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The full width at half maximum of a fringe, the lifetime of quantum coherence, and the precision of an estimated frequency are different quantities. In the ideal short-pulse limit, write the transfer probability with contrast $0<V\le 1$ and second-pulse oscillation phase $\phi$ asAt quadrature, $P=1/2$ and $|\partial P/\partial \nu |=\pi VT$. For $N$ independent, equally prepared particles with ideal binary readout, the binomial distribution gives ${\sigma }_P=\sqrt{P(1-P)/N}=1/(2\sqrt{N})$. Local propagation of uncertainty therefore yields the displayed ${\sigma }_{\nu }$. For chosen $T=5\mathrm{ms}$, $V=0.8$ and $N=10^4$, it is approximately $0.398\mathrm{Hz}$, although the ideal fringe width is $100\mathrm{Hz}$. This is an original counting-noise estimate at a known quadrature, not the accuracy of a historical clock or a guarantee of finding the correct fringe.

For the illustrative dephasing model $V(T)=e^{-T/T_2}$ with fixed $N$, minimizing ${\sigma }_{\nu }$ maximizes $T{e}^{-T/T_2}$ and gives $T=T_2$: its derivative is $e^{-T/T_2}(1-T/T_2)$. This conclusion changes if longer interrogation reduces the number of accepted particles or the repetition rate. Quantum decoherence, a distribution of flight times, reference oscillator noise and readout errors can reduce contrast; a fitted decay time need not identify one microscopic mechanism. More counting can improve precision while a reproducible cavity quantum phase offset still biases the frequency, as in atomic clock frequency shifts.

The Rayleigh criterion calls two equal, mutually incoherent optical point sources just resolved when one circular-pupil Airy disk's centre coincides with the other's first zero. It is a specified discrimination convention, distinct from full width at half maximum and from precision in estimating one source's position.

For this circular optical pupil, the Rayleigh criterion uses angular separation $s_{\mathrm{R}}={\theta }_1\simeq 1.22\lambda /D$, or $s_{\mathrm{R}}=r_1\simeq 1.22\lambda f/D$ in the paraxial approximation to the focal plane. The point spread functions still overlap.^[ref]

For the Airy disk design example with $D=5.00\mathrm{mm}$, $f=100\mathrm{mm}$ and $\lambda =550\mathrm{nm}$, the angular convention gives $s_{\mathrm{R}}=1.34164\times 10^{-4}\mathrm{rad}$, or about $27.67$ arcseconds, and the image-plane separation is $13.4164\mu \mathrm{m}$. Two equally bright, mutually incoherent optical point sources at separations $0.7$, $1$ and $1.5$ times this scale would have image separations $9.391$, $13.416$ and $20.125\mu \mathrm{m}$ respectively. These calculated cases are compared in incoherent two-source Airy profile.

The Rayleigh criterion is useful because it ties an easily recognized two-peak convention to a predictable optical scale. It does not assert that all information vanishes below that separation. Inferring two unknown positions also depends on photon shot noise, background, unequal brightness and uncertainty in the point spread function; locating a single isolated source is a different task. A theoretical analysis shows that alternative optical measurements can estimate the separation of two incoherent optical point sources below this scale.^[ref]

The mutual optical coherence condition is essential. In incoherent imaging, time-averaged light intensities add. In coherent imaging, wave amplitudes add before squaring, leaving a wave interference term controlled by relative wave phase. Two holes illuminated by the same coherent laser therefore do not implement the equal-incoherent-source convention merely because their individual patterns resemble Airy disks. The coherent single-aperture laboratory test establishes the response of each aperture; it does not establish the pair's mutual incoherence.

A recoil-free nuclear transition emits or absorbs a gamma ray while leaving the internal vibrational state of the surrounding solid unchanged. It belongs to the elastic part of the process: no phonon is created or destroyed. The photon still carries momentum $p=E_{\gamma }/c$, where $c$ is the speed of light. The emitting solid takes momentum $-\mathbf{p}$; the absorbing solid takes $+\mathbf{p}$. Conservation of momentum holds for each event.

Why does an isolated atomic nucleus pose a problem? Start with an excited state at rest and let $E_0$ be its transition energy, with nuclear mass $M$ and $E_0\ll M{c}^2$. Conservation of momentum gives a recoil momentum of magnitude $p$; its nonrelativistic kinetic energy is thereforeBy conservation of energy, emission spends part of $E_0$ on nuclear recoil, whereas absorption must supply both $E_0$ and the final kinetic energy. To leading order,Thus the relevant comparison is $2E_R$ against the overlapping spectral line widths, rather than $E_R/E_0$ alone. If $2E_R$ greatly exceeds the widths, the isolated-nucleus lines have negligible overlap. Finite Lorentzian line shape tails mean that this is suppression, rather than an absolute prohibition on absorption.

For a bound atomic nucleus, the recoil-free channel transfers momentum to the center of mass of the emitting or absorbing solid, with negligible kinetic energy $p^2/(2M_{\mathrm{solid}})$. Other channels exchange phonons and carry substantial energy into or out of the internal vibrations. A large $M_{\mathrm{solid}}$ makes the elastic channel's energy cost small; the vibrational quantum state determines its probability. Merely replacing $M$ by a large mass in the free-particle formula does not calculate that probability.

Here is a deliberately stipulated teaching system, independent of historical isotope data: $E_0=20\mathrm{keV}$, $M{c}^2=10^{11}\mathrm{eV}$, and a single-line natural linewidth ${\Gamma }_E=10^{-7}\mathrm{eV}$ measured as an energy full width at half maximum. These give $E_R=0.002\mathrm{eV}$ and $2E_R/{\Gamma }_E=40,000$. A solid of $10^{12}$ such nuclear masses has elastic recoil energy $2\times 10^{-15}\mathrm{eV}$, far below ${\Gamma }_E$. In this article ${\Gamma }_E$ is an energy width; the corresponding lifetime is $\tau =\hslash /{\Gamma }_E$, with $\hslash$ the Planck constant divided by $2\pi$.

In the Driven two-level Rabi model, the transition probability is $P(t)={\Omega }^2{\sin }^2(\sqrt{{\Omega }^2+{\delta }^2}t/2)/({\Omega }^2+{\delta }^2)$. Maximizing over time at each detuning gives a Lorentzian line shape, with first maximizing time $\pi /\sqrt{{\Omega }^2+{\delta }^2}$ and full width at half maximum $2|\Omega |$ in angular frequency. The amplitude parameter $\Omega$ must use the specified Rabi angular frequency convention.

This envelope differs from a spectrum obtained with a fixed pulse duration: optimizing the time separately removes the sine-squared factor, while fixing time retains it. A finite linewidth here describes detuning-limited transfer under coherent driving, without requiring dissipative decay.

An actual teaching experiment in LM-0300 Fabry Perot Spectrum Analyser by Walter Luhs, §§3.1.1–3.1.2, prepares a green laser beam by aligning it near and far along a rail, then centring back-reflected light from adjustable curved mirrors. An actuator using piezoelectricity moves one mirror. A screen first displays interference fringes; a focusing optical lens and photodiode then replace the viewing arrangement. The oscilloscope records the photodiode signal and the piezo drive reference on separate channels, using the latter as its trigger. Alignment and mirror separation are adjusted to suppress extra transverse normal modes; source temperature and drive electric current are controlled. Repeated peaks calibrate the scan using the appropriate free spectral range, and their spacing divided by their full width at half maximum estimates finesse.^[ref]

This experiment uses a confocal curved-mirror cavity, whose repeated peak families can have spacing $c/(4d)$ in air; it must not be calibrated blindly with the parallel-mirror $c/(2d)$. Extra transverse normal modes can mimic source features. A finite measured peak width bounds source sharpness only after instrumental broadening is considered; it does not determine the source's intrinsic width by itself.^[ref]

For an independent ideal scan calculation, hold a narrow laser wavelength fixed and vary the parallel-mirror gap $d$ at normal incidence. Advancing one resonance order requires $\Delta d=\lambda /(2n)$. If $d$ changes uniformly over the measured interval, the time between repeated peaks divided by their full width at half maximum in time equals finesse. The cancellation of scan rate follows from $d(t)=d_0+vt$: both peak spacing and peak width are divided by the same $|v|$. It fails if the scan rate changes appreciably over that interval. A practical control in the University of Washington's motor-driven flat-mirror experiment is to check that successive peaks are equally spaced before using the time axis this way; its handout also warns that table strain and aggressive low-pass filtering alter the trace.^[ref]

Single-slit diffraction is the spreading of light transmitted by one slit. Consider an ideal rectangular slit of width $a$, illuminated normally by a uniform plane wave with a single wavelength $\lambda$ in the surrounding medium. This monochromatic wave has uniform wave amplitude and equal incident wave phases across the slit, providing the required optical coherence. In the scalar Fraunhofer diffraction model, the normalized angular wave intensity isHere $\theta$ is measured from the forward normal in the plane across the slit width. The sinc function uses the unnormalized convention $\mathrm{sinc}u=\sin u/u$. The limit makes the central value one. This ideal envelope neglects additional direction-dependent propagation and detector factors; near the forward direction it describes the normalized screen irradiance. ^[ref].

The noncentral zeros satisfyThe restriction follows from the range of the sine of a real angle. The first minima have $\sin \theta =\pm \lambda /a$ when $\lambda /a\le 1$. If $a<\lambda$, this model has no noncentral zero in the forward hemisphere; equality puts the first zeros at grazing directions, outside the small-angle approximation and off any finite perpendicular screen. ^[ref].

For $0<\lambda /a<1$, define the central width as the separation between the two first minima. Its angular width and its width on a perpendicular screen areThe angular approximation uses radians and requires $\lambda /a\ll 1$; the screen approximation uses $y\approx L\theta$. The tangent relation is exact straight-line geometry, even though the underlying Fraunhofer diffraction field is approximate. This width is distinct from the full width at half maximum.

For $\lambda =500\mathrm{nm}$, $a=0.1\mathrm{mm}$ and $L=1\mathrm{m}$, the positive first minimum isThe dimensionless far-field parameter is $a^2/(\lambda L)=0.02$. Doubling the slit width to $a=0.2\mathrm{mm}$ gives $W\approx 5.00002\mathrm{mm}$ and parameter $0.08$; both curves below use the Fraunhofer diffraction approximation. Choosing $L=2\mathrm{m}$ halves those parameters and doubles both screen widths.

Each curve is normalized to its own $I_0$; the arrows measure between first minima. The calculation uses $\theta =\arctan (y/L)$ before evaluating the sinc function, rather than replacing $\sin \theta$ by $y/L$. The lower diagram shows the exact screen geometry with the vertical scale enlarged. Normalization removes the change in peak brightness when the slit width changes.

The wave interference explanation is especially simple: divide the slit into equal narrow strips and add their complex amplitudes, rather than their wave intensities. With wavenumber $k=2\pi /\lambda$, the continuous slit givesIntegrating the complex exponential gives the endpoint difference; its continuous value at zero is one. Taking the squared complex modulus gives the stated wave intensity. This is the Fourier transform of an interval indicator expressed in angular coordinates. Feynman's continuous-source argument obtains the same pattern by letting the number of sources tend to infinity. ^[ref].

At the positive first minimum, the wave phases span exactly $2\pi$ across the slit. Every strip in one half has a partner $a/2$ away, with equal wave amplitude and phase difference $\pi$, so the pairs cancel. At $\theta =0$, all the wave phases align instead. ^[ref].

The drawing uses $N=8$ equally weighted midpoint samples $x_j/a=-1/2+(j+1/2)/8$, with $j=0,\dots ,7$. At the first minimum their complex amplitudes are a common rotation of the eight roots of unity, so their head-to-tail polygon closes exactly. This finite sum illustrates cancellation; the continuous integral, rather than eight point sources, defines the slit pattern at general angles. The circle shows the continuous range of wave phases; coloured arrows show the eight samples.

A narrower aperture requires a larger angle to accumulate the cancelling phase difference, so narrowing the slit widens the beam. This is a useful limit on geometrical optics: confining light more tightly at an opening increases its subsequent angular spread. At fixed incident irradiance and slit height, reducing $a$ also reduces transmitted power, exposing a practical balance between diffraction-limited resolution and light throughput. In a diffraction grating, this single-slit diffraction envelope modulates the wave interference between different slits.

A motivating measurement used the McMath solar optical telescope at Kitt Peak. The Sun's image drifted across a $10\mathrm{mm}$-high, $0.2\mathrm{mm}$-wide optical slit; a double-pass spectrograph selected wavelength intervals from $0.0015\mathrm{nm}$ wide at $303\mathrm{nm}$ to $0.007\mathrm{nm}$ at $1099\mathrm{nm}$. An EMI 9750 QB photodetector or silicon photodiode supplied a digitized signal: $1\mathrm{kHz}$ samples were summed into 4096 positions per crossing, including dark and off-limb sky readings. ^[ref]

Controls included daily realignment of optical telescope mirrors, wavelength settings between spectral lines insensitive to the rotational Doppler shift, selection of detectors without detectable hysteresis, and measured electronic nonlinearity corrections below $0.1\%$ of maximum signal. Limb-position calibration combined centres of gravity of the edge derivative profiles with almanac semidiameters; the preliminary edge shifts were consistent with astronomical seeing of approximately $5$ arcseconds full width at half maximum. ^[ref]

The decreasing centre-to-limb readout gives a specific intensity profile, constraining stratification of the photosphere under a radiative transfer model. Astronomical seeing, instrumental scattering of light, and solar granulation limit this inference; brightness alone does not uniquely determine temperature versus geometric height. These observations need not follow the stipulated linear limb-darkening law.

For $b>0$, the Fourier transform of $f$ is ${\int }_{-\infty }^{\infty }f(t)e^{ixt}dt=2b/(b^2+x^2)$. A weak interaction $D_{0}f(t)\cos \omega t$ between energy levels of gap $\hslash {\omega }_0$ gives final probability amplitude $-idb\{[b^2+({\omega }_0-\omega {)}^2{]}^{-1}+[b^2+({\omega }_0+\omega {)}^2{]}^{-1}\}/\hslash$, where $d$ is the transition matrix element. Only when $b,|{\omega }_0-\omega |\ll {\omega }_0$ does the single resonant lobe give an approximately squared Lorentzian line shape excitation profile. Its full width at half maximum is $2b\sqrt{\sqrt{2}-1}$ in angular frequency.

Temporal optical coherence compares a field at one position with a delayed copy. In a Michelson interferometer, unequal round-trip lengths supply this delay. Even a source with rapidly changing wave phase can give high interferometric visibility close to zero delay, because both arms then deliver nearly the same source history. A source narrowed in optical spectrum generally retains correlation over larger delays; a single width does not describe sources with several separated lines. ^[ref].

For a stationary scalar field, choose the carrier convention $e^{-i2\pi \nu t}$, with frequency $\nu$ in cycles per second. Let $S(\nu )\ge 0$ be its spectral density per unit frequency, with $0<\int S(\nu )d\nu <\infty$. Its normalized field correlation is the Fourier transformThe optical spectrum weights the rotating contributions; a broad spread of frequencies allows them to cancel at short delays. This is the stationary field correlation–spectrum relation. The relevant spectrum is that of the optical field in the detected mode, rather than the low-frequency fluctuations of a detector's light intensity. ^[ref].

Here are two stipulated line profiles, each centered at ${\nu }_0$ with full width at half maximum $\Delta \nu >0$. Write $f=\nu -{\nu }_0$ and use normalized profiles on the whole detuning line:The first is a Lorentzian, mathematically a Cauchy distribution; the second is a Gaussian function. Taking their Fourier transforms gives, with the carrier factor $e^{-i2\pi {\nu }_0\tau }$ omitted from the magnitudes,These are calculated model results, not measured line profiles. The detuning approximation treats the narrow line as extending to arbitrarily large positive and negative $f$; for optical applications require $\Delta \nu \ll {\nu }_0$. Exact spectra of the complex positive-frequency field have no negative absolute frequencies. Changing to angular frequency replaces $\Delta \nu$ by $\Delta \omega /(2\pi )$; it must not silently preserve the same numerical constant.

Equal detected single-arm light intensities give interferometric visibility equal to the correlation magnitude. Doubling the Gaussian full width at half maximum from $1$ to $2\mathrm{GHz}$ halves every fixed-contrast delay. At $\tau =0.5\mathrm{ns}$ the model gives Gaussian interferometric visibility approximately $0.410686$ at $1\mathrm{GHz}$ and $0.0284471$ at $2\mathrm{GHz}$; the $1\mathrm{GHz}$ Lorentzian gives $0.207880$. The dashed curve instead keeps the $1\mathrm{GHz}$ Gaussian correlation and changes $I_2/I_1$ to $1/4$, multiplying all interferometric visibilities by $0.8$. Its lower contrast at zero delay therefore comes entirely from intensity imbalance.

The product of a specified temporal width and spectral frequency width. State whether each width refers to power or field amplitude and whether it is a full width at half maximum or a standard deviation.

Choose a closed two-level system, equal stimulated absorption and emission strengths, a monochromatic pump, and no extra dephasing. Let $p_g+p_e=1$ be the lower and upper populations, and let spontaneous emission remove upper population at rate $\Gamma =1/{\tau }_e$. In this convention the natural linewidth is full width at half maximum $\Gamma$ in angular frequency, its half width at half maximum is $\Gamma /2$, and its full width at half maximum in hertz is $\Gamma /(2\pi )$. All $\Delta$ below are angular detunings.

Let $s=I/I_{\mathrm{sat}}$ be the on-resonance saturation parameter. An effective stimulated transition rate that reproduces the monochromatic steady response isHere $R$ describes each stimulated direction, while $\Gamma$ describes irreversible spontaneous emission. The rate equation gives the steady populations; it is not a description of transient Rabi oscillations. Setting ${\dot{p}}_e=0$ givesand the population difference relevant to net weak-probe photon absorption isThe pump depletes the lower population by $p_e$, but reduces the population difference by $q=2p_e$: stimulated emission contributes as well as missing ground-state absorbers. On resonance, $s=1$ gives $p_e=1/4$, $p_g=3/4$, $d=1/2$; $s=9$ gives $p_e=0.45$, $p_g=0.55$, $d=0.10$. As $s\to \infty$, the two populations approach $1/2$, rather than producing a population inversion. Far off resonance, $p_e\to 0$ and $d\to 1$.

For moving atoms use $\Delta =\delta -kv$ in these pump-prepared populations. A weak counterpropagating probe interrogates them near $\delta +kv=0$. A simple population-only approximation to its absorption coefficient averages the weak-probe Lorentzian line shape times $d(\delta -kv)$ over the Maxwell-Boltzmann velocity distribution. It neglects interference between the beams, changes of velocity during illumination, and multilevel optical pumping. Those restrictions matter for real alkali chemical atoms.