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5 Optical Measurement Techniques
Fig. 5.10 Top: Sketch of the principle of photon statistics measurements in a Hanbury–Brownand-Twiss experiment. a Schematic diagram of the photon second-order temporal autocorrelation
function G (2) (τ ) for a pulsed single-photon Fock state (with no peak occuring at τ = 0) as well
as a sub-Poissonian state (exhibiting an additional peak at τ = 0, here dash-lined grey-shaded),
i.e. with G (2) (0) = 0 and G (2) (0) < 1, respectively. Dashed horizontal lines are guides to the eyes
indicating a constant long-delay count level. b Schematic G (2) (τ ) for a pulsed coherent state with
Poissonian distribution. Reproduced with permission. [50] Copyright 2009 Arash Rahimi-Iman
This function, which represents the correlation of intensities, is used as a measure of
the second-order temporal coherence of emitted light. It is often represented by the
following equation:
g
(2)
(τ ) =
I (t)I (t + τ )
I (t)
2
=
I R (t)I T (t + τ )
I R (t) I T (t + τ )
,
(5.2)
with I R (t) and I T (t + τ ) the measured incidence intensities of the reflected and
transmitted detection beam from a beam splitter (in the HBT setup), respectively,
at two times separated by a delay time τ . Quantum-mechanically represented,
it becomes g
(2)
(τ ) =
: ˆ
n T ˆ
n R :
/(
ˆ
n T
ˆ
n R
) =
: ˆ
a
†
T ˆ
a
†
R ˆ
a T ˆ
a R :
/(
ˆ
a
†
T ˆ
a T
ˆ
a
†
R ˆ
a R
), with
photon creation and annihilation operators ˆ
a
† and ˆ
a, respectively.