Interference and diffraction
Diffraction does not allow drawing details much smaller than the wavelength used to draw them. That is why the chip industry moved from 436 nm to 13.5 nm, and each jump cost a fortune.
01Two slits
Young’s experiment is one of the cleanest pieces of evidence that light is a wave. Two slits illuminated by the same source behave like two emitters in phase, and on the screen not two spots appear but a succession of fringes.
The decisive point is that coherence is required: the two slits have to emit with a stable phase relationship. That is why the experiment is done by illuminating both with the same source, and why two different lamps never produce fringes: their phases change randomly millions of times per second and the pattern averages out until it disappears.
02A single slit does something too
A slit of finite width diffracts: the light spreads out as it passes, and a pattern appears with a wide central maximum and minima on either side. It seems the opposite of what one would expect —the narrower the slit, the wider the beam— and it comes from the same superposition.
In a real setup both things happen at once: the interference between slits draws the fine fringes and the diffraction of each slit modulates them, giving them an envelope. That is why in a double-slit pattern some fringes are missing: they fall exactly where the envelope is zero.
The complete pattern, with the diffraction envelope drawn separately. Increase the number of slits and watch how the maxima sharpen without moving: that is what turns a grating into a spectrometer.
03The diffraction grating
With many slits instead of two, the maxima are still where \( d\sin\theta = m\lambda \) —the position does not change— but they become much narrower. And since their position depends on \( \lambda \), each color emerges in a different direction: that is a spectrometer.
The tracks of a disc are 1.6 µm apart on a CD and 0.74 µm on a DVD, which is the same order as the wavelength of light. It works as a reflection diffraction grating: each color is deflected differently and that is why the rainbow appears. The DVD, with tracks closer together, spreads out more, and that difference can be seen by eye by putting the two side by side.
04The limit that diffraction imposes
Every aperture diffracts, so the image of a point is never a point: it is a disk with rings. Two objects can be told apart as long as their disks do not overlap too much, and the Rayleigh criterion sets the threshold:
| Instrument | \( D \) | Limit | Consequence |
|---|---|---|---|
| Human pupil | 3 mm | 0.013° | Telling apart two headlights at about 6 km |
| 200 mm telescope | 200 mm | 0.7 arcseconds | Separating double stars |
| Deep-UV photolithography | — | Tens of nm | The minimum size of a transistor |
The last row is the reason the chip industry has always run toward shorter wavelengths: from 436 nm to 365, to 248, to 193 nm, and from there to the extreme ultraviolet of 13.5 nm. Each jump was enormously expensive and each was made for the same reason: diffraction does not allow drawing details much smaller than the wavelength used to draw them.
05In the lab
Project onto a distant wall the pattern of a laser through a double slit of known separation. Measure the fringe spacing and solve for \( \lambda \). Compare with the stated wavelength of the laser.
Use a CD as a reflection grating with a laser of known wavelength. Measure the angle of the first order and solve for the track spacing. Repeat with a DVD and compare.
Illuminate a hair with a laser and measure the diffraction pattern it produces. Apply the slit formula —a thin obstacle gives the same pattern, by Babinet’s principle— and estimate its diameter.
06Common mistakes
- Believing that two lamps can be made to interfere. Coherence is required: the same source, a stable phase relationship.
- Confusing \( d \) with \( a \). One is the separation between slits; the other, the width of each one.
- Expecting the beam to narrow when the slit is narrowed. It spreads: it is the other way around.
- Thinking that more slits shift the maxima. They do not move them: they sharpen them.
- Attributing the resolution limit to the lenses. It is diffraction, and no lens overcomes it.
- Forgetting the diffraction envelope. It is what explains the fringes that are missing.
07Self-assessment
What condition is needed to see interference fringes?
Coherence: the two sources must keep a stable phase relationship. That is why a single source is used for the two slits.
What happens to the fringes if the slits are brought closer together?
They move apart. The spacing is \( \lambda L/d \), inversely proportional to \( d \).
What changes when the number of slits is increased?
The maxima become much sharper but do not move. That is what makes a diffraction grating useful.
Why are some fringes missing in a double-slit pattern?
Because they fall where the diffraction envelope of each slit is zero.
What does the Rayleigh criterion say?
That the minimum resolvable angle is \( 1.22\lambda/D \). It is a physical limit, not one of optical quality.
Why did photolithography move to shorter wavelengths?
Because diffraction does not allow drawing details much smaller than the wavelength used.
08Further reading
- Eugene Hecht. Optics. 5th ed. (Spanish edition, “Óptica”), Pearson, 2017. Chapters 9 and 10: interference, coherence, Fraunhofer diffraction and gratings.
- Raymond A. Serway and John W. Jewett. Physics for Scientists and Engineers with Modern Physics. 10th ed. (Spanish edition, “Física para ciencias e ingeniería”), Cengage, 2019. Young, thin films, the single slit, gratings and the Rayleigh criterion with solved problems.
- Chris Mack. Fundamental Principles of Optical Lithography. Wiley, 2007. How the diffraction limit governs the minimum size of a transistor, and what is done to stretch it.