Let Q q be a section of a lens, and f A r its optical axis, or the line in which a ray of light passes unchanged in its direction through the lens, from its being normal to both surfaces, whether the lens be double-convex as above, or plano-convex (see [fig. 53]), then the principal focus f is that point where the rays from r r r, which fall parallel to the optic axis on the outer face of the lens, meet after refraction at the two faces,—or, to speak more in the language of the art which is under consideration, the principal focus f is the point whence the rays of light, proceeding in their naturally divergent course, fall on the inner surface Q A q of the lens, and are so changed by refraction there and at the outer face, that they finally emerge parallel to the optic axis in the directions Q r, q r. The position of this point depends partly on the refractive power of the substance of which the lens is composed and partly on the curvature of the surface or surfaces which bound it.
It would be quite beyond the scope of these Notes to attempt to present the subject of refraction at spherical surfaces before the reader’s view in a rigorous or systematic manner, and thus to advance, step by step, to the practical application of refracting instruments, as a means of directing and economising the light in a Pharos. This would involve the repetition, in a less elegant form, of what is to be found in all the works on optics; and instead of this, I am content to refer, where needful, to those works, and shall confine myself simply to what concerns Lighthouse lenses and their use. It would also be superfluous to determine the position of the principal focus of a plano-convex lens, in terms of the refractive index and radius of curvature,[55] as it can be very accurately found in practice by exposing the instrument to the sun, in such a manner that his rays may fall upon it in a direction parallel to its axis. The point of union between the converging and diverging cones of rays (where the spectrum is smallest and brightest), which is the principal focus, is easily found by moving a screen behind the lens, farther from or nearer to it as may be required. The path of the Lighthouse optician, moreover, generally lies in the opposite direction; and his duty is not so much to find the focal distance of a ready-made lens, as to find the best form of a lens for the various circumstances of a particular Pharos, whose diameter, in some measure, determines the focal distance of the instruments to be employed. All, however, that I shall really have to do is to give an account of what has been done by the late illustrious Fresnel, who seems to have devoted such minute attention to every detail of the Dioptric apparatus, that he has foreseen and provided for every case that occurs in the practice of Lighthouse illumination. His brother, Mons. Leonor Fresnel, who succeeded him in the charge of the Lighthouses of France, has, with the greatest liberality, put me in possession of the various formulæ used by his lamented predecessor, in determining the elements of those instruments which have so greatly improved the lighthouses of modern days.
[55] F = rm - 1 in which r is the radius of curvature, and m is the refractive index.—Coddington’s Optics, Chap. VIII. If the radiant be brought near the lens, so as to cast divergent rays on its surface, then the conjugate focus will recede behind the principal focus; and when the luminous body reaches the principal focus in front of the lens, the rays will emerge from its posterior surface in a direction parallel to its axis. If it be brought still nearer the lens, the rays would emerge as a divergent cone. Hence converging lenses can only collect rays into a focus, when they proceed from some point more distant than the principal focus.
Spherical lenses, like spherical mirrors, collect truly into the focus those rays only which are incident near the axis; and it is, therefore, of the greatest importance to employ only a small segment of any sphere as a lens. The experience of this fact, among other considerations, led Condorcet, as already noticed, to suggest the building of lenses in separate pieces. Fresnel, however, was the first who actually constructed a lens on that principle; and he has subdivided, with such judgment, the surface of the lens into a centre lens and concentric annular bands and has so carefully determined the elements of curvature for each, that no farther improvement is likely to be made in their construction. For the drawings of the great lens, I have to refer to [Plate XII.], which also contains a tabular view of the elements of its various parts. The central disc of the lens, which is employed in lights of the first order, and whose focal distance is 920 millimètres, or 36·22 inches, is about 11 inches in diameter; and the annular rings which surround it vary slightly in breadth from 2³⁄₄ to 1¹⁄₄ inches. The breadth of any zone or ring is, within certain limits, a matter of choice, it being desirable, however, that no part of the lens should be much thicker than the rest, as well for the purpose of avoiding inconvenient projections on its surface, as to permit the rays to pass through the whole of the lens with nearly equal loss by absorption. The objects to be attained in the polyzonal or compound lens, are chiefly, as above noticed, to correct the excessive aberration produced by refraction through a hemisphere or great segment, whose edge would make the parallel rays falling on its curve surface converge to a point much nearer the lens than the principal focus, as determined for rays near the optical axis, and to avoid the increase of material, which would not only add to the weight of the instrument and the expense of its construction, but would greatly diminish by absorption the amount of transmitted light. Various modes of removing similar inconveniences in telescopic lenses have been devised; and the suggestions of Descartes, as to combinations of hyperbolic and elliptic surfaces with plane and spherical ones, more especially fulfil the whole conditions of the case; but the excessive difficulty which must attend grinding and polishing those surfaces, has hitherto deprived us of the advantages which would result from the use of telescopic lenses entirely free from spherical aberration. In Lighthouse lenses, where so near an approach to accurate convergence to a single focus is unnecessary, every purpose is answered by the partial correction of aberration which may be obtained, by determining an average radius of curvature for the central disc, and for each successive belt or ring, as you recede from the vertex of the lens. In the lenses originally constructed for Fresnel by Soleil, the zones were united by means of small dowels or joggles of copper, passing from the one zone into the other; but the greater exactness of the workmanship now attained, has rendered it safe to dispense with those fixtures; and the compound lens is now held together solely by a metallic frame and the close union between the concentric faces of the rings, which, however, are in contact with each other at surfaces of only ¹⁄₄ inch in depth, as shewn in [Plate XII.] It is remarkable, that an instrument, having about 1300 square inches of surface, and weighing 109 lb., and which is composed of so many parts, should be held together by so slender a bond as two narrow strips of polished glass, united by a thin film of cement.
I now proceed to the formulæ employed by Fresnel, to determine the elements of the compound lens,[56] in the calculation of which two cases occur, viz., the central disc and a concentric ring. The focal distance of the lens and the refractive index of the glass are the principal data from which we start.
[56] It may be proper to mention that, while the formulæ given in the text are those of M. Fresnel, I am responsible for the investigations in the Notes; I have, at the same time, much pleasure in acknowledging my obligations, at various times (about ten years ago), to Mr Edward Sang, and (more recently) to Mr William Swan, for their kind advice on this part of the subject.
Fig. 54.
I begin with the case of the central disc or lens round which the annular rings are arranged. Its principal section is a mixtilinear figure ([fig. 54]) composed of a segment b a c, resting on a parallelogram b c d e, whose depth b d or c e is determined by the strength which is required for the joints which unite the various portions of the lens. Those particulars have, as I already stated, been determined with so much judgment by Fresnel and the dimensions of the lenses so varied to suit the case of various lights, that nothing in this respect remains to be done by others.