A Simplified Shack Hartmann Sensor
to Demonstrate Wavefront Analysis
Shannon Hicks and John Noé
Laser Teaching Center
Department of Physics &
Stony Brook University
At the beginning of the semester, frankly, I had no idea what I wanted to
do for a project. Well, that's not true; more accurately, I had so many
things I was interested in that it was extremely hard for me to narrow my
options down to just one! However, I knew that I wanted to directly relate
my work here in the Laser Lab back to the field of astronomy. Dr.
Noé and I looked at and considered various ideas, including:
measuring the angular resolution of lenses, modeling several kinds of
telescopes, studying various aspects of solar astronomy and the sun, and
looking at the very nature of light and how it travels. Finally, we were
able to narrow the topic down to something involving a little bit of
everything - Adaptive Optics (AO) and in particular the Shack-Hartmann
wavefront sensor that is a key element of AO systems.
In recent decades the technique of adaptive optics (AO) has
revolutionized ground-based observational astronomy. Due to the
unavoidable turbulence in the atmosphere wavefronts from distant light
sources such as stars become distorted and fluctuate many times a
second. Thus when these objects are viewed through a telescope the
image becomes blurred and constantly shifts, an effect astronomers
call "seeing." Adaptive optical systems allow astronomers to
these wavefront distortions in real time using deformable mirrors
whose shape can be rapidly varied under computer control.1
These diagrams, taken from Claire Max's AO course at UC
Santa Cruz, illustrate these ideas. Click on the image to see the
If you're an even more visual person (like me!) then check out this awesome
animation of how AO works! Watching this in my astronomy class is
what got me interested in AO in the first place.
Here are a few pictures from Claire Max's powerpoint demonstrating
just how much an image can be changed using Adaptive Optics. The
first pair of images is of the planet Neptune. The one on the left
was taken by the Hubble Space Telescope (HST) with visible light.
The picture on the right also shows Neptune, but the image is much
clearer even though it was taken from a ground-based telescope at a
longer wavelength (infrared), due to Adaptive Optics.
The second pair shows the Starburst Galaxy taken by the CFHT
(Canadian-France-Hawaii Telescope) recorded with AO (left) and
without it (right).
Wavefront Sensing, Greatly Simplified
A wavefront sensor is an optical device to rapidly evaluate the shape
of the incoming distorted wavefronts. The commonly-used Shack-Hartmann
sensor consists of a 2-dimensional array of hundreds of tiny lenses
("lenslets") that typically occupies an area smaller than a postage
stamp.3 Each lenslet creates a separate focal spot on a CCD
mounted behind the lens array. If the incoming wavefront is distorted
(i.e. is not a plane wave) these spots of light shift in position by an
proportional to the distortion. Computer programs capture images from
the camera and determine the centroid shift of each spot and hence the
overall shape of the wavefront.3 This information is then used
a feedback signal that controls the shape of the deformable mirror.
The goal of our project was to simulate and demonstrate the operating
principles of a Shack-Hartmann wavefront sensor and to gain experience
with the associated data analysis. Instead of an array of many lenslets,
we used a single "lenslet" that can be shifted to different positions
using a translation stage. This single "lenslet" is not actually a tiny
lens but rather a normal camera lens preceded by a centered pin-hole
aperture. The lens is fixed to the CCD camera, and both move together.
Thus the displacement of the focal spot depends only on the deviation of
the incident wavefront from a plane wave.
This scanning wavefront sensor arrangement was used in a previous student
project to study the wavefront tilt of an optical vortex laser beam
5, and scanning has been proposed for commercial wavefront
.4 A further simplification in the current project was to limit
study to a spherical wavefront, and to scan this along just a single line
rather than across the full 2-D plane. Using a known wavefront allowed us
to compare our results with a specific prediction, and hence test the
accuracy of our procedures.
Setup and Procedures
The setup used materials already in the lab. The spherical wave was
created by placing a strong halogen light source behind an iris aperture
closed down to about 1.0 mm diameter. The "lenslet" was a 500 μm dia.
pinhole attached to the front of a normal 16 mm focal length camera lens.
The CCD camera was an Electrim-1000N, with 7.4 μm square pixels. The
camera software saves the array of 8-bit pixel values as .tif files on a
Windows 95 computer. The camera-lens assembly was mounted on a translation
stage and could be moved horizontally in steps as small as one-thousandth
of an inch (25.4 μm).
The translation stage was placed in such a way that the lenslet aperture
was 1.035 meters away from the light source aperture.
Here are two drawings I made of the setup and some pictures taken by Dr.
Noe. The first four pictures show the spherical-wave light source and the
second four show the CCD camera with the pinhole mounted to the lens and
the one-inch micrometer that moved the stage. Click on any of the images
to enlarge it.
The procedure consisted of recording a set of images as the wavefront
sensor was scanned in 2.54 mm increments across the 25.4 mm range of
the translation stage. The exposure was adjusted at each step to keep
the maximum pixel value around 200, safely below the saturation value
The goal of the data analysis was to find the centroid of each
recorded focal spot and its displacement from the reference spot that
would be created by a plane wave. The centroid calculations were
completed using a spreadsheet program (Microsoft Excel). The first step
was to convert images to arrays of pixel values. The centroid values were
then obtained by matching 2-D Gaussian surfaces to these pixel
We transferred the pictures into the Linux comptuer and used xv and
the "convert" command to cut all the pictures to the same size. This
allowed us to conclude the number of pixels we needed to keep in the
picture. After converting them, I imported them into Excel - which
gave me a table of numbers that represented pixel intensities. Here's
an example of one of the cropped images and what it looks like in
Excel. Each cell is equivalent to one pixel in the picture, and the
number inside the cell corresponds to the pixel's intensity. The
smaller numbers along the edges correspond to background light, which
were eventually canceled out.
Using the intensities and Excel, I was able to calculate the centroid
position, width, and intensity by fitting them to a sumsquare formula.
The sumsquare formula was based on a Gaussian curve fit. By adding and
changing different perameters in the equation, we were able to get the
data that corresponded to the least amount of error in the system.
In the formula x_c and y_x are the x- and y-direction centroids, while
w_x and w_y are the corresponding width parameters.
The next step was measuring the Wavefront (WF) Tilt and its relation
to the displacement of the centroid. Calculating Wafrefront tilt is
what allows the Shack Hartmann sensor to map out the shape of the
distorted wavefront coming into the actual telescope.
The WF tilt is essentially the angle of a
tangent to the wavefront with respect to a reference plane wave (i.e. a
vertical line in this drawing). The OPD is the horizontal distance between
the wavefront and the reference plane wave; it can be expressed as a
distance (in nanometers) or as some multiple of a wavelength of a particular
color of light. For a spherical wave the OPD is proportional to the square
of the tilt.
The wavefront tilt was calculated using a method involving similar triangles, as
shown in the diagram below. The distance the centroid moved, divided by the focal
length is Theta 2, or the wavefront tilt. We were able to calculate the
displacement of the centroids by using the gaussian formulas to find their exact
location on each picture. The pixel location was then converted to meters by a
factor of 7.4 micrometers per pixel. This gave us the observed wavefront tilt. We
then calculated the predicted WF tilt by noting that according to the law of
similar triangles, Theta 1 and Theta 2 should be equal. This was done simply by
dividing the track displacement by R (the distance fron the source to the
Table of the WF Tilt - Observed and Predicted:
Data Analysis and the Final Results!
The following table summarizes our results ...
After calculating all of the WF tilt values in Excel, I was able to
create a graph of them, related to the track displacement. It plots the
predicted WF tilt and superimposed are the values that were actually
calculated from the data we collected. It's quite close, which means that
the measurements we took from scanning across the wavefront were fairly
[Stuff Shannon will write goes here ...]
The next step for this project could be to create a distorted
wavefront. The data we've got here is for a nearly perfect spherical
wavefront ; the next set of data will be generated by placing some
sort of distorting object (such as a plastic sheet or an imperfect
lens) in front of the point source. This intentional distortion will
simulate the turbulence in "seeing" and the scanning wavefront sensor
will be able to measure the displacement of the centroid positions.In
other words - the spot will actually be moving around, and displaced
from where it is supposed to be on the CCD screen (like the first
diagram in the background section). From these data the wavefront tilt
and OPD can be calculated and the shape of the wavefront deduced.
"Adaptive Optics on the 200-inch Hale Telescope at the Palomar
Caltech Astronomy. 20 Apr. 2009
Center for Adaptive Optics, Univ. of California
at Santa Cruz, http://cfao.ucolick.org.
Claire Max, "Introduction to Adaptive Optics."
Course notes for Astronomy 289C at UC Santa Cruz. 20 April 2009 http://www.ucolick.org/%7Emax/289C/
Benjamin Wells, "Revising the Hartmann Test,"
Wells Research and Development. 2009. 20 Apr. 2009 http://www.wellsresearch.com/library/hartmann.php
Laser Lab Report