Optics 820

OPTICS 820 which depends only on the object point (x, y, z). There is thus no aberration in the image of this point. Moreover the aggregate of these object points forms a surface. For E satisfies a homo geneous relation of the form 0(e, a, 13,7)=o, to which we can add three homogeneous equations On eliminating the ratios a, and y bear to E between these four equations we obtain a relation of the form 0( ea, 8,3, ay) = o, and if ea, Es, ey are replaced by their values in terms of x, y, z and of x', y', z' we obtain the equations of the object and image surfaces. From the linear character of a, (3, 7 it follows that the image surface can at most be a regular deformation of the object surface.

The 'converse process of constructing the eikonal which will yield given aplanatic surfaces can be carried out. It is merely a slight generalization of the process by which the eikonal was constructed for a given refracting surface.

In general a given optical system can have only one pair of aplanatic surfaces, for E can only be expressed in one way as a homogeneous function of the direction cosines. Spherically symmetrical systems are exceptional. For example if Thus with a sphere, since its surface is sell-conjugate, we may put p=i1r, q=p,'r, and the alternative solution shows that the concentric spheres of radii r,u'Au and rµ/µ' are respectively aplanatic conjugate object and image surfaces.

Asymmetrical Systems.

We have discussed at some length the properties of symmetrical lens systems because they form by far the most important section of geometrical optics. The expressions that have been given enable all the ordinary problems to be dealt with—for example the paraxial expressions are suffi ciently accurate to determine how large any simple lens must be to pass the rays the instrument should transmit. It is also a simple matter to derive a number of well-known conclusions from the general laws that have been given—for example that the use of an optical instrument will not enable a brighter image of an object subtending an appreciable angle to be formed on the retina of an observer's eye. The reader is not likely to encounter any difficulty arising from the use of prisms inserted for the re flection of light at plane surfaces into the system, for they are equivalent to the insertion of a thick plate of glass with plane parallel faces. Methods of designing prisms to produce desired results cannot be considered in detail here. As a rule a trigono metrical procedure is adopted, but algebraic methods employing matrices appear to offer decided advantages.

There remain systems of much importance without axial sym metry. The theory of such systems, though not difficult, is much more involved than that of axially symmetrical systems. Taking systems in which all the surfaces are met normally by some straight line, and yield symmetrical sections when cut by any plane through this line, which may be called the axis, we find that in place of the second degree matrix for the axially sym metrical system we have to adopt a square matrix of the fourth degree, with sixteen constituent elements. Between these sixteen quantities six independent identities subsist, so that at most there are ten degrees of freedom for paraxial rays. For an account

of these quantities, their connections with the positions of the rays and with various expressions giving the lengths of paths through the system, reference should be made to the Transactions of the Optical Society. When we proceed to higher order terms representing aberrations the complexity of the theory is enhanced. For example corresponding to the six coefficients for the lowest order monorythmic aberrations in axially symmetric systems we have in these unsymmetrical systems no less than thirty-five coefficients.

Experimental Methods.

The marked changes in the way optical instruments have come to be regarded in recent years is reflected in experimental applications of the theory. A good example is afforded by the use of modified types of Michelson interferometers for the testing of optical instruments. This application of interference is due to F. Twym,an. An instrument of this type, equipped for varied work, has been constructed by Messrs. Adam Hilger, Ltd., for the National Physical Lab oratory, at Teddington, England. Many other interference methods have recently been described. Space will not permit us to discuss many interesting points which arise in the use of these and other methods of investigating the properties of optical instruments experimentally. (See LENS.) all authorities deal with the geometrical theory only, and some are chiefly of value on account of miscel laneous articles.

See for historical interest mainly: H. Coddington, Treatise on the Reflexion and Refraction of Light; W. R. Hamilton, "Theory of Systems of Rays" (4 papers from the Proceedings of the Royal Irish Academy which serve as a starting point for modern work) ; R. Smith, Compleat System of Opticks.

More modern works are: S. Czapski and 0. Eppenstein, Grundziige der Theorie der optischen Instrumente; H. Geiger and K. Scheel, Handbuch der Physik; XVIII. H. Konen, Geometrische Optik; R. T. Glazebrook, Dictionary of Applied Physics; IV. Optics; R. S. Heath, Geometrical Optics; R. A. Herman, Geometrical Optics; M. von Rohr (trans. R. Kanthack), Formation of Images in Optical Instruments; J. P. C. Southall, Principles and Methods of Geometrical Optics; Steinheil and Voit (trans. J. W. French), Applied Optics; G. C. Steward: The Symmetrical Optical System (Tract) ; H. D. Taylor, A System of Applied Optics.

For the physical principles involved reference may be made to the optical papers of the late Lord Rayleigh. The writings of A. Gullstrand are important for the optics of spectacle lenses.

Most recent work has been described in journals devoted to optics, of which the chief are the Transactions of the Optical Society, Revue d'Optique, Journal of the Optical Society of America, and Rivista d'Ottice.

Valuable contributions will also be found in Zeitschrift fiir Instru mentenkunde, Die Naturwissenschaften, Phil. Trans., Proceedings of the Physical Society, and many others. An extensive bibliography is given by Czapski and Eppenstein. (T. SM.)