- Email: [email protected]
Electromagnetic imaging in the confocal microscope
1 June 2000
Optics Communications 180 Ž2000. 1–8 www.elsevier.comrlocateroptcom
Electromagnetic imaging in the confocal microscope C.J.R. Sheppard ) , J. Felix Aguilar 1 Physical Optics Department, School of Physics A28, UniÕersity of Sydney, Sydney NSW 2006, Australia Received 28 September 1999; received in revised form 23 March 2000; accepted 29 March 2000
Abstract A simplified theory for imaging in confocal microscope is introduced. This can be used to model images of various structures under different polarisation conditions, and with finite sized detector pinholes. In particular, simple expressions and numerical results are given for the axial resolution for a planar specimen. The effects of polarisation are investigated. q 2000 Published by Elsevier Science B.V. All rights reserved.
1. Introduction In a recent paper w1x Torok ¨ ¨ and Wilson described a rigorous theory for imaging of planar objects in confocal microscopes. They treated rigorously the electromagnetic behaviour on traversing the lens, together with Fresnel reflection from an interface. Previously, imaging for systems with plane polarised illumination without an analyser has been presented using a simple theory based on the angular spectrum of plane waves, also rigorous, for a perfect reflector w2x, for a dielectric interface w3x, and for different structures w4,5x. In this paper, we generalise this treatment to electromagnetic imaging with a finitesized pinhole, the treatment we believe giving more insight compared with that of Torok ¨ ¨ and Wilson w1x. Some details of our approach are the same as in a treatment for imaging in fibre-optic confocal micro-
)
Corresponding author. Fax: q61-2-9351-7727; e-mail: [email protected] 1 On leave from Instituto Nacional de Astrofısica Optica y ´ Electronica, Luis Enrique Erro No. 1, Sta. Marıa ´ ´ Tonantzintla, Pue. 72840, Mexico.
scopes w6,7x, but in the present paper the effect of finite-sized detector pinhole is investigated. The difference between these two types of optical systems is that the optical fibre acts as a coherent detector Žit integrates amplitude over its modal distribution., while the pinhole acts as an incoherent detector of finite size Žit integrates intensity over its area.. The axial resolution of the system for planar objects is presented, and the effects of polarisation investigated.
2. High aperture focusing Wolf w8x described an integral representation for the electromagnetic field at the point r in the focal region a of high aperture lens in the Debye approximation ŽFig. 1. Esy Hsy
i
HH l i
HH l
A Ž p,q . s B Ž p,q . s
0030-4018r00r$ - see front matter q 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 0 0 . 0 0 6 5 9 - 3
e i k pPr dpdq,
Ž 1a .
e i k pPr dpdq,
Ž 1b .
C.J.R. Sheppard, J.F. Aguilarr Optics Communications 180 (2000) 1–8
2
where A, B are the electric and magnetic strength vectors on the Gaussian reference sphere, k s 2 prl, and p, q, s are direction cosines in the x, y, z directions, so that s s Ž1 y Ž p 2 q q 2 . .
1r2
.
Ž 2.
The vector p p s pi q q j q sk,
Ž 3.
is a normalised wave vector. Consider a system ŽFig. 2. in which the aperture stop is situated in the back focal plane of the objective, so that the strength of the angular spectral components are directly determined and the Debye approximation is valid. Microscope objectives are designed according to this principle. The aperture stop is illuminated by a collimated beam, so that after passing through the aperture the electromagnetic field is Pe , Pm . Thus Pe , Pm can be considered as a vectorial pupil function. For an aplanatic system the equivalent refractive locus Žthe locus of points of intersection of corresponding incident and refracted rays. is identical to the Gaussian reference sphere. For given Pe , Pm , the corresponding angular spectra A, B, and three-dimensional pupils Q e , Q m can be calculated using vectors as done by Richards and Wolf w9x and Stamnes w10x, or as described by Torok ¨¨ et al., using matrices, w11x. Here we adopt the latter treatment as it is more suitable for treating systems of different polarisation properties.
Fig. 2. The optical system. This is a reflection-mode confocal microscope, where the aperture stop is in the back focal plane of the objective lens.
We further generalise to arbitrary incident polarisation, by writing the pupil P as a matrix: a Ps b 0
yb a , 0
Ž 4.
where a, b are the x and y components of electric field, and yb, a those of the magnetic field, which are in general functions of position within the pupil. This can be transformed to cylindrical coordinates by multiplying by the matrix cos f F s ysin f 0
sin f cos f 0
0 0 . 1
Ž 5.
Then the vectorial angular spectrum Žstrength vector. is A s FP cos1r2 u ,
Ž 6.
where cos1r2 u is the aplanatic apodization factor w9,12x. The first two rows of this matrix represent the s and p components of the angular spectrum, while the third row is zero. The subscripts s, p should not of course be confused with the components of the wave vector p. 3. Confocal imaging of a planar object Fig. 1. The geometry of the electromagnetic problem. The integration is performed over an angular spectrum, equivalent to a three-dimensional pupil which corresponds with the Ewald sphere.
A common method of examining the axial resolution of a confocal microscope is the measurement of
C.J.R. Sheppard, J.F. Aguilarr Optics Communications 180 (2000) 1–8
the image signal when a planar reflector which is oriented with its normal parallel to the optic axis is scanned axially w1–3x. Then the field after reflection from the surface is calculated by multiplying the angular spectrum matrix by the matrix R rs Rs 0 0
0 yr p 0
0 0 ,
Ž 7.
0
where r s , r p are the Fresnel coefficients for reflection, defined in the usual way. It should be noted that this matrix differs from that of Torok ¨ ¨ et al. w11x because the present matrix determines the change in the plane wave components rather than the cartesian components. Taking the vectorial pupil function for the incident radiation as P1 , and describing the combination of the collector lens pupil and an analyser placed in the back focal plane of the second lens by a vectorial pupil function P2 , the field after reflection and traversing the collector pupil and analyser is
[ EH] s P2 Fy1 RFP1 .
Ž 8.
For a system without an analyser, the electric field is evaluated as Es
ž
a 2
=
qb cos 2 f .
/ ž iq
b 2
2
2
2
Ž 12 .
E p s Ž I0 q I2 cos 2 f X . i q I2 sin 2 f X j,
/
Ž 9.
Ž Ž rs y r p . q Ž rs q r p . cos 2 f . i q Ž rs y r p . Ž 10 .
Ž Ž rs y r p . q Ž rs q r p . cos 2 f . i,
Ž 11 .
Ž 13 .
where
H0 Ž r y r . sin u cos u J Ž k r s
p
X
0
sin u .
=exp Ž 2 ikz cos u . du ,
Then if an analyser is introduced, for orientation in the x direction 1
Ž r s y r p . sin 2 f j.
Ž 14a.
a
j.
=sin 2 f j.
Es
2
Our method of calculating the axial response then differs from that of Torok ¨ ¨ and Wilson w1x. We consider the optical system of Fig. 2. It is assumed that polarisation effects of the beam splitter are negligible. If the tube lens has a long focal length the paraxial approximation can be assumed and the field in the plane of the detector is simply the Fourier transform of that in the back focal plane of the collector lens. As the field in the back focal plane is purely transverse, the axial component of the field in the pinhole can be neglected. Performing the Fourier transformation and evaluating the f integral in the same manner as in Richards and Wolf w9x, we then obtain for the electric field at a point with cylindrical coordinates r X , f X in the pinhole plane, for the case without an analyser
Ž rr y r p . q Ž a sin 2 f
Ž rs y r p .
1
a
For illumination polarised in the x direction, b s 0 and hence 1
Es
I0 s 2
Es
while for orientation in the y direction Žcrossed polarisers.
Ž r s y r p . q Ž a cos 2 f q b sin 2 f . Ž rs q r p .
3
I2 s
H0 Ž r q r . sin u cos u J Ž k r s
p
=exp Ž 2 ikz cos u . du .
2
X
sin u .
Ž 14b.
Here we have omitted a constant multiplier, and included the phase factor expŽ2 ikz cos u . to account for the defocus of the reflector w2x. The integral is performed over the angle u in the high aperture space, unlike the treatment of Torok ¨ ¨ and Wilson w1x which performs an integral in the low aperture space. The factor cos u can be regarded as the aplanatic factor, squared for two passes through the lens, or alternatively as resulting from the transformation of the integration variable from r to u w6,7x.
C.J.R. Sheppard, J.F. Aguilarr Optics Communications 180 (2000) 1–8
4
Fig. 3. Ža. The response of the microscope to defocusing. Different curves correspond with different pinhole radii from 5 mm to 40 mm. The numerical aperture is 0.9, the wavelength is 488 nm. The refractive index of the mirror is 0.24–2.78i. Žb. The same as Ža. but the intensity has been normalised to unity to allow comparison of the widths of the curves.
The intensity detected by a finite-sized pinhole radius R is then, again dropping an unimportant multiplier constant, Is
R
H0 Ž < I < 0
2
q < I2 < 2 . r X d r X ,
Ž 15 .
the term in I0 I2U cancelling on integration over f X . It has been shown previously that for a perfect reflector r s s yr p s 1 so that I2 vanishes, and I0 agrees exactly with that given in w2x. In addition, for an interface with a small change in refractive index, r s s yr p is also independent of u w4,13x.
C.J.R. Sheppard, J.F. Aguilarr Optics Communications 180 (2000) 1–8
5
Fig. 4. Ža. The same as Fig. 3Ža., but in this case the mirror has been taken as a perfect reflector. Žb. The same as Ža., with the normalised intensity set to unity.
We also have for a system with a parallel analyser
Is
R
H0
ž
2
< I0 < q
< I2 < 2 2
/
rX d rX ,
Ž 16 .
and for one with a perpendicular analyser
Is
R
H0
< I2 < 2 2
rX d rX .
Ž 17 .
6
C.J.R. Sheppard, J.F. Aguilarr Optics Communications 180 (2000) 1–8
For a perfect reflector, the signal for a parallel analyser is identical to that without an analyser, while for a perpendicular analyser the signal vanishes, as might be expected.
Numerical results have been computed and are illustrated in Figs. 3–5, for the case of objective of numerical aperture 0.9, a wavelength of 488 nm and a gold reflector Žrefractive index 0.24–2.78i .. A
Fig. 5. Ža. Comparison between different polarisations. The continuous line corresponds to the axial intensity when no analyser is considered and the black dots when a parallel analyser is introduced. The radius of the pinhole is 10 mm. Žb. The response with a crossed analyser. The very low level of the intensity is shown in this figure. However the shape of the curve is similar to that obtained with the parallel analyser.
C.J.R. Sheppard, J.F. Aguilarr Optics Communications 180 (2000) 1–8
magnification factor of 63 has been assumed, which determines the effective size of the pinhole for a given actual size. In Fig. 3Ža. the signal is given in arbitrary units, but signals are unnormalised so that those from different sizes of pinhole can be compared. In Fig. 3Žb. the curves are normalised to unity, thus showing the increase in width with pinhole radius. In Fig. 4Ža. and Fig. 4Žb. we have included results obtained for the case of a perfect reflector mirror: as can be seen there are almost no differences compared with the case of a gold mirror. Effects of an analyser are shown in Fig. 5. For small pinhole sizes, there is a very weak cross-polarisation component and we expect little effect when a parallel analyser is inserted. As seen in Fig. 5, even for larger pinhole sizes a parallel analyser results in little change in the axial image. For a crossed analyser ŽFig. 5Žb.., the signal detected is much weaker, but the shape of the curve is similar to those for no analyser, or a parallel analyser. The curves, unlike those of Torok ¨ ¨ and Wilson w1x, are symmetrical. This can also be predicted from the form of Eq. Ž20.. It is difficult to ascertain fully the reason for the discrepancies between the present work and that in w1x because there are a number of differences in the treatment. There are two differences in the system model. First, we have assumed the aperture stop to be situated in the back focal plane of the objective, as indeed it should be in a properly designed objective. This removes the effects of vignetting and variations of numerical aperture with defocus. These effects, which can be regarded as originating from a finite Fresnel number, are present in Torok ¨ ¨ and Wilson’s treatment w1x. Second, we have assumed a system telecentric on both object and image sides rather than a non-infinite tube length. We would expect this to result in negligible differences if the Fresnel number on the image side were very large. There is also a difference in the method of calculation: we have integrated over angle in the object space, whereas Torok ¨ ¨ and Wilson have integrated in image space. To do the latter the phase factor in image space must be determined w12x, which complicates the calculation greatly. Our approach thus has the advantage of simplicity, but the two methods of calculation should be equivalent. Overall therefore we attribute the differences in the results as
7
Fresnel number effects. We believe that asymmetries in observed axial responses are caused primarily by residual aberrations. Torok ¨ ¨ and Wilson w1x also consider a ‘‘coherent detector’’. A detector that responds to amplitude can result from a number of different optical arrangements including optical fibre collection and interference systems, but it is not clear what practical arrangement Torok ¨ ¨ and Wilson are considering. A system employing a single-mode optical fibre for collection is equivalent to one with a point detector Ž R s 0. and no analyser, except that the integrals I0 , I2 in Eq. Ž20. are evaluated with an appropriate pupil weighting given by the far-field amplitude distribution of the fibre w6,7x. For example, for a fibre for which the Gaussian approximation holds, we must introduce a weighting expŽyŽ Ar2.Žsin2 ursin2 a . where A is a constant for a given fibre and geometry w6,7,14x. 4. Imaging of non-planar objects The imaging of planar structures, and others such as spheres, can be calculated without consideration of the field in the focal region of the objective. However, this is no longer the case for other objects. For such calculations, a treatment in terms of the three-dimensional pupil function is convenient. McCutchen w15x showed how the amplitude in the focal region could be expressed in terms of a three-dimensional pupil function. This approach was generalised to the vectorial case by Sheppard and Larkin w16x, who introduced the concept of the vectorial pupil function. Then the electromagnetic field in the focus can be written simply as a three-dimensional Fourier transform ŽFig. 1. Esy
i
l i
HHHQ ( p) e e
i k pPr
dp,
Ž 18a.
Q m ( p ) e i k pPr dp. Ž 18b. l The three-dimensional vector pupil functions Q e , Q m are related to the vector angular spectra of plane waves A, B by Hsy
HHH
Q e ( p ) s A Ž p,q . d Ž K y 1 . , Q m ( p ) s B Ž p,q . d Ž K y 1 . ,
Ž 19a. Ž 19b.
C.J.R. Sheppard, J.F. Aguilarr Optics Communications 180 (2000) 1–8
8
where K s
Ž 20 .
or `
As
H0 Q ( p) K
dK ,
Ž 21a.
m
( p ) K 2 dK .
Ž 21b.
`
Bs
H0 Q
2
e
The vectors A, B, Q e , Q m are all perpendicular to the direction of propagation from the Gaussian reference sphere to the focal point, while A, B and Q e , Q m are mutually perpendicular according to the normal behaviour of plane electromagnetic waves. The three-dimensional vector pupil functions Q e , Q m are non-zero only on a sphere of radius unity, representing the Ewald sphere. The quantities AŽ p,q .rs, BŽ p,q .rs then represent the projection of the threedimensional pupils on the x, y plane. The treatment is readily extended to the case of non-monochromatic radiation, in which case different values of K correspond to different values of k, and we must integrate over different radii of Ewald sphere. On propagation through the lens, the radial field is rotated so that the electric field E is in the u direction, the circumferential component is unchanged, while an axial vector is rotated so that it is directed towards the focus, so that the three-dimensional pupil function, converted to cylindrical coordinates is Q s LFP cos1r2 ud Ž K y 1 . ,
Ž 22 .
where L is w11x Ls
cos u 0 sin u
0 1 0
ysin u , 0 cos u
Ž 23 .
and is orthogonal. 5. Conclusions A simplified theory for electromagnetic imaging in confocal microscopes with finite-sized pinhole has been presented. Matrix expressions are given for the
angular spectrum and the three-dimensional pupil function. The image of a planar surface is treated using the angular spectrum matrix. It is not necessary to consider the field in the focal region of the objective. Simple expressions are given for the axial response. For a system with no analyser, the axial image of a gold surface is found to be very similar to that of a perfect reflector. The insertion of a parallel analyser does not change the shape of the axial response appreciably. A crossed analyser reduces the signal strength substantially, but the axial response is still altered little in shape. All the axial images presented are symmetrical about the focused position, this being a result of assuming an optical system comprising 2f subsystems, so that the Fresnel number for the system is infinite. Acknowledgements The authors acknowledge assistance from the Australian Research Council and the Science Foundation for Physics with the University of Sydney. They also acknowledge Consejo Nacional de Ciencia y Tecnologia in Mexico. References w1x P. Torok, ¨ ¨ T. Wilson, Opt. Commun. 137 Ž1997. 127. w2x C.J.R. Sheppard, T. Wilson, Appl. Phys. Lett. 38 Ž1981. 858. w3x I.J. Cox, D.K. Hamilton, C.J.R. Sheppard, Appl. Phys. Lett. 41 Ž1982. 604. w4x C.J.R. Sheppard, J. Connolly, J. Lee, C.J. Cogswell, Appl. Opt. 33 Ž1994. 631. w5x C.J.R. Sheppard, M. Gu, Opt. Commun. 88 Ž1992. 180. w6x M.D. Sharma, C.J.R. Sheppard, Bioimaging 6 Ž1998. 98. w7x M.D. Sharma, C.J.R. Sheppard, J. Mod. Opt. 46 Ž1999. 605. w8x E. Wolf, Proc. Roy. Soc. London A 253 Ž1959. 349. w9x B. Richards, E. Wolf, Proc. Roy. Soc. London A 253 Ž1959. 358. w10x J.J. Stamnes, Waves in Focal Regions, Hilger, Bristol, UK, 1986. w11x P. Torok, ¨ ¨ P. Varga, Z. Laczik, G.R. Booker, J. Opt. Soc. Am. A 12 Ž1995. 325. w12x C.J.R. Sheppard, M. Gu, J. Mod. Opt. 40 Ž1993. 1631. w13x C.J.R. Sheppard, Pure and Appl. Opt. 4 Ž1995. 665. w14x M. Gu, C.J.R. Sheppard, J. Mod. Opt. 38 Ž1991. 1621. w15x C.W. McCutchen, J. Opt. Soc. Am. 54 Ž1964. 240–244. w16x C.J.R. Sheppard, K.G. Larkin, Optik 107 Ž1997. 79.
Optics Communications 180 Ž2000. 1–8 www.elsevier.comrlocateroptcom
Electromagnetic imaging in the confocal microscope C.J.R. Sheppard ) , J. Felix Aguilar 1 Physical Optics Department, School of Physics A28, UniÕersity of Sydney, Sydney NSW 2006, Australia Received 28 September 1999; received in revised form 23 March 2000; accepted 29 March 2000
Abstract A simplified theory for imaging in confocal microscope is introduced. This can be used to model images of various structures under different polarisation conditions, and with finite sized detector pinholes. In particular, simple expressions and numerical results are given for the axial resolution for a planar specimen. The effects of polarisation are investigated. q 2000 Published by Elsevier Science B.V. All rights reserved.
1. Introduction In a recent paper w1x Torok ¨ ¨ and Wilson described a rigorous theory for imaging of planar objects in confocal microscopes. They treated rigorously the electromagnetic behaviour on traversing the lens, together with Fresnel reflection from an interface. Previously, imaging for systems with plane polarised illumination without an analyser has been presented using a simple theory based on the angular spectrum of plane waves, also rigorous, for a perfect reflector w2x, for a dielectric interface w3x, and for different structures w4,5x. In this paper, we generalise this treatment to electromagnetic imaging with a finitesized pinhole, the treatment we believe giving more insight compared with that of Torok ¨ ¨ and Wilson w1x. Some details of our approach are the same as in a treatment for imaging in fibre-optic confocal micro-
)
Corresponding author. Fax: q61-2-9351-7727; e-mail: [email protected] 1 On leave from Instituto Nacional de Astrofısica Optica y ´ Electronica, Luis Enrique Erro No. 1, Sta. Marıa ´ ´ Tonantzintla, Pue. 72840, Mexico.
scopes w6,7x, but in the present paper the effect of finite-sized detector pinhole is investigated. The difference between these two types of optical systems is that the optical fibre acts as a coherent detector Žit integrates amplitude over its modal distribution., while the pinhole acts as an incoherent detector of finite size Žit integrates intensity over its area.. The axial resolution of the system for planar objects is presented, and the effects of polarisation investigated.
2. High aperture focusing Wolf w8x described an integral representation for the electromagnetic field at the point r in the focal region a of high aperture lens in the Debye approximation ŽFig. 1. Esy Hsy
i
HH l i
HH l
A Ž p,q . s B Ž p,q . s
0030-4018r00r$ - see front matter q 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 0 0 . 0 0 6 5 9 - 3
e i k pPr dpdq,
Ž 1a .
e i k pPr dpdq,
Ž 1b .
C.J.R. Sheppard, J.F. Aguilarr Optics Communications 180 (2000) 1–8
2
where A, B are the electric and magnetic strength vectors on the Gaussian reference sphere, k s 2 prl, and p, q, s are direction cosines in the x, y, z directions, so that s s Ž1 y Ž p 2 q q 2 . .
1r2
.
Ž 2.
The vector p p s pi q q j q sk,
Ž 3.
is a normalised wave vector. Consider a system ŽFig. 2. in which the aperture stop is situated in the back focal plane of the objective, so that the strength of the angular spectral components are directly determined and the Debye approximation is valid. Microscope objectives are designed according to this principle. The aperture stop is illuminated by a collimated beam, so that after passing through the aperture the electromagnetic field is Pe , Pm . Thus Pe , Pm can be considered as a vectorial pupil function. For an aplanatic system the equivalent refractive locus Žthe locus of points of intersection of corresponding incident and refracted rays. is identical to the Gaussian reference sphere. For given Pe , Pm , the corresponding angular spectra A, B, and three-dimensional pupils Q e , Q m can be calculated using vectors as done by Richards and Wolf w9x and Stamnes w10x, or as described by Torok ¨¨ et al., using matrices, w11x. Here we adopt the latter treatment as it is more suitable for treating systems of different polarisation properties.
Fig. 2. The optical system. This is a reflection-mode confocal microscope, where the aperture stop is in the back focal plane of the objective lens.
We further generalise to arbitrary incident polarisation, by writing the pupil P as a matrix: a Ps b 0
yb a , 0
Ž 4.
where a, b are the x and y components of electric field, and yb, a those of the magnetic field, which are in general functions of position within the pupil. This can be transformed to cylindrical coordinates by multiplying by the matrix cos f F s ysin f 0
sin f cos f 0
0 0 . 1
Ž 5.
Then the vectorial angular spectrum Žstrength vector. is A s FP cos1r2 u ,
Ž 6.
where cos1r2 u is the aplanatic apodization factor w9,12x. The first two rows of this matrix represent the s and p components of the angular spectrum, while the third row is zero. The subscripts s, p should not of course be confused with the components of the wave vector p. 3. Confocal imaging of a planar object Fig. 1. The geometry of the electromagnetic problem. The integration is performed over an angular spectrum, equivalent to a three-dimensional pupil which corresponds with the Ewald sphere.
A common method of examining the axial resolution of a confocal microscope is the measurement of
C.J.R. Sheppard, J.F. Aguilarr Optics Communications 180 (2000) 1–8
the image signal when a planar reflector which is oriented with its normal parallel to the optic axis is scanned axially w1–3x. Then the field after reflection from the surface is calculated by multiplying the angular spectrum matrix by the matrix R rs Rs 0 0
0 yr p 0
0 0 ,
Ž 7.
0
where r s , r p are the Fresnel coefficients for reflection, defined in the usual way. It should be noted that this matrix differs from that of Torok ¨ ¨ et al. w11x because the present matrix determines the change in the plane wave components rather than the cartesian components. Taking the vectorial pupil function for the incident radiation as P1 , and describing the combination of the collector lens pupil and an analyser placed in the back focal plane of the second lens by a vectorial pupil function P2 , the field after reflection and traversing the collector pupil and analyser is
[ EH] s P2 Fy1 RFP1 .
Ž 8.
For a system without an analyser, the electric field is evaluated as Es
ž
a 2
=
qb cos 2 f .
/ ž iq
b 2
2
2
2
Ž 12 .
E p s Ž I0 q I2 cos 2 f X . i q I2 sin 2 f X j,
/
Ž 9.
Ž Ž rs y r p . q Ž rs q r p . cos 2 f . i q Ž rs y r p . Ž 10 .
Ž Ž rs y r p . q Ž rs q r p . cos 2 f . i,
Ž 11 .
Ž 13 .
where
H0 Ž r y r . sin u cos u J Ž k r s
p
X
0
sin u .
=exp Ž 2 ikz cos u . du ,
Then if an analyser is introduced, for orientation in the x direction 1
Ž r s y r p . sin 2 f j.
Ž 14a.
a
j.
=sin 2 f j.
Es
2
Our method of calculating the axial response then differs from that of Torok ¨ ¨ and Wilson w1x. We consider the optical system of Fig. 2. It is assumed that polarisation effects of the beam splitter are negligible. If the tube lens has a long focal length the paraxial approximation can be assumed and the field in the plane of the detector is simply the Fourier transform of that in the back focal plane of the collector lens. As the field in the back focal plane is purely transverse, the axial component of the field in the pinhole can be neglected. Performing the Fourier transformation and evaluating the f integral in the same manner as in Richards and Wolf w9x, we then obtain for the electric field at a point with cylindrical coordinates r X , f X in the pinhole plane, for the case without an analyser
Ž rr y r p . q Ž a sin 2 f
Ž rs y r p .
1
a
For illumination polarised in the x direction, b s 0 and hence 1
Es
I0 s 2
Es
while for orientation in the y direction Žcrossed polarisers.
Ž r s y r p . q Ž a cos 2 f q b sin 2 f . Ž rs q r p .
3
I2 s
H0 Ž r q r . sin u cos u J Ž k r s
p
=exp Ž 2 ikz cos u . du .
2
X
sin u .
Ž 14b.
Here we have omitted a constant multiplier, and included the phase factor expŽ2 ikz cos u . to account for the defocus of the reflector w2x. The integral is performed over the angle u in the high aperture space, unlike the treatment of Torok ¨ ¨ and Wilson w1x which performs an integral in the low aperture space. The factor cos u can be regarded as the aplanatic factor, squared for two passes through the lens, or alternatively as resulting from the transformation of the integration variable from r to u w6,7x.
C.J.R. Sheppard, J.F. Aguilarr Optics Communications 180 (2000) 1–8
4
Fig. 3. Ža. The response of the microscope to defocusing. Different curves correspond with different pinhole radii from 5 mm to 40 mm. The numerical aperture is 0.9, the wavelength is 488 nm. The refractive index of the mirror is 0.24–2.78i. Žb. The same as Ža. but the intensity has been normalised to unity to allow comparison of the widths of the curves.
The intensity detected by a finite-sized pinhole radius R is then, again dropping an unimportant multiplier constant, Is
R
H0 Ž < I < 0
2
q < I2 < 2 . r X d r X ,
Ž 15 .
the term in I0 I2U cancelling on integration over f X . It has been shown previously that for a perfect reflector r s s yr p s 1 so that I2 vanishes, and I0 agrees exactly with that given in w2x. In addition, for an interface with a small change in refractive index, r s s yr p is also independent of u w4,13x.
C.J.R. Sheppard, J.F. Aguilarr Optics Communications 180 (2000) 1–8
5
Fig. 4. Ža. The same as Fig. 3Ža., but in this case the mirror has been taken as a perfect reflector. Žb. The same as Ža., with the normalised intensity set to unity.
We also have for a system with a parallel analyser
Is
R
H0
ž
2
< I0 < q
< I2 < 2 2
/
rX d rX ,
Ž 16 .
and for one with a perpendicular analyser
Is
R
H0
< I2 < 2 2
rX d rX .
Ž 17 .
6
C.J.R. Sheppard, J.F. Aguilarr Optics Communications 180 (2000) 1–8
For a perfect reflector, the signal for a parallel analyser is identical to that without an analyser, while for a perpendicular analyser the signal vanishes, as might be expected.
Numerical results have been computed and are illustrated in Figs. 3–5, for the case of objective of numerical aperture 0.9, a wavelength of 488 nm and a gold reflector Žrefractive index 0.24–2.78i .. A
Fig. 5. Ža. Comparison between different polarisations. The continuous line corresponds to the axial intensity when no analyser is considered and the black dots when a parallel analyser is introduced. The radius of the pinhole is 10 mm. Žb. The response with a crossed analyser. The very low level of the intensity is shown in this figure. However the shape of the curve is similar to that obtained with the parallel analyser.
C.J.R. Sheppard, J.F. Aguilarr Optics Communications 180 (2000) 1–8
magnification factor of 63 has been assumed, which determines the effective size of the pinhole for a given actual size. In Fig. 3Ža. the signal is given in arbitrary units, but signals are unnormalised so that those from different sizes of pinhole can be compared. In Fig. 3Žb. the curves are normalised to unity, thus showing the increase in width with pinhole radius. In Fig. 4Ža. and Fig. 4Žb. we have included results obtained for the case of a perfect reflector mirror: as can be seen there are almost no differences compared with the case of a gold mirror. Effects of an analyser are shown in Fig. 5. For small pinhole sizes, there is a very weak cross-polarisation component and we expect little effect when a parallel analyser is inserted. As seen in Fig. 5, even for larger pinhole sizes a parallel analyser results in little change in the axial image. For a crossed analyser ŽFig. 5Žb.., the signal detected is much weaker, but the shape of the curve is similar to those for no analyser, or a parallel analyser. The curves, unlike those of Torok ¨ ¨ and Wilson w1x, are symmetrical. This can also be predicted from the form of Eq. Ž20.. It is difficult to ascertain fully the reason for the discrepancies between the present work and that in w1x because there are a number of differences in the treatment. There are two differences in the system model. First, we have assumed the aperture stop to be situated in the back focal plane of the objective, as indeed it should be in a properly designed objective. This removes the effects of vignetting and variations of numerical aperture with defocus. These effects, which can be regarded as originating from a finite Fresnel number, are present in Torok ¨ ¨ and Wilson’s treatment w1x. Second, we have assumed a system telecentric on both object and image sides rather than a non-infinite tube length. We would expect this to result in negligible differences if the Fresnel number on the image side were very large. There is also a difference in the method of calculation: we have integrated over angle in the object space, whereas Torok ¨ ¨ and Wilson have integrated in image space. To do the latter the phase factor in image space must be determined w12x, which complicates the calculation greatly. Our approach thus has the advantage of simplicity, but the two methods of calculation should be equivalent. Overall therefore we attribute the differences in the results as
7
Fresnel number effects. We believe that asymmetries in observed axial responses are caused primarily by residual aberrations. Torok ¨ ¨ and Wilson w1x also consider a ‘‘coherent detector’’. A detector that responds to amplitude can result from a number of different optical arrangements including optical fibre collection and interference systems, but it is not clear what practical arrangement Torok ¨ ¨ and Wilson are considering. A system employing a single-mode optical fibre for collection is equivalent to one with a point detector Ž R s 0. and no analyser, except that the integrals I0 , I2 in Eq. Ž20. are evaluated with an appropriate pupil weighting given by the far-field amplitude distribution of the fibre w6,7x. For example, for a fibre for which the Gaussian approximation holds, we must introduce a weighting expŽyŽ Ar2.Žsin2 ursin2 a . where A is a constant for a given fibre and geometry w6,7,14x. 4. Imaging of non-planar objects The imaging of planar structures, and others such as spheres, can be calculated without consideration of the field in the focal region of the objective. However, this is no longer the case for other objects. For such calculations, a treatment in terms of the three-dimensional pupil function is convenient. McCutchen w15x showed how the amplitude in the focal region could be expressed in terms of a three-dimensional pupil function. This approach was generalised to the vectorial case by Sheppard and Larkin w16x, who introduced the concept of the vectorial pupil function. Then the electromagnetic field in the focus can be written simply as a three-dimensional Fourier transform ŽFig. 1. Esy
i
l i
HHHQ ( p) e e
i k pPr
dp,
Ž 18a.
Q m ( p ) e i k pPr dp. Ž 18b. l The three-dimensional vector pupil functions Q e , Q m are related to the vector angular spectra of plane waves A, B by Hsy
HHH
Q e ( p ) s A Ž p,q . d Ž K y 1 . , Q m ( p ) s B Ž p,q . d Ž K y 1 . ,
Ž 19a. Ž 19b.
C.J.R. Sheppard, J.F. Aguilarr Optics Communications 180 (2000) 1–8
8
where K s
Ž 20 .
or `
As
H0 Q ( p) K
dK ,
Ž 21a.
m
( p ) K 2 dK .
Ž 21b.
`
Bs
H0 Q
2
e
The vectors A, B, Q e , Q m are all perpendicular to the direction of propagation from the Gaussian reference sphere to the focal point, while A, B and Q e , Q m are mutually perpendicular according to the normal behaviour of plane electromagnetic waves. The three-dimensional vector pupil functions Q e , Q m are non-zero only on a sphere of radius unity, representing the Ewald sphere. The quantities AŽ p,q .rs, BŽ p,q .rs then represent the projection of the threedimensional pupils on the x, y plane. The treatment is readily extended to the case of non-monochromatic radiation, in which case different values of K correspond to different values of k, and we must integrate over different radii of Ewald sphere. On propagation through the lens, the radial field is rotated so that the electric field E is in the u direction, the circumferential component is unchanged, while an axial vector is rotated so that it is directed towards the focus, so that the three-dimensional pupil function, converted to cylindrical coordinates is Q s LFP cos1r2 ud Ž K y 1 . ,
Ž 22 .
where L is w11x Ls
cos u 0 sin u
0 1 0
ysin u , 0 cos u
Ž 23 .
and is orthogonal. 5. Conclusions A simplified theory for electromagnetic imaging in confocal microscopes with finite-sized pinhole has been presented. Matrix expressions are given for the
angular spectrum and the three-dimensional pupil function. The image of a planar surface is treated using the angular spectrum matrix. It is not necessary to consider the field in the focal region of the objective. Simple expressions are given for the axial response. For a system with no analyser, the axial image of a gold surface is found to be very similar to that of a perfect reflector. The insertion of a parallel analyser does not change the shape of the axial response appreciably. A crossed analyser reduces the signal strength substantially, but the axial response is still altered little in shape. All the axial images presented are symmetrical about the focused position, this being a result of assuming an optical system comprising 2f subsystems, so that the Fresnel number for the system is infinite. Acknowledgements The authors acknowledge assistance from the Australian Research Council and the Science Foundation for Physics with the University of Sydney. They also acknowledge Consejo Nacional de Ciencia y Tecnologia in Mexico. References w1x P. Torok, ¨ ¨ T. Wilson, Opt. Commun. 137 Ž1997. 127. w2x C.J.R. Sheppard, T. Wilson, Appl. Phys. Lett. 38 Ž1981. 858. w3x I.J. Cox, D.K. Hamilton, C.J.R. Sheppard, Appl. Phys. Lett. 41 Ž1982. 604. w4x C.J.R. Sheppard, J. Connolly, J. Lee, C.J. Cogswell, Appl. Opt. 33 Ž1994. 631. w5x C.J.R. Sheppard, M. Gu, Opt. Commun. 88 Ž1992. 180. w6x M.D. Sharma, C.J.R. Sheppard, Bioimaging 6 Ž1998. 98. w7x M.D. Sharma, C.J.R. Sheppard, J. Mod. Opt. 46 Ž1999. 605. w8x E. Wolf, Proc. Roy. Soc. London A 253 Ž1959. 349. w9x B. Richards, E. Wolf, Proc. Roy. Soc. London A 253 Ž1959. 358. w10x J.J. Stamnes, Waves in Focal Regions, Hilger, Bristol, UK, 1986. w11x P. Torok, ¨ ¨ P. Varga, Z. Laczik, G.R. Booker, J. Opt. Soc. Am. A 12 Ž1995. 325. w12x C.J.R. Sheppard, M. Gu, J. Mod. Opt. 40 Ž1993. 1631. w13x C.J.R. Sheppard, Pure and Appl. Opt. 4 Ž1995. 665. w14x M. Gu, C.J.R. Sheppard, J. Mod. Opt. 38 Ž1991. 1621. w15x C.W. McCutchen, J. Opt. Soc. Am. 54 Ž1964. 240–244. w16x C.J.R. Sheppard, K.G. Larkin, Optik 107 Ž1997. 79.