Polarization of light: Malus’ law, the Fresnel equations, and optical activity.

PHYS 3330: Experiments in OpticsDepartment of Physics and Astronomy,

University of Georgia, Athens, Georgia 30602

(Dated: Revised August 2012)

In this lab you will (1) test Malus’ law for the transmission of light through crossed polarizers;(2) test the Fresnel equations describing the reflection of polarized light from optical interfaces, and(3) using polarimetry to determine the unknown concentration of a sucrose and water solution.

I. POLARIZATION

You will need to complete some background readingbefore your first meeting for this lab. Please carefullystudy the following sections of the “Newport Projects inOptics” document (found in the “Reference Materials”section of the course website): 0.5 “Polarization” Alsoread chapter 6 of your text “Physics of Light and Optics,”by Peatross and Ware. Your pre-lab quiz cover conceptspresented in these materials AND in the body of thiswrite-up. Don’t worry about memorizing equations – thequiz should be elementary IF you read these materialscarefully. Please note that “taking a quick look at” thesematerials 5 minutes before lab begins will likely NOT beadequate to do well on the quiz.

II. MALUS’ LAW

When completely linearly polarized light is incident ona polarizer, Malus’ law predicts the transmitted intensityto vary proportionally as the square of the cosine of theangle between the transmission axes of the analyzer andthe direction of polarization of the incident light:

I(θ)

I0= cos2 θ

You will test this law using the polarized light of aHeNe laser, an assortment of polarizers, and a photo-diode – an electrical device which generates an electriccurrent proportional to the intensity of incident light.

FIG. 1: Schematic of setup for Malus’ law investigation.

III. PROCEDURE

1. Position the fixed “V” polarizer after mirror “M”to establish a vertical polarization axis for the laserlight.

2. Carefully adjust the position of the photodiode sothe laser beam falls entirely within the central darksquare.

3. Plug the photodiode into the bench top voltmeterand observe the voltage – it should be below 200mV; if not, you may need to attenuate the light by[anticipating the validity off Malus’ law!] insertingthe polarizer on a rotatable arm “R” upstream ofthe “V” polarizer, and rotating “R” until the read-ing on the photodiode is approximately 200mV.

4. Place the precision rotatable polarizer “P” just infront of the photodiode. Make sure the beam goesthrough cleanly.

5. Rotate “P” to maximize the transmitted power.Then rotate the polarizer so that the “0” of thevernier scale lines up with a tick mark of the fixedscale, to establish a nice place to start your system-atic study of intensity versus angle.

6. Starting with zero degrees, record your data inthe following pairs: (power through the polarizer,power with the polarizer removed). Rotate the an-alyzer in steps of 4 degrees, recording these twovalues at each step. Take data over at least 90 de-grees of rotation.

7. Estimate your uncertainty in each measurement bysitting at some fixed angle and repeating the pairsof measurement 5 times (physically take the po-larizer in and out for each pair!). Record all theindividual measurements for later processing.

IV. ANALYSIS

Make an array which is the ratio of the two measure-ments, at each theta. Calculate your uncertainty in theratio by finding the standard deviation of the 10 ratios

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you took at a fixed angle in the last step of the proce-dure. Plot this ratio versus angle, with error bars. Testthe Law of Malus by fitting your data to the followingmodel:

P (θ)

P0= cos2(θ − θ0) + C (1)

Give a physical explanation of why you might need theparameters θ0, C? Plot a best fit model overlaid witha plot of your data, with error bars. Plot your residu-als with error bars. Report the values and uncertaintiesof all fitted parameters, discuss the meaning (or arbi-trariness) of the values of each parameter with respect toverifying/contradicting the theory under test. Plot thefit residuals and discuss them. Report the chi-squaredper degree of freedom statistic obtained by your fit, anduse it to discuss the likelihood that, if the Malus theorywere true, you could expect to see data such as yours.Explain any discrepancy; note – saying “we must havemade an error....” is not an explanation.

V. FRESNEL EQUATIONS

The Fresnel equations give the transmission and reflec-tion coefficients at a dielectric interface. They dependupon the polarization and angle of incidence, and indicesof refraction of the media on both sides of the interface.For light crossing from a medium of index n1 to one ofn2, the fractional reflected intensity is predicted to be,for pure s−polarized light (refer to your text for the def-initions of s and p if you are unsure).

Rs(θi) =IsrIsi

=

n1 cos θi − n2

√1 − (n1/n2)2 sin2 θi

n1 cos θi + n2

√1 − (n1/n2)2 sin2 θi

2

whereas for pure p-polarized light the theory predicts:

Rp(θi) =IprIpi

=

n1

√1 − (n1/n2)2 sin2 θi − n2 cos θi

n1

√1 − (n1/n2)2 sin2 θi + n2 cos θi

2

A. Procedure

You will measure the reflection of polarized light off ofa BK7 borosilicate glass prism. There are several stepsto the alignment of the optical system.

1. Make the laser beam parallel to the surface of the ta-ble by adjusting M to achieve the same beam heightclose to and far from itself, marking the height ofthe beam on a white index card with a pen.

FIG. 2: Schematic of setup for Fresnel equation investiga-tion for vertically polarized light. Mirror M1 is the rightmostoptic. The “V” polarizer is shown in place.

FIG. 3: Schematic of setup for Fresnel equation investigationfor horizontally polarized light, with “R” and “H” polarizersinserted.

2. Set the angular readout of the rotational stage to 0degrees by releasing the stage lock screw, settingthe stage angle to 0 degrees, and tightening thestage lock screw again. Do not overtighten the setscrew.

3. Position the prism in its mount so that the laser beamenters on the prism face marked with the white dot,and hits in the middle of the face when the face isperpendicular to the beam.

You will now make the prism face perpendicular to thetable.

4. Look for the back reflection from the prism face by bypoking a small hole through the card at the beamheight, then holding it in front of M so that thebeam passes through the hole.

5. Slightly release the lock screw of the prism postholder, but not so much the post slips down. Asyou rotate the post back and forth, you will noticetwo back reflections on from the prism swingingacross the card. One of the spots is an external re-flection from the front surface – this is the one youwant; the other is due to multiple internal reflec-tions from the hypotenuse and opposite right sideof the prism (see diagram). To differential betweenthe two, drop a drop of methanol on the side ofthe prism indicated in the diagram, while observ-ing both spots. The spot you DO NOT want willacquire ripples, which speedily vanish before youreyes (why does this happen!!??)

6. To set the left-right adjustment, rotate the post in thepost holder until the reflected spot is centered (pos-

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sibly above or below) the hole in the card. Tightenthe post holder lock screw.

7. To set the up-down adjustment, use the adjusterscrews on the bottom of the prism stage. Yourgoal is to get the back-reflected spot centered onthe hole in the note card.

You will first measure vertically polarized light – youmust decide if this is s− or p− polarization!

8. Place the polarizer marked “V” about 5 in. awayfrom M1, this establishes vertical polarization forthe light incident on the prism.

9. To make your test of the Fresnel equations, rotateprism stage and measure the intensity of the re-flected light as a function of the angle of incidenceusing the photodiode. Starting with zero degrees,record your data in the following pairs: (power re-flected off the prism, power incident on the prism).Take data every 4 degrees or so, for angles of inci-dence as close to 90◦ and 0◦ practical. Rememberto use the correct spot identified in step 5 above!!You will have to move the photodiode each timeto be centered on the reflected spot. Put the pho-todiode right in front of the prism, at close to thesame spot each time, when you take the second datavalue in each pair. Make sure to keep your beamin the center of the face of the prism by makingsmall adjustments to the final-bounce mirror (don’tworry – this doesn’t change your angle of incidenceby much). Always ensure the entire beam spot fallswithin the black square in the photodiode package.Read out the voltage on the voltmeter to as manydigits as believe to be valid.

9a . Use the same technique as in the Law of Maluspart of the lab to estimate your uncertainty in thesemeasurements. Pick one angle of of reflection (say45 degrees or so) and take 10 measurement pairs ina row.

Now re-take the measurements for horizontally polar-ized light.

10. Place the “H” polarizer about 2 inches after the“V” polarizer. You will find that not much lightgets through, because the light incident upon it ismostly vertically polarized.

11. To get some light the “H” polarizer through, makeuse of the law of Malus – put the “R” polarizer in-between the “H” and “V” polarizers. You shouldnow find that light is transmitted through the “H”polarizer.

12. Repeat the data taking instruction of step 9 and 9a.

VI. ANALYSIS

Plot your data with error bars. Test the Fresnel equa-tions by fitting your data to the following model:

Ir(θi)

Ii= R(θi − θ0) + C

What is the physical reason for possibly needing theparameters θ0 and C? Plot a best fit model overlaid witha plot of your data, with error bars. Plot your residualswith error bars. Report the values and uncertainties ofall fitted parameters, discuss the meaning (or arbitrari-ness) of the values of each parameter with respect to ver-ifying/contradicting the theory under test. Plot the fitresiduals and discuss them. Report the chi-squared perdegree of freedom statistic obtained by your fit, and useit to discuss the likelihood that, if the Fresnel equationswere true, you could expect to see data such as yours.Explain any discrepancy; note – saying “we must havemade an error....” is not an explanation.

VII. OPTICAL ACTIVITY OF SUCROSESOLUTION

Many substances exhibit “optical activity,” meaningthey rotate the polarization of transmitted light. Sucrosedissolved in water is such a substance. Linearly polarizedlight passing through l (in units of decimeters) of a su-crose solution of concentration c ≡ msucrose/Vsolution in

units of [ grams of sugarml of total solution ] will be rotated through an an-

gle:

α = [α]lc (2)

where [α] is constant called the “specific rotation” of thesolution. The specific rotation depends strongly on thewavelength of the light, so that it is typically furtherspecified:

[α]Tλ

Because of a weak temperature dependence, T = 20 Cshould also be specified, and λHeNe = 632.8 nm for yourlasers. Under these conditions, the specific rotation ofyour sucrose in water solution is

[α]20632 = 5.72144deg

mm · gml

.

You will use polarimetry to measure the unknown con-centration of a water+sucrose solution. There are 8beakers of sucrose solution that have been pre-prepared.Each group will experiment with a different beaker–retrieve the one with your group’s number on it fromthe refrigerator in room 208. Your grade in this lab willdepend in part on how accurately you determine the trueconcentration.

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FIG. 4: Schematic of setup for optical rotation of a su-crose+water solution.

VIII. PROCEDURE

1. Clean your cuvette thoroughly with distilled water.

2. Fill your cuvette approximately half-full of yourgroups assigned sucrose solution, using a stirringrod to pour to prevent solution from getting on theclear sides of your cuvette (it is ok to get solutionon the the frosted sides.)

3. Place the “H” polarizer after mirror M, to definitelyset the polarization of the laser to be linear.

4. Adjust the height of the cell so that the beam passesthrough the top, unfilled part.

5. Place the precision polarizer “P” just after the cell.

6. Position the photodiode to record the light passingthrough “P”.

7. Turn the room lights off for all measurements inthis part of the lab, and take care that no straylight from any group’s desk lamps reaches on thephotodiode by placing black aluminum foil aroundthe photodiode.

8. Rotate the “P” polarizer until the voltage is approx-imately minimized.

9. Tighten the fine adjust lock screw (ask your instruc-tor for help with this), and use the fine-adjust tofurther minimize the voltage. Now, read off the an-gle to a precision of 10 arcseconds using the vernierscale on the mount. See the appendix for instruc-tions on how to read the vernier scale.

10. Now raise the cell so that the laser passes completethrough the sugar solution. Use the fine-adjust tofind a new angle which minimizes the voltage. Youwill not have to rotate it more than a few degrees!

IX. ANALYSIS

Estimate your error in determining both the angle ofthe polarizer. Calculate your estimate of the sugar con-

centration, and your uncertainty in this value, using Eq.(2). Note that path length in your cell is 10.0 mm.

X. APPENDIX: READING A ROTATIONALVERNIER SCALE

The ticks on the coarse scale (the one printed on therotating bezel) are 2 degrees apart. The ticks on thevernier scale (a.k.a. fine scale, the one on the top ofthe optic, which doesn’t move as rotate the polarizer)are 10/60ths of a degree apart. The complete reading ofthe scale is the sum of two readings, acquired as follows.In the picture, the “0” of the vernier scale is slightly tothe right of the 30 degrees tick of the coarse scale. Thecomplete angular reading is therefore 30 + V degrees,where V is an amount to be determined next. To findV , examine the ticks on the vernier scale lying to theright of the vernier “0”. Identify whichever vernier tickbest lines up with ANY tick on the coarse scale. In thispicture this happens to be 5th tick to the right of thevernier “0”, and it lines up with the 40 degree tick markof the coarse scale. We calculateV = (5th tick to the right of vernier “0”) × (10/60 de-grees per vernier tick) = 50/60 degrees = 0.83 degrees.Therefore the complete reading of the scale is 30.83 de-grees. Note that it is possible for V to be larger than 1degree, but never larger than 2 degrees. Another impor-tant note is to IGNORE the “30” and “60” labels on thevernier scale – they are misprinted, and should read “60”and “120”.

You will need to use a 10 cm convex lens as a magnify-ing glass to make accurate readings of your rotationalvernier scale. Beware of parallax – the phenomenonthat objects can look aligned or misaligned with one an-other depending on where you view them from. Here’sa demonstration. Line up your extended thumb withthe wall clock. Close your left eye, then your right eye.For one eye the objects will align, for the other eye theywill not. In the case of this vernier rotational scale, atick on the vernier scale is said to line up with a tickon the coarse scale ONLY if it lines up when lookingSTRAIGHT ALONG the tick. Note that the picture ofFig 5. was taken looking straight down the 5th tick mark.If you looked at this mount from a different angle, someother tick would have appeared to line up, but this wouldbe an incorrect reading.

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FIG. 5: This scale reads 30.83 degrees. Do you see it?