Faraday`s Effect
Contents
1. Table 4 Numerical integration of the magnetic field Gee le eee a BG ams BG ams jWGem l i You will now measure iqac the dc component of the detected signal in the ab sence of magnetic field with the sample in place Perform optical alignment uf required 9 Connect the photodiode to the oscilloscope select the dc input mode Re move the background reading from ambient light either by placing a black tube or by making the background level at the datum Open the laserhead shutter 10 Rotate the analyzer angle and find out the maximum and minimum in tensity Set the analyzer at angle of 45 approximately w r t polarizer Note 22 11 12 13 14 15 16 down the value of voltage at the oscilloscope and divide it by 1 MQ the input impedance of oscilloscope to get tge Neat you will determine iac the rms value of the ac component of the photocurrent x intensity using the lock in amplifier Activate the magnetic field Provide the reference signal from the oscillator to the lock in amplifier Select the reference frequency mode f Hz in the reference channel of lock in amplifier Select the current input mode J from the input section Make the offset equal to zero Connect the photodiode to the input BNC connector Make sure that no error indication unlock or overload occurs Turn on the band pass and line frequency filter The effect of each key press on the lock i2. a polarization Also show that linearly polarized light can be written as a sum of left and right circular light 1 2 1 Magneto optical effect in transmission geometry Michael Faraday observed the relationship between electromagnetism and light in 1845 Faraday s observation gave birth to the field of magneto optics the interaction of optical radiation with magnetic media or the interaction of light with an optically inactive medium placed inside a magnetic field 2 1 1 Birefringence Some substances are optically anisotropic t e their optical properties are di rection dependent An atom can be viewed as a positive charge surrounded by an electron shell with some binding forces the dipole oscillator model For an anisotropic substance the binding forces on the electron are anisotropic implying that the spring constant will be different in differ ent directions an electron displaced from its equilibrium position along one direction will oscillate with a differ ent frequency than another direction Since the electric field associated with light drives the electrons of medium at its frequency these electrons reradi ate The resulting secondary wavelets recombine and light propagates through a medium The speed of the wave through the medium is therefore determined by the difference in natural res onating frequency of electrons and the frequency of the applied electric field With anisotropy the whole process b
3. z 3 2 2 _ ee a 2 Gea n For N number of turns Eq 49 becomes axial 9 a 22 8 2 bo Ni 50 18 For the Helmholtz pair if one coil is placed at z 0 and the other at z a and if current flows through both the coils in same direction referred to as superposition condition Figure 12 by symmetry the radial component of magnetic field along the axis must be zero Hence the magnetic field on the common axis of the coils becomes 6 Figure 12 Pair of Helmholtz coil with separation equal to the common radius and carrying the current in same direction HUoNi r 2E 142 zg Mt 8 24 1 E 51 a a Show that at the point on the axis midway between the coils z a 2 the field is 4 bo Ni BS 52 TE 52 Using the binomial expansion 1 2 1 nz we show that Eq 51 can also be written as olN1 pa 1 cyz4 eg2 53 2a where c4 15 8a and cg 105 48a What do you conclude from equation 53 about the uniformity of the magnetic field How does the field in the middle of the coils compare with the field in the center of a single circular loop of the same radius In our experiment the Helmholtz coil is constructed from 18 gauge copper wire diameter 1 2 mm Each multilayer coil consists of 18 turns in 18 layers the coil s outer and inner diameters are 10 2 cm and 6 5 cm respectively The length of each coil is 2 7 cm and radius is 4 5 cm
4. 2m na 56 m 0 Let the amplitude of coefficient of the terms containing Qt and 2Qt be represented by s and s respectively Then A sj 5 21 J 26 sin 2 ay UN Be seo 7 q ylq 1 1 2 7 1 2s I 5 A gt TO 2 q 4 We T imeo e it At het gtis Aofi t S08 i sin 20 rs IT is the conventional symbol to generalize the factorial function Since we are using I for numerically integrated magnetic field therefore we have used y to denote the general form of factorial function 26 A S2 292 280 cos 2 yE eu qiy q 2 1 2 A P y as 1 9 _ 6 cos 2 y 3 yA 2ly 5 2 0 zoi a o l A Ao gp f Ta ga cos 2 59 32 a Since bs V Bad 60 Substituting equation 60 in 58 j 1 s A 2V Bd f VBod lV Bad sin 20 U V Bod f VB d lV Bad sin 2 61 where U is the steady power on photodetector when polarizers are set for maximum transmittance in the absence of applied magnetic field Substituting equation 60 into 59 we obtain A 1 VB d 1 VB d s SAV Bud SV Bad 5 2B d cos 2 VB a 1 5V Bol Zj Bod 0820 62 The f and 2f components are determined through lock in amplifier which displays rms values so from equation 61 the rms value of the first harmonic component of output current ignoring
5. higher order terms is e lsin n 2 U V Bd sin 2 63 Uy where B B V2 B represents the rms value of the field measured by the Gauss meter Similarly from 62 the rms value of the second harmonic component of output current is V Bod cos 26 64 U2 N 5 V Bd cos 2 65 Both equations 63 and 65 can be used to determine Verdet constant In short we have three three different means of measuring the Farday rotation at Method 1 The gradient of the plot of wu or tac against B for 45 results in the Verdet constant This is in fact the method you have used in previous section Since U 2ige and u1 iac Equation 63 is actually Eq 43 in disguise Method 2 Determine the gradient of the least squares fit line to a plot of u B against sin 2 for fixed U Equate the gradient to V BdU and find the Verdet constant 9 Method 3 Determine the gradient of a plot of u gt against B when 90 equate this to to V2d U 2 and find the Verdet constant Soo Find the Verdet constant for TGG at 405 nm using methods 2 and 3 28
6. is 15 minutes and its output is linearly polarized therefore remove the polarizer A use analyzer B only and repeat all the steps 23 4 1 Applications of Faraday rotation 4 1 1 Optical isolators An optical isolator acts as a photon valve passing radiation in one direction and blocking in the other An isolator is shown in Figure 14 Polarizer A is used to make the beam horizontally polarized which is then passed through a 45 Faraday rotator C followed by another linear polarizer analyzer B at 45 relative to A If any of the light is reflected or backscattered from analyzer it undergoes an extra rotation of 45 by C and thus is blocked by A In a LASER if any of the emitted light returns into the active medium through an unwanted reflection it can generate instabilities in the emission Optical isolators are used to prevent the unwanted reflection in lasers 7 3 AO 45 O Figure 14 Optical isolator backscattered radiation undergoes an additional 45 rotation by C thus is blocked by polarizer A 4 1 2 Domain Observation Light will have different characteristics after reflection or transmission by regions having different orientations of magnetic moments Let a sample be made up of three domains the magnetization of each domain is shown in Figure 15 Plane polarized light of wavelength A passing through domain A is rotated through some angle 01 while interacting with C is rotated 6 If the analyzer is
7. of circular current is given by Me tA il nr 4 where ew l 5 i 2 5 whereas the angular momentum of electron is given by L rxp L mvr mrw 6 Substituting Eqs 4 5 6 into 3 we get 2 wr GOE 7 8 5 showing that the Larmor frequency wy is independent of the orientation of the current loop and the overall effect is the rotation of electronic structure about the direction of applied magnetic field 3 2 1 4 Semi Classical description of induced birefringence You must have realized from Q 1 that plane polarized light is a combination of left and right circular l and r polarized light Now if light of vacuum frequency f is traveling through a medium whose electrons are rotating at the Larmor frequency then the l and r components will rotate the electron clouds with frequencies f fr and f fr Therefore in the dispersive medium refractive index is frequency dependent the functional dependence of the respective refractive indices can be written as n u J and Np n f fi If plane polarized light traverses a distance d then the optical path lengths for l and r light are nd and n d respectively so the optical path difference is n n d The difference of two refractive indices the induced birefringence is nr n f fr n f ft Using the Taylor Series d d m a AEFI O go 9 af 10 From Equation 8 O WE eb j Qn Arm Eq 10 become
8. splitting of spectral lines of the atom when placed in magnetic field is called the Zeeman effect after the name of discoverer P Zeeman From Eq 3 Lwy L eB e He Be gh Since the left and right l and r components of light carry an angular momentum of h and A respectively the l component drives electron into left circular motion and the r component drives electrons into right circularly motion resulting in different magnetic moments Interaction of the magnetic moment ue with the magnetic field B slightly shifts the energy of atomic level by an amount om 15 e AE A ueB Ape B h B 16 Thus under the application of an axial magnetic field dispersion curves for left and right circularly polarized light are identical but displaced by the frequency difference between the two Zeeman components AE e Aw SB h m which results in two different refractive indices n and n and therefore a different speed at a given w 2 Figure 5 Refractive indices for left and right circularly polarized components of plane wave in the presence of magnetic field The dispersion curves for the two components are shifted by Aw 2 2 Jones calculus Jones calculus invented by the American physicist R Clark Jones in 1941 is a useful formalism to understand the state of polarization of perfectly polarized light 8 as well as its transformation by various optical devices For example polarized li
9. the laser and sample holder to pass the beam through the center of the coils you might need to adjust the heights of optical components by translating the stainless steel posts Close the shutter again The height of 2l the sample holder will now be kept fixed Mark the end points of the sample on the scale 7 Remove the sample Fix the axial probe on another holder on either side of the Helmholtz coil and turn on the Gauss meter Select the 200 G range and ac mode Switch on the audio amplifier and tune the function generator to the resonating frequency 1 22 kHz Current is now passing through the Helmholtz coil Select some value of current and measure the corresponding magnetic field at the midpoint between the Helmholtz coils A field of 90 G rms is a reasonably good value Move the probe away from the center of coils on both sides Check that magnetic field is not reaching the polarizers and photodiode If required adjust the distances by moving the rail carriers along the length of the optical rail Map the magnetic field profile by moving the probe along the length of the sample with a step size of 0 5 cm for different values of current similar to the observation tabulated in 4 Remove the probe and turn off the magnetic field Tabulate the variation in magnetic field along the length of the sample as in Table 4 8 Estimate the numerically integrated magnetic field over the length of the sample d T X B z Az 54
10. Inductance of the coils determined using the LCR meter is found to be 7 mH with a resistance of 1 5 Q for each coil so the total inductance of the Helmholtz pair is 15 mH and the total resistance is 3 Q The Helmholtz coil pair constitutes a series RLC circuit At resonating frequency 19 Helmholtz coil Gaussmeter Figure 13 Instruments for creating and detecting the oscillating magnetic field wr the inductive reactance Xz is equal to the capacitive reactance Xz and total impedance is purely resistive The resonating frequency is _ fi tr VIC fr Or 1 2n LC Komna Calculate the resonating frequency when a capacitor of 0 97 uF is con nected in series with the coil Why is the Helmholtz coil made resonating 4 The Experiment 1 Assemble the setup according to Figure 10 Turn on the audio amplifier and the function generator Amplify an approximately 1 V 70 Hz sinusoidal signal through audio amplifier Apply this amplified output to the Helmholtz coil 2 Increase the frequency of the ac signal applied to the coil Tabulate the frequency against current passing through the Helmholtz coil Table 2 and plot the frequency response The current is measured with the help of a clamp meter or an ammeter 20 Frequency Hz Current A rms Table 2 Mapping the frequency response of the Helmholtz coil 3 Set the function generator at the resonating frequency Increase the current by increas
11. Phase Sensitive Faraday rotation Aysha Aftab Rabiya Salman and Sabieh Anwar LUMS School of Science and Engineering Tuesday February 09 2010 Can light propagating through a medium be influenced by the application of an external magnetic field You have observed optical activity in chiral molecules in your freshmen lab The present experiment extends these concepts to magnetically induced birefringence through the historically important Faraday Effect which reveals the rich interplay between optics and magnetism KEYWORDS Polarization Birefringence Faraday rotation Verdet constant Phase Sensitive Detection Jones Calculus Laser Helmholtz coil Resonance in RLC series circult APPROXIMATE PERFORMANCE TIME 1 week PRE REQUISITE EXPERIMENT Basic measurements with the Lock in amplifier 1 Objectives In this experiment we will 1 shed some light on the underlying mechanism of magnetically induced bire fringence 2 demonstrate the advantages of phase sensitive detection PSD 3 understand the mathematical formalism for polarized light and its manipu lation 4 build or use sources of uniform magnetic fields and measure the field strengths using a commercial magnetometer l 5 calculate numerical integrals 6 build resonant RLC series circuit and understand the resonance phenomenon 7 calculate the Verdet constant of terbium gallium garnet TGG and of a diamagnetic liquid Re
12. at 6 A may be dark C bright and D of intermediate shade For analyzer at 90 D will be dark A and C will be equally bright For analyzer set at 041 A will be bright C dark and D of intermediate shade i e polarization direction may be turned one way or the other depending on magnetization thus resulting in different intensities it is therefore possible to image magnetic domains 7 4 1 3 Circulator Optical circulators are used in fiber optics to separate light traveling in opposite directions Figure 16 shows one such circulator It is made up of two Foster Seely Prisms and a 45 rotator placed between the prisms In these prisms the rejected polarized light is internally reflected so that it exits perpendicular to the axis of prism Horizontally polarized light entering along a passes straight through the 24 hs Figure 15 The domain imaging through magneto optic rotation prism is rotated to 45 by the rotator using Faraday rotation and emerges from second prism at b However any light reflected back to the circulator entering at b with polarization azimuth 45 undergoes a 45 rotation through the rod thus polarized at 90 and exits from port c Similarly light entering at c emerges at d and entering at d exits at a 7 A circulator has at least three ports The 45 rotator a b lt O C 90 d 435 Figure 16 45 rotator placed between two Foster Seely prisms constitutes a circu lato
13. easuring instruments and transverse probes akeShore 410 Kyoritsu KEW SNAP 2017 QuadTech Tne Optical breadboard 90 x Thorlabs PBI51506 60 x 6 cm Rotation mount 2 de Thorlabs RSPO5 M gree resolution A i D Laser post 20 cm long Thorlabs P200 M V shaped laser housing Thorlabs C1502 M Glass cell 6 cm Tong Crescent shaped cell holder Teflon crystal holder Laser safety glasses OD 4 OD 7 Thorlabs LG4 LG10 M6 and M4 screws Thorlabs TOG crystal d 1 om Lock in amplifier Clamp meter Homemade 16 Helmholtz coil Photodetector analyzer B a Lens C Sample Laser polarizer A Laser power supply Audio amplifier De supply Signal generator Figure 10 Schematic of experimental setup for Faraday rotation Lock in amplifier it as the light source HeNelaser of wavelength 633 nm or an electrically pumped diode laser of wavelength 405 nm will be used in the experiment EO What is the basic principle of a laser How does a HeNelaser work 3 2 5 Mechanism for producing and detecting the magnetic field In principle both ac and dc magnetic field can be used in this experiment Dc sources include permanent magnets or solenoids having steady current in their windings Since Faraday rotation is small in magnitude of the order of microra dians so a large dc magnetic field of several kilo gauss will be required to achieve a sizeable rotation which in turn requir
14. ecomes direction dependent Since the refractive index n c v is a function of speed the anisotropy results in different refractive indices along different directions This so called birefringence manifests itself the in rotation of the plane of polarization 1 figure 1 Negatively charged shell bound to positive nucleus by pairs of spring hav ing different stiffness 2 1 2 Faraday rotation Chiral compounds exhibit rotation of linearly polarized light due to natural bire fringence but this birefringence can also be induced in otherwise optically inactive materials either by applying stress magnetic or electric field The Faraday effect is magnetically induced birefringence Linearly polarized monochromatic light while transmitting through an optically inactive material under the influence of an axial magnetic field is rotated by an angle 0 as shown in Figure 2 The angle of rotation 0 is given by 3 Direction of propagation Figure 2 Faraday rotation The plane of polarization of light is rotated under the action of an axial magnetic field 0 V Bd 1 provided the magnetic field remains uniform throughout the length d of sample For non uniform magnetic field is given by 0 V f Bae 2 The proportionality constant V is a characteristic of the material called the Verdet constant and is a function of the wavelength of light temperature and refractive index of the material It is the rotation per u
15. es large and bulky dc magnets or a large dc power supply to produce required field 8 However using an ac magnetic field the rotation becomes oscillatory and can be tracked by PSD For example in this experiment you will be provided with a Helmholtz coil capable of generating a field of approximately 120 G rms The Helmholtz coil A pair of Helmholtz coils is used to produce a uniform magnetic field over a large volume of space It consists of two identical coils such that separation d of the coils is equal to their common radius a Let us consider a single loop of conductor of radius a carrying a current t Using the Biot Savart rule magnetic induction at the point P at a distance r is Ho dB nye dl x u 45 where u is the unit vector connecting the conducting element with the point at Lg Figure 11 Magnetic field at point P due to single circular coil carrying a current Ve which the field is to be determined ju is permeability of free space 4r x 107 Hm This geometry is shown in Figure 11 Substituting a 46 r sin Z into Eq 45 we get dB sin a idl x u 47 Ata The axial component of magnetic induction is dBoriald dBsina Therefore oa oe sin a i dl x u 48 Since dl is perpendicular to u and integrating round the coil f dl 2ra we obtain the total axial field ob Baria n sin a 27a _ Hot 3 sin a 2a o bot a 2a a
16. ferences 1 Eugene Hetch and A R Ganesan Optics 4th edition Pearson Education Inc India 2008 pp 318 320 2 Daryl W Preston and Eric R Dietz The Art of Experimental Physics John Wiley and Sons 1999 pp 355 362 3 Frank L Pedrotti and Peter Bandettini Faraday rotation in the undergrad uate advanced laboratory Am J Phys 58 542 544 1990 4 Lock in amplifier user Manual Stanford Research System SR 510 http www thinksrs com 5 Aloke Jain Jaynat Kumar Fumin zhou and Lian Li A simple experiment for determining Verdet constants using alternating current magnetic fields Am J Phys 67 714 717 1999 6 David Jiles Introduction to Magnetism and Magnetic Materials 2nd edi tion CHAPMAN and HALL CRC 1998 pp 20 25 7 JAN SMIT Magnetic Properties of Materials Inter University Electronic Series Vol 13 McGRAW HILL New York p 197 8 Frank J Loeffier A Faraday rotation experiment for the undergraduate physics laboratory Am J Phys 51 661 663 1983 9 K Turvey Determination of Verdet constant from combined ac and de mea surents Am J Phys 64 1561 1567 1993 10 _ Eric W Weisstein Jacobi Anger Expansion from MathWorld A Wolfram Web Resource http mathworld wolfram com Jacobi AngerExpansion html 2 Theoretical introduction Q 1 What is polarization of light Write down the equation for linear and
17. ght given by E z t iE cos kz wt bz J Eoy cos kz wt dy 17 is represented in the Jones formalism as E z t Fone om Bist HENS one ell z wt 18 at E 2 t Eye 18 The two component column vector completely specifies the amplitude and phase of electric field and hence its state of polarization This is called the Jones vector Most of the times it is not necessary to know the exact phase but the phase difference between the x and y components Moreover e is always understood to be present Accordingly Jones vector can also be written as E z t a Ja 19 Ignoring the term et Bags oa 20 For linearly polarized light 0 or 180 therefore the general form of Jones vector for linearly polarized light is E z t 21 oy Jones vectors can be normalized such that the sum of the squares of their compo nents is 1 1 e Beb o r iay l This normalized form discards the amplitude information needed for absorption calculations but simplifies analysis in many other cases The normalized form of 21 at an angle a w r t an arbitrary reference axis is COS Q Ble 28 J where the angle a is defined such that Lo V Eor Eoy Hog COSQ E oy ia oy Ey sna 9 Write down the normalized Jones column vector for horizontally verti cally left and right circularly polarized light S
18. ing the gain of the amplifier Table 3 Measure the magnetic field using the Gaussmeter in ac mode LakeShore Model 410 equipped with transverse probe at the midpoint between the two coil Plot a graph between current and magnetic field Do you observe a linear relationship as predicted by equation 52 Current A rms Magnetic field G rms Table 3 Linear relationship between the current A and the magnetic field Gauss Next you will determine the Verdet constant for Terbium Gallium Garnet TGG crystal and a diamagnetic liquid such as carbon disulphide CS3 Never touch the lateral surfaces of the TGG crystal 4 Turn on the provided laser The HeNetakes about 30 mins to warm up and reach a stable value Never look directly at the laser light Always wear safety goggles when operating the laser For the time being keep the laser power switched on but close the shutter of the laser head 5 The TGG crystal is 1 cm long but CS is filled in a 6 cm long glass cell We cannot expect the magnetic field in between the coils of Helmholtz pair to be uniform over this large a distance So you need to map the magnetic field profile and perform numerical integration of the magnetic field as suggested in Equation 44 The next few steps will help you calibrate the magnetic field 6 Fix a scale with the edge of the sample holder Place the glass cell over the crescent shaped sample holder Open the laserhead shutter Adjust the height of
19. measured intensity is 7 Aoi c0s2 8 Ah cos 2 cos 20 sin 2 sin 20 2 1 a sin 20 13 For 45 and sin 20 S 26 A2 I amp z 1 26 37 The field is made oscillatory with an oscillating frequency Q B Bosin Ot and since the angle of rotation is directly dependent on the magnetic field 0 A sin Qt 38 therefore Eq 37 can be written as z A2 1 a sin Qt 39 3 2 2 Converting light intensities into photocurrents The photodiode converts the photon intensities into current thus i idet ihe where tde A and il oA sin Qt Modulated photocurrent due to Faraday rotation tac is measured through lock in amplifier which displays the rms values therefore the output of the lock in amplifier is i OA _ ac lee 7 V Taking the ratio of the modulated current shown by the lock in amplifier to the steady current we obtain ae OA 2 fe Ah 2 ao tde V2 A a gt 0 Al V2 lde and as far as the rms value of the Faraday rotation angle is concerned 0 ae ENEE PN 42 VA Dii The dc component is measured by an oscilloscope in the absence of magnetic field while the ac component is measured by the lock in amplifier in the presence of the magnetic field For a uniform magnetic field Verdet constant is determined from the experimental values of 0 B and d 9 V Bd 43 14 dc a ae Se ee 1 time background
20. n amplifier is indicated by a nearby LED Select a suitable sensitivity usually 200 nA and time constant 3 ms for the pre filter Check that no offset is introduced by the lock in amplifier at the selected sensitivity scale Adjust the phase located in reference section to make the output equal to zero Then introduce a phase shift of 90 bringing the reference and input signals in phase Rotate the analyzer angle in steps of 10 and tabulate tac for any fixed value of the magnetic field You will observe that the maximum rotation occurs when analyzer is at an angle of approximately 45 relative to polarizer What does the reading on the lock in amplifier physically repre sent Fix the analyzer at 45 relative to the polarizer Increase the magnetic field from an initial value of 10 Gauss in steps of 5 or 10 Gauss by increasing the current The transverse probe of Gaussmeter can be fixed to observe the magnetic field in the center of Helmholtz coil Tabulate the values for iac for each value of current and hence the corre sponding magnetic field passing through the coil Use the above results to calculate the Verdet constant of your sample Eon Clearly quantify your uncertainties What are the major sources of error Q 20 Can you measure tge with the help of the lock in amplifier Remove the He Nelaser from setup place AlGaN diode laser Turn the diode laser on its warm up time
21. nit path length per unit applied magnetic field In other words it quantifies the induced birefringence In this experiment you will measure this induced birefringence 2 1 3 Larmor precession of the electron cloud in an applied magnetic field We now try to posit some foundational arguments describing the underlying mech anism of Faraday rotation Consider an electron moving in a circle of radius r in a plane whose normal makes an angle a with an applied magnetic field B Since an electron is negatively charged its angular momentum L and magnetic moment He are opposite to each other The magnetic field exerts a torque T on the magnetic dipole pe T He X B ueB sina Referring to Figure 3 what is the direction of the torque on the magnetic dipole According to Newton s second law an angular impulse 7 produces a change in angular momentum Tdt dL Thus the attached vector L rotates in anticlockwise direction The resulting pre cession traced out by tip of the vector L is shown in Figure 3 The angle of rotation through which angular momentum s projection along the applied field L moves in time dt is LA ar plane of electron rotation Figure 3 Precession of angular momentum vector about the direction of the ap plied magnetic field d dL L Ttdt Lsina and the precessional or the Larmor angular velocity becomes do T UeBsina peb 3 WJ _ e II dt Lsina Lsin a L The magnetic moment
22. olarizer Figure 8 Rotation 0 of plane of polarization of light analyzer is oriented at relative to the polarizer 12 the transformation matrix is J 0 cos o et 31 sin cos sin Write the Jones vector for the combination of the polarizer sample and analyzer placed in the same order You can conclude from Q 8 that electric field of the light beam after emerging from the sample followed by the analyzer is E ete Peng Anexpilke wi 32 The intensity of light measured by the photodetector is I kAslcos 80 33 Derive the expression 33 What are the dimensions of the constant k In the subsequent discussion we will normalize k 1 HINT Use the concept of the Poynting vector 3 2 1 Optimization of the analyzer angle According to Eq 33 the rotation of the plane of polarization manifests as a change in intensity at the photodiode To get maximum change in intensity the analyzer angle needs to be optimized Differentiating the intensity w r t we get a A 2cos A sin 0 34 A sin2 9 35 Differentiating again d 5 Po 7 2A cos 2 8 36 Maximum change in intensity is obtained by maximizing 7 or by setting a 1 24A cos 2 0 0 since Ag 0 we have cos 2 0 0 6 0 45 Since the Faraday rotation 0 is much smaller than maximum AJZ is obtained when the analyzer is set at 45 relative to polarizer The
23. quency are shown in Figure 7 4 The large peaks at 50 Hz and its multiples are due 10 Display _ gt Detector Light source Sample Figure 6 A simple optical system oO v a 2 ke E z E T wo D c G ge no o 4 Cc i f a o O QL z O Z 50 100 150 frequency Hz 50 100 150 frequency Hz Figure 7 a Noise and signal amplitude as a function of frequency b Modulating the signal to a region of low noise to electrical interference from the mains power lines The noise power increases at lower frequencies remember this is due to 1 f noise Faraday rotation is extremely small in magnitude If such a small signal buried in noise is to be measured amplifying the signal will not improve the signal to noise ratio the noise is amplified with the signal A clever approach is to move the signal to a region of low noise to higher frequency For example in the present experiment we use an ac magnetic field for inducing Faraday rotation instead of a dc field produced by dc current or a permanent magnet This technique gives two real advantages e The weak signal of interest buried in noise can be extracted successfully through PSD e Faraday rotation can be observed at smaller values of magnetic field e g 80 G rms This circumvents the need for large expensive bulky water cooled electromagnets for producing large magnetic fields Can you think of a simple experiment that measu
24. r light beam if entering from one port after passing through circulator exits from the second Another light beam entering from the second port or light reflected from second port exits from third port and so on 5 OPTIONAL Measurement of the the Verdet constant using higher harmonic components The light rotated by the Faraday medium incident on the photodetector from analyzer contains fundamental as well as components at higher frequencies The rms values u and us at f and 2f respectively of these current components are 29 measured where f is the frequency of ac signal passing through Helmholtz coil The ratios u U and u2 U can also be used to determine the Verdet constant where U is the steady output from the photodiode under zero magnetic field and analyzer set for maximum transmittance 9 The power transmitted through a Faraday rotator is A L 5 ll cos 2 0 55 2 foh cos 2 4 cos Qt 2 A a 1 cos 2 cos 26 cos Nt sin 2 sin 24 cos Qt Using the Jacobi Anger expansion we obtain 10 cos 26 cosQt J 20 2 ae Jom 20 cos 2mQt sin 26 cosQt 2 S o 1 Jom 1 20 cos 2m 1 9t m 1 where the Bessel function is L a Lie SA E ee e o ee and y is the factorial function given by y n n 1 Therefore Eq 56 becomes T f cos 2 i 20 2 Do Jom 20 2 os 2m0 sin 2 2 N 1 Jam 1 200 cos
25. res the noise spectrum of laser light detected by a photodetector li Po What is Malus s law How does a polarizer work 3 2 Overview of the experiment The plane of polarization of linearly polarized monochromatic light traversing through the sample S of length d placed under the influence of an ac magnetic field is rotated Since the field is oscillatory the rotation angle is also oscillatory Another polarizer set at an arbitrary angle relative to input polarizer subsequent to the sample is required to analyze the rotation The analyzer converts the po larization modulation to an amplitude modulation by the way of Malus s Law The emerging light beam carrying the information in the form of amplitude vari ations is incident upon a photodiode whose output appears in the form of current proportional to the light intensity Let us suppose incident light polarized along the x axis is propagating in the z direction The electric field in terms of Jones vector is i H A exp i kz wt 28 where A corresponds to the amplitude of the electric field After passing through the sample S of length d placed in magnetic field the plane of polarization of light is rotated by an angle 0 so the Jones vector after emerging from the sample is cos 0 o 4 29 and the corresponding electric field is E o 4 A expi kz wt 30 sin 6 Rete Suppose the analyzer is set at an angle w r t the polarizer show that p
26. s eB dn Np n 2 tam ap Since phase change of a wave is k 27 X times the physical path traversed by the wave the phase change for the two components is b Yan 11 b ESen 12 a Figure 4 Superposition of left and right circularly polarized light into linearly polarized light a Entering the sample both the l and r components are moving with same speed and b while passing through the sample these components have travelled with different velocities When l and r waves enter the sample the phase difference is zero but the phase difference accumulates as light passes through the sample The vector sum of the two electric fields on emerging from the sample is shown as E with a net rotation 0 from its initial value Since E is an equal superposition of l and r components we see from Figure 4 that Opt Pr 0 Qi _ Pr dg 7 2 Thus the Faraday rotation angle is 1 27d 0 5 Vira nr Td eB dn Fy A 2am df e dn Bad 1 FSN if 13 Comparing Eq 1 and 13 the Verdet constant gt a Ff 2mA which is a function of wavelength and the dispersion 3 The Faraday rotation is a direct result of n n arising because of the frequency dependent refractive index 14 2 1 5 Description of dispersion based on the Zeeman Effect The physical reason behind the change in refractive index can also be explained through Zeeman splitting The
27. ss safe eee napnannnnne J Figure 9 Signal measured by photodiode is made up of two parts average light intensity 7g and modulated intensity at the frequency of ac magnetic field ihe The currents are proportional to the intensities whereas for non uniform magnetic field 0 is given by 0 V f Beye 44 What is the working principle of a photodetector What does the pho todetector measure The electric field or the intensity Can the photodiode measurement be affected by stray magnetic field HINT the Hall Effect 3 2 3 Schematic of the experiment Figure 10 shows the schematic diagram of the experimental setup for the observa tion of Faraday rotation The setup comprises these components a Light Source b Mechanism for producing and measuring an oscillating magnetic field c Detection devices 3 2 4 Light source Light from a lamp can be used after collimating it by a lens and passing through a color filter to make it monochromatic however since LASER is a source of highly directional and monochromatic light and is easily available it is convenient to use 15 Table 1 List of Equipment used in the experiment Component Supplier Component Supplier o Magnetic field production Audio amplifier 150 W_ CERWIN VEGA Helmholtz coil 120 G Honni rms Detection element Photodiode Newport 818 SL Stanford Fat Sys tem SR 510 WTth Gaussmeter with axial s LakeSh 4 M
28. uppose that the Jones vector for polarized incident beam E is represented by E after transmission through an optical element then the optical element can be represented as a 2 x 2 transformation matrix J called the Jones matrix given by where J a gt 23 J21 J22 Equation 22 can be written as Eix Ja e i 24 es Ga Ja E g If the beam passes through a series of optical elements represented by the matrices Ji Jo J3 sany da then E Ja J3 Jo J1 Ey 25 The matrices do not commute so they must be applied in proper order Ton Show that the transformation matrix J for horizontal linear polarizer is J k 4 26 HINT Write expression 25 for maximum and zero transmittance solve the simultaneous equations to get the coefficients of transformation matrix 3 Experimental Technique 3 1 Why PSD in Faraday rotation You have already performed an introductory experiment of using the lock in am plifier so without discussing the details of the technique and the instrumentation any further we will only focus on why are we using phase sensitive detection PSD in this experiment Consider a simple optical system used to measure the trans mission of light through a medium Let us suppose a small response obscured by overwhelming noise is to be measured The output signal in this case will be Vo VaT Vase 27 The noise and signal amplitudes for such a system as a function of fre
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