Past exam of the physics A course of the University of Cambridge / 2011 / Paper 1 / Section B / B8

At screen coordinate $X$, the propagation length from aperture coordinate $x$ is $[D^2+(X-x{)}^2{]}^{1/2}$. Its expansion has a constant term, a term linear in $x$ giving the observation angle, and a quadratic term of order $x^2/(2D)$. For $|x|\le d/2$, the quadratic wave phase is at most of order $\pi d^2/(4\lambda D)$. Requiring it to be much smaller than one gives the sufficient condition $D\gg d^2/\lambda$; higher terms are then smaller in the paraxial geometry. In the lens arrangement, put the source at the first lens's front focus, illuminate the grating with its collimated output, and put the screen at the second lens's back focal plane. This obtains the angular Fraunhofer diffraction pattern without a physically very distant screen.

The slit pitch is $p=1/(200\mathrm{m}{\mathrm{m}}^{-1})=5.00\mu \mathrm{m}$. Normal-incidence maxima satisfy $p\sin \theta =m\lambda$. Thus $\sin {\theta }_3=0.276$ and, using rectilinear focal-plane geometry,The paraxial result $x_3\simeq fm\lambda /p=\pm 82.8\mathrm{mm}$ is a roughly 4% lower approximation; an actual wide-angle lens's mapping depends on its design. The angular result does not depend on that convention.

For $N$ uniformly illuminated equal slits of width $w$, the diffraction intensity has the formHere $C$ is unchanged when the incoming intensity is fixed. The broad single-slit diffraction envelope multiplies the narrow grating orders. At a principal maximum,The fixed-illumination grating peak change with slit width gives the changes from $w=1.0$ to $1.5\mu \mathrm{m}$ areThe central peak increases by 125%, while the third-order peak decreases by about 89.4%. Separate normalizations to each central peak would hide this effect. The following width-specific solutions and profiles use a common incoming normalization.

With $w=1.0\mu \mathrm{m}$ the order positions are unchanged from the grating equation, and the envelope's first zeros satisfy $|\sin \theta |=\lambda /w$. Equivalently the first missing integer orders are $m=\pm 5$. The third-order peaks remain appreciable. The displayed order-coordinate $s=p\sin \theta /\lambda$ places all principal maxima at integers.

With $w=1.5\mu \mathrm{m}$ the same principal maxima sit inside a narrower envelope, whose first zeros are at $s=\pm 10/3$. The third order is now close to a zero and is strongly suppressed, whereas the central peak is taller. The first exact missing nonzero integer orders are $m=\pm 10$; $m=3$ is suppressed rather than identically missing.

A $5.0\mathrm{mm}$ beam illuminates $N=5.0\mathrm{mm}/p=1000$ slits. The resolving power of a diffraction grating in order $m=3$ is $mN=3000$, corresponding to a Rayleigh criterion separation $\delta {\lambda }_R\simeq 460/3000=0.153\mathrm{nm}$. The $0.5\mathrm{nm}$ separation therefore gives two well separated main peaks. The finite-grating incoherent doublet profile adds the mutually incoherent lamp intensities, not their amplitudes.

A $1.5\mathrm{mm}$ beam illuminates only $N=300$ slits, giving $mN=900$ and $\delta {\lambda }_R\simeq 0.511\mathrm{nm}$. The two main lobes broaden by $1000/300$ compared with the wider beam. Their $0.5\mathrm{nm}$ separation is slightly below the stated Rayleigh threshold: marginally unresolved by that criterion. This is not an assertion that the profile has only one maximum: the ideal equal-line sum still has two maxima and a modest central dip, as plotted. Inference from real data also requires adequate signal-to-noise ratio and a known instrumental profile.