A nonlinear interferometer for Fourier-transform mid-infrared gas spectroscopy using near-infrared detection

Nonlinear interferometers allow for mid-infrared spectroscopy with near-infrared detection using correlated photons. Previous implementations have demonstrated a spectral resolution limited by spectrally selective detection. In our work, we demonstrate mid-infrared transmission spectroscopy in a nonlinear interferometer using single-pixel near-infrared detection and Fourier-transform analysis. A sub-wavenumber spectral resolution allows for rotational-line-resolving spectroscopy of gaseous samples in a spectral bandwidth of over 700$\,$cm$^{-1}$. We use methane transmission spectra around 3.3$\,\mu$m wavelength to characterize the spectral resolution, noise limitations and transmission accuracy of our device. The combination of nonlinear interferometry and Fourier-transform analysis paves the way towards performant and efficient mid-infrared spectroscopy with near-infrared detection.


Introduction
Nonlinear interferometers offer new perspectives in metrology, relying on the quantum effects observed in the interference phenomena of correlated photons. A typical source of correlated photons is spontaneous parametric down-conversion (SPDC) inside a nonlinear-optical crystal. This effect can be described as the spontaneous decay of pump photons inside a nonlinear medium into two photons, called signal and idler. The sum of the energies of signal and idler photons equals the energy of the pump photon. If the pump, signal and idler photons of one SPDC source pass through a second, identical nonlinear crystal, interference can be observed between the two indistinguishable correlated photon sources for both signal and idler photons. Due to induced coherence [1], the visibility and phase of the interference intensity pattern of the signal photons depend on the transmission and phase of all three beams: pump, signal and idler [2][3][4][5]. Recently, nonlinear interferometers using a large spectral separation between signal and idler photons have sparked interest, being able to separate sample interaction and light detection into different spectral regions. Nonlinear interferometers have been demonstrated for applications such as imaging [6], sensing [7], microscopy [8,9], ellipsometry [10] and optical coherence tomography [11][12][13]. Interferometers using pairs of correlated mid-infrared and near-infrared/visible photons offer interesting measurement concepts for spectroscopy [14][15][16][17]. The use of silicon-based visible or near-infrared detectors promises lower dark noise and higher bandwidth in comparison to infrared detectors, which often require cooling.
One key element of spectroscopic nonlinear interferometers is the retrieval of the infrared spectral information via detection of the visible or near-infrared signal interference. Using a monochromator for spectral selective detection of the visible signal light, a spectral resolution of 5.2 cm −1 was demonstrated for idler wavlengths around 2200 nm [16]. In a recent publication, it was demonstrated that the infrared spectral information can also be obtained from a Fouriertransform of the detected signal interference pattern with a comparable spectral resolution of 6 cm −1 [17]. To this purpose, a series of signal interference patterns was recorded using a silicon-based camera while varying the delay between the signal and idler interferometer arm. Hereby, a bandwidth of 100 cm −1 was realized by using non-collinear SPDC emission in periodically-poled lithium niobate (PPLN). This approach required spatially resolved detection of the signal interference pattern with a camera and pixel-wise analysis due to a spatially varying phase and optical path difference for the non-collinear components. In this work, we overcome these limitations by using an SPDC source designed for broadband collinear emission (cf. [8,18]). This allows using a single-pixel (near-infrared) silicon-based detector to accurately analyze the mid-infrared transmission of a sample using a simple Fouriertransform analysis. In consequence, the possible spectral resolution is only limited by the maximum interferometer delay, opening the door for sub-wavenumber resolution, which enables spectroscopic analysis of gaseous samples resolving individual rotational absorption lines. In contrast to previous works, a spectral range of 3.1 µm to 4.0 µm wavelength (725 cm −1 ) is covered in a single measurement with a spectral resolution of up to 0.56 cm −1 . We demonstrate this approach with transmission spectra of a dilution of methane (CH 4 ) in nitrogen (N 2 ), which are used for a performance characterization of the presented setup. The spectroscopic accuracy is evaluated in a quantitative comparison to a simulation based on the HITRAN spectroscopic database.

Experimental realization
As a source of correlated photons we use SPDC in a 10-mm-long PPLN crystal with 5-mol%-MgO doping. In order to achieve broadband collinear signal and idler emission, we design the pump wavelength and poling period as described in reference [18]. We use a pump wavelength of 785 nm and a poling period of 21.5 µm for a broadband phase-matched idler emission range of 3.1 µm to 4.0 µm wavelength with correlated signal emission below 1.1 µm wavelength, so that silicon-based detectors with low dark noise can be used. A schematic setup of the nonlinear interferometer is shown in Fig. 1. As a pump source, we use a tunable continuous-wave Ti:sapphire laser with an output power of 700 mW, which is frequency stabilized using an external wavelength meter. The pump laser beam passes an optical isolator and is focused with lens L p to a beam diameter of about 70 µm at the center of the nonlinear crystal (PPLN). The pump beam is reflected by the dichroic mirror DM s and illuminates the nonlinear crystal. The crystal is stabilized to a temperature of 65°C using a Peltier element. The signal and idler photons created by SPDC are then separated by the dichroic mirror DM i . The mid-infrared idler beam is transmitted by DM i and then collimated with a CaF 2 lens (L i ) with a focal length of 100 mm. The plane gold mirror M i placed in 100 mm distance to L i reflects the idler light, which is imaged back into the center of the nonlinear crystal. The mirror M i is mounted on a voice-coil translation stage, which allows for a maximum displacement of ±10 mm in beam direction. The interferometer is adjusted so that the point of zero delay (ZDP) between the interferometer arms is at zero displacement of M i . The signal and pump beams, which are reflected by the dichroic mirror DM i , take an analogue path in the second interferometer arm, which has a fixed optical length. The signal and pump light is collimated with lens L s with 100 mm focal length and reflected by the plane dielectric mirror M s , which images the signal light back into the nonlinear crystal. The back-reflected pump laser causes a second SPDC process, leading to interference of the signal and idler photons generated by either pass through the crystal. Using a silicon-based photodiode power sensor, a total signal SPDC power of (20.0 ± 1.4) nW (double pass, same setting as for the spectroscopic measurements) was measured. It is to be noted that blocking the idler beam inside the nonlinear interferometer has no measurable effect on the total signal power of the second SPDC process (cf. [1,6]), which verifies that the two SPDC processes are indeed spontaneous and without any stimulated emission.  After the second pass, the pump beam is again reflected by DM s and removed by the optical isolator. The idler light is absorbed by the dichroic mirror. The overlapping signal light is transmitted by the dichroic mirror and collimated by lens L s2 with 100 mm focal length. Residual pump and ambient light is removed by an optical 850 nm long pass filter. The lens L det focuses the signal light onto the active area of the detector PD signal . As a detector we use a silicon avalanche photodiode (APD) with a bandwidth of 1 MHz and a noise equivalent power of 6 fW/ √ Hz (specified at 905 nm detection wavelength). At a gain factor of = 100, the responsivity is specified to 45 A/W at 1 µm wavelength, which corresponds to a quantum efficiency of 55.8 %. For noise reduction, the detector signal passes an electronic filter with 0.4 kHz to 5 kHz pass band. The signal is then digitized with an analog-digital converter. A maximum interference contrast of 15 % is measured as the peak AC/DC ratio of the detector signal without the electronic bandpass filter. According to simulations the achievable contrast is limited to about 20 % by dispersion inside the nonlinear crystal; additional losses are expected due to the non-ideal reflective and anti-reflective coatings of the optical elements and the imperfect beam overlap of the two SPDC processes. The sample placed in the idler beam is a gas cell filled with an analyte or pure nitrogen as reference. The cell consists of a small cylinder with 15 mm free aperture and about 20 mm of interaction length. For obtaining the spectrum of the interferometer with a Fourier-transform, the interferogram has to be sampled at equidistant positions of M i . Setups of classical Fourier-transform infrared (FTIR) spectrometers often use an additional HeNe-laser as reference, since the sinusodial interferogram of a near-monochromatic source has its zero-crossings at intervals which correspond to a retardation of half its wavelength [19]. In our setup, we take advantage of the residual transmission of the pump beam through the dichroic mirror DM i , which is shown as a dashed line in Fig. 1 Fig. 1(b) shows a zoom into recorded measurement signals, referenced to their acquisition time stamp. The nonlinear interference detector signal (orange curve) is evaluated at every second zero crossing (gray dots) of the reference signal (green curve) to obtain the required equally-spaced data points as input for the Fourier-transformation. Figure 2 gives an overview over the steps taken in the measurement and analysis procedure, which will be described in the following.

Measurement and analysis procedure
Measurements and interferogram analysis. As a first step, a number of interferograms are recorded with a nitrogen-filled sample cell (reference measurement). In an acquisition time of 9 s, the idler mirror is moved with a constant velocity of 2 mm/s. The available displacement range of the double-sided interferogram results to 18 mm, corresponding to a theoretical maximum resolution of 0.56 cm −1 . Each time-referenced interferogram is then transformed into a positionreferenced interferogram by using the zero-crossings of the pump interference. In order to increase the signal-to-noise ratio (SNR), several interferograms are added coherently. Since the position reference of each interferogram yields only a relative position information and the absolute position can be shifted by an unknown offset, the correlation of each interferogram is calculated. The interferograms are then shifted by the displacement offset value which maximizes the correlation and added. A normalized section of the resulting interferogram is shown in Fig. 3 as the light blue curve.
While a classical FTIR interferogram shows a clear and symmetric centerburst, the interferogram  Fig.2(b)), a reconstructed interferogram (dark blue) can be calculated by Fourier-transforming the power spectral density of the reference spectrum (Fig. 4).
presented here is broadened and has an asymmetric shape. This is caused by the dispersion of the nonlinear crystal which is inherent to the system and results in a wavelength(-pair) dependent optical path difference between signal and idler beam throughout the large emission bandwidth. For a 10-mm-long MgO-doped lithium niobate crystal, the optical path difference between signal and idler beam (both extraordinary polarized) can be calculated to 0.64 mm at 3200 cm −1 idler frequency (about 3.1 µm wavelength) and 1.03 mm at 2500 cm −1 idler frequency (4 µm wavelength) using refractive index equations from Ref. [20]. The offset of the points with zero path difference between signal and idler results to 390 µm, which is in agreement to the observed width of the interferogram. A chirped interferogram yields the same spectral information as an interferogram without dispersion, which has also been demonstrated for classical FTIR spectroscopy [19,21].
Fourier transform and non-apodized transmission spectrum. In the next step, the positionreferenced and averaged interferogram is Fourier-transformed. We use a discrete Fourier-transform (DFT) based on the Fast Fourier-Transform (FFT) algorithm to obtain the spectrum in spatial frequency space. The power spectral density (PSD) is calculated as the modulus of the complex amplitude resulting from the Fourier-transform. The normalized power spectral density of the reference spectrum (calculated from the interferogram shown in Fig. 3) is shown in Fig. 4 (blue curve). The spectrum exhibits a large bandwidth of over 725 cm −1 , centered around 2850 cm −1 .
The measurement and analysis procedure is then repeated with the sample cell filled with the gas sample. For a demonstration of a realistic measurement task using high spectral resolution, we chose a sample of 1 % methane in nitrogen at atmospheric pressure. The normalized power spectral density of the sample spectrum, measured with 100 scans, is depicted in Fig. 4 as the orange curve. The spectrum shows the same envelope and spectral width as the reference spectrum. In the spectral range around 3000 cm −1 , absorption lines are clearly visible.
The transmission of the sample can be calculated from the quotient of the power spectral densities of the reference and sample spectrum. Since the corresponding interferograms were not multiplied with any apodization function, they are truncated by a boxcar (rectangular) window function given by the measurement range. Figure 5 shows the non-apodized transmission spectrum (green curve) containing rotational absorption lines of the 3 band of methane calculated from the spectra shown in Fig. 4.

Apodization.
If the spectral resolution is lower than the width of the measured spectral features, spectra using a simple rectangular window function show pronounced sidelobes. In practical FTIR spectroscopy, these are often removed using apodization (multiplying the interferogram with a function which is unity at zero path difference and decreases with increasing delay), at the cost of a lower spectral resolution [19]. Direct apodization of a chirped interferogram poses difficulties, as the apodization function would need to be a function of both position and frequency [19,21]. Instead, we use a different procedure for calculating the apodized transmission spectrum, which is shown schematically in Fig. 2(b). First, we reconstruct the interferograms without chirp from the spectra shown in Fig. 4. By calculating the power spectral density (or modulus) of the complex amplitude of the spectrum, the phase information is erased. Therefore, a Fourier-transform of the spectrum (into the spatial domain) contains the same spectral information without any phase shift, in essence removing the chirp (cf. [21]). The resulting reconstructed normalized interferogram of the reference measurement is shown in Fig. 3 (dark blue curve). The width of the interferogram is only determined by the large bandwidth of the SPDC source. As a next step, the interferograms are then multiplied with an apodization function. For this demonstration we chose a Gaussian apodization function described by for the mirror position , the total interferogram length = 18 mm and parameter = 0.2. The apodized interferograms are then Fourier-transformed. Figure 5 shows the transmission calculated from the sample and reference spectrum using apodization (red curve), the right part of the graphic provides a detailed view on a single absorption feature. In comparison to the transmission spectrum without apodization (green curve), the spectrum shows no additional sidelobes and a smoother shape. Due to the decreased spectral resolution, the apodized transmission spectrum shows weaker absorption lines and less noise.

System Characterization
In the following, we use the methane transmission spectra for a characterization of the performance and limitations of the spectrometer. Since the signal-to-noise ratio and spectral resolution depend on the instrument function of the spectra (determined by the window function applied to the interferograms), we use the transmission spectrum without apodization for the analysis of these parameters.

Signal-to-noise ratio
An important characteristic of the nonlinear spectrometer is the signal-to-noise ratio (SNR), which determines the minimum transmission change that can be distinguished from random fluctuations. The signal-to-noise ratio can be quantified from the measured spectra in a spectral range without sample absorption. In the following, we limit the SNR-calculation to a spectral range of 2800 cm −1 to 2850 cm −1 , which is highlighted in Fig. 5 (I). In this range, the measured transmission values (green curve) are normally distributed around their expectation value 0 ≈ 1. The width and mean of the normal distribution allow a calculation of the SNR: For the non-apodized transmission spectrum shown in Fig. 5 (green), the SNR results to 180. This corresponds to a transmission change of about 1.6 % (3 width) that can be distinguished from noise. For light sources with low output power, shot noise due to random fluctuations of the photon number can become a fundamental limitation. In the following, we will derive an estimation for the SNR using parameters of the current setup, assuming shot noise as the only source of noise. Since the Poisson-distributed number of Photons has a standard deviation of √ , a noise-equivalent power of N can be defined as using the detector quantum efficiency , the SPDC power , the median energy of a single photon ℎ and the measurement time . The excess noise factor √ describes the increase in shot noise due to the avalanche effect of the photo detector [22,23]. For a silicon APD, the factor is typically specified with ≈ 5 [24]. The signal power available for a Fourier-transform measurement with a spectral resolution of Δc an be described by (cf. Ref. [19]) =˜Δ˜, with the interferometer efficiency and the spectral power density˜. The spectral power density of the SPDC source is approximated by˜=˜m ax −˜m in (5) with the SPDC bandwidth˜m ax −˜m in , which assumes a uniform distribution. Using these approximations, we can estimate a limitation to the SNR for averaged measurements: Estimating the efficiency by the measured maximum interference contrast of ≈ 0.15, for 100 averaged scans with a measurement time of 9 s per scan, a detector efficiency of = 55.8 %, a total SPDC power of 20 nW, a single photon energy of 1.24 eV (signal wavelength approximated to 1 µm), a spectral resolution of 0.56 cm −1 and a bandwidth of 725 cm −1 , the resulting SNR-limit amounts to SNR th ≈ 366. Taking into account the uncertainties in the above derivation, shot noise must be considered an essential contribution to noise in the current setup.

Spectral resolution
For a quantitative characterization of the spectral resolution, we compare the measured transmission spectrum to the expected transmission spectrum m calculated as a convolution of spectroscopic data th and the instrument function : Spectroscopic data on methane th (˜) is obtained from the HITRAN database [25,26], using a Voigt line profile and ambient conditions as parameters (room temperature of 24°C, 1 atm pressure). Under these conditions, the widths of individual absorption lines is at the order of 0.12 cm −1 (full width at half maximum, FWHM). As described before, we will analyze the transmission spectrum which was calculated using no apodization, which yields the highest spectral resolution. The instrument function (˜), which is defined as the Fourier-transform of the rectangular window function, can then be described by [19] with the spectral resolution Δ˜, which is limited to the inverse of the maximum interferometer delay 1/ ≈ 0.56 cm −1 .
In order to determine the spectral resolution of the measured transmission spectrum, the model function (Eq. 7) is fitted to the measurement data using a non-linear least square optimization. The optimization yields a spectral resolution of Δ˜= 0.56 cm −1 , which is in agreement to the theoretic value. This shows that the spectral resolution of the nonlinear interferometer is only limited by the maximum delay between the interferometer arms. curve) for the spectral ranges II and III marked in Fig. 5. Clearly visible are the sidelobes around the absorption feature in Fig. 6(b) due to the shape of the instrument function. A systematic influence of methane absorption features on the difference between measured transmission and fitted model function is still observed, which we attribute to small deviations in the shape of the instrument function.

Accuracy
In order to determine the transmission accuracy of the nonlinear interferometer, we use the apodized transmission spectrum (shown as red curves in Figs. 5,7), for which the amplitude of sidelobes is greatly reduced. For a quantitative evaluation, analogue to the procedure described above, we calculate the expected spectrum from a convolution (Eq. 7) of spectroscopic data [25,26] and a modified instrument function . The instrument function of the apodized interferogram is the Fourier-transform of the apodization function (Eq. 1), which can be approximated by a Gaussian function: with an expected width of f = 1/(2 ) ≈ 0.44 cm −1 . The model function is fitted to the measured data using the width of the gaussian instrument function as free parameter. The optimization yields f = 0.43 cm −1 , which is in good agreement to the expected value. The resolution of the spectrum (full width at half maximum of the gaussian instrument function) results to 2 √︁ 2 ln(2) f ≈ 1 cm −1 . Analogue to the procedure described in section 4.1, we can determine the signal-to-noise ratio from the transmission spectrum in a spectral range of 2800 cm −1 to 2850 cm −1 . The SNR of the apodized transmission spectrum results to 390, which corresponds to a transmission change of 0.8 % (3 width) which can be distinguished from noise. The apodized transmission spectrum (red curve) and the model function using the optimized parameter (black curve) are shown in Fig. 7. The transmission residuum (difference between simulation and measured values) is shown below the transmission spectrum. Over a broad spectral range, the residuum shows small deviations distributed around the baseline. For few absorption lines in the P-and Q-branch, systematic deviations between measured data and model function can be observed. The standard deviation of the residuum amounts to 0.34 %, from which we can estimate the accuracy of the transmission values to 1 % (3 width). The transmission spectrum is slightly less accurate than expected from noise contributions alone, due to few outliers. The overall good accuracy, in combination with its high spectral resolution and large spectral bandwidth, demonstrates the applicability of the nonlinear interferometer to quantitative spectroscopic tasks.

Conclusion
We have demonstrated mid-infrared transmission spectroscopy of a gaseous sample using nearinfrared single-pixel detection in a nonlinear interferometer. Our approach combines the data acquisition and analysis in analogy to classical FTIR-spectroscopy with a quantum effect: The phase and transmission dependence of the two-photon interference allows measuring mid-infrared information with near-infrared detection. For a performance characterization of the nonlinear interferometer using Fourier-transform analysis, the measured methane transmission spectrum is compared to a model based on HITRAN data. A broadband spectral coverage of 3.1 µm to 4.0 µm wavelength is demonstrated in a single measurement setting. The spectral resolution of the nonlinear interferometer is only limited by the available maximum delay between the interferometer arms. While the demonstrated sub-wavenumber resolution allows for practical gas analysis in its current state, the resolution can be further improved by using larger mirror displacement. The transmission spectrum using a Gaussian apodization function shows good agreement to the expected spectrum with a transmission residuum below 1 % (3 width), which demonstrates the photometric accuracy over a broadband spectral range.
Considering the signal-to-noise ratio of the transmission spectrum, photon shot noise is demonstrated to be a significant noise contribution. Future implementations of nonlinear interferometers may greatly benefit from more efficient sources for correlated photons, which could be realized using other nonlinear materials or waveguides [27]. Using a high-gain nonlinear interferometer (cf. [13]) or resonant pump enhancement may also yield increased measurement signal. On the other hand the infrared power used to illuminate the sample in the presented setup is six orders of magnitude weaker compared to a classical FTIR spectrometer with a black body radiator in the 3 µm to 4 µm wavelength range. This demonstrates the high efficiency of the nonlinear interferometer and can be advantageous for the spectroscopy of sensitive samples. The presented setup can be realized cost-efficiently: using no additional spectrometer or camera (in contrast to previous works on nonlinear interferometers), and no cooled mid-infrared detector (in contrast to classical infrared spectroscopy methods). The pump wavelength can also be addressed with cost-efficient single-frequency diode lasers reaching output powers of several hundreds of milliwatts. The measurement approach can be extended to other spectral ranges, such as the terahertz regime [7,28], or into the fingerprint infrared range from 7 µm to 12 µm wavelength using other nonlinear materials.
In comparison to other implementations of nonlinear interferometers for mid-infrared spectroscopy [14][15][16][17], a much higher spectral resolution and a larger bandwidth is demonstrated with residuals small enough for quantitative analysis. The presented measurement approach enables the novel measurement principle of nonlinear interferometers, allowing mid-infrared measurement via near-infrared detection, to be applied to spectroscopic tasks efficiently and accurately.

Disclosures
The authors declare no conflicts of interest.