The wave of a photon/Other experiments
The Hong-Ou-Mandel interferometer is a variant of the Mach-Zehnder interferometer in which two simultaneous photons are compared. A 351 nm laser radiates a KDP crystal, which because of SPDC emits now and then two entangled photons of about 702 nm. The KDP is propably a type I, in which both photons have the same polarization and the same phase. Both waves are mixed in a 50% reflecting mirror, filtered by a bandpass interference filter IF and detected with a coincidence of 7.2 ns.
It is supposed that this beam splitter is a glass plate with reflector on one side, causing (only) the reflection from the normal input having a phase shift of π. All other have no phase shift. At the detectors the waves are: Iu = 0.5sin(π+ωt+φ/2) + 0.5sinαsin(ωt-φ/2) and Il = 0.5sin(ωt+φ/2) + 0.5sinαsin(ωt-φ/2). With an ideal bandwidth filter d = Δω/ω, the probability is calculated:
Pu = 0.5 - 0.5cos(4πΔx/λ) sin(πd(2k+2Δx/λ)) / πd(2k+2Δx/λ) and Pl = 0.5 + 0.5cos(4πΔx/λ) sin(πd(2k+2Δx/λ)) / πd(2k+2Δx/λ)
These formula predicts the result of the measurement if the cosine term is ignored and k = 0. Then the formula will be:
Pu = 0.5 - 0.5sin(πd2Δx/λ) / πd2Δx/λ
Pl = 0.5 + 0.5sin(πd2Δx/λ) / πd2Δx/λ
coincident: P = Pu * Pl
In the experiment ω = 2πc/702nm = 2.7 1015. The bandwidth filter is 3 1013, d = 3 1013 / 2.7 1015 = 0.011. Then FWHM = 113 μm.