8
1 Introduction
If we consider the first term in Eq. 1.22 as a projection of the vector ηD onto a plane
perpendicular to u, we may define a projection operator P u such that
P u ηD = − u × ( u × ηD).
(1.23)
By using the relations k
2
0 = ω
2
μ 0 0 and n = k/k 0 , we can rewite Eq. 1.22 as an
eigenvalue equation of the operator P u η,
P u ηD =
1
n 2 D,
(1.24)
which has two eigenvectors D a and D b corresponding to the normal modes of propagation in the direction of u, with their corresponding eigenvalues being 1/n
2
a and
1/n
2
b .
The solutions to this eigenvalue equation can be visualised using either the dispersion relation or the index ellipsoid. In the form ω = ω(k 1 , k 2 , k 3 ), the dispersion
relation can be considered as the equation of a surface in k-space. For an arbitrary
direction of propagation u there are two intersections with the surface in k-space,
which correspond to the two normal modes of propagation along u. These normal
modes are demonstrated on the index ellipsoid in Fig. 1.1, with the plane perpendicular to u up to the boundary of the index ellipsoid and passing through the origin known
as the index ellipse. The major and minor axes of this index ellipse are the directions
of the eigenvectors D a,b , and the lengths of the semi-major and semi-minor axes are
the values of the refractive indices n a,b along the normal modes of propagation.
1.1.4 Effects of an Anisotropic Medium on the Polarisation
State of Light
In an isotropic medium, for any given propagation direction and orientation of E of
an incident plane electromagnetic wave, the electric field will experience the same
refractive index, and so the polarisation state will remain unchanged after propagating
through an isotropic medium. So what happens when we consider the case where
the principal refractive indices are not all the same? Having established the index
ellipsoid and the two orthogonal normal modes of propagation for a given direction,
we can now use these concepts to describe how the polarisation state of light evolves
upon propagating through an anisotropic medium.
For light with angular frequency ω propagating in an anisotropic crystal with an
arbitrary wavevector k there are two orthogonal normal modes, with polarisation
eigenvectors D a,b . These normal modes each have different refractive indices. If the
incident light is polarised along one of the eigenvectors of a normal mode, then
the wave experiences only one refractive index, and the polarisation state remains
unchanged after transmission through the medium. However, if the incident light
is linearly polarised with components of E along D a and D b , then each component
1 Introduction
If we consider the first term in Eq. 1.22 as a projection of the vector ηD onto a plane
perpendicular to u, we may define a projection operator P u such that
P u ηD = − u × ( u × ηD).
(1.23)
By using the relations k
2
0 = ω
2
μ 0 0 and n = k/k 0 , we can rewite Eq. 1.22 as an
eigenvalue equation of the operator P u η,
P u ηD =
1
n 2 D,
(1.24)
which has two eigenvectors D a and D b corresponding to the normal modes of propagation in the direction of u, with their corresponding eigenvalues being 1/n
2
a and
1/n
2
b .
The solutions to this eigenvalue equation can be visualised using either the dispersion relation or the index ellipsoid. In the form ω = ω(k 1 , k 2 , k 3 ), the dispersion
relation can be considered as the equation of a surface in k-space. For an arbitrary
direction of propagation u there are two intersections with the surface in k-space,
which correspond to the two normal modes of propagation along u. These normal
modes are demonstrated on the index ellipsoid in Fig. 1.1, with the plane perpendicular to u up to the boundary of the index ellipsoid and passing through the origin known
as the index ellipse. The major and minor axes of this index ellipse are the directions
of the eigenvectors D a,b , and the lengths of the semi-major and semi-minor axes are
the values of the refractive indices n a,b along the normal modes of propagation.
1.1.4 Effects of an Anisotropic Medium on the Polarisation
State of Light
In an isotropic medium, for any given propagation direction and orientation of E of
an incident plane electromagnetic wave, the electric field will experience the same
refractive index, and so the polarisation state will remain unchanged after propagating
through an isotropic medium. So what happens when we consider the case where
the principal refractive indices are not all the same? Having established the index
ellipsoid and the two orthogonal normal modes of propagation for a given direction,
we can now use these concepts to describe how the polarisation state of light evolves
upon propagating through an anisotropic medium.
For light with angular frequency ω propagating in an anisotropic crystal with an
arbitrary wavevector k there are two orthogonal normal modes, with polarisation
eigenvectors D a,b . These normal modes each have different refractive indices. If the
incident light is polarised along one of the eigenvectors of a normal mode, then
the wave experiences only one refractive index, and the polarisation state remains
unchanged after transmission through the medium. However, if the incident light
is linearly polarised with components of E along D a and D b , then each component