May 21, 2008

Photonic crystals are periodically structured electromagnetic media. The periodicity, whose lengthscale is proportional to the wavelength of light of interest, is the electromagnetic analogue of a crystalline atomic lattice, where the latter acts on electron wavefunction to produce the familiar band gaps, semiconductors, etc., of solid state physics. Like we can control the behavior of electrons in semiconductor, we can also control the properties of electromagnetic waves by modeling the structure of photonic crystals. In this report, we will study the physical origin of photonic band gap and discuss how we can manipulate the propagation of light.

To start with, let's study the propagation of light in
a photonic crystal by the Maxwell equations. If we restrict ourselves to
linear, lossless materials and assume that the Maxwell equations impose on
a field pattern that happens to vary sinusoidally (i.e. in harmonic modes)
with time, we can separate out the time dependence of electric and
magnetic fields, i.e., **H**(**r**,t) =
**H**(**r**)e^{-iωt}. Then the Maxwell
equations for the steady state are [1],

∇ ×

Express the equation in magnetic field only:

This is the master equation. Let operator Θ =
∇ × (1/ε(**r**)) ×, then the master equation
becomes

Then it becomes an eigenvalue problem, with
(ω/c)^{2} being the eigenvalues and **H(r)** the
eigenvectors (modes). We can further prove that Θ is a Hermitian. So
solutions **H(r)** are orthogonal and ω^{2} are real and
positive, and they can be obtained by a variational principle.

Now we use the variational theorem to study the EM distribution of modes in the dielectric media. The lowest-frequency mode is the field pattern that minimizes the EM energy functional:

Add a small variation δ**H(r)** to
**H(r)**, it can be proved that E_{f}(H + δ**H(r)**) =
E_{f}(**H**) when **H** is an eigenvector of Θ, i.e.,
E_{f}(**H**) is stationary with **H**. Therefore, solutions
(modes) to the master equation are stationary points of the functional.
And we can see from the last equation that E_{f} is minimized when
the field is concentrated in the region of dielectric constant. Therefore,
lower order modes tend to concentrate its displacement field in regions of
high dielectric constant. This conclusion provides us a simple way to
qualitatively understand the features of modes, i.e., band diagram, in the
dielectric media.

A photonic crystal has periodical dielectric constant
ε(**r**) = ε(**r** + ** R_{i}**),

Like the band gap of semiconductor, photonic crystals
also have band gaps in which there are no propagating solutions (real
**k**) of the master equation. We will discuss the origin of photonic
band gaps in the rest of the section. For simplicity, we examine the
one-dimensional system.

Consider a 1D system with uniform dielectric constant
ε. Then the eigenvectors are ω(k) = ck, as depicted in
Fig. 1a and the E-fields are E(x) ∼ e^{±kx}. We set
an artificial periodicity *a* to fold the bands to the Brillouin zone
(as shown by the dashed lines in Fig. 1a). So now the *k = -π/a*
mode lies at an equivalent wavevector to the *k = π/a* mode, and
at the same frequency. So there is a degeneracy at the zone boundary, *k
= π/a*. We rewrite the E-fields at the boundary to be *e(x)* =
cos(*πx/a*) and *o(x)* = sin(*πx/a*) (shown in
Fig.1(c)), a linear combination of E(x) ∼ e^{±kx}, both
at *ω = cπ/a*. Now, we perturb *ε* so that it
is nontrivially periodic with periodicity *a*; for example,
*ε(x)* = 1 + Δcos(*2πx/a*). The two standing
waves then distribute accordingly. There are two ways to center a standing
wave of this type. We can center its nodes in center of each low-ε
layer, or in each high-ε layer. Any other position would violate
the symmetry of the unit cell about its center. From the Fig.1(c),
cos(*πx/a*) has its peaks lying in the high-ε layers thus
has energy concentrated in high-ε regions, while sin(πx/a) has
energy concentrated in low-ε regions. From previous discussion of
electromagnetic variational theorem, we know the low-frequency modes tend
to concentrate their energy in the high-ε regions while
high-frequency modes the opposite way. Therefore, in the "perturbed"
system, *e(x)* wave has a lower frequency while *o(x)* has a
higher frequency. So the degeneracy is broken and a band gap at 1BZ
boundary forms. By the same arguments, it follows that any periodic
dielectric variation in one dimension will lead to a band gap, a larger
variation for a larger gap. We call the band below a gap as the
"dielectric band",and a band above a gap as the "air band", analogous to
the "valence band" and the "conduction band" in semiconductor.

Inside the band gap, there are no propagating (real
**k**) modes. But if the wavevector is a complex, say, k =
k_{0} + iκ, then the master equation has evanescent modes,
i.e., *H(x) = e ^{ikx}u(x) =
e^{ik0x}u(x)e^{-κx}*, decaying
exponentially. We want to see where the complex wavevector originates.
Suppose a complex wavevectors exist in the vicinity of the band gap. As
shown in Fig.1(b), suppose we try to approximate the upper band near the
gap by expanding

Δω =
ω_{2}(*k*) - ω_{2}(*π/a*) =
α(*k - π/a*)^{2} =
α(Δ*k*)^{2}

So for frequencies lower than
the top of the gap, i.e., within the gap, Δω < 0, Δk =
iκ is purely imaginary. As we traverse the gap, the decay constant
κ grows as the frequency reaches the mid-gap, then disappears at the
gap edges. For perfect photonic crystals, because of the translational
symmetry, evanescent modes cannot be excited. However, a defect might
sustain such a mode, since the corresponding translational symmetry may be
broken. In those cases, we can create a localized, evanescent mode within
the photonic band gap. This concept is the basis for photonic-crystal
cavities and photonic-crystal waveguides.
As discussed previously, by perturbing a single lattice
site (or several connected sites) properly, we can permit a single mode or
sets of closely spaced modes that have frequencies in the gap. These modes
decay exponentially away from the defect, i.e., they are localized
(confined). This is the concept of photonic-crystal cavity based on
photonic band gap confinement. There are various ways to perturb the
selected site, say, removing it from the crystal, replacing it with
another whose size or dielectric constant is different, etc. To study the
effects that a small variation in ε(**r**) has on the frequency
of a mode, we can still use the variational theorem,

As we can use point defects in photonic crystal to trap photons, we can also guide light from one location to another by using line defects, i.e., by changing the dielectric along a line in a photonic crystal to generate modes that lie within the band gap. Then lights of corresponding frequencies are confined to, and can be directed along the waveguide.

© 2007 B. Dai. The author grants permission to copy, distribute and display the work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.

[1] J. D. Joannopoulos *et al*, *Molding the
Flow of Light* (Princeton University Press, 1995).