This section is from the book "Telephotography: An Elementary Treatise On The Construction And Application Of The Telephotographic Lens", by Thomas Rudolphus Dallmeyer. Also available from Amazon: Telephotography and Telephotographic Lens.
Glass has a higher refractive index than air (roughly 1.5), thus the angle ф will be less than ф.
On meeting the second surface of the glass at c, b c forms the same angle ф' with the normal p'cp, and as the ray now passes from a medium whose refractive index is ц into air (refractive index unity),
ц sin ф' = sin ф, and hence the emergent ray c d forms the same angle q with the normal as did the incident ray ; hence c d is parallel to A B or c' d'.
If the second surface of the plate were not parallel to the first, and the ray formed an angle ф" with the normal to this surface, the angle at emergence, ф"' would similarly be found from the equation
ц sin ф" = sinф"' .
Let us now apply the laws of refraction to see how rays of light are affected in passing through a lens.
If we trace the passage of a single ray of light, we may always consider it as passing through a plate of glass whose sides are either parallel or inclined to each other. In the latter case, the particular ray passes through the lens as if it were a prism, the ray, on emerging, being always bent or refracted towards its base, or thicker part.
Let us trace the course of two rays parallel to the axis which meet either a positive or negative lens.
The ray a b in either figure meets the lens in the point b ; if we draw a line from the centre of curvature of the surface o to meet e, and draw a line p r through this point at right angles to it (termed a tangent), it is evident that the ray a b may now be considered as meeting a plane surface p b r at b, forming an angle ф with the normal o b (or perpendicular) to this surface. The ray from b takes an altered course b c making a smaller angle ф' with the normal bo, such that sin ф=ц sin ф'. On the ray arriving at c, join c to the centre of curvature of this surface c', and draw a line through c at right angles to o c, forming the tangent PCS. Before emerging from the lens bc makes an angle ф" with this normal ocn'; after emergence, as it passes from a dense, to a rarer medium, it will be bent or refracted from this normal, forming an angle ф'" with it, such that ц sin ф'=sin ф'".
In the case of the positive lens the ray then takes the course cf, meeting the axis in f. f is one of the focal points of the lens, or the locus of its real focus for parallel rays.
In the case of the negative lens, the ray takes the course c d, being bent or refracted from the axis, but proceeding as from a point v on the axis, in front of the lens, v is one of the focal points of this lens, and is the locus of its virtual focus for parallel rays.
The rays thus traced have passed through the lenses, just as though they had passed through the prisms p r s in drawings, being refracted in either case towards the base of the prism.
Similarly, if we trace the course of a parallel ray at a different distance from the axis, such as a! b' in either figure, it will be found that its course is the same as through a prism having sides inclined at a different angle, yet finally refracting the ray to the real focal point of the positive lens, and as from the virtual focal point of the negative lens.
From the above it will be easy to see that a ray of light meeting the lens in any direction will, in general, pass through its curved surfaces as if it passed through a plate of glass with sides more or less inclined to each other ; but when the ray in its passage through a lens passes through the centre, the direction of the ray, after refraction, will be parallel to its direction before refraction, or will be affected as if it had passed through a plane plate with parallel sides, as in Fig. 11.
We have drawn attention to the fact (Fig. 14) that there are in reality two "principal," or "focal," points in every lens (or combination of lenses), for it is only when a lens is infinitely thin that they can coincide, or become one "optical centre." These two points, and the two focal points, as has already been remarked, are termed the "four cardinal points" of a lens. In connection with them we define the planes that intersect the four cardinal points in the axis, perpendicular to the axis, as principal and focal planes.
For any given lens the principal points are a certain distance apart, depending on the thickness and form of the lens, and the material of which it is composed, and possess the property that light proceeding from any direction towards one of them passes out from the lens as though it had passed through the other.
If parallel rays are incident upon a lens in either direction, we have seen that f2 and f1 are the focal points of the lens, and that their distances from the optical centre of a thin lens are called the focal length of the lens. When the thickness of the lens is taken into consideration, with the two principal points, the true measure of the focal length is the distance between N2 and F2, or N1 and F1 The two focal points are always situated at equal distances from their corresponding principal points, or f2 n2=F1 n1. In other words, any lens or system of lenses has the same absolute focal length whichever surface is presented to incident rays; but it does not follow that the position of the focal points are equidistant from the surface of the lens nearest to them.
Fig. 27 represents a case familiar to many photographers. Here we have a meniscus lens with its principal planes P1 p2 passing through the principal points n2 n2 and f1 f2, the two focal points ; n1 F1 = N2F2.
It is evident that if the convex side of the meniscus were presented to the light, the distance between the back surface of the lens and the focal point in the same direction would be considerably shorter than if the concave surface of the meniscus were so presented ; both images, however, would be of the same size, because the true measure of the focal length is the same in either case, or N1 f1=n2 f2.
In order to calculate the distances between the principal points for a single lens or combination of lenses, the student is referred to any modern work on geometrical optics. Most positive lenses consist of two combinations. The following formula for calculating the true focal length f, the distance from the back lens to the focal point on the same side b f, and the resultant width w, between the principal points of the combination, may prove useful:
F = F1f2/f1+f2-a BF= f2(f1-a)/f1+f2-a.
W = W1 + W2 - a2/f1+f2-a, where f1 and f2 are the focal lengths of the two combinations, a their distance apart, and wl and w, the width between their principal points.
To determine the position and magnitude of the image of an object formed by a thick lens, or a combination of lenses, we have only to represent the/our cardinal points and planes of the lens by points and lines as before, setting the principal points and planes at a definite distance apart=w for a single lens, or w for a combination.
A ray from a parallel to the axis passes from p1 to p2 in a straight line ; from the point where it leaves p2, it passes through the focal point F3. Similarly, a ray a n„ passing towards one of the principal points, N1, emerges from the other principal point, n2, in a direction parallel to a Nl, as n2 a', meeting the former ray in a' and thus determining its position. So that a' b' is the position of the image of the object a b.
Again f1 N1 = F2 n2, and both are equal to the true focal length of the lens=f Calling b f1 =x: and b' F2 = y ; we know that: xy=f2, and that the size of a b : size of a' b' : : n1 b : n2 b'.
From this it is evident that we need not know the distance between the. principal points in order to determine the focal length of a lens, provided we know the positions of the two focal points.
The former method of determining the focal length of a lens illustrated in Fig. 23 will of course apply ; but we give another practical method, showing that the result can be ascertained without a knowledge of the separation of the principal points.
(1) Determine the position of the focus of the lens for a very distant object upon the screen of the camera and mark the position upon the base-board; this is the plane of the back focal point F1 (2) Reverse the lens in its flange, and find the position of the focus for the same distant object, measuring the distance of the screen from some fixed point in the lens mount, say the hood. We know that the position of the other focal point F, is the distance d from the hood of the lens. (3) Replace the lens in its normal position with the screen at f1 and then remove it an exact distance y further away - roughly about one-fourth of its distance from the lens for convenience. (4) Find the distance now necessary for the placing of an object o, so that its image is well defined at the new position of the screen at i. (Operations (3) and (4) may be reversed.)
The distance of the object from the hood of the lens, less d, will be its distances from the front focal point f2. Hence f=√xy ; or, if we multiply x and y together and extract the square root, we find the true focal length of the lens. Supposing x=2", y =50, then f2=100, or f=10.