Gratings: Dispersion and Resolving Power — Question 9

Question 9

Angular pointing noise creates a wavelength-calibration floor

A first-order grating has $d=2.00\,\mu\mathrm m$ and measures $\lambda=600\,\mathrm{nm}$ at normal incidence. The observed diffraction angle has standard uncertainty $\sigma_\theta=10.0\,\mu\mathrm{rad}$. Treat d and incidence angle as exact.

1. Propagate the angular uncertainty into wavelength uncertainty.

2. Find the illuminated slit count at which the ideal Rayleigh separation becomes equal in magnitude to this uncertainty scale.

Question 9 - Solution

Angular pointing noise creates a wavelength-calibration floor

(a) Method and calculation

$$\lambda=d\sin\theta\Longrightarrow\boxed{\sigma_\lambda=d|\cos\theta|\sigma_\theta}.$$$$\sin\theta=0.300\Longrightarrow\boxed{\sigma_\lambda=0.0190788\,\mathrm{nm}}.$$

(b) Interpretation and further calculation

$$\lambda/N_*=\sigma_\lambda\Longrightarrow N_*=\lambda/\sigma_\lambda.$$$$\boxed{N_*=31448.5\simeq3.14\times10^4}.$$

Physical check

The comparison is between two different width conventions, used here only as a scale estimate; adding grooves alone does not reduce pointing noise.