This chapter introduces the finite element method (FEM) as a tool for solution of classical electromagnetic problems. Although we discuss the main points in the application of the finite element method to electromagnetic design, including formulation and implementation, those who seek deeper understanding of the finite element method should consult some of the works listed in the bibliography section.

Introduction to the Finite Element Method

This tutorial presents a guided tour of laser feedback interferometry, from its origin and early development through its implementation to a slew of sensing applications, including displacement, distance, velocity, flow, refractive index, and laser linewidth measurement Along the way, we provide a step-by-step derivation of the basic rate equations for a laser experiencing optical feedback starting from the standard Lang and Kobayashi model and detail their subsequent reduction in steady state to the excess-phase equation We construct a simple framework for interferometric sensing applications built around the laser under optical feedback and illustrate how this results in a series of straightforward models for many signals arising in laser feedback interferometry Finally, we indicate promising directions for future work that harnesses the self-mixing effect for sensing applications

Laser feedback interferometry: a tutorial on the self-mixing effect for coherent sensing

Semiconductor Lasers Stability Instability And Chaos

In this paper, the inherent error as well as the robustness of a previously published displacement retrieval technique called the phase unwrapping method (PUM) is analyzed. This analysis, based on a detailed study of laser feedback phase behavior, results in a new algorithm that removes the PUM inherent error while maintaining its robustness. The said algorithm has been successfully tested on simulated and experimental self-mixing (SM) interferometric signals. Simulations in weak and moderate feedback regimes demonstrate that the said algorithm can reach a subnanometric precision compared with approximately 25 nm for PUM. For experimental SM signals affected by noise, the measured rms displacement error and the maximum absolute error are 14 and 37 nm, respectively, for the proposed algorithm and 34 and 123 nm for the PUM, which indicates a threefold displacement precision improvement over the PUM. Finally, it is explained that the precision can be further improved by a reduction of the noise level of experimental SM signals.

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Study of Laser Feedback Phase Under Self-Mixing Leading to Improved Phase Unwrapping for Vibration Sensing

This work describes the development of an optical sensor for measurement of vibration without contact. The realized vibrometer is based on real-time digital elaboration of the signal obtained by a self-mixing interferometer, with an embedded autofocus system. Two different algorithms are implemented, for the continuous working on diffusive surfaces, with different levels of optical reflectivity. Thanks to the autofocus and the digital processing, the proposed sensor is easy to use and requires no assistance of a skilled operator.

Self-mixing vibrometer with real-time digital signal elaboration.

Approaches that are, to our knowledge, novel, are proposed in this paper to improve the accuracy performance of self-mixing interferometry (SMI) for displacement measurement First, the characteristics associated with signals observed in SMI systems are studied, based on which a new procedure is proposed for achieving accurate estimation of the laser phase The studies also revealed the reasons for the inherent errors associated with the existing SMI-based techniques for displacement measurement Then, this paper presents a new method for estimating the optical feedback level factor (denoted by C) in real time Combining the new algorithms for estimating the laser phase and updating C value, the paper finally presents a novel technique for displacement measurement with improved accuracy performance in contrast to existing techniques The proposed technique is verified by both simulation and experimental data

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Improving the measurement performance for a self-mixing interferometry-based displacement sensing system

To achieve complete security of communication, there is a need for “ real-time ” ultrafast physical random bit generators (RBGs). In the available physical RBGs, random bit extraction is effected in the electrical domain and, therefore, cannot directly function beyond the frequencies of 10 GHz or so in real time . Here, we present a fully photonic strategy for real-time random number generation and report a proof-of-concept experimental demonstration of an integrated 10 Gb/s RBG system (prototype) based on laser chaos. The use of an ultrastable mode-locked laser, together with ultrafast optical interactions, gives our method the capability to develop super-high-speed RBGs in practice. The photonic implementation is fully compatible with optical communications so that well-established optical multiplexing techniques can be used to realize Tb/s real-time random bit generation.

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Ultrafast Fully Photonic Random Bit Generator

A self-mixing interferometry (SMI) system is a laser diode (LD) with an external cavity formed by a moving external target. The behavior of an SMI system is governed by the injection current J to the LD and the parameters associated with the external cavity mainly including optical feedback factor C, the initial external cavity length (L0) and the light phase (ϕ0) which is mapped to the movement of the target. In this paper, we investigate the dynamic behavior of an SMI system by using the Lang-Kobayashi model. The stability boundary of such system is presented in the plane of (C, ϕ0), from which a critical C (denoted as Ccritical) is derived. Both simulations and experiments show that the stability can be enhanced by increasing either L0 or J. Furthermore, three regions on the plane of (C, ϕ0) are proposed to characterize the behavior of an SMI system, including stable, semi-stable and unstable regions. We found that the existing SMI model is only valid for the stable region, and the semi-stable region has potential applications on sensing and measurement but needs re-modeling the system by considering the bandwidth of the detection components.

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Dynamic stability analysis for a self-mixing interferometry system.

A simple method for measuring the linewidth enhancement factor (LEF) of semiconductor lasers (SLs) is proposed and demonstrated in this paper. This method is based on the self-mixing effect when a small portion of optical signal intensity emitted by the SL reflected by the moving target re-enters the SL cavity, leading to a modulation in the SL's output power intensity, in which the modulated envelope shape depends on the optical feedback strength as well as the LEF. By investigating the relationship between the light phase and power from the well-known Lang and Kobayashi equations, it was found that the LEF can be simply measured from the power value overlapped by two SLs' output power under two different optical feedback strengths. Our proposed method is verified by both simulations and experiments.

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Simple method for measuring the linewidth enhancement factor of semiconductor lasers.

This paper presents comprehensive studies on the stability of a semiconductor laser (SL) with external optical feedback. In particular, based on numerical computation on the Lang and Kobayashi equations, the stability limit under the condition of minimum linewidth is investigated, revealing how it is influenced by three major parameters associated with SL operation conditions, including the feedback strength ( $k$ ), the external cavity length ( $L$ ), and the injection current ( $J$ ). In contrast to existing work in literature, the presented removed all the approximations and assumptions, hence resulting in relatively complete description of the stability limit. This paper presented leads to a number of important and interesting discoveries: 1) the possible stable area is broader than what is described by the existing work; 2) for long external cavities, the stability limit can be described by a linear relationship between $k$ and $L$ ; and 3) on the stability limit, the critical feedback strength ( $k_{c} $ ) and critical external cavity length ( $L_{c} $ ) are discovered, respectively, being proportional and inversely proportional to $J$ and $k$ .

Yuanlong Fan

Papers

Improving the measurement performance for a self-mixing interferometry-based displacement sensing system

Ultrafast Fully Photonic Random Bit Generator

Dynamic stability analysis for a self-mixing interferometry system.

Simple method for measuring the linewidth enhancement factor of semiconductor lasers.

Stability Limit of a Semiconductor Laser With Optical Feedback