**Figure 5.** The parameter 1 *L*2 will affect the relative position of the first image points of the o and e light. (**a**) Schematic diagram of light path. Additionally, the relative position difference distribution is shown with the crystal geometry parameters of (**b**,**d**) \* 14.14 θ*i* = ° , 6.50 θα = ° , and 0 θ β = ° , (**c**,**e**) \* 15.07 *i* = ° , 6.49 θα = ° , and 4.64 θ β =− ° . The red solid lines are the results of the 2nd AOTF device **Figure 5.** The parameter *L* 1 2 will affect the relative position of the first image points of the o and e light. (**a**) Schematic diagram of light path. Additionally, the relative position difference distribution is shown with the crystal geometry parameters of (**b**,**d**) *θ* ∗ *i* = 14.14◦ , *θα* = 6.50◦ , and *θβ* = 0 ◦ , (**c**,**e**) *θ* ∗ *i* = 15.07◦ , *θα* = 6.49◦ , and *θβ* = −4.64◦ . The red solid lines are the results of the 2nd AOTF device by the optimization strategy. *xoe* in (**d**,**e**) is the relative distance to the incident point *O*.

by the optimization strategy. *oe x* in (**d**,**e**) is the relative distance to the incident point *O* . *2.3. Optimization Strategy for Polarization Multiplexing Applications*  Traditional AOTF devices, which follow the parallel tangent principle, are typically designed for a single polarization state (o or e light). However, when the 2nd AOTF device is directly used for polarization multiplexing applications, there can be a substantial spectral difference in the ±1 order light, as depicted in Figure 6(a,c4). This issue can be mitigated by optimizing the crystal geometric parameters of the AOTF devices [29,30]. By optimizing these parameters, the normal incident o and e light of the same wavelength can satisfy the momentum matching principle at the same tuning frequency, resulting in a In this paper, two AOTF devices are presented for comparison. The first device (the 1st AOTF) has optimized crystal geometry parameters of *θ* ∗ *i* = 14.14◦ , *θα* = 6.50◦ , and *θβ* = 0 (the 1st AOTF) for non-polarization applications [29,30]. The second device (the 2nd AOTF) is used in the experiments and closes to the parallel tangent principle for e-light input application, with crystal geometry parameters of *θ* ∗ *i* = 15.07◦ , *θα* = 6.49◦ , and *θβ* = −4.64◦ [31]. As shown in Figure 5b,d, the optimized *L* 1 2 of the 1st AOTF device is 66.81 mm, and the minimum *δoe* is less than 1.0 µm. From Figure 5c,e, we can find that the optimized *L* 1 2 of the 2nd AOTF device is 67.36 mm, and the minimum *δoe* is 28.4 µm. As the optimization strategy is applied to the 2nd AOTF device, resulting in improved aberration values of *L* 1 2 = 67.06 mm and *δoe* = 16.7 µm. Additionally, the optimization strategy is described in detail in Section 2.3. It is observed that using AOTF devices with

significant improvement in the matched wavelength distribution of o and e light within

plane to achieve the same goal, as illustrated in Figure 2. This also enables the o and e light of the same wavelength to satisfy the momentum matching principle at the same tuning frequency, as depicted in Figure 6f. In this case, the matched wavelength distribution of o and e light within the real aperture angle is shown in Figure 6d. The difference between the optimization strategy and the optimized crystal geometric parameters is negligible, with differences of less than 0.01 nm. Moreover, AOTF devices are sensitive to the polarization state and incident angle of the light. It can be observed from Figure 6b,d that the optimized AOTF device can only ensure consistent matched wavelengths of o and e light at the given incident angle. As shown in Figure 6(c1), the optimized AOTF device has the same matched wavelengths of o and e light under normal incidence. Although the matched wavelengths at the edge have been improved, the difference is still about 9.7 nm, as shown in Figure 6(c2). Previous studies have reported the beneficial effects of the

θ

0 ) in the polar

tion strategy to rotate the traditional AOTF device at an off-axis angle ( \*

θ

unoptimized crystal geometry parameters results in large aberrations, which is not desirable for polarization multiplexing design. Additionally, the aberration will be improved with applying the optimization strategy. However, it is pointed out that the optimized AOTF device is a better choice and we are working on this for the future.

### *2.3. Optimization Strategy for Polarization Multiplexing Applications*

Traditional AOTF devices, which follow the parallel tangent principle, are typically designed for a single polarization state (o or e light). However, when the 2nd AOTF device is directly used for polarization multiplexing applications, there can be a substantial spectral difference in the ±1 order light, as depicted in Figure 6(a,c4). This issue can be mitigated by optimizing the crystal geometric parameters of the AOTF devices [29,30]. By optimizing these parameters, the normal incident o and e light of the same wavelength can satisfy the momentum matching principle at the same tuning frequency, resulting in a significant improvement in the matched wavelength distribution of o and e light within the real aperture angle, as demonstrated in Figure 6b,e. However, developing such a new AOTF device can be costly and time-consuming. Alternatively, we propose an optimization strategy to rotate the traditional AOTF device at an off-axis angle (*θ* ∗ 0 ) in the polar plane to achieve the same goal, as illustrated in Figure 2. This also enables the o and e light of the same wavelength to satisfy the momentum matching principle at the same tuning frequency, as depicted in Figure 6f. In this case, the matched wavelength distribution of o and e light within the real aperture angle is shown in Figure 6d. The difference between the optimization strategy and the optimized crystal geometric parameters is negligible, with differences of less than 0.01 nm. Moreover, AOTF devices are sensitive to the polarization state and incident angle of the light. It can be observed from Figure 6b,d that the optimized AOTF device can only ensure consistent matched wavelengths of o and e light at the given incident angle. As shown in Figure 6(c1), the optimized AOTF device has the same matched wavelengths of o and e light under normal incidence. Although the matched wavelengths at the edge have been improved, the difference is still about 9.7 nm, as shown in Figure 6(c2). Previous studies have reported the beneficial effects of the telecentric confocal design in suppressing optical sidelobe effects [27]. However, we have found that this design also has a great adaptability for polarization multiplexing applications. It enables the AOTF device to filter the arbitrary views equally, and the spectral response function is equal to the integral within the cone angle of the light entering the AOTF device as [32]:

$$\text{Respose} \propto \int \int\_{-\frac{\theta\_{\text{in}}}{2}}^{\frac{\theta\_{\text{in}}}{2}} T\_0^{\theta}(\theta, \varphi, \lambda) I(\eta\_{\text{re}}, L\_0, \Delta k\_0) d\theta d\varphi + \int \int\_{-\frac{\theta\_{\text{in}}}{2}}^{\frac{\theta\_{\text{in}}}{2}} T\_0^{\xi}(\theta, \varphi, \lambda) I(\eta\_{\text{re}}, L\_{\text{e}}, \Delta k\_{\text{e}}) d\theta d\varphi \tag{4}$$

where *θ* is polar angle of the incident light and *ϕ* is the azimuth angle as shown in Figure 2. *T o* 0 (*θ*, *ϕ*, *λ*) and *T e* 0 (*θ*, *ϕ*, *λ*) are the intensities of o and e light entering the AOTF device, respectively. Additionally, *I*(*η*, *L*, ∆*k*) is the diffraction efficiency equation, given as:

$$I = \eta \frac{\sin^2 \left[\eta + \left(\Delta k L / 2\right)^2\right]^{1/2}}{\eta + \left(\Delta k L / 2\right)^2} \tag{5}$$

where *L* is the acousto-optic interaction length that is affected by the length of the transducer. Additionally, the actual transducer length of the 2nd AOTF device used in the experiments is about 10 mm. The parameter ∆*k* describes the momentum mismatch and *η* represents the peak diffraction efficiency [33]. By employing telecentric confocal optics, the spectral response curves of o and e light are closer to each other as Figure 6(c3). In contrast, adopting the collimation design would cause the occurrence of "double peaks" in the spectral response curves at some image points as shown in Figure 6(c2), which is not conducive to spectral detection.

**Figure 6.** Matched wavelength distributions within the real aperture angle at 632.8 nm of (**a**) the 2nd AOTF device, (**b**) the 1st AOTF device, and (**d**) the 2nd AOTF device by the optimization strategy. (**c**) Spectral curves with the collimation design in which (**c1**) describes the normal incidence of the 1st AOTF device, and (**c2**) corresponds to the edge field. Spectral curves with telecentric confocal design that (**c3**) describes the integral response of the 1st AOTF device, and (**c4**) corresponds to the 2nd AOTF device. (**e**) The wave vector diagram of the 1st AOTF device, and (**f**) corresponds to the 2nd AOTF device by the optimization strategy. **Figure 6.** Matched wavelength distributions within the real aperture angle at 632.8 nm of (**a**) the 2nd AOTF device, (**b**) the 1st AOTF device, and (**d**) the 2nd AOTF device by the optimization strategy. (**c**) Spectral curves with the collimation design in which (**c1**) describes the normal incidence of the 1st AOTF device, and (**c2**) corresponds to the edge field. Spectral curves with telecentric confocal design that (**c3**) describes the integral response of the 1st AOTF device, and (**c4**) corresponds to the 2nd AOTF device. (**e**) The wave vector diagram of the 1st AOTF device, and (**f**) corresponds to the 2nd AOTF device by the optimization strategy.

The torsional strategy is introduced in detail below. The light with an off-axis angle *θ*0 obeys the Snell's law as:

$$n\_0 \sin \theta\_0 = n\_1 \sin \theta\_1 \tag{6}$$

where *θ*1 is the refraction angle in the crystal, and *n*0 and *n*1 are the refractive indices in the air and crystal, respectively. The *n*0 is generally 1.0 in the air, and *n*1 of the o and e light can be solved by [34]:

$$\begin{cases} \begin{array}{c} n\_1^o = n\_o \\ n\_1^\varepsilon = \frac{n\_o n\_\varepsilon}{\sqrt{n\_o^2 \sin^2 \theta\_2^\varepsilon + n\_\varepsilon^2 \cos^2 \theta\_2^\varepsilon}} \end{array} \tag{7} $$

with markers *o* and *e* to distinguish *o* and *e* light. *n*o and *ne* are the principal refractive indices, respectively. In the crystal coordinate system, the refraction angle *θ*2 is:

$$
\theta\_2 = \theta\_1 - \theta\_i^\* \tag{8}
$$

Using Equations (6)–(8), we can easily obtain *θ o* 2 by:

$$
\theta\_2^o = \arcsin(\frac{\sin \theta\_0}{n\_o}) - \theta\_i^\* \tag{9}
$$

while *θ e* 2 can be obtained by solving a quadratic equation:

$$\left(\cos^2\theta\_i^\* - \frac{\sin^2\theta\_0}{n\_\varepsilon^2}\right)\tan^2\theta\_2^\varepsilon + 2\sin\theta\_i^\*\cos\theta\_i^\*\tan\theta\_2^\varepsilon + \left(\sin^2\theta\_i^\* - \frac{\sin^2\theta\_0}{n\_o^2}\right) = 0\tag{10}$$

Now we can obtain *θ o* 2 and *θ e* 2 in the wave vector diagram (Figure 6f) at the incident angle of *θ*0, and *n o* 1 and *n e* 1 can also be obtained from Equation (7). The incident optical wave vectors can be solved by:

$$\begin{cases} \begin{vmatrix} \stackrel{\rightarrow}{k\_i^o} \\ \stackrel{\rightarrow}{k\_i^e} \end{vmatrix} = \frac{2\pi n\_1^o}{\lambda} \\ \begin{vmatrix} \stackrel{\rightarrow}{k\_i^e} \end{vmatrix} = \frac{2\pi n\_1^e}{\lambda} \end{cases} \tag{11}$$

and the coordinates of the intersection *A* are:

$$\begin{cases} A\_o = \frac{2\pi n\_1^o}{\lambda} (\cos \theta\_2^o, \sin \theta\_2^o) \\\ A\_\varepsilon = \frac{2\pi n\_1^\varepsilon}{\lambda} (\cos \theta\_2^\varepsilon, \sin \theta\_2^\varepsilon) \end{cases} \tag{12}$$

The linear equations of *AoBe* and *AeBo* are similar:

$$y - y\_A = \frac{1}{\tan \theta\_a} (\mathbf{x} - \mathbf{x}\_A) \tag{13}$$

The wave vector elliptic equations of o and e light are:

$$\begin{cases} \frac{\mathbf{x}^2}{n\_o^2} + \frac{\mathbf{y}^2}{n\_o^2} = \left(\frac{2\pi}{\lambda}\right)^2\\ \frac{\mathbf{x}^2}{n\_o^2} + \frac{\mathbf{y}^2}{n\_c^2} = \left(\frac{2\pi}{\lambda}\right)^2 \end{cases} \tag{14}$$

Then, using Equations (12)–(14), the coordinate of the point *Be*(*x e B* , *y e B* ) can be obtained by solving a quadratic equation:

$$\begin{aligned} \left(\frac{\tan^2\theta\_a}{n\_o^2} + \frac{1}{n\_c^2}\right) y\_B^{\varepsilon 2} &\quad + \frac{4\pi\tan\theta\_a}{\lambda n\_o} (\cos\theta\_2^o - \sin\theta\_2^o \tan\theta\_a) y\_B^{\varepsilon} \\ &\quad + \frac{4\pi^2}{\lambda^2} \left(\cos^2\theta\_2^o + \sin^2\theta\_2^o \tan^2\theta\_a - 2\sin\theta\_2^o \cos\theta\_2^o \tan\theta\_a - 1\right) = 0 \end{aligned} \tag{15}$$

and

$$\mathfrak{x}\_{B}^{\varepsilon} = \tan \theta\_{\mathfrak{a}} (y\_{B}^{\varepsilon} - y\_{A}^{\rho}) + \mathfrak{x}\_{A}^{\rho} \tag{16}$$

Similarly, we can obtain the coordinates of point *Bo*(*x o B* , *y o B* ) by:

$$\begin{bmatrix} \left(\tan^2\theta\_a + 1\right)y\_B^{\rho} + \frac{4\pi n\_1^{\epsilon}\tan\theta\_a}{\lambda}(\cos\theta\_2^{\epsilon} - \sin\theta\_2^{\epsilon}\tan\theta\_a)y\_B^{\rho} \\ + \frac{4\pi^2}{\lambda^2} \left[ \left(n\_1^{\epsilon}\right)^2 \cos^2\theta\_2^{\epsilon} + \left(n\_1^{\epsilon}\right)^2 \sin^2\theta\_2^{\epsilon}\tan^2\theta\_a - 2\left(n\_1^{\epsilon}\right)^2 \sin\theta\_2^{\epsilon}\cos\theta\_2^{\epsilon}\tan\theta\_a - n\_o^{\epsilon}\right] = 0 \end{bmatrix} \tag{17}$$

and

$$\mathbf{x}\_B^o = \tan \theta\_\mathbf{a} (y\_B^o - y\_A^\varepsilon) + \mathbf{x}\_A^\varepsilon \tag{18}$$

Finally, the fitting torsion angle *θ* ∗ 0 can be obtained by traversing method with → *AoBe*  = → *AeBo* . When using a normal AOTF device with e light input design, *θ* ∗ 0 is greater than 0. While for an AOTF device with o light input design, *θ* ∗ 0 is less than 0. In this study, the 2nd AOTF device used had a fitting *θ* ∗ 0 of 2.19◦ . The optimization strategy employed also allows for the same AOTF device to be used in o light input mode, which can be achieved through the calculation of the Bragg angle of the o light input application under the parallel tangent principle. Specifically, the Bragg angle of the o light input application should be 13.18◦ at *θα* = 6.49◦ , and the torsion angle required is 4.29◦ , switching to the o light input mode. We can now set the example system parameters as shown in Table 1.

**Table 1.** Example system parameters of the proposed design.

We have further calculated the theoretical throughputs of the two AOTF devices using Equation (4) with the parameters listed in Table 1. The results of these calculations are presented in Figure 7 and Table 2. In our calculations, we accounted for both polarization states and utilized an angular sampling step of 0.1◦ . As a result, we obtained an input intensity of 3362 (41 × 41 × 2) at each wavelength within the real aperture angle of 4.0◦ . Compared to the traditional single polarization detection, the polarization multiplexing design proposed in this paper can improve the throughput by about two-fold. Additionally, we observed a slight increase of 0.7% in the peak diffraction efficiency.

**Table 2.** Theoretical throughput data and comparison results.

Ratio of throughput

**Table 1.** Example system parameters of the proposed design.

0 2 () ω

1

1

AOTF θ

**Parameters The 1st AOTF The 2nd AOTF**

 ° ±7.5 ±7.5 input *D* (mm) 5.2 5.2

*f* (mm) 75 75

2 *L* (mm) 66.8 67.1 AOTF *D* (mm) 20 20

 ( )° 4.0 4.0 2*f* (mm) 75 75 block *D* (mm) 5.2 5.3 imager *D* (mm) 15.9 16.1

ally, we observed a slight increase of 0.7% in the peak diffraction efficiency.

We have further calculated the theoretical throughputs of the two AOTF devices using Equation (4) with the parameters listed in Table 1. The results of these calculations are presented in Figure 7 and Table 2. In our calculations, we accounted for both polarization states and utilized an angular sampling step of 0.1°. As a result, we obtained an input intensity of 3362 (41 × 41 × 2) at each wavelength within the real aperture angle of 4.0°. Compared to the traditional single polarization detection, the polarization multiplexing design proposed in this paper can improve the throughput by about two-fold. Addition-

**(Optimization Strategy)**

**Figure 7.** Spectral response curves. (**a**) The 1st AOTF device with optimization of crystal geometric parameters. \* 14.97 θ*i* = ° and \* 13.20 θ*i* = ° satisfy the parallel tangent principle of e and o light, respectively. (**b**) The 2nd AOTF device with the optimization strategy. **Figure 7.** Spectral response curves. (**a**) The 1st AOTF device with optimization of crystal geometric parameters. *θ* ∗ *i* = 14.97◦ and *θ* ∗ *i* = 13.20◦ satisfy the parallel tangent principle of e and o light, respectively. (**b**) The 2nd AOTF device with the optimization strategy. improvement 2.03 2.00 2.05 2.02 **3. Results and Discussion**

#### **3. Results and Discussion** *3.1. Test and Analysis of Optimization Strategy*

#### *3.1. Test and Analysis of Optimization Strategy* A spectral test optics setup, as shown in Figure 8, was constructed. The light source

A spectral test optics setup, as shown in Figure 8, was constructed. The light source was a wide-spectrum light source composed of an integrating sphere (Halogen lamp) and a collimator. The input aperture used is a circular aperture of 5.2 mm diameter. The AOTF device selected for the experiment was the SGL100-400/850-20LG-K, which was sourced from China. The commercial spectrometer used is the Ocean Optics USB2000+. Spectral sampling of o and e light was performed behind the light barrier (exit pupil). The results of spectral tests are shown in Figure 9. Additionally, the results show good agreement with the theoretical calculations. This demonstrates that spectral modulation of the AOTF device can be realized using the optimization strategy, making it suitable for polarization multiplexing applications. In addition, our previous work has revealed the consistent response of the incident light at a torsion angle of 2.19◦ [31]. was a wide-spectrum light source composed of an integrating sphere (Halogen lamp) and a collimator. The input aperture used is a circular aperture of 5.2 mm diameter. The AOTF device selected for the experiment was the SGL100-400/850-20LG-K, which was sourced from China. The commercial spectrometer used is the Ocean Optics USB2000+. Spectral sampling of o and e light was performed behind the light barrier (exit pupil). The results of spectral tests are shown in Figure 9. Additionally, the results show good agreement with the theoretical calculations. This demonstrates that spectral modulation of the AOTF device can be realized using the optimization strategy, making it suitable for polarization multiplexing applications. In addition, our previous work has revealed the consistent response of the incident light at a torsion angle of 2.19° [31].

**Figure 8.** Structure diagram of the desktop system for testing the spectral curves. **Figure 8.** Structure diagram of the desktop system for testing the spectral curves.

**Table 2.** Theoretical throughput data and comparison results. **AOTF Device The 1st AOTF The 2nd AOTF**

> **Single e Light**

**Single o Light**

A spectral test optics setup, as shown in Figure 8, was constructed. The light source was a wide-spectrum light source composed of an integrating sphere (Halogen lamp) and a collimator. The input aperture used is a circular aperture of 5.2 mm diameter. The AOTF device selected for the experiment was the SGL100-400/850-20LG-K, which was sourced from China. The commercial spectrometer used is the Ocean Optics USB2000+. Spectral sampling of o and e light was performed behind the light barrier (exit pupil). The results of spectral tests are shown in Figure 9. Additionally, the results show good agreement with the theoretical calculations. This demonstrates that spectral modulation of the AOTF device can be realized using the optimization strategy, making it suitable for polarization multiplexing applications. In addition, our previous work has revealed the consistent re-

PC

θ

θ

**Figure 8.** Structure diagram of the desktop system for testing the spectral curves.

620–650 nm 78090 38458 39119 77680 37979 38402

efficiency 33.3% 32.4% 32.6% 33.1% 31.8% 32.4%

improvement 2.03 2.00 2.05 2.02

sponse of the incident light at a torsion angle of 2.19° [31].

**±2°**

\* θ0

**Polarization Multiplexing**

**Single e Light**

**Single o Light**

Spectrometer

**Detection Mode Polarization**

Theoretical throughput within

Maximum diffraction

Ratio of throughput

Integrating sphere

Collimator

**Multiplexing**

**3. Results and Discussion**

*3.1. Test and Analysis of Optimization Strategy*

**Figure 9.** The tested and theoretical spectral response curves at (**a**) *θ* ∗ 0 = 0, (**b**) *θ* ∗ 0 = 2.19◦ , and (**c**) *θ* ∗ 0 = 4.29◦ . **Figure 9.** The tested and theoretical spectral response curves at (**a**) \* 0 = 0 , (**b**) \* 0 = ° 2.19 , and (**c**) \* 0 θ= ° 4.29 .

### *3.2. Test and Analysis of Throughput 3.2. Test and Analysis of Throughput*

Then, the actual throughput of the proposed polarization multiplexing design and the traditional single polarization light design are compared in this paper. The evaluation factor used is the average Digital Number (DN) index of the detector within the effective area (green box area of the images), while the image signal-to-noise ratio (SNR) is evaluated using the same size of the red box area of the image. In the tests, len1 and lens2 temporarily use a single lens here, which can be optically optimized in the future. Additionally, further details of the system parameters can be found in Table 1. In the traditional design, cross polarizers are used before and after the AOTF device as shown in Figure 10, and the baffle used is wider to eliminate interference from the other polarization light. The polarizer used in the system has a transmittance of approximately 86% (GCL-050003). The imager uses a camera (Basler acA640-120gm) with 50 mm focal length lens (PENTAX B5014A). The light source is also the wide-spectrum light source composed of an integrating sphere and a collimator. The target is a hollow square target. Throughout the experiment, the camera maintained a consistent exposure time of 20 ms (milliseconds) and a fixed gain of 300. The captured images are listed in Figure 11, which show that the proposed polarization multiplexing design can effectively improve the system throughput by over 2.6-fold, as indicated in Table 3. By eliminating the losses incurred by the cross polarizers, the resulting increase in throughput is nearly twice as much as predicted by the theoretical calculations in Section 2.3. Simultaneously, the efficiency of conventional designs in separately collecting o light and e light is comparable. Then, the actual throughput of the proposed polarization multiplexing design and the traditional single polarization light design are compared in this paper. The evaluation factor used is the average Digital Number (DN) index of the detector within the effective area (green box area of the images), while the image signal-to-noise ratio (SNR) is evaluated using the same size of the red box area of the image. In the tests, len1 and lens2 temporarily use a single lens here, which can be optically optimized in the future. Additionally, further details of the system parameters can be found in Table 1. In the traditional design, cross polarizers are used before and after the AOTF device as shown in Figure 10, and the baffle used is wider to eliminate interference from the other polarization light. The polarizer used in the system has a transmittance of approximately 86% (GCL-050003). The imager uses a camera (Basler acA640-120gm) with 50 mm focal length lens (PENTAX B5014A). The light source is also the wide-spectrum light source composed of an integrating sphere and a collimator. The target is a hollow square target. Throughout the experiment, the camera maintained a consistent exposure time of 20 ms (milliseconds) and a fixed gain of 300. The captured images are listed in Figure 11, which show that the proposed polarization multiplexing design can effectively improve the system throughput by over 2.6-fold, as indicated in Table 3. By eliminating the losses incurred by the cross polarizers, the resulting increase in throughput is nearly twice as much as predicted by the theoretical calculations in Section 2.3. Simultaneously, the efficiency of conventional designs in separately collecting o light and e light is comparable.

**Figure 10.** Structure diagram of the AOTF spectral imaging systems for testing the throughput. (**a**) Polarization multiplexing design. (**b**) Single polarization detection design. **Figure 10.** Structure diagram of the AOTF spectral imaging systems for testing the throughput. (**a**) Polarization multiplexing design. (**b**) Single polarization detection design.

Single o-polarized light

Single e-polarized light

Polarization multiplexing

Lens1 AOTF

aperture

(**b**)

(**a**)

Crossed polarizers

Polarization multiplexing design. (**b**) Single polarization detection design.

barrier Imager Input

Lens2 Light

**Figure 10.** Structure diagram of the AOTF spectral imaging systems for testing the throughput. (**a**)

**Figure 9.** The tested and theoretical spectral response curves at (**a**) \*

signs in separately collecting o light and e light is comparable.

Then, the actual throughput of the proposed polarization multiplexing design and the traditional single polarization light design are compared in this paper. The evaluation factor used is the average Digital Number (DN) index of the detector within the effective area (green box area of the images), while the image signal-to-noise ratio (SNR) is evaluated using the same size of the red box area of the image. In the tests, len1 and lens2 temporarily use a single lens here, which can be optically optimized in the future. Additionally, further details of the system parameters can be found in Table 1. In the traditional design, cross polarizers are used before and after the AOTF device as shown in Figure 10, and the baffle used is wider to eliminate interference from the other polarization light. The polarizer used in the system has a transmittance of approximately 86% (GCL-050003). The imager uses a camera (Basler acA640-120gm) with 50 mm focal length lens (PENTAX B5014A). The light source is also the wide-spectrum light source composed of an integrating sphere and a collimator. The target is a hollow square target. Throughout the experiment, the camera maintained a consistent exposure time of 20 ms (milliseconds) and a fixed gain of 300. The captured images are listed in Figure 11, which show that the proposed polarization multiplexing design can effectively improve the system throughput by over 2.6-fold, as indicated in Table 3. By eliminating the losses incurred by the cross polarizers, the resulting increase in throughput is nearly twice as much as predicted by the theoretical calculations in Section 2.3. Simultaneously, the efficiency of conventional de-

*3.2. Test and Analysis of Throughput*

\* 0 θ= ° 4.29 .

0 θ

= 0 , (**b**) \*

0 θ

= ° 2.19 , and (**c**)

**Figure 11.** The captured images of the square target at 632.8 nm.

**Table 3.** The response DN values of the captured images.

This paper also includes a comparison of the SNR of images captured by various systems. The SNR can be computed using the following formula:

$$R = 20 \lg \left( DN\_s \left/ DN\_n \right) \right. \tag{19}$$

where *DNs* and *DNn* are the DN values of the signal and noise, respectively. The variable *R* represents the magnitude of the image SNR. The results demonstrate that the proposed polarization multiplexing design improves the image SNR by approximately 8 dB, as illustrated in Table 3.

### *3.3. Test and Analysis of Spatial Resolution*

In addition, spatial resolution represents a crucial characteristic of the imaging system. To assess the image quality in the polarization multiplexing system, we conducted a comprehensive evaluation employing the USAF-1951 resolution target as the reference. Figure 12a displayed the captured image, revealing a notable improvement in quality compared to previously published works [18,35]. For quantitative assessment, we utilized the contrast transfer function (CTF) defined as follows:

$$F = \frac{DN\_{\text{max}} - DN\_{\text{min}}}{DN\_{\text{max}} + DN\_{\text{min}}} \tag{20}$$

where *DN*max and *DN*min are the maximum and minimum DN values along the orthogonal line direction as shown in Figure 12a, respectively. The variable *F* denotes the CTF value. Furthermore, we obtained the CTF analysis results in both horizontal and vertical directions, displayed in Figure 12b. The horizontal direction corresponds to the AOTF diffraction direction. The spatial resolution of the USAF-1951 target can be determined as follows:

$$\text{Resolution} = 2^{Group + \frac{(Element - 1)}{6}} \tag{21}$$

where *Group* is the group number and *Element* is the element number as Figure 12a. In the red dot wireframe (*Group* = 0, *Element* = 1), the corresponding single linewidth is 0.5 mm, indicating a resolution of 1 Lp/mm. The CTF analysis results show that the horizontal resolution is slightly higher than the vertical resolution as shown in Figure 12b. This discrepancy is primarily influenced by polarization multiplexing, resulting from the distinction between o and e light [36,37]. In future research, we aim to enhance this aspect by optimizing the geometric parameters of the AOTF device (as detailed in Section 2.2) and optimizing the lens group of the overall optical system. 0.5 mm, indicating a resolution of 1 Lp/mm. The CTF analysis results show that the horizontal resolution is slightly higher than the vertical resolution as shown in Figure 12b. This discrepancy is primarily influenced by polarization multiplexing, resulting from the distinction between o and e light [36,37]. In future research, we aim to enhance this aspect by optimizing the geometric parameters of the AOTF device (as detailed in Section 2.2) and optimizing the lens group of the overall optical system.

*Materials* **2023**, *16*, x FOR PEER REVIEW 15 of 17

#### **4. Conclusions 4. Conclusions**

62075202).

In summary, we propose a novel design to address the low throughput of traditional AOTF spectral imaging systems. Our design utilizes polarization multiplexing to simultaneously capture ±1 order light of the AOTF device, resulting in a more than two-fold increase in system throughput. The experimental results validate the effectiveness of our design in improving system throughput and enhancing imaging SNR by approximately 8 dB. Moreover, our proposed design offers simplicity in structure compared to previously published works. Additionally, it is optimized for concerns such as inconsistent spectral response, real aperture angle compression, and transmission stray light interference. This research offers a valuable contribution to the field of AOTF spectral imaging systems in target detection applications. However, it should be noted that the polarization multiplexing design may lead to a reduction in spatial resolution, especially in the AOTF diffraction direction, which necessitates further efforts to overcome in future studies. In summary, we propose a novel design to address the low throughput of traditional AOTF spectral imaging systems. Our design utilizes polarization multiplexing to simultaneously capture ±1 order light of the AOTF device, resulting in a more than two-fold increase in system throughput. The experimental results validate the effectiveness of our design in improving system throughput and enhancing imaging SNR by approximately 8 dB. Moreover, our proposed design offers simplicity in structure compared to previously published works. Additionally, it is optimized for concerns such as inconsistent spectral response, real aperture angle compression, and transmission stray light interference. This research offers a valuable contribution to the field of AOTF spectral imaging systems in target detection applications. However, it should be noted that the polarization multiplexing design may lead to a reduction in spatial resolution, especially in the AOTF diffraction direction, which necessitates further efforts to overcome in future studies.

In addition, AOTF devices used in polarization multiplexing applications require a special design that does not follow the parallel tangent principle. This paper introduces an optimization strategy for arbitrary AOTF devices, which can achieve similar spectral effects without incurring high costs or long manufacturing periods. The test results demonstrate that the optimization strategy effectively corrects the spectral inconsistency of ±1 order light from the AOTF device at the same tuning frequency. This work is not only applicable to the proposed polarization multiplexing system, but also beneficial to AOTF polarization spectroscopy detection systems [38] and other AOTF detection systems that collect ±1 order light simultaneously. In addition, AOTF devices used in polarization multiplexing applications require a special design that does not follow the parallel tangent principle. This paper introduces an optimization strategy for arbitrary AOTF devices, which can achieve similar spectral effects without incurring high costs or long manufacturing periods. The test results demonstrate that the optimization strategy effectively corrects the spectral inconsistency of ±1 order light from the AOTF device at the same tuning frequency. This work is not only applicable to the proposed polarization multiplexing system, but also beneficial to AOTF polarization spectroscopy detection systems [38] and other AOTF detection systems that collect ±1 order light simultaneously.

**Author Contributions:** Conceptualization, H.Z. (Hao Zhang) and H.Z. (Huijie Zhao); Methodology, H.Z. (Hao Zhang); Investigation, H.Z. (Hao Zhang) and D.X.; Validation, H.Z. (Hao Zhang) and

draft preparation, H.Z. (Hao Zhang); Writing—review and editing, H.Z. (Huijie Zhao); Supervision, H.Z. (Huijie Zhao) and Q.G.; Project administration, H.Z. (Huijie Zhao); Funding acquisition, H.Z. (Huijie Zhao) and Q.G. All authors have read and agreed to the published version of the manuscript. **Funding:** This research was funded by the National Natural Science Foundation of China (61875004,

**Institutional Review Board Statement:** Not applicable.

**Author Contributions:** Conceptualization, H.Z. (Hao Zhang) and H.Z. (Huijie Zhao); Methodology, H.Z. (Hao Zhang); Investigation, H.Z. (Hao Zhang) and D.X.; Validation, H.Z. (Hao Zhang) and W.T.; Formal analysis, H.Z. (Hao Zhang); Resources, H.Z. (Huijie Zhao) and Q.G.; Writing—original draft preparation, H.Z. (Hao Zhang); Writing—review and editing, H.Z. (Huijie Zhao); Supervision, H.Z. (Huijie Zhao) and Q.G.; Project administration, H.Z. (Huijie Zhao); Funding acquisition, H.Z. (Huijie Zhao) and Q.G. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the National Natural Science Foundation of China (61875004, 62075202).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** We would like to thank Xu Zefu for some discussions and Beihang Nano for generously providing access to their darkroom and some experimental equipment.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **References**

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## *Article* **Research and Application Validation of a Feature Wavelength Selection Method Based on Acousto-Optic Tunable Filter (AOTF) and Automatic Machine Learning (AutoML)**

**Zhongpeng Ji 1,2 , Zhiping He 1,2,\*, Yuhua Gui 1,2, Jinning Li 1,2, Yongjian Tan 1,2, Bing Wu 1,2, Rui Xu 1,2 and Jianyu Wang 1,2,\***

**Abstract:** Near-infrared spectroscopy has been widely applied in various fields such as food analysis and agricultural testing. However, the conventional method of scanning the full spectrum of the sample and then invoking the model to analyze and predict results has a large amount of collected data, redundant information, slow acquisition speed, and high model complexity. This paper proposes a feature wavelength selection approach based on acousto-optical tunable filter (AOTF) spectroscopy and automatic machine learning (AutoML). Based on the programmable selection of sub nm center wavelengths achieved by the AOTF, it is capable of rapid acquisition of combinations of feature wavelengths of samples selected using AutoML algorithms, enabling the rapid output of target substance detection results in the field. The experimental setup was designed and application validation experiments were carried out to verify that the method could significantly reduce the number of NIR sampling points, increase the sampling speed, and improve the accuracy and predictability of NIR data models while simplifying the modelling process and broadening the application scenarios.

**Keywords:** AOTF; AutoML; feature wavelength selection; near infrared detection system

### **1. Introduction**

NIR spectroscopy has many advantages, such as being nondestructive and accurate, and has been widely applied in areas such as food safety [1,2], drug analysis [3], agricultural testing [4], and basic chemistry [5]. The current common approach to NIR spectroscopy is to obtain the full continuous spectral data of the sample in the spectral range and to use the corresponding algorithms to model the correlation between the sample and the spectral data. However, the NIR spectral data of a sample has a relatively high dimensionality and also suffers from inter-spectral overlap, covariance, and noise, which negatively affects the performance of the NIR spectral model [6]. The selection of the effective feature wavelengths of the sample is extremely important at this point. Feature wave-length selection extracts spectrally valid variables and removes useless or interfering wavelength data, improving the accuracy and predictiveness of the data model. Only spectral data from a specific band or specific wavelength points are required to build a well-performing detection model, requiring significantly fewer wavelength sampling points.

Numerous algorithms exist to select the characteristic wavelengths of the collected NIR spectra of the samples to build reliable models [7], such as the adaptive reweighted sampling (CARS) [8], the random frog hopping algorithm (RF) [9], the PLS–genetic algorithm (PLS-GA) [10], etc. These algorithms have different performance for different data

**Citation:** Ji, Z.; He, Z.; Gui, Y.; Li, J.; Tan, Y.; Wu, B.; Xu, R.; Wang, J. Research and Application Validation of a Feature Wavelength Selection Method Based on Acousto-Optic Tunable Filter (AOTF) and Automatic Machine Learning (AutoML). *Materials* **2022**, *15*, 2826. https:// doi.org/10.3390/ma15082826

Academic Editors: Alexander S. Machikhin and Vitold Pozhar

Received: 4 March 2022 Accepted: 8 April 2022 Published: 12 April 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

and problems, and in practice modelling usually involves human experience in selecting the most suitable model, which increases the complexity of the modelling process. In contrast, the recently proposed automatic machine learning (AutoML) [11] allows for automated model selection without human intervention. The model automatically generates the network structure that is most efficient for the task at hand, and the model automatically searches for the best sequence of combinations of operations under the different structures produced by itself. This approach can effectively reduce the complexity of modelling and also ensures the robustness of the model.

After modelling the characteristic wavelengths of the sample, the current instruments still need to collect the full spectrum first and then select the characteristic spectra to input into the model to obtain the final results [12,13]. However, this full-spectrum acquisition method results in slow data acquisition and processing. Although common grating-based spectroscopy instruments can acquire data quickly, the cost is significantly higher due to the use of array detectors, and problems such as non-uniformity between detectors can also affect the signal-to-noise ratio of the acquisition. In some applications where fast real-time processing is required, such as industrial on-line analysis, faster spectral acquisition and higher data quality are often required. Therefore, an NIR spectrometer that can be coupled with a feature wavelength filtering algorithm to achieve variable wavelength acquisition is a solution that can guarantee both speed and data quality and model robustness. An AOTF spectrometer can change the diffraction wavelength by changing the frequency of the RF power signal added to it, which can achieve sub-nm-level central wavelength picking in the full spectrum and use a unit detector to obtain data, which can effectively improve the signal-to-noise ratio and is an ideal device that can be used with the feature wavelength selection algorithm.

AOTF-based NIR spectroscopy has been extensively studied in food inspection and agricultural applications. Several studies have been conducted to implement commercial AOTF-NIR spectrometers for nondestructive detection of dried apple and olive fruit [14–17]. Diffuse reflectance spectra were acquired in the wavelength range 1100–2300 nm at 2 nm intervals using the Luminar 5030 AOTF-NIR Miniature 'Hand-held' Analyzer (Brimrose Corporation, Baltimore, MD, USA). The feasibility of using AOTF-NIR spectroscopy in an intelligent drying system for nondestructive detection and monitoring of physicochemical changes in organic apple wedges during the drying process was investigated. Partial least squares (PLS) regression models were developed to monitor changes in water activity, moisture content, soluble solids content, and chroma during drying. The classification models were computed using K-means and Partial Least Squares Discriminant Analysis (PLS-DA) algorithms in sequence [14]. AOTF-NIR was also satisfactorily applied to predict phenolic compounds [15] and monitor the ripening of olives [16].  In addition to the PLS regression algorithm, a sensor software based on an artificial neural network (SS-ANN) was designed by Allouche et al. [18] for monitoring olive malaxation. In hyperspectral imaging, a prototype on-line AOTF-based hyperspectral image acquisition system (450–900 nm) has also been developed for tenderness assessment of beef carcasses [19]. However, in these previous studies, it is usually necessary to collect all spectral data before making predictions, and for AOTF spectroscopy more research is needed to optimize the sampling strategy according to specific applications.

Therefore, this paper utilizes the features of AOTF programming to acquire specific wavelengths and combines an AOTF-based spectrometer with an automatic machine learning–based feature wavelength selection algorithm to achieve rapid output of target substance detection results in the field. This form of application will effectively improve the efficiency of online NIR spectroscopy systems and bring broader prospects for industry applications, as it can significantly reduce the amount of data to be collected and reduce the time required for the cyclic data collection process during online inspection.

#### **2. AutoML-Based Feature Wavelength Selection** ance, such as data preprocessing, feature selection, algorithmic models, and the determi-

**2. AutoML-Based Feature Wavelength Selection**

*Materials* **2022**, *15*, x FOR PEER REVIEW 3 of 13

The efficiency of current machine learning algorithms often relies on human guidance, such as data preprocessing, feature selection, algorithmic models, and the determination of hyperparameters. With the complexity of machine learning algorithms, the number of available algorithms and processes is increasing. AutoML is a spatial search optimization method that can find the optimal solution in a finite spatial range in the shortest possible time, reducing time and labor costs while improving operational accuracy [11]. AutoML generally consists of two main components: an evaluator and a tuner. The evaluator is responsible for measuring the performance of the learning tool under the adoption number conditions provided by the tuner and for feeding the results to the tuner, while the tuner is responsible for making the feedback information for the learning tool to update the configuration information. In addition to algorithm selection, hyperparameter optimization [20], and neural network architecture searches, AutoML can also cover automatic data preparation, automatic feature selection, automatic flowline construction, automatic model selection, and integrated learning [21]. One of the solutions for AutoML is AutoGluon– Tabular architecture [22], which is very suitable for structured data regression problem solving with more accurate model robustness. The analytical modeling of NIR spectral data is a typical structured data regression problem, and the screening of feature wavelengths is also a variable selection problem in machine learning, so it is very suitable for AutoGluon–Tabular architecture, which is used in this paper for the analytical modeling and model-based feature wavelength screening of NIR spectral data. This architecture differs from the common hyperparameter search-based technique architecture in that it relies on fusing multiple models that do not require a hyperparameter search, thus avoiding hyperparameter search and increasing the number of trained models in a prescribed time. The operational steps and the flow chart for data feature wavelength selection of NIR spectral data using AutoGluon–Tabular architecture are as shown in Figure 1. nation of hyperparameters. With the complexity of machine learning algorithms, the number of available algorithms and processes is increasing. AutoML is a spatial search optimization method that can find the optimal solution in a finite spatial range in the shortest possible time, reducing time and labor costs while improving operational accuracy [11]. AutoML generally consists of two main components: an evaluator and a tuner. The evaluator is responsible for measuring the performance of the learning tool under the adoption number conditions provided by the tuner and for feeding the results to the tuner, while the tuner is responsible for making the feedback information for the learning tool to update the configuration information. In addition to algorithm selection, hyperparameter optimization [20], and neural network architecture searches, AutoML can also cover automatic data preparation, automatic feature selection, automatic flowline construction, automatic model selection, and integrated learning [21]. One of the solutions for AutoML is AutoGluon–Tabular architecture [22], which is very suitable for structured data regression problem solving with more accurate model robustness. The analytical modeling of NIR spectral data is a typical structured data regression problem, and the screening of feature wavelengths is also a variable selection problem in machine learning, so it is very suitable for AutoGluon–Tabular architecture, which is used in this paper for the analytical modeling and model-based feature wavelength screening of NIR spectral data. This architecture differs from the common hyperparameter search-based technique architecture in that it relies on fusing multiple models that do not require a hyperparameter search, thus avoiding hyperparameter search and increasing the number of trained models in a prescribed time. The operational steps and the flow chart for data feature wavelength selection of NIR spectral data using AutoGluon–Tabular architecture are as shown in Figure 1.

the efficiency of online NIR spectroscopy systems and bring broader prospects for industry applications, as it can significantly reduce the amount of data to be collected and reduce the time required for the cyclic data collection process during online inspection.

The efficiency of current machine learning algorithms often relies on human guid-

**Figure 1. Figure 1.** Feature wavelength selection using the AutoGluon-Tabular architecture. Feature wavelength selection using the AutoGluon-Tabular architecture.

(1) Data pre-processing

Firstly, the acquired transmission/reflection spectral data are converted into absorbance data, and the data are loaded into AutoGluon–Tabular. AutoGluon–Tabular will automatically check the label columns, determine the type of problem, and distinguish whether it is a classification problem or a regression problem; then the data are pre-processed, the feature data are classified, and the useless feature data are discarded.

(2) Model training

After the data pre-processing, the data will be trained by a series of machine learning models, using integration and stacking techniques to combine multiple models. AutoGluon– Tabular will select models for training in a unique order, firstly selecting the models with reliable performance and then gradually selecting the more computationally intensive but less reliable models. The current AutoGluon–Tabular architecture supports algorithms such as Random Forest [23], Super Random Tree, k Nearest Neighbor, LightGBM boosted tree, CatBoost boosted tree, AutoGluon–Tabular deep neural network, etc. In this paper, we will train all of the above algorithms. AutoGluon–Tabular ensembles multiple models and stacks them in multiple layers, which offers better use of allocated training time than seeking out the best. We set 'root\_mean\_squared\_error' as "eval\_metric". AutoGluon– Tabular tunes factors such as hyperparameters, early-stopping, ensemble-weights, etc. in order to improve this metric on validation data.

(3) Wavelength importance ranking

We calculated the permutation importance which measures the importance of a feature and ranked the wavelength variables according to this. The higher the permutation importance, the higher the contribution of the wavelength to the model and the more representative the wavelength is of the characteristics of the detection target.

### (4) Feature wavelength combination screening

After ranking the importance of wavelengths, we are not sure which wavelength combinations can achieve the best performance of the model. In order to filter the best combination of feature wavelengths, we select the wavelengths with the top X permutation importance to retrain the model, record the evaluation index of the model performance trained by X wavelengths, reduce the number of X by 1, and repeat the retraining process until X equals 0, then stop the training. Based on the number of different X's and the evaluation metrics of the model performance, we introduce the precision criterion to determine the optimal number of wavelength variables. If a precision value is entered, the model suggested according to the precision criterion is the model with the lowest number of variables among all models, whose evaluation metric differs from the minimum evaluation metric by no more than the precision.

Finally, we determined the optimal wavelength combination to achieve the minimum number of measurements for a more effective model.

### **3. Instrument Design and Working Principle**

After filtering the feature wavelengths of spectral data by AutoML, the construction of the excellent performance model and the accurate prediction results can be achieved via the collection of spectral data at specific wavelengths; however, the extraction of the feature wavelengths of the current conventional spectrometer spectral data requires us to obtain the full-band spectrum by scanning in advance, which results in sampling time waste and resource consumption. Thus, the above problems can be effectively solved if the wavelength filtering results can be targeted and finely sampled, while the AOTF near-infrared spectrometer has the characteristic of freely adjustable wavelength, which allows it to conduct discrete sampling and perform targeted and finely spectral sampling for specific filtering results.

AOTF is an electrically tunable filter based on the acousto-optic effect, which is mainly composed of a birefringent acousto-optic crystal (the most widely used material TeO2), a piezoelectric transducer, and an acoustic wave absorber, as shown in Figure 2a. In practical applications, the RF drive signal is converted to ultrasonic waves inside the crystal by a piezoelectric transducer fixed on the crystal surface, and the diffraction wavelengths are selected by changing the frequency of the RF signal, with a wide tuning range and fast scanning speed. Spectrometers based on AOTF spectroscopy have the advantages of small size, light weight, all-solid state, and strong environmental adaptability and have been widely used in many fields such as food inspection [24], environmental monitoring [25,26], and deep space exploration [27,28]. According to the condition of momentum matching, the

tuning relationship between the driving frequency *f*(*λ*) of AOTF and the output diffraction wavelengths is shown as Equation (1) [29,30]: output diffraction wavelengths is shown as Equation (1) [29,30]: () = [ ଶ + ௗ ଶ − 2ௗ ( − ௗ)]ଵ/ଶ/, (1)

TeO2), a piezoelectric transducer, and an acoustic wave absorber, as shown in Figure 2a. In practical applications, the RF drive signal is converted to ultrasonic waves inside the crystal by a piezoelectric transducer fixed on the crystal surface, and the diffraction wavelengths are selected by changing the frequency of the RF signal, with a wide tuning range and fast scanning speed. Spectrometers based on AOTF spectroscopy have the advantages of small size, light weight, all-solid state, and strong environmental adaptability and have been widely used in many fields such as food inspection [24], environmental monitoring [25,26], and deep space exploration [27,28]. According to the condition of momentum matching, the tuning relationship between the driving frequency () of AOTF and the

*Materials* **2022**, *15*, x FOR PEER REVIEW 5 of 13

$$f\_a(\lambda) = V\_a[n\_i^2 + n\_d^2 - 2n\_l n\_d \cos(\theta\_l - \theta\_d)]^{1/2} / \lambda,\tag{1}$$

where *λ* is the incident light wavelength, *Va* is the ultrasonic propagation velocity in the crystal, *fa*(*λ*) is the corresponding ultrasonic frequency, *θi* is the incident angle for a device, *θd* is the output beam angle, *ni* is the refractive index for the incident light, and *nd* is the refractive index for the diffracted light. The relationship curve between the diffraction wavelength of the crystal and the driving frequency is shown in Figure 2b. The curve data were obtained by measurement. A HORIBA iHR 320 spectrometer was used to generate monochromatic light. The wavelength of the monochromatic light was first set, and then the scanning was performed by changing the driving frequency in fine steps. The frequency at which the most intense diffracted light was measured is the driving frequency corresponding to that wavelength. This procedure was repeated for the entire operating spectral range to obtain all data. It is possible to realize both sequences in the time-fine scanning of wide spectrum bands by wavelength point by point and the high-precision diffraction center wavelength picking of specific wavelength bands by coding control of RF driving frequency when using a good correspondence between RF driving frequency and diffraction wavelength, with simple wavelength picking and good repeatability. vice, ௗ is the output beam angle, is the refractive index for the incident light, and ௗ is the refractive index for the diffracted light. The relationship curve between the diffraction wavelength of the crystal and the driving frequency is shown in Figure 2b. The curve data were obtained by measurement. A HORIBA iHR 320 spectrometer was used to generate monochromatic light. The wavelength of the monochromatic light was first set, and then the scanning was performed by changing the driving frequency in fine steps. The frequency at which the most intense diffracted light was measured is the driving frequency corresponding to that wavelength. This procedure was repeated for the entire operating spectral range to obtain all data. It is possible to realize both sequences in the timefine scanning of wide spectrum bands by wavelength point by point and the high-precision diffraction center wavelength picking of specific wavelength bands by coding control of RF driving frequency when using a good correspondence between RF driving frequency and diffraction wavelength, with simple wavelength picking and good repeatability.

**Figure 2.** AOTF spectroscopy schematic. (**a**) AOTF spectroscopy and wave vector diagram; (**b**) curve of AOTF driving frequency with diffraction wavelength. **Figure 2.** AOTF spectroscopy schematic. (**a**) AOTF spectroscopy and wave vector diagram; (**b**) curve of AOTF driving frequency with diffraction wavelength.

According to the characteristics of the AOTF spectrometer, the optical path structure of this system is shown in Figure 3a by combining an AutoML autonomous feature wavelength filtering algorithm and AOTF spectrometer-specific band sampling. The use of a dual optical path and a dual detector design, the use of an adjustable beam splitter to introduce the reference optical path, comprehensive data processing, and compensation of light intensity fluctuations caused by the instability of the light source and errors caused by environmental interference (in order to improve the accuracy of the instrument) allowed us to achieve a wide spectral scan range of 900–2400 nm. The actual spectra are sampled by a short-wave infrared AOTF, using both high-frequency and low-frequency drivers to achieve a wide spectral sampling range of 900–2400 nm, with a spectral resolution of 3.75–9.6 nm. The main performance parameters of the AOTF are shown in Table 1.

**Figure 3.** AOTF detection system and testing process. (**a**) schematic diagram of AOTF dual optical path and dual detector NIR spectrometer; P1, P2, polarizer; BS, adjustable beam splitter; D1, D2, InGaAs photodetector; (**b**) flow chart of AOTF detection based on AutoML. **Figure 3.** AOTF detection system and testing process. (**a**) schematic diagram of AOTF dual optical path and dual detector NIR spectrometer; P1, P2, polarizer; BS, adjustable beam splitter; D1, D2, InGaAs photodetector; (**b**) flow chart of AOTF detection based on AutoML.

spectrum. By transferring the obtained data to the model established by the combination of feature wavelengths, the detection and analysis of the characteristic components or contents of the sample can be realized. The flow chart of the system is shown in Figure 3b. Firstly, the full spectrum data of the substance is obtained through the AOTF spectrometer, and the filter modeling of the feature wavelength of the target is completed using the AutoML algorithm to extract the feature

and baijiu) were selected, and validation experiments were performed according to the detection system and testing process shown in Figure 3. The experimental results of the

**4. Application Results**

wavelength of the sample. After that, the RF drive function of the feature wavelengths is generated using the selected feature wavelength combinations combined with the drive frequency function of AOTF, and the drive function is stored in the RF controller. When the target is measured again, the system can take advantage of the flexible adjustment of the AOTF central wavelength at the subnanometer level to accurately locate the desired band and achieve the adjustable, high-precision, and fast acquisition of the feature spectrum. By transferring the obtained data to the model established by the combination of feature wavelengths, the detection and analysis of the characteristic components or contents of the sample can be realized.

### **4. Application Results**

To verify the effectiveness of the system, two different types of liquid samples (milk and baijiu) were selected, and validation experiments were performed according to the detection system and testing process shown in Figure 3. The experimental results of the two measurement methods for fat content (%) in milk samples and alcohol by volume (ABV, %) in baijiu samples are shown.

### *4.1. Sample Preparation*

For the milk samples, we chose nine different brands of liquid pure milk, and five samples of each milk brand were prepared for a total of 45 samples. The nine milk samples had different fat contents, as shown in Table 2, to establish the relationship between spectral data and fat content in the milk model using the nominal value of fat content.

**Table 2.** The fat contents of the milk samples.

For the liquor samples, we selected two brands and five types of liquor divided into 209 samples marked and sealed for storage. The determination of alcoholic content was entrusted to the Shenzhen Institute of Measurement and Quality Inspection using the national standard (GB 5009.225-2016) alcoholometer method, and the measurement result was the volume fraction of alcohol, i.e., alcoholic content (%vol). The range of alcoholic content of the samples was 42.0–56.2, with an accuracy error of 0.1, as shown in Table 3.

1 ABV = alcohol by volume.

The NIR transmittance spectra of all samples were collected using this system with a spectral sampling range of 900–2400 nm and a sampling interval of 5 nm, and the NIR spectral data for each sample was 301 wavelength points. For the milk samples, a 0.5 mm optical path length quartz cuvette was used to hold the samples, and the transmittance spectra of 45 samples were obtained. For the baijiu samples, the transmittance spectra of 209 samples were obtained using a quartz cuvette with a 2 mm optical path length. The NIR transmittance spectra of all samples were collected using this system with a spectral sampling range of 900–2400 nm and a sampling interval of 5 nm, and the NIR spectral data for each sample was 301 wavelength points. For the milk samples, a 0.5 mm optical path length quartz cuvette was used to hold the samples, and the transmittance spectra of 45 samples were obtained. For the baijiu samples, the transmittance spectra of 209 samples were obtained using a quartz cuvette with a 2 mm optical path length.

The absorbance spectra of all samples were obtained by taking log(1/T) of the transmittance spectra of all samples. Savitzky-Golay (SG) [31,32] smoothing was used to reduce the random noise. The absorbance spectra were processed using Multiplicative Scatter Correction (MSC) [33] to remove the influence of non-concentration factors such as baseline shift and granularity on the spectra. The spectral data with low signal-to-noise ratio at the head and tail were removed, and the final data were in the wavelength range of 1250–2300 nm and 5 nm sampling interval, with a total of 211 data points. The absorbance spectral data obtained by processing are shown in Figure 4. The absorbance spectra of all samples were obtained by taking log(1/T) of the transmittance spectra of all samples. Savitzky-Golay (SG) [31,32] smoothing was used to reduce the random noise. The absorbance spectra were processed using Multiplicative Scatter Correction (MSC) [33] to remove the influence of non-concentration factors such as baseline shift and granularity on the spectra. The spectral data with low signal-to-noise ratio at the head and tail were removed, and the final data were in the wavelength range of 1250–2300 nm and 5 nm sampling interval, with a total of 211 data points. The absorbance spectral data obtained by processing are shown in Figure 4. *Materials* **2022**, *15*, x FOR PEER REVIEW 9 of 13 1250–2300 nm and 5 nm sampling interval, with a total of 211 data points. The absorbance spectral data obtained by processing are shown in Figure 4.

**Figure 4.** Absorbance spectra of different samples. (**a**) milk samples; (**b**) alcohol samples. **Figure 4.** Absorbance spectra of different samples. (**a**) milk samples; (**b**) alcohol samples. **Figure 4.** Absorbance spectra of different samples. (**a**) milk samples; (**b**) alcohol samples.

#### *4.2. Modeling and Feature Wavelength Selection 4.2. Modeling and Feature Wavelength Selection 4.2. Modeling and Feature Wavelength Selection*

The spectra of milk and alcohol samples obtained by AOTF spectrometer were converted into spectral absorbance values as input features, and fat content and alcohol content were used as model target values. The data sets were randomly divided, with 70% (31 samples) of milk used as training data and the remaining 30% (14 samples) used as test data; 70% (146 samples) of baijiu was used as training data and the remaining 30% (63 samples) was used as test data, as shown in Tables 4 and 5. Only the training data was provided to AutoML framework at training time, while the test data was only provided at prediction time. The spectra of milk and alcohol samples obtained by AOTF spectrometer were converted into spectral absorbance values as input features, and fat content and alcohol content were used as model target values. The data sets were randomly divided, with 70% (31 samples) of milk used as training data and the remaining 30% (14 samples) used as test data; 70% (146 samples) of baijiu was used as training data and the remaining 30% (63 samples) was used as test data, as shown in Tables 4 and 5. Only the training data was provided to AutoML framework at training time, while the test data was only provided at prediction time. The spectra of milk and alcohol samples obtained by AOTF spectrometer were converted into spectral absorbance values as input features, and fat content and alcohol content were used as model target values. The data sets were randomly divided, with 70% (31 samples) of milk used as training data and the remaining 30% (14 samples) used as test data; 70% (146 samples) of baijiu was used as training data and the remaining 30% (63 samples) was used as test data, as shown in Tables 4 and 5. Only the training data was provided to AutoML framework at training time, while the test data was only provided at prediction time.

**Table 4.** Statistics on fat of milk samples. **Table 4.** Statistics on fat of milk samples. **Table 4.** Statistics on fat of milk samples.

**Table 5.** Statistics on ABV of baijiu samples.

**Data Sets Number of Samples Min Max Mean STD**  Total samples 209 42 56.2 52.66 3.60 Calibration set 146 42 56.2 52.49 3.80 Prediction set 63 42 56.2 53.05 3.07

The fit application programming interface (API) of AutoGluon–Tabular was used to train these models. Within the call to fit, AutoGluon automatically preprocesses the raw data, identifies what type of prediction problem this is (binary, multi-class classification, or regression), partitions the data into various folds for model-training vs. validation, individually fits various models, and finally creates an optimized model ensemble that out-

performs any of the individual trained models.

**Table 5.** Statistics on ABV of baijiu samples.

The fit application programming interface (API) of AutoGluon–Tabular was used to train these models. Within the call to fit, AutoGluon automatically preprocesses the raw data, identifies what type of prediction problem this is (binary, multi-class classification, or regression), partitions the data into various folds for model-training vs. validation, individually fits various models, and finally creates an optimized model ensemble that outperforms any of the individual trained models. *Materials* **2022**, *15*, x FOR PEER REVIEW 10 of 13 The performance of these models was adjudged based on root mean square errors

The performance of these models was adjudged based on root mean square errors (RMSE) of validation (RMSEV) and prediction (RMSEP). RMSEV is obtained from Auto-Gluon which is named as "score val," and RMSEP is calculated based on the prediction results of the test set. This is as shown in Figure 5a,c. (RMSE) of validation (RMSEV) and prediction (RMSEP). RMSEV is obtained from AutoGluon which is named as "score val," and RMSEP is calculated based on the prediction results of the test set. This is as shown in Figure 5a,c.

**Figure 5.** Results of the "score val" (RMSEV) and permutation importance of the two samples. (**a**) "score val" (RMSEV) of milk samples. We also calculated RMSEP as "score test." (**b**) degrees of permutation importance of the milk samples; (**c**) "score val" (RMSEV) of alcohol samples. We also calculated RMSEP as "score test." (**d**) permutation importance of alcohol samples. **Figure 5.** Results of the "score val" (RMSEV) and permutation importance of the two samples. (**a**) "score val" (RMSEV) of milk samples. We also calculated RMSEP as "score test". (**b**) degrees of permutation importance of the milk samples; (**c**) "score val" (RMSEV) of alcohol samples. We also calculated RMSEP as "score test". (**d**) permutation importance of alcohol samples.

The importance the ranking [34] of data variables was performed based on permutation importance for milk and baijiu samples, as shown in Figure 5b,d. Higher importance indicates that the feature variables have more influence on the model performance. The importance the ranking [34] of data variables was performed based on permutation importance for milk and baijiu samples, as shown in Figure 5b,d. Higher importance indicates that the feature variables have more influence on the model performance.

Based on the accuracy criteria, we calculated the difference between the RMSEV and the minimum RMSEV for all models, where the model with a difference not higher than 20% and with the least number of variables was selected. According to our precision criterion, eight characteristic wavelengths were selected for milk, and eight characteristic Based on the accuracy criteria, we calculated the difference between the RMSEV and the minimum RMSEV for all models, where the model with a difference not higher than 20% and with the least number of variables was selected. According to our precision criterion, eight characteristic wavelengths were selected for milk, and eight characteristic

wavelengths were selected for baijiu. The actual results of the selected wavelengths are

The AOTF-NIR spectroscopy system combining the model and the corresponding sampling strategy was deployed to a real application scenario to perform experiments on

**Samples Selected Wavelengths (nm)**

Milk 1280, 1265, 2170, 1290, 1305, 1260, 1310, 2040 Baijiu 1250, 1665, 1710, 1405, 1430, 1425, 1485, 1415

shown in Table 6.

*4.3. Experimental Results of the System*

wavelengths were selected for baijiu. The actual results of the selected wavelengths are shown in Table 6.

**Table 6.** Results of feature wavelength selection.

### *4.3. Experimental Results of the System*

The AOTF-NIR spectroscopy system combining the model and the corresponding sampling strategy was deployed to a real application scenario to perform experiments on the measurement of samples with unknown target value content. Using this instrumentation system with models, we were able to perform 211-point and 8-point measurements on the samples, respectively. The data obtained from the measurements could then be used to invoke the corresponding model for component content prediction. The analytical performance of the system was evaluated when the full-spectrum model and the characteristic wavelength model were applied separately. The performance of the prediction could be evaluated by calculating new RMSEPs.

The RMSEV shown in Table 7 is the "score val" obtained during the training phase. We also performed 211-point and 8-point spectral acquisitions for the above test set of samples, respectively. After data acquisition, the average spectrum calculated in the model building phase is used as the true spectrum for MSC processing. No smoothing operation is performed at this stage. The data are then input to the model to derive predicted values, which are displayed on the software interface. The RMSEP metric is obtained by calculating the error between the predicted and true values obtained from the inference of the input model after the spectral measurement; the inference time is automatically recorded by the program in the host computer; the sampling time is the estimated time taken by the instrument to acquire data at the desired wavelength point.

**Table 7.** The performance of the system for two datasets.

As shown in Table 7 above, the use of the characteristic wavelength model and the corresponding sampling strategy achieves a slight improvement in prediction performance, while the sampling time consumption is ~22-fold smaller. The prediction performance improvement indicates that there are many irrelevant wavelengths throughout the spectrum that do not contribute to effective prediction of the target values and negatively affect the performance of the prediction model.

Combined with the above experimental results, the automatic machine learning framework can accurately and robustly filter the wavelength combinations with the best prediction performance for the target values; combined with the eight-wavelength models and the corresponding new sampling mechanism that only collects data at the eight wavelength points, higher prediction performance and considerable improvement in detection efficiency can be obtained compared with the full-spectrum model and the full-spectrum sampling mechanism. This advantage is particularly critical for the application of AOTF-NIR analysis systems for online inspection. The combination of AOTF's flexibility and automatic machine learning provides a good reference case for the application of intelligent optical inspection in automated production by applying an automatic machine learning

framework to easily build a well-performing model for different application objectives and adaptively change the sampling strategy.

### **5. Conclusions**

In this study, an NIR detection system based on AOTF with AutoML feature wavelength screening was proposed and validated. Combining various chemometric algorithms, spectral detection based on the characteristic wavelengths instead of the full spectrum achieves the improvement of detection efficiency. Meanwhile, the model is simplified, interpretability is improved, and storage and computational resource usage are reduced. Taking milk and alcohol as examples, the validation experiments achieve a reduction in the number of sampling wavelengths while achieving accurate and nondestructive rapid determination of fat and alcohol content.

**Author Contributions:** Conceptualization, Z.J. and Z.H.; methodology, Z.J.; software, Z.J.; validation, Y.G., J.L. and Z.H.; formal analysis, Z.J.; investigation, Z.J. and Y.G.; resources, Y.T. and R.X.; data curation, Z.J. and B.W.; writing—original draft preparation, Z.J. and Y.G.; writing—review and editing, Z.H. and J.L.; visualization, Y.G., Y.T. and B.W.; supervision, J.W. and Z.H.; project administration, Z.H. and J.W.; funding acquisition, Z.H. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the National Natural Science Foundation of China (Grant No. 61605231), the Program of Shanghai Academic Research Leader (Grant No. 19XD1424100), and the National Science Foundation for Distinguished Young Scholars (Grant No. 62125505).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **References**

## *Article* **Analysis on the Influence of Incident Light Angle on the Spatial Aberrations of Acousto-Optical Tunable Filter Imaging**

**Kai Yu 1,\* and Huijie Zhao 1,2,3,\***

**Abstract:** Acousto-optical tunable filter (AOTF) does not conform to the pinhole model due to the acousto-optic interaction. A calculation method of AOTF aberrations under the condition of incident light with a large arbitrary angle is proposed to solve the problem of coordinate mapping between object space and image space of the AOTF system without refractive index approximation. This approach can provide accurate pointing information of the interested targets for the tracking and searching system based on AOTF. In addition, the effect of cut angle values of the paratellurite crystal on aberrations was analyzed to optimize the design of AOTF cutting according to different application requirements. Finally, distribution characteristics and quantitative calculation results of AOTF aberrations were verified by experiments with different targets, respectively. The experimental results are in good agreement with the simulations.

**Keywords:** AOTF spectrometer; aberrations; coordinate mapping

### **1. Introduction**

Photoelectric tracking and searching system is the most important way of passive detection at present. With the development of infrared technology, all-day and all-weather detection can be realized. At the same time, more advanced jamming technologies have emerged accordingly. These jamming techniques can make the target and the decoy highly similar in geometry, kinematics, and radiance within a wide spectral range. The consequence is the failure of traditional detection and identification methods. Therefore, it is necessary to introduce the characteristics of the spectral dimension. For example, the apparent temperature, the gradient of temperature, or the emissivity of the target can be obtained by multispectral detection [1–3]. The features of these kinds are very stable and different from the bait, so they can be used to distinguish the target from the bait. Thus, spectral detection plays an important role in anti-jamming imaging systems now and in the future.

Spectral imaging detection includes optical system design, mechanical design, photoelectric conversion technology, target detection, recognition algorithm, and many other fields. However, to adapt to the rapid changes in the scene, there is an urgent need for an imaging method with a flexible spectrum. The acousto-optic tunable filter (AOTF) is a kind of narrowband filter based on birefringent crystal material. The wavelength, direction, and intensity of diffracted light can be modified by modulating the sound field with a highfrequency sinusoidal signal. Moreover, it has many advantages, such as being electronically controlled with high speed, wide spectral range, and all-solid-state. As a result, AOTF staring imaging spectrometer is commonly used in spectral analysis [4], laser shaping [5,6],

**Citation:** Yu, K.; Zhao, H. Analysis on the Influence of Incident Light Angle on the Spatial Aberrations of Acousto-Optical Tunable Filter Imaging. *Materials* **2022**, *15*, 4464. https://doi.org/10.3390/ ma15134464

Academic Editors: Alexander S. Machikhin and Vitold Pozhar

Received: 10 May 2022 Accepted: 22 June 2022 Published: 24 June 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

spectral microscopic [7,8], and many other fields [9–13]. Similarly, the advantages of AOTF make it very suitable for tracking and searching system.

As we know, AOTF is mostly applied to the spectral imaging detection of static targets due to the different application requirements before. In addition, the field angle of view of the AOTF system is generally designed to be small (approximately equal to ±2 ◦ ) because of the constraint of the separation angle. Additionally, it is considered that the imaging aberration caused by AOTF is very small and can be ignored. Therefore, within a small angle range, many scholars only considered the impact of incident light angle on device performance. Zhang Zhonghua analyzed the influence of the incident light angle on the spectral bandwidth, diffraction efficiency, and the separation angle inside and outside the crystal; however, their analysis is limited to the interaction plane of AOTF [14–17]. Yushkov gave the analytical expressions of the central wavelength and the tuning frequency under oblique incidence [18], but for some special applications, the precise geometric information of the target is required. At this time, the aberration caused by AOTF can not be ignored even under the condition of a small field angle of view. Pozhar and Machikhin focused on the quantitative description of AOTF spatial aberrations. In their previous works, an approximation of small birefringence was made, and they only analyzed the aberrations in the direction of the optical axis and the direction perpendicular to the optical axis. Then they extended the analysis of AOTF spatial aberrations from two-dimensional space to three-dimensional space [19–23] and proposed a method of correcting spectral and spatial aberrations by AOTF pre-calibration for applications in which geometric characteristics shall be attained at the same time, such as microimaging [24]. Later, they used the developed DLL to combine the ray-tracing model with ZEMAX and applied it to the optical design of the stereoscopic spectral imaging system [25–27]. In addition, they used a single AOTF to obtain the three-dimensional information while obtaining the spectral information of the target [28]. Nonetheless, no quantitative experimental results of AOTF aberrations were given. In a word, the influence of incident light angle on AOTF can be summarized into several stages. From the analysis within the two-dimensional plane (acousto-optic interaction plane) to three-dimensional space, from the approximate analytical solution neglecting the change in refractive index to the accurate iterative solution. Until now, the acousto-optic interaction model based on momentum matching conditions has been relatively perfect.

However, if AOTF is used for tracking and searching systems, different issues need to be considered. The number of pixels occupied by the detected target on the image plane is relatively few, and the angle of light entering AOTF may no longer be constrained by the separation angle because there is no overlap between the zero-order light imaging and the diffracted light imaging. In other words, the zero-order light and the first-order diffracted light are allowed to be imaged at the same time under the condition that the imaging is not saturated. Moreover, zero-order light imaging can be eliminated through image processing because it always exists stably, and the intensity of diffracted light fluctuates with the change in driving frequency. Therefore, for AOTF used for tracking and searching system, there are two main problems to be solved: (1) How can the problem of central wavelength drift, that is, to ensure the accuracy of the spectral data of the target, be solved? (2) How can the accurate pointing information (azimuth and pitch angle) of the target be obtained? As photoelectric detection equipment, it is one of the most basic functions to give the target pointing information accurately. Otherwise, additional optoelectronic equipment is required for assistance.

As the field of view increases, the drift of the central wavelength of AOTF becomes more serious. Some scholars corrected the drift through the computational technique to achieve the purpose of expanding the field of view [29,30]. It means that problem (1) mentioned above can be solved for the AOTF system with a large field of view. On the other hand, the expansion of the field of view also means that the aberration of AOTF is more severe. The AOTF cannot use the pinhole model in ray tracing. We first conducted an in-depth study of AOTF aberration to solve the problem (2). If the calculation and analysis

of aberration of AOTF can ensure sufficient accuracy, AOTF can be extended to the tracking and searching system of moving targets. Therefore, it is necessary to analyze the influence of different incident angles on imaging aberrations in a large range of angles. As far as we know, the study of this kind is not complete enough and lacks experimental verification. In this article, we analyzed the influence of the angle of incident light on AOTF spatial aberrations in three-dimensional space by the ray-tracing method and show quantitative results of the experiments. Moreover, the influence of cut angle values of the paratellurite crystal on aberrations was simulated.

The main contribution of this paper was to deeply analyze the aberration of AOTF with a large field of view so that it can give the target pointing information accurately when AOTF is used for the detection of moving targets and put forward a new application direction of AOTF. For the AOTF imaging spectrometer with a small field of view, the aberration is very small. AOTF can be considered to conform to the pinhole model, and the method in this paper is not required for ray tracing. In addition, the algorithm needs to be optimized in the real-time system due to the use of the iterative algorithm.

### **2. Method**

At present, the AOTF imaging spectrometer can be classified into two types of structures according to the light entering the AOTF. One is with parallel entering light, and the other is with the convergent entering light. Both have their advantages; in theory, the lateral chromatic aberration of the confocal structure is less, while the axial chromatic aberration of the collimated optical path is less. Considering that lateral chromatic aberration can be corrected by the wedge angle of the AOTF rear surface (the rear wedge angle of AOTF we used is 4.83◦ ), so the collimated light structure is used for the simulations and the experimental design.

The analysis of aberration is essential to analyze the relationship between the incident light angle of AOTF and the exit direction of diffracted light. As a spectrometer, the raytracing method can no longer be limited to the crystal but needs to be converted to the world coordinate system for analysis. The advantage of this method is that it has better adaptability. It can still analyze and guide the design of the optical system when changing the cut angle values of AOTF or the angle of Bragg's diffraction.

### *2.1. Ray Tracing of Arbitrary Light in Three-Dimensional Space*

In the design and performance analysis of AOTF devices, monochromatic light is usually employed as the light source. It should be pointed out that in many analyses of AOTF device modeling, the incident light is usually vertical, and a certain diffracted light is selected by changing the driving frequency that matches the wavelength of the incident light. Momentum mismatch occurs when the incident light is not perpendicular to the AOTF front surface, or the wavelength of the incident light does not match the driving frequency. However, in actual applications of spectral imaging detection, the incident light is mostly polychromatic, and the diffracted light also essentially satisfies the momentum matching condition in another wave vector diagram in which the wavelength of incident light drifts.

Take e-polarized incident light and o-polarized diffracted light as an example. When the incident light is white, the driving frequency is fixed, and the angle of the incident light is changed, then the original incident wavelength (*λ*1) no longer satisfies the momentum matching condition, as the blue triangle shown in Figure 1a. Additionally, there is another wavelength (*λ*2) of incident light and the sound vector to meet the new momentum matching condition, such as the red triangle shown in Figure 1b, the wavelength and direction of the diffracted light can be calculated accordingly. In Figure 1a,b, the length and direction of *LAB* and *LDE* do not change, which means the sound vector remains unchanged. In other words, when the incident light is polychromatic, the diffracted light of a certain wavelength can always be found to meet the momentum matching condition. The momentum mismatch is only an artificially defined quantity to calculate the diffraction

efficiency or spectral bandwidth. If the intensity of the sound field is constant, the amount of momentum mismatch is only related to the direction and wavelength of the diffracted light. Momentum mismatch is not considered in the analysis of AOTF imaging aberrations. Therefore we need to find the wavelength and exit direction of diffracted light that meet the conditions.

**Figure 1.** Wave diagram of anisotropic AO diffraction. Momentum mismatching (**a**) and momentum matching (**b**) under oblique incidence.

In the world coordinate system, assuming the direction cosine of the incident light is (*L*0, *M*0, *N*0), in the crystal coordinate system, the direction cosine of the incident light is converted to (*L*1, *M*1 cos *θ* ∗ *i* − *N*1 sin *θ* ∗ *i* , *M*1 sin *θ* ∗ *i* + *N*1 cos *θ* ∗ *i* ) and *θ* ∗ *i* represents the angle between the front surface of AOTF and the 110 axes. The first surface [31] adopts the refraction theorem and the equation for calculating the refractive index of extraordinary light, as shown in Equation (1).

$$\begin{cases} n\_{a\dot{n}r} \sin \theta\_{\dot{n}t} = n\_{\varepsilon}(\theta) \sin(\theta - \theta\_{\dot{t}}\*), \\\ n\_{\varepsilon}^{2}(\theta) = \frac{n\_{\varepsilon}^{2} n\_{\varepsilon}^{2}}{n\_{\sigma}^{2} \sin^{2} \theta + n\_{\varepsilon}^{2} \cos^{2} \theta'} \end{cases} \tag{1}$$

where *nair* is the refractive index of the air medium, *θin* is the incident angle in the air medium, *n*e(*θ*) is the refractive index of e light corresponding to the refracted light in the crystal, *θ* and *θi* ∗ is defined as shown in the partially enlarged view of Figure 2, *θ* is the angle between the refracted light in the crystal and the crystal axis in the polar coordinate system, *θ* − *θi* ∗ is the angle between the refracted light in the crystal and the normal of the interface, and *n*o and *n*e represent the refractive index of o light and e light, respectively. Then, the phase velocity direction of the incident light in the crystal can be obtained. Point A (*xa*, *ya*, *za*) is the intersection point of the incident light, and the ellipsoid of the light wave vector shown in Figure 1. |**Ka**| can be obtained with driving frequency corresponding to the wavelength of the incident light and the direction of the sound vector **Ka** under the condition of momentum matching when the incident light is perpendicular to the front surface of AOTF. Since the driving frequency has not been tuned and **Ka** remains the same, even if the incident light angle has been changed, the coordinates of point B (*xa* − *k*ax, *ya* − *k*ay, *za* − *k*az) can be calculated from the coordinates of point A. The coordinate of point C (*xc*, *yc*, *zc*) is the intersection of the extension line of OB and o-wave vector sphere, and the length of the BC line, *LBC*, can be obtained.

**Figure 2.** Ray tracing of AOTF in three-dimensional space.

It can be found that the value of *LBC* is related to the incident wavelength. It can be considered that the incident light vector, diffracted light vector, and sound vector satisfy the momentum matching condition when *LBC* is close to 0 with the change in incident light wavelength. Therefore, the corresponding central wavelength at an arbitrary incident angle is expressed by Equation (2). The "argmin" in Equation (2) means that the center wavelength corresponds to the smallest *LBC* when the wavelength changes within a certain range.

$$
\lambda\_{\text{center}} = \underset{\lambda\_{\text{min}} \le \lambda \le \lambda\_{\text{max}}}{\text{argmin}} \left( L\_{\text{BC}}(\lambda) \right). \tag{2}
$$

The intersection point of the incident light and the E-light ellipsoid can be calculated, i.e., D (*xd* , *yd* , *zd* ), after we obtain *λ*center. It can be understood that in the process of wavelength change under the condition of a certain angle of the incident light, point A moves to point D. Thus far, the intersection point E (*xe* , *ye* , *ze*) of the diffracted light and the o-light sphere, and the exit direction of diffracted light in the crystal coordinate system can be obtained simultaneously. After the diffracted light passes through the AOTF exit surface and then changes from the crystal coordinate system to the world coordinate

system, the directional cosine of the diffracted light( −→ **GH** in Figure 2) can be obtained. The iterative method must be used to obtain more accurate ray tracing because the refractive index of tellurium dioxide changes with the wavelength, and the analytical solution is no longer applicable.

Through this method, the exit direction of diffracted light corresponding to incident polychromatic light at any angle can be calculated, and further its imaging position, then the corresponding relationship between AOTF spatial aberrations and the angle of incident light can be calculated. This method is also applicable to the cases where the incident light is o-polarized, the diffracted light is e-polarized, and cases about the different cut angle values of the paratellurite crystal.

### *2.2. Aberration Simulation of AOTF at Different Incident Angles*

Through the method above, it can be known that another wavelength of the incident satisfies the momentum matching condition with the original sound vector when the white light is incident obliquely. Select the incident light wavelength *λ* = 632 nm, driving frequency *f* = 73.25 MHz, and calculate the matching center wavelength under the condition of oblique incidence within the range of ±4.0◦ polar angle and azimuth angle. The ultrasonic cut angle of AOTF is 6.5◦ . The rear wedge angle is 4.83◦ . These parameter settings are consistent with the actual design parameters of the AOTF cutting.

As shown in Figure 3, the central wavelength drifts with the increase in the incident light angle. In the azimuth plane, the drift of the central wavelength is symmetrically distributed. In the polar angle plane, the drift of the central wavelength is asymmetric, and the drift of the central wavelength near the AOTF crystal axis is greater than that away

from the crystal axis when the incident light is inclined by the same angle from the vertical incidence to both sides.

The direction of diffracted light can be further calculated and compared with the image plane position of diffracted light under the condition of pinhole imaging after the central wavelength is obtained, and the aberrations of AOTF can be obtained accordingly. The focal length of the rear objective lens is 50 mm, and the pixel size is 4.5 µm.

The *L*(*AC*) in Figure 4c corresponds to the ordinate in Figure 4b, and the *L*(*BC*) in Figure 4c corresponds to the ordinate in Figure 4a. AOTF aberrations are different in polar direction and azimuth direction significantly. In the azimuth direction, the aberration is 180◦ rotational symmetry, and the aberration is less than two pixels within a ±4 ◦ incidence angle. In the polar direction, the aberration follows the symmetrical distribution of the azimuth plane, and the aberration caused by oblique incidence is considerable. The larger the tilt angle is, the greater the aberration is, up to 30 pixels within ±4 ◦ incidence angle. It can be discovered from the comparison between Figures 3 and 4 that the greater the center wavelength drifts, the more serious the AOTF aberration is.

**Figure 4.** The difference between the imaging position of diffracted light at different incident angles of AOTF and pinhole imaging: (**a**) the difference in azimuth direction and (**b**) in polar direction, and (**c**) the image plane.

The accuracy of the simulation mainly considers the difference between the design value and the theoretical value of the ultrasonic cut angle.

As shown in Figure 5, the error of edge field aberration does not exceed two pixels when the error of ultrasonic cut angle is ±0.2◦ . The aberration error in the acousto-optic action plane is mainly considered here because the error in the azimuth plane affected by the ultrasonic cut angle is very small, so it is not shown. In the simulation, the principle of parallel tangent is no longer satisfied due to the error of the ultrasonic cut angle.

**Figure 5.** The accuracy of the simulation (**a**) α = 6.3◦ (**b**) α = 6.7◦ .

In addition, cut angle values of the paratellurite crystal, i.e., *α*, affect the aberrations of AOTF. The aberrations in the polar direction of AOTF with different *α* are simulated, as shown in Figure 6. The results show that the aberrations of AOTF decrease with the increase in *α* within the range of (0 ◦ , 18.9◦ ).

**Figure 6.** Aberrations of different ultrasonic cut angles of AOTF.

### **3. Results**

*3.1. Experiment of the Aberration Distribution Characteristics with Integrating Sphere as the Target*

We chose a small area target, the integrating sphere, as the light source to analyze the distribution characteristics of AOTF distortion more intuitively. The integrating sphere had a diameter of 12 inches. Such target selection can make the diffracted light imaging not overlap with the zero-order light imaging so that the influence of incident angle on aberrations of AOTF can be analyzed in a wider range.

The experimental imaging setup of different incident angles of the integrating sphere is shown in Figure 7. ϕ means the half-angle aperture in the azimuth direction (plane perpendicular to the acousto-optic interaction plane). The distance between the integrating sphere and AOTF was 2.8 m, the outlet diameter of the integrating sphere was 4 inches, the aperture of AOTF was 20 mm, the ultrasonic cut angle was 6.5◦ , the rear wedge angle was 4.83◦ , the spectral range was 400–1000 nm, the change in polarization was *e* → *o* , the focal length of the rear objective lens was 50 mm, and the pixel size was 4.5 µm. AOTF was fixed on the precision turntable, the driving frequency of the AOTF was adjusted by the PC, and the camera could obtain images of diffracted light with different wavelengths. Images of diffracted light at different incident angles can be obtained through the precision turntable.

**Figure 7.** Integrating experimental sphere setup of AOTF based on different incident light angles: 1—integrating sphere; 2—polar plane; 3—azimuth plane; 4—AOTF; 5—rotatable stage; 6 and 7—polarizers; 8—imaging lens; 9—camera.

The distance between the target and AOTF is far in the experimental setting so that the pre-collimating mirror group can be omitted. It is only necessary to measure the aberration of the imaging mirror group and add the aberration parameters to the simulation model.

In Figure 8, the diameter of the large dots is 15 mm, the smaller is 8 mm, and the distance between the calibration points is 40 mm. The coordinates of the calibration point were obtained by the method of geometric center extraction, and the unsatisfactory calibration points were eliminated. Finally, the radial distortion coefficient of the imaging lens group could be obtained by Zhang Zhengyou's calibration method, and the result is shown below as Equation (3).

$$
\begin{pmatrix} k\_1 \\ k\_2 \\ k\_3 \end{pmatrix} = \begin{pmatrix} 0.37667 \\ -3.40922 \\ -0.00195 \end{pmatrix} . \tag{3}
$$

$$
\begin{array}{ccccccccc}
\bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet \\
\bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet \\
\bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet \\
\bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet \\
\end{array}
$$

**Figure 8.** Distortion test of imaging lens group.

As shown in Figure 9, the angle of incident light from left to right gradually approaches the AOTF crystal axis, and the image of diffracted light of the integrating sphere has changed. In general, the aberrations present an asymmetric distribution in the polar direction. By taking the image of *polar angle* = 0 ◦ (vertical incidence) as the benchmark, the imaging changes are small when the incident polar angle gradually increases in the direction away from the crystal axis; thus, the spatial aberrations of the AOTF in this direction are small. The imaging of the integrating sphere changed significantly when the incident polar angle increased in the direction close to the crystal axis; thus, the spatial aberrations of the AOTF in this direction are tremendous.

**Figure 9.** Simulated (upper row) and experimental (lower row) images with the AOTF tuned to *λ* = 632.8 nm (*f* = 72.84 MHz) for different angles of incident light in the polar plane.

The influence of incident azimuth angles on the aberrations of AOTF can be obtained with the same experimental method by rotating the AOTF by 90◦ around the optical axis. From Figure 10, the aberrations caused by the change in the azimuth incidence angle still exist mainly in the polar direction and are distributed symmetrically up and down based on the image of *azimuth angle* = 0 ◦ .

**Figure 10.** Simulated (upper row) and experimental (lower row) images with the AOTF tuned to *λ* = 632.8 nm (*f* = 72.84 MHz) for different angles of incident light in the azimuth plane. The solid red line in the upper row indicates the ideal positions of the integrating sphere with the assumption of pinhole imaging.

Therefore, we can conclude that the aberrations of AOTF mainly exist in the polar direction, the amount of aberrations in the azimuth direction is small, the aberrations are affected by the incident angle in the polar direction greatly, and the closer the incident angle to the AOTF crystal axis is, the greater the aberrations are.

### *3.2. Quantitative Experiment with the Square Grid as the Target*

We chose to quantitatively analyze the influence of incident angle on AOTF aberrations in the polar direction. The experimental target is vertically placed at a distance of 5 m from AOTF, and other experimental conditions are consistent with those in Section 3.1. The imaging of the target center point, (*x*55, *y*55) in Figure 11a, is located at the center of the image plane. The size of the target is 350 mm × 350 mm, the length of the small square grid is 9 mm, and the distance between adjacent squares is 40 mm.

**Figure 11.** The experiment of the square target: (**a**) actual image of diffracted light at a polar angle of −5.3◦ ; (**b**) the partially enlarged view of the square target.

In Figure 11b, the blue star points represent the ideal points of pinhole imaging, the solid green dots represent the simulated imaging points, and the red boxes indicate the actual imaging areas of the square grid. The coordinate of the actual image point is obtained by extracting the geometric center of the sampling points. We found that the simulated imaging points were located near the actual imaging areas. The reason for the errors may include the surface of the target is not strictly perpendicular to AOTF, there exists an error in distance measurement and target center extraction, and the sound field of AOTF is not ideal. The *errorRMS* can be calculated by Equation (4).

$$error\_{RMS} = \sqrt{\frac{1}{81} \sum\_{i=1}^{9} \sum\_{j=1}^{9} \left[ (\mathbf{x}\_{actual,ij} - \mathbf{x}\_{model,ij})^2 + (y\_{actual,ij} - y\_{model,ij})^2 \right]}.\tag{4}$$

where *xactual*,*ij* is the row coordinate of the diffracted light imaging position. *xmodel*,*ij* is the row coordinate of the calculated imaging position. *y* is the column coordinate. The *errorRMS* between the actual imaging points and the ideal imaging points is 39.70 ≈ 40 pixels, and the deviation mainly exists in the polar direction. The *errorRMS* between the actual imaging points and the modeled imaging points is 2.14 ≈ 2 pixels, in which the coefficient of fixing errors is fitted by the least square method. The pixel difference of each point is shown in Figure 12. The sequence of the labels of the sampling points is to take the upper left point as the first sampling point. The sampling is in the row direction.

**Figure 12.** The pixel difference of each point: (**a**) pixel difference between the actual target position and simulated target position; (**b**) pixel difference between the ideal target position and actual target position.

As can be seen from Figure 12a, the pixel difference of all 81 sampling points between the actual target position and the simulated target position is less than 3.5 pixels. In Figure 12b, the closer the sampling point is to the right of the image, the smaller the aberration is, which is consistent with the previous analysis.

We made the polar angle closer to the AOTF crystal axis, and the result is shown in Figure 13.

**Figure 13.** The experiment of the square target: (**a**) actual image of diffracted light at a polar angle of −7 ◦ ; (**b**) the partially enlarged view of the square target.

By comparing Figures 11b and 13b, it was found that the aberrations of AOTF became more severe with the increase in polar angle, and the black squares on the left were deformed, which is consistent with the results of the previous analysis and simulations.

### **4. Conclusions and Future Directions**

The angle of incident light influences the spatial aberrations of AOTF imaging, and the aberrations mainly exist in the direction of the polar angle. In addition, the influence of cut angle values of the paratellurite crystal on aberrations was analyzed. In the range (0 ◦ , 18.9◦ ), the larger the ultrasonic cut angle is, the smaller the spatial aberrations of AOTF imaging are.

In this article, we verified the simulation results through quantitative experiments. The results show that the *errorRMS* between the simulation results and the experimental results is about two pixels when the incident angle in the polar direction is −5.3◦ under the experimental parameter settings above. Moreover, it provides some guidance for the design of the AOTF system with a large field of view. The computational accuracy of AOTF aberration guarantees that AOTF has the potential to be used in tracking and searching system, which is still blank in the previous research.

Compared with other spectroscopic methods, the biggest advantage of AOTF lies in the staring spectral imaging. Therefore, AOTF can be used not only for the spectral detection of static targets but also for the detection of moving targets, which can expand the application range of AOTF in the future.

**Author Contributions:** Conceptualization, H.Z. and K.Y.; methodology, K.Y.; software, K.Y.; validation, K.Y. and H.Z.; formal analysis, K.Y.; investigation, K.Y.; resources, H.Z.; data curation, K.Y.; writing—original draft preparation, K.Y.; writing—review and editing, H.Z.; visualization, H.Z.; supervision, H.Z.; project administration, H.Z.; funding acquisition, H.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the National Key Research and Development Program of China (2016YFB0500505) and the National Natural Science Foundation of China (NSFC) (Grant No. 61227806).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** We are very grateful to Zefu Xu for his dedication to the related discussions.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **References**

## *Article* **Calibration of Acousto-Optic Interaction Geometry Based on the Analysis of AOTF Angular Performance**

**Hao Zhang 1 , Huijie Zhao 1,2, Qi Guo 1,\* and Yan Xuan 1**

**\*** Correspondence<EMAIL_ADDRESS>
**Abstract:** Acousto-optic interaction geometry determines the spectral and spatial response of an acousto-optic tunable filter (AOTF). The precise calibration of the acousto-optic interaction geometry of the device is a necessary process before designing and optimizing optical systems. In this paper, we develop a novel calibration method based on the polar angular performance of an AOTF. A commercial AOTF device with unknown geometry parameters was experimentally calibrated. The experimental results show high precision, in some cases falling within 0.01◦ . In addition, we analyzed the parameter sensitivity and Monte Carlo tolerance of the calibration method. The results of the parameter sensitivity analysis show that the principal refractive index has a large influence on the calibration results, while other factors have little influence. The results of the Monte Carlo tolerance analysis show that the probability of the results falling 0.1◦ using this method is greater than 99.7%. This work provides an accurate and easy-to-perform method for AOTF crystal calibration and can contribute to the characteristic analysis of AOTFs and the optical design of spectral imaging systems.

**Keywords:** AOTF; acousto-optic interaction geometry; polar angular analysis; tolerance analysis

**Citation:** Zhang, H.; Zhao, H.; Guo, Q.; Xuan, Y. Calibration of Acousto-Optic Interaction Geometry Based on the Analysis of AOTF Angular Performance. *Materials* **2023**, *16*, 3708. https://doi.org/10.3390/ ma16103708

Academic Editors: Virgil-Florin Duma, Alexander S. Machikhin and Vitold Pozhar

Received: 26 February 2023 Revised: 26 April 2023 Accepted: 9 May 2023 Published: 13 May 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

### **1. Introduction**

An AOTF device is a spectral-splitting device based on the acousto-optic effect [1]. When compared with traditional optical splitters, such as prism, grating and interferometer, it has advantages in terms of a large-angle aperture, high spectral resolution, arbitrary wavelength configuration, fast tuning speed and it does not require moving elements [2,3]. Therefore, it is widely used in many applications, such as spectral imaging [4,5], polarization analysis [6,7], stereoscopic imaging [8] and notch filtering [9], etc. In addition, in the case of monochromatic input light, AOTFs also have the abilities of spatial filtering and edge enhancement [10,11]. There are two basic configurations, collinear and noncollinear, for AOTFs. Furthermore, in the collinear configuration, the interacting optical and acoustic waves propagate in identical directions, while the directions of the optical and acoustic waves are different in the noncollinear configuration. The first AOTF with a collinear design was reported by Harris and Wallace in 1969 [12]. Subsequently, Chang described noncollinear AOTFs in 1974 [13], which are commonly used today. Noncollinear AOTFs have some advantages, as a larger angular aperture and more materials with a large acousto-optic figure of merit can be chosen. Among them, TeO2 crystals are very suitable for noncollinear AOTFs that cover the spectral range of 350–4500 nm [14,15].

Acousto-optic interaction geometry is a key inherent property of an AOTF device, which greatly affects the spectral and spatial response of the device. Previously, Voloshinov analyzed the acousto-optic effect under three kinds of acoustic cut angles, and the results showed that acousto-optic interaction geometry affects the angular apertures and spectral resolution of an AOTF device [16]. In addition, acousto-optic interaction geometry will also affect the tuning relationship, sound field distribution, aberration and chromatic aberration

characteristics of the device, thus greatly affecting all aspects of the AOTFs in characteristic analysis and optical system design [4,17]. Therefore, AOTFs must be designed in detail to meet the special output requirements. Chang first proposed the parallel tangent condition for the design of noncollinear AOTFs with large angular apertures in the 1970s [13]. Yano discussed some properties of AOTFs using the simplified treatment [18]. Moreover, Gass corrected the birefringence approximation for the accurate design of the acousto-optic interaction geometry in the analysis [19]. These theories are all typically under the parallel tangent condition. For the non-parallel tangent condition, Yushkov expressed an exact phase-matching calculation equation at an arbitrary incident angle and recently proposed an alternative method for analyzing the Bragg angle curve in wide-angle AOTFs [10,20]. Zhang discussed the function of the phase mismatching condition and proposed a new tuning method with a non-radio-frequency signal [21]. After the processes of analysis and design, an AOTF will be manufactured using a series of fabrication technologies. Many steps are involved in the fabrication of AOTFs, including X-ray orientation, cutting, polishing, transducer orientation and fabrication, mounting and grinding [22]. The fabrication technologies of an AOTF are so complex that they easily lead to machining tolerances between the designed and actual device. If the designed values were used for device characteristic analysis and optical system design, it would lead to inaccurate results. Therefore, it is vital to calibrate the acousto-optic interaction geometry of an actual AOTF before use. In the past, our team proposed an acousto-optic interaction geometry calibration method by the tested tuning frequency curve under the parallel tangent condition [23]. However, this multi-wavelength method is not conducive to calibration accuracy as it relies on more constant parameters. In addition, to find the parallel tangent condition, as shown in Figure 1a, the incident angles must be adjusted accurately, which requires high precision in the experiment.

**Figure 1.** Comparison of operating principles. (**a**) Calibration method by the tested tuning curves under the parallel tangent condition. (**b**) Calibration method using polar angular analysis, which has no specific requirement of incident angles.

To overcome these shortcomings, here we develop a calibration method based on the polar angular performance of an AOTF. Firstly, we establish an AOTF angular frequency relationship model that can be solved analytically. Moreover, based on this model, a novel method is developed to calibrate the acousto-optic interaction geometry of an actual AOTF device. It does not introduce principal refractive index errors between multiple wavelengths and works with a single monochromatic light source. Furthermore, this method does not depend on determining characteristic incident angles, as shown in Figure 1b. Finally, using the principle of the minimum root mean square error (RMSE) between the measured and theoretical data, the acousto-optic interaction geometry of the actual AOTF device can be calculated through the use of the parameter traversal method. This method is an improvement of the calibration process in terms of simplicity and robustness and has been tested with high precision in experiments. Simultaneously, the method analyzes

the influence of crystal constants on calibration results in the visible range. This work is significant and provides a database for a range of research related to AOTF devices.

### **2.  As shown in Figure 2, this is the top view of the AOTF device and corresponds to the polar plane. In the AOTF, an acoustic wave is generated by a transducer and absorbed by an absorber. By switching the radio frequency signals applied to the transducer, the AOTF can scan the spectral regions of interest [24]. Given that the polarization state of the incident light (*L*0) is inconsistent with one of the eigenwave modes in TeO2, four types of emitted light are produced, namely diffraction ordinary polarized light (*L o d* ), transmission extraordinary polarized light (*L e t* ), transmission ordinary polarized light (*L o t* ) and diffraction extraordinary polarized light (*L e d* ).

**Figure 2.** Acousto-optic interaction geometry of an AOTF. The *θ* ∗ *i* is the front facet angle between the front facet and the crystal axis [110]. The *θα* is the acoustic cut angle between the transducer surface and the crystal axis [001]. The *θβ* is the back facet angle between the back and front facets.

Two coordinate systems have been established for analysis in this paper: the optical axes coordinate system (*x*0*oy*0) and the crystal axes coordinate system (*xoy*), as shown in Figure 2. The optical axes coordinate system is a rectangular coordinate system, where the *y*0 axis is the intersection line of the incident surface and polar plane, while the *x*0 axis is perpendicular to the incident surface. The crystal axes coordinate system is also a rectangular coordinate system, wherein the *y* axis is the crystal axis [110], while the *x* axis is the crystal axis [001].

The model between acousto-optic interaction geometry and polar angular performance for AOTFs involves two processes: (a) the calculation of the refraction at the plane of incidence, shown in Section 2.1, and (b) wave vector analysis of acousto-optic interaction, shown in Section 2.2. In addition, the relationship between the incident polar angles and matching frequencies is independent of the back facet, which will be analyzed in Section 2.1. Therefore, we need two measurements to obtain complete calibration results by swapping the front and back facets, as shown in Figure 3. For the exchange of the input and output facets, the AOTF device needs to be rotated about 180◦ around the axis perpendicular to the *xoy* plane.

**Figure 3.** Two modes for analysis. (**a**) Positive mode, lights enter AOTF from the front facet. (**b**) Reverse mode, lights enter AOTF from the back facet.

### *2.1. Refraction at the Plane of Incidence*

Firstly, the refraction of light in the plane of incidence obeys Snell's law as follows:

$$n\_0 \sin \theta\_0 = n\_1 \sin \theta\_1 \tag{1}$$

where *θ*0 is the incident polar angle between the incident light and the normal of the incident plane. *θ*1 is the refraction angle in the crystal between the refracted light and the normal to the plane of incidence. *n*0 and *n*1 are the refractive indices in the air and crystal, respectively. Moreover, it is known that *n*0 is equal to 1 in the air. TeO2 crystal has the anisotropy and *n*1 can be solved by:

$$\begin{cases} \begin{aligned} n\_1^o &= n^o\\ n\_1^\varepsilon &= \frac{n^o n^\varepsilon}{\sqrt{\left(n^\varepsilon\right)^2 \cos^2 \theta\_2^\varepsilon + \left(n^o\right)^2 \sin^2 \theta\_2^\varepsilon}} \end{aligned} \tag{2}$$

with superscripts (*o* and *e*) used to distinguish ordinary (o-polarized) from extraordinary (epolarized) light. *n o* and *n e* , related to the wavelength *λ*, are the principal refractive indices of TeO2. *θ o* 2 and *θ e* 2 are the angles between refracted light and the crystal axis [001] for o-polarized and e-polarized lights, respectively. The difference between *θ*1 and *θ*2 is as follows:

$$
\theta\_2 = \theta\_1 - \theta\_i^c \tag{3}
$$

where *θ c i* is the angle between the incident plane and the crystal axis [110], for which *c* = 1 corresponds to the positive mode, as shown in Figure 3a, while *c* = 2 corresponds to the reverse mode, as shown in Figure 3b. When switching from the positive mode to the reverse mode, the AOTF device must be rotated for swapping the input and output facets. Furthermore, the relationships between *θ c i* and the acousto-optic interaction geometry (*θ* ∗ *i* , *θα* and *θβ*) of an AOTF are as follows:

$$\begin{cases} \theta\_i^1 = \theta\_i^\* \\ \theta\_i^2 = \theta\_i^\* - \theta\_\beta \end{cases} \tag{4}$$

which means that both the positive and reverse modes are necessary to obtain the complete acousto-optic interaction geometry of an AOTF device. From Equations (1)–(3), we can obtain:

$$\begin{cases} F^{\varepsilon}(\mathbf{x}) = \left(a\_1 a\_3 - a\_2 a\_3 a\_5^2\right) \mathbf{x}^2 - 2a\_2 a\_3 a\_4 a\_5 \mathbf{x} + a\_1 a\_2 - a\_2 a\_3 a\_4^2 = 0\\ F^{\varepsilon}(\mathbf{x}) = \left(a\_1 - a\_3 a\_5^2\right) \mathbf{x}^2 - 2a\_3 a\_4 a\_5 \mathbf{x} + a\_1 - a\_3 a\_4^2 = 0 \end{cases} \tag{5}$$

where *F e* and *F o* correspond to the conditions under which the incident lights are the e-polarized and o-polarized, respectively. Furthermore, *a*1 = sin2 *θ*0, *a*2 = (*n e* ) 2 , *a*3 = (*n o* ) 2 , *a*4 = sin *θ c i* , *a*5 = cos *θ c i* and *x* = tan *θ*2. Equation (5) contains both the quadratic equations and can easily be solved. Then, using Equations (1)–(5), the relationship between *θ*0 and *θ*2 can be expressed as follows:

$$
\theta\_2 = F\_1\left(\theta\_{0\prime}\lambda, \theta\_i^\*, \theta\_{\beta}\right) \tag{6}
$$

where *F*1 is an implicit function.

### *2.2. Wave Vector Analysis of Acousto-Optic Interaction*

Acousto-optic interaction in the AOTF is usually analyzed through a wave vector diagram [25]. When the momentum-matching condition is satisfied, the incident wave vector **ki** , the acoustic wave vector **k**α and the diffraction wave vector **kd** constitute a closed triangle (Figure 4), shown as **ki** ± **k**α = **kd**. Some are dependent on the incident wavelength and refraction indices, as follows:

$$\begin{cases} \left| \mathbf{k\_{i}} \right| = \frac{2\pi n\_{i}}{\lambda} \\ \left| \mathbf{k\_{d}} \right| = \frac{2\pi n\_{d}}{\lambda} \end{cases} \tag{7}$$

where *ni* and *nd* are the refractive indices of the incident and diffracted light in TeO2. They satisfy:

$$\begin{cases} \frac{x^2}{\left(\frac{\nu^o}{\left(\nu^o\right)^2} + \frac{y^2}{\left(\nu^o\right)^2} = \left(\frac{2\pi}{\lambda}\right)^2} = \left(\frac{2\pi}{\lambda}\right)^2\\ \frac{x^2}{\left(\nu^o\right)^2} + \frac{y^2}{\left(\nu^o\right)^2} = \left(\frac{2\pi}{\lambda}\right)^2 \end{cases} \tag{8}$$

**Figure 4.** Wave vector diagrams of the noncollinear AOTF. (**a**) Positive mode. (**b**) Reserve mode.

In addition, acoustic wave vector **k**α satisfies:

$$|\mathbf{k\_{a}}| = \left| \stackrel{\rightarrow}{\mathbf{AB}} \right| = \frac{2\pi f\_{a}}{V\_{a}} \tag{9}$$

where *fα* is the acoustic frequency and *Vα* is the acoustic wave velocity. In the crystal, *Vα* is given by [26]:

$$V\_{\mathfrak{a}}^2 = V\_{110}^2 \cos^2 \theta\_{\mathfrak{a}} + V\_{001}^2 \sin^2 \theta\_{\mathfrak{a}} \tag{10}$$

where *V*001 and *V*110 are the acoustic wave velocities along the respective crystal axes. According to Equation (9), in order to solve acoustic frequency *fα*, we need to calculate the → AB . As shown in Figure 4, point A (*x*A, *y*A)satisfies tan *θ*2 = *y*A *x*A and Equation (8), which can be solved by following:

> *o* = (1, tan *θ o* 2*πn o*

$$\begin{cases} \mathbf{A}^{\lozenge} = (1, \tan \theta\_2^{\lozenge}) \frac{2\pi \mathbf{n}^{\lozenge}}{\lambda \cdot \sqrt{1 + \tan^2 \theta\_2^{\lozenge}}}\\\mathbf{A}^{\ell} = (1, \tan \theta\_2^{\ell}) \frac{2\pi \mathbf{n}^{\lozenge} \mathbf{n}^{\lozenge}}{\lambda \cdot \sqrt{\left(n^{\nu}\right)^2 + \left(n^{\nu}\right)^2 \tan^2 \theta\_2^{\epsilon}}} \end{cases} \tag{11}$$

Point B (*x*B, *y*B) satisfies the linear equation of → AB and Equation (8). For both positive and reverse modes, the linear equations of → AB are the same one as:

$$\frac{\mathbf{x\_B} - \mathbf{x\_A}}{y\_B - y\_A} = \tan \theta\_a \tag{12}$$

Therefore, point B can be solved by following:

$$\begin{split} F^{\diamond \to \epsilon}(y\_B) &= \left( \frac{\tan^2 \theta\_a}{\left( {\mu^\*} \right)^2} + \frac{1}{\left( {\mu^\*} \right)^2} \right) y\_B^2 + \frac{2}{\left( {\mu^\*} \right)^2} \left( x\_A \tan \theta\_a - y\_A \tan^2 \theta\_a \right) y\_B \\ &+ \left( \frac{{\ge\_A \times\_A \times\_A \tan^2 \theta\_a - 2 \ge\_A y\_A \tan \theta\_a}{\left( {\mu^\*} \right)^2} - \frac{2 \pi}{\lambda} \right) = 0 \\ F^{\gets \to 0}(y\_B) &= \left( \frac{\tan^2 \theta\_b}{\left( {\mu^\*} \right)^2} + \frac{1}{\left( {\mu^\*} \right)^2} \right) y\_B^2 + \frac{2}{\left( {\mu^\*} \right)^2} \left( x\_A \tan \theta\_a - y\_A \tan^2 \theta\_a \right) y\_B \\ &+ \left( \frac{\ge\_A + y\_A^\* \tan^2 \theta\_a - 2 \ge\_A y\_A \tan \theta\_a}{\left( {\mu^\*} \right)^2} - \frac{2 \pi}{\lambda} \right) = 0 \end{split} \tag{13}$$

which are all quadratic equations and easy to be solved exactly. *F o*→*e* corresponds to the condition wherein the incident light is o-polarized and the diffraction light is e-polarized, while *F e*→*o* corresponds to the condition wherein the incident light is e-polarized and the diffraction light is o-polarized. From Equations (9)–(13), the relationship between *fα* and *θ*2 can be expressed as follows:

$$f\_{\mathfrak{A}} = \mathbb{F}\_2(\theta\_2, \lambda, \theta\_{\mathfrak{a}}) \tag{14}$$

where *F*2 is an implicit function.

In summary, with Equations (6) and (14), the model between the acousto-optic interaction geometry and the incident polar angular frequencies of AOTFs can be established as follows: