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Concept# Wavelength

Summary

In physics and mathematics, wavelength or spatial period of a wave or periodic function is the distance over which the wave's shape repeats. In other words, it is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, troughs, or zero crossings. Wavelength is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. The inverse of the wavelength is called the spatial frequency. Wavelength is commonly designated by the Greek letter lambda (λ).
The term "wavelength" is also sometimes applied to modulated waves, and to the sinusoidal envelopes of modulated waves or waves formed by interference of several sinusoids.
Assuming a sinusoidal wave moving at a fixed wave speed, wavelength is inversely proportional to the frequency of the wave: waves with higher frequencies have shorter wavelengths, and lower frequencies have longer wavelengths.
Wavelength depends on the medium (for example,

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The magnetic force microscope (MFM) is an established experimental tool for imaging stray fields with high spatial resolution and sensitivity. The MFM contrast can however contain contributions from the sample topography, variations in the surface Kelvin potential and magnetic contributions arising from grain-to-grain variations of the areal density of the magnetic moment, apart from the contrast generated by the micromagnetic pattern of the sample. Differential imaging techniques can be used to disentangle these contrast contributions. The calibration of the response of the MFM tip on different spatial wavelengths of the field allows a quantitative determination of the magnetic vector field in the plane parallel to the sample surface scanned by the tip. Generally, the tip becomes less sensitive for smaller spatial wavelengths. Obtaining a high spatial resolution thus requires a high measurement sensitivity that can be obtained by MFM operation in vacuum and by using & nbsp;high-quality factor cantilevers. As a result, field sensitivities better than 80 mu T/root Hz can be obtained, even with low magnetic moment tips.

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It is known that the pitchfork bifurcation of Kelvin-Helmholtz instability occurring at minimum gradient Richardson number Ri(m) similar or equal to 1/4 in viscous stratified shear flows can be subcritical or supercritical depending on the value of the Prandtl number, Pr. Here, we study stratified shear flow restricted to two dimensions at finite Reynolds number, continuously forced to have a constant background density gradient and a hyperbolic tangent shear profile, corresponding to the 'Drazin model' base flow. Bifurcation diagrams are produced for fluids with Pr = 0.7 (typical for air), 3 and 7 (typical for water). For Pr = 3 and 7, steady billow-like solutions are found to exist for strongly stable stratification of Ri(m) beyond 1/2. Interestingly, these solutions are not a direct product of a Kelvin-Helmholtz instability, having half the wavelength of the linear instability, and arising through a superharmonic bifurcation. These short-wavelength states can be tracked down to at least Pr approximate to 2.3 and act as instigators of complex dynamics, even in strongly stratified flows. Direct numerical simulations of forced and unforced two-dimensional flows are performed, which support the results of the bifurcation analyses. Perturbations are observed to grow approximately exponentially from random initial conditions where no modal instability is predicted by a linear stability analysis.

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.
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