Grating Resolving Power and Dispersion

Look at a sodium streetlamp through a good diffraction grating and the single wash of yellow splits cleanly into two razor-thin lines, 589.0 and 589.6 nanometres apart — a six-tenths-of-a-nanometre gap that the naked eye can never see. That splitting is the whole business of spectroscopy, and a grating is its workhorse: a periodic array of thousands of finely ruled lines that fans white light into a spectrum far more sharply than any prism. The question this article answers is quantitative. Exactly how close can two wavelengths lie and still be told apart, and what property of the grating decides it? Two numbers settle everything. The first is the angular dispersion, how many milliradians of angle a nanometre of wavelength buys — the scale over which colours march apart across the focal plane. The second is the resolving power $R=\lambda/\Delta\lambda$ , the smallest fractional wavelength difference the instrument can separate. The striking result, worked out by Rayleigh in the 1870s, is that it collapses to the almost embarrassingly simple $R=mN$ : the diffraction order $m$ times the number of grooves $N$ the beam actually illuminates. Not the ruling density, not the spacing, not the focal length — just how many lines cooperate, and in which order. That simplicity hides a competition. Dispersion pushes two wavelengths apart in angle; the finite width of each interference peak fights to keep them merged. The peak narrows as more grooves join in, and when the Rayleigh criterion is applied the grating spacing cancels exactly, leaving $mN$ . Understanding that cancellation — and the free-spectral-range price of going to high order — is the core of grating spectroscopy from the ultraviolet to the X-ray.

Key takeaways

A grating's angular dispersion is $\mathcal D=\mathrm d\theta_m/\mathrm d\lambda=m/(d\cos\theta_m)$ : it grows with order $m$ , with fine rulings (small $d$ ), and without bound as the diffracted beam approaches grazing.
Its chromatic resolving power is $R=\lambda/\Delta\lambda=mN$ — the order times the number of coherently illuminated grooves, independent of the spacing $d$ and of the incidence angle.
$R=mN$ follows from equating the Rayleigh criterion's angular shift $\mathcal D\,\Delta\lambda$ to the peak-to-first-zero half-width $\Delta\theta_{1/2}=\lambda/(Nd\cos\theta_m)$ ; the factor $d\cos\theta_m$ cancels.
On the detector the useful figure is the reciprocal linear dispersion (plate factor) $\mathrm d\lambda/\mathrm dx=d\cos\theta_m/(mf)$ , in nm per mm.
The absolute ceiling is $R_{\max}=W(\sin\theta_i+\sin\theta_m)/\lambda\to 2W/\lambda$ , fixed only by the ruled width $W$ and set by using near-grazing angles (the echelle strategy).
Key idea: a grating's resolving power is $R=mN$ — order times illuminated groove count, nothing else — because dispersion and peak width both carry the same $d\cos\theta_m$ , which cancels when the Rayleigh criterion is applied.

The core insight

The chromatic resolving power of a diffraction grating is $R=\lambda/\Delta\lambda=mN$ , the diffraction order times the number of coherently illuminated grooves. It does not depend on the line spacing or the mounting geometry, because the angular dispersion $\mathcal D=m/(d\cos\theta_m)$ and the angular half-width of a principal maximum $\Delta\theta_{1/2}=\lambda/(Nd\cos\theta_m)$ share the identical factor $d\cos\theta_m$ , which cancels the moment the Rayleigh criterion equates them. Resolving power counts the total path difference across the illuminated grating in units of a wavelength — nothing more.

Where this lives today

Every dispersive spectrograph is an exercise in $R=mN$ and $\mathcal D=m/(d\cos\theta)$ . Grating monochromators select a narrow band by rotating the grating, the plate factor fixing how finely the exit slit slices the spectrum. In astronomy the flagship instruments are cross-dispersed echelle spectrographs such as HARPS and ESPRESSO, which reach $R\sim10^5$ by combining high order with a large illuminated width, resolving stellar absorption lines finely enough to detect the reflex wobble a planet imposes on its star. The same relations run to the spectrum's extremes. At X-ray and extreme-ultraviolet wavelengths, where materials barely reflect except at grazing incidence, spectrometers work near $\theta\to90^\circ$ to exploit the $1/\cos\theta_m$ surge in dispersion — solar observatories use them to resolve the closely spaced emission lines of highly ionised iron in the corona. And in the laser lab a grating in Littrow mount ( $\theta_i=\theta_m$ ) forms the wavelength-selective end of a tunable external cavity, its $R=mN$ over the beam footprint setting the linewidth delivered to the experiment.

The physical picture

Treat the $N$ illuminated grooves as $N$ tiny coherent antennas. At a principal maximum every antenna radiates in phase, the amplitudes add straight up as $N$ , and the intensity peaks as $N^2$ . Swing the angle a hair off that peak and the antennas fan out in phase; the peak dies completely when the phase spread from the first groove to the last reaches $2\pi$ — when the extreme rays differ by exactly one wavelength across the full width $W=Nd$ . More grooves, a tighter tolerance, a sharper peak. It is a Fourier statement in disguise: a wider illuminated aperture always makes a narrower diffraction feature. Resolving two nearby wavelengths is therefore a tug-of-war. Dispersion, set by $m/d$ , drags the peaks apart in angle; the peak width, set by $\lambda/(Nd)$ , keeps them overlapping. Carry the algebra through and the spacing $d$ drops out of the ratio entirely, leaving the clean survivor $R=mN$ . The first figure catches the tug-of-war at a photo finish — the Rayleigh criterion, one peak sitting on its neighbour's first zero.

Figure

Rayleigh's criterion on a grating. Left: two wavelengths one order apart, separated by exactly the peak-to-first-zero half-width Δθ = λ / (N d cosθ) — the summed profile (ink) shows the 8/π² ≈ 19% saddle dip and the lines are just resolved. Right: at about half that spacing the dip fills in and the pair merges into a single unresolved blob.

Push the number of grooves up and the peaks turn into needles. That is why a laboratory grating a few centimetres wide, ruled at 600 lines per millimetre, splits the sodium doublet not just barely but with a factor of thirty to spare. The next figure carries the real numbers.

Figure

Punchline: the sodium D doublet split by a 600 line/mm grating ruled over 50 mm. The beam fills N = 30 000 grooves, so in first order R = mN = 30 000 and the resolution limit is δλ = λ/R = 19.6 pm — thirty times finer than the 0.597 nm gap between the D lines. Angular dispersion 𝒟 = 0.649 mrad/nm.

The core equations

The spectroscopic worth of a grating is captured by two independent numbers — how far apart it throws colours, and how finely it can split them — plus a ceiling that no mounting can beat. Start with the headline result. Applying the Rayleigh criterion (the peak of one line falling on the first zero of its neighbour) to the $N$ -groove interference pattern gives the chromatic resolving power, and the answer is spectacularly clean.

R \equiv \frac{\lambda}{\Delta\lambda} = mN \qquad (1)

where: $R$ — dimensionless resolving power; $\lambda$ — working wavelength; $\Delta\lambda$ — smallest resolvable wavelength difference; $m$ — diffraction order; $N$ — number of grooves coherently illuminated by the beam. Read physically, $R$ is the total optical path difference across the illuminated grating measured in wavelengths: order $m$ contributes $m$ wavelengths per groove, and there are $N$ of them. Where the colours actually land is governed by the angular dispersion, obtained by differentiating the grating equation $d(\sin\theta_i+\sin\theta_m)=m\lambda$ at fixed incidence. It tells you the rate at which the diffraction angle sweeps with wavelength.

\mathcal D \equiv \frac{\mathrm d\theta_m}{\mathrm d\lambda} = \frac{m}{d\cos\theta_m} \qquad (2)

where: $\mathcal D$ — angular dispersion (radians per unit wavelength); $d$ — groove spacing; $\theta_m$ — diffraction angle of the working order. Dispersion climbs with order, with finer rulings (smaller $d$ ), and diverges as $\theta_m\to90^\circ$ — the grazing regime that echelle gratings exploit. A spectrograph records position on a detector, not angle, so the practical quantity is the reciprocal linear dispersion, or plate factor: how many nanometres of spectrum fall on each millimetre of detector after a camera lens of focal length $f$ . It is simply the inverse of $f\mathcal D$ .

\frac{\mathrm d\lambda}{\mathrm dx} = \frac{d\cos\theta_m}{mf} \qquad (3)

where: $x$ — position in the focal plane, $x\approx f\theta_m$ ; $f$ — camera focal length. A small plate factor (few nm/mm) means a widely spread, finely sampled spectrum; a long camera and a high order both shrink it. Finally, resolving power cannot grow forever: since $R=mN$ and the grating equation caps $m(\lambda)$ through the ruled width $W=Nd$ , there is a hard ceiling reached only by pushing both angles toward grazing.

R_{\max} = \frac{W(\sin\theta_i+\sin\theta_m)}{\lambda} \;\xrightarrow[\;\theta_i,\theta_m\to90^\circ\;]{}\; \frac{2W}{\lambda} \qquad (4)

where: $W=Nd$ — total ruled (and illuminated) width; $\theta_i,\theta_m$ — incidence and diffraction angles. No grating can resolve better than twice its ruled width in wavelengths; the echelle configuration approaches this limit by working at large $\theta$ .

Derivation AResolving power R = mN from the width of a principal maximum

The aim here is to show that the tidy formula $R=mN$ is not an empirical fit but the exact outcome of a competition between two angular scales — how far dispersion separates two wavelengths, and how wide each interference peak is — and to watch the grating spacing cancel identically so that only the order and the illuminated groove count survive. Every quantity in the argument traces back to the $N$ -groove interference factor and the grating equation.

Step 1. Locate the principal maxima. The $N$ -groove interference factor $[\sin(N\gamma)/\sin\gamma]^2$ , with adjacent-groove phase $2\gamma=(2\pi d/\lambda)(\sin\theta_i+\sin\theta_m)$ , peaks where every groove is in phase, $\gamma=m\pi$ — precisely the grating equation. Why: constructive addition of all $N$ amplitudes requires the interslit phase to be a whole multiple of $2\pi$ , which is the same statement as an integer-wavelength path difference between neighbours.

Step 2. Find the first zero beside a peak. The numerator $\sin(N\gamma)$ vanishes when $N\gamma=(Nm\pm1)\pi$ , i.e. one step $\Delta\gamma=\pi/N$ away from the principal maximum. Converting phase to angle through $\mathrm d\gamma=(\pi d\cos\theta_m/\lambda)\,\mathrm d\theta_m$ gives the peak-to-first-zero half-width $\Delta\theta_{1/2}=\lambda/(Nd\cos\theta_m)$ . Why: the peak collapses when the extreme rays across the full width $W=Nd$ differ by one wavelength, so a wider grating makes a proportionally narrower peak.

Step 3. Impose the Rayleigh criterion. Two wavelengths $\lambda$ and $\lambda+\Delta\lambda$ are just resolved when the peak of one sits on the first zero of the other — when their angular separation $\mathcal D\,\Delta\lambda=(m/d\cos\theta_m)\,\Delta\lambda$ equals the half-width $\Delta\theta_{1/2}$ . Why: at exactly this spacing the summed profile shows the shallow $8/\pi^2\approx0.81$ saddle, the conventional threshold at which the human eye or a detector still registers two peaks rather than one.

Step 4. Equate and cancel. Setting $\dfrac{m\,\Delta\lambda}{d\cos\theta_m}=\dfrac{\lambda}{Nd\cos\theta_m}$ , the entire geometric factor $d\cos\theta_m$ divides out, leaving $\lambda/\Delta\lambda=mN$ . Why: because dispersion and peak width both scale as $1/(d\cos\theta_m)$ , the mounting geometry and ruling density are irrelevant to the resolving power — only the order and the illuminated groove count matter. *Verified: symbolic cancellation of $d\cos\theta_m$ and the numeric anchor N = 30000, R = 30000, δλ = 19.6 pm at 589.3 nm in verify_aq-optics-grating-resolving-power-dispersion.py.*

Go deeper · Derivation B · 1 minAngular dispersion, plate factor, and the resolving-power ceiling

Derivation BAngular dispersion, plate factor, and the resolving-power ceiling

This card builds the three geometric figures of merit — angular dispersion, its detector-plane counterpart the plate factor, and the absolute ceiling $R_{\max}$ — from a single differentiation of the grating equation, so that the reader can predict where each colour lands, how densely the spectrum samples a detector, and why even an infinitely finely ruled grating has a finite resolving power set only by its width. All three fall out of one line of calculus and one inequality.

Step 1. Differentiate the grating equation. Holding $\theta_i$ fixed in $d(\sin\theta_i+\sin\theta_m)=m\lambda$ and taking $\mathrm d/\mathrm d\lambda$ gives $d\cos\theta_m\,\mathrm d\theta_m=m\,\mathrm d\lambda$ , hence the angular dispersion $\mathcal D=m/(d\cos\theta_m)$ . Why: only $\theta_m$ and $\lambda$ vary within one order, and the $\cos\theta_m$ appears because equal steps in $\sin\theta_m$ map to ever-larger steps in $\theta_m$ as the beam tilts toward grazing.

Step 2. Convert angle to detector position. A camera of focal length $f$ maps angle to position by $x\approx f\theta_m$ , so the linear dispersion is $\mathrm dx/\mathrm d\lambda=f\mathcal D=fm/(d\cos\theta_m)$ ; its reciprocal is the plate factor $\mathrm d\lambda/\mathrm dx=d\cos\theta_m/(mf)$ . Why: spectroscopists quote nanometres per millimetre because it states directly how much spectrum a detector pixel subtends, the quantity that fixes sampling.

Step 3. Derive the ceiling. Multiply the grating equation by $N$ : $Nd(\sin\theta_i+\sin\theta_m)=Nm\lambda$ , i.e. $W(\sin\theta_i+\sin\theta_m)=(mN)\lambda=R\lambda$ . Solving, $R=W(\sin\theta_i+\sin\theta_m)/\lambda$ , which is largest when both angles approach $90^\circ$ , giving $R_{\max}=2W/\lambda$ . Why: the resolving power is the number of wavelengths spanned by the total path difference across the aperture, and no geometry can make that exceed twice the ruled width divided by the wavelength. *Verified: 𝒟 = 0.649 mrad/nm = 0.03716°/nm and plate factor 3.08 nm/mm at λ = 633 nm, θ = 22.32°, f = 0.5 m, and R_max = 1.70×10⁵ for W = 50 mm in verify_aq-optics-grating-resolving-power-dispersion.py.*

Limiting cases

Case	Result	Meaning
Two elements, $N=2$	$R=2m$	a double slit is a grating with almost no resolving power — broad $\cos^2$ fringes, no sharp orders
Small angle, $\theta_m\to0$	$\mathcal D\to m/d$	near normal incidence dispersion is just $m/d$ ; finer rulings disperse more per nanometre
Grazing / high order, $\theta_m\to90^\circ$	$\mathcal D\to\infty$ , $R\to R_{\max}=2W/\lambda$	dispersion diverges but resolving power saturates at the width-limited ceiling (echelle regime)
More grooves, $N\uparrow$	$R=mN$ grows, FSR fixed	enlarging the illuminated grating buys resolving power without shrinking the free spectral range
Finer ruling only, $d\downarrow$	$\mathcal D$ up, $R$ unchanged	dispersion is not resolving power — a finer grating spreads colours more but does not resolve them better unless $N$ grows

Common trapA finer grating — more lines per millimetre — always has higher resolving power

This is the single most common grating error, and it confuses two things. Ruling density (lines per millimetre, $1/d$ ) controls angular dispersion, $\mathcal D=m/(d\cos\theta_m)$ — how far apart colours fan out. Resolving power is $R=mN$ , which contains no $d$ at all; it depends only on the order and the grooves the beam actually bathes in coherent light. A coarse grating flooded across a wide aperture can out-resolve a fine one lit over a narrow patch, because it presents more cooperating grooves. Two further traps ride along. First, $N$ is the illuminated groove count, not the total ruled on the blank — underfill the grating and you throw away resolving power. Second, dispersion buys nothing on its own: spreading two broad peaks farther apart does not separate them. Always compute $R=mN$ for resolving power and reserve $\mathcal D$ for where the light lands.

Worked examples

1 · Splitting the sodium doublet (confidence builder)

A grating ruled at 600 lines/mm over a width of 50 mm is fully illuminated and used in first order at $\lambda=589.3$ nm. Find $N$ , the resolving power, the smallest resolvable wavelength difference, and check whether it separates the sodium D lines at 589.0 and 589.6 nm.

Step 1. Count the grooves. At 600 lines/mm over $W=50$ mm the beam fills $N=600\times50=\mathbf{30\,000}$ grooves. This is the only groove number that matters — the illuminated count.

Step 2. Resolving power. In first order, $R=mN=1\times30\,000=\mathbf{30\,000}$ . The order multiplies the groove count directly; nothing about the 600 lines/mm or the focal length enters.

Step 3. Resolution limit. The smallest resolvable step is $\Delta\lambda=\lambda/R=589.3\ \text{nm}/30\,000=0.01964\ \text{nm}=\mathbf{19.6\ \text{pm}}$ . That is the width of one resolution element at this wavelength.

Step 4. Compare with the doublet. The D lines lie precisely at 588.995 and 589.592 nm, a separation $\Delta\lambda_{\text{Na}}=0.597$ nm (about six-tenths of a nanometre), so resolving them needs only $R=589.3/0.597\approx\mathbf{987}$ . The grating delivers 30 000 — a factor $30\,000/987\approx\mathbf{30}$ in hand. The doublet is resolved roughly thirty times over: N = 30 000, R = 30 000, δλ = 19.6 pm, versus the 0.597 nm gap requiring only R ≈ 987.

2 · Dispersion, plate factor, and the ceiling (exam level)

The same 600 lines/mm grating ( $d=1666.67$ nm) is used in first order at $\lambda=633$ nm (a HeNe laser), where $\theta_1=22.32^\circ$ . Find the angular dispersion, the plate factor for a camera of focal length $f=0.5$ m, and the ultimate resolving power a 50 mm ruling could reach.

Step 1. Angular dispersion. With $\cos\theta_1=\cos22.32^\circ=0.925$ , $\mathcal D=m/(d\cos\theta_1)=1/(1666.67\times10^{-9}\times0.925)=6.49\times10^{5}\ \text{rad/m}$ . In friendlier units that is $\mathbf{0.649\ \text{mrad/nm}}=0.0372^\circ/\text{nm}$ : a nanometre of wavelength shifts the beam by two-thirds of a milliradian.

Step 2. Plate factor. The reciprocal linear dispersion is $\mathrm d\lambda/\mathrm dx=d\cos\theta_1/(mf)=(1666.67\times10^{-9}\times0.925)/(1\times0.5)=3.08\times10^{-6}$ , i.e. $\mathbf{3.08\ \text{nm/mm}}$ . Every millimetre of detector carries about 3 nm of spectrum, so a 25 µm pixel samples $\approx0.077$ nm — far coarser than the 19.6 pm resolution limit, a reminder that detector sampling, not just the grating, can be the bottleneck.

Step 3. The ceiling. Even pushed to grazing, this ruled width cannot beat $R_{\max}=2W/\lambda=2\times0.050/(589.3\times10^{-9})=\mathbf{1.70\times10^{5}}$ . 𝒟 = 0.649 mrad/nm, plate factor 3.08 nm/mm at f = 0.5 m, and a width-limited ceiling R_max ≈ 1.70×10⁵.

Physics of the portrait

Resolving power is not a property of any one instrument — it is the universal yardstick for how finely a telescope, microscope, or spectrograph distinguishes two neighbouring signals, and for a grating it takes its purest form.

Origins · 1 minresolving power and how this was discovered

resolving power

the figure of merit R = λ/Δλ, which for a grating equals the order times the illuminated groove count, mN · spectroscopy and instrument science

Resolving power began as a practical anxiety: given two spectral lines a whisker apart, will my instrument show them as two, or smear them into one? Lord Rayleigh gave the question its enduring answer in the 1870s by insisting on a definite convention — two features are "just resolved" when the diffraction peak of one falls on the first zero of the other — and computing the consequence for a grating. The result, $R=mN$ , is remarkable for what it omits: not the groove spacing, not the wavelength, not the focal length, only the order and the number of cooperating grooves. It is really bookkeeping on optical path difference: the grating resolves as many parts in $\lambda$ as there are wavelengths of path spread across its illuminated face. The same criterion on a circular aperture yields the $1.22\lambda/D$ of telescopes and the $\lambda/2\,\mathrm{NA}$ of microscopes — the grating is simply the version where the aperture is filled with a periodic ruling. And because $R=mN$ can be raised through $m$ (at the cost of free spectral range) or $N$ (at the cost of size), it turned spectrograph design into a clean engineering trade — the one that separates a teaching monochromator from an instrument measuring a star's radial velocity to a metre per second.

History

1786

— David Rittenhouse strings a grating of fine hairs across two screws and counts its diffracted orders

1821

— Joseph von Fraunhofer rules precision gratings and uses them to measure spectral wavelengths directly

1874

— Lord Rayleigh publishes his theory of the resolving power of diffraction gratings

1879

— Rayleigh sets out the resolution criterion (one peak on the neighbour's first minimum) in his "Investigations in optics"

1888

— Henry Rowland maps the solar spectrum with concave ruled gratings, cataloguing thousands of lines

1898

— Albert Michelson builds the echelon grating, reaching very high order and enormous resolving power

Going further

Texts. Hecht, Optics, 5th ed., §10.2.7 (the diffraction grating, dispersion and resolving power) · Foot, Atomic Physics, Ch. 2 and appendix on spectrographs · Demtröder, Laser Spectroscopy, Vol. 1, Ch. 4 (spectroscopic instrumentation) · Saleh and Teich, Fundamentals of Photonics, 3rd ed. (diffraction gratings) · Loudon, The Quantum Theory of Light, 3rd ed. (interference and diffraction).

Provenance. The grating equation, the $N$ -groove intensity, $R=mN$ , the angular dispersion $m/(d\cos\theta_m)$ , and the free-spectral-range coupling are standard textbook results. A verifier should reproduce: the cancellation of $d\cos\theta_m$ that yields $R=mN$ (Hecht §10.2.7); the sodium-doublet requirement $R\approx589/0.6\approx987$ (Foot); the angular dispersion and plate-factor numbers at $\lambda=633$ nm; and the ceiling $R_{\max}=2W/\lambda$ . All numbers here are recomputed from the locked cluster constants ( $d=1666.67$ nm, $\theta_1=22.32^\circ$ ); none are fabricated. Facts on order overlap and the resolving-power trade appear in Demtröder and in Saleh and Teich.

Prerequisites. Diffraction Grating and the Grating Equation · Single-Slit Diffraction · Rayleigh Resolution Criterion · Huygens–Fresnel Principle. Leads to. Free Spectral Range and Grating Orders · Blazed Gratings · Fabry–Perot Interferometer · Echelle Spectrograph.

Why does the resolving power not depend on the line spacing d?

Because $d$ enters the two competing angular scales identically and cancels. Angular dispersion carries a factor $1/(d\cos\theta_m)$ , so a finer grating spreads colours farther apart; but the angular half-width of each principal maximum also carries $1/(d\cos\theta_m)$ , so a finer grating gives correspondingly narrower peaks that would need to be spread farther to separate. When the Rayleigh criterion equates the separation to the width, the $d\cos\theta_m$ divides out on both sides, leaving $R=mN$ . Physically, $R$ counts the total optical path difference across the illuminated grating in units of a wavelength, and that count is $mN$ regardless of how the grooves are spaced.

What is the difference between dispersion and resolving power?

Dispersion is where the light goes; resolving power is whether you can tell two colours apart. Angular dispersion $\mathcal D=m/(d\cos\theta_m)$ says how many radians (or millimetres, after a camera) a nanometre of wavelength buys. Resolving power $R=mN$ says the smallest fractional wavelength difference the instrument can separate. A grating can have huge dispersion and poor resolving power (fine ruling, few illuminated grooves) or modest dispersion and superb resolving power (coarse ruling, an enormous illuminated width). They coincide only through the peak width: high dispersion helps resolution only if the peaks are narrow, which requires many grooves.

Is N the number of lines ruled on the grating or the number the beam illuminates?

Only the illuminated ones count. Resolving power measures how many grooves actually add coherently, so if your beam footprint covers 10 000 grooves of a grating ruled with 100 000, your resolving power is $m\times10\,000$ , not $m\times100\,000$ . This is why spectrograph designers work hard to fill the whole grating with a collimated beam, and why an oversized grating underfilled by a small beam is wasted glass. The practical rule: compute $N$ from the illuminated width, $N=W_{\text{illum}}/d$ , never from the manufacturer's total line count.

Why go to a higher order instead of just using more grooves?

Both raise $R=mN$ , but at different costs. Adding grooves (a larger, well-figured grating) raises resolving power while leaving the free spectral range $\lambda/m$ untouched — the clean way to gain capability, at the price of size and the difficulty of ruling a large surface accurately. Going to higher order raises $R$ just as effectively with a modest grating, but shrinks the free spectral range in exact proportion, so adjacent orders overlap and must be sorted with a filter or a cross-disperser. High order is the echelle strategy: cheap resolving power, paid for with order-overlap management.

Does the incidence angle change the resolving power?

Not the fundamental $R=mN$ . Both the angular dispersion and the peak half-width depend on the mounting through $\cos\theta_m$ , but that factor cancels, so the number of resolvable elements is fixed by order and groove count however you tilt the grating. What the geometry does change is the ceiling: $R=W(\sin\theta_i+\sin\theta_m)/\lambda$ shows that large incidence and diffraction angles push the maximum achievable $R$ toward $2W/\lambda$ . That is why echelle gratings work near grazing — not to change $mN$ at a given order, but to make very high orders geometrically accessible.

How does the plate factor relate to detector pixels and sampling?

The plate factor $\mathrm d\lambda/\mathrm dx=d\cos\theta_m/(mf)$ times the pixel pitch gives the wavelength interval per pixel. In the worked example, 3.08 nm/mm across a 25 µm pixel is about 0.077 nm per pixel — far coarser than the grating's own 19.6 pm resolution limit. When that happens the detector, not the grating, sets the achievable resolution, and you must lengthen the camera focal length (smaller plate factor) or raise the order. Good design keeps one resolution element spread over at least two pixels (Nyquist), so the plate factor and $R$ are matched rather than one wasting the other.