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Chapter 8 Experiment 6: Light Polarization

Chapter 8 Experiment 6: Light Polarization

Chapter 8

Experiment 6:

WARNING This experiment will employ Class III(a) as a coherent, monochromatic light source. The student must read and understand the safety instructions on page 90 before attending this week’s laboratory.

8.1 Introduction

We might recall that light is “transverse electro-magnetic ”. ’s electrodynamics equations describe these waves in detail. Transverse waves have the E and B vectors perpendicular to the direction of propagation. Out of 3-dimensional space, this still leaves two dimensions (or two degrees of freedom) in which the vectors might point; they can point in any radial direction around the light ‘rays’. These two degrees of freedom are represented mathematically by two orthogonal coordinate axes. These two coordinates are the for specifying the polarization of the light. The electric field vector E can be specified completely using only these two coordinates. Although the same is true for the magnetic field vector, the in get moved by the electric field; they experience an electric force −eE. Since their velocities are much smaller than light-speed, any magnetic force is much smaller and their random velocities generally average to zero quickly anyway. (The positively charged nuclei are thousands of times heavier than the electrons so that their response to the waves are neglected; their accelerations are thousands of times less.) This leads us to specify the polarization pˆ of the light to be the direction of the electric field, E(t) = E(t) pˆ. We also know that sinusoidal waves are equal parts positive and negative so that there is no difference between positive and negative polarization, pˆ ⇔ −pˆ. has the polarization vector pˆ a fixed unit vector in space and time.

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Figure 8.1: A absorbs oscillating electric fields parallel to its grain. This can polarize natural light or it can check whether light is already polarized. Polarized light will be absorbed when its electric field is parallel to the polaroid’s grain.

This is the most common form of polarization and the one we will be studying in this lab. If the plane of vibration rotates around the direction of propagation, then the light is said to be elliptically polarized. If the major and minor axes of an ellipse are equal, the ellipse is a circle and the same is true for circularly polarized light. Depending on the direction of rotation the light may be right- or left-circularly polarized. These two rotation directions are also independent of each other and form an alternate basis for specifying a general polarization. An ordinary light source consists of a very large number of randomly oriented atomic emitters. Each excited atom radiates a polarized wave train of roughly 10−8 s duration (in other words imagine a sinusoidal wave about 3 m long). New wave trains are constantly emitted and the overall polarization changes in a completely unpredictable fashion. These wave trains make up natural light. It is also known as unpolarized light, but this is a bit of a misnomer since, in actuality, every wave train is polarized but the wave trains’ polarization planes are randomly oriented in space (i.e. are different polarization states). The set of all wave trains has every possible polarization in equal abundance. In this lab you will

1) study the phenomenon of light polarization, 2) compare our observations to Malus’ law and to Brewster’s law 3) learn about the function and use of a semiconductor diode laser, and 4) learn about the function and use of a PIN photodiode light detector. Light’s polarization can be used to manipulate light. Just as changing its inside different dielectrics allows us to bend and to focus light, placing optical elements between two allows us to choose whether the light passes the second . For example, liquid (nematic fluids) rotate the polarization of light so that we can choose with an electric field whether the polarization matches the second polarizer and is

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Figure 8.2: A photograph of the light detector and shape wheel. We will use one of the circular apertures. transmitted. Many materials exhibit electro-optical or magneto-optical effects that can also be used in this manner. Birefringent (or dual index of ) crystals can be cut to exactly the right length to form half-wave plates or quarter-wave plates whose orientation relative to the incident polarization affects the polarization state of the transmitted light. Finally, light reflected from surfaces is partially polarized. Light reflected at the Brewster’s angle from a dielectric surface is completely polarized. We will observe the Brewster’s angle for Lucite as part of today’s experiment.

Checkpoint What property of electro-magnetic radiation determines its polarization?

Checkpoint What is a wavetrain? What is the plane of vibration of light? What is the difference between polarized and natural light?

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Checkpoint Is the light emitted by most light sources polarized? Explain. What is circularly polarized light?

8.2 Polarizers

An optical device whose input is natural light and whose output is some form of polarized light is known as a “polarizer.” We know a variety of ways to polarize light. All of these techniques have in common some form of asymmetry associated with the way they operate. Three such techniques are discussed next.

1. Wire-Gridiron Polarizer. Imagine that an unpolarized electromagnetic wave impinges on a gridiron of very fine wire from the left. (See Figure 8.1.) The electric field can be resolved into two orthogonal components; in this case, one chosen to be parallel to the wires and the other perpendicular to them. The y-component of the field drives the conduction electrons along the length of each wire, thus generating a current. This current heats the wires so that energy is transferred from the field to the grid. In contrast the electrons are not free to move very far in the x-direction and the corresponding field component of the waves is essentially unaltered as it propagates through the grid. The transmission axis of the grid is accordingly perpendicular to the axes of the wires. The green arrows in Figure 8.1 indicate the polarization axis. 2. A Polaroid is a molecular analog to the wire gridiron. A sheet of clear polyvinyl alcohol is heated and stretched in a single direction, its long hydrocarbon molecules becoming aligned in the process. The sheet is then dipped into an ink solution rich in iodine. The iodine impregnates the and attaches to the straight long-chain polymeric molecules, effectively forming a chain of its own. The conducting electrons associated with the iodine can move along the chain as if they were long thin wires. The component of E in an incident wave which is parallel to the molecules drives the electrons, does work on them, and is strongly absorbed. The transmission axis of the polarizer is therefore perpendicular to the direction in which the film was stretched. 3. A polarizing beam-splitter separates an incoming beam of light into a transmitted, horizontally polarized beam and a reflected, vertically polarized beam. These polarizers make use of Brewster’s law that will be studied today.

Checkpoint How can you produce a beam of linearly polarized light? How can you establish if a beam of light is linearly polarized? How does a polaroid film polarize a beam of light?

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8.2.1 Malus’s Law

How do you determine experimen- tally whether or not a device is ac- tually a linear polarizer? Consider Figure 8.1 where natural (unpolar- ized) light is incident on an ideal linear polarizer. Only light with a specific orientation (polarization state) will be transmitted. That state will have an orientation par- allel to a specific direction which we call the transmission axis of the polarizer. In Figure 8.1 the trans- mission axis is depicted by green arrows. In other words, only the component of the optical field par- allel to the transmission axis will pass through the device unaffected. If the polarizer is rotated about the z-axis the amount of transmitted light will be unchanged because of the complete symmetry of unpolar- ized light. Now suppose that we introduce a second possibly iden- tical ideal polarizer, or analyzer, whose transmission axis is vertical. If the amplitude of the electric field transmitted by the polarizer is E0, Figure 8.3: The polarization light perpendicular to only its component parallel to the the is unaffected by reflections at transmission axis of the analyzer dielectric surfaces. will be passed on (assuming 100% efficiency). This component will have magnitude E0 cos θ. The observed is the square of the amplitude, consequently

2 I(θ) = I0 cos θ. (8.1)

This is known as Malus’s law, having first been published in 1809 by Etienne Malus, military engineer and captain of the army of Napoleon. Observe that I(90◦) = 0. This arises from the fact that the electric field that has passed through the polarizer is perpendicular to the transmission axis of the analyzer (two devices so arranged are said to be “crossed”).

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Checkpoint What is the relation between polarizer orientation and transmitted light if the incident light is linearly polarized?

Checkpoint What happens to the polarization of light that doesn’t pass the polarizer?

Checkpoint What is Malus’ law?

8.2.2 Malus’ Law

First, place the lens in the beam path very near the laser and place one polaroid very near the lens. Remove everything else in the beam path except the detector. If the laser’s spot is not centered on the detector, ask your TA for assis- tance. Next, observe the polariza- tion of your laser beam by rotating the polaroid film placed in your laser beam, and then while noting changes in the beam intensity on the screen (Figure 8.2). The po- laroid film has the same function as the analyzer shown in Figure 8.1. If the intensity of the transmit- ted light does not change when rotating the analyzer, the beam exiting the laser is not linearly polarized. If it varies according to Malus’s law (Equation (8.1)) the light is linearly polarized. Enter your observations in your Data and adjust the polaroid for maximum transmission. You can the Figure 8.4: A sketch of a green light reflected by sensor’s response on the computer a dielectric surface. The polarization in the plane of as you rotate the polarizer. Briefly incidence is greatly reduced by the reflection.

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remove the polaroid and observe the sensor’s response now. Compute the polaroid’s efficiency Intensity with polaroid η = ; (8.2) Intensity without polaroid this is also known as “insertion loss” and the factory specifies η = 0.76. Replace the polaroid adjacent to the laser and lens and (if necessary) adjust it again for maximum transmission. Execute the Pasco Capstone configuration file from the lab’s website at

http://groups.physics.northwestern.edu/lab/polarization.html

Observe the polarization states of the light traversing the polarizer. As described before, light transmitted through a polaroid film is linearly polarized. Place the poloroid with the rotation sensor attached in front of the detector; this polaroid is your analyzer. Start recording and rotate the analyzer and encoder at least 360◦ = 6.3 rad and stop recording. Note the character of the transmitted light vs. angle in your Data. Select all of the data in the table and copy it to Windows’ clipboard. (Click a table entry, ctrl-a, and ctrl-c is the easiest way.) Now execute Vernier Software’s Ga3 graphical analysis template from the website, click row 1 under the angle header, and press ctrl+v to paste the data into Ga3. Increase or decrease ‘BG’ to align the bottom of the data to the angle axis. Draw a box around the data points on the graph and Analyze/Curve Fit. Fit the transmitted intensity to Malus’ law near the bottom of the list. This model will fit your data using

I1 * (cos(x - x0))^2.

Verify that the formula is correct and “Try Fit”. If the model line passes through your data, click “Done” and make sure the uncertainties in the fit parameters are shown. Rescale the axes so that the data and Figure 8.5: A graphical plot illus- model curve occupies most of the graph and there is trating how reflected light is polarized still room for the parameters box. Print your graph differently for different angles of in- for your notebook and copy and paste it into your cidence. MS Word R report.

8.2.3 Crossed Polarizers

Be careful not to disturb the laser, the lens, the detector, or the first polaroid until this next step is complete. Replace the encoder assembly with a second polaroid without the encoder attached near the detector. Rotate the second

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polaroid until it is “crossed” with the first; no light passes the second polaroid. Once again, you can observe the sensor’s response on the computer. Leave these polaroids alone and carefully insert the encoder assembly between the polaroids; note what happens as you rotate the encoder’s polaroid. How can an intermediate polaroid cause light to pass crossed polaroids? When rotating this third (middle) filter the transmitted intensity goes through a maximum which appears each time the filter in the middle is rotated by 90◦. The appendix shows how we might analyze this situation using mathematics, but students will learn more if they deduce this for themselves (in less detail, perhaps). For which orientations of the transmission axis of the filter in the middle do you observe a maximum or a minimum of transmitted light intensity? Record another set of data for at least 360◦ rotation of the encoder and middle polaroid. Copy and paste this data into Ga3 as before (Data/Clear all data in Ga3 before pasting). Adjust ‘BG’ up or down to align the minima with the horizontal axis. Fit this data to the “MalusToo” model near the bottom of the list; this model should agree with

0.76 * I1 * (sin(x - x0) * cos(x - x0))^2.

Compare I1 from this data to I1 from the first data set above. Why must we include the factor of 0.76? Light from the first experiment passed through two polaroids and light from this experiment passed through three polaroids. This data might also be fit to “Malus 2”

I2 * (sin(2*(x - x0)))^2,

4 I2 but in this case I1 = 0.76 . Why must we multiply by 4? (Hint: At what angle, θ, is the intensity maximum now? Alternately, consider a trigonometric identity.) If the detector or laser was inadvertently disturbed despite the warning above repeat the first experiment without disturbing the laser or detector. Just remove the last polaroid and execute the experiment again.

Checkpoint What is the efficiency of our polarizers?

8.2.4 Polarization By Reflection

One of the most common ways to polarize light is to reflect the light off of some dielectric medium. The glare spread across a window pane, a sheet of paper, or the sheen on the surface of a telephone or a book jacket are all partially polarized. One simple, although incomplete explanation of this polarization is supplied by the -oscillator model. Consider an incoming linearly polarized so that its field is perpendicular to the plane of incidence (see Figure 8.3). The wave is refracted at the interface entering

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the medium. Its electric field drives the bound electrons, in this case normal to the plane of incidence, and they in turn re-radiate a reflected component of the incident light. The radiation emitted by an electron oscillating along a straight line is called dipole radiation. Its electric field is largely parallel to the direction of motion of the charge and no radiation is emitted along the axis of the dipole (the direction of ). It is consequently clear from the geometry of the dipole radiation pattern that both the reflected and refracted waves must also be in the same polarization state as the incident radiation. On the other hand, if the incoming field is in the incident plane as shown in Figure 8.4 the electron-oscillators will vibrate under the influence of the refracted wave. The flux density of the reflected wave is relatively low because the reflected ray direction makes a small angle ϕ with the dipole axis. If we could arrange things so that ϕ = 0, or equivalently ◦ θrefl + θrefr = 90 , the reflected wave would vanish entirely. Consider an incoming unpo- larized wave represented by two components, one polarized normal to the incident plane and one in the incident plane. Under these circumstances only the component normal to the incident plane, and therefore parallel to the surface, will be reflected. The particular angle of incidence for which this situation occurs is designated by θp and is referred to as the polariza- tion angle or Brewster’s angle, so ◦ Figure 8.6: A photograph of the Lucite block on the that θp + θrefr = 90 . Hence from rotary table and the optical bench. Snell’s law

n1 sin θp = n2 sin θrefr (8.3)

◦ and from the fact that θrefr = 90 − θp it follows that

n1 sin θp = n2 cos θp (8.4)

and n2 tan θp = (8.5) n1

Checkpoint What is Brewster’s law? A certain reflected light beam satisfies Brewster’s law. In which plane is the light polarized?

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General Information The electronic bonds in the dielectric delays the radiated light by 90◦ shift in relative to the incident beam. These re-radiated waves add to the incident wave everywhere in the dielectric. The sum of all of these waves is the total electric field and appears to propagate slower inside the dielectric. This re-radiation is responsible for both the reflections and the refraction from dielectrics.

This is known as Brewster’s law after the man who discovered it empirically about Reflected from forty years before light phenomena were Dielectric Block described as electromagnetic waves. The degree of polarization of unpolarized light reflected from Lucite (n2/n1 = 1.5) is shown in Figure 8.5. The reflected beam is 100% polarized at 56.2◦. Polarizer to tune polarization

8.2.5 Confirm Brewster’s Law

We want to check Brewster’s Law by observ- o ing the polarization of the light reflected off 0 at normal the surface of a block of Lucite when the incidence angle of incidence is set at the Brewster’s ◦ angle (for Lucite θp = 56.2 ). For this part remove the lens from the optical bench. Set the Lucite block on the rotary table (Figure 8.6). Hold the rotary table at 0◦ while orienting the Lucite block perpendicular to the beam. The reflection will return to the laser when this is the case. (See Figure 8.7.) The surface off which you are reflecting the laser beam must be vertical and situated along a diameter of the rotary table. Figure 8.7: A photograph of the Lucite block ◦ The lasers have been oriented to emit aligned to θ1 = 0.0 incidence. The reflection mostly horizontal polarization but we need has the same horizontal location as the laser to eliminate the vertical component of polar- exit port. If the angle is not zero on the scale, ization completely from the beam incident subtract its value from the Brewster’s angle on the dielectric. See Figure 8.8 to guide later. this discussion. First rotate the polarizer to maximize the transmitted light. Next try to see a minimum intensity in the reflected beam as you rotate the turntable. If no minimum is seen, tweak the polarizer 2-3◦ and try again.

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Reflected Laser Spot

No Laser Spot

Figure 8.8: Three photographs illustrating how to observe Brewster’s angle. First, adjust the polarizer to pass maximum laser light. Next, rotate the dielectric to observe an intensity minimum. Finally, rotate the polarizer until the reflected minimum is dark.

Set the turntable at the minimum and fine-tune the polarizer again. Repeat these two steps until the reflection is too weak to see. Record this turntable angle as θp. How far can you rotate the turntable before the reflection is visible? Does this suggest what your measurement uncertainty should be? Record your observations in your Data. What would

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◦ happen at −θp. Might you use this to eliminate the error in your θ = 0 alignment? Be careful not to disturb your setup while you calculate the index of refraction. If n differs from 1.495 by more than 1%, move the view-screen across the optical bench and repeat your measurement. The best polarization angle is the average of (the absolute values of) ◦ ◦ these two. For θp > 180 subtract from 360 . Explain your need to do this in your Data or Analysis. The observed degree of polarization should vary as shown in Figure 8.5. Since the incident light is oppositely polarized, the observed intensity should vary as 100% minus the degree of polarization. In this configuration we are using the reflection as an analyzer instead of a polarizer. Slowly increase the angle and observe these intensity variations. Enter in your Data a brief description of what you have observed. Make a drawing. Using your measured Brewster’s angle (corrected for zero) calculate the index of refraction of your Lucite block. Remember that the index of refraction for air is very nearly 1.

Checkpoint Polarization by reflection determines the . Is it the vertical or the horizontal plane in which the vector of the reflected light oscillates? Explain.

8.3 Analysis

How well does Malus’ law predict your data? Do both models fit your data well? Use the strategy in Section 2.9 to compare the I1 for the first and second experiments. Is your measurement of Lucite’s index of refraction consistent with previous measurements (1.495)? You may expect that the manufacturer’s specified tolerance is (±0.005). What other sources of error did you notice while performing your experiments? How might each of these have affected your data? Do your models fit your data? Don’t forget that the period of the sine (cosine) function was predicted explicitly by the model and was not varied by the fitter.

8.4 Conclusions

What has your data demonstrated? Is Malus’ law supported? Contradicted? Is Brewster’s law supported? Contradicted? Define all symbols used in equations and communicate with complete sentences. Have you measured anything that might be of value in the future? If so state your measured values, units, and errors. Can you think of any applications for what we observed? What improvements might you suggest?

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8.4.1 References

Sections of this write up were taken from:

• D. Halliday & R. Resnick, Part 2, John Wiley & Sons. • R. Blum and D.E. Roller, Physics Volume 2, Electricity Magnetism and Light, Holden- Day. • E. Hecht/A. Zajac; , Addison-Wesley Publishing Company.

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8.5 APPENDIX

8.5.1 Crossed Polarizers

The second experiment in Section 8.2.3 has two crossed polarizers with a third polarizer placed between. In this appendix we learn what should happen when the intermediate polarizer is oriented to angle θ. To do this we define coordinates such that xˆ is parallel to the first polarizer’s transmission axis and that yˆ is parallel to the third polarizer’s transmission axis. In this coordinate system the angle between the first and second polarizers is θ and the angle between the second and third polarizers is 90◦ + θ.

Historical Aside It is evident that the light exiting the second polarizer has reduced intensity according to Malus’ law. But if this was all that happened, then the polarization would still be perpendicular to the third polarizer and no light would pass. It is evident, therefore, that the second polarizer changes the polarization state of the light that it passes. By extension, it is also evident that every polarizer changes the state of the light that it passes (unless it is already aligned with the incident polarization). Since adjacent crossed polarizers pass nothing, the exiting light must be aligned with each polarizer’s transmission axis.

Then the angle between the polarization of the light between the second and third polarizers and the third polarizer’s transmission axis is also 90◦ + θ. If the light exiting the first polarizer is I0, then Malus’ law predicts that the light exiting the second polarizer is 2 I1 = I0 cos θ. (8.6) Since cos(90◦ + θ) = − sin θ, we invoke the polarization state between the second and third polarizers to find the exiting intensity

2 ◦ 2 2 2 I2 = I1 cos (90 + θ) = I1 sin θ = I0 sin θ cos θ. (8.7)

8.5.2 Uncertainties

To determine the uncertainty in the calculated index of refraction, we must use calculus. The index of refraction in Brewster’s law (n = tan θp) is a function of only the polarization angle θp so dn 2 δn = δθp = sec θp δθp dθp

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where θp is in radian units. Since our goniometer reports θp in degrees, we must employ a unit conversion π rad 0.056 δn = sec2 θ δθ ≈ δθ (8.8) p 180◦ p 1◦ p where the uncertainty in this angle is in degrees.

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