Investigation of Angle Measurement Based on Direct Third Harmonic Generation in Centrosymmetric Crystals

: This paper proposed angle measurement methods based on direct third harmonic generation (THG) in centrosymmetric crystals. The principles of the intensity-dependent and the wavelength-dependent angle measurement methods were illustrated. In this study, three prospective centrosymmetric crystals and two different phase-matching types were investigated in a wavelength range from 900 nm to 2500 nm. For the intensity-dependent method, a dispersion-less wavelength range was found from 1700 nm to 2000 nm for α -BBO and calcite. Compared with rutile, α -BBO and calcite had relatively better measurement performance based on the angle measurement sensitivity calculation. The wavelength-dependent method was considered in a dispersive range of around 1560 nm. The results suggested that α -BBO and calcite were also suitable for wavelength-dependent measurement. In addition, the effects of focusing parameters were considered in the simulation, and the optimized focal length ( f = 100 mm) and the focused position (in the center of the crystal) were determined.


Introduction
Angle measurement is essential for controlling the quality of products in manufacturing processes [1]. Among angle measurement methods, optical angle measurement is widely adopted for the advantage of non-contact detection. In the decades, an amount of optical angle measurement techniques were developed [2], such as the well-used optical encoders mounted on the rotary shaft for covering the 360 • angle displacement range of rotation [3]. Accurate evaluation of small angular displacements is also required in many situations, such as the angular error motion of a moving stage and the tilt error motion of a rotating spindle [4]. For this purpose, angle sensors based on autocollimation and light interferometry were proposed [5][6][7]. In such an angle sensor, a reflective mirror is mounted on the measured target. The angular displacement of the reflective mirror induces changes in the sensor output for the measurement.
In the field of optical metrology, an appropriate light source should be selected. Femtosecond laser sources have not only the characteristics of conventional laser sources but also a series of stable, equally spaced modes in the frequency domain and femtosecondscale ultrashort pulses in the time domain [1]. Several angle measurement setups have been built employing the femtosecond laser source for high-resolution angle measurement [8][9][10]. For the optical lever sensor based on the femtosecond laser, a grating reflector was mounted on a rotary stage while the detector was kept stationary. In this case, the femtosecond laser beam was projected onto the grating reflector after the laser beam was modulated by using a Fabry-Pérot etalon. A group of first-order diffracted beams of the by the non-phase-matched second-order χ (2) optical processes [26]. The simulation results of investigating the angle measurement performance of different centrosymmetric crystals are reported in this paper. Firstly, the angle dependence principle of direct THG in the centrosymmetric crystal is illustrated for an intuitive understanding. Then, two angle measurement methods are introduced, including the intensity-dependent and wavelengthdependent methods. For theoretical investigation, three centrosymmetric crystals are selected, which could achieve a considerate THG conversion efficiency. Their material specifications are also supported by the literature. A wide wavelength range from 900 nm to 2500 nm is considered in the simulation for angle measurement. In this range, the measurement sensitivity is evaluated to choose the crystals that are suitable for angle measurement. In addition, the effects of focusing parameters, including the focal length and the focused position, are discussed. The optimized measurement configurations are found, and the reasons are briefly discussed. Figure 1a shows a brief schematic of the THG process in centrosymmetric crystal. The fundamental wave (FW) passes through the crystal and produces a third harmonic wave (THW) by three-order susceptibility χ (3) . The produced third harmonic wave has three times the frequency (ν 3 = 3ν 1 ) and the wave number (k 3 = 3k 1 ) of the fundament wave. Its wavelength becomes one-third of the FW accordingly (λ 3 = λ 1 /3). The angle dependence of THG is mainly related to the birefringence and dispersion of the crystal's refractive index, which makes the light propagate at different speeds in different directions and wavelengths. Therefore, we introduce the refractive index first. For the common uniaxial crystal, Figure 1b gives the refractive index ellipsoid of the uniaxial crystal, where X-Y-Z is the Cartesian coordinates of the crystal, and the Z axis is assumed to be parallel to the optical axis. θ is the angle between the wave vector k and the optical axis. ϕ denotes the azimuthal angle, which is the angle between the projection of k in the X-Y plane and the X-axis. N o and N e are the length of the refractive ellipsoid. There are two different propagated waves in crystal, called the ordinary wave (o-wave) and the extraordinary wave (e-wave), of which the electrical fields are perpendicular and parallel to the incident plane, respectively. It should be noted that the effect of the walk-off is ignored in this paper. Thus, we assume that the e-wave is perpendicular to the k. The refractive index can be calculated by Equation (1), where the numbers 1 and 3 correspond to the FW and THW. We can see that the refractive index of the e-wave is dependent on the angle θ, and it is also related to the wavelength λ. Because the refractive index of the e-wave is determined by both θ and λ, the angle θ can be used for the compensation of refractive index dispersion between FW and THW. That is to say, the refractive index n 3 of the third harmonic wave may be equal to that of the fundamental wave n 1 by propagating at one special angle and choosing a suitable polarization combination, which is so-called birefringent phase-matching. Figure 2 illustrates the type I phase-matching of positive uniaxial crystal as an example. In this figure, FW and THW are assumed as e-wave and o-wave, respectively. X'-Y'-Z' and X-Y-Z denote the stationary laboratory's coordinate and the rotated crystal coordinate, respectively. The Z' axis is parallel to the FW propagation directionk 1e (laser direction) in crystal. Figure 2a shows the refractive index surface of the phase-matching case. We can seek 1e is in the direction of phase-matching angle θ m , which makes the phase mismatching ∆k(θ m ) = 6π/λ 1 (n 3o − n 1e (λ 1 , θ m )) = 0 because of the refractive index difference ∆n(θ m ) = n 3o − n 1e (λ 1 , θ m ) = 0. In the phase-matching case, as given in Figure 2b, FW and THW have the same propagated speed in the crystal because ∆k = 0 and ∆n = 0. The THW generated in different positions (the first, second, and third, positions shown in Figure 2b) can be strengthened by constructive interference. Therefore, the detectable THG intensity is generated. However, as the crystal is rotated at an angle ∆θ away from the phasematching angle θ m shown in Figure 2c, d, the transmitted THG will be weakened because the generated THWs in different positions are reduced by the destructive interference where the ∆k(θ m + ∆θ) and ∆n(θ m + ∆θ) are not equal to zero. For the birefringent phase matching, there are three types for both the negative uniaxial crystal and the positive uniaxial crystal. Here, we only discuss the type I phase-matching of positive uniaxial crystal and the type II phase-matching of negative uniaxial crystal, taking into consideration that the highefficiency THG in the centrosymmetric crystals has been demonstrated in the literature. The concept is similar in both the type I phase-matching of positive uniaxial crystal and the type II phase-matching of negative uniaxial crystal. The only difference is that the phase-matching angle θ m should meet the condition of n 1o + n 1o + n 1e (λ 1 , θ m ) = n 3e (λ 3 , θ m ), and the incident fundamental wave should contain both the o-wave and the e-wave for the THG process. Combined with Equation (1) and the concept of phase-matching, the calculation of θ m can be easily induced, and they are displayed in Equation (2) and Equation (3). Equation (2) is for the type I phase-matching, and Equation (3) is for the type II phase-matching. It should be noted that Equation (3) only shows an implicit relation, which should be solved by numerical methods.

The Basic Principle of Angle Dependence of Direct THG in Centrosymmetric Crystal
Generally, a small incident angle deviation ∆θ can cause a large change in the THG intensity, which is especially beneficial for the small-angle measurement. Next, we propose to utilize the angle dependence of direct THG for angle measurement. Two measurement methods will be explained in detail.   Figure 2 illustrates the type I phase-matching of positive uniaxial crystal as an example. In this figure, FW and THW are assumed as e-wave and o-wave, respectively. X'-Y'-Z' and X-Y-Z denote the stationary laboratory's coordinate and the rotated crystal coordinate, respectively. The Z' axis is parallel to the FW propagation direction 1e (laser direction) in crystal. Figure 2a shows the refractive index surface of the phase-matching case. We can see 1e is in the direction of phase-matching angle θm, which makes the phase mismatching ∆k(θm) = 6π/λ1(n3o − n1e(λ1, θm)) = 0 because of the refractive index difference ∆n(θm) = n3o − n1e(λ1, θm) = 0. In the phase-matching case, as given in Figure 2b, FW and THW have the same propagated speed in the crystal because ∆k = 0 and ∆n = 0. The THW generated in different positions (the first, second, and third, positions shown in Figure 2b) can be strengthened by constructive interference. Therefore, the detectable THG intensity Generally, a small incident angle deviation ∆θ can cause a large change in the THG intensity, which is especially beneficial for the small-angle measurement. Next, we propose to utilize the angle dependence of direct THG for angle measurement. Two measurement methods will be explained in detail. Figure 3a shows the angle measurement schematic based on direct THG in centrosymmetric crystal. Because the magnitude of χ (3) for THG is very small, the femtosecond laser beam is needed to be focused into centrosymmetric crystal by a focused lens after passing through the polarizer 1. The generated third harmonic wave is collected by a detector. The polarizer 2 used here prevents the femtosecond laser beam from affecting the measured result. The crystal is mounted on a rotary stage. The angle θ changes as the target rotates. In a THG process, the incident fundamental wave (FW) and the transmitted THW generally contain a certain bandwidth in the frequency domain, which can be seen in Figure 3a. For a mode-locked femtosecond laser, the FW has N equally spaced modes in the frequency domain. It is also the case for the THW. The spaces between the i th mode and the i + 1th mode are νrep and 3νrep for FW and THW, respectively. The carrier-envelope offsets are νCEO and 3νCEO for FW and THW, respectively, because the THG process triples  Figure 3a shows the angle measurement schematic based on direct THG in centrosymmetric crystal. Because the magnitude of χ (3) for THG is very small, the femtosecond laser beam is needed to be focused into centrosymmetric crystal by a focused lens after passing through the polarizer 1. The generated third harmonic wave is collected by a detector. The polarizer 2 used here prevents the femtosecond laser beam from affecting the measured result. The crystal is mounted on a rotary stage. The angle θ changes as the target rotates. In a THG process, the incident fundamental wave (FW) and the transmitted THW generally contain a certain bandwidth in the frequency domain, which can be seen in Figure 3a. For a mode-locked femtosecond laser, the FW has N equally spaced modes in the frequency domain. It is also the case for the THW. The spaces between the i th mode and the i + 1th mode are ν rep and 3ν rep for FW and THW, respectively. The carrier-envelope offsets are ν CEO and 3ν CEO for FW and THW, respectively, because the THG process triples the frequency of FW in the crystal. The λ 1,3 are used to denote the corresponding wavelengths of FW and THW, respectively, in Figure 3a. In this paper, we assume the focused beam has a typical Gaussian distribution in free space, and the characteristics are displayed in Figure 3b. Here, w and b are the beam waist and the confocal parameter of the Gaussian beam, respectively. The Z L and Z R are the coordinate value in the incident plane and the output plane, respectively. The crystal length L = Z R − Z L . The d and f are the diameter of the incident laser beam and the focal length of the focused lens, respectively. the frequency of FW in the crystal. The λ1,3 are used to denote the corresponding wavelengths of FW and THW, respectively, in Figure 3a. In this paper, we assume the focused beam has a typical Gaussian distribution in free space, and the characteristics are displayed in Figure 3b. Here, w and b are the beam waist and the confocal parameter of the Gaussian beam, respectively. The ZL and ZR are the coordinate value in the incident plane and the output plane, respectively. The crystal length L = ZR − ZL. The d and f are the diameter of the incident laser beam and the focal length of the focused lens, respectively. The angle dependence of transmitted THG I3(λ3, θ) is given in Equation (4) [27], where c and ε0 are the light speed and dielectric constant of free space, respectively. I1 is the intensity of the fundamental wave. (3) eff χ is the third-order effective nonlinear coefficient, which is related to the phase-matching type and the χ (3) matrix of crystal. Here, the J3 function is crucial for the angle measurement. It is dependent on the phase-mismatching variation ∆k(λ1, θ). The equations for the calculation of ∆k, b1, and w1 are given as Equation (6) to Equation (8), respectively. The beam waist and the confocal parameter for THW are w3 = √3w1 and b3 = b1, respectively [28].  The angle dependence of transmitted THG I 3 (λ 3 , θ) is given in Equation (4) [27], where c and ε 0 are the light speed and dielectric constant of free space, respectively. I 1 is the intensity of the fundamental wave. χ (3) eff is the third-order effective nonlinear coefficient, which is related to the phase-matching type and the χ (3) matrix of crystal. Here, the J 3 function is crucial for the angle measurement. It is dependent on the phase-mismatching variation ∆k(λ 1 , θ). The equations for the calculation of ∆k, b 1 , and w 1 are given as Equation (6) to Equation (8), respectively. The beam waist and the confocal parameter for THW are w 3 = √ 3w 1 and b 3 = b 1 , respectively [28]. χ (3) eff will be calculated in the next section corresponding to the investigated centrosymmetric crystals.

The Angle Measurement Methods Based on Direct THG with a Femtosecond Laser
Because the actual rotated angle for the measurement target is the incident angle θ i , the relationship between θ i and θ should also be considered for the angle measurement. As shown in Figure 3c, θ r is the refractive angle, and θ spec is the angle between the optical axis and the normal direction, which is often around the phase-matching θ m . This enables the incident laser to easily achieve phase-matching conditions at the normal incidence. The relationship can be described by Snell's law as Equation (9), where n 1 can be calculated according to Equation (1), which is related to the dispersion and the angle θ.
As reported in our previous studies, there are two types of angle measurement methods based on the phase-matching principle of THG: the intensity-dependent method and the wavelength-dependent method. For the former, a single photodiode (PD) can be used as the detector (see Figure 3a), which converts the angle-related intensity change into an electrical current output. Then, the electrical current is amplified to a voltage by an I/V converter, and the voltage is displayed on an oscilloscope. The dispersion of angle dependence is very important because a single PD can only detect the total intensity I THW of THW, which has a band from λ 1 3 to λ N 3 . Therefore, the characteristics in the frequency domain cannot be measured. The angle dependence of measured total intensity I THW (θ) can be evaluated by Equation (10) [12]. Figure 4 is used to visually explain the effect of dispersion. Supposing the angle dependence I 3 (θ) is dispersion-less in the wavelength band as shown in Figure 4a, there is a less overlap area of angle dependence between the modes of λ 1 3 , λ i 3 , λ i+1 3 , and λ N 3 . That means there is also less possibility for the angle sensitivity to be reduced between the modes. A relatively stronger total intensity I THW (θ) can then be expected because the summation of Equation (10) will be large. Conversely, as shown in Figure 4b, if the angle dependence is dispersive, the expected angle measurement sensitivity will be weakened because of the large overlap area of angle dependence.  Different from the intensity-dependent method, the wavelength-dependent method is based on the angle dependence dispersion. An optical spectrum analyzer (OSA) is used for THG detection in the frequency domain. As shown in Figure 5a, the different blue lines correspond to the angle dependence curve of different THG wavelengths 1 Different from the intensity-dependent method, the wavelength-dependent method is based on the angle dependence dispersion. An optical spectrum analyzer (OSA) is used for THG detection in the frequency domain. As shown in Figure 5a, the different blue lines correspond to the angle dependence curve of different THG wavelengths λ 1 3 , λ i 3 , λ i+1 3 , and λ N 3 . Due to the dispersion, the angle dependence curve for each wavelength is different. The corresponding phase-matching angles θ m (λ 3 ) also change in the THG band. As the angle θ changes from θ m (λ 1 3 ) to θ m (λ N 3 ), the corresponding wavelength that has the highest intensity should also change from λ 1 3 to λ N 3 . Therefore, an obvious spectrum peak shift can be seen in the frequency domain as illustrated in Figure 5b. To quantitatively characterize the angle-dependent peak shift, we use the weighted wavelength λ w (θ i ) to denote peak wavelength [13], as given in Equation (11). Different from the intensity-dependent method, the wavelength-dependent method is based on the angle dependence dispersion. An optical spectrum analyzer (OSA) is used for THG detection in the frequency domain. As shown in Figure 5a, the different blue lines correspond to the angle dependence curve of different THG wavelengths 1 3 λ , 3 i λ , 1 3 i λ + , and 3 N λ . Due to the dispersion, the angle dependence curve for each wavelength is different. The corresponding phase-matching angles θm(λ3) also change in the THG band. As the angle θ changes from θm( 1 3 λ ) to θm( 3 N λ ), the corresponding wavelength that has the highest intensity should also change from 1 3 λ to 3 N λ . Therefore, an obvious spectrum peak shift can be seen in the frequency domain as illustrated in Figure 5b. To quantitatively characterize the angle-dependent peak shift, we use the weighted wavelength λw(θi) to denote peak wavelength [13], as given in Equation (11).

The Investigated Centrosymmetric Crystals
In this paper, three centrosymmetric crystals, Alpha Barium Borate (α-BBO), calcite, and rutile, are investigated considering their transparency, THG conversion efficiency, broad phase-matching range, and availability. Here, it should be clarified that α-BBO is a typical centrosymmetric crystal, which is different from the non-centrosymmetric Beta Barium Borate (β-BBO) in our previous study, although they have a similar refractive

The Investigated Centrosymmetric Crystals
In this paper, three centrosymmetric crystals, Alpha Barium Borate (α-BBO), calcite, and rutile, are investigated considering their transparency, THG conversion efficiency, broad phase-matching range, and availability. Here, it should be clarified that α-BBO is a typical centrosymmetric crystal, which is different from the non-centrosymmetric Beta Barium Borate (β-BBO) in our previous study, although they have a similar refractive index [26]. α-BBO and calcite are negative crystals (N e < N o ), and rutile is a positive crystal (N e > N o ). The dispersion of the refractive index can be empirically calculated by the Sellmeier relation as shown in Equation (12). The coefficients are given in Table 1 from the literature. The third-order effective nonlinear coefficient χ (3) eff should be considered with the phase-matching type and the formation of χ (3) matrix, which is dependent on the crystal class. Here, we discuss the type II phase-matching for α-BBO and calcite of crystal class 3m, and the type I phase matching for rutile of crystal class 4/mmm, because of the high THG conversion efficiency [26,29,30]. The calculation of χ (3) eff is given in Equation (12), and the parameters χ (3) eff (θ m ) can be found in the literature [30].

The Sensitivity Investigation of Angle Measurement Based on Direct THG
As illustrated before, the intensity-dependent and wavelength-dependent methods are based on the dispersion-less and dispersive properties of angle dependence. Here, we use the phase-matching angle θ m (λ 1 ) to denote the dispersion speed in a certain wavelength range because θ m represents the angle of the highest point in the angle dependence curve. We first consider a wavelength range from 900 nm to 2500 nm, which covers a commonly used range of femtosecond laser, to find a suitable wavelength range for intensity-dependent angle measurement. The simulation results are given in Figure 6a, where the lines correspond to α-BBO, calcite, and rutile. In the investigated wavelength range, the θ m (λ 1 ) of rutile is monotonically decreasing from 79.8 • to 34.2 • , which indicates the angle dependence of rutile has a very significant dispersion in this range. Therefore, rutile is not an ideal crystal for intensity-dependent measurement in the investigated range. Compared with rutile, both the θ m (λ 1 ) curves of α-BBO and calcite go down firstly from 900 nm to around 1700 nm, then keep stable from 1700 nm to 2000 nm, and finally slowly rise from 1700 nm to 2000 nm. We can see that the trends of α-BBO and calcite are very similar, and the range 1700 nm to 2000 nm is the ideal range for intensity-dependent angle measurement because of the low dispersion. In the dispersion-less area, the θ m (λ 1 ) of α-BBO and calcite are around 33.0 • and 27.1 • . It is easy to understand that the derivative of θ m (λ 1 ) in the dispersion-less area should be close to zero. Figure 6b shows the calculated phase-matching angle derivative in a unit of deg/nm. In this figure, the α-BBO and calcite correspond to the left vertical axis. It can be seen that the dispersion speed of θ m (λ 1 ) is in the order of 10 −3 deg/nm. Here, the red circles represent the positions having zero derivatives, which are 1803 nm and 1858 nm for calcite and α-BBO, respectively. As a comparison, the derivative result of rutile is also given in the figure. It corresponds to the right vertical axis. We see the dispersion speed of θ m (λ 1 ) is in the order of 10 −2 deg/nm, which is much higher than the dispersion-less cases. measurement because of the low dispersion. In the dispersion-less area, the θm(λ1) of α-BBO and calcite are around 33.0° and 27.1°. It is easy to understand that the derivative of θm(λ1) in the dispersion-less area should be close to zero. Figure 6b shows the calculated phase-matching angle derivative in a unit of deg/nm. In this figure, the α-BBO and calcite correspond to the left vertical axis. It can be seen that the dispersion speed of θm(λ1) is in the order of 10 −3 deg/nm. Here, the red circles represent the positions having zero derivatives, which are 1803 nm and 1858 nm for calcite and α-BBO, respectively. As a comparison, the derivative result of rutile is also given in the figure. It corresponds to the right vertical axis. We see the dispersion speed of θm(λ1) is in the order of 10 −2 deg/nm, which is much higher than the dispersion-less cases. To evaluate the angle measurement sensitivity of the intensity-dependent method, we calculate the angle dependence of ITHW(θi). In the simulation process, the parameters are set as: the focal length f = 70 mm, the Z positions of the crystal are ZL = −1 mm and ZR = 1 mm, respectively, the crystal length L = ZR − ZL = 2 mm, the diameter of incident femtosecond beam d = 3.6 mm. We assume the focused point (beam waist) is perfectly in the center of the crystal, and the azimuthal angle φ = 30°. The wavelength range of the incident femtosecond laser is set from 1800 nm to 1900 nm, covering the dispersion-less area. The space between the modes is 1 nm. Figure 7 shows the simulated results, where the index (a), (b), and (c) of subplots correspond to α-BBO, calcite, and rutile, respectively. In each subplot, the lines in different colors correspond to the angle dependence I3(θi) of different modes or the total intensity ITHW(θi). The horizontal axis denotes the angular displacement of the incident angle θi. The left and right vertical axes denote the intensity of I3 and ITHW, respectively. It is clear that I3(θi) does not have an obvious change between the wavelength of 1800 nm, 1850 nm, and 1900 nm for α-BBO and calcite, and the angle dependence of ITHW(θi) keeps the same shape as I3(θi). The maximum total intensities of ITHW are strengthened to 96.8 a.u. (α-BBO), 93.0 a.u. (calcite), respectively. Conversely, due to the dispersion, the rutile has a much lower maximum total intensity ITHW of 2.7 a.u. The measurement sensitivity is calculated from about 80% to 20% of ITHW, and they are −0.087 a.u./arcsec, −0.108 a.u./arcsec, 0.002 a.u./arcsec, for α-BBO, calcite, and rutile, respectively shown in Figure 7. To evaluate the angle measurement sensitivity of the intensity-dependent method, we calculate the angle dependence of I THW (θ i ). In the simulation process, the parameters are set as: the focal length f = 70 mm, the Z positions of the crystal are Z L = −1 mm and Z R = 1 mm, respectively, the crystal length L = Z R − Z L = 2 mm, the diameter of incident femtosecond beam d = 3.6 mm. We assume the focused point (beam waist) is perfectly in the center of the crystal, and the azimuthal angle ϕ = 30 • . The wavelength range of the incident femtosecond laser is set from 1800 nm to 1900 nm, covering the dispersion-less area. The space between the modes is 1 nm. Figure 7 shows the simulated results, where the index (a), (b), and (c) of subplots correspond to α-BBO, calcite, and rutile, respectively. In each subplot, the lines in different colors correspond to the angle dependence I 3 (θ i ) of different modes or the total intensity I THW (θ i ). The horizontal axis denotes the angular displacement of the incident angle θ i . The left and right vertical axes denote the intensity of I 3 and I THW , respectively. It is clear that I 3 (θ i ) does not have an obvious change between the wavelength of 1800 nm, 1850 nm, and 1900 nm for α-BBO and calcite, and the angle dependence of I THW (θ i ) keeps the same shape as I 3 (θ i ). The maximum total intensities of I THW are strengthened to 96.8 a.u. (α-BBO), 93.0 a.u. (calcite), respectively. Conversely, due to the dispersion, the rutile has a much lower maximum total intensity I THW of 2.7 a.u. The measurement sensitivity is calculated from about 80% to 20% of I THW , and they are −0.087 a.u./arcsec, −0.108 a.u./arcsec, 0.002 a.u./arcsec, for α-BBO, calcite, and rutile, respectively shown in Figure 7. For the wavelength-dependent method, the spectrum peak shift can be found except for the dispersion-less area (1700-2000 nm) of α-BBO and calcite. In our previous work, a commercial femtosecond laser source was used for producing harmonic waves, and the central wavelength of the laser was around 1560 nm. Therefore, the wavelength-dependent method will be discussed near 1560 nm for future experimental work. In the simulation results, the parameters are the same as those of the intensity-dependent method. Figure 8 shows the calculated results for three crystals. In this figure, the letters a, b, and c correspond to the results of materials of α-BBO, calcite, and rutile, respectively. The numbers 1 and 2 correspond to the 3-D plot and 2-D plot third harmonic intensity I3, respectively. For each subplot, the color bar is used to indicate the amount of intensity. Here, we fixed the range from −0.8° to 0.6° to compare the peak shifts. The results in Figure 8(a1,b1,c1) show that the peak shifts exist for all the investigated materials, and all the peak shifts toward the longer wavelength in the frequency domain as θi decreases. The peak shifts of α-BBO and calcite both cover a range from around 500 nm to 540 nm. In contrast, the peak shift of rutile is relatively slow because it even cannot cover the wavelength range from 519 nm to 521 nm. The essence of peak shift is the point movement, which makes ∆n(λ1, θ) = const. This movement can be described by a contour map of ∆n(λ1, θ). It has been analyzed in our previous work on SHG. Therefore, it is not repeated here because of the similarity. The weighted wavelengths λw(θi) are shown in Figure 8(a2,b2,c2) by the white dotted lines to quantitatively evaluate peak shifts. The sensitivity is calculated by a linear fitting function to get the slope of λw(θi). The calculated sensitivities are −0.0054 nm/arcsec, −0.0063 nm/arcsec, and −0.0002 a.u./arcsec for α-BBO, calcite, and rutile, respectively. The results suggest that α-BBO and calcite have better measurement performance in the wavelength around 1560 nm. For the wavelength-dependent method, the spectrum peak shift can be found except for the dispersion-less area (1700-2000 nm) of α-BBO and calcite. In our previous work, a commercial femtosecond laser source was used for producing harmonic waves, and the central wavelength of the laser was around 1560 nm. Therefore, the wavelength-dependent method will be discussed near 1560 nm for future experimental work. In the simulation results, the parameters are the same as those of the intensity-dependent method. Figure 8 shows the calculated results for three crystals. In this figure, the letters a, b, and c correspond to the results of materials of α-BBO, calcite, and rutile, respectively. The numbers 1 and 2 correspond to the 3-D plot and 2-D plot third harmonic intensity I 3 , respectively. For each subplot, the color bar is used to indicate the amount of intensity. Here, we fixed the range from −0.8 • to 0.6 • to compare the peak shifts. The results in Figure 8(a 1 ,b 1 ,c 1 ) show that the peak shifts exist for all the investigated materials, and all the peak shifts toward the longer wavelength in the frequency domain as θ i decreases. The peak shifts of α-BBO and calcite both cover a range from around 500 nm to 540 nm. In contrast, the peak shift of rutile is relatively slow because it even cannot cover the wavelength range from 519 nm to 521 nm. The essence of peak shift is the point movement, which makes ∆n(λ 1 , θ) = const. This movement can be described by a contour map of ∆n(λ 1 , θ). It has been analyzed in our previous work on SHG. Therefore, it is not repeated here because of the similarity. The weighted wavelengths λ w (θ i ) are shown in Figure 8(a 2 ,b 2 ,c 2 ) by the white dotted lines to quantitatively evaluate peak shifts. The sensitivity is calculated by a linear fitting function to get the slope of λ w (θ i ). The calculated sensitivities are −0.0054 nm/arcsec, −0.0063 nm/arcsec, and −0.0002 a.u./arcsec for α-BBO, calcite, and rutile, respectively. The results suggest that α-BBO and calcite have better measurement performance in the wavelength around 1560 nm.

The Focusing Parameter's Effects on Angle Measurement
The influence of the focusing parameters on the THG angle measurement is investigated by changing the focal length f and the focused position. Here, we only put the results of calcite crystal because of its better sensitivity performance compared with rutile. The results are similar to those of α-BBO. In the simulation, the crystal length L = 2 mm, d = 3.6 mm, λ1 = 1850 nm. The focused point is in the center of the crystal and the azimuthal angle φ = 30°. Figure 9 shows the results of angle dependence I3(θi), where the lines in different colors correspond to different focal lengths. In Figure 9a, the range of incident angle θi is from −13° to 1°. The peak intensities corresponding to the focal lengths are noted in the figure. The peak intensities are not monotonically decreasing as the focal length f increases, and the highest intensity is 1.00 a.u. appear as f = 70 mm. The overall trend of the peak

The Focusing Parameter's Effects on Angle Measurement
The influence of the focusing parameters on the THG angle measurement is investigated by changing the focal length f and the focused position. Here, we only put the results of calcite crystal because of its better sensitivity performance compared with rutile. The results are similar to those of α-BBO. In the simulation, the crystal length L = 2 mm, d = 3.6 mm, λ 1 = 1850 nm. The focused point is in the center of the crystal and the azimuthal angle ϕ = 30 • . Figure 9 shows the results of angle dependence I 3 (θ i ), where the lines in different colors correspond to different focal lengths. In Figure 9a, the range of incident angle θ i is from −13 • to 1 • . The peak intensities corresponding to the focal lengths are noted in the figure. The peak intensities are not monotonically decreasing as the focal length f increases, and the highest intensity is 1.00 a.u. appear as f = 70 mm. The overall trend of the peak intensity is that it increases from 0.74 a.u. (10 mm) to 1.00 a.u. (70 mm) then decreases from 1.00 a.u. (70 mm) to 0.51 a.u. (150 mm). We can also see the width of the angle dependence curve is reduced as the focal length becomes longer from f = 10 mm (11.9 deg.) to f = 150 mm (0.5 deg.). In Figure 9b, θ i only covers the range from −1 • to 0.5 • , and the sensitivity result is shown except for the f = 10 mm because of the relatively low value. We can see the maximum sensitivity result is −4.21 a.u./deg when f = 100 mm, rather than f = 70 mm, which has the highest peak intensity. This is because the f = 100 mm case has a similar high peak intensity (0.96 a.u.) but has a relatively narrower angle dependence range compared with the f = 70 mm case. Therefore, better angle measurement can be expected from the setting of f = 100 mm. Appl. Sci. 2023, 13, x FOR PEER REVIEW 13 intensity is that it increases from 0.74 a.u. (10 mm) to 1.00 a.u. (70 mm) then decr from 1.00 a.u. (70 mm) to 0.51 a.u. (150 mm). We can also see the width of the angl pendence curve is reduced as the focal length becomes longer from f = 10 mm (11.9 to f = 150 mm (0.5 deg.). In Figure 9b, θi only covers the range from −1° to 0.5°, and sensitivity result is shown except for the f = 10 mm because of the relatively low value can see the maximum sensitivity result is −4.21 a.u./deg when f = 100 mm, rather tha 70 mm, which has the highest peak intensity. This is because the f = 100 mm case h similar high peak intensity (0.96 a.u.) but has a relatively narrower angle depend range compared with the f = 70 mm case. Therefore, better angle measurement ca expected from the setting of f = 100 mm. We also give a qualitative explanation of the results of Figure 9a. It can be seen Equation (4) to Equation (8) that the angle dependence I3(θi) is related to both the b waist w1 and the confocal parameter b1. As the focal length f decreases, both the beam w w1 and the confocal parameter b1 are reduced in Equations (7) and (8). On the one h as in Equation (4), the decrease of w1 makes the I3 increase directly. On the other hand decrease of confocal parameter b1 will make the |J3| 2 change because J3 is related t term 1/(1 + 2iz/b1). After a Taylor expansion of 1/(1 + 2iz/b1), |J3| 2 has the following as shown in Equation (14), We also give a qualitative explanation of the results of Figure 9a. It can be seen from Equation (4) to Equation (8) that the angle dependence I 3 (θ i ) is related to both the beam waist w 1 and the confocal parameter b 1 . As the focal length f decreases, both the beam waist w 1 and the confocal parameter b 1 are reduced in Equations (7) and (8). On the one hand, as in Equation (4), the decrease of w 1 makes the I 3 increase directly. On the other hand, the decrease of confocal parameter b 1 will make the |J 3 | 2 change because J 3 is related to the term 1/(1 + 2iz/b 1 ). After a Taylor expansion of 1/(1 + 2iz/b 1 ), |J 3 | 2 has the following form as shown in Equation (14), where the decrease of b 1 will make the part of J (2) 3 . It will induce an extra phase in |J 3 | 2 , which is harmful to the phase-matching when ∆k(θ m ) = 0. Therefore, there should be a trade-off between the beam waist w 1 and confocal parameter b 1 . The focal length of 100 mm is the best choice, which has a Rayleigh length of 2.2 mm. In addition, as can be found in Equation (14), a longer focal length f makes the J (2) 3 smaller and |J 3 | 2 will degenerate to the sinc 2 (∆k(θ)L/2) function form. Then, the side peaks of sinc 2 (∆k(θ)L/2) in the Maker fringe area also can be seen in Figure 9c [33].
In the optimized case of f = 100 mm, the effect of the beam waist position is considered in Figure 10. Lines and circles are used to denote the beam waist positions of ±0.2 mm, ±0.4 mm, ±0.6 mm, ±0.8 mm, and ±1 mm relative to the crystal center. The maximum intensity is obtained in the center of the crystal at 1.00 a.u., which can be seen in the red line. This is because the central position makes the high-intensity area of the Gaussian beam to be fully used for the THG process. As the beam waist gradually moves away, the peak intensity falls from 0.95 a.u. (±0.2 mm) to 0.51 a.u. (±1 mm). The symmetric results for the positive/negative beam waist movements are the consequence of the Gaussian beam's symmetric shape. In addition, the angle dependence areas of I 3 are both close to 0.63 deg, as shown in the figure. Thus, the best measurement sensitivity should be obtained when the beam waist is in the center associated with the highest intensity.
In the optimized case of f = 100 mm, the effect of the beam waist position is considered in Figure 10. Lines and circles are used to denote the beam waist positions of ±0.2 mm, ±0.4 mm, ±0.6 mm, ±0.8 mm, and ±1 mm relative to the crystal center. The maximum intensity is obtained in the center of the crystal at 1.00 a.u., which can be seen in the red line. This is because the central position makes the high-intensity area of the Gaussian beam to be fully used for the THG process. As the beam waist gradually moves away, the peak intensity falls from 0.95 a.u. (±0.2 mm) to 0.51 a.u. (±1 mm). The symmetric results for the positive/negative beam waist movements are the consequence of the Gaussian beam's symmetric shape. In addition, the angle dependence areas of I3 are both close to 0.63 deg, as shown in the figure. Thus, the best measurement sensitivity should be obtained when the beam waist is in the center associated with the highest intensity.

Conclusions
In this paper, we have proposed the intensity-dependent and wavelength-dependent angle measurement methods by THG in centrosymmetric crystals. Three prospective crystals, including the α-BBO, calcite, and rutile, have been considered in this research for the two angle measurement methods. The effects of the focusing parameters on the angle dependence have also been considered. The optimized focusing parameters have been found for future experimental work.
For the intensity-dependent method, the type I phase-matching angle of rutile and the type II phase-matching angle of α-BBO have been calculated in the wavelength range from 900 nm to 2500 nm. The results show that the type II phase-matching angle has a good performance due to its dispersion-less property in the wavelength range of 1700-2000 nm. The sensitivities of angle measurement have been evaluated for calcite and α-BBO, which are −0.108 a.u./arcsec and −0.087 a.u./arcsec, respectively. They are much higher than the sensitivity of 0.002 a.u./arcsec for rutile. The sensitivity of the wavelengthdependent method has been evaluated in the dispersive range of around 1560 nm. The peak shift in the frequency domain has been shown for the three investigated crystals. The

Conclusions
In this paper, we have proposed the intensity-dependent and wavelength-dependent angle measurement methods by THG in centrosymmetric crystals. Three prospective crystals, including the α-BBO, calcite, and rutile, have been considered in this research for the two angle measurement methods. The effects of the focusing parameters on the angle dependence have also been considered. The optimized focusing parameters have been found for future experimental work.
For the intensity-dependent method, the type I phase-matching angle of rutile and the type II phase-matching angle of α-BBO have been calculated in the wavelength range from 900 nm to 2500 nm. The results show that the type II phase-matching angle has a good performance due to its dispersion-less property in the wavelength range of 1700-2000 nm. The sensitivities of angle measurement have been evaluated for calcite and α-BBO, which are −0.108 a.u./arcsec and −0.087 a.u./arcsec, respectively. They are much higher than the sensitivity of 0.002 a.u./arcsec for rutile. The sensitivity of the wavelength-dependent method has been evaluated in the dispersive range of around 1560 nm. The peak shift in the frequency domain has been shown for the three investigated crystals. The weighted wavelengths λ w have been used to characterize the peak shift speeds, which were −0.0054 nm/arcsec, −0.0063 nm/arcsec, and −0.0002 a.u./arcsec for α-BBO, calcite, and rutile, respectively. α-BBO and calcite also have fast peak shifts around 1560 nm.
In addition, the focal length f has been changed from 10 mm to 150 mm for investigating the effect of f on the angle measurement. The optimized focal length has been obtained as f = 100 mm. It has been demonstrated that the maximum sensitivity is achieved when the focus position is in the center of the crystal.

Data Availability Statement:
The data presented in this study are available on request from the corresponding author.