Wave amplitude gain within wedge waveguides through scattering by simple obstacles

Wave confinement, e.g., in waveguides, gives rise to a huge number of distinct phenomena. Among them, amplitude gain is a recurrent and relevant effect in undulatory processes. Using a general purpose protocol to solve wave equations, the boundary wall method, we demonstrate that for relatively simple geometries, namely, a few leaky or opaque obstacles inside a $\theta$ wedge waveguide (described by the Helmholtz equation), one can obtain a considerable wave amplification in certain spatially localized regions of the system. The approach relies on an expression for the wedge waveguide exact Green's function in the case of $\theta =\pi /M$ ($M=1,2,...)$, derived through the method of images allied to group theory concepts. The formula is particularly amenable to numerical calculations, greatly facilitating simulations. As an interesting by-product of the present framework, we are able to obtain the eigenstates of certain closed shapes (billiards) placed within the waveguide, as demonstrated for triangular structures. Finally, we briefly discuss possible concrete realizations for our setups in the context of matter and electromagnetic (for some particular modes and conditions) waves.

Figure 1

(a) Schematics of the wedge waveguide considered in this work. (b) The case of $\theta =\pi /2$. (c) The method of images construction for $\theta =\pi /2$. Each shown $G$ represents the expression for the Green's function in the 2D free space with the signals of $x,y$ changed as indicated. The $G^{\prime }\mathrm{s}$ with modified $x,y$ are then specular “images” of the Green's function in the first quadrant through the planes $x=0$ and $y=0$. The $\pm$ in front of the $G^{\prime }\mathrm{s}$ are the coefficients ${\alpha }_l$ defined in the paper. (d) The lines $l=0,1,...,2M-1$ (with $M$ a positive integer) form a starlike pattern in the plane. The $\theta =\pi /M$ wedge waveguide is delimited by the lines 0 and 1. (e) The specular image of a point within the waveguide with respect to the line 1 (top) and the successive specular images of this point about all the lines of the starlike structure of (d) (bottom). (f) With the help of parts of the wedge waveguide walls, a proper located segment of line wall $\mathcal{C}$ forms a billiard shape, as a right triangular billiard for the $\theta =\pi /4$ wedge waveguide. (g) For an acute wedge, examples of different walls $\mathcal{C}$ which can be placed inside the waveguide.

Figure 2

By properly inserting a straight segment $\mathcal{C}$ (dotted lines) within a $\theta$ wedge waveguide, it is divided into two distinct regions: a triangular billiard (labeled B) and a waveguide with a wall scatter at its end (labeled W). This is illustrated in the inset of (a). In (a)–(c) $\theta ={45}^{\circ }$ and in (d)–(l) $\theta ={60}^{\circ }$. The resulting billiard shapes are (a)–(c) right ${45}^{\circ }–{45}^{\circ }$, (d)–(f) equilateral, (g)–(i) right ${60}^{\circ }–{30}^{\circ }$, and (j)–(l) ${60}^{\circ }–{\gamma }^{\circ }–({120}^{\circ }-{\gamma }^{\circ })$ [with $\gamma =(2-3\sqrt{2}/\pi ){60}^{\circ }\approx 38.{97153}^{\circ }$, hence irrational], triangles. For all them, the base side lengths are $\ell =1$. The density plots correspond to the modulus square of the billiard eigenstates in B—existing depending on $k$—and of the scattering solutions in W. The darker (lighter) regions indicate a higher (lower) ${\left|\psi \right|}^2$. As it should be, ${\left|\psi \right|}^2$ is null along $\mathcal{C}$ as clearly observed in the plots. The $k^{\prime }\mathrm{s}$ of the incident waves considered are listed in Table 1. In particular, the $k$ values for the cases displayed in (c), (f), (i), and (l) do not correspond to any eigenwavenumber of the billiards. Therefore, the BWM leads to null ${\psi }^{\prime }\mathrm{s}$ within the related B regions.

Figure 3

Density plots of $|{\psi }_n{|}^2$ from the analytic solutions in the Appendix pp4 for (a),(b) isosceles, (c),(d) equilateral, and (e),(f) right ${60}^{\circ }$–${30}^{\circ }$ triangles. The (a)–(f) eigenstates correspond to the cases (a), (b), (d), (e), (g), and (h) of Fig. 2.

Figure 4

Modulus square of $A$ given in Eq. (24) as a function of $\lambda /R\ge 0.1$ for the fundamental mode $n=1$ and two values of $\theta$ and $P_t$. To maintain $P_t=4k^2/(4k^2+{\gamma }^2)$ constant, $\gamma$ assumes different values as $\lambda =2\pi /k$ varies.

Figure 5

(a) Two permeable circles of radii $R_1<R_2$ are placed close to the $\theta$ wedge waveguide vertex. Both circles have their centers along the waveguide bisector. The regions to be analyzed for wave amplitude gain are indicated as $A$ (between the vertex and the smaller circle) and $B$ (between the two circles). A relevant characteristic length of the region $A$ ($B$) is the distance between the waveguide vertex and the smaller circle border, $D-R_1$ [the borders of the two circles, $(R_1+R_2)(1/sin[\theta /2]-1)-D]$. (b) The circles segments used to calculate $g$ (main text): the dashed line ($M_A)$, the full lines ($M_B$), and the dotted line ($M_E$).

Figure 6

Density plots of (a) $g_A$ and (b) $g_B$ as function of $k$ and $D/R_1$ for $\gamma =50$. For selected values of $k$ and $D/R_1$ (as indicated), density plots of the scattering solutions ${\left|\psi \right|}^2$ assuming the incident $\phi$ with $\alpha =\pi /8$. The specific values of $k,D/R_1$ are (c) 18.1, 6.46729, (d) 18.40, 6.27787, (e) 18.22, 5.3849, (f) 17.86, 5.06018, (g) 17.8, 4.92488, (h) 18.02, 4.81665, (i) 18.5, 4.62723, (j) 18.52, 4.38369, (k) 18.26, 3.89661, (l) 18.02, 3.49071, (m) 17.82, 3.13894, and (n) 17.52, 2.65186.

Figure 7

Density plots of $g_A$ as function of $k$ and $D/R_1$ for distinct values of $\gamma$. From the top to the bottom panels, $\gamma$ is set to 20 [so that $P_t(k=25)=0.862]$, 50 $[P_t(k=25)=0.500]$, 100 $[P_t(k=25)=0.200]$, 200 $[P_t(k=25)=0.059]$, 400 $[P_t(k=25)=0.015]$, 1000 $[P_t(k=25)=0.002]$, and $\infty$ (for this last, complete opaque circles, $P_t=0$). Many amplification resonance families can be clearly identified in the distinct plots as darker lines forming “trajectories” in such a $k\times D/R_1$ space. The lighter vertical lines for greater ${\gamma }^{\prime }\mathrm{s}$ correspond to the small circular billiard eigenwavenumbers.

Figure 8

The same as in Fig. 7, but for $g_B$. As $\gamma$ increases (from top to bottom panels), there is the formation of two vertical lines, a behavior also observed in Fig. 7. They correspond to the eigenwavenumbers $k_n$ of the small circle.

Figure 9

(a) For $\gamma =200$, detail of $g_B$ in Fig. 8 in the region $18<k<21$ and $2.6<D/R_1<6.4$. It highlights three resonance families, labeled as (I), (II) and (III). In the inset, 3D plots of these three trajectories. For the latter two 3D graphs, an artificial offset has been implemented just to facilitate the visualization. Along the different resonance families, density plots of ${\left|\psi \right|}^2$, corresponding to particular values of the pair $(k,D/R_1$) (indicated by arrows), are displayed from (b) to (m) [for (n) and (o) see Fig. 10]. The specific values of $k$ and $D/R_1$ are (b) 18.28, 4.74, (c) 19.14, 5.30, (d) 19.66, 5.52, (e) 20.68, 5.82, (f) 18.34, 3.57, (g) 19.04, 4.22, (h) 20.00, 4.82, (i) 20.56, 5.14, (j) 18.16, 2.79, (k) 18.5, 3.39, (l) 18.89, 3.93, (m) 19.20, 4.19, (n) 20.06, 4.63, and (o) 20.36, 4.77.

Figure 10

Density plots of ${\left|\psi \right|}^2$ for the states (n) $k=20.06, D/R_1=4.63$ (left graph), and (o) $k=20.36, D/R_1=4.77$ (right graph) indicated in Fig. 9.

Figure 11

The distance between the points $(\rho ,\varphi )$ and $({\rho }_0,{\varphi }_0)$, both lying in the upper-half plane, is $R$. The specular image of $(\rho ,\varphi )$ about the $x$ axis is the point $(\rho ,-\varphi )$, whose distance to $({\rho }_0,{\varphi }_0)$ is $R_{-\varphi }$.