Zeiss LSM 880, Operating Manual

Page: 680 / 690

I
Pupil Illumination
to a truncation factor T=1.3. The lateral coordinate is nor-
malized in Airy units (AU). From T=3, the Airy character is
predominating to a degree that a further increase in the
truncation factor no longer produces a gain in resolution.
Because of the symmetry of the point image in case of
diffraction-limited imaging, the graph only shows the in-
tensity curve in the +X direction. Furthermore any trunca-
tion of the illuminating beam cross-section at the pupil
plane causes a certain energy loss and thus a decrease in
efficiency. Figure A (right) shows the percentage efficiency
as a function of pupil diameter in millimeter, with constant
laser beam expansion. The smaller the pupil diameter, the
higher the T-factor, and the higher the energy loss (i.e. the
smaller the efficiency). Example: If the objective lens utilizes
50% of the illuminating energy supplied, this means about
8% resolution loss compared to the ideal Airy distribution.
Reducing the resolution loss to 5% is penalized by a loss
of 70% of the illumina ting energy. In practice, the aim is to
reach an optimal approximation to a homogeneous pupil
illumination with a reasonable loss in efficiency. All descriptions in this monograph suggest a confocal LSM
with a ray geometry providing homogeneous illumination
at all lens cross sections. It is known that the intensity dis-
tribution generated in the focus of the objective lens, is the
Fourier transform of the intensity distribution in the objec-
tives pupil plane (backfocal plane). Hence it follows, that
a homogeneous distribution in the pupil plane results in a
focal distribution following the Airy function (also known
as Airy disk) [In Carl Zeiss microscope objectives, the pupil
diameter is implemented by a physical aperture close to the
mounting surface of the lens].
The Airy distribution is characterized by a smaller width at
half maximum and thus by a higher resolving power than
a Gaussian distribution. Figur A left shows the normalized
intensity distribution at the focal plane as a function of dif-
ferent truncation factor’s T (T is the ratio of the laser beam
diameter (1/e²) and the pupil diameter of the objective lens).
The red curve results at a homogeneous pupil illumination
with T > 5.2, while the blue one is obtained at a Gaussian
pupil illumination with T ≤ 0.5; the green curve corresponds
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.9
0.81
0.72
0.63
0.54
0.45
0.36
0.27
0.08
0.09
Lateral distance [AU]
T < 0.3 (Gauss)
T = 1.3
T > 5.2 (Airy)
Pupil diameter [mm]
Intensity distribution at the focus
Relative intensity
Relative efficiency
Efficiency
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 14 16 18 20
Fig. A
The trunction factor T is defined as the ratio of laser beam diameter (1/2
2
) and pupil diameter of the objective lens used:
T = the resulting efficiency is defined as
The full width at half maximum of the intensity distribution at the focal plane is definied as:
with
With T< 0.6, the Gaussian character, and with T>1 the Airy character predominates the resulting intensity distribution.
d
laser
d
pupille
(
-2
)
T 2 h
= 1 - e
FWHM = 0.71 ·
NA

·
 
= 0.51 + 0.14 · In (
1-n
1
)