CARLOS A. RIOS

GRUPO DE BIOFISICA

U. de Antioquia

INTRODUCTIONControlling the

Properties of MaterialsPhotonic Crystals

ELECTROMAGN. IN MIXED DIELECTRIC

MEDIA

The Macroscopic Maxwell Equations

Why Use the Magnetic Field, and Not the

Electric?

Electromagnetism as an Eigenvalue Problem

General Properties of the Harmonic Modes

Electromagnetic Energy and the

Variation Principle

SYMMETRIES AND SOLID-STATE EM

Symmetries in 1D photonic Crystals

BandGap

THE MULTILAYER FILM: A ONE

DIMENSIONAL PHOTONIC CRYSTAL

Transfer Matrix Simulations

o Mechanical, electrical and optical properties.

For reach this goal is so important to observe and study the natural “examples”.

o A crystal is a periodic arrangement of atoms or molecules. The pattern with which the atoms and molecules are repeated in space is the crystal lattice. The crystal presents a periodic potential to an electron propagate through it, and both the constituents of the crystal and the geometry of the lattice dictate the conduction properties of the crystal.

The Optical analogue is the photonic crystal, in which the atoms or molecule are replaced by macroscopic media with differing dielectric constants, and the periodic potential is replaced by a periodic dielectric function (periodic index of refraction). If the dielectric constants of the material in the crystal are sufficiently different, and if the absorption of light by the materials is minimal, then the refractions and reflections of light from all of the various interfaces can produce many of the same phenomena for photons (light mode) that the atomic potential produces for electrons. In particular, we can design photonic crystal with photonic band gaps, preventing light form propagating in certain directions with specified frequencies (i.e., a certain range of wavelengths, or “colors”, of light)

Some photonic crystals in the nature: Sea Mouse and its hair

Some photonic crystals in the nature: MorphoRethenor Butterfly

o We will restrict to propagation within a mixed dielectric medium, a composite of regions of homogeneous dielectric material as a function of position vector r, in which the structure does not vary with time, and there are no free charge or currents.

We are going to assume:

1. The field strengths are small enough so that we are in the linear regime.

2. The material is macroscopic and isotropic.

3. We ignore any explicit frequency dependence (material dispersion) of the dielectric constant.

4. We focus primarily on transparent material, which means we can treat ε(r) as purely real and positive.

refractive index is:

o The Maxwell equations become:

o harmonic modes.

We can change the temporal derivate for –iω, so the main equations are:

We can decouple these equations and obtain the called MASTER EQUATION, where the speed of light c=(0 ε0) -½ :

o The master equation together with the divergence equation, it tells us everything we need to know about H.

The strategy will be as follows:

1. For a given structure ε(r), solve the master equation to find the modes H(r) and the corresponding frequencies, subject to the transversality requirement.

2. Use the next equation to recover E (r)

o If we identify the left side of the master equation as an operator acting on H(r), , then it will look like a traditional eigenvalue problem:

Where the operator takes the curl, then divides by ε(r) , and then takes the curl again:

The eigenvectors H(r) are the spatial patterns of the harmonic modes, and the eigenvalues ω/c2 are squared frequencies of those modes.

This operator has the following properties, which demonstration are not difficult:

1. It is a linear operator.

2. It is an Hermitian Operator:

Having established that is Hermitian, we can show that :

1. The eigenvalues of this operator are real numbers.

2. ω2 is always nonnegative for ε0 .

3. For two Harmonic modes H1 (r) and H2(r), with different frequencies ω1

and ω2 , for the Hermiticity they have inner product of zero. This

implies that

if ω1 ω2 then (H1 ,H2)=0, and we say that they are orthogonal. If two

harmonic modes have equal frequencies ω1 = ω2 then we say they are degenerate and they are not necessarily orthogonal.

o Roughly, a mode tends to concentrate its electric-field energy in regions of high dielectric constant, while remaining orthogonal to the modes below it in frequency. If we use the VariationalTheorem , we will say that the eigenvalue ω0/c2 minimizes the functional:

o Uf is called the Rayleigh Quotient, and we will refer to it as the electromagnetic “energy” functional. Now we want to know what is the resulting small change Uf in the energy functional by adding a small-amplitude function H It should be zero if the energy functional is really at the minimum.

o Ignoring terms higher than first order in H, we can write Uf in the form:

o This G can be interpreted as the gradient of the functional with respect H. At an extremum, Uf , must vanish for all possible shifts H, which implies that G=0. This implies that the parenthesized quantity in the last equation is zero, which is equivalent to saying that H is an eigenvector of . Therefore, Uf is at an extremum if and only if H is a harmonic mode.

o The idea for solve a photonic crystal problem, is that for a given frequency, we could solve for H and then determine E. But we could choose to solve for E and then determine H. Why not?

By pursuing this alternate approach, one finds the condition on the electric field to be:

In this case, the operator is not Hermitian but if we stick to the generalized eigenproblem, however, then simple theorems analogous to those of the previous section can be developed because the two operators of the generalized eigenproblem are easily shown to be both Hermitian and positive semi-definite, but the calculations turn into in a very difficult problem.

“The Symetries of a system allow one to make general statement about thatsystem’s behavoir “

Whenever two operators commute, one can construct simultaneouseigenfunctions of both operators. In our case, the new operator is going tobe the symmetry operator.

A system with translational symmetry is unchanged by a translation through a displacement d. For each d, we can define a translation operator T which, when operating on a function f(r), shifts the argument by d.

A system with continuous translation symmetry in the x direction:

With the plane waves approximation we have:

If the system is invariant under all of the translation operators of the xy plane. We can classify the modes according to their in-plane wave vectors. So:

This is the case of an 1D photonic crystal. For this system we still have continuous translational symmetry in the x and in the y direction, but now we have discrete translational symmetry in the z direction. The basic step length is the lattice constant a. The basic step vector is called the primitive lattice vector a=ay. Because of this discrete symmetry

, by repeating this translation, we see that Where R=may, m is an integer.

As we saw before, because of the continuous translational symmetry:

And:

So:

All the modes with wave vectors of the form where n is an integer, form a degenerate set, they all have the same translational eigenvalue of . k// can assume any value but kz can be restricted to a finite interval, the one-dimensional Brillouin zone. Using the primitive reciprocal lattice vector , the Brillouin zone is

Results for some multilayer systems where each layer has a width of 0.5a{

Left: Homogeneous media Center: ε alternate between 13 and 12 Right: ε

alternate between 13 and 1 .

BandGap obtained in a multilayer film alternating layers with different

width. The width of ε=13 layer is 0.2a and the width of ε=1 layer is 0.8a

In a boundary b we have, taking the tangential components of E and H:

These fields at the other interface at the same instant and at a point with identical x and y coordinates can be determined by alterind the phase factors of the waves to allow for a shift in the z coordinate from 0 to -d

(1)

(2)

so:

Solving for Ea

Doing the same for Ha:

This can be written as a Matrix as follows:

If this process is repeating for q layers, we get:

Where:

For TM or p-polarization:

For TE or s-polarization:

With the transfer matrix we can calculate the reflectance, transmittance and absorptance, with the following equations:

Morpho Rethenor:

With normal incidence and with TE polarization (There is no a

significant difference between both polarizations in normal incidence)

Reflectancia y transmitancia

experimental y simulada

Morpho Adonis

Reflectancia y transmitancia

experimental y simulada

Morpho Achilleana

Reflectancia y transmitancia

experimental y simulada

Morpho Portis

Reflectancia y transmitancia

experimental y simulada

With an incidence of 45⁰:

TM

TE