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Concept# Geometrical optics

Summary

Geometrical optics, or ray optics, is a model of optics that describes light propagation in terms of rays. The ray in geometrical optics is an abstraction useful for approximating the paths along which light propagates under certain circumstances.
The simplifying assumptions of geometrical optics include that light rays:

- propagate in straight-line paths as they travel in a homogeneous medium
- bend, and in particular circumstances may split in two, at the interface between two dissimilar media
- follow curved paths in a medium in which the refractive index changes
- may be absorbed or reflected.

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Related lectures (72)

Removing geometrical details from a complex domain is a classical operation in computer aided design for simulation and manufacturing. This procedure simplifies the meshing process, and it enables faster simulations with less memory requirements. However, depending on the partial differential equation that one wants to solve in the geometrical model of interest, removing some important geometrical features may greatly impact the solution accuracy. For instance, in solid mechanics simulations, such features can be holes or fillets near stress concentration regions. Unfortunately, the effect of geometrical simplification on the accuracy of the problem solution is often neglected, because its analysis is a time-consuming task that is often performed manually, based on the expertise of engineers. It is therefore important to have a better understanding of the effect of geometrical model simplification, also called defeaturing, to improve our control on the simulation accuracy along the design and analysis phase.In this thesis, we formalize the process of defeaturing, and we analyze its impact on the accuracy of solutions of some partial differential problems. To achieve this goal, we first precisely define the error between the problem solution defined in the exact geometry, and the one defined in the simplified geometry. Then, we introduce an a posteriori estimator of the energy norm of this error. This allows us to reliably and efficiently control the error coming from the addition or the removal of geometrical features. We subsequently consider a finite element approximation of the defeatured problem, and the induced numerical error is integrated to the proposed defeaturing error estimator. In particular, we address the special case of isogeometric analysis based on (truncated) hierarchical B-splines, in possibly trimmed and multipatch geometries. In this framework, we derive a reliable a posteriori estimator of the overall error, i.e., of the error between the exact solution defined in the exact geometry, and the numerical solution defined in the defeatured geometry.We then propose a two-fold adaptive strategy for analysis-aware defeaturing, which starts by considering a coarse mesh on a fully-defeatured computational domain. On the one hand, the algorithm performs classical finite element mesh refinements in a (partially) defeatured geometry. On the other hand, the strategy also allows for geometrical refinement. That is, at each iteration, the algorithm is able to choose which missing geometrical features should be added to the simplified geometrical model, in order to obtain a more accurate solution.Throughout the thesis, we validate the presented theory, the properties of the aforementioned estimators and the proposed adaptive strategies, thanks to an extensive set of numerical experiments.

Romain Christophe Rémy Fleury, Nadège Sihame Kaïna, Bakhtiyar Orazbayev

Rapid progress in all types of communication systems imposes each time more strict requirements on the communication devices, requiring having the overall device's size as small as possible, but also increasing the demands on the robustness of the transmission channels to the disorders with an aim of achieving most efficient signal transmission. The existing schemes for transferring signals, based on the conventional materials, are tied to the operation wavelength of the propagating signal and therefore fundamentally limited by it. Moreover, in such schemes the absence of any sort of protection renders them vulnerable to possible defects in the channel, forcing the use of additional elements (for instance filters, amplifiers, etc.) and increasing the overall size and cost of the devices. However, recent developments in the field of artificial media, known as metamaterials [1], showed a great potential for achieving more control over the wave propagation and providing viable solutions for an efficient signal transmission. Unfortunately, since these artificial media consist of resonant inclusions - meta-atoms, they are inherently susceptible to geometrical imperfections and disorder-induced backscattering, which significantly reduces their performance and limits their real applications. In recent years, it has been demonstrated that the topological concepts, that originally have been derived in solidstate physics, can be also applied to photonic crystals [2] and locally-resonant crystalline metamaterials [3], providing a certain degree of protection against disorders. However, such photonic topological designs are based on preserving the time-reversal symmetry and, therefore, rely on the lattice structure of the media and frequency dispersion of the crystal. Thus, such time-reversal invariant topological designs are vulnerable to any disruption of the lattice symmetry that can couple time-reversed modes, which is the case for the most of disorders (in the location or in resonance frequency of the resonant inclusions). In this work, we experimentally demonstrate in the microwave regime that by exploiting a chiral metamaterial [4], [5] a robust-to-disorder subwavelength waveguiding can be achieved. Moreover, we quantitatively demonstrate the superiority of the proposed waveguiding scheme in terms of robustness to both spatial or frequency disorders to the previously proposed subwavelength waveguiding schemes: frequency defect lines, symmetry-based topological edge modes and valley interface states. To this end, we compare their performance in the presence of disorders by performing ensemble averages on disorder realizations along the path of the guided wave. The obtained results clearly demonstrate the superior robustness of the chiral metamaterial waveguide against both spatial and frequency designs while other analyzed designs are robust for one type of disorder [6].

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Recent advances in the field of metamaterials have shown that waves can be efficiently manipulated at the subwavelength scale through the interactions with an ensemble of resonant inclusions, opening new horizons in overcoming the size limits of devices which are often tied to the wavelength of operation [1]. Such size limit is crucial for many applications where the overall dimensions are required to be as small as possible, for instance cost-eﬃcient devices for satellite communications. Unfortunately, the resonant inclusions of these artificial media result in a large sensitivity of the propagation to geometrical imperfections and disorder-induced backscattering, reducing their performance. More recently, it has been demonstrated that the topological concepts which originated in solid-state physics can be transferred to not only to photonic crystals [3,4], which still scale with the operating wavelength, but also to locally-resonant crystalline metamaterials, which can have a deeply subwavelength structure [5]. However, since the topological properties in such time-reversal invariant designs heavily rely on the lattice structure of the media and frequency dispersion of the metamaterial, they are inevitably sensitive to any disruption of the lattice symmetries that can couple time-reversed modes. Moreover, since most of disorders (in the location or in resonance frequency of the resonant inclusions) will most likely break the lattice symmetry and disrupt the wave propagation, these time-reversal invariant topological designs are also in principle sensitive to defects. In this talk, we will show that a chiral metamaterial [6,7] can be exploited to create a robust-to-disorder subwavelength waveguide and we will demonstrate this possibility experimentally in the microwave regime. Moreover, we will quantitatively demonstrate the superior robustness of the proposed design to both spatial or frequency disorders by performing ensemble averages on disorder realizations along the path of the guided wave, and comparing them with previously proposed subwavelength waveguide designs: frequency defect lines, symmetry-based topological edge modes and valley interface states. [1] J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 2008), 2nd ed. [2] N. Kaina, F. Lemoult, M. Fink, and G. Lerosey, Negative Refractive Index and Acoustic Superlens from Multiple Scattering in Single Negative Metamaterials, Nature, 525, 77 (2015). [3] S. Raghu and F. D. M. Haldane, Analogs of Quantum-Hall-Effect Edge States in Photonic Crystals, Phys. Rev. A - At. Mol. Opt. Phys., 78, 1 (2008). [4] Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacić, Observation of Unidirectional Backscattering-Immune Topological Electromagnetic States., Nature, 461, 772 (2009). [5] S. Yves, R. Fleury, T. Berthelot, M. Fink, F. Lemoult, and G. Lerosey, Crystalline Metamaterials for Topological Properties at Subwavelength Scales, Nat. Commun., 8, 16023 (2017). [6] M. Goryachev and M. E. Tobar, Reconfigurable Microwave Photonic Topological Insulator, Phys. Rev. Appl., 6, 1 (2016). [7] J. E. Vázquez-Lozano and A. Martínez, Optical Chirality in Dispersive and Lossy Media, Phys. Rev. Lett., 121, 43901 (2018).

2019