Result
—
Max Order
—

What is a Diffraction Grating?

A diffraction grating is an optical component with a periodic structure that splits and diffracts light into several beams travelling in different directions. It consists of thousands of very thin, parallel, equally spaced slits. When light hits the grating, each slit acts as a secondary source of waves, and these waves interfere with each other.

Unlike a double-slit experiment which produces broad fringes, a diffraction grating produces very sharp, narrow maxima. This makes them extremely useful for spectroscopy, as they can separate different wavelengths of light with high precision.

d sin θ = nλ

Where:

  • d is the grating spacing (the distance between adjacent slits).
  • θ is the angle of diffraction from the normal.
  • n is the order of the maximum (n = 0, 1, 2, ...).
  • λ is the wavelength of the incident light.

Grating Spacing (d)

Gratings are often described by the number of lines per mm (or per meter). The spacing d is simply the reciprocal of this density:

d = 1 / (lines per mm)

If a grating has 600 lines/mm, then d = 1/600 mm = 1.67 × 10⁻³ mm = 1.67 × 10⁻⁶ m (or 1667 nm).

💡 Pro Tip: Higher line density (smaller d) results in larger diffraction angles. This increases the "resolving power" of the grating, making it easier to distinguish between different wavelengths.

Maximum Visible Order

Because the value of sin θ cannot exceed 1, there is a limit to how many orders (n) can be observed for a given wavelength and grating:

nmax < d / λ

Simply calculate d/λ and round down to the nearest whole number to find the highest visible order.

Deep Dive: Worked Examples

✅ Example 1: Calculating Grating Spacing

A diffraction grating is labeled as having 5000 lines per centimeter. Calculate the spacing 'd' in meters and nanometers.

Given: 5000 lines / cm = 500,000 lines / m
Formula: d = 1 / (lines per meter)
Solution:
d = 1 / 500,000 = 0.000002 m
d = 2.0 × 10⁻⁶ m (or 2000 nm)

✅ Example 2: Finding the Diffraction Angle

Light with a wavelength of 632.8 nm (He-Ne laser) shines on a grating with 300 lines/mm. At what angle does the first-order (n=1) maximum appear?

Given: λ = 632.8 nm, n = 1, d = 1/300 mm = 3333.3 nm
Formula: sin θ = nλ / d
Solution:
sin θ = (1 × 632.8) / 3333.3 = 0.1898
θ = arcsin(0.1898) = 10.94°

✅ Example 3: Determining Wavelength

A second-order (n=2) maximum is observed at an angle of 35° using a grating with 600 lines/mm. What is the wavelength of the light?

Given: n = 2, θ = 35°, d = 1/600 mm = 1666.7 nm
Formula: λ = d sin θ / n
Solution:
λ = (1666.7 × sin 35°) / 2
λ = (1666.7 × 0.5736) / 2 = 956.0 / 2
λ = 478.0 nm

✅ Example 4: Counting Visible Orders

Green light (λ = 532 nm) hits a grating with 800 lines/mm. How many orders of maxima are visible (on one side of the central maximum)?

Given: λ = 532 nm, d = 1/800 mm = 1250 nm
Condition: n < d / λ
Solution:
n < 1250 / 532=2.35
Since n must be an integer, the highest order is n = 2.
(Total visible maxima = 2 on left + 1 center + 2 on right = 5).

✅ Example 5: Angular Separation of Colors

White light containing wavelengths from 400 nm to 700 nm hits a 500 lines/mm grating. Calculate the angular width of the first-order spectrum.

Given: d = 1/500 mm = 2000 nm, n = 1
θ for 400 nm: sin θ₁ = (1 × 400) / 2000 = 0.2 → θ₁ = 11.54°
θ for 700 nm: sin θ₂ = (1 × 700) / 2000 = 0.35 → θ₂ = 20.49°
Angular Width: Δθ = 20.49° - 11.54° = 8.95°