In the half-wave slit model, we define the family of curves passing through the intensity field extrema behind the slit as 'extremal' (or 'stationary') lines.
The region of stationary points for the diffraction field intensity function of the half-wave model is shaded in purple.
In this model, each point represents the intersection of three base curves: a focal curve, a hyperbola, and a parabola. The existence of two branches of the parabola means that for every intersection point within the shaded region, there is a corresponding point outside it—also formed by the intersection of the same three base curves. For example, points A1 and A2 are shown here.
Intersection points of three families of curves at the stationary points of the half-wave slit model, overlaid on a Heatmap of the intensity distribution calculated via the Kirchhoff diffraction integral.●Red → local maxima ☐.
●Blue→local minima ☐.
●Saddle points. I distinguish two types of them: + and+.
Behaviour of "constructive" and "destructive" interference lines from 2 "phantom" sources at the edges of a slit (families of hyperbolic curves) for the case of its width b=6 in the far field
In the far field, as shown in the applet, the lines of "destructive" interference from the two 'phantom' sources at the edges of the slit roughly coincide with the directions of the diffraction Fraunhofer diffraction-maxima. The lines of "constructive" interference from these two 'phantom' sources coincide exactly with the directions of the minima of Fraunhofer diffraction (except for the axial direction).
The lines of "destructive" interference in the diffraction pattern pass through saddle points, while the lines of "constructive" interference pass through points of local maxima and minima.
![[size=85] The region of stationary points for the diffraction field intensity function of the half-wave model is shaded in [color=#9900ff]purple[/color].
In this model, each point represents the intersection of three base curves: a [color=#1e84cc]focal curve[/color], a [color=#6aa84f]hyperbola[/color], and a [color=#980000]parabola[/color]. The existence of two branches of the parabola means that for every intersection point within the [color=#9900ff]shaded region[/color], there is a corresponding point outside it—also formed by the intersection of the same three base curves. For example, points [color=#0000ff]A1[/color] and [color=#0000ff]A2[/color] are shown here.[/size]](https://www.geogebra.org/resource/tzqcng9h/wKpjs6F8n3NjLHUN/material-tzqcng9h.png)
![[size=85] Intersection points of three families of curves at the stationary points of the half-wave slit model, overlaid on a [url=https://www.geogebra.org/m/peazngue]Heatmap[/url] of the intensity distribution calculated via the Kirchhoff diffraction [url=https://www.geogebra.org/m/c5rhh2tz]integral[/url].[/size]
[size=85]●[b][color=#ff0000]Red[/color] [/b]→ local [color=#ff0000][b]maxima[/b] ☐[/color].
●[color=#0000ff][b]Blue[/b][/color]→local [color=#0000ff][b]minima[/b] ☐[/color].
●[b]Saddle points[/b]. I distinguish two types of them: [size=100][color=#ff0000]+[/color] [/size][/size][size=85]and[/size][size=85][color=#0000ff] [size=100]+[/size][/color].[/size]](https://www.geogebra.org/resource/pkh2vtep/x8k9LJ7syGKMHSDh/material-pkh2vtep.png)
