Gratings: Dispersion and Resolving Power — Question 8

Question 8

Adjacent-order coincidences give unequal exact wavelength gaps

A spectral line at $\lambda=600\,\mathrm{nm}$ is measured in order m=10. At the same angle, order 11 admits a shorter wavelength and order 9 a longer wavelength. Assume these orders are supported by the instrument.

1. Calculate both coincident wavelengths.

2. Calculate the blue and red gaps from 600 nm and compare with the large-order estimate $\lambda/m$.

Question 8 - Solution

Adjacent-order coincidences give unequal exact wavelength gaps

(a) Method and calculation

$$10\lambda=11\lambda_b\Longrightarrow\boxed{\lambda_b=6000/11=545.455\,\mathrm{nm}}.$$$$10\lambda=9\lambda_r\Longrightarrow\boxed{\lambda_r=6000/9=666.667\,\mathrm{nm}}.$$

(b) Interpretation and further calculation

$$\boxed{\lambda-\lambda_b=\lambda/(m+1)=54.5455\,\mathrm{nm},\quad\lambda_r-\lambda=\lambda/(m-1)=66.6667\,\mathrm{nm}}.$$$$\boxed{\lambda/m=60.0\,\mathrm{nm}}\quad\text{is only the large-m approximation}.$$

Physical check

The exact neighboring-order coincidences are not symmetrically placed in wavelength about the chosen line.