Diffraction applications

A diffraction grating is a periodic array of $N$ identical slits (or reflecting grooves), each of width $a$, separated by a distance $d$ called the grating period. Light diffracted from every slit interferes with light from every other slit, and the pattern on a distant screen is built from that mutual interference rather than from any one slit alone. A principal maximum — a bright fringe reinforced by all $N$ slits at once — appears in a direction $\theta$ whenever the path difference between light leaving adjacent slits equals a whole number of wavelengths:

where: $d$ is the grating period, $\theta$ is the angle measured from the straight-through direction, $\lambda$ is the wavelength, and $m$ is the order of the maximum — an integer labelling which reinforcement condition is satisfied. The order $m=0$ gives the undiffracted central beam, common to every wavelength and therefore carrying no colour information; the orders $m=\pm1,\pm2,\dots$ spread out on either side, each order landing at a different angle for each wavelength present. Two features distinguish a grating from a simple double slit, both arising because $N$ is large (often thousands, not two):

Sharper lines. With $N$ slits interfering, each principal maximum is squeezed into a narrow angular width that shrinks roughly as $1/N$, with faint secondary ripples crowded between successive principal maxima. As $N$ grows large, these secondary ripples become negligible and the principal maxima stand out as sharp, bright lines on an almost dark background — very different from the smooth, broad cosine-squared fringes of a two-slit pattern. The full multi-slit intensity formula behind this is beyond this course; only the qualitative picture — narrow bright lines, not broad fringes — is required.

Wavelength-dependent angle. Since $\sin\theta=m\lambda/d$, two different wavelengths present in the same order $m$ land at two different angles. A grating therefore spreads white light into a rainbow-like spectrum in each order, exactly as a prism does, but by interference between many diffracted beams rather than by refraction through glass. This wavelength sorting is what makes a grating disperse light, and it is the operating principle of a grating spectrometer: measuring the angle of a bright line tells you the wavelength that produced it.

A diffraction grating has $5000$ lines per centimetre. Light of wavelength $\lambda=589\ \text{nm}$ (the sodium D line) falls on it normally. Find the grating period $d$, the angles of the first- and second-order principal maxima, and the highest order the grating can produce for this wavelength.

Step 1. The grating period is the reciprocal of the line density: $d=\dfrac{1\ \text{cm}}{5000}=\dfrac{1\times10^{-2}\ \text{m}}{5000}=2.0\times10^{-6}\ \text{m}=2000\ \text{nm}$.

Step 2. For the first order ($m=1$), $\sin\theta_1=\dfrac{m\lambda}{d}=\dfrac{589}{2000}=0.2945$, so $\theta_1=\sin^{-1}(0.2945)\approx17.1^\circ$.

Step 3. For the second order ($m=2$), $\sin\theta_2=\dfrac{2\times589}{2000}=0.589$, so $\theta_2=\sin^{-1}(0.589)\approx36.1^\circ$.

Step 4. Since $\sin\theta$ can never exceed $1$, the highest possible order is $m_{\max}=\lfloor d/\lambda\rfloor=\lfloor2000/589\rfloor=3$. So orders $m=1,2,3$ appear on each side of the central beam; a fourth order would require $\sin\theta>1$, which is impossible. Answer: $d=2000\ \text{nm}$, $\theta_1\approx17.1^\circ$, $\theta_2\approx36.1^\circ$, and only three orders ($m=1,2,3$) exist on each side.

Competitive stretch

White light (roughly $400$–$700\ \text{nm}$) falls normally on a grating with $d=1.5\ \mu\text{m}=1500\ \text{nm}$. In the first order, violet ($400\ \text{nm}$) appears at $\sin\theta_v=400/1500=0.2667\Rightarrow\theta_v\approx15.5^\circ$, while red ($700\ \text{nm}$) appears at $\sin\theta_r=700/1500=0.4667\Rightarrow\theta_r\approx27.8^\circ$. The first-order spectrum spans $\Delta\theta=\theta_r-\theta_v\approx12.3^\circ$ — red bends farther than violet, the opposite ordering from a prism, since the grating disperses by diffraction, not refraction.

• Mixing up the grating equation $d\sin\theta=m\lambda$ (bright maxima, slit spacing $d$) with the single-slit minima condition $a\sin\theta=m\lambda$ (dark fringes, individual slit width $a$) — same algebraic form, very different physical roles.
• Assuming a grating disperses colours like a prism through refraction — it does so by diffraction and multi-slit interference, and typically sends red to a larger angle than violet in a given positive order, opposite to a prism.
• Forgetting to check $|\sin\theta|=|m\lambda/d|\le1$ before claiming an order exists — only a finite number of orders fit for any given $d$ and $\lambda$.
• Treating "more lines per unit length" as simply making a "bigger" spectrum, without separating sharper lines (an $N$ effect, qualitative here) from a wider angular spread (a $d$ and $m$ effect, which is quantitative and examinable).

Ask the tutor: Want to work through how many orders a different grating supports, or check an angular-spread calculation step by step?