Computation of diffracted fields for the case of high numerical aperture using the angular spectrum method

Abstract

The angular spectrum (AS) method is a popular solution to the Helmholtz Equation without the use of approximations. In this work, new criteria on sampling requirements are derived using the Wigner distribution (WD). It is shown that for the case of high numerical aperture the conventional AS method requires a very large amount of zero-padding, making it impractical due to requirements on memory and computational effort. This work proposes the use of a modified AS algorithm that evaluates only non-zero components of the field. The results obtained from the WD combined with the modified AS algorithm enable an accurate and efficient field computation for cases where the conventional AS method cannot be implemented.

Modified shifted angular spectrum method for numerical propagation at reduced spatial sampling rates

“…18, will increase the acceptable bandwidth of the input wave field, but at the cost of an increased number of samples. This has also been discussed in [2,5,6,17]. Especially, for the case of X-ray optics, where wavelengths are very small compared to the sampling window size, a sufficient sampling rate can be harder to obtain than a sufficient sampling window size.…”

“…For a given bandwidth of the input wave field a given size of the sampling window and a given offset between the sampling windows, the propagation can only be done up to a certain distance limit without loss of accuracy. Detailed discussions on this limit for the angular spectrum and shifted angular spectrum method, which is also true for this modified method, can be found for example in [4][5][6]17]. To enable propagation for distances above the distance limit one has to choose another method for example [4][5][6].…”

“…Detailed discussions on this limit for the angular spectrum and shifted angular spectrum method, which is also true for this modified method, can be found for example in [4][5][6]17]. To enable propagation for distances above the distance limit one has to choose another method for example [4][5][6]. Or, one has to increase the size of the sampling window, which according to Eq.…”

“…Compared to that, the method presented in section 2 is directly applicable for more general wave fields under the conditions given in section 4, and thus for example illumination with spherical waves, which is also shown in section 5. In [5,6], methods are presented which reduce the amount of zero padded samples while spatial sampling rate plays no role in these approaches. A combination of these methods with the method presented in section 2 could be possible.…”

Holographic tomography with scanning of illumination: space-domain reconstruction for spatially invariant accuracy

“…The accuracy of ORC-FBPP can be significantly improved [21] by replacing the approximated Rytov propagation with the rigorous angular spectrum (AS) propagation method [28], which enables obtaining ORC-EDOF-FBPP. While in [21] the solution is provided for ORC-HT, in this paper it will be shown that the same principle holds also for ISC-HT.…”