Download Calculation of vectorial three-dimensional transfer

A. Schönle and S. W. Hell
Vol. 19, No. 10 / October 2002 / J. Opt. Soc. Am. A
2121
Calculation of vectorial three-dimensional transfer
functions in large-angle focusing systems
Andreas Schönle and Stefan W. Hell
High Resolution Optical Microscopy Group, Max-Planck-Institute for Biophysical Chemistry,
D-37070 Göttingen, Germany
Received September 27, 2001; revised manuscript received May 3, 2002; accepted May 7, 2002
The optical transfer function (OTF) is used in describing imaging systems in the Fourier domain. So far the
calculation of the OTF of a large-aperture imaging system has been difficult because the vectorial nature of
light breaks the cylindrical symmetry of the pupil function. We derive a simple line integral solution for calculating the vectorial three-dimensional OTF. We further extend this approach to imaging through a planar
interface of two media with mismatched refractive indices. In general, our formalism allows for calculation of
the Fourier transform of any product of two arbitrary vector components of the electromagnetic field. Arbitrary phase or amplitude modifications of the pupil function can be taken into account. © 2002 Optical Society of America
OCIS codes: 110.4850, 180.6900.
1. INTRODUCTION
The imaging properties of large-angle focusing systems
such as objective lenses or high-angle mirrors are of great
scientific interest. They are usually described by the
spatial distribution of quantities that depend on products
of the vector components of the electromagnetic field.
For example, the modulus squared of the electric field is
proportional to the focal energy density, which in turn determines the intensity point-spread function (intensity
PSF) for incoherent imaging.
Three-dimensional image formation is often analyzed
in the Fourier domain by using the concept of transfer
functions.1,2
For incoherent imaging the threedimensional (3D) optical transfer function (OTF) is the
Fourier transform of the intensity PSF. It was introduced by Frieden,3 and its paraxial approximation was
derived in the same publication. However, for lenses of
high numerical aperture this approximation is not fully
valid, and the computation of the OTF for conventional,
confocal, and multiphoton microscopy has since been an
active field of research. The derivation of an analytical
expression allowing for fast and efficient calculation of
the scalar OTF has proved especially useful in numerical
calculations.4,5
At large focusing angles the vectorial nature of the electromagnetic field becomes increasingly significant and the
accuracy of the scalar description is compromised.6–8 In
addition, many applications involve imaging of anisotropic samples such as a distribution of oriented dipoles
in single-molecule experiments. The scalar theory cannot be applied in such cases, and an efficient way to calculate the vectorial OTF becomes necessary.
Richards and Wolf gave an integral representation for
the electromagnetic field in the focus.9 In principle, one
could use these results to determine numerically the spatial distribution of the PSF or of any other quantity depending on the fields and subsequently Fourier transform
the results. However the computation in two steps is ex1084-7529/2002/102121-06$15.00
tensive, and controlling the numerical error is difficult.
If the transform is not required on an equidistant grid,
this method is particularly inadequate, because fast Fourier transform algorithms can not be used efficiently.
Alternatively, the OTF can be expressed as the correlation or convolution of the vector components of the pupil
function. In two dimensions numerical correlation yields
the two-dimensional OTF on a discrete grid,10 but in
three dimensions this approach fails because the pupil
functions are nonzero on spherical shells only. However,
the correlation can be written as a single line integral
along the circular intersection of the two spheres.
Frieden used this method to derive the OTF in the
paraxial approximation, where the spheres are replaced
by parabolic surfaces.3 For high-angle focusing systems
this approach was used successfully to derive an analytical expression of the scalar OTF4 under the assumption of
cylindrical symmetry. However, the transverse nature of
light breaks this symmetry so that a more general formalism is needed in the vectorial theory. Here we derive a
theory that allows for arbitrary pupil functions and use
this theory to give a line integral representation for the
Fourier transform of an arbitrary product of two vector
components of the electromagnetic field and its complex
conjugate. We demonstrate how to use these results to
calculate the vectorial OTF.
Owing to the general form of the pupil function, it is
possible to extend our theory to describe spherical aberrations induced by focusing through an interface of two
media with mismatched refractive indices. The results
are again applied to the calculation of the vectorial OTF.
2. GENERAL FORMALISM
Let (r, ␪ , ␾ ) be dimensionless spherical coordinates
originating at the focal point and normalized with the
wave vector 2␲/␭. The spherical inverse coordinates are
(k, ␽ , ␸ ), and ␣ is the half-aperture angle of the lens.
© 2002 Optical Society of America
2122
J. Opt. Soc. Am. A / Vol. 19, No. 10 / October 2002
A. Schönle and S. W. Hell
Using the same assumptions as Richards and Wolf,9 we
write each component of the focal field as the Fourier
transform of the related component of the vectorial 3D
pupil function:
f 共 r, ␪ , ␾ 兲 ⫽
iA
␲
F 3D关 P 共 ␽ , ␸ 兲 a f 共 ␽ , ␸ 兲 ␦ 共 k ⫺ 1 兲兴 . (1)
The scalar pupil function P( ␽ , ␸ ) describes a relative amplitude and phase common to all components of the electromagnetic fields throughout the spherical exit pupil and
vanishes for ␽ ⬎ ␣ . A is a scaling factor, and the functions a f ( ␽ , ␸ ) are scalar strength factors. For the individual vector components of the electric and magnetic
fields they are given by9
a e x ⫽ ⫺cos ␽ cos2 ␸ ⫺ sin2 ␸ ,
a e y ⫽ cos ␸ sin ␸ 共 1 ⫺ cos ␽ 兲 ,
a e z ⫽ sin ␽ cos ␸ ,
(2)
and
a h x ⫽ cos ␸ sin ␸ 共 1 ⫺ cos ␽ 兲 ,
a h y ⫽ ⫺sin2 ␸ cos ␽ ⫺ cos2 ␸ ,
a h z ⫽ sin ␽ sin ␸ .
(3)
In the case of mismatched refractive indices these
strength factors will have to be modified owing to refraction at the interface. Defining
A f 共 k兲 ⫽
iA
␲
P 共 ␲ ⫺ ␽ , ␲ ⫹ ␸ 兲 a f 共 ␲ ⫺ ␽ , ␲ ⫹ ␸ 兲 , (4)
we rewrite Eq. (1) as an inverse Fourier transform:
⫺1
f ⫽ 共 2 ␲ 兲 3 F 3D
关 A f ␦ 共 k ⫺ 1 兲兴 .
(5)
The Fourier transform of a product of a field component
and the complex conjugate of another component are encountered, for example, in calculating the incoherent
OTF. Using the convolution theorem and Eq. (5) we can
now write such a Fourier transform as the correlation of
two weighted spherical shells:
F 3D关 f * g 兴共 k兲 ⫽ 共 2 ␲ 兲 3
冕
d3 k ⬘ A f* 共 k⬘ 兲 A g 共 k ⫹ k⬘ 兲 ␦ 共 k ⬘ ⫺ 1 兲
Fig. 1. 3D autocorrelation of a spherical shell. The figure illustrates the coordinate systems and the angles of the rotation matrix used in the text. The center of the first sphere is located at
the origin of the primed coordinate system, and the second center
is at ⫺k. Their circular intersection is shown as a white line.
The new, double-primed coordinate system is chosen such that
the centers of both spheres are on the new s ⬙ axis. This is accomplished by rotating the coordinate system first by an angle ␸
around the s ⬘ axis and then by an angle ␽ around the n ⬘ axis as
illustrated in (a). The intersection is now in the plane of constant s ⬙ ⫽ ⫺k/2 and centered about the s ⬙ axis as shown in (b).
F 3D关 f * g 兴共 k兲
⫽ 共 2␲ 兲3
冕
dk ⬙ k ⬙ 2 d cos ␽ ⬙ d␸ ⬙ A f* 共 k1 兲 A g 共 k2 兲
(6)
⫻ ␦ 共 k ⬙ ⫺ 1 兲 ␦ (冑共 k ⫹ k ⬙ cos ␽ ⬙ 兲 2 ⫹ 共 k ⬙ sin ␽ ⬙ 兲 2 ⫺ 1),
Next we introduce Cartesian coordinates in inverse space
(m, n, s) and a new coordinate system denoted by
double-prime coordinates as shown in Fig. 1. The transformation is described by an orthogonal matrix:
(8)
⫻ ␦ 共 兩 k ⫹ k⬘ 兩 ⫺ 1 兲 .
k⬘ ⫽
冋
cos ␽ cos ␸
⫺sin ␸
cos ␽ sin ␸
cos ␸
⫺sin ␽
0
sin ␽ cos ␸
册
sin ␽ sin ␸ k⬙ .
cos ␽
With k1 ⫽ k⬘ and k2 ⫽ k ⫹ k⬘ , Eq. (5) reads
and after radial integration and simplification of the delta
function’s argument we obtain
F 3D关 f * g 兴共 k兲 ⫽ 共 2 ␲ 兲 3
(7)
冕
d cos ␽ ⬙ d␸ ⬙ A f* 共 k1 兲 A g 共 k2 兲
⫻ ␦ 共 冑k 2 ⫹ 2k cos ␽ ⫹ 1 ⫺ 1 兲 .
Azimuthal integration yields
(9)
A. Schönle and S. W. Hell
Vol. 19, No. 10 / October 2002 / J. Opt. Soc. Am. A
F 3D关 f * g 兴共 k兲
⫽ 共2␲兲3
1
k
冕
␲
冏
m 1/2 ⫽ 共 b ⫾ a 兲 cos ␸ ⫺ d sin ␸ ,
d␸ ⬙ A f* 共 k1 兲 A g 共 k2 兲
⫺␲
2123
n 1/2 ⫽ 共 b ⫾ a 兲 sin ␸ ⫹ d cos ␸ ,
, (10)
s 1/2 ⫽ ⫾r 0 sin ␽ cos ␸ ⬙ ⫹ 共 k cos ␽ 兲 /2.
cos ␽ ⬙ ⫽⫺k/2,k ⬙ ⫽1
which indeed is a line integral along a circle of radius r 0
⫽ (1⫺k 2 /4) 1/2.
If we define a ⫽ r 0 cos ␽ cos ␸⬙, b
⫽ (k sin ␽)/2 and d ⫽ r 0 sin ␸⬙, we can use Eq. (7) to calculate the Cartesian components of k1 and k2 :
(16)
The integration range is now defined by ⫺s 1/2 ⭓ cos ␣, resulting in
␤ 2 ⭐ 兩 ␸ ⬙兩 ⭐ ␲ ⫺ ␤ 2 ,
m 1/2 ⫽ 共 a ⫿ b 兲 cos ␸ ⫺ d sin ␸ ,
␤ 2 ⫽ arccos关共 k cos ␽ ⫺ 2 cos ␣ 兲 / 共 2r 0 sin ␽ 兲兴 .
n 1/2 ⫽ 共 a ⫿ b 兲 sin ␸ ⫹ d cos ␸ ,
If the argument of the inverse cosine is larger than 1 or
smaller than zero, we have ␤ 2 ⫽ 0 and ␤ 2 ⫽ ␲ /2, respectively.
s 1/2 ⫽ ⫺r 0 sin ␽ cos ␸ ⬙ ⫿ 共 k cos ␽ 兲 /2.
(11)
Note that the functions A f depend merely on the sine and
cosine functions of the spherical angles. It is straightforward to express them as functions of the Cartesian components in Eqs. (11) by using the following relations:
2 ⫺1/2
,
sin共 ␲ ⫹ ␸ 1/2兲 ⫽ ⫺n 1/2共 1 ⫺ s 1/2
兲
cos共 ␲ ⫺ ␽ 1/2兲 ⫽ ⫺s 1/2 ,
(12)
For numerical calculations it is necessary to determine
the integration range in Eq. (10). We defined the pupil
functions to be nonzero only for ␽ ⭐ ␣ . Thus we have
cos ␽1/2 ⫽ ⫺s 1/2 ⭓ cos ␣, yielding
0 ⭐ 兩 ␸ ⬙兩 ⭐ ␤ 1 ,
␤ 1 ⫽ arccos关共 2 cos ␣ ⫹ k 兩 cos ␽ 兩 兲 / 共 2r 0 sin ␽ 兲兴
(13)
and ␤ 1 ⫽ 0 if the argument of the inverse cosine is larger
than unity. Now it is possible to numerically solve the integral in Eq. (10). However, it will be advantageous to
analytically extract the dependence on the polar angle
whenever possible.
So far, we have calculated the Fourier transform of a
product of the form f * g, which is needed for calculating
the OTF of a lens or the weak object transfer function of a
partially coherent transmission system.11,12 When we
describe a confocal microscope in reflection mode, the geometry is changed and the calculation of the coherent
transfer function will require the Fourier transform of the
product fg.13 In this case we have to substitute the correlation of Eq. (6) by a convolution:
F 3D关 fg 兴共 k兲 ⫽ 共 2 ␲ 兲
3
冕
d k ⬘ A f 共 k⬘ 兲 A g 共 k ⫺ k⬘ 兲 ␦ 共 k ⬘ ⫺ 1 兲
⫻ ␦ 共 兩 k ⫺ k⬘ 兩 ⫺ 1 兲 .
1
k
冕
⫺␲
(14)
冏
d␸ ⬙ A f 共 k1 兲 A g 共 k2 兲
⫽ Cx ⫹ Cy ⫹ Cz .
C x ⫽ 16␲
C y ⫽ 16␲
C z ⫽ 16␲
k
A2
k
A2
k
共 I 0 ⫹ I 1 cos 2 ␸ ⫹ I 2 cos 4 ␸ 兲 ,
共 I 3 ⫺ I 2 cos 4 ␸ 兲 ,
共 I 4 ⫹ I 5 cos 2 ␸ 兲 ,
(19)
冕
␤1
P *共 ␲ ⫺ ␽ 1 兲 P 共 ␲ ⫺ ␽ 2 兲 J id ␸ ⬙.
(20)
If we define the variables S 1/2 ⫽ s 2/1 /(1 ⫺ s 1/2) and S 3
⫽ (1 ⫺ s 1 ) ⫺1 (1 ⫺ s 2 ) ⫺1 , the integrands evaluate to
J0 ⫽
J1 ⫽
(15)
Using Eq. (7), we find the Cartesian components of the
new vectors k1 and k2 to be
A2
with
.
cos ␽ ⬙ ⫽k/2,k ⬙ ⫽1
(18)
We restrict ourselves to cylindrically symmetric scalar pupil functions and use Eqs. (4), (11), and (12) to rewrite the
integral in Eq. (10). Owing to the symmetric integration
range, we can remove all asymmetric terms from the integrand and restrict the integration range to positive ␸ ⬙ .
In a second step, one can use simple trigonometry to rewrite the angular dependence in terms of cos 2␸ and
cos 4␸, yielding
0
We redefine the vectors k1 ⫽ k⬘ and k2 ⫽ k ⫺ k⬘ and use
the same coordinate transformation as above to obtain
F 3D关 fg 兴共 k兲 ⫽ 共 2 ␲ 兲 3
C ⫽ F 3D关 e x e x* 兴 ⫹ F 3D关 e y e y* 兴 ⫹ F 3D关 e z e z* 兴
Ii ⫽
3
␲
3. VECTORIAL OPTICAL TRANSFER
FUNCTION
In fluorescence imaging the PSF is proportional to the focal intensity distribution and therefore the OTF is given
by the Fourier transform of the electric energy density.
It can be written as the sum of its vector components,
which correspond to the contribution of the three polarization directions:
2 ⫺1/2
cos共 ␲ ⫹ ␸ 1/2兲 ⫽ ⫺m 1/2共 1 ⫺ s 1/2
,
兲
2 1/2
sin共 ␲ ⫺ ␽ 1/2兲 ⫽ 共 1 ⫺ s 1/2
兲 .
(17)
J2 ⫽
S3
2
S3
2
S3
2
共 3J 42 ⫹ b 2 d 2 兲 ⫺ L ⫹ s 1 s 2 ,
关 d 4 ⫺ 共 a 2 ⫺ b 2 兲 2 兴 ⫺ 共 S 1 ⫹ S 2 兲 d 2 ⫹ L,
共 J 52 ⫺ a 2 d 2 兲 ,
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J. Opt. Soc. Am. A / Vol. 19, No. 10 / October 2002
J3 ⫽
J4 ⫽
J5 ⫽
L⫽
S3
2
1
2
1
2
A. Schönle and S. W. Hell
共 J 42 ⫺ b 2 d 2 兲 ,
共 a2 ⫺ b2 ⫹ d2兲,
共 a2 ⫺ b2 ⫺ d2兲,
S1
2
关共 a ⫺ b 兲 2 ⫹ d 2 兴 ⫹
S2
2
关共 a ⫹ b 兲 2 ⫹ d 2 兴 .
(21)
For asymmetric pupil functions, extraction of the OTF’s
polar dependence might still be possible, but Eqs. (19)
might take a more general form and the integrals in Eq.
(20) have to be recalculated.
The results of the scalar theory are reproduced if we
use the autocorrelation of A f (k) ⫽ iAP( ␲ ⫺␽ )/ ␲ in Eq.
(10), yielding
C 共 k, ␽ 兲 ⫽ 16␲
A2
k
冕
␤1
0
P * 共 ␲ ⫺ ␽ 1 兲 P 共 ␲ ⫺ ␽ 2 兲 d␸ ⬙ .
(22)
We now change to the usual system of cylindrical optical coordinates, which are defined by ␯ ⫽ sin ␣ r sin ␪, u
⫽ 4 sin2(␣/2)r cos ␪ and the polar angle ␾. When using
the inverse coordinate system
关 l ⫽ k sin ␽ /sin ␣ , s ⫽ k cos ␽ /4 sin2 共 ␣ /2兲 , ␸ 兴 , (23)
we have to scale the result by a factor 4 sin2 ␣ sin2(␣/2)
owing to the coordinate transformation. The result then
satisfies the scaling condition
⫺1
F 3D
关 C 兴共 0兲 ⫽
1
共 2␲ 兲3
冕
d ␸ dldslC 共 l, s, ␸ 兲
⫽ e共 0兲 • e* 共 0兲 .
(24)
In addition, we can choose the scaling factor A in such a
way that the Fourier back transform of the OTF at the
origin, e(0) • e* (0), becomes unity. The choice depends
on the pupil function. For aplanatic, uniform, and parabolic apodization we have P a ( ␽ ) ⫽ (cos ␽)1/2, P u ( ␽ ) ⫽ 1,
and P p ( ␽ ) ⫽ 2/(1 ⫹ cos ␽), respectively.
The corresponding A’s can be determined by evaluating Eq. (1) with
the integrals derived by Richards and Wolf9 and subsequent normalization:
16
A a ⫽ 共 15
⫺
2
3
cos3/2 ␣ ⫺
3
A u ⫽ 共 2 ⫺ cos ␣ ⫺
1
2
A p ⫽ 共 2 ⫺ 2 cos ␣ 兲 ⫺1 .
2
5
cos5/2 ␣ 兲 ⫺1 ,
cos2 ␣ 兲 ⫺1 ,
(25)
After implementing our integrals we checked their validity by Fourier transforming the components of the OTF
and comparing the result with the vectorial PSF. For
randomly polarized illumination, Eq. (19) must be averaged over the polar angle and only I 0 , I 3 , and I 4 need to
be solved. Additionally, transition moments oriented
along the x and y axes are now imaged identically, and we
are left with an axial and a lateral component of the OTF,
as presented in Fig. 2.
Fig. 2. (a) Lateral and (b) axial components of the vectorial OTF
for randomly polarized illumination with use of a lens with halfaperture angle ␣ ⫽ 1.1. The axial component corresponds to
imaging a molecular transition moment oriented along the optic
axis, and the lateral counterpart applies to an orientation parallel to the focal plane.
4. MISMATCHED REFRACTIVE INDICES
Focusing through an interface of two isotropic, homogeneous media with different refractive indices results in
spherical aberrations. This problem is frequently encountered in microscopy, e.g., when imaging a sample
mounted in glycerol with an oil immersion lens. Several
approaches exist to calculate the resultant electromagnetic field in the focal region.14,15 Starting from these results we shall write the focal field as an inverse Fourier
transform of a modified vectorial pupil function and apply
our formalism.
We assume that the light emerges from the lens as a
perfect spherical wave that initially propagates in medium 1. At a distance ⌬ from the Gaussian focus point, it
is refracted by a planar interface between medium 1 and
medium 2.
Let n 1 and n 2 be the refractive indices of the media and
(r, ␪ , ␾ ) spherical coordinates normalized with the wave
vector in medium 2 and centered at the Gaussian focus
point. We write again (k, ␽ , ␸ ) for the inverse coordi¯ ⫽ arcsin(n1 sin ␽/n2). Török
nates and define the angle ␽
and co-workers extended the theory of Richard and Wolf 9
to represent the field in medium 2 as a superposition of
A. Schönle and S. W. Hell
Vol. 19, No. 10 / October 2002 / J. Opt. Soc. Am. A
plane waves.15 In their solution for the fields, we change
¯ to ␽. The Jacobian of
the integration variable from ␽
this transformation is
¯ 兲 ⫽ n 1 ␥ /n 2 ,
n 1 2 cos ␽ / 共 n 2 2 cos ␽
(26)
where we defined the function ␥. The integrals can now
be identified as Fourier transforms in the form of Eq. (1)
but with modified strength factors and a modified scalar
pupil function. The strength vectors are
a e x ⫽ ⫺␶ p cos ␽ cos2 ␸ ⫺ ␶ s sin2 ␸ ,
a e y ⫽ cos ␸ sin ␸ 共 ␶ s ⫺ ␶ p cos ␽ 兲 ,
(27)
a e z ⫽ sin ␽ cos ␸ ␶ p ,
and
a h x ⫽ cos ␸ sin ␸ 共 ␶ p ⫺ ␶ s cos ␽ 兲 ,
a h y ⫽ ⫺␶ s sin2 ␸ cos ␽ ⫺ ␶ p cos2 ␸ ,
(28)
a h z ⫽ sin ␽ sin ␸ ␶ s ,
for the electric and the magnetic field, respectively. The
Fresnel coefficients are functions of ␽ and given by
2125
¯ /sin共 ␽
¯ ⫹ ␽ 兲 ⫽ 2/共 1 ⫹ ␥ 兲 ,
␶ s ⫽ 2 sin ␽ cos ␽
¯ ⫺ ␽兲
␶ p ⫽ ␶ s /cos共 ␽
⫽ ␶ s n 1 / 关 n 2 ⫹ n 2 cos2 ␽ 共 ␥ ⫺1 ⫺ 1 兲兴 .
(29)
The new pupil function is
¯ , ␸ 兲,
P 共 ␽ , ␸ 兲 ⫽ ␥ exp关 ⫺i⌬ 共 ␥ ⫺1 ⫺ 1 兲兴 P̄ 共 ␽
(30)
¯ , ␸ ) is the pupil function in the absence of rewhere P̄( ␽
fraction. If sin ␣ ⭐ n1 /n2 holds for the half-aperture
angle, the new pupil function is nonzero only for ␽
⭐ arcsin(n2 sin ␣/n1). For larger aperture angles total
internal reflection occurs, and we have ␽ ⭐ ␲ /2.
It is interesting to note that for small angles ␣ we have
␥ ⬵ n 1 /n 2 . The phase is then proportional to the inverse axial coordinate, which is equivalent to a shift of the
electrical field along the optic axis as predicted by geometrical optics.
To calculate the OTF we rewrite the trigonometric
functions of the azimuthal and polar angles in terms of ␸ ⬙
using Eq. (11) and (12) and solve the integral in Eq. (10).
If we assume a cylindrically symmetric pupil function, the
calculations are very similar to those in Section 2. When
the same definitions as in Eq. (18) and (19) are used, the
Fig. 3. Modulus of the vectorial OTF (a) for an unaberrated
system and (b), (c) with focusing into 80% glycerol (n 2
⫽ 1.45) with an oil (n 1 ⫽ 1.52) immersion lens. The Gaussian focus was assumed at (b) 20 ␮m and (c) 50 ␮m from the
interface; the half-aperture angle was ␣ ⫽ 1.1. With increasing focusing depths the OTF is suppressed at high axial
frequencies corresponding to a loss in axial resolution. The
phase for the aberrated case is shown in the insets. Its oscillation along the inverse optic axis corresponds to a shift of
the main maximum of the PSF, and the contortion of the resulting pattern is due to the spherical aberrations.
2126
J. Opt. Soc. Am. A / Vol. 19, No. 10 / October 2002
A. Schönle and S. W. Hell
result is again given by Eq. (20) but with modified integrands J i . If we redefine the S i ,
2
S 1/2 ⫽ ␶ p,2/1 s 2/1共 ␶ s,1/2 ⫹ ␶ p,1/2 s 1/2兲 / 共 1 ⫺ s 1/2
兲,
(31)
S 3 ⫽ S 1S 2 /共 s 1s 2␶ 兲,
where ␶ ⫽ ␶ p,1␶ p,2 , the integrands take almost the same
form as before:
J0 ⫽
J1 ⫽
J2 ⫽
J4 ⫽
L⫽
S3
2
S3
2
S3
2
␶
2
REFERENCES
关 d 4 ⫺ 共 a 2 ⫺ b 2 兲 2 兴 ⫺ 共 S 1 ⫹ S 2 兲 d 2 ⫹ L,
共 a2 ⫺ b2 ⫹ d2兲,
S1
The authors can be reached by fax, 49-551-201-1085, or
e-mail: [email protected] and [email protected].
共 3J 42 / ␶ 2 ⫹ b 2 d 2 兲 ⫺ L ⫹ s 1 s 2 ␶ ,
共 J 52 / ␶ 2 ⫺ a 2 d 2 兲 ,
关共 a ⫺ b 兲 2 ⫹ d 2 兴 ⫹
J3 ⫽
J5 ⫽
S2
␶
2
S3
2
The vectorial description is mandatory in imaging transition dipoles of single dye molecules or anisotropic
samples or when the accuracy of a scalar description is insufficient. Since no restrictions were imposed on the pupil function, our theory can be applied for arbitrary phase
and amplitude pupil filters used for point-spread-function
engineering.
1.
共 J 42 / ␶ 2 ⫺ b 2 d 2 兲 ,
2.
3.
共 a2 ⫺ b2 ⫺ d2兲,
关共 a ⫹ b 兲 2 ⫹ d 2 兴 .
4.
(32)
5.
For identical refractive indices in both media (n 1 /n 2
→ 1), the Fresnel coefficients become unity and Eqs. (21)
are reproduced. We introduce the same inverse optical
coordinates from Eq. (23) and therefore have to multiply
the result by the identical scaling factor 4 sin2 ␣ sin2(␣/2).
The prefactor A is given by Eqs. (25). In Fig. 3 we evaluate the OTF for the common situation of focusing from oil
into glycerol. The results for two different focusing
depths are shown in Figs. 3(b) and 3(c) and compared
with the vectorial OTF without aberrations in Fig. 3(a).
6.
2
2
5. CONCLUSION
We have developed a formalism to calculate the correlation or convolution of two functions defined on spherical
caps without assuming cylindrical symmetry. This allowed us to derive an integral solution for the vectorial
OTF of a high-angle imaging system. We further extended this solution to the case of mismatched refractive
indices. In general, our formalism can be applied to express the Fourier transform of products of two arbitrary
vector components of the electromagnetic field as a single
line integral. Therefore it should prove equally useful for
the description of confocal microscopes or partially coherent imaging systems in terms of vectorial transfer functions.
7.
8.
9.
10.
11.
12.
13.
14.
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