The Phase Associated With Laser Gain,
and the Kramers-Kronig Dispersion Relation
David Eimerl, Ph.D.
EIMEX Software and Consulting
4042 Camrose Avenue
Livermore California 94550
Abstract
We report a rigorous method of calculating the phase that laser gain imparts to a
laser pulse as a result of passing through the laser gain medium. The method is based on
the very fundamental principle of causality, and does not depend in any way on the
mechanism by which gain is achieved nor on the distribution of excited states responsible
for the gain. Thus for example, the phase associated with gain in a Nd:glass amplifier
does not depend in any way on the distribution of ions or ion species. The only data
required to determine the phase pulling associated with the existence of gain is the actual
gain spectrum as a function of frequency.
The Phase Associated With Laser Gain,
and the Kramers-Kronig Dispersion Relation
The dielectric polarizability near a resonance with a single decay constant is has the
familiar Lorentz profile h() .
h( )  i /(  i)
or
1

 1
h
i
The field gain (or loss rate) is proportional to Re(h). The presence of the imaginary terms
introduces a phase on any laser beam that experiences gain or loss. It is now is well-known1 that,
in general, the phase of the polarizability cannot be zero at all frequencies. This result was first
obtained by Kramers on the very general grounds of (a) the reality of the electric field and dipole
moment, and (b) causality, which requires that any response function for any system, considered
as a function of complex frequency, may have no singularities in the upper half complex plane.
Remarkably, these two conditions alone are capable of fixing the phase associated with any
response function such as a dielectric polarizability.
For optical response functions, this leads to the Kramers-Kronig relations relating the real and
imaginary parts of the polarizability or refractive index1. These are also referred to as dispersion
relations. According to the Kramers-Kronig theory, the real part of the refractive index arising
from an intensity gain () is as follows.
 R ( ) 
P



0
c  ( ' )
d '
 '2  2
where P denotes principal value. The gain-associated phase written on laser beam after traversing
a distance Z is
  k0 Z R
If the intensity gain is written in terms of its normalized spectrum,gh0(), where h is
normalized so that its peak value is 1.0, then this phase is
  gZ .h1 ()
and where
h1 ( ) 
P



0
 h0 ( ' )
d '
 '2 2
Thus on traversing a small distance Z the complex field increment is
E  E exp[ (1 / 2) gZ (h0  2ih1 ) ]
In laser glass, the gain (i.e. the imaginary part of the refractive index) may often be represented
adequately by a single species of ion with a gain cross-section spectrum given by the
experimentally observed gain cross-section. In this model, saturation of the gain follows a FrantzNodvik format with a single saturation fluence. Applying the Kramers-Kronig relations to the
single-ion model gives the phase-pulling associated with the gain. The (complex) field gain is
g E ( ) 
1
g 0 (n / n0 )( h0 ( )  i 2h1 ( ))
2
where g0 is the initial peak (intensity) gain, (n/n0) represents gain saturation, and the h’s are the
complex gain spectrum. For LSG8 the complex gain spectrum is plotted below.
Complex Gain
1.2
1
real
0.8
0.6
0.4
0.2
0
imaginary
-0.2
-0.4
-0.6
1.02
1.03
1.04
1.05
1.06
1.07
1.08
1.09
1.1
Wavelength
Figure 7: Real and Imaginary parts h0 and h1 of the complex gain for LHG8
1. See for example Panofsky and Phillips, Classical Electricity and Magnetism, p 416 (1962),
Addison-Wesley Publishing Company, NY