Geometric optics is arguably the most classical and traditional of the branches of physical science. The optical design of instruments and devices has been developed and improved over the centuries. From Galileo`s telescopes to the lens of the contemporary camera, advances, while impressive, have been largely scalable, with modern design benefiting enormously from the availability of fast and relatively inexpensive digital computers. However, in one important respect, conventional optical design of image formation is quite ineffective, namely in the simple concentration and collection of light. This is well illustrated by an example of the concentration of solar energy. The method of obtaining geometric optical equations by analyzing surfaces of discontinuities of solutions to Maxwell`s equations was first described in 1944 by Rudolf Karl Lüneburg. [10] It does not limit that the electromagnetic field has a special form required by the summer field runge method, in which the amplitude A ( k o , r ) {displaystyle A(k_{o},mathbf {r} )} and the phase S ( r ) {displaystyle S(mathbf {r} )} the equation lim k 0 → ∞ 1 k 0 ( 1 A ∇ S ⋅ ∇ A + 1 2 ∇ 2 S ) = 0 {displaystyle lim _{k_{0}to infty }{1 over k_{ 0}}left({ 1 over A},nabla Scdot nabla A+{1 over 2}nabla ^{2}Sright)=0}. This condition is met, for example, by plane waves, but is not additive. Beam optics theory also describes the conventional light capture structure of Fig. 2 in 2D. For the isotropic case, the gain factor is radiation optics or geometric optics, based on the short-wave approximation of electromagnetic theory. It is defined in relation to a set of rules (the rules of geometric optics) that can be obtained from Maxwell`s equations in a coherent approximation scheme called the eikonal approximation, which is briefly described in this chapter.
The basic concepts of the geometric wavefront, beam path and optical path length are elaborated and the rules for transporting field vectors along beam paths are explained. The laws of reflection and refraction are derived, and the approximation scheme leading to Fresnel formulas is briefly described. The basic ideas of the calculation of variations are explained. The characterization of beam trajectories within the meaning of Fermat`s principle is elaborated and the type of stationarity of the optical path length is explained with the help of examples. The Lagrangian and Hamiltonian formulations of geometric optics are sketched, the determination of the beam trajectory being considered analogous to a limited problem in mechanics. The concepts of caustics and conjugate points are explained using examples. The integral formulation of the path is described as a useful heuristic principle in geometric optics. The Luneburg-Kline approach, which gives geometric optics a safe foundation, is briefly mentioned. An optical lens works by refracting light at its interfaces.
In these examples, it is assumed that the lens is thin, in this case the thickness of the lens is negligible compared to its focal length. Lenses are basically of two types. A converging lens causes the parallel rays to converge and a divergent lens causes the parallel rays to dive. Figure 17 shows the trajectories of the rays through the lens and focal point for each case. The optical axis, focal point, and focal length definitions given for curved mirrors apply to lenses, with the addition that lenses have focal points on either side of the lens. The principles introduced by Hamilton lead to the idea of properties in geometric optics, the basic idea of which is briefly described in section 3.5. Geometric optics or beam optics is a model of optics that describes the propagation of light in the form of rays. The beam in geometric optics is a useful abstraction for approaching the paths on which light propagates under certain circumstances. CGO is a method based on geometric optics, but it accurately describes the wave phenomena related to the propagation of the Gaussian beam (along the central beam of the beam). Since CGO is a paraxial method, we describe spatially narrow beams that are located near the central rays. In inhomogeneous media, these central rays are curvilinear due to a gradient not equal to zero of the relative permittivity of the medium and can be described using Hamiltonian equations in the following form: In geometric optics, we represent the power density on a surface by the density of the beam intersections with the surface and the total power by the number of beams. This term, reminiscent of useful but outdated “lines of force” in electrostatics, works as follows.
We take N beams arranged uniformly above the inlet opening of a concentrator at an angle of incidence θ, as shown in Figure 2.1. Suppose that after following the rays by the system, only N′ exits through the outlet opening, the dimensions of which are determined by the desired concentration ratio. The remaining N-N′ rays are lost by processes that become clear in some examples. Then, the power transmission for the angle θ is assumed N′/N. This can be extended as needed to cover an angular range θ. It is clear that nitrogen must be taken large enough to ensure that a thorough study of possible radiation pathways is carried out in the concentrator. In this chapter, I present the basic ideas of determining radiation pathways in Gaussian optics with respect to 2 × 2 transfer matrices and apply them to cases of image formation by axially symmetrical systems consisting of reflective and refractive surfaces. The camera, telescope and microscope, the three classic optical instruments, generally use image formation according to the principles of Gaussian optics.
Subject to the conditions set by linear optics (the conditions of paraxiality), an axially symmetrical optical system forms a point image for each given point object, which makes it possible to see that for a point object at a certain position, the position of the image is completely determined by a series of parameters that characterize the optical system under consideration. For an optical system consisting of a series of component subsystems, these parameters can be determined in relation to those of the subsystems. A feature of image formation in Gaussian optics is the geometric similarity between a planar object and its image, both of which are perpendicular to the axis. These fields obey transport equations that correspond to the transport equations of the Sommerfeld–Runge approach. Light rays in Lüneburg theory are defined as orthogonal trajectories to discontinuity surfaces, and with proper parameterization, it can be shown that they obey Fermat`s principle of least time and thus establish the identity of these rays with standard optical light rays. Since the basic principle of geometric optics lies in the limit value λ o ∼ k o − 1 → 0 {displaystyle lambda _{o}sim k_{o}^{-1}rightarrow 0}, the following asymptotic series is assumed, Various consequences of Snell`s law include the fact that for light rays that migrate from a material with a high refractive index to a material with a low refractive index, It is possible that the interaction with the interface results in zero transmission.