ABCD Matrix Research Articles

Spatially controllable stimulate Brillouin scattering in the diamond Raman laser

Diamond Raman lasers offer a significant advantage over conventional inversion lasers in achieving high-power single-longitudinal mode (SLM) output, making them highly suitable for various potential applications. However, the presence of stimulated Brillouin scattering (SBS) introduces instability and degrades the SLM mode into a multimode longitudinal mode (MLM) as power scaling increases. In this research article, we present an analytical model that elucidates the transverse-mode behavior of the SBS field within the resonator to explore the mechanism of SBS inhibition. We further investigate the relationship between the cavity ABCD matrix and the transverse mode distribution of the SBS. To validate the model, we design and build laser cavities to observe the SBS beam profile. By utilizing this model, we achieve the controllable TEMmn (m + n ≤ 9) transverse modes of SBS. This model not only acts as a guide for suppressing SBS in diamond Raman lasers, thereby increasing the power limit of SLM output, but also provides a novel approach to attain flexible transverse modes with high power, narrow linewidth, and adjustable wavelength without the need for additional modulation elements.

Numerical Analysis of Optical Fiber Refractive Index in the Miniaturized Graded‐Index Fiber Probe

ABSTRACTWe analyze the impact of the refractive index of optical fibers on the focusing properties of a miniaturized graded‐index (GRIN) fiber probe. The ABCD ray transfer matrix and characteristic parameters are employed to characterize the working distance, focusing spot size, and the depth of field effectively. The three‐dimensional (3‐D) function diagrams and two‐dimensional (2‐D) graphs are used to illustrate the impact of the no‐core fiber (NCF) refractive index and the center refractive index of GRIN fiber on the focusing properties. Numerical analysis results show that when the length of the NCF is 0.36 mm and the GRIN fiber is 0.1 mm, the variation of the refractive index of the NCF between 1.44 and 1.52 leads to the variation of the working distance in the range of 0.02 mm, while the focusing spot size varies in the range of 5 μm. A comparison of 12 probe samples reveals that a 0.02 difference in the center refractive index of GRIN fiber could result in a 0.16 mm variation in working distance and a variation in focusing spot size of over 3 μm. The experimental results indicate that alterations in the length of the fibers have a considerable effect on the focusing properties of the probe. In contrast, the range of variation in focusing properties with fiber refractive index is relatively limited.

Analytical equation for camera imaging with refractive interfaces

Camera imaging through refractive interfaces is a crucial issue in photogrammetric measurements. Most past studies adopted numerical optimization algorithms based on refractive ray tracing procedures. In these studies, the camera and interface parameters are usually calculated iteratively with numerical optimization algorithms. Inappropriate initial values can cause iterations to diverge. Meanwhile, these iterations cannot efficiently reveal the accurate nature of refractive imaging. Therefore, obtaining camera calibration results that are both flexible and physically interpretable continues to be challenging. Consequently, in this study, we modeled refractive imaging by employing ray transfer matrix analysis. Subsequently, we deduced an analytical refractive imaging (ARI) equation that explicitly describes the refractive geometry in a matrix form. Although this equation is built upon the paraxial approximation, we executed a numerical experiment that shows that the developed analytical equation can accurately illustrate refractive imaging with a considerable object distance and a slightly tilted angle of the flat interface. This ARI equation can be used to define the expansion center and the normal vector of the flat interface. Finally, we also propose a flexible measurement method to determine the orientation of the flat interface, wherein the orientation can be measured rather than calculated by iterative procedures.

Estimating the astigmatic power of the crystalline lens and eye from ocular biometry.

To estimate the astigmatic power of the crystalline lens and the whole eye without phakometry using a set of linear equations and to provide estimates for the astigmatic powers of the crystalline lens surfaces. Linear optics expresses astigmatic powers in the form of matrices and uses paraxial optics and a 4 × 4 ray transfer matrix to generalise Bennett's method comprehensively to include astigmatic elements. Once this is established, the method is expanded to estimate the contributions of the front and back lens surfaces. The method is illustrated using two examples. The first example is of an astigmatic model eye and compares the calculated results to the original powers. In the second example, the method is applied to the biometry of a real eye with large lenticular astigmatism. When the calculated powers for the astigmatic model eye were compared to the actual powers, the difference in the power of the eye was (where T represents the matrix transpose) and for the crystalline lens, the difference was (power vector format). A second example applies the method to a real eye, obtaining lenticular astigmatism of -5.84 × 175. The method provides an easy-to-code way of estimating the astigmatic powers of the crystalline lens and the eye.

Ray-transfer matrix for GRIN lenses: application to the crystalline lens

We present a Hamiltonian optics approach to determine the paraxial propagation of rays in gradient-index (GRIN) media. A derivation of the evolution equation for the corresponding ray-transfer matrix is given. An analytical solution can be found for a certain class of GRIN media. The proposed matrix is shown to be a generalization of the well-known matrices for planar and cylindrical GRIN lenses. We provide the accuracy results when applying our analytical method to arbitrary refractive indices and verify its validity by comparing it to other matrices in the literature with satisfactory results.

Spheric Aberration Evaluation of Thin Lens by Ray Transfer Matrix

Spheric Aberration Evaluation of Thin Lens by Ray Transfer Matrix

Three-dimensional rainbow refractometry.

We propose a new, to the best of our knowledge, rainbow technique called three-dimensional rainbow refractometry (TDRR), with a cylindrical lens in the signal collecting system. With a TDRR model based on the ray transfer matrix developed, it is proved that the tilt angle of the rainbow signal is related to the axial position of the droplet, which helps to obtain the 3D position. By converting rainbow scattering angle calibration into the system parameter calibration, a new rainbow data processing program is written in combination with the model to obtain the refractive index and the particle size. With TDRR, we measured a monodisperse droplet stream of deionized water at room temperature for experimental validation and obtained the refractive index with an absolute error of less than 0.0015, the droplet size with an error within ±5%, and the axial position with an error within ±3%, which demonstrated a high accuracy of TDRR.

An open-source alignment method for multichannel infinite-conjugate microscopes using a ray transfer matrix analysis model.

Multichannel, infinite-conjugate optical systems easily allow implementation of multiple image paths and imaging modes into a single microscope. Traditional optical alignment methods which rely on additional hardware are not always simple to implement, particularly in compact open-source microscope designs. We present here an alignment algorithm and process to position the lenses and cameras in a microscope using only image magnification measurements. We show that the resulting positioning accuracy is comparable to the axial resolution of the microscope. Ray transfer matrix analysis is used to model the imaging paths when the optics are both correctly and incorrectly aligned. This is used to derive the corresponding image magnifications. We can then extract information about the lens positions using simple image-based measurements to determine whether there is misalignment of the objective lens to sample distance (working distance) and with what magnitude and direction the objective lens needs to be adjusted. Using the M4All open-source 3D printable microscope system in combination with the OpenFlexure microscope, we validate the alignment method and highlight its usability. We provide the model and an example implementation of the algorithm as an open-source Jupyter Notebook. This article is part of the Theo Murphy meeting issue 'Open, reproducible hardware for microscopy'.

Design of Hybrid Bandpass Filter Chips with High Selectivity and Wideband Using IPD and FBAR Technology

A novel hybrid bandpass filter (BPF) with wideband and high selectivity is proposed in this paper. The hybrid BPF is composed of two film bulk acoustic resonator (FBAR) cells and three lumped component resonators realized by the integrated passive device (IPD) technology. The ABCD matrix to S-parameters matrix method is used to calculate the frequency response of the BPF. Moreover, the expressions of transmission zeros (TZs) have been extracted. In the design process, an iterative design approach is proposed to improve the circuit and layout of the hybrid filter based on the packaged acoustic-electric hybrid simulation effect. Finally, two parts of the filter are packaged based on the flip-chip method, and two prototypes for the BPF are measured. The measured results of two chips with 3 dB fractional bandwidth of 13.7% and 15.8% are designed and fabricated, which verifies the validity of the proposed design principle.

Design of IPD-based wideband bandpass filter chips with controllable transmission zeros

ABSTRACT In this brief, two novel wideband bandpass filter (BPF) chips with independently controllable transmission zeros are proposed. BPF I uses bridge T-type highpass and lowpass structures cascaded to achieve wideband filtering. On this basis, BPF II adds three pairs of LC resonant circuits to add additional transmission zeros while retaining the lowpass structures. The analytical methods of odd-even mode and ABCD matrix to S-parameter matrix are used to analyze and calculate two designs. For validation, two prototypes of proposed BPFs in integrated passive device (IPD) have been designed, fabricated, and measured. One prototype operates at 5.3 GHz, providing a 3-dB fractional bandwidth (FBW) of 46.9%, while the other prototype operates at 4.7 GHz, providing a 3-dB FBW of 57.6%. Moreover, the two chips exhibit minimum insertion losses of 1.55 dB and 1.5 dB, respectively, while occupying compact sizes of 2.27 mm × 0.83 mm and 1.7 mm × 0.88 mm.

Design of optical telescopes for laser field wavefront reconstruction in gravitational-wave detectors

Pick-off beams in Interferometric Gravitational Wave Detectors are often used to derive information on the field circulating in the interferometer. In order to retrieve the wavefront of the field at some chosen location, an optical system has to be deployed so to image the desired field characteristics to an optical bench and ultimately to suitable sensors. In this paper, we detail the requirements in terms of Gouy Phase and ABCD matrix for such an optical system to provide the correct information. We show also that by designing the optical system with adequate Gouy Phase evolution, the same kind of transformation holds true not only for the fundamental Gaussian mode, but also for both even and odd Higher Order Modes (HOMs) of the main interferometer beam. This is important because these HOMs are excited by the aberrations of the optical elements in the interferometer, and carry the information one is interested to. Simulations with two tools (Finesse and Zemax OpticsStudio®) are provided to confirm the analytical results obtained using Gaussian modes expansion of a generic field.

Correction Method for Optical Scaling of Fundoscopy Images: Development, Validation, and First Implementation.

Although fundus photography is extensively used in ophthalmology, refraction prevents accurate distance measurement on fundus images, as the resulting scaling differs between subjects due to varying ocular anatomy. We propose a PARaxial Optical fundus Scaling (PAROS) method to correct for this variation using commonly available clinical data. The complete optics of the eye and fundus camera were modeled using ray transfer matrix formalism to obtain fundus image magnification. The subject's ocular geometry was personalized using biometry, spherical equivalent of refraction (RSE), keratometry, and/or corneal topography data. The PAROS method was validated using 41 different eye phantoms and subsequently evaluated in 44 healthy phakic subjects (of whom 11 had phakic intraocular lenses [pIOLs]), 29 pseudophakic subjects, and 21 patients with uveal melanoma. Validation of the PAROS method showed small differences between model and actual image magnification (maximum 3.3%). Relative to the average eye, large differences in fundus magnification were observed, ranging from 0.79 to 1.48. Magnification was strongly inversely related to RSE (R2=0.67). In phakic subjects, magnification was directly proportional to axial length (R2=0.34). The inverse relation was seen in pIOL (R2=0.79) and pseudophakic (R2=0.12) subjects. RSE was a strong contributor to magnification differences (1%-83%). As this effect is not considered in the commonly used Bennett-Littmann method, statistically significant differences up to 40% (mean absolute 9%) were observed compared to the PAROS method (P<0.001). The significant differences in fundus image scaling observed among subjects can be accurately accounted for with the PAROS method, enabling more accurate quantitative assessment of fundus photography.

A systematic approach for the development of wide band band pass filters using resonators

In this paper, for the first time a novel and systematic approach is presented to construct a wideband Band Pass Filters (BPFs) with high selectivity and stopband response. Two BPFs (Filter I and Filter II) were built on a Taconic (TLX-9) substrate using semi-circular tapered coupled lines, Defected Microstrip Structures (DMSs), Defected Ground Structures (DGS), and Stepped Impedance Resonators (SIRs). The precision of the design procedure is confirmed through 3-D full wave Electro-Magnetic (EM) simulation software where two wideband filters were designed for L, S, and C-bands. Additionally, parametric analysis has been done to evaluate the performance of both wideband filters with key features such as return loss, insertion loss, Band Width (BW), selectivity, and stopband response. Even-odd mode analysis has been implemented using the ABCD matrix that acts as a source for mathematical analysis of Filter I. Eventually, two wideband filters have been fabricated and tested for their effectiveness, and the measured results of two wideband filters are in line with the simulated ones with slight deviations due to fabrication tolerances and connector losses. The prototyped Filter I functions as a bandpass in the frequency range of 2.4 GHz to 4.8 GHz with a maximum insertion loss of 0.85 dB and return loss of 19.5 dB thereby providing 64% Fractional Band Width (FBW). Near 19 dB rejection is observed in the lower and upper stopband regions with BW (stopbands) greater than 2 GHz. Moreover, Filter II is also fabricated and tested. The experimental return loss of 20 dB and insertion loss (maximum) of 0.73 dB over the passband (1.9 GHz to 3.4 GHz) were found with 0.7 Selectivity Factor (SF). More than 20 dB attenuation of stopbands (lower and upper) was achieved. Thus, the fabricated wideband BPFs operate in L, S, and C-bands for radars, satellite and long-distance radio communications, and military telemetry applications.

A perturbation method for the ray-transfer matrix of the crystalline lens

Analytical expressions for the ray-transfer matrix have been proven useful for the understanding of ray propagation in gradient-index (GRIN) lenses. The determination of an exact analytical expression for the ray-transfer matrix of arbitrary GRIN lenses remains unsolved. We propose an approximation based on the perturbation method with highly accurate results for models of the crystalline lens, which outperforms existing methods.

A simple method for pulse contrast enhancement via self-focusing

Abstract Here we report on a simple-to-implement and cost-effective approach for laser pulse contrast enhancement, based on the ${\chi}^{(3)}$ nonlinear self-focusing effect. An intentionally induced and gently controlled self-focusing in a thin glass transforms the time-dependent intensity into variation in beam divergence. Followed by a spatial discriminating filter, only the strongly focused fraction traverses the setup, at the expense of efficiency. A numerical model, accounting for the pulse and material parameters via a Gaussian ABCD matrix, provides an estimate for the instantaneous beam waist and transmission efficiency, which enables us to evaluate the resulting contrast enhancement. The estimated contrast enhancement spans between 0.5 and 2.5 orders of magnitude, in conjunction with approximately 25%–90% estimated efficiency, depending on the pulse parameters. In a preliminary experiment we demonstrated the effect with 10s-μJ sub GW regime with approximately 40 $\%$ efficiency and a contrast improvement of more than or equal to 20 dB.

The Propagation Properties of a Lorentz–Gauss Vortex Beam in a Gradient-Index Medium

Based on the Huygens–Fresnel integral and ABCD matrix, the propagation equation for the Lorentz–Gauss vortex beam (LGVB) in a gradient-index medium (GRIN) is rederived. The evolution of the intensity and phase distributions of an LGVB through a GRIN medium are numerically calculated as a function of the gradient-index parameter with changes in the incident beam parameters. The results showed that the propagation path and intensity distributions changed periodically with increasing propagation distance. In contrast, phase distributions change at multiples of π/β or 2π/β, depending on whether the M values are odd or even, respectively. At the same time, the parameters of the gradient index determine the periodic values of the Lorentz–Gauss vortex beams during propagation, and as β increased, the period of evolution decreased. The Lorentz–Gauss vortex beam propagating through the gradient index will develop from a square beam to a Gaussian vortex beam more quickly with an increase of w0x=w0y. In addition, the topological charge affects the size of the dark spot at the center of the beam and the size of the beam, causing the phase distributions to change periodically in the medium. This study is beneficial for laser optics and optical communications.

Self-Focusing and Self-Phase Modulation of a Focused Femtosecond Laser Beam in Fused Silica at Near-Critical Peak Power

Self-focusing and self-phase modulation of focused femtosecond laser pulse were studied in bulk fused silica. Depending on pulse energy magnitude, three different regimes were observed in the experiment: (1) linear regime, when the self-focusing effect is insignificant, characterized by linear spectral broadening; (2) transition regime, when the self-focusing effect notably reduces a laser beam spot in a focal plane without considerable nonlinear absorption and beam collapse, characterized by quick increase in spectral broadening due to self-phase modulation; (3) nonlinear regime, filamentation, characterized by near-to- linear spectral broadening due to self-phase modulation. The slope for the linear and nonlinear regimes was the same within the measurement error bars. The numerical simulation, using the ray transfer matrix method, envisioned the self-phase modulation spectral broadening effect in all these regimes.

Integration for coaxial irradiation of fiber laser, oxygen jet, and oxy-fuel gas

This study introduces a convergence approach named laser oxyfuel and oxy-flame hybrid (LOXAFH) cutting, integrating laser cutting and oxyfuel cutting techniques for cutting clad steel plates. The LOXAFH method combines laser, oxyflame, and oxygen jet in a coaxial manner, harnessing 1 kW power concentration. The LOXAFH cutting head was meticulously developed on the basis of a ray transfer matrix and estimated refractive index by incompressible flow with a computational fluid dynamics program and an optical design program. The LOXAFH cutting is validated through three experiments on 15 mm clad steel plates: oxyfuel cutting, laser-assisted oxygen cutting (LASOX), and LOXAFH cutting.

3D trajectory, morphology and phase discrimination of burning Al/AP propellant particles with time-resolved astigmatic dual-beam interferometric particle imaging

This work implements astigmatic dual-beam interferometric particle imaging (ADIPI) in propellant combustion diagnostics to enable the measurement of the three-dimensional (3D) spatial position and particle morphology of opaque particles. The ADIPI allows the discrimination of different substances, such as metal droplets, flames, and pyrolysis residues. A general theoretical model based on the generalized Huygens-Fresnel integral and the ray transfer matrix model is developed to describe the identification of ADIPI signals for regular and irregular particles. An experimental system combining a 13 kHz time-resolved ADIPI with digital in-line holography (DIH) is built to enable simultaneous measurements of the propellant combustion field. The typical ADIPI signals presented by different substances in the combustion field are analyzed by comparing the experimental results of DIH. The time-resolved 3D trajectory reconstruction of burning particles and the measurement of irregular particle size and morphology are realized. Due to the effect of the flame, the average deviation of the depth locating of the particles is 1.4 mm with a standard deviation of 1.0 mm. The average deviation value of the depth position relative to the measured distance is 1%. In propellant combustion measurements, the motion of burning particles can be accurately determined, proving its feasibility and practicality.

Coupling of laser diode with cylindrical microlens fabricated on the tip of single-mode photonic crystal fiber: prediction of coupling optics by ABCD matrix

Abstract We report the theoretical prediction of the coupling optics involving a laser diode and photonic crystal fiber which contains on its tip cylindrical microlens. The analysis is based on relevant ABCD matrix formalism. Though photonic crystal fiber has very large single-mode region, we carry on the investigation for two commonly used wavelengths such as 1.3 µm and 1.5 µm. For the particular kind of photonic crystal having specific value of air filling ratio and hole pitch, we have optimized the distance of laser diode from the microlens in order to get maximum value of coupling efficiency for each value of radius of cylindrical microlens. Incidentally, we have got here maximum coupling efficiency to the extent of 86.99 % at the wavelength 1.3 µm when the radius of the cylindrical microlens is 8.5 µm and effective spot size of the fiber is 4.433909 µm. Further, the maximum efficiency becomes 91.78 % at 1.3 µm when the radius of the cylindrical microlens is 4.0 µm and effective spot size of the fiber is 1.859907 µm. It has been shown that wavelength 1.5 µm is not so coupling efficient like 1.3 µm. Moreover, it has been shown that with same set of relevant parameters, PCF is more coupling efficient compared to ordinary graded index fiber. Thus the present work generates scope for varying different photonic crystal fiber parameters and the wavelengths of source as well for enhancing the coupling efficiency. Accordingly, the present analysis will be extremely helpful for design of efficient coupler of this kind.