Interferometric Signal Research Articles

Self-Mixing Displacement Measured by a Two-Color Laser in 66-nm Steps

We demonstrate a new self-mixing interferometer (SMI) simultaneously working on two wavelengths, the fundamental at 1064 nm and the second harmonic at 532 nm delivered in the same beam. The beam is sent to the remote target in the usual way, and in the return path, two photodiodes detect the two self-mixing signals without crosstalk. As the source, we were able to use a diode-pumped neodymium-doped yttrium vanadate (Nd:YVO4 laser crystal) with potassium titanyl phosphate (KTP nonlinear crystal) two-color laser of a commercial pointer. With simple processing of the two self-mixing signals, we are able to obtain the interferometric signals <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\cos (2ks)$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\sin (2ks)$ </tex-math></inline-formula> at 532 nm, from which the unambiguous measurement of arbitrary target displacement follows with counts in units of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\lambda /8$ </tex-math></inline-formula> of the second-harmonic wavelength or 66 nm.

Two-Dimensional Phase Unwrapping Based on Residual Prediction Neural Network

Two-dimensional phase unwrapping (2-D PU) is a crucial step in interferometric signal processing. The accuracy of phase unwrapping (PU) will significantly affect subsequent processing. Existing algorithms perform poorly in areas with rapid terrain changes or with low coherence. In order to solve the above problems, this letter proposes a method of using a neural network to predict residuals, followed by the minimum <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L^{1}$ </tex-math></inline-formula> -norm algorithm for phase reconstruction. The network for predicting residuals is called residual prediction network (RPNet). To construct more complex loss terms, the continuous relaxation strategy is applied to its training. The proposed method overcomes the effects of noise and terrain changes due to the accurate prediction of residuals. The compensated phase gradients enable the minimum <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L^{1}$ </tex-math></inline-formula> -norm algorithm to reconstruct phases more accurately. The experimental results performed on simulation and interferometric synthetic aperture radar (InSAR) data demonstrate the effectiveness of the proposed method.

Developing an improved optical biosensing system based on gold nanoparticles acting as interferometric enhancers in Lactoferrin detection.

We report for the first time the whole development of a biosensing system based on the Interferometric Optical Detection Method (IODM) enriched with gold nanoparticles (AuNPs), acting as interferometric enhancers for improving the performance of immunoassays. For this purpose, the Lactoferrin sandwich immunoassay model was employed. We describe in detail the entire value chain from the AuNPs production, its functionalization, and characterization with anti-Lactoferrin (anti-LF), the biosensing response of these conjugates as well as their corresponding calculation of the kinetic constants, performance comparison of the readout interferometric signals versus Scanning Electron Microscopy (SEM) and the percentage of the sensing surface covered. Finally, a Lactoferrin sandwich immunoassay was carried out and correlated with Enzyme-Linked ImmunoSorbent Assay (ELISA), and the Limit of Detection and sensitivity figures were obtained. As a result, we demonstrate how the AuNPs act as interferometric amplifiers of the IODM for improving the biosensing response, opening the possibility of being applied in multiple biological detection applications.

Enhancing detection of low-coherence interferometry signals acquired with a spatial heterodyne detector

The use of low-coherence interferometry for industrial applications is constantly growing due to its non-destructive nature and high resolution. Fourier domain systems are usually preferred as they provide a scanning-free method. Recently, a spatial heterodyne detector has proved to be useful in extending the measuring range without losing the micrometric resolution. This system Fourier transforms the interferometric signal, giving the information of interest as fringes in the output image. However, for low light intensities, the images might need to be processed to facilitate the visualization and detection of these fringes. In this work we analyze two main approaches to achieve this: one that uses a cross correlation method and the other one that uses a specially designed two-dimensional adaptive filtering technique. Both methods showed an improved performance when compared with conventional image processing techniques, enhancing the fringes contrast. Additionally, we showed that combining normalized cross correlation and the image gradient is an effective method to locate low intensity signals.

Increased Spurious-Free Dynamic Range in Frequency Sweeping Interferometry by Suppression of Parasitic Amplitude Modulation

We propose an envelope detrending method to correct parasitic amplitude modulation (AM) in frequency sweeping interferometry (FSI). Parasitic AM arises from effects such as device spectral response and residual AM of laser. The periodic component in parasitic AM introduces spurious harmonics, which degrade the spurious-free dynamic range (SFDR) and interfere with information analysis. The proposed method employs a Hilbert transform to obtain a time-varying amplitude fluctuation curve from an auxiliary interferometer. This curve is then employed to detrend the parasitic AM in the main interferometric signal, thus restoring the point spread function and improving the SFDR. An optimal SFDR can be obtained when the optical path differences (OPDs) of the two interferometers are equal, while mismatched OPDs lead to a rapid degradation of the SFDR correction. Experimental results demonstrate the effectiveness of the proposed method and the optimal matching condition. A maximum SFDR of 90.2dB can be achieved when the OPD matching condition is satisfied. By applying the method to an optical frequency domain polarimeter, we realized a measured 81.1dB polarization extinction ratio. We believe that the proposed method can also help to improve the SFDR in other FSI systems.

Absolute distance measurements for in-situ interferometer characterisation using range-resolved interferometry

Range-resolved interferometry (RRI) allows the simultaneous demodulation of multiple interferometric signal sources and provides a tomographic view of all constituent interferometers that may be present in a setup. Through comparison with a reference distance of known length, absolute distance measurements can be performed. RRI is tailored to the use of laser frequency modulation through injection-current modulation of regular, monolithic laser diodes that are both cost-effective and highly coherent and therefore this approach promises broad applicability. In this paper, two methods for absolute distance measurement, one based on the direct evaluation of the signal peak positions and one based on the phase demodulation of an additional lock-in modulation signal, are experimentally demonstrated. Using an external verification displacement interferometer, both techniques are shown to achieve in-situ absolute distance measurements with systematic errors below over a 50 mm travel range. The aim of this paper is to establish the general suitability of RRI for absolute distance measurements and in-situ tomographic interferometer characterisation for precision engineering. In future, this approach could be used to diagnose interferometric setups for parasitic signal contributions, multiple reflections or to determine the dead path length for accurate environmental compensation, either for use during initial setup of, or for continuous operation alongside, a regular displacement measuring interferometer.

Pseudo Wigner-Ville distribution for 3D white light scanning interferometric measurement.

White light scanning interferometry is a commonly used optical measurement method for three-dimensional (3D) surface profiles. In the case of large phase errors, accurate height values can generally be obtained indirectly from the interferometric signal envelope information derived using various envelope extraction methods. However, the current envelope extraction algorithms have the disadvantages of low robustness, increasing the half-width of the envelope information, and requiring correct parameter settings in advance. In this study, the pseudo Wigner-Ville distribution is modified and applied to white light scanning interferometric 3D measurement to avoid the above-mentioned drawbacks. Simulations and experiments are performed in a single-frequency mode (only the approximate central wave-number is used to guide both the proposed and wavelet transform methods). The simulation results prove that the proposed method has a 31.7% higher reconstruction accuracy than the wavelet transform method under a 25 dB signal to noise ratio condition. Concurrently, the proposed method is insensitive to the change in the central wavelength with a constant central wave-number parameter and has a good extraction effect for a long coherent length. The experiments measure standard step objects (VLSI standard, 1.761 ± 0.01 µm), and the reconstruction height error of the proposed method is 0.0035 µm. Simulations and experiments show that the proposed method can adaptively provide accurate envelope information after half-width reduction under the condition that only the approximate central wave-number a priori knowledge is used. Simultaneously, the proposed method is shown to be robust and effective.

Hologram formation by using vertical antenna in a shallow water waveguide

Holographic signal processing method by using a vertical linear antenna in shallow water is presented in the paper. Offered method of moving source sound field hologram formation by using vertical antenna is based on holographic signal processing of each element of vertical antenna. Sound field of source moving in shallow water waveguide creates a stable interference pattern (interferogram) in the frequency-time domain on each element of vertical antenna. The 2D-FT (2D Fourier transform) of the interferogram is antenna element hologram (AEH). AEH allows to coherently accumulate the sound intensity of the interferogram in a relatively small area as focal spots for each element of antenna. It is shown that the focal spots coordinates on AEH depends upon the source range, velocity, and motion direction. The focal spots coordinates on AEH are same for each element of antenna. The output antenna hologram is formed by superposition AEH. Relationship between focal spot coordinates on the antenna hologram and the source range, velocity, and motion direction are derived in the paper. The results of the numerical experiment of antenna hologram formation for low-frequency (100200 Hz) sound field of source moving in shallow water are presented. The results presented in the paper significantly expand the efficiency of interferometric signal processing in shallow water. [This study was supported by RFBR 19-29-06075, МК-6144.2021.4, and МК-4846.2022.4.]

Enhancement of frequency scanning interferometry signal for non-cooperative target based on generative adversarial network

In non-cooperative target frequency scanning interferometry, the return optical power is low, the quality of the interferometric signal is poor, and the signal-to-noise ratio (SNR) is low. Moreover, the power change accompanying the use of the frequency scanning laser modulates the interferometric signal’s amplitude and shifts the amplitude centre. Traditional signal enhancement techniques, such as filtering, can only solve some of the problems affecting the measurement accuracy, and the full-factor processing of such signals is difficult. This paper proposes a non-cooperative target frequency scanning interferometry signal enhancement method based on a generative adversarial network. By learning the sample dataset, the SNR of the signal can be improved within a certain range, and the signal waveform can be corrected simultaneously. The simulation results reveal that the SNR of the non-cooperative target signal is improved and the signal waveform is satisfactorily corrected. Finally, the effectiveness of the enhancement method was experimentally confirmed.

Determining System Parameters and Target Movement Directions in a Laser Self-Mixing Interferometry Sensor

Self-mixing interferometry (SMI) is a promising sensing technology. As well as its compact structure, self-alignment and low implementation cost, it has an important advantage that conventional two-beam interferometry does not have, i.e., SMI signal fringe evolves into asymmetrical shape with increasing optical feedback level, which leads to discrimination of target movement directions for unambiguous displacement measurement possible by a single-channel interferometric signal. It is usually achieved by using SMI signals in moderate feedback regime, where the signals exhibit hysteresis and discontinuity. However, in some applications, e.g., in biomedical sensing where the target has a low reflectivity, it is hard for the SMI system to operate in a moderate feedback regime. In this work, we present comprehensive analyses on SMI signal waveforms for determining system parameters and movement directions by a single-channel weak feedback SMI signal. We first investigated the influence of two system parameters, i.e., linewidth enhancement factor and optical feedback factor, on the symmetry of SMI signals. Based on the analyses on signal waveform, we then proposed a method of estimating the system parameters and displacement directions. The method was finally verified by experiments. The results are helpful for developing sensing applications based on weak feedback SMI systems.

Investigating the spatio-temporal variation of vegetation water content in the western United States by blending GNSS-IR, AMSR-E, and AMSR2 observables using machine learning methods

Vegetation water content (VWC) is an important parameter when evaluating vegetation growth and climate change. The Global Navigation Satellite System Interferometric Reflectometry (GNSS-IR) technique has provided us with an effective approach for the monitoring of VWC. The normalized microwave reflection index (NMRI) was defined to reflect the change of VWC, based on the fact that the amplitude of the direct and reflected GNSS interferometric signal is related to the variation of VWC. However, the sparse distribution of the ground-site-based observation stations restricts the application of NMRI. Fortunately, microwave vegetation optical depth (VOD) is a dimensionless parameter that describes the rate of attenuation of microwaves as they pass through the vegetation canopy, which can provide spatially continuous vegetation information. In this study, we integrate the GNSS-IR NMRI dataset with Advanced Microwave Scanning Radiometer - Earth Observing System (AMSR-E) and Advanced Microwave Scanning Radiometer 2 (AMSR2) VOD datasets by using three machine learning models: the generalized regression neural network (GRNN) model, the back-propagation neural network (BPNN) and multiple linear regression (MLR) models. This method can overcome the drawbacks of sparse distribution and spatial discontinuity of GNSS-IR site data, and obtains spatio-temporally continuous NMRI products that can more intuitively and accurately reflect the variability of VWC in the study area. The results showed that the GRNN model has the best retrieval accuracy, with model fitting and cross-validation R values of 0.76 and 0.74, and root-mean-square error (RMSE) values of 0.034 and 0.035, respectively. Based on the GRNN model, we obtained a spatio-temporally continuous NMRI product to further analyze the changes in VWC from 2007 to 2018. The results indicated that the effects of ENSO events with different intensities on vegetation and precipitation are different, and the response of VWC with different vegetation types to ENSO are also different. The anomalies of VWC show a lag phenomenon and is more stable, not as sensitive as precipitation to the ENSO events.

Design of a phasemeter for real-time measurements of the average plasma density with the microwave interferometer of the tokamak T-15MD

Interferometry is one of the key diagnostics for fusion research. This diagnostic measures electron plasma density. Interferometers that work on an intermediate frequency are widespread nowadays. The phase shift between interferometer signals has to be measured. The use of a microwave interferometer on machines, such as a tokamak, leads to challenging requirements for the phasemeter operation. We present a method to provide real-time measurements of phase shifts with values much higher than 2π. The phasemeter has been designed for the microwave interferometer of the T-15MD tokamak. It is based on microcircuits AD8302. Two signals with frequencies of 5 and 10MHz are used as interferometer outputs. The phasemeter output will be used to create a feedback loop for the working gas injection into the tokamak chamber to control the plasma density. The phasemeter has been successfully tested. The typical phase error and non-linearity of the device are 1° and 5 × 10-4, respectively.

High-precision fiber-optic temperature sensor based on Fabry-Perot interferometer with reflective thin-film multilayer structures

An embodiment of a fiber-optic temperature sensor based on a Fabry-Perot interferometer and a scheme for interrogating an experimental sample of the sensor are proposed. The proposed solution makes it possible not to use expensive spectral measuring devices (spectrum analyzer, interrogator). The region of free dispersion and the phase sensitivity of the developed Fabry-Perot interferometer were determined in the temperature range from 20 °C to 590 °C. The accuracy of measuring the ambient temperature is calculated. The long-term stability of the measuring setup at room temperature has been evaluated. The phase shift of the Fabry-Perot interferometer with temperature change was registered. The design of the Fabry-Perot interferometer is implemented using reflective thin-film multilayer structures obtained by stage-by-stage electron-beam deposition in vacuum on polished end cleavages of an optical fiber. The interferometer interrogation method is based on the use of a vertical-cavity surface-emitting laser (VCSEL) operating in a pulsed mode. The principle of registering the phase shift of the interferometer with a change in temperature is based on the use of auxiliary modulation of laser radiation along the wavelength due to modulation (periodic change) of the duration of optical pulses. Auxiliary modulation makes it possible to obtain additional harmonic components in the interferometer signal, which are further used in homodyne demodulation to restore the interferometer phase shift signal proportional to the change in the optical path difference between the interferometer mirrors. The design of the high-temperature sensor is based on a Fabry-Perot interferometer the reflecting mirrors of which are five alternating layers of thin films of TiO2 and Al2O3. Based on the results of the temperature experiment, it was concluded that an increase in the ambient temperature leads to a decrease in the free dispersion region of the Fabry-Perot interferometer. The conclusion made is consistent with the theoretical data. According to the results of the experiment, it is shown that the phase sensitivity of the interferometer to temperature changes is 0.94 rad/K. The accuracy of temperature measurements at the 3σ level was 0.017 K. The results of the study may be of great importance in creating systems for monitoring temperatures above 300 °C. The use of such an interferometer makes it possible to carry out high-precision relative temperature measurements.

Construction of a vector-field cryogenic magnetic force microscope.

Owing to the high resolution of magnetic force microscopes (MFMs) operating at low temperatures and high-applied magnetic fields, they can be employed to study various phenomena observed in topological magnetic materials and superconductors. In this study, we constructed a low-temperature MFM equipped with a 2-2-9-T vector magnet and a three-axis fiber-optic alignment system. The three-axis alignment device enables in situ calibration of the scanner at low temperatures as well as optimizes the intensity and sensitivity of the interferometer signal. A massive homebuilt vibration isolation table lowers the resonance frequency of the system and minimizes mechanical noise. Consequently, the minimum detectable force gradient of our proposed model is close to the thermodynamic limit of the cantilever. To demonstrate the low-temperature capability of the MFM, we obtained magnetic domain images of the van der Waals ferromagnet Fe4GeTe2 and the Abrikosov superconducting vortices of an Nb film. Furthermore, we performed field angle-dependent MFM experiments in a van der Waals magnetic insulator Cr2Ge2Te6 to verify its vector-field functionality and observed a transition in the domains from the stripe to the bubble phase with respect to the magnetic field angle. The vector-field capability of our MFM can be useful for investigating various anisotropic magnetic phenomena in topological magnetic and superconducting materials.

SU(2) hyper-clocks: Quantum engineering of spinor interferences for time and frequency metrology

In 1949, Ramsey's method of separated oscillating fields was elaborated boosting over many decades metrological performances of atomic clocks and becoming the standard technique for very high precision spectroscopic measurements. A generalization of this interferometric method is presented replacing the two single coherent excitations by arbitrary composite laser pulses. The rotation of the state vector of a two-level system under the effect of a single pulse is described using the Pauli matrices basis of the SU(2) group. It is then generalized to multiple excitation pulses by a recursive Euler-Rodrigues-Gibbs algorithm describing a composition of rotations with different rotation axes. A general analytical formula for the phase-shift associated with the clock's interferometric signal is derived. As illustrations, hyper-clocks based on three-pulse and five-pulse interrogation protocols are studied and shown to exhibit nonlinear cubic and quintic sensitivities to residual probe-induced light-shifts. The presented formalism is well suited to optimize composite phase-shifts produced by tailored quantum algorithms in order to design a new generation of optical frequency standards and robust engineering control of atomic interferences in AMO physics with cold matter and anti-matter.

Arm locking performance with the new LISA design

The laser interferometer space antenna (LISA) is a future space-based gravitational wave detector designed to be sensitive to sources radiating in the low frequency regime (0.1 mHz to 1 Hz). LISA’s interferometer signals will be dominated by laser frequency noise which has to be suppressed by about seven orders of magnitude using an algorithm called time delay interferometry (TDI). Arm locking has been proposed to reduce the laser frequency noise by a few orders of magnitude to reduce the potential risks associated with TDI. In this paper, we present an updated performance model for arm locking for the new LISA mission using 2.5 Gm arm lengths, the currently assumed clock noise, spacecraft motion based on LISA Pathfinder data and shot noise. We also update the Doppler frequency pulling estimates during lock acquisition.

Geometric tilt-to-length coupling in precision interferometry: mechanisms and analytical descriptions

Tilt-to-length (TTL) coupling is a technical term for the cross-coupling of angular or lateral jitter into an interferometric phase signal. It is an important noise source in precision interferometers and originates either from changes in the optical path lengths or from wavefront and clipping effects. Within this paper, we focus on geometric TTL coupling and categorise it into a number of different mechanisms for which we give analytic expressions. We then show that this geometric description is not always sufficient to predict the TTL coupling noise within an interferometer. We, therefore, discuss how understanding the geometric effects allows TTL noise reduction already by smart design choices. Additionally, they can be used to counteract the total measured TTL noise in a system. The presented content applies to a large variety of precision interferometers, including space gravitational wave detectors like LISA.

Four-wavelength quadrature phase demodulation technique for extrinsic Fabry-Perot interferometric sensors.

In this Letter, we report a four-wavelength quadrature phase demodulation technique for extrinsic Fabry-Perot interferometric (EFPI) sensors and dynamic signals. Four interferometric signals are obtained from four different laser wavelengths. A wavelength interval of four wavelengths is chosen according to the free spectrum range (FSR) of EFPI sensors to generate two groups of anti-phase signals and two groups of orthogonal signals. The linear fitting (LF) method is applied to two groups of anti-phase signals to eliminate the dc component and ac amplitude to obtain two normalized orthogonal signals. The differential cross multiplication (DCM) method is then used to demodulate the phase signal from these two normalized orthogonal signals. The proposed LF and DCM (LF-DCM) based four-wavelength quadrature phase demodulation overcomes the drawback of the traditional ellipse fitting (EF) and DCM (EF-DCM) based dual-wavelength demodulation method that it is not suitable for weak signal demodulation since the ellipse degenerates into a straight line, which makes the EF algorithm invalid. Moreover, it also avoids the assumption that the dc component and ac amplitude of interferometric signals are identical, which is widely used in three-wavelength demodulation. An EFPI acoustic sensor is tested to prove the four-wavelength quadrature phase demodulation and experimental results show that the proposed phase demodulation method shows advantages of large dynamic range and wide frequency band. Linearity is as high as 0.9999 and a high signal-to-noise ratio (SNR) is observed from 1 Hz to 100 kHz.

Approaches for Joint Retrieval of Wind Speed and Significant Wave Height and Further Improvement for Tiangong-2 Interferometric Imaging Radar Altimeter

The interferometric imaging radar altimeter (InIRA) adopts a short baseline along with small incidence angles to acquire interferometric signals from the sea surface with high accuracy, thus the wide-swath sea surface height (SSH) and backscattering coefficient (σ0) can be obtained simultaneously. This work presents an approach to jointly retrieve the wind speed and significant wave height (SWH) for the Chinese Tiangong-2 interferometric imaging radar altimeter (TG2-InIRA). This approach utilizes a multilayer perceptron (MLP) joint retrieval model based on σ0 and SSH data. By comparing with the European Center for Medium-Range Weather Forecasts (ECMWF) reanalysis data, the root mean square errors (RMSEs) of the retrieved wind speed and the SWH are 1.27 m/s and 0.36 m, respectively. Based on the retrieved SWH, two enhanced wind speed retrieval models are developed for high sea states and low sea states, respectively. The results show that the RMSE of the retrieved wind speed is 1.12 m/s when the SWHs &lt; 4 m; the RMSE is 0.73 m/s when the SWHs ≥ 4 m. Similarly, two enhanced SWH retrieval models for relatively larger and relatively smaller wind speed regions are developed based on the retrieved wind speed with corresponding RMSEs of 0.19 m and 0.16 m, respectively. The comparison between the retrieved results and the buoy data shows that they are highly consistent. The results show that the additional information of SWH can be used to improve the accuracy of wind speed retrieval at small incidence angles, and also the additional information of wind speed can be used to improve the SWH retrieval. The stronger the correlation between wind speed and SWH, the greater the improvement of the retrieved results. The proposed method can achieve joint retrieval of wind speed and SWH accurately, which complements the existing wind speed and SWH retrieval methods for InIRA.

Modeling of fiber-coupled confocal and interferometric confocal distance sensors

Laser distance sensors are a widespread, fast and contactless approach for distance and surface topography measurements. Main characteristics of those sensors are given by resolution, measurement speed and sensor geometry. With decreasing sensor size, the alignment of the optical components in sensor setup becomes more challenging. The depth response of optical profilers is analyzed to obtain characteristic parameters and, thus, to value the alignment and the transfer behavior of those sensors. We present a novel miniaturized sensor setup comprising of confocal and interferometric confocal signals within one sensor in order to compare both principles simply by obscuring the reference arm by an absorber. Further, we introduce a theoretical signal modeling in order to analyze influences such as spatial coherence, Gaussian beam characteristics and tilted reflectors on depth response signals. In addition to this, we show that the coherent superposition significantly reduces the axial resolution due to the confocal effect in interferometric signals compared to simple confocal signals in measurement and simulation results. Finally, an appropriate fit function is presented, in order to figure out characteristic sensor parameters from the obtained depth response signal. In this context, a good agreement to simulated and measured signals is achieved.