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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.

Write $a=D/2$ and wavenumber $k=2\pi /\lambda$. The pupil Fourier transform, normalized by pupil area, gives the Fraunhofer diffraction wave amplitude. In pupil polar coordinates,The angular average is $J_0$ because $J_0(z)=(2\pi {)}^{-1}{\int }_0^{2\pi }{e}^{-iz\cos \varphi }d\varphi$. The Bessel function of the first kind identity $d\{z{J}_1(z)\}/dz=z{J}_0(z)$ then reduces the radial integral to $A(u)=2J_1(u)/u$. Taking the squared magnitude yields the Airy disk light intensity profile.^[ref]

The apparent division by zero is removable. The power seriesgives $A(u)=1-u^2/8+u^4/192+O(u^6)$ and $P(u)=1-u^2/4+5u^4/192+O(u^6)$. At the centre, every pupil contribution has the same wave phase, so their normalized sum is one. The first dark ring instead corresponds to $J_1(u)=0$ with $u>0$, rather than to the removable zero of $J_1$ at the origin.

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.

For uniform coherent plane-wave illumination, an ideal binary amplitude zone plate transmits alternate Fresnel half-period zones. If $N_o$ complete odd zones are open, every one contributes approximately twice the unobstructed axial complex amplitude, in the same phase, and $I_f/I_0\simeq 4N_o^2$. This is a local light intensity gain relative to removing the plate and outer stop. It differs from comparison with a finite clear aperture: an aperture containing an odd integer number of half-period zones has axial intensity $4I_0$, giving plate-to-aperture ratio $N_o^2$. Neither gain is a total power efficiency.

Beam divergence describes the angular spread of a beam away from its optical axis; specify the radius convention and whether the quoted angle is a half-angle or full angle. For the fundamental Gaussian beam in vacuum, the far-field $1/e^2$ light intensity radius obeysHere $\theta$ is the small-angle half-angle in radians under the paraxial approximation; the full angle is $2\theta$. The geometric asymptote has angle $\arctan (w_0/z_R)$, approximated by $w_0/z_R$ in this regime. Other beam profiles need their own beam divergence definitions and propagation laws.

The beam waist is the narrowest cross-section of a freely propagating Gaussian beam. Its radius $w_0$ uses the wave amplitude $1/e$, or light intensity $1/e^2$, convention; its diameter is $2w_0$. Specifying a “spot size” without saying radius or diameter can introduce a factor-of-two error. Within the scalar paraxial approximation, the wavefront at the beam waist is planar. The minimum radius sets both the Rayleigh range and the far-field beam divergence.

The carrier-envelope phase locates the carrier oscillation relative to the envelope of an optical pulse. In $E_{\mathrm{real}}(t)=F(t)\cos ({\omega }_{0}t+{\varphi }_{\mathrm{CE}})$ with envelope maximum at zero, it is the displayed phase modulo $2\pi$. Few-cycle dynamics can depend on it even when cycle-averaged light intensity is unchanged. A pulse train's phase increment is related to its carrier-envelope offset frequency; signs depend on the complex-field convention.

Chirped pulse amplification increases an optical pulse's energy while keeping its peak power lower during amplification: stretch the pulse, amplify it, then compress it. A chirp is a changing instantaneous frequency. Stretching separates the arrival times of different frequencies; pulse compression brings their contributions back together. The gain medium supplies energy, whereas the compressor concentrates the surviving energy into a shorter time.

Donna Strickland and Gérard Mourou demonstrated chirped pulse amplification at Rochester in 1985. Their experiment used an optical fiber, a regenerative amplifier and a diffraction grating compressor.^[ref]

The useful idea is to arrange where the large peak power occurs. Lower light intensity in the amplifier reduces unwanted self-focusing and self-phase modulation; the compressed output can still drive strong nonlinear optics. Stretching does not remove every damage mechanism, and compression does not manufacture energy.

Let $P(t)$ be the pulse's power integrated over its transverse section, and $U$ its energy. Its peak power is ${\max }_{t}P(t)$. For a chosen Gaussian power envelope with full width at half maximum $\tau$,The numerical shape factor is approximately $0.9394$. It matters when comparing quoted peak powers; the estimate $U/\tau$ assumes a shape and a width convention.

If an otherwise unchanged pulse is stretched by a factor $S$ without loss, its energy stays fixed and its peak power falls by $S$. For energy gain $G$ and compressor energy transmission $\eta$, ideal restoration of the original shape givesThese relations require a specified shape and preserved spectrum; they do not model amplifier saturation. For the diagram's chosen $1\mathrm{nJ}$, $30\mathrm{fs}$ seed, $S=9.2959$, $G=1000$ and $\eta =0.8$, the successive peak powers are $31.31\mathrm{kW}$, $3.369\mathrm{kW}$, $3.369\mathrm{MW}$ and $25.05\mathrm{MW}$. The last pulse contains $0.8\mu \mathrm{J}$: the compressor increases peak power while losing some energy.

Fluence is time-integrated light intensity, $F(x,y)=\int I(x,y,t)dt$, measured in $\mathrm{J}{\mathrm{m}}^{-2}$. For a uniform beam of area $\mathcal{A}$, $F=U/\mathcal{A}$ and $I_{\mathrm{pk}}=P_{\mathrm{pk}}/\mathcal{A}$. At fixed transverse profile, lossless temporal stretching lowers peak light intensity but leaves fluence unchanged. Absorbed energy, heat removal, beam size, coatings and pulse duration still constrain optical components. For nonoverlapping identical pulses at repetition frequency $f_r$, the average power is $U{f}_r$; it is a different quantity from peak power.

The optical coherency matrix records the second moments of transverse complex amplitudes. Normalize its trace to light intensity. Under a deterministic Jones matrix $M$, $C\mapsto MC{M}^{†}$. Unpolarized light has $C=(I_0/2)I_2$, so its output intensity is $(I_0/2)\mathrm{tr}(M{M}^{†})$; it cannot be represented by one fixed Jones vector.

Choose $N$ equal-amplitude, monochromatic odd components with orders $q_j=q_0+2j$, $j=0,\dots ,N-1$, and odd $q_0$. Fix the complex-field conventionHere $I$ is a normalized light intensity after averaging the fast carrier oscillations. Setting every ${\varphi }_j=0$ and summing a geometric series givesAt the removable singularities take the continuous limit. Maxima repeat every $T/2=\pi /\omega$; the first zeros on either side of a maximum lie at $t=\pm T/(2N)$. The zero-to-zero width is $T/N$, which is not the full width at half maximum.

Choose orders $13,15,17,19,21$ and driving wavelength $\lambda =800\mathrm{nm}$. Using $T=\lambda /c$ with the speed of light $c$ gives $T=2.66851\mathrm{fs}$ and burst spacing $1.33426\mathrm{fs}$. Solving the exact half-maximum equation gives a full width at half maximum of approximately $240.6$ attoseconds. These are values of the specified five-component model, not measurements from the 2001 experiment.

A deliberately chosen spectral phase ${\varphi }_q=q\omega t_0$ translates the whole attosecond pulse train by $t_0$ without changing its shape. Adding $\kappa (q-17{)}^2/2$ makes neighboring phase increments unequal: the same component intensities then produce a different temporal profile. For the choice $t_0=0.20\mathrm{fs}$ and $\kappa =0.08$ radians per squared harmonic order, the adjacent-component delay estimates arenamely $0.0981$, $0.1660$, $0.2340$ and $0.3019\mathrm{fs}$ for $q=13,15,17,19$. This finite difference estimates the group delay at the pair's midpoint. It is not a single common arrival time.

The infinite monochromatic sum is periodic forever and has infinite total energy. A real attosecond pulse train has a finite envelope and finite component linewidths. This calculation isolates coherent addition; it cannot describe an isolated burst or establish its contrast against satellites. As a further check, averaging $I(t)$ over one half-period gives $N|a{|}^2$ for any constant component spectral phases, because the cross terms integrate to zero by Fourier orthogonality. Spectral phase redistributes the light intensity in time while leaving that average fixed.

In coherent imaging, complex wave amplitudes add with their relative wave phases preserved. The measured light intensity includes wave interference between the contributions.

The following original model isolates the conversion mechanism. Use a periodic image window, unit illumination, ideal unit magnification, and a phase plate that changes only the zero spatial frequency. Write $c_0=⟨e^{i\varphi }⟩$ for the spatial mean. Multiplying that single Fourier coefficient by $a{e}^{i\theta }$, with $0<a\le 1$, gives the exact image field in this model:This formula follows by splitting $e^{i\varphi }=c_0+(e^{i\varphi }-c_0)$: the second term has zero mean and is left unchanged. A real annular, partly coherent instrument alters a finite band of directions, so this calculation is a mechanism model rather than its full image-formation law.

For a weak phase object with $⟨\varphi ⟩=0$, one has $c_0=1+O({\varphi }^2)$ and henceThe cross term is $2\mathrm{Re}(a{e}^{-i\theta }i\varphi )=2a\varphi \sin \theta$. Thus $\theta =\pi /2$ makes positive $\varphi$ brighter and $\theta =-\pi /2$ makes it darker under this stated phase convention for complex amplitudes. The sign comes from this convention and chosen plate; a manufacturer’s positive/negative contrast labels should not be inferred from the sign of $\theta$ alone.

For example, choose $a=0.5$ and a zero-mean periodic pattern whose peak phase is $0.084$. The first-order peak intensities are $0.25\pm 0.084$, compared with the uniform ideal bright-field microscopy value $1$. The relative change about the reduced background is approximately $2\varphi /a$ for a quarter-cycle shift. Attenuating the reference increases relative contrast while reducing the background photon count; it is not light amplification or a universal improvement of signal-to-noise ratio.

Two checks expose what the model can and cannot measure. A spatially constant phase gives $E_{\mathrm{out}}=a{e}^{i\theta }{e}^{i\varphi }$ and $I_{\mathrm{out}}=a^2$, so no absolute phase is recovered. Orthogonality of the constant and zero-mean parts also givesFor $a\le 1$ this cannot exceed the incident mean light intensity. The image redistributes brightness and loses some reference light rather than creating energy.

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.

The degree of polarization quantifies how strongly a transverse optical field departs from an unpolarized mixture. Here ${\mu }_1\ge {\mu }_2\ge 0$ are the eigenvalues of its coherency matrix. For two mutually uncorrelated orthogonal linear components of intensities $I_1,I_2$, $\mathcal{P}=|I_1-I_2|/(I_1+I_2)$. It is undefined when the total light intensity is zero. A signed difference is a separate convention that also identifies a preferred polarization direction.

Destructive interference occurs when coherent contributions partially or completely cancel their complex amplitudes. For two equal contributions, a phase difference of $\pi$ modulo $2\pi$ gives zero resultant wave amplitude. The missing local light intensity is consistent with conservation of energy because the full spatial interference pattern redistributes power.

A diffraction envelope is the angular factor imposed by the shape and width of each opening in an interference pattern. For two equally illuminated rectangular slits of width $b$ and separation $s$, Fraunhofer diffraction gives $I(\theta )=4I_s{\mathrm{sinc}}^2(\pi b\sin \theta /\lambda ){\cos }^2(\pi s\sin \theta /\lambda )$, where $\mathrm{sinc}u=\sin u/u$ and $I_s$ is one slit's on-axis light intensity. The first factor limits the observable angular range even when the rapid interference fringes remain perfectly coherent.

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 double-slit experiment compares propagation through two openings. Coherent illumination gives double-slit interference; independent incoherent source patches contribute light intensities whose displaced patterns can reduce the observed interferometric visibility.