Optic Axes and Elliptic Cone Equation in Coordinate-Invariant Treatment ()
A Remark to Notations
Three-dimensional vectors: bold letters, e.g.,
,
scalar products,
vector products,
volume products,
dyadic products, Three-dimensional operators: serif-less letters
,
operator products,
products of operators with vectors,
bilinear (and quadratic
) forms,
trace, second invariant and determinant of
.
In three-dimensional Euclidean spaces with a symmetric metric tensor
with
the dual anti-symmetric pseudo-tensor
to a vector
(
Levi-Cicita symbol) can be also seen as antisymmetric operator
and in coordinate-invariant form we can write this antisymmetric operator as
with the advantage that vector and also volume products can be written only by displacement of the squared brackets, e.g.,
,
,
.
In physical texts it makes sometimes difficulties to write vectors by small bold letters and operators by serif-less Capital letters. In case of Greek letters, 'Latex' (and also printing) does not provide serif-less letters. In these cases I write as a compromise operators by bold letters, e.g.,
such as for vectors to distinguish them, in particular, from scalars. This means that in present physical text one must know which kind of quantities one has: scalars, vectors or operators.
1. Introduction
The presence of optic axes in the general case of biaxial crystal causes very interesting effects for the propagation of light in the neighborhood of such directions and belongs to the exceptional cases which are more difficult to describe in comparison to the propagation into other directions. The reason is that the polarization and the group velocity in direction of optic axes is not uniquely determined and depends very much on small deviations of the refraction vectors from the optic axis. William Rowan Hamilton forecasted in 1828 the effect of conical refraction in direction of an optic axis or binormal. The conical approximation of the dispersion surface (surface of refraction vectors) in the neighborhood of an optic axis is sufficiently well understood concerning the plane spanned by the optic axis and the cone axis which in general case of biaxial crystals form an angle but not fully concerning the plane perpendicular to it containing the optic axis or the cone axis. In particular, some loose remarks in treatments make the impression that the mentioned cone approximation is evidently considered as approximation by a circular cone that, however, is not true. In reality it is an elliptic cone. Although in many cases if the main (or principal) permittivities in direction of the tensor axes are not very different the differences of the elliptic cone to a circular cone are small, but this is a principal question. Interesting treatments are to find in the comprehensive encyclopedic article of Szivessy [1] and in vol. 8 of the course of theoretical physics of Landau, Lifshits (complemented by Pitayevski) [2] although the last authors announce their own treatment as highly schematically. Further interesting representations including optic axes are to find, e.g., in books of Sommerfeld [3], of Born and Wolf [4], of Ditchburn [5] and of Sivukhin [6].
New in present article is that we derive the full three-dimensional elliptic cone equation for the approximation of refraction vectors near the optic axis in coordinate-invariant way and that we make general remarks how the wave-equation operator is connected in case of optic axes with the vanishing of the second invariant of this operator and with vanishing of its complementary operator. The vanishing of the second invariant is a necessary but not a sufficient requirement for the presence of an optic axis. In particular, we illustrate in most of the figures the surface of vanishing of the second invariant in connection with the necessary vanishing of the determinant of the wave-equation operator in three-dimensional space of refraction vectors for the cases of solutions near the optic axes. Furthermore, we calculate principal formulae for the polarization of the electric field near to propagation in direction of optic axes. Coordinate-invariant methods from the starting equations up to the results without using specialized coordinate systems which are not directly connected with vectors and operators with physical meaning for the problem were first described and applied in optics of anisotropic media by F.I. Fyodorov [7] [8] and later applied for the description of the Lorentz group in [9] where they are also very useful and the most of our articles (e.g., [10] [11]) in the seventieths work also with such methods. Depart from the introduction of the notion of axes of a second-rank tensor I did not find in [7] [8] and in other renowned sources a detailed treatment of the elliptic cone equation for optic axes. For this purpose I searched in Internet for it where though I found rich experimental material and also some known theoretical representations but not the mentioned equation and the relation to the second invariant of the wave-equation operator.
We begin in Section 2 with a general consideration and classification to three-dimensional operator equations together with its possible degenerative cases and specializes this in Sections 3 - 5 to the wave equation for anisotropic media. In Section 6 we derive the cone approximation for refraction vectors in the neighborhood of an optic axis and diagonalize the cone operator in Section 7. In Section 8 we describe the intersection of the plane containing the optic axis and the direction perpendicular to it. Section 9 provides a general formula for polarization of the electric field in the neighborhood of the optic axis. In Section 10 we discuss shortly the transition to uniaxial media. Section 11 considers possible degenerations to optic axis in the complex domain and discusses but cannot fully solve whether cases with vanishing of determinant and second invariant of the wave-equation operator are possible which are not connected with optic axes. In Section 12 we make short remarks to a duality between optics with refraction operators and ray optics in connection with the cone approximation in the neighborhood of an optic axis. Some more mathematical problems and an ethical problem are discussed in the Appendices.
2. Vanishing of the Second Invariant
of the Operator
of the Wave Equation and the Case of Optic Axes
First, we consider the general case of the algebra of three-dimensional operators
for which right-hand solutions
and left-hand solutions
of the equations
(2.1)
should be calculated. This may be also expressed in the way that
and
are eigen-solutions (eigenvectors) of the operator
to eigenvalue
. Usually, the operator
depends on parameters and this means that the parameters must be chosen in such way that
possesses at least one eigenvalue
to get a solution of
.
An arbitrary three-dimensional operator
satisfies the three-dimensional Hamilton-Cayley identity (e.g., Gantmakher [12]) with
the three-dimensional identity operator
(2.2)
where the invariants with respect to similarity transformation of the operator
(trace, second invariant, determinant) are defined by1
(2.3)
The complementary operator
to operator
is defined by
(2.4)
The eigenvalues
of the operator
to right-hand and left-hand eigenvectors
and
satisfy in accordance with the Hamilton-Cayley identity (2) the equation
(2.5)
Necessary condition for solutions of (2.1) or the presence of an eigenvalue
of
is the vanishing of the determinant of
that means
leading from (2.5) to the from
(2.6)
with, at least, one simple eigenvalue
(2.7)
The other two eigenvalues
of
satisfy then the equation
(2.8)
with the solutions
(2.9)
In case of
the Hamilton-Cayley identity (2.2) using definition (2.4) takes on the form
(2.10)
By successive application of the Hamilton-Cayley identity follows the general relation
(2.11)
which in case of vanishing determinant
can be written in the form
(2.12)
Thus
and
for arbitrary vectors
and
are right-hand or left-hand eigen-solutions of
to eigenvalue
, respectively, and
is projection operator to obtain such solutions. For
these projection operators are no more defined and these are degenerate cases.
Since in present article we mainly investigate degenerate cases such as optic axes we continue our further investigations at once with the case that apart from the determinant
also the second invariant
of the operator
vanishes that means
(2.13)
In this case the Hamilton-Cayley identity (2.2) takes on the following form
(2.14)
with the invariants with respect to similarity transformations defined in (2.3) and the complementary operator
in (2.4) which specializes here to
(2.15)
Due to vanishing of
in the denominator one-dimensional projection operators
to eigenvalue
of
are no more existing. We have to distinguish two subcases
(2.16)
In first subcase of
it is proportional to the dyadic product of
and can be normalized as following
(2.17)
In this case we find right-hand and left-hand solutions proportional to
and
which are orthogonal to each other. Their scalar product
cannot be normalized. This belongs to a Jordan normal form (e.g., [12]) of L to a twofold degenerate eigenvalue
.
In the second subcase
we find according to the definition (2.15) under the assumption
(2.18)
and we may define projection operators
to twofold degenerate eigenvalue
according to
(2.19)
and any vector
is right-hand vector and any vector
left-hand eigenvector of
to degenerate eigenvalue
according to
(2.20)
This is the case of optic axes of
with
the two-dimensional projection operator to determine to twofold degenerate eigenvalue
possible eigenvectors as solutions of Equation (2.1).
In case of
and
according to (2.8) and (2.9) the third, in general, non-degenerate eigenvalue of
is
and the projection operator
to this eigenvalue is
(2.21)
Any vector
is right-hand eigenvector and any vector
is left-hand eigenvector of
to the eigenvalue
that is also clear without the formalism. The vector
is perpendicular to all two-fold degenerate right-hand eigenvectors of
to eigenvalue zero and the analogous property is true for the left-hand eigenvalues. Since
is projection operator with trace equal to 1 it is proportional to a dyadic product
with scalar product
that means
(2.22)
The operator
is in case of
proportional to a dyadic product of two mutually normalized vectors
and
. All (twofold degenerate) solutions
and
of
and
according to
(2.23)
are perpendicular to
or
, respectively.
For completeness we will classify the last possible degenerate case, the vanishing of all three invariants of operator
with three subcases although it does not possess essential importance for optic axes
(2.24)
The first two subcases belong to two different Jordan normal forms. For example, in first subcase one may find a right-hand vector
with
,
,
and analogously for left-hand vectors. Orthogonality relations between these vectors are also easily to obtain. The third subcase is a rare case where for certain very special cases
vanishes and they possess importance as special case embedded into the neighborhood of more general cases (following Sections).
We try now to collect the results to eigenvalue
in a scheme. Our notations are the invariants of three-dimensional operators plus the complementary operators and projection operators as follows
(2.25)
This leads to the schematic tree:
(2.26)
It is possible that for a given operator
not all cases are realized, for example, the case
,
but
. From next Section on we consider a special operator
which concerns the optics of homogeneous anisotropic media.
3. Preparations for Treatment of Wave Propagation in Direction of Optic Axes
The energy and momentum propagation of quasiplane and quasimonochromatic waves becomes complicated for refraction vectors
in direction of
optic axes due to non-uniqueness of the group velocity
or ray vectors
. For reflection and refraction problems the refraction vectors
or the wave vectors
and (circular) frequency
are the most fundamental notions in comparison to the ray vectors or group velocities and we start with this concept. In this Section we begin to consider by coordinate-invariant methods exceptional and degenerate cases to which, in particular, belong the singular cases of optic axes. The presented approach is of principal importance since it can be applied also to other operator equations with operators different from our considered specialized operator
. Thus we continue the investigations of Section 2 to degenerate cases for the following special operator (1) of the wave optics of anisotropic media2
(3.1)
which by solution of the equation
provides solution for the amplitude
of the electric field (
are refraction vectors). A certain symmetry of the frequency-dependent permittivity tensor
we do not assume in this Section. The invariants of
are
(3.2)
and the complementary operator
to
defined in (4)
(3.3)
In case
, the projection operator
defined in (2.12) with explicitly given invariant
in (3.2) and complementary operator
in (3.3) does not exist. We can distinguish then two partial cases. The first case is
(3.4)
where
follows from (2.11) in connection with the requirement
and the supposition
from the general relation for arbitrary three-dimensional operators
(3.5)
However, from
in (3.4) not automatically follows also
. In the case (3.4) due to
there exist vectors
and
such that
and
and they are the only linearly independent right-hand and left-hand eigenvectors of
to eigenvalue 0, respectively, and thus their polarization is non-degenerate but their scalar product is vanishing and, therefore, they are orthogonal to each other
(3.6)
This case is not possible in lossless media (
,
is Hermitean operator) with real refraction vectors
since then the left-hand eigenvectors
of
are proportional to
and the right-hand eigenvectors
to
and
is non-vanishing in every case. However, this case may happen for evanescent waves with complex refraction vectors or in lossy media not dealt with in present paper.
We consider the first case, the vanishing of
and of
but
must not necessarily be vanishing
(3.7)
By multiplication of the first equation with
and the second equation with
one finds an equivalent equation for one of these equations
(3.8)
Two of the three Equations (3.7) and (3.8) are independent and must be solved to get the relations between the components of the refraction vectors
in case of
and
but
that is fairly complicated.
We consider now the second case which is the important case of twofold degeneration of the eigenvalue
of the operator
for a given refraction vector
and thus the case of optic axes or binormals. In this case not only
according to (4) is vanishing but
itself and we have
(3.9)
From (3.9) follows that in this case the, in general, non-vanishing and non-degenerate eigenvalue of
is
and from
that vectors
and
for arbitrary appropriate
and
are right-hand and left-hand eigenvectors of
to this eigenvalue
, respectively
(3.10)
From
follow for the special operator (3.1) by multiplication from right and from left with the refraction vector
the identities
(3.11)
and many other more complicated identities can be derived in similar way. By scalar multiplication of the vectorial Equation (3.11) with the vector
we get the scalar requirement
(3.12)
Using the dispersion equation
with
explicitly given in (3.2) we see that from (3.12) also follows
(3.13)
as a condition for optic axes which has to be satisfied plus in addition with a further independent one of the above derived equations from
and
, for example
(3.14)
This equation possesses the advantage that it is only quadratically in the components of the refraction vector
and, therefore, with only two solutions for one component in dependence on the other components. That the first of the considered two cases is more general as the second can be seen from the fact that it is obviously impossible to derive a second-degree equation for the components of the refraction vectors.
According to (2.19) and (2.21) the operators
and
for
are projection operators in case of
and together with (3.11) we find in our case more explicitly
(3.15)
Due to
right-hand eigenvectors
of
are polarized perpendicular to the plane determined by the normal vector
.
The necessary (but not sufficient) condition
for optic axes takes on the following form
(3.16)
By multiplication of (3.16) with
and elimination of
by means of (3.13) we find that the condition
(3.17)
is satisfied for the case of an optic axes. This is the eigenvalue equation for the tensor
with the eigenvalues
. Since the eigenvalues of
are the values
to eigenvectors
for an optic axis at least one of the equations
has to be satisfied. In next Section we see that this is the eigenvalue
in the ordered sequence (4.4) of eigenvalues. It is clear that not all of the derived scalar identities for the presence of optic axes are independent from each other and there are many possibilities to derive them and also further ones.
Due to (3.11) which can be written
(3.18)
the three vectors
are linear dependent. This can be also expressed by the vanishing of the volume product
(3.19)
Therefore the three vectors
lie in the same plane spanned by two of the three vectors.
4. Coordinate-Invariant Treatment in Case of Electrically Anisotropic Media with Real Symmetric Permittivity Tensor
We apply now the general discussions of last Sections to the classical crystal optics with the special operator (3.1) of the wave equation for the amplitude of the electric field
for refraction vectors
that means
(4.1)
and extend it by consideration of a symmetrical permittivity tensor
(or with indices
) in principal axes form with real unit vectors
(axes notations by letter
according to Fyodorov [7])
(4.2)
For the n-th powers of the tensor
we find from (4.2)
(4.3)
We suppose that the real-valued and non-negative principal (or main) values
are ordered according to3
(4.4)
The directions of the vectors
are called the principal axes of the tensor
. Depending on the symmetry of the medium the normalized vectors
in direction of the principal axes may or may not depend on the frequency. The permittivity tensor
is now by this proposition not only symmetric but becomes also a Hermitean one, i.e.
. Our further treatment here distinguishes from the very interesting considerations of Fyodorov [7] [8] insofar that we from beginning on involve more the operator
, its invariants and the complementary operator.
The invariants of
were already given in (3.2) but have now to be specialized according to (4.2) and are
(4.5)
and the complementary operator
to operator
with the identity
is
(4.6)
The general formulae for the invariants of a complementary operator
to a three-dimensional operator
are
(4.7)
If in addition to
which in our case is the dispersion equation for the refraction vectors also
then the complementary operator
to
can be vanishing
or non-vanishing but
according to (3.5) is in every case vanishing. The first case
is the case of optic axes.
We introduce the abbreviations
. The determinant
written in principal axes components takes on the form
(4.8)
plus further forms by permutation of the indices. In the planes perpendicular to the principal optic axes
from the dispersion equation
in the form (4.8) we find the well-known relations
(4.9)
that means the dispersion surface decomposes in every case in the plane perpendicular to a principal axis into a circle and an ellipse (but not in general). Due to ordering (4.4) only in the case
from the three principal planes (4.9) we have intersections of the circle and the ellipse and therefore real optic axes (Figure 1; similar pictures are, for example, also in [2] (§99, pp. 409, 410) and in [4] [7]. In cases of
and
the three cases (4.9) reduce to two different cases and the medium becomes optically uniaxial.
The invariant
in principal axes form takes on the following form
(4.10)
that can be reduced to
Figure 1. T he geometry of the dispersion surface
with an optic axis in connection with the principal values
of the permittivity tensor
. In particular, the left-hand figure is a well-known picture for illustration of the dispersion surface (e.g, [1] [2]) which in next Sections is complemented by more detailed figures.
(4.11)
In analogy to (9) the second invariant
can be factorized in the special cases
. We write here explicitly down only the special case
(4.12)
This is a non-rational factorization which is only unique with the additional requirement that the expression within the square root is positive (only positive sum terms) under the supposition (4.4) of ordering the permittivities but not in other case.
The general form of this non-rational factorization is
(4.13)
which easily can be specialized to
and
. It shows that in the real domain of this three-dimensional space the considered surface consists of two shells with no self-intersections but in the complex domain it possesses such points for the vanishing of the root.
The complementary operator
to
expressed in principal axes form is
(4.14)
It is easy to check
explicitly given in (4.10).
If
, we can determine according to (2.12) a “one-dimensional” projection operator
for the determination of polarization vectors proportional to solutions
in the following way
(4.15)
and we obtain explicitly specialized by (4.1)
(4.16)
An arbitrary vector
and an arbitrary vector
is a right-hand or left-hand eigenvector of
to eigenvalue zero, respectively, but usually non-normalized. Due to orthonormality of vectors
it is not difficult to calculate from representation (4.14) the products of the operator
with vectors
, for example
(4.17)
This is the sum of a vector proportional to vector
and a linear combination of vectors
and
in the plane perpendicular to
. Divided by
according to (4.10) the operator
is projection operators to the operator
in case of
but
. Different choice of vectors for the determination of eigenvectors lead to more or less favorable representation of the eigenvectors. Many special cases could be discussed.
5. The Case of Optic axes
and
with Real-Valued Refraction Vectors
We continue now the considerations to optic axes. Novel is that we integrate the vanishing
of the second invariant of the operator
and take it into account also in the illustrations. Both surfaces together are represented in Figure 2. The surface
is a two-shell surface of forth degree with no intersection in the real region and no rational factorization but the inner shell touches the dispersion surface
at the points of self-intersections that means at the optic axes. This is better to see in pictures of intersection of both these surface in three-dimensional space of refraction vectors
with the plane
and no intersections in the planes
and
in Figure 3.
Starting from Equation (4.9) we consider the special case
. This case leads to two real solutions for a circle and an ellipse with self-intersection and by multiplication of the separated equations with factors to the following two equations
(5.1)
Figure 2. Intersections of dispersion surface
(blue, green, yellow) and the surface
with the principal planes of the permittivity tensor
. The optic axes are in the plane
. The surface
on right-hand picture possesses two shells without intersection but only the inner shell intersects the dispersion surface
at the optic axes. The chosen values for the Figures are
,
,
, the same as in Figure 1. Pay attention that the scales in both partial pictures are different!
By forming the difference of these equations we find (agrees, e.g., with [2] (§99, Equation (99.5))
(5.2)
Thus the components
and
are related to each other with 4 possible solutions. One can check by (4.9) that besides
for the points of intersection also the second invariant vanishes, i.e.,
. In general,
can take on complex values and then for
result also complex values. The case (5.2) leads to optic axes which among others (see Section 11) are present for real refraction (and wave) vectors. Since this case is connected with a self-intersection of the dispersion surface in its neighborhood one has different non-degenerate polarizations of the two branches and the polarization for the optic axes becomes twofold degenerate. We investigate now the special case that
and therefore also
take on real values.
We denote the angle between the two possible solutions of
for
of the corresponding refraction vectors
by
and have then the relations (easily calculable from
but for convenience many are given; see Figure 1 and Figure 3)
(5.3)
Figure 3. Intersections of dispersion surface
(blue, green) and the surface
(red) with the planes perpendicular to vectors
(upper curves) and
and
(lower curves). The dispersion curves consist in every case of these planes (but not in general) of a circle (blue) and an ellipse (green) whereas the curves
are two-sheet forth-order curves (red) without self-intersections as two-dimensional surfaces in the three-dimensional space for real wave vectors. In case of optic axes one sheet of the two-sheet curves intersects the dispersion curves for the optic axes exactly in the point of intersection of the dispersion curves. We have chosen
,
,
from which results
,
.
or the Cosines and Sines, correspondingly
(5.4)
For the illustration in many figures in next Sections we choose the values
,
,
which are sufficiently different to show characteristic features but do not correspond to the majority of experimental values with smaller differences. For these values we obtain an angle
.
From the derived relations (e.g., (5.3)) together with
in (4.9) follows for the optic axes
(5.5)
By reason which become soon clear we make now a sidestep of our considerations by introduction of two unit vectors
in direction of the optic axes as follows
(5.6)
with the inversion
(5.7)
Their normalization and scalar products are
(5.8)
The representation of the symmetrical permittivity tensor
in (4.2) and of its inverse
by the vectors
are
(5.9)
We see that in these representations only the tensor
takes on a relatively simple form. This is a slightly varied variant of the method of Fyodorov [7] [8] where directly the optic axes of the refraction vectors are involved. The original method of Fyodorov of introduction of axes of a symmetric tensor we represent in Appendix B.
The entirety of possible refraction vectors
in (or in opposite) direction of the optic axes is
(5.10)
We omit now the trivial signs (±) and may calculate from (5.10) immediately
(5.11)
and furthermore
(5.12)
The last expression is the Cosine of the angle between the optic axes (see (5.3)). Since the right-hand sides of the three vectors on the left-hand side in (5.11) are linear combinations of only two vectors
and
they are linearly dependent and the relation is given. From (5.11) follows
(5.13)
For the following vector products we find
(5.14)
All these vector products are proportional to the vector
perpendicular to the plane spanned by the vectors
and
. Since
lies in last mentioned plane the volume product of three vectors
is vanishing
(5.15)
This means that the three vectors
are in the plane perpendicular to vector
6. The Neighborhood
of Refraction Vectors to the Refraction Vector
for an Optic Axis in the Cone Approximation
We investigate in this Section the dispersion surface in the neighborhood of an optic axis and denote the refraction vector to an optic axis by
and consider its neighborhood for small deviations
from
. The vector
can be any of the 4 calculated vectors
(6.1)
in (5.10). The quadratic forms given in (5.11) are unspecific which special vector
is meant and are
(6.2)
and with
we will denote one of the two possible vectors
according to
(6.3)
but do not change it in correlated calculations. Thus we have
(6.4)
For relations which are specific for the signs “±” in (6.4) such as for the vector product
(6.5)
we make the agreement to use the upper sign “±” in (6.4).
From this follows that the operator
degenerates in direction of the optic axes to a direct product of operators as follows represented by
(6.6)
In direction of an optic axis the eigenvectors of
to the refraction
(for the electric field) are not uniquely determined due to
and
.
The two-dimensional projection operator
for the determination of right-hand and left-hand eigenvectors
of
to the twofold degenerate eigenvalue
is
(6.7)
from which follows
(6.8)
All vectors
are proportional to possible solutions
of Equation (3.1) and they span the plane perpendicular to the vector
and thus we have two-fold degeneration of the possible polarization. Both last relations were already derived in Section 3 in more general form.
Thus the transition to optic axes
(6.9)
is connected with the conditions
(6.10)
We have to investigate now the dispersion surface
and the complementary operator
to operator
in a small neighborhood
of
and find in the first two terms of a Taylor series
(6.11)
Two not very simple problems wait for us to solve.
Our first problem which we now begin to investigate is to find from vanishing of the determinant
an approximate solution for the possible refraction vectors
in the neighborhood of an optic axis
. This will become a conical approximation for
. The second problem is to determine an approximation of the non-degenerate eigenvectors in the neighborhood of an optic axis that we solve in a later Section.
For the determinant of the sum of two three-dimensional operators
and
one finds the general formula
(6.12)
with
and
the complementary operators to
and
, respectively. This formula is specific for three-dimensional operators and has to be generalized for higher dimension. We apply this formula for
(6.13)
Since determinant
and complementary operator
of
vanish according to (6.10) we have to take into account only the last two terms
and
in (6.12). The term
is already in third order of
and can be also neglected because we want to take into account only the first non-vanishing addition to the dispersion surface which is of second order in
. For the same reason one may also neglect the terms from the second and higher-order derivatives of
in (6.11) and find
(6.14)
The written term on the right-hand side is the only additional term which has to be taken into account in second-order with respect to
(conical approximation). Thus we have now to calculate the complementary operator
that is made for the principal structure in Appendix A. With the substitutions
and
there one finds
(6.15)
from which follows
(6.16)
with the abbreviation
for the following symmetric second-rank tensor
(6.17)
The index 0 at
means that for the vector
is inserted one of the four possible solutions
from (6.3) which can be reduced to the two given solutions in (6.3) since they are involved in
only quadratically.
The homogeneous equation with second-order terms in
(6.18)
describes a cone in three-dimensional space and their solutions
provide the approximation of the dispersion surface
in direction to the chosen refraction vector
of the optic axis. This means that the whole solution for refraction vectors
in the neighborhood of a refraction vector
for the chosen optic axis is
(6.19)
where
is one of the possible solutions of the cone Equation (6.3). This is represented in Figure 4.
Due to the operator and scalar identities for optic axes (see (6.6)) we have
(6.20)
The cone tensor
is equivalent to other forms of this tensor. A good (likely the best) possibility is to combine (6.20) to the identity
(6.21)
and to subtract it from
in (6.17). Then one obtains the equivalent tensor
(6.22)
In comparison to (6.17) with 8 sum terms in biquadratic degree of
the form (6.18) seems to be simpler with only 5 sum terms in quadratic degree of
but this is not fully true since in calculating the powers of
from (6.17) some sum terms vanish due to
.
Our next problem is to go more into the details of the structure of the second-rank symmetric tensor
(can be also seen as operator) and to calculate its eigenvalues and eigenvectors that we make in next Section.
Figure 4. The geometry of coupled refraction vectors
in the neighborhood of optic axes in cone approximation for plane
. The tangential components
of coupled wave vectors are the same. In the right-hand figure two cases of tangential components
to refraction vectors
near the optic axis are drawn. In general, each possible tangential component
goes into the equations with a certain weight. The ray vectors which are perpendicular to the cone surface, properly speaking, do not belong to this picture and it cannot be generally said at which points in real space they have their origin in applications (for example at the inside to a boundary plane).
7. Eigenvalues and Eigenvectors of the Cone tensor
The cone tensor
was determined in two forms (6.17) and (6.22) which are equivalent if one takes into account the identity (6.21). We calculate now first its eigenvalues and then its eigenvectors. For this purpose one has first to calculate the powers
and
that we do not write down for their many terms and then their traces. However, we made a special approach. Since the vectors
are in the plane spanned by the vectors
and
(see (5.11)) one eigenvalue of
and the corresponding (right- and left-hand) eigenvector proportional to
is easily to obtain directly from (6.22) with the result
(7.1)
Therefore we needed only the traces of
and
and could later also find the trace of
by
which altogether are4
(7.2)
Using the general definitions of operator invariants (2.3) this is equivalent to
(7.3)
From the equation for the eigenvalues of
(7.4)
we obtained the following three eigenvalues
(7.5)
The half difference between
and
is
(7.6)
Under the assumption (4.4) that the principal values
are non-negative and are ordered we find the inequalities
(7.7)
and one eigenvalue
is non-negative and two eigenvalues
are non-positive. This follows from the inequality between arithmetic and geometric mean in the two special forms
(7.8)
For
all three eigenvalues (7.5) are genuinely different.
Knowing the eigenvalues
it is possible to determine the eigenvectors
directly by the operator equation, in our case from
. We give them at once in the form of projection operators
and
with the result (the prime at
is written because we have already defined different vectors
in (5.6) as unit vectors in direction of the optic axes)
(7.9)
with a form of the Cauchy-Schwarz-Bunyakovski inequality
(7.10)
The operators
and
as projection operators to determine eigenvectors in non-degenerate case satisfy the following relations
(7.11)
The cone tensor
can now be represented by
(7.12)
The combinations of permittivities in (7.5) and (7.9) can be also obtained and expressed in simple way by the quadratic forms
and some inequalities appear then as shown in (7.10) as the Cauchy-Schwarz-Bunyakovski inequality.
The cone approximation in combination with a part of the dispersion surface
at an optic axis is illustrated in Figure 5. In the left-hand partial picture one can see only the external part of the double cone. However, it is hardly to see that this cone is an elliptic cone since the chosen parameters (
,
,
) are not extremely enough different. In the right-hand partial picture an intersection with the plane
spanned by both optic axes is made and one may see also the internal part of the double cone and that the optic axes forms a “small” angle with the cone axis.
The complete approximate solutions for refraction vectors in the neighborhood of an optic axis is
. We now make for
the proposition
(7.13)
Figure 5. C one approximation of the dispersion surface
in the neighborhood of an optic axis. The left-hand picture shows only the cone approximation of the external part of the dispersion surface in the neighborhood of an optic axis. The right-hand picture shows in addition also the approximation of the internal part of the dispersion surface by the opposite part of the double cone.
The components
are not independent from each other since they have to satisfy the cone Equation (6.18). Inserting the proposition (7.13) into the cone equation one finds due to the orthonormality (7.9) of the vectors
(7.14)
This is the equation for an elliptic cone with axis in direction of the vector
. An elliptic cone with axis in z-direction in coordinates
can be written
(7.15)
Thus the cone Equation (7.14) can be resolved, for example (remind that
and
are negative in comparison to
)
(7.16)
The whole solution for
is only determined up a (small) real factor
and therefore is
(7.17)
but other forms are also possible. For this and also for other purposes we calculate now the inverse eigenvalues to
with the result
(7.18)
The angles between vectors which we calculated in coordinate-invariant form are now to obtain in convenient way.
The half cone angle
in the plane spanned by vectors
and
(plane
) is
(7.19)
The angle
between the optic axis and the cone axis of the refraction vectors is obtained from
(7.20)
with a result which equivalently can be expressed by
(7.21)
There is the following relation between the angles
and
(see also Figure 4, the left-hand partial figure)
(7.22)
The half cone angle
in the plane spanned by the unit vector
of the cone axis and the unit vector
perpendicular to the plane of the optic axes is
(7.23)
or for the corresponding Cosines
(7.24)
Some numerical values of the angles for the chosen permittivities
in the illustrations are5
(7.25)
The numerical value
for our choice of parameters in the figures is very near to the numerical value
and results here from the small numerical difference of the arithmetic and geometric mean of
and
in (7.5) for
and
which lead to
and are combined with further constants in
and
(moreover, in the formulae for the mentioned angles their square roots are relevant). It seems to be possible that in many experiments the small differences between the genuine elliptic cone and a conjectured circular cone in the conical refraction is hardly to see but this must not be the general case, in particular, since remarkable differences of the principal permittivities can be also artificially generated by external fields.
8. Plane Spanned by Optic Axis with Vector
and Vector
Perpendicular to Plane of Optic Axes
There is still another interesting plane of refraction vectors which is spanned by one of the two optic axes with the unit vector
and the principal axis of the permittivity tensor with the unit vector
(see (4.2)) and is perpendicular to the plane of both optic axes and possesses the normal unit vector
. A general wave vector
can be decomposed in a vector
in this plane and a vector
perpendicular to this plane according to
(8.1)
Since the component
is perpendicular to the considered plane it is omitted and the vector
in this plane possesses two perpendicular components according to
(8.2)
Both vectors
and
are unit vectors and are perpendicular to each other and also to the unit vector
. From (8.2) we find
(8.3)
The condition
makes in considered plane the transition to
(8.4)
and the condition
the transition to
(8.5)
The curves for
and
in considered plane are shown Figure 6. We see that the curves from
touches the curves
in the direction of optic axes. In the special case
these are exactly two circles with radii
and
. In general case the curves
are also not two circles displaced in considered plane perpendicular to the optic axis as it is shown in Figure 7 for two cases of extremely different principal permittivities. The graph of the dispersion surface in the intersection with considered plane is represented in Figure 8 together with two cases of small deviations
of the coupled refraction vectors to both sides of the optic axis and this is clearly
Figure 6. Intersections of dispersion surface
(blue, green) and the surface
(red) with the plane perpendicular to vector
. The dispersion curves consist in this case of two 4th-order curves which look similar to two circles if we change on the negative side on the axis
the colors blue and green but are not such. This can be seen if one choose essentially other parameters than in the picture which are here
. The optic axis is here vertical. The cone axis is not contained in considered plane.
Figure 7. Intersections of dispersion surface
(blue, green) with the plane perpendicular to vector
with extremely different permittivities. At the intersection points with the optic axes (vertical) the curves are not analytic. The permittivities are (
,
,
) (left) and (
,
,
) (right). This is more for illustration since the chosen values have little to do with really ones for media.
Figure 8. Refraction vectors in the neighborhood of the optic axis in plane with normal vector
. This plane is spanned by the vectors
of the optic axis and the principal vector
of permittivity tensor
. The cone axis does not lie in this plane and therefore also not the ray vectors and cannot really be drawn in this picture. Therefore, the angle
is (for our choice of parameters, angle between optic and cone axis
, see (7.25)) small different from the angle
and was not exactly calculated. The ray vectors, properly, also do not belong to the picture of refraction vectors and their origin in the real space depends on the considered device.
symmetric to optic axis.
The cone axis does not lie in considered plane spanned by the optic axis with vector
and the vector
and forms with it an angle
and therefore cannot be shown in this Figure 8. In analogy to (7.19) the half cone angle
in the plane spanned by the cone axis and the vector
is given by the relations
(8.6)
Its numerical value for the used permittivities is
as was already given in (7.25) in comparison to the corresponding angle
in the plane of the optic axes in Figure 6. Therefore, the angle
in Figure 7 is a little different from
and the ray vectors cannot be (exactly) drawn in this picture.
9. Polarization of Electric Field in the Neighborhood of an Optic Axis
In case of
and
the projection operator in (4.15) for the determination of polarization vectors
of the electric field becomes indeterminate and fails to act that is a consequence of the presence of an optic axis and the twofold degeneration of polarization vectors of the electric (and also magnetic) field in such directions. This twofold degeneration is removed for propagation in arbitrary small deviations from the optic axis. We deal with now these cases. Clearly, this is already made in the literature, in particular, in coordinate-invariant way in the monographs of Fyodorov [7] [8] but we are mainly interested for the neighborhood of optic axes. Our representation here distinguishes from that of Fyodorov in the way that we from beginning on involve more the operator
and its invariants. We make this for the classical crystal optics with only one frequency-dependent permittivity tensor
(or
written with indices) of the form (4.2) (for lossless case) and use as before the notation
for the refraction vectors in the neighborhood of the optic axis to the refraction vector
.
We write the non-degenerate projection operators (4.15) now in the form
(9.1)
and make an expansion of the refraction vectors in numerator and denominator in powers of the “small” deviations from the optic axis
. The general identity for the complementary operator of a sum of two operators
and
is
(9.2)
from which follows for its trace (can also independently be calculated from the second invariant of
)
(9.3)
Up to linear terms in
we find then from these identities or from (4.16) in the cone approximation6
(9.4)
and for its trace
(9.5)
As already said and derived in Section 7 the components of the small deviations of the refraction vectors from the optic axis are not independent from each other and are restricted by “cone” conditions of the form (7.16) or (7.17) which also have to be taken into account in (9.4). To get (non-normalized) polarization vectors of the electric field one has to apply these projection operators
to appropriately chosen vectors that can be made in various way. The formulae (9.4) and (9.5) are fairly complicated and it is usually a stony way to explain with them real experiments in detail. Besides conical refraction the polarization effects in direction of optic axes are diverse and difficult to calculate exactly but one may find many pictures from experiments in monographs and in Internet.
10. Optic Axes in Uniaxial Media
The special case of uniaxial crystal is very well known and we make only short remarks to classify them by the limiting transition from biaxial crystals to this uniaxial case. We have to distinguish two cases of the limiting transition, the case
and the case
. In both cases the primarily two optic axes coincide to one optic axis but in different positions if we preserve the ordering relations (4.4) for the principal values of the permittivities.
As first limiting case we consider
(10.1)
From (3) follows then
and the cone approximation degenerates to a tangential plane to the optic axis. This becomes a uniaxial positive medium (Figure 9).
The invariants of the tensor
are then
(10.2)
Furthermore we introduce the notations
(10.3)
The invariants of the operator
are
(10.4)
The invariant
can be also written in the form
(10.5)
from which results the following decomposition into a product, however, not with rational factors (two-shell surface)
(10.6)
For
we get the special product form
Figure 9. Limiting transition from optically biaxial to uniaxial media with dispersion surface
(blue, green) and surface
(red). The optic axes is vertical in first case and is horizontal in second case due to our limiting transition
and
in first and second case, respectively. The tangents of all considered surfaces at the touching points are orthogonal to the optic axes and the cones from biaxial crystals degenerate to tangential planes. From the two-shell surface
only the inner shell touches the optic axes and lies between the sphere and the rotation ellipsoid of the dispersion surface
. In left-hand picture the chosen permittivities are
and in right-hand picture
.
(10.7)
It touches the dispersion surfaces
at the optic axes.
As second limiting case we consider now
(10.8)
From (5.3) follows in this case
and the cone approximation degenerates also to a tangential plane at the optic axis. This becomes then a uniaxial negative medium (Figure 8).
The invariants of the permittivity tensor
are then
(10.9)
Furthermore we introduce the notations
(10.10)
The invariants of the operator
are the same as in (10.4). The invariant
can be written in the same two forms as in (10.5). The decomposition in product form is the same as in (10.6).
The cone in the cone approximation of refraction vectors near to an optic axis in biaxial crystals degenerates in uniaxial media to a tangential plane and there is then no difference between ordinary and extraordinary waves in this approximation. To get differences one would have go to the next higher-order approximation that, however, is not necessary since it is not very difficult to treat uniaxial media in full generality.
11. Optic Axes in the Domain of Complex Refraction Vectors
The optic axes which we considered up to now are present for real wave vectors
and they are the only ones in this domain. If we suppose real non-negative principal values
of the permittivity tensor
and preserve the ordering (4.4) that means
then in analogy to (5.1) we can may write down the following two solutions of the dispersion equations for the plane
(11.1)
and for the plane
(11.2)
If we form the difference of both equations we find in case of (11.1)
(11.3)
and in case of (11.2)
(11.4)
Contrary to the case
dealt with from Section 5 on, in the here considered two additional cases we cannot have at the same time only real components of the refraction vectors for the intersection points. These are the only cases with self-intersection and where at least one of the component
is vanishing.
We discuss now the more general problem to determine the simultaneous solutions of the two equations
and
in the complex domain and try to solve the question whether or not the case
but
and
is possible. For this purpose we admit also that the principal values
may possess complex values since this does not make the derivations more difficult. Self-intersections of the solutions of
are only possible if one of the three components
is vanishing and it seems that these cases are dealt with exhaustively by this. So we may suppose that in the searched case all three components
are non-vanishing.
The dispersion equation
for the refraction vectors
given in (4.8) in the principal-axis form resolved to the component, e.g.,
in dependence on
and
becomes the biquadratic equation
(11.5)
with the abbreviations
(11.6)
and the equation for vanishing
of the second invariant of
in (4.10) the biquadratic equation
(11.7)
with the abbreviations
(11.8)
The solutions of a biquadratic equation of the form
(11.9)
may be represented in the following ways
(11.10)
However, it is not necessary to solve immediately the special biquadratic equations (11.5) and (11.7). One may first form the difference of both these equations and obtains the following quadratic equation for
(11.11)
which expresses
in dependence of
and
and on the permittivities. The result for
can be inserted in one of the biquadratic Equations (11.5) or (11.7) which provide then relations where
is eliminated. Effectively this becomes a bicubic equation which has to be solved. It is hardly possible to make this without a computer. Alternatively, one can also solve by computer both the Equations (5) and (7) with respect to
that provides 4 solutions (2 solutions for
) for both equations with 4 possible combinations of establishing equalities to eliminate
. Due to voluminous coefficients in these equations the arising bicubic equation possesses solutions which even by computer are given in an extremely long form which we do not write down. The trials to simplify the expressions or to find interesting special cases were without success up to now. Therefore, we can say that, apparently, exist searched cases
and
but
in the complex domain but due to their difficulty they are probably little interesting. This problem is not solved. For real components
and symmetry
such solutions are absent due to
for left-hand and right-hand eigenvectors and
.
12. Remarks to a Duality between Refraction Vectors and Ray Vectors
In the more general consideration of spatial dispersion one may start from the wave-equation operator
and the vectorial wave equation
(12.1)
which in case of neglect of dispersion that means with constant
(12.2)
may be reduced using refraction vectors
to the operator
according to
(12.3)
as was assumed for the treatment of optic axes in present article.
In the concept (12.1) we may resolve the dispersion equation
to
(12.4)
The group velocity
can be obtained by differentiation of the last identity
which depends only on the wave vector
and under neglect of the dispersion and with introduction of refraction vectors
one obtains
(12.5)
From this follows with introduction of ray vectors
(e.g., [2], §97 and [6], §81)
(12.6)
In the concept (12.3) with the refraction vectors
and the operator
we find
(12.7)
and the ray vectors
can be directly determined by the operator
in the following way
(12.8)
This is part of the well-known duality between treatment of crystal optics in the space of refraction vectors and the treatment by ray vectors in the real vector space (e.g., [2], Eq. (97.19))
(12.9)
In the cone approximation in the neighborhood of an optic axis from the dispersion equation
(12.10)
follows for the ray vectors
using the symmetry of the cone tensor
(12.11)
Thus from the cone approximation for the refraction vectors
in the neighborhood of the optic axis
(12.12)
follows the equation for the ray vectors
in the neighborhood of an optic axis (see also [2], Eq. (97.14))
(12.13)
For the direction of an optic axis
the ray vectors
become indeterminate on the ray cone but the relation
remains true in the limit.
One must not forget that the derived relations are only true under neglect of the dispersion of the permittivity tensor
and here additionally concern the cone approximation in the neighborhood of an optic axis. In other case it is usually more difficult to derive a scalar equation describing a surface, for example, for the group velocity. We did not intend to consider in this article in detail the mentioned known duality.
13. Conclusion
Our main result for crystal optics is the coordinate-invariant derivation of the cone equation (6.18) with the cone tensor (6.17) as approximation of the dispersion equation in the neighborhood of optic axes and of the eigenvalues and eigenvectors of this cone tensor with the basic results in (7.5) and (7.9). This cone proved to be an elliptic cone. Although the differences of the experimentally determined principal values of the permittivities for some crystals which we found in different sources are small and the elliptic cone is then very near to a circular cone it is a basic question of its form since they can be enlarged by external fields. Polarization vectors for the electric field to the refraction vectors in the neighborhood of optic axes are calculated. They depend mainly on the small deviations
of the refraction vectors from the refraction vector
of the optic axis. Furthermore, the presence of optic axes as degeneration points of the dispersion equation for the refraction vectors is connected with the necessary vanishing of the second invariant of the three-dimensional wave-equation operator. This vanishing is not sufficient for the presence of an optic axis and involves also the case where the scalar product of right-hand and left-hand polarization vectors of the electric field is vanishing and cannot be normalized. Such cases cannot be present for real-valued refraction vectors. This problem was investigated but not fully solved. In Appendix A we calculate the eigenvalues and eigenvectors of a special three-dimensional operator. In a further Appendix B we consider shortly the introduction of axes in the sense of Fyodorov for an arbitrary second-rank symmetric tensor.
Appendix A. Operator
and Its Invariants and Complementary Operator
We consider the three-dimensional symmetrical operator
defined by
(A.1)
with general three-dimensional vectors
and
and calculate its invariants and its complementary operator. First, we have to calculate its second and third powers for which one easily finds
(A.2)
and from which follows for their traces
(A.3)
Using the general formulae (2.3) for three-dimensional operators one also easily finds the second invariant
and the determinant
(A.4)
with
the vector product of
and
. We see immediately that the determinant
vanishes in the two special cases
and
where the operator
specializes to
and to
, respectively, and which can be calculated in more easier way. For the complementary operator
to
, generally defined in (2.4), we find
(A.5)
Using the Hamilton-Cayley identity for the operator
and the definition (A.1) one may check that the relation
is satisfied and, furthermore,
.
Alternatively, in this and similar cases one may calculate the complementary operator
to
also from the eigenvalues
and then the eigenvectors (or their projection operators
in the sense
) of
for which follows from (A.4)
(A.6)
Then one gets the representations
(A.7)
One may also write
with
and
and then one may use all formulae derived for a sum of two three-dimensional operators
and
.
The above coordinate-invariant calculations for the operator (A.1) represent also a good example illustrating the advantages to use coordinate-invariant methods in such and similar cases.
Appendix B. Fyodorov’s Introduction of Axes of a Symmetrical Three-Dimensional Tensor
We consider here according to Fyodorov [7] [8] the introduction of axes
for a three-dimensional second-rank real symmetrical tensor
with the (trivial) extension to a Hermitean tensor
(B.1)
with the real eigenvalues
and make the following transformation of the vectors
and
to new vectors
(denoted by
in [7] [8])
(B.2)
with the inversion
(B.3)
Then we find for the n-th powers of
(B.4)
This formula is also correct for negative n-th powers
. The second sum term in braces vanishes only for
leading to
(B.5)
For all other powers of n the formula for
becomes complicated and does not bring an advantage in comparison to the first representation in (B.4). Fyodorov has the opposite sign at the first sum term in braces because he defined our vector
with the opposite sign (§ 26 of [8]). Furthermore, Fyodorov considers the case of real symmetric tensors
and calls the directions determined by the unit vectors
the axes of the tensor
. In accordance with general use he calls real symmetric tensors with
isotropic, with
but
uniaxial and general ones biaxial. In a later § 28 he considers also complex tensors in similar way.
The optic axes for the refraction vectors in the dispersion equation equivalent to the Fresnel equation are the axes (in the sense of F.) of the inverse permittivity tensor
. The application of the axes of a second-rank symmetric tensor according to Fyodorov is often of restricted use since all powers of the permittivity tensor involved, e.g. in the dispersion equation, have different axes and the same is the case in many other equations but the equation for the magnetic field contains only the inverse permittivity tensor that is used by Fyodorov in his monograph [7] in § 22.
Appendix C. Remarks to a Plagiarism of Fyodorov’s Coordinate-Invariant Methods
This Appendix reports in form of an example about a fraud in science by Hollis C. Chen from Ohio University in U.S.A. who published in 1983 a book with the title “Theory of Electromagnetic Waves, A Coordinate-Free Approach” at McGraw-Hill Book Company, New York. This book is a plagiarism of two books of Fyodor Ivanovich Fyodorov from Minsk with titles “Optika anisotropnykh sred (Optics of Anisotropic Media), Minsk 1958 (2000 exemplars) and of “Teoriya girotropii” (Theory of Gyrotropy), Minsk 1976 (1050 exemplars). Some of the chapters of the book of Chen are almost a literal translation of chapters of the two books of Fyodorov and of one of my papers about radiation in uniaxial media in two chapters without any citation of the genuine authors but with citations which have nothing to do with coordinate-invariant representation. Often he uses totally the same letters for notations without trying to conceal this. What Chen made is the change of notations in the algebra of vectors and tensors from that of Fyodorov (which partially is an older but very good one) to that of Gibbs later introduced but less appropriate for many purposes, in particular, anisotropic media. The first chapter of Chen “Linear analysis” (pp. 1-57, with omissions) is almost the rewritten last chapter of Fyodorov’s secondly mentioned book (pp. 362-440).
Due to present paper about degenerate and peculiar cases, in particular, optic axes I made a comparison of Fyodorov’s work of three-dimensional algebra with the similar in mentioned chapter (of my copy) of Chen’s book. From a general symmetric second-rank tensor of the form
(C.1)
with
it is easy to come to the representation (Fyodorov, Gyrotropy, p. 394) by the formula (Fyodorov usually omits identity operators)
(C.2)
The same formula of Chen (p. 35) is written (dyadic products without a point between two vectors)
(C.3)
Now follows a linear combination of the vectors
to new non-orthogonal vectors which leads to a result (Fyodorov, p. 395, Eq. (26.108) and the same Ch., p. 35, Eq. 1.179 a little differently written). H. Ch. used that likely the Minsk colleagues and also I could not travel at that time to Conferences in the West and that the mentioned misused books of Fyodorov were difficult to get (even in libraries in East-Berlin) and that they are now a rarity.
To the overall-plagiarism by Chen I want to mention in addition the following case. In the effort to rationalize his notations Fyodorov does not introduce a special symbol for the second invariant of a three-dimensional operator
which I denote by
and which he denotes by
in [7]
(from Russian “sled” which means “trace”; “c” is Russian letter for Latin “s”) and
in [8]. Apart from visibility and unfavorable distinction from vector indices this is not favorable since the complementary operator
to
must then be directly determined by
with
on both sides and the Hamilton-Cayley identity is then
but in such and similar
cases F. writes then in detail
. Ch. makes exactly the same without citation of Fyodorov in his book only with some other letter types.
NOTES
1Our notation of the invariants possesses the advantage that they are easily to see and to recognize as such in formulae and text. Furthermore, since letter
in
is not specific for a certain
we prefer the special notation
for the identity operator in (1). However, there are also formulae which are not specific for a certain dimension where it is then appropriate to write the identity operator, e.g.,
if
determines the dimension of
.
2We neglect in this article widely frequency dispersion and therefore we do not write the frequency
as additional variable in
. Taking it into account it is more appropriate to use the “three-dimensional” operator
with
the wave vector. In last case
and
become three-dimensional (hyper-) surfaces in the four-dimensional
-space.
3This means that we neglect absorption or, more generally, dissipation and also do not take into consideration cases of negative principal values of the permittivities.
4It is easy to check that the trace
obtained from (6.17) or (6.22) using (6.2) agrees with the form given here.
5The numerical coincidence
(see after (5.3)) is not a general equality and is incidentally caused by our choice of parameters with
in different formulae and was not foreseen and intended.
6We write the following and wrote already some formulae before preserving a certain left-right-symmetry that means a little more general as absolutely necessary for symmetrical permittivity tensors
that may become important in generalizations to Hermitean permittivity tensors and demonstrates more the symmetries.