Hair Diffraction Calculator

Estimate the diameter of a hair strand from a laser diffraction pattern using a·sinθ_m = mλ, based on the laser wavelength, screen distance, fringe order, and measured fringe position.

Quick Facts

Diffraction condition
a·sinθ_m = mλ
Dark fringes (minima) form where this holds; a is the hair diameter and θ_m is the angle to the m-th minimum.
Babinet's principle
Hair ≈ a slit of the same width
Away from the direct beam, a thin opaque strand produces the same diffraction pattern as a slit of equal width.
Typical human hair
≈17-181 μm diameter
Fine hair runs thinner and coarse hair thicker; the overall average is roughly 70 μm.
Better measurements
Longer screen distance L
A larger L spreads the fringes further apart, making the fringe position easier to measure precisely.

Your Results

Calculated
Hair Diameter
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a = mλ / sinθ_m, in micrometers
Diameter (mm)
-
Same result in millimeters
Diffraction Angle
-
Angle from center to the measured minimum
Compared to Typical Hair
-
Typical human range ≈ 17-181 μm

Ready

Enter your laser wavelength, screen distance, fringe order, and fringe position, then press Calculate.

Formula and Method for the Hair Diffraction Calculator

Shine a laser pointer at a single strand of hair held taut in front of the beam, and instead of a simple shadow you get a pattern of bright and dark bands spread out on a screen behind it. This happens because the hair is thin enough (tens of micrometers) to be comparable to the wavelength of light, so the beam diffracts around it. By Babinet's principle, a thin opaque obstacle produces (away from the central beam) the same diffraction pattern as a slit or gap of the identical width. That means the well-known single-slit diffraction equation can be solved backwards to find the diameter of the hair itself: a·sinθ_m = mλ, where a is the hair diameter, λ is the laser wavelength, m is the order of the dark fringe (1, 2, 3, …), and θ_m is the angle from the central bright band to that dark fringe.

How the calculation works

You measure the perpendicular distance y_m from the center of the pattern to the m-th dark fringe on a screen placed a known distance L behind the hair. From those two lengths the diffraction angle follows exactly from trigonometry: sinθ_m = y_m / √(y_m² + L²). Substituting into the diffraction condition and solving for the hair diameter gives a = mλ / sinθ_m = mλ·√(y_m² + L²) / y_m. This calculator uses the exact trigonometric form rather than the small-angle shortcut (a ≈ mλL/y_m), though the two agree closely whenever y_m is much smaller than L, which is typical for a tabletop setup.

Setting up the measurement

  • Mount the hair vertically in front of the laser, taped across a slot or frame so it stays flat and does not sag or twist during the measurement.
  • Use a known wavelength. Most inexpensive laser pointers are labeled with their wavelength (commonly ~650 nm red or ~532 nm green); an unknown or wrong wavelength shifts the computed diameter by the same percentage as the error.
  • Measure to a dark fringe, not the bright center. The central maximum is wide and its edges are fuzzy, while the dark minima are sharply defined and much easier to mark accurately with a ruler.
  • Use a longer screen distance (1-3 m) when possible — it spreads the fringes further apart on the screen, reducing the relative error in your y_m measurement.

Limits of this method

The result assumes a single, roughly uniform, opaque, cylindrical strand illuminated by coherent, monochromatic light — a curly, split, or non-uniform hair, stray reflections, or a light source with a spread of wavelengths will all blur the fringes and reduce accuracy. The math also assumes the screen is flat and perpendicular to the beam and that the hair diameter is much smaller than both the wavelength-scaled fringe spacing and the screen distance (the far-field, or Fraunhofer, regime), which holds for essentially any hair-and-tabletop setup.

Frequently Asked Questions

Why does a strand of hair create a diffraction pattern like a slit?
By Babinet's principle, a thin opaque obstacle such as a hair produces essentially the same far-field diffraction pattern as a slit (gap) of the same width, except right at the center of the direct beam. So the a sinθ = mλ minima condition used for single-slit diffraction also describes the shadow pattern cast by the hair, and can be solved for the hair's diameter.
What laser wavelength should I use?
Any laser pointer works as long as you know its wavelength. Common inexpensive pointers are red (about 630-650 nm) or green (about 532 nm). Check the packaging or spec sheet, since even a small wavelength error shifts the calculated diameter by the same percentage.
Why measure to a dark fringe instead of the bright center?
Dark fringes (minima) occur at sharply defined angles given by a sinθ_m = mλ, so their positions can be marked precisely. The bright central maximum is wide with fuzzy edges, which makes it a poor reference point for an accurate angle measurement.
What is a typical human hair diameter?
Human scalp hair typically ranges from about 17 to 181 micrometers (0.017-0.181 mm) in diameter depending on hair type and ethnicity, with fine hair near the low end and coarse hair near the high end. The overall average is roughly 70 micrometers.

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