Gaussian Beam Optics

Introduction
One usually thinks of a laser beam as a perfectly collimated beam of light rays with be beam
energy uniformly spread across the cross section of the beam. This is not an adequate picture for
discussing the propagation of a laser beam over any appreciable distance because diffraction
causes the light waves to spread transversely as they propagate, Fig. 1.
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Figure 1. Divergence of a Laser Beam
Additionally, the energy (irradiance) profile of a laser beam is typically not uniform. For the
most commonly used He-Ne lasers (operating in the TEM00 mode) the irradiance (the power
carried by the beam across a unit area perpendicular to the beam = W/m2) is given by a Gaussian
function:
( ) 2 2 / 2 2 2 / 2
0 2
I r I e r w 2P e r w ,
? w
= ? = ? (1)
where w is defined as the distance out from the center axis of the beam where the irradiance
drops to 1/e2of its value on axis. P is the total power in the beam. r is the transverse distance from
the central axis. w depends on the distance z the beam ahs propagated from the beam waist. w0 is
the beam radius at the waist. [The beam waist is defined as the point where the beam wave front
was last flat (as opposed to spherical at other locations).] For a hemispherical laser cavity such as
the one used for the He-Ne laser used in the lab, the waist is located roughly at the output mirror.
w0 is related to w by
Experimentally, one can use a CCD detector array to measure how the irradiance various across
the beam for several values of z>>zR. Then fit the data for each z using Eq. (1), which will yield
values for w(z). Then one can use Eqs. (3) and (4) to determine w0 and calculate zR.
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plane (i.e. the two lenses are separated by f0+fe). The rays reaching the eye are again parallel,
but appear to subtend a much larger angle than the original object. From Fig. 2 it is easy to see
that the angular magnification is
Figure 2. The Astronomical Telescope
Beam expander:
Because Gaussian beams do not follow the rules of ray optics, we cannot use the lens equation to
design a beam expander. However, as discuss in the Melles Griot Optics Guide, if you consider
the object to be the beam waist of the incoming beam and the image to be the beam waist after
the beam passes through the lens, then you can use a modified lens equation:
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Lets’ now apply this to an inverted astronomical telescope with the focal length of the first lens
being 5 cm and the second 40 cm. In the astronomical telescope the two lenses are separated by
the sum of the focal lengths of the two lenses – 45 cm in this case. We assume a red He-Ne laser
(633nm) with beam waist radius of 0.4 mm. We first use Eq. (3) to get zR=0.80 m. For the first
lens, f = 5 cm, and the beam waist for the laser is close to the exit of the laser. We put the lens as
close to the laser as possible and assume s=0. The using Eq. (5) we get
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