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The Shack-Hartmann Sensor

The Shack-Hartmann sensor is a device used to assess distortions or aberrations in an optics system. Light coming into the sensor travels through an array of lenslets, which each project an image of the source onto a detector. If the wavefront coming in is not parallel, then the images created on the detector will not be directly on their ideal center. (The ideal center is where the center of the image would be if the light was coming in parallel.) The displacement of the center of the actual image from its ideal center is calculated. A computer program then mathematically determines the error values. This information can then be analyzed to determine what distortions are present and how to correct them. There are two parts to the Shack-Hartmann sensor, image formation and analysis.¹

Figure 1: The Shack-Hartmann Sensor

http:///www.polytec-pi.fr/wavefrontsciences/ophthalmic/hartmannsensor.html, 7/30/01

Image Formation

In the Shack-Hartmann sensor, incoming light enters an array of lenslets. Each lenslet then projects an image of the object onto a detector, at the focal point of the lenslets. Usually this detector is a CCD camera. If the incoming light is not parallel, this affects the position of the image on the detector and it will move the image in the x-axis, the y-axis, or both. The movement of each image is unique to the shape of the section of the wavefront going through its lenslet. The detector is able to locate the center of each image and find its deviation from its idea center. The cross-correlation between the two centers is then computed..¹

Figure 2: A Wavefront Entering a Shack-Hartmann Detector.

http://www.aoainc.com/technologies/adaptiveandmicrooptics/aostutorial.html, 6/25/01

(Adaptive Optics Associates, Inc.)

Analysis

By using the X and Y deviations from the ideal center and the focal length of the lenses, the wavefront slopes for each image are calculated. These slopes are the partial derivatives of the wavefront and are integrated to find the original wavefront and in doing so, the error of each image. In addition to finding the error values, the information found in the wavefront analysis is often fitted to a Zernike polynomial; Zernike polynomials have been used to describe aberrations in optical systems since 1934. Zernike polynomials are useful because they break the wavefront into different components that represent specific aberrations. Each of these individual components can then be analyzed independently. Once the wavefront information is analyzed, the information is used to determine any problems in the optical system.²

Figure 3: Analysis of the Images in a Shack-Hartmann Detector.

http://www.sunspot.noao.edu/AOWEB/correlatingshack.htm, 8/1/01

History of the Shack-Hartmann Sensor

The principles used in the Shack-Hartmann sensor are a result of the original Hartmann test, which took place in 1900. Hartmann used this method to test telescope optics. In the Hartmann test a radial screen with holes in a specific pattern is put in front of the object being tested. The positions of the holes in the screen are mapped onto a detector (like a CCD camera). Then, the positions of these spots are mapped back to the object and the differences between the holes and the spots are computed.³

Physicist and author I. Ghozeil concisely describes the theory behind the Hartmann test as "The premise is that a portion of a wavefront, when tilted relative to the ideal wavefront in that region, causes light to come to a focus at a place other than the intended focus, or to intercept some plane at a location other than the one that would be obtained with light coming from the ideal wavefront and from that region." He goes on to explain that "The converse can be used to determine the tilt error in a portion of a wavefront, by determining where the light from that region intercepts some plane and what difference there is between that intersection and the one expected from the perfect wavefront of interest."⁴

Screen tests were commonly used to test optical equipment because they could be easily analyzed on a polar coordinate system and they are able to detect the most common flaws in mirrors, such as zonal errors and hills and valleys resulting from the polishing and grinding of the mirrors. Then, when the sample points are related to one another, the wavefronts can be reconstructed. Hartmann was the first to develop this kind of test. Since then, other screen patterns have been used, the most effective was a square pattern. The biggest problem with the radial pattern is that the area sampled by the holes farthest from the center of the circle is larger than the area sampled by the holes near the center of the screen. On a screen test, each sample point is assumed to give the average tilt for that area on the surface, and if the surfaces are not equal sizes this is a large source of error.⁵

Figure 4: The Classical Hartmann Radial Pattern.⁵

Ben Platt and Roland Shack expanded on Hartmann's test in 1970 while they were working on a classified laser project for the U. S. Air Force trying to obtain clear photographs of Soviet satelites.⁸ Platt and Shack replaced Hartmann's screen with an array of lenses. The lenses increase the sensitivity of the test and according to Spot-Optics "The Shack-Hartmann system is easier to calibrate and is of much higher precision compared to the Hartmann method."³ Currently, the Hartmann test has been abandoned in favor of the Shack-Hartmann test.

Figure 5: The Difference Between the Hartmann and the Shack-Hartmann Test

www.spot-optics.com/Hartmann_SH.htm, 8/1/01

Adaptive optics had other military applications as well. Soon after Shack and Platt started their work, the Strategic Defense Initiative (Stars Wars program) began to study adaptive optics "as a way to compensate for atmospheric distortions when focusing a ground-based laser weapon at an incoming missile." ⁸

In the 1980s, adaptive optics was introduced to astronomy as a means to remove the "twinkling" effect on the stars. While the Hartmann test had previously been used to test optical equipment, the Shack-Hartmann sensor became an integral part of the technology that made it possible to obtain much clearer and higher resolution pictures of the universe.

In 1994, Liang introduced the Shack-Hartmann sensor as a way of testing the eye for aberrations. Since then, clinical Shack-Hartmann sensors have been developed that are now being used to detect many more aberrations in the eye, at a much earlier state, than ever before. Since Shack and Platt made their improvements on Hartmann's test, the Shack-Hartmann sensor has been used for adaptive optics, wavefront aberration measurements, applications in telescopes, lasers and optical systems, and vision science.

The Shack-Hartmann Sensor in an AO System

The Shack-Hartmann sensor plays a very important role in the adaptive optics system. According to the Center for Adaptive Optics, "Adaptive Optics refers to optical systems which adapt to compensate for optical effects introduced by the medium between the object and its image." ⁶ The Shack-Hartmann sensor detects the "optical effects introduced by the medium between the object and its image." The Shack-Hartmann sensor detects the optical effects caused by the Earth's atmosphere. Light from stars and planets coming to Earth travels for light years in a straight line, until it goes through the Earth's atmosphere where the turbulence and temperature differences in the atmosphere distort the wavefronts.

The Shack-Hartmann sensor detects the distortions in the wavefront as explained in Image Formation, by using an array of lenses and a detector. The information it detects is analyzed as explained in Analysis. Once the incoming wavefront is reconstructed, it is inverted and that information is sent to the deformable mirror. The mirror then deforms to the shape of the inverse wavefront. As wavefronts hit the deformable mirror and reflect off of their inverses, they reflect off parallel. Because the system needs to respond very quickly to changes in the Earth's atmosphere, the mirror needs to be readjusted at least several hundred times a second.⁷

As you can see in the picture below, light enters an adaptive optics system through a telescope. The light then travels through a collimating beam onto the deformable mirror. As the image reflects off of the deformable mirror it is split and half of it goes into a scientific instrument, such as a camera. The other half of the light travels into the wavefront sensor (usually the Shack-Hartmann sensor), where the distortion of the wavefront is found. The information is analyzed and sent to the deformable mirror, which changes shape to cancel the distortions caused by the Earth's atmosphere.

Figure 6: The Shack-Hartmann in an Adaptive Optics System

http://op.ph.ic.ac.uk/ao/overview.html#adaptive, 8/2/01

The Shack-Hartmann Sensor and Vision Science

The Shack-Hartmann has been used for wavefront aberration measurements, applications in telescopes, lasers and optical systems, and vision science, in addition to adaptive optics. The Shack-Hartmann has many applications and possible applications for vision science ranging from super-human contacts to detecting rips in the retina while they can still be fixed.⁸

The application of the Shack-Hartmann sensor to the eye was first presented by Liang in the early 1990s. Previous to Liang, it was only possible to test for a few aberrations in the eye, such as defocus, astigmatism, and prism. Using the Shack-Hartmann method it is possible to test for many more of the aberrations in the eye.⁸ It is interesting to note that in his paper, Liang referred to the method as the Hartmann-Shack method instead of Shack-Hartmann. As a result most of the information on vision science uses "Hartmann-Shack" while most of the adaptive optics information uses "Shack-Hartmann." Others, as can be seen in figure 7, are not happy with the name Shack-Hartmann or Hartmann-Shack; they attribute the original idea of the sensor to Scheiner in 1619.⁹

The following is an explanation of how the Shack-Hartmann is used to test the eye. A laser beam shines into the eye, which reflects off of the retina and travels back out through the cornea.¹⁰ A pair of relay lenses is used to focus the image of the lenslets back onto the pupil. That way, when the light exits the pupil it is split into many individual beams of light (from the lenslets).¹¹ This position of each beam is recorded onto a detector. In the perfect eye, each of the beams of light would be evenly spaced. By analyzing the placement of each beam of light, aberrations in the eye can be found.¹⁰Zernike polynomials are used to analyze the wavefront because certain polynomials represent certain aberrations as shown below.

Figure 7: The Shack-Hartmann Sensor in Vision Science

http://www.osa.org/homes/vision/resources/vsia-tutorial/sld018.htm, 8/2/01

Figure 8: Examples of some Zernike Models.

http://arapaho.nsuok.edu/~salmonto/ShackHartmannSlides/sld016.htm 8/8/01

Demonstrating the Shack-Hartmann Sensor

Our demonstration shows the image formation process of the Shack Hartmann sensor. In the demonstration a light source is projected at an array of 16 lenses. A target is placed at the focal length of the lenses, with the ideal centers for each image marked. When there is nothing between the light source and the lenses, the centers of all 16 images are lined up with their ideal centers. A clear tray filled with mineral oil is used to simulate the atmosphere. As surface waves are created in the tray between the light source and the array of lenses, the images on the target can be clearly seen moving in different directions at different speeds.

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