Designing a point diffraction interferometer

In this tech talk, we will dive a little deeper into the design of a point diffraction interferometer, or PDI for short. We have talked about it a little superficially in an earlier tech talk but this time, there should be enough for anybody who may consider building one. This application is well supported by the WaveMe platform which lowers the threshold the an abosolute minimum for those interested in exploring this technology.

The PDI is not a generic tool. It contains elements adapted to the numerical aperture and focal lengths of the system in which it operates. That said, it can be used for other designs which share some parameters, albeit not all. So, for example, two telecentric systems where the rear group share focal length but differ in numerical aperture, if the pinhole (PH)is small enough for the larger numerical aperture, the interferometer will work for the smaller aperture but with lower contrast. However, the contrast requirements are rather benign when the interferometer is augmented with the phase-shifting technique and most probably, the results will still be very good. A well designed PDI will reach λ/300 (RMS) accuracy so there is accuracy to spare for most cases.

The Parts

There are three core parts to a phase-shifting (or phase stepping) PDI, an actuator, a coarse transmission grating and a PDI plate or diffraction mask. Depending on the light source, an additional pinhole (PH) to generate an aberration free reference wavefront to illuminate the optics under test may be needed but a single-mode fiber, if its numerical aperture is large enough, is also a very nice source if the fiber coupling is done well.

The drawing above shows only one of many possibilities to arrange phase-shifting, by placing the grating at the Fourier plane of the system. The grating can be placed as the last element before the PDI plate, as was shown in the images in a previous tech talk, or it can even be part of the reference illumination, which then becomes a two-PH reference illumination if one wants to avoid, or cannot, put anything inside the optical system itself.

The PDI plate can also be arranged in many other ways, especially if a more flexible design is the goal, but this is a tech talk about the principles, therefore the simplest one is shown.

Spatial Resolution

If we look at the optical system between grating and the image plane of the aperture projection optics, there is another 4F system with the PDI plate with its PH and test window (or test aperture) in the Fourier plane, and from this we can draw the conclusion that the size of the test window determines that optical resolution of the interferometer. This determines how sharp the aperture of our system under test will look on the camera, which is in the image plane of the aperture projection optics, and how small wavefront features we can resolve.

There are choices to be made here. If we make the test window large, the tails of the spot illuminating the PH at the center will become increasingly visible through the test window. If we make it smaller, we lose resolution. As a rule of thumb, setting the diamter of this aperture to distance between the projected grating diffraction orders is a good choice. Some interference will be visible but if one is interested in mapping the aberrations on the Zernike polynomials, accurate results can be obtained with this choice.

The PDI plate is not an off-the-shelf item but pinholes are and to add the test aperture to a standard PH of appropriate size can even be done by drilling, but there are many other possibilities. If the PDI plate is printed on chrome, an optical density (OD) of at least 5 is needed, which is larger than one can expect using standard masks which are at best OD 4 or typically OD 3, which is insufficient. Precision pinholes from Thorlabs, Edmund Optics and several othe sources can be obtained down to 1 μm. If the desired diameter of of the PH,

cannot be found, consider measuring the system in the opposite direction, if the magnification is not 1, and use the largest diameter that is still smaller then the obove diameter for the reference PH. The standard texts would have a factor of 0.5 in front where I have put 0.4. This is not a big effect but, depending on aberration, countless simulations show that this choice is less likely to give results outside the expected accuracy.

We are talking about small probabilities here, like one in a thousand measurements, but if high and reliable accuracy is the target, 0.4 seems to be a better choice. As rule of thumb, the PH diameter will be about the same as the diffraction limited resolution of the system, which is a number usually well imprinted into an optical designers mind.

Before we can start a measurement, the pinholes must be aligned to the diffracted spots. Here is a list to follow,

1. Identify the 0th order by its power. It is the strongest one. We are assuming here that the light will be going through the test aperture.
2. Move the spot of the 0th order toward the edge nearest to the pinole.
3. Use the Foucalt knife-edge technique to focus the PDI plate. The aperture projection optics is already perfectly setup for this purpuse.
4. Translate the PH (with its projection optics) the nominal distance between the edge and the center of the PH. This distance we may not know that well the first time, but it is worthwhile to remember once we have it.
5. Maximize the fringe contrast to make sure that foucs and centering are correct.

The foucsing stage using the Foucault test is very nice here and actually often provides a first hint about the aberration of the system. One can, for instance, identify astigmatism down to 0.1 wave with a little training. In practical application of the PDI, one usually doesn’t need to do all the steps in the list, but at least every time when the system under test is exchanged. After that, it is usually enough to know the magfinication and move the source and PH (or rather the entire aperture projection optics which should be mounted on a common mechanical base) to their nominal positions and do the fine adjustments once there.

One final word about the PDI plate, if you have done really good job at aligning everything, there will often be camera reflex bouncing back to the PH. Either make sure, the side facing the camera is black or add a quarter-wave plate and a linear polarizer before the camera. A black marker applied to the backside of the PH can improve results quite significantly. This is not always that easy to see in the measured results so making sure the backside is black is a simple way not having to worry about it.

The Grating

The grating needs to get some (engineering) attention. After all, this is the heart of the phase-shifting part of the inteferometer since it provides both beam splitting and the phase shifting. Since the targetted accuracy of this interferometer is quite high, the straightness of the grooves and positioning accuracy of the actuator (in a broad sense) needs to be about 0.1% of the grating period, although it depends on a few factors such as, if the test window is at the 1st or 0th order, or even part of a two-PH setup illumination reference. Also, one must take care so that the actuator does not rotate the grating about the optical axis. Usually, it would be the pitch specification one has to keep an eye on, while yaw and tilt are not a problem for an actuator that manages to keep the pitch.

In the image above, the grating is in the plane given by the direction of travel and the normal of the actuator plane. Usually, unless the grating is very small, one will not find an actuator that has a specification which meets the requirements, and under all circumstances, the grating should be placed as close as possible to the actuator, but those specifications are always for the entire travel range while in our case, we usually do not use that much of the travel range, but one must keep an eye on this number.

One can detect potential problems by making a measuremnt that also captures the 360° phase shift. In WaveMe, this is referred to as the 5-point measurement. It will cancel pitch errors where the actuator surface normal rotates around an axis at a finite distance.

Just to make sure we are all on the same page, the number of fringes that will be visible on the camera is the same as the system aperture diameter times the groove frequency, but the question is now, how many should we have? If we choose a lower number, the resolution of the PDI will be lower but the beams will be closer to each other and the error contributions that are derived from the path difference, typically in the aperture projection part, will be smaller. And then, converseley, with a denser grating, resolution will increase and so will errors related to the difference between paths. Now, those errors can be more or less difficult to remove by calibration which will be quite easy when measuring through the 1st order and more difficult when measuring through the 0th order. If the rear group of the aperture projection optics is centered and aplanatic, this problem goes away to a large extent. Again, this all depends on the accuracy we are targetting.

So, how many grooves over the system aperture? No hard rules but 50 is on the low side and 100 is maybe a little high. As always, depending what one is looking for. With the lower number, a plano-convex singlet will even work between the pinhole and the camera, especially if we measure through the 1st order and calibrate by isolating the two beams which otherwise would go through the test aperture and the pinhole.

Two-Beam Reference

When it comes to accuracy, the grating does require some attention. Even when printed on a fused silica wafer of high quality, there is the possibility that the lack of planarity will prevent the application to reach the desired accuracy. For visible and UV wavelengths, a grating made of single crystal silicon waver, etched all the way through avoids this problem entirely but in order to keep the straightness of the grooves, the gratings have to be handled with great care.

One solution that avoid all those problems is to make the phase-shifting process part of a two-reference PH illumination where the grating is now part of the illumination. All information that remains after the pinhole is their mutual phase difference and the system under test remains untouched during the measurement.

Some personal reflections on this technology

My own experience is mostly from KrF (248nm) and 355nm wavelengths. What I most enjoy when deploying and using it is that there are no ifs and buts. Optics, by its nature, is sensitive and this sensitivity usually depends on thermal and mechanical drifts. The pinhole is not going to change, espepially a pinhole manufactured on a 0.1mm stainless steel foil. One can assemble this interferometer from scratch and know that the results are at least accurate to milliwaves.

When assembling a complex optical system for the first time, a lot of materials and physics is combined to produce a desired result. Things will move, with temperature and with time due to material creep. Since the repeatability of this tool is so good, one does not have to wait a week for the potential problems to present themselves. One will see them the next day, at least, and we can trust our observations, if they appear to be changing, are not due to the interferometer.

Although this is a tool primarily for the diffraction limited region and below, it is useful for more aberrated wavefronts although for those cases, one cannnot claim nanometer accuracy. It is the high-end tool for high-performing optical machines, one that will take all the guessing out of any performance review.