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3.3. Conics and aberrationsTelescope aberrations can be divided in two main groups: (1) those intrinsic to optical elements in their proper alignment, and (2) aberrations induced by some external factor. This division is not exclusive; most major aberrations can be caused by both, intrinsic surface properties and misalignment or thermal factors. Rather, its purpose is to identify and relate aberrations according to their origin. Among the aberrations caused by intrinsic properties of optical elements, most important are those characteristic of a conic surface of revolution in monochromatic (single wavelength) light. While all conic surfaces of revolution have perfect radial symmetry, that alone does not enable them to form perfect wavefronts. In order for this to happen, a surface must have specific form determined by both, its optical properties (index of reflection or refraction) and form of the incoming wavefront. For instance, spherical reflecting surface will form a perfect spherical wavefront if the object is placed at the mirror center of curvature (obviously, the emitted wavefront, the surface and the reflected wavefront all coincide at the surface). However, as the object distance increases and the wavefront flattens, spherical surface advances wavefront points toward the edges more than those toward the center. As a result, the reflected wavefront is aberrated, curving in more strongly toward the edges than its perfect reference sphere. For object at infinity, the incoming wavefront is practically flat, and the extent of aberration reaches its maximum. Thus, a surface curving less toward the edges - a paraboloid - is needed to form a perfect spherical wavefront. For intermediate object distances, the "perfect" surface is an appropriate prolate ellipsoid. 3.3.1. Conic sections, geometric propertiesIn other words, for any reflecting conic surface there is a single pair of conjugate points (i.e. object and image locations) that will produce perfect axial image (such surface is called Cartesian reflecting surface). We'll call these two points (conic) specific conjugates, or specific/geometric foci. This pair of points is, in fact, what optically defines the conical curve, as a set of points for which the optical path length via any point on the conic is constant (geometric focus of a conic is different from the Gaussian focus, determined by focal length and independent of the conic form). Obviously, that is the condition for zero aberrations. For the sphere, the geometric foci coincide at its center of curvature; for parabola, one is at the Gaussian focus, and the other at infinity; for hyperbola, one is on the opposite side of surface and the other one is inside the focal point (this merely means that it is free from spherical aberration for object distance smaller than its focal length, or for converging incoming beam). For prolate ellipse - the one with horizontal axis longer than vertical axis - specific foci lie at the longer (horizontal axis), equally separated from the center point (F1 and F2 in FIG. 11). And for oblate ellipse, the two spherical aberration-free conjugates, or foci, are also on the longer axis, which is this time vertical (note that the ellipse here is categorized as oblate or prolate based on its form around the optical axis, which is horizontal; when placed in a coordinate system and categorized with respect to the vertical axis, ellipse is oblate when flattened vertically, and prolate when vertically extended).
Rotating conic section 360° around its axis of symmetry produces a conic surface of revolution centered around its vertex: rotating the circle produces sphere, oblate ellipse produces oblate ellipsoid, prolate ellipse produces prolate ellipsoid, parabola paraboloid, and hyperbola hyperboloid.
The main parameters of these surfaces are
given in the table bellow, followed by more detailed description.
*True eccentricity/conic in the brackets; true conic constant is given as positive, to differentiate between the two types of ellipsoidal surface; from the relations defining the squares of ellipse's semi-axes and center-to-focus separation, oblate ellipse's eccentricity value used for in the ellipse function (not a true eccentricity) is an imaginary number of the negative sign when squared In terms of conics' shape, for given vertex radius, oblate ellipses are most strongly curving away from the vertex (center). As their vertical foci converge closer, oblate ellipse becomes less strongly curved, converging to a circle when its foci merge in a single point. As the center of a circle separates into a pair of points on horizontal axis, they define the next less strongly curved conic, prolate ellipse. Further increase in separation of the two foci results in still less strongly curved conics: parabola, for which the farther away focus lies at infinity, and hyperbola, whose far focus is beyond infinity - at the opposite side of conic's vertex. The measure of how strongly curved is a conic section is its eccentricity (ε). In optical terms, it can be expressed as the ratio of its inter-focal separation to the optical path length (OPL) from one specific focus to another, via any point of a conic (any ray originating at one of the two foci is reflected to the other one). Since both parameters are constant for given conic shape, so is its eccentricity (note that eccentricity of conic sections is usually expressed in geometric terms - as the ratio of a conic's point distance from its near focus to that point's distance to a fixed straight line, called directrix - but defining it in optical terms is more appropriate here). For oblate ellipse, the inter-focus separation is finite, and ranges from approaching zero to approaching the optical path length between the foci, via point belonging to a conic section. Thus, its eccentricity range is 0<ε<1, as in prolate ellipse. Note that this approach is uncommon, in that the eccentricity of oblate ellipse is expressed in the same manner as that of prolate ellipse, that is, as the ratio of its inter-focus separation vs. longer (in this case, vertical) semi-axis. As a consequence, identical eccentricity here implies identical shape of oblate and prolate ellipses: their longer vs. shorter axis ratio is identical, with the oblate ellipse only being smaller for given vertex radius. Such approach also allows to unify oblate ellipse with other conic sections in defining their shape by using optical path length between the two specific foci. However, more common approach is to express the eccentricity of oblate ellipses as the ratio of their inter-focus separation vs. horizontal, shorter axis (this value is not a true eccentricity, but allows to define a form of the oblate ellipse with Eq. 4). Thus, to convert from the eccentricity as expressed here, to the more common form, the eccentricity value needs to be multiplied by the ratio of longer vs. shorter axis of the oblate ellipse. This ratio is given by (1-ε2)1/2/(1-ε2), or 1/(1-ε2)1/2, with ε being the eccentricity calculated in the same manner as for prolate ellipses. Thus, for instance, for ε=0.95 here, the common eccentricity equivalent for oblate ellipse would be 1.93. For the circle, the inter-focus separation is zero, and so is its eccentricity. For prolate ellipse, the inter-focus separation is finite, always smaller than the OPL; thus, its eccentricity range is 0<ε<1. Inter-focus separation for parabola, with one specific focus at infinity is infinity, and so is the OPL; thus its eccentricity equals 1. Finally, for hyperbola, with one specific conjugate (focus) located inside its Gaussian focus, and the other, imaginary, on the opposite (positive) side of the conic's vertex, the inter-focus separation, which is positive, is always greater than the OPL. Here, the inter-focus separation is identical to the OPL geometrically, but is numerically smaller, being given by a sum of the larger, positive portion measured from the surface to the imaginary specific focus to the right, and smaller negative portion measured from the surface to the real specific focus located to the left of the vertex, inside the Gaussian focus. Conic shape is commonly expressed as the conic, or Schwarzschild constant, defined as K=-e2. Since the eccentricity e is a positive number for all conics but oblate ellipses, the corresponding conic is negative for prolate ellipses, parabola, and hyperbolas. It is positive for oblate ellipses (this originates from mathematical formalism, defining ellipse center-to-focus separation as c2=a2-b2, a, b being horizontal and vertical semi-axis, respectively; since in prolate ellipses the latter is longer, their c2 is a negative value, and so is ε2= c2/a2). Denoting the inter-focus separation as S, and the optical path length from one specific focus to a conic point and to the other specific focus as OPL, the eccentricity is given by ε=S/OPL. It defines the conic section as: (1-ε2)x2-2Rx+y2=0 (4) with x and y being the horizontal and vertical coordinate, respectively. Specific near focus distance L1 from the conic's vertex, in terms of the vertex radius of curvature R, is given by: L1=R/(1+ε), or L1=R/(1+√|K|) (4.1) That gives L1=R for sphere, L1=R/2 for parabola, and intermediate values for prolate ellipses. Near focus distance is smaller than R/2 for hyperbolas and oblate ellipses. Note that the latter are a special case, in that their specific foci are not located on the axis, rather on a straight line perpendicular to it. Separation from the axis, and the conic vertex is equal for both foci. The vertex-to-perpendicular-line separation is R/(1-ε2), with ε2 being, by definition, negative for oblate ellipses, and the vertex-to-focus separation is R/(1-ε2)1/2. The inter-focus separation S, measured from the near focus with a conic oriented to the left, for all conics except oblate ellipses is given by: S=2{1/(1-ε2)-[1/(1+ε)]}R, or S=2{1/(1+K)-[1/(1+√|K|)]}R (4.2) For oblate ellipses, it is: S=2{1/(1-ε2)1/2-[1/(1+ε)]}R, or S=2{1/(1+K)1/2-[1/(1+√|K|)]}R (4.3) where the eccentricity ε is positive. This gives S=0 for the circle, S=-∞ for parabola, -∞<S<0 for prolate ellipse, 0.46R<S<0.42R (inter-focal separation here reaches maximum of ~0.46R for the eccentricity ε~0.62, decreasing for both, higher and lower values of ε, flattening into a vertical line for ε=1 and S~0.42R), and 0>S>∞ for hyperbolas. Obviously, the far focus distance from the vertex is L2=L1+S for prolate ellipsoid and parabola. For oblate ellipses, as mentioned, both foci are at an equal distance from the vertex. The optical path length (OPL) between the two specific foci, via point of a conic, is given by: OPL = 2R/(1-ε2) = 2R/(1+K) (4.4) This gives OPL=2R for a sphere, OPL=-∞ for parabola, -∞<OPL<2R for prolate ellipses, 2R<OPL<0 for oblate ellipses, and 0<OPL<∞ for hyperbolas (note that Eq. 4.3 gives the geometric OPL; nominal OPL for hyperbolas, determining their eccentricity, is smaller - and of positive sign - given by the difference between the larger positive vertex-to-far and smaller negative vertex-to-near specific focus). The eccentricity expressed as ε=S/OPL is, therefore, of a positive sign for all conic types (zero for the circle). The OPL as given by Eq. 4.3 is identical to the longer axis of either prolate or oblate ellipses. An alternate set of relations for ellipses is based on the vertex radius of curvature R, horizontal semi-axis a and vertical semi-axis b, as shown in FIG 11. With a=R/(1-ε2) and b=R/(1-ε2)1/2, the eccentricity is given by ε2=1-(b/a)2, and the vertex radius of curvature R=b2/a.
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