1. Now look at the astigmatism chart. All of the lines are printed the same thickness and brightness, but a person with astigmatism sees some lines as darker than others. Cover one eye and look at the chart both with and without your glasses, if you wear them. Do some of the lines look darker than others? If they do, rotate the figure 90° to convince yourself that the lines are actually the same and it is only your eye that causes the effect. Try rotating your glasses in front of your face while looking at the chart through one of the lenses. Describe what you see.
2. Set the eye model to normal, near vision (put the +62 mm lens in the SEPTUM slot, remove other lenses, and put the retina screen in the NORMAL position). With the eye model looking at the nearby light source, adjust the eye-source distance so that the image is in focus.
3. Place the -128 mm cylindrical lens in slot A. The side of the lens handle marked with the focal length should be towards the light source. Describe the image formed by the eye with astigmatism.
4. Rotate the cylindrical lens. What happens to the image? This shows that astigmatism can have different directions depending on how the defect in the eye’s lens system is oriented.
5. Eyeglasses can be outfitted with compensating cylindrical lenses to correct astigmatism. Now place the other astigmatic lens in the eyeglass holder slot and move it until the image is again focused. What manipulation did you have to do to bring the image into focus again? Describe how you could find out experimentally if a pair of eye glasses have astigmatic corrections.
Optical Instruments and the Human Eye
Optical instruments enhance vision by forming an image that is a different size or at a different position from the object:
A microscope is a combination of two or more lenses that forms a distant image of a near object. The lens nearest the object observed is called the objective lens, and its aperture determines the light-collection angle and hence the system’s spatial resolution. In the example illustrated below, a system of two converging lenses forms an upright, real image at infinity.
A telescope forms a larger image of a distant object. In the example of a simple refractor telescope illustrated below, a system of two converging lenses forms an inverted, real image. Note that the image is not closer to the eye than the object; this telescope forms a distant image which can be viewed by the eye in its relaxed state, when it is focused at infinity. Because its objective lens, or first lens, usually has a larger area than the pupil of the eye, a telescope gathers more light than the unaided eye, and thus can allow the eye to see objects that would normally be too dim to detect.
Refractor telescopes have the advantage that they form upright images, rather than inverted images which appear upside-down to us. They are still used for many terrestrial applications, such as piracy. However, all modern optical research telescopes use mirrors to provide a much larger diameter aperture for superior light-collecting and for spatial resolution. (You will learn more details about the latter issue in the Physical Optics Lab.) One popular design is the Newtonian reflector, which uses a curved (spherical or, ideally, parabolic) primary mirror to collect light, and focuses that light onto an eyepiece or camera using a flat secondary mirror at an angle to the primary. One disadvantage of this design is that the larger aperture the mirror, the longer and heavily the tube required to mount it.
By contrast, a Schmidt-Cassegrain telescope uses two mirrors: a large primary and a smaller secondary, both of which are curved. The primary mirror has an aperture in its center that allows the light to be focused by the secondary onto the eyepiece or a camera. Note that this design folds the light pathway into a more compact distance even for a large aperture primary mirror, so the tubes can be shorter than for a Newtonian reflector. Our large Observatory dome telescopes have this design.
Telescopes are also classified by their mounts. Either refractors or reflectors can use equatorial mounts equipped with angular adjustments that allow tracking of objects in the sky. This is because equatorial mounts allow the telescope’s optical axis to be pointed at the North Star, and a second angular adjustment allows celestial objects to be tracked about the Earth’s axis of rotation so objects in the sky appear at a constant location in the telescope’s field of view. All research telescopes on Earth have this design. By contrast, Dobsonian mounts are only used for reflectors. In a Dobsonian (named for its hippie monk inventor), a Newtonian reflector tube is mounted on an “altazimuth” base with two angular adjustments: pivoting about a horizontal axis and rotation about the vertical axis. Because Dobsonians can be built to pivot smoothly and support a very large mirror inexpensively, they are popular with amateur astronomers. The bases are low profile and pivot and point easily by hand, although they do not track the motion of celestial objects. Dobsonians provide a way to stably mount and move about a very large diameter mirror, since the mirror is mounted stably near the base. For this reason they are popular with amateur astronomers.
Experiments with Optical Instruments (Try to do at least one if you have time!)
The resolution limit for microscopes and telescopes is expressed in terms of the Rayleigh resolution limit, which you’ll learn about in the Physical Optics lab. Basically, it corresponds to the limit of how fine a spatial feature you can resolve given limits imposed by diffraction due to the wave-like nature of light. The easiest way to understand this is to note that even a perfect mathematical point will emit light that’s focused to a blurry spot by a lens or mirror because of interference and diffraction effects. If two near-by objects are imaged, then the angle, in terms of the distance, d, between two nearby objects and the distance L from the observing optics is d/L, for small angles. The smallest resolvable feature is then given by the Rayleigh resolution limit: .1 For a telescope, D is the diameter of the objective lens (the one closest to the object) or primary mirror. You can relate the resolution limit by noting the distance to a remote object and the smallest feature resolvable: for small angles, ~ (feature size)/(distance). For the microscope, there is an additional complication, since resolution is expressed in terms of NA or numerical aperture (see Wikipedia image at left).
where n is the index of refraction of the medium between the lens and the object, and D and f are the objective lens’s diameter and focal length. (The index of refraction appears because microscopy sometimes is done using an intervening layer of high index oil between the sample and objective, so that refraction allows the objective to capture a wider range of rays from the object. Telescopes always operate in air or vacuum, so the index of refraction is always 1.00 in these cases and can be ignored.) Each microscope objective is marked both with its magnification and its NA. This gives the effective size resolution limit of the smallest feature resolvable by a light microscope as, where the wavelength is a typical value for visible light (somewhere between 400 and 700 nm.)