ias 2013- abhishek

Indian Institute of Space Scienceand Technology

Department of Physics

Project Report

Author:Surya Abhishek SingarajuIMSc 2nd year,

University of Hyderabad

Supervisor:Dr.C.S.Narayanamurthy

Professor and Head,

Department of Physics, IIST

Investigation on Some OpticalInterferometers using Coherence and

Geometric Phase

July 15, 2013

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Contents

1 Declaration 3

2 Acknowledgements 4

3 Abstract 5

4 Multiple Beam Interferometry 54.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54.2 Fabry Perot Interferometer . . . . . . . . . . . . . . . . . . . . 5

4.2.1 Results and Discussions . . . . . . . . . . . . . . . . . 74.3 Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

5 Geometric Phase Interferometry 105.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.2 Michelson Interferometer . . . . . . . . . . . . . . . . . . . . . 11

5.2.1 Results and Discussions . . . . . . . . . . . . . . . . . 155.3 Sagnac Interferometer . . . . . . . . . . . . . . . . . . . . . . 17

5.3.1 Results and Discussions . . . . . . . . . . . . . . . . . 205.4 Mach Zehnder Interferometer . . . . . . . . . . . . . . . . . . 22

5.4.1 Results and Discussions . . . . . . . . . . . . . . . . . 245.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

6 Conclusion 28

7 Appendix 297.1 Appendix.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297.2 Appendix.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297.3 Appendix.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

8 References 31

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1 Declaration

I,Surya Abhishek Singaraju, a student of the University of Hyderabad study-ing Integrated MSc 2nd year(11ICMC13), and an Indian Academy of Sciences2013 summer fellow with application number PHYS1530 hereby declare thatall the work done under the topic ”Investigation on Some optical Interfer-ometers using Coherence and Geometric Phase” is done by me under theguidance of Prof.C S Narayanamurthy, Professor and Head,Department ofPhysics,Indian Institute of Space Science and Technology in his AdaptiveOptics labaratory from 15 May 2013 to 15 July 2013.

Place:

Date:

Signature(Supervisor): Signature(Summer Fellow):

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2 Acknowledgements

Firstly,I thank my parents who always supported me regardless of my achieve-ments without which I would not have been into science.I thank my sis-ter,Gayathri for giving nice valuable suggestions from her research experi-ences.I thank my cousin,Sudhir and my friends, Vamsi, Ritu, Divya, Harish, Sahithi,Vivek, Satwik, Nithin, Dinesh, Praneeth and all other school and collegemates who stayed in touch with me,making sure I didnot miss home for thetwo months.I also thank Mrs Jayasree for welcoming me to their home and helping mewhenever I needed.I thank my lab mates,Remya,Nikhil,Richa,Vinu and Anand for the wonderfulcompany in the laboratory.I also thank the IIST canteen and maintenance staff for providing me witha comfortable stay for two months.Lastly, I thank my guide,Prof.C.S.Narayanmurthy who always encouragedme to try new things and helped me through constant discussions.

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3 Abstract

Optical Coherence and phase are integral part of interferometry and play acrucial role in the same. In this project, we investigated the effect of op-tical coherence on interferometry for different light sources in Fabry Perotinterferometer and the effect of Geometric(Pancharatnam) phase on interfer-ometry in various amplitude splitting interferometers like Michelson,Sagnacand MachZehnder interferometers. Also,in these experiments,Pancharatnamphase is observed even in the presence of only a single phase retarder. De-tailed experimental analysis is carried out.

4 Multiple Beam Interferometry

4.1 Introduction

Optical Interference corresponds to the interaction of two or more light wavesyielding a resultant irradiance that deviates from the sum of the componentirradiance.[1] The most important conditions for optical interference are thatthe light waves should be of the same wavelength and should be coherent.Optical Coherence is the attribute of two or more waves, or parts of a wave,whose relative phase is nearly constant during the resolving time of the ob-server.[2]. It is related to the stability or predictability of the phase. Coher-ence is classified into temporal and spacial Coherence. Temporal Coherencedescribes the relation between the waves in terms of time difference whereasSpacial coherence describes the same as a function of distance. The influenceof coherence is studied in Fabry Perot etalon, a multiple beam interferometerusing different optical sources.

4.2 Fabry Perot Interferometer

Fabry Perot Interferometer consists of two parallel mirrors which are usedto produce multiple beam interference. When the mirrors are held fixed andadjusted to be parallel, it is called an etalon.It also consists of a convex lenswhich is used to collimate the beam on to the mirrors and another focusinglens.The etalon setup is as seen in Fig.1.

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Fig.1. Schematic of the Fabry perot etalon

In this Fabry Perot setup, the influence of optical coherence is studiedusing three light sources: Helium-Neon(He-Ne) laser, Sodium light and Mer-cury light. The Etalon spacing,d(Fig.1) is varied from 1mm to 17mm andat each value of the spacing, the three sources are used to check accuracyat the corresponding etalon spacing. A Charge Coupled Device(CCD) cam-era is used to view the interference pattern and IMAGE J software is usedto analyse the pictures. It is observed that for large etalon spacing(12mmto 17mm), He-Ne laser is precise and the calculated value of the spacingis best matching with the real value for He-Ne laser at the value.This isbecause He-Ne laser has high coherence length. For small spacing(1mm to4mm),Mercury(White) light gave accurate values of spacing from the calcu-lations which were matching the real values better than for other sources.Fringes are not observed after a spacing of 9mm.Sodium light is most accu-rate for intermediate values(4mm to 11mm). Fringes are not observed forSodium light after a spacing of 14mm. Pictures of the fringes observed forthe different light sources were taken while varying the etalon spacing.

Fig.2. Fringes observed for He-Ne laser source

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Fig.3. Fringes observed for Sodium source

Fig.4. Fringes observed for Mercury source

Figures 2 to 4 are the fringes observed at an etalon spacing of 5mm forHe-Ne laser,Sodium vapour and Mercury vapour lights respectively. It isclear from the three pictures that Sodium source has very clear and morenumber of fringes at 5mm spacing. He-Ne laser has lesser number of fringeswhich increased as the etalon spacing increased. But for mercury light, thefringes became more and more distorted as the spacing increased.

4.2.1 Results and Discussions

The values of the radii of the fringes are observed at every etalon spacing foreach light source. Starting from 1mm, the etalon spacing was increased till17mm and the number of fringes and their radii are measured correspondinglyand can be seen in the tables below. Tables 1,2 and 3 give measurements ofradius of fringes with respect to etalon spacing for He-Ne laser, Sodium lightand Mercury source respectively.

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Table 1: Radii of fringes observed for He-Ne laser

Fringes Radius of fringes(µ m)Etalon Spacing(mm) 17 15 13 11 10 8 5

r0 180 180 225 252 351 351 396r1 252 261 315 414 513 576 657r2 306 423 459 576 657 711 819r3 396 558 612 702 783 846 963r4 477 648 729 801 855 963 1116r6 576 738 801 882 945 1224r7 873 1035 1341

Table 2: Radii of fringes observed for Sodium Light

Fringes Etalon Spacing(µ m)Etalon Spacing(mm) 14 12.3 10 8 6 5

r0 243 162 198 261 252 189r1 396 288 351 486 549 432r2 513 504 558 657 810 594r3 639 612 684 828 1026 855r4 801 693 819 927 1179 1026r5 891 801 891 1026 1314 1170

Table 3: Radii of fringes observed for Mercury Light

Fringes Radius of Fringes(µ m)Etalon Spacing(mm) 9 8 5.14 5

r0 243 234 180 180r1 423 486 522 423r2 603 666 774 711r3 756 819 936 927r4 882 927 1098 1071r5 990 1035 1242 1224r6 1089 1116 1341 1332

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From tables,1,2 and 3 it can be seen that as the etalon spacing increased,the number of fringes for He-Ne laser increased but for sodium and mercurylights, fringes could not be seen after 14mm and 9mm respectively. Thissensitivity of fringes is plotted in Plot.1 with respect to etalon spacing forlaser, Sodium lamp and Mercury source.

Plot 1: Comparisons in Fabry Perot Setup

In the plot, the green curve represents mercury light while the red andblue curves represent sodium and He-Ne laser sources respectively. Fromthe graph, we can see that upto an etalon spacing of around 2.5mm, whitelight has highest number of fringes(10). From 5 to 7mm, sodium light hasmost fringes(16) and from thereon, He-Ne laser has the maximum fringes. At15mm etalon spacing, the number of fringes for He-Ne laser is maximum(35)followed by Sodium(25).

4.3 Result

From the above graph and calculation of spacing from the fringe width andwavelength, it is observed that for a spacing of 1-3mm,Mercury(white) light

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gave better readings and so did Sodium light for spacing of 4-8mm and He-Nelaser is best suited for spacing beyond 8mm.So for small spacings white lightis best suited since its coherence is small and for large spacing, He-Ne laseris best since its coherence is large.For medium spacing, Sodium lamp suitsthe best.In interferometry, phase difference can be obtained through different means.Itcan be obtained by having a path difference between the interfering wavesas seen in the above Fabry Perot interferometer. But for the waves withouta path difference, the phase difference can be obtained by introducing cer-tain objects in the path of the wave. Geometric Phase interferometry is anexample of such a method.

5 Geometric Phase Interferometry

5.1 Introduction

Geometric phase interferometry is a method of introducing a phase differ-ence between the waves without the need of path difference and by justcreating few changes in the polarisation states of the light waves. When alight is taken from one polarisation state through few changes and broughtback along the same path to it’s original polarisation state, it is still not inthe same state as it was initially. It undergoes a phase shift. This phaseshift is known as Panchartnam phase[3]. This phenomenon is similar toa cyclic adiabatic process where the system comes to the initial state butwith a energy change. This phase shift can be measured from interferenceexperiments.Many experimental and theoritical attempts have been done suc-cessfully to prove the pancharatnam phase for polarised light and to plot iton the Poincare sphere.[4]-[14]. While the Pancharatnam phase was mostlyproved for cyclic polarisation state changes, it was also proved to be trueeven for non cyclic changes.[15]. We report experimental investigation ofthe Pancharatnam phase for different experimental setups with certain in-teresting changes in them. We did the experiments in amplitude-splittinginteferometers like Michelson,Sagnac and Mach Zender Interferometers byusing quarter wave plates and half wave plates. We tried to solve the mathe-matics of the experiments using Jones matrices[16]-[17] as it was done earlierand to plot it on the poincare sphere. The Jones matrices of each objectnamely the polariser, Quarter Wave plates(QWP), Half wave plates(HWP)and the mirror were taken into account[18]-[21] and the resultant matricesfor the setups were calculated.The jones matrices of few objects are listed below

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Jones matrix for a mirror

J =

(1 00 −1

)(1)

Jones matrix for Linear polarizer with axis of transmission vertical

J =

(0 00 1

)(2)

Jones matrix for a rotatable waveplate which can be rotated at an angle θ to the initial polarisation axis and creates

a phase difference of φ.[18]

J =

(cosφ/2 + isinφ/2cos2θ isinφ/2sin2θ

isinφ/2sin2θ cosφ/2− isinφ/2cos2θ

)(3)

For a quarter wave plate which induces a phase difference of π/2, the Jones matrix

J =1√2

(1 + icos2θ isin2θisin2θ 1− icos2θ

)(4)

For a halfwave plate which induces a phase difference of π on the light wave, the Jones matrix

J =

(cos2θ sin2θsin2θ −cos2θ

)(5)

This matrix information is used in the determination of the output ma-trices in each of the following interferometers.Experiments are carried outusing Michelson, Sagnac and Mach Zehnder interferometers.

5.2 Michelson Interferometer

The basic setup is simple and consists of two mirrors and a beam splitter.Thefigure is seen below(Fig.5)

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Fig.5. Basic michelson setup

The light from the source(He-Ne laser) is split into two beams by thebeam splitter and these are again reflected at the respective mirrors.SinceHe-Ne laser is a coherent source,both these reflected beams interfere and thisis seen at the detector.The Pancharatnam phase can be created in one of thebeams and to achieve this, a polariser and quarter wave plates, one at a fixedangle of 45 degrees to the initial polarisation and the other rotatable wereused[22].A photodetector is used to detect the intensity variation.(Fig.6)However, it is observed that even when QWP 1 is not present also there isPancharatnam phase.(Fig.7) and when both QWPs are on either arms(Fig.8)

Fig.6.Schematic of expt with 2 QWPs Fig.7.Expt with one QWP

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Fig.8.Schematic of another expt done with two QWPs on two arms

The Schematic of the closed path followed by the polarisation state of thelight for Fig.6 is given on the poincare sphere below.The initial light reach-ing the Quarter wave plate 1 is linearly polarised and is represented by pointA on the Poincare sphere.This light once passing through the QWP1 getscircularly polarised and is shown as point B on the poincare sphere.Afterpassing through QWP2(rotatable),it again becomes linearly polarised(pointC on poincare sphere).The light is then reflected from mirror M1 and retracesits path through the Quarter wave plates thus completing the loop throughD to A on the poincare sphere.The complete loop is ABCDA.

If θ and φ are the polar and azimuthal coordinate angles of any point onthe Poincare sphere,respectively, the 2 X 2 polarization or coherence matrixJ, normalized to unit trace, representing an arbitrarily (fully) polarized planewave[23] is of the form

Eqn.6. Coherence matrix of arbitrarily polarised plane wave

J =1

2

(1 + cosθ sinθeiθ

sinθe−iθ 1− cosθ

)(6)

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Fig.9. Poincare sphere schematic of the Pancharatnam phase in Michelson setup

From the knowledge of the Jones matrices, the matrix of each experimentis solved i.e for each arm in the Michelson setup.

The Jones matrix for the arm containing the single QWP in Fig.7(Appendix.1)

Eqn.7 Jones matrix for single QWP

J =

(sin22θ + icos22θ − sin4θ

2

− sin4θ2

−sin22θ − icos22θ

)(7)

The Jones matrices for each arm in Fig.8 containing Quarter wave platesis as following:

For the arm containing QWP fixed at 45 degrees to the initial polarisationaxis, the Jones matrix is written as

J =

(1 00 −1

)(8)

This is same as the matrix of a mirror.(see Appendix.1)

And for the arm containing the rotatable quarterwave plate, the matrixis same as Eqn.7 and is equal to

J =


2

− sin4θ2


)(9)

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In Eqn.8, we have seen that the matrix of the arm containing QWP fixed at45 degrees to the original polarisation state is same as that of mirror. So,this means the experimental setups in Fig.7 and Fig.8 are the same.To provethe pancharatnam phase with a single quarterwave plate, intensity measure-ments are taken for every 10 degree rotation of the rotatable quarterwaveplate for 360 degrees and are tabulated in Table.4 below.

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Table 4: Geometric Phase in Michelson setup

Rotation of HWP Intensity(mA) Intensity(mW)2 QWPs on same arm only one QWP 2 QWPs on either arm

0 0.5 3.08 2.77510 0.7 2.822 2.91420 0.8 2.67 3.12730 0.9 2.7 3.30540 1 2.92 3.35450 1.1 3.11 3.26460 1 3.28 3.02770 0.8 3.33 2.82880 0.6 3.291 2.74490 0.7 3.23 2.771100 0.3 2.95 2.952110 0.9 2.85 3.159120 1.1 2.94 3.294130 0.5 3 3.31140 1 3.2 3.179150 0.7 3.45 2.984160 0.9 3.55 2.798170 1.1 3.364 2.713180 1.6 3.18 2.786190 1.4 2.9 2.955200 1.3 2.735 3.206210 1.2 2.716 3.333220 1.1 2.835 3.333230 1 3.03 3.216240 0.9 3.141 3.053250 0.8 3.19 2.839260 0.5 3.087 2.744270 0.3 2.875 2.809280 0.2 2.7 2.982290 0.7 2.628 3.183300 1.4 2.706 3.297310 1.2 2.895 3.255320 0.8 3.071 3.21330 1 3.225 3340 1.2 3.208 2.73350 1.6 3.084 2.72360 1.9 2.902 2.7

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The graph of the angle of rotation of the quarterwave plate in degreesas it is rotated for 360 degrees and the resulting intensity of the interferencepattern in mA and mW is plotted for the three different setups namely withboth QWPs on same arm i.e the blue curve for which the intensity is plottedin mA, with a single QWP on the red curve and with both the QWPs oneither arm seen as the green curve for which the intensity is plotted in mW.

Plot.2.Geometric phase in Michelson setup

From the graph, it is clear that even the experiment with a single QWP isshowing Pancharatnam phase. Also it is shown that the quarter wave platefixed at 45 degrees and a mirror together act as a simple mirror reflection only.

5.3 Sagnac Interferometer

In the michelson interferometer, there could be an error in the geometricphase due to a possible change in the length of one of the optical paths lead-ing to a change in phase(dynamic phase).This problem is avoided by using a

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sagnac interferometer in which the two beams traverse the same optical pathin opposite senses. The phase difference between the beams thus dependsonly on the Pancharatnam phase[24].Sagnac Interferometer is similar to theMichelson interferometer but the mirrors are tilted in such a way that theyface each other so that both the beams reflected on either of them travelalong the same path.

Fig.10 Basic Sagnac setup

In the sagnac setup, two QWPs and a Half Wave Plate(HWP) in themiddle are kept along the beam between the two mirrors(Fig.11).In thissetup too, it is very clear that even in the absence of the two QWPs, thereis Pancharatnam phase(Fig.12).

Fig.11 Schematic of expt with QWPs,HWP Fig.12 Expt with only HWP

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The geometric phase is set on the path between both the mirrors usingquarter wave plates and a half wave plate(Fig.11).For detecting intensity,aphotodetector is used.The operation of the interferometer is given on thepoincare sphere below.Initially,the p-polarised light which is linear is con-verted to left circular polarisation state by QWP1 and is represented bypoint A1 on the poincare sphere.Then, the half wave plate whose axis is atan angle θ to the principal axis of QWP1 shifts this to right circular polari-sation state.Then the QWP2 brings the polarisation state back to the initiallinear polarisation.Thus the pancharatnam loop on the poincare sphere forp-polarised light is A1SA2NA1. Similarly for the s-polarised light it isB1SB2NB1.This arrangement gives rise to an additional 2θ phase differ-ence thus give=ing a total phase difference of 4θ.[24]-[25]

Fig.13.Poincare sphere schematic of the Pancharatnam phase in Sagnac setup

The Jones matrices of the experimental setups with only halfwave plateare found out.(see Appendix.2)

The jones matrix for one arm(p arm) in sagnac setup with only HWP

J =

(cos2θ −sin2θ−sin2θ −cos2θ

)(10)

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For the other arm(s arm) in sagnac setup with only HWP, Jones matrix

J =


)(11)

Jones matrix for 2 QWPs in sagnac setup

J =

(0 −i−i 0

)(12)


Even without the two quarter wave plates in the original setup, Pancharat-nam phase is observed. This is just by rotating the halfwave plate withoutthe quarter wave plates. To prove the pancharatnam phase with only halfwave plate, intensity measurements are taken for every 10 degree rotation ofthe rotatable quarterwave plate for 360 degrees and are tabulated in Table.5and Table.6 below.

Table 5: Geometric Phase with 2 QWPs and 1 HWP

θ Intensity(µA) θ Intensity(µA) θ Intensity(µA) θ Intensity(µA)0 4.7 100 3.9 190 1.9 280 5.410 2.2 110 7.2 200 0.9 290 8.920 1 120 10.7 210 0.9 300 11.830 0.8 130 13.4 220 1.2 310 14.340 1.2 140 14.4 230 1.3 320 14.850 1.5 150 13.6 240 1.1 330 13.260 1.4 160 11.2 250 0.9 340 10.170 1.1 170 7.4 260 1.3 350 6.980 1.1 180 4.1 270 3 360 3.490 2

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Table 6: Geometric Phase with only HWP

θ Intensity(µA) θ Intensity(µA) θ Intensity(µA) θ Intensity(µA)0 99.8 100 28.1 190 19 280 23.510 24.7 110 7.6 200 5.1 290 7.920 5 120 43.8 210 40.4 300 42.130 43.2 130 116.1 220 115.5 310 114.940 112.1 140 186.9 230 184 320 189.750 189.1 150 200 240 200 330 20060 200 160 200 250 200 340 20070 200 170 154.2 260 156.3 350 142.780 165.8 180 78.6 270 83.7 360 62.790 89.8

The graph of the angle of rotation of the halfwave plate in degrees as it isrotated for 360 degrees and the resulting intensity of the interference patternin µA is plotted for the two different setups namely with HWP between twoQWPs on the blue curve and with only HWP as the red curve.

Plot.3.Geometric phase in Sagnac setup

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From the graph, it is clear that even the experiment with only the HWPis showing Pancharatnam phase. In fact it is showing a better reult as themaximum and minimum intensities are observed clearly.

5.4 Mach Zehnder Interferometer

The next experiment done is to measure the Pancharatnam phase in a MachZehnder interferometer. This was done previously by Piotr Kurynomski,Wladyslawand Szarycz[26]. In a Mach Zehnder interferometer which is used to measurethe path shift caused by a medium(Fig.14), the Pancharatnam phase interfer-ometry was performed using quarter wave plates and half wave plate(HWP).The setup is such that the path travelled by the two beams split by a beamsplitter is the same.

Fig.14 Basic Mach Zehnder setup

In the Mach Zehnder setup, two quarter wave plates with a half wave platein the middle are inserted(Fig.15). However even in this setup, it is clear thatthere is Panchratnam phase even without the quarter wave plates(Fig.16).

Fig.15.Schematic of expt with QWPs,HWP Fig.16 Expt with only HWP

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On one arm, Pancharatnam phase is setup by introducing quarter waveplates and a half wave plate and the two beams are passed through anotherbeam splitter and then allowed to interfere. There is no path difference be-tween the two light beams but there is a phase shift on one arm due to theQuarter wave plates and the half wave plate(Fig.15). Each of these platestransform the polarisation state into B,C and come back to A in the figureshown below. Regardless of the intermediate states B and C the outgoinglight has same polarisation as input light A thus following the loop ABCA.However, there is a phase shift and following Pancharatnam,we can calculatephase shift as a half of the solid angle designated by the geodesic triangleon the Poincare sphere[26]. The Pancharatnam triangle to calculate the ge-ometric phase on the Poincare sphere for this is shown in Fig.17

Fig.17.Pancharatnam’s triangle on Poincare sphere in Mach Zehnder setup

The Jones matrices for the Mach Zehnder experiment done are found.(seeAppendix.3)

Jones matrix of arm with single halfwave plate

J =

(cos2θ −sin2θsin2θ cos2θ

)(13)

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Jones matrix of the arm with only two QWPs(see Appendix.3)

J =

(0 −ii 0

)(14)


Even without the two quarter wave plates in the original setup, Pancharat-nam phase is observed. This is just by rotating the halfwave plate without thequarter wave plates. To prove the pancharatnam phase with only half waveplate, intensity measurements are taken for every 10 degree rotation of therotatable quarterwave plate for 360 degrees and are tabulated in Table.7(2QWPs and HWP) and Table.8(only HWP) below.

Table 7: Geometric Phase with 2 QWPs and 1 HWP

θ Intensity(mW) θ Intensity(mW) θ Intensity(mW) θ Intensity(mW)0 3.32 100 4.138 200 5.184 300 4.15310 3.189 110 4.018 210 4.975 310 4.36320 3.394 120 3.896 220 4.88 320 4.96530 3.615 130 3.92 230 4.5 330 4.87540 3.855 140 4.072 240 4.793 340 5.18350 4.03 150 4.252 250 5.08 350 5.44360 4.08 160 4.3 260 5.28 360 4.99470 3.968 170 4.618 270 5.1880 4.076 180 5.034 280 4.71990 4.125 190 5.337 290 4.472

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Table 8: Geometric Phase with only one HWP

θ Intensity(mW) θ Intensity(mW) θ Intensity(mW) θ Intensity(mW)0 2.623 100 3.159 200 3.062 300 2.96510 2.696 110 3.155 210 2.99 310 2.9520 2.732 120 3.101 220 2.943 320 3.08630 2.924 130 3.073 230 2.896 330 3.12840 2.836 140 3.003 240 2.821 340 3.04750 2.804 150 2.786 250 2.896 350 3.10560 2.84 160 2.89 260 2.912 360 3.18570 2.888 170 2.998 270 3.00580 2.994 180 3.092 280 3.06490 3.103 190 3.108 290 3.021

The graph of the angle of rotation of the halfwave plate in degrees as it isrotated for 360 degrees and the resulting intensity of the interference patternin mW is plotted for the two different setups namely with HWP between twoQWPs on the blue curve and with only HWP as the red curve.

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Plot.4.Geometric phase in MachZehnder setup

From the graph, it is clear that even the experiment with only the HWPis showing Pancharatnam phase.Thus, in the three different interferometers,it is seen that even if the quarter wave plates fixed at 45 degrees are notpresent, there is Pancharatnam phase.

5.5 Results

In the geometric phase experiments done with Michelson,Sagnac and MachZehnder interferometers, it is observed that geometric phase is observed evenwhen the Quarter wave plates at 45 degrees are not present.In the experiments done in Michelson, Sagnac and Mach Zehnder interferom-eters,the Quarter wave plates fixed at 45 degrees were responsible for circularpolarisation and the construction of a Pancharatnam triangle was possible.However when there is only one phase retarder present, Pancharatnam phasecan still be constructed.It might look like the following

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Fig.18.Schematic of the closed path followed by polarisation states of light on the Poincare sphere

Figure.15 is the construction of Pancharatnam loop on the poincare sphereto calculate the geometric phase. The light which is initially linearly po-larised(represented by A in the above Poincare sphere) before entering theQuarter wave plate is elliptically polarised on coming out of the Quarter waveplate and is represented by point B on the Poincare sphere. After reflectionat the mirror, the light retraces its path through C to A. Thus the closedloop ABCA is constructed on the Poincare sphere thus concluding that thisis indeed a Pancharatnam Phase.

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6 Conclusion

In part one, i.e the multiple beam interferometry we concluded that as theetalon spacing increased, the accuracy of Mercury light and Sodium lightdecreased and He-Ne laser increased thus proving that white light is moreaccurate only for small spacing, Sodium light for medium spacing and He-Nelaser for longer spacings.In the second part, i.e in the Geometric Phase inter-ferometry, we could prove that Pancharatnam phase could be observed evenin the presence of a single phase retarder.As seen in Figure.15, Pancharat-nam phase is possible even with a single phase retarder. This might be dueto the change in polarisation introduced by mirror.[27]

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7 Appendix

7.1 Appendix.1

In the Michelson experiment,the Jones matrix for the arm containing thesingle QWP as seen in Fig.7 can be found by multiplying the matrices ofeach component i.e the rotatable quarterwave plate, mirror and again thequarterwave plate.

J =1√2


)×(

1 00 −1

)× 1√

2


)

So, for the arm containing only one quarterwave plate whose axis is ro-tatable with respect to the initial axis of polarisation,the matrix reduces to

J =


2

− sin4θ2


)(15)

If the axis of this quarterwave plate is fixed at 45 degrees, then the matrixfor the combination of only the quarterwave plate fixed at 45 degrees andmirror is given by

J =1√2

(1 ii 1

)×

(1 00 −1

)× 1√

2

(1 ii 1

)=

(1 00 −1

)(16)

Thus, the matrix of a mirror and a quarterwave plate infront of a mirror is

the same.

7.2 Appendix.2

In the Sagnac experiment,the Jones matrix for the arm containing the singleHWP as seen in Fig.12 can be found by multiplying the matrices of each com-ponent i.e the mirror, rotatable halfwave plate, and again the other mirror.

J =

(1 00 −1

)×


)×(

1 00 −1

)

29

Hence, the matrix for the p-arm with only halfwave plate is written as,

J =

(cos2θ −sin2θ−sin2θ −cos2θ

)(17)

For s-arm, there is a π difference. So, the matrix reduces to

J =


)(18)

For the s-arm with two quarterwave plates, the matrix is written as,

J =

(1 00 −1

)× 1√

2

(1 ii 1

)× 1√

2

(1 ii 1

)×(

1 00 −1

)=

(0 −i−i 0

)(19)

7.3 Appendix.3

In the Mach Zehnder experiment,the Jones matrix for the arm containingthe single HWP as seen in Fig.16 can be found by multiplying the matricesof each component i.e the rotatable halfwave plate and the mirror.

J =


)×(

1 00 −1

)=

(cos2θ −sin2θsin2θ cos2θ

)(20)

Similarly for the arm containing only two quarterwave plates, the matrix

formation is equal to

J =1√2

(1 ii 1

)× 1√

2

(1 ii 1

)×(

1 00 −1

)=

(0 −ii 0

)(21)

30

8 References

1.Optics,Fourth Edition by Hecht,Eugene2.http://dukemil.bme.duke.edu/Ultrasound/k-space/node83.S.Pancharatnam,”Generalized Theory of Interference, and Its Applications.Part I. Coherent Pencils”,Proc,Indian Acad.Sci.Sect.A 44,247(1956)4.Ernesto DeVito,Alberto Levrero,”Pancharatnam’s phase for polarized light”,Journalof Modern Optics,Volume 41, Issue 11, 19945.Berry, M V,J.Mod.Optics, 34, 1401-1407, ’The adiabatic phase and Pan-charatnam’s phase for polarized light’(1987)6.Berry, M V, Proc. R. Soc. A 392, 45-57, ’Quantal phase factors accompa-nying adiabatic changes’(1984)7.A. Tomita and R.Y. Chiao,Observation of Berry’s Topological Phase byUse of an Optical Fiber,Phys. Rev. Lett. 57, 937 - 940 (1986)8.Y.Aharonov and J.Anandan,Phase change during a cyclic quantum evolu-tion.,Phys.Rev.Lett.58,1593(1987)9.R.Bhandari and J.Samuel,”Observation of topological phase by use of alaser interferometer,” Phys.Rev.Lett.60,1211-1213(1988)10.S.Ramashesan,”The Poincare sphere and the Pancharatnam phase-somehistorical remarks,” Curr.Sci.59,1154-1158(1990)11.R.Bhandari,”Polarisation of light and topological phases,” Phys.Rep.281,1-64(1997)12.Y.Ben-Aryeh,”Berry and Pancharatnam topological phases of atomic andoptical systems,” J.Opt.B:Quantum Semiclass.Opt.6,R1-R18(2004)13.P.Hariharan,S.Mujumdar and H.Ramachandran,”A simple demonstrationof Pancharatnam phase as a geometric phase,” J.Mod.Opt.46,1443-1446(1996)14.J.Courtial,”Wave plates and the Pancharatnam phase,” Opt.Commun.171,179-183(1999)15.T Van Dijk, HF Schouten, W Ubachs, TD Visser, ”The Pancharatnam-Berry phase for non-cyclic polarisation changes,” Opt.Express 18, 10796-10804(2010)16.Julio C. Gutierrez-Vega,”Pancharatnam-Berry phase of optical systems,”April 1, 2011 / Vol. 36, No. 7 / OPTICS LETTERS,1143-114617.Fymat, A. L, ”Jones’s Matrix Representation of Optical Instruments. I:Beam Splitters”. Applied Optics 10 (11): 2499–2505. (1971).18.E. Collett, ”Field Guide to Polarization,” SPIE Press, Bellingham, WA(2005).19.Jones,R.C.,”New Calculus for the Treatment of Optical Systems”,J.Opt.Soc.Am.A,31,488-493 (1941).20.Fowles, G. (1989). ”Introduction to Modern Optics (2nd ed.)”. Dover. p.35.

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21.P.Yeh and C.Gu,”Jones matrix formulation,” Optics of liquid Crystal Dis-plays (Wiley, 2010), Chap. 4.1, pp. 173-289 22.R.Simon,T.H.Chyba,L.J.Wang,and L.Mandel, ”Measurement of the Pancharatnam phase for a light beam,”OPTICS LETTERS,Vol.13,No.7,July,(1988)23.M. Born and E. Wolf,”Principles of Optics,” 5th ed.(Pergamon,Oxford,1975),Chap.10.24.P.Hariharan,”The geometric phase,” Progress in Optics,Volume 48, 2005,Pages 149–20125.M.Martinelli and P.Vavassori,”A geometric(Pancharatnam) phase approachto the polarisation and phase control in the coherent optics cicuits,” Opt.Commun. 80, 166-176(1990)26.Piotr Kurzynowski, W ladys law A. Wozniak, and Ma lgorzata Szarycz,”GeometricPhase: two trianlges on the Poincare sphere,” J.OSA A, Vol. 28, Issue 3, pp.475-482 (2011)27.instructor.physics.lsa.umich.edu/int-labs/Chapter4.pdf

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