Model
A monochromatic scalar field U(x, y, z) e−iωt in vacuum obeys the Helmholtz equation (∇² + k²)U = 0 with k = 2π/λ.
Writing the field at z = 0 as a superposition of plane waves (its angular spectrum A(fx, fy) = ℱ{U(x, y, 0)}),
each plane wave simply acquires a phase on propagation:
For fx² + fy² > 1/λ² the square root is imaginary, H = exp(−κz) with κ = 2π√(f² − 1/λ²): these
evanescent components decay and carry no power to the far field. The tool either drops them or lets them decay.
Expanding the square root to second order gives the paraxial Fresnel transfer function
H ≈ eikz exp(−iπλz f²), whose inverse transform is the impulse response h = eikz/(iλz) · exp[iπ(x² + y²)/(λz)].
When NF = a²/(λz) ≪ 1 the quadratic phase across the aperture is negligible and the field is the
Fraunhofer pattern, a scaled Fourier transform of the aperture evaluated at f = x/(λz).
- λ
- vacuum wavelength (medium index n = 1)
- t(x, y)
- complex aperture transmittance (thin-element model)
- a
- in the formulas: aperture half-width or radius (for zone plates the outer radius), i.e. half the width/diameter set by the “Width / diameter a” control
- N, Δx, L
- samples per side, grid pitch and window extent L = NΔx
- Np, Lp
- padded grid size and extent (padding factor × N)
- zc
- critical distance LpΔx/λ: Fresnel TF is well sampled for z ≤ zc, IR and single-FFT Fresnel for z ≥ zc
- flimit
- band limit 1/[λ√((2z/Lp)² + 1)] (Matsushima–Shimobaba) that prevents the ASM transfer function from aliasing
- I₀
- incident intensity; all maps are |U|² in units of I₀ (plane wave: |U| = 1)
Assumptions: scalar, monochromatic, fully coherent illumination; thin aperture (Kirchhoff boundary values
U = t·Uinc immediately behind it); homogeneous medium; periodic boundary conditions of the discrete Fourier transform.
The phase maps show the reduced field U·e−ikz.
Derivation: angular spectrum, Fresnel and the single-FFT transform
Insert U = ∬ A(fx, fy; z) ei2π(fxx + fyy) dfx dfy into the Helmholtz equation.
Each component satisfies d²A/dz² + (2π)²(1/λ² − f²)A = 0; keeping only the forward-going (or decaying) solution gives
A(z) = A(0) exp[i2πz√(1/λ² − f²)] with Im √ ≥ 0. This is exact for the scalar problem and equals the first Rayleigh–Sommerfeld solution.
Paraxial: for λf ≪ 1, √(1/λ² − f²) ≈ 1/λ − λf²/2, giving the Fresnel TF. Its inverse Fourier transform is the chirp h(x, y).
Writing the Fresnel convolution U(x) = ∫ U(ξ) h(x − ξ) dξ and expanding (x − ξ)² = x² − 2xξ + ξ² gives
U(x) = eikz eiπx²/(λz)/(iλz) · ℱ{U(ξ) eiπξ²/(λz)} at f = x/(λz). A discrete FFT over Np samples of
pitch Δx returns frequencies spaced 1/(NpΔx), i.e. output positions spaced Δx₂ = λz/(NpΔx).
Dropping the inner chirp (NF ≪ 1) gives Fraunhofer diffraction.
Sampling: the TF chirp exp(−iπλz f²) changes phase by 2πλz f Δf between frequency samples; requiring ≤ π at f = 1/(2Δx) gives
Δx ≥ λz/Lp, i.e. z ≤ zc. The IR chirp in space gives the opposite condition. Parseval's theorem makes the ASM
and TF unitary on propagating frequencies, so power is conserved exactly unless the band limit or evanescent treatment removes some.
Worked example — designing a Fresnel zone plate
Design an amplitude zone plate with focal length f = 0.2 m for λ = 633 nm. Zone boundaries sit where the path to the focus grows by
λ/2: rn = √(nλf), so r₁ = √(633 × 10⁻⁹ × 0.2) m = 0.356 mm and with 8 zones the outer radius is r₈ = 1.006 mm.
The outermost zone width is r₈ − r₇ ≈ r₈/(2·8) ≈ 65 µm, so a 4 mm window needs Δx ≲ 16 µm, i.e. N = 256 (Δx = 15.6 µm; the readout reports 4.2 samples per feature).
Each open zone contributes amplitude 2 (relative to the unobstructed wave) with the same sign, so the four open zones of eight give |U| ≈ 8 and a focal intensity
of ≈ Nzones² = 64 I₀. The “Fresnel zone plate” preset gives about 63 I₀. Replacing the opaque zones by π phase steps doubles the amplitude: ≈ 250 I₀.
Higher-order foci appear at f/3, f/5, …; try z = 66.7 mm. The peak there is still ≈ 53 I₀ (paraxial theory gives the same N² peak), but the spot is 3× narrower, so it carries only about 1/9 of the power.