The wave propagation of harmonic fields through homogeneous media is an essential simulation technique in optical modeling and design by field tracing, which combines geometrical and physical optics. For paraxial fields the combination of Fresnel integral and the Spectrum of Plane Waves (SPW) integral solves the problem. For non-paraxial fields the Fresnel integral cannot be applied and SPW often suffers from a too high numerical effort. In some situations the far field integral can be used instead, but a general solution of the problem is not known.

In 2012 we have developed without further physical approximations three new algorithms for the fast propagation of non-paraxial vectorial optical fields containing smooth but strong phase terms. Dependent on the shape of the smooth phase term different propagation operators are applied.

The first method for the efficient propagation of fields, which are containing smooth spherical phase terms, is based on Mansuripur's extended Fresnel diffraction integral [M. Mansuripur, J. Opt. Soc. Am. A 6(5), 786-806 (1989)] using fast Fourier Transformations. This concept is improved by Avoort's parabolic fitting technique [C. van der Avoort et al., J. Mod. Opt. 52(12), 1695-1728 (2005)]. Furthermore we have introduced the inversion of the extended Fresnel operator for the fast propagation of non-paraxial fields into the focal region.

Propagated non-paraxial field
Figure 1: Propagated non-paraxial field after strong-aberrated high-NA Gaussian-to-Top-Heat-beam-shaper

Secondly we have developed a new semi-analytical spectrum of plane waves (SPW) operator for the quick propagation of fields with smooth linear phase terms. The method is based on the analytical handling of the linear phase term and the lateral offset, which reduces the required computational window sizes in the target plane.

Finally we have generalized our semi-analytical SPW operator concept to universal shapes of smooth phases by decomposing non-paraxial fields into subfields with smooth linear phase terms (Fig. 2). In the target plane, all propagated subfields are added coherently where the analytical known smooth linear phase terms are handled numerical efficient by a new inverse parabasal decomposition technique (PDT).

a nummerical effortd
Figure 2: The numerical effort to sample the phase in (a) can be drastically reduced by extracting the inherent smooth linear phase term. The remaining residual phase, which needs drastically reduced sampling effort, is shown in (b). The propagation of the extracted linear phase terms can be handled analytically by our new propagation technique.

With these three new rigorous techniques we are able to propagate even high non-paraxial electromagnetic fields through homogeneous media quite efficient.

For further information please check our following publications:

1.    Daniel Asoubar, Site Zhang, Frank Wyrowski, Michael Kuhn, "Semi-analytical techniques for efficient electromagnetic
       field propagation", Proceedings of SPIE Vol. 8550, 85503F (2013)  
2.    Daniel Asoubar, Site Zhang, Frank Wyrowski, Michael Kuhn, "Parabasal field decomposition and its application to
       non-paraxial propagation", Optics Express Vol. 20, Issue 21, pp.23502-23517 (2012)