A fast-Fourier-transform method of topography and interferometry is proposed. By computer processing of a noncontour type of fringe pattern, automatic discrimination is achieved between elevation and depression of the object or wave-front form, which has not been possible by the fringe-contour-generation techniques. The method has advantages over moire topography and conventional fringe-contour interferometry in both accuracy and sensitivity. Unlike fringe-scanning techniques, the method is easy to apply because it uses no moving components.

Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry

A self-scanned 1024 element photodiode array and minicomputer are used to measure the phase (wavefront) in the interference pattern of an interferometer to lambda/100. The photodiode array samples intensities over a 32 x 32 matrix in the interference pattern as the length of the reference arm is varied piezoelectrically. Using these data the minicomputer synchronously detects the phase at each of the 1024 points by a Fourier series method and displays the wavefront in contour and perspective plot on a storage oscilloscope in less than 1 min (Bruning et al. Paper WE16, OSA Annual Meeting, Oct. 1972). The array of intensities is sampled and averaged many times in a random fashion so that the effects of air turbulence, vibrations, and thermal drifts are minimized. Very significant is the fact that wavefront errors in the interferometer are easily determined and may be automatically subtracted from current or subsequent wavefrots. Various programs supporting the measurement system include software for determining the aperture boundary, sum and difference of wavefronts, removal or insertion of tilt and focus errors, and routines for spatial manipulation of wavefronts. FFT programs transform wavefront data into point spread function and modulus and phase of the optical transfer function of lenses. Display programs plot these functions in contour and perspective. The system has been designed to optimize the collection of data to give higher than usual accuracy in measuring the individual elements and final performance of assembled diffraction limited optical systems, and furthermore, the short loop time of a few minutes makes the system an attractive alternative to constraints imposed by test glasses in the optical shop.

Digital wavefront measuring interferometer for testing optical surfaces and lenses

We demonstrate a simplified dual-comb LIDAR setup for precision absolute ranging that can achieve a ranging precision of 2 μm in 140 μs acquisition time. With averaging, the precision drops below 1 μm at 0.8 ms and below 200 nm at 20 ms. The system can measure the distance to multiple targets with negligible dead zones and a ranging ambiguity of 1 meter. The system is much simpler than a previous coherent dual-comb LIDAR because the two combs are replaced by free-running, saturable-absorber-based femtosecond Er fiber lasers, rather than tightly phase-locked combs, with the entire time base provided by a single 10-digit frequency counter. Despite the simpler design, the system provides a factor of three improved performance over the previous coherent dual comb LIDAR system.

/pdf/sub-micron-absolute-distance-measurements-in-sub-millisecond-uy3sgpcqgr.pdf

Sub-micron absolute distance measurements in sub-millisecond times with dual free-running femtosecond Er fiber-lasers.

Laser frequency combs emit a spectrum with hundreds of thousands of evenly spaced phase-coherent narrow lines. A comb-enabled instrument, the dual-comb interferometer, exploits interference between two frequency combs and attracts considerable interest in precision spectroscopy and sensing, distance metrology, tomography, telecommunications, etc. Mutual coherence between the two combs over the measurement time is a pre-requisite to interferometry, although it is instrumentally challenging. At best, the mutual coherence reaches about 1 s. Computer-based phase-correction techniques, which often lead to artifacts and worsened precision, must be implemented for longer averaging times. Here with feed-forward relative stabilization of the carrier-envelope offset frequencies, we experimentally realize a mutual coherence over times approaching 2000 s, more than three orders of magnitude longer than that of state-of-the-art dual-comb systems. An illustration is given with near-infrared Fourier transform molecular spectroscopy with two combs of slightly different repetition frequencies. Our technique without phase correction can be implemented with any frequency comb generator including microresonators or semiconductor lasers.

/pdf/a-phase-stable-dual-comb-interferometer-4aa8h1dgj9.pdf

A phase-stable dual-comb interferometer

Absolute distances were measured using two femtosecond lasers of different pulse repetition rates by revisiting the dual-comb interferometric method proposed by Coddington et al (2009 Nature Photon. 3 351–6). The apparatus built for experiments was designed to eliminate the dead zones in the measurement range by separating the measurement pulses from the reference pulses using orthogonal polarization. In addition, the pulse repetition rate of the signal laser was made tunable in order to extend the non-ambiguity range (NAR) by adaptively adjusting the synthetic wavelength in consideration of the de facto measurement stability in the air. Actual tests performed in the open air proved that a target distance of 69.3 m is measured without interruptions at a 200 µs update rate in the presence of a ~170 µm drift of the optical path length caused by the fluctuation of the refractive index of air. The proposed hardware system design for effective NAR extension will facilitate the use of dual-comb interferometry for various terrestrial applications.

https://iopscience.iop.org/article/10.1088/0957-0233/24/4/045201/pdf

Absolute distance measurement by dual-comb interferometry with adjustable synthetic wavelength

We propose a distance meter that utilizes an intermode beat of terahertz frequency in an optical frequency comb to perform high resolution and high dynamic range absolute distance measurements. The proposed system is based on a novel method, called multiheterodyne cross-correlation detection, in which intermode beat frequencies are scaled down to radio frequencies by optical mixing of two detuned optical frequency combs with a nonlinear optical crystal. Using this method, we obtained a 1.056 THz intermode beat and achieved a distance resolution of 0.820 μm from its phase measurement. Absolute distance measurement using 1.056 THz and 8.187 GHz intermode beats was also demonstrated in the range of 10 mm, resulting in a precision of 0.688 μm.

A distance meter using a terahertz intermode beat in an optical frequency comb.

We present a high-resolution heterodyne interferometric system using a high-output-power, three-longitudinal-mode He-Ne laser. In this system, an interferometric wavelength of λ/2 is synthesized with the super-heterodyne detection of the three modes, resulting in twice the sensitivity for the target displacement in comparison with the conventional two-frequency interferometer. In the experiment, the achievement of subnanometre resolution has been confirmed.

https://iopscience.iop.org/article/10.1088/0957-0233/12/2/305/pdf

A subnanometre heterodyne interferometric system with improved phase sensitivity using a three-longitudinal-mode He-Ne laser

Previously, the authors have developed a subnanometre heterodyne interferometric system known as a three-mode heterodyne interferometer (TMHI). By converting the target displacement into twice the phase variation of a conventional interferometer, the TMHI has achieved position resolution of 0.044 nm even with a single-pass interferometer. However, the heterodyne beat frequency in TMHI is quite low, of the order of several hundred kilohertz, which limits the measurable target speed to several tens of millimetres per second. In this paper, we present a method to extend the measurable target speed of the TMHI without increasing the heterodyne beat frequency of a light source. Keeping the excellent position resolution of the TMHI, the proposed method provides sufficient response to high speed movement of the target. The system comprises a pair of TMHI and utilizes two conjugate signals generated by them selectively according to the direction of movement of the target. Possible errors in switching the two signals are analysed by introducing an error zone model. The switching function was demonstrated using the mechanical vibration of the interferometer and proved to be free from erroneous counts. It was also confirmed that the system can respond at a speed that exceeds that determined by the beat frequency.

https://iopscience.iop.org/article/10.1088/0957-0233/16/9/017/pdf

High-speed subnanometre interferometry using an improved three-mode heterodyne interferometer

A novel interferometric method, called double shearing interferometry, is presented for profiling large-scale quasi-planar surfaces, such as semiconductor wafers, optical flats and x-ray mirrors. The surface profile is measured even if there is an inclination in the scanning stage. By adopting the common-path optical configuration and heterodyne detection, the proposed method achieves excellent resolution at the subnanometre scale and allows a robust measurement in the presence of unwanted disturbances in the measurement environment. A height resolution of 0.1 nm was achieved experimentally. The standard deviation of the analytical surface profiles was found to be 1.3 nm, even when using a conventional screw-lead scanning stage. The measured results show that the surface profile and stage inclination are determined separately. We confirmed that the resultant profile was quite consistent with that measured with a Fizeau interferometer.

/pdf/sub-nanometre-double-shearing-heterodyne-interferometry-for-22lyw3p69s.pdf

Sub-nanometre double shearing heterodyne interferometry for profiling large scale planar surfaces

A simple frequency stabilization method for frequency-doubled output (532 nm) of an Nd:YAG laser is described. The frequency shift caused by an acousto-optic modulator is effectively applied to stabilize the laser frequency. For the frequency stabilization, there are two kinds of error signal: (1) the first-derivative of absorption of iodine molecules and (2) the first-derivative of dispersion were extracted to examine which signal is suitable for the actual stabilization of the laser frequency. Based on this examination, the stabilization was executed utilizing the error signal (1), resulting in a frequency instability of .

Toshiyuki Yokoyama

Papers

A distance meter using a terahertz intermode beat in an optical frequency comb.

A subnanometre heterodyne interferometric system with improved phase sensitivity using a three-longitudinal-mode He-Ne laser

High-speed subnanometre interferometry using an improved three-mode heterodyne interferometer

Sub-nanometre double shearing heterodyne interferometry for profiling large scale planar surfaces

A simple frequency lock of green YAG laser to Doppler-broadened spectrum of iodine molecules utilizing an acousto-optic frequency shifter