Critical points such as the tops and bottoms of lakes play a key role in the overall organization and structuring of flow lines. S-field flow lines in the vicinity of a perfectly conducting surface are studied.
Preliminaries
- The Poynting Vector
- Flow Lines and Critical Points
- Plane Poynting Vector Fields
- Index of Rotation
- Wave Momentum and Poynting Vector
In what follows, the critical points of the Poynting vector fields Se, generated by //-polarized fields given by (9), will be studied. There is an interesting interpretation of the Poynting vector in terms of wave momentum for this field.
Classification
Taylor Series Expansion at Critical Points
Therefore, in a small area around the origin, the components of the Poynting vector can be written as. This can be recognized as the Taylor series expansion of the Poynting vector field around the origin.
Non Degenerate Critical Points
The form of the function g(θ} is not important for this discussion, except that it is continuous and differentiable, which is guaranteed from (26). It is of interest to investigate the magnetic field in the vicinity of a Poynting critical point -vector In the previous discussions, it was assumed that isolated critical points exist in the Poynting vector field.
This proves the claim that the rotation index of the Poynting vector field is in the far zone. There are not many examples of the critical points of the Poynting vector field in the published literature. From the above expression, it is clear that the streamlines of the Poynting vector field are curves in the y-z plane.
Structurally stable critical points of the Poynting vector field have been categorized in the previous section.
Index of Rotation at Non Degenrate Critical Points
Degenerate Critical Points
Behavior of the S-field streamlines near a critical direction of the truncated Poynting vector. Again it is assumed that the behavior of the Poynting vector field is fully characterized by the approximate field given in (47).
Critical Points Due to Arbitrary Electric Field
The rotation index of the degenerate critical point cannot be calculated just from knowing m and n of the truncated Poynting vector field. With the help of m and n of the canonical electric field derived above, it will be possible to classify the behavior of the current lines.
Magnetic Field at Critical Points
Therefore, information about the polarization of the magnetic field can also be obtained from the curl of the Poynting vector. As a result of the above discussion, it becomes clear that the morphology of flux lines at a critical point can be determined from a knowledge of the sense of rotation of the magnetic field on a small circle around the critical point. The sense of rotation of the magnetic field is measured on a circle with a small radius, centered on the critical point.
As a bonus, based on such a measurement, one can also calculate the rotation index of the critical point using the expression. If m = 0 or n = 0, then the magnetic field is linearly polarized near the critical point.
Elementary Critical Points
Hoe's lines and lines of constant phase near an elementary center. The energy density of the electric field, We, is zero at the critical point and increases nearby. How's lines and lines of constant phase near an elementary saddle point.
The polarization of the magnetic field is linear in the vicinity of this critical point. At the saddle point the electric field is not zero and therefore current will be induced in the wire but no average force will be exerted on it.
Wave Interference: Example of Critical Points
It can be noted that the device described above is sensitive to the curl of the Poynting vector field, which is limited and non-zero for the center point and zero for the saddle point. The reason is that when the dipole is aligned with Ez and charges are induced on its ends, there is no transverse part of the electric field to act on those charges. If α = 0, then the Poynting vector is zero everywhere and all the points are critical points, but they are not isolated.
An explicit expression for the exact lines of the Poynting vector field can be obtained by integrating the differential equation (6). These changes and how they come about will be the next topic of conversation. a) The exact lines of the Poynting vector apply to the case of three-wave interference when 0 < ∣α∣ < 1. b) The exact lines of the Poynting vector ßeld for the case of three-wave interference when ∣α∣ = 1.
Structural Stability of Critical Points
A sketch of the streamlines for the Poynting vector field for this situation is given in Figure 17(a) . In the previous section, the behavior of the Poynting vector field near a perfectly conducting surface was determined. The rotation index of the Poynting vector field on the bipartite scattering surface is
A Taylor series expansion of the Poynting vector field is needed to study the behavior of its critical points. This fact can be used in the construction of quality plots of flow lines.
Creation, Annihilation and Reorgnization of Critical Points
Examples of Structural Instability
As the value of b increases, the saddle location moves up along the j∕ axis. The rotation index of the source point is +1 because all streamlines emanate radially outward from the line source. It was pointed out in Section 14 that under rotation of the Poynting vector field, saddle-point interconnections are structurally unstable.
Critical points resulting from the interference of a plane wave and a cylindrical wave, (a) when the amplitude of the cylindrical wave is small, (b) when the amplitude of the cylindrical wave becomes greater than a certain value. This example therefore serves to illustrate the vulnerability of the saddle point to saddle point connections.
Applications
Poynting Vector and Perfect Conductors
The streamlines of the Poynting vector field in the vicinity of a regular point and a half saddle point on the perfect conductor are depicted in figures 19(b) and 19(c) respectively. From equation (30) it then follows that α2 = π∕2∙ The leading term of the Poynting vector is then calculated as The behavior of streamlines is immediately recognized as similar to the behavior of the streamlines in figure 19(c).
It is possible that the leading term in the Taylor series expansion of the Poynting vector field is of order greater than 3. In the case of //-polarization, the streamlines are perpendicular to the family of curves φ∏ = constant.
Critical Points in Resonant Cavities
This has been shown by Courant and Hilbert [16]. that the nodal lines of the mode Rq divide the cavity into no more than q areas. Since the mode function is real, the Poynting vector is identically zero inside the cavity. When the cavity is configured in this way, some of the energy flows from the source to the load and the Poynting vector field will not be zero at every point in the cavity.
The rotation index of the Poynting vector field on a closed curve parallel to the cavity walls and across both slits is clearly 91 + 92· If the mode function Rq has qt saddle points in the cavity, then there are qt elementary saddle points of the Poynting vector field inside the cavity. Therefore, there must exist critical points of the Poynting vector field inside the cavity with positive rotation indices.
Critical Points in Electromagnetic Scattering
Therefore, it is concluded that elementary critical points of the Poynting vector field can exist within a tuned cavity resonator. Thus, the critical points of the Poynting vector field, if any, exist only in the vicinity of the scatterer. The index of the Poynting vector field on a closed curve with a very large radius around the scatterer is zero.
The Poynting vector field near the scatterer is homotopic up to a critical point with the indices m - 4 and n = 0 shown in (b). This difference is compensated by the rotation index of the critical points in the near zone of the scatterer.
Examples of Critical Points in Diffraction and Scattering
Let the algebraic sum of the indices of the critical points in the near zone be denoted by 7κt. The Seed lines of the Poynting vector field for a perfectly conducting cylinder of radius a = ∖∣π immersed in a standing wave. It can be remembered that half. a) The streamlines of the Poynting vector Apply near the diffraction half-plane.
A plane wave is incident along the x-axis on two perfectly conducting cylinders. Using expression (101) and the methods described above, the diagrams for the streamlines of the Poynting vector field are drawn.
Other Examples of Critical Points
The behavior of the flow lines can then be studied with frequency as the disturbing parameter. As in the previous case, a linear transformation T of the coordinate system can be found such that. A sketch of the behavior of flow lines near this type of critical point is given in figure 33(b).
These authors considered the energy flow in the vicinity of the focus of a coherent beam. The main objective of this thesis is to classify the critical points of the plane Poynting vector fields according to the behavior of the streamlines in the vicinity of these points.
Generalization
Critical Points of Three Dimensional Poynting Vector Fields
Poynting vector fields are the phase of only one complex function that controls the direction of the Poynting vector. Many authors have persistently found a representation of the Poynting vector field in such a way that the streamlines should be perpendicular to surfaces of constant phase. We can write down the parametric equations for the streamlines in the transformed coordinate system.
The spiral streamlines are a hyperbola in any plane that is perpendicular to the x-y plane. As in the previous case, the parametric equation (119) can be transformed back into the original coordinate system, but such a transformation does not change the qualitative behavior of the streamlines.
Example of Critical Points in Three Dimensional Fields
The streamlines near this critical point are qualitatively similar to the streamlines in the ï-ÿ plane in Fig. 33(b). Near these critical points, the streamlines behave like near-saddle streamlines. These lines are of the form that will satisfy the expression (117) with 2o = 0∙ These lines are also qualitatively similar to the streamlines in the ÿ-z plane in Fig. 33(a).
The energy in the beam flows in the direction of the z-axis, which is called the optical axis. The Poynting vector is zero on concentric circles in the focal plane perpendicular to the optical axis.
Conclusion
Summary and Reflections
Although critical points of an infinite variety are possible, only elementary critical points, that is, the critical points of the lowest order, occur most frequently. The overall structure of the streamlines may change under this perturbation, but the total index of rotation is preserved. The behavior of the flux lines on a perfectly conducting surface and its surroundings was studied for both types of polarizations.
The number of these points on the conducting surface determines the rotation index of the Poynting vector field on a curve near the surface. Some problems can be solved qualitatively as far as the structure of the flow lines is concerned using the concepts developed in this thesis.
