The present state of high-resolution displacement measuring interferometry is reviewed. Factors which determine the accuracy, linearity and repeatability of nanometre-scale measurements are emphasized. Many aspects of interferometry are discussed, including general metrology and alignment errors, as well as path length errors. Optical mixing and the nonlinear relation between phase and displacement are considered, as well as the influence of diffraction on accuracy. Environmental stability is a major factor in the repeatability and accuracy of measurement. It is difficult to obtain a measurement accuracy of 10-7 when working in air. Several approaches to improving this situation are described, including multiwavelength interferometry. Recent measurements of the short- and long-term frequency stability of lasers are summarized. Optical feedback is a subtle, but important source of frequency destabilization, and methods of detection and isolation are reviewed. Calibration of phase measuring electronics used for subfringe interpolation is included. Progress in 'in situ' identification of error sources and methods of validating accuracy are emphasized.

http://iopscience.iop.org/article/10.1088/0957-0233/4/9/001/pdf

Recent advances in displacement measuring interferometry

As a part of the program to establish a thermal expansion standard, the linear thermal expansion coefficients of single-crystal silicon have been determined in the temperature range 293 to 1000 K using a dilatometer which consists of a heterodyne laser Michelson interferometer and gold versus platinum thermocouple. The relative standard deviation of the measured values from those calculated from the best least-squares fit was 0.21%. The relative expanded uncertainty in the measurement was estimated to be 1.1 to 1.5% in the temperature range, based upon an analysis of thirteen standard uncertainties. The present data are compared with the data previously obtained by similar dilatometers and the standard reference data for the thermal expansion coefficient of silicon, recommended by the Committee on Data for Science and Technology (CODATA). The present data are in good agreement with the most recently reported data but not with the earlier high-temperature data used to evaluate the standard reference data, which suggests a need for the reevaluation of the standard reference data for the thermal expansion coefficient of silicon at temperatures above 600 K.

Linear Thermal Expansion Coefficient of Silicon from 293 to 1000 K

We demonstrate a new heterodyne Michelson interferometer design for displacement measurements capable of fringe interpolation accuracy of one part in 36 000. Key to this level of accuracy are the use of two acousto-optic modulators for heterodyne frequency generation and digital signal processing demodulation electronics. We make a direct comparison of our interferometer to a commercial interferometer based on a Zeeman-stabilized laser, and show that the residual periodic errors in ours are two orders of magnitude lower than those in the commercial unit. We discuss electronically induced optical cross talk and optical feedback as sources of periodic error. Our new interferometer is simple, robust, and readily implemented.

Michelson Interferometry With 10 PM Accuracy

A new, to our knowledge, heterodyne interferometer for differential displacement measurements is presented. It is, in principle, free of periodic nonlinearity. A pair of spatially separated light beams with different frequencies is produced by two acousto-optic modulators, avoiding the main source of periodic nonlinearity in traditional heterodyne interferometers that are based on a Zeeman split laser. In addition, laser beams of the same frequency are used in the measurement and the reference arms, giving the interferometer theoretically perfect immunity from common-mode displacement. We experimentally demonstrated a residual level of periodic nonlinearity of less than 20 pm in amplitude. The remaining periodic error is attributed to unbalanced ghost reflections that drift slowly with time.

Heterodyne interferometer with subatomic periodic nonlinearity.

A Heterodyne Interferometer With Subatomic Periodic Nonlinearity

Heterodyne laser interferometry and its accuracy at subnanometer levels are discussed. The simplest optical nonpolarization heterodyne interferometer is tested experimentally to understand and reduce fringe distortion. An accuracy of 0.1 nm for the 633-nm He-Ne laser interferometer is achieved. >

Linear interpolation of periodic error in a heterodyne laser interferometer at subnanometer levels (dimension measurement)

Photonic accelerators have been intensively studied to provide enhanced information processing capability to benefit from the unique attributes of physical processes. Recently, it has been reported that chaotically oscillating ultrafast time series from a laser, called laser chaos, provides the ability to solve multi-armed bandit (MAB) problems or decision-making problems at GHz order. Furthermore, it has been confirmed that the negatively correlated time-domain structure of laser chaos contributes to the acceleration of decision-making. However, the underlying mechanism of why decision-making is accelerated by correlated time series is unknown. In this study, we demonstrate a theoretical model to account for accelerating decision-making by correlated time sequence. We first confirm the effectiveness of the negative autocorrelation inherent in time series for solving two-armed bandit problems using Fourier transform surrogate methods. We propose a theoretical model that concerns the correlated time series subjected to the decision-making system and the internal status of the system therein in a unified manner, inspired by correlated random walks. We demonstrate that the performance derived analytically by the theory agrees well with the numerical simulations, which confirms the validity of the proposed model and leads to optimal system design. This study paves the way for improving the effectiveness of correlated time series for decision-making, impacting artificial intelligence and other applications.

Theory of Acceleration of Decision Making by Correlated Times Sequences

Quantum walks (QWs) exhibit different properties compared with classical random walks (RWs), most notably by linear spreading and localization. In the meantime, random walks that replicate quantum walks, which we refer to as quantum-walk-replicating random walks (QWRWs), have been studied in the literature where the eventual properties of QWRW coincide with those of QWs. However, we consider that the unique attributes of QWRWs have not been fully utilized in the former studies to obtain deeper or new insights into QWs. In this paper, we highlight the directivity of one-dimensional discrete quantum walks via QWRWs. By exploiting the fact that QWRW allows trajectories of individual walkers to be considered, we first discuss the determination of future directions of QWRWs, through which the effect of linear spreading and localization is manifested in another way. Furthermore, the transition probabilities of QWRWs can also be visualized and show a highly complex shape, representing QWs in a novel way. Moreover, we discuss the first return time to the origin between RWs and QWs, which is made possible via the notion of QWRWs. We observe that the first return time statistics of QWs are quite different from RWs, caused by both the linear spreading and localization properties of QWs.

/pdf/directivity-of-quantum-walk-via-its-random-walk-replica-1afdyev9.pdf

Directivity of Quantum Walk via Its Random Walk Replica

Multiscale entropy (MSE) has been widely used to examine nonlinear systems involving multiple time scales, such as biological and economic systems. Conversely, Allan variance has been used to evaluate the stability of oscillators, such as clocks and lasers, ranging from short to long time scales. Although these two statistical measures were developed independently for different purposes in different fields, their interest lies in examining the multiscale temporal structures of physical phenomena under study. We demonstrate that from an information-theoretical perspective, they share some foundations and exhibit similar tendencies. We experimentally confirmed that similar properties of the MSE and Allan variance can be observed in low-frequency fluctuations (LFF) in chaotic lasers and physiological heartbeat data. Furthermore, we calculated the condition under which this consistency between the MSE and Allan variance exists, which is related to certain conditional probabilities. Heuristically, natural physical systems including the aforementioned LFF and heartbeat data mostly satisfy this condition, and hence, the MSE and Allan variance demonstrate similar properties. As a counterexample, we demonstrate an artificially constructed random sequence, for which the MSE and Allan variance exhibit different trends.

Information-theoretical analysis of statistical measures for multiscale dynamics.

In this paper, we demonstrate that a common underlying structure—a skeleton structure—is present behind discrete-time quantum walks on a one-dimensional lattice with a homogeneous coin matrix. More speciﬁcally, we examine the transition probabilities of random walks that replicate the probability distribution of quantum walks. We show that the transition probability contains a skeleton structure by considering the weak limit that excludes the oscillatory behavior. Remarkably, the skeleton structure does not depend on the coin matrix or the initial conditions of the quantum walk. Furthermore, we propose a random walk whose transition probabilities are deﬁned by the skeleton structure, which we call a “quantum-skeleton random walk.” We demonstrate that the resultant properties of the walkers are similar to the original quantum walk.

T. Yamagami

Papers

Linear interpolation of periodic error in a heterodyne laser interferometer at subnanometer levels (dimension measurement)

Theory of Acceleration of Decision Making by Correlated Times Sequences

Directivity of Quantum Walk via Its Random Walk Replica

Information-theoretical analysis of statistical measures for multiscale dynamics.

Skeleton structure inherent in discrete-time quantum walks