Physics · Ch 9 — Optics

Refraction at a spherical surface and lenses

9.7

Refraction at a spherical surface and lenses

Section 9.5 derived the real-depth/apparent-depth relation only for a FLAT (plane) refracting interface. Many optical situations instead involve one or more CURVED (spherical) refracting surfaces -- liquid drops, lenses, and ellipsoidal paperweights among them -- where the plane-surface relation no longer applies, and the radius of curvature $R$ must be included alongside the refractive indices.

A LENS is essentially built from the intersection of two spherical surfaces (of radii $R_1$ and $R_2$ ) or of one sphere and a flat plane ( $R=\infty$ ). A CONVEX lens is thicker in the middle and narrows towards the edges -- visualised as the INTERNAL cross-section common to two intersecting spheres; a CONCAVE lens is thicker at the edges and narrows towards the centre -- the EXTERNAL cross-section of two intersecting spheres. Concavo-convex and convexo-concave (meniscus) shapes are the ones commonly used for spectacles of positive and negative 'number' respectively. For a lens made of material optically DENSER than its surroundings, convex lenses have positive focal length (converging) and concave lenses have negative focal length (diverging), by the Cartesian sign convention; for a lens in a RARER surrounding medium instead (e.g. an air bubble embedded in glass, or an air-filled lens shape submerged in water), this reverses -- a geometrically convex shape then DIVERGES light and a geometrically concave shape CONVERGES it. A THIN lens (maximum thickness at least 50 times smaller than $u,v,f,R_1,R_2$ ) lets both of its refracting surfaces share a single common pole, or optical centre.

For any thin lens, $\dfrac{1}{f} = \dfrac{1}{v}-\dfrac{1}{u}$ . If several thin lenses of the same axis are held in close contact, their focal POWERS simply add algebraically: $\dfrac{1}{f}=\dfrac{1}{f_1}+\dfrac{1}{f_2}+\dfrac{1}{f_3}+\dots$ , i.e. $P=P_1+P_2+P_3+\dots$ . For just two thin lenses separated by a distance $d$ in air, $\dfrac{1}{f}=\dfrac{1}{f_1}+\dfrac{1}{f_2}-\dfrac{d}{f_1f_2}$ , i.e. $P=P_1+P_2-dP_1P_2$ .

REFRACTION AT A SINGLE SPHERICAL SURFACE: consider a spherical refracting surface of radius $R$ separating media of index $n_1$ (containing the object) and $n_2$ . Taking two paraxial rays from a point object O on the axis -- one travelling straight through the pole undeviated, one striking the surface obliquely and refracting per Snell's law -- and using the small-angle (paraxial) approximation to relate the various angles at incidence, at the normal, and at refraction, back to the distances $u$ (object), $v$ (image) and $R$ (radius), one arrives at $\dfrac{n_2}{v}-\dfrac{n_1}{u}=\dfrac{n_2-n_1}{R}$ . Although derived here for one particular case (a rarer-to-denser, convex surface, real object), this SAME single equation holds for ANY spherical surface and ANY real or virtual image, PROVIDED every quantity is substituted with its correct Cartesian sign; the one fixed rule throughout is that $n_1$ is always the refractive index of the medium containing the real object, and $n_2$ is the OTHER medium (where a real image, if one forms, will actually be located; a virtual image instead forms back in the $n_1$ medium).

Worked illustration: a glass paperweight ( $n=1.5$ , radius 3 cm) has a trapped air bubble 2 cm from its near surface (hence 4 cm from the far surface, through which it is viewed). With $n_1=1.5$ (glass, the object-side medium), $n_2=1$ (air), $u=-4$ cm and $R=-3$ cm (the far surface, as seen from that side): $\dfrac{1}{v}-\dfrac{1.5}{-4}=\dfrac{1-1.5}{-3}\Rightarrow\dfrac{1}{v}=\dfrac{1}{6}-\dfrac{3}{8}=-\dfrac{5}{24}\Rightarrow v=-4.8$ cm -- so the bubble appears 4.8 cm inside the glass, FARTHER than its actual 4 cm depth, precisely because of the surface's curvature (unlike a flat interface, where apparent depth is always LESS than real depth, a curved surface can push it the other way). …

Figure 9.11Figs 9.11(a)-(f): The six common lens shapes as intersections of spheres

What this figure shows. A set of six small cross-section diagrams of common lens shapes, each shown as the intersection of two spherical surfaces (or one spherical surface and a flat plane) with the lens's cross-section shaded/outlined. (a) Convex lens: shown as the shape common to (the INTERNAL overlap of) two intersecting circles, thicker at the centre, tapering to a point at each edge -- symmetric double-convex outline. (b) Concave lens: shown as the shape formed by the EXTERNAL region between two intersecting circles' arcs curving inward, thinner at the centre and thicker at the edges -- symmetric double-concave outline, both surfaces curving inward toward the centre. (c) Plano-convex lens: one face perfectly FLAT (a straight vertical line), the other face bulging outward in a single convex arc. (d) Plano-concave lens: one face flat, the other face curving inward (concave arc), thinner at the centre than at the edges. (e) Concave-convex (meniscus) lens: both faces curve the SAME way (same sign of curvature), one surface concave and the other convex, giving a crescent-like cross-section, thicker at one region. (f) Convex-concave lens: the mirror-image men …

Figure 9.12Fig 9.12: Refraction at a single spherical refracting surface

What this figure shows. A spherical refracting surface YPY' (an arc, convex toward the left/object side) with radius of curvature R and centre of curvature C, separating two transparent media of refractive index n1 (on the left, containing the object) and n2 (on the right), with n1 less than n2. P is the pole (vertex) of the surface and the principal axis X'PX passes through P and C. A point object O lies on the axis at object distance -u from P, in medium n1. Two rays from O are drawn: the axial ray OP, which travels straight through undeviated along PX; and an oblique paraxial ray OA, which strikes the surface at point A, where the normal CAN (drawn from the centre of curvature C through A) makes angle i with the incident ray OA. After refraction at A (bending toward the normal, since n1 is less than n2), the ray travels along AZ into medium n2 at angle of refraction r, crossing the principal axis at point I, which is the real image of O -- the angles that OA, CA and AZ each make with the principal axis (labelled …

Misc Ex.9.6Apparent position of an air bubble in a glass paperweight, viewed through the curved surface

Worked out. A spherical glass paperweight of refractive index 1.5 and radius 3 cm has a tiny air bubble trapped inside it, at a closest distance of 2 cm from the near surface (hence 3+3-2=4 cm from the FAR surface, through which it is viewed, since the bubble sits off-centre). Applying the single-spherical-surface formula n2/v - n1/u = (n2-n1)/R with the object (bubble) now effectively in the glass viewed from the far, farther surface (u = -4 cm, n1 = 1.5 for glass as the object-side medium, n2 = 1 for air, R = -3 cm for that far surface as seen from inside), the bubble is found to appear at v = -4.8 cm, i.e. it appears 4.8 cm inside the glass when viewed from the far end -- deeper than its actual 4 cm distance from that surface, showing that (unlike a flat interface, where apparent depth is always LESS than real depth) a curved refracting surface can mak …

Figure 9.13Fig 9.13: Derivation of the lens maker's equation using two refracting surfaces

What this figure shows. A thin lens with two spherical refracting surfaces of radii of curvature R1 (first surface, facing the object) and R2 (second surface), both surfaces sharing a single common pole P since the lens is thin, kept in a surrounding medium of index 1 with the lens material of index n. A point object O lies on the principal axis at distance u from P, in front of the first surface. The axial ray OP travels undeviated. A paraxial ray OA strikes the FIRST surface at A, refracts, and (in the ABSENCE of the second surface) would converge to an intermediate image point I1 on the axis at distance v1=PI1 from P -- this partial ray path OA-to-I1 is drawn with I1 marked explicitly. Before actually reaching I1, this same ray is intercepted by the SECOND refracting surface at a point B (drawn close to the shared pole P since the lens is thin), where it refracts a second time and finally crosses the axis at the true final image point I, at distance v=PI from P -- both the intermediate (virtual construction) im …

Misc Ex.9.7Radii of curvature of a dense double-convex lens from given object/image distances

Worked out. A dense glass double-convex lens of refractive index n=2, with the ratio of the magnitudes of its two radii of curvature |R1|:|R2|=1:5, forms a real image at 7.5 cm of a point object placed 15 cm in front of it. Using the thin lens formula 1/f=1/v-1/u with u=-15 cm and v=+7.5 cm, the focal length works out to f=5 cm. Since the lens is double convex, R1 is positive and R2 is negative with |R2|=5|R1|; substituting into the lens maker's equation 1/f=(n-1)(1/R1-1/R2) with n=2 and solving the resulting equation in R1 gives R1=6 cm and R2=-30 …

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