Biomedical Engineering Reference
In-Depth Information
N
N
y
2
π
λ
x
∑
∑
Γ( , )
x y
=
C
x y
α
β
.
(9.14)
α β
,
α
=
0
β
=
0
hen,
C
α,β
defines a set of aberration coefficients, relating to the linearly independent polynomials
of
x
and
y
and each coefficient corresponds to a unique aberration type. For example, coefficient
C
0,0
defines a uniform phase offset, or piston aberration, while
C
1,0
and
C
0,1
respectively describe tilts along
x
and
y
. In an off-axis holographic microscope, some tilt aberrations are expected due to the off-axis angle
between the reference and the object waves that introduces a linear phase gradient
2
π
λ
(9.15)
Γ
TILT
( , )
x y
=
(
C x C y
+
)
=
k
x
+
k
y
,
1 0
,
0 1
,
0
,
x
0
,
y
where the lateral coordinates
x
and
y
are expressed in physical distance units. Correcting for this linear
phase gradient demodulates the carrier frequency of an off-axis hologram, thus shifting the hologram
spectrum so that one initially modulated imaging term becomes centered in the spatial frequency space.
This correction can be easily implemented and automated, since
k
0,
x
and
k
0,
y
can be deduced from the
carrier frequency of the hologram (Equation 9.10).
As for coefficients
C
2,0
and
C
0,2
, they can, in a parabolic approximation, compensate for possible cur-
vature mismatch between the object and the reference waves. If such a mismatch exists, the interference
fringes will no longer be straight lines, but rather will have some hyperbolic shape. This can also be
observed by a broadening of the carrier frequency in the hologram spectrum.
In summary, all these aberrations are easily corrected by numerical parametric lenses and other,
higher-order aberrations can be corrected similarly. It is however generally rare that a microscope pres-
ents a significant higher-than-second-order aberrations. Finally, an automated procedure for aberration
correction with numerical parametric lenses can be found in Colomb et al
.
(2006a).
9.4.3.2 Reference Hologram correction
This method is based on a two-step recording process. In one step, a reference hologram is recorded
without the presence of the specimen and one of its imaging term is reconstructed to quantitatively
characterize the influence of optical aberrations on the retrieved phase. In another step, the desired
hologram is recorded with the specimen, and one of its associated imaging term is reconstructed. The
method of reference hologram correction then simply consists in dividing the wavefront reconstructed
in the second step by the one reconstructed from the reference hologram.
Mathematically, let us define the object waves
o
=
ϕ
, respectively, with and with-
out the presence of the specimen. Reconstructing the reference hologram provides the imaging term
ψ
0
=
g
o,r
o
0
r
. Similarly, imaging term ψ
1
=
g
o,r
o
1
r
is retrieved from reconstruction of the second hologram.
Dividing ψ
1
by ψ
0
yields:
=
A e
if
ϕ
and
o
A e
if
1
0
1
1
0
0
g
o r
ψ
ψ
o
o
A
A
e
(
)
o r
o r
,
,
1
0
1
0
1
0
1
0
if
ϕ ϕ
−
(9.16)
=
=
=
.
1
0
g
o r
In principle,
A
0
/
A
1
compensates for nonuniformities in the illumination profile, while φ
1
− φ
0
cor-
rects the phase aberrations.
While reference hologram may seem to complicate the recording process, it needs to be performed
only once for a given microscope. It is therefore not much time consuming, and does not limit the
technique in any way. Thanks to its simplicity and its excellence at compensating all aberrations, this
method is widely used in digital holography. Unfortunately, it is not compatible with background-
free techniques such as holographic SHG imaging, in which the object wave contains no signal if the
