I
Pupil Illumination
Details
All descriptions in this monograph suggest a confocal LSM
with a ray geometry providing homogeneous illumination
at all lens cross sections. The focus generated in the object
has an Airy distribution, being a Fourier transform of the
intensity distribution in the objective’s pupil plane. However,
the truncation of the illuminating beam cross-section need-
ed for an Airy distribution causes a certain energy loss (a
decrease in efficiency). [In Carl Zeiss microscope objectives,
the pupil diameter is implemented by a physical aperture
close to the mounting surface].
The Airy distribution is characterized by a smaller width at
half maximum and a higher resolving power. Figure 21 (left)
shows the intensity distribution at the focus as a function of
the truncation factor T (the ratio of laser beam diameter
(1/e
2
) and pupil diameter).
The graph presents the relative intensity distributions at the
focus (each normalized to 1) for different truncation fac-
tors. (The red curve results at a homogeneous pupil illumi-
nation with T > 5.2, while the blue one is obtained at a
Gaussian pupil illumination with T
≤
0.5; the green curve
corresponds to a truncation factor T = 1.3). The lateral coor-
dinate is normalized 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 intensity curve in the +X direction). Figure 21 (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 utilizes 50 % of the illuminating
energy supplied, this means about 8 % resolution loss com-
pared to the ideal Airy distribution. Reducing the resolution
loss to 5 % is penalized by a loss of 70 % of the illuminating
energy. In practice, the aim is to reach an optimal approxi-
mation to a homogeneous pupil illumination; this is one
reason for the fact that the efficiency of the excitation
beam path in a confocal LSM is less than 10 %.
Fig. 21
The trunction factor T is defined as the ratio of
laser beamand pupil diameter of the objective lens used: T =
d
laser
;
the resulting efficiency is defined as
= 1 - e
T
The full width at half maximum of the intensity
distribution at the focal plane is definied as
With T< 0.6, the Gaussian character, and with T>1 the Airy character predominates the resulting intensity distribution.
Relative efficiency
Relative intensity
Lateral distance [AU]
2
4
6
8 10
12
14 16 18 20
Pupil diameter [mm]
Efficiency
Intensity distribution at the focus
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
d
pupille
(
2
)
-2
NA
FWHM
= 0.71 . .
= 0.51 + 0.14 .
In ( )
1
1-n
0.1
0.2 0.3 0.4 0.5 0.6 0.7 0.8
0.9 1
T < 0.3 (Gauss
)
T = 1.3
T > 5,2 (Airy)
,
with
0
0
337_Grundlagen_Infoboxen_e 25.09.2003 16:17 Uhr Seite 1
