Optical Instruments

Perhaps the simplest optical instrument is the lens magnifier. Without optical aid, we cannot “see” things close up. The eye will simply not focus closer than about 0.25 m (unless you are nearsighted!). But an object places just inside the focal point of a converging lens will produce a large virtual image that can be viewed more easily. Let’s look at our Convex Lens again.

Her we can see that the ratio of the heights of the subject and image, the magnification M is

Usually we are able to get good magnification and place the image near 0.25 m if the object is close to the focal point on the object side of the lens. Using i ~ 0.25 m and s ~ f, we get

as long as f is measured in meters. (For f in cm, the constant in the numerator is 0.25 x 100 = 25).

The standard optical microscope consists of two lenses (each can be a compound lens). By placing the object to be observed very close to the focal point of the first or objective lens, a larger real (but inverted) image will be produced. This real image is then observed with a second lens, the eyepiece, which acts as a magnifier to make the image even larger. If you wear eyeglasses or reading glasses, removing them will allow you to view the microscope properly.

The net magnification of the entire system is the product of the magnifications of the objective and eyepiece. For these we just use the magnifications given for a simple lens and a magnifier:

Because the image is much larger than the object, it usually requires that the object be brightly lit, or it will be too dark to see well.

Another practical limit on an optical microscope comes from the fact that the wavelength of visible light is so “large”. The fineness of detail that can be observed, measured in radians, is given by Rayleigh’s criterion:

where d is the diameter of the opening through which the light passes (such as the objective), and l is the wavelength of light used. This limitation comes from the wave properties of light. Light passing through a narrow opening undergoes diffraction, which spreads the beam out. Diffraction is basically just the interference pattern of a light wave with other portions within the same opening instead of a different opening.

Better resolution can be obtained using UV light, but that only helps a little.

Electrons have much smaller wavelengths than visible light, and so can be used to see smaller details. This is the basis of the electron microscope, which uses magnetic fields to focus the electrons. However, this is getting off the subject of light & color….

Telescopes come in many different designs. Those that use a lens as the objective to gather and focus the light are refracting telescopes, while those that do this with mirrors are reflecting telescopes.

The first telescopes were refractors that used a convex lens to form an inverted image of a distant object, and used a concave lens to invert this image to an upright orientation as well as provide some additional magnification. Because this was the sort of device Galileo used to make the first important telescopic discoveries in astronomy, it is usually referred to as a Galilean telescope.

Galilean Telescope

However, Kepler found that greater magnification could be obtained using an eyepiece that was a convex lens, at the expense of keeping the inverted orientation of the image formed by the objective.

Keplerian Telescope

Refracting telescopes used in astronomy are of the Keplerian type, while the upright image of the Galilean design is usually preferred for terrestrial observing.

The critical optical parameters for an astronomical telescope are its light-gathering power, angular resolution, magnification, and image brightness.

The greater the diameter of the objective, the greater the surface that will intercept the light from an object. If one is dealing with faint sources, this is usually the single most important criterion. For this reason, astronomers “label” telescope sizes by the diameter of their objectives (lens or mirror) and try to make it as large as possible. If the diameter of the telescope is D meters, then the surface area intercepting the light is

The largest optical telescopes today have diameters of 8-10 m. By contrast, the inner diameter of the iris in the human eye, after dark adaptation, is about 1/2 cm.

Although the diameter D is generally much larger than the wavelength of light, it is not infinitely bigger, and Rayleigh’s criterion still applies. Larger diameter objectives and smaller light wavelengths improve the situation. (As a practical matter, atmospheric turbulence will dominate over diffraction in degrading the image for telescopes larger than about 10 cm. Compensating for this phenomenon using adaptive optics or placing the telescope above the atmosphere improve the situation).

Because the actual image distance and size are often unknown (in astronomy usually unknown), we will deal with angular magnification instead.

The angular magnification is simple the ratio of the focal lengths of the objective & eyepiece:

So with a telescope of some fixed objective, higher magnification is just a matter of using small focal length eyepieces., The magnification is usually described as magnifying “power”, often just designated with an “X”. Thus a telescope with a magnification of 100 would be described as “100 power” or “100 X”.

When using the human eye to observe with a telescope, care must be paid to how the beam of light enters the eye. The so-called exit pupil should roughly match that of the eye. Making it too big directs light outside the pupil of the eye, so it is wasted. Making it smaller is done at the cost of a smaller effective objective opening, and the image is fainter (the light-gathering power of the eye is wasted).

Up to a limit then, the larger “cone” the brighter the source. The number often used to describe this is the f-ratio defined as

If the focal length is 5 times the diameter, the lese is said to be an f/5 lens. In both astronomical photography and terrestrial photography, the eye is replaced by a camera (which may or may not have a lens in it). The smaller the f-number, the brighter the image formed on the detector (film, plate, CCD chip, etc). Getting small f/ requires highly curved lenses, and this requires the use of multiple lenses of differing shapes & materials in order to minimize chromatic (and other) aberration. Every surface that is required costs money to make…..

In order to avoid chromatic aberration, Newton invented the reflecting telescope. The strengths of reflectors over refractors are:

avoidance of chromatic aberration

only one surface to shape precisely

the glass need not be Perfect” in its interior

its weight can be supported from the back

The light convergence is done by using a curved surface for the mirror. If that were all that were use, however, the eyepiece and the observer’s head would have to be placed in front of the telescope, blocking the light! So Newton used a flat secondary mirror to redirect the light to the side. This design is still called a Newtonian telescope.

A spherical mirror will produce noticeable spherical aberration unless the f-ratio is large. Spherical aberration is eliminated by using a parabolic mirror, at the cost of introducing coma. Another way to produce a better image is to use a curved secondary mirror designed to remove some of the aberrations of the objective or primary mirror. The Cassegrain telescope is one example.

Other designs are possible. Placing a weak lens in front can help reduce the problems of coma and spherical aberration without introducing significant coma, as in the Schmidt-Cassegrain design.

Binoculars are just 2 refracting telescopes joined side-by-side. They include a pair of special reflecting prisms in the light path to:

1. shorten the physical length of the tubes holding the objectives & eyepieces

2. narrow the distance between incoming light paths to match eye separation

3. invert the inverted image so that it is upright

Cheap imitations (“field glasses”) without the prisms use Galilean telescopes instead.

A single telescope of this design is called a monocular.

One clever design uses a monocular and magnifier to make a small microscope!

Film and slide projectors consist of 4 basic components:

1. a lamp (often with a rear reflective mirror to add more light
2. a condenser lens to direct as much light through the film/slide as possible and form an image of the filament of the lamp where it won’t be visible
3. the film/slide
4. the projection lens

The projection lens forms an image of the film/slide on a screen. The image of the filament, formed inside the projection lens is so out of focus as not to be visible.

The design of projection screens is a science of its own. Screens can be made to reflect the light back in a number of ways. (There is more to reflective surfaces than Lambertian and specular reflection!).

The standard “overhead projector” used in classrooms uses a Fresnel lens as the condenser.

A similar design I used for some automobile headlights, but the “grooved” surface is inside. If it were on the outside, it would collect dirt more easily and be a nightmare to clean!