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.

Figure 1

Analogy between the decay of a compound nucleus (a) and a bound state in a bent waveguide (b). A compound nucleus is formed by bombarding a nucleus $A$ with a particle $a$ and eventually decays into a residual nucleus and particles, $B+b$, $C+c$, etc. This coupling to the exterior region is modeled by attaching different formation and decay channels to a confined, interior region. Panel (b) shows a sketch of a quantum wire or electromagnetic waveguide with a sharp bend of angle $\alpha$, the straight parts of which have a length $l$ and a width $W$. The border between them and the corner region is indicated by dashed lines. Since bound states form in the latter, we also denote it as the interaction region. The coordinate along the leads is denoted by $x_L$.

Figure 2

Numerical results for the rescaled eigenfrequencies $f/f_c$ with $f$ the eigenfrequencies of the bound states and $f_c$ the cutoff frequency vs the bending angle $\alpha$ of the waveguide. Part (a) shows the ratios for $0.2^{\circ }\le \alpha \le 4^{\circ }$, while (b) shows those for $4^{\circ }\le \alpha \le {60}^{\circ }$. Each curve corresponds to the eigenfrequency of one bound state in units of $f_c$, which is a continuous function of the bending angle $\alpha$, starts at $\alpha =0^{\circ }$, and increases until it reaches the value $f/f_c=1$ at a certain bending angle. There the bound state turns into a propagating mode. The corresponding angles are called critical angles ${\alpha }_{\mathrm{crit}}$. The arrows in panel (b) indicate the critical angles considered in the experiments with microwave waveguides. The number of bound states increases rapidly as $\alpha$ approaches $0^{\circ }$. Indeed, for $\alpha \approx 0.2^{\circ }$ the curves are barely distinguishable. The eigenfrequency of the first bound state, which corresponds to the lowest curve in both panels, converges to $f/f_c=1/2$. This is the cutoff frequency of a waveguide of double the width of the leads. The dots mark the eigenfrequencies of the four bound states at $\alpha =9.{65}^{\circ }$, whose intensity distributions are shown in Fig. 3.

Figure 3

Square of the numerically computed wave functions associated with the four bound states at $\alpha =9.{65}^{\circ }$. The corresponding eigenfrequencies are marked by dots in Fig. 2b. Panel (a) shows the intensity distribution for $f/f_c=0.631$, panel (b) that for $f/f_c=0.748$, panel (c) that for $f/f_c=0.851$, and panel (d) that for $f/f_c=0.946$. The blue (darkest) color corresponds to the lowest, the yellow (brightest) to the highest intensity.

Figure 4

Number of bound states $\mathcal{N}\left(\alpha \right)$ in its dependence on the bending angle $\alpha$. The circles show the numerical results; the solid line is obtained from the analytical expression Eq. (3). The agreement between the numerical results and the analytical expression is very good. The inset shows a magnification for ${\alpha }_{\mathrm{crit}}\le 1^{\circ }$.

Figure 5

Photo of a bent waveguide with angle $\alpha =26.{50}^{\circ }$. The three small copper blocks soldered on top of the waveguide are used to attach the antenna ports (not shown) labeled by numbers 1, 2, and 3, where microwave power is coupled into and out of the waveguide. Additional straight waveguide parts can be attached to the two flanges.

Figure 6

Bound states of a bent waveguide, here with bending angle $\alpha =16.{40}^{\circ }$, identified by the measured resonance spectrum shown in logarithmic scale. The gray bar indicates the frequency range where the cutoff frequency $f_c$ is expected. The broad resonances observed above $f_c$ are due to partial reflections of the microwaves at the open ends of the waveguide. The intensity distributions for the resonances labeled (a)–(d) are shown in the corresponding panels of Fig. 7. The arrow indicates the position of the resonance (d), which is above the cutoff frequency $f_c$ and thus broad. The inset is a magnification around $f_c$ including resonance (c).

Figure 7

Intensity distributions of three bound states and one propagating mode for the waveguide with $\alpha =16.{40}^{\circ }$ and length $l=411.4$ mm measured with the perturbation body method. The electric field intensity $I\propto |E_z{|}^2$ is plotted in false colors. The blue (darkest) color corresponds to the lowest, the yellow (brightest) to the highest intensity. Panels (a)–(d) correspond to the resonances indicated in Fig. 6. In panels (a)–(c), i.e., for $f<f_c$, the intensity distribution is localized around the inner corner, whereas for $f>f_c$ it periodically extends over the whole waveguide as shown for one example in panel (d). In panels (a)–(c), yellow (bright) stripes are observed, and the intensity does not seem to vanish at the walls of the waveguide. This is an experimental artifact due to a lag of the perturbation body behind the guiding magnet during the measurements. Furthermore, the intensity distributions do not show the expected symmetry. This again is an experimental artifact resulting from the choice of the rectangular grid of positions of the perturbation body, one side being parallel to, respectively, the lower lead of the waveguides in the figure.

Figure 8

Measured (crosses) eigenfrequencies of the bound states in units of the cutoff frequency in their dependence on the bending angle $\alpha$. The solid lines correspond to the eigenfrequencies computed in Sec. 2 with a conventional numerical method [same data as in Fig. 2b, but shown only in the range $8^{\circ }\le \alpha \le {30}^{\circ }$]. When an eigenfrequency crosses the dashed line, it reaches the value of the cutoff frequency. The occasional, slight deviations between the experimental and the numerical results are attributed to imperfections in the construction of the waveguides resulting from the soldering process. Also small differences between the critical angles computed in Sec. 2 and the experimental ones are observed.

Figure 9

Grid of the conformal coordinates $(x(u,v),y(u,v))$ obtained from the map $z=F\left(\xi \right)$ given in Eq. (8) for $\alpha ={180}^{\circ }$, ${126}^{\circ }$, and ${54}^{\circ }$. For a fixed $v=\mathrm{const}$ the coordinate lines are perpendicular on the boundaries $(x(u=1,v),y(u=1,v))$ and $(x(u=0,v),y(u=0,v))$, while for fixed $u=\mathrm{const}$ the coordinate lines are parallel to the latter for large values of $v$. An abrupt change of the contour density occurs near the corners which is related to the Jacobian of the transformation $z=F\left(\xi \right)$ as expressed by Eq. (15). This feature stands out most clearly in the example shown on the right with an accumulation of lines at the inner corner and a blank space to the right of the outer corner.

Figure 10

(a) Effective potential ${\tilde{V}}_{n=1}\left(v\right)=1-V_{n=1}\left(v\right)$ and (b) the derivative $\frac{d{\tilde{V}}_{n=1}\left(v\right)}{dv}=-\frac{d{V}_{n=1}\left(v\right)}{dv}$ for $\alpha =1^{\circ }$ (dotted line), $\alpha ={25}^{\circ }$ (dashed-dotted line), $\alpha ={49}^{\circ }$ (dashed-dotted-dotted line), $\alpha ={73}^{\circ }$ (dashed-dashed-dotted line), $\alpha ={97}^{\circ }$ (dashed line), and $\alpha ={121}^{\circ }$ (full line).

Figure 11

(a) Comparison between the rescaled eigenfrequencies $f/f_c$ of the bound states computed with the conventional numerical method outlined in Sec. 2 (full lines) with those obtained from the corner-corrected and ground-state shifted WKB approximation (dots). (b) Relative deviations of the critical angles evaluated numerically and with the WKB approximation vs ${\alpha }_{\mathrm{crit}}^{\mathrm{WKB}}$. Both panels demonstrate that the agreement between the numerical and the WKB results is already good for the third bound state.

Figure 12

Decay of a bound state into the leads represented by the projection of the measured field distribution (gray line) onto the $x_L$ axis parallel to the waveguides [see Fig. 1b]. Here we use the second bound state at $9.364$ GHz for $\alpha =27.{65}^{\circ }$ with lead length $l_0$ and show this projection in linear scale (top panel) and logarithmic scale (bottom panel). In the regime from $20\le x_L\le 100\mathrm{mm}$, the intensity distribution decays exponentially, indicated by the dashed line, with the decay length ${\delta }_{\mathrm{expt}}=(16.2\pm 0.4)$ mm. The inset displays the complete intensity distribution, but parts of the waveguides are not shown.

Figure 13

Influence of the short leads on the bound states as expressed by the leakage of the measured intensity distributions of the third bound state into the leads for angles (a) $\alpha =14.{55}^{\circ }$, (b) $\alpha =15.{95}^{\circ }$, and (c) $\alpha =16.{40}^{\circ }$. The electric field intensity $I\propto |E_z{|}^2$ is plotted in false colors. The blue (darkest) color corresponds to the lowest, the yellow (brightest) to the highest intensity. In contrast to Fig. 7, the waveguides now have short leads with length $l_0$ summarized in Table . With increasing bending angle, the intensity penetrates deeper into the leads.

Figure 14

Resonant tunneling in a bent waveguide observed in measured microwave transmission spectra. (a) Sketch of the setup. The crosses marked 2 and 3 indicate the positions of the wire antennas also used in the experiments presented in Sec. 3 and the crosses marked 4 and 5 denote the positions of the waveguide-to-coaxial adapters. (b) Frequency spectrum of the waveguide with $\alpha =16.{40}^{\circ }$ and adapters attached. The spectrum was measured both with the adapters (black thin line) and with antennas 2 and 3 located near the tip of the waveguide (thick gray line). The light gray vertical bars and their width indicate the cutoff frequency $f_c$ and its error bar, respectively. The two resonances at $6.49$ and $8.15$ GHz observed in $|S_{32}{|}^2$ are not seen in $|S_{54}{|}^2$ because the corresponding intensity distributions decay rapidly in the leads and thus cannot couple to the adapters. Only the third resonance below $f_c$ at $9.506$ GHz is observed in $|S_{54}{|}^2$, since its intensity distribution extends far into the waveguide (see the inset in the bottom panel).

Figure 15

Definition of the two coordinate systems used in the analysis of a sharply bent waveguide. 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 2, respectively.