Fourier

Fourier-domain operations and filtering.

Overview

The Fourier module provides tools for applying operations in the frequency domain, including spatial filtering, frequency-domain masks, and wrapping spatial components to operate on Fourier transforms.

Examples

Frequency-Domain Filtering

using FluxOptics

xv, yv = spatial_vectors(128, 128, 1.0, 1.0)
u = ScalarField(Gaussian(20.0)(xv, yv), (1.0, 1.0), 1.064)

# Sharp low-pass filter
f_cutoff = 0.02  # 1/μm
lowpass = FourierMask(u, (fx, fy) -> sqrt(fx^2 + fy^2) < f_cutoff ? 1.0 : 0.0)

# Apply filter
u_filtered = propagate(u, lowpass, Forward)

# Check power conservation (should be < 1 due to filtering)
power(u_filtered)[] / power(u)[]

0.9659322082218439

# Gaussian filter (soft cutoff)
sigma_f = 0.15  # 1/μm
gaussian_filter = FourierMask(u, (fx, fy) -> exp(-(fx^2 + fy^2)/(2*sigma_f^2)))

u_smooth = propagate(u, gaussian_filter, Forward)

power(u_smooth)[] / power(u)[]

0.9944025531935685

Frequency-Domain Phase

# Parabolic phase in frequency (quadratic chirp in spatial domain)
fourier_phase = FourierPhase(u, (fx, fy) -> π * 0.01 * (fx^2 + fy^2))

u_chirped = propagate(u, fourier_phase, Forward)

# Phase modulation doesn't affect power
power(u_chirped)[]

0.9999999996946143

Wrapping Spatial Components

# Apply spatial phase mask in Fourier domain
phase_spatial = Phase(u, (x, y) -> 0.01 * x^2; trainable=true)
phase_in_fourier = FourierWrapper(u, phase_spatial)

# Equivalent to: FFT → phase_spatial → IFFT
istrainable(phase_in_fourier)

true

Using in Systems

# Combine spatial and frequency filtering
source = ScalarSource(u)
lowpass = FourierMask(u, (fx, fy) -> sqrt(fx^2 + fy^2) < 0.02 ? 1.0 : 0.0)
phase = Phase(u, (x, y) -> 0.01 * (x^2 + y^2))
prop = ASProp(u, 1000.0)

system = source |> lowpass |> phase |> prop

result = system()
power(result.out)[]

0.965932207926862

Key Types

FourierOperator: FFT/IFFT transformation (low-level)
FourierWrapper: Apply component in Fourier domain (FFT → component → IFFT)
FourierPhase: Phase mask in Fourier space (convenience)
FourierMask: Amplitude/complex mask in Fourier space (convenience)

Coordinate Convention

Functions receive frequency arguments (fx, fy) in units of 1/length:

Frequency grid: fx = fftfreq(nx, 1/dx)
Zero frequency at center after fftshift

Example: For dx = 1.0 μm and nx = 128, fx ranges from -0.5 to +0.5-1/128 μm⁻¹.

Component Relationships

FourierPhase and FourierMask

Convenience constructors that internally use FourierWrapper:

# These are equivalent:
FourierPhase(u, (fx, fy) -> φ(fx, fy))
FourierWrapper(u, Phase(u, ...))  # Phase evaluated in frequency

FourierMask(u, (fx, fy) -> m(fx, fy))
FourierWrapper(u, Mask(u, ...))   # Mask evaluated in frequency

FourierWrapper

General wrapper that applies: u → IFFT[component(FFT[u])]

Useful for applying spatial-domain components to frequency content.