External Magnetic Field Research Articles

Dichotomous dynamics of magnetic monopole fluids

A recent advance in the study of emergent magnetic monopoles was the discovery that monopole motion is restricted to dynamical fractal trajectories [J. N. Hallén et al., Science 378, 1218 (2022)], thus explaining the characteristics of magnetic monopole noise spectra [R. Dusad et al., Nature 571, 234 (2019); A. M. Samarakoon et al., Proc. Natl. Acad. Sci. U.S.A. 119, e2117453119 (2022)]. Here, we apply this novel theory to explore the dynamics of field-driven monopole currents, finding them composed of two quite distinct transport processes: initially swift fractal rearrangements of local monopole configurations followed by conventional monopole diffusion. This theory also predicts a characteristic frequency dependence of the dissipative loss angle for AC field-driven currents. To explore these novel perspectives on monopole transport, we introduce simultaneous monopole current control and measurement techniques using SQUID-based monopole current sensors. For the canonical material Dy2Ti2O7, we measure [Formula: see text], the time dependence of magnetic flux threading the sample when a net monopole current [Formula: see text] is generated by applying an external magnetic field [Formula: see text] These experiments find a sharp dichotomy of monopole currents, separated by their distinct relaxation time constants before and after t ~[Formula: see text] from monopole current initiation. Application of sinusoidal magnetic fields [Formula: see text] generates oscillating monopole currents whose loss angle [Formula: see text] exhibits a characteristic transition at frequency [Formula: see text] over the same temperature range. Finally, the magnetic noise power is also dichotomic, diminishing sharply after t ~[Formula: see text]. This complex phenomenology represents an unprecedented form of dynamical heterogeneity generated by the interplay of fractionalization and local spin configurational symmetry.

Cu(II)-Ln(III) (Ln = Gd, Tb and Dy) complexes of an unsymmetrical N2O3 donor ligand: field induced SMM behaviour of Cu(II)-Tb(III) complexes.

Three new hetero-metallic CuII-LnIII complexes [(CuL)Gd(NO3)3(CH3OH)]n (1), [(CuL)Tb(NO3)3(H2O)]·[CuL] (2) and [(CuL)Dy(NO3)3(H2O)]·[CuL] (3) have been synthesized using a mono-nuclear Cu(II) complex, [CuL], of an unsymmetrically di-condensed N2O3 donor Schiff base ligand, N-(3-methoxysalicylidene)-N-(salicylidene)-1,2-ethylenediamine (H2L). Single crystal X-ray crystallography revealed that complex 1 is a nitrate bridged 1D chain of dinuclear Cu(II)-Gd(III) units whereas in 2 and 3, the dinuclear Cu(II)-Ln(III) units are co-crystallized with a [CuL] unit. The Ln(III) centers are nine coordinated with the geometry of a spherical capped square antiprism for Gd and spherical tricapped trigonal prism for Tb and Dy. The geometry of the Cu(II) center is distorted octahedral for complex 1 and distorted square planar for complexes 2 and 3. Temperature-dependent molar magnetic susceptibility measurements in 1-3 revealed the presence of overall ferromagnetic coupling between the Cu(II) and Ln(III) centers. Notably, field induced single-molecule magnet behavior was witnessed in the Tb(III) derivative (2). The ab initio calculations indicated that upon application of an external magnetic field, the tunneling in the ground state of complex 2 gets reduced and thereby field-induced SMM behaviour is observed. Besides, in the case of complex 1, BS-DFT calculations were carried out to gain further insights into the magnetic exchange coupling interactions between the Cu(II) and Gd(III) centers.

Synthesis, modifications, and applications of iron-based nanoparticles.

Magnetic nanoparticles (MNPs) are appealing materials as assistant to resolve environmental pollution issues and as recyclable catalysts for the oxidative degradation of resistant contaminants. Moreover, they can significantly influence the advancement of medical applications for imaging, diagnostics, medication administration, and biosensing. On the other hand, due to unique features, excellent biocompatibility, high curie temperatures and low cytotoxicity of the Iron-based nanoparticles, they have received increasing attention in recent years. Using an external magnetic field, in which the ferrite magnetic nanoparticles (FMNPs) in the reaction mixtures can be easily removed, make them more efficient approach than the conventional method for separating the catalyst particles by centrifugation or filtration. Ferrite magnetic nanoparticles (FMNPs) provide various advantages in food processing, environmental issues, pharmaceutical industry, sample preparation, wastewater management, water purification, illness therapy, identification of disease, tissue engineering, and biosensor creation for healthcare monitoring. Modification of FMNPs with the proper functional groups and surface modification techniques play a significant role in boosting their capability. Due to flexibility of FMNPs in functionalization and synthesis, it is possible to make customized FMNPs that can be utilized in variety of applications. This review focuses on synthesis, modifications, and applications of Iron-based nanoparticles.

Development and characterization of lipid nanocapsules loaded with iron oxide nanoparticles for magnetic targeting to the blood-brain barrier.

Brain drug delivery is severely hindered by the presence of the blood-brain barrier (BBB). Its functionality relies on the interactions of the brain endothelial cells with additional cellular constituents, including pericytes, astrocytes, neurons, or microglia. To boost brain drug delivery, nanomedicines have been designed to exploit distinct delivery strategies, including magnetically driven nanocarriers as a form of external physical targeting to the BBB. Herein, a lipid-based magnetic nanocarrier prepared by a low-energy method is first described. Magnetic nanocapsules with a hydrodynamic diameter of 256.7 ± 8.5nm (polydispersity index: 0.089 ± 0.034) and a ξ-potential of -30.4 ± 0.3mV were obtained. Transmission electron microscopy-energy dispersive X-ray spectroscopy analysis revealed efficient encapsulation of iron oxide nanoparticles within the oily core of the nanocapsules. Both thermogravimetric analysis and phenanthroline-based colorimetric assay showed that the iron oxide percentage in the final formulation was 12 wt.%, in agreement with vibrating sample magnetometry analysis, as the specific saturation magnetization of the magnetic nanocapsules was 12% that of the bare iron oxide nanoparticles. Magnetic nanocapsules were non-toxic in the range of 50-300μg/mL over 72h against both the human cerebral endothelial hCMEC/D3 and Human Brain Vascular Pericytes cell lines. Interestingly, higher uptake of magnetic nanocapsules in both cell types was evidenced in the presence of an external magnetic field than in the absence of it after 24h. This increase in nanocapsules uptake was also evidenced in pericytes after only 3h. Altogether, these results highlight the potential for magnetic targeting to the BBB of our formulation.

Microscopic structure of electromagnetic whistler wave damping by kineticmechanisms in hot magnetized Vlasov plasmas

Abstract Electromagnetic transverse perturbations propagating parallel to the external magnetic field in a
warm electron plasma, specifically the warm electron whistler waves, are simulated in κ distributed
electrons. The Vlasov-Maxwell phase-space continuum simulations are applied to the stable and
unstable (i.e., isotropic and anisotropic) VDFs including those with energetic tail, i.e., κ distribution.
The variation of real frequency from both numerical solution of dispersion relation and simulations
show limited sensitivity to electron temperature in low wave-number regime as compared to high
wave number regime, however the opposite holds for the imaginary frequency or the decay rate. For
large amplitude, nonlinear cases, simulated with anisotropic κ distributions the very low anisotropy
cases find the growth rate increasing with increasing the κ while this trend reverses for higher
anisotropy. The phase-space portraits of the these cases show that after the linear damping phase
of the evolution, the particles are trapped in the wave magnetic field leading to the wave amplitudes
oscillating about a mean value which follow the theoretical analysis [1].

Biocompatible polyethersulfone membrane modified by hydrophilic polymer decorated magnetic nanoparticles gilded by facile external magnetic field

AbstractPolymer‐based membranes such as polyethersulfone are widely used for hemodialysis and bio‐related applications; however, this type of membrane still fails to satisfy optimum performance in terms of clearance, safety, and bio‐adaptability. Herein, a polymer brush of polyvinylpyrrolidone is decorated on the surface of magnetic nanoparticles by using reversible addition‐fragmentation chain transfer polymerization (RAFT), and under the guidance of an external magnetic field, the prepared magnetic nanoparticles with the polymer brush system are blended with a polyethersulfone membrane during a liquid‐liquid phase inversion process. The prepared membranes with different loads of the pure magnetic nanoparticle and polyvinylpyrrolidone@magnetic nanoparticle polymer brushes are prepared, compared with the pristine membrane, the modified membrane shows a high flux value at 349 L m−2 h−1 and 99.9% bovine serum albumin protein rejection. This membrane also demonstrated improved hydrophilicity and pore structure compared with the pristine membrane. Concerning the blood biocompatibility analysis, the modified membranes acquire prolonged time for clotting factors such as prothrombin time, activated partial thromboplastin time, and thrombin time while exhibiting less fibrinogen concertation at 0.4 g L−1. Furthermore, for the first time, an innovative method for fouling detection based on magnetic field attenuation due to protein accumulation on the membrane surface is reported; the method in this work could be a positive booster for the safe use of polymeric membranes in hemodialysis and bio‐based applications.

Comparison of Various Theoretical Measures of Aromaticity within Monosubstituted Benzene

The effects of monosubstitution on the aromaticity of benzene are assessed using a number of different quantitative schemes. The ability of the mobile π-electrons to respond to an external magnetic field is evaluated using several variants of the NICS scheme which calculate the shielding of points along the axis perpendicular to the molecule. Another class of measures is related to the drive toward the uniformity of C-C bond lengths and strengths. Several energetic quantities are devised to approximate an aromatic stabilization energy and the tendency of the molecule to maintain planarity. There is a lack of consistency in that the various measures of aromaticity lead to differing conclusions as to the effects of substituents on the aromaticity of the ring.

Efficiency of magnetic zeolitic imidazolate framework-8 (ZIF-8) modified graphene oxide for lead adsorption.

Simple and efficient removal of Pb(II) ion from aqueous solution through adsorption has accelerated the development of many new composites to improve this popular method. In this study, the composites of graphene oxide (GO), zeolitic imidazolate framework-8 (ZIF-8), and magnetic materials were synthesized via coprecipitation method utilizing a different molar ratio between FeCl2 and FeCl3 of 1:0.5, 2:1, 3:1.5, and 4:2. The ZIF-8/GO was prepared via room temperature synthesis method prior to its further modification with magnetic materials for ease of separation. It was observed that the MZIF-8/GO2 of molar ratio 2:1 showed the best performance in adsorbing Pb(II) ion. As confirmed by FESEM image, it appeared to be ZIF-8 particles that have grown all over the GO platform and overlayed with Fe3O4 granular-shaped particles. The MZIF-8/GO2 successfully achieved 99% removal of Pb(II) within 10min. The optimum values obtained for the initial concentration of Pb (II) were 100mg/L, pH of 4 to 6, and adsorbent dosage used was 10mg. The Langmuir isotherm and the pseudo-second-order kinetic model were deemed suitable to evaluate the adsorption of Pb(II) using MZIF-8/GO2. Results showed that MZIF-8GO2 achieved a maximum adsorption capacity of 625mg/g of Pb(II) adsorption. All parent materials demonstrated a good synergistic effects, while exhibiting a significant contribution in providing active sites for Pb(II) adsorption. Therefore, this ternary composite of MZIF-8/GO2 is expected to be a promising adsorbent for Pb(II) adsorption from aqueous solution with an added value of ease of post phase separation using external magnetic field.

Field-Induced Slow Magnetization Relaxation of a Tetrahedral S=2 FeIIS4-Containing Complex.

In the work described herein, the spin relaxation properties of the mononuclear tetrahedral S = 2 [Fe{(SPiPr2)2N}2] complex (1) were studied by employing static and dynamic magnetic measurements al liquid helium temperatures. In the absence of an external direct current (DC) magnetic field, 1 exhibits fast magnetization relaxation. However, in the presence of external magnetic fields of a few kOe, slow relaxation is induced as monitored by alternating current (AC) magnetic susceptibility measurements up to 10 kHz, in the temperature range 2-5 K. Analysis of the temperature dependence of the corresponding relaxation time reveals contributions by Quantum Tunnelling of Magnetization, and the Direct and Orbach processes in the magnetization relaxation mechanism of 1. The energy barrier, Ueff, of the Orbach process, as determined by this analysis, is compared with that related to the zero field splitting parameters of 1 which were previously determined by high- frequency and -field electron paramagnetic resonance and Mössbauer spectroscopies.

Two-dimensional quantum droplets in binary quadrupolar condensates

Abstract We study the stability and characteristics of two-dimensional (2D) quasi-isotropic quantum droplets (QDs) of fundamental and vortex types, formed by binary Bose-Einstein condensate with magnetic quadrupole-quadrupole interactions (MQQIs). The magnetic quadrupoles are built as pairs of dipoles and antidipoles polarized along the x-axis. The MQQIs are induced by applying an external magnetic field that varies along the x-axis. The system is modeled by the Gross-Pitaevskii equations including the MQQIs and Lee-Huang-Yang correction to the mean-field approximation. Stable 2D fundamental QDs and quasi-isotropic vortex QDs with topological charges S ≤ 4 are produced by means of the imaginary-time-integration method for configurations with the quadrupoles polarized parallel to the system’s two-dimensional plane. Effects of the norm and MQQI strength on the QDs are studied in detail. Some results, including an accurate prediction of the effective area, chemical potential, and peak density of QDs, are obtained in an analytical form by means of the Thomas-Fermi approximation. Collisions between moving QDs are studied by means of systematic simulations.

Impact of heat and mass transfer on the magnetohydrodynamic two-phase flow of couple stress fluids through a porous walled curved channel using Homotopy Analysis Method

The analysis of heat and mass transfer in the movement of immiscible couple stress fluids through a curved channel whose walls are at two different temperatures is the focus of the current work. Immiscible fluids are subjected to an external magnetic field that is taken normal to the fluid’s flow. For suction or injection in the flow, the channel walls are presumed to be porous. The buoyancy force, which is modelled using the Boussinesq approximation, and a constant pressure gradient cause the flow of the couple stress fluids in the curved channel. The highly non-linear coupled differential equations associated with the proposed work are solved with the use of the semi-analytic Homotopy Analysis Method. The semi-analytical expression of the various flow variables such as flow velocity, temperature, concentration, etc. is obtained by solving these non-linear differential equations after they have been non-dimensionalized by taking into account non-dimensional parameters. From the obtained results authors have noted how the two immiscible couple stress fluids’ flow velocity, temperature, and concentration behaviour depend on the flow-effecting parameters, including the couple stress parameter, curvature parameter, Schmidt number, and so on. Further, the authors of the proposed work have analysed the effect of various flow parameters on heat and mass transfer. Since the authors have studied the two-phase flow of couple stress fluids in a porous walled curved channel together with the effect of magnetic field, the current model has various applications in the fields of crude oil extraction, filtration, nuclear reactor, and bio-mechanics.

Superconductivity above 105 K in Nonclathrate Ternary Lanthanum Borohydride below Megabar Pressure.

Hydrides are promising candidates for achieving room-temperature superconductivity, but a formidable challenge remains in reducing the stabilization pressure below a megabar. In this study, we successfully synthesized a ternary lanthanum borohydride by introducing the nonmetallic element B into the La-H system, forming robust B-H covalent bonds that lower the pressure required to stabilize the superconducting phase. Electrical transport measurements confirm the presence of superconductivity with a critical temperature (Tc) of up to 106 K at 90 GPa, as evidenced by zero resistance and Tc shift under an external magnetic field. X-ray diffraction and transport measurements identify the superconducting compound as LaB2H8, a nonclathrate hydride, whose crystal structure remains stable at pressures as low as ∼ half megabar (59 GPa). Stabilizing superconductive stoichiometric LaB2H8 in a submegabar pressure regime marks a substantial advancement in the quest for high-Tc superconductivity in polynary hydrides, bringing us closer to the ambient pressure conditions.

Direct measurement of repulsive and attractive pair potentials using pairs of optical traps

We present a technique for measuring the interactions between pairs of colloidal particles in two optical traps. This method is particularly suitable for measuring strongly attractive potentials, an otherwise challenging task. The interaction energy is calculated from the distribution of inter-particle separations by accounting for the contribution from the optical traps with arbitrary trap profiles. The method is simple to implement and applicable to different types of pair potentials and optical trapping geometries. We apply the method to measure dipolar pair interactions in experiments with paramagnetic colloids in external magnetic fields. We obtain consistent and accurate results in all regimes, from strongly attractive to repulsive potentials. By means of computer simulations, we demonstrate that the proposed method can be successfully applied to systems with complex pair interactions characterized by multiple attractive and repulsive regimes, which are ubiquitous in soft and biological matter.

Investigation of Fiber Orientation of Fused Filament Fabricated CFRP Composites via an External Magnetic Field

Abstract For carbon fiber-reinforced plastic (CFRP) composites, controlling the interior fiber distribution and orientation during the manufacturing process is a common approach to optize the structural performance of fabricated parts. However, few studies have been conducted to investigate fiber alignment during the additive manufacturing of CFRP composites. This study proposes a magnetic field controlled (MFC) method to control the fiber orientation during the fused filament fabrication (FFF) of nickel-coated carbon fiber (NCF) reinforced polymer composites. Firstly, a theoretical analysis model is established to explore the suitable magnetic field intensity for fiber rotation. Secondly, a customized FFF system with MFC components is implemented, and a polylactic acid matrix composite containing 10 wt% NCF is printed to validate the feasibility of the proposed approach. The microstructure of the printed samples is examined to assess the effectiveness of the method. Finally, uniaxial tensile tests are performed to investigate the impact of fiber orientation adjustment on mechanical properties. The experimental results reveal that the MFC method can effectively align the interior fiber orientation of CFRP composites, leading to a significant increase in the tensile strength (approximately 8.8%) and Young's modulus (around 10.5%) of the printed samples.

Theoretical characterization and its application of electromagnetic anisotropy in high-temperature superconducting bulks under rotated postures

The electromagnetic anisotropy of high-temperature superconducting (HTS) bulks limits their levitation ability in the applied magnetic fields from the permanent magnet guideway (PMG), thus impeding the enhancement of load-carrying capacity in HTS pinning maglev systems. Developing a suitable matching scheme between bulk orientation and magnetic field direction is a valuable way to relieve this restriction. In this paper, a method for characterizing bulk anisotropy in a rotating coordinate system is proposed to explore the best bulk orientation. The method is based on the concept of equivalent resistivity tensor and its eigenvectors, and includes an extended description of two types of anisotropy: conductivity anisotropy and magnetic field angle dependence. It provides a theoretical foundation for simulating anisotropic bulks under any rotated posture. Experimental investigations on the levitation force distribution of cylindrical bulks with different c-axis orientation were conducted, through which the accuracy of the characterization method and calculated results were validated. Analysis of current distribution reveals that aligning the c-axis parallel to the external magnetic field helps achieve the best match between the bulk and the PMG. Additionally, considering that the two types of anisotropy have opposite effects on levitation force distribution trends, prioritizing conductivity anisotropy when analyzing anisotropic bulk is recommended. This research not only offers a theoretical framework for simulating the anisotropy of rotated HTS bulks but also provides guidance for matching the optimal bulk orientation in applied magnetic fields.

Dimensional and Doping Engineering of Chiral Perovskites with Enhanced Spin Selectivity for Green Emissive Spin Light-Emitting Diodes.

Chiral perovskites play a pivotal role in spintronics and optoelectronic systems attributed to their chiral-induced spin selectivity (CISS) effect. Specifically, they allow for spin-polarized charge transport in spin light-emitting diodes (LEDs), yielding circularly polarized electroluminescence at room temperature without external magnetic fields. However, chiral lead bromide-based perovskites have yet to achieve high-performance green emissive spin-LEDs, owing to limited CISS effects and charge transport. Herein, we employ dimensional regulation and Sn2+-doping to optimize chiral bromide-based perovskite architecture for green emissive spin-LEDs. The optimized (PEA)x(S/R-PRDA)2-xSn0.1Pb0.9Br4 chiral perovskite film exhibits an enhanced CISS effect, higher hole mobility, and better energy level alignment with the emissive layer. These improvements allow us to fabricate green emissive spin-LEDs with an external quantum efficiency (EQE) of 5.7% and an asymmetry factor |gCP-EL| of 1.1 × 10-3. This work highlights the importance of tailored perovskite architectures and doping strategies in advancing spintronics for optoelectronic applications.

The research on triple-coil inductive debris sensor with electromagnetic shielding

A high sensitivity inductive debris sensor is of great significance for oil condition monitoring and the fault diagnosis of mechanical equipment. However, inductive sensors are susceptible to interference from external electromagnetic fields. In this study, an inductive debris sensor with a shielding layer is studied, which can effectively reduce the influence of external magnetic fields on the sensor. Theoretical and experimental results verify the effectiveness of the shielding layer, and also show that the shielding layer impacts the magnetic field of the sensor, resulting in a decrease in the amplitude of the output signal. Detailed analysis was conducted to investigate the effect of structural parameters on the performance of inductive sensors. These parameters include the distance between the shielding layer and the coils, the thickness of the shielding layer, and the shielding layer material. Based on the theoretical and experimental results, reference suggestions are provided for the design of the shielding layer. This study ensures the performance and reliability of inductive sensors in practical applications.

Magnetic Microrobot Swarms with Polymeric Hands Catching Bacteria and Microplastics in Water.

The forefront of micro- and nanorobot research involves the development of smart swimming micromachines emulating the complexity of natural systems, such as the swarming and collective behaviors typically observed in animals and microorganisms, for efficient task execution. This study introduces magnetically controlled microrobots that possess polymeric sequestrant "hands" decorating a magnetic core. Under the influence of external magnetic fields, the functionalized magnetic beads dynamically self-assemble from individual microparticles into well-defined rotating planes of diverse dimensions, allowing modulation of their propulsion speed, and exhibiting a collective motion. These mobile microrobotic swarms can actively capture free-swimming bacteria and dispersed microplastics "on-the-fly", thereby cleaning aquatic environments. Unlike conventional methods, these microrobots can be collected from the complex media and can release the captured contaminants in a second vessel in a controllable manner, that is, using ultrasound, offering a sustainable solution for repeated use in decontamination processes. Additionally, the residual water is subjected to UV irradiation to eliminate any remaining bacteria, providing a comprehensive cleaning solution. In summary, this study shows a swarming microrobot design for water decontamination processes.

Geant4 simulation of secondary photons generated by laser–solid interaction in the presence of external magnetic field

A parametric study of the average energy and scattering angle distribution of Bremsstrahlung x-ray photons generated in over-dense fuel were investigated using the Geant4 simulation toolkit in fast ignition concept. In our simulation, over-dense fuel with the mass ρc∼(300−1000)g.cm−3, and fast ignitor with the intensity I=(1021,1023)W.cm−2 and wavelength λ=(0.35,0.53)μm were considered in the presence of an external magnetic field with the strength Bext=(0−10)kT. Further, for the produced MeV electrons in pre-plasma, two types of energy distribution functions, i.e., exponential and quasi two-temperature, were considered. The simulation results show that by applying an external magnetic field strength Bext∼(1−2)kT, a smooth increase was observed in the x-ray photon average energy and a considerable increase in scattering angle and photon counts in over-dense plasma obtained, so that the peak values mostly obtained for Bext∼1kT. Meanwhile, between the two energy distribution functions for electrons, the optimal average energy and scattering angle values were obtained by considering the quasi-two-temperature energy distribution function and for the fast igniter intensity and wavelength, I=1021W.cm−2,λ=0.35μm respectively, and the fuel density ρc∼300g.cm−3. Finally, regarding to the obtained optimal average photon energy (i.e., Ebph<0.4MeV), which could be considered as minimum photon energy loss and maximum electron penetration toward the fuel, the minimum thickness of lead and concrete shields required to reduce the flux of photons to its one-tenth value were found to be 1.5 cm and 30.7 cm, respectively.

Exact closed forms for the transmittance of electromagnetic waves in one-dimensional anisotropic periodic media

In this work, we obtain closed expressions for the transfer matrix and the transmittance of electromagnetic waves propagating in finite one-dimensional anisotropic periodic stratified media with an arbitrary number of cells. By invoking the Cayley–Hamilton theorem on the transfer matrix for the electromagnetic field in a periodic stratified media formed by N cells, we obtain a fourth-order recursive relation for the matrix coefficients that defines the so-called Tetranacci polynomials (TPs). In the symmetric case, corresponding to a unit-cell transfer matrix with a characteristic polynomial where the coefficients of the linear and cubic terms are equal, closed expressions for the solutions to the recursive relation, known as symmetric TPs, have recently been derived, allowing us to write the transfer matrix and transmittance in a closed form. We show as sufficient conditions that the 4×4 differential propagation matrix of each layer in the binary unit cell, Δ , a) has eigenvalues of the form ±p1 , ±p2 , with p1≠p2 , and b) its off-diagonal 2×2 block matrices possess the same symmetric structure in both layers. Otherwise, the recursive relations are still solvable for any 4×4 matrix and provide an algorithm to compute the Nth power of the transfer matrix without carrying out explicitly the matrix multiplication of N matrices. We obtain analytical expressions for the dispersion relation and transmittance, in closed form, for two finite periodic systems: the first one consists of two birefringent uniaxial media with their optical axis perpendicular to the z-axis, and the second consists of two isotropic media subject to an external magnetic field oriented along the z-axis and exhibiting the Faraday effect. Our formalism applies also to lossy media, magnetic anisotropy or optical activity.