# L-2 THIN LENSES

1. PURPOSE:

To investigate the relation between focal length, object distance, and image distance for converging and diverging thin lenses; and to determine their focal lengths. The experiment also provides experience in using parallax methods to locate images in space.

2. APPARATUS:

Optical bench, three lens holders, four carriages, illuminated object box, screen, positive and negative lenses, rod or pencil for use as a reference pointer, object-image screen and holder, mirror.

3. NOTES ON THEORY:

(1) The lens equation: Notation and conventions:

(a) For convenience of discussion we assume that the light passes through the lens from left to right. Ray diagrams will follow this convention.

(b) The focal point of a lens is found by allowing a bundle of mutually parallel rays to enter the lens (i.e., from an object infinitely far from the lens). The lens alters the direction of these rays, making them emerge as a convergent or divergent bundle. The point to which they converge (or from which they diverge) is called the focal point. The diagrams illustrate this; F labels the focal points.

(c) The focal length, f, is the distance of the focal point from the lens. Its sign is determined as described below.

(d) A lens which converges a bundle of parallel rays is called a converging lens, or positive lens (its focal length is taken as positive.) The converging lens is thicker at its center than at its edge.

(e) A lens which diverges a bundle of parallel rays is called a diverging lens, or a negative lens (its focal length is taken as negative.) The diverging lens is thicker at its edge than at its center.

(f) Light rays from a point source (object) passing through a lens emerge convergent to a point or divergent from a point. In either case, that point is called the image of the source.

(g) When the emergent rays converge to a point, the image is called real.

(h) When the emergent rays diverge from a point, the image is called virtual. Such images can be seen only by looking through the lens, toward the light source. By our convention, with the rays passing through the lens from left to right, you must have your eye to the right of the lens and look through the lens to see the image which is to the left of the lens.

(i) The distance from the object to the lens is designated p.

(j) The distance from the image to the lens is designated q.

(k) When rays diverge as they enter a lens, the object (source) is real.

(l) When the rays converge as they enter a lens, the object is located at the point toward which they converge. Such an object is virtual.

(m) When the object is to the left of the lens, and the image to the right, both p and q are positive. If either one is on the other side of the lens, the distance of that one is negative.

(n) Finally, using these conventions, all thin lenses obey the lens equation:

```
1   1   1
- + - = -
p   q   f
```

The graph of the thin lens equation is an hyperbola. Its asymptotes cross the p and q axes at values equal to the focal length. Fig. 6 shows the object-image relation for a converging (positive) lens. The hyperbola has two branches, symmetric about the point (F,F).

Question 1: How does the graph for a diverging (negative) lens differ from this?

(2) Images. A luminous point emits light energy radially outward in all directions. This is described by a ray model which depicts light rays emanating radially from the point. Some of these rays pass into the lens, which alters their direction (refracts them). Rays emerging from the lens are either divergent, convergent, or parallel. If they are convergent, one may place a screen at the point of convergence, and the image is seen on the screen as a luminous point. Such a point of convergence is called a real image.

If rays emerging from a lens are divergent, one may see the image by looking back through the lens, toward the source. The image appears as a luminous point, floating in space, but located at a different place than the actual object. Such an image is called a virtual image. A virtual image can not be made visible on a screen because the image is only an apparent source of divergence. The light rays don't actually pass through the image.

A luminous surface may be thought of as an infinite array of luminous points. Lenses can form images of such surfaces. Whether the source is a point, or a surface, it is called the object. The image of a surface object is another surface, generally at a different location. A "perfect" lens forms a flat image of a flat object; patterns on the object surface are simply reduced or enlarged on the image. The ratio of image to object size is the image magnification. [We assume that the object surface lies perpendicular to the symmetry axis of the lens system.)

(3) Parallax. For objects nearer than about 20 meters, we have the ability to distinguish their relative distances. The brain does this by using the fact that each of our two eyes sees objects from a slightly different point of view. Close, or cover, your eyes alternately and observe this difference. The position of nearby objects relative to more distant objects is different for each eye. This difference, due to different viewpoints, is called parallax.

The brain processes these two images, translating the parallax differences into a sensation of depth, solidity, and relative location of objects in space. This process is called stereoscopic vision. While stereoscopic parallax gives very good relative distance information, it is nearly useless for absolute distance estimation. Our brain is easily fooled when we look at an object of unknown size against a featureless background.

We also use parallax in another way. When we move about, our eyes see our surroundings from a continually changing point of view. This provides the brain with additional depth information, and is particularly striking when we observe a scene from a moving vehicle. Even a one-eyed person can, by moving the head, observe objects from different angles to make depth and distance judgments.

A demonstration of parallax will be set up in lab, consisting of several objects of unknown size at different distances. With no size or shadow clues, you cannot tell their relative distances using only one eye keeping your head stationary. But when you are allowed to move your head back and forth you see the objects shift relative to each other. If, as you move your head to the right, object A moves to the right relative to object B, then object A is farther way from you than object B.

Objects at the same distance from the eye show no relative shift; they maintain a constant lateral separation as your head moves. Our eyes are quite good at detecting the presence or absence of relative motion, so we can use parallax as a sensitive means for locating two objects at the same distance, even when one object is a "phantom" like a virtual image. To do this, we arrange a reference object on a scale, and move it until there is no parallax shift between it and the image. We say we have then "eliminated parallax" and our reference object is at the same distance as the image. It may also be used for real images, but there are simpler and more accurate methods available for locating them.

When this method is used for locating images seen through lenses, the lateral motion of the eye must be kept very small, so the eye is always looking through the center portion of the lens. If the eye sweeps across the lens from edge to edge, the image appears to warp in shape due to the distortions and aberrations of the lens. This may easily be confused with the parallax shift.

4. APPARATUS:

An optical bench is a device for testing optical components and systems. It allows positioning optical elements, moving them along a straight line, and measuring their positions accurately. An object box, usually with internal illumination, provides a luminous pattern on a plane surface. Lens holders position and center lenses. Screens are flat surfaces which may be opaque (either painted black or white), or they may be frosted glass. Screens are used to locate real images. All of these components are attached to sliding carriages which move them along the bench and provide a reference pointer which indicates the carriage position along a metric scale.

It is important that all lenses be aligned along a common axis parallel to the bench scale. For most purposes, the plane of the object and the plane of the screen should be parallel to each other and perpendicular to the lens axis. This alignment and centering should be attended to first, for the ultimate accuracy of your results depends upon it.

NOTE: You will measure focal lengths of lenses using several independent methods. To meaningfully compare results of different methods, it is essential that the inherent uncertainties be studied. The uncertainties are due primarily to imprecision in determining when as image is sharpest, or when parallax is least. Several components of the optical system are movable, and movement of any one alters the location of the image. Be consistent by setting all except one component at fixed location, then make final adjustments by varying only the one remaining component. Thus the adjustment uncertainty can be considered to be localized entirely in this component, the errors in the other components being merely the uncertainty of reading the carriage position with respect to the optical bench scale (and probably is small enough to be negligible).

5. PROCEDURE:

(1) REAL IMAGES. Position the object near the zero end of the bench and the screen at the other end. Put a converging lens in a lens holder between them, and move it along the bench until you see a clear image on the screen. If no image can be found, you may have a lens of inconveniently long focal length for the length of your bench. For a one meter long bench, you should use a lens of focal length between 10 and 20 cm.

There are two lens locations where a distinct image appears on the screen. The image is small in one case and large in the other. In this, and all subsequent sections, record the object, lens, and image positions where the sharpest images are obtained. Note that the image is inverted up-down and right-left compared to the object.

Move the screen about 5 cm toward the object and repeat. Continue bringing the screen closer in increments, locating two images for each screen position. Eventually the screen is so close to the object that only one image is found, and if the screen is moved any closer, no real image is found.

Question 2: Explain why only one real image is found at the critical object- screen distance, and none at smaller distances. What is the mathematical relation between this distance and the lens focal length? Use the lens equation to prove this relation algebraically.

(2) VIRTUAL IMAGES. If the positive lens is brought closer to the object, the real image moves farther away, Try to move the image beyond the end of the bench, even to cast it on a wall. When the lens is one focal length from the object, the image is infinitely far away. (Don't bother to send someone with a screen to find it.) As the lens is moved still closer, less than a focal length from the object, the image becomes virtual, and may be seen by looking back through the lens, toward the object-box. Note that the image is not inverted, and it is not the same size as the object. By moving your head, observing parallax, you can judge whether the image is nearer or farther than the object-box.

Observation of virtual images is much easier if you move the lens to the rightmost end of the bench, and have that end near the edge of the table so you can comfortably place your eye near the lens.

A pencil or rod mounted upright in a bench carriage can serve as a reference-pointer to aid in locating the image by the method of parallax. Move the object closer to the lens, less than one focal distance. Accurately locate a few images in order to test whether they fit the lens equations. If the image is beyond the end of the bench, obviously you can't get an accurate location, so move the lens forward or back until you get a case where the image falls within the limits of the bench.

Place the reference-pointer about where you think the image is. Place your eye so you can see the image through the lens, and also see the pointer directly. Ignore the image of the pointer seen through the lens. Fig. 8 shows approximately what you should see. Now move your head back and forth slightly while adjusting the position of the lens until parallax is eliminated. You have now moved the image to coincide with the pointer. This is easier than moving the pointer to coincide with the image, for your arm may not be long enough to reach the pointer.

This process is easier to do than to describe. If in doubt whether you are doing it correctly, ask your instructor to demonstrate it and check your results.

THE PARALLAX METHOD FOR LOCATING IMAGES

CONVERGING LENS. In the diagram above, the relative positions of the pointer P, object box O, lens O, and the eye E are shown. The observer's eye E is to the right of the lens looking toward the left. The diagram at the right shows approximately what the observer sees when looking toward the lens.

Light from the object O passes from left to right through the lens L and to the observer's eye E. The image is virtual and to the left of the object. Its position is located by the pointer P.

The image of the object as seen through the lens is larger than the object itself. If the pointer is correctly located at the image position, the distance d between the reference pointer and the lateral position of the image should remain constant as the observer's head is moved from left to right.

DIVERGING LENS. The image of the object as seen through the lens is smaller than the object itself. It lies between the lens and the object. If the pointer is correctly located at the image position, the distance d between the reference pointer and the image should remain constant as the observer's head is moved from left to right.

You can see an image of the pointer (P') in the lens. IGNORE IT; it plays no role in any measurements you need to make.

Stereo 3D versions of these figures will be found in the appendix, for those who are comfortable with free viewing.

Note that it is not necessary that the upright bar of the image arrow coincide with the reference-pointer, only that their lateral separation remain constant as you shift your head.

(3) DIVERGING LENSES: Obtain a diverging (negative) lens. A diverging lens always produces a virtual image of real objects. Locate a few of these images by the parallax method. Fig. 9 shows approximately what you see. Be especially careful here to ignore the image of the pointer seen through the lens. Always concentrate on the pointer seen directly, just past the edge of the lens. If the pointer blocks the center of the image, use an off-center portion of the image.

(4) VIRTUAL OBJECTS: In steps (1) through (3) the object was always real, producing divergent rays. If rays are convergent as they enter the lens, we say the object is virtual. Such an object is located at the point toward which the rays converge. The lens refraction prevents them from actually reaching that point, which is why the object is called virtual.

To create such a situation we may use a converging (positive) lens to re-converge the rays coming from an object box. The lens being studied is then placed in this convergent beam. Set this up as shown in Fig. 10.

Lens B is the one being studied; use the same one you studied in parts (1) and (2). Lens A is another positive lens; it must have sufficiently long focal length that its lens-to- object distance can be made about twice the focal length of lens B. This allows sufficient range of movement for lens B so both real and virtual images may be produced.

The first lens (A) creates a real image in space. Measure its position when lens B is removed from the system. You needn't measure the positions of lens A and the object box, you only need the position of the image they produce, for it is the object for lens B, the one we are studying. Remember that p is negative now.

(5) In similar fashion, use the negative lens of part (3) with a virtual object. Use this arrangement to study two cases: (i) real final image, and (ii) virtual final image. See whether the results agree with the lens equation and with your previously determined focal length for this lens. The real image data is probably the best data for this lens, from which you can get the best determination of its focal length.

(6) LOCATING IMAGES WITH A TELESCOPE: (Instructor's option) A short working distance telescope can be used to locate real or virtual images. Simpler methods are available for locating real images, so the telescope's greatest usefulness is in locating virtual images.

The telescope is rigidly mounted on a sliding carriage and aimed parallel to the bench to receive the light emerging from the lens system. With left-to-right passage of light, the telescope must be to the right of the image. Focus the telescope on the image, then adjust its cross hairs or reticle to eliminate parallax. Now, without changing the telescope setting or position, remove the lens system which produced the image. Place a reference target on a sliding carriage and move it until you see a parallax- free image of it in the telescope. Now this reference object is in the same plane as the image you wished to locate. This takes more time, but may be more precise, than the method previously used.

(7) THE OBJECT-IMAGE SCREEN is a small device with a cut-out arrow on one side and a white screen on the other. It is illuminated from behind and used as an object for a lens. A plane mirror is placed near the lens, as shown in Fig 11, and the lens and mirror are moved until an image appears on the white screen portion of the object-image device. The object and image now lie in the same plane, and the distance from this plane to the lens is the focal length of the lens. Use this to check the focal length of your positive lens.

(8) FURTHER CHECKS OF FOCAL LENGTHS: There are several quick ways to determine approximate focal lengths.

a) Use the object-image screen (see part 6). This works only for positive lenses. b) The focal length of a positive lens may be found by aiming it at a distant scene, allowing the image of that scene to fall on a screen. The lens-screen distance is then approximately the focal length.

c) The focal length of a negative lens may be checked by aiming it at the sun. DO NOT LOOK THROUGH THE LENS AT THE SUN! Refer to Fig. 12. Cast the light emerging from the lens onto a screen S (or sheet of paper.) The lens has diameter D. Move the screen until the circle of light on it is 2D, twice the diameter of the lens. The lens-to-screen distance is then equal to the the focal length of the lens. (Notice we did not say the lens- to-screen distance is the focal length.)

d) Theory predicts that the diopter rating of two thin lenses in contact is the sum of their individual diopter ratings. Prove this by geometry. The focal length of a negative lens may therefore be found approximately by placing it very near a stronger positive lens so the combination forms a real image. Place the lenses close together, but in such a way that they don't scratch each other.

6. ANALYSIS:

Graphs of p vs. q (like Fig. 6) are pretty, but not much use for analysis. Data is generally easier to analyze if it can be transformed to a linear relation. The lens equation is of such a form that a graph of 1/p vs. 1/q is a straight line! Furthermore its intercepts are at 1/p = 1/q = 1/f, providing a neat way to use all of your best data for a particular lens to determine the focal length. Do this two ways:

(1) Directly plot the reciprocals of the data points and use a ruler to obtain the best straight line fit. Average the intercepts and take the reciprocal to obtain f. (Fig. 13)

(2) Do a least squares fit (linear regression) of the reciprocals of the data points. A computer program is available which makes this job easy. It gives the intercept and slope of the best straight line fit, from which you may obtain the other intercept, then average these to get the reciprocal of f.

Do the above for each lens. Use all of the data, for both real and virtual images.

7. QUESTIONS:

(1) How does the graph for a diverging (negative) lens differ from that of the converging lens shown in Fig. 6? Sketch the graph, accurately showing shape and its intercepts. Do the same for the reciprocal graph, like Fig. 13.

(2) Explain why only one real image is found at the critical object-screen distance, and none at smaller distances. What is the relation between this distance and the lens focal length? Use the lens equation to prove this relation algebraically.

(3*) Explain why the method described in section 8 (c) works for finding the focal length of a negative lens. Use geometry (draw a correct and complete ray diagram) to derive the relation which validates this method. What is special about the choice of a circle twice the diameter of the lens? Fig. 12 was deliberately shown in a schematic fashion so as not to `give away' too much, and no attempt was made to show all of the rays from sun to screen. You must improve on that! Do not overlook the fact that the sun is not a point source, in fact it's diameter in the sky is about 1/2 degree of arc. This fact will figure into your error analysis.

(4) What optical device has a focal length of infinity? Describe it, and explain the reasoning which led you to discover it. Likewise, describe an optical device which has a focal length of negative infinity.

(5*) Make sketches like Figs. 6 and 12 for both positive and negative lenses (four sketches total). Label the segments of each graph by kinds of objects and images (real or virtual) obtained. Note the correspondence of points between the hyperbolic and straight line graphs. Points at infinity are mapped into finite points and vice versa by the reciprocal transformation. Sometimes mathematicians say that +infinity and -infinity are not two points, but the same point"the point at infinity. Consider that when you have a real image at +infinity you also have a virtual image at -infinity. Remember, in high school you were told that "straight lines meet at infinity?" Discuss.

(6) Prove that a diverging lens always gives a virtual image of a real object and can only produce a real image when the object is virtual and inside the focal point.

(7) Prove that a converging lens always gives a real image of a virtual object, and can only produce a virtual image when the object is real and inside the focal point.

8: APPENDIX:

The 3d drawing above shows light rays from object to image. The object (O) is seen in the picture at the left rear. The lens (L) projects an image (I) onto a screen at the near right. This stereoscopic drawing has the image for the right eye in the middle. Those who use parallel (wall-eyed) viewing must use the left two drawings. Those who can do cross-eyed free viewing may use the right two drawings, or, they may use the enlarged inverted stereo pair shown below. Those who need instruction in 3d viewing may consult my 3d illusions document. The drawing is in perspective, so even if you can't view stereo, the arrangement of components should be apparent. Here's a larger version just for cross-eyed viewers.

Figures 8 and 9 are reproduced below in stereo 3D for those who are comfortable with free-viewing.

Text and diagrams © 1994, 2004 by Donald E. Simanek