Biomedical Engineering Reference
In-Depth Information
In the above set of equations,
W
0
is the beam waist,
z
0
is the Rayleigh length describing the axial
extent of the focal volume,
θ
0
describes the angular spread of the incident light (related to the numerical
aperture), and
k
is the wavevector, given by 2
πn
/
λ
.
5
This expression differs from plane-wave propaga-
tion by two notable effects. First, there is a
z-
dependent Gaussian radial scaling of the field amplitude,
the width of which is minimized for
z
= 0 (corresponding to the focal plane). Second, there is an overall
additional phase change of
π
that varies smoothly upon traversing the focal volume from the tan
−1
(
z
/
z
0
)
term in the exponent, which is known as the Guoy phase shift.
Although the amplitude of the field changes significantly across the focal volume from Equation 1.15,
both the
x-
and
y
-polarization components are equally affected by the change. As such, the normalized
Jones vector describing the local polarization state remains identical for each individual position across
the focal volume, scaled in magnitude and absolute phase by the axial and radial position. Since the
polarization state of light described by Jones vectors only includes relative phase, the change in absolute
phase between the fundamental and SHG across the focal volume is inconsequential in the limit of a
thin sample. Consequently, the polarization transfer matrices connecting the local Jones vector to the
far-field Jones vector and vice versa are simply given by identity matrices and can be largely ignored,
with one caveat. This analysis
only
holds in the limit of thin or point samples relative to the focal vol-
ume. Because
z
0
depends on wavelength, the absolute phase between the fundamental and the second-
harmonic beams can shift across the focal volume. For samples in which the SHG source extends axially
across distances approaching or exceeding the Rayleigh length, coherent interference between the SHG
generated at each axial slice can influence the detected intensity and polarization, described in more
detail in Section 1.3.3.
Formally, this outcome can be expressed as a Jones matrix by recasting Equation 1.14 using a position-
dependent complex scaling factor,
C
ω
(
ρ,z
). Since the formalism introduced herein is based on express-
ing the local tensors within the laboratory frames, it is arguably most convenient to express the problem
solving for the far-field polarization state, rather than in the local frame.
ω
E
( , )
( , )
( , )
ρ
ρ
ρ
z
ω
x
1 0 0
0 1 0
e
e
1
x
0
(1.16)
E
z
=
y
C
ω
( , )
ρ
z
y
0
E
z
z
If it is assumed that the detected SHG radiated by the sample can also be reliably modeled as a
Gaussian beam, the paraxial approximation yields an analogous expression connecting the far-field and
local field components for the SHG.
2
ω
E
( , )
( , )
( , )
ρ
ρ
ρ
z
2
ω
x
1 0 0
0 1 0
e
1
x
0
E
z
=
(1.17)
y
C
2
ω
( , )
ρ
z
e
y
0
E
z
z
From inspection of Equations 1.16 and 1.17, all the field components are equally scaled in both ampli-
tude and phase within the paraxial approximation in the limit of a thin SHG-active source. Assuming
the SHG source mirrors the square of the driving fundamental field,
W
2
ω
and
θ
ω
2
are both reduced
accordingly. Interestingly,
z
2
ω
is unchanged. Consequently, the situation is even simpler to treat math-
ematically than the local field correction factors described in the preceding section. In studies focused on
interpreting polarization-dependent effects, the overall scaling factors for treating the perturbations to the
polarization-dependence upon focusing simply disappear upon normalization!
o
