 Solution

The first-order grating equation is $d\sin \theta =\lambda$, so $d=2.879\mu \mathrm{m}$, about $347$ lines per millimetre. For $N$ equally illuminated lines,Principal maxima stay at the same angles, their peaks scale as $N^2$ for fixed illumination per line, and their angular widths shrink as $1/N$. The resolving power of a diffraction grating is $\lambda /\Delta \lambda =mN$. Thus first order needs $N=1000$ and illuminated width $Nd\simeq 2.88\mathrm{mm}$.

The grating dispersion follows by differentiation: $d\theta /d\lambda =1/(d\cos \theta )$. With a detector centred on the chief ray, one pixel corresponds approximately toThis exceeds the nominal $0.5\mathrm{nm}$ grating width. The practical resolution scale is therefore a few nanometres; two-pixel sampling gives about $4.5\mathrm{nm}$. Exact resolvability also depends on slit width and image sampling. Taking the detector coordinate as $f\tan \theta$ about the original axis instead changes the single-pixel estimate only to $2.20\mathrm{nm}$.

For ideal photon counting, Poisson distribution statistics give SNR $=\sqrt{N_{\gamma }}$. Hence $N_{\gamma }=10^4$ and $t_{\min }=10^4/(2\times 10^5)=0.050\mathrm{s}$.

Use a lock-in amplifier for phase-sensitive detection: chop or otherwise modulate the source before the entrance slit, retaining a synchronous reference. At each grating setting amplify the photodiode output, multiply it by a reference and low-pass filter. For a signal $s\cos ({\omega }_{r}t+\varphi )$ and reference $2\cos {\omega }_{r}t$, the low-pass output is $s\cos \varphi$; quadrature demodulation also recovers $s\sin \varphi$. Choose the modulation frequency above slow drift while within the source/detector bandwidth and away from strong electrical interference. Average for many cycles before changing the grating setting.

This rejects steady backgrounds and reduces low-frequency noise through narrow-band detection. It does not remove random dark-current/stray-light shot noise within that bandwidth, nor a background modulated coherently with the source. The chopping geometry must therefore distinguish wanted light from stray paths.^[ref]