Lens Combinations and Optical Aberrations

1Why does a camera lens weigh so much?

The "lens" on a good camera or in a school microscope is not one piece of glass but a tube holding several, sometimes more than ten. One perfect lens would be lighter and cheaper, so why use many? There are two reasons. Combining lenses lets a designer choose the overall focal length and magnification, and it lets one lens cancel the faults of another. Those faults, called aberrations, are the reason a single spherical lens or mirror never gives a perfectly sharp image. This lesson treats both ideas.

2Two thin lenses in contact

Put two thin lenses of focal lengths $f_{1}f\_1$ and $f_{2}f\_2$ in contact, and place an object at distance $uu$. Lens 1 alone would form an image at $v_{1}v\_1$:

Before the light reaches that image, it meets lens 2. The rays are converging towards a point $v_{1}v\_1$ beyond lens 2, so for lens 2 this is a virtual object at distance $-v_1-v\_1$ (real is positive). Lens 2 then forms the final image at $vv$:

Adding the two equations, the $\frac{1}{v_1}\backslash frac\{1\}\{v\_1\}$ terms cancel:

The pair behaves as one lens of focal length $FF$: powers in contact add. This is why an optician can test your eyes by dropping trial lenses into one frame and simply adding their powers.

Lenses separated by a distance $dd$. Treat them one at a time: the image formed by lens 1 is the object for lens 2 (real if it lies in front of lens 2, virtual if behind it), and the total magnification is $m=m_1\times m_{2}m\; =\; m\_1\; \backslash times\; m\_2$. For parallel light the pair has a single equivalent power, which can be shown to be

which reduces to $P_1+P_{2}P\_1\; +\; P\_2$ when $d=0d\; =\; 0$.

Figure 4.1: Lenses in sequence. The image I₁ formed by lens 1 is the object for lens 2.

3Optical aberrations

The mirror and lens formulas were derived for paraxial rays of a single wavelength. Real light is neither. The departures from a perfect point image are called aberrations. The two most important are spherical aberration and chromatic aberration.

Spherical aberration (lenses and mirrors)

You saw in lesson 1 that a ray striking a spherical mirror at angle $\theta \backslash theta$ to the normal crosses the axis at $\frac{r\mathrm{/}2}{\cos \theta }\backslash frac\{r/2\}\{\backslash cos\backslash theta\}$ from C, closer to the mirror than F when $\theta \backslash theta$ is large. A spherical lens behaves in the same way: its outer zone is too strongly curved for the rays it receives, so marginal rays (near the edge) are focused nearer the lens than paraxial rays. The light never meets at one point; the smallest patch it forms is called the circle of least confusion.

Figure 4.2: Spherical aberration in a wide concave mirror, drawn with the law of reflection applied exactly to each ray.

Reducing it: (1) use a stop (aperture) to block the marginal rays, at the cost of a dimmer image; (2) for mirrors, use a parabolic surface, which sends every ray parallel to the axis through one focus (car headlamps, satellite dishes and solar cookers use this shape); (3) for lenses, share the bending between surfaces, for example by turning a plano-convex lens so its curved face meets the parallel light, or by grinding aspheric surfaces; (4) combine a converging and a diverging lens whose aberrations are opposite.

Spherical aberration can be very costly. In 1990 the Hubble Space Telescope sent back blurred images; NASA's investigation found that its primary mirror had been polished to the wrong curve, too flat near its outer edge, an error ten times the allowed tolerance. In December 1993 astronauts installed corrective optics that cancelled the error, like a pair of spectacles for the telescope.

Chromatic aberration (lenses only)

The refractive index of glass is slightly larger for violet light than for red light (lesson 5). By the lens-maker's formula, $\frac{1}{f}=(n-1)(\frac{1}{R_1}+\frac{1}{R_2})\backslash frac\{1\}\{f\}\; =\; (n\; -\; 1)\backslash left(\backslash frac\{1\}\{R\_1\}\; +\; \backslash frac\{1\}\{R\_2\}\backslash right)$, violet light therefore has the shorter focal length. White light from a point is focused into a spread of coloured images along the axis, and a photograph shows coloured fringes at sharp edges (Figure 4.3a). The distance $f_r-f_{v}f\_r\; -\; f\_v$ is the longitudinal chromatic aberration.

A mirror has no chromatic aberration: the law of reflection does not depend on wavelength. This is one reason why large astronomical telescopes use mirrors rather than lenses.

The achromatic doublet. Cement a converging lens of crown glass to a weaker diverging lens of flint glass, which spreads colours more strongly (Figure 4.3b). For a thin lens, $\frac{1}{f}\propto (n-1)\backslash frac\{1\}\{f\}\; \backslash propto\; (n\; -\; 1)$, so a small change $\delta n\backslash delta\; n$ between blue and red light changes the power by $\delta \text{\hspace{0.17em}⁣}\left(\frac{1}{f}\right)=\frac{\delta n}{n-1}\cdot \frac{1}{f}=\frac{\omega }{f}\backslash delta\backslash !\backslash left(\backslash frac\{1\}\{f\}\backslash right)\; =\; \backslash frac\{\backslash delta\; n\}\{n\; -\; 1\}\backslash cdot\backslash frac\{1\}\{f\}\; =\; \backslash frac\{\backslash omega\}\{f\}$, where $\omega =\frac{n_b-n_r}{n-1}\backslash omega\; =\; \backslash frac\{n\_b\; -\; n\_r\}\{n\; -\; 1\}$ is the dispersive power of the glass. For the pair in contact to have the same power for both colours, the two changes must cancel:

Since ${\omega }_1\backslash omega\_1$ and ${\omega }_2\backslash omega\_2$ are both positive, one lens must be converging and the other diverging, and the lens with the larger $\omega \backslash omega$ (flint) must be the weaker.

Figure 4.3: Chromatic aberration (a) and its cure (b). The colour spread is exaggerated.