Abstract

We present an analysis of scanning near-field optical microscopy in terms of the so-called communication modes using scalar wave theory. We show that the number of connected modes increases when the scanning distance is decreased, but the number of modes decreases when the size of the scanning aperture is decreased. In the limit of small detector aperture the best-connected mode reduces effectively to the Green function, evaluated at the center of the scanning aperture. We also suggest that the resolution of a scanning optical near-field imaging system is essentially given by the width of the lowest-order communication mode.

© 2006 Optical Society of America

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References

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  1. V. Westphal and S. W. Hell, "Nanoscale resolution in the focal plane of an optical microscope," Phys. Rev. Lett. 94, 143903 1-4 (2005).
    [CrossRef]
  2. J. M. Vigoureux, F. Depasse, and C. Girard, "Superresolution of near-field optical microscopy defined from properties of confined electromagnetic waves," Appl. Opt. 31, 3036-3045 (1992).
    [CrossRef] [PubMed]
  3. D. Courjon, Near-Field Microscopy and Near-Field Optics (Imperial College Press, London, UK, 2003).
  4. D. A. B. Miller, "Communicating with waves between volumes: evaluating orthogonal spatial channels and limits on coupling strengths," Appl. Opt. 39, 1681-1699 (2000).
    [CrossRef]
  5. R. Piestun and D. A. B. Miller, "Electromagnetic degrees of freedom of an optical system," J. Opt. Soc. Am. A 17, 892-902 (2000).
    [CrossRef]
  6. A. Thaning, P. Martinsson, M. Karelin, and A. T. Friberg, "Limits of diffractive optics by communication modes," J. Opt. A: Pure Appl. Opt. 5, 153-158 (2003).
    [CrossRef]
  7. J. A. Veerman, A. M. Otter, L. Kuipers, and N. F. van Hulst, "High definition aperture probes for near-field optical microscopy fabricated by focused ion beam milling," Appl. Phys. Lett. 72, 3115-3117 (1998).
    [CrossRef]
  8. A. Walther, The Ray and Wave Theory of Lenses (Cambridge University Press, Cambridge, UK, 1997).
  9. T. Habashy, A. T. Friberg, and E. Wolf, "Application of the coherent-mode representation to a class of inverse source problems," Inverse Probl. 13, 47-61 (1997).
    [CrossRef]
  10. W. Streifer, "Optical resonator modes — Rectangular reflectors of spherical curvature," J. Opt. Soc. Am. 55, 868-877 (1965).
    [CrossRef]
  11. D. Porter and D. S. G. Stirling, Integral Equations—A Practical Treatment from Spectral Theory to Applications (Cambridge University Press, Cambridge, UK, 1990).
  12. M. Bertero, C. de Mol, F. Gori, and L. Ronchi, "Number of degrees of freedom in inverse diffraction," Opt. Acta 30, 1051-1065 (1983).
    [CrossRef]
  13. C. Lanczos, Linear Differential Operators (Van Nostrand, London, 1961).
  14. B. R. Frieden, "Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions," in Progress in Optics, E. Wolf, ed., (North-Holland, Amsterdam, 1971), Vol. VIII pp. 311-407.
  15. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, eds., Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, Cambridge, UK, 1992).
  16. L. Novotny, D. W. Pohl, and P. Regli, "Light propagation through nanometer-sized structures: the twodimensional-aperture scanning near-field optical microscope," J. Opt. Soc. Am. A 11, 1768-1779 (1994).
    [CrossRef]
  17. D. A. Christensen, "Analysis of near field tip patterns including object interaction using finite-difference timedomain calculations," Ultramicroscopy 57, 189-195 (1995).
    [CrossRef]
  18. J. Lindberg, T. Setälä, M. Kaivola, and A. T. Friberg, "Degree of polarization in light transmission through a near-field probe," J. Opt. A: Pure Appl. Opt. 6, S59-S63 (2004).
    [CrossRef]

2004 (1)

J. Lindberg, T. Setälä, M. Kaivola, and A. T. Friberg, "Degree of polarization in light transmission through a near-field probe," J. Opt. A: Pure Appl. Opt. 6, S59-S63 (2004).
[CrossRef]

2003 (1)

A. Thaning, P. Martinsson, M. Karelin, and A. T. Friberg, "Limits of diffractive optics by communication modes," J. Opt. A: Pure Appl. Opt. 5, 153-158 (2003).
[CrossRef]

2000 (2)

1998 (1)

J. A. Veerman, A. M. Otter, L. Kuipers, and N. F. van Hulst, "High definition aperture probes for near-field optical microscopy fabricated by focused ion beam milling," Appl. Phys. Lett. 72, 3115-3117 (1998).
[CrossRef]

1997 (1)

T. Habashy, A. T. Friberg, and E. Wolf, "Application of the coherent-mode representation to a class of inverse source problems," Inverse Probl. 13, 47-61 (1997).
[CrossRef]

1995 (1)

D. A. Christensen, "Analysis of near field tip patterns including object interaction using finite-difference timedomain calculations," Ultramicroscopy 57, 189-195 (1995).
[CrossRef]

1994 (1)

1992 (1)

1983 (1)

M. Bertero, C. de Mol, F. Gori, and L. Ronchi, "Number of degrees of freedom in inverse diffraction," Opt. Acta 30, 1051-1065 (1983).
[CrossRef]

1965 (1)

Bertero, M.

M. Bertero, C. de Mol, F. Gori, and L. Ronchi, "Number of degrees of freedom in inverse diffraction," Opt. Acta 30, 1051-1065 (1983).
[CrossRef]

Christensen, D. A.

D. A. Christensen, "Analysis of near field tip patterns including object interaction using finite-difference timedomain calculations," Ultramicroscopy 57, 189-195 (1995).
[CrossRef]

de Mol, C.

M. Bertero, C. de Mol, F. Gori, and L. Ronchi, "Number of degrees of freedom in inverse diffraction," Opt. Acta 30, 1051-1065 (1983).
[CrossRef]

Depasse, F.

Friberg, A. T.

J. Lindberg, T. Setälä, M. Kaivola, and A. T. Friberg, "Degree of polarization in light transmission through a near-field probe," J. Opt. A: Pure Appl. Opt. 6, S59-S63 (2004).
[CrossRef]

A. Thaning, P. Martinsson, M. Karelin, and A. T. Friberg, "Limits of diffractive optics by communication modes," J. Opt. A: Pure Appl. Opt. 5, 153-158 (2003).
[CrossRef]

T. Habashy, A. T. Friberg, and E. Wolf, "Application of the coherent-mode representation to a class of inverse source problems," Inverse Probl. 13, 47-61 (1997).
[CrossRef]

Girard, C.

Gori, F.

M. Bertero, C. de Mol, F. Gori, and L. Ronchi, "Number of degrees of freedom in inverse diffraction," Opt. Acta 30, 1051-1065 (1983).
[CrossRef]

Habashy, T.

T. Habashy, A. T. Friberg, and E. Wolf, "Application of the coherent-mode representation to a class of inverse source problems," Inverse Probl. 13, 47-61 (1997).
[CrossRef]

Kaivola, M.

J. Lindberg, T. Setälä, M. Kaivola, and A. T. Friberg, "Degree of polarization in light transmission through a near-field probe," J. Opt. A: Pure Appl. Opt. 6, S59-S63 (2004).
[CrossRef]

Karelin, M.

A. Thaning, P. Martinsson, M. Karelin, and A. T. Friberg, "Limits of diffractive optics by communication modes," J. Opt. A: Pure Appl. Opt. 5, 153-158 (2003).
[CrossRef]

Kuipers, L.

J. A. Veerman, A. M. Otter, L. Kuipers, and N. F. van Hulst, "High definition aperture probes for near-field optical microscopy fabricated by focused ion beam milling," Appl. Phys. Lett. 72, 3115-3117 (1998).
[CrossRef]

Lindberg, J.

J. Lindberg, T. Setälä, M. Kaivola, and A. T. Friberg, "Degree of polarization in light transmission through a near-field probe," J. Opt. A: Pure Appl. Opt. 6, S59-S63 (2004).
[CrossRef]

Martinsson, P.

A. Thaning, P. Martinsson, M. Karelin, and A. T. Friberg, "Limits of diffractive optics by communication modes," J. Opt. A: Pure Appl. Opt. 5, 153-158 (2003).
[CrossRef]

Miller, D. A. B.

Novotny, L.

Otter, A. M.

J. A. Veerman, A. M. Otter, L. Kuipers, and N. F. van Hulst, "High definition aperture probes for near-field optical microscopy fabricated by focused ion beam milling," Appl. Phys. Lett. 72, 3115-3117 (1998).
[CrossRef]

Piestun, R.

Pohl, D. W.

Regli, P.

Ronchi, L.

M. Bertero, C. de Mol, F. Gori, and L. Ronchi, "Number of degrees of freedom in inverse diffraction," Opt. Acta 30, 1051-1065 (1983).
[CrossRef]

Setälä, T.

J. Lindberg, T. Setälä, M. Kaivola, and A. T. Friberg, "Degree of polarization in light transmission through a near-field probe," J. Opt. A: Pure Appl. Opt. 6, S59-S63 (2004).
[CrossRef]

Streifer, W.

Thaning, A.

A. Thaning, P. Martinsson, M. Karelin, and A. T. Friberg, "Limits of diffractive optics by communication modes," J. Opt. A: Pure Appl. Opt. 5, 153-158 (2003).
[CrossRef]

van Hulst, N. F.

J. A. Veerman, A. M. Otter, L. Kuipers, and N. F. van Hulst, "High definition aperture probes for near-field optical microscopy fabricated by focused ion beam milling," Appl. Phys. Lett. 72, 3115-3117 (1998).
[CrossRef]

Veerman, J. A.

J. A. Veerman, A. M. Otter, L. Kuipers, and N. F. van Hulst, "High definition aperture probes for near-field optical microscopy fabricated by focused ion beam milling," Appl. Phys. Lett. 72, 3115-3117 (1998).
[CrossRef]

Vigoureux, J. M.

Wolf, E.

T. Habashy, A. T. Friberg, and E. Wolf, "Application of the coherent-mode representation to a class of inverse source problems," Inverse Probl. 13, 47-61 (1997).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

J. A. Veerman, A. M. Otter, L. Kuipers, and N. F. van Hulst, "High definition aperture probes for near-field optical microscopy fabricated by focused ion beam milling," Appl. Phys. Lett. 72, 3115-3117 (1998).
[CrossRef]

Inverse Probl. (1)

T. Habashy, A. T. Friberg, and E. Wolf, "Application of the coherent-mode representation to a class of inverse source problems," Inverse Probl. 13, 47-61 (1997).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (2)

A. Thaning, P. Martinsson, M. Karelin, and A. T. Friberg, "Limits of diffractive optics by communication modes," J. Opt. A: Pure Appl. Opt. 5, 153-158 (2003).
[CrossRef]

J. Lindberg, T. Setälä, M. Kaivola, and A. T. Friberg, "Degree of polarization in light transmission through a near-field probe," J. Opt. A: Pure Appl. Opt. 6, S59-S63 (2004).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Acta (1)

M. Bertero, C. de Mol, F. Gori, and L. Ronchi, "Number of degrees of freedom in inverse diffraction," Opt. Acta 30, 1051-1065 (1983).
[CrossRef]

Ultramicroscopy (1)

D. A. Christensen, "Analysis of near field tip patterns including object interaction using finite-difference timedomain calculations," Ultramicroscopy 57, 189-195 (1995).
[CrossRef]

Other (7)

D. Porter and D. S. G. Stirling, Integral Equations—A Practical Treatment from Spectral Theory to Applications (Cambridge University Press, Cambridge, UK, 1990).

C. Lanczos, Linear Differential Operators (Van Nostrand, London, 1961).

B. R. Frieden, "Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions," in Progress in Optics, E. Wolf, ed., (North-Holland, Amsterdam, 1971), Vol. VIII pp. 311-407.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, eds., Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, Cambridge, UK, 1992).

A. Walther, The Ray and Wave Theory of Lenses (Cambridge University Press, Cambridge, UK, 1997).

V. Westphal and S. W. Hell, "Nanoscale resolution in the focal plane of an optical microscope," Phys. Rev. Lett. 94, 143903 1-4 (2005).
[CrossRef]

D. Courjon, Near-Field Microscopy and Near-Field Optics (Imperial College Press, London, UK, 2003).

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Figures (9)

Fig. 1.
Fig. 1.

Real and imaginary parts of the first four transmitting modes, computed for sample width 10λ, detector aperture size λ/10, and distance from sample to detector za =λ/20.

Fig. 2.
Fig. 2.

Real and imaginary parts of the first four receiving modes. The parameters are the same as in Fig 1.

Fig. 3.
Fig. 3.

Normalized coupling strengths for various distances za between sample and receiving aperture. The curve with squares corresponds to the modes shown in Figs. 1 and 2.

Fig. 4.
Fig. 4.

Zero-order mode, calculated for the detector placed over the center of the sample (bottom), and at 2λ away from the center in the lateral direction (top). The sample width is 10λ, detector size is λ/10, and the distance is za =λ/20.

Fig. 5.
Fig. 5.

Variation of the zero-order mode (absolute value) as a function of distance za , for a sample width 10λ and aperture size λ/10.

Fig. 6.
Fig. 6.

Absolute value of the zero-order transmitting mode, calculated for detector aperture sizes A between λ/2 and λ/50. The Green function is plotted for reference. The distance to the detector is λ/20.

Fig. 7.
Fig. 7.

First five normalized coupling strengths for detector aperture sizes A between λ/2 and λ/50. The distance between sample and detector is λ/20. For small detector sizes the coupling coefficients fall off rapidly as n increases.

Fig. 8.
Fig. 8.

Intensity profile obtained by scanning the detector, as a function of lateral distance from a point object, for different numbers of modes in the expansion. Sample and aperture widths are 10λ and λ/10, respectively, and the distance to the detector is λ/20.

Fig. 9.
Fig. 9.

Detector-scanned total optical intensity profiles for two point objects of separations δρx ranging between 0.05λ and 0.2λ. The system parameters are as in Fig. 8.

Equations (28)

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U ( x ) = S G ( x , ρ x ) U 0 ( ρ x ) d ρ x ,
G ( x , ρ x ) = i k z a 2 H 1 ( 1 ) { k [ ( x ρ x ) 2 + z a 2 ] 1 2 } [ ( x ρ x ) 2 + z a 2 ] 1 2 .
G ( x , ρ x ) = n = 0 g n ϕ n ( x ) ψ n * ( ρ x ) ,
S G ( x , ρ x ) ψ n ( ρ x ) d ρ x = g n ϕ n ( x ) ,
A G * ( x , ρ x ) ϕ n ( x ) d x = g n * ψ n ( ρ x ) .
g n 2 ψ n ( ρ x ) = S K s ( ρ x , ρ x ) ψ n ( ρ x ) d ρ x ,
g n 2 ϕ n ( x ) = A K a ( x , x ) ϕ n ( x ) d x ,
K s ( ρ x , ρ x ) = A G * ( x , ρ x ) G ( x , ρ x ) d x ,
K a ( x , x ) = S G ( x , ρ x ) G * ( x , ρ x ) d ρ x .
S ψ n * ( ρ x ) ψ m ( ρ x ) d ρ x = A ϕ n * ( x ) ϕ m ( x ) d x = δ nm ,
G ( x , ρ x ) G ( x 0 , ρ x ) ,
K s ( ρ x , ρ x ) G * ( x 0 , ρ x ) G ( x 0 , ρ x ) A d x = A G * ( x 0 , ρ x ) G ( x 0 , ρ x ) .
g n 2 ψ n ( ρ x ) = G * ( x 0 , ρ x ) A S G ( x 0 , ρ x ) ψ n ( ρ x ) d ρ x .
ψ n ( ρ x ) = ψ 0 ( ρ x ) = C 1 2 G * ( x 0 , ρ x ) ,
C = S G ( x 0 , ρ x ) 2 d ρ x ,
g n 2 = g 0 2 = A S G ( x 0 , ρ x ) 2 d ρ x = C A .
U 0 ( ρ x ) = n = 0 a n ψ n ( ρ x ) ,
a n = S U 0 ( ρ x ) ψ n * ( ρ x ) d ρ x .
U ( x ) = n = 0 a n S G ( x , ρ x ) ψ n ( ρ x ) d ρ x = n = 0 a n g n ϕ n ( x ) .
a n = S ψ n * ( ρ x ) δ ( ρ x ρ x 0 ) d ρ x = ψ n * ( ρ x 0 ) .
I a = A n = 0 N a n g n ϕ n ( x ) 2 d x = n = 0 N a n 2 g n 2 = n = 0 N ψ n ( ρ x 0 ) 2 g n 2 ,
I a ( ξ ) = n = 0 N ψ n ξ ( ρ x 0 ) 2 g n ξ 2 ,
ψ n ξ ( ρ x 0 ) ψ n 0 ( ρ x 0 ξ ) = ψ n ( ρ x 0 ξ ) ,
g n ξ g n 0 = g n ,
I a ( ξ ) = n = 0 N ψ n ( ρ x 0 ξ ) 2 g n 2 .
I a ( ξ ) = ψ 0 ( ρ x 0 ξ ) 2 g 0 2 G ( ρ x 0 ξ ) 2 ,
U 0 ( ρ x ) = δ ( ρ x ρ x 1 ) + δ ( ρ x ρ x 2 )
I a ( ξ ) ψ 0 ( ρ x 1 ξ ) + ψ 0 ( ρ x 2 ξ ) 2 .

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