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This is the basic principle behind fiber optics in which light is guided along a high index glass core in a lower index glass cladding (Figure d). Ray optics only gives a rough picture of how waveguides work. Maxwell's equations can be solved by analytical or numerical methods for a full-field description of a dielectric waveguide.
The space and time scales over which E 0 varies are generally much longer than the spatial wavelength and temporal period of the carrier wave. A numerical solution of the envelope equation thus can use much larger space and time steps, resulting in significantly less computational effort.
An electromagnetic wave propagating in the +z-direction is conventionally described by the equation: (,) = [()], where E 0 is a vector in the x-y plane, with the units of an electric field (the vector is in general a complex vector, to allow for all possible polarizations and phases);
Numerical aperture is commonly used in microscopy to describe the acceptance cone of an objective (and hence its light-gathering ability and resolution), and in fiber optics, in which it describes the range of angles within which light that is incident on the fiber will be transmitted along it.
Both spatial domain methods, and frequency (spectral) domain methods are available for the numerical solution of the discretized master equation. Upon discretization into a grid, (using various centralized difference, Crank–Nicolson method, FFT-BPM etc.) and field values rearranged in a causal fashion, the field evolution is computed through iteration, along the propagation direction.
Kirchhoff's integral theorem, sometimes referred to as the Fresnel–Kirchhoff integral theorem, [3] uses Green's second identity to derive the solution of the homogeneous scalar wave equation at an arbitrary spatial position P in terms of the solution of the wave equation and its first order derivative at all points on an arbitrary closed surface as the boundary of some volume including P.
Physical optics is also the name of an approximation commonly used in optics, electrical engineering and applied physics. In this context, it is an intermediate method between geometric optics , which ignores wave effects, and full wave electromagnetism , which is a precise theory .
Fourier optics begins with the homogeneous, scalar wave equation (valid in source-free regions): (,) = where is the speed of light and u(r,t) is a real-valued Cartesian component of an electromagnetic wave propagating through a free space (e.g., u(r, t) = E i (r, t) for i = x, y, or z where E i is the i-axis component of an electric field E in the Cartesian coordinate system).