# Earle Wilson(Morehouse College)

## Diffractive Wavefront Control Using a Digital Micro-mirror Device

 MOTIVATION SET-UP THEORY RESULTS CONCLUSION ACKNOWLEDGEMENTS FINAL COMMENTS

### Theory

For our experiment we chose to put a Fresnel Zone Plate pattern on our DMD chip. The Fresnel Zone Plate is a radially symmetric pattern with variable spacing that alternates between opaque and transparent. An example is shown below.

By using this pattern we can apply a focus aberration to the incoming plane wavefront, which essentially adds curvature to the plane wavefront - bringing the wavefront to a focus. The Fresnel Zone Plate focuses light waves by way of diffraction. For our purposes, we can interpret the Fresnel Zone Plate as a diffraction grating with variable spacing. The diagram below illustrates this concept.

The diagram above shows light traveling through a diffraction grating of multiple spacing. The spacings are varied such that transmitted light comes to focus at multiple focal points on the optical axis. In essence, the Fresnel Zone Plate works as 2-dimensional diffraction grating of multiple spacing. So for an incoming light beam, we would expect the Fresnel Zone plate to diffract the light rays to multiple foci are on the optical axis. By using the grating equation sinα + sinβ = mλ/d (where 'α' is the angle of incidence, 'β' is the angle of diffraction, 'm' diffractive order, 'λ' wavelength and 'd' distance between each spacing), we can predict the angle of diffraction and the corresponding focal lengths by way of the grating equation.

The arrangement of the micro-mirrors would also contribute to the diffraction pattern. The square-grid array of mirrors acts a 2-dimensional diffraction grating. On it's own, the micro-mirror array would create a square grid array of diffraction spots. To predict the final diffraction pattern, we use principles from the Fraunhofer Diffraction theory.

For our approximations, we took the object plane to be the surface of the micro-mirror array and image plane to be surface of CCD. Both these planes are related by a Fourier transform. Since the Fresnel Zone Plate pattern and micro-mirror array pattern are both in the object plane, they are convoluted in the image plane. The diagram below illustrates this concept.

Hence we would expect to see a square grid array of Fresnel Diffraction Patterns.

It is worth mentioning that other factors will contribute to the observed diffraction pattern. Such factors include the shape of each mirror, the surface of each mirror, and the surface of the entire array. However, the Fresnel Zone Pattern and square grid array are the primary contributors to the final diffraction pattern observed in the image plane.

### Making the Fresnel Zone Plate pattern

For the purposes of our experiment, we designed a Fresnel Zone Plate pattern that could apply a focus aberration to an optical wavefront. We started out by expressing the intended wavefront and its aberrations as Zernike polynomials, which are a sequence of polynomials that are orthogonal on a unit disk. Using a MATLAB script, we converted these Zernike Polynomials to a specific surface aberration pattern - called an interferogram. Below is an example of a fresnel zone plate interferogram.

This interferogram (hologram) would take the form of the Fresnel Zone Plate pattern. We would then upload this pattern as bitmap image, by way of the DMD's user software, unto the DMD surface.We would then use the principles of holographic reconstruction to produce the intended wavefront. So by illuminating the zone plate pattern with a coherent light source, a focus aberrated wavefront will be created.

 MOTIVATION SET-UP THEORY RESULTS CONCLUSION ACKNOWLEDGEMENTS FINAL COMMENTS