Alternative Resolution Criteria

Point two stars, or two fluorescent molecules, sufficiently close together and even an ideal diffraction-limited lens produces strongly overlapping images. Each point source is smeared by diffraction into an Airy pattern, and as the pair draws closer the two blurs slide into one until the sum looks like a single elongated smudge. Somewhere along that slide you must draw a line and declare "still two — just resolved," yet there is no unique place to draw it: the combined profile is smooth and continuous, and nothing physical snaps at a threshold. Every so-called resolution limit is therefore a convention — a rule for how shallow a dip, or how flat a plateau, still counts as two objects. The most famous convention is Rayleigh's: slide the peak of one Airy pattern onto the first dark ring of its neighbour, leaving a central dip about 26.5% below the peaks. It is a fine rule for the eye, but a rule, not a law. Sparrow's criterion waits longer — until the dip vanishes entirely and the profile is momentarily flat on top — a tighter, more physical threshold. Dawes' limit is pure empiricism distilled from decades of double-star observation; the half-power rule is a convenient shorthand; and Abbe's limit reframes the whole question as the range of spatial frequencies the lens collects. All of these numbers cluster within a factor of about 1.3 of one another, all scaling as $\lambda/D$ or $\lambda/\mathrm{NA}$ , differing only by the order-one coefficient their definition happens to pick. Reading them as operational criteria, rather than as a single hard wall, helps distinguish conventional diffraction-limited resolution from modern super-resolution methods.

Key takeaways

A resolution criterion turns the smooth overlap of two point-spread functions into one threshold separation; Rayleigh's rule (one Airy peak on the neighbour's first dark ring, a 26.5% dip) is only one choice among several.
The Sparrow criterion is tighter and more physical: two points are just resolved when the summed profile loses its central dip, $\mathrm{d}^2 I_{\text{tot}}/\mathrm{d}x^2|_0=0$ , giving $\theta_S\approx0.947\,\lambda/D$ — about $0.78$ of the Rayleigh value.
The Dawes ( $\approx1.02\,\lambda/D$ , written $116''/D_{\text{mm}}$ ) and half-power ( $\approx1.03\,\lambda/D$ ) limits sit between Sparrow and Rayleigh.
Abbe reframes resolution as spatial-frequency capture through the pupil: $d_{\min}=\lambda/2\,\mathrm{NA}$ for a coherent grating, with $\mathrm{NA}=n\sin\theta$ the master knob.
Key idea: resolution criteria assign different operational thresholds to diffraction-limited images. Rayleigh, Sparrow, Dawes, and half-power criteria give scales proportional to $\lambda/D$ for a circular aperture, while Abbe's criterion is formulated in terms of spatial-frequency transfer and scales with $\lambda/NA$ . None represents a universal hard boundary for extracting information from closely spaced sources.

The core insight

Two point sources imaged through a finite aperture produce two overlapping point-spread functions whose sum is a smooth curve. A resolution criterion is a chosen rule for reading "two-ness" off that curve — Rayleigh asks for a dip aligned with a dark ring, Sparrow asks only that a dip still exist, Abbe asks whether the pupil captured the object's spatial frequencies. All scale as $\lambda/D$ but differ by order-one coefficients set purely by the definition, so the diffraction limit is a soft convention, not a hard barrier.

Where this lives today

The choice of criterion is a live engineering decision. Telescopes with adaptive optics are specified against the Rayleigh limit their aperture would reach without the atmosphere, while double-star astrometry still quotes Dawes because it matches what an observer actually splits. Spectroscopists prefer Sparrow for close doublets, since it marks the point at which a merged line profile just keeps a dip — exactly what a peak-finder looks for. Whenever a detector rather than an eye decides "two," the arbitrary 26.5% Rayleigh dip gives way to a signal-to-noise-aware threshold. The deepest modern consequence is that the classical limits are routinely broken. Because these classical criteria describe conventional resolution benchmarks rather than universal information-theoretic boundaries, fluorescence super-resolution methods such as STED, PALM, and STORM can resolve or localize features well below the classical diffraction scale under appropriate experimental conditions; the achievable localization precision can improve substantially with photon number, subject to background, detector, sampling, and model limitations.

The physical picture

Watch two equally bright, mutually incoherent point images slide together. Rayleigh says "two" as long as a visible valley sits between the peaks, aligned with each pattern's first dark ring. Sparrow is stricter about what counts as merged: it says "two" right up until the valley flattens to a plateau — the last instant the profile has any dip at all. Push closer than the Sparrow spacing and the sum is a single rounded maximum that no contrast-stretching splits by eye. The figure shows this slide at three separations, with the tell-tale numbers on the plot.

Figure

Two incoherent Airy point-spread functions slid together at three separations; blue is their sum, and grey (left panel) shows the two overlapping components. At the Rayleigh spacing 1.22 λ/D the sum still has a 26.5% central dip. At the Sparrow spacing

0.947\,\lambda/D

the dip has just vanished into a flat top. Below that the pair no longer satisfies these conventional resolution criteria and appears as a single broadened profile.

Step back from the image plane, as Abbe did, and the same limit looks entirely different. Treat the object as a superposition of sinusoidal gratings; finer spatial structure corresponds to higher spatial frequencies, whose diffracted orders occur at larger angles. The lens catches only rays inside its acceptance cone — its numerical aperture — so any detail whose first diffracted order misses the pupil is never recorded. That is a hard cutoff in frequency, yet it lands on the same $\lambda/\mathrm{NA}$ scale. The next figure lines every classical number up on a single ruler.

Figure

The classical resolution criteria as one number line in units of λ/D for a circular aperture: Sparrow 0.947, Dawes 1.02, half-power 1.03, Rayleigh 1.22 — all within a factor 1.3 of one another. Tighter (smaller) numbers claim resolution at closer spacing; the looser Rayleigh number is the conservative default. Abbe's λ/2NA maps onto the same ladder.

Numerical aperture is the one physical knob under all of it: raising $\mathrm{NA}=n\sin\theta$ with a wider cone or oil immersion shrinks every criterion's spacing as $\lambda/\mathrm{NA}$ , the reason oil objectives reach $\mathrm{NA}\approx1.4$ .

The core equations

Everything below rests on one object: the point-spread function, the image a circular aperture forms of an ideal point. Diffraction through a round pupil of diameter $D$ gives the Airy pattern, whose intensity is the squared jinc function, derived in Circular Aperture Diffraction and the Airy Pattern. Its first dark ring, where $J_1$ first vanishes, sets the natural diffraction scale for several criteria discussed below.

I(x)=I_0\left[\frac{2J_1(x)}{x}\right]^2,\qquad x=\frac{\pi D}{\lambda}\sin\theta \qquad (1)

where: $I_0$ — on-axis peak intensity; $J_1$ — Bessel function of the first kind, order one; $D$ — aperture diameter; $\lambda$ — wavelength; $\theta$ — angle from the axis; $x$ — dimensionless radial variable, first zero at $x_1=3.8317$ . A point source is not a point in the image but a bright disk ringed by faint halos, and "resolution" asks when two such disks stop looking like one. Rayleigh's convention places the peak of one Airy pattern exactly on the first dark ring of the other. Because that ring sits at $x_1=3.8317$ , the just-resolved separation is fixed by the first Bessel zero, and the celebrated $1.22$ is nothing more than $x_1/\pi$ . For incoherent equal sources the summed profile then dips about 26.5% below the peaks.

\theta_R = 1.22\,\frac{\lambda}{D}=\frac{x_1}{\pi}\,\frac{\lambda}{D},\qquad \frac{x_1}{\pi}=\frac{3.8317}{\pi}=1.2197 \qquad (2)

where: $\theta_R$ — Rayleigh just-resolved angular separation; $x_1=3.8317$ — first zero of $J_1$ . The 26.5% dip is the visible consequence of the convention, not part of its definition: Rayleigh simply chose a dip the eye can catch. Sparrow refused to wait for a visible dip and asked for the last moment a dip exists at all — the separation at which the summed profile is exactly flat at its centre. That is a condition on curvature: the second derivative of the total intensity at the midpoint vanishes. It yields a tighter number, about $0.78$ of Rayleigh's.

\left.\frac{\mathrm{d}^2 I_{\text{tot}}}{\mathrm{d}x^2}\right|_{x=0}=0 \;\Longrightarrow\; \theta_S \approx 0.947\,\frac{\lambda}{D}\approx0.777\,\theta_R \qquad (3)

where: $I_{\text{tot}}(x)=I(x-s/2)+I(x+s/2)$ — the incoherent sum of two shifted Airy patterns; $s$ — their separation in $x$ -units; the Airy solution is $s=x_{\text{sep}}=2.976=0.947\pi$ . Sparrow marks the resolution at which two peaks have just merged into one flat-topped hump. Two conventions bracket the gap between Sparrow and Rayleigh. The Dawes limit is not derived but distilled empirically from Victorian double-star observations, famously written $116''$ over the aperture in millimetres; the half-power (Houston) rule sets the separation to the half-maximum radius of one Airy pattern. Both land near $1.03\,\lambda/D$ .

\theta_{\text{Dawes}}\approx1.02\,\frac{\lambda}{D}=\frac{116''}{D_{\text{mm}}},\qquad \theta_{\text{FWHM}}\approx1.03\,\frac{\lambda}{D}=\frac{2x_{1/2}}{\pi}\frac{\lambda}{D} \qquad (4)

where: $D_{\text{mm}}$ — aperture diameter in millimetres; $x_{1/2}=1.616$ — radius where one Airy pattern falls to half its peak, giving $2x_{1/2}/\pi=1.029$ . These are convenience rules: Dawes for what a trained eye actually splits, half-power for a quick estimate. Abbe's limit lives in the microscope and asks whether the objective collects enough spatial frequency to reconstruct a periodic object; the answer is set entirely by the numerical aperture. For a coherently illuminated grating the smallest resolvable period is $\lambda/2\,\mathrm{NA}$ ; the equivalent two-point radius is $0.61\,\lambda/\mathrm{NA}=x_1/2\pi\cdot\lambda/\mathrm{NA}$ .

d_{\min}=\frac{\lambda}{2\,\mathrm{NA}},\qquad \theta_{\min}^{\text{(radius)}}=0.61\,\frac{\lambda}{\mathrm{NA}},\qquad \mathrm{NA}=n\sin\theta \qquad (5)

where: $d_{\min}$ — smallest resolvable grating period; $\mathrm{NA}=n\sin\theta$ — numerical aperture; $n$ — index of the medium; $\theta$ — half-angle of the acceptance cone. The full two-step diffraction picture behind $\lambda/2\,\mathrm{NA}$ is developed in Abbe Theory of Image Formation; here it is one more coefficient on the same $\lambda/\mathrm{NA}$ ruler.

Derivation AThe Sparrow criterion from a vanishing central curvature

This is the machine-provable result that sets Sparrow apart from the eyeball conventions: the just-resolved separation is fixed not by a chosen dip depth but by the moment the summed intensity stops being concave-up at its midpoint. The goal is to turn "the dip has just vanished" into a precise equation, solve it for the circular Airy pattern to get the coefficient $0.947$ , and show the identical logic gives $2.606$ for slit-like line sources.

Step 1. Write the combined image of two equal, mutually incoherent sources centred at $\pm s/2$ as the sum of intensities, $I_{\text{tot}}(x)=I(x-s/2)+I(x+s/2)$ . Being even in $x$ , its first derivative vanishes at the midpoint, so the centre is always an extremum — valley or hump. Why: incoherent sources add in intensity with no interference cross-term, and the symmetry forces a stationary point at the centre.

Step 2. Classify that extremum by the second derivative. While $I''_{\text{tot}}(0)>0$ the centre is a minimum — a real dip, resolved; when it turns negative the sources merge. The threshold is $I''_{\text{tot}}(0)=0$ , and by evenness $I''_{\text{tot}}(0)=2\,I''(s/2)$ , so the condition is $I''(s/2)=0$ : the half-separation lands on the PSF's own inflection point. Why: the sum's central curvature is inherited entirely from each component evaluated half a separation away.

Step 3. Solve numerically for $I(x)=[2J_1(x)/x]^2$ . The root is $s=x_{\text{sep}}=2.976=0.947\pi$ ; converting through $x=(\pi D/\lambda)\sin\theta$ gives $\theta_S=(x_{\text{sep}}/\pi)\,\lambda/D=0.947\,\lambda/D$ , and $\theta_S/\theta_R=2.976/3.8317=0.777$ . Why: the Sparrow separation is a fixed multiple of the Airy scale, so like Rayleigh it inherits the pure-diffraction $\lambda/D$ dependence — only the coefficient changes.

Step 4. Repeat for a one-dimensional slit, whose line-spread function is $I(u)=\mathrm{sinc}^2 u$ . The same condition $I''(u_s/2)=0$ gives $u_s=2.606$ ; the coefficient shifts because the sinc and Bessel profiles curve differently near their inflections, but the criterion is word-for-word identical. Why: the rule is a statement about the shape of the spread function, so it transfers between one and two dimensions, only re-tuning the number. *The Airy root $x_{\text{sep}}=2.976\Rightarrow0.947\,\lambda/D$ , ratio $0.777$ , and the sinc² root $u_s=2.606$ are each confirmed by solving $\mathrm{d}^2 I_{\text{tot}}/\mathrm{d}x^2|_0=0$ numerically for the respective spread function.*

Go deeper · Derivation B · 2 minAbbe's diffraction limit from spatial-frequency capture

Derivation BAbbe's diffraction limit from spatial-frequency capture

Abbe's route to the same $\lambda/\mathrm{NA}$ scale abandons the two-point picture and asks a different question: can the lens gather enough of the object's diffracted light to reconstruct its finest period at all? The goal is to derive $d_{\min}=\lambda/\mathrm{NA}$ on-axis and its halving to $\lambda/2\,\mathrm{NA}$ under oblique illumination, then place that limit beside Rayleigh's so their order-one difference is transparent.

Step 1. Model the finest object detail as a sinusoidal grating of period $d$ . Coherently illuminated, it diffracts light into orders at $\sin\theta_m=m\lambda/(nd)$ in a medium of index $n$ : the zeroth order goes straight through, the $\pm1$ orders fan to $\pm\theta_1$ , and finer periods throw those first orders to steeper angles. Why: a periodic transmittance is a sum of spatial-frequency harmonics, each radiating a plane wave whose transverse frequency is the grating's inverse period.

Step 2. Demand the minimum needed for a fringe. A single transmitted beam has uniform intensity and no periodicity; only two overlapping beams beat into a spatial pattern carrying $d$ . On-axis illumination puts the zeroth order down the axis, so the first order must fit the cone: $\sin\theta_1=\lambda/(nd)\le\mathrm{NA}/n$ , i.e. $\lambda/d\le\mathrm{NA}$ , giving $d\ge\lambda/\mathrm{NA}$ . Why: fewer than two orders means no beating and a featureless image, so resolution has a hard geometric cutoff.

Step 3. Recover a factor of two with oblique illumination. Tilt the beam so the zeroth order enters at one edge of the cone, $-\theta$ ; then the first order need only reach $+\theta$ , and the condition relaxes to $\lambda/d\le2\sin\theta$ , i.e. $d\ge\lambda/2\,\mathrm{NA}$ . Filling the full condenser aperture achieves this, which is why $\lambda/2\,\mathrm{NA}$ is quoted as the Abbe limit. Why: spreading illumination over all incidence angles doubles the span of object frequencies whose first order still lands inside the pupil.

Step 4. Set Abbe beside Rayleigh. Rayleigh's $1.22\,\lambda/D$ (radius $0.61\,\lambda/\mathrm{NA}$ ) counts two incoherent point images merging; Abbe's $\lambda/2\,\mathrm{NA}$ counts frequency capture of a coherent periodic object. Different objects, different illumination — yet both are diffraction through the same pupil, so both scale as $\lambda/\mathrm{NA}$ and land within an order-one factor. Why: all resolution limits are the aperture's diffraction spread in different clothes, sharing the $\lambda/\mathrm{NA}$ scaling while their coefficients record which question was asked. *At $\mathrm{NA}=1.4$ and $\lambda=550$ nm these formulas give $d_{\min}=\lambda/\mathrm{NA}=393$ nm (on-axis Abbe), $\lambda/2\,\mathrm{NA}=196$ nm (oblique Abbe), and Rayleigh radius $0.61\,\lambda/\mathrm{NA}=240$ nm.*

Limiting cases

For a circular pupil imaging two equal, incoherent point sources, the just-resolved separations rank as follows ( $D$ = aperture diameter, $\mathrm{NA}$ = numerical aperture).

Case	Result	Meaning
Sparrow	$\theta_S\approx0.947\,\dfrac{\lambda}{D}$	Central curvature vanishes; tightest "still has a dip" limit, $\approx0.78\,\theta_R$ .
Dawes	$\theta\approx1.02\,\dfrac{\lambda}{D}=\dfrac{116''}{D_{\text{mm}}}$	Empirical double-star split; matches what a trained eye resolves.
Half-power	$\theta\approx1.03\,\dfrac{\lambda}{D}$	Separation = one Airy pattern's half-maximum radius; quick estimate.
Rayleigh	$\theta_R=1.22\,\dfrac{\lambda}{D}$	Peak on first dark ring, 26.5% dip; the conservative default.
Abbe (microscope)	$d_{\min}=\dfrac{\lambda}{2\,\mathrm{NA}}$	Spatial-frequency cutoff for a coherent grating, full condenser.
High signal-to-noise	below every entry	With enough photons, estimation and super-resolution beat all thresholds.

Common trapThe Rayleigh criterion is a hard physical limit on resolution

It is not. Every criterion here is a convention imposed on a smooth, continuous point-spread-function overlap — Rayleigh's 26.5% dip is a rule for the eye, Sparrow's flat top a rule for curvature, Dawes' number a rule fit to observations. None is a wall set by physics. They all assume mutually incoherent equal point sources, an ideal circular pupil, and resolution by inspection. Change any of those and the numbers move: coherent sources interfere and can look merged or split at the same spacing, and with high signal-to-noise a maximum-likelihood two-versus-one test, deconvolution, or a dedicated super-resolution method resolves far below Rayleigh and even below Sparrow. The only genuine floor is photon noise, through the Cramér–Rao bound on how precisely a separation can be estimated — and that floor keeps dropping as you collect more light. Treat the classical criteria as design targets and sanity checks, never as a barrier that cannot be crossed.

Worked examples

1 · One telescope, three "resolutions" (confidence builder)

A telescope has aperture $D=100$ mm and observes at $\lambda=550$ nm. What does each classical criterion give for the closest resolvable double star?

Step 1. Rayleigh: $\theta_R=1.22\,\lambda/D=1.22\times(550\times10^{-9})/0.100=6.71\times10^{-6}$ rad. Converting, $6.71\times10^{-6}\times206265=$ 1.38″ — the textbook diffraction limit of a 10 cm aperture.

Step 2. Sparrow: $\theta_S=0.947\,\lambda/D=0.947\times(5.50\times10^{-6})=5.21\times10^{-6}$ rad $=$ 1.07″, tighter by the factor $0.947/1.22=0.78$ .

Step 3. Dawes: the empirical rule $116''/D_{\text{mm}}=116/100=$ 1.16″, sitting between Sparrow and Rayleigh — which is exactly why nineteenth-century observers splitting stars by eye landed near the theoretical Sparrow value.

Step 4. For one and the same telescope the "resolution" is 1.07″ (Sparrow), 1.16″ (Dawes), or 1.38″ (Rayleigh) — a spread of a factor $1.3$ . None is wrong; each answers a different question. The number is meaningless without naming the criterion.

2 · The microscope diffraction limit under three definitions (exam level)

An oil-immersion objective has $\mathrm{NA}=1.4$ and works at $\lambda=550$ nm. Give the resolution by the Abbe (oblique and on-axis) and Rayleigh definitions.

Step 1. Abbe, full condenser (oblique): $d_{\min}=\lambda/2\,\mathrm{NA}=550/(2\times1.4)=$ 196 nm — the classical diffraction limit of the light microscope, the figure that shaped Zeiss objective design.

Step 2. Abbe, on-axis coherent: only the zeroth order runs down the axis, so the first order must fit the whole cone and the limit doubles to $d=\lambda/\mathrm{NA}=550/1.4=$ 393 nm.

Step 3. Rayleigh two-point radius: $0.61\,\lambda/\mathrm{NA}=0.61\times550/1.4=$ 240 nm, the Airy first-zero radius — the number for two separate fluorescent points rather than a grating.

Step 4. Collect them: 196 nm (Abbe oblique), 240 nm (Rayleigh), 393 nm (Abbe on-axis) — a factor-of-two spread for one objective at one wavelength. A bare "0.2 µm resolution" hides which you mean. Resolution is a number plus a convention, never a number alone.

Physics of the portrait

Every alternative in this article is measured against one reference convention — the Rayleigh criterion, the choice that one Airy peak shall sit on the other's first dark ring. It earns the portrait precisely because it is a choice: understanding why it was made, and what it deliberately ignores, is understanding why Sparrow, Dawes and Abbe could all answer the same question differently.

Origins · 1 minRayleigh criterion and how this was discovered

Rayleigh criterion

one Airy peak on the other's first dark ring — the reference convention of resolution · astronomy · spectroscopy · instrument specification

The Rayleigh criterion is resolution theory's standard metre — not a law of nature but a convention so well chosen it has anchored the field since 1879. Rayleigh, analysing the spectroscope, proposed calling two lines resolved when the peak of one falls on the first zero of the other; carried to the circular pupil it reads $\theta_R=1.22\lambda/D$ , and the composite profile of two point-spread functions at that separation shows a comfortable 26.5% central dip that any detector or eye can catch. Its genius is operational: the first zero is a sharp, computable landmark on the Airy pattern, so the criterion needs no statement about noise, detector or observer. Its arbitrariness is equally plain — nothing physical happens at exactly that separation — and every alternative in this article is a different reading of the same overlapping profile: Sparrow (1916) pushes down to the flat-top separation $0.947\lambda/D$ where the dip just vanishes, Dawes (1867) found visual double stars splitting at about $1.02\lambda/D$ , and Abbe (1873) abandoned the two-point picture in favour of spatial-frequency capture, giving the classical criterion $\lambda/(2\,\mathrm{NA})$ . Modern estimation theory completes the arc: with a known point-spread function and enough photons, model fitting resolves below every classical criterion — the logic behind the 2014 Nobel Prize for super-resolution microscopy.

History

1867

— William Rutter Dawes fixes his empirical double-star limit at 4.56 arcseconds per inch of aperture

1873

— Ernst Abbe derives the microscope diffraction limit λ/2NA in his theory of image formation

1874

— Hermann von Helmholtz independently obtains the same microscope resolution limit

1879

— Lord Rayleigh states the peak-on-first-zero criterion for the spectroscope and telescope

1916

— Carroll M. Sparrow proposes the vanishing-central-dip (undulation) criterion

2014

— Betzig, Hell and Moerner share the Nobel Prize in Chemistry for far-field super-resolution microscopy

Going further

Texts. Hecht, Optics, 5th ed., §10.2 (Fraunhofer diffraction and resolution) · Born & Wolf, Principles of Optics, 7th ed., §8.6 · Saleh & Teich, Fundamentals of Photonics, 2nd ed., Ch. 4 · Foot, Atomic Physics, §11 · Demtröder, Laser Spectroscopy, Ch. 4. Key numerical anchors: $x_1=3.8317$ (first zero of $J_1$ ) with $3.8317/\pi=1.2197\approx1.22$ (Rayleigh coefficient); Sparrow-to-Rayleigh ratio $\approx0.77$ for a circular aperture, giving $\theta_S\approx0.947\,\lambda/D$ ; the 1D Sparrow root $u_s\approx2.606$ from $\mathrm{d}^2[\mathrm{sinc}^2]/\mathrm{d}u^2=0$ ; Abbe $d_{\min}=\lambda/(2\,\mathrm{NA})$ from the grating equation and the OTF-autocorrelation doubling; and the oil-immersion figures $\mathrm{NA}\approx1.4$ , $d_{\min}\approx0.2\,\mu\mathrm{m}$ at $\lambda=0.55\,\mu\mathrm{m}$ .

Prerequisites. Circular Aperture Diffraction and the Airy Pattern · Rayleigh Resolution Criterion · Fraunhofer Diffraction · Single-Slit Diffraction. Leads to. Optical Transfer Function · Confocal and Super-Resolution Microscopy · Fisher Information and the Cramér–Rao Bound.

Which resolution criterion should I actually use?

It depends on who — or what — judges "two." For a human splitting double stars, Dawes ( $116''/D_{\text{mm}}$ ) matches real performance best. For a conservative specification, Rayleigh ( $1.22\,\lambda/D$ ) is the safe default. For a detector recording the whole profile where you want the tightest honest threshold, Sparrow ( $0.947\,\lambda/D$ ) marks the last separation with any dip. For a microscope imaging periodic structure, Abbe's $\lambda/2\,\mathrm{NA}$ is the natural language. With plenty of photons and a model to fit, none of these is your true limit — use estimation theory. Always state which one you mean.

Where does the 26.5% Rayleigh dip come from?

It is the depth of the valley when two equal incoherent Airy patterns are separated by exactly the first-dark-ring spacing. At that spacing the midpoint intensity comes from one pattern's peak plus the other's first zero (which is zero) plus small tails; working it out, the saddle sits at about 73.5% of the peak, so the dip is 26.5%. Rayleigh did not derive this depth as fundamental — he chose the peak-on-first-zero geometry because the resulting dip is comfortably visible, and the 26.5% simply falls out. It is a consequence of the convention, not its justification.

Does the coherence of the two sources change the criterion?

Strongly. The classical numbers assume mutually incoherent sources, so the two point-spread functions add in intensity with no cross-term. Coherent sources add in amplitude first, and the interference term can either fill the dip (in-phase sources look merged where incoherent ones resolve) or deepen it (out-of-phase sources show a dark central fringe and look split where incoherent ones do not). So resolution under coherent illumination, as in a scanning laser microscope, needs its own analysis and cannot borrow the incoherent number directly.

If they are only conventions, is there any real resolution limit?

Yes, but it is set by photons, not the pupil. Given a known point-spread function, "how far apart are these two sources?" is a parameter-estimation problem, and the Cramér–Rao bound gives the best possible precision for a fixed number of detected photons. That bound has no hard floor: collect more light and you can, in principle, distinguish arbitrarily close sources. The classical criteria correspond roughly to signal-to-noise of order one on the dip; super-resolution simply pours in far more signal. So the genuine limit is statistical, it depends on how many photons you gather, and it lies below every classical threshold.

Why is the one-dimensional Sparrow number 2.606 but the circular one 2.976?

Because the two spread functions have different shapes. The Sparrow condition — the separation at which a single profile sits at its own inflection — depends on how sharply that profile curves near its shoulder. A slit gives the line-spread function $\mathrm{sinc}^2 u$ , with inflection at $u_s/2$ , $u_s=2.606$ ; a circular aperture gives $[2J_1(x)/x]^2$ , with inflection at $x_{\text{sep}}/2$ , $x_{\text{sep}}=2.976$ . Same criterion, same logic, two curvatures, two numbers — a reminder that the coefficients in resolution formulas encode the aperture shape, not a deep constant.

How does immersion oil sharpen resolution without changing the vacuum wavelength?

Through the index $n$ in $\mathrm{NA}=n\sin\theta$ . Filling the space between object and front lens with oil of $n\approx1.515$ lets the objective collect diffracted beams at steeper effective angles — equivalently the wavelength in the medium is $\lambda/n$ , so finer periods throw their first order within the same cone. The numerical aperture climbs from below 1 (dry) to about 1.4 (oil), and since every criterion scales as $\lambda/\mathrm{NA}$ , resolvable detail shrinks in proportion — why oil objectives reach roughly $0.2\ \mu\mathrm{m}$ for green light.