By H. A. Buchdahl
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Extra resources for An Introduction to Hamiltonian Optics
25) are the basic formulas for the realization of the 1D FDTD evolution scheme. 4 FDTD METHOD IN 3D Extension of the FDTD method to 3D problems is straightforward. 4. This grid was originally proposed by Kane Yee in 1966 , the author of the FDTD method. The Yee grid realizes leapfrog time update of the fields on a staggered 3D mesh and thus represents the core of the FDTD method. Components of the electric fields are defined in the middle of the edges of the cube and magnetic field components in the centres of cube faces.
Since the propagation constants now feature both a real and an imaginary part, a clear distinction between propagating and evanescent modes is not possible. 43). 55). 59) for the ez component, the lateral and longitudinal components of the magnetic field become h⊥ = 1 ∇⊥ · (εe⊥ ) zˆ × βe⊥ − ∇⊥ μ0 ω βε hz = , zˆ · ∇ × e⊥ . 61) Let us now consider the relationship between the fields of a forward propagating (E+ , H+ ) and a backward propagating (E− , H− ) eigenmodes derived from the same solution e⊥ to the eigenvalue problem.
D+ u ≡ = u (x) + u (x)h u (x)h2 uIV (x)h3 + + + O(h4 ) = u (x) + O(h). 2! 3! 4! 6) which tells us that the error of such approximation is of the first order of magnitude with respect to the grid step h. Consequently, this scheme serves as the first-order approximation to the derivative u (x). In the same way, we can prove that the backward difference scheme works pretty much with the same precision: u (x) = D− u + O(h). 7) 26 Numerical Methods in Photonics The accuracy of the derivative approximation can be improved if the central difference scheme is employed: u(x + h) − u(x − h) 2h u (x)h2 u (x)h3 uIV (x)h4 1 u(x) + u (x)h + + + + O(h5 ) = 2h 2!
An Introduction to Hamiltonian Optics by H. A. Buchdahl