Young's double slit: λ = ax / D

The main symbols used in this lesson:

Shine monochromatic light through a single narrow slit, and then through two narrow slits close together. A screen some distance away shows a row of equally spaced bright and dark bands.

The single slit matters: it makes the light arriving at the double slit coherent. The two slits then act as two coherent sources, and their waves overlap on the screen.

This was strong evidence that light behaves as a wave — something particles could not explain.

Look at the geometry. The two slits are a distance $a$ apart, the screen is a distance $D$ away, and the point P is a distance $x$ from the centre of the pattern.

The ray from the lower slit travels slightly further. For $x$ much smaller than $D$, that extra distance is

$\text{path difference} \approx \frac{ax}{D}.$

A bright fringe occurs where the waves arrive in phase, so the path difference must be a whole number of wavelengths:

$\frac{ax}{D} = n\lambda.$

For the spacing between neighbouring bright fringes, the path difference changes by one wavelength, so $n$ increases by 1 and

$x = \frac{\lambda D}{a}, \qquad \text{or} \qquad \lambda = \frac{ax}{D}.$

From $x = \dfrac{\lambda D}{a}$:

• Longer wavelength → wider spacing. Red fringes are further apart than blue ones.
• Larger slit separation → narrower spacing.
• Further screen → wider spacing.

With white light, each wavelength makes its own pattern at a different spacing. The central fringe is white (all colours in phase there), and the fringes on either side are coloured and soon overlap.

The fringes are equally spaced near the centre because the small-angle approximation holds there. Further out they become slightly uneven.