Investigation and compensation of the nonlinearity of heterodyne interferometers

Hou, Wenmei; Wilkening, G.

doi:10.1016/0141-6359(92)90054-z

Cited by 182 publications

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“…Since the periodic errors of the laser interferometers were first predicted [2], research pertaining to theoretical models and compensation techniques has been investigated and reported [3][4][5]. Recently, a real-time first-order periodic error correction technique was validated under various experimental conditions indicating that the remaining periodic error is at the subnanometer level [6].…”

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Simple heterodyne laser interferometer with subnanometer periodic errors

et al. 2009

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We describe a simple heterodyne laser interferometer that has subnanometer periodic errors and is applicable to industrial fields. Two spatially separated beams can reduce the periodic errors, and the use of a right-angle prism makes the optical configuration much simpler than previous interferometers. Moreover, the optical resolution can be enhanced by a factor of 2, because the phase change direction is opposite between reference and measurement signals. Experiments have demonstrated the periodic errors are less than 0.15 nm owing to the frequency mixing of the optical source. The improvements for reducing the frequency mixing of the optical system are also discussed. © 2009 Optical Society of America OCIS codes: 120.3180, 120.3930, 120.3940.The periodic errors limit the accuracy of heterodyne laser interferometers because they deteriorate the purity of the interference signals [1]. Since the periodic errors of the laser interferometers were first predicted [2], research pertaining to theoretical models and compensation techniques has been investigated and reported [3][4][5]. Recently, a real-time first-order periodic error correction technique was validated under various experimental conditions indicating that the remaining periodic error is at the subnanometer level [6]. In addition to compensation methods, nonpolarizing optical configurations to eliminate the periodic errors have been designed [7][8][9]. The common feature of these interferometers is to use spatially separated beams that have different frequency components. Essentially, the polarization states are never mixed; thus the nonlinearity can be significantly reduced. However, the optical setups are complicated because both beams propagate separately, which requires additional optical components [7]. Although several designs have been reported with heterodyne interferometers using an acousto-optic frequency shifter (AOFS) as a beam splitter [8,9], the small AOFS diffraction angle and their specific configurations limit the applicability for displacement measurements.In this research, a simple and industry-adaptable heterodyne laser interferometer designed to significantly reduce periodic errors is proposed and tested. The periodic errors are essentially caused by splitting and recombining two nonideal beams using polarizing optics that are both nonideal and are sensitive to alignment. To eliminate the periodic errors in the optical configuration, two beams with different optical frequencies should be spatially separated to avoid the frequency and polarization mixing. Figure 1 shows the optical configuration that was employed in this investigation. An optical source provides two parallel beams to the interferometer, where each beam has the same polarization state but has different optical frequencies f 0 and f 0 + f s , respectively.The two beams propagate to a nonpolarizing beam splitter (NPBS) where they are split into two sets of beams, reference and measurement beams. The reference beams are reflected by a right-angle prism (RAP), and the meas...

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Simple heterodyne laser interferometer with subnanometer periodic errors

et al. 2009

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“…This reduction approach is based on the quality of the coaxial beam's source frequency orthogonality together with the quality of the linear polarizations. If these are of high quality, it is also essential that highquality polarized optics are applied for beam splitting, combined with alignment [5][6][7][8][9].A less expensive approach is to digitally compensate for the PNL through analytical modeling [5,[9][10][11]. The advantage of this method is that it typically ensures PNL below 1 nm for small system imperfections, without changing the interferometer configuration.…”

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“…A less expensive approach is to digitally compensate for the PNL through analytical modeling [5,[9][10][11]. The advantage of this method is that it typically ensures PNL below 1 nm for small system imperfections, without changing the interferometer configuration.…”

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“…When operating at this level, a heterodyne interferometer system is hampered by many error sources. The main error sources are the frequency stability of the laser source, noncommon optical pathway variations due to variations in the refractive indices of optical transport media, system alignment (i.e., cosine error, Abbé error), optical wavefront quality combined with beam walkoff [4], and periodic nonlinearity (PNL) in the measured phase [5][6][7][8][9][10]. Operating in vacuum will improve the obtained measurement error.…”

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Heterodyne displacement interferometer, insensitive for input polarization

Meskers¹,

Spronck²,

Schmidt³

2014

Opt. Lett.

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Periodic nonlinearity (PNL) in displacement interferometers is a systematic error source that limits measurement accuracy. The PNL of coaxial heterodyne interferometers is highly influenced by the polarization state and orientation of the source frequencies. In this Letter, we investigate this error source and discuss two interferometer designs, designed at TU Delft, that showed very low levels of PNL when subjected to any polarization state and/or polarization orientation. In the experiments, quarter-wave plates (qwps) and half-wave plates (hwps) were used to manipulate the polarization state and polarization orientation, respectively. Results from a commercial coaxial system showed first-order PNL exceeding 10 nm (together with higher order PNL) when the system ceased operation at around 15°hwp rotation or 20°qwp rotation. The two "Delft interferometers," however, continued operation beyond these maxima and obtained first-order PNLs in the order of several picometers, without showing higher order PNLs. The major advantage of these interferometers, beside their high linearity, is that they can be fully fiber coupled and thus allow for a modular system buildup. Laser interferometry is an often applied measurement method in several fields of research, since it allows for noncontact measurements. It is especially preferred in the field of metrology, because of its direct traceability to the length standard [1]. Interferometry is also used in lithography machines in the semiconductor industry, gravitational wave detection [2], coordinate measuring machines, and as a calibration tool for other measurement devices, such as capacitive sensors, inductive sensors, and optical encoders. Many types of displacement interferometer systems can be distinguished; this publication deals with heterodyne displacement interferometry using a stabilized He-Ne laser (λ 632.8 nm) combined with two acoustooptic modulators for generating two (fixed-offset) source frequencies.Industrial manufacturing processes currently operate with measurement errors at the subnanometer level and will require even smaller errors in the near future [3]. When operating at this level, a heterodyne interferometer system is hampered by many error sources. The main error sources are the frequency stability of the laser source, noncommon optical pathway variations due to variations in the refractive indices of optical transport media, system alignment (i.e., cosine error, Abbé error), optical wavefront quality combined with beam walkoff [4], and periodic nonlinearity (PNL) in the measured phase [5][6][7][8][9][10]. Operating in vacuum will improve the obtained measurement error. However, PNL in the measurand remains a substantial error source.PNL manifests itself primarily in traditional heterodyne systems with coaxial beams. Such beams contain two linearly polarized frequencies that are orthogonally oriented. These frequencies may mix (due to "frequency leakage"), resulting in periodic errors that are superimposed on the obtained displacement data. This type ...

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A method for measuring two‐dimensional refractive index distribution by using Fresnel equations and phase‐shifting interferometry

Hsieh

Jian

Chen

et al. 2008

Phys. Status Solidi (c)

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Based on Fresnel equations and the phase‐shifting interferometry, an alternative method for measuring the two‐dimensional refractive index distribution of a material is presented. A linearly polarized light passes through a quarter wave‐plate and is incident on the tested material. The reflected light propagates through an analyzer, and then the interference signal can be obtained. The special equations to estimate the phase of the interferometric signal can be derived by using Fresnel equations. Next, the associated two‐dimensional phase distribution is measured by the four‐step phase‐shifting interferometry. Then, the measured data are substituted into the special equations derived previously, and the two‐dimensional refractive index distribution of the tested material can be obtained. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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Investigation and compensation of the nonlinearity of heterodyne interferometers

Cited by 182 publications

References 5 publications

Simple heterodyne laser interferometer with subnanometer periodic errors

Simple heterodyne laser interferometer with subnanometer periodic errors

Heterodyne displacement interferometer, insensitive for input polarization

A method for measuring two‐dimensional refractive index distribution by using Fresnel equations and phase‐shifting interferometry

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