CHAPTER II.

REFRACTION AT TWO SURFACES IN SUCCESSION.

40. Many of the properties which belong to a ray of light when refracted at one surface only, may be extended almost directly to the case in which the ray is bent a second time in traversing a second surface.

We assume that the second surface is related in position to the first in the manner described in fig. 1; that it is wholly independent of it as far as curvature is concerned; and that the two surfaces are at any distance whatever apart.

Two such surfaces combined form an ordinary thick lens, the character of which depends upon the curvatures of the two surfaces, the directions in which their concavities are turned, and the refractive index of the medium between them.

The form of what we may call our standard lens is given in fig. 14; for, in accordance with our convention, both the radii of curvature are there positive. It is therefore the simplest case to demonstrate, as well as the one from which particular cases can most easily be deduced.

FIG. 14.

We will also assume, as a rule, that the first and last media are similar-air, for example-and we will denote

the refractive indices of the successive media by Hoμ1 Mo respectively.

If, then, r, and r, be the radii of curvature of the two surfaces, the properties of any particular lens may be deduced from this general case by assigning to r, and r, their proper values, and to μ, the value of the refractive index of the particular material.

41. If rays proceed from a luminous point and traverse two refracting surfaces in succession, they will, after emergence, meet again in one and the same point.

Suppose P to be the luminous point.

It has been proved in Chap. I. that the rays from P will, after refraction at the first surface, meet again in a certain point P.

The point P1, or the image of P with respect to the first surface, may be considered as a source of light from which rays proceed across the second surface.

All these rays, after refraction at the second surface, will, in consequence of the same law, meet again in a certain. point P', the point P' being the image of P, with respect to the second surface.

Hence, all rays which proceed from a luminous point P and traverse two refracting surfaces in succession will, on emergence, meet again in one and the same point P'. This result holds whatever be the position of P, and whether the plane of incidence contain the axis of the lens or not.

42. It is obvious that, just as in the case of one surface only, P and P' are reciprocally related to one another, and that if we were to consider P' as the source of light, all rays from it which traverse the two surfaces would on emergence meet together at the point P.

Hence P and P' are conjugate to one another with respect to the lens considered; or, in other words, P' is the image of P.

43. The definition of conjugate points leads directly to the two following propositions:

(i) If s and denote any two incident rays, and s' and σ' o' the corresponding emergent rays, the point of concurrence of 8 and is conjugate to the point of concurrence of s' and o'.

(ii) If P and Q be a pair of conjugate points, and p and q another pair, a ray which before incidence passes through P and p will after emergence pass through Q and 2.

44. If a number of points P lie upon a plane perpendicular to the axis, all the points P' conjugate to them will also lie upon a plane perpendicular to the axis.

For the points which are conjugate to the system P with respect to the first surface lie upon a certain plane perpendicular to the axis. This was proved in the former chapter.

We will call this system of points P1, and we may consider the points as sources of light from which rays traverse the second surface.

Again, we know that all the points P' which are conjugate to the points P, with respect to the second surface also lie upon a certain plane perpendicular to the axis.

But the points P' are conjugate to the points P with respect to the lens.

Whence the proposition follows.

45. If P and P' be conjugate points, and planes pass through them perpendicular to the axis, it follows that any point on one of the planes has its conjugate on the other. Two such planes are said to be conjugate to one another with respect to the lens, and are called briefly Conjugate Planes.

Also the points where the planes meet the axis of the lens are called Conjugate Foci.

46. If any two conjugate planes be taken, and any number of points on one plane be joined to their conjugate points on the other, all these straight lines will meet the axis in the same point.

Let PN, P,N1, P'N' be planes such that PN and P,N, are conjugate to one another with respect to the first surface, and PN, and P'N' conjugate with respect to the second surface; and let C1, C2 be the centres of curvature of the two surfaces respectively (fig. 15).

Let P' be conjugate to P with respect to the lens, and let the straight line PP' meet the axis at the point C'. We will show that C' is a fixed point, for different positions of Pin the plane PN.

If P, be conjugate to P with respect to the first surface, and therefore conjugate to P' with respect to the second surface, it has been proved in Art. 32, that the straight lines PP, and PP' pass through C, and C, respectively.

therefore C' is a fixed point for all positions of P in the plane PN.

47. If the point N move along the axis to an infinite distance from the lens, the rays which proceed from it, in the limiting position, will before incidence be parallel to the axis, and after emergence will meet at a certain point N' on the axis.

Again, if the point N' move along the axis to an infinite distance from the lens, the emergent rays which converge to N' will, in the limiting position, be parallel to the axis, and must before incidence have issued from an origin of light at a point N situated upon the axis.

The limiting position of N' as N moves off to an infinite. distance, and the limiting position of N as N' moves off to an infinite distance, are called the Principal Foci of the lens. They are commonly referred to simply as the Foci, and are denoted by the letters F" and Frespectively.

Hence all rays which before incidence are parallel to the axis will after refraction pass through the point F", and all rays which after emergence are parallel to the axis must have proceeded before incidence from the point F.

48. If we denote by I and I' respectively the infinitely distant points towards which N and N' move, it follows that I and F', and F and I' are pairs of conjugate points.

49. The planes through the foci F and F" perpendicular to the axis are called the Focal Planes.

50. The planes conjugate to the Focal Planes are at an infinite distance.

Hence, if the luminous point be on a Focal Plane, it follows that all the rays which proceed from it will, on emergence, be parallel to one another.

Also, if an image fall on a Focal Plane, it follows in the same way that the incident rays must all have been parallel to one another.