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Summary

This image presents the dispersion relation of phonon polaritons according to a simple theory[1], giving the two following dispersion relations for the coupled optical phonon/electromagnetic field system :

  1. longitudinal optical phonons (LO phonons)
  2. polaritons (transverse em field + TO phonons)

The dashed red curves correspond to the uncoupled modes (photons and phonons: Ω = 0), and the black curves represent solutions of the above equations with parameters chosen to fit the dispersion relation of polaritons in GaP[2], giving

Uncoupled optical phonon frequency
ω0 = 45.4 meV
Coupling parameter
Ω = 61 meV
Electric susceptibility
χ = 7.5

Note that k is very small compared to π/a.

  1. Max Born, Kun Huang, Dynamical Theory of Crystal Lattices, Oxford University Press, 1954 (ISBN 9780198503699), « Chapter 8 : Infrared Dispersion and the Retardation Effect on Lattice Vibrations »
  2. C. H. Henry and J. J. Hopfield, Raman Scattering by Polaritons, Phys. Rev. Lett. 15, 964 - 966 (1965).

(further minor editing and conversion to svg done with inkscape)

Summary

Description
English: Dispersion relation of phonon polaritons with parameters corresponding to GaP.
Français : Relation de dispersion des polaritons avec des paramètres correspondant au GaP.
Date
Source Own work
Author Mathieu Perrin
SVG development
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The SVG code is valid.
 
This diagram was created with MATLAB.
Source code
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MATLAB code

e=1.602e-19;
hbar = 1.05457168e-34;
c=2.9978e8;
ang=1e-10;
omega0=45.4e-3*e/hbar; %conversion meV->rd/s
Om=61e-3*e/hbar;
chi=7.5;

k=linspace(0,0.0004/ang)';
y1=omega0*ones(size(k)); %uncoupled phonon dispersion
y2=c*k/sqrt(1+chi);      %uncoupled photon dispersion
y3=sqrt(y1.^2+Om^2/(1+chi));%longitudinal phonons
%solve for omega in
%k^2*c^2=omega^2*(1+chi+Om^2/(omega0^2-omega^2))
a1 = (1+chi)*ones(size(k));
a2 = -((1+chi)*omega0^2+k.^2*c^2+Om^2);
a3 = k.^2*c^2*omega0^2;
D = sqrt(a2.^2-4*a1.*a3);
y4=sqrt((-a2+D)./(2*a1));%upper polaritons
y5=sqrt((-a2-D)./(2*a1));%lower polaritons

plot(k*ang, [y1 y2]*1000*hbar/e,'r--','Linewidth',1.8);
hold on
plot(k*ang, [y3 y4 y5]*1000*hbar/e,'k','Linewidth',1.8);
hold off
axis([0 0.0004 35 60]);
set(gca,'FontSize',18,'TickLength',[.02 .02],'LineWidth',1.5);
xlabel('k (A^{-1})');    %A to be replaced
ylabel('h\omega (meV)'); %h to be replaced

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15 November 2008

14,738 byte

450 pixel

436 pixel

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b41136dd99b0036511511b85f8c3b3b850a59a02

File history

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Date/TimeThumbnailDimensionsUserComment
current11:20, 13 February 2010Thumbnail for version as of 11:20, 13 February 2010436 × 450 (14 KB)Cepheidenfiex rendering error of one exponent
11:15, 13 February 2010Thumbnail for version as of 11:15, 13 February 2010436 × 450 (15 KB)Cepheidenfixed font scaling
23:19, 15 November 2008Thumbnail for version as of 23:19, 15 November 2008450 × 436 (21 KB)Mathieu Perrin
22:55, 15 November 2008Thumbnail for version as of 22:55, 15 November 2008450 × 436 (21 KB)Mathieu Perrin
22:53, 15 November 2008Thumbnail for version as of 22:53, 15 November 2008450 × 436 (21 KB)Mathieu Perrin
22:42, 15 November 2008Thumbnail for version as of 22:42, 15 November 2008450 × 436 (22 KB)Mathieu Perrin
22:33, 15 November 2008Thumbnail for version as of 22:33, 15 November 2008450 × 436 (20 KB)Mathieu Perrin{{Information |Description={{en|1=Dispersion relation of phonon polaritons with parameters corresponding to GaP.}} {{fr|1=Relation de dispersion des polaritons avec des paramètres correspondant au GaP.}} |Source=travail personnel (own work) |Author=[[Use

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