www.sciencemag.org/cgi/content/full/1125907/DC1
Supporting Online Material for
Controlling Electromagnetic Fields
J. B. Pendry,* D. Schurig, D. R. Smith
*To whom correspondence should be addressed: j.pendry@imperial.ac.uk
Published 25 May 2006 on Science Express
DOI: 10.1126/science.1125907
This PDF file includes:
SOM Text
Figs. 1 to S3
Supporting online material for:
Controlling Electromagnetic Fields
J. B. Pendry1, D. Schurig2 and D. R. Smith2
Department of Physics, The Blackett Laboratory,
Imperial College London, London SW7 2AZ, UK,
2
Department of Electrical and Computer Engineering,
Duke University, Box 90291, Durham, NC 27708, USA.
1
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Scaling of ε, µ for a
General Coordinate Transformation
The following is taken from reference (14) of the main text with corrections for a few
typographical errors.
We start from Maxwell's equations in a system of Cartesian coodinates:
∇ × E = −µµ0 ∂H ∂t ,
∇ × H = +εε0 ∂E ∂t
(1)
where both ε and µ may depend on position. Let us transform to a general system defined
by,
q1 ( x, y, z ) , q2 ( x, y, z ) , q3 ( x, y, z )
(2)
Lines of constant q2 , q3 define the generalized q1 axis, and so on. Thus if we define a set
of points by equal increments along the q1, q2 , q3 axes, the mesh will appear distorted in
the original x, y, z co-ordinate frame: see figure 1.
Figure 1. Simple cubic lattice of points in one co-ordinate system (left) maps into a
distorted mesh in the other co-ordinate system (right). Choosing the co-ordinate
transformation correctly give can control the distortions and produce a mesh tailored
to our requirements.
Maxwell's equations as defined above are written in the original Cartesian system. What
form does the new set of equations take when expressed in terms of q1, q2 , q3 ? The answer
is surprisingly simple. Maxwell's equations in the new system of co-ordinates become
ˆ = −µ µˆ ∂H
ˆ ∂t ,
∇q × E
0
ˆ = +ε εˆ ∂E
ˆ ∂t
∇q × H
0
(3)
ˆ, H
ˆ are renormalized electric and magnetic fields.
where εˆ , µˆ are in general tensors, and E
All four quantities are simply related to the originals. In other words the form of Maxwell's
equations is preserved by a co-ordinate transformation: the co-ordinate transformation does
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Scaling of ε, µ for a General Coordinate Transformation
not change the fact that we are still solving Maxwell's equations, it simply changes the
definition of εˆ , µˆ .
We define three units vectors, u1, u2 , u3 , to point along the generalized q1, q2 , q3 axes. The
length of a line element is given by,
ds 2 = dx 2 + dy 2 + dz 2 = Q11dq12 + Q22 dq22 + Q33dq32
+2Q12 dq1dq2 + 2Q13dq1dq3 + 2Q23dq2 dq3
(4)
where,
Qij =
∂x ∂x ∂y ∂y
∂z ∂z
+
+
∂qi ∂q j ∂qi ∂q j ∂qi ∂q j
(5)
In particular we shall need the length of a line element directed along one of the three axes,
dsi = Qi dqi
(6)
Qi2 = Qii
(7)
where for shorthand,
To calculate ∇ × E consider a small element, small enough that it resembles a
parallelepiped (figure 2). In this we assume that the transformation has no singularities such
as points or lines where the co-ordinate system suddenly heads off in a different direction.
Figure 2. Small element resembling a parallelepiped.
Figure 3. Integration path for finding ∇ × E .
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Scaling of ε, µ for a General Coordinate Transformation
First we calculate the projection of ∇ × E onto the normal to the u1 − u 2 plane by taking a
line integral around the u1 − u 2 parallelogram and applying Stokes’ theorem (figure 3). We
define,
E1 = E ⋅ u1,
E2 = E ⋅ u 2 ,
E3 = E ⋅ u3
(8)
so that,
( ∇ × E ) ⋅ ( u1 × u 2 ) dq1Q1dq2Q2 = dq1
∂
∂
( E2 dq2Q2 ) − dq2 ( E1dq1Q1 )
∂q1
∂q2
(9)
or,
( ∇ × E ) ⋅ ( u1 × u 2 ) Q1Q2 =
∂Eˆ 2 ∂Eˆ1
ˆ
−
= ∇q × E
∂q1 ∂q2
(
)
3
(10)
where we use the conventional superscript notation for contravariant components of a
vector. We have defined,
Eˆ1 = Q1E1,
Eˆ 2 = Q2 E2 ,
Eˆ3 = Q3 E3
(11)
Note that the right-hand side of equation (10) is simple 'component 3' of curl evaluated in
the new co-ordinate system. Now applying .Maxwell,
( ∇ × E ) ⋅ ( u1 × u 2 ) Q1Q2 = −µ0µ
∂H
⋅ ( u1 × u 2 ) Q1Q2
∂t
(12)
We write H in terms of the contravariant components,
H = H 1u1 + H 2u 2 + H 3u3
(13)
which in turn can be expressed in terms of the covariant components,
⎡ H1 ⎤
⎢ ⎥ ⎡ u1 ⋅ u1 u1 ⋅ u 2
g −1 ⎢ H 2 ⎥ = ⎢⎢u 2 ⋅ u1 u 2 ⋅ u 2
⎢ 3⎥
⎢ H ⎥ ⎢⎣ u3 ⋅ u1 u3 ⋅ u 2
⎣ ⎦
⎡ 1⎤
u1 ⋅ u3 ⎤ ⎢ H ⎥ ⎡ H1 ⎤
u 2 ⋅ u3 ⎥⎥ ⎢ H 2 ⎥ = ⎢⎢ H 2 ⎥⎥
⎢ ⎥
u3 ⋅ u3 ⎥⎦ ⎢ H 3 ⎥ ⎢⎣ H 3 ⎥⎦
⎣ ⎦
(14)
where the first part defines g , and
H1 = H ⋅ u1,
H 2 = H ⋅ u2 ,
H 3 = H ⋅ u3
(15)
Inverting g gives
3
H i = ∑ g ij H j .
(16)
j =1
Substituting equation (13) into (12) gives
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Scaling of ε, µ for a General Coordinate Transformation
∂H
⋅ ( u1 × u 2 ) Q1Q2
∂t
3
∂H j
u3 ⋅ ( u1 × u 2 ) Q1Q2
= −µ0µ ∑ g 3 j
∂t
j =1
( ∇ × E ) ⋅ ( u1 × u 2 ) Q1Q2 = −µ0µ
(17)
Define
(
µˆ ij = µg ij u1 ⋅ ( u 2 × u3 ) Q1Q2Q3 Qi Q j
)
−1
(18)
and
Hˆ j = Q j H j
(19)
so that,
3
( ∇ × E ) ⋅ ( u1 × u2 ) Q1Q2 = −µ0 ∑ µˆ 3 j
∂Hˆ j
j =1
∂t
(20)
Hence on substituting from equation (10),
(
3
)
ˆ i = −µ ∑ µˆ ij
∇q × E
0
j =1
∂Hˆ j
(21)
∂t
and by symmetry between E and H fields,
(
)
i
3
ˆ = +ε ∑ εˆ ij
∇q × H
0
j =1
∂Eˆ j
(22)
∂t
where
(
εˆ ij = εg ij u1 ⋅ ( u 2 × u3 ) Q1Q2Q3 Qi Q j
)
−1
(23)
Note that these expressions simplify considerably if the new co-ordinate system is
orthogonal, e.g. cylindrical or spherical, when
g ij u1 ⋅ ( u 2 × u3 ) = δij
(24)
The orthogonal version is quoted in the main text.
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