Biomedical Engineering Reference
In-Depth Information
E 0
E
R
χ (3) ( x , y , z )
f
Φ
θ
r ( ρ, φ, z )
Z
θ max
x
z
y
Backward
propagating
(B-THG)
Forward
propagating
(F-THG)
FIgurE 3.20 Geometry for modeling coherent microscopy. Illustration of the notations used in this section. See
text. (Adapted from Novotny L, Hecht B 2006. Principles of Nano-Optics . Cambridge University Press; Cheng JX,
Xie X 2002. J. Opt. Soc. Am. B 19:1604-1610; Olivier N, Beaurepaire E 2008. Opt. Express 16:14703-14715.)
absence of resonance effects. The linear index mismatches are also neglected, although they may have a
significant influence on beam propagation.
Description of Focused Fields
The angular spectrum representation [13] is used to calculate the field distribution obtained by propa-
gating an initial field distribution (amplitude and phase) at the back aperture of a microscope objective
under a given set of focusing conditions (NA, index of refraction). In this representation, the field is
described as a sum of plane waves with variable amplitudes and propagation directions, so that the
excitation field near focus is calculated by propagating all the plane waves and then summing them up
coherently. Assuming a homogeneous isotropic linear medium, we have
θ
2
π
m
ik fe
ik f
ω
ω
E ( ,
ρ φ
, )
z
=
e
ik z
cos
( )
θ
e
ik
ρ
sin
( )
θ
cos
(
Φ φ
)
sin
( )
θ
E
( ,
θ
Φ d d
)
θ
ω
(3.40)
2
π
0
0
with
sin
cos
Φ
Φ
n
cos
si
Φ
Φ
E
( ,
θ
Φ
)
=
(
cos
θ
)
1 2
/
E
( ,
θ
Φ
)
0
0
0
cos
sin
Φ
Φ
cos
Φ
Φ
cos
θ
θ
+
(
cos
θ
)
1 2
/
E
( ,
θ
Φ
)
sin
cos
sin
(3.41)
0
0
θ
where E 0 (θ,Φ) describes the field distribution at the back aperture of the objective, k = k ω = 2 π ω/ n ω is the
wavenumber, f is the focal length of the objective, n ω is the refractive index at frequency ω, (ρ,ϕ, z ) are
cylindrical coordinates near focus, and θ max = sin −1 ( NA / n ) is the maximum focusing angle of the objec-
tive, as illustrated in Figure 3.20.
This integral can then be integrated numerically on each point of a 3D grid representing the focal
volume.
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