e ε ¼
ε ξξ ε ξη ε ξζ
ε ηξ ε ηη 0
ε ζξ 0 ε ζζ
0
B
@
1
C
A,
where e ε is expressed in the ξηζ-system. In terms of the matrix algebra, the principal
submatrix with respect to the η- and ζ- components should be diagonalized (see Sect.
14.5). On this condition, the electric flux density of the electromagnetic wave is
polarized in the η- and ζ-direction. A half of the reciprocal square root of the
principal axes, i.e., 1=
ffiffiffiffiffiffi
ε ηη
p or 1=
ffiffiffiffiffiffi
ε ζζ
p , represents the anisotropic refractive indices.
Suppose that the ξηζ-system is reached by two successive rotations starting from
the abc
à -system in such a way that we first perform the rotation by α around the c
à -
axis that is followed by another rotation by δ around the ξ-axis (Fig. 9.12). Then we
have [7]
ε ξξ ε ξη ε ξζ
ε ηξ ε ηη 0
ε ζξ 0 ε ζζ
0
B
@
1
C
A ¼ R
À1
ξ δ
ð ÞR
À1
c à α
ð Þ
ε aa
0 ε ac Ã
0 ε bb
0
ε c à a 0 ε c à c Ã
0
B
@
1
C
ARc à α
ð ÞR ξ δ
ð Þ, ð9:114Þ
where R c à α
ð Þ and R ξ (δ) stand for the first and second rotation matrices of the above,
respectively. In (9.114) we have
R c à α
ð Þ ¼
cos α À sin α 0
sin α
cos α 0
0
0
1
0
B
@
1
C
A, R ξ δ
ð Þ ¼
1
0
0
0 cos δ À sin δ
0 sin δ
cos δ
0
B
@
1
C
A: ð9:115Þ
With the matrix presentations and related coordinate transformations for (9.114)
and (9.115), see Sects. 11.4, 17.1, and 17.4.2. Note that the rotation angle δ cannot
be decided independent of α. In the literature [7], furthermore, the quantity ε ξη ¼ ε ηξ
was ignored as small compared to other components. As a result, (9.111) is
converted into the following equation described by
rot H ¼ ε 0
ε ξξ 0 ε ξζ
0 ε ηη 0
ε ζξ 0 ε ζζ
0
B
@
1
C
A
∂E
∂t
:
ð9:116Þ
Notice here that the electromagnetic quantities H and E are measured in the ξηζcoordinate system.
9.5 Lasers
367
ε ξξ ε ξη ε ξζ
ε ηξ ε ηη 0
ε ζξ 0 ε ζζ
0
B
@
1
C
A,
where e ε is expressed in the ξηζ-system. In terms of the matrix algebra, the principal
submatrix with respect to the η- and ζ- components should be diagonalized (see Sect.
14.5). On this condition, the electric flux density of the electromagnetic wave is
polarized in the η- and ζ-direction. A half of the reciprocal square root of the
principal axes, i.e., 1=
ffiffiffiffiffiffi
ε ηη
p or 1=
ffiffiffiffiffiffi
ε ζζ
p , represents the anisotropic refractive indices.
Suppose that the ξηζ-system is reached by two successive rotations starting from
the abc
à -system in such a way that we first perform the rotation by α around the c
à -
axis that is followed by another rotation by δ around the ξ-axis (Fig. 9.12). Then we
have [7]
ε ξξ ε ξη ε ξζ
ε ηξ ε ηη 0
ε ζξ 0 ε ζζ
0
B
@
1
C
A ¼ R
À1
ξ δ
ð ÞR
À1
c à α
ð Þ
ε aa
0 ε ac Ã
0 ε bb
0
ε c à a 0 ε c à c Ã
0
B
@
1
C
ARc à α
ð ÞR ξ δ
ð Þ, ð9:114Þ
where R c à α
ð Þ and R ξ (δ) stand for the first and second rotation matrices of the above,
respectively. In (9.114) we have
R c à α
ð Þ ¼
cos α À sin α 0
sin α
cos α 0
0
0
1
0
B
@
1
C
A, R ξ δ
ð Þ ¼
1
0
0
0 cos δ À sin δ
0 sin δ
cos δ
0
B
@
1
C
A: ð9:115Þ
With the matrix presentations and related coordinate transformations for (9.114)
and (9.115), see Sects. 11.4, 17.1, and 17.4.2. Note that the rotation angle δ cannot
be decided independent of α. In the literature [7], furthermore, the quantity ε ξη ¼ ε ηξ
was ignored as small compared to other components. As a result, (9.111) is
converted into the following equation described by
rot H ¼ ε 0
ε ξξ 0 ε ξζ
0 ε ηη 0
ε ζξ 0 ε ζζ
0
B
@
1
C
A
∂E
∂t
:
ð9:116Þ
Notice here that the electromagnetic quantities H and E are measured in the ξηζcoordinate system.
9.5 Lasers
367