Collimation is a crucial process in aligning the various elements of a telescope, especially the optical surfaces, according to their intended positions in a given optical layout. Without accurate alignment, the performance of the telescope can significantly deteriorate. Collimation involves adjusting the 3D position of the elements, including tilt, tip, offset, and spacing, which are all part of the alignment process. Each optical element has five degrees of freedom.

Since collimation directly affects the optical performance of the telescope, any misalignment can result in aberrations. These aberrations can be used as indicators to determine the alignment state of the telescope, providing feedback during the collimation process. In short, collimation is a numerical approach to alignment.

The initial level of collimation involves measuring the mechanical alignment of the optical surfaces. This is usually done using a combination of optical tools like eyepieces, collimation telescopes, and laser beams. Some telescopes, such as the Ritchey–Chrétien (RCT), provide reference marks like dark spots and donuts to aid in the process. However, this first step of collimation is typically a coarse adjustment and should be followed by a more precise optical alignment, which involves analyzing the telescope’s aberrations. It’s important to note that to achieve optimal telescope performance (Strehl’s ratio at or above 80%), aberrations should be limited to a fraction of a wavelength. For visible light with an average wavelength of 550nm, this means aiming for wavefront errors in the range of 75nm root mean square (rms).

Wavefront analysis offers an accurate and quantitative method to measure the aberrations in a telescope, indicating its departure from perfect collimation by comparing the errors in phase between an ideal optical system and the actual telescope. This analysis provides information about aberrations, their types, amounts, as well as the scope’s Strehl’s ratio (SR) and Optical Transfer Function (OTF), including the Modulation Transfer Function (MTF). These quantitative indicators reflect the telescope’s performance and the quality of the images it produces. Typically, wavefront analysis requires a specialized device called a wavefront sensor, with the Shack-Hartmann wavefront sensor being the most common type.

Innovations Foresight has developed a cutting-edge technology, currently pending a patent, that utilizes a defocused image of a star (natural or artificial) to retrieve the telescope’s wavefront using artificial intelligence (AI). This allows us to determine the telescope’s aberrations without the need for a dedicated wavefront sensor. Our technology works with just a single frame, even under challenging atmospheric conditions. While the most common approach involves using an on-axis star, it is not limited to that; a star field can be used to obtain field-dependent aberrations all at once. This capability is particularly useful for perfect collimation of specific telescope types, such as the RCT. Although primarily used for telescope collimation, our technology can be easily adapted and customized for various other applications. Please don’t hesitate to contact us for any specific requirements or applications you may have.

In the figure above, we can observe two similar imaging systems (let’s call them telescopes) on the left side. The top one represents a perfect system, known as diffraction limited (DL), where the resulting spherical wave converges precisely at the focal plane, where the Point Spread Function (PSF) is located. The rays, which are perpendicular to the wavefront by definition, intersect the focal plane at a single point (as predicted by ray tracing in geometric optics). These rays are parallel to the radii of the spherical wave, pointing toward the center of the sphere (the image point).

The yellow disk, displayed as a “heat” map, represents the color-coded wavefront error (or phase error), indicating deviations from a perfect spherical wave. In the case of the DL system, there is no error, resulting in a uniform yellow disk. The color bar on the right provides a qualitative indication of the wavefront errors. Reddish colors indicate peaks in the wavefront error surface, suggesting that the actual wavefront leads the expected (perfect) wavefront. On the other hand, blueish colors represent valleys (troughs) in the wavefront error, indicating that the actual wavefront lags behind the perfect one. Typically, wavefront errors are expressed in nanometers (nm) or as a fraction of a wavelength for a specific reference wavelength.

Since electromagnetic radiation, such as light, follows a periodic sinusoidal pattern, a deviation of one wavelength corresponds to a phase shift of 360 degrees (or 2π radians). This is why we refer to it as a phase error. When there is no phase error, the PSF is perfect, and the system is diffraction limited. In the system shown below the DL one, we observe distortions in the wavefront, resulting in various color shades in the wavefront heat map. Notably, we can clearly see a peak and a trough in its 3D representation next to it. This characteristic pattern corresponds to an aberration called coma, which gives the PSF a comet-like shape.

On the right side of the figure, examples of basic aberrations are shown along with their names, wavefront errors (heat maps), and the corresponding PSFs.

Under specific conditions, a mathematical relationship exists between the irradiance (intensity profile) of an image and the corresponding wavefront (WF) error. This relationship is described by a non-linear differential equation known as the irradiance transfer equation, also referred to as the transport of intensity equation (TIE).

The TIE provides a link between the irradiance (I) and the wavefront (W). It represents how changes in the wavefront affect the intensity distribution in an image. However, it is important to note that the TIE cannot be solved analytically. Instead, numerical methods are employed to solve this equation and obtain the desired information.

By applying numerical techniques, scientists and researchers can approximate the wavefront error from the observed intensity distribution in an image, allowing for the analysis and characterization of optical systems. These numerical methods provide valuable insights into the wavefront properties and help improve our understanding of the system’s aberrations and performance.

It is worth noting that the TIE serves as a fundamental tool in various applications, such as wavefront reconstruction, aberration correction, and adaptive optics, enabling us to extract valuable information about the wavefront from intensity measurements.

To estimate the wavefront (WF) phase error in real-time, the Transport of Intensity Equation (TIE) is solved using numerical optimization algorithms. This process involves solving the TIE for each new image captured, allowing for the estimation of the WF phase error on-the-fly.

In order to minimize computing time and power consumption, certain approximations are often employed, typically based on linearity assumptions. These approximations are valid under specific defocus conditions and for a limited amount of aberrations. This approach, known as the direct model approach, enables efficient estimation of the WF phase error.

At Innovations Foresight, leveraging our extensive knowledge and expertise in machine learning (ML) and artificial intelligence (AI), we have developed a novel approach [3]. Our approach focuses on learning the inverse model, which captures the relationship between the intensity of defocused star images and the corresponding wavefront. This relationship is often expressed parametrically using Zernike polynomials, although other representations of the wavefront and related aberrations can also be learned.

By using a defocused star image, a bias known as phase diversity (PD) is introduced to the wavefront. This allows for the retrieval of the phase error without ambiguity, as long as the measured aberration magnitudes are smaller than the diversity term. The figure below illustrates this concept:

By utilizing this phase diversity technique, we can overcome the ambiguity in the wavefront phase error estimation and accurately characterize aberrations, even under challenging conditions. This approach significantly enhances the efficiency and effectiveness of wavefront analysis and provides valuable insights for optimizing optical systems.

As mentioned earlier, due to the absence of an analytic solution for the Transport of Intensity Equation (TIE), we employ a numerical approach. However, there is a fundamental distinction between the direct model discussed previously and the inverse model. The inverse model is precomputed beforehand, requiring only straightforward and rapid calculations during runtime. This enables us to achieve high processing speeds, even at video rates, which is particularly valuable for applications such as adaptive optics (AO).

In the case of telescope collimation, where we are primarily interested in characterizing the telescope’s aberrations rather than those induced by atmospheric seeing, longer exposures ranging from 30 to 60 seconds are used. These longer exposures allow us to average out the effects of seeing. Unlike AO, there are no real-time constraints for solving the TIE in this context.

The advantage of not having real-time constraints is that we can dedicate extensive time to learning the inverse model. This can span days, weeks, or even longer if necessary. Consequently, we are not compelled to make any approximations, assumptions, or limit the technique to specific subsets of aberrations or defocus ranges. This makes the method highly versatile, flexible, and applicable to a wide range of scenarios. Moreover, it allows us to consider multi and extended sources, expanding the applicability of the technique.

The process involves training a feed-forward artificial neural network (NN) exclusively on synthetic (simulated) data specific to a particular class of optical systems, such as telescopes in the context of astronomy. The training database consists of simulated defocused star images, including the effects of atmospheric seeing, with known levels and types of aberrations. During the learning process, the NN is trained to output the corresponding wavefronts from these images. Typically, the wavefronts are expressed in the form of Zernike polynomial coefficients, which are commonly used to represent aberrations, or in other relevant formats. The figure below illustrates the overall process:

This scope is a 17″ Corrected Dall-Kirkham before proper collimation. It exhibits 0.2 wave rms (200 mw) of total WF error leading to a Strehl ratio (SR) of 20% dominated by misaligned mirror (collimation errors) induced third order coma and astigmatism aberrations. Below a summary of the analysis:

The first left defocused star image (B&W FIT file) is the raw image from the telescope taken through a red filter, the second most left image is the simulated images reconstructed using the Zernike coefficients and related polynomials outputted by the NN after adding the estimated seeing and the spider diffraction patterns. This is a monochromatic simulation, while the raw one is polychromatic will less contrast. Both images are structurally identical which tell us how good the NN analysis and retrieved WF as well as aberration are. Indeed our technology has been tested against a high-precision interferometer (PhaseCam 4D Technology model 6000) and shown a rms accuracy in the order o 10mw rms (~5nm), or better [4]. The two last images are, from left to right, the scope PSF without any seeing (in space condition) and the WF error heat plot.

While using SKW the computing time itself (stars location, extraction, pre-processing and WF analysis) is in the order of few seconds on an average laptop.
The time to load the frame from the imaging camera is of course up to the user’s imaging software used.
SKW, on request, can watch a given directory (folder) for new FIT images auto-load (this includes a filter for the file name, if any). When a new frame is available it will auto load and analysis it if it was set to do so by the user.
Alternatively the user can load a frame manually too. SKW does not connect to any hardware and therefore does not need to deal with any driver, it works only with .FIT files.
Therefore SKW works with any imaging/acquisition software as long as it can output monochromatic (B&W) FIT files (8 bits, 16 bits, or float format). When using a one shot color (OSC with Bayer’s filter) camera one would convert one of the color channel, usually red to minimize seeing, in a monochromatic image. In most cases a luminance (L) frame can be used too but we recommend using a color filter to narrow the raw image bandwidth increasing its contrast when doable.
One should understand that running the trained neural network (NN) is very fast, the longest time in the process after a frame has been acquired and uploaded is to locate and preprocess the defocused star(s) in the frame as discussed above. The NN is a feed-forward one the computing time in an average laptop (Windows 7 one CPU) is less than 100ms.
On the other hand building the training databases for a class of telescope/optics (learning, validation and test data) as well as the actual training of the NN is a different story. Those tasks are done beforehand by Innovations Foresight. It may takes days to weeks to do so, depending of the size of the databases and the computing resources allocated. The NN is usually trained with at least several 100,000 to millions of samples.
However this is totally transparent for the user. When somebody buy from us a mathematical model for their scope to be run with SKW this one come in the form of an encrypted file which is decoded using your SKW local machine license key. The resulting process extracts the model data used by SKW for the related scope. The size of the model data file is in the order of few hundred MB only, and NN run time inside SKW is negligible.

It is worth noting that the SKW can analysis, depending of the software version, multi star (actual or artificial) at once, within the same frame. This approach provides field dependent (on and off axis) aberrations, like field curvature, aberrations maps (2D and 3D), and field height related wave front aberration functions (Wxxx).
This is a very powerful capability of our AIWFS technology, a single frame replaces several wave front sensors, or the need to scan the field when using a single one.

Below an example of a field curvature analysis, including the Petzval’s surface, while testing on the bench in double pass an achromatic lens using 7 artificial stars, figure extracted from the SPIE proceedings [2].