Wigner-Smith (WS) time delay concepts  have been adopted in many branches of wave physics, including quantum physics, electromagnetics, optics, and acoustics. Applications include the characterization of principle modes in fibre optics , the study of molecular ionization , wavefront shaping in disordered media , and the construction of particle-like wave packets with minimal dispersion ; for other recent applications, see [8, 3, 10, 2, 7, 21, 6, 5]. Different definitions of the WS time delay matrix for a system with scattering matrix have appeared in the literature, including 
For lossless and reciprocal electromagnetic systems, the scattering matrix is unitary, i.e. , implying . Recently, Patel and Michielssen [16, 17] showed that for such systems may be expressed in terms of volume integrals of energy-like densities plus correction terms that depend on the guiding, scattering, or radiating nature of the system. They furthermore developed efficient techniques for computing using integral equation methods, reducing the dimensionality of the integrals to be evaluated from three to two. Finally, they illustrated that so-called WS modes obtained by diagonalizing oftentimes untangle wave and scattering phenomena experiencing well-defined group delays corresponding to
’s eigenvalues. Unfortunately, the restriction of the methods in[16, 17] to systems composed of nondispersive and lossless materials limits their use in many real-world applications. Few studies to date have extended the use of WS methods to such systems. An exception is recent work by Chen et al.  that analyzes WS time delays in lossy systems by utilizing the pole expansion of and analyzing the average of ’s eigenvalues through Eq. (2).
This study extends  to systems with material dispersion and losses. Its specific contributions are threefold:
It shows that defined via Eqs. (1) can be expressed in terms of the volume integrals of energy-like quantities in  plus additional correction terms that account for the dispersive and lossy nature of the system. Unfortunately, the eigendecomposition of produces limited insights into the system; specifically, its eigenvalues incorporate effects of both time delays and losses.
It shows that the eigenvectors and the real parts of eigenvalues ofalso diagonalize , and hence retain the physical interpretation of WS modes and time delays, respectively. The simultaneous diagonalization of and implies that the WS modes are fully decoupled and frequency stable.
It uses analytical and numerical examples to illustrate the effects of material dispersion and losses on the eigenstates of , and further highlights the difference between and .
Contributions 1, 2, and 3 are detailed in Sections II, III, and IV below. Throughout this paper, denotes frequency and denotes angular frequency. In addition, denotes , and , , and denote conjugate, transpose, and conjugate transpose operations.
Ii WS Relationship
Let and denote the volume and port surfaces of a guiding system composed of cavities and waveguides with perfect electrically conducting (PEC) walls (Fig. 1). Let and denote the permittivity and permeability of the dispersive and lossy medium that fills and let denote a Cartesian parameterization of . It is assumed that is removed from all waveguide material and geometric discontinuities, and that and are homogeneous, lossless, and nondispersive near . With these assumptions, waveguide fields near can be expanded in a set of propagating modes with real, frequency independent, and orthonormal transverse profiles , propagation constants , and impedances , (See Appendix A in ).
Assume the system is excited by the -th incoming mode with transverse field components near given by
where is the distance away from outside . Going forward, spatial and frequency dependencies will be omitted whenever possible to simplify notation. The outgoing transverse fields near are
where is the system’s scattering coefficient characterizing coupling from mode to mode . For near , the total transverse fields are
Ii-B Expressions for and
Let and denote total fields in excited by the -th incoming mode. The frequency derivatives of Maxwell’s equations for fields and are
Likewise, the complex conjugates of Maxwell’s equations for fields and are
) and making use of standard vector identities results in
Integrating the left- and right-hand side (LHS and RHS) of Eq. (10) over and applying the divergence theorem while accounting for the fact that electric fields tangential to all PEC walls vanish, yields
The subscripts V suggest that the result is computed directly by a volume integral and the superscripts , , indicate energy, dispersion, and loss, respectively – this terminology is further explained below. Eq. (11) relates a surface integral of transverse field components across to volume integrals of fields across .
Taking the complex conjugate and frequency derivative of Eqs. (5a) and (5b) and substituting the resulting expressions for and into the LHS of Eq. (11) allows the LHS of Eq. (11) to be expressed as 
where the superscript stands for base, and
In matrix form, Eq. (18) reads
To arrive at Eq. (2), define
The evaluation of the WS time delay matrix for lossy and dispersive systems involves the following steps.
The contributions of dispersion and losses generalize expressions derived in  for single port antennas to a multi-port setting.
In the absence of loss and dispersion, is unitary (). It follows that , and the above derivation reverts to the base formulation applicable to lossless and dispersion-free systems in .
The expression for in Eq. (14) involves and , frequency derivatives of fields, seemingly increasing the computational cost of evaluating the WS time delay matrix for lossy systems. These quantities however can be easily evaluated via an approach that incurs only marginal additional computational cost, detailed in Appendix A. With this approach, the computation of only involves the numerical solutions of Maxwell’s equations at a single frequency and it remains convenient to obtain via .
Iii Time Delay Interpretation
Iii-a Time Delays
The manipulation of Maxwell’s equations and Gauss theorem produces Eqs. (19)–(21) and facilitates the volume integral evaluation of . However, ’s physical meaning and properties are not yet clear. To interpret , consider the following setting.
For a linear, time-invariant and lossy system with only one port, assume a time-harmonic incoming signal
where is the distance from the port and is positive outside the system; denotes the propagation constant (outside the system) at frequency and is purely real. Correspondingly, the system generates an outgoing signal
where is the scattering coefficient. due to the loss and represents the system’s phase delay.
In order to characterize the system’s group delay, consider a narrow-band incoming pulse signal with center frequency , bandwidth and real-valued envelop ,
The system generates a corresponding outgoing pulse
where Taylor expansions for , , at are made use of. For notational convenience, and are defined. It is worth mentioning that is generally nonzero even for nondispersive but lossy media and it results in the deformation of the original wave packet. However, is smooth enough for a narrow-band signal and the second term on the right-hand side (RHS) of Eq. (25) can be neglected.
Iii-B Simultaneous Diagonalization
For a nondispersive and lossless multiport system where and are matrices, their properties are discussed in . In the absence of loss, is symmetric and unitary; is Hermitian, which follows from the symmetry of . The diagonalization of , i.e. , yields fully decoupled WS modes and well-defined time delays, given by columns of and diagonal elements of , respectively .
In the presence of loss, is symmetric but not unitary; therefore is not Hermitian. Suppose is diagonalized by , i.e.
Note that would only hold for Hermitian in lossless systems. By defining
Since is symmetric,
where is diagonal. Now Eq. (2) is completely diagonalized. The simultaneous diagonalization of the matrices , , indicates that columns of are so-called WS modes associated with well-defined group delays given by elements of the diagonal matrix .
It is worth mentioning that and do not simultaneously diagonalize except for a lossless system. This can be shown using similar procedures as above.
Iii-C WS modes
Denote and Eq. (29) is rewritten as
Consider an excitation with the incoming signal
which is define as the -th incoming WS mode. The above excitation generates outgoing signal where . Therefore
is defined as the -th outgoing WS mode. One can observe that each incoming WS mode excites only the corresponding outgoing WS mode with an attenuation factor . This suggests that WS modes are completely decoupled for lossy systems at a fixed frequency.
Finally, the field distribution of a WS mode is formally defined to be the total field excited by the corresponding incoming WS mode, i.e.
Iii-D Frequency Stability of WS modes
WS modes not only stay decoupled at a fixed frequency, but also exhibit minimal dispersion with respect to a frequency perturbation. This property is demonstrated below for a lossy system.
Assume that and are constant matrices defined as , ; at a close enough perturbed frequency , define
Then, applying a first-order forward Euler approximation to yields
A perturbed incoming “WS mode” constructed by eigenvectors at ,
would excite outgoing signal
where denotes the perturbed outgoing “WS mode”, defined as
This implies that WS modes are frequency stable with minimal dispersion, i.e. they react to a frequency perturbation primarily by collecting phase delays while remaining decoupled. The difference between WS modes at frequency and is much smaller compared to that between any other bases including waveguide modes.
Iv Illustrative Examples
This section analytically and numerically validates WS relationships Eqs. (19) and (21), and illustrates the effects of loss and dispersion on the WS characterization of a system. Throughout, matrices and are constructed by volume integration of fields via Eqs. (12)–(14), (16)–(17) and (20). Fields and scattering matrices are computed using a high-order mode expansion-based finite-element code . When needed, is computed using a finite-difference approximation with frequency step size . Numerical errors incurred in the computation of and are defined as and where denotes the Frobenius norm.
Iv-a Waveguide Homogeneously Filled with Low-Loss Material
To analytically validate the formalism from Section II, consider a PEC-terminated rectangular waveguide of length and transverse dimensions that is filled with a homogeneous, nondispersive, and lossy material with complex permittivity and real permeability . The waveguide is excited by a unit-power TE mode with cut-off wavenumber and mode profile
The wavelength, speed of light, and wavenumber inside the waveguide are , , and , respectively. To simplify expressions for , and , it is assumed that , , and , which imply that and . Below, “” implies equality up to an error of . The propagation constant for the -th mode therefore is
where , and .
The RHS of Eq. (19) therefore equates to
To evaluate the LHS of Eq. (19), note that the total fields inside the waveguide can be expressed as
Comparing this result with Eq. (48), it follows that
Furthermore, using , it follows from Eq. (21) that
Iv-B Waveguide Inhomogeneously Filled with a Weakly Lossy Absorber
To investigate the similarities and differences between and for low-loss structures, consider the air-filled and PEC-terminated waveguide shown in Fig. 2, which contains a block of weakly absorbing dielectric material with relative permittivity . At GHz, the waveguide supports 27 propagating TE modes with profiles
The skin depth in the absorber is 350 mm, significantly larger than the absorber’s dimensions. The relative errors and of the computed matrices and are and , respectively.
is diagonalized as , and the real parts of its eigenvalues are converted to “spatial shifts” by multiplying them with the speed of light in air. is diagonalized similarly as . With some abuse of terminology, the products of the real parts of its eigenvalues and the speed of light in air also are termed spatial shifts.
Fig. 3 shows that spatial shifts obtained from and largely coincide. The similarity in the spatial shifts does not extend to the eigenvectors (WS modes) and , however. To illustrate this, consider WS mode #1, shown in Fig. (a)a. This mode experiences a spatial shift of 60.04 mm, which maps onto a round-trip between the port and waveguide termination. In contrast, the field associated with the first eigenvector of , shown in Fig. (c)c, is characterized by a spatial shift of 56.83 mm. This spatial shift and the field profile reflect the fact that the eigenvectors of minimize time delays while maximizing losses by preferentially traveling through the absorber block in the upper part of the waveguide. Similar observations hold true for WS mode #2 and the field derived from ’s second eigenvector, shown in Figs. (b)b and (d)d, respectively. The values below Figs. (a)a and (b)b are a measure of the attenuation that the WS modes experience; for this low-loss system, the magnitudes of all scattering parameters are close to one.
Iv-C Waveguide Inhomogeneously Filled with a Strongly Lossy Absorber
To investigate differences between and for highly lossy structures, consider the air-filled and PEC-terminated waveguide shown in Fig. 5. The waveguide and its excitation are identical to those in the previous section, except for the position and material properties of the lossy block, which now has a relative permittivity of and a skin depth of 7.0 mm. The relative errors and of the computed matrices and once again are and , respectively.
Spatial shifts obtained by diagonalizing and are shown in Fig. 6. Also shown are spatial shifts obtained by diagonalizing WS time delay matrix for the waveguide without the absorber. Clearly, the absorber does not noticeably affect the time delays of most WS modes relative to those observed in the empty waveguide, as the real part of its permittivity is identical to that of the surrounding air. The eigenvalues of , instead, no longer match those of as was the case for the low-loss structure. This fact can be ascribed wholesale to the large material losses.
Fig. 7 shows several WS modes constructed from the eigenvectors of .
WS modes #1 and #6 are fields that primarily reflect off the waveguide termination instead of the waveguide’s sidewalls, and experience very similar spatial shifts equating to roughly twice the waveguide length. Despite the similarity in spatial shifts, these modes’ ’s differ substantially because mode #1 involves waves that mostly bypass the material block whereas those of mode #6 pass straight through it.
WS mode #15 travels along a zigzag pattern, reflecting off both the termination and the PEC walls, resulting in a larger spatial shift. The value is smaller compared to that of mode #6, as mode #15 excites waves traveling diagonally through the absorbing block and suffering from bigger losses.
WS mode #27 excites waves that travel almost parallel to the port surfaces. This waves travel a much longer distance than Ws mode #6 prior to exiting via the port, thus experiencing the longest time delay among all WS modes. Both the field distribution and the small indicate this modes experiences significant attenuation. That said, Fig. 6 shows that mode #27’s time delay (red) is smaller than that of the corresponding mode in the absence of the absorber (dashed blue), suggesting that the presence of the absorber causes reflections limiting field penetration deep into the waveguide.
Iv-D Waveguide with Material with Anomalous Dispersion
To illustrate the effect of dispersion on WS modes, consider the (somewhat artificial) two-port air-filled waveguide containing a dispersive NaCl insert shown in Fig. 8. At frequency GHz, NaCl exhibits anomalous dispersion; its relative permittivity is and Hz [19, 18].
The matrix is constructed with numerical error . Spatial shifts obtained upon diagonalizing , are shown in Fig. 9; also shown are spatial shifts assuming the material insert has no dispersion, i.e. . The first four time delays for the system with dispersion are negative, indicating negative group delays, while all the time delays for the system without dispersion are positive. The negative group delays can be attributed to the anomalous dispersion. Indeed, for the system with dispersion, the overall group delay experienced by a WS mode is a combination of positive and negative group delays as the wave travels outside and inside the NaCl insert.
Several representative WS modes are shown in Fig. 10. The most negative group delay is experienced by WS mode #1 (Fig. (a)a), which travels through the NaCl insert obliquely to maximize its interaction with the latter. WS mode #3 also exhibits a negative group delay with a smaller magnitude by traveling nearly normal to the surface of the insert (Fig. (b)b). These two modes have very small as their energies are effectively absorbed by the insert. In contrast, WS modes #10 and #17 (Figs. (c)c and (d)d) experience positive group delays, tend to avoid interactions with the insert, and have large .
Iv-E Three-port Waveguide with GaAs Material
To illustrate the use of the methods of Section II on a slightly more complicated geometry, consider the three-port air-filled waveguide loaded with a dispersive and lossy GaAs cylinder shown in Fig. 11. At GHz, GaAs has a relative permittivity , and Hz [19, 15].
Ports #1, #2, and #3 support 20, 23, and 18 propagating TE modes, respectively; and therefore both are 61-by-61. The relative errors of the matrices