PHYSICAL REVIEW B
VOLUME 44, NUMBER 15
15 OCTOBER 1991-I
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relations in optical data inversion
K.-E.Peiponen and E.M. Vartiainen
(Received 21 March 1991)
University ofJoensuu, P.O. Box 111,SF80100J-oensuu, Finland
Some remarks are made on the use of Kramers-Kronig
the tails of the absorption
and extinction curves.
relations in optical data inversion.
It is shown
on the optical constant should be taken into account when modeling
II. DISPERSION REI.ATIONS
where x'is a pole on the real axis, x is a real variable, and
only the mathematical
class of functions
rather wide. There is a relatively
~z~~ ao. Note that there are no symmetry
posed on f.
In linear optics the dispersion
given using the energy, angular
of the incident
light as a variable.
on the variable:
of this restriction
tions of Eq. (1) are usually
linear optical constants.
from the fact that a real-valued
duce a real-valued
polarization of the charges.
If we give the symmetry
=n(co)+ik(co), we can write
i.e.,the real refractive index n is even and the extinction
coeKcient k is an odd function of angular
a system with the aid of measured data. In optical phys-
ics, data inversion
can be derived
theory of complex-variable
have found applications
to describe analytic signals,
the case of linearly-polarized-light
basically using Hilbert-transform-type
method for inverting
has been introduced
is based on the use of conjugate
and the symmetry
properties of linear optical constants.
has been found effective in the analysis of
color centers in mixed alkali-halide
method to that of King has been
in the study of the real and imaginary
has been applicable
in the interpretation
In addition to the normal data
derive sum rules for the linear optical constants'
In this paper we consider some basic mathematical
that are important
in optical data inversion.
of some properties of a system, may give er-
roneous results in connection
used in physics to ob-
some of the properties of
based on the use of Hilbert
linear optical constants
the data of linear
the aid of the squared
of the susceptibility."This kind of formalism
of nonlinear Ra-
have been exploited to
are well estab-
%'e show that
to be an analytic
function of a complex variable z=x+iy, are as follows
u (x')=— —P,dx,
of the function f,the
of f: f(z)—~z~, 5)0, as
If we consider
relations of Eq. (1) are
There is one physical
it must be positive.
in a diFerent
The basic idea of
is based on the sym-
electric field must pro-
for the complex-
co )=—k (co),
QC 1991The American
curve (A =1, coo=10, W=2)
true (solid line) and erroneous
FIG. 1. Gaussian
the change of the real refractive
form, we have
If we write the Kramers-Kronig
index, hn, in its usual
observe that k (—co)A—kG(co). We calculated
fractive index change using the conventional
(3) and the exact form in this case as follows:
A is the amplitude,
W is the full width at half maximum.
coo is the central
form of Eq.
b,n (co')=—PI,dco .
It was observed
we used Eq. (3) the ratio
However, this relation is valid only if k is an odd function
one has to use a relation
first relation of Eq. (1). If we know nothing
of the data to be inverted,
the data, using some model, to be given as a
sum of even and odd parts. Such a decomposition
for any arbitrary
In such a case we may write
using a physically
from zero to inanity.
since we know by measurement
sum function but not the functional
ble even and odd parts.
As an example, in Fig. 1 we demonstrate
ical demand of k(co) being an odd function may affect the
result of a Kramers-Kronig
coefficient is chosen to be a Gaussian
rapidly at its tails. It can be given in the form
similar to the
behavior of its possi-
how the phys-
line, which falls o6'
coo/W was crucial
coo/W' is large, then the integral of Eq. (3) gives a good
On the other hand, if the ratio has a rela-
tively low value, then Eq. (3) gives an erroneous result, as
in Fig. 1. It is worth noting that the zero
is shifted from the peak position
Eq. (3) in the present
coo/ W decreases.
One also has to take care when
From this it follows that the absorp-
a(—co)=a(co), in order to make use of Eq. (3). As an ex-
that a widely
band shape in color-center
shape, is not consistent with the above symmetry
cases the tails of the extinction
coefficient or absorption
coefficient usually have to be ap-
somehow beyond the measured
to perform the Kramers-Kronig
mend that the symmetry
relations imposed on the optical
constant are taken into account when approximating
tails in order to get reliable approximations
lated optical constant.
to obtain correct
values of An .If
The shift increases
using the absorption
of co, i.e.,
data in order
for the calcu-
In this paper we have drawn attention to the fact that
one should be careful when choosing a particular
results, which was demonstrated
ing a Gaussian
line shape for the extinction
One has to make sure that the measured
imated in the low- and high-frequency
Only in such a case may one expect a good
of the refractive index change or other op-
tical constant calculated using Kramers-Kronig
If the application
of sum rules gives strange results, the
of the line model should be taken into
in cases where the extinction
In this case the dispersion
curve is rather
and the location of zero dispersion
as it may deserve. It is computationally
fast and reliable method for calculating
case of a single band.
Finally, we wish to emphasize
to those above hold also for the inter-
aid of Hilbert
There is a possibility
data are approx-
limits so that they
using also King's model
has a Gaussian
has not attracted
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