This package provides the functions for diffraction calculations based on the angular spectrum method (ASM). In addition to the standard diffraction calculation with ASM, various diffraction calculations with band-limited, scalable, scaled, shifted, and tilted ASM are available. In-place versions of each function are also provided. When using the in-place version functions, append an ! to the end of each function name.

To install this package, open the Julia REPL and run

julia> ]add AngularSpectrumMethod

or

julia> using Pkg
julia> Pkg.add("AngularSpectrumMethod")

Import the package first.

julia> using AngularSpectrumMethod

The function ASM returns the diffracted field by the angular spectrum method (ASM). Evanescent waves are not eliminated but attenuated as $\exp(-2{\pi}wz)$. Without attenuation, the total energy $\iint|u|\mathrm{d}x\mathrm{d}y$ is conserved.

julia> v = ASM(u, λ, Δx, Δy, z; expand=true)

• u: input field.
• λ: wavelength.
• Δx: sampling interval in the x-axis.
• Δy: sampling interval in the y-axis.
• z: diffraction distance.
• expand=true: if true (default), perform 4× expansion and zero padding for aliasing suppression.

  
    input field             diffracted field by ASM

The function BandLimitedASM returns a diffracted field by the band-limited ASM (see Ref. 1).

julia> v = BandLimitedASM(u, λ, Δx, Δy, z; expand=true)

1. Kyoji Matsushima and Tomoyoshi Shimobaba, "Band-Limited Angular Spectrum Method for Numerical Simulation of Free-Space Propagation in Far and Near Fields," Opt. Express 17, 19662-19673 (2009)

  
ASM with a long propagation distance       Band-limited ASM      

The function ScalableASM returns automatically scaled diffraction field by the scalable ASM (see Ref. 2). The sampling pitch in the destination plane $\Delta_{d}$ is $\Delta_{d}=\dfrac{\lambda z}{pN\Delta_{s}}$, where $\Delta_{s}$ is the sampling pitch in the source plane, $N$ is the number of pixels in the source or destination plane, and $p=2$ is the padding factor.

julia> v = ScalableASM(u, λ, Δx, Δy, z; expand=true)

2. Rainer Heintzmann, Lars Loetgering, and Felix Wechsler, "Scalable angular spectrum propagation," Optica 10, 1407-1416 (2023)

  
      input field          diffracted field by Scalable ASM

The function ScaledASM returns a scaled diffraction field according to the scale factor $R$ by the scaled ASM (see Ref. 3).

julia> v = ScaledASM(u, λ, Δx, Δy, z, R; expand=true)

3. Tomoyoshi Shimobaba, Kyoji Matsushima, Takashi Kakue, Nobuyuki Masuda, and Tomoyoshi Ito, "Scaled angular spectrum method," Opt. Lett. 37, 4128-4130 (2012)

  
Scaled ASM with R = 2.0            R = 0.5    

The function ShiftedASM returns a shifted diffraction field with the shift distance $x_{0}$ and $y_{0}$ by the shifted ASM (see Ref. 4).

julia> v = ShiftedASM(u, λ, Δx, Δy, z, x₀, y₀; expand=true)

4. Kyoji Matsushima, "Shifted angular spectrum method for off-axis numerical propagation," Opt. Express 18, 18453-18463 (2010)

  
       ASM            Shifted ASM with x direction shift

The function TiltedASM returns a tilted diffraction field for a rotation matrix $T$ by the tilted ASM (see Ref. 5, 6). If weight=true, a diagonal weighting matrix is used as the Jacobian determinant (default false). In this case, the energy conservation improves, but the computational cost is high (see Ref. 7).

julia> v = TiltedASM(u, λ, Δx, Δy, T; expand=true, weight=false)

5. Kyoji Matsushima, Hagen Schimmel, and Frank Wyrowski, "Fast calculation method for optical diffraction on tilted planes by use of the angular spectrum of plane waves," J. Opt. Soc. Am. A 20, 1755-1762 (2003)
6. Kyoji Matsushima, "Formulation of the rotational transformation of wave fields and their application to digital holography," Appl. Opt. 47, D110-D116 (2008)
7. James G. Pipe and Padmanabhan Menon, "Sampling density compensation in MRI: Rationale and an iterative numerical solution," Magn. Reson. Med. 41, 179-186 (1999)

  
        input field        Tilted ASM with rotation around the x-axis