Subject: Summary: Absorption Cross Section

From: <gustavo~at~mercury.chem.pitt.edu>
Date: Thu, 23 Oct 2003 10:20:37 -0400 (EDT)

Dear CCL readers,
     here you have the summary to my question about absorption
cross sections. I thank Prof. Egbert Zojer for his answer.
Basically, he too is looking for a reference describing the
derivation of the equation that relates the two-photon absorption
cross section to Im[gamma].
     sincerely yours,
                         Gustavo L.C. Moura
                     gustavo~at~mercury.chem.pitt.edu

Original question:

Dear CCL readers,
      I have read several papers about the calculation of the cross
sections for two-photon absorption (d). The value of d as a function of
the frequency of the incident photon (w) is related to the
imaginary part of the second hyperpolarizability gamma(-w;w,w,-w)
through

        d=(8*pi^2*hbar*w^2)/(n*c)^2*Im[gamma(-w;w,w,-w)]

where n is a refractive index and c is the speed of light.
      None of the papers I have read so far say where the equation
above came from. So my question is: Does anybody on this list know the
origin of the equation above? Reference? I am also curious to know if
there is a similar equation relating the imaginary part of the
polarizability alpha(-w;w) to a cross section for one-photon
absorption. Is this cross section simply the oscillator strength
for the one photon transition?
      Thank you very much in advance. I will summarize if I receive
any answer.
                                  Gustavo Moura
                            gustavo~at~mercury.chem.pitt.edu

Answer:

Dear Gustavo,

You can, e.g., get this equation by solving the wave equations for a
damped  non-linear medium. I am pretty sure that it has to be somewhere
in the  literature, but I also could not find it, so I did the
derivation myself  starting from the un-damped equations as you can find
them e.g. in Boyd  "Nonlinear Optics".

The linear absorption coefficient is proportional to Energy times
Im(alpha). This you should, however, find in most textbooks about
electrodynamics and you can simply get it from the linear damped wave
equations - just include a complex k-vector in your plane wave, put it
into  the wave equations and you will see that the amplitude decreases.

regards,

egbert

P.S.: If anyone can point you to a derivation of delta prop. Im(gamma)
in  the literature, would be very interested to know it.

At 10:28 PM 10/19/2003 -0400, you wrote:
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A.o. Univ.-Prof. Dr. Egbert Zojer
Tel.:++43/316/873-8475 Advanced Materials Division
      Fax.:++43/316/873-8478 Institute of Solid State Physics
Graz University of Technology                  e-mail.:
egbert.zojer~at~tugraz.at Petersgasse 16, 8010 Graz, AUSTRIA
http://www.cis.TUGraz.at/if/egbert.htm
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currently at the
School of Chemistry and Biochemistry
Georgia Institute of Technology
Atlanta, GA 30332-0400
Tel.: ++1/404/385-4984
FAX.: ++1/404/894-7452
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