DIFFERENTIAL FORM APPROACH TO
THE ANALYSIS OF ELECTROMAGNETIC
CLOAKING AND MASKING
F. L. Teixeira
ElectroScience Laboratory and Department of Electrical and
Computer Engineering, The Ohio State University, Columbus, OH
43210, USA
Received 26 January 2007
ABSTRACT: We discuss the relationship between (a) electromagnetic
masking or cloaking of objects produced by some metamaterial blue-
prints and (b) some invariances of Maxwell equations obviated by means of
differential forms (exterior calculus). © 2007 Wiley Periodicals, Inc.
Microwave Opt Technol Lett 49: 2051–2053, 2007; Published online in
Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.
22640
Key words: electromagnetic theory; maxwell equations
1. INTRODUCTION
It has been recently shown that it is in principle possible to achieve
(under certain idealized conditions) electro-magnetic cloaking of
objects by using metamaterial coatings with properly designed
constitutive parameters [1]. The basic feature underlying this re-
markable possibility is the invariance of Maxwell equations under
diffeomorphisms (metric invariance) [1–5], i.e., the fact that a
change on the metric of space can be mimicked by a proper change
of the constitutive tensors. In this work, we discuss how this
feature of Maxwell equations is obviated using differential forms
and the exterior calculus framework [3, 5–14]. We then illustrate
how this feature also allows for generic masking of objects (again,
under idealized conditions) via appropriate metamaterial coatings.
2. MAXWELL EQUATIONS AND DIFFERENTIAL FORMS
In terms of differential forms, Maxwell equations in four-dimen-
sional spacetime write in a very concise manner as [6]
dF = 0
d F = J (1)
where d is the four-dimensional exterior derivative, F is the elec-
tromagnetic strength 2-form, J is the source density 3-form, and *
is the four-dimensional Hodge star operator. By performing a 3 +
1 splitting and treating time as a parameter, we can write F in terms
of the electric field intensity 1-form E and the magnetic flux
density 2-form B, and write J in terms of the (electric) current
density 2-form J and the (electric) charge density 3-form as
F = E dt + B,
J = P - J dt. (2)
where, d is the three-dimensional (spatial) exterior derivative, and
is the exterior product. The exterior derivatives d and d are
both metric-free and nilpotent, and related through
d = d + dt
t
.
It is interesting to note that the nilpotency of d implies dJ = 0,
a statement of charge conservation.
On each spatial slice, three-dimensional (spatial) Hodge star
operators
and
can be defined such that [9], [10]
B =
H,
D =
E, (3)
where D is the electric flux density (twisted) 2-form, and H is the
magnetic field intensity (twisted) 1-form. In terms of
and
, the
four-dimensional Hodge * can be written as
G = F =-
-1
B dt +
E =- H dt + D.
From dF = 0, we obtain
d + dt
t
B + E dt = 0,
and hence
dB +
dE +
B
t
dt = 0. (4)
Moreover, from dG = d* F = J, we have
d + dt
t
D - H dt = - J dt ,
and hence
dD - +
J - dH +
D
t
dt = 0 (5)
From (4) and (5), Maxwell equations in 3 + 1 dimensions can be
decomposed as [5]
dE =-
B
t
,
dB = 0,
dH =
D
t
+ J ,
dD = , (6)
where the only spatial operator present is d. When applied to 0-, 1-, or
2-forms, respectively, d is equivalent to the more familiar grad, curl,
and div operators of vector calculus, distilled from their metric struc-
ture. Indeed, since d is metric-free, none of the equations in (6) depend
on the metric of space (i.e., they are invariant under diffeomorphisms)
[5]. All metric information is incorporated into the spatial-slice con-
stitutive Eq. (3) that map 1-forms into 2-forms in 3-space [more
generally, Hodge operators map p-forms into (n - p)-forms in n-
space] together with the material properties.
3. DIFFEOMORPHISMS ON THE METRIC OF SPACE
Under a differentiable but otherwise generic transformation (dif-
feomorphism) on the metric of space,
g g ˜ ,
DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 8, August 2007 2051