Abstract

We present an algorithm for calculating the field distribution in the focal region of stratified media which is fast and easy to implement. Using this algorithm we study the effect on the electric field distribution of an air gap separating a solid immersion lens and a sample, where we analyse the maximum distance for out-of-contact operation. Also, we study how the presence of a metallic substrate affects the field distribution in the focal region; the interference effects of the reflected field could be used as an alternative for 4Pi-microscopy.

© 2004 Optical Society of America

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References

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  1. V.S. Ignatowsky, �??Diffraction by a lens of arbitrary aperture,�?? Tr. Opt. Inst. Petrograd 1(4), 1-36 (1919).
  2. B. Richards and E. Wolf, �??Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,�?? Proc. Roy. Soc. A 253, 358-379 (1959).
    [CrossRef]
  3. H. Ling and S. Lee, �??Focusing of electromagnetic waves through a dielectric interface,�?? J. Opt. Soc. Am. A 1, 965-973 (1984).
    [CrossRef]
  4. P. Török, P. Varga, Z. Laczik and G.R. Booker, �??Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,�?? J. Opt. Soc. Am. A 12, 325-332 (1995).
    [CrossRef]
  5. A. Egner and S.W. Hell, �??Equivalence of the Huygens-Fresnel and Debye approach for the calculation of high aperture point-spread functions in the presence of refractive index mismatch,�?? J. Microsc. 193, 244-249 (1999).
    [CrossRef]
  6. D.P. Biss and T.G. Brown, �??Cylindrical vector beam focusing through a dielectric interface,�?? Opt. Express 9, 490-497 (2001), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-10-490">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-10-490</a>.
    [CrossRef] [PubMed]
  7. A.S. van de Nes, P.R.T. Munro, S.F. Pereira, J.J.M. Braat, and P. Török, �??Cylindrical vector beam focusing through a dielectric interface: comment,�?? Opt. Express 12, 967-969 (2004). <a href ="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-5-967">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-5-967</a>.
    [CrossRef] [PubMed]
  8. D.G. Flagello and T. Milster, �??3D Modeling of high numerical aperture imaging in thin films,�?? in Design, modeling, and control of laser beam optics, Y. Kohanzadeh, G.N. Lawrence, J.G. McCoy and H. Weichel, eds., Proc. SPIE 1625, 246-276 (1992).
  9. D.G. Flagello, T. Milster and A.E. Rosenbluth, �??Theory of high-NA imaging in homogeneous thin films,�?? J. Opt. Soc. Am. A 13, 53-64 (1996).
    [CrossRef]
  10. P. Török and P. Varga, �??Electromagnetic diffraction of light focused through a stratified medium,�?? Appl. Opt. 36, 2305-2312 (1997).
    [CrossRef] [PubMed]
  11. R. Kant, �??Vector diffraction problem of focussing a category 1 aberrated wavefront though a multilayered lossless medium,�?? J. Mod. Opt. 51, 343-366 (2004).
  12. W. Welford, Aberrations of optical systems (Adam Hilger, Bristol, 1986).
  13. G. N. Watson, A treatise on the theory of Bessel functions (Cambridge University Press, Cambridge, 1966).
  14. W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical recipes in C: The art of scientific computing, 2nd ed. (Cambridge University Press, Cambridge, 1992).
  15. P. Török, C. J. R. Sheppard and P. Varga, �??Study of evanescent waves for transmission near-field microscopy,�?? J. Mod. Opt. 43, 1167-1183 (1996).
    [CrossRef]
  16. A. Egner, M. Schrader and S. W. Hell, �??Refractive index mismatch induced intensity and phase variations in fluorescence confocal, multiphoton and 4Pi-microscopy,�?? Opt. Commun. 153, 211-217 (1998).
    [CrossRef]
  17. C. J. R. Sheppard and M. Gu, �??Axial imaging through an aberrating layer of water in confocal microscopy,�?? Opt. Commun. 88, 180-190 (1992).
    [CrossRef]
  18. L. E. Helseth, �??Roles of polarization, phase and amplitude in solid immersion lens systems,�?? Opt. Commun. 191, 161-172 (2001).
    [CrossRef]
  19. I. Ichimura, S. Hayashi and G. S. Kino, �??High-density optical recording using a solid immersion lens,�?? Appl. Opt. 36, 4339-4348 (1997).
    [CrossRef] [PubMed]
  20. S.W. Hell, �??Increasing the resolution of far-field fluorescence light microscopy by point-spread-function engineering,�?? in Topics in fluorescence spectroscopy, Vol. 5, J.R. Lakowicz ed. (Kluwer Academic/Plenum, New York, 1997), pp. 361-426.
    [CrossRef]
  21. K.S. Youngworth and T.G. Brown, �??Focusing of high numerical aperture cylindrical-vector beams,�?? Opt. Express 7, 77-87 (2000), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-7-2-77">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-7-2-77</a>.
    [CrossRef] [PubMed]
  22. S. Quabis, R. Dorn, M. Eberler, O. Glöckl and G. Leuchs, �??Focusing light to a tighter spot,�?? Opt. Commun. 179, 1-7 (2000).
    [CrossRef]
  23. R. Dorn, S. Quabis, and G. Leuchs, �??Sharper Focus for a Radially Polarized Light Beam,�?? Phys. Rev. Lett. 91, 233901 (2003).
    [CrossRef] [PubMed]
  24. M. Paulus, P. Gay-Balmaz and O.J.F. Martin, �??Accurate and efficient computation of the Green�??s tensor for stratified media,�?? Phys. Rev. E 62, 5797-5807 (2000).
    [CrossRef]

Appl. Opt. (2)

J. Microsc. (1)

A. Egner and S.W. Hell, �??Equivalence of the Huygens-Fresnel and Debye approach for the calculation of high aperture point-spread functions in the presence of refractive index mismatch,�?? J. Microsc. 193, 244-249 (1999).
[CrossRef]

J. Mod. Opt. (2)

R. Kant, �??Vector diffraction problem of focussing a category 1 aberrated wavefront though a multilayered lossless medium,�?? J. Mod. Opt. 51, 343-366 (2004).

P. Török, C. J. R. Sheppard and P. Varga, �??Study of evanescent waves for transmission near-field microscopy,�?? J. Mod. Opt. 43, 1167-1183 (1996).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Commun. (4)

S. Quabis, R. Dorn, M. Eberler, O. Glöckl and G. Leuchs, �??Focusing light to a tighter spot,�?? Opt. Commun. 179, 1-7 (2000).
[CrossRef]

A. Egner, M. Schrader and S. W. Hell, �??Refractive index mismatch induced intensity and phase variations in fluorescence confocal, multiphoton and 4Pi-microscopy,�?? Opt. Commun. 153, 211-217 (1998).
[CrossRef]

C. J. R. Sheppard and M. Gu, �??Axial imaging through an aberrating layer of water in confocal microscopy,�?? Opt. Commun. 88, 180-190 (1992).
[CrossRef]

L. E. Helseth, �??Roles of polarization, phase and amplitude in solid immersion lens systems,�?? Opt. Commun. 191, 161-172 (2001).
[CrossRef]

Opt. Express (3)

Phys. Rev. E (1)

M. Paulus, P. Gay-Balmaz and O.J.F. Martin, �??Accurate and efficient computation of the Green�??s tensor for stratified media,�?? Phys. Rev. E 62, 5797-5807 (2000).
[CrossRef]

Phys. Rev. Lett. (1)

R. Dorn, S. Quabis, and G. Leuchs, �??Sharper Focus for a Radially Polarized Light Beam,�?? Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

Proc. Roy. Soc. A (1)

B. Richards and E. Wolf, �??Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,�?? Proc. Roy. Soc. A 253, 358-379 (1959).
[CrossRef]

Proc. SPIE (1)

D.G. Flagello and T. Milster, �??3D Modeling of high numerical aperture imaging in thin films,�?? in Design, modeling, and control of laser beam optics, Y. Kohanzadeh, G.N. Lawrence, J.G. McCoy and H. Weichel, eds., Proc. SPIE 1625, 246-276 (1992).

Topics in fluorescence spectroscopy (1)

S.W. Hell, �??Increasing the resolution of far-field fluorescence light microscopy by point-spread-function engineering,�?? in Topics in fluorescence spectroscopy, Vol. 5, J.R. Lakowicz ed. (Kluwer Academic/Plenum, New York, 1997), pp. 361-426.
[CrossRef]

Tr. Opt. Inst. Petrograd (1)

V.S. Ignatowsky, �??Diffraction by a lens of arbitrary aperture,�?? Tr. Opt. Inst. Petrograd 1(4), 1-36 (1919).

Other (3)

W. Welford, Aberrations of optical systems (Adam Hilger, Bristol, 1986).

G. N. Watson, A treatise on the theory of Bessel functions (Cambridge University Press, Cambridge, 1966).

W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical recipes in C: The art of scientific computing, 2nd ed. (Cambridge University Press, Cambridge, 1992).

Supplementary Material (4)

» Media 1: AVI (209 KB)     
» Media 2: AVI (231 KB)     
» Media 3: AVI (240 KB)     
» Media 4: AVI (277 KB)     

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Figures (7)

Fig. 1.
Fig. 1.

Schematic overview of the studied configuration. An aplanatic imaging system, denoted by the operation M, maps the light distribution defined in the entrance pupil E 0(kr , kϕ ,0) to the exit pupil E 1(kr , kϕ , kz1 ). The field distribution in the focal region E i (r,ϕ,z) is obtained by an integration over the exit pupil distribution, where several medium transitions can be encountered at z=di . The first medium at the exit pupil has electric permittivity ε 1 and the final medium has electric permittivity εN.

Fig. 2.
Fig. 2.

Absolute value of the field distribution in the focal region for the (a) radial, (b) azimuthal and (c) longitudinal components. An equally weighted combination of radially and azimuthally polarised light is used to illuminate an imaging system with NA=0.933. The medium transition from air (n 1=1.0) to glass (n 2=1.5) takes place at two wavelengths in front of the geometrical focus and has been indicated by the dotted line.

Fig. 3.
Fig. 3.

Absolute value of the field distribution in the focal region for the (a) radial, (b) azimuthal and (c) longitudinal components. An equally weighted combination of radially and azimuthally polarised light is used to illuminate an imaging system with NA=1.4. The medium transition from glass (n 1=1.5) to air (n 2=1.0) takes place at two wavelengths in front of the geometrical focus and has been indicated by the dotted line.

Fig. 4.
Fig. 4.

Amplitude (a) and phase (b) of the Fresnel reflection Ri,s/p and transmission Ti,s/p coefficients, where the index i=1 corresponds to the transition from air to glass, and i=2 to the transition from glass to air, as a function of the angle 0≤θ≤69° (sin-1NA/n 1) for both polarisations TE (s) and TM (p).

Fig. 5.
Fig. 5.

Movies showing the square root of the electric energy in the focal region when an air gap is translated in the focal region. The gap widths are (a) [208 KB] 1 8 λ , (b) [230 KB] 1 4 λ , (c) [240 KB] 1 2 λ and (d) [320 KB] λ. The air gap (n 2=1.0) is located between both dotted lines and separates a solid immersion lens (NA=1.4, n 1=1.5) and a substrate (n 3=1.5). In the four images above the air gap ends at z=-2λ.

Fig. 6.
Fig. 6.

(a) Electric energy distribution in the focal region for radially polarised light illuminating a lens with NA=0.95, whose centre (NA<0.80) is blocked. The air to photo-resist (n 2=1.5) transition takes place at z=0, the photo-resist to aluminum (n 3=0.49+4.87i) transition at z=20 nm and the aluminum to glass (n4=n2) transition at z=120nm. (b) The electric energy density in the transverse direction after averaging in the longitudinal direction over the resist thickness of 20nm.

Fig. 7.
Fig. 7.

Schematic overview of the projection operator P, rotation operator R. The Cartesian basis (,ŷ,ẑ) and natural basis (,,) have been introduced to describe the lens operator L=P -1 RP.

Equations (36)

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E ( r ) = i 2 π Ω a ( k x , k y ) k z e i k · r d k x d k y ,
E ( r ) = i 2 π Ω a ( k x , k ϕ ) k z e ir k r cos ( k ϕ ϕ ) + i k z z k r d k r d k ϕ .
E ( r ) = iR 2 π Ω k z k M · E 0 ( k r , k ϕ ) k z e ir k r cos ( k ϕ ϕ ) + i k z z k r d k r d k ϕ .
E i ( r ) = iR 2 π Ω e ir k r cos ( k ϕ ϕ ) k z 1 k 1 [ e ik zi z M i + + e ik zi z M i ] · E 0 ( k r , k ϕ ) k r d k r d k ϕ .
M i ± = ( g i 0 ± g i 2 ± cos 2 k ϕ g i 2 ± sin 2 k ϕ 0 g i 2 ± sin 2 k ϕ g i 0 ± + g i 2 ± cos 2 k ϕ 0 g i 1 ± cos k ϕ g i 1 ± sin k ϕ 0 ) ,
0 2 π e ink ϕ e irk r cos ( k ϕ ϕ ) d k ϕ = 2 π i n J n ( rk r ) e in ϕ .
E ( r ) = A e ± i k i ± · r .
P = ( cos k ϕ sin k ϕ 0 sin k ϕ cos k ϕ 0 0 0 1 ) ,
R = 1 k 1 ( k z 1 0 k r 0 k 1 0 k r 0 k z 1 ) ,
L = P 1 RP
= 1 k 1 ( k z 1 cos 2 k ϕ + k 1 sin 2 k ϕ ( k z 1 k 1 ) sin k ϕ cos k ϕ k r cos k ϕ ( k z 1 k 1 ) sin k ϕ cos k ϕ k z 1 sin 2 k ϕ + k 1 cos 2 k ϕ k r sin k ϕ k r cos k ϕ k r sin k ϕ k z 1 ) ,
k ̂ i ± = 1 k i ( k r cos k ϕ k r sin k ϕ ± k zi )
l ̂ i ± = l ̂ = ( sin k ϕ cos k ϕ 0 )
m ̂ i ± = 1 k i ( ± k zi cos k ϕ ± k zi sin k ϕ k r )
k ̂ i ± k ̂ 1 = 1 k 1 k i ( k r 2 cos 2 k ϕ k r 2 sin k ϕ cos k ϕ k r k z 1 cos k ϕ k r 2 sin k ϕ cos k ϕ k r 2 sin 2 k ϕ k r k z 1 sin k ϕ ± k r k zi cos k ϕ ± k r k zi sin k ϕ ± k z 1 k zi ) ,
l ̂ l ̂ = ( sin 2 k ϕ sin k ϕ cos k ϕ 0 sin k ϕ cos k ϕ cos 2 k ϕ 0 0 0 0 ) ,
m ̂ i ± m ̂ 1 = 1 k 1 k i ( ± k z 1 k zi cos 2 k ϕ ± k z 1 k zi sin k ϕ cos k ϕ k r k zi cos k ϕ ± k z 1 k zi sin k ϕ cos k ϕ ± k z 1 k zi sin 2 k ϕ k r k z 1 sin k ϕ k r k z 1 cos k ϕ k r k z 1 sin k ϕ k r 2 ) ,
M i z ± = f i z ± · k ̂ i ± k ̂ 1 · L = f i z ± k i ( 0 0 k r cos k ϕ 0 0 k r sin k ϕ 0 0 ± k zi ) ,
M i s ± = f i s ± · l ̂ l ̂ · L = f i s ± ( sin 2 k ϕ sin k ϕ cos k ϕ 0 sin k ϕ cos k ϕ cos 2 k ϕ 0 0 0 0 ) ,
M i p ± = f i p ± · m ̂ i ± m ̂ 1 · L = f 1 p ± k i ( ± k zi cos 2 k ϕ ± k zi cos k ϕ sin k ϕ 0 ± k zi cos k ϕ sin k ϕ ± k zi sin 2 k ϕ 0 k r cos k ϕ k r sin k ϕ 0 ) ,
M i ± = M i p ± + M i s ± = ( g i 0 ± g i 2 ± cos 2 k ϕ g i 2 ± sin 2 k ϕ 0 g i 2 ± sin 2 k ϕ g i 0 ± + g i 2 ± cos 2 k ϕ 0 g i 1 ± cos k ϕ g i 1 ± sin k ϕ 0 ) ,
g i 0 ± = 1 2 ( f i s ± ± f i p ± k zi k i ) ,
g i 2 ± = 1 2 ( f i s ± f i p ± k zi k i ) ,
g i 1 ± = f i p ± k r k i ,
ε i E i = ε i + 1 E i + 1 , E i = E i + 1 , B i = B i + 1 , 1 μ i B i = 1 μ i + 1 B i + 1 .
A i s + e ik zi d i + A i s e ik zi d i = A i + 1 s + e ik zi + 1 d i + A i + 1 s e ik zi + 1 d i ,
k zi A i s + e ik zi d i k zi A i s e ik zi d i = k zi + 1 A i + 1 s + e ik zi + 1 d i k zi + 1 A i + 1 s e ik zi + 1 d i ,
ε i k i A i p + e ik zi d i + ε i k i A i p e ik zi d i = ε i + 1 k i + 1 A i + 1 p + e ik zi + 1 d i + ε i + 1 k i + 1 A i + 1 p e ik zi + 1 d i ,
k zi k i A i p + e ik zi d i k zi k i A i p e ik zi d i = k zi + 1 k i + 1 A i + 1 p + e ik zi + 1 d i k zi + 1 k i + 1 A i + 1 p e ik zi + 1 d i .
( A A + ) i s p = F i , i + 1 s p e ik zi + 1 d i + ( A A + ) i + 1 s p e ik zi + 1 d i e ik zi + 1 d i + ( A A + ) i + 1 s p F i , i + 1 s p e ik zi + 1 d i e 2 ik zi d i ,
F i , i + 1 s = k zi k zi + 1 k zi + k zi + 1 ,
F i , i + 1 p = ε i + 1 k zi ε i k zi + 1 ε i + 1 k zi + ε i k zi + 1 .
f i + 1 s p + = γ i , i + 1 s p [ f i s p + e ik zi d i f i s p F i , i + 1 s p e ik zi d i ] e ik zi + 1 d i ,
f i + 1 s p = γ i , i + 1 s p [ f i s p + F i , i + 1 s p e ik zi d i + f i s p e ik zi d i ] e ik zi + 1 d i ,
γ i , i + 1 s = k zi + k zi + 1 2 k zi + 1 ,
γ i , i + 1 p = k i + 1 k i ε i + 1 k zi + ε i k zi + 1 2 ε i + 1 k zi + 1 .

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