Annular pupils, radial polarization, and superresolution Colin J. R. Sheppard and Amarjyoti Choudhury
An annular pupil, which can be used to produce a Bessel beam, when combined with radially polarized illumination promises improvements in microscope resolution, increased packing density for optical storage, and finer optical lithography. When combined with a circular detection pupil in confocal microscopy a point-spread function 112 nm wide results 共 ⫽ 488 nm兲. Radially polarized annular illumination of a solid-immersion lens can yield a focal spot smaller than 100 nm for ⫽ 488 nm. Use of radially polarized illumination with pupil masks is diskussed. © 2004 Optical Society of America OCIS codes: 050.1960, 110.0180, 210.4770.
1. Introduction
Investigation of the use of an annular pupil in optical imaging has a long history. First investigated by Airy1 in the 1840s, Rayleigh2 showed that the pointspread function for a narrow annulus could be expressed in terms of the Bessel function J02 共hence the name Bessel beam兲, thus the central spot of the Airy disk is sharpened but accompanied by much stronger outer rings. After annular pupils had been studied in detail by Steward,3 Steel,4 and Linfoot and Wolf,5 the greatly improved depth of focus was investigated by Welford6 and McCrickerd.7 The optical transfer function for a lens with annular aperture was presented by O’Neill.8 Tschunko9,10 calculated the point-spread function far from the optic axis for annular pupils. The use of an annular condenser in a microscope was studied by McKechnie.11 The axicon12 共conical prism兲 is a way of generating an approximation to a narrow annulus, and an approximation may also be accomplished by a diffractive optical element 共grating兲.13 The authors14,15 investigated the use of annular pupils in scanning microscopes. They showed that use of an annular pupil together with an unobstructed circular pupil allowed the strength of the outer rings to be reduced. This property has been C. J. R. Sheppard 共
[email protected]兲 is with the Division of Bioengineering, National University of Singapore, 9 Engineering Drive 1, Singapore 117576 and with the Department of Diagnostic Radiology, National University of Singapore, 5 Lower Kent Ridge Road, Singapore 119074. A. Choudhury is with the Nuclear Science Centre, P.O. Box 10502, New Delhi 110067, India. Received 22 December 2003; revised manuscript received 22 April 2004; accepted 27 April 2004. 0003-6935兾04兾224322-06$15.00兾0 © 2004 Optical Society of America 4322
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the basis for many papers on superresolving pupils in confocal microscopes,16 –20 based on Toraldo di Francia’s21,22 proposal of arrays of annuli, either narrow or contiguous. The transfer function for confocal imaging with an annular pupil has been described.23 The relative dimensions of an annular and circular pupil have been shown to be critical in achieving good imaging of an edge object.24 Brakenhoff et al.25 showed that the improvement in resolution predicted for a confocal system with one annular pupil does not occur in practice with microscope objectives of high numerical aperture 共NA兲. This property was also demonstrated theoretically with the vectorial diffraction theory for a highaperture system.23 The intensity in the focal region of a high-aperture lens with annular pupil 共i.e., a vectorial Bessel beam兲 with plane-polarized illumination has been investigated.26 –28 A fluorescence microscope with annular illumination was described by Arimoto and Kawata.29 Although the annular pupil gives a narrow central lobe in the point-spread function, the pupil that maximizes the intensity at the focus for a given input energy in a paraxial system 共the Luneberg apodization criterion兲 is the unobstructed pupil.30 For a high-aperture system with plane-polarized illumination, the polarization of the focused radiation is the same as crossed electric and magnetic dipoles. Then the apodization that gives the greatest total energy density at the focus is also the same as that for crossed electric and magnetic dipoles.31 However, usually it is the electric energy density that is the important quantity. Then for the electric energy density to be maximized at the focus, the polarization of the focused radiation should be the same as an electric dipole.32,33 This electric dipole can be oriented either in a transverse or in an axial direction.
The latter corresponds to a radially polarized 共transverse-magnetic兲 field. The transverse-electric dipole polarization maximizes the electric energy density at focus for any given angular semiaperture smaller than 90°. The axial electric dipole polarization gives a greater electric energy density than crossed dipoles for an NA greater than 0.993 共for a dry lens兲. Radial polarization has been the subject of some attention over the years. Kogelnik and Li34 diskussed how radially polarized beam modes can be synthesized from plane-polarized Laguerre–Gauss modes. Radial polarization is well known in the output of various laser systems.35–37 Many papers have considered the propagation of radially polarized beams.38 – 45 In particular, Bouchal and Olivik46 considered radially polarized Bessel beams. Recently, several groups have been investigating tight focusing of radially polarized beams.47–50 Radially polarized fields exhibit a strong longitudinal field component. Cicchitelli et al.51 diskussed the use of the longitudinal field component for particle acceleration. Xie and Dunn52 have used the different field components to excite different orientations of fluorescent molecules. Imaging with three differently polarized beams can be used to investigate the three-dimensional orientation of molecules. 2. Focusing in High-Aperture Systems with Annular Pupil
The intensity in the focal region in the paraxial approximation for a lens with narrow annular pupil 共a Bessel beam兲 is I共兲 ⫽ J 02共兲,
(1)
where is the normalized radial optical coordinate ⫽ krn sin ␣
(2)
with k ⫽ 2兾, ␣ being the angular semiaperture, and n being the refractive index of the immersion medium. This annular case compares with the case of an unobstructed pupil 共Airy disk兲 I共兲 ⫽
冋 册
2
2J 1共兲 .
(3)
For the high-aperture case with plane-polarized 共in the x direction兲 illumination, the intensity 共timeaveraged electric energy density兲 for the Bessel beam in polar coordinates , is27 I共, 兲 ⫽ J 02共兲 ⫹ 2 tan2共␣兾2兲J 12共兲 ⫹ tan4共␣兾2兲J 22共兲 ⫹ 2 cos共2兲tan2共␣兾2兲 ⫻ 关 J 12共兲 ⫹ J 0共兲J 2共兲兴.
(4)
This result follows directly from Richards and Wolf.53 For circularly polarized illumination, Richards and Wolf53 showed that the intensity is half the sum of
Fig. 1. Electric energy density in the focal region for an NA of 1.4 with a narrow annular pupil, for circularly polarized and radially polarized illumination. For the circularly polarized annulus, the central lobe is slightly narrower 共in normalized coordinates兲 than for the paraxial case for an unobstructed aperture, and the sidelobe level is much higher. For the radially polarized 共transversemagnetic兲 annulus, the response is quite close to that of the paraxial annulus except that the minima are no longer zeros.
those for x and y polarizations. The result is axially symmetric: I共兲 ⫽ J 02共兲 ⫹ 2 tan2共␣兾2兲J 12共兲 ⫹ tan4共␣兾2兲J 22共兲. (5) For radially polarized illumination 共transversemagnetic or axial electric dipole field兲, the radial field is rotated through an angle ␣ by the lens so that after resolving into axial and transverse components and normalizing, I共兲 ⫽ J 02共兲 ⫹ cot2 ␣J 12共兲.
(6)
In Fig. 1, we show a comparison of the different responses. For the circularly polarized annulus, the central lobe is almost the same width 共in normalized coordinates兲 as that for the paraxial case for an unobstructed aperture, and the side-lobe level is higher. For the transverse-magnetic annulus, the response is quite close to that of the paraxial annulus case, except that the minima are no longer zeros. The circularly polarized annulus gives such a broad response because of the effect of the longitudinal field component. This response is illustrated further in Fig. 2. Along the y axis, the annulus gives a much sharper response than for an unobstructed pupil, but along the x axis the response is broad because of the longitudinally polarized field component. 3. Confocal Imaging
In scalar theory of confocal imaging, the overall effective point-spread function is given by the product of those for illumination and detection.15 However, we should really consider the polarization behavior.23 For fluorescence microscopy the role of polarization in the imaging process is very important.54,55 We must distinguish between the cases of permanent and of induced dipole moments on the one hand and between fixed and freely rotating dipoles on the other.56,57 For the case of freely rotating induced dipoles, the point-spread function is 1 August 2004 兾 Vol. 43, No. 22 兾 APPLIED OPTICS
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Fig. 3. Intensity of the confocal image of a point object for an NA of 1.4. The detection pupil is an unobstructed circular pupil. The illumination is a circularly polarized circular pupil, a circularly polarized narrow annulus, or a radially polarized annulus.
4. Reflection Contrast, Total-Internal-Reflection Fluorescence, Solid-Immersion Lenses, and Photon-Tunneling Microscopy
Fig. 2. Intensity in the focal plane along the x and y axes for plane-polarized illumination and for circularly polarized illumination for 共a兲 a circular pupil and 共b兲 a narrow annular pupil. Along the y axis, the annulus gives a much sharper response, but along the x axis the response is broad because of the longitudinally polarized field component. The NA is 1.4.
identical to that for an isotropic 共in three dimensions兲 point object, and in addition, the illumination and detection processes are effectively decoupled from one another.56 We restrict our attention here to this simple case, but, of course, a complete treatment can yield information on molecular orientation and rotational diffusion by fluorescence correlation techniques. In the paraxial case, the minima of the diffraction pattern of a circular aperture coincide with the maxima of that of an annular aperture.15 Examination of Fig. 2 shows that this is approximately true for the high-aperture case. However, as the depth of field of an annular lens is large, the confocal optical sectioning effect is also reduced. Figure 3 shows the intensity of the confocal image of a point object for an NA of 1.4 for different confocal systems, assuming the wavelength of the fluorescence is equal to that of the illumination. The detection pupil is taken to be an unobstructed circular pupil. Illumination is circularly polarized light from a circular pupil, circularly polarized light from a narrow annulus, or radially polarized light from an annulus. The radially polarized annulus gives the sharpest response. The full-width half-maximum 共FWHM兲 of various cases is given in Table 1. Use of radially polarized illumination in nonlinear microscopy, such as multiphoton fluorescence, harmonic generation, and CARS imaging is also attractive.58 4324
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The size of a focused light spot decreases in proportion to the refractive index of the immersion medium. If focused radiation is incident on an interface between a solid high-index medium and a low-index medium, total-internal reflection occurs for rays incident beyond the critical angle, and a focused evanescent field is produced in the low-index medium. The foregoing is the basis of several different microscopical techniques. Reflection contrast is used to image attachment sites of a specimen to a substrate59 where the specimen comes close to the substrate reflection from the interface reduced. In totalinternal-reflection fluorescence microscopy, a fluorescent molecule in the low-index medium, close to the interface, is excited by the evanescent field.60 Solid-immersion lenses61 共SILs兲 can be used with either reflected or fluorescent light collected in the reflection direction. The production and properties of evanescent Bessel beams have been studied.62,63 Evanescent Bessel beams can be generated with a SIL or a grating.62 The photon-tunnelling microscope64 is a transmission- or reflection-mode nearfield microscope in which an evanescent field produced by total-internal reflection is used. If the incident light has a large angular spectrum, then the transmitted evanescent components suffer a phase change depending on angle of incidence and polarization direction.65 This change is equivalent to an aberration and increases the size of the focused spot in the low-index medium. For plane-polarized illumination, aberrations equivalent to spherical aberration and astigmatism are present. In addition, the longitudinal field component is large and enlarges the focal spot. This latter problem is overcome by use of radially polarized illumination63 that also eliminates the astigmatism aberration. In addition, if the range of angles of incidence is small, the spherical aberration becomes small, and the size of the spot is conserved in the low-index medium. The field in the focal region can be calculated as in
Table 1. FWHM of the Point-Spread Function for Different Optical Systems for an NA
System
FWHM 共兲
FWHM 共nm兲 for ⫽ 488 nm 共1.4 NA兲
Paraxial Airy disk theory Paraxial annulus Circular polarized Circular polarized annulus TM annulus Confocal, two circular pupils Confocal, one circularly polarized annulus Confocal, one TM annulus
3.24 2.26 3.68 3.30 2.36 2.63 2.44 2.02
180 125 204 185 131 146 135 112
Measured in terms of and in nanometers for ⫽ 488 nm. TM, transverse-magnetic.
a
To¨ ro¨ k et al.,65 except that the functions in Eq. 共8兲 are different for the radially polarized case. In particular to the case of the radiation being focused onto the interface and the observation point also being on the interface, the intensity 共time-averaged electric energy density兲 in the focal plane is given by
冉
I共兲 ⫽ J 02共兲 ⫹ 1 ⫺
冊
n 22 cosec2 ␣ J 12共兲. n 12
(7)
The only difference from Eq. 共6兲 is the factor determining the strength of the second term, which thus fills in the minima of the response. Figure 4 shows the intensity in the focal spot for the cases of diamond–water and silicon–air SILs illuminated with a cone of light with semiangle 68°, correspond-
Fig. 4. Intensity in the focal spot for a solid-immersion system with diamond–water or silicon–air illuminated with radially polarized light with a cone of semiangle 68°, corresponding to an NA in air of 0.927. The curves are plotted against the optical coordinate , defined for the high-index medium.
ing to an NA in air of 0.927. Note that the curves are plotted against the optical coordinate , defined for the high-index medium. Other cases are similar to those shown. Table 2 gives the FWHM of the focal spot for different combinations of high-index media and immersion media. For As2Se3–air, the FWHM is 94 nm. For biological microscopy, As2Se3–water, with a FWHM of 89 nm, and silicon–water at 1.3 m, with a FWHM of 198 nm, could prove useful. 5. Superresolving and Apodizing Pupil Filters
In paraxial optics, pupil filters are often constructed from arrays of rings. In particular, Toraldo di Francia’s21 original proposal for superresolving masks consisted of an array of narrow annuli. To determine the strength of the rings requires solving a set of simultaneous equations for the positions of assumed zeros in the point-spread function. This method can be also applied to a high-NA scalar theory, in which case the strength of the rings should be increased with angle by a factor sec1兾2 to cancel the apodization factor.66 For radial polarization, if the same method is used to design a superresolving filter so that the longitudinal field is the same as the amplitude in the paraxial case, the relative strength of the rings should be increased with angle by a factor sec1兾2 cosec , in which the first part cancels the apodization factor, and the second part cancels a geometric polarization dependence. However, there will also be a transverse field component, and this will dominate the focal intensity. Nevertheless, it seems likely that suitable designs will be easier to develop for the radially polarized case than for the circularly polarized case, when there are three field
Table 2. NA and FWHM for a SILa with Radially Polarized Annular Illumination
Media
Refractive Indices
NA
FWHM 共兲
FWHM 共nm兲
Diamond–air 共 ⫽ 488 m兲 Diamond–water 共 ⫽ 488 m兲 As2Se3–air 共 ⫽ 488 m兲 As2Se3–water 共 ⫽ 488 m兲 Silicon–air 共 ⫽ 1.3 m兲 Silicon–water 共 ⫽ 1.3 m兲
2.38兾1.00 2.38兾1.33 2.77兾1.00 2.77兾1.33 3.5兾1.00 3.5兾1.33
2.21 2.21 2.57 2.57 3.25 3.25
3.04 2.84 3.12 2.96 3.20 3.30
107 100 94 89 204 198
a
With a cone of semiangle 68°, corresponding to an NA in air of 0.927. 1 August 2004 兾 Vol. 43, No. 22 兾 APPLIED OPTICS
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components and also an angular field dependence. General expressions for the performance parameters in the paraxial approximation for real-valued masks have been proposed.67 These have been partially extended to the high-aperture case for plane-polarized illumination.66 A similar treatment could be applied to the radially polarized case, but diskussion of this treatment is beyond the scope of this paper. 6. Conclusion
Although annular pupils give a reduced focal spot size for paraxial optical systems, for plane-polarized illumination if the NA is large, this improvement is canceled by the effects of the strong longitudinal field component. A way around this problem is to use radially polarized illumination, in which case the longitudinal field dominates. However, for an unobstructed aperture the transverse field then broadens the response. Using radial polarization and an annular pupil reduces the transverse field, and we obtain a sharp focal spot, very similar to the paraxial annular case. Use of this arrangement for illumination, together with a circular pupil for detection, results in a very narrow point-spread function 共112 nm FWHM at ⫽ 488 nm兲. Annular radially polarized illumination of a SIL can yield an FWHM of less than 100 nm for ⫽ 488 nm. Finally, we expect radial polarization to be the preferred approach for pupil masks for superresolution and apodization at high NA. C. J. R. Sheppard thanks R. Dorn, S. Quabis, G. Leuchs, S. J. van Enk, J. J. M. Braat, A. S. van den Nes, and S. F. Pereira for useful diskussions. References 1. G. B. Airy, “The diffraction of an annular aperture,” Philos. Mag. Ser. 3 18, 1–10 共1841兲. 2. Lord Rayleigh, “On the diffraction of object glasses,” Mon. Notes R. Astron. Soc. 33, 59 – 63 共1872兲. 3. G. C. Steward, “IV Aberration diffraction effects,” Philos. Trans. R. Soc. London Ser. A 225, 131–198 共1926兲. 4. W. H. Steel, “Etude des effets combine´ s des aberrations et d’une obturation centrale de la pupille sur le contraste des images optiques,” Revue d’Optique 32, 4, 143, 269 共1953兲. 5. E. H. Linfoot and E. Wolf, “Diffraction images in systems with an annular aperture,” Proc. Phys. Soc. B 66, 145–149 共1953兲. 6. W. T. Welford, “Use of annular apertures to increase focal depth,” J. Opt. Soc. Am. 50, 749 –753 共1960兲. 7. J. T. McCrickerd, “Coherent processing and depth of focus of annular aperture imagery,” Appl. Opt. 10, 2226 –2230 共1971兲. 8. E. L. O’Neill, “Transfer function for an annular aperture,” J. Opt. Soc. Am. 46, 285–288 共1956兲. 9. H. F. A. Tschunko, “Imaging performance of annular apertures,” Appl. Opt. 13, 1820 –1823 共1974兲. 10. H. F. A. Tschunko, “Annular apertures with low and high obstruction,” Appl. Opt. 20, 168 –169 共1981兲. 11. T. S. McKechnie, “The effect of condenser obstruction on the two point resolution of a microscope,” Opt. Acta 19, 729 –737 共1972兲. 12. J. H. McLeod, “The axicon: a new type of optical element,” J. Opt. Soc. Am. 44, 592–597 共1954兲. 13. J. Dyson, “Circular and spiral diffraction gratings,” Proc. R. Soc. London Ser. A 248, 93–106 共1958兲. 4326
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14. C. J. R. Sheppard, “The use of lenses with annular aperture in scanning optical microscopy,” Optik 48, 329 –334 共1977兲. 15. C. J. R. Sheppard and A. Choudhury, “Image formation in the scanning microscope,” Opt. Acta 24, 1051–1073 共1977兲. 16. Z. S. Hegedus, “Annular pupil arrays—application to confocal scanning,” Opt. Acta 32, 815– 826 共1985兲. 17. Z. Hegedus and V. Sarafis, “Superresolving filters in confocally scanned imaging systems,” J. Opt. Soc. Am. A 3, 1892–1896 共1986兲. 18. I. J. Cox, C. J. R. Sheppard, and T. Wilson, “Reappraisal of arrays of concentric annuli as superresolving filters,” J. Opt. Soc. Am. 72, 1287–1291 共1982兲. 19. C. J. R. Sheppard, G. Calvert, and M. Wheatland, “Focal distribution for superresolving Toraldo filters,” J. Opt. Soc. Am. A 15, 849 – 856 共1998兲. 20. M. Martinez-Corral, P. Andre´ s, C. J. Zapata-Rodriguez, and C. J. R. Sheppard, “Improvement of three-dimensional resolution in confocal scanning microscopy by combination of two pupil filters,” Optik 107, 145–148 共1998兲. 21. G. Toraldo di Francia, “Nuovo pupille superrisolvente,” Atti. Fond. Giorgio Ronchi 7, 366 –372 共1952兲. 22. G. Toraldo di Francia, “Supergain antennas and optical resolving power,” Nuovo Cimento Suppl. 9, 426 – 435 共1952兲. 23. C. J. R. Sheppard and T. Wilson, “The image of a single point in microscopes of large numerical aperture,” Proc. R. Soc. London A379, 145–158 共1982兲. 24. X. S. Gan and C. J. R. Sheppard, “Imaging in a confocal microscope with one circular and one annular lens,” Opt. Commun. 103, 254 –264 共1993兲. 25. G. J. Brakenhoff, P. Blom, and P. Barends, “Confocal scanning light microscopy with high aperture immersion lenses,” J. Microsc. 117, 219 –232 共1979兲. 26. A. Yoshida and T. Asakura, “Electromagnetic field in the focal plane of a coherent beam from a wide-angular annularaperture system,” Optik 40, 322–331 共1974兲. 27. C. J. R. Sheppard, “Electromagnetic field in the focal region of wide-angular annular lens and mirror systems,” IEE J. Microwaves, Opt. Acoust. 2, 163–166 共1978兲. 28. S. R. Mishra, “A vector wave analysis of a Bessel beam,” Opt. Commun. 85, 159 –161 共1991兲. 29. R. Arimoto and S. Kawata, “Laser-scan fluorescence microscope with annular excitation,” Optik 86, 7–10 共1990兲. 30. R. K. Luneberg, Mathematical Theory of Optics 共University of California Press, Berkeley, Calif., 1964兲. 31. C. J. R. Sheppard and K. G. Larkin, “Optimal concentration of electromagnetic radiation,” J. Mod. Opt. 41, 1495–1505 共1994兲. 32. V. Dhayalan and J. Stamnes, “Focusing of electric-dipole waves in the Debye and Kirchhoff approximations,” Pure Appl. Opt. 6, 347–372 共1997兲. 33. C. J. R. Sheppard and P. To¨ ro¨ k, “Electromagnetic field in the focal region of an electric dipole wave,” Optik 104, 175–177 共1997兲. 34. H. Kogelnik and T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 共1966兲. 35. J. J. Wynne, “Generation of rotationally symmetric TE01 and TM01 modes from a wavelength tunable laser,” IEEE J. Quantum Electron. QE-10, 125–127 共1974兲. 36. Y. Mushiake, K. Matsumura, and N. Nakajima, “Generation of radially polarized optical beam modes by laser oscillation,” Proc. IEEE 60, 1107–1109 共1972兲. 37. T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. Anderson, and M. J. Rooks, “Circularly-symmetric operation of a concentriccircle-grating, surface emitting, AlGaAs兾GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 共1992兲. 38. L. W. Davis and G. Patsakos, “TM and TE electromagnetic beams in free space,” Opt. Lett. 6, 22–23 共1981兲. 39. L. W. Davis and G. Patsakos, “Comment on “Representation of
40.
41. 42.
43.
44. 45.
46. 47.
48.
49.
50. 51.
52. 53.
vector electromagnetic beams”,” Phys. Rev. A 26, 3702–3703 共1982兲. R. H. Jordan and D. G. Hall, “Free-space azimuthal paraxial wave equation: the azimuthal Bessel-Gauss beam solution,” Opt. Lett. 19, 427– 429 共1994兲. D. G. Hall, “Vector-beam solutions of Maxwell’s wave equation,” Opt. Lett. 21, 9 –11 共1996兲. P. L. Greene and D. G. Hall, “Diffraction characteristics of the azimuthal Bessel-gauss beam,” J. Opt. Soc. Am. A 13, 962–966 共1996兲. A. A. Tovar, “Production and propagation of cylindrically polarized Laguerre-Gaussian laser beams,” J. Opt. Soc. America A 15, 2705–2711 共1998兲. S. R. Seshadri, “Electromagnetic Gaussian beam,” J. Opt. Soc. Am. A 15, 2712–2719 共1998兲. C. J. R. Sheppard and S. Saghafi, “Transverse-electric and transverse-magnetic beam modes beyond the paraxial approximation,” Opt. Lett. 24, 1543–1545 共1999兲. Z. Bouchal and M. Olivik, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 共1995兲. S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 共2000兲. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77– 87 共2000兲, http:兾兾www.opticsexpress.org. S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “The focus of light—theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B: Lasers Opt. 72, 109 – 113 共2001兲. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 共2003兲. L. Cicchitelli, H. Hora, and R. Postle, “Longitudinal components for laser beams in vacuum,” Phys. Rev. A 41, 3727–3732 共1990兲. X. S. Xie and R. C. Dunn, “Probing single molecule dynamics,” Science 265, 361–364 共1994兲. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London A 253, 358 –379 共1959兲.
54. G. Weber, “Polarized fluorescence,” in Fluorescence Techniques in Cell Biology, A. A. T. a. M. Sernatz, ed. 共Springer-Verlag, Berlin, 1973兲. 55. D. Axelrod, “Carbocyanine dye orientation in red cell membrane studied by microscopic fluorescence polarization,” Biophys. J. 26, 557–573 共1979兲. 56. C. J. R. Sheppard and P. To¨ ro¨ k, “An electromagnetic theory of imaging in fluorescence microscopy, and imaging in polarization fluorescence microscopy,” Bioimaging 5, 205–218 共1997兲. 57. P. To¨ ro¨ k and C. J. R. Sheppard, “The role of pinhole size in high-aperture two- and three-photon microscopy,” in Confocal and Two-Photon Microscopy: Foundations, Applications and Advances, A. Diaspro, ed. 共Wiley-Liss, New York, 2002兲, pp. 127–152. 58. D. P. Biss and T. G. Brown, “Polarization-vortex-driven second harmonic generation,” Opt. Lett. 28, 923–925 共2003兲. 59. C. S. Izzard and L. R. Lochner, “Cell-to-substrate contacts in living fibroblasts: an interference reflection study with evaluation of the technique,” J. Cell Sci. 21, 129 –155 共1976兲. 60. D. Axelrod, T. Thompson, and T. Burghardt, “Total internal reflection fluorescence microscopy,” J. Microsc. 129, 19 –28 共1983兲. 61. S. M. Mansfield and G. S. Kino, “Solid immersion microscope,” Appl. Phys. Lett. 57, 2615–2616 共1990兲. 62. S. Kawata, “Grating-probe designing for evanescent-field microscope,” in Proceedings of the OITDA Second International Forum on Optical Scanning Microscopy, S. Minami, ed. 共Optoelectronic Industry and Technology Development Association, Tokyo, Japan, 1992兲, p. 73–77. 63. S. Ruschin and A. Leizer, “Evanescent Bessel beam,” J. Opt. Soc. Am. A 15, 1139 –1143 共1998兲. 64. J. M. Guerra, “Photon tunneling microscope,” Appl. Opt. 29, 3741–3752 共1990兲. 65. P. To¨ ro¨ k, C. J. R. Sheppard, and P. Varga, “Study of evanescent waves for transmission near-field optical microscopy,” J. Mod. Opt. 43, 1167–1183 共1996兲. 66. J. Campos, J. C. Escalera, C. J. R. Sheppard, and M. J. Yzuel, “Axially invariant pupil filters,” J. Mod. Opt. 47, 57– 68 共2000兲. 67. C. J. R. Sheppard and Z. S. Hegedus, “Axial behavior of pupil plane filters,” J. Opt. Soc. Am. A 5, 643– 647 共1988兲.
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