# Collimation Testing with the Talbot and Moiré Effects

## Introduction

This research came about from my interest in moiré effects. After I learned of the moiré early on in my summer at Stony Brook I read many papers about moiré patterns and created various simulations in Mathematica. I presented these models on my Simons program poster. Only near the end of the summer program did I learn that the moiré effect has an interesting application in optical testing, more specifically, in determining the collimation (wavefront curvature) of a beam of light. This method also relies on the Talbot effect, a very interesting optical phenomenon.

The moiré effect appears in a wide range of situations, but in the context of this setup, a simple definition will suffice. Moiré patterns can be observed when two linear gratings of different period are overlapped. The disparity in period causes a "beat" pattern to emerge from the differing amounts of overlap, known as a "moir pattern. If one grating is simply placed on top of the other, so that grating lines are parallel, the difference in period manifests in the number of moiré fringes observed in a given area. The closer in period the two gratings are, the fewer the number of fringes. If, however, one grating is overlapped and then tilted with respect to the other, then period difference affects the tilt of the observed moiré fringes.

The Talbot effect occurs when a laser beam is shone through a grating; a self-image of the grating is formed at fixed distances (the Talbot distance) from the original grating, so long as the distance from the grating is still in the near field. In the far field, the self-image is replaced with the expected textbook diffraction pattern.

Together, these two effects allow for a simple, but effective, method of determining the collimation of a Gaussian beam.

## Setup

A 150 micron pinhole is placed 8 centimeters away from a 633 nm HeNe laser light source. 15 centimeters from the pinhole, two lenses are arranged so that they magnify the beam. The first has a focal length of 25.4 millimeters, and the second, the collimating lens, has a focal length of 330 millimeters. In order for the laser beam to be collimated, the collimating lens must be placed so that its focus coincides with the focus of the first lens. If the collimating lens is moved closer to the first lens, then the beam will diverge, and vice versa. A Ronchi grating (50 lines/ inch) tilted a minute amount is placed about 99 centimeters from the collimating lens. A retroreflector is placed at half of the grating's Talbot distance away from the grating.

Incident light passes through the grating and forms a self-image as it hits the retroreflector, half the Talbot distance away. The self image is reflected twice, so that the returning light forms the self-image of the grating reflected about the y-axis. This oppositely tilted self-image passes once more through the original grating, creating moirfringes. A mirror placed just before the grating is angled so that it does not block any light, but reflects the fringes off to the side, where they are photographed through white paper with a camera. A collimated beam has a radius of curvature approaching infinity-- or rather, a wavefront that approaches planar. As collimation decreases, the radius of curvature also decreases and the wavefront becomes spherical, creating either a divergent or a convergent beam. If the wavefront is spherical rather than planar, it follows that the resulting Talbot self-image will be either magnified or diminished in size. This magnification of the image of the grating, or lack thereof, is equivalent to placing two gratings of different period on top of each other. Furthermore, the self image is tilted with respect to the original grating, so that the resulting moirfringes will similarly tilt as the period of the self image changes. Following the formula specified by Creath and Wyant (1992), the tilt of the moirfringes can be used to determine the difference in period between the grating and the self-image. This information can be used to find the radius of curvature of the incident light, which can then be used to find out how far the collimating lens must be moved in order to achieve collimation. When the incident light is perfectly collimated, the period of the self-image will be equal to that of the original grating. In this case, the fringes produced will be horizontal. As the beam moves away from collimation, the fringes will acquire a greater tilt from the horizontal; furthermore, fringes produced by divergent light will tilt in an opposite direction from fringes produced by convergent light.