◄ 3.5. Aberration function   ▐    4.2. Coma ►
 

As mentioned, these are the aberrations caused by inherent properties of properly positioned optical elements. They include the five primary aberrations intrinsic to conical surfaces - spherical, coma, astigmatism, field curvature and distortion - as well as chromatism and aberrations resulting from fabrication errors.

     2.1. Spherical aberration

Spherical aberration - or correction error - is the only form of monochromatic axial wavefront aberration produced by rotationally symmetrical surfaces centered and orthogonal in regard to the optical axis. The attribute spherical is probably due to this aberration being inherent to the basic optical surface - spherical - for object at infinity. However, spherical aberration will appear whenever the surface profile doesn't mach the object distance or, with multi-surface objectives, also from deviations in proper spacing. Spherical aberration affects the entire image field, which makes its correction first priority.

FIG. 16 illustrates under-corrected (negative) form of spherical aberration, characteristic of a spherical mirror for object at infinity. Due to the actual wavefront being not spherical, rays projected from it do not meet at the same point; the wavefront becoming more strongly curved toward the edge causes the foci for rays projected from its outer zones to fall progressively closer to the mirror.

With over-corrected (positive) spherical aberration, marginal rays focus farther away than paraxial rays. In either case, geometrical structure of the defocused zone remains identical in regard to the paraxial focus. For the longitudinal aberration normalized to 2 (Lg0<Λ<2), the geometric (ray) spot has a constant relative size and structure for a given value of Λ, as illustrated in FIG. 17. In units of the paraxial blur diameter (Λ=0), the blur size is 0.385 for Λ=2 (marginal focus), 0.25 for Λ=1.5 (circle of least confusion) and 0.5 for Λ=1 (diffraction focus). The relative wavefront error - either P-V or RMS - for any point between the two foci, in units of the error at the paraxial or marginal focus, is also constant, given by:

                                                                        ŵ=√1+0.9375Λ(Λ-2)                                 (6)

It gives the minimum relative aberration of 0.25 for Λ=1, which is the mid point between marginal and paraxial focus.

Note that the parameter Λ is related to peak aberration coefficients for spherical aberration and defocus, S and P, respectively, as Λ=-P/S. Thus, in terms of defocus error, spherical aberration is minimized, or balanced, for P=-S, or for spherical aberration at paraxial focus combined with identical P-V of defocus aberration.

Wave aberration function (i.e. the P-V wavefront error as optical path difference) of spherical aberration at diffraction focus is:

                                                                    Wb= sd4(r4- r2)                                                   (7)

with s being the aberration coefficients for spherical aberration, d the pupil radius and r the ray height at the pupil in units of the pupil radius. The quantity S=sd4 is the peak aberration coefficient for spherical aberration, equal to the p-v wavefront aberration at the paraxial focus. Quantity in the brackets determines aberration maximum at best (diffraction) focus in terms of the peak aberration coefficient as S/4 for ρ=√0.5.

You will notice that this function form for spherical aberration differs from one given for primary aberrations (Eq. 5.1). This is because Eq. 5.1 gives the aberration function for paraxial focus, or so-called classical aberrations. Advance in calculation methods revealed that the Gaussian image point is not the best focus location for the three aberrations affecting point-image quality - spherical, coma and astigmatism. Each of them requires shift from the Gaussian image point to their respective best focus location, where the central intensity of diffraction pattern is at its maximum (thus, best focus location is also called diffraction focus). Primary aberrations evaluated at the best focus location are called orthogonal or balanced. For spherical aberration, defocus from the paraxial focus needed for the shift to diffraction focus location is given by P=-S, with P and S being the peak aberration coefficients for defocus and spherical aberration, respectively (FIG. 18). Since for identical longitudinal aberration defocus P-V error is double the P-V error of spherical aberration at paraxial focus (or P=-2S), best focus for spherical aberration is at the mid point of its longitudinal aberration, as already stated.

Another difference between classical and orthogonal spherical aberration is in the sign of the P-V wavefront error. Since with the classical spherical aberration the point of maximum aberration lies closer to the focus than the perfect reference point (that is, the aberrated point has smaller optical path length), it is of negative sign. On the other hand, the point of maximum P-V deviation for balanced (best focus) spherical aberration is farther away than its perfect reference point, giving it a positive sign. Hence, the aberration coefficient for the classical aberration is negative, and that for the orthogonal (best focus) aberration is positive, due to its specific factor (ρ4-ρ2) being also negative (for ρ>0).

 [1] Bright stars will display more pronounced diffraction rings, especially the first, which may appear nearly as bright as the central disc even with the perfect pattern. This is due to a logarithmic intensity response of the eye: a 56 times lower intensity of the first bright ring may appear to the eye less than twice fainter. Only as the first ring intensity drops close to the threshold of perception, the central disc begin to appear much brighter.

In order to obtain specific wavefront aberration, we need to calculate the aberration coefficient s. General expression for the aberration coefficient of spherical aberration of a thin lens is:

                               s = - [n3 + (n+2)q2 + (3n+2)(n-1)2p2 + 4(n2-1)pq]/32n(n-1)2ƒ3            (8)

with n being the refractive index, q=(R2+R1)/(R2-R1) the lens shape factor, and p=1-2ƒ/i the lens position factor (ƒ is the lens focal length, and i the image-lens separation).

Suffice to say, single lens cannot be free from spherical aberration, except for the object inside the focal point of a positive or negative meniscus with specific values of p and q (in other words, lens forming a real image cannot have spherical aberration cancelled). The aberration is at its minimum for q=-2p(n2-1)/(n+2). More detailed evaluation of the lens primary spherical aberration is given in 2.8 Chromatic aberration.

Fortunately, calculating the aberration coefficient for mirror surface is quite simple. Its general form is given by:

                                                       sG= n[K+(m+1)2/(m-1)2]/4R3                                        (9)

with n being the index of incidence, K being the mirror conic, m the magnification and R the radius of curvature. Magnification is given by m= -(i/o), which is the image (i) to object (o) distance from the objective. For the object distance o known it can easily be obtained from m=-ƒ/(o-ƒ). For distant objects mg0 and the coefficient reduces to:

                                                               s = n(K+1)/4R3                                                         (9.1)

This gives the peak aberration coefficient for mirror surface and object at infinity as:

                                          S = n(K+1)D4/64R3  =  -n(K+1)D/512F3                             (9.2)

(for mirror oriented to the left, n=1). The P-V wavefront error of spherical aberration S and the RMS wavefront error ω relate as ω = S/√180. As mentioned, the peak aberration coefficient S equals the P-V wavefront error at the paraxial focus, and is larger than the P-V error at best (diffraction) focus by a factor of 4. Relationship between the P-V and RMS wavefront error in units of the wavelength, and the blur diameter at best focus in Airy disc diameters (for the same wavelength) is given by  Bs=8S/2.44 and Bs= ω√11,520/2.44, respectively.

    Spherical aberration can also be expressed in terms of the peak aberration coefficient S as ray aberrations:

                                     L= 16SF2r2,        T= 16SFr3     and      Ta= 16Sr3/D                 (10)

for the longitudinal, transverse and angular aberration, respectively, with D being the pupil diameter, and 0<r≤1 the relative ray height in the pupil with the radius normalized to 1 (for spherical mirror, ray aberrations are simply: L=D/32F, T=D/32F2 and Ta=1/32F3). The r parameter shows that longitudinal spherical aberration changes with the square of the ray height, while transverse and angular aberration change with the cube of ray height in the pupil. The transverse and angular aberration are for the blur diameter at the paraxial focus; blur at best focus is smaller by a factor of 0.5, and the circle of least confusion by a factor of 0.25. The sign of peak aberration coefficient determines the sign of ray aberration as negative, indicating that rays forming the blur boundary focus shorter with respect to the image plane (this also holds for best focus location, but not for the marginal focus location, where the blur boundary is formed by converging rays - that is, rays that focus farther away from the image plane - and all three ray aberrations are positive).

What could be of interest is the RMS blur radius for various locations within the span of longitudinal spherical aberration. It can also be expressed in terms of normalized longitudinal aberration Λ, and peak aberration coefficient S, as rRMS=8FS[0.25-(Λ/3)+(Λ2/8)]1/2. In units of the paraxial blur radius, it is just the value in the brackets, [0.25-(Λ/3)+(Λ2/8)]1/2. Location of the smallest RMS blur radius is determined for Λ=4/3 (between best focus and circle of least confusion), where it is smaller than (geometric) paraxial blur radius by a factor of 0.167 (in comparison, best focus and circle of least confusion RMS blur radii are smaller than paraxial blur radius by a factor of 0.204 and 0.177, respectively). Relative wavefront aberration for Λ=4/3 is ŵ=0.408 (Eq. 6), which is smaller than that at the location of the circle of least confusion (ŵ=0.545), but still significantly higher than at the best focus location (ŵ=0.25). It shows that the RMS ray blur size, while generally somewhat more meaningful than the geometric ray blur size, still lacks the accuracy required of a reliable indicator of optical quality.

Spherical aberration in either wavefront or ray form, is independent of the position of aperture stop[2]. That makes it relatively simple to find out the combined spherical aberration coefficient for two or more mirrors. For a pair of mirrors, the combined peak aberration coefficient is given by Sc= S1 + S2, (Eq.9), with the aperture diameter D for the second surface being determined by the height of the marginal ray at it. Since the object for the second surface is the object-image formed by the preceding surface, the magnification m for the second surface is greater than zero, given by m=R2/(R2-kR1), R1 and R2 being the radii of curvature of the first and second surface, respectively.

Considering the importance of axial system correction, higher-order spherical aberration does have some significance in certain types of amateur telescopes with typically more strongly curved optical surfaces. More details on the characteristics and effect of this aberration form are given in sections about apochromatic refractors and Maksutov-Cassegrain telescope.

[2] For object at infinity and constant effective surface diameter.

                                                     
◄ 3.5. Aberration function   ▐    4.2. Coma ►

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