36-5 Diffraction Gratings

36-5 Diffraction Gratings#

Prompts

  • How does a diffraction grating differ from a double-slit setup? Why are the bright fringes called lines and why are they narrower?

  • Write the condition for bright lines in a grating. What are order numbers \(m\)? Why is there a maximum order for a given grating?

  • Derive or explain the half-width of the central line. Why does increasing the number of rulings \(N\) make the lines narrower?

  • A grating has \(N = 10{,}000\) rulings over width \(w = 25\) mm. What is the grating spacing \(d\)? At what angle does the first-order maximum appear for \(\lambda = 600\) nm?

  • How does a grating spectroscope use a diffraction grating to analyze the wavelengths of light from a source?

Lecture Notes#

Overview#

  • A diffraction grating is a series of many parallel slits (rulings)—often thousands per millimeter—that separate an incident wave into its component wavelengths.

  • Bright lines occur at angles \(\theta\) satisfying \(d\sin\theta = m\lambda\) (\(m = 0, 1, 2, \ldots\)), where \(d\) is the grating spacing.

  • Unlike double-slit interference, grating lines are very narrow and widely separated, making wavelength measurement practical.

  • Line half-width decreases as the number of rulings \(N\) increases; more rulings produce sharper, more resolvable lines.

Diffraction gratings#

A diffraction grating is like a double-slit arrangement but with a much larger number \(N\) of slits (called rulings)—often several thousand per millimeter. Light can be transmitted through open slits or scattered back from grooves on an opaque surface.

Pattern. With monochromatic light, as \(N\) increases from 2 to many:

  • The intensity plot evolves from the broad double-slit pattern to one with narrow peaks separated by wide dark regions.

  • The bright maxima are called lines because they appear as thin bright bands on the screen.

  • Each line is labeled by an order number \(m\): zeroth-order (central), first-order, second-order, etc.

Grating vs double-slit

The same condition \(d\sin\theta = m\lambda\) applies to both. The difference is that with many slits, destructive interference between adjacent rulings is much more effective in the dark regions, so the bright fringes become very narrow. This makes gratings useful for wavelength determination—different wavelengths produce lines at different angles that do not overlap.

Condition for bright lines#

Assume the viewing screen is far enough that rays from the rulings to a point \(P\) are approximately parallel. The path length difference between adjacent rays is \(d\sin\theta\), where \(d\) is the grating spacing (separation between rulings). If \(N\) rulings occupy total width \(w\), then \(d \approx w/N\).

A bright line occurs when the path length difference between adjacent rays is an integer number of wavelengths:

(308)#\[ d\sin\theta = m\lambda, \quad m = 0, 1, 2, \ldots \]

  • \(m = 0\): central line (all rays in phase).

  • \(m = \pm 1, \pm 2, \ldots\): first-order, second-order lines on either side.

Determining wavelength. Rewriting as \(\theta = \arcsin(m\lambda/d)\) shows that for a given grating, the angle to any line depends on \(\lambda\). Measuring angles to higher-order lines allows determination of unknown wavelengths—even multiple wavelengths can be distinguished.

Maximum order. Since \(|\sin\theta| \leq 1\), we need \(|m\lambda/d| \leq 1\), so \(|m| \leq d/\lambda\). The maximum order is the largest integer \(m\) satisfying this. For a given \(\lambda\), gratings with larger \(d\) can produce more orders (but lines are spread less in angle).

Width of the lines#

A grating’s ability to resolve (separate) lines of different wavelengths depends on how narrow the lines are.

Half-width. The half-width \(\Delta\theta_{\text{hw}}\) of the central line is the angle from its center (\(\theta = 0\)) to where the line effectively ends at the first minimum. The full width is \(2\Delta\theta_{\text{hw}}\).

At the first minimum, the \(N\) rays from the \(N\) rulings cancel. The path length difference between the top and bottom rays is \(Nd\sin\Delta\theta_{\text{hw}}\) (since the top and bottom rulings are separated by \(Nd\)). For cancellation, this must equal \(\lambda\):

(309)#\[ Nd\sin\Delta\theta_{\text{hw}} = \lambda \]

For small \(\Delta\theta_{\text{hw}}\), \(\sin\Delta\theta_{\text{hw}} \approx \Delta\theta_{\text{hw}}\) (in radians), so

(310)#\[ \Delta\theta_{\text{hw}} \approx \frac{\lambda}{Nd} \quad \text{(central line)} \]

For other lines (at angle \(\theta\)), the half-width depends on position:

(311)#\[ |\Delta\theta_{\text{hw}}| \approx \frac{\lambda}{Nd\cos\theta} \]

Effect of \(N\). Larger \(N\) gives narrower lines—more rulings produce sharper peaks and better wavelength resolution. Dispersion and resolving power (section 36-6) formalize these ideas.

Grating spectroscope#

A grating spectroscope uses a diffraction grating to analyze the wavelengths of light from a source. Light from the source passes through a slit, is collimated, strikes the grating, and the diffracted light is focused onto a detector or viewing screen. Each wavelength produces lines at angles given by Eq. (308). By measuring these angles, the emission spectrum (e.g., of a gas or star) can be determined.

Example: hydrogen lines

The visible emission lines of hydrogen (e.g., red, blue-green) appear as separate lines in each order. Higher orders are spread out more in angle. Some orders may be incomplete—e.g., if \(\sin\theta = m\lambda/d > 1\) for a given \(m\) and \(\lambda\), that order does not appear for that wavelength.

Summary#

  • Diffraction grating: many rulings; bright lines at \(d\sin\theta = m\lambda\) (\(m = 0, 1, 2, \ldots\)).

  • Order numbers \(m\) label the lines; maximum \(|m|\) limited by \(d/\lambda \geq |m|\).

  • Half-width of central line: \(\Delta\theta_{\text{hw}} \approx \lambda/(Nd)\); larger \(N\) gives narrower, more resolvable lines.

  • Grating spectroscope: uses a grating to separate and measure wavelengths in a light source.