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Mapping Focal Branches via Implicit Equations: The Half-Wave Zone Slit Model

저자:Roman Chijner
Applet Description: Focal Curves in Near-Field Slit Diffraction This applet derives explicit equations for the focal curves within the Half-Wave Zone Model of slit diffraction. 1. Visualization of Stationary Points The ap plet identifies the stationary points of the 2D distribution J(x, y) using a distinct color-coded system:
  • Red & Blue Points: Represent local maxima and minima, respectively.
  • Crosses (Multiple Colors): Represent saddle points positioned between the extrema.
  • Symmetry: Note that the distribution of these points is symmetric with respect to the y-axis.
2. From Implicit Equations to Cubic Roots The derivation begins with the implicit equation (eq00) for the focal curves. By eliminating irrational terms, this is transformed into a cubic polynomial equation (eq0). We solve the general cubic form: a(z)y3 + p(z)y2 + c(z)y + d(z) = 0 where z is treated as a complex variable. This allows us to express the roots in the explicit form y = f(z). All symbolic and numerical computations were performed directly within GeoGebra. 3. Analysis of the Solution Branches
  • Curve Cuf1: Corresponds to the real part of the first complex function, f1(z). This branch coincides perfectly with the original implicit function eq00.
  • Curve Cuf2, Curve Cuf3 : Correspond to the real part of the complex functions f2(z), f3(z), respectively, and represent subsequent transformations and additional branches of the cubic solution.
Notes: Rendering complex functions is computationally intensive. Please be patient or use a desktop computer for a smoother experience. Below the applet, you will find images of the CAS transformations and the behavior of the curves related to the solutions f1(z), f2(z), and f3(z).

Obtaining the implicit equation (eq00) of the focal curve:

Obtaining the implicit equation (eq00) of the focal curve:

Obtaining the implicit equation of the focal curve (eq0) from the equation (eq00) by eliminating irrationality from it:

Obtaining the implicit equation of the focal curve (eq0) from the equation (eq00) by eliminating irrationality from it:

1. The transformation to eliminate irrationality in the focal curve introduces additional branches to the implicit function

1. The transformation to eliminate irrationality in the focal curve introduces additional branches to the implicit function
Implicit functions before (a) and after (b) transformation. As illustrated in the secondary plot, the transformation to eliminate irrationality in the focal curve introduces additional branches to the implicit function.

2. Branches of the complex solutions for the cubic polynomial

2. Branches of the complex solutions for the cubic polynomial
(a) The implicit function following transformations, reduced to a cubic polynomial form. (b)–(d) Branches of the complex solutions for the cubic polynomial. As illustrated, the imaginary components of these complex solutions vanish along the real axis. While the first branch, f1(z), satisfies the original implicit function, the remaining two branches emerge as a result of the subsequent transformations.

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