= Solution
Differentiate the <grating equation> with incidence fixed:
$$
\frac{d\beta}{d\lambda}=\frac{m}{d\cos\beta},\qquad
q=\frac{mf_{\rm cam}}{d\cos\beta}.
$$
The second formula is local to a spectrum centered on the <optical camera> <optical axis>, where $dx=f_{\rm cam}\,d\beta$ to first order. Since the illuminated surface width $W$ projects to <optical camera> beam width $A_{\rm cam}=W\cos\beta$,
$$
\boxed{L=\frac{\lambda qA_{\rm cam}}{f_{\rm cam}}=\frac{Wm\lambda}{d}.}
$$
The expression takes positive <diffraction order> and width magnitudes. Off-axis, a <photodetector> mapping $x=f_{\rm cam}\tan(\beta-\beta_0)$ also has a $\sec^2(\beta-\beta_0)$ factor; the exam's compact formula uses the local paraxial mapping.
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