Advanced Solid State Physics

13 Optical Processes

In chapter 10.1 (Linear Response Thoery) we derived the Kramers-Kronig relations. The Kramers-
Kronig relations relate the real component (χ′) of a generalized susceptibility to the imaginary com-
ponent (χ′′):

χ′′(ω) =

1

π

∫∞

−∞

χ′(ω′)

ω′−ω

dω′ (209)

χ′(ω) = −

1

π

∫∞

−∞

χ′′(ω′)

ω′−ω

dω′ (210)

The characteristics of the two parts of the generalized susceptibility parts are:

χ′(ω) = χ′(−ω)

χ′′(ω) = −χ′′(−ω)

This form (eqn. (209) and (210)) of the Kramers-Kronig relations should be used for mathematical
calculations. For experimental data the second form of the Kramers-Kronig relation should be used
(eqn. (211) and eqn. (212)). Now it is possible to transform the first form of the Kramers-Kronig
relation into the second:
First step is to split the integral and change the sign ofω:

=⇒

χ′(ω) = −

1

π

∫ 0

−∞

χ′′(ω′)

ω′−ω

dω′−

1

π

∫∞

0

χ′′(ω′)

ω′−ω

dω′

=⇒

χ′(ω) = −

1

π

∫∞

0

χ′′(ω′)

ω′+ω

dω′−

1

π

∫∞

0

χ′′(ω′)

ω′−ω

dω′

With the relationship

1

ω′+ω

+

1

ω′−ω

=

2 ω′

ω′^2 −ω^2

the second form of the Kramers-Kronig relation is derived:

χ′(ω) =

2

π

P

∫∞

0

ω′χ′′(ω′)

ω′^2 −ω^2

dω′ (211)

χ′′(ω) = −

2 ω

π

P

∫∞

0

χ′(ω′)

ω′^2 −ω^2

dω′ (212)