Biomedical Engineering Reference
In-Depth Information
=
α
1
2
I r z
( , )
cos
θ
sin
2
θ
J kr
(
sin exp
θ
)
(
i
kz
cos d
θ θ
)
(1.18c)
1
1
0
α
1
2
I
( , )
r z
=
cos
θ
sin
θ
(
1
−
cos
θ
)
J kr
(
sin exp
θ
)
(
i
kz
cos d
θ θ
)
(1.18d)
2
2
0
r
= (
x
2
+
y
2
)
1/2
, α is the aperture half-angle,
k
is the wave vector (2π
n
/λ), and the
J
n
terms are Bessel
functions of the first type with order
n
.
11
The subscript of 0 on the cylindrical coordinates indicates the
particular location of the SHG-active point source. Unlike in the paraxial case, the local fields change
significantly as a function of position close to the focal point, such that the polarization components
experienced by the local SHG-active source can also change while imaging.
The calculations shown in Equation 1.18 for the optical fields present within the volume of a focused
Gaussian beam have interesting implications for polarization-dependent measurements in nonlinear
optical imaging. For an
x
-polarized incident light, the
y-
and
z
-components of the optical field calcu-
lated using Equation 1.18 exhibit nodes along the optical axis. Even for high-NA excitation, the field
strengths are at least two orders of magnitude lower than the
x
-polarized fields (for
x
-polarized incident
light) in that region, and may safely be assumed to be negligible. Interestingly, the
z-
polarized fields
exhibit two side lobes of opposite polarity positioned along the
x
-axis, such that objects adjacent to the
beam center can couple to these additional field components. These field components drop off quickly
from a maximum intensity of 0.25 relative to the maximum intensity of
E
x
for an NA of 1.4 to 0.05 for
an NA of 0.8 and are therefore only generally significant when using oil or water immersion objectives.
For measurements of moderate precision (~1-2 significant figures), the use of the much simpler paraxial
approximation expressions can still be reasonably accurate for measurements obtained with an NA < 1.
However, explicit numerical calculations may be necessary for interpreting polarization-dependent data
obtained at higher NA.
12
As expected, explicit calculations of the local fields also generate an accumulated Guoy phase-shift
of 180° upon traversing the focal volume, qualitatively similar to that described within the paraxial
approximation detailed in Section 1.3.2.
The implications of nonnegligible
z-
polarized field components are tractable, but nontrivial, since the
emerging SHG light can now couple through a whole new set of local frame tensor elements contain-
ing
z
indices. Indeed, all 27 possible nonzero laboratory-frame tensor elements can now contribute to
the detected response. To systematically build up to this most general cases, we will first consider SHG
from a point source, or more precisely, a source that is thin axially relative to the wavelength of light
(i.e., <~100 nm in thickness along the
z
-axis). Starting with this limit has the distinct advantage of
largely eliminating the need to consider complicating interference effects, described in greater detail in
Section 1.4. In this limit, the local electric field experienced at any specific location within the vicinity of
the confocal volume can be related back to the incident polarization through the polarization transfer
matrix described in Equation 1.18. Unlike a plane wave, the local fields change significantly as a func-
tion of position close to the focal point, such that the polarization components experienced by the local
SHG-active source can also change while imaging.
Just as within the paraxial approximation in Section 1.3.2, the matrix in Equation 1.18 needs to be
inverted to isolate the Jones vector and allow the local tensor to be expressed within the laboratory
frame. Since
G
is a 3 × 2 matrix and inversion requires a square matrix, the simplest way to perform the
inversion numerically is to left multiply both sides of the equation by
G
T
to recover an invertible matrix
on the right, then left-multiply both sides by (
G
T
G
)
−1
to recover the identity matrix on the right side of
the equality. The local field vector is then multiplied by Γ ≡ (
G
T
G
)
−1
G
T
, which is a 2 × 3 matrix. This
same set of operations was performed implicitly in Section 1.3.2, but with a much simpler analytical
