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Time-dependent simulation and analytical modelling of electronic Mach-Zehnder interferometry with edge-states wave packets

We compute the exact single-particle time-resolved dynamics of electronic Mach-Zehnder interferometers based on Landau edge-states transport, and assess the effect of the spatial localization of carriers on the interference pattern. The exact carrier dynamics is obtained by solving numerically the time-dependent Schroedinger equation with a suitable 2D potential profile reproducing the interferometer design. An external magnetic field, driving the system to the quantum Hall regime with filling factor one, is included. The injected carriers are represented by a superposition of edge states and their interference pattern reproduces the results of Y.Ji et al.[Nature 422, 415 (2003)]. By tuning the system towards different regimes, we find two additional features in the transmission spectra, both related to carrier localization, namely a damping of the Aharonov-Bohm oscillations with increasing difference in the arms length, and an increased mean transmission that we trace to the energy-dependent transmittance of quantum point contacts. Finally, we present an analytical model, also accounting for the finite spatial dispersion of the carriers, able to reproduce the above effects.


I. INTRODUCTION
Edge states (ESs) are chiral one-dimensional conductive channels which arise in a 2D electron gas (2DEG) in the Integer Quantum Hall regime (IQHE) 1,2 . They are essentially immune to backscattering and are characterized by very long coherence lengths 3 . Besides their remarkable interest for basic solid-state physics and coherent electronic devices, they are ideal candidates for both the so-called "electron quantum optics" 4 and "flying qubit" implementations of quantum computing architectures 5,6 . Indeed, several attempts to demonstrate single-qubit gate operations and electronic interferometers in coupled quantum wires were hampered by scattering and decoherence processes [7][8][9] . On the other hand, experimental realizations of electronic interferometry based on edge-channel transport seem more mature, to the point of demonstrating not only single-electron 10,11 but also two-electron interference 12 . Also, "whichpath" detectors or quantum erasers [13][14][15][16][17] have been implemented and the formation of quantum entanglement between indistinguishable particles has been demonstrated 12,[18][19][20][21] . While specific features of electronic Mach-Zehnder Interferometers (MZIs) 22 are still under investigation and involve possibly many-particle effects 21,23,24 , their basic functioning can be explained in terms of stationary electron waves interference, like their more common optical counterpart. However, contrary to photons, the time that a conduction electron takes to cross the interferometer can be comparable to switching times of typical microelectronic devices, and its spatial localization can be much less than the dimensions of a single arm of the interferometer. Thus, understanding the detailed quantum dynamics of the carriers, beyond the simple plane wave model, is critical for the design of novel devices based on ES transport. This is even more relevant if the device is intended to process the information encoded in a single particle at a time, a task which requires a high resolution both in time and space. However, the vast majority of the literature on electron interferometry in ESs deals with electronic currents 10,12,13,17,22,25 , and the theoretical models for the transport are based upon delocalized scattering states [14][15][16]18 , with few notable exceptions 26,27 .
In this work we address, both from numerical and analytical points of view, the interference properties of a Mach-Zehnder device based on ES channels tuned to filling factor ν=1 and quantum point contacts used as splitting elements, when the electron travelling inside it is strongly localized. We first use a numerical approach, which, unlike other recent works 28,29 , is based upon the direct solution of the effective-mass time-dependent Schrödinger equation, in presence of an external magnetic field. The electrons travelling inside the device are localized wave packets (WPs) of ESs, and are propagated by a generalized split-step Fourier method [30][31][32] . The effects of the WP size on the transport process is analyzed. Then, we develop an analytical model also accounting for the finite spatial dispersion of the carriers. The transmission coefficient of the device subject to Aharonov-Bohm (AB) oscillations 33 is obtained as a function of the magnetic field and of the geometrical parameters of the interfering paths, and the results are compared with the numerical simulations. Specifically, in Section II, we describe the initial electronic wave function and the physical device used in our simulations. In Section III we give some details on the numerical algorithm that we adopt and we describe the results of the numerical simulations. Section IV is devoted to the developement of an analytical model for the transport, which takes into account the energy dispersion of the wave packet. A comparison between the predictions of the latter model and the numerical simulations is then presented in Section V. Finally, the conclusions are drawn in Section VI.

II. PHYSICAL SYSTEM
We consider a conduction-band electron in a 2DEG on the x-y plane, with charge −e and effective mass m * . A uniform magnetic field B = (0, 0, B) is applied along the z direction and a non-uniform electric field induces a local potential energy V (x, y). The latter will represent the field generated by a polarized metallic gate pattern above the 2DEG that will define energetically forbidden regions. The generic Hamilto-nianĤ = (−i ∇ + e A) 2 /(2m * ) + V can be rewritten in a more explicit form by substituting the magnetic vector potential A with its expression in the Landau gauge A = (0, Bx, 0). The 2D Hamiltonian for the electron in the 2DEG then readŝ Our time-dependent numerical simulations and analytical model are based on Eq. (1), as we explain in the following. Before that, we must recall briefly the derivation of Landau states and of corresponding ESs 34 . Let us consider a region where the electric potential is invariant along y, i.e. V (x, y) = V (x). In the Landau gauge, the Hamiltonian (1) shows an explicit translational symmetry along the y direction, and the quantum evolution of the particle can be factorized along the two axes. In this case, the eigenstates ofĤ have the form Ψ(x, y) = ϕ(x)e iky . In fact, the y-dependent part of the wave function (WF) is a plane wave, while the x-dependent part (which depends also on the wavevector k) is a solution of H ef f L ϕ n,k (x) = E n (k)ϕ n,k (x), with the following 1D effective Hamiltonian where and where ω c = −eB m * is the cyclotron frequency. Note that the parameter x 0 representing the center of the effective parabolic confinement along x depends on k, i.e. the wavevector in the y direction. The discrete energy levels E n (k) associated withĤ ef f L are the socalled Landau levels (LLs). If V (x) = 0, the system eigenfunctions are the wellknown localised Landau states. However, if V (x) is a step-like function, defining a sub-region in which the electrons are confined, the Landau states with x 0 close to the edge have higher energy, and show a finite dispersion in k. They become edge states, which are delocalized WFs associated with a net probability density flux, and which can act as 1D conductive channels.
Since we want to model a carrier as a propagating wave packet, beyond the delocalized scattering state model, we need to construct such WF as a proper combination of ESs, rather than considering a single ES. This is similar to representing a free electron by a minimum uncertainty wave packet rather than a plane wave, thus including a finite uncertainty in both its position and its kinetic energy. However, in our case the magnetic field couples the two directions and some care has to be taken, since the position along x and the velocity along y are not independent. We suppose that the injected electron lies in the first LL and we represent it as a linear combination of ESs with n = 1, with gaussian weights 35 where F (k) is the Fourier transform of a 1D gaussian function along the y axis: This leads to a localized WF along both directions: in y, the gaussian envelope gives a finite extension around the central position y 0 , in x, the functions ϕ 1,k (x) are always localized around x 0 (k), and the wave vector k is, in turn, localized around k 0 by our choice of F (k). With the inital condition Eq. (4), our simulations will be able to take into account the effects of the size σ of the wave packet (WP) on the interference phenomena. In order to use Eq. (4), we will be careful to initialize the WP Ψ 0 where the potential V (x, y) does not depend on y. The specific form of the ESs depends on the the shape of the potential barrier V (x), and in general there is no closed-form expression for them. In order to have a realisitc model of the smooth edges between the allowed 2DEG region and the depleted one, defined e.g. with the split-gate technique, we take where F τ (x) = (exp(τ x)+1) −1 is a Fermi distribution with a "broadening" parameter τ , and we compute numerically the corresponding ϕ n,k (x) states. The shape of the depleted regions of our 2DEG is chosen in order to mimic, with the ES channels, the Mach-Zehnder interferometer (MZI) of Ref. 10. This is depicted in Fig. 1, where the dark and light regions represent V (x, y) = 0 and V (x, y) = V 0 , respectively. Two narrowings, where two areas with high potential are close to each other, form two quantum point contacts (QPCs), both with a square area 36 of δ 2 : their dimension δ is tuned in order to give a trasmittivity of 50% in the two output channels. The localized electron is injected from the top of Fig. 1 (where the initial WF is centered around x = −20 nm and y = 700 nm), in the first LL, whose ESs follow the boundary between the high-and low-potential regions. After the first QPC, each ES is split into two parts that constitute the two arms ("left" and "right") of the interferometer. They are rejoined at the second QPC. As a consequence, the two components of the WP that follow the two arms, interfere at the second QPC. Indeed, two outputs are available there to the electron: one part is reflected inside the MZI, and then absorbed by a contact (modeled as an absorbing imaginary potential V abs , as detailed later), the other part is transmitted towards the bottom of the device. The norm of the latter part gives a measure of the transmission coefficient T of the device. The geometrical parameters of the MZI are indicated in Fig. 1. Specifically, the "right" arm and the "left" arm have two horizontal sections each, of length L 0 and L 0 + a L , respectively. The vertical sections of the two arms have obviously the same length, H 0 . The absorbing lead is modeled by a purely imaginary potential 32,37 where x c and y c are the coordinates of the contact, while δ c and d are, respectively, the contact extension along the x and y axes. By changing the parameters of the device, such as a L (which controls the relative lenght of the two arms and the area) or the magnetic field B, we observe AB oscillations in the transmission coefficient T . With the introduction of the absorbing lead V abs , the calculation of T is straightforward, since it is the norm of the final wave function after the absorption process. Differently to other works 38 focusing on the dynamics of carriers moving along Hall ESs, our confining potential V does not depend on time.
The "center" x 0 (k 0 ) of the initial WP (t = 0) is fixed at a distance ∆ from the inflexion point of the potential barrier V (x), and consequently k 0 can be calculated by Eq. (3). The resulting local bandstructure of the 1st LL around k = k 0 can be determined numerically, and from a parabolic fit we get the values of the parameters E 0 , m * B and k 1 in its expression: The parameters k 0 and k 1 are gauge-dependent quantities, since they depend by the origin of the coordinate system: however, with a proper choice of this degree of freedom and of the energy zero, we can set k 1 = 0 and E 0 = 0 without any physical change in the description of the system, in order to get a simpler description of the system. Now k 0 is directly related to the group velocity v g = k 0 /m * B of the WP, which behaves like a free particle of mass m * B in 1D. In the simulations we use for the electron the effective mass of GaAs m * = 0.067 m e . The parameters of the potential and of the absorbing contact are

III. NUMERICAL SIMULATIONS
The time evolution of the WP is realized with the split-step Fourier method 32,37,40 . In summary, the evolution operatorÛ (δt) = e − i δt·Ĥ is applied N times to the initial wave function Ψ(x, y; 0), each leading to the evolution of a short time step δt: The Hamiltonian of Eq. (1) can be written asĤ = By applying the Trotter-Suzuky factorization and the split-step Fourier method, the operatorÛ (δt) can be approximated with where F x(y) and F −1 x(y) denote, respectively, the direct and inverse Fourier transform with respect to the variable x(y). By using the above expression, the nu- merical solution of Eq. (9) is reduced to an alternating application of discrete Fourier transforms and array multiplications, since each operator acts only in its diagonal representation. Within this scheme, we can reproduce the exact dynamics of the electronic WF inside the device. As an example, the evolution of the WP at four different times is shown in Fig. 1 for the specific case described in the caption. The simulations have been performed with a time-step δt = 0.1 fs. As expected 10,12,17 , we obtain AB oscillations of the transmission T (see Figure 2) with respect to both the variation of the area (which is controlled through a L ) and the variation of the magnetic field B. This is consistent with the results of Refs. 10 and 12, where the tailoring of the length of one of the two paths is achieved through a modulation gate. However, we observe two extra features, namely (a) a damping of the AB oscillations with a L and (b) an increase of the average T as the initial spatial localization σ of the WP is increased, with a consequent decrease of the visibility 10,12,13,22 of the AB oscillations (of the transmitted current I), which is always smaller with respect to the ideal case v M ZI = 1. The two issues above are addressed in the following.
(a) The AB oscillations as a function of a L are modulated by a gaussian envelope, whose extension is directly correlated with the size σ of the initial WP. A fitting of T vs σ is reported in Fig. 3  50%, ending up with a total transmission of T 0.5. In general, the larger the time offset at which the centers of the two WPs reach the second QPC, the less effective is the quantum interference. Therefore, we expect a saturation value of ∼ 0.5 for T (a L ), and an oscillation amplitude of ∼ 0.5. However, our numerical simulations show a different trend, i.e. issue (b). (b) For smaller values of σ, the average (or saturation value with a L ) of the transmission T is higher and the amplitude of the oscillations is lower (Fig.  3). These effects are due to the energy-dependent features of the scattering process at the QPCs, which are included automatically in the numerical simulations based on the direct solution of the Schrödinger equation. In fact, contrary to a delocalized scatteringstate model, where the particle is represented by a single-energy state, our WP is composed by different ESs, as given in Eq. (4), each with a slightly different energy. As a consequence, the two parts of the WP, transmitted and reflected by the QPC, have different spectral weights, which depend on σ. Only in the limit of σ → ∞, the WP is split into two identical parts that give an ideal 50% splitting with T = 0.5. In order to have a better physical insight and to give a quantitative assessment of this effect, we will include the energy-dependent trasmittivity of the QPCs in the analytical model presented in the following section.

IV. THEORETICAL MODEL FOR THE INTERFERENCE
Following Weisz et al 17 , we will represent a single ES as a 1D plane wave, and we will describe the scattering process and the transport inside the MZI with the scattering matrix formalism 2 . Of course, the effects of the scattering on the WP can be determined from the linear superposition of the scattering of single plane waves. The device can be divided into three regions -I, II and III -as indicated in Fig. 4, where the plane-wave states are given by: II.
The initial wave packet is prepared in the state and the scatterings at the two QPCs are described by the scattering matrices with j = 1 (j = 2) labeling the first (second) QPC, and where r j and t j are the transmission and reflection amplitudes. After the first QPC, the WF is given by: where F L (k) = r 1 (k)F (k) and F R (k) = i t 1 (k)F (k). While travelling in the two arms of the MZI inside region II, the WF acquires a different phase in each arm, which is given by 2,41,42 : where h = L (h = R) labels the left (right) arm and the integral is a line integral along the arm h. The first contribution, the dynamic phase ξ h , is due to the canonical momentum p, while the second contribution, the magnetic phase φ h , accounts for the vector potential A. The matrix describing the phase acquired is given by and therefore the final WF in region III is given by wheret = r 1 r 2 e iϕ L − t 1 t 2 e iϕ R andr = ir 1 t 2 e iϕ L − it 1 r 2 e iϕ R . It is easy to see that the transmitted part of the WP is given by where D 2 (k )|D 2 (k) = 2πδ(k − k ), and thereforẽ If the scattering amplitudes r j and t j are energyindependent, Eq. (24) can be integrated analytically, to give:T where T 0 = |r 1 r 2 | 2 + |t 1 t 2 | 2 and T 1 = 2r 1 r 2 t 1 t 2 . T 0 is the mean value of the oscillations, while T 1 is their maximum amplitude. In the previous formula, ∆l 2a L is the length difference between the two arms of the interferometer, while ∆φ e BH 0 (a L0 + 2L) = π Φ B Φ0 is the well-known Aharonov-Bohm phase 2,34,41 (Φ B is the flux of the magnetic field through the area of the MZI and Φ 0 = h 2e is the magnetic flux quantum) 43 .

V. DISCUSSION
As one can see, Eq. (25) provides the correct trend for the results of the simulations, with a sinusoidal oscillation enveloped by a Gaussian and with a fixed offset in the transmission. Indeed, by substituting the expressions for ∆φ and ∆l into Eq. (25), we see that the transmission coefficient should oscillate as a function of a L like cos(k e a L + ϕ 0 ), where k e = e BH − 2k 0 4.80 · 10 9 m −1 , which is consistent within 2% with the values of k e obtained by fitting the simulation data (see Table I). For what concerns the amplitude and the saturation value of T , if we suppose that the QPCs have a perfect half-reflecting behavior for all incoming energies -that is t j = r j = 1 √ 2 for all k -then our model predicts a saturation value T 0 = 0.5 and an oscillation amplitude T 1 = 0.5, as we previously stated. This prediction, however, does not match with the numerical simulations, where we observe higher values for T 0 and lower values for T 1 (see Table I). The model also predicts that the oscillations of T (a L ) should be modulated by a Gaussian with a standard deviation Σ = σ, i.e. with the same spatial dispersion of the initial wave packet. However, the actual values of Σ obtained from numerical fitting the data of Fig. 3 are bigger than σ by more than a factor of two in the case with smaller σ. As σ increases, the values of Σ (always obtained by fitting the results of the simulations) get closer to σ (see Table I). Indeed, a better agreement can be obtained by enhancing our model. This simplified model is valid under certain approximations, that are: (i) the scattering process can be treated as a quasi-1D problem; (ii) all the edge channels involved in the transmission follow the same path, i.e. we can consider the area enclosed by the interfering paths and the difference in length ∆l as they were independent on k; (iii) the reflection and transmission amplitudes r j and t j of the QPCs are energyindependent (that means also k-independent). While assumptions (i) and (ii) are in general verified, the approximation (iii) must be dropped in order to reproduce the results of the simulations. Indeed, our results can be explained considering that the scattering amplitudes of the QPCs are not energy-independent, as it has been already pointed out in the literature 19 . From a physical point of view, the edge states of higher energy are closer to the barrier, and therefore they should tunnel easier through the QPCs. This prediction can be easily verified, since the energy-dependent scattering process should produce different values in the weights F L (k) and F R (k) of the WP after the first QPC. This can be linked directly to the differences in the values of T 0 and T 1 with respect to the ideal case, as we will see briefly (a model for the correct estimation of Σ, T 0 and T 1 is given in Appendix A). The amplitudes t(k) and r(k) (which are equal for both QPCs, due to their identical shape) can be calculated numerically from |Ψ II . In Eq. (19), the squared modulus of the weights |F L (k)| 2 and |F R (k)| 2 has been calculated from a simulation on the wave packet with σ = 40 nm by projecting |Ψ II over the local eigenstates |L(k) and |R(k) 44 . Then, using a phenomenological approach, the corresponding data have been fitted with the functions f L (k) = F αr (k − k r ) |F (k)| 2 and f R (k) = F αt (k − k t ) |F (k)| 2 , thus giving a Fermidistribution expression for |r(k)| 2 and |t(k)| 2 . The results of the fits shown in Fig. 5 are in good agreement with the numerical simulations and are also consistent with the constraint |r(k)| 2 + |t(k)| 2 = 1. From the same figure, we also deduce that the QPCs are more transparent at higher energies (i.e., lower values of k), as we predicted. With the hypothesis that t(k) and r(k) are real 17 , we can insert the transmission and reflection amplitudes calculated numerically into Eq. (24) and find the exact transmission coefficientT . The results for a WP of σ = 20 nm are shown in Fig. 6: here, the simplified transmission coefficient of Eq. (25) (where t and r are energy-independent and both equal to 1/ √ 2) is compared with the exact energy-dependent calculation of Eq. (24), which reproduces the result of numerical simulations (see Fig. 3 for comparison). Then, as we see, the energy dependence of the scattering amplitudes t(k) and r(k) is able to account for the values of T 0 and T 1 and also for the broadening of Σ (the region in which we observe interference) with respect to the original spatial extension σ of the WP. Therefore, taking into account these corrections, it is still possible to use the simplified model of Eq. (25) to describe the interference, provided that one uses the energydependent values for T 0 , T 1 and Σ, that we nameT 0 , T 1 andΣ. They can be calculated from our model as: (see Appendix A forΣ and for the details of the derivation ofT 0 andT 1 ). In summary, we obtained an analytical formula for the visibility v M ZI of the device, which is Indeed, we see that the device has a reduced visibility with respect to the expected ideal value v M ZI = T 1 /T 0 = 1, due to phase-averaging effects between different energies involved in the WP (and not because of  Table I). Note how the energy-dependent analytical model is able to reproduce the numerical results of Fig. 3 (top curve) decoherence effects), as some experimental works already pointed out 10 . Interestingly, these remarkable effects, which are comparable with experimental observations, are not explicitly related to fluctuations of the area enclosed by the two paths of the interferometer, because we used the same values of ∆l and ∆φ for all edge states composing the WP. As we can see from the model, in the mono-energetic plane-wave limit (σ → ∞) the energy-dependent effects become less important, and we recover the ideal behavior with T 0 = T 1 = 1 2 (notice that r(k 0 ) = t(k 0 ) ≈ 1 √ 2 ) and unitary visibility. The gaussian modulation of the AB oscillations of the transmission coefficient disappears, since |F (k)| 2 → (2π) −1 δ(k − k 0 ): This is consistent with scattering-states models used in the literature 10,12,17 .

VI. CONCLUSIONS
In conclusion, our time-dependent simulations allow us to reproduce the interference pattern of electrons transmitted through Landau edge-states defining a MZI. Specifically, we first used the split-step Fourier method to solve exactly the time-dependent Schrödinger equation, and then we calculated the dynamics and the transmission coefficient of the interferometer, which shows Aharonov-Bohm oscillations. We exposed the effects of the finite size of the electronic WP. In fact, the longitudinal extension (i.e. along the edge state) of the WP is connected to its energy uncertainty which, in turn, is determined by the injection process of the carriers: the larger the energy uncertainty, the more localized the WP. In Ref. 10, for example, the carrier injection in the device is achieved by an ohmic contact operating as the source terminal. However, the specific dynamics of the electrons mov-ing from the Fermi sea of the metallic contact to the semiconductor edge state is unknown, in general. Indeed it is strongly affected by the atomistic details of the junction, by the temperature (i.e. the broadening of the Fermi level) and, in the case of Schottky contact, by the metal-semiconductor barrier. Furthermore, if the injection of the single carrier is realized via cyclic voltage pulses 45,46 , its WF can be partially tailored by the pulses timing and shape. With our approach, the real-space spreading of the carriers can be estimated a posteriori from the interference pattern 26 . In addition to simulating the exact dynamics, we developed an analytical model that incorporates the physics needed to reproduce the main features of the transport spectrum. Such a model must go beyond the standard delocalized scattering-state approach. Indeed, we derived Eqs. (24) and (25) by including the effects of the finite size of the electronic WP. Furthermore, the values of T 0 and T 1 in Eq. (25) must depend on the different wave vectors k composing the WP, in order to account for the energy-dependence of the scattering at the QPCs and, finally, to reproduce the offset value of the transmission, as a function of the spatial dispersion σ. Also, the model leading to Eq. (24) can justify the reduction of the visibility of the interferometer with respect to the ideal case. Finally, we stress that the understanding of the dynamics of single electrons in Landau ESs, possibly including 4-way splittings in QPCs and a non-trivial geometry, is of utmost importance for several proposals of quantum computing architectures based on edgechannel transport. Usually, in these proposals, the quantum bits are encoded into two ESs of the same LL that are physically separated in space. However, some recent studies also focused on the possibility of coupling ESs belonging to different LLs [47][48][49] , between ESs belonging to two different 2DEGs through tunneling 50 or between spin-resolved ESs of the same LL, which could also be used in the future to realize a different type of MZI 51,52 or scalable quantum gates. Also in the above systems, the effects of the spatial localization of the WP are pivotal for a realistic modeling of the devices. (where α = 1.13 · 10 −7 m is obtained by fitting the numerical simulations). We now replace these functions with gaussian profiles, since near the turning point k 0 and in the decaying tail of the Fermi distribution it holds: (here e is the Euler number). Therefore, since Eq. (24) for r 1 = r 2 = r and t 1 = t 2 = t can be written as × r(k) 4 + t(k) 4 + 2r(k) 2 t(k) 2 cos(−k∆l + ∆φ) , we deduce that, from each term in the square brackets, we can factor out exp(−2 α 2 4e (k − k 0 ) 2 ), which can be multiplied with |F (k)| 2 ∝ exp(−2σ 2 (k − k 0 ) 2 ) to give an "effective" gaussian weight with an increased standard deviation. We conclude that the energydependent value for Σ is given by: With this approximated model we obtain, for σ = 20 nm,Σ = 39.7 nm, which is in excellent agreement with the value Σ = 40.98 nm obtained from the numerical fitting of the exact simulations (see Table I).
A similar good agreement is obtained for σ = 40 nm and σ = 60 nm. Concerning the calculation ofT 0 andT 1 , we see from Eq. (A4) that the oscillating part ofT and the gaussian damping are given by the third term in the square brackets. Therefore, for large values of ∆l = 2a L , the rapid oscillations of the cosine cancel out, and only the first two terms survive, to give the saturation value of the transmission. Theñ = 2πˆdk |F (k)| 2 r(k) 4 + t(k) 4 .
As we did forT 0 , we can estimateT 1 by comparing Eq. (A4) with Eq. (25). As a first approximation, T 1 is the maximum amplitude of the oscillating part, which is obtained when all the oscillating functions are maximized and the gaussian damping is zero, i.e. when the argument ∆ϕ of all the cosines in the integral vanishes (and also ∆l = 0). Therefore, in this limit, we can subtract the backgroundT 0 fromT to getT 1 :T For example, by using the above formulae, we obtaiñ T 0 = 0.760 andT 1 = 0.240 for the WP with σ = 20 nm, that are in good agreement wit the values of Table I. It is straightforward to generalize these results to the cases when r 1 = r 2 and t 1 = t 2 , to get the expressions of Eqs. (26) and (27).