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### INTRODUCTION

Zernike modes are an infinite series of polynomials that can be used to describe surface shapes on the unit disk. They are often used in optics to describe and quantify wavefront aberrations in mirrors and lenses with a circular aperture.

Zernike polynomials are defined in a polar coordinate system with radius and angle . Each Zernike polynomial is uniquely defined by its radial order and angular order .

Where:

- : Normalization term
- : Radial polynomial
- : Angular term
- : Zernike coefficient

Each mode consists of an order polynomial over the radius of the unit disk. This polynomial is multiplied with a sinusoidal term to obtain a surface shape of a given mode:

The normalization term gives the Zernike mode an RMS value of 1.

With the Kronecker delta function, which is for and in all other cases. The Zernike coefficient defines the amplitude of the given mode. The double indexing is often replaced by a single index according to the Noll or Wyant expansion schemes.

Much like a Fourier transform, any wave front shape can be described by the sum of an infinite number of Zernike polynomials. A vector containing the Zernike coefficients uniquely describes the wave front. Since the polynomials are orthonormal, the cumulative RMS wave front error is given by the Euclidian norm of the vector :

Since the modes are orthogonal, they are mutually independent. Expanding the approximation with higher order terms does not influence the coefficients of the lower order terms.

Zernike polynomials are often used in beam optics, since they represent common optical aberrations. For instance, the first three modes (piston, tip & tilt) represent the orientation of an optical component in the system, and by manipulating the mode, the focus of a system can be controlled.

At JPE we use Zernike polynomials in our Cryogenic “Set-and Forget” Deformable Mirror. Optical components like lenses and mirrors create static aberrations in the wave front due to manufacturing and alignment tolerances. These errors can only be measured once the system is installed. Infrared systems are cooled down to cryogenic temperatures, making the access to iterative re-alignment very difficult.

A detector measures the shape of the wave front error and a controller determines the first 15 Zernike coefficients. The required surface shape of the deformable mirror is determined. PiezoKnob actuators are used to deform the mirror surface and correct the wave front error. Once the desired shape is reached, the actuators lock on friction and do not dissipate any energy.

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