1. Introduction

Since the first demonstration of the single-mode, room temperature and continuous wave operating quantum cascade laser (QCL) less than a decade ago [1], a number of applications were triggered that use their advantages as monolithic mid-infrared (mid-IR) semiconductor lasers. One of the most significant applications is the spectroscopic molecular sensing, which can be several orders of magnitude more sensitive than in the near infrared. These monolithic, tunable mid-IR lasers allow development of both compact and highly sensitive trace-gas detection systems. Tunable laser spectroscopy with semiconductor laser sources inherently relies on the wavelength tuning capability through variation of laser injection current. However, since this process is not arbitrarily fast this imposes some limitations on the applications.

There were only a few reports in the literature on the experimental results of the (small-signal) modulation response of directly modulated QCLs. Examples include [2] (intensity modulation, IM) [3], (frequency modulation (FM) up to 100 kHz), and [4] (only THz QCLs). Significantly more extensive are reports on the FM response in diode lasers [5,6].

Main differences between QCLs and conventional interband DFB diode lasers (e.g., those operating at telecom wavelengths) are expected from the thermal implications of significantly thicker active regions and the fundamentally different carrier dynamics in QCLs. The latter is due to the ultrashort intersubband carrier lifetimes, according to which QCLs can have relaxation frequencies in the 100 GHz range. This has been shown theoretically by Shore et al. [7,8]. Hence, it is very interesting to have experimental studies in the high frequency regime. Furthermore, the electronic tuning (at f >>10MHz) could be quantified.

In this paper the FM to IM ratio for a CW-DFB-QCL emitting at ~9 µm is presented for a wide range of modulation frequencies up to 1.7 GHz. The variation of the FM-IM ratio with bias current and its influence on the optical spectrum of the laser radiation is analyzed.

2. Optical frequency tuning mechanisms and emitted spectrum

The tuning of optical frequency of a semiconductor laser through injection current is mainly caused by two physical effects [9,6]: firstly the Joule heating of the active region results in thermally induced changes of the refractive index, which causes frequency tuning of the emitted laser radiation, and secondly the refractive index variation from the electro-optic light-carrier interaction also affects the laser output frequency. Since heat conduction is an inert process, the low frequency behavior is mostly dominated by thermal tuning, whereas the electronic tuning is governed by the carrier dynamics and for diode lasers dominates in the MHz and GHz range.

In the small-signal regime (i.e., ΔI<<I₀) one can assume the laser is a linear time-invariant system with respect to optical power and frequency of emitted radiation. Thus, a sinusoidal modulation at frequency f_M of the laser injection current

 

Fig. 1 Explanation of symbols of the intensity P_L(t) and frequency f_L(t) of the emitted light and its spectrum. To be measured: the FM-IM ratio Δf /ΔP and phase-shift θ, the sideband ratio E₁/E_-1, and the FM response Δf /ΔI.

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In the general case, when the laser current is modulated with a frequency f_M in the order of the laser linewidth, discrete sidebands, separated by f_M, are observed in the optical spectrum. In principle infinite number of harmonic sidebands are generated, but for small modulation indices (m<<1 and β<<1) only two sidebands at f₀ ± f_M dominate and their electric fields are

 

Fig. 2 Schematic spectra of the E-field under different IM and FM conditions.

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Any modulation case can be conveniently characterized using the sideband ratio (defined as SR = |E_-1| / |E₁|) which for the small signal modulation is only dependent on FM-IM index ratio β / m and FM-IM phase-shift θ and can be expressed as:

Whether the laser operation is close to one of these special cases is an interplay between device properties and device operating conditions. Since the FM index β depends on the modulation frequency f_M and the IM index m depends on optical DC output power P₀ (and consequently on laser bias current I₀), operating parameters such as I₀ or f_M can be used to achieve different SR-s. It should be noted that in the small-signal regime the FM-IM ratio β / m does not depend on the injection current amplitude ΔI because it affects the Δf and ΔP in the same way.

The experimental characterization of laser modulation capabilities requires separate measurements for IM and FM. While the IM (i.e., ΔP and ϕ_IM) can be measured using a high-speed photodetector, measurement of the FM is more challenging and usually requires additional means for FM-to-IM conversion. Commonly used approaches for FM-to-IM conversion include an interferometer setup [11,6] or the slope of a narrow gas absorption line [12]. The latter method is significantly simpler; however in order to adopt it for high frequency measurements we had to address some technical issues. First, in conventional implementations the laser emission frequency has to be stabilized to the side slope of the absorption line and the slope has to be well known. The second issue is related to dispersion of the gas sample that has to be accounted for when the modulation frequency is comparable to the width of the absorption line (this issue is nonexistent in case of low frequency measurements [12]). These two issues have been addressed by implementing a full spectral scan (see appendix A for details on data acquisition) with a complete spectral modeling of absorption and dispersion of the sample used to retrieve the laser IM and FM parameters. The details of the theoretical model that involves computation of the laser emission spectrum, multiplying it by the transfer function of the gas and computation of the intensity variation are outlined in appendix B.

3. Experimental setup

In this paper we used the _qR¹(4)_a NH₃ transition at 9.618 µm to perform the FM-to-IM conversion. In addition to the high speed sinusoidal modulation at f_M, the laser was slowly scanned over the target absorption line (with a 19.2 s sawtooth current ramp). The schematic of the experimental setup is shown in Fig. 3.

 

Fig. 3 Experimental setup. The setup can be switched between a high frequency and a low frequency mode (LF and HF labeled switches). The spectrum analyzer is phase synchronized to the RF signal generator and operated at the same center frequency. It records the amplitude and phase of the signal at PD1 or PD2. The oscilloscope is used to capture the DC components of the photodetector signals and the DC and AC amplitude (RF DET) of the laser injection current. At each modulation frequency and laser temperature setting a single ramp (duration 19.2 s, or frequency of 52 mHz) is captured and the data stored for signal post-processing.

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The laser is a commercial single-mode and continuous-wave QCL (Hamamatsu) in a high heatload (HHL) package operating at λ = 9.6 µm and providing ~30 mW laser power. The photodetector PD1 is a fast ac-coupled (> 10 kHz with 3 dB bandwidth of 1000 MHz), thermo-electrically cooled mid infrared mercury cadmium telluride photodiode (VIGO System S.A. PVI-3TE-10.6) with integrated preamplifier (VPAC-1000F, G = 8000 V/A). Additionally it is connected to an amplifier (Minicircuits ZFL-1000LN + ) which provides a gain of 10 (not shown in Fig. 3). The sensitivity of the module at the operation wavelength, including the additional amplifier, is 430 V/mW and the noise equivalent power is NEP = 9.1 pW/Hz^1/2. The photodetectors PD2 and PD3 are DC coupled MCT detectors (VIGO System S.A. PVI-4TE-8) with 5 MHz bandwidth, which are used to record the light transmitted through a reference Etalon for wavelength calibration (PD3) and through the gas cell (PD2). Two bias tees (Minicircuits ZFBT-6GW + , RF port 100 kHz – 6 GHz) are incorporated into the laser anode and cathode path for injection of the sinusoidal RF modulation and for detection of the actual RF current with a RF power detector (Herotek, Schottky diode DHM020BB).

Two wedged beam splitters used to create the reference branch and to perform low-frequency measurements with PD2 are made from ZnSe (50R:50T) and CaF₂ (45R:15T) respectively. The attenuator with ~6% transmission placed at the laser output is used to prevent photodetector saturation and potential nonlinear saturation effects in the low pressure gas cell. The gas cell (10 cm glass tube with wedged CaF₂ windows) is filled with 13 mbar of pure NH₃. The laser temperature control is realized using the thermo-electric cooler integrated in the HHL package driven by an Arroyo 5305 temperature controller. The laser driver (LDX-3232, ILX Lightwave) used in the experiment which is equipped with a 100 kHz bandwidth modulation input, is utilized to deliver slow current ramp for the optical frequency scan, as well as for the small signal sinusoidal current modulation in the low-frequency regime (< 100 kHz) directly through the DC port of the bias tees. Both control signals (ramp and sine-wave) are generated with an external signal generator (Tektronix AFG3102). Since the fast detector is AC coupled (with a low cut-off frequency of ~10 kHz) the DC coupled detector PD2 is used in the low-frequency mode instead of PD1 for signal recording. In the high-frequency regime (>100 kHz) the current modulation sine wave is generated using a RF signal generator (Stanford research SG 382) which is delivered to the laser through bias tees and PD1 is used for signal recording. The switch (Minicircuits) at the input of the spectrum analyzer (Tektronix RSA 5103A, 1 Hz to 3 GHz) used for selection of low- and high-frequency modes maintains a 50 Ω termination even for the unused ports assuring stable measurement conditions.

4. Experimental results

A series of measurements have been done to characterize the IM and FM behavior of the QCL. Once the laser temperature and f_M are set, the laser is first tuned to the spectral range away from the absorption line and the pure IM is measured. Subsequently, spectral scans are performed to observe a combination of IM and FM, which then is analyzed using the theoretical model (see appendix B). Figure 4 shows an example of measurement data and corresponding theoretical model fits of RF amplitude and phase (Figs. 4(a) and 4(b) respectively) at f_M = 716 MHz and the corresponding DC component (Fig. 4(c)) measured without RF modulation.

 

Fig. 4 Measured signals (circles) and theoretical model fit (solid line) for RF amplitude (a), RF phase (b) at f_M = 716 MHz, and the corresponding DC power measured with PD2 with no RF modulation. For better visibility the traces in (a) and (b) have been offset for viewing purposes (DC signal offsets are due to actual optical power changes).

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Since the laser temperature shifts the emission wavelength, the absorption feature appears at different injection currents in the data recorded at different heatsink temperatures. Families of curves are collected for a series of modulation frequencies and heatsink temperatures followed by the retrieval of the FM and IM parameters using the model developed in appendix B. The details of the data acquisition and processing can be found in appendices A and B respectively.

The retrieved FM-IM ratio and FM-IM phase-shift are shown in Figs. 5(a) and 5(b), respectively. The transition from thermal tuning to electronic tuning is clearly observed in both graphs. Thermal tuning, which causes red-shift with increasing current corresponds to a FM-IM phase of 180°, clearly shows dominance at low frequencies with a gradual roll-off in the few-kHz range. The electronic tuning, with its blue-shift trend corresponds to a FM-IM phase of 0°, dominates at high frequencies and has a strong bias current dependence (Fig. 5(b)). Since all curves correspond to the same internal laser temperature of ~48°C (see appendix A), the observed variation with laser bias is a direct effect of increased current injection into the active region and secondary effects such as static self-heating can be excluded. It should be noted, that the difference of the curves in Fig. 5(a) at low frequencies (below a few MHz) is primarily caused by different laser optical powers P₀ for each curve.

 

Fig. 5 Experimental results on the QCLs FM tuning behavior and comparison with behavior from a standard DFB diode laser. Due to their low FM-IM ratio QC lasers allow for quasi SSB operation. (a) The ratio of absolute frequency modulation Δf and intensity modulation index ΔP/P₀ at different laser bias currents. Between DC and 300 Hz an interpolation has been made and the qSSB operation points are indicated as a dashed line. (b) The FM-IM phase-shift θ between frequency modulation and intensity modulation at different laser bias currents.

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If a frequency independent IM response is assumed (i.e., ΔP/ΔI is frequency independent), the laser FM response (i.e., Δf /ΔI) can be extrapolated from the FM-IM index ratio. This is a reasonable assumption given the high intrinsic relaxation frequency of QCLs. Based on this assumption the FM response of the QCL used in the experiment is shown in Fig. 6. The electronic tuning is clearly visible as the constant region in the high frequency regime of Fig. 6 with FM response values varying with laser bias current between 0.5 to 1.7 MHz/mA. The rise of the FM tuning coefficient with the current is consistent with the non-zero/rising phase shift observed in Fig. 5(b), which additionally supports the measurement conclusions.

 

Fig. 6 The FM response (current tuning coefficient vs. modulation frequency) at different laser bias currents. This is obtained by extrapolation with an assumption of a constant IM response (i.e., ΔP / ΔI is frequency independent). At low frequencies the thermal tuning is observed whereas in the RF domain (> 10 MHz) the inversely acting tuning by carriers is seen.

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As predicted by Eq. (5), the sideband ratio (SR) depends on the FM-IM ratio and FM-IM phase-shift so it is interesting to see the influence of the device behavior on the SR. Hence, the SR shown in Fig. 7(a) is calculated from the FM-IM ratio (according to Eq. (5)) and its variation with frequency and laser bias current.

 

Fig. 7 The SR as a function of modulation frequency f_M (a).The frequencies where quasi-SSB is achieved (green curve, b) correspond to the minima in plot (a) with SR exceeding 15 dB for ~200 MHz < f_M < 1 GHz. The SR is limited because the FM-IM phase shift at the qSSB operation point is θ_min = ~18° (blue curve, b), which limits SR_min = tan(θ_min / 2) ≈ −16 dB.

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As theoretically predicted, the SSB modulation occurs when the FM-IM ratio crosses the line of β = m /2 (cf. Figure 4). Since in practical implementation the FM-IM phase-shift at this point is non-zero, only quasi-SSB instead of pure SSB can be obtained. Since the minimum FM-IM phase-shift is around θ_min = ~18° (see Fig. 7(b)), the best case sideband ratio becomes SR_min = tan(θ_min / 2) ≈-16 dB (cf. Equation (5)). Despite this non-ideal behavior, the qSSB modulation of directly modulated QCL is achievable over a wide range of frequencies from 200 MHz to 1 GHz by adjusting the laser bias current. This represents a unique and new property for semiconductor lasers. Note, that despite the indirect character of this measurement, there is a direct evidence for the DSB and qSSB behavior in the raw spectra. For low bias currents the red curve in Fig. 4(a) clearly shows three copies of the absorption line with two, nearly equal side-features separated from the central peak by the modulation frequency of 716 MHz. This is a result of the carrier and each sideband interacting with the absorption line during the slow scan. Similarly the qSSB case indicated in Fig. 7(a) is evident in the pink curve in Fig. 4(a), which clearly shows one of the side-features (side-bands) strongly suppressed.

For comparison with diode lasers, the FM-IM ratio for a standard DFB laser operating at ~1.5µm [5] is shown in Fig. 4. In order to achieve SSB/qSSB mode the FM-IM index ratio plot would need to cross the dashed quasi-SSB line in Fig. 5(a). Since the FM-IM plot for the diode laser does not cross the qSSB line, it cannot achieve any significant sideband suppression. Even at higher modulation frequencies the DFB laser curve will not cross the qSSB line, because the FM-IM index ratio β / m (this is the curves in Fig. 5(a) divided by f_M, see also Fig. 11 in the appendix) for all lasers is decreasing monotonously with frequency until it will plateau off at α_H/2 with α_H being the linewidth enhancement factor [5] (see also appendix B6). Hence, one can infer that for all semiconductor lasers β ≥ α_H m / 2 holds. Since non-QC semiconductor lasers have α_H in the range of 3 - 9, the regime of qSSB (β = m / 2) modulation requiring α_H ≤ 1 is inaccessible. This clearly demonstrates that the low linewidth enhancement factor of QCLs, causes their unique modulation behavior with a potential use in applications.

5. Conclusion

Measurement results of the FM-IM ratio of a directly modulated DFB QCL are presented within the modulation frequency range of 300 Hz - 1.7GHz and for a wide range of bias currents. The RF measurements of QCLs are of importance to many applications, but such data have been difficult to find in the literature most likely due to technical limitations and availability of RF optoelectronics in the mid-IR. Hence, the developed measurement method that uses direct modulation of QCLs is described in detail (see also the appendices) and its practical implementation to QCL characterization has been discussed in this work. The discussed method gives high quality measurement results that are acquired at constant internal laser temperature, which allows for separation of the direct effect of the bias current from the secondary effects due to self-heating. The retrieved laser parameters as a function of modulation frequency (including the tuning coefficient in GHz/mA) clearly show the transition from thermal tuning to electronic tuning and the gradual variation of FM-IM phase-shift from 180° to nearly 0° is clearly observed. The electronic tuning reveals information about the value of the alpha factor (see appendix B.6). Full analysis of the electronic tuning is beyond the scope of this paper and will be addressed elsewhere [23].

The comparison of QCL modulation parameters with a diode DFB laser reveals that at comparable conditions diode lasers exhibit much higher frequency modulation than QCLs. This is a unique feature of QCLs caused primarily by their low linewidth enhancement factors. This feature allows for optical quasi-SSB modulation of QCLs directly through the injection current. The SSB modulation capability makes QCL unique among semiconductor lasers, and enables many new applications such as telecommunication links utilizing SSB [21] or high speed laser spectroscopy where SSB represents an advantage [13].

Appendices

Data acquisition

The slow current ramp (duration 19.2 s) is applied to the laser current scan from sub-threshold to the levels that allow scanning over the entire gas absorption line used as a probe for laser parameters. The sub-threshold data is mainly collected to determine the offset voltages of the DC coupled detectors as well as any RF pick-up in the high speed photodetector circuit PD1. Although the high speed detector PD1 was put in a well shielded metal box (dimensions 30 cm × 20 cm × 8 cm) and far away (~1.5 m) from the laser and RF generator, there is still residual electrical pick-up observed in the recorded signals (see Fig. 9), which distorts the measurement of amplitude and phase. The measured sub-threshold RF amplitude of −64 dBm is still above the noise level, but well below the above-threshold signal level of −45 dBm. After vector subtraction of the sub-threshold offset from the data acquired above threshold, these signals are used in the subsequent processing. A prerequisite for this method is simultaneous measurement of amplitude and phase, otherwise the vector subtraction cannot be implemented.

Since the commercial laser and its package used in this work are not designed for RF driving, the injection efficiency was rather low at high frequencies (cf. Fig. 8). In the LF mode and at frequencies well below the cutoff frequency of the laser driver (100 kHz) the amount of current that is injected into the laser is very well determined, since the laser driver is a current source. However, in HF mode this is not the case because the RF signal generator has a finite output impedance of 50 Ω, making it neither a current source (infinite impedance), nor a true voltage source (zero impedance). With the laser impedance Z_LD and neglecting parasitics the laser modulation current is theoretically given by ΔI = 2 U_gen/(100 Ω + Z_LD), which is the formula used to generate the theory trace in Fig. 8.

 

Fig. 8 Laser injection current amplitude measured using the Schottky diode (labeled RF DET in Fig. 3) (purple) and the expected current assuming a Z_LD = 7 Ω differential laser impedance (blue). The strong deviations are due to parasitic inductance/resonances of the HHL laser housing which is not designed for RF driving.

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Due to RF impedance issues and parasitics, accurate measurement of the FM or IM response separately (i.e. the FM or IM behavior with respect to injection current) is difficult. However, the FM-IM ratio is not affected. In our experiment the amplitude of the RF signal generator is chosen specifically so that sufficient RF current is injected into the laser for the measurement (the RF power detector RF DET is used to monitor the current injected into the laser at high frequencies). At low frequencies we selected −6 dBm of driving RF power which is increased to 12 dBm between 100 MHz and 400 MHz (cf. Fig. 8). Please note that even though the RF ports of the bias-tees have a cutoff frequency of 100 kHz, the amplitude roll-off is small enough to allow measurement down to 30 kHz, which provided enough overlap for the LF and HF measurements to test consistency of the results (cf. Fig. 5 and Fig. 6).

The absorption feature occurring at time around 17.5 s in the scans shown in Fig. 9 is used to probe QCL parameters. A zoom of the relevant data in this region can be seen in Fig. 4.

 

Fig. 9 Raw signals recorded during one ramp (For T_HS = −6°C and f_M = 808 MHz). The absorption feature of interest is marked in red (cf. Fig. 4).

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To extract the laser parameters at different operating bias currents the laser heat sink temperature is adjusted between −6 °C and 8 °C degrees in 2 °C steps with one additional point at 7 °C. This method allows shifting the gas absorption line probe to different locations on the laser light-current (LI) curve, thus providing retrieval of laser parameters at different bias currents. With an assumption that the internal temperature determines the effective refractive index in the laser waveguide and thus affects the lasing wavelength, by probing the same absorption line the measurement is guaranteed to always be performed at the same internal temperature. This is an advantage of this method, because it eliminates temperature-dependent effects and makes interpretation of the experimental laser data significantly easier. The internal temperature of the laser can be determined from the measurement data by extrapolation [14], which is demonstrated in Fig. 10. For each f_M and heatsink temperature, two scans are recorded: one with and one without sinusoidal modulation. The spectrum analyzer is synchronized with the modulation signal generator using a 10 MHz clock timebase which enables accurate measurement of signal phase. The detection bandwidth is set to 305 Hz which results in 7564 data points during the 19.2 s scan. To avoid influences of long-term drifts of laser parameters, the second scan (without modulation) is repeated in each measurement cycle. The data acquired in both scans are simultaneously fitted with the theoretical model described in the next section.

 

Fig. 10 Determination of internal temperature by extrapolation of the laser currents values corresponding to the absorption line center (after [14]).

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Theoretical model and data processing

Extraction of the laser FM and IM parameters requires accurate mathematical model for analysis of the measured spectra. To develop the model we relate the laser injection current I₀ (which is varied during the slow ramp) with both the first harmonic P_D,1 measured by PD1 and the DC component P_D,0 measured by PD2. We assume that I₀ is varying slowly compared to the applied sinusoidal modulation (this is fulfilled by scanning the laser with the 52 mHz ramp while the modulation frequency is ≥ 300 Hz). The model consists of three independent parts covering the laser emission spectrum, gas transfer function and the demodulation process. For each I₀ the detector signal is computed in frequency domain. First the emission spectrum of the laser is determined then multiplied with the transfer function of the gas (which models its attenuation and dispersion) and finally the calculation of the first harmonic of the intensity variation from the detected spectrum.

B.1. Laser emission spectrum

The light emitted by the laser with instantaneous power P_L(t) and instantaneous optical frequency f_L(t) is described by the electric field of [15]

(6)$$E_{\text{L}}\left(t\right)=\sqrt{P_{\text{L}}\left(t\right)}\cos \left(2\pi \underset{-\infty }{\overset{t}{{\displaystyle \int }}}{f}_{\text{L}}\left(t\right)\text{d}t\right)$$

(7)$$=\sqrt{P_0\left(1+m\cos \left(2\pi f_{\text{M}}t-{\varphi }_{\text{IM}}\right)\right)}\times \cos \left(2\pi f_{0}t+\beta \sin \left(2\pi f_{\text{M}}t-{\varphi }_{\text{IM}}-\theta \right)\right).$$

Since the modulation is periodic with frequency f_M the spectrum of the electric field consists of discrete lines E_L,n at frequencies f_n = f₀ + nf_M (where n = 0, ± 1, ± 2,…), with f₀ being the frequency of the optical carrier (lasing frequency). Hence we can write

(8)$$E_{\text{L}}\left(t\right)=\mathrm{Re} \left\{{\displaystyle \sum }_n{E}_{\text{L},n}{\text{e}}^{\text{i}2\pi \left(f_0+n{f}_{\text{M}}\right)t}\right\}.$$

For small intensity modulation (i.e.,m<<1) the E_L,n have a closed form expression [16]:

(9)$$E_{\text{L},n}=\sqrt{P_0}{\text{e}}^{-\text{i}n\left(\theta +{\varphi }_{\text{IM}}\right)}\left(J_n\left(\beta \right)+\frac{m}{4}\left(J_{n-1}\left(\beta \right){\text{e}}^{\text{i}\theta }+J_{n+1}\left(\beta \right){\text{e}}^{-\text{i}\theta }\right)\right).$$

With J_n denoting the Bessel function of the first kind [17]. The IM sideband amplitudes are complex with specified amplitude (|E_L,n|) and phase (∠E_L,n), the latter is specified relative to the injection current modulation.

The optical carrier frequency f₀ depends on the laser bias current I₀ through thermal tuning. Since the range of I₀ needed to scan over the target absorption line is relatively small (cf. Fig. 9), we can assume linear tuning and neglect the quadratic term in the current tuning relationship. With an assumption of the static current tuning coefficient k_I (unit: GHz/mA) the emission frequency and current are linked by:

(10)$$f_0-f_{\text{C}}=-\left(I_0-I_{\text{C}}\right)k_{\text{I}}.$$

The current I_C, which corresponds to the position of the absorption line center within the scan, is a fit parameter. The actual line center f_C can be found in the spectral database (e.g., HITRAN 2012 [21]), but is not needed for the model targeting a single transition, since I_C is a fit parameter. This approach is sufficient to describe the static laser tuning. The tuning coefficient k_I is determined from the reference Etalon scan. It could also be extracted from the data recorded at modulation frequencies well above the gas absorption linewidth, because the separation of the observed features is directly related to the sideband separation determined by the modulation frequency applied (cf. Figure 4).

A linear model is used to determine the laser output power during the scan. With the output power at the absorption line center P_C and slope efficiency k_P the output power around the spectral feature of interest can be modeled as

(11)$$P_0=P_{\text{C}}+\left(I_0-I_{\text{C}}\right)k_{\text{P}}.$$

Both P_C and k_P are used as fit parameters retrieved by fitting the model to the DC data generated with PD2. In the small-signal regime this simple linear model is sufficient (as also proven by the data fits shown in Fig. 4).

The modulation ΔP and IM phase is also assumed to be current dependent and is modeled as:

(12)$$\text{Δ}P{\text{e}}^{\text{i}{\varphi }_{\text{IM}}}=\text{Δ}{P}_{\text{C}}{\text{e}}^{\text{i}{\varphi }_{\text{IM},\text{C}}}+\left(I_0-I_{\text{C}}\right)k_{\text{Δ}P}$$

With ΔP_C being the intensity modulation amplitude, and ϕ_IM,C being the IM phase-shift at the line center. The k_ΔP parameter represents a complex coefficient accounting for changes in amplitude and phase of the laser IM around the absorption line. The k_ΔP was introduced to account for IM variations that manifested themselves as a sloped baseline of both RF amplitude and RF phase scans (see Fig. 4 for details). All three parameters are determined in the curve-fit.

B.2. Gas sample transfer function

The optical transfer function of a gas sample with length $L$ is given by

(13)$$H_{\text{gas}}\left(f\right)={\text{e}}^{-\text{i}\tilde{k}\left(f\right)L}={\text{e}}^{-\left(\text{i}k\left(f\right)+\alpha \left(f\right)/2\right)L}$$

with the complex wavenumber $\tilde{k}\left(f\right)=k\left(f\right)-\text{i}\alpha \left(f\right)/2$ containing the real (angular) wavenumber k(f), and absorption coefficient α(f) of the sample. Note, that when the intensity transmission is calculated the Lambert-Beer law | H_gas(f) |² = e^-α(f)L is obtained.

Both the angular wavenumber (which depends on refractive index) and the absorption coefficient are related to each other through the Kramers-Kronig relations. For a Voigt shaped absorption line, they can be expressed as real and imaginary part of the Faddeeva function. The general expression is given by

(14)$$\tilde{k}\left(f\right)=\frac{2\pi f}{c_0}-\text{i}\frac{{\alpha }_{\text{C}}}{2}\times \{\begin{array}{ll}\frac{W\left(\frac{i{\gamma }_{\text{L}}-\left(f-f_{\text{C}}\right)}{{\gamma }_{\text{G}}/\sqrt{lg 2}}\right)}{W\left(\frac{i{\gamma }_{\text{L}}}{{\gamma }_{\text{G}}/\sqrt{lg 2}}\right)}\hfill & {\gamma }_{\text{G}}>0\hfill \\ \frac{i{\gamma }_{\text{L}}}{i{\gamma }_{\text{L}}-\left(f-f_{\text{C}}\right)}\hfill & {\gamma }_{\text{G}}=0\hfill \end{array},$$

with the Lorentzian halfwidth γ_L, Gaussian half-width γ_G, peak absorption coefficient${\alpha }_{\text{C}}$at the line center frequency f_C and W the Faddeeva function [17–19]. The equation is to be understood in such a way that k and α/2 are given as real and imaginary part of the right hand side of Eq. (14). The first summand (2πf / c₀) is the vacuum wavenumber, which models the phase-shift the light experiences during propagation through vacuum. The second summand (with α_C / 2 pre-factor) is the contribution of the resonant transition in the gas to the total wavenumber (real part) and to attenuation (imaginary part). The expression after the curly bracket can be split into real and imaginary part to obtain the Voigt, Lorentzian (γ_G = 0), or Gaussian (γ_L = 0) absorption function and their dispersive counterparts.

The four absorption line parameters (α_C, γ_L, γ_G and f_C), which capture the effects of both dispersion and absorption of the gas, are determined in the fit. For correct retrieval it was necessary to include the dc data recorded with PD2 into the fit.

The Gaussian halfwidth is given by

(15)$${\gamma }_{\text{G}}=f_{\text{C}}\sqrt{2\lg 2N_{\text{A}}\frac{k_{\text{B}}T}{M/1000}}$$

with Boltzmann’s constant k_B, Avogadro’s number N_A, gas temperature T and the molecular mass of the gas molecule M (in [g/mol]). For the Ammonia absorption line used in the experiment (M = 17.031 g/mol) at f_C = c₀ × 1036 cm⁻¹ (λ = ~9.653μm) we obtain γ_G ≈45.1 MHz at room temperature (T = 296 K). Hence the Gaussian linewidth is known and is considered as a constant for all fits.

B.3. Demodulation process

The output of the RF demodulator provides the measured RF amplitude and phase. The E-field at the detector is periodic with frequency f_M and hence can be expressed as series of harmonics E_D,n which describe the amplitude and phase of the electric field spectral component at frequency f₀ + n f_M:

(16)$$E_{\text{D}}\left(t\right)=\mathrm{Re} \left\{{\displaystyle \sum }_n{E}_{\text{D},n}{\text{e}}^{\text{i}2\pi \left(f_0+n{f}_{\text{M}}\right)t}\right\}.$$

The detected spectral lines E_D,n result from interaction of the laser emission spectral lines E_L,n with the gas sample having transfer function H_gas(f) (Eq. (13) and are given by

(17)$$E_{\text{D},n}=E_{\text{L},n}{H}_{\text{gas}}\left(f_0+n{f}_{\text{M}}\right).$$

The photodetector is a square-law detector that detects the power envelope P_D(t) of the electric field E_D(t). This is given by |E_D(t)|² but with the frequency components at 2f₀ neglected. In a formally more stringent approach, the envelope is obtained as the absolute value of the complex envelope of Eq. (16). The complex envelope of a narrowband signal at a carrier f₀ (such as Eq. (16)) is its analytic signal divided by ${\text{e}}^{\text{i}2\pi f_{0}t}$. The analytic signal of Eq. (16) is just the expression inside the Re{…} clause (Note: The analytic signal of a given real signal has the same spectrum as the real signal for positive frequencies but zero spectral components at negative frequencies. Hence, if the original signal can be written as a real part of a complex signal which has no negative frequency components the expression inside of the real part is the analytic signal, which is fulfilled here because f₀ + n f_M is always positive). So we have:

(18)$$P_{\text{D}}\left(t\right)={\left|{\displaystyle \sum }_n{E}_{\text{D},n}{\text{e}}^{\text{i}2\pi n{f}_{\text{M}}t}\right|}^2=\mathrm{Re} \left\{{\displaystyle \sum }_{k=0}^{\infty }{P}_{\text{D},k}{\text{e}}^{\text{i}2\pi k{f}_{\text{M}}t}\right\}$$

with the harmonic coefficients

(19)$$P_{D,k}={\epsilon }_k{\displaystyle \sum _{n=-\infty }^{\infty }{E}_{D,n+k}{E}_{D,n}^{*}}$$

and ε_k = 1 for k = 0 and ε_k = 2 for k ≠0. Since the detected envelope (light power) is periodic with frequency f_M it also can be decomposed into a Fourier series. Although the detected power is real valued, the harmonic coefficients P_D,k are complex because they describe amplitude and phase. The spectrum analyzer is set to measure the amplitude |P_D,1| and phase (∠P_D,1)of the first harmonic.

Combining everything the final expression (valid for m<<1) for the first harmonic is obtained:

(20)$$\begin{array}{l}{P}_{\text{D},1}=P_0{\text{e}}^{-\text{i}\frac{2\pi f_{\text{M}}L}{c_0}-\text{i}{\varphi }_{\text{IM}}}{\displaystyle \sum }_{n=-\infty }^{\infty }\cdots \\ 2\left(J_{n+1}\left(\beta \right){\text{e}}^{-\text{i}\theta }+\frac{m}{4}\left(J_n\left(\beta \right)+J_{n+2}\left(\beta \right){\text{e}}^{-\text{i}2\theta }\right)\right)\left(J_n\left(\beta \right)+\frac{m}{4}\left(J_{n-1}\left(\beta \right){\text{e}}^{-\text{i}\theta }+J_{n+1}\left(\beta \right){\text{e}}^{\text{i}\theta }\right)\right)\\ \times \exp \left(-\frac{{\alpha }_{C}L}{2}\frac{W\left(\frac{\text{i}{\gamma }_{\text{L}}-\left(f_{n+1}-f_{\text{C}}\right)}{{\gamma }_{\text{G}}/\sqrt{lg 2}}\right)+W^{*}\left(\frac{\text{i}{\gamma }_{\text{L}}-\left(f_n-f_{\text{C}}\right)}{{\gamma }_{\text{G}}/\sqrt{lg 2}}\right)}{W\left(\frac{\text{i}{\alpha }_{\text{L}}}{{\gamma }_{\text{G}}/\sqrt{lg 2}}\right)}\right).\end{array}$$

where the emission power P₀, emission frequency $f_n-f_{\text{C}}=f_0-f_{\text{C}}+n{f}_{\text{M}}$ and baseline parameters (m = ΔP/P₀ and ϕ_IM) must be inserted from Eqs. (11), (10) and (12) respectively.

In the Lorentzian/near-Lorentzian case (γ_G << γ_L) the Faddeeva function W(z) can be replaced by its asymptote 1/z, to avoid numerical instability and/or to omit Doppler broadening in the model. In this case γ_G will cancel out and the exact expression for Lorentzian absorption and dispersion is obtained (cf. Eq. (14)). However, the experimental data acquired in this work require the complete Voigt profile to be applied.

Since the Bessel functions J_n(β) will approach 0 very quickly when |n| > β, the sum in Eq. (20) can be truncated and the summands with |n| >> β be omitted. In the present work n_max = max(10,3β) was used, with the sum running from -n_max to n_max.

It should be noted, that in the case of $m\approx 1$ the above formula is not valid, and the (infinite expansion) formula in [16] (Eq. (3) with $r_n=e^{\text{i}n\left({\varphi }_{\text{IM}}+\theta \right)}{E}_{\text{L},n}$ and $\text{Ψ}=\theta +\pi /2$) has to be used for E_L,n. This is because the expression for E_L,n, that was inserted in Eq. (17), neglects the higher harmonics generated by the IM. For the experimental data presented in this work the condition m << 1 is always fulfilled and the approximate expression Eq. (20) can be used instead of the exact but more complex approach from [16].

When the modulation frequency is decreased, the FM index β of a typical semiconductor laser grows very fast. This is because β=Δf /f_M and Δf rises due to thermal tuning while f_M decreases (cf. Fig. 6). In the case of β → ∞, or, equivalently, f_M →_{0, the limiting case of wavelength modulation spectroscopy (WMS) is attained, for which one can compute PD,k as:}

(21)$$P_{D,k}={\epsilon }_k{f}_M{\displaystyle \underset{-\frac{1}{2f_M}}{\overset{\frac{1}{2f_M}}{\int }}{P}_L(t){\left|H_{gas}\left(f_L(t)\right)\right|}^2{e}^{-ik2\pi f_{M}t}dt}$$

Al low modulation frequencies the condition for n_max in Eq. (20) returns a large number of necessary sidebands, and numerical evaluation of Eq. (20) becomes impractical. Therefore in this work for modulation frequencies below 1 MHz numerical evaluation of the Eq. (21) integral was used to simulate the spectra instead of using formula Eq. (20) involving sidebands.

B.4. Detected optical power (no modulation)

Without modulation the transmission of the gas is measured by PD2. The received optical power is given by

(22)$$P_{\text{D},0}=P_0|H_{\text{gas}}\left(f_0\right){|}^2=P_0{\text{e}}^{-\alpha \left(f\right)L}.$$

This expression contains parameters of P₀ (i.e. offset and slope efficiency of the LI curve, see Eq. (11)) and the gas absorption line parameters (Lorentzian line width γ_L, peak absorbance α_cL and the line center current I_C), which are retrieved in the fitting process.

B.5. Fitting procedure

As mentioned before, each measurement at a specific modulation frequency consists of two current scans: one with and one without sinusoidal modulation. The first is used to record the amplitude and phase of the intensity modulation received with PD1 (or PD2 at low frequencies). The second scan is used to record the reference Etalon signal for determination of the tuning coefficient k_I and the DC optical power received with PD2. All signals (except for the Etalon signal that is analyzed a-priori) are fitted simultaneously with the same set of parameters. The RF amplitude data is entered as a logarithm (essentially in unit dBm) so that the least squares fit weights deviations of the model and the data in a relative sense, i.e. deviations at low amplitudes contribute more to the least squares error than deviations at high amplitudes. All three traces (log amplitude, phase and DC intensity) are weighted so that their full scale variation divided by number of points is equal. This ensures that each trace contributes equally to the curve-fit. The DC model and optical power data from PD2 are then multiplied by a factor of 10 to account for the significantly higher signal to noise ratio of the DC scan compared to the RF scan used for amplitude and phase retrieval. Ideally the traces should be weighted according to their exact noise amplitudes (“equalization”). Note that the noise amplitude on the log amplitude (with respect to base e) and the phase in radians are of the same magnitude. All parameters of the model are listed in Table 1.

Table 1. List of model parameters and how they are obtained.

View Table

At the laser currents close to threshold, corresponding to laser temperatures of 7°C and 8°C a quadratic RF laser background was observed. For these conditions Eq. (12) was used with second order polynomial correction instead of the linear one, which introduces one additional complex parameter in the model (not shown in Table 1). This modification of Eq. (12) was necessary because of the strong variation of laser IM behavior with bias current; however for all currents that are larger than 1.1I_th the linear model with Table 1 parameters was used. The final output parameters of the fitting routine are the FM-IM phase-shift θ and the FM-IM index ratio β/m with m=ΔP_C /P_C.

B.6. α_H retrieval method

As mentioned above, the low linewidth enhancement factor of QCLs allows reaching the quasi-SSB operation conditions, which requires an FM-IM index ratio of β/m=1/2. The link between the FM-IM index ratio and the linewidth enhancement factor can be shown with a simple rate-equation approach [22]. Starting from the rate equation for the photons $\dot{S}\left(t\right)=\left(G\left(t\right)-1/{\tau }_p\right)S\left(t\right)$ with S the number of photons in the laser cavity, G the rate of stimulated emission and τ_p the photon lifetime, one can derive the desired relationship. By using a harmonic excitation with a small-signal approximation $S\left(t\right)=S_0+\text{Re}\left\{\text{Δ}S{e}^{\text{-i}2\pi f_{M}t}\right\}$ and $G\left(t\right)=G_0+\text{Re}\left\{\text{Δ}G{e}^{\text{-i}2\pi f_{M}t}\right\}$ (second order terms are neglected), and $\text{Δ}f={\alpha }_{\text{H}}/4\pi \text{Δ}G$ for the induced laser frequency change, the well-known relationship β/m=α_H /2 can be obtained (with β=|Δf |/ f_M and m=|ΔS |/ S₀). This simplified derivation assumes that the refractive index change, which is responsible for the laser emission frequency change, is solely caused by the change in optical gain (i.e., Δf =α_H / 4πΔG). This neglects several effects including thermal tuning, gain compression and possibly others, so the resulting formula can only be used to reproduce the so-called intrinsic transient chirp [22], which is described byβ/m=α_H /2. However, the transient chirp dominates at high-frequencies (i.e., Δf ∝ f_M, or β→const), because thermal and other tuning effects typically show low-frequency or constant behavior (i.e., Δf →1/f_M or Δf →const). By plotting β/m versus modulation frequency, one observes these three regions (see Fig. 11). The plateau at high frequencies (β→const) corresponds to the transient chirp. The α_H factor is obtained from the constant, asymptotic value at high frequencies, or α_H ≈ 0.2 in Fig. 11.

 

Fig. 11 The FM-IM index ratio versus modulation frequency. In the limit of high frequency ($f_M\to \infty )$, a plateau at ${\alpha }_H/2$ is observed which yields ${\alpha }_H\approx 0.2$ for this laser.

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Acknowledgments

The authors would like to acknowledge financial support by the NSF CAREER award CMMI-0954897, by the U.S. Environmental Protection Agency Grant No. RD-83513701-0, and by the NSF ERC MIRTHE award EEC-0540832.

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