. INTRODUCTIONhotonic crystal (PC) [1,2] is a special class of materialsith a periodic modulation in the dielectric property. Pre-ious studies of three-dimensional (3D) PC have revealedn enhancement in Bragg diffraction [3] and changes inhe symmetry of the diffraction pattern [4] compared withn ordinary diffraction grating. In a two-dimensional (2D)C, the in-plane diffraction was shown to exhibit spectralegions with very small diffraction efficiency [5] and thelazing effect [6]. The out-of-plane diffraction of a 2D PC,or example, a 2D PC with a weak dielectric modulation,ehaves similarly to a diffraction grating [7]. Although forCs with a finite dielectric modulation, the out-of-planeropagation properties have been reported with some newharacteristics in the zeroth-order beam [8–10], there iso report of the generic behavior of out-of-plane diffrac-ion.
In this paper, the out-of-plane diffraction of a 2D PC isnvestigated both theoretically and experimentally. In-tead of a 2D equal-frequency surface (EFS), which isypically used for an in-plane problem [11], we employ aD EFS to treat the out-of-plane problem. The 3D EFS of2D PC can be represented as a function �=h�kx ,ky ,kz�,here � is the normalized frequency, and kx, ky, kz are the
hree independent components of the recipirocal spaceith kx–ky representing the 2D periodic reciprocal plane.or a 2D PC with a weak dielectric modulation (i.e., a dif-
raction grating), the light incident normal to the 2D pe-iodic plane will be diffracted into discrete spots resem-ling the reciprocal lattice of the 2D PC [8], and if theitch is of the same order of the incident wavelength, aattern with discrete spots but a very large dispersiontill can be expected [12]. However, when the refractiveodulation of the PC is finite, we have found that the dis-
rete diffracted spots for a light incident normal to a dif-raction grating [8,12] become a continuous pattern in theiffraction plane. In addition, such a diffraction could oc-ur beyond the parameter region as limited by Bragg’srating equation.
. THEORYor a light normally incident on a 2D periodic plane, thez component, kzl, of the wave vector must be real in theC for the light to propagate inside the PC. Without lossf generality, we assume a structure with periodic holes intriangular lattice pattern with an air filling ratio of f.
he light propagation directions in the PC are determinedsing its EFS and applying the momentum conservationule across the interface [11], and thus the diffraction be-avior is essentially dependent on the shape and the cur-ature of the EFS.
The dispersion properties of a PC are best studied us-ng a ring beam [12], for which the wave vectors can beritten as k=g��sin � sin � ,sin � cos � , cos ��, where � is
he azimuthal angle, � is the angle between the wave vec-or and the kz axis [refer to Fig. 1(c) of [12]], g=2� /a, and=a /� with a and � being the pitch and the wavelength,
espectively. For a circular ring beam, � takes all the val-es from 0° to 360°, while the angle � is fixed. The experi-ental setup for realizing ring beam diffractions can be
ound in [12]. For a given wave vector at an angle of �� ,��,he momentum conservation line (MCL), which is parallelo the kz axis, conserves the transversal component acrosshe PC–air interface �k�sin �. If the MCL intersect withhe EFS of the PC, then the intersection point constitutesphotonic state, with the corresponding kz component of
he intersection point equal to k . The gradient of the
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FS at the intersection point between the MCL and theFS of the PC will determine the propagation directionsf the out-of-plane diffracted beams, which can be denotedy a similar set of angles ��o ,�o�. In the following subsec-ions, we will describe EFS and diffraction analyses forhe out-of-plane light propagation in a 2D PC.
. Equal-Frequency Surfacee first consider a 2D PC with a weak dielectric modula-
ion (WPC). PCs with a very small refractive index con-rast or a very small f are examples of WPCs. For the fol-owing discussion, we consider a WPC with a very small fhose homogenized refractive index is nav. To predict theiffraction pattern, the traditional amplitude gratingsan be considered WPCs with nav=1. The photonic bandsnd the EFS of a WPC can be constructed using Harri-on’s principles [13,14]. When the modulation is small,he frequency component of the periodic dielectric func-ion contains only one significant term [12]; hence theFS can be assumed to be spheres of radius �2� /a�nav�ttached to each reciprocal lattice point G. When � is verymall (i.e., in the long wavelength limit), there is only oneand involved. The EFS is given by a single sphere at-ached at the reciprocal lattice point G=0, and thepheres of adjacent lattice points do not intersect. If theCL intersects the EFS of the PC, the corresponding
tate is a transmitted state. On the other hand, when �ncreases until the sphere radius is of the order of Bril-ouin zone (BZ) dimensions, the adjacent spheres corre-ponding to different G points intersect and produceany intersection surfaces that form the bands for a
iven �.
ig. 1. (Color online) (a) Reciprocal lattice and BZ of a hexago-al lattice. (b) Cross section of EFS of WPC with nav=1.75, for=0.75 along kx axis (within the BZ). Bands 2, 4, and 6 overlapith bands 3, 5, and 7, respectively, along kx axis. (c) 3D plot ofFS ��=h�kx ,ky ,kz�� corresponding to band 2. (d) 3D plot of EFS
�=h�k ,k ,k �� corresponding to band 4.
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Figure 1(a) shows the reciprocal space of a hexagonalattice and the corresponding BZ. Figure 1(b) shows thex–kz cross section of the EFS (�=0.75, nav=1.75) wheny=0 with the band orders indicated. The parameters arehosen for convenience in matching the experiment. Itan be seen from Fig. 1(b) that there are seven bands con-ributing to the EFS of �=0.75. The EFS of band 1 origi-ates only from the sphere at G=0. As for the long wave-
ength limit, the state due to this band is a transmittedtate. The EFSs of the other six bands are made up of sixpheres originating from reciprocal lattice points G1hrough G6 [Fig. 1(a)]. These six bands interact with eachther to form several doubly degenerate bands along thex axis. This double degeneracy occurs only at symmetri-al axes of the lattice, and therefore in general there areix distinct bands. Moreover, there is a sixfold degeneracyt the point kx=ky=0. The 3D EFS plot of the second andourth bands are shown in Figs. 1(c) and 1(d), respec-ively. The EFSs clearly exhibit sharp edges along theymmetrical directions of the lattice, which are caused byntersections of six spheres originating from six differenteciprocal lattice points [i.e., G1–G6 shown in Fig. 1(a)].his is in fact the basic cause of the discrete nature of theiffracted spots seen in WPCs, including amplitude grat-ngs [8,12].
For a PC of finite dielectric modulation (FPC) (i.e., fi-ite f and finite refractive index contrast), the correspond-
ng dielectric function contains many frequency compo-ents, and thus the EFSs cannot be simply constructedsing spheres as for a WPC. In fact, a 3D plane-wave ex-ansion method has to be used [14,15] to compute thehotonic band structure of a FPC. From the photonicand structure, EFSs can be obtained by calculating allossible 3D wave vectors for a particular frequency [16].In Fig. 2(a), we show the kx–kz cross section of the
FS��=0.75� when ky=0 with the band orders indicatedor a FPC �f=0.25� with the averaged refractive indexav=1.75 [17], which equals that of the WPC in Fig. 1(b)
for comparison purposes). It can be seen from Fig. 2(a)hat the seven bands in the corresponding WPC with the
ig. 2. (Color online) (a) Cross section of EFS of FPC with f0.25 and an averaged index of 1.75 for �=0.75 along kx axis
within the BZ). (b) 3D plot of EFS ��=h�kx ,ky ,kz�� correspondingo band 3. (c) 3D plot of EFS ��=h�kx ,ky ,kz�� corresponding toand 7.
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ame nav split into separate bands due to polarizationoupling [9]. Furthermore, the sixfold degeneracy is liftednd pseudobandgaps are formed. There are still degener-te points, especially at the symmetrical points of the lat-ice, and such degenerate points are consequences of theymmetry possessed by the lattice and can be analyzedsing a group theoretical approach [18]. The 3D EFS plotsimilar to Fig. 1(c) and 1(d) (i.e., bands 3 and 7—the orderhanges after consideration of band splitting) for thePC are shown for the FPC in Figs. 2(b) and 2(c), respec-
ively. Obviously, the bands become flattened and moreontinuous for a FPC compared with those of a WPC.
. Diffraction Analysest �=0.75, a group of incident wave vectors is contained
n the ring beam [12], and the corresponding MCL of aPC will pass through seven different bands. The inter-
ection of the MCL with band 1 produces a transmittedtate, while the intersections with the rest of the bandsause diffraction states. We can see from Fig. 1(b) that theteepness of the band increases with the increase in theand order, which excites six different sets of wave vec-ors with different sets of (�o, �o) distribution for dif-racted light. The angles �o and �o can be obtained by nu-erically calculating the gradients of the EFS at the
oint where it intersects the MCL. When �=0, apart fromand 1, the MCL passes through the sixfold degenerateoint at which the other six bands intersect. The smooth-ess of the EFS at this point is identical for all bands,ausing the �o angle to be the same for all bands. Thiseans all diffracted rays resulting from a single degener-
te point have the same diffraction order.The number of degenerate points is large for a large �,
ince spheres from a large number of reciprocal latticeoints will be involved. Thus for a large �, many diffrac-ion orders are allowed. On the other hand, as opposed to
continuous � in the incident ring, the diffracted ringsake only discrete values of �o. This indicates physically
ig. 3. (Color online) Discrete curves: �o versus � for a WPCith nav=1.75 ��=0.75� for bands 2 (center, blue) and 4 (uppernd lower, red) shown in Figs. 1(c) and 1(d), respectively. Con-inuous curves: �o versus � for a FPC with f=0.25 and an aver-ged index of 1.75 ��=0.75� for band 3 (straight curve, green)nd 7 (sinusoidal curve, pink) shown in Figs. 2(b) and 2(c),espectively.
hat the diffraction spots are discrete in the diffractionlane. This is true for all bands in the case of a WPC, ands an example, the relationship between � and �o ishown in Fig. 3 for bands 2 and 4 of a WPC withav=1.75 and �=2°. This can be understood by consider-
ng the sharp edges observed in the bands of the WPCFigs. 1(c) and 1(d)]. The sharp edges mean the EFS is notifferentiable mathematically; consequently �o becomes aoncontinuous function for a WPC (Fig. 3). The relation-hip between � and �o for a FPC (nav=1.75, f=0.25, �2°) is also shown in Fig. 3, and as we can see, the dis-rete nature of �o turns out to be more continuous; theeason is that the corresponding bands of a FPC [Figs.(b) and 2(c)] are smoother. Figure 3 suggests that themoothness variation is also band-dependent in a FPC.
In a WPC, when the sphere radius is of the order of theZ dimensions, there is more than one transmitted state
or a given incident wave vector. These states are pro-ided by the higher-order bands of the PC (diffractiontates), and their number decreases as � decreases. Forxample, for band 2 of the WPC in Fig. 1(b), the EFS crossection in the kx–kz plane �ky=0� for various � is shown inig. 4(a), and the corresponding MCL for �=0 is alsohown.
As we can see from the figure, near �=0 the steepnessf the EFS increases as � decreases. For large �, the pho-onic states have a small gradient, indicating a smalleriffraction angle ��o�. As � decreases, the gradient in-reases and hence the diffraction angle �o increases. Ifhe interface medium is air, diffraction will be observedoth in the air and the WPC until �ca=2/�3, when theradient angle reaches the critical angle of the WPC–airnterface; i.e., diffraction in air happens only for a
2/�3�. This value of �ca is just a result of a 2D versionf the grating equation in air for normal incidence [12].ence, for ��2/�3, diffraction occurs only in the WPCnd ceases in air. The diffracted light in the WPC will notscape to air because of total internal reflection. The dif-raction in the WPC continues until a value of �cw=0.66 iseached, at which point the corresponding EFS gradient
ig. 4. (Color online) (a) Variations of band 2 [Fig. 1(c)] with re-pect to � with a step of 0.02 in a WPC along kx axis and withinhe BZ. (b) Variations of band 3 [Fig. 2(b)] with respect to � withstep of 0.02 in a FPC along kx axis and within the BZ. (c) kz asfunction of � for WPC and FPC, respectively, at �=0, showing
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Alagappan et al. Vol. 25, No. 5 /May 2008/J. Opt. Soc. Am. A 1101
s 90° to the MCL, with a zero kzl [kzl is the kz value at thentersection point of the MCL with the EFS].
As we can see from Fig. 4(a), there are no diffractiontates at �=0 for ��0.66. Thus for a normal incidence,he diffraction in a WPC with nav=1.75 is prohibited for�0.66. The �cw of triangular lattice WPC is given by/ �nav�3�. It has to be noted that �cw is again a 2D ver-ion of the diffraction condition for a medium with refrac-ive index nav at normal incidence.
As previously discussed, in a WPC, band 1 correspondso a transmitted state. When the modulation is increasedi.e., FPC), band 1 splits into two bands. In a FPC, thesewo split states correspond to two transmission states. Inhe long wavelength limit (i.e., ��1), they exhibit bire-ringence [19]. The diffraction states start to form fromand 3 onward in a FPC. For a FPC with f=0.25 and theame nav (1.75), the EFS cross section of band 3 in thex–kz plane for various � is given in Fig. 4(b).Apart from being flatter than the corresponding EFS ofWPC, there are states below the frequency specified by
he Bragg condition, �cw=2/ �nav�3�=0.66. There aretates (with a real kzl) existing until the correspondingutoff frequency of �cf=0.58. The plots of kzl as a functionf � are shown for both WPC and FPC in Fig. 4(c), wherehe diffraction cutoff condition is lowered from �cw=0.66o � =0.58.
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. EXPERIMENTinally, we provide experimental proof of the diffractioneyond the conventional parameter region with the con-inuous diffraction behavior. We fabricated a FPC withhe same f and nav as discussed (f=0.25 and nav=1.75) in
slab form on a silicon substrate. The slab contains ahemical-vapor-deposited silicon nitride (SiN) layer (re-ractive index=2) with a thickness of 250 nm on a 2 milica �SiO2� spacer layer. The 2D triangular lattice pat-ern with a=400 nm was created using deep-UV lithogra-hy in a square area of 2 mm by 2 mm. Subsequently, re-ctive ion etching was used to produce 2D periodic holesith f=0.25 and a depth of 350 nm. The deep etching was
o reduce radiation loss to the substrate [8–10]. Acanning-electron-microscopy (SEM) image of the samples shown in Fig. 5(a). The inset in the figure illustrateshe cross section of the sample.
The experiment was performed using a He–Ne laser20 mW output, wavelength=632 nm), and hence the cor-esponding experimental frequency is �expt=0.63 ��cf�expt=0.63��cw��ca�. The patterned slab was sand-iched between two media of different refractive indicesith periodic variation of reflectivity on the surface.ence, it is natural to show a phenomenon similar to anmplitude diffraction grating. As discussed before, such aiffraction is similar to WPC diffractions with n =1, and
av
ig. 5. (Color online) (a) SEM image of the sample (viewed at 45° to the surface normal). The inset in (a) shows a sketch of the crossection of the sample. The thickness of the SiN and the silica layer are 250 nm and 2 m, respectively. The depth of the air holes is50 nm. (b) Simulated diffraction pattern at �=0.63. (c) Experimental photograph of the diffraction pattern at �=0.63. (d) �o versus � forand 7–8.
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1102 J. Opt. Soc. Am. A/Vol. 25, No. 5 /May 2008 Alagappan et al.
ence will not occur, since �expt��cw=2/ �nav�3��1.15. Sohe diffraction pattern is due solely to the bands of thePC. Meanwhile, because of the deep etching, the holesxtend to the silica layer. Hence, the interface betweenhe SiN layer and the silica layer becomes a PC-(SiN–airoles PC)-to-PC (silica–air holes PC) interface. We com-uted for the silica–air holes PC that �cf=0.8 and �cw0.86. Since �expt=0.63��cf��cw, the diffraction due to
he silica–air holes PC should not occur, i.e., the diffrac-ion observed experimentally comes solely from theiN–air hole PC.For �expt=0.63 with nav=1.75, there are eight bands
resent (some are degenerate) at the symmetrical pointx=ky=0. Making use of such degeneracies, a ring beamith a small numerical aperture ���1° � was used to re-uce the calculation cost. The laser and the sample wereeparated by a screen with a small hole at the center. Theiffracted light beam was reflected from the sample androjected onto the screen and captured by a CCD [refer toig. 1(c) of [12] for the setup detail].The simulated [calculated from (�o, �o)] and experi-ental diffraction patterns are shown in Fig. 5(b) and
(c), respectively, exhibiting striking similarities forands 3 and 7–8. The diffraction pattern corresponding toand 4–6 is blocked by the sample and therefore cannote compared with that of the simulation [refer to Fig. 1(c)n [12] for the arrangement of the sample and the setupetails]. Unlike the simulation, the experimental diffrac-ion pattern for band 3 seems to be more diffuse comparedith that for band 7–8. Furthermore, the relative posi-
ions of the diffraction patterns for band 3 and band 7–8re different in Fig. 5(b) and 5(c). There are several rea-ons for the discrepancies between the simulated and ex-erimental diffraction patterns: (1) the ring beam widthas not considered in the simulation, (2) the silica–airole PC was treated as a homogenous silica layer withoutir holes, and (3) the confinement effect due to the finitehickness of the SiN–air hole PC was ignored in the simu-ations [20].
It is worth mentioning that the theoretically computedo angles for all bands are smaller than the estimatedritical angle for the FPC–air (33°), and hence light wasble to escape from the slab. The theoretically computedo versus � relationship for the band with the starlike dif-
raction pattern is plotted in Fig. 5(d); it is oscillating andontinuous, as expected from our analysis. For the bandith the circular diffraction pattern, the relationship be-
ween �o and � is linear. The experimental results pre-ented confirm the distinct diffraction behavior of a 2DPC.
. CONCLUSIONn conclusion, we have presented a generalized picture ofut-of-plane diffractions in a 2D photonic crystal usinghotonic bands. The discrete spots of diffraction in aeakly modulated photonic crystal, including those of
onventional diffraction gratings, gradually become con-inuous when the modulation becomes finite. Further-ore, in the finite-modulated photonic crystals, the dif-
raction can take place even in the region prohibited by
he famous Bragg law, since there are available states foriffraction to occur that would be evanescent in the casef diffraction gratings.
CKNOWLEDGMENTShis project is supported by Info-Comm Cluster grant
Idea-Bank) 2006ICTG03 of the Nanyang Technologicalniversity, Singapore. G. Alagappan thanks the Agency
or Science, Technology and Research �A*STAR�, Sin-apore, for its support.
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