The diffraction grating

2. Key Definitions

• Diffraction: The spreading of a wave as it passes through a gap or past the edge of an obstacle.
• Diffraction Grating: An optical component consisting of a large number of parallel, equidistant, and closely spaced slits used to produce high-resolution interference patterns.
• Grating Spacing ($d$): The distance between the centres of adjacent slits (or the distance between the centres of adjacent lines) on a diffraction grating.
• Order of Diffraction ($n$): An integer ($0, 1, 2, \dots$) that represents the number of wavelengths of path difference between light rays from adjacent slits that interfere constructively to form a maximum.
• Zero-order Maximum ($n=0$): The central bright fringe formed in the direction of the incident light ($\theta = 0^\circ$) where the path difference for all wavelengths is zero.
• Monochromatic Light: Light consisting of a single wavelength or frequency.
• Coherent Sources: Sources that have a constant phase difference and the same frequency. In a grating, the slits act as multiple coherent sources by diffracting a single incident wavefront.

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3. Content

3.1 The Physics of the Diffraction Grating

A diffraction grating typically contains between 100 and 1000 lines per millimetre. When a wavefront of light strikes the grating, each slit acts as a point source of secondary wavelets (Huygens' Principle). These wavelets spread out (diffract) and superpose in the space beyond the grating.

Why use a Grating instead of a Double Slit?

While a Young’s double-slit setup produces a pattern of fringes, a diffraction grating is superior for measurement for three reasons:

1. Sharpness: The maxima (bright fringes) are extremely narrow. Because there are thousands of slits, even a tiny change in angle $\theta$ away from the maximum causes the waves from the many slits to cancel each other out through destructive interference.
2. Brightness: Since more light passes through thousands of slits than through just two, the resulting maxima are much more intense.
3. Resolution: The maxima are spaced much further apart, allowing for more precise measurement of the diffraction angle $\theta$.

3.2 Calculating Grating Spacing ($d$)

The grating is usually described by the number of lines per unit length, $N$. To use the grating equation, you must first calculate the distance $d$ between the slits.

Equation: $$d = \frac{1}{N}$$

Important Unit Conversions:

• If $N$ is in lines per mm: $d = \frac{1 \times 10^{-3}}{N}$ metres.
• If $N$ is in lines per cm: $d = \frac{1 \times 10^{-2}}{N}$ metres.
• If $N$ is in lines per m: $d = \frac{1}{N}$ metres.

3.3 Derivation of the Grating Equation: $d \sin \theta = n\lambda$

Consider a beam of monochromatic light of wavelength $\lambda$ incident normally (at $90^\circ$) on a grating with spacing $d$.

1. Light diffracts at each slit. We consider rays emerging at an angle $\theta$ to the normal.
2. For a maximum to be observed at angle $\theta$, the light from all slits must interfere constructively.
3. Constructive interference occurs when the path difference between light from adjacent slits is an integer number of wavelengths ($n\lambda$).
4. By drawing a right-angled triangle between two adjacent slits:
   • The hypotenuse is the grating spacing $d$.
   • The angle opposite the path difference is $\theta$.
   • The side opposite the angle $\theta$ is the path difference.
5. Using trigonometry: $$\sin \theta = \frac{\text{Path Difference}}{d}$$
6. Substituting the condition for a maximum (Path Difference = $n\lambda$): $$\sin \theta = \frac{n\lambda}{d}$$
7. Rearranging gives the Diffraction Grating Equation: $$d \sin \theta = n\lambda$$

3.4 Determining the Wavelength of Light

The diffraction grating is the standard tool for measuring the wavelength of light from lasers or gas discharge lamps.

Experimental Procedure:

1. Setup: Place a monochromatic light source (e.g., a laser) behind a diffraction grating. Ensure the beam is incident normally on the grating.
2. Observation: Position a screen at a large, measured distance $D$ from the grating.
3. Measurement:
   • Identify the central zero-order maximum ($n=0$).
   • Measure the distance $x$ from the $n=0$ maximum to the $n=1$ or $n=2$ maximum.
   • Optimization: To reduce percentage uncertainty, measure the distance between the $+n$ and $-n$ orders (e.g., from the 1st order on the left to the 1st order on the right) and divide by 2 to find $x$.
4. Calculation of $\theta$:
   • Use the geometry of the setup: $\tan \theta = \frac{x}{D}$.
   • Therefore, $\theta = \tan^{-1}\left(\frac{x}{D}\right)$.
   • Note: Do not use the small-angle approximation ($\sin \theta \approx \tan \theta$) unless $\theta < 5^\circ$. In grating experiments, $\theta$ is usually large.
5. Final Calculation: Calculate $d$ from $N$, then use $\lambda = \frac{d \sin \theta}{n}$.

3.5 Maximum Number of Orders

There is a physical limit to the number of orders ($n$) that can be observed. Since the maximum possible angle of diffraction is $90^\circ$, and $\sin(90^\circ) = 1$:

$$n\lambda = d \sin \theta$$ $$n_{max} \leq \frac{d}{\lambda}$$

Rules for $n_{max}$:

1. $n$ must be an integer.
2. Always round down the result of $d/\lambda$ to the nearest whole number (e.g., if $n = 3.8$, the maximum order is $n=3$).
3. To find the total number of maxima visible: $$\text{Total} = (2 \times n_{max}) + 1$$ (The '+1' accounts for the $n=0$ central maximum).

3.6 Diffraction of White Light

If white light is incident on a grating, a continuous spectrum is observed for every order where $n > 0$.

• Central Maximum ($n=0$): Appears white. This is because at $\theta = 0^\circ$, the path difference is zero for all wavelengths, so all colours interfere constructively at the same spot.
• Higher Orders ($n \geq 1$): Since $\sin \theta \propto \lambda$, different colours diffract at different angles.
  • Violet/Blue light has the shortest wavelength, so it diffracts through the smallest angle.
  • Red light has the longest wavelength, so it diffracts through the largest angle.
• Result: For each order, a spectrum is formed with violet closest to the centre and red furthest away. At very high orders, these spectra may overlap.

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4. Worked Examples

Worked Example 1 — Calculating Wavelength

A laser beam is directed normally at a diffraction grating with $6.00 \times 10^5$ lines per metre. The third-order maximum is observed at an angle of $48.4^\circ$ to the normal. Calculate the wavelength of the laser light in nanometres.

Step 1: Calculate the grating spacing $d$ $$d = \frac{1}{N} = \frac{1}{6.00 \times 10^5 \text{ m}^{-1}}$$ $$d = 1.667 \times 10^{-6} \text{ m}$$

Step 2: Use the grating equation $$d \sin \theta = n\lambda$$ $$(1.667 \times 10^{-6}) \times \sin(48.4^\circ) = 3 \times \lambda$$

Step 3: Solve for $\lambda$ $$\lambda = \frac{(1.667 \times 10^{-6}) \times 0.7478}{3}$$ $$\lambda = 4.155 \times 10^{-7} \text{ m}$$

Step 4: Convert to nanometres $$\lambda = 416 \text{ nm}$$

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Worked Example 2 — Determining Line Density

Light of wavelength $589 \text{ nm}$ is incident on a grating. The first-order maximum is found at an angle of $15.5^\circ$ from the central zero-order maximum. Calculate the number of lines per millimetre on the grating.

Step 1: Identify variables $\lambda = 589 \times 10^{-9} \text{ m}$ $n = 1$ $\theta = 15.5^\circ$

Step 2: Calculate $d$ $$d = \frac{n\lambda}{\sin \theta}$$ $$d = \frac{1 \times 589 \times 10^{-9}}{\sin(15.5^\circ)}$$ $$d = \frac{589 \times 10^{-9}}{0.2672}$$ $$d = 2.204 \times 10^{-6} \text{ m}$$

Step 3: Calculate $N$ in lines per metre $$N = \frac{1}{d} = \frac{1}{2.204 \times 10^{-6}} = 453,720 \text{ lines m}^{-1}$$

Step 4: Convert to lines per millimetre $$N = \frac{453,720}{1000} = 454 \text{ lines mm}^{-1}$$

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Worked Example 3 — Total Number of Maxima

A diffraction grating has 400 lines per mm. It is illuminated by monochromatic light of wavelength $633 \text{ nm}$. Determine the total number of bright spots that can be observed.

Step 1: Calculate $d$ $$d = \frac{1 \times 10^{-3}}{400} = 2.50 \times 10^{-6} \text{ m}$$

Step 2: Find the maximum possible order $n$ $$n \leq \frac{d}{\lambda}$$ $$n \leq \frac{2.50 \times 10^{-6}}{633 \times 10^{-9}}$$ $$n \leq 3.949$$

Step 3: Determine the highest integer order The highest order visible is $n = 3$. (The 4th order would require $\sin \theta > 1$, which is impossible).

Step 4: Calculate total spots Total spots = $(2 \times 3) + 1 = 7$. (These are the $n = -3, -2, -1, 0, 1, 2, 3$ orders).

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5. Key Equations

Equation	Symbols	SI Units	Data Sheet?
$d \sin \theta = n\lambda$	$d$: grating spacing; $\theta$: angle; $n$: order; $\lambda$: wavelength	$d$ (m); $\theta$ ($^\circ$); $\lambda$ (m)	Yes
$d = \frac{1}{N}$	$d$: grating spacing; $N$: lines per unit length	$d$ (m); $N$ (m⁻¹)	No
$n_{max} < \frac{d}{\lambda}$	$n_{max}$: highest possible order	Dimensionless	No
$\tan \theta = \frac{x}{D}$	$x$: distance to fringe; $D$: distance to screen	$x$ (m); $D$ (m)	No

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6. Common Mistakes to Avoid

• ❌ Wrong: Using $N$ as $d$ in the formula.
  • ✓ Right: $N$ is the number of lines (e.g., 300). $d$ is the distance between them. Always use $d = 1/N$ and ensure the units are converted to metres.
• ❌ Wrong: Rounding $n$ up when finding the maximum order.
  • ✓ Right: If $n = 2.1$, the maximum order is 2. If $n = 2.99$, the maximum order is still 2. The 3rd order cannot exist as it would require a $\sin \theta$ value greater than 1.
• ❌ Wrong: Forgetting to double the orders when counting total fringes.
  • ✓ Right: Maxima appear on both sides of the centre. Total = $2n + 1$.
• ❌ Wrong: Using the small-angle approximation ($\sin \theta \approx \theta$).
  • ✓ Right: In Young's Double Slit, angles are small. In Diffraction Gratings, angles are often $20^\circ$ to $60^\circ$. You must use $\sin \theta$ and $\tan \theta$ explicitly.
• ❌ Wrong: Calculator in Radians mode.
  • ✓ Right: Ensure your calculator is in Degrees (DEG) mode for these problems.

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7. Exam Tips

1. The "Lines per..." Trap: Read the units for $N$ carefully. "Lines per mm" is common, but "lines per metre" or "lines per cm" also appear. Convert to $d$ in metres immediately to avoid power-of-ten errors.
2. Measuring $2\theta$: In some practical-based questions, you are given the angle between the first-order maximum on the left and the first-order maximum on the right. This is $2\theta$. You must divide by 2 before using $d \sin \theta = n\lambda$.
3. White Light Spectra: If asked about the appearance of the pattern with white light, always mention:
   • The central fringe is white.
   • The other fringes are spectra.
   • Blue is closest to the centre, Red is furthest away.
4. Significant Figures: Grating spacing $d$ is often a very small number (e.g., $1.67 \times 10^{-6}$ m). Keep at least 4 significant figures in your intermediate $d$ calculation to ensure your final $\lambda$ or $\theta$ is accurate to 3 s.f.
5. Path Difference Logic: If a question asks why a maximum is formed at a specific angle, the mark scheme usually requires:
   • Mention of diffraction at the slits.
   • Mention of superposition or interference of waves.
   • The condition that the path difference is an integer number of wavelengths ($n\lambda$).