Diffraction, Gratings and Resolving Power: Quick Revision

A brief note on Diffraction, Gratings and Resolving Power. Read it before a test. Try each question at the end before you open its answer.

Single slit

• Dark places: $a\sin\theta = n\lambda$, with $n = \pm1, \pm2, \dots$ The centre is not dark.
• Central maximum: $w = \dfrac{2\lambda D}{a}$. It is twice as wide as each weak maximum.
• Intensity: $I/I_0 = (\sin\alpha/\alpha)^2$, with $\alpha = \pi a\sin\theta/\lambda$. The weak maxima have $0.047$, $0.016$ and $0.008$ of the centre.
• Narrower slit, wider pattern.
• Fresnel distance: $z_F = a^2/\lambda$. Ray optics holds up to about $z_F$.

Double slit with diffraction, and the grating

• Fringes inside the central maximum: orders $m < d/a$. A missing order has $m = n\,d/a$.
• Grating: $d\sin\theta = m\lambda$, with $d = 1/N_1$. The highest order has $m < d/\lambda$.
• More slits: narrower, brighter principal maxima.
• White light: the zero order is white. Each higher order is a spectrum, with violet inside and red outside. Orders 2 and 3 overlap.

Resolving power

• Circular aperture: first dark ring at $1.22\lambda/D$. Rayleigh limit: $\theta_{\min} = 1.22\lambda/D$.
• Resolving power: $1/\theta_{\min}$. It rises with a bigger aperture and a shorter wavelength.
• Microscope: $d_{\min} = 0.61\lambda/\text{NA}$.
• Grating: $R = \lambda/\Delta\lambda = mN$.

A worked example

A slit $0.25$ mm wide is lit by light of wavelength $650$ nm. The screen is $1.20$ m away. Find the width of the central maximum.

1. First minimum. $y_1 = \dfrac{\lambda D}{a} = \dfrac{650 \times 10^{-9}\ \text{m} \times 1.20\ \text{m}}{0.25 \times 10^{-3}\ \text{m}} = 3.12\ \text{mm}$.
2. The width. $w = 2y_1 = 6.24\ \text{mm}$.

Practise with new numbers

Check yourself

1. A grating has $6000$ lines in each centimetre. Light of $650$ nm falls on it. What is the highest order?

2. Why can a bigger telescope tell apart two closer stars?

3. The slit of a single-slit experiment is made narrower. What happens to the pattern?

Sources

• OpenStax, University Physics Volume 3, Chapter 4 (Diffraction).
• NCERT Physics Class 12, Chapter 10 (Wave Optics).