Physics · Ch 6 — Optics

Lens Maker's Formula and Lens Equation

6.6.3

Lens Maker's Formula and Lens Equation

For a thin lens of refractive index $n_2$ in a surrounding medium of index $n_1$ , with radii of curvature $R_1,R_2$ for its two surfaces, applying the single-spherical-surface refraction equation once at each surface in turn (first surface: $\frac{n_2}{v'}-\frac{n_1}{u}=\frac{n_2-n_1}{R_1}$ ; second surface, using the first image as its object: $\frac{n_1}{v}-\frac{n_2}{v'}=\frac{n_1-n_2}{R_2}$ ) and adding the two equations eliminates the intermediate image distance $v'$ , giving $\dfrac{1}{v}-\dfrac{1}{u}=\left(\dfrac{n_2}{n_1}-1\right)\left(\dfrac{1}{R_1}-\dfrac{1}{R_2}\right)$ . Setting $u\to\infty$ , $v\to f$ (the very definition of focal length) gives the lens maker's formula, for a lens of refractive index $n$ in air, $\boxed{\dfrac{1}{f}=(n-1)\left(\dfrac{1}{R_1}-\dfrac{1}{R_2}\right)}$ -- telling a lens manufacturer exactly what curvatures to grind, for a chosen glass, to achieve a desired focal length; it holds for concave as well as convex lenses once the radii are correctly signed. Comparing the two deriv …

Figure 6.34Refraction through thin lens

What this figure shows. A thin lens made of a medium of refractive index n2, surrounded by a medium of refractive index n1, has two spherical surfaces of radii R1 and R2 sharing a common pole P (since the lens is thin). A ray from a point object O refracts first at the near surface, forming an intermediate (unrealised) image I', and then refracts a second time at the far surface, forming the final image I -- applying the single-spherical-surface equation once at each face in turn, then adding the two resulting equations together and eliminating the intermediate image distance, is exactly h …

Misc Example 6.14Focal length of a biconvex lens, and its invariance when the lens is flipped

Worked out. A biconvex lens of refractive index 1.5 has radii of curvature 20 cm and 15 cm on its two faces (R1 = +20 cm, R2 = -15 cm by the sign convention for a biconvex shape). Substituting into the lens maker's formula 1/f = (n-1)(1/R1 - 1/R2) gives 1/f = 0.5 times (1/20 + 1/15) = 0.5 times 7/60 = 7/120, so f = 120/7 = 17.14 cm, a converging lens since f is positive. Flipping the lens end-for-end swaps the roles so now R1 = +15 cm and R2 = -20 cm; recomputing gives exactly the same 1/f = 0.5 times (1/15 + 1/20) = 7/120, so f = 17.14 cm again -- confirming that a lens's focal length never changes when it is turned back to front, a general result tr …

Misc Example 6.15Focal length of a convexo-concave lens

Worked out. A convexo-concave lens of refractive index 1.52 has R1 = 10 cm and R2 = 20 cm (both surfaces curving the same way, giving this meniscus-like shape). Substituting into the lens maker's formula gives 1/f = (1.52-1) times (1/10 - 1/20) = 0.52 times (1/20) = 0.026, so f = 1/0.026 = 38.46 cm. Since f is positive, this convexo-concave lens is still net converging overall, despite having one concave face, because the convex face's curvature do …

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