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Calculating Photonic Band StructureJ. Phys. [Condensed Matter] 8 1085-1108 (1996).

JB PendryBlackett Laboratory

Imperial CollegeLondon SW7 2BZ

UK

Abstract

Photonic materials structured on the scale of the wavelength of light, have become anactive field of research fed by the hope of creating novel properties. Theory plays acentral role reinforced by the difficulty of manufacturing photonic materials: unusuallywe are better able to design a photonic material than to build it. In this review weexplore the application of scattering theory to Maxwell’s equations that has enabledtheory to make such a central contribution: implementation of Maxwell on a discretemesh, development of the electromagnetic transfer matrix, order N methods, andadaptive meshes. At the same time we present applications to a few key problems byway of illustration, and discuss the special circumstances of metallic photonicstructures and their unique properties.

PACS numbers: 41.20.Jb, 71.25.Cx, 84.90.+a

Calculating Photonic Band Structure


1. Introduction

As optics merges with electronics to fuel a revolution in computing andcommunication, it is natural to compare and contrast electrons and photons. Theformer are for the moment the more active members of the partnership, but for futureprogress we look increasingly to photons. Electrons have enjoyed the advantage in thiscontest because of the ease with which they can be controlled: piped down wires,stored in memories, or switched from one channel to another. Semiconductors are themedium in which they operate, and semiconductors enjoy their control of electronicproperties because of diffraction effects in the atomic lattice. Structure and functionoperate harmoniously together in these materials to deliver the properties we need.

Photons also have a wave-like nature but on too large a scale to be diffracted byatomic lattices. Therefore to a large extent they have escaped detailed engineering oftheir properties. However there is no reason that we should not artificially structurematerials on the scale of the wavelength of light to produce band gaps, defect states,delay lines and other novel properties by analogy with the electron case. This agendaof reworking semiconductor physics for the photon has been seized upon, largely bycondensed matter theorists well versed in the intricacies of band theory.

It is a field for the moment chiefly populated by theorists as experiment struggles todevelop the complex technology required to give three dimensional structure to amaterial on the micron scale, even though there are some interesting optical structuresprepared by ‘self assembly’ methods [1]. Theorists have been very active over the pastfive years in developing the methodology of photonic band theory, and applying it to avariety of systems. In this short review we shall scrutinise the new techniques from thestand point of a condensed matter theorist, following an agenda very much inspired byelectronic band theory.

We begin by discussing the peculiarities of Maxwell’s equations viewed from acomputational standpoint, and present some of the early solutions to this problem.Next we show how to make a consistent adaptation of Maxwell to a discrete mesh anddevelop perhaps the most powerful approach to the problem. Finite differenceequations can be solved in several different ways. Transfer matrix methods are widelyused and are efficient and accurate. More recently ‘order N’ methods have beendeveloped and are extremely efficient in some circumstances. One problem forphotonic calculations is that materials can be structured in a wide variety of differentconfigurations in contrast to atomic lattices all of which have the same basic geometry.This leads us to considering adaptive mesh calculations in which the real space mesh isadjusted to the geometry of the material. Finally we discuss the special case of metallicphotonic structures which have some remarkable properties.

2. Maxwell on a computer

We are fortunate in photonic band theory that the equations governing the photon’sbehaviour are somewhat better defined than for electrons. Maxwell’s equations in SIunits are (see Panofsky and Phillips [2], or Jackson[3]),

∇⋅ = ∇⋅ =

∇ × = − ∇ × = + +

D B

EB

HD

j

ρ∂∂

∂∂

, ,

,

0

t t

(1)



where D is the electric displacement vector, E the electric field vector, B the magneticflux density, H the magnetic field intensity, ρ the charge density, and j the electriccurrent density. Most commonly we work at a fixed frequency so that,

D r D r E r E r

B r B r H r H r

r r j r j r

, exp , , exp ,

, exp , , exp ,

, exp , , exp

t i t t i t

t i t t i t

t i t t i t

b g b g b g b g b g b gb g b g b g b g b g b gb g b g b g b g b g b g

= − = −

= − = −

= − = −

ω ω

ω ω

ρ ρ ω ω

(2)

Note the sign in the exponential: a common source of confusion between physicistsand engineers. We adopt the physicists notation here, of course. Substituting intoMaxwell’s equations gives,

∇⋅ = ∇⋅ =∇ × = + ∇ × = − +

D BE B H D j

ρω ω

, ,,

0i i

(3)

These equations taken together with the constitutive equations,

D E B H j E= = =εε µµ σ0 0, , (4)

define the problem.

All the physics is contained in ε µ σ, , and for nearly all the problems of interest thesecan be taken as local functions, that is to say the most general linear form reduces to,

D r r r E r r r E rb g b g b gu b g b g= =ε ε ε ε, ' ' ' ,03

0d etcetera (5)

Non-locality results from atomic scale processes and is only important when photonicmaterials have relevant structure on an atomic scale. There are exceptions to this rule,for example when excitonic effects are important, but these specialist topics are beyondthe scope of this review.

For some materials such as metals the dielectric constant, ε , depends strongly on thefrequency in all frequency ranges. For many insulators ε is constant in the opticalregion of the spectrum. However it is an inevitable consequence of the properties ofmatter that all materials show dispersion in some range of frequencies: commonly inthe ultraviolet.

The first calculations of photonic band structure [4, 5, 6, 7, 8] addressed the problemof finding the dispersion relationship ω kb g in a periodic structure often with thespecific aim of finding a ‘photonic insulator’ as first proposed in [9] (an excellentsummary of some of these early calculations will be found in [10, 11]). It is usual toeliminate either the electric or the magnetic component from Maxwell’s equations. Forexample, on the common assumption that the permeability, µ , is everywhere unity, wecan derive,

∇ × ∇ × = −∇ + ∇ ∇⋅ = + −E E E E2 20

2b g ω εc (6)

where,

c00 0

1=

ε µ(7)

is the velocity of light in free space.

For condensed matter theorists the resemblance of (6) to the Schrödinger equation isillusory. Certainly the familiar ‘del squared’ term is there, but the additional presenceof ∇ ∇⋅Eb g transforms the equation into a decidedly awkward customer from the



computational point of view. This second term is responsible for the elimination of thelongitudinal mode from the spectrum and ensures that any wave of the form,

E k k rl = ⋅A iexpb g (8)

is always a solution of (6) provided that,

ωl kb g = 0 (9)

thus removing all longitudinal components from the finite frequency spectrum.However under certain circumstances this longitudinal mode can return to haunt an illconstructed computation.

The next step was to observe that the solutions for a periodic medium must take theBloch wave form and therefore the electric field can be expanded in a discrete Fourierseries,

E r e e k g rk g gg

b g n s b g= + + ⋅∑ E E is s p p exp (10)

where g is a reciprocal lattice vector and the two unit transverse vectors are definedby,

ek g n

k g n k g ns =

+ ×

+ × ⋅ + ×

b gb g b g (11)

ek g k g n

k g k g n k g k g np =

+ × + ×

+ × + × ⋅ + × + ×

b g b gb g b g b g b g (12)

Each of the two components of the field, ‘s’ and ‘p’ is constructed to be transverse tothe relevant wave vector. The Fourier expansion contains an infinite summation but, asfor electronic band structure, is truncated at some finite value of k g+b g chosen largeenough to ensure convergence. Conventionally the unit vector, n, is the normal to asurface of the material, but could be chosen to be any arbitrary vector.

In this basis equation (6) becomes,

k g g g g gg

+ = + − ∑2 20

2E c Et t t tt

ω ε ; ' ' ' '' '

(13)

where,

ε εt t t t i dg g e k g e k g r g g r r; ' ' ' ' exp '= + ⋅ + − ⋅−b g b g b g b guΩ 1 3 (14)

and the integration is over the volume, Ω , of the unit cell.

Provided that ε does not depend on ω equation (13) is a generalised eigenvalueequation and can be solved for ω kb g by conventional techniques.

This technique and closely related ones were used to calculate the first photonic bandstructures. For example we see in figure 2 the band structure of the ‘Yablonovite’structure (figure 1) calculated by this technique using several thousand plane waves inthe expansion [6].



Figure 1. ‘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 angle35.26° away from normal, and spread out 120° on the azimuth. The resultingcriss-cross of holes below the surface of the slab, suggested by the cross-hatchingshown here, produces a fully three-dimensionally periodic fcc structure.

Figure 2. Frequency versus wavevector dispersion along various directions in fcck-space, where c0 /b is the speed of light divided by the fcc cube length. The ovalsand triangles are the experimental points for s and p polarisation respectively. Thefull and broken curves are the calculations for s and p polarisation respectively.The dark shaded band is the totally forbidden band gap. The lighter shaded stripesabove and below the dark band are forbidden only for s and p polarisationrespectively.

This was the first structure to show an absolute band gap and gave great impetus tothe subject. So far the structure has not been realised experimentally at visiblewavelengths, but a microwave version has been built and measured. This is alreadysufficient confirmation of the validity of the theory because Maxwell’s equations obeya scaling law. If a structure ε rb g sustains a solution of Maxwell’s equations, E rk b g ,



−∇ + ∇ ∇ ⋅ = + −2 20

2E r E r r E rb g b gc h b g b gω εc (15)

then we can deduce a solution to a different problem by substituting,

r r' /= a (16)

so that,

−∇ + ∇ ∇ ⋅ = + −' ' ' ' ' ' '2 2 20

2E r E r r E ra a a c a ab g b gc h b g b gω ε (17)

The transformed equation tells us that a new photonic structure in which all the lengthscales have been expanded by a sustains a solution whose frequency is less than theoriginal by the same factor a. So that if an experimental structure has an absolute gapin the GHz band, the frequency of that gap will scale upwards as the lattice spacing ofthe structure is scaled down. Our argument assumes that ε is not a function offrequency.

This method of solution has a number of advantages:

• It gives a stable and reliable algorithm for computing ω kb g• The computer programming required is minimal and can draw extensively on

standard methodologies such as Fourier transformation and matrix diagonalisation.

• The longitudinal mode causes no numerical problems because it is projected out ofthe equations at an early stage.

• An arbitrary ε rb g can be handled since no assumptions are made about the shape ofobjects.

Therefore it was an excellent way to start a serious study of the subject. Howeverthere are some situations in which the method runs into difficulties. For example:

• If ε happens to depend on ω then equation (13) is no longer an eigenvalueequation, but something much more complicated and difficult to handle. Theproblem stems from the methodology of fixing k, then calculating ω .

• Experiments commonly measure quantities such as reflection coefficients at a fixedfrequency and to calculate these we need all the Bloch waves at that frequency i.e.again we need to fix ω and calculate k, not the other way round.

• A plane wave expansion of the wave field is rather inefficient and renderscalculations for the more complex structures impossibly time consuming. (Matrixdiagonalisation scales as the cube of the number of plane waves).

• The latter problem is particularly acute in situations where several different lengthscales are forced upon us. For example when an electromagnetic wave is incident ona metal surface the scale of the penetration depth and the wavelength in vacuum candiffer by many orders of magnitude, and yet the Fourier components have toreproduce both to good accuracy.

These are difficulties which are also encountered in electronic band structure and havebeen countered with techniques which can also be applied to the photonic case.

3. Maxwell on a Lattice

The two classic models of electronic band structure are the ‘nearly free electronapproximation’ and the ‘tight binding model’. They start from completely differentstand points: the former assumes that plane waves are a good approximation to the



electron wave function, the latter that electrons interact strongly with the atoms andonly occasionally hop across the space between one atom and the next. Both have beendeveloped as the basis for more accurate and sophisticated models. The plane waveexpansions discussed above are the obvious counterparts of the nearly free electronmodel. But it is also possible to find a tight binding analogy for the photon case. We dothis by choosing to represent the photon wavefield on a discrete lattice of points in realspace. This has several pitfalls and has to be done carefully, but when successfullycompleted it has many rewards in terms of speed of computation.

The original derivation by MacKinnon and Pendry [12, 13] was made with a view toextending MacKinnon’s successful tight binding models of disordered electronicsystems to the photon case, but has in fact had much wider application to orderedsystems. Of course, the main problem in representing differential equations on a latticeis in approximating the derivatives. For Maxwell’s equations we have another problem:they have the property that all longitudinal modes are ‘dead modes’, appearing only aszero frequency solutions. This fundamental property, which has at its heart theconservation of charge, must be preserved in a valid scheme of approximations.

We start from equation (3) and Fourier transform to give,

k E H

k H k k E k k

× = +

× = − zωµ

ωε ε

0

03, ' ' 'b g b gd (18)

where we have assumed that we are inside a dielectric medium where,µρ

== =

10j

(19)

Note that the ‘dead mode’ property holds for a more general class of equations,

κ ωµ

κ ωε ε

E

H d

k E H

k H k k E k k

b gb g b g b g

× = +

× = − z0

03, ' ' '

(20)

since from above,

κ κ κ κ ω εH E H E c dk k E k k k E k k k E k kb g b gc h b g b g b g b g b g b g⋅ − ⋅ = + − z20

2 3, ' ' ' (21)

and choosing,

E k klb g b g= A Hκ (22)

always gives the result,

ω l kb g = 0 (23)

The idea is that if we choose κ E kb g and κ H kb g so that they have simplerepresentations in real space in terms of difference equations, and so that theyapproximate the exact equations, we shall have obtained a real space approximationthat exactly fulfils the ‘dead mode’ requirement. In k-space the differential operatorstransform exactly to,

∂∂E r

E kb g b gx

ik etceterax↔ , (24)

However there is a closely related transformation,

a a ia ik a etceterax− −+ − ↔ −1 1 1E r x E r E k$ exp ,b g b g b g b g b g (25)



So that if we choose

κ

κ

Ex x x x

Hx x x x

ia ik a k O k a

ia ik a k O k a

k

k

b g b g b g e jb g b g b g e j

= + + − ≈ +

= − − − ≈ +

−

−

1 2

1 2

1

1

exp

exp(26)

substitute into equation (20), and transform back into real space,

+ + − − + − = +

+ + − − + − = +

+ + − − + − = +

[ ] [ ]

[ ] [ ]

[ ] [ ]

E E E E ia H

E E E E ia H

E E E E ia H

z z y y x

x x z z y

y y x x z

r b r r c r r

r c r r a r r

r a r r b r r

b g b g b g b g b gb g b g b g b g b gb g b g b g b g b g

ωµ

ωµ

ωµ

0

0

0

(27a)

− − − + − − = −

− − − + − − = −

− − − + − − = −

[ ] [ ]

[ ] [ ]

[ ] [ ]

H H H H ia E

H H H H ia E

H H H H ia E

z z y y x

x x z z y

y y x x z

r b r r c r r r

r c r r a r r r

r a r r b r r r


ωε ε

ωε ε

ωε ε

0

0

0

(27b)

and we retrieve a set of finite difference equations. In these equations a b c, , representdisplacements through a distance a along the x, y, z axes respectively. Furtherinspection will reveal that the lattice on which we sample the fields is a simple cubiclattice: fields on each site are coupled to the six neighbouring sites of this lattice.

Equations (27a,b) are the counterparts of the nearest neighbour tight bindingapproximation familiar in quantum mechanics, but derived from Maxwell’s equations.In fact we can formulate them to resemble a Hamiltonian even more closely,

H F Ftt tt

t' '' '

, ' 'r r r rr

b g b g b g∑ = ω (28)

where,

F r

rrrrrr

b g

b gb gb gb gb gb g

=

L

N

MMMMMMM

O

Q

PPPPPPP

EEEHHH

x

y

z

x

y

z

(29)

In the limit of very small a, relative to the wavelength of radiation, they are an exactrepresentation. Note that the approximations have all to do with the differentialoperators therefore we can get an impression of how accurate the equations are byasking how well they represent free space where,

ε = 1 (30)

In this case equation (21) gives,



aEEE

cEEE

Ey Hy Ez Hz Ex Hy Ex Hz

Ey Hx Ex Hx Ez Hz Ey Hz

Ez Hx Ez Hy Ex Hx Ey Hy

x

y

z

x

y

z

−

−

+ − −− + −− − +

L

NMMM

O

QPPPL

NMMM

O

QPPP

=L

NMMM

O

QPPP

2

20

2

κ κ κ κ κ κ κ κκ κ κ κ κ κ κ κκ κ κ κ κ κ κ κ

ω

(31)

If we calculate the dispersion for k directed along the x-axis,

κ κ κ

κ κ κ

Ex x Ey Ez

Hx x Hy Hz

ia ik a

ia ik a

k k k

k k k

b g b g b g b g b gb g b g b g b g b g

= + + − = =

= − − − = =

−

−

1

1

1 0 0

1 0 0

exp , , ,

exp , ,(32)

we get,

ω202 2 2 1

24= −c a k axsin e j (33)

Obviously this reduces to the correct limit when a is small,

ω 202 2 1

122 21= + +c k k ax x L (34)

This real space formulation of the problem as set out in (27) can be used as analternative to the k-space version described in the last section when computing theband structure. However this is not the best way forward as the real space formulationoffers some efficient alternatives to straightforward matrix diagonalisation. Efficiencyof these new methods is owing to the sparse nature of equations (27).

As we point out above, the accuracy of the real space formulation is limited by themesh size. The situation can be improved at, the expense of simplicity, by devising amore accurate approximation to k. For example we can improve on (26) as follows,

κ

κ

' exp exp

' exp exp

Ex x x x x

Hx x x x x

ia ik a ik a k O k a

ia ik a ik a k O k a

k

k

b g b g b g b g e jb g b g b g b g e j

= + − + + + − ≈ +

= − − − + − − ≈ +

−

−

1 12

32

3 2

1 12

32

3 2

2 2

2 2(35)

Transformation into real space of the resulting equations gives difference equationscoupling beyond the nearest neighbours to second nearest neighbours of a simple cubiclattice. Taking higher orders of the expansion of kx in terms of exp + −ik axb g 1 givesmore accurate approximations and more complex formulae. Of course the computerhas no objection to complex formulae, but the benefits of a more accurateapproximation that allows us to use less sampling points must be balanced against thedisadvantage that the matrix grows less sparse as the number of nearest neighbourcouplings rises.

Finally in this section, a word about the type of lattice on which we sample the fields. Itmight be imagined that we could make a more accurate approximation by choosing amore complex lattice of sampling points: fcc instead of sc perhaps? Although it is truethat we can do better in terms of improved approximations to k k kx y z, , , the morecomplex lattices usually have some subtle but fatal disadvantages. A detaileddiscussion is beyond the scope of this review, but has to do with introduction ofspurious solutions in addition to the ones we seek. These mix with the desiredsolutions to introduce numerical instabilities. In this instance a simple cubic lattice hasvirtues of stability and robustness. It should not be abandoned lightly.



4. How to Solve the Discrete Maxwell’s Equations Using Transfer Matrices

Many of the experiments with which we shall be concerned are made at fixedfrequency. This certainly makes life simpler in materials where the dielectric functiondepends on ω : ε ωb g can immediately be specified. And there is a further advantage inthat we can employ one of the efficient ‘on shell’ methodologies. In this context theterm ‘on shell’ means that the calculation is confined to a surface or shell inconfiguration space on which all the states have the same frequency. On shell methodshave commonly been employed in low energy electron diffraction theory [14]. Theanalogy with LEED has also been used by Stefanou et al [15], and the transfer matrixapproach has been addressed in a different context by Russell [16].

Looking back to equations (27): if ω is fixed the equations specify a relationshipbetween fields on different sites of the sampling mesh. For example in figure 3 we see

Figure 3. Plan view of the mesh on which the fields are sampled showing theorientation of the axes. Two planes of points are marked by dashed lines: at afixed frequency the fields on these two sets of planes are related by Maxwell’sequations to that if we know the fields on one of the planes, we can calculate thefields on the other.

a plan view of the sampling mesh. Two sets of planes are marked by dashed lines and ifwe know the fields on one of these planes we can infer the fields on the other by virtueof equations (27). In fact if we consider a slab of material, then specifying fields on asingle plane on one side of the sample infers the fields at all points within the sample.At fixed frequency, given the fields on one side of a sample we can transfer the fieldsthroughout the whole sample by successive applying Maxwell’s equations. This resultis no more that the discrete equivalent of the well know result for differentialequations: if we specify the boundary conditions, then we can integrate the equationsthroughout the sample.

The detailed derivation of this result from (27) is elementary but tedious and the readeris referred to [12] for details. The result is,



E r c E r r H r

r

H r a H r

H r b H r

r a

H r H r a

H r a b H r a

x x y

y y

x x

y y

x x

ic

ia

ia

+ = +

+ + − −

− − −

LNMM

OQPP

−+

+ − +

− + − − +

LNMM

OQPP

b g b g b g b g

b gb g b gn sb g b gm r


ωµ µ

ωε ε

ωε ε

0

0

0

(36a)

E r c E r r H

r

H r a H r

H r b H r

r b

H r a b H r b

H r H r b

y y x

y y

x x

y y

x x

ic

ia

ia

+ = −

+ − −

− − −

LNMM

OQPP

−+

+ − + − +

− − +

LNMM

OQPP

b g b g b g



ωµ µ

ωε ε

ωε ε

0

0

0

(36b)

H r c H r r c E r c

r a c

E r c E r a c

E r a b c E r a c

r c

E r a c E r c

E r b c E r c

x x y

y y

x x

y y

x x

ic

ia

ia

+ = − + +

+− +

+ + − − +

− − + + − − +

LNMM

OQPP

−+

+ + + − +

− + + − +

LNMM

OQPP

b g b g b g b g



ωε ε

ωµ µ

ωµ µ

0

0

0

(36c)

H r c H r r c E r c

r b c

E r a b c E r b c

E r c E r b c

r c

E r a c E r c

E r b c E r c

y y x

y y

x x

y y

x x

ic

ia

ia

+ = + + +

+− +

+ + − + − − +

− + − − +

LNMM

OQPP

−+

+ + + − +

− + + − +

LNMM

OQPP

b g b g b g b g



] ωε ε

ωµ µ

ωµ µ

0

0

0

(36d)

Notice that we work with only two components of each of the fields. The thirdcomponent is inferred from the ‘dead longitudinal mode’ which we carefully built intoour equations,

− − − + − − = −

+ + − − + − = +

− −

− −

( ) [ ] ( ) [ ]

( ) [ ] ( ) [ ]

ia ia

ia ia

y y x x z

y y x x z

1 10

1 10

H r a H r H r b H r r E r

E r a E r E r b E r r H r

b g b g b g b g b g b gb g b g b g b g b g b g

ωε ε

ωµ µ(37)

Equations (36) define the transfer matrix,

E r cE r cH r cH r c

T r r T r r T r r T r rT r r T r r T r r T r rT r r T r r T r r T r rT r r T r r T r r T r r

x

y

x

y

++++

L

N

MMMMM

O

Q

PPPPP=

L

N

MMMMM

O

Q

b gb gb gb g

b g b g b g b gb g b g b g b gb g b g b g b gb g b g b g b g

11 12 13 14

21 22 23 24

31 32 33 34

41 42 43 44

, ' , ' , ' , ', ' , ' , ' , ', ' , ' , ' , ', ' , ' , ' , '

PPPPP

L

N

MMMMM

O

Q

PPPPP∑r

E rE rH rH r

'

''''

x

y

x

y

b gb gb gb g

(38)

where the summation is over all points in a plane.

In the case that we have a periodic structure which repeats after Lz cells we can relatethe fields on either side of the unit cell and exploit this to calculate the band structure,




T

E rE rH rH r

x z

y z

x z

y z

z

x

y

x

y

LLLL

L

++++

L

N

MMMMM

O

Q

PPPPP=

L

N

MMMMM

O

Q

PPPPP

b gb gb gb g

b gb gb gb gb g

,

''''

0 (39)

where,

T TL j jzj

Lz, ,0 1

1b g b g= −

=∏ (40)

hence on applying Bloch’s theorem to the periodic structure,


E rE rH rH r

x z

y z

x z

y z

z z

x

y

x

y

LLLL

iK L a

++++

L

N

MMMMM

O

Q

PPPPP=

L

N

MMMMM

O

Q

PPPPP

b gb gb gb g

b gb gb gb gb g

exp

''''

(41)

we identify the Bloch waves as eigenvectors of the transfer matrix,

T

E rE rH rH r

E rE rH rH r

L iK L az

x

y

x

y

z z

x

y

x

y

,

''''

exp

''''

0b gb gb gb gb g

b gb gb gb gb g

L

N

MMMMM

O

Q

PPPPP=

L

N

MMMMM

O

Q

PPPPP(42)

Thus we can find all the Bloch waves at a given frequency. Contrast this with equation(13) which gives all the Bloch waves at a fixed wave vector. Obviously (42) has theadvantage if ε depends on ω since the eigenvalue equation remains unchangedbecause ω has already been specified so that we know what value of ε to write in theequations.

Transfer matrix methods have another advantage in that they generally work withsmaller matrices than would otherwise be required. Suppose that we have a periodicsystem in which the unit cell is subdivided into a mesh of 10×10×10 sampling points.We could diagonalise equations (27) to find the eigen-frequencies of the system. Therelevant matrices would have dimensions 6000×6000 reflecting the 6000 independentfield components of the system. This would be a huge numerical calculation requiringthe services of a super computer. Transforming into k-space would bring no reliefbecause to give a similar level of precision (13) would have to employ a similar numberof plane waves in the Fourier expansion. In contrast transfer matrices for this systemhave dimensions 400×400 and can easily be diagonalised on a workstation or even ahighly specified personal computer.

We can calculate the scaling of computing times with size of system. For off-shellmethods,

τoff shell x y zL L L~ d i3 (43)

whereas for on shell methods,

τon shell x y zL L L~ d i3 (44)



representing a considerable saving of time in systems which require many layers ofmesh points to describe their structure. In our example above the saving isapproximately a factor of 102=100.

Because they calculate all the states at a given frequency, transfer matrix methods arevery well suited to calculating the scattered wavefield from a complex object, and itwas transfer matrix methods which opened up this aspect of the subject. Robertson etal [17] have measured the response of a photonic system in the microwave band. Theirstructure consisted of an array of cylinders and is shown below.

Figure 4 Schematic version of the microwave experiment by Robertson et al.Microwaves are incident on a simple cubic array of cylinders, ε = 8 9. , 7 layersdeep but effectively infinite in the directions perpendicular to the beam. Theamplitude of the transmitted beam is measured.

In reference [13] the unit cell of the system was divided into a 10×10×1 mesh: for eachcell an average was taken over the dielectric constant within that cell. The transfermatrix was found by multiplying the matrices for each of the ten slices. Its eigenvaluesgive K K Kz x yω , ,d i. For the transmission coefficient a similar division of the cell was

made, but multiplying transfer matrices for the seven layers of unit cells would have ledto numerical instability so instead multiplication was halted after integrating throughone unit cell at which point the transmission and reflection coefficients were calculatedfor a slab one cell thick. Slabs were then stacked together using the multiple scatteringformula familiar in the theory of low energy electron diffraction [14]. Convergence wastested by repeating the calculation for a 20×20×1 mesh: changes of the order of 1% at80GHz were found in the band structure. The results are shown below compared tothe experiment. The computer codes used in this calculation have been published [18].



Figure 5. Left: dispersion relation for propagation of electromagnetic waves alongthe (10) direction of a 2D array of dielectric cylinders. (a) E perpendicular to thecylinders (b) E parallel to the cylinders. The experimental results are shown asblack dots and have a resolution of 5GHz. Note that very narrow bands are notresolved experimentally. Right: transmitted power for an array of seven rows ofdielectric cylinders. The dotted curve shows the instrument response in the absenceof the cylinders. Upper curve (a): E perpendicular to the cylinders, lower curve (b):E parallel to the cylinders.

5. How to Solve the Discrete Maxwell’s Equations Using ‘Order N’ Methods

When handling very large systems the way in which computing times scale with systemsize is the determining factor in their efficiency. Obviously the optimum scaling of asystem with N independent components must be proportional to N because it wouldtake at least this time to write down the problem. As computers have become morepowerful and able to treat very large systems the importance of achieving an ‘order N’methodology has been realised.

In the electron band structure problem such methodologies have been realised for tightbinding systems in which the Hamiltonian reduces to a sparse matrix. Formallycontaining N2 elements, the Hamiltonian would appear to lead to computation times atleast as large as N2 but is saved from this fate by only order N components being non-zero. Therefore any method based on a finite number of matrix×vector multiplications,which are now of order N, gives the desired optimum scaling.

In the context of photonic band structure the formulation of order N methodologiesawaited construction of a tight-binding formalism. The real space formulationdescribed in §3 gives rise to sparse matrices in exactly the desired manner. Each matrixdefined by (27) contains O Nb g components where N is the number of mesh points inthe system. These possibilities were first exploited by Chan, Yu, and Ho [19].

The idea is as follows: if we start with a set of fields, arbitrary except in that they obeythe Bloch condition for wave vector K,

E HK Kt t= =0 0b g b g, (45)

and use Maxwell’s equations to calculate their time evolution, we can in effect find theband structure of the system. Since we know that we can expand the arbitrary fields interms of the Bloch waves:

E E

H H

K K K

K K K

t b i t

t b i t

j j jj

j j jj

b g d i

b g d i

= −

= −

∑

∑

exp ,

exp

ω

ω

(46)

Hence Fourier transforming the fields with respect to time picks out a series of deltafunctions located at the Bloch wave frequencies. The resolution of these frequenciesdepends on the time interval over which integration proceeds. One must be careful thatthe starting fields contain a finite component of the desired Bloch wave.



We need to reformulate Maxwell’s equations so as to represent them not only on amesh which is discrete in space, but on one that is discrete in time. To do this we goback to (27) and further approximate these equations so that when transformed backinto the time domain, they result in finite difference equations. We use the same tricksthat we used in the spatial case. The approximation,

w iT i T

w iT i T

E

H

= − − − ≈

= + + − ≈

−

−

b g b gb g b g

1

1

1

1

exp ,

exp

ω ω

ω ω(47)

gives a set of equations that conserve energy as before,

+ + − − + − = +

+ + − − + − = +

+ + − − + − = +

[ ] [ ]

[ ] [ ]

[ ] [ ]

E E E E iaw H

E E E E iaw H

E E E E iaw H

z z y y H x

x x z z H y

y y x x H z

r b r r c r r

r c r r a r r

r a r r b r r

b g b g b g b g b gb g b g b g b g b gb g b g b g b g b g

µ

µ

µ

0

0

0

(48a)

− − − + − − = −

− − − + − − = −

− − − + − − = −

[ ] [ ]

[ ] [ ]

[ ] [ ]

H H H H iaw E

H H H H iaw E

H H H H iaw E

z z y y E x

x x z z E y

y y x x E z

r b r r c r r r

r c r r a r r r

r a r r b r r r


ε ε

ε ε

ε ε

0

0

0

(48b)

and in the time domain,

+ + − − + − = + − −

+ + − − + − = + − −

+ + − − + − = + −

−

−

−

[ , , ] [ , , ] , ,

[ , , ] [ , , ] , ,

[ , , ] [ , , ] ,

E t E t E t E t aT H t T H t

E t E t E t E t aT H t T H t

E t E t E t E t aT H t

z z y y x x

x x z z y y

y y x x z

r b r r c r r r

r c r r a r r r

r a r r b r r

b g b g b g b g b g b gb g b g b g b g b g b gb g b g b g b g

10

10

10

µ

µ

µ T H tzb g b g− r,(49a)

− − − + − − = + + −

− − − + − − = + + −

− − − + − − = +

−

−

−

[ , , ] [ , , ] , ,

[ , , ] [ , , ] , ,

[ , , ] [ , , ]

H t H t H t H t aT E t T E t

H t H t H t H t aT E t T E t

H t H t H t H t aT

z z y y x x

x x z z y y

y y x x

r b r r c r r r r

r c r r a r r r r

r a r r b r

b g b g b g b g b g b g b gb g b g b g b g b g b g b gb g b g b g b g

10

10

1

ε ε

ε ε

ε ε0 r r rb g b g b gE t T E tz z, ,+ −

(49b)

These equations can be used to advance the fields in time steps of T.

Using methods similar to the one described above, Chan et al calculated for a structureshown in figure 6: an array of cylinders packed in a diamond lattice with periodicboundary conditions imposed on the starting wavefunction. The sampling grid was a48×48×48 mesh. The time interval used was T b c= 0 003 0. / where b is the latticeconstant, and the integration continued through 105 steps



Figure 6. Unit cell of the photonic structure consisting of a series of cylinders,ε=12.96, connecting the points of a diamond lattice. The diameter of the cylindersis adjusted so that they occupy 20% of the volume. To simplify the figure thelattice points are shown as circles, and the cylinders are not drawn to scale.

The time sequence was the Fourier analysed to give the spectrum shown in figure 7when the boundary conditions are set for the X point in the Brillouin zone. Repeatingthe exercise for other K points results in the band structure shown in figure 8.

Figure 7. Spectral intensity at the X-point of the diamond structure. Thefrequency is given in units of c0/b where b is the lattice constant.



Figure 8. The photonic band structure of a diamond structure generated with thespectral method. The frequency is given in units of c0/b where b is the latticeconstant.

These results were checked against calculations made using a plane wave expansionand agreed very well. The main difference was the length of time taken to make thecalculation.

The time evolution method has a further advantage over the plane wave expansionmethods, one which it shares with transfer matrix methods: it can be used to calculatereflection coefficients of objects by choosing the initial condition to be a pulse ofradiation incident on the object in question, allowing the pulse to evolve with time untilit has finished interacting with the object, then Fourier transforming the result to getthe response at each frequency.

The time taken for calculations using the time evolution operator is proportional to,

τ time evol = ×number of mesh points number of time stepsb g b g (50)

and hence is ‘order N’ if N is taken to be the number of mesh points and the number oftime steps is independent of N. The latter condition is fulfilled for a wide class ofsystems:

• when absorption is finite so that any incident radiation is removed after a finiteinterval of time.

• when the system scatters only weakly so that radiation escapes from the systemafter a finite length of time.

However there are situations where the number of time steps does depend on thenumber of mesh points. For example in a large system in which no absorption ispresent, radiation can scatter to and fro within the system defining structure on anincreasingly fine frequency scale, and requiring many time steps before it can beresolved. This happens in so called ‘Anderson localised’ systems where radiation canonly escape from a disordered system in a time which increases exponentially withsystem size. It also is the case for ‘defect states’ enclosed within a photonic band gapmaterial - at least if we wish to estimate the lifetime of the defect which growsexponentially with system size. For this class of system methods based on the timeevolution operator have a disastrous scaling law,



τ time evol localised system

number of mesh points number of mesh points

b gb g b g= × exp

(51)

Not surprisingly these methods are not used to study Anderson localisation wheretransfer matrix techniques are the order of the day.

In a system which is not localised, in which there are no photonic band gaps andradiation diffuses around a complex structure, the time to escape will be determined bythe diffusion equation and be proportional to the square of the linear dimensions,

τ time evol delocalised

number of mesh points number of mesh points

number of mesh points

b gb g b gb g

= ×

=

23

43

(52)

Although not ‘order N’, this is still a very favourable scaling law.

6. Maxwell on a Non-Uniform Mesh

By and large electronic structure calculations all deal with the same basic structure: anassembly of atomic potentials. Although this presents its own difficulties, it sits incontrast to the very wide variety of structures and of materials with which we work inphotonic band structure. Most of the methods described above work well in situationswhere there is only one length scale. For example in the array of dielectric cylindersshown in figure 4, electric and magnetic fields can be described accurately by samplingon a uniform mesh.

In contrast systems which contain metals with finite conductivity have two lengthscales: the wavelength of radiation outside the metal, and the skin depth inside, oftendiffering by many orders of magnitude: see figure 9. Unless the wavefield is correctlydescribed inside the metal, estimates of energy dissipation will be in error. Thisrequires either an impossibly dense uniform mesh, or a non-uniform mesh adapted tothe shape of the metal.

Figure 9. At a vacuum-metal interface there is a scale change in the wavefield. Inthe vacuum scale is determined by the wavelength, and in the metal by the skindepth.

Another instance where a uniform mesh is inappropriate occurs when the material hassome symmetry which is relevant to its function but which is incompatible with thesymmetry of a uniform simple cubic mesh. For example in an optical fibre cylindrical



symmetry is a central concept. Here we may wish to choose a cylindrical mesh adaptedto this symmetry as shown in figure 10.

Figure 10. An adaptation of the simple cubic mesh to the geometry of a fibre. Thez-component of the mesh can be taken to be uniform.

At first sight it may seem that we have a very tiresome task ahead of us: that ofreworking all our formalism for every different mesh we encounter. In fact we aresaved by a most elegant result: if Maxwell’s equations are rewritten in a newcoordinate system they take exactly the same form as in the old system provided thatwe renormalise ε and µ according to a simple prescription [20]. This affords a hugesaving in programming effort because it reduces what appears to be a new problem toan old one: that of solving for the wavefield on a uniform mesh in a non uniformmedium.

Consider a general coordinate transformation,

q x y z q x y z q x y z1 2 3, , , , , , , , ,b g b g b g (53)

then after some algebra we can rewrite Maxwell’s equations in the new system,

∇ × = −=

∑qi i j j

j

H

t$ $

$Ed i µ µ

∂

∂01

3(54a)

∇ × = +=∑q

i i j j

j

E

t$ $

$Hd i ε ε

∂

∂01

3(54b)

which as promised are identical to the familiar equations written in a Cartesian systemof coordinates. The renormalised quantities are,

$ε εi j i ji jg Q Q Q Q Q= ⋅ ×

−u u u1 2 3 1 2 3

1b g d i (55a)

$µ µi j i ji jg Q Q Q Q Q= ⋅ ×

−u u u1 2 3 1 2 3

1b g d i (55b)

where u u u1 2 3, , are three unit vectors pointing along the q q q1 2 3, , axes respectively,and,

gu u u u u uu u u u u uu u u u u u

− =⋅ ⋅ ⋅⋅ ⋅ ⋅⋅ ⋅ ⋅

L

NMMM

O

QPPP

11 1 1 2 1 3

2 1 2 2 2 3

3 1 3 2 3 3

(56)

Qxq

xq

yq

yq

zq

zqij

i j i j i j= + +

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

(57)

Q Qi ii2 = (58)



This result has great value in calculations based on real space methods as it enables allthe original computer programs written for a uniform mesh to be exploited withoutchange when an adaptive mesh is employed. Less obviously it is also of value in theplane wave expansion methods too: all that is necessary is to apply the plane waveexpansion in q- space to the new Maxwell’s equations and all the benefits of theadaptive mesh accumulate in enhanced convergence of the Fourier expansion.

We remark in passing that similar transformations can be used in the case of theSchrödinger equation which also retains its form under a coordinate transformation.However in this case one must consider the most general form which includes a vectorpotential as well as a scalar potential because curvature of the coordinate systemimplies an effective magnetic field.

Equations (54a,b) raise an intriguing possibility. Suppose that we set up a coordinatesystem in free space that is non-uniform in some finite region of space, but far awayfrom this region reverts to a uniform Cartesian system. Obviously the results of anyscattering experiment in which radiation enters the region in question from infinitycannot depend on the choice coordinate system. Whatever system we choose, mustreturn the result that radiation is not scattered by what after all remains free space.

Now let us change our viewpoint. Suppose that the ‘real system’ contains a materialfor which,

$ ,

$

ε

µ

i j i ji j

i j i ji j

g Q Q Q Q Q

g Q Q Q Q Q

= ⋅ ×

= ⋅ ×

−

−

u u u

u u u

1 2 3 1 2 31

1 2 3 1 2 31

b g d ib g d i

(59)

Then the electromagnetic influence of the medium can be undone by the reversecoordinate transformation: we can find a system in which ε and µ are everywhereunity and which has no observable influence on electromagnetic radiation. The objectin question is therefore invisible.

In fact the restriction that the coordinate transformation be embedded in a flat 3-spaceis a considerable one, and for most objects we cannot completely undo their scatteringin this way even if the other condition we require,

ε µ= (60)

is met. However there are instances where this is possible: a sheet of material isperfectly non reflecting to normally incident radiation provided that condition (60) istrue. This observation is the basis of much stealth technology since only radar signalsat normal incidence have a chance of returning to the transmitter.



Figure 11. The lowest three modes of a 10-4m radius cylindrical metal waveguidefor angular momentum m=0, plotted against wave vector [20]. The full lines arethe results of a numerical calculation using a cylindrical mesh; the dotted lines arethe exact results of analytic theory.

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 are compared to the analyticresult and show that the distorted mesh shows the same excellent convergence as theoriginal uniform version.

7. Metallic Photonic Structures

Metals are different. Although their properties can be represented by an effectivedielectric function they are unique amongst materials in that this function can beessentially negative. For example nearly free electron metals such as aluminium can bemodelled by,

ε ωω

ω ω γb g b g= −

+1

2p

i(61)

where ω p is the plasma frequency, and γ is related to the conductivity, σ , by,

γω ε

σ= p

20 (62)

If the conductivity is large, and ω is below the plasma frequency, then ε ωb g isessentially negative. This seemingly innocent property of a metal has a profoundimpact on its electromagnetic properties.

Let us work in the electrostatic limit, ignoring retardation for the moment. We havethe requirement that,



∇⋅ = ∇⋅ ∇φ =D r r rb g b g b gε 0 (63)

where D is the displacement field and φ is the scalar potential. Normally this impliesthat there are no localised electrostatic modes in the absence of external charge, andthat in the presence of external charges the fields are unique: see Panofsky and Phillipsp42 [2]. Consider a system containing no external charges, but which contains alocalised region where ε is non-unity. Enclose the region by a surface Γ which liesarbitrarily far from the region of interest. Normally we can show that if the potentialand its gradient on this surface are zero, then the fields everywhere are zero. In otherwords, fields inside the surface have to be induced by sources outside the surface,whose fields will pass through the surface itself. Consider

φ ε ε∇φ0 0b g.dSΓz = (64)

which is zero in the limit that the surface is infinitely far from the region containing theelectrostatic activity. From Gauss’ theorem this implies,

0 0 0

0 0

= ⋅ + ∇φ ⋅ ∇φ

= ⋅ + ⋅ = ⋅

zz z

φ∇ ε ε∇φ ε ε

φ∇ ε ε ε ε

b g b g b gdV

dV dV

V

V VD E E E E

(65)

If ε is everywhere positive, then the implication is that,

E = 0, everywhere (66)

However if the surface Γ encloses a photonic metallic structure within which ε isalternately negative and positive in different parts of the structure, then the theorem nolonger holds. Metallic systems may support localised resonances whose fields areentirely confined to the photonic region, at least in the electrostatic limit. In fact thesemodes can be thought of as originating in surface plasmons [21] decorating thesurfaces of the photonic structure, often shifted in frequency by strong interactionbetween one surface and another. These localised resonances are a dominant feature ofstructured metallic systems as discussed by Lamb et al [22].

One further general result is of interest. It concerns the frequencies of these localisedresonances. If we have a non zero solution to equation (63), then by substituting,

r r= α ' (67)

we can find another solution,

0

1 1

= ∇ ⋅ = ∇⋅ ∇φ

= ∇ ⋅ ∇− −

D r r r

r r

b g b g b gb g b g

ε ω

α ε ω α α φ α

,

' , ' ' '(68)

and therefore,

∇⋅ ∇φ =ε ω α α, r rb g b g 0 (69)

so that having found a localised resonance for one structure at a frequency ω , weknow that all structures with the same shape have a resonance at exactly the samefrequency. This is because Laplace’s equation contains no natural length scale.

If we now generalise away from the electrostatic limit, the localised resonances will ingeneral couple to externally incident electromagnetic radiation. If the photonicstructure is on a scale much finer than the wavelength, the coupling will be relativelysmall and therefore the size of the structure does in fact change the nature of the



resonances if we go beyond the electrostatic limit: it alters the external coupling of thelocalised modes.

A good illustration of these effects is seen in an ‘ordered colloid’ as shown infigure 12.

Figure 12. An ordered simple cubic structure composed of 16Å diameteraluminium spheres, lattice spacing 52.9 Å.

Aluminium has a plasma frequency of 15eV and in the first instance we shall assume nodissipation from electrical resistance so that we can substitute into (61),

ω γp = =15 0eV, (70)

We are now in a position to calculate the photonic band structure of this object whichis shown in figure 13, taken from [12].

Figure 13. The electromagnetic band structure of a simple cubic array ofaluminium spheres calculated for loss-free metal with a dielectric constantgiven by (61). Note the characteristic structure which consists of a largenumber of extremely flat bands: these are surface plasma modes of the metalspheres, found in the range 15/√3eV to 15/√2eV for isolated spheres, but inthis instance spread to a lower range of energies by interaction betweenspheres. The free space dispersion relation is shown as a dashed line, andclearly interacts strongly with the surface modes.



As we predicted, the presence of structured metal introduces a host of resonances intothe system, here manifesting themselves as very flat bands which, again as predicted,hybridise weakly with the transverse modes which intersect them. A structure withtwice the lattice spacing and twice the sphere diameter would show very similarstructure but would have stronger interaction with the transverse modes.

Figure 14. Reflectivity and transmissivity of solid aluminium, left, and that of acolloidal array of 16Å radius aluminium spheres filling 12% of the sample volume.The thickness of material considered is 3385Å.

Reintroducing the electrical resistance of the metal means that the localised resonancescan now dissipate energy and, since there are a great number of resonances, this willlead to strong absorption of incident light. In fact metallic colloids, such as thecolloidal silver familiar from photographic plates, are extremely black in appearance.We can show this quantitatively by calculating the reflection and transmissioncoefficients of a thin layer of this colloidal structure. The results are shown in figure14. Clearly the colloid, in contrast to the metal, neither transmits nor reflects in a wideband of frequencies from the near infra red into the ultra violet. This is a directconsequence of the new resonant modes introduced by the colloidal structure.

These resonances are also believed to be responsible for surface enhanced Ramanscattering observed at rough silver surfaces [23, 24].

An interesting aside is that this effect has been exploited for some time in solar heatingpanels. Coating a layer of polished metal with metal colloid suspended in glass gives acomposite which is strongly absorbing in the visible and near infra red, but reflecting inthe far infra red. hence the panel absorbs solar radiation efficiently, but retains the heatbecause it is a bad emitter in the far infra red. Figure 15 shows a schematicarrangement.

Figure 15. A schematic figure of a solar panel showing how the strong absorptionin the visible, and weak absorption in the infra red, of a metal colloid is exploitedto give an efficient solar panel.



Figure 16. The stopping power, as a function of the energy quantum lost,observed by Howie and Walsh [25] for 50keV electrons passing through analuminium colloid, compared to our theoretical estimate based on the bandstructure shown in figure 13.

This high density of states means that colloids are very effective at extracting energyfrom fast charged particles [26, 27]. Essentially this can be regarded as Cerenkovradiation. Reference to figure 13 shows that the group velocity of the resonant modes,as defined by d dkω / is much less than the velocity of light, given by the slope of thedotted lines. Therefore a moderately fast particle such as a 50keV electron willstrongly Cerenkov radiate. We show in figure 16 some electron energy loss data takenby Howie and Walsh [25] in experiments on aluminium colloids compared to theory byMartín-Moreno, and Pendry [28, 29]. There is very strong absorption in the region ofthe resonant modes.

8. Conclusions

I hope that this review has shown photonic theory to be in good shape, despite itsrapid development. Certainly we have an excellent capability for calculation of bandstructure in periodic dielectrics and the main challenge in photonic band gap materialsis their manufacture and integration with electronic technology. There are manypromising developments in this respect and it cannot be long before the first structureshaving band gaps at visible frequencies emerge.

For the moment the microwave region of the spectrum is the most accessibleexperimentally and provides a testing ground for theory. However there are manypotentially valuable applications to microwave technology particularly of complexmetallic structures and this area of application should be recognised as valuable in itsown right.

Another experimental tool which has much to offer is the electron microscope. Longused to study microstructured materials, electron microscopes have also developedsophisticated methods for measuring electron energy loss of the order 10eV or 20eVand provide a valuable tool for study of density of electromagnetic states in structuredmaterials. A community of theorists concerned with the impact of radiation on solids



[26] has for some time been studying what are essentially photonic effects and havemuch to contribute to the debate.

For the future theory still has many challenges. This is particularly true in for metallicstructures which as we have indicated show unique and complex effects. Thecomputational methodology for handling such systems has only recently becomeavailable through the adaptive mesh approach [20] and will be applied in the next fewyears to probing new structures, especially those of microwave significance.

Finally these is the link between electromagnetic properties and forces acting withinphotonic structures. These vary from the relatively weak Van der Waals forces actingbetween dielectric spheres, which may play a role in self assembly, to the much morepowerful dispersion forces acting between metallic surfaces separated by distances ofonly a few Ångstroms. This electromagnetic analogue of the electronic total energycalculation remains to be investigated in detail. The computational techniques are all inplace and await application.

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