Analysis of the extended defects in 3C-SiC.pdf - Nelson Mandela ...
Analysis of the extended defects in 3C-SiC.pdf - Nelson Mandela ...
Analysis of the extended defects in 3C-SiC.pdf - Nelson Mandela ...
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44<br />
Substituted <strong>in</strong>to equation 4.18 and rearranged to give,<br />
2<br />
d T 2 2<br />
T 0<br />
(4.23)<br />
2<br />
dz<br />
with <br />
s<br />
2<br />
g<br />
1<br />
<br />
q q<br />
g<br />
g<br />
This is <strong>the</strong> differential equation for <strong>the</strong> harmonic oscillator with general solution,<br />
T ( z)<br />
Acos(<br />
z) B s<strong>in</strong>( z)<br />
(4.24)<br />
From <strong>the</strong> boundary condition at <strong>the</strong> entrance surface T(z = 0) = 1 <strong>the</strong> value <strong>of</strong> A is<br />
found to be 1. The value for B is found by comput<strong>in</strong>g dT/dz and substitut<strong>in</strong>g <strong>in</strong>to <strong>the</strong><br />
equation for S above and apply<strong>in</strong>g <strong>the</strong> boundary condition S(z = 0) = 0 which leads to<br />
B = -isg/σ and thus,<br />
is g<br />
T ( s,<br />
z)<br />
cos( z) s<strong>in</strong>( z)<br />
; (4.25)<br />
<br />
i<br />
S(<br />
s,<br />
z)<br />
s<strong>in</strong>( z)<br />
; (4.26)<br />
q <br />
g<br />
These are <strong>the</strong> general solutions to <strong>the</strong> dynamical two-beam equations <strong>in</strong>clud<strong>in</strong>g<br />
absorption. Note also that σ is complex.<br />
4.2.5 The Two-Beam Scatter<strong>in</strong>g Matrix<br />
In Section 4.2.4 <strong>the</strong> solutions to equations 4.17 and 4.18 was found by us<strong>in</strong>g <strong>the</strong><br />
follow<strong>in</strong>g <strong>in</strong>itial conditions:<br />
T ( s,<br />
z 0)<br />
1;<br />
S ( s,<br />
z 0)<br />
0 ; (4.27)