ABCD Matrix Research Articles

The hollow Gaussian beam propagation on curved surface based on matrix optics method

In this paper, based on the Euler–Lagrange equation, an ABCD matrix is constructed out to study the paraxial transmission of light on a constant Gaussian curvature surface (CGCS), which is the first time to our knowledge. Then, by using the method of matrix optics, we extend the CGCS matrix to a general transfer matrix which is suitable for a gently varying curvature. As a beam propagation example, based on the Collins integral and the derived ABCD matrix elements, an analytical propagation formula for the hollow Gaussian beams (HGBs) on the CGCS is deduced. The propagation characteristics of HGBs on a CGCS are illustrated graphically in detail, mainly including the change of dark spot size and splitting rays. Besides its propagation periodicity and diffraction properties, a criterion for convergence and divergence of the spot size is proposed. The area of the dark region of the HGBs can easily be controlled by proper choice of the beam parameters and the shape of CGCS. In addition, we also study the special propagation properties of the hollow beam with fractional order. Compared with propagation characteristics of HGBs in flat space, these novel propagation characteristics of HGBs on curved surface may further expand the application range of hollow beams.

A quint mode fractal architecture based resonator and its efficacy in the design of multiband bandpass filters

This paper presents a new approach to design multiband bandpass filters (BPF) using multimode resonator topology inspired from Bamileke fractal architecture in the form of fractal shaped resonator. The proposed resonator architecture exhibits quint mode resonance behavior where resonance frequencies are determined using ABCD matrix. It offers flexibility in terms of choosing passband center frequencies precisely. Initially a second order dual band BPF is presented by suppressing higher order modes using asymmetric feed line. The same structure is used further to implement a triband and a quad band BPF where the input/output coupling circuit is modified to extract the higher order modes. Two stub loaded resonators are also embedded to improve the passband selectivity and to improve upper stopband characteristics. The design guidelines are demonstrated through the external quality factor (Qe) and coupling coefficient (k) synthesis approach. The developed filters are fabricated and measured to verify the predicted and simulated results. A close agreement is observed between them. The fabricated multiband BPFs has a very compact size, less insertion loss and good return loss profile.

State evolution formula and stability analysis of a paraxial optical system.

By analyzing the phase vector evolution of a paraxial optical system (POS) with a variational background refractive index, we obtain a continuous dynamic equation, called state evolution formula (SEF), which simultaneously gives the phase vector transformation and ray trajectory inside and outside the optical elements. Compared with ray transfer matrix method, this phase-vector equation is universal in treating problems about propagation and stability of paraxial rays, since it extends the linear and discrete matrix equation to a differential equation. It takes a consistent form for both continuous and discontinuous cases without considering the special rays, even the input and output states present a nonlinear relation. Based on the SEF, we further propose a rigorous criterion about whether a continuous and non-periodic POS is stable. This formula provides a reference model for the theoretical analysis of ray dynamics in geometric and physical optical systems.

Power divider with wideband harmonic suppression for center‐fed antenna arrays

AbstractIn this paper, a center‐fed four‐way power divider with wideband harmonic suppression is proposed. A distinguishing feature is the use of sectorially modified Gielis curves (SMGCs) for open stubs, shorted stubs, and stub‐loaded coupled‐line structures, providing more degrees of freedom in the designs to suppress the wideband harmonics. The expressions for the SMGCs, along with the ABCD matrix elements of the stub‐loaded coupled‐line structure having different even and odd mode electrical lengths (θec ≠ θoc) are another distinguishing features presented in this work. The novelty is to use two stages (first and second) with a combination of open, shorted, coupled‐line stubs and tapered transmission lines to achieve good impedance matching and wideband harmonic suppression for center‐fed antenna arrays. The proposed center‐fed four‐way power divider achieves a wideband harmonic suppression of up to 4fo and a relatively wide −10 dB impedance bandwidth of 52%. The proposed power divider is used in a horizontally‐polarized antenna array for wireless L‐band applications.

Universal approach for local higher-order wavefront tracing equations for complex optical systems.

Analytical wave tracing, including higher-order aberrations, is the state of the art, but at present it is only provided for single propagation or refraction. A complex optical system being a sequence of many refractive surfaces and gaps in between can only be treated by successive application of elementary wave-tracing steps, which is time-consuming for a large number of surfaces or even impossible if a system specification by surfaces is not available. Provided the ray transfer properties of a system are summarized as a nonlinear function f whose Jacobian is the ABCD matrix of the system, by multiple derivative we obtain wave-tracing equations for the wavefront's local derivatives of any desired order. The outgoing wavefront derivative of any order can be written as a sum of multinomials of derivatives of the incoming wavefront, weighted by system-dependent coefficients and by powers of the factor β=(A-BE2)-1, where E2 is the second-order aberration of the incoming wavefront. Compared to stepwise wave tracing, this approach is extremely efficient when tracing many different wavefronts through one fixed optical system.

확장된 백오프 출력에서 고효율 동작을 위한 최적화된 대칭 이단 도허티 전력 증폭기 설계

This article presents a symmetric two-stage Doherty power amplifier (DPA) for the extended output power back-off level, which can be attained even with a reduced circuit size by changing the output impedance of the peaking amplifier to be inductive without the offset line. The ABCD matrix is employed to design the output matching network of the peaking amplifier with a simple transmission line. Consequently, we can reduce the design complexity by applying the same circuit to the carrier amplifier as the DPA is symmetric. To deal with reduced output power of the class-C peaking amplifier, a two-stage DPA structure was applied. The power amplifier exhibited a drain efficiency (DE) of 63.5 %, and a power added efficiency (PAE) of 50.6 % at 7.5 dB output backoff was obtained using the 3.5 GHz pulsed continuous wave (CW) signal. Furthermore, a DE of 53.3 %, a PAE of 43.2 % and a power gain of 18 dB were obtained at the power level of 41 dBm using a 20 MHz LTE signal with a peak-to-average power ratio of 7.5 dB.

Laser Beam Pointing Stabilization Control through Disturbance Classification

In laser systems, beam pointing usually drifts as a consequence of various disturbances, e.g., inherent drift, airflow, transmission medium variation, mechanical vibration, and elastic deformation. In this paper, we develop a laser beam pointing control system with Fast Steering Mirrors (FSMs) and Position Sensitive Devices (PSDs), which is capable of stabilizing both the position and angle of a laser beam. Specifically, using the ABCD matrix, we analyze the kinematic model governing the relationship between the rotation angles of two FSMs and the four degree-of-freedom (DOF) beam vector. Then, we design a Jacobian matrix feedback controller, which can be conveniently calibrated. Since disturbances vary significantly in terms of inconsistent physical characteristics and temporal patterns, great challenges are imposed to control strategies. In order to improve beam pointing control performance under a variety of disturbances, we propose a data-driven disturbance classification method by using a Recurrent Neural Network (RNN). The trained RNN model can classify the disturbance type in real time, and the corresponding type can be subsequently used to select suitable control parameters. This approach can realize the universality of the beam stabilization pointing system under various disturbances. Experiments on beam pointing control under several typical external disturbances are carried out to verify the effectiveness of the proposed control system.

Generalized calculation model of different types of optical multi-pass cells based on refraction and reflection law

In this work, a generalized and accurate calculation model of the multi-pass cell is proposed. This novel calculation model without any errors can describe the ray’s propagation、reflection and refraction in the multi-pass cell, which will be suitable for kinds of multi-pass cells, such as the plane mirror multi-pass cell, ring multi-pass cell and the traditional multi-pass cell consisted by different optical elements including spherical mirror, lens or cylindrical mirror. By the numerical calculation, it is found that the non-paraxial ray can create a set of intricate spot patterns with respect to a standard Herriot circle spot pattern from the traditional ABCD matrix. The detailed numerical calculations of the different multi-pass cells are carried out, thus the improvement of the utilization efficiency of the mirror surface and production of a longer total optical path length by non-paraxial ray in a compact volume can be verified.

Mathematical Quad Cross-Stub Stepped Impedance Resonator Application

This simulation study involved the use of a QC-SSIR (quad cross-tub stepped impedance resonator) to examine the behavior and performance of the MW-BPF (multi-wideband band pass filter). To evaluate the performance of the proposed model, it was compared to the case of the conventional resonator. In the results, MW-BPF was found o exhibit a superior performance in terms of the ease of fabrication, good transmission coefficients, and a wider fractional bandwidth. For the filter structure analysis, this study incorporated the ABCD matrix, with the design of the MW-BPF also based on the FR4 microstrip. The experimental conditions and parameters were set in such a way that for the substrate, tan = 0.0265, thickness = 1.6 mm, and r = 4.4, and thickness h = 1.6 mm. at 2.58GHz, 1.71 GHz, and 0.81 GHz, the proposed QC-SSIR-based MW-BPF achieved transmission coefficients/fractional bandwidths of 1.93dB/13.9%, 1.49dB/18.7%, and 0.60dB/49.3%, respectively. Relative to the filter size reduction, the FQC-SSIR (folded quad cross-tub impedance resonator) was incorporated. The resultant observations indicated a possibility of BPF size reduction up to 46%. Also, the proposed framework was found to yield (at 2.62, 1.80, and 0.82 GHz) 1.76dB/12.5%, 1.21dB/17.7%, and 0.57dB/49.6% transmission coefficient/fractional bandwidths, respectively. It is also worth indicating that the filter employed LTE2600, WCDMA1800, and GSM800.

Laser cavity with a Van der Pol dynamics

In this article, a beam within a ring phase conjugated laser is described by means of a Van der Pol bidimensional dynamic map using an ABCD matrix approach. Explicit expressions for the intracavity chaos-generating matrix elements were obtained; furthermore, computer calculations for different values of Van der Pol map’s parameters were made. The rich dynamic behavior displays periodicity when the parameter ¹ (which determines the non-inearity term) takes values around zero. These results were observed in phase diagrams and in diagrams of the optical thickness of the intracavity element.

Analysis of a model of a natural gas pipeline—a transfer function approach

Abstract A framework for natural gas pipelines is developed in a context similar to the theory of electric transmission lines. The system of semi-linear partial differential equations describing the time-dependent flow of natural gas is linearized around the steady-state flow. Additional approximations lead to a constant coefficient linear system that is equivalent to an electrical circuit that is analytically solvable and admits an ABCD matrix representation of input and output. The sinusoidal steady-state operation of natural gas pipelines is analysed including the distortion of waves. It is shown that the timing of the propagation of phases and other events is accurately represented in the approximation. The quantitative accuracy for flux and gas density of the approximation depending on different operating scenarios and depending on the frequency of the disturbances is documented.

Misalignment considerations in laser diode to single-mode graded index fiber coupling via upside down tapered parabolic microlens on the circular core fiber

We report the theoretical analysis of the coupling optics involving laser diode to single-mode circular core graded index fiber excitation via upside down tapered parabolic microlens on the fiber in presence of possible transverse and angular misalignments. We use the relevant ABCD matrix appropriate for such type of coupling device in order to prescribe analytical expressions of coupling efficiencies in presence of the said misalignments. The prescribed expressions are employed for estimation of the concerned coupling optics. The execution of our formalism involves little computation and as such the present formalism will prove user friendly to the system engineers. The results obtained will be useful to the designers and packagers in the context of maximum launch optics of such coupling device.

Wigner matrix formalism for phase-modulated signals.

Laser beams can carry multi-scale properties in space and time that ultimately impact their quality. The study of their evolution along complex optical sequences is of crucial interest, especially in high-intensity laser chains. For such analysis, results obtained with standard numerical methods strongly depend on the sampling. In this paper, we develop an analytic model for a sinusoidal phase modulation inside a sequence of first-order optics elements based on the Wigner matrix formalism. A Bessel decomposition of the Wigner function gives pseudo-Wigner functions that obey the general ABCD matrix law transformation without approximations and sampling considerations. Applied to a Gaussian beam, explicit expressions are obtained for the projections of the Wigner function in the sub-spaces and give a powerful tool for analyzing the laser beam properties. The formalism is established in the spatial and temporal domains and can be used to evaluate the impact of the phase noise on the beam properties and is not limited to small modulation depths. For the sake of illustration, the model is applied to the Talbot effect with the analysis of the propagation in the spatial and phase-space domains. A comparison with full numerical calculations evidences the high accuracy of the analytic model that retrieves all the features of the diffracted beam.

On the Reciprocity Relation in General Multiport Microwave Circuits and Errata to Vector-Short-Open-Calibration Deembedding

The matrix form of the reciprocity relationship for the ABCD matrix of a general distributed microwave circuit featuring multiple ports is revisited. The correct reciprocity relationship is used to identify mistakes previously made in the derivation of key relations in the vector-short-open-calibration (VSOC) deembedding methodology, which is commonly used for removing the effect of port discontinuities in the Method-of-Moments-based electronic design automation (EDA) models of microwave circuits, digital interconnects, and other structures. Numerical results demonstrating the impact of the incorrect reciprocity relation on the accuracy of VSOC deembedding are provided.

The design and simulation of a novel ring multi-pass optical cell for detection of environmental trace gas

A ring multi-pass optical cell which consists of three spherical mirrors to form a ring cell has been proposed to achieve a long path length in compact volumes. As with the astigmatic multi-pass optical cell, the dense beam spot pattern from the ring multi-pass optical cell more evenly fills the interior surface of the spherical mirror. The disadvantage of the astigmatic cell is that it requires extremely high-precision mirrors to ensure that the pattern will close. However, with the ring multi-pass optical cell, adjusting the inclination angle of spherical mirrors can make up for errors of the mirrors, and the self-consistent condition always can be established. The desirable patterns are based on the given base length, the inclination angle of spherical mirrors, the coupling window from the optical axis and the incident angle of the beam. In consideration of the astigmatism of the off-axis mirror, the matrix optics is employed in describing and analyzing the behaviors of the ring multi-pass optical cell. The new ring multi-pass cell presented in the paper achieves a 120 m optical path length in a volume of 717 cm3. The optical simulation based on the ABCD matrix and optical software shows the validity and the usefulness of the techniques proposed.

Laser diode to single-mode graded index fiber coupling via cylindrical microlens on the fiber tip: evaluation of coupling efficiency by ABCD matrix formalism

Abstract We use ABCD matrix formalism for estimation of the coupling optics involving laser diode to cylindrical lensed circular core graded index fiber excitation. The investigation has been carried on for some typical step, parabolic, and triangular index fibers as examples of graded index fiber. It is relevant to mention in this connection that we have shown that the coupling optics is excellent along the vertical plane only while the coupling along the horizontal plane produces poor performance. Analytic expressions for coupling efficiencies are prescribed. The investigations have been made for two commonly used wavelengths, namely 1.3 and 1.5 µm. Our formulations require little computation for the necessary execution. The wavelength of 1.3 µm has been found to be more efficient in respect of coupling. Our formalism takes care of the limited aperture allowed by the cylindrical lens and thus provides the realistic prediction of coupling optics. The results will prove extremely useful to the designers and packagers working with cylindrical microlens.

Cascaded nonlinearity induced spatial domain effects in a high power femtosecond optical parametric oscillator.

We have investigated the effect of cascaded optical nonlinearity on the spatial beam properties of a femtosecond optical parametric oscillator (OPO). The OPO was operated with a tunable phase mismatch by varying the angle of the nonlinear crystal. The cascaded nonlinearity induced self-focusing and defocusing changed resonator's stability and impacted mode properties. With tuning of a phase mismatch, the calculated parabolic part of cascaded nonlinearity lens focal length changes from f ∼ 30 mm (D ∼ 33 m-1 at Δθ ∼ -0.5o) to infinity and back to f ∼ -110 mm (D ∼ -9 m-1 at Δθ ∼ 0.9o) in the LBO nonlinear crystal. Such high power nonlinear lenses in a cavity operated near its stability limit promoted the generation of axially asymmetric or pass-to-pass unstable resonator modes. It was shown that phase mismatched optical parametric oscillation changes the physical character of the resonator from linear to ring-like with two nonlinear crystals having two different focusing powers. Calculations showed that the QCN induced spatial nonlinear phase should lead to severe longitudinal chromatic aberrations for broad spectrum pulses. A numerical simulation in XYZ spatial domain and calculations using ABCD matrix approach confirmed the physical mechanisms underlying the experimental results and allowed for the interpretation of the observed effects.

Analysis of numerical diffraction calculation methods: from the perspective of phase space optics and the sampling theorem.

Diffraction calculations are widely used in applications that require numerical simulation of optical wave propagation. Different numerical diffraction calculation methods have their own transform and sampling properties. In this study, we provide a unified analysis where five popular fast diffraction calculation methods are analyzed from the perspective of phase space optics and the sampling theorem: single fast Fourier transform-based Fresnel transform, Fresnel transfer function approach, Fresnel impulse response approach, angular spectrum method, and Rayleigh-Sommerfeld convolution. The evolutions of an input signal's space-bandwidth product (SBP) during wave propagation are illustrated with the help of a phase space diagram (PSD) and an ABCD matrix. It is demonstrated that all of the above methods cannot make full use of the SBP of the input signal after diffraction; and some transform properties have been ignored. Each method has its own restrictions and applicable range. The reason why different methods have different applicable ranges is explained with physical models. After comprehensively studying and comparing the effect on the SBP and sampling properties of these methods, suggestions are given for choosing the proper method for different applications and overcoming the restrictions of corresponding methods. The PSD and ABCD matrix are used to illustrate the properties of these methods intuitively. Numerical results are presented to verify the analysis, and potential ways to develop new diffraction calculation methods are also discussed.

Propagation properties of partially coherent modified Bessel-Gauss beams through the gradient-index medium.

Based on the ABCD matrix method and Collins diffraction integral formula, the general analytical expression for the partially coherent modified Bessel-Gauss beam propagating in a gradient-index medium is derived. The propagation trajectory, intensity, and phase distribution of such a beam are numerically investigated. The effects of the topological charge, the coherence parameter, and the coefficient of the gradient refractive index on propagation properties are considered. Results show that the propagation trajectory of such beam focuses and diverges periodically, which is different from free-space propagation. The period of intensity distribution is consistent with that of phase distribution under different cases. As propagation distance increases, the dark core always exists and the phase singularities remain stable and do not split. The dark core can be modulated by topological charge and coherence parameter, and the periodical distance can be modulated by the coefficient of the gradient refractive index. These results will help to explore such beams and find applications in optical communication and optical trapping.

Propagation of hollow sinh-Gaussian beams in strongly nonlocal nonlinear media

Based on the Snyder-Michell model and using the ABCD matrix description of a strongly nonlocal nonlinear media (SNNM), the propagation properties of hollow sinh-Gaussian beams (HsGBs) through SNNM are investigated. The propagation formula, the second-order intensity moment beam width and the critical power for an on-waist incident HsG beam in SNNM are derived. The evolution of the intensity distribution pattern and the beam width of the HsGB are illustrated, graphically and analyzed with numerical examples. It is found that the HsGB pattern evolves periodically during propagation in SNNM, and its evolution behavior is strongly affected by the initial beam power and beam parameters. For a critical input power, the beam width keeps invariant during propagation, which can be regarded as a HsGB-soliton.