Pavel Pavliček

Spatial coherence profilometry on tilted surfaces

The influence of tilted surfaces on the measurement of shape by spatial coherence profilometry is investigated. Based on theoretical analysis and experimental results, the systematic measurement error caused by surface tilt is determined. The systematic measurement error depends not only on the tilt angle but also on the parameters of the experimental setup. The theoretical analysis and the experiments show the similarities and differences between spatial coherence profilometry and white-light interferometry. We also suggest the conditions to obtain correct measurements by use of spatial coherence profilometry.

Methods for Optical Shape Measurement and their Measurement Uncertainty

There is a wide spectrum of optical 3-D sensors based on only a few basic principles. We calculate the measurement uncertainty for several “paradigm” methods, by means of the Cramér-Rao lower bound. It turns out that when all other sources of noise are neglected, the shot noise becomes the ultimate limit of the measurement uncertainty and the measurement uncertainty can be traced back to a Heisenberg uncertainty product. Consequently, making one of the factors uncertain, the other factor will become more precise. The different methods vary only by the way how this “uncertainty” is generated.

White-light interferometry on rough surfaces—measurement uncertainty caused by noise

White-light interferometry on rough surfaces is an optical method for the measurement of the geometrical form of objects. The longitudinal coordinate of the measured surface is obtained from the measured interferogram by means of an evaluation method. However, the longitudinal coordinate cannot be determined completely accurately because the interferogram is affected by noise. We calculate the lower limit of the longitudinal measurement uncertainty caused by noise by use of the Cramer-Rao inequality. Additionally, we calculate the lower limit of the longitudinal measurement uncertainty caused by shot noise only.

LCD-based low spatial coherence interferometer

An interferometer based on the spatial coherence detection has been developed for absolute height measurement. The location of a coherence peak resulting from the illumination of an extended quasi-monochromatic spatially incoherent light source with zone-plate-like spatial structure gives the longitudinal depth information. Besides simplifying the control of the light source structure, LCD-SLM eliminates the mechanical movement. Experiments of step height detection and profilometry of an object with a rough surface are presented that demonstrate the validity of the principle.

Height profile measurement by means of white light interferometry

White light interferometry is an established method for height profile measurement of objects. This method, unlike classical interferometry, can be used for measurement of objects with rough surface which is an important advantage. The white light interferometer is in principle a Michelson interferometer with a broad-band light source and a CCD camera as a detector. The Michelson interferometer has the object to be measured in one arm and the reference mirror in the other arm. Due to the reflection on the rough surface, a speckle pattern arises in the detector plane. This pattern is superimposed on the reference wave. The phase in particular speckle is random, but it remains approximately constant within one speckle. This renders the white light interference observable, if the optical path lengths of the two arms differ less than the coherence length. The object to be measured is mounted on a micropositioner for translating in the longitudinal direction. Gradually, as parts of the object surface cross the reference plane, the white light interference is observable in thc corresponding speckles. The position of the micropositioner in which the interference is maximal is stored for each pixel. This value for each pixel of the object image describes the geometrical shape of thc measured object. The measurement range is theoretically unlimited, practically it is limited by the range of the micropositioner. Thc longitudinal uncertainty does not depend on the parameters of the optical setup, its value is given by the roughness of the measured surface. The height profile of the object is measured during one measurement process, unlike the scanning profilers. The illumination and the observation are coaxial which avoids shadows.

Measurement Uncertainty of Optical Methods for the Measurement of the Geometrical Shape of Objects

There are various optical methods for the measurement of the geometrical shape of objects. Some of them are suitable for the measurement of the shape of objects with optically smooth surfaces. They are, for example, classical interferometry and phase measuring deflectometry [1, 2].

Measurement of the shape of objects by two wavelength interferometry

We propose a fast and precise optical 3D measurement method. The principle is similar to that of white-light interferometry. The broad-band light source of white-light interferometry is replaced by two lasers with different wavelengths. The object to be measured is placed into one arm of a Michelson interferometer and moved along the optical axis. The intensity measured at the output of the interferometer is equal to the field autocorrelation. In the case of two wavelengths, the autocorrelation is a periodical function with peaks as a result of their beating. The period can be adjusted by the choice of the wavelength difference. By choosing a short period, a fast and precise measurement is performed in the range of a single beat. However, such a measurement is ambiguous if the object has structures deeper than the beat period. The ambiguity is removed by a fast auxiliary measurement with a long beat period covering the whole depth range of the object. The auxiliary measurement need not be precise and can be completed quickly with a large sampling step.

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?

Influence of surface roughness on the measurement uncertainty of white-light interferometry

White-light interferometry measuring on rough surface commonly does not resolve the lateral structure of the surface. Thus the height differences within one resolution cell of the imaging system can exceed one-fourth of the wavelength of the used light. Consequently the phase of the interferogram, which is recorded at the interferometers output during the depth scan, becomes a random variable. In the image plane of the imaging system, a speckle pattern arises. Therefore in white-light interferometry on rough surface, the phase is not evaluated and the zero path difference is determined by seeking the maximal contrast of the interference fringes. Because of the rough surface, the measured value of the height coordinate is the result of a statistical process. In this way the rough surface gives rise to a measurement error. By means of numerical simulations we determine, how the surface roughness influences the measurement uncertainty of whitelight interferometry.

Spatial coherence profilometry

Spatial coherence profilometry is a method for measurement of the geometrical form of objects. In addition to the two lateral coordinates x and y, it measures the longitudinal coordinate z. In this way the complete 3D description of the objects surface is acquired. The main piece of the presented method is a Michelson interferometer illuminated by a monochromatic spatially extended light source. The surface of the object whose geometrical form should be measured is used as one mirror of the Michelson interferometer. By moving of the measured object along the optical axis, the intereference is observable only if the objects surface occurs in the vicinity of the so-called reference plane. The reference plane is given by the position of the object mirror when the Michelson interferometer is balanced. The described effect follows from the form of the spatial coherence function originated by the spatially extended light source. If the intensity at the output of the interferometer is recorded as a function of the position of the measured object, a typical correlogram arises. This correlogram is similar to that known with white-light interferometry. From the maximum of the correlogram, the z coordinate of the objects surface can be determined. Usually a CCD camera is used as the detector at the output of the Michelson interferometer. Then z coordinates of many surface points are parallel measured in the course of one measurement procedure and the 3D description of the objects surface is acquired. The scanning in the lateral direction is not necessary. Thus the described method provides a spatial coherence analogy to white-light interferometry which is based on temporal coherence. Unlike white-light interferometry, the described method does not require a broadband light source, the interferometer is illuminated by a monochromatic light source, usually a laser.