Geometrical Optics

The term geometrical optics refers to the study of light propagation in the limit as the wavelength of light is much smaller than any of the optical components of the system (e.g., apertures, lenses, or mirrors). Another simplifying assumption is that each medium through which the light travels (e.g., air, water, glass) is homogeneous and that all changes between media are abrupt at the interfaces. A consequence of these assumptions is that light travels in straight lines through each medium and that changes in the direction the light travels occur only at the interfaces between media. The direction light travels is conveniently described by the term rays.

In this lab, we will do the following:

  • study image production by mirrors and lenses;
  • apply Snell’s law;
  • test the relationship among surface curvature and focal length;
  • test the relationship among focal length, objective and image distances; and
  • construct a telescope and determine its magnification.

Introduction


Law of reflection

When light strikes a mirror and reflects from its surface, the angle of reflection is equal to the angle of incidence (with both angles being measured from the normal to the mirror surface). Also, the incident ray, the reflected ray, and the normal to the surface all lie in the same plane.

Law of refraction (Snell's law)

When light passes from one transparent medium into another, in general the light will change speed at the interface between the two media. This change in speed is accompanied by a change in direction or refraction of the light. The angle through which the light changes direction depends on the angle of incidence at which the light strikes the surface and a characteristic of the media at the interface. This characteristic is known as the index of refraction, $n$, which is defined as

$n = \dfrac{c_{\textrm{vacuum}}}{c_{\textrm{medium}}}$ (1)

where $c_{\textrm{vacuum}}$ is the speed of light in a vacuum and $c_{\textrm{medium}}$ is the speed of light in the medium.

The relationship between the direction of travel of light and the indices of refraction of the media is known as Snell's law,

$n_1\sin\theta_1 = n_2\sin\theta_2$, (2)

where the angles $\theta_1$ and $\theta_2$ are measured between the light rays and the normal to the surface in each medium.

Focal point, focal length, and images

When reflecting or refracting materials like mirrors or clear glass are shaped in special ways, they can be used to re-direct light to form images. If the reflecting or refracting surfaces are spherical, this geometry (together with the laws of reflection and refraction) give rise to the ray diagrams illustrated in Fig 1. The lenses shown in Fig 1 are considered thin lenses for simplicity and it is assumed that all of the refraction takes place at the center of the lenses.

Figure 1: Parallel light rays entering from the left, striking lenses and mirrors, and continuing on. The focal points (fp) and focal lengths (f) are shown.

Note that the double convex lens and the concave mirror of Figs. 1(a) and 1(b) redirect the light so that the light rays converge at the focal points. Images formed in this way are called real images, since light actually passes through them. Real images can be projected onto a screen. 

Note also that the double concave lens and the convex mirror of Figs. 1(c) and 1(d) cause the rays to diverge. Images formed this way must be inferred by extending the light rays back to where they appear to have come from as the dashed lines show. Since no light actually passes through these images they are referred to as virtual images and they cannot be projected onto a screen.

The magnification of a lens or mirror is defined as the ratio of the image diameter to the object diameter.

A consequence of the laws of reflection and refraction and the spherical shape of the mirror or lens surface is the relationship

$\dfrac{1}{f} = \dfrac{1}{OD} + \dfrac{1}{ID}$, (3)

where $f$ is the focal length, $OD$ is the object distance (the distance from the object to the lens or mirror), and $ID$ is the image distance (the distance from the lens or mirror to the image). It is remarkable that Eq. (3) applies both to mirrors and lenses with spherical surfaces even though the physics of refraction and reflection is quite different.

Parallax

When nearby objects are viewed from different vantage points, they seem to shift positions relative to more distant objects. This phenomenon is called parallax and is illustrated in Fig. 2.

Figure 2: Parallax between near and distant objects.

Conversely if two objects are observed from different vantage points and they do not appear to change relative positions, then the objects must be at the same position. (See Fig. 3.)

Figure 3: No parallax between objects in the same plane

We shall make use of parallax to judge when two objects are at the same position in several parts of this experiment.

Experimental procedure


NOTE: There are two distinct experiment stations: the optical rail (one setup for each group) and the pins and ray-tracing station (only a few in each room). These stations may be completed in either order.

Lab notebook template

One member of the group should click on the link below to start your group lab notebook. (You may be asked to log into your UChicago Google account if you are not already logged in.) Make sure to share the document with everyone in the group (click the “Share” button in the top right corner of the screen) so each member has access to the notebook after you leave lab.

Station 1: Pins and ray-tracing

Plane mirror

Find the image of a pin by the method of parallax. To do so, place a piece of paper on the rubberized board and set up the apparatus shown in Fig. 4 on top of the paper.

Figure 4: Plane mirror with object pin and its image (parallax method)

Stick a pin (the object pin) into the rubberized board a few centimeters in front of the mirror. While looking at the image of this pin in the mirror, try to place a second pin on the board behind the mirror so that the second pin and the image of the object pin stay together while you move your head from side to side. Once you have found the position of the image stick the second pin in securely.

What kind of image has been formed in the mirror (real or virtual)? 

How does the distance from the object pin to the rear surface of the mirror (where the reflection takes place) compare with the distance from the image to the rear surface of the mirror?

Snell's Law

Find the index of refraction, $n$, of water by setting up the apparatus shown in Fig. 5.

Figure 5: Testing Snell’s law with a plastic box and water

Place a piece of paper on a rubberized board and a water box on the paper. Trace the edges of the water box onto the paper. Sight through the clear edges of the box. Stick pins 1, 2, 3 and 4 into the board so that all four pins appear to lie in a straight line. (Note that you will be looking at pins 1 and 2 through the water while viewing pins 3 and 4 directly.)

Remove the box and draw a line from pin 1 through pin 2 and stopping at the edge of the box. Do the same for pins 3 and 4. 

What does the line from pin 1 to pin 2 represent? What does the line from pin 3 to pin 4 represent?
What path did the light take through the water? Draw that line on your paper, too.

Using a protractor and straight edge, construct the normals to the water box where lines 1,2 and 3,4 intersect the refracting surfaces.

Measure the angles of incidence and refraction at both interfaces. Using Snell's law, calculate the index of refraction of water. (Assume the index of refraction of air is 1.0).

Index of refraction of water by the apparent depth of an object pin

Here we shall use the same water box and rubberized board as in Fig. 5. This time, however, place a single object pin at the rear of the water tray as shown in Fig. 6.

Figure 6: Index of refraction of water from the apparent depth of a pin

Next, place two pins at the front surface of the water tray near the center (a few centimeters apart). Look through the water at the object pin from two different points of view as in Fig. 6. Place additional pins as shown so that they appear to be in-line with the object pin and with the pin you placed at the front of the tray. The pins thus added actually are in line with the virtual image of the object pin. Trace the water box onto the paper and remove the box.  Use the ray tracing pins to find the position of the virtual image of the object pin. Trace the rays and use Snell’s law to find the index of refraction of water.

At small angles, the index of refraction inside the water is $n = h/h^{\prime}$. Determine the index of refraction of the water in this way.
Where does this approximation come from?

In Fig. 6a, we we draw a line normal to the edges of the tray that passes through the object pin in order to form two right triangles – one with height $h$ (with a hypotenuse formed by the refracted ray) and one with height $h^{\prime}$ (with a hypotenuse formed by the unrefracted (virtual) ray).

Figure 6a: The two right triangles formed by the refracted and unrefracted (virtual) rays.

Using geometry, we have $sin\theta = x/\sqrt{x^2 + h^2}$ and $sin\theta^{\prime} = x/\sqrt{x^2 + (h^{\prime})^2}$. In the limit that both angles are small, $h \gg x$ and $h^{\prime} \gg x$, so this simplifies to $sin\theta \approx x/h$ and $sin\theta^{\prime} \approx x/h^{\prime}$.

From Snell's law, we therefore have

$n^{\prime}\sin\theta^{\prime} = n\sin\theta$

which becomes (in the limit of small angles)

$n^{\prime}x/h^\prime = nx/h$

or

$n = h/h^{\prime}$.

Station 2: Optical rail

Convex lens and focal length

Using an optical rail, set up the apparatus as shown in Fig. 7.

Figure 7: Optical bench with light source, lens, and projection screen

Use the thinner, red-edged lens here. Move the lens and plastic screen along the optical rail until a sharp image is formed on the screen. Note that there is an infinite number of such configurations which will produce images.

Measure the object distance (the distance from the light source to the lens), and the image distance (the distance from the lens to the screen). 

Using Eq. (3), calculate the focal length of the lens. (Watch the sign conventions for the object and image distances!)

Is the image upright or inverted? Is the image real or virtual? 

Measure the magnification for this configuration. How is the magnification related to the image and object distances?

A more direct measurement of the focal length may be made by observing the image formed by a distant object. Use a distant light source to form an image on some convenient surface.

Measure the image distance for this special case. Apply Eq. (3) to this limiting case of large object distance to find the focal length of the convex lens.

Auto-collimation

Recall that light emanating from a point in the focal plane of a convex lens and then passing through the lens will emerge in a parallel bundle of rays. It follows that if we place a plane mirror after the lens so as to reflect that parallel bundle back through the lens, the light will be brought to a focus in the focal plane once more.

Set up the optical bench as shown in Fig. 8.

Figure 8: Auto-collimation apparatus

For this measurement, use the thicker, shorter focal length lens with the blue edge. The white metal mask has three small holes which will serve as point light sources. Place a mirror a few centimeters to the right of the lens. While moving the lens back and forth along the optical rail, look for the image of the point light sources on the white mask surface.

The ray diagram in Fig. 9 shows a technique of ray tracing to locate the image.

Figure 9: Auto-collimation ray tracing to locate the image.

Ray 1, drawn parallel to the optic axis, is refracted as it passes through the lens. Ray 2 passes through the focal point on the right side of the lens and strikes the mirror at angle $\theta$. Ray 3 is reflected from the mirror at angle $\theta$ and returns to the lens. Imaginary Ray 4 is drawn passing through the focal point, parallel to Ray 3. Imaginary Ray 5 is drawn parallel to the optic axis. Since Rays 3 and 4 are parallel to each other, they must intersect in the focal plane. Thus, we can draw Ray 6, locating the image.

What is the focal length of the lens? 

Lensmakers' formula

A consequence of Snell's law is that there is a predictable relationship among the focal length of a lens, the index of refraction of the glass, and the radii of curvature of the lens surfaces. This relationship is called the lensmakers' formula,

$\dfrac{1}{f} = (n-1)\left(\dfrac{1}{R_1}+\dfrac{1}{R_2}\right)$. (4)
Figure 10: Using a spherometer to measure the radius of curvature

Use the spherometer to find the radii of curvature of the plano-convex lens with the blue edge. (See Fig. 10.). To do so, first place the spherometer on a flat glass plate and adjust the micrometer so that all four points contact the glass surface. Record the micrometer reading. Now place the spherometer onto one surface of the lens and re-adjust the micrometer until all four points contact the lens surface. Using the geometry of Fig. 10, one may derive the relationship

$R = \dfrac{S^2}{6a} +\dfrac{a}{2}$, (5)

where $R$ is the radius of curvature, $S$ is the distance between any two of the spherometer legs which form an equilateral triangle, and $a$ is the elevation of the spherometer screw necessary to make contact with the lens surface at all four points. Repeat the measurement for the other side of the blue-edged lens. The index of refraction of the blue-edged lens has been independently measured and is $n = 1.53\pm0.02$, which is typical for many glasses.

Calculate the focal length of the lens using the lensmakers' formula. 

Measure the focal length of the blue-edged lens by forming an image of a distant light source.

Are these focal lengths consistent, within uncertainties?

Telescope

Measure the focal length of the thinner, red-edged lens by forming an image of a distant light source. To do so, remove the lens from its holder and hold the lens near a wall so that the light from a distant window or other light source forms an image on the wall. The red lens should have a longer focal length than the blue lens which you used earlier. This shorter focal length lens will be the eyepiece of your telescope.

Figure 11: (top) Telescope apparatus, (bottom) telescope magnification ray diagram

Arrange the two lenses, light source and screen as shown in the top portion of Fig. 11. First use the longer focal length lens (objective lens) to form the sharpest possible image of the light source on the ground plastic screen. Then place the shorter focal length convex lens (eyepiece) to act as a magnifier of the image on the screen. The position of the eyepiece will be a bit subjective, since your eye will try to accommodate to form the magnified image on your retina.

The magnification ray diagram is shown in the bottom portion of Fig. 11. In this diagram, $h$ is the object height, $h^{\prime}$ is the height of the image formed by the objective lens, $s^{\prime}$ is the distance from the image to the eyepiece, and $OF$ is the distance from the object to the focal point of the eyepiece. The dashed lines form an angle  at the eyepiece and represent the angular size of the un-magnified object. The magnification is defined to be $M \equiv \theta\,' /\theta$.

Measure the object ($h$) and image ($h^{\prime}$) sizes, and the distances from the object to where it appears in focus($OF$) and from the screen to the eyepiece($s'$) as defined in Fig. 11.

Calculate the angles $\theta$ and $\theta^{\prime}$ and thus the expected magnification.

While looking through the eyepiece, remove the plastic screen.

What do you see now? What was the function of the ground plastic screen? 

Note that the crosshairs of a telescope would be placed at the location of the plastic screen.

Next, try to estimate the observed magnification. To do so, look at the light source directly with one eye and through the telescope with the other eye. With practice you can superimpose the two images and compare their relative sizes and estimate the magnification.

What magnification do you estimate using this method? Compare this observed magnification with the geometric analysis done above.

Is the image formed by the telescope upright or inverted? Is the image real or virtual?

Submit your lab notebook

Make sure to submit your lab notebook by the end of the period. Download a copy of your notebook in PDF format and upload it to the appropriate spot on Canvas. Only one member of the group needs to submit to Canvas, but make sure everyone's name is on the document!

When you're finished, don't forget to log out of both Google and Canvas, and to close all browser windows before leaving!

Post-lab assignment


Write your conclusion in a new document (not your lab notebook) and submit that as a PDF to the appropriate assignment on Canvas when you are done. You should write your conclusion by yourself (not in a group), though you are welcome to talk to your group mates to ask questions or to discuss.

The conclusion is your interpretation and discussion of your data. In general, you should ask yourself the following questions as you write (though not every question will be appropriate to every experiment):

  • What do your data tell you?
  • How do your data match the model (or models) you were comparing against, or to your expectations in general? (Sometimes this means using the $t^{\prime}$ test, but other times it means making qualitative comparisons.)
  • Were you able to estimate uncertainties well, or do you see room to make changes or improvements in the technique?
  • Do your results lead to new questions?
  • Can you think of other ways to extend or improve the experiment?

In about one or two paragraphs, draw conclusions from the data you collected today. Address both the qualitative and quantitative aspects of the experiment and feel free to use plots, tables or anything else from your notebook to support your words. Don't include throw-away statements like “Looks good” or “Agrees pretty well”; instead, try to be precise.

REMINDER: Your post-lab assignment is due 24 hours after your lab concludes. Submit a single PDF on Canvas.