Electromagnetic density of modes for a finite-size three-dimensional structure
Giuseppe D’Aguanno,
1,2,3,
*
Nadia Mattiucci,
1,2,4
Marco Centini,
2,3
Michael Scalora,
2
and Mark J. Bloemer
2
1
Time Domain Corporation, Cummings Research Park, 7057 Old Madison Pike, Huntsville, Alabama 35806, USA
2
Weapons Sciences Directorate, Research Development and Engineering Center, U.S. Army Aviation & Missile Command,
Building 7804, Redstone Arsenal, Alabama 35898-5000, USA
3
INFM at Dipartimento di Energetica, Università di Roma “La Sapienza,” Via A. Scarpa 16, I-00161 Rome, Italy
4
Università “RomaTre,” Dipartimento di Fisica “E. Amaldi,” Via Della Vasca Navale 84, I-00146 Rome, Italy
(Received 29 September 2003; published 24 May 2004)
The concept of the density of modes has been lacking a precise mathematical definition for a finite-size
structure. With the explosive growth in the fabrication of photonic crystals and nanostructures, which are
inherently finite in size, a workable definition is imperative. We give a simple and physically intuitive defini-
tion of the electromagnetic density of modes based on the Green’s function for a generic three-dimensional
open cavity filled with a linear, isotropic, dielectric material.
DOI: 10.1103/PhysRevE.69.057601 PACS number(s): 42.70.Qs, 41.20.-q
Several attempts have been made to generalize the notion
of the local density of modes (LDOM)—or local density of
states—and the density of modes (DOM)—or density of
states—to the case of open cavities—i.e., structure of finite
size where electromagnetic energy can flow in and out of the
volume bounded by the surface S of the cavity [1–6]. Quite
surprisingly, the concept of the density of states for a finite-
size structure still lacks a simple, concise definition. In the
words of Felbacq and Smaali, “Such notions as that of the
density of states or local density of states, which are crucial
in the description of the coupling between field and matter,
cannot be straightforwardly defined for finite structures” [5].
In our estimation, the most likely reason why the concepts of
the LDOM and DOM have not yet found straightforward
extensions to the case of three-dimensional (3D), finite struc-
tures is probably due to the fact that most approaches have
focused on the mathematical rather than the basic physical
aspects of the problem. Furthermore, due to their basic sim-
plicity, 1D and 2D open cavities filled with nonabsorbing
materials remain the subject of choice of most researchers.
If we have a closed cavity such that the field vanishes at
the edges or a cavity where periodic boundary conditions can
be applied (such as a multilayer stack of infinite length),
filled with a nonabsorbing medium, we can expect that the
electromagnetic energy will be conserved inside the cavity,
and the problem in Hermitian. In this case, the LDOM stands
for the number of eigenmodes per unit volume and unit fre-
quency at a point r
inside the cavity. If the electromagnetic
field can be specified by a single field component (TE or TM
polarization), then the scalar Green’s function can be ex-
panded in terms of the eigenmodes of the cavity and the
LDOM can be calculated through the imaginary part of the
scalar Green’s function:
r
-ImG
r
, r
[4,7,8]. The
DOM is then defined as the average LDOM inside the vol-
ume V of the cavity. So we ask (i) what happens if the cavity
is open, such that electromagnetic energy can flow in and out
of the volume bounded by the surface S of the cavity? (ii)
What happens if the cavity is filled with an absorbing mate-
rial? These questions have no easy answers fundamentally
because the electromagnetic problem is no longer Hermitian
and the cavity does not admit eigenmodes in the usual sense
of the word. As a consequence, the Green’s function does not
admit a straightforward expansion in terms the cavity modes
[7,9] and the very notion of the LDOM would seem to lose
its validity. In other words, can a LDOM still be defined
when the problem is not Hermitian (the case of open cavity
or material absorption), and what is its physical meaning in
this case [10]?
To answer these deceptively simple but crucial questions
we go back to the usual starting place—that is, Maxwell’s
equations, which we write in MKSA units, in the frequency
domain, assuming a harmonic time dependence of the type
e
-i t
and nonmagnetic materials
r
1:
E
= iB
, 1a
B
=
0
J
r
- i
c
2
r
E
, 1b
where J
r
is a complex current density,
r
is the spa-
tially dependent, relative, complex dielectric function of the
material, and
r
=
R
r
+ i
I
r
. Note that we use a ge-
neric, linear, and isotropic dielectric material. Equations (1)
describe the steady-state case; i.e., we assume that both the
electromagnetic field and the source J
r
oscillate with a
harmonic time dependence. Taking the curl of Eq. (1a) and
using Eq. (1b), we arrive at the following equation for the
electric field:
-
E
+
2
c
2
r
E
=- i
0
J
. 2
Let us now consider a cavity of volume V and surface S filled
with a dielectric material of dielectric function
r
, where
a known current density J
r
is present. Multiplying Eq. (2)
by E
*
, integrating over the volume V, using the vector ana-
log of Green’s first identity [11], and using Eq. (1a), we
arrive at the equation *Electronic address: giuseppe.daguanno@timedomain.com
PHYSICAL REVIEW E 69, 057601 (2004)
1539-3755/2004/69(5)/057601(4)/$22.50 ©2004 The American Physical Society 69 057601-1