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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.

Peak normalization $P(0)=1$ does not make the integral of a point spread function equal to one. In the paraxial approximation to the focal plane, define $b=\pi D/(\lambda f)$ and $u=br$. The unit-integral point spread function normalization isHere $p$ has units of inverse area; multiplying by the detected radiant flux gives image-plane irradiance. Its encircled energy, meaning the fraction inside radius $R=U/b$, isThis is an integral over area, so its radial weight is $u$. An unweighted integral of a line through the centre is a different quantity.

A short proof uses the Bessel function of the first kind identities $J_0^{\prime }=-J_1$ and $J_1^{\prime }=J_0-J_1/u$. They giveBoth expressions for $E$ vanish at zero. Since these Bessel functions of the first kind tend to zero as $U\to \infty$, $E(\infty )=1$, establishing the normalization. At the first dark ring,A circular detector region ending at the first zero therefore misses about $16.22\%$ of the ideal radiant flux, even though the surrounding rings have low peak light intensity. This matters when converting measured counts into total source brightness or choosing an image aperture for a photodetector.

The amplitude spread function is the complex field response to an optical point source. Its squared magnitude, with a specified point spread function normalization, gives the intensity point spread function.

Angular differential imaging keeps the instrument pupil fixed while the sky rotates. A stellar reference point spread function is subtracted, then the residuals are derotated and combined. Insufficient rotation and imperfect reference selection cause differential-imaging self-subtraction.

Astronomical seeing is the atmospheric blurring of a stellar image, conventionally expressed as the angular full width at half maximum of its long-exposure point spread function, usually in arcseconds. For a large aperture in the ideal Kolmogorov model, it is approximately $0.98\lambda /r_0$ radians, with Fried parameter $r_0$.

Coma makes an off-axis optical point source appear asymmetric, often with a compact head and a flaring tail like a comet. Different parts of the optical pupil send its light rays to different transverse image positions. A sharp central star image therefore does not guarantee sharp images across an optical telescope's field of view. In an idealized, aligned system with rotational symmetry, coma vanishes on the optical axis; misalignment can introduce coma even at the nominal field centre.^[ref]

Coma differs from spherical aberration, whose dependence on optical pupil radius can blur an on-axis point without selecting a transverse direction. Astigmatism gives different optical foci in two perpendicular sections of an off-axis bundle; a through-focus sequence can change its elongated image's orientation. Diffraction persists when these optical aberrations vanish. A real asymmetric point spread function may mix several effects, so its appearance alone is not a unique diagnosis of coma.

The classical primary description belongs to the nineteenth-century Seidel aberration classification of five primary monochromatic image errors.^[ref] It remains useful in modern design: OpticStudio's Seidel Coefficients by ANSYS, Inc. documentation lists coma as $W_{131}$ and tabulates each surface's contribution as well as the system sum (Discussion). Cancellation of contributions can reduce coma without making every surface individually free of it. The same documentation limits this calculation to specified systems with axially symmetric surfaces; higher-order terms and broken rotational symmetry require more general ray tracing.

For an isolated fluorescent molecule at displacement $\mathbf{r}$ from the scanning optical focus, write $E(\mathbf{r})$ for relative excitation light intensity and $D_a(\mathbf{r})$ for the fraction of its collected emission passed by a centred detection pinhole of radius $a$. The expected response, apart from a common brightness and collection factor, isHere $\mathbf{u}$ is position in the detection-pinhole plane and $p_{\mathrm{em}}$ is the emitted light intensity distribution there conditional on collection, with units of inverse area. This point spread function normalization retains throughput: $D_a$ is a fraction, not a curve rescaled to have peak one. The product expresses excitation followed by detection; it is an intensity/probability product, not multiplication of coherent wave amplitudes from different molecules. ^[ref]

Assume linear, unsaturated fluorescence, independent emitters whose light intensities add, a stationary specimen, fixed collection efficiency and quantum efficiency, and negligible changes from scattering, absorption or bleaching during a scan. Different excitation and emission wavelengths generally give different excitation and detection responses. For a sufficiently small pinhole, $D_a$ is approximately its area times $p_{\mathrm{em}}$ at the centre; this is the familiar excitation-times-detection point spread function. It is not universally the square of one point spread function. A spatially invariant response further allows the image to be expressed as a convolution with the emitter distribution; depth-dependent optical aberrations can invalidate that approximation.

For an explicitly synthetic axial example, put $t=z/z_0$ and choose an on-axis excitation $E(t)=1/(1+t^2)$, inspired by a Gaussian beam. Stipulate a circular Gaussian function for the detection-plane light intensity, with standard deviation $s(t)=s_0\sqrt{1+t^2}$. In dimensionless coordinates $\mathbf{v}=\mathbf{u}/s_0$, put $\alpha =a/s_0$. ThenThe expression for $D_{\alpha }$ follows by integrating $p$ over a disk using polar coordinates. The assumptions specify a toy model with matched axial scales; its parameters are not a calibration for a real objective optical lens. It omits Airy disk rings, vector diffraction, and unequal excitation and emission wavelengths.

For a vanishingly small pinhole, the peak-normalized synthetic response tends to $(1+t^2{)}^{-2}$, whose full width at half maximum is $2\sqrt{\sqrt{2}-1}{z}_0\simeq 1.2872z_0$. With all collected emission accepted it tends to $(1+t^2{)}^{-1}$, of width $2z_0$. Meanwhile $H_{\alpha }(0)=1-e^{-{\alpha }^2/2}\to 0$ as the pinhole closes. The narrower normalized curve therefore accompanies a vanishing detected signal; peak normalization must not conceal that loss.

At one depth, scan the illumination spot across $x$ and $y$ and assign the detector reading during each dwell interval to a pixel. Move the specimen or objective optical lens axially to acquire the next depth. A stack of these optical sections samples three-dimensional structure. With beam scanning, the returning fluorescence commonly retraces the scanning optics: this descanning holds the image of the currently illuminated point at the detection pinhole. With stage scanning, the fixed spot and pinhole remain aligned while the specimen moves. ^[ref]

An optical section is a depth-weighted image of an intact specimen, with a finite axial response and tails. A physical cut actually separates a slice of material; its thickness and cutting damage are properties of the preparation. Changing optical focus selects another optical section without removing a slice, although illumination can still damage a specimen. Neither a thin physical slice nor a sharply displayed optical section makes the imaging point spread function infinitely narrow. ^[ref]

Point response and section thickness answer different questions. For a spatially uniform, very thin fluorescent sheet, its response versus axial displacement is proportional to the lateral integral of the three-dimensional confocal response,Consequently a trace through a single bead, $H_a(0,0,z)$, does not by itself give the sheet's optical-section thickness. State which object and which full width at half maximum are being measured. A smaller axial sampling step gives more samples of the same response, rather than automatically improving diffraction-limited resolution. ^[ref]

Two planes are optically conjugate when an optical lens system images one onto the other. In confocal microscopy, an illumination pinhole is imaged onto the selected specimen point, and that specimen point is imaged onto the detection pinhole. Thus the two apertures select corresponding positions in different planes; they need not occupy the same physical plane. Modern laser instruments can supply the illumination point with an optical fiber or a suitable focused laser mode instead of a separate metal illumination pinhole. ^[ref]

A fluorescent emitter at the selected optical focus forms a compact point spread function at the detection aperture. An emitter displaced axially produces a broadened spot there through defocus; a smaller fraction of its collected light passes. The emitter still exists and still emits photons. Rejected photons strike the opaque mask or other parts of the instrument instead of contributing to the accepted detector channel. Neither a finite pinhole nor diffraction makes the accepted out-of-focus contribution exactly zero.

A coronagraph is an optical instrument that suppresses a bright source's light before detection while transmitting some light from nearby directions. Its purpose is to reveal faint structure buried in the bright source's point spread function, as in exoplanet direct imaging. A Lyot coronagraph combines a coronagraphic focal plane mask with a Lyot stop in a reimaged optical pupil; blocking the central image alone does not remove its diffraction rings. ^[ref].

The original motivation was the solar corona: hiding the bright photosphere would allow longer, repeated observations of faint surrounding light. In his 1930 paper, Lyot explained that short total eclipses restricted observing time and prevented some long exposures for spectral lines. The attraction is an instrument that makes the faint neighborhood accessible on demand. ^[ref]. The Sun is an extended source; the point-source calculation below illustrates imaging of an unresolved star and is not a quantitative model of the solar corona.

A coronagraphic focal plane mask modifies the wave amplitude or wave phase in an intermediate focal plane. Here it is a perfectly opaque circular disk centered on the bright source. A finite aperture produces an extended point spread function, so a finite disk blocks only part of the stellar light. Its sharp edge also changes the transmitted field and hence the later diffraction pattern. The exact model below makes both effects visible.

Schmid et al. tested the Lyot stop on sky using the close binary $\alpha$ Hydri with SPHERE/ZIMPOL at the Very Large Telescope. They kept the CLC-S-WF coronagraphic focal plane mask, the NB730 filter and pupil-stabilized imaging mode, and compared observations without a pupil stop and with stopB1_2. Each configuration used ten 8-second exposures. One camera imaged the optical pupil while the other simultaneously imaged the focal plane, connecting the blocked pupil features to the image improvement. ^[ref].

Bright rings occurred at the inner and outer pupil boundaries; the stop covered them and other unwanted features. To compare point spread functions, the authors multiplied the no-stop image by $0.68$, accounting for lost off-axis transmission. The companion's reported detection significance rose from $5.2$ to $9.7$. This demonstrates a useful gain despite lost companion light. It measures the complete stop, including obstruction, support-vane and faulty-actuator blockers; it does not isolate outer-rim clipping alone or establish a universal suppression factor. The figures here are independent toys, not reproductions of those observations. ^[ref].

Defocus is the mismatch between the receiving plane and the intended optical focus. Changing the plane changes the point spread function even when the optical lens is otherwise ideal.

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.

A diffraction-limited system has an optical response close to the ideal diffraction response for its specified optical pupil, wavelength and optical focus. Improving the optical aberrations further makes little difference to that response. This describes the optics: a poorly sampled or noisy detector can still deliver an image with much worse usable diffraction-limited resolution.

The optical pupil determines which wave amplitudes and wave phases contribute to an image point. In a scalar, paraxial approximation at the focal plane of an optical lens in air, write the pupil transmission as $T(u,v)e^{i\varphi (u,v)}$, after removing the ideal focusing phase. The amplitude spread function is proportional to its scaled Fourier transform:Here $(u,v)$ are pupil coordinates, $(x,y)$ are image coordinates, $f$ is focal length, and $\lambda$ is the wavelength in air. For a uniformly transmitting, unobscured circular optical pupil at ideal optical focus, the result is an Airy disk with surrounding rings. A blocked centre, support structures, or nonuniform transmission changes the ideal point spread function; “diffraction limited” does not specify one universal spot shape. The existing Fourier model of a Lyot coronagraph illustrates how pupil and focal masks deliberately alter this response.^[ref]

Optical aberrations add pupil wave phase errors. Defocus contributes a quadratic pupil wave phase in the paraxial approximation, redistributing light intensity from the best-focused spot. The Strehl ratio compares the peak with the ideal response for the same pupil and total radiant flux. A broad optical spectrum produces a weighted sum of wavelength-dependent point spread functions; chromatic aberration can additionally change optical focus with wavelength. Thus the operating conditions belong with any claim that a system is diffraction limited.

The dirty beam is the point spread function obtained by directly inverting the weighted sampled complex visibilities. For sampling positions $u_k$ and nonnegative weights $w_k$ with $W={\sum }_k{w}_k>0$, chooseInclude each $u_k$ with $-u_k$ and the same weight so that $B$ is real. Its unit peak convention means that an isolated unit point source contributes a shifted unit-height response. It is not a unit-integral convention. Sparse sampling can leave substantial subsidiary peaks, even when the longest interferometric baseline is large.

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.

Encircled energy is the fraction of total image radiant flux within a circle of radius $R$ around the centre of a point spread function.

The precision of a fitted single-emitter position is distinct from its diffraction-broadened point spread function width. An ideal Gaussian function image of width $s$ with $N$ independent detected photons gives ${\sigma }_{\mathrm{loc}}\simeq s/\sqrt{N}$. Pixel size, background, model error, sample motion and drift limit practical performance; precision alone does not guarantee accuracy.^[ref]

A useful motivating experiment moves a bead emitting fluorescence through the selected optical focus and records the detector signal against position. Measurement of point-spread function (PSF) for confocal fluorescence microscopy by InCheon Song and others (2005, §2 and Fig. 1) describes a $543.8\mathrm{nm}$ laser delivered through a single-mode optical fiber, collimation, beam-size adjustment and a dichroic beam splitter. A $100\times$, numerical aperture $1.3$ oil-immersion objective optical lens excites slide-mounted fluorescent beads and collects their emission. Emission passes a filter and collecting optical lens, then a pinhole, before an avalanche photodiode. A three-dimensional positioning stage moves the specimen. The apparatus also contains a phase plate for deliberately modified responses.^[ref]

The published comparison uses nominal bead diameters $0.17,0.5,1\mu \mathrm{m}$: larger beads give broader measured spots. This is a control on the finite emitter, because the image is the convolution of bead emission with the confocal point spread function. The smallest bead has weaker signal relative to noise; incorrect bead size, defocus, bleaching and optical aberrations limit inference from a restored response. The reported bead-size comparison concerns lateral responses; it is not a measured pinhole-size series or a measured sheet-sectioning curve.^[ref]

To test axial pinhole selection with this kind of apparatus, the following is a proposed extension, not an additional result from that paper. Choose an isolated bead small relative to the response, centre it laterally, then step $z$ through optical focus while recording background-subtracted signal at several pinhole settings. Hold excitation power, dwell interval, detector gain, emission band and sampling fixed. Preserve raw peaks before also plotting peak-normalized traces. A lower peak and faster relative decay with axial displacement would support the paired-selection mechanism; fitting only normalized widths would miss the throughput cost.

Measure blocked-excitation background, include a bead-free region, avoid detector saturation, and repeat scans in reversed order to reveal drift or bleaching. Check pinhole centring separately, since misalignment can mimic signal loss. Record bead diameter, immersion medium, refractive index, numerical aperture, wavelengths and axial step. A finite bead is an approximation to an optical point source; a point trace does not certify confocal optical sectioning of a whole fluorescent sheet. Depth-dependent spherical aberration, scattering and sample motion can broaden or distort the response. Increasing magnification or making a display brighter does not correct those effects.