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Photonic Gap MaterialsJ.B. Pendry

The Blackett LaboratoryImperial College

London SW7 2BZUK

Abstract

The concept of a photonic material is explained in terms of Bragg diffraction. Analogiesare also made with electrons in semiconductors. Possible routes to manufacture of thesestructures are explored as some naturally occurring micro periodic structures. Thearticle concludes by giving some background to computational modelling of thesematerials.

published in:

Current Science 76 1311-6 (1999)

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1. The Concept of a Photonic Crystal.

As the world demands ever more of computers and communications we turnincreasingly to optical devices whose band width and speed of execution offer greatpotential. However materials science has lagged behind in this rush into the opticaldomain: optical properties of materials are not always well matched to the functions weseek. This is in contrast to the vast range of electronic properties available to us. In theelectronic domain we can almost make materials to order. The root cause of thisrichness in electronic properties is the interaction of electrons with the periodic structureof the materials. It is this interaction that decides whether a material will be a metal, asemiconductor, or an insulator, and can be further exploited to fine tune the detailedelectronic properties. Change the structure, change the properties.

It was this concept that led Yablonovitch [1] to propose that we try the same trick withlight. Interaction of electromagnetic waves with periodic structures goes back to Braggand his observation that planes of atoms can act like perfect mirrors to Xrays when theBragg condition is met:

λ θ δ= ±2d sinb g (1)

See figure 1. In fact the perfect mirror behaviour holds over a range of angles, ± δ ,determined by how strongly the atoms scatter Xrays. Any given arrangement of atomswill contain many possible Bragg planes reflecting Xrays from different directions. As acrystal is turned in a beam of Xrays it will briefly ‘glint’ as each Bragg condition is met.Figure 2 illustrates the point.

Figure 1. A regular array of atoms diffracts Xrays when the Bragg condition ismet. For incident Xrays of a given wavelength different planes reflect at differentBragg angles.

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Figure 2. Bragg Scattering for Xrays. Each set of planes acts like a mirror in asmall range of angles which scarcely overlap.

Thus we know that a crystal rejects Xrays for certain limited angles of incidence. Theconcept of Bragg reflection applies equally well to visible radiation except that wecannot rely on atoms to do our work for us. We must arrange that the material has someperiodic structure on the scale of the wavelength of light: a fraction of a micron. In factnature has made this arrangement for us in the case the gemstone opal.

Some years ago it was shown using an electron microscope that the brilliant coloursobserved in precious opal was due to mesoscale structure within the mineral. Thestructure had eluded previous workers because the scale was too fine to be seen withoptical instruments, and too coarse to be observed by Xrays. However it is now possibleto make beautiful electron micrographs that reveal the face centred cubic close packedarrangement of the microscopic silica spheres that constitute opal. Figure 3 shows sucha micrograph.

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Figure 3. Precious opal comprises sub micron sized silica spheres arranged in aface centred cubic close packed structure. In this electron micrograph the individualspheres are clearly visible under a layer of surface clutter.

Nearly all of biology is ‘engineered’ on the micron scale and with the help of DNA verycomplex structures are manufactured. Not surprisingly the optical properties arefrequently exploited, nowhere with more spectacular effect than in the butterfly. Manyspecies show iridescent green or blue patches and these owe their colouring todiffraction from periodic material in the scales of the wing. Figure 4 shows a photographof a Peacock butterfly where the ‘eyes’ exhibit the characteristic blue metallic sheen of aphotonic material. Figure 5 shows a somewhat enlarged section of an almost entirelyblue butterfly, the Adonis Blue, in which the tiny sub-millimetric scales are visible, and infigure 6 we see an electron micrograph of a scale, the entity responsible for the colour.Diffraction from a three dimensional array of holes in the scale give rise to itscharacteristic colour. Man has not yet completely mastered the art of manufacturingthree dimensional optical structures which nature produces with such ease.

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Figure 4. Many butterflies owe their colour to diffraction by three dimensionalmicro structures within the wings. This is particularly true of blues or greenswhich exhibit a metallic sheen, as in the ‘eyes’ of this Peacock (Inachis Io).

Figure 5. The wing of a butterfly consists of a series of scales each a fraction of amillimetre long. In this photograph of an Adonis Blue (Lysandra Bellargus) thesmall scales are visible as striations of the blue section of the wing. The black spotsare caused by the occasional missing scale.

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Figure 6. An electron micrograph of a broken scale taken from the butterflyMitoura Grynea revealing a periodic array of holes responsible for the colour. Thebar in the bottom left is one micron long. Taken from Helen Ghiradella’s paper on‘light and color on the wing: structural colors in butterflies and moths’ [2]

So, just like electrons, photons can be pushed around by the structure of a material.However Yablonovitch’ aim was a more extreme one. He had in mind that just as asemiconductor has a forbidden band of energies within which no electron could enter thecrystal, so it should be possible to make a periodic dielectric such that in a forbiddenrange of frequencies no photon could enter the crystal. What we need to do is find aperiodic material in which the scattering of light is so strong that the rejection windowsof the different Bragg planes overlap, see figure 7. When this happens photons will berejected whatever direction they attempt to penetrate the material. In fact such amaterial could truly be called a ‘photonic insulator’.

Figure 7. Bragg scattering for a photonic material. Each set of planes acts like amirror in a large range of angles which overlap completely. Under thesecircumstances no radiation can penetrate into the material which is a photonicinsulator. For a limit range of frequencies, the band gap, radiation is rejectedwhatever the incident angle.

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2. Realising the Concept

Although the concept is a simple one, realising a material structured on a sub micronscale in three dimensions is a highly non trivial exercise. In opals nature does the workfor us by close packing spheres. Unfortunately this material, although beautiful, fallsshort of perfection: it is not a photonic insulator. The dielectric contrast between silicaand air is too small, and the structure consisting of close packed spheres is notsufficiently favourable. However Yablonovitch has shown [3,4] that there is a structuremanufactured by drilling holes that does behave as a photonic insulator. This structurehas been christened ‘Yablonovite’ in honour of its inventor, and the recipe for itsconstruction is given in figure 8.

Figure 8. ‘Yablonovite’: a slab of material is covered by a mask consisting of atriangular array of holes. Each hole is drilled through three times, at an angle 35.26°away from normal, and spread out 120° on the azimuth. The resulting criss-cross ofholes below the surface of the slab produces a fully three-dimensionally periodic fccstructure.

Provided that the refractive index of the host material is of the order of 3.6 or greater,this material is a photonic insulator. This was demonstrated in an elegant manner byobserving that all structured materials obey a very general scaling law: if a givenstructure is a photonic insulator for incident light of wavelength λ , then anothermaterial structured on double the length scale of the first, will also be a photonicinsulator, but at incident wavelength 2λ . This is true provided that the dielectricfunction does not depend on frequency. Therefore it should be possible to make a proofof principle experiment on a much larger structure, active in the microwave range, andmuch more easily constructed, then to invoke the scaling theorem to prove that if only asmall structure could be manufactured, it would function as an insulator in the visible.

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Figure 9. The transmission coefficient to microwaves of a sample of Yablonovitethat shows a large band gap at frequencies around 14GHz.

Figure 9 shows Yablonovitch’ measurements for a sample of Yablonovite made withunit cell dimensions of a few millimetres and shows the band gap clearly visible.

There still remains the challenge of sub micron engineering which is currently beingaddressed in laboratories around the world. One tool that has shown great promise isreactive ion etching. Ions incident an a sample will react with the first surface atom theymeet and carry it away into the vacuum. Thus shaping the beam by passing it through ashadow mask, and choosing its orientation, we can drill holes of a chosen cross sectionan any direction we choose. Figure 10 illustrates the principle of this technique.

Figure 10. Hole drilling by reactive ion etching: reactive ions, SiCl4 in this case,erode a surface at the point where they strike. A mask selectively protects thesurface from unwanted erosion. Holes as small as 100nm in diameter can be drilled.

This technique has been employed by the Glasgow group of Krauss, De La Rue, andBrand [5] to make the structures shown below in figure 11.

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Figure 11. A photonic structure after Krauss, De La Rue, and Brand, drilled into a0.4µm thick sample of AlGaAs. On the left an overview is given with the photonicstructure in the centre surrounded by connecting waveguides. On the right we see a10x enlargement of the central photonic region.

In fact the Glasgow structure, though a technical tour de force, is still only a twodimensional material, and therefore is not a true photonic insulator. Neverthelesstechnical progress is sufficiently encouraging that we can expect these structures to bepart of a materials scientist’s resources in the not too distant future.

3. Why Photonic Materials are Special.

Why should a material that excludes all light in a given frequency range be special? Afterall there are other ways of excluding light: a metal box is quite dark inside! I want toemphasis how very special is a photonic material: it rejects light not because the lightcannot penetrate through the surface, but because there is just nowhere for the light togo once it is inside. There are no electromagnetic states available inside the material.

Take a practical illustration of this fact. The interior of a sealed metal box is dark, butcan be illuminated by, say, striking a match inside. However if we introduced a suitablyminiature match down one of the excavated cylinders in Yablonovite and attempted tostrike it, we could not do so. The structure would return the photons to the reactingatoms refusing to accept the energy.

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Figure 12. Trying to make light inside a photonic insulator by striking a match: thematch will not light because photons are rejected by the photonic insulator. It is sodark that not even the quantum zero point fluctuations are present. We can exploitthis effect to preserve atoms in their excited state to make lasers more efficient.

Although no one is interested in striking matches inside these structures, they areinterested in manufacturing lasers from them where similar principles may help theefficiency of the laser. In most lasers there is a power threshold for lasing action,because there are other ways of emitted light than into the desired lasing mode. Thedemands of these spurious modes must be met before the critical amplitude can beestablished in the lasing mode. Encasing the laser in a suitably structured photonicmaterial can forbid the power sapping modes and increase the laser efficiency moving usfurther towards thresholdless lasers.

4. The Contribution of the Theorist

At this challenging time when the photonic concept is established, but practicalrealisation remains so difficult, it is extremely valuable to have computer programs thatcan predict the properties of these materials, avoiding costly experimentation withdifficult new technology.

Perhaps the most successful approach to computation has been the discrete sampling ofthe electric and magnetic fields on a fine lattice of points. Maxwell’s equations presentparticular difficulties in that they are concerned with vector fields and geometry is anever present subtlety. However one successful approach [6, 7] is to define two simplecubic lattices - unit cells shown in figure 13. Along the edges of one arrange the E fields,along the edges of the other, the H fields.

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Figure 13. The unit cells of the H and E lattices. Note that the origin is different ineach case: at the top right of the cell of the H-field, and at the bottom left for the E-field.

Figure 14. The H and E fields comprise two interpenetrating simple cubic lattices:the origin of the H-lattice is displaced by (a+b+c)/2 relative to the origin of the E-lattice. Note that each bond carrying a component of the E-field is encircled by fourbonds defining the side of one of the H-field cells, and correspondingly for eachbond of the H-field.

Next arrange the two lattices so that they interpenetrate: one lattice centres the other aswe see in figure 14. Note that each face of the H lattice is pierced by a line of force ofthe E lattice.

Applying Stokes theorem around the edges of the face and using Maxwell’s equation,

∂∂

ε εt 0 ⋅ = +∇ ×E H

gives the discrete form of first Maxwell equation. Similarly the second equation can beobtained by considering faces of the E lattice and applying,

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∂∂

µ µt 0 ⋅ = −∇ ×H E

The resulting difference equations have been much exploited to calculate photonicproperties and to publish computer codes that are freely available [8]. One of the firstsuch calculations was a ‘proof of principle’ for an array of dielectric cylinders shown infigure 15. The calculations [7] shown in figure 16 are in good agreement withexperiment [9].

Figure 15. Schematic version of the microwave experiment by Robertson et al.Microwaves are incident on a simple cubic array of cylinders, ε = 8 9. , 7 layers deepbut effectively infinite in the directions perpendicular to the beam. The amplitude ofthe transmitted beam is measured.

Figure 16. Transmitted power for an array of seven rows of dielectric cylinders. Thedotted curve shows the instrument response in the absence of the cylinders. E isparallel to the cylinders.

Even if the lattice is distorted, we can still apply Stokes and get some equations. Thesecan be transformed back to the original Maxwell's equations on the undistorted latticeprovided that we modify ε and µ and the fields [10],

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∂∂

ε ε

∂∂

µ µ

t

t

0

0

$ $ $

$ $ $

⋅ = +∇ ×

⋅ = −∇ ×

E H

H E

Remarkably we find that the form of Maxwell’s equations is invariant to co-ordinatetransformations. We can choose any co-ordinate system, adapted to the particulargeometry in which we are working, and the only effect on the equations we solve is tomodify ε and µ. This is a remarkable and valuable simplification. It means that once wehave written our computer codes for one co-ordinate system, all we have to do to workwith another system is to plug in different values of ε and µ.

For example suppose that we are examining structured optical fibres: we can choose acylindrical co-ordinate system compatible with the cylindrical symmetry of the fibres,rather than try to fit a square peg in a round hole which is what the original cubic latticewould require us to do. We show the new co-ordinate system below in figure 17.

Figure 17. An adaptation of the simple cubic mesh to the geometry of a fibre. Thez-component of the mesh can be taken to be uniform.

Finally we show this scheme in action for a cylindrical wave guide with perfect metalboundaries. One advantage of the cylindrical mesh is that angular momentum is now agood quantum number in the computation, and in addition cylindrical symmetry can beexploited to speed up the calculation. The computations taken from [10] and presentedin figure 18 are compared to the analytic result demonstrating that the distorted meshshows the same excellent convergence as the original uniform version.

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Figure 18. The lowest three modes of a 10-4m radius cylindrical metal waveguidefor angular momentum m = 0, plotted against wave vector. The full lines are theresults of a numerical calculation using a cylindrical mesh; the dotted lines are theexact results of analytic theory.

5. Conclusions

The idea of photonic materials is at an exciting stage: the concept is well established andat the same time powerful numerical techniques enable us to foresee many newapplications. However we have still to meet the challenge of manufacturing thesestructures. Existing sub-micron technology is well developed for two dimensionalobjects, but photonic materials demand a fully three dimensional facility. When it arriveswe can expect the rapid incorporation of this new technology into optical devices. In themeantime we can draw inspiration from the manner in which nature structures her workson exactly the scale we seek.

[1] Yablonovitch, E., 1987, Phys. Rev. Lett., 58, 2059.

[2] Ghiradella, H., 1991, Applied Optics, 30 3492-500.

[3] Yablonovitch, E., Gmitter, T.J., and Leung, K.M., 1991, Phys. Rev. Lett., 67,2295.

[4] Yablonovitch, E., 1993, J. Phys.: Condensed Matter 5, 2443.

[5] Krauss, TF, De La Rue, RM, and Brand, S., 1996, Nature, 383, 699.

[6] Pendry, J.B., and MacKinnon, A., 1992, Phys. Rev. Lett. 69 2772.

[7] Pendry, J.B., 1994, J. Mod. Optics 41 209.

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[8] Bell, P.M., Pendry, J.B., Martín-Moreno L., and Ward, A.J., 1995, Comp. Phys.Comm., 85, 306.

[9] Robertson, W.M., Arjavalingam, G., Meade, R.D., Brommer, K.D., Rappe, A.M.,and Joannopoulos, J.D., 1992, Phys. Rev. Lett. 68 2023.

[10] Ward, A.J. and Pendry, J.B., 1996, Journal of Modern Optics, 43 773.