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758 A. Diaspro et al.

From this point of view, a linear space invariant (LSI) model is a good

choice, pliable enough to obtain important insights and develop suit-

able mathematical tools for the analysis of most concrete situations. Let

us consider the situation sketched in Figure 11–3, showing a typical

confocal setup.

A monochromatic point light source is focused onto a sample in the

focal plane through a lens L

(condenser) and the emitted radiation

from the sample (which is also supposed to be monochromatic) is col-

lected through a second lens L

(objective) by a point detector. Let h

and h

be, respectively, the impulse response of L

and L

, i.e., the lens

response to a point-like light source. Under this hypothesis it can be

written that U

(x) = (h

⊗ δ

)(x) = h

(x), where the excitation light

source is modeled by a Dirac impulse.

It can be shown that with U

being the signal reaching the sample,

the emitted signal scales with the ﬂ uorescent dye density D according

to U

= D ⋅ U

. The emitted radiation is then focused on the point

detector through L

This leads to U

det

(x) = [(h

⊗ U

) ⋅ δ

](x) where the point detector

function is assumed to be a Dirac impulse. The overall signal collected

by the detector is

IUxdxdxxhUx

dx x dyh x y D y

tot det d em em

dem

== ⊗

=−

∫∫

∫

() ()( )()

() ( )(

δ ))()

() () () ( )

() () (

dyD y h y dx x h x y

dyD y h y h y

ex d em

ex em

∫

∫∫

=−

))

∫

If we now limit ourselves to a point-like sample:

Uxdx dyyhyh y h h

det ex em ex em

() () () ( ) () ()

∫∫

=−=δ 00

where h

= h

under the hypothesis of L

= L

and λ

= λ

Since an x–y–z scanning process is generally coupled to the imaging

one, it is natural to write, for a general point P(x,y,z): I

tot

= h

(x,y,z), which

is the general expression for the point spread function (PSF), that is, the

system impulse response. A mathematical expression for h(x,y,z) can be

obtained through the scalar electromagnetic waves theory.

The formulation, based on Fraunhofer diffraction, leads to

huv J v e d

(,) ( )

∝

−

∫

ρρρ

(2)

Where u and v are suitable dimensionless variables deﬁ ned according

to the following:

u ∝ z

vxy∝+

The equivalence of h

= h

is valid only for pinhole sizes ≤ 0.25 AU. AU is

the so-called Airy unit, which represents the diameter of the Airy disk.

Chapter 11 Two-Photon Excitation Fluorescence Microscopy 759

By limiting the discussion to the points along the optical axis and in

the focal plane, we have:

(,)

()

(,)

(/)

∝













∝













sin

that is

tot tot

sin

(,)

()

(,)

(/)

∝

























Compared to a conventional microscope, where I

tot

≈ h(u,v), the calcula-

tion of the fall width at half-maximum (FWHM), representing the

system resolution, leads to an improvement in resolution by a factor

1.4 (Brakenhoff et al., 1979; Wilson and Sheppard, 1984; Diaspro et al.,

1999a; Jonkman and Stelzer, 2002; Torok and Sheppard, 2002).

3.4 Remarks and Comments

Comparisons between the ideal PSF in the case of strict confocality

with that of conventional microscopes account for improvements in

resolution. However, despite the use of this mathematical formalism

for concrete situations, further drawbacks need to be highlighted. First,

there is a natural relation between the pinhole size and the PSF: the

more the pinhole size is increased, the more the confocal microscope

response tends to ﬁ t conventional responses.

This means that in the case of dim or highly photosensitive speci-

mens some compromise has to be found between the resolution and

the amount of the collected signals, according to the kind of analysis

to be performed (whether a morphometric one or intensity one).

Second, the PSF is obviously dependent on many physical para-

meters, inter alia the refractive index of the sample, immersion medium,

its turbidity, the degree of homogeneity of the sample, and the photo-

chemical properties of the dyes used. For these reasons the develop-

ment of complicated computations often leads to poorly applicable

results in practice, since conditions often change dramatically for the

different measurements.

One of the most meaningful factors on which PSF depends is the

refractive index mismatch between the objective immersion medium

and that of the sample solution. This results in a loss of axial resolution

and a corruption of the shape (Diaspro et al., 2002a).

Table 11–1 reports the value of theoretical and experimental FWHM

of confocal PSFs using different pinhole sizes. A sample of subresolu-

Table 11–1. FWHM of confocal PSFs for different pinhole sizes.

Oil (n = 1.5)

Lateral (nm) Axial (nm)

Pinhole 20 mm Pinhole 50 mm Pinhole 20 mm Pinhole 50 mm

Experimental 186 ± 6 215 ± 5 489 ± 6 596 ± 4

Theoretical 180 210 480 560

760 A. Diaspro et al.

tion beads [Polyscience, diameter = (0.064 ± 0.009) µm] has been

imaged by means of a 100× NIKON oil-immersion objective (NA = 1.3;

WD = 0.20 mm) under argon laser excitation (λ = 488 nm). Theoretical

values are those expected (in the absence of mismatch) and are cal-

culated by means of web-based deconvo lution software (http://www.

powermicroscope.com).

As can be seen from the reported values, the system resolution is

worse along the optical axis and is strictly dependent on the pinhole

size: these results are in accordance with what is expected from the

above theory. Asymmetry of the plots in the real case is typical and

becomes even more evident when focusing through different stratiﬁ ed

media (Diaspro et al., 2002a).

The theory, developed within the context of electromagnetic waves

focusing across stratiﬁ ed media, suggests a progressive broadening of

the PSF with respect to the focusing depth, becoming even more notice-

able under refractive-index mismatch conditions. Furthermore, on a

higher level of complexity, the coexistence of different refractive indices

within the sample and the resulting artifacts can be considered.

As a consequence of this, the largest percentage of variation of the

lateral FWHM, with respect to the focusing depth, goes from 6% (oil-

immersed PSF), to 48% (air-immersed PSF), whereas the axial FWHM

varies up to 130% (air-immersed PSF) (see Table 11–2).

This phenomenon is related to a subsequent weakening of the signal

with respect to the focusing depth, which turns out to be more evident

in the case of refractive index mismatch. Table 11–3 gives typical

observed values of the percentage of variation of the PSF intensity peak

under different mismatch conditions and at different focusing depths

(referred to the coverslip).

Table 11–3. Variations in PSF intensity with mismatch conditions

and focusing depths.

Medium % at 30 µm depth % at 60 µm depth % at 90 µm depth

Oil 3 6 7

Glycerol 17 27 34

Air 44 51 60

Table 11–2. Variation of lateral FWHM with focusing depth.

Air Glycerol Oil

Depth (mm) Lateral (nm) Axial (nm) Lateral (nm) Axial (nm) Lateral (nm) Axial (nm)

0 187 ± 8 484 ± 24 183 ± 14 495 ± 29 186 ± 6 489 ± 6

30 244 ± 10 623 ± 9 221 ± 5 545 ± 12 197 ± 10 497 ± 21

60 269 ± 11 798 ± 10 252 ± 7 628 ± 9 186 ± 12 496 ± 19

90 277 ± 5 1063 ± 24 268 ± 8 797 ± 26 191 ± 9 484 ± 12

Chapter 11 Two-Photon Excitation Fluorescence Microscopy 761

3.5 Resolution and Three-Dimensional Optical Sectioning

Three-dimensional reconstruction of an object starting from the acqui-

sition of 2D confocal slices is one of the most powerful procedures for

morphological analysis and volume rendering, especially within bio-

logical sciences, where optical slicing allows information to be obtained

from different planes of the specimen without being invasive, thus

preserving the structure and functionality of the different parts.

Three-dimensional optical sectioning is intrinsically coupled with

the axial resolution of the confocal microscope. For pinhole diameters

smaller than 1 AU the approximation of a point-like pinhole is used.

For this case the FWHM of the total PSF in the z direction can be

expressed as

nnNA

−−

064

. λ

(3)

This expression describes the axial resolution and the effective

optical slice thickness for the sectioning of the specimen. Here λ

–

λλλλλ≈+

()

em ex ex em

: a mean wavelength. This technique is

essentially based on an automatic ﬁ ne z stepping either of the objective

or of the sample stage, coupled with the usual x–y point-to-point scan-

ning of the focal plane and image capturing.

The synchronous x–y–z scanning allows the collection of a set of

in-focus 2D images, which are less affected by signal cross-talk from

other planes of the sample as more strictly confocal conditions are

respected.

This means that when a set of 2D images is acquired at various focus

positions and under certain conditions, in principle the 3D shape of the

object can be recovered. However, the observed image o(x,y,z), pro-

duced by the true intensity distribution i(x,y,z), is corrupted by the

characteristic PSF of the image formation system s(x,y,z), by noise stem-

ming from different sources n(x,y,z), and by cross information coming

from different planes rather than from the focus one.

At a certain plane of focus z

within the sample or, at discrete planes

along the z axis, the simplest way to describe such a process for the jth

plane can be regarded as the following:

= i

⊗ s

+ i

j−1

⊗ s

j−1

+ i

j+1

⊗ s

j+1

+ (other plane contributions

if relevant) + n (4)

where the subscripts on i and o refer to the z plane numbers, while the

subscripts on s refer to the number of interplane spacings z away from

the “in-focus” position at the actual jth plane.

This relationship is usually transferred to the Fourier frequency

domain, where the convolution operator becomes an algebraic multi-

plication (Diaspro et al., 1990). Image restoration algorithms (deconvo-

lution) aim to invert such equations in order to extract the true measured

quantity i(x,y,z). Thus a 3D sample reconstruction is possible directly

by piling up 2D images. Therefore further scale corrections are per-

formed accounting for axial distortion phenomena linked to the refrac-

tive index mismatch.

762 A. Diaspro et al.

The generic considerations for this method evidently demonstrate

the crucial importance of getting to know the system performance

under different working conditions, by means of its PSF. This knowl-

edge will extend the possibilities of applying the confocal technique to

a wide range of studies.

In practice, one wants to ﬁ nd the best estimate, accordingly to some

criterion, of i(x,y.z) through the knowledge of the observed images, the

distortion or PSF of the image formation system, and the additive noise

within a restoration scheme classical for space invariant linear systems

(Diaspro et al., 1990; Bertero and Boccacci, 1998; Boccacci and Bertero,

2002). Figure 11–4 shows an example of digital restoration of micro-

scopic data obtained after solving the appropriate set of equations. So

far, this can be computationally done starting from any data set

of optical slices. Recently a WWW service, named Power-Up-Your-

Microscope (Diaspro et al., 2002c; Bonetto et al., 2004), has become

available that produces the best estimate of i(x,y,z) accordingly to the

acquired data set of optical slices. Interested readers can ﬁ nd informa-

tion and check the service through the webpage http://www.power-

microscope.com for free (Figure 11–4).

Image restoration is needed only to correct PSF distortions that are

less than in the conventional case. Unfortunately, a drawback occurs.

In fact, during the excitation process of the ﬂ uo rescent molecules the

whole thickness of the specimen is harmed by every scan, within an

hourglass-shaped region (Bianco and Diaspro, 1989). This means that

even though out-of-focus ﬂ uorescence is not detected, it is generated,

with the negative effect of potential induction of those photobleaching

and phototoxicity pheno mena previously mentioned. The situation

becomes particularly serious when there is the need for 3D and tem-

poral imaging coupled to the use of ﬂ uorochromes that require excita-

Figure 11–4. Comparison between 3D views of a helicoidal biological sample before (left) and after

(right) image processing utilizing 3D deconvolution strategies as implemented at http://www.power-

microscope.com as the web-based computational facility (Difato et al., 2004).

Chapter 11 Two-Photon Excitation Fluorescence Microscopy 763

tion in the ultraviolet regime (Stelzer et al., 1994; Denk, 1996). As

earlier reported by König and colleagues (1996a,b), even using UVA

(320–400 nm) photons may modify the activity of the biological system.

DNA breaks, giant cell production, and cell death can be induced at

radiant exposures of the order of magnitude of a few J/cm

, accumu-

lated during 10 scans with a 5-µW laser scanning beam at approxi-

mately 340 nm and a 50-µs pixel dwell time. In this context, TPE of

ﬂ uorescent molecules provides an immediate practical advantage over

confocal microscopy (Denk et al., 1990; Potter, 1996; Centonze and

White, 1998; Gu and Sheppard, 1995; Diaspro, 1998; Piston, 1999;

Squirrel et al., 1999; Diaspro and Robello, 2000; So et al., 2000; Wilson,

2002). In fact, reduced overall photobleaching and photodamage are

generally acknowledged as major advantages of TPE in laser scanning

micro scopy of biological specimens (Brakenhoff et al., 1996; Denk and

Svoboda, 1997; Patterson and Piston, 2000) even though photobleach-

ing in the focal plane can be accelerated (Patterson and Piston, 2000).

However, the excitation intensity has to be kept low considering a

regime under 10 mW of average power as a normal operation mode.

When laser power is increased above 10 mW some nonlinear effects

might arise, evidenced through abrupt rising of the signals (Hopt and

Neher, 2001). Moreover photothermal effects should be induced, espe-

cially when focusing on single molecule detection schemes (Chirico

et al., 2003a).

4 Two-Photon Excitation

Let us now move from conventional excitation of ﬂ uorescence

as used in computational optical sectioning and confocal

microscopy to a special case of multiphoton excitation, i.e., TPE.

All considerations can be easily extended to the TPE. The physical

suppression of contributions from adjacent planes is realized in a

completely different way, thus moving again to 3D optical sectioning

ability.

In TPE, two low-energy photons are involved in the interaction with

absorbing molecules. The excitation process of a ﬂ uorescent molecule

can take place only if two low-energy photons are able to interact

simultaneously with the very same ﬂ uorophore. As mentioned in the

introduction, the time scale for simultaneity is the time scale of molecu-

lar energy ﬂ uctuations at photon energy scales, as determined by the

Heisenberg uncertainty principle, i.e., 10

−16

–10

−15

s (Louisell, 1973). These

two photons do not necessarily have to be identical, but their wave-

lengths, λ

and λ

, have to be such that

λλ

≅+

(

)

−

(5)

where λ

is the wavelength needed to prime ﬂ uorescence emission in

a conventional one-photon absorption process according to the energy

relation given in Eq. (1). This situ ation, compared to the conventional

one-photon excitation process shown in Figure 11–5, is illustrated using

764 A. Diaspro et al.

a Perrin–Jablonski-like dia gram. It is worth noting that for practical

rea sons the experimental choice is usually such that (Denk et al., 1990;

Diaspro, 2001; Girkin and Wokosin, 2002)

= 2λ

= λ

≈ 2λ

(6)

and

∆E

(7)

Considering this as a nonresonant process and assuming the existence

of a virtual intermediate state, the resident time, τ

virt

, in this intermedi-

ate state should be calculated using the time–energy uncertainty con-

sideration for TPE:

∆E

⋅ τ

virt

≅ h/2 (8)

where h

= h/2π. It follows that

virt

≅ 10

−15

–10

−16

s (9)

This is the temporal window available to two photons to coincide in

the virtual state.

So far, in a TPE process it is hence crucial to combine sharp spatial

focusing with temporal conﬁ nement of the excitation beam.

The process can be extended to n-photons requiring higher photon

densities temporally and spatially conﬁ ned. Thus, near infrared (circa

680–1100 nm) photons can be used to excite UV and visible electronic

transitions producing ﬂ uorescence. The typical photon ﬂ ux densities

are of the order of more than 10

photons cm

−2

−1

, which implies intensi-

ties around MW–TW cm

−2

(Goppert Mayer, 1931). An elegant treatment

in terms of quantum theory for two-photon transition has been pro-

posed by Nakamura (1999) using perturbation. He clearly described the

process by a time-dependent Schrödinger equation, where the Hamil-

tonian contains electric dipole interaction terms. Using a perturbation

expansion, the ﬁ rst-order solution is found to be related to one-photon

excitation while higher order solutions are related to n-photon ones

(Esposito et al., 2004).

The dependence of TPE on I

should be evident and is demonstrated

by using simple arguments (Diaspro and Sheppard, 2002).

Figure 11–5. Perrin–Jablonsky-like diagram illustrating the difference

between conventional (left) and two-photon (right) excitation (Esposito et al.,

2004). In the second case, photons delivering one-half the energy convention-

ally needed for bringing the ﬂ uorescent molecule to an excited state are used

(see text).

Chapter 11 Two-Photon Excitation Fluorescence Microscopy 765

The ﬂ uorescence intensity per molecule, I

(t), can be considered to

be proportional to the molecular cross-section δ

(λ) and to the square

of I(t) as follows:

It It Pt

() () ()

()

∝⋅ ∝⋅













δδπ

(10)

where P(t) is the laser power and (NA) is the numerical aperture of

the focusing objective lens. The last term of Eq. (10) simply takes care

of the distribution in time and space of the photons by using the par-

axial approximation in an ideal optical system (Born and Wolf, 1980).

It follows that the time-averaged two-photon ﬂ uorescence intensity per

molecule within an arbitrary time interval T, 〈I

(t)〉, can be written as

Itdt

hc T

Pt dt

() ()

()

()=∝













∫∫

δπ

(11)

in the case of continuous wave (CW) laser excitation.

Now, because the present experimental situation for TPE is related

to the use of ultrafast lasers, we consider that for a pulsed laser T = 1/f

where f

is the pulse repetition rate. This implies that a CW laser beam,

where P(t) = P

ave

, allows transformation of Eq. (11) into

It P

f,cw ave

()

∝⋅













δπ

(12)

For a pulsed laser beam with pulse width, τ

, repetition rate, f

, and

average power

ave

= D ⋅ P

peak

(t)

where D = τ

⋅ f

, the approximated P(t) proﬁ le can be described as

Pt P D t

Pt t f

() /

() (/ )

=<<

ave p

for

(13)

We can write Eq. (12) as:

hc T

f,p

ave

()

∝













∫

1 (()NA













(14)

The conclusion here is that CW and pulsed lasers operate at the very

same excitation efﬁ ciency, i.e., ﬂ uorescence intensity per molecule,

if the average power of the CW laser is kept higher by a factor of

1/ τ⋅ f

. This means that 10 W delivered by a CW laser, allowing the

same efﬁ ciency of conventional excitation performed at approximately

−1

mW, is nearly equivalent to 30 mW for a pulsed laser (Diaspro and

Chirico, 2002).

5 Fluorescent Molecules under TPE Regime

The above steps lead to the most popular relationship reported below,

which is related to the practical situation of a train of beam pulses

focused through a high numerical aperture objective, with a duration

and repetition rate f

. In this case, the probability, n

, that a certain

766 A. Diaspro et al.

ﬂ uorophore simultaneously absorbs two photons during a single pulse,

in the paraxial approximation, is (Denk et al., 1990)

ave

∝

⋅













τλ

(15)

where P

ave

is the time-averaged power of the beam and λ is the excitation

wavelength. Introducing 1 GM (Goppert-Mayer) = 10

−58

⋅s), for a δ

approximately 10 GM per photon (Denk et al., 1990; Xu, 2002), focusing

through an objective of NA > 1, an average incident laser power of

≈1–50 mW, and operating at a wavelength ranging from 680 to 1100 nm

with 80–150 fs pulsewidth and 80–100 MHz repetition rate, would satu-

rate the ﬂ uorescence output as for one-photon excitation. This suggests

that for optimal ﬂ uorescence generation, the desirable repetition time

of pulses should be on the order of a typical excited-state lifetime, which

is a few nanoseconds for commonly used ﬂ uorescent molecules. For this

reason the typical repetition rate is around 100 MHz. A further condi-

tion that makes Eq. (15) valid is that the probability that each ﬂ uorophore

will be excited during a single pulse has to be smaller than one. The

reason lies in the observation that during the pulse time (10

−13

s of dura-

tion and a typical excited-state lifetime in the 10

−9

s range) the molecule

has insufﬁ cient time to relax to the ground state. This can be considered

a prerequisite for absorption of another photon pair. Therefore, when-

ever n

approaches unity saturation effects start to occur. The use of Eq.

(15) makes it possible to choose optical and laser parameters that maxi-

mize excitation efﬁ ciency without saturation. In case of saturation the

resolution declines and the image becomes worse (Cianci et al., 2004). It

is also evident that the optical parameter for enhancing the process in

the focal plane is the lens numerical aperture, NA, even if the total ﬂ uo-

rescence emitted is independent of this parameter as shown by Xu

(2002). This is usually conﬁ ned around 1.3–1.4 as the maximum value.

Now, it is possible to estimate n

for a common ﬂ uorescent molecule like

ﬂ uorescein that possesses a two-photon cross-section of 38 GM at 780 nm

(Diaspro and Chirico, 2003).

To this end, we can use NA = 1.4, a repetition rate at 100 MHz, and

a pulse width of 100 fs within a range of P

ave

values of 1, 10, 20, and

50 mW, and substituting the proper values in Eq. (15) we get n

≅

5930 ∗ P

ave

. This result for P

ave

= 1, 20, as a function of 1, 10, 20, and

50 mW, gives values of 5.93 × 10

−3

, 5.93 × 10

−1

, 1.86, and 2.965, respec-

tively. It is evident that saturation begins to occur at 10 mW (Diaspro

and Sheppard, 2002).

The related rate of photon emission per mole cule, at a nonsaturation

excitation level, in the absence of photobleaching (Patterson and Piston,

2000; So et al., 2001), is given by n

multi plied by the repetition rate of

the pulses. This means approximately 5 × 10

photons s

−1

in both cases.

It is worth noting that, when considering the effective ﬂ uorescence

emission, a further factor given by the so-called quantum efﬁ ciency of

the ﬂ uorescent molecules should also be considered. At present, the

quantum efﬁ ciency value is usually known from conventional one-

photon excitation data (Diaspro, 2002).

Chapter 11 Two-Photon Excitation Fluorescence Microscopy 767

Now, even if the quantum-mechanical selection rules for TPE dif-

fer from those for one-photon excitation, several common ﬂ uorescent

molecules can be used. Unfortunately, knowing the one-photon

cross-section for a speciﬁ c ﬂ uorescent molecule does not allow any

quantitative prediction of the two-photon trend, except for a sort of

“rule of thumb.” This simple rule states that, in general, a TPE cross-

section may be expected to peak at double the wavelength needed for

one-photon excitation. However, the cross-section parameter has been

measured for a wide range of dyes (Xu et al., 1995; Albota et al., 1998b;

Diaspro and Chirico, 2003). It is worth noting that due to the increasing

dissemination of TPE microscopy, new “ad hoc” organic molecules,

endowed with large engineered two-photon absorption cross-sections,

have recently been developed (Albota et al., 1998; Abbotto et al., 2005).

Figure 11–6 summarizes the pro perties of some commonly used ﬂ uo-

rescent molecules under two-photon excitation (Xu et al., 1995; So

et al., 2000). TPE ﬂ uorescence from NAD(P)H, ﬂ avoproteins (Piston

et al., 1995; So et al., 2000), tryptophan, and tyrosine in proteins

(Lakowicz and Gryczynski, 1992) has been measured. In addition, the

autoﬂ uorescent biological proteins such as the GFP and its mole cular

variants are important molecular markers (Chalﬁ e et al., 1994; Chalﬁ e

and Kain, 1998; Potter, 1996; Zimmer, 2002). Their TPE cross-sections

are between 6 and 40 GM (Blab et al., 2001). As a comparison consider

that the cross-section for NADH, at the excitation maximum of 700 nm,

is approximately 0.02 GM (So et al., 2000). Moving to quantum dots

there is an increase of cross-section up to 2000 GM.

6 Optical Consequences of TPE

In terms of optical consequences the two-photon effect limits the excita-

tion region to within a subfemtoliter volume. The 3D conﬁ nement of the

TPE volume can be understood with the aid of optical diffraction theory

Rhodamine B

Bodipy Fl

Dil

Coumarin

Lucifer Yellow

Fluorescein

Cascade Blue

Pyrene

Bis-MSB

Dansyl

DAPI

600 700 800

Two photon Excitation

Cross Sections (GM)

Wavelength (nm)

Ti:Sapphlie

SHG of Cr:YAG

SHG of Cr:Forsterite

Cr:LiSGAF

Cr:LiSAF

Nd:YLF or Nd:glass

–1

–2

–3

900 1000 1100

Figure 11–6. Two-photon cross-sections for popular ﬂ uorescent molecules as a

function of the excitation wavelength. Red bars indicate the emission range of

some common laser sources utilized in TPE micro scopy and spectroscopy. (See

color plate.)