7/06 Geometric Optics GEOMETRIC OPTICS JUPITER THROUGH A REPLICA OF GALILEO'S TELESCOPE

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1 GEOMETRIC OPTICS JUPITER THROUGH A REPLICA OF GALILEO'S TELESCOPE The four Galilean moons of Jupiter (from left: Europa, Callisto, Io and Ganymede): a 6 sec exposure taken on Nov. 28, 2002 at approximately 2:0 am PST. For the moons to be visible, the disk of Jupiter must be highly overexposed bringing out the halo of violet light that results from chromatic aberration. A normal exposure of Jupiter's disk has been pasted over the original overexposed image to indicate the true scale. Jupiter has a large number of satellites. Of these, four are comparable to the Earth's Moon in size; the rest are orders of magnitude smaller. When Jupiter is at opposition and closest to the Earth, the stellar magnitude of its four large moons is between 5 and 6. [] This means that, were it not for the shielding brightness of Jupiter, these bodies would be visible with the naked eye. The aperture of the telescope used by Galileo in 60 and its magnification thus brought these four "Galilean" satellites within his grasp. But first Galileo had to make adjustments to the instruments. When viewing bodies that are very bright and very small, the optical defects of the telescope can be crippling. By trial and error Galileo learned to stop down the aperture of his instrument until he could begin to make useful observations. At the end of 609, as he was finishing his series of observations of the Moon, Jupiter was at opposition and the brightest object in the evening sky (not counting the Moon). When he had made the new adjustment to his instrument, he turned his attention to Jupiter. On 7 January 60 he observed the planet and saw what he thought were three fixed stars near it, strung out on a line through the planet. This formation caught his attention, and he returned to it the following evening. Galileo's expectation was that Jupiter, which was then in its retrograde loop, [2] would have moved from east to west and had left the three little stars behind. Instead, he saw all

2 three stars to the west of Jupiter. It appeared as though Jupiter had not moved to the west but rather to the east. This was an anomaly, and Galileo returned to this formation again and again. Over the next week he found out several things. First, the little stars never left Jupiter; they appeared to be carried along with the planet. Second, as they were carried along, they changed their position with respect to each other and Jupiter. Third, there were not three but four of these little stars. By the 5th of January he had figured it out: these were not fixed stars but rather planetary bodies that revolved around Jupiter. Jupiter had four moons. His book, Sidereus Nuncius, in which his discovery was described, came off the press in Venice in the middle of March 60 and made Galileo famous. About this lab Galileo's discovery relied on the transparency of glass and on the general phenomenon of refraction (bending) when waves of any type pass a boundary between two materials having different propagation speeds. Most materials exhibit wave dispersion different propagation velocities for different wavelengths. This gives rise to prismatic color separation and also to chromatic aberration in focusing devices variation of image position with color (wavelength). A shaped glass surface can be used to form either real and virtual images. A metalized, reflecting surface is little penetrated by electromagnetic waves, which are reflected. Thus, images are formed only on the incident side of the mirror. But mirrors have the great advantage that they are non-dispersive, since only the incident medium is involved. Their focusing properties are purely geometric, independent of wavelength. And only one surface is needed. For this reason, modern astronomy with large telescopes (for light collection from faint sources) is based on mirrors, sometimes of many smaller facets independently adjustable by piezoelectric drives to compensate for air turbulence. Other examples of mirror telescopes include automobile headlights, satellite dish receivers and radio telescopes. Smaller telescopes may be glass refractors. And, of course, everyday objects such as eyeglasses, binoculars, magnifying glasses etc. depend on refraction, not reflection Glass materials have come into their own in modern communications. Optical fibers, key to modern data communication, depend on total internal reflection to confine laser light pulses and reduce signal weakening to tolerable level over long distances. The material below involves a number if illustrations of the behavior of wave front rays in formation of an image. Appendices contain more detail and derivations of predictive formulae, some numerical examples, and discussion of experimental procedure. References Serway & Beichner, v2, 5 th ; Chapter 36 Cutnell & Johnson, 6 th : Chapters 25 and 26

3 Apparatus: Optical bench, lenses, concave mirror, bent clipboard mount, telescope eye chart, light source with projection object, special bar or paper target, meter stick, desk lamp. Figure The red tick marks are at distances f and 2f from the lens center. s is the distance to the object and s' is the distance to the inverted, reduced real image (the rays really get there). But this lens can also produce an enlarged, inverted real image if f < s < 2f, as well as an erect virtual image. The geometric construction above, involving principal rays, leads to the thin lens algebraic formula Equation and also the the magnification expression Equation 2 s + s' = f m= - s' s..

4 Object-Image Relations: In geometrical optics we can get a good intuitive feeling for the behavior of lenses using these basic simplified assumptions: () All rays incident on a converging lens parallel to its axis are refracted in such a way that they cross the axis at a common point called the focal point or focus of the lens. (Departures are referred to as aberrations.) The distance from the lens center to the focus is called the focal length, f. Each lens has two focal points, at equal distance on opposite sides. (2) After refraction by the lens, all non-parallel rays coming from a single object point will either converge to form a real image point (as in a camera) or diverge so that they appear as if they come from a "virtual" image point (as in a magnifying glass). (3) If the distance from the object to the lens is called s and the distance from the lens to the image s' (where s' is positive for a real image and negative for a virtual image) then the "thin lens" equation (derived from the geometrical construction discussed above) holds:

5 Figure 2 Imaging by a convex lens. Refraction at the lens reduces the angle of the rays, except for central ray. The image position can be determined using only the principal rays, as shown. A convex lens can produce an erect, enlarged virtual (green) image (the rays don't really come from their apparent source)of an object (blue), if the object lies inside the focal point of the lens (s<f). Figure 3 A concave lens produces an erect, reduced virtual image. Refraction at the lens increases the angle of the rays (except for central ray), in contrast to the action of a convex lens.

6 Figure 4 A commercial parabolic dish antenna for satellite reception. The detector is located at the focus of the parabola. Figure 5 The concave mirror produces an inverted, magnified real image. It can also produce an inverted, reduced real image.

7 Figure 6 The convex mirror produces an erect, reduced virtual image (virtual because the rays don't really get there - they don't penetrate the mirror. Instead they are reflected.) Appendix : Refractive geometrical optics The relationship between object, real image and focal length for a converging lens is again shown below. Two rays are drawn from the object to the lens. One ray is incident parallel to the axis and is refracted toward the focus, f. The other ray passes through the center of the lens, and will not be deviated because the surfaces of a "thin" lens are essentially parallel at the center. Using this successively you can construct any optical system formed of thin lenses.

8 Real Real Figure 7 Principal ray geometric derivation of the thin lens formula

9 From the figure, the ratio of the distance of the image from the axis to the distance of the object from the axis (called the lateral magnification, m) is equal to -(s'/s). The magnification is positive for an upright image and negative for an inverted image. The thin-lens equation can be used to show that Equation 3 m = s ' s = ( s ' f f ) A special case of the thin lens equation occurs when the object distance, s, is very large (s >> s') compared to the image distance s'. Under these circumstances the (/s) term will be small enough to be neglected and the thin lens equation reduces to Equation 4 s + s' approches s' = f or s' = f when s >> s'.. We will use this special case to estimate the focal length of the lenses used in the experiment. In the table below, we have summarized the sign convention for m and s '. Sign m, magnification s', image distance + upright image real image - upside down image virtual image Example : The converging lens and the simple magnifier Case a): s > f. Suppose f = (+) 20 centimeters for the converging lens of figure above. Then, if the object distance s = + 30 cm, the thin lens formula is s + s' = f s' = + 20 s' = = 60 and s' = + 60 cm.

10 where the + sign for s' means the image is real and on the other side of the lens from the object. The lateral magnification is m = ( height of image / height of object is: m = h image. In h object this case it is equal to - s' s = - 60 = - 2, indicating an enlarged image. The minus sign 30 for m means that the image is inverted. For a real image it is possible to place a screen at the focal plane and directly measure the height of the image, but not so for a virtual image. Case b): s < f. Suppose the object distance s = + 5 cm for the converging lens with f = 20 cm. Then the thin lens equation becomes s ' += 20 s' = 20-5 = - 60 and s' = 60 cm. The - sign for s ' means that the image is virtual (on the same side of the lens as the object, impossible to view or measure on a screen). The magnification is m = - s' s [-60] = = + 3. The + sign for m here means that [+ 20] the image is upright. We see that a converging lens can give either real or virtual images, depending on the location of the object relative to the focal point. Example 2: The diverging lens

11 Figure 8 Image formation by a diverging lens

12 Suppose the focal length of a diverging lens is f = - 20 cm. If the object is at s = + 30 cm, solving the lens equation gives s' = -2, the - sign again indicating a virtual image: s' = -20 The magnification is s' = = = - 2 m = - s' s = = and s' = 2 so the image is reduced and erect (see figure 2). A diverging lens, by itself, can produce only virtual images. Appendix 2: Reflective optics

13 Figure 9 Image formation by a concave mirror Focal length = R / 2. Once you understand refractive optics, reflective optics is easy. In this case both object and image distances (s and s', respectively) are measured on the same side of the reflective surface. The thin lens formula and magnification expressions for lenses can be applied to the spherical mirror, with f = R 2 Consider the concave, spherical mirror in Fig. 3. Note that the radius of curvature, R, is R not the focal length, which is. A point source of light placed at the center of 2 curvature would have the light focused back on itself, whereas it would produce parallel rays if placed at the focal point. (Small angle rays ( paraxial rays focus nearly to a point, whereas larger angle rays exhibit increasing spherical abberation. Compound lenses attempt to compensate by use of more than one lens element.). Appendix 3: Procedure for lenses Look at the aluminum optical bench on the lab table. There should be a light source, a small projection screen, and a lens mounted on black lens holders, all on the bench. Other lenses, a concave mirror, a meter stick and a special ruled bar target (fits in the black holders) should be on the table too.

14 Figure 0 Optical bench with lamp and object, lens and image screen. Positions are adjustable.

15 Figure Lamp and object. Figure 2 Optical bench, showing image formation by a simple lens. The two bright spots are just reflections.

16 . For lenses A, B, and C estimate their focal lengths by projecting an overhead light (try looking for the pattern of the light louvres) on the floor or table (adjust the lens-floor distance to find the sharpest image). Can you do this with lens D? Why? 2. For lenses A, B and C study and measure the relation between the image and object distances. Switch on the light source to shine light onto lens which is then focused on the projection screen. Adjust the object (your object will be the 2cm by cm figure taped onto the light source, shown below) and image (screen) distance by moving the lens and/or screen back and forth until you obtain a sharp image on the screen. Do this for three cases: ) s > s ', 2) s < s ', 3) s = s '. (Lens C has a long focal length. Its image is usually beyond the optical bench. Turn the paper clipboard sideways and move it until a sharp, real image is obtained. Then measure s and s' as usual.) 3. Use lens B to determine the negative focal length of lens D. See the Negative Lens section of the report form for the detailed procedure. Appendix 4: Procedure for Mirror Make the same determinations as in Part 2 given above, but use a converging mirror. The special "bar" target must be inserted between the lamp (object) and the mirror. The image is very small and distorts easily if your alignment is poor. See the converging mirror section of the report for the detailed procedure. Appendix 5: Procedure for telescopes Refracting telescopes

17 Astronomical Telescope Objective Eyepiece Eye sees a large inverted image. f o f e Figure 3 Astronomical telescope Two converging lenses are placed at a separation ( f + f 2 ). f is the focal length of the lens which collects light from the distant object (objective lens) and produces a real, inverted image, which is the object for lens 2. The function of lens 2 (ocular or eyepiece) is to produce a magnified virtual image (still inverted) for convenient viewing. The eyepiece acts like a simple magnifying lens. The theoretical magnification is: f objective f eyepiece. A converging ("positive') objective lens ( f ) and a diverging ("negative") ( f 2 ) ocular are used to make a short telescope (Figure 6, Terrestrial telescope) with an upright image. The lenses are spaced a distance approximately ( f + f 2 ) apart (note that. f 2 is negative since the lens is concave) and give an angular magnification of - f f 2

18 Terrestrial or Galilean Telescope Objective Eyepiece Upright Image Intermediate Image f o -f e Figure 4 Terrestrial telescope These telescopes are compound instruments consisting of an objective and an eyepiece, which produces a virtual image of its object (the image formed by the objective). The eyepiece can be either converging s < f or diverging (always produces a virtual image). Your eye does the rest. 5. Construct an Astronomical and a Terrestrial (Galilean) telescope. For the astronomical and Galilean telescopes start with lens separation of ( f objective + f eyepiece ) as in the diagrams, but remember that, in the Galilean telescope, the eyepiece is a diverging lens (this gives an erect image, in contrast to the astronomical telescope) and its f is negative. ("Right side up" or "upside down" is not meaningful for a star or galaxy.) For the astronomical telescope use lens C as the objective and lens A as the ocular. For the Galilean telescope use lens C as the objective and lens D as ocular. Adjust your telescope to focus on a distant eye chart or other object. Record the lens separation and give the ratio to the theoretical separation ( f objective + f eyepiece ) for an object at infinity. (It is helpful if the eye chart is illuminated. View a wall mounted eye chart or make a reciprocal arrangement with students across the room, whose desk you can see. Put charts in the paper clipboards on each desk and illuminate with the desk lamp.) How to Estimate Magnification: Measuring magnification is not so easy; you should try to use resolution rather than height estimation. Using the wall chart, note the smallest figure or letter that you can distinguish with the naked eye, Then do the same using your telescope, and take the ratio of actual heights.