The rest of the paper is structured as follows. In the Results section, we first give a brief introduction to the theory of 2D acoustic waveguides. Then, the channel-basis and point-basis S-matrices are formally derived, followed by a detailed discussion of the basis conversion relations, which is the core result of this paper. We then validate the conversion relations by considering focusing through a random medium. We then focus on the non-unitary characteristics of the conversion relations by considering the cause of the absence of an open channel in the point-basis S-matrix. Lastly, we wrap up with a miscellaneous discussion.
Our discussion starts with the theory of acoustic waveguides. The waveguide is formed by sound-hard boundaries on both the and -planes, extending infinitely in the -direction, as depicted in Fig. 1a. The waveguide has a uniform rectangular cross-section with a height of in the z-direction and a width of in the -direction. The height () is chosen to be smaller than half the wavelength of the sound wave being considered, such that the sound field is approximately uniform in the -direction. The sound field in the waveguide can thus be treated as two-dimensional (2D). In this study, sound waves are confined within the waveguide by sound-hard boundaries, resulting in the presence of only guided modes. The number of these modes is determined by the waveguide's lateral size (as detailed later). Since acoustic waves are scalar longitudinal waves, there are no bulk modes or transverse electric/vertical (TV) modes like those found in electromagnetic waves. In the absence of scatterers, sound propagation within the waveguide is described by a 2D Helmholtz equation
Here, represents the pressure field in the frequency domain, is the magnitude of the wavevector, is the speed of sound, and is the angular frequency. The solution of Eq. (1) can be expressed as a series of waveguide modes
where is a positive integer that represents the modal number, and and are the complex amplitudes of the th mode. The -component wavevector for the th waveguide mode can be determined using , with as the y-component wavevector. In this paper, we focus solely on modes where has a real value, indicating propagating waves. The positive (negative) sign in front of in Eq. (2) denotes the left-propagating (right-propagating) waves. The distribution in the -direction of the th waveguide mode, , is given by
valid for all positive integers . Two fundamental differences between acoustics and optics (or electromagnetism) are immediately obvious here. First, owing to the longitudinal nature of acoustic waves in fluids, is a plane-wave mode that does not have a cutoff frequency. Second, sound-hard boundary conditions dictate that the pressure fields peak at the boundary in the normal direction ( in Eq. (3)), so are cosine functions. These waveguide modes are both normalized and orthogonal, satisfying
where is the Kronecker delta symbol. The requirement for to be real constrains the total number of propagating waveguide modes to
where denotes floor operation, i.e., rounding down to the closest integer.
Next, the scattering medium occupies a region of , as shown in Fig. 1a. Unlike an empty waveguide, different propagation modes in the waveguide will be mixed due to the influence of the scattering medium. The explicit form of the S-matrix depends on the chosen basis. One common choice of basis is called 'the channels', which are associated with waveguide modes
Here, the superscript L(R) denotes the left (right) lead. It should be noted that this decomposition slightly differs from the waveguide modes described in Eq. (2) due to the additional normalization factor , whose purpose is to ensure equal energy flux in different scattering modes (see Supplementary Note 1 for more details). These adjusted waveguide modes are usually referred to as the scattering channels. The amplitudes of incoming and outgoing scattering channels are represented by the column vectors and , respectively. The channel-basis S-matrix is defined as
where the dimensions of and are both and the S-matrix has dimensions of . This indicates a total of scattering channels on the left and right leads of the waveguide. The entries specify the scattering amplitude from the -th incoming channel to the th outgoing channel.
It is useful to lay down some general properties of the channel-basis S-matrix. In the absence of loss and gain, the total outgoing energy flux must equal to the total incident energy flux, that is . Therefore, the channel-basis S-matrix must be unitary
where is the identity matrix of dimensions . Additionally, in the presence of time-reversal symmetry, there is . Here, the superscript '' represents the conjugate. Substituting this relationship into Eq. (8) leads to . Therefore,
This implies that in this context, the S-matrix is a symmetric unitary matrix. The S-matrix can be expressed as a combination of four block matrices
Here, the diagonal blocks, and , correspond to the reflection matrices for left and right incidence, respectively. The off-diagonal blocks, and , correspond to the left-to-right and right-to-left transmission matrices, respectively. Naturally, these block matrices are also transpose invariant, i.e., , , . The traces of and , denoted as and , are
where and are the eigenvalues of and , respectively. is the total number of propagating waveguide modes in the system. Obviously, . Singular value decomposition can be applied to the transmission matrix in the channel basis as , where the unitary matrices and (of size ) contain the left () and right () singular vectors of as columns, respectively. The singular vectors have a norm of 1, and any two left (or right) singular vectors are orthogonal. The matrix is a diagonal matrix with the diagonal elements being the singular values , arranged in descending order. Hence, there exists the relationship , with being a diagonal matrix with as diagonal entries. According to random matrix theory, obeys a bimodal distribution as the number of channels approaches infinity, where is the thickness of complex medium along the waveguide, is the transport mean free path. The effective number of open channels can be estimated as , which means the effective number of channels contributing to the transmitted field.
The channel-basis S-matrix has been widely applied in theoretical analyses due to its advantageous characteristics. In addition, in Supplementary Note 2, we provide a detailed explanation for retrieving the channel-basis S-matrix through numerical simulations.
The S-matrix can also be expressed in the point-to-point basis, or simply the point basis, denoted as . On this basis, the S-matrix connects the input and output physical fields at a set of coordinate points. In this paper, points are strategically selected at the boundaries and on both sides of the scattering region, totaling points. The selection of points follows two rules: First, the condition must be satisfied to ensure that the point-basis S-matrix captures all channel information. Second, all points must be symmetrically distributed with respect to the centerline. This symmetry ensures accurate conversion between the point-basis and channel-basis S-matrices (see Supplementary Note 4). For simplicity, the sequence of y-coordinates of the points on both leads is chosen to be the same, and given by , which can be expressed as
where, . The separation between two adjacent points is , with representing the width of the waveguide. Monopole sources are placed at each point for excitation, with each monopole source emitting a field governed by the inhomogeneous Helmholtz equation . Here denotes the source strength with units of Pa, and is the position of the monopole source. The resulting scattered sound pressure is recorded at points to construct the output vector, i.e., . The point-basis S-matrix reads
This equation, relates the set of monopole source to the scattered sound pressure at the 2M points. Naturally, generally does not exhibit unitary properties. The dimension of is , which can also be expressed in a block form as
where the diagonal blocks correspond to the reflection matrices under left () and right () incidence, and the off-diagonal blocks correspond to the transmission matrices from left to right () and from right to left (). All four blocks are in dimension.
The theoretical foundation for this conversion lies in the connection between the system Green's function and the S-matrix. The Green's function describes the response of a system to a point source or point perturbation. In the case of wave propagation within a waveguide containing multiple scattering media, the Green's function can be used to calculate the sound field at any point in the system, given the source and boundary conditions. Once the sound field is determined by Green's function, it can be decomposed into scattering channels. Thus, the conversion of channel-based and point-based S-matrix is achieved. The detailed description is presented in Supplementary Note 3.
Here, we focus only on converting the left-incident reflection matrix () and the left-to-right transmission matrix () from the channel basis to the point basis. The conversion relations are the same for the case of incidence from the right side. To simplify the expression, we omit the subscript ''. Same as the previous section, the -coordinates of the input and output planes are and , which also serve as the reference zero phase surface for the left and right ports, respectively. This choice aligns with the definition of the scattering channels as presented in Eq. (6). It is worth noting that this choice of the zero-phase plane also simplifies the relationship between the S-matrix in the channel basis and the point basis. Since we are considering the left-incidence case, we first assume that a monopole source with strength is placed at the left boundary of the scattering region with coordinates , and then the scattered sound pressure is detected at positions and , denoted as and , respectively. According to Eqs. (13) and (14), we can define the entries of the point-basis reflection matrix as , and the entries of the point-basis transmission matrix as . They can be expressed using the Green's function. The Green's function for the empty waveguide (Please refer to Supplementary Note 3 for details) can be expanded using the waveguide modes
where represents the response at () by excitation at , and is the th waveguide mode as defined in Eq. (3). It follows that the sound field excited by a monopole source at can be expressed as
Then, we can express the left incident field as
The left incident field can then be straightforwardly expressed using the scattering channels
Here, the summation is truncated at to omit the waveguide modes that are evanescent in . The expression of is obtained by computing the inner product of and , that is
We arrive at
The coefficients of the left-outgoing waves in scattering channels are given by
where is the reflection coefficient from the th to th channels. Therefore, the left scattered pressure
By substituting Eq. (21) into Eq. (22), we obtain
Here, we have defaulted . To simplify, we define matrix with entries , having dimensions . According to the orthogonality condition, . We also define a diagonal matrix with entries , having dimensions . Here, is the x-component wavevector for the -th waveguide mode. Furthermore, we introduce a transformation matrix , with its entries given by and a dimension of (see Supplementary Note 5 for details). Eq. (23) can be rewritten in matrix form as
where, and , respectively represent the reflection matrix in the channel basis and point basis. Similarly, we can obtain similar formulas for the left-to-right transmission matrix in the point basis () as
where is the transmission coefficient from the th to th channels. And we arrive at
We remark that the dimensions of and can be different, so the matrix is generically a rectangular matrix.
The conversion from point to channel basis is equivalent to obtaining the pseudo-inverse of , which is , such that . (The pseudo-inverse matrix of a column full rank matrix is given by . has a dimension of with , and it is column full rank). The entries of are . Similarly, it can be shown that matrix is the pseudo-inverse matrix of , as . Consequently, the conversion from point basis to channel basis is given by
It is important to note that this conversion is accurate only when the number of points is sufficient () and they are selected symmetrically about the central line of the waveguide, i.e., with locations following Eq. (12). For further details and a proof, please refer to Supplementary Note 4 and Note 5.
We remark that an implicit assumption for the accurate conversion is that the effect of evanescent modes is negligible. The channel basis, by definition, considers only propagative modes and ignores any evanescent modes. But the point basis can indeed pick up evanescent modes from the scattering. So, the measurement plane needs to be sufficiently far () from the scattering media for evanescent modes to be negligible.
This section presents a numerical case study of focusing spatially modulated acoustic waves through a complex medium. The purpose of this case study is to validate the basis conversion relations and to demonstrate the application of S-matrices in channel and point bases.
The complex medium consists of 100 rigid cylinder scatterers, each with a diameter of 0.02 m, randomly distributed within a 2D air-filled waveguide of width (see Fig. 2a, b). The sound frequency employed is , corresponding to a sound wave wavelength of approximately , so the waveguide sustains 50 propagative modes, i.e., the number of channels is . The complex medium occupies a rectangular region of in the waveguide, with extending along the -direction. To safely neglect the evanescent waves, the measurement planes are positioned at a distance of approximately away from both the left and right-hand sides of the complex medium. The separation between the two measurement planes is . For simplicity, the number of sampling points is chosen to be equal to the number of channels, i.e., . Our calculation indicates , and the effective number of open channels can be estimated as , while the remaining channels are classified as closed (i.e., for them). Given these parameters, the transmission mean free path is estimated to be approximately . Additionally, the scattering mean free path of the complex medium can be estimated about (see Methods for details).
In Fig. 2b, the scattering behavior of a plane wave is depicted, showcasing the notable distortion of the wavefront due to the presence of the complex medium. The transmission matrices are directly acquired in both the channel and point bases, and the real components of these matrices are depicted in the left two panels of Fig. 2c. Then, the channel-basis transmission matrix is transformed into the point basis using Eq. (26). And conversely, the point-basis transmission matrix is transformed into the channel basis using Eq. (28). The corresponding results are shown in the right two panels of Fig. 2c. Excellent agreement is seen. This successful alignment validates the accuracy of our basis conversion relationships.
Next, based on the transmission matrices in the two different bases, we use the technique of inverse filtering to synthesize acoustic wavefronts to achieve focusing through the complex medium. The target is to generate one focal spot at a distance away from the output plane. The wavefield () at the output plane ( is chosen as
Expand this equation using the channels
Here, represents the output channel coefficients. By multiplying both sides of Eq. (30) by and integrating across the cross-section, with the orthogonality, we arrive at
Utilizing the relationship , where and . Therefore, we can express the amplitudes of the incident channels as
where the matrix is the same as shown in Fig. 2c. The resulting focusing effect based on the channel basis can be observed in the upper panels of Fig. 3b, c.
A similar focusing can be obtained using the point-based transmission matrix. To this end, we discretize the output wavefield (Eq. (29)) at a set of discrete points , obtaining the point-basis output vector , which is the result of a set of monopole sources at the input plane , given by
The focusing effect based on the point-basis method is shown in the lower panels in Fig. 3b, c.
Examining the results in Fig. 3b-e, excellent agreement is seen between the results obtained using the channel and point bases in both the overall field map and the pressure distribution on various planes. The slight degree of discrepancy in Fig. 3d, e will diminish with the further increase of the number of channels and sampling points.
Here, we compare open channels in the two types of bases and analyze their differences in transmission efficiency. According to the theory of random matrices, channels with transmittance close to 1 almost always exist in complex media. However, these channels are typically excited only within the transmission matrix in channel basis (). The open channel (full transmission) takes the first right singular vector as the input wavefront, which is associated with the largest singular value , and its transmittance approaches unity in the channel basis. This results in the first left singular vector as the output wavefront, i.e.,
Equation (26) describes a mapping from channel to point basis, noting that
Due to the varying propagation wave vectors across different modes in multimode waveguides, is not a unitary matrix, and is also non-unitary (see Note 5 in the Supplementary Information for details). Consequently, the conversion alters the magnitudes of the singular values, which may also shuffle the order of the singular vectors. To be more specific, when is the first right singular vector of , the corresponding channel coefficient is not necessarily the first right singular vector of (or proportional to it). Therefore, using the singular vectors associated with the largest singular value for excitation, the transmission matrix in point basis generally cannot achieve full transmission (see Note 6 in the Supplementary Information for details).
Here, we employ the same complex medium utilized in the focusing to validate this conclusion. In Fig. 4, we present the acoustic fields generated by a left-hand side input vector with the first singular value of the transmission matrices in the two bases. In the case of channel basis, this is an open channel with a singular value > 0.99 (Fig. 4a). Total transmission is seen. In contrast, in the point basis scenario, the maximum singular value is only ~0.32, so transmission is much weaker (transmitted flux ~0.56) and there is notable reflection (Fig. 4b). Consequently, open channels generally cannot be obtained using the point basis.