Hawkes P.W., Spence J.C.H. (Eds.) Science of Microscopy. V.1 and 2

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798 S.W. Hell and A. Schönle

and Stelzer, 1992b; Gustafsson et al., 1995) if light is detected coherently

through both lenses. That is, the intensity maxima for excitation and

detection are located at different points in space.

Three major types of 4 Pi microscopy have been reported (Hell and

Stelzer, 1992a). They differ on whether the spherical wavefronts are

coherently added for illumination, for detection, or for both simultane-

ously; they are referred to as type A, B, and C, respectively. Usually

the detection has been confocalized, but in conjunction with TPE suc-

cessful axial separation with nonconfocal detection has also been

reported. Here we will concentrate on the TPE 4 Pi (type A), the 4 Pi

type C, and the TPE 4 Pi type C confocal microscopes. Of these three,

the TPE 4 Pi confocal microscope has been applied to the largest number

of imaging problems. It uses the very effective lobe-reducing measure

of TPE combined with “point-like” detection. In reality the size of the

“point-like” detector amounts to about the size of the main maximum

of the diffraction-limited ﬂ uorescence spot (Airydisk), when imaged

into the focal plane of the objective lens.

Clearly, nonconfocal wide-ﬁ eld detection and regular illumination

would make 4 Pi micro scopy more versatile. Therefore, the related

approach of I

M (Gustafsson et al., 1995, 1996, 1999; Gustafsson, 1999)

conﬁ nes itself to using the simultaneous interference of both the excita-

tion and the (Stokes-shifted) ﬂ uorescence wavefront pairs; the latter

are spherical as in a 4 Pi microscope. The potential beneﬁ ts of I

M are

readily stated: single-photon excitation with arguably less photobleach-

ing, an additional 20–50% gain in ﬂ uorescence signal, and lower cost.

This method has so far yielded 3D images of actin ﬁ laments with an

axial resolution slightly better than 100 nm in ﬁ xed cells (Gustafsson

et al., 1999). To remove the side-lobe artifacts, I

M-recorded data are

deconvolved ofﬂ ine. While the consideration of the OTF support in

Figure 12–3 suggests that this single mechanism is indeed sufﬁ cient, it

turns out that the relaxation of the side-lobe suppression comes at the

expense of an increased vulnerability to sample-induced aberrations,

especially with nonsparse objects (Nagorni and Hell, 2001a, 2001b).

Thus I

M imaging, which has so far relied on oil immersion lenses, has

required mounting the cell in a medium with n = 1.5 (Gustafsson et al.,

1999). Live cells inevitably necessitate aqueous media (n = 1.34). More-

over, water immersion lenses have an inferior focusing angle and

therefore larger lobes to begin with (Bahlmann et al., 2001). Potential

strategies for improving the tolerance of I

M are the implementation of

a nonlinear excitation mode and its combination with pseudoconfocal

or patterned illumination (Gustafsson, 2000). While these measures

again add physical complexity, they may have the potential to render

M more suitable for live cells.

However, at this stage, the implementation of at least two of the

mechanisms above proved more reliable: After initial demonstration

of TPE 4 Pi confocal microscopy (Schrader and Hell, 1996), superre-

solved axial separation was applied to ﬁ xed cells (Hell et al., 1997). The

image quality could be improved further by applying nonlinear resto-

ration (Holmes, 1988; Carrington et al., 1995; Holmes et al., 1995). Under

biological imaging conditions, this typically improves the resolution

up to a factor of two in both the transverse and the axial direction.

Chapter 12 Nanoscale Resolution in Far-Field Fluorescence Microscopy 799

Therefore, in combination with image restoration, TPE 4 Pi confocal

microscopy has resulted in a resolution of ∼100 nm in all directions, as

ﬁ rst witnessed by the imaging of ﬁ lamentous actin

and immunoﬂ uo-

rescently labeled microtubules (Nagorni and Hell, 1998; Hell and

Nagorni, 1998) in mouse ﬁ broblasts.

While a very useful and explanative comparison of the OTF supports

has been published (Gustafsson, 1999) it is obvious from the above that

consideration of the supports alone is not sufﬁ cient to understand the

respective beneﬁ ts and limitations of SWM, I

M, and 4 Pi microscopy.

Rather than comparing their technical implementation (Gustafsson et

al., 1999), a quantitative analysis of their PSFs and OTFs is needed to

clarify under which conditions these microscopes will be able to deliver

3D-resolved images with superior resolution. The success of increasing

the axial resolution with coherently used opposing lenses not only

depends on the achievable bandwidth, but also on the strength with

which the respective systems transfer the spatial frequencies within

this bandwidth. Gaps and weak parts that occur in some systems

(Gustafsson et al., 1995; Krishnamurthi et al., 1996) must be quantiﬁ ed

for a particular optical setting because they may render the removal of

artifacts impossible, thus precluding an increase in axial resolution.

Below we demonstrate that these gaps are intimately connected with

the optical arrangement and therefore are inherent to some of the

methods described.

3.1 The Optical Transfer Function of 4 Pi-Microscopy and

Related Systems

Before expanding on the analysis of PSFs and OTFs, we quickly review

scope, as well as the detection PSF of the wide-ﬁ eld, confocal, SWM, and

4 Pi-type microscopes, are regular intensity PSFs. Assuming that exci-

tation and emission involve a dipole transition of the dye, the excitation

and emission PSF of a single lens is well approximated by the absolute

square of the electric ﬁ eld in the focal region:

h = |E

(z, r, φ)|

(3)

r, respectively) and the polar angle φ. The ﬁ eld is usually calculated

using the vectorial theory of Richards and Wolf (1959). In our case, we

assumed circular polarization of the light. In frequency space the Fourier

transform of the electrical ﬁ eld is then simply given by a spherical cap

of radius k

(Figure 12–2). This is equivalent to approximating the

spherical wavefronts emerging at the exit pupil as plane waves close to

the focal spot. The absolute square in Eq. (3) corresponds to an autocor-

relation in frequency space. When calculating the OTF, this convolution

can be carried out directly (Schönle and Hell, 2002; Sheppard et al., 1993)

or the OTF can be determined by Fourier transformation of Eq. (3).

For the excitation PSF of the 4 Pi confocal microscopes (type A and

C) and for the detection PSF of the I

M and the 4 Pi confocal (type B

and C), a calculation of constructive interference between the two

spherical wavefronts is required. The PSF is therefore given by the

coherent addition of two beams

It depends on the axial and radial distance to the geometric focus (z and

their theoretical derivation. The excitation PSF of the confocal micro-

800 S.W. Hell and A. Schönle

h = |E

(z, r, φ) + E

(z, r, φ)|

(4)

The ﬁ eld of the opposing lens is given by

(r) = ME

−1

r) (5)

M is the coordinate transform from the system of lens number 2 to lens

number 1 and is a diagonal matrix inverting the z-components for a

triangular cavity and the y- and z-components for a rectangular cavity

(Bahlmann and Hell, 2000). In frequency space, the Fourier transform

of the electric ﬁ eld is now given by two caps corresponding to the two

focused wavefronts. Consequently, the OTF consists of an autocorrela-

tion part equivalent to that for single lens excitation or detection and

a cross-correlation of two opposite spherical caps represented by the

outer “brackets” in Figures 12–2, 12–3, and 12–4. For TPE the excitation

PSF is simply given by squaring the one-photon PSF scaled to the

4% 100%

PSF OTFPSF OTF

SWM I

M 4Pi C

PSF OTF

det

TPE 4Pi CTPE 4Pi A

Widefield Confocal

Figure 12–4. Overview of the excitation (bottom), detection (center), and effective (top) OTFs’ modulus

and the corresponding PSFs of the wide-ﬁ eld, confocal, standing wave (SWM), I

M, 4Pi confocal type

C, TPE 4Pi confocal type A, and TPE 4Pi confocal type C microscopes. The color look-up table (LUT)

has been designed to emphasize the important weak OTF regions. The OTFs are shown in the squares

above the corresponding PSFs; the zero frequency point is in the center and the largest frequency

displayed is 2π/80 nm

−1

. The circles represent the maximum possible carrier as explained in Figure

12–3. For TPE, the excitation OTFs slightly extend over these circles because the excitation wavelength

of 800 nm is less than double the one-photon excitation wavelength of 488 nm. While all these methods

extend the OTF along the axial direction, they fundamentally differ in contiguity and absolute strength

within the support region. For example, there are pronounced frequency gaps for the SWM and

depressions for the I

M. The rectangular images of the PSFs represent a region of 5 × 2.5 µm with the

geometric focus in the center. (See color plate.)

Chapter 12 Nanoscale Resolution in Far-Field Fluorescence Microscopy 801

appropriate TPE wavelength and similarly the OTF is the autoconvolu-

tion of the scaled one-photon OTF (Denk et al., 1990).

The excitation intensity of the SWM is given by a plane standing

wave along the optic axis,

h = I

cos

(k ) (6)

constructive interference to occur at the common geometric focus. The

excitation OTF is its Fourier transform and given by

o = I

[δ(k) + δ(k − 2k

)/2+ δ(k + 2k

)/2]/2 (7)

Again, this is the result of auto-correlating the electric ﬁ eld’s Fourier

transform that consists of delta peaks at ±2k

While the 4 Pi microscope uses a spatially coherent point-like laser

illumination, in the I

M microscope wide-ﬁ eld illumination is used,

normally in the Köhler mode, either with a lamp or a laser. The physical

consequences of this difference are best explained as follows. If the two

spherical wavefronts of the 4 Pi illumination are decomposed into

plane waves incident from different angles and corresponding to dif-

ferent points of the illumination apertures, all plane waves of the aper-

ture interfere with each other in the focal region. In the I

M the

illumination light is not coherent throughout the aperture. Therefore

only pairs of plane waves originating from corresponding points (mirror

images about the focal plane) of the illumination apertures are mutu-

ally coherent, forming a standing wave in the focal region. The period

of these standing waves scales with the cosine of the azimuth angle θ.

The excitation PSF of the I

M can then be calculated by adding the

intensity of these plane standing waves. Assuming uniform intensity

throughout the exit pupil of the lens, the PSF is given by

hz I d d k z

Id kz

()=

()

∫∫

φθθ θ

πθθ θ

sin cos cos

∫

(8)

The support of the excitation OTF is readily inferred: For each θ the

integrand in Eq. (8) is the excitation PSF of the SWM and thus the total

OTF consists of a superposition of expressions of Eq. (6) for wave

vectors ranging from k

cos(α) to k

. Loosely speaking, this incoherent

superposition smears out the delta peaks at the sides, forming the lines

in Figures 12–3 and 12–4. This excitation mode contributes to avoiding

the gaps that remain in the SWM’s support after convolution with the

detection OTF. If, on the other hand, the incoherent light source is

imaged into the focal plane of the lens, i.e., critical illumination, mutu-

ally coherent points form wavefronts focused onto and interfering at

the conjugate point in the image of the light source. The individual 4 Pi

PSFs produced by each point of the light source as a result are incoher-

ently summed up, giving an integral of the 4 Pi excitation PSF over the

ﬁ eld of view in the focal plane. The OTF becomes nonzero exclusively

on the inverse optical axis where it is given by the values of the 4 Pi

OTF, altogether leading to a result not very much different from that

predicted by Eq. (8). However, critical illumination is problematic due

where I denotes a constant,

is the wavenumber; and we assumed

802 S.W. Hell and A. Schönle

to potential nonuniformities in the light source and will be omitted in

our analysis.

In any case, once the ﬂ uorescence light is generated in the sample,

the I

M collects the spherical ﬂ uorescent wavefronts just as in a 4 Pi

microscope of type B or C. The two counterpropagating spherical

wavefronts of ﬂ uore scence collected by each lens interfere construc-

tively in a common point on the camera. Disturbance of the interfer-

ence pattern of neighboring points of ﬂ uorescence emission is damped

by the spatial incoherence in the focal plane: the radius of spatial coher-

ence is largely given by the Airy disk associated with the ﬂ uorescence

light at the aperture in use. Thus the I

M implements the highest pos-

sible degree of parallelization of 4 Pi detection.

Figure 12–4 shows the numerically calculated excitation, detection,

and effective PSFs/OTFs of the 4 Pi, I

M, and SWM setups, along with

those of the conventional epiﬂ uorescence and confocal microscope. The

epiﬂ uorescence micro scope features uniform illumination intensity

throughout the sample volume; its OTF is a single deltapeak at the

origin. To obtain a practically relevant comparison, we assumed a

numerical aperture of NA = 1.35 and oil immersion with a refractive

index n = 1.51. In the case of single-photon excitation of the dye, an

excitation and detection (i.e., central ﬂ uorescence) wavelength of 488 nm

and 530 nm, respectively, is assumed. For TPE, an excitation wavelength

of 800 nm was chosen. Finite-sized pinholes can be taken into account

by convolving the detection PSF with the image of the pinholes in the

focal plane: a disk of radius r

. In frequency space this corresponds to

a multiplication with the disk’s Fourier transform given by

= J

(kr

)/k (9)

The ﬁ rst root of the Bessel function is at ∼3.83 and thus the detection

OTF becomes zero at k = 3.83/r

. However, the largest frequency

present in the detection OTF is given by half the wavelength and

therefore its support is unaltered if the pinhole radius is smaller than

3.83 λ/(4πNA) ≅ 0.3 λ/NA, which corresponds to half the Airy disk

radius. Pinholes smaller than this can virtually be neglected in the

computation, while for sizes around this value and larger, the PSF will

widen laterally, suppressing the OTF at higher lateral frequencies. The

effect of the pinhole size on axial resolution remains small as long as

it does not exceed that of the Airy disk. We will therefore neglect the

pinhole in our further analysis. All PSFs were numerically computed

in a volume of 128 × 128 × 512 pixels in x-, y-, and z-directions, respec-

tively, for cubic pixels with 20 nm length. The OTFs were calculated

by Fourier transformation; of the 512 pixels in the z-direction only data

based on the central 256 pixels are shown in Figure 12–4. The color

look-up-table (LUT) has been chosen so that the regions of weak signal

are emphasized for both PSF and OTF. This reveals important differ-

ences between the systems. Areas of low but nonnegligible intensity

are important since they cover a large volume and substantially contri-

bute to the image formation.

Let us ﬁ rst consider the PSFs (Figure 12–4, narrow columns). Imme-

diately, some differences between the various approaches become

Chapter 12 Nanoscale Resolution in Far-Field Fluorescence Microscopy 803

apparent. While the excitation modes of the SWM and I

M are similar,

the local minima are not zero for the latter due to the incoherent addi-

tion of the standing wave spectrum. The most important difference is

observed when comparing the 4 Pi microscopes. As a result of focus-

ing, their PSFs are conﬁ ned in the lateral direction so that contributions

from the outer lateral parts of the focal region are reduced. The conﬁ ne-

ment has important consequences. Whereas the 4 Pi confocal micro-

scope, especially its two-photon version, exhibits only two pronounced

but rather low lobes, the I

M and even more so the SWM feature a

multitude of lobes and fringes on either side of the focal plane, despite

the fact that all of them rely on the same aperture. The second conse-

quence is that due to its quadratic or cubic dependence on the excitation

distributions, the 4 Pi confocal PSFs can be separated into an axial

and a radial function in good approximation (Hell et al., 1995; Schrader

et al., 1998):

h(r, z) ≅ c(r)h

(z) (10)

Separability is a particular feature of the 4 Pi confocal and multiphoton

arrangements and we shall see later that it is the prerequisite for simple

online removal of sidelobe effects in the image. TPE leads to a further

suppression of the outer parts of the excitation focus and thus of the

side lobes of the 4 Pi illumination mode. Figures 12–3 and 12–4 also

reveal that in conjunction with coherent detection (type C), TPE 4 Pi

confocal microscopy features an almost lobe-free PSF. Its OTF almost

ﬁ lls the maxi mum support region. In the SWM and the I

M, the number

and relative heights of the lobes increase dramatically when moving

away from the focal point because an effective suppression mechanism

is missing. In the SWM the lobes become even higher than the central

peak itself. In the I

M the secondary maxima are as high as the ﬁ rst

maxima of the single-photon 4 Pi confocal microscope of type C.

3.2 Removing Periodic Artifacts through Deconvolution

Even if they are small, side lobes and resulting ghost images in the raw

data remain a common feature of all methods employing two lenses

coherently. Next, we turn to the OTF to understand the circumstances

under which image processing can effectively be used to remove the

artifacts induced by the lobes. It is obvious from Eq. (2) that if the OTF

were nonzero everywhere, we could divide the Fourier transform of

the image by it. Subsequent Fourier back-transformation of the data

would render the object. In practice, the OTF is limited in bandwidth

and has weak regions. As division outside the OTF support is impos-

sible, these frequencies are lost. But even in regions where the OTF is

small, division strongly ampliﬁ es noise-producing artifacts.

Image restoration techniques, whether linear or nonlinear, aim at

restoring as many frequencies as possible while trying to avoid this

effect. Linear deconvolution is based on the division approach but

introduces a special treatment for frequencies not transmitted by the

OTF. It is capable of restoring frequencies only where the OTF is above

the noise level. If frequencies are missing, a correct representation of

804 S.W. Hell and A. Schönle

the object in the image can be given only if these frequencies are

extracted from a priori knowledge of the object. This extraction is

mathematically more complex and often not viable. Linear deconvolu-

tion, on the other hand, is computationally facile and fast. Speed is of

particular importance because the interference artifacts are ideally

removed online making the ﬁ nal image readily accessible. Therefore,

one of the most prominent advantages of a uniformly strong OTF is

the possibility of applying a linear deconvolution.

The comparison of the effective OTFs in Figure 12–4 highlights the

severe gaps in the SWM, making deconvolution impossible. Linear

deconvolution is reportedly possible in the I

M (Gustafsson et al., 1999).

However, as the gaps in the I

M are ﬁ lled with rather low amplitudes,

this method will create image artifacts for objects that are not sparse,

not very bright, or objects containing spatial frequencies coinciding

with the gap. Owing to the contiguity of its support and the strong

amplitudes of the OTF, 4Pi confocal microscopy fulﬁ lls the precondi-

tions for linear deconvolution. In fact, linear deconvolution along the

axial direction based on the separability of the PSF has been applied

for the removal of interference artifacts arising in the recording of

complex objects. Thus superresolved axial imaging has been shown in

the dense ﬁ lamentous actin

and in the microtubular network (Nagorni

and Hell, 1998) of a mouse ﬁ broblast cell. We will have a closer look at

this important procedure in the subsequent section.

Approximating the 4Pi confocal PSF as in Eq. (10) allows us to

perform a computationally inexpensive one-dimensinal linear deconvo-

lution that simply eliminates the effect of the lobes in the image. The

axial factor of the PSF can basically be decomposed into the convolu-

tion of a function h

(z) describing the shape of a single peak, and a lobe

function l(z) containing the position and relative heights of the lobes:

(z) ≅ h

(z) ⊗ l(z) (11)

quantiﬁ es the blur and l describes the replication that is responsible

for the “ghost images” in unprocessed 4Pi images. The effect of the

lobe function can be eliminated using algebraic inversion. We use a

discrete notation, where each lobe is represented by a component l

the vector l with l

denoting the strength of the central lobe and the

index running from −n to n. Negative indices denote lobes to the left

of the central peak. If the lobe distance in pixels is denoted by d, the

values of the object along the line are given by O

and those of the

image by I

. Thus, the convolution is given by

IlO

jkjdk

−

∑

(12)

Looking for a ﬁ lter l

−1

inverting this convolution we need

OlI llO

jsjdsskjdsdk

−

−−

∑∑

(13)

for all possible objects. The inverse ﬁ lter needs to fulﬁ ll the condition

sjs j

−

∑

(14)

Chapter 12 Nanoscale Resolution in Far-Field Fluorescence Microscopy 805

At this point we can arbitrarily choose the length of the inverse ﬁ lter,

assuming an index running from −m to m. The index j in Eq. (13) can

take values from −m − n to m + n. Therefore we have a system of 2(m

+ n) + 1 equations with 2m + 1 unknowns, which is usually not solvable.

An approximation is found by considering the equations for j =

−m . . . m only. Now, the problem is equivalent to solving a linear

Toeplitz problem with the Toeplitz matrix given by the vector l (Nagorni

and Hell, 2001; Press et al., 1993). The approximation is good if the

edges of the inverse ﬁ lters are small, since the remaining equations are

nearly satisﬁ ed. This holds if the ﬁ rst-order lobes are <45%; in this case,

the error is practically not observable. A typical length of the inverse

ﬁ lter is 11. In conjunction with a lobe height of 45%, the edges of the

inverse ﬁ lter feature a modulus below 1% of the ﬁ lter maximum. For

lobes of 35% relative height, this value drops to a value of only about

0.08%. Thus, using this technique, the separability of the 4Pi confocal

PSF provides a quick way to obtain a ﬁ nal image that is equivalent to

imaging with an “ideal” optical microscope that has a single narrow

main maximum at the focal point. The inverse ﬁ lter is discrete and

nonzero only at a few points. This very effective side lobe removal

method is therefore referred to as point deconvolution. Exploiting the

characteristics of the PSF of the 4Pi microscope, it is applicable only in

conjunction with this method.

Axial lobes in the PSF entail suppressed OTF regions along the optic

axis. Thus, point deconvolution restores suppressed frequencies by

linear deconvolution even though the actual calculation takes place in

real space. For PSFs that cannot be written as in Eq. (10) point decon-

volution is insufﬁ cient, because the transverse directions (x, y) have to

be involved as well. In this case, it is mathematically clearly preferable

to pass the data through the frequency domain. Established linear

deconvolution algorithms rely on inverting Eq. (2) but also accommo-

date for the vanishing regions of the OTF. An estimate of the frequency

spectrum of the object can be obtained by using

(k) = I

(k)o*(k)/(|o(k)|

+ µ) (15)

The regularization parameter µ (Bertero et al., 1990) sets a lower thresh-

old on the denominator to avoid ampliﬁ cation of frequencies where the

modulus of the OTF is so small that the frequency spectrum of the

image is dominated by noise. If the OTF is a convex function that is in

the absence of “gaps,” the effect of regularization is similar to smooth-

ing. In most cases considered in this chapter, however, the OTF is not

convex and the situation is more complicated. Small OTF values are

found not only at the OTF boundaries, but also within its region of

support, e.g., in the vicinity of the minima (Figure 12–4) or at the fre-

quency gaps (Figure 12–3). If the lobes are too high (typically >50%),

the level of the minima of the OTF is comparable or smaller than the

noise level. This is deﬁ nitely the case in the SWM, but also in the pres-

ence of slight aberrations in the I

M, as well as in an aberrated 4Pi

microscope. Therefore, the necessity of implementing a certain value

of µ is closely connected with the lobe height and with the potential

artifacts induced by the lobes.

806 S.W. Hell and A. Schönle

If the PSF is separable in a peak function and an axial lobe function,

the lobe removal can be elegantly targeted:

h(r) = h

(r) ⊗ l(z) (16)

Contrary to the decomposition in Eq. (11), the peak function h

(r) also

contains the dependence of the peak on the lateral coordinates. We no

longer require the PSF to separate in a radial and an axial part, yet the

lobe function still gives the relative height of the lobes as well as their

location:

lz l z ds

()=−

(

)

∑

(17)

with d now being the lobe distance in units of length. The frequency

spectrum of the image is then given by



IGoGhlk

()

() ()

()

()kkkkk=⋅=⋅ ⋅ (18)

If the lobe function is symmetric with respect to the focal plane (i.e.,

for constructive interference), we can write

() ( )lk l sdk

zsz

∑

cos

(19)

This decomposition immediately discloses why the critical lobe height

is 50%: If the PSF consists of a main maximum and two primary lobes

of 50%, the right-hand side of (19) vanishes for the axial frequencies

associated with the distance d. So, if the lobes are >50% the frequency

represented by the lobes is not contained in the OTF, and hence not

transferred to the image. In reality, the critical lobe height is slightly

above 50%. The reason is the inﬂ uence of the secondary lobes that have

been neglected in our reasoning. Nonetheless, the 50% threshold is an

excellent rule for the critical lobe height, which applies equally to

SWM, I

M, and 4Pi micro scopy, for fundamental reasons.

Equation (18) implies that the effect of the lobes can be removed by

direct Fourier inversion. This is the case if the Fourier transform of the

lobe function remains above the noise level throughout the relevant

frequency spectrum. The spectrum of the lobe-free image is then

obtained by dividing the image spectrum by the Fourier transform of

the lobe function. Figure 12–5 shows a typical data set acquired with

a TPE type A 4Pi confocal microscope and illustrates lobe removal both

in the spatial and the frequency domain. The avalanche photodiode

used as a detector had a typical dark count rate <1 count/pixel, which

is negligible (Hell et al., 1997). Hence, the only signiﬁ cant source of

noise was the Poisson noise of the photon-counting process, manifest-

ing itself as white noise that is independent of the spatial frequency.

Since the primary lobes are well below 50%, the ﬁ rst minima of the

OTF are at 19%, which is well above the noise level of typically 0.5–1%.

Direct lobe removal is straightforward in this example, underscoring

the importance of nonvanishing amplitudes in the OTF. If the OTF

exhibited regions close to zero, as is predicted for the I

M, the multi-

plication with a high number in this region would result in a strong

ampliﬁ cation of noise, and compromise the obtained image. In the

Chapter 12 Nanoscale Resolution in Far-Field Fluorescence Microscopy 807

1 µm

a) b) c)

l(k

)

l(k

)

-1

FFT

confocal

4pi (raw)

4pi

FFT

e) f)

Figure 12–5. Lobe removal and deconvolution in 4Pi microscopy. The ﬁ gure shows images of the same

pair of actin ﬁ bers in a ﬁ xed mouse ﬁ broblast cell recorded in the TPE confocal (a) and TPE 4Pi type

A (b and c) mode. The corresponding Fourier transform along the optic axis is also shown (d, e, and

f). The ﬁ ve-fold axial resolution increase (a vs. b) and the correspondingly extended OTF (d vs. e) are

immediately visible. The side lobes are well below 50% and the factorization of the PSF’s axial and

lateral dependence is possible in 4Pi microscopy. Therefore, an inverse discrete ﬁ lter can be found and

its application to the raw data (c) yields a valid and almost artifact-free image (b). Alternatively, lobe

removal can be performed in the frequency domain. Equation (16) indicates that the Fourier transforms

of the raw data (f) is given by the product of the Fourier transform of the lobe-free image (e) and the

lobe function



). Thus, (2) Fourier transforming the raw data, (3) multiplying with the inverse of

the lobe function’s Fourier transform,



−1

), and (4) Fourier backtransforming lead to almost the

same lobe-free image. This method can be applied even if the separation of axial and lateral depen-

dence is impossible for the PSF. The Fourier transforms along the axial direction (3 and 4) merely have

to be replaced by their 3D counterparts. (See color plate.)