36-6 Gratings: Dispersion and Resolving Power

36-6 Gratings: Dispersion and Resolving Power#

Prompts

  • What is dispersion \(D\) of a diffraction grating? How does it differ from resolving power \(R\)? Which depends on \(N\)? Which on \(d\)?

  • Derive the dispersion formula \(D = m/(d\cos\theta)\) from \(d\sin\theta = m\lambda\). Why does smaller \(d\) give higher dispersion?

  • Two gratings have the same \(d\) but different \(N\). Which produces narrower lines? Which can resolve wavelengths that are closer together?

  • A grating has \(N = 10{,}000\), \(d = 2\,\mu\text{m}\), used in first order at \(\theta \approx 17°\). Estimate the resolving power \(R\) and the dispersion \(D\) (in rad/m) for \(\lambda = 600\) nm.

  • For the sodium doublet (589.00 nm and 589.59 nm), how would you use the dispersion to find the angular separation of the two lines in the first order?

Lecture Notes#

Overview#

  • Dispersion \(D\) measures how much the grating spreads lines of different wavelengths in angle. Higher \(D\) means greater angular separation for a given \(\Delta\lambda\).

  • Resolving power \(R\) measures the grating’s ability to distinguish two close wavelengths. Higher \(R\) means lines can be resolved even when \(\Delta\lambda\) is smaller.

  • \(D\) depends on \(d\) and \(m\) (not \(N\)); \(R\) depends on \(N\) and \(m\) (not \(d\)). Do not confuse them.

Dispersion#

To distinguish wavelengths in a spectroscope, a grating must spread apart the diffraction lines associated with different wavelengths. This spreading is called dispersion, defined as

(312)#\[ D = \frac{\Delta\theta}{\Delta\lambda} \]

where \(\Delta\theta\) is the angular separation of two lines whose wavelengths differ by \(\Delta\lambda\). The greater \(D\) is, the farther apart the lines appear for a given wavelength difference.

From the grating equation \(d\sin\theta = m\lambda\). Treating \(\theta\) and \(\lambda\) as variables and differentiating:

(313)#\[ d\cos\theta\,d\theta = m\,d\lambda \quad \Rightarrow \quad D = \frac{d\theta}{d\lambda} = \frac{m}{d\cos\theta} \]

  • Higher dispersion: use smaller grating spacing \(d\) and/or work in higher order \(m\).

  • Dispersion does not depend on \(N\)—only on \(d\) and \(m\).

Units: rad/m or °/m (or °/µm for practical spectroscopes).

Resolving power#

To resolve two lines (make them distinguishable), each line must be narrow enough that they do not overlap. The resolving power is defined as

(314)#\[ R = \frac{\lambda_{\text{avg}}}{\Delta\lambda} \]

where \(\lambda_{\text{avg}}\) is the mean wavelength of two lines that can barely be recognized as separate, and \(\Delta\lambda\) is their wavelength difference. The greater \(R\) is, the closer two wavelengths can be and still be resolved.

By Rayleigh’s criterion, two lines are just resolved when their angular separation equals the half-width of each line. Combining the dispersion relation \(\Delta\theta = (m/(d\cos\theta))\,\Delta\lambda\) with the half-width \(\Delta\theta_{\text{hw}} = \lambda/(Nd\cos\theta)\) (section 36-5) yields

(315)#\[ R = Nm \]

  • Higher resolving power: use more rulings (larger \(N\)).

  • Resolving power does not depend on \(d\)—only on \(N\) and \(m\).

Dispersion vs resolving power#

Quantity

Formula

Depends on

Physical meaning

Dispersion \(D\)

\(m/(d\cos\theta)\)

\(d\), \(m\)

Angular spread per unit \(\Delta\lambda\)

Resolving power \(R\)

\(Nm\)

\(N\), \(m\)

Smallest \(\Delta\lambda\) that can be resolved

Do not confuse

  • Same \(d\), different \(N\): gratings have the same dispersion but different resolving power. The one with larger \(N\) produces narrower lines and can resolve closer wavelengths.

  • Same \(N\), different \(d\): gratings have the same resolving power but different dispersion. The one with smaller \(d\) spreads lines more in angle.

Example (from textbook Table 36-1, \(\lambda = 589\) nm, \(m = 1\)): Grating \(A\) (\(N=10{,}000\), \(d=2540\) nm) and \(B\) (\(N=20{,}000\), \(d=2540\) nm) have the same \(D\) but \(B\) has higher \(R\)—narrower lines. Gratings \(A\) and \(C\) (\(N=10{,}000\), \(d=1360\) nm) have the same \(R\) but \(C\) has higher \(D\)—greater angular separation.

Example: sodium doublet

Sodium doublet: 589.00 nm and 589.59 nm. Grating: \(N = 1.26\times 10^4\) rulings over \(w = 25.4\) mm \(\Rightarrow\) \(d = w/N \approx 2.02\,\mu\text{m}\).

(a) First-order angle for 589.00 nm: \(\theta = \arcsin(m\lambda/d) \approx 16.9°\).

(b) Angular separation: \(\Delta\theta \approx D\,\Delta\lambda = (m/(d\cos\theta))\,\Delta\lambda\). With \(m=1\), \(\Delta\lambda = 0.59\) nm: \(\Delta\theta \approx 0.00030\) rad \(\approx 0.017°\).

Summary#

  • Dispersion \(D = \Delta\theta/\Delta\lambda = m/(d\cos\theta)\): spread lines in angle; increase by smaller \(d\) or higher \(m\).

  • Resolving power \(R = \lambda/\Delta\lambda = Nm\): distinguish close wavelengths; increase by larger \(N\).

  • \(D\) and \(R\) are independent: a grating can have high dispersion without high resolving power, and vice versa.