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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]

Quadrature detection measures a signal against two reference oscillations separated by a quarter cycle. The resulting in-phase and quadrature components retain both amplitude and relative phase; a single component can vanish merely because the signal and reference are out of phase. A two-channel lock-in amplifier is a concrete implementation.^[ref]

For a signal $s(t)=A\cos (\omega t+\varphi )$ with constant $A$ and $\varphi$, choose the normalized references $2\cos (\omega t)$ and $-2\sin (\omega t)$. Multiplication followed by a low-pass filter that removes terms at $2\omega$ givesThus $A=\sqrt{I^2+Q^2}$, while the pair $(I,Q)$ specifies phase including its quadrant. For $\varphi =\pi /2$, the in-phase reading is zero but the quadrature reading is $A$: the signal has not disappeared. The sign of $Q$ depends on the chosen reference convention.

In an actual detector, finite filter bandwidth admits nearby frequencies and noise. Synchronization does not reject an unwanted signal at exactly the same reference frequency. If the reference is detuned, the output components rotate at the difference frequency rather than remaining constant.^[ref]