Thin-Film Lithium Niobate Based Acousto-Optic Modulation Working at Higher-Order TE 1 Mode

: Acousto-optic modulation (AOM) is regarded as an effective way to link multi-physical ﬁelds on-chip. We propose an on-chip AOM scheme based on the thin-ﬁlm lithium niobate (TFLN) platform working at the higher-order TE 1 mode, rather than the commonly used fundamental TE 0 mode. Multi-physical ﬁeld coupling analyses were carried out to obtain the refractive index change of the optical waveguide (>6.5 × 10 − 10 for a single phonon) induced by the enhanced acousto-optic interaction between the acoustic resonator mode and the multimode optical waveguide. By using a Mach-Zehnder interferometer (MZI) structure, the refractive index change is utilized to modulate the output spectrum of the MZI, thus achieving the AOM function. In the proposed AOM scheme, efﬁcient mode conversion between the TE 0 and TE 1 mode is required in order to ensure that the AOM works at the higher-order TE 1 mode in the MZI structure. Our results show that the half-wave-voltage-length product ( V π L ) is <0.01 V · cm, which is lower than that in some previous reports on AOM and electro-optic modulation (EOM) working at the fundamental TE 0 mode (e.g., V π L > 0.04 V · cm for AOM, V π L > 1 V · cm for EOM). Finally, the proposed AOM has lower loss when compared with EOM because the electrode of the AOM can be placed far from the optical waveguide.


Introduction
Acousto-optic modulation (AOM) connecting the electric, mechanical, and optical fields, takes a vital role in the microwave, lightwave, and quantum photonic signal processing [1][2][3][4][5]. Comparing with integrated electro-optic modulations (EOMs) [6][7][8][9][10], AOM typically works with multiple physical fields and its basic working process can be described as follows. Firstly, the input microwave signal is converted to the acoustic wave transmitted along the acoustic medium through a piezoelectric transducer (e.g., interdigital transducer (IDT)). Secondly, the generated acoustic wave would interact with the optical mode of the waveguide, leading to a change in the mode effective index of the optical waveguide. Finally, the mode effective index change is used to modulate the output spectrum via waveguide structures, such as a Mach-Zehnder interferometer (MZI) [1,3] or microring resonator (MRR) [1,4]. On-chip integrated AOM can enhance the AOM performance and reduce the device dimension, but requires an ideal material platform to ensure efficient couplings between the different physical fields.
Recently, various materials such as aluminum nitride (AlN) [11,12], gallium arsenide (GaAs) [13,14], gallium phosphide (GaP) [15,16], silicon [17,18], and lithium niobate to be placed close to the optical waveguide, while close electrode placement to the optical waveguide is necessary for EOM. The proposed AOM scheme would contribute to the development of on-chip low power, low loss, and compact AOM. AOM-related applications and components, such as frequency shifter [1,5], tunable filter [1,5], beamforming [3], and cavity optomechanics [33][34][35] would also be benefited, contributing to the building of on-chip photonic systems. Figure 1a shows the schematic layout of the proposed AOM working at the higherorder TE 1 mode. The key modulation part is integrated at one arm of the MZI structure together with the mode converters. The whole device is based on a 600 nm-thick X-cut TFLN. Owing to the excellent piezoelectric properties of TFLN, energy conversion from the applied microwave electric field to surface acoustic wave (SAW) can be realized with the use of IDT [12,14,25,26,32,33,35]. Accordingly, we choose the electrode parameters of the IDT to match with the wavelength of the SAW and deposit the IDT on the surface of the TFLN directly (parallel to the optical waveguide). For the generated SAW, it will transmit along the TFLN and further interact with the optical mode, where the propagation directions of the SAW and optical wave are perpendicular with each other. To enhance the AOI effect, we use the higher-order TE 1 mode as the working optical mode (the two mode spots have a π phase difference between them) within the waveguide region. As a result, the optical refractive index change of the TFLN optical waveguide is effectively increased when compared with that of AOM working at the fundamental TE 0 mode. The relatively large optical refractive index change can be utilized to modulate the output spectrum by using an optical MZI structure. This is the key idea of the present work. Figure 1b plots the cross-sectional view of the AOM along the x-z direction, where the waveguide width and etching depth are w and h 1 (=300 nm), respectively, and the electrode width, gap and thickness of IDT are a (=300 nm), b (=300 nm) and 100 nm, respectively. The gap width between the optical waveguide and IDT is G. Moreover, the gap width between the IDT and the left side of the acoustic resonator is chosen as L t (=1.2 µm), to match with the wavelength of the generated SAW. Considering the current fabrication abilities, the slant angle of the TFLN waveguide sidewall is set to 70 • for the proposed AOM design [3,[19][20][21][22][23][24]26]. Note that we also etch two relatively large slots on both sides of the AOM along the y-z direction as shown in Figure 1a (light gray rectangular) to make the corrosive liquid contact with the buried oxide (BOX) layer below the AOM region and remove it via the corrosion method. As a result, the generated SAW is well-confined and transmitted in the top LN layer of the TFLN wafer and has stronger interaction with the optical waveguide when compared with some of the BOX unremoved schemes [5,25], which contributes to the on-chip low-V π L modulation. More important, the etched two slots form an acoustic resonator to enhance the SAW energy through the resonance effect [25,[33][34][35]. The width of the acoustic resonator is set to 12 µm. Figure 1c illustrates the enlarged view of the mode converters from the TE 0 mode to the TE 1 mode and vice versa, where the widths and gap of the two-coupled waveguides of the asymmetrical directional coupler are w 1 , w and d, respectively. The coupling length is L C and the length between these two mode conversion regions is L R . Using the proposed structure, the generated SAW and higher-order TE 1 mode interact well with each other within the modulation region, leading to the change of the optical refractive index of the TFLN waveguide, which will modulate the output spectrum of the MZI structure. The multi-physical field effects acting on the overall AOI should be analyzed first before we optimize the structure of the AOM. In typical AOMs, the generated elastic wave (SAW) introduces the optomechanical coupling which leads to the structural distortion of the optical waveguide because of the photo-elastic effect and moving boundary effect of TFLN [36,37]. The structural distortion can also induce the generation of electric field through the piezoelectric effect of TFLN. The generated electric field will further affect the optical field based on the electro-optic effect of TFLN. Therefore, the overall modulation performance of the proposed on-chip AOM is determined by a combination of the moving boundary effect, photo-elastic effect, and electro-optic effect [26]. This combination effect will ultimately change the refractive index of the optical waveguide and modulate the output spectrum of the AOM embedded MZI structure.

Materials and Methods
The following theoretical analysis are based on the two-dimensional (2D) structure (x-z) of the AOM (along the SAW propagation direction), the results of which can be used to evaluate the coupling strength of the mode field interaction for the on-chip AOM. Firstly, the moving boundary effect due to the elastic deformation of the TFLN acting on the change of the refractive index of the optical waveguide is given by [26,36,37] where is the normal vector of the boundary facing outward, n is the optical refractive index along the z-direction of the TFLN, is the electric field distribution, ∥ is the electric field component parallel to the interface, is the electric displacement field component perpendicular to the interface, ϵ is the permittivity of the optical electric field, Δ = − , and Δ = − , respectively. is the zero-point motion of the mechanical mode representing the coupling between the electromagnetic field and mechanical field ( = ℏ/2 ) [36], where is the resonant frequency of the acoustic mode and ℏ is the reduced Planck constant. S defines the 2D simulation region shown in Figure 1b and the coordinate ∈ . The effective mass of the acoustic mode is given by [36,37] = ( ) / max ( ( ) ) , where is the length of the acoustic resonator along the y-direction, is the material mass density of LN, and is the solid displacement field. Note that the acoustic mode amplitude is normalized to a single phonon occupying the acoustic resonator and the same normalization will be used in the analyses of multi-physical effects. The multi-physical field effects acting on the overall AOI should be analyzed first before we optimize the structure of the AOM. In typical AOMs, the generated elastic wave (SAW) introduces the optomechanical coupling which leads to the structural distortion of the optical waveguide because of the photo-elastic effect and moving boundary effect of TFLN [36,37]. The structural distortion can also induce the generation of electric field through the piezoelectric effect of TFLN. The generated electric field will further affect the optical field based on the electro-optic effect of TFLN. Therefore, the overall modulation performance of the proposed on-chip AOM is determined by a combination of the moving boundary effect, photo-elastic effect, and electro-optic effect [26]. This combination effect will ultimately change the refractive index of the optical waveguide and modulate the output spectrum of the AOM embedded MZI structure.
The following theoretical analysis are based on the two-dimensional (2D) structure (x-z) of the AOM (along the SAW propagation direction), the results of which can be used to evaluate the coupling strength of the mode field interaction for the on-chip AOM. Firstly, the moving boundary effect due to the elastic deformation of the TFLN acting on the change of the refractive index of the optical waveguide is given by [26,36,37] wheren is the normal vector of the boundary facing outward, n is the optical refractive index along the z-direction of the TFLN, E is the electric field distribution, E is the electric field component parallel to the interface, D ⊥ is the electric displacement field component perpendicular to the interface, is the permittivity of the optical electric field, ∆ = LN − air , and ∆ −1 = −1 LN − −1 air , respectively. χ zpf is the zero-point motion of the mechanical mode representing the coupling between the electromagnetic field and mechanical field ( χ zpf = √ /2m eff ω m ) [36], where ω m is the resonant frequency of the acoustic mode and is the reduced Planck constant. S defines the 2D simulation region shown in Figure 1b and the coordinate r ∈ S. The effective mass m eff of the acoustic mode is given by [36,37] where L is the length of the acoustic resonator along the y-direction, ρ is the material mass density of LN, and Q is the solid displacement field. Note that the acoustic mode amplitude χ zpf is normalized to a single phonon occupying the acoustic resonator and the same normalization will be used in the analyses of multi-physical effects. Secondly, the refractive index change of the optical waveguide induced by the photoelastic effect for the X-cut TFLN is given by [26,36,37] where 0 is the vacuum permittivity, E x , E y , E z are the electric field components along the x-, y-, z-direction, respectively. dB 1 ∼ dB 6 are the optical indicatrix coefficients which are closely related with the generated strain s 1 ∼ s 6 by the SAW [26], and the transformation matrix between them should be rotated according to the X-cut Z-propagation TFLN. The transformation methods and related elasto-optic coefficients can be found in [38,39]. Finally, the electro-optic effect acting on the refractive index change of the optical waveguide (∆n EO ) has the same form with that of ∆n PE shown in Equation (3). The only difference lies in the optical indicatrix coefficients dB 1 ∼ dB 6 , which are related to the electric fields (ξ x , ξ y , ξ z ) of the generated SAW due to the piezoelectric effect of the TFLN [26]. Similarly, the transformation matrix also needs to be rotated for the X-cut Z-propagation TFLN and the electro-optic coefficients can be obtained from [38,39]. Therefore, the total refractive index change of the optical waveguide (∆n tot ) due to the combination of moving boundary effect, photo-elastic effect, and electro-optic effect for a single phonon is given by The in-cavity phonon number N p for the input power P in has the following relationship [26], where γ and γ e are the decay and external coupling rates, respectively, V p is the peak voltage of the input power, and R in is the input impedance (50 Ω). To accumulate a π phase shift in the MZI structure, the phase shift relation should satisfy, where λ is the optical wavelength (λ = 1.55 µm). Using Equations (5) and (6), the key modulation parameter of V π L for the proposed device is Thus, by enhancing the AOI through working at the higher-order TE 1 mode in the proposed on-chip AOM, the obtained ∆n tot could be increased effectively, leading to the reduction of V π L for the device.

Results
Two-dimensional cross-sectional analyses (x-z direction) are required in order to determine the key parameters of the AOM shown in Figure 1. We use the finite element simulation tool COMSOL Multiphysics to study the multi-physical coupling processes of the proposed AOM, in which the microwave electric field, mechanical field, and optical field can be obtained for analyses and optimization of the AOM structure [40]. Figure 2a illustrates the working process of the AOM from the 2D cross-sectional view. The generated SAW acts as a bridge connecting the microwave input and the optical mode index. The three multi-physical effects together induce the change in the refractive index of the optical waveguide based on the TFLN platform. Figure 2b plots the calculated strain displacement field, piezoelectrically induced electric field, and optical electric field on the optical waveguide which is a multimode waveguide (w = 2.1 µm) supporting the higher-order TE 1 mode. Based upon these mode field distributions and the material properties of the TFLN, we can analyze and calculate the field interaction and provide effective ways to optimize the AOM structural parameters.
Photonics 2022, 9, x FOR PEER REVIEW 6 of 14 optical waveguide based on the TFLN platform. Figure 2b plots the calculated strain displacement field, piezoelectrically induced electric field, and optical electric field on the optical waveguide which is a multimode waveguide (w = 2.1 μm) supporting the higherorder TE1 mode. Based upon these mode field distributions and the material properties of the TFLN, we can analyze and calculate the field interaction and provide effective ways to optimize the AOM structural parameters. Mode distributions of the optical electric field, strain displacement field, and the piezoelectrically induced electric field for the same cross-sectional structure of AOM. The working wavelength is 1550 nm and the frequency of the acoustic mode is 4.08 GHz.
Next, we perform structural analyses and optimizations of the proposed on-chip AOM. Figure 3a,b show the calculated refractive index change of the optical waveguide (Δ ) induced by the AOI as a function of the gap width G between the optical waveguide and IDT, when the waveguide works at the fundamental TE0 mode and the higherorder TE1 mode, respectively. The single-mode and multimode waveguide widths are w = 1.0 μm and w = 2.1 μm, respectively. From the calculations, the generated SAW has different eigenfrequencies corresponding to different strain displacement field distributions [4,25]. We use 30 frequency points around the estimated eigenfrequency of the SAW to determine the optimal microwave working frequency (which matches with the eigenfrequency of SAW) at different gap widths. For the TE0 mode, we observe that the generated refractive index change of the optical waveguide for most of the acoustic modes is within the range from −3 × 10 to 3 × 10 for a single phonon. For the TE1 mode, the range increases to from −5 × 10 to 5 × 10 for a single phonon. We then extract the maximum change in the refractive index of the optical waveguide (|Δ |) at every gap width G for both the TE0 and TE1 modes. The results are shown in Figure 3c. The performance of the refractive index change of the TE1 mode is clearly better than that of the TE0 mode because of the enhanced AOI. The maximum |Δ | is lower than (higher than) 4 × 10 (6.5 × 10 ) for the TE0 mode (TE1 mode) for a single phonon. The increase in the refractive index change when the optical waveguide works at the TE1 mode will reduce the key modulation parameter VπL of the proposed AOM. From Figure 3c, we can also determine two optimum gap widths between the optical waveguide and IDT, G = 1.7 μm and G = 3.5 μm, corresponding to the highest |Δ |. The respective microwave working frequencies are 4.06 GHz and 4.08 GHz. In addition, since large gap width G reduces the metal absorption loss of the optical waveguide and the whole AOM device [1][2][3][4][5], large gaps are preferred in on-chip AOM under the same fabrication processes and requirements. Next, we perform structural analyses and optimizations of the proposed on-chip AOM. Figure 3a,b show the calculated refractive index change of the optical waveguide (∆n tot ) induced by the AOI as a function of the gap width G between the optical waveguide and IDT, when the waveguide works at the fundamental TE 0 mode and the higher-order TE 1 mode, respectively. The single-mode and multimode waveguide widths are w = 1.0 µm and w = 2.1 µm, respectively. From the calculations, the generated SAW has different eigenfrequencies corresponding to different strain displacement field distributions [4,25]. We use 30 frequency points around the estimated eigenfrequency of the SAW to determine the optimal microwave working frequency (which matches with the eigenfrequency of SAW) at different gap widths. For the TE 0 mode, we observe that the generated refractive index change of the optical waveguide for most of the acoustic modes is within the range from −3 × 10 −10 to 3 × 10 −10 for a single phonon. For the TE 1 mode, the range increases to from −5 × 10 −10 to 5 × 10 −10 for a single phonon. We then extract the maximum change in the refractive index of the optical waveguide (|∆n tot |) at every gap width G for both the TE 0 and TE 1 modes. The results are shown in Figure 3c. The performance of the refractive index change of the TE 1 mode is clearly better than that of the TE 0 mode because of the enhanced AOI. The maximum |∆n tot | is lower than (higher than) 4 × 10 −10 (6.5 × 10 −10 ) for the TE 0 mode (TE 1 mode) for a single phonon. The increase in the refractive index change when the optical waveguide works at the TE 1 mode will reduce the key modulation parameter V π L of the proposed AOM. From Figure 3c, we can also determine two optimum gap widths between the optical waveguide and IDT, G = 1.7 µm and G = 3.5 µm, corresponding to the highest |∆n tot |. The respective microwave working frequencies are 4.06 GHz and 4.08 GHz. In addition, since large gap width G reduces the metal absorption loss of the optical waveguide and the whole AOM device [1][2][3][4][5], large gaps are preferred in on-chip AOM under the same fabrication processes and requirements. Photonics 2022, 9, x FOR PEER REVIEW 7 of 14 first and then decreases. The maximum value locates at the width of w = 2.1 μm for both of the gap width considered. The optimal waveguide width implies that the AOI effect reaches the maximum at w = 2.1 μm, a width that has also been used in the aforementioned gap width analyses. We etch two slots on both sides of the AOM to form an acoustic resonator. We use the acoustic resonance to enhance the SAW energy [25,[33][34][35]. The resonance acoustic mode will interact further with the higher-order TE1 mode. Thus, the acoustic resonator width should also be analyzed. Figure 5 shows the refractive index change of the optical waveguide Δ as a function of the gap width G between the optical waveguide and IDT for different widths Wd of the acoustic resonator. The multimode waveguide width is still fixed to w = 2.1 μm and we choose Wd = 8, 9, 10, 11, and 13 μm. From the results, we can always find the maximum value of Δ for different widths of the acoustic resonator by tuning the gap width G, and the obtained maximum values of |Δ | are all higher than 6 × 10 for a single phonon. The observations indicate that the field distribution of the acoustic resonator mode is closely related to the acoustic resonator width, and we need to reoptimize the position of the multimode waveguide on the TFLN for different widths Wd in order to obtain the highest AOI, which corresponds to the maximum value of |Δ |. Moreover, through additional calculations using different acoustic resonator widths, we find that the maximum value of |Δ | can still be increased slightly if the width Wd is further decreased. For example, the maximum value of Apart from gap width G, the multimode waveguide width w is also an important parameter affecting the on-chip AOI effect. Here, we consider two cases: G = 1.7 µm and G = 3.5 µm, and choose the maximum refractive index change of the optical waveguide |∆n tot | from the calculations at different multimode waveguide widths as shown in Figure 4. Within the calculation range from w = 1.8 to 2.2 µm, the value of |∆n tot | increases first and then decreases. The maximum value locates at the width of w = 2.1 µm for both of the gap width considered. The optimal waveguide width implies that the AOI effect reaches the maximum at w = 2.1 µm, a width that has also been used in the aforementioned gap width analyses. We etch two slots on both sides of the AOM to form an acoustic resonator. We use the acoustic resonance to enhance the SAW energy [25,[33][34][35]. The resonance acoustic mode will interact further with the higher-order TE 1 mode. Thus, the acoustic resonator width should also be analyzed. Figure 5 shows the refractive index change of the optical waveguide ∆n tot as a function of the gap width G between the optical waveguide and IDT for different widths W d of the acoustic resonator. The multimode waveguide width is still fixed to w = 2.1 µm and we choose W d = 8, 9, 10, 11, and 13 µm. From the results, we can always find the maximum value of ∆n tot for different widths of the acoustic resonator by tuning the gap width G, and the obtained maximum values of |∆n tot | are all higher than 6 × 10 −10 for a single phonon. The observations indicate that the field distribution of the acoustic resonator mode is closely related to the acoustic resonator width, and we need to reoptimize the position of the multimode waveguide on the TFLN for different widths W d in order to obtain the highest AOI, which corresponds to the maximum value of |∆n tot |. Moreover, through additional calculations using different acoustic resonator widths, we find that the maximum value of |∆n tot | can still be increased slightly if the width W d is further decreased. For example, the maximum value of |∆n tot | can be even Photonics 2022, 9, 12 8 of 14 higher than 7 × 10 −10 for a single phonon if W d is less than 8 µm. However, such a small acoustic resonator width will lead to a decrease in the gap width between the optical waveguide and IDT, which will then increase the metal absorption loss of the AOM. From the above calculations, we determine the key cross-sectional dimension of the proposed AOM including the gap width G, multimode waveguide width w, and acoustic resonator width W d , where the AOM length along the y-direction is set as 100 µm since the BOX layer below the AOM region needs to be removed [33,34]. In summary, the structural parameters of the proposed AOM can be obtained with higher AOI effect working at the higher-order TE 1 mode when compared with that working at the fundamental TE 0 mode. Photonics 2022, 9, x FOR PEER REVIEW 8 of 14 |Δ | can be even higher than 7 × 10 for a single phonon if Wd is less than 8 μm. However, such a small acoustic resonator width will lead to a decrease in the gap width between the optical waveguide and IDT, which will then increase the metal absorption loss of the AOM. From the above calculations, we determine the key cross-sectional dimension of the proposed AOM including the gap width G, multimode waveguide width w, and acoustic resonator width Wd, where the AOM length along the y-direction is set as 100 μm since the BOX layer below the AOM region needs to be removed [33,34]. In summary, the structural parameters of the proposed AOM can be obtained with higher AOI effect working at the higher-order TE1 mode when compared with that working at the fundamental TE0 mode.  Another key component of the proposed scheme is the mode converter, which realizes the conversion between the TE 0 and TE 1 mode efficiently. The mode converter is implemented by the asymmetrical directional coupler (ADC) shown in Figure 1c. From the ADC theory, the required mode conversion can be achieved via evanescent coupling under the phase matching condition [41,42]. To achieve phase matching, we calculate the mode features of the TFLN optical waveguide and make the effective index of the TE 0 mode of the single mode waveguide match with that of the TE 1 mode of the multimode waveguide. The optimized results are the single mode waveguide width w 1 = 870 nm, multimode waveguide width w = 2.1 µm, and gap width between the two waveguides d = 330 nm. The working wavelength is λ = 1.55 µm. Then we conduct transmission analyses to determine the optimum coupling length L C [43], which can be found in Figure 6. It is noted that the highest mode transmission from the input TE 0 to output TE 1 mode is nearly 0.98 (normalized to the input power) corresponding to the coupling length L C = 52 µm as shown in Figure 6a. We note that the mode transmission is still higher than 0.95 even if L C changes from 46 to 61 µm, thus relaxing the fabrication tolerance of the TFLN waveguide. The transmission wavelength spectrum of the mode converter is also analyzed. From Figure 6b, the bandwidth is 123 nm for transmission higher than 0.95 (202 nm for transmission higher than 0.90). The large bandwidth can be utilized for broadband AOMs. Figure 6c shows the field evolution through the whole conversion region from the input TE 0 mode to the converted TE 1 mode and then to the output TE 0 mode, where the generated TE 1 mode is used for the AOM. The mode patterns of the input TE 0 mode and the converted TE 1 mode are also shown in the insets. We find that the converted TE 1 mode has a high mode purity and low mode crosstalk, contributing to the efficient operation of the AOM working at the higher-order TE 1 mode. In summary, the key structures of the AOM and the mode converter have been obtained. The rest are typical MZI structure shown in Figure 1a, the waveguide width of which is 870 nm. Another key component of the proposed scheme is the mode converter, which realizes the conversion between the TE0 and TE1 mode efficiently. The mode converter is implemented by the asymmetrical directional coupler (ADC) shown in Figure 1c. From the ADC theory, the required mode conversion can be achieved via evanescent coupling under the phase matching condition [41,42]. To achieve phase matching, we calculate the mode to the converted TE1 mode and then to the output TE0 mode, where the generated TE1 mode is used for the AOM. The mode patterns of the input TE0 mode and the converted TE1 mode are also shown in the insets. We find that the converted TE1 mode has a high mode purity and low mode crosstalk, contributing to the efficient operation of the AOM working at the higher-order TE1 mode. In summary, the key structures of the AOM and the mode converter have been obtained. The rest are typical MZI structure shown in Figure 1a, the waveguide width of which is 870 nm.

Discussion
The device fabrication process can be divided into three steps; they are (1) define the TFLN waveguide structure, (2) fabricate the IDT electrode on the TFLN, and (3) open the acoustic resonator slot and remove the BOX layer below the AOM region. We can start from the X-cut TFLN wafer with a 600 nm-thick top LN layer. First, the designed optical waveguide structure of the AOM is patterned and etched on the TFLN wafer using E-beam lithography (EBL) and reactive ion etching (RIE) [2][3][4][5]25], where the waveguide widths (w = 2.1 µm, w 1 = 870 nm), gap (d = 330 nm), and etching depth (h 1 = 300 nm) should be well controlled. Second, we use typical metal lift-off process to fabricate the IDT electrode [2][3][4][5]25], where the electrode width (a = 300 nm), gap (b = 300 nm), and thickness (100 nm) need to be satisfied. Third, two rectangular slots on both sides of the AOM are etched on the slab layer with etching depth h 2 = 300 nm to form the acoustic resonator. Finally, we employ BOX etchant contact with the BOX layer and then remove it through two completely etched slots of the TFLN layer [33,34]. For these fabrication steps, we calculated the refractive index change of the optical waveguide ∆n tot as functions of the IDT electrode thickness and waveguide etching depth h 1 . We note that no obvious variation of ∆n tot is observed as the IDT electrode thickness changes from 80 to 120 nm. In contrast, the waveguide etching depth variation has a stronger effect on ∆n tot since it is directly related to the mode features of the generated SAW and the optical waveguide, leading to variation of the AOI effect. Therefore, the waveguide etching depth (h 1 = 300 nm) should be carefully controlled during device fabrication process.
Finally, we calculate the key modulation parameter V π L for the proposed device due to refractive index change of the optical waveguide where Equation (7) is used to estimate the value. The AOM length along the y-direction is chosen to be 100 µm and the beam length of the MZI structure is 500 µm. The typical Q factor of the acoustic resonator Q m = 2000, γ = ω m /Q m and γ e /γ = 0.15 (from experimental results) are used in the following calculations [26]. Table 1 lists the calculation results at the TE 0 and TE 1 modes. The value of ∆n tot is chosen as the maximum one generated by the specific acoustic mode frequency chosen from various acoustic mode frequencies for every device structure mentioned above. We find that the use of the higher-order TE 1 mode reduces the key modulation parameter V π L for the proposed AOM. The minimum value V π L is 0.0084 V·cm, which is obviously lower than previous reports [25][26][27][28][29]. Figure 7 shows the refractive index change ∆n tot of the waveguide versus the slant angle of the TFLN waveguide sidewall. Other waveguide parameters are the same as that in Figure 3b, where G = 3.5 µm. Note that the allowable slant angle can be varied from 64 • to 70 • , while keeping |∆n tot | higer than 6.5 × 10 −10 for a single phonon. Such relatively large tolerance is useful in the etching process of the TFLN waveguide. Thus, the AOI effect is enhanced by using the higher-order TE 1 mode, leading to the performance improvement of on-chip AOM.

Conclusions
In conclusion, we have proposed an on-chip AOM based on the

Conclusions
In conclusion, we have proposed an on-chip AOM based on the typical TFLN platform, in which the optical waveguide is a multimode waveguide supporting the higher-order TE 1 mode instead of the commonly used fundamental TE 0 mode. To analyze the multiple physical field features and determine the optimum structural parameters of the proposed AOM, multi-physical field coupling analyses were carried out by considering the photoelastic effect, moving boundary effect, and electro-optic effect. We observed that the change of the refractive index of the optical waveguide increases (>6.5 × 10 −10 for a single phonon) because of the enhanced AOI effect. We then added the proposed AOM structure to an MZI circuit to form an MZI-type AOM, where the mode converters between the TE 0 and TE 1 mode have been designed and embedded at both sides of the AOM. From the results, the key half-wave-voltage-length product V π L is calculated to be less than 0.01 V·cm, which can be used for low-power modulation. In addition, the metal absorption loss is quite low for the proposed AOM since the electrode can be placed far from the optical waveguide when compared to typical EOM structures in which the electrode must be placed close to the optical waveguide. The proposed device would contribute to the development of on-chip high-performance multimode AOM.