In the first part of this lab, we will investigate the basic principles of ray optics, Snell's law and the Law of Reflection.

In the second part of this lab, we will study the application of these laws to lenses.^{Hoveroverthese!}

• Optics Kit:
  • Laser Apparatus
  • Triangular Prism
  • Mirror
  • Graph Paper
  • Laminated Protractor Paper
  • Mini-ruler
• Lens Kit:
  • Optical Bench with Lamp
  • "Object" to display (arrow printed on glass) and holder
  • 5cm Converging Lens
  • 10cm Diverging Lens
  • 3 Optical Component Holders
  • 1 Screen
• Record data in this Google Sheets data table

Ray Optics

There are two phenomena that occur when a light beam hits an interface between two different kinds of material: refraction and reflection.¹

Refraction occurs when the light is transmitted into the new material, but at a different angle than it came in. Reflection occurs when the light reflects back into the material whence it came.

In both cases, we first draw a line perpendicular to the interface, which we call the normal. The basic angles we deal with in optics are the angles between a beam of light and the normal.

For reflections, the law of reflection states that the incoming angle (with respect to the normal) and the outgoing angle (w.r.t. normal) are the same. This is true regardless of the details of the reflection (whether you are going "inside" a material to "outside," vice-versa, or bouncing off a mirror).

For refractions, the angles are instead governed by Snell's law:

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

Here, we label the two sides of the interface by \(1\) and \(2\) (with, usually, \(1\) being the side of the incoming beam, and \(2\) being the side of the refracted beam). The quantities \(\theta_1\) and \(\theta_2\) are then the angles (w.r.t. normal) of the light beam on the respective sides of the interface.

The quantities \(n_1\) and \(n_2\) are the indices of refraction of the materials on each side of the interface. The index of refraction is a property of a material which describes how fast light moves in a material (\(v\)) vs. in vacuum (\(c\)):

$$n=\frac{c}{v}$$

Hence, \(n=1\) for vacuum, and approximately so for air. Otherwise (generally), \(n>1\).¹

When a beam of light moves from an area of higher \(n\) to lower \(n\), depending on \(\theta_1\), it is possible that there is no \(\theta_2\) that will cause Snell's law to hold. Thus, no light can be refracted, and we have what is called total internal reflection.

A full formula for the intensity of the reflected vs. refracted beams are beyond the scope of this course. Nevertheless, one basic principle that governs these powers can be understood.

The intensity of the beam is the power (energy per unit time) transmitted per unit area. Therefore, by conservation of energy, if no light is absorbed (which is a separate possible phenomena), the total incoming intensity has to equal the total outgoing intensity.

Lenses

If a piece of refracting material is shaped appropriately, it will focus or defocus the light. If the light rays are focused to (approximately) a point, or defocused so they approximately look like they're coming from a single point, then we call that a lens.²²

Suppose the rays come into the lens parallel, and consider the outgoing rays (extending them backward behind the lens as well). These rays will all meet at some point, known as the focal point.

The forward-displacement to the focal point is known as the focal length, \(f\). (This means that if the focal point is behind the lens, as occurs for a diverging lens, then the focal length is negative.)

In general, understanding how lenses make images requires a whole bunch of complicated geometry. With lens is thin, however, one can make an approximation³ that leads to the thin lens equation:

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

Here, \(d_o\) is the distance to the object (always defined as positive) and \(d_i\) is the distance to the image (positive if the image is "forward" from the lens and real, negative if it is "behind" and virtual).

The image from the lens may be magnified (larger) or demagnified (smaller). The magnification is defined as the ratio of image size \(h_i\) to object size \(h_o\) (with a negative sign if the image is flipped vertically, calculated as a negative \(h_i\)):

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

Some geometry (using similar triangles) also allows us to calculate the magnification in terms of the distances (with the same sign conventions as above):

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

This lab has two separate apparatuses: the optics kit (Part I) and the lens kit (Part II). Each one will occupy half the lab benches; there are insufficient setups to fill the whole room with both.

Begin by performing the portion of the lab that corresponds to the setup at your bench. Halfway through the lab (or once you are done with your part and a group with the opposite setup is done with theirs), switch setups, and do the other part.

If you are waiting around to switch, play around with the equipment provided. What else do you observe? (There's plenty to see beyond what we actually measure in this lab, especially with the optics kit.) Write down if you notice anything significant!

Part IA: Refraction

In this part, we are going to measure all of our angles by measuring distances. This will enable us to measure them with a reasonable degree of precision.

The angles and distances we will look at¹ are shown in the following diagrams. Red lines are the laser beams; in the left diagram, blue lines are normals. (Note \(Y_2\) extends downwards to the location of the light beam at that x-coordinate.)

Take the \(45^\circ\) triangular prism, your laser, and one sheet of graph paper. Plug in your laser, and set it to output one beam (middle setting on the switch). Press the button and make sure it shines a beam.²

Place the triangular prism on the graph paper such that one of the shorter edges aligns with one of the grid lines. Slide sideways until the \(45^\circ\)-angle-corner of that side is on a grid line intersection. (See diagram below if these instructions are confusing.)

Now, take the laser and shine it on the prism perpendicularly to the edge you lined up, from the hypotenuse side. Line the laser up so that its beam travels along the gridline which has three boxes inside the prism.

Schematically, your setup should be illustrated by the following figure (the location of the left edge of the prism may vary):

Now, measure the distances \(X_1\), \(Y_1\), \(X_2\), and \(Y_2\). Measure as precisely as you can here, ideally to sub-millimeter precision - you'll get better results that way. Estimate uncertainties.

Part IB: External Reflection

Set aside the graph paper, and take the giant laminated protractor paper from the kit. In this part, we will measure angles directly.

Begin by lining up the hypotenuse of the trianglular prism with the 90-270 line on the protractor (such that the normal is along the 0-degree axis).

Put the laser off at some angle (less than 90 degrees from zero), and point it towards the center of the protractor. Measure both the angles with respect to the normal of both the incoming and reflected outgoing beam using the protractor. Estimate uncertainties.

Also take note of the relative intensity of the reflected beam compared to the incident beam.

Take out the mirror from your kit. Replace the prism with the mirror (laser in the same place, mirror along the 90-270 line).

Take note of the reflected intensity (compared to the incident intensity) for this component. (Think: why a difference?)

Part IC: Total Internal Reflection

In this part, we will again measure angles; however, they will be determined with significantly less precision than the last part.³

Shine the laser on the hypotenuse of the prism, and rotate the prism. Observe that at some point, the beam refracted outwards from the back edge prism "folds into" the prism and disappears. We want to measure the internal reflection angle at this point.

Line up your laser, prism, and protractor so that you are at the angle where the outside beam just disappears, and the point where it reflects on the back edge is at the center of the protractor.⁴

You should now have something that looks like the following:

Some angles are exaggerated for illustrative effect. Note the only thing that needs to be "lined up" with the protractor is the point of reflection.

Now: using your protractor, as best you can, measure both the incoming and outgoing angles (simply as the numbers on your protractor - don't worry about "with respect to the normal" yet; just measure with respect to zero on the protractor for now).⁵

Estimate your uncertainty on the above measurements. (This will depend on how much care you took in making those measurements, and will not be primarily determined by the "half-degree" uncertainty of the protractor itself!)

Now: rotate your prism so it's just outside the angle at which total internal reflection happens (so that you have a refracted beam that is almost gone). Wiggle it slightly around this critical angle.

How do the incident, internally reflected, and refracted beams vary in intensity as you approach the critical angle (from the direction where the refracted beam appears), if at all? (This is most visible when you're really close to the critical angle.)

Part IIA: Convergent Lenses

Begin by selecting a 5cm converging lens,⁶ and place it in the center holster on the optical track.

Ensure that the arrow is in between the lamp and the lens, and the screen is on the other side. Place the arrow and the screen some distance greater than 20cm apart from each other (say, ~30cm).

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

Adjust the height of the arrow and/or, if necessary, to make sure that you have something you can definitely measure the size of on the screen. (E.g.: that you can see both sides of the "head" of the arrow.) You need something you can see all of in order to measure "heights" for magnification.

Assuming everything still looks focused, we will now take our first set of measurements. Measure the distance from the arrow 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.)

Now we need to slightly adjust those numbers. In fact, the lens is slightly offset from the center of the holder, by 0.6cm. Take note of which direction it is offset, and add/subtract 0.6cm to the two previously-calculated distances to account for this. Record these as the object distance \(d_o\) and the image distance \(d_i\), respectively (with appropriate signs if needed: \(d_o\) is always positive, but is \(d_i\) positive or negative here?). Also, since this adjustment varies between lenses, take the uncertainty in position as 2mm.

Then, take height measurements. Take a distance you can measure on both the object and image, 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 IIB: Divergent Lenses

Replace the lens with a 10cm diverging lens.⁷

Shift it back and forth. Does it make a focused image on the screen at any point? Take note of this.

Now, you will look through the diverging lens at your arrow directly. To avoid blinding yourself, either place a piece of paper between yourself and the lamp (to diffuse the light), or just remove both the lens and arrow from the holder and look at it in ambient light (perhaps in the hallway outside the classroom, assuming the lights are off).

Now, make observations about the location, size, and orientation of the image of the arrow that you (should) observe by looking through the lens, in accordance with the questions on the data sheet.

Warning: Excel/Sheets/etc.'s trig functions assume inputs of radians, and the inverse trig functions output angles in radians. Pay attention to this when doing calculations involving trigonometric functions.

Part IA: Refraction

First, calculate angles \(C\), \(D\), and \(E\). For simplicity, here are the formulas for them (although you should verify that they are true):

$$C=45^\circ-D$$ $$D=\tan^{-1}\left(X_1/Y_1\right)$$ $$E=\tan^{-1}\left(X_2/Y_2\right)$$

Then, calculate the sines of these angles (and \(A\)). Keep in mind (for both of these calculations) the warning above.

Note that, for both of these steps, we did the uncertainty propagation for you (because propagating uncertainty through trigonometric formulas is somewhat messy and uses specialized formulas that require calculus to motivate).

Then, using those sines, calculate the index of refraction, assuming \(n_\text{air}=1\), twice, once at each boundary (i.e., once using angles A and C, and once using angles D and E).

Finally, answer the questions on the data sheet about whether these agree with the expected value for \(n\).

Part IB: External Reflection

Answer the questions about whether your results agree/make sense.

Part IC: Total Internal Reflection

Based on the incident and reflected angle you measured, calculate the critical angle.⁸ Propagate uncertainties.

Calculate the sine of the critical angle (as above, keep in mind the warning, and the trig error propagation is done for you).

From this, calculate index of refraction, and propagate uncertainty.

Finally, answer the questions about whether your results agree/make sense

Part IIA: Convergent 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 IIB: Divergent Lenses

Answer the question about the nature of the image.

Answer the two questions about magnification. (The first is intended to look at the sign of \(m\). The second is intended to look at whether \(|m|\) is larger than, smaller than, or equal to \(1\). For both, you should combine the definition of magnification with the calculation involving the \(d_o\) and the \(d_i\).)

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

Experimental Questions:

1. Suppose you had some random error in \(X_1\) or \(Y_1\). Would this error have a same-direction or opposite-direction impact on angles \(C\) and \(D\)? A same-direction or opposite-direction impact on your two measurements of \(n\) in part IA? Would you consider this to be a good thing or a bad thing from the perspective of getting an accurate measurement of \(n\), and why?

Theoretical Questions:

1. For part IIA, take the "Ray Tracing Diagram Paper" from the bottom of this manual and draw the ray tracing diagram for each converging lens configuration. Set an x scale and a y scale; these can be different (e.g., horizontally 1 square = 5cm but vertically 1 square = 1cm), but be sure you are to scale in each direction (and record your scale).
2. Why was it important that we placed the screen and the object at least 20cm apart in part IIA? (Use the thin lens equation, and remember that we have a 5cm converging lens.)
3. If we had used a 2cm converging lens instead of a 5cm converging lens in part II, would the first point at which the image focuses (going from the lamp towards the screen) be closer to or farther from the lamp than it was for the 5cm lens? Why? (This is a trickier question than it looks, since both \(d_0\) and \(d_i\) change simultaneously.)
4. If we had put the arrow and the lamp really far away, what would be the distance between the (5cm converging) lens and the image? What would the magnification be? What would the orientation be? (Hint: there's both a mathematical and physical way to understand this.)

For Further Thought:

1. The Earth's ionosphere has an index of refraction less than 1 (due to it being a plasma)³. Draw what a radio wave passing from the stratosphere (with \(n=1\)) into the ionosphere might look like, based on Snell's law, for a large angle w.r.t. the relevant normal. (Hint: it's going from an area of higher index of refraction to one of lower index of refraction... what can happen in such a case?) How might this aid long-distance radio transmission?
2. Investigate the difference between "phase velocity" and "group velocity." Why does an index of refraction of less than 1 (so \(v\gt c\)) not violate relativity, which says no information can move faster than the speed of light?

Guide to Uncertainty Propagation & Error Analysis (Quick Reference)

Ray Tracing Diagram Paper