Biomedical Engineering Reference
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(a)
(b)
1
χ (3) = 1
χ (3) = 0
χ (3) = 0
THG
0.5
THG from all χ (3) tensor elements
THG from ordinary axis elements
d
0
0
1
2
3
4
5
6
Birefringent slab width d (µm)
FIgurE 3.16 Calculation of THG from a birefringent medium. (a) Geometry. (b) Comparison between THG
from a slab of calcite-like medium considering either only tensor elements in the ordinary−ordinary axis (gray) or
also taking into account the ordinary−extraordinary elements (black) Conditions: NA = 1.2, n o = 1.66, n e = 1.49.
Figure 3.16 shows numerical simulations of THG from a slab of calcite-like medium considering an
incoming beam polarized along the ordinary axis. The dispersion is neglected, and linear indices are
taken to be n o = 1.66 and n e = 1.49. The gray curve corresponds to a calculation considering only the
tensor elements that produce a harmonic field along the ordinary axis. The size response in this case
is the typical single-peaked curve that reaches its maximum for a size approximately equal to the
coherence length, and then decreases until it reaches zero. However, if we now take into account the
tensor elements that induce a nonlinear polarization along the extraordinary axis (black curve), we
see the negative dispersion compensates the Gouy phase shift and the THG signal increases until it
reaches a plateau.
THG can therefore be observed from homogeneous birefringent media [44].
3.3.4 Quasi-Phase Matching
We consider here the possibility of quasi-phase matched THG from an axially periodic medium. Quasi-
phase matching (QPM) is an idea developed in nonlinear optics [50,51], where the conversion efficiency
is an important parameter, and stems from the consideration that it is sometimes easier to change the
structure of the sample (typically a nonlinear crystal) than to change the structure of the excitation
(usually a focused Gaussian beam). QPM is usually implemented using a stacked noncentrosymmetric
media exhibiting a periodic arrangement of alternative perpendicular orientations, with a period corre-
sponding to the coherence length of the nonlinear process. That way, interference is always constructive
along the optical axis.
Here, we will not consider the usual arrangement, as it usually implies birefringence, but we will
consider instead a medium that exhibits a sinusoidal variation of χ (3) from 0 to 1 with a period δ e , while
keeping a constant linear index (see Figure 3.17a).
1
2
2
π
δ
z
e
χ
( ) ( )
3
z
=
1
+
sin
(3.27)
The F-THG and B-THG obtained with a focused Gaussian beam from this geometry are shown in
Figure 3.17b as a function of the period δ e . The signal is here normalized to that obtained in the forward
direction from a single interface perpendicular to the z -axis, a geometry that corresponds to the same
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