refracting surface. Let such a surface divide media of refractive
indices n and n′, the former being to the left. The point where the
axis intersects the surface is the vertex S (fig. 7). Denote the
distance of the axial object-point O from S by s; the distance from
O to the point of incidence P by p; the radius of the spherical
surface by r; and the distance OC by c, C being the centre of the
sphere. Let u be the angle made by the ray with the axis, and i
the angle of incidence, i.e. the angle between the ray and the normal
to the sphere at the point of incidence. The corresponding quantities
in the image-space are denoted by the same letters with a dash.
From the triangle O′PC we have sin u = (r/c) sin i, and from the
triangle O′PC we have sin u′ = (r/c′) sin i′. By Snell’s law we have
n′/n = sin i/sin i′, and also φ = u′ + i′. Consequently c′ and the
position of the image may be found.
Fig. 7.
To determine whether all the rays proceeding from O are refracted through O′, we investigate the triangle OPO′. We have p/p′ = sin u′/sin u. Substituting for sin u and sin u′ the values found above, we obtain p′/p = c′ sin i/c sin i′ = n′c′/nc. Also c = OC = CS + SO = −SC + SO = s − r, and similarly c′ = s′ − r. Substituting these values we obtain
| p′p = n′(s′ − r)n(s − r), or n(s − r)p = n′(s′ − r)p . | (6) |
To obtain p and p′ we use the triangles OPC and O′PC; we have p2 = (s − r)2 + r2 + 2r(s − r) cos φ, p′2 = (s′ − r)2 + r2 + 2r(s′ − r) cos φ. Hence if s, r, n and n′ be constant, s′ must vary as φ varies. The refracted rays therefore do not reunite in a point, and the deflection is termed the spherical aberration (see Aberration).
Developing cos φ in powers of φ, we obtain
and therefore for such values of φ for which the second and higher powers may be neglected, we have p2 = (s − r)2 + r2 + 2r(s − r), i.e. p = s, and similarly p′ = s′. Equation (6) then becomes n(s − r)/s = n′(s′ − r)/s′ or
| n′s′ = ns + n′ − nr . | (7) |
This relation shows that in a very small central aperture in which the equation p = s holds, all rays proceeding from an object-point are exactly united in an image-point, and therefore the equations previously deduced are valid for this aperture. K. F. Gauss derived the equations for thin pencils in his Dioptrische Untersuchungen (1840) by very elegant methods. More recently the laws relating to systems with finite aperture have been approximately realized, as for example, in well-corrected photographic objectives.
Position of the Cardinal Points of a Lens.—Taking the case of a single spherical refracting surface, and limiting ourselves to the small central aperture, it is seen that the second principal focus F′ is obtained when s is infinitely great. Consequently s′ = -f ′; the difference of sign is obvious, since s′ is measured from S, while f ′ is measured from F′. The focal lengths are directly deducible from equation (7):—
| f ′ = n′r / (n′ − n) | (8) |
| f = nr / (n′ − n). | (9) |
By joining this simple refracting system with a similar one, so that the second spherical surface limits the medium of refractive index n′, we derive the spherical lens. Generally the two spherical surfaces enclose a glass lens, and are bounded on the outside by air of refractive index 1.
The deduction of the cardinal points of a spherical glass lens in air from the relations already proved is readily effected if we regard the lens as a combination of two systems each having one refracting surface, the light passing in the first system from air to glass, and in the second from glass to air. If we know the refractive index of the glass n, the radii r1, r2 of the spherical surfaces, and the distances of the two lens-vertices (or the thickness of the lens d) we can determine all the properties of the lens. A biconvex lens is shown in fig. 8. Let F1 be the first principal focus of the first system of radius r1, and F1′ the second principal focus; and let S1 be its vertex. Denote the distance F1 S1 (the first principal focal length) by f1, and the corresponding distance F′1 S1 by f ′1. Let the corresponding quantities in the second system be denoted by the same letters with the suffix 2.
By equations (8) and (9) we have
| f1 = | r1 | , f ′1 = − | nr1 | , f2 = − | nr2 | , f ′2 = | r2 | , |
| n − 1 | n − 1 | n − 1 | n − 1 |
f2 having the opposite sign to f1. Denoting the distance F′1F2 by Δ, we have Δ = F′1F2 = F′1S1 + S1S2 + S2F2 = F′1S1 + S1S2 − F2S2 = f ′1 + d − f2. Substituting for f ′1 and f2 we obtain
Writing R = Δ(n − 1), this relation becomes
We have already shown that f (the first principal focal length of a compound system) = −f1f2/Δ. Substituting for f1, f2 and Δ the values found above, we obtain
| f = | r1r2n | = | r1r2n | , |
| (n − 1)R | (n − 1) {n (r2 − r1) + d(n − 1)} |
which is equivalent to
| 1 | = (n − 1) | 1 | − | 1 | + | (n − 1)2 d | . |
| f | r1 | r2 | r1r2n |
If the lens be infinitely thin, i.e. if d be zero, we have for the first principal focal length.
| 1 | = (n − 1) | 1 | − | 1 | . |
| f | r1 | r2 |
By the same method we obtain for the second principal focal length
| f ′ = | f ′1f ′2 | = − | nr1r2 | = −f. |
| Δ | (n − 1)R |
Fig. 8.
The reciprocal of the focal length is termed the power of the lens and is denoted by φ. In formulae involving φ it is customary to denote the reciprocal of the radii by the symbol ρ; we thus have φ = 1/f, ρ = 1/r. Equation (10) thus becomes
| φ = (n − 1) (ρ1 − ρ2) + | (n − 1)2 dρ1ρ2 | . |
| n |
The unit of power employed by spectacle-makers is termed the diopter or dioptric (see Spectacles).
We proceed to determine the distances of the focal points from the vertices of the lens, i.e. the distances FS1 and F′S2. Since F is represented by the first system in F2, we have by equation (2)
| x1 = | f1f ′1 | = | f1f ′1 | = − | nr12 | , |
| x′1 | Δ | (n − 1)R |
where x1 = F1F, and x′1 = F′1F2 = Δ. The distance of the first principal focus from the vertex S, i.e. S1F, which we denote by sF is given by sF = S1F = S1F1 + F1F = −F1S1 + F1F. Now F1S1 is the distance from the vertex of the first principal focus of the first system, i.e. f1 and F1F = x1. Substituting these values, we obtain
| sF = − | r1 | − | nr12 | = − | r1 (nr1 + R) | . |
| n − 1 | (n − 1)R | (n − 1)R |
The distance F′2F′ or x′2 is similarly determined by considering F′1 to be represented by the second system in F′.
We have
| x′2 = | f2f ′2 | = − | f2f ′2 | = | nr22 | , |
| x2 | Δ | (n − 1)R |
so that
| sF′ = x′2 − f ′2 = | r2 (nr2 − R) | , |
| (n − 1)R |
where sF′ denotes the distance of the second principal focus from the vertex S2.
The two focal lengths and the distances of the foci from the vertices being known, the positions of the remaining cardinal points, i.e. the principal points H and H′, are readily determined. Let sH = S1H, i.e. the distance of the object-side principal point from the vertex of the first surface, and sH′ = S2H′, i.e. the distance of the image-side principal point from the vertex of the second surface, then f = FH = FS1 + S1H = −S1F + S1H = −sF + sH; hence sH = sF + f = −dr1/R. Similarly sH′ = sF′ + f ′ = −dr2/R. It is readily seen that the distances sH and sH′ are in the ratio of the radii r1 and r2.
The distance between the two principal planes (the interstitium) is deduced very simply. We have S1S2 = S1H + HH′ + H′S2, or HH′ = S1S2 − S1H + S2H′. Substituting, we have
The interstitium becomes zero, or the two principal planes coincide, if d = r1 − r2.
We have now derived all the properties of the lens in terms of its elements, viz. the refractive index, the radii of the surfaces, and the thickness.
Forms of Lenses.—By varying the signs and relative magnitude of the radii, lenses may be divided into two groups according to their action, and into four groups according to their form.
According to their action, lenses are either collecting, convergent
