Page 1

PHYSICAL REVIEW B

VOLUME 44, NUMBER 15

15 OCTOBER 1991-I

Brief Reports

B«efReports «e accou«s ofcompleted

not warrant

regular articles

same publication

schedule asfor regular articles is followed, and page proofs are sent to authors

research

which, while meeting

the usual Physical Review standards ofscientific quahty,

do

A.Brief Report may be no longer than four printed pages and must be accompanied

by an abstract.

The

Kramers-Kronig

relations in optical data inversion

K.-E.Peiponen and E.M. Vartiainen

Department ofPhysics,

(Received 21 March 1991)

Vaisala Laboratory,

University ofJoensuu, P.O. Box 111,SF80100J-oensuu, Finland

Some remarks are made on the use of Kramers-Kronig

that symmetry

relations

imposed

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

I. INTRODUCTION

II. DISPERSION REI.ATIONS

u(x')=—Pf,dx,

1

~

'll

—oo X

where x'is a pole on the real axis, x is a real variable, and

P denotes

the Cauchy

principal

only the mathematical

properties

class of functions

that

fulfill

rather wide. There is a relatively

the asymptotical

behavior

~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.

restriction

on the variable:

of this restriction

on measurement,

tions of Eq. (1) are usually

known

as Kramers-Kronig

providing

more applicable

relations

metry

properties,

known

as crossing

linear optical constants.

The symmetry

from the fact that a real-valued

duce a real-valued

polarization of the charges.

If we give the symmetry

angular-frequency-dependent

=n(co)+ik(co), we can write

co)=n(co),

—

k (—

i.e.,the real refractive index n is even and the extinction

coeKcient k is an odd function of angular

Dispersion

tain indirectly,

a system with the aid of measured data. In optical phys-

ics, data inversion

is usually

which

can be derived

theory of complex-variable

contour integration.

sion relations

have found applications

to describe analytic signals,

the case of linearly-polarized-light

circularly-polarized-light

interactions.

method

was presented

describing

ties

with

several

complex-angular-frequency

basically using Hilbert-transform-type

An alternative

method for inverting

optical constants

has been introduced

method

is based on the use of conjugate

and the symmetry

properties of linear optical constants.

This method

has been found effective in the analysis of

the

frequency-dependent

refractive-index

color centers in mixed alkali-halide

A basically

similar

method to that of King has been

applied

in the study of the real and imaginary

nonlinear

susceptibilities

with

modulus

has been applicable

in the interpretation

man susceptibilities.

In addition to the normal data

inversion,

the dispersion

relations

derive sum rules for the linear optical constants'

nonlinear

In this paper we consider some basic mathematical

physical

requirements

that are important

with Kramers-Kronig

relations,

lished relations

in optical data inversion.

some theoretical

line-shape

description

of some properties of a system, may give er-

roneous results in connection

tions.

relations

via calculations,

are widely

used in physics to ob-

some of the properties of

based on the use of Hilbert

by means

transforms,'

of the

Disper-

optics

'

and also

in quantum

linear optical constants

interactions,

in

Recently,

susceptibili-

variables,

integrals.

the data of linear

by King.

Fourier series

a

nonlinear

'

His

change

of F

crystals.'

parts of

the aid of the squared

of the susceptibility."This kind of formalism

' '

of nonlinear Ra-

have been exploited to

and

susceptibilities. "''

and

in connection

are well estab-

%'e show that

appropriate

which

models,

in the

with Kramers-Kronig

rela-

The

Hilbert

transforms

which

for

a

to be an analytic

function

f,

f(z)=u(z)+iu(z),

function of a complex variable z=x+iy, are as follows

~

u(x)

—oo X

X

u(x)

is assumed

1

7T

u (x')=— —P,dx,

X

value.

of the function f,the

the Hilbert

transforms

weak assumption

of f: f(z)—~z~, 5)0, as

If we consider

is

about

properties

im-

relations of Eq. (1) are

frequency,

or wavelength

There is one physical

it must be positive.

the dispersion

given

in a diFerent

relations.

The basic idea of

is based on the sym-

relations,

properties

electric field must pro-

Because

rela-

form,

of the

follow

relations

refractive

for the complex-

index,

lt (co)

n(

co )=—k (co),

frequency

co,

8301

QC 1991The American

Physical Society

Page 2

8302

BRIEFREPORTS

0.5—

3

~0.0

3

-1.0

10

curve (A =1, coo=10, W=2)

true (solid line) and erroneous

15

20

FIG. 1. Gaussian

and corresponding

dispersion curves.

extinction

(dashed line)

respectively.

the change of the real refractive

form, we have

An(co')=PJ—

If we write the Kramers-Kronig

relation for

index, hn, in its usual

dco .

2

cok (co)

CO

6)

(3)

kG(co)=A exp

where

and 2(ln2)'

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

frequency,

We

the re-

form of Eq.

kG(co)

b,n (co')=—PI,dco .

It was observed

—oo

CO

CO

(4)

that when

we used Eq. (3) the ratio

However, this relation is valid only if k is an odd function

of co,'otherwise,

one has to use a relation

first relation of Eq. (1). If we know nothing

symmetry

of the data to be inverted,

decompose

the data, using some model, to be given as a

sum of even and odd parts. Such a decomposition

for any arbitrary

function.

In such a case we may write

the dispersion

relation

using a physically

tegration

interval

from zero to inanity.

however,

a problem,

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

about the

we may

try to

is valid

observable

There exists,

in-

the

behavior of its possi-

how the phys-

calculation.

The extinction

line, which falls o6'

coo/W was crucial

coo/W' is large, then the integral of Eq. (3) gives a good

approximation.

On the other hand, if the ratio has a rela-

tively low value, then Eq. (3) gives an erroneous result, as

demonstrated

in Fig. 1. It is worth noting that the zero

dispersion

is shifted from the peak position

Eq. (3) in the present

example.

coo/ W decreases.

One also has to take care when

coefficient

in data

inversion.

known

relation,

the absorption

a(co)=2cok(co)/c.

From this it follows that the absorp-

tion

coefficient

must

be an

a(—co)=a(co), in order to make use of Eq. (3). As an ex-

ample

we mention

that a widely

band shape in color-center

shape, is not consistent with the above symmetry

ment.

In many

practical

cases the tails of the extinction

coefficient or absorption

coefficient usually have to be ap-

proximated

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

when using

The shift increases

as

using the absorption

According

coefficient

to the

is given

well-

by

even

function

of co, i.e.,

accepted

the Gaussian

absorption

physics,

line

require-

data in order

We recom-

calculations.

the

for the calcu-

III. DISCUSSION

In this paper we have drawn attention to the fact that

one should be careful when choosing a particular

to

describe

optical

properties

Kramers-Kronig

relations.

taining erroneous

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

are consistent

with symmetry

cal constants.

Only in such a case may one expect a good

approximation

of the refractive index change or other op-

tical constant calculated using Kramers-Kronig

If the application

of sum rules gives strange results, the

broken symmetry

of the line model should be taken into

account.

We calculated

the dispersion

in cases where the extinction

In this case the dispersion

curve is rather

mated

and the location of zero dispersion

correct.

King's

model,

however,

much attention

as it may deserve. It is computationally

fast and reliable method for calculating

case of a single band.

Finally, we wish to emphasize

similar arguments

to those above hold also for the inter-

pretation

of nonlinear

susceptibilities

aid of Hilbert

transforms

of several

variables.

model

with

of ob-

byus-

in

connection

There is a possibility

coefficient.

data are approx-

limits so that they

imposed

relations

on opti-

relations.

using also King's model

has a Gaussian

line shape.

well approxi-

was always

has not attracted

as

a

dispersion

in the

that

obtained

angular-frequency

with the

Page 3

BRIEFREPORTS

8303

tp. M. Morse and H. Feshbach, Methods ofTheoretical Physics

(McGraw-Hill,

New York, 1953).

H. M.

Nussenzweig,

Causality

(Academic, New York, 1972).

3J.R. Klauder and E. C. G. Sudarshan,

turn Optics (Benjamin, New York, 1968).

~L.D. Landau and E. M. Lifshitz, Electrodynamics

ous Media (Addison-Wesley,

Reading, MA, 1960).

5D.Y. Smith, in Handbook ofOptical Constants ofSolids, edited

by E.D. Palik (Academic, Orlando, 1985).

D. Y. Smith, in Theoretical Aspects and New Deuelopments

Magneto-Optics,

1980)~

7K.-E.Peiponen, Phys. Rev. B35, 4116(1987).

F.W. King, J.Phys. C 10, 3199(1977).

F.W. King, J.Opt. Soc. Am. 68, 994 (1978).

P. Ketolainen, K.-E.Peiponen, and K. Karttunen,

B43, 4492 (1991).

and

Dispersion

Relations

Fundamentals

of Quan

ofContinu

in

edited by J.T. Deversec (Plenum, New York,

Phys. Rev.

'K.-E.Peiponen, Phys. Rev. B37, 6463 (1988).

~E. M. Vartiainen, K.-E.Peiponen, and T. Tsuboi, J. Opt. Soc.

Am. B7, 722 (1990).

~E. M. Vartiainen

and K.-E. Peiponen,

Solids, edited by O. Keller, Springer Series on Wave Phenom-

ena Vol. 9 (Springer-Verlag,

Berlin, 1990).

~4M. Altarelli, D. L. Dexter, H. M. Nussenzweig,

Smith, Phys. Rev. B6, 4502 (1972).

M. Altarelli and D. Y.Smith, Phys. Rev. B9, 1290(1974).

F.W. King, J.Math. Phys. 17, 1509 (1976).

F.W. King, J.Chem. Phys. 71,4726 (1979).

K.-E.Peiponen, Lett. Nuovo Cimento 44, 445 (1985).

K.-E.Peiponen, J.Phys. C 20, 2785 (1987).

D. Y. Smith, Phys. Rev. B 13, 5303 (1976).

K.-E.Peiponen, J.Phys. C 20, L285 (1987).

K.-E. Peiponen, E. M. Vartiainen,

A 41, 527 (1990).

D. L. Dexter, Phys. Rev. 111,119(1958).

in ¹nlinear

Optics in

and D. Y.

and T. Tsuboi, Phys. Rev.