Experimental Realization of Slowly Rotating Modes of Light

Claremont Colleges Scholarship @ Claremont HMC Senior Theses HMC Student Scholarship 2014 Experimental Realization of Slowly Rotating Modes of Light Fangzhao A An Harvey Mudd College Recommended Citation An, Fangzhao A., "Experimental Realization of Slowly Rotating Modes of Light" (2014) HMC Senior Theses 53 https://scholarship.claremont.edu/hmc_theses/53 This Open Access Senior Thesis is brought to you for free and open access by the HMC Student Scholarship at Scholarship @ Claremont It has been accepted for inclusion in HMC Senior Theses by an authorized administrator of Scholarship @ Claremont For more information, please contact scholarship@cuc.claremont.edu Experimental Realization of Slowly Rotating Modes of Light Fangzhao Alex An Theresa Lynn, Advisor May, 2014 Department of Physics c 2014 Fangzhao Alex An Copyright The author grants Harvey Mudd College and the Claremont Colleges Library the nonexclusive right to make this work available for noncommercial, educational purposes, provided that this copyright statement appears on the reproduced materials and notice is given that the copying is by permission of the author To disseminate otherwise or to republish requires written permission from the author Abstract Beams of light can carry spin and orbital angular momentum Spin angular momentum describes how the direction of the electric field rotates about the propagation axis, while orbital angular momentum describes the rotation of the field amplitude pattern These concepts are well understood for monochromatic beams, but previous theoretical studies have constructed polychromatic superpositions where the connection between angular momentum and rotation of the electric field becomes much less clear These states are superpositions of two states of light carrying opposite signs of angular momentum and slightly detuned frequencies They rotate at the typically small detuning frequency and thus we call them slowly rotating modes of light Strangely, some of these modes appear to rotate in the direction opposing the sign of their angular momentum, while others exhibit overall rotation with no angular momentum at all! These findings have been the subject of some controversy, and in 2012, Susanna Todaro (HMC ’12) and I began work on trying to shed light on this “angular momentum paradox.” In this thesis, I extend previous work in theory, simulation, and experiment Via theory and modeling in Mathematica, I present a possible intuitive explanation for the angular momentum paradox I also present experimental realization of slowly rotating spin superpositions, and outline the steps necessary to generate slowly rotating orbital angular momentum superpositions Contents Abstract iii Acknowledgments xi Introduction Angular Momentum of Light 2.1 Spin Angular Momentum of Light 2.2 Orbital Angular Momentum of Light 3 Slowly Rotating Mode Theory 13 3.1 Ket Notation: Angular Momentum of Slowly Rotating Modes 14 3.2 Electric Field and Rotation 20 Mathematica Simulation and Modeling 4.1 Simulating the | g− i mode 4.2 Simulating the | g+ i mode 4.3 Simulating the |b± i modes 4.4 Simulating the |h− i mode 4.5 Simulating the |h+ i mode 4.6 Simulating the |c± i modes 27 29 32 36 39 42 47 Experimental Work 5.1 Generation of |b± i 5.2 Detection of |b± i 5.3 Results 53 53 55 60 Future OAM Work 6.1 Generation of initial OAM superpositions 6.2 Generation of |c± i Modes 69 69 74 vi Contents 6.3 Detection and Measurement of |c± i Modes Conclusion A Mathematica Code 75 77 79 List of Figures 2.1 2.2 2.3 2.4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 Rotating Polarization Vector for Right Circularly Polarized Light Carrying SAM of +h¯ Electric Field Pattern for Light with SAM Quantum Number s=1 Electric Field Pattern for Linearly Polarized Light with OAM Quantum Number ` = Helical Phase Surface of OAM-carrying LG Modes and Time Averaged Intensity Profiles 3D Plot of Base Bessel Mode E(r ) 3D Plot of Base Bessel Mode E(r ) Vector Plot of Real Part of ~Eg− Magnitude and Angular Velocity of Real Part of ~Eg− as Functions of Time Real Part of ~Eg+ showing Amplitude Variation Maximum Norm of Real Part of ~Eg+ Vector Plot of Real Part of ~Eb+ Vector Plot of Real Part of ~Eb− Vector Plot of Real Part of ~Eh− Real Part of ~Eh− using ParametricPlot3D Real Part of ~Eh+ showing Amplitude Variation using VectorPlot Maximum Norm of Real Part of ~Eh+ using VectorPlot Real Part of ~Eh+ showing Amplitude Variation using ParametricPlot3D Maximum Norm of Real Part of ~Eh+ using ParametricPlot3D Vector Plot of Real Part of ~Ec+ Vector Plot of Real Part of ~Ec− Real Part of ~Ec− using ParametricPlot3D 10 28 28 30 31 34 35 37 38 40 41 43 44 45 46 48 49 50 viii List of Figures 4.18 Real Part of ~Ec+ using ParametricPlot3D 51 Old Experimental Setup for |b+ i mode Effect of Rotating Half-wave Plate on Incident Horizontal Polarization Past Experimental Data of the |b+ i Mode with 0.33Hz HWP Rotation Revised Experimental Setup for |b+ i mode “Stovall Rotator” - New Rotation Mount/Motor Voltage on Oscilloscope vs Power Measured New Experimental Data of the |b+ i Mode with HWP in Newport Rotator at 0.338Hz Power Data of |b+ i Mode with HWP in Newport Rotator at 0.112Hz and Linear Polarizer Rotated by Hand at 0.224Hz Power Data of |b− i Mode with HWP in Newport Rotator at 0.112Hz and Linear Polarizer Rotated by Hand at 0.224Hz New Experimental Data of the |b+ i Mode with HWP in Stovall Rotator at 1.05Hz New Experimental Data of the |b+ i Mode with HWP in Stovall Rotator at 30.9Hz Frequency of HWP vs Dial Value on Stovall Motor, Averaged over Trials Amplitude vs Frequency of Fitted Data, Averaged over Trials Power Meter Output of |b+ i Mode with HWP Rotating at 3.1Hz Power Meter Output of |b+ i Mode with HWP Rotating at 11.3Hz 53 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 6.1 6.2 6.3 6.4 Transverse Hermite-Gaussian and Laguerre-Gaussian Modes Possible Experimental Setup to Generate HG1,0 and HG0,1 from separate LG0±1 beams A Dove Prism Flipping an Input Image Apparatus to Measure Rotation of |c± i Modes 54 56 57 58 59 60 61 62 64 64 65 66 68 68 71 72 74 76 List of Tables 3.1 3.2 3.3 3.4 Angular Momentum Expectation Values for General States |q± i and |r± i Angular Momentum Expectation Values for Frequency Dependent States | g± i and |h± i Angular Momentum Expectation Values for Equal Superpositions |b± i and |c± i The Family of Slowly Rotating Modes of Light 16 17 18 19 ... Momentum of Light 2.1 Spin Angular Momentum of Light 2.2 Orbital Angular Momentum of Light 3 Slowly Rotating Mode Theory 13 3.1 Ket Notation: Angular Momentum of Slowly Rotating. . .Experimental Realization of Slowly Rotating Modes of Light Fangzhao Alex An Theresa Lynn, Advisor May, 2014 Department of Physics c 2014 Fangzhao Alex An Copyright... detuning frequency and thus we call them slowly rotating modes of light Strangely, some of these modes appear to rotate in the direction opposing the sign of their angular momentum, while others