Gaussian Beam Optics

Introduction
In this experiment, you will observe the Gaussian properties of laser beams. The measurements will
be made by partially blocking a beam with a smooth, straight edge (a razor blade) and measuring the
fraction of light that is not blocked relative to the position of the blade. This fraction will range from
1.0 when the blade is outside the beam to zero when it’s completely blocking the beam. How far
you’ll have to move the blade depends on the size of the beam, and you’ll measure beams that are as
large as 1 cm and as small as 0.01 cm (perhaps even less). Thus the blade movement has to be very
precise and controllable.
2 Theory
The strength of the electric field of the simplest mode of a laser is radially symmetric and given by
the Gaussian profile E(r) = E0e?(r/w)2 where r is the radial distance from the beam axis and w is the
characteristic beam radius. Note that there is no sharp boundary of the beam. Since the intensity I
is proportional to E2, the radial dependence of the intensity is given by I(r) = I0e?2(r/w)2 . Such a
Gaussian beam profile is illustrated in Fig. 1, where the distance from the beam axis is measured in
units of the beam radius.
x / beam radius -2 -1.5 -1 -0.5 0 0.5 1 1.5 2
y / beam radius
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
intensity / intensity at x=y=0
0
0.2
0.4
0.6
0.8
1
r / beam radius
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
intensity / intensity at r=0
0
0.2
0.4
0.6
0.8
1
r = w
I = 0.135 I
0
Figure 1: Gaussian beam profile in the x-y plane perpendicular to the beam axis (left side) and a
projection of the profile on the radial distance r = ?x2 + y2 from the beam axis (right side). The
beam radius depends on the position along the beam axis if it is focussed or diverging.
As the laser beam propagates its diameter may change: the beam may either converge or diverge.
The minimal size spot, where the beam has a radius w0, is called the beam waist (see Fig. 2). In
general, the beam radius is w(z) = w0q1 + (z/z0)2 where z0 ? w2
0/ and is called the Rayleigh
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PHY 300 Lab 4, Fall 2008 Nov. 13, 2008
length. In the far field, i.e. at a distance z ? z0 from the minimum at z = 0, the beam radius is
given by w(z) = z/(w0). A measurement of the beam radius as function of z therefore allows
the determination of w0 for known wavelength or, if the waist can be measured, determination of the
wavelength.
2w0
Figure 2: Schematic illustration of the
profile of a Gaussian beam near a focus.
3 Setup
Measuring the transverse shape of the beam from the Helium-Neon laser (HeNe) is too difficult to
start with - you should do that last. To start, set up your beam line with three lenses as shown in
Fig. 3. The purpose of the first lens is to diverge the beam so that the second lens can form a larger
beam that’s easier to measure. Thus it could even be a diverging lens, but its shown as converging
in Fig. 3. The second lens focusses the beam to a small spot, and you’ll measure the beam’s profile
at several places along its length, starting from near this lens. The third lens is simply an auxiliary,
because the laser beam itself may be too large to fit into the ?7 mm hole in the photometer where the
light sensor is. So this lens should be arranged to refocus the beam onto the detector’s light sensor.
You should arrange it so the beam is not focussed to a tiny spot there, but a disk a few mm in diameter.
Once you have focussed the light onto the detector you may find that the intensity of the laser
light is larger than the range of your photometer allows for. But you know from a previous lab about
a simple way to reduce the light intensity. Using the two polarizers from your optics kit you can first
polarize the laser light linearly (right after the beam leaves the laser), and then adjust the polarization
axis of the second polarizer, mounted just behind the first one, such that the laser light intensity
passing the polarizer pair is just below the maximum range of the photometer. Use of this scheme
may result in periodic drift if the laser has not had a long enough warmup time. Be sure to watch for
such behavior.
You need to measure the intensity profile of this beam near the focus to determine the waist, and
far from the focus to determine the divergence angle. You should therefore make about 10 or more
cuts across the beam at various distances from the focusing lens (see Fig. 3), and each cut should
comprise at least 20 points, so there is a lot of data to take. Moreover, after you finish, you will need
to either move the lenses to change something, or swap the lenses, and then repeat the measurements.
The heart of the experiment is the blade motion. To accomplish this we have disassembled the
Michelson interferometer bases, turned them over, and will use their precise micrometer motion control
to move the blade. First you need to mount the blade on the aluminum arm with the tiny magnets
as shown in Fig. 4 (use two above to hold the blade and one below to anchor the whole assembly). Be
exquisitely careful with these magnets - they’re easily lost and not easily replaced.
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PHY 300 Lab 4, Fall 2008 Nov. 13, 2008
HeNe laser
Diverging lens
Beam-forming lens
Focussing lens
Detector
Polarizers
Blade
Figure 3: This shows the overall optical setup. The HeNe should be mounted on the table, independent
of the other components. The first two lenses should be on the optical bench, near the end far from
the HeNe. The invertedMichelson base and blade should be separately sitting on the table beyond the
end of the bench. The focusing lens and detector should be independently mounted beyond the end
of