Terahertz near-field nanoscopy based on detectorless laser feedback interferometry under different feedback regimes terahertz microscopy of modal field distributions in micro-resonators

Near-field imaging techniques, at terahertz frequencies (1–10 THz), conventionally rely on bulky laser sources and detectors. Here, we employ a semiconductor heterostructure laser as a THz source and, simultaneously, as a phase-sensitive detector, exploiting optical feedback interferometry combined with scattering near-field nanoscopy. We analyze the amplitude and phase sensitivity of the proposed technique as a function of the laser driving current and of the feedback attenuation, discussing the operational conditions ideal to optimize the nano-imaging contrast and the phase sensitivity. As a targeted nanomaterial, we exploit a thin (39 nm) flake of Bi 2 Te 2.2 Se 0.8 , a topological insulator having infrared active optical phonon modes. The self-mixing interference fringes are analyzed within the Lang–Kobayashi formalism to rationalize the observed variations as a function of Acket’s parameter C in the full range of weak feedback (C

In near-field nanoscopy, the standard experimental strategy to measure the near-field scattering is based on the operation of the near-field probe in tapping mode, at frequencies of the order of hundreds of kHz, to modulate the near-field component of the scattered intensity. Accordingly, the employed photodetectors need to have high sensitivity (NEP < 10 −12 W/Hz 1/2 ) and simultaneously operate on timescales (<μs) fast enough to follow the signal modulation induced by the tapping. Standard THz detectors employed in THz nanoscopy include room-temperature Schottky diodes and Golay cells or hot-electron bolometers. The slow time response of thermal detectors (pyrometers and Si bolometers) conversely hinders their use for high resolution near-field experiments.
Recently, an alternative compact detection scheme, 18,22 relying on the intracavity re-injection of the radiation emitted from a THz quantum cascade laser (QCL), 26 through self-mixing interferometry (SMI), [27][28][29][30][31] has been demonstrated. The nonlinear interference of the beam reflected from a nanostructure with the optical field LETTER scitation.org/journal/app inside the laser cavity depends on the relative phase difference of the two superimposed e.m. fields and causes perturbation of all lasing parameters, including the lasing frequency and the threshold gain, which, in turn, results into a variation of the laser voltage under constant current operation. The latter process is inherently fast since the maximum response speed to optical feedback is determined by the frequency of the relaxation oscillations in the laser. In the case of THz-QCLs, the lifetime of the upper state of the lasing transition is limited by elastic and inelastic scattering mechanisms to 5-10 ps, 26,32 enabling, in principle, response frequencies of the order of 100 GHz. A wide variety of applications of self-mixing interferometry in diode lasers 30,33 and QCLs 18,22,[34][35][36][37][38] has been demonstrated including metrology, 30,33 sensing, 34 and imaging. 16,[34][35][36][37][38] The laser is typically operated slightly above the lasing threshold 30 and for interferometric applications, the amplitude of the re-injected field is kept low in order to work in the limit of the so-called very weak feedback regime in which the SMI fringe line shape is simply sinusoidal. 22,39,40 Here, we employ scattering near-field optical microscopy (s-SNOM) in a detectorless configuration at THz frequencies to investigate the near-field SMI signal as a function of the feedback amplitude and the operation current of the employed QCL source and to identify the preferable operating conditions [giving high signal-to-noise ratio (SNR), high speed, and large optical contrast] for near-field nano-imaging applications. By exploiting the intrinsic stability of THz-QCL to large feedback intensity, 40 we demonstrate that even by driving the QCL source at currents much larger than the lasing threshold, we can achieve high-contrast and low noise crucial ingredients for high resolution and fast nano-imaging.
By modeling the THz-QCL in the presence of the optical feedback through the formalism set by the Lang-Kobayashi equations, 39 we discuss the observed dependence of the self-mixing signal on the laser feedback and the laser driving current in terms of modifications of the optical feedback Acket's parameter C. We then explore the dependence of the phase sensitivity of SMI on the feedback intensity and QCL currents and show that the firstorder Taylor expansion of the solution of the Lang-Kobayashi equations, 39 for laser frequencies around the free running one, provides a good approximation of the signal amplitude and phase in the socalled weak feedback regime where the SMI fringes are asymmetric and cannot be described by an analytical function. Based on that the optical sampling of SMI fringes for phase determination can be strongly reduced. Finally, we validate the proposed method by applying the SMI nanoscopy to a topological insulating bismuth compound, which is endowed with resonant excitations in the THz range.
These results are of paramount importance for the development of coherent THz near-field nanoscopy systems based on THz-QCL sources and, more generally, pave the way for exploring novel domains for a more efficient detection based on the self-mixing process.
The experimental setup employed for THz near-field nanoscopy is shown in Fig. 1(a). It exploits a set of THz-QCLs fabricated with a single plasmon waveguide, 26 operating in continuous wave (CW) at the driving current I QCL , with output frequencies of either ω = 2.0 THz or ω = 2.7 THz and peak powers at 15 K of P = 5 mW and P = 4.2 mW, respectively. The QCL beam is collimated with a 90 ○ off-axis parabolic (OAP) gold mirror with 50 mm focal length, and it is coupled to a commercial scattering scanning near-field microscope (s-SNOM) from Neaspec. The THz beam is focused on a metallic AFM tip with a 60 ○ incident angle relative to the tip axis by a 25 mm OAP.
In the experiments reported in this work, we have employed the same tip. The strength of the near-field signal strongly depends on the tip geometry. The latter determines the field enhancement at the tip apex and how efficiently the near-field is converted into the far-field to be detected through self-mixing interferometry, i.e., the net dipole moment associated with the tip. For experiments that do not require sub-100 nm spatial resolution, higher sensitivity can be attained by choosing tips of larger radius. Recent numerical simulations 25 indeed predict that by increasing the radius from 25 to 750 nm, the s-SNOM signal strength is expected to increase by one order of magnitude at the expense of only a factor four reduction in the spatial resolution. We use PtIr coated AFM tips (Rocky Mountain Nanotechnology) with 20 nm apex radius and an 80 μm-long tip to exploit the enhanced scattering efficiency at THz frequencies of tens of μm s long cantilever tips. 41 A portion of the THz field, which is backscattered by the tip, re-injected into the laser cavity and detected through the induced variations of the laser voltage ΔV due to a self-mixing effect. 22 The tip works in tapping mode with frequency Ω = 70-110 kHz such that the signal includes oscillating near-field components ΔV = ΣnΔVne iΩn , which can be isolated from the background due to the far-field illumination of the tip shaft and the sample by lock-in detection at the harmonics Ωn = nΩ of the tapping frequency with n = 1, . . ., 5 demodulation order. The tapping motion modulates the amplitude of the backscattered field, which is re-injected into the laser cavity without varying the optical path length. The incident power and the feedback intensity are controlled by inserting a variable attenuator (THz attenuator set from Microtech instruments) with transmittance T = 10%, 30%, and 60% in the optical path, along segments of the collimated propagating beam.
To detect the SMI fringes, we vary the relative phase between the incident and backscattered fields by changing the length of the external cavity formed by the tip and the output facet of the THz-QCL with an optical delay line consisting of two 45 ○ flat mirrors mounted on a linear translation stage (Physik Instrument model M-413) with 0.16 μm resolution. This later corresponds to a minimum variation of the cavity length L = 0.32 μm, which allows for a fine sampling of the SMI fringes that are characterized by λ/2 periodicity equal to ∼37-150 μm in the range of 1-4 THz.
We first investigate the imaging capability of the experimental setup with a THz-QCL emitting at 2.7 THz by raster scanning a 75 nm thick gold film evaporated on a Si undoped substrate with a 300 nm SiO 2 top layer, whose topography map is reported in Fig. 1(b). The tip is brought in close proximity to the sample, which is scanned by moving the sample holder with a three-axes piezomotor. The signal intensity is expected to increase with the tapping amplitude, 44 which is kept constantly equal to 280 nm for the measurements on the gold marker. The near-field maps collected at different demodulation orders n = 2, . . ., 5 are shown in Fig. 1(c); the maps are acquired by keeping fixed the cavity length at a value chosen to maximize the SMI signal. The scattered signal amplitude reflects the dielectric properties of the sample and increases at the gold marker, as expected, based on the higher reflectivity 43,44 of gold at 2.7 THz, compared to that of the silica/silicon substrate. The line scans extracted from the maps in Fig. 1(c), integrating along the direction orthogonal to the gold-substrate interface, are reported in Fig. 1(e). We estimate the spatial resolution of the setup by analyzing the signal step-like variation at the gold/substrate interface with the empirically found asymmetric fit function from Ref. 42, which takes into account that the different light confinement at the metal and the substrate sides gives rise to non-point symmetric line profiles. The step in the topography at the boundary complicates a reliable determination of the spatial resolution due to tip-sample convolution, which would require the analysis of the signal variation at topography-free sharp boundaries. 42,44,45 The derivative of the fitting function, which corresponds to the line spread function of our imaging system, is a piecewise Lorentzian with width δx at the metal side. As expected, due to the tip-sharpening effect, 14 both the signal amplitude and δx decrease with the demodulation order n [see Fig. 1(e)]. Remarkably, the width δx = 28 nm at the n = 5 demodulation order would correspond to a spatial resolution of λ/4000, significantly exceeding the limit posed by diffraction (∼λ/2 = 55.5 μm). The near-field nature of the signal is reflected by the strong monotonic decrease of signal amplitude as the tip-sample distance is increased (see the supplementary material, Fig. S1). The third demodulation order is chosen in the following discussion as a good compromise between signal intensity and far-field background suppression.
The SMI fringes, acquired at a fixed position at the gold marker, are reported in Fig. 1(g) as a function of L, which is used to tune the phase difference between the laser field and the retro-injected field. The asymmetric line shape indicates that, despite the fact that the tip is scattering only a reduced portion of the incident radiation, the laser does not exhibit a response typical of a very weak feedback regime, with its characteristic sinusoidal dependence on L. The Fourier spectra in Fig. 1(h) describe the spatial periodicity of the SMI fringes (λ/2) that translate into the fundamental and the harmonics of the QCL emission frequency. The presence of higher harmonics is directly related with the characteristic asymmetric line shape of SMI fringes for field feedback values of the order of 10 −4 -10 −3 .

LETTER scitation.org/journal/app
Interestingly, we observe that for high demodulation orders n > 1, the fringes have the same normalized line shape. The first demodulation order instead has a distinct line shape and distinct phase frequency spectrum that can be attributed to residual contributions from the far-field background. To analyze the self-mixing fringes, we apply the Lang-Kobayashi model 39 with the feedback strength parameter describing the fraction of the backscattered light that efficiently couples to the lasing mode, which depends on sample reflectivity. The best agreement [ Fig. 1(h)] is obtained with Acket's characteristic parameter C = 0.5 and Henry's linewidth enhancement factor α = 0.5, which is in good agreement with reported values for a similar QCL in the THz range. 46 With the aim of identifying the regime providing the best compromise of the SNR, phase sensitivity, and scan speed, we investigated the dependence of the near-field signal on two key experimental parameters: the QCL driving current I QCL and the feedback intensity. We performed two sets of experiments employing two different THz-QCLs emitting at 2.0 and 2.7 THz. The scattering efficiency of the AFM tip is strongly wavelength dependent 41 and a reduction of the scattered field is indeed expected for an increasing photon energy. Figures 2(a) and 2(b) show a selection of near-field self-mixing maps measured at the third demodulation order ΔV 3 for increasing I QCL , acquired at the two probing frequencies.
To quantify the variation in the near-field signal amplitude, we extract The change in the signal amplitude reflects a mode hopping 47 unveiled by Fourier transforming the self-mixing fringes acquired for a 30 mm-long L-scan (see the supplementary material, Fig. S3). Above I QCL = 650 mA (at 2.7 THz) and I QCL = 660 mA (at 2.0 THz), both lasers operate in the regime of C ≥ 0.5, and the self-mixing signal remains stable and constant with I QCL , as expected, given that the ratio between the output and the reinjected field amplitudes is not changing. We estimate the ΔV 3 amplitude on gold and on the substrate in Figs. 2(e) and 2(f) by averaging along the 1 μm-long portion of the line scans of Figs. 2(c) and 2(d), where the topography is flat. Despite the variation of the absolute signal with increasing the current I QCL , the ratio between the signal measured on gold and that measured on the substrate remains constant with a maximum 5% variation at 2.0 THz. The image contrast is indeed an intrinsic property related to the dielectric constants of the probed materials. Within the framework of analytic 48,49 and numerical models 50-52 of the tip-sample interaction, the image contrast measured with THz s-SNOM allows for quantitative determination of the complex-valued dielectric permittivity, which, in turn, contains information on the material's vibrational modes 22 and on the charge-carrier density beyond its Drude-like response. [53][54][55] We measure a constant contrast of about 4, at both pumping frequencies, as a consequence of the flat spectral response of gold and SiO 2 /Si in the probed frequency range.
Since the image contrast is preserved while varying I QCL , we can analyze the noise in the maps. The increase in the signal is accompanied by an increase in the root-mean-square (rms) [Fig. 2(i)], evaluated on bi-dimensional regions of 20 × 20 pixels 2 in the maps shown in Figs. 2(a) and 2(b). However, the SNR in Fig. 2(f) is optimized at high I QCL values due to the relatively higher improvement in terms of the signal amplitude. Based only on the SNR, the high-current high-power working condition seems preferable.
A very-good quantitative agreement with the experiments is obtained for calculated fringes with the α = 0.5 and C parameter in the range of 0.05-0.63 for 2.7 THz and 0.01-0.2 for 2.0 THz (see the supplementary material). The very weak feedback regime is reached when the fringes have a sinusoidal dependence on L, corresponding to C < 0.1. At the threshold current, the fringes are sinusoidal at 2.7 and 2.0 THz corresponding to C < 0.1. For higher currents, we have to use higher values of the C parameter (C > 0.1) to describe the fringes of the 2.7 THz laser due to the asymmetry of the asymmetry of their lineshape.
We then consider the effect of the feedback intensity on the near-field self-mixing signal by keeping constant the driving current I QCL , above the threshold, while varying the filter transmission T, as shown in Figs. 3(a)-3(d). Higher T transmission corresponds to higher feedback intensity, and T = 100% is achieved by removing the filter wheel from the optical path. The signal amplitude on gold and the substrate is first evaluated by extracting the line scans shown in Figs. 3(c) and 3(d) and then by averaging along portions of flat topography as we have done for the analysis of the self-mixing signal as a function of I QCL discussed before. The average signal in Figs. 3(e) and 3(f), at both 2.0 and 2.7 THz, drops with the reduction in the feedback intensity, following a linear dependence. However, the contrast between gold and the substrate in Figs. 3(g) and 3(h) remains constant within a 1% variation, as expected.
In close analogy to the aforementioned dependence on I QCL , the SNR follows the correlation of the signal amplitude with feedback intensity such that despite the increase in the rms [see Fig. 3(i)], the SNR is larger in the high-feedback regime [see Fig. 3(j)].
The effect of attenuation on the fringes line shape is similar to that observed when decreasing I QCL , as reported in Figs. 3(k) and 3(l) where the experimental results are superimposed to the numerical ones. The SMI fringes become gradually more symmetric by reducing the feedback intensity and sinusoidal interference fringes appear in the very weak regime, attained for T = 30% at both 2.0 and 2.7 THz. Accordingly, the C parameters that better describe the experimental data decrease linearly with the filter transmission from C = 0.4 to C = 0.03 (see the supplementary material). The linear dependence of C on the attenuator transmission is expected since with the attenuator, we are varying the ratio between the output power and the re-injected field amplitude.
To explore the dependence of phase sensitivity on the feedback intensity, we investigate the near-field response of a thin (39 nm) flake of Bi 2 Te 2.2 Se 0.8. This topological insulator material has infrared active optical phonon modes 56 in the range of 1.6-2.8 THz. In this frequency range, phase variations of the near-field scattered field are expected, as induced by the interaction of the impinging light with these optical phonons. The phase variation can be detected as a shift of the SMI fringes as a function of L, as previously shown in CsBr. 22 Figures 4(a)-4(c) show the topography of the Bi 2 Te 2.2 Se 0.8 flake together with the corresponding near-field SMI maps, acquired at 2.0 THz while driving the QCL at I QCL = 700 mA, keeping L fixed and using a tapping amplitude of 130 nm. We compare the maps of the third-order SMI signal acquired under two different feedback levels, obtained by using an attenuator transmission T = 100% and T = 30%. In both the cases, we observe that the near-field SMI signal is enhanced at the flake with respect to the undoped silicon/silica substrate, with signal peaked at the flake's edges, signature of edge resonances of phonon-polariton modes. 57 To retrieve the phase of the SMI signal from the flake, we move on the sample along a line orthogonal to the flake/substrate interface, following the path indicated in Fig. 4(b), with 30 nm steps, acquiring SMI fringes as a function of L at each position X on the sample. The fringe maps as a function of L and X are reported in Figs. 4(d)-4(f) for three different attenuator transmission T = 10%, 30%, 100%. The fringes change the line shape with the feedback attenuation, as previously observed for gold, becoming increasingly asymmetric with increasing T, as shown in Fig. 4(g), by the comparison of the fringes at X = 150 nm for T = 30% and 100%.
In the very weak feedback limit, reached at T = 30%, the voltage change at the QCL terminals can be described 3

by the relation
where ω 0 is the unperturbed laser frequency, L is the external cavity length, c is the light speed in vacuum, and s 3 and φ 3 are the amplitude and phase of the SMI fringes, respectively. The phase and amplitude can be thus retrieved by a fitting algorithm, based on the least squares method, using a sinusoidal fitting function. The outcome of the fitting procedure is shown in Figs. 4(h) and 4(i), as obtained by fixing ω 0 = 2.0 THz and setting the amplitude s 3 and the phase φ 3 as fitting variables. A finite phase shift is observed for the field scattered by the flake indicating the activation of phonon-polariton modes, predicted for Bi 2 Se 3 in this frequency range. 58 For higher feedback strength, i.e., moving from the very weak to the weak regime, a sizable deviation from the sinusoidal dependence is observed as shown by the SMI fringes acquired with attenuator transmission T = 100% in Fig. 4(g). In the latter case, the scattering amplitude and phase retrieval becomes more complex since, to account for the experimental line shape of the SMI fringes, additional components at harmonics of ω 0 should be considered [ Fig. 1(h)]. We observe that the amplitude and phase retrieved from a sinusoidal fit in Figs Each of the analyzed fringes consists of about 150 points in the cavity length. We integrate the signal for 10 ms per point for a total of ∼2 s per fringe at a fixed position on the sample to be compared with the 13.5 s acquisition time per spectrum achieved with hyperspectral near-field imaging based on THz time domain spectroscopy (TDS) 59 and with 0.3 s with nano-FTIR holography. 60 In order to fulfill the antialiasing theorem, the sampling rate must not exceed the Nyquist rate, which is equal to two times the highest frequency of the periodic signal. Accordingly, the presence of high frequency components in the high-feedback regime imposes a finer sampling of the SMI fringes. This later implies, in turn, a longer acquisition time per pixel, which would be especially detrimental for measurements that require high spectral resolution and larger sampling of the fringes, as is the case for those based on multimode sources 61 for hyperspectral imaging.
In order to identify a faster routine for retrieving the amplitude and phase of the SMI fringes, valid even in the presence of non-sinusoidal fringes, we consider the first-order Taylor expansion of the perturbed laser frequency (with feedback), around the unperturbed frequency, and use the expansion in the expression of the contact voltage ΔV(L) as a function of the cavity length L in the weak regime (see the supplementary material). In the approximated formulas, given by Eqs. (15) and (16)  . The deviation observed for φ 3 extracted at the highest attenuation (T = 10%) for X < 0 at the substrate side may be attributed to the low SNR (<2) of the data, affecting more the analysis with the approximated formula that is based on only four points in L and has been introduced to describe the signal at higher feedback levels. At the flake side, where the signal increases, the agreement with the sinusoidal fits and with the estimations for T > 10% is recovered. Accordingly, we conclude that the sinusoidal fit provides a reliable determination of s 3 and φ 3 also in the presence of a higher feedback level, i.e., in the weak feedback regime. Moreover, we got the evidence that a faster routine relying on the measurement of only four values of the signal ΔV 3 (L), as described above, can be exploited to extract the same type of information (amplitude and phase) with a higher degree of accuracy, being valid even in the presence of non-sinusoidal fringes.
In conclusion, we exploit self-mixing interferometry under different feedback regimes to study a detectorless THz near-field optical microscope built on THz-QCLs. Its operational performances are then investigated in terms of the image contrast and SNR as a function of the THz-QCL driving current and the feedback intensity. The observed signals are interpreted within the Lang-Kobayashi formalism under operational conditions, ranging from the very weak (C < 0.1) to the weak (0.1 < C < 1) feedback regimes. The best imaging conditions in terms of the SNR are obtained in the high feedback regime. Despite the more-complex non-sinusoidal line shape of the LETTER scitation.org/journal/app self-mixing fringes, the latter leads to optimized performances in terms of the SNR. A first-order approximated method is used to evaluate the self-mixing signal phase and amplitude even for nonsinusoidal fringes, allowing for an efficient, in terms of scan speed, and reliable image reconstruction.
The supplementary material describes the approach curves, the emission spectrum, the evolution of Acket's parameter C as a function of the current and the feedback, and the signal increase with tapping amplitude. It also presents the Lang-Kobayashi model for SD s-SNOM and the first-order reconstruction of the scattering coefficient harmonics.

DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request.