Figure 1: Shack-Hartman sensor

A less well corrected lens will also form a converging wave front, but one with deviations from a perfect sphere. The resulting PSF will be larger than the theoretical minimum Airy disk. Such deviations may be due to limitations of the basic lens design, or due to imperfect realization of the design. In most real-world situations it is a combination of both.

A lens bench is an apparatus that allows us to view the PSF. On older equipment, a visual microscope was used for the evaluation [A] . More modern equipment can also display the image on a computer screen. In some cases, the bench can make measurements of image quality such as contrast or MTF.

In any case, the PSF gives only limited information about the wavefront. If the PSF size approaches the limiting size set by diffraction, we can deduce that the wavefront must be perfectly spherical. However, if the PSF is larger we may wonder why the spot is larger. Knowledge of the wavefront can give helpful insight into this question.

When a small aperture is introduced, only a small portion of the lens contributes to forming the PSF image.

This technique works even in the face of discontinuous jumps in wavefront phase, so long as the jumps are only a fraction of a wavelength. By sampling at a large number of places across the lens aperture, we can build up a complete wavefront slope map.

We began with a commercial lens-testing bench [B] . A halogen light source with a narrow band filter illuminates a reticle, typically a 50 micron pinhole. For the tests described in this paper we used a 550 nm filter. The pinhole lies in the focal plane of a well-corrected [C] 200 mm lens. The lens under test follows the collimator, and forms a real image of the pinhole in space

Figure 6: experimental apparatus

We a constructed a small motorized XY stage to carry the moveable aperture over a 50 mm square area. Because high precision is not required, we were able to use an inexpensive “digital linear actuator.” This clever device combines a fixed leadscrew with a stepper motor whose rotor is threaded. The positional accuracy of the stage is approximately 1/20 mm, which is adequate for this application.

A 1.5 mm sampling aperture was used for the tests described here, picked to be about 1/50 of the lens aperture. For convenience, we have chosen place the scan aperture immediately before the lens, although this is not an essential requirement.

Control software

We created software to step the aperture through an XY pattern with programmable step size. At each point, an image is grabbed, and the location of the PSF is measured in the X and Y. If the grid spacing is 2 mm or less, the system captures and analyzes approximately 2 images per second. At the end of the pattern, the locations are displayed as an arrow plot of “ray” errors.

Experimental results

We measured a molded plastic triplet intended for use in an inexpensive 35 mm camera. The specific lens measured for this paper was an early shot from the mold, selected to represent less than sterling performance. Such a lens would be difficult or impossible to measure on an interferometer because of the large number of fringes.

Focal length of the lens was 76 mm. The design aperture was 6.3 mm, or F12. The error map is shown below. The x-direction MTF curve is also shown for reference

Figure 7: ray error plot for plastic triplet

Figure 8: MTF plot for same lens

When testing a lens on-axis it is desirable to identify and extract the radial component of ray error. The purely radial component can be compared to the sagital ray fan generated by a lens design program. After removing the radial component, any non-symmetric residual must be due to fabrication error

To demonstrate this technique we measured an 80 mm achromatic doublet, intentionally mounted with the flatter surface towards the collimator. This mounting position would be expected to generate a relatively large amount of spherical aberration. The raw error map is shown below, together with the best fit to the radial ray error [D] .

Figure 9: ray error plot for reversed doublet

Figure 10: best fit of radial ray error.
Note large amount of spherical aberration.

After removing the radial component, the residual fabrication error looks like this:

Figure 11: residual ray error plot after removing radially symmetric content

Figure 12: Wavefront that would produce the ray errors shown in figure 11.

In many situations, the plot of ray errors is the most useful way to present the information. However, for some purposes it is desirable to reconstruct the wavefront phase map itself.

The literature describes two main techniques for reconstructing the wavefront: modal and zonal.

In modal reconstruction [4] , the wavefront is viewed as series of orthogonal functions such as Zernike polynomials. An attempt is made to find the set of polynomials that best fits the measured slope data.

In zonal reconstruction [5] , an attempt is made to find the actual wavefront that would produce the measured slope data. Because the system is overdetermined, we seek a solution that minimizes the squared error terms. Matrix math can be used to solve directly using SVD. (singular value decomposition) Alternately SOR (successive over relaxation) can be used to generate a series of successively better approximations. The approximations can be shown to converge on the desired wavefront. We used Southwell’s SOR method to compute the wavefront map shown in figure 12.

Fundamental accuracy limitations

We are aware of several factors that can limit the precision and accuracy of this technique. The first is mechanical vibration. Any motion of the microscope translates directly to error in the ray error plots. We quickly learned not to touch the fixture during a measurement cycle. However, once we exercised reasonable care we found vibration was not the dominant error.

We were concerned with camera noise, but this turned out not to be a significant noise source. The Airy PSF spots are quite large because of the Hartman aperture results in a large F-number. Noise in individual pixels is averaged out because so many pixels are involved in the image.

We also realize that it would be difficult to separate (for example) spherical aberration in the microscope lens from spherical aberration in the lens under test. We believe the microscope lenses to be of high quality because of off-line testing [E] .

The remaining error source is air path disturbance. Even heat from our hands was enough to create small thermal disturbances that could be seen on the when measuring PSF spot locations. This problem is not unique to our system: It should be familiar to anyone who has used an interferometer to evaluate lenses or flats. Nevertheless, we had to take special care to shield the test setup from thermal effects.

Comparison to Shack-Hartman sensor

It should be apparent that the scanning aperture technique is not appropriate when dynamic events must be captured. Capturing several hundred data points with our stage takes one or two minutes, and this rules out many applications.

Even when making static measurements the Shack Hartman sensor has some advantage because the effects of thermal air path disturbances are less severe over short time spans. However, air path disturbances are a serious issue for both measurement techniques.

On the other hand, in the context of lens evaluation, speed is seldom the driving issue. The ability to make wavefront measurements in-situ on the lens bench is of considerable value. Transferring the lens to a separate (but faster) instrument could possibly consume as much time as it saves.

Moreover, the scanning technique has some outright advantages over the Shack Hartman sensor

• The number of apertures is completely adjustable, and not constrained by the lenslet array.
• With a moving aperture it is simple to over-sample as shown in figure 14. The Shack-Hartman sensor is limited to adjacent-aperture sampling, as shown schematically in figure 13. This may be of value when the wavefront has high frequency components that would be aliased by adjacent aperture sampling [6]
• Closely spaced apertures do not force a tradeoff between dynamic range and sensitivity [F] .
• High sensitivity measurements are less sensitive to camera noise because the PSF spot covers many pixels.

Figure 13: S-H apertures limit sampling

Figure 14: oversampling

* The author may be contacted by email at ben@wellsresearch.com, or by phone at 781-258-8667

[A] Purists (including the author) occasionally prefer an eyepiece to an image on a computer screen. The human eye has remarkable dynamic range, and can see subtle detail that is hard to capture or display with a video camera.

[B] Wells Research OpticStudio model OS-100, distributed by ThorLabs

[C] This lens was separately tested on an interferometer to verify wavefront error was less than 1/20 wave.

[D] The fit was done in Microsoft Excel using the wonderful “solver” tool. In the future, we intend to use SVD matrix math to automate the least-squares fit process.

[E] The next logical step is to use the scanning technique to measure the microscope lens, but we have not done this yet.

[F] In a Shack-Hartman sensor, PSF spots from adjacent lenslets may interfere or even overlap if the wavefront is highly aberated. Shorter focal length lenslets minimize this problem, but at the expense of lower angular sensitivity.

This paper prepared for presentation at SPIE Annual meeting in San Diego, August 2003

[1] Malacara devotes a full chapter to the Hartman test, and concludes with an extensive bibliography: D. Malacara, Optical Shop Testing, Wiley Interscience, New York, 1992.

[2] See for example: D. Kwo, et al, A Hartman-Shack wavefront sensor using a binary optic lenslet array. SPIE Vol 1544: Miniature and Micro Optics: Fabrication and System Applications.

[3] For example: Wavefront Sciences “CLAS –2D Wavefront sensor.” www.wavefrontsciences.com

[4] Guang-ming Dai, Moddified Hartman-Shack Wavefront Sensing and Iterative Wavefront Reconstruction, SPIE Vol 2201: Adaptive Optics in Astronomy, 1994.

The classic, albeit terse, reference is: W. H. Southwell, Wave-front estimation from wave-front slope measurements, JOSA Vol 70,No. 8, August 1980. Kwo, et al (above) gives an easier to follow presentation of Southwell’s method

[6] J.D. Mansel et al Sub-Lens Resolution Shack-Hartman Wavefront Sensing, NSF grant 2WMF572, article available at http://fastloki.stanford.edu/~jmansell/kewfs/kewfs5.pdf