Optical-feedback cavity-enhanced absorption spectroscopy in a linear cavity: model and experiments

Abstract

Optical-feedback cavity-enhanced absorption spectroscopy is a highly sensitive trace gas sensing technique that relies on feedback from a resonant intracavity field to successively lock the laser to the cavity as the wavelength is scanned across a molecular absorption with a comb of resonant frequencies. V-shaped optical cavities have been favoured in the past in order to avoid additional feedback fields from non-resonant reflections that potentially suppress the locking to the resonant cavity frequency. A model of the laser–cavity coupling demonstrates, however, that the laser can stably lock to a resonant linear cavity, within certain constraints on the relative intensity of the two feedback sources. By mode mismatching the field into the linear cavity, we have shown that it is theoretically and practically possible to spatially filter out the unwanted non-resonant component in order for the resonant field to dominate the feedback competition at the laser. A 5.3 $\upmu \hbox {m}$ cw quantum cascade laser scanning across a $\hbox {CO}_2$ absorption feature demonstrated stable locking to achieve a minimum detectable absorption coefficient of $2.7\,\times \,10^{-9}\,\hbox {cm}^{-1}$ for 1-s averaging. Detailed investigations of feedback effects on the laser output verified the validity of our theoretical models.

1 Introduction

Cavity-enhanced spectroscopic techniques, which have become widely used in both laboratory and field applications for sensitive gas detection, rely on trapping light between two or more high reflectivity mirrors to enhance the effective optical path length through the gas sample by several orders of magnitude [1–4]. Optical-feedback cavity-enhanced absorption spectroscopy (OF-CEAS) is a variant that uses the optical cavity to reduce the linewidth of the laser source emission and thereby increase laser–cavity coupling [5–12]. In essence, the resonant cavity field retraces back to the laser facet and acts as an external source of injection seeding. In addition to the cavity-enhanced path length, these feedback effects can significantly enhance the signal intensity transmitted through the cavity, making the technique attractive for mid-infrared spectroscopy where optics and detector technology are intrinsically limited. The 3–10 $\upmu \hbox {m}$ region of the electromagnetic spectrum is particularly useful for trace gas measurements due to the presence of many well-separated transitions with large absorption cross sections.

Recent advances in laser technology have increased the availability of mid-infrared sources. In particular, quantum cascade lasers (QCLs) are becoming increasingly popular due to their high power (100 s of mW of cw or pulsed infrared light) and growing commercial availability [13]. QCLs are semiconductor chips which can be designed to probe the 3–15 $\upmu \hbox {m}$ spectral region by altering the composition and structure of the semiconductor layers that create a series of potential wells for electrons to cascade down [13–17]. Linewidths are intrinsically narrow (100 s of kHz) due to a single-carrier lasing mechanism, which leads to QCLs having negligibly small linewidth enhancement factors [18–20]. Many studies of optical-feedback effects on diode lasers have reported instability when the feedback intensity becomes too high [18, 19, 21–24]. The single-carrier lasing mechanism of QCLs, however, eliminates this weakness making them stable even under high feedback conditions, and therefore highly suitable sources for OF-CEAS [17, 20, 25–27].

Several studies have demonstrated the potential for QCLs to serve as effective infrared sources for sensitive OF-CEAS experiments. Most work has been conducted using V-shaped optical cavities where the output of the laser is injected at the central mirror. In this set-up, any light not injected into the cavity is reflected away from the incident laser propagation direction ensuring that only light leaking out of the resonant optical cavity can return to the laser facet [6, 8, 28]. Using this geometry, Maisons et al. reported a minimum detectable absorption coefficient ($\alpha _{\mathrm{min}}$) of $3\times 10^{-9}\, \hbox {cm}^{-1}$ for 1-s averaging measurements of $\hbox {N}_2\hbox {O}$ and $\hbox {CO}_2$ with a 4.46 $\upmu \hbox {m}$ QCL, while Hamilton et al. went to longer wavelengths to probe $\hbox {CH}_4$ and $\hbox {N}_2\hbox {O}$ at 7.84 $\upmu \hbox {m}$ and achieved an $\alpha _{\mathrm{min}}$ of $5.5\times 10^{-8}$ $\hbox {cm}^{-1}$ for 1-s averaging [9, 10]. Recently, Gorrotxategi-Carbajo et al. [11] reported the best mid-infrared OF-CEAS sensitivity of $1.6\times 10^{-9}\, \hbox {cm}^{-1}$ for a single scan (100 ms) probing formaldehyde with a 5.6 $\upmu \hbox {m}$ QCL.

While the V-shaped cavity eliminates competition between multiple feedback fields, there are some disadvantages in using this geometry. For example, customised cells are sometimes required to accommodate the optical path shape within a reasonably small volume. For an optimal arrangement, both cavity arms have to be set to precisely the same length, usually equal to the laser–cavity distance, further adding to the experimental constraints. In this arrangement, the resonant field will have a node or anti-node on the folding mirror depending on whether the excited longitudinal mode is an odd or even harmonic, which results in cavity modes exhibiting alternating intensity due to subtle differences in dielectric coating reflectivity and thus requiring additional data analysis [6, 10]. Alternatively, if the cavity arm lengths are set to half the laser–cavity distance, only one set of harmonics will be excited.

Linear optical cavities are commonly used in other cavity-enhanced experiments and can be easily set up with commercially available mirror mounts, cavity ring-down mirror sets, and conventional sample tubes requiring small volumes. There is also no need to separate even and odd harmonic modes. The linear geometry notably suffers from one major weakness, which is the feedback competition between the resonant cavity field and the non-resonant reflection off the first cavity mirror when the beam is on-axis, as both retrace the incident trajectory and re-enter the laser cavity. Despite this difficulty, our group demonstrated that on-axis linear cavity OF-CEAS is possible and can achieve similar sensitivities to the more traditional V-shaped set-up with a 1-s averaged $\alpha _{\mathrm{min}}$ value of $2.4\times 10^{-8}\,\hbox {cm}^{-1}$ for measurements of NO with a low power $(\le$3 mW) 5.3 $\upmu \hbox {m}$ QCL [12]. This paper uses theoretical models to explain why it is possible for the resonant feedback to dominate the feedback competition and allow the laser frequency to lock to a resonant linear cavity. We then compare these predictions to new linear cavity OF-CEAS measurements obtained using a more powerful 5.3 $\upmu \hbox {m}$ QCL for exemplar detection of $\hbox {CO}_2$.

2 Theory

In this paper, we consider a linear optical cavity formed by two mirrors, designated $\hbox {M}_1$ (input coupler) and $\hbox {M}_2$, with the same effective reflectivity ($R$). In discussions of feedback, we describe the field emitted from the resonant cavity towards the laser as the resonant feedback as opposed to the non-resonant feedback field reflected off the inner curved surface of $\hbox {M}_1$.

It has long been known that external fields entering a single-mode semiconductor laser cavity act as injection seeding and stimulate emission, causing the laser emission to deviate from the free-running condition dictated by the current and temperature changes being applied [29, 30]. In OF-CEAS, the external feedback field arises due to reflections that allow photons from the source laser to re-enter the laser cavity after traversing through an external optical system. The feedback field must be in phase with the laser field in order for injection seeding to occur. The strength of feedback is described by a feedback rate ($\beta$), defined as the ratio of the feedback intensity to the laser output intensity.

Non-resonant feedback—feedback from a single reflection off a surface—has been studied extensively in semiconductor diode lasers as it can be an unwanted effect [24]. This type of feedback results in a characteristic sawtooth shape of the laser output power and frequency as a function of injection current where discontinuous jumps occur at regular intervals depending on the distance between the laser and external reflector [22, 30]. The tuning rate of the coupled laser is reduced relative to the free-running scan rate in between these hops. The seminal work of Lang and Kobayashi examines this effect in great detail, noting multistability arising from interfering effects—due to a broad gain spectrum and sensitive dependence of laser medium refractive index to temperature and carrier density—that lead to the observed discontinuities in laser output [30].

While this behaviour can be problematic when it causes unwanted changes in laser tuning, feedback effects can also be exploited to change the laser behaviour in a favourable way. When the laser is aligned with an optical cavity, a resonant feedback field will arise when the laser wavelength is in resonance with the cavity. If the cavity has a high finesse, this returning field will have a narrow linewidth (~10–100 s kHz) relative to the free-running linewidth of the laser (~few MHz for QCLs, typically determined by the quality of the current controller). Thus, a resonant cavity field in phase with the laser will induce a narrowing of the emitted linewidth and stimulate emission at the resonant wavelength [6, 8, 23]. The laser becomes “locked” to the cavity frequency; we note that this term can be misleading as it applies in this context to the condition when the laser frequency is being tuned very slowly due to feedback rather than remaining stationary at a single value. The passive locking of the laser to the resonant cavity allows a build-up of intracavity intensity—and correspondingly an increase in the measured transmission signal—and the narrowed linewidth enhances injection efficiency into the resonant cavity field as the portion of laser power within the cavity linewidth increases. The cavity remains in resonance much longer than the average photon lifetime of the system—or cavity ring-down time ($\tau$)—before discontinuously jumping to its free-running condition.

As the laser frequency is scanned, it will come into and out of resonance with the cavity forming a comb of frequencies that can be used to probe molecular absorptions of gases within the cavity. The resonant cavity modes are separated by the free spectral range (FSR) of the cavity, which is simply determined by its physical length ($L$), the refractive index of the intracavity medium ($n$), and the speed of light ($c$): $\hbox {FSR} = \frac{c}{2\,\it {nL}}$. Typically the FSR is on the order of the Doppler linewidth of the sample gases, making this technique best suited for pressure-broadened systems typical of sensing applications.

A simple model of the laser frequency behaviour in the presence of feedback can be constructed by considering the nature of the electric fields returning to the laser facet. Emitted photons have three possible routes, ignoring losses due to diffraction: reflection off the flat, uncoated surface of the near cavity mirror ($\hbox {M}_1$); reflection off the curved, dielectric-coated surface of the mirror (non-resonant feedback); or coupling into the optical cavity (resonant feedback). In the first case, the field reflected off the first mirror will virtually all be lost before reaching the laser as will be shown later. Separate transfer functions ($h_i$) can be defined to describe the remaining fields. The magnitude and phase of the reflected non-resonant field are functions of laser frequency ($\omega$), with the former determined solely by the cavity mirror reflectivity ($R$) and the latter by the laser–cavity distance ($L'$):

$$\begin{aligned} h_1=\frac{E_{\mathrm{refl}}}{E_\mathrm{o}}=\sqrt{R}\mathrm {e}^{-2i\omega L'/c}. \end{aligned}$$

(1)

The transfer function describing the field leaking from the resonant linear cavity is given by:

$$\begin{aligned} h_2=\frac{E_{\mathrm{res}}}{E_\mathrm{o}}=\frac{T\sqrt{R}\mathrm {e}^{-2i\omega L/c}}{1-R\mathrm {e}^{-2i\omega L/c}}\mathrm {e}^{-2i\omega L'/c}. \end{aligned}$$

(2)

Assuming that the only losses are from the mirror transmission ($T=1-R$) and the total feedback is the sum of the fields, the combined transfer function ($h_{\mathrm{fb}}$) is:

$$\begin{aligned} h_{\mathrm{fb}}=\left( C_1\sqrt{R}+C_2\frac{(1-R)\sqrt{R}\mathrm {e}^{-2i\omega L/c}}{1-R\mathrm {e}^{-2i\omega L/c}}\right) \mathrm {e}^{-2i\omega L'/c} \end{aligned}$$

(3)

where $C_1$ and $C_2$ are scaling factors for the non-resonant and resonant feedback, respectively. Following the analysis outlined by Morville et al. [6, 28], we can develop an expression relating the free-running laser frequency ($\omega _\mathrm{o}$) and the coupled laser frequency ($\omega _{\mathrm{fb}}$):

$$\begin{aligned} \omega _\mathrm{o}=\omega _{\mathrm{fb}}+\frac{\sqrt{\beta }(1-R_\mathrm{o})c}{\ell n_\mathrm{o} \sqrt{R_\mathrm{o}}} \times \left[ \mathfrak {R}(h_{\mathrm{fb}})\sin \left( \frac{2L'\omega _{\mathrm{fb}}}{c}\right) -\mathfrak {I}(h_{\mathrm{fb}})\cos \left( \frac{2L'\omega _{\mathrm{fb}}}{c}\right) \right] . \end{aligned}$$

(4)

The parameters in this equation include laser cavity length ($\ell$), laser medium refractive index ($n_\mathrm{o}$), laser facet reflectivity ($R_\mathrm{o}$), and feedback rate ($\beta$).

Fig. 1

a Feedback model of coupled laser frequency compared to free-running with varying non-resonant feedback contribution. In all simulations, $\beta = 0.001$, $C_2=1$, and $R=0.9999$. Three values of $C_1$ were modelled: 0.0 (green), 0.1 (red), and 0.5 (blue). The thicker lines show the emitted frequency as an upward ramp is applied for the scenarios without or with (high) non-resonant feedback rates in the same colours. The laser frequency increases monotonically, leading to jumps as shown. b Zoom-in of a cavity resonance for the same conditions

Fig. 2

a Feedback model of coupled laser frequency compared to free-running with different phase shared by both feedback contributions. In all simulations, $\beta = 0.001$, $C_1=0.5$, $C_2=1$, and $R=0.999$. Three phase conditions ($\varPhi$) were modelled: 0 rad (red, solid), 0.16 rad (blue, dotted), and 1.0 rad (green, dashed). b Simulation of cavity transmission for same settings as above. Note that the longest locking range and optimal mode symmetry do not correspond to zero phase offset

To ensure that the feedback from the cavity is in phase with the laser field at each successive resonance, the cavity length must be an integer multiple of the laser–cavity distance and so the two distances are usually set to be equal ($L'=L$). This also means that the resonant and non-resonant feedback fields are in phase when the distances are precisely equal, as each experiences an odd number of $\pi$ phase shifts due to reflections off dielectric-coated surfaces [31].

Figure 1a shows simulations of a typical 5.3 $\upmu \hbox {m}$ QCL frequency tuning over two cavity resonances under different feedback conditions to highlight how the non-resonant field affects the laser behaviour. In all simulations, $R=0.9999$, $C_2 = 1$, and $\beta =0.001$ with varying non-resonant feedback strengths $(C_1 = 0-0.5)$. This range of feedback rates was chosen based on estimates reported in previous studies [6, 11, 20]. The green line ($C_1 = 0$) is analogous to a V-shaped cavity where only resonant feedback returns to the laser. The blue and red lines indicate the tuning with different feedback rates for the non-resonant field and show that the frequency locking due to the high-finesse optical cavity is superimposed on an oscillating frequency tuning of varying amplitude due to the non-resonant reflection. In all cases, there are jumps in the coupled laser frequency as the laser monotonically tunes up or down. Without non-resonant feedback, these jumps only occur when the laser discontinuously hops from the resonant cavity frequency to its free-running condition. The presence of the non-resonant field, on the other hand, leads to frequency jumps at regular intervals that are unrelated to the optical cavity. These give rise to the sawtooth-shaped laser output seen in previous studies with diode lasers [30]. While the addition of the non-resonant field imposes certain constraints on the ratio of feedback rates ($C_1:C_2$) in order for the laser to emit at the resonant frequency, the figure shows that it is still theoretically possible for this locking to occur under the correct conditions. One can also see that the amount of non-resonant feedback becomes irrelevant once the laser is locked to the cavity.

The phase of the feedback is also an important consideration in simulations. Figure 2a shows similar feedback models of tuning rate for different phase conditions, set by adding a phase offset ($\varPhi$) to Eq. 3 to account for small changes in laser–cavity distance:

$$\begin{aligned} h_{\mathrm{fb}}=\left( C_1\sqrt{R}+C_2\frac{(1-R)\sqrt{R}\mathrm {e}^{-2i\omega L/c}}{1-R\mathrm {e}^{-2i\omega L/c}}\right) \mathrm {e}^{-2i(\omega L'/c + \varPhi )} \end{aligned}.$$

(5)

The transfer function ($h_3$) describing the field exiting the cavity through $\hbox {M}_2$, which is slightly different from the feedback through $\hbox {M}_1$, can be used to show the effect of phase on the locking:

$$\begin{aligned} h_3=\frac{E_{\mathrm{trans}}}{E_\mathrm{o}}=\frac{T\mathrm {e}^{-i\omega L/c}}{1-R\mathrm {e}^{-2i\omega L/c}}\mathrm {e}^{-i(\omega L'/c+\varPhi )}. \end{aligned}$$

(6)

This equation can be used to simulate cavity transmission for different phase conditions, as shown in Fig. 2b, similar to previous analysis for V-shaped OF-CEAS [6, 28, 32]. The simulations are all scaled to a maximum amplitude of unity and therefore do not account for changes in cavity transmission intensity due to increased coupling of the laser into the resonant cavity field.

3 Experiment

The linear OF-CEAS set-up, shown in Fig. 3, is similar to one described previously but utilises a new, more powerful QCL [12]. A cw distributed feedback QCL (Maxion, M575AY), which emits light between 1883 and 1896 $\hbox {cm}^{-1}$ with a power of up to 220 mW, was placed in a custom-made mount and Peltier-cooled to an operating temperature selected from within the range $-$30 to 9 $^{\circ }\hbox {C}$. Both the temperature of the chip and current supplied were controlled by custom electronics built in-house. The current was modulated by a function generator which applied a triangular ramp to scan over the selected transitions at a rate of 10 Hz. The laser was normally operated well above the threshold current (429 mA at $-$10 $^{\circ }\hbox {C}$ under free-running conditions).

Fig. 3

Experimental set-up for linear OF-CEAS with a cw QCL. The laser–cavity distance is controlled by the delay line and PZT, which is also connected to the computer for active control of the feedback phase. (Note: PD photovoltaic detector, Cam infrared camera)

The output beam was collimated by an aspheric lens (4 mm focal length, Thorlabs) and focused down to an intermediate beam waist by an off-axis parabolic mirror (OAPM). An iris was placed at the beam waist to spatially discriminate reflections coming off the first cavity mirror from light exiting the cavity. A piezoelectric transducer (PZT) attached to a steering mirror that directed the beam into the cavity allowed fine control over the phase of the feedback fields. A polariser (Thorlabs, RSP1C/M) was placed in the beam path before the cavity to tune the feedback rate and simultaneously allow measurement of the laser output with a photovoltaic detector (VIGO, PVMI-3TE-10.6) collecting the reflection as shown. The 77-cm-long optical cavity was formed using two high reflectivity ZnSe mirrors (Los Gatos) with radii of curvature of 50 cm. A $\hbox {CaF}_2$ lens ($f=15\,\hbox {cm}$) was set at a slight angle to the beam axis and focused light into the cavity. The cavity was enclosed by a 1-in.-diameter glass tube connected to a vacuum pump, pressure gauge, and inlet for gas samples. For these experiments, the cavity was filled with 30–165 mbar of a 10 % ($\pm$0.5 %) $\hbox {CO}_2$ mixture with $\hbox {O}_2$ and $\hbox {N}_2$ (BOC). The beam exiting the cavity was focused by an OAPM onto a second photovoltaic detector (VIGO, PVMI-3TE-10.6). This OAPM was placed on a flip mirror to allow imaging of the transmitted beam profile by an infrared silicon microbolometer camera (WinCamD-FIR2-16-HR). The laser–cavity distance was set using an optical delay line and the PZT to be equal to the cavity length. The detector output was sent to a data acquisition card (National Instruments, PCI-6221), and a LabView program recorded the signal. Resonant cavity modes were maintained by applying a DC error signal generated in a LabView program to the PZT to correct for optimal phase matching and symmetric cavity mode transmission [10, 12]. Simultaneous measurements from both detectors were recorded on an oscilloscope (LeCroy WaveRunner 6100A, 1 GHz bandwidth).

4 Results

The high-power QCL allowed sufficient transmitted power for monitoring of the intracavity transverse mode profile by the beam imaging camera. Small adjustments to the cavity alignment allowed different $\hbox {TEM}_{mn}$ modes to be selectively resonant in the cavity. As expected, the signal amplitude and locking range decreased when higher modes were preferentially excited due to increased diffraction losses leading to a lower feedback rate. All data presented here were collected with $\hbox {TEM}_{01}$ or $\hbox {TEM}_{02}$ Hermite-Gaussian transverse modes excited in the cavity and measured on the camera as these allowed stable locking to successive longitudinal modes over a range of powers incident on the cavity. $\hbox {TEM}_{00}$ modes could also be selected but required a much greater attenuation of light incident on the cavity to ensure that the feedback rate maintained feedback locking for less than one FSR.

Fig. 4

a Sample cavity transmission from a single laser scan with the frequency decreasing from left to right. b The lower panel shows a subsection in higher detail in order to highlight the symmetric cavity mode shapes and high signal-to-noise ratio. Each cavity mode is separated by the FSR of the cavity, which is 0.0065 $\hbox {cm}^{-1}$ (195 MHz). This is indicated by the dotted lines showing the rising edge of consecutive modes. The oscillation in the signal is due to an etalon between the detector and cavity

Figure 4 shows a sample cavity transmission measurement as the laser is scanned over a $\hbox {CO}_2$ absorption feature. The feedback rate was attenuated with the polariser to ensure excitation of consecutive cavity modes. No frequency calibration is required, as each cavity mode is separated by one FSR $(0.0065\,\hbox {cm}^{-1})$. Measurements of the laser output indicated that the power of the laser was not significantly affected by changes in feedback rate due to absorption in the cavity, and so the signal was not normalised to output power.

Based on the cavity transmission transfer function (Eq. 6) and analysis by Bergin et al. [12], an equation for absorption coefficient ($\alpha$) based on the transmission intensity of each cavity mode with ($I$) and without ($I_\mathrm{o}$) an absorbing gas sample is given by:

$$\begin{aligned} \alpha = \frac{(1-R)}{L}\left[ \sqrt{\frac{I_\mathrm{o}}{I}}-1\right] . \end{aligned}$$

(7)

This equation also applies for OF-CEAS with a V-shaped cavity. In the discussion that follows, we define the absorption ($A$) as the expression in the square brackets. Similar to previous studies, the baseline of the spectrum ($I_\mathrm{o}$) was fit to a fifth-order polynomial due to the nonlinearity of the laser output with a sinusoidal contribution to account for an etalon from the cavity–detector alignment [9, 11].

Fig. 5

A sample plot of absorption data calculated from a single 100-ms cavity scan over a $\hbox {CO}_2$ absorption feature (red circles) with corresponding fit (black line) based on the transitions listed in Table 1. For this measurement, 135 mbar of a gas mixture containing 10 % $\hbox {CO}_2$ was in the cavity, and the laser was scanned over the range 1887.65–1888.15 $\hbox {cm}^{-1}$. Below the spectrum is a plot of the residual to the fit. The 1$\sigma$ standard deviation of the residual, which is an estimate of the minimum detectable absorption of the measurement, is $3.1\times 10^{-3}$ corresponding to an $\alpha _{\mathrm{min}}$ of $1.6\times 10^{-8}\,\hbox {cm}^{-1}$ for a single scan

The spectral region around 1888 $\hbox {cm}^{-1}$ covers five $\hbox {CO}_2$ transitions, including transitions of $^{18}\hbox {O}$ and $^{17}\hbox {O}$ isotopologue species (see Table 1). Measurements collected with the frequency increasing or decreasing yielded the same results. The example spectrum shown in Fig. 5 was collected with 135 mbar of the 10 % $\hbox {CO}_2$ mixture in the cavity and the laser scanned over 0.5 $\hbox {cm}^{-1}$ with a 10-Hz triangular ramp. Absorption values were calculated as described above, with each point representing one cavity mode. Data were fit to a combination of Voigt profiles to account for each transition. In order to accurately simulate the physical system, some constraints were applied to the Voigt fits: the relative areas of the $^{16}\hbox {O}^{12}\hbox {C}^{16}\hbox {O}$ transitions were set according to HITRAN values, and the relative centre frequencies of significantly overlapped transitions were fixed.

Table 1 Details from HITRAN of the main $\hbox {CO}_2$ transitions within the spectral region scanned

The reflectivity of the cavity mirrors was determined by pressure series measurements to be 0.999603 $\pm \,6\times 10^{-6}$. Measurements repeated for different incident powers (set by the polariser) demonstrated that no saturation effects influenced results. A plot of the full-width at half-maximum of the transition at 1887.980 $\hbox {cm}^{-1}$ verifies that measured pressure broadening coefficients closely match predictions from HITRAN [33]. The relative isotopic abundances of $^{16}\hbox {O}^{12}\hbox {C}^{16}\hbox {O},\, ^{16}\hbox {O}^{12}\hbox {C}^{18}\hbox {O}$, and $^{16}\hbox {O}^{12}\hbox {C}^{17}\hbox {O}$ were determined to be 1343:4.9:1, which is close to the natural relative abundances expected in this gas sample (1348:5.3:1) [33].

The absorption spectrum shown (Fig. 5) was converted to absorption coefficient using Eq. 7 with the $R$ value determined from the pressure series. The standard deviation of the residual—shown in the lower panel—is $3.12\times 10^{-3}$, which is equivalent to $\alpha _{\mathrm{min}}=1.6\times 10^{-8}\,\hbox {cm}^{-1}$ for a single 100-ms scan. This may be reduced further by using an anti-reflection coating on the flat face of the second cavity mirror to minimise etalon effects. An alternative method for determining absorption coefficient based on a single ring-down time during the measurement was not used because the feedback rate was set such that the locking range was nearly equal to one FSR, leading to insufficient time between modes to accurately measure a ring-down time [10, 28].

The full potential of the technique is better represented by the consistency of repeated measurements. After fitting multiple absorption spectra from individual scans, an Allan-Werle variance analysis was performed to quantify the variability in the measured area of a strong $\hbox {CO}_2$ transition at 1890.763 $\hbox {cm}^{-1}$ [34, 35]. Using the same reflectivity as above yields $\alpha _{\mathrm{min}} = 2.7 \times 10^{-9}\,\hbox {cm}^{-1}$ for an average over 1 s, corresponding to a minimum detectable concentration of 250 ppm $\hbox {CO}_2$. This value reflects the time-dependent variability of the fit, rather than the quality of each individual fit across the region. While this minimum detectable concentration is relatively high, using a laser that can emit at higher wavenumber to access the stronger transitions around 2350 $\hbox {cm}^{-1}$ should improve this by six orders of magnitude.

An Allan-Werle analysis was also used to assess the temporal stability of the amplitude of individual modes. For a 1-s average, the normalised standard deviation of the amplitude is $8.6\times 10^{-4}$ V, equivalent to $\alpha _{\mathrm{min}} = 1.1\times 10^{-9}\,\hbox {cm}^{-1}$. This is not an accurate indicator of the sensitivity of the measurement, but rather shows that fluctuations in mode intensity are not a limiting factor in the spectrometer’s performance.

In order to test that the feedback model accurately simulates the experiment, the relative power of resonant and non-resonant light coming back towards the laser while the cavity is in resonance must be determined. To estimate these values, the power transmitted through the polariser, the intensity of the combined feedback fields from the cavity, and the power of the light transmitted through the cavity were measured. The latter could be accurately measured using a calibrated power meter due to the ability of the laser to remain locked to a cavity mode for a time (seconds) longer than the response time of the meter when not applying a current ramp to the laser. The feedback light—a combination of both resonant and non-resonant fields—was measured using a photovoltaic detector placed on the laser side of the polariser in order to collect a small reflection of the light returning to the laser from the cavity. Based on these measurements, feedback coming from the cavity is heavily dominated by the non-resonant contribution, which accounts for approximately 85 % of the total returning power. In order for the resonant feedback to dominate at the laser facet, mode mismatching and spatial filtering are required to separate the two feedback fields and preferentially attenuate the non-resonant contribution [36]. Despite the reduction in coupling efficiency due to mode mismatching, reasonable intracavity build-up was observed. For the experiments shown, the incident laser intensity on the cavity was attenuated to 25 mW using the polariser in order to ensure excitation of consecutive modes spaced by one FSR; typically the power of the transmitted beam through an evacuated cavity was 3 mW. Thus, we conclude that the decrease in coupling efficiency is not a significant problem in this case.

The OAPM and lens were positioned between the laser and cavity to allow for mode mismatching of the incident light into the intracavity field. A Gaussian ray trace matrix analysis of the experiment was used to model the size and curvature of the beams for the region between the laser and cavity [31]. The collimated beam waist after the collimating lens was estimated to be 1.5 mm based on measurements with the infrared camera. Figure 6 shows the modelled beam size for various fields as a function of position between the laser facet and inner surface of the input cavity mirror. A comparison is shown between the case where the beam is mode-matched into the cavity (a) and the set-up used here (b) where the mode is mismatched in order to separate the focal points of the non-resonant and resonant fields for spatial filtering. An iris was placed at the focal point of the output beam (33 cm) in order to preferentially attenuate the non-resonant feedback. As evident in this figure, the beam size of the non-resonant feedback was much larger than the resonant at the iris position, and so a greater fraction of the resonant power was transmitted through the iris aperture (2 mm). It is also worth noting that the reflection off the flat, uncoated surface of the mirror diverges quickly and is virtually all lost either at the iris or off the edges of the optics. Alternatively, an anti-reflection coating on the flat face of the cavity mirror or fabrication of a mirror substrate with a wedge shape would help eliminate this unwanted reflection.

Most of the spatial filtering, however, occurred at the laser facet, which is specified to be 5.8 $\times$ 2 $\upmu \hbox {m}$ (Maxion). The ray trace analysis was extended beyond the collimating lens to predict the size of the various fields at the facet, located approximately 4 mm in front of the aspheric lens. Due to the mode mismatching conditions set, both the resonant and non-resonant fields were clipped at the facet but the fraction of non-resonant beam intensity filtered out was more than ten times greater than for the resonant beam. This attenuation is sufficient such that the resonant field feedback rate was greater than the non-resonant field at the laser cavity. Given that the total feedback rate is in the range 10⁻⁴ – 10⁻³, it is likely that diode lasers would experience coherence collapse and not be usable under these experimental conditions [18].

Fig. 6

These plots show beam size as a function of position along the laser–cavity path for four different beams: laser output (green), reflection off the flat face of $\hbox {M}_1$ (black), non-resonant feedback reflection off the curved face of $\hbox {M}_1$ (blue), and resonant cavity field (red). In a, the off-axis parabolic mirror and lens are positioned such that the incident beam is nearly mode-matched into the cavity. In b, which simulates the experimental set-up used here, the incident beam is mismatched in order to change the spatial distribution of the beams to allow for preferential filtering of the non-resonant feedback field. The arrow indicates the position and aperture of the spatially discriminating iris, and the laser facet is located at $x=-0.4$ cm with the collimating lens positioned at $x = 0$ cm. Note, beam size is defined as the half-width at $\frac{1}{\mathrm {e}}$ magnitude of the field assuming a spherical Gaussian profile

Based on these calculated feedback rates, the coupled laser feedback model was used to simulate the experiment. To verify the model, the expected behaviour of the coupled laser was compared with experimental measurements of changes in laser output. Figure 7a shows a theoretical coupled laser frequency tuning curve given the estimated feedback rates of the system. Again, it is clear that the sharp feature due to the high-finesse optical cavity is superimposed on the oscillatory tuning curve characteristic of the multistable laser cavity experiencing non-resonant feedback from an external reflector [30]. Figure 7b shows experimental measurements of the laser output power (red) overlaid on the cavity transmission signal (grey) collected simultaneously to indicate when the laser is resonant with the cavity as the frequency is scanned. The laser output follows a similar sawtooth-like pattern, as seen previously in diode laser studies, as well as a clear locking range where the output power remains stable. Thus, we can interpret the laser output measurement based on the theoretical feedback model. Starting at point A, the laser frequency comes into resonance with the optical cavity and stays locked at this frequency until reaching point B where the laser discontinuously jumps out of resonance with the cavity to a frequency dictated by the non-resonant feedback field. The tuning rate of the laser frequency continues to be slower than for the free-running system, and at point C, the laser suddenly mode hops to another point on the tuning curve, as seen in previous studies [28, 30]. The laser continues tuning at an attenuated rate until reaching point D, at which time the optical cavity is resonant and the laser responds to this new, dominant field until the end of the locking range at point E. Measurements of the laser output filtered through a Ge etalon verified that frequency, as well as amplitude, did not measurably vary through the duration of locking to the resonant linear cavity.

The threshold current of a semiconductor laser decreases with increased feedback rate, and so measurements of this threshold reduction also provide some insight into the amount of feedback the laser experiences [22, 30]. With the laser operating at $-10\,^{\circ }\hbox {C}$, the threshold current decreased by 3.8 % from 429 to 411 mA with non-resonant feedback only (i.e. no cavity transmission signal). When the laser frequency was locked to a resonant cavity, this change increased to 4.2 % indicating that the total feedback rate increased, as expected.

Following the analysis of Morville et al. [6], the effect of tuning rate on cavity injection efficiency was examined by simultaneously measuring laser output, feedback light intensity, and cavity transmission at varying feedback rates and free-running frequency tuning rates. The amplitude of cavity transmission for optimal phase matching was used as a proxy for the injection efficiency. The critical tuning rate—the maximum frequency tuning rate at which feedback is still observed—increases with increased feedback rate as expected. For a typical feedback rate used for the experiments shown, there is a sharp decrease in injection efficiency as the tuning rate exceeds 700 GHz/s, which is similar to previous observations for similar feedback rates with V-shaped cavities and diode lasers. All data shown were collected with a much slower tuning rate to allow stable feedback locking.

Fig. 7

a Feedback model and b experimental measurements demonstrating the effect of non-resonant and resonant feedback on the laser. Based on the experimental parameters and Gaussian ray trace analysis, the two feedback rates were estimated to be $C_1\beta =1.2\times 10^{-4}$ and $C_2\beta =2.6\times 10^{-4}$ for this measurement. The dashed lines indicate corresponding changes in the laser output at the labelled points in the coupled frequency plot. In b, the grey trace is a cavity transmission signal measured after the exit mirror of the cavity, with the laser output intensity (red) measured simultaneously from a portion of light reflected off the polariser. As noted earlier, the change in laser output power due to feedback from the resonant cavity is insignificant

Though feedback models are mainly relevant to experiments where the laser frequency is being scanned, we also consider the case where the frequency is set to a single value with no current or temperature changes applied. If the emitted wavelength is resonant with the cavity, the length of the lock is solely determined by noise or drifts in the system, either mechanical or electrical. In fact, we have observed that the cavity transmission could be maintained passively for seconds. With a simple error loop to maintain the correct phase and frequency, it should therefore be possible to actively maintain intracavity excitation and frequency locking for significantly longer times. Given that the intracavity power is quite high (up to 250 W), this capability may be attractive for applications requiring continuous measurements under high powers with high frequency resolution [37, 38].

5 Conclusions

Linear optical cavity geometries have been unpopular for OF-CEAS experiments due to the persistent competition between feedback fields from the resonant cavity and reflection off the input cavity mirror. Despite this, we have established stable feedback with high cavity coupling efficiencies in measurements of $\hbox {CO}_2$ using a 5.3 $\upmu \hbox {m}$ DFB QCL. For 1-s averaging, we achieved an $\alpha _{\mathrm{min}}$ of $2.7\times 10^{-9}\,\hbox {cm}^{-1}$, which is on par with previous studies using V-shaped cavities [9–11]. Complementary laser feedback and Gaussian beam profile models demonstrated that it is feasible for the resonant feedback rate to dominate the feedback competition if spatial filtering selectively attenuates the non-resonant field. This can be achieved by careful mode mismatching to ensure that the non-resonant beam size is enlarged at the laser facet while the resonant field is focused more tightly onto the laser facet.