Biomedical Engineering Reference
In-Depth Information
phase distribution is defined modulo 2π, meaning that phase unwrapping techniques have to be applied
for specimens producing optical path differences larger than one wavelength.
In reflected-light holography, and for a specimen immersed in a uniform medium of known refrac-
tive index
n
(λ), the phase can be expressed as
=
4
π
λ
(9.18)
∆
ϕ
n
( )
λ
∆
z
,
and is directly related to the surface topography Δ
z
of the specimen.
In transmitted-light holography, the detected phase is related to the optical length of the path traveled
by the light and can be expressed as
z
F
2
π
λ
∫
∆
ϕ
=
n z
( ,
λ
)
d
z
,
(9.19)
z
I
where
n
is the refractive index and
z
I
,
z
F
are respectively the coordinates of the light source and the
detector. For transmitted light, the phase cannot be easily related to either the refractive index distribu-
tion or the specimen thickness and advanced schemes to decouple the two are needed. Such schemes
include changing the known value of refractive index of the surrounding medium (Rappaz et al
.
, 2008),
confining the specimen in a microchannel of known depth (Lue et al
.
, 2006) or using air bubbles in the
medium surrounding the specimen to determine its refractive index (Kemper et al
.
, 2006).
We have seen that digital holography is especially suited for imaging moving specimens. It is therefore
no surprise that phase imaging finds applications in topographic measurements of MEMS operating in
the MHz frequency range and in live cell analysis. But phase imaging is also uniquely appropriated for
characterization of microlenses and micro-optics in general.
9.5.3.2 Holographic SHG Phase imaging
The interpretation of the phase signal in holographic SHG imaging is rather complicated, as the wave-
length of the light changes due its interaction with the specimen. But while the wavelength changes, the
phase remains continuous since SHG preserves the coherent nature of light. The detected phase thus
reflects the optical path length, but is not as readily related to specimen height or thickness, as it is the
case in bright-field digital holography.
An important aspect to consider is that the SHG phase is only determined where second harmonic is
generated (Figures 9.9a and 9.9b). While this is perfectly logic, it has an undesired consequence: wher-
ever no second harmonic is generated, the reconstruction algorithms introduce physically meaningless
random phase fluctuations that might disturb the observer. Fortunately, regions of random phase can
be eliminated by applying a binary mask, based on thresholding of the SHG amplitude, to the image
(Figure 9.9c).
Let us consider the case of transmitted-light holographic SHG phase imaging. The detected SHG
phase φ depends on the phase of the fundamental wave at the location
z
SHG
of SHG, as well as on the
optical path length (at SHG wavelength) from that point to the detector, located at
z
H
. he SHG phase
can thus be expressed as
z
z
H
SHG
2
π
λ
2
π
λ
∫
∫
0
∆
ϕ
=
n z
( ,
λ
)
d
z
+
n z
,
d
z
.
0
(9.20)
λ
/
2
2
0
0
z
z
0
SHG
Quantitatively relating the SHG phase to physical properties like refractive index, specimen
thickness, or surface topography is not trivial. Some cases are nonetheless rendered quite accessible by
making simple assumptions.
