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White-light interferometry, Hilbert transform, and noise

White-light interferometry is an established and proved method for the measurement of the geometrical shape of objects. The advantage of white-light interferometry is that it is suitable for the measurement of the shape of objects with smooth as well as rough surface. The information about the longitudinal coordinate of the surface of the measured object is obtained from the white-light interferogram. The interferogram is the intensity at the detector expressed as the function of the position of the object. (The object is moved along the optical axis during the measurement process.) If the shape of an object with rough surface is measured, the phase of the interferogram is not evaluated because it is a random value. The information about the longitudinal coordinate is obtained from the center of the interferogram envelope. A classical method for the calculation of the envelope of white-light interferogram is the demodulation by means of Hilbert transform. However, the electric signal at the output of the camera is influenced by the noise. Therefore, as expected, the calculated envelope is also influenced by the noise. The result is that the measured longitudinal coordinate of the surface of the object is affected by an error. In our contribution, we look for the answer on following questions: How does the noise of the evaluated envelope differ from the noise of the interferogram? What is the minimal measurement uncertainty that can be achieved?