Modes are solutions of Maxwell’s equations which are separable in the propagation direction, z. They have the form
E(x, y, z) = F(x, y) exp( -j \beta z)
where E is any of the field components, electric or magnetic. When this equation is substituted into Maxwell’s equations, the resulting equation for F is an equation with material parameters appearing in the coefficients. If those materials are loss-less, they are represented by real numbers. The solution for those equations are usually real valued functions. That is why the modes of loss-less waveguides are represented as real functions.
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