2.2.1 Johnson noise

      The photodetector circuit is shown in Fig. 2.2. A photodiode PD converts the incoming photons (with energy hv) into photoelectrons which are then collected in a capacitor C of the photodiode. A resistor R₀ is introduced into the circuit (Fig. 2.2) in order to get a current i. The RC circuit (electronic filter) introduces a limited bandwidth B for the detection. The photodiode is supplied by a bias U and the output voltage produced by the incoming light is U_out.

      Johnson noise (or thermal noise) is caused by the statistical behavior of the electrons (fluctuations produced by thermal motion) in the electronic circuit. It is independent of the detected optical power. The electrical power of the thermal noise at the output of the detector is given by [12-14]

      (2.7)

      where k = 1.3810^-23 J/K is the Boltzmann constant and T is the absolute temperature (in Kelvin).

2.2.2 Shot noise

      Shot noise, or quantum noise, is a fundamental limit of the nature of light and is characterized by the fluctuation of the photons arriving at the detector. The resulting photoelectrons, produced by the conversion of photons (with an energy hv) into electrons, can be described by Poisson's statistics [15]. The variance of the shot noise is therefore equal to the mean number n_e of the collected photoelectrons during a characteristic time interval, called integration time t = 1/(2B), where B is the detection bandwidth [15]. The ratio of the number of photoelectrons n_e created to the number of incoming photons n_p is called the quantum efficiency h = n_e/n_p (h_max = 1). The ratio of the current i (proportional to the photoelectrons produced per unit time) and the optical power P_opt gives the spectral sensitivity

      (2.8)

      where e = 1.60210^-19As is the electron charge, h = 6.626 10^-34Js Planck's constant and v is the light frequency. The variance (i²_SN) of the shot noise current is given by the number of electrons generated in the photodiode and the bandwidth B through (i²_SN) = 2eBn_e. These electrons are produced by the conversion of photons and by the dark-current i_d, which is the remaining current when the photodetector is not exposed to light (in the dark). It contributes to the fluctuations of the detected number of electrons, and thus to the shot noise current. Considering the photodetector circuit of Fig. 2.2, the electric power corresponding to the shot noise, for frequencies within the detection bandwidth B, is [12]

      (2.9)

      where P_tot = P_r + P_o is the sum of the reference and the object power. Therefore, if the reference power is large, the photocurrent SP_opt dominates the dark current i_d. In this case, the electrical power corresponding to the shot noise at the output detector becomes

      (2.10)

2.3 Signal to noise ratio of the heterodyne detection

      The signal to noise ratio (SNR) of the heterodyne signal is defined by the ratio of the heterodyne signal power (ac power) to the total noise power. From Eq. (2.6), we get for the optical power seen by the photodiode

      (2.11)

      where P_o is the optical power of the object and P_r the optical power of the reference. This equation expresses the signal produced in the ideal case. In reality, the amplitude of the signal is reduced by a factor m, called the relative interference amplitude. The factor m (0 £ m £ 1) is a characteristic of the interference quality, i.e. temporal and spatial coherence, polarization and wave superposition. The optical power of the interference is therefore given by

      (2.12)

      and the resulting output current i(t) of the photodiode becomes

      (2.13)

      where i_dc = SP_tot is the dc current and

      is the amplitude of the ac current i_AC(t) = i_ac cos(2pDft+f). The electrical power of the heterodyne signal is found to be

      (2.14)

      According to the definition, we get for the electronic signal to noise ratio

      (2.15)

      Heterodyne detection offers an elegant way to increase the electronic signal for a given object power P_o (or more precisely the signal to noise ratio) up to a point where shot noise dominates Johnson noise. In this case, we have shot noise limited detection. To get P_SN larger than P_TN (and thus to be shot noise limited), the power of P_tot has to be enhanced by increasing the reference power P_r for fixed P_o.

      To make sure of being limited by shot noise only, the reference power has to be larger than the minimum reference optical power P^min_r. This lower limit is derived from Eqs. (2.7) and (2.10). In fact, for P^min_SN.= P_TN. and assuming that P_r >> P_o we get

      (2.16)

      By increasing P_r so that P_r > P^min_r , P_SN >> P_TN and the SNR reaches a constant value of

      (2.17)

      Assuming that m  = 1 (ideal case) and P_r >> P_o (which is always possible), the signal to noise ratio is given by the relation

      (2.18)

      Let us introduce n^o_e the number of photoelectrons and n^o_p the number of photons corresponding to the power P_o. The number of photons received during the integration time
t = 1/(2B) is n^o_p = t P_o /hv and Eq. (2.18) can be expressed by

      (2.19)

      This equation demonstrates that the signal to noise ratio is only limited by the shot noise of the object power P_o. The factor of 2 with respect to direct detection is known as heterodyne gain. To get the best SNR, h has to be as large as possible (h _max = 1). This is the main reason why we use a standard silicon photodiode (PD) rather than a photomultiplier (PM). The quantum efficiency of a PD is larger than that of a PM (h_Si 70%, h_PM 10%).

      The bandwidth B considered so far is the bandwidth of the photodetector circuit. However, after the detection, the bandwidth can be decreased by means of the integration time of a spectrum analyzer or a lock-in amplifier. It is this post-detection bandwidth which is relevant for the final SNR of the measurement. An example of measured SNR from an optical heterodyne system with a spectrum analyzer for B = 62.5 Hz will be presented in section 3.3.

      The accuracy of the amplitude and phase of the optical field can be expressed via the measured SNR. From Eq. (2.15), the amplitude

      of the heterodyne signal, which is proportional to the amplitude of the optical field under investigation, is proportional to the square root of the SNR. Thus, the relative standard deviation of the amplitude is given by

      (2.20)

      and the accuracy of the phase measurement by [16]

      (2.21)

      For example, in case of shot noise limited detection, for an object power P_o = 1 pW, a bandwidth B = 62.5 Hz, a quantum efficiency  = 70% and a wavelength of l = 532 nm, we get from Eq. (2.19) n^o_e = 15'000 and a SNR of about 45 dB (Eq. (2.18)). The accuracy for the measured amplitude is then dA/A = 610^-3 and for the measured phase df = 6 mrad ( =2p/10³ or 0.33°).

      It is interesting to know the limit of the optical heterodyne detection. Theoretically, the lower limit of detection is given by SNR = 1 for the heterodyne signal. Following Eq. (2.19) this corresponds to one half photoelectron (n^o_e = 1/2) contributed by the object power P_o. Then, the minimum detectable optical power becomes

      (2.22)

      With the previous values B = 62.5 Hz, h = 0.7, l = 532 nm, the minimum detectable optical power with heterodyne detection is found to be

      As the photon energy is hv = 3.710^-19 J, this power corresponds to about 90 photons per second. An example of low level signal detection is presented in chapter 4.2.2.

3.2.1 Laser

      The laser which has been mainly used is a 150 mW single mode (TEM00) frequency-doubled Nd:YAG diode-pumped solid-state laser (l = 532 nm) (Coherent Inc., Model COLCOMPASS 315M-150). A 1 mW single mode frequency-stabilized He-Ne laser
(l = 633 nm) (ReniShaw, Model SL10) laser has also been used. Both of these lasers have a long coherence length (l_Nd:YAG > 60 m and l_He-Ne > 300 m) and are stable in intensity.

3.2.2 Detector

      The detector of Fig. 3.5 consists of a silicon photodiode (Vishay Telefunken, Model BPW 34) followed by a transimpedance amplifier circuit, using a high-speed and low current noise operational amplifier (Burr-Brown, Model OPA 602).

      The photodiode is supplied by a bias U = 15 V. The capacitor C takes into account the photodiode capacitance C_d = 15 pF and the amplifier input capacitance C_a = 3pF
(C = C_d + C_a). A serial resistor (R_out = 50 W) is added at the output of the detector to adapt its impedance to the 50 impedance of the coaxial-cables used. If the input impedance R_in of the measuring devices (oscilloscope, lock-in amplifier or spectrum analyzer) is large compared with R_out, the input voltage measured by the device is the same as the voltage at the output of the detector. The total resistance R_L = R_in + R_out is called the load resistance.

      The photocurrent of the photodiode is given by

      (3.1)

      where P_opt is the incident optical power and S the response (spectral sensitivity) of the silicon photodiode; S = 0.4 A/W at l = 633 nm and S = 0.33 A/W at l = 532 nm. Since the photocurrent i is small, an amplifier circuit is needed to detect the signal in a convenient way. The operational amplifier also controls the voltage at the photodiode boundary and regulates the current flux through the feedback resistance R₀ = 4.710⁵ W. The contribution of the noise by the amplifier can be neglected because it is smaller than the thermal noise of the feedback resistor R₀ at T = 300 K. In this case, the SNR at the amplifier output is the same as at the photo-diode output. A feedback capacitor C₀ = 1.5 pF is added to minimize phase distortion of the frequency response at the cut-off frequency.

      The relation between the measured dc voltage at the output of the detector and the incident optical power is given by

      (3.2)

      The output voltage ranges from 0 to 10 V. If the optical power is modulated at a frequency p, the output voltage of the transimpedance amplifier can be expressed by [10]

      (3.3)

      where R(w) is the transfer function of the detection circuit. The frequency response |R(w)| of the specific detector (of Fig. 3.5) was measured using a modulated light emitting diode (LED). By varying the frequency of the LED current between 10⁴ Hz and 10⁷ Hz, we acquired the frequency response of the detection circuit with a spectrum analyzer (Hewlet Packard, Model 4195A). The result is shown in Fig. 3.6. The detector pass-band (at -3 dB) is 300 kHz. The beat frequency has been chosen at 70 kHz which is well below the cut-off frequency. The response U_out to an optical input power P_opt is then given by U_out = R₀SP_opt. Calculation of the amplification gain of the detector with the given parameters agrees with the measurement.

      Taking into account the feedback resistor R₀ and the load resistor R_L, Eq. (2.7) for thermal noise becomes

      (3.4)

      In Fig. 3.7, we measured the noise level of the detector in the dark with a FFT spectrum analyzer (Stanford Research Systems, Model SR770 FFT Network Analyzer), with an input impedance of R_in = 1 MW and working in the frequency range of 0 to 100 kHz). With
R₀ = 470 kW, R_L = 1 MW (R_out = 50 W), B = 62.5 Hz and T = 300 K, the calculated electric power corresponding to the thermal noise is R_TN = 510^-19 W.

      The FFT analyzer measures the spectrum in dBV units (U_dBV = 10log(U_V/U_ref ), where U_V is the input voltage of the signal and U_ref = 1 V is the reference voltage. The corresponding electric power is obtained through

      (3.5)

      The measured electric power corresponding to the thermal noise of the detector is
P_TN = 110^-18 (Fig. 3.7). The measured noise power is a factor 2 larger than the calculated thermal noise power, which is quite satisfactory.

3.2.3 AFM/PSTM microscope and fiber probes

      Approach of the probe to the sample is controlled by a commercial atomic force microscope (AFM) (Park Scientific Instruments, Model BioProbe) (Fig. 3.3). A laser diode beam is deflected by a mirror (M1) and focused on the top of the fiber cantilever. Then, the beam is deflected by the cantilever and directed by a second mirror (M2) onto a four-quadrant position-sensitive photodetector (PSPD) (Fig. 3.8). The PSPD can measure displacements of light as small as 1 nm. Usually, flat silicon cantilevers with pyramidal tips are used. Due to the cylindrical shape of the fiber, the beam reflected from the fiber cantilever spreads out in a line perpendicular to the fiber. The displacement of this line light is measured with good accuracy by the detector. As the tip approaches the surface, inter-atomic forces (van der Waals repulsion forces [33]) between the tip apex and the surface cause the cantilever to bend, deflecting the laser beam. The position of the deflected signal is sent to computer feedback to control the approach before touching the surface.

      a)

      b)

      In general, atomic-force microscopes are used to image the surface topography on an atomic scale by scanning a sample area. However, we did not use the fiber cantilever for this purpose. In fact, fiber cantilevers are quite bad AFM tips, because of their rigidity. The force constant is about 4 N/m compared with 0.02 N/m for a typical silicon cantilever. However, the AFM regulation allows control of the approach of the fiber probe to the surface. Once the approach done, the feedback control of the AFM is switched off and scanning is carried out at constant height above the surface.

      In conventional photon scanning tunneling microscopes (PSTMs), the sample is illuminated by total internal reflection. The tip-sample distance approach is controlled by the optical measurement of the exponential rise of the evanescent wave above the sample. By combining the AFM approach system with the PSTM optical detection (of the evanescent field), the resulting device gives birth to the AFM/PSTM microscope [34-36].

      The probes used in this work are bent fiber tips, made from single-mode fibers (3.4 mm mode diameter), with a Germanium doped core of 3 mm diameter and a pure silica cladding of 125 mm diameter [37]. The working wavelength is 515 nm and the cut-off wavelength is 460 nm (± 40 nm). The fiber end is bent by laser heating in order to be used as a conventional AFM cantilever. The cantilever is 400 to 500 mm long and the curved part is 120 to 200 mm (Fig. 3.9a). The fiber apex is heated and pulled to produce sharp tips (Fig. 3.9b). A thin metal layer is then deposited on the fiber probe and covers the apex of the tip entirely.

      a)

      b)

      The metal thickness (about 20-50 nm) determines the aperture at the apex. In Fig. 3.10a, we present a 2-D model of the tip apex using commercial software MAFIA4 [38, 39]. The apex has been modeled by a hemispherical shape. The core of the probe is a dielectric material (glass of dielectric constant e_diel = 2.25) and the apex has an internal curvature radius of
r = 50 nm. The dielectric apex is coated with Chromium (e_Cr = -12.3+ i24.5 at l = 516.6 nm [40]). The apex has an external curvature radius of R = 70 nm so that the metal thickness is 20 nm at the end of the tip. Chromium is preferred to Aluminum because Al forms large aggregates of particles and the apex is not always well defined. In this model, we see that the light tunnels through the metal layer and determines a well-defined light localization. In Fig. 3.10b, a cross-section of the electric field (indicated by "b" in Fig. 3.10a) is shown. We can see the extension of the field at the apex. The full-width at half maximum (FWHM) gives the "aperture" of the tip, which is about 16 nm.

      a)

      b)

      In Fig. 3.11, we present a fiber tip model (with the same properties as before) with an aperture. The apex has been cut in order to make a physical hole of 80 nm. We observe that the field is not better defined than in the previous case. In fact, we can see two halos at each part of the metal extremity.

      These tip models demonstrate clearly that no physical hole is required in near-field fiber probes.

3.2.4 Signal processing and acquisition

      The signal processing is mainly performed with a lock-in amplifier (Stanford Research Systems, Model SR530). The two quadrature signals of the electrical input signal (proportional to the optical beat signal) with respect to the reference signal (at the beat frequency of 70 kHz) are obtained from the analog outputs (Rcos f and Rsin f) of the lock-in amplifier. The amplitude and the phase are then extracted from these two signals by numerical calculation (Eqs. (3.13) and (3.14)). The range of the output signals is 0-10 V. The full-scale sensitivity ranges from 100 nV to 500 mV. The input impedance of the signal is R_in = 100 MW with a frequency range from 0.5 Hz to 100 kHz. The noise voltage specification for the SR530 model is

      at 1 kHz. The measured thermal noise (Eq. (3.4) and Fig. 3.7) of the detector is P_TN = 10^-18 with R_L = 1 MW and B = 62.5 Hz. The corresponding voltage noise density is then

      (3.6)

      With the same R_L and B we get

      which is much larger than the voltage noise density of the lock-in amplifier. Thus, the noise of the lock-in amplifier is negligible.

      The signal acquisition can be accomplished in different ways. If the scan is controlled by the AFM commercial software, the two signal outputs of the lock-in amplifier are sent to the AFM electronics through a signal access module (SAM). The driving AFM software ("Dp.exe" software of Park Scientific Instruments) saves the two inputs (channels Raw1AM2 and Raw1AM3) into non-standard "hdf" (hierarchical data format) files. These files are converted into simple text format files (using "ScanActiveX"). Then, with e.g. Matlab, an image is built from the data.

      In the final set-up configuration, where the scanning is done by an external x-y-z piezoelectric translation stage, the signal is acquired with software written in Labview (cf. chapter 8.1). The tip approach is done by means of the AFM software. Then, the AFM electronics feedback is switched off and the raster scan, as well as the image acquisition, is accomplished by the software. A linear voltage ramp is applied to the piezo-stage to scan in one direction and the data acquisition is performed simultaneously. The number of samples N_s, the scan dimension D_s and the integration time t of the lock-in amplifier are chosen. For each sampling point, the signal is acquired during t, which is typically 10-30 ms. Thus, the time needed to scan one line is

      (3.7)

      The scanner displacement D_s in one direction (e.g. in the x-direction) is synchronized with the scan time T. For instance, for a number N_s of 128 pixels, scanning on one line will take 3.8 seconds. For a 2-D image (128 x 128 pixels), the acquisition will take about 8 minutes.

3.4.1 Amplitude and phase measurement of a plane wave

      As a test of the heterodyne scanning probe microscope we measured the amplitude and the phase of a plane wave. By sending the object beam directly to the fiber tip with an incident angle q (compared to the zdirection), we measured the optical amplitude and phase by varying the position of the tip in the zdirection.

      A plane wave propagating in air is described by

      (3.15)

      where E₀ is the amplitude, k the wave number and F₀ the phase shift. As a function of z, the amplitude is constant (E₀) and the phase is

      (3.16)

      The measured amplitude and phase as a function of z are presented in Fig. 3.13. In the ideal case, when the incoming beam is perfectly parallel to the tip movement (i.e. q = 0), the period for the phase is l. In our measurement we had q = 25, which gives a period of
L = l/cosq = 586 nm (Fig. 3.13b).

      a)

      b)

      The measured phase fits very well the theoretical curve (Eq. (3.16)), but the measured amplitude shows an unexpected periodic modulation. This can be explained by an interference phenomenon in the system. During the measurement, another beam is reflected (or scattered) and the tip detects the interference.

      The interference of two plane waves of same frequency and with intensities I₁ and I₂, respectively, can be described as Eq. (2.6), but in the space domain instead of the time domain. The total intensity of the interference can be expressed by

      (3.17)

      where I₁ + I₂ = I_dc is the dc level,

      the amplitude of the intensity modulation, L the interference modulation period and D_F is the phase difference.

      From the measured amplitude (Fig. 3.13a), we get the dc level I_dc = I₁ + I₂ = 7.45 (a.u.) and

      amplitude of the optical signal (Fig. 3.14a). We can thus calculate the respective intensities to be I₁ = 7.42 (a.u.) and I₂ = 0.03 (a.u.). The intensity ratio of the two interfering waves is
I₂/I₁ = 0.4%. The visibility of the interference, defined by

      (3.18)

      is then 3%. We can thus conclude that even with a small visibility, the interference is well detected thanks to heterodyne detection.

      a)

      b)

      A two-dimensional scan (in the X-Z plane) is presented in Fig. 3.15. This type of 2-D scans are often used throughout this work. We will show in chapters 4 and 6 such 2D measurements in more detail. The propagation of a plane wave, given by the phase, becomes more clearly visible in this representation.

      a)

      b)

      The iso-phase lines in Fig. 3.15 allow to see the phase fronts of the wave field. In fact, the tilt angle q becomes more evident in this representation. Besides this, we can also observe a small curvature of the wave fronts due to the collimating lens (section 3.1 and Fig. 3.4).

4.1.1 Total internal reflection

      Let us consider two media of refractive index n₁ and n₂. A plane wave illuminates the interface of the two media with an incident angle q₁. If n₁ < n₂, there is always a refraction of the light (Fig. 4.1a). This means that the incoming beam will be split into a reflected beam, and a refracted (transmitted) beam in the second medium with a direction of propagation given by the Snell-Descartes law [42], or also called refraction law [43]:

      (4.1)

      where q₂ is the angle between the refracted wave and the normal at the interface.

      Let us now consider the opposite case where n₁ > n₂. The Snell-Descartes law allows a refracted wave only as long as the angle of incidence is smaller than a critical angle (Fig. 4.1b)

      (4.2)

      If this limit is exceeded, propagation of the transmitted wave is no longer possible and all the light is reflected (Fig. 4.1c). This phenomenon is called total internal reflection.

      a)

      b)

      c)

      We usually speak of TE (transverse electric), or s-polarization, when the electric vector E subscript of the incident plane wave is normal to the incident plane and TM (transverse magnetic), or p-polarization, when E subsript is parallel to the incident plane. Let us describe the system in Fig. 4.2 with k_j, k_r, k_t the incident, the reflected and the transmitted wave vectors, respectively. The refractive index n₁ is higher than n₂ and the angle of incidence q is larger than q_c.

      In total internal reflection, the refracted wave vector has a real component in the x-direction and a purely imaginary z-component. Thus, the transmitted wave propagates parallel to the surface (i.e. along the x-axis) and is attenuated in the z-direction. In this case, we get an inhomogeneous plane wave or an evanescent wave.

      a)

      b)

      Let us now consider the evanescent field in medium n₂ (z ³ 0). The boundary conditions associated with Maxwell's equations at the interface between the two homogeneous media with refractive indices n₁ and n₂ (in the z = 0 plane) have to be preserved [44]. Taking the boundary conditions into consideration, we get for the z-variation of the evanescent electric field in medium n₂ for s-polarization [45]

      (4.3)

      where n = n₂/n₁ is the relative refraction index and for p-polarization,

      (4.4)

      where q is the incident angle, Eⁱ_s,p the amplitude of the electric field at z = 0 and
(e_x, e_y, e_z) are the normalized vectors of the x-y-z-direction. Equations (4.3) and (4.4) describe an exponential decrease of the evanescent amplitude in the media n₂ with a penetration depth of

      (4.5)

      The penetration depth expresses the values of z where the amplitude of the electric field Eⁱ has reduced by a factor 1/e. The variable d_p is independent of the polarization of the incident light and decreases with increasing q. For instance, if l = 532 nm, n₁ = 1.5 (glass), n₂ = 1 (air), the critical angle is q_c = 41.8°. By taking an angle of incidence of q = 45°, the penetration depth is d_p = 239 nm. For an incident angle of q = 80°, the penetration depth is d_p = 78 nm. Figure 4.3 shows the amplitude of the exponentially decreasing evanescent electric field with increasing distance from the interface (in TE-mode and TM-mode).

      a)

      b)

4.1.2 Frustrated total reflection

      The mean energy associated with an evanescent wave is given by the Poynting vector

      Energy flows in the x-direction, parallel to the surface, but not in the z-direction because the Poynting vector in z is imaginary [44]. To access this energy, it is necessary to transform one part of this evanescent wave into a propagating wave (k_t) which is then detectable. Let us look at Fig. 4.4 to understand the transfer of the light to a third medium. The incident wave (k_i) is totally internal reflected (k_r) at the interface between the media n₁ and n₂. By approaching a third medium n₃ close to the surface, the evanescent wave is frustrated and a wave propagates into the third medium with a wave vector k_t. This process is called frustrated total reflection. Newton made such an experiment by approaching a lens close to a prism. He observed that the area where the light was transmitted into the lens was larger than a simple contact point [46].

      If we assume n₁ = n₃, the complex amplitudes E⁵₃ and E^p₃ of the electric fields in the third medium at z ³ d for s- and p-polarization respectively, are given by [46]

      (4.6)

      where E^s,p₁ are the incident amplitudes of the electric field, q is the incident angle, K = 1/d_p and d is the distance between media 1 and 3. The factors A_s,p are

      (4.7)

      for s-polarization, with

      (4.8)

      and

      (4.9)

      for p-polarization, where

      (4.10)

      From Eq. (4.6), the normalized electric field amplitudes in medium 3 are then given, for both polarizations, by

      (4.11)

      From Eq. (4.6) we also get

      (4.12)

      where a = n₁kcos q and b = kd. From this equation, we can deduce the phase as

      (4.13)

      The amplitude of the electric field at the interface between the air and the third medium depends on the wavelength l, the thickness d of the second medium and the incident angle q. The variation of the field with distance is no longer exponential as in the total internal reflection case (Fig. 4.5). This is due to coupling between the two media at small distances. The slope at the interface (z = 0) is zero.

      a)

      b)

4.1.3 Frustrated total reflection in more complex systems

      The most basic form of frustrated total reflection is based on three media only. Study of more complicated systems generally has to consider more media. In frustrated total reflection, the frustrated light intensity in the third medium decreases with the distance between the media 1 and 3. If we put a fourth medium in the system, the behavior can be strongly perturbed. It has been shown that the frustrated intensity is a periodic function of the thickness of the second media [47].

      If we consider a metallic layer (e.g. Chromium layer of 30 nm thickness, with a complex refractive index of n_Cr = - 12.3 + i24.5 at l = 516.6 nm [40]) at the surface of the third medium, a system with up to four different refractive indices is created. Figure 4.6 shows the frustration of the total internal reflection with four different media.

      Even if the amplitude of the transmitted field in the fourth medium is weak, the evanescent wave is transmitted into a propagating wave in the last medium and becomes then detectable at large distances. The system can be extended to five media by coating the surface of the medium n₁ with another metal layer for instance.

      In the special case where media 2 and 4 are the same, the maximum of the transmission in the last medium can be equal to 1 and does not necessarily correspond to a zero thickness of air (medium 2). The maximum of the transmission is a periodic function of the thickness of medium 2 and describes the system resonance [48].

4.2.1 Set-up and measurements

      The set-up for evanescent wave measurements is presented in Fig. 4.7a. From the illumination fiber, the light is collimated to a plane wave. The polarization of the light is controlled by a fiber polarization controller, a quarter waveplate and a Glan-Thomson polarizer. The evanescent wave is created by total internal reflection (TIR) in the prism. The entire optical system is mounted on a xyz translation stage (section 3.1). We illuminate the internal surface of a prism at an angle q₁ = q_c = 41.5°. The refractive index of the prism is
n₁ = 1.5; n₂ = 1 is the refractive index of the air and n₃ is the refractive index of the tip
(n₃ = 1.5 if the probe is a dielectric fiber tip).

      a)

      b)

      By approaching a fiber tip to the surface (Fig. 4.7b), the evanescent field is converted into a propagating wave in the fiber. Frustrated total internal reflection is concerned with a change of refractive index and occurs in PSTM by the introduction of a probe tip (medium n₃) close to the boundary. The distance dependence of the measured amplitude reveals the expected quasi-exponential decay of the evanescent wave (Fig. 4.8). Although the tip is modeled here by a semi-infinite plane (Figs 4.4 and 4.6), it yields, in spite of its simplicity, information on the distance effect of the evanescent field frustration. Thus, if the perturbation of the tip is week, the measured field is the evanescent field itself. The exponential curve agrees well with the theory.

      The situation in Fig. 4.7b is system with three media (Fig. 4.4) composed of a prism, air and fiber probe. By coating the tip with a metal layer for instance, it becomes a four media system (Fig. 4.6).

      In order to have an idea of light magnitude detected, it is necessary to calibrate the voltage coming from the detector U_in into an absolute optical power P_o. In this case, we will see the quantitative limitation of optical power detection. In section 3.2.4, we presented the properties of the lock-in amplifier. We will show how the signal from the detector is processed through the lock-in amplifier (Fig. 4.9).

      The signal at the output of the detector, as obtained from Eqs. (2.12) and (3.2), is a sinusoidal voltage

      (4.14)

      composed of a dc offset

      (4.15)

      and of an ac part with amplitude

      (4.16)

      Since P_r >> P_o, the optical power P_r corresponding to the reference can be obtained from U_dc, measured with an oscilloscope (Hewlet Packard, Model 54600A, input impedance R_in = 1 MW).

      The lock-in amplifier processes only the ac part of the signal. The two output signals of the lock-in amplifier (RcosF and RsinF of Fig. 4.9), measure directly the two quadrature components U_ac cos f and U_ac sin f in Eq. (4.14). These components contain all the information of the measured optical field.

      From Eq. (4.16), we get the optical power corresponding to the field detected by the probe,

      (4.17)

      where U_r is the dc voltage of the detector output measured for the optical power P_r of the reference and U²_ac  is obtained from (Rcos  f)² + (Rsin  f)². As R₀ and S are known, assuming that m = 1 and measuring the voltage of the reference U_r, we get the conversion of the measured ac voltage amplitude into an optical power. It is interesting to see that Eq. (4.17) is independent of the load impedance R_L and the post-detection bandwidth B. For the following experiments, the measured U_ac has been converted into the optical power collected by the tip using Eq. (4.17) with U_r = 1.3 V, R₀ = 470 kW, S = 0.33 A/W and m = 1. The integration time t of the lock-in amplifier has been chosen to be t = 10 ms. The corresponding post-detection bandwidth is therefore B = (2t)^-1 = 50 Hz.

      The theoretical prediction of the frustrated total reflection (Eq. (4.11)) and the measurements are shown in Fig. 4.8 on a linear scale. In Fig. 4.10, the corresponding optical power of the measured amplitude is plotted on a logarithmic scale to view better the variations at low level. The theoretical limit of detection, given by the shot noise of P_o, (Eq. (2.22)), is P^min_o = 2.710^-17 W (for l = 532 nm, B = 50 Hz, h = 0.7).

      The angle of incidence is q = 50° and thus the penetration depth is d_p = 150 nm. We can see the very good agreement of the measurement with the theory up to a distance of about 600 nm from the surface. Theoretically, the distance limit (intersection of the theoretical curve and the shot noise limit), where the evanescent would be detectable, is 700 nm. The signal after this limit should be at the noise level. However, the detected signal beyond this distance does not only result from the evanescent wave. In fact, there is an optical background due to another light source. This undesirable background signal arises from scattered light at the surface (surface defects, dust, etc). This optical background is superposed on the evanescent wave. The maximal level of the scattered light seems to be at about P_o = 210^-15 W, which corresponds to the power of evanescent field at z = 500 nm. However, it should be noted that the dynamic range of the measured optical power is about 10⁴. From Eq. (2.18), the corresponding SNR is about 19 dB (for B = 50 Hz).

      The phase of the frustrated evanescent wave is shown in Fig. 4.11. The incident angle is q_inc = 50°. For an angle of incidence of q_inc= 57.8° the slope of the phase would be constant. We can see a good agreement of the experimental data with the theoretical curve given by Eq. (4.13). The absolute value of z and the measured phase offset have been adapted to fit best the theoretical and the experimental results.

      First, we showed that the measured signal from the evanescent field is corrupted by scattered light beyond a distance of 500 nm. We can see in Fig. 4.11 that the phase accuracy decreases also beyond 500 nm. From Eq. (2.21), the standard deviation of the phase is

      which yields df = 6.6° for SNR = 19 dB at this point. The SNR of the maximum detected power of P_o = 4.610^-13 is 42 dB, yielding df = 0.4°.

4.2.2 Lower limit of signal detection

      The aim of this section is to demonstrate the sensitivity of heterodyne detection. Figure 4.12 shows the results for evanescent waves measured with two different illumination intensities, with a ratio of 20 dB. The angle of incidence is q_inc = 59°. The curve for the lower intensity is obtained by inserting a ND (neutral density) filter in the illumination beam. The relation between the object power and the signal to noise ratio is given by Eq. (2.18). Theoretically, the minimum detectable optical power is P^min_o = 2.710^-17 W for B = 50 Hz (Eq. (2.22)), in both cases.

      The experimental data of the evanescent wave amplitude follows the theoretical curves, taken from Eq. (4.11). At distances larger than 400 nm for the higher power and 180 nm for the lower power, an undesirable background signal is detected because of scattered light at the surface. This background noise is clearly optical. Indeed, for the lower power, the background is also reduced, disappearing into the shot noise.

      a)

      b)

      For comparison, in the case of direct detection (without reference), the signal to noise ratio is limited by the Johnson noise. The minimum detectable optical power (SNR = 1 in Eq. (4.18)), with the same detector of Fig. 3.5, would be

      (4.18)

      With B = 50 Hz, R₀ = 470 kW, S = 0.33 A/W and T = 300 K, the minimum optical power detectable without heterodyne detection is thus only P^min_o = 410^-12, instead of
P^min_o = 2.710^-17 with heterodyne detection. In a direct measurement, a photomultiplier or an avalanche photodetector would therefore be more suitable, even with a smaller quantum efficiency h.

4.2.3 2-D evanescent field measurement

      We have presented one-dimensional curves of the evanescent field. It is interesting to investigate now two-dimensional plots of such fields by using the same set-up as in Fig. 4.7a. Indeed, we will demonstrate that the low-level background noise is really optical and is due to scattering points at the surface. We can eliminate the field due to the scattering effects from the evanescent field [49] for better image interpretation. By scanning the surface in the X-Z plane we get a mapping of the field above the surface. The angle of incidence, q = 43° (close to the critical angle q_c = 41.8°), is smaller than in the previous measurements of Figs 4.10 and 4.11. The penetration depth, d_p = 392 nm, is therefore larger, allowing the evanescent field to be detected up to large distances. Figure 4.13a shows the 2D amplitude, and Fig. 4.13b a 1D phase cross-section measurement of the evanescent wave amplitude and power, respectively. In this case, background light perturbation is not observed.

      a)

      b)

      The measured phase is presented in Fig. 4.14a as an iso-phase contour plot. The wavevector k only has a component k_x in the x-direction. Thus, the measured phase is the evanescent component of the field. Two "bold" lines of the phase contour plot are separated by p, corresponding to L = l/n sin(qXXXnm. However, the measurement accuracy in the x-direction is not very high, because the scans with the probe were done in the z-direction in order to getnm precise measurements of the optical power as a function of z. The steps in x are 20 nm and displacement errors may be accumulated after each step. At 1 µm above the surface, the evanescent wave can still be measured with an SNR of 23 dB (corresponding to an optical power of 6 fW for B = 50 Hz) and thus with a phase resolution of 3.8°. Beyond this limit, the phase is strongly perturbed because of noise; the limit of signal detection is reached.

      a)

      b)

      A measurement revealing otherwise well-distinguished scattering points is presented in Fig. 4.15. The surface of the glass sample had been made deliberately dirty. We can see the evanescent field (with a wavevector k_x) mixed with a propagating field (k_t), produced by scattering points. Detection is possible up to 3 mm above the surface. The influence of the scattered light becomes visible at about 500 nm above the surface. Above 1 mm it completely obscures the evanescent wave field.

      a)

      b)

      Another measurement of scattered light is shown in Fig. 4.16. Here the scattering is due to surface defects and not by dust. The evanescent wave field is already completely disturbed at the surface of the prism.

      a)

      b)

4.2.4 Water meniscus formation between the tip and the sample surface

      When the tip is brought close to the sample surface, some interactions have to be taken into consideration. In addition to van der Waals interaction between the tip and the sample at nanometric distances, one can be confronted with other forces such as Coulombian or capillary forces [33]. When contact between the tip and sample is made, a friction force results. If we work in a liquid medium, viscosity forces have also to be considered. The van der Vaals forces are useful for the AFM approach mechanism. Coulombian forces are pure electromagnetic interactions. As we have not worked in a liquid medium, the viscosity forces are negligible. One interaction that has been observed is the formation of a water meniscus between the tip and the surface, caused by capillary forces. Experimental measurement reveals this effect.

      In a relatively humid ambient environment, water condensation leads to the formation of a thin water layer both on the tip and the sample surfaces [50, 51]. A water meniscus is formed when the tip is close to the sample. The capillary force arises by decreasing the tip-surface distance and can apply strong attractive forces that hold the tip in contact with the surface. This effect is demonstrated in the following by experimental observations.

      In addition to relative humidity, the meniscus formation depends strongly on the hydrophility of the tip as well as the sample. For the geometrical considerations of Fig. 4.17 (i.e. by approximating the tip apex by a sphere), the attractive capillary force between the sphere and the surface due to the liquid is given by [33]:

      (4.19)

      where g_L is the specific surface energy of the liquid, q is the contact angle between the tangent to the sphere and the meniscus, d is the sphere penetration depth into the water layer and D is the sphere-surface distance. The capillary force is vertical.

      The strength of the capillary force has been estimated taking into account the corresponding values for surface energy of water g_L = 72 mJ/m² [33], a tip apex radius of R  = 50 nm and tip-sample distances of D  = 10 nm and 50 nm. The penetration depth of the sphere into the liquid has been estimated to be d  = 15 nm

      and the contact angle equal to q = 10°. The contribution of the capillary force in air, when the tip approaches a glass substrate can be thus estimated to |F_cap|= 10^-8 N for D = 50 nm and
|F_cap|- 2.710^-18 N for D = 10 nm.

      With the same set-up as in section 4.2, we measure an evanescent wave by frustrated total internal reflection (Fig. 4.7) in a prism (n_prism = 1.5). The incident angle is q = 73°. The mechanical movement of the tip and the optical signal are shown in Fig. 4.18.

      As the tip is far from the surface, the optical signal is low (Fig. 4.18/1). When the tip is moved towards the surface, light is coupled into the fiber by photon tunneling and the optical signal begins rising exponentially (Fig. 4.18/2). The medium between the prism and the probe is air. When the surface-tip distance becomes smaller (z = 50 nm from the surface), the humidity produces a water particle aggregation around the tip, forming a meniscus between the tip and the surface (Fig. 4.18/3). This causes an abrupt attraction of the tip and a corresponding cantilever deformation. In fact, at D = 50 nm, the capillary force is about
10^-8 N. With a fiber cantilever force constant of about 4 N/m (section 3.2.3), the deformation (in the z-direction) of the cantilever is about 25 nm. As the tip-surface distance becomes smaller, because of this deformation, the optical signal increases (Fig. 4.18/3). However, this is not the only explanation. In fact, the water meniscus causes a change of refractive index of the medium between the prism and the probe. The refractive index of air (n_air = 1) is replaced by the refractive index of water (n_water = 1.3) in Eq. (4.11) for the frustrated total internal reflection. The change of refractive index from 1 (air) to 1.3 (water) results in a modification of the coupling into the fiber probe. The theoretical consequence of the refractive index change is shown in Fig. 4.18 (dashed curve). Due to the water meniscus, the optical signal increases then drastically (Fig. 4.18/3). It is interesting to observe that the theoretical curve (Eq. (4.11) for n_water = 1.3) passes through the maximum of the discontinuity at z = 50 nm. This strengthens the theory of water meniscus formation.

      By continuing the approach, the tip makes the meniscus break up (Fig. 4.18/4) because of pressure and the signal decreases abruptly. Indeed, the medium between the surface and the tip is air again. Then, the coupled light follows again the theoretical curve (Eq. (4.11)) for n_air = 1.

      In order to avoid this perturbation of the measurements, we have to control the humidity. We have built a plexiglas box around our experimental set-up (Fig. 3.3), enclosing the SNOM (tip and sample). Silica gel is placed inside the box to reduce the humidity.

6.1.1 Theoretical background of grating theory

      It is well-known that a periodic scatterer illuminated by a plane wave generates a discrete set of propagating plane waves. The main goal of grating theory is to predict the complex amplitudes of these plane waves if the grating structure is known. The opposite case is presented here: by measuring the diffracted field, we would like to know the structure.

      The property of a grating to diffract an incident beam into clearly distinguished directions is given by the grating equation [58]:

      (6.1)

      where q_i and q_m are the angles between the incident (and the diffracted) wave directions and the normal to the grating, l is the wavelength, n₁ and n₂ are the refractive indices of the media, L is the grating period and m = 0, ±1, ±2, ... is an integer numbering the diffracted orders, so that the specular one is numbered as 0 (Fig. 6.1). If m = 0, Eq. (6.1) is equivalent to the Snell-Descartes law of Eq. (4.1). Two grating configurations are possible: reflection and transmission gratings. Equation (6.1) is the transmission grating equation (Fig. 6.1).

      Another notation uses a = k_x/|k| and b = k_z/|k|, where k = 2p/l is the wavenumber
(k_y = 0). The propagating diffraction orders are thus expressed by a = nsin q and b = ncos q. The quantity K = 2p/L is called the grating wavenumber. The corresponding grating vector K has the components K_x = 2p/L and K_y = K_z = 0.

      A solution of Eq. (6.1) is possible only when

      (6.2)

      Diffraction orders with number m fulfilling Eq. (6.4) are called propagating orders. The vertical (zdirection) wavevector component can be found from [59]:

      (6.3)

      so that

      (6.4)

      The x and z variation of the propagating orders

      (6.5)

      represents a plane wave which propagates in the direction of k_m.

      In the opposite case where |sin q_m| > 1, Eq. (6.6) implies that the vertical component of the wavevector is imaginary and these orders decrease exponentially with the distance from the grating surface. Their amplitudes are given by

      (6.6)

      with

      (0.7)

      These orders are called evanescent orders. They can not be detected at a distance greater than a few wavelenghts from the grating surface. Fig. 6.2 represents the directions of the diffracted orders formed by adding or substracting a multiple of the grating wavevector K from the zeroth order. The wavevectors of the propagating orders have moduli equal to 2p/l. Evanescent orders lie outside of this area. The representation of Fig. 6.2 is called Ewald-sphere [60].

      The grating equation determines the direction of propagation of the different diffracted orders but says nothing about the amount of light which is distributed in each order. The ratio of the optical power of a particular order (parallel to the z-axis) to the incident optical power is called diffraction efficiency.

6.1.2 Interference of three diffracted waves from a grating

      In this sub-section, we are interested in the special case of a 1 mm pitch surface relief grating. With a wavelength of l = 0.532 mm, which is smaller than the pitch of L = 1 mm, we get for normal incidence (Fig. 6.3) 3 propagating wavevectors k₀, k₊₁ and k_-1. The diffraction angles (given by sin q = l/L) are q₀ = 0° and q_±1 = 32.1°. The orders k_±m for m ³ 2 are evanescent.

      a)

      b)

      As the wavelength is smaller than the period of the grating, we will always get at least one propagating non-zero order. In this case, the evanescent orders are "hidden" by the diffracted orders in the far-field (at distances larger than l). As evanescent orders do not contribute to the far-field [61], the situation here can be assimilated to three-wave interference. We have to distinguish two polarization states. If the incident wave is linearly polarized and the electric field vector is perpendicular to the plane of incidence (X-Z plane), all diffracted orders have the same polarization, called s- or TE-polarization (Fig. 6.3a). The other case, where the electric wavevector is parallel to the incident plane, is called p- or TM-polarization (Fig. 6.3b). Any other polarization state can be expressed by a linear combination of these two fundamental states. In the TE case, only one component E_y of the electric vector is present, whereas in the TM case two components E_x and E_z have to be considered.

      For TE-mode illumination, the total electric field of the -1, 0 and +1 diffraction orders is given by

      (6.8)

      where a₀ and a_±1 are the (complex) amplitudes of the three propagating plane waves, q is the diffraction angle of the -1 and +1 orders and k = 2p/l is the wavenumber. By assuming that
a_-1 = a₊₁ = a₁, we get the symmetric case

      (6.9)

      and thus the amplitude of the electric field is

      (6.10)

      and the phase is

      (6.11)

      The pitches of the interference pattern in the x- and z-direction are given by

      and

      (6.12)

      The resulting amplitude and phase of the interference of the three plane waves is shown in Fig. 6.4. Depending on the position in z, we get a modulation in the x-direction (z constant) either of L = L_x = 1 mm or L = L_x/2 = 0.5 mm. In the z-direction, the periodicity of the optical feature is L_z = 3.48 mm (Fig. 6.4a). In Fig. 6.4b a phase singularity is shown inside the circle. Phase singularities are isolated points where the intensity of the optical field is zero. Interference of at least three plane waves is needed to give birth to phase singularities. Properties of these special phase-points are presented in section 6.4.

      For the TM-mode illumination (Fig. 6.3b), the total electric field vector of the -1, 0 and +1 diffraction orders is given by

      (6.13)

      By assuming again that a_-1 = a₊₁ = a₁, we get for the x-component of the total electric field vector

      (6.14)

      and for the z-component of the total electric field vector

      (6.15)

      Amplitude and phase of E_x and E_z following Eqs. (6.14) and (6.15) are presented in Fig. 6.5.

      a)

      b)

      We see that the x-component is very similar to the case of the TE-mode, but that the z-component is totally different. Since E_x is perpendicular, but E_z parallel to the tip, the problem will be to know what is really detected by the SNOM tip: only E_x, or a combination of E_x and E_z? This is a crucial question, because the answer explains the behavior of the tip probe and might lead to know its transfer function [62, 63] for the electrical field vector.

6.1.3 Rigorousely calculated field diffracted by gratings

      In the previous sub-section, we have studied the propagation of the light diffracted by a grating of 1 mm pitch assuming a superposition of 3 plane waves of identical amplitude and polarization, interfering in free space. Analysis of the diffraction problem has to consider the exact shape of the grating that diffracts the incoming light. However, the distribution of the amplitude and the phase behind a grating depends strongly on the relief of the grating (shape, height, fillfactor, period). Moreover, the near-field close to the surface (less than one wavelength) is quite different from the far-field, because of the contribution from the evanescent waves.

      For a correct analysis, rigorous diffraction theory has to be employed. Thanks to computation time improvement, rigorous methods have been developed to calculate exact solutions of Maxwell's equations. Rigorous methods take into account the state of polarization of the light whereas it is neglected in approximation methods such as thin-element. In this section, we will give a very short description of the Fourier Modal Method [64], which is a differential method based on mode expansion of the field inside the grating structure.

      The Fourier Modal Method (FMM) is based on variable separation of the Helmholtz equation in homogenous media. A 2-D software based on FMM calculations has been developed [65] for periodic optical structures in TE- and TM-mode. The elementary structure, which is then repeated periodically, is approximated by a number of different homogenous layers (Fig. 6.6).

      The larger the number of layers, the better the approximation of the shape of the structure, but this demands more computer time. The input parameters are the shape of the grating, the incident monochromatic wave (wavelength, incident angle and polarization), the number of diffraction orders and the refractive indices (at the different interfaces). The result of a rigorous calculation of gratings will be shown in sections 6.4 and 6.5.