Download Imaging properties of high aperture multiphoton fluorescence

Journal of Microscopy, Vol. 193, Pt 2, February 1999, pp. 127–141.
Received 7 August 1998; accepted 15 October 1998
Imaging properties of high aperture multiphoton
fluorescence scanning optical microscopes
P. D. HIGDON, P. TÖRÖK & T. WILSON
Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, U.K.
Key words. Confocal microscopy, fluorescence microscopy, imaging.
Summary
A theory for multiphoton fluorescence imaging in high
aperture scanning optical microscopes employing finite
sized detectors is presented. The effect of polarisation of the
fluorescent emission on the imaging properties of such
microscopes is investigated. The lateral and axial resolutions are calculated for one-, two- and three-photon
excitation of p-quaterphenyl for high and low aperture
optical systems. Significant improvement in lateral resolution is found to be achieved by employing a confocal
pinhole. This improvement increases with the order of the
multiphoton process. Simultaneously, it is found that, when
the size of the pinhole is reduced to achieve the best possible
resolution, the signal-to-noise ratio is not degraded by more
than 30%. The degree of optical sectioning achieved is
found to improve dramatically with the use of confocal
detection. For two- and three-photon excitation axial full
width half-maximum improvement of 30% is predicted.
1. Introduction
Conventional and confocal fluorescence microscopes are
routinely used, primarily in biological sciences, to investigate three-dimensional structures of tissues. The image
contrast in these microscopes results from measuring the
radiation emitted from the individual fluorochrome molecules with which the specimen has been labelled. In the
conventional microscope we essentially measure the
intensity of this emission, whereas in the confocal microscope the emitted fluorescence radiation is focused down
onto a confocal pinhole. Since the individual fluorochrome
molecules are responsible for the image formation we shall
present a detailed theory of the imaging of single fluorescent
molecules in conventional and confocal microscopes. We
shall consider the role of the polarisation state of the
exciting illumination as well as the finite size of the confocal
pinhole.
Correspondence to: P. Török. Tel: þ44 1865 273099; fax: þ44 1865 273905;
e-mail: [email protected].
q 1999 The Royal Microscopical Society
We shall begin by determining the electric field which
excites the fluorescent molecule in the focal region of a high
aperture, high Fresnel number, aplanatic microscope
objective lens using the integral formula of Richards &
Wolf (1959). The fluorescent molecule will be treated as a
harmonically oscillating dipole whose probability of excitation (strength) is proportional to the intensity (modulus
square of the electric field) of the illumination to the power
of the order of the multiphoton process (Gryczynski et al.,
1996a). In general, the fluorescent emission is partially
polarised (Perrin, 1929; Soleillet, 1929) and in this paper
the two limiting cases of polarised and unpolarised emission
are considered. The emitted field passes back through the
high aperture system to a finite sized detector. The theory is
used to describe the axial and lateral resolution of multiphoton fluorescent microscopes employing linearly and
circularly polarised illumination and finite sized detectors.
The effect of the partial polarisation of the fluorescent
emission on the imaging properties is analysed. The signalto-noise ratio is calculated for finite sized confocal pinholes
and used to find an ‘optimum’ pinhole size. The theory is
also applicable to low aperature lenses and it is found that,
as expected, the state of polarisation of the fluorescent
emission may be ignored in such cases.
A brief introduction to multiphoton fluorescence imaging
is given, followed by a detailed description of the theoretical
considerations, and finally numerical results describing the
imaging properties of one-, two- and three-photon fluorescent microscopes are presented. It is well known that twoand three-photon imaging results in optical sectioning
without the use of a confocal pinhole. However, as will be
shown, a significant improvement in the optical sectioning
is obtained if a confocal pinhole is employed.
2. Multiphoton fluorescence imaging
A fluorescent molecule is excited by absorbing the energy of
a photon and subsequently irradiating another photon as it
returns to a lower energy state. The wavelength of the
emission for Stoke’s fluorescence is longer than that of the
127
128
P. D. H I GDO N E T A L .
excitation (the emitted photon possesses less energy than
the absorbed photon owing to energy lost in the excitation
process). Alternatively, the energy of two photons may be
absorbed simultaneously by the fluorescent molecule and,
again, the energy is irradiated as a single photon. In this
case the energy of the emitted photon is limited by the
combined energy of the two absorbed photons. Hence the
emission spectrum has a lower limit of approximately half
the excitation wavelength. In theory, any number of
photons, N, may be simultaneously absorbed and a single
photon emitted. By similar consideration of the energy
involved, the wavelength of the emission must be greater
than 1/N of the illumination wavelength.
When a single absorbed photon causes the emission of a
single photon the process is called one-photon(1-p) fluorescence. When N simultaneously absorbed photons cause the
emission of a single photon the process is called N-photon
(N-p), or multiphoton, fluorescence. Practical two-photon
scanning optical microscopy was demonstrated successfully
in 1990 (Denk et al., 1990) as was three-photon scanning
microscopy in 1995 (Hell et al., 1995). In these papers it
was shown that 2-p and 3-p fluorescence microscopy have
distinct advantages over the 1-p case: for example,
photobleaching is confined to the illumination volume of
the specimen, and UV dyes can be excited with visible or
near-infrared light. More importantly, perhaps, optical
sectioning can be achieved without a confocal pinhole. It
is not surprising that a great deal of work has been carried
out on the theory of image formation of multiphoton
fluorescent microscopy. Most notable is the paraxial
fluorescence theory presented by Sheppard & Gu and their
co-workers. These authors have analysed the lateral and
axial resolution of two-photon (Sheppard & Gu, 1990; Gu &
Sheppard, 1995) and three-photon (Sheppard, 1996)
microscopes and the three-dimensional image formation
for finite sized (Gu & Sheppard, 1993) and infinitely small
detectors (Gu & Sheppard, 1995; Gu & Gan, 1996).
However, the theories presented are only applicable to low
aperture objective lenses and no account is made for the
polarisation state of the illumination.
There are two possible ways to model the mechanism of
fluorescent emission. The first is to consider the fluorescent
emission as being randomly polarised. In previous publications (Van der Voort & Brakenhoff, 1990; Hell et al., 1994;
Visser & Wiersma, 1994) this led to mathematical formulae
in which the point spread function of the illumination and
excitation optical paths were simply multiplied. The other,
more rigorous, approach is to consider the mechanism of
fluorescence emission as a polarisation dependent process.
Extensive attention has been given to the study of the
relationship between absorbed and emitted polarisation
states in the literature since the theories of Perrin (1929)
and Soleillet (1929) were presented. In general, fluorescent
emission is partially polarised and the number of
parameters that determine the state of the polarisation is
extremely large (Soleillet, 1929). The relationship between
illumination and emission polarisation direction also
depends on a number of parameters such as the fluorescent
molecule’s mobility (Axelrod, 1979 and its individual
characteristics (Gryczynski et al., 1996a). Accounting for
the polarisation states in 2-p and 3-p fluorescence is a more
involved process (Friedrich, 1981; Callis, 1993, Gryczynski
et al., 1996b) but the conclusion is still that the fluorescent
emission is partially polarised. As the state of the polarisation varies with the order of the multiphoton process, the
fluorescent molecule used and many experimental conditions it would be of little value to model any one specific
case. Rather, in the current work the limiting cases of
polarised and non-polarised emission will be considered,
either of which can be used as an approximation to given
experimental conditions. Recently, Sheppard & Török
(1997) dervied a high aperture vectorial theory for
conventional and confocal one-photon fluorescence microscopes. However, they did not account for polarised emission
nor did they account for finite sized pinhole detectors.
The following fluorescent excitation and emission processes are considered. The fluorescent material is illuminated by the electric field produced by a high aperture, high
Fresnel number condenser lens. The probability of excitation is proportional to the intensity to the power N (i.e. the
modulus square of the electric field to the power 1, 2 or 3 for
1-p, 2-p or 3-p processes, respectively). Once the molecule is
excited it irradiates as a harmonically oscillating electric
dipole. The light emitted from the fluorescent dipole is
incoherent with respect to the illumination. The direction of
the dipole moment defines the polarisation state of the
emission. Three cases will be considered throughout this
paper:
Case A: Unpolarised emission. All of the incident electric
field is absorbed. The electric dipole moment of the
emission is randomly orientated.
Case B: Unpolarised emission. The electric dipole moment
of the emission is randomly orientated in a fixed
direction. Only the component of the incident
electric field parallel to the emission dipole moment
is absorbed.
Case C: Polarised emission. All of the incident electric field
is absorbed. The electric dipole moment of the
emission coincides with the incident field.
It should be noted that, while Cases A and B are
physically plausible, Case C should be treated with care.
Gryczynski et al. (1996a) obtained expressions for the
angular displacement of the emission transition moment
from the direction of the polarised excitation for 1-p, 2-p
and 3-p fluorescence. In Case C it is assumed that there is
zero angular displacement between excitation and emission,
which is never actually the case but, rather, a limiting
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H I G H A P E RT U R E S CA N N I N G O P T I CA L M I C ROS C O P E S
model which is a good approximation to a highly polarised
emission. A fourth polarised emission case that is not
considered in this paper is that of fixed dipoles that are
aligned in a given direction. In such a case the direction of
the fixed dipole alignment with respect to the incident
polarisation direction is the important factor in image
formation. As it is not possible to consider all possible dipole
orientations in this paper and because a theory for image
formation of 1-p fluorescence confocal and conventional
microscopes with dipoles fixed in three orthogonal directions has already been developed (Sheppard & Török, 1997)
this fourth case is omitted.
3. Theory
In order to study the effects of incident polarisation on the
image formation we chose to analyse the practically
important case in which a Babinet–Soleil compensator is
introduced into the illumination optical system. This allows
the study of the microscope response to a range of incident
illumination polarisation states between linear and circular.
The plane wave exiting the Babinet–Soleil compensator is
then incident upon a high aperture, high Fresnel number
objective lens. The spherical wavefront produced by this lens
may be represented as a coherent superposition of plane
waves. The integral formula of Richards & Wolf (1959) then
gives the electric field in the focal region. From this function
it is possible to calculate the intensity (modulus square of
the electric field) at any position in the focal region and the
unit vector of polarisation at that point. The fluorescent
molecule is excited and subsequently irradiates as a
harmonically oscillating electric dipole. The relationship
between this incident field and the dipole moment of the
fluorescent emission is presented for Cases A, B and C as
discussed in Section 2. In the detection system a lens
identical to that used for the illumination recollimates the
light. The collimated light is not analysed by further
polarising components in this treatment but is simply
129
focused by a detector lens onto a finite sized pinhole
detector.
3.1. Fluorescent dipole moment
A schematic representation of the illumination system is
shown in Fig. 1. The illumination, E0, is linearly polarised in
the x direction with unit electric vector, i.e.
0 1
1
B C
E0 ¼ @ 0 A:
0
ð1Þ
The electric vector, Ei, of the field after propagating through
the Babinet–Soleil compensator and the high aperture lens
is given by:
ð2Þ
Ei ¼ T f : BS : E0 ;
where Tf is the matrix transformation representing the high
aperture objective lens and may be written as Török et al.,
1998a:
p
T f ¼ cos vi
0
1
cos2 fi cos vi þ sin2 fi sin fi cos fi ðcos vi ¹ 1Þ ¹ sin vi cos fi
B
C
@ sin fi cos fi ðcos vi ¹ 1Þ sin2 fi cos vi þ cos2 fi ¹ sin vi sin fi A;
sin vi cos fi
sin vi sin fi
cos vi
ð3Þ
where the spherical polar coordinate system is √defined such
that r > 0, 0 # vi < p and 0 # fi < 2p and the cos vi term
describes the apodisation function of an aplanatic lens. BS
represents the generalised Jones Matrix of the Babinet–
Soleil compensator given by (Török et al., 1998a):
0
1
i sin 2g sin 2d
0
cos 2d þ i cos 2g sin 2d
B
C
BS ¼ @
i sin 2g sin 2d
cos 2d ¹ i cos 2g sin 2d 0 A;
0
0
1
ð4Þ
where d is the relative retardation and g is the orientation of
the optical axis to the x direction. If the Babinet–Soleil
compensator is taken to have its optical axis at 458 to the
direction of incident polarisation the form of the matrix BS
is greatly simplified, becoming:
0
1
A B 0
B
C
ð5Þ
BS ¼ @ B A 0 A;
0
0
1
where the functions A and B are given by:
A ¼ cos
Fig. 1. Schematic diagram of the illumination system.
q 1999 The Royal Microscopical Society, Journal of Microscopy, 193, 127–141
d
d
and B ¼ i sin :
2
2
ð6Þ
The electric vector Ei is given by the substitution of Eqs (1),
130
P. D. H I GDO N E T A L .
(3) and (5) into Eq. (2). This results in:
p
Ei ðv1 ; fÞ ¼ cos vi
0
A½ð1 þ cos vi Þ ¹ ð1 ¹ cos vi Þ cos 2fÿ ¹ Bð1 ¹ cos vi Þ sin 2fi
1
B
C
×@ B½ð1 þ cos vi Þ þ ð1 ¹ cos vi Þ cos 2fÿ ¹ Að1 ¹ cos vi Þ sin 2fi A:
2 sin vi ðA cos fi þ B sin 2fi Þ
ð7Þ
The electric field, ei, in the focal region is given by the
integral formula of Richards & Wolf (1959) with Ei taken as
the strength vector. Hence:
… a1 … 2p
Ei ðvi ; fi Þ exp½iki ðrs sin vi cosðfi ¹ fs Þ þ zs cos vi Þÿ
ei ¼
0
0
× sin vi d vi d fi ;
ð8Þ
where a1 is the convergence semi-angle of the illuminating
high aperture lens, (rs,fs,zs) is the position of the fluoresent
molecule in cylindrical polar co-ordinates and ki is the
wavenumber of the illumination. After performing the
integration in fi Eq. (8) gives:
1
0 1 0
AI0 þ I2 ðA cos 2fs þ B sin 2fs Þ
eix
C
B C B
ei ¼ @ eiy A ¼ @ BI0 þ I2 ðA sin 2fs ¹ B cos 2fs Þ A ð9Þ
eiz
¹2iI1 ðA cos fs þ B sin fs Þ
where the functions I0, I1 and I2 are given by:
I0 ¼
… a1 p
cos vi sin vi ð1 þ cos vi ÞJ0 ðki rs sin vi Þ expðiki zs cos vi Þdvi ;
0
I1 ¼
… a1 p
cos vi sin2 vi J1 ðki rs sin vi Þ expðiki zs cos vi Þdvi ;
ð10Þ
0
I2 ¼
… a1 p
cos vi sin vi ð1 ¹ cos vi ÞJ2 ðki rs sin vi Þ expðiki zs cos vi Þdvi ;
Integrating the intensity in the detector plane due to each
dipole orientation (vdp,fdp) will then give the response of an
‘average’ dipole moment.
In Case C the electric dipole moment of the emission, pC,
is taken to be the unit vector in the direction of the incident
electric field at the fluorescent molecule. The unit vector of
polarisation at position rs in the object space is given by:
êi ðrs Þ ¼
ei ðrs Þ
:
jei ðrs Þj
ð12Þ
The Cartesian components of the electric dipole moment are
thus given by:
0 1
0 1
eix
px
1
B C
B C
C
ð13Þ
p ¼ @ py A ¼ êi ðrs Þ ¼
@ eiy A:
jei ðrs Þj
pz
eiz
The probability of excitation is given by the modulus square
of the electric field absorbed to the power of the multiphoton
process order. For Cases A and C all of the incident field is
absorbed, hence:
PAN ¼ PCN ¼ jei j2N ;
ð14Þ
where N is the order of the process, i.e. 1-p (N ¼ 1), 2-p
(N ¼ 2), etc. From Eq. (9) it can be seen that:
d
jei j2 ¼jI 0 j2 þ 2jI 1 j2 þ jI2 j2 þ 2 cos cos 2fs
2
× ½RefI0 I2¬ g þ jI1 j2 ÿ
ð15Þ
where Re{.} denotes the real part of a complex number and
* denotes the complex conjugate. For Case B only the
component of the incident electric field parallel to the
emission dipole moment is absorbed, hence:
PBN ¼ jpB :ei j2N :
ð16Þ
Having expressions for the probability of excitation and the
dipole moments for each case, all that remains is to consider
the detection arm of the microscope.
0
where Jn(.) is the nth order Bessel function of the first kind.
Equations (9) and (10) represent the general solution to the
electric field in the focal region of a high aperture lens
illuminated by an arbitrary polarisation state between linear
and circular. If the Babinet–Soleil compensator is set to zero
retardance (d ¼ 0), Eq. (9) reduces to the corresponding
equations of Richards & Wolf (1959), for linearly polarised
illumination.
For cases A and B the electric dipole moment of the
emission is randomly orientated. The unit vector, and hence
the dipole moment, orientated at angles vdp and fdp to the z
and x axes, respectively, are given by:
0
1
cos fdp sin vdp
B
C
ð11Þ
pA ¼ pB ¼ @ sin fdp sin vdp A:
cos vdp
3.2. The detected signal
The light emitted from the fluorescent molecule is collected
Fig. 2. Schematic diagram of the detection system.
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H I G H A P E RT U R E S CA N N I N G O P T I CA L M I C ROS C O P E S
by a high aperture objective lens and imaged onto a finite
sized pinhole, as shown in Fig. 2. The expression for the
electric field in the detector plane due to a radiating dipole
in the focal region is given by Török et al. (1998b) as:
0
px K0I
B
I
ed ¼B
@ p y K0
1
¹ 2ipz K1I cos fd þ K2I ðpx cos 2fd þ py sin 2fd Þ
C
¹ 2ipz K1I sin fd þ K2I ðpx sin 2fd ¹ py cos 2fd Þ C
A;
II
II
pz K0 ¹ iK1 ðpx cos fd þ py sin fd Þ
For Case C the intensity distribution in the detector plane is
then given by:
IdC ¼ PC jeCd j2 :
ð23Þ
Substituting the probability functions from Eqs (14) and
(16) gives:
… p … 2p
jeAd j2 sin vdp dvdp dfdp ;
IdA ¼ jei j2N
ð17Þ
IdB ¼
… p … 2p
0
0
0
0
jpB :ei j2N jeBd j2 sin vdp dvdp dfdp ;
ð24Þ
IdC ¼ jei j2N jeBd j2 :
where the K functions are given by:
KnI;II ¼
… a2 p
cos v1 sin v2 OI;II
n Jn ðkf rd sin v2 Þ expðikf zd cos v2 Þdv2 :
0
ð18Þ
where (v2,f2) are spherical co-ordinates in the low
aperature space with:
OL0 ¼ 1 þ cos v1 cos v2
Substituting for the electric dipole moment and electric
fields ei and ed for Case A and after performing the
integrations in vdp and fdp Eq. (24) gives:
IdA ¼
8p 2N I 2
je j ðjK0 j þ 2jK1I j2 þ jK2I j2 þ 2ðjK0II j2 þ jK1II j2 ÞÞ:
3 i
ð25Þ
For Case B it is not possible to obtain a general solution for
multiphoton process as the process order, N, is within the
integral in Eq. (24). The first three orders have been
evaluated analytically as:
OI1 ¼ sin v1 cos v2
IdB ¼
OI2 ¼ 1 ¹ cos v1 cos v2
8p
ð2ðjK0I j2 þ jK1I j2 þ jK2I j2 þ jK0II j2 þ 2jK1II j2 Þ
15
× ðjeix j2 þ jeiy j2 Þ
OIIO ¼ sin v1 sin v2
OII1 ¼ cos v1 sin v2
131
þ ðjK0I j2 þ 6jK1I j2 þ jK2I j2 þ 6jK0II j2 þ 2jKIII j2 Þjeiz j2
ð19Þ
1-p:
where kf is the wavenumber of the fluorescent emission and
(rd,fd,zd) is the position in the detector plane in cylindrical
polar co-ordinates. The angular aperture of the detector
lens, a2, and the co-ordinate v2 are related to the angular
aperture of the collimating objective lens, a1, and the coordinate v1 via:
n1 sin a1 n1 sin v1
¼
¼ b;
n2 sin a2 n2 sin v2
ð20Þ
¹4RefK1I ðK0I¬ þ K2I¬ Þ þ 2K0II K1II¬ g
× Refeix e¬iz cos fd þ eiy e¬iz sin fd g
þ 2ðRefK0I K2I¬ g þ jKiII j2 Þððjeix j2 ¹ jeiy j2 Þ cos 2fd
þ 2Refeix e¬iy g sin 2fd ÞÞ;
ð26Þ
for 1-photon. For a 2-photon process this becomes:
2-p:
where b is the nominal magnification of the objective lens
and n1 and n2 are the refractive indices in the object and
image space, respectively. The defocus of the image is given
by (Török & Wilson 1997):
zd ¼ 2zs b2 :
0
q 1999 The Royal Microscopical Society, Journal of Microscopy, 193, 127–141
8p 2
je j ðð3jK0I j2 þ 2jK1I j2 þ 3jK2I j2 þ 2jK0II j2
35 i
þ 6jK1II j2 Þðjeix j2 þ jeiy j2 Þ
þ ðjK0I j2 þ 10jK1I j2 þ jK2I j2 þ 10jK0II j2 þ 2jK1II j2 Þjeiz j2
ð21Þ
For Cases A and B the integration must be performed over
all possible dipole orientations, i.e.:
… p … 2p
PA;B jedA;B j2 sin vdp dvdp dfdp :
ð22Þ
IdA;B ¼
0
IdB ¼
¹8RefK1I ðK0I¬ þ K2I¬ Þ þ 2K0II K1II¬ g
Refeix e¬iz cos fd þ eiy e¬iz sin fd g
þ4ðRefK0I K2I¬ g þ jK1II j2 Þððjeix j2 ¹ jeiy j2 Þ cos 2fd
þ2Refeix e¬iy g sin 2fd ÞÞ
ð27Þ;
132
P. D. H I GDO N E T A L .
and for 3-photon excitation:
3-p:
IdB ¼
8p 4
je j ð2ð2jK0I j2 þ jK1I j2 þ 2jK2I j2 þ jK0II j2 þ 4jK1II j2 Þ
63 i
× ðjeix j2 jeiy j2 Þ þ ðjK0I j2 þ 14jK1I j2 þ jK2I j2 þ 14jK0II j2
þ 2jK1II j2 Þjeiz j2
¹ 12RefK1I ðK0I¬ þ K2I¬ Þ þ 2K0II K1II¬ g
× Refeix e¬iz cos fd þ eiy e¬iz sin fd g
þ 6ðRefK0I K2I¬ g þ jK1II j2 Þððjeix j2 ¹ jeiy j2 Þ cos 2fd
þ 2Refeix e¬iy g sin 2fd ÞÞ:
ð28Þ
Substituting for the electric dipole moment and electric
fields ei and ed in Case C gives:
IdC ¼ jei j2ðN¹1Þ ððjK0I j2 þ jK2I j2 þ 2jK1II j2 Þðjeix j2 þ jeiy j2 Þ
þ 4ðjK1I j2 þ jK0II j2 Þjeiz j2
¹ 4RefK1I ðK0I¬ þ K2I¬ Þ þ 2K0II K1II¬ g
× Refeix e¬iz cos fd þ eiy e¬iz sin fd g
þ 2ðRefK0I K2I¬ g þ jK1II j2 Þððjeix j2 ¹ jeiy j2 Þ cos 2fd
þ 2Refeix e¬iy g sin 2fd ÞÞ:
ð29Þ
The detected signal of a fluorescence microscope employing
a pinhole detector of radius R is given by integrating the
intensity over the area of the detector, i.e.:
… R … 2p
IdA;B;C rp drp dfp ;
ð30Þ
I A;B;C ¼
0
0
where (rp,vp) is a co-ordinate system centred on the pinhole
detector. It should be noted that the In in Eq. (10) are
functions of rs whereas the Kn in Eq. (18) are functions of rd.
The vector relationship between the co-ordinates of the
detector plane and those of the pinhole are given by Török
et al. (1998b) as:
rp ¼ rd ¹ rs b:
that, for example, a 3-p fluorescence microscope exhibits
better lateral and axial resolution than a 2-p microscope
when characteristic quantities are computed for the same
excitation wavelength and for different fluorescence wavelengths (Lindek et al., 1995). Conversely, when the system
has different excitation wavelengths and the same fluorescence wavelength the 3-p microscope exhibits worse lateral
and axial resolution than a 2-p microscope. This is due to
the increase in the spatial distributions of the illuminating
field due to the longer wavelength inherent with a higher
order process. For this reason one has to decide whether the
first or the second case is computed numerically. Gryczynski
et al. recently demonstrated multiphoton excitation of pquaterphenyl (p-QT) dyes (Gryczynski et al., 1996b). In this
study the emission spectrum of the fluorescence was
measured at different excitation wavelengths for one-,
two- and three-photon excitation. For p-QT the same
emission spectrum was observed for three-photon excitation
at 850 nm, two-photon excitation at 586 nm and onephoton excitation at 283 nm. The peak emission is at
around 360 nm. All numerical results presented in this
paper will be calculated for these wavelengths.
In previous publications on the theory of multiphoton
fluorescence microscopes numerical results were presented
in terms of normalised optical co-ordinates. In two papers in
particular (Sheppard, 1996; Gu & Gan, 1996) results of
similar theories may appear to be contradictory purely
because of the choice of normalisation wavelength
employed in the two cases. To avoid such confusion in
this paper and because it is not possible to express the
results of high aperture theory in universally applicable
normalised optical co-ordinates, results will be presented for
specific examples in terms of absolute dimensions. However,
pinhole radii will also be expressed in terms of Airy units
(A.u.) where 1 A.u. is the radial distance of the first zero of
the Airy distribution in the detector plane of the 1·4 NA,
100 × microscope objective lens used in the numerical
simulations (1 A.u. ¼ 19·6 mm).
ð31Þ
Equation (30) is used to compute numerical results with the
appropriate substitution for each case.
4. Numerical results
The theory presented in the preceding section is generally
applicable to any multiphoton fluorescence microscope.
However, certain parameters must be chosen before
numerical results can be presented. Firstly, it is well known
4.1. Lateral and axial resolution
We will begin by considering the two limiting cases of linear
and circular illumination. It will be useful to clarify the
physical meaning of the three polarisation dependent
fluorescent emissions (Cases A, B and C) described in
Section 2. Cases A and B, the two limiting models of
unpolarised emission, describe much the same process
whatever the polarisation state of the illumination. In Case
A the circularly polarised illumination is absorbed and a
Fig. 3. Image of a fluorescent molecule illuminated with circularly polarised light with unpolarised emission for 1-p, 2-p and 3-p for pinhole
radii of 1 and 100 mm. The image size is 0·8 mm × 0·8 mm.
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134
P. D. H I GDO N E T A L .
randomly orientated dipole emits. In Case B the dipole
moment of emission is fixed in a random direction and only
the component of the illumination parallel to the dipole
moment is absorbed. In Cases A and B averaging over all
dipole directions leaves the fluorescent emission completely
unpolarised. In Case C all of the incident illumination is
absorbed and the dipole moment of the emission is aligned
with the electric vector of the illuminating field. This
statement is equally true for circularly as well as linearly
polarised illumination.
In Fig. 3 the image of a fluorescent molecule scanned in
the focal plane for the specific example of unpolarised
emission (Case A) in a microscope employing a 100×,
1·4 NA oil immersion (noil ¼ 1·518) objective lens for 1-p, 2-p
and 3-p processes is presented. The left-hand images were
computed for a 100 mm (5·1 A.u.) radius pinhole, which
will later be shown to be equivalent to conventional
imaging, whilst the right-hand images were calculated for
a microscope employing a confocal pinhole of 1 mm
(0·051 A.u.) radius. The R ¼ 100 mm plots give an idea of
the relative image size for the first three multiphoton
processes in a conventional scanning fluorescence microscope imaging a single molecule. It is seen that the image of
the single molecule increases dramatically in size, and
hence the lateral resolution worsens, as the order of the
multiphoton process increases. Such a worsening in lateral
resolution, caused by the longer excitation wavelength
required with higher order processes as illustrated in Fig. 3,
is a high price to pay for the advantages that are inherent
with the higher order (2-p and 3-p) processes. However,
when the 1 mm pinhole is employed, and hence the imaging
is confocal, the process order has a much smaller effect on
the lateral resolution. The image size still increases with the
number of photons but by a significantly smaller amount.
This is an important result which suggests that, although a
confocal pinhole is not required to achieve optical sectioning
in 2-p and 3-p fluorescence imaging, considerably higher resolution imaging will result if a confocal pinhole is employed.
Each of the images in Fig. 3 is circularly symmetric. This
is due to the fact that the circularly polarised illumination
results in a circularly symmetric electric field in the focal
region when focused by the high aperture lens and, in Case
A, the emission is unpolarised. Cases B and C also result in
circularly symmetric images: Case C because the emission,
as well as the illumination, is circularly polarised, and Case
B for exactly the same reasons as Case A. To show how the
limiting cases of the fluorescent emission affect the
resolution properties of the microscope Fig. 4 is presented.
In this figure the lateral and axial half width half-maximum
(HWHM) curves are plotted as a function of pinhole radius
for 1-p, 2-p and 3-p and for Cases A, B and C. Figure 4
shows the transition between the two limiting cases of
confocal and conventional imaging, which were illustrated
in Fig. 3, as the pinhole size is increased. The improvement
in lateral resolution as the pinhole size is reduced is clearly
seen to increase with the process order for Cases A, B and C.
In fact, the dependence of the lateral resolution on the
pinhole radius is similar for Cases A, B and C. So when
circularly polarised illumination is employed the state of
polarisation of the fluorescent emission does not greatly
affect the imaging properties of the microscope. The most
significant resolution difference (of less than 7%) is seen for
the 1-p process and is between Case C (polarised emission)
and Case A (unpolarised with all illumination absorbed).
Case C is seen always to exhibit the best lateral resolution
and Case A the worst. Hence, the resolution of the
Fig. 4. Lateral and axial half width half-maximum (HWHM) curves
for 1-p, 2-p and 3-p as a function of pinhole radius for Cases A, B
and C with circularly polarised illumination. 100 ×, 1·4 NA.
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microscope always falls between these two limits and is
determined by the state of polarisation, which depends on
many parameters as discussed in the introduction to this
paper. In general, the higher the state of polarisation of the
fluorescent emission, the better the resolution.
Figure 4 also shows that, for lateral resolution, pinhole
radii greater than 30 mm (1·53 A.u.) correspond to conventional imaging whereas pinhole radii smaller than
10 mm (0·51 A.u.) give confocal imaging of each of the
multiphoton processes. It should be noted that these values
are applicable only to the current numerical parameters.
Although results have also been given for the reader’s
information in normalised units for the current numerical
parameters, it is not possible to make general statements
that are independent of lens aperture or the wavelengths of
illumination or emission when considering fluorescence
processes in high aperture systems.
A measure of the axial resolution of the multiphoton
fluorescence microscope imaging a single fluorescent
molecule is also given in Fig. 4 in terms of the axial
HWHM as it is scanned along the optical axis. It is seen that
the polarisation state of the emission has little influence on
the axial HWHM and that reducing the size of the pinhole
reduces the axial HWHM. This improvement in the axial
resolution due to the introduction of a confocal pinhole is
seen to increase with process order (as with lateral
resolution). For axial resolution confocal imaging is
obtained for pinhole radii less than 20 mm (1·2 A.u.) and
for conventional imaging greater than 60 mm (3·05 A.u.).
Although it is known that two- and three-photon
fluorescence imaging exhibits optical sectioning without
the need to use a confocal point detector it is important,
particularly in light of the results presented in Fig. 3, to
determine whether any advantage results from the use of
confocal detection. To investigate this we introduce the
integrated intensity, Iint, as:
… ∞ … 2p
Iðrs ; fs ; zs Þrs drs dfs ;
ð32Þ
Iint ðzs Þ ¼
0
0
which is essentially the total energy in the image as a
function of defocus. In the case of conventional singlephoton fluorescence imaging it is well known that this
function is constant, whereas for a confocal geometry it
decays, i.e. the system exhibits optical sectioning. This is
illustrated in Fig. 5 for Case A and circularly polarised
illumination. Similar trends are to be expected for Cases B
and C. The 2-p and 3-p cases illustrate the optical sectioning
capability of a conventional multiphoton fluorescence
microscope. However, as also shown in this figure, the
introduction of a confocal pinhole leads to a further
improvement in the optical sectioning for both 2-p and 3-p
fluorescence scanning microscopes with a decrease of the
integrated intensity HWHM of ,30%. This result, together
with those of Fig. 3, suggest that considerable advantage is
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135
Fig. 5. Normalised integrated intensity as a function of axial position for 1-p, 2-p and 3-p for conventional (dashed curves) and confocal (full curves) detection. 100 ×, 1·4 NA.
likely to result from implementing confocal rather than
conventional multiphoton fluorescence imaging.
4.2. Image asymmetry
It has been shown previously (Wilson et al. 1997; Török
et al., 1998b) that when a point scatterer is imaged by a
linearly polarised incident light, the lateral asymmetry in the
electric field in the focal region causes a laterally asymmetric
image of the scatterer. This effect is now investigated for
multiphoton fluorescence imaging as follows. A measure of
the image asymmetry is introduced as the ratio of the
HWHM of a scan along the x direction to the HWHM of a
scan along the y direction. For circularly polarised illumination (d ¼ 908) the asymmetry is zero and so the ratio of the
HWHMs in the two orthogonal directions is unity. If the
136
P. D. H I GDO N E T A L .
function of pinhole radius and it is seen that, in general, the
asymmetry is smaller when the pinhole radius is reduced.
The images in Fig. 7 were plotted for the case of unpolarised
fluorescent emission (Case A) and hence the asymmetry is a
‘mapping’ of the asymmetry in the illuminating electric
field. When the fluorescent emission dipoles are randomly
fixed (Case B) for the emission is polarised (Case C) the
asymmetry becomes a more complicated combination of
multiple asymmetries in the high aperture system.
4.3. Paraxial approximation
Fig. 6. Asymmetry in the image of a fluorescent molecule for 1-p,
2-p and 3-p excitation as a function of the relative retardance setting of the Babinet–Soleil compensator for R ¼ 100 mm for Case A.
Babinet–Soleil is set at any relative retardance other than
908 (08 < d < 1808) then the electric field in the focal region
becomes asymmetric. In general the incident polarisation is
elliptical, and for the special cases of d ¼ 08 and d ¼ 1808 the
incident illumination is linearly polarised in the x and y
directions, respectively. Figure 6 shows the asymmetry ratio
as a function of relative retardation setting d for the case of
polarised 1-p, 2-p and 3-p fluorescent emission detected with
a 100 mm pinhole. There is a smooth transition in the
symmetry ratio with maxima and minima corresponding to
the two linear illumination cases as expected.
Figure 7 shows the equivalent images to those presented
in Fig. 3 but for linearly polarised illumination (d ¼ 0), i.e.
Case A, 100 × 1·4 NA lens, 1-p, 2-p and 3-p, for R ¼ 100 mm
and R ¼ 1 mm and field of view 0·8 mm × 8 mm. These lateral
images are asymmetric as expected as a consequence of
employing a high aperture microscope objective lens and
linear polarisation. Although the form of the image is more
complicated for linearly polarised light, the general dependences of image size on process order and pinhole size are
the same as for circular polarisation, i.e. the improvement in
lateral resolution due to the confocal pinhole increases with
the order of the multiphoton fluorescence process. The
symmetry properties in Fig. 7 are seen to vary from image to
image, with process order and pinhole radius. As the image
asymmetry is of practical importance, Fig. 8 is presented.
This figure shows the variation in image asymmetry as a
All of the asymmetry effects are due to the combination of
high aperture objective lens and the linearly polarised
illumination used for the preceding numerical results.
Figure 9 shows axial and lateral HWHM curves calculated
for a low aperture, 10 ×, 0·15 NA, lens. Again the dashed
and full lines in the lateral plot correspond to orthogonal
scan directions. Each of the Cases A, B and C are plotted in
this figure and, although causing a slight widening of some
of the curves, they cannot be distinguished from one
another, indicating that the state of polarisation does not
affect the axial or lateral resolution in low aperture systems.
The asymmetry in the image is also seen to be negligible for
each case. It should also be noted that when plotted for
circularly polarised illumination almost identical results to
those in Fig. 9 were obtained. These results may also be
obtained from the, less complicated, paraxial theory for
multiphoton fluorescence as the polarisation state of the
emission and high aperture effects may be ignored. However,
in practice, such low numerical aperture lenses are seldom
used. To determine the deviation of the results of the paraxial
theory from those of the high aperture, polarised emission
theory Fig. 10 is presented. Figure 10(a) shows the
percentage image asymmetry as a function of lens magnification for a conventional 3-p fluorescence microscope with
unpolarised emission. It can be seen that there is a 10%
difference in the results of the paraxial and high aperture
theories for a 40×, 0·6 NA lens increasing to about a 50%
difference for a 100×, 1·4 NA lens. Figure 10(b) shows the
percentage difference in lateral resolution for emission
processes that are polarised and unpolarised. In this case
there is a 10% difference between the two theories for a 60×,
0·8 NA lens increasing to a difference of about 15% for a
100×, 1·4 NA lens lens. Hence for lenses of less than 40×,
0·6 NA magnification the paraxial theory may be employed
at the expense of a maximum 10% error. For lenses
of greater magnification, high aperture effects become
important. However, it is not until the lens magnification
Fig. 7. Image of a fluorescent molecule illuminated with linearly polarised light (in the direction indicated) with unpolarised emission (Case
A) for 1-p, 2-p and 3-p for pinhole radii of 1 and 100 mm. The image size is 0·8 mm × 0·8mm.
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138
P. D. H I GDO N E T A L .
Fig. 8. Asymmetry in the image of a fluorescent molecule for 1-p,
2-p and 3-p excitation as a function of pinhole size for linearly
polarised illumination for Case A.
exceeds 60×, 0·8 NA that the polarisation of the fluorescence emission becomes an important factor. Alternatively,
a lens of magnification greater than 60×, 0·8 NA should be
employed to derive information regarding the partial
polarisation of the fluorescence emission.
The percentage improvement in axial and lateral resolution for an unpolarised emission (Case A) of a fluorescent
molecule illuminated with circular polarisation between the
conventional and confocal imaging limits is given in Table 1
for 1-p, 2-p and 3-p.
The axial resolution improvement is approximately the
same for the two lenses. However, there is seen to be a
significantly greater improvement in the lateral resolution
due to a confocal system for the 10×, 0·15 NA lens than for
the 100×, 1·4 NA lens. The results for the 10×, 0·15 NA
lens were very close to those of the paraxial theory. Since the
paraxial theory may be written in normalised co-ordinates
its results are simply scaled according to the lens. Therefore,
the percentage difference in lateral resolution results from
the paraxial theory are the same for any lens. Considering
the data in Table 1 this means that, for the high aperture
case, the paraxial theory overestimates the increase in
lateral resolution due to a confocal pinhole by approximately 100% for 1-p, 2-p and 3-p processes. By the same
considerations the paraxial theory estimates the axial
resolution for the high aperture lens very accurately for
the 1-p, 2-p and 3-p processes.
4.4. Signal-to-noise ratio
In the previous subsections it has been shown that
employing a confocal pinhole in 2-p and 3-p fluorescence
Fig. 9. Lateral and axial HWHM curves for 1-p, 2-p and 3-p as a
function of pinhole radius for Cases A, B and C with linearly
polarised illumination. The dashed curves indicate scans in the
direction perpendicular and the full curves parallel to the direction
of incident polarisation. 10×, 0·15 NA.
microscopes leads to great improvements in the lateral and
axial resolution. However, it is important to consider that in
such microscopes the light budget can be limited. It is
therefore the aim in this section to determine an optimum
pinhole radius in terms of maximising the signal-to-noise
ratio of the microscope.
The approach is similar to that employed by Wilson et al.
(1998) and Sandison et al. (1995) but employing the high
aperture theory. That is, shot noise is assumed to be
proportional to the area of the detector, of radius R, and a
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139
Fig. 11. Signal-to-noise ratio as a function of pinhole radius for 3-p
fluorescence imaging and for Case A. 100×, 1·4 NA.
Fig. 10. (a) Percentage image asymmetry for Case A with 3-p and
R ¼ 100 mm as a function of numerical aperture (assumed to be
linearly related to magnification) for an oil immersion lens. (b) Percentage difference in lateral resolution for polarised (Case C) and
unpolarised (Case A) fluorescent emission.
measure of the signal-to-noise ratio is given by:
S
IðRÞ
ðRÞ ¼ p ;
N
IðRÞ þ yR2
ð33Þ
pinhole radius is given in Fig. 11. These curves are plotted
for 3-p fluorescence on a log–log scale. Figure 12 shows the
optimum pinhole radius plotted as a function of the noise
level (y) for the two limiting cases of polarised (Case C) and
unpolarised (Case A) emission for 1-p, 2-p and 3-p
fluorescence in the high aperture system. In the high noise
limit it is seen that the optimum pinhole radius varies
between approximately 14 mm and 16 mm (0·71–0·81 A.u.)
depending of the state of polarisation and is independent of
the order of the multiphoton process. In the low noise limit
the optimum pinhole radius varies between approximately
17 mm and 25 mm (0·86–1·27 A.u.) for unpolarised and
polarised emission, respectively. In this case there is seen to
be a small variation in the curves corresponding to the 1-p,
2-p and 3-p processes but in practice these differences may
be ignored. These optimum pinhole radii correspond to
imaging in the region between the conventional and
where the constant y represents the noise level. A typical
example of how the signal-to-noise ratio changes with
Table 1. Percentage increase in resolution due to a confocal pinhole over conventional imaging for 1-, 2- and 3-p fluorescence
for low and high aperture lenses.
Resolution improvement (%)
Lens
10×, 0·15
100×, 1·4
Lateral
Axial
Lateral
Axial
1-p
2-p
3-p
25
25
12
24
48
49
24
48
69
70
35
68
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Fig. 12. Pinhole radius corresponding to maximum signal-to-noise
ratio as a function of noise level for fluorescence imaging for 1-p,
2-p and 3-p excitation and for Cases A and C. 100×, 1·4 NA.
140
P. D. H I GDO N E T A L .
Fig. 13. Signal-to-noise ratio for a confocal pinhole (R ¼ 10 mm) as
a percentage of the maximum signal-to-noise ratio. Plotted as a
function of noise level for fluorescence imaging for 1-p, 2-p and
3-p excitation and for Cases A and C. 100×, 1·4 NA.
confocal limits, i.e. employing the optimum pinhole, already
leads to a certain improvement in the axial resolution over
the conventional imaging case. However, to obtain the
maximum possible improvement in resolution an even
smaller pinhole should be employed.
Figure 13 is plotted to determine the cost in terms of the
signal-to-noise ratio of employing a pinhole which leads to
true confocal imaging. For these calculations a pinhole of
radius 10 mm (0·051 A.u.) is used. Such a pinhole would
lead to resolution properties only slightly worse than an
ideal confocal microscope (R → 0). Figure 13 shows the
signal to noise ratio with the 10 mm pinhole as a percentage
of the signal-to-noise ratio obtained by employing the
optimum pinhole radius as a function of the noise level. It is
seen that in the high noise limit the signal-to-noise ratio is
reduced from its maximum value only by 8–12% depending
on the state of polarisation of the fluorescent emission. In
the low noise case the decrease in signal-to-noise ratio is
more significant at 17–32% of its maximum value. Again,
the extent of the reduction in signal-to-noise ratio is due
mainly to the state of polarisation rather than the order of
the multiphoton process. However, it should be noted that
3-p fluorescence, which benefits the most in terms of
resolution from confocal imaging, actually suffers the least
in terms of loss in signal-to-noise ratio. Ultimately, even the
largest loss in the signal-to-noise ratio mentioned above is a
relatively small price to pay for the vast improvement in the
lateral resolution of the 2-p and 3-p microscopes achieved
by employing the confocal pinhole.
5. Conclusion
In this paper a theory for 1-p, 2-p and 3-p multiphoton
polarisation dependent fluorescence processes has been
presented. From analysis of the numerical results the
following conclusions are drawn.
In conventional imaging the lateral resolution is reduced
significantly as the order of the multiphoton process
increases. However, the improvement in lateral resolution
due to a confocal pinhole increases significantly with
increasing process order, i.e. for confocal imaging the
lateral resolution is not degraded significantly as the process
order increases. When employing circularly polarised
illumination, although the state of polarisation of the
fluorescent emission does not affect the lateral resolution
by more than 7% even for the 100×, 1·4 NA lens
considered, polarised emission always produces slightly
better resolution than unpolarised. The improvement in
axial sectioning due to a confocal pinhole increases
significantly with increasing process order, suggesting that
confocal detection should be used whenever possible in
multiphoton imaging.
When employing linearly polarised illumination the
degree of image asymmetry depends strongly on the state
of polarisation of the emission. Unpolarised emission leads
to greater asymmetry than polarised emission. This image
asymmetry is greatly reduced by the introduction of a
confocal pinhole and this reduction in asymmetry increases
with the multiphoton process order.
For the high aperture system considered with low noise
the optimum pinhole radius varies between 15 mm and
25 mm (0·75–1·28 A.u.) depending on the state of polarisation. For high noise the optimum pinhole radius is
independent of the state of polarisation at 15 mm (0·75
A.u.). The order of the multiphoton process has very little
effect on the optimum pinhole size. To achieve true confocal
imaging the signal-to-noise ratio is reduced by between 8
and 30% from its maximum value depending on the noise
level and polarisation of emission.
Acknowledgement
Peter Török is an EPSRC Advanced Fellow at the University
of Oxford.
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