PHYSICAL REVIEW B 15 OCTOBER 2000-IIVOLUME 62, NUMBER 16

Theory of light propagation in strongly modulated photonic crystals:Refractionlike behavior in the vicinity of the photonic band gap

M. Notomi*NTT Basic Research Laboratories, 3-1 Morinosato-Wakamiya, Atsugi, Kanagawa 243-0198, Japan

~Received 7 April 2000!

Although light propagation in weakly modulated photonic crystals is basically similar to propagation in adiffraction grating in which conventional refractive index loses its meaning, we demonstrate that light propa-gation in strongly modulated two-dimensional~2D!/3D photonic crystals becomes refractionlike in the vicinityof the photonic bandgap. Such a crystal behaves as a material having an effective refractive index controllableby the band structure. This situation is analogous to the effective-mass approximation in electron-band theory.By utilizing this phenomenon, negatively refractive material can be realized, which has interesting opticalproperties such as mirror-image refraction.

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I. INTRODUCTION

A photonic crystal is a structure whose refractive indis periodically modulated, and the resultant photonic dispsion exhibits a band nature analogous to the electronic bstructure in a solid. The one-dimensional~1D! versionof a photonic crystal has long been known as a multilareflector, but 2D/3D photonic crystals have only recenstarted to attract attention after the appearance of a predicthat photonic insulators can be developed by photocrystals.1 Since then, photonic crystals have become a masubject of today’s photonic engineering research.2 So far,most of the concern has been focused on their potentiaphotonic insulators.3 However, they can also be photonconductors whose conductance is determined by their bstructure.4 In photonic crystals, light travels as Bloch wavein a similar way to plane waves in continuous materiBloch waves travel through crystals with a definite propation direction despite the presence of scattering, but thpropagation is complicated because it is influenced byband structure.

In this paper, we investigate the situation shown in FigA light beam is traveling through different media. If the mdium is a dielectric material, then we observe conventiorefraction phenomenon. If the medium is a diffraction gring, then we observe diffraction phenomenon. Then, wkind of light propagation phenomena would be observedthe medium is a photonic crystal? There have been sworks related to this issue in the literature, but systemand consistent way of understanding is still lacking. Thefore, we will now start from the simple cases of a dielectmaterial and a diffraction grating and then examine photocrystals with weak and strong periodic modulation effectsorder to obtain a systematic view for propagation in periostructures and, in particular, photonic crystals. We wclarify features of light propagation in photonic crystals, ashow how a strongly modulated photonic crystal exhibremarkably interesting propagation characteristics whichbe understood as refractionlike phenomenon in standardmetrical optics with unusual refractive index.

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II. ANOMALOUS LIGHT PROPAGATION IN PHOTONICCRYSTALS?

Unusual light propagation in photonic crystals has beremarked upon by several authors.5–9 Lin et al. reported thatrefraction angle becomes anomalous near the bandgap.6 Ko-sakaet al. reported that, under certain conditions, the ligpropagation direction in photonic crystals becomes very ssitive to the incident angle and wavelength, and large besteering is observed, which they call superprisphenomenon.7 However, almost the same phenomena wobserved in 1D and 2D grating wave guides,5 and the dis-tinction between behavior in photonic crystals and thatgrating wave guides is not clear. In addition to that, thehave been some theoretical reports that predict unusuafractive index for photonic crystals.8,9

The essential explanation of these phenomena shouldin the photonic band structure because the direction of lipropagation inside the photonic crystal is determined byequifrequency surface of the photonic bands in thstructures.10 Although this feature of photonic crystals habeen frequently discussed, there have been very few repabout quantitative comparison between theory and expment so far.11 Considering this feature, it might be possibto reconstruct the photonic band structure from measuments of the light propagation inside the photonic crysWe recently demonstrated such an experiment3D Si/SiO2 photonic crystals that were fabricated by autcloning technology,12 and a very detailed photonic banstructure was successfully obtained by the measureme13

This experiment directly shows that the light propagation

FIG. 1. Schematic diagram of light propagation phenomenthrough different media.

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FIG. 2. ~a! Photonic band diagram of a S(n53.5)/SiO2(n51.46) multilayer~1D photoniccrystal! structure with a period ofd. The thick-ness is 0.15d and 0.85d, respectively.~b! De-duced phase refractive index versus frequencythe multilayer structure.~c! Photonic band dia-gram of a 1D photonic crystal with a vanishinglsmall index modulation~empty lattice!. The av-erage index is the same as~a!. ~d! Deduced phaserefractive index versus frequency for the emplattice. Frequency is normalized asvd/2pc.

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indeed determined by the photonic band structure. Tmeans that, if we want to investigate the light propagationphotonic crystals, what we have to do is just to calculatecorresponding photonic band structure. However, photoband structures are fairly complicated and it is thus not eto understand the light propagation phenomena in photocrystals in qualitative terms. Moreover, the relation betwethe light propagation in photonic crystals and that in convtional dielectric materials or gratings has yet to be cleademonstrated. We believe that a simpler way of understaing light propagation in photonic crystals is possible ashould be established, which will clarify the difference btween behavior in photonic crystals and conventional refrtion or diffraction phenomena.

As mentioned above, light propagation in photonic crytals is represented by Bloch waves. Bloch waves have a dnite propagation direction in spite of strong scattering byperiodic structure. This character leads us to consider ametrical optic approach to understand the propagation inIn conventional geometrical optics in dielectric materialight propagation—as shown in Fig. 1—is described byphase refractive index and Snell’s law. Therefore, in ordepursue geometrical approaches for photonic crystals, weexamine the concept of phase refractive index for photocrystals.

The phase refractive index of photonic crystals has bdiscussed by several authors in the long wavelenlimit.14–16 They have homogenized the periodic structuand deduced an appropriate phase index in the lfrequency limit. However, such a result cannot be extento higher frequencies of which wavelength becomes comrable to, or smaller than, the lattice period. Since mostinteresting phenomena, including unusual beam propagaoccur outside the low-frequency limit, we are not satisfiwith this homogenization method to understand the lipropagation in photonic crystals.

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Lin et al. investigated the lowest band of a 2D photoncrystal near its first gap, and argued that the refractive inis modified from the low-frequency limit value near the gbecause the slope of the dispersion curve is reduced.6 Thiseffect is not significantly large because the control rangeindex is limited within refractive indices of materials. Thpresent photonic crystal effect simply arises as a modifition of the mixing ratio of index values, similar to the waywhich the effective refractive index of a conventional slwave guide is derived.17 Furthermore, their argument did noshow whether the index they deduced could be meaninoutside the low-frequency limit~we will show later that suchindex is generally meaningless except under a certain cotion!, and it is not clear how this index is related to propgation direction.

To investigate the phase index of periodic structures oside the low-frequency limit, we must first consider thband-folding effect. To see this, we plot a photonic badiagram of a multilayer~1D photonic crystal! structure as inFig. 2~a!. If we simply use the textbook formulan5ck/v forthe phase refractive index, we obtain the result shown in F2~b!. The resultant phase index exhibits very unusual behior as can be seen in the figure. Dowlinget al. used essen-tially the same argument to predict an ultrasmall indexphotonic crystals.8 This effect is due to the reduction of wavvectork near the zone center as a result of the band foldiHowever, this argument leads to an ultrasmalln even for anempty lattice with the same crystal structure. Figure 2~c! is aband diagram of a 1D photonic crystal with a infinitely smindex modulation. The corresponding phase index is shoin Fig. 2~d!. We know that light propagation in such aempty-lattice photonic crystal~at least when its frequencdoes not satisfy the Bragg condition! should be normal; how-ever, this model still predicts abnormal phase index. Tapparent contradiction shows that the deduced smalln doesnot posses real meaning and that the band folding itself d

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not lead to unusual beam propagation. This contradicarises mainly because we have only consideredk in theabove analysis. We must also consider the group velovector to study beam propagation in photonic crystals.thus need to investigate the equifrequency surface~EFS! ofthe photonic band structure.

III. LIGHT PROPAGATION IN DIELECTRIC MATERIALSAND DIFFRACTION GRATINGS

In this section, we reexamine light propagation in dieletric materials and diffraction gratings by using EFS ploFirstly, we show a very simple example of EFS analysisFig. 3~a!, which describes a light incident problem from ato a dielectric material. A circle in the figure is an EFS of tphotonic band of a dielectric, namely,v5ck/n. The k vec-tor in the dielectric medium is determined by the continuof tangential components of thek vector across the interfaceand light always propagates parallel to thek vector in thiscase. This is an EFS expression of conventional refracphenomenon, and this plot is a graphical representatioSnell’s law ink space

n1 sinu15n2 sinu2 . ~1!

In Fig. 3~b!, we depict light propagation in a diffractiograting with a period ofd. In this case, equifrequency circleare repeated along the periodic axis due to the grating’sriodicity, and thek-conservation rule has to be generalizedsatisfy the periodic boundary condition. As a result of thapplying thek conservation rule, we see that more than ononidentical beams can be excited in a grating. In the figwave A ~on a circle centered at the origin! corresponds to atransmitted wave and wave B~on a circle centered at a reciprocal lattice point! is a diffracted wave. This is nothingbut beam decomposition by a diffraction grating. Note ththe light propagation direction is not parallel to thek vectorfor a diffracted wave, but it is oriented normal to the dfracted wave circle. This is a graphical representation offormula for a diffraction grating

ml5d~sinu11sinu2!. ~2!

Other than beam decomposition, a few points candrawn from this figure. At point C~this point can be excited

FIG. 3. ~a! EFS plot for light incident problem from air to adielectric material.~b! EFS plot for a diffraction grating.

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under certain conditions!, k is very close to the origin, whichleads to very small phase index. However, this phase incannot be used to express the light propagation direcbecause the propagation direction~indicated by the arrow! isnot parallel to thek vector. In other words, this index ialmost meaningless in terms of the light propagation prlem. This directly demonstrates why the previous modincluding Dowling’s8,9 are insufficient to describe the refraction phenomena. To analyze propagation phenomena,have to examine the curvature of EFS.

Next, look at point D, which is an intersecting point otwo circles. This point is a kind of singular points ink space.At such a point, the propagating direction becomes unfined, and generally anticrossing occurs between interseccircles. Therefore, the propagation direction switches frone circle to the other in the vicinity of this point. That iwhen we vary the incident angle or wavelength in the vicity of this point, the beam propagation direction changvery rapidly. If we carefully choose the wavelength andcident angle to excite the vicinity of this intersecting pointhe beam propagation direction becomes very sensitive toincident angle and wavelength. This is the origin of the larbeam steering observed in the grating wave guide. Thisgeneral phenomenon, which occurs at singular points ikspace, thus we call it singular point diffraction. Conicrefraction17 in anisotropic media apparently has the same ogin as singular point diffraction. The similar conical singlarity has also been discussed in the electron band theorthe case of weak periodic modulation.18 In addition to that, tobe precise, a gap opens up at such singular points, andwill discuss its influence in the next section.

From these discussions, it is now clear that we candefine a phase index for a grating in terms of Snell’s lawwe define a phase index, the index is strongly dependenthe incident angle ork-vector angle. Therefore, Snell’s lawloses its meaning. This means that the discussed phenomin a grating cannot be understood within a refraction pictuand must be understood as diffraction.

IV. LIGHT PROPAGATION IN WEAKLY MODULATEDPHOTONIC CRYSTALS

We now move on to the case for photonic crystals. Fiwe examine a 2D photonic crystal with a weak periodmodulation effect. Hereafter, we use a plane-wave expanmethod,19 to calculate a photonic band diagram and EFS,which Bloch waves are expanded by approximately 10plane waves. Frequency is normalized asva/2pc ~a is thelattice constant!. We examine mainly TE modes~magneticfield lies perpendicular to the 2D plane! of the structure, butthe result obtained in this paper is not specific to TE mod

The EFS of a hexagonal 2D photonic crystal with a vaishingly small index modulation is plotted in Fig. 4~a!. TheEFS in the first Brillouin zone is expanded to the outerciprocal space. The EFS consists of repeated circles refling the 2D hexagonal periodicity, by the same mechanismthat in a 1D diffraction grating. If a plane wave is launchto this photonic crystal from air at a certain incident angseveral phenomena are expected from this figure. Firslight beam is decomposed into more than one nonidentwaves. In the situation in Fig. 4~a!, two waves A and B are

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excited. Wave A corresponds to a transmitted wave,wave B is a diffracted wave. Second, the propagation dirtion is apparently not parallel to thek vector for diffractedwaves. The propagation direction is oriented to the grovelocity vector vg5gradkv which is normal to the EFSThird, the propagation angle is very sensitive to the incidangle and wavelength if the launched beam excites the wnear intersecting points~e.g., point C!, which leads to largebeam steering. This is the origin of the superprism effecreported in Ref. 7. Fourth, in some regions of the EFS nthe G point, k vector becomes very small and it leads tovery small index value. But such an index is not meaningby the same reason as for a grating. As readers alreadyhave noticed, the situation as a whole is similar to that fograting. The anomalous beam propagation in photonic ctals can be explained by the mechanism outlined above fdiffraction grating. Consequently, we still cannot defineproper phase refractive index for a weakly modulated phonic crystal which precisely reflects the light propagation.

In the above discussion, we investigated the effect of bfolding for light propagation, which is mostly sufficient tunderstand gratinglike diffraction phenomena. If, howevthe periodic modulation increases, another effect needs ttaken into account; the gap appears in EFS at the intersepoints. This is the seed for the photonic band gap. Iffurther enlarge the periodic modulation, this gap eventuacomes to dominate the whole ofk space and photonic bangaps will emerge. For now, however, we still stay with retively small index modulation, which is actually the case fmost of grating wave guides or weakly modulated photocrystals in which anomalous beam propagation wasserved. We plot an EFS of a 2D photonic crystal with a finbut small periodic modulation effect in Fig. 4~b!. ComparingFig. 4~b! to Fig. 4~a!, one can observe small gaps that app

FIG. 4. ~a! Schematic EFS plot for a hexagonal 2D photoncrystal with a vanishingly small index modulation. The first Brlouin zone~BZ! is shown as a hexagon.~b! EFS plot for a hexago-nal 2D photonic crystal with finite index modulation. This EFScalculated for TE mode in a 2D hexagonal GaAs (n53.6) air-holephotonic crystal atv50.35 which is far from the gaps, but this typof EFS is general for photonic crystals at frequencies far fromgaps or photonic crystals with a small index modulation.~c! Sche-matic of anomalous diffraction near the singular point.

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near the intersecting points although the overall structurmuch the same. The effect of gap opening is that some ofdiffracted waves are not excited due to this gap. In otwords, we can selectively pick up or exclude some of dfracted waves. This mechanism can be applied to excludtransmitted wave and excite a diffracted wave only, as illtrated in Fig. 4~c!. Figure 4~b! shows that, ak-conservationline is passing through a small gap formed around theKpoint and it intersects with a circle~dotted! centered at theorigin. This circle represents aG50 plane wave corresponding to a transmitted wave. In this small gap region, a tramitted wave@e.g., wave A in Fig. 4~a!# becomes evanescenand thus only a diffracted wave is excited. Normally, tpropagation direction of a transmitted wave does not difgreatly from that of conventional refraction in a dielectrmaterial, and the propagation direction of a diffracted wacan be very different from that of conventional refractioSo, excluding the transmitted wave, the situation itself mseem as if the light beam is beingrefractedin a very strangeway. However, this is not a correct view. Such a situatican only be realized at certain incident angles whentransmitted wave falls into a gap. Otherwise, conventiorefraction or diffraction should occur. This is obvious frothe EFS shape in Fig. 4~b! that is almost the same as the EFwithout gaps shown in Fig. 4~a! except the vicinity of theintersecting points. Therefore, we still cannot define a refrtive index for such a situation. This discussion of the gopening also applies for a 1D grating.

To summarize the preceding discussion, what we hshown is that light propagation in weakly modulated phonic crystals is quite similar to that in 1D grating. The beadecomposition and large beam steering observed are duthe band folding and singularities at the intersecting pointsk space. The anomalous light propagation reported for ptonic crystals, for example the superprism effect, is easyunderstand within this picture. Therefore, the light propation in such photonic crystals cannot be analyzed in termthe phase refractive index, in the way that conventionalfraction in a dielectric can.

V. LIGHT PROPAGATION IN STRONGLY MODULATEDPHOTONIC CRYSTALS

What then happens when the periodic modulationcomes large? We know that propagation is still governedEFS andk conservation across the interface, but the situatis now qualitatively different from the weakly modulateone. Although the gaps influence occurs only near the sinlar point in the former case, the gap opening now comesdominate the overall EFS shape. This breaks up the gratlike picture in Sec. IV, which we will show in this section

Bloch modes in photonic crystals are expressed as a mture of transmitted plane wave and diffracted waves havinreciprocal vectorsG:

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In the preceding sections, we assumed that we can chaterize excited waves as transmitted waves or as diffracwaves with a certainG. That is, we categorized Bloch wavein terms of their dominantG component since the degree

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FIG. 5. ~Color! Hexagonal~a! and square~b!2D GaAs air-hole photonic crystals~with a holediameter of 0.7a,! and the resulting EFS’s at several frequencies. Frequency is normalizedva/2pc. vgap>0.48 for ~a! and vgap>0.34 for~b! wherevgap is a gap edge frequency.

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mixing is not strong. However, excited waves in strongmodulated photonic crystals should be a strong mixturemany diffracted wave components with differentG. In suchcases, the light propagation angle does not follow themula for a grating diffraction@Eq. ~2!#. In general, EFS cannot be decomposed into a simple ensemble of circles, whexcludes a simple gratinglike description or refractionlidescription. Seemingly in such cases, the situation canbe characterized as chaotic, and a simple qualitative behacannot be extracted.

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To examine this, we plot EFS of 2D hexagonal and squGaAs air-hole photonic crystals~with a hole diameter of0.7a! at several frequencies near one of the band gaps in5. In both cases, the EFS shape becomes rounded afrequency approaches to the band gap. This is a rathereral effect for periodic modulation. The Fermi surface~EFSof the electronic band structure for a solid! of metal exhibitsa similar starlike shape reflecting the symmetry of the crtal, which is sometimes referred to as a ‘‘monster.’’ Aswell known in solid-state physics, the crystal effect alwa

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FIG. 6. ~Color! ~a! EFS plot of TE modes in a2D GaAs pillar hexagonal photonic crystal~n1

53.6,n251, 2r 50.7a! atv50.5620.635~fromouter to inner!. The colors represent frequencindicated in~c!. The first BZ of a hexagonal lat-tice and the symmetry points are also shown.~b!Refraction angle versus incident angle atv50.575 and 0.61.~c! Effective refractive indexas a function of the angle of thek vector at vari-ous frequencies.

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round out the monster’s sharp corners.20 This can be ex-plained by the fact that the mixing among differentG com-ponents becomes more pronounced near the bandgap earound symmetry points in the reciprocal space, which evtually pushes out the sharp corners. This tells that whenperiodic modulation effect is not large, the EFS exhibits gerally a starlike shape consisting of arcs belonging tofracted waves; but when it becomes strong, the EFS shbecomes circular or spherical near the gap.21

We will now investigate this effect in more detail by eamining TE modes of a 2D hexagonal GaAs pillar photocrystal ~with a pillar diameter of 0.7a! in air. Figure 6~a!shows the EFS for this 2D photonic crystal atv50.56– 0.635. Asv approaches 0.635 that correspondsone of the gap frequenciesv(G3), the shape of EFS becomes rounded and finally becomes circular. In the casFig. 6~a!, the gap width aroundv(G3) is very small, but thisdoes not mean the periodic modulation effect is small.fact, the modulation effect is significantly large as seen infigure. This means that the gap width itself is generallydirectly related to the strength of the periodic modulatieffect but the periodic modulation effect~monster rounding!is most pronounced near the gap frequency~including a gapwith very small gap width: In such a case, it might be betreferred as symmetry point frequency!. We will later exam-ine the strength of the monster rounding for various photocrystals having different gap width.

From the shape of EFS, we can deduce the propagaangle using the relationvg5gradkv. We calculated the relation between the incident and propagation angles forv50.61 andv50.575 as shown in Fig. 6~b!. Though thepropagation angle atv50.575 is complicated as a result othe complicated shape of EFS, the curve is very simplev50.61. At v50.61, the EFS consists of a single circlNote that the EFS plot looks similar to that of a conventiodielectric material as shown in Fig. 3~a!. The excited wave isdetermined by the circular EFS within the first Brillouzone, and the propagation angle thus should follow Snelaw. This means that we can define an effective refracindex neff from the radius of the EFS using Snell’s law.Fig. 6~b!, we confirmed that the curve forv50.61 followsSnell’s law usingneff520.50. This curve shows that thpropagation angle follows Snell’s law at u in,arcsin(uneff /n0u), and no waves are excited in the photoncrystal atu in.arcsin(uneff /n0u), which corresponds to total internal reflection for a dielectric material. Note that total iternal reflection does not occur when a light beam is incidfrom air to a conventional material. In this way, the ligpropagation in this case is properly described by the effecindex derived by using Snell’s law, suggesting that the bepropagation is refractionlike.

We check the applicable range of this effective indeFigure 6~c! shows the deduced effective index as a functof the in-plane angle ofk at various values ofv. This graphindicates that the deduced index does not depend on tkangle in the range 0.59,v,0.645. This means that the deduced index is well defined over this range. This range cresponds to 140 nm at 1.55mm ~the typical wavelength foroptical communication!, which is wide enough for considering real applications.

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Although we have pointed out similarity to conventionrefraction, there is a striking difference from conventionrefraction, which we have not pointed out in the precedparagraph. Note that asv increases up tov50.635 the ra-dius of the monster shrinks. Consideringvg5gradkv, thismeans that propagation direction is inward for this monsthough it is always outward for a conventional dielectrThis results in a negative propagation angle for all incidangles as already shown in Fig. 6~b!, which leads to a negative effective index. This unusual negative refraction isdirect consequence that the gap opening dominates the oall EFS structure. In contrast to Fig. 4~b! where the EFS canbe approximated by the EFS of an empty lattice, we obsethat this EFS in Fig. 6~a! cannot be traced back to the EFcircles of an empty lattice. That is, a conventional transmted wave does not exist at any angle, and the excited Blwave cannot be approximated by any single diffracted whaving specificG, but is a strong mixture of diffractedwaves.

Figure 7~a! is a plot of effective index againstv. In therange where the index is well defined, the effective indvaries from20.7 to 0.5. We compare thisv dependence tothe photonic band diagram for this structure as shown in F7~b!. Notice that the sign of the effective index is reversedv50.635 which corresponds to thev(G3) point in the bandstructure. Band I@v.v(G3)# has a positive index and banII @v,v(G3)# has a negative index. The same charactetics are found for bands nearv(G2). Although the positiveand negative index bands are almost touching in boththese cases, touching is not essential. We have confirthat the effective index becomes well defined in the vicinof open gaps as well. This is trivial because the monsrounding generally occurs in the vicinity of gaps arousymmetry points as mentioned earlier. We show anotherample that is TM~electric field lies perpendicular to the 2Dplane! photonic band diagram of a 2D GaAs air-hole phonic crystal in Fig. 8~a!. In this case, band I~III ! and band II~IV ! near the open gap betweenv(G1) @v(G3)# andv(G2) @v(G4)# are well-defined positive and negative indestates, respectively. Figure 8~b! shows the effective index obands I and II againstv.

FIG. 7. ~a! Effective index versus frequency of TE modes of2D GaAs pillar hexagonal photonic crystal~n153.6, n251, 2r50.7a!. The frequency range where the index becomes non-wdefined is indicated by the broken line.~b! Photonic band structureof the same photonic crystal.

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It is generally known that an air-hole-type 2D photoncrystal has a larger gap in TE modes, and a pillar typephotonic crystal has a larger gap in TM modes. We obsethat the formation of well-defined effective index states hthe opposite tendency. The air-hole-type crystal prefersmodes and the pillar type crystal prefers TE modes,22 whichis probably because available propagating bands do notwhen the periodic modulation effect is strong for the caoptimized for wider gaps due to the efficient gap formatiobut such modes still exist for the case optimized for wideffective index region even when the periodic modulationstrong.

Concerning these characteristics, there is an interesanalogy with the electronic band in semiconductors,shown in Fig. 9. In a semiconductor, a negative effectmass state~the hole band! appears below the energy gap, aa positive effective mass state~the electron band! appearsabove the gap, which is quite similar to the manifestationeffective index states in photonic crystals. This analomakes sense if we note that the sign of effective massemiconductors and the sign of effective index in photocrystals are both derived from the band curvature. Furthmore, the effective mass approximation is only valid nearbandgap in the electron band theory. This is similar tocase where the effective index state is only valid near

FIG. 8. ~a! Photonic band structure of TM modes for a 2D Gaair-hole hexagonal photonic crystal~n253.6,n151, 2r 50.8a!. ~b!Effective index versus frequency.

FIG. 9. Analogy between effective mass approximationBloch electron bands and effective index approximation for Blophoton band.

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photonic gap. A Bloch electron in a solid is generally vedifferent from a free electron, but it becomes free-electrolike near the bandgap or band minimum and the effectmass approximation can be applied in such regions. Incontext, the present situation can be understood that a Bphoton comes to resemble a free photon~that is, a planewave! having an effective refractive index near the bandgdespite of large scattering by the periodic lattice, that is,geometrical optic approach can work near the photonic bagap.

These results indicate that, if there is a substantiastrong periodic modulation in a photonic crystal, it behavas a continuous isotropic material having an effective refrtive index at a certain frequency range near the band edgsuch cases, exotic light propagation phenomena insidephotonic crystal can be simply described by Snell’s lawing an effective refractive index. In contrast to the weakmodulated case where the definition of refractive index ismeaningful, this effective refractive index has a clear corspondence to the true phase refractive index as far asindex defined in Snell’s law is concerned. The sign andsolute value of the effective index can be artificially variby frequency, crystal structure, and refractive indicescomposing materials; it is not limited by the range of tindices of the materials themselves. The effective indexbe negative or less than unity. The fundamental limit on teffective index isuneffu<max(n1,n2). Note that this effectiveindex is not limited by the Hahin-Shtrikman bounds23 whichonly holds in the effective medium regime, correspondingphotonic crystals in the long-wavelength limit.

Although the monster rounding generally occurs in tvicinity of any gaps, the strength of this rounding diffefrom band to band. Some of the bands retain anisotrrather close to the band edge. For example, the lower bantheG2 point ~band V! in Fig. 8~a! forms a surprisingly accu-rate hexagon in the vicinity of the bandgap. In such casenormal effective index cannot be defined but light propation phenomenon is interesting in itself. Due to its hexagoshape, the beam propagation direction is frozen at eitheor 60° over a wide incident angle range. In other words, sa photonic crystal becomes a network of straight waguides oriented towards its symmetry axes, and propagain other directions is prohibited by a partial photonic gap24

Such freezing of the propagation direction cannot beplained by the model for weakly modulated photonic crystwhich we used in Sec. IV, where the EFS can be appromated as a sum of circles. It only occurs for strongly modlated photonic crystals where EFS is a strong mixturemany diffracted waves. This anisotropy of the effectivedex near the gap seems to resemble the anisotropy of etive mass in semiconductors. In semiconductors, this aniropy originates in the anisotropy of the atomic orbitals whicompose a particular electron band~such ass, p, or d or-bital!. In the usual photonic crystal case, however, suchatomic-orbital-like character25 does not significantly influ-ence the photonic band because of the weak confinemethe lattice point~photonic atom!. Thus, the anisotropy ismainly due to the character of the band itself determinedthe crystal symmetry at least when the wavelength is coparable to the lattice constant, which could be analyzedthe group theoretical approach.26

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PRB 62 10 703THEORY OF LIGHT PROPAGATION IN STRONGLY . . .

Although the main concern of this paper is phase indhere we would like to comment on the group refractive indfor these photonic crystals. In order to examine it, we haveinvestigate the frequency dispersion of photonic crystApparently seen from the band diagram, the photonic crtals we are discussing have a strong frequency disperswhich significantly influences the group velocity index. Athough the complicated band curvature is indicating a coplicated group velocity, it can be reduced to a considerasimple relation in the vicinity of the bandgap. This is easunderstood if one compares again the present situationthat of electron bands in a solid, in which the band shohave a parabolic dispersion near the band edge. This pbolic dispersion does not result from the parabolic dispersof free electronsE5\2k2/2m, but is attributed to the facthat the periodic modulation induces the energy gap insecond-order perturbation. In the case of photonic crysthis can be approximately expressed as follows:

v22v02'hk2 ~4!

~v0 is the band edge frequency, andh is an expansion coefficient!, which can be reduced to

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This shows that Bloch photons have an electronlike pabolic dispersion near the band edge.v (52v0c/h) containsall information on the band near the gap within this conteand could be seen as an effectivemassof Bloch photonsbecause it represents a parabolic~rather than a linear! disper-sion of bands. This leads to the following dispersion relatof group indexng :

ng5v/2k. ~6!

A expression for the phase index dispersion is not simbut if we use a scaled frequencyv85v2v0 in analogy tothe electron band case, we will get the following anothphase index dispersion:

np85ck/v85v/k5ng/2, ~7!

which is simply related to the group indexng by a factor of2 ~due to the parabolicity!. This new expression of phasindex is mostly useless in the present situation, but it cobe meaningful when we discuss the light propagationtween different photonic crystals sharing a similar crysstructure in common. Some of readers might notice that this some similarity to the nonparabolicity problem in thsemiconductor heterostructures consisting of materials wdifferent effective mass.27

To see how this parabolic representation is relevant,replot Fig. 7~b! as v versusk2 in Fig. 10. This shows thabands near the band edges are very close to parabolic,the expressions~5! and~6! are relevant under such a regimA detailed discussion of frequency dispersion is beyondscope of this paper, but this already shows that althoBloch photons propagate like free photons at fixed fquency, as far as frequency dispersion is concerned theyhibit a unique parabolic dispersion which is quite differefrom that of free photons.

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In this section we have shown that, the monster roundgenerally occurs near the bandgap, and the effective inbecomes well defined in such regions, and the sign ofindices differs between the upper and lower bands. Befmoving on to a next section, we would like to point out ththis discussion can naturally be extended to 3D photocrystals and it is thus basically possible to control thepropagation of light. However, realization of such states3D photonic crystals are more limited than in 2D photoncrystals, since it requires the existence of full photonic bagaps that are known to be more difficult to obtain in 3structures.28

VI. NEGATIVE REFRACTION

The fact that we can realize an arbitrary refractive indstate leads to many possibilities for the control of ligpropagation. The most interesting point is that this realinegative refraction, as illustrated in Fig. 11~a!. This negativerefraction leads to many anomalous light propagation pnomena. We show some examples: an imaging effect@Fig.11~b!# and an open cavity formation@Fig. 11~c!#. In the latter

FIG. 11. Schematic diagrams of light propagation in negativerefractive photonic crystals:~a! negative refraction,~b! mirror-inverted imaging effect, and~c! formation of an open cavity.

FIG. 10. Replotted version of the band diagram shown in Fig~b! as a function ofk2. Straight lines are guide to the eye foindicating the parabolic character of bands near the singular po

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10 704 PRB 62M. NOTOMI

case, there exist many closed optical paths running acrosfour interfaces which form a kind of an open cavity despthe fact that there is no reflecting wall surrounding the city. In the former case, light is emitted from a point sourcea negatively refractive photonic crystal. Within the convetional paraxial-ray treatment, the refracted wave convergeanother point in the photonic crystal. This means that objein the left-hand space produce real images in the right-hspace. This imaging is fundamentally different from convetional imaging by a lens. Figure 12 schematically illustratwo types of imaging. Imaging by a lens is describedNewton’s formula, in which the focal length is an importaparameter. Magnification depends on the relative distancan object from the lens and focus point. Therefore it oproduces a 2D image on the focal plane and does notduce a 3D image. On the contrary, a negatively refracphotonic crystal produces a 3D image~if it is a 3D nega-tively refractive photonic crystal! by the mirror-inversiontransformation (x,y,z)→(x,y,2bz) where b5abs(neff /n0), which is different from Newton’s formula. Inaddition, the lens imaging has a definite principal axis,the present imaging has translational symmetry in the bouary plane. In this sense, this imaging is rather close to iming by a mirror. The apparent difference between a photocrystal and a mirror is that the former produces a real imbut the latter only produces a virtual image. This uniqproperty is suggesting possibilities of 3D photographinguse of negatively refractive photonic crystals.

FIG. 12. Schematics of imaging by a negatively-refractive ptonic crystal and imaging by a lens.

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We again point out the difference between this negatrefraction and the situation for a weakly modulated photocrystal shown in Fig. 4~c!. Although the propagation anglcan become negative at a certain incident angle in a wemodulated photonic crystal, this does not lead to real iming because negative refraction only occurs over a limiincident angle range and even within this region light raemitted from the same point but traveling in the differeorientation do not converge to the same point.

In the bulk of this paper, we have discussed the ligpropagation using EFS plots; in other words we examinthe wavevector conservation across the interface. This trment is adequate for determining the propagation angle,it is not still clear to what extent such Bloch waves will bexcited by a plane wave incidence since we have not qutitatively discussed the amplitude continuity across the inface. In principle, this can be done by a proper handlingthe amplitude connection between the allowed Bloch wain the photonic crystal and the outside plane waves conering the effect of the periodic boundary condition in tinterface plane. This treatment was the same as that hasestablished in election diffraction theory.29,30 Instead of do-ing this, we have numerically simulated electromagnewave propagation~in TE mode! in a hexagonal GaAs photonic crystal by a 2D finite-difference time-domain method31

using a real refractive index distribution profile of the peodic structures. We used a perfectly matched-layer absorcondition for the outer boundaries.32 Figure 13~a! showsnegative refraction where an angled Gaussian beam isdent to the interface. The negative propagation angletracted from this result coincides with that is obtained frothe EFS calculated by the plane wave expansion. Fig13~b! shows mirror-inverted imaging. A point source is located in the conventional material, and focusing is cleaobserved in the negative-index photonic crystal. Thesemerical calculations directly solved Maxwell equations icluding appropriate amplitude continuity without any simpfication, and thus these results clearly demonstrateexperimental observability of the phenomena discussedthis paper.

VII. SUMMARY

We have systematically analyzed the light propagatphenomena in periodic structures and photonic crystals w

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FIG. 13. ~Color! Numericalsimulation of electromagneticwave propagation in negativelyrefractive photonic crystals. Magnetic field (Hz) distribution of TEmode after 23104 steps of calcu-lation. The lower half is a hexagonal photonic crystal with 100357 unit cells. ~a! A slightlytilted Gaussian beam is launchefrom air (n051). ~b! A pointsource emitting atv50.62 is lo-cated in continuous material withn050.5. @If we replace it with air(n51.0), the lower half resultwill merely be compressed vertically.

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PRB 62 10 705THEORY OF LIGHT PROPAGATION IN STRONGLY . . .

the help of the band theory and numerical simulations.motivation is to examine whether light propagation in ptonic crystals can be understood by simple geometricalanalogies. First, we have examined the effect of the bfolding that is directly related to the periodic boundary cdition of the structure. It has been shown that the propagcharacteristics of diffraction gratings and weakly modulaphotonic crystals are much alike and can be explained wa similar framework. This explains anomalous propagaphenomena in grating waveguides and photonic crysThese studies have clarified that the light propagatiothese media is fundamentally different from conventionafraction, and therefore we cannot define appropriate retive index.

However, subsequent studies showed that the light pgation in strongly modulated photonic crystals near thetonic bandgaps, in which the second effect—theopening—dominates EFS shape, comes to resemble rtion phenomenon in a dielectric material even in the presof strong multiple diffraction. In these cases, we can deeffective phase refractive index to explain the propagainside the photonic crystal using the conventional Sn

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law. Since such effective index is determined by the phonic band structure, it can be negative or less than unwhich leads to unusual refraction phenomena includinegative refraction. The basic mechanism is similar to teffective mass model in electron band theory. A Bloch phton becomes free-photon-like in the vicinity of the bandgapand can be considered to be refracting with an effectivefractive index. Such effective index states only exist nearphotonic bandgap, in a similar way to the effective mastates in a semiconductor. We have shown that negativedex states lead to interesting propagation phenomena, sas imaging effect, which have been confirmed by numerisimulation of electromagnetic wave propagation. Theunique properties of refracting Bloch photons have the ptential to drastically change the form of optical componen

ACKNOWLEDGMENTS

The author would like to acknowledge T. Nakahara,Taniyama, W. Lui, T. Yamanaka, Y. Yoshikuni, H. Kosakaand S. Kawakami for helpful discussions, and T. Tamamufor his encouragement throughout this work.

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22This does not mean that the monster rounding does not occurthe opposite case~such as TE modes of an air-hole-type photonic crystal!. Note that it actually does occur as shown in Fig. 5

23Z. Hashin and S. Shtrikman, J. Appl. Phys.33, 3125~1962!.24Similar, almost perfectly hexagonal EFSs also appear in the v

cinity of the edge of the range where the effective index approximation works, such asv50.58 in Fig. 6~a!.

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