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Contribution of evanescent waves to the effective medium of disordered waveguides

We consider a wave propagating through a thin disordered slab inside a wire or waveguide of finite width. In the dense weak scattering limit, the statistics for the complex reflection and transmission coefficients (the coherent field) is found to depend dramatically on the contribution of evanescent modes or closed channels, leading to an effective refractive index whose real part is quite sensitive to the closed channels inclusion. In contrast, evanescent modes play no role in the statistical average of transmittances and reflectances. The theoretical predictions, based on the perturbative Born series expansion, are in excellent agreement with numerical simulations in disordered wires.


I. INTRODUCTION
The coherent transport of electromagnetic, electronic and acoustic waves through random media has long been a central issue in Physics [1]. In recent years, there has been a revival of interest in the impact of inhomogeneities and defects in the transmission properties in quasi-onedimensional (Q1D) systems [2] and is the subject of intense research including the electronic conductance of nanowires, nanotubes and nano ribbons [3][4][5], the propagation of slow light through disordered photonic-crystal waveguides [6][7][8][9] or acoustic propagation [10].
A correct description of wave transport through inhomogeneous media involves not only "propagating" traveling modes ("open" channels) but it is also very sensitive to near-field "evanescent" modes ("closed" channels) [11,12]. The contribution of evanescent modes has been known for a long time in the analysis of inhomogeneous waveguides and periodic structures like multilayered frequency selective surfaces [13], being particularly important near the thresholds of new propagating modes where the coupling between open and closed channels lead to geometric resonance effects [14][15][16]. However, the statistical properties of transport through disordered Q1D systems, obtained from numerical calculations, do not depend on the evanescent modes [17,18] and they are in good agreement with theoretical scaling approaches [2,17,[19][20][21][22][23], including the celebrated Dorokhov-Mello-Pereyra-Kumar (DMPK) [20] and nonlinear sigma-model [21] approaches, which do not take into account explicitly evanescent modes. Why evanescent modes are critical for a given configuration while it seems they play no role after statistical averaging over different configurations? One of our main goals here is to provide an answer to this open question.
In this Letter, we use numerical simulations and a perturbative analytical approach to analyze the influence of evanescent modes on the statistical properties of the transport coefficients of a thin disordered slab inside a wire or waveguide. The influence of the evanescent modes is analyzed in the asymptotic region for the far field and no near to the physical interface of the disordered region, where is well known that the evanescent modes affect strongly the statistics of the near-field [11]. The wave number k is considered halfway between the threshold of the last propagating mode and the first evanescent mode; this avoids the strong influence of the first evanescent mode on the statistical scattering properties when k is near to the threshold of a new propagating mode [14].
Assuming a weak scattering, non-dissipative, random medium with white-noise statistics, we find that the statistical averages of (power) transmittances and reflectances are solely determined by the mean free paths (MFPs) (in agreement with scaling theory). As we will show, in the so-called dense weak scattering limit (DWSL) [17,23], the MFPs themselves do not depend on the evanescent modes. In striking contrast, the statistical average of the scattered field (coherent field) present a completely different behavior (which was not captured by previous scaling approaches). The propagation of the coherent wave field [24] is characterized by an effective wave number, whose real part determines the speed of propagation, while its imaginary part represents the losses due to scattering (often known as waveguide extrinsic losses). As the wave propagates, the amplitude of the coherent part decays exponentially, whose rate decrease is the scattering MFP. We will see that the scattering MFP is insensitive to evanescent modes, while changes in the phase of the coherent field (i.e. in the real part of the effective wave number) are solely related to evanescent modes.

II. MODEL SYSTEM AND NUMERICAL RESULTS
Consider a wave propagating along the x-direction in the two-dimensional (2D) waveguide sketched in Fig. 1. In this work we restrict ourselves to the problem of scalar waves following the equation ∇ 2 Ψ+k 2 Ψ = U Ψ. For electron transport, Ψ and U represent the wave-function and the random potential (U (ρ) = 2mV (ρ)/ 2 ), respectively and k = 2π/λ is the wave number in absence of fluctuation (k 2 = 2mE F / 2 for free electrons with effective mass m and Fermi energy, E F ). For electromagnetic waves, Ψ would be the electric field normal to the (x, y) plane propagating through a waveguide filled with a material whose refractive index fluctuates around an averaged value, e.g. a fluid with spatial density fluctuations or a dielectric with real and positive relative permittivity ǫ h (in general, ǫ h > 1) and random inclusions with fluctuating permittivity ∆ǫ(x, y) in the region 0 ≤ x ≤ L. U would then be given by U (ρ) = −k 2 ∆ǫ(ρ)/ǫ h with k = (ω/c) √ ǫ h (ω the frequency and c the speed of light). The lateral confinement define a set of waveguide eigenfunctions where the integer number b labels the wave modes, also referred to as scattering channels. Modes with b ≤ N (≤ kW/π), are propagating modes with longitudinal The fluctuating potential is considered as a sequence of thin random slices (b) with zero mean scattering potential (for electromagnetic waves, this would correspond to random inclusions with permittivity slightly larger and lower than the background permittivity).
wave number k b = k 2 − (bπ/W ) 2 real; N denotes the number of traveling modes supported by the waveguide. Those modes with b > N have an imaginary wave number and represent evanescent modes. The disordered slab is constructed as a sequence of n (≫ 1) statistically independent and identically distributed thin scattering units of thickness δ, with kδ ≪ 1. The scattering units are separated one each other by a fixed distance d ≫ δ in the wave propagation direction x, with L = nd + δ as it is sketched in Fig. 1. The r-th scattering unit (r = 0, 1, 2, · · · , n), centered at x r = rd, is specified by its potential u r (y) is again constructed as a sequence of m (= W/δ ≫ 1) segments of length δ centered at y s = (s − 1/2) δ, where the potentials in each scattering unit u rs are statistically independent and uniformly distributed between − √ 3u 0 and + √ 3u 0 , i.e. with The angle brackets represent the ensemble average over different disorder realizations. If the limits δ → 0 and d → 0 are considered, our model approaches the standard Gaussian white noise-like potential models. For each microscopic realization of disordered slab, the (Fresnel) complex transmission, t aa0 , and reflection, r aa0 , coefficients (for a wave incident in channel a 0 and scattered in channel a) are numerically obtained by using a generalized scattering matrix (GSM) technique see Refs. [13,17,25] and App. A 1. We also obtain the corresponding (flux/power) transmittances and reflectances T aa0 ≡ |t aa0 | 2 and R aa0 ≡ |r aa0 | 2 , the total channel transmittance T a0 = N a=1 T aa0 , total reflectance R a0 = N a=1 R aa0 as well as the dimensionless conductance of the waveguide, g ≡ N a0=1 T a0 . Although the transport coefficients link (far-field) incoming and outgoing propagating channels, their actual values are known to be strongly dependent on the number of closed channels N ′ (in practice, N ′ is increased until convergence of the numerical results). Independently of N ′ the GSM approach guaranties flux conservation (T a0 + R a0 = 1) for each realization.

III. EVANESCENT MODES AND STATISTICAL AVERAGES
In order to illustrate the effect of the evanescent modes on the statistical averages of different quantities, we consider a waveguide that supports N = 2 propagating modes, with kW/π = 2.5 far away from a new propagating mode; in this case, the scattering matrix of the disordered system is a 4 × 4 matrix, while its transmission t and reflection r blocks are 2 × 2 matrices themselves. For this system, we perform four numerical simulations, each one considering a different number of evanescent modes N ′ = 0, 1, 2, 3.
The results for a 0 = a = 2 are summarized in Fig. 2 (a complete summary for all the other coefficients can be found in App. A 2), where the statistical averages are plotted as a function of the slab thickness L in units of the mean free path ℓ. The mean free path ℓ, is numerically obtained (kℓ ≈ 100) from the slope of the conductance curve at L = 0 ( g L ≈ N (1 − L/ℓ + · · · )) see Ref.
[5] and App. A 3. All the expectation values involve an ensemble average of 10 6 different realizations of the microscopic random potential with kd = 0.1, kδ = 0.001 and u 0 δ 2 = kδ/(4 √ 3). Our results show that the ensemble averages of both transmittance, T aa0 (Fig. 2a) and reflectance, R aa0 (Fig. 2b) are not sensitive to the number of closed channels that are taken into account in the calculations, a remarkable unexplained result already pointed out in previous numerical simulations for those quantities [17,18]. In contrast (Fig. 2(c-f )), the complex transmission and reflection coefficients are strongly dependent on the evanescent modes field but in a peculiar way: while the real (imaginary) part of the transmission (reflection) coefficients are rather insensitive to the number of evanescent modes (at least for small thicknesses), (Color online) Numerical results. a-b) Transmittance T22 , Reflectance R22 and their coherent intensities | t22 | 2 , | r22 | 2 . c-f) Complex Coefficients t22 and r22 . Each simulation considers N = 2 propagating modes and different evanescent modes (N ′ = 0, 1, 2, 3).
both Im t 22 and Re r 22 show a clear dependence with the number of evanescent modes. Before discussing the origin of these, apparently, puzzling results, it is interesting to notice that the intensity of the averaged transmitted and reflected fields (the so-called "coherent" intensity), defined as | t aa0 | 2 and | r aa0 | 2 , is also independent on the evanescent modes as shown in Fig. 2a, b (the difference T aa0 − | t aa0 | 2 = |∆t aa0 | 2 corresponds to the diffuse field intensity). This suggest that evanescent modes only affect the phase of the averaged (coherent) fields: see App. A 4.
In order to understand the numerical findings, let us first consider the transmission, t r , and reflection, r r , matrices for a single slice centered at x = x r . Expanding the wave function in transverse eigenfunctions and assuming that the wave function inside the slice is constant along x, it is easy to obtain where ϕ ± r are diagonal matrices, (ϕ ± r ) bb ′ = ϕ ±b (x r )δ bb ′ and T r is the transition T -matrix of a thin scattering unit being the matrix elements In contrast with previous approaches based on point or "delta"-scatterers [14,16], our approach (with slices having a finite thickness δ) converges for any number of evanescent modes and does not require any regularization scheme.
In absence of dissipation, flux conservation leads to the Q1D version of the Optical Theorem (OT) [27] for a single slice: which relates the imaginary part of the forward scattering amplitude to the total scattered intensity.
In the limit L → 0 (in the macroscopic sense, i.e. the slab still contains a macroscopic number of scatterers n ≫ 1), we can estimate the effective wave number k eff inside a thin slab by comparison between the average transmitted coherent field, Ψ , to the result of an homogeneous uniform media [24,28] For weak scattering units, Eq. (12) leads to an effective longitudinal wave number inside the thin slab given by (k eff ) a0 ≈ k a0 + ∆k a0 , being In Eq. (13), we have identified ℓ −1 a0 = Im(k eff ) a0 as the inverse of the scattering MFP for the incoming open channel a 0 . ℓ a0 can be related to the channel-channel mean free paths ℓ aa0 , which are associated with the incoherent sum of reflections from channel a 0 to channel a, The last identity, in agreement with Eq. (13), is a direct consequence of the OT for a single slice: see Eq. (10). Notice that our definition of the scattering MFP, ℓ a0 , commonly used in the scaling theory of transport [2], differs by a factor of 2 from that of the MFP ℓ * a0 of kinetic theory, i.e. ℓ a0 = 2ℓ * a0 .

IV. WEAK SCATTERING LIMIT
In the weak scattering limit, T r ≈ δu r −δu r Gδu r +· · · (the two first terms corresponding to the second order Born approximation [16]), so from Eq. (13), we easily obtain Near the onset of a new propagating channel, G bb diverges and the weak scattering approximation is not longer valid. However, as we have mentioned before, the wave number has been considered halfway for the threshold from the last open channel and the first closed channel, i.e., kW/π = N + 1/2. In this regime, while the scattering MFP ℓ −1 a0 = Im(k eff ) a0 is independent on the evanescent modes, changes in 1/ℓ ′ a0 come solely from the evanescent modes. This result, valid for arbitrary transversal correlations inside each slab, is a key important outcome of the present work. If we restrict ourselves to short range correlations (see Eqs (3) and (4)), k eff can be rewritten as which, in the limit W → ∞, is equivalent to that obtained for a homogeneous infinite medium, with sub wavelength short range correlations [29].
In order to obtain a physical description of the macroscopic statistics beyond the L → 0 limit, we make use of the Born series in the DWSL. Since the Born approach generates a double series expansion in powers of L/ℓ and in inverse powers of kℓ (here k denotes symbolically any possible wave number k b , while ℓ does represent any physical parameter ℓ aa0 , ℓ a0 or ℓ ′ a0 ), we work in the short-wavelength approximation (SWLA) [17], which fix the relation between the characteristic lengths: kδ ≪ 1 ≪ kL ≪ kℓ. After a lengthly calcu- lation, we find: which shows that the transmittances and reflectances only depend on the MFP's and, as a consequence, they are independent on the number of evanescent modes. As it is shown in Fig. 3(a), in the ballistic regime L < ℓ (with 1/ℓ = N −1 N a=1 1/ℓ a ), there is an excellent agreement between the theoretical predictions and the numerical results of Fig. 2(a, b). The complex coefficients, given by are also in excellent agreement with the numerical results as it can be seen in Fig. 3(b − d). The results for t aa0 suggest an exponential behavior ∼ δ aa0 e i∆kaL = δ aa0 e iL/ℓ ′ a 0 e −L/ℓa 0 which differs from the simple exponential decay predicted by the scaling theory given in Ref. [23]. In contrast, r aa0 oscillates (∝ e i2kaL ) around a constant background ∼ −δ aa0 ∆k a /2k a which explains the peculiar behavior of the reflection coefficients in Fig.  2(e, f ): see details in App. A 3.

V. CONCLUSIONS
The numerical and theoretical results of the present work demonstrate that even in the asymptotic region and far away from the threshold of the first evanescent mode, the evanescent modes could affect the statistical scattering properties of disordered waveguides. Equations (A5f), (20) together with (13), (16) and (17) summarize the main theoretical results of this Letter. They show that in the dense weak scattering limit, the mean free paths ℓ aa0 do not depend on the evanescent modes which provide a simple explanation of the numerical results and the success of the scaling approach to wave transport. In contrast, we have shown that the so-called coherent field and its effective wave number depend both on propagating and evanescent modes which could be specially relevant to understand the influence of disorder in the propagation of slow waves in a waveguide or when dealing with a device where interferences between different wires or waveguide arms are relevant. Our predictions could be tested, for example, in microwave waveguides where analysis of transport coefficients and intensities is currently performed to obtain accurate values of the permittivity of complex solid and liquid dielectrics [30] or by direct measurement of both real and imaginary part of the electric fields [31]. Equivalent measurements in the visible or infrared would require waveguide heterodyne methods [32].  This supplemental material gives technical support and complementary details to the information presented in the main text of the Letter. The supplemental material is organized as follows: In Sec. A 1 it is introduced the concept of the generalized scattering matrix that is used to obtained the numerical results shown in the Letter. In Sec. A 2, it is shown the complete set of numerical results for the expectation values of the transmittance, reflectance, conductance and the complex coefficients. In addition, the complementary information presented in Sec. A 2 shows that the numerical results are consistent with the flux conservation property. In Sec. A 3, we present a technical discussion of the Born series prediction given in the Letter for the expectation values of the observables on interest. Finally, in Sec. A 4 we give a brief discussion of the coherent and diffuse fields contributions.

Generalized scattering matrix method
As it is mentioned in the text, the numerical results for the expectation values of the complex reflection r aa0 and transmission t aa0 coefficients as well as the reflectance R aa0 and transmittance T aa0 , are obtained by using the extended or generalized scattering matrix (GSM) technique. In this section we briefly introduce the concept of the generalized scattering matrix S and how this one was implemented to obtain the numerical results of the Letter. The interested reader can find details of the GSM method in Refs. [2,13,25].
In order to illustrate the implementation of the GSM technique, we first consider the most general situation of the scattering problem in a waveguide, which is shown in Even though the scattering problem is completely described by the S matrix, it is important to notice that the closed channel amplitudes b (−) Q decrease exponentially as we move away from the disordered system, which is shown schematically in Fig. 4; moreover, those amplitudes do not contribute to the flux density current. For this reason, the scattering problem is usually described in terms of the well known open channels or reduced scattering matrix S, that relates open channel outgoing-and incoming-wave amplitudes in the asymptotic region, i.e., We should notice, however, that the reduced matrix S of a given realization of the microscopic disorder, is extracted from the generalized matrix S, which in turn is Q , are the corresponding outgoing closed channel vectors, whose dimensionality N ′ is, in principle, infinite.
calculated by combining the generalized scattering matrices s r ∼ T r (r = 1, 2, · · · , n) of the individual scattering units: see Eqs. (5) and (6) in the Letter. This combination captures any possible open or closed channel transition, which are result of the multiple scattering processes inside the disordered system. Once the reduced matrix S is known, the complex reflection r aa0 and transmission t aa0 coefficients as well as the corresponding reflectance R aa0 = |r aa0 | 2 and transmittance T aa0 = |t aa0 | 2 , are easily calculated. This procedure is repeated numerically for each microscopic realization of the disordered system, which allows us to generate an ensemble of matrices S and consequently to obtain, numerically, the ensemble averages r aa0 , t aa0 , R aa0 and T aa0 .
The implementation of the GSM method described above, exhibits that the closed channel or evanescent waves contribute implicitly to the S matrix given in Eq. (A2). For this reason the S matrix of any microscopic realization of the microscopic disorder -and consequently the statistical properties associated to S -depend formally on the closed channels, which in general are neglected in the theoretical studies of quasi one dimensional (Q1D) disordered systems. It is important to mention that, the GSM method also guaranties the flux conservation S † S = I. This means that for any microscopic configuration of the disorder, the total transmittance T a0 = N a=1 T aa0 and reflectance R a0 = N a=1 R aa0 satisfy the condition T a0 + R a0 = 1; consequently, the numerical results shown in the Letter are consistent with the flux conservation property.

Complete set of numerical results
The numerical results shown in Fig. 2 Here in Fig. 5, we present the complete set of numerical expectation values of the transmittance T aa0 and reflectance R aa0 . That figure exhibits the same behavior reported in the Letter: the inclusion of the closed channels does not have an important effect in T aa0 and R aa0 . In a similar way, Fig.  6 shows that the expectation values of the total transmittance T a0 = N a=1 T aa0 and reflectance R a0 = N a=1 R aa0 as well as the dimensionless conductance g = N a,a0=1 T aa0 = N a0=1 T a0 are also insensitive to the closed channel contributions. In Fig. 6, we can also appreciate that the numerical expectation values T a0 and R a0 satisfy the flux conservation property, i.e., T a0 + R a0 = 1; therefore, the numerical results are consistent with the flux conservation property.
The numerical evidence shown in Figs. 5 and 6, confirms previous numerical results, where the GSM method was implemented [17]: the closed channels do not contribute in the statistics of transport coefficients. This evidence justifies the omission of the closed channels in the description of the statistical properties of the transport coefficients T aa0 and R aa0 , which is a common approximation in the theoretical study of wave transport through disordered systems; however, there is not any theoretical explanation for this numerical evidence.

b. Complex Transmission and Reflection Coefficients
The numerical results of Fig. 7 show a decreasing behavior with L/ℓ for Re t a0a0 , which becomes more notorious as the number of closed channels (N ′ = 0, 1, 2, 3) considered in the calculations is increased. This figure also shows that the influence of the closed channels in Im t a0a0 is even more dramatic: if closed channels are not included in the numerical simulations (N ′ = 0), Im t a0a0 is small (∼ 10 −2 ), but not strictly zero, however, when the closed channels are considered (N ′ = 1, 2, 3), Im t a0a0 increases one order of magnitude.
In Fig. 8 we can appreciate a remarkable oscillatory behavior for r a0a0 , which is rapidly attenuated as L/ℓ increases, regardless the number of closed channels that are considered in the calculation. The phase and amplitude of r a0a0 depend little on the closed channels contributions. However, the most notorious dependence on the closed channel contributions is exhibited by Re r a0a0 , whose oscillations are given around a "background" that depends on the number of closed channels considered in the calculations; in contrast, the four simulations of Im r a0a0 seem to oscillates around the same "background", no matter how many closed channels were used in the calculations.
Previous theoretical models cannot describe the nu-merical evidence shown in Figs. 7 and 8 for t a0a0 and r a0a0 , what is due to the absence of closed channels in those descriptions. As an example, we consider MT prediction [23], for the expectation values of the complex coefficients r aa0 where ℓ a0 is the scattering mean free path of the incoming open channel a 0 : see Eq. (14) in the Letter. Since in the weak scattering limit, the scattering mean free path ℓ a0 does not depend on the closed channels inclusion [see Eq. (16) in the Letter], then Eq. (A3) cannot describe the influence of the closed channels in t a0a0 , that is shown in Fig. 7. We have verified that the MT prediction t a0a0 (MT) = e −L/ℓa 0 is indistinguishable from the numerical result Re t a0a0 , when the closed channels are not considered in the calculation. However, once the closed channels are considered (N ′ = 1, 2, 3), Re t a0a0 decreases faster than the MT prediction Eq. (A3). In addition, the behavior Im t a0a0 = 0 is not predicted by MT result. On the other hand, although the numerical results of Fig. 8 show that the amplitude of the oscillation of r a0a0 is small (∼ 10 −2 ), this expectation value is not zero, even when the closed channels are not included; therefore, the MT prediction r a0a0 (MT) = 0 is not able to describe the numerical results of Fig. 8.

Theoretical results
In the Letter, the Born series predictions for expectation values are use to understand the role of the closed channels in the expectation values T aa0 , R aa0 , t aa0 and r aa0 . In the ballistic regime (L/ℓ ≪ 1) and in the SWLA (kℓ ≫ 1), this perturbative approach shows an   (19) and Fig. 2 in the Letter. In this section we give some complementary details above Born series prediction and also present its comparison with the complete set of numerical expectation values. To do this it is convenient to rewrite Eqs. (18) and (19) of the Letter as follows where the real and imaginary parts of t a0a0 and r a0a0 have been written explicitly.
As it is discussed in the Letter, the role of the closed channels in the statistical parameters ℓ aa0 , ℓ a0 and ℓ ′ a0 , allows us to understand theoretically the numerical results shown in Figs. 5-8: • The expectation values of the transport coefficients T aa0 and R aa0 are insensitive to the closed channels inclusion, what is due to the absence of ℓ ′ a0 in their lowest order contributions in powers of 1/kℓ.
• The dominant contributions in powers of 1/kℓ of expectation values for the expectation values t aa0 and r aa0 depend strongly on the closed channels through ℓ ′ a0 .

a. Complex Transmission and Reflection Coefficients
In Eq. (A5a), we can appreciate that the linear term L/ℓ a0 of Re t a0a0 , depends only on the scattering mean free path ℓ a0 , so that the first closed channel contributions appear in the quadratic term (L/ℓ ′ a0 ) 2 . This fact explains qualitatively the numerical behavior shown in Fgs. 7a and 7c, where the four curves (N ′ = 0, 1, 2, 3 closed channnels) for Re t a0a0 are quite similar for small values of L/ℓ. On the other hand, the theoretical prediction given in Eq. (A5b) shows that, at the origin, the slope of Im t a0a0 depends strongly on the closed channels through the ℓ ′ a0 . This dependence on ℓ ′ a0 explains qualitatively, in the ballistic regime (L ≪ ℓ), the numerical evidence shown in Fgs. 7b and 7d for Im t a0a0 .
In Fig. 9 the Born series prediction for t a0a0 , Eqs. (A5a) and (A5b), it is compared with the numerical simulations shown in Fig. 7. The comparison is done when N ′ = 3 closed channels are taken into account in the calculations: the agreement is excellent in the ballistic regime.
Equations (A5c) and (A5d) show that, both Re r a0a0 and Im r a0a0 have an oscillatory behavior with e 2ka 0 L , whose amplitudes are modulated by factors in powers of 1/k a0 ℓ a0 , 1/k a0 ℓ ′ a0 and in powers of L/ℓ a0 , L/ℓ ′ a0 and L/ℓ a0a0 . To understand the numerical results shown in Fig 8, it is important to notice that, Re r a0a0 and Im r a0a0 contain a length independent contribution: −1/2k a0 ℓ ′ a0 and −1/2k a0 ℓ a0 , respectively. As we know, the mean value of the sine and cosine functions is zero, so that the theoretical curves of Re r a0a0 and Im r a0a0 shall oscillate, in the ballistic regime, around their corresponding L-independent contribution. Since ℓ ′ a0 is extremely sensitive to the number of closed channels considered in its evaluation [see Eq. 16 in the Letter], Re r a0a0 suffers a displacement as we increase the number of closed channels. In contrast, ℓ a0 is insensitive to the closed channels contributions, so that Im r a0a0 does not show any displacement with the inclusion of the closed channels. This analysis, allows to understand qualitatively, in the ballistic regime, the numerical evidence shown in Fig. 8 for r a0a0 . Figure 10 shows the comparison between the Born series prediction for r a0a0 , Eqs. (A5c) and (A5d), and the numerical simulations of Fig. 8. The comparison is done when N ′ = 3 closed channels are taken into account in the calculations. The agreement is good enough in the ballistic regime, where the amplitude and the phase of oscillation show quantitative agreement. However, as the L/ℓ increases, the agreement in the phase ceases to be good, suggesting that the oscillation not only depends on e 2ka 0 L .

b. Transmittance and Reflectance
Equations (A5e) and (A5f) show the perturbative results for the expectation values T aa0 and R aa0 . It is easy to verify, up to the order (L/ℓ) 2 , that those perturbative expressions are consistent with the flux conservation property, i.e., As it is mentioned in the Letter, the dominant contributions in powers of 1/kℓ of the expectation values T aa0 and R aa0 only depend on the channel-channel ℓ aa0 and scattering mean free ℓ a0 paths, which are insensitive to the closed channels inclusion. The closed channels contribute to the expectation values T aa0 and R aa0 at the order 1/kℓ or higher, where characteristic length ℓ ′ a0 appears; for instance, the first contribution in Born series expansion of R aa0 , is given in the following way: (A7) The second term of the last equation is of the order (1/kℓ) 2 , so that, in the SWLA (kℓ ≫ 1) and in the ballistic regime (L/ℓ ≪ 1), the linear term L/ℓ aa0 is the leading term of the first Born approximation; therefore, the closed channel contributions do not significantly modify the behavior of the expectation values of the transport coefficients T aa0 and R aa0 . This explains qualitatively, at least in the ballistic regime, the numerical evidence shown in Fig. 5.
In Fig. 11 we compare the Born series prediction of T aa0 and R aa0 , Eqs. (A5e) and (A5f), with the numerical results given in Fig. 5. The comparison is done when the waveguide supports N = 2 open channels and N ′ = 3 closed channels are taken into account in the calculations. Once again, the Born series prediction gives an excellent agreement with the numerical simulations in the ballistic regime L ≪ ℓ.
b. Influence of the diffuse fields in the expectation values of the transport coefficients Equation (A10) shows that the ensemble average of the transmittance T aa0 and reflectance R aa0 are constituted of two different contributions: the coherent field intensities | t aa0 | 2 , | r aa0 | 2 and the diffusive field contributions |∆t aa0 | 2 , |∆r aa0 | 2 . In this section we briefly analyze the contributions of those kind of contributions.
The structure given in Eq. (A10), is numerically illustrated in Figs. 2a-b) of the Letter, where the ensemble average of the transport coefficients T 22 , R 22 and the corresponding coherent intensities | t 22 | 2 , | r 22 | 2 are plotted: the difference between T 22 , R 22 and the corresponding coherent intensities exhibit the influence of the diffusive fields |∆t 22 | 2 , |∆r 22 | 2 . In the Letter, Fig. 2a) shows that the coherent field intensity | t 22 | 2 becomes less important for large values of L/ℓ, where the diffuse field governs the behavior of T 22 . In contrast, Fig. 2b) shows that the diffuse field |∆r 22 | 2 (Num) dominates the behavior of R 22 (Num) ∀L/ℓ. In order to understand the numerical evidence shown in Figs. 2a-b) of the Letter, in this we use the Born series prediction to identify the coherent and diffuse fields contributions.
From Eqs. (A5a) and (A5b), it is easy to obtain, up to the order (L/ℓ) 2 , the coherent field contributions to the expectation value of the transmittance T aa0 , i.e., | t aa0 | 2 = δ aa0 1 − 2 L ℓ a + 2 L ℓ a 2 + · · · + O 1 kℓ (A12a) Since the last expression is exactly the same to the first square parenthesis in Eq. (A5e), then we can identify the second term of Eq. (A5e), as the diffusive field contributions to T aa0 , i.e., A similar procedure allows to demonstrate that the coherent field contribution of the reflectance R aa0 is of the order (1/kℓ) 2 , i.e., while the diffusive field contribution is given by Equations (A12) and (A13) show that expectation value of the off diagonal coefficients T aa0 (a = a 0 ) and each R aa0 are dominated by their corresponding diffuse fields. The coherent fields are only relevant for the expectation value of the diagonal transmission coefficient T a0a0 . The numerical results for T 22 and R 22 shown in Figs. 2a-b) of the Letter, are in good agreement with the last theoretical prediction.