Measurement of discontinuous surfaces using multiple-wavelength interferometry
01 Jul 2009-Optical Engineering (International Society for Optics and Photonics)-Vol. 48, Iss: 7, pp 073603
TL;DR: In this paper, a three-wavelength interferometric technique is used with a phase-shifting phase evaluation procedure, which gives wrapped phase at any pixel corresponding to these wavelengths.
Abstract: Interferometric surface profilers using a single wavelength offer excellent vertical resolution, but have an ambiguity-free range of less than half a wavelength. Multiple-wavelength or white light interference techniques are used to overcome the problem. We discuss a three-wavelength interferometric technique used with a phase-shifting phase evaluation procedure. The phase evaluation at the three wavelengths gives wrapped phase at any pixel corresponding to these wavelengths. We use the fact that the variation of phase with wavenumber for a given profile height is linear to determine the absolute value of the profile height. The height is then used to ascertain the fringe order. The fringe order, along with the wrapped phase, gives the profile height with a resolution given by the phase-shifting technique. Experimental results for large step height measurement on etched silicon samples are presented.
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TL;DR: In this article, the authors proposed a method based on the Huygens-Fresnel principle for computing the diffracted intensity from measured or modeled profiles, which can be extended to optical systems with an arbitrary number of reflections.
Abstract: Context. The imaging sharpness of an X-ray telescope is chiefly determined by the optical quality of its focusing optics, which in turn mostly depends on the shape accuracy and the surface finishing of the grazing-incidence X-ray mirrors that compose the optical modules. To ensure the imaging performance during the mirror manufacturing, a fundamental step is predicting the mirror point spread function (PSF) from the metrology of its surface. Traditionally, the PSF computation in X-rays is assumed to be different depending on whether the surface defects are classified as figure errors or roughness. This classical approach, however, requires setting a boundary between these two asymptotic regimes, which is not known a priori.Aims. The aim of this work is to overcome this limit by providing analytical formulae that are valid at any light wavelength, for computing the PSF of an X-ray mirror shell from the measured longitudinal profiles and the roughness power spectral density, without distinguishing spectral ranges with different treatments.Methods. The method we adopted is based on the Huygens-Fresnel principle for computing the diffracted intensity from measured or modeled profiles. In particular, we have simplified the computation of the surface integral to only one dimension, owing to the grazing incidence that reduces the influence of the azimuthal errors by orders of magnitude. The method can be extended to optical systems with an arbitrary number of reflections – in particular the Wolter-I, which is frequently used in X-ray astronomy – and can be used in both near- and far-field approximation. Finally, it accounts simultaneously for profile, roughness, and aperture diffraction.Results. We describe the formalism with which one can self-consistently compute the PSF of grazing-incidence mirrors, and we show some PSF simulations including the UV band, where the aperture diffraction dominates the PSF, and hard X-rays where the X-ray scattering has a major impact on the PSF degradation. The results are validated with ray-tracing simulations, or by comparison with the analytical computation of the half-energy width based on the known scattering theory, where these approaches are applicable. Finally, we validate this by comparing the simulated PSF of a real Wolter-I mirror shell with the measured PSF in hard X-rays.
65 citations
TL;DR: A review focused on surface profile measurement techniques of optical interferometry is introduced with a detailed classification sorted by operating principles, and examples in each category are discussed and analyzed for better understanding.
60 citations
TL;DR: In this article, a white light interferometer with a colour CCD camera is discussed, where interference intensity information from the three channels of the color CCD simulating three-wavelength measurement is accessed.
53 citations
TL;DR: In this paper, a technique that encodes a 3D surface shape into a single 24-bit color image was proposed to store and transmit the 3D data in real time.
Abstract: With recent advancements in 3-D imaging and computational technologies, acquiring 3-D data is unprecedentedly simple. However, the use of 3-D data is still limited due to the size of 3-D data, especially 3-D video data. Therefore, the study of how to store and transmit the 3-D data in real time is vital. We address a technique that encodes a 3-D surface shape into a single 24-bit color image. In particular, this image is generated by advanced computer graphics tools with two primary color channels encoded as sine and cosine fringe images, and the third chan- nel encoded as a stair image to unwrap the phase obtained from the two fringe images. An arbitrary 3-D shape can then be recovered from a single image. We test 3-D shapes with differing levels of complexity along with various image formats. Experiments demonstrate that, without significantly losing the shape quality, the compression ratio can go up to 1:36.86, compared with the native smallest possible 3-D data represen- tation method. © 2010 Society of Photo-Optical Instrumentation Engineers. DOI: 10.1117/1.3456632
46 citations
TL;DR: This paper presents a method based on the use of an image sensor for obtaining the complex amplitudes of beams diffracted from an object at two different wavelengths by the Doppler phase-shifting method.
Abstract: This paper presents a method based on the use of an image sensor for obtaining the complex amplitudes of beams diffracted from an object at two different wavelengths. The complex amplitude for each wavelength is extracted by the Doppler phase-shifting method. The principle underlying the proposed method is experimentally verified by using the method with two lasers having different wavelengths to measure the surface shape of a concave mirror.
44 citations
References
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TL;DR: The phase modulation in an interferometer can be induced by moving a mirror, tilting a glass plate, moving a grating, rotating a half-wave plate or analyzer, using an acoustooptic or electro-optic modulator, or using a Zeeman laser as mentioned in this paper.
Abstract: Publisher Summary This chapter describes the phase-measurement interferometry techniques. For all techniques, a temporal phase modulation is introduced to perform the measurement. By measuring the interferogram intensity as the phase is shifted, the phase of the wavefront can be determined with the aid of electronics or a computer. Phase modulation in an interferometer can be induced by moving a mirror, tilting a glass plate, moving a grating, rotating a half-wave plate or analyzer, using an acousto-optic or electro-optic modulator, or using a Zeeman laser. Phase-measurement techniques using analytical means to determine phase all have some common denominators. There are different equations for calculating the phase of a wavefront from interference fringe intensity measurements. The precision of a phase-measuring interferometer system can be determined by taking two measurements, subtracting them, and looking at the root-meansquare of the difference wavefront. The chapter discusses the simulation results. The elimination of the errors that reduce the measurement accuracy depends on the type of measurement being performed. Phase-measurement interferometry (PMI) can be applied to any two-beam interferometer, including holographic interferometers. Applications can be divided into: surface figure, surface roughness, and metrology.
1,340 citations
TL;DR: In this article, a scanning white-light interferometer for high-precision surface structure analysis is described, where the interferogram for each of the image points in the field of view is generated simultaneously by scanning the object in a direction perpendicular to the object surface, while recording detector data in digital memory.
Abstract: We describe a scanning white-light interferometer for high-precision surface structure analysis. Interferograms for each of the image points in the field of view of the instrument are generated simultaneously by scanning the object in a direction perpendicular to the object surface, while recording detector data in digital memory. These interferograms are then transformed into the spatial frequency domain and the surface height for each point is obtained by examination of the complex phase as a function of frequency. The final step is the creation of a complete three-dimensional image constructed from the height data and corresponding image plane coordinates. The measurement repeatability is better than 0·5 nm r.m.s. for a surface height range of 100 μm.
401 citations
01 Jan 1984
TL;DR: A technique that combines ideas of phase shifting interferometry (PSI) and two-wavelength interferometer (TWLI) to extend the phase measurement range of conventional single-wa wavelength PSI is described.
Abstract: This paper describes a technique that combines ideas of phase shifting interferometry (PSI) and two-wavelength interferometry (TWLI) to extend the phase measurement range of conventional single-wavelength PSI. To verify theoretical predictions, experiments have been performed using a solid-state linear detector array to measure 1-D surface heights. Problems associated with TWLPSI and the experimental setup are discussed. To test the capability of the TWLPSI, a very fine fringe pattern was used to illuminate a 1024 element detector array. Without temporal averaging, the repeatability of measuring a surface having a sag of ~100 μm is better than 25-A (0.0025%) rms.
376 citations
TL;DR: In this article, a technique that combines ideas of phase shifting interferometry (PSI) and two-wavelength interference (TWLI) to extend the phase measurement range of conventional singlewavelength PSI is described.
Abstract: This paper describes a technique that combines ideas of phase shifting interferometry (PSI) and two-wavelength interferometry (TWLI) to extend the phase measurement range of conventional single-wavelength PSI. To verify theoretical predictions, experiments have been performed using a solid-state linear detector array to measure 1-D surface heights. Problems associated with TWLPSI and the experimental setup are discussed. To test the capability of the TWLPSI, a very fine fringe pattern was used to illuminate a 1024 element detector array. Without temporal averaging, the repeatability of measuring a surface having a sag of ~100 μm is better than 25-A (0.0025%) rms.
365 citations