<title>High Precision Angle Measurements Off Small Surfaces</title>

Abstract

High Precision Angle Measurements Off Small SurfacesRussell A. ChipmanOptical Sciences Center, University of ArizonaTucson, Arizona 85721AbstractA method is described for measuring the orientation of small plane surfaces several tens ofmicrometers on a side with a resolution of several seconds of arc. A laser beam waist islocated on the test surface which underfills the surface. The reflected beam is detected by aquadrant detector to determine the offset of the beam from its nominal position. Thisdisplacement determines the relative angular difference from a predetermined angular standard.Advantages of this method for precise angle determination are the ability to measure smallsurfaces, insensitivity to translations of the test piece, large test piece clearance forinaccessible surfaces, and compensation for laser drift.IntroductionPrecise angle measurements from small surfaces present several problems. Conventionalmethods of angle measurement have difficulty measuring the angle of small plane surfaces in theten to one hundred micrometer range. Such measurements are of interest for submillimetergratings, small optical elements for integrated optics, corner cube arrays for traffic safetyequipment, and precision machinery. Because of the small size, substantial diffraction isassociated with the surface. Autocollimators work poorly because of the small amount of lightreturned. Interferometers have limited accuracy due to the small baseline over which fringesmay be counted.The method described here has several desirable features and does not require a criticallyaligned setup.ConfigurationThe angle measurement setup is depicted in Figure 1. The output of a laser is divergedthen focused by two simple lenses. The beam is folded by a beam splitter and is incident atvery nearly normal incidence on the test surface. The lenses are arranged so that the lasers'Gaussian beam waist is at the test piece and underfills it by a factor of two or three. Thelight reflects off the test surface back through the beamsplitter and onto the quadrantdetector.The quadrant detector is a mosaic of four square detectors abutted along their sides. Theoutputs of the separate quadrants are added or subtracted in combinations to characterize thelocation of the laser beam with respect to the vertex. Adding the output voltages from all fourquadrants yields a voltage proportional to the total power incident on the detector:VT = V1 + V2 + V3 + V4. The voltage Vg = V1 - V2 - V3 + V4 describes the x - coordinate of thebeam relative to the vertex. Similarly, Vy = 172 + V2 - V3 - V4 characterizes the y - coordinateposition of the beam. If Vx = Vy = 0, then the median of the beam is centered on the quadrantdetector.ProcedureThe procedure for measuring angle is outlined in Figure 2.First a reference angle must be established with reference tooling and a reference mirror.The reference mirror is inserted into the test setup and the quadrant detector is positioned sothat Vx = Vy = O. The median of the reference beam is located at the center of the quadrantdetector, and now the reference tooling is removed.Next, tooling holding the test piece at the reference angle is positioned in the setup.If the test piece has the desired angle, its reflected beam will be centered on thequadrant detector. The total power incident on the quadrant detector is measured to correct forvariations in test piece reflectance. Next the voltages Vx and Vy are measured. Let L be theseparation from the test piece to the quadrant detector and w be the beam radius at thedetector. Then the relative angular deviation is: