The thin-lens approximation assumes rays refract one time, at the lens plane. In fact, as FIGURE 34.41 shows, rays refract twice, at spherical surfaces having radii of curvature $R_1$ and $R_2$. Let the lens have thickness $t$ and be made of a material with index of refraction $n$. For simplicity, we’ll assume that the lens is surrounded by air.

FIGURE 34.41 Image formation by a lens.

The object at point P is distance $s_1$ to the left of the lens. The first surface of the lens, of radius $R_1$, refracts the rays from P to create an image at point $\text{P}\prime$. We can use Equation 34.20 for a spherical surface to find the image distance $s_1^{\prime }:$

where we used $n_1=1$ for the air and $n_2=n$ for the lens. We’ll assume that the image $\text{P}\prime$ is a virtual image, but this assumption isn’t essential to the outcome.

With two refracting surfaces, the image $\mathbf{\text{P}}\prime$ of the first surface becomes the object for the second surface. That is, the rays refracting at the second surface appear to have come from $\text{P}\prime$. Object distance $s_2$ from $\text{P}\prime$ to the second surface looks like it should be $s_2=s_1^{\prime }+t$, but $\text{P}\prime$ is a virtual image, so $s_1^{\prime }$ is a negative number. Thus the distance to the second surface is $s_2=\left|s_1^{\prime }\right|+t=t-s_1^{\prime }$. We can find the image of $\text{P}\prime$ by a second application of Equation 34.20, but now the rays are incident on the surface from within the lens, so $n_1=n$ and $n_2=1$. Consequently,

For a thin lens, which has $t\to 0$, Equation 34.22 becomes

Our goal is to find the distance $s_2^{\prime }$ to point $\text{P}″$ the image produced by the lens as a whole. This goal is easily reached if we simply add Equations 34.21 and 34.23, eliminating $s_1^{\prime }$ and giving

The numerical subscripts on $s_1$ and $s_2^{\prime }$ no longer serve a purpose. If we replace $s_1$ by $s$, the object distance from the lens, and $s_2^{\prime }$ by $s\prime$, the image distance, Equation 34.24 becomes the thin-lens equation:

where the focal length of the lens is

Equation 34.26 is known as the lens maker’s equation. It allows you to determine the focal length from the shape of a thin lens and the material used to make it.

We can verify that this expression for $f$ really is the focal length of the lens by recalling that rays initially parallel to the optical axis pass through the focal point on the far side. In fact, this was our definition of the focal length of a lens. Parallel rays must come from an object extremely far away, with object distance $s\to \text{∞}$ and thus $1/s=0$. In that case, Equation 34.25 tells us that the parallel rays will converge at distance $s\prime =f$ on the far side of the lens, exactly as expected.

We derived the thin-lens equation and the lens maker’s equation from the specific lens geometry shown in Figure 34.41, but the results are valid for any lens as long as all quantities are given appropriate signs. The sign convention used with Equations 34.25 and 34.26 is given in TABLE 34.3.

EXAMPLE 34.12 Focal length of a meniscus lens

What is the focal length of the glass meniscus lens shown in FIGURE 34.42? Is this a converging or diverging lens?

FIGURE 34.42 A meniscus lens.

FIGURE 34.42 Full Alternative Text

SOLVE If the object is on the left, then the first surface has $R_1=-40\text{cm}$ (concave toward the object) and the second surface has $R_2=-20\text{cm}$ (also concave toward the object). The index of refraction of glass is $n=1.50$, so the lens maker’s equation is

$$\begin{array}{ccc}\hfill {\displaystyle \frac{1}{f}}& =\hfill & \left(n-1\right)\left({\displaystyle \frac{1}{R_1}}-{\displaystyle \frac{1}{R_2}}\right)=\left(1.50-1\right)\left({\displaystyle \frac{1}{-40\text{cm}}}-{\displaystyle \frac{1}{-20\text{cm}}}\right)\hfill \\ \hfill & =\hfill & 0.0125{\text{cm}}^{-1}\hfill \end{array}$$

Inverting this expression gives $f=80\text{cm}$. This is a converging lens, as seen both from the positive value of $f$ and from the fact that the lens is thicker in the center.