Hanbury Brown and Twiss Effect - Wave Mechanics

Wave Mechanics

The HBT effect can in fact be predicted solely by treating the incident electromagnetic radiation as a classical wave. Suppose we have a single incident wave with frequency on two detectors. Since the detectors are separated, say the second detector gets the signal delayed by a phase of . Since the intensity at a single detector is just the square of the wave amplitude, we have for the intensities at the two detectors

which makes the correlation

\langle i_1i_2\rangle = \lim_{T\rightarrow\infty}\frac{E^4}{T}\int^T_0 \sin^2(\omega t)(\sin(\omega t)\cos(\phi)+\sin(\phi)\cos(\omega t))^2\,dt
= \frac{E^4}{4}+\frac{E^4}{8}\cos(2\phi),

a constant plus a phase dependent component. Most modern schemes actually measure the correlation in intensity fluctuations at the two detectors, but it is not too difficult to see that if the intensities are correlated then the fluctuations, where is the average intensity, ought to be correlated. In general

\langle\Delta i_1\Delta i_2\rangle = \langle(i_1-\langle i_1\rangle)(i_2-\langle i_2\rangle)\rangle =\langle i_1i_2\rangle-\langle i_1\langle i_2\rangle\rangle -\langle i_2\langle i_1\rangle\rangle +\langle i_1\rangle \langle i_2\rangle
=\langle i_1i_2\rangle -\langle i_1\rangle \langle i_2\rangle,

and since the average intensity at both detectors in this example is ,

\langle \Delta i_1\Delta i_2\rangle=\frac{E^4}{8}\cos(2\phi),

so our constant vanishes. The average intensity is because the time average of is 1/2.

An evaluation of a degree of the second-order coherence for complementary (anti-correlated) outputs of an interferometer leads to behaviour like "anti-bunching effect". For example a variation in reflectivity (and thus also in transmittance) of a beam splitter, where

results in the negative correlation of fluctuations

\langle \Delta i_1\Delta i_2\rangle=-\frac{E^4}{8}\cos(2\phi),

i.e. a dip in the coherence function .

Read more about this topic:  Hanbury Brown And Twiss Effect

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