# Technical articles

Matching of the wavefront corrector to the statistics of aberrations in the optical system can drastically improve the performance of adaptive optical system and reduce its price and complexity.

## Introduction

Since adaptive optics was suggested by Babcock [1], a great number of wavefront correctors has been proposed, realized and used to improve the performance of adaptive optics in science, medicine, and industry. However, many applications of adaptive optics still belong to researchers. Scientists, in their vast majority, are not interested in adaptive optics as a technology. They just want to use it as an invisible tool that helps to reach their goals, usually unrelated to adaptive optics.

Adaptive optics is just a means to improve the performance of otherwise impaired system. Ideally, it should be invisible, cheap, and do exactly what is expected, causing no trouble. But in real life, introduction of adaptive optics usually means a considerable increase in the system complexity, bringing advantages and problems in a single package.

The mutual goal of the developers and users of an AO system is the optimization of the performance, keeping the complexity low. Less complexity means higher reliability, simpler, cheaper and easier to set up and more robust to control adaptive optical system. The goal of high performance and low complexity in one package can be reached by tuning the parameters of the adaptive optical system to the expected aberrations. Such a tuning can save a lot of effort and money, and the parameters to tune are simple and easy to understand.

## Wavefront correction

Adaptive optics can be used to optimize almost any measurable parameter of an optical system. A very short list includes the output power of a laser, laser beam quality, Strehl number, image sharpness, pulse shape, pulse duration, beam focus, etc. The improvements are generally achieved by changing the wavefront of the light wave. Commonly, the correction consists in the approximation of the aberration function A by a combination of N modes $\psi_i, i=1,\dots,N$ of the wavefront corrector, minimizing the residual aberration

where $R$ is the residual error, $a_i$ are the control signals and $\psi_i$ are the modes of the corrector.

In practice, first the aberration $A$ is minimized by aligning the optical system. If the result is not satisfactory, adaptive optics can be introduced. The control loop of the AO system minimizes $R$ by a choice of the best possible combination $a_i$. If the correction is not satisfactory, the common wisdom is to increase the number of degrees of freedom $N$. Large number of control channels usually means a complex and expensive adaptive optical system. The described approach leads to a commonly accepted wisdom that a good adaptive optical system should have many control channels and, as a consequence, be expensive.

However, in many cases, a very good result can be achieved with only a small number of correction channels, by statistically matching the influence functions $\psi_i$ to the most expected aberrations.

For instance, if the most expected aberrations are tip, tilt, and defocus, it is sufficient to use an AO system with only three influence functions - also represented by tip, tilt and defocus - to reduce the residual aberration practically to zero. Of course, a system with a larger numbers of control channels can be also used, and certain improvement will be achieved, however, if all these functions are not statistically matched to the aberrations, the result can be far from satisfactory. A good example of such a situation is given by deformable mirrors with uncoupled piston actuators. Approximation of smooth shapes, such as tip, tilt, and defocus with step-like function is either unsatisfactory, or requires a very large number of control channels $N$.

## Statistics of aberrations

The expected aberration can be represented as a series over some set of orthogonal functions with random weights. The best possible fit, reaching maximum precision with the minimum number of modes is given by the Karhunen-Loeve functions, with eigenvalues corresponding to the statistical weights of the corresponding terms. In general, each optical system has its own aberration statistics, requiring building a specific set of Karhunen-Loeve functions, which is usually impossible, because the statistics is unknown. Fortunately, a well-studied case of the atmospheric turbulence gives some useful clues. It was found that the Karhunen-Loeve functions of the aberration statistics of the atmospheric turbulence are very close to Zernike polynomials. On the other hand, classical theory of optical system also uses Zernike polynomials to describe the most important aberrations of optical systems: tip, tilt, defocus, astigmatism, coma, trifoil, and spherical aberration. Thus, we can assume that in most cases, the aberrations described by the first Zernike terms will be the most prominent in a general optical system, regardless of their nature. This is a rather strong sentence and there is a great chance that it is not always correct, but we just do not have any better.

## Quasi-optimal correctors

As explained earlier, the tip-tilt (misalignment) and third-order Zernike terms are the most statistically significant aberrations in the majority of optical systems. Correction of these aberrations dynamically is the primary goal of adaptive optics. Fast correctors of tip-tilt (scanners) are available from a number of companies. They close the gap between the "traditional" and the "adaptive" optics, in a sense that they provide fast dynamic correction, but are not really adaptive. One has to expect the tip-tilt correction to be a simple task, however the dynamic correctors are quite expensive, especially if they work in a calibrated feed-forward mode.

And there is a problem of two separate correctors: usually a scanner stage is used for tip and tilt, and a separate deformable mirror is used to correct higher order aberrations. Since both correctors should be positioned in the system pupil, an additional imaging telescope is applied to achieve the pupil conjugation, resulting in a more complex, less compact system with higher losses and scattering. In some cases it is possible to use the deformable mirror in a scanner mode - most deformable mirrors can correct small amounts of tip and tilt, however it restricts the correction range for other aberrations, and the quality of correction is usually unsatisfactory as tip and tilt do not belong to the eigenfunctions of a typical deformable mirror. To solve this problem, we, in OKO Tech, have developed a special corrector, featuring a membrane deformable mirror with 17 actuators [2] for correction of all 3-rd order aberrations, mounted on a fast tip-tilt stage. The mirror is shown in Fig. 1.

Figure 1: 17-ch membrane deformable mirror with integrated tip-tilt capabil-ity (left) and 19-ch piezoelectric mirror optimized for correction of low-orderZernike terms.

The mirror uses 15 mm micromachined membrane with 10 mm working aperture, that provides 3 mrad of tip and tilt in a frequency range of 200 Hz and also corrects all third order aberrations with maximum amplitudes of several wavelengths in a frequency range of up to 600 Hz.

Appropriately coated micromachined membrane mirrors can work with continuous laser power of up to hundreds of Watts, however they are not suitable for laser applications in the kiloWatt range. For these applications, we have developed a piezoelectric deformable mirror with a special actuator configuration providing very good correction of low-order Zernike terms in the aperture of 20 mm (30 mm mirror), and 35 mm (50 mm mirror). Response optimization is achieved by positioning of most of the actuators outside the clear aperture of the mirror [3], providing the best possible match between the mechanical response of the mirror, and the statistically most expected aberrations, as seen in Fig. 2. These mirrors can be specially coated for high-power applications with continuous power densities of up to several kW/cm2.

Figure 2: Interferometric patterns corresponding to low-order aberrations formed with the optimal 19-ch piezoelectric mirror: defocus, astigmatism,coma, trifoil, spherical aberration, Z24 (left to right, top to bottom).

## Conclusions

In many scenarios the goal of simple, efficient, and inexpensive adaptive optics can be achieved with a minimum effort by using correctors matched to the statistics of the expected aberrations. Such correctors, combining deformable mirror and tip-tilt stage in a single device, offer good performance for a fraction of the price that should be paid for a more complicated multichannel systems with separate subsystems for tip-tilt and adaptive optics.

## References

[     1] H. W. Babcock. The Possibility of Compensating Astronomical Seeing. PASP, 65, October 1953.

[     2] Gleb Vdovin and P. M. Sarro. Flexible mirror micromachined in silicon. Appl. Opt., 34(16):2968-2972, 1995.

[     3] Gleb Vdovin, Oleg Soloviev, Alexander Samokhin, and Mikhail Loktev. Correction of low order aberrations using continuous deformable mirrors. Opt. Express, 16(5):2859-2866, 2008.