In this lab, we will investigate the properties of lenses and lens system. We will begin with a single lenses, then make more complicated instruments with multi-lens systems.^{Hoveroverthese!}

• 2 Optical Rails, long (~1.2 meters) and short
• Lamp assembly with object (a number "4")
• Aperture assembly
• Lenses:
  • 10cm Converging Lens
  • 20cm Converging Lens
  • 15cm Diverging Lens
• Screen
• Record data in this Google Sheets data table

For students: this background section is unusually long in part because this is an undercovered subject in a typical physics curriculum, and some TAs may find the more thorough overview helpful. Feel free to skim more than usual if you feel you understand lenses fairly well.

Basics of Lenses

There are two basic components to build optical instruments: mirrors and lenses. In this lab, we will only be dealing with lenses.¹

A converging lens takes incoming parallel light (resulting from an object "at infinity") and focuses it at some pont, known as the focal point. The distance to this focal point is the focal length, and our sign convention is that this is positive. Geometrically, a converging lens is "convex" (bulging outwards in the middle).

A diverging lens takes incoming parallel light and splits it apart, as though the light came from a fixed point behind the lens. That point is the focal point of the diverging lens, and negative the distance from the lens to the focal point is the focal length, \(f\). Geometrically, a diverging lens is "concave" (wider on the edges than in the middle).

In general, understanding how lenses make images requires a whole bunch of complicated geometry, because you have to consider the refraction of the light as it comes in, the motion of the light inside the lens, and the refraction of the light when it leaves.

A customary approximation is to assume the lens is very thin, and therefore to neglect that middle part, treating the lens as "plane-like" (except insofar as the changed angle from refraction is concerned).¹

In this case, if you place an object on one side of a parabolic lens,² there will somewhere be an "image" formed. This is where the object appears to be if you look at it through the lens.

The image can appear either in front of or behind the lens. If the image appears in front of the lens (i.e., opposite side of the lens from the object), then the image is real - if you put a screen at that location, you will actually see the image there, because the light beams pass through that point. If the image is behind the lens (i.e., same side as the object), then the image is virtual - although the image appears to be there, no light rays actually pass through that point, and hence you cannot project such an image.

The distance between the lens and the object defines the object distance \(d_o\), and the distance between the lens and the image defines the image distance \(d_i\). Similarly, the height of the object we call \(h_0\), and the height of the object we call \(h_i\).

These distances and heights can be positive or negative; we choose the signs on these quantities according to the following conventions:

• Object distance, \(d_o\):
  • This is positive if the object is "behind" the lens (same side as the light comes from), and negative if it is "in front" of the lens. [The latter is uncommon, but can "effectively" occur in composite lens systems, as you will see in this lab.]
  • Why this makes sense: The "normal" position for the object is what we call "positive."
• Image distance, \(d_i\):
  • This is positive if the image is "in front of" the lens (opposite side of the lens from the light source), and hence a real image. It is negative if it is "behind" the lens, hence virtual.
  • Why this makes sense: A real image (that we can actually project) is what we call "positive."
• Focal length, \(f\):
  • This is positive if the focal point is in front of the lens (i.e., if the lens is convex/converging), and negative if the focal point is behind the lens (i.e., if the lens is concave/diverging).
  • Why this makes sense: A real focal point (where light actually focuses) is what we call "positive."
• Object height, \(h_o\):
  • This is basically always positive, by definition.³
  • Why this makes sense: Usually in pictures our "object" is an upright arrow, and upright is naturally "positive."
• Image height, \(h_i\):
  • This is positive if the orientation is the same as the object (the image is "upright"), and negative if the orientation is flipped (the image is "inverted").
  • Why this makes sense: Hopefully obvious.

Be aware that several of these conventions change for mirrors.⁴ We won't be dealing with curved mirrors in the lab, but you will probably see them in the course. (Those conventions also make sense if you think about them, though.)

Properties of Single-Lens Images

Under the thin lens approximation that we made, one can derive (with ray-tracing diagrams) the thin lens equation:

$$\frac{1}{d_o}+\frac{1}{d_i}=\frac{1}{f}$$

This equation only works if \(d_0\) and \(d_i\) use the sign convention defined above. (It also works with mirrors, if you use the mirror sign conventions.)

The image from the lens may be magnified (larger) or demagnified (smaller), and may be upright or inverted. The magnification for a projected image is defined as the ratio of image size \(h_i\) to object size \(h_o\) (again, with signs as defined above):

$$m=\frac{h_i}{h_o}$$

Some geometry also allows us to calculate the magnification in terms of the distances (again, noting signs):

$$m=-\frac{d_i}{d_o}$$

Ray Tracing Diagram

The basic way one understands lenses (and lens systems) is with ray-tracing diagrams.

One imagines a fixed point the object emitting rays of light in all directions, and traces them to their destination. We will deduce the location of the image from the ultimate location of these rays.

Three of the emitted light rays have easy-to-determine behaviors, and we typically understand the image as a whole via these three rays:

• One light ray passes from the point on the object, straight through the center of the lens, and onwards. (This happens regardless of converging/diverging.)
• One light ray comes from the object and is emitted horizontally. The behavior on the non-object side of the lens depends on the type of lens:
  • For a converging lens, this ray goes through the focal point (on the non-object side).
  • For a diverging lens, this acts as though it is emitted by the focal point on the object side.
• The third light ray is the "opposite" of the second one: it is the light ray that ends up horizontal. What happens on the object side of the lens (and hence where this ray comes out of the lens) depends again on the type of lens:
  • For a converging lens, this ray either passes through or comes from the focal point on the side of the object. (If the object is outside the focal length, the ray passes through that focal point. If the object is inside, the ray acts as though it is emitted by that point.)
  • For a diverging lens, this goes towards the focal point on the image side.

If all three outgoing light rays intersect at some point, then (for an ideal lens) all light rays will intersect at that point, and this point is where a focused (real!) image will form.

If the rays do not intersect, but are not parallel, you can still find a place where the "lines" intersect, if you "project back" to where it seems like the light came from. This is where the "virtual image" appears to be.

If the outgoing light rays are all parallel, then the object appears (to the eye) to be infinitely far away, like a star in the night sky. (This is best for viewing, because looking at closer objects creates more eye strain.)

Using these methods, prototypical converging and diverging lens ray-tracing diagrams look like those in your textbook² and can be found in online simulations, such as those at OPhysics.

Eye Vision: Size Vs. Apparent Size

A fairly subtle point is the notion of whether an image "looks bigger." This is, after all, not just a matter of the image's height - we have the notion that a distant image "looks smaller" than a closer one, despite both being the same physical height!

The way to understand how big something "looks" is via its angular size: how much of your field of view it occupies. If the image (or object) is small compared to its distance away, we can make a small angle approximation and calculate its angular size as:

$$\theta=\frac{h_i}{d_\text{image to eye}}$$

For a lens intended to be viewed by eye (as occurs in telescopes, microscopes, etc.), we use a definition of magnification based on these angular sizes, rather than ordinary sizes (since this is relevant to our eye):

$$\text{MA}=\frac{\theta_i}{\theta_o}$$

More precisely, \(\theta_o\) is determined under "ideal" conditions: with the object taken as close to our eye as our eye can focus. For "typical" vision, this point (known as the near point of the eye) is 25cm away.

Therefore, our expression can also be written more explicitly (where \(m\) is the "linear magnification" defined above):

$$\text{MA}=\theta_i\frac{\text{25cm}}{h_o}=m\frac{\text{25cm}}{d_\text{image to eye}}$$

Most optical instruments are designed to have the final light rays end up parallel, with the image "at infinity." In this case, we can ignore the difference between \(d_\text{image to eye}\) and \(|d_i|\), at which point (using our earlier relations for magnification) we have:

$$\theta_i=\frac{h_i}{|d_i|}=\pm\frac{h_o}{d_o}$$

(The sign requires some thought as to whether the image will appear upright or inverted based on the ray tracing diagram, especially for more complicated optical instruments.) The end result for such instruments is thus simply:

$$\text{MA}=\frac{\text{25cm}}{d_o}$$

Note that in these more complicated setups \(d_o\) is the distance to whatever is the object for your last lens, which may or may not be the "actual" object (and depends on the lens configuration, in general).

Multi-Lens Systems

When you have a system with a sequence of lenses, there is a simple rule: the first lens makes an image (real or virtual), and the second lens treats that image as its object.

It is possible that this can lead to a negative object distance, per the definition we gave above. This can produce results you may find unintuitive! (In this case, you have to adapt the ray-tracing rules you have above, because the "object" is on the wrong side of the lens.)

The linear magnifications \(m\) also have a nice rule you can use: you multiply the magnifications for each lens together to get the "final" magnification (i.e., height of the final image over height of the initial object).

Basic Optical Apparatuses

There are two "classic" optical apparatuses we will be constructing in this lab: a microscope, and a telescope. They operate on similar principles, and are both made from two converging lenses.³

In each case, you have two lenses. The first lens through which the light passes, known as the objective, has as its goal to make an image at some reasonable distance that the second lens can see. (In more complicated setups, they can be composed of multiple lenses.)

The second lens, known as the eyepiece, is then placed so that this image made by the objective is at its focal point. This allows the eyepiece to take that image and "parallelize" the rays so they appear as coming from infinity.⁵⁶

The major difference between these two instruments is the distance that the object is designed to be at.

For a telescope, the object is designed to be infinitely far away. This focuses the light rays at the focal point of the objective, so the distance between the objective and eyepiece is the sum of the focal lengths.

For a microscope, the object is designed to be just outside the focal length of the objective. You then have to calculate where the image is depending on this distance to determine where the eyepiece goes. (It gets further away the closer you get to the focal length, which means your microscope needs to get bigger!)

Angular Magnifications of Optical Apparatuses

Using ray-tracing diagrams and the definition of angular magntification, one can directly compute the angular magnification that a telescope provides in terms of the focal length of the eyepiece \(f_e\) and the focal length of the objective \(f_o\):

$$\text{MA}=-\frac{f_o}{f_e}$$

The microscope computation is more complicated, for two reasons.

Firstly: for a telescope, the object was at infinity, so had a well-defined angular size to begin with. For a microscope, the object is some finite distance away, and the angular size depends on what that distance is.

The usual assumption (for practical reasons) is that we are holding the object as close to our eye "as possible." In order for a human eye to focus on an object, the object needs to be above a certain distance from the eye. That distance, which (for a typical person) has a nominal value of 25cm, is called the near point of the eye.

The second complication is that, unlike a telescope, a microscope has multiple possible configurations, depending on the distance from the object to the objective (or equivalently the separation difference between the two lenses). Each of these has a different magnification, and we need a convenient way to characterize this.

The simplest characterization is to define the tube length \(L\) as the separation distance between the objective and the focal point of the eyepiece (i.e., the image distance of the objective). In this case, one can compute the angular magnification of a microscope:

$$\text{MA}=-\frac{L}{f_o}\frac{25\text{cm}}{f_e}$$

To begin, your TA will turn off some of (but not all of) the lights in the room.

Ensure that your optical track is set up correctly: the rail has location markers on it. The numbers should be on the top and facing you, so you can read them. If not, fix this.

All "holsters" should be placed on the track so that the little arrow points toward the numbers, which will tell you the location of the holster (to a fairly high degree of precision).

In fact, there is actually a slight (~0.6cm) displacement from this position: the lenses are offset in the holders, and similarly the alligator clips hold things slightly displaced from the location of the stand. However, provided all these shifts are in the same direction (which requires you to orient things appropriately), these impacts should cancel out (to within uncertainty).¹

Due to residual impacts of those (not precisely measured) displacements, we take an uncertainty of 2mm on all distances (where uncertainties are considered).

Part I: Converging Lenses

First, grab the lamp assembly, and place it at one end of the optical rail. Ensure that the light is directed to shine on the rest of the track.

Then, select the 10cm converging lens and place it in the middle of the optical track.

Now, place the screen on the other side of the lens. Move the lamp assembly and the screen until they are some distance greater than 40cm apart from each other (say, ~50cm - not too far, or the light won't reach well).

Now, we will only move around the lens on the optical track. Shift it around until you see a focused image on the screen.

We will now take our first set of measurements. Measure the distance from the lamp to the lens holder, then from the lens holder to the screen. (You can do this by looking at the optical track, which has tick marks on it.)

Then, take height measurements. Take a distance you can measure on both the object and image (the number "4" has plenty of "features" to look at!), and measure that distance on both. Record these as \(h_o\) and \(h_i\) (again, with appropriate signs if needed: \(h_o\) is always positive, but is \(h_i\) positive or negative here?).

If you slide the lens around, you should find another place where the image focuses. Repeat the measurements you did for the previous position with this new focal point.

Part II: Diverging Lenses

Replace the 10cm converging lens with a 15cm diverging lens.

Shift it back and forth. Does it make a focused image on the screen at any point? Take note of this. Then, turn off the lamp.²

Now, take the lamp and object assembly off the rail and look at the object directly through the 15cm diverging lens.

Make observations about the orientation and (angular) size of the image of the "4" that you (should) observe by looking through the lens, in accordance with the questions on the data sheet.

Part III: Telescopes

Begin by removing the lamp and screen optical track. For this part, we will use only lenses on the short track.

First, grab the 20cm converging lens and 15cm diverging lens. We will first use these to make a Galilean telescope, with which we will be looking at an eye chart hung on the opposite wall.

Place the 20cm and -15cm lenses on the short track, with the 20cm lens closer to the opposite wall. Shift them until they are 5cm apart. This is the theoretical configuration of a Galilean telescope: they are separated by the sum of their focal lengths. (Note the diverging focal length is negative.)

Hang the eye chart on the opposite wall, somewhere you can see through your telescope. (Coordinate with the group across the room.)

Look through your telescope at the eye chart, and adjust the location of the -15cm lens until the chart is in focus. (This may be a slightly different position than the theoretical configuration indicates; that's fine.) Observe: is the eye chart upright or inverted?

Using your phone, take a picture through the telescope of the eye chart. Then, without changing the zoom on your phone, remove the lenses from your holsters and take a picture of the eye chart, with your phone in the same place you had it before. (This is to have a picture that mimics looking at the eye chart "with the naked eye" - in this case, with a naked phone camera.)

We'll now modify this to be a (stronger) Keplerian telescope. Change the 15cm diverging lens for a 10cm converging lens (red tape), and place it 30cm away from the 20cm lens.

Now, again: look through, and focus on the eye chart. Identify: upright or inverted? What is the lowest row on the chart that you can read now?

Mentally compare the two telescopes. (They have different magnitudes of focal length, which means the comparison isn't exactly 1:1, but we can draw some partial conclusions.) Which one is more compact?

Part IV: Microscopes

Now, we'll make two different microscope configurations. As before, we'll start with one using the diverging lens for the eyepiece.

Begin by placing the wingdings slide (which we'll use as our object) on one end of the track. (Tape it to the screen.)

Next, take your 10cm converging lens and place it right around 11cm away from teh screen.

Take your 15cm diverging lens and place it on the track. Start with it approximately 25cm beyond the converging lens.

Now: look through the diverging lens towards the wingdings slide. Adjust the position of the middle, 10cm lens (the objective) until you find the location where things focus.⁷

Record the locations of the slide and lenses when you find the focal point. Note: are the wingdings magnified or demagnified (to your eye)? Do you see them upright or inverted?

Then, as you did for the telescopes above, take phone pictures of the wingdings slides both with and without the lenses in the holsters (one picture "through the microscope," one "with the naked eye"). Remember: don't change your zoom between the pictures!

Now, repeat the process with a 20cm converging lens instead of the 15cm diverging lens (black tape), where the 20cm lens is placed ~50cm beyond the 10cm lens. Focus it, and record positions and your qualitative observations.

Part I: Converging Lenses

For each focusing point:

• Calculate the magnifications using the distances \(d_o\) and \(d_i\). Propagate uncertainties.
• Calculate the magnifications using the heights \(h_o\) and \(h_i\). Propagate uncertainties.
• Answer whether they agree (including signs!).
• Calculate the focal length using the thin lens equation. Propagate uncertainties. (The extra boxes should help you with that.)
• Answer whether your focal length agrees with expectation.

Part II: Diverging Lenses

Answer all questions on the data table.

Part III: Telescopes

For each telescope:

• Draw the ray tracing diagram (use the theoretical focal length and separation distance of the lenses rather than the "actual" positions for this diagram), to scale.
• Compute the theoretical angular magnification of the telescope using the focal lengths of the lenses. (You can ignore uncertainties here.)

Part IV: Microscopes

For each microscope:

• Assuming a 11cm object distance, compute the image distance of the objective. Then, compute the distance between the objective and the eyepiece. Consider: how well does this match up with your "actual" objective-to-eyepiece distance?
• Draw the ray tracing diagram (use the theoretical focal length and separation distances of the lenses rather than the "actual" positions for this diagram), to scale.
• Compute the theoretical angular magnification (using the assumed 6cm object distance) of the microscope using the focal lengths of the lenses. (You can ignore uncertainties here.)

Your TA will ask you to answer some of the following questions (they will tell you which ones to answer):

Experimental Questions:

1. Measure your experimental magnifications for each telescope and microscope configuration using your pictures. Then, compare this to the theoretical magnification you computed. Use the following procedure to analyze each telescope/microscope configuration:
   1. Take the two pictures for that configuration. Identify some feature that is visible (and identifiable) in both pictures (say, the height of a particular letter/wingding).
   2. Measure the height of the object in each picture. You can do this either using pixel measuring software, or by printing out both pictures (at the same photo size!) and using rulers on the photograph. (Either will work, so long as you are consistent in your method.)
   3. Those heights are proportional to the angular size of the object with and without the telescope, hence the ratio of those heights (with an appropriate ± sign, as necessary) tells you the magnification. Compute this.
2. Suppose we set up our diverging lens with the object some known distance \(d_o\) away. Then, we look through the lens (as in Part II), except now we measure the angular magnification that the lens provides (using some unspecified procedure - perhaps the one given above, for instance). How could we compute the focal length of the diverging lens using that information (the object distance \(d_o\) and the angular magnification)?

Theoretical Questions:

1. Why was it important that we placed the screen and the object at least 20cm apart in part I? Why is 20cm specifically special? (Hint: if it's not clear to you, try using the thin lens equation to solve for \(d_i\) in terms of \(d_o\) [with a 5cm converging lens], then plotting \(d_o+d_i\) vs. \(d_o\). What do you notice about this plot, and how does that correspond to our problem?)
2. Suppose we had used a 2cm converging lens instead of a 5cm converging lens in part I while keeping the screen-to-object distance the same. Would the first point at which the image focuses (as we move the 2cm lens from the object towards the screen) be closer to or farther from the object than it was for the 5cm lens? Why? (In other words: would the smallest value for \(d_o\), with the same object-to-screen distance, be smaller or larger when \(f\) is reduced from 5cm to 2cm?) Be careful: both \(d_o\) and \(d_i\) are changing as we move the lens!
3. If we had put the arrow and the lamp really far away in Part I, what would be the distance between the (5cm converging) lens and the image? What would the magnification be? What would the orientation be (upright or inverted)? Would you have a real or virtual image? (Hint: there's both a mathematical and physical way to understand this.)

For Further Thought:

1. In this lab, we saw a key advantage to using a diverging eyepiece for a microscope; what was it? What might be an advantage (from a practical perspective) for using a converging eyepiece instead?
2. In Part IV, if we had a moderate uncertainty on the 6cm object distance, we would find a fairly large uncertainty on the image distance (even assuming \(f\) of 5cm without unceratainty). Why? (You may want to compute the propagation formula yourself to have a sense of how the numbers work out.)
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