Decomposable bianisotropic media: Factorization, equivalent media, Green dyadics, and Hertz potentials
 The aim of this paper is to provide the reader with a rigorous, well-structured, and complete study of the most general decomposable bianisotropic media. These media were introduced elsewhere. We will give full expressions for the factorization of the Helmholtz determinant operator, for the equivalent media, for the Green dyadics of these equivalent media, and for the generalized Hertz potentials. This will be done efficiently using a six-vector formalism. Finally, we will treat, as an example, an anisotropic medium that has not been treated before.
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 The authors of the present analysis have been actively studying basic electromagnetic field solutions in bianisotropic media during the past decade [Olyslager and Lindell, 2002]. In a general bianisotropic medium, characterized by the constitutive relations
analytic field solutions such as plane waves or Green dyadics are difficult to obtain. However, certain subclasses of these media allow some progress in such analytic solutions. A well-known example is the uniaxial anisotropic medium where we can decompose the fields in TE and TM parts. These TE and TM parts experience the uniaxial bianisotropic medium as a simpler affine isotropic medium [Lindell, 1995]. We say that a uniaxial anisotropic medium is a decomposable medium. After some efforts we were able to obtain [Olyslager and Lindell, 2001a; Olyslager and Lindell, 2001b], the most general bianisotropic medium that allows a decomposition of the fields. In general the slowness surface that describes plane wave solutions in a medium is a fourth-order surface. However, media that allow a decomposition of the fields are also so-called factorizable media. In these media the slowness surface consists of two second-order surfaces, i.e., two quadrics. Olyslager and Lindell [2001c] have shown that in these decomposable media it is possible to introduce two scalar potentials from which the fields in sourceless regions can be derived. These potentials are generalizations of the Hertz potentials in isotropic media [Jones, 1986]. We have earlier studied these generalized Hertz potentials in decomposable anisotropic media [Lindell, 2000] and in simple decomposable bianisotropic media [Lindell and Olyslager, 2001].
 In the present contribution we focus on the most general decomposable bianisotropic medium as introduced by [Olyslager and Lindell, 2001a]. Olyslager and Lindell [2001a] argued that this medium was also factorizable but an explicit factorization was not given. The reason for that was that this would have been an extremely complex calculation. To illustrate this complexity we refer to Lindell and Olyslager  where a “simpler” special case was treated. Throughout this paper we will use a six-vector formalism, i.e., a formalism where we group the electric and magnetic field in one vector. We will show that this six-vector formalism allows a very efficient treatment that allows us to solve the factorization problem for the most general decomposable medium. The decomposed parts of the field see the original medium as simpler media so-called equivalent media. We will derive the Green dyadics for these equivalent media in closed form. Again this can be done very elegantly using the six-vector formalism. We will also, unlike Olyslager and Lindell [2001c], use the six-vector formalism to introduce generalized Hertz potentials. Finally we will illustrate the whole theory by considering a particular medium. This medium will be an anisotropic medium with interesting properties that has never been studied before and that has not been known to be decomposable. Before reading this paper we refer the reader to Appendix A to get acquainted with the six-vector notations and rules.
 In six-vector notation we can write the Maxwell curl equations compactly as
with six-vector field (r) and six-vector source (r) defined as
and with six-dyadic operator (∇) given by
The six-dyadic groups the medium dyadics of a bianisotropic medium as
We opted for giving all the components in the six vectors and six dyadics the same dimensions. This will keep subsequent equations much simpler.
 A general solution of (3) can be derived from the Green six-dyadic G(r) that satisfies
with the unit six-dyadic
Formally we can write the solution of (7) as
The six-vector field due to an arbitrary source can then be obtained from
 By using a so-called duality transformation [Jackson, 1975; Lindell and Ruotanen, 1999] it is possible to transform field solutions in one medium to those in another medium. The duality transformation is defined by an angle θ as
If we define the dual source and dual medium as
then it follows from (3) that dual fields and sources in the dual medium are related by
These are again Maxwell's equations. The Green six-dyadic in the dual is easily derived from the Green six-dyadic in the original medium through
3. Decomposable Media
 In the work by Olyslager and Lindell [2001a] it was shown that the most general bianisotropic medium that allows a decomposition of the fields has a medium six-dyadic of the form
where is an arbitrary anti-symmetric six-dyadic that can be expanded as
and where en are two arbitrary six-vectors
In (16)T is a trace free matrix given by
It was shown by Olyslager and Lindell [2001a] that the fields in this medium decompose in two parts as
where each of the parts satisfy
If one applies a duality transformation on a medium of the type (16) then a medium of the same type is obtained, indeed
It is easily seen that indeed d is again antisymmetric, hence we can expand d as
 In order to simplify the medium let us now choose the angle θ such that
With this angle one easily checks that
We note that the dual medium is indeed simpler since αd = 0. Let us now proceed with this simpler medium. This medium was treated by Lindell and Olyslager  without using six-vector notation.
 The operator d(∇) for this medium can be written as
with and . From Appendix A it follows that the inverse six-dyadic operator d−1(∇) for the dual medium can be written as
From (31) we note that the denominator is a product of two second-order operators. This means that the dual medium is a so-called factorizable medium. From (14) it follows that denominator of −1(∇) will also be a product of two second-order operators and that the original medium (16) will also be factorizable. The denominator in (31) corresponds, up to some proportionality factor, to the Helmholtz determinant operator [Lindell, 1995] . The factorizability of the medium is a necessary condition to be able to derive the Green dyadics in closed form, however it is not a sufficient condition. The slowness surface describing the propagation of plane waves is determined by with k the wave vector. The fact that factorizes in two second-order operators means that the slowness surface consists of two second-order surfaces, i.e., of two quadrics, that are easily handled analytically.
4. Equivalent Media
 We have argued in (20) that the fields in sourceless regions of the medium (16) can be decomposed in two parts that satisfy the conditions (21). We now demonstrate that each of the components see the original medium (16) as a simpler so-called equivalent medium. For the a component we have that
with the equivalent medium six-dyadic a given by
It is clear that the equivalent medium is simpler than the original since a is now the product of T with an antisymmetric six-dyadic. For the b component of the fields we have analogously that
with the equivalent medium six-dyadic b given by
In the sequel we will use the subscript c as a generic notation for the subscripts a or b.
 Let us now look at the Green six-dyadics for the equivalent media. These Green six-dyadics are given by
To simplify the equivalent media we invoke again a duality transformation with the same angle θ as defined by (25). This results in dual equivalent media with medium six-dyadics c,d of the form
The Green six-dyadics for the equivalent media are
with and .
 In Appendix A it is shown that
This means that we can derive Gc,d(r) from scalar Green functions Gc,d(r) through
where the scalar Green functions Gc,d(r) satisfy
Since detc(∇) are second-order operators it is possible to derive these scalar Green functions in closed form. The results are [see, e.g., Puska and Lindell, 1999]
We have assumed that . When other forms of the scalar Green functions can be obtained an example of which will be given below. From the scalar Green functions the Green six-dyadics Gc,d(r) can be obtained using (49). Finally, by using the inverse duality transformation, the Green six-dyadics Gc(r) for the equivalent media are obtained in closed form:
 It is interesting to remark that the Helmholtz determinant operators for the dual equivalent media are given, up to some proportionality factor, by [detc(∇)]2. It can also be shown that
5. Hertz Potentials
 In this section we will show that it is possible to derive the fields in a sourceless region in the general decomposable medium (16) from two scalar potentials. These scalar potentials are solutions of second-order differential equations. These potentials are generalized Hertz potentials. Let us assume that the two decomposed parts c(r) can be derived from two potentials ϕc(r), c = a or b
where c(∇) are six-vector operators that we have to determine. Outside the sources the two potentials satisfy the equations
where we also still have to determine the scalar operators lc(∇). To do this we start from the knowledge that outside the sources the fields c(r) satisfy (cf. (3) for the equivalent media)
Inserting (57) yields
Comparing this with (58) shows that
with c(∇) six-vector operators that are for the moment still arbitrary. Solving (61) for c(∇) and using (44) and (46) yields
This urges us to chose lc(∇) = detc(∇) such that
Note that lc(∇) are indeed second-order differential operators. Now we still have to impose that · c(r) = 0 or
These are conditions for the six-vector operator c(∇) that can be written compactly as
Since c,d(2)(∇) are asymmetric a possible solution of (65) is
where sc(∇) are arbitrary scalar operators. This results in
The easiest choice for this scalar operators sc(∇) are constants sc.
 Let us now look at the source problem. Suppose that we know that c(r) are sources for c(r), i.e.,
or using (57)
Using (61) we can recast this further as
Let us now multiply this by arbitrary six-vector operators c(∇). That leads to
If, for simplicity, we take c(∇) and sc(∇) constants then we find
a general solution of which can be written as
 Let us now apply the previous theory on an example. We will consider an anisotropic medium
In the work by Lindell and Olyslager  a study was made of all possible anisotropic media that allow factorization of the Helmholtz determinant operator or equivalently of all anisotropic media that are decomposable. Nevertheless, although the medium (76) and (77) is a special case of (16) it is not a special case of the factorizable media in [Lindell and Olyslager, 1998]. The reason for this is that the initial ansatz (8) of Lindell and Olyslager  is not complete.
 For the medium (76) and (77) we have that
and α = 1, η = τ = 0.
 The angle θ of the duality transformation is 3π/4. Hence
which leads to ηd = τd = 1 and
This results in
The two factors in the Helmholtz determinant operator, i.e., the denominator of (31), are
 The medium parameters for the equivalent media are
where the upper sign is for a and where the lower sign is for b. The medium parameters for the dual equivalent media are
It is now easily checked that
This indeed corresponds with the two factors (86) and (87). The scalar Green functions Gc(r) for these operators cannot be obtained from (51) since in this case. Let us assume, without loss of generality, that f × g is located in the xz plane. One can always orient the y axis such that this is the case. Let us write f × g as (axux + azuz) and fz2 − gz2 as ν2. From (86) and (87) it follows that the Green functions Gc(r) are solution of
The solution of this equation can be written as
where Hc(r) satisfies
Let us now use a spatial Fourier transformation in the z direction to eliminate the z derivatives. The spectral functions c(x, y, kz) satisfy
a solution of which can be written as
where A is a coefficient that we still have to determine and where u(x) is the Heaviside step function. Using an inverse spatial Fourier transformation it is possible to obtain Hc(r) in closed form as Prudnikov et al. 
In order to keep the inverse Fourier transformation convergent we have to choose A = 1 when ±(ax) > 0 and A = 0 when ±(ax) < 0. If ax is real then we are free to choose A, a symmetric choice would be A = 1/2.
 We have illustrated the power of the six-vector formalism to treat field problems in bianisotropic media. We have derived the factorization of the most general decomposable media and provided closed form expressions for the Green dyadics of the equivalent media. We have introduced generalized Hertz potentials for decomposable media and provided expressions for the sources of these potentials. Finally we have studied a new anisotropic medium with the techniques developed in this paper.
 In this appendix we give a short introduction in six-vector and six-dyadic formalism. We assume that the reader is familiar with regular dyadic formalism [Lindell, 1995]. Six-vector is an ordered pair of two three-vectors a1, a2 and a six-dyadic is an ordered quadruple of four three-dyadics
The transpose of a six-vector is not shown explicitely. The transpose of a six-dyadic is defined as
From two six-vectors a and b a six-dyad is constructed as follows
 The dot product between six-vectors and/or six-dyadics is defined as
The unit six-dyadic is defined as
 Let us now concentrate on anti-symmetric dyadics since they play a crucial role in the analysis in this paper. The general antisymmetric six-dyadic has the form
where p, q are general vectors and where is a general dyadic.
 In the study by Olyslager and Lindell [2001a] it is shown that the inverse of is given by
which is again an antisymmetric six-dyadic.
 We define a double-cross operation between two antisymmetric six-dyadics as
which is another antisymmetric six-dyadic. In particular we define
The determinant of antisymmetrix six-dyadic is given by Lindell and Olyslager 
which shows that
 Let us now consider the inverse of the six-dyadic + where is an arbitrary six-dyadic by expanding
Taking to be the symmetric six-dyadic + we have
Because of the antisymmetry of −1 we have · −1 · = 0 and · −1 · = 0, whence the expression can be reduced to
These two six-dyadics can be expanded as
Thus the inverse can be expressed as the sum of three terms,
This can be expanded as
The inverse (A23) can be cast in another useful form by expressing it as
Let us now apply a relation for an antisymmetric six-dyadic derived by Lindell and Olyslager 
to the last term in (A25)
This leaves us with
Of the three six-dyadic terms here the middle one is symmetric and the first and third ones are antisymmetric. The inverse has thus the form