Two-dimensional defect modes in optically induced photonic lattices

WANG, YANG, AND CHENBloch surface, k 1 , k 2 lies **in** the first Brillou**in** 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 Brillou**in** zone,n=1,2,... form a complete set 31.In addition, the orthogonality condition between these Bloch**modes** 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 **in**to bandgaps. Asymptotic analysis of these DMs for ≪1 will bepresented below.Consider bifurcations of **defect** **modes** from an edge po**in**t= c of the nth Bloch diffraction surface. If two or moreBloch diffraction surfaces share the same edge po**in**t, 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 po**in**t of this diffraction surface is located at a symmetry po**in**t of the Brillou**in** zone where k 1 ,k 2 =0,0.If the bifurcation po**in**t is located at other po**in**ts such as M orX po**in**ts, the analysis would rema**in** the same.When 0, **defect** **modes** can be expanded **in**to 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 rema**in**der of this subsection, unless otherwise**in**dicated, **in**tegrations for dk 1 and dk 2 are always over thefirst Brillou**in** zone, i.e., the lower and upper limits for k 1 andk 2 **in** the **in**tegrals 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 **in**to 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 def**in**ed 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 Brillou**in** zone for all values of nand . When solution expansion 3.4 is further substituted**in**to the right-hand side of Eq. 3.5 and orthogonality conditions3.3 utilized, we f**in**d that n k 1 ,k 2 satisfies the follow**in**g**in**tegral 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 def**in**ed 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. 3.8Aga**in**, s**in**ce 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 po**in**ts **in** the Brillou**in** zone. The fact of functions n and W m,n be**in**g uniformly bounded is important **in** thefollow**in**g calculations. Otherwise when fx,y is not 2Dlocalized,for **in**stance, the results below would not be valid.At the edge po**in**t = c , n /k 1 = n /k 2 =0. For simplicity,we also assume that 2 n /k 1 k 2 =0 at this edgepo**in**t — an assumption which is always satisfied for Eqs.2.1–2.3 we considered **in** Sec. II due to symmetries of its**defect**ive lattice. If 2 n /k 1 k 2 0 **in** some other problems,the analysis below only needs m**in**or modifications. Underthe above assumption, the local diffraction function near a-symmetry edge po**in**t 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 220,0S**in**ce = c is an edge po**in**t, which should certa**in**ly be alocal maximum or m**in**imum po**in**t 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 determ**in**ed next.When Eqs. 3.9 and 3.11 are substituted **in**to Eq. 3.7,we see that only a s**in**gle term **in** the summation with **in**dexm=n makes O n contribution. In this term, the denom**in**ator− n kˆ1,kˆ2 is very small near the symmetry po**in**tbifurcation po**in**t 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 denom**in**ators 013828-4

TWO-DIMENSIONAL DEFECT MODES IN OPTICALLY…− m kˆ1,kˆ2 **in** such terms are not small anywhere **in** the firstBrillou**in** zone. Thus n k 1 ,k 2 =/ 2 2 nkˆ1,kˆ2W n,n k 1 ,k 2 ,kˆ1,kˆ2dkˆ1dkˆ2 − n kˆ1,kˆ2+ O n . 3.12S**in**ce 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 **in**the calculations below. In order for the denom**in**ator **in** the**in**tegral of Eq. 3.12 not to vanish **in** the Brillou**in** zone, wemust require that = − sgn 1 = − sgn 2 .3.13This simply means that the DM eigenvalue must lie **in**sidethe band gap as expected.Now we substitute expressions 3.9 and 3.11 **in**to Eq.3.12, and can easily f**in**d 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 **in**tegral **in** the above equation, we **in**troducevariable scal**in**gs: 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 **in**tegration is over the scaled Brillou**in** zone withk˜1 1 /T 1 and k˜2 2 /T 2 . This **in**tegration 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 **in**tegral **in** Eq. 3.16 can be easilycalculated us**in**g polar coord**in**ates 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 **in**to Eq. 3.11, we f**in**allyget 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. S**in**ce we have shown that and 1 , 2 haveopposite signs, we conclude that the condition for **defect**modebifurcations from a -symmetry edge po**in**t is thatsgnW n,n 0,0,0,0 = − sgn 1 = − sgn 2 . 3.21Under this condition, the DM eigenvalue bifurcated fromthe edge po**in**t c is given by formula 3.19. Its distancefrom the edge po**in**t, 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 po**in**t of DM bifurcations is not at a symmetrypo**in**t, 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 underly**in**gedge po**in**t **in** the Brillou**in** zone.Now we apply the above general results to the specialsystem 2.1–2.5 we were consider**in**g **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−**in**f**in**itebgap1st−gap 2nd−gap4 6 8 10 12 Γ 14FIG. 2. Color onl**in**e Bifurcations of **defect** **modes** with the**defect** described by Eq. 2.2 at E 0 =15 and I 0 =6. Solid l**in**es: Numericalresults. Dashed l**in**es: Analytical results. The shaded regionsare the Bloch bands. Profiles of DMs at the circled po**in**ts **in** thisfigure are displayed **in** Figs. 3 and 4.It should be mentioned that at some band edges, two l**in**early**in**dependent 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 weak**defect**s 2.3, two different DMs of forms ux,y and uy,xbut with identical eigenvalues will bifurcate out from thesetwo Bloch **modes** of edge po**in**ts, respectively. This can bepla**in**ly seen from the above analysis. S**in**ce these DMs havethe same propagation constant, their l**in**ear superposition rema**in**sa **defect** mode due to the l**in**ear nature of Eq. 2.5.Such superpositions can create more **in**terest**in**g DM patterns.This issue will be explored **in** more detail **in** the nextsubsection.B. Numerical results of **defect** **modes** **in** localized **defect**sof various depthsThe analytical results of the previous subsection hold underweak localized **defect**s. If the **defect** becomes strong i.e., not small, the DM formula 3.19 will be **in**valid. In thissubsection, we determ**in**e DMs **in** Eq. 2.5 numerically forboth weak and strong localized **defect**s, and present varioustypes of DM solutions. In addition, **in** the case of weak **defect**s,we will compare numerical results with analytical ones**in** the previous subsection, and show that they fully agreewith each other. The numerical method we will use is apower-conserv**in**g squared-operator method developed **in**30.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 l**in**es. First, we look at these numerical results underweak **defect**s ≪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 **in**Fig. 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 exam**in**ed the numerical values for ≪1, andfound that they are **in**deed well described by functions of theform 3.19. Data fitt**in**g of numerical values of **in**to theform 3.19 gives values which are very close to the theoreticalvalues of Eq. 3.20. For **in**stance, for the DM branch**in** the semi-**in**f**in**ite band gap **in** Fig. 2, numerical data fitt**in**gfor 0≪1 gives num =0.1289 C num =0.4870. The theoretical value, obta**in**ed 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 **in**Fig. 2. To do so, we first calculate the theoretical value from Eq. 3.20 at each band edge. Regard**in**g the constant C**in** the formula 3.19, we do not have a theoretical expressionfor it. To get around this problem, we fit this s**in**gleconstant from the numerical values of . The theoretical formulasthus obta**in**ed at every Bloch-band edge are plotted **in**Fig. 2 as dashed l**in**es. Good quantitative agreement betweennumerical and analytical values can be seen as well nearband edges.As **in**creases, DM branches move away from bandedges toward the left when 0 and toward the right when0. Notice that **in** attractive **defect**s 0, thesebranches stay **in**side their respective band gaps. But **in** repulsive**defect**s 0, DM branches march to edges of higherBloch bands and then reappear **in** higher band gaps. For **in**stance,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 repulsive**defect**s, when DM branches approach edges of higherBloch bands, the curves become tangential to the verticalband-edge l**in**e. This **in**dicates that these 1D **defect** **modes**cannot enter Bloch bands as embedded eigenvalues **in**sidethe cont**in**uous spectrum. This fact is consistent with thestatement “discrete eigenvalues cannot lie **in** the cont**in**uousspectrum” **in** 32 for locally perturbed periodic potentials **in**1D l**in**ear Schröd**in**ger equations. In the present 2D repulsive**defect**s, on the other hand, the curves seem to **in**tersect thevertical band-edge l**in**es 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 Va**in**berg 33 showsthat for separable periodic potentials perturbed by localized**defect**s, 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 onl**in**e Profiles of **defect** **modes** **in** repulsive **defect**s **in** Fig. 2. a–c DMs at the circled po**in**ts a,b,c **in** Fig. 2, with,=−0.6,8.33, −0.8,10.73, and −0.6,12.31, respectively; d coexist**in**g DM of mode c; e, f **in**tensity and phase of the vortexmode obta**in**ed by superimpos**in**g **modes** c and d with /2 phase delay, i.e., **in** the form of ux,y+iuy,x; g, h **in**tensity and phaseof the diagonally-oriented dipole mode obta**in**ed by superimpos**in**g **modes** c and d with no phase delay, i.e., **in** the form of ux,y+uy,x.Second, our prelim**in**ary numerical computations support thisnonexistence of embedded DMs. We f**in**d 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 be**in**g a DM anymore.Now we exam**in**e profiles of **defect** **modes** on DMbranches **in** Fig. 2. For this purpose, we select one representativepo**in**t from each DM branch, mark them by circles, andlabel them by letters **in** Fig. 2. DM profiles at these markedpo**in**ts are displayed **in** Figs. 3 and 4. The letter labels forthese **defect** **modes** are identical to those for the markedpo**in**ts on DM branches **in** Fig. 2. First, we look at Fig. 3,which shows DM profiles **in** repulsive **defect**s 0. Wesee that DMs at po**in**ts a,b of Fig. 2 are symmetric **in** both xand y, with a dom**in**ant hump at the **defect** site, and satisfythe relation ux,y=uy,x. These DMs are the simplest **in**their respective band gaps, thus we will call them fundamentalDMs. Notice that these fundamental DMs are sign**in**def**in**ite,i.e., they have nodes where the **in**tensities arezero, because they do not lie **in** the semi-**in**f**in**ite band gap.Although DMs **in** Figs. 3a and 3b look similar, differencesbetween them ma**in**ly **in** their tail oscillations do existdue to their resid**in**g **in** different band gaps. These differencesare qualitatively similar to the 1D case, which has been carefullyexam**in**ed before see Figs. 8b,d **in** 10 and Figs.4b,d **in** 8. The DM branch of po**in**t c is more **in**terest**in**g.At each po**in**t on this branch, there are two l**in**early **in**dependentDMs, the reason be**in**g that this DM branch bifurcatesfrom the right edge of the second Bloch band where twol**in**early **in**dependent Bloch **modes** exist this band edge is aX symmetry po**in**t; see Fig. 1. These two DMs are shown **in**Figs. 3c and 3d. One of them is symmetric **in** x and antisymmetric**in** 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 conf**in**ed **in**sidethe **defect** site. They are closely related to certa**in** s**in**gle-Bloch-mode solitons reported **in** 22.S**in**ce the DM branch of po**in**t c admits two l**in**early **in**dependent**defect** **modes**, their arbitrary l**in**ear superpositionwould rema**in** a **defect** mode s**in**ce Eq. 2.5 is l**in**ear. Suchl**in**ear superpositions could lead to new **in**terest**in**g DM structures.For **in**stance, 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 **in**tensityand phase structures are shown **in** Figs. 3e and 3f. Thisvortex DM is strongly conf**in**ed **in** the **defect** site and lookslike a familiar vortex r**in**g. It is qualitatively similar tovortex-cell solitons **in** periodic **lattices** under defocus**in**g 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 onl**in**e Profiles of **defect** **modes** **in** attractive**defect**s **in** Fig. 2. i–l DMs at the circled po**in**ts 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 obta**in**ed by superimpos**in**g mode l andits coexist**in**g 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 CHENl**in**earity as reported **in** 22, as well as l**in**ear vortex arrays **in****defect**ed **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** withfocus**in**g nonl**in**earity, 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 **in**tensityand phase profiles are shown **in** Figs. 3g and 3h.This dipole DM is largely conf**in**ed **in** the **defect** site andresembles the DM of Figs. 3c and 3d, except that itsorientation is along the diagonal direction **in**stead. This dipoleDM is qualitatively similar to dipole-cell solitons **in**uniform **lattices** under defocus**in**g nonl**in**earity as reported **in**22. If the two DMs **in** Figs. 3c and 3d are superimposedwith phase delay, i.e., **in** the form of ux,y−uy,x, theresult**in**g 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 **defect**ed lattice.Now we exam**in**e DMs **in** attractive **defect**s, which areshown **in** Fig. 4. At po**in**t i **in** the semi-**in**f**in**ite band gap, theDM is bell-shaped and is strongly conf**in**ed **in**side the **defect**site see Fig. 4i. This mode is guided by the total **in**ternalreflection mechanism, which is different from the repeatedBragg reflection mechanism for DMs **in** higher band gaps.On the DM branch of po**in**t j, there are two l**in**early **in**dependentDMs 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. L**in**ear superpositions of thismode ux,y with its coexist**in**g mode uy,x would generatevortex- and dipolelike DMs similar to those **in** Figs.3e–3h. In particular, the vortex DM at this po**in**t j wouldbe qualitatively similar to the gap vortex soliton as observed**in** 20. On the DM branch of po**in**t k, a s**in**gle DM exists.This is because the left edge of the third Bloch band wherethis branch of DM bifurcates from is located at the M symmetrypo**in**t of the Brillou**in** zone see Fig. 1 and admitsonly a s**in**gle Bloch mode. The DM at po**in**t k is displayed **in**Fig. 4k. This mode is largely conf**in**ed at the **defect** site andis quadrupolelike. On the DM branch of po**in**t l, two l**in**early**in**dependent 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 dom**in**ant humps. Unlike DMs at po**in**ts c, j, this DM,superimposed with its coexist**in**g mode with /2 phase delaywould not generate vortex **modes**, s**in**ce this DM has nonzero**in**tensity 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 dom**in**ant hump **in** thecenter of the **defect**, surrounded by a negative r**in**g, 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** underfocus**in**g nonl**in**earity as reported **in** 22.Most of the DM branches **in** Fig. 2 bifurcate from edgesof Bloch bands. Even the branch of po**in**t 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 po**in**t 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 exam**in**ations show that DMson this branch closely resemble Bloch **modes** at the lowest-symmetry po**in**t **in** the third Bloch band see Fig. 1. Thusthis l branch should be considered as bifurcat**in**g from thatlowest -symmetry po**in**t when ≪1. However, we see fromFig. 1 that this lowest -symmetry po**in**t is not an edge po**in**tof the third Bloch band. Even though it is a local edge m**in**imumpo**in**t of diffraction surfaces **in** the Brillou**in** zone, itstill lies **in**side the third Bloch band. Let us call these type ofpo**in**ts “quasiedge po**in**ts” of Bloch bands. Then the l branchof DMs bifurcates from a quasiedge po**in**t, not a true edgepo**in**t, of a Bloch band. To illustrate this fact, we connectedthe l branch to this quasiedge -symmetry po**in**t at =0 bydotted l**in**es through a fitted function of the form 3.19 **in**Fig. 2. This dotted l**in**e lies **in**side the third Bloch band. Onequestion we can ask here is what is the nature of this dottedl**in**e? S**in**ce it is **in**side the Bloch band, the mathematicalresults by Kuchment and Va**in**berg 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 po**in**t along this route? This question awaits further**in**vestigation. We note by pass**in**g that the and Mpo**in**ts **in**side the second Bloch band are also quasiedgepo**in**ts see Fig. 1a. Additional DM branches may bifurcatefrom such quasiedge po**in**ts as well.IV. DEPENDENCE OF DEFECT MODES ON THEAPPLIED dc FIELD E 0In the previous section, we **in**vestigated 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 the**defect** strength. So **in** this section, we **in**vestigate how DMsare affected by the value of E 0 **in** Eqs. 2.1 and 2.2 underlocalized **defect**s of Eq. 2.3. When E 0 changes, so does thelattice potential. We consider both attractive and repulsive**defect**s 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 rem**in**ded that throughout thissection, the I 0 value is fixed at I 0 =6.A. Case of repulsive **defect**sIn 2D **photonic** **lattices** with s**in**gle-site repulsive **defect**swe 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 correspond**in**g lattice profile of I L x,y is shown**in** Fig. 5a, which still closely resembles the **defect**s **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 onl**in**e 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 correspond**in**g DM branches **in** the ,E 0 parameterspace. Shaded: Bloch bands. DMs at the circled po**in**ts are displayed**in** Fig. 6.For this **defect**, we found different types of **defect** **modes**at 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 0**in**creases from zero, a DM branch appears **in** the secondband gap at E 0 8.5. When E 0 **in**creases further, this branchmoves toward the third Bloch band and disappears at E 015. But when E 0 **in**creases 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 **in**creases even further, DMbranches will cont**in**ue 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 po**in**ts a,b,c are “fundamental” DMs. At po**in**t d,there are two l**in**early **in**dependent 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. L**in**earµdcFIG. 6. Color onl**in**e a–d Defect **modes** supported by therepulsive localized **defect** of Fig. 5a at the circled po**in**ts 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** obta**in**ed by superpos**in**g mode d and itscoexist**in**g 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 **defect**sFor attractive **defect**s, we choose =0.9 follow**in**g theabove choice of =−0.9 for repulsive **defect**s. 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 **defect**s, DMs **in** this attractive **defect**exist **in** all band gaps **in**clud**in**g the semi-**in**f**in**ite andfirst band gaps. In addition, DM branches here stay **in** theirrespective bandgaps as E 0 **in**creases, contrast**in**g the repulsivecase. DM profiles at marked po**in**ts **in** Fig. 7b are displayed**in** Fig. 8. The DM **in** Fig. 8a is bell-shaped and allpositive, and it is the fundamental DM **in** the semi-**in**f**in**iteband gap. This DM is similar to the one on the i branch ofFig. 2, and is guided by the total **in**ternal reflection mechanism.The DM **in** the first band gap is dipolelike and is displayed**in** Fig. 8b. This DM branch is similar to the j branchof Fig. 2. On this branch, there is another coexist**in**g DMwhich is a 90° rotation of Fig. 8b. L**in**ear 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 s**in**gle l**in**early **in**dependent **defect** mode.The d branch is similar to the l branch of Fig. 2. On thisbranch, there are two l**in**early **in**dependent 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 onl**in**e Defect **modes** supported by attractivelocalized **defect**s of Fig. 7a at the circled po**in**ts 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 **defect**s.One is that **in** nonlocalized **defect**s, DM eigenvaluescan bifurcate out from edges of the cont**in**uous spectrum algebraically,not exponentially, with the **defect** strength . Theother one is that **in** nonlocalized **defect**s, DMs can be embedded**in**side the cont**in**uous spectrum as embedded eigen**modes**.For simplicity, we consider the follow**in**g l**in**earSchröd**in**ger equation with separable nonlocalized **defect**s,FIG. 7. Color onl**in**e 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 correspond**in**g DM branches **in** the ,E 0 parameterspace. Shaded: Bloch bands. DMs at the circled po**in**ts are displayed**in** Fig. 8.superposition of these DMs with zero phase delay creates ahump-r**in**g-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 **defect****modes** **in** 2D-localized **defect**s. We found that DM eigenvaluesbifurcate out from Bloch-band edges exponentially withthe **in**crease of the **defect** strength see Eq. 3.19. Inaddition,DMs cannot be embedded **in**side Bloch bands i.e.,the cont**in**uous spectrum. In this section, we briefly discussDMs **in** nonlocalized **defect**s, and po**in**t out two significantu xx + u yy + V D x + V D yu =−u,5.1where function V D is a one-**dimensional** **defect**ed 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 separable**defect** **in** Eq. 5.1 because all its eigen**modes** bothdiscrete and cont**in**uous can be obta**in**ed analytically seebelow. The non-localized nature of this separable **defect** canbe visually seen **in** Fig. 9a, where the **defect**ed 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-**in**dex function **in** optics. We see that along thex and y axes **in** Fig. 9a, the **defect** extends to **in**f**in**ity 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 eigen**modes** of the 2D **defect** equation 5.1 below.S**in**ce the potential **in** Eq. 5.1 is separable, the eigen**modes**of this equation can be split **in**to the follow**in**g 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 onl**in**e a Profile of a 2D potential with nonlocalized**defect**s 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 nonlocalized**defect** **in** the , parameter space. Shaded: the cont**in**uous spectrum.DMs at the circled po**in**ts are displayed **in** Fig. 10.ux,y = u a xu b y, = a + b , 5.3where u a , u b , a , b satisfy the follow**in**g one-**dimensional**eigenvalue 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 1D**defect** equation we studied comprehensively **in** 10. Us**in**gthe splitt**in**g 5.3 and the 1D results **in** 10, we can constructthe entire discrete and cont**in**uous spectra of Eq. 5.1.Before we construct the spectra of Eq. 5.1, we need toclarify the def**in**itions of discrete and cont**in**uous eigenvaluesof this 2D equation. Here an eigenvalue is called discrete ifits eigenfunction is square-**in**tegrable thus localized alongall directions **in** the x,y plane. Otherwise it is called cont**in**uous.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 cont**in**uous, 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 cont**in**uous-spectrum **in**tervals1D 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, ....5.7Us**in**g the relation 5.3, we f**in**d that the discrete eigenvaluesand cont**in**uous-spectrum **in**tervals 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 cont**in**uum = 1 + I 1 ,2I 2 , 1 + I 3 , = 4.6628,5.8986,6.8400, . 5.9Note that at the left edges of the two cont**in**uous-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 cont**in**uous-spectrumbands as Bloch bands.Repeat**in**g 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 cont**in**uous spectrum.This figure shows two significant features which are dist**in**ctivelydifferent from those **in** localized **defect**s. One is thatseveral DM branches such as the c and d branches areeither partially or completely embedded **in**side the cont**in**uousspectrum. This means that **in** nonlocalized **defect**s, embeddedDMs **in**side the cont**in**uous spectrum do exist. Thiscontrasts the localized-**defect** case, where the mathematicalresults of Kuchment and Va**in**berg 33 and our numericsshow nonexistence of embedded DMs. Another feature ofFig. 9b is on the quantitative behavior of DM bifurcationsfrom edges of the cont**in**uous spectrum at small values of the**defect** 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, us**in**g the relation 5.3, wecan readily see that when DMs bifurcate out from edges ofthe cont**in**uous spectrum see the a,b,d branches, for **in**stance,the distance between DM eigenvalues and the cont**in**uumedges 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 onl**in**e Defect**modes** **in** nonlocalized **defect**s5.1 and 5.2 at the circled po**in**tsa,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 **defect**s. To demonstrate, we picked fourrepresentative po**in**ts on the DM branches of Fig. 9b. Thesepo**in**ts are marked by circles and labeled by letters a,b,c,d,respectively. Profiles of DMs at these four po**in**ts are displayed**in** Fig. 10. Compar**in**g these DMs with those of localized**defect**s **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 correspond**in**gbranches 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 **in**stance. This is simply because **in** thenonlocalized **defect**, we chose E 0 =6, while **in** the localized**defect**, we chose a much higher value of E 0 =15. Higher E 0values **in**duce deeper potential variations, which facilitatesbetter conf**in**ement of DMs. We need to po**in**t out that, eventhough the DMs at po**in**ts c,d are embedded **in**side the cont**in**uousspectrum, their eigenfunctions are perfectly 2Dlocalizedand square-**in**tegrable 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 **in**vestigated **defect****modes** **in** two-**dimensional** **photonic** **lattices** with localized ornonlocalized **defect**s. When the **defect** is localized and weak,we analytically determ**in**ed **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 **in**terest**in**g phenomenon we foundwas that, some **defect**-mode branches bifurcate not fromBloch-band edges, but from quasiedge po**in**ts with**in** Blochbands. When the **defect** is localized but strong, **defect** **modes**were studied numerically. It was found that both the repulsiveand attractive **defect**s can guide various types of **defect****modes** such as fundamental, dipole, tripolar, quadrupole, andvortex **modes**. These **modes** reside **in** various band gaps ofthe **photonic** lattice. As the **defect** strength **in**creases, **defect****modes** move from lower band gaps to higher ones when the**defect** is repulsive, but rema**in** with**in** each band gap whenthe **defect** is attractive. The same phenomena are observedwhen the **defect** is held fixed while the applied dc field **in**creases.If the **defect** is nonlocalized i.e., the **defect** does notdisappear at **in**f**in**ity **in** the lattice, we showed that DMsexhibit two new features: i DMs can be embedded **in**sidethe cont**in**uous spectrum; ii DMs can bifurcate out fromedges of the cont**in**uous spectrum algebraically rather thanexponentially. These theoretical results pave the way for furtherexperimental demonstrations of various type of DMs **in**a 2D lattice.ACKNOWLEDGMENTSWe thank Dr. Dmitry Pel**in**ovsky 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. W**in**n, PhotonicCrystals: Mold**in**g the Flow of Light Pr**in**ceton UniversityPress, Pr**in**ceton, 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. Mart**in**, 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. 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