T Smith

The changes in aberrations when the object and stop are moved

If the aberrations of any centered optical system are known both for an object which intersects all rays transmitted by the system and also for the centre of the effective stop, the position in the image space of the emergent portion of a given incident ray is known, and the aberrations in the image of any other object for any stop position can be expressed in terms of those for the first object. The investigation aims to express the relations in the second case in terms of those present in the first when the objects are planes normal to the axis of symmetry whatever the order of the aberration may be. The investigation is based upon a convenient potential function, and the Eikonal, with the principal foci as reference points, is chosen as the standard function. First a distinction is drawn between aberrational and non-aberrational terms, and the discussion leads to the general form taken by the sine-condition in the outer parts of the field of view; it also becomes evident that there is only one stop aberration of each order to be considered, and that this is of spherical aberration form. The transformation is achieved by a change of variables, and to secure simple forms the aberrations of any order are arranged in series, each series forming a separate group as regards transformations. The rays in a plane containing the axis belong entirely to one series, which is called the zero series; in this series the coefficients in general must have finite values depending on the position of the object and of the stop if the aberrations are to be removed. In all the other series the coefficients should be zero for freedom from the aberrations, the conditions thus being independent of the positions of the stop and object. As the number of the series increases the expressions for the calculation of the coefficients become more simple. These facts are illustrated by the conditions for first aberrations, all of which are members of series 0 except the Petzval sum which belongs to series 1. The formulae obtained enable the aberrations to be expressed in terms of the coefficients of the standard Eikonal or vice versa, or alternatively the aberrations for any object and stop positions to be given in terms of those for standard positions. For use in the latter case certain advantages are secured by choosing for the stop the position where the magnification is + 1 and for the object the surface for which the magnification is - 1. As a very simple application the principles underlying the use of the formulae to obtain with the minimum of labour the effects of small variations in the constructional data of the system are illustrated.

On systems of plane reflecting surfaces

An algebraic method is evolved of finding the coordinates of the image of any point and of the direction of the emergent portion of any given incident ray after reflection at any number of plane reflecting surfaces. Systems of reflectors are classified according to the nature of the self-conjugate region of the field. A method of designing a system having any assigned properties is described. Suitable criteria are given to determine whether with a prismatic system the whole is non-dispersive, and whether total internal reflection takes place at any given surface; also the boundary conditions at each surface are found. The calculations are simple and free from any ambiguity of sign.

The hiding power of diffusing media

From theoretical considerations an expression is constructed as a numerical measure of the power of a sheet of a diffusing medium to hide the brightness contrasts of a surface on which it is laid. The dependence of this factor on the transmission and reflection factors of the sheet is exhibited and the effect of varying the thickness of the sheet is discussed. A comparison is made of these theoretical results with published experimental observations. The definition adopted for the hiding power is 50/(residual percentage contrast). The properties of all sheets may be expressed in terms of two constants, of which one is the reflection factor for an infinitely thick sheet, and the other is a factor for converting sheet thicknesses to the proper numerical scale. For a non-absorbing medium the absolute hiding power is 1 + 2xt, and the reflection factor xt/(1 + xt), where t is the thickness of the sheet. For an absorbing medium the absolute hiding power is 1 + λ sinh 2kt + λ (κ - λ) (cosh 2kt - 1) and the reflection factor λ/(κ + coth kt) where κ = cosh 2, λ = sinh 2, and the reflection factor for a very thick sheet is tanh . Expressions are given for the hiding power with respect to any assigned ground contrast. In experimental determinations the measurement of diffuse reflection factors in a photometric sphere is suggested.

The addition of aberrations

The evaluation of the aberration coefficients of a complex system in terms of those of its components is investigated on a general basis which is independent of special properties of the system. For example, in an axially symmetrical system the refracting surfaces may be any surfaces of revolution, or the media may be non-homogeneous as long as they are homogeneous for all circles normal to and centered on the axis. Two methods are employed. The first, by successive approximation applied to equations of reduced order, is utilized to obtain the well-known results for the six first order aberrations and the comparatively unknown results for the ten second order aberrations. For higher orders this method becomes almost impracticable. The second method is based upon the evaluation of a function when it is stationary by Lagranges theorem, and gives a theoretical solution to the problem for all orders. The results already obtained by the previous method are checked by this more general method, and the addition formulae for the fifteen third order aberrations are also obtained. These are given in a simplified form free from all removable terms by the definition of a new function in terms of which the eikonal is expressed. The conditions for freedom from aberrations are given in terms of this function.

On toric lenses

A system of toric lenses having a common normal to all their surfaces possesses in general ten independent primordial coefficients. A single surface has only three degrees of freedom, and this number also holds for any system of negligible axial depth. Formulae are given for the calculation of the ten coefficients, which are only all independent when the system includes at least three separated toric refracting surfaces with their planes of principal curvature finitely inclined to one another. An eye with both its cornea and its crystalline lens astigmatic and the meridians of principal curvature different has more independent coefficients than a spectacle lens has effective degrees of freedom. The nature and importance of the unavoidable residual errors in central vision when such an eye is corrected by a toric lens are discussed.

The primordial coefficients of asymmetrical lenses

An easily calculable system of sixteen magnitudes is constructed for the representation of the properties of asymmetrical lenses by the addition of four lengths to the magnitudes previously used. The equations connecting this system and the coefficients of the eikonal and of the characteristic function, the equations for combining systems or moving their reference points, and the identities between the coefficients are expressed in matrix form. It is shown that the eleven variables present in a system of three separated astigmatic lenses only yield nine degrees of freedom, and such a system cannot, if the reference points are placed where the outer lenses meet the axis, represent the general system, which has ten degrees of freedom.

Graphical constructions for a refracted ray

Two groups of spherical or plane conjugate surfaces can be found for refraction at any point of a sphere, and they give rise to distinct methods of constructing a refracted ray. Snells construction is a special case of one group, and Youngs a special case of the other group. Other cases of interest are examined. By inverting these two groups other constructions are obtained.

Imagery around a skew ray

The one-one correspondence between rays in the object and image spaces of an optical instrument only yields a corresponding relation between points, even for a narrow ray pencil, in exceptional circumstances. The usual criterion by which conjugate points are defined fails unless neighbouring rays intersect in both spaces. In general this condition is not satisfied, and the criterion is therefore extended to include the points of nearest approach of non-intersecting rays. It is shown that pairs of conjugate points on a given chief ray determined according to this definition are coincident for a fan of near rays with one degree of freedom, that is the light path between the conjugate points is constant to the second order for the routes corresponding to all rays of this fan, and that the enlarged definition is applicable to all near rays. The relation connecting conjugate points is of the same form as for the simpler cases generally recognised, and may be determined by parallel projection through fixed points depending on the chief ray and on the particular fan under consideration. Apart from the exceptional case of complete anastigmatism due to the coincidence of the projective centres for all the fans, there can at most be two pairs of conjugate points free from astigmatism. Three conditions must be satisfied for stigmatic imagery at a given pair of points. The pencil around any skew ray has ten degrees of freedom, and the fundamental coefficients which determine the imagery completely are elements of a square matrix of the fourth order. Formulae are given for the construction of these matrices for refraction at each surface and for the transference from one surface to another. The matrix for the complete system is the product of the matrix factors for the successive elementary events taken in their proper order. Each refracting surface may be replaced by the osculating surface of the second degree at the point of refraction of the chief ray. It is assumed that the curvature of the surfaces is everywhere finite and continuous. A more natural matrix for refraction in space of three dimensions would be of the sixth order. Such matrices are derived from those of the fourth order. For theoretical investigations and perhaps also for routine numerical work, these matrices are preferable to those of lower order since they contain no quantities which are not relevant to the problem. The coefficients of the eikonal and of the characteristic function can be derived from the elements of either type of matrix. Fifteen independent relations between the coefficients of these matrices are obtained in two distinct forms.

The theory of aplanatic surfaces

The necessary and sufficient condition that an optical system should have a pair of aplanatic surfaces is that the eikonal of the system can be expressed as a homogeneous function of the first order of three variables, each of which is a linear function of the direction cosines of the ray in the object and image spaces. Methods are given for finding the equations of these surfaces when the eikonal is given, and for finding the eikonal when the surfaces are given. Conjugate aplanatic surfaces are similar apart from arbitrary uniform unidirectional extensions or compressions. In general only one pair of aplanatic surfaces is possible, but in spherically symmetrical systems two pairs are found. In addition there may be aplanatic imagery at isolated points.

The back vertex power of a combination of lenses

The general relations between the constants of a compound instrument and those of its components are obtained when the imagery is collinear. From these relations the expression for the vertex power of a compound instrument is built up in the form convenient for application in spectacle calculations. The formulae have other applications.