Understanding "Walking the Beam"
Simone Agha and Daniel Minkin
Stony Brook Laser Teaching Center, July 2007
Introduction
"Walking the beam" is the process of aligning a laser beam using two
adjustable mirrors in such a way that it will reach a specific point in
space with a specific angle. Since laser beams are straight lines, this
is equivalent to making the beam pass through two consecutive apertures.
This process is used to make two laser beams precisely collinear when
aligning an interferometer, for example, or when coupling light into
an optical fiber.
It can be a difficult and frustrating task to properly adjust the four
knobs that control the horizontal and vertical angles of the
mirrors. Without making the correct sequence of adjustments it is even
possible to move further away from the goal instead of towards it!
Fortunately there is a specific procedure that can be followed that is
sure to work eventually. The procedure involves considering the two
mirrors as having (partly) separate functions. The first mirror seen
by the beam is called the position mirror; it primarily affects
the position of the beam at the first aperture. The second mirror is
the angle mirror; it primarily affects the angle of the beam
at the first aperture.
The purpose of this project was to analyze mathematically the systematic
process of steering a beam through two pinholes. The equations derived to
describe the algorithm were graphed to provide a visual interpretation
of what it means to "walk the beam."
Setup
The setup used in this project consisted of a HeNe laser emitting
a beam that was reflected off mirror 1 to mirror 2. The beam passed
through two pinholes. The pinholes were irises that could be opened to
so that the misaligned beam was able to pass. The distances between
the components ranged from a few centimeters to a meter.
Line Diagram
To more easily represent the setup, it can be drawn as a straight
line. This is because the distances between the optics components are
not changed. The misaligned beam can be drawn at an angle from the
laser. When the beam is reflected off mirror 1, its trajectory is
altered by the addition of the angle of reflection. The angle is
called alpha 1. The same is true for the beam at mirror 2, and the
angle is called alpha 2. When the beam is misaligned, the distance the
beam strays from the center of the pinhole is d1. The distance that
the beam strays from the center of pinhole 2 is labelled d2.
In this setup, only the offset along the xaxis was measured and
calculated because the procedure is the same for the beam
along the yaxis. Since the location of the beam in the two axes is
independent of each other, it was only necessary to consider one
axis.
Mirrors
Using the line schematic, the purposes of a position and angle
mirror are clearly delineated. Mirror 1, the position mirror of this
setup, moved the beam a greater d1 (here represented as x1) than
mirror 2 for the same angle, alpha. Mirror 2, the angle mirror, moves
the beam a lesser distance than mirror 1 for the same alpha. However,
it moves the beam a greater distance from pinhole 2 (x2') than it does
from pinhole 1 (x2) for the same angle. Therefore, it is more
effective for adjusting the offset of the beam at the second
pinhole. While this is true for mirror 1 as well, it is more
pronounced in mirror 2 because mirror 2 cannot move the beam as far as
mirror 1 along pinhole 1 for the same angle.
In practice, this requires mirror 1 to be used to adjust the
location of the beam along pinhole 1 while mirror 2 is used to adjust
the beam's location along pinhole 2. In the first iteration, mirror 1
is adjusted so that d1 = 0. Therefore, d2 must not be 0 (or the beam
wiil already be aligned). In the "half" iteration, mirror 2 is adjusted
so that d2 = 0. D1 no longer is 0. In the second iteration, d1 is set
again to 0. D2 should now have decreased as compared to d2 in
iteration 1.
Hypothetical Plot
Following the method outlined above, it was originally thought
that the offset distances of the beam from iteration to iteration
could be plotted as a spiral. A hypothetical spiral plot was created
using the spreadsheet program Quattro Pro:
Plotted along the xaxis is d1 and along the yaxis is d2. Before
aligning, the offset from both pinholes is an arbitrary value. Mirror
1 is moved so that d1 = 0 and the first value is plotted along the
yaxis only (d2 does not equal 0). For the half iteration, d2 = 0, and
the second value is plotted along the xaxis only (d1 does not equal
0).
However, after performing the procedure, it was determined that
the spiral plot was wrong. For a spiral to be plotted from the data,
the offset distances must alternate on which side of the pinhole they
are found (positive and negative xaxis). This does not occur because
the beam is never moved beyond its "target" (i.e. the
pinhole). Instead, the graph appears a zigzig since the offset
distances always remain either in the positve or the negative xaxis
and yaxis; the plot is located in one quadrant only.
Iterations
To derive the equations, the initial coordinates of the
beam were (r,theta) where r and theta are the initial horizontal
displacement and displacement angle of the laser from the first
mirror, respectively. The beam was then reflected off mirror one,
translated to mirror 2, reflected off mirror 2, and translated through
pinholes 1 and 2. The final coordinates of the beam were r + L1theta +
(L2+L3)(theta + alpha1 + alpha2), theta + alpha1 + alpha2.*** This is
the distance, d2 from pinhole 2. The coordinates at d1 were r +
L1theta + (L2)(theta + alpha1 + alpha2), theta + alpha1 + alpha2.***
Using the values for L1, L2, L3, rinit., and thetainit., d1 and d2
were calculated. Then, d1 was set to 0 and alpha 1 and d2 were
calulated. D1 was the offset of the first iteration. For the half
iteration, d2 = 0 and alpha 2 was solved for. D1 was then able to be
calculated, the offset of the first half iteration. To calculate d2
for the second iteration, d1 was set again to 0 and the procedure was
repeated.
The equations were inserted into a Quattro Pro spreadsheet to
calculate the offset distances. When L1 = 0.6, L2 = 0.6, L=3,
rinit. = 0.1, and thetainit. =0.1, the convergence plot was:
Convergence
The rate of convergence was found to be related only to distances
L1 and L2 of the setup. When L1 is greater than L2, it serves as a
better postion mirror because the difference in the offsets of the beam
from pinhole 1 when mirror 1 and mirror 2 are set at the same angle is
alphaL1***. When L1 is nearly 0, the difference in the offsets is
nearly 0, and mirror 1 does not act as an effective position
beam. Conversely, when L2 is nearly 0, it acts as a better angle
mirror because it moves the beam farther from pinhole 2 than it does
from pinhole 1. The rate of convergence increases since moving mirror
2 to set d2 = 0 does not significantly move the beam across pinhole
1.
The rates of convergence were simulated in Quattro Pro using the
derived equations. The plot below exhibits a fast rate of
convergence. For the plot, L1 = 1, L2 = 0.2, L = 3, rinitial = 0.01,
and thetainitial = 0.01. L1 is greater than L2 by a factor of
five.
For the next plot, L1 = 0.2, L2 = 1. All other dimensions are kept
constant. L2 is greater than L1 by a factor of five. The rate of
convergence is slow.
Overshooting
With the procedure in this report, it is impossible to perfectly
align the beam. D1 and d2 will never equal 0 at the same time unless
one overshoots the "target" and moves the beam past the
pinhole. Graphically, this is explained by extending the line past the
d1 axis (past the second pinhole). The next line is drawn parallel to
the line that would have been drawn from the d1 axis to the d2
axis. The offset along the d2 axis is less for the overshoot plot than
it is for the original plot. Therefore, the rate of convergence is
faster when the beam passes the pinhole during alignment.
Conclusion
In this project, the equations for laser alignment through two pinholes were derived. The beam's convergence was analyzed and then related to the concept of a position and angle mirror. Using the results of this report, the rate of convergence can be maximized to align the beam in the least number of steps.
