One-way acoustic beam splitting in spatial four-waveguide couplers designed by adiabatic passage

In this work, we introduce quantum-mechanical adiabatic passage into the design of spatial acoustic four-waveguide (WG) couplers. Thanks to the agreement in form between the Schrödinger equation in quantum mechanics and the coupled-mode equation of classical wave, the behavior of propagating wave in coupled WGs is capable of mapping to quantum states driven by external fields. By coupling the input and output WGs with a mediator WG in space, an apparent beam splitting is realized and the ratio of intensity can be customized arbitrarily by altering the space-dependent coupling strengths. Moreover, a one-way propagation feature is exhibited in the spatial coupler when an appropriate loss is introduced in the mediator WG owing to the existence of dark state. This work builds a bridge between quantum adiabatic technology and acoustic beam splitter, which may have potential applications in acoustic communication, filtering and detection.


Introduction
As a technique to achieve efficient and selective population transfer between quantum states, adiabatic passage (AP) has been widely concerned for its valuable applications in quantum optics [1][2][3], solid-state physics [4,5] and physical chemistry [6]. By performing quantum-classical analogs, the technique of AP has been applied to classical physics as well, showing surprising merits for designing optical devices with attractive functions, such as, to name a few, optical polarization control [7], frequency conversion [8], and energy transfer [9][10][11][12][13][14][15][16]. Despite numerous achievements in optics, the applications of AP in acoustics have not been widely reported so far [17][18][19][20], especially its ability to obtain arbitrary beam splitting in waveguide (WG) couplers.
Acoustic beam splitter is a device that can split incident wave into multiple parts with the same or different intensities, which has potential applications in acoustic communication, sensing and detection. In past several years, phononic crystals [21] and acoustic metasurfaces [22][23][24] were demonstrated to be good candidates to achieve beam splitting by taking advantages of line defect and generalized Snell's law [25], respectively. Both of these two schemes, however, are suffering from the limitation of bandwidth or robustness. For phononic crystals, the dependency on bandgap property needs a perfect match between the structure size and incident wavelength, which inevitably results in a narrow band of the device. For acoustic metasurface, resonance cavities are always demanded to meet the requirement of phase shift [26,27], making the device can only work near the resonant frequency and has a poor robustness. Recently, the combination of acoustic system with parity-time symmetry [28] has exhibited a fascinating ability for constructing beam splitter with single-sided response owing to its extremely asymmetric scattering feature. Nevertheless, it is quite difficult to realize gain/loss modulations in a passive acoustic system, hindering the application of beam splitter designed by parity-time symmetry. Given the excellent performance of optical and terahertz WG couplers designed by adiabatic technology [29][30][31][32][33][34][35], building a bridge between AP and acoustic WG systems is expected to be an alternative to achieve efficient acoustic beam splitting with robust and broadband features.
In this work, by performing an analog of the quantum-mechanical AP in an acoustic system, we present an approach to achieve acoustic beam splitting in spatial four-WG couplers. The behaviors of beam splitting with arbitrary ratio as well as complete energy transfer are capable of realizing in the same coupler by selecting different input ports. Owing to the existence of dark state, we further show that a one-way propagation feature can be exhibited in the beam splitter when an appropriate loss is introduced in the mediator WG. The agreements between analytical calculations and numerical simulations verify the feasibility of our design, extending the application of AP in acoustics and providing a direct visualization in space of typical ultrafast phenomena in time.

Acoustic four-WG coupler designed by AP
We start by considering the model of a four-level quantum system, as shown in figure 1(a), quantum states |1⟩ , |3⟩ , and |4⟩ are coupled to |2⟩ by time-varying external fields Ω P (t), Ω S1 (t), and Ω S2 (t), respectively, which can be described by a Schrö dinger equation [36] i d dt where being the probability amplitude of the state |k⟩ (k = 1, 2, 3, 4). Under the rotating-wave approximation [36], the Hamiltonian H (t) can be written as: When the time-varying fields Ω S1 (t) = Ω S2 (t), the four eigenvalues of H (t) can be calculated as , and the corresponding adiabatic states are where the mixing angle θ (t) = arctan √ 2Ω S1 (t) /Ω P (t) . The state function ψ (t) is able to be mapped into the eigenvector space by [36] In the adiabatic limit [6] (see appendix A), with being connected to the initial state |ψ (t 0 )⟩. If we set a 1 (t 0 ) = 1 and a 2 (t 0 ) = a + (t 0 ) = a − (t 0 ) = 0, the state function will follow Thus, the results of |ψ (t 0 ) = |1⟩ and ψ t f = − 1 √ 2 (|3⟩ + |4⟩ ) can be achieved when the mixing angle θ (t) changes from π /2 to 0, where t 0 and t f represent the initial time and final time, respectively. Note that the state |2⟩ does not get involved in the evolution process from |1⟩ to |3⟩ and |4⟩ , indicating the existence of a dark state [6]. Although both the adiabatic states described in equations (3a) and (3b) are dark states, the adiabatic evolution here is driven by state |B 1 (t) because it involves |1⟩ , |3⟩ and |4⟩ , which is important for the beam splitting behavior from WG 1 to WGs 3 and 4 in the designed acoustic four-WG system shown in On the other hand, if we set a 1 (t 0 ) = a 2 (t 0 ) = 0 and a + (t 0 ) = a − (t 0 ) = 1 √ 2 , the evolution of the state function will along the superposition of |B + (t) and |B − (t) . In the adiabatic limit [6], the amplitudes are By setting the mixing angle θ (t) changes from π /2 to 0, and then making the phase β (t) vary from 0 to (2k + 1) π /2 (k is an integer), we can obtain the results of |ψ (t 0 ) = 1 √ 2 (|3⟩ + |4⟩ ) and ψ t f = −i |2⟩ . Compare the evolution processes achieved by equations (5) and (6), it can be concluded that population transfer is realized from |1⟩ to |3⟩ and |4⟩ , while from |3⟩ and |4⟩ to |2⟩ under the same condition of mixing angle θ (t), showing an asymmetric transfer behavior in this four-level quantum system. Actually, the final population ratio of states |3⟩ and |4⟩ can be customized arbitrarily by utilizing different time-varying fields Ω P (t), Ω S1 (t), and Ω S2 (t). Taking final population ratio of 2:1 for states |3⟩ and |4⟩ as an example, the time-varying fields need to satisfy the requirement of Ω S1 (t) = √ 2Ω S2 (t). In this case, the four eigenvalues of H (t) can be calculated as ε 1 (t) = ε 2 (t) = 0, ε + (t) = ε − (t) = ± Ω 2 P (t) + 3 2 Ω 2 S1 (t), and the corresponding adiabatic states are where the mixing angle θ 1 (t) = arctan √ 3ΩS1(t) √ 2ΩP(t) . Following the equations (4) and (7), for a given condition of a 1 (t 0 ) = 1 and a 2 (t 0 ) = a + (t 0 ) = a − (t 0 ) = 0, the state function will follow Consequently, by setting the mixing angle θ 1 (t) changes from π /2 to 0, the results of |ψ (t 0 ) = |1⟩ and can be achieved, which indicates a final population ratio of 2:1 for states |3⟩ and |4⟩ . Note that other population ratios of states |3⟩ and |4⟩ can be achieved as well by altering the ratios of the corresponding peak Rabi frequencies.
By mapping the time-dependence into space-dependence [9,12], as shown in figure 1(b), our analogy in an acoustic system is proposed by a spatial four-WG coupler. Here, WGs 1, 2, 3 and 4 correspond to the quantum states |1⟩ , |2⟩ , |3⟩ and |4⟩ , respectively, and WG 2 serves as an intermedium WG spatially coupled with WGs 1, 3 and 4. The coupling strengths between WGs 2 and i (i = 1, 3, 4) are C P (x), C S1 (x) and C S2 (x), which play the roles of Rabi frequencies Ω P (t), Ω S1 (t) and Ω S2 (t), respectively. In optical WG system, the coupling strength can be modulated by changing the distance between the WGs [12,16]. Optical Isolators are needed to avoid the unwanted coupling actions between the WGs [16]. In acoustic WG system we proposed in this work, the coupling action can only occur when two WGs are connected by air slits. As illustrated in figure 1(b), the WG 2 is coupled with WGs 1, 3 and 4 by air slits, while no air slits are adopted to connect WG 1 with WGs 3 and 4. Therefore, the direct coupling actions only exist between WGs 1 and 2, WGs 3 and 2, and WGs 4 and 2. Since acoustic velocity c and the propagation distance x has a relation of x = ct, the couplings between WGs can be considered as space dependent behaviors. Due to the agreement in form between Schrö dinger equation in quantum mechanics and the coupled-mode equation of classical waves, the evolution of the wave amplitudes in a four-WG system can be described as [36] where being the amplitude of the acoustic pressure in WG k (k = 1, 2, 3, 4), and the corresponding acoustic intensity is I k (x) = |A k (x)| 2 . By solving the coupled mode equation (equation (9)) with the coupling matrix being The acoustic pressure distribution along each WG can be obtained. Before constructing the four-WG coupler, the coupling strength between two WGs needs to be modulated flexibly to achieve space-dependent coupling actions. As shown in figure 2(a), we propose a two-WG acoustic coupler, where WGs A and B are coupled by period slits with cycle length being p = 7 mm. Since the slit width (d) along the propagation direction is fixed, the coupling strength can be viewed as a constant C, and the coupling matrix has the form of M (x) = 0 C C 0 . Supposing the pressure waveforms in WGs A and B take the forms of A (x) = a 1 e −iqx + a 2 e iqx and B (x) = b 1 e −iqx + b 2 e iqx , the following relations are deduced by solving the coupled mode equation: From equation (11), we can obtain q = C, a 1 = b 1 and a 2 = b 2 . Combined with initial conditions of A (0) = a 1 + a 2 and B (0) = b 1 + b 2 , the pressure waveforms in WGs A and B are further calculated as . If acoustic waves are injected from left port of WG A, the initial conditions can be set as A (0) = A 0 and B (0) = 0, and the acoustic power flows along the propagation direction of WGs A and B are From equation (12), it can be seen that the acoustic waves will funnel back and forth in two composing WGs, resulting in a periodic oscillation of intensity distribution. The distance between two adjacent peaks in the intensity field is defined as a coupling length L, which is inversely proportional to the coupling strength (L = π /C). As an effective parameter to modulate the coupling strength, as shown in figure 2(b), the coupling length will decrease when the slit width d increases at the same working frequency. By scanning the value of d from 0.2 mm to 6.4 mm, we obtain a fitted function L = a/d with a = 518 mm 2 to connect the slit width with coupling length. Actually, the minimum precision of the slit width can hardly be smaller than 0.4 mm for the machine tool [18]. Therefore, the minimum and maximum of the coupling strength can be calculated by C x = π d/a with d min = 0.4 mm and d max = 7 mm − 0.4 mm = 6.6 mm, and the available coupling strength (C x ) is 2.42 m −1 <C x < 40 m −1 in the proposed acoustic system. In addition, it can be seen from figure 2(b) that for a fixed slit width, the coupling length has little change at three different operating frequencies of 8500 Hz, 9500 Hz and 10 500 Hz, providing a broadband response for the modulation of coupling strength.
Next, we intend to construct a one-way spatial four-WG coupler by adopting well-tailored coupling slits, through which a desirable beam splitting from WGs 3 and 4 can be achieved for WG 1 incidence, while no apparent intensity from WG 1 can be obtained for WGs 3 and 4 simultaneous incidence. Analogous to the quantum state transfer process illustrated in equation (5), the mixing angle θ (x) = arctan √ 2C S1 (x) /C P (x) in acoustic system needs to change from π /2 to 0 smoothly so that the acoustic energy can be transferred equally from WG 1 to WGs 3 and 4, which manifests that the space-dependent coupling action C S1 (x) requires to precede C P (x) at the starting position x 0 and vanish earlier than C P (x) at the ending position x f . Therefore, we adopt the Gaussian-shaped coupling pulses [34,35] CP(x) = 0, where the parameters σ = 127, x 0 = 430 mm, s = 55 mm, and C max = 31.836 m −1 . The influence of maximum coupling strength C max on the performance of acoustic coupler is illustrated in appendix B. According to the fitted function L = a/d, as shown in figure 2(c), the corresponding space-dependent slit widths can be further obtained by d P (x) = aC P (x) /π and d S1 (x) = d S2 (x) = d S (x) = aC S (x) /π . To reduce the thermal viscosity loss caused by micro-slit, the smallest slit width is obliged to be larger than 0.4 mm [17], which means that both ends (d < 0.4 mm) of the slit width pulses shown in figure 2(c) are ignored for the construction of the four-WG acoustic coupler.

Performance of the acoustic four-WG coupler
The performance of the designed acoustic beam splitter is numerically simulated by COMSOL Multiphysics. Owing to the huge impedance difference between rigid materials and air background, the rigid materials can be set as hard boundary conditions in the numerical simulations. In practical, the rigid materials can be selected as metal materials or resin materials. The ports of the four WGs are selected as planar wave radiation conditions, and the acoustic pressure value of the incident plane wave is set as 1 Pa in the numerical simulations. The operating frequency is set as 8800 Hz, and the mass density and sound velocity of the air are ρ air = 1.21 kg m 3 and c air = 343 m s −1 . Given the simulated acoustic intensity fields shown in figures 3(a) and (b), it can be observed that the acoustic waves output from WGs 3 and 4 have the same intensity for WG 1 incidence, and a complete energy transfer to WG 2 is realized for WGs 3 and 4 simultaneous incidence. The simulated acoustic intensity fields of the coupler at other operating frequencies are illustrated in appendix C, showing a broadband response for beam splitting and energy transfer. The red, green, blue and cyan points in figures 3(e) and (f) illustrate the simulated acoustic intensity distributions along WGs 1, 2, 3 and 4, respectively, which agree well with the analytic results solved by Schrö dinger-like equation (equation (9); solid lines). The asymmetric energy transfer behavior in a four-WG acoustic coupler can be interpreted by two different evolution processes. For acoustic beam splitting from WG 1 to WGs 3 and 4, similar to equation (5), it follows For complete acoustic energy transfer from WGs 3 and 4 to WG 2, similar to equation (6), it follows Compared with equations (13) and (14), it can be seen that the WG 2 does not get involved in the evolution of beam splitting, while it undergoes obvious oscillation in the evolution of complete energy transfer, which can be confirmed by the green lines or points shown in figures 3(e) and (f). Since the acoustic intensity distribution along WG 2 for WG 1 incidence is quite lower than that for WGs 3 and 4 simultaneous incidence, the acoustic energy attenuation for WG 1 incidence will slower than that for WGs 3 and 4 incidence when a certain wave dissipation is introduced in WG 2. In this case, the coupling matrix (equation (10)) becomes a non-Hermitian form where γ = 0.13C max is the constant damping rate. The acoustic intensity fields of lossy coupler are shown in figures 3(c) and (d) for WG 1 incidence and WGs 3 and 4 simultaneous incidence, respectively, from which it can be observed that an equal-weighted beam splitting can be generated for WG 1 incidence, while the energy transfer from WGs 3 and 4 to WG 2 is hardly achieved under this circumstance. The excellent agreement between simulated and analytical intensity distributions along four WGs shown in figures 3(g) and (h) further verifies that a one-way beam splitting is realized in the proposed lossy acoustic coupler. When acoustic waves are incident from WG 2, as shown in figure 4(a), a complete energy transfer to WG 1 is obtained. The simulated acoustic intensity distributions along WGs 1, 2, 3 and 4 shown in figure 4(b) are agree well with the analytic results solved by Schrö dinger-like equation (equation (9); solid lines). Due to the reversibility of acoustic propagation path, the evolution of the propagating wave can also be regarded as the transfer from WG 1 to WG 2 when the acoustic waves are incident from the right port of WG 1. Therefore, the acoustic energy could transfer from WG 1 to WGs 3 and 4 for the left port of WG 1 incidence (shown in figure 3(a)), while that could transfer from WG 1 to WG 2 for the right port of WG 1 incidence (shown in figure 4(a)). The asymmetric behavior of energy transfer for WG 1 incidence can be explained by two different evolution processes. For the left port of WG 1 incidence, the evolution follows equation (13), and the mixing angle θ (x) is set to change from π /2 to 0 smoothly, thus leading to a result of I 1 (x 0 ) → 1 2 I 3 x f + 1 2 I 4 x f . For the right port of WG 1 incidence, the evolution follows equation (14), and the mixing angle changes from 0 to π /2 in this case, exhibiting another result of I 1 (x 0 ) → I 2 x f . When a certain wave dissipation (γ = 0.13C max ) is introduced in WG 2, as shown in figure 4(c), the acoustic energy is attenuated completely for WG 2 incidence. The simulated acoustic intensity distributions along WGs 1, 2, 3 and 4 shown in figure 4(d) are agree well with the analytic results solved by Schrö dinger-like equation (equation (9); solid lines). Therefore, combined with the results illustrated in figures 3(g) and (h), it can be concluded that for acoustic waves propagating along the positive direction of the x-axis, only WG 1 incidence can achieve a desirable energy localization at the output ports. When acoustic waves incident from WGs 2, 3 or 4, no apparent energy transfer can be obtained at the output ports owing to the wave dissipation in WG 2.
Actually, the ratio of the beam splitting can be customized arbitrarily by utilizing corresponding space-dependent coupling actions C P (x), C S1 (x), and C S2 (x). For instance, the equation (8) in spatial dimension can be expressed as where 2CP(x) and C S1 (x) = √ 2C S2 (x). We adopt the Gaussian-shaped coupling pulses of )+s] 2 /σ 2 to determine the space-varying slit widths. As shown in figure 5(a), the acoustic waves are output from WGs 3 and 4 together for WG 1 incidence. From the simulated acoustic intensity distributions along four WGs shown in figure 5(b), it can be seen that the normalized intensities of output wave in WGs 3 and 4 are around 0.66 and 0.33, respectively, which agrees well with the analytic results calculated by equation (9) and confirms a high conversion efficiency of beam splitting. In addition to the ratio of 2:1, figures 5(c) and (d) also illustrate a 3:1 beam splitting behavior by utilizing the Gaussian-shaped coupling pulses of C P (x) = C max e −[(x−x0)−s] 2 /σ 2 and C S1 (x) = √ 3C S2 (x) = C S (x) = C max e −[(x−x0)+s] 2 /σ 2 , fully confirming that beam splitting with arbitrary ratio can be customized by altering the peak value of the coupling strengths C S1 (x) and C S2 (x).

Discussion
Although the proposed acoustic system provides an effective solution for the design of one-way acoustic beam splitter, there are two limitations in the current scheme. Firstly, the precision of discrete coupling slits should be considered while mapping the time-dependent driving pulses into spatial dimension. The period length of the slit needs to be as short as possible to guarantee enough discrete points in a fixed device length. On the other hand, the slit width and the period length should satisfy the relationships of d ⩾ 0.4 mm and (p − d) ⩾ 0.4 mm owing to the limitation from fabrication craft [18]. Consequently, the acoustic coupler designed in our work is a compromise consideration. Secondly, the available design philosophies in acoustics are relatively less than those in optics because the modulation of space-dependent acoustic detuning has not been realized so far, hindering the application of AP and its variants for the design of various acoustic functional devices. Therefore, the modulation of detuning parameter in acoustic systems needs to be investigated in the future.
Note that there are numerous achievements in the design of optical or terahertz adiabatic two-WG couplers [11,[37][38][39]. Compared with adiabatic two-WG couplers, the four-WG coupler proposed in our work has an advantage of one-way propagation feature owing to the utilization of dark state. In addition, compared with acoustic adiabatic three-WG couplers in previous works [17,18], the four-WG coupler proposed in our work has an advantage of multifunctional responses. By altering the input ports, the function of the device can be switched from beam splitting to complete energy transfer conveniently, improving the utilization of the device.
For optical WG couplers, the technique of AP can usually be extended to a shortcut version [14,39]. Actually, the acoustic four-WG couplers can be designed by shortcut to AP (STAP) as well. The physical mechanism of the acoustic four-WG coupler designed in this work can be interpreted by the so-called Morris-Shore (MS) transformation [16,40]. The MS basis contains the mediator WG 2, two dark superpositions of output WGs d 3 and d 4 , and a bright superposition b of output WGs, which is coupled to the mediator 2 with a coupling strength C S (x) = C 2 S1 (x) + C 2 S2 (x). The dark superpositions are of no significance in the present context, and the bright superposition has the form of b (x) = [0, 0, C S1 (x) , C S2 (x)] T /C s (x). Therefore, the output-WG components are proportional to the coupling strengths C S1 (x) and C S2 (x). For the special case of C S1 (x) = C S2 (x), b (x) will be an equally weighted superposition of output WGs 3 and 4. In fact, the three-WG ladder 1 → 2 → b in the MS basis is the subspace in which the beam splitting takes place via STIRAP-like adiabatic transfer from 1 to b. Consequently, we can design the four-WG acoustic beam splitter by performing an analog of the quantum three-level system. In our previous work [19], we have demonstrated that the STAP technique can be adopted to achieve complete energy transfer from input WG to the target WG efficiently in a three-WG acoustic system. The approach in our previous work [19] is applicable to the design of compact acoustic four-WG couplers as well. To shorten the device length, a dressed state |Ψ ] is adopted to realize the desired energy transfer instead of the AP pathway in a three-WG acoustic system. Two modified space-varying coupling pulses C ′ P (x) and C ′ S (x) are needed by adding two extra coupling actions into C P (x) and C S (x), so that the transitions from |Ψ s (t) to other orthogonal dressed states can be eliminated. By using the STAP method of dressed states [19,41], the modified coupling strengths can be obtained as C ′ . Therefore, the acoustic four-WG beam splitter based on STAP can be designed by adopting C P (x) = C ′ P (x) and C S1 (x) = C S2 (x) = C ′ S (x).

Conclusion
An approach is exhibited in this work for acoustic beam splitting by connecting AP with the design of a spatial four-waveguide (WG) acoustic coupler. The transmission wave inside the coupler can mimic the evolution of AP for a quantum four-level system by adopting the Gaussian-shaped pulses to determine the space-varying coupling slits between composing WGs. The behaviors of beam splitting with arbitrary ratio as well as complete energy transfer are capable of realizing in the same coupler by selecting different input ports. Moreover, a one-way propagation feature is exhibited in a spatial coupler when an appropriate loss is introduced in the mediator WG owing to the existence of dark state. Our work builds a bridge between quantum adiabatic technology and acoustic WG coupler, which may have profound impacts on designing advanced acoustic functional devices and provide a direct visualization in space of typical ultrafast phenomena in time.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).

Date availability statement
All data that support the findings of this study are included within the article.

Appendix A. The criterion of the adiabatic limit
The criterion of the adiabatic limit [6] can be obtained from the following derivations with the new basis vectors {|B Through the unitary transformation, the transient state function in the new basis vectors obeys the transformed Schrö dinger-type equation where we introduce a unitary transformation matrix Therefore, the systematic Hamiltonian on the new basis vectors can be derived as Following the equation (A3), it can be found that the off-diagonal terms of Hamiltonian H a (t) can be ignored for a given condition Owing to the existence of derivative in equation (A4), the adiabatic condition requires the smoothness of the time-varying fields Ω P (t), Ω S1 (t), and Ω S2 (t).

Appendix B. Influence of maximum coupling strength on the performance of acoustic coupler
To explore the influence of maximum coupling strength (C max ) on the performance of acoustic coupler, as shown in figure 6, the relations between the output intensities of four WGs and C max are plotted for different incident ports.
For WG 1 incidence, as shown in figure 6(a), a beam splitting from WGs 3 and 4 can be achieved when C max > 15 m −1 . As illustrated in equation (13), the acoustic coupler follows the evolution of a dark state for WG 1 incidence, making it is robust to the change of C max in the adiabatic limit [ 1 For WG 2 incidence and WGs 3 and 4 simultaneous incidence, as shown in figures 6(b) and (c), the output ports of the acoustic coupler are closely related to the C max , which is attributed to the existence of phase ε + (x ′ ) dx ′ in the evolution process illustrated in equation (14). Therefore, the output ports of the coupler can be customized flexibly by selecting an appropriate value of C max . The C max in our design is selected as C max = 31.836 m −1 , according to the figures 6(a)-(c), the beam splitting from WGs 3 and 4 can be achieved for WG 1 incidence, complete energy transfer to WG 1 can be obtained for WG 2 incidence and complete energy transfer to WG 2 can be realized for WGs 3 and 4 simultaneous incidence, indicating a switchable energy transfer behavior by selecting different incident ports.

Appendix C. The performance of acoustic four-WG coupler at different incident frequencies
In fact, the working band of the device for cases of incidences from distinct inputs are different because the acoustic coupler follows distinct evolution paths in these cases. For WG 1 incidence, it follows a dark state described in equation (13). For WG 2 incidence or WGs 3 and 4 simultaneous incidence, it follows a superposition state described in equation (14). Compared with figures 6(a)-(c), it can be found that the performance of the coupler for WG 1 incidence is robust to the change of coupling strength, while that for WG 2 incidence or WGs 3 and 4 simultaneous incidence is sensitive to the change of coupling strength. Therefore, the working band of the device for WG 1 incidence is broader than that for WG 2 incidence or WGs 3 and 4 simultaneous incidence because the alteration of the working frequency leads to the change of coupling strength. Figure 7 illustrates the performance of acoustic four-WG coupler at different incident frequencies. The acoustic waves are propagating along the positive direction of the x-axis. For WG 1 incidence, acoustic beam splitting from WGs 3 and 4 can be achieved in a broadband of 7500 Hz-13 500 Hz ( figure 7(a)). For WG 2 incidence, complete energy transfer from WG 2 to WG 1 can be obtained in a broadband of 8500 Hz-10 500 Hz ( figure 7(b)). For WGs 3 and 4 simultaneous incidence, complete energy transfer to WG 2 can be realized in a broadband of 8500 Hz-10 500 Hz (figure 7(c)). Thus, for a one-way acoustic beam splitter, the working band is 8500 Hz-10 500 Hz because the two evolution paths described in equations (13) and (14) need to be considered simultaneously. Appendix D. The performance of acoustic four-WG coupler with different device lengths for WG 1 incidence It can be found that the efficiency of beam splitting in the case of l 0 = 0.35 m is lower than that in the case of l 0 = 1.12 m because the adiabatic condition cannot be well satisfied in short devices. For the case of l 0 = 1.12 m, as shown in figure 8(c), the desired beam splitting can be generated in a broadband of 7500 Hz-13 500 Hz, which is the same as the performance in the case of l 0 = 0.857 m illustrated in figure 7(a). Therefore, the working band of the beam splitter for WG 1 incidence is 7500 Hz-13 500 Hz when the device length is longer than 0.75 m. The parameter of l 0 = 0.857 m adopted in our work is a suitable choice, because the bandwidth cannot be further expanded significantly if we continue to increase the device length after the adiabatic condition is well satisfied, and the longer device needs higher cost.