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A Michelson interferometer divides light at a beam splitter, sends it to two mirrors, and recombines the returned beams. A compensating plate can equalize glass traversal. For approximately axial rays the optical path difference is $\Delta =2(n_1{L}_1-n_2{L}_2)$, plus fixed instrument phase, andfor coherent, mode-matched fields. Moving one mirror by $\delta L$ in air changes the round-trip path by $2\delta L$, so one fringe corresponds to ${\lambda }_0/2$ mirror motion. It measures displacement, wavelength and refractive-index changes; scanning the delay enables Fourier-transform spectroscopy. Mirror alignment, polarization overlap, vibration and source coherence control visibility. A finite coherence length limits useful arm mismatch.

A Michelson interferometer divides a source beam at a partially reflecting beam splitter. The two perpendicular arms terminate in mirrors whose reflected beams return to the splitter and overlap at the detector. The recombined fields must have overlapping wavefronts and compatible polarization of waves. Moving one mirror by $\Delta L$ changes the round-trip optical path difference by $2\Delta L$, so $N$ passing fringes determine $\Delta L=N\lambda /2$. Conversely a known mirror displacement measures wavelength. Scanning the delay can determine temporal optical coherence and, by Fourier transform spectroscopy, spectral structure. A compensating plate may equalize glass traversal, particularly for broadband illumination.

For equal real peak field amplitudes, the superposition isTime averaging the square over many optical cycles gives $⟨E^2⟩=2a^2{\cos }^2(\varphi /2)=a^2(1+\cos \varphi )$. With $\varphi =kx$ after absorbing a constant port-dependent phase into the zero of $x$,Here $I_0$ is the sum of the two separate-arm intensities; maximum intensity is $2I_0$ and the ideal minimum is zero. Equal amplitudes give unit interference visibility.

For a stationary source with a flat intensity spectrum, different spectral components add in time-averaged intensity. Keeping their individual two-arm interference terms and defining $\mathrm{sinc}z=\sin z/z$ givesOne must not coherently add unrelated source frequencies as if they had a fixed common phase. The sinc envelope is the Fourier transform of the rectangular spectral intensity.

Numerically, ${\nu }_0=5.00\times 10^{14}\mathrm{Hz}$ and $\Delta \nu =1.00\times 10^{13}\mathrm{Hz}$. The annotated interferogram usesIt is even in $x$, has central value 2 and background 1, and its carrier-fringe period in optical path difference is $c/{\nu }_0=0.600\mu \mathrm{m}$. The first envelope zeros are at $x=\pm c/\Delta \nu =\pm 30.0\mu \mathrm{m}$, corresponding to mirror shifts of $\pm 15.0\mu \mathrm{m}$ from equal paths. Side lobes persist beyond the zeros; a negative sinc reverses fringe phase. The intensity remains between $1-|\mathrm{sinc}|$ and $1+|\mathrm{sinc}|$, not between the signed sinc curves with an assumed fixed ordering.

Let $x$ be mirror displacement from equal optical paths in a Michelson interferometer, so the optical path difference is $2x$. Different spectral components add detected light intensities; each component interferes with its own delayed copy. For equal line strengths and ideally balanced arms, writeThe trigonometric identity for the sum of cosines givesThus the sketch consists of rapid fringes within a slowly varying contrast envelope. With $\bar{\lambda }$ the mean wavelength and $|\Delta \lambda |\ll \bar{\lambda }$, the mirror-displacement fringe spacing is $\bar{\lambda }/2$. The interference visibility isThe first contrast zero is at $x=1/(4\Delta s)$; successive visibility maxima are separated byBetween successive contrast zeros the signed envelope changes sign, shifting the rapid fringes by half a cycle. Unequal line strengths leave a nonzero minimum contrast. Finite linewidth adds a longer-range coherence envelope, while arm imbalance and polarization mismatch can reduce contrast without indicating a different spectral spacing.

In division of amplitude, one incident beam is split into two or more parts at a partially reflecting interface; they follow distinct paths and are recombined. The Michelson interferometer and thin-film interference use this principle. In division of wavefront, spatially distinct parts of the same wavefront are selected and allowed to overlap, as in double-slit interference. Both require adequate optical coherence; wavefront division particularly exposes the need for spatial optical coherence between the selected regions.

Take the two interfering complex amplitudes to have equal intensities $I_0/2$ and relative phase $kx$. The interference cross term then givesThus the mean intensity is $I_0$, the maximum is $2I_0$, and the minima areThe optical path difference period is $\lambda$ and the mirror-displacement period is $\lambda /2$. The assumed equal intensities refer to the two fields at the chosen output, with matching polarization of waves and full optical coherence. They set the normalization; they should not additionally be interpreted as all incident light entering that output port. A physical lossless Michelson interferometer has a complementary second output.

A Michelson interferometer splits a collimated beam at a partially reflecting beam splitter. Two mutually perpendicular arms reflect from mirrors and recombine at the same beam splitter; the superposed returning beams produce interference fringes at the detector. Translate one mirror along its arm while holding the other fixed.

If that mirror moves a measured distance $\Delta x$, the round-trip optical path difference changes by $2\Delta x$ in air, approximated as vacuum. Every change of one wavelength moves one complete bright-dark-bright fringe cycle past a fixed detector point. Count $N$ such cycles to obtainUse a spectral line selected with a suitable optical filter, and count many interference fringes to reduce the fractional uncertainty of the distance and count. Keep the two arms within the source's optical coherence length and maintain overlap and alignment. If the vacuum wavelength is required for an arm filled with air of known refractive index $n$, the corrected relation is ${\lambda }_0=2n\Delta x/N$ at stable $n$.

A Michelson interferometer uses a beam splitter to divide collimated input light between a fixed mirror and a moving mirror. Returning beams recombine at the beam splitter; one output reaches a photodetector. A compensator plate can equalize the glass traversed in the two arms. Moving one mirror by $x$ changes the optical path difference by $2x$.

The usual nominal Fourier transform spectroscopy specification is $\delta \nu \simeq 1/d_{\max }$, where $d_{\max }$ is the maximum optical path difference measured from equal arms. It givesThis is a nominal spectral resolution element convention.^[ref] A symmetric rectangular scan from $-d_{\max }$ to $d_{\max }$ instead has a sinc instrumental line shape with first-zero offset $1/(2d_{\max })$. Quoting that first-zero criterion would give $d_{\max }=1$ cm; interferogram apodization and other linewidth criteria change the numerical factor. An exact mathematical minimum requires such a criterion.

In the measurement chain, the Michelson interferometer is the spectral analyzer or modulator, between the collection optics and the photodetector. It encodes the incoming radiation spectrum into an interferogram; detector electronics digitize that signal and a Fourier transform reconstructs the radiation spectrum. It does not itself turn radiant power into an electrical signal.

Measure the absolute spectral radiance over a range of frequencies and fit the Planck spectrumFor example, the COBE FIRAS blackbody spectrum experiment used a differential Michelson interferometer to compare the sky with an internal blackbody; an external calibrated blackbody could cover the sky entrance. Cold bolometers measured the difference spectrum, and foreground separation distinguished Milky Way emission from the nearly isotropic background. The fitted background temperature is approximately $\boxed{T_0=2.73\mathrm{K}}$.^[ref] The result relies on an absolute spectral comparison, rather than merely measuring a map's hot-to-cold contrast.

A photon gas is a collection of photons described statistically as radiation. Here the equilibrium model is homogeneous, isotropic electromagnetic radiation in vacuum at temperature $T>0$: each photon has energy $E=c|\mathbf{p}|=h\nu$, with momentum $\mathbf{p}$ and frequency $\nu$, and there are two independent transverse states of electromagnetic polarization. An absorbing cavity with walls at a common temperature provides a way to establish this thermal equilibrium. The bulk formulas assume cavity dimensions much larger than the typical wavelength $hc/(k_{B}T)$, so the electromagnetic mode density in a box can replace individual cavity modes by a continuum. ^[ref]

The crucial constraint is whether photon emission and photon absorption, or other number-changing processes, can equilibrate the photon population. When they can, photon number is not an independently conserved quantity and the equilibrium photon chemical potential is ${\mu }_{\gamma }=0$. For each polarization and frequency, the mean occupation number then follows the Bose-Einstein distribution:If number-changing processes are ineffective but number-conserving interactions redistribute energy, a constrained Bose-Einstein distribution can instead have $f=[\exp ((h\nu -{\mu }_{\gamma })/(k_{B}T))-1{]}^{-1}$ with chemical potential ${\mu }_{\gamma }\le 0$ fixed by the number constraint. Thus thermal equilibrium with respect to energy exchange alone does not establish ${\mu }_{\gamma }=0$. The zero value can also persist in an already prepared Planck photon distribution during suitable subsequent evolution; continual photon emission is not required to preserve it. ^[ref]

Combining $f$ with the two-polarization density of states, $g_{\nu }/V=8\pi {\nu }^2/c^3$, gives the Planck photon distribution and the spectral photon energy density:Here $B_{\nu }$ is the spectral radiance in Planck's law; $u_{\nu }$ is stored energy density per unit frequency, not outgoing power per area. Their different units are connected by the full-sphere solid angle and the speed of light $c$. The Boltzmann constant is $k_B$, and $h$ is the Planck constant. ^[ref]

The equilibrium equation of state is particularly simple:The radiation constant $a_R$ is the same constant called $a_{\mathrm{rad}}$ in black-body radiation; $\sigma$ is the Stefan-Boltzmann constant. An outline of the energy density calculation substitutes $x=h\nu /(k_{B}T)$ and uses ${\int }_0^{\infty }{x}^3/(e^x-1)dx={\pi }^4/15$. The radiation pressure follows because the angular average of the squared directional component of photon momentum is $1/3$ for isotropy; the thermodynamic derivation of blackbody radiation pressure supplies an independent route. These formulas describe the radiation contribution, not the cavity walls. The isotropic massless relation $p=u/3$ survives a different isotropic spectrum, whereas the coefficient in $u=a_R{T}^4$ requires the zero-chemical potential Planck photon distribution.

For a computed example with stipulated $T=2.725\mathrm{K}$, the photon energy density is $u\simeq 4.17174\times 10^{-14}\mathrm{J}{\mathrm{m}}^{-3}$ and the radiation pressure is $p\simeq 1.39058\times 10^{-14}\mathrm{Pa}$. The photon number density in the International System of Units isHere $\zeta$ is the Riemann zeta function. The mean photon energy differs from the energy at the peak of a spectral curve: counting photons and weighting them by energy emphasize different frequencies. The calculations use the exact International System of Units values of the Planck constant, Boltzmann constant and speed of light. ^[ref]

A motivating observation is the cosmic microwave background spectrum measured by the Far Infrared Absolute Spectrophotometer (FIRAS) aboard COBE. Its apparatus used a polarizing Michelson interferometer, with one input viewing a roughly seven-degree sky patch and the other an internal blackbody. Scanning mirrors changed the optical path length difference; heat-sensitive detectors recorded the varying power, and Fourier transform spectroscopy recovered the spectral difference. Operation outside the terrestrial atmosphere removed atmospheric emission and absorption, while shielding and cooling the optics in a helium cryostat suppressed the instrument's own radiation. ^[ref]

Absolute calibration came from moving an external blackbody into the sky aperture, filling the same beam used for the sky. This was an iron-loaded epoxy absorber shaped as a re-entrant cone. Repeated reflections made it a close approximation to a blackbody, while thermometers supplied its temperature. Mather and colleagues explicitly evaluated reflections, temperature gradients, radiation leaking through the material and leakage around its edge. These are limits on how faithfully the calibrator realizes thermal equilibrium radiation, rather than details that can be eliminated merely by calling it black. ^[ref]

During sky observations the internal blackbody temperature was adjusted to nearly null the spectral difference, reducing gain and drift errors. Fixsen and colleagues' 1996 analysis used calibrated spectra over wavenumbers $2$ to $21\mathrm{c}{\mathrm{m}}^{-1}$, about $60$ to $630\mathrm{GHz}$. They separated the sky's spatially uniform component, its dipole and Galactic emission, and varied sky cuts and foreground templates as controls. The resulting cosmic microwave background spectrum agreed with a single Planck function, with reported RMS deviations below 50 parts per million of its peak brightness. This is a residual statistic over the analyzed spectrum, not a pointwise relative error bound at every frequency. ^[ref]

The observation motivates using a photon gas as a physical model rather than merely a formal counting exercise. Its interpretation has a clear boundary: agreement with Planck's law constrains departures from an equilibrium spectral shape within the calibrated band and foreground assumptions; it does not directly measure radiation pressure or entropy density of a photon gas. Nor does it establish that today's cosmic microwave background is continually re-equilibrating with matter. A previously thermal spectrum can retain its Planck function shape as expansion of the universe lowers every photon frequency in the same proportion. The usefulness of the model is that one measured spectral temperature, together with its equilibrium assumptions, predicts energy density, radiation pressure, photon number density and entropy density of a photon gas consistently.

A Michelson interferometer turns displacement into a count of interference fringes: a moving mirror changes both the outward and return paths. This lets a wavelength act as a ruler whose divisions need no engraving. Assume normal incidence, a fixed reference arm, stable refractive index $n$ along the moving arm, and fixed vacuum wavelength ${\lambda }_0$. Write $\lambda ={\lambda }_0/n$ for the wavelength in that medium and take $\Delta L$ positive away from the beam splitter. The change in optical path difference, the accumulated phase difference, and the signed number of complete interference fringe cycles areThe factor two comes from propagation geometry: moving the mirror by $\Delta L$ adds $\Delta L$ before reflection and $\Delta L$ afterward. One bright-to-bright cycle means mirror travel $\lambda /2$; bright-to-dark means $\lambda /4$, half a cycle. All lengths must use compatible units, and $\Delta N$ is dimensionless. A constant reflection phase difference changes the initial fringe position but cancels from this change formula.

For an original hypothetical example, suppose $\lambda =600\mathrm{nm}$ in the actual surrounding medium and a continuous scan gives $40.25$ cycles, including the final fractional cycle. Then $\Delta L=(600\mathrm{nm})(40.25)/2=12075\mathrm{nm}=12.075\mu \mathrm{m}$. With the wavelength treated as exact, a fringe-reading measurement uncertainty of $0.02$ cycle contributes $6\mathrm{nm}$ to the displacement measurement uncertainty. An uncertainty of $0.01\mathrm{rad}$ in a local phase difference instead corresponds to $600(0.01)/(4\pi )\mathrm{nm}\approx 0.477\mathrm{nm}$. This illustrates measurement resolution, not a complete claim of accuracy: wavelength calibration, air changes, mirror tilt and missed cycles can dominate.

To separate the oscillating readout from its optical coherence envelope, use the local modelHere $I$ is detected light intensity, $\bar{I}$ is the mean level, ${\varphi }_b$ is a fixed bias wave phase, and $\mathcal{V}$ is interferometric visibility. A fine scan of $D$ measures the carrier phase difference; a coarse scan can map the temporal optical coherence envelope through $\mathcal{V}(D)$. The envelope maximum locates matched delays only under the source and apparatus assumptions; it is not a bright fringe identifying an arbitrary mirror position. Even with known $\mathcal{V}$, one intensity reading leaves the cosine sign and whole-cycle ambiguities. Continuous motion of known direction, or an independently obtained pair of signals in phase quadrature, is needed to recover signed travel through reversals.

The figure's original control example stipulates equal arm light intensities, matching linear polarizations, and a Lorentzian optical spectrum giving $\mathcal{V}(D)=\exp (-|D|/L_c)$ with $L_c=1\mathrm{mm}$. In vacuum, arm mismatches $|L_2-L_1|=0.60\mathrm{mm}$ and $0.10\mathrm{mm}$ give $\mathcal{V}=e^{-1.2}\approx 0.301$ and $e^{-0.2}\approx 0.819$. Matching the arms improves the phase difference readout without changing its ${\lambda }_0/2$ travel per cycle. A fine mirror scan of one cycle changes $D$ by only $600\mathrm{nm}$ for the stipulated $600\mathrm{nm}$ source, much less than $L_c$. These are calculated values, not historical data.

A proposed measurement sequence follows from this model. Overlap the returning beams, match their linear polarizations, find a high-contrast delay and use a small calibrated mirror scan to determine the sign and fringe period. Read a fixed spatial sample of the interference fringes with a photodiode; record each arm separately and the dark signal, avoid detector saturation, and retain enough time resolution to count every cycle. Approach mechanical settings consistently and repeat the return scan to expose lost motion or drift. For very small changes, bias near phase quadrature, where $|dI/d\varphi |=\bar{I}\mathcal{V}$ is largest; an intensity extremum has zero first-order sensitivity. A known drive voltage alone does not provide measurement calibration for mirror travel: compare the actual travel with the fringe count using an independently known wavelength.

Control temperature, pressure, vibration and air currents because the instrument measures optical path difference, not mirror motion alone. For a fixed reference arm and fixed ${\lambda }_0$, the first-order change of the moving arm's round-trip path is $\Delta D\approx 2n\Delta L+2L\Delta n$. Thus an unrecognized $\Delta n$ mimics $\Delta L_{\mathrm{app}}=L\Delta n/n$. A hypothetical $L=0.10\mathrm{m}$, $n\approx 1$, and $\Delta n=10^{-7}$ give $10\mathrm{nm}$ apparent travel. Holding the mirror still while varying or monitoring the air conditions tests this confounder. Separate checks of alignment, wavelength stability and thermal expansion are necessary before sub-fringe measurement resolution becomes a defensible measurement uncertainty.

For two equal-intensity mutually incoherent narrow spectral lines with wavenumbers $k_1,k_2$, a Michelson interferometer with optical path difference $x$ has normalized intensity $I/I_0=1+\cos (\bar{k}x)\cos (\Delta kx/2)$, where $\bar{k}=(k_1+k_2)/2$ and $\Delta k=|k_1-k_2|$. With mirror displacement $d=x/2$, the fast period is $\pi /\bar{k}\simeq \bar{\lambda }/2$; successive interference visibility revivals are separated by $\pi /\Delta k\simeq {\bar{\lambda }}^2/(2\Delta \lambda )$. The first visibility zero is halfway to the first revival. A signed envelope has twice the period of its absolute value. Unequal line strengths prevent perfect zeros.

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.

Instrument overview of FIRAS, describing its differential Michelson interferometer, internal reference and external calibrated blackbody. It reports the absolute cosmic microwave background temperature inferred from the spectral comparison.

The University of Florida's Rotation Vibration Spectrum of the HCl Molecule, University of Florida laboratory protocol, printed pp. 2–3 and 10–11, uses a scanning Michelson interferometer for Fourier transform spectroscopy. An air-filled silicon-window cell supplies the background. Supervised preparation adds concentrated hydrogen chloride liquid; the cleaned cell subsequently receives DCl liquid. Acquisition settings are identical.

The readout is transmission or absorbance against spectroscopic wavenumber, resolving molecular rotation alongside stretching. Background addresses instrument and cell contributions; cleaning addresses carryover. Excessive absorption impairs analysis. Assigning isotopes and rotational branches precedes fitting band parameters to test isotope scaling of diatomic vibration. These are prescribed measurements, not numerical observations supplied here.^[ref]

For a transparent ideal model, add the linear rigid rotor to the harmonic vibrational energy levels and assume the same frequency-valued rotational constant $B$ in both vibrational states:For a spin-free diatomic molecule with no electronic angular momentum, electric-dipole selection rules give $\Delta J=\pm 1$ for this vibrational absorption. Subtracting initial from final energy eigenvalues yieldsThese are two rotational branches about the vibrational band origin, not a single stretch line. For stipulated dimensionless values ${\tilde{\nu }}_{\mathrm{harm}}=100$ and $\tilde{B}=1$, the first three R-branch positions are $102,104,106$, and the first three P-branch positions are $98,96,94$. There is no $J=0\to -1$ transition. The numbers illustrate level subtraction; they are not HCl measurements. Real upper and lower rotational constants differ, and centrifugal distortion shifts lines further.