Wavefront model software

Aberrations are deviations errors from this expected spherical wave due to limitations, imperfections, tolerances, and miss-alignment of the optics. Meaning the actual wave converging to the focal plane is not a prefect spherical one anymore. The figure above, on the left side, shows two similar imaging systems say telescopes. The top one is perfect diffraction limited , DL , the resulting spherical wave converges to the focal plane where the PSF is located.

The rays, which are by definition perpendicular to the wave front, intercept the focal plane at one point ray tracing from geometric optics , they all are parallel to the spherical wave radii in direction of the sphere center the image point. In the DL case there is no error, hence a uniform yellow disk. The color bar on the right indicates, from a qualitative view point here, the wavefront errors. Reddish colors mean peaks in the wave front error surface, meaning the the actual wavefront leads the expected perfect wavefront one, while blueish colors mean valleys troughs in the wave front error, the actual wave front lags behind the perfect one.

One usually expresses those wave front errors in term of manometers nm or in term of wave for a given reference wavelength. Since the EM radiation light in our case amplitude is a periodic sinusoidal function a departure of one wave means a phase shift of degrees 2 Pi.

Hence the name phase error. No phase error means that the PSF is perfect and the system is diffraction limited. The system shown below the DL one exhibits some distortions in the wave front, leading to wave front errors expressed by the various color shades in the wave front heat map. One can clearly see one peak and one trough on its 3D version next to it.

This is the fingerprint of coma, which makes the PSF looking like a comet shape. On the right part of the figure there is examples of basic aberrations with their names, wave front errors heat maps and related PSFs. This technique is known as curvature sensing, or CS in short. CS relies on taking two near instantaneous images few ms apart of the same star from two different locations in the optical path, near the focal plane, in order to cancel out seeing scintillation for AO.

It has been shown that a single defocused image of a star is enough for retrieving WF phase information [2]. The figure below shows, in the context of a refractor, the concept, finding the WF error from the irradiance image intensity of a defocused star.

It exists, under the proper conditions, a mathematical relationship between the irradiance intensity profile and the WF described by a non-linear differential equation known as the irradiance transfer equation, or transport of intensity equation TIE.

In order to minimize computing time as well as power one usually relies on some approximations typically linearity which may be valid only under some defocus conditions and limited amount of aberrations. This method for solving the TIE at run time is know as the direct model approach. By leveraging Innovations Foresight extensive knowledge and long experience in machine learning ML and AI we designed a novel approach [3] which learns the inverse model , the relationship between the defocused star image intensity and the WF, often expresses in a parametric way using the Zernike polynomials.

T his is not a limitation other representations of the WF and related aberrations can be learned too. When using a defocused star one essentially adds a bias, known as phase diversity PD , to the WF which in turn allows for retrieving the phase error without any ambiguity as long as the measured aberration magnitudes are less than the diversity term.

Both images are identical, meaning WF sign error is lost, there is ambiguity. As stated before because we do not have an analytic solution for the TIE differential equation we use a numerical approach. The main and key difference between the direct model discussed above and the inverse model is that the latter is computed beforehand once for all, therefore the run time calculations are straightforward and very fast, we can process at video rates if needed for applications such as AO.

OF telescope collimation on the other hand we use longer exposures in the order of 10 to 30 seconds in order to average out the seeing since we are only interested, unlike for AO, in the telescope aberrations, not the ones induced by the seeing. Since we are not restricted by any real time constraint for solving the TIE we can spend a lot more time learning the inverse model, days, weeks, or more if needed.

This also means that we do not have to make any approximation, assumption, or limit the technique to any subset of aberrations and defocused ranges, making this method much more generic, flexible and useful than solving at run time for the WF using the TIE. Multi and extended sources can be considered as well.

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