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The 1999 experiment used an approximately $200\mathrm{fs}$, $394\mathrm{nm}$ pump pulse and a beta barium borate crystal. Two polarization-entangled photon pairs supplied four photons. A trigger and three other photodetectors selected a GHZ state through fourfold coincidence counting. Narrow interference filters lengthened the photon coherence time to approximately $500\mathrm{fs}$, exceeding the pump duration and suppressing timing information about which pair supplied which photon. The optical mode matching was essential to coherence, not merely to routing photons.^[ref]

The observed state was selected by all four detections, which also measured the photons. A trigger alone did not demonstrate a guaranteed surviving three-photon output. This distinction connects postselection to what an apparatus actually supplies, rather than identifying a fourfold coincidence with an unconditional source.^[ref]

Optical mode matching makes the spatial field of an incident beam equal to a desired optical cavity normal mode. For fundamental circular Gaussian beams, matching the complex beam parameter fixes the beam waist size and location; the optical axis, polarization and frequency must also match. A single converging optical lens connects waist Rayleigh ranges $z_A,z_B$ only when $f^2\ge z_A{z}_B$. With $s=\sqrt{f^2/(z_A{z}_B)-1}$, waist-to-lens distances obey $u=f\pm z_{A}s$, $v=f\pm z_{B}s$ with correlated signs, followed by physical distance constraints.

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.

Propagate from one beam waist to the other, using the optical lens ray transfer matrix:Optical mode matching requires $q_A=i{z}_A$ to transform to $q_B=i{z}_B$. Multiplying the denominator across givesEquating real and imaginary parts and multiplying by $f$ yieldsSet $\delta =u-f$. The second equation gives $v=f+(z_B/z_A)\delta$. Substituting this into the first cancels the terms $f\delta$:Consequently the two algebraic branches areEquivalently, $u=f+\delta$. The signs in $u$ and $v$ are the same. Real solutions require $\boxed{f^2\ge z_A{z}_B}$; a converging optical lens therefore needs $f\ge \sqrt{z_A{z}_B}$. At equality $u=v=f$. For the stated arrangement, additionally require $u,v\ge 0$ and that the chosen optical lens position is accessible outside both optical resonators. A negative $v$ is an algebraic virtual-waist solution, not a downstream beam waist.

For a symmetric confocal optical resonator, the mirror separation equals either mirror's curvature, $R_2=L_2$, so its symmetric fundamental Gaussian beam hasThe same optical mode matching calculation applies, now with $f_{\min }=\sqrt{z_A{L}_2/2}$. Keeping the example's $L_2=10\mathrm{m}$ gives $z_B=5\mathrm{m}$ and $f_{\min }=0.707107\mathrm{m}$, compared with $0.930605\mathrm{m}$ for the nonconfocal target. With $f=1.5\mathrm{m}$, the two branches becomeAgain only the positive-$v$ branch fits the stated downstream geometry. The confocal target's smaller Rayleigh range means a smaller fundamental beam waist at the same wavelength and a less stringent minimum $f$.

A confocal optical resonator also has transverse normal mode degeneracies: in the ideal paraxial spectrum, increasing transverse order by two shifts the resonance by one longitudinal spacing. Such a cavity can transmit combinations of higher transverse normal modes at a fundamental resonance, so transmitted power alone is a weaker diagnostic of pure fundamental optical mode matching. The degeneracy does not remove the complex beam parameter matching requirement if the objective is specifically the symmetric fundamental Gaussian beam.

A postselection scheme in which two polarization-entangled photon pairs pass through polarizing beam splitters, a half-wave plate and a nonpolarizing beam splitter. One photon at each of a trigger and three output detectors selects a GHZ state in the three output polarizations, provided optical mode matching removes distinguishable pair labels. All four detections are needed for the selected event; the trigger alone does not guarantee the three other output occupations.