Power in path

This issue is just one of precision of simulations. In waveguide.bpd, the Power overlap integral varies between 1 and 0.974. In the same project with finer mesh, larger window, and mode solver tolerance made tighter (waveguide3.bpd), the variation is from 1 to 0.996. This simulation is showing the accuracy of the simulation under the given mesh and mode solver tolerances.

There are two normalizations for power, Global and Local. If we write the power in the transverse mesh at point z as Tm(z), and put the Input Plane at z = 0, then the local Power in Path is
Power In Path(z) = (power in waveguide)(z) / Tm(z),
and the global one is
Power In Path(z) = (power in waveguide)(z) / Tm(0)

The Power Overlap Integral is an overlap integral. In the mathematical sense, it is an inner product in an L^2 Hilbert space. The Wikipedia entry on Hilbert space can tell you a lot about the background behind that notion. The reason Hilbert space is relevant in optical waveguide theory is because for waveguides, the Maxwell equations can be reduced to an eigenvalue problem, LE = (lambda) E, where lambda is actually the square of the propagation constant. L is a differential operator, and for loss-less materials at least, it turns out to be self-adjoint. That means the eigenvectors (the modes) are mutually orthogonal in the sense of the L^2 inner product.

The power overlap integral is usually used in the case where you have a waveguide in the space Z > 0, and some random field in the space Z < 0. You want to know how much optical power in the space Z < 0 gets into the waveguide mode in the Z > 0 space. Let’s suppose the waveguide mode field pattern is E_2. Then imagine E_1 to be a linear combination of the orthogonal modes to be found in the Z > 0 space (an eigenvalue expansion). Note that E_2 is both an eigen mode in that expansion and the fundamental mode of the waveguide. Since the modes in the expansion are orthogonal, forming the inner product with one of the eigenmodes of that space (E_2) will yield the coefficient of the field that is applied to the fundamental mode of the waveguide. The power in the waveguide is the magnitude of that coefficient.

That explains the numerator. The denominator has the normalizations. The E_2^2 factor is required to normalize the E_2 function to be an eigenfunction of unit size, i.e. the inner product with itself is 1. If the normalization with E_1 is not used, you’ll get the total power transferred to the waveguide in Z > 0. If you divide by the E_1 factor, you’ll get a transmission coefficient instead. The power overlap integral gives you the power transmission, the amplitude of the transmission coefficient. (Insertion loss).