1. Introduction

Dispersive Fourier-transform spectroscopy (dFTS) is a powerful technique which allows an optical characterization of both, absorptive and dispersive properties of solid, liquid, and gaseous samples in a direct measurement [1–4]. A fundamental difference to conventional Fourier-transform infrared (FTIR) devices is the position of the sample: placing the specimen inside one arm of the Michelson-type interferometer accesses the phase delay caused by the sample. Ideally, the accuracy of the measured dispersion is limited by random phase noise, which is, among other factors, determined by detector noise.

Nonlinear interferometers based on spontaneous parametric down-conversion (SPDC) allow measuring the spectral information of undetected photons [5]. SPDC sources can be designed for emission of correlated mid-infrared (idler) and near-infrared (signal) photon pairs. In a nonlinear interferometer, the emission of two SPDC sources is superimposed. If the SPDC sources are indistinguishable, quantum interference of both, signal and idler light can be observed. Due to an effect called induced coherence without induced emission [6], the transmission and phase of all three beams (pump, signal and idler) determine the visibility and phase of the interference pattern of the signal photons [7]. Measuring the interference contrast of the near-infrared signal photons therefore allows to determine the transmission and phase shift for the mid-infrared idler photons between the two SPDC sources without any mid-infrared detection of the idler photons. Instead, silicon-based near-infrared detectors with lower noise-equivalent power than typical mid-infrared detectors can be used. This allows accurate mid-infrared measurements with extremely low light exposure. Metrology with nonlinear interferometers based on SPDC has been demonstrated for various applications, such as quantum imaging [5,8,9], optical coherence tomography [10] and spectroscopy [11–15].

Analogue to classical interferometry, measuring the interference modulation as a function of spatial delay between the two SPDC sources allows determining the spectral information by Fourier-transform analysis [12,14]. Such a quantum Fourier-transform spectrometer has been demonstrated for high-resolution transmission spectroscopy of a gaseous sample [13].

This approach requires no dispersive spectrometer for spectral analysis and allows sub-wavenumber spectral resolution.

A nonlinear interferometer can also access the dispersive properties of a sample. Paterova et al. [11] demonstrated measurements of refractive indices by directly evaluating the spatial shift of the interference pattern in the case of narrowband SPDC emission. Kaufmann et al. [15] demonstrated a phase scanning approach. Hereby, the spectral information was measured using an external spectrometer. The spatial delay of the nonlinear interferometer was varied over few wavelengths, and the resulting sinusoidal interference modulation of each spectral bin was fitted to determine the phase shift. A measurement at varying sample lengths then allowed to reconstruct the dispersion of carbon dioxide at a spectral resolution of 1.5 cm$^{-1}$. Using a Fourier-transform approach similar to this work, Mukai et al. [14] evaluated the refractive index of a non-absorbing sample around 1.5 µm wavelength at a spectral resolution of 25 cm$^{-1}$.

In this work we demonstrate a method for highly accurate measurement of the mid-infrared dispersion of a sample without mid-infrared but only near-infrared detection. The measurement and analysis procedure in analogy to classical Fourier-transform spectroscopy allows high spectral resolution. Here, we present the first (to the best of our knowledge) dispersion spectra measured with undetected photons capable of resolving rotational lines. The simple measurement approach requires no additional spectrometer – a single pixel detector is sufficient – and is proven to be robust against other sources of dispersion within the nonlinear interferometer.

2. Realization

A schematic setup of the quantum Fourier-transform spectrometer is shown in Fig. 1. The measurement concept has also been detailed in Ref. [13]. The pump light source is a Ti:sapphire laser with an output power of 700 mW, which is stabilized to a wavelength of 785 nm with an accuracy of 0.06 pm. The pump laser beam passes an optical isolator and is focused to a beam diameter of 70 µm at the center of the nonlinear crystal. The pump beam is coupled into the nonlinear interferometer with a dichroic mirror (DM$_\text {s}$) and passes through the nonlinear crystal, which is a 10-mm-long periodically poled lithium niobate (PPLN) crystal.

 

Fig. 1. Schematic view of the nonlinear interferometer for Fourier-transform spectroscopy. The pump laser beam (green) is reflected on a dichroic mirror (DM$_\text {s}$) and passes through the nonlinear crystal (PPLN) causing SPDC emission of signal (orange) and idler (red) light. The beams are separated using another dichroic mirror (DM$_\text {i}$), collimated and back-reflected by plane mirrors (M$_\text {s}$ and M$_\text {i}$). The superimposed signal light is transmitted by DM$_\text {s}$, passes an optical filter and is focused onto the detector PD$_\text {sig}$. The residual transmission of the pump beam through DM$_\text {i}$ (dashed green line) is used for an accurate position reference of the idler mirror and detected by PD$_\text {ref}$. Collimation and focusing lenses are omitted for clarity.

Download Full Size | PDF

For broadband collinear emission, the phase matching condition is designed for signal and idler wavelengths with matching group indices, as described in Ref. [16]. In this experiment, signal light with a wavelength around 1 µm and idler light around 3.6 µm is used. Quasi-phase-matching is achieved with a poling period of 21.5 µm and a crystal temperature of 65 $^{\circ }$C, which is stabilized using a Peltier element. As detailed in Ref. [13], the idler light covers a spectral range of about 2500–3200 cm$^{-1}$. The SPDC source emits signal light with a total power of about 10 nW (measured using a photodiode power sensor), from which an effective mid-infrared power on the sample of about 6 nW can be estimated. The correlated signal and idler photons generated by SPDC are then separated by the dichroic mirror DM$_\text {i}$.

The mid-infrared idler beam is transmitted by DM$_\text {i}$ and collimated with a CaF$_2$ lens (omitted in Fig. 1) with a focal length $f=$ 100 mm. The idler light passes a sample cell, which can be filled with pure nitrogen for reference measurements or with an analyte for sample measurements. The cell consists of a small cylinder with antireflective coated BaF$_2$ windows and an interaction length of 20 mm. Behind the sample cell, the idler light is reflected on a plane gold mirror (M$_\text {i}$). The idler light is imaged back into the center of the nonlinear crystal in an effective 4$f$ relay optic. The mirror M$_\text {i}$ is mounted on a voice-coil translation stage, which allows a maximum displacement of $\pm$10 mm. The signal and pump beam, which are reflected on the dichroic mirror DM$_\text {i}$ take an analogue path in the second interferometer arm, which has a fixed 4$f$ optical path length.

The back-reflected pump beam causes a second SPDC process. Due to the low emission probability of SPDC, which is on the order of 1$\cdot$10$^{-8}$, induced emission can be neglected (cf. Reference [5,6]). By careful alignment, emission of the two SPDC processes (forward and backward direction) becomes indistinguishable, and both, signal and idler light is modulated by interference. Behind the nonlinear crystal, the pump beam is again reflected by DM$_\text {s}$ and removed by the optical isolator. The idler light is discarded, and only the superimposed signal light (of the first and second SPDC process) is transmitted by the dichroic mirror. The signal light is collimated, passes an optical longpass filter, and is focused onto the detector PD$_\text {sig}$. As a detector, a single-pixel silicon avalanche photodiode is used. The detector signal passes an electronic band pass filter and is digitized.

For accurate Fourier-transform spectroscopy, the intensity modulation of the signal light needs to be sampled at precise equidistant positions of the moving mirror M$_\text {i}$. Instead of an additional reference laser, which is often used in classical FTIR devices, in this set-up, we can take advantage of the residual transmission (<1 %) of the pump beam through the dichroic mirror DM$_\text {i}$, which is shown as a dashed green line in Fig. 1. The dichroic mirror DM$_\text {i}$ and the end mirrors M$_\text {s}$ and M$_\text {i}$ effectively form a classical Michelson-type interferometer for the pump beam with high amplitude contrast between the interferometer arms. The travel distance of the mirror M$_\text {i}$ between the zero-crossings of the pump interferogram corresponds to the known pump wavelength. The weak pump interference pattern is detected with a photodiode PD$_\text {ref}$, the electronic signal passes a high-pass filter and is digitized.

An interferogram is measured by recording the signal intensity as the idler mirror is moved with constant velocity. In an acquisition time of 9 s, the idler mirror is moved by up to 9 mm in either direction, measuring a double sided interferogram. This results in a maximum optical path difference of 18 mm, which corresponds to a spectral resolution of about 0.56 cm$^{-1}$. The time-referenced interferogram is transformed into a position-referenced interferogram by using the zero-crossings of the pump interference. Several measurement scans can be added coherently to increase the signal to noise ratio. In the following, an interferogram measured with 10 averaged measurement scans will be analyzed, which corresponds to a total measurement time of 90 s.

As a first step, a reference measurement is taken, wherein the sample cell is filled with pure nitrogen. Then, the measurement procedure is repeated for a sample measurement. A dilution of 1 % methane in nitrogen is filled into the sample cell at atmospheric pressure. We can then use the complex Fourier-transform amplitudes to determine the phase spectra and the sample dispersion in the region of the $\nu _3$ absorption band centered at 3020 cm$^{-1}$.

3. Results

3.1 Phase spectra

A Fourier-transform analysis is used to retrieve the spectral and phase information. The complex Fourier-transform amplitude $S(\tilde {\nu })$ for a wave number $\tilde {\nu }$ from an interferogram $I(\delta )$ can be written as [17]:

We use a discrete Fourier-transform based on the Fast Fourier-Transform algorithm. The magnitude spectra of the reference (black) and sample (red) spectra in a spectral range of 2870–3170 cm$^{-1}$ are shown in Fig. 2(a). The envelopes of both spectra are in good agreement, methane absorption lines are visible in the sample spectrum. An exemplaric evaluation of magnitude spectra for quantum Fourier-transform spectroscopy is detailed in Ref. [13]. The phase of the complex Fourier-transform amplitude can be calculated by [17]:

 

Fig. 2. Evaluation of the complex Fourier-transform amplitude for a reference and sample (1% methane) measurement: a) Normalized power spectral density (PSD) of the magnitude spectra of sample (red) and reference (black) measurement. b) Phase of the complex Fourier-transform amplitudes as calculated using Eq. (2). c) Unwrapped phase spectra. d) Phase difference (red) between sample and reference measurement. A fitted linear baseline is shown in gray.

Download Full Size | PDF

Equation 2 only returns the principal values within $-\pi /2<\phi <\pi /2$. Therefore the phase spectra show periodic jumps by $\pi$. The phase spectra can be unwrapped using an algorithm which minimizes the difference between consecutive points by adding $\pm \pi$. The unwrapped phase spectra are shown in Fig. 2(c). Both phase spectra show a rapidly varying phase over the broad spectral range. This phase progression is due to dispersion inside the nonlinear interferometer, mostly caused by the nonlinear crystal itself. As we evaluate the phase difference between sample and reference measurement, the dispersive properties of the nonlinear interferometer have no influence on the dispersion analysis. The phase difference is shown in Fig. 2(d) as a red curve. Next to pronounced features at frequencies with absorption lines, the phase difference shows a linear baseline with small slope, a linear fit is shown in gray.

For an evaluation of the phase difference, we can use the Fourier-shift theorem: If an interferogram $I(\delta )$ is shifted by a constant or frequency-dependent spatial delay $\epsilon$, the complex Fourier-transform amplitude $S(\tilde {\nu })$ will be modified by [3]:

The spectrum therefore receives a phase shift of

Note that the sign of the phase shift is given by the sign of the Fourier-transform (Eq. (1)), which can differ for other conventions. A constant shift in spatial domain therefore corresponds to a linear slope of the phase spectrum. We can attribute the linear baseline of the phase difference in Fig. 2(d) (gray line) to a constant offset in spatial delay between sample and reference measurement. This offset corresponds to a shift of 0.57(3) µm in optical path difference between the sample and reference interferogram. The small shift is most likely caused by a limited reproducibility of the starting position of the measurement, due to the limited position resolution of the voice coil stage and timing errors in the electronic data acquisition. For further analysis, the fitted linear baseline will be subtracted from the measured phase difference.

3.2 Dispersion analysis

For determining the dispersion of the sample, the formalism as described in classical dispersive Fourier-transform spectroscopy can be used. As can be easily seen from Eq. (4), for a sample cell with an interaction length $d=$ 2 cm, which is passed twice, the dispersion can be calculated by [1,3]:

The dispersion of methane as calculated from the measured phase difference is shown in Fig. 3(a) as red dots. Panel b) provides a detailed view on single absorption features, which shows the high spectral resolution of the measured dispersion.

 

Fig. 3. Dispersion of methane: Measured values (red dots) are evaluated from the phase difference of sample and reference measurement (Fig. 2(d)) using Eq. (5). Theory values (black curve) are calculated from the Kramers-Kronig dispersion relation (Eq. (7)) based on absorption data from the HITRAN-database [18].

Download Full Size | PDF

For a quantitative comparison, we use the Kramers-Kronig dispersion relation to calculate the dispersion from well-characterized absorption data. As reference data we use the absorption coefficient as provided by the HITRAN-database [18] (atmospheric pressure and room temperature) and calculate the extinction coefficient $\kappa$ in a spectral range of 2800 cm$^{-1}$ to 3300 cm$^{-1}$. To account for the finite spectral resolution of our measurement, the extinction coefficient is convolved with an instrument line function (cf. Reference [17]):

The measured data and the theory values are in good agreement. A minor offset and baseline error is visible over the broad spectral range. These are likely caused by small drifts of the pump wavelength below the accuracy of the frequency stabilization of the pump laser and other phase drifts between the measurements.

Figure 3(c) highlights the accuracy of the measured dispersion. The oscillations visible in the theory curve are caused by the shape of the instrument line function (Eq. (6)). The standard deviation of the measured dispersion in a spectral range without absorption features (2800–2850 cm$^{-1}$) can be determined to $\sigma _{\Delta n}=0.23(2)\cdot 10^{-6}$. This is in agreement with the deviations from the theory curve shown in Fig. 3(a)–3(c). We would like to note that some deviations between measured dispersion and theory values may also arise from the uncertainty of the reference data (absorption line intensity) and the inherent numerical difficulty of calculations using the Kramers-Kronig relations.

4. Discussion

We have evaluated the phase spectra of a Fourier-transform measurement with a nonlinear interferometer based on spontaneous parametric down-conversion. The set-up is capable of measuring the dispersion of a gaseous sample with high spectral resolution and accuracy, which is comparable to measurements demonstrated with classical FTIR devices [1,2,20]. In contrast to classical devices, the quantum Fourier-transform spectrometer allows measuring the mid-infrared information with a detection of near-infrared correlated photons. This allows using silicon-based detectors, which typically have lower noise and require no cooling. This enables accurate measurements with extremely low mid-infrared power < 10 nW, which is several orders of magnitude lower than the light exposure in typical FTIR devices. A low light dose might be advantageous for the characterization of light-sensitive samples in biology or life science. Instead of detector noise, photon shot noise becomes the dominant noise factor (cf. Ref. [13]). The measured dispersion spectrum is limited by random noise, which allows increasing the sensitivity by investing more measurement time (or a higher number of averaged scans) or reducing the spectral resolution.

Previous implementations of nonlinear interferometers typically use a dispersive spectrometer for determining the spectral information [11,15]. The Fourier-transform approach allows measuring with high spectral resolution, which is only limited by the maximum optical path difference, without using an external spectrometer. The analysis presented in this work is robust against other sources of dispersion inside the nonlinear interferometer, such as the nonlinear crystal. In addition, the setup is capable of measuring the dispersion of samples with large optical path lengths (e.g. highly refractive but sufficiently transparent materials), as each interferogram is measured over a spatial delay of 18 mm. This is highly relevant for an optical characterization of absorption and dispersion of solid samples such as polymers.

The measurement approach can be easily adapted to the full infrared transparency range of lithium niobate using a different pump wavelength and poling period. In future implementations, dispersive Fourier-transform spectroscopy with undetected photons may also be extended to the fingerprint infrared range (cf. Ref. [21]), using other nonlinear materials.

Analyzing the dispersive properties of a sample requires no modification of the simple measurement procedure of a Fourier-transform nonlinear interferometer, as presented in Ref. [13]. The evaluation of the phase spectrum, and its modification by the spectroscopic sample completes the information of sample interaction. In addition, the measured dispersion may be used for optical inversion using the Kramers-Kronig dispersion relation [19], obtaining additional information on the absorptive properties of the sample.

5. Conclusion

We have demonstrated accurate and high-resolution dispersive Fourier-transform spectroscopy based on the interference effects of correlated photon pairs. In a nonlinear interferometer, the information of undetected, mid-infrared photons can be measured by detecting the interference modulation of the correlated near-infrared photons. A Fourier-transform analysis of the measured interferogram allows obtaining accurate information on both, absorptive and dispersive properties of a spectroscopic sample. In future implementations, this may allow differentiating between absorption, scattering and reflection losses. We believe that the presented approach is a valuable addition to the methods for optical characterization in quantum Fourier-transform spectrometers.