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Concept# Electromagnetic wave equation

Summary

The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous form of the equation, written in terms of either the electric field E or the magnetic field B, takes the form:
\begin{align}
\left(v_{\mathrm{ph}}^2\nabla^2 - \frac{\partial^2}{\partial t^2} \right) \mathbf{E} &= \mathbf{0} \
\left(v_{\mathrm{ph}}^2\nabla^2 - \frac{\partial^2}{\partial t^2} \right) \mathbf{B} &= \mathbf{0}
\end{align}
where
v_{\mathrm{ph}} = \frac{1}{\sqrt {\mu\varepsilon}}
is the speed of light (i.e. phase velocity) in a medium with permeability μ, and permittivity ε, and ∇2 is the Laplace operator. In a vacuum, vph = c0 = 299,792,458m/s, a fundamental physical constant. The electromagnetic wave equation derives from Maxwell's equations. In mo

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Karim Achouri, Andrei Kiselev, Olivier Martin

In this paper, we aim at unveiling the underlying physical mechanism for transversal optical forces, appearing due to the simultaneous illumination of a spherical object with two plane waves possessing different polarizations. The appearance of such a transversal force is quite counterintuitive since it seems to contradict the law of momentum conservation. We consider the cases of perfect electric conductor (PEC) and silver spheres illuminated by two orthogonally polarized plane waves propagating obliquely with respect to each other. Interestingly, the Poynting vector in these cases acquires a nonzero component transverse to the plane of propagation. Since the momentum transfer is related to the energy transfer, or equivalently, to non-negligible Poynting vector pointed in a particular direction, an arbitrary object placed in such external field is expected to experience a transversal force. To cast light upon this peculiar effect, we use a surface integral equation method and, along with the Maxwell stress tensor formalism, find the optical force acting on various spheres. We observe this effect for PEC spheres of different sizes and find that they are indeed subject to such transversal force. We find an explanation for this phenomenon via interference effects between selected multipoles excited in the structure. With recently developed methods, we expand the optical force into contributing pairs of selected multipoles and show that, depending on the phase between each multipole pair, the sign and direction of the force can be controlled. We also compare the results for silver and PEC spheres and find that the transversal force magnitude in silver has higher values for more limited range of sphere radii, as compared to PEC.

2021Collective spin excitations can propagate in magnetically ordered materials in the form of waves. These so-called spin waves (SWs) or magnons are promising for low-power beyond-CMOS information processing, which does not rely anymore on the lossy movement of electric charges. SWs in the few GHz frequency regime possess nanoscale wavelengths about five orders of magnitude smaller than electromagnetic waves of the same frequency. This property makes SWs ideally suited for application in microwave technology, essential for on-chip processing of wireless telecommunication signals. In this thesis, three crucial challenges relevant for the technological application of SWs are addressed:
First, to functionalize SWs and exploit their small wavelengths, it is necessary to control them at the nanoscale. Here, periodically nanostructured materials, denoted magnonic crystals, are promising, as they allow to tailor the band structure of SWs. We report on SWs propagating in a prototypical one-dimensional magnonic crystal consisting of dipolarly coupled magnetic nanostripes. The remanent magnetization of individual stripes was designed to be bistable along the long axis. By magnetizing an individual stripe in opposite direction to the others, we created a magnetic defect. We measured by means of all-electrical spin wave spectroscopy and Brillouin light scattering microscopy phase and amplitude of SWs trespassing the defect. We found that SW phases and amplitudes were modified at the nanoscale, and phase shifts could be tuned by an applied bias magnetic field. Using micromagnetic simulations, we identified specific bias fields for which phase shifts of Pi are achieved without suppressing SW amplitudes. This result is highly relevant for the implementation of logic gates based on interference of phase-controlled SWs. We further measured propagation of short-waved SWs in an antiferromagnetically ordered one-dimensional magnonic crystal, where every second stripe was magnetized in opposite direction. We found a band gap closing at the Brillouin zone boundary when no magnetic bias field was applied. Our observations are promising for reprogrammable microwave filters capable of adjusting stop- and passband.
Second, we address how long-waved electromagnetic waves can be coupled efficiently to nanoscale SWs. We demonstrate by space- and time-resolved scanning X-ray transmission measurements, that excited nanogratings allow to transfer their reciprocal lattice vector and multiple of it to an underlying magnetic thin film, in which nanoscale propagating SWs are launched. Additionally, we discovered a second method for short-waved SW generation based on magnetic microwave guides. This approach is easy to fabricate and relies on the adaption of the SW wavelength to a changing effective magnetic field. Efficient coupling of electromagnetic waves to nanoscale SWs promises an unprecedented miniaturization of microwave components.
Third, we found that the magnetization direction of bistable nanomagnets can be switched by propagating SWs in an underlying magnetic thin film when a threshold amplitude is reached. This discovery is promising for the realization of a non-volatile magnonic memory, which stores SW amplitudes. A possible application are SW logic gates, which encode the outcome of a logic operation in the output SW amplitude. Magnonic memory would allow for storing these amplitudes directly, without requiring lossy conversion into the electrical domain.

Jilei Chen, Dirk Grundler, Haiming Yu

Electromagnetic metasurfaces modulate a material's response to electromagnetic waves by specifically arranged elements with dimensions below the wavelength. They have opened new fields of research, including flat optics and nanophotonics on a chip. Ferromagnetic metasurfaces could become the building blocks for manipulation of both microwaves and spin waves (magnons). So far, the functionality of magnonic devices has been limited by high intrinsic damping of the materials employed, suppressing long‐distance spin‐wave propagation. Here ferromagnetic metasurfaces are reported, which are created from periodic arrays of either 15 nm thick Co20Fe60B20, Ni80Fe20 or Co nanodisks on ferrimagnetic yttrium iron garnet (YIG) hosting topologically protected vortex states. This device, a reconfigurable spectral filter, operates in the microwave regime near 0.9 GHz and manipulates long‐distance spin wave transmission in thin YIG. An efficiency of 98.5% is demonstrated, with the metasurface covering only 15% of the microwave antenna. This first dem-onstration of a ferromagnetic metasurface opens unprecedented possibilities for on-chip control of microwaves in low-damping ferrimagnetic insulators.

2022