Note on transformation to general curvilinear coordinates ...

arXiv:physics/0307029v1 [physics.optics] 3 Jul 2003 Note on transformation to general curvilinear
coordinates for Maxwell’s curl equations

D M Shyroki

Department of Theoretical Physics, Belarusian State University,
Fr. Skaryna Avenue 4, 220080 Mensk, Belarus
E-mail: [email protected]

Abstract. Maxwell’s curl equations are formulated in curvilinear non-orthogonal
coordinates in three dimensions in a manner that enables to preserve conventional
Cartesian mesh for the finite-difference schemes by means of mere redefinition of the
permittivity and permeability coefficients. It is highlighted that in the previous works
on this subject (Ward and Pendry 1996 J. Modern Opt. 43 773; 1998 Phys. Rev. B
58 7252) inaccurate transformation to curvilinear coordinates, rooted in questionable
transformation laws assigned to electric and magnetic field vectors, led to physical
dubiety of the final formulae. In this note the amended equations are presented,
capable of being easily adopted for the accurate finite-difference time-domain (FDTD)
modelling of electromagnetic propagation and scattering problems in complicated
geometries within the existing computational codes.

PACS numbers: 03.50.De, 02.70.Bf

Note on transformation to general curvilinear coordinates for Maxwell’s. . . 2

1. Introduction

Maxwell’s equations span a long life of nearly one and a half century, yet it often
remains a real challenge to the theorist to solve, say, the problem of light scattering
from a complex surface or propagation in state-of-the-art photonic band-gap material.
To tackle all those numerous problems, a vast array of numerical methods have been
developed, in particular the finite-difference time-domain (FDTD) technique [1, 2] called
otherwise the order-N method [3] due to its linear scaling with the size of the system. In
its framework, the problem is formulated as an initial value problem leaned on discretized
Maxwell’s curl equations which can be integrated numerically.

The FDTD algorithm was originally given by Yee [4] in rectangular Cartesian
coordinates, i.e., on the cube mesh of points constituting the computational space.
Meanwhile, many objects of interest have curved surfaces which are purely represented
in a rectilinear mesh. Furthermore, severe problems associated with a uniform mesh arise
when the refractive index varies strongly within the structure, for example in metallo-
dielectric components. To circumvent these shortcomings, Holland [5] attempted
to formulate the FDTD algorithm in general curvilinear coordinates, but extreme
complexity of curvilinear mesh that “exacts a toll in computer time” [6] and makes
erecting a system of discretized equations “a programmer’s nightmare” [7] restrained
the practical usage of Holland’s result. Even in a two-dimensional case [6, 8], the
awkwardness of the formulae leaves one beyond any hope to qualitatively analyse the
equations.

Fortunately, a few years ago Ward and Pendry [7, 9] noticed that considerable
simplification of programming efforts can be achieved since transformation to curvilinear
coordinates can be regarded as redefining the permittivity and permeability coefficients,
ǫ and µ, so that the well-developed algorithm of computing on a rectangular mesh can
be translated almost verbatim to the case of a non-uniform mesh, provided the mesh
structure “encoded” in the effective ǫ and µ. This idea, being no doubt a really fruitful
and natural one, was implemented afterwards in a computer program for calculating
photonic band structures, Green’s functions and transmission–reflection coefficients [10].
An agreement with alternative numerical computations on orthogonal mesh and with
plane-wave expansion method was believed confidently good if not perfect.

However, one relevant inaccuracy in the cited works has been left out of sight:
in Maxwellian electrodynamics, the electric and magnetic fields and displacements E,
B, D, and H can be treated as vectors only with respect to pure rotations in three-
dimensional (3-D) Euclidean space, but care should be taken when transforming to 3-D
Riemannian space referenced to arbitrary curvilinear coordinate system. Meanwhile,
Ward and Pendry in a manner of early Holland’s misconstruction grounded upon the
highly questionable representation of affine-invariant Maxwell’s equations and in fact
came to a physically dubious formulation of final analytic results. The scope of this
note is to clarify the origin of that misconception and present the amended formalism
aimed to facilitate the non-orthogonal curvilinear FDTD technique.

Note on transformation to general curvilinear coordinates for Maxwell’s. . . 3

2. Maxwell’s equations in non-orthogonal curvilinear coordinates

The sound theoretical basis for any problem within the scope of classical electrodynamics

is constituted by Maxwell’s equations. They describe electromagnetic field as a unified

quantity invariant with respect to Lorentz group, i.e., rotations in Minkowski space,

and hence their utmost consistent formulation involves four-dimensional electromagnetic

field and flux tensors. However, if the medium in the laboratory frame is at rest, it is

often found convenient to decompose Maxwell’s equations into fully vectorial system for

three-dimensional quantities E, B, D, and H, written in cgs units as

∇ × E = − 1 ∂B , ∇ · B = 0, (1)
c ∂t (2)

∇ × H = 1 ∂D + 1 4πρv, ∇ · D = 4πρ,
c ∂t c

where ρ is the charge density, v is the charge velocity, c is the vacuum speed of light.

The accompanying constitutive relations are

D = ǫE, B = µH. (3)

Naturally the above equations are expected to be coordinate-free, that is preserving the

formal structure in any coordinate system, under any spatial coordinate transformation.

A principal point to be understood here is that, scholarly speaking, E, B, D, and H
are vectors with respect to rotations in three-dimensional Euclidean space referenced
to rectangular Cartesian system only. Even the point reflection is known to brake the
seeming symmetry between, say, E and B vectors in free-space counterparts of equations
(1), (2), since under r → −r operation we have E → −E for electric “polar” vector

versus B → B for magnetic “pseudovector.” Another example on the surface is r → ar
conversion: in contrast with “true” vectors, E should transform like E → a−1E to keep
the work of electromagnetic force δA = eE · dr invariant.

The vague ideas of “pseudoquantities” etc. find an elegant and comprehensible
mathematical conceptualization in the framework of conventional tensor calculus. To
be specific, for more than half a century the affine-invariant form of Maxwell’s equations

(1), (2) is known in both research [11] and textbook [12] literature:

2∂[α E β] = 1 ∂Bαβ , ∂[α B βγ] = 0, (4)
c ∂t (5)

∂αHˆ αβ = 1 ∂Dˆ β − 1 4πρˆvβ , ∂αDˆ α = 4πρˆ.
− c

c ∂t

Here Eα is covariant vector, Bαβ = B[αβ] is covariant bivector, Hˆ αβ = Hˆ [αβ] is
contravariant bivector-density, and Dˆ α is contravariant vector-density; the square

brackets denote alternation in a usual manner: ⋆[αβ] = 1 (⋆αβ − ⋆βα), etc. Note that
2!

despite of someone’s contradictory claims [13], the above equations are actually metric-

free; all the metric information is hidden in constitutive relations (see [12, ch 6])

Dˆ α = −ǫ g 1 gαβ Eβ , Bαβ = µ g− 1 gαγ gβδ Hˆ γδ , (6)
2 2

Note on transformation to general curvilinear coordinates for Maxwell’s. . . 4

where gαβ is the metric tensor of Riemannian space referenced to the right-handed
coordinate system xα; α, β, γ, δ = 1, 2, 3; g = det(gαβ).

In FDTD method, only the curl equations are needed, provided the divergence

equations satisfied initially. Eliminating the flux quantities via (6) yields

2∂[α E β] = µ g − 1 gαγ gβδ ∂Hˆ γδ , ∂αHˆ αβ = ǫ g 1 g βγ ∂ Eγ (7)
c 2 ∂t 2
c ∂t

for source-free region. Allowing for invariant correspondence between bivector-density

Hˆ αβ and pseudovector Hγ [12], one can rewrite (7) as

∂Eβ eαβγ = µγδ ∂Hδ , ∂Hβ eαβγ = ǫγδ ∂Eδ , (8)
∂xα c ∂t ∂xα − ∂t

c

where eijk is the fully skew-symmetric Levi-Civita symbol, and in a manner of [7, 9] I

introduce the effective permittivity and permeability quantities

ǫγδ = ǫg 1 g γδ , µγδ = µg − 1 gγ δ . (9)
2 2

Thus obtained equations are isomorphous to Maxwell’s curl equations on a conventional

Cartesian mesh. To restore fields in Euclidean space referenced to Cartesian system

xi (designated by Roman indices in contrast with Greek ones reserved for curvilinear

coordinates), the transformation matrix ∂ixα should be used:

Ei ⊜ Ei = ∂xα Hi ⊜ Hi ⊜ Hi = ∂xα (10)
∂xi Eα, ∂xi Hα.

The circle above the equality sign (⊜) means that this equality is not invariant but

holds in the given coordinate system. I also assume here that coordinate transformation

preserves the handedness of the system, which is likely to comprise all the practicable

cases of interest.

3. Comparison with Ward and Pendry

Formulae (8), (9), and (10) constitute the core of generalized non-orthogonal FDTD
method. In [7, 9] Ward and Pendry came to some principally similar results, but for the
effective permittivity and permeability they obtained (in current notation and units)

′ǫγδ = ǫg 1 g γ δ , ′µγδ = µg 1 gγδ . (11)
2 2

Note that the powers of g are different in (9), namely, 1 in the formula for ǫ vs. − 1 in
2 2
1
that for µ, while in (11) they are 2 both. This discrepancy is because Ward and Pendry

assumed with no argument the “symmetry between E and H fields” [7, p 777] which is

actually a questionable point when general coordinate transformations are considered.

The difference between the powers of g in (9) definitely seems to be more reasonable and

physically-intuitive due to the distinct difference between E and H fields as for their

transformation characteristics. Altogether I should remark that the discrepancy between

(9) and (11) vanishes when we consider the group of volume-preserving transformations,

e.g., introducing the skew lattice instead of orthogonal one, but is essential for the general

coordinate transformations with g = 1, especially in the case of magnetic media with

permeability µ comparable to or even exceeding the permittivity ǫ.

Note on transformation to general curvilinear coordinates for Maxwell’s. . . 5

4. Conclusion

The formulae by Ward and Pendry [7, 9] for the Maxwell’s curl equations in curvilinear
non-orthogonal coordinates are amended to allow for the marked difference between
electric and magnetic fields with respect to their transformation laws. The formulae
are considerably simple and can be easily adopted for an accurate non-orthogonal
FDTD modelling of electromagnetic propagation and scattering problems in complicated
geometries within the existing computational codes, since a mere cosmetic revision
is actually needed. The hitherto demonstrated agreement between numerical results
computed by Ward and Pendry and others may be accounted for two reasons: (i) special
character of coordinate systems used, namely those with affine volume g = 1, and (ii)
prevailing impact of effective ǫ over µ in dielectric media with high ǫ and unit µ.

Acknowledgments

I’d like to thank Dr A V Lavrinenko who acquainted me with Ward’s and Pendry’s
work, and Dr M N Polozov for a qualified check to the key ideas of this note.

References

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