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

Transcription

1 Polarzaton of lght: Malus law, the Fresnel equatons, and optcal actvty. PHYS 3330: Experments n Optcs Department of Physcs and Astronomy, Unversty of Georga, Athens, Georga 3060 (Dated: Revsed August 0) In ths lab you wll () test Malus law for the transmsson of lght through crossed polarzers; () test the Fresnel equatons descrbng the reflecton of polarzed lght from optcal nterfaces, and (3) usng polarmetry to determne the unknown concentraton of a sucrose and water soluton. I. POLARIZATION III. PROCEDURE You wll need to complete some background readng before your frst meetng for ths lab. Please carefully study the followng sectons of the Newport Projects n Optcs document (found n the Reference Materals secton of the course webste): 0.5 Polarzaton Also read chapter 6 of your text Physcs of Lght and Optcs, by Peatross and Ware. Your pre-lab quz cover concepts presented n these materals AND n the body of ths wrte-up. Don t worry about memorzng equatons the quz should be elementary IF you read these materals carefully. Please note that takng a quck look at these materals 5 mnutes before lab begns wll lkely NOT be adequate to do well on the quz.. Poston the fxed V polarzer after mrror M to establsh a vertcal polarzaton axs for the laser lght.. Carefully adjust the poston of the photodode so the laser beam falls entrely wthn the central dark square. 3. Plug the photodode nto the bench top voltmeter and observe the voltage t should be below 00 mv; f not, you may need to attenuate the lght by [antcpatng the valdty off Malus law!] nsertng the polarzer on a rotatable arm R upstream of the V polarzer, and rotatng R untl the readng on the photodode s approxmately 00mV. II. MALUS LAW 4. Place the precson rotatable polarzer P just n front of the photodode. Make sure the beam goes through cleanly. When completely lnearly polarzed lght s ncdent on a polarzer, Malus law predcts the transmtted ntensty to vary proportonally as the square of the cosne of the angle between the transmsson axes of the analyzer and the drecton of polarzaton of the ncdent lght: I(θ) = I 0 cos θ You wll test ths law usng the polarzed lght of a HeNe laser, an assortment of polarzers, and a photodode an electrcal devce whch generates an electrc current proportonal to the ntensty of ncdent lght. 5. Rotate P to maxmze the transmtted power. Then rotate the polarzer so that the 0 of the verner scale lnes up wth a tck mark of the fxed scale, to establsh a nce place to start your systematc study of ntensty versus angle. 6. Now, rotate the analyzer n steps of 4 degrees, recordng the transmtted power at each step. Take data over at least 80 degrees. Estmate (and explan/justfy for estmates of) the uncertantes n your measurements of voltage and the angle. (Hnt: to estmate your voltage uncertanty, try montorng the photodode voltage for a few mnutes wthout changng anythng, to see how stable t s over a duraton as long as t wll take you to measure around 80 degrees...) IV. ANALYSIS Plot your data wth error bars. Test the Law of Malus by fttng your data to the followng model: FIG. : Schematc of setup for Malus law nvestgaton. P (θ) = kv 0 cos (θ θ 0 ) + C ()

2 What are the parameters V0, θ0, C? What s k and what are ts unts? Consder that volts s not a unt of (optcal) power. To calculate the optcal power n watts, you need to know that the photodode takes every ncdent optcal photon and converts t nto one electron. Ths current passes over a XX Ohm resstor to produce the voltage you measure on the voltmeter. Calculate the current and you know the photons/s; calculate the energy (Joules!) per photon, and combne these results, and you know the optcal power (Watts=Joules/s) ncdent the photodode. Use these deas to calculate the constant k n Eq. (). Plot a best ft model overlad wth a plot of your data, wth error bars. Plot your resduals wth error bars. Report the values and uncertantes of all ftted parameters, dscuss the meanng (or arbtrarness) of the values of each parameter wth respect to verfyng/contradctng the theory under test. Plot the ft resduals and dscuss them. Report the ch-squared per degree of freedom statstc obtaned by your ft, and use t to dscuss the lkelhood that, f the Malus theory were true, you could expect to see data such as yours. Explan any dscrepancy; note sayng we must have made an error... s not an explanaton. Now endeavor to drectly establsh the values of V0, θ0, and C through auxlary measurements (of your own devsng). Ft your prevous data to a model n whch these parameters are no longer treated as free parameters of the ft, but rather are hardwred constants. Plot the new best ft model (overlad wth your data and the prevous model). Report the values and uncertantes of all ftted parameters, dscuss the meanng (or arbtrarness) of the values of each parameter wth respect to verfyng/contradctng the theory under test. Plot the ft resduals and dscuss them. Report the ch-squared per degree of freedom statstc obtaned by your ft, and use t to dscuss the lkelhood that, f the Malus theory were true, you could expect to see data such as yours. V. FRESNEL EQUATIONS The Fresnel equatons gve the transmsson and reflecton coeffcents at a delectrc nterface. They depend upon the polarzaton and angle of ncdence, and ndces of refracton of the meda on both sdes of the nterface. For lght crossng from a medum of ndex n to one of n, the fractonal reflected ntensty s predcted to be, for pure s polarzed lght (refer to your text for the defntons of s and p f you are unsure). FIG. : Schematc of setup for Fresnel equaton nvestgaton for vertcally polarzed lght. Mrror M s the rghtmost optc. The V polarzer s shown n place. FIG. 3: Schematc of setup for Fresnel equaton nvestgaton for horzontally polarzed lght, wth R and H polarzers nserted. whereas for pure p-polarzed lght the theory predcts: q sn θ n cos θ p n (n /n ) I Rp (θ ) = rp = q I n (n /n ) sn θ + n cos θ A. Procedure You wll measure the reflecton of polarzed lght off of a BK7 boroslcate glass prsm. There are several steps to the algnment of the optcal system.. Make the laser beam parallel to the surface of the table by adjustng M to acheve the same beam heght close to and far from tself, markng the heght of the beam on a whte ndex card wth a pen.. Set the angular readout of the rotatonal stage to 0 degrees by releasng the stage lock screw, settng the stage angle to 0 degrees, and tghtenng the stage lock screw agan. Do not overtghten the set screw. 3. Poston the prsm n ts mount so that the laser beam enters on the prsm face marked wth the whte dot, and hts n the mddle of the face when the face s perpendcular to the beam. You wll now make the prsm face perpendcular to the table. Rs (θ ) = Irs Is q (n /n ) sn θ q = n cos θ + n (n /n ) sn θ n cos θ n 4. Look for the back reflecton from the prsm face by by pokng a small hole through the card at the beam heght, then holdng t n front of M so that the beam passes through the hole.

3 3 5. Slghtly release the lock screw of the prsm post holder, but not so much the post slps down. As you rotate the post back and forth, you wll notce two back reflectons on from the prsm swngng across the card. One of the spots s an external reflecton from the front surface ths s the one you want; the other s due to multple nternal reflectons from the hypotenuse and opposte rght sde of the prsm (see dagram). To dfferental between the two, drop a drop of methanol on the sde of the prsm ndcated n the dagram, whle observng both spots. The spot you DO NOT want wll acqure rpples, whch speedly vansh before your eyes (why does ths happen!!??) 6. To set the left-rght adjustment, rotate the post n the post holder untl the reflected spot s centered (possbly above or below) the hole n the card. Tghten the post holder lock screw. 7. To set the up-down adjustment, use the adjuster screws on the bottom of the prsm stage. Your goal s the get the back-reflected spot centered on the hole n the note card. You wll frst measure vertcally polarzed lght you must decde f ths s s or p polarzaton! 8. Place the polarzer marked V about 5 n. away from M, ths establshes vertcal polarzaton for the lght ncdent on the prsm. 9. To make your test of the Fresnel equatons, rotate prsm stage and measure the ntensty of the reflected lght as a functon of the angle of ncdence usng the photodode. Take data every 4 degrees or so, for angles as close to 80 and 0 practcal. Remember to use the correct spot dentfed n step 5 above!! You wll have to move the photodode each tme to be centered on the reflected spot. You must also make sure to keep your beam n the center of the face of the prsm, by small adjustments to the fnal-bounce mrror (don t worry ths doesn t change your angle of ncdence by much). Ensure the entre beam spot falls wthn the black square n the photodode package. Read out the voltage on the voltmeter to as many dgts as beleve to be vald. To estmate your voltage measurement uncertanty, measure the same spot three tmes, each tme movng the photodode out of the beam, then back n. Now re-take the measurements for horzontally polarzed lght. 0. Place the H polarzer about nches after the V polarzer. You wll fnd that not much lght gets through, because the lght ncdent upon t s mostly vertcally polarzed.. To get some lght the H polarer through, make use of the law of Malus put the R polarzer nbetween the H and V polarzers. You should now fnd that lght s transmtted through the H polarzer.. Repeat the data takng nstructon of step 9. VI. ANALYSIS Plot your data wth error bars. Test the Fresnel equatons by fttng your data to the followng model: I r (θ ) = I R(θ ) + C Plot a best ft model overlad wth a plot of your data, wth error bars. Plot your resduals wth error bars. Report the values and uncertantes of all ftted parameters, dscuss the meanng (or arbtrarness) of the values of each parameter wth respect to verfyng/contradctng the theory under test. Plot the ft resduals and dscuss them. Report the ch-squared per degree of freedom statstc obtaned by your ft, and use t to dscuss the lkelhood that, f the Fresnel equatons were true, you could expect to see data such as yours. Explan any dscrepancy; note sayng we must have made an error... s not an explanaton. Now endeavor to drectly establsh the values of V 0, θ 0, and C through auxlary measurements (of your own devsng). Ft your prevous data to a model n whch these parameters are no longer treated as free parameters of the ft, but rather are hardwred constants. Plot the new best ft model (overlad wth your data and the prevous model). Report the values and uncertantes of all ftted parameters, dscuss the meanng (or arbtrarness) of the values of each parameter wth respect to verfyng/contradctng the theory under test. Plot the ft resduals and dscuss them. Report the ch-squared per degree of freedom statstc obtaned by your ft, and use t to dscuss the lkelhood that, f the Fresnel equatons were true, you could expect to see data such as yours. VII. OPTICAL ACTIVITY OF SUCROSE SOLUTION Many substances exhbt optcal actvty, meanng they rotate the polarzaton of transmtted lght. Sucrose dssolved n water s such a substance. Lnearly polarzed lght passng through l (n unts of decmeters) of a sucrose soluton of concentraton c m sucrose /V soluton (n unts of grams of sugar /00 ml of soluton) wll be rotated through an angle: α = [α]lc ()

4 8. Rotate the P polarzer untl the voltage s approxmately mnmzed. 4 FIG. 4: Schematc of setup for optcal rotaton of a sucrose+water soluton. where [α] s constant called the specfc rotaton of the soluton. The specfc rotaton depends strongly on the wavelength of the lght, so that t s typcally further specfed: [α] T λ Because of a weak temperature dependence, T = 0 C should also be specfed, and λ HeNe = 63.8 nm for your lasers. Under these condtons, the specfc rotaton of your sucrose n water soluton s [α] 0 63 = deg dm g/00ml. You wll use polarmetry to measure the unknown concentraton of a water+sucrose soluton. There are 8 beakers of sucrose soluton that have been pre-prepared. Each group wll experment wth a dfferent beaker retreve the one wth your group s number on t from the refrgerator n room 08. Your grade n ths lab wll depend n part on how accurately you determne the true concentraton. 9. Tghten the fne adjust lock screw (ask your nstructor for help wth ths), and use the fne-adjust to further mnmze the voltage. Now, read off the angle to a precson of 0 arcseconds usng the verner scale on the mount. See the appendx for nstructons on how to read the verner scale. 0. Now rase the cell so that the laser passes complete through the sugar soluton. Use the fne-adjust to fnd a new angle whch mnmzes the voltage. You wll not have to rotate t more than a few degrees! IX. ANALYSIS Estmate your error n determnng both the angle of the polarzer. Calculate your estmate of the sugar concentraton, and your uncertanty n ths value, usng Eq. (). Note that path length n your cell s 0.0 mm. VIII. PROCEDURE. Clean your cuvette thoroughly wth dstlled water.. Fll your cuvette approxmately half-full of your groups assgned sucrose soluton, usng a strrng rod to pour to prevent soluton from gettng on the clear sdes of your cuvette (t s ok to get soluton on the the frosted sdes.) 3. Place the H polarzer after mrror M, to defntely set the polarzaton of the laser to be lnear. 4. Adjust the heght of the cell so that the beam passes through the top, unflled part. 5. Place the precson polarzer P just after the cell. 6. Poston the photodode to record the lght passng through P. 7. Turn the room lghts off for all measurements n ths part of the lab, and take care that no stray lght from any group s desk lamps reaches on the photodode by placng black alumnum fol around the photodode.

5 5 X. APPENDIX: READING A ROTATIONAL VERNIER SCALE The tcks on the coarse scale (the one prnted on the rotatng bezel) are degrees apart. The tcks on the verner scale (a.k.a. fne scale, the one on the top of the optc, whch doesn t move as rotate the polarzer) are 0/60ths of a degree apart. The complete readng of the scale s the sum of two readngs, acqured as follows. In the pcture, the 0 of the verner scale s slghtly to the rght of the 30 degrees tck of the coarse scale. The complete angular readng s therefore 30 + V degrees, where V s an amount to be determned next. To fnd V, examne the tcks on the verner scale lyng to the rght of the verner 0. Identfy whchever verner tck best lnes up wth ANY tck on the coarse scale. In ths pcture ths happens to be 5th tck to the rght of the verner 0, and t lnes up wth the 40 degree tck mark of the coarse scale. We calculate V = (5th tck to the rght of verner 0 ) (0/60 degrees per verner tck) = 50/60 degrees = 0.83 degrees. Therefore the complete readng of the scale s degrees. Note that t s possble for V to be larger than degree, but never larger than degrees. Another mportant note s to IGNORE the 30 and 60 labels on the verner scale they are msprnted, and should read 60 and 0. You wll need to use a 0 cm convex lens as a magnfyng glass to make accurate readngs of your rotatonal verner scale. Beware of parallax the phenomenon that objects can look algned or msalgned wth one another dependng on where you vew them from. Here s a demonstraton. Lne up your extended thumb wth the wall clock. Close your left eye, then your rght eye. For one eye the objects wll algn, for the other eye they wll not. In the case of ths verner rotatonal scale, a tck on the verner scale s sad to lne up wth a tck on the coarse scale ONLY f t lnes up when lookng STRAIGHT ALONG the tck. Note that the pcture of Fg 5. was taken lookng straght down the 5th tck mark. If you looked at ths mount from a dfferent angle, some other tck would have appeared to lne up, but ths would be an ncorrect readng.

6 FIG. 5: Ths scale reads degrees. Do you see t? 6