[size=85] In the half-wave slit model, we define the [url=https://www.geogebra.org/m/dzspjpfj]family of curves[/url] passing through the intensity field extrema behind the slit as 'extremal' (or 'stationary') lines.[/size]
[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].[br] 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]
[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][br][size=85]●[b][color=#ff0000]Red[/color] [/b]→ local [color=#ff0000][b]maxima[/b] ☐[/color]. [br]●[color=#0000ff][b]Blue[/b][/color]→local [color=#0000ff][b]minima[/b] ☐[/color]. [br]●[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]
[size=85] In the far field, as shown in the [url=https://www.geogebra.org/m/jwtpe5ps]applet[/url], the lines of "[color=#333333]destructive[/color]" interference from the two '[color=#ff0000]phantom[/color]' sources at the edges of the slit roughly coincide with the directions of the diffraction [url=https://en.wikipedia.org/wiki/Fraunhofer_diffraction]Fraunhofer diffraction[/url]-[color=#ff7700]maxima[/color]. The lines of "[color=#ff0000]constructive[/color][color=#ff7700]" [/color]interference from these two '[color=#ff0000]phantom[/color]' sources coincide exactly with the directions of the [color=#0000ff]minima[/color] of [i]Fraunhofer diffraction (except for the axial direction)[/i].[br] The lines of "destructive" interference in the diffraction pattern pass through [b]saddle[/b] points, while the lines of "[color=#ff0000]constructive[/color]" interference pass through points of local [b][color=#ff0000]maxima[/color][/b] and [b][color=#0000ff]minima[/color][/b].[/size]
