Biomedical Engineering Reference
In-Depth Information
(a)
1
(b)
π
0
a.u. 0
- π rad.
FIgurE 9.12 SHG amplitude (a) and phase (b) generated by a BaTiO 3 nanoparticle and retrieved by holographic
SHG imaging. Scale bars are 5 μm. (Reprinted with permission from Shaffer, E. and Depeursinge, C., 2010. Digital
holography for second harmonic microscopy, in: Proceedings of the SPIE: Multiphoton Microscopy in the Biomedical
Sciences X . Copyright 2010, Society of Photo-Optical Instrumentation Engineers.)
space and time. To this date, nanocrystals of ZnO (Kachynski et al . , 2008), polar Fe (IO 3 ) 3 (Bonacina
et  al . , 2007), KNbO 3 (Nakayama et al . , 2007), KTiOPO 4 ( KTP ) (Le Xuan et al . , 2008), BaTiO 3 (Hsieh
et al . , 2009; Pantazis et al . , 2010; Shaffer et al . , 2010a), and of different types have been investigated for
this purpose.
To be of interest, nanoprobes such as the one of Figure 9.12 must be used in conjugation with a
technique capable of localizing them through space and time. Over the years, many algorithms were
proposed for determining the lateral position of nanoparticles at presumably nanometer (or at least
sub-pixel) precision—for a quantitative comparison, see Cheezum et al . (2001). But the real challenge
has always been and remains the determination of the axial position, which is generally no better than
the micrometer range, unless the full diffraction field can be accessed, in terms of amplitude and phase.
Even in these cases, it does not reach the sub-micrometer without a priori knowledge of shape or size of
the particle. Precise sub-micrometer tracking of nanoprobes is therefore a need to be addressed.
Holographic SHG imaging proposes two different methods for determination of the axial position of
nanoprobes in the appropriate precision range. One relies on numerical field propagation and the other
is based on SHG phase measurement.
Method 1: Determination of axial position by numerical field propagation : We have seen in Section
9.4.1 that numerical reconstruction of digital holograms, by allowing to bring in focus scatterers located
at various depths in the specimen, gives holographic SHG imaging an extended depth of field that can
be used to determine the axial position of these scatterers (Hsieh et al . , 2009; Shaffer et al . , 2010a). Here,
we show how this is possible and discuss the performance of this method.
To make things as clear as possible, let us consider a distribution of SHG-emitting nanoparticles,
spread in both lateral and axial positions, and imaged by a lens, for example, a microscope objective,
at different positions along the optical axis, as illustrated in Figure 9.13. Let us suppose that a digital
sensor records an hologram of the out-of-focus images (or Fresnel zone plates) in the indicated plane.
Numerically reconstructing the hologram at distance d would bring in focus the image of one nanopar-
ticle, while reconstructing the same hologram, but at a greater distance (here d + Δ d ) would bring in
focus the image of the other nanoparticle. The Δ d change in the reconstruction distance needed to bring
the image of the second particle in focus is directly related to the Δ z difference in the axial position of
the objects by
d M z M z
L
=
=
2
,
(9.24)
T
where M L and M T are, respectively, the longitudinal and transverse magnification of the imaging system.
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