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Bound States in Sharply Bent Waveguides: Analytical and Experimental Approach

Quantum wires and electromagnetic waveguides possess common features since their physics is described by the same wave equation. We exploit this analogy to investigate experimentally with microwave waveguides and theoretically with the help of an effective potential approach the occurrence of bound states in sharply bent quantum wires. In particular, we compute the bound states, study the features of the transition from a bound to an unbound state caused by the variation of the bending angle and determine the critical bending angles at which such a transition takes place. The predictions are confirmed by calculations based on a conventional numerical method as well as experimental measurements of the spectra and electric field intensity distributions of electromagnetic waveguides.


I. INTRODUCTION
The opinion that a binding potential is necessary to obtain a bound state of a particle in quantum mechanics [1] was already exposed as being incomplete in the early days of quantum mechanics. Indeed, in 1929 John von Neumann and Eugene P. Wigner [2] showed that even a strongly repulsive potential can support a bound state with a normalizable wave function. Since then many examples of unusual bound states in various quantum and classical analogue systems with different bindung mechanisms have been investigated. For an overview see for example Ref. [3].
In quantum mechanics it is by now well-known that a binding potential in two space dimensions always supports a bound state no matter how shallow the potential, whereas in three space dimensions the potential needs to have a certain depth. The reason is that in two space dimensions the state of vanishing angular momentum, that is, a rotationally symmetric state, brings in an additional attractive potential which decays quadratically with the inverse of the distance. This anti-centrifugal potential [4] arises from the adaption of the Laplacian to the symmetry of the state and appears in many other situations where the boundary conditions or the preparation mechanism dictate the symmetry of the state. In the present paper we develop an approximate but analytical technique to describe the occurrence of such geometry-induced bound states for the case of a bent waveguide. We compare and contrast our analysis with numerical calculations based on a conventional method as well as with measurements of these bound states.

A. From compound nucleus to bent waveguides
Systems of the type considered in the present paper in fact serve as models for the understanding of properties of the scattering processes associated with compound nucleus reactions. Within such a scattering approach the compound nucleus is modeled by a confined region of quasibound states, and a number of open reaction channels attached to it. These represent the coupling of the quasibound states to the continuum, i.e., their formation and decay [5][6][7][8][9][10] [see Fig. 1(a)].
The universal scattering properties of a compound nucleus reaction, in turn, can be simulated in microwave experiments, where electromagnetic waves propagate via antennas or waveguides into a cavity corresponding to the interior region which allows for bound or quasibound states [10][11][12][13]. The most elementary example is a bent waveguide with a corner as depicted in Fig. 1(b) -a model for strong interactions in nuclei similar to that one was proposed in Ref. [14]. States trapped around the inner corner were predicted to exist below the energy of the first propagating mode [14][15][16][17][18] and observed experimentally in bent microwave waveguides [19][20][21] and in quantum wires [22,23,26]. Note that the Helmholtz equation describing the physics of the former and the Schrödinger equation which is applicable to the latter are mathematically identical in the energy range of the bound states. This analogy was used in the numerical computation of bound states in bent waveguides [20] and quantum wires [24][25][26]. The experimental and the numerical studies revealed that with the variation of the waveguide's bending angle α [see Fig. 1(b)] a transition of the modes from bound to propagating ones takes place. The bending angles at which such a transition occurs are called critical angles in the following.

B. Formulation of the problem
In this paper we present a detailed theoretical and experimental study of that transition.
To test the analytical approach, existing results [20] were extended down to bending angles α = 0.2 • using an improved conventional numerical method which is based on a scattering approach. Its main achievement is an analytical formula for the number of bound states at a given bending angle derived from a fit to the numerical result. The experiments performed to thoroughly test these results comprised measurements of the eigenfrequencies (energies) and the associated field intensity distributions (squared wave functions) of the bound states for several bending angles. Furthermore, we present experimental results concerning the effect of the finite lead lengths on the bound states and the occurrence of resonant tunneling. These investigations go beyond the numerical and theoretical ones which assume waveguides of infinite lengths. Only recently the existence of trapped modes in finite waveguides, as used in the experiments, was studied theoretically [27].
The numerical computations of the bound states and the critical angles of a waveguide or quantum wire involve the solution of the Schrödinger equation of a free particle with Dirichlet boundary conditions imposed on its wave functions at the walls of the system [15,17,18,20,24,25,28,29].
The theoretical treatment of a problem of this kind presented in Ref. [30] encompasses a conformal mapping applied to the geometry of the bent waveguide which induces in the Schrödinger equation an additional effective potential given by the associated Jacobian.
Starting from this equation the bound states at a given bending angle and the critical angles were determined using a Wentzel-Kramers-Brillouin (WKB) approximation developed in Ref. [31] for systems containing sharp corners. The results agree well with those obtained experimentally with electromagnetic waveguides and from the numerical computations based on a scattering approach.

C. Relation to other work
It is worthwhile to mention again that some of the salient features of the spectrum of a bent waveguide were already understood in previous experiments with electromagnetic waveguides [19][20][21] and quantum wires [22,23]. Nevertheless, we feel that it is important to obtain a complete solution, by improving the conventional numerical and experimental methods, which encompasses all the important features and providing a general method which does not rely on numerical calculations. For example, one may pose the following question: What is the number of bound states supported by an orthogonal cross-wire configuration? The numerical answer is two [15], but the reason for this should be revealed by a suitable treatment relating the shape of a wire to its binding capabilities.
These effects, rather than being just accidental, are linked to deeper concepts in undulatory physics. The confinement of a particle to a restricted domain by introducing walls is directly related to a proper formulation of the d'Alembert principle in quantum mechanics [32,33]. This problem must be carefully tackled in the light of well-known results such as the trapping of waves in smoothly bent quantum wires [16,18,28,34] and the emergence of a so-called curvature-induced quantum potential [35]. For instance, it is striking that in bent waveguides or quantum wires with open ends as depicted schematically in Fig. 1(b), semiclassical methods such as trace formulas derived for closed [36,37] and open systems [38][39][40] fail because there are no closed orbits and there is no classical counterpart including a binding force. We present an analytical approach which yields a Hamiltonian whose spectrum comprises the bound states. The conclusion to be drawn from this analysis is that the mere presence of curved boundaries may induce a binding potential, with significant implications close to sharply bent boundaries.

D. Outline of the article
The article is organized as follows: in Sec. II we shortly review the results of our calculations of the bound states in a bent waveguide and of the critical angles based on a conventional method. Section III comprises the description of the microwave experiments and the comparison of the experimental and the numerical results from Sec. II. Section IV presents analytical estimates of the critical angles and the number of bound states in a bent waveguide. The experimental and the numerical results presented in Secs. II and III demonstrate that for a given bending angle the wave functions of the bound states with excitation frequency close to the cut-off frequency penetrate considerably into the leads. Similarly, the influence of the binding potential revealed in the theoretical approach of Sec. IV reaches into the leads. In Sec. V this phenomenon is investigated experimentally. The conclusions are summarized in Sec. VI.

II. NUMERICAL RESULTS
In this section we compute the eigenenergies and wave functions for the bound states of sharply bent quantum wires, that is, the eigenfrequencies and the associated electric field strengths for those of sharply bent rectangular waveguides using a conventional method.
For this we use the fact that below the frequency f max = c/(2h), with c the speed of light and h the height of the waveguide, only the TE 1,0 -mode is excited [41]. There, the Helmholtz equation associated with a bent waveguide is mathematically identical to the stationary Schrödinger equation of an open quantum billiard or a quantum wire of corresponding shape [11,42,43]. The eigenfrequencies of the bound states are all below the cut-off frequency f c = c/(2W ) of the first propagating mode, where W is the width of the waveguide. Consequently, since rectangular waveguides with the width larger than the height were used in the experiments, such that f max > f c , the numerical solutions of the stationary Schrödinger equation obtained in this section apply to bent quantum wires [23][24][25][26] and also to bent microwave waveguides.
The stationary Schrödinger equation of an open quantum billiard with the shape of a bent waveguide corresponds to that for a free particle with Dirichlet boundary conditions along the boundary ∂Ω defined by the walls of the waveguide in Fig. 1(b). Here, Ω denotes the interior of the waveguide. A coordinate system (x, y) is introduced in the plane of the waveguide with origin at the outer corner and the x-axis along the symmetry axis of the waveguide to write for x, y ∈ Ω where we use unitsh 2 /(2m) = 1, the energy E corresponds to the square of the wave number k, E = k 2 , the wave function φ(x, y) denotes the z-component of the electric field vector in the waveguide and is the two-dimensional Laplacian. The width of the waveguide is set to W = 1 in all calculations.
The procedure for the numerical determination of the eigenenergies and eigenfunctions of Eq. (1) is outlined in Appendix A. It is based on the scattering approach described in Refs. [24,25,39,44]. For this purpose the waveguide is subdivided into an interior region and an asymptotic region defined by its straight leads with parallel walls as indicated in Fig. 1(b). The procedure is similar to that used for the description of compound nucleus reactions in the framework of scattering theory, where the interior region corresponds to the compound nucleus and the leads to the formation and decay channels [9,45]. Indeed, as can be observed in Fig. 2(a), it is hardly possible to resolve all curves around α ≈ 0.2 • due to their large number. For α → 0 • the eigenfrequency of the lowest bound state approaches from above the value f /f c = 1/2, so all eigenfrequencies obey the inequality f /f c ≥ 1/2. Note that the frequency f = f c /2 corresponds to the cut-off frequency of a straight waveguide with twice the width of the leads. Indeed, such a waveguide is obtained by decreasing the angle of α to 0 • in Fig. 1(b). Figure 3 shows the intensity distributions corresponding to the four bound states existing at α = 9.65 • . These are marked by dots in Fig. 2

III. MEASUREMENT OF BOUND STATES
In the preceding section we have presented numerically determined eigenfrequencies and associated wave functions, i.e., electric field strengths, of the bound states in bent rectangular waveguides as function of the bending angle and determined the critical angles, where a bound states turns into a propagating mode. These results were tested experimentally with bent microwave waveguides as described in the present section.

A. Experimental setup
Experiments with microwave cavities are widely used to investigate universal properties of quantum systems such as the spectral statistics of chaotic quantum billiards [46], chaotic scattering [10,47] or fidelity decay [48]. This approach is possible due to the analogy noted already above between the stationary Schrödinger equation and the Helmholtz equation for flat cylindrical microwave cavities [11,42]. Similarly, the properties of quantum wires have been investigated in experiments with microwave waveguides of rectangular cross section [19][20][21]. Here we report on microwave experiments performed to validate the numerical and the theoretical results concerning the critical angles and the dependence of the bound states on the bending angle.
Nine different waveguides with different bending angles α were investigated experimentally. Figure 1(b) shows the geometry of the bent waveguides. They were constructed by soldering together two tapered WG18 brass waveguides (by Flann Microwave) as shown in Below f max = c/(2h) ≈ 19 GHz the electric field is perpendicular to the waveguide plane defined by the x and y coordinates and described by the scalar Helmholtz equation [43,49] (∆ x,y + k 2 )E z (x, y) = 0, Microwave power was coupled into and out of the waveguides with two small wire antennas aligned perpendicularly to the waveguide plane. They protruded about 1.5 mm into the waveguide through holes with a diameter of 3 mm in its top wall at three different positions, one in the interior region (label 1 in Fig. 5) and two close to it in the leads (labels 2 and 3 in Fig is the ratio of the power P out, b coupled out via antenna b and the power P in, a coupled in via antenna a at the excitation frequency f .

B. Resonance spectra and frequencies and field intensities
The measured resonance spectrum of the waveguide with bending angle α = 16. To obtain a better understanding of the resonances their electric field intensity distributions were measured using the perturbation body method [50,51]. A cylindrical perturbation body with diameter 2 mm and height 2.6 mm made of a rubberlike plastic combined with magnetic barium ferrite powder was introduced into the cavity and moved by an external guiding magnet to different positions (x, y) on a grid with 2.5 mm spatial resolution. For each position of the perturbation body, the shift ∆f of the resonance frequencies was measured. It is proportional to the electric field intensity [52], I ∝ |E z (x, y)| 2 , at the position of the perturbation body, The measured intensity distributions at the frequencies of the resonances with labels (a)-(d) in Fig. 6 are plotted in Fig. 7. In panels (a)-(c) they exhibit yellow (bright) stripes, the color of which does not change to blue (dark) towards the walls of the waveguide, that is, the intensity does not seem to vanish there. This is an experimental artifact resulting from the friction between the perturbation body and the surface of the brass waveguides, which can lead to a lag of it behind the guiding magnet yielding an inaccuracy in its position.
The intensity distributions shown in panels (a), (b) and (c) of Fig however, is localized in the corner region, which demonstrates that it also is a bound state.
Still, it extends significantly into the leads. This is further discussed in Sec. V. Note that the length l of the leads was chosen such that it is much larger than the decay lengths of the electric field intensities associated with the bound states.

IV. EFFECTIVE POTENTIAL FOR SHARPLY BENT WAVEGUIDES
In this section we present an analytical approach to determine the bound states in the waveguides and the values of α crit . It involves the solution of the Schrödinger equation for a Hamiltonian with an effective binding potential, induced by a conformal map, based on an extension of the WKB method to systems comprising sharp corners [31]. A first work on this subject was published in Ref. [30], which unfortunately contains misprints. For this reason we revisit the approach and derive an improved approximation for the effective binding potential.

A. The conformal map
The essential idea of the theoretical approach in Ref. [30] was to transform the Schrödinger equation (1) with the coordinates x, y ∈ Ω confined to the interior of a sharply bent waveguide to one with coordinates 0 ≤ u ≤ 1, −∞ < v < ∞ restricted to an infinite strip.
One procedure to obtain such a transformation is to introduce complex variables z = x + iy and ξ = u + iv and a transformation z = F (ξ), which maps the infinite straight strip in the complex plane defined by the range of values of u and v onto the interior of the bent waveguide in the (x, y) coordinate system.
There are many possibilities to define F (ξ). However, we impose the requirements that it should be smooth everywhere except at the corners of the bent waveguide and that the For α > 0 • a transformation with these properties was constructed in Ref. [30] using a combination of two Schwarz-Christoffel mappings [53], which images the infinite strip ξ = u + iv onto whereα = α 180 • and B(x, p, q) is the incomplete Beta function [54]. The image z(u, v) = F (ξ) of the point (u, v) = (0, 0) is the outer corner of the bent waveguide at (x, y) = (0, 0), that of the point (u, v) = (1, 0) is its inner corner at (x, y) = (csc[α/2], 0), while the line (0, v) is mapped onto its outer wall and the line (1, v) onto its inner wall.  The change of coordinates implies the transformation where the inverse of the Jacobian appears as a prefactor and the Schrödinger equation (1) takes the form To compute the Jacobian we perform the variable transformation t = sin 2 π 2 ξθ in the integral of Eq. (8) and obtain the expression which with Eq. (10) yields = cosh(πv)+cos(πu) Forα = 1, i.e., α = 180 • we find J (u, v;α = 1) = 1 and the Schrödinger equation reduces to that of a straight waveguide, the solutions of which are known.
The abrupt change of the contour density observed in Fig. 9 for α < 180 • close to the corners is related to the behavior of the Jacobian in their vicinity. In fact, near the outer corner, where |ξ| ≃ 0, the Jacobian can be approximated as that is, it has a singularity of the type x −γ with the order 0 ≤ γ = 2 − 2α ≤ 2. On approaching the inner corner, i.e., for |ξ − 1| → 0, becomes vanishingly small. For large values of v ≫ 1 the Jacobian can be approximated as and converges to unity. This behavior is expected since there the transformation from ξ to z = F (ξ) simply corresponds to a rotation of the coordinate grids by an angle π 2 (1 −α) with respect to each other around the inner corner.

C. Emergence of the potential
Multiplication of the Schrödinger equation (11) with the Jacobian J (u, v;α) yields that is, the mapping Eq. (7) entails the emergence of an effective binding potential depending on the Jacobian J (u, v;α), i.e., the area density. To obtain an approximate analytical solution of this boundary value problem it is turned into an effective potential problem in the v coordinate by choosing an appropriate orthonormal basis for the u coordinate.
The u-coordinate lines approach the v-coordinate lines at the boundaries of the waveguide perpendicularly (see Fig. 9). Thus a suitable choice that obeys the boundary conditions imposed on ψ(u, v) [see Eq. (11)] is given by with Furthermore, since the only singularity of the Jacobian, which is located at (u, v) = (0, 0), has an order less than 2, products of the form sin(mπu) sin(nπu)J (u, v;α) are regular functions at all points inside the waveguide. Therefore, we may expand the function ψ(u, v) in Eq. (19) in terms of the orthonormal functions √ 2 sin(nπu), n = 1, 2, . . . and obtain the system of coupled equations with where are the coupling matrix elements. In this equation the energy E appears parametrically as the strength of the coupling matrix elements V nm (v) which define an infinitedimensional matrix. The numerical evaluation of the integral Eq. (24) yields three characteristic properties of V nm (v) which may also be deduced from those of the conformal map: (i) the matrix elements V nm (v) decrease as one moves away from the diagonal, where the exact range of validity depends on n and on α, whereas for v > ∼ 1.5 Accordingly, for small and large values of v we may replace Eq. (22) by withṼ where we use the notation V n (v) = V nn (v).
In terms of a Hamiltonian H(v) the Schrödinger equation would read at v = ±ǫ with ǫ ≪ 1, which is comparable to a jump κ = V ′ max − V ′ min for small bending angles α.

E. Semiclassical approximation
In Ref. [31] an extension of the WKB approximation to systems described by a potential which displays a corner and has a discontinuous derivative is proposed. We now apply this approach to the present situation and determine approximately the bound states.

Quantization condition
Due to the minimum of the effective potential, bound states may exist for E n ≤ EṼ n (v) ≤ 0. The value of v, where equality E n = EṼ n (v) holds, defines the turning point v t = |v|.
For n = 1 the inequality is not satisfiable for E ≥ π 2 . This value defines the cut-off energy E c above which no bound states exist. For E = 0.9999π 2 we obtain v t ≃ 1.5 for n = 1.
For n ≥ 2 the value of the turning point is smaller than 0.25. Furthermore, the extended WKB approximation involves the potential around the corner at v = 0. Accordingly, we may replaceṼ n (v) by 1 − nV n (v) in Eq. (27) using Eq. (28).
The corner-corrected WKB approximation yields for a symmetric potential the quantization condition [31] ∆ where we defined the action and introduced the notations For n = 1, 2, . . . , N b (α), with N b (α) denoting the total number of bound states at α, this equation yields the rescaled eigenvalues e n (α) of the bound states of a bent waveguide.
Here, the values of e n (α) increase with their index, i.e., e n (α) ≤ e n+1 (α). They are bounded from above by the condition e ≤ N b (α). Beyond that value all states are unbounded. Thus, the cut-off value of the rescaled energies is given by e c (α) = N b (α).

Number of bound states
We determined the number of bound states N b (α) with the help of [55,56].
The value of p max is obtained from the classical analog of Eq. (19), which can be written as E class = p 2 2 + (1 − J (u, v;α)) E π 2 , and satisfies the inequality E class ≤ 0 for the bound states. Thus, p 2 Note, that for finite lead lengths, i.e., for v restricted to some value v ≤ v max , where integration over the Jacobian, A(α), yields the area of the finite bent-waveguide. The integral already approaches the valueÃ(α) for v max ≃ 1.5. In fact,Ã(α) can be interpreted as the effective area covered by the potential, as it corresponds to the area of an unbounded bent waveguide minus the area of the leads where the potential becomes vanishingly small. In the interval considered, 0.7 • ≤ α ≤ 30 • , it is well described byÃ(α) = csc(α/2)−a 0 , with a 0 ≃ 0.91, that is, to a good approximaton the integer part of N b (α) coincides with N (α) given in Eq. (3). It equals 0 for α = 180 • .

Approximate critical angles and rescaled energies
For α < 180 • the rescaled energies, i.e., the ratio of the solutions e n (α) of Eq. (34) and the cut-off energy e c (α) is given byẽ To determine the critical angles, where a bound state turns into an unbound one, we furthermore have to take into account that Eq. (34) does not yield the correct ground state energy [31]. Indeed, for α → 0 • the rescaled eigenenergyẽ 1 (α) = e 1 (α)/e c (α) does not converge to the expected value e/e c = f 2 /f 2 c = 0.25 (see Fig. 2). However, when introducing a constant shift e 0 ≈ 0.25 of the rescaled energiesẽ n (α), which ensures the correct convergence for α → 0 • , a good agreement between the solutions α crit of the equation e 0 +ẽ n (α crit ) = 1 and the numerical results of Sec. II and thus also the experimental ones of Sec. III is achieved. This property is also true for the bound states as demonstrated in Fig. 11(a). Already for the second bound state the agreement between the numerical values of f /f c of Sec. II and those obtained with the corner-corrected and ground-state shifted WKB approximation is good.
The relative deviations of the critical angles obtained with the two methods are shown in Fig. 11(b). They are largest for large critical angles, however less than 2%, and decrease rapidly to a relative deviation of less than 1% for α crit < ∼ 1 • . Thus the WKB approximation proposed in Ref. [31] for systems with a potential displaying a kink provides a good prediction of the critical angles. However, it is necessary to include the known result for the behavior of the rescaled energies, respectively, frequencies in the limiting case α → 0 • .

F. Summary
In summary, the conventional numerical methods, also the one presented in Sec. II, used In this case, the eigenfrequency of the bound state is well below the cut-off frequency and the associated intensity distribution extends not very far into the waveguides. Generally, good agreement with Eq. (38) was found for states with eigenfrequencies f ≪ f c . Next, the transmission of microwave power from port 4 at one lead end through the waveguides to port 5 at the other one in Fig. 14(a) was measured in order to investigate the coupling of the bound states to the continuum. For this, waveguide-to-coaxial adapters (model 4609 by Narda Safety Test Solutions) were attached to the flanges at the ends of the waveguides (see Fig. 5). The adapters close the system but provide a strong coupling to the attached coaxial cables. The geometry of this setup is shown in Fig. 14(a). If the coupling to the exterior is non-negligible, then the bound states are observable as resonances below f c in the measured spectra like in Refs. [22,23,60].

B. Coupling to the continuum
The transmission spectrum obtained by coupling in and out microwave power via the two adapters is shown as black thin line in Fig. 14 This can be attributed to its large decay length in the leads. Thus the coupling of the third bound state to the continuum indeed implies a resonance in the transmission spectrum due to resonant tunneling, whereas the first and second bound state do not. This is explainable with their considerably smaller decay lengths.
Note that the resonance frequency of the third bound state differs slightly for the measurements with the antennas and the adapters, respectively. This is attributed to the fact that the coupling of the adapters to the waveguide is much stronger than that of the small wire antennas, as can be deduced from the average value of |S ba | 2 above f c , which is larger in the former case.
C. Summary Figure 14(b) shows that the bound states can indeed modify the transmission spectrum of the bent waveguides (see also Ref. [59]) due to resonant tunneling as observed in quantum wires [22]. However, this was seen in just two cases for all nine waveguides, and only if no extensions for the leads were used. In both cases, the associated resonance frequency was less than 5 MHz apart from the cut-off frequency. Apparently, for resonant tunneling to be observed, the decay length δ of the corresponding state must be about the same as the lead length. However, only states close to the cut-off frequency in waveguides with bending angles close to a critical angle satisfy this condition which emphasizes the importance of the knowledge of the critical angles.
Furthermore our experiments show that especially for these cases the finite lead length and the details of the coupling can become important, which requires further theoretical investigations. First results regarding the influence of the lead length were published in Ref. [27].

VI. CONCLUSIONS
We have studied experimentally and numerically the occurrence of bound states in sharply bent waveguides below the threshold for propagating modes. In some of the experimental realizations the bending angle was chosen close to a critical angle. The measured eigenfrequencies of the bound states and the values of the critical angles were predicted to a high accuracy with two independent methods: The first method consists of the numerical solution of the Schrödinger equation for a free particle with Dirichlet boundary conditions at the walls of the waveguide based on a scattering approach.
The second one encompasses the derivation of a Hamiltonian containing a binding potential by introducing a suitable conformal mapping. The associated Schrödinger equation was solved semiclassically using corner-corrected WKB approach [31]. These results are of particular relevance for the transmittance of electromagnetic waveguides and quantum wires. The theoretical treatment is, to our knowledge, the first one to work out the problem by analytical methods, even though it relies on an approximation. The demonstration of its applicability to sharply bent waveguides renders possible the analytical treatment of two-dimensional problems with non-trivial boundaries through the use of a conformal mapping and an extended WKB approximation.
In further experiments, the intensity distributions of the bound states were measured.
They exhibit interesting features close to the cut-off frequency and for bending angles close to a critical one. Especially the influence of the finite lead lengths on their boundedness, that is, the size of their coupling to the exterior, was investigated. These studies revealed that predominantly bound states with resonance frequencies close to the cut-off frequency are affected if the length of the leads is of the same size as their decay length, the relevant mechanism being resonant tunneling. We introduce in each lead a coordinate system with origin at the inner corner of the bent waveguide and one axis along the inner wall of the respective lead and the other one perpendicular to it (see Fig. 15). Let L = 1 denote the lead with y > 0 in Fig. 15 and L = 2 that with y < 0, W the width of the leads and d = W csc(α/2) the distance between the outer and the inner corner of the bent waveguide.
We introduce the notation where m = 1, 2, . . . denotes the transverse mode numbers in the leads. The wave number k m is purely imaginary for k < mπ W and thus does not correspond to a propagating mode. For a given k the number M o of the latter equals the integer part of W k/π, i.e., Below the cut-off wave number k c of the first propagating mode, i.e., for k < k c = π W , no propagating modes exist in the leads.
In both equations the first term corresponds to an incoming wave with mode number In the interior region, the solutions of Eq. (A1), which are symmetric with respect to the x-axis and vanish along its outer boundary, that is, along its common boundary with the bent waveguide, can be written explicitly in polar coordinates where we introduced the notationμ = π α (2µ−1). Due to the property of the Bessel functions Jμ(x) that they become vanishingly small forμ ≫ x, the infinite expansion can be truncated at a certain value µ = M t , which depends on that of kr.
Combining Eqs. (A17) and (A20), the matrix elements ofB are obtained aŝ For k > k c , that is, if the leads allow for propagating modes, we obtain with Eq. (A16) setting M c = 0 the final result for the scattering matriceŝ The full scattering matrix is symmetric according to Eq. (A23). Using thatĈ andD are real and thatÂ is purely imaginary, it can be shown thatŜ is unitary.
Below the cut-off frequency, that is, for k < k c the first term on the right hand side of         The origin of the coordinate system (x, y) coincides with the outer corner and the x-axis with the symmetry line of the waveguide. Moreover, we identify two asymptotic regions defined by the leads and one interior region around the inner and the outer corner of the waveguide which suggests to introduce two coordinate systems (x 1 , y 1 ) and (x 2 , y 2 ) defined in the leads L = 1 and L = 2, respectively.