Characteristics and mechanism of coupling effects in parallel-cladded acoustic waveguides

The characteristics and mechanism of coupling effects between parallel cladded acoustic waveguides (PCAWs) are essential when considering their applications in acoustic wave control and signal processing. We investigated its characteristics and revealed the nature of the coupling effect using a theoretical model of two-dimensional PCAWs and simulation experiments. We derived the eigenmode equation describing the behavior of a single waveguide based on the wave acoustic theory and derived analytic expressions for the coupling effects in the PCAWs using the coupled mode theory. Using the finite-element method, we analyzed the waveguide coupling exhibited by this structure given different configurational and acoustic parameter settings. Both theoretical and simulated results indicate that the input wave directed into one of four ports of this structure propagates and tunnels alternately between the two waveguides. Our theoretical model established yields analytic relations between the coupling lengths as well as the dependence on parameters of the evanescent wave and the structure. Analyses indicate wave coupling in the two PCAWs is essentially mediated by the evanescent wave. The unique evolution of the acoustic wave in PCAWs can be employed to develop pure acoustic devices such as frequency-selective filters, directional couplers, and acoustic switches.


Introduction
When a wave is inputted into one of four ports of a pair of parallel cladded waveguides, the wave propagates and tunnels alternately between the pair waveguides rather than being confined to propagate along the original incident waveguide. In the fields of optics, this coupling phenomenon is referred to as waveguide coupling [1][2][3][4][5]. Subsequent research focused on establishing an accurate theoretical model for both weak and strong waveguide coupling and revealed that the coupling is mediated by evanescent waves [5][6][7][8][9][10]. Optical waveguide coupling is widely employed in optical wave steering [11][12][13][14], pure optical switches [15,16], directional couplers [17], and frequency-selective filters [18,19]. The theoretical research and exploration of other applications of waveguide coupling in optics have continued unabated [20][21][22].
Similar to the coupling effect in optical waveguides, an analogous acoustic waveguide coupling effect is also widely applicable for acoustic wave control and signal processing. In 1994, coupled wave propagation in two guides through perforations was discussed theoretically by Kergomard and collaborators [23]. Later, in 2016, they analyzed the coupling between two waveguides consisting of periodic lattices of finite length in 2016 [24]. In 2005, Pennec and collaborators studied the resonant tunneling of acoustic waves in phononic crystal waveguides. Acoustic waves of a specific frequency (f) tunnel from one waveguide to another through a resonance cavity [25]. In the same year, Sun and colleagues analyzed the wave coupling between two parallel phononic crystal waveguides and evaluated the coupling length (CL) using numerical experiments [26]; here, CL is defined as the propagation distance necessary for an acoustic wave to transfer from an initial waveguide to an adjacent waveguide. In 2015, Maksimov and collaborators investigated the coupled-mode theory (CMT) of the acoustic resonance cavity [27]. In 2016, an acoustic add-drop filter was fabricated from two line-defect waveguides coupled through a resonator cavity based on a phononic crystal platform [28]. These studies focused mainly on the coupling between the phononic crystal waveguides and the resonance cavity.
Recently, the coupled waveguide network has been employed in the acoustic topological structure [29][30][31]. Shen and collaborators investigated super-mode propagation in a coupled waveguide complex [32] and demonstrated that with properly tailored couplings the sound was completely localized in one targeted intermediate cavity [33]. Molerón and collaborators investigated discrete propagation of trapped modes in airborne acoustic-waveguide arrays and showed the possibility of generating diffraction-free acoustic beams as well as focusing acoustic energy within a single lattice site [34]. However, little has been reported in studies of waveguide coupling in structures of cladded acoustic waveguides (CAWs) [35].
In 2018, we reported waveguide coupling in a CAW structure formed by two parallel ducts and realized acoustic focusing based on this effect in nested waveguides [36]. We found that when an acoustic wave is fed into one port of a waveguide of a pair of parallel ducts with fixed spacing, it propagates and tunnels back and forth between the ducts, with CL proportional to the wave frequency. In the same year, we studied the waveguide coupling in a structure of three parallel ducts and found that CL for two adjacent waveguides solely depends on the wave frequency and the spacing between the two cores but independent of the input mode [37]. Based on wave coupling in parallel waveguides, we can develop more acoustic devices analogs to those in optics. For example, tailoring the length of one waveguide of the structure equal to CL of a particular frequency, a pure acoustic device can be designed to realize frequencyselective filtering or even demodulating. The essential basis for those devices is the theoretical relationships between CL and the parameters such as wave frequency, core spacing, and material properties. The above studies demonstrated the phenomena and applications of waveguide coupling in the CAWs, but little is known of its mechanism and characteristics of the coupling effect from a theoretical point of view.
In this paper, we present a detailed theoretical model of waveguide coupling and reveal the mechanism and the characteristics of the coupling effects in two-dimensional parallel CAWs (PCAWs). First, we derive the analytic solutions of the eigenmode equation for a single CAW and reveal the propagation characteristics of an acoustic wave in both the cladding and core zones. Then, we derive the coupling equations for PCAWs and the parameter dependence of the coupling coefficients on both wave and structure using the CMT. Next, to verify theoretical results, the finite-element simulation is used to demonstrate and analyze waveguide coupling phenomena in the PCAWs for different core spacings and frequencies. The characteristics of evanescent waves in the cladding zone and the dependence of CL on core spacing and frequency are further compared and analyzed with the results. Finally, the characteristics of coupling effects are discussed and, its mechanism is revealed. The theoretical model may serve in applications of waveguide coupling to acoustic wave regulation and signal processing.
2 Theoretical model

Eigenmode equation in Single CAW
The structure of a single CAW is a core with cladding on both sides [35]. A cross-section of the two-dimensional model of the CAW (Fig. 1) was configured for the purpose of simulations having a width w and density q 1 for the core zone, and a cladding density q 0 . For the analysis, c 1 , c 0 , and c a denote respectively the velocity of sound in the core, cladding, and air, respectively. The spatial distribution of the refractive index throughout the CAW may be written as Beginning with the equation governing the acoustic wave in two-dimensional free space, where p denotes the acoustic pressure at position (x, z) at time t; we assume the solution of wave equation in the waveguide as where w(x) is the modal function of the wave in the waveguide, and x = 2pf is the angular frequency. Substituting equation (3) into equation (2), we have where k a denotes the wave number of the wave in air. For a single CAW, the modal function is [4] wðxÞ ¼ On substituting equation (5) into equation (4), we obtain Given that the sound pressure p and the vibration velocity v are continuous at x = 0 and x = w, we obtain the modal eigenvalue equation of the guided mode in a single CAW, tan hw ¼ Setting the normalized amplitude to unity (A = 1), the modal function of nth-order can be written as cos h n x þ q n q 1 h n q 0 sin h n x; ð0 x wÞ cos h n w þ q n q 1 h n q 0 sin h n w e Àq n ðxÀwÞ ; ðw xÞ Equation (9) describes evanescent waves that strictly decay exponentially with penetration depth and q n . The value of q n depends on f and w in accordance with equations (6) and (8). Further analysis indicates that the evanescent wave also decays exponentially with f in a first approximation. Moreover, from equations (6) to (8), the cut-off frequency for the nth-order mode is As long as f < f nc , the nth-order mode does not exist. That is, when f is less than f 1c of first-order mode, only one mode exists in the CAW regardless of the source condition.

Coupling equation for a PCAW
For the PCAW structure in two-dimensional crosssection (Fig. 2), we denote the densities of the cladding, core 1, and core 2 by q 0 , q 1 , and q 2 , respectively; in addition, c 0 , c 1 , and c 2 denote the respective velocity of sound in these zones. The spatial distribution of the refraction index is written where l = 1, 2 denotes the core index; here, Án 2 l is only non-zero in the corresponding core.
To ensure that there is only one mode in the waveguides, a specific frequency range is chosen f < f 1c . In consequence, the expression for the wave front in the structure is expanded in the separable form, Substituting equation (12) into equation (2), we have

See the Equation (13) bottom of the page
Assuming that the wave amplitude of the coupled mode varies slowly with z, and combining with equation (4), equation (13) Multiplying both sides of equation (14) by w 1 (x) and w 2 (x) and integrating with respect to x while applying mode orthogonality, we obtain where i, j represents number of the waveguide, i, j = 1, 2, and i 6 ¼ j. The coefficient of self-coupling is   ¼ 0 ð13Þ and the coefficients of mutual-coupling is Here, we consider a symmetrical PCAW for which q 1 = q 2 and c 1 = c 2 . Then, by calculating the integrals of equations (16) and (17)

directly, we obtain
See the Equations (18) and (19) bottom of the page Given initial conditions n 1 (0) = 1 and n 2 (0) = 0, the analytical solutions of equation (15) is In view of equations (18) and (19), K is much smaller than j and has no theoretical dependence on CL. We therefore omit it in equations (20a) and (20b) and hence obtain for the variation in acoustic power of the two waveguides, Therefore, CL, defined as the propagation distance necessary to realize the maximum power transfer from an initial waveguide to an adjacent waveguide, is

Theoretical analyses and simulations of PCAW coupling
To verify the theoretical model and investigate the coupling effects systematically, we studied the series of scenarios for a structure with fixed configurational settings of w = 2 mm, q 0 = 2700 kg/m 3 , c 0 = 6260 m/s, q 1 = 998 kg/m 3 , and c 1 = 1480 m/s. The frequency varies from 200 to 350 kHz, and the core spacing varies from d = 1.4 mm to 2.2 mm.
The results confirm the prediction that K is much smaller than j by about three orders of magnitude, which is the assumption imposed in obtaining equations (21a) and (21b). Moreover, using equation (22) and the j values listed (Tab. 1), theoretical values of CL for specific f and d were determined (Tab. 2).

Simulation experiments
Using the finite-element method, we conducted a series of simulations to evaluate wave coupling appearing in various PCAW structures. The configuration and parameter settings were the same as those in the theoretical studies. The amplitude of the acoustic wave was fixed at P A = 10 5 Pa.
In snapshots of the acoustic wave propagating in the PCAW structure for various f and fixed d, we plotted the effective intensity distribution I e (Fig. 3a) and the corresponding curves along the central axis of the input waveguide (Fig. 3b).
In a similar analysis, we simulated acoustic waves with fixed f propagating in PACW structure of distinct d. We plotted the distribution of I e (Fig. 4a) and the corresponding effective intensity curves along the central axis of the input waveguide (Fig. 4b).
Rather than being confined to the input waveguide, the wave alternates periodically between the two waveguides along the propagation direction with a fixed period (Figs. 3a and 4a). The intensity distributions and curves indicate clearly that CL is proportional to f and d. Moreover, by analyzing the intensity curves along the input waveguide obtained from a series of simulations, we evaluated CL for different f and d (Tab. 3).

Characteristics of wave coupling in the PCAW structure
To compare theoretical and simulation results, we plotted the CL values of Tables 2 and 3 (Fig. 5). The coupling length CL increases approximately exponentially for both f and d in accordance with the dependence of the evanescent wave on f and d. Furthermore, discrepancies in the plot are evident between the theoretical and simulation results. Therefore, we analyzed the relative error in CL between these results (Tab. 4).
Viewing Table 4, the max error is 12.03% and the average error is 5.66%. The smallest error is 0.20% and most errors are smaller than 10.00%. Neglecting the quadratic terms in equation (13) and the accuracy of the modal eigenvalues for a single CAW, we found that both contribute to    those errors. Indeed, in conducting more analysis on an input wave of f = 375 kHz, the errors become even larger. The reason is that the wave frequency exceeds the cut-off frequency for the first-order mode and the wave propagate along the waveguide as two modes, a scenario that is not considered in our theoretical analysis. Nevertheless, the theoretical model presented helps to reveal the nature of the wave coupling in the PCAW structures and predicts CL values for a single mode accurately.

Nature of wave coupling in a PCAW structure
Both coefficients of self-and mutual-coupling, equations (16) and (17), were determined using the overlap integrals of the guide modes and calculated by integral segmentation. Because Dn l 2 is non-zero only in this part of the core, the integration for j 11 is non-zero in this section of core 2 and depends on the evanescent wave of waveguide 1. Analogously, j 22 depends on the evanescent wave of waveguide 2, j 12 on the evanescent wave of waveguide 1 and the guided wave of waveguide 2, and j 21 on the evanescent wave of waveguide 2 and the guided wave of waveguide 1. For symmetrical parallel waveguides, K describes the leakage of the evanescent wave and j describes the interaction between the evanescent wave of one waveguide and the guided wave in the adjacent waveguide.
The evanescent wave decays exponentially with penetration depth and with f in an approximate exponential form; similarly, j (CL) decays (increases) with d in strict exponential form and with f in an approximate exponential form. Considering the above analyses, we are assured that the coupling effect in a PCAW structure is essentially an evanescent wave coupling. When the evanescent wave is weaker, the wave coupling is weaker; for a complete wave coupling process, CL is longer. Enlarging the wave frequency, widening the spacing between cores, and, of course, increasing the material difference between the cladding and core weakens the evanescent wave and certainly suppresses waveguide coupling.

Conclusions
Applying the wave acoustic theory, we presented the propagation characteristics of acoustic waves in both the cladding and core zones of a single CAW by imposing appropriate continuous boundary conditions. A theoretical model of wave coupling in a pair of PCAWs was established by combining the modal functions of the single waveguide and CMT.
Both the nature and characteristics of the wave coupling were analyzed based on results obtained from theoretical modal and simulations. The analysis revealed that: (1) evanescent waves exist in the cladding zone of the single CAW and decrease exponentially with increasing d and f, the latter only approximately; (2) CL increases exponentially with increasing d and f approximately; (3) and from those two observations, we concluded that as d and f  increased, the evanescent wave became weaker, resulting in a longer propagation distance (CL) required for the acoustic wave to transfer from one waveguide to the adjacent waveguide. Thus, the acoustic wave coupling in the two waveguides is mediated by the evanescent wave in the cladding. In theory, the coupling process does not break down. Circumstances may weaken the coupling and increase the CL. When the structure length is much smaller than CL, the coupling process is incomplete and weak, and it could be barely observed.
Our results regarding the waveguide coupling support their further application in the field of acoustic wave control and signal processing. Potentially, pure acoustic devices such as frequency selectors, focusing devices, and coupling switches may be designed given appropriate settings of the CL characteristics.