Derivative

Summary

In mathematics, the derivative shows the sensitivity of change of a function's output with respect to the input. Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable. Derivatives can be generalized to functions of several real variables. In this generalization, the derivative is reinterpreted as a linear transformation whose graph is (after an appropriate tra

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In this thesis, we study two distinct problems. The first problem consists of studying the linear system of partial differential equations which consists of taking a k-form, and applying the exterior derivative 'd' to it and add the wedge product with a 1-form 'a'. The study of this differential operator is linked to the study of the multiplication by a two form, that is the system of linear equations where we take a k-form and apply the exterior wedge product by 'da', the exterior derivative of 'a'. We establish links between the partial differential equation and the linear system. The second problem is a generalization of the symmetric gradient and the curl equation. The equation of a symmetric gradient consists of taking a vector field, apply the gradient and then add the transpose of the gradient, whereas in the curl equation we subtract the transpose of the gradient. Both can be seen as an equation of the form A * grad u + (grad u)t * A, where A is a symmetric matrix for the case of the symmetric gradient and skew symmetric for the curl equation. We generalize to the case where A verifies no symmetry assumption and more significantly add a Dirichlet condition on the boundary.

EPFL2018

Binary dielectric metasurfaces are arrays of sub-wavelength structures that act as a thin layer of artificial material. They are generally lossless and relatively simple to fabricate since only a single structuring step is required. By carefully designing the metasurface, the phase, amplitude and the polarization of the incident light can be controlled at will. In practice, fabrication constraints and the limited choice of materials reduce what can be done with metasurfaces. But a wide range of functionalities can be implemented with the proper design techniques and knowledge. This thesis contributes to both. The modes are key to understand the phenomena occurring inside a metasurface. To facilitate the analysis of the modes, the Poynting operation, which is related to the Poynting vector, is introduced. We describe how this operation can be used to reformulate the boundary condition in order to estimate the reflection and transmission coefficients with reduced knowledge on the modes involved, and to orthonormalized the modes. The Fourier modal method, which is the method used for the rigorous simulation of metasurfaces in this work, is improved in order to facilitate the access to valuable information that can be used to better understand the phenomena occurring inside a metasurface, and to speed up the design and optimization process. This method computes the eigen-modes present in the metasurface. To better analyze them, the eigen-modes are orthonormalized using the Poynting operation. We show that most of the modes can be filter out in order to simulate a metasurface with different thicknesses in milliseconds. From the analysis of the modes propagating in metasurfaces, two types of metasurface are identified: single-mode metasurfaces and multi-mode metasurfaces. For single-mode metasurfaces, we provide design techniques that translates the desired response into internal properties of the metasurfaces. For multi-mode metasurfaces, the concept of self-coupling mode is developed. We show that, based on this concept, the angular and spectral response of a metasurface can be interpolated safely with a few simulations even if high-Q resonances are present. For both types of metasurfaces, examples of design are provided. Gradient-based optimization methods allow to obtain the optimal metasurface in a few iterations, but the derivative of the merit function is necessary. The adjoint method computes the functional derivative of the merit function with respect to the permettivity and permeability. We provide the equations of the adjoint method and apply them to diffractive optical elements such that they can be used in conjunction with the Fourier modal method. This thesis contributes to design challenges for complex electromagnetic problems,and it does not only provides concepts, but also the tools to put them into operation.

EPFL2020

Complex magnetoelectric effect in multiferroic composites: the case of PFN- PT/(Co,Ni)Fe2O4

Hadi Papi

We report a systematic study of the magnetoelectric (ME) voltage coefficient as a complex quantity in the particulate composite of ferroelectric solid solution 0.94Pb(Fe1/2N1/2)O-3-0.06PbTiO(3)(PFN-PT) with CoFe2O4(CFO) and NiFe2O4 (NFO) ferrites. The results show that the real part of the ME voltage coefficient (alpha') is highly influenced by the magnetostrictive phase through lambda(H). NFO produces larger alpha' at a lower magnetic field, which originates from the softer magnetic properties. In addition, alpha' was found to be positive for NFO composite, while CFO composite shows a negative ME voltage coefficient at high magnetic fields. We argue that the field dependence of alpha' can be interpreted using the dynamic piezomagnetic coefficient, q(ac) = partial derivative lambda(ac)/partial derivative H. The imaginary part of the ME voltage coefficient (alpha '') was also determined for all composites. Both the alpha' and alpha '' show a peak at the same magnetic field, which is attributed to the maximum dynamic piezomagnetism when the magnetic domains are collectively rotated by the dc bias magnetic field. Our results show that the ME voltage coefficient in the composites of PFN-PT/(Co,Ni)Fe2O4 is highly influenced by the content, magnetic softness and field dependence behaviour of magnetostriction of the piezomagnetic phase.

2019

EPFL2020