Definition of light rays

Transcription

1 Geometrical optics In this section we study optical systems involving lenses and mirrors, developing an understanding o devices such as microscopes and telescopes, and biological systems such as the human eye. In the previous section we were studying physical optics, ocusing on the undamental description o electromagnetic waves, and on basic phenomena such as relection, reraction, and diraction. In geometrical optics, we apply the principles and equations o relection, reraction, and dispersion to understand optical systems. The irst ew slides bring orward the topics we need rom the previous section.

2 Recall Deinition o light rays The motion o a point on a wave ront may be indicated by a direction vector perpendicular to the ront. These direction vectors may in turn be connected into single lines or curves, called rays. to show the overall motion o a wave ront. Typically, a given problem will ocus on the behavior o either the wave ronts or the rays, and oten only one o the two will be shown. One can always be constructed rom the other by drawing perpendicular segments and connecting them.

3 Recall Electromagnetic waves in materials Electromagnetic waves travel more slowly in materials than they do in vacuum. For visible light, we are most interested in transparent materials such as glasses, plastics, liquids (especially water), and gases. The ratio o the speed o light in vacuum to the speed in the given material, is called the index o reraction, n: n c Material Index o Reraction Vacuum.0000 <--lowest optical density The table at right lists the index o reraction or a number o common (and uncommon) materials. You can see the trend that the index o reraction rises with density. I we want to calculate the speed o light in these materials, we solve or v above: Fundamentally: K K 0 m 0 K 0 0 c K c n c n Air.0003 Ice.3 Water.333 Ethyl Alcohol.36 Plexiglas.5 Crown Glass.52 Light Flint Glass.58 Dense Flint Glass.66 Zircon.923 Diamond 2.47 Rutile Gallium phosphide 3.50 <--highest optical density

4 Recall Laws o Relection The Laws o Relection: () The angle o incidence equals the angle o relection: a = r. (2) The incoming and outgoing rays, and the normal, are all in one plane.

5 Recall Law o Reraction Light traveling rom a medium o lower index o reraction to higher bends toward the normal. From higher to lower (eg. glass to air) the bending is away. n < n 2 n > n 2 n n 2 n n 2 The angles are calculated rom the Law o Reraction: n sin n2 sin 2 The critical angle, beyond which total internal relection occurs, or n > n 2 only: sin c n n 2

6 How a lens ocuses light Light rays going through glass plates emerge at their incident angles. But overall bending can be achieved by creating wedges, or prisms, o glass. A stack o prisms can bend light toward one line. A symmetric stack o prisms can bend (ocus) to a line on its symmetry plane. Smoothing the ront and back suraces into spherical sections will create a lens that can ocus incoming parallel rays to a sharp point, known as the ocal point.

7 Focal point o a converging lens As the picture shows at right, rays rom ininity (parallel rays), going through a converging lens, will pass through a ocal point. This is the way we deined ocal point on the previous slide. The second picture shows that a point source o light placed at a ocal point will cause parallel rays (plane waves) to emerge rom the other side o the lens. () Every lens has two ocal points one on each side, the same distance rom the center, and (2) the lenses pictured here are called converging lenses since they bend light toward their optical axis, causing the rays to converge. Notice that, beyond F 2 in the top picture, they are diverging.

8 The object-image relation or a thin lens As we know, a lens can ocus light to orm an image on a screen. In the diagram at right, the object is the arrow on the let, and the image is the inverted red arrow on the right. I we have been given the position and size o the object, the problem is to ind the position and size o the image. The method we are using here is called ray tracing. There are light rays leaving the top o the object in all directions. By knowing any two rays, we can ind where they cross, and that is the position where the light rom the top o the object ocuses. We know that () the ray traveling parallel to the lens axis will pass through the ocal point and, (2) the ray passing through the center o the lens will be straight (or a thin lens). We can calculate the crossing position, Q, rom similar triangles: OPQ sim OP Q : Equate y /y: s s s y s y s s y y s s s OAF 2 sim P Q F 2 : s s s y Object-image relation y s then m y s y y y s s Magniication

9 Images created by light rays passing through lenses From our everyday experience, we know that images we see in mirrors are inverted let-right ( mirror images ). Look at your let hand in a mirror some time, and you ll see a right-handed image o your let hand. This does not happen with images created by light passing through a lens. The ray tracing construction on the previous slide showed that a converging lens inverts an image in the y direction. It does the same thing in the x direction. As the picture below shows, the net result is that the image is rotated, but that its handedness is preserved. This is still true in systems with multiple lenses, since rays pass through the lenses in succession, and each lens preserves handedness.

10 Sign rules or object and image distance, and radii o curvature Sign rule or the object distance: When the object is on the same side o the relecting or reracting surace as the incoming light, the object distance s is positive; otherwise, it is negative. Sign rule or the image distance: When the image is on the same side o the relecting or reracting surace as the outgoing light, the image distance s is positive; otherwise, it is negative. Sign rule or the radius o curvature o a spherical surace: When the center o curvature C is on the same side as the outgoing light, the radius o curvature is positive; otherwise, it is negative.

11 Focal point o a diverging lens When we introduced the terminology converging lens, you may have suspected that lurking in the shadows would be a diverging lens. Well, here it is! Any lens that is thinner at the middle than at the edges is a diverging lens, which causes incoming light rays to spread (diverge) instead o ocusing (converging). Oddly enough, diverging lenses still have a ocal point. It is the point rom which parallel light seems to come when passed through the lens (upper picture). Also, light ocusing to a point can be made parallel i its ocal point coincides with that o the lens (lower picture). Fortunately, no new derivations are required. We can use the object-image relation with a diverging lens simply by making the ocal length,, negative! ( We ll discuss this again later.) Consider a converging lens,, ollowed by a diverging lens with = -.

12 The lenses at the top are all examples o converging lenses, and at the bottom, diverging lenses. Consider () how this creates convenience, (2) why certain combinations o a and b clearly have no net eect.

13 Real images and virtual images In the upper picture, light rom the object at P is passing through a converging lens and brought to a ocus at P, where it creates a real image on a screen. Diuse relection rom the screen is delivering light to the eye. In the lower picture, light rom the object at P is passing through a diverging lens. I we look through the lens we see a virtual image that appears to be at location P. This time the light comes directly rom the lens to the eye. As we shall see, however, under the right geometrical conditions, either type o lens can lead to either type o image!

14 Ray tracing & object-image relation or diverging lenses Recall the rays we constructed to analyze the converging lens, then construct analogous rays or the diverging lens. Note: we have drawn three rays in each case, but any two are enough to speciy the image location. I we repeated the derivation o the object-image relation or this lens, we would see that we get the same equation as beore as long as we speciy the ocal length o the diverging lens as a negative number. Diverging lenses are sometimes called negative lenses. s s Object-image relation y m y s s Magniication

15 MEPI Images produced with a converging lens

16 The lensmaker s equation Imagine that you have a piece o glass, with index o reraction n, that you want to orm into a lens. The problem involves determining the radii o curvature or the spherical suraces needed to achieve a certain ocal length. For this we use (without derivation) the lensmaker s equation: ( n ) R R 2 Note that radii with C on the outgoing side, such as R,, have a positive sign, whereas those on the incoming side have a negative sign. I the two radii are equal in magnitude, this simpliies to: 2( n ) R

17 The pinhole camera Image rom a cylindrical oatmeal box camera

18 The human eye Near-sightedness Far-sightedness

19 Correcting astigmatism Operator s manual Lens power, in diopters For lenses used to correct myopia or hyperopia in the eye, instead o speciying the ocal length o the corresponding diverging or converging lens, it is conventional to speciy the same inormation as the power o the lens in units o diopters. The conversion is simple: the power is the reciprocal o the ocal length in meters. Eg. =.4 m 2.5 diopters or a converging lens, with negative numbers or diverging lenses (since is negative).

20 A magniying glass A normal human eye can ocus at a nearest distance o about 25 cm. Usually we use a converging lens to create a real image on a screen. But i that screen is our eye, placed just inside the ocal point, we will see a large virtual image appearing nearby. The virtual image looks much larger than the original because it covers a larger angular range. This causes the image projected on our retinas to be larger by (approximately) the ratio o these angles, which we see as magniication, M: y / 25 cm M y / 25 cm A magniying glass cannot be used or M greater than about X3 to X4 because the lens becomes thick and introduces aberrations. For higher magniications, we need an optical microscope. (Below.)

21 Two lens system For optical systems with more than one optical element (lens, mirror), the image position and magniication are ound by stepping through the elements one at a time, inding the intermediate image at each step, until light rays have reached the inal image position. At each step, the previous image becomes the new object, and or each element in the system one applies the object-image relation or that particular element: s s

22 Optical microscope An optical microscope, otherwise known as a compound microscope has two lenses. The one nearest the object ( objective lens ) orms a magniied real image at location F, then the lens nearest the eye ( eyepiece lens ) acts as a magniying glass to magniy it urther. The total magniication M is the product o the magniications o the objective, m, and eyepiece, M 2. Once again, the near ocus limit o the eye at 25 cm is the limiting criterion. Magniication o the objective lens: m s s s Magniication o the eyepiece: M 2 25 cm 2 MEPI Combined magniication: M m M 2 (25 cm) s 2

23 Reracting telescope Like the compound microscope, the reracting telescope uses an objective lens to create a real image at a location within the tube, and this image is then viewed with an eyepiece lens to magniy the image urther. Like the magniying glass, the total magniication or a telescope is best understood in terms o angular magniication. F 2 angular range o object at ininity: tan y Eye angular range o image at F 2 : tan y 2 MEPI Magniication, rom ratio o angles: M y/ y/ 2 2

24 Images in a plane mirror To analyze lens optics we used the Law o Reraction, but or mirrors o any shape in an optical system, we use the Law o Relection. Diuse light coming rom any point on an object relects rom the mirror surace with a r. Some o the light rom this point enters our eye, orming a virtual image. For a plane mirror, the image distance behind the mirror surace is equal to the object distance in ront o the mirror. For an extended object, in a plane mirror, the virtual image is the same size as the object. So the magniication is. Discuss the rays we have constructed to ind the image position.

25 Chromatic aberration Dispersion, the variation o the index o reraction with requency in transparent materials, causes problems with the ocal properties o lenses. As we saw, in prisms, blue light bends more than red light. So the same eect must happen in lenses where one assumes that ray paths are independent o color. The irst picture below shows how lenses will have slightly dierent ocal lengths or dierent colors. This eect is called chromatic aberration and, when noticeable, the colors are seen to separate at the edges o images. The second picture shows one way to reduce this eect: an achromatic doublet, made o two lenses with dierent indices o reraction (oten ound in cameras).

26 MEPI Spherical mirrors, and their ocal points In many optical systems, concave or convex spherical mirrors may take the place o lenses. As seen in the pictures at right, parallel rays entering a concave spherical mirror will ocus at a point. And parallel rays shining on a convex spherical mirror will appear to come rom a ocal point. So concave spherical mirrors are similar in their action to converging lenses, and convex, to diverging lenses. In act, ollowing the sign rules listed in an earlier slide, we may use the object-image relation to ind images ormed by spherical mirrors. Notice: = R/2 Mirrors have several advantages over lenses. Among them: () they are based on the Law o Relection, so there is no chromatic aberration, and (2) it is practical to make them VERY large. (More later on the latter.)

27 Discuss The images ormed by concave spherical mirrors

28 The ocal length o a spherical mirror Two slides back, we saw (without any proo), that the ocal length o a spherical mirror is = R/2, where R is the radius o curvature. The text goes through a rather lengthy derivation o this result, using the two pictures below. We won t reproduce the proo here. This simple ormula is ine or doing homework problems. But it is based on the small angle approximation, and is not adequate or real optical systems. In act, most large, high quality concave or convex mirrors are not spherical sections, they are parabolic sections. We ll see why on the next slide.

29 Spherical aberration mirrors should be parabolic or perect ocus This demonstration shows that the geometry o a spherical mirror does not cause all rays to cross its axis at exactly the same point. Mathematics shows that the surace giving a perect ocus must be parabolic. (Try sketching rays entering the edge o a hemisphere to see why.) A sphere approximates a parabola at small angles, the domain where the simple ormula is acceptable. Spherical Parabolic

30 The U o A Mirror Lab Roger Angel, a Proessor o Astronomy at the University o Arizona, invented the method used to produce all the gigantic telescope mirrors being installed in observatories worldwide. The mirrors are spin-cast in a rotating oven at the Mirror Lab, below the ootball stadium seats. The oven melts the borosilicate glass as it rotates, causing the glass to low into a paraboloid shape that requires very little grinding and polishing to create the inal mirror.

31 The U o A Mirror Lab A mirror, ollowing spin-casting, being readied or removal rom the oven.

32 The U o A Mirror Lab Ater grinding, the mirrors are polished to a surace accuracy o a raction o a wavelength.

33 The U o A Mirror Lab 8.4 meter diameter mirror in place at the Large Binocular Telescope on Mount Graham Roger Angel

34 Examples