Appendix H
Intensity and Power of Gaussian
Beams
H.1
Peak Intensity
In order to determine the peak intensity of a beam from the total power in that
beam, consider a elliptical Gaussian beam with an intensity profile given by,
2
2
I = I0 e−(x/a) e−(y/b) ,
(H.1)
where, a is the long axis and b is the short axis.
Make the substitutions
x
a
du
du =
a
u=
=⇒
y
,
b
dv
d=
.
b
:
v=
:
(H.2)
(H.3)
The total power in the beam, P , is given by,
P =
!∞ !∞
I dx dy ,
=
!∞ !∞
I0 a b e−(u
0
0
(H.4)
0
2 +v 2
) du dv .
(H.5)
0
In order to integrate equation H.5 , make the substitution r2 = u2 + v 2 and
switch to circular co-ordinates.
185
Appendix H. Intensity and Power of Gaussian Beams
186
Thus,
!2π !∞
P = I0 a b
2
e−r r dr dθ ,
0
"0
e−r
= 2πI0 a b −
2
2
#∞
.
(H.6)
(H.7)
0
Hence the peak intensity is related to the power within the beam by,
I0 =
H.2
P
.
πab
(H.8)
Mean Intensity
In some circumstances it will be useful to consider a mean intensity, I, as
opposed to the peak intensity, I0 . As Gaussian beams have an infinite extent
obviously some limit on the extent of the beam will have to be considered.
Suppose that the mean is calculated such that a fraction Z of the total power
is taken into account, this will give an approximation to the mean, I, of IZ .
Hence,
PZ
.
π xZ y Z
IZ =
(H.9)
For the purposes of this calculation consider a circularly symmetric beam such
that, a = b = r0 , and xZ = yZ = rZ .
Then,
PZ = I0
!2π !rZ
0
2
e−(r/r0 ) r dr dθ ,
(H.10)
0
where rZ is the radius encompassing the power PZ .
It follows that,
PZ = −πr02 I0
= −πr02 I0
$
$
e−(r/r0 )
2
%r Z
e−(rZ /r0 )
,
(H.11)
%
−1 .
(H.12)
0
2
Appendix H. Intensity and Power of Gaussian Beams
187
But PZ = Z P ,
∴
=⇒
PZ = πr02 I0 · Z ,
(H.13)
2
Z = 1 − e−(rZ /r0 ) .
(H.14)
Thus it follows that the beam radius encompassing a fraction, Z, of the total
beam power is given by,
rZ2
=
r02
&
&
& 1 &
&
& .
ln &
1−Z&
(H.15)
Now substituting equation H.15 back into equation H.9 , allows the mean intensity to be determined in terms of either the peak intensity or the total power.
=⇒
IZ =
∴
IZ =
πr02
ZP
& 1 & ,
&
ln & 1−Z
I0
& .
1 &
Z
&
ln & 1−Z
(H.16)
(H.17)
Thus the mean intensity depends only on the peak intensity and the fraction of
total power in the beam over which the mean is to be calculated.