Parametric oscillation of optical phonons in magnetoactive III-V semiconductors by IRJET Journal

Parametric oscillation of optical phonons in magnetoactive III-V semiconductors

Sudhir Dalal

Department of Physics, A.I.J.H.M. College, Rohtak 124001, Haryana, India ***

Abstract Using the hydrodynamicmodelofsemiconductor plasmas and coupled mode theory of interacting waves, we have analytically studied parametric oscillation of optical phonons in magnetizeddopedIII Vsemiconductors.Theorigin of nonlinear interaction is assumed to lie in the complex effective second order optical susceptibility arising from the nonlinear polarization created by inducedcurrentdensityand by interaction of the pump wave with molecular vibrations generated within the medium.Expressionshavebeenobtained for threshold pump intensity and conversion efficiency of the proposed single resonant parametric oscillator. Numerical analysis has been made for a representative n InSb crystal 2 laser. The effects of some important parameters such as transverse magnetostaticfield, mirror reflectivity, crystal length etc. on the threshold pump intensity and conversion efficiency of the single resonant parametric oscillator have been analyzed in detail. The analysis establishes the technological potentiality of transversely magnetized doped III V semiconductors as the hosts for parametric devices like parametric oscillators.

Key Words: Nonlinearoptics,Opticalparametricoscillation, Opticalphonons,III Vsemiconductors

1. INTRODUCTION

Nonlinear Optics is a branch of optics that deals with the behavior of light in nonlinear media. It deals with the interactions of applied electromagnetic fields in various materialstogeneratenewelectromagneticfieldsalteredin phase,frequency,amplitudeorotherphysicalproperties[1 3].Thefieldofnonlinearopticsisgivenincreasingattention due to its wide application in the area oflaser technology, optical communication and data storage technology [4, 5] For the design and operation of a wide range of potential devicesinvolvinglight,understandingtheconceptsofvarious nonlinearinteractionsareessential.Amongthese,parametric interactionisthefundamentalinteraction.Theoriginofthis interactionliesinsecond (2) of themedium.Inthisprocess,anintenselaserbeam,hereafter referred as ‘pump’, interacts with nonlinear medium and results into generation of waves at new frequencies. This occursduetomixingorcontrolledsplittingofwaveswhich mayundergoamplificationorattenuationdependingonthe material properties and geometry of externally applied electric and magnetic fields. Parametric interactions have beenemployedsuccessfullytostudythephotonamplifiers

[6]andtogeneratehighpeakpowersub picosecondoptical pulse[7].

Thedevicesbasedonparametricinteractionssuchasoptical parametricamplifiersandoscillatorsoccupiesaspecialplace in nonlinear optics due to their potential applications [8]. They have been extensively used in fabrication of tunable coherentradiationsourcewithconsiderablyhighgainand highconversionefficiency[9].Opticalparametricoscillators are used for the generation of coherent light and mode lockedpulsetrainsoveracontinuousrangeoffrequencies, usually in frequency bands where there is a paucity of tunablelasersources[10,11].

Anopticalparametricoscillator(OPO)isadevicebasedon optical parametric oscillation which oscillates at optical frequencies.Itconvertsaninputpumpwaveatfrequency 0 intotwooutputwavesoflowerfrequency( s  , i )bymeans ofsecondordernonlinearopticalinteraction.Thesumofthe output waves frequencies is equal to the input wave frequency: 0 si  .Forhistoricreasons,thetwooutput wavesarecalled"signal"and"idler",wherethewavewith higher frequency is called signal. A special case is the degenerateOPO,whentheoutputfrequencyisone halfthe pumpfrequency, 0 /2 si 

Themainapplicationofan OPOistogeneratelongerwavelengths,likepulsedlasersat around 1.55 µm, from a 1.064 µm pump. The former wavelengthliesintheeye safebandandiscommonlyused by range finding devices. But an OPO can also be used to generatelongerwavelengths,orvariouswavelengthsinthe visibleregimewhenpumpedat355nm[12].

Recentlymajorenhancementsintheoverallperformanceof continuous wave single resonant parametric oscillator (SRPO)throughfiniteoutputcouplingoftheresonantwave andgainwidthandrisetimestudiesofpulsedunstableOPOs have been reported [13, 14]. Various effective methods of generating eye safe radiations and light in the yellow and blue spectral region based on OPO characteristics are also being developed [15 17]. The developments in the field of OPOandtheirusagearerestrictedtoalargeextentduetothe inherent material limitations, viz., low optical damage threshold,inadequatebirefringenceandopticaltransparency range.

The advent of new nonlinear materials with high optical damagethreshold,wideopticaltransparencyrangeandlaser

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pumpsourcesofimprovedspectralandspatialcoherencehas created new vistas for generating widely tunable optical coherentradiationusingOPO.Amongstthevariousnonlinear materials, the semiconductor crystals (especially III V semiconductors) are substantially transparent for photon energiesmuchlessthantheband gapenergiesandundergo optical damage at considerably large excitation intensities [18,19].Inaddition,semiconductorshaveaddedadvantages overothermaterialsintermsofcompactness,controlover thematerialrelaxationtimes,observedlargenonlinearitiesin opticalpropertiesundernearresonantlaserirradiationand sophisticatedfabricationtechnology[20,21].Furthermore,in these crystals, there exist a number of optically excited coherentcollectivemode(suchasacousticalphononmode, opticalphononmode,polaronmode,polaritonmodeetc.)and using the coupled mode scheme a strong tunable electromagnetic Stokes mode may be achieved as a signal waveattheexpenseofthepumpwave.

Up to now, the optical parametric oscillation caused by optically excited coherent collective modes, in III V semiconductor crystals have been reported by several researchgroups[22 24].Itappearsfromavailableliterature thatnotheoreticalformulationhasbeendevelopedtillnow tostudytheopticalparametricamplificationinmagnetized dopedIII VsemiconductorslikeInSb,GaAs,GaSb,InAsetc. withopticalphononsactingastheidlerwave.Suchcrystal classesareusuallypartiallyionicand,therefore,piezoelectric scattering is overshadowed by optical phonon scattering mechanisms [25, 26]. The study of the propagation characteristicsofcoherentopticalphononsareofsignificance importantinthestudyoffundamentalpropertiesofcrystals. Thestudyoflaser longitudinalopticalphononinteractionsin III Vsemiconductorsiscurrentlyoneofthemostactivefields of research due to its vast potentiality in fabrication of optoelectronicdevices.

Keeping in view the possible impact of parametric interactions,inthispaper,byusinghydrodynamicmodelof (one component) semiconductor plasma and adopting the coupled mode approach, attention is focused on analytical investigations of optical parametric oscillation of optical phononmodeinmagnetizeddopedIII Vsemiconductors.

2. THEORETICAL FORMULATIONS

AnOPO,initssimplestform,consistsofanonlinearcrystal (hereIII Vsemiconductorcrystal)placedwithinanoptical resonatorandilluminatedbyanintenselaser(pump)beam at 0 . The pump field gives rise an idler wave at optical phononfrequency op  andasignalwaveatStokesfrequency s  .Thegeneratedfieldsat op  and s  getamplifiedasthey travelthroughthenonlinearcrystal.Theamplificationat op  and s  occursviaparametricinteractionamongthreeoptical fieldsviz.pumpfield,idlerfieldandsignal fieldwithinthe

nonlinear crystal. This interaction is a result of nonlinear polarization exhibited by the crystal, which is noncentrosymmetric in nature. For a noncentrosymmetric III V semiconductor crystal of length L, we will determine some operational characteristics such as threshold characteristics and conversion efficiency of the proposed OPO.

Parametricinteractionprocesseshavebeenstudiedbymany researchersbothclassicallyaswellasquantummechanically. Iftherearelargenumberofphotonsintheradiationfield,it canbewelldescribedclassically.Inthetreatmentofcoupled wave problems, the classical description is found more appropriatesincethenthedecayoramplificationofwaves dependonrelativephasesamongthem,whereasinquantum mechanicaldescription,ifthenumberofquantaisprescribed, thephaseswillbeundeterminedasrequiredbyuncertainty principle. Thus, here, classical discussion on parametric interaction among pump field, idler field and signal field withinthesemiconductorcrystalisgiven.Weconsiderthe hydrodynamic model of homogeneous semiconductor plasma.Thismodelrestrictsouranalysistobevalidonlyin thelimit kl <<1(k isthewavenumberand l isthecarrier meanfreepath).

Inordertostudysingleresonantparametricoscillationdue to excitation of optical phonon mode, the coupled mode schemeisused.Weconsiderthepropagationofapumpwave 0000 ˆ exp[()]ExEikxt  (1) inaIII Vsemiconductorcrystalplacedinmagnetostaticfield 00 ˆ BzB  ;normaltothepropagationvectors 0k r , op k r and s k r (all parallel to the x axis) of three interacting waves, viz, pump ( 00 ,k  r ), idler ( , opop k  r ) and signal ( , ss k  r ) respectively.Themomentumandenergyexchangebetween thesewavescanbedescribedbyphase matchingconditions: 0 ops kkk  rrr hhh and 0 ops  hhh ;knownasmomentum andenergyconservationrelations,respectively.Thisyield 0 sop  and 2 ops kkk  rrr (since 0 s kk  rr ).TheStokes mode( , ss k  r )arisesduetocouplingofthepumpfieldwith densityperturbationsatanopticalphononfrequency op  in thecrystal.ThecouplingofthepumpStokesfieldgenerates a strong electrostatic (space charge) field oscillating at frequency op  ;whichpossessesbothlongitudinalaswellas transversecomponents.Intheweaklypolarsemiconductors (viz.,GaAs,InSb),thelongitudinaloptical(LO)phononsare associatedwithlongitudinalelectricfieldwhiletransverse optical (TO) phonons are accomplished by transverse electrostatic field. The intensities of LO and TO phonons scatteringbecomeunrelatedforfinite k r becausetheyare determined by two independent parameters, viz., the electro opticaleffectandthedeformationpotentialRaman

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tensor,respectively[27].Thepolarizationassociatedwith thelongitudinalvibrationismuchlargerthanthetransverse component in III V crystals like InSb, GaAs, etc. with zinc blend structures [28]. Consequently, we neglect the transverse electric field and treat op  as the longitudinal opticalphononfrequency. We consider the weakly polar III V semiconductor crystal consisting of the two atoms in the molecule vibrating in oppositedirections.Thisdiatomicmoleculeischaracterized byitspositionandnormalvibrationcoordinates (,)uxt  .In the presence of pump wave, the equation of motion for a singleoscillatorunderonedimensional(1 d)configuration isgivenby[29,30]

(2a) where ()suu

istherelativedisplacementsofpositive and negative ions, respectively. op E is the macroscopic internal electrostatic field oscillating at op  . In equation (2a), opo  istheopticalphononfrequencyat 0 op k  r op  is introducedphenomenologicallyasaconstantparameterand it takes into account the phonon decay. We consider 2 10 opopo  [31 33]. sq istheSzigetieffectivecharge[34]. Itisconvenienttowritetherelativedisplacementintermsof new parameter w defined as [35]: 1/2() wNMs  Accordingly,equation(2a)canbeexpressedas: 1/2 2 2 2 2 opoopsop wwqwqE tM t  

    . (2b) Since, the ions in the diatomic molecule possess opposite charges;thereforetherewillbeanetdipolemomentinthe crystal andhenceitinduces a polarization( s Pqw  ).The inducedlongitudinalphononelectrostaticfield( op E ),arising due to the induced polarization can be determined from Poission’sequation.Consequently,wemay express op E in termsofthepolarizationas:

, (3) where 0  , op n is the perturbed carrier density oscillatingat op  and e iselectroniccharge.Attheentrance window 0 x  ,weconsiderthatthedensityperturbation op n isnegligiblysmallsuchthattheterm (/) op ne  inequation (3)maybeneglectedsafely.Weobtain

s op Nq w E kx      , (4)

wherewehavetaken op kk  r (say).Usingequation(2b),the valueof op E canbeestimatedapproximatelyintheweakly polar III V semiconductor crystals from the knowledge of strain (/) wx [36, 37]. Assuming 6(/)10 wx  , we get 5(0)2.1510 op E  Vm 1 in n InSb crystal with 29 2.710kg M  , 327()3.710 lNa m 3 , 20 1.210 sq  , 61.8610 k  m 1 , 17.54 s  and 15.68   Theotherbasicequationsconsideredintheformulationof (2) are:

(8b)

cyclotron frequency. Substitution

k nnvnv   (8a) E Nq ek nvnv x M        ,

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   

 

2 2 2 2 s opoopop q

sE tM t





opop s Ene Nq w xx   





0 111 010 0 v nnv vnn txxx  

 (5) 0 000000.[()]() eff v ee vvvEvBE txmm      (6) 1 10110110 ..[()] v e vvvvvEvB txxm      (7)

and * 00 () sops s k nnvnv  

Here, 22 () [()] opop op cop iE e v m i    and 22 () [()] ss s cs iE e v m i    , with 1 ops nnn  and 1 ops vvv  . 0 (/) c emB  is the electron

ofvaluesof u and op n

These equations and the symbols used have been well described in Ref. [22, 23]. In order to obtain the coupled modeequationfortheopticalphononflux,weconsiderthe densityperturbationstovaryas: exp[()]opopop nikxt  and exp[()]sss nikxt  . Using equation (5), these density perturbationscanbeobtainedas: 00 () opsop op yield 2 00 2 1() op s sop op pp

fromequations(2a)and (8a)respectively,inequation(3)andsimplemathematical simplification

(9)

where 222 2 ppopoopopop i  . Equation (9) reveals that the coupling of the perturbed electrondensityandoscillatoryelectronfluidvelocityactas a disturbed source that can feed energy to the induced opticalphononfield( op E )leadingtotheamplificationofthis fieldwithalargegaincoefficient.Wemayexpressequation (9)intermsofthecoupledelectricfields.Usingequations (4) (6) and (9), the induced longitudinal optical phonon fluxcanbeexpressedas 0()() op opopops

theelectron plasmafrequency.

 

(10a) Here opop E  isintroducedphenomenologicallytotakeinto accounttheabsorptionprocessesinthecrystal; op  being the absorption coefficient in the off resonant regime. op may be treated as the coupling parameter which can be obtainedfromtheaboveformulationsas 1 2 22 0 22222 0

mMi

Using equations (11) and (15) and employing SVEA, the equations describing the collinear OPO fields can be representedintermsofcomplexamplitudesas * 0 000 ops E EiEE x    (16a) and * 0 s sssop E EiEE x    . (16b)

Equations(16a)and(16b)representthecouplingbetween thepumpandthebackwardscatteredStokeswaveviathe optical phonon mode in the medium. 0 and s  are the inherentbackwardabsorptioncoefficientscorrespondingto pump wave frequency ( 0 ) and Stokes wave frequency ( s  ),respectively.Thesearedefinedas:

(1) 0 0 i c       (17a) and (1) s si c    , (17b) where (1) i is the imaginary part of linear optical susceptibility [Eq. (13)] and  is the crystal background refractiveindex.Moreover 0 0k c   and s s k c  

In equations (16a) and (16b), 0 and s are the coupling parameterdefinedas:

ps i sss     (13) and 22 (2) 22 000 sp sspp iekNq mM    , (14) where 2 0 2 0 1 c    , 2 2 1 c s s    and 2 22 p opop s i    ,in which 1/2 2 0 p ne m      is

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E EiExEx



() 1 ()[()] pop s op opcppopopc





, (10b) Inderivingequation(10b),weassumed 0, s  ? . Theinducednonlinearpolarization NLP arisingduetothe nonlinearcurrentdensitygivenby * 1200 ()()NLsop PJdtJJdtnevnevdt   rrrr , (11) where 01JJJ  rrr represents the total induced current densityinthecrystalanditcomprisesofzeroth order( 0J r ) as well as perturbed components oscillating at the Stokes frequency 1J r The complex optical susceptibility ri i  can be obtainedfrom: (1)(2) 00[] NL PEEE  (12)

Inequation(12),wehavetruncatedtheexpressionuptothe second ordernonlinearopticalsusceptibility( (2) )sincethe originofthethree waveparametricinteractionsliesin (2) ofthemedium.Followingthemethodadoptedinchapter3 andusingequations(2b) (6),(11)and(12),weobtain 2 (1) 3222 [(/)]

Using the slowly varying envelope approximation (SVEA) [38], let us now obtain an expression for threshold pump intensityofSRPOincludingtheattenuationlossescausedby mirror transmission as well as absorption and scattering insidethecavity.Inordertoobtainthesteadystatecoupled mode equations for the pump and the Stokes waves, we employ Maxwell’s equations in the presence of a finite induced nonlinear polarization NLP arising due to the nonlinearcurrentdensitygivenby 2 2 2 0 222 1 NLP E E ctt    . (15) where c isthevelocityoflightinthecrystal.

  . (20) Weobtain 0 2 s    , (21) where 1/2 0 2 0 0 8 opos I c  i EEL         .

8 opos I c    2 (0)(0)sinh() op op

Inordertostudythethresholdpumpintensityandthebasic operational characteristic of OPO, we consider that the crystalsatisfyingequations(24a)and(24b)iskeptinsidean opticalcavitysuchthatwecanhavefeedbackofStokesfield. Iftheparametricamplifiedpowergainissufficientlyhigh,a parametric oscillator can be constructed [39]. The considerationoffeedbackmechanismmakesitpossiblefor theparametricgaintoovercomethelossesandsubsequently parametricoscillationsoccur.

ordertostudytheparametric

intheweakly polar semiconductor crystal,

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    ,inwhich 2 000 1 2 IcE  is the pump field intensity. For simplicity, we consider ops   . Also we have assumed 00ss for 0 ~ sop  ? Consequently,formequations(19a)and(19b),weget 00 11 ()expexpexp222 op xx x ExCC             (22a) and 00 22 ()expexpexp222 s xx x ExCC             .(22b) In

oscillations

let us

(0) op E

100 0 1 ()(0)2(0)(0) 2 opops CEiEE    m

and * 2000 0 1 ()(0)2(0)(0) 2 ssop CEiEE   

Wealsoobtain 00 0 ()expcosh()sinh()(0) 2 opop L ELLLE            00 0 2 (0)(0)sinh() op s i EEL        

(2) 0 0 c       (18a) and (2) s s c       , (18b) where (2) is the second order optical susceptibility [Eq. (14)] of semiconductor crystal in the presence of magnetostaticfield. We now address ourselves to the analytical study of thresholdpumpintensityandoperationalcharacteristicsof SRPO.Weemployequations(10a)and(16b)andsolvethem assumedsolutionoftheforms 1 ()exp() op ExCx  (19a) and 2 ()exp() s ExCx  , (19b) where C1and C2arethearbitraryconstantsand  isthegain coefficient.Onemayrecallthattheparametricinteractions yieldidenticalgaincoefficientsforboththeidlerandsignal waves. The gain coefficient can be determined by using equations (10a) and (16b) through equations (19a) and (19b).Simplificationyield 1/2 ,0 2 0 0 and 00 0 ()expcosh()sinh()(0) 2 ss L ELLLE            * 00 0

consider

and (0)

E tobefiniteduetofinitenessoflatticevibrationsand spontaneousnoisefield,respectively.Theseconsiderations enableonetodeterminethesignalandtheidlerfieldsatthe exitwindow x = L with L beingthesamplelength.Usingthe aboveboundaryconditions,weobtain

(23a)

(23b)

(24a)

(24b) Equations (24a) and (24b) describe the output signal and idlerwaves x = L (i.e.endofthecrystaloflength).

Inthepresentchapter,wehavemadeanattempttoestablish theweaklypolarIII Vsemiconductorcrystalsastheclassof materialssuitableforthedevelopmentofSRPO.TheSRPO hasmanyadvantagesoverthedoubleresonantparametric oscillator (DRPO) except that it requires large threshold pump intensity. Therefore, we have chosen SRPO for the presentstudy.TheopticalcavityconfigurationforSRPOhas been chosen such that the second mirror located at exit windowofthecavityisstronglyreflectiveattheStokeswave frequency s  whereas, for the pump and optical phonon frequencies 0 and op  , both the mirrors are nearly transparent.Wehavetaken Rs astheStokesfieldreflectivity parameter of the second mirror. In this analysis, we have neglectedthephaseshiftsinphononandStokesfielddueto traversalandreflectioninthecavity[40].Forsimplicityitis also assumed that the cavity length is equal to the crystal length L.Inordertoreducethethresholdvalueofthepump intensity and to enhance the power gain of the optical phononmodeswellabovethethreshold,wehaveemployed theround tripmechanism. Undersuchcircumstances, the thresholdconditioncanbegivenby

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RELE

()(0)

(25) Physically, equation (25) illustrates that the Stokes field undergoesneitheramplificationnorattenuationduringone round trip. While considering the attenuation loss due to absorptionandscatteringinthecavityaswellasthemirror transmission loss, the total loss in the cavity may be expressedas[41]:

(27)

In this sub section, we will obtain address the analytical investigation of operational characteristics such as conversionefficiencyoftheproposedSRPO.

Theconversion efficiency k isdefinedastheratioofthe idler(signal)outputenergytotheincidentpumpenergy.In ordertoestimatetheconversionefficiencyoftheparametric oscillator, relation between the idler (signal) power to constantsignal(idler)powermustbeknown.Inthepresent article,thisrelationcanbederivedfromthesetofcoupled mode equations (10) and (16a). In doing so, we have assumedthesignalfield(Stokesmode)asconstantduring single round trip. Solving equations (10) and (16a) and obtainedsimplifiedsolutionoftheidlermodeas

() (0) op k

2 2 0

EL E  . (30)

Equation (30) may be used to determine the conversion efficiencyoftheproposedSRPO,respectively.

3. RESULTS AND DISCUSSION

Inthissection,weaddressthedetailednumericalanalysisof the threshold and operational characteristics of SRPO consistingofmagnetizeddopedIII Vsemiconductorcrystals. Thefollowingsetofmaterial parametershasbeenusedto performnumericalappreciationoftheresultsobtainedand thecrystalisassumed 2 laser [24,31]: 0 0.015 mm  , 35.810 kgm 3 , 17.8 L  , 15.68   , 113.510 s 1 , 133.710 op  s 1 , 1924 0 1010 n  m 3 , 29 2.710 M  kg, 281.4810 N  m 3 , 20 1.210 sq  C, 16 1.6810 u  SIunits.

This set of data is related to a typical n InSb crystal, however, the results obtained in previous section may be appliedtoanyIII Vsemiconductorcrystal.

Thenatureofdependenceofthethresholdpumpintensity 0,th I of SRPO on different parameters such as externally appliedmagnetostaticfield 0B ,mirrorreflectivity R,crystal length L etc.maybestudiedfromequation(28).Theresults areplottedinFigs.1 3. MagnetostaticField,T) B0 (

=1020m–3

Fig 1:Variationofthresholdpumpintensity I0,th ofSRPO withmagnetostaticfield B0 with R =95%, L =0.3mmand n0 =1022 m 3 fortwodifferentdopingconcentrations n0

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L   

'' 000 0

expRe()(0)sinh()2(0) s T ssop E L REL E       ,

where 1/2 00 '2 0 0 8Re() 4 ops T I c         ; 0 (0)(0) s EE = The

0 as 2 12 0 0 2 0 11 cosh4 8Re() thT ops c I Q L               ,

where 0 0 (0) (0)Re()exp (0)2 T sops s E L QLRE E       

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sss



1 (ln)

(26) UsingEqs.(24b),(25)and(26),

obtain

(0)

thresholdpumpintensityforSRPOcanbeobtainedby differentiatingequation(27)withrespectto

(28)

22 1 ()exp()cosh()sinh()(0) T opTop ELLLLE            02 1 2 (0)(0)sinh() op s i EEL         , (29) where 12 T  , 1/2 2 2 20Re() Topss E   Usingequation(29)andthedefinitionof k asgivenabove, wecanhave

0 T h r e s h o l d P u m p I n t e n s i t y , × 1 0 1 2 W m 2 ) I 0 th ( 10 12

8 6 4

2 4 6 8 10 12 14 16 18

14 16 18

2 20 n0 =1022m–3 n0

T h r e s h o l d P u m p I n t e n s i t y , × 1 0 1 2 W m 2 ) I 0,

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0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.1

T h r e s h o l d P u m p I n t e n s i t y ,

× 1 0 1 2 W m 2 ) I 0, th ( 10

20 n0 =1022m–3

18 8 6 4 2

n0 =1020m–3

0.8 1.2 1.4 2.0 2.4 2.8 3.2 3.6 4.0 0.4

Fig.2showsthevariationofthresholdpumpintensity 0,th I ofSRPOwithmirrorreflectivity R with B0 =14.2T(because atthisparticularvalueof 0B , 0,th I islowest)and n0=1022m 3 for three different values of crystal length L. It can be observedthatfor R =15%, 0,th I isveryhigh,independentof L and all the curves coincide. With increasing mirror reflectivity, 0,th I decreases very rapidly for R < 40%; the increase in mirror reflectivity increases the net gain per round trip at the signal wavelength resulting in lower oscillationthresholdintensityforhighreflectivity.For R > 40%, the rate of fall of threshold intensity is very small. Moreover, with increasing crystal length, the rate of thresholdintensityisfaster.Hence,weconcludefromthis figure that threshold pump intensity of SRPO in III V semiconductorscanbeeffectivelyreducedbyincreasingthe crystallengthandmirrorreflectivity R ~40%. CrystalLength,×10-4m) L(

Fig 3:Variationofthresholdpumpintensity 0,th I ofSRPO withcrystallength L withmirrorreflectivity R =95%, B0 = 14.2Tfortwodifferentvaluesofdopingconcentration n0

Fig.3showsthevariationofthresholdpumpintensity 0,th I of SRPO with crystal length L with mirror reflectivity R = 95% and B0 = 14.2T for two different values of doping concentration n0.Itcanbeobservedthatforagivendoping concentration, 0,th I is very large for 50m L  and decreasesquitesharplyfor 50m<120m L  .Beyondthis valueofcrystallength, 0,th I becomesnearlyindependentof L.Moreover,thehighlydopedsemiconductorcrystalyields lowerofthresholdpumpintensityforsamecrystallength.

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th (

12 14

Fig.1showsthevariationofthresholdpumpintensity I0,th of SRPOwithmagnetostaticfield B0 with R =95%, L =0.3mm fortwodifferentvaluesofdopingconcentration(n0=1020m 3and n0=1022m 3).Itcanbeobservedthatinboththecases, I0,th starts with a high value in the absence of B0; remains nearlyconstantfor 0 2 B  Tanddecreasessharplyattaining a minimum value ( 12 0, 910 th I  Wm 2) at 0 3 B  T. With increasing B0 beyondthisvalue, 0,th I increasessharplyand remainsconstantoverawiderangeof 0B ( 410 T).With furtherincreasing 0B , 0,th I againdecreasessharplyattaining aminimumvalueat 0 11 B  T.Withfurtherincreasing 0B beyondthisvalue, 0,th I increasessharplyandagainremains constant over a narrow range of 0B ( 1213 T). With further increasing 0B beyond this value, 0,th I again decreasessharplyattainingaminimumvalue at 0 14.2 B  T. Beyond this value of 0B , 0,th I againincreases sharply and becomes independent of magnetostatic field. The dip at 0B  3T, 11T and 14.2T may be attributed to resonance conditions: 22 cop  , 22 cs  and 22 0 c  respectively.A comparison between two cases reveals that except at resonance conditions, 223203 00 0,0,10m10m ()() ththnnII   . At resonance conditions, 0,th I becomes independent of 0n Hence, an external magnetostatic field plays an important roleinreducingthethresholdpumpintensityofSRPOinIII Vsemiconductorsaroundresonanceconditions. MirrorReflectivity,%) Rs( 8

16 18

6 4 2 20 L=200 m L =300 m L =400 m

Fig 2:Variationofthresholdpumpintensity 0,th I ofSRPO withmirrorreflectivity R with B0 =14.2T(becauseatthis particularvalueof B0, I0,th isminimum), n0 =1022m 3 for threedifferentvaluesofcrystallength L.

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Henceweconcludefromfigures2and3thatthresholdpump intensityofSRPOcanbeloweredbyincreasingcrystal(or cavity)length,mirrorreflectivityanddopingconcentration. Usingthematerialparameters(forn Insb)givenabove,the natureofdependenceofconversionefficiencyondifferent parameters such as externally applied magnetostatic field 0B ,dopingconcentration 0n ,pumpfieldintensity 0I ,crystal lengthetc.wellabovethethresholdpumpintensitymaybe studied from equations (30) and (31), respectively. The resultsareplottedinFigs.4 9.

5 10 15 20 25 30 n0 =1020m–3 n0 =1022m–3

2 4 6 8 10 12 14 16 18 0 % ) k (

Fig -4:Natureofdependenceofconversionefficiency k onexternallyappliedmagnetostaticfield B0 with R =95%, L =0.3mmand 13 0 6.510 I  Wm 2 fortwodifferentvalues ofdopingconcentration n0.

 

Moreover,weobservedthatwithincreasing n0from1020m 3 to 1022m 3 , k becomes almost 1.5 times. Hence, we concludefromthisfigurethatconversionefficiencyofSRPO canbeenhancedbyproperselectionofmagnetostaticfield

C o n v e r s i o n E f f i c i e n c y ,

% ) k ( 5

B0 =14.2T B0 =11T

B0 =3T

30 B0=0T

Fig 5:Natureofdependenceofconversionefficiency k onpumpintensity I0 with R =95%, L =0.3mmandn0= 1022m 3 fortwodifferentcases,viz.(i)absenceof magnetostaticfield(B0 =0T),and(ii)presenceof magnetostaticfield(B0 =3T,11T,14.2T). C o n v e r s i o n E f f i c i e n c y ,

42 n0=1020m 3

% ) k ( 32

n0 =1022m–3 CrystalLength,×10-4m) L (

0.8 1.2 1.4 2.0 2.4 2.8 3.2 3.6 4.0 0.4

Fig 6:Natureofdependenceofconversionefficiency k oncrystallength L with R =95%, B0 =14.2Tand 13 0 6.510 I  Wm 2 fortwodifferentvaluesofdoping concentration n0

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MagnetostaticField,T) B0 ( C o n v e r s i o n E f f i c i e n c y ,

Fig. 4 shows the nature of dependence of conversion efficiency k on externally applied magnetostatic field B0 with R = 95%, L =0.3 mm and 13 0 6.510 I  Wm 2 fortwo different values of doping concentration n0. We observed thatinboththecases, k isnearlyindependentof B0(from0 to18T)exceptat3T,11Tand14.2T.Attheseparticular values of B0, k shows sharp peaks due to resonance conditions( 22 ~ cop  , 22 ~ cs  and 22 0 ~ c  respectively).On comparingthethreepeaks,wenotethat 000 14.2T11T3T ():():()1.43:1.17:1 kBkBkB

(around resonance conditions: 22 cop  , 22 cs  and 22 0 ~ c  )andincreasingdopingconcentration. 4.5 5 5.5 6 6.5 7 7.5 8 8.5 4 PumpIntensity,×1013Wm–2) I0(

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Fig. 5 shows the nature of dependence of conversion efficiency k onpumpintensity I0 with R =95%, L =0.3mm and n0 = 1022m 3 for different cases, viz. (i) absence of magnetostatic field (B0 = 0T), and (ii) presence of magnetostaticfield(B0=3T,11T,14.2T).Weobservedthat intheabsenceofmagnetostatic field, k isverysmalland increaseslinearlywith I0.Inthepresenceofmagnetostatic field, k increaseslinearlywith I0 ( 13710 Wm 2).Beyond thispoint, k increasesquadricallywith I0.Suchabehavior agreeswellwiththetheoreticallycalculatedvalues[42,43] and experimental observations [44, 45]. Moreover, 000 14.2T11T3T()()() kBkBkB  and the effect of magnetostatic field is more pronounced at higher pump intensities. We obtained 33% k  at B0 = 14.2T, which is limitedbyavailablepumppowerratherthancrystaldamage orintra cavitylosses.Henceweconcludefromthisfigurethat conversionefficiencyofSRPOcanbeenhancedbyincreasing the pump intensity and externally applied magnetostatic field;theeffectofmagnetostaticfieldismorepronouncedat highpumpintensities.

Fig. 6 shows the nature of dependence of conversion efficiency k oncrystallength L with R =95%, B0=14.2Tand 13 0 6.510 I  Wm 2 for two different values of doping concentration n0. We observed that in both the cases k increasesquadricallywith L.Thegrowthrateof k incaseof heavy doping is more. Hence, it is clear that the effect of dopingconcentrationismorepronouncedforlargercrystal lengths.

4. CONCLUSIONS

In the present study, a detailed numerical analysis of the thresholdandoperationalcharacteristicsofSRPOconsisting ofmagnetizeddopedIII Vsemiconductorcrystalshavebeen undertaken. The hydrodynamic model of semiconductor plasmahasbeensuccessfullyappliedtostudytheinfluence of different parameters such as externally applied magnetostatic field, doping concentration, mirror reflectivity,crystallengthetc.onthresholdpumpintensity and conversion efficiency of SRPO consisting of III V semiconductor crystal duly irradiated by slightly off resonant laser pulsed laser. An externally applied magnetostaticfieldplaysanimportantroleinreducingthe thresholdpumpintensityofSRPOinIII Vsemiconductors aroundresonanceconditions.Thethresholdpumpintensity of SRPO can be lowered by increasing crystal (or cavity) length, mirror reflectivity and doping concentration. The conversion efficiency of SRPO can be enhanced by proper selection of magnetostatic field (around resonance conditions) and increasing doping concentration. The technologicalpotentialityoftransverselymagnetizeddoped III V semiconductor crystals as the hosts for parametric deviceslikeparametricoscillatorshasbeenestablished.

Factor value:

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