Optical Measurements

CHAPTER

Optical Measurements

12

12.1  HAVE A LOOK AT NANOSCALE As mentioned in Chapter 1, the main difference between scanning probe microscopes and conventional optical instruments lies in fact, that SPM does not expand our sight but our touch. However, light is delivering approximately 80% of information about the surrounding world to a human. It would be therefore unreasonable to leave out light—and generally electromagnetic field interaction—while developing SPMs. As the light interaction with a nanostructure is physically very far from classical concept of optical microscopy, use of optical methods in SPM is still relatively rare, comparing to other techniques and comparing to amount of information this interaction can carry. If we leave out the fact that light is used in most of the SPMs for optical detection of force acting on a cantilever there are several other areas where light takes an important or even a key role in scanning probe microscopy: • super-resolution microscopes based on breaking diffraction limits—scanning near field optical microscopes (SNOM) [1]; • light beams used for SPM cantilever actuation [2, 3]; • local luminescence or Raman scattering measurements using tip enhancement of electrostatic field (e.g. Tip Enhanced Raman Scattering—TERS). At present, most of the instrumentation development is focused on scanning near field optical microscopy techniques [4], both for the purposes of sub-wavelength imaging (where aperture SNOM is usually applied) and for local luminescence and Raman measurements (where apertureless SNOM takes place mostly). Even if SNOM is still rather complex and expensive from the point of ­instrumentation, in the past twenty years it was used for a wide range of applications, including nearly all the fields of surface science and engineering, like optoelectronic devices studies (photonic crystals, laser diodes, optical waveguides) [5], single ­molecule spectroscopy measurements [6], magneto-optical measurements [7] or even near-field optical lithography [8]. It should be however noted that due Quantitative Data Processing in Scanning Probe Microscopy. http://dx.doi.org/10.1016/B978-1-45-573058-2.00012-7 © 2013 Elsevier, Inc. All rights reserved.

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to the complex nature of interaction of light and sample in nano- and microscale region, quantitative aspect of SNOM measurements are probably least developed for ­practical use of all the methods discussed in this book. In this chapter, we would like to describe basic phenomena related to an interaction of light with nanostructures (Section 12.2) and basics of scanning near-field optical microscopy methods (Section 12.3). Detailed description of available modeling techniques is provided in Section 12.4, as modeling is again understood as a crucial step in quantitative SPM data processing. The key part of this chapter is Section 12.5, describing methods that can help us to perform more quantitative measurements, both from the point of proper instrument calibration, measurement, and successive data processing. As for every chapter, the tips and tricks for maximizing amount of quantitative information are listed in Section 12.6.

12.2  FUNDAMENTAL PHENOMENA Interaction of an electromagnetic wave with a solid object can be described as a mixture of several optical phenomena, with a ratio depending on size of the object and light wavelength. Generally, any tiny material inhomogeneity in wave path leads to a scattering. Here the word scattering can mean simply any changes from regular propagation of wave. There are many theories describing different scattering effects of different inhomogeneities and their origin. The results of scattering processes for interaction with concrete material geometries are often called diffraction effects and these phenomena are already macroscopic enough to be seen and employed in ­practical life. Similarly, scattering in a volume of material (often completed with absorption) leads to material opacity that can be simply observed as well. In order to treat the electromagnetic field interaction with structures similar to its wavelength, we need to fully employ the wave structure and therefore to i­ ncorporate all the scattering, diffraction, and interference processes. All this is generally governed by Maxwell equations forming relation between unsteady electric and magnetic fields and material properties. Given unsteady electric field intensity E and magnetic field induction B we can write the Maxwell equations in vacuum as ρ ∇ ·E= , ε0 ∇ · B = 0, ∂B ∇ ×E=− , ∂t ∂E ∇ × B = µ0 J + µ0 ε0 , ∂t where ρ is electric charge density and ε0 and µ0 are free space electrical permittivity and magnetic permeability, respectively. If material properties are introduced properly, Maxwell’s equations can describe all electromagnetic field phenomena, both microscopic and macroscopic.

12.2  Fundamental Phenomena

With proper knowledge of all the related information, i.e. namely material p­ lacement and its properties we can in principle calculate all the effects in both the near- and far-field. However, to solve partial differential equations like Maxwell ones is not an easy task. Fortunately, for larger objects and inhomogeneities, the results of all these processes can be treated in much simpler approaches, i.e. using geometrical optics approach, so in most of the practical applications in optics Maxwell equations do not need to be solved. However, in scanning probe microscopy we treat with dimensions in order of nanometers or micrometers where wave properties need to be incorporated properly, and Maxwell equations need to be solved using some of numerous analytical or numerical approaches. In any scanning probe microscopy technique, the key concept is a high resolution. The basic SPM task in field of optics would be therefore to detect scattered light with the same resolution as other physical quantities described in this book (interatomic forces, electrical fields, temperature, etc.). Light that is absorbed by a material can be also transformed, e.g. into heat (exciting atom or crystalline lattice vibrations), or can excite other particles, as photons in case of material luminescence or electrons for photoelectric materials. To quantify these effects on nanoscale and microscale level is another basic task of optical methods in SPM. From calculations of diffraction on an aperture it is known that no instrument can overcome resolution given by the diffraction limit R=

0.61λ , NA

(12.1)

where R is resolution (determining smallest object resolved), λ is wavelength of light and N A is numerical aperture of an objective, usually a number slightly below 1 for measurements in air. In order to overcome the diffraction limit it is necessary to use completely different measurement concept. Locally scattered waves with i­maginary components of wave vector, called evanescent waves need to be collected by microscope optics. This is a rather complex task, as evanescent waves decay very rapidly and are not part of light radiated from the illuminated sample. The only solution is to bring a nanoscale detector in close proximity to the sample, e.g. to the near field region. Fortunately, scanning probe microscopy is based on technologies that can help with this task. Both nanoscale source or detector can be easily brought close to sample surface by using SPM feedback loop (based again on tip-sample force detection) and piezoceramic actuators. A typical configuration of an optical SPM measurement is given in Figure 12.1. Here a nanoscale source is formed by the probe apex due to local field enhancement process (for details, see the next section). From Figure 12.1, we can see that in the field of scanning probe microscopy we usually deal with wavelengths of size ­similar to typically employed objects—cantilevers, tips, and surface irregularities. Wave optics approach needs to be used in order to model all the related effects. Moreover, it is necessary to treat with a large number of different materials, from transparent dielectrics, up to metals. Finally, as in whole SPM instrumentation, all the used components are nearly at manufacturing technological limits, therefore neither geometry

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Figure 12.1 Typical ratio between employed objects and wavelength for optical SPM ­measurements (apertureless measurement of nanoparticles on absorbing substrate covered by thin non-absorbing film).

of involved bodies nor its proper material properties are usually known with a high precision. Therefore, understanding light interaction within a SPM system is still big challenge for near-field optics and computational electrodynamics.

12.3  BASIC TECHNIQUES As mentioned in previous section, scanning near-field optical microscopy is based on use of nanoscale light source, detector or scatterer scanned in close proximity to sample surface. In this way the diffraction limit of conventional optics can be overcome. Each SNOM instrument basically consists of positioning apparatus with nanometric resolution in all three axes, feedback mechanism to set and measure probesample distance and optical devices used to generate, guide, scatter, and detect light. For feedback, a modification of contact or non-contact probe-sample force sensing approach known from atomic force microscopy is used. This modification can mean different source and direction of oscillations (e.g. tuning fork shear force measurements), use of another laser to monitor fiber movement, creation of cantilevers from optical fibers, etc. There are many different experimental devices described in literature that employ the basic SNOM principle described in the previous section. Here we focus on two main branches of instrumentation development—aperture and apertureless measurements, as those two principles are most frequently found in commercial instruments. Aperture SNOM is using small aperture of optical fiber as a detector or source, apertureless SNOM is based on metallic scaterrer formed by AFM tip creating a nanoscale source. Sources of light are high intensity lasers nearly in all the cases as the throughput of whole optical system is very small. Photomultiplier tubes or cooled PIN diodes

12.3  Basic Techniques

Figure 12.2 Typical aperture SNOM geometry, using tuning fork feedback mechanism (based on straight fiber probe) and far field detector.

are mostly used as detectors. For spectroscopy applications also spectrophotometers can be used. A special family of measurement methods is tip enhanced Raman scattering technique, that is also based on apertureless SNOM combined with a Raman spectrophotometer.

12.3.1  Aperture SNOM As aperture SNOM we call a wide range of techniques using pointed optical fiber as SPM probe. Fiber can be used as nanoscale source, detector, or both. It is usually prepared by a pulling or etching technique. Nanoscale aperture is often shielded by a metal layer in commercial instruments in order to prevent light leaking close to aperture. Fiber is glued on the feedback element (tuning fork, cantilever) and its aperture is scanned in close proximity of sample. Fiber probe can be used as a source or a detector. Typical configuration of aperture system is plotted on Figure 12.2. Depending on light direction, aperture SNOM is working in illumination mode (light coming from the aperture) or collection mode (light passing to the aperture). Rest of optical path is usually created by conventional optics in order to maximize light throughput (using aperture for both illumination and collection is rare). As the fiber aperture is smaller than outcoming light wavelength, there should be only evanescent wave coming through and even out from the probe in illumination mode. This is however not true as the end of probe (aperture) acts itself as a scatterer and light is therefore radiated out even if the probe is far from the sample. Light therefore tunnels through the probe aperture and is radiated again when it leaves the probe (for most of the probes we can see it by eye). While brought to close proximity with the sample (several nm) probe acts as nanoscale source or detector.

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Probe is scanned pixel by pixel forming the final image. During probe movement, constant gap between probe and surface is usually maintained. For some theoretical consideration also constant height mode (with respect to mean sample plane) would be of interest, however, stability of feedback loop and thermal drifts often prevent doing this without breaking the probe.

12.3.2  Apertureless SNOM and TERS As the fiber probe is still relatively large to obtain really nanometer resolution, there are attempts to decrease probe size even more than is possible using tapered fiber technology. This can be done by using a small conductive probe as a scatterer. Probe therefore acts as nanoscale source of scattered light that is collected in far-field. Apertureless measurements [9] are typically based on use of metallic AFM probe scanning in small distance from measured sample surface. Direct illumination or illumination using total reflection inside sample volume can be used. Tip-surface distance modulation and lock-in detection needs to be used in order to suppress background from other scatterers usually found on surface. Big advantage of this method is no need for fiber technology use and increased resolution, featuring even single molecule detection possibilities. On the other side, signal to noise ratio is not ideal for many configurations and sample geometries. Typical geometry of an apertureless measurement is shown in Figure 12.3. A special application of apertureless SNOM is use of this technique in Raman scattering measurements. In this special case the electric field enhancement and confinement at probe apex (mixture of electron waves excitation and lighting rod effect) is known to produce local Raman signal increase up to several orders of intensity for some materials [10]. However, due to its instrumentation price and complexity this method is not as widely used as it would be expected. Local field enhancement is also highly dependent on tip properties and its preparation can be relatively ­complicated [11].

12.4  NUMERICAL ANALYSIS As the probe-sample system is rather complex in all SNOM variants, both from the point of geometry and from the point of materials used, proper modeling is not a simple task. Even if for some cases we are able to construct analytical models, for most of the practical calculations we need to use numerical techniques. In this Section we describe the available methods and then focus on Finite Difference in Time Domain technique that represents one of possible universal solutions for SNOM modeling.

12.4.1  Classical Electrodynamics Classical optics and electrodynamics approach was the first to prove theoretical abilities of SNOM to overcome diffraction limit [1]. However, as already mentioned, for

12.4  Numerical Analysis

Figure 12.3 Typical apertureless SNOM geometry, using long working distance objective and metallic AFM probe (whisker type).

most of the realistic SNOM simulations, the probe-sample geometry is too complicated to use it and only special cases can be modeled. Apertureless metallic probes can be modeled using Mie’s theory modeling ­interaction of an electromagnetic field with metallic particles. Modeling tip apex as metallic ellipsoid particle as shown in Ref. [12] and taking into account both tip dipole and virtual dipole induced in the sample can be used to model the approach curves for flat surface under various conditions. Similarly, simple models can be constructed even for fields used in Tip Enhanced Raman Scattering, as done in Ref. [11], again based on mirrored dipoles. More complicated systems, however, need much more effort to be modeled using the classical electrodynamics approach. For example, interaction of apertureless probes with metallic nanoparticles was studied using aggregate Mie theory in Ref. [13]. Apertureless images of single nanoparticle or cluster of nanoparticles were simulated. Using even more developed formalism of Green’s tensor technique, SNOM signals measured on different artificial structures (steps, islands) were investigated [14] both for collection and illumination mode. It was suggested to use light polarization in order to minimize topographical artifacts. The Green’s tensor technique formalism used for this was published in Ref. [15].

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Figure 12.4 Placement of electric and magnetic field vector components in FDTD ­computation grid (reprinted from Wikipedia).

12.4.2  Finite Difference in Time Domain modeling There are also several numerical modeling methods for electrodynamics that can be used to simulate complicated probe-sample interaction geometry. As an example, we choose here Finite Difference in Time Domain method, which is one of the most universal methods in numerical electrodynamics field. Finite Difference in Time Domain is a widespread method for complete ­numerical solution of Maxwell equations in discretized space. For linear, isotropic, non-dispersive, and eventually lossy material Maxwell equations describing time evolution of magnetic and electric field intensity vectors E and H can be written in the following form ∂H 1 1 (12.2) = − ∇ × E − (M + σ ∗ H), ∂t µ µ ∂E 1 1 = ∇ × H − (J + σ ∗ E), ∂t ε ε

(12.3)

where µ is magnetic permeability, ε electrical permittivity, σ electric conductivity, σ ∗ equivalent magnetic loss, and M and J are electric and magnetic source current densities. These equations are the basis of the FDTD method, calculating electric and magnetic field intensities in a staggered grid as shown in Figure 12.4. Using very dense discretization (at least λ/10 where λ is incident wavelength) and solving directly electric and magnetic field in a leap-frog scheme leads to a very stable algorithm suitable to many problems in physics and engineering. Method, together with many surrounding algorithms, can be used for nearly any material (including media with realistic optical response or gain media) and their results can be very precise. Unfortunately, the dense discretization in both space and time forms a bottleneck for many of the problems that could be simulated using FDTD. Realistic size of

12.4  Numerical Analysis

volume and time that can be modeled on regular PC nowadays can be seen as a cube with side length of several tens of wavelengths; with several thousands of time steps. As the probe-sample region responsible for SNOM contrast is usually very small, it fits perfectly FDTD possibilities. In Ref. [16] the FDTD method was used to determine SNOM transmission mode profiles over metallic edge, stripe or island laying on dielectric substrate. Conical probe is scanned in constant height regime and fed by soft circular source. Intensity profiles are evaluated using near-field to far-field (NFFF) transform. Different incident polarization effects are discussed as well. It is seen that several experimentally observed effects, like contrast inversion, can be modeled using FDTD in a satisfactory way. FDTD can be used for both aperture and apertureless SNOM modeling, including effects of local field enhancement in Tip Enhanced Raman Scattering [17]. FDTD possibilities are not limited on SNOM imaging, however. FDTD was used also to model thermal dissipation in SNOM probes [18] and to model heating of both probe and sample during scanning. It is shown that even if the probe itself can be heated significantly (known also from other literature), the thermal flux between probe and sample is negligible compared to optical flux through aperture. 2D FDTD simulations of corrugated metal coated tip can be found also in Ref. [19] in order to show corrugation effects on final probe resolution. To show the suggested computational approach and its potential drawbacks more in detail, we refer here to our previous simulations [20]. For quantitative data ­processing it is important to model whole probe-sample system on the basis of real morphologies, including all the irregularities like probe defects, surface and probe roughness, etc. In order to obtain all this information we need to employ relatively large effort—determine probe shape, aperture size, obtain high quality morphology data for surface, etc. For modeling aperture SNOM instrument image in a realistic way, we need to perform the following two steps: 1. Run optical fiber probe analysis based on geometry obtained using scanning electron microscope and data-sheet material properties of fiber. Geometry of this computation is illustrated by a cross-section presented in Figure 12.5 and in this way we can compute the field propagation inside SNOM probe and eventually at its apex. 2. Run probe-surface geometry analysis using the SNOM probe fields computed in step 1 and AFM topography of the grating surface; using NFFF computation of the far-field limit. Note that this step is run pixel by pixel, producing the final simulated SNOM image. The described model fits to illumination mode SNOM; for collection mode setup we can use plane wave for illumination and place detector inside the probe. For apertureless SNOM modeling, the situation is even simpler, as we do not need to run the probe analysis. On the other side, as we need to model tip enhancement of electric field, metal properties need to be treated properly, e.g. using Drude formula with corresponding parameters.

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Figure 12.5 Geometry of the computational domain for the fiber modeling (step 1, ­cross-section) and for the complex SNOM geometry modeling (step 2, Cross-section).

To compute above-mentioned electromagnetic field distributions, we need to use FDTD with the following extensions: • uni-axial perfectly matched layer (UPML) to allow waves to leave computational domain, • conformal modeling of the material boundaries or subgridding algorithms, • proper material handling, namely for metals in apertureless SNOM, • near- to far-field transform (NFFF) for evaluation of the far-field limit of the electromagnetic field distribution (e.g. Ref. [21]), • computational domain stepping in one dimension to model elongated structures (for probe analysis), • plane wave source in total/scattered field formalism (for illumination mode aperture or apertureless SNOM). In Figure 12.6a typical image of morphology of calibration grating used for atomic force microscope calibration is presented. In Figure 12.6b and c, two modeled illumination mode aperture SNOM images of this grating are presented, simulated for two different positions of far-field detector (that is placed off-axis). An example of modeling apertureless measurements is presented in Figure 12.7. We simulated light propagation in whisker-like probe based apertureless SNOM, using a Gaussian beam for illumination. Sample is formed by micro-crystalline silicon embedded in amorphous silicon (with thin non-absorbing film on the top). Here, we plot a

12.5  Quantitative Measurements

Figure 12.6 Part of the calibration grating (topography used for simulation taken from AFM image), and simulated SNOM reflection images for two different positions of far-field detector.

Figure 12.7 Light propagation in aperture SNOM measuring silicon crystal in amorphous silicon film (cross-section): (a) schematics, (b) snapshot of electric field distribution in single time step, and (c) averaged electric field intensity.

cross-section of the simulation volume, showing both sample geometry, ­snapshot of modeling and averaged intensity in the material. If metallic material is treated properly (with right frequency dependent permittivity model, e.g. Drude formula) even tip enhancement effects can be modeled using FDTD as reported in the literature [17]. It should be noted that FDTD is very slow if run for so detailed probe-sample geometry as suggested here. Calculation of SNOM image can therefore last s­ everal weeks on a single computer and several days on a distributed computing system. ­Fortunately, an alternative for this can be using graphics cards as computation devices [22] instead of computer processor. As shown in Ref. [22], this can lead to a ­significant speedup of SNOM image modeling (up to 70 times, depending on card used), and single image can be modeled in few hours only.

12.5  QUANTITATIVE MEASUREMENTS As the electromagnetic field propagation in probe-sample region is very complex, the possibilities of quantitative optical measurements using SNOM are still very limited. It needs to run both measurement and modeling incorporating full geometry of probe and sample, and even in this case it can be used only if interpreted very carefully.

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There are however many easier tasks that can be done in converting our measurements to more quantitative ones, limiting all artifact sources, and preventing possible misinterpretations or obtained results. In this section we first discuss these issues, starting with instrument calibration and continuing with all possible error and uncertainty sources. Second, we discuss how modeling can help us in data interpretation, from simplified tools up to complete modeling using FDTD as described in previous section.

12.5.1  Instrument Calibration General remarks for instrument calibration given in Chapter 3 are valid for SNOM use as well. There are however several other tasks that need to be addressed for proper SNOM calibration: • Light source and detector calibration: as the light intensity is mostly interpreted as a relative quantity (directly comparable only for one probe and the same sample setup), the key issue is long term stability of both source and detector. Even small variations in long time source or detector performance can lead to wrong results. Similarly, alternative optical paths in the instrument need to be minimized. Using sample with ideally homogeneous response (e.g. silicon wafer) can help for characterizing measurement background. As we usually try to collect extremely low light intensities (using extremely large gains and small signals), we can easily get results correlated to any other signal in our SNOM, e.g. voltage on piezos when the instrument electrical design is not ideal. • State of polarization: for apertureless measurements the desired polarization of incident light can be setup easily. However to set or measure state of polarization in fiber optic system like aperture SNOM is much more complex. Both fiber bending and presence of fiber aperture change the polarization state of light. It is therefore crucial not to move fiber between successive measurements. Polarization state of light coming from the aperture can be also measured in far-field, however to do this while not changing the system geometry between probe characterization and SNOM measurements is rather complicated. It should be also noted that due to very small light intensity the SNOM measurements are usually very slow (up to several hours for single image). This increases demands on scanner drifts, feedback stability, etc. Unfortunately, namely for aperture probes, the feedback stability is not always ideal and long term measurement can be often very problematic from point of preserving the probe quality and constant probe-sample distance.

12.5.2  Artifacts and Uncertainty Sources As for any quantitative measurements we need to know all the uncertainty sources and we need to discuss all the effects leading to possible errors in interpretation and

12.5  Quantitative Measurements

evaluation of SNOM images. As an example, we can think on two different kinds of measurement—first measurement of dimensional quantities on basis of SNOM data, and second, measurement of local optical properties of material using SNOM data. Uncertainty budget for optical dimensional measurements using near-field probe is rather complicated, but still understandable. We can discuss the following effects that could in principle influence dimensional measurement results, for example measuring lateral size of a region having refractive index different from rest of the sample: 1. calibration of the microscope stage, similarly to AFM and other SPM techniques (discussed in Section 3), 2. thermal effects during measurement, again similar to AFM case, adding thermal effects due to light coming from probe aperture or illuminating optics of SNOM, 3. probe-sample convolution, effect of different location of optically active part of probe and part responsible for feedback; topography related artifacts, 4. probe geometry influence on optical results, effect of its aperture size and shape, 5. far-field detection or illumination system influence, 6. illumination light polarization state influence. Effects mentioned above will be discussed in next sections. Even if the scientific understanding of many of them is still quite weak, it can be said that the uncertainty budget for dimensional measurement using optical probe can be assembled and most of the uncertainty sources can be computed, modeled, or at least estimated. An opposite case is quantitative optical measurement, for example determination of refractive index profile of a planar waveguide by aperture SNOM measurement. We will refer to this case as to local optical measurements. Here the uncertainty budget would be as follows: 1. source and detector calibration, being probably the only simple task in this example, 2. effect of fiber guiding laser light into the probe (polarization by strain, attenuation), 3. power dissipation and polarization changes inside probe, 4. effects of roughness at the probe aperture, effects of metal shielding of probe, 5. scattering in probe-sample region, image formation mechanism. The amount of information that we can know from theory, or that can be modeled or estimated is here dramatically lower than in the first example. However, big progress in SNOM image formation mechanism studies observed in last years, together with large speedup of computers allows us to hope that even this case could be easily solvable in near future. In the following text we present discussion of main uncertainty sources discussed above.

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Figure 12.8 Effects of sample morphology on volume between probe aperture and sample surface. Note that the distance between closest parts of sample and probe is always the same.

Topography Errors As the dependency of scattered field intensity on probe-sample distance is relatively steep, it can be expected that all topography effects leading to change of p­ robe-sample distance will alter SNOM images significantly [23, 24]. As stated in Chapter 3, SPM imaging process can be modeled (for most of the SPM modes) by convolution of probe and sample geometry. As the probe is usually far from ideal, delta function shape, namely for relatively large aperture probes, the resulting tip trajectory forming ­convolved image can be also far from “real” surface morphology. This can lead to a fact that the distance between optically active part of probe and sample is not constant. Also, the geometry of region between probe and sample where most of the image forming mechanisms play role is changing while scanning. These effects are illustrated in Figure 12.8. As mentioned, these effects are pronounced namely for aperture probes, being relatively large and producing very large tip convolution effects. Moreover, for these probes we often encounter different morphological defects close to aperture (apexes, protrusions) after some measurements. They can arise from random collisions with sample surface due to fails of feedback mechanism and can make convolution ­artifacts even more complicated. Topographic artifacts can easily obscure real optical information contained in the SNOM image as the effects caused by locally different optical properties can be similar to effects of morphology [25]. For apertureless measurements the tip shape is usually much simpler and closer to data-sheet values. However, even here we can observe morphology artifacts related to different materials forming surface and to excited surface plasmons [9]. All the optical information that is correlated with morphology changes, ­sidewalls, steep slopes, particles, etc., should be therefore treated as suspicious [26]. ­Unfortunately, for most of the measurements the important information is expected in such problematic regions and topography free surfaces having local optical ­properties changes are rather sporadic.

12.5  Quantitative Measurements

In order to minimize topography artifacts, it is recommended to use well-defined probes, which in the case of aperture SNOM measurements usually means “new” probes. If there is a need for any quantitative analysis from the measured image, or qualitative analysis at the regions in image with highly pronounced morphology, it is always necessary to do some further experiments in order to discover topography artifacts influence on measured data. More probes with different geometries can be used to measure the same part of the sample. The sample can be measured twice while we modify the sample after first measurement in order to be optically homogeneous (covering it by thin metal film, for example). Finally, we can do numerical or ­analytical modeling of topography artifacts and compare modeled results for ­optically homogeneous and heterogeneous sample. For good measurement practice it is important to perform a set of measurements of different simple structures (step, line) homogeneous from optical point of view. These structures can be found e.g. on calibration gratings. Measurement should be done for each mode of microscope used in laboratory (reflection, transmission, etc). This can help to discover the main effects of topography on measured optical data.

Feedback Mechanism Errors SNOM feedback can be basically formed by various mechanisms. Even if there is steep near-field scattering dependence on distance between probe and sample, this mechanism is usually not used for feedback, namely due to bad signal-to-noise ratio and by presence of many other scattering sources in the observation space. Using synchronous detection of light the signal-to-noise ratio can be improved highly and modulating probe-sample distance the scattering from probe-sample interaction can be selected from background as well. However, both these techniques are relatively complicated and not being standard part of most commercial SNOMs. Feedback is therefore in most of the cases similar to contact or non-contact atomic force microscopy. For apertureless measurements the feedback mechanism is often the same as for AFM or STM as the probe is formed more or less by a standard metallic or metal coated AFM tip. In case of AFM feedback mechanism the error signal acquisition is based on reflection of laser light from the top of cantilever and its detection using quadrant photodiode. This mechanism is well understood, can be very fast, can be used both for contact and non-contact mode and generally there is no estimation that could produce any special artifacts in SNOM data acquisition if working correctly. To prevent influence of detection laser diode it is also possible to use a self-sensing cantilever, usually manufactured from a piezoresistive material. For aperture probes, there are several basic mechanisms observed in practice: • Tuning fork mechanism based on use of piezoelectric element (that can be selfexcited or mechanically excited) sensing its own vibrations via piezoelectric current. • AFM like mechanism using second laser and a special probe (e.g. probe with hole in the middle or bent fiber forming AFM cantilever. For commercial aperture SNOMs, the tuning fork approach is probably the mostly used and mostly discussed in literature [27]. The main advantage of this approach

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is no need of another laser source in SNOM and relatively easy design of the probe and SNOM head. The only complicated issue—mounting of fiber to tuning fork—is expected to be done by probe manufacturer. Both fiber mounted and unmounted tuning forks can be bought, prepared by manufacturer for different SNOMs, and bare tuning forks can be easily obtained for much lower price from low frequency crystal oscillators bought in any electronics devices shop (e.g. 32768 Hz crystal in MTF32 package). Tuning fork is usually used with mounted fiber probe as presented in Figure 12.2. Probe aperture vibrates laterally above sample surface and forces arising from probe-sample interaction lead to shift of tuning fork resonance frequency. Amplitude or phase change detection can be used to detect this shift indirectly, otherwise the shift can be measured e.g. using self oscillation circuit preserving probe in its own ­resonance frequency. The mechanism of feedback using lateral vibrations is still the subject of ­discussions. In Ref. [28] the model of probe-sample interactions for tuning fork including laterally vibrating probe (as a damping element) is suggested and damping-distance dependencies modeled and compared to curves measured in UHV conditions. Power law with power of 3.8 is found for the damping-distance dependency. Direct measurement of forces between probe and sample was performed in Ref. [29] using AFM cantilever as force sensor. The measurements were performed both for resonance and off-resonance mode. Large influence of static attractive force (Coulombic and van der Waals) on overall force-distance curve shape was found; these forces can easily exceed the force caused by probe lateral movement. From practical point of view, it is known that the feedback mechanism is much weaker than for example for AFM using laser light deflection. In combination with relatively brittle probes used in SNOM it leads often to probe changes. However, this is probably not the only problematic issue of the tuning fork feedback. It was observed also that limited performance of feedback mechanism can cause even contrast (similar to morphology related artifacts), caused by incorrectly working PID mechanism [30]. In order to prevent unwanted effects caused by feedback mechanism the error signal must be measured every time and minimized. This can be sometimes hard as for pronounced topography the weak shear-force response can be inefficient. In this case we can use recorded map of error signal to increase uncertainty of measurement in problematic regions. Due to weaknesses of shear force feedback, there are also attempts to use ­oscillations normal to sample plane by some manufacturers (e.g. same as for AFM noncontact mode). Here the feedback control is much simpler, however, probe-sample distance varies during the measurement as the probe oscillates. Moreover, as the probe is brittle, higher forces that can be reached if amplitude of oscillations is too high can lead to fast probe wear.

Probe Geometry Effects For apertureless measurements, the probes are usually formed by sharp metallic or metal coated tips (usually treated as having apex in paraboloid or sphere shape).

12.5  Quantitative Measurements

As the feedback mechanism used for apertureless measurements is usually very stable (similarly to AFM measurements), probe changes within single measurement can be minimized easily. Moreover, AFM tip manufacturing technology is repeatable enough, therefore using different new tips for apertureless SNOM measurements can lead to repeatable results. On the other side, for aperture probes the basic probe geometry is unknown. Even for brand new probes the nominal probe aperture varies largely within tens of percents and all the changes that can probe undergo during measurement (or even during first approach) can alter the probe aperture shape even more. Laser pulling technique, most frequently used for probe preparation, is based on local heating of optical fiber using laser (e.g. CO2 laser) [31]. Fiber is pulled at the same time, which leads to formation of conical probe with aperture, typically between 50–150 nm. Several tens of nanometers of aluminum are then evaporated on probe sides in order to prevent light leaking from probe. During evaporation, probe is rotated along its axis to allow aluminum to form homogeneous film on cone sides and tilted by a small angle to prevent aluminum reaching top of probe (aperture). Sometimes, aperture probes are also used uncoated [32]. Even this approach can (at some conditions) lead to satisfactory resolution due to focusing effect of cylindrical shape of fiber probe as was demonstrated in literature. However, the amount of parasitic light leaking from probe is much higher than for coated probes (where there are ideally no leaks) and resolution is still much worse than for coated probes (even if it is breaking diffraction limit as well). Tips of well defined geometry can be prepared using etching technique, as shown in Ref. [33]. Using variable temperature bath consisting of hydrofluoric acid and ammonium fluoride, fiber tips with different cone angles can be produced. Cladding and core etch rate that is responsible for tip formation process was measured as well. Both single mode and multimode fibers can be used for tip formation. Even more sophisticated apparatus, in order to get better probe shape control was presented in [34] finally resulting in possibility of controlled preparation of probes with taper angle between 20° and 55° and aperture between 20 and 300 nm. Even if the geometry of probe is well defined, which is key from the point of metrology, we can expect this geometry is going to change during measurement. In Figure 12.9 taken from Ref. [35] scanning electron microscopy images of the four different SNOM probes (A–D) are presented. The ideal probe is shown in Figure 12.9a. There are several typical defects, seen on the probes B–D, all probably caused by a tip crash during scanning. First of all, a broken apex can be seen on Figure 12.9B, this is a feature that can be observed on SEM images of SNOM probes very frequently and cannot be distinguished within SNOM measurement easily. Second, in Figure 12.9c there is a probe with similar, but a slightly larger defect, having some more material around the apex coming probably also from the measurement. Finally, really broken probe is presented in Figure 12.9d. The probe damage at this level can be determined using video microscope installed on the SNOM or simply by looking at its total light output. Here, Probes A, B, and C could be used for normal SNOM operation (see

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Figure 12.9 Scanning electron microscopy images of four different SNOM aperture probes [35]. The magnification is same for all the images.

Figure 12.10 Topography (left) and SNOM reflection data (right) of standard sample, measured with probe b, c, and d [35].

Figure 12.10), and produced typical images of the standard sample (hexagonally packed aluminum islands). It can be seen that the topography images are very similar and the hexagonal structure of the aluminum islands can be easily seen on both the topography and the SNOM image. Topography measured using probe B is slightly affected by worse feedback operation (probably caused by the complicated structure of its apex). The SNOM image produced by probe B is however slightly better than using probe C and corresponds to a resolution of approximately 100 nm for probe B and 130 nm for probe C. Finally, the data measured using probe D are very bad, both the topography and the SNOM image. Probe D does not have the resolution abilities sufficient to measure this kind of sample. From this example it is seen that a simple probe characterization method that could be used before measurement could save a lot of experimental time.

12.5  Quantitative Measurements

For an estimation of the SNOM probe geometry we can use three methods generally used in scanning probe microscopy (see Chapter 3 for details): • Scanning over known structures—imaging the tip directly using a tip characterization sample containing known topographical structures much sharper than the tip itself. • Blind tip estimation—using random roughness or other topographic features on measured sample together with blind tip estimation algorithm [36, 37]. • Scanning electron microscopy—imaging the SNOM tip in a scanning electron microscope before or after measurement as in previous example. Even if the methods mentioned here are used regularly for atomic force microscopy tip shape determination, they are not very suitable for SNOM probe analysis. Tips must be moved to a different instrument (usually quite far from SNOM) or special sample consisting of sharp structures (that can lead to breaking the probe itself) must be used. Another way how to determine probe quality is to measure its radiation. Far-field measurement can be used to determine aperture SNOM probe basic characteristics. In Ref. [38] a spherical mirror imaging tip radiation onto CCD chip is used to ­visualize angular dependence of probe radiation. It is shown that probes of different ­aperture size can be clearly distinguished by this method. Similar measurements using a goniometer for measuring angular dependence of far-field radiation is presented in [39]. Far-field measurements were accompanied by near field intensity-distance measurements in range of 10–500 nm and show a steep decay of the near field signal as expected. Polarization state measurements in far-field were published in Ref. [40]. Here, 2D measurement of probe radiation was performed and served basically to test hypothesis of possibility using multipole analysis in further SNOM data modeling. A simple device consisting of two goniometers was used for probe analysis in our work [35], shown in Figure 12.11. As an example of radiation measurement probe analysis, the radiation distribution diagrams of all the probes presented in

Figure 12.11 Probe radiation diagram measurement device [35].

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Figure 12.12 Probe radiation diagrams for probe a, b, c and d.

Figure 12.9 are plotted in Figure 12.12. For all the diagrams it can be seen that the mean direction of probe radiation is not optimally aligned with respect to the measuring instrument hemisphere, which is probably caused by the geometry of the tuning fork probe holder (in the ideal case, the maximum would be at the bottom horizontal line of the diagram). However, all the probes are aligned in the same way. From this and the previous figures, it can be stated that the regularity of the radiation diagram corresponds with the probe quality. More defects on the probe apex produce more inhomogeneity in the radiation into different directions. Using far-field radiation measurements we can check the probe quality and minimize uncertainty sources related to wrong estimation of aperture size and shape. In some cases, this can help us to find even probe morphology defects that would lead to large tip convolution effects and image distortion. To determine real probe geometry from far-field radiation diagrams is a complicated task, that could be in principle solved only by numerical modeling. Generally, for good measurement practice, it is important to use the probe after experiment to image some standard sample in order to get some comparison to other probes. It is also better to perform sets of measurements (within one experiment) that should be mutually comparable with one probe, optionally measuring standard sample in between as well.

Far-Field Detector/Source Effects For most of the SNOM systems, at least one key element in optical path is in the far-field region. For aperture SNOM this can be source or detector (for collection or illumination mode), for apertureless SNOM usually both. In principle, it can be expected, that all optical elements position (including far-field source or detector) will influence both optical dimensional measurements and local optical measurements significantly. On and off-axis detector effects on reflection measurements were studied in Ref. [41]. The effect of blocking light at steep edges in case of off-axis detection was described here. It is also noted that off-axis geometry can lead to bigger signal to noise ratios, which is probably also the reason of its present popularity. As an illustration, there is part of diffraction grating measured presented in Figure 12.13. We can see light and dark regions close to side walls of depressions

12.5  Quantitative Measurements

Figure 12.13 Typical aperture SNOM reflection image of calibration grating (illumination mode), (left)—shear force topography, (right)—optical signal.

forming the grating. In Figure 12.6 there was FDTD simulation of optical signal obtained from grating already presented, corresponding to two different positions of the far-field detector. We can see that blocking light effects are depending simply on far-field detection position. Of course, for more complicated morphology, the interpretation of off-axis detector influence can be much more complicated. Here, a numerical modeling is probably the only possible source of information.

Polarization Effects Polarization state of light used in SNOM is one of the key parameters being varied from the first beginnings of SNOM development. It is well known from measurement practice that changing the polarization state of incident light alters resulting image significantly, namely in case of larger morphological features found on surface. There were attempts to use polarization for enhancing or suppressing topography artifacts [42]. In Ref. [43] there was found both experimentally and from modeling, that for case that probe is used both for illumination and collection of light, vertically polarized light (normal to sample surface) can produce better resolution for most of the surface structures than horizontally polarized light. A similar situation can be observed for apertureless measurements, where polarization of incident light is also related to final contrast [44]. While tuning the polarization state of incident light can be used effectively to improve contrast on final SNOM image, a quantitative analysis based on polarized light use is much more complicated. For aperture probes both fiber that is leading light from laser to SNOM and probe itself can change the polarization state. Usually, this is prevented by using fiber optic polarizer in the setup and tuning the system experimentally to get desired polarization state. However, it can be expected that there is also a strong depolarization effect at the probe aperture and this can alter dramatically any quantitative results. Similarly, depolarization at metallic tips in apertureless SNOM were observed and studied in Ref. [45]. Measurement of depolarization by sharp metallic tips was

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performed in back-scattered light, using same objective as for sample illumination. Rotating analyzer was used to measure polarization state and strong cross-polarized component compared to illumination polarization was observed in back-scattered light. To prepare an experiment using polarized light with well-defined state is therefore not an easy task. Generally, there are attempts to create new approaches to improve polarization state in probes, using special probes acting as nano-polarizers [46] or using polarization preserving optical fibers [47].

12.5.3  Quantitative Data Interpretation—Image Modeling Fast Determination of Problematic Parts As seen from the previous text, there is no way of simple understanding to SNOM data on most of the surfaces. The only solution is to run numerical simulation for whole tip-sample geometry and determine all the possible artifacts by comparison of simulated and real results. Of course, for many measurements this is too complex approach. Simulation tools are not always available and employed bodies geometry and material properties are not always fully known. In these cases a simple determination of problematic parts of image can be done on the basis of local morphology. As a simplest tool a local curvature analysis can be used, similarly as for other techniques discussed in this book. In the regions where slope of sample is changing rapidly we can expect large changes of material configuration in probe and sample region. For aperture probes, for example, the distance between probe and aperture will vary significantly. If we therefore combine areas corresponding to multiple probe touches (­see ­certainty map construction in Chapter 3) with areas featuring high local curvature, we can obtain rough estimate of problematic regions in our SNOM data. We can then exclude these regions from our quantitative analysis or run full calculation on them as shown in the next section. An example of presented approach is given in Figure 12.14.

Figure 12.14 (a) Calibration grating morphology, and (b) problematic parts in aperture SNOM measurements, probe apex radius 80 nm (flat cap).

12.5  Quantitative Measurements

Full Calculation For rigorous calculation of SNOM image, the full electromagnetic field propagation calculation in probe sample region as described in Section 12.4.2 need to be performed. To do this, it is important to preserve all the information available during the experiment: • Sample morphology obtained for sharpest possible probe. As the tip convolution strongly influences the shape of problematic steep slopes, it is important to minimize this effect. For large aperture SNOM probes it is necessary to use another tip, usually AFM, to characterize morphology independently. • Sample and probe material properties obtained from datasheets, bulk values, or other measurements. • Probe morphology obtained from SEM measurement or tip characterizer sample. • Illumination or detection device position with respect to sample and probe. • Probe aperture obtained from far-field radiation measurements or from determined resolution on standard samples (for aperture measurement). Combining all this information, we can set up FDTD model according to ­Section 12.4.2 and run simulated SNOM measurement pixel by pixel. Using conventional computing approaches this can be still quite slow, however with application of ­distributed computing or graphics cards [22] it can be completed in time similar to real experiments. In the following examples, these parameters of FDTD models were used: 1. Optical fiber probe analysis was based on geometry obtained using scanning electron microscope and data-sheet material properties of fiber; computed in space of 20 × 3 × 3 wavelengths (λ), space discretization of λ/20, using stepping in one dimension and conformal modeling. 2. Probe-surface geometry analysis using the SNOM probe fields computed in step 1 and AFM topography of the grating surface; computed in space of 4 × 4 × 4λ, space discretization λ/40, using NFFF computation of the far-field limit. Three examples of modeling aperture SNOM data are given here. In Figure 12.15 an atomic force microscopy image of typical probe resolution test sample designed for SNOM is presented. This sample is formed by hexagonally ordered aluminum islands (created by aluminum evaporation on closely packed polystyrene spheres and their dissolution). Simulated SNOM image (illumination reflection mode) for probe aperture of approximately 120 nm is presented in Figure 12.15 as well. We can see that FDTD approach can be used even for structures much smaller than wavelength, which means higher discretization than required λ/10 (here λ/80). Much more complicated structure computation is presented in Figure 12.16. The sample studied consists of a silicon substrate covered by an amorphous hydrocarbon thin film (a-C:H) grown using the PECVD technique using a pulsed glow discharge

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Figure 12.15 (a) SNOM probe resolution test sample morphology measured using AFM, (b) simulated SNOM image for 120 nm probe aperture.

Figure 12.16 (a) Shear-force SNOM topography of delaminated thin film sample, (b) reflection SNOM data measured simultaneously to the shear-force data, and (c) simulated SNOM image based on same shear-force data (central part only).

in a mixture of acetylene and argon. After deposition, a local ­delamination of film occurred, forming network of blisters. In the SNOM image showing part of one blister, interference effects caused by local film geometry are clearly seen. Similar effects can be seen in simulated SNOM image (only central part was ­simulated in order to minimize boundary effects on computation). In contrast to previous examples, SNOM shear-force morphology was used here as a basis of computations in order to get data comparable to real measurement. This, however, leads to doubling the probe-sample convolution effect which can change the resulting simulated images slightly. For thickness of film and its optical properties the values known from previous measurements using ellipsometry were taken. Large computation volume was used for FDTD computation (approximately 10 times larger than for previous examples) and simulation seen in Figure 12.16c was completed within three days even if run on graphics card. It should be noted that similar simulation would be run on our CPU cluster in time scale of months. As SNOM is often used for localized luminescence measurements, we have chosen a third example from this field. This sample is formed by montmorillonite with intercalated dye—rhodamine B. Aperture SNOM measurements were performed in illumination mode, measuring both excitation signal (488 nm) and luminescence signal output (between 550–800 nm). We have observed that luminescence signal is correlated to typical morphological structures—particles on flat sample surface.

12.5  Quantitative Measurements

Figure 12.17 Localized dye luminescence experiment and simulation: (a) measured excitation signal SNOM image, (b) measured luminescence SNOM image, (c) simulated excitation signal SNOM image, (d) simulated luminescence SNOM image under assumption of a homogeneous luminescence, (e) simulated luminescence SNOM image under assumption of a luminescence localized in particle.

This could be by both effect of real luminescence of particles or some kind of topographical artifact. We used FDTD to distinguish these cases. In Figure 12.17, the results of the modeling of the excitation and luminescence signal SNOM image are presented and compared with experimental results. A detail of the measured shear force data (corresponding to a single particle) was used as a substrate for the modeling, and the same detail of the excitation and luminescence image was used for the comparison between modeled and experimental data. In Figure 12.17a the SNOM excitation signal (wavelength 488 nm) obtained with direct SNOM measurement in illumination mode (no filters applied) is presented. In Figure 12.17b a SNOM luminescence signal (wavelength above 550 nm) is presented. The following images (c–e) represent the modeled excitation signal (c), signal which corresponds to the whole sample gain computation (d) and particle gain luminescence image (e). From Figure 12.17a and c, it can be seen, that our approach can be used to simulate the SNOM signal in a satisfactory way; topography related artifacts that are ­typically caused by a combination of varied aperture-sample distance and far-field detector position effects can be easily observed on the simulated image. Slight differences between simulated and real signal are caused by errors in SNOM probe geometry characterization and tip convolution while obtaining AFM topography. The tip-sample geometry used for the computation is therefore not exactly the same as tip-sample geometry of the experiment. In Figure 12.17d and e, the similarly modeled images of the luminescence signal are presented for two assumptions—homogeneous spread of the gain media and gain media localized in the particles. We can see that there is no observed topography artifact that would lead to a signal similar to localized luminescence at the location of particle on the surface as observed in the experiment (Figure 12.17b). The particle itself exhibits therefore much larger luminescence than the flat part of the film upper ­boundary. Note that without FDTD analysis we could hardly identify this sample property.

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12.6  TRY IT YOURSELF For fast check of problematic parts on your data, an approach described in one of the previous sections can be used. You can use a data processing module prepared for Gwyddion open source software (http://gwyddion.net) and available on the attached CD-ROM. Loading your topography data and peforming the fast local curvature check can simplify first guess of what parts of your image might be corrupted by topography artifacts. In order to determine influence of artifacts on your data properly, it is recommended to run a simulation of the SNOM data acquistion process including modeling of electromagnetic field propagation. If FDTD is selected for these purposes, as suggested in this chapter, there are many possibilities of how to start. First of all, there are several open source or free FDTD codes available publicly, like Meep or pFDTD and of course also commercial tools, like Lumerical. The key problem for running a FDTD analysis (not being an expert on FDTD) is how to construct proper model and how to process the results. Here we present an open source software GSvit that is dedicated for SNOM modeling and should be able to minimize your effort in this direction. It is the software that was used for all the simulations shown in this chapter. GSvit is a Finite Difference in Time Domain solver dedicated for fast computing, namely on the field of nanoscale optics. It can optionaly use a graphics card (using Nvidia CUDA technology) can speed up the computation by factor up to 70 [22]. Therefore, calculations that would need a supercomputer or a distributed computing system can be performed on a regular computer and with good graphics card a full FDTD simulation of SNOM image can be performed in an order of few hours. GSvit is available from http://gsvit.net as a set of different tools—command line based FDTD solver (Svit) and set of graphical user interfaces dedicated to different tasks. For analysis of SNOM data you can use GSvit-SNOM graphical user interface in Gwyddion that allows you to load your surface topography (obtained using your atomic force microscope or using shear force topography data from your SNOM), to model your SNOM tip (aperture or apertureless), illumination mode and near-field or far-field collection points for output intensities. Comparing the simulation with your data can help you to distinguish the artifacts from real information contained on your sample. For example, if your sample should be optically inhomogeneous, you can compare real measurement results with simulated images generated under assumption of homogeneous material. This allows you to separate topography artifacts from material contrast related data.

12.7  TIPS AND TRICKS After taking care for all the error sources, SNOM can be very efficient for obtaining unique metrology information about sample structure and optical properties. However, for many probe and sample geometries the measurement needs to be complemented with modeling of complete probe-sample system and light propagation in it.

References

To harvest most of the available information from optical data available in a SNOM instrument, it is recommended to follow the following basic steps: • Do not believe probe datasheet values: probe aperture and geometry can differ by several tens of percents from values stated in the datasheet. Always think on measuring the probe outer geometry, e.g. using a tip characterizer, or step sample (see Chapter 3). For inner geometry (in aperture SNOM) determine resolution using sample with enough small objects or determine aperture size from far-field measurements. • Always measure background: there are certainty alternative light paths in your instrument and there could be also some electrical cross-talks in the electronics. Measuring with blank sample (e.g. flat silicon for reflective measurements, glass block for transmission) will help to determine these effects. • Measure with different probes: none probe is ideal, if using two different probes with same nominal geometry you can determine effect of tiny variations in probe apex on your data. • Avoid using too small signals: signals needing extremely large photomultiplier gains or similar amplification are very sensitive to any electric cross-talks. This is in particular true for luminescence or Raman measurements. Try to get the same image with more light. For aperture SNOM, try to use fiber with larger aperture—even if it does not provide the fine resolution it would help with determining whether you see real effect and what is circuitry artifact. • Never believe results on steep slopes: try to detect slopes as described in previous section, if possible, try to exclude parts of optical image that are connected to sloping regions from data processing. • Try to get maximum information: could you change wavelength, far-field detector position, illumination angle or any other parameter? Try to measure more images with different settings for the same part of sample. • If possible, run full modeling: use FDTD or any other full calculation method to process all available information, simulate the measurement and compare it to reality.

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