1. Introduction

Precise, fast and cost effective phase measurement is an important issue for many optical instruments and sensors. Among the large number of interferometric modulation/demodulation techniques [1], phase modulations based on optical mirror vibration or piezo-strecher are particularly interesting in the exciting context of phase-sensitive microscopy and near-field imaging [2–4]. The advantages of mirror based modulation compared to acousto-optics frequency modulators, also used in optical microscopy [5–7], include excellent achromaticity which is needed for spectroscopic application, simplicity, low cost while keeping a low phase noise.

The general principle consists in modulating the phase in the reference arm of an interferometer by adding a variable delay length and measuring the resulting interferences. When the phase modulation is linear an harmonic beating with the signal arm occurs, and any phase change in the signal arm produces a phase lead or delay which can be recorded with an electronic phase-meter. Because a phase modulation can be described as a Doppler frequency modulation, such methods are often referred to as pseudo-heterodyne although this term was initially used to describe a detection scheme where the optical source itself is swept in frequency [8, 9] or in phase [10] in unbalanced interferometers. Since it is much easier to obtain a sinusoidal phase modulation by vibrating a mirror or stretching a fiber, than a linear one, pseudo-heterodyne methods have been generalized to such non linear periodic phase modulation. However, in these cases the detected intensity presents multiple harmonics and the phase is not retrieved with a conventional phase meter.

Different approaches have been proposed to tackle this issue, most notably in the context of fiber optic sensors [11,12] and near-field optics [3,4,13] where the signal is particularly weak. For a sinusoidal modulation function, lock-in amplifier (LIA) detections on two consecutive harmonics [3] present in the detected signal have been used successfully. However, since the useful signal is spread over a larger number of frequencies, more efficient or simpler methods are considered [14,15]. In this paper, we propose a generalized LIA approach to recover amplitude and phase with an optimum signal-to-noise ratio for any modulation function. Instead of using multiple LIA at selected harmonic references (M-LIA), our lock-in detector is fed by references having all the frequencies of interest. The basic operation principle of this Generalized Lock-In Amplifier (G-LIA) is first presented and applied to phase sensitive spectroscopy. The approach is then extended to the case where the signal field presents an additional amplitude modulation. This extension is notably important in the context of near-field nanoscopy where a nano-probe provides a weak modulated signal in the signal arm. The practical implementation of the extended G-LIA in near-field nanoscopy is then detailed, and the method advantages are presented. Some calculation details and a summary table for different modulation functions are given in appendices.

2. Interferometry using G-LIA

2.1. Principle

We consider a monochromatic two-arm interferometer with a phase modulated reference field. We denote by ω the angular frequency of the light, $E_S(t)=\sqrt{2}{E}_S\text{cos}(\omega t+{\varphi }_S)$ is the signal field with unknown amplitude term E_S and phase ϕ_S, and $E_R(t)=\sqrt{2}{E}_R\text{cos}(\omega t+{\varphi }_R)$ is the reference field with a constant amplitude and a time-modulated phase ϕ_R(t). In practice ϕ_R(t) can be produced by either a phase, or an equivalent frequency modulation [8, 9]. With these notations the detected intensity proportional to 〈(E_S(t) + E_R(t))²〉 is:

Here, we propose another approach where the detected intensity is not multiplied by consecutive single harmonic references such as in the M-LIA method [3]. Instead the signal is multiplied by two orthogonal references having the same set of harmonic frequencies than I_mod, namely cos(ϕ_R) and sin (ϕ_R). As for the serrodyne modulation, this operation can be performed with an electronic multiplier and integrator:

In order to work with any modulation function, the DC terms in I_det can be simply filtered using a high pass filter. Amplitude and phase information are then obtained via the magnitude and angle of (X_ϕR (Ĩ_det)/k̃_X + i * Y_ϕR (Ĩ_det)/k̃_Y) where Ĩ_det denotes the filtered detected intensity and (k̃_X, k̃_Y) the associated proportionality constants. In this way, all the available modulated power is used to recover the signal field information. Moreover, many phase modulation functions can be considered: sinusoidal, symmetric or asymmetric triangular functions, etc., regardless of the phase modulation amplitude a. For such simple modulation functions the constants have simple analytical expressions and a proper choice of a actually leads to k_X = k_Y = 1 as for the perfect serrodyne case. The expressions of these constants are given in appendices A and B for sine and triangular modulation functions.

More generally, the actual phase modulation ϕ_R = ϕ_real can deviate from the required drive excitation. This real phase modulation induced by the modulator can usually be measured during the experiment, for example, using piezoelectric modulator equipped with strain gauges or other displacement sensors. Empirically, the same approach can also be used in this case: the integrals (8–9) are numerically evaluated with the measured ϕ_R in order to determine the proportionality constants (k̃_X, k̃_y).

2.2. Test and application to phase resolved spectroscopy

The Michelson interferometer shown in Fig. 1(a) was built to assess the validity of the technique for precise phase measurements in presence of arbitrary phase modulation. To create the phase modulation, the reference mirror, mounted on a piezoelectric crystal was excited sinusoidally at an angular frequency of 200π rad/s and an arbitrary amplitude a = 2.4 rad. For demonstration purposes, the second mirror mounted on a piezoelectric crystal was used in the signal arm to obtain controllable changes of the phase ϕ_S, monitored via a capacitive displacement sensor. The detected signal I_det containing many harmonics was filtered from its DC component by a high pass filter and the resulting signal Ĩ_det was sampled with a 16-bit acquisition card. The G-LIA operation expressed by (8–9) was then performed numerically using LabVIEW [17], and the phase and (constant) amplitude were retrieved using the expression of (k̃_X, k̃_Y) given in the appendix A.

 

Fig. 1 Test and application of the G-LIA technique. (a) Michelson interferometer operating with arbitrary phase modulation ϕ_R in the reference arm. (b) Phase measurement obtained with the setup (a) for a sine modulation of the reference mirror. The phase of the signal field follows a triangular function monitored by capacitive sensors (thick line). The simultaneous interferometric measurements obtained with an integration time of 0.05 s are marked with circles. (c) Same setup as (2) with a gas cell in the signal arm and a tunable laser diode. (d) Example of phase resolved spectroscopy obtained with setup (c).

Download Full Size | PDF

Figure 1(b) shows the phase measurements obtained for a symmetric triangular displacement of the signal mirror. A remarkable agreement is observed between the G-LIA based interferometric measurement and the the actual phase change obtained through the displacement sensor. With this approach similar results were obtained with sine phase modulation frequencies of few kHz, only limited by the bandwidth of the loaded piezoelectric actuator.

Figure 1(c) shows a similar setup where a single-mode laser diode operating near λ = 650.0 nm was used to perform phase-resolved spectroscopy. A gas cell of iodine was added in the signal arm and the wavelength was tuned over few picometers across an absorption line of the gas with ramps of current. As for the previous experiment an arbitrary sine phase modulation was created in the reference arm (2000π rad/s, a = 1.2 rad) and amplitude and phase were retrieved through the G-LIA operation. The corresponding results are shown in Fig. 1(d). The observed magnitude and phase spectra actually reflect the expected variation of the real and imaginary parts of the gas refractive index across a Lorentzian absorption peak slightly convoluted with the source linewidth.

3. Extension of the G-LIA to amplitude modulated signals

We consider now the situation where both the signal field amplitude and the phase ϕ_R of the reference field are modulated. An amplitude modulation is particularly interesting to recover a weak modulated signal field, and to get rid of unwanted (and not modulated) coherent light impinging on the detector in addition to E_S and E_R. This approach is notably used successfully in near-field nanoscopy, where heterodyne and pseudo-heterodyne interferometric detection schemes are used to determine the amplitude and phase of the field scattered by sharp oscillating nano-probes. In such experiments the angular frequency Ω_A of the amplitude modulation is typically one or few order of magnitudes higher than the characteristic angular frequency Ω of ϕ_R.

To simplify the mathematical description, we note Ω_A=nΩ, where n is typically much greater than one. The part of the signal field modulated at Ω_A is then noted to be $E_P(t)=\sqrt{2}{E}_P\text{cos}(n\mathrm{\Omega }t)\text{cos}(\mathrm{\Omega }t+{\varphi }_S)$. With this expression, the useful modulated term I_mod of relation (2) is now

4. Test and application to infrared nanoscopy

In the second configuration depicted in Fig. 2(a), the signal field is modulated in amplitude which is particularly useful to extract a weak signal from noise. To test and assess the quality of the phase measurement in this second mode, an amplitude modulation was achieved by placing in the signal arm a tuning fork vibrating at Ω_TF. In this way the signal intensity < $E_S^2$> is sinusoidally modulated and therefore the signal field also presents modulation at the frequency Ω_TF (and harmonics). The signal mirror was then moved back and forth in a controlled triangular motion (±0.80μm corresponding to ±1.0 rad). To recover the corresponding signal phase, the sine phase modulation ϕ_R = acos(Ωt) and the tuning fork excitation signal proportional to cos(Ω_At) were used to build the references C(t) and S(t) with Ω_A = Ω_TF. The corresponding phase measurement obtained with such extended G-LIA method is shown as an insert in Fig. 2(b), proving the validity of the approach.

 

Fig. 2 Extended G-LIA operation performed in the mid-IR. (a) Experimental setup used for testing. (b) Phase recorded when a triangular phase modulation of about 1 rad is added in the modulated signal arm. (c–f) Phase-sensitive infrared nanoscopy experiments. (c) Schematic of the experimental setup and investigated samples. (d) Amplitude and phase of the signal field scattered by the nano-probe on oxidized copper lines embedded in Si (sample 1). The phase profile along the white dashed line is plotted below (raw signal averaged on 5 lines). (e) Similar experiments performed with a CO₂ laser for decreasing probe sizes as shown in the inserts. (f) Amplitude (blue), phase (gray) and topography (dashes) profiles obtained across the gold/polymer grating surface (sample 2) showing a fast signal drop when the gold-probe distance increases.

Download Full Size | PDF

The same setup (including the source) was then used in a near-field optics experiment where the bare tuning fork and signal mirror were replaced by a scanning probe microscope. The system is sketched in Fig. 2(c). In this case, the light periodically scattered by a nano-probe produces the modulated signal field interfering with the phase-modulated reference field. As for the previous experiment, shown in Fig. 2(a), an acquisition card featuring high resolution (22 bits) was used for the G-LIA operation to offer the performance level of a high quality digital LIA. A tunable infrared quantum cascade laser [18] emitting near 10.1μm with a rather low but highly stable power of few mW was used. The laser polarization was set along the probe axis with an incident angle of about 60°. More details on the optical system itself can be found in [19].

An example of highly resolved phase sensitive nanoscopy measurement is shown in Fig. 2(d). The images were obtained on a highly sub-wavelength surface grating made of slightly oxidized copper lines (native oxidation) embedded in silicon as depicted in the insert of Fig. 2(c). Two copper lines imaged in the mid-infrared are clearly visible, both in the amplitude (dark lines) and in the phase (bright lines). While a higher signal is typically obtained above metal structures in infrared near-field nanoscopy due to an increased effective polarizability of the probe [20], very sharp probes are actually mostly sensitive to the first nm of the sample material (CuO₂). Additional experiments were carried out on the same sample with a similar setup using a more powerful CO₂ laser and different probe sizes [21]. A stronger amplitude signal was indeed obtained for bigger probes as shown in Fig. 2(e). We point out also that given the size of the probe, phonon confinement effects could also play an important role on the permittivity of the sharpest probe and impact the observed signal [21, 22]. The description of possible specific probe-sample interaction leading to the observed contrasts should be the subject of a separate paper.

The extended G-LIA method was also tested on a second subwavelength grating made of gold and polymer (S1813 photopolymer). This structure is referred as sample2 in Fig. 2(c), and the signal profiles are given in Fig. 2(f). The G-LIA amplitude signal exhibits an expected higher signal on the gold material [23] compared the the polymer signal and a sharp drop as the probe moves away from the metal surface.

Several precautions were taken to minimize any background contribution. Experimentally, one of the remarkable advantages of the mounted setup is the implementation of an aberration-corrected and diffraction limited Schwarzschild objective in the signal arm. This afforded the ability to highly focus the beam to ≈ λ² (as measured by an IR-camera) on the tip apex. The probe was electrochemically etched from a straight tungsten wire [24], and was attached to a tuning fork operating in tapping mode. The tip end diameter was smaller than 30 nm with an angle of about 10°. The small size of the probe body and the absence of cantilever was found to be useful to strongly reduce eventual background light modulation. Another crucial issue, was the usage of an ultra-small oscillation amplitude for the probe (≤ 5 nm, as measured by interferometry). This unconstrained the necessity to use detection at the probe’s higher harmonics frequencies [25, 26] often required to suppress modulated background light, the two approaches being actually nearly equivalent [27]. The setup was built to operate in the mid-IR range where the scattering cross-section is essentially low. The G-LIA method, however, can be easily adapted if required to detect the signal at the k^th harmonic of the probe oscillation frequency by setting Ω_A = k × Ω_tip, i.e. simply by multiplying the factor n by k.

In these experiments, the amplitude modulation frequency was imposed by the experimental resonance frequency of the tuning fork (≃ 30kHz) on which the scattering nano-probe is glued to operate in tapping mode (cf. Fig. 2(b)). The frequency of the vibrating reference mirror was arbitrarily chosen equal to 1 kHz leading to a frequency ratio of n = Ω_A/Ω ≃ 30. Although the frequency ratio could be set to an integer value with high precision, we point out that the benefit is actually negligible for such high values of n, as exemplified in Fig. 3(a).

 

Fig. 3 (a) Example of simulated relative error on the phase using G-LIA detection for increasing real values of n. The simulation was made with ϕ_S = π/4, Ωt_int = 20π, a sampling interval equals to 2π/1000, and a = 3. (b) Background field attenuation for increasing values of a and identical background and modulated signal fields values. In this simulation, n = 30, Ωt_int = 20π, and the sampling interval is equal to 2π/1000. The background contribution drops to zero for a corresponding to a zero of J₀(a) and large n. (c) Simulated comparison of the signal level between G-LIA and a M-LIA approach where the two first sideband harmonics are detected (Ω_A + Ω_R, Ω_A + 2 * Ω_R)

Download Full Size | PDF

The two references signals C(t) and S(t) were built from the normalized modulation signals e_i(t):

Finally, it is important to note that the phase of the mechanical probe oscillation can be strongly phase shifted with respect to the excitation signal, notably for cantilever probes. While such shift does not affect the measured signal phase it attenuates the retrieved amplitude. Therefore it is better to use the measured mechanical modulation signal rather the excitation one. Alternately, it is clear that the G-LIA can be applied twice using cos(Ω_At) and sin(Ω_At) as modulation functions to recover the full amplitude signal whatever the mechanical phase shift. This approach also provides the mechanical phase shift and is described in appendix D. A general summary table is given in appendix E.

5. Conclusion

In conclusion we have described a modified lock-in detection method to recover amplitude and phase in optical interferometers, where the reference beam can be non linearly modulated in phase or frequency. This method is optimum in the sense that it fully exploits the detailed spectrum of the detected beating between signal and reference, thus providing the highest signal-to-noise ratio possible. The principle was detailed for sinusoidal and triangular phase modulations and was experimentally tested, showing high stability and reliability. The method was extended to amplitude-modulated signals in the common case of a sinusoidal phase modulation. This second approach, useful in the the case of very low signal experiments, was successfully tested in mid-IR near-field nanoscopy, leading to a high signal to noise ratio under low illumination power conditions. In addition, the conditions to obtain efficient background light suppression in this second approach were derived. Although the tests were made using vibrating mirrors, much faster phase modulation can be achieved, for example using a slight wavelength modulation of the laser source in an unbalanced interferometer.

Appendices

A. Sine wave phase modulation

We start from Eqs. (8–9) where the references cos(ϕ_R) and sin (ϕ_R) contain all the frequencies of interest contained in the intensity. In the case of a sinusoidal modulation ϕ_R = a sin(Ωt), the two integrals have well known analytical solutions based on the integral representation of Bessel functions [16]. Neglecting for now the contribution of the constant term I₀, i.e. considering that it can be subtracted, we have:

(19)$$X_{a\text{sin}(\mathrm{\Omega }t)}(I_{\mathit{mod}})=k_X{E}_R{E}_S\text{cos}({\varphi }_S)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\text{with}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{k}_X=(1+J_0(2a)),$$

(20)$$Y_{a\text{sin}(\mathrm{\Omega }t)}(I_{\mathit{mod}})=k_Y{E}_R{E}_S\text{sin}({\varphi }_S)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\text{with}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{k}_Y=(1-J_0(2a)),$$

where J₀ is the 0 – th Bessel function. Although the phase can be retrieved from these two Eqs., it is convenient to choose a phase modulation a such that J₀(2a)= 0 (e.g. for a ≃1.20 rad, 2.76 rad, etc.). The outputs are then identical to the perfect serrodyne case cf. relations (4–5), i.e. k_X = kY = 1.

While it is possible to obtain I_mod by subtracting I₀, such operation complicates the experimental setup, unless the signal is small enough to consider $I_0\simeq <E_R^2>$. The constant term I₀ is easily filtered by a simple DC filter, but such filter can affect I_mod since the reference cos (ϕ_R) may exhibit a non negligible DC component as can be seen in Fig. 4(c).

 

Fig. 4 Orthonormal reference signals (cosϕ_R, sinϕ_R) and their Fourier transforms for selected phase modulations ϕ_R(t): (a) Sawtooth with an amplitude 2π, (b) Sinusoidal with a phase amplitude a₁ corresponding to the first zero of J₀, and (c) Triangular with an amplitude of π/2 rad.

Download Full Size | PDF

This is also the case for a sinusoidal modulation if we consider an arbitrary amplitude a (cf. Eqs. (6) with m = 0). For this reason we evaluate the signal outputs for a filtered detected signal Ĩ_det:

(21)$$X_{a\text{sin}(\mathrm{\Omega }t)}({\tilde{I}}_{\text{det}})={\tilde{k}}_X{E}_R{E}_S\text{cos}({\varphi }_S)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathit{with}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\tilde{k}}_X=\left(1+J_0(2a)-2J_0^2(a)\right)$$

(22)$$Y_{a\text{sin}(\mathrm{\Omega }t)}({\tilde{I}}_{\text{det}})={\tilde{k}}_Y{E}_R{E}_S\text{sin}({\varphi }_S)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathit{with}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\tilde{k}}_Y=(1-J_0(2a))$$

from where (E_s, ϕ_s) can again be retrieved directly. We note that for 2a corresponding to a k-th zero of J₀, the term $J_0^2(a)$ is actually small for k even, and especially for large value of k, so that we can have k̃_X ≃ k̃Y ≃ 1. The corresponding expressions for a triangular modulation are given in appendix B hereafter. Examples of orthonormal references are given in Fig. 4 for serrodyne 4(a), sinusoidal 4(b) and triangular 4(c) modulation in direct and frequency space.

B. Triangle wave phase modulation

For a triangle wave modulation ϕ_R = aTr(Ωt), the modified lock-in operation performed on the modulated intensity term I_mod = 2E_RE_S cos (ϕ_S − ϕ_R) is

(23)$$X_{aTr(\mathrm{\Omega }t)}(I_{\mathit{mod}})=k_X{E}_R{E}_S\text{cos}({\varphi }_S)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathit{with}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{k}_X=\left(1+\frac{\text{sin}(a)\text{cos}(a)}{a}\right)$$

(24)$$Y_{aTr(\mathrm{\Omega }t)}(I_{\mathit{mod}})=k_Y{E}_R{E}_S\text{sin}({\varphi }_S)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathit{with}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{k}_Y=\left(1-\frac{\text{sin}(a)\text{cos}(a)}{a}\right)$$

where Tr(Ωt) is the triangle wave ranging from −1 to 1 in phase with sin(Ωt). It is then convenient to choose a phase modulation a equals to a positive integer number times π/2 in order to obtain the same output as the ideal serrodyne given by (4–5), and simple reference signals cf. Fig. 4(c). Since generally I_mod can not be obtained by a DC filtering of the detected intensity I, it is more convenient to consider a modified lock-in operation on the filtered detected intensity Ĩ_det :

(25)$$X_{aTr(\mathrm{\Omega }t)}({\tilde{I}}_{\text{det}})={\tilde{k}}_X{E}_R{E}_S\text{cos}({\varphi }_S)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathit{with}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\tilde{k}}_X=\left(1+\frac{\text{sin}(a)\text{cos}(a)}{a}-\frac{2{\text{sin}}^2(a)}{a^2}\right)$$

(26)$$Y_{aTr(\mathrm{\Omega }t)}({\tilde{I}}_{\text{det}})={\tilde{k}}_Y{E}_R{E}_S\text{sin}({\varphi }_S)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathit{with}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\tilde{k}}_Y=\left(1-\frac{\text{sin}(a)\text{cos}(a)}{a}\right)$$

The additional term $\frac{2{\text{sin}}^2(a)}{a^2}$ is small for large values of the phase modulation amplitude a, or values close to an integer number times π, so that we can also have k̃_X ≃ k̃_Y ≃ 1.

C. Background elimination for amplitude-modulated fields

In addition to E_R(t) and E_P(t) we consider now that we may have a third additional (unwanted) coherent background field $E_{Bg}(t)=\sqrt{2}{E}_{Bg}\text{cos}(\omega t+{\varphi }_{Bg})$, comparable or stronger than the modulated signal field $E_P(t)=\sqrt{2}{E}_P\text{cos}(n\mathrm{\Omega }t)\text{cos}(\omega t+{\varphi }_S)$. This constant background field can interfere with E_P(t) and gives an additional intensity term modulated at radial frequencies nΩ:

(27)$$I_{\mathit{mod}}^{\mathit{Bg}}=2E_{\mathit{Bg}}{E}_P\text{cos}(n\mathrm{\Omega }t)\text{cos}({\varphi }_{\mathit{Bg}}-{\varphi }_S).$$

Since C(t) and S(t) also have frequency components at nΩ, we have to evaluate the contribution of this term in the extended G-LIA operation:

(28)$$X_{n\mathrm{\Omega }t\pm {\varphi }_R}\left(I_{\mathit{mod}}^{\mathit{Bg}}\right)=\frac{1}{\mathrm{\Omega }{t}_{\mathit{int}}}{\int }_0^{\mathrm{\Omega }{t}_{\mathit{int}}}{I}_{\mathit{mod}}^{\mathit{Bg}}C(t)d(\mathrm{\Omega }t)$$

(29)$$Y_{n\mathrm{\Omega }t\pm {\varphi }_R}\left(I_{\mathit{mod}}^{\mathit{Bg}}\right)=\frac{1}{\mathrm{\Omega }{t}_{\mathit{int}}}{\int }_0^{\mathrm{\Omega }{t}_{\mathit{int}}}{I}_{\mathit{mod}}^{\mathit{Bg}}S(t)d(\mathrm{\Omega }t).$$

The contribution of these two integrals are found to be proportional to J_2n(a)+ J₀(a). From this result we see that the contribution of the unwanted terms modulated at nΩ can be canceled out by setting a modulation amplitude a corresponding to a zero of the 0th Bessel function, J_2n(a) being extremely small for a sufficiently large n.

It is worth noting that the additional unwanted interference term between E_Bg(t) and E_R(t), and the self interference term proportional to $E_{\mathit{Bg}}^2$ are not modulated at nΩ, and are therefore efficiently filtered without special precaution.

D. Double generalized lock-in detection

We consider an amplitude modulated signal field E_s(t) at an angular frequency nΩ, where Ω is still the phase modulation frequency. In general, the amplitude modulation of this field can be phase shifted with respect to the excitation signal. We denote by $E_s(t)=E_s^{\mathit{full}}(t)\text{cos}(\omega t+{\varphi }_s)$ the full signal field. A quite general expression for the modulated amplitude is $E_s^{\mathit{full}}(t)=\mathit{const}.+E_s\text{cos}(n\mathrm{\Omega }t+\psi )$ + harmonics. The phase ψ accounts for the eventual phase shift existing between the fundamental term and the driving signal in phase with cos(nΩt). To obtain an optimal determination of the amplitude and phase (E_S, ϕ_S) without knowing ψ, the G-LIA can be applied twice as part of the fundamental term is modulated proportional to cos(nΩt) and the other part proportional to sin(nΩt):

(30)$$X={\int }_0^{t_{\mathit{max}}}{I}_{\mathit{det}}\text{cos}(n\mathrm{\Omega }t)\text{cos}({\varphi }_R)dt\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\text{and}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Y={\int }_0^{t_{\mathit{max}}}{I}_{\mathit{det}}\text{cos}(n\mathrm{\Omega }t)\text{sin}({\varphi }_R)dt$$

(31)$$X^{\prime }={\int }_0^{t_{\mathit{max}}}{I}_{\mathit{det}}\text{sin}(n\mathrm{\Omega }t)\text{cos}({\varphi }_R)dt\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\text{and}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{Y}^{\prime }={\int }_0^{t_{\mathit{max}}}{I}_{\mathit{det}}\text{sin}(n\mathrm{\Omega }t)\text{sin}({\varphi }_R)dt$$

Evaluation of the terms gives:

(32)$$X=k_X{E}_R{E}_s\text{cos}(\psi )\text{cos}({\varphi }_s)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\text{and}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Y=k_Y{E}_R{E}_s\text{cos}(\psi )\text{sin}({\varphi }_s),$$

(33)$$X^{\prime }=-k_X{E}_R{E}_s\text{sin}(\psi )\text{cos}({\varphi }_s)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\text{and}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{Y}^{\prime }=-k_Y{E}_R{E}_s\text{sin}(\psi )\text{sin}({\varphi }_s),$$

where k_X, k_Y are the already known proportionality constants corresponding to a sine phase modulation in Eqs (15) and (16). The amplitude, the phase ϕ_s but also the unknown phase shift ψ are then calculated from these four outputs.

E. Summary table

The values of all the proportionality constants (k_X, k_Y) and (k̃_X, k̃_Y) corresponding to the different modulation functions mentioned in this paper are summarized in Fig. 5.

 

Fig. 5 Summary table

Download Full Size | PDF

Acknowledgments

Authors gratefully acknowledge the DRRT and the Champagne-Ardenne (G-mini) for financial funding, the Nano’mat platform for the infrared laser source and the National Research Agency (ANR) for financing complementary near-field and interferometric equipments ( ANR-09-BLAN-0168-01 and ANR-09-BLAN-0124-01).

References and links

1. Y. Shizhuo, F. T. Yu, and P. Ruffin, Fiber Optic Sensors (CRC Press, 1998).

2. P. S. Carney, B. Deutsch, A. A. Govyadinov, and R. Hillenbrand, “Phase in nanooptics,” ACS Nano , 6(1), 8–12 (2012). [CrossRef]   [PubMed]  

3. N. Ocelic, A. Huber, and R. Hillenbrand, “Pseudoheterodyne detection for background-free near-field spectroscopy,” Appl. Phys. Lett. 89(10), 101124 (2006). [CrossRef]  

4. M. Vaez-Iravani and R. Toledo-Crow, “Phase contrast and amplitude pseudoheterodyne interference near field scanning optical microscopy,” Appl. Phys. Lett. 62(10), 1044–1046 (1993). [CrossRef]  

5. R. Hillenbrand and F. Keilmann, “Complex optical constants on a subwavelength scale,” Phys. Rev. Lett. 85(14), 3029–3032 (2000). [CrossRef]   [PubMed]  

6. R. Debdulal, S. Leong, and M. Welland, “Dielectric contrast imaging using apertureless scanning near-field optical microscopy in the reflection mode,” J. Korean Phys. Soc. 47, 140–146 (2005).

7. Y. Sasaki and H. Sasaki, “Heterodyne detection for the extraction of the probe-scattering signal in scattering-type scanning near-field optical microscope,” Jpn. J. Appl. Phys. 39(4A), L321 (2000). [CrossRef]  

8. D. Jackson, A. Kersey, M. Corke, and J. Jones, “Pseudoheterodyne detection scheme for optical interferometers,” Electron. Lett. 18(2), 1081–1083 (1982). [CrossRef]  

9. G. Conforti, M. Brenci, A. Mencaglia, and A. G. Mignani, “Fiber optic vibration sensor for remote monitoring in high power electric machines,” Appl. Opt. 28(23), 5158–5161 (1989). [CrossRef]   [PubMed]  

10. A. Kersey and D. Jackson, “Pseudo-heterodyne detection scheme for the fibre gyroscope,” Electron. Lett. vol. 20(2), 368–370 (1984). [CrossRef]  

11. A. Dandridge, A. Tveten, and T. Giallorenzi, “Homodyne demodulation scheme for fiber optic sensors using phase generated carrier,” IEEE Trans. Microwave Theory Tech. 30(10), 1635–1641 (1982). [CrossRef]  

12. B. Lee and Y. Jeong, Fiber Optics Sensors (CRC Press, 2008), Chap. 7.

13. S. Pilevar, W. Atia, and C. Davis, “Reflection near-field scanning optical microscopy: an interferometric approach,” Ultramicroscopy. 61(1), 233–236 (1995). [CrossRef]  

14. L. Stern, B. Desiatov, I. Goykhman, G. M. Lerman, and U. Levy, “Near field phase mapping exploiting intrinsic oscillations of aperture NSOM probe,” Opt. Express 19(13), 12014–12020 (2011). [CrossRef]   [PubMed]  

15. B. Deutsch, R. Hillenbrand, and L. Novotny, “Near-field amplitude and phase recovery using phase-shifting interferometry,” Opt. Express 16(2), 494–501 (2008). [CrossRef]   [PubMed]  

16. H. J. Weber, L. Ruby, and G. B. Arfken, Mathematical methods for physicists (Harcourt/Academic, 2000).

17. C. Elliott, V. Vijayakumar, W. Zink, and R. Hansen, “National instruments LabVIEW: A programming environment for laboratory automation and measurement,” Journal of the Association for Laboratory Automation 12(1), 17–24 (2007). [CrossRef]  

18. Y. Yao, A. Hoffman, and C. Gmachl, “Mid-infrared quantum cascade lasers,” Nat. Photonics 6, 432–439 (2012). [CrossRef]  

19. Z. Sedaghat, A. Bruyant, M. Kazan, J. Vaillant, S. Blaize, N. Rochat, N. Chevalier, E. Garcia-Caurel, P. Morin, and P. Royer, “Development of a polarization resolved mid-IR near-field microscope,” Proc. SPIE 7946, 79461N (2011). [CrossRef]  

20. S. Amarie and F. Keilmann, “Broadband-infrared assessment of phonon resonance in scattering-type near-field microscopy,” Phys. Rev. B 83(4), 045404 (2011). [CrossRef]  

21. Z. Sedaghat, “A near-field study of the probe-sample interaction in near and mid-infrared nanoscopy,” Ph.D. thesis, University of Technology of Troyes (2012).

22. M. Kazan, A. Bruyant, Z. Sedaghat, L. Arnaud, S. Blaize, and P. Royer, “Temperature and directional dependences of the infrared dielectric function of free standing silicon nanowire,” Phys. Status Solidi C 8(3),1006–1011 (2011). [CrossRef]  

23. B. Knoll and F. Keilmann, “Enhanced dielectric contrast in scattering-type scanning near-field optical microscopy,” Opt. Commun. 182(4), 321–328 (2000). [CrossRef]  

24. M. Kulawik, M. Nowicki, G. Thielsch, L. Cramer, H.-P. Rust, H.-J. Freund, T. P. Pearl, and P. S. Weiss, “A double lamellae dropoff etching procedure for tungsten tips attached to tuning fork atomic force microscopy/scanning tunneling microscopy sensors,” Rev. Sci. Instrum. 74(2), 1027–1030 (2003). [CrossRef]  

25. A. Apuzzo, M. Fvrier, R. Salas-Montiel, A. Bruyant, A. Chelnokov, G. Lerondel, B. Dagens, and S. Blaize, “Observation of near-field dipolar interactions involved in a metal nanoparticle chain waveguide,” Nano Lett. 13(3), 1000–1006 (2013). [CrossRef]   [PubMed]  

26. R. Salas-Montiel, A. Apuzzo, C. Delacour, Z. Sedaghat, A. Bruyant, P. Grosse, A. Chelnokov, G. Lerondel, and S. Blaize, “Quantitative analysis and near-field observation of strong coupling between plasmonic nanogap and silicon waveguides,” Appl. Phys. Lett.100(23), (2012). [CrossRef]  

27. A. Bruyant, “Etude de structures photoniques en champ proche par microscopie optique à sonde diffusante,” Ph.D. thesis, University of Technology of Troyes, (2004).