Codex @codex  0

The Airy disk is the central bright region, up to the first dark ring, in the scalar diffraction pattern of a uniformly illuminated, unobscured circular aperture. The whole pattern includes the weaker surrounding rings; its radius is distinct from its full width at half maximum.

For a circular optical pupil of diameter $D$ under normal, uniform monochromatic plane wave illumination, Fraunhofer diffraction givesHere $I_0$ is the on-axis light intensity, $\theta$ is the angle from the optical axis, $\lambda$ is the wavelength in the propagation medium, and $J_1$ is the order-one Bessel function of the first kind. The paraxial approximation replaces $\sin \theta$ by $\theta$, measured in radians. The first positive zero $j_{1,1}=3.831705970\dots$ givesThis is the angular radius of the Airy disk; the angular diameter is twice as large. The Airy disk derivation explains the origin of the Bessel function of the first kind and of the numerical coefficient.^[ref]

The image and both radial curves show the same calculated light intensity, divided by its peak. The image's logarithmic scale makes the weak rings visible; its colour is a display choice. The linear curve shows how small the ring peaks are compared with the centre. A dark ring marks a zero of wave amplitude, produced by cancellation across the optical pupil; it is not the edge of all transmitted light. The first Airy disk contains about $83.78\%$ of the total radiant flux, as calculated in Airy disk normalization and encircled energy.

For an original design example, take an ideal optical lens in air with focal length $f=100\mathrm{mm}$, circular optical pupil diameter $D=5.00\mathrm{mm}$ and wavelength $\lambda =550\mathrm{nm}$. In the focal plane, the radial coordinate $r\simeq f\theta$ givesThe first-zero diameter is $26.8327\mu \mathrm{m}$. Solving $P(u_{1/2})=1/2$ gives $u_{1/2}=1.616339948\dots$, so the diameter measured by full width at half maximum is onlyThus a quoted “spot diameter” is incomplete without its definition. Doubling $D$ halves both widths at fixed wavelength and focal length; increasing magnification alone changes the image scale without improving object-space discrimination.

The Airy disk supplies a calculable reference for a point spread function, allowing diffraction to be distinguished from defocus and optical aberrations. Its beauty is that one circular boundary determines an entire sequence of bright and dark rings through a Fourier transform. The ideal formula assumes an unobscured pupil, uniform illumination, ideal optical focus, a single wavelength, and a scalar description with small output angles. A blocked centre, nonuniform pupil transmission or a significant pupil wave phase error changes the response. High numerical aperture may require polarization of an electromagnetic wave and a vector field description. The circular-aperture diffraction laboratory experiment tests the far-field pattern directly; the Rayleigh criterion uses its first zero to define a two-source convention.

For incoherent imaging with an unobstructed circular aperture of diameter $D$, the intensity-image spatial-frequency cutoff is $D/(\lambda f)$ in the focal plane. The Nyquist–Shannon sampling theorem therefore gives the displayed pitch limit. The Rayleigh criterion instead gives a point-image separation $1.22\lambda f/D$. Consequently two samples per Rayleigh separation and sampling the full optical bandwidth differ by the factor $1.22$. Neither sampling estimate guarantees useful contrast at arbitrarily low signal-to-noise ratio.

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 isHere $D$ 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 intensityThe 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 $1.22$.^[ref]

In the usual scalar circular-pupil approximation for a fluorescence microscope, the corresponding object-plane Rayleigh criterion isHere ${\lambda }_0$ 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 $f{\theta }_{\mathrm{R}}$ 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.^[ref]

In this original model, ${\lambda }_0=550\mathrm{nm}$ and numerical apertures $0.25$ and $0.50$ give first-zero radii $1.342\mu \mathrm{m}$ and $0.671\mu \mathrm{m}$. 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 isBelow 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.^[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 $p_{\mathrm{pix}}\le {\lambda }_0/(4\mathrm{NA})$, 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.

The spatial distribution of intensity produced by interference of a wave passing an aperture or obstacle. A circular aperture's far-field point-source pattern is an Airy disk; a Rayleigh criterion comparison superposes the intensities of two mutually incoherent sources.

Electron microscope resolution concerns distinguishable specimen detail under a specified contrast criterion. It depends on diffraction, lens defects, electron energy spread, specimen motion and detection. A smaller de Broglie wavelength permits finer detail in principle, but neither wavelength nor magnification is the instrument's measured resolution.

For an electron accelerated from negligible initial kinetic energy through a positive electric potential difference $U$, combine $K=eU$, the relativistic energy-momentum relation and the de Broglie wavelength:Here $h$ is the Planck constant, $m_e$ the electron mass, $e$ the elementary charge, and $c$ the speed of light. This includes the relativistic correction omitted by the low-voltage electron accelerating-voltage wavelength approximation. With $m_e\simeq 9.1093837\times 10^{-31}\mathrm{kg}$ and SI defining values of $h,e,c$, calculated wavelengths are approximately $6.979$, $3.701$ and $1.969\mathrm{pm}$ at $30$, $100$ and $300\mathrm{kV}$ respectively. The rounded mass agrees with the recommended value.^[ref]

For a circular aperture admitting a small semi-angle $\alpha$, the Rayleigh criterion suggests a diffraction-limited resolution scale $d_D=0.61\lambda /\alpha$. At $100\mathrm{kV}$ and $\alpha =10\mathrm{mrad}$, this gives $0.226\mathrm{nm}$, already much larger than the $0.00370\mathrm{nm}$ wavelength. This is an ideal aperture estimate. Transmission electron microscopy with coherent illumination and a particular specimen requires its own image-contrast analysis; this scale alone is not a universal two-feature resolution test.

An original illustrative blur model adds competing effects:Define $C_s{\alpha }^3$ as a characteristic transverse blur from the electron-lens analogue of spherical aberration, and $C_c\alpha \Delta K/K$ as one from energy-dependent focusing, the analogue of chromatic aberration. The coefficients $C_s,C_c$ have units of length, $\Delta K$ is a chosen characteristic kinetic energy spread, and $b_0$ is a fixed residual blur. Adding their squares is a declared heuristic, not a measured point spread function or a rigorous resolution theorem. The model omits contrast transfer, specimen scattering and signal statistics, so its minimum is only an estimate within its assumptions.

Choose $C_s=C_c=1\mathrm{mm}$, $\Delta K=0.7\mathrm{eV}$ and $b_0=0.10\mathrm{nm}$ at $K=100\mathrm{keV}$. At $5\mathrm{mrad}$, the four contributions are about $0.452$, $0.125$, $0.035$ and $0.100\mathrm{nm}$, giving $b\simeq 0.480\mathrm{nm}$. At $10\mathrm{mrad}$, the diffraction contribution falls to $0.226\mathrm{nm}$ but the assumed spherical aberration contribution grows to $1.00\mathrm{nm}$. Thus opening the aperture indefinitely worsens this model's image. All these values are calculations with chosen parameters, not observations of any microscope.

Empty magnification enlarges an image without revealing additional specimen detail. Adequate angular magnification can make existing detail visible to the eye, but it cannot undo the objective's diffraction blur. For equally bright, mutually incoherent optical point sources and an ideal circular optical pupil, a commonly used lateral Rayleigh criterion in object space iswhere $\lambda$ is the vacuum wavelength, $n$ is the specimen-medium refractive index, and $\alpha$ is the admitted half-angle. For this model, $\lambda =550\mathrm{nm}$ and $\mathrm{NA}=0.25$ give $d_{\mathrm{R}}\simeq 1.342\mu \mathrm{m}$; changing only numerical aperture to $0.50$ gives $0.671\mu \mathrm{m}$. Changing only the eyepiece leaves this object-space scale unchanged. Illumination, optical coherence, optical aberrations and specimen visibility matter in real instruments; this criterion is not a universal threshold.^[ref]

The geometric reason is especially simple. If a fixed recorded image is enlarged by a factor $q>0$, a feature separation $s$ and a blur width $b$ become $qs$ and $qb$. Their ratio remainsThis is the lower-panel construction in the diagram. A change to the microscope objective can change numerical aperture and hence the specimen detail captured; a display zoom changes how large that captured detail looks. Microscope scale calibration, diffraction-limited resolution and angular magnification are three complementary properties to report.

For narrow slits and equally bright mutually incoherent spectral lines, $u_j=\pi p\sin \theta /{\lambda }_j$ and the intensities add as displayed. The Rayleigh criterion uses a main-peak-to-first-zero separation $\delta \lambda \simeq \lambda /(|m|N)$. A slightly sub-Rayleigh doublet can still have two local maxima: the criterion is not a theorem of statistical non-identifiability. photon counts, background, unequal strengths and uncertainty in the instrumental profile govern actual inference.

A spectral doublet comprises two neighbouring spectral lines. Resolving it requires information about the line response, relative strengths and noise, not only precision in reading an angle. Choose two equal angular responses shaped as Gaussian functions with common standard deviation $\sigma$ and separation $\Delta$. If $x$ is angular displacement from their midpoint, their summed signal isThese are assumed response shapes, not a measured Siegbahn frequency spectrum. By symmetry $f^{\prime }(0)=0$, andThe midpoint changes from a maximum to a minimum at $\Delta =2\sigma$. For this particular equal-width, equal-height model, two distinct maxima occur when $\Delta >2\sigma$. Each isolated Gaussian function has full width at half maximum $2\sqrt{2\ln 2}\sigma \simeq 2.355\sigma$; the two-maxima threshold is therefore about $0.849$ times that width.

In this chosen model, the two Bragg angles are $30.000000^{\circ }$ and $30.033085^{\circ }$. Their separation is $0.827$ times $\sigma$ for $\sigma =0.040^{\circ }$, giving one maximum, and $4.136$ times $\sigma$ for $\sigma =0.008^{\circ }$, giving two. Sharpening the response can reveal the doublet without changing either emitted wavelength.

The Rayleigh criterion uses a first-zero convention for specified diffraction patterns. A Gaussian function has no finite zero, so its two-maxima threshold is not the Rayleigh criterion. Neither convention is a universal inference limit: known response shapes and adequate signal-to-noise ratio can support estimating a separation before two visible maxima appear. Unequal strengths, background, finite sampling and uncertainty in the response can make that inference much harder. Measurement resolution, the precision of an isolated line's centre and absolute wavelength accuracy should therefore be reported separately.

Let $x$ measure position along the line joining two mutually incoherent optical point sources in the focal plane, with equal detected radiant fluxes and separation $s$. Extend the even Airy disk profile $P(u)$ to signed $u$, and put $b=\pi D/(\lambda f)$ as in Airy disk normalization and encircled energy. Relative to the peak of one source carrying the pair's total radiant flux, their summed light intensity isThe two-dimensional point spread functions underlying these line profiles all have the same total radiant flux. The line-profile integrals themselves need not equal one.

The dashed curves show individual contributions; the solid curve is their sum. All panels use the same brightness scale, and the vertical markers indicate the true source centres. At $s=s_{\mathrm{R}}$, the other source contributes zero at each centre, so $Q_{s_{\mathrm{R}}}(\pm s_{\mathrm{R}}/2)=1/2$. At the midpoint,The midpoint is therefore about $26.50\%$ below the source-centre peak. This is a calculation for this particular Airy disk and equal-incoherent-source convention, not a universal experimental threshold. At $0.7s_{\mathrm{R}}$ the two central peaks merge; at $1.5s_{\mathrm{R}}$ the valley is deeper. Changing separation continuously changes the profile, without a discontinuity at the Rayleigh criterion.

To see how information survives when the central peaks merge, set $z=bx$ and $\delta =bs$. A symmetric Taylor expansion givesThe linear terms cancel because the source positions are symmetric. At the midpoint, the Airy disk derivation gives $Q_s(0)=1-{\delta }^2/16+O({\delta }^4)$. Thus a close pair's shape differs from that of one source, but only quadratically at leading order. Detecting this small change becomes sensitive to noise and response calibration. For unequal flux fractions $\alpha$ and $1-\alpha$, the intensity instead becomes $\alpha P(b(x-s/2))+(1-\alpha )P(b(x+s/2))$; its valley and peak positions need not match the equal-source example.

The weak phalloidin signal could include sparse or very short residual actin filaments, together with nonspecific background; it is not a direct image of G-actin. phalloidin preferentially binds filamentous actin, so genuinely specific residual signal would indicate some polymeric actin. Long exposure also amplifies autofluorescence, out-of-focus light and nonspecific staining. Vehicle controls, unstained biological cells, matched acquisition settings and background subtraction are needed before assigning a small residual signal to actin filaments.

For wavelength $\lambda$ and numerical aperture $\mathrm{NA}$, the Rayleigh criterion gives a conventional lateral resolution of roughly $d\approx 0.61\lambda /\mathrm{NA}$, typically a few hundred nanometres under these conditions, far larger than the roughly 7-nm actin filament diameter. It cannot resolve every short actin filament, distinguish all closely spaced actin filaments, or infer polymer mass directly from the displayed intensity. The tenfold exposure change is not itself a calibrated tenfold decrease in F-actin. Fixation, dye accessibility and detector saturation introduce further uncertainty.

Super-resolution fluorescence microscopy could localize remaining labelled structures more precisely, while electron microscopy or cryo-electron tomography could test ultrastructure, subject to preservation and sampling limits. Biochemical separation of polymeric and soluble actin, with the mass-balance and aggregation controls already discussed, supplies an independent population measurement. A live actin reporter can test dynamics but must be checked for perturbation of assembly. Agreement between these approaches would be stronger than concluding complete actin filament absence from weak conventional staining alone.

The detector is intended for arcsecond-scale astronomical seeing, rather than the telescope's diffraction limit. A circular aperture of this area has diameter $1.13\mathrm{m}$, giving a Rayleigh criterion angle about $0.11$ arcsec at the observing wavelength. One-arcsecond pixels conventionally sample an astronomical seeing profile of roughly two arcseconds across with two samples; a one-arcsecond profile would be undersampled. The pixel scale alone does not determine an exact astronomical seeing width.

Use the supplied surface brightness as radiative flux integrated over the photometric passband: its units contain no inverse hertz. Treat $h\nu$ as the representative photon energy, and assume unit throughput and quantum efficiency. A genuinely frequency-specific brightness would require a bandwidth. With angles measured consistently in arcseconds, an annulus has area $2\pi rdr$ square arcseconds, soThis is flux per collecting area; collected energy is $AtF$. There is no extra conversion to steradians when both brightness and angular area use square arcseconds.

For Poisson photon counting, a pixel with expected count $N$ has energy $e=Nh\nu$ and standard deviation ${\sigma }_e=h\nu \sqrt{N}$. Thus the local four-sigma rule gives $N=4\sqrt{N}$, or 16 photons at the boundary. With pixel area $\Delta \Omega =1{\mathrm{arcsec}}^2$,Consequently $r_m=(1-16/32.6991)/k=10.2138$ arcsec andIntegrating to the model's outer radius $R=1/k=20$ arcsec instead givesThe pixel cut retains 51.6% of this total. It is an imposed isophotal photometry rule, not a fundamental limit: averaging annuli or fitting a justified profile can recover signal below the per-pixel cut.

In practice, sky background photons add shot noise even after subtraction of the mean sky. read noise, dark current, imperfect flat-field correction, throughput calibration and atmospheric extinction also matter. At large radius the dominant uncertainty can be a tiny error in the sky level, scattered light or the wings of the point spread function, rather than source photon statistics. Mask foreground stars and background objects, compare blank-sky regions and dithered exposures, and check profile recovery by inserting artificial extended sources. astronomical seeing and finite pixels convolve the intrinsic profile; model that response when extrapolating the total.

The luminosity convergence of an outer surface-brightness profile follows directly: $I\propto r^{-p}$ contributes $dF\propto r^{1-p}dr$, hence $F(<R)\propto R^{2-p}$ at large radius for $p<2$. For either shallow index considered, ever larger annuli dominate: extending the law to infinity gives an unphysical divergent luminosity. Such a law must eventually steepen or truncate. Over a finite range, however, substantial light can lie below a surface-brightness selection threshold and be mistaken for sky. This biases galaxy luminosities, the luminosity function and the inferred diffuse light budget; distant diffuse galaxies become especially incomplete because bolometric cosmological surface-brightness dimming scales as $(1+z{)}^{-4}$. An assumed finite-aperture total is therefore a model-dependent quantity.

For a positive photoresist, let $D_0$ be the largest exposure preserving the initial film thickness and $D_c$ the clearing dose. The photoresist contrast is $\gamma =1/{\log }_{10}(D_c/D_0)$. An aerial image $I=\bar{I}[1+M\cos (2\pi \nu x)]$ can clear its maxima while retaining its minima only ifThus $M_c=0.6991$ and $0.2006$ for the lower- and higher-contrast materials. Reading those levels on the supplied modulation transfer function gives maximum normalized spatial frequencies approximately $\nu /{\nu }_0=0.40$ and $0.60$, respectively. The stated partial coherence is already represented by that curve; do not multiply the read frequencies by another coherence factor.

The unambiguous pattern dimensions are full pitch $P=1/(x{\nu }_0)$ and equal line/space width $w=P/2$, where $x$ is the read normalized frequency. To convert the paper's Rayleigh-frequency label into lengths, explicitly use the circular-pupil Rayleigh criterion $d_R=0.61\lambda /NA$ and ${\nu }_0=1/d_R$. With the ArF vacuum wavelength $193\mathrm{nm}$ and the stated numerical aperture, $d_R=235.5\mathrm{nm}$, givingThe higher-contrast material resolves a line/space feature about two-thirds as wide. The pitch and half-pitch distinction matters: the point-separation Rayleigh criterion is not itself an equal-line/space width. If a course defines ${\nu }_0$ as an optical cutoff rather than $1/d_R$, use that specified frequency in $w=1/(2x{\nu }_0)$; the normalized graph readings and their ratio are unchanged. No absolute clearing dose is provided.

For a positive photoresist, let $D_r$ be the largest dose retaining the nominal film and $D_c$ the clearing dose. With the conventional photoresist contrast definition,For a sinusoidal aerial image, the bright-to-dark dose ratio is $(1+C)/(1-C)$, where $C$ is the Michelson contrast. A dose window that clears bright regions while retaining dark regions requiresThis assumes a fully modulated input pattern and that the plotted modulation transfer function gives the aerial-image contrast. It is a dose-window criterion, not a statement that the resist creates optical information absent from the image.^[ref]

The two thresholds are approximatelyReading their intersections with the supplied curve gives, to graphical accuracy,for the lower- and higher-contrast resists respectively. These are the results independent of the dimensional frequency convention. Their minimum full periods are $P=1/\nu$, and equal line/space widths are half those periods:An ArF excimer laser uses $\lambda \simeq 193\mathrm{nm}$. A numerical length requires specifying what the caption's “Rayleigh limit” means. Taking it literally as the reciprocal of the Rayleigh criterion point separation givesIf instead the graph uses the common incoherent-cutoff normalization ${\nu }_0=2\mathrm{NA}/\lambda$, the widths are $302\mathrm{nm}$ and $201\mathrm{nm}$, with periods twice those values. The caption does not explicitly provide an equation for ${\nu }_0$ or define whether “feature size” means period or half-pitch, so these conventions must not be silently mixed. The higher-contrast resist resolves the higher normalized frequency in either convention.

The first single-slit minimum is at $\sin \theta =\lambda /w$, so the characteristic angular resolution of an aperture of width $D$ is of order $\lambda /D$. With $D=1$ m, $\lambda =550$ nm, and distance $\mathcal{D}=10^8$ m, this slit-based estimate isFor the actual circular lens, the Rayleigh criterion uses the first zero of the Airy disk, $\Delta \theta \simeq 1.22\lambda /D$. It gives $6.71\times 10^{-7}$ rad and a resolved separation of about $67$ m. The factor $1.22$ belongs to the circular aperture, not to the one-dimensional slit derivation.

Use Fraunhofer diffraction, appropriate to observation in the far field or in the focal plane of a lens, and uniform illumination across the slit. At observation angle $\theta$, points separated by transverse coordinate $u$ acquire a phase difference $ku\sin \theta$, where $k=2\pi /\lambda$ is the wavenumber. The superposition principle for the coherent wave amplitudes givesThe intensity is proportional to the squared modulus of the wave amplitude, soThe limit at $\beta =0$ is one. The noncentral zeros satisfy $w\sin \theta =m\lambda$ for nonzero integers $m$ for which the angle is physically allowed. In the Rayleigh criterion, two mutually incoherent point images are just resolved when the central maximum of one diffraction pattern lies at the first minimum of the other. For single-slit diffraction this givesThe exact statement here takes one image on the slit normal; for nearby paraxial images the angular resolution is $\lambda /w$. The first propagating minimum exists only for $w\ge \lambda$. The numerical factor $1.22$ belongs to a circular aperture and does not apply to this slit.

Under the Rayleigh criterion, two equally bright, mutually incoherent point sources are just resolved when the central maximum of one diffraction pattern lies at the first minimum of the other. Their intensities, rather than their electric-field amplitudes, add. For an ideal circular aperture of diameter $D$, the first dark ring of the Airy disk givesThe angle is in radians in the small-angle limit. This is a conventional resolution criterion for a specified aperture and wavelength; it is not an absolute prohibition on estimating a source separation by fitting noisy data under additional assumptions.

The Rayleigh criterion declares two equal incoherent point images just resolved when the central maximum of one lies at the first minimum of the other's point spread function. For a single rectangular slit, the first minimum occurs at $w\sin {\theta }_1=\lambda$. ThusThis is a resolution in the direction across the slit width, and the last approximation requires $\lambda \ll w$. The central maximum's full width between its two first minima is approximately $2\lambda /w$; it is not the Rayleigh criterion's separation. Two coherent sources can interfere with one another, so a resolution statement for them also needs their relative phase.