(14) |
We can write the extrapolation wavenumber as a first-order Taylor expansion relative to a reference medium:
(15) |
The first part of mixed.3d, corresponding to the extrapolation wavenumber in the reference medium ,is implemented in the Fourier () domain, while the second part, corresponding to the spatially variable medium coefficients, is implemented in the space () domain.
If we make the further simplifying assumptions that and , we can write
(16) |
where
. s|_0 &=&
2s_0
4_0s_0^2 - _0^2 ,
. |_0 &=&
- i_0 2_0^2 +
_0^2-2_0 s_0^2
2_0^2 4_0s_0^2 - _0^2 ,
. |_0 &=&
i 2_0 -
_0
2_0 4_0s_0^2 - _0^2 .
By ``0'', I denote the reference medium (s0,cj0). In principle, we could also use many reference media, followed by interpolation, similarly to the phase-shift plus interpolation (PSPI) technique of (38).
For the particular case of Cartesian coordinates (), mixed.3d.explicit reduces to
(17) |