Читать книгу Optical Engineering Science - Stephen Rolt - Страница 27

Figure 1.14 shows a lens made up of two spherical surfaces, of radius, R₁ and R₂. Once again, the convention is that the spherical radius is positive if the centre of curvature lies to the right of the relevant vertex.

So, in the biconvex lens illustrated in Figure 1.14, the first surface has a positive radius of curvature and the second surface has a negative radius of curvature. The lens is made from a material of refractive index n₂ and is bounded by two surfaces with radius of curvature R₁ and R₂ respectively. It is immersed totally in a medium of refractive index, n₁ (e.g. air). In addition, it is assumed that the lens has negligible thickness (the thin lens approximation). Of course, as for the treatment of the single curved surface, we assume all angles are small and θ ∼ sinθ. First, we might calculate the angle of refraction, φ₁, produced by the first curved surface, R₁. This can be calculated using Eq. (1.14):

Figure 1.14 Refraction by two spherical surfaces (lens).

Of course, the final angle, φ, can be calculated from φ₁ by another application of Eq. (1.14):

Substituting for φ₁ we get:

(1.15)

As for Eq. (1.14) there are two parts to Eq. (1.15). First, there is an angular term that is equal to the incident angle. Second, there is a focusing contribution that produces a deflection proportional to ray height. Equation (1.15) allows the tracing of all rays in a system containing the single lens and it is straightforward to calculate the Cardinal points of the thin lens: Cardinal points for a thin lens

Both Principal Points: At centre of lens
Both Nodal Points: At centre of lens

Since both object and image spaces are in the same media, then both focal lengths are equal and the principal and nodal points are co-located. One can take the above expressions for focal length and cast it in a more conventional form as a single focal length, f. This gives the so-called Lensmaker's Equation, where it is assumed that the surrounding medium (air) has a refractive index of one (i.e. n₁ = 1) and we substitute n for n₂.

(1.16)