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Thermal radiation from samples of Au layers patterned on GaAs, SiO2, and SiC at 300 K are studied with a scattering-type scanning near-field optical microscope (wavelength: ~14.5 μm), without applying external illumination. Clear near-field images are obtained with a spatial resolution of ~60 nm. All the near field signals derived from different demodulation procedures decrease rapidly with increasing probe height h with characteristic decay lengths of 40 ~60 nm. Near-field images are free from any signature of in-plane spatial interference. The findings are accounted for by theoretically expected surface evanescent waves, which are thermally excited in the close vicinity of material surfaces. Outside the near-field zone (1 μm < h), signals reappear and vary as a sinusoidal function of h, exhibiting a standing wave-like interference pattern. These far-field signals are ascribed to the effect of weak ambient radiation.
©2011 Optical Society of America
1. Introduction
Thermally activated random motion of conduction electrons in metals or of ion cores in dielectrics generates strong electromagnetic surface waves in the close vicinity of material surfaces. Thermally excited such electromagnetic surface waves at room temperatures have recently attracted considerable theoretical attention [1–3]. Experimentally, however, relevant studies have so far been restricted mostly to the proximity effect of heat transfer between two materials placed closely to each other [4,5]. To achieve deeper understanding of such evanescent waves, a scattering-type scanning near-field optical microscope (s-SNOM) is expected to provide a powerful tool [2]. In conventional studies of s-SNOMs [6–9], however, weak thermal excitation at room temperature can be easily spoiled by much stronger external illumination. Study of emission from objects at 170 °C via s-SNOM without using external radiation source was first reported by De Wilde et al. in 2006 [10]. Recently we have developed an improved s-SNOM system in a long wavelength infrared (LWIR) region (wavelength λ ~14.5 μm), which can detect thermal radiation at room temperature without utilizing external illumination [11]. In this passive s-SNOM, a sharp metal tip scatters evanescent surface waves towards a novel ultra-highly sensitive detector called the charge-sensitive infrared phototransistor (CSIP) [12,13]. The detectivity of CSIP detectors is by a factor more than two orders of magnitude higher than any other conventional LWIR detectors like HgCdTe. The unprecedented detector sensitivity, together with a carefully designed home-made confocal optical system [14], makes the equipment particularly suitable for the study of thermally excited surface waves.
Here we apply our system to study thermally excited surface waves at room temperature for twofold purpose. The first is to obtain surer experimental evidence of the detection of thermally excited surface waves at 300 K. The second is to highlight an important effect of background radiation (close to but lower than 300 K) in the measurements: In conventional s-SNOM studies, samples are illuminated with much stronger external lights so that effects of thermal background radiation are negligibly small. In the present extremely sensitive measurements, however, external illumination is not adopted and careful attention has to be paid on the thermal background radiation because relevant signals are far smaller than those in most s-SNOM measurements.
The experimental findings are briefly summarized as in the followings. Two dimensional scan of patterned Au layers deposited on top of GaAs, SiO2, or SiC substrates at 300 K gives near-field images (λ ~12 μm and 14.5 μm) similar to those found in our previous work [11] with an improved spatial resolution of ~60 nm. Here we focus our attention to the dependence of the image on the distance h of the metal probe apex from the sample surface. In the close vicinity of the surface (h < 100 nm), detected signals are found to rapidly decrease with increasing h regardless of the manner of signal demodulation: A characteristic decay length is ~50 nm and the near-field signal completely vanishes at h ~150 nm. These features strongly support the interpretation in terms of thermally excited surface waves. A different class of signals appears as the probe tip is moved further away from the surface (h > 1 μm). As a function of h, the signals exhibit a standing-wave like interference pattern. The amplitude of the pattern is material specific (Au, GaAs, SiO2, or SiC), but the interval of interference pattern is independent of material, being several micrometers. When the probe tip is laterally scanned while h is maintained at a few μm’s, two-dimensional (2D) images are obtained with a spatial resolution on the order of wavelength. These signals are successfully analyzed by a simple model describing interference effect of background radiation.
2. Passive near-field signal
The experimental setup (Fig. 1(a) ) is similar to the one described in our previous work [11]. A CSIP detector is placed at liquid helium temperature [12,13]. In this study, we use two CSIP detectors with narrow spectral bands centered at λ = 14.5 ± 0.7 μm and 12.0 ± 0.7 μm as shown in Fig. 1(b) [15]. A tungsten probe tip (the radius of apex curvature being R = 40 ~60 nm) is placed at the focal point of a Ge objective lens (numerical aperture, N.A. = 0.60, or an aperture half-angle of 37°) of a confocal microscope with a geometric optical resolution of 15 μm, which has been improved from the previous system [11,14] by reducing the pinhole size from 125 μm to 62.5 μm. The tungsten probe is attached to a tuning fork operated at a resonance frequency of f TF ~32.7 kHz, and the distance between the probe tip and the sample surface, h, is controlled in shear-force mode [16] of a home-made atomic force microscope (AFM) system. For modulation of radiation signal, the tungsten probe is lifted up and down vertically at a low frequency of f M = 10 Hz with an amplitude of Δh = 50 ~600 nm as illustrated in the lower right of Fig. 1(a) [17]. In what follows h will denote the bottom position of the tip.
Fig. 1 (a) Schematic diagram of the passive s-SNOM equipped with a CSIP detector. The tungsten probe is vertically modulated at f M = 10 Hz. (b) Spectra of CSIP detectors for λ = 14.5 μm and 12.0 μm.
Figure 2(a) is a microscope image of a sample with 3 μm pitch gratings of a 100 nm-thick Au layer deposited on a SiC substrate. Figure 2(b) shows a passive near-field image (λ = 14.5 μm) with h ~10 nm and Δh = 200 nm. The scan is made over a 20 × 20 μm area with a 0.3 s time constant for each 200 nm-step: Totally it takes around an hour. Signal demodulation is made at the fundamental frequency, f M = 10 Hz.; a signal designated later as I f. In Fig. 2(b) Au gratings on SiC are clearly distinguished. Though not shown here, clear images are also obtained in similar samples with Au stripes on GaAs and on SiO2, where the signal intensity on Au is found to be distinctly stronger than on these substrates: The relative intensities of the substrates to Au (λ = 14.5 μm) are, roughly, 0.4 ~0.5 (SiC), 0.2 ~0.3 (GaAs), and 0 ~0.2 (SiO2), where uncertainty arises from different probe tips and different surface treatments. The distance of the probe tip from the material surfaces is kept unchanged to h ~10 nm during the scan across the stripes; viz., the distance between the probe and each material (Au, SiC, GaAs, and SiO2) is always ~10 nm. In these passive measurements, all the components (the sample, the probe tip and the Ge objective lens etc.) are kept at room temperature and the signals are taken with no external illumination. The spatial resolution of the images is derived to be ~60 nm as shown in Fig. 2(c), which displays an example of the edge profile at the Au-SiO2 boundary. The resolution is found to be probe-tip dependent but independent of the sample material.
Fig. 2 (a) Optical microscope image of the sample with Au/SiC grating. (b) Passive near-field image (λ = 14.5 μm) at room temperature. (c) The spatial resolution ~60 nm shown in the step edge between Au and SiO2.
We note that the 2D-image in Fig. 2(b) is free from any systematic structure ascribable to spatial coherence of radiation. We have confirmed that the structure-less image is a general feature observed in all the samples. Though not shown here we have also confirmed that systematic patterns in 2D-image never show up in samples with differently wide stripes (3 μm ~50 μm) nor in samples with different shapes (round discs or rectangular stripes). The structure-free feature is observed also in the 2D-images taken with the signals, I 2f, demodulated at the second harmonic frequency, 2f M.
Figure 3 shows how signals I 0, I f, and I 2f decay as the tip is moved vertically (10 nm < h < 500 nm) away from the sample surface, where the probe tip moves along the axis normal to the sample surface at the center of a 25 μm-wide Au stripe (on a GaAs substrate). Here, I 0 is a non-modulated detector signal obtained without modulation (Δh = 0 nm), proportional to the total radiation power reaching the detector. The signals I f and I 2f are the demodulated signals, respectively, at fundamental frequency f M and the second harmonic frequency 2f M with Δh = 100 nm. We note in Fig. 3 that all the signals decrease rapidly with increasing h, with characteristic decay lengths being 60 nm, 50 nm, and 40 nm, respectively, for I 0, I f, and I 2f [18]. (For deriving the decay length of I 0, we set 276.3 nA as null reference.) Here we define the decay length as the probe height at which the signal amplitude deceases to 1/e. Figure 2 (c) shows that the spatial resolution measured with I f is ca. 60 nm. These features are distinctly different from those widely known in standard s-SNOMs with external illumination [19]. In standard s-SNOMs, radiation from an external light source is focused onto the sample with a spot size on the order of the wavelength. Since the wavelength is much larger than the size of a probe tip, a large area of the probe scatters the incident beam, yielding spurious signals in I 0 and I f with long decay components. Hence in the vast majority of s-SNOM measurements, I 0 and I f do not correctly represent near-field signal, but only higher order demodulation signals, In f with n ≥ 2, give reliable near-field signals. Differently from such standard s-SNOMs, all of I 0, I f, and I 2f exhibit similarly rapid decay. The 2D images obtained from I f and I 2f signals are similar to each other as mentioned already in the above.
Though not shown here similar approach curves are obtained also on the surface of dielectrics. The facts that (i) all of the I 0, I f, and I 2f signals exhibit similarly rapid decay on Au and on dielectrics (SiC, GaAs, SiO2) and (ii) similar 2D image with excellent spatial resolution is obtained both with I f (Fig. 2) and I 2f, strongly indicate that the detected radiation in these passive measurements are those strictly confined in the close vicinity of the material surface (mainly h ≤ 60 nm), much narrower than the radiation wavelength. We hence suppose that only very small part of the probe tip is exposed to the strong surface evanescent waves of the material, while the probe shaft high above the tip is free from strong ambient radiation. It follows that all the signals, I 0, I f, and I 2f, correctly represent the near-field signal in the present passive measurements [20].
The experimental features described in the above are consistently interpreted in terms of the theoretically expected thermally excited surface evanescent waves [1–3]. According to the theories, high energy density is stored in the form of thermally excited electromagnetic evanescent waves and its absolute intensity increases dramatically as the surface is approached. In most metals, the energy density reaches a level by a factor more than four orders of magnitude higher than that of Planck’s blackbody radiation for h < 60 nm at λ ~14.5 μm and T = 300 K [2]. In dielectrics as well [3], the energy density rapidly increases as the surface is approached, but the absolute intensity is not so high as in most metals except in the resonant region of surface phonon polariton (SPhP) mode: In GaAs, SiO2, and SiC, λ = 12 ~14.5 μm is outside the resonant region. The higher radiation intensity on Au beyond the substrate materials is thus consistent with theoretical predictions [21].
Unlike the surface plasmon polariton (SPLP) mode with a well defined dispersion relation described by the wave number K ~2π/λ in the LWIR region, dominant components of thermally excited evanescent waves are not normal modes and dispersion relation is not defined. At a given radiation frequency ω, the excitation is featured by a broad range of wave numbers (K >> 2π/λ), much higher than that of the SPLP mode. The experimentally found structure-less 2D image is reasonably ascribed to this broad range of higher spatial frequencies with K >> 2π/λ.
As demonstrated in Fig. 3 for Au, it is the common feature in all the materials studied (Au, GaAs, SiC, and SiO2) that the near-field signal completely vanishes as the tip is away from the surface more than h = 100 ~200 nm. This is exactly what the theories predict [1–3]. The evanescent wave energy density is predicted to sharply drop with increasing h, down to a level below that of Planck’s blackbody at h > 300 nm in most metals [2] and in dielectrics [3]. As a brief summary we have experimental results that strongly indicate the detection of thermally excited evanescent waves.
3. Interference effect of background radiation
Surprisingly signals reappear as the tip is moved further away from the material surface as shown in Fig. 4(a) , where approach curves of I 0 and I f (Δh = 100 nm) are displayed for larger values of h up to 20 μm [22]. Sinusoidal wave pattern is recognized, where the amplitude is not largely different from those of the near-field components as marked by the arrows in a dotted ellipsoid. Signals of such standing-wave like pattern have been seen on all the materials (Au, GaAs, SiC, and SiO2). The period of the interference pattern is found to be material independent, but the amplitude is material dependent. The amplitude ratio, however, is different from that of near-field signals. It is hence certain that the physical origin of the signals appearing in h > 1 μm is completely unrelated to that of near-field signals.
Fig. 4 (a) Approach curves of detector signals I 0 and I f (Δh=100 nm) above Au surface for h < 20 μm. (b) A schematic explanation: Specularly-reflected radiation at the sample surface (E r) interferes with the scattered radiation from the probe apex (E s).
In standard s-SNOMs where a probe tip is illuminated by a coherent external source, similar standing wave patterns are known to arise from interference between scattered lights of incident beam [23,24]. In our passive s-SNOM, external illumination source is not applied. Nevertheless, the probe tip is exposed to background radiation. Although the background radiation is of a broad thermal spectrum, the detected band width in the measurements is relatively narrow (Fig. 1(b)). We hence expect that the standing wave pattern can result from the interference between the scattered light of the background radiation at the probe apex and the specularly-reflected light at the sample surface as illustrated in Fig. 4(b). Here θ is the angle of incidence, h’ is the distance of the scattering center from the probe apex. Letting Eb = E 0cos(ωt) be the electric field of incident radiation, those of the specularly-reflected radiation and the scattered radiation are represented, respectively, by E r = rE 0cos(ωt−α r −α path) and E s = sE 0cos(ωt−α s) with ω being the angular frequency of radiation. Here (r, α r) and (s, α s) are the amplitude and the phase shift, respectively, of the reflection and the scattering, and α path ={4π(h+h’)/λ}cosθ refers to the difference in optical path length. Since the detector signal, I 0(h) is proportional to the time-average of
we have andRelations (1) and (2) indicate that I 0 and I f exhibit sinusoidal waveform with period
Note that |E s| << |E r| in general and that Relations (1)-(3) describe the homodyne detection of E s.
Since the radiation outside the aperture of the Ge objective lens (Fig. 1 (a)), θ > 37°, do not reach the detector in the present confocal system, only the radiation within the cone θ < θ Max=37° should be considered. In Fig. 4(a), θ= 21.6° is derived from d = 7.8 μm with Eq. (4), which certifies that the radiation is from the Ge lens at room temperature. The effective radiation temperature, however, is lower than the room temperature (estimated to be ~250 K [25]) because the emissivity of Ge lens is substantially smaller than unity.
In Fig. 4(a), the reduction in amplitude of interference pattern is noted to be small but finite, roughly being 8% over h = 0 ~20 μm. The spectral bandwidth (Δλ = ± 0.7 μm) at λ = 14.5μm yields a coherence length of ~λ 2/Δλ = 150 μm [26], accounting for ~4%-reduction. Additional decay may be attributed to a finite spread of the angle Δθ of the dominant radiation. If we assume Gaussian distribution of θ with Δθ ~± 4°, 8%-amplitude reduction of the interference pattern is roughly expected at h = 20 μm.
The solid line in Fig. 4(a), drawn according to Relation (2), correctly reproduces the experimental data points of I f. We emphasize that practically no arbitrary parameters except E 0 are used in the comparison, where h’ = 50 nm is taken to be the radius of tip apex, Δh = 100 nm, αr = π – 0.021 is the Fresnel reflectance of Au for λ = 14.5 μm, and θ = 21.6°. While αs = π is expected in the Rayleigh limit for the scattering, αs = π + 0.11 is taken for better fitting in the theoretical curve of Fig. 4(a) [27]. In the following analyses, the same parameter values (θ = 21.6°, αr = π – 0.021, αs = π + 0.11, and h’ = 50 nm) will be used unless otherwise stated.
Figure 5(a) displays data taken with Δh = 100, 200, and 600 nm to show that the amplitude of interference pattern (I f) is nearly proportional to the modulation amplitude Δh, while the near field signal intensity does not vary so significantly with Δh (> 100 nm) as the interference signal amplitude. Respective solid lines are drawn according to Relation (2) by using identical parameters to show excellent agreement with the respective data points. The period of interference pattern is reduced when the wavelength is shorter (λ = 12.0 ± 0.7 μm) as shown in Fig. 5(b), where θ = 26.9° is derived from d = 6.7 μm. The theoretical values shown by the solid line in Fig. 5 (b) reproduce nicely the experimental results.
Fig. 5 (a) I f signal on Au with Δh = 600 nm, 200 nm, and 100 nm for λ = 14.5 μm. (b) I f signal on Au with Δh = 600 nm for λ = 12.0 μm.
The interpretation in the above is made more certain in the geometry where the sample surface is tilted by 45° as illustrated in Fig. 6(a) . Figure 6(b) displays the I f versus h curve (Δh = 600 nm) for λ = 14.5 μm on Au. It is found that period d increases to 8.7 μm, from which θ = 33.6° is derived. If the dominant radiation relevant to the interference is the one exactly propagating along the optical axis, θ= 45° is expected. The smaller angle than θ= 45° may reflect the form factor of the scattered radiation at the probe tip. Another feature seen in Fig. 6(b) is a remarkable decay of the interference pattern with increasing h. In the tilted geometry, the probe tip moves away from the focal point (optical axis) by hcos(π/4) with increasing h. Assuming a Gaussian-type image of the focal point, we expect that the signal intensity decreases as exp[-{hcos(π/4)/ΔX}]2 with ΔX = 15 μm being the spatial resolution of the present confocal optics. This decay is taken into account in the solid line in Fig. 6(b), which agrees well with the experimental results.
Fig. 6 (a) Sample surface is tilted at 45° from the horizontal plane. (b) I f signal against h on Au for λ = 14.5 μm with Δh = 600 nm in the tilted geometry.
Figure 7(a) compares the I f-signals (Δh = 600 nm) between Au and GaAs for h > 200 nm. The phase of the sinusoidal wave is substantially the same between Au and GaAs, but the amplitude of GaAs is smaller (around 60%) than that of Au. This ratio is distinctly different from that of the near-field signal (I f,GaAs/I f,Au = 20% ~30%) as noted already in the above. The dependence on the material is definitely accounted for by the reflectance parameters (r, αr ) in Relation (2): That is, the similar phase between Au and GaAs is explained by αr ~π for both materials, and the ratio of amplitude is accounted for by the ratio in reflectance r GaAs/r Au ~0.54, where r GaAs=0.53 for GaAs (refractive index ~3.25) [28] and r Au = 0.99 for Au [29]. All the features are thus interpreted consistently in terms of Relations (1) and (2).
Fig. 7 (a) I f signals (Δh = 600 nm) on Au and on GaAs for h > 200 nm and λ = 14.5 μm. (b) One dimensional profiles scanned over a 25 μm-wide Au stripe on GaAs substrate at fixed heights of h = 2, 4, and 6 μm. (c) 2D image of the Au/GaAs pattern taken at h = 6 μm.
Figure 7(b) displays I f signals obtained by scanning the probe in the horizontal direction at fixed probe heights of h = 2, 4, and 6 μm. As expected from Fig. 7(a), negative, flat, and positive contrasts are obtained for Au with respect to GaAs, respectively, at h = 2, 4, and 6 μm. The spatial resolution is on the order of wavelength because the propagating far-field radiation is being observed. Figure 7(c) is a 2D image recorded at the extremity of a 25 μm-wide Au stripe on GaAs, where the tip height is kept at h = 6 μm. The image is much less sharp compared to the near-field image in Fig. 2(b), but the material contrast is still visible. Au is brighter than GaAs at h = 6 μm, but the contrast reverses its polarity when the probe height is at h = 2 μm (not shown).
4. Discussion
The sample and the metal probe are exposed to ambient radiation in our passive s-SNOM as has been explicitly demonstrated in Sec. 3. It should be recognized that the interpretation of the near-field signal in Sec. 2 in terms of the thermally excited surface evanescent waves is unaltered. We have directly studied the influence of illumination by introducing weak incoherent radiation in additional experiments. We have confirmed that the radiation incident at angle θ outside the cone, θ> θ Max = 37° causes appreciable effect neither in near-field nor in far-field signals. The radiation within the cone, θ< θ Max = 37°, was found to influence the absolute size of the observed signals, which will be reported in detail elsewhere. The effect, however, does not affect qualitative features described in Sec. 2 so that the interpretation is not altered.
It is evident that the absolute amplitude of interference patterns described in Sec. 3 is directly influenced by the effective temperature of incident radiation (T radiation ~250K). Additional experiments with weak external radiation source revealed that the sample temperature (T sample = 300 K) is also important in determining the absolute amplitude of the interference pattern. It turned out, however, that the qualitative feature is kept unchanged and the discussion given in Sec. 3 is valid so long as the inequality T sample > T radiation holds. Detailed results will be reported elsewhere [30].
Finally we mention a pioneering work [10], where samples of Au patterns deposited on SiC and SiO2, heated up to 170 °C, were studied via s-SNOM. LWIR radiation scattered at a metal probe tip was detected with a HgCdTe detector. Interestingly, I 2f-signal on SiC is reported to be stronger than on Au in the resonance region of SPhP (λ = 10.9 ± 0.5 μm). As a general feature, however, the decay lengths of near-field signals are reported to be largely different between I f and I 2f. That is, I f exhibits surprisingly slow decay with a decal length of ~3 μm [31] and 2D-images taken with I f are featured by fringe structures showing spatial coherence over a few micron meters on Au stripes: The spatial coherence is interpreted in Ref. 10 to be a demonstration of SPLP excitation. These findings and the interpretation are different from ours in the present work.
5. Summary
Patterned Au layers deposited on top of GaAs, SiO2, and SiC at 300 K are studied with an improved s-SNOM (λ ~12 μm and 14.5 μm) without applying external illumination. In all the samples clear images of near-field signals with sharp material contrast, similar to those reported in our previous work [11], have been obtained with an improved spatial resolution of ~60 nm. The non-modulated detector signal I 0 as well as the signals I f and I 2f demodulated at f M and 2f M decrease rapidly with increasing the distance h between the metal probe apex and the sample surface with characteristic decay lengths of 60 nm (I 0), 50 nm (I f), and 40 nm (I 2f). Near-field images recorded on a number of different samples have made it certain that (i) the images obtained from I f and I 2f are similar to each other, (ii) the intensity on Au is stronger than on SiC, SiO2, and GaAs in the spectral region studied, and (iii) the near-field images are free from any systematic interference patterns. These findings have been consistently interpreted in terms of theoretically expected thermally excited surface evanescent waves strongly confined in the close vicinity of material surfaces [1–3].
Another group of signals has been found to appear when the probe tip is moved away from the surface at distances h larger than 1 μm. As a function of h, the signals (I 0, I f and I 2f) exhibit sinusoidal interference pattern, where the amplitude is material specific but the period of interference pattern is material-independent. These signals are ascribed to far-field effect of propagating waves; viz., scattered radiation of the incident background light at the probe tip is superposed with the reflected wave at the sample surface and is amplified through homodyne detection mechanism.
Acknowledgments
This work was supported by CREST project of Japan Science and Technology Agency (JST) and Grant-in-Aid for Challenging Exploratory Research of Japan Society for the Promotion of Science (JSPS).
References and links
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20. As discussed in [19], scattering amplitude of a probe rapidly decreases with h in a range 0 < h < R (radius of apex curvature), which would cause the signals I0, If, and I2f to rapidly decrease. Nevertheless, if the near-field contains a long-extending component, I0 and If will have additional slow decay components.
21. The LDOS is theoretically discussed in [2], which predicts the LDOS (z = 50 nm) on Au is more than 10 times larger than those on the dielectrics (SiC, GaAs, SiO2). The discrepancy with the experimental values arises from the isotropic model of [2]. Larger experimental values certainly arise from the piezoelectric acoustic phonon modes that are not considered in the theoretical treatment.
22. The near-field signal If near the surface in Fig. 4(a) is a little smaller than that in Fig. 3. The difference is attributed mainly to the probe condition. The signal change is not the problem because the signal characteristics like signal ratio between different materials are independent of the probe condition.
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25. The effective radiation temperature, Tradiation is estimated by comparing the far-field radiation intensities from 77K-liquid nitrogen, 300K-Au and 300K-SiO2. We can derive Tradiation by knowing the emissivity and the reflectivity of each material and noting that Tradiation and Tsample are respectively relevant to the reflected and the emitted radiations.
26. M. Born, and E. Wolf, Principles of Optics, 7th edition, (Cambridge Univ. Press, Cambridge, 1999).
27. s ~π + 0.0005 should be taken if a sphere dipole model is assumed as in [19].
28. J. C. Brice, Properties of Gallium Arsenide, 2nd edition, (INSPEC, London, 1990).
29. M. Wakaki, K. Kudo, and T. Shibuya, Physical Properties and Data of Optical Materials, (CRC Press, Boca Raton, 2007).
30. Thermodynamics requires that the interference pattern is visible when Tsample ≠ Tradiation but vanishes in thermal equilibrium, and that it reverses its sign according as Tsample > Tradiation or Tsample < Tradiation. The experimental values shown in Figs. 4–6 are opposite in sign to Relation (2) because Tsample > Tradiation.
31. The probe tip in [10] is modulated in tapping mode at a frequency much higher than 10 Hz. It is difficult, however, to ascribe the discrepancies to the different method of tip modulation.
