Lens L₁ maps angles at the object plane to positions in the Fourier plane (FT). Lens L₂ maps positions in the filtered Fourier plane back to the image plane (inverse FT). With spacing f between each element, the image is magnified ×1 and inverted.

What does a low-pass filter do?

It blocks high spatial frequencies (large |u|,|v|), which sharpens edges and fine structure in the object. The image looks blurred — equivalent to convolving with a wide point-spread function.

What does a high-pass filter do?

It removes the DC and low-frequency content, highlighting edges and subtracting a uniform background. Periodic objects may show only their higher harmonics.

Finite lens aperture, aberrations, polarization, phase-only filters, and pixel fill factor. The FFT is paraxial, scalar, and monochromatic.

Fourier Optics 4f System

In a standard 4f optical processor, a transparency g(x,y) at the front focal plane of lens L₁ (focal length f) produces a complex field whose Fourier transform G(u,v) appears at the back focal plane — the shared Fourier plane halfway between two lenses separated by 2f. A spatial filter H(u,v) (iris, slit, or phase plate) multiplies the spectrum; lens L₂ performs an inverse transform so the rear focal plane carries g′(x,y) ∝ ℱ⁻¹{H·G}, inverted and scaled. Low-pass apertures blur fine detail (remove high spatial frequencies); high-pass apertures emphasize edges and periodic structure. This simulator uses a discrete 2D FFT on 64×64 object presets (grating, slit, cross, grid), circular low-pass and high-pass filters with adjustable radius, and displays object, |G|, filtered spectrum, and image intensity |g′|² in a four-panel layout with a ray schematic. Focal length f labels the 4f geometry (total length 4f) but the FFT model uses normalized spatial coordinates — appropriate for teaching the convolution theorem in Fourier optics.

Who it's for: Undergraduate optics and photonics students after Fraunhofer diffraction and before spatial light modulators or digital holography.

Key terms

4f system
Fourier optics
Spatial filter
Fourier plane
Low-pass filter
High-pass filter
Convolution theorem
Optical processing

Live graphs

How it works

Fourier optics 4f system: object → lens → Fourier plane (spectrum) → spatial filter → second lens → image; low-pass and high-pass apertures.

Key equations

G(u,v) = ℱ{g(x,y)} at Fourier plane; g′ = ℱ⁻¹{H·G} at image

Low-pass: |u| < u_c · High-pass: |u| > u_c · 4f spacing: L₁–L₂ = 2f

Frequently asked questions

Why two lenses?: Lens L₁ maps angles at the object plane to positions in the Fourier plane (FT). Lens L₂ maps positions in the filtered Fourier plane back to the image plane (inverse FT). With spacing f between each element, the image is magnified ×1 and inverted.
What does a low-pass filter do?: It blocks high spatial frequencies (large |u|,|v|), which sharpens edges and fine structure in the object. The image looks blurred — equivalent to convolving with a wide point-spread function.
What does a high-pass filter do?: It removes the DC and low-frequency content, highlighting edges and subtracting a uniform background. Periodic objects may show only their higher harmonics.
What is left out?: Finite lens aperture, aberrations, polarization, phase-only filters, and pixel fill factor. The FFT is paraxial, scalar, and monochromatic.

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Live graphs

How it works

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Frequently asked questions

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Key equations

Frequently asked questions