(See in conjunction the enhanced visualization via Geogebra animation https://www.geogebra.org/m/exufcmbz )
Noted at this time, the effort described below is limited to the unique geometric qualities of the Paraboloid regardless of reflected wave form. The word “focus” is used in the context of both a focal point and ring (radially displaced). As well, dimensional applicability is interchangeable unless otherwise noted (i.e., the 2D parabola and 3D paraboloid are reciprocal).
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At its most fundamental, the lack of sufficient energy density necessitates the reflector (i.e., a reflector acts to reduce a signal’s entropy). In the mode of reception, a reflector collects and concentrates wave energy by means of aperture size and geometric shape. As well, during transmission, the distribution and density of wave energy is facilitated by the same. When used as a reflector, a paraboloid offers optimum efficiency for this process. Perhaps most critically, the paraboloid allows for on-axis wave interaction to remain in-phase throughout. Thus, a parabolic reflector is frequency agnostic. Another superlative characteristic is the paraboloid’s natural filtration attribute which is exemplified in the context of reception: Only a planar, incident wave front, traveling in a direction parallel to the paraboloid’s axis of rotation (on-axis), reflects off the paraboloid and collapses spherically, in-phase, to a singularity (or ring if radially displaced) at the focus. Of course, in the case of transmission, the reverse occurs resulting in a collimated wave front emanating from the focal plane.
The parabola itself is defined as the locus of a displaced point such that its distance to a fixed point (focus) is the same to that of a fixed straight line (directrix), and in the most basic standard form, it is represented by the quadratic y = x 2. For some two-dimensional context, the parabola and the circle, as defined by conic section, exists as singular entities. This, unlike the ellipse1 and hyperbola which are infinite in shape variation. Our ability to manipulate the parabola and circle is limited strictly to dilation. However, while the circle can be offset with a parallel path creating another circle, the parabola cannot. A locus of points offset parallel to the parabola does not, and cannot, constitute a new parabola. Thus, a distinguishable uniqueness of the parabola. Simply put, any point not on the parabola does not possess its qualities of reflection.2 As a result, three-dimensional real-world efforts in paraboloid fabrication most certainly result in an overall hyperboloid, ellipsoid or combination of the two but absolutely never the paraboloid.
I should note my disagreement with the GD&T use of Profile of a Surface (⌓) to characterize the allowable deviation of the paraboloid’s surface. This parallel tolerance band is analogous to RMS and thus, my equivalent discontent with it as a measure of reflector goodness.
Anecdotally, GD&T is deficient regarding the conic sections (less the circle). The circle is never characterized by (⌓ ) “ Profile of a Surface ” but rather by its own unique callout of “ Roundness ” or “ Circularity ” (◯). An ellipse, for example, is characterized by its eccentricity value. In my view, conic sections should be characterized by their eccentricity and evaluated in a manner consistent with a departure from those values. For the parabola, eccentricity offers the possibility of a tolerance band that is a non-parallel region, straddling the nominal, and tapering to convergence at the vertex. This, however, necessitates viewing deviation in the context of a dilated focal length or in a slightly different reference frame (elliptical or hyperbolic) resulting from a change in the sectioning plane.
Given, as stated above, that the purity of the paraboloidal shape is impossible to achieve, perhaps we need to better define the criteria for expected conformance with its intended purpose. To that end, I propose that for a given applications, the use of a maximum allowable phase error as criteria.3 This means working backwards to calculate a reflector surface profile tolerance for first-order effects. It is an effort to more correctly define and convey the geometric precisions necessary without over-constraint.
Metrology of a reflector should therefore be constrained by its vertex (found empirically), and the nominal focus. In an orthogonal (Cartesian) reference frame, surface deviations are then measured as departures from the quadratic form y(focal axis) = a ( ( x2 + z2 ) .5 ) 2 via the resultant changes to the exponent 2.4 Much in the way sphericity represents how geometrically close an object is to a sphere, this methodology provides a sense of how accurate a surface is to that of the paraboloid. Thus, deviations can be evaluated in terms of the mathematical surface model without specific regard to their eccentricity characteristics. Additionally, this metric provides a high degree of deviation sensitivity allowing truly anomalous data to be easily identified and eliminated from the data set.
Next, we need to establish an applicable maximum allowable phase error. For discussion purposes, let’s use π / 8 or 1 / 16 λ to be the maximum allowable for parabolic reflectors.
We can now calculate a surface deviation tolerance.
Notational References:
x
y = cartesian coordinate values
z
nom = nominal
act = actual or measured
res = resolved
mmc = maximum material condition
lmc = least material condition
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Therefore, for any measured point on the surface of the reflector, a ± tolerance zone can represented via the quadratic’s, or now quasi-quadratic’s, exponent as a function of the maximum allowable phase error. By doing this, the constraints of the vertex and nominal focal length are preserved. Exponent representation also offers a delineation of the measured point’s geometric tendency where ℝ > 2 is ellipsoidal, and ℝ < 2 is hyperboloidal relative to these constraints. As well, since each deviated point on the surface of a reflector has an associated phase error , one could consider a reflector more specular or “clear” if the aggregate phase error is more consistent. Inconsistency, regarding the same, and the reflector might be considered more diffuse or “fuzzy”.
Concurrent with exponent representation, the more conventional representation of a tolerance band is available. Within the orthogonal (Cartesian) reference frame, and in compliance with the controlling mathematical surface function of the paraboloid
ynom = anom ( xnom2 + znom2 ) .5 ,
a linear ± tolerance is correctly applied in a direction parallel to the .5 Note that as a result of the boundary limits, the quadratic bounding functions provide a non-parallel tolerance band which naturally tapers toward the vertex. This necessitates that as xact and zact change, so too does the ± yact tolerance zone serving to mitigate the application of artificial weighting to the metrology data.
In either regard, we need only define the reflector constraints as the vertex being the origin, the focal length as a basic value, and the maximum allowable phase error.
e.g.,
-A- , defines Y0
-B- , defines X0 , Z0
[ 12.9874 ]” , defines the focal length
mnom = .0242” , defines the maximum allowable phase error
Here, a note on the easy expansion of metrology analysis and metrics. Contact applications such as ball probing or laser ball tracking may provide normal i , j , k vectors associated with discrete surface deviations. These vectors, in conjunction with a Rx signal’s incident vector, establish angles of reflection for use in calculations to ascertain proximity of 2f termination to the focus (or focal ring if radially displaced) and focal axis. The same in reverse provides a picture of the wave front’s “ flatness ” at the focal plane in the case of Tx . Additional metrics such as eccentricity, wave front deviation, focal length deviation, density along the focal axis, focal axis parallelism, and the conventional RMS can be derived from this information. We should keep in mind that the vectors involved, by definition, have both magnitude and direction. While magnitude can be viewed as an in-phase terminus that either overshoots or undershoots the focus, direction represents a trajectory or “off-target” amount to the focus. Together, they represent a deviation from the focus by some spherical radius. From this, a focal “cloud ” can be visualized where its location and density can be analyzed in terms of sphericity (a torus for the focal ring).
If we consider our intention is to construct a reflector via a surface of revolution, then it may not be unreasonable to analyze deviated points in a two-dimensional resolved reference frame. This where xres = ( xres2 + zres2 ) .5 or xres = ( xres2 + zres2 ) .5 – r if radially displaced and yres = yact . In this context, the following derived from a deviated point on the surface of the paraboloid:
Eccentricity6 –
A unitless value where for the parabolic ecc = 1 , the hyperbolic ecc > 1 , and the elliptical1 0 < ecc < 1 .
Wave Front Deviation –
This calculation is viewed as a reflected, spherical wave front, from the deviated point (see associated animation). The result is the in-phase over or under-shoot of the nominal focus / focal plane.
Focal Length Deviation –
Applying y = ax 2 to a deviated point while constraining the vertex allows dilation resulting in a deviated focus.
Focal Density –
A measure of Focal Length Density along the focal axis.
The examples of Wave Front and Focal Length deviation yield density graphs. A histogram to depict a normalized probability density, with frequency specific bin width equal to the allowable phase shift and a KDE plot to show normalized statistical density. In each case, graphs are centered about the nominal focus. For the KDE plot of the Wave Front Deviation, the area under the phase shift curve represent over-shot and under-shot densities relative to the nominal focus. In the KDE plot of Focal Length Deviation, the area under the phase shift curve represents data densities along the optical axis relative to the nominal focus with the highest peak indicating optimal focus. In-phase data in both case is represented directly underneath any specific point along the phase shift curves.
Focal Axis Parallelism –
Here, reflection from the deviated point is considered where Tx is originated at the nominal focus. Positive values converge with the focal axis and negative values diverge.
Traditional RMS –
this from:
. . .
1 Special exception for the circle ecc = 0 .
2 At the expense of focal length and phase shift, a deviated point may reside on a different, dilated parabola in the same reference frame.
3 Nothing here attempts to preclude the validity or value of the Ruze equation for RF specific evaluation of surface efficiency.
4 Changes in the integer power of 2, while benign to the geometric plane, manifest mathematically in a complex plane.
5 While this tolerance could be converted to a value that represents a normal-to-nominal, it would directly contradict the fundamental equation.
6 Where θ is the displaced angle in degrees from the axis of a double napped conic
† Best case – reflected wave front (Tx – normal intercept with focal plane, Rx – intercept with resolved focal axis)
‡ Best case – fnom is resolved origin for Tx and terminus for Rx
* Deviation is normal to nominal
