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 bright fringes. 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.

Its full width on a screen a distance $D$ away is the gap between the two first minima:

w = \dfrac{2\lambda D}{a}

The weaker secondary maxima sit roughly halfway between minima, at $a\sin\theta \approx \left(n + \tfrac{1}{2}\right)\lambda$ , and each is only half as wide as the central one.

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}

The central maximum spans from $-4.5$ mm to $+4.5$ mm, so its width is $\dfrac{2\lambda D}{a} = 9$ 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

Since $\sin\theta$ can't exceed 1, the highest order you can see is the largest whole number $n$ with $n < \dfrac{d}{\lambda}$ . (If $\dfrac{d}{\lambda}$ were a whole number, that order would sit at $\theta = 90^\circ$ , along the grating itself, and couldn't be seen.)

Worked example: diffraction grating

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

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

For the highest order, $\dfrac{d}{\lambda} = \dfrac{2000}{589} \approx 3.4$ , so orders up to $n = 3$ appear on each side of the centre. A 4th order would need $\sin\theta \approx 1.18$ , which is impossible.

Single slit, double slit and grating compared

The condition that looks the same means different things

Setup	Condition	What it gives
Single slit, width $a$	$a\sin\theta = n\lambda$ , $n \neq 0$	Dark fringes (minima)
Double slit, spacing $d$	$d\sin\theta = n\lambda$	Bright fringes, evenly spaced near the centre
Grating, spacing $d$	$d\sin\theta = n\lambda$	Very sharp, bright principal maxima

Common mistakes

Treating $a\sin\theta = n\lambda$ as a bright fringe. For a single slit it marks the dark fringes.
Halving the central maximum. Its width runs from the first minimum on one side to the first minimum on the other: $\dfrac{2\lambda D}{a}$ , not $\dfrac{\lambda D}{a}$ .
Including $n = 0$ as a minimum. At $\theta = 0$ every part of the slit is in phase, which is the brightest point of the pattern.
Forgetting to convert lines per centimetre. 5000 lines/cm means $d = 2\times10^{-4}$ cm, which has to be converted to the same unit as $\lambda$ before dividing.

Where this comes up in exams

Diffraction due to a single slit and the width of the central maximum are listed in the JEE Main 2026 and NEET (UG) 2026 syllabi, and single-slit diffraction in the JEE Advanced 2026 syllabus. None of the three lists the diffraction grating.

Checked against JEE (Main) 2026 syllabus (NTA), NEET (UG) 2026 syllabus (NMC, published by NTA) and JEE (Advanced) 2026 syllabus. Syllabi can change from year to year, so confirm with the latest official notification for your exam.

Fit it into your week with the free timetable generator.
Mark it off as you revise in the syllabus tracker.
See how long you have left: JEE Main 2027 (Session 1) countdown, JEE Advanced 2027 countdown and NEET UG 2027 countdown.