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A formal grating order must have a real outgoing angle. For normal incidence, $|m|\lambda \le d$ is necessary. Oblique incidence can access orders outside that normal-incidence range; reflection Littrow geometry has $m\lambda =2d\sin \theta$. The ideal resolving power of a diffraction grating applies only to an order that exists and has sufficient signal.

A grating spectrometer places an illuminated narrow slit at the front focal plane of a collimator. The resulting parallel beam reaches the diffraction grating at known incidence, here taken normal. A rotatable telescope focuses each outgoing diffracted parallel beam onto its crosshair or detector. Align the slit with the grating rulings, focus the collimator at infinity, and illuminate many grooves across a suitable aperture. A narrow slit supplies sufficient spatial coherence and prevents slit width from blurring the angular separation; the telescope must also resolve the two images.

The groove spacing is $d=(2000\times 100{)}^{-1}\mathrm{m}=5.00\mu \mathrm{m}$. The grating equation at normal incidence is $d\sin {\theta }_m=m\lambda$. For diffraction order $m=2$,ThereforeThe ideal resolving power of a diffraction grating is $|m|N$, so the Rayleigh criterion for this doublet requires about $N\ge 145$ illuminated grooves, spanning roughly $0.73\mathrm{mm}$. This is an ideal lower bound; finite slit width, aberrations and detector resolution can require a larger usable aperture.

One instrument is a diffraction grating spectrometer. Its principal maxima satisfy $d\sin \theta =m\lambda$ at normal incidence. N coherently illuminated grooves give a narrow principal peak; applying the Rayleigh criterion to the next wavelength yields the resolving power of a diffraction gratingHigher order and more illuminated grooves improve resolution, but finite entrance-slit width, aberrations, detector sampling and overlap between spectral orders limit useful resolution. Angular dispersion $d\theta /d\lambda =m/(d\cos \theta )$ is distinct from the ability to resolve two finite-width peaks.

A second instrument is a Fabry–Pérot interferometer. For plate spacing L, refractive index n and internal angle theta, transmission resonances satisfy $2nL\cos \theta =m\lambda$. Multiple coherent reflections produce an Airy transmission profile. For identical lossless mirrors of intensity reflectivity $\mathcal{R}$, the high-finesse limit givesThe frequency free spectral range is approximately $c/(2nL\cos \theta )$ for nondispersive n. A larger optical spacing narrows a line but also shrinks the unambiguous frequency interval; nearby orders need sorting. Mirror flatness, absorption, alignment and aperture broadening lower the effective finesse. This resolving-power formula assumes separated narrow lines and a stated line-separation convention, rather than equating transmission peak spacing with peak width.

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.

Near order $m$, the first angular zero from a principal maximum is $\delta \theta =\lambda /(ND\cos \theta )$. Two wavelengths have peak separation $\Delta \theta =m\Delta \lambda /(D\cos \theta )$. Equating them applies the Rayleigh criterion and gives the resolving power of a diffraction grating,Here the required resolving power is $400/0.2=2000$, so in second order $N\ge 1000$. The illuminated width is thereforeWhether one counts the full illuminated window or the first-to-last slit-centre distance changes the width by one pitch and is immaterial at this precision. The result assumes negligible aberrations and adequate slit illumination and coherence.

For $N$ equally spaced illuminated slits, with $N\simeq W/D$, summing a geometric series of complex amplitudes givesHere $I_1(\theta )$ is the one-slit single-slit diffraction wave envelope. The grating equation for principal maxima iswith height $N^2{I}_1({\theta }_m)$ and much weaker subsidiary maxima between orders. The first zeros around order $m$ are $\beta =m\pi \pm \pi /N$, so the angular distance from a maximum to its first minimum is, for large $N$,Changing wavelength by $\delta \lambda$ moves the same diffraction order byThe Rayleigh criterion places the maximum for one wavelength at the first zero for its neighbour. Equating these separations givesThis is the ideal resolving power of a diffraction grating, for an illuminated and transmitted order with sufficiently narrow source and instrumental broadening. The zeroth order has no wavelength separation and therefore no useful spectral resolving power. The distinction between $W=ND$ and the span $(N-1)D$ of the outer slit centres is a small edge convention; the exact state count in the resolving-power formula is $N$.

The required resolving power of a diffraction grating isFor $N=260$ illuminated slits, $\mathcal{R}=260|m|$. In orders $1,2,3,4$, the ideal resolvable wavelength differences are respectively $2.267$, $1.133$, $0.756$ and $0.567\mathrm{nm}$. ThusFourth order gives $\mathcal{R}=1040>982.2$, only a modest margin. It must also exist: for normal incidence the grating equation requires $D\ge 4\lambda \simeq 2.36\mu \mathrm{m}$, with $D>4\lambda$ needed to avoid an exactly grazing order. The relevant order must transmit appreciable light intensity rather than lie at a single-slit diffraction zero, and the source and detector widths must preserve this resolution.

The original question gives only the slit count, not the line spacing, slit width, usable diffraction order or instrumental broadening. It therefore supports the conditional conclusion “yes in fourth or higher usable order”, but not an unconditional historical claim that this particular instrument resolved the sodium absorption line doublet in its observed configuration.

Represent each wave by a complex phasor. Add phasors before squaring: two amplitudes give $I\propto |A_1+A_2{e}^{i\delta }{|}^2=A_1^2+A_2^2+2A_1{A}_2\cos \delta$. For $N$ equally illuminated narrow slits, adjacent phasors differ by $\delta =2\pi d\sin \theta /\lambda$, soAll phasors align when $\delta =2\pi m$, giving the grating equationThe nearest zero has adjacent-phase increment $2\pi /N$: the phasor polygon closes once. Thus $d\cos {\theta }_m\Delta \theta \simeq \lambda /N$, givingThe angular linearization is valid away from grazing directions; the underlying exact first-zero condition is $d(\sin {\theta }_{\mathrm{zero}}-\sin {\theta }_m)=\pm \lambda /N$.

For the first subsidiary maximum of a large slit array, use the stipulated $1.5$ phasor wraps: $N\delta /2\simeq 3\pi /2$ relative to a principal maximum. Its amplitude is approximately $2N/(3\pi )$ times one slit's amplitude. ThereforeThis uses the question's approximate maximum position, not the exact extremum of the slit-array intensity.

The Rayleigh criterion places one line's principal maximum at the other's first minimum. Grating dispersion then gives $|m|N\gtrsim \lambda /\delta \lambda =1.674\times 10^6$. Combining with normal incidence, $|m|\lambda =d|\sin \theta |$, gives the normal-incidence grating width bound:The $1.29\mathrm{m}$ value is an ideal lower bound approached near grazing diffraction. A usable finite angle needs greater width; for example $\theta =60^{\circ }$ requires about $1.49\mathrm{m}$. The exact sine first-zero relation still gives the lower bound when the local angular approximation fails near grazing.

Such a coherently illuminated grating is very large and needs accurate groove placement and suitable collection optics; an unaided far-field screen would require distances much greater than $W^2/\lambda \sim 2\times 10^6\mathrm{m}$. Focusing optics avoid that screen distance but not the required resolving power of a diffraction grating. Real linewidth, slit width and aberrations can require more resolution. The unequal isotope abundances also make this ideal Rayleigh estimate less decisive than for equally bright lines.

First intensity minimum. In Fraunhofer diffraction, the path difference between adjacent slits is $d\sin \theta$, so the relative wave phase is $\delta =2\pi d\sin \theta /{\lambda }_1$. Summing the $N$ equal complex amplitudes as a geometric series gives the diffraction grating interference factorThe central maximum has $\delta =0$. Its nearest zero occurs at $N\delta /2=\pi$, where the denominator is nonzero. ThusThere is a symmetric first zero on each side of the incident direction. The narrow slits are treated as having a sufficiently broad single-slit diffraction envelope.

Wavelength resolution and the missing order specification. Adopt the Rayleigh criterion: the principal maximum of one wavelength lies at the adjacent minimum of the other in the same nonzero diffraction order $m$. For ${\lambda }_2>{\lambda }_1$ and $m>0$, those directions obeyConsequently the ideal resolving power of a diffraction grating is ${\lambda }_1/\Delta \lambda =mN$, givingThe usual first-order reading therefore gives $64\mathrm{nm}$. Zero diffraction order cannot separate the two wavelengths, because both central maxima coincide.

The paper does not specify which diffraction order is observed, so $64\mathrm{nm}$ is not an order-independent minimum. In the ideal infinitely narrow-slit model, the largest usable order for this positive wavelength separation must satisfy $(m+1/N){\lambda }_1\le d$. Since $d/{\lambda }_1=125$, this permits $m\le 124$, giving the formal all-order resultThat pair meets the criterion at $\theta \simeq 83.1^{\circ }$. Order $125$ of ${\lambda }_1$ is at grazing incidence and the corresponding longer-wavelength peak cannot propagate. In a real device, finite slit width, weak high-order intensity and screen acceptance may prevent such an extreme order from being used; those data are absent. Thus the specified-order expression, and $64\mathrm{nm}$ when first order is intended, are the defensible resolution answers under the stated criterion.

For a diffraction grating with $N$ illuminated lines in order $p$, its ideal spectral resolving power isThe two spectral lines have separation $\Delta \lambda =3.4\mathrm{nm}$ and mean wavelength $\overline{\lambda }=395.1\mathrm{nm}$. In first order, $p=1$, so the Rayleigh criterion requiresWith line density $500\mathrm{m}{\mathrm{m}}^{-1}$, the ideal continuous estimate for the illuminated width isRounding the required number of illuminated grating lines upward gives $117$ lines, conventionally an illuminated width $W\simeq 117/(500\mathrm{m}{\mathrm{m}}^{-1})=\boxed{0.234\mathrm{mm}}$. Using the longer wavelength rather than the mean also gives the same integer requirement. This is the ideal limit set by the resolving power of a diffraction grating; additional broadening from the illumination or observing apparatus could require a larger width.

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]

A normal-incidence diffraction grating of pitch $D$ has order $m$ at $D\sin \theta =m\lambda$ and angular separation $\Delta \theta \simeq m\Delta \lambda /(D\cos \theta )$. A lens whose axis follows the selected order maps a small angular interval to $\Delta x\simeq f\Delta \theta$. Detector pixels substantially smaller than this separation are needed; two pixels per resolution interval correspond to $t\lesssim \Delta x/2$. If instead detector coordinates obey $x=f\tan \theta$ relative to the original axis, the scale is $\Delta x\simeq f{\sec }^2\theta \Delta \theta$. A numerical pixel limit therefore requires an optical layout and a sampling criterion as well as the resolving power of a diffraction grating.

For $N$ coherently illuminated grating lines used in diffraction order $m$, the Rayleigh criterion yields the displayed ideal resolving power of a diffraction grating. The angular maximum-to-first-zero separation is $\lambda /(ND\cos {\theta }_m)$, while nearby wavelengths are separated by $|m|\delta \lambda /(D\cos {\theta }_m)$. The order must exist, carry appreciable light intensity and remain free from dominant instrumental broadening. Line count alone therefore does not determine whether a particular instrument resolves a specified doublet.