1.1 Crystal Optics
3
where the constant of proportionality is the dielectric constant of the medium, which
is independent of the orientation and direction of propagation of the wave. However,
in an anisotroptic medium the structure is no longer the same for any arbitrary
direction of propagation, and as such the optical properties will vary depending
on the relative orientations of the crystallographic directions and the electric field.
Each of the components of D will now be made up of a linear combination of the
components of E, such that
D x = xx E x + xy E y + xz E z ,
(1.2)
D y = yx E x + yy E y + yz E z ,
(1.3)
D z = zx E x + zy E y + zz E z .
(1.4)
Hence the optical properties of the medium can now be described by a 3 × 3 matrix,
the dielectric tensor , where
⎡
⎣
D x
D y
D z
⎤
⎦ =
⎡
⎣
xx xy xz
yx yy yz
zx zy zz
⎤
⎦ ·
⎡
⎣
E x
E y
E z
⎤
⎦ ,
(1.5)
which can alternatively be written
D i =
j
i j E j ,
(1.6)
where i, j represent the x, y and z coordinates. The dielectric tensor is symmetric,
such that i j = ji , and can therefore be described by six independent values. The
symmetry of the crystal structure in some materials can further reduce the number
of independent components of the dielectric tensor.
1.1.2 Principal Axes and the Index Ellipsoid
The particular values of the components of the dielectric tensor depend on the choice
of coordinate system relative to the crystal structure. For any given crystal, it is
possible to choose a frame of reference such that the off-diagonal components vanish,
and the dielectric tensor becomes
=
⎡
⎣
1 0 0
0 2 0
0 0 3
⎤
⎦ ,
(1.7)
where 1 = xx , 2 = yy and 3 = zz . This coordinate system is known as the principal axis system, and defines the directions in which D and E must be parallel to each
3
where the constant of proportionality is the dielectric constant of the medium, which
is independent of the orientation and direction of propagation of the wave. However,
in an anisotroptic medium the structure is no longer the same for any arbitrary
direction of propagation, and as such the optical properties will vary depending
on the relative orientations of the crystallographic directions and the electric field.
Each of the components of D will now be made up of a linear combination of the
components of E, such that
D x = xx E x + xy E y + xz E z ,
(1.2)
D y = yx E x + yy E y + yz E z ,
(1.3)
D z = zx E x + zy E y + zz E z .
(1.4)
Hence the optical properties of the medium can now be described by a 3 × 3 matrix,
the dielectric tensor , where
⎡
⎣
D x
D y
D z
⎤
⎦ =
⎡
⎣
xx xy xz
yx yy yz
zx zy zz
⎤
⎦ ·
⎡
⎣
E x
E y
E z
⎤
⎦ ,
(1.5)
which can alternatively be written
D i =
j
i j E j ,
(1.6)
where i, j represent the x, y and z coordinates. The dielectric tensor is symmetric,
such that i j = ji , and can therefore be described by six independent values. The
symmetry of the crystal structure in some materials can further reduce the number
of independent components of the dielectric tensor.
1.1.2 Principal Axes and the Index Ellipsoid
The particular values of the components of the dielectric tensor depend on the choice
of coordinate system relative to the crystal structure. For any given crystal, it is
possible to choose a frame of reference such that the off-diagonal components vanish,
and the dielectric tensor becomes
=
⎡
⎣
1 0 0
0 2 0
0 0 3
⎤
⎦ ,
(1.7)
where 1 = xx , 2 = yy and 3 = zz . This coordinate system is known as the principal axis system, and defines the directions in which D and E must be parallel to each