Results
Using the diffraction grating equations and Zernike Polynomials, we were able to make quantitative predictions for the focal lengths and diffraction angles for each diffraction order. As shown before, we also used our theories to make qualitative predictions on the the structure of the diffraction pattern.
For our first experiment, we uploaded a zone plate pattern unto to the DMD. The surface of the DMD was positioned normal to the incoming beam of light.
Below are graphs representing the predicted angles of diffraction and focal lengths with the measured angles of diffraction and focal lengths.
As the diagram above shows, the observed angles diffraction correlated with the predicted angle of diffraction.The observed focal lengths also correlated with the predicted the focal lengths.
We found that the diffraction pattern took the structure of a 2-dimensional array, which was expected based on our theory. The diffraction spots outside the horizontal plane of DMD chip were significantly dimmer than those on the horizontal plane.
We also found multiple foci along the axis of each order, which also corresponds the Fresnel Zone Plate pattern.
Since the DMD was positioned normal to the incident light, the angle of specular reflection was 25 degrees (due to the 12.5 degree tilt of the individual micro-mirrors). The 5th, 6th and 7th diffraction orders were the most pronounced because they were closest to the angle of reflection.
Below is a graph showing the relative intensities for each of these orders.
As the diagram above shows, most of the energy was deposited in the 5th order but the 6th and 7th order still received signifcant amounts.
However, this broad distribution of energy is very inefficient. Ideally, we would want almost all the energy to be deposited in one specific order. This would ensure that almost all of the energy would go into one beam of light, creating a single distinct LGS.
So we repeated our experiment by making the angle of reflection equal to the angle of diffraction for 9th order. This is known as the blaze condition for ninth order diffraction. The blaze condition was acheived by rotating the DMD such that the reflected light coincided with 9th order diffraction angle.
The subsequent relative intensities for the 7th, 8th, 9th and 10th diffraction orders are shown below:
The above diagram shows that almost all the incident light energy was deposited the 9th order. Therefore, we managed to make our technique very efficient.
Our next step was to improve on the image quality of the diffraction spots. The image below is an example of one of the diffraction spots.
Ideally, the diffraction spot should focus down to the size of pixel but instead our image had a very scattered linear point spread distribution.
This vertical point spread distribution is due to the orientation of the DMD with respect to the CCD camera. The surfaces of the DMD and CCD were not parallel to each other, so the CCD had a slanted view of the Zone Plate pattern.
Hence from the perspective of the CCD camera, the circular zone plate pattern would appear more like ellipses. This effect causes an aberration known as astigmatism, which can be corrected using the appropriate Zernike polynomials.
We then re-made the zone plate interferogram by altering our old Zernike expression to compensate for the astigmatic point spread distribution. The images below show the actual interferograms loaded unto the DMD, before and after the correction for astigmatism.
The new interferogram is identical to the first except that now it looks a bit stretched. Therefore, the zone plate pattern will look circular instead of ellipic when view from the perspective of the CCD camera.
Below is a comparison the point spread distribution with and without astigmatic correction.
By correcting for astigmatism, we managed to reduce the linear point spread distribution by a factor of 10. More importantly, we successfully used the DMD device to correct for a specific optical aberration!
Nonetheless, at 0.1 mm in length, the diffraction spot is far from it's ideal size. Ideally, we want to the diffraction spot to come to focus on a single CCD pixel.
This means we want the spot to focus in an area of 10 microns by 10 microns.
This last image is a contour map of the DMD's surface taken by our phase shifting interferometer.
This image shows that the surface of the micro-mirror array has many irregularities. These surface irregularities account for significant distortions of the diffracted wavefront.
To correct for these aberrations, we planned to use our Phase Shifting interferometer to decompose the diffracted wavefront into it's Zernike polynomials, making for a much more accurate measurement of the wavefront distortions. The decomposition could be done using the interferomter's software.
The Zernike functions would give us a quantitative expression for the aberrations caused by the irregularities of the DMD surface. With this information we should be able to create a new Zone Plate pattern that would exactly compensate for these aberrations.
Unfortunately, our REU program came to an end before we could carry out this next step.
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