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Thermal emission and design in

2D-periodic metallic photonic crystal

slabs

David L. C. Chan, Marin Soljaˇ ci´ c and J. D. Joannopoulos

Department of Physics and Center for Materials Science and Engineering,

Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

dlcchan@mit.edu

Abstract:

derstand some of the physical phenomena that drive thermal emission in

2D-periodic metallic photonic crystal slabs, emphasizing phenomenology

and physical intuition. Through detailed numerical calculations for these

systems, we find that periodicity plays a key role in determining the types

of physical phenomena that can be excited. We identify two structures

as good candidates for thermal design, and conclude with a discussion

of how the emissive properties of these systems can be tailored to our needs.

We present a useful framework within which we can un-

© 2006 Optical Society of America

OCIS codes: (000.6800) Theoretical physics; (160.4670) Optical properties; (350.5610) Radi-

ation.

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1. Introduction

A blackbody is defined as an object of perfect absorption. Its entropy is maximized, and

in that sense it exemplifies utter disorder. The physics of blackbodies has both fascinated

and intrigued scientists for well over a century now [1]. In practice, most objects have only

finite absorption, and are thus referred to as ‘graybodies’. However, graybodies are of in-

terest because their thermal emission spectra can be changed by altering the geometry of

the system or the materials used. The ability to modify or tailor the thermal emission pro-

file of an object is of great importance and interest in many areas of applied physics and

engineering. It has been noted that periodic sub-wavelength scale patterning of metallo-

dielectric systems, i.e. photonic crystals, can modify their emission spectra in interesting

ways [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. Thermal radiation from 2D-

periodic photonic crystals has been studied within the contexts of spectral and directional con-

trol [19, 8, 20, 21, 22], guided resonances [23], thermophotovoltaic generation [13], resonant

scattering [24, 25], laser action [26], Kirchhoff’s law [15], coherence [9, 22], and spontaneous

emission enhancement [5, 7, 20].

In this article, we focus on some of the most important physical phenomena that give rise

to many of the features observed in thermal emission spectra of 2D-periodic metallic photonic

crystal slabs, with the intention of developing physical intuition and understanding of features

of emission spectra. We demonstrate through detailed numerical studies the key role played by

periodicity in determining the types of physical phenomena that can be thermally excited in 2D-

periodic metallic photonic crystals. We develop understanding and physical insight using two

illustrative examples, before applying them to hybrid structures. Such structures exhibit strong

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Received 24 April 2006; revised 13 August 2006; accepted 27 August 2006

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thermal emission peaks which can be used as building blocks in thermal design. We show how

one can tailor the emissive properties of these structures to one’s design needs by changing two

simple physical parameters.

2.Description of numerical methods

Kirchhoff’s law states that for an object in thermal equilibrium with the surrounding radiation

field, its absorptivity and emissivity are equal, for every frequency, direction, and polarization.

Thus, to study thermal emission of an object, we need simply calculate its absorptivity spec-

trum, knowing that the object’s absorptivity and emissivity spectra are identical. Moreover, for

the purposes of developing an intuitive understanding of the physics behind thermal emission,

it is often more helpful to think in terms of absorption rather than emission, and it is on this

basis that we will proceed.

Fig. 1. Schematic illustrating the geometry of a typical system. The x- and y-axes are de-

fined in the plane of the slab, with the z-direction coming out of the slab. We study the

thermal radiation being emitted in the perpendicular direction.

Figure 1 is a schematic illustrating the geometry of a typical system under investigation. It

is important to note that because of the mirror symmetry of the system in a plane perpendic-

ular to x and y, the modes of the system can be separated into transverse electric (TE) and

transverse magnetic (TM) modes with respect to the mirror plane. As a result of this symme-

try, x-polarized modes do not mix with y-polarized modes. Thus, we can analyze these two

polarizations completely separately, and this is what we do in all our calculations.

Numerical simulations in our work are performed using a finite-difference time-domain

(FDTD) algorithm [27]. These are exact (apart from discretization) 3D solutions of Maxwell’s

equations, including material dispersion and absorption. We choose a computational cell with

dimensions 40×40×240 grid points, corresponding to 40 grid points per lattice constant a.

The faces of the cell normal to the x and y axes are chosen to have periodic boundary conditions,

while the faces normal to the z-axis (i.e. the top and bottom ones) have perfectly matched layers

(PML)topreventreflection.Inotherwords,thisisa3Dsimulationofa2D-periodicsystem.The

PhC slab is in the middle, and flux planes are placed on either side of it at least 2a away. We run

the simulation for a total of 40,000 time steps, chosen to be sufficiently large to allow resolution

of peaks with quality factors (Q) up to 250. We illuminate the photonic crystal slab with a nor-

mally incident, temporally Gaussian pulse. We record the fields going through flux planes on ei-

ther side of the slab and perform a discrete Fourier-transform on the time-series of fields, which

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we use to calculate fluxes as functions of frequency, Φ(ω) =1

We run the simulation once with the slab in place, and again with vacuum only, such that

Eslab= Evac+Eref, with Erefbeing the field due to reflection. The reflectance is given by

2Re{?E∗(r,ω)×H(r,ω)·dS}.

R(ω) ≡Φref

Φvac

=−1

2Re{?

A1[Eslab(r,ω)−Evac(r,ω)]∗×[Hslab(r,ω)−Hvac(r,ω)]·dS}

1

2Re{?

where A1is the flux plane corresponding to ‘1’, and the minus sign in the numerator is there

to make the reflected flux positive. This expression can be shown to simplify to R(ω) =

[Φvac

1

(ω)]/Φvac

flux plane further from the light source is ‘2’. (One can show that the numerator becomes

Φvac

1

for incoming and outgoing plane waves in vacuum, for which E and H are proportional.) Sim-

ilarly, the transmittance is given by T(ω) = Φslab

A(ω)=1−R(ω)−T(ω). This way, we obtain reflectance, transmittance and absorbance spec-

tra for PhC slabs. We incorporate absorption into our simulations by means of the Drude model,

according to the following equation:

A1E∗

vac(r,ω)×Hvac(r,ω)·dS}

(1)

1(ω)−Φslab

1(ω) where the flux plane closer to the light source is ‘1’, and the

1(ω)−Φslab

(ω)+1

2Re{?

A1(E∗

vac×Href−Hvac×E∗

ref)·dS} but the cross term vanishes

(ω)/Φvac

2

2(ω) and the absorbance is simply

ε(ω) = ε∞+

σ

(ω2

0−ω2−iγω)

(2)

where ε∞, γ, ω0and σ are input parameters. In our case, we are concerned with metals, for

which ω0= 0. By Kirchhoff’s law, the absorbance spectra so calculated are identical to the

emittance spectra of these objects, for each polarization, frequency and observation angle.

3.Holes and dips

Let us now turn our attention to real systems and the physical effects that are manifested therein.

The goal is to develop an understanding of the physical processes that drive emittance in these

systems. The first structure we will examine is a simple metal slab with holes (see Fig. 2). If we

illuminate the structure with light incident from the top of the cell, the light propagates down

the holes which act as metallic waveguides. Waveguide cut-offs arise from the requirement that

the parallel component of the electric field be continuous across a boundary. Inside a perfect

metal, the electric field is strictly zero. For such a material, E?is constrained to vanish at the

surface, and this leads to the well-known cut-off frequency corresponding to a half-wavelength

oscillation. Below this frequency, no propagating mode can be supported within the waveg-

uide, because the boundary condition cannot be satisfied. For a realistic metal (i.e. one that

permits some penetration of fields), the fields are not required to exactly vanish at the surface,

but must decay away rapidly and exponentially once inside the material. Such boundary con-

dition matching leads to a similar cut-off as in the case of the perfect metal, except that the

penetration of field into the metal produces a cut-off with a slightly lower frequency, because

the effective width of the waveguide is slightly larger. Cut-off frequencies depend on the width

of the waveguide. The wider the waveguide, the lower the cut-off frequency.

We present emittance and transmittance spectra for this system. Figure 2(a) shows how the

spectrachangewithholeradius.Thepeaksbelow1.0(indicatedbyblackarrows)arewaveguide

cut-offs arising from propagation of light through the holes. These peaks decrease in frequency

with increasing radius, a clear signature of waveguide cut-offs. They correspond to modes that

fit approximately half a wavelength across the hole in the x-direction. As we discussed, the

electric field has to be continuous as we cross media boundaries in the x-direction (because Eyis

paralleltothemediaboundary)butnotinthey-direction.Thus,thesemodeshaveone‘hump’as

we cross the holes in the x-direction, and decay exponentially inside the metallic bulk between

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01234

Frequency (c/a)

0

0.2

0.4

0.6

0.8

1

Emittance/Transmittance

t = 1.0a

t = 2.0a

0

0.2

0.4

0.6

0.8

1

Emittance/Transmittance

r = 0.1a

r = 0.2a

r = 0.3a

r = 0.4a

?

?

?

?

???

???

?

?

?

?

Fig. 2. (Color) Here we show emittance (solid lines) and transmittance (dotted lines)

spectra for a 2D-periodic metal slab with circular holes, viewed at normal incidence

and for y-polarized light. The Drude parameters used for the metal are ε∞= 1, ω0= 0,

γ = 0.3(2πc/a) and ωp=√10(2πc/a). In Panel (a), we fix the thickness of the slab at

1.0a (where a is the lattice constant of the slab) and vary the radius of the holes. The black

arrows indicate the peaks produced by the waveguide cut-off in the x-direction. In Panel

(b), we keep the hole radius constant at 0.4a and vary the thickness of the slab. Here, we

use arrows to indicate the peaks produced by diffraction.

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Received 24 April 2006; revised 13 August 2006; accepted 27 August 2006

18 September 2006 / Vol. 14, No. 19 / OPTICS EXPRESS 8789