1. Introduction

Near-field scanning optical microscopy (NSOM) has been a field of rapid growth for the past ten years [1–2]. By detecting the evanescent wave in the vicinity of the sample surface with a minuscule probe-tip, NSOM is able to generate an optical image with sub-wavelength resolution. Typically two kinds of probes are used in NSOM, those with apertures and those without apertures. The aperture-probe [1] is usually a tapered monomode optical fiber, while the apertureless probe [2] is usually the probe tip of a conventional atomic force microscope locating and scanning the sample surface in the near-field region. Particularly, the NSOM accompanying an apertureless probe is commonly called a scattering near-field scanning optical microscopy (s-NSOM). For the conventional s-NSOM, the detector is used only to receive the energy of the scattered light from the probe-surface system, irrelevant to the polarization state of the scattered light, i.e. only the intensity of the near field was measured. However recently, new techniques successfully realized integrating an interferometer on the base of an s-NSOM, where the detector receives the interference between the scattered light and the reference light, measuring the near-field intensity and the near-field phase simultaneously [3–7]. In comparison to the conventional s-NSOM, the s-NSOM interferometer has many advantages, such as high image resolution, improved image contrast, and high immunity to artifacts. It is well known that the interference between two beams of light strongly depends on their polarization states, and only the components that are polarized in the same direction can interfere. Thus for the s-NSOM interferometer, only the component with the same polarization direction to the reference light will interfere at the detector. Understanding the basic principles, a series of images can be obtained by simply changing the polarization-direction of the reference light for a given probe-sample system. Comparing the relation between the polarized images is of great interest, since it can help to accurately interpret the images and develop better ways to construct the s-NSOM interferometer of high accuracy. To our knowledge, not many researches have been reported on this subject up to date. For a clear insight into the physics of this question, the present paper uses a rigorous numerical method to study the relation and the difference between the diversely polarized images of a specimen. It was found that for a given specimen the near-field polarized images were remarkably different from each other depending on their polarizations and the position of the detector. We have found that in order to realize high-accuracy imaging, the reference light should be polarized in the detecting plane, which is vertical to the sample plane and contains both the detection point and the probe-tip.

2. Theory

In this section, the principle and the simulation model for the s-NSOM interferometer are described in sub-section 2.1, and the calculation formulas are briefly introduced in sub-section 2.2. The main purpose of this paper is to find the fundamental properties of the s-SNOM interferometer using the point-dipole simulation model, which has been successfully used to illustrate common s-NSOM. Both the calculation formulas and the simulation model itself are only briefly outlined, where details can be found in cited references.

2.1 Principle and simulation model

The principle of the s-NSOM interferometer is schematically shown in Fig. 1(a), where the light from a laser is divided into two beams. One beam of the light is incident on a prism to form an evanescent wave for the illumination of the specimen, and another beam of light is incident on the far-field detector surface, acting as the reference light. The near-field light on the surface of the sample, which contains the information of the sub-wavelength features of the specimen, interacts with the probe tip in the near-field region and is scattered as propagating waves expressed as Sexp(iφ). S and φ are the amplitude and the phase of the scattered light. The scattering light is focused by an objective lens to the detector surface, where it interferes with the reference light. If the probe tip dithers vertically to the sample surface at frequency Ω, the intensity-signal received by the detector will be modulated at sum frequencies of nΩ (n=1,2,…) [8]. When the detected signal is measured by a lock-in amplifier operating at the frequency of Ω (or the sum frequency nΩ), the phase φ and the amplitude S of the scattered near field are simultaneously determined.

 

Fig. 1. Setup of the s-SNOM interferometer (a) and the simulation model (b)

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Demonstrated by Dr. Mufei and his coworkers [9–11], the setup shown in Fig. 1(a) can be modeled with Fig. 1(b), where the surface features with sub-wavelength details are represented by a group of tiny spheres lying contact with the surface, and the probe is represented by a tiny sphere scanning at a constant distance d from the surface. The interference between the radiation of the surface-probe system and the reference light is assumed to be detected by a far-field point detector, which is placed at a macroscopic distance D from the probe and with a view angle of (α,β,θ) to the x, y, and z axes respectively. Since the scanning probe sphere and the surface spheres are all in the near-field range, they interact with each other and change the original incident field. For a given probe-position, the induced near field signal can be rigorously calculated with a coupled-dipole model, and all polarization components of the induced far-field light received by the detector can be determined. Hence, when the whole the specimen is numerically scanned by the probe sphere in the near-field region, its polarized near-field intensity and phase images generated by a s-NSOM interferometer can be rigorously simulated, revealing some basic properties of s-NSOM inerferometry.

2.2. Calculation formulas

Assuming that the probe-surface system is illuminated with a monochromatic incident light with an angular frequency denoted by ω, the induced electric field E(r _j,ω) of the jth sphere is [10–12]:

where µ₀is the permeability of vacuum and N is the number of the discrete spheres including the probe sphere. E ⁰(r _j,ω) is the field that would prevail at r _j if the surface spheres and the probe sphere were absent. G(r,r ^′) is the well-known screened electromagnetic Green’s function, which describes the field propagating from source point at r’ to a view point at r. α _i(ω) is the local conductivity response tensor of the ith sphere, if both the probe sphere and the surface spheres are assumed to be isotropic. The local conductivity response tensor of the probe sphere has the form of α _prob(ω)=-i4πε₀${\mathrm{a}}_{\text{prob}}^3$[ε(ω)-1]/[ε(ω)+2]U, where the a _prob is the radius of the probe sphere, ε(ω) is the electric function of the probe material, and the U is a unite tensor. For the surface spheres, since the influence of the plate surface on their conductivity response tensor must be taken into consideration, α_sur(ω) has the form [9]

where δ is the center to surface distance between the bulk and the surface-dipoles, and ε_b(ω) is the bulk electric function; r ^p(ω) is the p-polarized reflection coefficient for the surface, which can be calculated as r^p..£ω£©=[ε_b(ω)-]/[ε_b(ω)+1]. has the form of α(ω)=-i4πε₀${\mathrm{a}}_{\text{sur}}^3$[ε_b(ω)-1]/[ε_b(ω)+2] and a_sur is the radius of the surface spheres.

To resolve the self-consist equations of Eq. (1), we can rewrite it as follows

where, ξ and ξ⁰ are the super-vectors corresponding E(r_j,ω) to and E⁰(r_j,ω) in Eq. (1) respectively. ℘ is a super-tensor determined by G(r _j , r _i , ω)·α_i (ω). The exact solution of ξ can then be expressed in the form

Eq. (4) gives the rigorous induced electronic field E(r _j,ω) of all the tiny spheres, including the probe sphere and the surface spheres in Fig. 1(b). Then, the three polarized components (in x, y and z directions) of the scattering field at the detector position r (a point detector is assumed) can be calculated with the point-dipole model.

The phase and the intensity distributions of the near-field polarized image of the specimen can be directly determined by computing the phase and the absolute values of the E_x (r,ω), E_y(r,ω) and E_z (r,ω) respectively. The total intensity of the radiation, which is measured by conventional s-NSOM, can also be determined by computing |E_x(r,ω)|²+|E_y(r,ω)|²+|E_z(r,ω)|².

In the computation of Eq. (5), both the propagating and the standing terms are included in the Green-tensor G(r,r_j,ω), but because the standing term decays exponentially with the distance, the calculated electronic field of Eq. (5) essentially contains the contribution of the propagating terms only, because the distance (r in Eq. (5)) between the detector and the probe-surface system is much larger than the wavelength.

3. Numerical results and remarks

3.1 Simulation Parameters

This section presents the numerical results calculated by using the formulas introduced in Section 2 for a sample system shown in Fig. 2, where 25 gold spheres are distributed in a rectangular lattice with an area of 140nm×140nm above a gold substrate. The radius of each sphere is 10nm, and the surface-to-surface distance between neighboring spheres is 10 nm. The topography of this sphere system is shown in Fig. 2(b). The probe sphere (assumed to be glass) of 10nm radius scans the specimen under a so-called constant-distance model. The geometric relation between the probe and the surface spheres is schematically shown in Fig. 2(c), where the surface-surface distance between the probe sphere and the sample sphere was always kept 10 nm in the vertical direction. The detector is placed at macroscopic distance (D=2cm) from the probe-surface system, with a view angle of (α,β,θ) to the three axes. The phase and the intensity distributions of different polarized components of scattering field at the detector point (E _x, E _y and E _z in Eq. (5)) are considered as the signals detected by the s-NSOM interferometers. For each given detector position, we have seven images, namely, one image formed by the intensity of the total radiation, three intensity images and three phase images formed by the x, y and z polarized components respectively.

 

Fig. 2. (a) distribution of the surface-spheres. (b) The topography of the spheres. (c) geometric relation between the probe and the surface spheres

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In Fig. 1(a), the sample is illuminated by an evanescent field established by the total internal reflection. The evanescent field is exponentially decaying in the direction normal to the surface and polarized vertically to the surface if the surface spheres are absent. Since our sample surface is assumed to have topographic features it is difficult to calculate the evanescent field accurately. To simplify the problem, we assume a unit field at the sites of the surface spheres as E ⁰=(0,0,1). The incident field at the probe is assumed to be zero, as the field decays rapidly, and the wavelength is set at 632.8 nm.

3.2. Simulation results

Assuming that the detector is located at 2cm away from the probe with a view angle of (85°, 90°, 5°), we obtained seven polarized images with Eq. (5). The intensity of the total scattered field is shown in Fig. 3(a), and the intensity and phase images the x, y, and z plane components are shown in Figs. 3(b), 3(c) and Figs. 3(d), 3(e) and Fig. 3(f), 3(g), relatively. From Fig. 3, it can be found that the total-intensity image (Fig. 3(a)) has the closest resemblance to the practical topography of the specimen (see Fig. 2(b)). All the other images cannot represent the practical surface features of the specimen, though they contain some sub-wavelength details of the specimen. We can also find in Fig. 3 that the images of the x and z 140 nm polarized components have almost the same distributions, and their phase images have a higher resolution only in the x direction. (In Fig. 3(c) and Fig. 3(g), the images of the gold spheres adhere to each other only in the y direction) However, the phase image of the y component has a higher resolution in y direction. Noting that all these images can not exactly represent the practical surface features, we may conclude that, for a practical interferometer, the detector should not be located at a position with a small view angle to the z-axis.

 

Fig. 3. Polarized images obtained by a detector with angles of (85°, 90°, 5°)

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By increasing the view angle θ to 35°, that is when the detector is located 2cm away from the probe-surface system and at a view angle of (65°, 90°, 35°), we obtained seven more images, which are shown in Fig. 4. Compared to Fig. 3, no remarkable changes can be found, except for some improvement in the resolution of the x and z component images for both the phase and intensity. It is obvious that the seven images in Fig. 4 still can’t contentedly represent the physical distribution of the sample surface.

 

Fig. 4. Polarized images obtained by a detector at view angle of (65°, 90°, 35°)

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Increasing the view angle of the detector to (35°, 90°, 65°), we obtained the near field polarized images of the specimen shown in Fig. 5. Compared to Fig. 3 and Fig. 4, improvement in the resolution can be found in the images of the x and z polarized components, especially in their intensity images. However the images of the y component remain almost unchanged. Though we can retrieve some sub-wavelength details of the specimen from Fig. 5(b), Fig. 5(c), Fig. 5(f) and Fig. 5(g), there still exists distinct difference between the images and the practical distribution of the specimen (see Fig. 2)

 

Fig. 5. Polarized images obtianed by a detector at view angle of (35°, 90°, 65°)

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Increasing the view angle further to (5°, 90°, 85°), we obtain another seven images shown in Fig. 6. Significant improvement in the image resolution can be found in the total intensity images, the x-polarized images, and the z-polarized images. All the images sufficiently represent the practical distribution of the surface spheres with sub-wavelength resolution. The phase images (see Fig. 6(c) and Fig. 6(g)) and the actual topography image of the specimen (see Fig. 2(b)) can be seen to have good correlation.

 

Fig. 6. Ploarized images obtianed by a detector at view angle of (5°, 90°, 85°)

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In the above analysis, we studied the difference between the various polarized images of a specimen and their dependence on the detector-position by fixing the detector in z-x plan and adjusting its view angle to the z-axis. It was found that the near-field images of the z and the x components always have the same distributions and can retrieve the physical features of the specimen if the detector is placed at a large enough view angle from the z-axis, which is in accordance with the results of Dr. Mufei [9]. Due to the symmetry of the probe-surface system shown in Fig. 1, we can deduce that for a detector placed in other detection planes with a sufficiently large detecting angle θ, all the components polarized in the detection plane will have nearly the same intensity and phase distributions, enabling us to accurately retrieve the surface features. To verify this conclusion, we computed the polarized images of the specimen by assuming that the detector is placed at a view angle of about (45°, 45°, 85°) shown in Fig. 7. We can find that the intensity and the phase images of the z-polarized component (Fig. 7(e) and Fig. 7(f)) can represent the physical distribution of the specimen since it is polarized in the detecting-plane. All the images of the x and y polarized components (from Fig. 7(a) to Fig. 7(d)) can not retrieve the practical surface-features of the specimen well, as in this case these images are not polarized in the detecting-plane. On the other hand, the intensity and the phase images of the polarized component of E_x+E_y (Fig. 7(g) and Fig. 7(h)), which is polarized in the detecting plane, can accurately represent the practical distribution of the specimen.

 

Fig. 7. Images obtianed by a detector at view angle of (45°, 45°, 85°)

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In this paper, we only studied the polarized images by comparing their resemblance to the practical distribution the specimen and left their absolute intensity value unconsidered. According to the principles of the interferometry, a higher intensity of the detected light always means higher noise immunity and higher measurement accuracy. Thus, it is meaningful for us to study the intensity distribution among different polarization components in the detection plane. Fig. 8(a) and Fig. 8(b) are the three-dimension displays of the intensity of Fig. 6(b) and Fig. 6(f). From these two images, we can find that the intensity of the z component is almost one hundred times of that of the x component. For practical s-SNOM interferometers, this means keeping the reference light polarized in the z-direction will lead to higher measurement accuracy. On the other hand, according to related near-field electromagnetic theories, the near-field distribution of a weak-scattering specimen can be seriously predicted and its phase distribution is always determined by the so-called efficient-surface of the specimen, and in the case of a uniform material the near-field phase of a specimen is exactly its surface profile [13]. In our analysis the probe-surface system is simplified to a series of spheres, and their induced field is calculated under point-dipole approximation. In this case, the quantitative relation between the calculated phase distribution and the practical surface profile of the specimen is very complicated [11]. For simplicity, in this paper the polarized images are discussed only by analyzing their resemblances to the actual distribution of the specimen rather than by analyzing the rigorous relation between their exact values and the specimen topography. This kind of treatment is quite rough for quantitative study, but sufficient for our qualitative discussion of the properties of the s-SNOM interferometer.

 

Fig. 8. Three-dimensional display of Fig. 6(b) and Fig. 6(f)

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4. Summaries and conclusion

The polarized images of the s-NSOM interferometer were numerically studied by modeling the interferometer as a coupled point-dipole system. For a given specimen, it was shown that both the intensity and the phase images show strong dependence on polarization and detector-position. In particular, for a given specimen, obvious differences were found in the images of various polarization components and detector positions. In the case of evanescent wave illumination, highly accurate images can be obtained only when the detector is placed at a large view angle with the z-axis (vertical to the sample surface) and the reference light is polarized in the detection plane. This numerical result could be of importance for the design and the use of the s-NSOM interferometers, and for the interpretation of their experimental result.