Table Of ContentRapidlyreconfigurableopticallyinducedphotoniccrystalsinhotrubidiumvapor
BethanyLittle1,∗ DavidJ.Starling1,2,JohnC.Howell1,RaphaelCohen3,† DavidShwa3,andNadavKatz3
1Department of Physics and Astronomy, University of Rochester, Rochester NY 14620
2Division of Science, Pennsylvania State University, Hazleton, PA 18202 and
3The Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel
(Dated:January29,2013)
Throughperiodicindexmodulation,wecreatetwodifferenttypesofphotonicstructuresinaheatedrubidium
vaporforcontrolledreflection, transmissionanddiffractionoflight. Themodulationisachievedthroughthe
useoftheACStarkeffectresultingfromastanding-wavecontrolfield. Theperiodicintensitystructurescreate
translationally invariant index profiles analogous to photonic crystals in spectral regions of steep dispersion.
Experimentalresultsareconsistentwithmodeling.
3
1 I. INTRODUCTION thesesstructuresoccuroverlargecontrolbeamdetunings(po-
0 tentially answering some questions posed in Ref. [16]) and
2 All-opticalstructureswhichcanbedynamicallycontrolled can be modified at a rate exceeding Ref. 14. This leads to
n areanimportantresourceformanyapplications,suchasultra- manyinterestingpossibilitiesforfuturework,suchasrapidly
a fast optical switches [1] or nonlinear processes in waveg- creatingacavityatthelocationofashortpulseoflighttotrap
J uides [2]. The photonic crystal [3, 4] is one such all-optical it,quicklyreflectingabeamofacertainfrequency,ormaking
7 structure, in which light propagating in a periodic dielectric useofslowlightsuchasin[17].
2 medium has a band structure of allowed and forbidden fre- In the following section we introduce the theory based on
quencies. AnotherusefulstructureisthethickBragggrating theACStarkeffectinrubidiumgas,andinsectionsIIIandIV
]
s or more generally the volume hologram which may be used wedescribetwodifferentexperiments,bothofwhichdemon-
c for information storage [5]. By carefully constructing a di- strate Stark effect induced structural behavior. Finally, con-
i
t electric medium, light can be guided and shaped for use in clusionsandfutureworkarediscussedinsectionV.
p
o informationtechnologyandotherapplications[6].
. Muchprogresshasbeenmadeinthisfield[4],includingthe
s II. THEORY
c dynamical control of the structure by using the electro-optic
i effect[7]andfastpulses[8–10]. However, asArtoniandLa
s
y Roccapointout[11], alloftheseschemesarelimitedbythe Thephaseshiftandabsorptioncausedbyanatomictransi-
h originalspecificationsofthestructure;whiletheindexofthe tionarerelatedtotherealandimaginarypartsofthesuscepti-
p mediummayberapidlychanged,thespatialstructurecannot bilityofthattransition[18].Asufficientlystrongcontrollaser
[ bemodifiedquickly. will,viatheACStarkeffect,causeashiftinthetransitionfre-
1 One interesting solution to this problem is to “write” a quency. Theresultingchangeinthesusceptibilitywilldepend
v structure onto a medium using interfering laser beams, such upontheintensityofthecontrollaserandthetemperatureand
0 as in Ref. [12]. By making use of electromagnetically in- densityoftheatoms. Thischangeinthesusceptibilitycauses
8 duced transparency (EIT), a periodic structure with alterna- anadjustmenttotherefractiveindexandtransmission;theex-
3 tively transparent and opaque regions can be created with pectedmodificationforatypicalsetupisshowninFigure1a.
6
counter-propagatingbeams. Ithasbeenshownthatthiseffect For a given propagation length, it can be seen that there is a
.
1 could also be used to create structures that behave like pho- trade-off between absorption and change in refractive index.
0 tonic crystals [13] which can be modified in time [14]. No- Forafixedcontrolbeamintensity(andhenceafixedACStark
3
tably,inadditiontobeingrapidlyreconfigurable,thesestruc- shift),agreaterrefractiveindexchangecanbeachievedbyin-
1
turesalsohavehighspatialfidelity,avoidingthelocaldisorder creasing the density of the rubidium atoms; however, this in
:
v problemsofsolidcrystals[11]. turnleadstogreaterabsorption.
Xi However,allofthesesystemswhichutilizeEITarelimited TheACStarkshiftcausedbyacontrolfieldoffrequencyωc
by coherence times and are more readily realizable in cold maybecalculatedbyconsideringthefieldtobeconstantover
r
a atomsthanatroomtemperatureowingtotheballisticordif- theregionofagivenatomandusingthestandardHamiltonian
fusivebehavioroftheatoms. Anotherpossibilityistocreate foratwo-levelatomwithresonancefrequencyω0 interacting
structuresusingopticalpumping[15],butthebandwidthwill withanelectricfieldgivenby(cid:126)E= 21(E(cid:126)ce−iωct+E(cid:126)c∗eiωct). We
belimitedbythedecaytime. assumethelaserisnearresonanceandusetherotatingwave
In this paper we propose and demonstrate photonic struc- approximation,wheretheHamiltonianbecomes
ture behavior within a cell of hot rubidium gas by using the
(cid:18) (cid:19)
0 Ω/2
ACStarkeffect.Inadditiontoavoidingtheuseofcoldatoms, H =−h¯ , (1)
R Ω∗/2 ∆
withRabifrequencyΩ=d(cid:126)·E(cid:126) /h¯,detuning∆=ω −ω and
c c 0
∗[email protected] d(cid:126) is the transition dipole moment. The shifted levels can be
†[email protected] foundfromtheeigenvaluesofH .
R
2
(a) (b)
88
1100
77
52P F' = 1, 2, 3, 4
-510) 5656 1 80 %) 3/2
x
( (
ex 44 60 on δε/ħ
n Ind 33 missi 52P1/2 F' = 2, 3
i 22 0 40 s
nge 11 0.6 0.9 1.2 ran ω ω ω
ha 00 20 T 0 p c
C
F = 3
--11
52S
00 1/2
--22 F = 2
--44 --22 00 22 44
δε/ħ
Δ
Probe Detuning (GHz)
FIG.1. (Coloronline)(a)Themodificationoftherefractiveindexandtransmissionforatypicalexperimentalsetup(controlbeam: P=30
mW,abeamwaistof810×85µm2,T=85◦Candacelllengthof7.5cm).Thetransmissioncurvesforaprobebeamwith(dashedgreen)and
without(solidblue)thecontrolfieldpresentareshownalongwiththeresultingchangeintherealpartoftherefractiveindex(solidblack).(b)
Therelevantlevelsforrubidium85areshown. Thecommongroundstate(52S F=2)isStark-shiftedbythecontrolbeameither780nm
1/2
(blue,solid)or795nm(red,dashed).ThemagnitudeoftheStarkshiftisincludedasδε/h¯ fortwolevels.
Ifthecontrolfieldischosentointeractonlywiththeground where I is the intensity of the coupling field. It is therefore
c
state(whichiscommontoboththeprobeandcontrolfields), advantageous to increase the coupling field intensity and re-
asinthe780nmfieldofFigure1b,thentheexpectedshiftof duce the detuning (while still avoiding other processes such
theproberesonancecanbefoundbyconsideringtheshifton asopticalpumping).
justthelowerlevel. Thisisgivenby
h¯ (cid:18) (cid:113) (cid:19)
δε=− ∆− ∆2+|Ω|2 . (2) III. PHOTONICBANDGAP
2
Forlargedetuning,thisbecomes A. Experiment
h¯|Ω|2
δε≈− . (3) Onesuchstructurecanbeinducedbyusingacrossedbeam
4∆
setup, shown in Figure 2a, which creates a set of disallowed
Byinterferingthecontrolbeamwithitself,aperiodicstand- frequencies for a probe beam. We chose to use a crossed-
ingwavecanbecreatedwithinaheatedcellofrubidium.This beamsetupratherthanacollineardesigntoenhancefiltering,
standingwaveprovidesaperiodicintensityandthusaperiodic toavoidcoherenteffects,andtopreventthecontrolbeamfrom
changeinrefractiveindex,analogoustoaphotonicstructure, creatingabandgapforitself. Wechoosetheanglesbetween
butcreatedinahotcloudofrubidiumgas. thebeamssuchthatthedistance∆pbetweennodesofthecon-
ForsmallACStarkshiftsthechangeinrefractiveindex∆n trolbeamishalfthewavelengthoftheprobebeam. Sincethe
willbeproportionaltotheshiftinsusceptibility; butthesus- index contrast ∆n=n1−n2 between the node and antinode
ceptibility is proportional to the AC stark shift δε/h¯, hence of the control field is small, this creates the equivalent of a
linear in control intensity and inversely proportional to the quarter-wavestack,withn1a1≈n2a2anda1=a2=∆p/2.
control detuning [via Eq. (3)]. The efficiency of such struc- Fortwocontrolbeamswithwavenumbersk =k =2π/λ
1 2 c
tureswillbeproportionaltothesquareofthisrefractiveindex wecancalculatetherequiredangle2θbetweenthebeamsfor
contrast[Ref.5,Eq.(45)];thatistosay,theefficiencyhasthe abandgaptooccur. Treatingthetwocrossingcontrolbeams
behavior asplanewavesoftheform(cid:126)E =A(cid:126) ei((cid:126)k·(cid:126)r−ωt)+c.c.(j={1,2})
j j
e∝Ic2 and e∝ ∆12, (4) ainntdenassistyumasinagftuhnecatimonploitfuddiesstaonfceeaxchawbeaaymfroAm1=theAi2r=croAs,sitnhge
3
(a) 100 14
Probe Pump
Reflected Transmitted %) 80 12
R
Probe 50:50 θ Probe on ( 60 10 efle
si 8 c
Rubidium Cell mis 40 6 tion
InPcriodbeen t Trans 20 4 (%)
2
(b)
Diffracted Probe 0 0
Probe --44 -2 0 2 4 6
Detuning (GHz)
FIG. 3. (Color online) Plot of transmitted (dashed) and reflected
He ated (solid) probe light. Since the index contrast created by the Stark
Rubidium Cell
PBS effectisgreatestnearresonance,thebandgapismoredominantin
PBS
theseregions.
Control
PBS
The percent reflected was calculated by subtracting back-
ground from the reflected power and dividing by the input
Transmitted Probe power. The control beam was detuned from resonance by
at least 1 GHz to limit optical pumping. As expected, the
frequency of the band gap depended on the frequency of the
FIG. 2. (Color online) (a) Schematic of the photonic band gap
controllight,butreflectionwasstillpresentfordetuningsex-
experimentalsetupofsectionIII.Probelightat795nmisincident
ceeding50GHz. Temperatureadjustmentswereusedtoopti-
onarubidiumvaporcellandappropriatedetectors. AStark-shifting
controlbeamat780nmanddetuning∆ofatleast1GHzissetupina mizetheopticaldepth. Theanglebetweenthecontrolbeams
crossedconfigurationinordertocreateaninterferencepatternwitha wastunedtomaximizereflectionforagivenprobefrequency,
periodicitythatcreatesabandgapinthemediumfortheprobebeam. consistentwithEq.(6).
(b)SchematicoftheBragggratingexperimentalsetupofsectionIV.
Polarizingbeamsplitters(PBS)areusedtocombineandseparatethe
controlandprobefields. B. Results
Theresultsoftheexperimentareconsistentwiththemodel
of a periodic index contrast induced by the AC Stark effect.
pointforagivenangleθis
This is evident through observations of certain disallowed
I=2A2{1−cos[2θ]cos[2kxcos(θ)]}. (5) probefrequencies(exhibitedbyreflectionfromthegas). The
disallowed probe frequencies depend on both the angle and
Solvingforthespacing∆pbetweenpeaks,wefindthat∆p= the frequency of the control field, despite the lack of coher-
λ /[2cos(θ)],whereλ isthewavelengthofthecontrolfield. encepresentintheexperiment.
c c
Letting the spacing be half the wavelength of the probe field Eq. (2) can be used to find the expected Stark shift from
(λ )suchthat∆p=λ /2,wefindthesimplecondition thecontrolfield. Foracontrolpowerof120mW,agaussian
p p
beam diameterof 600 µm anda detuning of2 GHz, thepre-
cos(θ)=λ /λ . (6) dictedshiftis5.7×107Hz. Baseduponprobescanswithand
c p
without the control field, the average shift was found to be
Forλc=780nmandλp=795nm,thisgivesθ=11.2◦. 2×107 Hz. However,thisisonlyanapproximationthatdoes
Theexperimentalsetupconsistsofa780nmexternalcav- nottakeintoaccountthedistortionofthetransmittedfieldplot
itydiodelaserthatproducesahorizontallypolarizedstanding whenthestandingwavecontrolfieldispresent.
waveina1mmlongcellwithnaturallyabundantrubidium.A The reflection and transmission spectra of the probe are
verticallypolarizedprobebeamgeneratedbya795nmexter- showninFigure3. Thesteepestdispersionoccursinregions
nalcavitydiodelaserisincidentona50:50beamsplitter.Half near resonance; therefore, the reflectivity is strongest there,
of the light passes through the cell to a detector, where the consistent with Figure 1. This is also where the greatest ab-
absorption spectra is measured. The other half is measured sorptionoccurs,accountingforthelowpercentageofreflected
at a second detector so that percentage-of-input calculations light. Depending on the frequency and angle of the con-
can be made. Reflected probe light passes back through the trol beams, the reflection occurs strongly at the correspond-
beamsplittertoathirddetector. ingprobefrequencies. Inthedatashown,theanglewastuned
Absorption,transmissionandreflectionspectraoftheprobe so that the reflected peak was maximized for the D transi-
1
beam were recorded for various control beam frequencies. tionof87Rb,fromtheF =1groundstate. Itisnotsurprising
4
1.0 ) 0.6
%
ntensities00..68 II(pcI c(()CR2oenfletrcotle) d Probe) DiffractionEfficiency ( 00..24
d I 0
e 1
Normaliz00..42 rmalizednsmission 000...468
oa
Nr 0.2
T
50 100 150 200 250 0
Time (ns)
0 2 4 6 8
FIG. 4. (Color online) Plot of probe (black, thin) and control Probe Detuning (GHz)
field(blue,thick)normalizedintensitiesalongwiththecontrolfield
squared(red,dashed)duringpulsedoperation. Arapidresponseof FIG. 5. (Color online) (a) Plot of measured (blue) and simulated
thebandgapeffecttotheturn-onofthecontrolfieldisevident. (red, dashed) diffracted probe light. (b) Measured (blue) and sim-
ulated(red, dashed)transmittedprobe. Probedetuningisfromthe
D line of the 87Rb F =1→F(cid:48) =2 transition. Cell and control
1
parametersaresameasinFigure1andthegratingspacingis55µm.
that the effect occurs at the other resonances (although more
weakly),sinceaccordingtoEq.(2)aStarkshiftisstillpresent
fornonzerodetuning. Itisalsoimportanttonotethatthere-
moval of one of the control beams results in no frequency-
in Figure 2b. A standing wave is produced in the plane per-
dependent reflections; this behavior supports our conclusion
pendiculartotheaveragedirectionofpropagationofthetwo
thatthestandingwaveisresponsibleforthebandgap.
controlbeams.Ifthecontrolandprobewavelengthsareclose,
The time response of the system was then investigated by
thentheBraggangleforsuchagratingisidenticaltotheangle
pulsingthecontrolfieldandobservingthereflectedlight.Fig-
ofincidenceofthecontrolbeam.Thepathoftheprobeandits
ure 4 shows a plot of the amplitude of both the control field
diffractionthereforeliealongthepathsofthecontrolbeams.
andthereflectedprobefieldasafunctionoftime.Thepeakof
Thus, weareabletouseorthogonalpolarizationstoseparate
thereflectedpulseoccursatthesametimeasthepeakofthe
the probe from the control. In contrast to the experiment of
control field pulse, within the precision of our measurement
section III, here we use an isotopically pure 85Rb vapor cell
(≈5ns). Fasterpulsinganddetectingapparatuswouldbere-
oflength7.5cm. Also,inthisconfiguration,itisdesirableto
quiredtodetectadelayintheresponseofthebandgap,which
usenearlydegeneratecontrolandprobefields(inthiscasethe
should be proportional to the Rabi frequency of the control
D lineofrubidium).
1
field. This will be the subject of future work. However, we
Wesimulatetheeffectofthisgratingbyfirstcalculatingthe
note here that the delay of the reflected pulse in Ref. 14 was
expectedchangeintherefractiveindexcausedbythecontrol
ontheorderofamicrosecond.
field using classical susceptibility equations, calculated from
It may also be noted that the width of the reflected probe
[18],withtheaddedACStarkshiftofEq.(3)andtakinginto
pulseisshorterthanthecontrolfieldpulse(thinblackversus
considerationholeburningandopticalpumpingcausedbythe
thick blue traces in Figure 4). However, by comparing the
probe. Theseresults, alongwithexperimentalmeasurements
widthofthesquareoftheintensitytothewidthofthereflected
ofthebeamsizesandangles,arethenincorporatedintoasim-
probepulse(dashedredversusthinblacktracesinFigure4),
ulationofthe3Dpropagationoftheelectromagneticfield.
wefindtheresultsconsistentwiththediscussionofEq.(4).
This first demonstration is the all-optical equivalent of a If the intensity profile of the interfering control beams
quarter wave stack in a dielectric medium. In the following I(x,y,z) is known, the accumulated phase difference of the
section,weturntoasecondarrangementwhereweagainem- probefieldcanbeexpressedas
ploytheACStarkeffect,butinaco-propagatingconfiguration
leadingtoathickBragggrating. 2π
φ(x,y,z)= δnI(x,y,z)z+i[α+δαI(x,y,z)]z, (7)
λ
IV. BRAGGGRATING wherenistherealpartofthetotalindexofrefractionn+in(cid:48)(cid:48),
α is the absorption coefficient (given by α=2πνn(cid:48)(cid:48)/c), and
A. Experiment δnandδαarethecorrespondingchanges. Theintensityofthe
diffractedandtransmittedprobefieldwasfoundbysumming
To create the Bragg grating we interfere two nearly co- theabsolutesquareofthefieldoverthepixelsintheappropri-
propagatingbeamstakenfromthesamelasersource,asshown atediffractionorder.
5
1.00 slopeis−1.99±0.02. Whenthecontrolisclosetothe85Rb
D F =2transition,opticalpumpingdominatesasexpected.
a.u.) On1 theothersideofthetransitionthenewdressedstatecaused
y ( 0.75 bythecontrolbeamcausestheprobetolieinaplaceofvery
sit 100 high absorption; thus no diffracted signal is observed at this
n
e probefrequency.
nt 10-1
e I 0.50
b
o 100 101
Pr V. CONCLUSIONS
d 0.25
e
ct Both systems show optically induced structures that are
a
Diffr 0.00 oItnilsycelfefeacrtifvroemwhtheensethereisnutletsrfethriantgthceonAtrColSfitealrdkseafrfeecptreissetnhte.
causeofthesestructuresandthattheresponsetimeis(cid:46)5ns.
-2 0 2 4 6 8 The results of Section IIIB are consistent with the model
Control Field Detuning (GHz) of a periodic index contrast induced by the AC Stark effect,
which creates the photonic band gaps seen in Figure 3. The
FIG. 6. (Color online) Plot of measured diffracted probe with results of Section IVB demonstrate that we are able to very
changeinthedetuningofthecontrolfield(blue). Theinsetshowsa closelysimulatetheseprocesses,andthattofullyunderstand
log-logplotofthemeasureddata(blue)andastraightlinefit(red) thebehaviorofthesestructuresoneneedstotakeintoaccount
withaslopeof−1.99±0.02.Thecontrolfielddetuningismeasured thechangeinboththerealandimaginarypartsofthesuscep-
fromtheD1lineofthe85RbF=2→F(cid:48)=3transition. tibilitycausedbytheACStarkeffect. Infuturework,weplan
to extend the periodic structure into two dimensions. This
could be achieved with the use of a digital micro-mirror de-
vicecapableofgeneratingperiodicpatternsresemblinga2D
B. Results
photoniccrystal. Thispatterncouldbeeasilymanipulatedat
highspeed,effectivelycreatingafast,controllableopticalcir-
We first measure the diffracted signal, shown in Figure 5,
cuit.
by tuning the control beam to the D line of the 87Rb F =
1 Insummary,wehaveshownthatavarietyofstructuresmay
1→F(cid:48) =2 transition, 2.7 GHz from the closest 85Rb tran-
besetupinhotrubidiumgasbyuseoftheACStarkeffect.By
sition; this ensures that there is no optical pumping. (Again,
turningthecontrolfieldson(off)wecanenable(disable)the
thereflectedprobefieldisnotpresentwhenweblockeitherof
photonic band gap, thereby producing a fast optical switch.
thecontrolbeams,thusverifyingthatoursignalisduetothe
Importantly,thecontrolandprobefieldscanbederivedfrom
inducedgrating). Weseethatthediffractedsignalisstrongest
differentsourcesandtheatomiccloudisheated;thisisavast
ontheedgesoftheabsorptionasexpected(seeFigure1),and
simplification over designs that utilize EIT and cold atoms.
isobservedtobelargestonthe85RbD F=2transitionwhere
1 Theseresultsopenupmanyinterestingpossibilitiesinquan-
there is the smallest detuning. The results of the simulation
tum computing and alternatives to photonic crystal technol-
confirmthatthegratingisproducedbytheACStarkshift.
ogy.
Finally, we vary the control field frequency while holding
theprobeataconstantdetuningof2.2GHzfromtheD line
1
ofthe87RbF =1→F(cid:48)=2transition. Thediffractedprobe
signalisshowninFigure6andisvisibleforpumpdetunings ACKNOWLEDGEMENTS
up to 10 GHz. The diffracted probe signal falls off with an
inverse square law relationship consistent with an AC Stark This research was supported by the ARO Grant No.
effect induced grating as predicted by the theory [Eq. (4)]. W911N-12-1-0263, ISF Grant No. 1248/10 and the Univer-
This behavior is shown in the inset of Figure 6, where the sityofRochester.
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