Introduction

The idea for this project came from a general discussion of diffraction. Looking at Young's double slit experiment, I was tasked to derive a function that would describe intensity of light with respect to the distance, y, from the center of the image plane to the point in question. The experiment and the derivation both make the assumption that the distance between the two slits and the image plane is very large compared to the size of the slits themselves. This is the Fraunhofer approximation.

The Fraunhofer approximation offers a limited simplistic model for the study of diffraction, and so we decided to take a trek away from this. We wanted to examine the more intriguing and complex nature of diffraction in the near-field, where (for a circular aperture) an observer will find alternating patterns of bright and dark centers and an incrementally increasing number of rings.

Theory

Diffraction is the phenomenon in which waves encounter an obstruction such as an aperture and are thereby redirected in new directions according to a principle first described by Huyghens over 300 years ago. While this project is concerned with visible light waves, the same principles apply to other types of waves, including sound and water waves. Interference between these diffracted waves results in distinctive patterns, called diffraction patterns. (If there is more than one aperture the patterns are generally called "interference patterns," but regardless of the number of diffracting objects both diffraction and interference are involved in forming the patterns.) Relatively close to the obstruction or aperture (in the "near field") the patterns vary with distance to the observing screen, but at large distances (in the "far field") the patterns remain the same except for an overall increase in size in proportion to distance. The transition from near-field to far-field can be quantified with the Fresnel number F = a ² / λ L, where a is the radius of the aperture, λ is the wavelength, and L is the distance between the aperture and the image plane. The near-field has Fresnel number roughly equal to or greater than 1, while the far-field has F << 1. Diffraction is often demonstrated with laser light incident on small apertures (eg 100 microns diameter). In such experiments the far-field pattern is easily observed by eye at a distance of several meters, but the rich complexity of the near-field patterns isn't evident as these occur just a few mm from the aperture on a microscopic scale.

The goal of our project is to explore these fundamental ideas with simple experiments in which coherent light from a red HeNe laser (wavelength 633 nm) is incident on a variety of pinhole apertures, using various techniques to record the resulting diffracton patterns at a range of distances from the near field into the far field. Our best results so far have been obtained with a 0.50 mm diameter commercial aperture - with this aperture the near-field transition (F=1) occurs at about 400 mm from the aperture. We magnified the near-field patterns with a microscope objective lens and recorded them with a normal consumer camera. A distinctive feature of the patterns is the appearance of a dark central spot at certain distances L; at other distances a bright central spot is surrounded by one or more dark rings.

Experimental Setup

The experimental set up we used went through a few stages of evolution. Initially, we intended to use a ThorLabs CMOS camera to image the Fresnel diffraction. However, once installation was complete, we found that the quality of the images to be very poor. This was disappointing, since we had set up mirrors to reflect the HeNe laser beam around the table, allowing it to expand in order to saturate the iris of the camera. We scrapped this setup and considered the use of an objective lens to magnify the otherwise minute images.

Using a 20x objective lens, we managed to form easily photographable images at a mere 2 meters away. While these images weren't very clean, this was clearly a suitable substitution for the cameras. After further experimentation, we wound up using a 10x objective lens which we had fixed to a 1-inch translation stage. This gave clearer images, and with a transfer between holes on the translation stage, we were able to run through about Fresnel numbers 2 through 11, as well as an image of the aperture itself, taken at a distance of 8mm between the lens and the aperture.

 

Data

Fresnel Numbers 2 through 11 in order: The final two pictures are An unmagnified airy disk and the aperture image: The images were recorded with a DSLR consumer camera placed behind the observing screen.

Results and Analysis

Our recorded data consists of translation readings Rn at which we observed patterns with dark or bright centers, corresponding to even or odd integer Fresnel numbers. The separation of these R values is relatively accurate, but they share a (very nearly) common distance offset due to the unmeasured mechanical offset of the lens from the aperture and uncertainty in the position of its optical principal plane. We accounted for this in the spreadsheet analysis by introducing a parameter that relates the measured and absolute lens positions. The positions of our data points in the figure are very sensitive to this parameter and for d = 1.75 +/- 0.01 inches we obtain excellent agreement with the prediction given by the Fresnel equation (straight line through the origin).