We exploit the coupled emission-states of a single-chip semiconductor InAs/GaAs quantum-dot laser emitting simultaneously on ground-state (λGS = 1245 nm) and excited-state (λES = 1175 nm) to demonstrate coupled-two-state self-mixing velocimetry for a moving diffuse reflector. A 13 Hz-narrow Doppler beat frequency signal at 317 Hz is obtained for a reflector velocity of 3 mm/s, which exemplifies a 66-fold improvement in width as compared to single-wavelength self-mixing velocimetry. Simulation results reveal the physical origin of this signal, the coupling of excited-state and ground-state photons via the carriers, which is unique for quantum-dot lasers and reproduce the experimental results with excellent agreement.
© 2014 Optical Society of America
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 . 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 . 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 . 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 . 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  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  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 . 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 .
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.
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  embedded in a 440 nm GaAs waveguide surrounded with Al0.35Ga0.65As 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 λES = 1175 nm and on the QD GS at a wavelength of λGS = 1245 nm at an injection current of 54 mA and at a 16 °C . 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.
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 . 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.
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 fbeat = vrefl/c(ωES − ωGS)/π signal with the velocity vrefl of the reflector, c the speed of light and ωGS,ES the GS, ES emission frequencies. This Doppler beat frequency corresponds to a beat wavelength of λbeat = 21 μm and the reflector velocity is estimated from fbeat and amounts to 2.96 mm/s together with a measurement error of only 1.3 %. This observed narrow linewidth of fbeat 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/λbeat = 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 λGS/λbeat = 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.
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).
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 ηF. 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 . 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 ηF we assume that the sole effect of the feedback is a variation of the cavity GS and ES photon lifetime through the effective reflectivity . 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  and is out of the scope of this paper.
In our work the effective reflectivity is calculated asEq. (2). In our model the reflector motion changes only the effective photon lifetimes in the GS and ES photon rate equations reported in  and also presented here for the sake of clarity Eq. (3) the terms on the right hand side are the fraction of spontaneous emission rate coupled to the coherent photons, the stimulated emission term and the cavity photon loss rate, respectively. Hereby, is the occupation probability of the electron and hole states and τg,GS,ES is a time constant proportional to the differential gain . The effective photon lifetime is (3) coupled with the carrier density rate equation reported in .
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 . 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  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, ηF = 0.3 corresponding to about −10 dB coupling loss. With this set of values the feedback parameter C (, ) amounts to 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 vrefl of 3 mm/s, in Fig. 6(a) the numerically calculated total photon density (sGS + sES) 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.
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 ωGS and the other one emitting on the ES at ωES, 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 δsGS,ES with respect to photon density of the sole laser s0,GS,ES with the variation of the equivalent reflectivity . 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 with respect to while the corresponding variation of the hole occupations 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 with respect to the photon lifetime of the sole laser τp,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 . 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 δsGS,ES and the variation of the total photon density (δstot) can be calculated asEq. (6) can be approximated by Eq. (7) the first two terms depend only on the variation of the GS and ES photon lifetime, respectively whereas the last term is a nonlinear contribution indicated in the following as δscoupling that couples and . From the GS threshold condition can be written as function of the variation of the GS photon lifetime: Eq. (9) it is shown that the variation of the photon lifetime is directly proportional to the variation of the effective reflectivity. Substituting Eq. (8) in Eq. (7) yields Eq. (10) shows that the coupling term contains the product of the two effective facet reflectivities via the term . This coupling is proportional to due to the fact that the GS lasing controlled by is fed by the carriers from the ES, which itself is also simultaneously lasing and controlled by . From Eq. (2) it can be observed that this product leads to a time domain trace with a low frequency component according to Eq. (6) is therefore responsible for the low frequency component at about ≈ 300 Hz as observed in Fig. 6(b). We also note that the first two right hand terms in Eq. (6) are the origin of the frequency components at and therefore of the two peaks around ≈ 5 kHz observed in Fig. 6(b). This coupling term is intrinsically present in any QD rate equation model containing the cascade relaxation path for electrons [23, 28, 29] while we followed .
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  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 are shown and are correlated to the numerically obtained time traces in Fig. 6(a) of section 4. It can be observed that 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 . 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 .
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 induced by the reflector motion; these variations occur at different frequencies for the GS and ES such that 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.
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.
This work was supported by the EU FP7 (FAST DOT project Grant no. 224338). The authors thank Prof. D. Bimberg and H. Schmeckebier for fruitful discussions on the idea behind this concept. S. B. also gratefully acknowledges support by the Adolf-Messer Foundation, Germany.
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