Diffraction

Diffraction is the spreading of a wave as it passes the edge of an obstacle or goes through an opening. Through an opening much wider than its wavelength a wave travels on almost in a straight line, spreading a little at the edges; through an opening comparable to its wavelength it fans out over a wide angle. The angle of spread is roughly the wavelength divided by the width of the opening, $\lambda \mathrm{/}a$ in radians. A 1 kHz tone, whose wavelength in air is 34 cm, passing through a 90 cm doorway spreads over about 0.38 rad, more than twenty degrees, which is why sound is heard around a corner; green light at 550 nm passing through the same doorway spreads by $6\times {10}^{-7}$ rad, and the edge of its shadow looks sharp. Every kind of wave diffracts: water waves, sound, light, X-rays and electrons. For light, diffraction sets the smallest spot a lens can focus and the finest detail a microscope or telescope can resolve.

Why diffraction happens

In the Huygens–Fresnel picture, every point on a wavefront acts as a source of secondary wavelets, and the wave farther on is their sum. For an unobstructed plane wave the wavelets add up to a plane wave moving straight ahead and cancel in every other direction. An aperture or obstacle removes part of the wavefront, and the wavelets from the remaining part no longer cancel off axis. For a slit of width $a$, the wavelets from its upper and lower halves arrive out of step, and cancel in pairs, at the angle where the path difference between the two edges, $a\sin \theta$, equals one wavelength. That is the first dark fringe, at $\sin \theta =\lambda \mathrm{/}a$, and the narrower the slit, the wider the angle.

Diffraction is therefore interference among the parts of a single wavefront. The physics is the same superposition in both cases; by convention "interference" is used for a few separate beams and "diffraction" for a continuous wavefront cut off by an edge or aperture.

Diffraction and refraction

Refraction and diffraction both change the direction of a wave, for different reasons. Refraction happens at a boundary between two media in which the wave travels at different speeds and follows Snell's law, $n_1\sin {\theta }_1=n_2\sin {\theta }_2$; the size of the boundary does not enter. Diffraction happens within one medium whenever part of a wavefront is blocked, and its angle depends on the ratio of wavelength to aperture. Reflection, the third way a wave changes direction, returns it into the medium it came from at an angle equal to the angle of incidence.

A prism and a grating both spread white light into a spectrum, and the order of the colors tells the two effects apart: a glass prism, by refraction, deviates violet the most, while a grating, by diffraction, deviates red the most.

Examples of diffraction

• Sound around a doorway. The 1 kHz example above. Low notes have longer wavelengths and spread more than high ones, so music heard around the corner from an open door sounds muffled.
• The colors of a CD or DVD. The spiral track acts as a reflection grating. With the CD's 1.6 µm track pitch, first-order light at normal incidence leaves at 16° for 450 nm blue and at 24° for 650 nm red; a DVD, with a 0.74 µm pitch, sends the same colors out between 37° and 61°.
• A laser pointer through a slit or past a hair. A 650 nm beam through a 100 µm slit makes a central bright band 26 mm wide on a screen 2 m away, flanked by fainter bands. A hair of the same width gives the same pattern outside the central spot (Babinet's principle), a common classroom way to measure its thickness.
• Star images in a telescope. A star is imaged as an Airy pattern, a bright disk with faint rings, and by the Rayleigh criterion a 200 mm telescope just resolves two stars 0.69 arcsecond apart at 550 nm. The struts that hold a reflecting telescope's secondary mirror add diffraction spikes.
• Softer photographs at small apertures. The Airy disk of a camera lens has a diameter of $2.44\lambda N$ at f-number $N$: for green light, 5.4 µm at f/4 and 21.5 µm at f/16. On a sensor with 4 µm pixels the f/16 disk spans about 5.4 pixels, and fine detail is lost.
• X-ray diffraction by crystals. Atomic planes a few tenths of a nanometer apart diffract X-rays of similar wavelength at angles set by the Bragg condition, $2d\sin \theta =m\lambda$: copper Kα X-rays (0.154 nm) reflect from the silicon (111) planes ($d$ = 0.3136 nm) at 14.2°, which is how X-ray diffraction determines crystal structures.
• Electron diffraction. At 54 eV an electron has a wavelength of 0.167 nm. In 1927 Davisson and Germer observed such electrons diffracted by a nickel crystal, confirming de Broglie's proposal that matter has wave properties.

Fresnel and Fraunhofer diffraction

Close to an aperture the pattern changes shape with distance, which is Fresnel, or near-field, diffraction; far away it keeps its shape and only grows in proportion to distance, which is Fraunhofer, or far-field, diffraction. The Fresnel number $F=a^2\mathrm{/}(\lambda z)$, for an aperture of radius $a$ viewed from a distance $z$, separates them: Fresnel diffraction applies for $F\gtrsim 1$ and Fraunhofer diffraction for $F\ll 1$. For a 1 mm hole ($a$ = 0.5 mm) at 550 nm, $a^2\mathrm{/}\lambda$ is 0.45 m, so the far-field pattern forms well beyond half a meter. A lens brings the far field to its focal plane, so the focal spot of a lens is the Fraunhofer pattern of its aperture.

In the Fraunhofer regime the field is the Fourier transform of the aperture function $A(x^{\mathrm{\prime }},y^{\mathrm{\prime }})$:

$$I(x,y)\propto {\mid \iint A(x^{\mathrm{\prime }},y^{\mathrm{\prime }})e^{-i2\pi (x{x}^{\mathrm{\prime }}+y{y}^{\mathrm{\prime }})\mathrm{/}(\lambda z)}d{x}^{\mathrm{\prime }}d{y}^{\mathrm{\prime }}\mid }^2,$$

evaluated at the spatial frequencies $f_x=x\mathrm{/}(\lambda z)$ and $f_y=y\mathrm{/}(\lambda z)$. Fourier optics builds on this result.

Common diffraction patterns

The slit diffraction calculator draws the far-field pattern of one, two or many slits, with the minima, fringe spacing and missing orders on a screen at a chosen distance.

Resolution and the diffraction limit

A lens images every point as a small diffraction pattern rather than a point, so diffraction sets the finest detail an optical instrument can resolve, its diffraction limit:

• Microscope. Abbe's limit $d=\lambda \mathrm{/}(2\text{NA})$ is the finest periodic detail an objective of numerical aperture NA can image when the condenser fills its aperture: 196 nm at 550 nm and NA 1.4. Two point sources must be $0.61\lambda \mathrm{/}\text{NA}$ apart by the Rayleigh criterion, 240 nm.
• Telescope. The angular resolution is $1.22\lambda \mathrm{/}D$: 0.69 arcsecond for a 200 mm aperture at 550 nm.
• Focused spot. The radius to the first dark ring is $1.22\lambda f\mathrm{/}D$: 2.7 µm for a 100 mm lens at 25 mm aperture (f/4) and 550 nm, a diameter of 5.4 µm. The Airy Disk Calculator evaluates these for any wavelength and aperture.

Diffraction gratings

A diffraction grating, a surface carrying many parallel grooves, sends each wavelength into orders at angles $\sin {\theta }_m=m\lambda \mathrm{/}\mathrm{\Lambda }$ at normal incidence, so a spectrometer reads wavelength from angle. Its resolving power is

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

where $N$ is the number of illuminated grooves. The sodium D lines at 589.0 and 589.6 nm need $R$ of about 980, so about a thousand illuminated grooves separate them in first order; a grating with $N={10}^5$ resolves 5 pm at 500 nm in first order. The Diffraction Grating Calculator computes order angles, dispersion and spectrometer resolution.

Laser beams and fibers

A laser beam is limited by diffraction in the same way. A Gaussian beam with waist radius $w_0$ spreads at the half-angle

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

A beam of 1 mm waist radius at 1550 nm diverges at 0.49 mrad; its Rayleigh range is 2.0 m, and 10 m from the waist its radius is 5.0 mm. A wider beam spreads less, which is why beam expanders are used before long free-space paths. Light leaving a single-mode fiber diverges quickly for the same reason: SMF-28, with a 10.4 µm mode field diameter ($w_0$ = 5.2 µm), emits into a half-angle of 95 mrad (5.4°) at 1550 nm, which sets the focal length of a lens that collimates it. The Gaussian Beam Calculator computes divergence, Rayleigh range and the focused waist behind a lens.

Beyond the diffraction limit

Super-resolution fluorescence microscopy (STED, PALM and STORM, recognized by the 2014 Nobel Prize in Chemistry) images cells at a resolution of tens of nanometers. These methods do not focus light more tightly than diffraction allows. They switch fluorescent molecules on and off so that neighboring molecules do not emit at the same time: STED confines emission to a spot smaller than the diffraction limit, and PALM and STORM locate each isolated molecule to better than the width of its diffraction spot. Near-field scanning microscopes avoid the limit in a different way, by collecting light within a fraction of a wavelength of the surface, before it has spread.

Common questions

What is diffraction in simple terms?

The spreading of a wave when it passes an edge or goes through a gap. It is strongest when the gap is about as wide as the wavelength, which is why it is obvious for sound and water waves, while with light it takes a narrow slit or a laser to see it clearly.

What is the difference between diffraction and interference?

Interference is the adding of waves that overlap; diffraction is the spreading of a single wave limited by an aperture, and its pattern is the interference of the wavelets from the open part of the wavefront. The double-slit experiment shows both: each slit's diffraction sets a broad envelope, and interference between the two slits sets the fine fringes inside it.

Why does sound diffract more than light?

Diffraction depends on the ratio of wavelength to the size of the opening. Audible sound has wavelengths from about 17 mm to 17 m, comparable to doors and furniture; visible light has wavelengths from about 380 to 750 nm, 460,000 to 900,000 times shorter than the 34 cm wavelength of a 1 kHz tone.

Does diffraction depend on wavelength?

Yes. The diffraction angle is proportional to the wavelength, so red light spreads more than blue, a grating deviates red the most, and shorter wavelengths focus to smaller spots, which is why photolithography moved to 193 nm and then 13.5 nm light.

References: E. Hecht, Optics, 5th ed. (Pearson, 2017), ch. 10. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2005). B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019), ch. 4. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999), ch. 8. C. Davisson and L. H. Germer, "Diffraction of electrons by a crystal of nickel," Physical Review 30, 705 (1927). IEC 60908 and ECMA-267 for the CD and DVD track pitches.