1. Introduction

The mid-infrared frequency region is referred to as the fingerprint region of molecules because of the presence of many absorption lines specific to molecular species in the region, and spectroscopy in this region has many potential applications, including environmental measurements, the investigation of combustion phenomena, and human breath analysis [1–3]. Mid-infrared spectroscopy has thus been extensively researched to achieve a wider spectral bandwidth, higher resolution, faster acquisition rate, and higher signal-to-noise ratio (SNR) [4–13]. All of these are in trade-off relationships, and various approaches have been studied to push the limits of the trade-offs.

In applications where detecting trace amounts of gases is important, such as human breath analysis, it is crucial to detect small changes in a spectrum. Taking the example of methane, which has strong absorption lines in the mid-infrared, the measurement of a 0.2% change in the transmission is required to detect 100 ppb methane in exhaled breath using a 1-m-long gas cell. Detecting such minute changes is often challenging owing to fluctuations in the light source, detector, and other environmental factors [14]. This is especially an issue when detecting trace gases whose concentrations change rapidly in time, such as in engine startup or human exhalation cycles. In addition, a high spectral resolution and wide bandwidth are required to identify molecular species.

One common approach in mid-infrared spectroscopy is to measure the temporal changes in light intensity resulting from spectral interference and then reconstruct them into spectra by Fourier transformation. A typical example is, as the name implies, Fourier transform infrared spectroscopy (FTIR). The advantages of this approach are its wide spectral bandwidth and high spectral resolution; in principle, the only limiting factors are the spectral bandwidth of the light source and detector. A well-engineered laser-based FTIR spectrometer achieved an SNR of approximately 1000 in the mid-infrared region [15], with the data acquisition taking tens of seconds to several minutes [4,15]. In the past decade, the use of dual combs or fast optomechanics to control spectral interference at higher speeds has been intensively studied [16–21]. These techniques can achieve fast spectral acquisition rates of hundreds of Hz to tens of kHz. In addition, the dual combs have a high spectral resolution of tens to hundreds of MHz. On the other hand, the SNR is limited to several tens because a single-pixel detector has to handle a broad spectral bandwidth and high resolution. At present, it is challenging to achieve a high dynamic range with a fast single-pixel detector in the mid-infrared region. With well-designed digital signal processing, an SNR of 6500 has been reported for an 80-minute integration [20]. In addition, an SNR of 2000 has been realized with a measurement time of 10 ms by using a quantum cascade laser comb as the light source, which has a 10-GHz longitudinal mode spacing [22].

Another approach is to use diffractive optics to spatially separate light by wavelength and measure the light with a detector. The advantage of this traditional spectroscopic method is that the light intensity at the detector directly corresponds to the spectral intensity at the target wavelength. No other wavelength components are input to the detector, and the dynamic range of the detector can thus be fully utilized for the specific wavelength. More to the point, in contrast to the former approach, spectral information is not encoded in the time domain, making the approach robust against fluctuations in the light source or environment.

Meanwhile, the diffractive-optics approach imposes additional limitations on the spectral bandwidth and resolution. The length of diffractive optics is proportional to the resulting spectral resolution, and a large grating is thus necessary to achieve high spectral resolution. As an example, a 30-cm-long grating is required to achieve resolving power of 100,000 at 3 µm, corresponding to spectral resolution of 1 GHz. In addition, for real-time measurements with a wide spectral bandwidth and high spectral resolution, we have to adopt a two-dimensional array detector with a large number of pixels, but the use of such detectors requires that the spectrum be spatially separated in two dimensions, horizontally and vertically.

Higher-order diffraction optics such as an echelle grating [23] and a virtually imaged phased array [24] have been used in combination with a low-diffraction grating to map spectra in two dimensions and thus achieve a high spectral resolution and wide bandwidth [25–30]. Nugent-Glandorf et al. reported an SNR of 2400 at a data acquisition rate of 120 Hz using a virtually imaged phased array for 3-µm mid-infrared light generated by a 1.5-W optical parametric oscillator, where the acquisition rate was limited by the camera frame rate [28]. They found that the improvement in the SNR due to averaging over multiple frames saturated at 6$\times$10³ at 10 frames, which they attributed to flicker noise in the laser's output. Iwakuni et al. reported an SNR of 3$\times$10³ for a 10-s integration in the 8–10-µm region [30].

In this study, we developed a high spectral mid-infrared spectrometer with a high SNR adopting an immersion-type echelle grating made of germanium with an extremely high diffraction order, which we refer to as an extremely-high-order germanium immersion grating (EGIG) [29,30]. Owing to germanium’s high refractive index of 4, a high spectral resolution was achieved despite the grating length of 13 cm. The extremely high diffraction order of 1122 at 3.3 µm allowed the pixels of the mid-infrared camera to be fully utilized, while the diffraction efficiency was as high as 70%. The spectrometer achieved a bandwidth of 2 THz with a spectral resolution of 1 GHz. The high spectral resolution is not only valuable in itself but also beneficial in removing signal fluctuations. We devised a noise reduction technique, which we call a reference spectrum estimation, and achieved an SNR of 3000 at 125 Hz in the presence of fluctuations in the light source, detector, and environment. Integration over 0.1 s improved the SNR to 10,000 even with a differential-frequency-generation-based light source having output power of 100 µW.

2. Experimental setup

The spectroscopic system comprises three sections: a mid-infrared light source, gas cell, and spectrometer, as shown in Fig. 1(a).

 

Fig. 1. (a) Experimental setup. DCF: double-clad fiber. PCF: photonic crystal fiber. DM: dichroic mirror. CD: cross-disperser. (b) Schematics of an echelle grating and an echellegram. (c) Typical echellegram. (d) Extracted spectra.

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2.1 Mid-infrared light source

Mid-infrared light was provided through difference-frequency generation (DFG) as follows. The seeding mode-locked oscillator was a 100-MHz Yb-fiber oscillator with a nonlinear polarization rotation scheme. The seed pulses from the oscillator were amplified to 2 W using a Yb-doped double-clad fiber and compressed by a pair of transmission gratings. The pulse width after the compression was 140 fs. The amplified laser pulses were divided into two paths by a beam splitter for supercontinuum generation and DFG. For supercontinuum generation, a 35-cm-long photonic crystal fiber was used to generate 1.3–1.6-µm pulses. Mid-infrared pulses at 3–5 µm were then generated by DFG between the fundamental and the supercontinuum. Periodically poled MgO-doped stoichiometric LiTaO₃ crystals with length of 1 cm were used for DFG. The crystal had a fan-out periodically poled structure, and the quasi-phase matching condition could be adjusted by the position of light passing through the crystal [31]. The power of the mid-infrared light was estimated to be 100 µW using a HgCdTe photodetector. Figure S1 shows the total intensity fluctuation and spectral fluctuations of the mid-infrared light. The fluctuation of the entire mid-infrared light is 0.5% in the root-mean-square error.

2.2 Gas cell and spectrometer

The mid-infrared light passed through a gas cell with an optical path length of 1.0 m and was guided to the spectrometer. The gas cell was evacuated before gas measurement. After isolating the gas cell from the vacuum pump, gas was introduced by opening a gas valve.

The spectrometer comprised the EGIG, a cross-disperser, and an InSb-based mid-infrared camera. The EGIG, which was manufactured with nanometer-precision machining by Canon Inc. [32,33], enabled the spectrum to be mapped to the 640 ${\times} $ 512 pixel mid-infrared camera with a designed spectral resolution better than 0.6 GHz. The line spacing of the EGIG was 476 µm, and the corresponding free spectral range was 82 GHz. The free spectral range was given by $c/({2dn\sin \alpha } )$, where c is the speed of light in a vacuum, d was the line spacing, $\textrm{n} \approx 4.04$ was the refractive of germanium at the target wavenumber, and $\mathrm{\alpha } = 75^\circ $ was the angle between the incident face and grating face of the EGIG. The InSb-based camera had a well capacity of 7.2 M electrons, a pixel pitch of 15 µm, and a dynamic range of 14 bits. The exposure time was set at 300 µs. The camera operated at a maximum frame rate of 125 Hz. The EGIG divided the spectrum of the incoming beam into segments in intervals of the free spectral range of the grating and gave each spectral segment angular dispersion in the vertical direction, as shown in Fig. 1(b). Each spectral segment corresponded to a different diffraction order and was horizontally separated by a low angular dispersion grating, commonly called a cross-disperser. We used a grating with 300 grooves/mm as the cross-disperser. The beam with angular dispersion in the vertical and horizontal directions was projected onto the two-dimensional array detector through an achromatic lens having a focal length of 200 mm. The projected spectral image is called an echellegram.

A typical echellegram is shown in Fig. 1(c). Each vertical line corresponds to a different order of the EGIG, and the upper right corresponds to shorter wavelengths. The intensity distribution along the vertical lines corresponds to the spectral intensity. The intensity distribution of the entire echellegram was circular because the beam was clipped by the window of the cryogenic cell of the array detector. Figure 1(d) shows the spectra extracted from the echellegram. Each bell-shaped spectrum corresponded to a different diffraction order of the EGIG, and each data point of the spectra was an average value for an area of 1 × 7 pixels vertically and horizontally in the echellegram. Background counts of approximately 3500 originating from dark currents and ambient blackbody radiation were subtracted for all the presented results. From the sum of the signal counts, the total power of the mid-infrared light incident on the sensor was estimated to be 40 µW.

The wavelength range that fits in a single camera frame was approximately 80 nm. To measure a different wavelength range, we adjusted the diffraction angle at the cross-disperser. The cross-disperser was mounted on a computer-controlled motorized rotation stage. In addition, the position of the wavelength conversion crystal was adjusted to match the phase matching conditions. The wavelength range of 3–5-µm was limited by the InSb detector and optical coatings.

2.3 Signal processing procedure

The procedure for obtaining a spectrum from an echellegram is as follows. (1) Calculate the traces of the diffractions on the image from the angular dispersion of the EGIG and the cross-disperser. The tuning parameters in this process were the focal length of the focusing lens and the uniform horizontal displacement in the image. (2) Average 7 pixels horizontally for each vertical position of the calculated traces. This operation maps the echellegram to the pixel number as shown in Fig. S2. (3) Assign pixel numbers to wavenumbers. As shown in Fig. 1(d), two or more pixels may correspond to the same wavenumber, and a pixel may span two bins of wavenumbers. In order to accommodate both cases, the correspondence was made using fractional weights determined by the overlap between the pixels and the wavenumbers. In this process, the tuning parameters were the uniform vertical displacement in the image and the aforementioned focal length. Each is adjusted by comparing the absorption lines of the molecule to the HITRAN database. The latter had been roughly determined in (1) and was a fine adjustment.

3. Results

3.1 Evaluation of the effective spectral bandwidth and resolution

Figure 2(a) shows the transmission spectrum of ethylene at a concentration of 20 ppm introduced into the 1-m gas cell. The transmission spectrum was obtained by dividing the spectrum with gas integrated for 5 seconds by the spectrum without gas integrated for 1 second. Note that the proposed noise reduction method was not applied in Fig. 2. Wavenumber ranges with little or no original spectral intensity (below 200 counts), as in Fig. 1(d), were excluded from the calculation. HITRAN spectra are obtained by convolving absorption spectra calculated at room temperature and pressure of 2 Pa with an instrumental function with full width at half maximum (FWHM) of 3.4 GHz. The upper panel shows transmission spectra simulated using the HITRAN's line-by-line data [34], and the lower panel shows the measurements. The transmission spectrum with a bandwidth exceeding 60 cm⁻¹ was obtained and found to be consistent with the HITRAN spectrum. Figure 2(b) shows an expanded spectrum from 3008 to 3018 cm⁻¹. While the results were qualitatively consistent, there was a quantitative discrepancy. The residuals obtained by subtracting the HITRAN spectra from the experimental spectra are shown in Figs. S3(a) and (b). Two factors may be responsible for the deviation. First, the point spread function on the image sensor may not be Gaussian due to chromatic aberration or clipping by apertures in the optical path. As shown in Fig. S3(c), the horizontal line width of the echellegram varied with the position in the image sensor. In addition, due to the low purity of the introduced gas, absorption by other gases was mixed in, as in the peak at 2980 cm.

 

Fig. 2. (a) Transmission spectrum of ethylene and the corresponding spectrum from the HITRAN [33]. (b) Magnified view of (a). (c) Spectral resolution estimation using methane.

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For spectral resolution evaluation of the system, we used the characteristic three peaks of methane at 3057.7 cm⁻¹. In this measurement, the overlap of the beams in the DFG was carefully adjusted so that the mid-infrared beam mode was as clean as possible. The measured transmission spectrum and corresponding HITRAN line-by-line data are shown in Fig. 2(c). The three peaks were separated, showing that the spectral resolution was as good as 1 GHz. The FWHM of the instrument function in this experimental condition was 0.83 ± 0.04 GHz.

3.2 Reference spectrum estimation to improve the SNR

The SNR is an important measure of a spectrometer in detecting trace gases. Suppose we want to detect a gas that has an absorption peak with absorbance $A$:

Assuming that the absorption by the gas is sufficiently small (i.e., ${\mathrm{\Delta }_{\textrm{gas}}} \equiv {\textrm{I}_{\textrm{ref},0}} - {\textrm{I}_{\textrm{gas},0}} \ll {\textrm{I}_{\textrm{ref},0}})$, we get

In the last equality, we made a general assumption that noise at different times was uncorrelated and the SNR is defined as $\textrm{SNR} \equiv {\textrm{I}_{\textrm{ref}}}/\sqrt {{\mathrm{\delta }^2}} $.

The situation drastically changes if $\mathrm{\delta }({{t_{ref}}} )$ can be replaced by $\mathrm{\delta }({{t_{gas}}} )$ because the second term in the middle of Eq. (3) vanishes. Such a substitution is possible if the spectrum without gas can be estimated from the spectrum with gas. Although such estimation should not be allowed if the noise was completely random, we found that there were several patterns in the spectral fluctuations and that the contribution from this noise could be estimated from the spectrum of one video frame.

Figure 3(a) shows the noise reduction procedure. We first prepared a time series of reference spectra. By stacking the series of the spectra along the time axis, we obtained a matrix whose rows correspond to times and columns to spectra, as shown in Fig. S2. Singular value decomposition (SVD) was then adopted to extract characteristic spectra from the matrix. SVD is a generalization of eigenvalue decomposition and factorizes an arbitrary m × n real matrix into two orthonormal bases, one comprising m m-dimensional vectors and the other comprising n n-dimensional vectors. SVD has a number of applications and has been used to analyze spectra having multiple components [35].

 

Fig. 3. (a) Schematics of the reference spectrum estimation. (b) Raw signal data at the absorption peak of ethylene located at 2994 cm⁻¹. The shaded area indicates the first 100 frames, which were used to compose a basis of the reference spectrum. (c) Changes in signal relative to the estimated reference spectrum at each time. (d–e) Magnified views of (b) and (c) at the timing of the gas introduction.

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We obtained an orthonormal basis for the reference spectra via SVD, as shown in Figs. 3(a) and S2. If a small subset of this basis can well constitute the reference spectra, then fluctuations originating from the light source, detector, or environment should have characteristic spectral structures. As we will see later, this assertion was valid. Using the subset of the orthonormal basis to decompose the spectra in the presence of the target gas, we estimated the reference spectra, including the fluctuations at the moment. See the following section for the detail.

Figure 3(b) shows the time variation of the spectral intensity at the position of an absorption peak when ethylene was introduced into the gas cell. The gas was introduced by opening the gas valve as quickly as possible. The target absorption peak is located at 2994 cm⁻¹, to the right of the strongest absorption peak, with a weak absorption of approximately 1.5%. For a light intensity of approximately 4000 counts, there was a fluctuation of 20 counts in the root-mean-square value and approximately 80 counts peak-to-peak in a period of 1 s. The decrease in light intensity at 1.1 s, which was presumed to be the timing of the gas introduction, was approximately 50 counts, and it is difficult to estimate the magnitude of absorption by the gas in a single video frame.

Figure 3(c) shows the change in light intensity ${\mathrm{\Delta }_{gas}}$ due to the introduction of the gas with respect to the reference spectrum ${I_{ref}}({{t_{gas}}} )$ estimated from the original spectrum ${I_{gas}}({{t_{gas}}} )$. The first 100 frames, indicated by shaded areas, were used as data in constructing the basis for estimating the reference spectrum. The estimation of the reference spectrum reduced the noise, and it was clearly seen that the gas was introduced rapidly at 1.1 s, then slowly over a period of 0.5 s, after which the gas concentration was constant. Note that the reference estimation was made independently for each spectrum at each time and did not use any information before or after the time. For comparison, a box-averaged version of the original data with a time window of 0.1 s is plotted as dots in Fig. 3(c). To make the comparison easier, the trace was shifted so that the time region without the gas was at the zero point.

Figure 3(d) and (e) shows magnified data at the timing of the gas introduction. The raw data were dominated by high-frequency noise, whereas the data obtained using the reference spectrum estimation clearly showed that the gas introduction took 80 ms. These fast fluctuations were presumably due to the light source, whereas light intensity changes on the scale of 1 s near the gas introduction were likely due to human movement or deformation of the gas cell windows. Both fast and slow fluctuations were successfully removed by the procedure in Fig. 3(a).

Figure 4 shows transmission spectra obtained from a single frame using the reference spectrum estimation and its time evolution. The single-shot spectra were consistent with the HITRAN results, as in Fig. 2. Also, the spectra showed little variation with time, which was consistent with the results in Fig. 3.

 

Fig. 4. (a) Time evolution of transmission spectrum obtained by the reference spectrum estimation method. (b) Single-frame spectra cut from (a) after gas introduction.

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4. Analysis & discussion

4.1 Detailed procedure for the reference spectrum estimation

As in the previous section, we consider an $m \times n$ matrix M with m time-ordered reference spectra of n elements. SVD factorizes M by an $\textrm{m} \times m$ orthogonal matrix T, an $\textrm{n} \times \textrm{n}$ orthogonal matrix S, and a rectangular diagonal matrix W:

If the spectrum is wide enough and the resolution is high enough, the spectral changes due to absorption by the gas or unexpected noise and the spectral basis vectors of the reference are nearly orthogonal (i.e., ${w_k} \approx \overrightarrow {{s_k}} \cdot \mathrm{\vec{\sigma }}$), which yields

In reality, each element of $\mathrm{\vec{\Delta }}$ and $\overrightarrow {{s_1}} $ is positive and their dot product is not zero, which leads to underestimation of the absorption. To compensate for this, additional prior information is needed. Let us assume that we know that there is no absorption at a specific frequency ${\mathrm{\omega }_c}$. The transmission at ${\mathrm{\omega }_c}$ should be unity at any time, and because the dominant contribution of underestimation comes from the first basis vector, we obtain a first-order approximation of the weights of the reference spectrum by:

This approximation pushes all nonzero components of the second term on the right side of Eq. (6) to ${w_1}$. Figure S4(a) shows the transmission spectrum before and after the correction. The effect of the correction was a simple vertical shift of the spectrum. Before the correction, ${w_1}$ was underestimated because of the inability to distinguish between the reduction in light intensity due to the introduction of the gas and the reduction in light intensity of the light source. A more accurate estimation of the weighting can be made if transmissions at multiple frequencies are used in the correction. In cases where it is difficult to find a spectral region with no absorption, e.g., for molecules with broad absorption or for mixtures of unknown, ${w_1}$ could be corrected by monitoring the total power of the light source using a separate mid-infrared detector.

In this paper, the transmission at 3018 cm⁻¹ was used for the correction in Figs. 3–5. This choice of correction wavelength shifts the absolute value of transmittance but has little effect on SNR as shown in Fig. S4(b). Note that the correction itself increased the noise by approximately a factor of $\sqrt 2 $ because it spread the unexpected noise at a frequency ${\mathrm{\omega }_c}$ throughout the spectrum. The need for higher-order corrections to compensate for possible remaining biases depends on the degree to which spectral fluctuations and absorption have similarities. In principle, higher spectral resolution and wider spectral bandwidth of a spectral system allow higher-order influences to be neglected.

 

Fig. 5. (a) First four basis vectors of spectra and time variations. (b) SNR as a function of the number of basis vectors used to estimate the reference spectrum. (c) SNR as a function of the number of reference frames.

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After the correction, averaging was performed for four adjacent spectral elements (four vertical pixels in the echellegram), corresponding to a frequency of 0.92 GHz. This operation slightly improves SNR without compromising spectral resolution. For a fair comparison, the same averaging was done for the raw data in Fig. 3(b), but almost no improvement was observed. This is thought to be due to the unexpected noises such as read noise in the analog-to-digital conversion being almost negligible in the raw data.

In this study, signal processing is performed after the measurement. Since the calculations for each frame are only matrix multiplication, vector subtraction, and division, a high-end personal computer can perform signal processing simultaneously with data measurement.

4.2 Characteristics of the reference spectra

Figure 5(a) shows the first four basis vectors of the reference spectrum and the corresponding basis vectors of time variations for each vector. The horizontal is the number of pixels corresponding to the concatenated spectra as shown in Fig. S2, and they are arranged vertically in order from the highest singular value to the lowest. Note that the basis vectors of the time variations are shown for reference purposes only and were not used in the analysis. The first spectra basis vector comprised bell-shaped spectra corresponding to different diffraction orders and was nearly constant in time. The subsequent components had characteristic spectral fringes and were highly fluctuating in time. Such fluctuating spectral fringes are thought to be due to fluctuations in the light source or unintentional interference among optics.

As the extracted basis vectors show, this method is effective as long as the spectral fluctuations have common modes and as long as they are orthogonal to the molecular absorption in the frequency domain. In other words, the method should work even if the target molecules have broader absorption features or a large spectral fluctuation exists in the environment.

In case long-term spectral drift exists in the light source, the reference spectrum should be retaken accordingly. In our experimental system, coupling to the PCF was the source of the long-term drift, which changed the mid-infrared spectrum on a time scale of several minutes. Figure S5 shows the basis vectors of the reference spectrum and their time variation in a measurement made 30 minutes after the measurement in Fig. 5. While the first component is nearly identical to that in Fig. 5(a), the rest of the fluctuation pattern has changed. If the reference spectra are taken over a long enough period, the drift of the light source spectrum should also be included in the basis vectors, but the computational cost of the SVD of large matrices would be an issue.

4.3 Characterization of the SNR

Figure 5(b) shows the relationship between the number of basis vectors used to estimate the reference spectrum and the resulting SNR. The SNR was calculated from the signal fluctuation at the small absorption peak in the inset of Fig. 3(b). More specifically, the standard deviation for a 5-s period beginning at 1.6 s, when the gas introduction had ended, was used as the noise, and the corresponding light intensity was used as the signal. The first basis vector alone did not reduce the noise, whereas the use of seven basis vectors drastically improved the SNR. The first basis vector would be the time-averaged spectrum, while the subsequent basis vectors would be attributed to light source fluctuations and other factors. Thus, the SNR dependence on the number of basis vectors suggests that the fluctuations in total light intensity were not the dominant origin of the noise but that the finer spectral structure was the limiting factor of the SNR. In this paper, 30 basis vectors were used to estimate the reference spectrum, and the SNR was 3010 at the absorption peak at 2994 cm⁻¹. The SNR was improved by a factor of 14. The number of vectors, 30, was chosen with margin.

The number of frames used to learn the reference spectra also significantly impacts the SNR. Figure 5(c) shows the dependence of SNR on the number of reference frames. A larger number of reference spectra improved the SNR because more spectral variation patterns could be captured in the reference spectrum estimation. Figure S6 shows the temporal changes at 2994 cm^-1. Meanwhile, when the number of basis vectors used was insufficient, the increased information could not be used effectively, and the SNR improvement was limited.

A pixel-by-pixel evaluation of the SNR over the entire spectral range showed power laws with respect to the signal counts ${I_{sig}}$, with $SNR = 1.3 \times I_{sig}^{1.0}$ for counts below 1000 and $SNR = 37 \times I_{sig}^{0.54}$ for counts above 1000 as shown in Fig. 6(a). The existence of the scaling laws indicates that SNR was determined only by spectral intensity. A comparison with the raw spectrum as in Fig. 1(d) shows that the region where the SNR exceeded 2000 for spectral ranges over 1.5 THz. The transition in the exponent suggests that the proposed noised reduction method suppressed noise originating from light source fluctuations to a level of 0.1% of the spectral intensity. When the spectral intensity is less than 1000 counts, the SNR increases linearly with spectral intensity, indicating that the detector was the primary source of noise, mainly due to the detector's read noise. The transition to the square root scaling above 1000 counts suggests the presence of white noise-like light source fluctuations that cannot be removed by the proposed method. Under our experimental condition, the four spectral elements to be averaged had 10⁸ incoming photons per frame. Therefore, the upper limit of SNR due to shot noise should be 10⁴ in a single frame at 4000 signal counts.

 

Fig. 6. (a)Signal-to-noise ratio vs. signal count over the entire spectrum. The power exponents of the dash lines are 1.0 and 0.54. (b) SNR as a function of the integration time. The power exponent of the dashed line is 0.5.

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Figure 6(b) shows the plot of the SNR versus the integration time, reaching an SNR of 10,000 at 0.1-second integration. The exposure time was 300 µs, and a camera with a higher frame rate could thus achieve an SNR of 10,000 in a shorter time. As shown in the dotted line in Fig. 6(b), the signal-to-noise ratio improved with the square root of the integration time up to approximately 0.06 seconds, indicating that white noise was dominant. At longer integration times, the power exponent became approximately 0.3, possibly due to long-term drift that was not included in the training data, i.e., the first 100 frames.

5. Conclusion

We developed a 3–5-µm spectroscopic system for trace gas detection and achieved an SNR of 3000 at a data acquisition rate of 125 Hz and 10,000 at 10 Hz. Noise reduction by the reference spectrum estimation suppressed noise to less than one-tenth under our experimental conditions. The method should be effective in other systems, especially in a spectroscopy system where the light is spatially dispersed and measured with a multi-array detector. Even for a spectroscopy system that encodes spectra in the time domain, such as a dual comb, the method would be effective if the timescale of sources of spectral fluctuations is longer than the single-shot measurement time. For this method to be effective, the spectral fingerprints of the target molecules and the spectral fluctuations must be orthogonal in the frequency domain. For this reason, a high spectral resolution and a broad spectral bandwidth are desirable, and higher-order dispersive elements such as an EGIG are well suited for this purpose.