An involute is the curve traced by the end of a taut string as it is unwound from a fixed curve. For a circle of radius $R$, one parametrization is $\mathbf{x}(\theta )=R(\cos \theta +\theta \sin \theta ,\sin \theta -\theta \cos \theta )$. The straight free length is $R\theta$, and differentiation shows that the traced velocity is perpendicular to the taut string. Reversing the motion winds the string back around the circle.

Photon antibunching means suppression of close pairs compared with pairs at some later delay: $g^{(2)}(0)<g^{(2)}(\tau )$ for some $\tau$. If correlations decay to the independent-count reference, $g^{(2)}(0)<1$ is a sufficient signature. It violates the classical intensity bounds in photon bunching. For the specified single-mode number measurement, second-order optical coherence also shows that $g^{(2)}(0)<1$ is equivalent to $\mathrm{Var}N<\mu$. This variance criterion concerns that mode or gate; it must not be applied unchanged to arbitrarily long integrated counting intervals.

Photon bunching is enhanced near-coincident detection relative to an appropriate independent-count reference. It need not require nonclassical light. In a classical intensity model with nonnegative instantaneous light intensity $I$, ideal detection with a conditional Poisson distribution givesA fluctuating intensity therefore bunches counts. For a stationary classical intensity with finite second moment, the Cauchy-Schwarz inequality further gives $⟨I(t)I(t+\tau )⟩\le ⟨I^2⟩$. A zero-delay correlation smaller than one at a later delay cannot be explained by this model.

For a single mode in thermal equilibrium, its quantum harmonic oscillator energy ladder gives a geometric distributionIts probability generating function is $F(z)=(1-q)/(1-qz)$. Differentiating twice gives $F^{\prime \prime }(1)=2q^2/(1-q{)}^2=2{\mu }^2$, hence $g^{(2)}(0)=2$. At the chosen mean $\mu =2$, $q=2/3$, $p_0=1/3$, $p_1=2/9$, $p_2=4/27$, and $\mathrm{Var}N=6$. Rare high occupations increase the pair count even though the mean matches the other preparations. The geometric weights also arise from a Gaussian mixture of coherent states.^[ref]

Single-mode selection is essential to this factor of two. If $M$ independent equally populated thermal modes are collected together, let $N={\sum }_{j=1}^M{N}_j$ with $⟨N_j⟩=\mu /M$. Adding their variances gives $\mathrm{Var}N=\mu +{\mu }^2/M$, hence $g^{(2)}(0)=1+1/M$. Averaging over many unresolved modes can make thermal pair counts approach the independent-count reference.

For a coherent state, write $a|\alpha ⟩=\alpha |\alpha ⟩$ and $|\alpha ⟩={\sum }_n{c}_n|n⟩$ in number states. The annihilation operator gives $c_{n+1}\sqrt{n+1}=\alpha c_n$, so $c_n=c_0{\alpha }^n/\sqrt{n!}$. Normalization yields $|c_0{|}^2=e^{-|\alpha {|}^2}$. Thus photon statistics follows a Poisson distribution with $\mu =|\alpha {|}^2$, and the normal ordering of every moment gives $⟨(a^{†}{)}^k{a}^k⟩={\mu }^k$. This factorization is the defining virtue of coherent states for optical calculations.^[ref]

As an original comparison, choose $\mu =2$. Then $p_0=e^{-2}$, $p_1=p_2=2e^{-2}$, $\mathrm{Var}N=2$, and $⟨N(N-1)⟩=4$. A two-photon Fock state also has mean two but has $\mathrm{Var}N=0$ and pair moment two. ConsequentlyIn particular, $|1⟩$ cannot supply a pair. Nevertheless, photon statistics alone does not determine a quantum state: averaging $|\alpha ⟩⟨\alpha |$ over a uniform quantum phase removes its off-diagonal entries while retaining exactly the same $p_n$ and these equal-time moments.

Independent loss alone leaves normalized factorial moments invariant: if every input photon is retained with probability $\eta$, binomial thinning conditional on a total gives $⟨K⟩=\eta \mu$ and $⟨K(K-1)⟩={\eta }^2⟨N(N-1)⟩$. Both powers cancel from $g^{(2)}(0)$. Fewer coincidences still mean poorer statistical precision.

Now let each channel contain a signal rate $s_i$ and an independent Poisson process background rate $b_i$, independent also of the signal. Put ${\rho }_i=s_i/(s_i+b_i)$. The pair density is $s_1{s}_2{g}_{\mathrm{sig}}^{(2)}+s_1{b}_2+b_1{s}_2+b_1{b}_2$. Dividing by $(s_1+b_1)(s_2+b_2)$ proves the displayed formula, for the same bin and timing response. For chosen equal signal fractions ${\rho }_1={\rho }_2=4/5$, ideal signal values $0,1,2$ become $9/25,1,41/25$. Background pulls both dips and peaks toward one.

This calculation assumes additive, mutually independent background counts. Scattered coherent light that interferes with the signal, correlated electronic noise, or recovery effects require a different model. Correcting a dip without independently estimating the backgrounds cannot establish photon antibunching.

For stationary continuous detection, let $C_j$ count all cross-channel pairs with $t_2-t_1$ in a bin of full width $\Delta \tau$, and let $r_1,r_2$ be observed single-count rates. Let $T_j$ be their joint live observation time at that delay, approximately the common duration $T_{\mathrm{obs}}$ when delays are short. Independent streams have expected pair count $r_1{r}_2{T}_j\Delta \tau$, giving the displayed estimator. It estimates a bin average of second-order optical coherence, further blurred by detector timing response. For a finite uninterrupted run the exact overlap weighting is $T_{\mathrm{obs}}-|\tau |$ inside the delay integral, rather than a constant at long delays.

For an original numerical example, choose $r_1=2000{\mathrm{s}}^{-1}$, $r_2=3000{\mathrm{s}}^{-1}$, $T_j=100\mathrm{s}$, and $\Delta \tau =10\mathrm{ns}$. The independent reference is six pairs per bin. Chosen counts $C_j=3,6,12$ give estimates $0.5,1,2$; they are teaching inputs with large counting fluctuations, not historical measurements. For pulsed sources, compare properly normalized same-pulse and separate-pulse coincidences instead of assuming this continuous-source baseline.

Detector dead time suppresses nearby same-detector counts and can manufacture an apparent dip in a single-detector record. Two independent detector channels avoid that particular same-channel exclusion, but shared electronics can still reject pairs. Timing spread and wide bins wash out narrow features. A device recording only the first stop after each start measures a waiting-time distribution; interpreting it as all-pair correlations requires a sufficiently low stop probability over the relevant delay range.

The photoelectric effect is the release of electrons from matter following photon absorption. Here the focus is emission from a metal surface into vacuum: a photon supplies energy, the work function pays the escape cost, and the remainder can become the emitted electron's kinetic energy. A photocurrent counts collected electric charge; a stopping potential probes the upper kinetic energy limit. These are different measurements.

In 1905, Albert Einstein proposed that an individual light quantum could transfer its whole energy to one electron. His §8 allowed losses before escape, distinguished the fastest emitted electrons from the rest, and predicted a straight stopping potential versus frequency relation with a material-independent slope. He separately predicted that changing light intensity changes the number emitted without changing their velocity distribution under otherwise identical conditions. He also discussed incomplete transfer and limits to the account.^[ref]

The useful modern starting model is a clean, homogeneous metal with independent single-photon events, negligible initial thermal broadening, negligible lowering of the surface barrier by an applied electric field, and negligible electric fields from the emitted electrons themselves. Specify the surface and its work function; it is not enough to name the bulk material. The Einstein photoelectric equation describes the upper edge of the emission energy distribution. It neither predicts that every absorbed photon liberates an electron nor supplies the whole photocurrent curve. Its appeal is that a complicated surface process still exposes the universal Planck constant through a simple slope.

The photovoltaic effect is the production of an electric potential difference or an electric current by photon absorption in a material or device. In a semiconductor, absorbed photons generate electrons and electron holes; asymmetric transport and selective collection can maintain distinct quasi-Fermi levels and deliver electrical power to a load. An equilibrium built-in potential alone supplies no continuous electrical power.

An image sensor with active circuitry at its optical detector pixels to buffer or amplify collected signals. CMOS implementations can select pixel signals without transporting every charge packet through the entire image.

A bolometer detects absorbed electromagnetic radiation through the resulting change in temperature, often read as a change in electrical resistance. Its response depends on absorbed power, so interpreting an optical spectrum requires accounting for the incident wavelength band and detector calibration.

A photodetector in which light releases electrons from a cathode into a vacuum, and an anode collects them. A positive collecting-anode bias relative to the emitting cathode drives collection across the vacuum gap.

Phosphoenolpyruvate supplies a phosphate group to ADP in the final glycolysis step catalyzed by pyruvate kinase. The free ionic form used in the worked balance has charge $-3$.

Vaporization changes a liquid into a gas. It includes evaporation at a surface and boiling within the liquid. Below the liquid-gas critical point, equilibrium conversion along the liquid-vapour phase coexistence curve involves latent heat; the liquid and vapour have equal chemical potential there.

Conversion of vapor into liquid.

Conversion of a gas to a liquid.

A change between a material behaving as an electrical conductor and as an electrical insulator. The distinction concerns the limiting electronic response as temperature tends to zero; finite thermally activated electrical conductivity alone does not establish a metallic state. A Peierls transition opens a band gap through a change in the crystal structure.

Digital optical phase conjugation uses a wavefront sensor and a spatial light modulator to measure an incident field and launch a beam with the opposite spatial wave phase. A phase-only spatial light modulator illuminated uniformly generally does not reproduce the incident wave amplitude as well; exact $\gamma U^{\ast }$ requires the appropriate wave amplitude and polarization of an electromagnetic wave too. The experimentally useful question is how much of the original optical mode can be recovered with the measured and controlled field components.

In 2010, Meng Cui and Changhuei Yang demonstrated digital optical phase conjugation through a prepared multiple scattering sample. They dried mixed-size polystyrene microspheres on cover glass, added immersion oil, and sealed them with another cover glass. A 532 nm laser supplied signal and reference beams. The signal crossed the sample between two objectives; the reference interfered with it on a charge-coupled device. Phase-stepped wave interference recovered the spatial wave phase; a calibrated spatial light modulator imposed its negative on the reference-derived return beam. The reference supplied playback power, without nonlinear mixing pumps. An optical camera read out the recovered signal. With the same sample and a zero-phase spatial light modulator control, the return instead produced a scattering of light pattern. This comparison supports mode recovery by spatial wave phase control, not merely increased illumination. A separate ordinary-mirror control tested source displacements. Finite sampling and collection limited recovery; the system controlled wave phase rather than the complete vector field. These restrictions distinguish demonstrated reconstruction from exact reversal of every scattered component.^[ref]

Model a thin transparent plate by a lossless reciprocal phase screen with transmission $t(x)=e^{i\psi (x)}$ at the chosen angular frequency. Both passages multiply the local field by the same $t(x)$: reversing travel does not by itself change the sign of an accumulated optical path length. Assume the plate is stationary, the two passages use corresponding transverse positions, and the plate is adjacent to the return surface or perfectly reimaged onto it. Ignore propagation between these reference surfaces, wave absorption, clipping and intensity-dependent changes. Take the incident boundary envelope to be $U_0(x)=A(x)e^{i{\varphi }_0(x)}$. Outward transmission gives $t{U}_0$. Ideal phase conjugation and the second transmission then giveThis is a complete short proof of cancellation in the specified phase screen model. For an ideal ordinary mirror with constant reflection factor $r$, the corresponding envelope is $rt(x{)}^2{U}_0(x)$: the extra wave phase is $2\psi (x)$, rather than zero. A constant reflection wave phase changes neither conclusion.

For an original toy optical aberration, let $\xi =x/a$ on $-1\le \xi \le 1$, take ${\varphi }_0=0$, and chooseThe even term adds symmetric curvature and the odd term makes the two sides unequal. The outward wave phase is $\psi$; immediately after ideal phase conjugation it is $-\psi$; after the second passage it is zero. At $\xi =-1,0,1$, the outward values are $\pi /4,0,3\pi /4$, while an ordinary mirror's completed return has values $\pi /2,0,3\pi /2$. All wave phases here are unwrapped relative to the central ray and are otherwise equivalent modulo $2\pi$. At a stipulated vacuum wavelength $\lambda =532\mathrm{nm}$, these outward errors correspond to optical path length errors $h=\lambda \psi /(2\pi )$ of $66.5,0,199.5\mathrm{nm}$. These are chosen values; they are not observations or physical plate thicknesses.

The wave phase plots make a subtle distinction visible: the field is still curved immediately after phase conjugation, and becomes flat only after crossing the same plate again. More generally, conjugation restores $U_0^{\ast }$, not a universally flat wavefront. If the plate instead has transmission $t=\alpha (x)e^{i\psi (x)}$ with real $0\le \alpha \le 1$, the return is $\gamma \alpha (x{)}^2{U}_0^{\ast }$. Its plate-induced wave phase cancels but its lost wave amplitude does not recover. If the return plate has changed to wave phase ${\psi }_{\mathrm{r}}$, the residual factor is $e^{i({\psi }_{\mathrm{r}}-\psi )}$. Those two variations distinguish correction of a stationary optical aberration from universal reversal of loss or material evolution.