The Optical Bench

An optical bench is a long rail on which lenses, mirrors and screens slide along a common axis. Every arrangement on it, however complicated, can be worked out with the method of Chapter 5: find the vergence of the light arriving at the first element, add the element’s power, carry the result to the next element, and repeat. On the benches in this chapter the figures do that bookkeeping for you, element by element, and list it under each diagram.

#One lens and a screen

Start with the simplest arrangement: an object, a +5.00 D lens and a screen. Drag the screen until the image on it is sharp, or move the object:

With the object 40 cm away, light reaches the lens with a vergence of −2.50 D and leaves it with +2.50 D, converging to an image 40 cm beyond the lens. Anywhere else, the light that meets the screen is either still converging or already spreading again, and the screen shows a blur. The list under the diagram is the reasoning the figure used: the vergence that arrives, the vergence that leaves, and where the light is heading.

#Two lenses

Add a second lens, and the image formed by the first becomes the object for the second. Drag the second lens, or the object:

Between the lenses, the light’s vergence changes just by travelling, as we saw in Chapter 5. Arriving at the second lens, a distance $d$ on, it has $U_2=V_1/(1-d{V}_1)$. With the second lens beyond the first image, that light is diverging again, from a real image 1 in between. Slide the second lens in front of image 1 and the light is still converging when it gets there, heading for a point beyond the lens. The second lens then works on a virtual object, and the same equation still holds.

#A lens and a mirror

Mirrors fit into the same scheme; the light just turns round. Here a +5.00 D lens has a plane mirror behind it, and the object sits at the lens’s focal point:

The light leaves the lens parallel, the mirror sends it back still parallel, and the lens focuses it again in the object’s own plane, so the image comes back upside down beside the object, wherever the mirror is. This is autocollimation, a classic way to find the focal point of a lens or to check that an instrument is focused for infinity. Move the object off F and its returning image moves the other way, so the two separate quickly.

#The Badal optometer

One more classic arrangement. Put the eye at the second focal point of a +10.00 D lens, and look through the lens at a target:

Light from a target exactly at the lens’s first focal point reaches the eye parallel. Move the target 1 cm nearer the lens and the light reaching the eye has a vergence of −1.00 D; 1 cm farther away, +1.00 D. With a +10.00 D lens the vergence at the eye changes by exactly one dioptre for every centimetre, because it equals $P^{2}x$, where $x$ is how far the target lies beyond the focal point. And the ray from the tip of the target that travels parallel to the axis is bent through the lens’s second focal point, just where the eye is, always at the same angle: $hP$, for a target of height $h$. So wherever the target is, its angular size at the eye, and with it the size of its retinal image, stays the same.

That makes a simple optometer. Slide the target until the patient, with accommodation relaxed, sees it sharply, and read their refractive error straight off the scale. The myopic eye here sees the target sharply at −4, nearer than the focal point; the hyperopic eye needs converging light, and sees it sharply at +2, beyond the focal point. This is Badal’s principle, still at work inside many optometers and autorefractors.

#Your own bench

The last figure is a bench of your own, with a lens and a screen to start with. Add lenses, prisms, mirrors, stops (openings that limit the beam, like a pupil), screens and eyes, drag them along the axis, and select one to change it. The list underneath follows the light from element to element, whatever you build. When you make something worth keeping, copy a link to it. Elements can’t share a place on the rail, but they can sit as close as 5 mm.

A few things to try:

• Two +5.00 D lenses 5 mm apart act almost exactly as one +10.00 D lens. Pull them apart and watch the image move (Chapter 5). Set it up
• A Galilean telescope: parallel light arriving at a slight angle, a +5.00 D lens and a −20.00 D lens 15 cm apart. The light leaves parallel, at four times the angle. Set it up
• An eye with 8.00 D of myopia, and a lens 10 mm in front of it. A −8.00 D lens leaves the focus short of the retina; what power puts it on the retina? (About −8.75 D: a minus lens loses effect as it moves away from the eye, Chapter 6.) Set it up
• A +5.00 D concave mirror with an object 30 cm in front of it: where does the image form, and how big is it? Set it up
• A stop just in front of a lens, to see how an aperture limits the light that gets through, without moving the image. Set it up

That is the toolkit of geometric optics: light described by its vergence, surfaces and lenses that add their power to it, distances that change it, and an eye at the end of it all that needs it to be just right. Everything else is arrangement. Part Two puts it to work measuring and correcting real eyes, and Part Three turns to light as a wave, lasers and the instruments of the clinic.

#Recap

• Any arrangement can be followed one element at a time: the vergence that arrives, plus the element’s power, is the vergence that leaves.
• Between elements the vergence changes with distance: $U_2=V_1/(1-d{V}_1)$. An image formed by one element is the object for the next, real or virtual.
• Mirrors fit in by turning the light round, with distances measured in the direction it travels.
• With the eye at the second focal point of a Badal lens, the vergence at the eye is $P^{2}x$, proportional to the target’s distance beyond the focal point, and the retinal image keeps its size.

#Test yourself

#Further reading

• M. H. Freeman and C. C. Hull, Optics. Butterworth-Heinemann.
• M. P. Keating, Geometric, Physical, and Visual Optics. Butterworth-Heinemann.
• D. A. Atchison, A. Bradley, L. N. Thibos and G. Smith, Useful variations of the Badal optometer. Optometry and Vision Science 72 (1995) 279–284.