## Abstract

In our paper, we present a new design for a single-grating tunable spatial heterodyne spectrometer (SHS). Our design simplifies the change of the center wavelength (Littrow wavelength) thus one can quickly tune the system to an arbitrary spectral range. Furthermore, we introduce a new calibration method that provides superior calibration accuracy over the generally used formulas involving small angle approximations. We also present considerations about the general usability of the SHS technique in broadband measurements and propose different strategies to improve the signal-to-noise ratio.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Spatial heterodyne spectroscopy (SHS) is a fairly new spectroscopic measurement method first demonstrated by Harlander et al. [1] in 1992. SHS employs interference spectrometers which have many advantages over slit spectrometers: much higher throughput and compact size with high spectral resolution. The concept of SHS introduces additional advantages compared to conventional interference spectrometers (Fabry-Perot, Michelson). It utilizes a detector array (CCD, CMOS device) instead of a point detector and acquires the interferogram in the space domain across the detector plane. Therefore it does not require a moving element during the data acquisition. This feature allows to use it in measurements with pulsed sources [1–3]. In addition the etendue of an SHS instrument can be further increased by the incorporation of field-widening prisms [1,2]. The basic SHS configuration resembles a Michelson interferometer. The difference is that mirrors are replaced in the two arms by reflective diffraction gratings. The schematic representation of the basic design is shown in Fig. 1(a) and the angle notation in Fig. 1(b).

The light from the source is collected and collimated with a lens, and a plane wave (ideally) enters the spectrometer. The beams transmitted and reflected from the beamsplitter then hit the diffraction gratings which are tilted by an angle $\Theta _L$ (Littrow angle) with respect to the optical axis. When the incoming wavelength is matching the Littrow wavelength set by the gratings, the beams are going to be reflected back along the optical axis, exiting the interferometer through the exit optics and their interference will give a uniformly lit field on the detector. However, when the wavelength of the incoming light is different from the Littow wavelength, the beams from the two arms will meet under an angle of $2\gamma$, forming a Fizeau interference pattern on the detector plane with a frequency of $f=2k\sin {\gamma }$, where $k=1/\lambda$ is the wavenumber of the light.

The angle $\gamma$ can be derived from the grating equation which has the form:

where $m$ is the diffraction order which is used in the measurements and is generally $1$ or $-1$. From the grating equation considering only small angles for $\gamma$ with respect to the optical axis, one can get a straightforward formula for the fringe frequency: where $k_L$ is the Littrow wavenumber. As one can see, the fringe frequency depends on the wavenumber difference between the Littrow wavenumber and the wavenumber of interest. This relation determines the heterodyne type of the method. Every fringe pattern for each wavenumber will then sum up and form the final interferogram. After the interferogram is acquired, the fringe frequency spectrum can be calculated via Fourier transform. The final spectrum in wavelength units is then calculated from the fringe frequencies.This technique has been demonstrated to have high potential in a variety of optical measurements. It was used to monitor hydroxyl radicals in the Earth’s middle atmosphere [4], to observe the limb-scattered sunlight to study water vapour [5], and to sense carbon dioxide [6]. It was also innovatively used in Raman spectroscopy [7–11] and LIBS [12] measurements. It has been demonstrated that this technique can be used in the infrared region [13]. The concept is also suitable to build it as a full monolithic design, thanks to the lack of moving elements [12,14] and it can also work in Echelle grating configuration [15].

## 2. Single-grating design

In the case of the original SHS design (Fig. 1) one difficulty is the alignment of two separate gratings. Both gratings have to be set to the same Littrow-wavelength. This feature makes the center wavelength tuning of the device cumbersome. The single-grating design aims to reduce the requirement for precise and synchonous operation of the moving parts during center wavelength tuning as the change of Littrow-angle happens simultaneously in both arms. This comes very handy when the user cannot monitor the live interferogram for example in low light applications and makes the expansion of the measurable bandwidth easier. Furthermore, the misalignment of the two gratings can also cause frequency shifts in the measured fringe spectra.

Some ideas have been already presented in the literature to create conveniently tunable design like using one-mirror-one-grating layout [7,12] or gluing the two gratings together back-to-back and placing them into a common out-folded optical path [3]. In case of the one-mirror-one-grating design [7,12] the diffraction happens only in one arm of the interferometer therefore the maximum phase retardation for a given wavelength across the detector surface is only half than that of the original design and thus results in lower resolution. Compared to that in Lenzer’s design [3] the diffraction happens in both interferometer arms but it utilizes two gratings glued together back to back. This implies that the two gratings has to be glued very carefully in order to be their grooves parallel. Our design eliminates these problems as we show it in this section.

#### 2.1 New single-grating design

The conceptual drawing of the new design can be seen in Fig. 2. The main concept is based on the folding of the interferometer arms to have a Sagnac-like interferometer. Then we placed a roof-top mirror (RTM) into the common path to turn the beams outward parallel to each other. After the roof-top mirror the parallel beams reach the diffraction grating (G) which is tilt-tuned to the desired Littrow angle. The beams that were diffracted are then relayed onto the detector surface with the exit optics to form a high visibility interferogram.

When the grating is tilted, the optical path difference is not zero on the optical axis (see Fig. 2 - blue dotted line is the plane of zero path difference) which causes the interferogram to shift in the direction of dispersion. This path difference can be easily compensated by mounting the roof-top mirror and the grating onto a common platform and adjusting its position as drawn in Fig. 2.

The path difference correction that we have to introduce is $\delta =\Delta \tan {\Theta _L}$. When the spectrometer is tuned to another Littrow angle we have to compensate again for the path difference change in the arms, but this can be done easily even by hand while we are monitoring the interferogram with the imaging detector to keep the interferogram in the middle.

In our experiment we used commercially available components from Thorlabs. M1 and M2 are 1" x 1" Protected Silver Mirrors (PFSQ10-03-P01), BS is a 25 mm x 36 mm 50:50 UVFS Plate Beamsplitter (BSW26R) and RTM is a Knife-Edge Right-Angle Prism Protected Silver Mirror (MRAK25-P01). For the broadband measurement we utilized a 300 lines/mm Ruled Reflective Diffraction Grating (GR50-0305) and for the high resolution measurements we replaced the low density grating with a 1200/mm Reflective Holographic Grating (GH50-12V).

For the exit optics we adopted a bi-telecentric imaging system design consisting of two achromatic doublet lenses (Thorlabs AC254-100-A-ML) with the same effective focal length to keep the magnification unity. This bi-telecentric configuration images the grating surface to the detector array and keeps the ray angles exactly as they left the diffraction grating.

#### 2.2 Detection and bandwidth

The detector is a common CMOS camera, ZWO ASI 183MM Pro. The sensor itself is a Sony IMX183, 1" sensor with resolution of 5496$\times$3672 and a pixel size of 2.4 $\mu m$. The pixel size is a crucial parameter of an SHS instrument, because it sets the sampling frequency of the interferogram. According to the Nyquist theorem, at least two pixels are needed to resolve one fringe pattern. This limit leads to the maximum fringe frequency we are able resolve as $f_{max}=1/2d_{pix}$, where $d_{pix}$ is the size of the pixel in the direction of the dispersion (x-direction). Knowing the Littrow wavelength one can calculate the maximum and the minimum wavelength that can be acquired from the fringe frequency spectrum. We define the bandwidth of the instrument as the difference of the two wavelength extrema $\Delta _B\lambda = \lambda _{max}-\lambda _{min}$. With the help of the fringe frequency expression from the small angle approximation (Eq. (2)) one can formulate the spectral bandwidth of the instrument depending on the pixel size:

Figure 3(a) shows a typical interferogram of a deuterium lamp acquired with our new configuration. We used the deuterium arc lamp of an Analytical Instruments System Model 2000 UV lamp for this demonstration (with the halogen lamp turned off). The lamp is fiber coupled and the transmitted light from the fiber was collimated with a Thorlabs F810SMA-543 collimator. The collimated light then was feed directly into our heterodyne spectrometer. The core diameter of the multimode optical fiber was $0.2 \mu m$ which can be viewed as an entrance aperture of the system. In Fig. 3 we display the full spectrum calculated from the interferogram. This spectrum was taken with a 300 lines/mm grating. The spectral range can be read from the wavelength axis to be $459.3$ nm. This value agrees well with $\Delta _B\lambda =448.6$ nm calculated from Eq. (3). The small difference comes from the fact that for the formula we used the small angle approximated expressions while the spectrum shown in Fig. 3 has been calibrated via a numerical method described in Section 3.

Here we want to emphasize the importance of tunability. For low light applications, high sensitivity detectors are available only with larger pixel size than that we used here. In those cases the larger $d_{pix}$ causes the bandwidth to drop dramatically. If one wants to examine a large spectral range with high resolution, a feasible option is to scan trough the whole range.

It is important to note here that the wavelength components below and above the Littrow-wavelength are heterodyned symmetrically, which leads to an ambiguity in the fringe frequency spectrum [1–3]. This ambiguity can be resolved by introducing a phase retardation in the direction perpendicular to the plane of dispersion (y-direction). This breaks the symmetry of the frequency spectra in a way that negative and positive fringe frequencies will separate in the 2D spectrum [1,5,9]. A tilt between beams can be ideally managed by rotating the roof-top mirror by a small angle around the optical axis. When the roof-top mirror is turned around the right axis the tilt of the beams happens in the opposite direction symmetrically in the two arms. The consequence of this tilt can be seen in the interferogram in Fig. 3(a) as the fringe patterns get tilted by the additional retardation introduced in the perpendicular direction. Applying 2D Fourier transform results in a 2D spectral map (Fig. 3(b)) where the spectrum and the conjugate spectrum are separated from each other and therefore there is no overlap of the wavelength components below and above the Littrow wavelength. One can extract the 1D spectra (Fig. 3(c)) from this power spectral density map. In this example we extracted the points along the dashed line shown in Fig. 3(b). The formula in Eq. (3) assumes that we record a 2D spectrum in order to take advantage of the full spectral range.

One bottleneck which can limit the measured spectral range with an SHS instrument is the size of lens L1 of the exit optics (a feature of every design). Light beams that have diffracted into larger angles can be apertured down at that lens. This effect is limiting the range of the spectrum passing through the system.

It is also a possible scenario to have wavelength components passing through the system that would create interference patterns with spatial frequencies too large for the detector to resolve. This would lead to serious frequency artifacts in the spectrum. This problem can be solved with an aperture (AP) placed at the focal plane of the first lens (L1) serving as a spatial filter to sort out the spectral components that would be otherwise undersampled by the CCD.

#### 2.3 Resolution

Beside the bandwidth the other critical parameter of a spectroscopic instrument is the resolving power $R=\lambda /\Delta \lambda$. It has been shown that the resolving power of an SHS system is [16]:

which is the theoretical resolving power of a grating system. Here $W$ is the width of the illuminated part of the grating. The wavelength resolution that can be achieved can also be approximated by dividing the full bandwidth by the number of detector elements. Both approaches give very close results. However, Eq. (4) overestimates the resolving power, because it does not takes account the lateral confinement of the interferograms. Lenzer et al. introduced correction in the calculation of the resolving power which gives more closer result at very high Littrow-angles [3].The resolution in practice becomes lower than what is given by the theoretical formulas. Apodization of the interferogram is crucial in order to have good signal-to-noise ratio (SNR) but it broadens the spectral lineshapes. Beside the effect of apodization the lineshape is also affected by the distortion of the ideal plane wavefronts. This kind of optical error distorts the fringe patterns and can be described as a phase distortion in the harmonic interference patterns [17]. Fortunately this error can be corrected with the method described by Englert et al. [17] and also neural networks were effectively used for this task [18,19].

Figure 4(a) shows an interferogram of a Neon glow lamp measured with a 1200 lines/mm grating. Figure 4(b) displays the spectrum based on that interferogram. We applied the phase correction method for the 2D spectrum in order to achieve resolution close to the theoretical maximum. Figure 4(c) compares the resolution of our system with the resolution of a Renishaw inVia QONTOR Raman spectrometer with a 2400 lines/mm grating and a $65 \mu m$ entrance slit. Our entrance aperture can be considered to be $200 \mu m$ as we used a multimode optical fiber to couple the light of the neon lamp into our system.

Despite the simple tunability of our SHS based on the new single-grating design, it has also disadvantages. First, the size of the system cannot be as compact as in the case of the original concept. The second disadvantage is related to the first one very closely. Since the interferometer arms are folded out and are relatively long, we have to use longer focal length lenses in the exit optics to image the gratings on the detector. This makes the exit optics also longer than what would be needed in a close-packed design. Notwithstanding these facts, the size is still small compared to a high-resolution slit spectrograph and the ease of tuning and usability compensate for the somewhat larger packaging and the design is very well suited as a general-purpose high-resolution spectrometer.

As demonstrated here and in the literature, SHS provides high resolution with a small footprint. High resolution is meaningful only if it comes also with high accuracy. In the processing of the raw data one can lose accuracy easily. The use of the right formulas and methods are crucial in the optimal operation of the SHS spectrometer. In the next chapter we revisit the commonly used calibration and propose a better solution that fulfills the requirements of such a calibration.

## 3. High-accuracy calibration method

In the case of SHS technique, spectral calibration means assigning wavelength and/or wavenumber values to the measured fringe frequency (spatial frequency) points in the spectra. The commonly used method was proposed by Englert et al. [20] and used widespread in the measurements one can find in the literature. This calibration uses the measured fringe frequency of two emission lines with well-known wavelengths to determine the Littrow angle ($\Theta _L$) and to calculate the wavelength axis corresponding to the measured frequency axis. While this method is commonly used, it assumes the small-angle approximation. This approximation leads to a deviation in the calculated wavelengths from the real wavelengths along the spectra. Furthermore this calibration does not take into account the uncertainty in the groove density of the grating. For a broad-range and high resolution spectrometer this difference can grow huge compared to the offered resolution.

Here we propose a numerical calibration method which can take into account the deviation of groove density from the nominal value and excludes the usage of any approximation.

The idea is to use the grating equation (Eq. (1)) directly without any approximation to examine the $f(\lambda )$ relation. With simple algebraic transformations one can express the fringe frequency as the function of the wavelength:

With the help of this equation the calibration can be treated as an optimization problem (or curve fitting). We have two optimization variables namely $\Theta _L$ (Littrow angle) and G (groove density). For a given wavelength $\lambda _c$ ($c$ stands for calibration) we want to change $\Theta _L$ and $G$ in a way to get back $f_c$ (optimization operand) the measured calibration frequency, so we specify $\phi = (f-f_c)^2$ as an error function. However, if we plot this error function depending on $\Theta _L$ and $G$, we observe that this function does not have a local minimum. For that reason we have to use at least two known wavelength-fringe frequency combinations and define the final objective function to be minimized as: where $w_i$ are the weights for each operand but we consider those to be equally one. Figure 5(a) shows the main objective function constructed with the help of two well-known emission lines of a hydrogen arc lamp. These emission line are $\lambda _1=486.13615\, nm$ (blue) and $\lambda _2=656.45377\, nm$ (red). Both are components of the Balmer series and their wavelengths depend only on physical constants. In the measurement with the current arrangement the corresponding fringe frequencies were $f_1=58.38225\, mm^{-1}$ and $f_2=-111.96908\, mm^{-1}$.Figure 5(a) also displays the Littrow angle acquired from Englert’s method considering the nominal groove density of $300 mm^{-1}$. It clearly shows how inaccurate it is to use the formulas from the small angle approximation. The blue and a red dashed curves in Fig. 5 display the Littrow-angle calculated as a function of the groove density for the blue and the red hydrogen emission line separately in the small angle limit. The Littrow angle for both lines should be the same, consequently their cross-over point would give the calibrated groove density. However, because those curves apply the small angle approximation, they still cannot retrieve the real Littrow angle and groove density. The perfect calibration can be achieved only by numerical optimization. Figure 5(b) shows the $f(\lambda )$ function calculated with the acquired parameters.

The next problem after the calibration is to calculate the wavelengths from the fringe frequencies. This calculation can be done with the help of small angle approximation formulas like Eq. (2), but then we would introduce further errors. Instead of Eq. (2) we can apply optimization for this task using directly the grating equation (Eq. (1)). If we subtract the right side and express the angle $\gamma$ with a wavelength, it will form an objective function naturally:

This calibration based on numerical optimization does not have large computational costs and we suggest this method to obtain the most accurate spectra which the unique SHS technique can provide.

In Fig. 6(b) we present an emission spectrum of a neon glow lamp. The dashed vertical lines represent the position of the emissions acquired from the NIST database [21].

We would like to note that there are other numerical calibration methods for FTS instruments, like utilizing neural networks [18,19]. These techniques directly output the phase corrected and wavelength calibrated spectra. When serious distortions are present and the instrument has fixed setup, a trained neural network can be the method of choice for calibration as they reduce computational complexity by integrating the different post-processing steps under the umbrella of a neural network. However this requires additional equipment and procedures for measuring the training data.

Our spectral calibration method only requires two known spectral line in the measured bandwidth. When the phase correction is done or there is no considerable distortion in the spectrum our method gives accurate result over the widely used small angle approximation demonstrated by the achievement of a perfect match with the database values.

## 4. SHS of broadband sources

In previous works measurements concentrated mostly on line spectra where the bandwidth of the instrument was not fully covered with the spectral components. However, if we have continuous, broad light-emission from the object of interest, an SHS measurement suffers from noise problems which were not sufficiently discussed in connection with this method.

The cause behind the noise issues comes from the shrinking of the interferogram. When we have such a broad spectrum, all the wavelength components will add up and form a narrow interferogram located on a small region of the detector. This way the information is stored in just a few pixels and strongly exposed to the effect of detector noise present on those pixels.

A narrower spectrum would result in a broader interferogram with better signal-to-noise ratio. Figure 7 demonstrates the increase of the noise level with the broadening of the measured spectrum.

We used the (AP) aperture to filter out spectral components as a bandpass filter. In Fig. 7(a) we show the effect of closing the aperture. As a consequence the measured spectra will be narrower and the noise level drops as 7(b) demonstrates. All spectra were measured by capturing 2D interferograms. Each spectrum was then extracted from the 2D Fourier-transform spectrum as it was introduced and shown in Fig. 3 and Section 2.2. This way one can acquire the spectrum below and at the Littrow-wavelength also. All spectra were normalized to the value at zero frequency and the noise was calculated by taking the standard deviation at the negative frequency end of the spectra where no spectral components are present. The noise level has linear tendency with the width of the spectra. Taking advantage of the ease of tunability one can close the aperture to have better SNR and scan through the spectra in few steps to measure the whole range.

Another obvious way to deal with the noise is to use a very low noise detector and use low gain during the image acquisition if enough light is present during the measurement. This solution on the other hand increases the cost of the device and cannot further decrease the noise than the detector limit.

Another option is to take multiple (N) interferograms and average the spectra calculated from those images as it is widely used in FTS instruments. This method decreases the noise in the spectra by $1/\sqrt {N}$ but in the case of SHS it requires to take $3\cdot N$ images because flat-fielding [22] is very important in case of broad range interferograms.

We also propose a different approach which takes advantage of broadening the interferogram on the detector. Instead of filtering the spectra, one could magnify the interference pattern. This magnification can be introduced by angular demagnification. This way the angle between the two plane waves from the two arms of the spectrometer will be smaller and the resulting interference pattern will have longer spatial pattern length.

Figure 8(a) presents the effect of interferogram magnification. The spectrum in fringe frequency space will shrink while we keep all the spectral components. As we increase the magnification, the noise level decreases with respect to the signal as $1/M$. In Fig. 8(b) we show frequency spectra with different magnification $(1-5)$. We note that these spectra were calculated from simulated interferograms with added noise. The frequency axis for all spectra were also calculated back with respect to the original frequency with $M=1$. However, this method has one important disadvantage, namely that the resolution of the spectrum will be lower because it is distributed on a lower number of frequency points.

We showed here that the SHS technique has limited capabilities when measuring broadband light sources with a very broad bandwidth. However, one can filter the spectrum around the Littrow wavelength to increase the SNR and take advantage of the easy tunability to swipe through the whole spectrum. The other way around is to sacrifice resolution and increase SNR by using magnification of the interferogram.

## 5. Conclusion

We presented a new design for spatial heterodyne spectrometry which can enhance the potential of this method in a wide range of spectroscopic applications. The single-grating design makes it very easy to tune the spectral range to the desired spectral region. Moreover we introduced a new calibration method for the SHS technique. Through numerical optimization we achieved perfect accuracy throughout a wide spectral range. We compared this method to the generally used calibration and we found that numerical optimization yields superior performance for high accuracy, high resolution spectroscopy. We also addressed noise issues of the SHS technique when measuring broadband light sources. We showed how the shrinking of the interferogram causes the decrease of spectral SNR. We showed that by reducing the spectral range by an aperture and taking advantage of the easy tuning and scanning through the spectra with that reduced acquisition range, one can enhance the SNR of the measurement. We also showed that the magnification of the interferogram enhances the SNR and can be utilized with the exit optics. We think that our findings are important to the further advancement of spatial heterodyne spectrometry.

## Funding

Nemzeti Kutatási Fejlesztési és Innovációs Hivatal (FK-125063, PD-121320, VEKOP-2.3.2-16-2016-00011).

## Acknowledgments

The authors thank Gábor Erdei for the enlightening discussions and Katalin Kamarás for the useful practical views and advises during the project.

## Disclosures

The authors declare no conflicts of interest.

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