49. Tangential and Radial Images. The fact that a stigmatic pencil gives a double image, viz, two straight lines (focal lines) in different planes when refracted by an astigmatic system, gives the following easily-observed phenomenon. If an object consists of circles, concentric with the axis, and radii thereto, the elements R of each point of a radius will merge into one another and give a sharp image (at least for a certain length), while the elements 7' will give a blurred image (Fig. 25). Conversely, the elements T will merge into one another and give a sharp image of the circles (or short tangents to them). For this reason the images R and T are often called radial (or sagittal) and tangential images respectively. If the two focal lines are not widely separated from one another, or if the angular aperture of the beam is sufficiently small, a more or less homogeneous image will be produced if the screen is placed at an inter mediate position where the pencil gives a circular patch (circle of least confusion, C, Fig. 24).
The locus of the radial images of infinitely distant points (e.g. stars) given by a lens is a surface S,. (Fig. 26), which (at least in the central region) is generally concave to the lens ; the tangential images lie on another surface generally less curved than These two sur faces (focal surfaces) have a point of contact at the focus F.
The radial and tangential images of points in any plane perpendicular to the optical axis form analogous surfaces.' In order to represent the astigmatism of a Correction of astigmatism is only possible by the employment of at least three separated lenses, or, if the lenses are to be cemented in groups, at least four lenses of different material. Two at least of the glasses must form what is lens, a graphic method is used similar to that already employed for spherical aberration (Fig. 22). The displacements of the two focal surfaces (multiplied by four to facilitate reading of the curves) for a lens of 4 in. focal length are plotted called an abnormal pair in which the refractive index varies in the opposite direction to the dispersion, thus behaving in a contrary manner to the old glasses. 1 A lens corrected for astigmatism is said to be stigmatic, or, more usually (in spite of the pleonasm), anastigmatic or an anastigrnat.
5o. Coma. Coma is due to the difference of refraction in oblique rays between the central and marginal zones of a lens ; it may thus be said to be spherical aberration of pencils travers ing the lens obliquely. On account of dissym metry between the path of the rays in the on the horizontal scale, while the angle made by the secondary axis with the principal axis is plotted vertically on the scale of ox in. to the degree. Figs. 27A and 27B, from von Rohr, show respectively the astigmatism curves for a, lens partly corrected for astigmatism (Ortho stigmat type II) and for one well-corrected (Planar).
meridian and sagittal sections (already referred, to in the explanation of astigmatism), an un- symmetrical patch is formed instead of a point image, the appearance somewhat resembling the image of a comet (whence the name), the tail of which is generally directed away from the optical axis (outward coma).
Coma is often associated with astigmatism, but whilst in the case of a lens incompletely corrected, astigmatism attains a maximum and then decreases as the inclination of the rays to the axis increases, coma steadily increases. Also, being of zonal origin, coma is much more rapidly reduced by the use of a small diaphragm than is astigmatism.
Coma is seen in its characteristic form chiefly when long exposures are made on an object having a number of brightly illuminated points off the axis, the tails sometimes stretching a great distance.
Fig. 28, taken from S. P. Thompson, shows the cross-section of the beam of light by a plane perpendicular to the axis in the neighbourhood of the normal position of the image of a point formed by a piano-convex lens, having a dia phragm like that of Fig. 20, but containing several annular apertures. 1 51. Curvature of the Field. For reasons of symmetry, it is easy to see that the images of infinitely distant points given by a sphere of glass would lie on a spherical surface concentric with that of the spherical lens, and of radius equal to the focal length. In these circumstances the image of a near plane would be a surface of still greater curvature.
The focal surface of a lens of old type (achrom atics, rectilinears, symmetricals) always has a very ma flied concavity towards the lens, the mean radius of curvature being between 1-c times and twice the focal length.' In an astigmatic objective the surface that is to be con sidered as the locus of the image is neither the radial nor the tangential surface, but an intermediate surface containing the circles of least confusion (C, Fig. 24).
The practical consequence of curvature of the field is that. if a plane held perpen dicular to the axis is dis placed relatively to the lens, the position corresponding with maximum sharpness of the central region of the image is more or less distant from that corresponding with maximum sharpness of the marginal regions of the the fact that (as we shall see) there is a latitude in the position of the focussing screen or photo- graphic plate (depth of focus), in focussing the image sharp, curvature of the field sets a limit in every case to the useful angle of field of the lens. Fig. 29, which is essentially only diagram matic , shows the impossibility of having on a plane P a sharp image formed on the surface S.