Curvature measurement using phase shifting in-line interferometry, single shot off-axis geometry and Zernike's polynomial fitting

Abstract

This paper describes the difference between phase shifting in-line interferometry, single shot off-axis geometry and Zernike's polynomial fitting methods for measuring the curvature of a spherical smooth surface by using the Michelson interferometer. In phase shifting in-line interferometry, four interferograms shifted by a piezoelectric actuator (PZT) were captured by a digital detector and corrected by using the flat fielding method. In off-axis geometry, single shot off-axis interferogram was obtained by tilting the reference and the object wave of the off-axis interferogram was reconstructed in the central region of the observation plane by using the digital reference wave concept. The demodulated phase map was obtained and unwrapped to remove the 2π ambiguity. The unwrapped phase map was converted to height and the sagittal length that used for curvature measurement was calculated accurately. The results extracted from phase shifting in-line interferometry and single shot off-axis geometry methods were compared with the results extracted from single shot Zernike's polynomial fitting method and the results were in excellent agreement. A new trial was done to overcome the fringes produced from the object interfaces. Some factors of uncertainty which affected on the measurement were estimated in the order of 6.0 × 10⁻⁵ mm or 0.003 dioptre (▽).

Introduction

Optical methods have been used as metrological tools for a long time. They are non-contacting, nondestructive and highly accurate. In combination with computers and other electronic devices, they have become faster, more reliable, more convenient and more robust. Among these optical methods, interferometry has received much interest for its shape measurement of optical and non-optical surfaces. Information about the surface under test can be obtained from interference fringes which characterize the surface. Two-beam interference fringes have been used to investigate the shape of optical and non-optical surfaces for a long time [1]. The extracted phase from a single closed fringe pattern is ambiguous [2]. This phase ambiguity can be easily removed by using the phase shifting technique [3].

Fourier-transform method [4], [5], [6] can extract phase information very quickly, because it only needs a single interferogram to demodulate the unknown phase distribution. However, when an interferogram includes closed fringe patterns without a tilt, i.e. without a carrier frequency, this method has difficulty in determining the complex fringe amplitude because the Fourier spectra of the interferogram are not separated completely. In this paper, the phase ambiguity from a single closed fringe pattern captured from the Michelson interferometer was removed by using three techniques. The first was performed by the phase shifting technique. In this technique, four different interferograms with 0, π/2, π and 3π/2 radian phase shifts were captured and corrected with the flat fielding method. The demodulated phase map was obtained and unwrapped to remove the 2π ambiguity [7]. The unwrapped phase map was converted to height and the sagittal length that used for curvature measurement was calculated accurately. The second technique was fulfilled by using the off-axis geometry. In this technique, a single closed fringe pattern was processed numerically to reconstruct the surface object using computer programs [8], [9], [10], [11], [12], [13], [14]. The captured interferogram of the surface object was processed using Matlab codes for getting a reconstructed object wave (amplitude and phase). The digital reference wave in the reconstruction algorithm should match as close as possible to the experimental reference wave. This was done in this paper by selecting the appropriate values of the two components of the wave vector k_x = 0.00180 mm⁻¹ and k_y = 0.011538 mm⁻¹. The reconstructed phase map of the object surface was unwrapped and the unwrapped phase map was converted to height and the sagittal length that used for curvature measurement was calculated accurately. The third technique was accomplished by using the Zernike's polynomial fitting method. In this technique, a single closed fringe pattern was processed through a written matlab code until the fringes were thinned and assigned. The data were fitted to a polynomial set to reconstruct the surface profile and the sagittal length that used for curvature measurement was calculated accurately [15], [16], [17], [18]. From the measured sagittal length and half the chord length of the tested surface, the curvature was calculated accurately from the aforementioned techniques and the results were in excellent agreement. The uncertainty budget due to some factors affected on the measurement was estimated to be of the order of 6.0 × 10⁻⁵ mm or 0.003 ▽.