10.3 Diffraction

Wave nature of light

Diffraction only makes sense once you accept that light behaves as a wave. When light encounters an obstacle or passes through an opening, it doesn't just travel in straight lines. It bends, spreads, and overlaps with itself, producing patterns that only wave behavior can explain.

Huygens' principle

Huygens' principle is the starting point for understanding diffraction. It states that every point on a wavefront acts as a source of tiny secondary wavelets. These wavelets spread out in all directions and combine to form the new wavefront.

This idea explains why waves can bend around corners and spread after passing through a slit. It applies to all types of waves: light, sound, and water waves alike.

Young's double-slit experiment

Young's double-slit experiment provided some of the first strong evidence that light is a wave. The setup is straightforward: shine light through two narrow, closely spaced slits and observe the pattern on a distant screen.

Instead of two bright spots (which you'd expect if light were purely a particle), you see alternating bright and dark fringes. Bright fringes appear where waves from the two slits arrive in phase and constructively interfere. Dark fringes appear where they arrive out of phase and cancel.

The wavelength of light can be calculated from the fringe pattern:

$\lambda = \frac{yd}{L}$

• $y$ = fringe spacing on the screen
• $d$ = distance between the two slits
• $L$ = distance from the slits to the screen

Interference vs diffraction

These two terms come up together constantly, so it's worth being clear about the distinction:

• Interference is the superposition of waves from discrete, separate sources (like two slits). It produces sharp, evenly spaced fringes.
• Diffraction is the bending and spreading of waves when they encounter an obstacle or aperture. It produces broader patterns with a dominant central maximum.

In practice, the two always occur together. A double-slit experiment, for example, shows an interference pattern modulated by a single-slit diffraction envelope. The interference controls the fine fringe spacing; the diffraction controls the overall intensity envelope.

Single-slit diffraction

When light passes through a single narrow slit, it spreads out and produces a characteristic pattern on a distant screen: a bright central band flanked by progressively dimmer side fringes.

Intensity distribution

The intensity at any angle $\theta$ from the center is given by:

$I = I_0 \left(\frac{\sin \alpha}{\alpha}\right)^2$

where

$\alpha = \frac{\pi a \sin \theta}{\lambda}$

• $a$ = slit width
• $\theta$ = angle from the central axis
• $\lambda$ = wavelength of light

This is the sinc-squared function. It peaks at $\alpha = 0$ (the center) and drops off with oscillations on either side. The intensity of higher-order maxima falls off rapidly.

Central maximum

The central maximum is the brightest part of the pattern, centered directly in line with the slit. It contains about 84% of the total diffracted light energy.

A key relationship: the central maximum gets wider as the slit gets narrower. For small angles, the full angular width of the central maximum is:

$\Delta\theta = \frac{2\lambda}{a}$

This inverse relationship between slit width and pattern width is a hallmark of diffraction. A very narrow slit produces a very broad central peak.

Secondary maxima and minima

On either side of the central maximum, you'll find alternating dark and bright bands.

Minima (dark fringes) occur at angles satisfying:

$a \sin \theta = m\lambda \quad (m = \pm 1, \pm 2, \pm 3, \ldots)$

Note that $m = 0$ is not a minimum; it's the center of the bright central maximum.

Secondary maxima sit roughly halfway between adjacent minima. Their intensities drop off quickly:

• First secondary maximum: about 4.7% of the central maximum intensity
• Second secondary maximum: about 1.7%

This rapid falloff is why the central maximum dominates the pattern visually.

Multiple-slit diffraction

When light passes through many slits instead of one, the pattern sharpens dramatically. You get the broad single-slit diffraction envelope combined with narrow, bright interference peaks.

Diffraction gratings

A diffraction grating is an optical element with hundreds or thousands of equally spaced parallel slits (or grooves). The large number of slits produces very sharp, well-separated bright lines called principal maxima.

The grating equation gives the angles where these bright lines appear:

$d \sin \theta = m\lambda$

• $d$ = spacing between adjacent slits (the grating spacing)
• $m$ = order number (0, ±1, ±2, ...)
• $\lambda$ = wavelength

Because different wavelengths diffract to different angles, a grating separates white light into its component colors. This is why gratings are the core component in spectrometers.

Gratings come in two main types: transmission gratings (light passes through) and reflection gratings (light bounces off grooved surfaces).

Resolving power

The resolving power of a grating tells you how well it can distinguish two wavelengths that are very close together. It's defined as:

$R = \frac{\lambda}{\Delta\lambda} = mN$

• $\Delta\lambda$ = smallest wavelength difference the grating can resolve
• $m$ = diffraction order
• $N$ = total number of slits illuminated

Two ways to increase resolving power: use more slits, or work in a higher diffraction order. A grating with 5000 slits operating in second order has $R = 2 \times 5000 = 10{,}000$, meaning it can resolve wavelengths differing by as little as 1 part in 10,000.

Blazed gratings

A standard grating spreads light across many orders, which wastes energy. A blazed grating has grooves cut at a specific angle (the blaze angle) so that most of the diffracted light is concentrated into one particular order.

This makes blazed gratings much more efficient. They're the standard choice in spectrometers and monochromators where you need maximum signal in a specific spectral order.

Circular aperture diffraction

Real optical instruments like telescopes, cameras, and microscopes use circular lenses and apertures. When light passes through a circular opening, the diffraction pattern is circular too: a bright central disk surrounded by faint concentric rings.

Airy disk

The central bright spot in a circular aperture diffraction pattern is called the Airy disk. It contains about 84% of the total light, with the remaining 16% spread across the surrounding rings.

The radius of the first dark ring (the edge of the Airy disk) is:

$r = 1.22 \frac{f\lambda}{D}$

• $f$ = focal length of the lens
• $\lambda$ = wavelength
• $D$ = diameter of the aperture

A larger aperture produces a smaller Airy disk, which means sharper images.

Rayleigh criterion

The Rayleigh criterion defines the minimum angular separation at which two point sources (like two stars) can be distinguished. Two sources are considered "just resolved" when the central maximum of one falls on the first minimum of the other.

This gives a minimum resolvable angle of:

$\theta_{min} = 1.22 \frac{\lambda}{D}$

For example, a telescope with a 10 cm aperture observing visible light ($\lambda \approx 550$ nm) has a resolution limit of about $6.7 \times 10^{-6}$ radians, or roughly 1.4 arcseconds.

Angular resolution

Angular resolution is the practical measure of how much detail an optical system can distinguish. Smaller values mean better resolution.

Two factors control it:

• Larger aperture $\rightarrow$ better (smaller) angular resolution
• Shorter wavelength $\rightarrow$ better (smaller) angular resolution

This is why radio telescopes need to be enormous (radio waves have long wavelengths) and why electron microscopes can resolve atomic-scale features (electrons have very short de Broglie wavelengths).

X-ray diffraction

X-rays have wavelengths on the order of 0.1 nm, which is comparable to the spacing between atoms in a crystal. This makes X-ray diffraction a powerful tool for probing atomic structure.

Bragg's law

When X-rays hit a crystal, they scatter off parallel planes of atoms. Constructive interference occurs when the path difference between rays reflecting from adjacent planes equals a whole number of wavelengths:

$2d \sin \theta = n\lambda$

• $d$ = spacing between crystal planes
• $\theta$ = angle of incidence (measured from the plane, not the normal)
• $n$ = integer (1, 2, 3, ...)

By measuring the angles at which strong reflections occur, you can determine the spacing between atomic planes and work out the crystal structure.

Crystal structure analysis

X-ray crystallography uses the full set of diffraction angles and intensities to reconstruct where atoms sit within a crystal. The process involves:

1. Illuminate a crystal with a monochromatic X-ray beam
2. Record the pattern of diffracted spots on a detector
3. Measure the angles and intensities of each spot
4. Use mathematical techniques (including Fourier transforms) to build an electron density map
5. Determine atomic positions from the density map

This technique has been used to determine the structures of everything from simple salts to DNA and complex proteins.

Powder diffraction method

Not all samples come as nice single crystals. The powder diffraction method handles polycrystalline materials by grinding them into a fine powder. With millions of tiny crystallites oriented randomly, every possible Bragg angle is represented simultaneously.

Instead of discrete spots, the pattern consists of concentric rings (called Debye-Scherrer rings). Each ring corresponds to a specific set of crystal planes. This technique is especially useful for:

• Identifying unknown crystalline materials by matching ring patterns to databases
• Detecting different phases in a mixed sample
• Applications in materials science, geology, and pharmaceutical quality control

Applications of diffraction

Spectroscopy

Diffraction gratings are at the heart of most spectrometers. By dispersing light into its component wavelengths, they allow precise measurement of spectral lines. This is used in chemical analysis (identifying elements by their emission spectra), astronomy (determining the composition and motion of stars), and medical diagnostics.

Optical instruments

Every lens-based instrument is ultimately limited by diffraction. Telescope designers use the largest feasible apertures to push angular resolution as far as possible. Microscope designers maximize numerical aperture for the same reason. Techniques like adaptive optics (which correct for atmospheric turbulence in real time) and phase contrast microscopy both rely on a deep understanding of how diffraction affects image formation.

Holography

Holography records not just the intensity of light (like a photograph) but also its phase information. A laser illuminates an object, and the reflected light interferes with a reference beam on a recording medium. The resulting interference pattern encodes a full 3D representation of the object.

When the hologram is illuminated again, diffraction reconstructs the original wavefront, producing a three-dimensional image. Applications include data storage, security features on credit cards and banknotes, and 3D display technology.

Fresnel vs Fraunhofer diffraction

Diffraction patterns look different depending on how far you are from the aperture. The two limiting cases have different names and require different mathematical approaches.

Near-field diffraction

Fresnel diffraction occurs when the observation screen is relatively close to the aperture (or when the aperture is large). In this regime:

• The wavefronts reaching the screen are curved, not flat
• The patterns are complex and change shape as you move the screen
• The math requires Fresnel integrals, which are more involved

You'll encounter Fresnel diffraction when working close to a diffracting edge or with large apertures.

Far-field diffraction

Fraunhofer diffraction occurs when the screen is very far from the aperture (or equivalently, when a lens focuses the diffracted light). In this regime:

• The wavefronts are effectively flat (plane waves)
• The patterns are stable and don't change shape with distance
• The math simplifies to Fourier transforms of the aperture function

All the single-slit and grating formulas discussed earlier in this guide assume Fraunhofer conditions. Most practical optical systems operate in this regime.

Transition zone

The boundary between Fresnel and Fraunhofer regimes isn't sharp. A useful estimate for the transition distance is:

$z \approx \frac{a^2}{\lambda}$

where $a$ is the aperture size. For distances much greater than this, you're safely in the Fraunhofer regime. For distances much less, you're in the Fresnel regime. In between, the pattern gradually transitions from one to the other.

Diffraction limits

Optical resolution

Every imaging system has a fundamental resolution limit set by diffraction. No matter how perfectly you grind a lens, you can't focus light to a point smaller than the Airy disk. The resolution depends on the wavelength of light and the numerical aperture (NA) of the system.

Microscopy limitations

For a microscope, the smallest resolvable feature size is approximately:

$d_{min} = \frac{\lambda}{2 \cdot NA}$

where $NA = n \sin \theta$ ($n$ is the refractive index of the medium, $\theta$ is the half-angle of the cone of light entering the objective).

For visible light and the best oil-immersion objectives ($NA \approx 1.4$), this gives a resolution limit around 200 nm. To see smaller features, you need shorter wavelengths. Electron microscopes achieve sub-nanometer resolution because electrons can have wavelengths thousands of times shorter than visible light.

Super-resolution techniques like STED and PALM/STORM cleverly circumvent the classical diffraction limit using fluorescence tricks, but the underlying physics of diffraction still applies.

Telescope performance

For telescopes, diffraction limits the angular resolution (as described by the Rayleigh criterion). Larger mirrors or lenses yield finer resolution. Space-based telescopes like Hubble avoid atmospheric blurring entirely, allowing them to operate near their diffraction limit. Ground-based telescopes use adaptive optics to correct atmospheric distortion in real time, approaching diffraction-limited performance.

Mathematical treatment

Fraunhofer diffraction formula

For a single slit of width $a$, the electric field amplitude in the far field is:

$E(\theta) = E_0 \frac{\sin(ka \sin \theta / 2)}{ka \sin \theta / 2}$

where $k = \frac{2\pi}{\lambda}$ is the wave number. The intensity is proportional to the square of this expression, giving the sinc-squared pattern discussed earlier. This formula is derived by summing contributions from all points across the slit using the Huygens-Fresnel principle, under the assumption that the screen is far away.

Fourier analysis

There's an elegant connection between diffraction and Fourier transforms: the far-field diffraction pattern of any aperture is the Fourier transform of the aperture's transmission function. A rectangular slit gives a sinc pattern. A circular aperture gives an Airy pattern. A grating (periodic array of slits) gives sharp peaks at discrete angles.

This Fourier relationship is a powerful tool for designing diffractive optical elements and predicting patterns from complex aperture shapes.

Diffraction patterns

The key mathematical functions you'll encounter:

• Single slit: sinc-squared function, $I \propto \left(\frac{\sin \alpha}{\alpha}\right)^2$
• Double slit: product of the single-slit envelope and a cosine-squared interference term
• Diffraction grating: product of the single-slit envelope and a sharply peaked function with $N$ slits
• Circular aperture: Airy function involving first-order Bessel functions

More complex apertures can be analyzed by breaking them into simpler components using convolution, then applying Fourier techniques to predict the resulting pattern.

Diffraction in nature

Structural coloration

Many of the most vivid colors in nature come not from pigments but from nanoscale structures that diffract light. Butterfly wings, peacock feathers, and certain beetle shells all contain periodic structures with spacings on the order of visible light wavelengths. These structures act like natural diffraction gratings, selectively reinforcing certain wavelengths and producing iridescent, angle-dependent colors.

Atmospheric phenomena

Diffraction plays a role in several familiar atmospheric effects. The corona (a colored ring sometimes visible around the sun or moon) is caused by diffraction of light by small water droplets in thin clouds. Glories, the colorful rings sometimes seen around an airplane's shadow on clouds, also arise from diffraction.

Note: the blue color of the sky is primarily due to Rayleigh scattering, which is a related but distinct phenomenon from diffraction.

Biological structures

Some organisms have evolved structures that exploit diffraction. Moth eyes, for example, have nanoscale bumps that create a graded refractive index, reducing reflection (an antireflective "coating" produced by diffraction effects). Certain deep-sea organisms use diffraction-grating-like structures for camouflage or signaling. These biological examples demonstrate that the physics of diffraction has practical relevance well beyond the laboratory.