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Virtual Journal for Biomedical Optics

Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editor: Gregory W. Faris
  • Vol. 2, Iss. 8 — Aug. 10, 2007
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Analytical model for quantitative prediction of material contrasts in scattering-type near-field optical microscopy

A. Cvitkovic, N. Ocelic, and R. Hillenbrand  »View Author Affiliations


Optics Express, Vol. 15, Issue 14, pp. 8550-8565 (2007)
http://dx.doi.org/10.1364/OE.15.008550


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Abstract

Nanometer-scale mapping of complex optical constants by scattering-type near-field microscopy has been suffering from quantitative discrepancies between the theory and experiments. To resolve this problem, a novel analytical model is presented here. The comparison with experimental data demonstrates that the model quantitatively reproduces approach curves on a Au surface and yields an unprecedented agreement with amplitude and phase spectra recorded on a phonon-polariton resonant SiC sample. The simple closed-form solution derived here should enable the determination of the local complex dielectric function on an unknown sample, thereby identifying its nanoscale chemical composition, crystal structure and conductivity.

© 2007 Optical Society of America

1. Introduction

Apertureless or scattering-type scanning near-field optical microscopy (s-SNOM) [1–5

1. F. Zenhausern, M. Oboyle, and H. Wickramasinghe, “Apertureless near-field optical microscope,” Appl. Phys. Lett. 65, 1623–1625 (1994). [CrossRef]

] is a nondestructive optical imaging technique which circumvents the diffraction imposed resolution limit with the aid of strong field concentration at the apex of illuminated scanning probe tips [6–9

6. O. J. F. Martin and C. Girard, “Controlling and tuning strong optical field gradients at a local probe microscope tip apex,” Appl. Phys. Lett. 70, 705–707 (1997). [CrossRef]

]. Optical imaging is performed by recording the electromagnetic field E sca scattered by the vibrating probe placed in the proximity of the specimen surface (Fig. 1a). The amplitude and phase of E sca depend on the near-field interaction between the probe and the sample surface. Owing to this effect, highly resolved optical maps of the sample surface can be obtained by scanning a s-SNOM probe along the specimen surface. The attainable spatial resolution is determined by the sharpness of the probing tip, thus being independent of the illumination wavelength [10

10. T. Taubner, R. Hillenbrand, and F. Keilmann, “Performance of visible and mid-infrared scattering-type near-field optical microscopes,” J. Microsc.-Oxf. 210, 311–314 (2003). [CrossRef]

]. The remarkable material specificity of the near-field optical maps - particularly at infrared frequencies - makes s-SNOM a highly valuable analytical method for non-destructive mapping of chemical [5

5. B. Knoll and F. Keilmann, “Near-field probing of vibrational absorption for chemical microscopy,” Nature 399, 134–137 (1999). [CrossRef]

, 11–13

11. T. Taubner, R. Hillenbrand, and F. Keilmann, “Nanoscale polymer recognition by spectral signature in scattering infrared near-field microscopy,” Appl. Phys. Lett. 85, 5064–5066 (2004). [CrossRef]

], structural[14

14. N. Ocelic and R. Hillenbrand, “Subwavelength-scale tailoring of surface phonon polaritons by focused ion-beam implantation,” Nat. Mater. 3, 606–609 (2004). [CrossRef] [PubMed]

, 15

15. A. Huber, N. Ocelic, T. Taubner, and R. Hillenbrand, “Nanoscale resolved infrared probing of crystal structure and of plasmon-phonon coupling,” Nano Lett. 6, 774–778 (2006). [CrossRef] [PubMed]

], and conduction [16–18

16. B. Knoll and F. Keilmann, “Infrared conductivity mapping for nanoelectronics,” Appl. Phys. Lett. 77, 3980–3982 (2000). [CrossRef]

] properties on the nanometer scale.

In order to explain and predict the observed s-SNOM contrast, it is crucial to understand the optical near-field (NF) interaction between the probing tip and the sample surface. Moreover, a quantitative model of the NF interaction would provide a key to unambiguous material identification based on near-field spectra and optical resonances. Earlier attempts at describing the NF interaction are based on a point-dipole model [5

5. B. Knoll and F. Keilmann, “Near-field probing of vibrational absorption for chemical microscopy,” Nature 399, 134–137 (1999). [CrossRef]

, 19–24

19. J. Gersten and A. Nitzan, “Electromagnetic theory of enhanced Raman-scattering by molecules adsorbed on rough surfaces,” J. Chem. Phys. 73, 3023–3037 (1980). [CrossRef]

] which is sufficient for a qualitative explanation of many phenomena experimentally observed in s-SNOM. They include material contrasts [11–15

11. T. Taubner, R. Hillenbrand, and F. Keilmann, “Nanoscale polymer recognition by spectral signature in scattering infrared near-field microscopy,” Appl. Phys. Lett. 85, 5064–5066 (2004). [CrossRef]

, 17

17. J. S. Samson, G. Wollny, E. Brundermann, A. Bergner, A. Hecker, G. Schwaab, A. D. Wieck, and M. Havenith, “Setup of a scanning near field infrared microscope (SNIM): Imaging of sub-surface nano-structures in gallium-doped silicon,” Phys. Chem. Chem. Phys. 8, 753 – 758 (2006). [CrossRef] [PubMed]

, 25–27

25. M. B. Raschke and C. Lienau, “Apertureless near-field optical microscopy: Tip-sample coupling in elastic light scattering,” Appl. Phys. Lett. 83, 5089–5091 (2003). [CrossRef]

] as well as the phonon-polariton resonant NF interaction [28

28. R. Hillenbrand, T. Taubner, and F. Keilmann, “Phonon-enhanced light-matter interaction at the nanometre scale,” Nature 418, 159–162 (2002). [CrossRef] [PubMed]

] and its blue-shift with increasing probe-sample distance [29

29. T. Taubner, F. Keilmann, and R. Hillenbrand, “Nanomechanical resonance tuning and phase effects in optical near-field interaction,” Nano Lett. 4, 1669–1672 (2004). [CrossRef]

]. However, the point-dipole model does not provide a good quantitative description of the measured near-field contrasts. Notable failings include the incapability to correctly reproduce near-field approach curves [30

30. L. Stebounova, B. B. Akhremitchev, and G. C. Walker, “Enhancement of the weak scattered signal in apertureless near-field scanning infrared microscopy,” Rev. Sci. Instrum. 74, 3670–3674 (2003). [CrossRef]

] as well as the spectral position, width and magnitude of polariton-resonant probe-sample NF interaction [14

14. N. Ocelic and R. Hillenbrand, “Subwavelength-scale tailoring of surface phonon polaritons by focused ion-beam implantation,” Nat. Mater. 3, 606–609 (2004). [CrossRef] [PubMed]

, 15

15. A. Huber, N. Ocelic, T. Taubner, and R. Hillenbrand, “Nanoscale resolved infrared probing of crystal structure and of plasmon-phonon coupling,” Nano Lett. 6, 774–778 (2006). [CrossRef] [PubMed]

, 28

28. R. Hillenbrand, T. Taubner, and F. Keilmann, “Phonon-enhanced light-matter interaction at the nanometre scale,” Nature 418, 159–162 (2002). [CrossRef] [PubMed]

] which permits nanoscale-resolved optical analysis of polar crystals [14

14. N. Ocelic and R. Hillenbrand, “Subwavelength-scale tailoring of surface phonon polaritons by focused ion-beam implantation,” Nat. Mater. 3, 606–609 (2004). [CrossRef] [PubMed]

, 15

15. A. Huber, N. Ocelic, T. Taubner, and R. Hillenbrand, “Nanoscale resolved infrared probing of crystal structure and of plasmon-phonon coupling,” Nano Lett. 6, 774–778 (2006). [CrossRef] [PubMed]

], nanocomposites and semiconductor devices [18

18. A. Huber et al., “Simultaneous infrared material recognition and conductivity mapping by nanoscale near-field microscopy,” Adv. Mater. (in press).

].

Numerical studies of the NF interaction between the tip and the sample employ different computational techniques, including finite element [7

7. R. Fikri, D. Barchiesi, F. H’Dhili, R. Bachelot, A. Vial, and P. Royer, “Modeling recent experiments of aperture-less near-field optical microscopy using 2D finite element method,” Opt. Commun. 221, 13–22 (2003). [CrossRef]

] and boundary element [31

31. A. Cvitkovic, N. Ocelic, J. Aizpurua, R. Guckenberger, and R. Hillenbrand, “Infrared imaging of single nanopar-ticles via strong field enhancement in a scanning nanogap,” Phys. Rev. Lett. 97, (2006). [CrossRef] [PubMed]

] methods, the multiple multipole method [32

32. J. L. Bohn, D. J. Nesbitt, and A. Gallagher, “Field enhancement in apertureless near-field scanning optical microscopy,” J. Opt. Soc. Am. A-Opt. Image Sci. Vis. 18, 2998–3006 (2001). [CrossRef]

, 33

33. J. Renger, S. Grafstrom, L. M. Eng, and R. Hillenbrand, “Resonant light scattering by near-field-induced phonon polaritons,” Phys. Rev. B 71, (2005).

], multipole expansion [34

34. S. V. Sukhov, “Role of multipole moment of the probe in apertureless near-field optical microscopy,” Ultrami-croscopy 101, 111–122 (2004). [CrossRef]

] and the finite difference time-domain approach [35

35. H. Hatano and S. Kawata, “Applicability of deconvolution and nonlinear optimization for reconstructing optical images from near-field optical microscope images,” J. Microsc.-Oxf. 194, 230 – 234 (1999). [CrossRef]

]. However, numerical modeling of the tip-sample interaction usually requires great experience in carefully selecting compromises which are needed to make such demanding simulations computationally feasible. Numerical calculations also rarely take into account the probing tip vibration and higher harmonic signal demodulation, although they represent essential aspects of the s-SNOM signal detection [22

22. B. Knoll and F. Keilmann, “Enhanced dielectric contrast in scattering-type scanning near-field optical microscopy,” Opt. Commun. 182, 321–328 (2000). [CrossRef]

, 36–38

36. M. Labardi, S. Patane, and M. Allegrini, “Artifact-free near-field optical imaging by apertureless microscopy,” Appl. Phys. Lett. 77, 621–623 (2000). [CrossRef]

].

To our knowledge, no analytical or numerical model has so far quantitatively reproduced the experimentally observed material contrasts and spectra of resonant NF interaction [14

14. N. Ocelic and R. Hillenbrand, “Subwavelength-scale tailoring of surface phonon polaritons by focused ion-beam implantation,” Nat. Mater. 3, 606–609 (2004). [CrossRef] [PubMed]

, 15

15. A. Huber, N. Ocelic, T. Taubner, and R. Hillenbrand, “Nanoscale resolved infrared probing of crystal structure and of plasmon-phonon coupling,” Nano Lett. 6, 774–778 (2006). [CrossRef] [PubMed]

, 28

28. R. Hillenbrand, T. Taubner, and F. Keilmann, “Phonon-enhanced light-matter interaction at the nanometre scale,” Nature 418, 159–162 (2002). [CrossRef] [PubMed]

]. With the constant improvement in the modelling techniques and computing power, it is likely that numerical methods will eventually succeed in this task. However, it is not likely that they will be fast enough to routinely determine the local dielectric function of a sample at each pixel in a recorded s-SNOM image. We therefore regard a model allowing a fast and accurate calculation of s-SNOM contrasts as an essential step towards this goal.

In this article we propose a novel model for describing material contrasts in infrared s-SNOM. Although several approximations are necessary to obtain an analytical solution, the model calculations yield an unprecedented agreement with experimental data.

In our finite-dipole model the s-SNOM probe (Fig. 1(a)) is represented by an isolated spheroid [19

19. J. Gersten and A. Nitzan, “Electromagnetic theory of enhanced Raman-scattering by molecules adsorbed on rough surfaces,” J. Chem. Phys. 73, 3023–3037 (1980). [CrossRef]

, 32

32. J. L. Bohn, D. J. Nesbitt, and A. Gallagher, “Field enhancement in apertureless near-field scanning optical microscopy,” J. Opt. Soc. Am. A-Opt. Image Sci. Vis. 18, 2998–3006 (2001). [CrossRef]

, 33

33. J. Renger, S. Grafstrom, L. M. Eng, and R. Hillenbrand, “Resonant light scattering by near-field-induced phonon polaritons,” Phys. Rev. B 71, (2005).

, 39–42

39. A. Wokaun, “Surface enhancement of optical-fields - mechanism and applications,” Mol. Phys. 56, 1 – 33 (1985). [CrossRef]

] since this is the best known approximation that can be solved analytically. The illuminating field E 0 is assumed to be uniform and perpendicular to the sample surface and the spheroid much smaller than the wavelength (Fig. 1(b)). To derive an analytical expression for the NF interaction strength and the amplitude of the scattered field, in Section 2.1 we first examine the behavior of a perfectly conducting spheroid (PCS) in a uniform electric field E 0 in the absence of a sample. Electrostatic calculations indicate that the spheroid near-fields induced by E 0 can be approximated by the field of two point charges (monopoles) Q 0 and -Q 0 located close to its ends, as can be seen in Fig. 2.

In Section 2.3, the approximate value and location of Qi and -Qi are derived, enabling us to find a set of equations which determine the NF interaction strength. In Section 2.4 an expression for the electric field scattered by the probe is provided, from which the s-SNOM signals can be derived. In Section 3 the results of our model are compared to near-field approach curves and spectra of phonon-polariton resonant NF interaction obtained with an infrared s-SNOM. We further compare the results with the commonly used point-dipole model (Fig. 1(c)). While the point-dipole model provides a good qualitative description of the near-field interaction, it cannot satisfactorily reproduce the exact spectral position and contrast obtained in the experiments [15

15. A. Huber, N. Ocelic, T. Taubner, and R. Hillenbrand, “Nanoscale resolved infrared probing of crystal structure and of plasmon-phonon coupling,” Nano Lett. 6, 774–778 (2006). [CrossRef] [PubMed]

, 28

28. R. Hillenbrand, T. Taubner, and F. Keilmann, “Phonon-enhanced light-matter interaction at the nanometre scale,” Nature 418, 159–162 (2002). [CrossRef] [PubMed]

, 29

29. T. Taubner, F. Keilmann, and R. Hillenbrand, “Nanomechanical resonance tuning and phase effects in optical near-field interaction,” Nano Lett. 4, 1669–1672 (2004). [CrossRef]

]. In contrast, the new finite-dipole model solves this problem by introducing an elongated tip and approximating it by a more accurate and versatile distribution of charge near its end.

Fig. 1. Comparison of a typical s-SNOM probe-sample configuration with the finite-dipole and the point-dipole model. (a) The probing tip is typically an elongated structure illuminated by plain waves from the side. (b) In the finite-dipole model the tip is approximated by a spheroid in a uniform electric field E 0. (c) In the point dipole model the tip is first reduced to a small sphere in an uniform electric field E 0. The sphere is then further reduced to a point dipole located in its center. The scattered field is finally calculated from a dipolar near-field coupling with the sample.

2. Finite-dipole model of the probe-sample near-field interaction

2.1. Monopole approximation of the probing tip near fields

As a first step towards an analytical description of the NF interaction between the spheroid and the sample surface, it will be shown that the near fields at the PCS apex can be approximated by the field of a monopole located in the center of curvature of the spheroid apex. Note that the illumination can be represented by a homogeneous field only for spheroids much smaller than the wavelength. For larger spheroids the strength of the near fields would be significantly overestimated in an electrostatic calculation due to the neglected dephasing effects. This fact restricts the possible length of the spheroids in our model to below λ/4 [32

32. J. L. Bohn, D. J. Nesbitt, and A. Gallagher, “Field enhancement in apertureless near-field scanning optical microscopy,” J. Opt. Soc. Am. A-Opt. Image Sci. Vis. 18, 2998–3006 (2001). [CrossRef]

, 43

43. Y. C. Martin, H. F. Hamann, and H. K. Wickramasinghe, “Strength of the electric field in apertureless near-field optical microscopy,” J. Appl. Phys. 89, 5774–5778 (2001). [CrossRef]

].

In the electrostatic case, the electric field outside the spheroid and along its major axis is given by E(D) = Es(D)+E 0 where E 0 is the illumination field, Es the field generated by the spheroid, and D the distance from the spheroid apex. In the absence of a sample, the electrostatic calculation for an isolated spheroid yields

Es(D)=2F(L+D)D2+L(2D+R)+lnLF+DL+F+D2F(LεtR)LR(εt1)lnLFL+FE0,
(1)

with F being half the distance between the spheroid foci, R the radius of curvature at the apex and εt the dielectric function of the tip material [41

41. N. Calander and M. Willander, “Theory of surface-plasmon resonance optical-field enhancement at prolate spheroids,” J. Appl. Phys. 92, 4878–4884 (2002). [CrossRef]

, 42

42. N. Ocelic, “Quantitative near-field phonon-polariton spectroscopy,” Ph.D. thesis, Technical University Munich (2007).

]. As well known, the field Es increases with L/R.

When the spheroid is located above a surface, it is additionally illuminated by its reflected near fields (Eq. 1). Unfortunately, a closed-form solution describing the response of a spheroid in such a situation in not known. However, as this is necessary for calculating the near-field interaction, we have to find a simpler representation of the spheroid near fields. We therefore take a closer look at the electric field distribution of a spheroid.

Fig. 2. (a) Electric field Es around a perfectly conducting spheroid (ε→∞) with R = 0.1L located in a homogeneous external field E 0 oriented along the spheroid’s major axis. (b) Electric field of the spheroid as a function of the distance D from the spheroid apex along the z-axis. Shown are the exact result Es from Eq. 1 (dashed line), the monopole field Em (red line), point-dipole field Epd (blue line), and field of a point dipole at the center of the spheroid, Ecd (green line), all scaled to the exact value Es at the spheroid surface (D = 0).

From Fig. 2(a) it is obvious that the near field around the spheroid looks much like a field of a spatially extended dipole with the constituent charges residing close to the opposite ends of the spheroid. Consequently, the near field close to the apex can be approximated by a monopole field, as we show below. We note here that in contrast to the corresponding far field [44

44. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons Inc, 1998). [CrossRef]

], the near field of an elongated spheroid cannot be described by a point dipole located in the spheroid center unless the spheroid is actually a sphere.

In Fig. 2(b) we compare the exact solution Es(D) with the field Ecd = Acd/(R + D)3 of a point dipole located in the center of a spheroid (R/L = 0.1), the field Em = Am/(R + D)2 of a monopole located in the center of curvature of the spheroid apex and the field Epd = Apd/(R + D)3 of a point dipole located in the center of a sphere inscribed into the tip apex. The constants Acd, Am and Apd have been chosen such that fields at the spheroid apex (D = 0) match the exact value Es(0), i.e. such that Epd(0) = Ecd(0) = Em(0) = Es(0). We find that both dipole fields significantly deviate from Es while the monopole field Em yields a good quantitative approximation of the spheroid near field Es. In the following we describe therefore the near field at the spheroid apex by a monopole to which we assign a charge Q 0.

2.2. Spheroid in an external monopole field

In order to calculate the near-field interaction between the probe and the surface, we describe the response of the sample to the spheroid’s near fields by a mirror charge Q0 = -βQ 0, where the electrostatic reflection factor

β=εs1εs+1
(2)

is a function of the dielectric function εs of the sample [45

45. J. D. Jackson, Classical Electrodynamics (Wiley & Sons, 1998).

]. The field generated by the charge Q0 acts back on the probe, thereby polarizing it [46

46. I. V. Lindell, G. Dassios, and K. I. Nikoskinen, “Electrostatic image theory for the conducting prolate spheroid,” J. Phys. D-Appl. Phys. 34, 2302–2307 (2001). [CrossRef]

]. The amount and distribution of the induced charge needs to be known before we can calculate the near-field interaction strength. To simplify the mathematical treatment of the problem, we first consider the behavior of a spheroid that is grounded. When an external point charge Qe (later being the mirror charge induced in the sample) is residing at a position ze > L (Fig. 3(a)) , the totally induced charge Qt on a grounded spheroid is given by [42

42. N. Ocelic, “Quantitative near-field phonon-polariton spectroscopy,” Ph.D. thesis, Technical University Munich (2007).

, 47

47. D. V. Redzic, “An electrostatic problem - a point-charge outside a prolate dielectric spheroid,” Am. J. Phys. 62, 1118 – 1121 (1994). [CrossRef]

]

Qt=QelnL+F+DLF+DlnL+FLF.
(3)
Fig. 3. (a) The electric potential outside a grounded perfectly conducting spheroid in the presence of an external charge Qe is equivalent to the potential of a certain non-uniform line charge distribution qi. (b) Line charge distribution qi induced along the axis of RL=0.1 spheroid as calculated from the first 10 terms in Eq. 4. The external point charge is located at D = 2R (dashed line) and D = 3R (full line).

From Fig. 4(a) we learn that Qt increases towards the value of Qe when the charge Qe approaches the spheroid boundary. Furthermore, at a given distance D, more charge Qt is induced when the ratio L/R increases. Thus, with increasing L not only the field Es increases (Eq. 1), but also the near-field coupling with the sample surface becomes stronger.

To quantify the contribution of the induced charge Qt to the NF interaction, it is necessary to investigate how Qt is distributed over the spheroid. In [46

46. I. V. Lindell, G. Dassios, and K. I. Nikoskinen, “Electrostatic image theory for the conducting prolate spheroid,” J. Phys. D-Appl. Phys. 34, 2302–2307 (2001). [CrossRef]

, 48

48. J. C. E. Sten and I. V. Lindell, “An electrostatic image solution for the conducting prolate spheroid,” J. Electro-magn. Waves Appl. 9, 599 – 609 (1995).

] it has been shown that the response of a grounded, perfectly conducting spheroid to Qe is equivalent to that of a line charge distribution qi(z) along the line connecting the two spheroid foci (Fig. 3(a)). This charge distribution is given by the series

qi(z)=QeΘ(F2z2)2Fn=0(2n+1)Qn((L+D)F)Pn(LF)Qn(LF)Pn(zF)
(4)

with Θ being the unit step function, Qn the n-th order Legendre functions of the second kind, and Pn the Legendre polynomials [46

46. I. V. Lindell, G. Dassios, and K. I. Nikoskinen, “Electrostatic image theory for the conducting prolate spheroid,” J. Phys. D-Appl. Phys. 34, 2302–2307 (2001). [CrossRef]

]. The two examples of qi(z) displayed in Fig. 3(b) clearly show that the amount and distribution of the induced charge depend on the distance D of the external charge Qe while the highest charge density is found at the spheroid focus (z = F), i.e. at a distance R/2 (for RL) from the spheroid apex.

2.3. Spheroid-sample near-field interaction

To be able to calculate the near-field interaction between the spheroid and the sample surface analytically, we replace the exact charge distribution qi by an effective point charge Qi. Since the density of the line charge qi is highest in the spheroid’s focus, we fix the location of Qi to z = F. When the spheroid is in contact with the sample surface, mainly the charges in the immediate proximity of the contact area (i.e. in the spheroid apex) contribute to the near-field interaction. However, with increasing distance D between the spheroid and the sample surface, the relative contribution of charges further away from the tip apex increases. We thus assign to Qi the charge located within the spheroid between z = L and z = L-(R+D):

Qi=gQt=L(R+D)Lqi(z)dz
(5)

The fraction g = Qi(D)/Qt(D) is shown in Fig. 4(b). We observe a similar behavior for different spheroid shapes and furthermore that g is roughly constant for D > R/2. A reasonable estimate for this constant seems to be g = 0.7±0.1.

Fig. 4. (a) Totally induced charge Qt, calculated for spheroids with RL=0.2 (full line), RL=0.1 (dashed) and RL=0.05 (dotted). (b) Fraction g of the totally induced charge Qt found within the range R + D from the tip end. Three different shape factor are shown: RL=0.2 (full line), RL=0.1 (dashed), and RL=0.05 (dotted). For D/R ≳ 1.5 the calculation was done according to Eq.4. As for D/R ≲ 1.5 the solution of Eq.4 exhibits convergence problems, an improved theory with better numerical behavior was used [46]. There the charge distribution is represented by a line charge plus a point charges located between the tip and the focus of each spheroid apex. Note that the shaded region between D = 0 and D = 0.5R represents distances which are not encountered in the model.

Up to now we have considered grounded spheroids, where the surplus charge -Qt is assumed to flow out. For an isolated spheroid (as assumed in our model, Fig. 1(b)) Qi has to be modified since the surplus charge cannot leave the spheroid, but rather distributes in such a way that the entire surface of the spheroid remains equipotential. This requirement is satisfied when any slice of thickness Δz perpendicular to the z-axis contains an equal amount of charge, ΔQ = -QtΔz/(2L) [49

49. D. V. Redzic, “Image of a moving spheroidal conductor,” Am. J. Phys. 60, 506–508 (1992). [CrossRef]

]. To account for this charge, we have to replace the factor g by g′ = g - (R + D)/(2L). We obtain

Qi=gQt=(gR+D2L)Qt.
(6)

It should be noted that when the spheroid above a flat surface is illuminated with an external field E 0 the induced charge Qi actually consists of two parts:

Qi=Qi,0+Qi,1.
(7)

For Q i,0, the external charge Qe is the mirror image Q0. For Q i,1, the external charge Qe is the mirror image of Qi itself which is denoted in the following by Qi. The corresponding distances D 0 and D 1 are easily read from Fig. 5:

D0=2H+Rand
D1=2H+R2,
(8)

where H denotes the distance between the spheroid apex and the sample surface.

Fig. 5. Charges participating in the near-field interaction between the probe and the sample, together with their positions according to the finite-dipole model.

Inserting Eqs. 8 into Eq. 6 and neglecting R/2 and D in comparison to 2L, the following relations between the image charges Q0 and Qi and the monopoles Q i,0 and Q i,1 they induce in the probe are obtained:

Qi,0=f0Q0=(gR+HL)ln4L4H+3Rln4LRQ0
Qi,1=f1Qi=(g3R+4H4L)ln2L2H+Rln4LRQi.
(9)

The total induced charge Qi is a measure for the strength of the near-field interaction between the probe and the sample.

The image charges Q0 and Qi can be written as

Q0=βQ0
Qi=βQi.
(10)

As the monopole Qi comprises Q i,1 which is proportional to the mirror image of Qi itself, we have obtained a recursive definition for Qi which can be solved by searching for a self-consistent solution. Inserting Q0 and Qi from Eq. 10 into Eq.9 we get

Qi=βf01+βf1Q0.
(11)

This allows a dimensionless “near-field contrast factor” η = Qi/Q 0 to be defined as an illumination-independent measure of the s-SNOM signal, comparable between different samples and different measurements:

η=QiQ0=β(gR+HL)ln4L4H+3Rln4LRβ(g3R+4H4L)ln2L2H+R.
(12)

2.4. Light scattering by the probe and effective probe polarizability

In a typical s-SNOM experiment, the light scattered from the probe is detected. We thus have to describe how the scattered field Esca depends on the near-field interaction i.e. the near-field contrast factor η. In this regard we again note that the electric neutrality of an isolated spheroid requires the existence of a charge - Qi somewhere on the spheroid. For this reason we can assign a dipole moment pi to the total charge distribution oscillating with the driving field frequency. pi radiates light according to the general expression Esca ~ p = αE 0 with α being the dipole polarizability.

The dipole moment pi can be easily derived by considering that the charge -Qi must not break the equipotential property of the spheroid. For symmetry reasons we thus can locate the total amount of - Qi into the spheroid center yielding pi = QiL. Subsequently, the effective dipole moment of the perfectly conducting spheroid is given by p eff = p 0 + pi, where p 0 = 2Q 0 L is the dipole moment induced by the external field E 0 and pi = η Q 0 L the dipole moment arising from the near-field interaction. From the monopole approximation of the spheroid field Es (Eq. 1) we obtain Q 0 = R 2 Es(0) and the effective probe polarizability α eff in the case RL:

αeff=peffE0=R2L2LR+lnR4eLln4Le2(2+β(gR+HL)ln4L4H+3Rln4LRβ(g3R+4H4L)ln2L2H+R).
(13)

Eq. 13 can be directly compared to the effective probe polarizability

αeffpd=α1αβ16π(R+H)3
(14)

obtained from the point-dipole model (Fig. 1(c), [4

4. F. Keilmann and R. Hillenbrand, “Near-field microscopy by elastic light scattering from a tip,” Philos. Trans. R. Soc. Lond. Ser. A-Math. Phys. Eng. Sci. 362, 787–805 (2004). [CrossRef]

, 22

22. B. Knoll and F. Keilmann, “Enhanced dielectric contrast in scattering-type scanning near-field optical microscopy,” Opt. Commun. 182, 321–328 (2000). [CrossRef]

]). Here α= 4πR 3 is the polarizability of a perfectly conducting sphere, R the radius of the sphere, and H distance between the sphere and the sample surface.

To obtain the final expression for the scattered field we have to take into account that the probe in an experimental situation is illuminated by the incoming field Einc directly and indirectly due to its reflection at the sample surface (Fig. 6). Furthermore, the scattered field is also reflected at the sample surface [25

25. M. B. Raschke and C. Lienau, “Apertureless near-field optical microscopy: Tip-sample coupling in elastic light scattering,” Appl. Phys. Lett. 83, 5089–5091 (2003). [CrossRef]

]. For simplicity, we shall assume that both the plane wave illumination and the backscattering are in the same direction. Although the incoming field may have both x- and z-components, due to the strong depolarization by the probing tip we can neglect the x-, and consider only the z-component of the incoming field, as assumed in the model. According to this assumption, we can write the field scattered from the probe (i.e. from the effective dipole p eff) as

Esca(1+rp)2αeffEinc,
(15)

with rp being the Fresnel reflection coefficient for p-polarized light.

We should point out here that in former applications [4

4. F. Keilmann and R. Hillenbrand, “Near-field microscopy by elastic light scattering from a tip,” Philos. Trans. R. Soc. Lond. Ser. A-Math. Phys. Eng. Sci. 362, 787–805 (2004). [CrossRef]

, 14

14. N. Ocelic and R. Hillenbrand, “Subwavelength-scale tailoring of surface phonon polaritons by focused ion-beam implantation,” Nat. Mater. 3, 606–609 (2004). [CrossRef] [PubMed]

, 15

15. A. Huber, N. Ocelic, T. Taubner, and R. Hillenbrand, “Nanoscale resolved infrared probing of crystal structure and of plasmon-phonon coupling,” Nano Lett. 6, 774–778 (2006). [CrossRef] [PubMed]

, 22

22. B. Knoll and F. Keilmann, “Enhanced dielectric contrast in scattering-type scanning near-field optical microscopy,” Opt. Commun. 182, 321–328 (2000). [CrossRef]

, 26

26. Z. H. Kim, B. Liu, and S. R. Leone, “Nanometer-scale optical imaging of epitaxially grown GaN and InN islands using apertureless near-field microscopy,” J. Phys. Chem. B 109, 8503–8508 (2005). [CrossRef]

, 28

28. R. Hillenbrand, T. Taubner, and F. Keilmann, “Phonon-enhanced light-matter interaction at the nanometre scale,” Nature 418, 159–162 (2002). [CrossRef] [PubMed]

, 30

30. L. Stebounova, B. B. Akhremitchev, and G. C. Walker, “Enhancement of the weak scattered signal in apertureless near-field scanning infrared microscopy,” Rev. Sci. Instrum. 74, 3670–3674 (2003). [CrossRef]

, 50

50. Z. H. Kim and S. R. Leone, “High-resolution apertureless near-field optical imaging using gold nanosphere probes,” J. Phys. Chem. B 110, 19,804–19,809 (2006).

] of the point-dipole model the scattered field was described differently, by

Esca~α(1+β)1αβ16π(R+H)3Einc=(1+β)αeffpdEinc
(16)

where the factor (1 + β) appears instead of (1 + rp)2. First, the indirect probe illumination by reflection at the sample surface is not taken into account in Eq. 16, which was already noted in [25

25. M. B. Raschke and C. Lienau, “Apertureless near-field optical microscopy: Tip-sample coupling in elastic light scattering,” Appl. Phys. Lett. 83, 5089–5091 (2003). [CrossRef]

, 33

33. J. Renger, S. Grafstrom, L. M. Eng, and R. Hillenbrand, “Resonant light scattering by near-field-induced phonon polaritons,” Phys. Rev. B 71, (2005).

]. Secondly, the scattered light reflected at the sample surface, E sca,r ~ eff E inc, is represented by light scattering of the mirror dipole peff = βp eff yielding E sca,r ~ β α effpd E inc. Here it is important to note that the electrostatic reflection coefficient β can be applied for calculating the near-field interaction between the probe dipole and the sample, as well as the resulting near fields. However, it can not be applied for describing the reflection of the propagating fields emitted by the effective probe dipole [42

42. N. Ocelic, “Quantitative near-field phonon-polariton spectroscopy,” Ph.D. thesis, Technical University Munich (2007).

, 51

51. J. Aizpurua, personal communication (2005).

]. As β can be larger than unity, this situation would not conserve energy. Therefore Eq. 15, rather than Eq. 16 should be employed in the future, independent of the underlying model used for describing the near-field interaction.

2.5. Parameters of the model

In the absence of other means to obtain g and L, they were determined by searching for values which are in a good agreement with the available experimental data presented in Section 3, including approach curves and near-field spectra of phonon-polariton resonant samples. According to our inspection, the best values found are g = 0.7e 0.0 6i and L = 300 nm. It is thereby important to note that the same values of these parameters will be subsequently used for all comparisons to the measured data, i.e. they are not adjusted to each experiment separately.

3. Experiment vs. finite-dipole model

In this section we compare our model quantitatively with mid-infrared s-SNOM data. After describing the microscope and experimental details we analyze approach curves on a flat gold surface and spectra of phonon-polariton resonant near-field interaction with a flat SiC surface.

3.1. Microscope setup and experimental details

Our infrared s-SNOM is based on a non-contact atomic force microscope (AFM, tapping frequency Ω, tapping amplitude Δz ≈ 20 nm, tip radius a ≈ 20 nm) where cantilevered Pt-coated tips act as scattering probes [4

4. F. Keilmann and R. Hillenbrand, “Near-field microscopy by elastic light scattering from a tip,” Philos. Trans. R. Soc. Lond. Ser. A-Math. Phys. Eng. Sci. 362, 787–805 (2004). [CrossRef]

]. It operates with different CO2-lasers covering the spectral region from 880 cm-1 to 1100cm-1. A parabolic mirror is used for illuminating the tip with the laser radiation and to collect the back scattered light. Interferometric detection of the scattered light together with demodulation of the detector signal at a frequency nΩ with n > 1 (higher harmonic demodulation) allows the unavoidable background scattering to be suppressed. By phase modulation of the reference beam (pseudo-heterodyne detection) we obtain amplitude sn and phase ϕn signals simultaneously and avoid the interferometric background effect [52

52. N. Ocelic, A. Huber, and R. Hillenbrand, “Pseudoheterodyne detection for background-free near-field spec-troscopy,” Appl. Phys. Lett. 89, (2006). [CrossRef]

].

In our model the higher harmonic demodulation [36

36. M. Labardi, S. Patane, and M. Allegrini, “Artifact-free near-field optical imaging by apertureless microscopy,” Appl. Phys. Lett. 77, 621–623 (2000). [CrossRef]

, 37

37. R. Hillenbrand and F. Keilmann, “Complex optical constants on a subwavelength scale,” Phys. Rev. Lett. 85, 3029–3032 (2000). [CrossRef] [PubMed]

, 53

53. R. Bachelot, G. Wurtz, and P. Royer, “An application of the apertureless scanning near-field optical microscopy: imaging a GaAlAs laser diode in operation,” Appl. Phys. Lett. 73, 3333–3335 (1998). [CrossRef]

, 54

54. B. Knoll and F. Keilmann, “Electromagnetic fields in the cutoff regime of tapered metallic waveguides,” Opt. Commun. 162, 177–181 (1999). [CrossRef]

] can be taken into account by Fourier decomposition of the time-dependent scattered field E sca = E sca[z(t)], yielding the demodulated optical signal

En=sneiϕn~(1+rp)2αeff,nEinc.
(17)

where z(t) = Δz sinΩt describes a sinusoidal tip motion.

From the s-SNOM amplitude and phase images we usually extract material contrasts rather than the absolute values. A typical experimental situation is shown in Fig. 6 where the tip scans a sharp boundary between the materials A and B. In the case where we measure sn and ϕn very close to the material boundary, we can assume that the reflection coefficient rp does not change when the probing tip moves from surface A to surface B. The amplitude and phase contrasts are then given by

sn,Asn,B=(1+rA)2αeff,n,A(1+rA)2αeff,n,B=αeff,n,Aαeff,n,B
(18)

and

ϕn,Aϕn,B=arg((1+rA)2αeff,n,A(1+rA)2αeff,n,B)=arg(αeff,n,Aαeff,n,B).
(19)

Note that the calculation with the finite-dipole model can be simplified for n > 0 by substituting the ratio α eff,n,A/α eff,n,B by η n,A/η n,B since α eff (Eq. 13) and η (Eq. 12) differ by constant term and constant factor only.

Fig. 6. Choosing the far-field reflection coefficient rp. (a) Probe close to the material boundary: rpr p,A, (b) Probe far from the material boundary: rp = r p,B.

3.2. Approach curves on a Au sample

In Fig.7 we show approach curves measured by s-SNOM on a flat Au film. They are obtained by switching off the closed-loop distance control of the AFM and moving the sample away from the vibrating tip. The symbols in Fig. 7(a) show the amplitude signal s 2 averaged over 100 single approach curves. The measurements were carried out with a Pt-coated Si tip (Nanosensors, PPP-NCHPt) which radius R = 20nm was estimated from the topographical resolution. The measured tip vibration amplitude was A = 18nm. The theoretical curve (solid line in Fig. 7(a)) was calculated according to our model using g = 0.7e0.06i and L = 300 nm. We find an excellent quantitative agreement. The point-dipole model in contrast predicts a much faster decay of the signal with D (dashed line in Fig. 7(a)). This is not surprising as the decay of the dipole field (D -3) is much faster than that of a monopole (D -2).

We note that the approach curves can be also fitted by a point-dipole model when the polar-izability of a spheroid is assigned to the probe dipole located in its center. The ratio between the spheroid’s major and minor axes is the fit parameter, while the apex curvature R is taken from the experiment [31

31. A. Cvitkovic, N. Ocelic, J. Aizpurua, R. Guckenberger, and R. Hillenbrand, “Infrared imaging of single nanopar-ticles via strong field enhancement in a scanning nanogap,” Phys. Rev. Lett. 97, (2006). [CrossRef] [PubMed]

]. Figure 7(b) shows the result of such a successful fit procedure. However, using the same fit parameter, this model fails to quantitatively reproduce the spectra of phonon-polariton resonant near-field interaction while our new model provides again an excellent agreement (Section 3.3).

Fig. 7. Approach curves on a gold surface demodulated at the second harmonic (n = 2) of the modulation frequency. (a) Predictions by the finite-dipole model (full line) and the point-dipole model (dashed line) compared to the experimentally obtained values (dots). All values are normalized to the signal at H = 0. (b) Experimental approach curve (red dots) fitted by a dipole model, with ratio between the spheroid’s semi-major and semi-minor axis being the fit parameter.

3.3. Near-field spectra of a phonon-polariton resonant SiC sample

One intriguing theoretical prediction of the point-dipole model is the existence of a near-field resonance when the real part of the dielectric function of the sample is close to ε′ ≈ -2, while the imaginary part is ε″ ≪ 1. Because of β = (ε - 1)/(ε + 1) the denominator in Eq. 14 becomes very small, which is also the case in our new model (Eq. 12). Such near-field resonances have been indeed observed with crystalline SiC samples when tuning the illumination wavelength from 9.2 μm to 11.2 μm [14

14. N. Ocelic and R. Hillenbrand, “Subwavelength-scale tailoring of surface phonon polaritons by focused ion-beam implantation,” Nat. Mater. 3, 606–609 (2004). [CrossRef] [PubMed]

, 15

15. A. Huber, N. Ocelic, T. Taubner, and R. Hillenbrand, “Nanoscale resolved infrared probing of crystal structure and of plasmon-phonon coupling,” Nano Lett. 6, 774–778 (2006). [CrossRef] [PubMed]

, 28

28. R. Hillenbrand, T. Taubner, and F. Keilmann, “Phonon-enhanced light-matter interaction at the nanometre scale,” Nature 418, 159–162 (2002). [CrossRef] [PubMed]

]. The observed phenomenon can be attributed to phonon-enhanced near-field interaction as it relies on the resonant near-field excitation of crystal lattice vibrations in SiC. Similar resonances can be also observed with doped semiconductor samples where the near fields of the tip excite resonant electron oscillations [16–18

16. B. Knoll and F. Keilmann, “Infrared conductivity mapping for nanoelectronics,” Appl. Phys. Lett. 77, 3980–3982 (2000). [CrossRef]

]. As the position and the strength of those near-field resonances depend on the energies of the phonon (plasmon) modes in the examined crystals (semiconductors) they are highly specific to the chemical composition of the crystal and its structure (doping concentration), just like far-field infrared spectra. This provides the opportunity for infrared identification of crystals and their structural properties with a spatial resolution of about 20 nm (λ/500) [14

14. N. Ocelic and R. Hillenbrand, “Subwavelength-scale tailoring of surface phonon polaritons by focused ion-beam implantation,” Nat. Mater. 3, 606–609 (2004). [CrossRef] [PubMed]

, 15

15. A. Huber, N. Ocelic, T. Taubner, and R. Hillenbrand, “Nanoscale resolved infrared probing of crystal structure and of plasmon-phonon coupling,” Nano Lett. 6, 774–778 (2006). [CrossRef] [PubMed]

, 28

28. R. Hillenbrand, T. Taubner, and F. Keilmann, “Phonon-enhanced light-matter interaction at the nanometre scale,” Nature 418, 159–162 (2002). [CrossRef] [PubMed]

] or the mapping of nanoscale doping gradients [18

18. A. Huber et al., “Simultaneous infrared material recognition and conductivity mapping by nanoscale near-field microscopy,” Adv. Mater. (in press).

]. Although the point-dipole model qualitatively predicts such resonances, the quantitative agreement is not yet good enough for measurement of the local dielectric function.

Our results displayed in Fig. 8 show a comparison of the theoretically predicted (lines) and experimentally observed (symbols) near-field spectra of a 4H-SiC crystal which c-axis is oriented parallel to the surface. In contrast to the point-dipole model (dashed line, Eq. 14 and Eq. 15), the finite-dipole model (solid line, Eq. 13 and Eq. 16) shows an excellent agreement in both amplitude and phase, s 2 and ϕ 2. The only deviation is observable in the phase ϕ 2 at frequencies where the amplitude s 2 is low. It is difficult to tell whether this discrepancy should be attributed to an error in the theory or in the experiment because the phase is very susceptible to random and systematic errors when the signal amplitude is low. In particular, the unsup-pressed part of the background scattering may significantly alter the measured signal phase if the near-field signal is weak.

We note that in Fig 7(b) the point-dipole model could be fitted to approach curves on a Au sample. However, as an increased distance between the probe dipole and the surface is required to achieve this, the calculation of the SiC near-field spectra (Fig. 8, doted curve) yields a resonance shift to higher frequencies [29

29. T. Taubner, F. Keilmann, and R. Hillenbrand, “Nanomechanical resonance tuning and phase effects in optical near-field interaction,” Nano Lett. 4, 1669–1672 (2004). [CrossRef]

]. For this reason the point-dipole model can not fit approach curves and near-field spectra of polariton-resonant near-field interactions simultaneously.

Fig. 8. Near-field spectra of a 4H-SiC crystal cut parallel to the c-axis. Signal demodulation was done at the second harmonic (n=2) of the tapping frequency. Shown are the experimental data (red dots) and the predictions by the finite-dipole model (solid line), the point-dipole model (dashed black line), and the point-dipole model employing the fit parameter as described in Section 3.2 (dashed blue line).

Having shown the result, we now provide more details of the experimental data acquisition and the calculation procedure. Both the experimental spectra and the calculations are normalized to a Au surface such that Eq. 18 and Eq. 19 apply. To perform a corresponding experiment we evaporated a 30 nm thick Au film onto the SiC crystal and acquired s-SNOM images of a sample area where the Au was partly removed by scratching. Amplitude s 2 and phase ϕ 2 values of SiC and Au were extracted within 200 nm distance from the sharp boundary between the two materials. To minimize the possibly disturbing excitation of surface phonon-polaritons on SiC at the material boundary [55

55. A. Huber, N. Ocelic, D. Kazantsev, and R. Hillenbrand, “Near-field imaging of mid-infrared surface phonon polariton propagation,” Appl. Phys. Lett. 87, (2005). [CrossRef]

], the sample was oriented such that the illumination direction was towards the Au film. The situation corresponds to Fig. 6(a) in case material A is the SiC and material B the Au film.

The calculation of the spectra was performed according to Eq. 13 (finite-dipole model, solid line) and Eq.14 (point-dipole model, dashed line) employing experimental parameters for the tip radius R ≈ 35nm (MikroMasch, CSC37/Ti-Pt) and the tip vibration amplitude A = 25nm. Note that the parameters g = 0.7e 0.06i and L = 300nm were taken over from Section 3.2. They were not adjusted separately for the spectra. For the dielectric function of the undoped 4H-SiC we use the single harmonic oscillator model

ε(ω)=ε+ε(ωLO2ωTO2)ωTO2ω2iγω
(20)

where ωLO is the longitudinal optical phonon frequency, ωTO the transverse optical phonon frequency and γ the damping parameter. Because of the anisotropy of the 4H-SiC crystal, we have to consider that the electrostatic reflection factor β depends on the electric field orientation [56

56. F. Engelbrecht and R. Helbig, “Effect of crystal anisotropy on the infrared reflectivity of 6H-SiC,” Phys. Rev. B 48, 15,698 – 15,707 (1993). [CrossRef]

]. When the c-axis is parallel to the surface (as with our 4H-SiC sample), the evanescent fields with polarization perpendicular to the c-axis reflect with

β=ε1ε+1,
(21)

while the evanescent fields parallel to the c-axis reflect with

β=εε1εε+1.
(22)

Due to the symmetry of the monopole/dipole field, all polarizations in the surface plane contribute equally to the near-field interaction. For the reflection factor β we thus assume an average electrostatic reflection factor

β=β+β2.
(23)

We calculate the values of ε and ε from the literature data shown in Table 1:

Table 1. Literature data used for calculating the SiC dielectric function.

table-icon
View This Table

We note that Eq. 23 is an approximation of how the electrostatic field of a point charge reflects off an anisotropic crystal. A more complete treatment of this problem is given in Refs. [60–62

60. I. V. Lindell, K. I. Nikoskinen, and M. J. Flykt, “Electrostatic image theory for an anisotropic half-space slightly deviating from transverse isotropy,” Radio Sci. 31, 1361 – 1368 (1996). [CrossRef]

] which could also be applied to our model in the future, especially for largely anisotropic materials. Due to a moderate anisotropy level in SiC, this is not deemed necessary here.

4. Conclusion

So far, nanoscale optical material characterization using s-SNOM has been limited by quantitative discrepancies between calculated and measured near-field optical contrasts. To overcome this problem, a new analytical model of the probe-sample interaction in s-SNOM has been derived in this article. Treating the probe as a prolate spheroid much smaller than the wavelength, we find that its near field is much better described by a finite dipole consisting of point charges Q 0 and -Q 0 rather than by the field a point dipole. The charge Q 0 residing close to the sample surface is assumed to induce the near-field interaction which further polarizes the spheroid. Approximating the corresponding charge redistribution by a point charge Qi in the spheroid’s focus and a uniform distribution of charge -Qi along the spheroid, we obtain a closed-form solution reproducing the experimental data. The degree of compliance with the experiment thereby depends on the model spheroid length, which was taken as a parameter and fitted to the experimental results. Interestingly, it was found to be 2L = 600 nm which is smaller than the maximum length permitted by the underlying electrostatic approximation (λ/4 ≈ 2.5μm). Future experiments or numerical studies are necessary to determine whether this is due to the neglected retardation and damping effects or whether the effective probe length is indeed much shorter than length of commonly used AFM tips (L ≥ 10μm). In the latter case, employing shorter tips would be preferable as they would produce less background scattering [42

42. N. Ocelic, “Quantitative near-field phonon-polariton spectroscopy,” Ph.D. thesis, Technical University Munich (2007).

]. The other parameter used in the model is the factor g. It determines the fraction of total charge induced in the spheroid that participates in the near-field interaction. While the real part of g was found by comparison with the exact solution for a spheroid in a monopole field, the imaginary part of g was introduced to account for the finite resistance of the probe and the radiation damping. Being very small (im(g) ≈ 0.04), in practice it only affects the magnitude of the resonant probe-sample near-field interactions, but not their spectral position. By extending our model to dielectric probes based on [46

46. I. V. Lindell, G. Dassios, and K. I. Nikoskinen, “Electrostatic image theory for the conducting prolate spheroid,” J. Phys. D-Appl. Phys. 34, 2302–2307 (2001). [CrossRef]

] and explicitly including the radiation damping, it should be possible to completely dispense with the imaginary part of g as a fit parameter. This would also enable the application of the model to visible wavelengths and to polariton-resonant probes. We note that both parameters g and L likely depend on the actual geometry of the probing tip. In our experiments we used Pt-coated Si tips rather than solid metal tips. As the skin depth of Pt at mid-IR frequencies (δ = λ/(2π im √ε Pt) ≈ 50 nm with ε Pt = - 1000+500i [63

63. M. A. Ordal et al., “Optical properties of the metals Al, Co,Cu,Au,Fe,Pb,Ni,Pd,Pt,Ag,Ti and W in the infrared and far infrared,” Appl. Opt. 22, (1983). [CrossRef] [PubMed]

]) is larger than the thickness of the Pt-coating (≈20 nm), we expect g and L also being dependent on the thickness of the coating.

Even in its present form, the finite-dipole model already promises the quantitative analysis of infrared s-SNOM images. Applied in the inverse way, it could allow the generation of nanoscale resolved maps of the dielectric function of the sample surface. Besides the chemical and structural identification, this could enable non-invasive measurements of properties such as the local strain and doping concentrations which are amongst the most important physical parameters in novel (opto-) electronic nanodevices. Another interesting application of the model would be the description of the probe-sample near-field interaction in the THz range. As typical AFM probes in that case are much shorter than the wavelength, their length could be directly employed in the model.

Acknowledgments

The authors acknowledge stimulating discussions with J. Aizpurua (San Sebastian) and P.S. Carney (Urbana-Champaign). We would also like to thank N. Issa, A. Huber, F. Keilmann (all Martinsried) and T. Taubner (Stanford University) for critical comments on the manuscript. Financial support by BMBF grant no. 03N8705 and the German Excellence Initiative via the “Nanosystems Initiative Munich (NIM)” is gratefully acknowledged.

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A. Cvitkovic, N. Ocelic, J. Aizpurua, R. Guckenberger, and R. Hillenbrand, “Infrared imaging of single nanopar-ticles via strong field enhancement in a scanning nanogap,” Phys. Rev. Lett. 97, (2006). [CrossRef] [PubMed]

32.

J. L. Bohn, D. J. Nesbitt, and A. Gallagher, “Field enhancement in apertureless near-field scanning optical microscopy,” J. Opt. Soc. Am. A-Opt. Image Sci. Vis. 18, 2998–3006 (2001). [CrossRef]

33.

J. Renger, S. Grafstrom, L. M. Eng, and R. Hillenbrand, “Resonant light scattering by near-field-induced phonon polaritons,” Phys. Rev. B 71, (2005).

34.

S. V. Sukhov, “Role of multipole moment of the probe in apertureless near-field optical microscopy,” Ultrami-croscopy 101, 111–122 (2004). [CrossRef]

35.

H. Hatano and S. Kawata, “Applicability of deconvolution and nonlinear optimization for reconstructing optical images from near-field optical microscope images,” J. Microsc.-Oxf. 194, 230 – 234 (1999). [CrossRef]

36.

M. Labardi, S. Patane, and M. Allegrini, “Artifact-free near-field optical imaging by apertureless microscopy,” Appl. Phys. Lett. 77, 621–623 (2000). [CrossRef]

37.

R. Hillenbrand and F. Keilmann, “Complex optical constants on a subwavelength scale,” Phys. Rev. Lett. 85, 3029–3032 (2000). [CrossRef] [PubMed]

38.

J. N. Walford, J. A. Porto, R. Carminati, J. J. Greffet, P. M. Adam, S. Hudlet, J. L. Bijeon, A. Stashkevich, and P. Royer, “Influence of tip modulation on image formation in scanning near-field optical microscopy,” J. Appl. Phys. 89, 5159–5169 (2001). [CrossRef]

39.

A. Wokaun, “Surface enhancement of optical-fields - mechanism and applications,” Mol. Phys. 56, 1 – 33 (1985). [CrossRef]

40.

W. Denk and D. Pohl, “Near-field optics - microscopy with nanometer-size fields,” J. Vac. Sci. Technol. B 9, 510–513 (1991). [CrossRef]

41.

N. Calander and M. Willander, “Theory of surface-plasmon resonance optical-field enhancement at prolate spheroids,” J. Appl. Phys. 92, 4878–4884 (2002). [CrossRef]

42.

N. Ocelic, “Quantitative near-field phonon-polariton spectroscopy,” Ph.D. thesis, Technical University Munich (2007).

43.

Y. C. Martin, H. F. Hamann, and H. K. Wickramasinghe, “Strength of the electric field in apertureless near-field optical microscopy,” J. Appl. Phys. 89, 5774–5778 (2001). [CrossRef]

44.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons Inc, 1998). [CrossRef]

45.

J. D. Jackson, Classical Electrodynamics (Wiley & Sons, 1998).

46.

I. V. Lindell, G. Dassios, and K. I. Nikoskinen, “Electrostatic image theory for the conducting prolate spheroid,” J. Phys. D-Appl. Phys. 34, 2302–2307 (2001). [CrossRef]

47.

D. V. Redzic, “An electrostatic problem - a point-charge outside a prolate dielectric spheroid,” Am. J. Phys. 62, 1118 – 1121 (1994). [CrossRef]

48.

J. C. E. Sten and I. V. Lindell, “An electrostatic image solution for the conducting prolate spheroid,” J. Electro-magn. Waves Appl. 9, 599 – 609 (1995).

49.

D. V. Redzic, “Image of a moving spheroidal conductor,” Am. J. Phys. 60, 506–508 (1992). [CrossRef]

50.

Z. H. Kim and S. R. Leone, “High-resolution apertureless near-field optical imaging using gold nanosphere probes,” J. Phys. Chem. B 110, 19,804–19,809 (2006).

51.

J. Aizpurua, personal communication (2005).

52.

N. Ocelic, A. Huber, and R. Hillenbrand, “Pseudoheterodyne detection for background-free near-field spec-troscopy,” Appl. Phys. Lett. 89, (2006). [CrossRef]

53.

R. Bachelot, G. Wurtz, and P. Royer, “An application of the apertureless scanning near-field optical microscopy: imaging a GaAlAs laser diode in operation,” Appl. Phys. Lett. 73, 3333–3335 (1998). [CrossRef]

54.

B. Knoll and F. Keilmann, “Electromagnetic fields in the cutoff regime of tapered metallic waveguides,” Opt. Commun. 162, 177–181 (1999). [CrossRef]

55.

A. Huber, N. Ocelic, D. Kazantsev, and R. Hillenbrand, “Near-field imaging of mid-infrared surface phonon polariton propagation,” Appl. Phys. Lett. 87, (2005). [CrossRef]

56.

F. Engelbrecht and R. Helbig, “Effect of crystal anisotropy on the infrared reflectivity of 6H-SiC,” Phys. Rev. B 48, 15,698 – 15,707 (1993). [CrossRef]

57.

H. Mutschke, A. C. Andersen, D. Clement, T. Henning, and G. Peiter, “Infrared properties of SiC particles,” Astron. Astrophys. 345, 187–202 (1999).

58.

M. Hofmann, A. Zywietz, K. Karch, and F. Bechstedt, “Lattice-dynamics of SiC polytypes within the bond-charge model,” Phys. Rev. B 50, 13,401–13,411 (1994). [CrossRef]

59.

H. Harima, S. Nakashima, and T. Uemura, “Raman-scattering from anisotropic LO-phonon-plasmon-coupled mode in n-type 4H-SiC and 6H-SiC,” J. Appl. Phys. 78, 1996–2005 (1995). [CrossRef]

60.

I. V. Lindell, K. I. Nikoskinen, and M. J. Flykt, “Electrostatic image theory for an anisotropic half-space slightly deviating from transverse isotropy,” Radio Sci. 31, 1361 – 1368 (1996). [CrossRef]

61.

I. V. Lindell, K. I. Nikoskinen, and A. Viljanen, “Electrostatic image method for the anisotropic half space,” IEE Proc.-Sci. Meas. Technol. 144, 156 – 162 (1997). [CrossRef]

62.

S. C. Schneider, S. Grafstrom, and L. M. Eng, “Scattering near-field optical microscopy of optically anisotropicsystems,” Phys. Rev. B 71, (2005). [CrossRef]

63.

M. A. Ordal et al., “Optical properties of the metals Al, Co,Cu,Au,Fe,Pb,Ni,Pd,Pt,Ag,Ti and W in the infrared and far infrared,” Appl. Opt. 22, (1983). [CrossRef] [PubMed]

OCIS Codes
(110.3080) Imaging systems : Infrared imaging
(120.5820) Instrumentation, measurement, and metrology : Scattering measurements
(180.5810) Microscopy : Scanning microscopy
(300.6340) Spectroscopy : Spectroscopy, infrared

ToC Category:
Microscopy

History
Original Manuscript: March 28, 2007
Revised Manuscript: May 6, 2007
Manuscript Accepted: May 10, 2007
Published: June 25, 2007

Virtual Issues
Vol. 2, Iss. 8 Virtual Journal for Biomedical Optics

Citation
A. Cvitkovic, N. Ocelic, and R. Hillenbrand, "Analytical model for quantitative prediction of material contrasts in scattering-type near-field optical microscopy," Opt. Express 15, 8550-8565 (2007)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-14-8550


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References

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  31. A. Cvitkovic, N. Ocelic, J. Aizpurua, R. Guckenberger, and R. Hillenbrand, "Infrared imaging of single nanoparticles via strong field enhancement in a scanning nanogap," Phys. Rev. Lett. 97, (2006). [CrossRef] [PubMed]
  32. J. L. Bohn, D. J. Nesbitt, and A. Gallagher, "Field enhancement in apertureless near-field scanning optical microscopy," J. Opt. Soc. Am. A-Opt. Image Sci. Vis. 18, 2998-3006 (2001). [CrossRef]
  33. J. Renger, S. Grafstrom, L. M. Eng, and R. Hillenbrand, "Resonant light scattering by near-field-induced phonon polaritons," Phys. Rev. B 71, (2005).
  34. S. V. Sukhov, "Role of multipole moment of the probe in apertureless near-field optical microscopy," Ultramicroscopy 101, 111-122 (2004). [CrossRef]
  35. H. Hatano and S. Kawata, "Applicability of deconvolution and nonlinear optimization for reconstructing optical images from near-field optical microscope images," J. Microsc.-Oxf. 194, 230 - 234 (1999). [CrossRef]
  36. M. Labardi, S. Patane, and M. Allegrini, "Artifact-free near-field optical imaging by apertureless microscopy," Appl. Phys. Lett. 77, 621-623 (2000). [CrossRef]
  37. R. Hillenbrand and F. Keilmann, "Complex optical constants on a subwavelength scale," Phys. Rev. Lett. 85, 3029-3032 (2000). [CrossRef] [PubMed]
  38. J. N. Walford, J. A. Porto, R. Carminati, J. J. Greffet, P. M. Adam, S. Hudlet, J. L. Bijeon, A. Stashkevich, and P. Royer, "Influence of tip modulation on image formation in scanning near-field optical microscopy," J. Appl. Phys. 89, 5159-5169 (2001). [CrossRef]
  39. A. Wokaun, "Surface enhancement of optical-fields - mechanism and applications,"Mol. Phys. 56, 1 - 33 (1985). [CrossRef]
  40. W. Denk and D. Pohl, "Near-field optics - microscopy with nanometer-size fields," J. Vac. Sci. Technol. B 9, 510-513 (1991). [CrossRef]
  41. N. Calander and M. Willander, "Theory of surface-plasmon resonance optical-field enhancement at prolate spheroids," J. Appl. Phys. 92, 4878-4884 (2002). [CrossRef]
  42. N. Ocelic, "Quantitative near-field phonon-polariton spectroscopy," Ph.D. thesis, Technical University Munich (2007).
  43. Y. C. Martin, H. F. Hamann, and H. K. Wickramasinghe, "Strength of the electric field in apertureless near-field optical microscopy," J. Appl. Phys. 89, 5774-5778 (2001). [CrossRef]
  44. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons Inc, 1998). [CrossRef]
  45. J. D. Jackson, Classical Electrodynamics (Wiley & Sons, 1998).
  46. I. V. Lindell, G. Dassios, and K. I. Nikoskinen, "Electrostatic image theory for the conducting prolate spheroid," J. Phys. D-Appl. Phys. 34, 2302-2307 (2001). [CrossRef]
  47. D. V. Redzic, "An electrostatic problem - a point-charge outside a prolate dielectric spheroid," Am. J. Phys. 62, 1118 - 1121 (1994). [CrossRef]
  48. J. C. E. Sten and I. V. Lindell, "An electrostatic image solution for the conducting prolate spheroid," J. Electromagn. Waves Appl. 9, 599 - 609 (1995).
  49. D. V. Redzic, "Image of a moving spheroidal conductor," Am. J. Phys. 60, 506-508 (1992). [CrossRef]
  50. Z. H. Kim and S. R. Leone, "High-resolution apertureless near-field optical imaging using gold nanosphere probes," J. Phys. Chem. B 110, 19,804-19,809 (2006).
  51. J. Aizpurua, personal communication (2005).
  52. N. Ocelic, A. Huber, and R. Hillenbrand, "Pseudoheterodyne detection for background-free near-field spectroscopy," Appl. Phys. Lett. 89, (2006). [CrossRef]
  53. R. Bachelot, G. Wurtz, and P. Royer, "An application of the apertureless scanning near-field optical microscopy: imaging a GaAlAs laser diode in operation," Appl. Phys. Lett. 73, 3333-3335 (1998). [CrossRef]
  54. B. Knoll and F. Keilmann, "Electromagnetic fields in the cutoff regime of tapered metallic waveguides," Opt. Commun. 162, 177-181 (1999). [CrossRef]
  55. A. Huber, N. Ocelic, D. Kazantsev, and R. Hillenbrand, "Near-field imaging of mid-infrared surface phonon polariton propagation," Appl. Phys. Lett. 87, (2005). [CrossRef]
  56. F. Engelbrecht and R. Helbig, "Effect of crystal anisotropy on the infrared reflectivity of 6H-SiC," Phys. Rev. B 48, 15,698 - 15,707 (1993). [CrossRef]
  57. H. Mutschke, A. C. Andersen, D. Clement, T. Henning, and G. Peiter, "Infrared properties of SiC particles," Astron. Astrophys. 345, 187-202 (1999).
  58. M. Hofmann, A. Zywietz, K. Karch, and F. Bechstedt, "Lattice-dynamics of SiC polytypes within the bondcharge model," Phys. Rev. B 50, 13,401-13,411 (1994). [CrossRef]
  59. H. Harima, S. Nakashima, and T. Uemura, "Raman-scattering from anisotropic LO-phonon-plasmon-coupled mode in n-type 4H-SiC and 6H-SiC," J. Appl. Phys. 78, 1996-2005 (1995). [CrossRef]
  60. I. V. Lindell, K. I. Nikoskinen, and M. J. Flykt, "Electrostatic image theory for an anisotropic half-space slightly deviating from transverse isotropy," Radio Sci. 31, 1361 - 1368 (1996). [CrossRef]
  61. I. V. Lindell, K. I. Nikoskinen, and A. Viljanen, "Electrostatic image method for the anisotropic half space," IEE Proc.-Sci. Meas. Technol. 144, 156 - 162 (1997). [CrossRef]
  62. S. C. Schneider, S. Grafstrom, and L. M. Eng, "Scattering near-field optical microscopy of optically anisotropic systems," Phys. Rev. B 71, (2005). [CrossRef]
  63. M. A. Ordal et al., "Optical properties of the metals Al, Co,Cu,Au,Fe,Pb,Ni,Pd,Pt,Ag,Ti and W in the infrared and far infrared," Appl. Opt. 22, (1983). [CrossRef] [PubMed]

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