## Scanning optical near-field resolution analyzed in terms of communication modes

Optics Express, Vol. 14, Issue 23, pp. 11392-11401 (2006)

http://dx.doi.org/10.1364/OE.14.011392

Acrobat PDF (164 KB)

### 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

## 1. Introduction

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]

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]

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]

## 2. Near-field communication modes

*x*. A planar sample of extent

*S*is located in the plane

*z*=0 and the resulting field ismeasured over a narrow aperture

*A*in the plane

*z*=

*z*

_{a}. Themeasurement takes place with a bucket detector that spatially integrates the optical intensity over its whole planar aperture

*A*. The propagation of light, at frequency

*ω*, from the sample to the detector then is described by the one-dimensional diffraction integral

*U*

_{0}(

*ρ*

_{x}) is the sample field and

*G*(

*x*,

*ρ*

_{x}) is the half-space Green function [8, 9

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]

*k*=

*ω*/

*c*=2

*π*/

*λ*is the wave vector of the light and

*z*) is the Hankel function of the first kind and of order one. In practical physical situations the Green function can be expanded bi-orthogonally as [10

10. W. Streifer, “Optical resonator modes — Rectangular reflectors of spherical curvature,” J. Opt. Soc. Am. **55**, 868–877 (1965). [CrossRef]

*g*

_{n},

*ϕ*

_{n}(

*x*), and

*ψ*

_{n}(

*ρ*

_{x}) are the so-called singular values and singular functions of Eq. (1), introduced in the following way [12

12. M. Bertero, C. de Mol, F. Gori, and L. Ronchi, “Number of degrees of freedom in inverse diffraction,” Optica Acta **30**, 1051–1065 (1983). [CrossRef]

*K*

_{s}(

*ρ*

_{x},

*ρ*′

_{x}) and

*K*

_{a}(

*x*,

*x*′) are compact Hermitian kernels; hence the eigenvalues (|

*g*

_{n}|

^{2}) are real and positive, and the eigenfunctions

*ψ*

_{n}(

*ρ*

_{x}) and

*ϕ*

_{n}(

*x*) form complete sets of functions that can be taken orthogonal and normalized in their respective domains [11], i.e.,

*g*

_{n}|), obtained in this way for a sample extent of 10

*λ*and a receiving aperture width of 0.1

*λ*, are illustrated in Figs. 1 and 2. The distance between the sample and the detector is

*z*

_{a}=0.05

*λ*. The corresponding coupling coefficients are shown in Fig. 3.

## 3. Limit of small detector size

*A*of the receiving aperture, centered at a position

*x*

_{0}, becomes very small, we evidently may approximate

## 4. Field propagation

*n*, as explicitly demonstrated in Figs. 3 and 7, and at some

*n*=

*N*the coupling is too weak to give a significant contribution to the field at the detector (compared to background noise, for instance). Hence the sum in Eq. (19) in practice only contains terms up to

*n*=

*N*.

## 5. Near-field resolution

*U*

_{0}(

*ρ*

_{x})=

*δ*(

*ρ*

_{x}-

*ρ*

_{x}0). The expansion coefficients

*a*

_{n}from Eq. (18) then are

*z*=

*z*

_{a}, of the strongly contributing modes, i.e.,

*ξ*,

*z*

_{a}) of the tip as

*ξ*denotes the dependency of the modes and the coupling coefficients on the detector location. Technically both the modes and the coupling strengths will have to be recalculated for each lateral position of the detector aperture.

*n*increases is relatively modest. And this widening is counter-balanced by the decrease in the coupling strengths, as demonstrated in Figs. 3 and 7. Rigorous numerical results, shown in Fig. 8, suggest that the width of the intensity profile when the detector is scanned over the point source is fairly insensitive to the number of modes present but, strictly, it is smallest when only the zero-order mode contributes to the expansion.

*N*=0 and obtain from Eq. (25) for the detector-scanned total intensity the approximation

*ψ*

_{0}(

*ρ*

_{x}-

*ξ*) is centered under the receiving aperture. The limited extent of this mode has the effect that the microscope only interrogates a small domain of the sample directly underneath the detector aperture. A point object at

*ρ*

_{x}=

*ρ*

_{x}0, which in the lateral direction is farther away from the aperture than about the half-width of

*ψ*

_{0}(

*ρ*

_{x}), is not detected. Hence we find that the transverse near-field resolution is, effectively, given by the width of the zero-order communication mode, or the Green function as one might expect on physical grounds.

*z*

_{a}=

*λ*/20,

*A*=

*λ*/10, and

*S*=10

*λ*. The scanning was simulated by calculating the total intensity collected by the aperture, Eq. (28), for different positions of the zero-order mode functions, with the distance between them unaltered. It is seen that in this particular case the Rayleigh criterion gives a near-field resolution of about δρx≈0.136λ. This value is in agreement with the well-known result that the resolution of an aperture SNOMis approximately equal to the aperture diameter, when the aperture is sufficiently close to the sample. The dependence of the zero-order mode on

*z*

_{a}, shown in Fig. 5, combined with Eq. (28), implies that the resolution improves strongly when decreasing the sample-detector distance, which is also in agreement with standard SNOM results. The resolution calculated here compares reasonably with the numerical results obtained from more elaborate models of the sample-tip system, such as those based on multiple-multipole (MMP) [16

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]

## 6. Conclusions

## Acknowledgments

## Footnotes

* | Permanent address: Department of Physics and Mathematics, University of Joensuu, P.O. Box 111, FI-80101 Joensuu, Finland. |

## References and links

1. | V. Westphal and S. W. Hell, “Nanoscale resolution in the focal plane of an optical microscope,” Phys. Rev. Lett. |

2. | J. M. Vigoureux, F. Depasse, and C. Girard, “Superresolution of near-field optical microscopy defined from properties of confined electromagnetic waves,” Appl. Opt. |

3. | D. Courjon, |

4. | D. A. B. Miller, “Communicating with waves between volumes: evaluating orthogonal spatial channels and limits on coupling strengths,” Appl. Opt. |

5. | R. Piestun and D. A. B. Miller, “Electromagnetic degrees of freedom of an optical system,” J. Opt. Soc. Am. A |

6. | A. Thaning, P. Martinsson, M. Karelin, and A. T. Friberg, “Limits of diffractive optics by communication modes,” J. Opt. A: Pure Appl. Opt. |

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, |

9. | T. Habashy, A. T. Friberg, and E. Wolf, “Application of the coherent-mode representation to a class of inverse source problems,” Inverse Probl. |

10. | W. Streifer, “Optical resonator modes — Rectangular reflectors of spherical curvature,” J. Opt. Soc. Am. |

11. | D. Porter and D. S. G. Stirling, |

12. | M. Bertero, C. de Mol, F. Gori, and L. Ronchi, “Number of degrees of freedom in inverse diffraction,” Optica Acta |

13. | C. Lanczos, |

14. | B. R. Frieden, “Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions,” in |

15. | W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, eds., |

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 |

17. | D. A. Christensen, “Analysis of near field tip patterns including object interaction using finite-difference timedomain calculations,” Ultramicroscopy |

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. |

**OCIS Codes**

(060.4510) Fiber optics and optical communications : Optical communications

(180.5810) Microscopy : Scanning microscopy

(260.1960) Physical optics : Diffraction theory

**ToC Category:**

Physical Optics

**History**

Original Manuscript: June 13, 2006

Revised Manuscript: August 22, 2006

Manuscript Accepted: August 23, 2006

Published: November 13, 2006

**Virtual Issues**

Vol. 1, Iss. 12 *Virtual Journal for Biomedical Optics*

**Citation**

Per Martinsson, Hanna Lajunen, and Ari T. Friberg, "Scanning optical near-field resolution analyzed in terms of communication
modes," Opt. Express **14**, 11392-11401 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-23-11392

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### References

- 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]
- 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]
- D. Courjon, Near-Field Microscopy and Near-Field Optics (Imperial College Press, London, UK, 2003).
- 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]
- 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]
- 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. 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]
- A. Walther, The Ray and Wave Theory of Lenses (Cambridge University Press, Cambridge, UK, 1997).
- 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]
- W. Streifer, "Optical resonator modes — Rectangular reflectors of spherical curvature," J. Opt. Soc. Am. 55, 868-877 (1965). [CrossRef]
- D. Porter and D. S. G. Stirling, Integral Equations—A Practical Treatment from Spectral Theory to Applications (Cambridge University Press, Cambridge, UK, 1990).
- 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]
- 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).
- 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]
- D. A. Christensen, "Analysis of near field tip patterns including object interaction using finite-difference timedomain calculations," Ultramicroscopy 57, 189-195 (1995). [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]

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