The spatial heterodyne spectrometer (SHS) was proposed by T. Dohi and T. Susuki in the 1970s [1]. In 1991, J. M. Harlander in his PhD thesis clarified the theory of SHS in detail and constructed the corresponding SHS model, which made the Fourier transform technology with high spectral resolution develop rapidly [2]. Compared with traditional Fourier transform spectrometers [3], a Michelson interferometer [4] and a Fabry Perot interferometer [5], the SHS without moving parts has low requirements for component manufacturing technology, and has advantages of high-throughput, the same as a Fourier transform spectrometer. Therefore, the SHS is widely used in the detection of trace elements in atmosphere [6], the vapor in the upper atmosphere [7], and the identification of other weak targets in laboratory astrophysics observations [8, 9]. There are many contradictions in SHS. Firstly, it has a contradiction between the limit of resolution and the spectral range. Secondly, it has a contradiction between the spectral resolution and the limited numbers of sampling points. Thirdly, it has a contradiction among the diffraction efficiency of the grating, the limit of the detector’s resolution, the detection performance and the detection accuracy of the instruments, which limits the application domain of SHS greatly. So it is very important to explore a high performance and high accuracy of the continuous wide spectral spatial heterodyne technology to improve the efficiency of the instruments.

At present, research on the spectral broadening of SHS is in full swing. There are two main schemes: First one is to use the technology of common-path SHS, namely using blazed grating as the dispersion and beam split optical elements at the same time, and using a plane mirror and a roof mirror to realize common-path system structure. In order to achieve the purpose of spread spectrum, this system controls the rotation angle of a mirror to scan the center wave numbers of the system gradually [10]. Another way is to use an echelle grating to replace the blazed grating of the SHS system, which uses the characteristics of the echelle grating, such as low groove density, large blaze angle, and high diffraction efficiency, that can realize multilevels differential interference, and obtain the wide spectral SHS (WS-SHS) system [11]. The advantage of the first scheme is that the interferometer system is composed of reflective elements, which makes the system avoid the dispersion spectrum correction problems. However there are moving parts, which require very strict precision control technology and stability for the system. The second scheme proposed by J. M. Harlander used multistage diffraction to realize the spectral broadening. It overcomes the problems in the moving parts of the first scheme, but the system has the effect of chromatism and band interference between orders. Considering these schemes proposed above, although they can solve the problems of narrow spectral range of SHS effectively, both of them have not considered the influence of grating diffraction efficiency, limited detector resolution and spectral continuity of the system.

The paper is arranged as follows: Firstly, the characteristics of SHS and WS-SHS systems are introduced briefly. Then the theoretical relationship among resolution, spectral range, diffraction efficiency and detector resolution are analyzed through the perspective of design of the system. Finally, a kind of alternative continuous WS-SHS (ACWS-SHS) system is put forward. Simulation results show that not only the resolution limit, spectral range and diffraction efficiency, but also the interference fringes contrast ratio is coincident with the theoretical design value. And corresponding relations among diffraction stages of the system are discussed, which can provide a reference for the application of SHS.

2.1.1. Structure of Spatial Heterodyne Spectrometer System

The structure of the SHS system (Fig. 1(a)) is mainly composed of three parts: the collimating system, the interferometer and the imaging system. The role of the collimating system is to make the source light passing through the collimating lens group into parallel light projected onto the interferometer. The interferometer is mainly composed of optical components such as a beam splitter and gratings. Its main purpose is to divide the collimated light into two coherent lights by beam splitter and grating diffraction, and to produce interference fringes. The role of the grating is to make the light with different wavelengths disperse in the space, so that in the absence of a mobile device the interferometer can produce an optical path difference. The function of the imaging system is to obtain the interference fringes imaging in the detector (CCD).

2.1.2. Theory of Spatial Heterodyne Spectrometer

Diffraction grating G1 and G2 take the place of the two planar reflected mirrors of the traditional Michelson interferometer in the SHS. The light beam that has gone through the collimating system is incident on the beam splitter, and the one reflected by the splitter is incident on grating G1, returns after G1 diffraction and goes down through the beam splitter; the other light beam passing through the splitter is incident on grating G2, returns to the beam splitter after G2 diffraction and is reflected downward. Then the two beams of light produce localized interference fringes on the detector by use of the imaging system. The input light source spectrum curve can be obtained by recording the interference fringes and using certain algorithms.

As shown in Fig. 1, axis y is along grooves of the gratings, axis z is the optical axis. The surface of emergent waves of different frequencies from the grating have small angle ± γ with the optical axis, the angle γ is determined by the grating equation,

where σ is the wavenumber of light, m is the order of diffraction, θ is the Littrow angle and 1/d is the grating groove density. If wavenumber σ₀ of the light beam on the axis meets Littrow auto-collimation conditions, i.e., the angle of the corresponding two emerging is zero (2γ = 0), this wavelength σ₀ is called the Littrow wavenumber. The difference of emergent angles between the light beam with Littrow wavenumber σ₀ and those with any other wavenumbers σ is γ. The difference of the two grating emergent light surface angles is 2γ, with the first-order approximation, the spatial frequencies of the two beams of interference light with wavenumbers σ is [2],

When the spectrum density function of the incident light is B(σ), the distribution of system interference is [2],

where x is measured on the detector in the dispersion plane of the gratings. The inverse Fourier transform of I(x) may recover the input spectrum B(σ), within the spectral range σ₀ ±Δσ.

A WS-SHS is a SHS with the blazed gratings replaced by echelle gratings (Fig. 1(b)), with the aid of multistage diffraction characteristics of an echelle grating to achieve spectrum extension. An echelle grating with high dispersion properties may achieve high spectral resolution, which makes the WS-SHS system have a series of Littrow wavenumbers ∑_mσ_0m corresponding to different orders of diffractions, and each diffraction order nearby σ₀ with a small spectral range can meet difference frequency interference conditions. Then we can obtain the recuperative spectrum within this free spectral range, and reuse more diffraction orders to detect an objective spectrum within a wide spectral range, which achieves the goal of improving the performance and application scope of detection.

Comparing with SHS, the WS-SHS expands single Littrow wavenumbers σ₀ of the original system to a series of Littrow wavenumbers ∑_mσ_0m. According to Eq. (3), the distribution about interference fringe of the WS-SHS was derived [2] as,

where m denotes the class of diffraction, σ_0m is the Littrow wavenumber for order m, F_m(σ) gives the diffractive efficiency of grating for order m.

A schema for two-dimensional WS-SHS is shown in Fig. 2, where each of the two echelle gratings rotates an angle α/2 around x. With such a rotation, the angle of interferometer emergent light along the vertical direction becomes α. Then the distribution of the interference figure for this system is [2],

Analysis of Eq. (5) reveals that the incident spectrum with different diffraction classes along the direction of dispersion has the modulation of difference frequency interference, and along the direction of vertical dispersion has the interference modulation related to the input wavenumber. Two-dimensional recuperative spectra whose orders are separated can be obtained by two-dimensional inverse Fourier transform of Eq. (5).

3.1.1. Diffraction Efficiency of Echelle Grating

The diffraction efficiency of grating is the ratio of the monochrome diffraction light flux and the incident light flux in a given spectral diffraction order. The definition of grating diffraction efficiency measured by the method of single slit diffraction is as follows [12],

where σ₀₁ is the Littrow wavenumber for order 1, and σ_0m is the Littrow wavenumber for order m. With the grating equation, σ₀₁ and σ_0m can be defined as,

where θ is the Littrow angle and 1/d is the grating groove density.

Plots of the diffractive efficiency predicted by Eq. (6) for order m − 1, m and m + 1 are shown in Fig. 3.

The difference of Littrow wavenumber between adjacent diffraction orders is Δσ_0m = σ_0(m+1) − σ_0m = σ₀₁, so the effective spectral range for each diffraction order is approximately σ_0m ± σ₀₁/2.

3.1.2. Limit of Detector’s (CCD) Resolution

The limit of a detector’s resolution is the minimum value of the detector pixels whereby the detector can just distinguish the contrast of input information. For this system, it is the minimum value of the detector which can just distinguish the contrast of interference fringes. The definition of contrast is [13],

where Δ gives the optical path difference, L_C is the coherence length of input light source. V = 1 means input light is coherent light completely; 0 < V < 1 means coherent light partially; V = 0 means incoherent light, and it will not produce interference fringes at this time. Obviously, the condition of producing an interferogram is that the optical path difference for this system is less than the coherence length of input light source Δ≤ L_C.

3.2.1. Limit of Resolution [2]

The limit of spectral resolution for WS-SHS depends on the maximum optical path difference of interferogram along the direction of dispersion. As shown in Fig. 2, the maximum optical path difference along axis x is U_{x max} = 4x_max tanθ with x_max = (W cosθ)/2 . Then the limit of resolution of WS-SHS is,

where W is the effective length of the grating, θ is the diffractive angle of the grating.

3.2.2. Resolving Power [2]

The resolving power of WS-SHS is determined by the minimum frequency of the system. When the system is in a minimum frequency spectrum, the detector can only receive one interference fringe. Then the resolving power of the system is,

3.2.3. Spectral Range [2]

① Considering the diffraction efficiency of an echelle grating

By Eqs. (6) and (7), the effective spectrum for each diffraction order of grating is σ₀₁, while the best spectrum that may not produce spectrum overlapping between adjacent diffraction orders is σ₀₁, and the lowest diffraction efficiency is about 40% at this time (cf. Fig. 3). So the spectral range of WS-SHS is,

In Eq. (11), when m is from zero to hundreds, the spectrum detection range of SHS will be from ultraviolet to infrared.

② Considering the contrast of interference fringe

From Eq. (8), it can be found that Δ≈ L_C results in V ≈ 0, the maximal Δλ corresponds to the minimal L_C. So for a given diffraction order, the largest range of spectrum is restricted by the system optical path difference. Only if the optical path difference is minimum, that is Δ_min = 2sin θ · d, and Δ≈ L_C, may the maximum spectral range of corresponding diffraction order be obtained.

where λ = 1/σ_0m, Δλ = 1/σ_0m − 1/σ_0m'. From Eqs. (7) and (12), the spectral range for the m^th diffraction order is denoted by Δσ_0m = σ_0m' − σ_0m, as shown in

where m' is the diffraction order which is adjacent to diffraction order m. In Eq. (13), the spectral range obtained is Δσ_0m > σ₀₁, so the interference fringes can be formed within the spectral range described by Eq. (11). In this sense, the spatial heterodyne spectrometer based on an echelle grating is a continuous wide spectrum system (bold lines shown in Fig. 3).

3.2.4. Numbers for Pixel of Detectors [2]

With 3.2.1 ~ 3.2.3 above, it has been proved that the SHS based on an echelle grating is a continuous wide spectrum system in terms of diffraction efficiency of the grating and contrast of interference fringes. While for continuous WS-SHS, the spectral range of the system should not be greater than the spectral range which can be detected by detectors with given number of pixels. So if the pixel number of a detector along axis x is N_x, the spectral range Δσ_N determined by pixels numbers should be:

While the spectral range for every order diffraction of the gratings is Δσ = 2σ₀₁, and the lower limit of the numbers for pixel of detectors along the direction of x, which is obtained via Δσ ≤ Δσ_N is

In the same way, the resolution of the system in axis y must be high enough to distinguish adjacent orders from one to another by at least one fringe. If the maximum diffraction order is m_max, then a lower limit of the numbers for pixel of detectors along the direction of y is [2]:

In order to separate adjacent diffraction orders by at least one fringe, the rotary angle of an echelle grating is α [2]:

where W_y is the effective width of the grating in the direction of y.

In Section 3 we showed theoretically that the WS-SHS may get the interferogram from continuous wide spectrum light source. But in practical application, a WS-SHS is mainly restricted by the following two aspects.

Limit 1: The diffraction efficiency of an echelle grating cannot reach 100% (usually the diffraction efficiency of an echelle grating in the peak is 50 ~ 60%), and each of the critical diffraction spectra corresponding to the diffraction efficiency of the grating is far less than 40% that reduces the resolving ability of the system.

Limit 2: Due to the restriction of CMOS technology, the resolution limit of the detector (CCD) is restricted. Namely, the CCD is sensitive to the interference fringes only when the contrast is greater than a certain value. But the analysis of Section 3 presumes that the detection accuracy of the CCD may be very low, which is not in accordance with reality.

Based on the above restrictions, the problem whether we can realize continuous WS-SHS without changing the performance of equipment is worth studying. So, we proposed the following kind of alternative continuous wide spectrum SHS (ACWS-SHS) system.

The proposed ACWS-SHS system is shown in Fig. 4 where two groups of similar echelle gratings are loaded into the system. The two groups of gratings are with the same blaze angle and physical size, but different groove density. The operation steps are as follows:

Step 1: Collecting interference fringes of input light source

First of all, the first group of echelle gratings is used to obtain the interference fringes image 1 of input light. Then the first group of echelle gratings is replaced by the second group of echelle gratings, and interference fringes can be obtained again as the image 2.

Step 2: Obtaining recuperative spectral diagram

Calculate inverse Fourier transform of the two interference fringes images 1, 2, respectively, and corresponding recuperative spectra of the two groups of grating systems may be obtained.

Step 3: Obtaining the spectrum

Superposing corresponding recuperative spectrum obtained in step 2, all of spectrum information of input light source can be shown completely.

The system parameters are assumed as follows: The blaze angle of the two groups of blazed grating is θ₁, physical size is l × w × t, and the groove density of the two groups of blazed gratings are f_G1 and f_G2, respectively. With no special claim, the spectral range of CCD is assumed to be not affected by the numbers of pixel. So the theoretical analysis of the ACWS-SHS is as follows.

4.3.1. Analysis of Grating Diffraction Efficiency

If f_G1 = 2 f_G2, according to the analysis of diffraction efficiency of the grating (cf. Eq. (6)), we take the 100th ~ 101st orders diffraction for the first group of gratings as an example; the 200th ~ 202nd orders diffraction for the second group of gratings as another example to be analyzed. First of all, the center wavenumbers of the two groups of gratings is obtained through the grating equation 2σ_0m · sin θ = m/f_G as,

Since the groove density of the grating meets f_G1 = 2f_G2, then the wavenumbers of the two sets meet σ_0m1 = 2σ_0m2. So when the center wavenumbers of the system are the same, the diffraction order of the first grating group is 2 times that of the second grating group. The 201st diffraction order of the second grating group will appear in the region between the 100th and 101st diffraction order of the first grating group as shown in Fig. 5(a).

Therefore, the main idea is to compensate the spectral range by a second group of gratings when the diffraction efficiency of the first group of gratings is low, remaining the limit of resolution δσ = 1/4 Wsin θ the same. Assume the diffraction efficiency of grating is η, i.e.,

From Eq. (11) and Fig. 3 one can find that the effective range of adjacent diffraction spectrum is Δσ = σ₀₁ = f_G/2 sin θ, then the spectral range of any two adjacent diffraction orders for the two grating groups are Eq. (20), respectively,

So Δσ1 = 2Δσ2 is obtained by f_G1 = 2f_G2, and σ₀₁1, σ₀₁2 are the spectral range corresponding to the first grating group and the second grating group of the basic SHS system, respectively. Taking Δσ1 and Δσ2 into Eq. (6), the diffraction efficiency of the grating obtained is 40%.

Suppose that the continuous spectrum is corresponding to the two groups of grating, the half spectral ranges of each diffraction are Δσ1' = σ−σ_0m1 and Δσ2' = σ−σ_0m2, respectively, then Δσ1' = 2Δσ2' is obtained by the relationship of groove density between the two group of gratings, while the groove density f_G , the physical size and blaze angle of SHS is the same as that of the first group of ACWS-SHS. To ensure the ACWS-SHS to be a system of continuous wide spectral system, it is necessary to ensure that the summation of spectral ranges of the two groups of grating system is equal to the spectral range of WS-SHS, i.e., the following equation must be satisfied (as shown in Fig. 5),

where σ₀₁ means the spectral range of each level of diffraction corresponding to SHS, and there is σ₀₁ = σ₀₁1. The half spectral range of the two groups of echelle gratings obtained by Eqs. (18), (20) and (21) are Δσ1' and Δσ2', respectively,

Substituting Eqs. (22), (23) into Eq. (19), the two groups of grating diffraction efficiencies at the system critical spectrum are obtained as follows,

Compared with Fig. 5, we proved that an ACWS-SHS system proposed in this paper will raise the diffraction efficiency of the grating from 0.4 (Fig. 5(a)) in the SHS to 0.6839 (bold lines shown in Fig. 5(b)).

4.3.2. Detector Influenced by the Contrast Degree

Section B analyzed the relationship between the contrast ratio, the difference of optical path, the relationship between coherence length and the spectral range. Now we analyze the coherence length and contrast ratio of the interference fringes assuming other parameters of the system remain the same (for note, the half spectrum ranges of each group of grating system are shown as Eqs. (22) and (23),

① Coherence length

Eq. (12) describes the relationship among coherence length, the center wavelength and spectral range, take the m^th order of diffraction as an example, the coherence length is analyzed below.

Firstly, the wavelength range of the m^th order of diffraction is calculated according to the spectral range of the system, Δσ₀₁1, is

where σ'_0m means the critical wavenumber of order m. And σ'_0m = σ_0m − Δσ₀₁1 is brought into Eq. (8), then the coherence length of this system is,

② Contrast degree

The coherence length of the ACWS-SHS is calculated by Eq. (27), but the relative difference of optical path is constant (Δ = 2 sin θ/f_G), so the contrast ratio of the system is,

4.3.3. Performance Analysis of ACWS-SHS

We assume that the groove densities of the first and second groups of echelle gratings are f _G1 = 31.6 l/mm and f _G2 = 15.8 l/mm, respectively; the two groups of echelle gratings have the same physical size which is 12.5 × 25 × 9.5 mm; and the blaze angle is 63°. Then the basic parameters of the system are as follows,

(1) Limit of resolution

According to Section 3B, the resolution limit of the SHS system is δσ = 1/4Wsin θ, it can be seen that the resolution limit is related to the effective size of the grating (W) and the diffraction angle (θ). Other parameters of the two group gratings in the ACWS-SHS system are the same except for the groove density of the echelle grating. So the resolution limit of the two systems should be the same, i.e.,

(2) Spectral range

According to Eqs. (22), (23) and the above parameters, the spectral ranges of the first and second groups of gratings, Δσ1, Δσ2, can be obtained respectively,

Assume that when the groove density is f _G1, the corresponding standard spectral range is σ₀₁ with diffraction efficiency being above 40%. Then the actual spectral ranges of the first and second groups of grating systems are 2σ₀₁/3 and σ₀₁/3, respectively. At this time, the spectral range of the two groups of systems is equal to the spectral range σ₀₁ of an ideal system (without considering the diffraction efficiency), the minimum diffraction efficiency of the grating is 68.39%, which is much higher than that of a single group of gratings (40%).

(3) Analysis of contrast for system interference fringes

When the band of the system is from ultraviolet to infrared wavelengths (m > 20), as shown in Eq. (28), the contrast of the system may be approximated as

Analysis of (2) and (3) shows that using the ACWS-SHS may:

① Avoid aliasing of adjacent diffraction orders in the wide spectrum scenario effectively;

② Narrow the working spectral region of grating in each of diffraction order, which will take advantage of high diffraction efficiency region of grating effectively;

③ Improve the contrast of interference fringes, and reduce the limitation to the resolution of image acquisition devices (CCD) effectively.

(4) An example

Here we give an example:

① Limit of resolution and spectral range without considering the effects of diffraction efficiency and contrast

Suppose that the parameters of the basic SHS system are the same as the first group of the system of ACWS-SHS. Without considering the influence of grating diffraction efficiency and resolution, the resolution and spectral range of the system may be obtained as follows,

② Spectral range of ACWS-SHS system

By Eq. (30) the spectral range for the first group of the system may be obtained as,

Similarly, the spectral range for the second group of the system may be obtained as,

So the spectral range of ACWS-SHS is,

In our simulation, the main parameters of the ACWS-SHS system are listed in Table 1.

Table 1. Basic parameters of the ACWS-SHS

In Table 1, f_G means the groove density of grating, θ means the blaze angle of grating, Size means the physical size of grating l × w × t, Tiltangle means the rotary angle of grating to be rotated around the axis of x, δσ means the resolution, N_x means the numbers of pixels in x axis, N_y means the numbers of pixels in y axis, and d_pixel means the size of single pixel.

According to the above parameters, the center coordinate of the spectrum for the first group of the system is (N_x1/2, N_y1/2) = (1581,561). By the same token, the center of the spectrum of the second group of the system is (N_x2/2, N_y2/2) = (1581,1121). And the angle of the grating should be α ≥ 0.44° (cf. Eq. (17)), which ensures that there are at least two sampling intervals between the adjacent diffraction orders. To improve the accuracy, here the resolution limit of the system is set to 0.3366 cm⁻¹.

4.4.1. Calibration Experiment of the Two Groups of Grating System

We won’t describe the principle of calibration experiment [14] any more since it was clarified in reference. Here we also selected a sodium light source as a calibration source. In Fig. 6(b), the spectral lines are A1 (wavelength: 589.6 nm; coordinate: (x, y, z) = (911, 922, 0.1684)) and B1 (wavelength: 589 nm; coordinate: (x, y, z) = (1096, 922, 0.1177)); In Fig. 6(d), the spectral lines of the recuperative spectra are A2 (wavelength: 589 nm; coordinate: (x, y, z) = (1857, 1839, 0.1731)) and B2 (wavelength: 589.6 nm; coordinate: (x, y, z) = (2041, 1839, 0.1539)).

Figures 6(a) and 6(c) are the interference patterns of the sodium light corresponding to the first group of grating system 31.6 l/mm and the second group of grating system 15.8 l/mm, respectively. Figures 6(b) and 6(d) are the recuperative spectra of the two groups of systems, respectively, where axis x reflects the spectral distribution of each diffraction order, axis y reflects the distribution of the input light source, axis z reflects the spectral intensity.

The characteristic wavelengths of the sodium light are 589 nm and 589.6 nm, respectively. Corresponding to the first group of grating system, its diffraction order is m = 100; while for the second group of grating system, corresponding diffraction order is m = 200. The sampling interval for the first and second groups of grating system obtained according to the relationship of coordinates and the incident wave numbers are calculated as follows [15].

① For groove density of SHS system being (31.6 l/mm):

where σ_A1 is the wavenumber corresponding to A1, σ_B1 is the wavenumber of corresponding to B1, x_A1 is the coordinate corresponding to axis x for A1, y_A1 is the coordinate corresponding to axis y for B1. Δσ_1ˍsample is the sampling interval of the first group of grating system.

② For groove density of SHS system being (15.8 l/mm):

where σ_A2 is the wavenumber corresponding to A2, σ_B2 is the wavenumber corresponding to B2, x_A2 is the coordinate corresponding to axis x for A2, y_A2 is the coordinate corresponding to of axis y for B2. Δσ_2ˍsample is the sampling interval of the first group of grating system.

Comparing the sampling interval of systems ① and ②, it can be found that although the groove densities of the two systems are different, the sampling interval of the systems are equal, i.e., the resolution limits of the system are the same.

(1) Spectrum distribution of the first and second group of system

The diffraction orders were selected according to Section 4.3.3, and the spectrum range of the two groups of system are calculated and listed in Tables 2 and 3, respectively.

The spectrum range of order 99 diffraction in the first group of grating system is [1749.6318, 1761.4536], the spectrum range of order 100 diffraction is [1767.3645, 1779.1864]. While the spectrum range of order 199 diffraction in the second group of grating system is [1761.4536, 1767.3645]. Comparing the two groups of data, we can find that the spectral range of the second group of grating system in the 199^th diffraction order is just inserted between the spectral region of the 99^th and 100^th order diffraction of the first group of grating system.

(2) Spectral range of the first and second group of grating system

Figures 7(a) and 7(c) are the interference patterns of the sodium light corresponding to the first group of grating system 31.6 l/mm and the second group of grating system 15.8 l/mm, respectively. Figures 7(b) and (d) are the recuperative spectrum of the two groups of systems, respectively, where axis x reflects the spectral distribution of each diffraction order, axis y reflects the distribution of the input light source, axis z reflects the spectral intensity.

Figure 7(b) reflects the spectral range of adjacent two diffraction orders for the first group of grating system with spectral lines being A (950, 937, 0.1194), B (950, 933, 0.1257), C (2212, 937, 0.1194), D (2212, 933, 0.1257), respectively; Fig. 7(d) reflects the spectral range of the second group of grating system with the spectral lines being A (1272, 1869, 0.145) and B (1890, 1869, 0.1405), respectively. These parameters are analyzed as follows:

① Analysis for the first group of data

The sampling interval of frequency spectrum of the system calculated by Section 1 is 0.0094 mm⁻¹. The two points of B and D can be judged to be of the 99^th diffraction order by the coordinates of A, B, C, D, while the two points of A and C are of the 100^th diffraction order. The characteristic wave numbers corresponding to these locations are,

Therefore, the corresponding wavelengths of the four points are λ_A = 565.8187 nm, λ_c = 562.0502 nm, λ_B = 571.5534 nm, λ_D = 567.7084 nm. For the first group of grating system, its corresponding spectral range of the 99^th diffraction order is [567.7084, 571.5534], corresponding spectral range of the 100^th diffraction order is [562.0502, 565.8187].

② Analysis for the second group of data

Similarly, the sampling interval of frequency spectrum for the system obtained by Section 1 is 0.0094 mm⁻¹. The characteristic wave numbers corresponding to the two points coordinates of A, B are calculated as,

So the wavelengths corresponding to the two positions are λ_A = 567.6955 nm, λ_B = 565.8315 nm. For the second group of grating systems, corresponding spectral range of the 199^th diffraction order is [565.8315, 567.6955].

③ Spectral range comparison of the first and second group of grating systems

The spectral range of the 99^th and 100^th diffraction orders for the first group of grating system obtained by ① are [562.0502, 565.8187] and [567.7084, 571.5534], respectively. The spectrum region between the two adjacent orders is [565.8187, 567.7084], while the spectral range of the 199^th diffraction order for the second group of systems is [565.8315, 567.6955]. So the two groups of systems are complementary in the spectrum region.

(3) Diffraction efficiency and contrast of interference fringe of echelle grating used in interactive SHS system

① Diffraction efficiency of echelle grating

The diffraction efficiency of the system obtained by Eq. (6)

② Contrast degree of interference fringes

In the same way, contrast degree of the system obtained by Eq. (8) is

4.4.3. Multiple Diffraction Orders Light Sources for the Two Group of Grating Systems

In this section, the simulation analysis is carried out on the wide spectrum light sources.

Figures 8(a) and 8(c) are the interference patterns of the sodium light corresponding to the first group of grating system 31.6 l/mm and the second group of grating system 15.8 l/mm, respectively. Figures 8(b) and 8(d) are the recuperative spectrum of the two groups of systems, respectively, where axis x reflects the spectral distribution of each diffraction order, axis y reflects the distribution of the input light source, axis z reflects the spectral intensity.

Figures 8(b) and 8(d) are the spectrogram of the first and second groups of grating system with wide spectrum range of input light sources, respectively. The corresponding locations of the spectrum for the first group of grating systems are A (1154, 1080, 0.1549), B (1901, 1012, 0.1977), C (1154, 1080, 0.1549), D (1688, 749, 0.162); corresponding diffraction orders are 50, 80, 120 and 160; the corresponding locations of the spectrum for the second group of grating systems are A (1368, 2162, 0.1377), B (1688, 2019, 0.1653), C (1794, 1718, 0.1111), D (1688, 1493, 0.1673); the corresponding diffraction orders are 319, 239, 159 and 99, respectively. Detailed data are listed in Table 4.

In Table 4 Coor. means the coordinate of the recuperative spectra, WN means the computational wavenumber, WL means the wavelength for the computational wavenumber, WLI means the theoretical wavelength of the spectral line, Error-WL means the error between WL and WLI, and W-band means the attribute of input light.

Analyzing the data in Table 4, it can be found that by using ACWS-SHS system, the maximum error between wavelengths calculated and the actual input wavelength is 0.0061 nm, which is less than the resolution limit of system 0.03366 mm⁻¹, and it is proved that the system can be applied to wide spectrum light sources. The error is mainly a result of the cut-off of data.

Combining Section 1 to 3, it is proved that the proposed ACWS-SHS system not only has a high resolution limit, but also can be applied to continuous wide spectrum light sources. It is more important that compared with SHS system or a WS-SHS system the system not only has a higher resolution limit (particularly in the edge of the spectral range) and diffraction efficiency of the grating, but also can get a relatively high contrast of interference fringes, which reduces the requirement on the detector’s resolving power.

Firstly, we demonstrated that the WS-SHS is suitable for the detection and identification of the characteristic information of the light sources with continuous wide spectrum (ultraviolet to infrared wavelengths) theoretically. Then considering that conventional WS-SHS has the disadvantages of low diffraction efficiency of grating and low contrast of interference fringes, an ACWS-SHS system is proposed in this paper. And it is proved that this system is a continuous wide spectrum system. The relationship among double gratings system, diffraction efficiency of grating and contrast of interferogram fringes are also analyzed. Simulation results show that the diffraction efficiency of the selected grating in the SHS proposed is more than 68.39%, and the ratio of the interference fringes is 0.4135. The results show that the average resolution obtained with light sources of different characteristic spectral lines (sodium light source, adjacent diffracted multiple wavelengths light sources and multistage single-wave light source) is 0.0314 mm⁻¹, which is similar to the resolution limit of the actual system. It can be found that the error mainly originates from the cut-off of data in the calculation process, so the current resolution limit of the system is suitable for equipment. And the ACWS-SHS system improves the performance of the system greatly and expands the application fields of SHS.