```Lab #4 - Gaussian Beams
ECE/PH 482/582
Goals:
1. Verify that the transverse beam intensity for a laser operating in the fundamental or
TEMoo mode follows a Gaussian profile.
2. Calculate and experimentally verify the beam waist and divergence angle for a typical
HeNe laser using a commercial beam profile instrument, the Photon, Inc. Beam Scan.
3. Build a beam-expanding telescope, expand the HeNe beam to a larger diameter, and
measure its expansion.
4. Verify the focusing of a Gaussian beam and how it varies with incident beam waist.
5. Examine and compare the beam shapes of the HeNe laser and a collimated visible
diode laser and understand why they are different.
Kuhn, Laser Engineering, Chapter 4: sections 4.3 – 4.4
Other Resources:
Two classic papers on the subject – both are posted on the class web site under “Links”.
Kogelnik and Li, Laser Beams and Resonators, Proc. IEEE, vol. 54, no.10, pp. 13121329, Oct.1966
Kogelnik, On the Propagation of Gaussian Beams of Light Through Lenslike Media
Including those with a Loss or Gain Variation, Applied Optics, Vol. 4, no. 12, pp. 15621569, Dec.1965
Yariv, Quantum Electronics, 3rd Edition, (Wiley and Sons), Section 6.6
Preparatory Questions:
Consider a HeNe laser (λo = 632.8 nm) with a beam waist wo = 0.5 mm:
1. Assume the beam waist is located at the output mirror of the laser. Calculate the size of
the beam waist w(z) after the beam travels across the laboratory. (Assume the lab is ~5
meters long.)
2. Now assume you focus the beam down into a spot 0.1 mm in diameter. Calculate the
size of the beam waist after the new Gaussian beam travels across the laboratory.
(Assume the lab is ~5 meters long.)
3. Now, assume you build a small Keplerian telescope and expand the beam to 5 mm in
diameter. Calculate the size of the beam waist after this new Gaussian beam travels
across the laboratory. (Assume the lab is ~5 meters long.)
4. In the lab, you will use a short focal length and a longer focal length (f.l.) lens to make
a Keplerian telescope to expand the beam diameter by 5x. Sketch such a telescope
design and specify the longer f.l. lens focal length required if the short f.l. is 1 inch.
5. The equation for the radial variation of intensity for a TEMoo Gaussian beam is given
below in Eqn. (1). Rewrite this equation so that the x-axis is “r2” and it gives a linear
relationship in r2 from r2 = 0 to ∞. [Note: This would “prove” that the Gaussian
relationship holds.]
Laboratory Experiment:
The transverse intensity profile of the TEMoo mode is a Gaussian distribution and is given by
I (r )  I 0e
2 r 2
 ( z )2
(1)
where Io is the peak beam intensity at r = 0, r is the transverse beam radius, n is the refractive
index of the medium (usually air so n = 1), and w(z) is the spot size at a distance z. The spot
size represents the beam radius at which the intensity has decreased to 1/e-2 times its peak
value. The peak value I 0 occurs at r = 0. Note that as z approaches infinity, w(z) varies as z,
and hence I ( r ) varies as 1/z2. In the far field then, the beam acts as an expanding spherical
wave with a finite angular extent.
There is another parameter appearing in the above equations, which is of particular interest.
The beam waist at a distance z is given by:
2
w( z )  w0
 z 
1 
 z 
 R
(2)
Where zR is the “Rayleigh range” or “confocal parameter” and
zR 
nw0 2
0
(3)
The minimum beam radius is wo. It is a function of the cavity length of the laser d, the radii of
curvature of the mirrors R1 and R2, and the wavelength of the light λo.
(
0 2 n d ( R1  d )( R2  d )( R1  R2  d )
)
0
( R1  R2  2d )2
(4)
The last major beam property we are interested in is the far field divergence angle. This is
given by
1 / 2 
0
n 0
(5)
where 1/2 is the half angle shown in the Figure 4.3 in the text (Kuhn) as θa. Using the “1/2”
subscript reminds us that it is only the half-angle of divergence and not the full angle. The
divergence angle can be used to estimate the spot size of a focused beam.
Procedure:
1. Make measurements of the HeNe transverse beam intensity at a fixed value of z of ~ 1
m. This is done by carefully aligning the center of the HeNe laser beam reflecting off
the rotating mirror to pass directly across the pinhole in front of the Si PIN detector.
ω (rev/s)
HeNe Laser
r
PIN Detector
Pinhole
The scan speed across the pinhole is 2(2π r ω).
[Why the “2” in front? Derive this.]
2. Plot the oscilloscope trace and use the oscilloscope cursors to measure x at several
points on the trace. These measurements will be used to calculate the radius r of the
beam. Measure and record your distance of z.
3. Now shine the laser beam against a far wall and calculate the far-field divergence angle.
[Note: Measure the beam "diameter" of the visible spot you can easily see. This is a
good approximation of 2w.] Use Eqn. (2) in the far-field limit to estimate the
minimum beam waist of the laser wo. From wo calculate zR. Were you really
measuring in the “far-field,” ie. z >> zR ?
4. Use the Photon Inc. "Beam Scan" to record the transverse profile and beam diameter of
the HeNe laser with z equal to the distance chosen for the part1 measurement. Look at
the “signal out” trace from the Beam Scan on the oscilloscope, measure the full width
at the 1/e2 power points, and verify that this width agrees with the meter reading of the
13.5% diameter. [The Beam Scan scans a slit across the beam at a linear rate of
______ cm/s based on the diameter of the slit cylinder (1.622” for 10 μm slit; 1.642”
for 1 μm slit) and its rate of rotation.]
5. Rotate the Beam Scan head by 90o to measure the HeNe beam parameters in the
perpendicular direction. Are they they same? Is the beam round?
6. Once you have verified that the Beam Scan is giving the correct value of 2w use
mirrors to measure the HeNe laser beam diameter for 5 different values of z ranging
from close to the laser to across the room. If the beam gets too large for the Beam
Scan (~7 mm diameter), just measure with a ruler. Using these values, you will plot
w(z) later.
7. Now place the 10 cm focal length lens in front of the laser to transform to a new
Gaussian beam and re-measure the beam diameter as a function of z for the focused
beam. What is the minimum spot size you measure at the focus of the lens? What is
the new divergence angle of the beam? Which slit (Aperture #1 or #2) did you use on
the Beam Scan? Why?
8. Now use the two lenses – one short f.l. and one longer f.l. – to construct a beam
expanding telescope to expand the HeNe laser beam to ~5 mm diameter leaving the
longer f.l. lens … and collimated. Measure the far-field (across the lab) diameter of
the beam.
9. Finally use the Beam Scan to measure the “collimated” beam dimensions of the diode
laser module in the lab. Is this beam round like the HeNe laser? What are the beam
dimensions you measure in the two perpendicular directions? Measure the far-field
divergence angles in both axial directions.
1. Make a plot of the ln(I/Io) vs. r2 for the HeNe laser beam a distance of 1 m from the
laser which you measured using the rotating mirror.
2. The PIN detector outputs a signal voltage. Is this signal voltage proportional to the
optical beam’s electric field (V/cm) or to its intensity (W/cm2)? Explain your answer
clearly. [This is a critical point for all these measurements.]
3. Using your plot in part 1, verify that the beam profile is indeed Gaussian and determine
the “spot size” w(z). For the remaining values of z, you can just read off the 1/e2 power
point from the Beam Scan once you have used a scope to check this value from the
Beam Scan unit.
4. Knowing w(z), calculate the minimum beam waist wo and beam divergence angle θfull
for the laser without a lens. Repeat for the measurements taken using the 10 cm f.l. lens
and for the expanded beam. Sketch all three cases showing where the minimum beam
waist is located.
5. Why must the pinhole used on the detector be much smaller than the beam diameter?
6. Describe the beam from the collimated diode laser giving its waist dimension(s) and
divergence angles.
7. Comment on anything you observed that didn't quite fit the "conventional theory".
Conclusion:
After this lab you should have a good understanding of how Gaussian beams behave and
how to calculate and measure important beam parameters. A solid knowledge of Gaussian
beams is very important when working with lasers and optical systems that use lasers, such
as fiber optic communications, free space communications, LIDAR, spectroscopy, etc.
Many times a laser must be focused to do machining or to be input into an optical fiber or
nonlinear crystal, so it is important to be able to calculate what the spot size and the
Rayleigh range are for the focused beam.
Revised Oct 2010; T.K. Plant
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