Gratings: Dispersion and Resolving Power — Question 7

Question 7

Pixel sampling and optical resolution impose separate wavelength scales

A first-order grating has $d=1.00\,\mu\mathrm m$, N=5000, and central wavelength $600\,\mathrm{nm}$. A centered camera has $f=0.200\,\mathrm m$ and pixel pitch $p=10.0\,\mu\mathrm m$. Require at least two pixels across a resolved spectral interval.

1. Find the wavelength interval corresponding to two pixels.

2. Compare it with the ideal diffraction resolution and identify the larger limiting scale.

Question 7 - Solution

Pixel sampling and optical resolution impose separate wavelength scales

(a) Method and calculation

$$\sin\theta=0.600,\quad\cos\theta=0.800\Longrightarrow\mathcal D=1.25\times10^6\,\mathrm{rad/m}.$$$$\boxed{\Delta\lambda_{2p}=2p/(f\mathcal D)=0.0800\,\mathrm{nm}}.$$

(b) Interpretation and further calculation

$$\boxed{\Delta\lambda_d=\lambda/N=600/5000=0.120\,\mathrm{nm}}.$$$$\Delta\lambda_d>\Delta\lambda_{2p}\Longrightarrow\boxed{\text{diffraction is the larger limit; its width spans }3\text{ pixels}}.$$

Physical check

Fine pixels can sample an optical line well, but they cannot recover spectral detail removed by the optical response.