Laser Beam Coherence

1
Laser Beam Coherence

• Purpose To determine the frequency separation
  between
• the axial modes of a He-Ne Laser

Theory of Measurement
All sources of light, including lasers, contain a
distribution of different wavelengths. The
interference of this non-monochromatic light with
itself creates a pattern of fringes. The
contrast of these fringes is a function of
distance from the maxima of contrast, which are
equidistant from each other. The distance
between a maximum and the nearest minimum is
called the coherence length of the laser. The
frequency spread ?? is easily calculated from
this length.
2
Experimental Setup

• Setup
• Mount laser assembly (LA) to the far side of the
  optics table (OT). Adjust the position so that
  the beam is parallel to the edge and along the
  tapped holes in the OT.
• Mount a beam steering assembly (BSA) along the
  beam path at the next corner of the OT and insert
  a mirror mount. Adjust the height of the mirror
  mount until the beam intersects the center of the
  mirror. Rotate the post until the laser beam is
  reflected at a 90 angle.
• Place a second BSA in line with the laser beam at
  the opposite corner of the OT. Adjust the mirror
  mount until the laser beam is parallel to the
  surface of the OT and rotated 90.
• Insert a short focal length (25.4 mm) negative
  lens (LP3) into a lens chuck assembly and mount
  it five inches from the first BSA-I. Align the
  lens so that the diverging beam is centered on
  the mirror of the second BSA-I.
• Insert a longer focal length (200mm) lens (LP2)
  into an LCA and place is 225 mm from the first
  lens in the diverging beam. Again, center the
  beam on the second mirror.
• Rotate the second BSA such that the beam returns
  back through the two lenses just to either side
  of the laser output aperture.
• Carefully adjust the position of the last lens by
  moving it back and forth along the beam until the
  returning beam is the same size as the output
  beam.

• Mount a 50/50 beam splitter into a lens chuck
  assembly (LCA) and rotate the assembly 45 to the
  optical path.
• Mount a BSA with its mirror centered about the
  path of the reflected beam five inches from the
  beam splitter. Adjust the mirror until its beam
  is directed back to the laser.
• Mount a BSA on a stepper motor assembly with its
  mirror centered about the path of the transmitted
  beam. Adjust the mirror so that the beam is
  retro-reflected back to the laser.
• Mount an index card as an observation screen on
  the other side of the beam splitter.

3
Experimental Procedure

• Mount a camera as close to the observation screen
  as possible without obstructing the laser beam.
  Make sure the camera is as secure as possible.
  Ideally, the camera lens should be parallel to
  the observation screen.
• Adjust the mirror position (using the stepper
  motor) so that the path lengths are equal. Note
  the light has to pass through the glass to
  reflect off the beam splitter this additional
  path length is approximately times the
  thickness of the beam splitter. Record this
  position.
• At each position, adjust the fixed mirror so that
  there are about five fringes.
• Take pictures of the fringes with different
  shutter speeds. Be sure not to move the camera
  between pictures.
• Using the stepper motor, move the mirror away
  from the beam splitter in 1cm increments. At
  each point, take another set of pictures with the
  same camera settings.

4
Photographing the Fringes

• Before photographing the fringes, we determined a
  range of exposure times we wanted to use and
  found, by an iterative process, the f-stop (f
  11) that yielded the sharpest images across the
  range of exposure times.
• For each centimeter, a series of five photographs
  was taken, each with a different shutter speed.
  The f-stop was held constant at f11. Bracketing
  the exposures in such a way increased the dynamic
  range of the cameras CCD allowing us to better
  analyze the contrast of the fringes.
• The photographs were straightened and cropped
  using Photoshop.
• After straightening, the images were analyzed
  using ImageJ. For each image, a rectangular
  selection was made about the center of the image,
  as shown, and the profile was plotted (Analyze-gt
  Plot Profile).
• From this profile, the list of points was copied
  into Excel and plotted with the rest of the
  profiles from the series. The maximum and minimum
  intensities listed for each series was used to
  calculated the percent contrast of the fringes.

5
Data

• We exported intensity data from ImageJ to Excel
• Using the min and max of the center fringes, we
  calculated contrast for each plot

93 Contrast
Used this range to find min/max
73 Contrast
1/2500 s
1/2500 s
1/2500 s
6
Analysis I

• We calculated the contrast for each series of
  photographs then plotted the results as a
  function of position
• This variation in contrast is due to the
  coherence of the beam
• The distance from max to min in contrast is the
  coherence length
• We fit the plot with a Sin function
• From the period of the Sin we extracted the
  coherence length

7
Analysis II

• The coherence length gives us the frequency
  spread of the laser
• We can compare this to the dimensions of the
  laser to see how realistic our results are
• where L is the length of the laser cavity and ?L
  is the coherence length
• Doubling the coherence length gives an estimate
  of the laser cavity length
• The difference in estimated cavity lengths is
  likely due to an under-estimation of the internal
  components of the laser

Measured Coherence Length Measured Laser Length
?L (m) L (m)
0.0885 0.24
  Correction for Optics/ Electronics
  0.03
Estimated Cavity Length (From Coherence Measurement) Estimated Cavity Length (From Laser Dimensions)
0.177 0.21
Frequency Spread Frequency Spread
?? (Hz) ?? (Hz)
8.5E08 7.1E08
8
Error Discussion

• Range of motion
• Because of the range of the stepper motor we were
  unable to fully explore the maxima because they
  occurred near the ends of its track
• Vibrations in the room
• Oscillations of the walls cause visible fringe
  vibrations where time dependence was not expected
• Uncertainty in path length
• Uncertainty in distance between beam splitter and
  movable mirror is a few millimeters which gives
  about 1 error in frequency spread
• Time dependence
• Fringes fade in and out near minima position, and
  because we took 5 separate exposures, it is
  probable that the images were taken at different
  relative phases
• Incident camera angle
• We took the photographs at an angle relative to
  the index card, shrinking the image 5
  horizontally (cos(?inc)0.95)
• Image compression
• Images stored as JPEG files, which results in
  some compression. This should not be a large
  source of error, as the JPEG algorithm mostly
  removes higher frequency brightness variations,
  which are likely noise.