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Optical Aberrations in Lens and Mirror Systems

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Optical Aberrations in Lens and Mirror Systems . v. .c " Yol. 9. No. 10 301 '. OPTICAL ABERRATIONS IN LENS AND MIRROR SYSTEMS by W. de ,GROOT. 535.317.6 Whèn dealing with technical-optical problems. such as X-ray screen photography and tel- evision projection. the need is felt of an optical system with large aperture • .Apart from the special lens systems developed for this purpose. Schmidt's mirror system, con- sisting of an aspherical correction plate, demands attention. In t1lls article the principal aberrations (so-called third order aberrations) of optical systems in general and of the spherical mirror in particular are discussed, and it is shown how with one exception (the curvature of field) all the third order aberrations of a spher- ical mirror can be eliminated with the aid of a correction plate. ' Intr6ductión In the field of illumination, photography, picture diaphragm' expressly :fitted for the purpose. As a ' projection, etc. innumerable, technical problems rule the aberrations due tó the aspherical shape of arise where the need of an optical system with large the wave surfaces become greater' the larger the aperture is felt. Certain special applications, as .for diameter of the limiting, apertures. , instance X-ray screen photography and television , On the other hand, as a result of this limitation projection, have emphasized the need of such a so-called diffraction phenomena arise which also system in the Philips laboratory. cause lack of definition in the image. The effect of Complicated lens systems with large relative these phenomena is the 'greater, the smaller the aperture have already been developed for this diameter of the limiting aperture relative to the purpose, but now a second solution, the application wavelength of the light used. It therefore depends' of mirror systems, is demanding more and more upon the circumstances whether the first or the attention. second cause predominates in the resulting lack of Following the invention by Schmidt, who for definition. With photographic lenses and with the astronomical purposes ·fitted a. spherical mirror mirror 'systems that will be dismissed i_n ,this article with an aspherical correction plate eliminating the one may, as a rule, 'ignore the diffraction effects. most important aberrations, Philips have developed It is thex{ permissible to substitute for the concep- a mirror system with a correction plate which can tion of light waves that of light rays, that is' to. say be applied for the purpose mentioned above and very narrow pencils of light which pass thro~gh, which is simple to construct. the system mdependently of each other and are The object of this' article is to examine more everywhere at right-angles to the wave surfaces: closely the aberrations 'of optical systems' in general ',.In other. words one is satisfied with the approxi- and of spherical mirrors' in partienlar. The proper- mation offered by ge ome t r ic al optics. ties and construction of the correction plate will he First-order aberrations discussed in a subsequent article. ' Fig.lis a representation .of the well-known Causes of the aberrations elementary construction of the passage .of rays It is already known that, with the aid of a lens through an optical system with rotational symmetry. or system of lenses whose limiting surfaces are The rays emitted from point P at a distance Xl in centred spherical surfaces, and also with aspherical front of the first focus F'i and a distance Yl below . concave' mirrors, waves of light emitted from a po~t the axis converge at a point Q at a distance x; source can he approximately éoncentrated at an- behind the second focus F2 and a distance Y2 above other point (the image point], The fact that 'this the axis, where the equa~ons imaging is imperfect is due to two causes. Y2 :Yl Xl = X2 sT= v. In the, first place after refraction or reflection =r: from a convex surface the spherical light waves apply. Thus the plane through P at right-angles emitted from the point source will generally be no to the axis is uniformly projected with a constant longer truly spherical and consequently will not linear magnification 11 upon a plane through Q at converge into -one point. right-angles to the .axis, This is only correct to a The wave surfaces are always limited' by thè sufficient approximation when all the. rays pass so. edges of the lenses or mirrors and often also by a close to the axis ("paraxial") that the angles formed, 302 PHILIPS TECHNICAL REVIEW 1947/1948 by all parts of the rays 'with the axis are so small W!_}may now cally-y 0 and z-zo (= z, since Zo= 0) that their sines and tangents may be interchanged .. t~e aberrations for the ray determined by Ju, Zu The quantity v is called the paraxial enlargement. and Q. These aberrations result from the choice of the plane W. In practice they may occur, for in- stance, with a photographic camera when the dis- tance between the frosted glass plate and the lens is incorrectly adjusted. As a matter of fact the aberrations described by equations (la, 1b) are sometimes called first-order aberrations, because they are described by an expression of the first Fig. 1. Elementary construction of the image Q of a point degree in Hand Yo' There are two kinds of first- ~ource P in the case of paraxial passage of the rays. Ht and H2 . repreaent. the principal planes of the system; the focal dis- order aberrations. The term with J J still remains .tance is f. even when the magnitude of H ~s zero. It then results in an increased distance between the image. Even in the simple case of the paraxial passage point and the axis, the increase being in the propor- of rays represented above one may speak of aberra- tion of (1 + a2) : L This may be called the first- tions. Let us suppose (fig. 2) that a narrow pencil order enlargement aberration. The terms governed of rays converges at a point Q in the plane V and by H increase ~th H, thus with the opening of the a distance Yo from the axis, If the rays are collected cone of rays. These may be called first-order aper- in another plane W not coinciding with V then ture aberrations. Obviously both enlargement aber- instead of a point of light a small patch is observed. rations and the aperture aberrations are propor- in this plane, displaced with respect to the point Q. tional to the distance d between V and Wand they In particular, a ray intersecting a plane U at a disappe~r when W is coincident with V (d = 0) .. point D having as coordinates J = H cos rp, Z = Higher-order aberrations H sin (/J, will strike the plane Wat a point having the coordinates If (fig. 3) the point source P is chosen on the axis of the optical system, but the condition that the Y = Jo + al H cosç + aaro' (la) .ray ~ust make a very small angle with the à:ds is , z = al H sin (/J, • • • • •• • (lb) in which ~ and a2 are proportional to the distance d between V and W. p Let (/J assume all values between. 0 and 2 :n, then . the point Yu, Zu describes a circle with radius H in .the plane U. The ray DQ thus describes a conic 51012 surface. Next let H assume all the values between Fig. 3. A ray throughP in the plane of the diagram (cp = 0), making a large angle with the axis, intersects the axis not at zero and a maximum value Hl' Then we get a cone the paraxial image point Q but at Ql and the plane V at S, . entirely filled with rays. One mayalso imagine this where to a first approximation QS is' proportional to H3. His· .the distance from the point of intersection of the ray with the plane U to the axis. dropped, then owing to the rotation-symmetry the ray will lie entirely in a plane through the axis. After refraction, however, it will no longer interseet the axis in. the paraxial' image point Q but in a 51011 point Ql and upon extension meet the plane V at a Fig. 2. Construction in the xy and yz projections of a cone of point S. if, to determine the ray, we again choose a .rays with apex Q (projections Q'. Q") intersecting the plane U in plane U and a point of intersection therein having a circle with radius H. This. cone intersects the plane W in a circle with respect to which the yz projection is' displaced the coordinates Ju = H cos (/J, Zu . H 'sin (/J, then from Q to Q" (adjustment aberration). the coordinates of S will be y = A cos (/J, z = A sin (/J, (2) having been brought about by the introduction Ui the plane U of a circular diaphragm with radius Hl' where A is a function of H, which may be devel- We shall therefore call the plane U the diàphragm oped in a series of the form' plane. (3) Vol.. 9, No. 10 OP,TICAL ABERRATION 303 The series necessarily has odd terms only,.because when H The fact that the equations for third-order aberrations are is replaced by -H and rp by rp + n, which does not cause y" . indeed such as given in (4a) and (4b) can most easily be and Zu to change in value, y and z must also remain unchanged, "deduced by a reasoning that originated with Conrady 1). Let us consider for this purposc (fi8.
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