Biomedical Engineering Reference
In-Depth Information
aberrations, and the ield-dependent specimen-induced aberrations. Static aberrations of the optical
system may originate from misalignment or slight imperfection of the optics. he specimen-induced
static aberrations can be caused by focusing into media with mismatched refractive indices while the
ield-dependent specimen-induced aberrations are due to the variation in refractive index of the sample.
We are interested in measuring both types of specimen-induced aberrations. It is feasible to sepa-
rate these contributions to the aberration by a two-step calibration. First, a reference wavefront*, 1, is
recorded from a microscopic slide with a coverslip on top without any sample ensuring no additional static
specimen-induced aberrations are present. For this measurement, the focus is set directly below the cover-
slip and the correction ring of the condenser lens is adjusted to minimize remaining aberrations. In a sec-
ond step, the sample with the actual biological specimen within a water-based solution is inserted without
alteration of the optical setting of the lenses. hen a reference wavefront, 2, is recorded at a position where
the light traverses the homogenous part of the sample next to the specimen. hen the stage is scanned
in the
x
/
y
plane and wavefronts are recorded in a raster manner. Now the diference between the refer-
ence wavefronts 1 and 2 delivers the static specimen-induced aberration whereas the diference between
the wavefronts recorded during the scan and the reference wavefront 2 gives the ield-dependent aberra-
tion component. Concerning the static fraction of the aberration, it should be mentioned that it depends
directly on the diference between the two absolute measurements of the specimen slide and a reference
slide. his assumes that both the coverslip and the microscopic slide of the measured specimen have identi-
cal thickness and refractive index. Despite nominal standardization, these glass slides may slightly difer.
4.8 Phase Extraction
he complex wavefront,
P
(
x
,
y
), in the pupil plane of the objective can be expressed in terms of its ampli-
tude
A
(
x
,
y
) and phase ψ(
x
,
y
) as follows:
P x y
( , )
=
A x y
( , )
exp
ψ
(
j
( , ))
x y
(4.7)
One wavefront measurement consists of a set of three interferograms. Assuming unity amplitude for
the reference beam, each interferogram has the form
[
]
I x y
( ,
,
∆
ψ
)
= +
1
A x y
2
( , )
1 2
+
cos(
ψ
( , )
x y
+
∆
ψ
)
(4.8)
where Δψ is the phase shift introduced in the reference arm. Since three interferograms
I
1
=
I
(
x
,
y
,0),
I
2
=
I
(
x
,
y
,2π/3), and
I
3
=
I
(
x
,
y
,4π/3) for known values of Δψ are recorded, one can solve for the phase
of the wavefront:
3
(
I
−
I
)
φ
( , )
x y
=
arctan
3
2
(4.9)
2
I
−
I
−
I
1
2
3
Figure 4.8a
through c show the three raw images recorded during the phase stepping procedure and
Figure 4.8d
is a representation of the combined phase and amplitude calculated from the raw images.
he phase is calculated for every pixel position (
x
,
y
) and yields the wrapped phase ϕ(
x
,
y
), which is
the absolute phase of the wavefront ψ(
x
,
y
) modulo 2π. Now a fast Fourier transform (FFT)-based phase
unwrapping technique (Ghiglia and Pritt 1998) is applied to recover the unwrapped phase ψ(
x
,
y
) or its
equivalent ψ(
r
,θ) in polar coordinates. hen the Zernike modal content is extracted from this function
through the Zernike mode analysis described in
Section 4.3
.
*
he measurement of one wavefront requires the recording of three interferograms as explained in Section 4.8.
