Method
Setup
The interferometer is used to produce a collimated light beam. The light beam is diffracted off the micromirror array on the DMD. The CCD camera is used without lenses to observe focused images that are diffracted off of the DMD. The translation stages allow the camera to observe different angles of diffraction and different focal lengths.
The DMD used in the experiment was a Texas Instruments Discovery 1100 chipset. The DMD contains a 1024 by 768 square grid of micromirrors with a pitch of 13.68 μm, each of which can be tilted at ±12.5 degrees. A Texas Instruments software GUI allows the DMD to be controlled via USB 2.0 connection.
Diffraction Gratings
A diffraction grating consists of a repeating pattern, such as grooves or mirrors, which causes diffracted light to interfere constructively or destructively at different angles. The diagram below schematically illustrates this process.
Incident light comes in at an angle α, and diffracted light goes out at an angle β. The condition for constructive interference is that the optical path length difference between light diffracted by two adjacent mirrors must be an integer multiple of the wavelength. In the above diagram, the green and magenta segments highlight the path length difference. From this, we can derive the grating equation as shown above.
In the above diagram, the mirrors are not parallel to the grating surface. If the mirrors are oriented such that the angle of diffraction coincides with the angle of specular reflection, we will have maximum diffraction efficiency in that order. In this case we say that the blaze condition is satisfied, and nearly all of the energy will be concentrated into a single diffraction order.
Fresnel Zone Plates
A Fresnel zone plate, as to the right, can be thought of as a diffraction grating with a variable spacing. As we move radially outward from the optical axis, the spacing changes in such a way that the diffraction angle directs the light toward the focal point (see diagram below).
The Fresnel zone plate acts as a diffractive focusing lens. An important consequence of the diffractive nature of the zone plate is that higher orders of diffraction create images with shorter focal lengths.
Holographic reconstruction
The principles of the Fresnel zone plate can be extended to create a zone plate pattern that, when illuminated, diffracts a wavefront with an arbitrary surface aberration. We start by expressing the desired surface aberrations in terms of a set of functions that are orthogonal on the unit disk called Zernike polynomials. Then, a MATLAB script we wrote converts the surface into an aberrated zone plate pattern using the principles of holography. Light incident on this aberrated zone plate pattern will then diffract into a wavefront with the desired aberrations.
Fraunhofer diffraction theory
A Fresnel zone plate pattern was programmed onto the micromirror array. The DMD will act as a diffractive focusing lens, which can easily be observed by placing the camera in the focal plane. However, there are two diffraction effects occurring simultaneously. First, the zone plate pattern causes diffraction. Second, the micromirror array, acting as a two dimensional diffraction grating, causes a second layer of diffraction. Because these two effects are inextricable, a rigorous electromagnetic treatment would be required in order to quantitatively calculate what the result would be. However, we can use the principles of the Fraunhofer theory of diffraction to make some qualitative predictions as to what we will see.
Under the appropriate approximations, the object plane (where the DMD is) and the image plane (where the camera is) are related by a Fourier transform. Since the zone plate pattern and micromirror array are multiplied in the object plane, they are convolved in the image plane. The diagram below illustrates this schematically.
Therefore we expect a regular array of spots, each of which is the image of a Fresnel zone plate. An important consequence of this is that we expect each spot to contain multiple foci.