Classial and Quantum Properties of
Optial Parametri Osillators
M. Martinelli
,C. L. GarridoAlzar
,P. H. SoutoRibeirox, and P.Nussenzveig
Institutode Fsia, UniversidadedeS~aoPaulo
CaixaPostal66318,S~aoPaulo,SP05315-970, Brasil
xInstituto deFsia,UniversidadeFederaldoRio deJaneiro
CaixaPostal68528,Riode Janeiro, RJ21945-970, Brasil
Reeivedon16May,2001.
WepresentareviewoftheOptial ParametriOsillator (OPO),desribingitsoperationandthe
quantumorrelationbetweenthelightbeamsgeneratedbythisosillator. Weshowtheonstrution
ofanOPOusingaPotassiumTitanylPhosphaterystal (KTP),pumpedbyafrequenydoubled
Nd:YAG laser, and disuss the stability of the system and related thermal eets . We have
measuredthequantumorrelationofsignalandidlerbeamsinatransientregime,obtaininganoise
orrelationlevel39%belowtheshotnoiselevel.
I Introdution
Theadventof thelaserasahigh poweroherentlight
soure markedthe beginof a wide study of nonlinear
interationsbetweenlightandmatter. Althoughmany
nonlinear optial phenomena were already known
be-forethelaser,itwasonlyafter theavailabilityofthese
intense light beams that the eets of up and down
onversionof lightould bestudied. We an mention
SeondHarmoniGeneration(SHG)astherststudy
in this subjet,involvingtheuponversionofapump
beam. In the rst experiment performed by Franken
et. al. in 1961[1℄ abulkquartz rystal, pumped by a
rubylaserat 694nm,generatedaseond beamathalf
thewavelength.
Thiseet anbedesribedbyanonlinear
susep-tibility(E),dependingontheeletrield. This
non-linear suseptibility will originatea quadrati termof
the polarization on the eletri eld. This nonlinear
term is responsible for parametri onversionof light,
likethesumand diereneoffrequenies,SHG,
spon-taneous down onversion, whih an beseen in many
textbooks[2℄.
ParametriDownConversionanbeseenasthe
in-verseproessofSHG.Pumpedbyafundamentalbeam
ofangularfrequeny!
0
,anonlinearrystalan
sponta-neously emit photons. These photons, produed from
the annihilation of a photon in the pump beam, are
produedin pairs, havingstrongorrelation of energy
resene,and thisspontaneousemissionanbe
under-stoodastheemissionoflightstimulatedbythevauum
utuationsoftheeld.
Thislightemissionanbeenhanedusinga
oher-entbeamoffrequeny !
1
, whihwill atas aseedfor
thelightgeneration. We obtainastimulated emission
oflight[3℄produing nowanintense beam. Similar to
the eet of apumped laser medium, this stimulated
emissionwillamplifytheinputbeam,alledsignal. As
aonsequeneofthestrongorrelationbetweenthe
sig-naland idler photons, an intense idler beamwill also
appear. Energyonservationimpliesthattheenergyof
thephotonemitted in theidler beamisthe dierene
of energy between the pump and the signal photons,
thereforewehavefortheidlerfrequeny!
2 =!
0 !
1 .
Everystimulatedphotonemittedinthesignalwillhave
atwin partner in the idlerbeam. Thiseet isalled
parametriampliation[4℄,andwaswellknownin
ele-troniiruitsandmirowavesystems[5℄.
Like the laser ampliation, this nonlinear rystal
pumped by an intense beam is a gain medium. The
gaindependsonthepumpintensity. Whenitisputinto
aavity, forappropriateonditionsof detuning, losses
and pump intensity, osillation will our, and an
in-tenseoutputbeamwithlaserharateristisanbe
ob-tained. ThisdevieisalledanOptialParametri
Os-illator(OPO).Fromearlywork[6℄,itwaspresentedas
alaser soureofwidetunability withpromising
areommeriallyavailable OPO'sfordownonversion
ofpulsedlaserbeams,usedaslightsoureintherange
between 330 and 2000 nm. Many reent eorts are
madetousethemasCWtunablelightsoures[8℄,with
thesearh ofnewmaterialsand theappliationof
mi-romanufaturing in the prodution of high eÆieny
nonlinearmedia[9℄, dierentavityongurations,use
ofdiodelasersaspumpsoures,et. Thesedeviesare
usedasoherentlightsouresin regionsofthespetra
where no eetive laser medium is available,
onvert-inglightofNd:YAGlasersintothemidinfraredregion.
Theyarealsostudied inthemetrologydomain,as
ele-mentsin lokinghainsforgeneration ofhighly stable
frequeny/time bases. An overview oftheeld anbe
seenin ref. [10℄.
On the other hand, quantum properties of light
emitted from these soures beame soon an intensely
studied subjet. Nonlassialstatesof theeldanbe
produedeitherbynonlinearproesses[11℄or
stabiliza-tion of light emission, as in diode lasers[12℄. Owing
to the quantum nature of light, there is an intrinsi
noise in its detetion that annot be avoided even in
the ase of the generation of the eld by a "lassial
urrent"[13℄. Thislimitisoftenalledshotnoise,sine
it an be understood as the noise reatedby the
de-tetionofindividualphotons,generatingawhitenoise
spetrum for the measured intensity of the beam like
theoneproduedbyarandomuxofpartileshitting
asurfae.
IntheaseoftheOPO,thereationofpairsof
pho-tonsinsteadoftheemissionofsinglephotonsbyalaser
attrated theattentionof manyresearhers, and soon
it beame an important soure of non-lassial states
of light. Earlywork hasshownthat squeezed vauum
utuations are obtained from its output when
work-ingbelowthethreshold[14℄. Lateron,thesqueezingof
the twin beams generated wasobserved[15℄, and they
represent the up-to-date reord of noise ompression
in nonlinear optial eets (-8.6 dB)[16℄. They work
also as "noise eaters". The ompression in the noise
of the pump beam reeted by a avity was reently
observed[17℄.
Itisourpurpose topresentin this paperageneral
overviewoftheOPOoperation,andsomeofthe
quan-tum features observed in the twin beams. We begin
byageneralpresentationoftriplyresonantOPO's,for
type I and type II phase mathing. Next, we make a
review of the quantum aspets of orrelationbetween
signalandidler beams. Weused atypeII triply
reso-nantOPO,builtwith aKTP(KTiOPO4) rystaland
pumpedat 532nm, todemonstrate itsoperation,
dis-ussingalsothe thermalbistability that ours in this
orrelation between the signal and idler beams, and
omparethe measuredompressionofnoiseto the
ex-petedvaluesobtainedintheOPOharaterization.
II OPO: Classial desription
There is a great variety of OPO ongurations[18℄.
Thesimplest oneistheuseof asingleresonantavity
(SROPO),where the avity is resonant for the signal
beam, and there is a single pass of the pump beam
through the rystal. The generated idler beam will
exit the rystalwithout anyfeedbak. The osillation
threshold anbe reduedbytheuse ofadoubly
reso-nantavity(DROPO),wherebothsignalandidlerare
kept resonant with the avity. Whilethe SROPO
al-lowsabroadontinuousvariationofthesignalandidler
wavelengthinthephasemathingrange,thethreshold
powerismuh higherwhenomparedto theDROPO.
Thisonehasthedrawbakofthewavelengthlimitation
to the ne tuning of the modes of the avity[19℄.
In-deed, the double resonantondition of theavityand
theenergyonservationonditionwilllimittheoutput
modes toadisrete setof values,as wewill see in the
presentdesription.
Inbothases,thethresholdpoweranbereduedif
insteadofusingasinglepassofthepumpbeamweusea
avity,whihwillinreasetheinidentintensityonthe
rystal. This isalled the Pump-enhanedOPO
(PE-OPO),also known astriply resonant OPO(TROPO)
when usedto redue thethresholdofaDROPO.This
ongurationallowedthresholdpoweraslowas1mW
for CW operation in KTP[17℄. The useof new
mate-rials redued this power to 300 W (CW) with quasi
phasemathed(QPM)rystals[20℄.
We present now an overview of the TROPO,
fol-lowingthetreatmentpresentedinref.[19℄foratypeII
phase mathing rystal, and extend the disussion to
thetypeIphasemathing,givingastraightforward
in-terpretation of thedetailed treatment given in ref.[7℄.
Consideranonlinearrystaloflength`plaedinsidea
ringavity, pumped byalaserbeamwithangular
fre-queny!
0
. ThefreeavitylengthisL,soL
av
=L+`
isthetotalavitylength(Fig. 1).
Wewanttoalulatethemeanamplitudeoftheeld
insidetheavityforasteady-stateoperation. Therefore
wewilladd upallthelosses, phaseshiftsand
ampli-ationsin around trip ofthe beamsinside the avity,
beginningwith the ouplingmirror. Thelosses of the
othertwomirrorsanbenegleted,andaddedlaterto
ree-R=1
R=1
r
0
r
1
r
2
α
0
in
α
0
out
α
1
out
α
2
out
Crystal
Figure1. BasiOPOonguration.
If the transmission through the oupling mirroris
small, weanexpress theoeÆientsof reetionand
transmissionas r j =e ( j ) '1 j ; t j =(2 j ) 1=2 ;
j =f0;1;2g!pump, signal,idler: (1)
where wehaveused
j
1. Thus, thetotal
transmit-taneforagivenmodeisT
j =2
j .
Insidethenonlinearrystal,thethreeeldsinvolved
in theOPOoperation(pump, signaland idler)willbe
oupled by the seond order nonlinear suseptibility.
Fromthis ouplingterm, we obtainaparametrigain
for thesignaland idlerelds, depending onthepump
eldamplitude. Onewealulatealltheontributions
inaroundtrip,weanseethatwhenthisgain
ompen-satesalltheround-triplosses,weobtaintheosilation
of theavityforthesignalandidlerbeams.
Itismoreonvenienttoexpresstheeldamplitude
asanormalizedvariable
j
, wheretheux ofphotons
per seond integrated in the beam transverse area is
j
j j
2
. Using this normalization, we an express the
amplitude variation of the elds at a point z inside
thenonlinearrystal,onsideringthatwehaveollinear
propagation,and that theslowlyvarying envelope
ap-proximationisvalid[2℄
d
0
=dz= 2 eff 2 (z) 1 (z)e ik z d 1
=dz=2
eff 0 (z) 2 (z)e ik z d 2
=dz=2
eff 0 (z) 1 (z)e ik z (2)
where k =k
0 k
1 k
2
is thephasemismath, and
theamplitudeofthewavevetorisjk
j j=!
j n j =. The oupling term eff
depends on the nonlinear
susep-tibility of the rystal, and on the shape of the three
beams involved. Sine we are working in avities, it
is reasonableto expet that the osillation modes are
gaussian. Inthisasethe eetiveouplingoeÆient
isgivenby[21℄
eff = (2) w 0 w 1 w w 2 0 w 2 1 +w 2 0 w 2 2 +w 2 1 w 2 2 ~! 0 ! 1 ! 2 0 3 n 0 n 1 n 2 (3) where w j
is thespot size inside the rystal(beam
ra-dius), and (2)
is the eetive seond order
susepti-bility,takinginto aountthepolarizationoftheelds
andtherystalorientation[2,38℄.
Tointegrateeq. 2,weanonsiderthat thegainis
small, so in arst order approximationtheamplitude
hangeinasinglepassbytherystalisgivenby
0 (`)= 0 (0) 2 1 (0) 2 (0) 1 (`)= 1
(0)+2
0 (0) 2 (0) 2 (`)= 2
(0)+2
0 (0) 1 (0) (4) with = eff ` sin(k`=2) k`=2 e ik `=2 : (5)
Ineq.4 weansee that thegainforthe signaland
idler modes inreases with the pump amplitude. So
doesthedepletionofthepumpforaninreasingvalue
ofthesignalandidler. Theamplitudeof theoupling
oeÆientvarieswithasinfuntionof(k`=2),
giv-inganeetivebandwidthforthegainonthevalueof
k.
Toalulatetheaumulatedphaseinaroundtrip,
we onsider the propagation in free spae and inside
therystal. Thelastonewilldepend ontherefrative
indexforeahbeam,giventhebeampolarization(due
to the rystal birefringene) and the wavelength
dis-persion. Thus the aumulated phasein around trip
is ' j = ! j (n j
`+L)=2p
j +Æ'
j ;p
j
=integer (6)
whereÆ'
j
isthedetuningoftheeldjfromtheavity
resonane. Thephaseshiftinthereetionofthe
mir-rorsanbeonsidered later, and will be negletedfor
themoment.
The losses of the beam in a round trip must
in-lude the reetion through the oupling mirror, and
alsootherspurious lossesinotheravitymirrors,
rys-talabsorption, et. Consideringthose lossesasan
ele-mentwithtransmissiont=e
,wehaveanadditional
energylossof2
j
in eah roundtrip. Thetotallossis
thereforegivenby 0 j = j + j .
Consideringthat we are near thetriple resonane,
whih is a onditionfor low osillationthreshold, the
detuning is very small (jÆ'
j
j 2). Therefore, the
totalamplitudein aroundtrip forasteady-state
on-ditionanbenally expressedas
0 0 0 (1 i 0
)= 2 1 2 + p 2 0 in 0 (7) 1 0 1 (1 i 1
)=2
0 2 2 0 (1 i 2
)=2
0
whereweusedthenormalizeddetuning j =' j = 0 j : (8)
Weseein thersttermofeq. 7theexternalpump
eld of theOPO as in
0
oupled by theinput mirror.
Thelasttwoequationsgiveustheoperationondition
foravitydetuningofsignalandidlerandthethreshold
valueoftheTROPO.
Thesolutionofeq.7fortheamplitudes oftheelds
gives us dierent operation regimes for a triply
reso-nantOPO[22℄, andthe stabilityof eah solutionmust
be studied. One simple solution is the trivial, with
1 =
2
= 0. This solution remainsstable for pump
values lower thana given limit. Abovethis threshold
the trivial solution beomes unstable and the system
willosillate.
An immediateonditionfor theOPOosillationis
obtainedmultiplyingtheseondequationineq.7bythe
omplexonjugateofthethird. Toassuretheresulting
equality 0 1 0 2 (1 i 1
)(1+i
2
)=4jj 2 j 0 j 2 (9)
wehavefromtheimaginaryparttherstonditionto
thesteady-stateoperation
1 =
2
=: (10)
II.1 Osillation threshold
Fromtherealpartofeq.9weanalulatethe
intra-avity pump powerthresholdfor theOPOosillation,
obtaining j 0 j 2 = 0 1 0 2 (1+ 2 ) 4jj 2 : (11)
Therefore,onetheOPOosillates,thepumppower
willbelimitedtothethresholdvalueintheaseofthe
assumed approximations (small gain, small detuning,
smalllosses),lampingtheintraavityamplitudetothe
valuegivenbyeq.11,dependingonthedetuningofthe
signalandidlerand ontheirlosses. Thethreshold
de-pendeneforthephasemathingk isinludedinthe
valueofjj 2
.
Inthease ofthe TROPO, weanuse the rstof
eqs.7to obtainthethresholdvalueoftheinput power
fortheOPOosillation. Thuswehave,for
1 = 2 =0 andj 0 j 2
givenbyeq. 11,thethresholdinputpower
j in 0 j 2 th = 0 2 0 0 1 0 2 (1+ 2 )(1+ 2 0 ) 8jj 2 0 : (12)
The threshold will be learly minimum in the
ex-at resonaneforall three modes. It will alsodepend
on the ratio of the intraavity losses to the oupling
denenowthepumpratioasthepumppower
normal-ized to the minimum threshold at zero-detuning and
zero-phasemismath = j in 0 j 2 j in 0 j 2 res = P in P th (13)
whih will be used laterin theoutput powerand
eÆ-ienyalulations.
II.2 Signal and idler frequenies
Theosillationondition(eq.10)willdeterminethe
possible wavelengths for osillation in the OPO,
on-sideringalsotheavitylenghtandthephasemathing
ondition. Itwillbelearerwhenexpressingthe
refra-tiveindexesinadierentform,astheaverageandthe
diereneofsignalandidler refrativeindex
n=(n1+n2)=2; Æn=(n1 n2)=2 (14)
and the beat note frequeny betweensignal and idler
as!=!
1 !
2
. Wewill rstassumethat thereis a
smalldierenebetweenthelossesineahmode,whih
anbeausedby dierentreetionoeÆientsof the
mirrorsatsignalandidlerwavelengths,orrystal
bire-fringeneontheabsorptionoeÆient. Expressingthe
lossesinthis symmetrialformwehave
0 =( 0 1 + 0 2 )=2; Æ 0 =( 0 1 0 2 )=2: (15)
It iseasier to work with thesum and the
subtra-tion of the phases of the elds in a round trip in the
avity. Fromeq. 6,weansee that
'
2 '
1
=2m+(Æ'
2 Æ' 1 ) ' 2 +' 1
=2q+(Æ'
2 +Æ'
1
) (16)
where m = p
2 p
1
and q = p
2 +p
1
are integers,
and neessarily have the same parity. It is
straight-forwardto alulatethebeatnotefrequenyusing the
denitions above andeqs.6, 8and 10. Assuming that
n!
0
Æn!,wehave
!=
Æn`!
0
+2m
L+n` + Æ 0 0 ! 0 2q
L+n`
(17)
where isthespeedoflightin vauum.
If the signal and idler losses are equal (
1 = 2 ), then Æ' 2 =Æ' 1
. We obtainthe equationexpressed in
ref. [19℄
!=
Æn`!
0
+2m
L+n`
= 2D Æn ` 0 +m (18)
withD==(L+n`). Soweansaythat:
a)Thebeatnotefrequenyhasadisretesetof
val-ues,determinedbyqandm(whihhavethesame
neigh-freespetralrangeatthesubhamonifrequenyofthe
pump !
0 =2.
b)This frequenyanbenelytuned byadjusting
Æn.
)Consideringdierentlossesforeahmodedoesn't
hangethebehaviorpreditedin[19℄loseto
degener-ay, sine the seond term in the right side of eq.17
is redued lose to the perfet phasemathing in the
quasi degeneratease, as wewill see later. Forhigher
valuesof!,itbeomesimportant,espeiallybeause
thenarrowspetralrangeoftheoatingforhigh
ree-tivityavitymirrorswill leadto unbalaned losses for
signalandidlermodes.
II.3 Resonane of signal and idler
AnotheronditionfortheOPOoperationisasmall
detuning for signal and idler. We will alulate now
the avitylength L where the OPOwill osillate,
be-ginningbytheexatresonantonditionÆ'
1 =Æ'
2 =0.
Clearlythisonditionisnotabsolutelyneessaryanda
smalldetuninganbeallowed,butitwilltelluswhih
longitudinalmode(m,q)islosesttoosillationdueto
itslowerthreshold.
Fromthephase'
2 +'
1
=2q(qinteger,positive),
weanalulate theavitylengthforexatsignaland
idlerresonane
L+n`=q
0
Æn`!=!
0
: (19)
Thisequationbeomesquiteumbersomewhen we
rememberthat ! also depends on L (eq.18). But a
simpleassumptionanbemade. Toperformtheavity
length sanning,searhingthe resonaneposition, the
displaementofLisin therangeofthewavelength,so
thehangeinthevalueofD willbenegligible,and we
anworkwithanaveragevalueofLinthedenominator
of eqs.17and18.
Asweanseefromtheseequations,wehaveadense
ombofresonanepositionsforsignalandidler. Fora
avity length given by q, we havea full range of
val-ues m that will dene around this point the possible
lengths for osillations, separated by adistane muh
smallerthan
0
. Thisdistane betweenadjaent
osil-lation points, remembering the parity of q and m, is
givenby
L=Æn` 4D
!
0 =Æn
2`
L+n`
0
: (20)
The existene of this dense series of points omes
fromtheadditionaldegreeoffreedomgivenbythebeat
frequeny that isn't found,for instane, in theaseof
SHG. As wewill see, the possible values for the beat
frequenyarelimitedbythephasemathingondition.
II.4 Phase mathing
Thephasemismathanbeviewedasaredutionin
in threshold power. From the phase mathing
ondi-tion we have k = [(n
0 n)!
0
Æn!℄=, so exat
phasemathingisobtainedfor!=!
0 (n
0
n)=Æn. A
valuedierentfromthisonewillinreasethethreshold
power. Tounderstandtheeetofmismathingonthe
seletionof thevalues! for osillation, wewill
on-siderthat theavityhasazerodetuning forthe three
elds.
Thethresholdvalueis thereforegivenbyeqs.5 and
12,anditanbeexpressedas
j in
0 j
2
pm =
0
0 2
0
1
0
2
8 2
eff `
2
0 sin
2
k`
2
: (21)
Thenewnormalizedpump poweran be expressedas
pm =j
0 j
2
=j in
0 j
2
pm .
Sine the osillation ondition for a given input
poweris that
pm
1,weansayfrom thisondition
andthe pump powerdened in eq.13that the
osilla-tionwill ourif
sin 2
k`
2
1
: (22)
Fromthis onditionwe obtainthe range of values
!thatareallowedtoosillateforagivenpumppower.
Observethatthisbandwidthishigherforhigherpump
power. Forinstane, onsiderthat =2. Inthisase,
wehavejk`j2:8asalimitondition. Therangeof
valuesof thebeat notefrequenyis 5:8=(Æn`)around
thevalueof!forexatphasemathing. Thenumber
ofpossibleosillatingmodesalulatedfromeq. 18will
be
N =2:8 L+n`
Æn`
(23)
so we expet to have typially more than ten modes
seletedby agiven phasemathing bandwidth, owing
tothesmallvalueofÆn.
II.5 Pump resonane
ATROPOisexpetedtoosillatelosetothepump
resonane,onditionwhihwillinreasetheintraavity
pumppower. Theavitylengthforexatresonanefor
thepump eldisgivenby
L+n
0 `=
0
s;s=integer: (24)
Combining the pump resonant ondition with
sig-nal and idler resonane ondition given by eq.19, we
analulatethebeatnotefrequeny!forthetriple
resonaneondition
!=2
Æn`
(q s)+!
0 n
0 n
Æn
: (25)
When we have exat phase mathing we an see
the value of q, and inside this group we have many
modesm ofosillation.
The same proedure used above for the phase
mathingonditionanbeappliedtoobtainthewidth
oftheregionwheretheOPOanosillatearroundthe
exatpump resoane. Inthis ase,foraperfet phase
mathing and signal resonane, the osillation
ondi-tion is given by 1+ 2
0
. For aentral valueof
avitylengthLgivenbyeq. 24,wehaveawidth
L
p =
0
0
0 p
1 (26)
ofpositionswheretheosillationispossible. Itislearly
neessarythatL
p
>Ldenedineq.20fortheOPO
osillation. Otherwise, beside the positioning of the
avitylength atthe pump resonane,weneedto tune
thevalueofÆnforexatresonaneofthethreemodes,
whih is inonvenient for a ontinuous operation and
stabilizationoftheOPO.
II.6 Type I phase mathing
The disussion presented above an be applied to
typeIIphasemathing,wherethevalueofÆndepends
muhmoreon therystalorientation thanonthe
sig-nal and idler wavelengths, sine the dierene of the
refrativeindexdue to therystalbirefringeneis one
orderofmagnitudehigherthanthevariationdueto
dis-persion. This assumptionis no longervalid for type I
phasemathing. Inthis ase,signalandidlerhavethe
same polarization, and the refrative index dierene
isaused onlybydispersion. Thealulatedequations
remain avalid approximationonly forwavelength
op-eration far from degeneray. Close to degeneray we
anexpressthedependene ofÆnonthebeatnote
fre-quenyinarstorderapproximation
Æn= n
! ?
?
?
?
!
0 =2
!
2 =n
0
(!) !
2
: (27)
Thederivativen 0
(!)oftherefrativeindexwith
re-spettofrequenywillplayamajorrolein the
osilla-tiononditionsfortheOPO.Applyingthis relationto
theequationsalulatedbeforewehave:
II.6.1 Beat note modes
Fromeq. 18wehave
!=
2m
L+ n+n 0
(!) !
0
2
`
=
2Dm (28)
asagoodapproximation,sinenn 0
(!)!
0
=2.
There-forethebeatnotefrequenyseparationisstillgivenby
theavitylength. But here,degenerateoperation an
be ahieved, sineÆn beomes zero in the degenerate
ase. IntypeIIphasemathing, thedegenerate
ondi-II.6.2 Signal and idler resonane
Theavitylengthforsignalandidlerresonanehas
nowaquadratidependeneon!. Fromeqs. 19and
27weansee that
L+n`=q
0 n
0
(!)`
2!
0 !
2
(29)
andthedistanebetweentworesonanepositionsis
L=
`
L+n`
0 n
0
(!)!: (30)
Asaonsequene,thedierenebetweenmirror
sep-arationforwhihthereisosillationislinearlyredued
withthebeatnotefrequeny. Theosillatingmodesan
easily overlaploseto degeneray. This leadsto mode
jumpsandaninreasingdiÆultytostabilizetheavity
at oneoperationmode!forquasi-degeneratetypeI
phasemathing.
II.6.3 Phase mathing
Exat phase mathing is assured if ! 2
=
2!
0 ( n
0
n)=n 0
. Near degeneray, this takes us bak
tothestraightforwardonditionn
0
=n. Wean
alu-latefromeq. 22therangeofvaluesof!forthetype
I phasemathing. Forthesameonditionof =2we
obtainabandwidthof p
5:8=n 0
`, biggerthantheone
obtainedinthetypeII rystal.
II.6.4 Pump resonane
Close to degeneray, n
0
= n, therefore s = q will
limit us to a group q of osillating modes. This will
givean interestingondition,oming fromeq.29. The
avitylengthforosillationhasaquadratidependene
on !. Sanning the avity in the pump resonane,
wewill haveadensegroupofosillationpointsforthe
OPO,butbelowagivenavitylength,theseosillations
will be interrupted. In typeII quasidegenerate
oper-ation this wouldn't our,sinenegativevaluesof !
wouldstillsatisfytheosillationondition,thatannot
befullled in typeI quasidegenerateoperation[23℄.
II.7 Frequeny tuning
As we an see, the osillation of the OPOwill be
determined by the onditions of avity resonaneand
phasemathing. Sotheseletionofthewavelengthwill
inludetheseletionofthemode,givenbytheintegers
m andq,andane tuningusingthevariationofÆn.
InFig. 2weseetheseletionoftheosillatingmode
in atypeII phasemathing. Thediagonallines
repre-sent thevalues of ! for exatresonane for agiven
avitylengthL
av
asdesribedin eq.19. Eahline
or-respondstoamodeq,andisinfatomposedofalose
set ofpoints,representingthedierentvaluesofmfor
and the avity length seletionby the pump resonant
ondition. In fat, if theavityhas a high nesse for
thepump, thewidthoftheavitylengthseletionan
besosmallthatonlyasinglevalueofmisallowedfora
givenvalueofq,but dierentvaluesofqanosillate,
depending on the phase mathing region. Therefore,
if wesan theavity lengthL
av
wewillsee the
osil-lation at dierent pointsof thepump resonanepeak.
If theirosillation lengthare loselyspaed, theOPO
aneasilyjumpfromonemodetoanother,andthiswill
representanadditionaldiÆultyinitsuseasatunable
lasersoure.
q+0
q+1
q+2
q+3
-0.004
-0.002
0.000
0.002
0.004
Pump Resonance
Phase Matching Region
L
cav
/
λ
0
∆ω/
ω
0
Figure2. SeletionofmodesfortypeIIphasemathing.
q+0
q+1
q+2
q+3
-0.06
-0.04
-0.02
0.00
0.02
0.04
0.06
Pump Resonance
Phase Matching Region
L
cav
/
λ
0
∆ω/
ω
0
Figure3. SeletionofmodesfortypeIphasemathing.
In Fig. 3, wehave the same analysis for atype I
phase mathing. As wehave seen,therangeof values
of ! that an osillate satisfying the phase
math-ing ondition is muh wider than in the ase of type
II phasemathing. Onthe otherhand, fromeq.29 we
observethattheseparationofthebeat notefrequeny
! for dierent valuesof q at the sameposition L is
inreased. Therefore,itisunlikelytohavemodejumps
An interesting limit ours when the pump
reso-nane oinides with the resonane of the degenerate
ase. Inthissituation,theosillationaroundthepump
resonanepeak is limitedto a singleside ofthe peak.
Forinreasingvaluesof L,wewillhaveosillation
lim-itedtotheondition!=0,verydierentofthetype
II phasemathing. Thissituation maybeomfortable
toobtaindegenerateoperationinthisOPO,lokingthe
avityto theside oftheresonanepeakat degenerate
ondition.
II.8 Output power
CalulatingtheoutputpowerofanOPOusingeq.7,
weanobtainanexpressionrelatingtheintraavity
sig-naleldandthepumpamplitudeexpressedbyeq. 13.
Thereforewehavefromeqs.7and12that
=
1
0 +
4jj 2
j
1 j
2
0
2
0
0
2
+(+
0 )
2
: (31)
From this equation, weobserve that for
0
1
and (1+ 2
)(1+ 2
0
),there isanonzerosolution
fortheavityelds. Forthetriplyresonantaseitwill
begivenby
j
j j
2
=
0
k
0
0
4jj 2
p
1
; (32)
withj;k=1;2ej6=k.
The output beams, oming out from the avity
throughtheouplingmirror,arethen
j out
j j
2
=
j
0
k
0
0
2jj 2
p
1
: (33)
This is the stable solution above threshold, when
the trivial solution (
j
= 0) beomes unstable. For
high values of detuning and pump intensities,
unsta-ble operationwill begin, with self-pulsing and haoti
operationoftheOPO[22℄.
For
0
>1,therearetwosolutionsforeq.31,
be-ingone stable and the other unstable. Together with
thetrivialsolution,wehavearangeof bistable
opera-tionin theOPO,asseenin theoretial[24℄and
exper-imentalworks[25℄.
It is interesting to ompare the eld intensities
(I
j =j
out
j j
2
) of signaland idler outputs. From their
meanvalueswehave
I
1
I
2 =
1
0
2
2
0
1
: (34)
Therefore,forequallosses,theaveragephotonux
will be equalforeah beam. Although this is notthe
noiseompressionmentionedintheintrodution,sine
wearedealingonlywithaveragevaluesoftheeldand
Thetotaloutputpoweranbealulated,in order
toobtaintheeÆienyoftheproess. Thetotaloutput
powerisgivenbythesumofthesignalandidleroutput
powers. For!
1 =! 2 , 1 = 2 , 1 = 2
,thispowerwill
begivenby
P out =~! 0 0 0 0 2jj 2 p 1 =4 max p
PP
th P th : (35)
SotheeÆieny willdependonthefator, being
maximum for =4 in the TROPO.Here we dened
themaximumeÆieny as
max = 0 0 0 0 = 0 (36) where j = j = 0 j
is the ratio of the avity losses
throughtheoupling mirrortransmitaneto the total
lossofthe avity. Aswewillsee, this isan important
parameterforthenoiseorrelationbetweensignaland
idlerbeams.
III Quantum properties in an
OPO
TheOPOplaysamajorrolein Quantum Optis,
pro-duingmanynon-lassialstatesoflight. Itbeganwith
thegenerationofsqueezedvauum[14℄,followedbythe
prodution of marosopi quantum orrelated beams
[15℄ and quantum noise ompression of the reeted
pump beamfrom aTROPOavity[17℄. Reently,the
generatedsqueezedvauumwasappliedinexperiments
for Quantum Teleportation[26℄, and the useof
quan-tumtomographyallowedthereonstrutionofthe
out-putsqueezedstateanOPO[27℄.
Onthe otherhand, the produtionof marosopi
beamswithorrelatedintensitieswasusedinhigh
sen-sitivityspetrosopy[28℄,andthisintensityorrelation
anbe usedto ontrolthe intensityin thesignalfrom
theintensity utuationsin theidlerbyafeedbakor
afeedforwardsystem[29℄, obtainingnoiseompression
belowthevauumlevel.
Wewill present ageneral desriptionof the
quan-tumpropertiesOPO,beginningwiththeMaster
Equa-tionofthedensitymatrixofthethreeelds insidethe
OPO avity. After that, we will present the
equiva-lent Fokker-Plank equation for the Wigner
distribu-tion, and then the Stohasti Dierential equations.
Fromtheseequations,alltheompressionrelationsfor
thesignal,idlerand pumpanbeobtained. Wenish
the presentation obtaining, from these equations, the
intensity orrelation for the signaland idler beams in
theoutputoftheOPO.
III.1 Master equation
Toobtain theMasterEquation, wewill followthe
forthedensityoperatorofthethreemodeeldinside
theavityisgivenby
d dt = i ~ [H f +H i +H ext ;℄+( 0 + 1 + 2 ): (37)
Thersttermin theommutator(H
f
)omesfrom
thefreemodesoftheeldinsidetheavity. Sowehave
H f = ~ 0 0 0 a y 0 a 0 ~ 1 0 1 a y 1 a 1 ~ 2 0 2 a y 2 a 2 (38) wherea + i ,a i
arethereationandanihilationoperators
oftheeletrield [13℄, istheroundtriptimein the
avity(assumedequalforallthethreemodes),and
j
isthedetuning,asdenedin eq. 8.
Theseondtermintheommutatoristheeetive
interation between the elds, given by the nonlinear
mediuminside theavity. Sowehave
H i =i~ 2 a y 1 a y 2 a 0 a 1 a 2 a 0 y (39)
where the oupling onstant is the same dened in
eq. 5. Thelasttermoftheommutatororrespondsto
thepumpdrivingeld (")injetedin theavity
H ext =i~ 0 " a y 0 a 0 : (40)
The last term of the Master Equation gives the
lossesoftheavityineahmode
j = 0 j 2a j a y j a y j a j a y j a j : (41)
It is shown in many referenes how to transform
from the density matrix formalism into the
quasi-probability distributions, e.g. ref. [31℄. We will use
this proedure to obtain a Fokker-Plank equation of
theeldsinside theavity.
III.2 Wigner representation and
Fokker-Plank equation
Thedensity matrixanbetreated in amuh
sim-pler way as a quasi-probability distribution in phase
spae. WeanusetheWignerdistribution oftheeld
in thephase spae,where wehave theappropriate
re-plaementoftheoperators(a +
i ;a
i
)oftheeldsby
om-plex amplitudes (
i ;
i
). The density matrix is also
replaed by a quasi-probability distribution in phase
spae, leadingto aFokker-Plankequation. This
pro-edureisompletelydesribedin ref. [31℄,andherewe
will present theoutline of this method applied to the
OPOavity.
Oneinterestingharateristi ofthe Wigner
repre-sentationisthattheoperatorsarereplaedbylassial
variables that anbe interpreted asan average value
plus autuation term. This leads to asimple
va-to those of a oherent state, and from the resulting
varianes of theoutput elds wean demonstrate the
quantum noise ompression, or the orrelation of the
elds amplitudes.
Making the substitution of the Master Equation
operators by the operators in the Wigner
representa-tion, we have the equivalent dierential equation for
theWignerdistribution
t W(f j g)= 3 X j=1 0 j " i j j j j j ! + j j + j j !# W(f j g) + 2 1 2 0 + 1 2 0 W(f j g) 2 0 1 2 + 0 1 2 W(f j g) 2 0 2 1 + 0 2 1 W(f j g) 0 " 0 + 0 W(f j g) + 3 X j=1 0 j 2 j j W(f j g) 2 3 0 1 2 W(f j g) (42)
where thevetorf
j
ghassix termsforthe eldsand
their omplex onjugates (
0 ; 0 ; 1 ; 1 ; 2 ; 2 ).
In-deed, in this eq. we an reognize a Fokker-Plank
equation, exept by the last triple derivative. For a
regular,wellbehaveddistribution(e. g. gaussian),this
termanbenegleted. Soeq.42anbeexpressedas
t W( f
j g)= X j j A j W(f j g) + 1 2 X j;k j k BB T jk W(f j g) (43)
where the vetorA is alled the drift vetor, and the
matrixprodutBB T
isthediusion matrix.
III.3 Stohasti dierential
equa-tions
TheFokker-Plankequationisequivalenttoasetof
StohastiDierentialEquations(SDE),alsoknownas
Langevinequations d dt j =A j +[B(t)℄ j (44)
where(t)isavetorofutuatingvariables
j
(t)with
azeromeanvalue,andthepropertythat
h i (t) j (t 0
)i=Æ
ij Æ(t t
0
): (45)
A rst onsequeneof this formulationan be
ob-servedin theanalysis ofthemean valuesoftheelds,
alulatedfrom the Langevinequations. Onean
im-mediatelyseethat theset ofequations
d dt j =hA j i (46)
is idential, in the steady state regime (d=dt = 0)
toeq. 7, presentedbefore, obtainedfrom thelassial
valuesof theelds. Therefore eq.46gives theaverage
valuesoftheeldandthestabilityofthese solutions.
Theset ofthe Langevinequations(eq.44) presents
manyquadratitermsthatavoidasimpleand
straight-forward treatment for the eld utuations. In this
ase,itismoreonvenienttouse alinearized
approxi-mationoftheeld utuations. Replaing theeld by
itsmeanvalueandasmall utuation
j (t)=
j
(t)+Æ
j
(t) (47)
andnegletingallsmallquadratitermsonthe
utu-ation,wehaveaset ofSDEdesribingalinearproess
with a onstant diusion, that an be treated to
ob-tainthespetraof the utuations of theelds. This
isthe proedure used in ref.[30℄, using thedesription
ofref.[31℄.
Althoughthis desriptionoftheeld angiveusa
diret expression for the noisespetraof the elds in
anOPO,allowingtheobservationofnoiseompression
andorrelationofeld amplitudes,theresulting
treat-mentof the66matrix is umbersome, and for our
purposes a simple substitution of variables will
sepa-ratetheset ofsix dierential equationsin twogroups
ofunoupledSDE's. Thenoisespetraofthereeted
pump utuations and of the signal and idler beams,
arealulatedandpresentedinref. [34,35℄. Up tothe
moment, only the reeted pump eld squeezing has
beenexperimentallyobserved[17℄.
Ifwereplaetheelds
1 ,
2
bythelinear
ombi-nation + = 1 + 2 p 2 = 1 2 p 2 (48)
in eq.44, we an obtain a new set of SDE equations
fortheelds utuations. Linearizingthe eldsabove
aroundtheirmeanvaluesweobtainthefollowing
equa-tionsfortheutuatingtermsÆ ,Æ
0
(1+i)Æ + + p 2 0 2 (t) d dt Æ = 2 0 Æ 0
(1 i)Æ +
p 2 0 3 (t) d dt Æ = 2 0 Æ 0
(1+i)Æ + p 2 0 4 (t): (49)
WeadoptheretheoperatingonditionsoftheOPO,
1 =
2
=,andassumethatthelossesarethesame
forbothsignalandidlermodes(
1 =
2
=). Thetwo
last equations below form a separate systemof
equa-tions. So theproblem of the utuationsof the elds
forthepumpbeamisreduedtotheevaluationofthe
44matrixoftheeldvarianes.
Theeldrelationalulatedfortheintraavityeld
doesn't giveimmediately the utuationsof the elds
omingoutfromtheavity. Wemustonsiderthe
out-putouplerasabeamsplitter,wherewehavethe
ini-deneof the intraavity eld andof the vauum led.
Thistreatmenthasbeenextensivelyusedbymany
au-thors[32,33℄. Thevauumutuationsanbe
onsid-eredintwoseparateterms. Oneomes intotheavity
from the oupling mirror, and the seond oneis
ou-pledbyotherspuriouslossesoftheavity. Deningthe
vetorfÆ (t)g =
Æ (t);Æ
(t)
, the eld
utu-ations an be alulated from the following equation
[32℄
d
dt
fÆ g= MfÆ (t)g
+ p 2 f a (t)g+ p 2 f b (t)g (50) wheref a
(t)gandf
b
(t)g arerespetivelythevauum
inputsfromtheouplingmirrorandthespuriouslosses,
and M= " 0 (1 i) 2 0 2 0 0
(1+i) #
: (51)
Using the Fourrier transform of the eld,
fÆ ()g = R fÆ (t)ge it dt = Æ();Æ ( ) , weobtain
fÆ ()g=
(M+iI) 1 p 2 f a ()g+ p 2 f b ()g (52)
withI beingthe22identitymatrix.
Theoutput utuationsare thengivenby thesum
oftheintraavityeld transmittedbytheoutput
mir-ror andthe reetedvauum utuationsonthis
mir-ror. Makingtheapproximationofthereetivityofthe
ouplingmirrorto1,wehave
fÆ
out
()g=f
a ()g
p
2fÆ ()g (53)
for thediereneof thesignaland idler elds
utua-tionintheoutputoftheavity.
III.4 Intensity orrelation
The amplitude orrelation of the signal and idler
beams anbe measured by the dierene of the
pho-tourrentsproduedbyeahbeam. Theresulting
out-put(onsideringthedetetionquantumeÆienytobe
100%)is I (t)=I
1 (t) I
2
(t). Theeld I
i
(t) analso
be expressedas amean valueand autuating term,
orrespondingtothenoiseofthebeam. Foraeld
am-plitude j (t) = j +Æ j
(t), the eld utuationswill
be
ÆI
j (t)=I
j (t) I j = j Æ j (t)+ j Æ j (t) : (54)
Itismoreonvenient,andequivalentinform,to
ro-tate thephaseof the beam, obtainingareal valuefor
themeaneld. Inotherwords,multiplyingtheeld
j
bye i
j
,wehave
ÆI
j
(t)=j
j j
Æ
j
(t)+Æ j (t) =j j jÆp j (t) (55) whereÆp j
(t)istheamplitudeutuationoftheeld.
Oneweareworkingwithabalanedavity,where
the losses of signal and idler elds are the same, the
meanamplitudeoftheeldswillbeequal(j
1 j=j
2 j).
Theutuationsoftheintensitydierenewillbegiven
by
ÆI (t)=jj [Æp
1
(t) Æp
2
(t)℄=j
j j
Æ (t)+Æ
(t)
:
(56)
When we measure the eld utuations, using a
SpetrumAnalyzer,weobtainthenoisespetraof the
photourrentutuationsin time[36℄. Fromthe
Four-riertransformof thephotourrentdierene ÆI(t),we
havethenoisespetra
S()=jj 2
hÆp ( )Æp ()i: (57)
Wehaveto alulate theamplitudedierene
u-tuationsÆp (t)fromeq.53. Thematrixpresentedineq.
51dependsonthepumpamplitude,whoseintensityis
givenbyeq.11. Makingthehoieofthephaseinorder
to simplify thealulationoftheeld utuations,we
have 0 = 0 2
(1 i): (58)
This will give us the amplitude utuation in the
frequenydomain
Æp ()= 1 2 2 0 i Æp a () 2 p 2 0 i Æp b () (59) whereÆp a (t),Æp b
(t)aretherealpartutuationsofthe
Onean immediately alulate, from the equation
above,thenoisespetrumS(),rememberingthe
nor-malization of the amplitude utuations to the shot
noise
hÆp
i
( )Æp
j
()i=Æ
ij
: (60)
Thenoiseutuationsarethengivenby[37℄
S()=jj 2
"
1
1+(=
0 )
2 #
(61)
where the avity bandwidth for the signal and idler
modeis
0 =2
0
=,and isthesamevalueusedineq.
36. Thisgivesusadiretphysialinterpretationofthe
noisespetrum. Forashotnoisemeasurement,likethe
division ofa singlemonomode beam in a50/50beam
splitter,andthesubtrationofbothphotourrents,we
havethenormalizednoiseS()=I =1. Butinthease
oftwinbeams,thetwooutputshaveanoiseorrelation
below the shotnoise. This gives us a noiseredution
oftheutuationswithalorentziandependeneonthe
analysis frequeny . The weighting fator of the
noiseompressiongivesthemaximumorrelation,that
isobtainedatfrequeny=0.
Sine the photons are reated in pairs inside the
avity from the annihilation of a pump photon, the
beamintensitiesare expetedto haveahigh degreeof
orrelation. Forthespontaneousemission ofa
nonlin-earrystal(parametriuoresene),wehaveapreise
time orrelation betweenthetwoemitted photons. In
theOPO,sinethegeneratedphotonwillhaveadeay
time to ome outfromthe avity,this orrelationwill
onlybeobservedifwewait atimelongenoughforthe
photontoomeoutfromtheavity. Thatisthereason
whywehavealorentzianurveforthedierenenoise,
withthefrequenynormalizedbytheavitybandwidth.
Thefator givestheratioofphotonsthat,oming
outfromtheavity,reahthedetetor. Ifthereareno
other losses than the oupling mirror, this fator will
reahunityand thebeamswill beperfetly orrelated
in intensity. Every time that there is the loss of one
photon from oneof thebeams,the orrelationwill be
degraded. Anotherimportantpointinthis exampleof
sub-shot noisemeasurement omes from the
indepen-deneon pumputuations. Ifthesystemis well
bal-aned,forbothoptiallosses andeletroni detetion,
the"lassial"amplitudeutuationswillanelatthe
subtration. Thisisnolongertrueforunbalaned
av-itylosses,asanbeseenin[35℄,butthesub-shotnoise
levelanstillbeattained,orevenreoveredforan
ex-ternal balaneofthebeamintensity.
IV Experimental results
We havebuilt an OPOfor thestudy of noise
orrela-tion,usinga10mmlong,typeIIphasemathedKTP
Æ Æ
Aording to the manufaturer (Cristal Laser S. A.),
itsabsorptionis2%at 532nmand0.05%at1064nm.
Thefaes oftherystalwereARoated,withresidual
reetionof0.5%at532nmand0.1%at1064 nm. In
thisbiaxialrystal,therefrativeindexforthez
orien-tationis1.83at1064nmand1.89at532nm. Forthe
xandyorientation,thedierenein therefrative
in-dexissmall,andforourpurpose,weanonsidertheir
valuesas1.74at1064nmand1.78at532nm[38℄.
Theavityis madeby two spherialmirrorsof
ra-dius13 mm. The avity length is 18 mm, assuringa
FSR=6GHzandavalueofz
0
=7mm. Wehavehere
aonfoal avity, what will simplify thealignmentof
themirrorsand theinjetionof thepump beam. The
Rayleigh length value isn't optimized to the relation
givenbyBoydandKleinman[21℄foroptimal
paramet-ri interation between gaussian beams (` = 5:6z
0 ).
The avity length is ontrolled by individual
transla-tionstagesforeahmirrorandnetuningismadewith
piezo-eletritransduers.
The rstalignmentand implementationwasmade
with two mirrors with high reetivity for 1064 nm
(99.8%)andalowerreetivityfor532nm(81%). The
measured avity nesse for the pump beam was F =
12and the estimated nesse forthe subharmoni was
around200.
A seond avitywasemployed replaingthe initial
outputouplerbyamirrorofR=97%at1064nmand
R=99.8%at532nm,assuringameasurednesseF=
24at 532 nmand an estimatednesse of 100at 1064
nm.
L2
M2
DM
OPO
VIS
SA
W3
PBS1
+/-PBS2
IR
IR
W2
M1
F R
LASER
W1
L1
Figure4. SetupoftheOPO.
Thesetup isshownin Fig. 4. Thelaser is adiode
pumped frequeny doubled Nd:YAG (Lightwave142),
with 200 mW output power. The power ontrol over
thesetupismade bythehalf waveplate(532 nm) W1
and the input polarizer of the optial isolator. This
se-instabilityinlaseroperationausedbytheOPOavity
reetion. Theseond523nmwaveplate(W2)rotates
thebeampolarization totheextraordinaryaxisofthe
KTPrystal(xyplane). LensesL1 andL2enablethe
modemathingofthelaserbeamwiththeavity. The
rystal is mounted in an oven, with temperature
on-trolledupto0.1 Æ
Candwitharangefrom20to160 Æ
C.
The polarization oftheoutput isontrolled bythe
1064 nm half waveplate W3. The polarizing ubes
PBS1 and PBS2 (Newport 10BC16PC.9) provide an
extintionratiobetterthan1:1000intransmission,
giv-ing agood separationof signal and idlerbeams. The
infrared beams are foused by a pair of lenses into
thephotodetetorsETX300(Epitaxx),withhomemade
ampliedeletronisforthehighfrequeny(HF)
om-ponents of the photourrent. The DC omponent is
alsomeasuredusingadigitalosillosope. TheHF
out-putsaresubtratedinanativeiruit,andthe
result-ingnoisepowerismeasuredintheSpetrumAnalyzer
(HP 8560). The remaining green light is transmitted
throughthedihroimirror,beingdetetedinavisible
photodetetorPDA55(Thorlabs).
IV.1 Threshold and eÆieny
Sanningthe avity lengthusing the PZT, we an
put it on resonane with the pump wavelength and
observe the osillation either by the light generation
around1064nm orbythedepletion oftheintraavity
power at thefundamental wavelength, whih is
moni-toredbythephotodetetorVIS.
Thebest value obtainedfor the osillation
thresh-oldwas8mWofpumppower,hangingwiththe
align-ment of the mirror in the avity. A typial value
ob-tained,aftersomerunningtimeandslowrystal
degra-dation(possibly due tograytraking [39℄),is 30mW.
InFig. 5weanseein thenormalizedpump intensity
that the depletion of the beam is inreased with the
pumppower,andthevaluefortheosillationondition
is lipped to amaximumvalue when we have
osilla-tion,asdesribedineq.11. Theintraavitypump eld
presentsaparabolishape,andfrom thettingofthis
equationtheavitylosses
1 ;
2
anbeevaluated,
giv-ingthevalueofthenesseforthesignalandidlerelds.
Weanalsoseethatasweinreasethepumppower,
many peaks appear. These peaks represent dierent
modes(m,q)ofosillation,andtheirrelavitepositions
intheavitylenghtsanningaregivenbyeq.19.
Measuringthepeakintensityofthegeneratedbeam
for dierent pump powers, we ould measure the
eÆ-ieny of ourOPO.The tof the outputpowervalue
given by eq.35 to the measuredvalues is presented in
Fig. 6. From the measuredvalue of
0
= (433)%
and the tted value of
m
ax = 0.8 % we have =
(2:00:2)%. Thevalueof
0
isobtainedfromthe
mea-surement of the transmittane of the input mirror at
532nm(pump)andthemeasurednesseoftheavity
Weansee learlyfromeq.61thatthenoisesqueezing
will be verysmall, and this system isn't adequate as
atwin beamgenerator,remaininganinterestingsetup
forthedesriptionoftheOPOoperation.
-1,5
-1,0
-0,5
0,0
0,5
80 mW
48 mW
Output (Signal + Idler)
16 mW
Time (ms)
Pump
Figure5. Normalizedoutputfortherstavity.
0
20
40
60
80
100
120
140
160
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
P
th
=(34.6 ± 0.6) mW
η
max
= (0.841 ± 0.005)%
O
u
tp
u
t P
o
w
e
r (m
W
)
Pump Power (mW)
0
20
40
60
80
100
120
140
160
0.000
0.002
0.004
0.006
0.008
0.010
E
ffic
ie
n
c
y
(
P
out
/P
in
)
Pump Power (mW)
This small value of omes from the high
ree-tivity of the oupling mirrors. In this situation, the
rystallossesareimportant,andthesurfaequalityof
the optial omponents involved also limits the
ee-tive losses. Mirrors and rystal surfaes have surfae
qualitybetterthan=10. Previousmeasurementswith
lowersurfaequalitymirrorspreventedtheosillation,
dueto aninreaseinthethresholdvalueduetohigher
losses.
After the initial study, we haveused another
av-ity onguration with an output oupler for the 1064
nm astheavitymirror. Thisonguration inreased
the overall losses at the subharmoni mode, giving a
higher threshold,but theratio is muh improvedby
inreasingtransmissivityoftheouplingmirror.
Asanbeseenin Fig. 7,for apump powerabout
twiethethreshold,weouldobserveonlyasinglepeak
for the OPOosillation. Whenwedon't have
osilla-tion atexatpump resonane(zerodetuning), wean
observeapairofpeaks,likethoseexpetedfordierent
modesm of osillationin the onditionobservedfrom
eq.20, L
av
> L. It is also lear that the
deple-tionofthepumpbeamhasaparabolibehavioronthe
signaldetuning,as givenbyeq.11.
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
0.0
0.4
0.8
1.2
1.6
2.0
2.4
2.8
3.2
Signal + Idler
Pump
In
tensi
ty (arb.
un.
)
Time (ms)
Figure7. Resonanefortheseondavity. Pumppower=
110mW.
Repeating the measurement of the peak power in
the OPOoutput sanningtheavity, we obtainedthe
maximumeÆieny from thettingof eq.35(Fig. 8).
From the measured value of
0
= (76 3)%, and
the adjusted value
max
= (372)%, we expet that
=(513)%.
Therefore,hehopeto observeanoiseorrelation3
dB belowtheshotnoiselevelin thisOPO.Theavity
bandwidthlimitstherangeofthespetrumwhere
om-pression anbe observed. Inour ase,the bandwidth
is 200 MHz. Sine the eletroni iruit is frequeny
limitedto50MHz,thehosenfrequenyfornoise
mea-surementis10MHz. Inthisase,weanapproximate
In spiteof many advantages regardingthe
squeez-ing,thehigherpump powerinvolvedisalimitationin
thisset-up,duetothermalbistabilityandrystal
degra-dation. We will disuss briey these aspets, before
proeedingtothenoisemeasurement.
60
80
100
120
140
160
0
10
20
30
40
50
60
P
th
= (59.3 ± 2.1) mW
η
max
= (37.5 ± 1.7) %
O
u
tp
u
t P
o
w
e
r (m
W
)
Pump Power (mW)
60
80
100
120
140
160
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
E
ffic
ie
n
c
y
(
P
out
/P
in
)
Pump Power (mW)
Figure8. OutputpowerandeÆienyfortheseondavity.
IV.2 Thermal bistability
KTPrystalsareommonlyusedinOPOandSHG
owingto their small absorption at 1064 nm. In spite
oftheir higher absorptionat 532 nm, theydon't pose
problemsinpulsedSHG,whenitisoftenusedina
sin-glepassondition. Owingto thetypeII phase
math-ing, itis hardly usedin adouble resonant avity
on-guration. Inthis ase, LiNbO
3
rystalsarepreferred
for SHG from CW Nd:YAG lasers using avity
on-guration to enhane the onversion eÆieny. In a
triply resonant OPO the KTP absorption will ause
someproblemsduetothethermalbistability.
In a KTP rystal, the value of n=T is about
310 5
K 1
[38℄. Therefore, the heating of the
sam-ple aused by absorption will produe an inrease in
therefrativeindexandin theeetiveoptialpathof
the beam. If the avity length is redued, using the
PZTsanning,aswegetneartheresonanethe
tempera-willinreasetheeetiveoptialpathinsidetherystal,
providing anegativefeedbakto themehanial
san-ningofthePZT.Thepulsewidth inaavitysanwill
inrease. On the other hand, if we inrease the
av-ity length, when we approah the resonane peak the
heating of therystal will inrease again theeetive
path of the beam, resulting in apositive feedbak to
thisdisplaement. Thiswill reduethepeakwidth.
These eets an be seen in Figs. 9 and 10. For
fastersanningspeeds,thereisthe broadeningor
nar-rowingofthepeakdependingonthesanningdiretion.
Ifthesanspeedisslowenoughto providea
stabiliza-tionofthetemperatureinsidethesampletherewillbe
abistable behaviorforonvenientvaluesof the pump
power. Foragiven avitylength, there aretwo
possi-bleavityintensities, dependingonwhethertheavity
lengthwasinreasedorredued.
-75
-50
-25
0
25
50
75
0,0
0,4
0,8
1,2
1,6
2,0
2,4
2,8
Signal
Pump
92
µ
m/s
9.2
µ
m/s
0.92
µ
m/s
In
te
n
s
ity
(
a
. u
.)
∆
L (nm)
Figure9. Bistabilityfor L=t<0for dierent sanning
speed.
-100
-75
-50
-25
0
25
50
0,0
0,4
0,8
1,2
1,6
2,0
2,4
2,8
Signal
Pump
92
µ
m/s
9.2
µ
m/s
0.92
µ
m/s
In
te
n
s
ity
(
a
. u
.)
∆
L (nm)
Figure10. BistabilityforL=t>0for dierentsanning
speed.
This fatoris a majorproblem in avity
stabiliza-tion. Oneproblem is thatif weloktheavity in one
signalandidlerpeak,theheatingoftherystalwill
pro-pump resonane. This displaement will inrease the
pump detuning, and with this variation the threshold
valueanbeomehigherthanthepumppower,andthe
osillationwilldisappear. Thiswilllimitthenessefor
the pump for apratial operation of theOPO. This
problemisoftennegletedinaDROPO[15℄wherethe
displaement of the signal and idler resonane due to
therystalheatingwon'taetthethresholdondition.
Aseondproblemthatisfaedistherystal
degra-dation. When loked at high pump power, the
intra-avity intensity anreah valuesof 150 kW/m
2
. An
inreaseoftherystalabsorptionat532nmisobserved
at thisvalue,and aonsequentinreasein thermal
ef-fets. This inreasing rystal absorption redues the
pumpnesse,andafteratimelongenoughthe
osilla-tionispreventedasaresultofthehighthresholdvalue.
Onepossiblereason for this eet is graytraking
[39℄. This kind of defet has been observed in KTP
rystals,andomesfromtheformationofolorenters
by the reation of displaed Ti
+3
ions in the rystal.
It wasobservedthat these defets are notalways
per-manent, and our in the pulsed regime. One they
appear, the annealing of the rystal at 200
Æ
C an, in
someases,restoretherystaltotheinitialabsorption
value [40℄. Reent studies have shown also the gray
trakingin theCWregimeforintensitiesrangingfrom
25 to 125 kW/m
2
, the limiting value depending on
rystal manufaturingquality[41℄. Although the
pre-viousindiationoftemperatureontrollingofthegray
traking,wehavenotobservedanyredutionofthe
de-fetformationintherystalwhenworkingintherange
from 20 to 170
Æ
C. Therefore, for the available pump
power of 200 mW a longterm operation of the OPO
was impossible in view of the inreasing value of the
threshold.
IV.3 Noise orrelation
Theombinationof thermalbistabilityand rystal
degradationavoidedastableoperationoftheOPO
av-ity, and it wasn't possible to keepit stable fora long
time(>1s). Butatransientoperationlastinghundreds
ofms ouldbemaintained. Inthis situation,weould
performthemeasurementof theeld orrelation.
Usinganeletronidisriminator,weouldgenerate
aTTLdigitalpulse whentheOPOstartedosillating.
Thispulsewasusedtotriggeradigitalosillosopeand
thespetrumanalyzersimultaneously. Sowehada
syn-hronousmeasurementintimeofthephotourrentand
thenoiseutuationsmeasuredbythephotodetetors.
An exampleof this measurement is shown in Fig.
11. As weansee, themeasurednoiseisreduedwith
the beam intensity,nearing the eletroni noiseof the
aquisitionsystemasthephotourrentreaheszero. As
wehave seen in eq.61, the noise level of the intensity
proportionalto the normalizednoiseof theamplitude
orrelationofthebeamsS().
To obtain this normalization, we subtrated from
themeasurednoisepowertheeletroni noiselevel(of
-85 dBmfor thegiven SA parameters). Thenthe
or-reted noisepoweris normalizedbythe photourrent.
Ifthisnormalizationisvalid,theresultingaveragevalue
shouldn't hange with time, and the value utuates
around thisaverage,due to thestohasti behaviorof
thenoisemeasurement. InFig. 12wesee the
normal-ized noise. There isn't anyobservablevariationin the
meanvalueofthenoisein time,andweobserveonlya
smallhangeinthedispersion,duetotheinreasing
in-ueneoftheeletroninoiseintheorrelationnoiseas
theintensitydereases. Wehaveagoodtofthisdata
intoagaussianurve,assuringusthatthismethodan
beusedtoobtaintheorrelationnoiseandompareit
to thevauum (shotnoiselevel).
0
5
1 0
1 5
2 0
2 5
3 0
-8 8
-8 4
-8 0
-7 6
-7 2
-6 8
T im e (m s)
No
is
e
(
d
B
m
)
0
4
8
1 2
1 6
2 0
O
u
tp
u
t P
o
w
e
r (m
W
)
Figure11. Synhronousaquisitionof noiseand
photour-rent at10MHz,RBW =300kHz,VBW =30kHz.
Ele-troninoiselevelat-85dBm.
0
5
10
15
20
25
30
-90
-88
-86
-84
-82
-80
-78
-76
-74
-72
No
is
e
(
d
B)
Time (ms)
-90
-85
-80
-75
-70
-65
0
20
40
60
80
100
Noise (dB)
Figure12. Distributionofthenormalizednoise.
Onewayto hek the noisesqueezing is observing
the normalizednoisevariation asthe inidentbeamis
attenuated. Asiswellknown[13℄,squeezingisdegraded
by absorption. So if we use neutral density lters in
our beam, inreasing the vauum input, wewill have
aninreaseinthenoise. Thisdependene inthe
trans-mission anbeexpressed as
S =1 (1 S
0
)T (62)
where S is the normalized noise, orresponding to 1
for the shot-noise level, S
0
is the normalized noise of
the measurement, and T is the transmittane of the
beam throughthe attenuator. ForT =0, weget the
shotnoiselevel. Iftheinputbeamisshotnoiselimited
oftheattenuator.
Weperformedasmallsequeneofmeasurementsof
thesqueezingdegradationfordierent neutraldensity
ltersintheOPOoutput. AsobservedinFig.13,fora
dereasingattenuation, wehaveaninreasing
normal-izednoise. Weanonludethatthereisnoise
ompres-sionin theintensityorrelation, 33%belowshot-noise
(41%orretingforlosses).
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
S = 0.666 ± 0.028
N
o
rm
a
liz
e
d
N
o
is
e
Transmittance
Figure 13. Squeezing degradation for inreasing
attenua-tion. TheOPOoutputpolarization isrotated100 Æ
bythe
halfwaveplate.
Makingthediereneofthehighfrequeny
ompo-nentin eahphotodetetorintheeletroni iruit,we
ouldobtaintheintensitynoiseorrelationinthesignal
andidlerbeams. Whenseparatedbytheorret
orien-tationofthehalfwaveplate,wehaveinoneportofthe
iruitthenoiseofthesignalandintheotherthenoise
oftheidlerintensity. Ifweturnthiswaveplate,mixing
thewavesin thepolarizingbeamsplitter, wewill have
adivisionofhalftheintensityofeahmodeineah
de-tetor. Asommonlyused,thephotourrentdierene
of asinglemode dividedin a50/50beam splitterwill
giveusthevauumutuationlevel,ortheutuation
expeted for aoherent state of the same average
in-tensity. With twomodes,whosebeatfrequenyiswell
abovethedetetorsbandwidth,theresultingnoisewill
orrespondtotheoherentstateutuations.
Therefore, when rotating the beam polarization,
hangingfromaompleteseparationofbothmodesin
the detetors to their mixture in a 50/50
beamsplit-teronguration,weexpetthat thenoisewill hange
fromtheorrelationnoisetotheshotnoise. Ifthereis
ompression inthe orrelationnoise,itwill beseenas
anoiselevelredutioninomparisontotheshotnoise,
whihis measuredfora45 Æ
rotationofthesignaland
idleratthepolarizingube(seeAppendix).
Werepeatedthenoisemeasurementfordierent
an-glesofthehalf waveplate,hangingthemixtureofthe
beamsinthepolarizingbeamsplitter. Theresults,
pre-sentedin Fig. 14, show that a maximum of squeeing
idler separation is 60.9%. Foran overalldetetion
ef-ieny of 81.1% (photodetetor eÆienies, losses in
lenses,beam-splitters,et.),weonludedthatwehave
a noise ompression down to 52%, or -2.8 dB of the
shot noise level. This agrees with the expeted value
obtained from the measurement of
max and
0 , that
gaveusaexpetedvalueofS=1 =(0:490:03).
0.0
22.5
45.0
67.5
90.0
112.5
135.0
157.5
180.0
0.6
0.7
0.8
0.9
1.0
1.1
S = 0.609 ± 0.016
N
or
m
al
iz
ed noi
se
Rotation Angle (degrees)
Figure14. Correlationnoisefordierentanglesof
polariza-tionrotation.
V Conlusions
We presentedin this paperageneral overview for the
CW operation of an OPO. We inluded in our
treat-mentthedesriptionoftheOPOtuningpropertieswith
type I and type II rystals and the alulationof the
onversioneÆieny of the pump into the signal and
idler beams. Wepresentedalso areview of the
quan-tum properties of theOPO,and studied theintensity
utuationsofthetwinbeamsinatypeIIOPO,
show-ing quantum orrelation in twin beams. The
tehni-al problemsofheating andrystaldegradationwhere
also treatedin thepresentwork,and the synhronous
measurementof noiseandintensityis presented asan
eÆientwaytoirumventtheseproblemsduring
mea-surements.
TheOPOremains, after more thanthree deades,
afasinating subjetof studyin quantum and
nonlin-ear optis. The renewed interest in the OPO omes
from new materials manufaturing, and the
ommer-ialappliationofthiswidebandtunablelaser-likelight
soure. Ontheother hand, sinethebeginningof
ex-perimental quantum optis, it provides a reliableand
simplesoure of intense beams withquantum
orrela-tions. Amongothersouresofsqueezedlightin
nonlin-earrystals,theOPOprovidesthebest resultin noise
ompression. For that reason, there is now a raising
