Wavefront Metrology

Optical wavefront metrology is a big subject. In this tech talk, we will focus on two topics which could even be considered dichotomies, phase-shifting interferometry and Shack-Hartmann wavefront metrology.

With phase-shifting inteferometry, accuracy is key and every nanometer is significant. Speed and versatility comes second.

Quite the opposite can be said for Shack-Hartmann wavefront metrology where versatility, ease of use and speed are its primary strengths while accuracy, although not at all bad, will have to take the back seat, especially when compared to phase-shifting interferometry. With interferometry, we can also address phase properties of the optical system that a Shack-Hartmann sensor will be blind to by its design principle.

Both, however, can be formidable alies when hunting for peak performance when designing and building complex optical engines that require optical performance well beyond the classical diffraction limit, such as for example pattern generators, a topic of particular interest here at Senslogic which is also the reason why the WaveMe software was developed with its focus on speeding up phase-shifting interferometry and integrating Shack-Hartmann metrology into day to day alignment tasks.

Phase-Shifting Interferometry: Harnessing Speed and Precision

In the world of nanoscale measurements, phase-shifting interferometry stands as a game-changer, offering unmatched precision by enhancing our interpretation of interference patterns. Here’s a deeper look:

When a photon is split into two paths and later recombined and absorbed by a detector, their mutual path length difference can be captured in the form of an inteference pattern. To truly understand these patterns, phase-shifting interferometry, which systematically tweaks the path length difference between these photons in several successive measurements, offers answers to questions that cannot otherwise be resolved by simply observing a single interference pattern because in a single interferogram, the interference term is provided by the cosine of the phase difference and the cosine hides the sign of its argument.

This technique, which requires at least three images with 120° phase shift between them, offers a direct solution to the three unknowns (two amplitudes and a phase difference) in a two-beam interference setup. However, the sweetspot, is clearly found using 90° phase shifts. Why 90°? The answer to that is error suppression. With 90°, the solution explicitly eliminates static background light and detector sensitivity. Less obvious is that it also suppresses second order detector non-linearities and by extending the method to four 90° shifts, also capturing the semingly redundant 360° phase shift, the influence of the actuator scale can be strongly reduced. All of this contributes to making this particular approach the go-to method for high accuracy interferometry.

Before leaving this introduction, it must be mentioned that phase-shifting interferometry is not a fringe analysis method. Nowhere in the math will one find an attempt to identify a fringe and try to follow it. It is entirely a local, detector pixel by detector pixel phase analysis. If there are regular fringes somewhere, then the math will reveal them, let’s call it, purely by coincidence. If we want, we can setup a Twyman-Green interferometer (which incidentally was referred to as useless by A.A. Michelson and is often confused with the interferometer named after him) and align it to display no fringes and analyze the wavefront phase at the same accuracy that we would have with any other amount of fringes, and actually probably even better because at zero fringes, we can have the combined beams going through the same glass before reaching the detector and eliminate phase error contributions from the projection optics that is often needed in a Twyman-Green interferometer.

The Need for Speed

As mentioned in the introduction, phase-shifting offers accuracy by design, but speed, one has to work for. If the target system is a single piece of optic, or a single mirror, waiting one second to get a wavefront is clearly not a deal-breaker, but how about measuring 10 million mirrors? And where do we find that many mirrors?

Perhaps a very short seque into pattern generators is in order here. The resolution limit for coherent illumination using spatial light modulators that only can access the positive real axis is set by 3 beam image formation, something we simply cannot avoid because we cannot remove the 0th order with such a modulator. This sets the angular distance between the interfering beams to an angle limited by the numerical aperture of the optics. If we could, however, extinguish the 0th order, then we could form images by interfering beams at +NA and -NA, combined that would mean 2xNA giving twice the resolution, but this is something we can only do with a modulator that can offer (at least) 180° phase shift, and in the case of an analog modulator, we can make good use of all the phase shifts in between. Once such phase modulator is described in the next paragraph and if we would want to use if for advanced lithography, we need to accurately calibrate the electro-optical properties of this modulator, and we need to do it quickly.

The RealHolo project is crafting an ambitious 9.6 million micro mirror piston array that has a bus bandwidth sufficinetly high to make it a very interesting component for a high-capacity pattern generation engine for advanced lithography. But it wouldn’t be advanced nor lithography is we couldn’t support it with accuracy that is the corner stone of every opto-lithographic writing engine, at least not today. And since we are dealing with pure phase measurements, we need interferometry both at high resolution and at high speed.

However, high-resolution MEMS is not the only application for a fast phase-shifting interferometer. The characterization of a deformable mirror with using a Twyman-Green interferometer offers another compelling application where interferometry offers results that Shack-Hartmann metrology simply cannot touch.

Pushing the Boundaries with Standard Tools

One might assume that such precision would require highly specialized equipment. However, that’s not the case. With proper software support even standard cameras can be deployed for this intricate task, achieving both the remarkable precision and measurement speed.

In essence, phase-shifting interferometry offers cutting edge accuracy, providing 10’s of wavefronts per second, or more, to revolutionize the way we look at interference patterns. With real-world applications like the RealHolo project underscoring the method’s importance, it’s evident how this technique is shaping the future of nanoscale measurements and beyond.

Phase-Shifting Using Fringe Analysis

Just to offer some perspective on phase-shifting based on the use of a physical phase alternating actuator, let us have a look at an alternative approach where we recover phase information from a single image. To do that, we have to arrange our measurement with a reference beam interfering at a sufficiently high angle. What “sufficiently” means depends on what we are currently measuring but generally speaking, it has to be higher than the highest spatial frequency of our object. If we would attempt to extract phase information using a Fourier method, then the fringe frequency would have to be twice the highest spatial frequency of our object, but there are cases where we can get away with less such as when we measure a piecewise flat surface as for example a TI-PLM spatial light modulator or the PLV from Silicon Light Machines. In this case, we can arrange the camera so that the fringes sample various positions of the (approximately) flat micro-mirrors. Without diving into too much detail, it should be fairly obvious that the fringe period must be smaller than the micro-mirror size in order to sample the intensity at the various phase shifts provided by the tilted reference beam.

Generally speaking, when using phase extraction from a fringe image, the fringe frequency must be much higher than the highest spatial frequency of the object if we want somewhat accurate results. In addition, one has to consider that cameras are not sampling the image at a point, they are instead integrating the intensity over a pixel and when we try to push the fringe density up in order to increase accuracy, the camera pixel density must follow because we must over-sample the fringe image quite substantially, and unless we have a very special area sampling device, the pixel count of the camera increases with the square of fringe density. This will make the measurement either limited to small objects or slow even when we consider that we have to extract 4 or 5 images using traditional phase shifting, assuming that our phase shifting method operates at a limit set by the interface speed (USB or Ethernet) of the camera which is true for the WaveMe toolbox. In addition, we will no longer have the error suppression provided by sampling the four or five different phase shifts using the same pixel and thus cancel error sources such as illumination uniformity and pixel response non-uniformity (PRNU).

Shack-Hartmann Sensors: A Work Horse of Wavefront Metrology

Imagine trying to understand the direction and intensity of thousands of tiny arrows pointing in different directions all at once. Shack-Hartmann sensors work similarly, deciphering the “direction and intensity” of wavefronts at specific isolated points. By capturing these minute details, or “gradients,” of the wavefront, these sensors offer a nuanced insights into dynamic wavefront.

What’s So Special About Shack-Hartmann Sensors?

Simplicity & Speed: While they may not be the kings of precision, Shack-Hartmann sensors are undeniably fast and user-friendly. Have a camera that snaps up 300 frames every second? These sensors can be made to keep the pace, sometimes even using a regular laptop.

And the accuracy is nothing to look down on. 10-15nm requires calibration to a known reference, but those can be made using the ligth emerging from a single-mode fiber or an edge-emitting laser diode.

The Technical Details

Venturing behind the scenes, deploying Shack-Hartmann sensors does call for considerable programming effort. Breaking down countless spots on an image, especially when aiming for real-time wavefronts (think 60 frames per second live views), presents its challenges. And while it’s not quite as intricate as rocket science—or, in Elon Musk’s words, rocket engineering—it still demands some attention to details.

Senslogic’s WaveMe Platform: A Step Ahead

Here at Senslogic, we’re not just observing; we’re innovating. Meet WaveMe, our pioneering platform. WaveMe isn’t tethered to a single technique. Instead, it offers the best of both worlds, the accuracy of phase-shifting interferometry and the rapid-fire capabilities of Shack-Hartmann wavefront metrology. From nanometer-focused precision to rapid frame-by-frame analysis, WaveMe is crafted to be your trusted companion, compatible with regular vision cameras. We offer one of the, if not the fastest, phase-shifting interferometry platform and the simplest shack-hartmann wavefront sensing solution. How can we say that? Well, can you require less than turning the program on? It’s made possible by WaveMe’s 3D Shack-Hartmann sensor calibration.

Intrigued? Stay connected with Senslogic’s trailblazing projects. Head to our contact page and subscribe for the freshest updates and announcements. Senslogic will offer the Shack-Hartmann software together with a camera and we are always open to OEM customers to adapt the phase-shifting and Shack-Hartmann softwares to their specific needs.