Geoscience Reference
In-Depth Information
c
n
.
ν
=
(6)
Implementing this equation to the phase and group velocities, the formulae for
the phase and group refractive indices
n
ph
and
n
gr
read
c
n
ph
,
ν
ph
=
(7)
c
n
gr
.
ν
gr
=
(8)
λ
Differentiating Eq.
7
with respect to
yields
d
ν
ph
d
c
n
ph
dn
ph
d
=−
.
(9)
λ
λ
Substituting Eqs.
9
,
8
, and
7
into Eq.
5
yields
c
n
gr
=
c
n
gr
+
λ
c
n
ph
dn
ph
d
,
(10)
λ
or
1
.
1
n
gr
=
1
n
ph
1
n
ph
dn
ph
d
+
λ
(11)
λ
=
+
ε)
−
1
Using the approximation
(
1
1
−
ε
, valid for small quantities of
ε
,Eq.
11
is inverted to
n
ph
1
1
n
ph
dn
ph
d
n
gr
=
−
λ
.
(12)
λ
Thus the group refractive index follows
dn
ph
d
n
gr
=
n
ph
−
λ
,
(13)
λ
Equation
13
is the modified Rayleigh equation (Hofmann-Wellenhof et al.
1993
).
A slightly different form is obtained by differentiating the relation
c
=
λ
f
with
respect to
λ
and
f
, that is
d
λ
λ
df
f
,
=−
(14)
and by substituting the results into Eq.
13
, the group refractive index yields
dn
ph
df
.
n
gr
=
n
ph
+
f
(15)
