Heterodyne phase-sensitive detection for calibration-free molecular dispersion spectroscopy

Pedro Martín-Mateos; Pablo Acedo

doi:10.1364/OE.22.015143

1. Introduction

When light travels through a gaseous sample, it experiences absorption and dispersion in amounts dependent of the composition and characteristics of the sample. As a result, there are different spectroscopic techniques for gas sensing and analysis that are capable of identifying the gases which are present in a medium and estimate parameters like its concentration, temperature or pressure. Several of the most important of those techniques are based on the use of tunable lasers. Between them, Tunable Laser Direct Absorption Spectroscopy (TLDAS) [1] is the most straightforward method and only requires the measurement of the power transmitted through the sample. When a spectral feature of a molecule is swept by a laser, TLDAS uses the amount of optical absorption to estimate the gas concentration. However, its major drawback is the existence of a baseline that requires normalization of the transmitted power and, additionally, this baseline is usually several orders of magnitude higher than the signal variation that is to be measured, limiting the sensitivity of the system.

Wavelength Modulation Spectroscopy (WMS) [2,3] overcomes some of the limitations of TLDAS by slowly modulating the laser wavelength and detecting the power transmitted through the gas sample (generally) at the second harmonic of the modulation frequency. This technique eliminates the baseline problem and improves the signal to noise ratio of the measurement by displacing the signal to a higher frequency region. However, an inherent problem of WMS is that the amplitude of the extracted lineshape depends on the total power received, making necessary the normalization of the measured waveform. This normalization process can be complicated by the presence of fluctuations of the emitted power or by pointing instabilities and also if the optical path is partially blocked by floating particles for example. Nevertheless, Rieker et al. have recently presented an interesting solution [4] to the problem of normalization.

A different approach is Frequency Modulation Spectroscopy (FMS) [5–7]. In this case, the wavelength modulation deviation of the laser is greater than the linewidth of the absorption feature and the high modulation frequencies decrease the intensity noise of the laser with respect to WMS. In addition, the technique is inherently baseline free and, even more important, FMS allows to measure not only optical absorption, but also wavelength dispersion. The disadvantages of this method are that high speed modulators and receivers are needed and, since the information is encoded in the amplitude of the transmitted signal, the normalization process is still indispensable. The ability of FMS to measure the dispersion spectrum has been exploited by Noise Immune Cavity Enhance Optical Heterodyne Molecular Spectroscopy (NICE-OHMS), a technique first proposed by Ye et al. [8] that is of interest for different groups [9,10]. Two Tone Frequency Modulation Spectroscopy [11–13] is a variant of FMS in which the laser wavelength is modulated using two high frequency signals separated a few megahertz. The beat note of the two signals can therefore be acquired using a low speed photodetector to give absorption information equivalent to that of FMS. As drawbacks, the ability of FMS to recover the dispersion spectrum is partially lost and power normalization remains necessary.

Tough these classic absorption spectroscopic methods are widely spread nowadays, the so called dispersion spectroscopy techniques are an interesting alternative for gas sensing and analysis. Dispersion spectroscopy methods are based on the measurement of optical dispersion associated to refractive index variations with wavelength that are inherent to molecular transitions and present significant advantages over classical methods. For example, with certain dispersion-based techniques, it is possible to directly overcome baseline and normalization problems typical of absorption-based approaches. Besides this, the refractive index changes are linearly dependent on the species concentration giving a better dynamic range to dispersion spectroscopy sensors when compared with absorption measurement instruments [14].

The first accurate measurements of molecular dispersion in the vicinity of spectral features were carried out a century ago when Roschdestwensky proposed the hook method [15]. This is a classical interferometric method that has been revised and improved over the years [16], [17] and employed in different scenarios [18,19]. The hook method has also served as the basis for the development of other techniques that use different information extraction setups [20] or illumination sources [21]. Similarly, phase shifts associated with spectral features have also been characterized combining tunable lasers and other interferometric schemes [22,23]. In a different line, Wysocki et al. recently proposed Chirped Laser Dispersion Spectroscopy (CLaDS) [24], a method that takes advantage of a frequency-chirped laser to transform optical phase changes into frequency shifts that can be demodulated to recover the dispersion spectrum. This technique can be combined with photonic processing leading to compact and easily integrable solutions [25].

In this paper, we present an alternative technique for dispersion spectroscopy, in which the optical phase shift produced by a spectral feature is measured using phase-sensitive detection. A tunable laser is intensity modulated at high frequency to generate an optical spectrum composed of three tones that is swept across the spectral range of interest. The changes of the refractive index in the vicinity of molecular transitions produce a relative phase shift between the three optical waves that is reflected in the phase of the beat note when the optical signals impinge on a square law photodetector. The phase shift of the beat note can be measured and used to recover the dispersion spectrum of the sample and, from this, the gas species present in the sample and their characteristics. Since the phase of the beat note is immune to power fluctuations, the technique inherently allows normalization-free operation. In combination, an analytical model for the propagation of light in gaseous samples has been developed enabling calibration-free molecular dispersion spectroscopy using spectral data from HITRAN [26].

In the next section, a simple analytical model for the propagation of light that describes the fundamentals of the method is presented. Then, two possible receiver schemes are introduced and experimentally validated using a ro-vibrational transition of methane at 1.65 $μ m$ . Finally, the performance of the new dispersion spectroscopy technique is discussed and compared with other existing methods.

2. Molecular dispersion spectroscopy using phase-sensitive detection

2.1 Operating principles

It is well known that the refractive index in the vicinity of a molecular transition of center $ω_{c}$ [cm⁻¹] has the form:

n (ω) = n_{0} + s \frac{ω_{c} - ω}{{(ω_{c} - ω)}^{2} + {(\frac{γ}{2})}^{2}}

where s is a variable dependent of the spectral line and

γ

[cm⁻¹] is the Full Width at Half Maximum (FWHM) of the spectral feature. The graphical representation of Eq. (1) is shown in Fig. 1(a). The dispersion profile is thus characterized by

ω_{c}

and FWHM.

Fig. 1 (a) Refractive index in the vicinity of a molecular transition with center $ω_{c}$ . (b) Spectrum of an intensity modulated optical carrier centered in the dispersion line.

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When a molecular transition is swept by a multi-tone optical signal like the one shown in Fig. 1(b), each optical tone experiences a slightly different refractive index. The dispersion spectroscopy technique proposed in this paper takes advantage of this effect to recover the refractive index profile and, consequently, the characteristics of the gas in the sample. The basic scheme for a dispersion spectroscopy system based on this idea is presented in Fig. 2.

Fig. 2 Block diagram of the phase-sensitive detection technique.

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A continuous wave tunable laser source is intensity modulated at a frequency $Ω_{1}$ [rad/s] to generate an optical spectrum composed of a carrier (E₁) and two sidebands (E₂ and E₃) (Fig. 1(b)):

E_{1} = A_{1} \cos (ω_{0} t)

E_{2} = A_{2} \cos [(ω_{0} + Ω_{1}) t]

E_{3} = A_{3} \cos [(ω_{0} - Ω_{1}) t]

where

ω_{0}

[rad/s] is the optical frequency. In the vicinity of a molecular transition, electromagnetic waves of different frequencies will experience different refractive indexes. Hence, as the optical waves propagate through a dispersive medium, each component travels at a slightly different speed producing changes in their relative optical phases. At a distance L [m] from the emitter, the three optical signals can be expressed as:

E_{1} = A_{1} \cos (ω_{0} t - φ_{1})

E_{2} = A_{2} \cos [(ω_{0} + Ω_{1}) t - φ_{2}]

E_{3} = A_{3} \cos [(ω_{0} - Ω_{1}) t - φ_{3}]

being

φ_{1} = \frac{ω_{0} L}{c} [n (ω_{0}) - 1]

φ_{2} = \frac{ω_{0} L}{c} [n (ω_{0} + Ω_{1}) - 1]

φ_{3} = \frac{ω_{0} L}{c} [n (ω_{0} - Ω_{1}) - 1]

where

n (ω)

is the refractive index of the medium at $ω$ and c is the speed of light in vacuum. When this three component signal impinges on a square law photodetector (with adequate electrical bandwidth) two beat notes are generated as a result:

I_{12} \propto \frac{A_{1}^{2}}{2} + \frac{A_{2}^{2}}{2} + A_{1} A_{2} \cos [Ω_{1} t - (φ_{2} - φ_{1})]

I_{13} \propto \frac{A_{1}^{2}}{2} + \frac{A_{3}^{2}}{2} + A_{1} A_{3} \cos [Ω_{1} t - (φ_{1} - φ_{3})]

Then, the phase of the resulting beat note (of frequency $Ω_{1}$ ) can be expressed as:

φ_{o} = \tan^{- 1} \frac{A_{2} \sin (φ_{2} - φ_{1}) + A_{3} \sin (φ_{1} - φ_{3})}{A_{2} \cos (φ_{2} - φ_{1}) + A_{3} \cos (φ_{1} - φ_{3})}

The expression for the phase of the beat note can be greatly simplified if absorption in the gaseous sample is considered negligible (low gas concentration). In this case, the amplitudes of the optical tones are equal, and hence, the amplitudes of two beat notes are also equal (A₂ = A₃) resulting in Eq. (14). As a reference, the low gas concentration approximation introduces an accuracy error of 0.85% in the experimental validation performed in this paper, nonetheless, it provides a simplified model that better illustrates the basics of the heterodyne phase-sensitive detection method.

φ_{o} = \frac{φ_{2} - φ_{3}}{2}

Replacing Eqs. (9) and (10) into Eq. (14), an Eq. relating the beat note phase and the refractive indexes at the two optical frequencies is obtained:

φ_{o} = \frac{ω_{0} L}{2 c} (n (ω_{0} + Ω_{1}) - n (ω_{0} - Ω_{1}))

Thus, if the optical signal is swept across the spectral transitions of a gaseous sample, the dispersion profile can be retrieved from the measurement of the phase $φ_{o}$ of the beat note. From $φ_{o}$ , and using Eq. (15), it is possible to directly obtain the refractive index profile of the medium as a function of the wavelength, and from this, the species concentration and other different parameters of the gas.

The ability to recover the dispersion spectrum is subject, nevertheless, to the proper selection of the modulation frequency $Ω_{1}$ . When the three tone signal is swept across a spectral line, the waveform and the amplitude of the detected phase signal $φ_{o}$ are a function of the modulation frequency $Ω_{1}$ , and the optimum modulation frequency (that maximizes the peak to peak phase output) can be obtained combining Eqs. (1) and (15). The relationship between modulation frequency $Ω_{1}$ and the peak to peak output phase $φ_{o}$ is shown in Fig. 3. It must be noted that the optimum modulation frequency is equal to the FWHM of the line multiplied by 0.58.

Fig. 3 Relationship between modulation frequency (normalized by the FWHM) and the maximum output phase. The modulation frequency that optimizes the measurement of dispersion is equal to the FWHM multiplied by 0.58.

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At the optimum modulation frequency, when the spectral feature is swept, $φ_{o}$ has the form shown in Fig. 4. If the modulation frequency is not at its optimum point, the peak to peak phase difference is reduced, and only for frequencies far from the optimum frequency the waveform is distorted.

Fig. 4 Measured phase at a function of the optical frequency for the optimum modulation frequency.

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To operate at optimum performance, the modulation frequency (and thus, the beat note frequency) must be in the MHz to GHz range for most gases depending on its pressure. Therefore, to obtain a compact system, heterodyning is necessary to down-shift the frequency of the beatnote. Two different heterodyne frequency conversion architectures have been used in this paper, the first architecture is based on radio-frequency heterodyning while in the second architecture most of the signal processing is done taking advantage of optical heterodyning.

2.2 Heterodyne phase sensitive detection for calibration-free molecular dispersion spectroscopy

As stated before, the basic scheme of the sensor presented in Fig. 2 requires an additional heterodyning stage for phase information recovery. The most obvious solution is the architecture based on radio-frequency down-conversion heterodyning shown in Fig. 5. The light of a tunable laser source is intensity modulated and detected by a high speed photodiode after traveling through the gaseous medium. The resultant beatnote must be, usually, amplified before being fed to the RF input of a mixer. A second signal generator is connected to the OL input of this mixer to obtain an intermediate frequency within the range of the lock-in amplifier that isolates the phase of the beat note. Another mixer is necessary to extract the frequency difference between the two signal generators that will be used as the reference by the lock-in amplifier.

Fig. 5 Block diagram of the dispersion spectroscopy sensor based on electrical frequency conversion.

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Alternatively, it is also possible to adopt an optical heterodyning approach similar to that of [25] to simplify the implementation requirements of the sensor and increase the robustness to noise and interference of the system. The block diagram of the phase-sensitive detection sensor based on optical frequency down conversion is shown in Fig. 6. The transmitter of this scheme is equal to the transmitter of the electrical frequency conversion architecture, a tunable laser source intensity modulated by an external modulator. However, in this approach, after travelling through the gas sample, the light is intensity modulated again by a second intensity modulator at a frequency $Ω_{2}$ (a few kHz lower than $Ω_{1}$ ). This second modulator generates two extra tones as a consequence of the modulation of the optical carrier:

E_{4} = A_{4} \cos [(ω_{0} + Ω_{2}) t - φ_{1}]

E_{5} = A_{5} \cos [(ω_{0} - Ω_{2}) t - φ_{1}]

and four extra tones of smaller amplitude as a consequence of the modulation of the two sidebands.

Fig. 6 Block diagram of the dispersion spectroscopy sensor based on optical frequency conversion.

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E_{6} = A_{6} \cos [(ω_{0} + Ω_{1} + Ω_{2}) t - φ_{2}]

E_{7} = A_{7} \cos [(ω_{0} + Ω_{1} - Ω_{2}) t - φ_{2}]

E_{8} = A_{8} \cos [(ω_{0} - Ω_{1} + Ω_{2}) t - φ_{3}]

E_{9} = A_{9} \cos [(ω_{0} - Ω_{1} - Ω_{2}) t - φ_{3}]

The spectrum of the nine tone signal is shown in Fig. 7. It is now possible to adjust the electrical bandwidth of the photodetector (higher than $Ω_{1}$ - $Ω_{2}$ , but much lower than $Ω_{1}$ and $Ω_{2}$ ), to capture only the beat notes between the sidebands E₂-E₄ and E₅-E₃ obtaining a phase signal equal to that of Eq. (15) but at a frequency $Ω_{1}$ - $Ω_{2}$ (that lies directly within the operation range of the lock-in amplifier). The beat notes between E₇-E₁ and E₁-E₈ (all of them of frequency equal to $Ω_{1}$ - $Ω_{2}$ ) will also contribute to the beat note, even though they do not affect the output phase. Besides this, it is necessary to add a correction factor k that accounts for unbalanced sidebands amplitudes due to mismatches between the transfer function characteristics of the optical intensity modulators. This results in a new expression for the output phase of the optical frequency conversion:

Fig. 7 Optical spectrum at the output of the second intensity modulator.

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φ_{o} = \frac{ω_{0} L}{2 c} (k * n (ω_{0} + Ω_{1}) - n (ω_{0} - Ω_{1}))

In our setup, k was found to be equal to 1.07. It is worthwhile to note that with this value of k, the optimum modulation frequency moves from 0.58 to 0.62 multiplied by the FWHM.

3. Experimental validation

The performance of the proposed dispersion spectroscopy architectures was experimentally demonstrated using as a target the molecular transition of methane at 1650.96 nm. To evaluate the calibration-free operation capabilities, the results are compared to simulations of the expected signals obtained using spectral data from HITRAN.

A custom build high-pressure (2150 Torr) reference gas cell of methane at a concentration of 7.5% and a path length of 70 mm was used in the tests. The tunable laser source selected was a VCSEL from VERTILAS (VERTILAS GmbH, Munich, Germany) with a tunable range of 1644.5 to 1651.5 nm. This VCSEL, the first optical intensity modulator (LN65S-FC, Thorlabs Inc., New Jersey, USA) and the gas cell were used in the tests of both architectures with equal configuration. Similarly, a SR-830 (Stanford Research Systems Inc., California, USA) lock-in amplifier was employed in both setups using a wideband balanced mixer (CMB40400609M, CERNEX Inc., California, USA) for the generation of the reference signal.

For the implementation of the electrical frequency conversion approach, a high speed XPDV2020 photodetector from u²t Photonics AG (Berlin, Germany), a 2-8 GHz wideband amplifier (ZRON-8G, Mini-Circuits Inc., New York, USA) and a ZMX-10G (Mini-Circuits Inc., New York, USA) mixer were employed. Measurements were carried out with a modulation frequency approximately equal to the FWHM of the spectral line multiplied by 0.58 ( $Ω_{1}$ = 7.5 GHz and $Ω_{2}$ = 7.4999 GHz, resulting in an intermediate frequency of 100 kHz). The measured and simulated dispersion spectra for this setup are shown in Fig. 8 where it is possible to appreciate the high correspondence between them. With an integration time of 1 s, a value of SNR = 19.3 dB was calculated by dividing the peak to peak phase amplitude by the standard deviation of the phase far from the absorption line. With this value of SNR, we can estimate a detection limit of 60 ppm*m/Hz^1/2. From the calibration-free operation point of view, the accuracy was calculated as the standard deviation of the difference between the measured and the simulated dispersion spectra, a figure of 4.5% was obtained.

Fig. 8 Dispersion spectrum obtained.

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On the other hand, in the implementation of the optical frequency conversion scheme, a second LN65S-FC optical intensity modulator was used in conjunction with a pigtailed G8195-12 InGaAs photodiode (Hamamatsu Photonics K. K., Iwai, Japan) directly connected to the lock-in amplifier current input. A modulation frequency approximately equal to 0.62 times the FWHM of the spectral feature ( $Ω_{1}$ = 8 GHz and $Ω_{2}$ = 7.9999 GHz, the intermediate frequency is 100 kHz) was used in the measurements. The dispersion spectrum obtained and the results of the simulations using HITRAN data are shown in Fig. 9 (it is possible to see the unbalance in the symmetry of the waveform produced by the effect of the correction factor of Eq. (22)). The SNR was calculated as before to obtain a value of 26.5 dB for an integration time of 1 s, this gives an estimated detection limit of 11.7 ppm*m/Hz^1/2, almost an order of magnitude better than for the electrical scheme. Finally, the calibration-free accuracy was calculated obtaining a value of 3.5%.

Fig. 9 Dispersion spectrum obtained using optical heterodyne frequency conversion.

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As a difference with the electrical frequency conversion scheme, in the optical conversion setup the only limitations in the bandwidth of modulation frequencies are (apart from the signal generators themselves) those related to the optical intensity modulators. The modulators used in this test have an effective bandwidth ranging from DC to 18 GHz. Therefore, with the optical conversion approach, it was possible to fully characterize the dependence of the peak to peak phase with the modulation frequency and the results are shown in Fig. 10. As expected, the behavior presented is in agreement with the results of previous simulations.

Fig. 10 Measurements of the maximum peak to peak output phase as a function of the modulation frequency.

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As stated in the introduction, one of the main advantages of the described dispersion spectroscopy technique presented in this paper, when compared with absorption based methods, is the immunity of the measurement to power fluctuations. This feature was tested by increasingly obstructing the optical path of the sensor and the results are shown in Fig. 11. It is possible to see that the output phase waveform is immune to power variations of 10 dB, and only a small degradation of the SNR is noticeable.

Fig. 11 Output phase signal for different power levels. The three signals are shifted vertically for viewing purposes.

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4. Conclusions

In this paper, a heterodyne phase-sensitive detection technique for molecular dispersion spectroscopy has been presented, tested and experimentally validated. The method is baseline free, immune to fluctuations of the received optical power and offers an output linearly dependent of the gas concentration that leads to a high dynamic range. Moreover, two different system designs have been proposed. The high correlation between the output signal for the electrical and optical frequency down conversion schemes and the results of the analytical model makes possible calibration-free operation with an accuracy of 4.5% and 3.5% respectively. Nevertheless, these figures could be further improved mainly with a better analytical model and, in this respect, there is already work in progress.

In comparison with other dispersion spectroscopy techniques, heterodyne phase-sensitive detection sensors are far simpler to implement than interferometric approaches without penalizing the detection limit. Unlike FMS and NICE-OHMS, the presented technique offers immunity to optical power fluctuation and easier operation, nevertheless, the sensitivity is far from the figures obtained by NICE-OHMS. By the range of application, the performance of the technique presented in this paper is directly comparable to CLaDS. The optical heterodyne phase sensitive detection setup implemented in this paper obtained a detection limit of 11.7 ppm*m/Hz^1/2 for methane in the 1.65 $μ m$ region. This figure is in the same order of magnitude that the 2.7 ppm*m/Hz^1/2 obtained by classical CLaDS sensors [27] and the 6.43 ppm*m/Hz^1/2 obtained by an optical heterodyne CLaDS system [25], both detecting methane in the same spectral region. Nevertheless, in general terms, the control and information extraction procedures using phase-sensitive detection are not nearly as complex as required by CLaDS and problems like the linearity and repeatability in the wavelength swept are simplified or eliminated.

Acknowledgments

The authors would like to thank Dr. Markus Ortsiefer from VERTILAS GmbH for providing the VCSEL used in this paper.

References and links

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Heterodyne phase-sensitive detection for calibration-free molecular dispersion spectroscopy

Abstract

1. Introduction

2. Molecular dispersion spectroscopy using phase-sensitive detection

2.1 Operating principles

2.2 Heterodyne phase sensitive detection for calibration-free molecular dispersion spectroscopy

3. Experimental validation

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (11)

Equations (22)

Optics Express