436
OPTICS LETTERS / Vol. 22, No. 7 / April 1, 1997
Optical transfer functions of 4Pi confocal
microscopes: theory and experiment
M. Schrader, M. Kozubek, S. W. Hell, and T. Wilson
Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UK
Received January 10, 1997
We measure the point-spread function in the two main configurations of 4Pi confocal microscopy as well as
in the traditional confocal arrangement and derive the optical transfer functions from the experimental data.
The optical transfer functions are in good agreement with their theoretical counterparts. We find a 3.5- to
5-fold increased axial bandwidth of the 4Pi confocal microscope and hence confirm the enhanced spatialfrequency content of 4Pi images.  1997 Optical Society of America
The basic aim of 4Pi microscopy is to increase the axial resolution of a confocal microscope. One achieves
this by illuminating the specimen from one or both
sides and detecting the backscattered and (or) the
forward-scattered light. The principal ideas of 4Pi microscopy are already well explained in the literature.1 – 3
In essence, interference between two counterpropagating probe or signal beams of high aperture results in a
sharpening of the main lobe of the point-spread function in the axial direction. The systems work equally
well in either the bright-field or the f luorescence imaging mode. The image-formation process may be described in terms of either an effective point-spread
function or a three-dimensional transfer function.4 In
this Letter we obtain the three-dimensional optical
transfer functions of different types of 4Pi microscopes
from experimental data.
We begin by reviewing the various imaging geometries.2 The first is the traditional confocal arrangement for which the bright-f ield image intensity is given
by
(1)
I ­ jh1 h2 ≠ tj2 ,
where h1 represents the amplitude point-spread function of the illuminating light, h2 is the amplitude pointspread function of the detection light, the symbol ≠
denotes the convolution operation, and t represents
the amplitude transmission (or ref lection) function of
the object. If the object consists of a single point, then
Eq. (1) predicts that
(2)
I sr, zd ­ jh1 sr, zdh2 sr, zdj2 ,
where r and z represent radial and axial coordinates,
respectively. If identical lenses are used to illuminate and image the point, then in a ref lection system we can set h1 sr, zd ­ h2 sr, zd ­ hsr, zd, whereas
in transmission h2 sr, zd ­ hsr, 2zd. However, since4
hsr, zd ­ hp sr, 2zd,5 the image intensity in both cases
is given by
I sr, zd ­ jhsr, zdj4 .
(3)
It is a property of confocal imaging systems that
Eq. (2) is very general; thus, if we elect to illuminate
the point from both sides with equal amplitude beams
that are arranged to interfere in phase in the focal
plane, then we can write h1 sr, zd ­ hsr, zd 1 hsr, 2zd,
0146-9592/97/070436-03$10.00/0
and hence the image intensity if we elect to detect the
scattered light from only one side is given by
I ­ jhsr, zd 1 hsr, 2zdj2 jhsr, 6zdj2 .
(4)
This is the geometry of the 4Pi(A) confocal microscope. If, however, we elect to illuminate the point
from only one side but detect the constructive interference between the forward-scattered and the backscattered light, then the result is the 4Pi(B) geometry,
which is, of course, equivalent to the 4Pi(A) geometry.
The f inal form that we consider is that of the 4Pi(C)
microscope in which the object is illuminated from both
sides by two constructively interfering beams. The
signal is detected by constructive interference of the
forward-scattered and the backscattered light. In this
case the image intensity is given by
I ­ jhsr, zd 1 hsr, 2zdj4.
(5)
To realize the 4Pi geometries that we have just
discussed we constructed an optical arrangement
with two opposing high-numerical-aperture (NA)
lenses whereby suitable beam stops were introduced
as necessary to vary the illumination and detection
conditions. Constructive interference was ensured
by the piezo-driven mirrors. A He –Ne laser operating at a wavelength of 633 nm was used. Crossed
polarizers were used to suppress the transmission
light. To measure the image of a point scatterer, we
used 80-nm-diameter colloidal gold particles immersed
in immersion oil and mounted between two cover
slips. We chose this size of bead to obtain as high
a signal-to-noise ratio as possible while still getting
a (Mie) scattered f ield similar to that obtained from
an inf initesimally small point scatterer. In all the
experiments a single bead was scanned in a plane containing the optic axis, and the scattered-light intensity
was measured as a function of position. Scanning
was performed by a piezoelectric stage scanner (Melles
Griot/Photon Control, Cambridge, UK) with a specified positioning accuracy of 5 nm. The bead was
scanned approximately 2.5 mm along the optic axis and
2.5 mm in the lateral direction. Two 100 3 1.4-NA
Leica Planapochromat oil immersion objectives were
used for the imaging. The diameter of the detection
pinhole was 5 mm, which is approximately 1y6 of the
 1997 Optical Society of America
April 1, 1997 / Vol. 22, No. 7 / OPTICS LETTERS
437
more than a five-fold increase over the confocal case.
The price one pays, of course, is the increased sidelobe
level. However, the sharpness and the well-def ined
nature of the response in this case make it a far more
suitable candidate for numerical deconvolution techniques than the confocal response of Fig. 1(b).
Equation (1) describes the image-formation process
in a bright-f ield confocal microscope. If the system
were imaging in f luorescence, the image-formation
properties of the system would be completely different
and would take the general form
I ­ jh1 h2 j2 ≠ f ,
(6)
where f represents the distribution of f luorophores.
The optical transfer function describing this process is
now given by the three-dimensional Fourier transform
as
ZZ
Csl, sd ­
jh1 h2 sr, zdj2
3 exps jszdJ0 srldrdrdz ,
(7)
where Csl, sd represents the three-dimensional transfer
function and l and s are spatial frequencies in the
lateral and the axial directions, respectively.
We obtained Eq. (7) by assuming that the effective point-spread functions are radially symmetric.
Our measured point-spread functions are x –z slices
through the r, z volume. It was therefore necessary to
average equivalent points in the four quadrants of each
Fig. 1. Theoretical [(a), (c), (e)] and experimental [(b), (d),
(f )] point-spread functions for the confocal [(a), ( b)]; the
4Pi(A) [(c), (d)]; and the 4Pi(C) [(e), (f )] geometries. In all
the cases the optic axis is shown horizontally; the lateral
axis, vertically. The scan region is a 2.5-mm square.
back-projected Airy disk. The results are shown in
Fig. 1.
Figures 1(a) and 1(b) show the theoretical and the
experimental confocal point responses, respectively.
The experimental results are the raw data. They were
not smoothed, averaged, or processed in any way. The
theoretical point-spread functions were obtained by use
of the vector theory of Richards and Wolf.6 The experimental result exhibits an axial FWHM of 496 nm,
which compares well with the theoretical value of
500 nm. Figures 1(c) and 1(d) show the 4Pi(A) responses, and Figs. 1(e) and 1(f) represent the 4Pi(C) responses, respectively. The 4Pi(A) response, Fig. 1(d),
has an axial FWHM of 140 nm together with welldef ined zeros. The form of this response has been
obtained previously,7 but the result presented here is
superior to that of Ref. 7 in the sense that it more
closely resembles theoretical predictions. The theoretical axial half-width is 130 nm in this case. Figure 1(f ) shows the 4Pi(C) response, which exhibits an
axial FWHM of 95 nm in the central peak compared
with a theoretical value of 95 nm. This represents
Fig. 2. Theoretical [(a), (c), (e)] and experimental [( b), (d),
(f)] transfer functions for the confocal [(a), (b)], the 4Pi(A)
[(c), (d)], and the 4Pi(C) [(e), (f )] cases.
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OPTICS LETTERS / Vol. 22, No. 7 / April 1, 1997
The high signal-to-noise ratio provided by the gold
beads is very advantageous for measuring the pointspread function and hence the OTF. There are, of
course, fundamental differences in imaging in f luorescence and scattered light. One is that in f luorescence
imaging phase is lost, whereas in scattering it is not.
We were therefore careful to image only an isolated
scatterer so as to avoid interference effects. Apart
from slight differences in wavelength, a single, ideal
subresolution scattering particle gives the same pointspread function as a single subresolution f luorescent
particle. This fact, as well as the fact that the Stokes
shift, i.e., the difference between excitation and average f luorescence wavelength, is only approximately
10%, allows us to use the present experimental results
to predict the form of the transfer functions in f luorescence 4Pi confocal microscopy.
Figures 2 and 3 clearly show pronounced minima
in the optical transfer functions that are not desired,
of course. An interesting version of 4Pi confocal microscopy is given by the combination with two- or even
three-photon excitation.8 In this case, the lobes of the
4Pi confocal point-spread function are reduced. This
is equivalent to filling the pronounced minima in the
4Pi optical transfer function.9
The present study shows remarkable agreement between experiments and theory; the 3– 5-fold increased
axial bandwidth makes the application of deconvolution techniques more attractive than with standard
confocal or conventional techniques. This applies not
only to fast three-point-deconvolution methods10 but
also to deconvolutions with the entire point-spread
functions.11 Both methods should result in an unprecedented resolution in far-field microscopy.
Fig. 3. Axial [(a), (c), (e)] and focal-plane [( b), (d), (f )] cuts
through the three-dimensional transfer functions shown in
Fig. 2. The theoretical data are shown as solid curves and
the experimental data as dashed curves. Graphs (a), ( b)
correspond to the confocal case; (c), (d), to the 4Pi(A) case;
and (e), (f ) to the 4Pi(C) case.
M. Kozubek acknowledges the support of the Soros
Foundation, New York. M. Schrader and S. W. Hall
were on leave from the Department of Medical Physics,
University of Turku, Finland.
of Figs. 1(b), 1(d), and 1(f) to estimate h1 h2 sr, zd from
the measured h1 h2 sx, zd data. It was also necessary to
determine accurately the center of mass of the experimental data. This was achieved most conveniently by
raising of the data to a high power (8 was adequate
although not critical) so as to minimize the effects of
noise in the outer regions of the scan. Figures 2 and
3 show the results and the remarkable agreement between theory and experiment.
We notice that in all the cases the 4Pi system
shows the same lateral resolution as the pure confocal
geometry. This is evident from Eqs. (1) –(6) as well
as from the transfer functions, where we can see
that Figs. 3(b), 3(d), and 3(f ) are essentially identical.
However, it is the improvement in the axial resolution
that is provided by the 4Pi geometries. The strongly
increased axial bandwidth is predicted by theory and
for the first time, to our knowledge, also confirmed by
these optical transfer functions of measured 4Pi spread
functions, as shown in Figs. 3(c) and 3(e).
1. S. W. Hell, European Patent Application 91121368.4
(1992).
2. S. W. Hell and E. H. K. Stelzer, J. Opt. Soc. Am. A 9,
2159 (1992).
3. S. W. Hell, S. Lindek, C. Cremer, and E. H. K. Stelzer,
Appl. Phys. Lett. 64, 1335 (1994).
4. M. Gu and C. J. R. Sheppard, J. Opt. Soc. Am. A 11,
1619 (1994).
5. T. Wilson and C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, London,
1984).
6. B. Richards and E. Wolf, Proc. R. Soc. London Ser. A
253, 349 (1959).
7. S. W. Hell, E. H. K. Stelzer, S. Lindek, and C. Cremer,
Opt. Lett. 19, 222 (1994).
8. M. Schrader and S. W. Hell, J. Microsc. 183, 189 (1996).
9. M. Gu and C. J. R. Sheppard, J. Microsc. 177, 128
(1994).
10. P. E. Hänninen, S. W. Hell, J. Salo, E. Soini, and C.
Cremer, Appl. Phys. Lett. 66, 1698 (1995).
11. H. T. M. van der Voort and K. C. Strasters, J. Microsc.
178, 165 (1995).
References