Polarization I: Malus’s law

PHYS430-530, Department of Physics, Northern Illinois University(Dated: October 18, 2019)

The polarization of light − conventionally taken as describing the direction of the electric field intime − plays an important role in many electromagnetic phenomena. The purpose of this lab is toshow the effect of several polarizers on a laser-beam intensity.

I. INTRODUCTION & OBJECTIVES

The purpose of this lab is to show the effect of severalpolarizers on a laser-beam intensity.

Consider an incoming wave is polarized and its fieldis given by EEE0. The wave encounter a polarizer with itstransmission axis making an angle θ with respect to theincoming electric field; see Fig. 1(a). The field can bedecomposed in term of the polarizer transmission axis(x′, y′) as E1,x′ = E0 sin θ, and E1,y′ = E0 cos θ. Taking,e.g., the axis y′ to be the transmitting1 axis we obtainedthe wave intensity downstream of the polarizer to be

I1(θ) = I0 cos2 θ, (1)

where I0 ≡ aE20 is the peak intensity achieved when θ = 0

and a is a constant. The latter equation is often referredto as Malus’ law after E.L. Malus (1775-1812). This lawactually applies to any two polarizing elements whosetransmission directions make an angle θ with each other(the first polarizer will in essence fix the incoming polar-ization reaching the second polarizer).

As an extension of the previous problem, we nowconsider a setup with three polarizers as depicted inFig. 1(b). Taking the case of an incoming unpolarizedwave, the first polarizer (P1) defined the transmittedlight polarization. We further assume that the first (P1)and last polarizer (P3) have their transmission axis ro-tated such that θ = 90◦. In such a case when the secondpolarizer (P2) is absent no light is transmitted. Whenwe insert P2 and take its transmission axis to make anangle φ with the transmission axis of P1 we find that thetransmitted intensity after P2 to be I2 = I1 cos2 φ. Con-sequently since the P3 transmission axis makes an angleπ/2 − φ with the transmission axis of P2, the intensitytransmitted downstream of P3

I3 = I2 cos2(π

2− φ

)= I1 cos2 φ cos2

(π2− φ

)=I14

sin2(2φ) (2)

The main goal of this laboratory is to measured theintensity transmitted through the two configuration de-scribed above. You will also use a laser and a photodi-ode to measure the signal. One technique implemented

1 this choice is arbitrary we can take the x′ axis to be the trans-mission axis and get Mallus law written as I1(θ) = I0 sin

2 θ

FIG. 1. Configuration and notations associated to the twocases considered in the theory section: a single polarizer (a)and a setup consisting of three polarizers(b).

FIG. 2. Photograph of the experimental setup used in thislaboratory.

in the setup is the use of an optical chopper to modulatethe laser beam and provide a baseline to the diode foran intensity measurement. Such optical-chopping tech-nique is commonly used when charactering continuous-wave (CW) beams or sources.

II. EXPERIMENTAL APPARATUS &METHODS

A. Overview

The experimental setup consists of a source (alaser with associated collimation system), a set of in-sertable polarizers (two polaroid polarizer and one Glan-Thompson prism), and a detection system comprising adiode and associated electronics; see Fig. 2. Each of the

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system is described below.

B. Source

The source include a HeNe laser followed by acollimator-iris system. A assembly consisting of a lensand an iris collimate the beam to a large radius (5 m)and the iris clip the laser beam to the desired size.

C. Polarizers

In this lab you will use a couple of polarizers picturedin Fig. 3:

• A Glan-Thompson prism consists of two right-angled calcite prisms that are cemented together bytheir long faces. The optical axes of the calcite crys-tals are parallel and aligned perpendicular to theplane of reflection. Birefringence splits light enter-ing the prism into two rays, experiencing differentrefractive indices; the p-polarized ordinary ray istotally internally reflected from the calcite?cementinterface, leaving the s-polarized extraordinary rayto be transmitted. The prism can therefore be usedas a polarizing beam splitter.

• A polaroid polarizer consists of many microscopiccrystals of iodoquinine sulphate (herapathite) em-bedded in a transparent nitrocellulose polymerfilm. The needle-like crystals are aligned duringmanufacture of the film by stretching or by apply-ing electric or magnetic fields. With the crystalsaligned, the sheet is dichroic: it tends to absorblight which is polarized parallel to the direction ofcrystal alignment, but to transmit light which ispolarized perpendicular to it. If the wave interactswith a line of crystals as in a sheet of polaroid,any varying electric field in the direction parallelto the line of the crystals will cause a current toflow along this line. The electrons moving in thiscurrent will collide with other particles and re-emitthe light backwards and forwards. This will cancelthe incident wave causing little or no transmissionthrough the sheet. The component of the electricfield perpendicular to the line of crystals howevercan cause only small movements in the electrons asthey can’t move very much from side to side. Thismeans there will be little change in the perpendic-ular component of the field leading to transmissionof the part of the light wave polarized perpendicu-lar to the crystals only, hence allowing the materialto be used as a light polarizer.

FIG. 3. Photograph of the Glan-Thompson-prism (left) andtwo polaroid polarizers.

D. Detection system

The detection system consists of four items: A pho-todiode which detects the laser beam and produce anelectrical signal. The diode is connected to an oscillo-scope where the output can be measured. In addition anamplifier is inserted between the oscilloscope and diode.To continuously get a reference signal the laser beam in-tensity is mechanically modulated using an optical chop-per (consisting of a rotating wheel with holes). in theabsence of any component the signal measured on thescope consists of square-tooth trace with it maximumand minimum respectively corresponding to the cases oflaser on and off (i.e. blocked by the chopper). Thereforea measure of the peak-to-peak height provide a quantityproportional to the laser intensity I ∝ E2. In the fol-lowing we will call this signal the diode signal and youwill record its value in Volt using the oscilloscope. Anabsolute measurement of the power on the diode couldin principle be possible but would required a precise cal-ibration of the diode (not available for this lab).

III. EXPERIMENTAL PROCEDURE

Before starting, you should familiarize yourself withthe experimental setup and identify each of the system(laser, collimator with iris, optical chopper, and detectionsystem). In a first step you should make sure none of thepolarizer are in the path. Check the beam laser beam isalign all the way and hit the detector area of the diode.Adjust the iris size to ensure the full beam is containedwithin the diode’s detection area. Make sure your turnon the amplifier (and dial a rotation frequency for thechopper).

A. Data taking

1. Single polarizer: Insert the Glan-Thompsonprism in front of the diode and align it to makesure the laser beam passes through its center.

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(a) Set the angle for a value and measure the in-tensity of the diode output using the oscillo-scope.

(b) Repeat the previous setup for for various angleof the Glan-Thompson prism and record thediode output I and prism angle θ.

2. Two polarizers: Insert one of the polaroid po-larizer downstream of the optical chopper (do notremove the Glan-Thompson prism) and align it tohave the laser-beam center passing on the polarizeraxis.

(a) record the value of the angle θ0 on the polaroidpolarizer.

(b) vary the Glan-Thompson prism θ and for eachpoint record the light intensity as done in step(1) above. Record the value I versus θ.

(c) repeat steps 2a, and 2b for few (at least 5)values of the angle θ0.

3. Three polarizers: we now are going to add a thirdpolarizer in our setup. Before, set the polaroid po-larizer to an angle θ0 and adjust the angle of theGlan-Thompson prism θ to ensure the diode sig-nal is zero no laser beam is transmitted). Recordyour value. Now insert a second polaroid polar-izer downstream of the first one and position it toensure the laser passes through its center.

(a) record the value of the angle θm on the 2ndpolaroid polarizer.

(b) vary the polaroid polarizer 2 angle θm andrecord the diode intensity you should have atable giving the I versus θm.

At the end your measurement you should have severaltables giving the transmitted laser intensity as a functionof angles. The analysis consists in fitting this data withthe equations derived in the theory section above.

B. Analysis

From the data taken you should be able to recover n(λ)along with its error bar (please detail the error propaga-tion and analysis).

1. Single polarizer:

(a) Plot the recorded diode signal as a function ofthe angle of the Glan-Thompson prism.

(b) Discuss your observations and discuss the im-plication on the polarization of the laser.

2. Two polarizers:

(a) Plot the recorded diode signal as a function ofthe angle of the Glan-Thompson prism I(θ)for the five value of θ0 you selected

(b) Fit your data with an Malus equation usinga form I = A cos2(θ − B). For each set youwill obtain a value for A and B (so in total5 values for each parameters). How do theAi compares to each other. Is that expected?What about Bi? Can you relate Bi to θ0?

3. Three polarizers:

(a) Plot the diode intensity I as a function of φand fit the resulting data with the Eq. ??adapted to introduce the proper offset (asdone for Malus’ law in the previous step).

C. Further discussion

Do you see any practical application associated to thecrossed-polarizer setup introduced in the ”Three polar-izer” setup?