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Diffraction of Light

Problem 5.143

The ultimate resolving power λ/δλ\lambda / \delta \lambda of the spectrograph's trihedral prism is determined by diffraction of light at the prism edges (as in the case of a slit). When the prism is oriented to the least deviation angle in accordance with Rayleigh's criterion, λ/δλ=b∣dn/dλ∣ \lambda / \delta \lambda=b|d n / d \lambda| where bb is the width of the prism's base (Fig. 5.28),), and dn/dλd n / d \lambda is the dispersion of its material. Derive this formula.

Reveal Answer
5.143. According to Rayleigh's criterion the maximum of the line of wavelength λ\lambda must coincide with the first minimum of the line of wavelength λ+δλ\lambda+\delta \lambda. Let us write both conditions for the least deviation angle in terms of the optical path differences for the extreme rays (see Fig. 5.28): bn−(DC+CE)=0,b(n+δn)−(DC+CE)=λ+δλ∙ b n-(D C+C E)=0, \quad b(n+\delta n)-(D C+C E)=\lambda+\delta \lambda_{\bullet} Hence, bδn≈λ.b \delta n \approx \lambda . What follows is obvious.
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