Biomedical Engineering Reference
In-Depth Information
calculation of the nonlinear Polarization
The nonlinear polarization induced by the focused fundamental beam can then be expressed in each
point of the grid as
∑
P
i
THG
(
)
(
3
ω
)
=
χ
( )
3
⋅
E
(
ω
)
E
(
ω
)
E
(
ω
)
(3.42)
ijkl
j
k
l
ij kl il
,
,
where
χ
ijkl
( 3
is the third-order nonlinear tensor of the medium. In the case of isotropic media, it can be
expressed as [43]
χ
( )
3
=
χ δ δ
(
+
δ δ
+
δ δ
)
(3.43)
ijkl
0
ij kl
ik
jl
il
jk
where δ
ab
= 1 if
a
=
b
, and δ
ab
= 0 if
a
≠
b
.
The last step is to propagate all these sources of THG to the far field, taking into account their phase
to determine the resulting interference.
Propagation to the Far Field Using Green's Functions
To calculate the far-field interference pattern created by the nonlinear polarization, each point of the
grid is considered as a dipole which electric field can be computed using Green's function, and all the
contributions of the sources located in the focal volume are then summed up in the far field.
The harmonic field originating from all positions
r
in the focal region and propagated to a position
r
in the collection optics aperture can be expressed as [13,15]
∫
E
(
R
)
=
P
(
nl
)
( )
r G R r
(
−
)
dV
(3.44)
FF
FF
V
where
V
spans the excitation volume and
g
FF
is the far-field Green's function:
exp ikR
R
(
)
[
(3.45)
G
=
I RR
−
/
R
2
]
FF
4
π
where
r
is the coordinate of a point in the far field,
R
= |
r
| and
I
is the third-order identity tensor. To
calculate the emitted THG intensity,
E
FF
(
r
) is finally evaluated on a grid describing a 2D surface (usu-
ally corresponding to a solid angle defined by the detection NA) and all the intensities are added up.
References
1. Terhune RW, Maker PD, Savage CM 1962. Optical harmonic generation in calcite.
Phys. Rev. Lett.
8:404-406.
2. Maker PD, Terhune RW 1965. Study of optical effects due to an induced polarization third order in
the electric field strength.
Phys. Rev.
137:A801-A818.
3. New GHC, Ward JF 1967. Optical third-harmonic generation in gases.
Phys. Rev. Lett.
19:556-559.
4. Ward JF, New GHC 1969. Optical third harmonic generation in gases by a focused laser beam.
Phys.
Rev.
185:57-72.
5. Bey PP, Giuliani JF, Rabin H 1967. Generation of a phase-matched optical third harmonic by intro-
duction of anomalous dispersion into a liquid medium.
Phys. Rev. Lett.
19:819-821.
6. Kajzar F, Messier J 1985. Third-harmonic generation in liquids.
Phys. Rev. A
32:2352-2363.
7. Tsang T 1995. Optical third-harmonic generation at interfaces.
Phys. Rev. A
52:4116-4125.
