Chapter 1 Rays and paraxial approximations - OED

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2 CHAPTER 1. RAYS AND PARAXIAL APPROXIMATIONS In many cases of optics, where the wavelength of the electromagnetic field is much smaller that the scale of variations in ε and µ, a more convenient (although approximate) description of the behavior of electromagnetic field can be given in terms of rays or beams. The Gaussian beam will be discussed in Chapter 2. We define rays in the following way. Definition 1 An electromagnetic ray is an orthogonal trajectory of surfaces of constant phase of a time- harmonic wave. The main characteristic of plane waves is that the surfaces of constant phase are planes orthogonal to the vector k. Hence, for plane waves the ray direction coincides with the direction of the vector k. The description in terms of rays gives rise to geometrical optics, since the transformation of rays by optical elements, such as lenses and mirrors, is entirely determined by the geometry of the configuration. 1.1.1 Reflection at an interface The reflection and refraction of plane waves at a planar interface between two homogeneous media can be solved analytically and gives rise to Snell’s laws. It can be shown that Snell’s laws also hold for rays, owing to the fact that plane waves that almost propagate in the ray direction mainly determine the behavior of the ray. Although Snell’s laws hold exactly for plane interfaces, they can also be applied for mildly curved interfaces, under the condition that the (local) radius of curvature of the interface is much larger than the wavelength. Snell’s law of reflection is θi = θr, (1.3) where θi is the angle of incidence, which is the angle between the normal of the interface and the ray direction in the medium where the ray originates, and θr is the angle of reflection, which is the angle between the normal of the interface and the ray direction in the same medium as where the ray originates. The angle of incidence and the angle of reflection are opposite to each other with respect to the normal of the interface.
1.1. RAYS AND PLANE WAVES 3 Snell’s law of refraction is given by Figure 1.1: Demonstration of Snell’s law. sin θt sin θi = n1 , (1.4) n2 where θi is again the angle of incidence and θt is the angle of refraction that is given by the angle between the normal of the interface and the ray direction in the medium in which the ray is transmitted. Further, n1 and n2 are the refractive indices of medium 1 and 2, respectively. A geometrical description of Snell’s laws is given in Figure 1.1. It can be seen that if the refractive index in medium 1 is smaller than in medium 2, the ray bends towards the normal, whereas in the opposite case, the ray bends away from the normal. Further, if
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Page 5 and 6: 1.2. RAYS IN SYSTEMS WITH RADIAL SY
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Page 9 and 10: Chapter 2 Gaussian beams In Chapter
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Page 13 and 14: 2.2. GAUSSIAN BEAM IN A HOMOGENEOUS
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Page 19 and 20: 3.2. MODAL DESCRIPTION: HIGHLIGHTS

Chapter 1 Rays and paraxial approximations - OED

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