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Keywords:

  • bianisotropic media;
  • Green dyadics;
  • Hertz potentials

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Basics
  5. 3. Decomposable Media
  6. 4. Equivalent Media
  7. 5. Hertz Potentials
  8. 6. Example
  9. 7. Conclusions
  10. Appendix A.
  11. References

[1] 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.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Basics
  5. 3. Decomposable Media
  6. 4. Equivalent Media
  7. 5. Hertz Potentials
  8. 6. Example
  9. 7. Conclusions
  10. Appendix A.
  11. References

[2] 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

  • equation image
  • equation image

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].

[3] 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 [1999] 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.

2. Basics

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Basics
  5. 3. Decomposable Media
  6. 4. Equivalent Media
  7. 5. Hertz Potentials
  8. 6. Example
  9. 7. Conclusions
  10. Appendix A.
  11. References

[4] In six-vector notation we can write the Maxwell curl equations compactly as

  • equation image

with six-vector field equation image(r) and six-vector source equation image(r) defined as

  • equation image

and with six-dyadic operator equation image(∇) given by

  • equation image

The six-dyadic equation image groups the medium dyadics of a bianisotropic medium as

  • equation image

We opted for giving all the components in the six vectors and six dyadics the same dimensions. This will keep subsequent equations much simpler.

[5] A general solution of (3) can be derived from the Green six-dyadic G(r) that satisfies

  • equation image

with the unit six-dyadic

  • equation image

Formally we can write the solution of (7) as

  • equation image

The six-vector field due to an arbitrary source can then be obtained from

  • equation image

[6] 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

  • equation image

If we define the dual source and dual medium as

  • equation image

then it follows from (3) that dual fields and sources in the dual medium are related by

  • equation image

with

  • equation image

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

  • equation image

3. Decomposable Media

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Basics
  5. 3. Decomposable Media
  6. 4. Equivalent Media
  7. 5. Hertz Potentials
  8. 6. Example
  9. 7. Conclusions
  10. Appendix A.
  11. References

[7] 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

  • equation image

where equation image is an arbitrary anti-symmetric six-dyadic that can be expanded as

  • equation image

and where equation image en equation image are two arbitrary six-vectors

  • equation image

In (16)T is a trace free matrix given by

  • equation image

It was shown by Olyslager and Lindell [2001a] that the fields in this medium decompose in two parts as

  • equation image

where each of the parts satisfy

  • equation image

If one applies a duality transformation on a medium of the type (16) then a medium of the same type is obtained, indeed

  • equation image

with

  • equation image

It is easily seen that indeed equation imaged is again antisymmetric, hence we can expand equation imaged as

  • equation image

[8] In order to simplify the medium let us now choose the angle θ such that

  • equation image

With this angle one easily checks that

  • equation image

with

  • equation image
  • equation image

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 [1999] without using six-vector notation.

[9] The operator equation imaged(∇) for this medium can be written as

  • equation image

with

  • equation image

with equation image and equation image. From Appendix A it follows that the inverse six-dyadic operator equation imaged−1(∇) for the dual medium can be written as

  • equation image

with

  • equation image
  • equation image

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 equation image−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] equation image. 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 equation image with k the wave vector. The fact that equation image 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

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Basics
  5. 3. Decomposable Media
  6. 4. Equivalent Media
  7. 5. Hertz Potentials
  8. 6. Example
  9. 7. Conclusions
  10. Appendix A.
  11. References

[10] 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

  • equation image

with the equivalent medium six-dyadic equation imagea given by

  • equation image

with

  • equation image

It is clear that the equivalent medium is simpler than the original since equation imagea is now the product of T with an antisymmetric six-dyadic. For the b component of the fields we have analogously that

  • equation image

with the equivalent medium six-dyadic equation imageb given by

  • equation image

with

  • equation image

In the sequel we will use the subscript c as a generic notation for the subscripts a or b.

[11] Let us now look at the Green six-dyadics for the equivalent media. These Green six-dyadics are given by

  • equation image

with

  • equation image

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 equation imagec,d of the form

  • equation image

The Green six-dyadics for the equivalent media are

  • equation image

where

  • equation image

with

  • equation image

with equation image and equation image.

[12] In Appendix A it is shown that

  • equation image

with

  • equation image

and

  • equation image

This means that we can derive Gc,d(r) from scalar Green functions Gc,d(r) through

  • equation image

where the scalar Green functions Gc,d(r) satisfy

  • equation image

Since detequation imagec(∇) 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]

  • equation image

with

  • equation image
  • equation image

We have assumed that equation image. When equation image 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:

  • equation image

[13] It is interesting to remark that the Helmholtz determinant operators for the dual equivalent media are given, up to some proportionality factor, by [detequation imagec(∇)]2. It can also be shown that

  • equation image

and

  • equation image

5. Hertz Potentials

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Basics
  5. 3. Decomposable Media
  6. 4. Equivalent Media
  7. 5. Hertz Potentials
  8. 6. Example
  9. 7. Conclusions
  10. Appendix A.
  11. References

[14] 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 equation imagec(r) can be derived from two potentials ϕc(r), c = a or b

  • equation image

where equation imagec(∇) are six-vector operators that we have to determine. Outside the sources the two potentials satisfy the equations

  • equation image

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 equation imagec(r) satisfy (cf. (3) for the equivalent media)

  • equation image

Inserting (57) yields

  • equation image

Comparing this with (58) shows that

  • equation image

with equation imagec(∇) six-vector operators that are for the moment still arbitrary. Solving (61) for equation imagec(∇) and using (44) and (46) yields

  • equation image

This urges us to chose lc(∇) = detequation imagec(∇) such that

  • equation image

Note that lc(∇) are indeed second-order differential operators. Now we still have to impose that equation image · equation imagec(r) = 0 or

  • equation image

These are conditions for the six-vector operator equation imagec(∇) that can be written compactly as

  • equation image

with

  • equation image

Since equation imagec,d(2)(∇) are asymmetric a possible solution of (65) is

  • equation image

where sc(∇) are arbitrary scalar operators. This results in

  • equation image

and

  • equation image

The easiest choice for this scalar operators sc(∇) are constants sc.

[15] Let us now look at the source problem. Suppose that we know that equation imagec(r) are sources for equation imagec(r), i.e.,

  • equation image

or using (57)

  • equation image

Using (61) we can recast this further as

  • equation image

Let us now multiply this by arbitrary six-vector operators equation imagec(∇). That leads to

  • equation image

If, for simplicity, we take equation imagec(∇) and sc(∇) constants then we find

  • equation image

a general solution of which can be written as

  • equation image

6. Example

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Basics
  5. 3. Decomposable Media
  6. 4. Equivalent Media
  7. 5. Hertz Potentials
  8. 6. Example
  9. 7. Conclusions
  10. Appendix A.
  11. References

[16] Let us now apply the previous theory on an example. We will consider an anisotropic medium equation image

  • equation image
  • equation image

In the work by Lindell and Olyslager [1998] 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 [1998] is not complete.

[17] For the medium (76) and (77) we have that

  • equation image
  • equation image

and α = 1, η = τ = 0.

[18] The angle θ of the duality transformation is 3π/4. Hence

  • equation image

which leads to ηd = τd = 1 and

  • equation image

with

  • equation image

This results in

  • equation image

where

  • equation image

and

  • equation image

The two factors in the Helmholtz determinant operator, i.e., the denominator of (31), are

  • equation image

and

  • equation image

[19] The medium parameters for the equivalent media are

  • equation image

where the upper sign is for equation imagea and where the lower sign is for equation imageb. The medium parameters for the dual equivalent media are

  • equation image

It is now easily checked that

  • equation image

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 equation image 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 equation image (axux + azuz) and fz2gz2 as ν2. From (86) and (87) it follows that the Green functions Gc(r) are solution of

  • equation image

The solution of this equation can be written as

  • equation image

where Hc(r) satisfies

  • equation image

Let us now use a spatial Fourier transformation in the z direction to eliminate the z derivatives. The spectral functions equation imagec(x, y, kz) satisfy

  • equation image

a solution of which can be written as

  • equation image

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. [1992]

  • equation image

In order to keep the inverse Fourier transformation convergent we have to choose A = 1 when ±equation image(ax) > 0 and A = 0 when ±equation image(ax) < 0. If ax is real then we are free to choose A, a symmetric choice would be A = 1/2.

7. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Basics
  5. 3. Decomposable Media
  6. 4. Equivalent Media
  7. 5. Hertz Potentials
  8. 6. Example
  9. 7. Conclusions
  10. Appendix A.
  11. References

[20] 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.

Appendix A.

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Basics
  5. 3. Decomposable Media
  6. 4. Equivalent Media
  7. 5. Hertz Potentials
  8. 6. Example
  9. 7. Conclusions
  10. Appendix A.
  11. References

[21] 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 equation image is an ordered pair of two three-vectors a1, a2 and a six-dyadic equation image is an ordered quadruple of four three-dyadics equation image

  • equation image

The transpose of a six-vector is not shown explicitely. The transpose of a six-dyadic is defined as

  • equation image

From two six-vectors a and b a six-dyad is constructed as follows

  • equation image

[22] The dot product between six-vectors and/or six-dyadics is defined as

  • equation image
  • equation image
  • equation image
  • equation image

The unit six-dyadic is defined as

  • equation image

[23] 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

  • equation image

where p, q are general vectors and where equation image is a general dyadic.

[24] In the study by Olyslager and Lindell [2001a] it is shown that the inverse of equation image is given by

  • equation image

which is again an antisymmetric six-dyadic.

[25] We define a double-cross operation between two antisymmetric six-dyadics as

  • equation image

which is another antisymmetric six-dyadic. In particular we define

  • equation image

The determinant of antisymmetrix six-dyadic is given by Lindell and Olyslager [2002]

  • equation image

which shows that

  • equation image

[26] Let us now consider the inverse of the six-dyadic equation image + equation image where equation image is an arbitrary six-dyadic by expanding

  • equation image

Taking equation image to be the symmetric six-dyadic equation image + equation image we have

  • equation image

Because of the antisymmetry of equation image−1 we have equation image · equation image−1 · equation image = 0 and equation image · equation image−1 · equation image = 0, whence the expression can be reduced to

  • equation image

with

  • equation image
  • equation image

These two six-dyadics can be expanded as

  • equation image
  • equation image

Thus the inverse can be expressed as the sum of three terms,

  • equation image

This can be expanded as

  • equation image

The inverse (A23) can be cast in another useful form by expressing it as

  • equation image

with

  • equation image

Let us now apply a relation for an antisymmetric six-dyadic equation image derived by Lindell and Olyslager [2002]

  • equation image

to the last term in (A25)

  • equation image

This leaves us with

  • equation image

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

  • equation image

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Basics
  5. 3. Decomposable Media
  6. 4. Equivalent Media
  7. 5. Hertz Potentials
  8. 6. Example
  9. 7. Conclusions
  10. Appendix A.
  11. References
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  • Jones, D. S., Acoustic and Electromagnetic Fields, Clarendon, Oxford, UK, 1986.
  • Lindell, I. V., Methods for Electromagnetic Field Analysis, 2nd ed., Oxford Univ. Press, New York, 1995.
  • Lindell, I. V., Potential representation of electromagnetic fields in decomposable anisotropic media, J. Phys. D, 33, 31693172, 2000.
  • Lindell, I. V., and F. Olyslager, Factorization of the Helmholtz determinant operator for anisotropic media, Arch. Elektron. Uebertraeg., 52, 261267, 1998.
  • Lindell, I. V., and F. Olyslager, Factorization of Helmholtz determinant operator for decomposable bi-anisotropic media, J. Electromagn. Waves Appl., 13, 429444, 1999.
  • Lindell, I. V., and F. Olyslager, Potentials in bi-anisotropic media, J. Electromagn. Waves Appl., 15, 318, 2001.
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  • Olyslager, F., and I. V. Lindell, Electromagnetic fields in bianisotropic media, in Proceedings of the International Conference on Electromagnetics in Advanced Applications, pp. 703706, Politec. di Torino, Torino, Italy, 2001c.
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  • Puska, L. H., and I. V. Lindell, Electromagnetic source decomposition for generalized decomposable bi-anisotropic media, J. Electromagn. Waves Appl., 14, 14771491, 1999.