Diffraction

Diffraction is the bending and spreading of waves as they pass around obstacles or through apertures, with characteristic interference patterns appearing in the resulting wave field. It is the wave-optics counterpart to geometric optics and explains every effect that depends on the finite wavelength of light — including the resolution limit of optical instruments, the spreading of laser beams, and the colors seen on optical surfaces.

Huygens-Fresnel principle. Each point on a wavefront acts as a source of secondary spherical wavelets. The total wave at any subsequent point is the superposition of all secondary wavelets, with appropriate amplitude and phase. This principle, dating to Huygens (1690) and refined by Fresnel (1818), provides the framework for all diffraction calculations.

Two regimes.

Regime	Geometry	Description
Fresnel (near-field)	Aperture-to-observation distance comparable to aperture size	Wavefront curvature important; pattern evolves with distance
Fraunhofer (far-field)	Observation distance >> aperture size	Plane-wave approximation; pattern is the Fourier transform of the aperture

The transition between regimes is characterized by the Fresnel number:

$$F \;=\; \frac{a^2}{\lambda z},$$

where $a$ is the aperture radius and $z$ is the observation distance. Fresnel diffraction applies for $F \gtrsim 1$; Fraunhofer for $F \ll 1$.

Fraunhofer diffraction formula. For a 2D aperture function $A(x', y')$, the far-field diffracted intensity is proportional to:

$$I(x, y, z) \;\propto\; \left| \int A(x', y') \, e^{-i 2\pi (x x' + y y')/(\lambda z)} \, dx' \, dy' \right|^2.$$

The diffracted field is the Fourier transform of the aperture function, evaluated at spatial frequencies $f_x = x/(\lambda z)$ and $f_y = y/(\lambda z)$.

Standard diffraction patterns.

Aperture	Far-field pattern	Notable feature
Circular aperture (radius $a$)	Airy pattern	Central spot at $\theta = 1.22 \lambda / D$ first null
Square aperture (side $a$)	sinc² × sinc²	Cross-shaped lobes
Slit (width $d$)	sinc²	Lobes along axis perpendicular to slit
Gaussian aperture (waist $w$)	Gaussian	Far-field Gaussian (self-similar)
Periodic grating (period $\Lambda$)	Discrete orders	Orders at $\sin\theta_m = m\lambda/\Lambda$
Edge	Half-period Fresnel	Oscillating intensity near geometric shadow edge

Resolution limits. Diffraction sets the fundamental resolution limit of optical instruments:

• Microscope resolution (Abbe limit): $d_\text{min} = \lambda / (2 \text{NA})$ for incoherent illumination
• Telescope angular resolution (Rayleigh criterion): $\theta_\text{min} = 1.22 \lambda / D$ for circular aperture
• Lens spot size: $d_\text{min} = 1.22 \lambda f / D$ where $f$ is focal length and $D$ is aperture diameter
• Fiber acceptance: limited by numerical aperture; related to diffraction at the fiber input

For visible light at $\lambda = 550$ nm:

• Microscope at NA = 1.4 (oil immersion): $d_\text{min} \approx 200$ nm
• Telescope, 8-inch (200 mm) aperture: $\theta_\text{min} \approx 0.7$ arcsec
• Diffraction-limited spot of 100 mm focal lens with 25 mm aperture: $d_\text{min} \approx 2.7$ μm

Diffraction grating spectrometry. A periodic structure with period $\Lambda$ diffracts an incident wave into discrete orders at angles $\sin\theta_m = m\lambda/\Lambda$. Different wavelengths diffract to different angles — providing the principle of grating spectrometers. The resolving power is:

$$R \;=\; \frac{\lambda}{\Delta\lambda} \;=\; m N,$$

where $N$ is the total number of illuminated grating lines. A high-quality grating ($N \sim 10^5$, $m \sim 1$) resolves $\Delta\lambda \sim 5$ pm at 500 nm — sufficient for atomic spectroscopy.

Diffraction in laser beam propagation. A laser beam of waist $w_0$ propagates with diffraction-limited divergence:

$$\theta \;=\; \frac{\lambda}{\pi w_0}.$$

A 1 mm-waist beam at 1550 nm diverges at $\sim 0.5$ mrad — visible spreading over $\sim 10$ m. To collimate the beam (small divergence), large beam diameter is needed; this is why telescopes and beam expanders are used in long-distance laser applications.

Diffraction in fiber and waveguide optics. Light emerging from a small-mode-field-diameter fiber diverges rapidly. SMF-28 with MFD 10.4 μm emits with half-angle:

$$\theta \;\approx\; \frac{\lambda}{\pi \cdot \text{MFD}/2} \;=\; \frac{1.55}{\pi \cdot 5.2} \;\approx\; 95 \text{ mrad} = 5.4°.$$

This sets the requirement for the focal length of any external collimating optics.

Anti-aliasing / sub-wavelength imaging. Many modern imaging techniques (STED, STORM, PALM) circumvent the diffraction limit by exploiting molecular states or sparse-sampling techniques — they do not violate diffraction theory but cleverly bypass it.

References: Saleh & Teich, Fundamentals of Photonics (3rd ed., 2019), Ch. 4 (Fourier optics); Goodman, Introduction to Fourier Optics (3rd ed., 2005) for the canonical comprehensive treatment; Born & Wolf, Principles of Optics (7th ed., 1999), Ch. 8 for the rigorous formalism.