1. Introduction

The spatial heterodyne spectrometer (SHS) is a type of Fourier transform spectrometer (FTS) that is widely used in applications including atmospheric observations [1–4], mineralogical analysis [5–7], astrophysics [8–10], and chemical composition identification [11–13]. The SHS operates under the diffraction limit of the grating to produce a wavenumber-dependent Fizeau fringe pattern, which can then be recorded on a charge-coupled device (CCD) detector [14]. The spectral resolution of the SHS is determined by the smallest shift in the wavenumber (or wavelength) that causes a change of one half of a fringe period in the fringe pattern on the CCD. In a dispersive spectrometer, the spectral resolution is directly proportional to the slit width at the focal length [15–17]. Therefore, when compared with dispersive spectrometers of the same approximate volume, the advantage of the SHS is that it can acquire high resolution measurements while also ensuring high throughput and a large acceptance angle. For remote sensing applications such as interstellar emission line studies, an SHS with a large field-of-view (FOV) is much easier to align than a dispersive spectrometer.

However, because of the multiplexing disadvantage of the SHS, the throughput advantage of the SHS does not usually lead to increased sensitivity. As both a Fourier transform spectrometer and a multiplex spectrometer, the available observation area for the SHS on the CCD detector must be shared among all the spectral elements. This means that when the broadband spectrum enters the CCD, the interferograms corresponding to different spectral regions will interfere with each other; then, if there is light with a sufficiently high brightness in a certain wavelength and this light intensity exceeds the saturation threshold of the CCD pixels, the resulting oversaturation will interfere with the spectral restoration from a spatial Fourier transform of the recorded interferograms. Simultaneously, for a spectrum that contains N_re resolution elements, the shot noise on the CCD is generated in proportion to the square root of N_re [18], and the simultaneous observation of all resolution elements raises the signal level by a factor of N_re for a uniform spectrum [19]. The increased noise and the raised signal level will reduce the signal-to-noise ratio (SNR), thus limiting the detection of weak spectral features and undermining the Jacquinot (throughput) advantage by the ratio of the square roots of the throughputs. With the performance of the CCD detector having reached its limit, more efficient use of the existing sensitivity and quantum efficiency characteristics of the CCD represents the only way to achieve any significant increase in performance [20,21]. Typically, this is achieved using a combination of a short-pass filter, a razor-edge long-pass filter [22–25], and a notch filter [8,22–26] to suppress the parts of the spectrum that are not related to the band to be measured. Although these SHS instrument designs have limited the shot noise and solved the problem of spectral interferogram aliasing to a certain extent, they have undermined both the Jacquinot advantage and the wavelength coverage by limiting the use of the instruments to only a narrow wavelength band. Other methods have been proposed to solve the multiplexing disadvantage of SHS instruments, with particular focus on replacement of conventional gratings with higher order echelle gratings [24,27–29]. This type of system requires a rotating mask to select the specific orders of the two echelle gratings to eliminate shadows and ghosts, and two CCD frames are required to construct a complete spectrum. However, the shadows and ghosts can still be observed. Qiu et al. modified their echelle grating SHS by replacing the echelle grating in one arm with a mirror [22]. Although this system can acquire signals over a broad spectral range without the requirement for a mixture of the different orders, the resolving power of the SHS was reduced by half and it lost its resolution advantage. Lamsal et al. modified their SHS using a custom plate beam splitter (BS) that provided higher light transmission than the conventional cube BS [30]. Hosseini et al. developed a “tunable” cyclical SHS with a narrow bandpass to shift between the target spectral features [31]. The data processing methods used for spectral restoration have also been improved, e.g., by extending the processing methods of filling zeroes and flat fielding [32,33], and using row-by-row Fourier transforms [34,35], which can improve the quality of the recovered Raman spectrum when obtaining interference data with a lower SNR. The methods described above only reduce the number of multiplexed spectral bands or perform compensation processing on the interference data obtained, but do not fundamentally solve the problem of the effects of spectral multiplexing.

This paper reports a new spectrometer variant called the cross-dispersed spatial heterodyne spectrometer (CDSHS), which pairs a reflection grating and a cylindrical lens in series with an SHS. The CDSHS can be designed to record specific narrow wavelength spectral bands using different given pixel rows on the CCD and thus can attain a higher SNR in broadband spectrum measurements while also eliminating the multiplexing disadvantage of traditional SHS instruments. In this paper, we provide a detailed description of the principle of the CDSHS, after which the theoretical design is presented, and numerical simulations are performed using an example for the spectral range from 620 nm to 700 nm. The simulation results are then shown as interferograms and detailed spectra, and our conclusions are finally presented.

2. Principle

Figure 1 illustrates the optical layout of the CDSHS, which integrates an SHS, a reflection grating G0, and a cylindrical lens. Figure 2 and Fig. 3 show the equivalent light path diagrams of the spectrometer in the meridian plane and the sagittal plane, respectively. The SHS consists of a single beam splitter and two gratings, designated G1 and G2, in a Littrow configuration. The cylindrical lens is set after the SHS, and the two-dimensional detector is located at a distance on the focal plane of the cylindrical lens. The collimated beams are dispersed in the longitudinal direction (in the y-axis direction) by grating G0 before the SHS, and are then divided into two groups of beams by the beam splitter. After they are diffracted by gratings G1 and G2 on the two light arms of the SHS, the two groups of beams are then dispersed in the lateral direction (i.e., in the x-axis direction) and leave the beam splitter. The beams are then collimated in the longitudinal direction by the cylindrical lens, but are not affected in the lateral direction. Finally, the beams are received by the two-dimensional detector as independent interferograms corresponding to narrow spectral regions. The optical path difference (OPD) of the interferograms is created by gratings G0, G1, and G2. Because the incident light rays are dispersed in two perpendicular directions, the proposed spectrometer is named the cross-dispersed SHS or CDSHS.

 

Fig. 1. System layout of the CDSHS with Littrow wavenumbers (blue) and non-Littrow wavenumbers (red).

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Fig. 2. Equivalent light path diagram of the spectrometer in the meridian plane. CL: cylindrical lens; F: focal plane of the cylindrical lens; D: detector.

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Fig. 3. Equivalent light path diagram of the spectrometer in the sagittal plane.

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The CDSHS combines two sets of reflection gratings with dispersion directions that lie perpendicular to each other, thus eliminating the disadvantages of the multiplex spectrometer. The formulation for the multiplex gain was summarized previously in the literature [19,36–38] as follows:

In Fig. 2 and Fig. 3, typical incident light rays in the meridian plane and in the sagittal plane of the CDSHS are drawn. The following mathematical derivation will determine the wavenumber range, the spectral resolution, the optical element characteristics and positions. These light rays are first diffracted by grating G0, and it can thus be shown that:

After the first-order diffraction from the grating G0, the angle of incidence of the SHS in the meridian plane can be expressed as:

Obviously, the angle of incidence φ_in is related to the wavenumber σ. Here, we let the optical axis in the meridian plane overlap with the light rays at the zero angle of incidence. From Eq. (2), we find that when the wavenumber σ₀ can satisfy the condition φ_in(σ₀) 0, the angle of incidence φ_in(σ) will take φ_in(σ₀) as the origin, and σ₀ can be given by:

After being divided in the meridian plane, the light rays then travel to gratings G1 and G2. The grating equations of the SHS are given by [14]:

Substitution of β_in = 0 into Eq. (5) and solving the equation in the grating G1 light arm for β_1out and φ_1out, and then expanding Eq. (5) to the second order in terms of angles and the first order in terms of (σ − σ_L)/σ gives:

For the angles of the outgoing rays from grating G2’s light arm, substitution of β_2out = −β_1out and φ_2out = −φ_1out into Eq. (6) gives:

With regard to the second order of the angles, the phase of the SHS can be written as δ ≈ 2πσ (β_1out − β_2out) x. Using Eq. (6) and Eq. (7), the phase can then be expressed as:

When the light rays propagate to gratings G1 and G2, the position of the center ray in the y-axis direction and the height of the light ray h_G can be expressed as:

After being diffracted by the SHS’s gratings, the outgoing rays from the SHS in the meridian plane can be seen to maintain the same angle as the incoming rays, and these light rays are then converged by the cylindrical lens. Therefore, the position of the center in the y-axis direction y_CL and the height of the outgoing ray on the cylindrical lens h_CL are given by:

After they are converged by the cylindrical lens, the light rays travel to the detector. Then, according to the lens characteristics and the geometry, it can be shown that:

As shown above, the height and the position of the light rays received at the detector plane are dependent on the wavenumber σ of the incident light. The height h(σ) is nearly z/f times the height of the incident light rays and is slightly higher for a larger wavenumber. Therefore, light rays with different wavenumbers can be separated on the detector plane.

Because the source spectrum covers a range from σ_min to σ_max, the height of each column (in the y-axis direction) of the detector must be greater than |y(σ_max) − y(σ_min)|, which means that the height of the detector determines the spectral range of the instrument. Let N denote the number of pixels in each column of the detector, and let a denote the size of each pixel in the detector. Therefore, to obtain the separated interferograms simultaneously, the following requirements can be found:

In practice, when the interference of the detector’s noise with the image data and the calibration of the optical path are considered, the height of each interferogram should be higher. From Eq. (17), we noted that selection of the appropriate values of z and f will allow us to achieve a minimum height for the interferogram that is no less than n times the pixel size. Therefore, Eq. (18) and Eq. (19) become:

At the same time, we also would like the utilization of the detector to be as high as possible, and in the extreme case, we would like the interferograms to simply fill the detector. As Fig. 4(b) shows, in this case, the interferograms of σ_min and σ_max are symmetrical about the x-axis and at the edge of the detector, which means that the positions of the center ray for σ_min and σ_max must satisfy the condition y(σ_max) + y(σ_min) 0, and the equation will determine the parameter of the grating G0. Substitution of Eq. (3) into Eq. (16) allows this condition to be expressed as:

 

Fig. 4. (a) Schematic diagram of the interference of two beams composed of light rays on the detector in the sagittal plane. (b) Interferogram of σ_min and σ_max on the detector

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Equation (4) and Eq. (22) can help us determine the values of the grating deflection α₁ and the incline angle of the incident light α₂ when the spectral range and the grating groove density d₀ are confirmed.

In Fig. 3, typical incident light rays with width w_L are drawn in the sagittal plane. According to the geometry shown, the center ray positions with respect to gratings G1 and G2 in the light arms in the x-axis direction x_1,2 and the widths of the light beams at the detector plane w_1,2 can be written as:

By substituting Eq. (6) and Eq. (7) into Eqs. (23)–(26), we find that:

From Eq. (27) and Eq. (28), we see that the diffracted light rays from gratings G1 and G2 spread symmetrically, with the optical axis (z-axis) as the axis of symmetry, and the widths of these light rays are equal. Therefore, as we can see in Fig. 4(a), the width of the interference area for each interferogram at the detector plane can be expressed as:

This is an inherent problem in the traditional SHS, because the absolute value of the angle of the outgoing rays |β_1,2out| increases as the wavenumber moves farther away from the Littrow wavenumber. Because we separated the interferograms using different wavenumbers, this issue will become clearer. To obtain interferograms that contain sufficient data for the analysis, we require the interference area for each wavenumber in each row to be as large as possible. From Eqs. (27)–(29), we see that when the Littrow wavenumber and angle are determined, the interference area width can be extended by enlarging the width of the incident light beam w_L and reducing the distance from the grating in each light arm to the detector plane (l₂ + f + z). Let M denote the number of pixels in each row (in the x-axis direction) of the detector; then, at least the width of the interference area should satisfy the condition w_A(σ) ≥ Ma/m, where 1/m is the proportion of the interference area width with respect to the width of each row. By a process of substitution using Eqs. (23)–(29), the position in the x-axis direction and the width of the light beam at the detector plane for σ_min and σ_max must satisfy the following conditions:

The intensity distribution produced by the spectrometer can be given by:

By adopting the definition for the resolution limit, the spectral resolution δ_σ and the resolution power R in each wavenumber of the CDSHS are given by:

From Eq. (33) and Eq. (34), the spectral range that can be detected is determined by:

Therefore, the number of interferograms that can be received on the detector is equal to Δ_σ/δ_σ = M/2, which means that the number of pixels in each row of the detector determines the number of the interferograms contained in the CDSHS.

Since the distances among the optical elements in the optical path are limited by the positions and the sizes of the optical elements and detector, substitution of Eq. (3) into Eqs. (9)–(17) and Eqs. (20)–(31) indicates that the distances l₁ and l₂, f, and z must satisfy the following conditions:

In the experiments, if we have determined the wavenumber range from σ_min to σ_max and have also selected the optical elements, Eq. (37) can then help us to adjust the distances between each of the optical elements on the premise of ensuring that the CDSHS can obtain the high quality interferograms simultaneously according to our actual requirements.

3. Theoretical design and numerical simulation

Assume here that the light source spectrum covers the wavelength range from 620 nm to 700 nm, i.e., the wavenumber range from 14285.7 cm⁻¹ to 16129.0 cm⁻¹, and set the Littrow wavenumber σ_L to be equal to 16129.0 cm⁻¹; the Littrow angle of gratings G1 and G2 in the SHS can then be expressed as θ = sin⁻¹(2/dσ_L) 2.6652^°. The spectral range Δ_σ = σ_max − σ_min = 1843.3 cm⁻¹ must then be smaller than Mδ_σ/2. Because the number of pixels for the detector was selected to be 1024×1024, the spectral resolution δ_σ was limited by:

Substitution of this value into Eq. (36) allows the width of gratings G1 and G2 to be expressed as:

Therefore, we set the effective width W of gratings G1 and G2 to be 14.93 mm, and from Eq. (35), the height and width of the light beam can be determined using h_L = w_L = Wcosθ = 14.91 mm.

Table 1 shows the wavelength difference versus the wavenumber difference for several different wavelengths. If the desired spectral resolution in terms of the wavenumber is δ_σ = 3.6 cm⁻¹, then the designed spectral resolution in terms of the wavelength is around 0.14 nm at 620 nm and 0.18 nm at 700 nm.

Table 1. Wavelength Difference versus Wavenumber Difference for Several Wavelengths

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Figure 5 shows the relationship between the angle of incidence φ_in and the wavelength at different values of the incident light ray incline angle α₂, the deflection angle α₁ of grating G0, and the groove density 1/d₀ when y(σ_max) + y(σ_min) 0. We can obtain some useful information from these curves. First, a smaller grating groove density 1/d₀ will produce a smaller maximum value for the angle of incidence φ_in(σ). When combined with Eq. (16) and Fig. 2, the small angular separation means that a longer distance (l₁ + l₂) will be required to separate the interferograms, which will then adversely affect the compactness of the instrument. However, as Fig. 5(c) and 5(d) show, a larger value of 1/d₀ means that the angular separation will also be larger, while the curve changes will also become more uneven. At small wavelengths, φ_in(σ) changes slowly, but at larger wavelengths, this angle changes more sharply, causing insufficient interferogram separation; this will result in sparse utilization of the detector’s pixels at small wavelengths, but the pixels will be his will result in sparse utilization of the detector’s pixels at small wavelengths, but the pixels will be multiplexing to lose their ability to separate the interferograms at larger wavelengths. to lose their ability to separate the interferograms at larger wavelengths. Therefore, comprehensive consideration of the selection of the appropriate groove density 1/d₀, deflection α₁, and incline angle α₂ will be necessary.

 

Fig. 5. Relationships between angle of incidence φ_in(σ) and wavelength for different α₂ and α₁ when 1/d₀ is equal to (a) 150 mm⁻¹, (b) 300 mm⁻¹, (c) 450 mm⁻¹, and (d) 600 mm⁻¹.

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To obtain both compactness and even interferogram separation simultaneously, the groove density must be high and the zero point of the curve (1/σ₀, 0) must be near the center of the wavenumber range (1/σ_c, 0). Table 2 shows the key parameters for the components used in the CDSHS in this theoretical design. The center of the wavenumber range σ_c = 15151.5 cm⁻¹ (λ_c = 1/σ_c = 660 nm) and then σ₀ can be calculated using Eq. (4) to be σ₀ = 15084.9 cm⁻¹ (λ₀ = 1/σ₀ = 662.9 nm). From Eq. (16), we know that y(σ₀) 0, which means that the wavelength at the center of the detector λ₀ is 2.9 nm from the center wavelength of the spectrum range λ_c, thus proving that the interferograms are distributed approximately uniformly and symmetrically.

Table 2. Key Parameters of the Components Used in the Theoretical Design of the CDSHS

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Figure 6 shows the y-axis coordinates and widths of the interferograms for different wavelengths at the detector plane. As verified using Eq. (21) and Eq. (31), we know that the minimum height of the interferogram h(σ_min) 0.13303 mm > 10a = 0.13 mm, and the minimum width of the interference area of the interferogram w_A(σ_min) 7.2020 mm > 1024a/2 = 6.65 mm.

 

Fig. 6. (a) The y-axis coordinates and (b) widths of the interferograms for different wavelengths at the detector plane.

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Based on Eq. (32), several interferograms (e.g., interferograms 1, 128, 256, 384, and 512, as shown in Table 3) were generated by the CDSHS simultaneously, as shown in Fig. 7(a). Figure 7(b) shows the detailed spectrum obtained from the Fourier transform of the interferograms shown in Fig. 7(a). Because these interferograms are separated completely, the number of resolution elements N_re in Eq. (1) for each interferogram is equal to one and the mean intensity I_m(σ) is equal to the intensity I(σ) in each interferogram, thus allowing the multiplex gain for each spectral feature (each interferogram) to be written as G_CDSHS = [I(σ)/I_m(σ)]^1/2 = 1. In a traditional SHS, because of the overlapping of the interferograms, N_re = 9; if we assume that the intensity I(σ) of each wavenumber is the same, then the multiplex gain for each spectral feature (mixed in one interferogram) is G_SHS = [I(σ)/9I_m(σ)]^1/2 =1/3. If a weak spectral feature was present, the G_SHS of this feature would be more or less than 1/3, while the value of G_CDSHS would still be 1. Furthermore, we simulated the interferograms and the spectrum obtained with interference from high-intensity light with the results as shown in Fig. 8, which obviously confirms that the high-intensity light rays at 680 nm do not interfere with the interferogram inversion at the other wavelengths. In the traditional SHS, the high-intensity light interference would mean that the spectrum for all wavelengths could not be restored.

 

Fig. 7. (a) Multiple interferograms generated simultaneously by the CDSHS. (b) Detailed spectrum obtained from the Fourier transform of these interferograms.

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Fig. 8. (a) Multiple interferograms generated simultaneously by the CDSHS in the presence of high-intensity light interference. (b) Detailed spectrum obtained from the Fourier transform of these interferograms.

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Table 3. Parameters of Selected Interferograms on the Detector in the Theoretical Design

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At the beginning of the numerical simulation, we have demonstrated the spectral resolution around 620 nm, 660 nm, and 700nm in Table 1 based on Eq. (39). To provide a better illustration of the spectral resolution, the interferograms and the detailed spectra obtained for the different wavelengths in Table 1 are shown in Fig. 9. Based on Eq. (32), three interferograms were generated by the CDSHS simultaneously, as shown in Fig. 9(a), 9(c) and 9(e). Figure 9(a) shows the interferogram contain only the wavelengths 620 nm, 620.1385 nm, and 620.2770 nm. Figure 9(c) shows the interferogram contain only the wavelengths 659.9396 nm, 660.0965 nm, and 660.2535 nm. Figure 9(e) shows the interferogram contain only the wavelengths 699.6905 nm, 699.8668 nm, and 700.0433 nm. Figure 9(b), 9(d) and 9(f) show the detailed spectra obtained from the Fourier transform of the interferograms shown in Fig. 9(a), 9(c) and 9(e), respectively.

 

Fig. 9. Interferograms and detailed spectra obtained from the Fourier transform of the interferograms for the wavelengths in Table 1: (a) interferogram and (b) detailed spectrum contain only the wavelength of 620 nm, 620.1385 nm, and 620.2770 nm; (c) interferogram and (d) detailed spectrum contain only the wavelength of 659.9396 nm, 660.0965 nm, and 660.2535 nm; (e) interferogram and (f) detailed spectrum contain only the wavelength of 699.6905 nm, 699.8668 nm, and 700.0433 nm.

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In the obtained spectrum shown in Fig. 9(b), the three peaks in order from left to right correspond to the spectra of the wavelengths of 620 nm, 620.1385 nm, and 620.2770 nm, and the wavelength difference of two adjacent peaks is around 0.14 nm. In the obtained spectrum shown in Fig. 9(d), the three peaks in order from left to right correspond to the spectra of the wavelengths of 659.9396 nm, 660.0965 nm, and 660.2535 nm, and the wavelength difference of two adjacent peaks is around 0.16 nm. In the obtained spectrum shown in Fig. 9(f), the three peaks in order from left to right correspond to the spectra of the wavelengths of 699.6905 nm, 699.8668 nm, and 700.0433 nm, and the wavelength difference of two adjacent peaks is around 0.18 nm. All nine spectral peaks are high and are clearly visible. It can be seen from these results that the spectral resolution of the CDSHS is around 0.14 nm at 620 nm, 0.16 nm at 660 nm, and 0.18 nm at 700 nm.

4. Conclusion

The cross-dispersed SHS (CDSHS) was proposed in this work and a theoretical CDSHS design was presented with numerical simulations of the device. The CDSHS solves the problems of overlapping and mutual interference of the interferograms that occurs in the traditional SHS by recording the interferograms corresponding to different narrow spectral regions using different rows on the CCD, and the equations for the width, height, and location of each of these cross-dispersed interferograms were derived.

An example of the CDSHS is provided in which the spectral resolution is 0.139 nm at 620 nm and 0.176 nm at 700 nm. A uniform interferogram distribution can be obtained by analyzing the curves and selecting appropriate parameters for the key components, and appropriate image sizes can be obtained by adjusting the optical element spacing via the design equations. The simulation results show that the designed CDSHS can record interferograms corresponding to different narrow spectral regions simultaneously, and that the spectral features are clear in each detailed spectrum. Additionally, it is shown that high-intensity light rays at a specific wavelength of the CDSHS do not interfere with the detailed spectra of the other wavelengths. Because the interferograms are separated completely, the CDSHS has the ability to attain a higher SNR than the conventional SHS. At the same time, the CDSHS offers advantages that include high resolution, high throughput, physical compactness, and zero moving parts. The CDSHS will be suitable for measurement of broadband, multi-spectral features and weak spectral features in a wide range of potential applications, and it also has a prospect for developing future multi-bandpass or tunable SHS instruments.