Hollow Laser Beam Web Report

by
Matthew Eardley
Department of Physics,
SUNY at Stony Brook
Advisor: Luis Orozco

I. Introduction

The goal for this project is to use spiral zone plates to generate a propagating beam of laser light that has a dark region near the center, whereas the normal profile of a laser beam is guassian. This builds directly on the work of Lucas Finco , which can be viewed here. I extended Finco's work by inceasing the paramater "l" in the equations that generate the spiral zone plates, generating six different zone plates that produced interesting results.

II. The Setup
A. Apparatus

Below is a sampling of the spiral zone plates, with the associated value of "l". The spirals were reduced to a size of about 0.72cm square, and laser printed onto a transparency.

l=2

l=6

l=12

The picture above shows the experimental setup. The the two lenses after the HeNe form a telescope that increase the size of the beam by a factor of 8, and the shutter is used to make sure the entire area of the zone plate is illuminated without spilling over the edges. The focusing lens creates the propagating hollow beam, and a CCD camera captures the image. If we think of the spiral as a Fresnel zone plate, then it has a primary focal length given by:
f=R^2/(m × L)
where R is the radius of illumination of the zone plate, m is the number of rings in the zone plate, and L is the wavelength of the light. (Hecht, p.446). In our case, R=0.36cm, m=23, and L=633nm, giving f=85cm. The original idea was to place the focusing lens at a distance equal to the focal length of the lens plus the focal length of the zone plate, thus creating a telescope and a parallel propagating beam. This did not work in practice; instead I adjusted the position of the lens to maximize the hole in the output beam. For most of the experiments, the distance between the zone plate and the focusing lens was 48cm, and the distance between the focusing focusing lens and the CCD camera was 107cm. The result was a beam with a hole in the center that was propagating but getting larger with distance, as can be seen in the Results section.

III. Results

A. Spirals of Increasing l

Below are the images generated by the spiral zone plates. Click on an image to view the associated intensity plot. The images show that the hole in the beam gets larger as l increases.

To plot the intensity of the images across a line going through the "hole" in the image, I used an image editing program. Below is a plot of the ratio of average intensity of light inside the hole to the average intensity outside the hole as a function of the parameter l. The plot shows that not only does the hole get larger with l, it also becomes darker.

B. Polarization

The spiral zone plates do not appear to affect polarization. We used two linear polarizers, one on either side of the zone plate, to prove this.

C. Propagation

The images below were taken with the CCD camera at different distances from the focusing lens, to show that the hollow beam is propagating in space. This also shows that it is getting larger, a fact that we were unable to compensate for.

l=12 small

d=76cm

d=107cm

d=145cm

D. Further Work

The most obvious route for further work is to use even higher values of the paramter l. This is only limited by the computer power for generating the images, and by the resolution of the printer. Further investigation into the focusing properties of the beam is warrented, to see if there is a way to make a propagating beam that is not diverging.

Conclusion

The images on these pages show that we were able to generate a propagating, diverging beam with a dark region in the center. This is an interesting application of classical optics that gave very good results. This project was not only interesting but useful to me in that I learned several basic techniques of experimental optics (building a telescope, using zone plates, polarizers, etc).

References

Hecht, Eugene; Optics, 2nd Edition (1987).
Finco, Lucas; "Hollow Beams" http://grad.physics.sunysb.edu/~lfinco/Hollow/one.html
Heckenberg, N.R. et al; Optics Letters 17, #3, 221(1992).

Acknowledgements
I would like to thank Professor Luis Orozco for advise and direction with this project. I would also like to thank Lucas Finco for getting me started by giving me all the materials and explanations from his project.