Biomedical Engineering Reference
In-Depth Information
where
P
(
·, ·,z
) describes the overall complex field distribution in the pupil of a
non-aberrated objective lens. The pupil function is a description of the magnitude
and phase of the wavefront that a point source produces at the exit pupil of the
imaging system. In simple terms, the Eq. (
4.3
) states that the field distribution at a
point (
x, y, z
) in the image space can be obtained by applying Fourier transform on
the overall pupil function. For a microscope with defocus, the pupil function can be
written as [
75
],
⎧
⎨
exp
i
2
πz
(
n
2
(
k
x
+
k
y
))
2
,
if
k
x
+
k
y
2
<
N
λ
,
λ
2
−
P
(
k
x
,k
y
,z
;
λ,
NA) =
⎩
0
,
otherwise
,
(4.4)
where the defocus
z
takes a value between [
1)
Δ
z
] and
Δ
z
is the step size between two slices (or axial sampling). When the
z
=0(at the
focal plane), the function in Eq. (
4.4
) is the disk in Fig.
4.4
a. Just like the one-
dimensional Fourier transform of a unit step function gives a
sinc
function, the
2-D Fourier transform of a disk should inductively give the Airy disk in Fig.
4.4
b.
In [
52
], the Debye's scalar diffraction model for a lens system was derived that
serves as the basis for obtaining an analytical expression for the CLSM PSF. As
standard detectors does not measure the coherent PSF but only the intensity PSF or
the incoherent PSF, we write the model as:
−
(
N
z
/
2)
Δ
z
,
(
N
z
/
2
−
|
2
×|h
A
(
|
2
,
(4.5)
h
clsm
(
x
;
λ
ex
,λ
em
,
NA)
∝|Π
(
x
)
∗ h
A
(
x
;
λ
ex
,
NA)
x
;
λ
em
,
NA)
where
Π
(
) is the pinhole model. If we assume the pinhole to be very small, it
can be modeled as a Dirac, and the PSF can be simply calculated by multiplying
the squared coherent excitation and the emission PSFs. The WFM PSF can also be
calculated from Eq. (
4.5
) by simply taking the square-root of the ideal CLSM PSF
(Algorithm
4
). If the pinhole is larger, it is modeled as a simple uniform circular
disc with its radius in nm or AU.
x
Algorithm 4
Theoretical confocal PSF calculation
Input:
Voxel sizes, peak wavelengths
λ
ex
λ
em
, numerical aperture
NA
, magnification
M
,
refractive index
n
.
Output:
PSF
h
(
x
)
.
1: Calculate defocus
z
.
2: Pupil bandlimit:
k
max
←
(NA
/λ
ex
).
3:
if
(
k
2
x
+
k
2
y
)
1
/
2
<k
max
then
P
(
k
x
,k
y
,z
;
λ
ex
,
NA)
←
exp
i
2
πz
(
n
/λ
ex
2
− λ
ex
2
(
k
2
x
2
y
))
1
/
2
4:
+
k
5:
else
6:
P
(
k
x
,k
y
,z
;
λ
ex
,
NA)
←
0
.
7:
end if
8: For every
z
,
h
ex
(
x, y, z
)
←
IFFT(
P
(
k
x
,k
y
,z
);
λ
ex
,
NA)
.
9: Repeat steps
2:7
for
λ
em
, and for every
z
,
h
em
(
x, y, z
)
←
IFFT(
P
(
k
x
,k
y
,z
);
λ
em
,
NA)
.
10: For ideal pinhole,
h
clsm
(
x
)
←|h
ex
(
x
)
|
2
2
×|h
em
(
x
)
|
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