In this seminar, we dive into Fourier optics, a framework connecting spatial frequencies to diffraction patterns. You will analyze how apertures shape light through the Fourier transform, design spatial filters, and understand diffraction-limited imaging. These concepts are central to microscopy, imaging systems, and holography.
Pen & Paper Problems
Problem 1: Rectangular Aperture Diffraction
A rectangular aperture of width \(a\) is illuminated by a plane wave of wavelength \(\lambda\). The diffraction pattern is observed on a screen at distance \(L\) from the aperture.
Tasks: 1. The Fourier transform of a rectangular aperture (rect function) is a sinc function: \(\mathcal{F}[\text{rect}(x/a)] = a \, \text{sinc}(u)\) where \(u = \pi a \sin\theta / \lambda\). 2. At what angles do the first zeros of the diffraction pattern occur? (Hint: \(\text{sinc}(x) = 0\) for \(x = \pm n\pi\), \(n \neq 0\).) 3. For \(a = 100\) μm and \(\lambda = 633\) nm (He-Ne laser), calculate the angular positions of the first three minima. 4. At distance \(L = 1\) m, what are the linear positions of these minima on the screen? 5. Sketch the diffraction pattern (intensity vs. angle), noting the central bright spot (Airy rectangle) and side lobes.
Problem 2: Circular Aperture & Airy Pattern
A circular aperture of diameter \(D\) produces the famous Airy pattern in the Fraunhofer (far-field) limit.
Tasks: 1. The intensity of the Airy pattern is: \[I(\theta) = I_0 \left[ \frac{2 J_1(x)}{x} \right]^2, \quad x = \frac{\pi D \sin\theta}{\lambda}\] where \(J_1\) is the first-order Bessel function.
The first minimum occurs at \(x \approx 3.83\). Calculate the angular radius of the Airy disk (first dark ring) for:
Optical microscope objective with NA = 0.65 and \(\lambda = 500\) nm
Expression: radius \(\approx 1.22 \lambda / D\) or \(\approx 0.61 \lambda / \text{NA}\) for a lens.
For a microscope with NA = 0.65 and λ = 500 nm, what is the Airy disk radius in μm?
Compare with the Rayleigh criterion: two point sources are resolved if separated by more than one Airy disk radius. What is the minimum resolvable distance?
Problem 3: Double-Slit Interference & Diffraction
Two slits, each of width \(a\), separated by distance \(d\), are illuminated by a plane wave. The pattern combines single-slit diffraction (envelope) and two-slit interference (fringes).
Tasks: 1. The combined intensity is: \[I(\theta) = I_0 \left[ \frac{\sin(\beta)}{\beta} \right]^2 \cos^2(\delta)\] where \(\beta = \pi a \sin\theta / \lambda\) (single slit) and \(\delta = \pi d \sin\theta / \lambda\) (slit separation).
For \(a = 100\) μm, \(d = 0.5\) mm, \(\lambda = 633\) nm, identify:
Spacing between interference fringes (given by the cosine term)
Period of the diffraction envelope (given by the sinc term)
Sketch the combined pattern, showing fringes and envelope.
Spatial frequency: The spacing between fringes corresponds to what spatial frequency (cycles per mm)?
Problem 4: Spatial Filtering in 4f System (Challenge)
A 4f system (two lenses of focal length \(f\), separated by distance \(2f\)) acts as a Fourier transform processor. A spatial filter can be placed at the Fourier plane to selectively remove or enhance features.
Tasks: 1. Explain how a 4f system creates a Fourier plane at the midpoint between the two lenses. 2. A low-pass filter (opaque except near the center) removes fine details. Describe what you would see on the output screen if a small aperture is placed at the Fourier plane. 3. A high-pass filter blocks low spatial frequencies (large features). What happens to a grating pattern or text? 4. Design a notch filter to remove a periodic grating (e.g., unwanted stripes in an image). Where would you place the filter?
Problem 5: SLM as Programmable Spatial Filter
A spatial light modulator (SLM) can imprint an arbitrary phase or amplitude pattern on a wavefront.
Tasks: 1. Explain how an SLM can replace fixed spatial filters in a 4f system. 2. A programmable phase mask \(\phi(x, y)\) on an SLM modulates the incident field to \(E'(x,y) = E(x,y) e^{i\phi(x,y)}\). 3. If the SLM imprints a blazed grating phase pattern, what is the effect on the output? (Hint: a grating diffracts light into different orders.) 4. How could an SLM implement a generalized wavefront shaping to focus light onto arbitrary 3D patterns or generate spatial modes?
Python Exercises
Setup
Exercise 1: Aperture Diffraction Patterns
Simulate diffraction from rectangular and circular apertures using the Fourier transform.
4f system simulation complete.
Smaller filter radius → more low-pass filtering (blur)
Larger filter radius → more high-frequency content preserved (sharp)
Solutions Guide
Problem 1: The first zeros of the sinc function occur at \(u = \pm \pi, \pm 2\pi, ...\) For a 100 μm aperture and 633 nm, the first minimum is at approximately 3.6°.
Problem 2: The Airy disk radius is \(r \approx 1.22 \lambda / D\) or equivalently \(r \approx 0.61 \lambda / \text{NA}\). For NA = 0.65 and λ = 500 nm, this gives approximately 0.47 μm.
Problem 3: The fringe spacing is determined by slit separation, while the envelope width is determined by slit width. With values given, fringes are very closely spaced compared to the slow diffraction envelope.
Problem 4: A 4f system maps spatial frequencies to spatial positions in the Fourier plane. Low-pass filtering (small aperture) removes high spatial frequencies and blurs the image. High-pass filtering (large opaque region near axis) removes low frequencies and enhances edges.
Problem 5: SLMs enable programmable spatial filtering, allowing dynamic wavefront shaping, beam steering, and even calculation of arbitrary Fourier transforms in real time.
Summary
Fourier optics unifies diffraction theory with spatial frequency filtering, providing powerful tools for image processing and optical design. Understanding these concepts is essential for advanced topics in microscopy, holography, and optical communication.