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5) interface. The graphs show that both the reflectance and transmittance are between 0 and 1 for all incident angles. By combining Eqs. 31 with the Fresnel formulae, we find that R+T=1, as we intuitively know that must be the case.

This solution represents a wave that is polarized in a direction orG thogonal to the direction of propagation as defined by the wave vector k . The G wave is uniform in planes orthogonal to k , and has a shape given by f(x) in the diG rection of k . 30) where f” is the second derivative of f with respect to its whole argument. 31) This is the dispersion relation for plane waves in a homogeneous medium. In free k . space or vacuum the dispersion relation is a simple straight line ω = μ0ε 0 In homogeneous materials, the dispersion relation is non-linear, due to the frequency dependence of the permittivity (and sometimes also the frequency dependence of the permeability).

With only In1 present, the outputs are Eout1 and Eout2=K12Eout1, where K12 a complex number. Similarly, with only In2 present, the outputs are Eout2 and Eout1=K21Eout2. Express K21 in terms of K12. 5 – Crossing waveguides Consider the symmetric single-mode waveguide crossing in the figure below. The crossing is well designed, so that all the power of the inputs stays in one or the other of the outputs. If there is only power in Input 1, then 75 % goes to Output 1 and 25 % to Output 2, and by symmetry, if there is only power in Input 2, then 25 % goes to Output 1 and 75 % to Output 2.