Issue 
Acta Acust.
Volume 6, 2022



Article Number  8  
Number of page(s)  8  
Section  General Linear Acoustics  
DOI  https://doi.org/10.1051/aacus/2022002  
Published online  08 February 2022 
Scientific Article
Characteristics and mechanism of coupling effects in parallelcladded acoustic waveguides
^{1}
Key Laboratory of Modern Teaching Technology, Shaanxi Normal University, Xi’an 710062, China
^{2}
School of Physics and Information Technology, Shaanxi Normal University, Xi’an 710119, China
^{*} Corresponding authors: jhan2012@snnu.edu.cn; guojz@snnu.edu.cn
Received:
8
October
2021
Accepted:
5
January
2021
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 twodimensional 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 finiteelement 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 frequencyselective filters, directional couplers, and acoustic switches.
Key words: Coupling effect / Cladded acoustic waveguide / Evanescent wave / Coupling length
© G. Yin et al., Published by EDP Sciences, 2022
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 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 [1–5]. 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 [5–10]. Optical waveguide coupling is widely employed in optical wave steering [11–14], pure optical switches [15, 16], directional couplers [17], and frequencyselective filters [18, 19]. The theoretical research and exploration of other applications of waveguide coupling in optics have continued unabated [20–22].
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 coupledmode theory (CMT) of the acoustic resonance cavity [27]. In 2016, an acoustic adddrop filter was fabricated from two linedefect 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 [29–31]. Shen and collaborators investigated supermode 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 acousticwaveguide arrays and showed the possibility of generating diffractionfree 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 twodimensional 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 finiteelement 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
2.1 Eigenmode equation in Single CAW
The structure of a single CAW is a core with cladding on both sides [35]. A crosssection of the twodimensional model of the CAW (Fig. 1) was configured for the purpose of simulations having a width w and density ρ_{1} for the core zone, and a cladding density ρ_{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
Figure 1 Schematic of a single CAW. 
Beginning with the equation governing the acoustic wave in twodimensional free space,
(2)where p denotes the acoustic pressure at position (x, z) at time t; we assume the solution of wave equation in the waveguide as
(3)where ψ(x) is the modal function of the wave in the waveguide, and ω = 2πf is the angular frequency. Substituting equation (3) into equation (2), we have
(4)where k_{a} denotes the wave number of the wave in air. For a single CAW, the modal function is [4]
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,
Setting the normalized amplitude to unity (A = 1), the modal function of nthorder can be written as
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 cutoff frequency for the nthorder mode is
As long as f < f_{nc}, the nthorder mode does not exist. That is, when f is less than f_{1c} of firstorder mode, only one mode exists in the CAW regardless of the source condition.
2.2 Coupling equation for a PCAW
For the PCAW structure in twodimensional crosssection (Fig. 2), we denote the densities of the cladding, core 1, and core 2 by ρ_{0}, ρ_{1}, and ρ_{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
(11)where l = 1, 2 denotes the core index; here, is only nonzero in the corresponding core.
Figure 2 Crosssectional schematic of the parallel cladded acoustic waveguide (PCAW) structure. 
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
Assuming that the wave amplitude of the coupled mode varies slowly with z, and combining with equation (4), equation (13) is simplified to
Multiplying both sides of equation (14) by ψ_{1}(x) and ψ_{2}(x) and integrating with respect to x while applying mode orthogonality, we obtain
(15)where i, j represents number of the waveguide, i, j = 1, 2, and i ≠ j. The coefficient of selfcoupling is
(16)and the coefficients of mutualcoupling is
Here, we consider a symmetrical PCAW for which ρ_{1} = ρ_{2} and c_{1} = c_{2}. Then, by calculating the integrals of equations (16) and (17) directly, we obtain
Given initial conditions ξ_{1}(0) = 1 and ξ_{2}(0) = 0, the analytical solutions of equation (15) is
In view of equations (18) and (19), K is much smaller than κ 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
3 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, ρ_{0} = 2700 kg/m^{3}, c_{0} = 6260 m/s, ρ_{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.
3.1 Theoretical analyses
Using equations (8), (18), and (19), the values of κ and K were calculated (Tab. 1).
Coefficients of self and mutual coupling obtained from analytical expressions.
The results confirm the prediction that K is much smaller than κ by about three orders of magnitude, which is the assumption imposed in obtaining equations (21a) and (21b). Moreover, using equation (22) and the κ values listed (Tab. 1), theoretical values of CL for specific f and d were determined (Tab. 2).
Coupling lengths obtained from theoretical model (mm).
3.2 Simulation experiments
Using the finiteelement 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).
Figure 3 Simulated results of wave coupling in the PCAW structure for various f and fixed d = 1.4 mm: (a) acoustic intensity distribution, (b) acoustic intensity curves along the centralaxis of the input waveguide. 
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).
Figure 4 Simulated results of wave coupling in the PCAW structures for various d and fixed frequency f = 240 kHz: (a) acoustic intensity distribution, (b) acoustic intensity curves along the centralaxis of the input waveguide. 
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).
Values of the coupling length CL (mm) obtained from the simulations.
4 Analyses and discussion
4.1 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).
Figure 5 CL values obtained from theory (colored surface) and simulation (red stars). 
Relative errors between CL from the theoretical and simulated results (%).
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 cutoff frequency for the firstorder 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.
4.2 Nature of wave coupling in a PCAW structure
Both coefficients of self and mutualcoupling, equations (16) and (17), were determined using the overlap integrals of the guide modes and calculated by integral segmentation. Because Δn_{l}^{2} is nonzero only in this part of the core, the integration for κ_{11} is nonzero in this section of core 2 and depends on the evanescent wave of waveguide 1. Analogously, κ_{22} depends on the evanescent wave of waveguide 2, κ_{12} on the evanescent wave of waveguide 1 and the guided wave of waveguide 2, and κ_{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 κ 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, κ (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.
5. 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.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant No. 12004237, 11727813, 12034005 and 12174004), the China Postdoctoral Science Foundation (Grant No. 2020M683416), the Fundamental Research Funds for the Central Universities (Grant No. GK202003090), and the Key Laboratory of Ultrasound of Shaanxi Province, China.
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Cite this article as: Yin G. Li P. Yang X. Tian Y. Han J, et al. 2022. Characteristics and mechanism of coupling effects in parallelcladded acoustic waveguides. Acta Acustica, 6, 8.
All Tables
All Figures
Figure 1 Schematic of a single CAW. 

In the text 
Figure 2 Crosssectional schematic of the parallel cladded acoustic waveguide (PCAW) structure. 

In the text 
Figure 3 Simulated results of wave coupling in the PCAW structure for various f and fixed d = 1.4 mm: (a) acoustic intensity distribution, (b) acoustic intensity curves along the centralaxis of the input waveguide. 

In the text 
Figure 4 Simulated results of wave coupling in the PCAW structures for various d and fixed frequency f = 240 kHz: (a) acoustic intensity distribution, (b) acoustic intensity curves along the centralaxis of the input waveguide. 

In the text 
Figure 5 CL values obtained from theory (colored surface) and simulation (red stars). 

In the text 
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