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

Per Martinsson; Hanna Lajunen; Ari T. Friberg

doi:10.1364/OE.14.011392

1. Introduction

In conventional optical far-field microscopy the resolution is limited by the Abbe diffraction barrier to approximately half the wavelength of the light. Although advanced methods to break this barrier have been developed [1], in most far-field imaging systems the Abbe-Rayleigh criterion substantially holds. Optical near-fields, on the other hand, deal with light waves that exist at sub-wavelength distances from material (emitting or scattering) bodies. By making use of the high spatial-frequency content of the field, near-field optical microscopy is known to provide a resolution far beyond the Abbe limit [2, 3]. In classical resolution theory these evanescent (or non-radiating) components are disregarded, since they do not contribute to the final image. To analyze the near-field resolution, such as in scanning near-field optical microscopy (SNOM), the conventional Abbe theory does not apply and the evanescent waves must duely be taken into account.

The so-called communication modes are a useful technique for the studies of free-space wave propagation, spatial channels, and optical information [4, 5]. This approach, which is applicable in the far-field and near-field geometries alike, is less intuitive than Abbe’s method, but it gives a more complete and accurate description of the information capacity and transfer of the field. For example, it can yield the best possible approximation for a given target field distribution [6]. Moreover, the communication-modes method does not exclude evanescent waves, even though the requirement of a finite signal-to-noise ratio cuts off the weakest-coupled modes, i.e., not all evanescent components necessarily contribute significantly to the near-field image.

In this paper we analyze the resolution of a scanning optical near-field system on the basis on the rigorous theory of communication modes. Although the interaction of the electric field with the sample and the tip obviously are of fundamental importance in SNOMimaging [7], we are more interested in the spatial information-theoretic properties of the field itself which are primarily determined by the imaging geometry. Therefore we choose a characteristic SNOM situation and make use of the optical scalar-wave theory in two dimensions, which contains the salient features of light. The communication modes can also be formulated in electromagnetic theory [5], and our results then correspond to SNOM imaging using s polarized light. The main goal of this work is to find the relation of the achievable scanning near-field resolution to the system’s scalar communication modes.

2. Near-field communication modes

To bring out the essential properties of the communication modes in a compact form we make use of the scalar theory of optics and consider a system that depends only on one transverse coordinate, say 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 (x) = \int_{S} G (x, ρ_{x}) U_{0} (ρ_{x}) d ρ_{x},

where U ₀(ρ_x ) is the sample field and G(x,ρ_x ) is the half-space Green function [8, 9]

G (x, ρ_{x}) = \frac{i k z_{a}}{2} \frac{H_{1}^{(1)} {k {[{(x - ρ_{x})}^{2} + z_{a}^{2}]}^{\frac{1}{2}}}}{{[{(x - ρ_{x})}^{2} + z_{a}^{2}]}^{\frac{1}{2}}} .

Here k=ω/c=2π/λ is the wave vector of the light and $H_{1}^{(1)}$ (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, 11]

G (x, ρ_{x}) = \sum_{n = 0}^{\infty} g_{n} ϕ_{n} (x) ψ_{n}^{*} (ρ_{x}),

where the asterisk denotes the complex conjugate and 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]:

\int_{S} G (x, ρ_{x}) ψ_{n} (ρ_{x}) d ρ_{x} = g_{n} ϕ_{n} (x),

\int_{A} G * (x, ρ_{x}) ϕ_{n} (x) d x = g_{n}^{*} ψ_{n} (ρ_{x}) .

Equation (3) is known as the Schmidt expansion of a non-Hermitian kernel and Eqs. (4) and (5) are called the shifted eigenvalue equations [13].

On direct substitutions between Eqs. (4) and (5) we readily find the integral equations for the sample and detector singular functions,

{∣ g_{n} ∣}^{2} ψ_{n} (ρ_{x}) = \int_{S} K_{s} (ρ_{x}, ρ_{x}^{'}) ψ_{n} (ρ_{x}^{'}) d ρ_{x}^{'},

{∣ g_{n} ∣}^{2} ϕ_{n} (x) = \int_{A} K_{a} (x, x') ϕ_{n} (x') d x',

respectively, where the kernels are

K_{s} (ρ_{x}, ρ_{x}^{'}) = \int_{A} G * (x, ρ_{x}) G (x, ρ_{x}^{'}) d x,

K_{a} (x, x') = \int_{S} G (x, ρ_{x}) G * (x', ρ_{x}) d ρ_{x} .

Both functions 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.,

\int_{S} ψ_{n}^{*} (ρ_{x}) ψ_{m} (ρ_{x}) d ρ_{x} = \int_{A} ϕ_{n}^{*} (x) ϕ_{m} (x) d x = δ_{nm},

where δ_nm is the Kronecker delta function. Within the usual Fresnel and Fraunhofer diffraction regimes the eigenfunctions are related to prolate spheroidal wave functions (PSWFs) [14].

In near-field geometries, the modes and coupling coefficients can be calculated numerically by representing the Green function in Eq. (2) as a finite (discretized) matrix and performing on it the singular value decomposition [15]. In our computations we used the standard Matlab routine svd. Examples of the best-connected modes (largest values of |g_n |), obtained in this way for a sample extent of 10λ and a receiving aperture width of 0.1λ, are illustrated in Figs. 1 and 2. The distance between the sample and the detector is z_a =0.05λ. The corresponding coupling coefficients are shown in Fig. 3.

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 z_a =λ/20.

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Fig. 2. Real and imaginary parts of the first four receiving modes. The parameters are the same as in Fig 1.

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For a typical scanning near-field geometry, in which the receiving aperture (detector) size is much smaller than the sample width, the lateral position of the detector has little effect on the shape of the transmitting modes, due to the small width of these modes. This is illustrated more explicitly in Fig. 4 for the zero-order mode. The effects of the distance between the sample and the detector aperture are illustrated in Figs. 3 and 5. The coupling coefficients fall off towards zero more slowly as the distance decreases, while the modes become ever narrower.

Fig. 3. Normalized coupling strengths for various distances z_a between sample and receiving aperture. The curve with squares corresponds to the modes shown in Figs. 1 and 2.

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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 z_a =λ/20.

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3. Limit of small detector size

When the width A of the receiving aperture, centered at a position x ₀, becomes very small, we evidently may approximate

G (x, ρ_{x}) \approx G (x_{0}, ρ_{x}),

in which case

K_{s} (ρ_{x}, ρ_{x}^{'}) \approx G * (x_{0}, ρ_{x}) G (x_{0}, ρ_{x}^{'}) \int_{A} d x = A G * (x_{0}, ρ_{x}) G (x_{0}, ρ_{x}^{'}) .

Consequently, the integral equation for the transmitting modes, Eq. (6), then takes on the form

{∣ g_{n} ∣}^{2} ψ_{n} (ρ_{x}) = G^{*} (x_{0}, ρ_{x}) A \int_{S} G (x_{0}, ρ_{x}^{'}) ψ_{n} (ρ_{x}^{'}) d ρ_{x}^{'} .

Fig. 5. Variation of the zero-order mode (absolute value) as a function of distance z_a , for a sample width 10λ and aperture size λ/10.

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We see that in this case the only mode solution is

ψ_{n} (ρ_{x}) = ψ_{0} (ρ_{x}) = C^{- \frac{1}{2}} G^{*} (x_{0}, ρ_{x}),

with the normalization

C = \int_{S} {∣ G (x_{0}, ρ_{x}) ∣}^{2} d ρ_{x},

and the corresponding coupling coefficient is

{∣ g_{n} ∣}^{2} = {∣ g_{0} ∣}^{2} = A \int_{S} {∣ G (x_{0}, ρ_{x}) ∣}^{2} d ρ_{x} = C A .

Our analysis thus shows the important result that when the detector size becomes increasingly small, as in scanning near-field optical imaging, only one transmitting aperture mode survives and it becomes effectively identical with the normalized form of the complex conjugate of the Green function between the sample aperture and the small detector. These results are confirmed by more rigorous numerical calculations in Figs. 6 and 7, which illustrate the variation of the zero-order mode and the coupling coefficients as the detector size is reduced.

4. Field propagation

On having examined the nature of the communication modes and coupling coefficients we may now consider the field propagation and optical resolution in a scanning near-field configuration. The field at the sample (object) is first expanded in the communications modes as

U_{0} (ρ_{x}) = \sum_{n = 0}^{\infty} a_{n} ψ_{n} (ρ_{x}),

where

a_{n} = \int_{S} U_{0} (ρ_{x}) ψ_{n}^{*} (ρ_{x}) d ρ_{x} .

Using Eqs. (1) and (4), the wave propagation then reduces to a simple multiplication, i.e.,

U (x) = \sum_{n = 0}^{\infty} a_{n} \int_{S} G (x, ρ_{x}) ψ_{n} (ρ_{x}) d ρ_{x} = \sum_{n = 0}^{\infty} a_{n} g_{n} ϕ_{n} (x) .

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.

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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.

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We note that Eqs. (17)–(19) are fully consistent with the bi-othogonal expansion of the Green function in Eq. (3).

In general, the modulus of the coupling coefficients gn decreases with increasing mode number 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

Let the sample field now be a point object, i.e., the field in the transmitting aperture is U ₀(ρ_x )=δ(ρ_x -ρ_x 0). The expansion coefficients a_n from Eq. (18) then are

a_{n} = \int_{S} ψ_{n}^{*} (ρ_{x}) δ (ρ_{x} - ρ_{x 0}) d ρ_{x} = ψ_{n}^{*} (ρ_{x 0}) .

The detector is taken to be a simple bucket detector that measures the total optical intensity, integrated over its planar aperture at a distance z=z_a , of the strongly contributing modes, i.e.,

I_{a} = \int_{A} {∣ \sum_{n = 0}^{N} a_{n} g_{n} ϕ_{n} (x) ∣}^{2} d x = \sum_{n = 0}^{N} {∣ a_{n} ∣}^{2} {∣ g_{n} ∣}^{2} = \sum_{n = 0}^{N} {∣ ψ_{n} (ρ_{x 0}) ∣}^{2} {∣ g_{n} ∣}^{2},

where, in the second step, we have used the orthonormality of the modes. When the detector is scanned over the sample, the intensity in Eq. (21) is given as a function of the position (ξ, z_a ) of the tip as

I_{a} (ξ) = \sum_{n = 0}^{N} {∣ ψ_{n}^{ξ} (ρ_{x 0}) ∣}^{2} {∣ g_{n}^{ξ} ∣}^{2},

where the superscript 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.

However, if the sample is sufficiently large (as is the case in scanning near-field microscopy), it is reasonable to assume that the modes are displaced but otherwise do not change significantly with the position of the detector. This was explicitly demonstrated in Fig. 4 for the lowest-order mode. Likewise, the coupling coefficients remain substantially unchanged during the detector movement. Hence, we may make the approximations

ψ_{n}^{ξ} (ρ_{x 0}) \approx ψ_{n}^{0} (ρ_{x 0} - ξ) = ψ_{n} (ρ_{x 0} - ξ),

and

g_{n}^{ξ} \approx g_{n}^{0} = g_{n},

and the scanned intensity in Eq. (22) then becomes

I_{a} (ξ) = \sum_{n = 0}^{N} {∣ ψ_{n} (ρ_{x 0} - ξ) ∣}^{2} {∣ g_{n} ∣}^{2} .

Let us next consider the effects of the mode widths and the coupling coefficients in Eq. (25). As Fig. 1 illustrates, the lowest-order mode is the narrowest but the mode broadening as 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.

It was shown in Section 3 that only a single (the lowest-order) mode appears, in particular, when the detector aperture is small, as is the case in SNOM imaging. Moreover, the mode then by and large is the Green function, as is illustrated in Fig. 6. In optical near-field imaging we may therefore take N=0 and obtain from Eq. (25) for the detector-scanned total intensity the approximation

I_{a} (ξ) = {∣ ψ_{0} (ρ_{x 0} - ξ) ∣}^{2} {∣ g_{0} ∣}^{2} {\propto ∣ G (ρ_{x 0} - ξ) ∣}^{2},

where in the proportionality we also used Eq. (14). The transmitting mode 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.

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.

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To determine the lateral two-point resolution of a near-field imaging system more precisely, we consider a source field consisting of two point sources, i.e.,

U_{0} (ρ_{x}) = δ (ρ_{x} - ρ_{x 1}) + δ (ρ_{x} - ρ_{x 2})

The point sources are taken mutually correlated and in phase, since normally the sample is illuminated by an external light wave and we consider sub-wavelength resolution. As above in the case of a single point object, we now find for the detector-scanned total optical intensity the expression

I_{a} (ξ) \propto {∣ ψ_{0} (ρ_{x 1} - ξ) + ψ_{0} (ρ_{x 2} - ξ) ∣}^{2} .

The resolution is thus given, approximately, by the distance at which two overlapping squared zero-order modes can be distinguished. The exact value of this distance for a given geometry and wavelength is not well defined, but in analogy with the Rayleigh criterion we may choose as our resolution condition that the central minimum is less than 81% of the maxima. In view of Eq. (26), the resolution is given by the overlap of two Green functions when the detector size approaches zero.

In Fig. 9 we present a numerical calculation of the detector-scanned optical intensity in the case of two point objects for z_a =λ/20, A=λ/10, and S=10λ. 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], finite-difference time-domain (FDTD) [17], or integral techniques [18].

Fig. 9. Detector-scanned total optical intensity profiles for two point objects of separations δρ_x ranging between 0.05λ and 0.2λ. The system parameters are as in Fig. 8.

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6. Conclusions

We have presented the so-called communication modes for a simple one-dimensional model of an aperture SNOM. The small width of the detector aperture (fiber tip) results in narrow modes on the sample side. The resolution of the near-field imaging system can be related to the width of these modes. In the small-aperture limit the lowest-order mode coincides with the Green function, evaluated at the center point of the aperture, and therefore the resolution is ultimately limited by the Green function.

It is interesting that the information contained in the spatial variation of the object field is contained in the excitation of the sample modes. Analysis of detailed imaging by an aperture SNOM therefore necessarily includes the object properties and the contributions and properties of the higher-order modes become important. For such an imaging arrangements the signalto-noise ratio would be an important factor. The limiting factor for two-point resolution of an aperture SNOM, however, does not appear to be the coupling strengths (and their magnitudes compared to S/N), but rather the width of the transmitting modes.

Acknowledgments

P. Martinsson thanks the Swedish Research Council (VR) for financial support. H. Lajunen acknowledges the Academy of Finland (project 111701), the Network of Excellence on Micro-Optics (NEMO, http://www.micro-optics.org), and grants from the Helsingin Sanomat Centennial Foundation and the Vilho, Yrjö, and Kalle Väisälä Foundation of the Finnish Academy of Science and Letters. A. T. Friberg is grateful for funding from the Swedish Foundation for Strategic Research (SSF).

Footnotes

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

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

Abstract

1. Introduction

2. Near-field communication modes

3. Limit of small detector size

4. Field propagation

5. Near-field resolution

6. Conclusions

Acknowledgments

Footnotes

References and links

Cited By

Figures (9)

Equations (28)

Optics Express