Question 3 of 6: Design of a 0.5 m sodium-D monochromator
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
Paper format. National Examinations, December 2013, 98-Phys-A7 Optics. Three hours, closed book, one approved Casio or Sharp calculator. Question 1 (20 short parts) is mandatory; the rubric asks for any four of Questions 2–6 — all six are worked here, because the set is a study resource.
Reference texts. E. Hecht, Optics, 5th ed.; F. L. Pedrotti, L. M. Pedrotti and L. S. Pedrotti, Introduction to Optics, 3rd ed.
Question 3: Design of a 0.5 m sodium-D monochromator (20 marks)
Given. Sodium D doublet $\lambda_1=589.6$ nm, $\lambda_2=590.0$ nm ($\lambda\approx589.8$ nm, $\Delta\lambda=0.4$ nm); first order $m=1$; a “0.5 m” monochromator, i.e. collimating and focusing mirror focal lengths $f=500$ mm; extended source, so the resolution is not photon-starved and can be set by the grating/slits.
Find. A complete, self-consistent Czerny–Turner design (grating, angle, slit width) that resolves the doublet, with every dimension justified.
Approach. Size the grating from the required resolving power, fix the diffraction (Littrow) angle from the grating equation, propagate the angular dispersion through the 500 mm focal length to a linear dispersion at the exit plane, and set the slit width from that linear dispersion and the required $\Delta\lambda$.
Required resolving power. $$R=\frac{\lambda}{\Delta\lambda}=\frac{589.8}{0.4}=\boxed{1475}$$ (rounded up); since $R=mN$ at $m=1$, at least 1475 grating lines must be illuminated.
Grating choice. Take a stock ruled grating of 1200 grooves/mm ($d=1/1200\text{ mm}=833.3$ nm) illuminated over $W=25$ mm (comfortably within a standard 1 in.×1 in. blank): $$N=W\times1200\text{ mm}^{-1}=\boxed{30\,000\text{ lines}}$$ giving $R_{actual}=30\,000$, a $\times20$ margin over the 1475 required so the design is not sitting on the edge of resolving the doublet.
Grating angle (near-Littrow). Using the grating equation in the autocollimating (Littrow) form $m\lambda=2d\sin\theta_L$: $$\sin\theta_L=\frac{(1)(589.8\text{ nm})}{2(833.3\text{ nm})}=0.3540\Rightarrow \boxed{\theta_L=20.7^\circ}$$
Angular and linear dispersion. $$\frac{d\theta}{d\lambda}=\frac{m}{d\cos\theta_L}=\frac{1}{(833.3\text{ nm})\cos20.7^\circ}=1.283\times10^{-3}\text{ rad/nm}$$ $$\frac{dx}{d\lambda}=f\frac{d\theta}{d\lambda}=(500\text{ mm})(1.283\times10^{-3}\text{ rad/nm})=\boxed{0.642\text{ mm/nm}}$$ (reciprocal dispersion 1.56 nm/mm).
Line separation and slit width. The two D lines land $$\Delta x=\frac{dx}{d\lambda}\,\Delta\lambda=(0.642\text{ mm/nm})(0.4\text{ nm})=\boxed{257\ \mu\text{m}}$$ apart at the exit plane. Matched entrance and exit slits of width $\approx130\ \mu$m (half the line separation) give a clearly resolved dip between the two peaks; the grating’s own diffraction-limited resolution element at $R=30\,000$ is only $\lambda/R\approx0.020$ nm, so the design is slit-limited, not diffraction-limited, with ample margin.
Check: engineering assumption — groove density (1200/mm), illuminated width (25 mm) and slit width (130–260 μm) are a self-consistent design choice, not a unique solution; any grating/width combination giving R ≥ 1475 and a matched slit width satisfies the specification.
Part (b). Mirrors, not lenses, are used precisely because reflection is wavelength-independent — a spherical (or off-axis parabolic) mirror focuses every visible wavelength to the same point, so this design has essentially zero chromatic aberration. Spherical aberration (and, off-axis in a Czerny–Turner, coma) is the dominant defect: rays striking the outer part of a spherical mirror’s aperture — exactly the marginal rays needed to fill the full 25 mm grating width for the resolving power computed above — focus slightly short of the paraxial focus, blurring the slit image and setting a practical (aberration-limited) resolution that can be worse than the diffraction/slit estimate above unless the mirrors are used near normal incidence or replaced by off-axis parabolas/toroids.
