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39. Topology and geometry of BP:Considering a closed surface enclosing the BP at its center the direction of unit magnetization vector field (m0in Equation 4) covers an integer number of times a whole solid angle. Expressed in spherical coordinates [23]this vector field is parameterized by a polar component expressible aspθ and an azimuthal component written asqϕ + γ. Sincem0is single valued qandpare integer number and the topological nature of the BP is then set by the quantityQ=pq called the topological charge.[11]γ determines the geometry: It is an azimuthal tilt with regard to the radial direction causing the BP to twist aroundz as discussed in the text and shown in Figure 3. Such as behavior and how the system goes from oneq pstate to another is illustrated for example in [40] for vortices and skyrmions using micromagnetic simulations. The discussions and numerical calculations presented in this work assume a specific γ. The actual value of γ for a given material depends on terms in the magneto‐static energy such as the dipolar energy [23]that are not in the focus of this work.
41. Energy scale: In the region around the BP the dominant energy contribution stems from exchange energy∫d3rJ2a(∇S)$\int d^3\mathbf {r} \frac{J}{2a} (\nabla \mathbf {S})$ whereJis the exchange interaction ais the lattice constant and ℏSis the molecular spin which is proportional to the magnetizationMappearing in (2). The energy scale is typically an order of magnitude higher than that due to dipolar interactions which govern magneto‐static excitations (spin waves) in uniform magnetic media.
47. Kozawa Y. · Photonics Review · 2025 · 2025
65. M.Luttmann M.Fanciulli P.Carrara M.Sacchi T.Ruchon arXiv preprint2025 arXiv:2506.15232.
75. Directly at the BP where the magnetization vanishes (cf. Figure 1) atomistic models are needed. The theory presented here does not include this particular point. A reasonable length beyond which our model is expected to hold is set by the exchange lengthJ/(Ms2a)$\sqrt {J/(M_s^2 a)}$ the value of this length is ≈6 nm for Permalloy. Since our focus is not on describing BPs as such but on the behavior of the EM fields scattering from BPs and the fields do not scatter whenMs= 0 this issue is less critical for this study.
77. Our formulation based on Equations (1 2) captures the physics of BPs. It does not include the magnetic material well beyond BP. This part enters our theory/simulations through a finite but constant relative permeability. Field matching is performed between the BP region and the surrounding rendering the EM fields well‐defined in the whole space. Note typically the response of the exchange‐dominated BP is at a different (higher) frequency than the magneto‐statically‐coupled region with a response in the GHz. Mathematically this means that within our formulation and boundary conditions we cannot realize BP as localized solutions in a vector field which is spatially homogeneous meaning collinear at infinity.
88. V.Jandieri K.Yasumoto Ch. 21 in book: “Advances in Mathematical Methods for Electromagnetics ” Institution of Engineering and Technology (IET) 2020.