Hair Diffraction Calculator
Shine a laser at a single strand of hair and a pattern of bright and dark bands appears on the wall behind it. This calculator uses the single-slit diffraction equation to turn those measurements into your hair diameter - or to reverse-solve for the fringe position, screen distance, or wavelength you need for a lab experiment. Choose a solve mode, enter three of the four variables, and the fourth is calculated instantly with a full worked-out solution.
Formula
Worked example
Red laser (650 nm), screen 100 cm away, 1st dark fringe at 10 mm from center: d = (1 x 650e-9 x 1.00) / 0.010 = 65 micrometers. Angle = arcsin(650nm / 65um) = arcsin(0.01) = 0.57 degrees, well within the small-angle limit. Max order = floor(65000/650) = 100 fringes theoretically, though only the closest few are bright enough to see.
How hair diffraction works
When a narrow laser beam strikes a single human hair, the hair acts as a thin obstacle. By Babinet's principle in wave optics, the diffraction pattern produced by an opaque obstacle is identical to the pattern produced by a slit of the same width. The laser light bends around both edges of the hair and the two wavefronts interfere on a screen behind it. Where the path difference equals a whole number of wavelengths, you get destructive interference (dark fringes). The spacing of those dark bands tells you how wide the hair was. The governing equation is d sin(theta) = m * lambda, where d is the hair diameter, theta is the angle to the m-th dark fringe, and lambda is the laser wavelength. When the screen is far away relative to the fringe spacing (small-angle limit, theta < about 10 degrees), sin(theta) is very close to tan(theta) = x / D, giving the simpler working formula d = m * lambda * D / x.
Setting up the experiment at home or in the lab
You need a laser pointer (a cheap red 650 nm pointer works fine), a single hair taped across a gap in a card or clamped between two books, a measuring tape, and a ruler or caliper. Shine the laser through the hair onto a flat wall or a sheet of white paper. The wall should be at least 50 cm away - 1 m or more gives a larger, more measurable pattern. Measure the distance from the hair to the wall (D) with a tape measure. Mark the center of the bright central spot, then measure the distance to the first (or second, third...) dark band on either side (x). Use the highest-order dark fringe you can cleanly see, because a larger x means the ruler measurement carries less relative error. Average the distances on both sides to cancel any tilt. Enter these three values and the laser wavelength (printed on the safety label) into the calculator to get the hair diameter.
Improving accuracy and reducing error
The single biggest source of error in this experiment is reading x with a ruler against a fuzzy fringe. Using higher orders (m = 2, 3) multiplies x, making the ruler reading more precise relative to the quantity being measured. Averaging multiple hairs from the same person helps because hair diameter varies along its length and between strands. Using a caliper instead of a ruler can resolve fringe positions to 0.1 mm. The small-angle approximation introduces less than 0.5% error when theta is below 10 degrees and less than 2% below 15 degrees; the calculator flags when this limit is approached so you know when to apply the exact arcsin formula. Temperature and humidity do not significantly affect visible-laser wavelengths, but keeping the paper screen flat and perpendicular to the laser reduces geometric distortion.
Six solve modes and when to use each
The default mode (solve for hair diameter) covers the classic lab experiment. The other five modes handle planning and reverse problems. Solve for wavelength if you have a hair of known diameter (measured under a microscope) and want to calibrate an unknown laser. Solve for screen distance if you need a specific fringe spacing for photography or a demonstration and want to know how far back to place the screen. Solve for fringe position predicts where a dark band will land before you run the experiment - useful for checking that it falls within your ruler range. Maximum observable order tells you how many dark fringes can theoretically appear: because sin(theta) cannot exceed 1, the highest possible order is floor(d / lambda). Central maximum width gives the full extent of the bright central blob on the screen, which is 2 * lambda * D / d.
Human hair diameter categories
| Type | Diameter range (μm) | Common in |
|---|---|---|
| Very fine / vellus | 17-40 | Infant hair, facial peach fuzz, people with fine hair |
| Fine | 40-60 | Caucasian and Asian fine hair types |
| Medium | 60-90 | Most adults; widest category |
| Thick / coarse | 90-120 | African hair textures, some East Asian hair |
| Exceptionally thick | 120-180 | Thick beard or animal guard hair |
Approximate diameter ranges for human scalp hair by thickness class. Measured in micrometers (1 μm = 0.001 mm).
Frequently asked questions
Why does a hair produce a diffraction pattern instead of just casting a shadow?
Light is a wave, and waves bend around obstacles through diffraction. A human hair is only about 50-100 times wider than the wavelength of visible light, so the bending effect is pronounced. By Babinet's principle, the interference pattern from the hair is identical to the pattern a slit of the same width would produce, just with the bright and dark regions swapped. The result is a series of alternating bright and dark bands flanking a central bright maximum.
What laser pointer do I need?
Any inexpensive laser pointer works. The wavelength is usually printed on the warning label or the barrel: red pointers are typically 635-660 nm, green pointers 520-532 nm, and violet pointers 395-410 nm. Avoid class 3B or class 4 lasers and never look into any laser beam. A class 2 red laser pointer (under 1 mW) is safe for this experiment.
Why should I use a higher-order dark fringe?
The fringe position x equals m * lambda * D / d, so measuring the 2nd fringe gives twice the distance from center as the 1st, and the 3rd gives three times. If your ruler has 1 mm divisions, the percentage uncertainty in measuring a 5 mm distance is much larger than in measuring a 15 mm distance. Using a higher order is a free accuracy upgrade - as long as the fringe is clearly visible and the angle stays under about 15 degrees.
Does it matter if the hair is not perfectly straight or circular?
A hair that is slightly oval in cross-section will produce slightly different fringe spacings when rotated. If you rotate the hair 90 degrees and the pattern changes, the hair is not circular. You can measure both orientations and average them for an effective diameter. Curvature along the length of the hair is less critical as long as the illuminated length is short enough to look uniform from the laser's perspective.
What is the small-angle approximation and when does it fail?
The exact formula is d = m * lambda / sin(theta). The approximation replaces sin(theta) with tan(theta) = x / D, giving the simpler d = m * lambda * D / x. This works because for small angles sin and tan are nearly equal. The approximation introduces less than 0.5% error below 10 degrees and less than 2% below 15 degrees. It fails (error exceeds 2%) when the screen is close or the hair is very thin, pushing the fringes far from center. The calculator shows the angle and flags when the exact formula is needed.
Why is the pattern identical to a slit even though a hair is an obstacle?
Babinet's principle in wave optics states that the diffraction patterns of complementary screens (one being the negative of the other) are identical except for the amplitude of the forward beam. A hair and a slit of the same width are complementary screens, so they produce the same set of dark fringes at the same angles. The only difference is the brightness of the very center: for a slit that is the transmitted beam, for a hair it is the part that diffracts around both edges and converges back.
What range of hair diameters can this method measure?
The method works whenever the hair width is much larger than the wavelength (so fringes exist) and the geometry keeps fringes in a measurable position. For a 650 nm red laser at 1 m screen distance, a 50 um hair puts the first dark fringe at about 13 mm from center - easy to measure with a ruler. A 200 um hair puts it at about 3.25 mm - still measurable. Human scalp hair (17-180 um) falls comfortably in the practical range. Wavelength-scale structures (under about 5 um) would push fringes to unmeasurably large angles.