fraunhofer¶

Computes Fraunhofer diffraction integral using Fourier transform method:

\[\begin{split}U'(x,y)&=\frac{\e^{\i kd}\e^{\i\frac{k}{2d}(x^2+y^2)}}{\i\lambda d} \iint U(u,v)\e^{-\i\frac{k}{d}(xu+yv)}\d u\d v \\ &=\frac{\e^{\i kd}\e^{\i\frac{k}{2d}(x^2+y^2)}}{\i\lambda d} \ft\{U(u,v)\}_{f_u=\frac{x}{\lambda d},f_v=\frac{y}{\lambda d}}\end{split}\]

where \(U\) and \(U'\) are the complex amplitude on source and target plane, respectively, \(d\) is the distance between two planes, \(\lambda\) is wavelength, \(\ft\) represents Fourier transform and \(k=2\pi/\lambda\).

Note that some intermediate computational steps don’t depend on pupil and pre-computing their results can avoid repeated computation if this function is expected to be called multiple times, as long as the arguments other than pupil don’t change. This can be done by calling init_fraunhofer() and passing its return value as intermediate argument. In this case, wl, distance, dx and dy cannot be passed. If all of them are given, intermediate will be ignored anyway.

Parameters:

pupil (Tensor) – Pupil function \(U\). A tensor of shape \((\cdots,N_d,N_\lambda,H,W)\).
wl (float, Sequence[float] or Tensor) – Wavelengths \(\lambda\). A scalar or a tensor of shape \((N_\lambda,)\).
distance (float, Sequence[float] or Tensor) – Propagation distance \(d\). A scalar or a tensor of shape \((N_d,)\).
dx (float or Tensor) – Grid spacing in horizontal direction to ensure correct scale for DFT. A scalar or a tensor of shape \((\cdots,N_d,N_\lambda)\). Default: ignored.
dy (float or Tensor) – Grid spacing in vertical direction, similar to dx. Default: same as dx. Note that dy will be also ignored if dx is None even though dy is given.

kwargs –

Optional keyword arguments:

Parameter	Type	Description
delta_mode	str	On which plane `dx` and `dy` are defined, source plane (`forward`) or target plane (`backward`). Default: `backward`.
phase_factor	bool	Whether to multiply \(\e^{\i kd}\e^{\i\frac{k}{2d}(x^2+y^2)}/\i\). It can be omitted to reduce computation if only amplitude matters. Default: `True`.
scale_factor	bool	Whether to multiply the factor \(1/(\lambda d)\). It can be omitted to reduce computation if relative scales across different \(d\) and \(\lambda\) do not matter. Default: `True`.
dft_scale	bool	Whether to apply DFT-scale. See Fourier transform. Default: `True`.
intermediate	dict	Cached intermediate results returned by `init_fraunhofer()`. This can be used to speed up computation. Default: omitted.

Returns:

Diffracted complex amplitude \(U'\). A tensor of shape \((\cdots,N_d,N_\lambda,H,W)\).

Return type:

Tensor