Disorder-Enhanced Imaging with Spatially Controlled Light

Optimizing light into a single channel 31
As a result we have
γ =
1
√
NIin
∑
γ 2 = 1
NI in
⎛
⎝ ∑
a∈a + Ẽ a , (2.35)
∣ ∣ ∣∣ ∣∣
2 ∑ ∑
Ẽ a +
a∈a + a∈a + a ′ ≠a
Ẽ a Ẽ ∗ a ′ ⎞
⎠ . (2.36)
〈
γ
2 〉 = 1 ( 〈 ∣∣∣ ∣ 〉 ∣∣
2 〉 ) 2
M Ẽ a + M(M − 1)
〈Ẽa , (2.37)
NI in
〉
where M is the cardinality of the set a + . Normally
〈Ẽa would average out to
zero, but due to the selective illumination this is no longer true. We find that
〉 〈Ẽa = 1 π
∫ π/2
−π/2
where we integrated over the relative phase θ ≡ arg
〈∣ ∣〉
∣∣ ∣∣
cos θ Ẽ a dθ = 2 〈∣ ∣〉
∣∣ ∣∣ Ẽ a , (2.38)
π
) (Ẽa − E a . The ensemble
averaged value of γ 2 then is
〈
γ
2 〉 = M 2 1
N 2 π + M (1
N 2 − 1 )
. (2.39)
π
If we assume that the phase of Ẽa is homogenously distributed between −π and
π so that M = N/2 we arrive at
〈 〉
γ
2 1
e =
4π + 1 (
1 − 1 )
, (2.40)
2N π
which converges to 1/4π for large values of N. So by only selectively blocking
parts of the incident wave front it is already possible to control a large fraction
of the transmitted light.
2.5 Optimizing light into a single channel
The first experimental demonstration of light control in disordered systems using
explicit knowledge of the transmission matrix elements was given by Vellekoop
and Mosk in 2007[18]. In this pioneering experiment they demonstrated that a
random scattering samples can focus scattered light by illuminating them with
the correct wave front. The light, after being scattered thousands of times,
interfered constructively into one of the transmitting channels, thereby creating
a tight focus behind the sample. In this section we will study the intensity
enhancement η in such a focus.
The enhancement in the focus is defined as
η ≡
Ĩβ
〈I β 〉 , (2.41)