This laboratory considers several concepts in geometrical optics and thin lenses: the assumptionof rectilinear propagation (light as rays) that deÞnes geometrical optics; the mapping from objectspace to image space by a thin lens (including real and virtual images, longitudinal and transversemagniÞcations, and the diﬀerential radiometry of the mapping); and the two types of cardinal pointsof lenses (focal and principal points). In particular, we will verify the thin lens formula and theformula for the focal length of two lenses combined, examine image size and Þdelity, and constructboth Galilean and Keplerian telescopes. As a side beneÞt, the lab will also demonstrate (though notquantify) the variation in image quality for on-axis and oﬀ-axis objects due to aberrations (primarilychromatic aberration, spherical aberration and coma), and demonstrate the eﬀect of stops and thelens shape factor on image quality.
Thin lenses are used in many optical systems, and understanding how they work involves many basicconcepts in geometrical optics. These include:
1. the assumption of “rectilinear propagation ”(light as “rays”),
2. the mapping from object space to image space
3. the variation in quality of images of “on-axis ”and “oﬀ-axis ”objects due to aberrations present
in the lens (primarily chromatic aberration, spherical aberration and coma),
4. the eﬀect of stops and the lens “shape factor ”on image quality.
Consider the imaging equation for a single thin lens:
where so and si are object and image distances from the lens, respectively, and f is the focal lengthof the lens. By simple rearrangement, we can express si as a function of so and f:
Note that this relationship is extremely nonlinear: the image distance is deÞnitely NOT proportionalto the object distance. This relationship will be demonstrated by imaging the white light source atvarious distances from the lens.
Ray diagram for a single thin lens: an input ray parallel to the optical axis passes through the rear
focal point, an input ray through the center of the lens emerges at the same angle, and an input ray
that passes through the front focal point emerges parallel to the optical axis.
The transverse magniÞcation of the image is given by the ratio of the image distance to the
(one can easily derive this result by analyzing similar triangles in Figure 1), where the negative
sign indicates that the image is inverted.
For combinations of two thin lenses in contact, we have seen that the focal length of the combi-
nation is given by an expression involving the individual focal lengths:
Combinations of two thin lenses can be used to make an extremely important optical device: therefracting telescope. Although it is not totally clear who invented the telescope, records in The Hague(in Holland) demonstrate that Hans Lippershey (1587-1619) applied for a patent on the device in1608. Soon after, Galileo built his Þrst telescope using one positive lens with convex surfaces andone negative lens with concave surfaces. Kepler also used a telescope, but favored an arrangementwith two convex lenses. In both cases, the lens where the light enters is called the “objective lens ”,while and the lens where the user places their eye is the “eye lens.” Both the Galilean and Kepleriandesigns are shown in Figure 2. They diﬀer in that the position of the eye lens as well as the type oflens. The Keplerian telescope forms a real intermediate image, while the Galilean does not. Notethat both designs are “afocal”; they do not form “images” because light from an object at z = −∞forms an image at z = +∞. A third lens (usually your eye lens or a camera) forms the real image.
The telescope is formed when the two lenses are separated by a distance d = f1 + f2. The systemfocal length is:
:The “power” of the system (reciprocal of the system focal length) is:
Optical layout of Galilean telescope: the “image space” (rear) focal point of the positive objective
lens coincides with the “object space” focal of the negative eye lens. Light rays that are incident
parallel to the axis emerge parallel to the axis. Parallel rays that enter the object at an angle to the
optical axis emerge parallel but at a “steeper” angle to the axis.
Optical layout of Keplerian telescope: the “image space” (rear) focal point of the positive objective
lens coincides with the “object space” focal of the positive eye lens. Light rays that are incident
parallel to the axis cross the axis at the focal point of the objective lens and emerge parallel to the
axis and on the “other” side. Parallel rays into the object at an angle relative to the optical axis
emerge parallel and at a “steeper” angle.
The Þrst part of this lab is a demonstration of images generated by a large plano-convex singlet lens,which illustrates the geometrical mapping of diﬀerent points in object space to diﬀerent locations inimage space by the action of the lens. The particular lens suﬀers from signiÞcant amounts of bothchromatic and monochromatic aberrations, and thus generates images that signiÞcantly diﬀer fromthe point-to-point character often assumed in geometrical optics.
1. The lens will be used Þrst to generate images of a string of colored Christmas lights stretched
along the optical axis. Use a white card for reßective viewing or a diﬀuser (ground glass ortissue paper) for transmissive viewing of the real images. The diﬀerent colored lights shouldbe helpful in locating the images corresponding to speciÞc locations in object space. Measurethe distances from the lights and from the images to the same point located as near as possibleto the center of the lens. Use the ensemble of measurements of the real object/image distancesto estimate the focal length of the lens and the error of this measurement. You may also try toestimate the longitudinal magniÞcation and the brightness of the images for diﬀerent locationsalong the optical axis.
2. If a string of Christmas lights is set up perpindicular to the optical axis, the locations of the
corresponding images are determined by the transverse magniÞcation of the imaging system.
Measure the distances along both axes (along and transverse to the optical axis) for boththe object and corresponding image. If time allows, you should repeat this step for severallines of lights at diﬀerent distances along the optical axis. From these measurements, you canconstruct a plot of the transverse magniÞcation at diﬀerent points in the image space. Also,note the visual quality of real images oﬀ axis; that is, whether the images of oﬀ-axis lightsdiﬀer signiÞcantly from those of on-axis lights. Sketch the diﬀerences.
3. The lens may be “stopped down” with an opaque shield to reduce the eﬀective diameter.
Examine the image quality that results when a stopped-down lens is used.
4. Cover the lens with the array of regularly spaced holes and look at the array of images at
diﬀerent locations along the optical axis. The aberrations of the lens should be apparent,particularly for images of oﬀ-axis objects. You might also try this with a laser source that hasbeen expanded with a spatial Þlter.
5. By using the white-light source, examine the eﬀects of chromatic aberration in both the lon-
gitudinal and transverse directions.
Obtain posts, post holders, and lens holders to mount three lenses on your optical rail, as well asan incandescent light source. Reserve a mount on which you can mount a white piece of paper as ascreen. Various lenses are available in lens kits. You’ll also need a lens for the CCD camera for thetelescope exercises. Although we will change the con Þguration from step to step in the lab, a basicconÞguration is shown.
1. For one of the “unknown ” lenses provided, measure seven or eight pairs of object and image
distances and the size of the real image. Then evaluate the transverse magniÞcation at eachobject distance. Use the ensemble of measurements to estimate the focal length of the lensand the error of this measurement. That is, calculate f for each pair of distances. Averagethe measurements to determine your Þnal result and estimate the standard deviation σ of themeasurements. Include as your uncertainty in the focal length σ
distance pairs you have measured. Repeat the procedure for a second “unknown lens.”
2. Next, take two positive lenses from your kit and put them in contact as closely as possible on
the optical rail. Two planar convex lenses with their ßat sides in contact will probably workbest. Use the same method as in Step 1, measure the focal length and transverse magniÞcationof the combination. The focal lengths of the individual lenses are labeled in the kits, so youcan calculate the expected value of feff . Within your uncertainty, is the formula satisÞed?
3. For one of the “very convex” lenses (short focal length, large power) in your kit, determine
image distances for two diﬀerent object distances for (a) white light, (b) blue light, and (c)red light. Use Þlters in front of the light source to change the color. What is the diﬀerence infocal length (mm) for red and blue light?
4. Place an aperture stop just in front of the lens. How does this aﬀect the diﬀerence in distance
between the blue and red focus points? Try the same experiment with a thinner lens. Howdoes the change aﬀect the diﬀerence in distance?
5. Use the “very convex” lens used in the previous step to study the image for an oﬀ-axis object
in a qualitative sense. Sketch the eﬀect on the image as the object is moved away from theoptical axis.
6. Using Figure 2 as a guide, set up a Galilean telescope. Use a lens with a “long ” positive
focal length for the objective and an “short ” negative focal length for the eye lens. Try toarrange things so your telescope points out the door of the lab. Take a look through it anddetermine if the image of a distant object is magniÞed. Since you won’t really be looking at anobject at inÞnity, you will have to adjust the position of the eye lens slightly to compensate.
Set up the CCD camera behind your telescope. The lens attached to the camera will act asthe third system lens to form the image. Focus the camera lens at “∞” (focusing on yourobject is probably suﬃcient). The camera will now focus parallel input rays. Grab an image ofsomething recognizable outside the door using the camera alone (without the telescope) — thisgives a “baseline” image to use to determine the angular magniÞcation. .(Try taping a sheet of
paper with some writing on it to the wall opposite the door.) Now insert the telescope and focusthe image by looking at the image from the CCD camera. Determine the angular magniÞcationof the telescope. Note the orientation of the image seen through the camera alone and by thecamera through the telescope. Save your images and include them with your report. Estimatethe Þeld of view of the telescope (you can compare it to the Þeld of view of the CCD camera withits lens. Also estimate the “brightness” of the image through the telescope compared to thatwith the CCD camera alone by measuring the mean gray value in the Photoshop “histogram”for the same regions of the images
7. Now repeat #6 with a diﬀerent eyelens and the same objective. Determine the angular magni-
Þcation with the new system and estimate the Þeld of view and “brightness”.
8. Repeat again using a diﬀerent objective and the original eyelens. Estimate the Þeld of view
9. Now make a Keplerian telescope using an objective with focal length at least twice that of the
eye lens. Grab the images as before and determine the angular magniÞcation. Estimate theÞeld of view and “brightness”.
10. Repeat for a diﬀerent objective lens with the same eyelens.
11. Repeat for a diﬀerent eyelens with the original objective lens.
In your lab write-up, be sure to include the following:
1. From your data in Step 1 of #2, plot si as a function of so. Estimate uncertainties in both
x - and y-coordinates and plot these as error bars. Overplot the curve that you get using theaverage value of the focal length you measured.
2. From your data in Step 2 of #2, again plot si as a function of so with error bars. This time,
overplot the curve you get using the value of f eff calculated from the equation. Do your datamatch the predicted curve?
3. Note that if we plug the right side of Equation 2 into Equation 3, we obtain
For the data obtained in steps 1 and 2, plot the measured image magniÞcation measured asa function so and then overplot the curve expected from the above equation using the focallengths you have measured. Do the data and the expected curve agree?
4. Compare the estimated magniÞcations, Þelds of view and image “brightnesses” for the diﬀerent
examples of Galilean and Keplerian telescopes.
1. Plot the graph of z2 vs. z1 for z1 in the range −10f ≤ z1 ≤ +10f.
(a) ConÞrm the (very useful) series identity:
un = u0 + u1 + u2 + · · · + uN + · · ·
1 + u + u2 + u3 + · · · if |u| < 1!
by computing a partial sum for a few values of u (e.g., u = 1 , 1 ,etc.).
(b) Use this result to Þnd a series expression for z2 in eq.(1.2) in terms of f and z1 to illustrate
the nonlinear relationship, assuming that |z1| > |f|.
(c) In the (usual) limit where z1 >> f (meaning that the object distance is much larger than
the focal length), the inÞnite series can be approximated by the sum of a few terms, where“few” is 2 − 4. Demonstrate the validity of the approximation. (This exercise, thoughnot speciÞcally optical, is intended to relate the subject to some of the mathematics thatis essential for many imaging problems.)
2. Plot the image space along the along the optical axis, i.e. locate the set of image points that
are generated by a set of object points that are equally spaced points along the optical axis.
3. Consider the case where Christmas lights are extended along the optical axis for the big lens.
Sketch the situation where a green lamp is closer to the lens and “obscuring” of a red lamp asseen by the lens. Do you expect that an image of the red lamp will be seen in image space (inother words, is the image of the red lamp “blocked”)? Explain.
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