[size=85][color=#333333] The half-period zone model is a conceptual framework used to explain light diffraction without relying on the direct calculation of Kirchhoff’s diffraction integrals. Instead, it is based on dividing the aperture into specific zones.While applying Fresnel's circular half-period zones to a circular aperture has proven successful, describing diffraction through a slit requires a slightly different approach. Specifically, in the slit model, the areas of the zones do not remain constant, meaning the number of secondary sources within each zone varies. My method for constructing these zones differs in the design of the central zone and can be found in the [url=https://www.geogebra.org/m/xMb3zVMx]applet[/url][b].[/b][br] In this applet, we identify specific observation points where the boundaries of the outer zones coincide with the physical edges of the slit. We propose that these points are "close" to the critical points of the intensity function J = J(x, y), as calculated via the [url=https://www.geogebra.org/m/c5rhh2tz]Kirchhoff integral[/url] for a slit. Comparisons with numerous [b]Heatmaps[/b] generated using this integral for different slits widths[/color][color=#333333] confirm the validity of these assumptions ([url=https://www.geogebra.org/m/bfm4ny82]1[/url]). Furthermore, [i]numerical calculations of the extrema[/i] align [i]with the model's predictions with a high degree of accuracy[/i] ([url=https://www.geogebra.org/m/djykrvyv]2[/url], [url=https://www.geogebra.org/m/mh4xu32k]3[/url]).[/color][/size][size=85][b][br] The primary objective[/b] is to use geometric transformations to determine the coordinates (x, y) of the intensity extrema J(x, y) within a diffraction pattern. These coordinates are expressed as functions of the integers [b]m1[/b] and [b]m2[/b], which define how the half-wave zones occupy the slit. For integer values of these filling numbers, the physical edges of the slit coincide with the boundaries of the full zones; in fact, the outermost boundaries of the zones should ideally coincide with the slit edges.[br] It has been established that these coordinates can be mapped using the [url=https://www.geogebra.org/m/dzzgh2fk]functions[/url]:[b] fx[/b] and [b]fy[/b]: [b][color=#1e84cc]A(x, y) [/color][/b]= ([b]fx[/b]([b]m1[/b], [b]m2[/b]), [b]fy[/b]([b]m1[/b], [b]m2[/b])) (see [b]Fig. 1[/b]). The derivation of these functions is based on a geometric analysis of the aspect ratios and coordinates within triangle [b]△[color=#1e84cc]A[/color][color=#ff0000]BC[/color][/b] (refer to the figures below):[br]● The vertex [b][color=#1e84cc]A(x, y)[/color][/b] is the observation point. The base [b][color=#1e84cc]a[/color][/b] is equal to the slit width [b]b[/b]. Sides [b][color=#ff0000]b[/color][/b] and [b][color=#ff0000]c[/color][/b] are defined as: [b][color=#ff0000]b[/color][/b]: [b]r1[/b]= [b]y[/b] + [b]Δ[sub]0[/sub][/b] + [b]m1[/b] * [b]λ[/b]/2 and [b][color=#ff0000]c[/color][/b]: [b]r2[/b]= [b]y[/b] +[b] Δ[sub]0[/sub][/b] + [b]m2[/b] * [b]λ[/b]/2, respectively.[br][b]● m1[/b] and [b]m2[/b]: These represent the number of full zones fitting into the right and left parts of the slit, respectively, relative to the middle of the central zone (Point [b][color=#1e84cc]A'[/color][/b]).[br]● We define [b]Δ[sub]0[/sub][/b] as the slit constant. To determine its value, we consider an observation point located [i]on the axis[/i] at the position of the final intensity [b]J=J(x)[/b] maximum, as calculated via the[url=https://www.geogebra.org/m/c5rhh2tz] [i]Fraunhofer diffraction integral[/i][/url] for the slit. This specific point is designated as the first focus ([b][color=#ff0000]F1[/color][/b]). In this configuration, the entire width of the slit is treated as the central zone. Consequently, [b]Δ[sub]0[/sub][/b] is defined as the optical path difference at this focal point between the central (axial) ray and the ray from the central zone boundary at the slit edge.[br] The results demonstrate that the extrema of the function [b]J(x, y)[/b] in this model belong to [b][i]three[/i][/b] distinct families of curves. The actual extremum points are located at the intersections of these curves -see [b]Fig. 6[/b].[br] [b][u]The resulting point system reveals three distinct directional groups, each following a characteristic curve[/u][/b].[/size]
[size=85][b] 1.[/b] Evidently, the points in the field for which the path difference [b]Δ=r1-r2[/b] remains [i]constant[/i] lie on a family of [b][u]hyperbolas[/u][/b]. Given that [b]Δ[/b] = ([b]m1[/b] - [b]m2[/b])*[b]λ/2[/b], where ([b]m1[/b] -[b] m2[/b]) is a constant integer, these curves represent lines of [i]constructive[/i] or [i]destructive interference[/i]. The [i]foci [/i]of these hyperbolas are the "[i]sources[/i]" [i]located at the ends of the slit, and the quantity ([b]m1[/b]-[b]m2[/b]) determines the specific branch of the family[/i][/size][size=85][b].[/b][/size]
[size=85][b][b]Fig. 1.[/b][u] Hyperbolic [/u][/b][u][b]"[color=#ff0000]Antin[/color][/b][/u][u][b][color=#ff0000]odal[/color][/b][/u][u][b]" [/b][/u][u][b]and [/b][/u][u][b]"N[/b][/u][color=#333333][u][b]odal[/b][/u][/color][u][b]" [/b][/u][u][b]lines [/b][/u][b]of [color=#ff0000]constructive[/color] and destructive interference of the [/b][color=#004dbb][i][b]two point source interference pattern [/b][/i][/color][i][b][color=#004dbb]from the edges [/color][color=#ff0000]B[/color][color=#004dbb] and [/color][color=#ff0000]C[/color][color=#004dbb] of a slit [/color][/b][/i][color=#333333][b]in single-slit diffraction pattern [/b][b]itself.[br][/b] Formulas for determining the coordinates [b]x[sub]k[/sub][/b] and [b]y[sub]k[/sub][/b] of the "extrema" of the diffraction field can be found at the bottom of the figure. [url=https://www.geogebra.org/m/jwtpe5ps]Applet[/url].[/color][/size]
[size=85][b]2. [/b]For the 'extreme' points of the model lying on each [b][u]parabolic line[/u][/b], one integer index remains constant, while the other changes by one.[/size]
[b][b]Fig. 2.[/b] [u]Please note[/u][/b]:[br][size=85]a.● For even values of the curve order [b][color=#ff0000]u = 0, 2..., 2*⌊b/λ}⌋-2[/color][/b]: the parabola of the diffraction [i]model under consideration[/i] passes through the "[b][color=#ff0000]local maxim ◇(max)[/color][/b]" and the "[b][color=#ff0000]saddle points + (maxsad)[/color][/b]" located between them. For all points [b]m2[/b]=[b][color=#ff0000]u[/color][/b].[br] ●The extrema for[b] "[color=#ff0000]◇max[/color]"[/b] and "[color=#ff0000]+[b]maxsad[/b][/color]" occur at points corresponding to [i]even[/i] and [i]odd[/i] values of [b]m1[/b], respectively. See Fig. 5.[br][br]b.● For odd values of the curve order [b]u = 1, 3, ...2*⌊b/λ}⌋-1[/b]: the parabola of the diffraction [i]model under consideration[/i] passes through the "[b][color=#0000ff]local minima ◇(min)[/color][/b]" and the "[b][color=#0000ff]saddle points +(minsad)[/color][/b]" located between them. For all points [b]m2[/b]=[b][color=#ff0000]u[/color][/b].[br] ●The extrema for[b] "[color=#0000ff]◇min[/color]"[/b] and "[color=#0000ff]+[b]minsad[/b][/color]" occur at points corresponding to [i]odd[/i] and even values of [b]m1[/b], respectively. See Fig. 5.[/size]
[b][size=85]3. Implicit equation of the [color=#ff00ff][u]focal curve[/u][/color].[/size][br] [/b][size=85]For any pair of extrema located symmetrically on the [color=#ff00ff]focal curve[/color] (e.g., points 0 and 4, or 1 and 3 from figure [b]a[/b]), the sum of the parameters [b]m1[sub]k[/sub][/b], [b]m2[sub]k[/sub][/b] is invariably equal to [b][color=#cc0000]u - 1[/color][/b]. Leveraging this symmetry, we can establish a system of equations (equations from figure [b]b[/b]) by introducing a parameter[b] [color=#ff00ff]k[/color],[/b] which, in this specific case, varies from [color=#ff00ff]4[/color] to [color=#ff00ff]0[/color]. By eliminating [b]k[/b] between these equations, we derive the desired [b][i][url=https://www.geogebra.org/m/qgg4ujte]implicit equation[/url][/i][/b].[/size][br] [size=85] Here the [color=#ff00ff]focal curve[/color] is defined as an implicit function. By adjusting the curve order (slider [b][color=#cc0000]u)[/color][/b], the curve is made to pass through all extrema corresponding to the focal points on the slit axis. The $15 and $18 lines (equations from figure b) describe the expressions for the focal curve in the cases of [i]vertical[/i] and [i]horizontal[/i] slit axes, respectively.[/size]
[b]Fig. 3a and Fig. 3b [/b]
[b]Applets for obtaining the focal curves in various mathematical forms[u]:[/u][/b][br][size=85][br][url=https://www.geogebra.org/m/qgg4ujte]1b. CAS Applet for Obtaining Implicit Equations of the Focal Curve via the Half-Wave Zone Model[/url][br] [br][url=https://www.geogebra.org/material/show/id/a8h6psbh]1c. CAS Applet for Transforming Implicit Functions to Eliminate Irrationality in Focal Curve Equations Derived via the Half-Wave Zone Model[/url][br] [br][url=https://www.geogebra.org/m/xafjz96h]Mapping Focal Branches via Implicit Equations: The Half-Wave Zone Slit Model [/url][br] [br][url=https://www.geogebra.org/m/bugexxbe]1d. CAS Applet for obtaining an explicit equation of the focal curve R(x), obtained in the half-wave zone model[/url][/size][br]
[size=85][b]Fig. 4.[/b] [u]Vector Orientation[/u]: The resultant vectors from the partitioned zones [i]are approximately parallel[/i]. Vectors aligned with the direction of the central zone are designated in [color=#ff0000][b]red[/b][/color], while antiparallel vectors are designated in [color=#0000ff][b]blue[/b][/color].[/size]
[size=85][b]5.[/b] [b]Formation of zones for the "critical points" in the proposed model of light diffraction behind a slit[br][/b][/size][size=85][b]●[b]Central Zone[/b]: [/b]A [i]central zone[/i] of action within the slit is present for all "[b]extreme points[/b]." [br]● [b][color=#ff0000]Max[/color]:[/b] A characteristic feature of zone formation is that all zones of opposing actions (excluding the central zone) are compensated: at the ends of the slit there are [b][color=#ff0000]red zones[/color][/b].[br]●[b][color=#0000ff]Min[/color]:[/b] A characteristic feature of zone formation is that all zones of opposing actions (excluding the central zone) on either side of the central zone are uncompensated: at the ends of the slit there are [b][color=#0000ff]blue zones[/color][/b].[br]●[b]Zone formation for "saddle points"[/b]:[br]-There are two distinct types of [i]saddle points[/i] based on their location and formation -See the note for [b]Fig. 2.[/b][br] -Common Features: For both types, there is a lack of compensation (excluding the central zone): at the ends of the slit there are multi-colored([color=#ff0000][b]red[/b][/color]-[color=#0000ff][b]blue[/b][/color]) zones.[br][br][/size]
[size=85][b]Fig. 6. Heatmap:[/b] Fragment of the diffraction field for a slit of width [b]b=20[/b].[br] Dependence [b]J=J(x)[/b] ([color=#ea9999]pink[/color]) for points of the [b]Focus curve[/b] ([color=#93c47d]light green[/color]), passing through [color=#ea9999][b]F₅[/b] [/color][color=#333333]and[/color] projected on a movable vertical screen (dashed and [color=#b6b6b6]beige[/color]), passing through point [color=#ff00ff][b]D[/b][/color].[/size]
An interactive exploration of the described model is available in the [url=https://www.geogebra.org/m/ne6edbww ]applet[/url]: "[b][i]Visualizing Near-Field Single-Slit Diffraction: A Tutorial Applet Based on the Half-Wave Zone Model[/i][/b]"
