Phase Portraits of Periodic Planar Waveguides
TO CYLINDRICAL RESONATOR
A waveguide is called periodic if it coincides with itself during the shift to a certain zero vector (we think that the waveguide has infinite length). Let us cut out the whole period from the waveguide using two segments, say, perpendicular to the vector as it is shown in Fig. 1. We will use that part for our resonator.
In order to simulate the behavior of light rays, the reflection of light rays from the resonator upper and lower boundaries will be considered to occur according to the law when "the angle of incidence equals the angle of reflection". At the same time during the reflection from a frontal segment both reflecting point and direction vector of the light ray are transferred to the opposite segment parallel to the vector . In other words, we identify frontal segments and convert the resonator into optical one at the same time.
Clearly, there is a one-to-one correspondence among the trajectories of light rays inside the waveguide and cylindrical resonator. However, it seems that it is easier to observe the behavior of light rays in the resonator with limited volume than in the elongated waveguide.
OF CYLINDRICAL RESONATOR
All information about the light ray trajectory is uniquely coded by means of its consecutive reflecting points. Due to a certain information loss we can a bit simplify the description of the light ray trajectory. That is why we will indicate only reflecting points from the frontal segment and light ray direction in those points on the trajectory.
The frontal segment will be considered to have a unit length. So, the reflecting point located on the same segment can be represented by the number x when 0 ≤ х ≤ 1 Also the light ray direction vector will be considered to be a unit one, then its projection on the frontal segment will be represented as –1 ≤ с ≤ 1. It is known that the light ray reflection from the frontal segment is characterized by those two numbers x and c. The space with coordinates х, с is called a resonator phase space. Every trajectory is displayed by the sequence of points (x1, c1), (x2, c2), ... (xn, cn), ... in the phase space.
For example, we will see further that the points which correspond to some trajectories fill the whole curves in the phase space, and stochastic clouds correspond to other trajectories in the phase space. If to transfer enough quantity of light ray trajectories at the same time, we can get an image known as the resonator phase space. It allows to obtain a lot of information about the behavior of all the light rays inside the cylindrical resonator as well as relevant information about the periodic waveguide.
It should be mentioned that periodic waveguides might include "trapped" trajectories inside the period, so they can intersect with frontal segments accordingly. Using our method of the trajectory displacement, they simply will not be displayed there. However, there is another type of waveguides where such situation is impossible. These are equidistant waveguides [1–4] which can have either optical or equivalent geometric definition.
Let us begin with the optical definition. Assume that there is a given initial wavefront set in two-dimensional homogeneous optical medium which is presented as a curve C0. Secondary wavefront sets are expanding in both sides from it. The area between two synchronous secondary wavefront sets C–r and Cr, which are located from the initial one at a distance r, is appeared to be the equidistant waveguide (Fig. 2).
At the same time the defining geometrical feature of the equidistant waveguide is that the perpendicular to one of its bounding curves is a perpendicular to the other curve as well, so the segments of those perpendiculars which are cut out by those curves are all equal (in our case their common length is equal).
As it was shown in [1,3], all light rays which are part of the equidistant waveguide move in one direction, and they usually reach the opposite end for very smooth waveguides. So, trajectories of almost all the light rays will be reflected in the phase space of the cylindrical resonator inside the appropriate equidistant waveguide. Obvious exception involves trajectories represented by the segments which are perpendicular to the waveguide boundaries.
FROM EQUIDISTANT WAVEGUIDE TO PLANAR RESONATOR
Let us describe one more alternative procedure of transition from the waveguide to the resonator. So, let us examine the equidistant planar waveguide. Assume that except the periodicity it additionally has an axial symmetry (say, axis of symmetry is one of two distinguished segments in Fig. 2). Clearly, the axis of symmetry is a common perpendicular to both boundary curves of the waveguide.
Let us choose two symmetry axis of the waveguide which are located from each other at a distance of its period, as for example, in Fig. 2, and cut the whole period using those axes. The cut-out area will represent the planar resonator corresponding to our waveguide. This time assume that the light ray reflection, which is inside the resonator in all the points without any exceptions, is carried out according to the standard law when "the angle of incidence equals the angle of reflection". So, we will obtain a planar resonator, and again it is obvious that there is a one-to-one correspondence among the trajectories of light rays inside both the waveguide and the planar resonator.
PHASE PORTRAIT OF PLANAR RESONATOR
The phase portrait is constructed in the same way as it was done for the cylindrical resonator. This time the first coordinate x of the reflecting point is the distance along the resonator boundary from the previously fixed point to the reflecting point. The second coordinate c is a value projection of the light ray unit direction vector on the tangent to the boundary at the reflecting point. At the same time the positive direction of the tangent line corresponds with the resonator counterclockwise orientation. The value x is represented as 0 ≤ x < L, where L is a length of the resonator boundary, and c is represented as –1 ≤ с ≤ 1.
Once again, every trajectory is displayed as a sequence of points (x1, c1), (x2, c2), ... (xn, cn), ... in the phase space. Also, the points which are responsible for the intersection with vertical segments (symmetry axes of the waveguide) are added to the points corresponding with the reflection of waveguide boundaries.
In order to construct the phase portraits of the model waveguide we choose an equidistant one as suggested by J. O. Rourke in . The medial line of this equidistant waveguide is a periodic curve C0 which consists of semicircles of the unit radius. The waveguide boundary lines C1 and C–1 are moved from it at a distance 1 (Fig. 3).
That is how one of the possible trajectories appears in such a waveguide (Fig. 4). In fact, it is difficult to judge the trajectory appearance about its asymptotic properties, for example, whether it is periodic, quasiperiodic or not. Then we depict this trajectory in the corresponding resonator and its phase space, so it will be easy to define what type it belongs to.
And now using the distinguished segments in Fig. 3, which are symmetry axes of the waveguides, let us cut out the whole period from it, a rough part for the corresponding resonators (Fig. 5). And now let us continue with the phase portraits.
As it turns out statistically almost every light ray trajectory has a horizontal line in the studied resonators. In other words, the whole phase space is filled with the trajectory image with horizontal lines. Although, as it was mentioned, half of the rays moves from left to right in the conditions of equidistance, and the other half moves in the opposite direction. Let us discuss the horizontal light rays with their starting point located at the left resonator boundary (Fig. 6). The rays with the starting point located on the segment are marked in red color, and rays with the beginning on the arc of circle are blue. The samples of their trajectories will be marked in the same color in the phase space. If any point of the phase space is located on both trajectory types at the same time, its color will be marked by a blend of red and blue, namely, purple shade according to the magenta RGB color code.
Left boundary segment will be directed from bottom to top, and the coordinate x, which is located there, will be changing from –1 to 0. If the light ray trajectory intersects with the left segment at the point with the coordinate x and has a unit direction vector , then с, –1 < с < 1 will indicate its projection on the segment. As it has been already mentioned, the light ray trajectory is represented by the sequence of points (x1, c1), (x2, c2), ... (xn, cn), ... in the phase space.
In our numerical experiment we have studied 58 trajectories moving from left to right, and their images fill the left part of the phase portrait shown in Fig. 7. The right part corresponds to the trajectories which move in the opposite direction. We capture for them the points of intersection x with the right boundary segment of the resonator directed from top to bottom, and here x changes from 0 to 1. The second coordinate c is also the projection of the light ray unit direction vector on the segment.
Generally speaking about the phase portrait, an island which consists of purple ovals stands out on it, so it means that they are images of the statistically corresponding light ray trajectories marked in red and blue colors (Fig. 6). Actually, the initial rays of those two mixed trajectories in the phase space are passing near the horizontal axis dividing in half the resonator area, and they are equally spaced from it. This island is surrounded by eight guides of pure red and blue colors. So, all this central area is also surrounded by stochastic clouds which occupy bigger part of the phase space. Much smaller fragments can be indicated, but the main ones are listed.
Now when we have two descriptions of the light ray trajectories in the resonator space and phase space, the visual connection should be indicated between these two descriptions in order to create some kind of an illustrated dictionary (Table).
The first column of Table shows the height of the initial horizontal ray emerging from the left boundary of the resonator. The second column depicts not only two trajectories emerging from the left boundary but also two trajectories emerging from the right boundary at the same height h in the phase space.
Table shows that red and blue trajectories coincide statistically till the height h approaching 0,5, and they are purple in the phase space. Also, one can observe quasiperiodic trajectories (in Table when h = ±0.24, h= ±0.37) as well as stable periodic ones (when h = ±0.35, h = ±0.46). For every height close to h = ±0.525 we observe two different trajectories, they are marked in pure red and blue colors in the phase space. Fig. 4 shows this type of trajectory. One can observe stochastic trajectories which fill bigger part of the phase space at higher heights.
Actually, we can see that the phase portrait contains a lot of information about the light ray behavior in the resonator as well as in the corresponding periodic waveguide. At the same time some information loss occurs regarding the behavior of the light ray trajectory during the phase portrait construction, because only points of intersection with boundary segments are captured but not the points of physical reflection.
We can avoid this loss by means of using the planar resonator. The space of this resonator will have the same appearance as in Fig. 5, and the reflection will occur from all the points of its boundaries including the boundary segments according to the law "when the angle of incidence equals the angle of reflection". Let us choose left bottom angle of the resonator as a starting point on its boundary, and the counterclockwise orientation will be assumed as the positive direction of the boundary bypass. Assume that the first coordinate x of the reflecting point is the distance along the boundary from the starting point to the very reflecting point when 0 ≤ x ≤ 2 + 2 π. The second coordinate c will be the projection of the reflected ray unit direction vector on the resonator boundary when –1 < с < 1.
Here, we launched the same 58 horizontal light rays moving from left to right and 58 rays moving in the opposite direction by means of computing visualization (Fig. 8). No wonder that the phase portrait of the planar resonator as a fragment consists of the phase portrait of the cylindrical resonator. Actually, part of the phase space located above the intervals and x ∈ ( 1 + 2 π, 2 + 2 π ) coincides with the reflections from the resonator boundary segments. And if we compare it with Fig. 7, it will be two halves of the phase portrait of the cylindrical resonator.