PHYSICAL REVIEW B 68, 205102 ~2003!

Holographic photonic crystals with diamond symmetry

D. N. Sharp,1 A. J. Turberfield,1,* and R. G. Denning21 Department of Physics, Clarendon Laboratory, University of Oxford,

Parks Road, Oxford OX1 3PU, United Kingdom2 Department of Chemistry, Inorganic Chemistry Laboratory, University of Oxford,

South Parks Road, Oxford OX1 3QR, United Kingdom~Received 15 April 2003; published 3 November 2003!

We explore the analytical design of high-symmetry photonic crystals made by holographic lithography. Weshow how holographic lithography may be used to produce diamond-like photonic crystals that have a full,three-dimensional photonic band gap at a refractive index contrast equal to the lowest yet published.

DOI: 10.1103/PhysRevB.68.205102 PACS number~s!: 42.70.Qs

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A photonic crystal is a dielectric medium in which perodic microstructure is used to create a very large periovariation of refractive index. Its microstructure is designedcontrol the dispersion of electromagnetic radiation~photonicband structure!, particularly for modes whose wavelengthare comparable to the lattice parameter.1,2 None of the avail-able techniques for the fabrication of visible optical or infrred three-dimensional~3D! photonic crystals allows complete freedom to design both lattice and basis to achievdesired photonic band structure. Self-assembly of collocrystals is the simplest but also the most restrictive fabrtion method: structures are generally limited to close-pacspheres in a face-centered-cubic~fcc! lattice. Some designfreedom is achieved by use of core-shell spheres with a nuniform distribution of refractive index,3,4 by using inversetemplated structures5–7 and by layer-by-layer growth of binary colloidal crystals, with two sphere sizes, to force diffent packing arrangements.8 Non-fcc packing has also beeobserved in colloidal structures grown within the physicconfines of lithographically defined pits,9,10 but only smallclusters of ;15 spheres are possible at present. Robassembly11 gives more control but it too is limited, so far, textremely small samples. Autocloning,12,13 which relies on abalance between sputter etching and sputter depositiodielectric layers on a patterned substrate, is more flexible,the structure is restricted to stacked layers with relatively lindex contrast. Use of semiconductor fabrication technoloto build up structures layer by layer14–16 gives independencontrol of the two-dimensional structure of each layer, bpractical structures are restricted to a few layers per unit~four in the case of a log pile structure17!. Holographiclithography18,19 is an alternative fabrication technique thuses a holographic interference pattern to create periodiccrostructure in photoresist. Holographic lithography allocomplete control of the translational symmetry of the phonic crystal and considerable freedom to design the unit cIn this paper we illustrate the flexibility, and some limittions, of this method by considering the design of photocrystals with diamond symmetry; we introduce a practitechnique for the fabrication of a structure with a photoband gap equal to the largest yet calculated.20

The structure of a photonic crystal produced by hographic lithography is defined by a three-dimensional, podic interference pattern that is used to expose a thick la

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of photoresist. In the case of a negative resist, underexpomaterial is then dissolved away in post-exposure processleaving a periodic network of material that has been mainsoluble by chemical changes initiated by exposure to hiintensity parts of the interference pattern. Post-exposurecessing thus imposes a threshold on the smoothly-varyintensity in the interference pattern; to a good approximatthe boundary between soluble and insoluble resist cosponds to a constant-intensity surface. To design a photcrystal it is therefore necessary to design a high-contrasttern containing a threshold-intensity surface of an approate form. Design criteria typically include connectivity of thdielectric and air components, dielectric filling fraction, asymmetry.

The electric field distribution in the interference patteformed at the intersection ofN mutually coherent lasebeams is given by

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where the amplitude and~in general! elliptical polarizationof thenth beam is represented by the complex vectoren andits wavevector bykn . In a linear, isotropic, homogeneoumedium, the time-averaged intensityI is

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The translational symmetry of the interference patternpends on its Fourier componentsgnm which are the differ-ences between laser wavevectors:gnm5kn2km . A four-beam interference pattern is described by 13 Foucomponents with coefficientsSnm5en•em* ~including the uni-form term S05(n50

3 Snn). As long as the Fourier componentsgnm,nÞm are noncoplanar, any three~say g10,g20,g30)may be chosen as primitive basis vectors to generate a thdimensional reciprocal lattice

Gi jk5 ig101 j g201kg30, i , j ,k integer.

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D. N. SHARP, A. J. TURBERFIELD, AND R. G. DENNING PHYSICAL REVIEW B68, 205102 ~2003!

All 13 Fourier components of the interference pattern liethis reciprocal lattice. A function whose discrete Fourcomponents lie on a reciprocal lattice is necessarily periothe general four-beam interference pattern has thdimensional translational symmetry.

The minimum number of beams required to createthree-dimensional, periodic interference pattern is four.cept in special cases of high symmetry, or when at leastpair of beams does not interfere with the others, the addiof more beams introduces incommensurate Fourier comnents that do not lie on the reciprocal lattice defined byfirst four; this destroys the periodicity of the interferenpattern. Aperiodic structures may have interesting photoproperties21,22 but only four-beam patterns with threedimensional translational symmetry will be considerbelow.

A four-beam interference pattern may be designed to pduceany Bravais lattice. The procedure is straightforwargenerate the corresponding reciprocal lattice;23 choose aprimitive basis set of three reciprocal lattice vectorsgn0; findthe reciprocal vectorq that is equidistant from the origin anfrom $gn0%. The four required optical wavevectors areq, q2g10, q2g20, andq2g30. For a given choice of basis vectors the lattice parameter scales with the laser wavelenFor a given Bravais lattice type there are an infinite numof different choices of primitive reciprocal lattice vectowhich give, in general, different lattice parameters. By stematically examining all$gn0% that lie sufficiently close tothe origin, we find that it is possible to generate fcc interfence patterns with 62 different lattice parameters less t3 mm using a 355-nm laser~although, as explained belowonly a few of them give useful interference patterns!.

The choice of polarization state for the interfering beais less trivial. Beam polarizations determine the FourierefficientsSnm5en•em* which are components of the structufactor that determines the crystallographic basis~the contentof a unit cell!. Freedom to design the basis is strongly costrained by the fact that the interference pattern has onlynonzero Fourier components; the structure factor at all oreciprocal lattice vectors is zero. All useful four-beam intference patterns that have been discussed to date have athat changes relatively slowly across the unit cell, withwhich it is possible to find threshold intensity contours thseparate connected regions of high and low intensity.achieve this, the nonzero Fourier components are clustnear the origin of the reciprocal lattice~that is why interfer-ence patterns with lattice parameters much greater thanlaser wavelength are unsuitable!. The further constraint thathe intensity is a real function@Snm5(Smn)* # means thatonly S0 ~real! and the six complex coefficientsSn,m,n ~13real parameters in all! completely define the intensity distrbution in a four-beam interference pattern. Three of thparameters determine the phases of the primitive Foucomponentsgn0 and therefore the position of the pattern retive to the origin of coordinates and one is an overall scalfactor for the intensity, leaving only nine ‘‘form’’ parameterthat specify the normalized intensity distribution withinunit cell. Control of these few form parameters, by adju

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ment of the polarization and relative intensities of the intfering beams, is the key to the design of the structure.

In an isotropic medium, the laser polarization and wavvector are orthogonal (en•kn50) so the four complex polarization vectorsen are defined by 16 adjustable parameteOne of these corresponds to the absolute phase of the elefield, which has no effect on the intensity distribution. Tresult of changing the relative phases of the beams~3 param-eters! is a uniform translation of the interference pattern; oparameter corresponds to an overall scaling of the intensthe remaining 11 polarization parameters are available toany target set of nine form parameters. We are not awarany general analytical procedure for choosing polarizatparameters to generate a desired interference pattern, orgeneral test that such a choice is possible. In this paperdiscuss this choice in special cases of high symmetry.

The full space group of the interference pattern maydeduced from the symmetry of its Fourier transform.24 Letr85Pr 1t be a space-group operation on the crystal, wheris a general point in real space,P is a point-group operatorand t is a translation vector. The reciprocal lattice has tsame point-group symmetries as the real-space lattice: ifG isa reciprocal lattice vector then so isG85PTG. The modulusof the structure factor has the same point grouuSG8u5uSGu. The phases of symmetry-related reflectiodiffer, in general, by a phase shift that is related to ttranslation component of the corresponding space grouperation and is nontrivial for screw and glide operationarg(SG8)5arg(SG)1G•t. ~The phase of the structure factohas the point-group symmetry of the crystal only for symorphic space groups which contain no translations otthan those of the Bravais lattice.!

Face-centered cubic photonic crystals with theFm3̄msymmetry of the Bravais lattice may be created usingbeam geometry shown in Fig. 1. In general the resultinterference pattern has nonzero Fourier components6@200#, 6@020#, and at all eight̂ 111& reciprocal latticevectors. The presence of only four out of the possible^200& components reduces the symmetry of the structuretor ~its point group cannot include the operation 3[111] , forexample! unless the polarizations are chosen such thatamplitudes of the6@200# and6@020# components are zeroHowever, this can be done by ensuring that the correspoing pairs of beams are orthogonally polarized.25 A set ofbeam polarizations that generates a fcc interference pawith Fm3̄m symmetry is given in Table I~a!; Table I~b!shows the corresponding structure factor which has comnents of equal amplitude at each^111& reciprocal latticepoint. This structure factor hasm3̄m point symmetry, and theinterference pattern~which may be considered as the supeposition of a fcc array of spherically symmetric function!

has the fullFm3̄m space group of the Bravais lattice. It hathe form

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HOLOGRAPHIC PHOTONIC CRYSTALS WITH DIAMOND . . . PHYSICAL REVIEW B68, 205102 ~2003!

where the sum runs over all^111& reciprocal lattice vectorsThe contrast of this interference pattern~defined as the ratioof maximum and minimum intensities! is infinite.

Figure 2 shows constant-intensity surfaces in this interence pattern. The spatially varying part of the intensity ptern changes sign under translation by half a lattice cons

FIG. 1. Reciprocal-space design of high-symmetry fcc hographic interference patterns.~a! Wavevectorskn of four laserbeams that produce a fcc interference pattern. By construcwavevector differenceskn2km form a basis that generates the breciprocal lattice shown.~b! Wavevector differenceskn2km corre-sponding to nonzero components of the structure factor. Thisincludes all eight of the family (2p/a)^111& ~solid vectors! and6(2p/a)@200#, 6(2p/a)@020# ~dashed vectors! but not6(2p/a)@002#.

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along any of the cubic axes: constant-intensity surfacexImax and (12x)I max, producing photonic crystals with dielectric filling fractionsn and 12n, are mapped onto eacother by this transformation. A surface corresponding tohigh threshold intensity@shown in Fig. 2~a!# defines identicaldisconnected regions of above-threshold intensity aroeach lattice point which in the high intensity limit are sphecal. A low-threshold-intensity surface defines isolated voiA startling consequence of the correspondence between hand low-intensity surfaces described above, and of the hsymmetry of the basis, is that a threshold exposure surfac12 I max creates the three-dimensional chequer board patshown in Fig. 2~b!. For the purposes of holographic lithography this interference pattern is impractical: if the threshintensity is set too high then insoluble regions are discnected and the structure falls apart; if it is set too low thsoluble regions are disconnected and solvent cannotetrate to remove underexposed material.~By sacrificingsome symmetry it is possible to engineer connectionstween above-threshold regions.25!

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FIG. 2. Constant-intensity surfaces in the interference patgenerated using the laser wavevectors shown in Fig. 1 and pizations given in Table I. The interference pattern has space gr

Fm3̄m corresponding to the space group of the fcc Bravais latt~a! I 50.93I max, ~b! I 50.53I max. Scale bar: laser wavelength.

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TABLE I. ~a! Choice of polarizations for the four beams shown in Fig. 1 that produces aFm3̄m

interference pattern.~b! Components of the structure factor that correspond to interference between pabeams. All^111& components are equal, and all other components are zero.@N.B. S2G5(SG)* .#

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D. N. SHARP, A. J. TURBERFIELD, AND R. G. DENNING PHYSICAL REVIEW B68, 205102 ~2003!

TABLE II. ~a! Choice of polarizations for the four beams shown in Fig. 1 that produces aFd3̄minterference pattern.~b! Corresponding structure factor.SG8 corresponds to a translation of the origin

coordinates bya@18

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Calculations have shown the local connectivity of a strture to be significant for photonic band gap formation, wgap formation appearing to prefer the low connectivity odiamond-like network.26 The widest calculated photoniband gaps are for structures with the diamond space gro20

~i.e., lower symmetry than structures shown in Fig. 2!. Oneway to generate a structure with the symmetry of diamoholographically would be to use a double exposure ofhighly symmetrical interference pattern shown in Fig. 2 wrelative phase shift between the four beams between e

sures to produce a translation of the pattern bya@ 14

14

14 #.36

A more elegant approach is to design the beam polartions to produce a structure withFd3̄m diamond symmetryin a single exposure. Table II shows an appropriate sebeam polarizations and the corresponding structure fact37

Again, the four^200& components have been set to zeroensuring that the corresponding pairs of beams are orthnally polarized. This pattern has an inversion center at

origin; if it is translated toa@ 18

18

18 # then all nonzero element

of the structure factor are seen to obey the conventionalfor diamond:Shkl}11exp@2i(p/2)(h1k1 l )# where@hkl#is a reciprocal lattice vector. Maximum contrast(A31A2)/(A32A2)59.90 is achieved foru5(4/3)(1/4);high contrast is a desirable feature as it minimizes constraon the reproducibility and nonlinearity of the photoresused.38 The interference pattern then has the form

I ~r!}2A32cosS 2p

a@111#•rD1cosS 2p

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Figure 3 shows constant-intensity surfaces in the interence pattern generated using polarizations given in TablA surface corresponding to a high threshold intensity@Fig.3~a!# defines two identical disconnected regions of abothreshold intensity in each primitive unit cell. These abovthreshold regions are spherical in the high intensity limit a

are centered at points6a@ 18

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~the coordinates of the carbon atoms in an atomic diamcrystal!. Again, the spatially varying part of the intensitpattern changes sign under translation by half a lattice cstant along any of the cubic axes: a low-threshold-inten

surface defines isolated voids at6a@ 38

38

38 # relative to each

lattice point. The constant-intensity surface at 0.73I max iscontinuous and divides continuously connected regiabove and below the threshold intensity. This is a suitathreshold-intensity surface for holographic lithography: svent can penetrate to remove material with below-threshexposure, leaving a robustly connected scaffold of insoluphotoresist that resembles the pattern of bonds in atodiamond.

Figure 4~a! shows the photonic band structure~calculatedusing the MIT Photonic-Bands Package27! of a photoniccrystal with air-filled voids and a dielectric of refractive index n52.5 occupying regions within the threshold intenssurface shown in Fig. 3~b!. There is a complete photoniband gap between the second and third bands. This isfundamental gap and is expected to be less susceptibstructural disorder than the higher gap that is open in inveopal structures.28 Figure 4~b! shows the fractional width ofthe complete gap as a function of filling fraction~which canbe varied by choosing different intensity threshold leve!

FIG. 3. Constant-intensity surfaces in the interference patgenerated using the laser wavevectors shown in Fig. 1 and pizations given in Table II. The interference pattern has the diam

space groupFd3̄m. ~a! I 50.973I max, ~b! I 50.73I max. Scalebar: laser wavelength.

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HOLOGRAPHIC PHOTONIC CRYSTALS WITH DIAMOND . . . PHYSICAL REVIEW B68, 205102 ~2003!

FIG. 4. ~a! Calculation of photonic band structure for an air-dielectric photonic crystal corresponding to the diamond structure3~b!, where dielectric with refractive indexn52.5 fills the region inside the threshold intensity surface.~b! Ratio of gap width~betweensecond and third bands! to mid-gap frequency for diamond structures with different dielectric filling fractions~given by different thresholdintensities in the interference pattern produced using parameters in Table II! and different refractive index contrasts. Positive ratio, abovehatched region, corresponds to a complete photonic band gap.

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and refractive index contrast. The minimum index contrrequired to produce a complete photonic band gap usingof this family of surfaces is slightly less than 1.9.29 The sameresult was obtained for a similar structure made of a netwof cylindrical dielectric rods;20 this is the lowest index contrast at which a full three-dimensional photonic band gapbeen reported. The epoxy photoresist that has been useholographic lithography18,30–33hasn51.6, so photonic crys-tals made directly in photoresist by holographic lithograpwould not have a complete gap. They may, however, be uas templates for the production of inverse replica structuby, for example, filling the voids with high-index dielectrand burning out or dissolving the photoresist master.18 Thesymmetry of these inverse structures is such that a photcrystal with dielectric filling factorn made by filling a holo-graphically defined template~filling factor 12n) has thesamesurface as the directly written structure with the safilling factor.

Most published examples of 3D holographic lithoraphy18,30,32,33use the beam configuration introduced in ofirst paper.18 This is a convenient arrangement becausefour beams are incident from the same half-space. It,

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creates a fcc interference pattern. Its nonzero Fourier cponents are less symmetrically distributed in reciprocal spthan in the configuration discussed above, the most smetrical pattern that can be generated has space g

F13̄2/m (R3̄m). These structures have rather small photoband gaps, and a large refractive index contrast (n*2.5) isnecessary to open an omni-directional gap. Other structthat have been reported are even less suitable, having hianisotropic bandstructure and no omni-directional gap31 ordisconnected voids that cannot be filled to create a hiindex replica.34 The diamond structure discussed above ismore suitable for the creation of an omni-directional phonic band gap than any other holographically defined photocrystal yet investigated.

By considering specific high-symmetry examples we hademonstrated that it is possible to understand and deholographic interference patterns that may be used to crphotonic crystals with optimized properties. We have shothat this technique is capable of defining structures whhave the highest photonic band gaps known for dieleccomposite media.

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piezoelectric control of the position of mirrors or beamsplitteor, in the case of nanosecond single-pulse exposure,18 by use ofa beamsplitter and optical delay line to produce a secondlayed pulse and fast electro-optic modulators to alter the phdelay in the arms of the interferometer.

37While preparing this manuscript we were made aware of a tunpublished paper containing a numerical solution for polaritions that generate a low-contrast interference pattern withmond symmetry.35

38When low-contrast interference patterns were used for hographic lithography30 it was found necessary to formulate thphotoresist specially to compensate for the uniform backgrointensity.

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