Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

3.1.1. Speckle amplitude
3.1. Optical speckle potential
The coherent light from a laser is widened and directed on a rough surface
or through a milky plate (diffusor). The roughness is assumed to be larger
than the laser wave length, such that the phases of the emitted elementary
waves are totally random. According to Huygens’ principle, each point on
the diffusor emits elementary waves proportional to eikL|r| /|r| =: h(r). The
electric field E(r) in the plane of observation in some distance L is the
superposition of these elementary waves with phase factors η(r ′ )
�
E(r) = d d r ′ η(r ′ )h(r − r ′ ), (3.1)
see figure 3.2. The complex random field η(r ′ ) has zero mean η(r ′ ) = 0 and
is uncorrelated in the sense that its spatial correlation is much shorter than
the laser wave length η ∗ (r)η(r ′ ) ∝ δ(r − r ′ )Θ(R − |r|). Here, the finite
radius R of the diffusor is expressed using the Heaviside step function Θ.
The integral over the random numbers can be regarded as a random walk
in the complex plane. According to the central limit theorem, the local
probability distribution is Gaussian
P (Re E, Im E) = 1
πI0
|E|2
− I e 0 , with I0 = |E| 2 . (3.2)
Due to the finite laser wave length and the finite optical aperture, there is
a spatial correlation length σ. The speckle intensity cannot vary on length
scales shorter than σ, because elementary waves from different points have
to acquire a certain path difference in order to switch from constructive to
destructive interference, see figure 3.2. Let us consider in more detail the
speckle interference pattern in the far-field. If the distance L is sufficiently
large, the Fresnel approximation [116, Chapter 4] can be made. It consists
in an expansion of |r − r| in h(r − r ′ ) in the small parameter |x − x ′ |/L,
where x and x ′ denote the components of r and r ′ in the diffusor plane and
the plane of observation, respectively:
h(r − r ′ ) ≈ 1
L eikLL exp
�
(x − x
ikL
′ ) 2�
2L
(3.3)
Inserting (3.3) into (3.1) and expanding the quadratic term in the exponent,
one finds
˜E(x) = 1
�
d
L
2 x ′ �
exp −i kL
�
x · x′ ˜η(x
L ′ ). (3.4)
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