Shack-Hartmann Wavefront Sensor Calibration in WaveMe

Calibration and Why it is Needed

The WaveMe toolbox has a rather exceptional calibration method that allows for much more flexible use of the Shack-Hartmann wavefront sensor. This tech talk will be about what it is and why it is different from what is available elsewhere. We will assume that you, the reader, are already familiar with the principles of the Shack-Hartmann wavefront sensor. However, most explanations one will find on the internet will not provide any numbers, I hope we can take a different view here.

A Simple Numerical Example

Let’s start with the typical placement accuracy of a micro-lens array, which is about 250 nanometres, and for simplicity, let’s assume that all mirco lenses have the same placement error, including direction. This is a little silly example because we have only shifted the entire array 250nm, but it helps us to connect some of the configuration parameters of a sensor with the integrated wavefront. To continue, if we also assume that the distance between the array and the sensor is the same as the size of the aperture, then after integration of this displacement error we will have 250 nm wavefront error after integration which will be seen as a wavefront tilt. If only half of the micro-lenses in the array are displaced by this amount, we will see half the wavefront error. This, however, is still a very large number relative the accuracy that we are aiming for.

But there is another point to be made here, which is the distinction between systematic and random errors. Our systematic error could easily be integrated to a large fraction of the basic displacement of 250 nanometres. Had those errors been random, then the integrated wavefront magnitude would be of the order of the same displacement divided by the square root of the number of illuminated micro-lenses.

Fine positions

While 250 nanometre sounds like a lot, even a mega-pixel sensor array will have a pixel size of several to sometimes north of 10 micrometres. Clearly, an accurate wavefront sensor needs to find the centre of light distribution with some accuracy.

There are several ways to do that, but a quite common method is to use the centroid, which sums the sampled intensity of a camera pixel multiplied by pixel coordinate and divided all by the sum of sampled pixel intensities. If our spots are large, meaning is intensity is varying slowly over the size of a pixel, this method is quite good. However, if the pixels are large compared to the spot size, this algorithm produces a substantial noise. Now, here’s something worth noting, if we are interested in low order wavefront errors, such as coma or astigmatism, which we often are, and we have a large number of spots, the numerical noise that we see in centroid position actually does not matter all that much and the results are actually still quite good and the reason for this is that the systematic error is still somewhat small. And here the other point about Shack-Hartmann sensor calibration, systematic errors must be kept really small and of the order the accuracy that we want to achieve.

Fine positions using templates

The Shack-Hartmann module in WaveMe does not compute centroids. Instead, something usually referred to as the template method is used. The template method requires that we figure out which model spot fits best to the current data. This results in a model spot which in turn is used to generate a set of model intensity profiles for a spot whose center is at different offsets within a pixel. The spot sampled by the sensor array is then correlated with the set of computed overlaps and the best fit is the best (intra) pixel offset. Since the target application for WaveMe is day-to-day alignment work, the sensor will need to handle both collimated and divergent beams. Therefore, the spot model and templates are constantly optimized. But the story does not end here, in fact, it is actually just the beginning.

Calibration Methods

There are many ways to calibrate a Shack-Hartmann sensor. Which one to use depends on the intended use case and how well the micro-lens array is aligned to the sensor. If the sensor is in a fixed setup, illuminated with a collimated beam a differential calibration may be more than enough. If a more unpredictable use case is to be expected, probably another calibration approach is to be preferred.

Differential Calibration

When it comes to selecting calibration methods, we may divide them into two, differential and absolute. When using the differential method, we would store the spot centers as a reference and present the wavefront “error” and the integrated deviation from this reference. If the reference was taken with plane wavefront, then our result is the actual wavefront illuminating the micro-lens array. If, however, we are measuring a divergent wavefront, several other systematic error contributions will appear that may easily dominate all other error sources. We’ll dive deeper into this topic later.

Absolute Calibration

Another calibration approach, which we can refer to as absolute calibration, uses spherical wavefronts, captured at different distances to the reference source. This base data set can then be (for example) interpolated into when generating a reference for the current source position. Here, we can quite immediately draw the conclusion that the common application of this data will be some sort of extrapolation and our model should therefore be a low-order polynomial. The advantage of this method is that spherical wavefronts are available at very high quality. Single mode fibres or proper pinholes will provide very high-quality spherical wavefronts. What they don’t provide is converging wavefronts, which means that when we use this method with converging wavefronts, the amount of extrapolation can be significant. For Shack-Hartmann sensors, there are limits how much of a converging wavefront we can measure anyway but still, this limits the useful range.

Synthetic Reference Calibration

WaveMe uses a different approach entirely, where the optical centres of the micro-lenses are extracted, in 3D, and a simple physical optics model is implemented to account the effects of a small Fresnel number. The only assumption is actually that the array is flat, but beyond that, it may be placed at mechanical tolerances. No alignment of the array is needed.

The relaxed alignment requirement of the micro-lens array can be quite an advantage because even small differential screws can be several millimetres in diameter and over a centimetre in length. This prevents any cost efficient integration of sensor and lens arrays as a single unit, while, together with the calibration used in WaveMe, the lens array can be soldered on the same mainboard as the sensor. The calibration and the math take care of the rest. This enables extremely compact and still very accurate solutions.

WaveMe assumes that the camera is always streaming images. This information is analyzed by an independent thread to extract the source position, and using this information, the Shack-Hartmann module creates its own calibration data based on the physical optics model and a virtual ideal point source which now may be at any position, generating both diverging , collimated, or converging (virtual) reference wavefronts. The virtual sources may also have any angle of incidence. This method could be thought of as an absolute calibration, but it is more descriptive to think of it as a vritual source calibration. The afore mentioned absolute calibration uses interpolated or extrapolated results from a larger number of explicit spot position measurements while the latter, infers a single set of optical centers for the microlenses from a larger number or measurements. The difference may seem subtle but it is not. WaveMe infers micro-lens positions and a physical optics model that explains all the data that are provided to it during the calibration process. In relative terms, the difficulty is moved to the calibration process while the application of the model is, at least numerically speaking, relatively simple and intrinsically covers all application cases. This is a choice that better fits the target application where no alignment of the incident wavefront can be expected since no alignment of the optics under test is to be expected.

Comparing with Other Implementations

Browsing the internet for Shack-Hartmann calibration and measurement techniques, one will encounter the concept of Area of Interest (AOI) or Region of Interest (ROI), which defines a set of square patches roughly aligned with the centers of spot positions.

This concept does not exist for WaveMe. WaveMe is based around the assumption of an image stream. When the Shack-Hartmann module is activated, it will try to identify the existence of a spot pattern. This pattern can have any rotation and the range of accepted spot spacings is larger than most practical wavefront sensing conditions. Once a spot pattern is identified, spots are enumerated and a process adapting a spot model to the current data is started in an independent thread. Once the spots have been enumerated, the following images are processed using the spot tracking.

Since there is no fixed space that the spot pattern must fit into, there is no explicit dynamic range for the sensor. There will, however, be one in the sense that when we the sensor is moved to close to the source, the relative sizes of the spots will change too much, and the accuracy will be somewhat degraded.

Another reason that WaveMe does not use a fixed grid to localize the spot pattern is that the source position is not assumed to be known. The wavefront can illuminate the sensor from a wide range of directions and distances. Fixed AOI’s would be too much of a limitation and the requirement of adapting them to the current spot positions is not acceptable for when using a sensor intended for interactive optical alignment, which is the target application for this module and WaveMe in general. After all, it is supposed to be point-and-shoot.