1. Introduction

Spectroscopy plays an important role in environmental science, biomedicine, aerospace, and other fields [1–5]. The application port presents higher requirements for spectral accuracy, spectral bandwidth detection, convenience, and detection time [6–8]. For example, in recent years, spectroscopy has been applied for home medical diagnosis, which has promoted the development of super-surface spectrometers [9,10]. In the field of atmospheric trace and gas detection, owing to the relatively weak gas spectral information and wide distribution of different gas-sensitive wavelengths, it is difficult for general spectrometers to simultaneously meet the measurement requirements of high luminous flux and wide spectral range under high spectral resolution [11]. However, the advantages of M-FTS are its high luminous flux, wide waveband, and high spectral resolution; thus, it has become the preferred method in the field of spectral gas detection in recent years.

In practice, the stability of the M-FTS moving mirror significantly influences the spectral quality. Even with air slides, it is difficult to guarantee the stability of horizontal movement [12]. The strict horizontal movement is destroyed by mechanical jitter, which causes the interference light reflected by the moving mirror to fail to return to the original optical path direction. After the two coherent lights passed through the focusing system, the interferometric modulation depth obtained by the detector was reduced. If the angle between the two coherent light beams is significantly large, interference is impossible. The errors in the interferogram caused by mechanical jitter have a non-negligible effect on the recovered spectrum. Therefore, many researchers aim to improve the stability of spectrometer operation by improving the moving mirror or designing a static spectrometer system with a loss of resolution [13–17]. An improved M-FTS, applied in Japan's Gosat series satellites [18,19], was designed with a structure of two arms that produce an optical path difference (OPD) by swing. The swinging motion was more stable than the horizontal motion. This advantage reduces interferogram errors. However, this spectrometer must place angle mirrors at the ends of both arms. To obtain a high-quality spectrum, the processing precision of the angle mirror must be high [20]. In addition, this spectrometer should perform a reciprocating motion, similar to a pendulum. During the operation of the spectrometer, reciprocating motion is constantly controlled by the servo system, and causes instability when the swing speed changes [21]. Therefore, we prefer a system in a stable regime to minimize the effect of servo operation error on spectral detection. For the M-FTS and double-arm spectrometer, controlling the reciprocating motion consumes stability and increases spectral detection time.

In this study, we propose a Fourier transform spectrometer based on a pair of rotating parallel mirrors (RPM-FTS). The spectrometer obtains the OPD by rotating a pair of parallel mirrors. During the operation, the servo control process is simplified, which only should maintain the pair of parallel mirrors in constant rotation at the same speed replaced the acceleration and deceleration in the M-FTS. This improvement increases the frame rate of spectral acquisition, to ensure that the instrument is suitable for transient spectrum measurement. Another advantage is that the instrument is highly stable. Provided the light path is established, regardless of the jitter of the machinery, ideal parallel mirrors placed at any position will not change the direction of the light path. The performance of the instrument was measured by the modulation depth [22]. The system modulation depth can be determined when the relative position relationship between a pair of parallel mirrors is determined. We provide a theoretical derivation of the OPD caused by the rotating parallel mirrors. Constraints on the engineering implementation of the spectrometer are discussed and analyzed. In addition, the interferogram tuning regime is not constant owing to errors in parallel mirror installation. Parallel mirror parallelism was modelled and analyzed, and installation tolerance was proposed based on the model deduced from the variation of the interferometric tuning regime during rotation. The results show that under the same installation level, in terms of interference measurement and modulation depth, the interference system of RPM-FTS is significantly improved compared to that of M-FTS.

2. Principle of RPM-FTS

Figure 1(a) shows the RPM-FTS and its basic components. The spectrometer consisted of a beam splitter, four reflective plane mirrors, pair of parallel mirrors rotating with a rotary motor (RPM-System), collimation system, and focusing system with a detector. The signal light was converted into collimated light through a collimation system in the instrument. To facilitate this explanation, we modeled the collimated optical axis of the incident system on the x-z plane, as shown in Fig. 1(b). In particular, Fig. 1(b) shows a schematic of the spectrometer in the x-z plane. The collimated light is divided into A-light (red light path) and B-light (blue light path) by beam splitter P₁. The OPD is compensated by compensation plate P₂. The plane reflectors G₁ and the plane mirror G₂ are symmetrical about the beam-splitting surface. The two beams are reflected in plane reflectors G₁ and G₂, respectively, and change direction into the RPM-System. The optical paths of A and B lights at the reflected positions A₀ and B₀ on G₁ and G₂ are equal. If the influence of the collimation and focusing systems on the optical path is not considered, the optical path can be calculated from G₁ and G₂.

 

Fig. 1. (a) Schematic diagram of the principle of the spectrometer. The red line represents the A light path. The blue line represents the B light path. The yellow line represents the common optical path of the two beams. The collimated beam enters the spectrometer system and interferes in the focal plane of the lens after generating the OPD. (b) Schematic diagram of the spectrometer in the x-z plane. The collimated light is obtained through the collimation system. P₁ is the beam splitter. P₂ is the compensation plate. The G₁, G₂, G₃ and G₄ are plane reflectors. M₁ and M₂ form a pair of circular parallel mirrors. The M₁ and M₂ are connected by a mechanical support structure, not shown in the figure. A mechanical structure is set at the center of M₁ as the connection to the parallel mirror and the motor on the X-axis. The rotation axis drives M₁ and M₂ to rotate around the X-axis. Parallel mirrors and rotary motors make up the RPM system. After the optical path is folded back, the focusing mirror and the detector form the focusing system.

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In the RPM-System, a pair of parallel mirrors was placed on the motor shaft at an artificial offset angle(k). The artificial offset angle is defined as the angle between the parallel mirror normal vector and x-axis at the initial rotation position when the normal vector has no y-component, as shown in Fig. 1(b). The artificial deflection angle causes the optical path of the A-light and B-light to change in the process of rotation without changing their direction after passing through the RPM-System based on Fresnel's law. According to the design of the optical path, the A-light and B-light should be perpendicular to G₃ and G₄ respectively, to fix G₃ and G₄. After the A-light and B-light are reflected on positions A₃ and B₃, they return along the same path.

Parallel mirrors M₁ and M₂ are driven by motors to rotate around the x-axis. Different rotation angles resulted in different light paths for A-light and B-light. The detector on the focal plane of the lens (L₂) collected light energy from different OPD at different sampling points. The intensity of the light at different moments constitutes an interferogram. According to the principles of Fourier spectroscopy [12], the spectrum is obtained using the Fourier transform of the interferogram.

3. Interference process model

The interferogram is composed of interference energies sampled at different OPD during rotation. To determine the instrument parameters and analyze the interference energy in this process, it is necessary to solve the relationship between OPD and the rotation angle. We used geometric light path tracing for calculation.

The parallel mirrors (M₁ and M₂) were rotated by one cycle around the x-axis to obtain two symmetrical interferogram samples. Therefore, any position could be used as the starting position. For convenience of calculation, the y-component of the normal vector at the starting position of the parallel mirrors was set to 0. Specifically, the x-z plane inclination was maximized, and the system obtained the ideal maximum OPD. Figure 1(b) shows the layout of the interference model with parallel mirrors placed at the starting position. The coordinate system was established with the intersection of the rotation axis and parallel mirror M₁ as the origin. Beam width should not be considered when calculating the optical path. Therefore, in this coordinate system, we considered the reflection points (A₁, A₂, A₃, B₁, B₂, B₃) of the beam center on each reflection surface when the motor is working. We proposed an optical path design in which the reflected light passing through G₁ and G₂ is incident at -45° (A-light) and 45° (B-light) in the x-z plane with respect to the x-axis on the rotating parallel mirror device. The installation angles of G₃ and G₄ were based on these two light incidence angles.

The key to tracking light is to calculate the direction of reflection of the light at any rotation angle. If the direction of incidence is known, the reflected light can be obtained from the normal reflected light and the law of reflection. The reflected light is normal to M₁ with respect to the angle of rotation. The vector [m, n, p] is the normal direction of M₁ when the rotation angle is u, as shown in Fig. 2. The intersection of the rotation axis with M₁ (the origin of the coordinate axis) is a fixed point. M₂ is determined by distance h from M₁. The entire rotation can be considered as the normal vector of M₁ rotating around the x axis. Assuming that the direction of rotation and the normal of the parallel mirror conform to the right-handed spiral rule, the normal vector can be calculated using the Rodriguez rotation in Eq. (1), where L is the direction of the rotation axis [1,0,0], and v = [1/tank,0,1], which is the normal vector corresponding to the initial position of M₁.

 

Fig. 2. Schematic diagram of the A-light trajectory of a parallel mirror rotating with rotation angle u. The green vector [m,n,p] indicates the vertical direction of the parallel mirror during the rotation. The red arrow indicates the light trajectory of A-light.

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The reflective point of the two lights is traced by the starting point position, start direction, and normal of the reflector, where the optical path is calculated. The optical path of the A-light is provided by Eq. (2), and the optical path of light B-light is denoted by Eq. (3). The OPD is expressed by Eq. (4).

By introducing these parameters into Eqs. (2)–(4), we can obtain the expressions for the optical path and OPD of the two beams. In Eqs. (5) and (6), x₀ and z₀ are the position parameters of G₁ and G₂. The position parameters of G₄ and G₃ are g_A and g_B. Owing to the symmetry of the two lights, the optical paths of the A-light and B-light have a phase difference of π. This physical process corroborates the results of the derived calculations. The results of Eqs. (5) and (6) show that the light path can be decomposed as a cosine function of the rotation angle u using the system parameters and a constant term. When g_A= g_B, the OPD is treated as a cosine-type function of rotation angle u, as shown in Eq. (7). The maximum value of OPD mainly determines the spectral resolution, which provides the basis for the design of instrument parameters.

4. Effect of parallel mirrors parallelism on the modulation depth

In engineering applications, RPM-FTSs experience many limitations. The k and structure size are limited based on the condition that each reflection point of the beam falls on the mirrors, and the beam is never blocked by M₁ after the secondary reflection. If the motor rotates at a constant speed, the rotation angle u = wt, where w is the speed of the motor and t is the time of motor rotation. According to Eq. (7), OPD is nonlinear with respect to time. Sampling at equal times causes interferogram sampling errors. To obtain accurate interferograms, reference laser interference-triggered acquisition is used to sample the interferogram with an equal OPD. This measurement approach directly targets the OPD and compensates for the optical path error of the instrument. This method has been widely used in high-precision M-FTSs [23].

In addition, there are many non-ideal factors in the manufacture and setup of instruments that limit high-precision spectroscopic measuring instruments, such as tilting of the moving mirror and misalignment of the setup of the fixed mirror. These errors cause tilting of the beam wavefront, resulting in non-ideal differences in the optical path of the beam in the interferometric plane owing to the beam width. The concept of modulation depth was defined to measure the intrusion of instrument errors into the interference energy [22,24]. The modulation depth of the M-FTS is expressed by Eq (8), where D is the beam diameter, v is the maximum wave number in the spectral range, and β_M-FTS is the inclination angle of the moving mirror. If the modulation depth is larger, the interferogram energy fluctuations are stronger and more interferogram detail is obtained. The induced nonequilibrium error in the image plane causes the integration result of interference intensity to deviate from the ideal intensity, resulting in a lower modulation depth and less energy detail. In discussing spectrometer tolerances, the modulation depth is an important indicator of the interferometric performance of the instrument.

Compared to M-FTS, the major enhancement of our spectrometer design is the replacement of horizontal motion by rotational motion and the replacement of flat mirrors by parallel mirrors, which achieves higher stability. Stability is not destroyed by high frame rate detection, unlike acceleration and deceleration movements. In the ideal case of perfect parallelism, if mechanical jitter occurs in the rotation of the RPM-FTS, only the direction of the parallel mirror (normal vector [m,n,p]) and the spatial position are affected, resulting in an optical path length change without changing the direction of the optical path. However, the optical path length error was eliminated using laser-triggered sampling. Therefore, under perfect parallelism, the moving parts do not affect interference. However, in the process of installing parallel mirrors, it is impossible to achieve the ideal conditions for perfect parallelism. Parallelism errors are inevitably introduced, resulting in the offset of the optical path. The parallelism error is transmitted to the optical system, causing the optical path to fail to interfere ideally according to the preset trajectory, resulting in a change in the modulation depth. The wavefront inclination of the two lights superimposes or offsets during rotation, which makes the problem more complicated. Therefore, we ignored other errors and discussed the influence of the parallel mirror error.

4.1 Analysis of parallelism effect on light direction

Parallelism is measured by the angle of the two plane mirrors relative to each other and the maximum orientation in space. Therefore, when establishing the parallelism error model, it is necessary to consider only the error caused by one of the planes. One plane connected to the rotation axis is chosen as the reference plane, and all the installation angle errors, acting only on another plane, cause parallelism deviation. The outgoing light after the secondary reflection of the parallel mirror is not incident perpendicular to the end mirror, as shown in Fig. 3. The A-light and B-light have the same direction offset relative to the vertical incidence of the end mirror in space. Compared with the ideal case, the inclination of the actual optical wavefront to the end mirror will be in different directions in space with the change in rotation u. It is necessary to consider the spatial direction of the wavefront when studying beam interference errors. This superposition with spatial direction is different from the two-dimensional wavefront superposition of M-FTS.

 

Fig. 3. Schematic diagram of optical path reaching the end mirror with parallelism error. The black dotted line indicates the optical path through M₂ reflection under the condition of perfect parallelism. The purple line represents the optical path reflected by M₂^′ (green solid line) with the included angle of parallelism. The red dotted line indicates the wavefront of the beam, which has a certain deviation angle from end mirror. (a) In the case of parallelism error, the schematic diagram of the optical path of the interference A-light from the parallel mirror to the end reflector. (b) Under the same parallelism, the schematic diagram of B-light from the parallel mirror to the end mirror.

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Owing to the complexity of analyzing a wide beam with a directional offset in three-dimensional space, we decomposed the wavefront vector in space corresponding to the three-dimensional coordinates into a y-component, and a component in the x-z plane. The decomposed y-component was observed in the x-y plane, as shown in Fig. 4. The initial interference light has no y-component, which indicates that the direction of the light is parallel to the x-direction in the x-y plane, even if it is A or B light. The y-component of the beam direction of the outgoing parallel mirror originates from non-ideal parallelism. The A-light and B-light obtain the same y-direction deflection angle upon arrival at the end mirror because the same parallel mirror is reflected. In particular, the additional OPD generated by the wavefront tilt angle in this direction can cancel each other out. Specifically, it can be assumed that no OPD is generated along the x and y directions. Thus, the RPM system has “adaptive” performance for the x-direction and y-direction parallelism errors.

 

Fig. 4. Schematic diagram of light path in x-y plane. The light enters the parallel mirror system along the x direction and produces the y-direction offset only due to the parallelism of the parallel mirror. The black dotted line represents the light path passing through the parallel mirror and the end mirror under the condition of perfect parallelism. When the parallel mirrors are not parallel, the M₂ deflection is M₂^′. At the same time, it causes the reflected light path to become a purple solid line. The red dotted line represents the wavefront of the purple light path.

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However, for the x-z plane, the situation is quite different, as shown in Fig. 5. In an ideal situation, the OPD of the corresponding points of the beams on reflectors G₁ and G₂ should be equal. However, owing to this error, the beam edge marked by the purple line has a larger optical path, and the beam edge marked by the orange line has a smaller optical path. This is because the incident direction of light is symmetrical but the parallelism is not. The asymmetry in the z-direction causes the inclination of the reflected wavefront to superimpose when interference occurs. Therefore, for RPM-FTS, the parallelism-induced degradation of the system modulation depth is attributed to the parallelism component in the z-direction of the rotating process, and is independent of the parallelism component of the x-y plane.

 

Fig. 5. Observe the optical paths of the two interfering beams in the rotating module in the x-z plane. At the initial positions of G₁ and G₂, the two beams are symmetrical. The purple line represents the edge of the beam with a longer optical path due to parallelism. The orange line represents the edge of the beam with a shorter optical path due to parallelism. The solid line is the trajectory of the optical path change after being affected, and the dashed line is the ideal optical path trajectory. The direction of two beams deflects after being reflected by M₂^′

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The problem of the effect of the system parallelism error on the system modulation depth is translated into solving for the parallelism component of the z-direction, thereby calculating the wavefront inclination angle and OPD. However, this error angle was not fixed during rotation, and its value varied with the rotation angle. Therefore, the problem of light direction during rotation should be solved.

4.2 Rotation parallelism model

The direction deviation of light is caused by the normal deviation of the reflecting plane; therefore, we analyzed the influence of parallelism on the normal of the reflecting plane. Because parallelism is a concept of the relative angle, we chose M₁ as a reference plane to study when measuring this angle. The normal vector of the reference plane is the parallelism reference vector v₀. We considered that the plane connected to the rotation axis as the reference plane, and all the installation angle errors only affected the deviation of the normal vector of another plane in the rotation process relative to the ideal case. The spectrometer installation method can be divided into two parts, as illustrated in Fig. 6. The first part combines the two independent plane mirrors into a group of parallel mirrors with a support structure. The parallelism generated by this process is expressed by the maximum angle α between the normal directions of the two planes, which is called static parallelism. The normal vector of error plane M₂^′ is v_α. The second part is to fix the parallel mirrors to the rotation axis with a certain artificial offset angle k. However, during the artificial bias installation process, the direction in which the bias angle k is located produces an angle θ with the direction of α. This will cause a phase change, making the installation and adjustment results more complex, as shown by the plane mirror normal vector marked by the purple arrow in Fig. 6(b).

 

Fig. 6. (a) Schematic diagram of the parallelism deviation caused by the first part of the assembly. The green vector v₀ represents the parallelism reference vector. The red vector ${\bf v}_{\boldsymbol{\alpha}}$ represents specular normal vector due to the α. (b) The second part of the assembly and adjustment, the schematic diagram of the parallelism deviation. On the basis of (a), the second part of the adjustment of the parallel mirror is carried out. The parallelism reference vector v₀ changed to yellow vector v₀^′due to artificial bias in the set-up. However, due to the included angle θ between k and $\boldsymbol{\alpha}$, the red vector after offset ${\bf v}_{\boldsymbol{\alpha}}{^{\boldsymbol{\prime}}}$ is deflected into a purple vector ${\bf v}_{\boldsymbol{\alpha}+\boldsymbol{\theta}}{^{\boldsymbol{\prime}}}$. The ${\bf v}_{\boldsymbol{\alpha}+\boldsymbol{\theta}}{^{\boldsymbol{\prime}}}$ is the result of the combined effect of the two parts of the parallelism error. The pink plane which the normal vector is ${\bf v}_{\boldsymbol{\alpha}}{^{\boldsymbol{\prime}}}$ is represents the M₂^′ when θ=0.

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The two error parameters α and θ obtained by this installation method affect the normal vector of M₂ during the rotation; thus, Eq. (1) can be upgraded to Eq. (9). The new parameter $ k^{\prime} $ and φ in Eq. (9) are adjusted owing to the influence of θ and α.

To calculate the relationship between the new parameter and error, we built a rotation vector parallelism model to analyze the angle and direction for each light. During rotation, the normal direction of M₂^′ also changes with the rotation angle. Figure 7(a) shows the ${\bf v}_{\boldsymbol{\alpha}+\boldsymbol{\theta}}{^{\boldsymbol{\prime}}}$ change during the rotation. In particular, during rotation, the area covered by the projection of ${\bf v}_{\boldsymbol{\alpha}+\boldsymbol{\theta}}{^{\boldsymbol{\prime}}}$ on the y-z plane is a circle whose radius is sink′. When rotating as u, the vector ${\bf v}_{\boldsymbol{\alpha}+\boldsymbol{\theta}}{^{\boldsymbol{\prime}}}$ becomes to ${\bf v}_{\boldsymbol{\alpha}+\boldsymbol{\theta}}{^{\boldsymbol{\prime\prime}}}$. For the parallel reference vector v₀, the radius of the projection circle is the sink. According to the analysis in the previous section, the degradation of the system modulation depth is attributed to the parallel component of the z-direction. Therefore, the deflection of M₂^′ in z-direction caused by parallelism is ${\beta} = \cos u({{\mathrm {sin}}k^{\prime} - {\mathrm {sin}}k} )$.

 

Fig. 7. Rotation vector parallelism model, which describes the effect of parallelism errors on the normal of the reflector during rotation. (a) The yellow planes M1 and M2 are the positions of the initial reflector plane. The normal vector of M₂^′ is ${\bf v}_{\boldsymbol{\alpha}+\boldsymbol{\theta}}{^{\boldsymbol{\prime}}}$ which is yellow arrow. When the angle of rotation is u, ${\bf v}_{\boldsymbol{\alpha}+\boldsymbol{\theta}}{^{\boldsymbol{\prime}}}$ turns into ${\bf v}_{\boldsymbol{\alpha}+\boldsymbol{\theta}}{^{\boldsymbol{\prime\prime}}}$. During rotation, the area covered by the projection on the y-z plane is a circle whose radius is sink′. By approximation sink∼k and ${\mathrm {sin}}k^{\prime} \sim k^{\prime} $, the projection vector on y-z plane shown in (b) and (c). The blue arrow vector k represents the projection of parallelism reference vector v₀. (b) There is only static parallelism, α≠0 and θ=0.k and α are superimposed in the same direction to obtain k${^\prime}$. (c) Both kinds of parallelism errors are exist where α≠0 and θ≠0. The vector k${^\prime}$ is obtained based on the law of vector summation. The angle between k${^\prime}$ and k is φ.

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When calculating the effect of the parallelism parameters θ and α on β, an approximation was made to simplify the calculation process: sink∼k and sink${^\prime}$∼k${^\prime}$. This approximation can be attributed to two factors. First, the value of k is considerably small that it does not have a significant effect on the results of the modulation depth system calculations. Second, the approximation will increase β, which corresponds to error amplification, because ${\mathrm {sin}}k^{\prime}- {\mathrm {sin}}k \lt k^{\prime}-k $. Therefore, the performance of the actual system was better than that of the calculated results.

In Fig. 7(b), the model describes the approximate change in the z-direction component of the M₂^′ normal vector in the y-z plane during rotation. In the model, the parameters k, $ k^{\prime} $ and α have a direction related to u, which is transformed into vectors. The radius of the blue circle in Fig. 7(b) and (c) is replaced by the magnitude of the artificial offset angle k, which is the length of vector k with no error. The direction of vector k is the rotation angle u. The length of vector k${^\prime}$ represents the M₂^′ normal vector in the y-z plane under error. During rotation, the two vectors rotate at the same rotation angle u. When static parallelism error α exists only without θ between vector k and vector ${\bf v}_{\boldsymbol{\alpha}}{^{\boldsymbol{\prime}}}$, as shown in the model in Fig. 7 (b), both vectors have the same rotation angle u with respect to the initial position. In this case, the normal vector of M₂^′ is v_α^′ as shown in Fig. 6(b). The difference β between the two vector projections in the z-direction is the effective mirror error angle affecting the system, which is expressed as Eq. (10).

When there is an angle θ between k and α, the composite vector and ideal vector are as shown in Fig. 7(c). In this case, the normal vector of M₂^′ is ${\bf v}_{\boldsymbol{\alpha}+\boldsymbol{\theta}}{^{\boldsymbol{\prime}}}$ as shown in Fig. 6 (b). The vector k${^\prime}$ is the sum of vectors $\boldsymbol{\alpha}$ and k, with an angle θ between them. k${^\prime}$ was obtained based on the law of vector summation. According to the corresponding geometric relationship, the vector length k${^\prime}$ and phase difference φ were obtained using Eq. (11). By considering the parameters calculated using Eq. (11) into Eq. (12), we can obtain the influence of parallelism on the normal vector during the rotation process. The difference β was obtained by projecting two vectors in the z-direction. Under the influence of θ, Eq. (10) is updated into Eq. (12). Equations (10) and (12) show the influence of the parallelism error on the z-direction of M₂ during the rotation. According to the reflection law, the change in the beam reflection angle is twice that in the normal angle. Therefore, the deflection of the wavefront returning to the interference plane is 2β_α,θ for each light, which is equivalent to the light reflection with β_α,θ tilted reflective plane mirror.

4.3 Modulation depth simulation analysis

Because Eq. (10) is a special case of Eq. (12), we employed Eq. (12) for the analysis. For the proposed RPM-FTS, the moving mirror tilt β_α,θ is the variable associated with u during rotation, which is calculated using Eq. (11) and Eq. (12). Parallelism influences both the optical paths of A-light and B-light in RPM-FTS, where the influence is superposed to each other. Therefore, in the calculation of RPM-FTS modulation depth, the tilt angle in Eq. (8) is replaced with 2β_α,θ. The modulation depth simulation analysis was based on Eq. (13), which is the improved modulation depth equation for the RPM-FTS.

First, we set the RPM-FTS system parameters. To meet general gas detection requirements, we designed RPM-FTS with a spectral resolution better than 0.5 cm^-1 and a beam diameter (D) of 2.4 cm, as shown in Table 1. The detection spectral range is 800 cm^-1∼3400cm^-1. The rotation angle of effective spectrum detection is limited to 1.18 rad ∼1.97 rad and 2.75 rad ∼3.54 rad, which is owing to occlusion and beam width not meeting the optical path during rotation. This range of rotation angles was obtained via beam tracing. The rotation angle boundary corresponds to the maximum value of OPD, which limits the interferogram sampling range. By substituting the parallel mirror parameters into Eq. (7), we can obtain the relationship between OPD and u without errors, as shown in Fig. 8. Two bilateral sampling interferograms were obtained by rotating each circle. When rotating at high speeds, the instrument exhibits a fast response performance.

 

Fig. 8. Schematic diagram of OPD as a function of rotation angle. The gray area represents the effective OPD range, and the orange dotted line corresponds to the boundary of the effective sampling range.

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Table 1. Parallel mirror size structure parameter setting

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According to the analysis in Section 4.1, the introduction of the error parameters θ and α changes β during the rotation. We first simulated the variation in the system modulation depth with different rotation angles, based on Eq. (13) under the conditions α = 20  μrad and θ = 0 rad. The maximum wave number v = 3400 cm^-1 is selected because the higher the wave number, the lower the modulation depth when other parameters remain unchanged. The simulation results are shown by the green curve in Fig. 9. In this case, the modulation depth of the interferogram is symmetric about its center. The system modulation depth is maximum when OPD = 0 and decreases symmetrically on both sides of the interferogram symmetry. The effect in the interferogram, similar to apodization, suppresses the sidelobe slightly. In the case of θ≠0, the different θ values correspond to curves of different colors. It can be observed that the influence for θ on system modulation depth lies in the phase shift of the curve. This changes the trend and magnitude of the system modulation depth for different u in the sampling range. In this case, the modulation depth of the interferogram is asymmetric with respect to the center of the interferogram. This phenomenon affects the intensity symmetry of the interferogram, causing errors in the spectral energy.

 

Fig. 9. Relationship between the regulating system and the rotation angle for different θ. The red dotted line marks the position of u = 1.57 rad, which is the position of OPD = 0. The simulation results are obtained at a wave number v = 3400 cm^-1 and a static parallelism of α=20 μrad. The unit of θ is rad.

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This light energy error can be adjusted and reduced effectively by installation, which makes θ as close to zero as possible. In addition, the algorithm can correct the error by using an interference signal with error to simulate the trend term. Therefore, we do not consider the fluctuation of the system modulation depth, but calculate the energy effect of the system on the entire interferogram in the sampling range.

By averaging the regimes for different rotation angles over the range of the effective rotation angle, we obtained the relationship between θ and the average modulation depth, as shown in Fig. 10. The black curve indicates the variation in the average modulation depth with θ within the sampling range. It can be observed that the average modulation depth exhibited fluctuating characteristics. When θ=0 rad, the average modulation depth was the largest at approximately 0.96. The red line indicates the modulation depth of the M-FTS, which is calculated using Eq. (8). The parameters D and v are the same as those of RPM-FTS and β_M-FTS=α. The calculated modulation depth was 0.83. When -0.46 rad < θ < 0.46 rad, the average modulation depth of the RPM-FTS is larger than that of M-FTS in the sampling range.

 

Fig. 10. Within the effective sampling range, the black solid line is the relationship between the average modulation depth degree of the rotating parallel mirror and θ. The solid red line is the modulation depth degree of the M-FTS.

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To make the energy of the rotating parallel mirror spectrometer interferogram at each position larger than that of the M-FTS, a higher requirement was imposed on θ. Figure 9 shows the variation of the modulation depth for -0.13 rad < θ < 0.13 rad where the modulation depth is always higher than 0.83, and the system energy performance will be better than that of M-FTS. From an installation point of view, the tolerance of θ is easier to realize. Therefore, the RPM-FTS has a greater advantage over the M-FTS in terms of the energy performance. Usually, the installation accuracy -0.1 rad < θ < 0.1 rad supports the mean value of the modulation depth greater than 0.95, where the energy fluctuation is 12% greater than that of the M-FTS.

5. Spectral simulation analysis

We simulated the spectral acquisition process using the RPM-FTS. We used an ideal collimated rectangular Gaussian spectrum for the simulation, whose spectral range was 2200 cm^-1 ∼ 3200 cm^-1. According to the instrument parameters in Table 1, we changed the OPD of the interference light intensity and simulated equal optical path sampling of the reference laser to obtain an ideal interferogram with an optical path interval of 632.8 nm, as shown by the gray line in Fig. 11. The interferogram of the M-FTS is obtained by integrating the beam intensity whose OPD change is caused by the ideal instrument, as well as by the change in tilt angle. Error β_M-FTS= 20μrad causes the OPD error to be 2Dβ_M-FTS, which changes the phase. The interferogram of the M-FTS was calculated by integrating the light intensity for different phases, as shown by the blue line in Fig. 11. This method was used to calculate the interferogram for the RPM-FTS. The difference is that β_α,θ is determined by the error parameters α = 20 μ rad, θ = 0.07 rad, and the rotation angle u. This causes the OPD to change to 4Dβ_α,θ. The interferogram of the RPM-FTS is shown by the red line in Fig. 11. The subplot shows the comparative relationship of energy between RPM-FTS and M-FTS at the position of OPD = 0. The RPM-FTS has a higher normalized amplitude of interferogram contrast for the same static parallelism. Therefore, with the same noise, the light energy masked by the noise is less, which leads to a higher accuracy and more details of the recovered spectrum.

The spectrum is recovered by a Fourier transform of the interferogram. A certain broadening is produced owing to the truncation function without considering the effect of parallelism. We used this spreading Gaussian recovery spectrum as the standard for comparison, as shown by the orange line in Fig. 12. The recovered M-FTS and RPM-FTS spectra were normalized to the standard recovery spectrum. The black line is the spectrum recovered by RPM-FTS at α = 20 μrad θ = 0.2 rad. The gray line represents the M-FTS-recovered spectrum. It can be observed that the spectral energy recovered by RPM-FTS is better than that of M-FTS under the same installation accuracy level.

 

Fig. 11. Normalized interferogram is obtained by simulating the interferometer work. The gray line is the interferogram under ideal conditions, and the intensity at the position of OPD = 0 is used as the normalization reference. The red line is the interferogram obtained by rotating the parallel mirror when the parallelism error parameters are α=20 μrad, θ=0.07 rad. The blue line is the interferogram obtained by the Michelson spectrometer at the same degree of parallelism. The sub-plot is a magnification of the normalized interference intensity when OPD = 0.

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Fig. 12. Energy-normalized recovery spectrum. The orange line is Gaussian standard recovery spectrum. The black line is the recovery spectrum of RPM-FTS in the case of θ=0.2 rad. The gray line is the M-FTS recovered spectrum.

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6. Conclusion

We proposed a design for RPM-FTS. Compared with the conventional M-FTS, rotational motion simplifies the servo control process, and the optical path design avoids the light direction offset caused by mechanical jitter during movement. We calculated the relationship between the OPD and u during rotation using optical tracing, and analyzed the influence of the parallelism of the rotating mirror on the interference. We established a rotation vector parallelism model, which describes the relationship between the static parallelism α, the phase angle θ between the manual offset angle and static parallelism, and the rotation angle u. Through this model, we demonstrated that the RPM-FTS has more advantages in energy than the M-FTS under the same installation accuracy level. We provided the structural parameters of the RPM-FTS and conducted a simulation where the energy fluctuation was 12% greater than that of the M-FTS. Therefore, the energy fluctuation is insensitive to parallelism; thus, the spectrometer is easier to install, and can adjust with higher stability during operation. The RPM-FTS has great potential in the field of transient high quality spectral detection.