Measuring the orbital angular momentum of light with time mapping and using it to probe higher ...

127 Pages·2014·8.1 MB·English

Measuring the orbital angular momentum of light with
time mapping and using it to probe higher dimensional
states.
by
PaulJoseph Bierdz
A dissertationsubmittedin partialfulfillment
oftherequirementsforthedegreeof
DoctorofPhilosophy
(Physics)
intheUniversityofMichigan
2014
DoctoralCommittee:
AssistantProfessorHui Deng,Chair
ProfessorPaul R. Berman
ProfessorC¸agliyanKurdak
ProfessorTheodoreB. Norris
ProfessorDuncan G. Steel
©PaulJosephBierdz
2014
In memoryofmy mother.
ii
Acknowledgments
First, Iwant to thankmy advisorHui. OutofBo, Rene´e and myselfwewere herfirsts.
During our first summer, she spent countless hours in the lab training us and having a
exorbitant amount of patience. I am embarrassed to think back about how long it took me
toalignalaserintoafiber forthefirst time. But shewas notembarrassed ofmeand rather
patiently encouraged me in all things. I also appreciate and am thankful for her allowing
meto pursuemy owncuriosityin thescientificrealm whilestillfundingme. Hui was also
immenselyapproachableand we hadmanyfruitful conversationabout whatit meansto be
a scientist. Apart from work, she was friendly and would host annual parties at her house
ortakeus outfor icecream every timetheundergraduates wouldleaveusat theend ofthe
summer. Sheisarolemodelofsomeonewhocanbeincrediblypatient,incrediblyavailable
and alsoincrediblybrilliantand successful.
I would also like to thank my committee members, Professor Paul Berman, Professor
C¸agliyan Kurdak, Professor Ted Norris and Professor Duncan Steel, for taking their time
to allow me to present my research to them and their many useful comments. I also want
tothank ProfessorLumingDuan forhisuseful commentsduringmyprelim.
IwanttothankProfessorMilesPadgettfromtheUniversityofGlasgowforallthecool
work that he has done with orbital angular momentum of light that has kept me interested
in this field when my work was slow. I had the pleasure of meeting him and learned a lot
duringourshortmeeting.
Now for past and present members of Hui’s group. Our optics tables are filled with
experiments, thanks to you all. I thank you for our conversations and your friendship.
Thank you Rene´e Harton, Lei Zhang, Seckin Senlik, Minchuan Zhou, Zhaorong “Joey”
Wang, Tyler Hill, Jinhai Chen, Hoon Kim and Glenn Leung. I want to thank Tyler in
particular for all the interesting mathematical problems we worked on. And also Lei for
iii
reminding me to register for the pre-defense meeting with Rackham. Glenn, I wish you
good luck. I also want to thank Connor Roncaioli and Minho Kwon for directly helping
withmealotofmyexperiments.
Oh wait, Iforgot someone! Bo Zhang! Hewouldbe really sad ifIdidn’tmentionhim,
soI’ll givehimhisownparagraph. Thankyou Bo.
ThankyousomuchforallthefriendsthatIhavemadehereattheUniversity: Michelle
Adan,Sarah Anderson,MattBales, KevinBergemann, AlexBurgers,MidhatFarooq,Jake
Ketchum, Jee Hyun Kim, Se Ryeon Lee, Yen Ting Lin, Gechuan Liu, Grant Meadors,
Justin Nieusma, Ben Norman, Max Radin, Ryo Saotome, Tim Saucer, Chao Shen, Chris
Trowbridge, Jia Xu, Yifan Zhang, Yuanyuan Zhang and Zhen Zhang. I want to especially
thankMichelleandMattfortheirpracticalhelpwithfindingajobandwritingathesis;and
thankYuanyuanforherhelp incritiquingmypresentations.
To all the support staff in the physics department, you have been wonderful and really
helped a panicked graduate student in times of need. I want to extend a special thanks to
the Graduate Coordinator, Christina Zigulis who should also be considered my therapist,
and JoeSheldon whoensured thesafety ofallmypackages includingfistbumps.
ToAdamLausche. IamsothankfulthatImetyouandthatyouwereseveralyearsolder
thanme. Youweremygraduatestudentmentor. Eventhoughyouwerenotinphysics,you
pavedthepathfor me.
To Dr. Don Lincoln, my advisor at Fermilab, who helped me apply to college and
graduateschool,whogavememuchneededadviceaboutfindingajobaftergraduateschool
and just being a general all around great guy and really inspired me to continue to be a
scientist,evenifnotin highenergy.
To JinJi,thecreatorofthisLATEX templateand alltheothercontributors1.
ToProfessorTheoLasserandMatthiasGeissbu¨hlerfortheirMorgenstemningandother
color-blindfriendlycolormapsforMATLAB[1].
1http://www-personal.umich.edu/˜dalle/codes/thesis-umich/
iv
Now I want to take some time to thank the various people that helped brought me to
and throughoutgraduateschoolwhereIcould completethiswork.
Tomymother. Shegavebirthtomeandraisedmeandinstilledinmetheimportanceof
education. She would take us to the library almost every night to hang out and read books
and let me check out any book, even if it seemed silly. So I would check a book half my
sizefilled withgloriousimagesofstars,galaxiesandnebulaealongwithGo,Dog. Go![2].
She would feed my penchant for mathematics by buying me a small bag of marbles for
every milestone I would make. She wanted all of my brothers and me to go to college.
Well Mom,I’m gettingaPhD; Ididit,thankyou forgettingmethisfar. Imissyou.
To my brothers, in our shared childhood and beyond. So many fights, but so many
laughs.
To my grandparents, who raised me after my parents’ deaths and also for being great
grandparentsbefore thatand great-grandparents now;what adifferenceahyphenmakes!
To theRuthenbergs. The best that I can say is that it’s complicated, but you knowwho
youare and whyyouare here.
To my wife Arina. Thank you for your all your support for these six long years. I
have discovered you are truly a marvelous person, which these acknowledgements are too
narrow tocontain.
To Zoe¨ and Mia. Withoutyou,thisthesis probablywouldhavebeen finished sooneror
withmorecontent,butifIhad thechancetogoback andmakethedecisionagain,I’dpick
you every time. Zoe¨ I am so happy that you made me a father. I love playing with you, I
loveteachingyouandIlovelovingyou. TheuniverseisamazingandIhopethatyou’llsee
thatforyourselfandbeabletodiscoverasectionforyourselftoshoweveryone. AndMia,
Ilookforward to meetingyou.
There have been countless others, but I must get on with the rest of the thesis. I still
thank you. And I thank you, yes you, the reader, for reading and I hope that you can learn
somethingaboutorbitalangularmomentumoflight.
v
TABLE OF CONTENTS
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
ListofFigures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
ListofTables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi
ListofAppendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
ListofAbbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix
Chapter
1 OrbitalAngularMomentum ofLight . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 HistoryofOAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Lightand momentum . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Paraxial waveequation . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.3 Laguerre-Gaussian and othersets ofmodes . . . . . . . . . . . . 3
1.2 QuantumNatureofOAM . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 EntanglementofOAM . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 ApplicationsofOAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.1 QuantumApplicationsofOAM . . . . . . . . . . . . . . . . . . 10
1.3.2 Enhanced ClassicalCommunication . . . . . . . . . . . . . . . . 12
1.3.3 ImagingwithOAM . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.4 Opticaltweezers . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.4 OutlineofThesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 Measurement ofthe OrbitalAngularMomentum ofLight . . . . . . . . . . . 19
2.1 Pattern matchingmethods . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.1 Modalmatching . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.1.2 Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1.3 Diffractionpatterns . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 Filteringmethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
vi
2.2.1 Cylindricallenses . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2.2 Forked diffractiongrating . . . . . . . . . . . . . . . . . . . . . 31
2.2.3 Vortexphaseplate . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2.4 Q-plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.3 OAMbeam-splittermethods . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3.1 Doveprism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3.2 QuantumZenoInterrogator . . . . . . . . . . . . . . . . . . . . 38
2.3.3 Imagereformater . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.4 Summary ofDetectionofOAM. . . . . . . . . . . . . . . . . . . . . . . 40
3 HighFidelity Detection of the Orbital Angular Momentum ofLight by Time
Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2 PrincipleoftheOAMspectrometer . . . . . . . . . . . . . . . . . . . . . 42
3.2.1 Energy distribution . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3 Experimentalimplementation . . . . . . . . . . . . . . . . . . . . . . . . 43
3.4 Analysisofthespectrometerperformance . . . . . . . . . . . . . . . . . 45
3.4.1 Laguerre-Gaussian modes . . . . . . . . . . . . . . . . . . . . . 46
3.4.2 Singlemodefibre . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4.3 VPP/SLM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.4.4 Limitingfactors offidelity . . . . . . . . . . . . . . . . . . . . . 51
3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4 Effects ofmisalignmentonOrbital AngularMomentum . . . . . . . . . . . . 53
4.1 Define overlapintegrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.1.1 DerivationofVPP tensor . . . . . . . . . . . . . . . . . . . . . . 54
4.1.2 Overlapbetween Gaussianand LG . . . . . . . . . . . . . . . . 55
4.1.3 Overlapbetween Gaussianand VPP . . . . . . . . . . . . . . . . 56
5 ACompactOrbitalAngularMomentumSpectrometerUsingQuantumZeno
Interrogation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.2 Imperfect OAMFilters . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.3 Pinholespatialfilter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.4 Discussionofresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6 GeneralizedQuantum Zeno Interrogation . . . . . . . . . . . . . . . . . . . . 68
6.1 OriginalQuantumZeno Interrogation . . . . . . . . . . . . . . . . . . . 68
6.2 Serial Binary QZI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.3 Weak ParallelQZI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.3.1 M = 1 bomb . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.3.2 M > 1 bombs . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.4 Strong Parallel QZI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
vii
7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
viii
LIST OF FIGURES
1.1 Thisfigureisdividedintotwosections: (left)intensityand(right)phase. More
precisely, three rows and 10 columns. The columns represent different ℓ’s
rangingfrom0to4,thenrepeatingforphase. Therowsrepresentdifferentp’s
ranging from 0 to 2. Negativevalues ofℓ will havethesame intensity,but the
phasewouldtwistin theoppositedirection. . . . . . . . . . . . . . . . . . . . 4
1.2 Top: The angular position ranging from all angles (open aperture), to narrow
angularrange. Bottom: Thecorrespondingorbitalangularmomentumoflight
(OAM)uncertaintywheretheoriginalstatebeforepassingthroughtheangular
aperturewas ℓ = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 A diagram of a vortex coronagraph. L1, L2, L3 are lens in the system. VC
is the vortex coronagraph (acts like vortex phase plate (VPP) that adds the
helically phase twist to the on-axis starlight (reddish-orange), but does not
addmuchphasetwisttotheoff-axisbrowndwarf’sorextrasolarplanet’slight
(greenish-blue). Sincetheon-axislightgetsahelical phasetwist,itwillfocus
toan annularshapeandthusblockedbytheLyotstop. Theoff-axisextrasolar
planet’slighttopassthroughand arriveat detectorD. . . . . . . . . . . . . . . 14
1.4 The image above is taken from a 1.5-meter portion of the Palomar Observa-
tory’s Hale Telescope in California. The star’s intensity (HR8799) has been
greatly suppressed and replaced with an X to indicate the position. The star
is 120 light years away. The three planets, called HR8799b, c and d respec-
tively are believed to be gas giants larger than Jupiter and orbiting the star
at 24, 38 and 68 AU. For comparison, Jupiter is about 5 AU away from our
Sun. This is the first image taken of extrasolar planets from a telescope this
size. Imagecredit: NASA andJPL-Caltech and PalomarObservatory . . . . . 15
1.5 A cartoon of a particle trapped by a focused laser beam. (Left) The particle,
as shown, is off-centered. There is a restoring force, the gradient force, that
moves the particle towards the region of high intensity. This force is repre-
sented by the red arrow. (Right) The particle is also off-centered, but is now
trappedbyanorbitalangularmomentumoflight(OAM)beam. Itmovesalong
the annulus because that is the region with highest intensity and will rotated
abouttheopticalaxis,thusimpartingtheparticlewithorbitalangularmomen-
tum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
ix