Gaussian Beam Properties
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The Gaussian beam is a radically symmetrical distribution
of coherent radiation whose intensity at radius r follows the following
equation:
I(r) = I0 * exp(-2 r2 /w
2)
The parameter
w
, usually called the Gaussian beam radius, is the radius at which the
intensity has decreased to 1/e2 or 0.136 of its axial, or
peak value I0. The total power, P(¥)
in watts, is related to the on-axis intensity, I0in watts/m2,
by:
P(¥) = I0
* (p w2/2)
The power contained within a radius r, P(r) can be obtained by
integrating the intensity distribution from 0 to r:
P(r) = P(¥)
* [ 1 - exp(-2 r2/ w2 ) ] |
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I (r =0.500w0) =
I0 * 0.607
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I (r =0.589w0) =
I0 * 0.500
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I (r =0.833w0) =
I0 * 0.250
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I (r =1.000w0) =
I0 * 0.136
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I (r =1.517w0) =
I0 * 0.010
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P(r =0.500w0)
=
P(¥) * 0.393
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P(r =0.589w0)
=
P(¥) * 0.500
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P(r =1.000w0)
=
P(¥) * 0.865
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P(r =2.000w0)
=
P(¥) * 0.9997
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A Gaussian beam remains Gaussian at every point along its
path of propagation; only its characteristic beam radius and the radius
of curvature of the wavefront changes. Propagation of Gaussian beam
through an optical system can be treated almost as simply as geometric
optics. |
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Imagine that we somehow created a Gaussian beam with a radius
w0 and a plane
wavefront at a position x = 0 (beam waist). The beam size and wavefront curvature will then
vary with x as the beam propagates.
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The beam size will increase, slowly at first, then faster,
eventually increasing proportionally to x. The wavefront curvature, which was infinite
at x = 0, will initially decrease with x. At some point it will reach a minimum value,
then increase with larger x, eventually becoming proportional to x.
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The propagation of a Gaussian beam is fully characterised
by its beam radius and wavefront radius of curvature R(x), as described by
the following two equations:
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w2(x) = w02 * { 1 +
[ (l x) / (p w02)
]2 } |
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R(x) = x * { 1 + [ (p w02)
/ (
l x) ]2 } |
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Where l is the wavelength
of the radiation of the Gaussian beam.
R(x) reaches its minimum value at the x location of
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xR = p w02 /
l
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This value of xR is also known as Rayleigh range.
At x = xR, the beam is (2)1/2 = 1.414 times larger than it is at the beam waist x = 0.
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At large distance (x
>>
xR) from the beam waist, the beam appears to
diverge as a spherical wave from a point source located at the center of the waist.
The diverging beam has a full angular width q
(defined by 1/e2 intensity points):
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q
» tan(q) = 4 l
/ (2 p
w0)
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A lens with a focal length of F will convert a parallel (low divergent)
Gaussian beam with a diameter D into a focusing beam with:
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q » D / F
2 w 0 = (4
l / p ) * (F / D)
2 xR = (8 l / p
) * (F / D)2 |
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2 xR is sometimes referred as Depth Of Focus
(DOF). |
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