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Scanning Optical Microscopy
3.1 "Classical" Optical Microscopy
•
Basic definitions
The term used in a title is rather uncertain. In
microscopy the recent progress of optical methods
resulted in the appearance of a large number of
techniques many of which can be considered as
classical. However, the purpose of this chapter is
not to review the wide capabilities of optical
microscopy but define some terms which will be
used further for the explanation of a confocal
microscope principle of operation. We introduce
these basic concepts taking the widefield optical
microscope as an example (Fig. 1).
Fig. 2. In a widefield microscope, various points
of an object are viewed simultaneously,
therefore points of planes, other than the object plane,
produce background illumination lowering the contrast
•
Numeric aperture and dimensionless
units
To describe parameters of optical instruments, some
special terms are adopted in optics. Namely,
numeric aperture NA is defined as
NA = n sin θ
(1)
where n – refractive index of a media, θ – halfangle of a cone within which light rays converge or
diverge. For a lens, this angle is defined by its
diameter D and focal length F .
sin θ =
Fig. 1. "Classical" optical microscope schematic diagram
In such a microscope, the image of an object placed
in a uniformly illuminated field of view is projected
by an optical system onto the retina of the eye or
onto a sensor (e.g. CCD array in a video camera)
plane. Generally, a sensor receives the light emitted
by different areas of a specimen which are both in a
focal point of an objective lens and beyond a focal
point (Fig. 2).
D
NA
=
2F
n
(2)
It is convenient to measure distances from the axis
in the object plane by the units of the light
wavelength in the media λ ′ = λ n , where p – light
wavelength in vacuum. Dimensionless radius unit
in this case will be written as
p=
2π
λ
NAr =
2πn
λ
r sin θ
(3)
while dimensionless distance ζ along the optical
axis will be
ζ =
Scanning Optical Microscopy
2π
2πn
NA 2 z =
z sin 2 θ
nλ
λ
(4)
•
What factors determine a microscope
resolution?
Images are formed by lenses or mirrors in
geometrically conjugate planes. In this case, for
rays emanating from every point of the object, the
Fraunhofer diffraction condition is met. Let, for
example, a parallel beam from the distant point
object converge in a lens focal plane (Fig. 3). Each
point in the focal plane corresponds to the point at
infinity, therefore, the Fraunhofer diffraction
condition is met in the focal plane. Diaphragm D
which confines the beam plays the role of an
obstacle for light diffraction. Such a diaphragm, in
particular, can be the lens mount. This is the case of
the diffraction at the optical system entrance
aperture.
corresponds to the diffraction pattern in the
telescope objective (or the eye), Fig. 4 – to the light
diffraction pattern in the microscope objective.
Field of view of conventional microscopes does not
exceed 1000 resolved picture elements.
Fig. 3
Fig. 4. Fraunhofer diffraction in a plane
geometrically conjugate with a source
•
The point spreading function (or the function of the
diffraction-limited
system
pulse
response)
determines the intensity distribution in the lens
focal plane due to the Fraunhofer diffraction from
the entrance aperture. As it was shown before,
exactly the same intensity distribution from a point
source is formed in the conjugate plane of a thin
lens.
Fig. 3. Fraunhofer diffraction in a lens focal plane
Similarly can be considered the case when the point
object is positioned at a finite distance a from the
lens and the image is formed at a distance b from
the lens on its right-hand side. Distances a and b
obey the lens formula:
1 1 1
+ =
a b F
(5)
To explain why Fraunhofer diffraction takes place
in this case also, we replace the single lens with
focal length F by two closely situated lenses with
focal lengths F1 and F2 (Fig. 4). Then the source will
be positioned in the front focal point of the first lens
and the image plane coincides with the rear focal
plane of the second lens. Condition (1) is
automatically met in this case because it is
equivalent to the optical power (i.e. inverse of a
focal length) sum rule of two closely situated
lenses. Between the two lenses light rays travel as a
parallel beam. Comparing Fig. 3 and Fig. 4 it can be
concluded that in the second case the Fraunhofer
diffraction occurs at the common lens mount and is
viewed in the rear focal plane of the second lens.
Scanning Optical Microscopy
Point spreading function (PSF)
PSF of the light beam limited by a circular aperture
with diameter D for the lens having focal length F
can be expressed in a general form as follows :
p (ζ , p ) = I 0 (ζ , p ) + 2 I1 (ζ , p ) + I 2 (ζ , p )
2
2
2
(6)
where
θ
(
)
I 0 (ζ , p )∫ J 0 ( p sin α sin θ ) cos α sin α (1 + cos α ) exp iζ cos α sin 2 θ dα
0
θ
(
)
I 1 (ζ , p )∫ J 1 ( p sin α sin θ ) cos α sin 2 α exp iζ cos α sin 2 θ dα
0
θ
(
)
I 2 (ζ , p )∫ J 2 ( p sin α sin θ ) cos α sin α (1 − cos α ) exp iζ cos α sin 2 θ dα
0
(7)
J k (x) – k-th order Bessel functions,
D
NA
.
sin θ =
=
2F
n
where
Here we introduce the more general function as
compared with that given before. Function p(ζ , p)
gives intensity distribution along radius p for
different planes ζ . This function has a remarkable
property for any plane ζ :
where λ ′ = λ n .
which means that energy flux through every plane
is constant.
It should be noted that on the system optical axis
( p = 0) : I1 (ζ ,0) = 0 and I 2 (ζ ,0) = 0 , therefore the
resolution along the optical axis is determined only
by contribution of I 2 (ζ ,0) . In the paraxial
approximation (small NA magnitudes), the relative
intensity distribution along the axis is given by:
In the paraxial approximation (small NA
magnitudes), the light intensity distribution in the
focal plane is given by:
⎛ sin(ζ / 4) ⎞
⎟⎟
p(ζ ,0) ≈ ⎜⎜
⎝ (ζ / 4) ⎠
∞
∫ p(ζ , p) pdp = const
(8)
0
⎛ 2 J (0, p ) ⎞
⎟⎟
p (0, p ) ≈ ⎜⎜ 1
p
⎝
⎠
2
(12)
2
(9)
where the normalization coefficient is selected so
that p(0,0) value in a focal point is equal to 1.
The diffraction pattern from a circular aperture is
concentric rings. A central bright spot is called the
Airy disk. The first bright ring maximum intensity
is about 2% of the intensity in the center of the Airy
disk. Distribution p(0, p) is shown in Fig. 5.
•
Microscope
criterion.
resolution,
Rayleigh
Resolution of the microscope generally means the
capability to distinguish two point objects of about
equal intensity. From the function of intensity
distribution in a focal plane p(0, p) it follows that
the resolution is determined by overlapping of Airy
disks of two point-like objects. Rayleigh proposed
the criterion which states that two points are
resolved if a "dip" in their images intensity is 26%
of the maximum intensity. Also, the separation
distance between two resolved points should be
more than the Airy disk radius (see previous
paragraph).
Summary
Fig. 5. Intensity distribution of light diffracted by a circular
aperture
The Airy disk radius is:
presel = 1.22π
(10)
or
r = 0.61
λ
n sin
= 1.22
λ′
D
F
Scanning Optical Microscopy
(11)
- The main characteristic of the objective lens is
its numeric aperture determined by its diameter
and a focal length.
- Resolution of a conventional optical microscope
is determined by the Fraunhofer diffraction at
the entrance aperture of the objective lens. The
minimum distance between resolved point
objects of equal intensity amounts to the Airy
- In present chapter, the expression for the point
spreading function (or the function of the
diffraction-limited system pulse response) is
derived which will be used further for the
explanation of the confocal microscope
operation.
References
1. Robert H. Webb, "Confocal optical microscopy"
Rep. Prog. Phys. 59 (1996) 427-471.
2. Richards B. and Wolf E., "Electromagnetic
diffraction in optical systems II. Structure of the
image field in an aplanatic system" Proc. R. Soc. A
253 (1959) 358-379.
Fig. 1a. Traces of light rays in conventional microscope.
The photodetector receives light from various points
of a specimen
3.2 Confocal Microscopy
•
Introduction
A confocal microscope differs from a "classical"
optical microscope (see chapter 3.1 "Classical"
Optical Microscopy) in the fact that every moment
of time there is formed an image of one object point
while a whole image is assembled by scanning
(moving a specimen or readjusting an optical
system). In order to register light from only one
point, a pinhole aperture is situated behind the
objective lens so that the light emitted by the
studied point (red rays in Fig. 1b) passes through the
aperture and is detected while light from the other
points (e.g. blue rays in Fig. 1b) is at most excluded.
The second feature is that the illuminator produces
not the uniform lighting of the field of view but
focuses light into the studied point (Fig. 1c). This can
be done by placing a second focusing system
behind a specimen; in this case, however, the
specimen should be transparent. Moreover, the
objective lenses are usually expensive enough, so
utilization of a second focusing system for
illumination is of little preference. An alternative is
the use of a beam splitter for the purpose of incident
and reflected light could be focused by the same
objective (Fig. 1d). Besides, such arrangement
Scanning Optical Microscopy
Fig. 1b. Aperture utilization allows to reduce sufficiently
background illumination from specimen points beyond the
studied area
Fig. 1c. Additional contrast increase is due
to illumination light focusing into the analyzable point
Fig. 1d. Arrangement with a beam splitter
simplifies the microscope construction and facilitates its
adjustment due to the objective two-fold use
(for illumination and reflected light collection)
It is clear that the application of the confocal
scheme should increase the image contrast because
"stray" light from points adjacent to the studied one
does not enter the detector. Note that the contrast
increase is achieved at the expense of complicated
scan by specimen or by light beam systems
utilization. Detailed examination of existing
confocal microscope designs is beyond the scope of
this chapter. Further information on the matter can
be found in reviews [1 - 11].
•
Resolution and contrast in confocal
microscopy
Now let us examine mathematically how and how
much the contrast is changed when utilizing the
confocal microscopy. Firstly, because the light in
the confocal microscope passes through the
objective twice, the point spreading function
(designated further as PSF, see definition in chapter
3.1 "Classical" Optical Microscopy) is given by
p conf (ζ , p ) = p (ζ , p )× p (ζ , p )
(1)
For the sake of convenience each PSF will be
qualified as a probability of a photon hitting the
point with coordinates (ζ , p ) or a photon detection
from the point with coordinates (ζ , p ) ; then the
confocal PSF is a product of independent
probabilities. Fig. 2 shows a representation of
conventional and confocal PSF.
Fig. 2. Confocal PSF
p conf (ζ , p ) = p (ζ , p )× p (ζ , p )
shown on the right, conventional PSF
as compared
microscope
rconf = 0.61
λ
n sin θ
with
= 1.22
the
λ′
D
conventional
F
optical
(3)
where λ ′ = λ n .
However, the major advantage of a confocal
microscope is a sufficient increase in the contrast
rather than resolution improvement in accordance
with the Rayleigh criterion. In particular, the
relation of the first ring maximum amplitude to the
amplitude in the center is 2% in case of
conventional PSF in a focal plane while in case of a
confocal microscope this relation is 0.04%. The
practical importance of this factor is illustrated in
Fig. 3. From the top part of the picture it can be seen
that a dim object (intensity 200 times less than of a
bright one) can not be detected in conventional
microscope though the separation distance between
objects exceeds that of the Rayleigh criterion. In a
confocal microscope (bottom part of Fig. 3) this
object should be well registered.
is
(ζ , p) – on the left
If we use the Rayleigh criterion for the resolution
(26% dip of the maximum intensity), the result is a
slight increase in resolution for the confocal
microscope:
rconf = 0.44
λ
n sin θ
= 0.88
Scanning Optical Microscopy
λ′
D
F
(2)
Fig. 3. Intensity profiles for conventional (top picture) and
confocal (bottom picture) microscopes. Intensity maximum
of the dim object is 200 times less than that of the bright one
The intensity distribution along the optical axis in a
confocal microscope is given by the following
expression:
⎛ sin(ζ / 4) ⎞
pconf (ζ ,0) ≈ ⎜⎜
⎟⎟
⎝ ζ /4 ⎠
4
(4)
Δz conf
nλ
⎛F⎞
= 1.5
= 1.5
= 6λ ′⎜ ⎟
2
2
n sin θ
NA
⎝D⎠
P ( p, ζ ) = p ⊗ S = ∫ p ( p − p s , ζ ) S ( p s ,ϕ s ) p s dϕ s dp s
(6)
Then, using the Rayleigh criterion for the resolution
in the direction along the optical axis we can write:
λ
In order to consider mathematically the presence of
an aperture and to obtain a new function of intensity
distribution one should perform convolution:
2
(5)
and for a confocal microscope to multiply the
obtained function P( p, ζ ) by p( p, ζ ) . The
resulting intensity distribution in case of the
aperture size of 5 Airy disks is shown in Fig. 4.
Notice that one should distinguish this resolution
and depth of focus in a conventional microscope.
Generally, the depth of focus is hundreds times
more than the resolution along the optical axis.
•
The effect of an aperture in a focal plane
One of parameters, that was not taken into
consideration above, is the size of an aperture in a
focal plane of illuminating and collecting lenses.
Notice that PSF for conventional and confocal
microscopes was calculated under assumption that a
source is point-like. Therefore the obtained PSF
describe properties of an objective lens and the
aperture image in the object plane determines areas
whose light is registered by the photodetector.
Lowering the aperture size obviously decreases the
amount of the passing light, increases noise level
and, finally, can reduce all advantages in the
contrast to nothing. Thus, the problem of choosing
the aperture optimal size and reasonable
compromise is quite relevant.
The use of aperture with the size which is less than
that of the Airy disk just results in intensity loss and
do not affect resolution. If aperture size is that of
the Airy disk, the objective lens resolution is
maximal. The best compromise, however, is the
aperture size which is 3-5 times more than that of
the Airy disk. The size considered here should be
understood as the image size in the object plane,
therefore, the actual aperture size depends on the
lens magnification. In particular, when lens with
100x magnification is used, the 1 mm orifice of the
diaphragm is projected onto the object plane as a
circle with radius 10 micron.
Scanning Optical Microscopy
Fig. 4. Point spreading functions for conventional microscope
with an aperture size of 5 Airy disks (top pictures) and for
confocal microscope (bottom pictures)
Summary
- Confocal microscopy provides an image
contrast increase due to the studied area
illumination via focusing objective lens and an
aperture placement in the image plane before
the photodetector. Such contrast improvement
allows for resolving objects having intensity
difference up to 200:1.
- In confocal microscopy, the resolution in the
object plane is slightly increased (1.5 times)
while the resolution along the optical axis is
high.
- These improvements are obtained at the expense
of the utilization of mechanisms for scanning
either by moving a specimen or by readjustment
of an optical system. Scanning application
allows to increase field of view as compared
with conventional microscopes.
References
1. Robert H. Webb "Confocal optical microscopy"
Rep. Prog. Phys. 59 (1996) 427-471.
2. Richards B. and Wolf E. "Electromagnetic
diffraction in optical systems II. Structure of the
image field in an aplanatic system" Proc. R. Soc. A
253 (1959) 358-379.
3. Kino G. S. and Corle T. R., 1989 Confocal
scanning optical microscopy Phys. Today 42 55–62.
4. Pawley 1991 J B Fundamental and practical
limits in confocal light microscopy Scanning 13
184–98.
5. Shotton D., (ed) 1993 Electronic Light
Microscopy—Techniques in Modern Biomedical
Microscopy (Wiley-Liss) p. 351.
6. Slater E. M. and Slater H. S., 1993 Light and
Electron Microscopy (Cambridge: Cambridge
University Press).
7. Stevens J. K., Mills L. R. and Trogadis J. (eds)
1993 Three-Dimensional Confocal Microscopy
(San Diego, CA: Academic).
8. Webb R. H., 1991 Confocal microscopes Opt.
Photon. News 2 8–13.
9. Wilson T. 1985 Scanning optical microscopy
Scanning 7 79–87.
10. Wilson T. (ed) 1990 Confocal Microscopy