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What Is Light?
Published in Travis S. Taylor, Introduction to Laser Science and Engineering, 2019
Pierre de Fermat (1607–1665), a French lawyer and mathematician, shown in Figure 1.18, is important to the development of humanity’s understanding of light for two main reasons. First, he made major contributions to mathematics, number theory, geometry, probability, and components of what would become calculus. His work in mathematics helped lay the groundwork for the new mathematical era that was about to begin. Second, Fermat expanded on Hero’s concept that light traveled through the shortest path, but instead explained that light would travel the path that takes the least amount of time to do so. This became known as Fermat’s principle. This was the first time it was proposed that light speed is finite and variable based on the medium it is traveling through. Fermat’s principle paved the way to truly formulate refraction and Snell’s law.
Reflection and Refraction
Published in Myeongkyu Lee, Optics for Materials Scientists, 2019
The refraction of light is quantitatively described by Snell’s law. This law represents the relationship between the angles of incidence and refraction, when a light wave passes through a boundary between two isotropic media with different refractive indices. Snell’s law can be derived from Fermat’s principle or the application of boundary conditions for electromagnetic waves. The detailed behaviors of reflection and refraction in a given situation are well explained by the Fresnel equations, which describe what fraction of the incident light is reflected and what fraction is refracted (i.e., transmitted) at a planar interface separating two optical media. They also describe the phase shift of the reflected light. It will be shown that all these quantities depend not only on the change in refractive index and the angle of incidence, but also the polarization state of the incident light. This chapter treats the general features associated with the propagation, reflection, and refraction of light in isotropic media. These media are assumed to be linear, homogeneous, and nonmagnetic. As discussed in Chapter 1, the refractive index of a substance varies with the wavelength of incident electromagnetic radiation. All materials have a refractive index very close to 1 at X-ray wavelengths. Therefore, no refraction occurs in this X-ray range. Here, we are concerned with the visible range, in which most transparent materials exhibit a nearly constant refractive index higher than unity.
Inhomogeneous Metamaterials: Super Quasicrystals
Published in Tie Jun Cui, Wen Xuan Tang, Xin Mi Yang, Zhong Lei Mei, Wei Xiang Jiang, Metamaterials, 2017
Tie Jun Cui, Wen Xuan Tang, Xin Mi Yang, Zhong Lei Mei, Wei Xiang Jiang
As we have known, Fermat’s principle is a very important rule in optics, which was formulated in 1662 by Pierre de Fermat, and was destined to shape geometrical optics [12]. Fermat’s principle describes the shortest optical path: light rays passing between two spatial points chose the optically shortest path. The optical path length s is related to the refractive index n and is defined as s=∫ndl. Fermat’s principle has profoundly influenced modern optics. The principle governs the path between two spatial points if it is known that light travels from one point to another. If the refractive index varies in space, for example, in inhomogeneous materials, the shortest optical path is no longer a straight line but a curved one. This bending of waves may generate many optical illusions.
Double effects between reflection and edge-diffraction in NURBS-UTD method
Published in Electromagnetics, 2018
Nan Wang, Fangfang Shi, Guiqi Chen, Yu Zhang, Changhong Liang
According to the Fermat’s principle, rays travel along the shortest path for convex structures. Therefore the tracing of the ray path becomes the optimization problem of the objective function with three variables, . The conjugate gradient method (CGM) which may not be the best but the most generally used method is chosen here to find the minimum of , the corresponding derivatives of used in CGM can be achieved by