Gratings: Dispersion and Resolving Power — Question 6

Question 6

A finite entrance-slit image can dominate the spectral line width

A centered camera has focal length $f=0.500\,\mathrm m$. Its grating uses $d=1.00\,\mu\mathrm m$, m=1, $\lambda=500\,\mathrm{nm}$, and N=10000 illuminated slits. The entrance slit is imaged to width $w=50.0\,\mu\mathrm m$ at the detector. For this engineering estimate, combine slit and diffraction wavelength widths in quadrature.

1. Calculate the slit-equivalent and ideal diffraction wavelength widths.

2. Find the combined width and effective resolving power under the stated model.

Question 6 - Solution

A finite entrance-slit image can dominate the spectral line width

(a) Method and calculation

$$\theta=30^\circ,\quad\mathcal D=1/(d\cos\theta);\quad\Delta\lambda_s=w/(f\mathcal D).$$$$\boxed{\Delta\lambda_s=0.0866025\,\mathrm{nm},\quad\Delta\lambda_d=\lambda/(mN)=0.0500\,\mathrm{nm}}.$$

(b) Interpretation and further calculation

$$\boxed{\Delta\lambda_{\mathrm{eff}}=\sqrt{\Delta\lambda_s^2+\Delta\lambda_d^2}=0.1000\,\mathrm{nm}}.$$$$\boxed{\mathcal R_{\mathrm{eff}}=500/0.1000=5000}.$$

Physical check

The quadrature rule is an explicitly supplied approximation; exact line shapes require convolution of the slit image with the diffraction response.