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The Aharonov-Bohm effect is the phase $q\Phi /\hslash$ acquired when a charge $q$ encircles magnetic flux $\Phi$, even when the magnetic field vanishes along the charge's path.

In the magnetic Aharonov-Bohm effect, a coherent electron wave takes two routes around inaccessible magnetic flux. Recombination converts their relative wave phase into displaced interference fringes. In 1959, Yakir Aharonov and David Bohm proposed a shielded solenoid geometry and identified the obstruction to removing the magnetic vector potential throughout the accessible region.^[ref]

Use International System of Units, electric charge $q$, and magnetic minimal coupling $H=(-i\hslash \nabla -q\mathbf{A}{)}^2/(2m)$. Let ${\gamma }_1,{\gamma }_2$ run from the same source $S$ to the same detector point $D$. The oriented closed curve $C$ follows ${\gamma }_1$ and then ${\gamma }_2$ backwards. For a static magnetic vector potential, the magnetic contribution to the relative wave phase isHere ${\Phi }_C={\int }_{\Sigma }\mathbf{B}\cdot d\mathbf{S}$ is the oriented magnetic flux through a spanning surface. Equal or calibrated nonmagnetic propagation phases are assumed. For an electron, $q=-e$, where $e>0$ is the elementary charge; reversing $C$ reverses the sign. The observable is the gauge-invariant holonomy $e^{i\delta }$, rather than a unique local value of the magnetic vector potential.

The accessible region is a multiply connected region: $C$ cannot contract to a point without entering the excluded flux region. Stokes theorem does not set the line integral to zero, because a spanning surface must cross that excluded region, where the magnetic field need not vanish. There is real magnetic flux in the apparatus; only the accessible paths are idealized as free of the magnetic field. The exterior of a finite straight solenoid also permits routes around its ends, so its exact topology and leakage differ from those of an infinite solenoid or an excluded solid torus.

An explicit model makes the obstruction visible. Outside an infinite flux tube of radius $a$, choose cylindrical coordinates $(r,\theta ,z)$ andIts curl vanishes there, but a loop with winding number $n$ has ${\oint }_C\mathbf{A}\cdot d\mathbf{l}=n\Phi$. Locally $\mathbf{A}=\nabla (\Phi \theta /(2\pi ))$; the angular coordinate changes by $2\pi$ on a circuit. For generic $q\Phi /\hslash$, that candidate gauge removal changes the wave function's single-valued boundary condition. On a simply connected region, a smooth static magnetic vector potential with zero curl instead has a globally defined scalar primitive and is a pure gauge potential. Thus local zero magnetic field and globally removable magnetic vector potential are different conditions.

For an original interference example, take mutually coherent scalar amplitudes with constant individual intensities $I_1,I_2$ and an imposed carrier phase $ky$, as in off-axis holography. ThenThe interference fringe spacing is $d=2\pi /k$. An electron enclosing $\Phi =h/(4e)$ has $\delta =-\pi /2$, shifting maxima by $d/4$ for this orientation. Increasing the magnetic flux by $h/e$ changes $\delta$ by $-2\pi$ and restores the same pattern: the electron Aharonov-Bohm effect has magnetic flux quantum $h/e$. This uses $h=2\pi \hslash$, the Planck constant. It does not require the source's magnetic flux to vary continuously during a single exposure.

For $I_1=1$, $I_2=1/4$, the extrema are $I_{\max }=9/4$, $I_{\min }=1/4$, and the interferometric visibility is $4/5$. A fixed Aharonov-Bohm phase translates these fringes without changing their contrast. Averaging unresolved fluctuating phases can reduce the contrast; a darkened pattern alone cannot identify enclosed magnetic flux. These formulas assume coherence, indistinguishable alternatives and stable beam envelopes.

The appeal of the Aharonov-Bohm effect is that a local Lorentz force calculation along either ideal path misses information carried by their global arrangement. It gives a physical example of holonomy and supplies the encircling phase used in Chern--Simons flux attachment. This alone does not establish intrinsic topological order. The static comparison also matters: changing magnetic flux can induce an electric field, so a time-dependent experiment requires the scalar-potential contribution as well as the displayed spatial line integral.

A physicist.^[ref]

The Michelson interferometer turns a change in an optical path into a shift of interference fringes. Controlling the returning beams and counting the shifts can connect very small distances to a wavelength.

For a film of refractive index $n$ and thickness $d$ in air, its round-trip propagation phase is $\delta =(4\pi d/{\lambda }_0)\sqrt{n^2-{\sin }^2{\theta }_i}$. Two neighbouring same-type interference fringes differ by $2\pi$. At small angular spacing in radians,The relation assumes slowly varying interface reflection coefficients and no rapidly varying reflection phase. At normal incidence the first derivative vanishes, so a local linear angular-spacing formula must be replaced by a quadratic expansion.

On 11 December 1924, Edward Victor Appleton and Miles Aylmer Fulton Barnett perform an experiment: the Bournemouth broadcasting radio transmitter slowly changes wavelength; an Oxford radio receiver records alternating maxima and minima from ground wave-skywave wave interference. The inferred equivalent height is about $90\mathrm{km}$. A separate comparison of simultaneous loop and vertical antenna signals tests the downward arrival direction. The E layer is established first; the higher F layer is found in winter 1926–27. These dates describe different findings.^[ref]

Why change the wavelength deliberately? Ordinary fading alone could arise from changing emission, receiver drift or a moving propagation path. A controlled sweep predicts a specific progression of phase difference between two arrivals. The directional antenna comparison supplies an independent kind of evidence for an overhead contribution. Agreement supports an atmospheric return; fitting a distance alone would not establish a unique electron density profile. In the model below, comparable received amplitudes make the interference fringes easier to see. A practical repetition would also calibrate the frequency sweep, track radio transmitter output and receiver gain, and compare repeated sweeps with fixed-frequency intervals to detect time drift. These are controls inferred from the measurement model, rather than additional procedures attributed to the historical account.

The radio receiver does not need to resolve two pulses to detect two paths. If one path takes longer, the two contributions reaching the receiver during a slow frequency sweep carry slightly different emission frequencies and can produce a slow beat in the detected amplitude. At strictly fixed frequency with stationary paths, the arrivals instead have a fixed phase difference: there is no continuing beat merely because one path is longer. This distinction connects the historical observation to ionospheric interference fringe spacing.

Suppose the two input beams share frequency, electromagnetic polarization and overlapping transverse profiles at each output, with stable relative wave phase during the photodetector's averaging interval. For the ideal beam splitter scattering matrix above, expanding the squared complex amplitudes givesThe imaginary part terms describe wave interference. Applying the single-input power split to each input separately and adding the results would discard these terms. Their opposite signs ensure $P_1+P_2=|a_1{|}^2+|a_2{|}^2$: conservation of energy survives even when one output is dark.

For an original example, let $a_1=\sqrt{P}$, $a_2=\sqrt{P}{e}^{i\delta }$ and $P=1\mathrm{mW}$. Then $P_1=P(1-\sin \delta )$ and $P_2=P(1+\sin \delta )$:

At $+\pi /2$, $a_2=i\sqrt{P}$ makes $b_1=(\sqrt{P}-\sqrt{P})/\sqrt{2}=0$ and $b_2=i\sqrt{2P}$; at $-\pi /2$ the dark output swaps. Thus the two inputs can feed all their combined power into one output without gain. At zero relative input wave phase, the splitter's own relative wave phase leaves the contributions in quadrature, so neither output is dark.

For fluctuating fields, average the cross term as $\mathrm{Im}⟨a_1^{\ast }{a}_2⟩$. It vanishes for mutually uncorrelated inputs, making the independent power-splitting rule valid for the averaged output. Partial first-order optical coherence reduces it. Orthogonal linear polarizations eliminate the cross term in a polarization-insensitive power readout, and spatial mismatch reduces the overlap. Detector averaging can also erase rapid relative wave phase changes. A fixed relation between two inputs can be prepared by splitting one laser beam and recombining the two arms.

An actual teaching protocol in Michelson Interferometer & Precision Wavelength Measurement — Week 1, University of Colorado Boulder, §§4–6, starts with an ideal-model prediction, then builds a Michelson interferometer using a helium-neon laser, a nonpolarizing beam splitter, adjustable mirrors, a micrometer translation stage and a photodetector. Steering mirrors align beam overlap at near and far planes; far-plane interference fringes refine alignment. Moving the arm mirror changes the returning relative wave phase; detector voltage is watched with a live acquisition display or oscilloscope. The guide compares measured arm powers with 50:50 splitting and asks for a revised field model. It tests electromagnetic polarization dependence, optionally changing one arm with a quarter-wave plate, and estimates noise from repeated readings at fixed mirror position. These controls test whether fringe changes reflect coherent recombination rather than splitter imbalance or misalignment. The guide supplies a protocol, not a numerical dataset. Its detector noise, imperfect overlap and electromagnetic polarization mismatch limit displacement inference.^[ref]

Additional background subtraction controls follow from the field model: block both arms to record the detector offset and ambient light, then unblock each arm separately to establish its detected power at fixed gain. With both open, scan the mirror and compare extrema after subtracting that background. Keep the detector below saturation and repeat the scan to check drift. A changing signal alone cannot distinguish source optical coherence from motion or changing overlap. For a vacuum round trip, a mirror displacement $x$ changes relative wave phase by $4\pi x/\lambda$; the useful signal is its modulation after detection, not an optical-frequency oscillation that the detector must resolve.

In division of wavefront, different spatial portions of an incident wavefront are selected and propagated so that their fields overlap. Double-slit interference is an example. The selected portions need sufficient spatial optical coherence for stable interference fringes.

Consider one selected polarization of waves, represented by complex fields $E_1(t)$ and $E_2(t)$ at two sampling positions. Suppose their second moments are finite and stationary: $J_j=⟨|E_j(t){|}^2⟩$ and the following uncentered wave-field correlation do not depend on the absolute time $t$. DefineThe star denotes the complex conjugate; brackets denote an ensemble average. A long detector average estimates this quantity only if it adequately samples the fluctuations. The superscript $1$ identifies a field correlation: it does not mean the autocorrelation of the measured light intensity. The one-position version is $g^{(1)}(\tau )=⟨E^{\ast }(t)E(t+\tau )⟩/⟨|E{|}^2⟩$, so $g^{(1)}(0)=1$ for every nonzero field covered by this definition; this alone says nothing about its correlation at nonzero delay. ^[ref].

The Cauchy-Schwarz inequality gives $0\le |g_{12}^{(1)}(\tau )|\le 1$. At a specified delay, magnitude one means $E_2(t+\tau )=a{E}_1(t)$ in mean square for a fixed nonzero complex $a$: the samples have a reproducible relation in both wave amplitude and wave phase. Magnitude zero means the averaged field cross term vanishes. It need not mean statistical independence. A deterministic phase difference changes the argument of $g^{(1)}$, not its complex modulus; a distorted but fixed wavefront can therefore remain fully coherent. ^[ref].

To see the observable, recombine the samples in the same detected spatial mode with matched wave polarizations. With a common positive conversion factor $C$, let $I_j=C{J}_j$ be the single-arm light intensities reaching that detector. Adding $E_1(t)+e^{i\varphi }{E}_2(t+\tau )$ givesThis follows simply by expanding the squared complex modulus and averaging, then taking the maximum and minimum as $\varphi$ spans a full cycle. The extra wave phase scan must leave $I_1,I_2$ and the magnitude of the correlation effectively unchanged. Thus interferometric visibility equals $|g^{(1)}|$ only for equal nonzero light intensities and ideal mode matching. For $I_2/I_1=1/4$, perfectly coherent beams have interferometric visibility $0.8$; a measured value $0.4$ would imply $|g^{(1)}|=0.5$ under these assumptions. Orthogonal linear polarizations, background light, unresolved interference fringes or vibrations can reduce measured contrast further. One absent arm gives zero contrast and makes the normalized two-sample correlation undefined.

In 1851 Hippolyte Fizeau detected a change in light propagation through moving water using interference fringes. Sunlight entered a telescope through a partially reflecting mirror; two slits selected beams passing through parallel glass tubes, each $1.487\mathrm{m}$ long with $5.3\mathrm{mm}$ internal diameter. A second telescope and focal mirror exchanged the tubes on return, and the beams recombined at a graduated eyepiece. Compressed air drove water between reservoirs, producing opposite flows; reversing both flows reversed the interference fringe displacement. The double transit suppressed static differences in temperature and pressure; filling only one tube with water tested this compensation. Separate supports reduced mechanical disturbance. Hippolyte Fizeau estimated flow speed from volumetric flow rate divided by tube cross section.^[ref]

The Fizeau experiment turns a tiny change in phase velocity into a visible fraction of an interference fringe. The Fizeau experiment path-length convention below explains why both tube passages contribute with the same sign for one beam and the opposite sign for the other. This makes the apparatus sensitive to the part of the phase difference that reverses with the flow, while rejecting disturbances shared by both beams.

Use the schematic below to count the water traversals explicitly. Each stationary tube has laboratory length $\ell$. In the displayed setting, upper-tube water flows right and lower-tube water flows left, each at speed $u>0$. Beam A goes right through the upper tube, exchanges tubes at the far optics, and returns left through the lower tube: it follows the flow twice. Beam B goes right through the lower tube and returns left through the upper tube: it opposes the flow twice. Each complete beam path therefore contains total water length $L=2\ell$, not $\ell$ or $4\ell$. The connecting air paths and fixed optics are assumed to contribute no flow-dependent phase difference.

Let $v_{+}$ and $v_{-}$ be positive phase velocity magnitudes along and against the water flow:For a laboratory source of frequency $\nu =c/{\lambda }_0$, where ${\lambda }_0$ is its vacuum wavelength, define the signed interference fringe offset $N$ by the propagation phase difference $\Delta \varphi =2\pi \nu \Delta t=2\pi N$. This is an equivalent phase delay between steady beams; no stopwatch resolves their transit-time difference. Subtract the zero-flow baseline and expand to first order:Thus one shifted fringe corresponds to one cycle of phase difference. If the stationary pattern has spatial interference fringe spacing $d$, a simple readout model is $I(x)=I_0[1+\mathcal{V}\cos (2\pi x/d+2\pi N)]$, where $\mathcal{V}$ is the interference visibility. The maxima shift by $-dN$ with this choice of screen coordinate. A sign assigned to “rightward” motion depends on the image orientation; the magnitude in fringe units does not.

Reversing both flows gives $N(-u)=-N(u)$. The displacement between equal and opposite settings therefore has magnitudeThe factor $2$ from comparing opposite flows is additional to the factor $2$ already included in $L=2\ell$. Repeating a round trip in just one flowing tube would instead put a beam with the flow once and against it once; that geometry does not produce the two same-sign contributions drawn here.

Choose $\ell =1.50\mathrm{m}$, $n=4/3$, ${\lambda }_0=550\mathrm{nm}$ and $u=6.00\mathrm{m}{\mathrm{s}}^{-1}$, with $c=299792458\mathrm{m}{\mathrm{s}}^{-1}$. These are illustrative choices, not measured values or a reconstruction of Hippolyte Fizeau's data. Each beam traverses $L=3.00\mathrm{m}$ of water. The nondispersive prediction isThe first number is the offset from rest to one flow setting; the second compares equal opposite settings. The very short equivalent delay explains why interference is useful: a measurable fraction of a wavelength replaces an unattainable direct timing measurement.

The Galilean transformation would give phase velocity magnitudes $c/n\pm u$, hence $N_{\mathrm{full}}\simeq 4\ell u{n}^2/(c{\lambda }_0)$. At the chosen values this is $0.38815$ fringe from rest, larger than the special relativity prediction by $n^2/(n^2-1)=16/7$. A model with no flow-induced change of phase velocity instead gives $N=0$. The three alternatives are experimentally distinguishable even when $u/c$ is tiny.

Hippolyte Fizeau's positive displacement favoured Augustin-Jean Fresnel's partial-drag law over zero or full drag. He explicitly distinguished support for that law from establishing its underlying luminiferous aether hypothesis.^[ref] The Fizeau experiment therefore supplies a nonzero relative-motion measurement, not a null test proving an luminiferous aether's absence. The derivation above exhibits the same first-order law within special relativity, showing why agreement with the coefficient alone cannot select an physical interpretation.

Two limitations matter for precision. The velocity profile sampled by the beam need not equal the cross-sectional average inferred from volumetric flow rate. Also, the Doppler effect gives opposite rest-frame frequencies for the counterpropagating beams. With optical dispersion, the coefficient $n^2-1$ in the interference fringe formula becomeswhere the refractive index and its derivative are evaluated for stationary water at the laboratory angular frequency $\omega =2\pi c/{\lambda }_0$. A wavelength-independent $n$ therefore defines an approximation, not a precision fit to real water.^[ref] Flow reversal removes a fixed phase difference, but a mechanical or temperature change that reverses with the flow can still imitate the signal and must be controlled.

Holography records the wave interference between a holographic object wave and a known holographic reference wave, then uses that record to reconstruct a wavefront. The record is a hologram. A photographic plate or photodetector measures light intensity; the relative wave phase is encoded in the positions and contrast of interference fringes, rather than measured directly as an optical oscillation. Reillumination makes the recorded pattern act as a diffraction grating that produces an image-bearing field.

The useful change of viewpoint is to preserve a field that can propagate again, instead of only a focused brightness picture. Recovered complex amplitude permits refocusing with the Fresnel propagator; a spatially extended hologram can also supply different viewing directions. The recoverable information is limited by the collected aperture, recorded spatial frequencies, optical coherence and separation of unwanted diffraction orders.

An interference fringe is a spatial band of enhanced or reduced wave intensity caused by a spatially varying phase difference. Uniform crossing plane waves produce straight parallel interference fringes; curved wavefronts can give curved bands. The interference fringes mark the relative wave phase, rather than the individual instantaneous wavefronts.

The distance between adjacent bright interference fringes, or between adjacent dark interference fringes, measured along a specified direction. For the symmetric crossing plane waves in wave interference, measured along $x$, $\Lambda =\lambda /(2\sin \theta )$. Bright-to-dark separation is $\Lambda /2$. As $\theta$ tends to zero the spacing diverges; at $\theta =0$ the phase difference is spatially constant and there are no periodic bands.

Wave interference is the reinforcement or cancellation that occurs when overlapping wave amplitudes add by the superposition principle. For two monochromatic waves of the same frequency, mutual optical coherence means that their relative wave phase is stable during the observation. Equal wave phases give constructive interference; a half-cycle phase difference gives the destructive interference intensity minimum. Wave amplitudes add before their squares determine wave intensity. ^[ref].

Consider two overlapping beams in a homogeneous medium, with the same polarization of an electromagnetic wave and individual local time-averaged light intensities $I_1,I_2$. Their phase difference $\delta$ is measured in radians. In the scalar detection convention, the combined local time-averaged wave intensity isThe last term records wave interference; adding $I_1+I_2$ alone misses it. The formula describes irradiance when both fields propagate along the same direction, or when the same detector geometry factor is included in all three intensities. For crossing beams it describes the squared electric field response, proportional to scalar wave intensity; a specified component of the Poynting vector also requires the propagation geometry. The ideal crossing plane waves below make that distinction explicit.

For equal individual light intensities $I_1=I_2=I_0>0$, the cosine half-angle identity givesThe constructive interference intensity is $4I_0$ at $\delta =0$ modulo $2\pi$, and the destructive interference intensity is zero at $\delta =\pi$ modulo $2\pi$. At phase quadrature, $\delta =\pi /2$, it is $2I_0$. With $I_1=1$ and $I_2=0.25$ in the same arbitrary wave intensity unit, the maximum is $2.25$, the minimum is $0.25$, and the phase quadrature value is $1.25$: unequal wave amplitudes cannot cancel completely.

Both curves use the same wave intensity unit and a horizontal angle in radians. The dashed levels are the averages over a full phase difference cycle. Equal beams have interferometric visibility one; the unequal example has interferometric visibility $0.8$.

A spatially varying phase difference turns the same law into interference fringes. Take two ideal plane electromagnetic waves with common wavelength $\lambda$, equal individual light intensities $I_0$, and propagation directions at angles $+\theta$ and $-\theta$ from the $z$ axis in the $xz$ plane, with $0<\theta <\pi /2$. Both have linear polarization along $y$, perpendicular to both propagation directions. Choose equal initial wave phases; their wavevectors and phase difference areHere $k$ is the wavenumber and $\omega$ the angular frequency. The common $z$ and time terms cancel, so the interference fringes stay at fixed $x$ for every $z$. Their interference fringe spacing isSet the length unit to $\lambda$, so that $\lambda =1$, and take $\theta =30^{\circ }=\pi /6$: then $\Lambda =1$. The constructive interference intensity bands lie at integer $x/\lambda$; the destructive interference intensity bands lie at half-integer $x/\lambda$.

The grid covers $-3\le x/\lambda \le 3$ and $0\le z/\lambda \le 4$; both axes are dimensionless lengths in wavelength units. The colour scale is $I/I_0$ from zero to four, using the same single-beam normalization as the equal-beam curve above. This is an ideal plane wave superposition, with no slits or point sources and no near-field approximation. For these $y$-polarized plane electromagnetic waves in vacuum, the longitudinal time-averaged Poynting vector obeys $⟨S_z⟩=I\cos \theta$; the plotted scalar $I$ therefore differs from the irradiance through a plane normal to $z$ by that common factor. This relation follows by adding the magnetic fields as well as the electric fields.

The bright peaks do not create energy. Averaging over one interference fringe spacing, or equivalently over one full phase difference cycle, givesbecause the integral of the cosine is zero, with sine as its antiderivative. Equal beams thus average to $2I_0$, despite bright bands at $4I_0$. For the crossing-wave geometry, the average longitudinal energy flux is correspondingly $2I_0\cos \theta$. Bright and dark regions redistribute the incoming energy; conservation of energy concerns the full energy flux, rather than the height of one peak.

Wave interference is useful because phase differences convert small changes in optical path length into measurable changes of wave intensity, as in a Mach-Zehnder interferometer. Its beauty is that the same superposition principle produces both complete cancellation and fourfold brightening; adding more coherent contributions also produces the narrow peaks of a diffraction grating. Feynman's treatment connects this many-source pattern directly to addition of complex amplitudes. ^[ref].

A short proof outline uses local real fields $E_j(t)=A_j\cos (\omega t+{\varphi }_j)$ with parallel linear polarizations. Write $I=C⟨(E_1+E_2{)}^2{⟩}_t$ and $I_j=C{A}_j^2/2$ for the same positive conversion factor $C$. Averaging over many wave periods while the phase difference stays fixed uses $⟨{\cos }^2(\omega t+{\varphi }_j){⟩}_t=1/2$ and $⟨\cos (\omega t+{\varphi }_1)\cos (\omega t+{\varphi }_2){⟩}_t=\cos \delta /2$, giving the stated law. Equivalently, add the complex amplitudes and take the squared complex modulus. ^[ref].

For different fixed linear polarizations in a common transverse plane, the cross term acquires the dot product of their real unit vectors, equal to $\cos \alpha$ for relative angle $\alpha$. Orthogonal linear polarizations remove the cross term in polarization-insensitive scalar detection; different orientations generally only reduce it. A common polarizer can select overlapping components and recover wave interference. Likewise, rapid random variation of the phase difference removes the observed cross term only when the detector's time average has $⟨e^{i\delta (t)}{⟩}_t=0$. Slowly varying wave phases can instead move resolved interference fringes, and biased or incompletely sampled wave phases can leave residual contrast. Finite optical coherence therefore depends on the observation time as well as the fields.

Interferometric visibility measures the contrast of a scanned interference fringe. For the mutually coherent, equally polarized two-beam model in wave interference, with fixed individual wave intensities and $I_1+I_2>0$, the constructive interference intensity and destructive interference intensity giveEqual nonzero wave intensities give $\mathcal{V}=1$; $I_1=1,I_2=0.25$ give $\mathcal{V}=0.8$. Unequal wave intensities can therefore reduce interferometric visibility even with perfect mutual optical coherence. Random phase differences or distinguishable wave polarizations can reduce it further; zero contrast alone does not identify the cause. Both intensities zero leave the ratio undefined.

For normalized paths with complex amplitudes $\alpha ,\beta$ and identical internal states, the output probability is $1/2+\mathrm{Re}({\alpha }^{\ast }\beta e^{i\theta })$, so $\mathcal{V}=2|\alpha \beta |$. A photon polarization fringe alone determines transverse quantum coherence but not the sign of the population difference.

// Target: optics.bigb

Consider an original two-path model with a locally flat ground baseline $d$ and a symmetric equivalent return height $h$. Replace the skywave by two straight segments traveling at the speed of light $c$, and replace the ground wave by a straight segment of length $d$. Their lengths and positive excess distance areThis is an equivalent geometric construction, not a metallic layer overhead. It neglects the curvature of Earth, refraction within the return region, ground-induced wave phase shifts and optical dispersion.

For received complex amplitudes $A_g$ and $A_s$, absorb their constant relative wave phase into ${\varphi }_0$ and choose their magnitudes nonnegative. An ideal detector reading proportional to squared received field hasAssume $\Delta L$ and ${\varphi }_0$ are independent of frequency across the sweep, with slowly changing amplitudes. Consecutive maxima differ by $2\pi$ in phase difference, soThe often-used approximation $\Delta f\simeq c/\Delta L$ therefore carries a frequency-independent-path assumption. Counting $N$ complete fringe periods between $f_1$ and $f_2$ gives $\Delta L\simeq Nc/|f_2-f_1|$; equivalently, $N=\Delta L|1/{\lambda }_2-1/{\lambda }_1|$ when the vacuum wavelength is $\lambda =c/f$. Count whole periods, not maxima plus minima as though each were a whole period.

Choose $d=150\mathrm{km}$ and $h=100\mathrm{km}$, with $c\simeq 3.00\times 10^8\mathrm{m}{\mathrm{s}}^{-1}$. Then $L_s=250\mathrm{km}$, $\Delta L=100\mathrm{km}$ and $\Delta f=3.00\mathrm{kHz}$. A $12.0\mathrm{kHz}$ sweep covers four full fringe periods. Conversely, the inferred equivalent height isThese chosen values illustrate the inversion; they are not a transcription of the 1924 measurements. With $A_g=1$ and $A_s=0.7$, the normalized reading ranges from $(1-0.7{)}^2=0.09$ to $(1+0.7{)}^2=2.89$: incomplete cancellation does not remove the distance information.

For a sweep with linearly changing frequency $f(t)=f_0+\dot{f}t$ and two constant delays ${\tau }_s>{\tau }_g$, the received instantaneous frequency difference has magnitude $|\dot{f}|({\tau }_s-{\tau }_g)$. In this model ${\tau }_s-{\tau }_g=\Delta L/c$, giving a beat frequency $|\dot{f}|\Delta L/c$. Equivalently, traversing each $\Delta f$ of the sweep crosses one fringe. This calculation requires a sufficiently slow sweep that propagation, tuning and amplitude changes can be treated consistently; it is not a beat between two independent transmitters.

More generally, measure the total phase difference $\Phi (f)$, including propagation and any reflection or apparatus contribution. Define the differential group delayFor a locally monotone wave phase with slowly varying derivative, the local spacing is $\Delta f\simeq 1/|\Delta {\tau }_g|$. If $\Phi =2\pi fD(f)/c+{\varphi }_r(f)$, this becomesHere $D(f)$ is a phase-equivalent optical path difference, not necessarily the geometric excess distance $\Delta L$. A drifting ionosphere, several simultaneous skywaves, changing wave polarization, a frequency-dependent reflection wave phase or significant loss can invalidate the simple constant-spacing inversion.

For a vertical outgoing and returning radio wave, define the virtual height $h^{\prime }=c{\tau }_g/2$ from its round-trip group delay ${\tau }_g$, after removing instrument delays. It is the height a fictitious reflector would have if both legs traveled at $c$. It differs from both the real turning height $z_t$ and a height fitted to an oblique constant-path interference fringe model.

An original solvable profile makes the difference explicit. Approximate the region below $z_0$ by $n=1$, and prescribe a linear rise of squared plasma frequency above it:Here the symbol $u_t$ denotes the penetration depth, and $a$ has units ${\mathrm{s}}^{-2}{\mathrm{m}}^{-1}$. In the propagating part of the ramp, $n=\sqrt{1-(z-z_0)/u_t}$. In the leading geometrical-ray delay model, with a frequency-independent turning connection wave phase, the round-trip group delay isThe endpoint singularity is integrable: substitute $s=\sqrt{1-u/u_t}$ to obtain $2u_t$. This approximation does not resolve the finite-wavelength structure at the turning region. It illustrates the effect of slow group velocity before the return, rather than adding extra geometric altitude.

As an independent phase-based derivation, the propagation contribution to the round-trip wave phase isIts derivative $d\Psi /d\omega =2(z_0+2{\omega }^2/a)/c$ agrees with the delay integral. Using the phase integral itself as a distance at speed $c$ would give $z_0+2u_t/3$ instead of $h^{\prime }$: wave phase, group delay and real turning height are three different quantities.

Choose $z_0=70\mathrm{km}$ and stipulate $f_p(z)=3\mathrm{MHz}\sqrt{(z-z_0)/(30\mathrm{km})}$, where $f_p={\omega }_p/(2\pi )$. ThenAt $f=3\mathrm{MHz}$, the real turning height is $100\mathrm{km}$ but the virtual height is $130\mathrm{km}$, corresponding to ${\tau }_g\simeq 0.867\mathrm{ms}$ with $c\simeq 3.00\times 10^8\mathrm{m}{\mathrm{s}}^{-1}$. These stipulated parameters describe neither a measured profile nor a separate physical reflector at $130\mathrm{km}$. Because the ramp has no density maximum, it also has no finite upper penetration threshold; a real bounded E layer or F layer requires a richer profile.

A Michelson interferometer divides light at a beam splitter, reflects the two arms at mirrors and recombines the returning fields. For vacuum arm lengths ${\ell }_1,{\ell }_2$, the delay magnitude is $|\tau |=2|{\ell }_2-{\ell }_1|/c$. At each coarse delay, a small mirror scan produces maxima and minima of the detected light intensity. A photodiode and its electrical readout resolve these changes; the scan should cover several carrier cycles while changing $|g^{(1)}|$ negligibly. MIT's laboratory guide uses a mirror actuator, photodiode and electrical drive to scan interference fringes and extract interferometric visibility.^[ref]

A verified motivating experiment is Shaoul Ezekiel's MIT demonstration of source-spectrum control. A polarizer selects one or several lines from a laser whose adjacent lines have orthogonal linear polarizations. A sampled beam goes to a scanning Fabry–Pérot interferometer to check the optical spectrum; the main beam enters a Michelson interferometer, and a lens projects its interference fringes onto a screen. With one selected line the fringes retain good contrast through the demonstrated delay range. With several lines their contrast falls and returns as the arm length changes. At a fixed delay with poor contrast, selecting one line restores it; at matched arms both selections give good contrast. This control isolates the role of spectral composition rather than simply the arm position. The presentation is a qualitative demonstration, not a calibrated linewidth measurement. ^[ref].

A deliberately simpler two-line model explains the return of contrast. Take two stationary, mutually uncorrelated monochromatic contributions of equal power at ${\nu }_0\pm \delta \nu /2$. Their normalized wave-field correlation isfor equal detected single-arm light intensities. At $\tau =1/(2\delta \nu )$ the two contributions' interference fringes cancel on averaging; at $\tau =1/\delta \nu$ they agree again. This is a deduction from the stipulated spectrum, not a fit to Ezekiel's two- or three-line source. Finite individual linewidths eventually damp these revivals.

For a quantitative version of the experiment, the visibility law suggests explicit controls. Measure each arm separately at every coarse delay, subtract the detector's dark signal and any measured background, then normalize by $2\sqrt{I_1{I}_2}/(I_1+I_2)$. Keep the returning beams aligned on the same detector region with parallel linear polarizations; keep detector gain fixed and avoid saturation. Resolve the small mirror scan electrically, rather than averaging over its moving interference fringes. Repeat scans to identify vibration or drift. These controls follow from the averaging and mode-matching assumptions in first-order optical coherence; loss of contrast alone cannot distinguish source decorrelation from apparatus motion, misalignment or insufficient spatial resolution. The finite travel range gives access only to a finite range of delays.

The practical gain is that wave interference measures field correlations even when the detector cannot follow the optical oscillation itself. A long optical coherence length allows unequal-arm measurements; a short one localizes high contrast near matched delays. The beautiful feature is that both fading and reappearance of interference fringes are sums of the same rotating complex amplitudes.

Albert Abraham Michelson's 1907 lecture, Recent Advances in Spectroscopy, pp. 171–172, connects counting interference fringes with a reproducible length standard. Broad or multiple spectral lines obscured fringes at large path differences; red cadmium emission supplied a suitable source. His 1907 Nobel Prize in Physics recognized optical precision instruments and their spectroscopy and metrology applications.^[ref]

His authored account of the metre comparison at Sèvres describes a heated, evacuated cadmium discharge tube and a spectrometer selecting one spectral line. A movable reference mirror was compared with pairs of silvered parallel surfaces defining intermediate standards. Counted interference fringes established a short interval; repeated comparisons extended it to a decimetre and then a metre. White-light fringes located surface coincidence, while monochromatic fringes supplied counts and fractional orders. A thermally shielding box, firm pier, calibrated translation screw and tested thermometers controlled drafts, vibration, travel and temperature. Uniform temperature mattered in comparisons; the final standard required an absolute temperature reading. Optical microscopes made the final comparison to the marked metre bar, limiting the whole result.^[ref]

This is a measurement chain, rather than one enormous unchecked fringe count: short intervals establish whole orders; fractional-order readings refine each longer comparison. The optical coherence envelope helps locate white-light coincidence, whereas the carrier phase difference measures small departures from it. Repetition checks counting mistakes, but agreement alone cannot exclude shared systematic errors. The general lesson of metrology is to preserve the link from the optical ruler through each transfer and control to the final length.

The Michelson stellar interferometer, associated with Albert Abraham Michelson, uses separated collectors and beam-redirection optics to produce interference fringes. The collector spacing determines sensitivity to a stellar angular diameter, independently of the final beam spacing at the combiner. The first null for a uniform disk is $D=1.21967\lambda /\alpha$, where $\alpha$ is the full angular diameter. Different surface-brightness laws, including limb darkening, change the null.