Excerpt

## Contents

1 Introduction

2 Physical Principles

2.1 Near and Far Field

2.1.1 Limitation of Far-Field Measurement

2.1.2 Evanescent Wave Measurement via Slit

2.2 Evanescent Waves and Plasmons

2.3 Plasmon Resonance from Theory

2.4 SNOM Theory

2.4.1 Aperture-Mode

2.4.2 Shear Force Feedback

2.5 Topographic Artifacts

3 Setup

4 Samples

5 Experimental

5.1 Preparation of Measurement

5.2 Measurement Procedure

5.3 Calibration and Tip Properties

5.4 Results

6 Discussion

6.1 Optical Correction by Height

6.2 Correlation between Topography & Near-Field Optics

6.3 Analysis of Contractive Effects - Emission / Absorption

7 Summarizing Conclusion

8 Outlook

9 Appendices

9.1 Equipment

9.2 Program for Data Correction and Display

## List of Figures

1 Visualization of Near and Far Field intensity^{4}

2 Airy Disks with logarithmic intensities. The middle shows two Airy Disks at resolution limit. Left and Right show fully resolvable and resolvable Disks.^{6}

3 Abstraction of two distinct particles by a double slit. Detection via classical microscope (a) and via SNOM (b)

4 Simulated Near-Field for an illuminated Silver Particle using COMSOL (*λ* = 532 *nm*)^{9}

5 Schematic representation of evanescent waves propagating along a metal- dielectric surface. The exponential dependence of the electromagnetic field intensity is shown on the right.^{10}

6 (a),(b) multipolar resonance wavelengths of silver spheres for free space (*nout* = 1) and in suspension (*nout* = 1 *.* 5). The used laser frequency of 532 nm is plotted in green^{13}

7 Schematic representation of an aperture tip and its evanescent field

8 Shear force feedback for height control^{15}

9 Topography illustrating Streak Artifacts and Tip Shape Repetitions

10 Visualization of the tip path when scanning over a spherical particle with a spherical tip. The real aperture path is actually translated upwards

11 Schematic Setup of the Scanning Near-Field Optical Microscope^{17}

12 Microscopic images of sample C

13 SEM images of samples C1 and C

14 TGQ1 Test Grating^{21}

15 Uncalibrated Test Grating TGQ1 Topography and Profiles

16 Topography of sample C5 with open and new tip

17 Topography Overview of Sample C5 (100x100 pixels)

18 Results of Sample C

19 Results of Sample C

20 Results of Sample C

21 Results of Sample C

22 Results of Sample C

23 Illustration of Discontinuous Optical Correction - Sample C

24 Continuous Corrected Optics - Sample C

25 Transmission and Reflection Contour - Sample C6 The contour ranges from blue (low) to red (high)

26 Relative Emission and Transmission Contour - Sample C

## List of Tables

1 Sample Properties

2 Scan Parameters

Acronyms

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## Abstract

Unlike conventional optics, scanning near-field optical microscopy (SNOM) over- comes the Rayleigh criterion and can therefore achieve better resolutions than con- ventional optical microscopes. This feature is utilized to measure the optical prop- erties of different silver particle distributions on a glass surface. This paper mainly lays focus on intensity correction of the optical data due to topographical artifacts, analysis of plasmonic behavior and a tentative representation of the optical data. The simple approach for optical artifact correction has been shown to yield qual- itative success, with necessity of improvement for quantitative results. Given the conditions of the experiment, it has also been observed that plasmonic coupling seems to have a greater impact on the small observed particles. The tentative rep- resentation of the optics suggests that the larger particles are able to emit light by absorption of electromagnetic energy from their surrounding.

## 1 Introduction

SNOM is a form of scanning probe microscopy, which can produce optical pictures that transcend the far field resolution limit given by the Rayleigh criterion. This technique is used to analyze the optical properties of random silver particle distributions on a glass substrate. Topography and optics are measured simultaneously with resolutions in the magnitude of 100 nm^{1}. Therefore, SNOM excels in optical images, but underperforms in topographical resolutions when contrasted to methods like atomic force - or scanning tunneling microscopy, which both can achieve atomic resolutions^{2}.

In order to theoretically understand and appreciate the SNOM, one must first look into the - usually applicable - optical resolution limit given by the Rayleigh criterion. The next step involves the comprehension of how the SNOM overcomes this limit by detection of the near field. This detected field however, is measured in the far field - demanding a comprehension of the near-field/far-field translation.

After the acquired optical pictures are freed from artifacts, they can be examined in terms of their plasmonic properties.

## 2 Physical Principles

In the chapter, the near field and its advantages over the far field shall be examined. The Rayleigh criterion is first needed to understand the reason in utilizing this near field. It will then be gone over to a theoretical approach revealing the plausibility of near field measurements via far field. After then showing the physical origin of the this field, the functioning principle of the setup will be gone over.

### 2.1 Near and Far Field

The near and far field describe regions in distance to a source of radiation. The border between near and far field lies at about one wavelength from the source^{3}.

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Figure 1: Visualization of Near and Far Field intensity^{4}

As illustrated above, electromagnetic waves directly behind a light source are very intense and closely correlate with shape and dimensions thereof. This distance of about one wavelength to a source is called near field.

After passing through a transition zone, the light reaches the far field which is located two wavelengths from the source until infinity. This light does not contain all the information needed to draw conclusions about the source.

The reason for this behavior is that the near field contains non-propagating waves which exponentially decay leaving only propagating waves for the far field.

The field of an electromagnetic wave *E* (*r, t*) can be written as

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As mentioned, waves of the far field propagate requiring all components of *k* to be real. In the near field however, non-propagating - evanescent - waves outweigh, describable with at least one imaginary component of *k*.

#### 2.1.1 Limitation of Far-Field Measurement

The Rayleigh criterion declares the smallest resolvable distance between two light-sources in the Far-Field.

To derive the criterion, one can use an illuminated pinhole as a light-source. The pattern emerging from this pinhole is called an Airy disk and is shown below^{5}.

Figure 2: Airy Disks with logarithmic intensities. The middle shows two Airy Disks at resolution limit. Left and Right show fully resolvable and resolvable Disks.^{6}

The criterion asserts, that two light sources are indistinguishable, when the center of one airy disk lies on the first minimum of the second disk. Figure (2b) shows two Airy Disks at resolution limit.

In order to mathematically derive the Rayleigh criterion, one must formulate the 2- dimensional Fourier transform of a an illuminated pinhole - merely using propagating waves - and then determine the first minimum of the emerging Airy disk in the far field.

The 2-dimensional Fourier transform is given by:

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*x*

The illuminated pinhole with radius R can be described by )= *r* (*cos* (*θ*)), by integrating

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The emitted propagating waves with wave vector *ξ* - traveling in the direction determined

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Using polar coordinates and normalizing the field intensity f(x,y) to 1 results in

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for the occurrent field. The trigonometric addition theorem

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was used for simplification.

The Bessel function )2 *m* +1

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The minimum intensity is given for *ξ R* = 3 *.* 83, keeping in mind that the intensity is the square amplitude of F.

When the geometrical relation

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where *θ* is the angular resolution, D the minimum resolvable distance and *λ* the wave- length of the used light. D is now minimized by maximizing *sin* (*θ*) which results in:

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Therefore, the best obtainable resolution is approximately 1 *.* 22 *λ*, meaning the lower the frequency of the used light, the better the resolution.

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#### 2.1.2 Evanescent Wave Measurement via Slit

This section shall show the plausibility of near-field detection via far field with a mathematical approach. It is necessary as our detectors are far over two wavelengths away from the illuminated particles thus only measure light in the far field.

Figure 3: Abstraction of two distinct particles by a double slit. Detection via classical microscope (a) and via SNOM (b).

As seen in figure (3), two separate particles with radius L, which shall be seen as sources of electromagnetic radiation, will be represented by a double slit with distance 2d. The double slit is illuminated by a monochromatic field *E* 0(*x, z*). In first approximation, the field just behind the aperture (z=0) is then given by^{7}:

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where *kx* is the wave vector in x direction.

The field at distance Z is determined by back-transformation into the position space with

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As shown in (2.1), the following statements are true for the integration interval considering a 2-dimensional wave vector k:

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As mentioned in (2.1), (*kz ϵ* C) implies evanescent waves. The exponentially decreasing amplitude of these waves cause them not to be detected if the distance Z is larger than the wavelength *λ*. In this case the integration interval only includes propagating parts:

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*x E* 1(*kx,z* = 0) *e − ik* 2 *− k x Z* (13)

Now a slit of size 2l, representing the SNOM-tip, will be placed in front of the double slit. The slit should be smaller than the wavelength of the light as this will overcome the Rayleigh criterion shown in chapter (2.1.1). It shall be centered at the point (z= *ϵ*,x=X), in very small distance *ϵ*. (*ϵ ≪ λ*)

By using the same approximation as above, the field just behind the slit is given by:

*E* 2(*x,z* = *ϵ*) = *Enear · C* (*x,X − l,X* + *l*) (14)

The Fourier transform right behind the slit is:

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After propagation to a detector in the far field at z = Z, the field in the position space is:

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If now the equations (13) and (16) are compared, it can be seen that the slit - or SNOM tip - causes an integration from *−∞* to + *∞* to appear in the equation and therefore

including complex values for * k* ^{2} *− k ′* 2 *x*. In other words containing evanescent waves.

### 2.2 Evanescent Waves and Plasmons

In contrary to the mathematical approach in (2.1.2), this section will focus on the physical origin and properties of the near field.

Similar to quantum tunneling - which results from the continuous solutions to the Schrödinger equation - evanescent waves are the consequence of the continuous electric and magnetic fields at a boundary^{8}. Naturally, evanescent waves are always present but only gain detectable amplitudes when generated by Plasmons.

6

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Figure 4: Simulated Near-Field for an illuminated Silver Particle using COMSOL (*λ* = 532 *nm*)^{9}

Plasmons are charge density oscillations which can be driven by light and described mathematically as a dipole p(t), with an electromagnetic field given by^{8}:

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R - distance of dipole to point of evaluation

Figure 5: Schematic representation of evanescent waves propagating along a metaldielectric surface. The exponential dependence of the electromagnetic field intensity is shown on the right.^{10}

The intensity decline after leaving the surface as seen in figure (5) can be formulated as

follows^{11}:

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The formula reflects the exponentially falling intensity when in distance Z from the surface. It also shows a dependency of the wavelength *λ*, the average angle *θ* of the surface and the reflective index * ns* of the substrate.

### 2.3 Plasmon Resonance from Theory

Plasmon resonance is dependent on many factors^{12} such as shape, size or the dielectric properties of an object and its surrounding. Even though the shape of the particles in this experiment is spherical, the plasmon resonance is nontrivial and must be calculated numerically.

Figure 6: (a),(b) multipolar resonance wavelengths of silver spheres for

free space (*nout* = 1) and in suspension (*nout* = 1 *.* 5). The used laser frequency of 532 nm is plotted in green^{13}

The plasmon resonance frequency of silver spheres in relation to the radius thereof can be seen in figure (6). On the left, the relation in air with a refractive index of 1 can be seen in contrast to glass with a refractive index of approximately 1.5 seen on the right. The used laser frequency of 532 nm is plotted in green. The average particle size will be shown in chapter (4) but has already been plotted as a dashed line. Most surface area of the silver particles is surrounded by air, only some of it is in contact with the glass surface suggesting the actual resonance frequency to be between the two graphs but strongly leaning towards the left graph with a refractive index of 1. As can be seen in the figure, dipole plasmons (l=1) and quadrupole plasmons (l=2) are of greatest interest, as small particles will hit the dipole resonance and large particle the quadrupole resonance.

**[...]**

- Quote paper
- Steven Kämmer (Author), 2014, Characterisation of metallic particle distributions by scanning near-field optical microscopy (SNOM) in simultaneous reflection and transmission mode, Munich, GRIN Verlag, https://www.grin.com/document/281073

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