There are several other ways to prove the Kramers-Kronig relations. For example,
a more direct way is to state the causality condition in terms of the signum function
sign
(t)
.
Indeed, because
u(t)
=
(
1
+
sign
(t)
)
/
2, Eq. (1.17.4) may be written in the
equivalent form
χ(t)
=
χ(t)
sign
(t)
.
Then, Eq. (1.17.7) follows by applying the same
frequency-domain convolution argument using the Fourier transform pair:
sign
(t)
P
2
jω
(1.17.10)
Alternatively, the causality condition can be expressed as
u(
−
t)χ(t)
=
0. This ap-
proach is explored in Problem 1.12. Another proof is based on the
analyticity
properties
of
χ(ω)
. Because of the causality condition, the Fourier integral in (1.17.3) can be re-
stricted to the time range 0
< t <
∞
:
χ(ω)
=
∞
−∞
e
−
jωt
χ(t)dt
=
∞
0
e
−
jωt
χ(t)dt
(1.17.11)
This implies that
χ(ω)
can be analytically continued into the
lower
half
ω
-plane,
so that replacing
ω
by
w
=
ω
−
jα
with
α
≥
0 still gives a convergent Fourier integral
†
The right-hand side (without the
j
) in (1.17.7) is known as a Hilbert transform. Exchanging the roles
of
t
and
ω
, such transforms, known also as 90
o
phase shifters, are used widely in signal processing for
generating single-sideband communications signals.

1.18.
Group Velocity, Energy Velocity
29
in Eq. (1.17.11). Any singularities in
χ(ω)
lie in the upper-half plane. For example, the
simple model of Eq. (1.11.7) has poles at
ω
= ±
¯
ω
0
+
jγ/
2, where
¯
ω
0
=
ω
2
0
−
γ
2
/
4.
Next, we consider a clockwise closed contour
C
=
C
+
C
∞
consisting of the real axis
C
and an infinite semicircle
C
∞
in the lower half-plane. Because
χ(ω)
is analytic in the
region enclosed by
C
, Cauchy’s integral theorem implies that for any point
w
enclosed
by
C
, that is, lying in the lower half-plane, we must have:
χ(w)
= −
1
2
πj
C
χ(w )
w
−
w
dw
(1.17.12)
where the overall minus sign arises because
C
was taken to be clockwise. Assuming that
χ(ω)
falls off sufficiently fast for large
ω
, the contribution of the infinite semicircle
can be ignored, thus leaving only the integral over the real axis. Setting
w
=
ω
−
j
and
taking the limit
→
0
+
, we obtain the identical relationship to Eq. (1.17.5):
χ(ω)
= −
lim
→
0
+
1
2
πj
∞
−∞
χ(ω )
ω
−
ω
+
j
dω
=
1
2
π
∞
−∞
χ(ω )
lim
→
0
+
1
j(ω
−
ω )
+
dω
An interesting consequence of the Kramers-Kronig relations is that there cannot
exist a dielectric medium that is purely lossless, that is, such that
χ
i
(ω)
=
0 for all
ω
,
because this would also require that
χ
r
(ω)
=
0 for all
ω
.
However, in all materials,
χ
i
(ω)
is significantly non-zero only in the neighborhoods
of the medium’s resonant frequencies, as for example in Fig. 1.11.1. In the frequency
bands that are sufficiently far from the resonant bands,
χ
i
(ω)
may be assumed to be
essentially zero. Such frequency bands are called
transparency bands
[164].
1.18
Group Velocity, Energy Velocity
Assuming a nonmagnetic material (
μ
=
μ
0
), a complex-valued refractive index may be
defined by:
n(ω)
=
n
r
(ω)
−
jn
i
(ω)
=
1
+
χ(ω)
=
(ω)
0
(1.18.1)
where
n
r
, n
i
are its real and imaginary parts. Setting
χ
=
χ
r
−
jχ
i
we have the condition
n
r
−
jn
i
=
1
+
χ
r
−
jχ
i
. Upon squaring, this splits into the two real-valued equations
n
2
r

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