Symmetric Optical Systems Research Articles

On the Optical Writings of Sir William Rowan Hamilton

The study of Geometrical Optics has not attracted much interest or attention for some years now. at all events until recently, save only among a quite small company of specialists, and, as but one particular result of this, the divorce between optical theory and optical practice has been lamentable, as, for example, in that problem of so great practical importance, the design of the symmetrical optical system. For the one part, the subject has been regarded as isolated and of little theoretical interest, and scarcely susceptible of wide and profound generalisations, then, too, and this more especially from the point of view of university teaching, geometrical optics has seemed to stand apart from the path to a general mathematical education and training, and, in consequence, in several universities the study of it has been almost abandoned, save only in its most elementary aspects.

On the Optical Writings of Sir William Rowan Hamilton

The study of Geometrical Optics has not attracted much interest or attention for some years now. at all events until recently, save only among a quite small company of specialists, and, as but one particular result of this, the divorce between optical theory and optical practice has been lamentable, as, for example, in that problem of so great practical importance, the design of the symmetrical optical system. For the one part, the subject has been regarded as isolated and of little theoretical interest, and scarcely susceptible of wide and profound generalisations, then, too, and this more especially from the point of view of university teaching, geometrical optics has seemed to stand apart from the path to a general mathematical education and training, and, in consequence, in several universities the study of it has been almost abandoned, save only in its most elementary aspects.

Book Review: The Symmetrical Optical System

Book Review: The Symmetrical Optical System

On the Lens Interferometer

The interferometer designed by Michelson has been adapted in recent years to the testing of various types of optical systems, one form being used for the testing of prisms and another form—the “lens interferometer”—for the examination of the symmetrical optical system. The details of the instrument may be found in a paper by Mr F. Twyman, in which are given some photographs of the results of its application to certain lenses of known aberration. Essentially the interferometer ensures that a plane and undistorted wave of light falls upon the lens or system to be examined, after refraction through which it falls upon a convex mirror whose centre is at the focus of the converging wave; by which, therefore, the wave is returned through the optical system and emerges as a plane wave if the system under examination be ‘perfect,’ or free from aberration. Otherwise the final wave is not plane, but is distorted by the aberrations impressed upon it by the system under test—through which, it will be noticed, it passes twice; and this emergent wave is superposed upon a portion of the incident (plane) wave, obtained by previous passage through a partly silvered mirror; with which, therefore, interference effects are observed. And these effects depend only upon the aberrations produced by the system under examination, for they measure, in fact, the departure of the finally emergent wave from the ideal plane form. Various interference contours are obtained in this way, and their form and size enable the observer to deduce the aberrations of the system and to measure their magnitude.

On Herschel's Condition and the Optical Cosine Law

§ 1. Sir John Herschel gave the condition which must be satisfied in order that a symmetrical optical system, free from spherical aberration for two conjugate axial points, may also be free from spherical aberration for two neighbouring and conjugate points upon the axis of the system; but Herschel's condition applies only to first order aberration, i.e. to aberration depending upon the cube of the inclination of the ray to the axis. Abbe shewed, later, that this condition could be included in a wider result, viz. that the spherical aberration, supposed zero, is stationary for axial variations provided that the incident and emergent rays for two conjugate axial points, associated with modified magnification m, satisfy the relationwhere θ and θ′ are their initial and final inclinations to the axis; and by ‘modified’ magnification is meant the ratio of the reduced sizes of the image and object.

On the Addition of the Primary Aberrations

In the application of the Characteristic Function of Hamilton, or of any allied function, to the computation of a symmetrical optical system three steps are necessary. The performance of the system as a whole must be considered, and from this it appears that the aberrations may be derived from certain ‘aberration coefficients’ which occur in the expansion of an ‘aberration function’. In the second place, relations must be obtained giving the properties of the complete system in terms of the properties of the component systems, which, in general, will be single refracting surfaces; and finally, an evaluation of the coefficients must be made for the simple system—a single surface. For the first and third of these steps reference may be made elsewhere, and also for a general investigation of the second; a simple derivation is given, in the present note, of the necessary relations between a composite system and its components in the case where first order aberrations only are considered.

Note on the Petzval optical condition

As a preliminary to an investigation of certain diffraction patterns I was led to consider, in some detail, the geometrical aberrations of a symmetrical optical system; and it appeared convenient then to classify the aberrations in orders according as they depend upon various powers of certain small quantities and to exhibit them as coefficients in the expansion of an ‘ Aberration Function.’ If aberrations of the first order only are considered, it becomes evident that one of them stands, in some sense, apart from the rest; I refer to the so-called ‘Petzval’ condition for flatness of field. It is of interest to notice that this condition was known to Coddington and to Airy before the time of Petzval—known at least as far as concerns systems of thin lenses. In the usual notation the condition is ΣΚ/μμ′ = 0; it is therefore independent of the positions of the object and pupil planes and in this respect it stands alone among the first order aberrations. But an increasing number of similar aberrations of higher orders will be found and it is of interest to examine these and to investigate their geometrical meaning. In the following note is given a proof of the Petzval condition, differing from that usually given and falling more into line with the general theory, and indicating also a general method of examining aberrations of this peculiar type.

IV. — Aberration diffraction effects

The image of a luminous point, given by a symmetrical optical system, will not itself be a point; and this will follow both from the nature of light and also from the necessary ‘ imperfections ’ of the system. A part only of the incident wave will pass through the system and diffraction phenomena will appear ; in addition the emergent wave will not be a portion of a sphere but will be distorted by the geometrical aberrations of the system. Diffraction theory would indicate that corresponding to a point source of light a system of luminous rings should be produced upon the image plane ; this was investigated by Airy in 1834 ; geometrical theory, on the other hand, leads to a consideration of several types and orders of aberration, the more common ones being better known as the 'Five Aberrations of Von Seidel.’ They are: Spherical Aberration, Coma, Astigmatism, Curvature of the Field and Distortion ; these are well known and they have been investigated by a number of writers. In the present paper a consideration is undertaken of the modification of the ‘ ideal ’ diffraction pattern produced by these geometrical aberrations. The method adopted depends upon the Eikonal Function of Bruns, and a summary of the properties of this function is given, therefore, in Part I of the paper ; Part II deals with the Aberration-Diffraction effects. Throughout Parts I and II of the paper it is assumed that the stops of the optical system are circular, with centres upon the axis of symmetry ; and this is generally the case. Occasionally, however, other stops are used, and in Part III of the paper is undertaken a consideration of the diffraction effects of such. The precise forms of aperture considered are the following :— 1. the usual circular aperture with the central portion stopped out—as suggested by Lord Rayleigh : 2. one (or two parallel) narrow rectangular apertures—as used for the determination of the diameters of large stars and for the separation of close double-stars : 3. a semi-circular aperture—as used in a heliometer.

II. On a theory of the second order longitudinal spherical aberration for a symmetrical optical system

If we consider a pencil of rays issuing from a point on the axis of a symmetrical optical system ( i. e ., a system of refracting spherical surfaces, the centres of which lie on a straight line called the axis of the system), it is well known that, if the pencil be a thin one, of which the mean ray is along the axis, the first approximation to the emergent pencil is another punctual pencil, of which the rays pass through an image point, also situated on the axis. The general method of treatment of such image points, which are usually referred to as “geometrical” images, is due to Gauss, and is developed in any text book of Geometrical Optics. When, however, the pencil considered is one of finite aperture, the outlying rays do not, after emergence, pass through the Gaussian image point, nor do they have the inclination assigned to them by the Gaussian calculation. The emergent rays lying in any one axial plane touch an envelope or caustic, which has one cusp at the Gaussian image, with the axis as proper tangent. The intercepts of any given emergent ray upon the axis and the image plane, measured from the Gaussian image, are known as the longitudinal and transverse spherical aberrations of that ray.

The Aberrations of a Symmetrical Optical System

The Aberrations of a Symmetrical Optical System

On a theory of the second order longitudinal spherical aberra­tion for a symmetrical optical system

1. The object of this investigation is to establish a formula for the longitudinal spherical aberration of rays which traverse a symmetrical optical system in an axial plane that shall be capable of fairly easy computation for any combination of lenses, and at the same time shall be accurate to the second order and free from certain important difficulties of convergency which occur in certain neighbourhoods when we attempt to use for the longitudinal aberration the method of aberration of successive orders. From the point of view of the optical designer, the usual theory of aberrations, which, for all practical purposes, is largely restricted to the first order, is known to give an unsatisfactory approximation. In practice, the designer adopts a semi-empirical method of tracing a number of rays through the system by means of the trigonometrical equations, a method which is laborious and lengthy, and which can at best give only incomplete informa­tion and very limited guidance for effecting improvements.

XLVII. A method of line-coordinates for investigating the aberrations of a symmetrical optical system

XLVII. A method of line-coordinates for investigating the aberrations of a symmetrical optical system

XX. Some properties of the nul point of thin axial pencils of light directly refracted through a symmetrical optical system

XX. <i>Some properties of the nul point of thin axial pencils of light directly refracted through a symmetrical optical system</i>