Gaussian Beam Modeling of the Radius of Curvature

A Gaussian model of the radius measurement of micro-optics has been developed and tested using simulations. The model is based on the propagation distances in the interferometer, a heretofore uninvestigated effect. The goal of the model is to determine the bias error in the radius due to the Gaussian Beam propagation model. After testing the model with varying conditions, we have concluded the following: the measured part is smaller than the input, the cat’s eye and confocal positions have approximately the same error, radius error increases with smaller test parts, decreasing the numerical aperture increases the errors, and the propagation distances do not affect the radius. The outline of the experimental plan to be used to verify the results is given. 1.0 Introduction Micro-optic components are the key to building compact optoelectronic systems and are used in many applications such as optical networks, medical imaging, and optical data storage. These components have aperture diameters from 10 micrometers up to fractions of a millimeter. A crucial step in the manufacturing of the refractive micro-optic lenses is the measurement of the radius of curvature (ROC). The interferometric measurement of ROC is defined as the distance between the confocal refection and the cat’s eye retro-reflection [1, 2, 3]. The confocal position occurs when the optic is placed in the test beam of the interferometer such that the wavefront curvature of the incoming beam and the curvature of the test optic match. In the geometric model, the cat’s eye position occurs when the surface of the optic is coincident with the focus of the test beam. Traditionally, a geometric model of each position is used to describe the ROC measurement [1]. For micro-optics and high precision macro-optics, this model is not adequate to describe the ROC measurement. We believe a Gaussian beam propagation model is required and will offer lower ROC measurement uncertainty due to eliminating a systematic bias. This paper describes Gaussian beam approximations used to model the radius measurement and proposed experiments to validate the model. 2.0 The Radius Measurement A schematic of the radius measurement is shown in Figure 1. The laser and collimating optics (not shown) produce a collimated beam with a well defined waist that enters the beam splitter where the light is split into two arms, the reference arm and test arm. In the reference arm, the beam propagates a distance of dr, reflects off the reference mirror (labeled Ref. mirror) and again propagates a distance of dr. In the test arm, the beam propagates a distance of dt, is focused by a lens with focal length, f, propagates a distance s, reflects from the test optic with radius r, propagates a distance s, is collimated by a lens with focal length, f, and is propagated a distance dt. At the beam splitter, the test and reference arms interfere and are focused onto the camera using imaging optics (not shown). Note that in Figure 1, the ray traces are shown to demonstrate the operation of the interferometer and do not indicate the Gaussian beam.