ABCD Matrix Research Articles

Micro-optical design of a three-dimensional microlens scanner for vertically integrated micro-opto-electro-mechanical systems

This paper presents the optical design of a miniature 3D scanning system, which is fully compatible with the vertical integration technology of micro-opto-electro-mechanical systems (MOEMS). The constraints related to this integration strategy are considered, resulting in a simple three-element micro-optical setup based on an afocal scanning microlens doublet and a focusing microlens, which is tolerant to axial position inaccuracy. The 3D scanning is achieved by axial and lateral displacement of microlenses of the scanning doublet, realized by micro-electro-mechanical systems microactuators (the transmission scanning approach). Optical scanning performance of the system is determined analytically by use of the extended ray transfer matrix method, leading to two different optical configurations, relying either on a ball lens or plano-convex microlenses. The presented system is aimed to be a core component of miniature MOEMS-based optical devices, which require a 3D optical scanning function, e.g., miniature imaging systems (confocal or optical coherence microscopes) or optical tweezers.

Two-frequency mutual coherence function for Gaussian-beam pulses propagating along a horizontal path in weak anisotropic atmospheric turbulence.

A theoretical formulation of the spherical-wave two-frequency mutual coherence function (MCF) for a propagation path characterized by a complex ABCD matrix with anisotropic atmospheric turbulence existing somewhere is developed. A specialization of this formulation leads to an expression for the two-frequency MCF of an equivalent pulsed Gaussian beam propagating in weak anisotropic atmospheric turbulence along a horizontal line-of-sight path; relevant closed-form analytical solutions under both near- and far-field conditions are obtained. The small- and large-scale solutions for both the plane- and spherical-wave spatial-coherence radii in either horizontal or vertical direction are derived. Analysis shows that the formula for the on-axis two-frequency MCF of a pulsed Gaussian beam under the weak-turbulence condition in both the near- and far-field regions is distinguished from that applicable in the strong-turbulence limit only by whether the turbulence-induced beam broadening can be thought of as negligible. Under both the near- and far-field conditions, the turbulence-induced increment of the mean-square temporal-pulse half-width is proportional to the effective anisotropy factor of turbulence. The MCF becomes statistically anisotropic due to the anisotropy of turbulence. For the spatial coherence radius of either a plane or spherical wave propagating along a horizontal line-of-sight path in anisotropic atmospheric turbulence, the corresponding small-scale solution is proportional to that for the plane-wave spatial-coherence radius in the isotropic-turbulence case with a proportionality coefficient depending only on the effective anisotropy factor of turbulence. The corresponding large-scale solution is proportional to that for the plane-wave spatial-coherence radius in the isotropic-turbulence case with a proportionality coefficient that depends on both the effective anisotropy factor and spectral index of turbulence.

Theory of angular-dispersive, imaging hard-x-ray spectrographs

A spectrograph is an optical instrument that disperses photons of different energies into distinct directions and space locations, and images photon spectra on a position-sensitive detector. Spectrographs consist of collimating, angular dispersive, and focusing optical elements. Bragg reflecting crystals arranged in an asymmetric scattering geometry are used as the dispersing elements. A ray-transfer matrix technique is applied to propagate x-rays through the optical elements. Several optical designs of hard x-ray spectrographs are proposed and their performance is analyzed. Spectrographs with an energy resolution of 0.1 meV and a spectral window of imaging up to a few tens of meVs are shown to be feasible for inelastic x-ray scattering (IXS) spectroscopy applications. In another example, a spectrograph with a 1-meV spectral resolution and 85-meV spectral window of imaging is considered for Cu K-edge resonant IXS (RIXS).

The ABCD matrix for parabolic reflectors and its application to astigmatism free four-mirror cavities

The ABCD matrix for parabolic reflectors is derived for any incident angles. It is used in numerical studies of four-mirror cavities composed of two flat and two parabolic mirrors. Constraints related to laser beam injection efficiency, optical stability, cavity-mode, beam-waist size and high stacking power are satisfied. A dedicated alignment procedure leading to stigmatic cavity-modes is employed to overcome issues related to the optical alignment of parabolic reflectors.

Modified coupled line trans‐directional coupler with arbitrary power divisions and its application to a 180° hybrid

A modified trans-directional (TRD) coupler for designing compact coupled line 180° hybrid with non-interspersed input/output ports is proposed. The modified TRD coupler which is composed of a coupled line and three different shunt capacitors features the characteristic of arbitrary power division and dc blocking. Design equations for the modified TRD coupler are derived using the ABCD matrix. Based on the modified TRD coupler and combined with the coupled line contra-directional (CTD) coupler, a novel coupled line 180° hybrid is constructed. It inherits the characteristics of the modified TRD coupler including arbitrary power division and dc blocking. Moreover, the input and output ports of the 180° hybrid are non-interspersed. In the topology of the proposed 180° hybrid, no bond wires, via holes or multilayer structures are required. The configuration of the 180° hybrid is analysed and the design equations are derived. Using the design equations, the coupling coefficients of the CTD and the modified TRD coupler can be flexibly chosen according to different fabrication conditions. As an example to validate the feature of the proposed 180° hybrids, two prototypes with 3 and 6 dB coupling coefficients are devised. The measured results are in good agreement with the simulated ones.

Analysis and Design of UWB Modified Two-Sections Wilkinson Power Splitter

In this paper, the analysis and design of a compact Ultra-Wideband (UWB) Wilkinson Power Splitter (WPS) is presented. The proposed splitter consists of two sections Wilkinson power splitter of equal characteristic impedances and unequal electrical lengths with a single stub on each section. The proposed power splitter examined analytically using the “Even Odd Mode” analysis method and the ABCD matrix. The simulation is done using Agilent ADS circuit and EM simulator. The proposed structure achieves an input return loss, S11 better than 12.2 dB, an output return loss, S22 and S33 better than 19.59 dB, output ports isolation, S23 and S32 better than 12.75 dB and insertion loss, S21 and S31 from 3.15 to 3.41dB through the whole UWB range from 3.1 GHz to 10.6 GHz. Furthermore, the power splitter has a compact size of 18.38 * 10.25 mm2 compared to other similar networks.

Matrix theory of wave propagation in hybrid electric/magnetic multiwire transmission line systems

Matrix methods to deal with multimodal propagation in electric multiwire transmission lines are quite well known. However, the matrix theory of magnetic multiwire transmission lines (MMTL) has been established only very recently. In this study, we advance the new subject of hybrid electric and MMTL, which has never been addressed in the literature. A complex system composed of N + 1 electric wires and n + 1 magnetic wires is theoretically analyzed from the viewpoint of their interaction/coupling, transmission line matrix equations, ABCD matrix formalism, and superposition of propagation modes.

MEMS filter’s design and modeling based on width-extensional mode plate resonator for wireless applications

This paper proposes a new filter's design process and modeling that can be employed in RF wireless communication systems. The filter is constructed by two similar rectangular piezoelectric plates vibrating in the width-extensional mode. Each plate is formed by a piezoelectric film and is sandwiched between two metallic electrodes. The vibrating plates are mechanically coupled by a beam linkage. The resonance frequencies of the vibrating structure are defined by the plate's lateral dimension which allows the integration of the multi-frequency filter banks on the same chip. The coupled resonators exhibit two mechanical resonance modes: in-phase and out-of-phase vibrating modes. These modes define the filter bandwidth. In the first part of this paper, we proposed a theoretical model allowing to define the desired filter bandwidth varying the coupling beams location. In the second part, we used the microwave circuit theory to calculate the generalized ABCD matrix and consequently the filter transmitted coefficient (S21). The filter bandwidth values extracted from the transmitted coefficient (S21) show a consistent agreement with the desired filter bandwidths determined by the theoretical model developed in the first part. As application, a filter with a center frequency of 217 MHz is able to operate with fractional bandwidth varying from very low value (close to 0 %) to 5.5 %. This modeling approach is very important for filter bandwidth control.

Direct measurement of particle size and 3D velocity of a gas-solid pipe flow with digital holographic particle tracking velocimetry.

The 3D measurement of the particles in a gas-solid pipe flow is of great interest, but remains challenging due to curved pipe walls in various engineering applications. Because of the astigmatism induced by the pipe, concentric ellipse fringes in the hologram of spherical particles are observed in the experiments. With a theoretical analysis of the particle holography by an ABCD matrix, the in-focus particle image can be reconstructed by the modified convolution method and fractional Fourier transform. Thereafter, the particle size, 3D position, and velocity are simultaneously measured by digital holographic particle tracking velocimetry (DHPTV). The successful application of DHPTV to the particle size and 3D velocity measurement in a glass pipe's flow can facilitate its 3D diagnostics.

ANN-based modeling of compact impedance-varying transmission lines with applications to ultra-wideband Wilkinson power dividers

The modeling of the physical and electrical characteristics of microstrip non-uniform transmission lines (NTLs) utilizing artificial neural networks (ANNs) is investigated. The fundamental equations and constraints for designing variable impedance transmission lines are first presented. Then, a proof-of-concept example of a compact non-uniform matching transformer and the counterpart modeled version is elaborated for source and load impedances Zs and Zl, respectively, at 0.5 GHz. For comparison purposes, weights and biases of the proposed ANN are established with three different training techniques; namely: backpropagation (BP), Quasi-Newton (QN), and conjugate gradient (CG); at which the ABCD matrix, impedance variations, input port matching (S11), and transmission parameter (S21) are set as benchmarks to examine the validity of the trained model. The concept is then extended to model a NTL ultrawideband (UWB) Wilkinson power divider (WPD) with three resistors for improved isolation. S-parameters derived from the trained ANN outputs are close to those obtained by the traditional time-consuming optimization procedure, and show input and output ports matching and isolation of below −10 dB, and acceptable values of transmission parameters over the 3.1 GHz to 10.6 GHz band. The resulting models outperform traditional optimizations in terms of simulation time and reserved resources with comparable accuracy. © 2015 Wiley Periodicals, Inc. Int J RF and Microwave CAE 25:563–572, 2015.

Propagation Properties of Finite Olver-Gaussian Beams Passing through a Paraxial ABCD Optical System

In this paper, an exact analytical propagation formula of Finite Olver-Gaussian Beams (FOGBs) passing through a paraxial ABCD optical system is developed and some numerical examples are performed. The propagation properties of the FOGBs through general optical systems characterized by given ABCD matrix are studied on the basis of the generalized Huygens-Fresnel diffraction integral, which permits to show the behavior of this laser beams family and its properties de-pending of the laser parameters. This research is of interest to prove some investigations done in the past by other researchers.

Aircraft electric system intermittent arc fault detection and location

Intermittent arc faults appear in aircraft power systems in unpredictable ways when the degraded wires are wet, vibrating against metal structures, or under mechanical stresses. They could evolve into serious faults that may cause onboard fires, power interruptions, system damage and catastrophic incidents, and thus have raised concern. Recent trends in solid state power controllers (SSPCs) motivated the development of nondestructive diagnostic methods for health monitoring of aircraft wiring. In this paper, the ABCD matrix (or transmission matrix) modeling method is introduced to derive normal and faulty load circuit models with better accuracy and reduced complexity compared to the differential equation approach, and an intermittent arc fault detection method is proposed based on temporary deviations of load circuit model coefficients and wiring parameters. Furthermore, based on the faulty wiring model, a genetic algorithm is proposed to estimate the fault-related wiring parameters such as intermittent arc location and average intermittent arc resistance. The proposed method can be applied to both the alternating current power distribution system (PDS) and the direct current (DC) PDS. Experiments using a DC power source have been conducted, and the experimental results have demonstrated effectiveness of the proposed method.

Origin of potential errors in the quantitative determination of terahertz optical properties in time-domain terahertz spectroscopy

We demonstrate theoretically and experimentally how changes of a terahertz (THz) beam induced by the sample affect the accuracy of the determination of THz dielectric properties in THz time-domain transmission spectroscopy (TDTS). We apply a Gaussian beam and the ABCD matrix formalism to describe the propagation of the THz beam in a focused beam setup. The insertion of the sample induces a focus displacement which is absent in the reference measurement without a sample. We show how the focus displacement can be corrected. The THz optical properties after focus displacement correction reported in this Letter are in quantitative agreement with those obtained using collimated beam THz–TDTS in previous work.

Propagation of Airy Gaussian vortex beams through slabs of right-handed materials and left-handed materials

By using the method of the ABCD matrix, the propagation of Airy Gaussian vortex (AiGV) beams through slabs of right-handed materials (RHMs) and left-handed materials (LHMs) is reported. Based on the Huygens diffraction integral, an approximate analytical propagation equation for AiGV beams is derived. Using numerical simulations, we study the intensity and phase distributions of the AiGV beams in RHMs and LHMs. We find that the optical vortex can destroy the center lobe of the AiGV beams, and the center lobe can reconstruct due to the fact that the acceleration of the vortex and the AiGV beam is not consistent. We also investigate the influence of different χ0 values on the propagation of the AiGV beams through slabs of RHMs and LHMs, which can cause the AiGV beams to tend toward an Airy vortex beam with a smaller value and a Gaussian vortex beam with a larger one. In addition, we elucidate the energy flow and the angular momentum of the AiGV beams.

Study on Transmission Characteristics of Carbon Nanotube Through Silicon Via Interconnect

In this letter, the Resistance Inductance Capacitance Conductance (RLCG) parameters of carbon nanotube through silicon via (CNT-TSV) are modeled and a transmission line (TL) model is established through ABCD matrix. The impact of their geometrical and material parameters on the TSV transmission characteristics is analyzed over a wideband of frequency using the proposed model. An optimization methodology using air gap insulator is proposed to improve the transmission performance. SPICE results show that a 30.83% reduction in insertion loss and a 40.72% increase in eye open area, respectively.

Effects of resonator input power on Kerr lens mode-locked lasers

Using the ABCD matrix method, the common stability region between the sagittal and tangential planes of a four-mirror Kerr lens mode-locked (KLM) laser cavity is obtained for different ranges of input power. In addition, the effect of the input power on the Kerr lens sensitivity is investigated. Optimal input power and position for highest Kerr lens sensitivity in the stability region are presented and self-starting regime has been achieved. Results show that the resonator input power has a great influence on designing the KLM lasers which can be used in fabricating an optimal femtosecond laser.

Design and Analysis of High Gain Diode Pre-Distortion Linearizer for TWTA

This paper presents the design and analysis of a high gain, broadband Schottky and PIN diode based RF pre-distortion linearizer for TWTA. The circuit is using ABCD matrix approach. The simulation is performed using Agilent ADS software. We have proposed a new linearizer circuit which can achieve a high gain compared to existing linearizer designs.

Research on propagation properties of controllable hollow flat-topped beams in turbulent atmosphere based on ABCD matrix

This paper studies the propagation properties of controllable hollow flat-topped beams (CHFBs) in turbulent atmosphere based on ABCD matrix, sets up a propagation model and obtains an analytical expression for the propagation. With the help of numerical simulation, the propagation properties of CHFBs in different parameters are studied. Results indicate that in turbulent atmosphere, with the increase of propagation distance, the darkness of CHFBs gradually annihilate, and eventually evolve into Gaussian beams. Compared with the propagation properties in free space, the turbulent atmosphere enhances the diffraction effect of CHFBs and reduces the propagation distance for CHFBs to evolve into Gaussian beams. In strong turbulence atmospheric propagation, Airy disk phenomenon will disappear. The study on the propagation properties of CHFBs in turbulence atmosphere by using ABCD matrix is simple and convenient. This method can also be applied to study the propagation properties of other hollow laser beams in turbulent atmosphere.

Calculation of the light intensity distribution reflected by a spherical cube corner retroreflector array

Because the light intensity distribution (LID) reflected by a spherical cube corner retroreflector array (SCCRA) is of great significance for the detection of the remote laser ranging to receive laser pulse echo precisely and to accomplish the function outstandingly, it is vital to find a method to calculate the LID reflected by SCCRAs with different structural parameters for the SCCRA design. Besides, considering that the reflected light goes through a complex optical system (COS) described by ABCD matrix before it reaches the detector, the method to calculate the LID should be suitable for the COS. This article derives formulae to calculate LID reflected by a SCCRA based on the lens-system diffraction formula of S.A. Collins. In the process of deriving, coordinate transformation matrixes and vector formulations are applied and the dihedral offsets are taken into account. Because the structural parameters of the SCCRA and the expression of the ABCD matrix of the COS are not limited in the deriving process, these formulae are suitable for calculation of the LID reflected by different SCCRA and going through different COS. In addition, computer simulations are applied to explain clearly how to get the LID of a SCCRA by the formulae derived in this paper.

Propagation of aberrated wavefronts using a ray transfer matrix.

A ray transfer matrix is used to calculate the propagation of aberrated wavefronts across a homogeneous refractive index. The wavefront is represented by local surface normals, i.e., by a ray bundle, and the propagation is accomplished by transferring those rays across the space. Wavefront shape is generated from the slopes and positions of the collection of rays. Calculation methods are developed for the paraxial case, for higher-order expansions, and for the exact tangent case. A numerical example is used to compare results between an analytical method and the methods developed here.