First symbiosis between rigorous ray tracing and the parabolic approximation

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Rigorous ray tracing originates from the trajectory representation of quantum mechanics while the (Leontovich–Fock) parabolic equation was originally applied to electromagnetic propagation. Rigorous ray tracing (and quantum trajectory representation) have had theoretical success while the parabolic approximation has been a computational success. The parabolic approximation inspires a similar equivalent approximation to the generalized Hamilton–Jacobi equation for rigorous ray tracing. For separable coordinates, the resulting approximations render the same approximate wavefunction. In return, the rigorous ray leads to a parabolic approximation incorporating reflection. The trajectory representation has rendered a synthesized single wavefunction that includes an incident wave of coefficient α and a reflected wave of coefficient β. This synthesized wavefunction can generate a new alternative parabolic equation that incorporates reflection, which for small reflection (β≪α) becomes Fzz+i2k[1−(2β/α)cos(2kx)]exp[i(2β/α)sin(2kx/B)]Fx+(κ2−k2)F=0.