-hatchingshown here, produces a fully three-dimensionally periodic fccstructure with unit cells as given by Fig. 10(b). The drilling canbe done by a real drill bit for microwave work or by reactive ionetching to create fcc structure at optical wavelengths.

E. Yablonovitch


u3

U3

Fig. 16. BZ of a fc structure, incorporating nonspherical atomsas in Fig. 10(b). Since the space lattice is not distorted, this issimply the standard fcc BZ lying on a hexagonal face rather thanthe usual cubic face. Only the L points on the top and bottomhexagons are threefold symmetry axes. Therefore they arelabeled L3. The L points on the other six hexagons are labeledL,. The U3 and K3 points are equivalent, since they are a recip-rocal lattice vector apart. Likewise, the Ui and K1 points areequivalent.

, 0.6

0.5 +~

0.4

LU

Z 03 6 L 3 3 L

ciU 0.2U-

0.1

Fg17reqec 3esswv etr( essk iseio

07

ao 0.5 sufce o

C)

LU

L 0.2-U-

0.1L K3 L U3 X U3 L

Fig. 17. Frequency versus wave vector ( versus k) dispersionalong the surface of the BZ shown in Fig. 16, where ca is thespeed of light divided by the fcc cube length. The ovals and thetriangles are the experimental points for s and p polarization,respectively. The solid and dashed curves are the calculationsfor s and p polarization, respectively. The dark shaded band isthe totally forbidden band gap. The lighter shaded stripes aboveand below the dark band are forbidden only for s and p polariza-tion, respectively.

meridian from the north pole to the south pole of the BZ.Along this meridian the Bloch wave functions separatenearly into s and p polarizations. The s- and p-polarizedtheory curves are the solid and dashed curves, respec-tively. The dark shaded band is the totally forbidden pho-tonic band gap. The lighter shaded stripes above andbelow the dark band are forbidden only for s and p polar-ization, respectively.

At a typical semiconductor refractive index, n = 3.6, thethree-dimensional forbidden gap width is 19% of its centerfrequency. Calculations2 indicate that the gap remainsopen for refractive indices as low as n = 2.1 for circularholes as in Fig. 15. We have also measured the imaginarywave-vector dispersion within the forbidden gap. Atmidgap we find an attenuation of 10 dB per unit cubelength a. Therefore the photonic crystal need not bemany layers thick to expel the zero-point electromagneticfield effectively. The construction of Fig. 15 can be im-plemented by reactive ion etching, as shown in Fig. 18.In reactive ion etching, the projection of circular maskopenings at 350 leaves oval holes in the material, whichmight not perform as well. Fortunately it was found,2 indefiance of Murphy's law, that the forbidden gap width foroval holes is actually improved by fully 21.7% of its centerfrequency.

4. DOPING THE PHOTONIC CRYSTALThe perfect semiconductor crystal is quite elegant andbeautiful, but it becomes ever more useful when it isdoped. Likewise the perfect photonic crystal can becomeof even greater value when a defect2 2 is introduced.

Lasers, for example, require that the perfect three-dimensional translational symmetry be broken. Eventhough spontaneous emission from all 4 sr should be in-hibited, a local electromagnetic mode, linked toa defect, to accept the stimulated emission is still neces-sary. In one-dimensional distributed-feedback lasers23 aquarter-wavelength defect is introduced, effectively form-ing a Fabry-Perot cavity as shown in Fig. 19. In three-dimensional PBS a local defect-induced structureresembles a Fabry-Perot cavity, except that it reflects ra-diation back upon itself in all 47r spatial directions.

The perfect three-dimensional translational symmetryof a dielectric structure can be altered in one of two ways.(1) Extra dielectric material may be added to one of theunit cells. We find that such a defect behaves much like adonor atom in a semiconductor. It gives rise to donormodes, which have their origin at the bottom of the con-duction band. (2) Conversely, translational symmetry

microfabrication byreactive ions

14144rotatingstage

Fig. 18. Construction of the nonspherical-void photonic crystalof Figs. 10(b) and 15-17 and of Plate III by reactive ion etching.

E. Yablonovitch


2

Left Mirror A Right Mirror

PhaseSlip

Fig. 19. One-dimensional Fabry-Perot resonator, made of multi-layer dielectric mirrors with a space of one half-wavelength be-tween the left- and right-hand mirrors. The net effect is tointroduce a quarter-wavelength phase slip defect into the overallperiodic structure. A defect mode is introduced at midgap.

< 1,1,1 >

t

Fig. 20. (1, 1, 0) Cross-sectional view of our fcc photonic crystal,consisting of nonspherical air atoms centered on the large dots.Dielectric material is represented by the shaded areas. Thedashed rectangle is a face-diagonal cross section of the unit cube.Donor defects consisted of a dielectric sphere centered on anatom. We selected an acceptor defect as shown centered in theunit cube. It consists of a missing horizontal slice in a singlevertical rib.

can be broken by removal of some dielectric material fromone of the unit cells. Such defects resemble acceptoratoms in semiconductors. The associated acceptor modeshave their origin at the top of the valence band. We shallfind that acceptor modes are particularly well suited toact as laser microresonator cavities. Indeed, it appearsthat photonic crystals made of sapphire or other low-lossdielectrics will make the highest-Q single-mode cavities(of modal volume -1A3), covering electromagnetic fre-quencies above the useful working range of superconduct-ing metallic cavities. The short-wavelength limit in theultraviolet is set by the availability of optical materialswith refractive index -2, the threshold index 8 '2 ' for theexistence of a photonic band gap.

Figure 20 is a (1, 1, 0) cross section of our photonic crys-tal of Fig. 10(b) of Figs. 15-17, and of Plate III, shown as acut through the center of a unit cube. Shading representsdielectric material. The large dots are centered on the airatoms, and the rectangular dashed line is a face-diagonalcross section of the unit cube. Since we could design thestructure at will, donor defects were chosen to consist of asingle dielectric sphere centered in an air atom. Like-wise, by breaking one of the interconnecting ribs, it is

easy to create acceptor modes. We selected an acceptordefect, as shown in Fig. 20, centered in the unit cube. Itconsists of a vertical rib that has a missing horizontalslice.

The heart of our experimental apparatus is a photoniccrystal embedded in microwave absorbing pads as shownin Fig. 21. The photonic crystals were 8-10 atomic layersthick in the (1, 1, 1) direction. Monopole antennas, con-sisting of 6-mm pins, coupled radiation to the defectmode. The HP 8510 network analyzer was set up to mea-sure transmission between the antennas. Figure 22(a)shows the transmission amplitude in the absence of a de-fect. There is very strong attenuation (-10-s) between13 and 16 GHz, marking the valence- and conduction-band edges of the forbidden gap. This is a tribute to boththe dynamic range of the network analyzer and the sizableimaginary wave vector in the forbidden gap.

A transmission spectrum in the presence of an acceptordefect is shown in Fig. 22(b). Most of the spectrum isunaffected, except at the electromagnetic frequencymarked deep acceptor within the forbidden gap. At thatprecise frequency, radiation hops from the transmittingantenna to the acceptor mode and then to the receivingantenna. The acceptor-level frequency, within the forbid-den gap, is dependent on the volume of material removed.Figure 23 shows the acceptor-level frequency as a func-tion of the defect volume removed from one unit cell.When a relatively large volume of material is removed, theacceptor level is deep, as is shown in Fig. 22(b). Asmaller amount of material removed results in a shallowacceptor level nearer the valence band. If the removedmaterial volume falls below a threshold volume, the accep-

HP 8510 NETWORK ANALYZER

MONOPOLE ANTENNAS

CO-AX CO-AX

ABSORBING PADS

CRYSTAL

Fig. 21. Experimental configuration for the detection of localelectromagnetic modes in the vicinity of a lattice defect. Trans-mission amplitude attenuation from one antenna to the other ismeasured. At the local mode frequency the signal hops by meansof the local mode in the center of the photonic crystal, producing alocal transmission peak. The signal propagates in the (1,1,1) di-rection through 8-10 atomic layers. Co-Ax, coaxial line.

E. Yablonovitch


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Frequency in GHzFig. 22. (a) Transmission attenuation through a defect-free pho-tonic crystal as a function of microwave frequency. The forbid-den gap falls between 13 and 16 GHz. (b) Attenuation through aphotonic crystal with a single acceptor in the center. The rela-tively large acceptor defect volume shifted its frequency to nearmidgap. The electromagnetic resonator Q was 1000, limitedonly by the loss tangent of the dielectric material. (c) Attenu-ation through a photonic crystal with a single donor defect, anoff-center dielectric sphere, leading to two shallow donor modes.

tor level falls within the continuum of levels below the topof the valence band, becoming metastable.

On an expanded frequency scale we can measure theresonator Q of the deep acceptor mode, which is Q - 1000,as limited by the loss tangent of the Emerson & CummingStycast material of which the photonic crystal was made.

The behavior of an off-center donor defect is shown inFig. 22(c). In this case the donor volume was only slightlyabove the required threshold for forming bound donormodes. Already two shallow donor modes can be seen inFig. 22(c). When the donor is centered in the WS unitcell, the two modes merge to form a doubly degeneratedonor level as in Fig. 23. Single donor defects seem to

produce multiple donor levels. Figure 23 gives the donor-level frequency as a function of donor volume. As in thecase of acceptors, there is a threshold defect volumerequired for the creation of bound modes below theconduction-band edge. However, the threshold volumefor donor defects is almost 10 times larger than the accep-tor threshold volume. Apparently this difference is due tothe electric-field concentration in the dielectric ribs at thetop of the valence band. Bloch wave functions at the topof the valence band are rather easily disrupted by themissing rib segment.

We have chosen in Fig. 23 to normalize the defect vol-ume to a natural volume of the physical system, (A/2n)3,which is basically a cubic half-wavelength in the dielectricmedium. More specifically, A is the vacuum wavelengthat the midgap frequency and n is the refractive indexof the dielectric medium. Since we are measuring a di-electric volume, it makes sense to normalize to a half-wavelength cube as measured at the dielectric refractiveindex. Based on the reasonable scaling of Fig. 23, ourchoice of volume normalization would seem justified.

The vertical rib with a missing horizontal slice, shownin Fig. 20, can be readily microfabricated. It should bepossible to create it in III-V materials by growing an Al-rich epitaxial layer and lithographically patterning itdown to a single dot that is the size of one of the verticalribs. After regrowth of the original III-V compositionand reactive ion etching of the photonic crystal, HF acidetching2 4 with a selectivity -108 will be used to removethe Al-rich horizontal slice from the one rib that containssuch a layer. The resonant frequency of the microcavitycan be controlled by the thickness of the Al-rich sacrificiallayer.

Therefore, by doping the photonic crystal, it is possibleto create high-Q electromagnetic cavities whose modalvolume is less than a half-wavelength cubed. Thesedoped photonic crystals would be similar to metallic mi-crowave cavities, except that they would be usable athigher frequencies, where metal cavity walls would be-come lossy. With sapphire as a dielectric, for example, itshould be possible to make a millimeter-wave cavity withQ 10'. The idea is not to compete directly with super-conducting cavities but rather to operate a higher frequen-cies, where the superconductors become lossy. Given therequirement for a refractive index >2, doped photoniccrystals should work well up to ultraviolet wavelengths atwhich diamond crystals and TiO2 are still transparent.

5. APPLICATIONS

The forthcoming availability of single-mode microcavitiesat optical frequencies will lead to a new situation in quan-tum electronics. Of course microwave cavities that con-tain a single electromagnetic mode have been known for along time. At microwave frequencies, however, sponta-neous emission of electromagnetic radiation is a weak andunimportant process. At optical frequencies spontaneousemission comes into its own. Now we can combine thephysics and technology of spontaneous emission with thecapability for single-mode microcavities at optical fre-quencies, where spontaneous emission is important. Thiscombination is fundamentally a new regime in quantumelectronics.

E. Yablonovitch


. CONDUCTION BAND EDGE

0

.

- 0

0 0

- 0

o S

0

0

o VALENCE BAND EDGE: - - -_ - - - - - - - - - - - - - - - - - - - - --_ _-_ _-_ _-_- -_- -_- -_- _ _

0 0o ACCEPTOR MODE

* DONOR MODE(DOUBLY DEGENERATE)

I I I

1 2 3 4 5 6 7

DEFECT VOLUME (IN UNITS OF [ /2n] 3)

Fig. 23. Donor and acceptor mode frequencies as a function of normalized donor and acceptor defect volume. The points are experimen-tal, and the corresponding curves are calculated. Defect volume is normalized to (A/2n)3 , where A is the midgap vacuum wavelength andn is the refractive index. A finite defect volume is necessary to bind a mode in the forbidden gap.

The major example of this new type of device is thesingle-mode light-emitting diode (SM-LED), which canhave many of the favorable coherence properties of laserswhile being a more reliable and thresholdless device.Progress in electromagnetic microcavities permits all thespontaneous emission of an LED to be funneled into asingle electromagnetic mode.

As the interest in the low-threshold semiconductor laserdiode has grown, e.g., for optical interconnects, its sponta-neously luminescent half-brother, the LED, has begun toreemerge in a new form. In this new form the LED issurrounded by an optical cavity. The idea is for the opti-cal cavity to make available only a single electromagneticmode for the output spontaneous emission from the semi-conductor diode. In fact the figure of merit for such acavity is 6, the fraction of spontaneous emission that isfunneled into the desired mode. What is new for this ap-plication is the prospective ability to make high-, cavitiesat optical frequencies that employ photonic crystals. Thethree-dimensional character of the cavities ensures thatspontaneous emission will not seek out those neglectedmodes that are found propagating in a direction wherethis is no optical confinement.

With all the spontaneous emission funneled into a singleoptical mode, the SM-LED can begin to have many of thecoherence and statistical properties normally associatedwith above-threshold lasing. The essential point is thatthe spontaneous-emission factor f3 should approach unity.(A closely related concept is that of the zero-thresholdlaser, in which the high-spontaneous-emission factor pro-duces a soft and indistinct threshold characteristic in thecurve of the light output versus the current input for laserdiodes.) The idea is to combine the advantages of theLED, which is thresholdless and highly reliable, with

those of the semiconductor laser, which is coherent andefficient.

The coherence properties of the SM-LED are illustratedin Fig. 24. In a laser single-mode emission is the result ofgain saturation and mode competition. In the SM-LEDthere is no gain and therefore no gain saturation, but the

Single-Mode Light-Emitting Diode

modulationMonochromatic speed > 10 GHz

Directional photonnumber-state

squeezedLight

'th Current

Fig. 24. Properties of the SM-LED, whose cavity is representedby the small circle inside the rectangular photonic crystal at left.The words Monochromatic and Directional represent the tempo-ral and spatial coherence of the SM-LED output, as is explainedin the text. The modulation speed can be >10 GHz, and thedifferential quantum efficiency can be >50%, which is competi-tive with that of laser diodes. But there is no threshold currentfor the SM-LED, as indicated by the curves for light outputversus the input current at the bottom. The regular streamof photoelectrons, e's, is meant to represent photon-number-state squeezing, which can be produced by the SM-LED if thespontaneous-emission factor P of the cavity is high enough.

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Table 1. Summary of Differences and Similaritiesbetween Photonic and Electronic Band StructuresCharacteristic EBS PBS

Underlying Parabolic Lineardispersionrelation

Angular Spin 1/2 Spin 1Momentum scalar wave vector wave

approximation character

Accuracy Approximate Essentiallyof band owing to exacttheory electron-electron

interactions

output is still single mode because only one mode is avail-able for emission. Since a single spatial mode can alwaysbe mode converted into a plane wave, the SM-LED can beregarded as having spatial coherence.

What about temporal coherence? The spectral line-width of the SM-LED is narrower than the luminescenceband of the semiconductor. All the radiation is funneledinto the narrow spectral band determined by the micro-cavity Q. Thus SM-LED's have both spatial and temporalcoherence, as represented by the words Directional andMonochromatic in Fig. 24.

What about the modulation speed of SM-LED's in com-parison with that of laser diodes under direct-currentmodulation? Generally, the modulation speed depends onthe carrier lifetime. Since electron-hole pairs in laserdiodes experience both spontaneous and stimulated recom-bination, they have an advantage. However, single-modecavities concentrate zero-point electric-field fluctuationsinto a smaller volume, creating a stronger matrix elementfor spontaneous emission. Detailed calculations indicatethat spontaneous emission can be sped up by a factor of-10 owing to this cavity quantum-electrodynamic (QED)effect. In Fig. 24 we indicate that a modulation speed>10 GHz should be possible for SM-LED's.

The same cavity QED effects can enhance the sponta-neous-emission efficiency of SM-LED's, since the radiativerate can then compete more successfully with nonradia-tive rates. External efficiency should exceed 50%, butthis can come most easily from intelligent LED design25

rather than from cavity QED effects.Plotted at the bottom of Fig. 24 is the light output

versus the input current of SM-LED's and laser diodes.SM-LED's can compete with laser diodes in terms of dif-ferential external efficiency, but the SM-LED's can havethe advantage of not demanding any threshold current.Lack of threshold behavior makes the output power andthe operating wavelength of an SM-LED relatively insen-sitive to ambient temperature. Combined with the inher-ent reliability of an LED, this should produce manysystem advantages for the SM-LED concept.

The final SM-LED property illustrated in Fig. 24 isphoton-number-state squeezing, as suggested by the regu-lar sequence of photoelectrons on the horizontal line.Stimulated emission is not required for those exoticsqueezing effects. The critical variable is absolute quan-tum efficiency. If the quantum efficiency of the SM-LEDis high, then these squeezing correlations will exist in thespontaneous output of the single-mode LED. This re-

quires, most of all, a high spontaneous-emission factor /3,our overall figure of merit for microcavities.

In the feature on photonic band gaps in this journalissue there are many other applications given for photoniccrystals, particularly in the microwave and millimeter-wave regimes. The applications are highly imaginative,and they have gone far beyond our initial goals for usingphotonic crystals in quantum optics.

6. CONCLUSIONS

It is worthwhile to summarize the similarities and the dif-ferences between photonic and electronic band structure.This is best done by reference to Table 1.

Electrons are massive, and so the underlying dispersionrelation for electrons in crystals is parabolic. Photonshave no mass, so the underlying dispersion relation is lin-ear. But as a result of the periodicity the photons developan effective mass in PBS, and this should come as littlesurprise.

Electrons have spin 1/2, but frequently this spin is ig-nored, and Schrodinger's equation is treated in a scalarwave approximation. In electronic band theory the spin1/2 is occasionally important, however. In contrast, pho-tons have spin 1, but it is generally never a good approxi-mation to neglect polarization in PBS calculations.

Finally, we come to the accuracy of band theory. It issometimes believed that band theory is always a good ap-proximation in electronic structure. This is not reallytrue. When there are strong correlations, as in the high-T, superconductors, band theory is not even a good zeroth-order approximation. Photons are highly noninteracting,so, if anything, band theory makes more sense for photonsthan for electrons.

The final point to make about photonic crystals is thatthey are nearly empty structures, consisting of 78%empty space. But in a sense they are much emptier thanthat. They are emptier and quieter than even the vac-uum, since they contain not even zero-point fluctuationswithin the forbidden frequency band.

ACKNOWLEDGMENTS

I thank John Gural for drilling all the holes and TomGmitter for making the measurements. Thanksare also due to Ming Leung and Bob Meade for their col-laborative work. The Iowa State group is thanked formaking me aware of its diamond structure results beforepublication.

*Work performed at Bell Communications Research,Navesink Research Center, Red Bank, New Jersey 07701-7040.

REFERENCES AND NOTES1. E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).2. S. John, Phys. Rev. Lett. 58, 2486 (1987).3. Y Yamamoto, S. Machida, and W H. Richardson, Science 255,

1219 (1992).4. E. M. Purcell, Phys. Rev. 69, 681 (1946).5. K. H. Drexhage, in Progress in Optics 12, E. Wolf, ed. (North-

Holland, Amsterdam, 1974), p. 165.6. V P. Bykov, Sov. J. Quantum Elec. 4, 861 (1975).

E. Yablonovitch


7. R. G. Hulet, E. S. Hilfer, and D. Kleppner, Phys. Rev. Lett.55, 2137 (1985).

8. A. Van der Ziel, Noise (Prentice-Hall, New York, 1954).9. There have been some reports recently of a photonic band gap

in a simple cubic geometry. H. S. Sozuer, J. W Haus, and R.Inguva, Phys. Rev. B 45, 13, 962 (1992).

10. C. G. Darwin, Philos. Mag. 27, 675 (1914).11. E. Yablonovitch and T. J. Gmitter, Phys. Rev. Lett. 63, 1950

(1989).12. S. Satpathy, Z. Zhang, and M. R. Salehpour, Phys. Rev. Lett.

64, 1239 (1990).13. K. M. Leung and Y F. Liu, Phys. Rev. B 41, 10188 (1990).14. S. John and R. Rangarajan, Phys. Rev. B 38, 10101 (1988).15. E. N. Economu and A. Z detsis, Phys. Rev. B 40, 1334 (1989).16. K. M. Leung and Y. R Liu, Phys. Rev. Lett. 65, 2646 (1990).17. Z. Zhang and S. Satpathy, Phys. Rev. Lett. 65, 2650 (1990).18. K. M. Ho, C. T. Chan, and C. M. Soukoulis, Phys. Rev. Lett.

65, 3152 (1990).

19. J. Maddox, Nature (London) 348, 481 (1990).20. C. T. Chan, K. M. Ho, and C. M. Soukoulis, Europhys. Lett.

16, 563 (1991).21. E. Yablonovitch, T. J. Gmitter, and K. M. Leung, Phys. Rev.

Lett. 67, 2295 (1991).22. E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe,

K. D. Brommer, and J. D. Joannopoulos, Phys. Rev. Lett. 67,3380 (1991).

23. H. A. Haus and C. V Shank, IEEE J. Quantum Electron. QE-12, 532 (1976); 5. L. McCall and P. M. Platzman, IEEE J.Quantum Electron. QE-21, 1899 (1985).

24. E. Yablonovitch, T. Gmitter, J. P. Harbison, and R. Bhat,Appl. Phys. Lett. 51, 222 (1987).

25. I. Schnitzer, E. Yablonovitch, C. Caneau, and T. J. Gmitter,Appl. Phys. Lett. (to be published).

E. Yablonovitch

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