We report on what we believe to be the first Doppler-free polarization spectroscopy with a quantum cascade laser. A hot-band CO2 transition around 4.3 μm wavelength has been used to test the potential of the technique for high-resolution spectroscopy and wide-bandwidth quantum cascade laser frequency stabilization.
©2009 Optical Society of America
Polarization Spectroscopy (PS) is among the highest sensitivity Doppler-free spectroscopic techniques. PS was first reported in 1976 by Wieman and Hänsch  as a useful Doppler-free method offering a considerably better signal-to-noise (S/N) ratio in comparison with standard saturation spectroscopy.
In a typical PS setup, a strong pump beam and a weak probe beam with different polarizations and counterpropagating through the target sample are tuned to the desired optical transition. The optical pumping induced by the polarized pump beam induces a birefringence in the medium and a consequent detectable polarization change in the weak probe beam. By changing the polarization of the pump, PS also offers the unique possibility to distinguish between the different types of molecular transitions, the P, Q or R branches [2, 3].
Several theoretical models have been proposed, both to discuss the expected spectral profiles and to evaluate the different contributions to the PS signal [4, 5, 6, 7]. Different approaches based on rate equations have been adopted to calculate the induced optical anisotropy in atoms and molecules [8, 9].
PS is widely used on atomic species in the visible spectral range, for high-sensitivity and high-resolution spectroscopy [4, 10, 11, 12], as well as for laser frequency stabilization . PS was first demonstrated for atomic transitions in the presence of degenerate Zeeman sublevels, but it works for all the molecular ro-vibrational transitions. In fact, it has also been applied to sub-Doppler spectroscopy of molecular rovibrational transitions in the visible [10, 13] and to atmospheric-pressure gas detection of combustion products [14, 15, 16, 17] in the near/mid-infrared region, by using coherent light mainly produced by difference-frequency generation of visible sources in non-linear crystals.
As widely demonstrated in the visible spectral range, PS can provide a great improvement both for high-resolution and high-sensitivity trace-gas detection. It also provides dispersion signals without any modulation of the laser: the PS signal can thus be efficiently used for laser frequency stabilization. The absence of any modulation signal applied to the laser, as well as the absence of any demodulation of the acquired signal, allows to exploit the whole available bandwidth for the servo loop. Over the years, this method has been applied to stabilize dye lasers  and diode lasers [19, 6].
The availability of high-power, narrow-linewidth and tunable IR laser sources like quantum cascade lasers (QCLs) offers the possibility to apply PS in the mid-IR spectral region (the so-called molecular fingerprint region) for high-sensitivity and high-resolution spectroscopy on IR-active fundamental rovibrational transitions of molecules like CO2. For long time, the mid-IR region has suffered for the lack of polarization optics but, thanks to the recent availability of wave-plates and polarizers, PS can now be efficiently used with QCLs in this crucial spectral region. In particular, by using wire-grid polarizers at a 45° angle of incidence like polarizing beam splitters, we can get differential balanced detection for the PS signal .
In our recent activity we have performed absolute frequency measurements on Doppler-broadened  and sub-Doppler  CO2 transitions around 4.3 and 4.4 μm wavelength with QCLs. The latter experiment, in particular, combining high-resolution spectroscopy to frequency lock of the laser, has pointed out the benefits coming from a frequency stabilization of the QCL and has allowed to improve the precision of frequency measurements to the kHz level. However, due to the first-derivative lock-in detection adopted, the feedback-loop bandwidth was intrinsically limited by the modulation frequency applied to the laser and the integration time constant of the lock-in amplifier. For this reason, the experiment could not provide a sensible reduction of the laser jitter: the upgrade to a PS setup can also be a crucial step towards a highly efficient stabilization of the laser, in order to reduce its frequency jitter and to allow high-precision frequency metrology with QCLs.
We report here what we believe to be the first PS measurements with a QCL. This technique has been applied to the hot-band (0111 – 0110) P(30) CO2 transition at 4.3 μm. We have studied different configurations and finally adopted a double-balanced detection scheme, both to increase the S/N ratio of the acquired spectra and to cancel the residual Doppler-broadened background. The result is a Doppler-free dispersion-like signal with a S/N ratio comparable to that obtained with standard wavelength modulation techniques and lock-in first derivative detection, but with no modulation of the laser and filtering of the detected signal. We have recorded Doppler-free signals narrower than those obtained in our previous experiment based on a standard first-derivative technique. Moreover, the absence of any modulation and filtering effectively makes the PS signal the ideal candidate as dispersive-like error profile for large bandwidth frequency stabilization of our QCL.
2. Experimental set-up
The experimental set-up is schematically drawn in Fig. 1.
The mid-IR radiation is provided by a cw DFB QCL emitting at 4.3 μm. The laser is housed in a liquid-nitrogen continuous-flow cryostat and stabilized at 86.00 ± 0.01 K. At this temperature the threshold current is 100 mA at a driving voltage of ~9 V. The current noise of the commercial driver is < 4 μArms (in the 10 Hz -10 MHz band). The frequency tunability with current and temperature are about 400 MHz/mA and 2 GHz/K respectively, and the maximum power is about 10 mW. The cryostat has been mechanically stabilized in order to minimize acoustic vibrations of the laser housing. A wire-grid polarizer (P1, extinction ratio ≃100) is used as a polarizing beam-splitter: the linear polarization of the QCL beam is rotated by a tunable half-wave plate in order to obtain a splitting of about 90%-10% for the transmitted (pump) and reflected (probe) beams respectively.
A counter-propagating geometry of the beams is adopted in this experiment. A tunable quarter-wave plate sets the pump beam polarization to circular. Two polarizers are placed on the probe beam: a high extinction-rate polarizer (P2: extinction ratio ≃60000) before the cell, in order to further clean the polarization, and a second wire-grid polarizer (the analysis polarizer P3, extinction ratio ≃100) in front of the detectors, in order to select the polarization components to reveal.
The pump and probe beam are overlapped inside the 12-cm-long spectroscopic cell, with a residual misalignement angle of at most 0.8°. The pump beam is almost collimated, has a diameter of about 3 mm and a power of 3.6 mW, corresponding to an intensity of about 510 W/m2. The probe beam is focused into the cell in order to get the largest overlap with the pump beam. Its diameter is not larger that 1 mm, and the corresponding transit time broadening is about 120 kHz (HWHM). The cell is at room temperature and filled with pure CO2 at pressures from 15 to 80 mTorr. For the latter pressure the highest signal amplitude has been measured. However, since the corresponding pressure broadening is about 280 kHz (HWHM), the measurements presented in the following have been taken at lower pressures (typically between 15 and 50 mTorr), where pressure broadening and transit time broadening are comparable.
In order to perform the differential detection of the two probe beam polarization components, the wire grid P3 polarizer is used at a 45° incidence angle and acts as a beam splitter, since it reflects the blocked polarization component, allowing to extract it and detect it as well as the transmitted one. The selection of the polarization components depends on the angle θ between P2 and P3 (the θ = 0° position is for crossed polarizers). In particular, it is possible to implement a saturated-absorption scheme (θ = 90°, only transmission acquired), a standard single detection polarization scheme (θ ~ 0°, only transmission acquired) and a double-balanced polarization detection (θ = 45°, both transmission and reflection acquired). Detection is made by two identical InSb liquid-N2 cooled photovoltaic detectors. The two photocurrents are resistively subtracted, and the difference is amplified by a home-made two-stages low-noise amplifier, with a 1 MHz bandwidth, and a 105 total gain (100 for the first stage, 1000 for the second stage).
The double-balanced differential scheme is the only one achieving a complete exploitation of the amplifier gain, since it produces a zero-offset signal allowing to amplify only the small polarization signal, as we will see in the next sections.
Although the chosen transition belongs to a hot band, we observe a strong absorption (about 70% of the probe available power) from atmospheric CO2, with the laser in resonant condition. In order to reduce this absorption, the optical bench has been enclosed inside a nitrogen-saturated atmosphere. This reduces the atmospheric absorption down to about 10%, and also allows to suppress the amplitude fluctuations of the signal caused by random airflows along the beam path.
3. Model of polarization spectroscopy
The theory needed for reproducing the profiles of polarization spectroscopy signals is well known, and presented in Ref.. It does not allow a quantitative calculation of the magnitude of the effect, which rather requires the solution of the time-dependent density matrix equations . Based on the Lorentzian nature of the effect, it leads to an analytical expression suitable for fitting the experimental data. In this section the most general form of this expression is presented: it can be easily manipulated with modern computer algebra software and, differently from its approximated versions, it allows to better take into account all the effects that contribute to the shaping of the detected signal.
Consider the linear polarization of the probe beam divided into its σ + and σ - components, which are differently affected by the circularly polarized pump beam. Let α + and n + (α - and n -) be the absorption coefficient and the refraction index experienced by the σ + (σ -) components. The difference ∆α(ω) = α +-α - represents the circular dichroism, while the difference ∆n(ω) = n + - n - is the optical birefringence. Let Φ be the complex phase factor taking into account both effects:
where L is the path length of the light into the sample.
In this expression the possible contributions to birefringence (∆br) and the dichroism (∆bi) coming from the cell windows have been added, both independent from frequency. The ∆n(ω) and ∆α(ω) terms, on the contrary, bring respectively a Lorentian dispersive and absorptive dependence on frequency:
where ∆α 0 L is the magnitude of the polarization effect, ω 0 is the center of the transition and γs its saturated linewidth (FWHM).
The conventional polarization spectroscopy scheme, as mentioned in the previous section, consists in the detection of the intensity transmitted by the nearly crossed analysis polarizer P3 (small θ). The most general expression for the intensity of the transmitted signal It is:
where α̅ = (α + + α -)/2 and the factor I 0 e -α̅L represents the saturated-absorption spectrum.
Indeed, this general expression can be plotted for any θ value: Fig. 2(a) shows several line profiles corresponding to θ values from 0° to 90°. At this latter angle, the signal is almost the same as in saturated absorption, while for θ = 0° only the symmetric Lorentzian contribution survives (see Fig. 2(b)). For small θ values the dispersive component arises, proportional to sin(2θ); for increasing angles, however, also the background arises, growing with sin2 θ and overriding the polarization signal. Two important constraints can thus be identified: first of all the need of keeping θ very small, which implies that the resulting signal is very small. Secondly, the unavoidable presence of an offset, comparable or even larger than the amplitude of the signal itself, and correlated to the laser intensity fluctuations. This latter point can be very critical when the polarization signal is used as the feedback signal in a frequency locking loop.
In order to overcome the limitations presented above, a double balanced acquisition of the two complementary components of the probe beam polarization has been implemented . It allows both to choose the ideal value θ = 45° for which the dispersive component of the signal is maximized and, at the same time, to obtain a zero-background signal. The polarization component Ir reflected by P3 and complementary to It is:
and the most general expression for the differential signal is:
The ξ parameter takes into account the non-ideal behavior of the analysis polarizer (finite extinction ratio and spurious reflection from the substrate), so that the selected components of polarization are not perfectly pure.
In Fig. 3 the differential signal is plotted for θ varying from 0° to 90°. For θ = 0° and θ = 90° the signal matches the saturated absorption one (with opposite signs), while the dispersive component is at a maximum for θ = 45°. In this configuration, for Φ ≪ 1, Eq. (5) becomes:
that is the expression appearing in Ref.. Only the dispersive component of the signal survives, as well as the birefringence contribution of the CaF2 cell windows. However, since CaF2 is optically isotropic, we can assume that the cell windows do not contribute to the PS signal. The approximated version of Eq. (5) is suitable for implementing fast fitting routines; however, as we will see in Sec. 4, only the general expression is able to reproduce the signals acquired in non-ideal experimental conditions. For example, small variations of θ or ξ parameters produce significant deformations of the profile, that cannot be described by the approximated form of Eq. (6), as shown in Fig. 3(b).
A further theoretical issue regarding the analysis of the experimental data is that the QCL source is broader than the expected linewidth of the investigated molecular transition. The latter results in 170 kHz (in experimental conditions), while the QCL emission linewidth varies in the range 1.5 ÷ 6 MHz, depending on the observation timescales. For this reason, all the fitting curves that will be shown in next sections are based on a Voigt transformation of Eq. (2). In Ref. some guidelines on how to obtain analytical expressions of the Voigt profiles using the complementary error function erfc(z)=1-erf(z) can be found. The erfc function is particularly suitable for implementation of efficient algorithms.
4. Spectroscopic measurements and discussion
In this experiment we mainly investigated the (0111 - 0110) P(30) rovibrational transition of CO2 (at ν=2310.5062 cm-1 and with a linestrength of 6.6×10-20 cm, as reported on the HITRAN database ), even if several much weaker transitions can be detected.
A first alignment between the pump and probe beams has been performed by maximizing the standard saturation signal as detected by D1, with the polarizer P3 transmitting all the probe beam (θ=90°). Figure 4 shows a recording of the saturated profile, with a CO2 pressure of 20 mTorr in the cell. The high gain of the amplifier has been tailored on the weak polarization signals: in this standard Lamb-dip acquisition the probe beam intensity has to be strongly attenuated to avoid saturation of the amplifier. The fit allows to determine some parameters (laser intensity, Doppler width) that will be used for fitting the PS data. Moreover, the measured Lamb-Dip width (1.5 MHz) is an evidence that the laser emission linewidth is larger than the expected molecular linewidth. This leads also to a rescaling of the effective saturation parameter. In fact, with a pump power of 3.6 mW (a factor 13 larger than the saturation intensity 40 W/m2 calculated according to ), a stronger saturation regime and a higher contrast factor of the Lamb dip would be expected. Considering that the frequency sweep time on the Lamb dip is about 20 μs (2 ms sweep time over the full range), the measured width is consistent with our independent measurement made in Refs.[20, 21] and in line with the values reported by other groups for free running QCLs linewidths [25, 26].
Single-detection PS is performed by moving the P3 polarizer in a nearly-crossed configuration with respect to P2 (small θ): the detected signal (again only the D1 detector is used) is now much smaller than before and the filters on the probe beam can be removed. The dispersive profile appears overlapped to the background saturated profile, as described in Sec. 3. Figure 5(a) shows the PS signal of the same transition obtained for the P3 angle θ=2°.
Due to the residual background, the signal is not symmetric and suffers from offset fluctuations. However, the signal-to-noise ratio is enhanced with respect to the saturated-absorption acquisition (see inset of Fig. 4). The curve fit performed with the It function of Eq. (3) shows a good agreement with the experimental data.
At this point the double-detection differential configuration is performed, as previously described: the analysis wire-grid polarizer is rotated at θ =45°, and both the transmitted and reflected components of the probe beam are detected. In this way the polarization effect is equally split into the two channels with opposite sign, and finally adds up thanks to the difference operation. On the other hand, the common background is equally split among D1 and D2 with the same sign: the difference operation cancels out its effects and the final signal has a flat (zero, in principle) background. The differential acquisition thus allows to cut off the background contribution, together with the related amplitude fluctuations. At the same time, the polarization signal is enhanced and, in principle, assumes a symmetric shape centered on the transition resonance. These improvements are clearly shown in Fig. 5(b).
Figure 6 shows two typical differential signals. As already pointed out, the condition of perfect balancing is difficult to achieve, since the dependance on θ is very critical and a manual regulation does not provide the required precision. Moreover, the non-ideal behavior of the analysis polarizer, together with the unavoidable difference between the gains of the two detectors, introduces some asymmetry and produces a signal offset that, at the moment, can be compensated only by tuning the angle θ itself. For all these reasons, the profile of the acquired signal is not perfectly symmetric, as we would have expected, even in the best case (see Fig. 6(a)). However, as anticipated in Sec. 3, the general expression used for data fitting takes into account all these additional effects, giving a good agreement with the data even in worse conditions (see Fig. 6(b)).
A quantitative analysis of the polarization signal will require a more strict control on the parameters that give rise to the effects cited above. However, for the purpose of using this signal in a frequency stabilization loop, the achieved improvements in comparison with the first-derivative dither-locking technique adopted in our previous work  are evident. In fact, the available bandwidth of the signal is now about 1 MHz (the bandwidth of the differential amplifier), in comparison to a few hundreds of Hz of the previous solution (limited by the lock-in time constant). The signal-to-noise ratio of about 250, increased of more than a factor 10 compared with the saturated absorption signal of Fig. 4, is very promising for an effective feedback loop. The differential PS signal has finally been used for investigating the QCL emission linewidth (∆νL, HWHM) over different timescales. The fit shown in Fig. 6(a) gives ∆νL = 1.5 MHz, that is consistent with the result of the fit on the Lamb-dip, since the time-scale is the same (20 μs). In Fig. 7, on the other hand, a zoom on the PS signal is shown, obtained by a 32-acquisitions average with a 1.25 kHz sweep rate and corresponding to an effective acquisition time of 26 ms. The broadening is evident, and the fit gives ∆νL = 6 MHz. This confirms the hypothesis made in our previous works: the QC laser emission is few-MHz broad over ms-timescales, but the main contribution to its frequency noise comes from the low-frequency spectral region (less than 1 MHz) and hopefully could be compensated by the locking loop bandwidth provided by the differential PS signal.
In this paper we have presented what we believe to be the first thorough application of sub-Doppler polarization spectroscopy to a QCL-based spectrometer. Both the basic PS configuration and a balanced detection have been applied to an hot-band CO2 transition around 4.3 μm. The balanced technique, in particular, allows to obtain sharp spectra with much higher signal-to-noise ratios with respect to standard saturation spectroscopy. Because of the absence of any modulation onto the laser, such signals are particularly suited for an efficient frequency stabilization of the laser to the molecular line. Finally, a detailed and complete theoretical discussion of the PS signal profile has been carried out: the match of the expected profiles with the experimental data in different experimental conditions is noteworthy.
The authors are grateful to Pablo Cancio, Giovanni Giusfredi, Davide Mazzotti, Iacopo Galli and Livio Gianfrani for the useful discussions and suggestions. We thank Livio Gianfrani also for lending us the QC laser. A special thank goes to the LENS electronic staff, in particular to Alessio Montori, for the valuable help in designing and fabricating the electronics used in the experiment. This work, as part of the European Science Foundation EUROCORES Program EUROQUAM-CIGMA, was partially supported by funds from CNR and the other participating national Funding Agencies and the EC Sixth Framework Program.
References and links
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