away from the axis. from this appearance it takes its name. the unsymmetrical form of the meridional pencil—formerly the only one considered—is coma in the narrower sense only; other errors of coma have been treated by arthur könig and moritz von rohr, and later by allvar gullstrand. curvature of the field of the image if the above errors be eliminated, the two astigmatic surfaces united, and a sharp image obtained with a wide aperture—there remains the necessity to correct the curvature of the image surface, especially when the image is to be received upon a plane surface, e.g. in photography. in most cases the surface is concave towards the system. distortion of the image fig. 3a: barrel distortion fig. 3b: pincushion distortion even if the image is sharp, it may be distorted compared to ideal pinhole projection. in pinhole projection, the magnification of an object is inversely proportional to its distance to the camera along the optical axis so that a camera pointing directly at a flat surface reproduces that flat surface. distortion can be thought of as stretching the image non-uniformly, or, equivalently, as a variation in magnification across the field. while "distortion" can include arbitrary deformation of an image, the most pronounced modes of distortion produced by conventional imaging optics is "barrel distortion", in which the center of the image is magnified more than the perimeter (figure 3a). the reverse, in which the perimeter is magnified more than the center, is known as "pincushion distortion" (figure 3b). this effect is called lens distortion or image distortion, and there are algorithms to correct it. systems free of distortion are called orthoscopic (orthos, right, skopein to look) or rectilinear (straight lines). figure 4 this aberration is quite distinct from that of the sharpness of reproduction; in unsharp, reproduction, the question of distortion arises if only parts of the object can be recognized in the figure. if, in an unsharp image, a patch of light corresponds to an object point, the center of gravity of the patch may be regarded as the image point, this being the point where the plane receiving the image, e.g., a focusing screen, intersects the ray passing through the middle of the stop. this assumption is justified if a poor image on the focusing screen remains stationary when the aperture is diminished; in practice, this generally occurs. this ray, named by abbe a principal ray (not to be confused with the principal rays of the gaussian theory), passes through the center of the entrance pupil before the first refraction, and the center of the exit pupil after the last refraction. from this it follows that correctness of drawing depends solely upon the principal rays; and is independent of the sharpness or curvature of the image field. referring to fig. 4, we have o'q'/oq = a' tan w'/a tan w = 1/n, where n is the scale or magnification of the image. for n to be constant for all values of w, a' tan w'/a tan w must also be constant. if the ratio a'/a be sufficiently constant, as is often the case, the above relation reduces to the condition of airy, i.e. tan w'/ tan w= a constant. this simple relation (see camb. phil. trans., 1830, 3, p. 1) is fulfilled in all systems which are symmetrical with respect to their diaphragm (briefly named symmetrical or holosymmetrical objectives), or which consist of two like, but different-sized, components, placed from the diaphragm in the ratio of their size, and presenting the same curvature to it (hemisymmetrical objectives); in these systems tan w' / tan w = 1. the constancy of a'/a necessary for this relation to hold was pointed out by r. h. bow (brit. journ. photog., 1861), and thomas sutton (photographic notes, 1862); it has been treated by o. lummer and by m. von rohr (zeit. f. instrumentenk., 1897, 17, and 1898, 18, p. 4). it requires the middle of the aperture stop to be reproduced in the centers of the entrance and exit pupils without spherical aberration. m. von rohr showed that for systems fulfilling neither the airy nor the bow-sutton condition, the ratio a' cos w'/a tan w will be constant for one distance of the object. this combined condition is exactly fulfilled by holosymmetrical objectives reproducing with the scale 1, and by hemisymmetrical, if the scale of reproduction be equal to the ratio of the sizes of the two components. zernike model of aberrations image plane of a flat-top beam under the effect of the first 21 zernike polynomials. the beam goes through an aperture of the same size, which is imaged onto this plane by an ideal lens. circular wavefront profiles associated with aberrations may be mathematically modeled using zernike polynomials. developed by frits zernike in the 1930s, zernike's polynomials are orthogonal over a circle of unit radius. a complex, aberrated wavefront profile may be curve-fitted with zernike polynomials to yield a set of fitting coefficients that individually represent different types of aberrations. these zernike coefficients are linearly independent, thus individual aberration contributions to an overall wavefront may be isolated and quantified separately. there are even and odd zernike polynomials. the even zernike polynomials are defined as z n m ( ρ , ϕ ) = r n m ( ρ ) cos ⁡ ( m ϕ ) ^(\rho ,\phi )=r_^(\rho )\,\cos(m\,\phi )\!} and the odd zernike polynomials as z n − m ( ρ , ϕ ) = r n m ( ρ ) sin ⁡ ( m ϕ ) , ^(\rho ,\phi )=r_^(\rho )\,\sin(m\,\phi ),\!} where m and n are nonnegative integers with n ≥ m , φ is the azimuthal angle in radians, and ρ is the normalized radial distance. the radial polynomials r n m ^ have no azimuthal dependence, and are defined as r n m ( ρ ) = ∑ k = 0 ( n − m ) / 2 ( − 1 ) k ( n − k ) ! k ! ( ( n + m ) / 2 − k ) ! ( ( n − m ) / 2 − k ) ! ρ n − 2 k if n − m is even ^(\rho )=\!\sum _^\!\!\!\,(n-k)!}\;\rho ^\quad n-m and r n m ( ρ ) = 0 ^(\rho )=0} if n − m is odd. the first few zernike polynomials, multiplied by their respective fitting coefficients, are: a 0 × 1 \times 1} "piston", equal to the mean value of the wavefront a 1 × ρ cos ⁡ ( ϕ ) \times \rho \cos(\phi )} "x-tilt", the deviation of the overall beam in the sagittal direction a 2 × ρ sin ⁡ ( ϕ ) \times \rho \sin(\phi )} "y-tilt", the deviation of the overall beam in the tangential direction a 3 × ( 2 ρ 2 − 1 ) \times (2\rho ^-1)} "defocus", a parabolic wavefront resulting from being out of focus a 4 × ρ 2 cos ⁡ ( 2 ϕ ) \times \rho ^\cos(2\phi )} "0° astigmatism", a cylindrical shape along the x or y axis a 5 × ρ 2 sin ⁡ ( 2 ϕ ) \times \rho ^\sin(2\phi )} "45° astigmatism", a cylindrical shape oriented at ±45° from the x axis a 6 × ( 3 ρ 2 − 2 ) ρ cos ⁡ ( ϕ ) \times (3\rho ^-2)\rho \cos(\phi )} "x-coma", comatic image flaring in the horizontal direction a 7 × ( 3 ρ 2 − 2 ) ρ sin ⁡ ( ϕ ) \times (3\rho ^-2)\rho \sin(\phi )} "y-coma", comatic image flaring in the vertical direction a 8 × ( 6 ρ 4 − 6 ρ 2 + 1 ) \times (6\rho ^-6\rho ^+1)} "third order spherical aberration" where ρ is the normalized pupil radius with 0 ≤ ρ ≤ 1 , ϕ is the azimuthal angle around the pupil with 0 ≤ ϕ ≤ 2 π , and the fitting coefficients a 0 , … , a 8 ,\ldots ,a_ are the wavefront errors in wavelengths. as in fourier synthesis using sines and cosines, a wavefront may be perfectly represented by a sufficiently large number of higher-order zernike polynomials. however, wavefronts with very steep gradients or very high spatial frequency structure, such as produced by propagation through atmospheric turbulence or aerodynamic flowfields, are not well modeled by zernike polynomials, which tend to low-pass filter fine spatial definition in the wavefront. in this case, other fitting methods such as fractals or singular value decomposition may yield improved fitting results. the circle polynomials were introduced by frits zernike to evaluate the point image of an aberrated optical system taking into account the effects of diffraction. the perfect point image in the presence of diffraction had already been described by airy, as early as 1835. it took almost hundred years to arrive at a comprehensive theory and modeling of the point image of aberrated systems (zernike and nijboer). the analysis by nijboer and zernike describes the intensity distribution close to the optimum focal plane. an extended theory that allows the calculation of the point image amplitude and intensity over a much larger volume in the focal region was recently developed (extended nijboer-zernike theory). this extended nijboer-zernike theory of point image or ‘point-spread function’ formation has found applications in general research on image formation, especially for systems with a high numerical aperture, and in characterizing optical systems with respect to their aberrations. analytic treatment of aberrations the preceding review of the several errors of reproduction belongs to the abbe theory of aberrations, in which definite aberrations are discussed separately; it is well suited to practical needs, for in the construction of an optical instrument certain errors are sought to be eliminated, the selection of which is justified by experience. in the mathematical sense, however, this selection is arbitrary; the reproduction of a finite object with a finite aperture entails, in all probability, an infinite number of aberrations. this number is only finite if the object and aperture are assumed to be infinitely small of a certain order; and with each order of infinite smallness, i.e. with each degree of approximation to reality (to finite objects and apertures), a certain number of aberrations is associated. this connection is only supplied by theories which treat aberrations generally and analytically by means of indefinite series. figure 5 a ray proceeding from an object point o (fig. 5) can be defined by the coordinates (ξ, η). of this point o in an object plane i, at right angles to the axis, and two other coordinates (x, y), the point in which the ray intersects the entrance pupil, i.e. the plane ii. similarly the corresponding image ray may be defined by the points (ξ', η'), and (x', y'), in the planes i' and ii'. the origins of these four plane coordinate systems may be collinear with the axis of the optical system; and the corresponding axes may be parallel. each of the four coordinates ξ', η', x', y' are functions of ξ, η, x, y; and if it be assumed that the field of view and the aperture be infinitely small, then ξ, η, x, y are of the same order of infinitesimals; consequently by expanding ξ', η', x', y' in ascending powers of ξ, η, x, y, series are obtained in which it is only necessary to consider the lowest powers. it is readily seen that if the optical system be symmetrical, the origins of the coordinate systems collinear with the optical axis and the corresponding axes parallel, then by changing the signs of ξ, η, x, y, the values ξ', η', x', y' must likewise change their sign, but retain their arithmetical values; this means that the series are restricted to odd powers of the unmarked variables. the nature of the reproduction consists in the rays proceeding from a point o being united in another point o'; in general, this will not be the case, for ξ', η' vary if ξ, η be constant, but x, y variable. it may be assumed that the planes i' and ii' are drawn where the images of the planes i and ii are formed by rays near the axis by the ordinary gaussian rules; and by an extension of these rules, not, however, corresponding to reality, the gauss image point