Codex @codex  0

Use the two-pair component of polarization-entangled down-conversion to account for identical bosons in the same collected modes. Write $a_H^{†}$ for a creation operator in mode $a$ with horizontal polarization, and similarly for the other modes. For the plus Bell state, the pair-creation operator is $K^{†}=a_H^{†}{b}_V^{†}+a_V^{†}{b}_H^{†}$. The normalized four-photon state isThese are normalized Fock states. In particular, $(a_H^{†}{)}^2|0⟩=\sqrt{2}|2H_a⟩$; omitting this factor would misnormalize the four-photon input.

The first polarizing beam splitter sends $a_H^{†}$ to $T_H^{†}$ and $a_V^{†}$ to $c_V^{†}$. Only the middle term can give exactly one trigger photon and leave another photon in $c$. In the first term both $a$ photons reach $T$ and none reaches $c$; in the last there is no trigger. Thus the relevant, still unnormalized component is

For the selected paths of the half-wave plate and beam splitter, takeHere $e$ labels the output towards $D_3$. An immaterial common reflection phase has been absorbed in the mode definitions. Keeping one photon at $D_3$ leavesAt the final polarizing beam splitter, the $H_{c^{\prime }}{H}_d$ alternative yields one $H$ at each of $D_1,D_2$, while $V_{c^{\prime }}{V}_d$ yields one $V$ at each. The two opposite-polarization alternatives send both photons to the same detector and fail fourfold coincidence counting. The accepted component is thereforeAfter conditioning and normalization, the three detected output modes haveThe optical mode matching must erase any timing, frequency or other record that distinguishes the two alternatives. Otherwise the same polarization populations could describe an incoherent mixed quantum state, rather than this quantum superposition. The Bouwmeester three-photon GHZ experiment uses this principle. The photonic GHZ preparation by fourfold coincidence is a measurement in the coincidence basis: detecting all four photons projects and consumes them. A click at $T$ by itself does not herald a guaranteed freely available three-photon state.

The optical success probability of a fourfold GHZ source depends on the photon-number statistics of the assumed two-pair input. Let $q$ be the probability that arm $a$ contains one $H$ and one $V$, given two pairs. The beam splitter then puts one of the two $b$ photons into each output with probability $1/2$, and the final polarization-parity selection succeeds with probability $1/2$. With ideal photodetectors,For the normalized, fully overlapping two-pair component of polarization-entangled down-conversion used above, $q=1/3$, so the squared norm of the accepted component gives $P=1/12$. Treating the two emissions instead as independently labelled Bell pairs gives the commonly used bookkeeping estimate $q=1/2$ and $P=1/8$. This is a different pair-statistics approximation; it must not be obtained by dropping labels from an input state without renormalizing it. Genuine GHZ state coherence additionally requires erasing information about the pair of origin.

Let $P_2$ be the two-pair probability per pulse and $f$ the pulse repetition rate. The coincidence rate is $R=f{P}_{2}q/4$. With the usual order-of-magnitude estimate $P_2\simeq p^2$, the supplied $f=10^6{\mathrm{s}}^{-1}$ and $p=10^{-3}$ give $f{P}_2\simeq 1{\mathrm{s}}^{-1}$. The fully overlapping model gives $R\simeq 0.083{\mathrm{s}}^{-1}$, about one event per twelve seconds; independent-pair bookkeeping gives $R\simeq 0.125{\mathrm{s}}^{-1}$, about one per eight seconds. Thus the estimate is of order $10^{-1}{\mathrm{s}}^{-1}$ under these idealizations.

An exact prefactor cannot follow from the single-pair probability alone. For example, the pair-number statistics of polarization-entangled down-conversion in a single collected mode pair give $P_2\simeq 3p^2/4$ at weak gain, and hence $R\simeq f{p}^2/16=0.0625{\mathrm{s}}^{-1}$ with $q=1/3$. Multiple temporal or spectral modes change those statistics. Real collection losses and detector efficiencies further reduce the rate; they are excluded by the ideal calculation.

The two output ports are orthogonal single-photon states. Conservation of total probability at a lossless beam splitter gives $2c^2=1$. With $c$ real and positive,In the ordered input basis $(0,1^{\prime })$ and output basis $(0^{\prime },1)$, the beam splitter scattering matrix isFor the usual reciprocal lossless beam splitter, reversing propagation uses $S^{-1}=S^{†}=S$. Consequently,The relative phase in the first rule is essential to the repeated-path calculation. Overall port phases may be changed consistently; they cannot be changed on one pass alone while retaining the original scattering rules. In the following, take unit mirror reflection amplitudes by incorporating fixed reflection phases into the propagation-mode convention, and keep any unremoved propagation phase explicitly if needed.

Starting at port 1, the first beam splitter encounter sends amplitude $c$ straight out through port 0 and stores amplitude $-c$ towards M2. A return from M2 enters port $1^{\prime }$; it sends the stored amplitude times $-c$ out through port 1 and times $c$ towards M1. A return from M1 enters port $0^{\prime }$; it sends amplitude times $c$ out through port 0 and times $c$ towards M2. Both internal continuations carry $+c$, so two additional encounters multiply the surviving amplitude by $+c^2$.

With the fixed mirror and propagation phases absorbed as in a, the nonzero coefficients of the outgoing single-photon state areand $c_{0,2j}=c_{1,2j+1}=0$. The first nonzero terms are $c_{0,1}=1/\sqrt{2}$, $c_{1,2}=1/2$, $c_{0,3}=-1/(2\sqrt{2})$ and $c_{1,4}=1/4$. If fixed reflection amplitudes $r_1,r_2$ are retained instead, multiply $c_{0,2j+1}$ by $(r_1{r}_2{)}^j$ and $c_{1,2j}$ by $r_2^j{r}_1^{j-1}$, as well as the appropriate delay propagation phases. These unit-modulus factors do not alter the following probabilities.

Distinct exit events are orthogonal because they differ in output port or arrival-time bin. This requires the incident wavepacket to resolve the delay. Summing the squared moduli as geometric series givesTheir sum is one. After $n$ encounters the probability of still being inside is $2^{-n}$, which tends to zero, so no trapped component is omitted.

The source drawing places the delay line in the M2 arm. A visit to M2 crosses that line in both directions. If $\tau$ is its one-way delay, then an exit after $n$ encounters has crossed it $2\lfloor n/2\rfloor$ times:If the phrase “delay time $\tau$” instead denotes the complete M2-arm round-trip delay, replace $2\tau$ by $\tau$ in the floor formula. The wording does not explicitly distinguish these conventions. Consecutive encounters can produce the same exit time in different ports: for example $n=2,3$ both follow one M2 visit. If the incident wavepacket were much longer than the arm delay, different histories in the same port would overlap and their probability amplitudes would have to be added before squaring; the asserted time-bin expansion would then not be an orthonormal event basis.

Let $S$ denote immediate transmission through the BS1 subsystem between Bob's lower-right port and Alice's upper-left port; let $L$ denote transmission after one visit to each internal mirror. From the same beam splitter rules as in b, the corresponding amplitudes in either direction areup to a common fixed two-mirror propagation phase. The minus sign in $L$ comes from the $1^{\prime }\to 1$ or reverse $1\to 1^{\prime }$ scattering. The two paths selected by D0's time gate are $L$ on the outward journey followed by $S$ on the return, and $S$ outward followed by $L$ on the return. They have the same total delay and fixed propagation phase, so their probability amplitudes interfere.

For the first history, Alice applies $e^{i{\varphi }_A}$ to the late component, while Bob's return-time modulation is not encountered in the return delay arm. For the second, Alice's early component is unmodulated and Bob applies $e^{i{\varphi }_B}$ in the return delay arm. Bob must time his modulation to the return passage, leaving the outward reference preparation unchanged; this is implicit in the stated encoding. Each enabled modulator is traversed twice, so its $\pi /2$ single-pass phase becomes the stipulated $\pi$.

Include the amplitude $c$ to enter BS1 through BS2, and the amplitude of modulus $c$ to reach D0 through BS2 on return. Both histories therefore contain six 50:50 beam splitter factors. Ignoring one common unobservable global phase,Hence the unconditional probability per emitted photon, with no renormalization after discarding paths, isFor the allowed values $0,\pi$, this is $1/16$ when ${\varphi }_A={\varphi }_B$ and zero otherwise. The ignored trajectories exit other ports or other time bins; excluding them from D0's acceptance does not increase the success probability. Equal mirror reflection phases, channel phases and Alice's mirror phase multiply both accepted histories equally and cancel from the interference condition. A residual relative phase would need calibration.

Use the symmetric lossless beam splitter convention with transmission amplitude $t=1/\sqrt{2}$ and reflection amplitude $r=i/\sqrt{2}$. The two mirrors contribute the same fixed quantum phase, which can be suppressed. At the rightward port $D_1$, the upper-arm route has one reflection at BS1 and one transmission at BS2, while the lower-arm route has one transmission at BS1 and one reflection at BS2. Their probability amplitudes therefore add asBy the Born rule,For the upward port, the two amplitudes are $r^2{e}^{i\theta }$ and $t^2$, giving $a_0=(1-e^{i\theta })/2$ and $P(D_0)={\sin }^2(\theta /2)$. They sum to one, as required for an ideal lossless Mach-Zehnder interferometer. Thus $D_1$ is bright at zero relative quantum phase and dark at quantum phase $\pi$. A fixed unequal reflection quantum phase or a swapped port labeling would shift the fringe; the stated convention fixes the port and quantum phase calibration used below.

Without the upper-arm rotation, the photon polarization still labels the path. Before BS2 the quantum state isAfter the balanced beam splitter, the polarization-valued output amplitudes areThe detectors do not resolve photon polarization. Since $⟨H|V⟩=0$, the cross terms vanish in the Born rule:These probabilities are independent of the tunable quantum phase and the input photon polarization, so there is no usable wave interference fringe. Equivalently, taking the partial trace over photon polarization removes the quantum coherence between the two path quantum states. The which-path information is retained in orthogonal photon polarizations even without anybody measuring it. Rotating the two occupied photon polarization quantum states to the same quantum state removes that distinction and allows the cross term in part (ii).

The beam splitter relation and the cavity output equation giveFor $\alpha \ne -1$, solve this scalar algebraic relation and substitute back:The cavity equation becomesFinally, use the other beam-splitter output and ${\alpha }^2+{\beta }^2=1$:These derivations retain the printed input $b_{min}$ in both beam-splitter equations; the converted TeX changes one of these subscripts. Define ${\gamma }_{meff}=\gamma (1-\alpha )/(1+\alpha )$ and $c=\beta \sqrt{\gamma }/(1+\alpha )$, so $\dot{a}=-{\gamma }_{meff}a/2-c{b}_{min}$ and $b_2=b_{min}+ca$. In the nondegenerate range $|\alpha |<1$, ${\gamma }_{meff}>0$ and $c^2={\gamma }_{meff}$, the expected passive input-output relation. This is instantaneous coherent cavity feedback through a beam splitter; it uses the zero-delay interconnection assumed by the equations.

The stated $M$ is specifically the transfer from the external input to the internal field $b_1$. Combining $b_1=G{b}_0$ with $b_0=\beta b_{min}-\alpha b_1$ givesFor $\gamma >0$ and a nondegenerate beam splitter $|\alpha |<1$, the closed-loop pole is $-{\gamma }_{meff}/2<0$. The return ratio for the Nyquist stability criterion is $L=\alpha G$. Its frequency locus is a circle of radius $|\alpha |<1$, so it cannot reach or encircle $-1$. The open cavity has no unstable poles; hence the closed loop has none. This agrees with the directly derived state damping rate.

The externally accessible output is different. From $b_2=\alpha b_{min}+\beta b_1$,It is again an all-pass cavity response. Confusing $b_1$ and $b_2$ would give the wrong output steady state.

For a constant coherent input mean $B=⟨b_{min}⟩$, the steady mean cavity amplitude follows by setting its derivative to zero:The input-output values also follow from $G(0)=-1$ and $M(0)=-\beta /(1-\alpha )$. For any initial mean amplitude, its difference from the stationary mean decays as $e^{-{\gamma }_{meff}t/2}$.

The stability statement needs endpoint qualifications. At $\alpha =1$, normalization forces $\beta =0$, and the cavity obeys $\dot{a}=0$: it is isolated and undamped, not asymptotically stable. Its external output is nevertheless the decoupled input. At $\alpha =-1$, the algebraic elimination divides by zero; the instantaneous feedback connection is singular and is not a well-posed instance of these reduced equations. Thus strict cavity stability holds for $|\alpha |<1$, not for every real pair satisfying only ${\alpha }^2+{\beta }^2=1$.

Finally, $b_{min}(t)$ is a quantum stochastic field, so arbitrary inputs do not approach a literal time-independent operator amplitude. The stationary expressions above describe a constant coherent drive's means, or the formal zero-frequency response. The full solution contains the filtered input-noise convolutionThe stationary cavity density operator also depends on the input noise statistics. For this ideal passive cavity with a coherent drive and vacuum noise, it is the coherent state with the amplitude just calculated; without such input statistics, the gain function alone does not specify a complete stationary quantum state.

In angular differential imaging, the instrument is operated in pupil tracking: its pupil and associated quasi-static speckle patterns remain nearly fixed on the detector, while the sky rotates with the parallactic angle. A reference stellar point spread function is formed from other exposures, preferably excluding frames where a companion remains at almost the same location. Subtract the reference from each exposure, then derotate the residuals into the sky frame and combine them. A companion adds coherently after derotation; the stellar residuals are reduced. This is the technique described in the original angular-differential-imaging analysis. At angular separation $\rho$, the approximate displacement is $\rho \Delta \psi$, so useful diversity normally requires $\rho |\Delta \psi |\gtrsim \lambda /D$.

In simultaneous spectral differential imaging, acquire nearby spectral-band images simultaneously, for example with a beam splitter and filters or an integral field spectrograph. Stellar speckle patterns approximately move radially in proportion to wavelength. Rescale each image by ${\lambda }_0/\lambda$ and normalize the stellar flux before subtracting bands. The speckles then approximately align, while a companion at a fixed sky position moves in the rescaled coordinates. A companion absorption band can also distinguish its spectrum from the star: methane bands are useful for cool companions, as in the TRIDENT instrument description, but methane is not a universal companion property. Positional diversity scales as $\rho |\Delta \lambda |/\lambda$.

The main optical path is telescope to deformable mirror to beam splitter to science camera. The splitter also directs reference light to a wavefront sensor; a controller reconstructs the wavefront error and feeds mirror commands back to the deformable mirror. The mirror is normally conjugate to a pupil so its actuators address the corresponding pupil phase. A separate steering mirror can handle overall image motion.

An optical squeezed vacuum has zero mean field, nonzero mean photon number ${\sinh }^{2}r$, and unequal noise in two principal optical quadratures. With $X_1=(a+a^{†})/2$ and $X_2=(a-a^{†})/(2i)$, the vacuum variance is $1/4$ and $[X_1,X_2]=i/2$. In axes aligned with the squeezing,The ideal pure state saturates $\Delta X_1\Delta X_2=1/4$. A below-threshold degenerate optical parametric amplifier or oscillator can generate it: a coherent pump drives pair creation into a mode, and the pump phase selects the squeezed quadrature. Loss mixes in vacuum noise, while technical phase fluctuations mix the squeezed and antisqueezed axes.

For the stated vacuum–one-photon superposition, normalization requires $|ϵ|\le 1$. Direct Fock-basis matrix elements giveUsing $a{a}^{†}=a^{†}a+1$,Consequently there is no squeezing of the specified quadrature:It equals the vacuum uncertainty only at $ϵ=0$ and reaches $\sqrt{3}/2$ at $ϵ=\pm 1$. The curve is even in $ϵ$. The orientation matters: the same real superposition has $(\Delta X_1{)}^2=1/4-{ϵ}^2/2+{ϵ}^4$, which is below $1/4$ for $0<{ϵ}^2<1/2$. This does not turn it into a Gaussian squeezed vacuum; it simply illustrates quadrature squeezing of a vacuum–one-photon superposition.

For the printed beam splitter convention, expansion gives the operator identitiesThuswhere different input modes commute. The source's equality mixing an expectation on one side with an operator on the other is understood as this operator identity, or with expectation brackets on both sides. The mean difference reveals phase-sensitive mutual coherence between inputs; it is not by itself a photon-number correlation measurement of the usual Hanbury Brown–Twiss type. Repeated fluctuations, rather than one mean value, are needed to infer quadrature uncertainty.

For the independent coherent local oscillator, replace $b,b^{†}$ by $\beta ,{\beta }^{\ast }$ in the mean. The effective signal operator isTherefore $\varphi =0$ measures $X_2$, and $\varphi =-\pi /2$ measures $X_1$, each multiplied by $2|\beta |$. Choosing $\varphi =\pi /2$ measures $-X_1$. These phase choices follow the specified complex beam-splitter matrix; importing a different beam-splitter convention would shift them.

This is balanced homodyne detection. A stable strong local oscillator supplies the phase reference, the signal and oscillator meet at a 50:50 beam splitter, and matched photodetector currents are subtracted. Varying the reference phase selects different quadrature projections. Recording their distributions supplies more information than their means and supports optical homodyne tomography.

The strong-reference limit also needs a noise qualification. Define $X_{\vartheta }=X_1\cos \vartheta +X_2\sin \vartheta$, with $\vartheta =\varphi +\pi /2$. For independent signal and coherent reference, an exact ideal-detector calculation givesOne derivation writes $b=\beta +\delta b$, where $\delta b$ is in vacuum. The reference-amplitude term gives the first contribution; the remaining vacuum term contributes $⟨a^{†}a⟩$. Thus the normalized variance approaches the signal quadrature variance when the reference is sufficiently strong relative to the signal. A finite reference is not exactly a classical noiseless field.

A single detector can also carry quadrature information, but its mean alone still does not determine the variance. It measures $n_C=(n_a+n_b+I_{CD})/2$, retaining the large reference background and its fluctuations. For fixed signal moments and a large coherent reference,The additional vacuum-level term and the uncancelled reference power must be calibrated. In practice the reference may also have much larger classical intensity noise. Balanced subtraction cancels the common total-power term and strongly rejects common-mode technical noise while retaining the phase-sensitive difference. Detector imbalance, unequal efficiencies and reference phase noise limit that advantage. It is therefore the more direct and usually more sensitive quadrature-noise measurement, rather than a claim that a single-port measurement contains no useful information.

Let the two arms deliver intensities $I_1,I_2$ with unit Jones vectors ${\mathbf{e}}_1,{\mathbf{e}}_2$. For a perfectly coherent input, the recombined intensity under relative phase $\varphi$ isConsequently the polarization overlap in interferometric visibility isFor equal returned intensities, two linear polarizations at angle $\theta$ have overlap $\cos \theta$ and thereforeChoose one linear direction as ${\mathbf{e}}_L=(1,0{)}^T$ and a circular polarization as ${\mathbf{e}}_C=(1,\pm i{)}^T/\sqrt{2}$. Their overlap has magnitude $1/\sqrt{2}$, givingA non-polarizing beam splitter does not by itself guarantee equal arm losses; if $I_1\ne I_2$, multiply either boxed polarization factor by $2\sqrt{I_1{I}_2}/(I_1+I_2)$. Perfect temporal coherence does not compensate for polarization or intensity imbalance.

A non-polarizing beam splitter divides the incident spectrum into a fixed arm and an arm with a translating mirror. Recombining their returned fields gives a detector signal as a function of mirror position $x$. The round-trip optical path difference is $D=2x$ relative to the equal-arm position, so the delay is $\tau =2x/c$ in air approximated as vacuum. Record both detector signal and calibrated position, subtract the constant background, and take a Fourier transform of the interferogram.

For matched arm polarizations and balanced throughput, write the ideal signal aswhere $S(\omega )$ is the incident spectral power density and $C$ is a constant set by the beam splitter and detector. This is the cosine-transform form of Fourier transform spectroscopy, or equivalently the Wiener-Khinchin theorem applied to delayed self-interference.

The delay-range resolution in Fourier transform spectroscopy follows because recording a finite delay range multiplies the correlation by a window. Its transform broadens each spectral line by an instrumental width of order the reciprocal delay range. A mirror travel range $L$ creates a delay range of order $2L/c$, soAt fixed wavelength, absolute wavelength resolution therefore improves inversely with travel range. The numerical coefficient depends on whether $L$ is a one-sided or total travel, on the window or apodization, and on whether resolution means first zero, half width or another criterion. The proportionality is the requested convention-independent result. Position sampling must also be fine enough to sample the fastest fringe variation without aliasing.

For each beam in vacuum, its magnetic field is ${\mathbf{B}}_j={\hat{\mathbf{k}}}_j\times {\mathbf{E}}_j/c$. Write the two complex scalar contributions, including the beam splitter factor, as $u_1=f(t-t_1)/\sqrt{2}$ and $u_2=f(t-t_2)/\sqrt{2}$. Since their electric fields are both along $z$,The optical-cycle and ensemble-averaged Poynting vector isThe mean individual light intensities are equal, soThus energy flows along the bisector of the two beam directions, crossing a detector plane normal to $x$. Such a sensor can in principle record the variation across its $y$ coordinate. Its actual normal energy flux is $I(y)/\sqrt{2}$ in the normalization used above; this fixed factor does not change interferometric visibility.

There is a practical resolution condition in addition to the direction of energy flow. The interference fringe spacing $\lambda /\sqrt{2}$ is very small, so direct pixels would need spacing no larger than $\lambda /(2\sqrt{2})$ as required by the Nyquist–Shannon sampling theorem, or optical magnification and sufficient angular aperture to preserve both incident directions. An ordinary optical camera with much larger effective pixels averages the interference fringes. The Poynting-vector argument establishes the allowed measurement geometry, not guaranteed resolution with any optical camera.

identical particles have the same intrinsic properties. Exchanging their bookkeeping labels cannot distinguish a physical preparation or change a measurable prediction. This is stronger than merely failing to track classical trajectories. For ordinary bosons and fermions, the many-particle wavefunction has respectively bosonic exchange symmetry or fermionic exchange symmetry under every particle exchange operator, acting on spatial and spin coordinates together. The Spin-statistics theorem connects integer spin to bosons and half-integer spin to fermions; this connection uses relativistic quantum-field assumptions, rather than following from the nonrelativistic Schrödinger equation alone.

For one-particle states $u,v$ satisfying orthonormality, the two alternatives form a quantum superposition:Its probability density contains an exchange interference term, absent from an incoherent mixture of labelled alternatives. The antisymmetric expression vanishes when both one-particle states are identical: one complete one-particle state can contain at most one fermion. This Pauli exclusion principle explains electron shell filling, the electronic Fermi sea and degeneracy pressure. Opposite spins are different one-particle states, so two electrons may share an atomic orbital. Spatially separated wave packets with negligible overlap can often be treated as distinguishable for local measurements; the underlying exchange symmetry remains.

In thermal equilibrium, occupation of a level of energy $ϵ$ obeysbosons can accumulate macroscopically in one state, giving Bose-Einstein condensation; fermions instead fill successive states up to the Fermi energy at zero temperature. When occupation is dilute, both expressions reduce to the Maxwell-Boltzmann distribution. The identical-particle classical partition function contains $1/N!$, avoiding spurious entropy from permutations and the Gibbs paradox. Exchange effects become conspicuous when the thermal de Broglie wavelength is comparable to the interparticle spacing.

An operational test is the Hong-Ou-Mandel effect. Down-converted photon pairs enter separate ports of a balanced beam splitter; indistinguishable alternatives for one photon per output cancel. Scanning the relative arrival delay restores coincidences away from overlap. Matching polarization, spectral filtering and spatial alignment controls distinguishability; measuring delayed accidental coincidences controls background. The observed dip tests two-photon interference, and its finite depth must not automatically be interpreted as a failure of bosonic symmetry. The original apparatus varied beam-splitter position and subtracted a separately measured accidental background.^[ref]

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.

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 $n$.

Two applications of optical interferometry are precision displacement or strain measurement, and Fourier transform spectroscopy, which recovers a spectrum from an interferogram.

For the round-trip phase in a Michelson interferometer, take a lossless 50:50 beam splitter with $t=1/\sqrt{2}$ and $r=i/\sqrt{2}$, and mirror reflection amplitude $-1$. At the depicted detector, each returning path has one beam-splitter transmission and one reflection. Their amplitudes are $-i{A}_0{e}^{2ikD}/2$ and $-i{A}_0{e}^{2ik(D+\delta )}/2$. Consequently,The factor 2 in the phase is the return journey: the labelled $\delta$ is a one-way arm-length difference. The other output port has the complementary intensity. The printed $|A_0{|}^2(1+2\cos k\delta )$ cannot be an intensity: it becomes negative, and does not match the depicted path difference. No choice of normalization fixes that negative minimum. The expression above is the physically consistent result for the drawn output port.

For mutually incoherent spectral components with incident power density $S(k)$,The plot labels are ${\lambda }_1=589.59\mathrm{nm}$ and ${\lambda }_2=590.00\mathrm{nm}$; these printed centres are used here. In the narrow-line, equal-power limit, each line has incident intensity $J$ andThere are rapid carrier fringes with approximately $0.2949\mu \mathrm{m}$ mirror-displacement spacing and a slowly varying Interferometric visibility envelope. The first visibility zero is at $0.2121\mathrm{mm}$, and successive visibility revivals are separated byFor the unequal-line visibility in a spectral doublet, with incident line powers $J_1,J_2$, replace the intensity by $[J_1+J_2+J_1\cos (2k_1\delta )+J_2\cos (2k_2\delta )]/2$. The visibility becomesThe visibility minima rise from zero to $|J_1-J_2|/(J_1+J_2)$. The finite widths drawn in the spectrum also make coherence decay with path difference. For example, Gaussian lines of common wave-number standard deviation ${\sigma }_k$ multiply the oscillatory term by $e^{-2{\sigma }_k^2{\delta }^2}$; the actual linewidth is not specified numerically. Thus the narrow-line plot gives the beat positions, while the true long-delay oscillations die away.^[ref]

The common-arm phase cancellation in a balanced interferometer means there is no smallest positive common mirror distance $D$ in a balanced interferometer. With $\delta =0$ the common factor $e^{2ikD}$ cancels from every intensity, and both paths interfere constructively at this detector for every $D>0$. Their infimum is zero. The $0.4242\mathrm{mm}$ beat length describes changes of arm difference, not of the common arm length. It therefore cannot answer the final request as printed.