Biomedical Engineering Reference
In-Depth Information
Anisotropic medium, linear polarization
We first consider the case of an interface between air and the uniaxial crystal, excited by a linearly polar-
ized Gaussian beam. θ denotes the angle between the incident polarization and the axis of the crystal.
Calculations show that the shape of the curve given in Figure 3.14 is conserved for every combination
of polarization angle and ratio between tensor elements (not shown).
Not unexpectedly, the signal is maximum when the polarization angle is along the axis of the crystal
that has the largest tensor element, much like SHG from a collagen fibril is maximized when the excitation
polarization and the fibril are co-aligned. This indicates that tensor anisotropy does not change the phase-
matching conditions: there is no THG signal from a homogeneous anisotropic sample, and the signal from
an interface is proportional to the signal expected from the interface between two isotropic media.
Anisotropic medium, circular polarization
In the case of a circularly polarized excitation, the THG signal scales as ((χ || − χ cr ) 2 + (χ cr − χ ) 2 ). Figure
3.15 illustrates the THG signal as a function of the value of χ || and χ cr (assuming χ = 1). he degenerate
situation χ || = χ cr = χ corresponding to an isotropic medium yields, as expected, no THG signal.
3.3.3 tHG from Birefringent Media
Birefringent media have polarization-dependent indices. When the material has a single axis of sym-
metry, the birefringence can be described by assigning two different refractive indices to the material for
different polarizations: one along the ordinary axis (perpendicular to the axis of symmetry) and another
along the extraordinary axis (parallel to the axis of symmetry).
If some tensor elements create a nonlinear field polarized perpendicularly to the excitation, we obtain
a situation where the excitation propagates along the ordinary axis, while the harmonic field propagates
along the extraordinary axis, thus producing an effective negative dispersion for the harmonic generation
process.
THG in highly birefringent structures such as calcite crystals (calcium carbonate (CaCO 3 )) or liq-
uid crystals [48] has been reported by Oron et al. [44,49]. Calcite is one of the most birefringent struc-
tures found in biological samples, and has been extensively used in nonlinear optics. At a wavelength
of 590 nm, calcite has ordinary and extraordinary refractive indices of 1.658 and 1.486, respectively.
16
12
Exc.
THG
8
4
0
4
3
3
2
2
χ cr
χ ||
1
1
FIgurE 3.15 Sensitivity to χ (3) anisotropy with circular polarization. (Left) Geometry considered. (Right) THG
signal obtained with circular incident polarization at the interface between an anisotropic medium and air, as a
function of the parameters χ || and χ cr with χ = 1 (see text). No signal is obtained in the case of an isotropic medium
|| = χ cr = χ = 1). (Reprinted from Olivier N et al. 2010. Harmonic microscopy of isotropic and anisotropic micro-
structure of the human cornea. Opt. Express 18:5028-5040. With permission of Optical Society of America.)
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