DIFFERENTIAL FORM APPROACH TO
THE ANALYSIS OF ELECTROMAGNETIC
CLOAKING AND MASKING
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 多⑀ and 多␮ can be defined such that [9], [10]
F. L. Teixeira
ElectroScience Laboratory and Department of Electrical and
Computer Engineering, The Ohio State University, Columbus, OH
43210, USA
B ⫽ 多 ␮H,
D ⫽ 多 ⑀E,
Received 26 January 2007
ABSTRACT: We discuss the relationship between (a) electromagnetic
masking or cloaking of objects produced by some metamaterial blueprints 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
(3)
where D is the electric flux density (twisted) 2-form, and H is the
magnetic field intensity (twisted) 1-form. In terms of 多⑀ and 多␮ , the
four-dimensional Hodge * can be written as
G ⫽ ⴱF ⫽ ⫺ 多 ␮⫺1B ⵩ dt ⫹ 多 ⑀E ⫽ ⫺ H ⵩ dt ⫹ D.
From dF ⫽ 0, we obtain
冉
Key words: electromagnetic theory; maxwell equations
d ⫹ dt ⵩
冊
⭸
共B ⫹ E ⵩ dt兲 ⫽ 0,
⭸t
and hence
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 remarkable 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-dimensional spacetime write in a very concise manner as [6]
(1)
where d is the four-dimensional exterior derivative, F is the electromagnetic 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
DOI 10.1002/mop
冊
⭸B
⵩ dt ⫽ 0.
⭸t
(4)
Moreover, from dG ⫽ d * F ⫽ J, we have
冉
d ⫹ dt ⵩
冊
⭸
共D ⫺ H ⵩ dt兲 ⫽ ␳ ⫺ J ⵩ dt,
⭸t
and hence
共dD ⫺ ␳ 兲 ⫹
冉
J ⫺ dH ⫹
冊
⭸D
⵩ dt ⫽ 0
⭸t
(5)
From (4) and (5), Maxwell equations in 3 ⫹ 1 dimensions can be
decomposed as [5]
dE ⫽ ⫺
dF ⫽ 0
dⴱF⫽J
冉
dB ⫹ dE ⫹
⭸B
,
⭸t
dB ⫽ 0,
dH ⫽
⭸D
⫹ J,
⭸t
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 structure. 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 constitutive Eq. (3) that map 1-forms into 2-forms in 3-space [more
generally, Hodge operators map p-forms into (n ⫺ p)-forms in nspace] together with the material properties.
3. DIFFEOMORPHISMS ON THE METRIC OF SPACE
Under a differentiable but otherwise generic transformation (diffeomorphism) on the metric of space,
g 哫 g̃,
MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 8, August 2007
2051
Figure 1 Decomposition of 3 ⫹ 1 Maxwell equations into a topological
part (top four equations) and a metric part (boxed equations), obviated by
using differential forms. Under diffeomorphisms, the metric tensor g gets
mapped to g̃. This can be translated to a modification in the Hodge
operators 多⑀ and 多␮ , which can be made equivalent to a change on the
(spatial-slice) constitutive properties of the underlying medium: 关⑀兴 哫 关⑀˜ 兴
and 关␮兴 哫 关␮˜ 兴
the spatial-slice constitutive equations transform to
B ⫽ 多̃ ␮H,
D ⫽ 多̃ ⑀E.
(7)
In other words, by using differential forms, a change on the
metric is manifestly equivalent to a change on the material properties of the medium, now encoded by 多̃⑀ and 多̃␮ . This is to say
that, for a given (doubly anisotropic, for generality) medium with
constitutive tensors [␧] and [␮], either one of the transformations
共关 ⑀ 兴,关 ␮ 兴,g兲 哫 共关⑀兴,关␮兴,g̃兲
共关⑀兴,关␮兴,g兲 哫 共关⑀˜ 兴,关␮˜ 兴,g兲
(8)
where 关⑀˜ 兴 and 关␮˜ 兴 are new particular constitutive tensors, produce
the same differential form solutions, as illustrated by the diagram
in Figure 1. In terms of vector fields, either of these two transformations produce the same field distributions outside the region
affected by the diffeomorphism g 哫 g̃ and up to a scale factor
inside that region [15]. More specifically, an arbitrary diffeomorphism on the metric g 哫 g̃ can be described by a differentiable
coordinate transformation (u1, u2, u3) 哫 共ũ1 , ũ2 , ũ3 兲 with [S] as the
Jacobian matrix of partial derivatives, Sij ⫽ ⭸ui /⭸ũj .
For an original medium with constitutive tensors [␧] and [␮],
this transformation on the metric produces identical differential
form solutions as those resulting from the following transformation on the constitutive tensors [15, 16]
fashion. The term masking denotes here the use of a metamaterial
blueprint for coating a physical scatterer S0 so that the resulting
scattered field (from the coated object) becomes identical to that
from a different (noncoated) scatterer S1 (mask). In the case of a
zero volume mask, masking recovers cloaking [1]. This is illustrated in Figure 2.
One example of such masking where a closed-form expression for
the final metamaterial blueprints has been derived in [17]. In that case,
a convex PEC object S0 with surface radii of curvature in the range r
⑀[rm, rM], with rM ⱖ rm, is coated by a metamaterial blueprint of
uniform thickness ␦ such that the resulting scattered field becomes
identical to that of a (noncoated) PEC S1 in a isotropic medium with
constitutive parameters ␧ and ␮ having radii of curvature in the range
r ⑀ [0, rM ⫺ rm]. At any point p at the boundary of S0, one can define
orthonormal vectors û1, û2, n̂, where û1 and û2 are tangent along the
principal lines of curvature, and n̂ ⫽ û1 ⫻ û2 is outward normal. The
local tangential coordinates are denoted as ␨1, ␨2 and the normal
coordinate is ␩, with ␩ ⫽ 0 parametrizing the surface of S0 itself. If
the principal radii of curvature at p are given by rs1 and rs2, then at a
point p⬘ along ␩ for fixed 共␨1,␨2兲, the radii of curvature of parallel
surfaces are ri(␨1, ␨2, ␩) ⫽ rsi(␨1, ␨2) ⫹ ␩, i ⫽ 1, 2. In particular, the
radii of curvature at a point p⬘ on the surface of the mask S1 are r0i(␨1,
␨2), i ⫽ 1, 2, with r0i(␨1, ␨2) ⫽ rsi(␨1, ␨2) ⫺ rm, and rm ⫽ minp[rs1(p),
rs2(p)]. The required diffeomorphism on the metric for this case is
represented by the map M: ␩ 僆 [⫺rm,␦] 哫 ␩˜ 僆 [0,␦], ␩˜ ⫽ 3 M(␩)
⫽ ␦(rm ⫹ ␩)/(rm ⫹ ␦) (for ␩ ⬎ ␦, we have ␩˜ ⫽ ␩) [17]. The
metamaterial coating that results is an inhomogeneous and anisotropic
material blueprint with constitutive tensors [⑀˜ ] ⫽ ⑀[⌳], [␮˜ ] ⫽ ␮[⌳]
where
关⌳兴共 ␨ 1, ␨ 2, ␩ 兲 ⫽ û 1û 1
1 h̃ 1h 2
1 h 1h̃ 2
h 1h 2
⫹ û 2û 2
⫹ n̂n̂s ␩
,
s ␩ h 1h̃ 2
s ␩ h̃ 1h 2
h̃ 1h̃ 2
(10)
⫺1
⫺1
where hi ⫽ rsi
(rsi ⫹ ␩) and h̃i ⫽ rsi
(rsi ⫹ ␩˜ ), i ⫽ 1, 2, and s␩
⫽ ⭸␩˜ /⭸␩ ⫽ ␦/共rm ⫹ ␦兲 [17]. The derivation of this metamaterial
blueprint has close analogy with the derivation of Maxwellian
perfectly matched layers (PMLs) for conformal boundaries [18].
This is because the latter can be derived in the various coordinate
systems by applying a complex coordinate stretching (i.e., complexification on the metric of space) and the above metric invariance [10, 18].
As another example, conformal maps [19] are a class of diffeomorphic maps that preserve both orthogonality of coordinates
and isotropy of the (transformed) constitutive parameters. In this
case, g 哫 g̃ ⫽ ⍀共u兲g, such that ⍀(u) is a strictly positive scalar
关 ⑀˜ 兴 ⫽ 共det关S兴兲⫺1 关S兴 䡠 关⑀兴 䡠 关S兴T ,
关 ␮˜ 兴 ⫽ 共det关S兴兲⫺1 关S兴 䡠 关␮兴 䡠 关S兴T ,
(9)
since the same Hodge operators 多̃⑀ and 多̃␮ are recovered. The
differentiability of the coordinate transformation implies continuity of [S]. This in turns implies that the intrinsic impedance is not
changed, akin to a PML transformation [15].
4. HODGE OPERATORS AND MASKING BLUEPRINTS
Using the above properties, metamaterial blueprints that produce
electromagnetic masking of objects can be derived in a systematic
2052
Figure 2 By coating an arbitrary PEC scatterer S0 (at the left) with a
properly designed masking metamaterial the scattered field becomes equal
to that of a (mask) object S1 (at the right) in the region outside the dashed
curve. If the mask is a zero volume object, cloaking is produced
MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 8, August 2007
DOI 10.1002/mop
function of position. In an orthogonal system (u1, u2, u3) it then
follows that Sij ⫽ ␦ij / 冑⍀ (where ␦ij is the Kronecker delta). Using
(9), one can then write:
关 ⑀˜ 兴 ⫽
冑⍀关 ⑀ 兴,
关 ␮˜ 兴 ⫽
冑⍀关 ␮ 兴.
(11)
5. CONCLUSIONS
perfectly matched absorber for electromagnetic waves, Microwave
Opt Technol Lett 17 (1998), 231–236.
19. C.N. Yang, Conformal mapping of gauge fields, Phys Rev D 16
(1977), 330 –331.
20. F.L. Teixeira, On aspects of the physical realizability of perfectly matched
absorbers for electromagnetic waves, Radio Sci 38 (2003), 8014.
21. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F.
Starr, and D. R. Smith, Metamaterial electromagnetic cloak at microwave frequencies, Science 314 (2006), 977–980.
© 2007 Wiley Periodicals, Inc.
We have used the differential forms (exterior calculus) framework to
provide an unified analysis of the derivation of metamaterial blueprints for cloaking and masking of electromagnetic scatterers. Differential forms are quite suited for this purpose because they factorize
Maxwell equations into a metric part and a (purely) tological part, and
hence obviate the invariance of topological part under diffeomorphisms on the metric of space. The discussion has been restricted to
the material blueprint level only [20]. Issues about the realizability
and limitations on manufacturability of such metamaterials have not
been addressed here. This is briefly touched upon elsewhere [1, 20].
The realization and experimental evaluation of approximate cloaks
have been considered in [21].
REFERENCES
1. J.B. Pendry, D. Schuring, and D.R. Smith, Controlling electromagnetic
fields, Science 312 (2006), 1780 –1782.
2. D. Van Dantzig, The fundamental equations of electromagnetism,
independent of metrical geometry, Proc Cambridge Phil Soc 37
(1934), 421– 427.
3. G. Deschamps, Electromagnetics and differential forms, Proc IEEE 69
(1981), 676 – 696.
4. A.J. Ward and J.B. Pendry, Refraction and geometry in Maxwell’s
equations, J Mod Opt 43 (1996), 773–793.
5. F.L. Teixeira and W.C. Chew, Lattice electromagnetic theory from a
topological viewpoint, J Math Phys 40 (1999), 169 –187.
6. K.S. Thorne, C.W. Misner, and J.A. Wheeler, Gravitation, Freeman,
San Fransisco, 1973.
7. C.V. Westenholz, Differential Forms in Mathematical Physics,
Elsevier Science, North-Holland, 1980.
8. P.R. Kotiuga, Helicity functionals and metric invariance in three
dimensions, IEEE Trans Magn 25 (1989), 2813–2186.
9. K.F. Warnick and D.V. Arnold, Green forms for anisotropic, inhomogeneous media, J Electromagn Waves Appl 11 (1997), 1145–1164.
10. F.L. Teixeira and W.C. Chew, Differential forms, metrics, and the
reflectionless absorption of electromagnetic waves, J Electromagn
Waves Appl 13 (1999), 665– 686.
11. A. Bossavit, Generalized finite differences in computational electromagnetics, Geometric Methods for Computational Electromagnetics
(F. L. Teixeira, ed.), PIER 32, EMW Publishing, Cambridge, Mass
(2001), 45– 64.
12. F.W. Hehl and Y.N. Obukhov, Foundations of classical electrodynamics: Charge, flux, and metric, Birkhauser, Boston, 2003.
13. P.W. Gross and P.R. Kotiuga, Electromagnetic theory and computations: A topological apporach, Cambridge University Press, Cambridge, UK, 2004.
14. B. He and F.L. Teixeira, On the degrees of freedom of lattice electrodynamics, Phys Lett A 336 (2005), 1–7.
15. F.L. Teixeira and W.C. Chew, General closed-form PML constitutive
tensors to match arbitrary bianisotropic and dispersive linear media,
IEEE Microwave Guided Wave Lett 8 (1998), 223–225.
16. G.W. Milton, M. Briane, and J.R. Wills, On claoking for elasticity and
physical equations with a transformation invariant form, New J Phys 8
(2006), 248.
17. F.L. Teixeira, Closed-form metamaterial blueprints for electromagnetic masking of arbitrarily shaped, convex PEC objects, IEEE Antennas Wireless Propag Lett, submitted.
18. F.L. Teixeira and W.C. Chew, Analytical derivation of a conformal
DOI 10.1002/mop
ERRATUM: TUNABLE ERBIUM-DOPED
FIBER RING LASER BASED ON
COUPLING BETWEEN SIDE-POLISHED
FIBER AND SLIDING TAPERED
PLANAR WAVEGUIDE
Dong-Ho Lee,1Kwang-Hee Kwon,2Jae-Won Song,1 and
Kwang-Taek Kim3
1
School of Electrical Engineering and Computer Science, Kyungpook
National University, Sankyuk-dong, Puk-gu Daegu 702-701, Republic
of Korea
2
LS Cable Ltd., 555 Hogye-dong, Anyang-si, Kyungki-do 431-080,
Republic of Korea
3
Department of Optoelectronics, Honam University, 59-1, SeobongDong, Gwangsan-Gu, Gwangju 505-714, Republic of Korea
Received 20 March 2007
ABSTRACT: Originally published in Microwave Opt Technol Lett 49:
887– 889, 2007. © 2007 Wiley Periodicals, Inc. Microwave Opt Technol
Lett 49: 2053, 2007; Published online in Wiley InterScience (www.
interscience.wiley.com). DOI 10.1002/mop.22642
It has been brought to our attention that the co-authors’ affiliations
were omitted from the original printed article. We apologize for
any inconvenience this may have caused.
© 2007 Wiley Periodicals, Inc.
ERRATUM: A NEW WIDEBAND PLANAR
ANTENNA WITH HYPERPOLARTRANSFORMED SIERPINSKI CARPET
Gabsoo Lee and Youngsik Kim
Department of Electrical Engineering, Korea University, Seoul, Korea
Received 15 September 2006
ABSTRACT: Originally published Microwave Opt Technol Lett 49:
1090 –1092, 2007. © 2007 Wiley Periodicals, Inc. Microwave Opt
Technol Lett 49: 2053, 2007; Published online in Wiley InterScience
(www.interscience.wiley.com). DOI 10.1002/mop.22711
It has been brought to our attention that Gabsoo Lee is affiliated
with KT, R&D Center, Seoul, Korea, and Youngsik Kim with the
Department of Electrical Engineering, Korea University, Seoul,
Korea. We apologize for any inconvenience this may have caused.
© 2007 Wiley Periodicals, Inc.
MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 8, August 2007
2053