Open Access
Scientific Article
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

© G. Yin et al., Published by EDP Sciences, 2022

Licence Creative CommonsThis 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 [15]. 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 [510]. Optical waveguide coupling is widely employed in optical wave steering [1114], 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 [2022].

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 [2931]. 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 frequency-selective 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

2.1 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 ρ1 for the core zone, and a cladding density ρ0. For the analysis, c1, c0, and ca 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

(1)

thumbnail Figure 1

Schematic of a single CAW.

Beginning with the equation governing the acoustic wave in two-dimensional 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 ka denotes the wave number of the wave in air. For a single CAW, the modal function is [4]

(5)

On substituting equation (5) into equation (4), we obtain

(6)

(7)

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,

(8)

Setting the normalized amplitude to unity (A = 1), the modal function of nth-order can be written as

(9)

Equation (9) describes evanescent waves that strictly decay exponentially with penetration depth and qn. The value of qn 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

(10)

As long as f < fnc, the nth-order mode does not exist. That is, when f is less than f1c of first-order 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 two-dimensional cross-section (Fig. 2), we denote the densities of the cladding, core 1, and core 2 by ρ0, ρ1, and ρ2, respectively; in addition, c0, c1, and c2 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 non-zero in the corresponding core.

thumbnail Figure 2

Cross-sectional 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 < f1c. In consequence, the expression for the wave front in the structure is expanded in the separable form,

(12)

Substituting equation (12) into equation (2), we have

(13)

Assuming that the wave amplitude of the coupled mode varies slowly with z, and combining with equation (4), equation (13) is simplified to

(14)

Multiplying both sides of equation (14) by ψ1(x) and ψ2(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 self-coupling is

(16)and the coefficients of mutual-coupling is

(17)

Here, we consider a symmetrical PCAW for which ρ1 = ρ2 and c1 = c2. Then, by calculating the integrals of equations (16) and (17) directly, we obtain

(18)

(19)

Given initial conditions ξ1(0) = 1 and ξ2(0) = 0, the analytical solutions of equation (15) is

(20a)

(20b)

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,

(21a)

(21b)

Therefore, CL, defined as the propagation distance necessary to realize the maximum power transfer from an initial waveguide to an adjacent waveguide, is

(22)

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, ρ0 = 2700 kg/m3, c0 = 6260 m/s, ρ1 = 998 kg/m3, and c1 = 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).

Table 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).

Table 2

Coupling lengths obtained from theoretical model (mm).

3.2 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 PA = 105 Pa.

In snapshots of the acoustic wave propagating in the PCAW structure for various f and fixed d, we plotted the effective intensity distribution Ie (Fig. 3a) and the corresponding curves along the central axis of the input waveguide (Fig. 3b).

thumbnail 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 central-axis 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 Ie (Fig. 4a) and the corresponding effective intensity curves along the central axis of the input waveguide (Fig. 4b).

thumbnail 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 central-axis 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).

Table 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).

thumbnail Figure 5

CL values obtained from theory (colored surface) and simulation (red stars).

Table 4

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 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.

4.2 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 Δnl2 is non-zero only in this part of the core, the integration for κ11 is non-zero in this section of core 2 and depends on the evanescent wave of waveguide 1. Analogously, κ22 depends on the evanescent wave of waveguide 2, κ12 on the evanescent wave of waveguide 1 and the guided wave of waveguide 2, and κ21 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 parallel-cladded acoustic waveguides. Acta Acustica, 6, 8.

All Tables

Table 1

Coefficients of self- and mutual- coupling obtained from analytical expressions.

Table 2

Coupling lengths obtained from theoretical model (mm).

Table 3

Values of the coupling length CL (mm) obtained from the simulations.

Table 4

Relative errors between CL from the theoretical and simulated results (%).

All Figures

thumbnail Figure 1

Schematic of a single CAW.

In the text
thumbnail Figure 2

Cross-sectional schematic of the parallel cladded acoustic waveguide (PCAW) structure.

In the text
thumbnail 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 central-axis of the input waveguide.

In the text
thumbnail 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 central-axis of the input waveguide.

In the text
thumbnail Figure 5

CL values obtained from theory (colored surface) and simulation (red stars).

In the text

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