Development of a Near-Infrared Spatial Heterodyne Spectrometer for Temporally Resolved Zeeman Effect Measurements on the Hydrogen Paschen-α Emission Line

Abstract

A near-infrared (NIR) spatial heterodyne spectrometer was developed for the hydrogen Paschen-α emission line (Paα, 1,875 nm) to enable temporally and spatially resolved measurements of hydrogen atomic density, temperature, and flow velocity in the edge region of magnetically confined fusion plasmas. To overcome the frame-rate limitations of two-dimensional NIR image sensors, an indium gallium arsenide (InGaAs) linear array was employed, and the interferogram was compressed into one dimension using a custom cylindrical imaging lens system. Spectral measurements of Ne I emission lines in a glow discharge tube confirmed an instrumental wavelength resolution of 125 pm. At zero magnetic field, the measured Paα interferogram in the glow discharge tube exhibited a contrast ratio of approximately 0.58, demonstrating agreement with the numerical model. Furthermore, spectral measurements under magnetic fields up to 3 T demonstrated that spectra influenced by the Zeeman effect were reproducible using fitting functions that incorporate the instrumental function and Doppler broadening. Despite a magnetic field-induced contrast-ratio degradation to approximately 0.4, a signal-to-noise ratio of approximately 20 was achieved at a 1-ms exposure time for a spectral radiance of 1.2 W m⁻² sr⁻¹ nm⁻¹, which is comparable to the emission expected in the ITER divertor region.

1.  Introduction

In fusion-edge plasmas, spatially and temporally resolved measurements of hydrogen atomic density, temperature, and flow velocity are essential for investigating fast phenomena, such as edge-localized modes (ELMs, ≤ 1 ms) [1] and turbulence-driven shear flows (≤ 10 ms) [2], and evaluating their impact on particle and energy transports. For instance, in JT-60U experiments, Type-I ELMs increased the line-integrated edge electron density by approximately 40% via the ionization of recycled atoms, which subsequently recovered to the pre-ELM levels on a millisecond timescale [3]. Furthermore, DIII-D and LHD experiments confirmed the charge-exchange generation of hot atoms in the edge region; these atoms exhibited temperatures closely matching those of the ions, suggesting their relation with transient phenomena, such as ELMs [4–6].

In these previous studies, hydrogen atomic density and temperature were measured via passive spectroscopy of the Balmer-α (Baα) emission line. Although model-based inversions have been applied in some measurements [5, 7], these observations are essentially line-of-sight (LOS)-integrated, limiting the resolution of the local spatial distribution along the LOS. To overcome this limitation, we employ emission-line Zeeman splitting to achieve spatially resolved measurements along the LOS.

A method based on the analysis of Zeeman splitting has been proposed to localize the emission region along a single LOS; this approach has been applied to JET [8], Alcator C-Mod [9], and other devices [10]. To apply this method with further improved spatial resolution, we measure the near-infrared (NIR) Paschen-α emission line (Paα, 1,875 nm) instead of the conventional visible Baα. This choice is based on the fact that Zeeman splitting exhibits a stronger wavelength dependence than Doppler broadening, becoming more pronounced at longer wavelengths [11, 12]. Our previous study demonstrated enhanced spatial resolution using this approach in a divertor-relevant plasma [11].

However, because the Paα emissivity is approximately 20 times lower than that of Baα in an ionizing plasma (assuming an electron temperature of 1 eV and a density of 10²¹ m⁻³), it will be challenging to achieve a 1-ms temporal resolution, which is the representative value required for the measurements of fast-edge phenomena, using conventional Czerny–Turner spectrometers (CTSs). Furthermore, Paα measurements necessitate the use of indium gallium arsenide (InGaAs) or mercury cadmium telluride image sensors whose nominal dark noise is typically more than three orders of magnitude higher than that of silicon image sensors used for Baα [13, 14], making it more challenging to acquiring spectra with a sufficiently high signal-to-noise ratio (S/N).

To overcome these limitations, we have developed an NIR spatial heterodyne spectrometer (SHS) [15, 16], a specialized class of Fourier transform spectrometer that operates without requiring mechanical scanning components. Compared with conventional CTSs, SHS provides one to two orders of magnitude higher geometric etendue [17, 18], i.e., enhanced light-collection efficiency, while maintaining a high wavelength resolution on the order of tens to hundreds of picometers. While visible-range SHS systems with temporal resolutions of less than 1 ms have been developed for fusion plasma spectroscopy [17–19], their adaptation to NIR applications remains constrained by the frame-rate limitations of standard two-dimensional image sensors [13, 14]. To address this issue, we adopted an InGaAs linear array detector, which generally enables higher frame-rate than two-dimensional detectors. Previous SHS implementations using linear array detectors relied on digital micromirror device–based scanning system [20, 21], which is incompatible with the high-time-resolution requirements considered here.

In this study, we present the design, numerical modeling, and validation of an NIR SHS optimized for the high-frame-rate measurements of Paα. The remainder of this paper is organized as follows: Section 2 describes the instrument and model, while presenting the performance evaluation. Section 3 describes the feasibility of Zeeman splitting measurements and Doppler broadening analysis in a glow discharge plasma under a fusion-relevant magnetic field.

2.  Near-Infrared Spatial Heterodyne Spectrometer

2.1  Design and specifications

Figure 1 shows a schematic optical layout of the developed NIR SHS. The emission from the plasma is collected and delivered through an optical fiber (1 mm core diameter, 0.22 NA). The collected light is collimated by a plano-convex lens (25.4 mm diameter, 50 mm focal length), exhibiting a beam diameter of approximately 22 mm and passing through a bandpass filter (1,870 nm central wavelength, 35 nm full width at half maximum (FWHM)). The collimated beam is bifurcated into two beams using a pellicle beamsplitter (45:55, 2 μm thickness). These beams are subsequently diffracted by two reflective gratings (600 grooves mm⁻¹, 28.68° blaze angle) configured at a Littrow angle (θ_L) of 34.19° for a Littrow wavelength of 1,873.0 nm. The diffracted beams are recombined and demagnified by an imaging lens system to form a wavelength-dependent crossover angle, thereby producing a Fizeau-type interferogram [22], which is detected by an InGaAs linear array (1 × 512 pixels, 25 × 250 μm pixel size, 15 bit, 30 pA dark current, 487 frames s⁻¹). The spectrum is obtained by applying a Fourier transform to the interferogram.

Fig. 1.  Optical layout of the developed near-infrared spatial heterodyne spectrometer (NIR SHS).

As the detector is a one-dimensional array, the two-dimensional interferogram is compressed along the vertical direction (y-axis in Fig. 1) to ensure that the entire light falls within the effective area of the detector. The imaging lens system comprises five cylindrical lenses whose spacing was optimized to minimize the vertical width and residual aberrations of the compressed interferogram. Lenses L1 (30 × 30 mm, 100 mm focal length) and L2 (30 × 30 mm, 50 mm focal length) are plano-convex cylindrical lenses configured as a telecentric pair to provide a twofold demagnification along the horizontal direction (x-axis in Fig. 1). The subsequent cylindrical lens group (L3) comprises a plano-convex cylindrical lens (25.4 mm diameter, 50 mm focal length), a plano-concave cylindrical lens (15 × 15 mm, 30 mm focal length), and an aspheric-cylindrical lens (12 mm diameter, 10 mm focal length). This combination yields a 175-fold demagnification along the vertical axis, reducing the interferogram height to 125 μm.

The wavelength resolution, as estimated from the FWHM of the instrumental function (described in Sec. 2.3), was 125 pm. The overall transmittance of the spectrometer was approximately 24%; it comprises the transmittances of the major optical components: bandpass filter (80%), pellicle beamsplitter (50%), and imaging lens system (60%). The geometrical etendue (2.8 × 10⁻⁸ m² sr) was obtained by multiplying the fiber etendue (1.2 × 10⁻⁷ m² sr) by the total transmittance.

2.2  Numerical model of the interferogram

Since vertical compression modulates the envelope and reduces the contrast ratio of the interferogram, a numerical model of the interferogram was developed to evaluate these effects.

The model was based on physical optics propagation (POP) simulations implemented in ZEMAX OpticStudio [23]. As POP simulations account for diffraction and interference effects associated with finite apertures, wavefront aberrations, and coherent light propagation, it is widely applied to simulate various interferometers (e.g., the Fabry–Pérot and Michelson interferometers) [24]. Unlike geometric ray tracing [23]. POP computes the evolution of the complex wavefront, thereby retaining the phase information. The wavefront is represented on discrete transverse planes, and the corresponding amplitude and phase relationships between these planes are computed using the Fresnel scalar diffraction theory or angular spectrum methods [22]. The detailed calculation procedure is described in Appendix A.

Figure 2 shows the calculated interferograms of the Paα emission lines for H and D atoms, assuming an intensity ratio of $I_{\mathrm{H}}/I_{\mathrm{D}}$ = 0.38, where $I_{\mathrm{H}}$ and $I_{\mathrm{D}}$ are the respective Paα peak intensities. The model-calculated interferogram is shown as the solid curve in Fig. 2(a). For comparison, Fig. 2(b) presents the interferogram calculated using the analytical expressions (Eq. (A2)); its corresponding pixel-binned profile is plotted as the dashed curve in Fig. 2(a). The contrast ratio $V$ of the analytical interferogram (see Appendix B) is approximately 0.9; this reduction from unity is due to the presence of the two Paα emission lines of the H and D atoms. In the POP simulation, $V$ further decreases to approximately 0.6 owing to the residual aberrations introduced by the imaging lens system. The fringes are visible only within the 100–400 pixels range because the residual aberrations increase for off-axis rays [22]. Only the interferogram within this pixel range was considered in the subsequent discussions.

Fig. 2.  Calculated interferograms of the Paα emission lines for the H and D atoms (the peak-intensity ratio is assumed to be $I_{\mathrm{H}}/I_{\mathrm{D}}$ = 0.38) based on the SHS theory and numerical model. (a) Comparison between the model-calculated interferogram (solid curve) and the pixel-binned interferogram of (b) (dashed curve). (b) Uncompressed interferogram calculated from Eq. (A2).

2.3  Calibration and performance evaluation

We adjusted the interferogram acquisition conditions according to the measurement objectives. The absolute-intensity and wavelength-calibration data were recorded using an exposure time of 100 ms and averaged over 10 frames. The Paα data were acquired with an exposure time of 1 ms without averaging. The same data-processing procedure was subsequently applied to all interferograms: the DC component (defined in Eq. (A2)) was removed [18], after which the Fourier transform was applied after zero-padding by a factor of 16, where the total data length was 8,192 pixels [25].

Absolute intensity calibration was performed by measuring the spectrum of a home-calibrated stabilized continuum light source (Thorlabs SLS203L/M, 500–9,000 nm). The calibration coefficient for converting the detector count rate to radiance was determined to be approximately 1.6 × 10⁻³ (W m⁻² sr⁻¹ nm⁻¹)/(counts ms⁻¹). The wavelength was calibrated by measuring two Ne I emission lines at 1,859.15 and 1,859.77 nm from a Ne glow discharge tube. Figure 3 shows the measured Ne I spectrum with the model-calculated spectrum. In the calculation, the shapes of the two Ne I emission lines were modeled as delta functions, and their intensities were set to the measured values. The calculated spectrum exhibits a FWHM of approximately 125 pm and reproduces the measured spectrum, with the exception of the side peaks. The larger side peaks observed in the measured spectrum were attributed to the misalignment of the optical components [26]. Consequently, in the subsequent discussion, the measured line shape of the peak at 1,859.77 nm in the wavelength range between the blue dashed vertical lines (0.22 nm width) in Fig. 3 was adopted as the instrumental function. The effect of omitting the side peaks was evaluated subsequently in the fitting analysis by extending the wavelength width to 0.32 and 0.42 nm, which are indicated by the gray dotted vertical lines.

Fig. 3.  Measured and calculated spectra of two Ne I emission lines at 1,859.15 and 1,859.77 nm from a Ne glow discharge tube. The experimental line shape between the two blue vertical dashed lines is utilized as the instrumental function.

The capability for temporally resolved measurements of the Paα emission line was validated using a deuterium glow discharge tube. Owing to a relatively high neutral pressure (~20 Pa), the radiance of Paα was estimated to be approximately 1.2 W m⁻² sr⁻¹ nm⁻¹, which is comparable to that expected during observation of the ITER-divertor plasma, assuming an integration length of 5 cm [27]. The measured and modeled interferograms are shown in Fig. 4(a). The modeled interferogram corresponds to that shown in Fig. 2(a). The contrast ratio $V$ of the measured interferogram is approximately 0.58, demonstrating close agreement with the numerical prediction. The larger residuals observed at the left and right edges of the interferogram were attributed to a misalignment of the optical components that influenced the interferogram envelope.

Fig. 4.  (a) Measured and calculated interferograms of the Paα emission lines of the D and H atoms from a deuterium glow discharge tube. (b) Measured Paα emission line and the fitting result (solid curve) obtained by convolving the instrumental function (dashed curve) with Doppler broadening (short-dashed curve).

Figure 4(b) shows the measured Paα emission lines of the H and D atoms. The experimental spectrum was fitted using two Gaussian functions corresponding to the Doppler broadening of the H and D atoms, convolved with the instrumental function. The fitting parameters included the peak-intensity ratio, $I_{\mathrm{H}}/I_{\mathrm{D}}$, and the atomic temperature $T$ of the H and D atoms, assuming thermal equilibrium between both species. The measured $I_{\mathrm{D}}$ was used to calculate the peak intensity in units of counts. The Doppler shift was neglected, and the standard deviation of the baseline intensity, $n_{\mathrm{base}}$, was fixed at 24 counts, which was determined from the intensity in the spectral range between 1,877 and 1,880 nm (see Fig. B2 for the spectrum over a wider wavelength range). Under the present deuterium glow discharge plasma, the contribution of Doppler broadening to the FWHM of the fitted result is small compared with that of the instrumental function. For the D-Paα peak (Fig. 4(b)), the FWHM of the fitted result exceeded that of the instrumental function by approximately 6 pm. The enlarged views around the half-maximum region of the D-Paα peak highlight this minor discrepancy between the fitted result and the instrumental function. The fitting yielded $I_{\mathrm{H}}/I_{\mathrm{D}}$ of 0.38 ± 0.01 and $T$ of 0.15 ± 0.03 eV. This temperature is comparable to the typical atomic temperature (0.03–0.14 eV) reported for low-pressure DC hydrogen glow discharge plasmas (40–670 Pa gas pressure, 1–50 mA discharge current) [28]. Additionally, fitting was repeated using instrumental functions with the wider wavelength widths described in Fig. 3. The estimated $T$ values were 0.18 ± 0.03 and 0.16 ± 0.03 eV for the 0.32 and 0.42 nm widths, respectively. Since these values agree within the fitting uncertainty, we confirm that neglecting the side peaks of the instrumental function does not introduce a significant bias into the estimated $T$ in this analysis.

As interferogram compression reduces $V$, it also degrades the spectrum S/N. Here, we define S/N as $(I_{\mathrm{D}}+I_{\mathrm{H}})/n_{\mathrm{base}}$ (see Fig. B2). S/N degradation was evaluated at a fixed exposure time (1 ms) and across varied Paα radiances by measuring the interferograms using neutral density (ND) filters at approximately 1.2 (no ND filter), 0.6, 0.3, 0.18, 0.1, and 0.06 W m⁻² sr⁻¹ nm⁻¹. The measured S/N values are plotted in Fig. 5 as a function of the average interferogram intensity, $S$, defined as the pixel-averaged interferogram intensity before DC removal within the 100–400 pixel range. The measured $S$ values were approximately 1,500, 750, 400, 250, 150, and 80 counts.

Fig. 5.  Measured S/N of the Paα spectra as a function of the average intensity of the interferogram. A fitting curve to the measured S/N (solid curve), a reference S/N for an aberration-free, pixel-binned interferogram (dashed curve, $V$ = 0.9), and a reference S/N for the Zeeman-split Paα spectra in Sec. 3 (dotted curve, $V$ = 0.4) are shown.

To describe the dependence of the measured S/N on the contrast ratio $V$, the experimental values were fitted using the following expression [29]:

  

$$\mathrm{S}/\mathrm{N}=\frac{I_{\mathrm{D}}+I_{\mathrm{H}}}{n_{\mathrm{base}}}=\frac{V\cdot S}{\sqrt{V\cdot S+n_{\mathrm{d}}^2}},$$(1)

where $n_{\mathrm{d}}$ is the standard deviation of the intensity between 1,877 and 1,880 nm measured from the spectra under no-light input. The product of $V$ and $S$ represents the maximum intensity of the interferogram after DC removal, i.e., the sum of the H and D peak intensities. The denominator, $n_{\mathrm{base}}$, is expressed as the quadrature sum of the shot noise associated with the two peaks, $\sqrt{V\cdot S}$ and $n_{\mathrm{d}}$. Accordingly, $n_{\mathrm{d}}$ was fixed to 5 counts, obtained by averaging 10 measurements, and $V$ was treated as the fitting parameter. Although in practice, $V$ decreases with the S/N, we assumed $V$ to be a constant for simplicity. The fitted result yields $V$ = 0.55 ± 0.05.

To assess the extent of S/N degradation caused by aberrations from interferogram compression, a reference case was calculated. The reference assumes an aberration-free imaging lens system corresponding to the pixel-binned interferogram in Fig. 2(a) ($V$ = 0.9). Comparison with this reference curve reveals that the compression resulted in an approximately 30% reduction in S/N. Coupled with the $V$ = 0.9 reference curve, another calculated curve ($V$ = 0.4) was also included in Fig. 5, corresponding to the measurement of the Zeeman-split Paα spectra (see Sec. 3).

3.  Measurements of Paschen-α Emission Lines under Fusion-Relevant Magnetic Fields

Under an external magnetic field $\mathit{B}$, the Paα emission line exhibits Zeeman splitting into 32 π and 48 σ components due to the lifting of the magnetic-sublevel degeneracy. The superposition of the interferograms generated by these components degrades the contrast ratio $V$ of the total interferogram as individual interferograms exhibit different spatial frequencies (defined in Eq. (A1)). This may produce false peaks and change the π-to-σ intensity ratio in the spectrum. Therefore, we assess whether the NIR SHS can measure the Zeeman-split Paα spectrum under fusion-relevant magnetic field conditions. Additionally, we examined whether Doppler broadening can be evaluated from the width of the spectral line under such conditions.

Figure 6 shows the experimental setup. A custom-fabricated hydrogen glow discharge tube (3 mm inner diameter, 260 mm length) was mounted inside a superconducting magnet (Cryogenic 1721; 7 T maximum field strength). The generated magnetic field was oriented parallel to the discharge-tube axis. The magnetic field inhomogeneity was less than 0.25% over a 10 mm-diameter sphere, including the observation volume; the field was consequently treated as uniform. The DC hydrogen glow discharge was operated at a voltage of 1.4 kV, a current of 12 mA, and a neutral pressure of 20 Pa. Under these conditions, pressure broadening was ignored because it was negligible compared with the instrumental width (125 pm) [28]. Emission was directly collected in the radial direction by the optical fiber that was positioned approximately 2 mm from the discharge-tube outer surface. The viewing direction was oriented approximately perpendicular to the magnetic field. At the plasma center, the cross-sectional diameter of the viewing cone was sufficiently smaller than the plasma-column diameter.

Fig. 6.  Experimental setup for measuring the hydrogen Paα emission line with the Zeeman effect.

The magnetic field strength was set to $|\mathit{B}|$ = 0, 1, 2, and 3 T, which are representative of values observed in fusion edge plasmas [7–9]. Because of the specification of the discharge tube, the Paα radiance at $|\mathit{B}|$ = 0 T was 0.05 W m⁻² sr⁻¹ nm⁻¹ and approximately 1/25 of that expected for the ITER divertor plasma. At $|\mathit{B}|$ ≠ 0 T, the Paα radiance decreased to approximately 0.035 W m⁻² sr⁻¹ nm⁻¹, probably owing to enhanced radial diffusion [30]. A longer exposure time (100 ms) was applied without averaging to accurately estimate the contrast ratio $V$.

The experimental spectra are shown in Fig. 7. Based on five independent measurements recorded under each magnetic field condition, the measured $V$ at $|\mathit{B}|$ = 0, 1, 2, and 3 T were 0.56 ± 0.03, 0.42 ± 0.02, 0.39 ± 0.02, and 0.39 ± 0.03, respectively; with the corresponding spectral S/N being 52 ± 3, 35 ± 3, 29 ± 3, and 28 ± 2, respectively. The denominator of S/N is defined as the sum of the peak intensities of the π and σ components. At $|\mathit{B}|$ = 0 T, the measured $V$ and S/N were consistent with the fitted result in Fig. 5, extrapolated to $S$ of approximately 6,000 counts. Conversely, at $|\mathit{B}|$ ≠ 0 T, $V$ and S/N decreased to approximately 0.4 and 30, respectively. Furthermore, the reference S/N curve for $V$ = 0.4 (see Fig. 5) matches these experimental cases. These results infer that, for a Paα spectral radiance comparable to the value predicted for the ITER divertor region (1.2 W m⁻² sr⁻¹ nm⁻¹), the S/N of the Zeeman-split spectrum would be on the order of 20 for a 1 ms exposure time, evaluated from the intersection point of the reference S/N curve ($V$ = 0.4) and the vertical dash-dot line (representing the expected Paα spectral radiance in ITER divertor plasma) in Fig. 5.

Fig. 7.  Measured and fitted Zeeman-split Paα emission lines at $|\mathit{B}|$ = 0, 1, 2, and 3 T. The determined atomic temperature $T$ is indicated.

The experimental spectra were fitted using a sum of Gaussian functions representing the Doppler broadening of all π and σ components, convolved with the instrumental function. The wavelengths of these components were obtained by calculating the energy levels of the hydrogen atom using first-order perturbation theory [31]. The relative intensities of these components were obtained by calculating their spontaneous emission coefficients and assuming the statistical weights of the upper-state population densities. We also considered the effect of integrating the angular distribution of the emissivity over the fiber acceptance cone [8]. The obtained relative π-to-σ intensity ratio was 0.49/0.51. The Doppler shift was neglected, and $n_{\mathrm{base}}$ of each spectrum was evaluated using the method described in Sec. 2. The determined atomic temperatures, $T$, were 0.12 ± 0.01, 0.14 ± 0.01, 0.16 ± 0.02, and 0.17 ± 0.02 eV for $|\mathit{B}|$ = 0, 1, 2, and 3 T, respectively. Large fitting residuals were observed in the wavelength range between the π and σ− components, appearing as small peaks adjacent to the latter. This behavior was attributed to the fact that as the wavelength separation from the Littrow wavelength (1,873.0 nm) increased, the interferogram spatial frequency also increased, thereby decreasing the interference fringe spacing. Such high-frequency interferograms are expected to exhibit greater sensitivity to misalignment-induced distortions [26]. This localized degradation of the fringe envelope ultimately introduced false peaks into the Fourier-transformed spectrum [32].

4.  Conclusion

We developed an NIR SHS for the temporally resolved measurements of the Zeeman effect on the hydrogen Paα emission lines. Employing an InGaAs linear array detector and a custom cylindrical imaging lens system, the two-dimensional interferogram was compressed to ensure that the entire light falls within the effective area of the detector while maintaining a high wavelength resolution.

A numerical model incorporating the SHS theory and POP simulation was constructed to quantitatively evaluate the compression impact on the wavelength resolution, contrast ratio $V$, and the spectral S/N. The calculated wavelength resolution was 125 pm, which correlated with the instrumental function measured from the Ne I emission line. The modeled Paα emission line interferogram correlated with the experimental one in a glow discharge tube at zero magnetic field, exhibiting an $V$ of approximately 0.58. This result indicates that compression decreases $V$ by 30%, leading to a 30% degradation in the spectral S/N.

Temporally resolved measurement of Paα emission line spectra under external magnetic fields up to 3 T demonstrated that spectra affected by Zeeman splitting can be reproduced using fitting functions that incorporate the instrumental function and Doppler broadening. Owing to $V$ degradation to approximately 0.4, an S/N of approximately 20 could be achieved for a spectral radiance comparable to that expected in the ITER divertor region (1.2 W m⁻² sr⁻¹ nm⁻¹) at an exposure time of 1 ms.

 Appendix

A.  Spatial heterodyne spectrometer theory and calculation steps of the numerical model

To elucidate the calculation steps of the numerical model, we first briefly introduce the SHS theory for calculating the interferogram. We define the xyz-coordinate system, as illustrated in Fig. 1. Employing the plane-wave approximation, as well as assuming monochromatic light incidence, the diffracted wavefront, ${\mathit{E}}_m$ (m = 1, 2), generated by a light ray emitted from a given point on the end-surface of the fiber core is given as follows [17]:

  

$${\mathit{E}}_m=\frac{1}{2}\left|{\mathit{E}}_0\right|\exp \left[i(k_{xm}x+k_{ym}y)\right],$$(A1)

where ${\mathit{E}}_0$ is the wavefront of the collimated light before the pellicle-beam splitter, $k_{xm}=\pm 4\pi \tan {\theta }_{\mathrm{L}}({\lambda }_{\mathrm{L}}-\lambda +{\alpha }^2{\lambda }_{\mathrm{L}}/2)/\lambda {\lambda }_{\mathrm{L}}$ and $k_{ym}=2\pi \sin {\varphi }_m/\lambda$ are the spatial frequencies of ${\mathit{E}}_m$. The plus and minus signs apply to ${k_x}_1$ and ${k_x}_2$, respectively; $\lambda$ and ${\lambda }_{\mathrm{L}}$ are the input and Littrow wavelengths, respectively; $\alpha$ is the angle between the light ray and optical axis (z-axis); and ${\varphi }_m$ is the projection of $\alpha$ onto the yz-plane. For on- and off-axis rays, $\alpha$ = ${\varphi }_m$ = 0, and ${\varphi }_1$ = ${\varphi }_2$ ≠ 0, $\alpha$ ≠ 0, respectively, where $\alpha$ exhibits a range within (0, $d_f/2f$], with $d_f$ being the diameter of the optical fiber core and $f$ the focal length of the collimator.

By neglecting the aberrations introduced by the optical components, the interferogram formed by all rays before the vertical compression by L3 is expressed as follows [17]:

  

$$\begin{array}{cc}I\left(x\right)\hfill & =2\pi {\int }_0^{d_{\mathrm{f}}/2f}{|{\mathit{E}}_1+{\mathit{E}}_2|}^2\alpha d\alpha \hfill \\ \hfill & =\frac{\pi d_{\mathrm{f}}^2}{4f^2}{\left|{\mathit{E}}_0\right|}^2\left\{\begin{array}{c}1+\mathrm{sinc}\left(\frac{\tan {\theta }_{\mathrm{L}}{d}_{\mathrm{f}}^2}{2\lambda f^2}x\right)\hfill \\ \times \cos [8\mathrm{\pi }\tan {\theta }_{\mathrm{L}}(\frac{1}{{\lambda }_{\mathrm{L}}}-\frac{1}{\lambda }-\frac{d_{\mathrm{f}}^2}{16\lambda f^2})x]\hfill \end{array}\right\},\hfill \end{array}$$(A2)

where the term before the plus sign represents the DC component, while sinc() is a normalized sinc function.

Regarding the numerical modeling procedure, since the phase modulation of a diffraction grating is not incorporated by the POP simulation, we separately applied the SHS theory to calculate ${\mathit{E}}_m$. Subsequently, we employed POP to propagate these phase-modulated wavefronts through the imaging lens system. First, Eq. (A1) was employed to calculate ${\mathit{E}}_m$ at the grating position using a plane-wave approximation. Thereafter, POP simulation was applied to compute the transfer function $\mathit{H}$ of the imaging lens system [22]. Finally, $\mathit{H}$ was used to calculate the wavefront propagating from the grating position to the sensor plane. The modeling employed the single-wavelength approximation at 1,875 nm, selecting POP sampling numbers as 2,048 to satisfy the Nyquist sampling requirement.

The specific computational steps are as follows:

(1) Calculate ${\mathit{E}}_m$ from Eq. (A1).

(2) As illustrated in Fig. A1, a sequence of imaging lens surfaces and the sensor plane was modeled in ZEMAX’s sequence mode (Ansys Zemax OpticStudio 2025, student version).

Fig. A1.  Sequence of imaging lens surfaces and the sensor plane in ZEMAX’s sequence mode.

(3) Set a monochromatic plane wave with unit intensity and zero phase at the grating position, denoted as ${\mathit{E}}_{\mathrm{in}}$.

(4) Propagate ${\mathit{E}}_{\mathrm{in}}$ through each lens surface to the sensor plane via POP simulation, with the propagated wavefront denoted as ${\mathit{E}}_{\mathrm{out}}$.

(5) Calculate the transfer function $\mathit{H}$ using the following equation:

  

$${\mathit{E}}_{\mathrm{out}}=F^{-1}\left\{\mathit{H}F\left\{{\mathit{E}}_{\mathrm{in}}\right\}\right\},$$(A3)

where $F\{\}$ and $F^{-1}\{\}$ denote the Fourier and inverse Fourier transforms, respectively.

(6) Calculate the wavefront ${\mathit{E}}_m^{\prime }$ on the sensor plane using Eq. (A3) by replacing ${\mathit{E}}_{\mathrm{in}}$ with ${\mathit{E}}_m$. The interferogram in Fig. 2(a) is obtained as $I^{\prime }\left(x\right)={|{\mathit{E}}_1^{\prime }+{\mathit{E}}_2^{\prime }|}^2$.

B.  Contrast ratio calculations

As illustrated in Fig. B1, we employed the measured interferogram (dotted curve) in Fig. 4(a) to elucidate the calculation of the contrast ratio, $V$, defined as follows:

Fig. B1.  Measured and calculated interferograms of the Paα emission lines of the D and H atoms from a deuterium glow discharge tube (same as Fig. 4(a)). The DC components and interference fringes are separated by frequency-domain filtering.

  

$$V=\frac{I_{max}-I_{min}}{I_{max}+I_{min}}=\frac{\left|I_{\mathrm{fringes}}\right|}{\left|I_{\mathrm{DC}}\right|},$$(B1)

where $I_{\max }$ and $I_{\min }$ are the maximum and minimum interferogram intensities, respectively, at the central pixel position, i.e., the position of $x$ = 0 in Eq. (A2); $I_{\text{fringes}}$ is the maximum intensity of the DC-removed interference fringes, which is equal to $I_{\mathrm{D}}+I_{\mathrm{H}}$ (Fig. B2); and $I_{\mathrm{DC}}$ is the averaged intensity of the DC components within the 100–400 pixel range.

Fig. B2.  Measured Paα emission lines for the D and H atoms from a deuterium glow discharge tube.

 Acknowledgment

This work was supported in part by JSPS KAKENHI (Grant No. 24K00604) and the NIFS Collaborative Research Program (Grant No. NIFS25KSPM001). The authors thank all co-workers in the research group for their assistance.

References

• [1]   A. Jansen van  Vuuren et al., Nucl. Fusion 61, 046001 (2021).
• [2]   Z.  Yan et al., Phys. Rev. Lett. 112, 125002 (2014).
• [3]   A.V.  Chankin et al., J. Nucl. Mater. 313–316, 828 (2003).
• [4]   S.R.  Haskey et al., Rev. Sci. Instrum. 93, 103504 (2022).
• [5]   B.A.  Grierson et al., Rev. Sci. Instrum. 87, 11E545 (2016).
• [6]   K.  Fujii et al., Nucl. Fusion 55, 063029 (2015).
• [7]   D.K.  Gupta et al., Rev. Sci. Instrum. 83, 10D534 (2012).
• [8]   P.G.  Carolan et al., Plasma Phys. Control. Fusion 27, 1101 (1985).
• [9]   J.L.  Weaver et al., Rev. Sci. Instrum. 71, 1664 (2000).
• [10]   T.  Shikama et al., Phys. Plasmas 11, 4701 (2004).
• [11]   M.  Murakumo et al., Rev. Sci. Instrum. 96, 043501 (2025).
• [12]   T.  Shikama, Rev. Mod. Plasma Phys. 6, 39 (2022).
• [13]   M.  Shah, Introduction to Scientific InGaAs FPA Cameras (Technical Note, Princeton Instruments, 2024).
• [14]   A.  Rogalski, Infrared Detectors, 2nd ed. (CRC Press, Boca Raton, FL, 2010).
• [15]   J.M.  Harlander et al., Astrophys. J. 396, 730 (1992).
• [16]   J.M.  Harlander, Appl. Opt. 62, 4288 (2023).
• [17]   M.  Xu et al., Nucl. Mater. Energy 42, 101850 (2025).
• [18]   M.G.  Burke et al., Rev. Sci. Instrum. 94, 033504 (2023).
• [19]   D.  Sheftman et al., Rev. Sci. Instrum. 92, 043541 (2021).
• [20]   Q.  Wang et al., Opt. Express 32, 22067 (2024).
• [21]   Q.  Wang et al., Opt. Commun. 507, 127595 (2022).
• [22]   E.  Hecht, Optics, 5th ed. (Pearson Education, Harlow, 2017).
• [23]   M.  Naeem and  T.  Imran, Opt. Contin. 4, 75 (2025).
• [24]   M.  Naeem and  T.  Imran, Photonics 12, 206 (2025).
• [25]   J.O.  Smith III, Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications, 2nd ed. (W3K Publishing, 2007).
• [26]   J.  Kauppinen and  P.  Saarinen, Appl. Opt. 31, 69 (1992).
• [27]   T.  Sugie et al., Rev. Sci. Instrum. 70, 351 (1999).
• [28]   J.  Amorim et al., Appl. Phys. 82, 2795 (1997).
• [29]   J.J.  Davenport et al., Appl. Opt. 54, 8135 (2015).
• [30]   F.F.  Chen, Introduction to Plasma Physics and Controlled Fusion, 3rd ed. (Springer, Cham, 2016).
• [31]   T.  Fujimoto and  A.  Iwamae, Plasma Polarization Spectroscopy (Springer, Berlin, 2008).
• [32]   L.  Wawrzyniuk, Appl. Opt. 60, 8792 (2021).