Existence and control of Shockley surface states of a one-dimensional defect chain in a photonic crystal

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N. Malkova and C. Z. Ning Vol. 24, No. 3 /March 2007/J. Opt. Soc. Am. B 707

Existence and control of Shockley surface states ofa one-dimensional defect chain in a

photonic crystal

N. Malkova and C. Z. Ning

Center for Nanotechnology, NASA Ames Research Center, Moffett Field, California 94035, USA, and Departmentof Electrical Engineering, Arizona State University, Tempe, Arizona 85287, USA

Received September 6, 2006; revised November 3, 2006; accepted November 6, 2006;posted November 9, 2006 (Doc. ID 74674); published February 15, 2007

We present detailed results of analytical and numerical investigation of the Shockley surface states in photoniccrystals doped with a chain of alternating s- and p-type defects. Conditions for the existence and control of suchsurface states are studied using the empirical tight-binding model and verified by the finite-difference time-domain technique. We show for the first time, to our knowledge, that, in contrast to the case of solids, theShockley states in photonic crystals with complete unit cells of defects do not appear simultaneously with theopening of the inverted bandgap between s and p bands. Rather, the width of the inverted bandgap must reacha critical value equal to the separation between the discrete levels. This results in a system size effect of theShockley states in photonic crystals. We show how such system size effect can be used to control the surfacestates. We also demonstrate the control of the Shockley states by controlling the overlap of s and p bands,achieved through change in either the radius of one of the defects or the refractive index via electro-opticaleffects. © 2007 Optical Society of America

OCIS codes: 240.6690, 230.3990, 230.7370.

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. INTRODUCTIONhotonic crystals (PCs) are promising for integrated pho-onic circuits. With the increase of integration density, theole of the PC interfaces becomes increasingly important.his fact motivates a great deal of interest in the surfacexcitations supported by PCs, which are referred to asurface Bloch states1–6 (SBSs). Interestingly, the role ofhe SBSs in PCs may be twofold. On the one hand, theBSs are an undesirable feature of the finite PCs, sincehey can disrupt light propagation and compromise deviceerformance7 due to localization at the device interface.n the other hand, existence of the SBSs might becomextremely beneficial under some circumstances. For ex-mple, the use of the surface modes to collimate light outf a PC waveguide has been recently demonstrated.8,9

herefore understanding and controlling the properties ofhe SBSs are of crucial importance for many applicationsf PCs.

As is well known about surface physics of solids, sur-ace states can be of either the Tamm or the Shockleyype.10,11 The Tamm states are entirely due to the changen the potential of the outermost cell of the crystal (sur-ace cell).12 In other words, the surface cell in Tamm’sodel is perturbed with respect to the bulk cells. Another

haracteristic feature of the Tamm’s model is that there iso band intersection in the entire spectrum. This is thease if the band spectrum is formed from a single orbital.n contrast, the surface cell in Shockley’s model is not per-urbed, but the energy bands can intersect because theyre formed by orbitals of different symmetry.13 Therefore,he wave function of electrons in the case of Shockleytates is expressed as a linear combination of different or-

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itals (hybridized orbitals). The coupling between the or-itals leads to the anticrossing effect. As a result, an in-erted bandgap opens up where the Shockley surfacetates may appear. Importantly, both the Tamm and thehockley states originate from the Bloch states of the pe-iodical structure.10,11 Hence both of them must be quali-ed as SBSs.On the basis of the analogy with solids, we recently

emonstrated that almost all the surface states discussedn the context of PCs are Tamm-like states.14,15 We havelso shown that the Shockley-like surface states can in-eed occur in PCs doped with a chain of alternating s anddefects,14 leading to hybrid orbitals. Such hybrid orbit-

ls can appear in a PC with a unit cell including morehan one dielectric element. It should contain either alter-ating rods supporting modes of different symmetry or al-ernating strong and weak bonds or both.14 In view of pos-ible future applications of surface states in PCs, it ismportant to know more about the conditions for the ex-stence of such surface states and, more importantly,hether one can actually control them.The goal of this paper is to investigate the properties of

he Shockley states in more detail and to establish criticalonditions governing their appearance. We will study twof the simplest structures supporting the Shockley sur-ace states, shown in Fig. 1. They contain the defects ofhe s and p types and differ by their termination. We ana-yze such structures theoretically and numerically. First,e use the model based on the empirical tight-binding ap-roach. Then the theoretical model is tested by the finite-ifference time-domain (FDTD) simulations. Throughonsistent theoretical and numerical analysis, we investi-

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708 J. Opt. Soc. Am. B/Vol. 24, No. 3 /March 2007 N. Malkova and C. Z. Ning

ate the evolution of the Shockley states when we gradu-lly change the parameters of the structure. We analyzehe critical conditions and discuss their application forontrol of surface states in PCs.

The paper is organized as follows. In Section 2 we de-cribe the model systems to be studied and discuss theight-binding analysis of the surface states in PCs. In Sec-ion 3, we present the FDTD simulations and explain theumerical results in terms of the tight-binding model.ection 4 concludes the paper with a summary of theain results.

. THEORY OF THE SHOCKLEY SURFACETATES OF THE DEFECT CHAINhe model systems shown in Fig. 1 represent periodic ar-ays of the infinitely long dielectric rods embedded in an-ther dielectric medium. This otherwise perfect two-imensional PC (gray circles) is doped with a chain of theefect rods with periodicity d. The unit cell of the chainontains two different defects supporting the nondegener-te s mode (small filled circles) and the double-degeneratemode (large open circles). To simplify the analysis, we

esign this structure in such a way that the chain wouldenerate the defect states (or the allowed band of thehain) inside the bandgap of the host crystal. Then, in therst-order approximation, we neglect the coupling be-ween the defect rods and other rods of the crystal. There-ore we can consider this structure as a periodic chain in-talled in another quasi-homogeneous medium, whoseole is to confine light in the direction perpendicular tohe chain and to generate defect states of the desiredype. We consider two structures: structure 1 with com-lete unit cells [Fig. 1(a)] and structure 2 with incompletenit cells [Fig. 1(b)].In the framework of the tight-binding theory,16 the

ave function of the chain (Fig. 1) is expressed as a linearombination of the s-type, �s�r−nd�, and two double-egenerate p-type, �px�y��r−md�, eigenmodes of the indi-idual defects located at the nth and mth sites, respec-ively, ��r , t�=�nan�t��s�r−nd�+�m,i=x,ybm

i �t��pi�r−md�.he tight-binding description includes the coupling ma-

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rix elements between the two nearest-neighbor s modes,s, and the nearest p modes, �p, along with the couplingatrix elements between the nearest s mode and pxode, �sp, and between the px and the s modes, �ps (seeig. 1). By symmetry, the coupling between the s and they modes vanishes. Therefore, for our problem, we can ig-ore the py mode and drop the x and y indices from nown. The coupling matrix elements �s and �p define theidth of the allowed bands for the chains including only

he equidistant s or p modes, respectively. The dynamicsf the field amplitudes al�t� (for s modes) and bl�t� (for podes) in the lth unit cell is described by the following set

f ordinary differential equations17:

id

dtal = �sal + �s�al−1 + al+1� + �spbl + �psbl−1,

id

dtbl = �pbl + �p�bl−1 + bn+1� + �spal + �psal+1, �1�

here �s,p=�s,p− i�s,p are the complex eigenvalues of thendividual s or p defects defined, in general, by the fre-uency, �s,p, and by the width, �s,p, of the s- or p-typeesonant peak. The coupling matrix elements are deter-ined as overlap integrals between the relevant defectodes.17 For an infinite chain of the defects, the solution

f the problem takes the form

an�t� = An exp�− i�t� = A exp�ikdn − i�t�,

bn�t� = Bn exp�− i�t� = B exp�ikdn − i�t�. �2�

ere k is the momentum defined by the dispersion rela-ion

�1,2�k� = 1/2��s + �p + 2��s + �p�c

± ��s − �p + 2c��s − �p��2 + 4�̃sp2 �, �3�

here �̃sp2 =�sp

2 +�ps2 +2�sp�psc and c=cos�kd�.

In the case of a finite chain with N unit cells, we haveN−2 evolution equations from Eqs. (1), with two bound-ry conditions. Using Eqs. (2), we derive the following ei-envalue problem:

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t is worth noting that Eq. (4) describes the chainith N complete unit cells including 2N defects. Byropping the last column and the last row inatrix (4), we obtain the characteristic matrix for the

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hain of 2N incomplete unit cells including 2N−1efects.Equation (4) forms the basics for the Shockley states ofone-dimensional system. As we discussed previously,14

he Shockley states appear due to the effects of terminat-ng the sp-hybridized chain rather than the perturbationt the end defects as in the case of Tamm states.10,11,15

ince our focus in this paper is the Shockley states, weill simplify the problem by assuming that the end de-

ects are not modified, or �s,p� =�s,p and �sp,s,p� =�sp,s,p. Inhe case of solids, the assumption of the nonperturbednds requires a careful justification. However, as we williscuss below, PCs allow for controlling the surface poten-ial, giving one the exciting opportunity of designing thetructures to support either the Shockley or the Tammurface states.

First, we analyze the eigenvalue problem (4) for struc-ures 1 [Fig. 1(a)] and 2 [Fig. 1(b)]. Both structures areharacterized by two bonds of identical strengths. Be-ause of different polarities of the two p lobes, the cou-ling matrix element between the s and p modes musthange sign when the direction of the bonds is changed, sohat �sp=−�ps. Moreover, the coupling matrix element be-ween the nearest s modes has a sign opposite to that ofhe coupling matrix element between the p modes, �s�p0. To facilitate the appearance of the Shockley states,e assume that the two defects have the same eigenval-es ��s=�p=��. For simplicity, we also assume that �s=�p=�. We will use the dimensionless units for energy,

�−�� /�, and coupling �=�sp /� in the following.Figure 2 demonstrates the emergence of the Shockley

tates for typical structure 1 of four unit cells when theoupling between s and p defects is gradually increasedrom �=0 to 3. In the case of uncoupled s and p defects�=0�, dispersion relation (3) is reduced to two intersect-ng cosinelike curves, ��1,2−�� /�= ±2c, shown by solidurves in Fig. 2(a). Note that the width of the allowed

ig. 1. (Color online) Coupled defect structures with unit cellsncluding one s defect (small filled circles) and one p defect (largepen circles) embedded in a perfect host PC (gray circles): (a)tructure 1 of six complete unit cells; (b) structure 2 of seven in-omplete unit cells. The source (S), port (P), perfect matched lay-rs (gray boxes), and reference system are shown.

and is 4�. The spectrum of the finite chain can be ob-ained from the eigenvalue problem (4). In the case of thencoupled chain with four complete unit cells, the dis-rete spectrum consists of four double-degenerate discreteevels, with each energy level related to two values of mo-

entum k [see stars in Fig. 2(a)]. The correspondingigenfunctions are presented in ascending order from bot-om to top in Fig. 2(b). We can see that there is no mixingf s and p defects. Each eigenstate is related to either s ordefects.In Figs. 2(c), 2(e), and 2(g) we show the allowed bands

f the infinite chain with coupled s and p defects (solidurves) in comparison with the allowed bands of the un-oupled chain (dashed–dotted curves). We note that theegeneracy of the spectrum is removed for ��0. As a re-ult, an inverted bandgap opens up. As follows from Eq.3) at small perturbation ��1, the bandgap is about�sp /�2, while at large perturbation the bandgap tends to�. The spectrum of the finite chain consists of eight dis-rete levels [dotted lines in Figs. 2(d), 2(f), and 2(h)]. Withncreasing coupling, the mixing of s- and p-defect statesncreases. Therefore, both s and p defects contribute toach eigenstate [compare Fig. 2(b) and Figs. 2(d), 2(f), and(h)]. From Fig. 2 we can see how, for �0.4, two statesplit from the allowed bands and move into the invertedandgap. These states become the surface modes ashown in Figs. 2(d), 2(f), and 2(h). The surface modes areescribed by the complex wave vector k=kr± i. The local-zation length of the surface mode is determined by ls

1/. The larger the value of is, the stronger the modes localized to the surface. By comparing Figs. 2(e) and(g), we note that, with increasing coupling �sp, the valuef kr shifts from the middle of the Brillouin zone kr� / �2d� toward its edge at kr=0.Appearance of the surface states for a chain of four in-

omplete unit cells is demonstrated in Fig. 3. Now, evenor the uncoupled chain, each eigenvalue corresponds to aingle eigenfunction [Figs. 3(a) and 3(b)], and all thetates are nondegenerate. We can see in Fig. 3(a) that forhe uncoupled chain one of the discrete levels is locatedxactly in the middle of an allowed band. At any finitealue of perturbation ��0�, this state gets inside the in-erted bandgap and forms the surface state as shown inigs. 3(c) and 3(d). With increase in coupling, anothertate falls from the allowed band into the forbidden bandnd becomes the surface state [see Figs. 3(e)–3(h)].In Fig. 2, we saw that surface states of the chain with

omplete unit cells do not appear simultaneously with thepening of the bandgap. Surface states appear only afterhe perturbation reaches a critical value, �cr�0.4. In con-rast, the surface states appear at any ��0 in the case oftructure 2 with incomplete unit cells as shown in Fig. 3.he explanation of this difference lies in the discrete na-

ure of the spectrum of the structures studied. Let us firstonsider the case of complete unit cells with equal num-er �N� of the s and p defects. Let us also assume that Ns even. In the case of the uncoupled chain ��=0�, N dis-rete levels of the chain should be distributed through thellowed bandwidth of 4�, so that the separation betweenhe levels is about �4� /N [see Fig. 2(a)]. Moreover, since

is even, no level gets into the middle of the band. Whatappens when we switch on the coupling? As was men-

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ioned, for weak coupling, the width of the inverted band-ap will increase as 4�sp /�2. When the inverted bandgapecomes as large as the separation between defects,�sp /�2=4� /N, the allowed bands of the hybridized sphain cannot accommodate all the states. As a result, twotates fall inside the forbidden gap [see Fig. 2(c)]. Fromhis general analysis we obtain a rough estimate for theritical value of perturbation �cr=�2/N, which is consis-ent with the critical value for the studied chain of fouromplete unit cells, �cr=�2/4�0.4.

We present in Fig. 4(a) the critical value �cr as a func-ion of the number of defects. The data calculated fromq. (4) are shown by a solid curve. For comparison we alsoresent the dependence �2/N by a dashed curve. We canee that the value of �cr��2/N can be used as a roughstimate. From Fig. 4(a) we note that for the short chainith N�10 the critical parameter is extremely sensitive

o the number of defects. If we design such a structureith parameters so that ��cr, then, depending on the

ize of the chain, the structure may or may not have theurface states. On the other hand, surface states of a simi-ar structure of fixed size can be controlled through the

odest change of parameters � or �sp or both. It is impor-ant to mention that such a size effect of the surfacetates cannot be observed in solids. Because of a hugeumber of atoms in solids, the Shockley surface states ap-ear simultaneously with the opening of the inverted
andgap.
In the above analysis we assumed that �s=�p. Now weant to understand what will happen if we gradually in-

rease the value of �s−�p . If �s−�p becomes as large ashe width of the allowed band of the uncoupled chain,�s−�p 4�, overlapping between the s and p bands is re-

oved. In this case, the Shockley surface states can ap-ear only if 4�sp reaches the width of the hybridizedands. In Fig. 4(b) we show the dependence of �cr as aunction of ��s−�p� /� calculated from Eq. (4) for the chainf four unit cells. We can see that, for ��s−�p� /��−4, thealue of �cr approaches infinity. Therefore, the surfacetates cannot appear at any perturbation �. In contrast,or ��s−�p� /�4, the value of �cr increases but stays fi-ite. This implies that the condition for the surface statesan be satisfied at some finite perturbation �. In the in-erval of �s−�p /�= �0 4�, function �cr is periodical, withodes at �s−�p /��1.25 and 3.2. This fact can be ex-lained by the discrete spectrum of the chain: the value ofcr tends to zero as soon as one of the discrete levels of thencoupled s chain overlaps with one of the levels of the phain. The number of nodes is equal to the number of unitells. Increasing the number of unit cells will smooth theteps of function �cr. In the limit N→ , �cr tends to 0 inhe interval �s−�p /�= �0 4�, in accordance with the well-nown result for solids.10,11

The situation is different for chains with incompletenit cells. In this case, we have an odd number of one typef defects (say, s defects) and an even number of another

ig. 2. (Color online) Energy spectrum of the defect chain with four complete unit cells for �=0 (a), (b); 0.4 (c), (d); 0.8 (e), (f); and 3 (g),h). (a), (c), (e), (g) Solid and dotted–dashed curves show dispersion relations of infinite coupled and uncoupled chains, respectively; thetars show the discrete spectrum of the finite chain with coupling. (b), (d), (f), (h) Wave functions for each eigenvalue of the discrete chainresented in ascending order from the bottom to the top of the figure.

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ype (say, p defects). Then, for the uncoupled chain, one ofhe discrete levels (associated with s defects) will be lo-ated exactly at the cross point between the s and the pands [Fig. 3(a)]. Hence at any finite perturbation ��0,his state falls inside the inverted bandgap and becomes aurface state [Fig. 3(c)]. However, this analysis corre-ponds to the ideal symmetrical situation (�s=−�p ands=�p). In a real situation with a small asymmetry in thepectrum of s and p defects (�s�−�p and �s��p), theritical value �cr may not be equal to zero for any chain. Itan be shown that in the case of small asymmetry in the

ig. 3. (Color online) Energy spectrum of structure 2 with four ina), (c), (e), (g) Solid and dotted–dashed curves show dispersiontars show the discrete spectrum of the finite chain. (b), (d), (f), (hrder from the bottom to the top of the figure.

ig. 4. Critical value �cr as (a) a function of the number of de-ects and (b) as a function of ��s−�p� /� for the chain of four unitells. The data calculated from Eq. (4) are shown by solid curves.he approximate value of the critical coupling �2/N is shown bydashed curve in (a).

pectrum, the estimate �cr��2/N is still a good approxi-ation for any number of defects in the chain.

. NUMERICAL RESULTSext, we will show that the theoretical analysis presented

n Section 2 can be verified by numerical simulation. Forhe numerical investigation we use the FDTDechnique.18 Our computational domain shown in Fig. 1as divided into uniform square mesh �x=a /40, where a

s the lattice constant of the host crystal. The computa-ional domain was surrounded by perfectly matched lay-rs (gray boxes in Fig. 1), with the thickness correspond-ng to ten layers of the discretization grid. The numericalimulations were performed with a total of 100,000 timeteps, with each time step �t=�x / �2c�. A Gaussian beamas launched at the input of the structure (S in Fig. 1).he spatial width of the beam was equal to 20 grid cells.To avoid the coupling between the allowed bands of the

ost crystal and the defect states, we have to take a PCith a large bandgap. We choose a square lattice of silicon

ods ��r=11.9� in vacuum ��o=1� with the radius of theods R=0.2a, which prohibits propagation of the TMode (electric field is parallel to the rods) in the frequency

ange as large as �̃=�a /2�c= �0.28,0.48�.19 Defects sup-orting the s and p modes can be created by the rods withadius Rs�R and Rp�R, respectively. We pick the twoefects with Rs=0.06a and Rp=0.3a, which have almosthe same eigenvalues �s=�p=0.358, close to the middle ofhe bandgap of the host PC. As shown in our recentaper,15 the eigenvalue of the defect decreases when the

ete unit cells for �=0 (a), (b); 0.1 (c), (d); 0.8 (e), (f); and 3 (g), (h).ns of infinite coupled and uncoupled chains, respectively, whilefunctions for each eigenvalue of the discrete chain in ascending

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efect is moved toward the surface of the crystal. The de-ect eigenfrequency was shown to decrease drasticallyhen the defect is located on the surface of the crystal.his effect eventually causes the Tamm-like surface state

n PCs. Since our focus in this paper is the Shockley sur-ace states, we want to minimize the effect of the Tammtates. Therefore, in the structures studied, we avoid de-ects on the surface of the crystal by adding one extraayer of the host crystal as shown in Fig. 1.

The results of our analysis for structures 1 and 2 arehown in Figs. 5 and 6, respectively. The schematics of thetructures are illustrated in the insets. The transmissionoefficients for these structures are presented in Figs. 5(a)nd 6(a), respectively. We can clearly see the opening ofhe inverted bandgap and the two surface modes (pointedut by the large arrows). The distributions of Ez for theseodes are presented in Figs. 5(d) and 5(e) and in Figs.

(c) and 6(d). To illustrate the hybridization of the spec-rum, in Figs. 5(c) and 5(f) we show the two modes corre-ponding to the central peaks of hybridized allowed bands

ig. 5. (Color online) Defect chain containing s and p modesith six complete unit cells. (a) Calculated transmission coeffi-

ient, where the large arrows indicate the two surface modes.he structure is shown in the inset. (b) Theoretical dispersion re-

ationship: the two crossed dashed–dotted curves show the spec-rum of the uncoupled chains of the s and p defects ��sp,ps=0�.he solid and dashed curves represent the supercell plane-wavealculation and the fitted dispersion relation �1,2�k� of Eq. (3) forhe infinite chain, respectively. The stars show the calculatedpectrum of the finite chain. (c), (d), (e), (f) The Ez distributionsor the modes pointed out by arrows in (a) are presented in de-cending order from top to bottom. (g), (h), (i), (j) The theoreticalave functions (dashed curves) in comparison with the ampli-

udes of the fields along the defect chain found from the FDTDimulations (solid curves) for the chains of (g), (h) six and (i), (j)en unit cells.

pointed out by the small arrows in Fig. 5(a)]. In agree-ent with the theoretical prediction [Figs. 2(g) and 2(h)]

or the structure with complete unit cells, the surfaceodes split from two different hybridized bands. It is in-

eresting to note that, in spite of strong hybridization [seeigs. 5(c) and 5(f)], one of the surface modes keeps theymmetry of the s band [Fig. 5(d)] and another keeps thatf the p band [Fig. 5(e)]. For the structure with incom-lete unit cells, both surface modes split from the sameand, holding the symmetry of the s band [Figs. 6(c) and(d)]. In both cases, the surface modes are not strongly lo-alized to the surface, since they lie close to the banddge.

It is instructive to analyze the results of the FDTDimulations in terms of the tight-binding model discussedn Section 2. This empirical model contains parameters,s,p, �s,p, and �sp,ps. We found these parameters by fittinghe dispersion relation of Eq. (3) with the exact (ab initio)pectrum of the infinite chain calculated by means of theupercell plane-wave technique. Using the parameters,e calculated the spectrum of the finite chain. The resultsf our analysis are summarized in Figs. 5(b) and 6(b). Theolid curves present the supercell plane-wave calculationf the spectrum for the infinite chain. The dashed–dottednd dashed curves show dispersion relation (3) for the in-nite chain of uncoupled s and p defects ��sp,ps=0� and forhe infinite chain of coupled defects ��sp,ps�0�, respec-ively. The stars show the discrete spectrum of the finitehain calculated from Eq. (4). The theoretical analysislearly demonstrates the two surface states close to theand edges. These states are characterized by the com-lex wave vector k=0± i. In the case of a structure withomplete unit cells, �0.3/d, which defines the localiza-ion length of the surface modes, ls�1/�3d. Using thearameters we have also calculated the wave functions ofhe surface states. These data are shown in Figs. 5(g) and(h) for the chain of six unit cells and in Figs. 5(i) and 5(j)or the chain of ten unit cells. We compare the theoreticalave functions (dashed curves) with the amplitudes of

he fields along the defect chain found from the FDTDimulations (solid curves). In spite of qualitative agree-ent between our theoretical model and numerical simu-

ations, we can see a noticeable discrepancy between theheoretical and the numerical results in the tails of theave functions. We explain this fact by the low intensityf the surface modes and their short lifetime, so that inhe tails the surface states overlap with the closest al-owed modes. Since in our experiment the excitation ofhe surface mode located at the crystal edge close to theource is much stronger than that of the mode located onhe opposite crystal side, the overlapping in the fieldsust be stronger for the latter than for the former [com-

are Figs. 5(g) and 5(i) and Figs. 5(h) and 5(j)].Finally, we analyze the critical conditions for the sur-

ace states. To test the relation of the surface states to thepectrum overlapping of the s- and p-defect chains, wearied the radius of the s defect in the range of Rs�0–0.16a� while keeping unchanged the radius of the pefect. In Fig. 7, we present the surface state energiesstars) for structure 1, identified from the FDTD simula-ions, as a function of Rs. We also show the band edges ofhe hybridized bands calculated using the supercell

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lane-wave technique (dashed line and curve). It is impor-ant to note that, owing to a finite discretization mesh ofhe FDTD simulations, there is an uncertainty in the ra-ius of defect rods. As a result, the relation between theDTD results and the plane-wave calculations is notnique. We found that in our simulations the uncertaintyas �R= ±0.003a. In Fig. 7 the uncertainty in Rs is

oughly specified by the horizontal size of star markers.Our calculations showed that when the radius of the s

efect is increased from Rs=0 to 0.15a the eigenvalue �secreases from 0.38 to 0.29 in comparison with the eigen-alue �p=0.358 of the p defect. Indeed, the dashed curvend line in Fig. 7 illustrate how the s band moves withespect to the p band. We can see that at Rs=Rs

0�0.08ahe spectrum of the chain becomes gapless. From disper-ion relation (3) at ��0, the gapless spectrum, � = �

ig. 7. (Color online) Surface states calculated from the FDTDimulations (star markers) and the band edges calculated usinghe supercell plane-wave technique (dashed line and curve) fortructure 1 of six unit cells as a function of Rs.

ig. 6. (Color online) Defect chain of s and p modes with seven inndicate the two surface modes. The structure is shown in the inso those in Fig. 5(b). (c), (d) The Ez distributions for the two surf

g 1

�2 =0, can be obtained only if �p−�s=2��s−�p�. Since byefinition ��0 for our structures, we have to assume thathis condition is satisfied at Rs=Rs

0. Then, taking into ac-ount that �s decreases with increasing Rs, we draw aonclusion that, at Rs�Rs

0, ��s−�p��−2��s−�p��−4�. Ase discussed in Section 2 [see Fig. 4(b)], the surface states

annot appear under such a condition. Indeed, as shownn Fig. 7, at Rs0.08a the surface states merge into thellowed bands. In contrast, at Rs=0a, a separation be-ween s and p states does not reach the critical value. Asresult, the surface states remain well defined at 0�Rs0.06a as shown in Fig. 7. We conclude that the surface

tates can be controlled, for example, by changing the ra-ius of one of the defects in the chain. Alternatively, aimilar effect can be achieved by one’s varying the dielec-ric constant of the corresponding defect.

In Section 2 we discussed another possibility for con-rolling the surface states. We showed that in the case of aeakly coupled system ��cr� the surface states can be

ontrolled by one’s changing the size of the structure. Asollows from the geometry of the structure studied, thealue of � cannot be less than 1. Indeed, � decreases fromto 1 when Rs increases from 0 to 0.14a. Such structures

orrespond to a system with strong coupling between theand the p modes. From Fig. 4(b) we might expect that��cr when ��s−�p��−4�. As we have just discussed,

uch a situation must correspond to the structure with aapless spectrum. Then it is reasonable to take a struc-ure close to the gapless one with Rs�Rs

0. We choose thetructure with Rs=0.075a. In Fig. 8(a) we show how, byncreasing the number of unit cells, the surface statesstars) move with respect to the edges of allowed bandsdashed lines). The uncertainty in Rs and Rp causesroadenings of the upper and lower band edges, respec-ively, which are shown by dashed areas in Fig. 8(a). Welso show the Ez distributions for the modes of high andow frequency in Figs. 8(b) and 8(c) and Figs. 8(d) and(e), respectively, for the structure of four [Figs. 8(b) and

ete unit cells. (a) Calculated transmission coefficient. The arrowsTheoretical dispersion relationship: all the notations are similardes indicated by arrows in (a).

complet. (b)

8tibtwsmtts

4WffwWmtmscwtawicrefioctSbtSabt

cttolscpAsaemc1dae

tfotmlhatucspats

tPpistp

FRE


(d)] and ten [Figs. 8(c) and 8(e)] unit cells. We can followhe evolution of the modes into surface states by increas-ng the system size: being located inside the allowedands in the short structure, the modes are hybridized be-ween the s and the p defects [see Figs. 8(b) and 8(d)];hen falling inside the bandgap in the long structure, the

ame modes become surface states, which experienceuch less hybridization and try to become purely either

he s or the p type [see Figs. 8(c) and 8(e)]. We concludehat the surface states in PCs can be controlled by theize of the structure.

. CONCLUSIONe have investigated in all detail the Shockley-type sur-

ace states of the defect chains embedded in a host PC. Weocused on the properties of the Shockley states in a chainith equidistant defects supporting the s and p modes.e analyzed this structure in terms of the tight-bindingodel, and then we verified theoretical predictions using

he FDTD simulations. We demonstrated a good agree-ent between the theoretical model and the numerical

imulations. Our particular interest was to understandritical conditions for the Shockley surface states in PCs,hich would allow for controlling them. We started from

he main condition for the Shockley states in solids, thenticrossing of bands. To facilitate the anticrossing effect,e designed the structure with the symmetrically cross-

ng s and p bands. For the structures studied, the anti-rossing of the bands is caused by the sp hybridization,esulting in the surface modes located close to the banddge of the inverted bandgap. We demonstrated for therst time, to our knowledge, that, in contrast to the casef solids, the surface states in PCs with complete unitells do not appear simultaneously with the opening ofhe inverted bandgap. The critical condition for thehockley states in PCs is that the width of the invertedandgap must reach a critical value of separation be-ween discrete levels. This results in a size effect of thehockley states in PCs, which implies that, in the case ofweakly coupled system ��cr�, the surface states can

e controlled by the size of the structure. We also showedhat the Shockley surface states can be controlled by

ig. 8. (Color online) (a) Calculated surface states (stars) and bs=0.075a as a function of the number of unit cells. The dashed

distributions for the modes of (b), (c) high and (d), (e) low freq
z
hanging the radius of one of the defects. Alternatively,he overlapping between the bands can be controlled byhe corresponding change in the dielectric constant of onef the defects through, say, electro-optical effects. We be-ieve that the surface states of a structure of a fixed sizepecially designed with ��cr can be highly efficientlyontrolled by a modest change of �. As mentioned, the �

arameter is extremely sensitive to the radii of defects.lternatively, it strongly depends on the dielectric con-tants of defects. The required small change of � could bechieved through the change of the defects’ radii or the di-lectric constants of defects made of electro-optical activeaterial. A preliminary analysis show that the required

hanges in radius and dielectric constant must be about%–5%. However, more detailed analysis of the depen-ence of coupling constants from the dielectric constantnd radius of the defects is needed to give a more precisestimate.

In this paper the surface states were illustrated usinghe FDTD simulations. In the structures studied the sur-ace modes are located close to the band edge and almostverlap with the allowed states. As a result, the peaks inhe transmission spectrum corresponding to the surfaceodes are similar to the peaks corresponding to the al-

owed states. Therefore, to identify the surface states, wead to use the transmission measurements along withnalysis of the field distributions of resonant peaks. Onhe other hand, the surface states can be easily identifiedsing near-field optics, when we put the collection pointlose to the edge defect. We point out that in the case of atructure with nonidentical bonds, discussed in our recentaper,14 the surface states fall deeply inside the bandgap,nd the peaks corresponding to these states are well dis-inguished from the allowed states even in the transmis-ion spectrum.

In this paper we studied the Shockley surface states inhe simplest case of the defect chain installed in a hostC. Besides a particular interest in the surface stateroblem in coupled defect structures, it would be interest-ng to understand if similar surface states can be ob-erved in a two-dimensional photonic crystal. In this con-ext, it is natural to generalize structures studied to aeriodical array of the defect chains or a PC with complex

dges of the hybridized bands (dashed lines) for structure 1 withat the band edges illustrate the uncertainty in defect radii. Thefor the structure of (b), (d) four and (c), (e) ten unit cells.

and eareasuency

bheStttsS

n

R

1

1

1

1

1

1

1

1

1

1


asis. We expect that such a PC can be also engineered toave the Shockley-like surface states with required prop-rties. Finally, we emphasize a crucial feature of thehockley states, which mostly distinguishes them fromhe Tamm states: the Shockley states are determined byhe bulk properties of the crystal and are independent ofhe surface perturbation. We believe that all these under-tandings are important for many applications of thehockley surface states in photonics.

The authors can be reached by e-mail [email protected] and asu.edu.

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Documents

Existence and control of Shockley surface states of a one-dimensional defect chain in a photonic crystal