= Diffraction-limited resolution
{title2=$\theta\sim\lambda/D$}
<Diffraction-limited resolution> is a scale for distinguishing details with a specified ideal <diffraction> response and a specified criterion. For two equally bright, mutually incoherent <optical point sources> imaged through a uniformly transmitting, unobscured circular <optical pupil>, the <Rayleigh criterion> places the centre of one <Airy disk> at the first zero of the other. At small <angles> in air this separation is
$$
\theta_{\rm R}=\frac{j_{1,1}}{\pi}\frac{\lambda}{D}\simeq1.22\frac{\lambda}{D},\qquad j_{1,1}\simeq3.831706.
$$
Here $D$ is pupil <diameter>, $\lambda$ is <wavelength> in air, and $j_{1,1}$ is the first positive zero of the order-one <Bessel function of the first kind>. This is the radius to the first zero, not the diameter of the central spot. <Fraunhofer diffraction> gives the radial peak-normalized <light intensity>
$$
P(q)=\left[\frac{2J_1(q)}q\right]^2,\qquad q=\frac{\pi D\theta}{\lambda},\qquad P(0)=1.
$$
The removable value at zero follows from $J_1(q)\sim q/2$. The pupil <Fourier transform> reduces to a radial <integral> by circular symmetry, which produces the <Bessel function of the first kind>; its first zero supplies the factor $1.22$.<Oscillations and Waves by Richard Fitzpatrick/Two-Dimensional Fourier Optics>{ref}
In the usual scalar circular-pupil approximation for a <fluorescence microscope>, the corresponding object-plane <Rayleigh criterion> is
$$
r_{\rm R}=\frac{j_{1,1}}{2\pi}\frac{\lambda_0}{\mathrm{NA}}\simeq0.61\frac{\lambda_0}{\mathrm{NA}},\qquad q=\frac{2\pi\mathrm{NA}\,r}{\lambda_0}.
$$
Here $\lambda_0$ is the vacuum emission <wavelength>, and the object-side <numerical aperture> already includes the medium's <refractive index>. Do not divide $\lambda_0$ by $n$ again. For small cone <angles> in air, $\mathrm{NA}\simeq D/(2f)$ and image-plane radius $f\theta_{\rm R}$ agrees with this form. The assumptions include a single <wavelength>, ideal <optical focus>, and mutually incoherent emitters. High <numerical aperture> can require a vector description with <polarization of an electromagnetic wave> and emitter orientation; <confocal microscopy> has its own excitation-and-detection response. The same $0.61$ formula therefore does not describe every microscope or illumination method.<Resolution by Michael W. Davidson>{ref}
\Image[/circular-pupil-point-spread-scale.png]
{title=Doubling numerical aperture halves the ideal circular-pupil point-response scale at fixed wavelength}
In this original model, $\lambda_0=550\,\mathrm{nm}$ and <numerical apertures> $0.25$ and $0.50$ give first-zero radii $1.342\,\mathrm{\mu m}$ and $0.671\,\mathrm{\mu m}$. Each <point spread function> is divided by its own peak to compare shape. The pupil axes are transverse <wave vector> components multiplied by $\lambda_0/(2\pi)$, so pupil radius equals <numerical aperture>. A wider pupil accepts a wider range of transverse <wave vectors>, producing a narrower image. Increasing <magnification> alone enlarges the same spot and does not reduce its object-plane radius.
The <Rayleigh criterion> is a useful visual convention, not an absolute information barrier. For equal incoherent sources with total detected <radiant flux> $F$ and separation vector $\mathbf s$, the ideal image is
$$
I(\mathbf x)=\frac F2\left[p(\mathbf x-\mathbf s/2)+p(\mathbf x+\mathbf s/2)\right].
$$
Below the <Rayleigh criterion> this still differs from $Fp(\mathbf x)$; for small $\mathbf s$, symmetry cancels the linear change and the first change is quadratic in separation. Estimating two unknown source positions is then increasingly sensitive to <noise>, background, relative brightness and uncertainty in the <point spread function>. Localizing one isolated source and recovering an arbitrary extended object are different inference tasks. Measurements that separate optical modes can also extract separation information differently from direct image <light intensity> measurement.<Quantum Theory of Superresolution for Two Incoherent Optical Point Sources by Mankei Tsang and others>{ref}
There remains a real limit for arbitrary images in this model: the <optical transfer function> is zero above the transmitted <spatial frequency> band. Its cutoff for scalar <incoherent imaging> is $\nu_c=2\mathrm{NA}/\lambda_0$ in object-plane cycles per unit length. The <Nyquist–Shannon sampling theorem> therefore suggests object-referred <detector pixel> pitch $p_{\rm pix}\leq\lambda_0/(4\mathrm{NA})$, with finer sampling providing margin. Finite <detector pixels> also integrate over their <area>, suppressing contrast; coarse sampling can cause <aliasing>. Finer sampling cannot alone recover <spatial frequencies> removed by the optics, and <Fourier deconvolution> amplifies <noise> where the <optical transfer function> is small. <Photon shot noise>, <read noise>, <optical aberrations> and <defocus> can all worsen practical discrimination. A useful resolution claim therefore reports the object model, criterion, <wavelength>, <numerical aperture>, sampling and <signal-to-noise ratio in photon counting>.
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