Tool · Imaging and diffraction

Slit Diffraction Calculator

Behind a slit of width a the first dark band sits at sin θ = λ/a, and two slits a distance d apart give fringes λL/d apart on a screen at distance L. From the wavelength, the slit width, the number and spacing of the slits and the distance to the screen, the calculator draws the far-field pattern and gives its minima, fringe spacing and envelope. Background: single-slit diffraction, double-slit experiment, and Fresnel number.

Slits and light

Wavelength λ (nm)

Slit width a (µm)

Number of slits N

Slit spacing d, center to center (µm)

Distance to the screen L (m)

The light arrives as a plane wave at normal incidence, as from a collimated laser. Set N = 1 for a single slit and N = 2 for Young’s double slit; larger N approaches a diffraction grating.

Presets

A hair or thin wire gives the same pattern as a slit of its width, apart from the bright spot on the axis (Babinet’s principle), which is how the laser-pointer measurement of a hair works.

Readouts

Intensity on the screen

Learn with it

Three short experiments. Each one sets the inputs, says where to look, and asks for a prediction before it shows the result.

Checked against

These checks run in your browser on every load. The closed forms are compared with values worked out by hand, the first side lobe of the single-slit pattern is found numerically, a missing order is confirmed to be dark, and the pattern is integrated to confirm that it carries the light that passed the slit.

Check	Expected	Computed	Tolerance

The expected values follow the Fraunhofer results in Hecht, Optics, and Born and Wolf, Principles of Optics, evaluated by hand for the stated cases. The tolerance is the largest relative difference from Expected that still passes.

The model

In the far field (Fraunhofer diffraction), a slit of width $a$ lit by a plane wave of wavelength $λ$ gives an intensity that depends on the angle $θ$ as the square of a sinc function. With $N$ identical slits at center spacing $d$ , that envelope is multiplied by the interference of the slits:

\frac{I}{I_{0}} = {(\frac{\sin β}{β})}^{2} {(\frac{\sin N γ}{N \sin γ})}^{2}, β = \frac{π a \sin θ}{λ}, γ = \frac{π d \sin θ}{λ}

The single-slit minima fall at $a \sin θ = m λ$ for $m = \pm 1, \pm 2, \dots$ , so the central maximum on a screen at distance $L$ is about $2 λ L / a$ wide. Each first side lobe peaks at 4.72 % of the central intensity and holds 2.36 % of the transmitted power, against 90.3 % in the central maximum (integrals of the sinc² curve). The bright fringes of the slits fall at $d \sin θ = m λ$ , spaced by $λ L / d$ on the screen for small angles. An order that lands on a zero of the envelope is missing: when $d / a = p / q$ in lowest terms, orders $\pm p, \pm 2 p, \dots$ vanish, so the 40 µm slits 250 µm apart of the worked example (d/a = 25/4) lose order 25. The far-field form holds when the Fresnel number $w^{2} / (L λ)$ , with $w$ the total width of the slits, is small: below about 0.1 the Fraunhofer pattern is accurate, between 0.1 and 1 it is an approximation, and above 1 the pattern is a Fresnel pattern and this model does not apply. For slits narrower than about 2λ the scalar theory used here is only qualitative. For a circular aperture the same physics gives the Airy disk (see the Airy disk calculator), and for many slits the diffraction grating.

Worked example

A helium-neon laser at 632.8 nm passes a 100 µm slit, and the screen is 1 m away. The first minimum is at 6.328 mrad, so the central maximum is 12.66 mm wide, and the Fresnel number is 0.0158, well into the far field. A double slit of 40 µm slits 250 µm apart at 532 nm gives fringes 2.128 mm apart, 13 of them inside the 26.6 mm central maximum.

References: E. Hecht, Optics, 5th ed. (Pearson, 2017), ch. 10. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999), ch. 8.