Two-dimensional defect modes in optically induced photonic lattices

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Two-dimensional defect modes in optically induced photonic lattices

WANG, YANG, AND CHENBloch surface, k 1 , k 2 lies in the first Brillouin zone, i.e.,−/T 1 k 1 /T 1 ,−/T 2 k 2 /T 2 , and G n x,y;k 1 ,k 2 is a periodic function in both x and y with periods T 1 and T 2respectively. All these Bloch modes B n x,y;k 1 ,k 2 ,k 1 ,k 2 the Brillouin zone,n=1,2,... form a complete set 31.In addition, the orthogonality condition between these Blochmodes is−−B m * x,y;k 1 ,k 2 B n x,y;kˆ1,kˆ2dx dy= 2 2 k 1 − kˆ1k 2 − kˆ2m − n. 3.3Here the Bloch functions have been normalized by0T 1dx 0T 2dyGn x,y;k 1 ,k 2 2T 1 T 2=1,is the -function, and the superscript * represents complexconjugation. When 0, localized eigenfunctions i.e.,DMs can bifurcate out from edges of Bloch bands into bandgaps. Asymptotic analysis of these DMs for ≪1 will bepresented below.Consider bifurcations of defect modes from an edge point= c of the nth Bloch diffraction surface. If two or moreBloch diffraction surfaces share the same edge point, eachdiffraction surface would generate its own defect mode uponbifurcation, thus one could treat each diffraction surfaceseparately. After bifurcation, the propagation constant ofthe defect mode will enter the band gap adjacent to the diffractionsurface. Without loss of generality, we assume thatthe edge point of this diffraction surface is located at a symmetry point of the Brillouin zone where k 1 ,k 2 =0,0.If the bifurcation point is located at other points such as M orX points, the analysis would remain the same.When 0, defect modes can be expanded into Blochwaves asx,y = n=1−/T 1/T 1dk1−/T 2/T 2dk2 n k 1 ,k 2 B n x,y;k 1 ,k 2 ,3.4where n k 1 ,k 2 is the Bloch-mode coefficient in the expansion.In the remainder of this subsection, unless otherwiseindicated, integrations for dk 1 and dk 2 are always over thefirst Brillouin zone, i.e., the lower and upper limits for k 1 andk 2 in the integrals are always −/T 1 ,/T 1 and−/T 2 ,/T 2 respectively, thus such lower and upper limitswill be omitted below.When solution expansion 3.4 is substituted into the lefthand side of Eq. 3.1, weget / nk 1 ,k 2 B n x,y;k 1 ,k 2 dk 1 dk 2 = fx,yx,y,n=1where n k 1 ,k 2 is defined as3.5 n k 1 ,k 2 n k 1 ,k 2 − n k 1 ,k 2 . 3.6Due to our localization assumption on the defect functionfx,y, the right-hand side of Eq. 3.5 is a 2D-localizedfunction; thus its Bloch-expansion coefficient n k 1 ,k 2 isuniformly bounded in the Brillouin zone for all values of nand . When solution expansion 3.4 is further substitutedinto the right-hand side of Eq. 3.5 and orthogonality conditions3.3 utilized, we find that n k 1 ,k 2 satisfies the followingintegral equation n k 1 ,k 2 =2 2 m=1/ mkˆ1,kˆ2 − m kˆ1,kˆ2W m,n k 1 ,k 2 ,kˆ1,kˆ2dkˆ1dkˆ2,where function W m,n in the kernel is defined asW m,n k 1 ,k 2 ,kˆ1,kˆ2 =−−fx,yB n * x,y;k 1 ,k 2 3.7B m x,y;kˆ1,kˆ2dx dy. 3.8Again, since fx,y is a 2D-localized function,W m,n k 1 ,k 2 ,kˆ1,kˆ2 is also uniformly bounded for all k 1 ,k 2 and kˆ1,kˆ2 points in the Brillouin zone. The fact of functions n and W m,n being uniformly bounded is important in thefollowing calculations. Otherwise when fx,y is not 2Dlocalized,for instance, the results below would not be valid.At the edge point = c , n /k 1 = n /k 2 =0. For simplicity,we also assume that 2 n /k 1 k 2 =0 at this edgepoint — an assumption which is always satisfied for Eqs.2.1–2.3 we considered in Sec. II due to symmetries of itsdefective lattice. If 2 n /k 1 k 2 0 in some other problems,the analysis below only needs minor modifications. Underthe above assumption, the local diffraction function near a-symmetry edge point can be expanded aswherePHYSICAL REVIEW A 76, 013828 2007 n k 1 ,k 2 = c + 1 k 1 2 + 2 k 2 2 + ok 1 2 ,k 1 k 2 ,k 2 2 ,3.9 1 = 1 2 n2 k20,0, 21= 1 2 n. 3.102 k 220,0Since = c is an edge point, which should certainly be alocal maximum or minimum point of the nth diffraction surface,clearly 1 and 2 must be of the same sign, i.e., 1 20. The DM eigenvalue can be written as = c + h 2 ,3.11where = ±1, and 0h≪1 when ≪1. The dependenceof h on will be determined next.When Eqs. 3.9 and 3.11 are substituted into Eq. 3.7,we see that only a single term in the summation with indexm=n makes O n contribution. In this term, the denominator− n kˆ1,kˆ2 is very small near the symmetry pointbifurcation point kˆ1,kˆ2=0,0, which makes it O n rather than O n . The rest of the terms in the summationgive O m contributions, because the denominators 013828-4


TWO-DIMENSIONAL DEFECT MODES IN OPTICALLY…− m kˆ1,kˆ2 in such terms are not small anywhere in the firstBrillouin zone. Thus n k 1 ,k 2 =/ 2 2 nkˆ1,kˆ2W n,n k 1 ,k 2 ,kˆ1,kˆ2dkˆ1dkˆ2 − n kˆ1,kˆ2+ O n . 3.12Since the first term on the right-hand side of the above equationis O n rather than O n , it can balance the left-handside of that equation. This issue will be made more clear inthe calculations below. In order for the denominator in theintegral of Eq. 3.12 not to vanish in the Brillouin zone, wemust require that = − sgn 1 = − sgn 2 .3.13This simply means that the DM eigenvalue must lie insidethe band gap as expected.Now we substitute expressions 3.9 and 3.11 into Eq.3.12, and can easily find that n k 1 ,k 2 =/ 2 2 n kˆ1,kˆ2h 2 + 1 kˆ1 2 + 2 kˆ2 2W n,n k 1 ,k 2 ,kˆ1,kˆ2dkˆ1dkˆ2 + O n .3.14The above equation can be further simplified, up to errorO n ,as n k 1 ,k 2 =2 2 n0,0W n,n k 1 ,k 2 ,0,01 /h 2 + 1 kˆ1 2 + 2dkˆ1dkˆ2 + O n .2 kˆ23.15To calculate the integral in the above equation, we introducevariable scalings: k˜1= 1 kˆ1,k˜2= 2 kˆ2. Then Eq. 3.15becomes n k 1 ,k 2 =2 2 n 0,0W n,n k 1 ,k 2 ,0,01 21 /h 2 + k˜1 2 + 2dk˜1dk˜2 + O n ,k˜23.16where the integration is over the scaled Brillouin zone withk˜1 1 /T 1 and k˜2 2 /T 2 . This integration regioncan be replaced by a disk of radius 1−h 2 in the k˜1,k˜2plane, which causes error of O n to Eq. 3.16. Over thisdisk of radius 1−h 2 , the integral in Eq. 3.16 can be easilycalculated using polar coordinates to be −2 ln h, thus Eq.3.16 becomes n k 1 ,k 2 =− ln h2 1 2 n 0,0W n,n k 1 ,k 2 ,0,0 + O n .3.17Lastly, we take k 1 =k 2 =0 in the above equation. In order forit to be consistent, we must have2 1 2ln h =−W n,n 0,0,0,0 + O1. 3.18When this equation is substituted into Eq. 3.11, we finallyget a formula for the DM eigenvalue as = c + Ce −/ ,where is given by Eq. 3.13,3.19 = 4 1 2W n,n 0,0,0,0 ,3.20and C is some positive constant. Obviously and musthave the same sign, thus and W n,n 0,0,0,0 must havethe same sign. Since we have shown that and 1 , 2 haveopposite signs, we conclude that the condition for defectmodebifurcations from a -symmetry edge point is thatsgnW n,n 0,0,0,0 = − sgn 1 = − sgn 2 . 3.21Under this condition, the DM eigenvalue bifurcated fromthe edge point c is given by formula 3.19. Its distancefrom the edge point, i.e., − c , decreases exponentially withthe defect strength . This contrasts the 1D case where suchdependence is quadratic 10. The constant C in formula3.19 is much more difficult to calculate analytically, thuswill not be pursued here.If the edge point of DM bifurcations is not at a symmetrypoint, the DM bifurcation condition 3.21 and the DMeigenvalue formula 3.19 still hold, except that 1 , 2 andW n,n in these formulas should now be evaluated at the underlyingedge point in the Brillouin zone.Now we apply the above general results to the specialsystem 2.1–2.5 we were considering in Sec. II. When≪1, this system can be written asu xx + u yy + + Vx,yu = fx,yu + O 2 ,where Vx,y is given in Eq. 2.8, andPHYSICAL REVIEW A 76, 013828 20073.22fx,y =− E 0I 0 cos 2 xcos 2 yF D x,y1+I 0 cos 2 xcos 2 y 2 . 3.23In this case, W0,0,0,0 is always negative, thus DM bifurcationcondition 3.21 reduces tosgn = sgn 1 = sgn 2 .3.24Thus, in an attractive defect, DMs bifurcate out from normaldiffractionband edges i.e., lower edges of Fig. 1a wherethe diffraction coefficients are positive; in a repulsive defect,DMs bifurcate out from anomalous-diffraction band edgesi.e., upper edges of Fig. 1a where the diffraction coefficientsare negative. This analytical result is in agreementwith the heuristic argument in the end of Sec. II.013828-5


WANG, YANG, AND CHENε10.50−0.5−1i j k lasemi−infinitebgap1st−gap 2nd−gap4 6 8 10 12 Γ 14FIG. 2. Color online Bifurcations of defect modes with thedefect described by Eq. 2.2 at E 0 =15 and I 0 =6. Solid lines: Numericalresults. Dashed lines: Analytical results. The shaded regionsare the Bloch bands. Profiles of DMs at the circled points in thisfigure are displayed in Figs. 3 and 4.It should be mentioned that at some band edges, two linearlyindependent Bloch modes exist. Due to the symmetryof the lattice in Eq. 2.2, these two Bloch modes are relatedas Bx,y and By,x, where Bx,yBy,x. Under weakdefects 2.3, two different DMs of forms ux,y and uy,xbut with identical eigenvalues will bifurcate out from thesetwo Bloch modes of edge points, respectively. This can beplainly seen from the above analysis. Since these DMs havethe same propagation constant, their linear superposition remainsa defect mode due to the linear nature of Eq. 2.5.Such superpositions can create more interesting DM patterns.This issue will be explored in more detail in the nextsubsection.B. Numerical results of defect modes in localized defectsof various depthsThe analytical results of the previous subsection hold underweak localized defects. If the defect becomes strong i.e., not small, the DM formula 3.19 will be invalid. In thissubsection, we determine DMs in Eq. 2.5 numerically forboth weak and strong localized defects, and present varioustypes of DM solutions. In addition, in the case of weak defects,we will compare numerical results with analytical onesin the previous subsection, and show that they fully agreewith each other. The numerical method we will use is apower-conserving squared-operator method developed in30.In these numerical computations, we fix E 0 =15, I 0 =6,and vary the defect strength parameter from −1 to 1. Wefound a number of DM branches which are plotted in Fig. 2solid lines. First, we look at these numerical results underweak defects ≪1. In this case, we see that DMs bifurcateout from edges of every Bloch band. When the defect isattractive 0, the bifurcation occurs at the left edge ofµcPHYSICAL REVIEW A 76, 013828 2007each Bloch band, while when the defect is repulsive 0,the bifurcation occurs at the right edge of each Bloch band.Recall that the left and right edges of Bloch bands in Fig. 2correspond to the lower and upper edges of Bloch bands inFig. 1a where diffractions are normal and anomalous respectively,thus these DM bifurcation results qualitativelyagree with the theoretical analysis in the previous subsection.We can further make quantitative comparisons between numericalvalues of and the theoretical formula 3.19. Thiswill be done in two different ways. One way is that we havecarefully examined the numerical values for ≪1, andfound that they are indeed well described by functions of theform 3.19. Data fitting of numerical values of into theform 3.19 gives values which are very close to the theoreticalvalues of Eq. 3.20. For instance, for the DM branchin the semi-infinite band gap in Fig. 2, numerical data fittingfor 0≪1 gives num =0.1289 C num =0.4870. The theoretical value, obtained from Eqs. 3.20 and 3.23, is anal =0.1297, which agrees with num very well. Anotherway of quantitative comparison we have done is to plot thetheoretical formula 3.19 alongside the numerical curves inFig. 2. To do so, we first calculate the theoretical value from Eq. 3.20 at each band edge. Regarding the constant Cin the formula 3.19, we do not have a theoretical expressionfor it. To get around this problem, we fit this singleconstant from the numerical values of . The theoretical formulasthus obtained at every Bloch-band edge are plotted inFig. 2 as dashed lines. Good quantitative agreement betweennumerical and analytical values can be seen as well nearband edges.As increases, DM branches move away from bandedges toward the left when 0 and toward the right when0. Notice that in attractive defects 0, thesebranches stay inside their respective band gaps. But in repulsivedefects 0, DM branches march to edges of higherBloch bands and then reappear in higher band gaps. For instance,the DM branch in the first band gap reaches the edgeof the second Bloch band at −0.70 and reappears in thesecond band gap when −0.76. These behaviors resemblethose in the 1D case see Fig. 4 in 10. One differencebetween the 1D and 2D cases seems to be that, in 1D repulsivedefects, when DM branches approach edges of higherBloch bands, the curves become tangential to the verticalband-edge line. This indicates that these 1D defect modescannot enter Bloch bands as embedded eigenvalues insidethe continuous spectrum. This fact is consistent with thestatement “discrete eigenvalues cannot lie in the continuousspectrum” in 32 for locally perturbed periodic potentials in1D linear Schrödinger equations. In the present 2D repulsivedefects, on the other hand, the curves seem to intersect thevertical band-edge lines nontangentially. This may lead oneto suspect that these DM branches might enter Bloch bandsand become embedded discrete eigenvalues. This suspicionappears to be wrong however for two reasons. First, themathematical work by Kuchment and Vainberg 33 showsthat for separable periodic potentials perturbed by localizeddefects, embedded DMs cannot exist. In our theoreticalmodel 2.5, the periodic potential is nonseparable, thus theirresult does not directly apply. But their result strongly suggeststhat embedded DMs cannot exist either in our case.013828-6


TWO-DIMENSIONAL DEFECT MODES IN OPTICALLY…PHYSICAL REVIEW A 76, 013828 2007(a)(b)(c)(d)−5−5−5−5y00005555−5 0 5x−5 0 5x−5 0 5x−5 0 5x(e)(f)(g)(h)−5−5−5−5y00005555−5 0 5x−5 0 5x−5 0 5x−5 0 5xFIG. 3. Color online Profiles of defect modes in repulsive defects in Fig. 2. a–c DMs at the circled points a,b,c in Fig. 2, with,=−0.6,8.33, −0.8,10.73, and −0.6,12.31, respectively; d coexisting DM of mode c; e, f intensity and phase of the vortexmode obtained by superimposing modes c and d with /2 phase delay, i.e., in the form of ux,y+iuy,x; g, h intensity and phaseof the diagonally-oriented dipole mode obtained by superimposing modes c and d with no phase delay, i.e., in the form of ux,y+uy,x.Second, our preliminary numerical computations support thisnonexistence of embedded DMs. We find numerically thatwhen a DM branch enters Bloch bands in Fig. 2, the DMmode starts to develop oscillatory tails in the far field andbecomes nonlocalized thus not being a DM anymore.Now we examine profiles of defect modes on DMbranches in Fig. 2. For this purpose, we select one representativepoint from each DM branch, mark them by circles, andlabel them by letters in Fig. 2. DM profiles at these markedpoints are displayed in Figs. 3 and 4. The letter labels forthese defect modes are identical to those for the markedpoints on DM branches in Fig. 2. First, we look at Fig. 3,which shows DM profiles in repulsive defects 0. Wesee that DMs at points a,b of Fig. 2 are symmetric in both xand y, with a dominant hump at the defect site, and satisfythe relation ux,y=uy,x. These DMs are the simplest intheir respective band gaps, thus we will call them fundamentalDMs. Notice that these fundamental DMs are signindefinite,i.e., they have nodes where the intensities arezero, because they do not lie in the semi-infinite band gap.Although DMs in Figs. 3a and 3b look similar, differencesbetween them mainly in their tail oscillations do existdue to their residing in different band gaps. These differencesare qualitatively similar to the 1D case, which has been carefullyexamined before see Figs. 8b,d in 10 and Figs.4b,d in 8. The DM branch of point c is more interesting.At each point on this branch, there are two linearly independentDMs, the reason being that this DM branch bifurcatesfrom the right edge of the second Bloch band where twolinearly independent Bloch modes exist this band edge is aX symmetry point; see Fig. 1. These two DMs are shown inFigs. 3c and 3d. One of them is symmetric in x and antisymmetricin y, while the other is opposite. They are relatedas ux,y and uy,x with ux,yuy,x. These DMs aredipolelike. However, these dipoles are largely confined insidethe defect site. They are closely related to certain single-Bloch-mode solitons reported in 22.Since the DM branch of point c admits two linearly independentdefect modes, their arbitrary linear superpositionwould remain a defect mode since Eq. 2.5 is linear. Suchlinear superpositions could lead to new interesting DM structures.For instance, if the two DMs in Figs. 3c and 3d aresuperimposed with /2 phase delay, i.e., in the form ofux,y+iuy,x, we get a vortex-type DM whose intensityand phase structures are shown in Figs. 3e and 3f. Thisvortex DM is strongly confined in the defect site and lookslike a familiar vortex ring. It is qualitatively similar tovortex-cell solitons in periodic lattices under defocusing non-−5y05−5y05(i)−5 0 5x(l)−5 0 5x−505−505(j)−5 0 5x(m)−5 0 5x−505−505(k)−5 0 5x(n)−5 0 5xFIG. 4. Color online Profiles of defect modes in attractivedefects in Fig. 2. i–l DMs at the circled points i, j,k,l in Fig. 2,with ,=0.8,4.39, 0.8,8.12, 0.8,11.96, and 0.8,12.57, respectively.m, n DMs obtained by superimposing mode l andits coexisting mode with zero and phase delays, i.e., in the formof ux,y+uy,x and ux,y−uy,x, respectively.013828-7


WANG, YANG, AND CHENlinearity as reported in 22, as well as linear vortex arrays indefected lattices as observed in 7. However, it is quite differentfrom the gap vortex solitons as observed in 20,where the vortex beam itself creates an attractive defect withfocusing nonlinearity, while the vortex DM here is supportedby a repulsive defect. If the two DMs in Figs. 3c and 3dare directly superimposed without phase delays, i.e., in theform of ux,y+uy,x, we get a dipole-type DM whose intensityand phase profiles are shown in Figs. 3g and 3h.This dipole DM is largely confined in the defect site andresembles the DM of Figs. 3c and 3d, except that itsorientation is along the diagonal direction instead. This dipoleDM is qualitatively similar to dipole-cell solitons inuniform lattices under defocusing nonlinearity as reported in22. If the two DMs in Figs. 3c and 3d are superimposedwith phase delay, i.e., in the form of ux,y−uy,x, theresulting DM would also be dipolelike but aligned along theother diagonal axis. Such a DM is structurally the same asthe one shown in Figs. 3g and 3h in view of the symmetriesof our defected lattice.Now we examine DMs in attractive defects, which areshown in Fig. 4. At point i in the semi-infinite band gap, theDM is bell-shaped and is strongly confined inside the defectsite see Fig. 4i. This mode is guided by the total internalreflection mechanism, which is different from the repeatedBragg reflection mechanism for DMs in higher band gaps.On the DM branch of point j, there are two linearly independentDMs which are related as ux,y and uy,x. One ofthem is shown in Fig. 4j. This mode is dipolelike and resemblesthe one in Fig. 3c. Linear superpositions of thismode ux,y with its coexisting mode uy,x would generatevortex- and dipolelike DMs similar to those in Figs.3e–3h. In particular, the vortex DM at this point j wouldbe qualitatively similar to the gap vortex soliton as observedin 20. On the DM branch of point k, a single DM exists.This is because the left edge of the third Bloch band wherethis branch of DM bifurcates from is located at the M symmetrypoint of the Brillouin zone see Fig. 1 and admitsonly a single Bloch mode. The DM at point k is displayed inFig. 4k. This mode is largely confined at the defect site andis quadrupolelike. On the DM branch of point l, two linearlyindependent DMs exist which are related as ux,y anduy,x. One of them is shown in Fig. 4l. This mode issymmetric in both x and y directions and is tripolarlike withthree dominant humps. Unlike DMs at points c, j, this DM,superimposed with its coexisting mode with /2 phase delaywould not generate vortex modes, since this DM has nonzerointensity at the center of the defect. However, their superpositionswith zero and phase delays, i.e., in the forms ofux,y+uy,x and ux,y−uy,x would lead to two structurallydifferent defect modes, which are shown in Figs. 4mand 4n. The DM in Fig. 4m has a dominant hump in thecenter of the defect, surrounded by a negative ring, and withweaker satellite humps further out. The DM in Fig. 4n isquadrupolelike, but oriented differently from the quadrupolelikeDM in Fig. 4k. These DMs resemble the spike-arraysolitons and quadrupole-array solitons in uniform lattices underfocusing nonlinearity as reported in 22.Most of the DM branches in Fig. 2 bifurcate from edgesof Bloch bands. Even the branch of point b in the secondPHYSICAL REVIEW A 76, 013828 2007band gap of Fig. 2 can be traced to the DM bifurcation fromthe right edge of the first Bloch band. But there are exceptions.One example is the DM branch of point l in the upperright corner of the second band gap in Fig. 2 we will call itthe l branch below. This l branch does not bifurcate fromany Bloch-band edge. Careful examinations show that DMson this branch closely resemble Bloch modes at the lowest-symmetry point in the third Bloch band see Fig. 1. Thusthis l branch should be considered as bifurcating from thatlowest -symmetry point when ≪1. However, we see fromFig. 1 that this lowest -symmetry point is not an edge pointof the third Bloch band. Even though it is a local edge minimumpoint of diffraction surfaces in the Brillouin zone, itstill lies inside the third Bloch band. Let us call these type ofpoints “quasiedge points” of Bloch bands. Then the l branchof DMs bifurcates from a quasiedge point, not a true edgepoint, of a Bloch band. To illustrate this fact, we connectedthe l branch to this quasiedge -symmetry point at =0 bydotted lines through a fitted function of the form 3.19 inFig. 2. This dotted line lies inside the third Bloch band. Onequestion we can ask here is what is the nature of this dottedline? Since it is inside the Bloch band, the mathematicalresults by Kuchment and Vainberg 33 suggest that it cannotbe a branch of embedded DMs. On the other hand, the ‘l’DM branch in the second band gap does bifurcate out alongthis route. Then how does the ‘l’ branch bifurcate from thequasiedge point along this route? This question awaits furtherinvestigation. We note by passing that the and Mpoints inside the second Bloch band are also quasiedgepoints see Fig. 1a. Additional DM branches may bifurcatefrom such quasiedge points as well.IV. DEPENDENCE OF DEFECT MODES ON THEAPPLIED dc FIELD E 0In the previous section, we investigated the bifurcation ofDMs as the local-defect strength parameter varies. In experimentson photorefractive crystals, the applied dc field E 0can be adjusted in a wide range much more easily than thedefect strength. So in this section, we investigate how DMsare affected by the value of E 0 in Eqs. 2.1 and 2.2 underlocalized defects of Eq. 2.3. When E 0 changes, so does thelattice potential. We consider both attractive and repulsivedefects at fixed values of = ±0.9 the reason for thesechoices of values will be given below. For other fixed values, the dependence of DMs on E 0 is expected to be qualitativelythe same. The reader is reminded that throughout thissection, the I 0 value is fixed at I 0 =6.A. Case of repulsive defectsIn 2D photonic lattices with single-site repulsive defectswe experimentally created in 7, −1. At this value of ,however, we found that DMs may exist in high band gaps,but not in low band gaps see Fig. 2 where E 0 =15. Thus forpresentational purposes, we take =−0.9 in our computationsbelow. The corresponding lattice profile of I L x,y is shownin Fig. 5a, which still closely resembles the defects in ourprevious experiments 7.013828-8


TWO-DIMENSIONAL DEFECT MODES IN OPTICALLY…PHYSICAL REVIEW A 76, 013828 2007(a)(b)(c)−5−5y0−50−50y05−5 0 5x5−5 0 5x5−5 0 5x5−5(d)−5(e)−5(f)(a)−5 0 5xy050505601stgap2ndgap3ndgap−5 0 5x−5 0 5x−5 0 5xE 0(b)453015ab00 10 20 30 40 50FIG. 5. Color online a Profile of the lattice I L x,y in Eq.2.2 with a repulsive localized defect of Eq. 2.3 and =−0.9 I 0=6. b The corresponding DM branches in the ,E 0 parameterspace. Shaded: Bloch bands. DMs at the circled points are displayedin Fig. 6.For this defect, we found different types of defect modesat various values of E 0 . The results are summarized in Figs.5b and 6. In Fig. 5b, the dependence of DM eigenvalues on the dc field E 0 is displayed. As can be seen, when E 0increases from zero, a DM branch appears in the secondband gap at E 0 8.5. When E 0 increases further, this branchmoves toward the third Bloch band and disappears at E 015. But when E 0 increases to E 0 23.4, another DMbranch appears in the third band gap. This branch movestoward the fourth Bloch band and disappears at E 0 36.7. Inthe fourth gap, there are two DM branches. The upper branchexists when 45.9E 0 64.2, and the lower branch existswhen 31.6E 0 40.8. As E 0 increases even further, DMbranches will continue to migrate from lower band gaps tohigher ones. These behaviors are qualitatively similar to the1D case see Fig. 7 in 10. Some examples of DMs markedby circles in Fig. 5 are displayed in Fig. 6. We can see thatDMs at points a,b,c are “fundamental” DMs. At point d,there are two linearly independent DMs related by ux,yand uy,x, one of which is displayed in Fig. 6d. TheseDMs are dipolelike, similar to the c branch in Fig. 2. LinearµdcFIG. 6. Color online a–d Defect modes supported by therepulsive localized defect of Fig. 5a at the circled points a,b,c,dof Fig. 5b, with E 0 ,=10.8,9.61, 27.6,22.06, 50.0,37.88,and 35.0,32.34, respectively. e, f Intensity profiles of the vortexand dipole modes obtained by superposing mode d and itscoexisting mode with /2 and zero phase delays.superpositions of these DMs with /2 and 0 phase delayswould generate a vortex DM and a diagonally-oriented dipoleDM, whose amplitude profiles are shown in Figs. 6eand 6f their phase fields are similar to those in Figs. 3fand 3h and thus not shown here.B. Case of attractive defectsFor attractive defects, we choose =0.9 following theabove choice of =−0.9 for repulsive defects. The latticeprofile at this value is shown in Fig. 7a. The dependenceof DM eigenvalues on the dc field E 0 is shown in Fig. 7b.Unlike DMs in repulsive defects, DMs in this attractive defectexist in all band gaps including the semi-infinite andfirst band gaps. In addition, DM branches here stay in theirrespective bandgaps as E 0 increases, contrasting the repulsivecase. DM profiles at marked points in Fig. 7b are displayedin Fig. 8. The DM in Fig. 8a is bell-shaped and allpositive, and it is the fundamental DM in the semi-infiniteband gap. This DM is similar to the one on the i branch ofFig. 2, and is guided by the total internal reflection mechanism.The DM in the first band gap is dipolelike and is displayedin Fig. 8b. This DM branch is similar to the j branchof Fig. 2. On this branch, there is another coexisting DMwhich is a 90° rotation of Fig. 8b. Linear superpositions ofthese two DMs could generate vortexlike and diagonallyorienteddipolelike DMs, as Fig. 3e–3h shows. In the secondband gap, there are two DM branches: The c branch andthe d branch. DMs on the c branch are quadrupolelike, andthis branch is similar to the k branch of Fig. 2. On thisbranch, there is a single linearly independent defect mode.The d branch is similar to the l branch of Fig. 2. On thisbranch, there are two linearly independent DMs which aretripolarlike and orthogonal to each other see Fig. 4l. A013828-9


WANG, YANG, AND CHENPHYSICAL REVIEW A 76, 013828 2007(a)(b)(c)−5−5y0−50−50y05−5 0 5x5−5 0 5x5−5 0 5x(d)(e)(f)5−5−5−5(a)−5 0 5xy050505E 0(b)60453015a b cde00 10 20 30 40 50µf−5 0 5x−5 0 5x−5 0 5xFIG. 8. Color online Defect modes supported by attractivelocalized defects of Fig. 7a at the circled points a,b,c,d,e, f ofFig. 7b, with E 0 ,=15,4.29, 15,8.00, 15,11.84,15,12.48, 30,24.23 and 42,36.69, respectively.differences between DMs in localized and nonlocalized defects.One is that in nonlocalized defects, DM eigenvaluescan bifurcate out from edges of the continuous spectrum algebraically,not exponentially, with the defect strength . Theother one is that in nonlocalized defects, DMs can be embeddedinside the continuous spectrum as embedded eigenmodes.For simplicity, we consider the following linearSchrödinger equation with separable nonlocalized defects,FIG. 7. Color online a Profile of the lattice I L x,y in Eq.2.2 with an attractive localized defect of Eq. 2.3 and =0.9 I 0=6. b The corresponding DM branches in the ,E 0 parameterspace. Shaded: Bloch bands. DMs at the circled points are displayedin Fig. 8.superposition of these DMs with zero phase delay creates ahump-ring-type structure which is shown in Fig. 8d seealso Fig. 4m. A different superposition would produce aquadrupole-type structure as the one shown in Fig. 4n. Inthe third and fourth band gaps, there are additional DMbranches. These branches exist at E 0 values higher than 15,thus have no counterparts in Fig. 2 where E 0 =15. Profilesof these DMs are more exotic as can be seen in Figs. 8e and8f.V. DEFECT MODES IN NONLOCALIZED DEFECTSIn previous sections, we exclusively dealt with defectmodes in 2D-localized defects. We found that DM eigenvaluesbifurcate out from Bloch-band edges exponentially withthe increase of the defect strength see Eq. 3.19. Inaddition,DMs cannot be embedded inside Bloch bands i.e.,the continuous spectrum. In this section, we briefly discussDMs in nonlocalized defects, and point out two significantu xx + u yy + V D x + V D yu =−u,5.1where function V D is a one-dimensional defected periodicpotential of the formV D x =−E 01+I 0 cos 2 x1+F D x , 5.2F D x=exp−x 8 /128 is a defect function as has been usedbefore 10, and is the defect strength. We chose this separabledefect in Eq. 5.1 because all its eigenmodes bothdiscrete and continuous can be obtained analytically seebelow. The non-localized nature of this separable defect canbe visually seen in Fig. 9a, where the defected potentialV D x+V D y for I 0 =3,E 0 =6, and =−0.7 is displayed. Notethat this potential plotted here differs from the usual conceptof quantum-mechanics potentials by a sign, and it resemblesthe refractive-index function in optics. We see that along thex and y axes in Fig. 9a, the defect extends to infinity thusnonlocalized. The above I 0 and E 0 parameters were chosenbecause they have been used before in the 1D defect-modeanalysis of 10, and such 1D results will be used to constructthe eigenmodes of the 2D defect equation 5.1 below.Since the potential in Eq. 5.1 is separable, the eigenmodesof this equation can be split into the following form:013828-10


TWO-DIMENSIONAL DEFECT MODES IN OPTICALLY…ε−0.5(b)10.50y(a)−10010−10 0 10a b c4 6 8 10FIG. 9. Color online a Profile of a 2D potential with nonlocalizeddefects V D x+V D y in Eqs. 5.1 and 5.2 with E 0 =6,I 0 =3, and =−0.7. b DM branches supported by this nonlocalizeddefect in the , parameter space. Shaded: the continuous spectrum.DMs at the circled points are displayed in Fig. 10.ux,y = u a xu b y, = a + b , 5.3where u a , u b , a , b satisfy the following one-dimensionaleigenvalue equation:u a,xx + V D xu a =− a u a ,5.4u b,yy + V D yu b =− b u b .5.5These two 1D equations for u a and u b are identical to the 1Ddefect equation we studied comprehensively in 10. Usingthe splitting 5.3 and the 1D results in 10, we can constructthe entire discrete and continuous spectra of Eq. 5.1.Before we construct the spectra of Eq. 5.1, we need toclarify the definitions of discrete and continuous eigenvaluesof this 2D equation. Here an eigenvalue is called discrete ifits eigenfunction is square-integrable thus localized alongall directions in the x,y plane. Otherwise it is called continuous.Note that an eigenfunction which is localized alongone direction say x axis but nonlocalized along anotherxµdPHYSICAL REVIEW A 76, 013828 2007direction say y axis corresponds to a continuous, not discrete,eigenvalue.Now we construct the spectra of Eq. 5.1 for a specificexample with E 0 =6,I 0 =3 and =0.8. At these parametervalues, the discrete eigenvalues and continuous-spectrum intervals1D Bloch bands of the 1D eigenvalue problem 5.4are see 10 1 , 2 , 3 , ... = 2.0847,4.5002,7.5951, ... ,5.6I 1 ,I 2 ,I 3 ,I 4 ,I 5 ,I 6 , ... = 2.5781,2.9493,4.7553,6.6010,7.6250,11.8775, ....5.7Using the relation 5.3, we find that the discrete eigenvaluesand continuous-spectrum intervals of the 2D eigenvalueproblem 5.1 are 1 , 2 , 3 , 4 , ... = 2 1 , 1 + 2 ,2 2 , 1 + 3 , ...= 4.1694,6.5849,9.0004,9.6798, ... ,5.8and continuum = 1 + I 1 ,2I 2 , 1 + I 3 , = 4.6628,5.8986,6.8400, . 5.9Note that at the left edges of the two continuous-spectrumbands =4.6628 and 6.8400, the eigenfunctions are nonlocalizedalong one direction but localized along its orthogonaldirection, thus they are not the usual 2D Bloch modes whichwould have been nonlocalized along all directions. This iswhy for Eq. 5.1, we do not call these continuous-spectrumbands as Bloch bands.Repeating the same calculations to other values, wehave constructed the whole spectra of Eq. 5.1 in the ,plane for I 0 =3 and E 0 =6. The results are displayed in Fig.9b. Here solid curves show branches of DMs discrete eigenvalues,and shaded regions are the continuous spectrum.This figure shows two significant features which are distinctivelydifferent from those in localized defects. One is thatseveral DM branches such as the c and d branches areeither partially or completely embedded inside the continuousspectrum. This means that in nonlocalized defects, embeddedDMs inside the continuous spectrum do exist. Thiscontrasts the localized-defect case, where the mathematicalresults of Kuchment and Vainberg 33 and our numericsshow nonexistence of embedded DMs. Another feature ofFig. 9b is on the quantitative behavior of DM bifurcationsfrom edges of the continuous spectrum at small values of thedefect strength . We have shown before that in the 1D case,DMs bifurcate out from Bloch-band edges quadratically with 10. For the 2D problem 5.1, using the relation 5.3, wecan readily see that when DMs bifurcate out from edges ofthe continuous spectrum see the a,b,d branches, for instance,the distance between DM eigenvalues and the continuumedges also depends on quadratically. This contraststhe localized-defect case, where we have shown in Sec.III013828-11


WANG, YANG, AND CHENPHYSICAL REVIEW A 76, 013828 2007y−10010(a)−10 0 10x−10010(b)−10 0 10x−10010(c)−10 0 10x−10010(d)−10 0 10xFIG. 10. Color online Defectmodes in nonlocalized defects5.1 and 5.2 at the circled pointsa,b,c,d in Fig. 9b, with ,=0.8,4.17, 0.8,6.59,0.8,9.00, and −0.7,8.61,respectively.that DMs bifurcate out from Bloch-band edges exponentially.Even though the defect in Eq. 5.1 is nonlocalized ratherthan localized, the DMs it admits actually are quite similar tothose in localized defects. To demonstrate, we picked fourrepresentative points on the DM branches of Fig. 9b. Thesepoints are marked by circles and labeled by letters a,b,c,d,respectively. Profiles of DMs at these four points are displayedin Fig. 10. Comparing these DMs with those of localizeddefects in Figs. 3 and 4, we easily see that they are quitesimilar. In particular, the a,b,c,d branches in Fig. 9b resemblethe i, j,k,a branches in Fig. 2, as DMs on the correspondingbranches are much alike. Notice that DMs in thenonlocalized defect case are a little more spread out thantheir counterparts in the local-defect case compare Figs.4k and 10c, for instance. This is simply because in thenonlocalized defect, we chose E 0 =6, while in the localizeddefect, we chose a much higher value of E 0 =15. Higher E 0values induce deeper potential variations, which facilitatesbetter confinement of DMs. We need to point out that, eventhough the DMs at points c,d are embedded inside the continuousspectrum, their eigenfunctions are perfectly 2Dlocalizedand square-integrable see Fig. 10d in particular.Thus the embedded nature of a defect mode does not necessarilycause it to spread out more.VI. SUMMARYIn summary, we have thoroughly investigated defectmodes in two-dimensional photonic lattices with localized ornonlocalized defects. When the defect is localized and weak,we analytically determined defect-mode eigenvalues bifurcatedfrom edges of Bloch bands. We found that in an attractiverepulsive defect, defect modes bifurcate out fromBloch-band edges with normal anomalous diffraction coefficients.Furthermore, distances between defect-mode eigenvaluesand Bloch-band edges are exponentially small functionsof the defect strength, which is qualitatively differentfrom the 1D case. Another interesting phenomenon we foundwas that, some defect-mode branches bifurcate not fromBloch-band edges, but from quasiedge points within Blochbands. When the defect is localized but strong, defect modeswere studied numerically. It was found that both the repulsiveand attractive defects can guide various types of defectmodes such as fundamental, dipole, tripolar, quadrupole, andvortex modes. These modes reside in various band gaps ofthe photonic lattice. As the defect strength increases, defectmodes move from lower band gaps to higher ones when thedefect is repulsive, but remain within each band gap whenthe defect is attractive. The same phenomena are observedwhen the defect is held fixed while the applied dc field increases.If the defect is nonlocalized i.e., the defect does notdisappear at infinity in the lattice, we showed that DMsexhibit two new features: i DMs can be embedded insidethe continuous spectrum; ii DMs can bifurcate out fromedges of the continuous spectrum algebraically rather thanexponentially. These theoretical results pave the way for furtherexperimental demonstrations of various type of DMs ina 2D lattice.ACKNOWLEDGMENTSWe thank Dr. Dmitry Pelinovsky for helpful discussions.This work was partially supported by the Air Force Office ofScientific Research and National Science Foundation.1 P. St. J. Russell, Science 299, 358 2003.2 J. D. Joannopoulos, R. D. Meade, and J. N. Winn, PhotonicCrystals: Molding the Flow of Light Princeton UniversityPress, Princeton, NJ, 1995.3 D. N. Christodoulides, F. Lederer, and Y. Silberberg, NatureLondon 424, 817 2003.4 Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibersto Photonic Crystals Academic Press, New York, 2003.5 J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N.Christodoulides, Nature London 422, 147 2003.6 H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides,Phys. Rev. Lett. 92, 123902 2004.7 I. Makasyuk, Z. Chen, and J. Yang, Phys. Rev. Lett. 96,223903 2006.8 X. Wang, J. Young, Z. Chen, D. Weinstein, and J. Yang, Opt.Express 14, 7362 2006.9 F. Fedele, J. Yang, and Z. Chen, Opt. Lett. 30, 1506 2005.10 F. Fedele, J. Yang, and Z. Chen, Stud. Appl. Math. 115, 2792005.11 A. A. Sukhorukov and Y. S. Kivshar, Phys. Rev. Lett. 87,083901 2001.12 J. Yang and Z. Chen, Phys. Rev. E 73, 026609 2006.13 U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg,and Y. Silberberg, Appl. Phys. Lett. 75, 1348 1999.14 H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, andJ. S. Aitchison, Phys. Rev. Lett. 81, 3383 1998.013828-12


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