Biomedical Engineering Reference
In-Depth Information
FIgurE 9.9
Holographic SHG imaging of a very small portion of
Caenorhabditis elegans
worm shell. Amplitude
image is normalized, and phase images are wrapped, that is, they scale from −π to π over the linear grayscale color-
map. (a) SHG amplitude, (b) SHG phase, and (c) SHG phase (masked). Scale bars are 5 μm.
9.5.3.3 Uniform, nondispersive Medium of Known Refractive index
Let us suppose that the refractive index of the medium surrounding the specimen is uniform and wave-
length-independent, that is,
n
=
n
(λ
0
) =
n
(λ
0
/2). This is notably the case for vacuum and air environ-
ment, but could become valid in other materials, given a judicious choice of wavelengths λ
0
and λ
0
/2.
Even if this assumption is not exactly verified, the following will provide a very good approximation,
since refractive index changes due to dispersion at typical wavelengths of ultrafast lasers are generally
very low and rarely exceed a few percent, unless the medium has a strong absorption band in that spec-
tral region, in which case it is intrinsically not suited for imaging.
Under the assumption of nondisperive medium, Equation 9.20 simplifies a little and the variations in
the observed SHG phase can be directly related to variations in the axial position of SHG:
=
2
π
λ
∆ ∆
ϕ
(
z
)
n z
∆
,
(9.21)
0
Interestingly, this relation also corresponds to the phase advance term of the fundamental field at
wavelength λ
0
in a medium of refractive index
n
.
9.5.3.4 Uniform Medium of Known Refractive index
Now, let us suppose that the medium surrounding the specimen is still uniform but dispersive. Its
refractive index therefore is not the same for electromagnetic waves of wavelength λ
0
or λ
0
/2, but can
always be expressed as
λ
n
(
λ
)
=
an
0
2
,
(9.22)
0
where
a
is a proportionality factor. Different
(
λ
0 0
/
couples will yield different coefficients
a
, but for a
given λ
0
, there will always be a coefficient
a
that satisfies the previous equation. Under this assumption,
Equation 9.20 leads to
,
)
2
π
λ
(9.23)
∆ ∆
ϕ
(
z
)
=
(
2
a
−
1
)
n z
∆
.
0
This more general equation suits any uniform medium. Water, for instance, has a refractive index
n
(800 nm) = 1.339 and
n
(400 nm) = 1.329, yielding a coefficient
a
of 1.0075. Here, the influence of dis-
persion is very small: it makes the phase-position relation differ by only 1.5% compared to Equation 9.21
that was developed for the case of nondispersive medium.
