1. Introduction

Laser self-mixing velocimetry is a compact, versatile laser Doppler velocimetry technique to determine the velocity or vibratory motion of reflecting surfaces e.g. in biomedical and structural sensing and yields high accuracy and resolution in practical applications [1–5]. In laser interferometry the light from the interferometer reference arm and the Doppler-shifted light reflected from the moving reflector interfere resulting in a velocity-dependent Doppler frequency [6]. Self-mixing velocimetry redundantizes the interferometer reference arm, where the laser irradiates a diffuse-reflecting target and a small fraction of the back-scattered electrical field directly interferes with the intra-cavity electrical field of the semiconductor laser. This causes a modulation of the laser field in the cavity both in frequency and amplitude, depending on the delay-length of the target distance and hence the phase of the back-scattered light [7]. In both techniques, however, speckle-induced signal fluctuations cannot be avoided [8, 9]. To circumvent this issue, an improvement in signal quality has been reported by using two optically injection-locked semiconductor-lasers emitting at different wavelengths and utilizing a free-space interferometer [10, 11]. This yields a signal with a frequency corresponding to the difference frequency of the two signals stemming from the two individual wavelengths namely the Doppler beat frequency. Such an signal can also be attained by exploiting the semiconductor laser nonlinear dynamics [12, 13] involving two tunable master lasers injection-locking a slave semiconductor laser [14]. However, besides the two semiconductor lasers, phase-locking by an external reference microwave source is required as well as a careful control of an optimum beam overlap for the free-space interferometer which considerably increases system complexity [10]. Semiconductor-laser self-mixing velocimetry utilizing two wavelengths would be a potentially attractive alternative to these approaches. However, as in dual-wavelength laser interferometry where the detector signal results from the cross-interference of the two direct and the two delayed interferometer beams [15] four-beam-cross-interference of the two wavelengths would still be required to generate a Doppler beat frequency signal.

In order to realize the necessary coupling in semiconductor-laser self-mixing velocimetry to obtain the Doppler beat frequency signal, in this letter a new concept is proposed exploiting the peculiar properties of semiconductor quantum-dot (QD) based lasers. We experimentally realize and numerically study coupled two-state self-mixing velocimetry using a single QD laser that emits simultaneously on two QD emission-states. Here, the QD active medium is expected to provide the necessary nonlinear coupling between excited-state (ES) and ground-state (GS) photons inside the active medium by means of the intersubband carrier dynamics [16, 18, 19]. This is expected to obviate the need for four-beam interference on an external detector to attain an intensity beat signal. Simulations based on a state-resolved rate equation model described in [16] allow us to study numerically the presented two-state self-mixing velocimetry concept in order to identify the basic physical principle behind the Doppler beat frequency signal and, in addition, allow to emphasize the functionality of the two-state QD laser as a phase-demodulator. We focus in fact on the novel physical mechanism provided by the two-state QD laser and to demonstrate a proof-of-concept system to potentially access advantages reported in [10]. It is not the aim of this paper to provide an detailed review of the various self-mixing velocimetry systems aready implemented and succesfully optimized in solid-state lasers [17].

The paper is organized as follows. At first, the two-state QD laser is presented. Then, the experimental two-state self-mixing velocimetry set-up is introduced before the mechanism of coupling between GS and ES is experimentally studied by state-selective and phase-varying weak optical feedback. We then present proof-of-concept results from a coupled two-state self-mixing velocimetry experiment validating the presence of Doppler signals originating from both the two different QD laser wavelengths and the Doppler beat frequency signal. Numerical simulation results are finally presented that identify that the origin of the Doppler beat frequency signal is based on the particular carrier coupling within the QD active medium. The achievements are summarized and strategies for exploitation of the method are sketched in the outlook.

2. Experiment

The QD laser has been grown by molecular beam epitaxy on a GaAs (100) substrate. The active region contains 5 layers of self-assembled InAs QDs [20] embedded in a 440 nm GaAs waveguide surrounded with Al_0.35Ga_0.65As claddings. The cavity length amounts to 1 mm and the ridge width is 4 μm. Both facets are as cleaved. The laser emits continuous-wave emission simultaneously on the QD ES at a wavelength of λ_ES = 1175 nm and on the QD GS at a wavelength of λ_GS = 1245 nm at an injection current of 54 mA and at a 16 °C [21]. As depicted schematically in the experimental set-up in Fig. 1 a part of the beam is reflected onto a diffuse reflector. A retro-reflecting film is attached to a piezo translator that is mounted on a high-precision motorized linear translation stage to provide a controllable velocity. The diffuse reflector provides very weak reflection with a ratio of below 2 · 10⁻⁵. Hereby, waveguide coupling losses are not included. The piezo is driven by a sinusoidal voltage with a frequency of 50 Hz and a spatial deflection of 10 μm. The feedback length from the laser facet to the reflector amounts to 0.5 m. State-selective feedback and detection is attained by inserting an optical band pass filter before the diffuse reflector. The influence of this state-selective feedback on GS and ES emission can be analyzed either by using a GS and ES band pass filter before the fiber coupling or by using two band pass filters and two detectors after fiber-coupling. Hereby, the piezo is used for reflector movement to provide a trigger for the oscilloscope which allows synchronized recording of the ES and GS time-traces. The transmission of the GS band pass filters are centered at 1260 nm with a transmission width of 35 nm and the transmission of the ES band pass filters at 1180 nm with a transmission width of 20 nm. For analysis of the self-mixing velocimetry signals, a part of the laser beam is coupled into a single-mode fiber which is connected to a fiber-coupled InGaAs photo-detector. Details on the emission diagnosis path are not shown in the schematic. Time-domain representation of the GS- and ES-self-mixing velocimetry signals as well as the two-state self-mixing velocimetry are investigated using an oscilloscope whereas radio-frequency analysis is performed using an electrical spectrum analyzer. A double-stage optical isolator with an isolation > 60 dB prevents unwanted back-reflections prior to the fiber coupling. Two diffraction gratings are inserted in the set-up to spectrally select individual modes of the GS and ES, forming a double Littrow-configuration. The distance between laser and gratings amounts to around 1.5 m. From the existence of the following results it is clear that the gratings improve signal quality and do not disturb self-mixing velocimetry operation. In addition, variable neutral density filters in each grating arm are used to control the feedback strength.

 

Fig. 1 Schematic of the experimental coupled two-state self-mixing velocimetry set-up realized by a two-state QD laser. Emission diagnosis equipment is not shown.

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3. Experimental results and discussion

In this section we present experimental results obtained with the coupled two-state self-mixing velocimetry concept focusing on two cases: In the first case the ES emission reaches the reflector and is reflected into the two-state QD laser by using an ES-transmitting band pass filter before the reflector; whereas the GS emission does not reach the reflector because it is blocked by the band-pass filter. It will be shown that this ES-feedback influences the emission of the GS which does not experience feedback. In the second case, two-state feedback is investigated by using no band pass filter before the reflector and using ES/GS band pass filters before the fiber coupling. In both cases, grating stabilization is used and neutral density filters are adjusted such that the laser system, without reflector feedback operates in a equal power condition as shown by the optical spectrum in Fig. 2(a). A spectral separation of 72 nm, corresponding to the energy difference of GS and ES is chosen exemplarily. It is expected that a wide tuning range is exploitable due to the large gain bandwidth of the QD laser medium [22]. However, investigations on the spectral tuning is not within the scope of this paper. In order to substantiate our hypothesis of a nonlinear mixing or cross-interference mechanism via the carriers by the carrier relaxation and escape between the two QD states in the following we perform state-selective ES-self-mixing velocimetry and analyze the influence on the emission characteristics of both states. The results of the first case are shown in Fig. 2(b) where the ES-self-mixing velocimetry signal is displayed as a function of delay, the latter being proportional to the delay length of the reflector position. Such an anti-phase behavior is the consequence of the strong coupling between the GS emission and the ES emission via the carrier relaxation from the ES to the GS: when ES stimulated emission increases due to constructive feedback, less carriers relax to the GS and therefore the GS stimulated emission is reduced. The results of the second case are shown in Fig. 3(a) where the detected time-domain signal for equal spectral GS and ES power is shown for a reflector velocity of 3 mm/s. Here, fast oscillations resulting from the fundamental Doppler signals of each ES and GS wavelength are visible together with a beat envelope that already indicates a frequency corresponding to the Doppler beat frequency.

 

Fig. 2 Experimentally obtained two-state emission: (a) Optical spectrum and (b) time domain signal of state-resolved ES-self-mixing velocimetry.

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Fig. 3 Experimentally obtained two-state self-mixing velocimetry signals. The reflector velocity is 3 mm/s. (a) Two-state self-mixing velocimetry and low-pass filtered two-state self-mixing velocimetry time-signal. (b) Radio-frequency spectrum measured simultaneously with the time-signal of Fig. 3(a) and Fourier transform of time domain signal.

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The time-signal is the sum of GS and ES time-series. The modulation depth of the envelope is almost maximal which indicates almost equal output power of ES and GS states. However, such an envelope also could result from a simple superposition of two independent self-mixing velocimetry signals. In order to identify the Doppler beat frequency signal in the time domain signal of Fig. 3(a), a 500 Hz low pass filter is applied to the measurement data (red curve). In fact, an intensity modulation can already be observed. In contrast, a simple superposition of two Doppler signals would yield no residual Doppler beat frequency signal after applying the low pass filter. Complementary to the time domain signal, in the measured radio-frequency domain a single broad signal at the fundamental self-mixing velocimetry frequency of ≈ 4.8 kHz with a −3 dB-width of 860 Hz is evident, as shown in Fig. 3(b). This broad spectrum consists of an overlap of ES and GS Doppler signals. In addition, a considerably narrow peak with a −3 dB-width of only 13 Hz is evident at a frequency of 317 Hz representing the Doppler beat frequency f_beat = v_refl/c(ω_ES − ω_GS)/π signal with the velocity v_refl of the reflector, c the speed of light and ω_GS,ES the GS, ES emission frequencies. This Doppler beat frequency corresponds to a beat wavelength of λ_beat = 21 μm and the reflector velocity is estimated from f_beat and amounts to 2.96 mm/s together with a measurement error of only 1.3 %. This observed narrow linewidth of f_beat can be explained as follows. The linewidth represents the velocity error that is relative with respect to the measured frequency. Assuming that the velocity error of GS and ES are fully correlated and assuming a common noise source, the resulting beat linewidth yields the same relative error consequently leading to a very narrow absolute linewidth. In addition to the radio-frequency domain results, in Fig. 3(b) the Fourier transform of the time-series from Fig. 3(a) is included. A good qualitative agreement between the measured radio-frequency domain and Fourier transform of the time domain signal is evident. In contrast to the radio-frequency domain results, the beat signal in the Fourier transform trace indicates a broader width and a lower spectral power. This is attributed to the limited time span in the oscilloscope and thus the lower resolution bandwidth.

In the following, we study the dependence of self-mixing velocimetry signals in the radio-frequency domain as a function of the reflector velocity. In Fig. 4, the reflector is moved unidirectionally with a velocity that is increased from 1.5 mm/s to 4.5 mm/s with an acceleration of 0.2 mm/s². The upper part of the contour plot of the resulting electrical spectra depicts the evolution of the Doppler signatures of ES and GS, in the lower part the evolution of the Doppler beat frequency signal is shown. Both GS and ES Doppler signals overlap forming a broad distribution within a frequency range covering 2.5 kHz to 6 kHz. Remarkably, a clear signature of the beat signal emerges at low frequencies across the investigated velocity range as depicted in the lower part of Fig. 4. This signature evolves linearly with velocity leading to a slope of 98.4 Hz/(mm/s), in good agreement with the expected slope of 2/λ_beat = 96.95 Hz/(mm/s). The realized broad spectral separation of GS and ES results in a velocity-independent Doppler signal frequency reduction by a factor of λ_GS/λ_beat = 17 as evident from Fig. 3. This reduction allows the detection of very high reflector velocities with low bandwidth electronic equipment by detecting only the slow Doppler beat frequency signal.

 

Fig. 4 Experimentally obtained electrical spectra of the coupled two-state self-mixing velocimetry for increasing reflector velocity displayed as a contour plot.

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4. Discussion supported by numerical modeling

In the following section we aim for validating, explaining and discussing these experimental observations by numerical simulations to reveal the physical origin of the Doppler beat frequency signal. In the simulations, the experimental set-up is reduced to the laser medium and the moving reflector. The two laser facet reflectivities are denoted with r₁ and r₂ and the reflector reflectivity with r₃ as schematically depicted in Fig. 5(a).

 

Fig. 5 (a) Simplified set-up considered in the modelling. (b) Simulated photon density versus current characteristic of the laser without external cavity feedback. GS photon density (red), ES photon density (blue) and total photon density (black) are calculated with the model presented in [16].

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The light emitted from the two-state QD laser is fed back by the reflector which represents the moving reflector. The field coupling efficiency into the QD laser waveguide is denoted by η_F. The reflector position is the distance L between the reflector r₃ and the facet r₂. The laser is assumed to emit single-mode on both GS and ES. The dynamics of the carriers in the QDs (i.e. electrons and holes in the GS, ES and wetting layer) and of the GS and ES photons are described through the set of rate equations reported in [16]. As shown in Fig. 5(b) this model allows to reproduce the simultaneous two-state lasing for the QD laser without external optical FB as well as the experimentally measured GS power reduction after the ES threshold [16, 21]. Since the feedback is very weak because of the small value of r₃ and the small η_F we assume that the sole effect of the feedback is a variation of the cavity GS and ES photon lifetime through the effective reflectivity $r_{2,\mathit{GS},\mathit{ES}}^{\mathit{eff}}$ [24]. We therefore neglect any variation of the GS and ES lasing frequency caused by the feedback as well as the external cavity modes. Our focus is indeed on the non-linear interaction between GS and ES photons in the laser active region with the aim to study how this interaction generates the Doppler beat frequency signal observed experimentally in Fig. 3 and Fig. 4. A more detailed numerical treatment of the mode competition induced by the feedback in a two-state QD laser has been recently presented in [23] and is out of the scope of this paper.

In our work the effective reflectivity $r_{2,\mathit{GS},\mathit{ES}}^{\mathit{eff}}$ is calculated as

Figure 6 depicts the simulation results obtained by the steady state numerical solution of the rate equations. The QD laser material and cavity parameters are the same as in [16]. Since the linewidth enhancement factor of the QD lasers should be quite small, zero linewidth enhancement factor is assumed for simplicity. The non zero α-parameter changes the shape of the traces as discussed in [27] for the quantum well lasers but it does not change the key concepts and conclusions we want to emphasize in this paper. For the feedback we assume a length L₀ of 50 cm, r₃ = 10⁻², η_F = 0.3 corresponding to about −10 dB coupling loss. With this set of values the feedback parameter C ([27], [24]) amounts to $C=\frac{(1-r_2^2){\eta }_F{r}_3{\tau }_{\mathit{ext}}}{r_2{\tau }_{\mathit{in}}}=0.28$ and is therefore consistent with a weak feedback regime with C < 1 and with the experimental conditions. In the simulations the laser is biased with a current of 232 mA to achieve equal GS and ES power emission of the solitary laser according to Fig. 5(b) and Fig. 2(a). For a reflector velocity v_refl of 3 mm/s, in Fig. 6(a) the numerically calculated total photon density (s_GS + s_ES) is shown. The low-frequency component of this trace, obtained by low-pass filtering the calculated traces with a filter cut-off frequency at about 600 Hz, is plotted in red in Fig. 6(a). A clear low frequency modulation is visible. This component is also evident in the calculated Fourier transform in Fig. 6(b). The spectrum evidences the double-peaked high frequency component at around 5 kHz and the low-frequency component at 317 Hz which represents the Doppler beat frequency. These results therefore substantially support the experimental findings by a good qualitative agreement with the measurements shown in Fig. 3.

 

Fig. 6 (a,c) Simulation results of total normalized photon density (s_GS + s_ES) versus time and (b,d) corresponding Fourier transform. The red line is the low frequency component obtained by low pass filtering the time-domain traces. The results in (a) and (b) are obtained by the two-state emitting laser, whereas in (c) and (d) they are obtained by two independent lasers emitting from the GS and ES, respectively. Letters A and B in (a) indicate time frames discussed in the following and in Fig. 7. In (b) also the Fourier transform from the experimental result of Fig. 3(a) is included.

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To verify that these results in Fig. 6(a) and (b) are only obtainable due to the two-state emission from GS and ES within the same QD active medium, in Fig. 6(c) the calculated self-mixing trace are plotted as obtained by summing the photon density of two independent and uncoupled lasers, one emitting on the GS at ω_GS and the other one emitting on the ES at ω_ES, with the same power. Although the time traces in Fig. 6(a,c) seem qualitatively similar, in Fig. 6(c) no residual modulation remains after low pass filtering. In contrast to the coupled case shown in Fig. 6(a,b) no low-frequency component is present in the frequency spectrum thus validating the necessary coupling by the carriers in the QDs.

5. Analytical derivation of the coupling term

To get further insight into the results shown in Fig. 6(a,b) and to understand the most relevant physical effects causing them, in the following we analyze how the equivalent facet reflectivity, depending on the reflector motion according to Eq. (2), affects the GS and ES emission. We want to find an analytical expression that relates the variation of the photon density δs_GS,ES with respect to photon density of the sole laser s_0,GS,ES with the variation of the equivalent reflectivity $\delta R_{2,\mathit{GS},\mathit{ES}}^{\mathit{eff}}$. Hereby, some approximations are necessary. These approximations are supported by the analysis of the numerical results obtained in section 4. We assume that the variation of the photon lifetime affects only the electron GS and ES occupations $\delta {\rho }_{\mathit{GS},\mathit{ES}}^e$ with respect to ${\rho }_{0,\mathit{GS},\mathit{ES}}^e$ while the corresponding variation of the hole occupations $\delta {\rho }_{\mathit{GS},\mathit{ES}}^h$ is considered negligible. We neglect the escape rates from GS to ES because it was observed that the GS stimulated emission is practically supported by the capture rate from the ES. We assume constant carrier injection in the wetting layer because the current injection in the laser does not change and we neglect the escape rate from ES to wetting layer because it was observed that any variation of the electron density in the ES is followed by an almost equivalent and opposite variation of the wetting layer carrier density. Since the variation of equivalent photon lifetimes $\delta {\tau }_{p,\mathit{GS},\mathit{ES}}^{\mathit{eff}}$ with respect to the photon lifetime of the sole laser τ_p,GS,ES is small, the GS and ES electron density rate-equations can be linearized around the steady state condition when the GS/ES stimulated emission rate equals the GS/ES photon loss rate:

Hereby, notations are the same as in [16]. In the first equation of Eq. (5) the term on the left hand side is the carrier capture rate from ES to GS that equals the GS stimulated emission rate on the right hand side. In the second equation the terms on the left hand side are the capture rate from wetting layer to ES and the relaxation rate from ES to GS, respectively. They equal the ES stimulated emission rate reported on the right hand side. Canceling the steady state solution of the sole laser, the variation of the photon densities δs_GS,ES and the variation of the total photon density (δs_tot) can be calculated as

Our analytical solution supplies the key for the interpretation of the numerical and experimental results previously discussed and it also demonstrates the important conclusion that the simulated and measured low frequency time traces can be interpreted as the phase detection signal of the instantaneous phase difference between the GS and ES external cavity round-trip phases. An analogy can be found in the field of electronic phase detector devices used to detect the phase difference between two input signals: one practical realization of these electronic devices is indeed performed by using an analog multiplier or mixer followed by a low pass filter. In our experiment the mixer is the intrinsic multiplication between the GS and ES effective reflectivity and the low pass filter is the filtering operation performed on the time traces. In our experiments the mixer operation is therefore simply obtained thanks to the coupling between GS and ES carriers inside the laser itself, whereas in [10] the mixing operation is performed on the detector.

The analytically estimated coupling term in Eq. (10) can now be used to highlight the importance of the phase relation between GS and ES effective reflectivity on the total output power in Fig. 7(a) and (b) the calculated effective reflectivities ${\left|r_{2,\mathit{GS},\mathit{ES}}^{\mathit{eff}}\right|}^2$ are shown and are correlated to the numerically obtained time traces in Fig. 6(a) of section 4. It can be observed that ${\left|r_{2,\mathit{GS},\mathit{ES}}^{\mathit{eff}}\right|}^2(T-T_0)$ are periodic signals. In the time frames when these signals are nearly in phase (Fig. 7(a) and time range marked with A in Fig. 6(a)) the filtered low frequency component of the photon density (Fig. 7(c)) has a minimum. Therefore it detects the minimum phase difference between GS and ES effective reflectivity. In this case it can be seen that the variations of these effective reflectivities is almost equal making the coupling term in Eq. (10) close to zero because $\delta {\tau }_{p,\mathit{GS}}^{\mathit{eff}}\simeq \delta {\tau }_{p,\mathit{ES}}^{\mathit{eff}}$. In this time frame the total time trace is dominated by the first two terms of Eq. (6) and it is similar to the trace of the two independent lasers. On the contrary, in the time frames when these signals are practically out of phase (shown in Fig. 7(b) and marked with B in Fig. 6(a)) the filtered low frequency component of the photon density (Fig. 6(d)) has a maximum since it detects the maximum phase difference between the GS and ES reflectivity. In this time frame the variations of GS and ES photon lifetimes have opposite signs and therefore the coupling term of Eq. (10) is maximized because $\delta {\tau }_{p,\mathit{GS}}^{\mathit{eff}}\simeq -\delta {\tau }_{p,\mathit{ES}}^{\mathit{eff}}$.

 

Fig. 7 (a,b) Simulation results of the equivalent GS (blue) and ES (red) facet reflectivity calculated according to Eq. (2). The dashed line is the equivalent reflectivity with $\mathrm{\Delta }{R}_{\mathit{GS},\mathit{ES}}^{\mathit{eq}}=0$. (c,d) Normalized photon density (black line) and filtered low frequency component (red line). The time frames of (a,c) and (b,d) correspond to the ones indicated with letter A/B in Fig. 6(a).

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From this analysis supported by numerical simulations in section 4 and analytical calculations in section 5 three concomitant mechanisms can be deduced and are therefore required for the nonlinear coupling and for the Doppler beat frequency based ≈ 300 Hz signal component. The first one is the change of the effective photon lifetime ${\tau }_{p,\mathit{GS},\mathit{ES}}^{\mathit{eff}}$ induced by the reflector motion; these variations occur at different frequencies for the GS and ES such that ${\tau }_{p,\mathit{GS},\mathit{ES}}^{\mathit{eff}}$ have an instantaneous phase difference proportional to the reflector speed. Such instantaneous phase difference can be detected measuring the low frequency component of the total photon density of the laser. The second mechanism is the fact that GS stimulated emission is only fed by the carriers relaxing down from the ES, which is concomitantly lasing. And the third mechanism is the independent GS and ES gain, which means that the variation of GS carrier density does not change the ES gain and vice-versa.

6. Conclusion

A coupled two-state self-mixing velocimetry concept based on coupled emission states of a two-state QD laser has been studied experimentally and by simulations. The two-state QD laser is grating-stabilized for spectral purity with a chosen spectral separation of 72 nm. By translating a diffuse reflector, the fundamental self-mixing velocimetry-fringes originating from each state could be observed together with a common beat envelope. By radio-frequency domain analysis the Doppler beat frequency signal could be verified experimentally as a 13 Hz-narrow peak with a frequency 17 times smaller than the GS or ES-self-mixing velocimetry frequency. The existence and robustness of the Doppler beat frequency signal under a unidirectionally increased reflector velocity from 1.5 mm/s to 4.5 mm/s has been subsequently experimentally investigated. Simulation results based on a rate-equation model substantiated the experimentally observed fundamental Doppler frequencies of GS and ES at 4.8 kHz and 5.1 kHz, respectively together with the Doppler beat frequency at 317 Hz both in the time domain and radio-frequency domain for a selected translation velocity of 3 mm/s. Time-domain signals revealed the anti-phase dynamics of GS and ES in case of feedback on one state only. An analytical expression for the nonlinear coupling has been derived where a product between the GS and ES effective reflectivities was identified as the origin for the low frequency component. Calculated photon densities as a function of feedback-delay modeled with and without carrier coupling indicated the strong necessity of this coupling to enable the existence of the Doppler beat frequency signal. This coupling is uniquely found in QD-active media. The Doppler beat frequency signal can not be obtained by two independent laser oscillators (i.e. two semiconductor lasers emitting on the GS- and ES wavelength, respectively). In summary, our investigations suggest the direct application potential of two-state QD lasers as compact semiconductor sources for laser-Doppler velocimetry towards the detection of high-velocity dynamics e.g. of micro-particle flows.