1.1 Crystal Optics
9
will propagate through the crystal with a different velocity. This introduces a relative
phase φ a = ωdn a /c and φ b = ωdn b /c to each component of E, depending on the
distance d the wave has propagated through the medium. After transmission through
a crystal of thickness L, the two components of E will have acquired a phase delay
φ = φ b − φ a =
ωL
c
(n b − n a ) ,
(1.25)
where the difference in refractive index n b − n a = n is the birefringence of the
medium. The polarisation state of the transmitted light will therefore depend on the
phase delay introduced: a superposition of two orthogonally polarised waves with
a phase delay of φ = π/2 will form a circular polarisation state, whereas for any
other value of 0 < <φ < π/2 the polarisation state will be elliptical.
1.2 Describing the Polarisation State of Electromagnetic
Waves
Having established that propagation through an anisotropic medium can alter the
polarisation state of light, a crucial consideration in the spectroscopy of anisotropic
materials is having a method of describing the polarisation state at a given time or
frequency. This section will outline some of the descriptions available which will be
used in this thesis: the ellipticity and orientation angle; the ellipsometric parameters;
Jones matrices; and Stokes parameters.
1.2.1 Ellipticity and Orientation Angle
The polarisation state of an electromagnetic wave can be parameterised by two quantities, the ellipticity angle χ and the orientation angle ψ [19]. This parameterisation is
shown schematically by the polarisation ellipse in Fig. 1.3, with the viewer oriented
such that they are looking into the direction of propagation of the electromagnetic
wave. For an arbitrary polarisation state the oscillation of the electric or magnetic
field when viewing from this position can be visualised as forming an ellipse. The
ellipticity angle is defined as
χ = tan
−1
b
a
,
(1.26)
where a and b are the lengths of the semi-major and semi-minor axes of the polarisation ellipse respectively. An ellipticity angle of zero corresponds to a linear polarisation state, whereas an ellipticity of ±45
◦ corresponds to right- and left-handed
circular polarisation, respectively. The orientation angle is defined as the angle of
9
will propagate through the crystal with a different velocity. This introduces a relative
phase φ a = ωdn a /c and φ b = ωdn b /c to each component of E, depending on the
distance d the wave has propagated through the medium. After transmission through
a crystal of thickness L, the two components of E will have acquired a phase delay
φ = φ b − φ a =
ωL
c
(n b − n a ) ,
(1.25)
where the difference in refractive index n b − n a = n is the birefringence of the
medium. The polarisation state of the transmitted light will therefore depend on the
phase delay introduced: a superposition of two orthogonally polarised waves with
a phase delay of φ = π/2 will form a circular polarisation state, whereas for any
other value of 0 < <φ < π/2 the polarisation state will be elliptical.
1.2 Describing the Polarisation State of Electromagnetic
Waves
Having established that propagation through an anisotropic medium can alter the
polarisation state of light, a crucial consideration in the spectroscopy of anisotropic
materials is having a method of describing the polarisation state at a given time or
frequency. This section will outline some of the descriptions available which will be
used in this thesis: the ellipticity and orientation angle; the ellipsometric parameters;
Jones matrices; and Stokes parameters.
1.2.1 Ellipticity and Orientation Angle
The polarisation state of an electromagnetic wave can be parameterised by two quantities, the ellipticity angle χ and the orientation angle ψ [19]. This parameterisation is
shown schematically by the polarisation ellipse in Fig. 1.3, with the viewer oriented
such that they are looking into the direction of propagation of the electromagnetic
wave. For an arbitrary polarisation state the oscillation of the electric or magnetic
field when viewing from this position can be visualised as forming an ellipse. The
ellipticity angle is defined as
χ = tan
−1
b
a
,
(1.26)
where a and b are the lengths of the semi-major and semi-minor axes of the polarisation ellipse respectively. An ellipticity angle of zero corresponds to a linear polarisation state, whereas an ellipticity of ±45
◦ corresponds to right- and left-handed
circular polarisation, respectively. The orientation angle is defined as the angle of