Diffraction: Single Slit and Diffraction Grating

A single slit produces minima where two slits produce maxima

This is the part that trips people up: for a single slit of width $a$ , the condition $a\sin\theta = n\lambda$ gives dark fringes, the opposite of the double-slit formula on Young's Double Slit, where the same-looking condition gives brightfringes. The reason: a single slit isn't two point sources interfering: it's infinitely many point sources across the slit width interfering with each other, and at these specific angles they cancel out in pairs.

\text{Single-slit minima: } a\sin\theta = n\lambda \qquad (n = \pm1, \pm2, \ldots)

The central band between the first minima on either side is the central maximum, far brighter and wider than any of the side bands.

Worked example: single-slit minimum

Light of wavelength 600 nm passes through a slit 0.2 mm wide. Find the angle to the first minimum, and its position on a screen 1.5 m away.

Solution: For the first minimum, $n=1$ :

\sin\theta = \dfrac{\lambda}{a} = \dfrac{600\times10^{-9}}{0.2\times10^{-3}} = 3\times10^{-3}

For such a small angle, $\sin\theta \approx \theta$ in radians, and the position on screen is $y \approx D\sin\theta$ :

y \approx (1.5)(3\times10^{-3}) = 4.5\times10^{-3}\text{ m} = 4.5\text{ mm}

Diffraction grating

A grating is many equally-spaced slits, spacing $d$ apart. Unlike a single slit, this produces sharp, bright principal maxima, the same form as the double-slit condition, just with far more slits reinforcing each other:

d\sin\theta = n\lambda

Worked example: diffraction grating

A grating has 5000 lines per centimetre. Find the angle of the first-order maximum for sodium light, $\lambda = 589\text{ nm}$ .

Solution: The line spacing is the reciprocal of the line density:

d = \dfrac{1}{5000\text{ lines/cm}} = 2\times10^{-4}\text{ cm} = 2000\text{ nm}

For the first order, $n=1$ :

\sin\theta = \dfrac{n\lambda}{d} = \dfrac{589}{2000} \approx 0.2945 \;\Rightarrow\; \theta \approx 17.1^\circ