Hair Diffraction Calculator

Background.

Shine a laser pointer at a single strand of hair and a wall a couple of metres away shows not a crisp shadow but a row of bright and dark bands spreading sideways from the beam. The hair is diffracting the light, and how far those bands spread depends on how thick the hair is: the thinner the strand, the wider the pattern. That turns a width of a few hundredths of a millimetre into a pattern you can measure with a tape measure.

The method rests on Babinet's principle. A narrow opaque strip and a slit of the same width are complementary screens, and away from the direct beam they produce the same diffraction pattern. The hair therefore behaves like a single slit, whose dark fringes fall where d sin θ = mλ for whole numbers m. At the small angles involved, neighbouring dark fringes on the screen sit Δy = λL ÷ d apart, which rearranges to d = λL ÷ Δy.

Two of the three inputs are easy. The wavelength is usually printed on the laser's warning label, and the screen distance is a tape-measure job. The fringe spacing needs care, because the central bright band is twice as wide as the others. Measure between matching dark fringes on either side of it and divide by twice their order, rather than judging a single gap by eye.

What comes out is a small-angle, far-field estimate. It treats the hair as a straight strip of uniform width square to the beam, whereas real hairs taper, are often slightly oval in cross-section and drift out of alignment. No uncertainty is attached to the figure, so repeat the measurement at a few points along the strand if the number matters.

What is hair diffraction calculator?

Hair diffraction is the spreading of a laser beam around a thin fibre into a pattern of alternating bright and dark fringes. Measuring the distance between neighbouring dark fringes, together with the laser wavelength and the hair-to-screen distance, gives the hair's diameter through the single-slit relation d ≈ λL ÷ Δy. All inputs and the result are in metres. It is a classic teaching experiment because it turns a microscopic width into a centimetre-scale pattern.

How to use this calculator.

1. 01Enter the Laser wavelength in metres. Red diode pointers are commonly 650 nm or 635 nm (6.5e-7 or 6.35e-7 m), a helium-neon laser is 632.8 nm and a green pointer 532 nm; check your laser's label.
2. 02Enter the Screen distance from the hair to the wall in metres. Two metres or more spreads the fringes enough to measure comfortably.
3. 03For the Adjacent dark-fringe spacing, measure from the nth dark fringe on one side of the central band to the nth on the other, then divide by 2n. A 12 cm span between the third dark fringes on each side means 0.02 m.
4. 04Read the hair diameter in metres and multiply by 1,000,000 for micrometres.

The formula.

d = λL ÷ Δy, where λ is the laser wavelength, L the distance from hair to screen and Δy the spacing between adjacent dark fringes, all in metres. It comes from the single-slit condition for darkness, d sin θ = mλ. Under the small-angle approximation sin θ ≈ tan θ = y ÷ L, the mth dark fringe sits at y = mλL ÷ d, so neighbouring dark fringes are λL ÷ d apart; Babinet's principle lets the opaque hair stand in for the slit. The diameter is inversely proportional to the fringe spacing, so misreading a 20 mm spacing by 1 mm shifts the answer by about 5%. The approximation is excellent at the angles a hair produces, about 0.01 radian in the example, and the setup is in the far field when L is much larger than d² ÷ λ, only a few millimetres for a hair.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, University Physics Volume 3, section 1.4, Diffraction.
2. [2]Hecht, Optics, 5th ed. (PRINT).
3. [3]BIPM, The International System of Units (SI Brochure), 9th ed., version 3.01, coherent derived units and quantity equations.