J. Eur. Opt. Society-Rapid Publ. 2025, 21, 32 Ó The Author( s), published by EDP Sciences, 2025 https:// doi. org / 10.1051 / jeos / 2025028 Available online at: https:// jeos. edpsciences. org
EOSAM 2024 Guest editors: Luca De Stefano and Raffaele Velotta
Journal of the European Optical Society-Rapid Publications
RESEARCH ARTICLE
2-dimensional in-plane displacement measurement system at fast sampling rate of 5 kHz using sinusoidal phase modulation interferometer
Masato Higuchi 1, Taku Sato 2, and Masato Aketagawa 2,*
1 National Institute of Technology, Gunma College, 850, Toriba Maebashi, Gunma, 371-8530, Japan 2 Nagaoka University of Technology, 1603-1, Kamitomioka, Nagaoka, Niigata 940-2188, Japan
Received 31 January 2025 / Accepted 6 June 2025
Abstract. Dynamic and high-resolution air fluctuation( wavefront aberration) measurements are necessary for precision interferometry and astronomical observations. In this paper, to solve the problem, we propose a measurement system that uses sinusoidal phase modulation interferometry to observe 2-dimensional( 2-D) in-plane displacements with sub-nm resolution and fast speed sampling rate of 5 kHz. The interferometer consists of a Michelson type interferometer incorporating an electric-optic modulator with a modulation frequency of 5 kHz and a high-speed camera synchronized to a clock signal at a frequency of 60 kHz, 12 times the modulation frequency. Phase demodulation of each pixel in the camera is performed by acquiring the interference signal to that pixel synchronously with the sampling signal and performing a specific addition and subtraction between the signals obtained synchronously. By applying this procedure to all pixels in the camera, 2-D in-plane displacements can be obtained. In this paper, we report on the measurement equipment, the demodulation principle, the pre-filter for noise reduction and experimental results. In the experiments, 2-D in-plane displacements with sub-nm resolution are confirmed at a sampling rate of 5 kHz. This technique has the potential to measure fast, dynamic deformation of object surfaces and dynamic wavefront aberrations due to air fluctuations. Based on the proposed demodulation method, interferometers with MHz-class sampling rates are possible by using faster electro-optical modulator and high-speed camera.
Keywords: Sinusoidal phase modulation, Interferometer, 2D in-plane displacement, Sub-nanometer.
1 Introduction
Along with the developments of the semiconductor industry, the minimum circuit width has been expected to be a few nanometers [ 1 ]. These trends lead to the shape error of mirrors in semiconductor manufacturing machine( EUV lithography machine) must be within 50 pm [ 2 ]. For such high-resolution measurement, the laser interferometry has been a good candidate because the measurement result has traceability to the definition of SI unit“ meter”.
The resolution of the displacement measuring interferometer has reached a few tens of picometer. Here, some examples(“ Heterodyne”,“ Homodyne” and“ Sinusoidal phase modulation( SPM)” interferometers) which measure physical mirror displacements in laboratory environment are shown.
1. Heterodyne interferometer: Hsu et al. reported a demodulation( phase-meter) algorithm using a digital
* Corresponding author: masatoaa @ vos. nagaokaut. ac. jp phase-locked loop( PLL) implemented in field programmable gate array( FPGA) [ 3 ]. Yokoyama et al. reported spatially separated interferometer to reduce polarization mixing effect included in the interference signal [ 4 ]. Nguyen et al. reported a digital phaselocked loop implemented in FPGA for improving sensitivity of the phase-meter [ 5, 6 ]. Kokuyama et al. reported introducing an oversampling for noise reduction using FPGA [ 7 ].
2. Homodyne interferometer: Pisani et al. reported a multi-folded configuration for increasing the basic resolution [ 8, 9 ]. Hori et al. reported a high sensitivity optic configuration and a correction of Lissajous diagram [ 10 ].
3. SPM interferometer [ 11, 12 ]: Higuchi et al. reported a high-speed modulation and a filtering for band limiting [ 13 ]. Through the development of high precision heterodyne interferometry [ 5, 6 ] and SPM interferometry [ 13 ] by our group, it was found that air fluctuations in the interferometer optical path are serious problems. The fluctuations were considered
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