A diffraction grating has \(N\) lines, a separation \(d\) apart. The spectrum is projected on a screen a distance \(D(\gg d)\) from the grating. Two lines are \(\lambda\) and \(\lambda+\Delta \lambda\) apart. How far apart are they on the screen?

A diffraction grating is a tool used to separate light into its constituent wavelengths. It consists of many closely spaced slits or lines. The core formula governing the behavior of light in a diffraction grating is \( d \, \text{sin}(\theta) = m \, \text{λ} \). In this formula:

• \( d \): the distance between adjacent slits in the grating
• \( \theta \): the angle at which light is diffracted
• \( m \): the diffraction order (an integer)
• \( \text{λ} \): the wavelength of the light

This relationship shows that constructive interference (resulting in peaks of light intensity) occurs at specific angles depending on the wavelength of the light and the order of the diffraction.

In many practical situations, the screen is placed far from the diffraction grating, making the angle \( \theta \) usually very small. For small angles, the sine and tangent functions can be approximated as equal to the angle itself, measured in radians. Therefore, \( \text{sin}(\theta) \approx \text{tan}(\theta) \approx \theta \).
This simplifies our diffraction formula to:
\( d \, \theta = m \, \text{λ} \) or \( \theta = \frac{m \, \text{λ}}{d} \).
The small angle approximation significantly simplifies calculations and is particularly valid when the distance from the grating to the screen (\( D \)) is much larger than the slit separation (\( d \)).

To find how the diffracted wavelengths appear on a screen, we need to translate the diffraction angle into a linear displacement. For a given order \( m \), the position \( y \) on the screen is given by:
\( y = D \, \text{tan}(\theta) \approx D \, \theta \).
Substituting \( \theta = \frac{m \, \text{λ}}{d} \) from the small angle approximation, we get:
\( y = D \, \frac{m \, \text{λ}}{d} \).
This formula tells us that the position of the wavelength \( \text{λ} \) on the screen directly depends on:

• The distance to the screen (\( D \))
• The order of diffraction (\( m \))
• The wavelength of the light (\( \text{λ} \))
• The separation of the grating lines (\( d \))

Now, for two closely spaced wavelengths \( \text{λ} \) and \( \text{λ} + \text{∆λ} \), the separation on the screen \( ∆y \) can be found as:
\( ∆y = D \frac{m \, \text{∆λ}}{d} \).
This shows how even small differences in wavelength can produce noticeable separations on the screen.

Diffraction gratings can produce multiple orders of diffraction, represented by the integer \( m \). Each order represents a different set of angles where constructive interference takes place.

• \( m = 0 \) — Zeroth order: This corresponds to the direct transmission of light without any diffraction.
• \( m = 1, 2, 3, \text{and so on} \) — First, second, third, etc., orders: These orders correspond to increasingly higher angles of diffraction.

The larger the order \( m \), the greater the angular separation between different wavelengths.
For a given wavelength \( \text{λ} \), the position of bright lines will be further out on the screen for higher orders.
However, higher-order diffraction is usually less intense because the light gets spread over wider angles, diminishing its brightness.