The method of excess fractions (EF) is well established to resolve the fringe order ambiguity generated in interferometric detection. Despite this background, multiwavelength interferometric absolute long distance measurements have only been reported with varying degrees of success. In this paper we present a theoretical model that can predict the unambiguous measurement range in EF based on the selected measurement wavelengths and phase noise. It is shown that beat wavelength solutions are a subset of this theoretical model. The performance of EF, for a given phase noise, is shown to be equivalent to beat techniques but offers many alternative sets of measurement wavelengths and therefore EF offer significantly greater flexibility in experimental design.
Multiwavelength interferometry (MWI) is a well established technique in the field of optical metrology. Previously, we have reported a theoretical analysis of the method of excess fractions that describes the mutual dependence of unambiguous measurement range, reliability, and the measurement wavelengths. In this paper wavelength, selection strategies are introduced that are built on the theoretical description and maximize the reliability in the calculated fringe order for a given measurement range, number of wavelengths, and level of phase noise. Practical implementation issues for an MWI interferometer are analyzed theoretically. It is shown that dispersion compensation is best implemented by use of reference measurements around absolute zero in the interferometer. Furthermore, the effects of wavelength uncertainty allow the ultimate performance of an MWI interferometer to be estimated.
In the last 10 years, freeform optics has enabled compact and high-performance imaging systems. This article begins with a brief history of freeform optics, focusing on imaging systems, including marketplace emergence. The development of this technology is motivated by the clear opportunity to enable science across a wide range of applications, spanning from extreme ultraviolet lithography to space optics. Next, we define freeform optics and discuss concurrent engineering that brings together design, fabrication, testing, and assembly into one process. We then lay out the foundations of the aberration theory for freeform optics and emerging design methodologies. We describe fabrication methods, emphasizing deterministic computer numerical control grinding, polishing, and diamond machining. Next, we consider mid-spatial frequency errors that inherently result from freeform fabrication techniques. We realize that metrologies of freeform optics are simultaneously sparse in their existence but diverse in their potential. Thus, we focus on metrology techniques demonstrated for the measurement of freeform optics. We conclude this review with an outlook on the future of freeform optics.
For several years, scientific, industrial, and biological fields have benefited from knowledge of phase information, which allows for the revealing of hidden features of various objects. An alternative to interferometry is single-beam phase retrieval techniques that are based on the transport of intensity equation, which describes the relation between the axial derivative of the intensity and the phase distribution for a given plane in the Fresnel region. The estimation of the axial intensity derivative is obtained from a series of intensity measurements, where the accuracy is subject to an optimum separation between the measurement planes depending on the number of planes, the level of noise, and the actual object phase distribution. In this Letter, a quantitative analysis of the error in estimated axial derivative is carried out and a model is reported that describes the interdependence between these parameters. The results of this work allow for estimation of the optimum separation between measurement planes with minimal error in the axial derivative.
We present an optimized method for multiwavelength interferometry that allows measurements beyond the largest beat wavelength. The approach exploits wavelength coincidence between two beat wavelengths in order to measure unambiguously over an extended range. Performance of the approach has been validated both through simulations and experimentally by means of a fiber interferometer for four measurement wavelengths. Initial results have demonstrated 1/200th of a fringe phase resolution, giving absolute metrology over 18.16 mm, or a dynamic range of 1 part in 2.4x10(6). With improved phase resolution the method has the potential to range over >100 m using femtosecond laser frequency comb sources.
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