Parametric amplification of optical phonons in magnetoactive III-V semiconductors

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Parametric amplification of optical phonons in magnetoactive III-V semiconductors

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

Abstract Using the hydrodynamicmodelofsemiconductor plasmas and following the coupled mode approach, the parametric amplification of optical phonons is analytically investigated in magnetized doped III V semiconductors. The origin of nonlinear interaction is assumed tolieinthecomplex 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. Expressions are obtained for threshold pump amplitude for the onset of parametricprocess and parametric gain coefficient (well above the threshold pump field). Numerical analysis are made for a representative n 2 laser. The proper selection of doping concentration and externally applied magnetic field (around resonance conditions) lowers the threshold pump amplitude and enhances the gain coefficient for the onset of parametric process. The analysis confirms the chosen nonlinear medium as a potential candidate material for the fabrication of parametric devices like parametric amplifiers and oscillators.

Key Words: Nonlinear optics, Optical parametric amplification,Opticalphonons,III Vsemiconductors

1. INTRODUCTION

Nonlinear optics is the branch of optics that describes the laser matterinteraction.Asaconsequenceofthisinteraction, there arise phenomena such as parametric interactions, modulation interactions, stimulated scatterings etc [1, 2]. Among these, parametric interaction is the fundamental interaction.Itisasecondordernonlinearopticaleffect(i.e. the origin of this interaction lies in second order optical (2)ofthemedium).Inthisprocess,anintense laserbeam(continuousorpulsedtype),hereafterreferredas ‘pump’, interacts with nonlinear medium and results into generationofwavesatnewfrequencies[3].Thisoccursdue tomixingorcontrolledsplittingofwaveswhichmayundergo amplification or attenuation depending on the material propertiesandgeometryofexternallyapplied electric and magneticfields.

In a nonlinear medium, the dissection of superposition principle leads to interaction among waves of different frequencies.Thereexistanumberofnonlinearinteractions whichcanbeallocatedasparametricinteractionofcoupled modes. Parametric interaction of coupled modes are the

nonlinearopticalphenomenainwhichpumpfieldenergyis transferredtothegeneratedwavesbyaresonantmechanism undertheconditionthatthepumpfieldamplitudeislarge enoughtocausethevibrationsofcertainphysicalparameters ofthemedium[4].

The phenomena of parametric interactions have played a distinctive role in nonlinear optics. Parametric processes have been widely used to generate tunable coherent radiationatafrequencythatisnotdirectlyavailablefroma lasersource;thesefrequencyconversiontechniquesprovide animportantmeansofextendingthespectralrangecovered by coherent sources [5, 6]. Optical parametric amplifiers, optical parametric oscillators, optical phase conjugation, pulsenarrowing,squeezedstategenerationetc.aresomeof theoutcomesofopticalparametricinteractionsinanonlinear medium.Amongthese,opticalparametricamplifiersareof specialinterestduetotheirvastapplicationsinscienceand technology[7,8].

Whilesurveyingongoingglobalresearchactivitiesonoptical parametric amplification, it has been observed that the manipulationsofthresholdpumpfieldandgaincoefficient have been important issues to improve the efficiency and functionality of optical parametric amplifiers due to unavailabilityofdesirednonlinearopticalmedia.Amongthe various nonlinear optical materials, the doped III V semiconductor crystals are advantageous hosts for fabricationofopticalparametricamplifiers.

Up to now, the optical parametric amplification caused by optically excited coherent collective modes, in III V semiconductor crystals have been reported by research groupsofSinghet.al[12,13],Bhanet.al.[14]andGhoshet.al. [15].Itappearsfromavailableliteraturethatnotheoretical formulationhasbeendevelopedtillnowtostudytheoptical parametric amplification in magnetized doped III V semiconductorslikeInSb,GaAs,GaSb,InAsetc.withoptical phonons acting as the idler wave. Such crystal classes are usuallypartiallyionicand,therefore,piezoelectricscattering isovershadowedbyopticalphononscatteringmechanisms [16].Thestudyofthepropagationcharacteristicsofcoherent opticalphononsareofsignificanceimportantinthestudyof fundamental properties of crystals. The study of laser longitudinal optical phonon interactions in III V semiconductorsiscurrentlyoneofthemostactivefieldsof research due to its vast potentiality in fabrication of optoelectronicdevices.

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Keeping in view the possible impact of parametric interactionsinvolvinganopticalphononmode,wepresent hereananalyticalstudyofopticalparametricamplification using the n type doped III V semiconductor crystals immersed in a large magnetostatic field following the hydrodynamic model for the semiconductor plasma. The influences of material parameters and externally applied magnetostaticfieldonthresholdpumpamplitudeandgain coefficientofparametricprocesshavebeenstudiedindetail. Weaklypolarn typeIII Vsemiconductorcrystalshavebeen chosenasnonlinearopticalmaterialssubjectedtoapump wave. The pump photon energy is taken well below the band gap energy of the sample. This allows the optical properties of the sample to be influenced considerably by freecarriersandkeepsitunaffectedbythephoto induced inter bandtransitionmechanisms[17 19].Usingthecoupled mode theory for semiconductor plasma, the complex effective second order optical susceptibility (2) eff of the crystalandconsequentthresholdpumpamplitude E0,th for the onset of parametric process and parametric gain coefficient gpara wellabovethethresholdfield(E0> E0,th)are obtained.

2. THEORETICAL FORMULATIONS

Let us consider the hydrodynamic model of an n type semiconductorplasma.Thismodelprovestobesuitablefor thepresentstudyasitsimplifiesouranalysis,withoutany loss of significant information, by replacing the streaming electrons with an electron fluid described by a few macroscopicparameterslikeaveragecarrierdensity,average velocity,etc.However,itrestrictsouranalysistobevalidonly inthelimit(kop.l <<1; kop theopticalphononwavenumber, and l thecarriermeanfreepath).

In order to obtain an expression for (2) eff , the three wave coupledmodeschemehasbeenemployed[20].Theoriginof (2) eff liesincouplingbetweenthepumpandsignalwavesvia density perturbations in the crystal. Let us consider the parametriccouplingamongthreewaves:(i)theinputstrong pump wave E0 (x, t) = E0 exp [i(k0x 0t), (ii) the induced opticalphononmode(idler) u(x, t)= u0 [i(kop 0t),and(iii) the scattered Stokes component of pump electromagnetic wave(signal) Es (x, t)= Es exp[i(ksx st).

Themomentumandenergyconservationrelationsforthese modes should satisfy the phase matching conditions:

0 sop kkk  and 0 sop  . We consider the semiconductorcrystaltobeimmersedinatransversestatic magneticfield 00 ˆ BzB  (i.e.perpendiculartothedirectionof inputpumpbeam).

InaweaklypolarIII Vsemiconductor,thescatteringofhigh frequency pump wave is enhanced due to excitation of a

normalvibrational(opticalphonon)mode.Weconsiderthat thesemiconductormediumconsistsof N harmonicoscillators per unit volume; each oscillator being characterized byits position x, molecular weight M and normal vibrational coordinates u(x, t).

The equation of motion for a single oscillator (optical phonon)isgivenby[21] 2 2 2 t uuF u tM t     , (1a)

where  is the damping constant equal to the phenomenological phonon collision frequency ( 2 ~10 t ) [22]; t being the un damped molecular vibrational frequencyandistakentobeequaltothetransverseoptical phonon frequency. F is the driving force per unit volume experienced by the medium can be put forward as: (1)(2)FFF  ,where (1) s FqE  and (2)20.5(,) u FExt  representtheforcesarisingduetoSzigetieffectivecharge sq and differential polarizability 0(/) u u (say), respectively. 0  ; 0 and  aretheabsoluteandhigh frequencies permittivities, respectively. After substituting thevalueof F,themodifiedequationofmotionfor u(x, t)of molecularvibrationsinasemiconductorcrystalisgivenby 2 22 2 11 (,) 2 tsu

(5)

(6) These equations are well described in Ref. [12 15]. The molecularvibrationsatfrequency op  causesamodulation of the dielectric constant of the medium leading to an exchange of energy between the electromagnetic fields separatedinfrequencybymultiplesof op

(i.e.,( 0 op p  ), where p =1,2,3,…).Themodesatfrequencies 0 op p  are knownasanti Stokesmodes;whilethoseat 0 op p  are Stokes modes. In the forthcoming formulation, we will consideronlythefirst orderStokescomponentoftheback scatteredelectromagneticwave.

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kkkEE ii

E Mi



      ,inwhich ,, cxzsxz e B m    , 22 2 2 pl r t

      l isthelongitudinalopticalphononfrequencyandisgiven by / lBD k  h ,where Bk and D areBoltzmannconstant andDebyetemperatureofthelattice,respectively. L isthe latticedielectricconstant. Thecomponentsofoscillatoryelectronfluidvelocityinthe presenceofpumpandthemagnetostaticfieldsareobtained fromEq.(2)as:

(9a) and

(9b) NowtheresonantStokescomponentofthecurrentdensity duetofinitenonlinearpolarizationofthemediumhasbeen deducedbyneglectingthetransientdipolemoment,which canberepresentedas:

cdssx Jnev 

 (10) where 222 roprop 

ueffx ropop

ekkkEE mi     2 2 2 2 1() 2() su ueffx ropop    

* 11 2 ()() () opeffx sop rss      

iekkE nn mi   (8) InEq.(8), 222 rsrs  ,where 22 22 22 cx rr c    

Usingtherelation

ekkk mi     2 2 2 2 1() 2() su ueffx ropop

q N E Mi             (12)

Here, it is worth pointing out that in addition to the polarization ()cdsP  , the system also possesses a polarization created by the interaction of the pump wave withthemolecularvibrationsgeneratedwithinthemedium, obtainedfromequations(1)and(5)as: 22 0 01 2 () 2() u mvsx ropop

N PEE Mi    . (13)

rsropc th op cxcz m E ek 

() [()]

[] s paraeffi g c

The nonlinear parametric gain of the signal as well as the idlerwavescanbepossibleonlyif para  isnegativeforpump field 00,th EE  .

3. RESULTS AND DISCUSSION

To have a numerical appreciation of the results, the semiconductorcrystalisassumedtobeirradiatedby10.6 m pulsedCO2laser.TheotherparametersaregiveninRef.[19].

Impact Factor value: 7.529

T h r e s h o l d P u m p A m p l i t u d e,

18

17

× 1 0 7 V m 1 ) E 0 th ( 15

20 B0=0T B0=14.2T

19 14 13 12 11

16

1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 1.5

Thenatureofdependenceofthethresholdpumpelectricfield E0,th necessary for the onset of parametric process on different parameters such as wave number kop, externally appliedmagnetostaticfield B0,dopingconcentration n0 etc. maybestudiedfromequation(16).Theresultsareplottedin Figs.1and2. OpticalPhononWaveNumber,×106m –1) k op (

Fig -1:Variationofthresholdpumpamplitude E0,th with opticalphononwavenumber kop intheabsence(B0 =0T) andpresenceofmagnetostaticfield(B0 =14.2T)with n0 = 1020m 3 .

Fig.1showsthevariationofthresholdpumpamplitude 0,th E with wave number op k in the absence ( 0 0 B  T) and presence of magnetostatic field ( 0 14.2 B  T) with 20 0 10 n  m 3.Itcanbeobservedthatinboththecases, 0,th E iscomparativelylargerfor 61.510 op k  m 3.Withincreasing op k , 0,th E decreases parabolically. This behaviour may be attributed to the fact that 1 0,thopEk  as suggested from equation(16).Acomparisonbetweenthetwocasesreveals that for the plotted regime of op k , for 0 14.2 B  T, 0,th E is comparativelysmallerthanthatfor 0 0 B  T.Thisisdueto thefactthataround 0 14.2 B  T, 22 0 ~ c  and 22 0 ()0 c  [Eq.(16)],thusloweringthevalueof 0,th E

4 3 2 1

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n0 =1020m–3

4 6 8 10 12 14 16 18 20 2

Fig 2:Variationofthresholdpumpamplitude E0,th with magnetostaticfield B0 forthreedifferentvaluesofdoping concentration n0.

Fig.2showsthevariationofthresholdpumpamplitude 0,th E with magnetostatic field 0B for three different values of dopingconcentration 0n .Itcanbeobservedthatinallthe threecases, 0,th E showsadipataparticularvalueof 0B (i.e. 14.2 T for 20 0 10 n  m 3, 11 T for 22 0 10 n  m 3, 3 T for 24 0 10 n  m 3).Thisbehaviourcanbeexplainedasfollows:(i) For 20 0 10 n  m 3, the dip at 0 14.2 B  T (corresponding 0 ~ c  )isdue tothe factor 22 0 ()0 c  [Eq. (16)].(ii) For 22 0 10 n  m 3,thedipat 0 11 B  Tisduetofactor rs  (i.e. 22 cs  )[Eq.(16)].(iii)For 24 0 10 n  m 3,thedipat 0 3 B  T is due to parameter rop  (i.e. 22 ~ cop  ) [Eq. (16)]. A comparison among the three cases reveals that with increasing 0n , the dip in the value of 0,th E becomes more deeper and shifts towards lower values of 0B . Hence, we conclude from this figure that externally applied magnetostaticfieldplaysanimportantroleinloweringthe threshold pump amplitude for the onset of optical parametric amplification in III V semiconductors. The increase in doping concentration further lowers the thresholdpumpamplitudeandshiftsthediptowardslower valuesofmagnetostaticfield.

Usingthematerialparameters(forn Insb)givenabove,the natureofdependenceofparametricgaincoefficient para g on differentparameterssuchaswavenumber op k ,externally applied magnetostatic field 0B , doping concentration 0n , pumpelectricfield 0E etc.wellabovethethresholdpump

electricfieldmaybestudiedfromequation(17).Theresults areplottedinFigs.3 6. P a r a m e t r i c G a i n C o e f f i c i e n t,

105

104

103

106 B0=14.2T B0=11T B0=0T

m 1 ) g para ( 101

102

1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 1.5

OpticalPhononWaveNumber,×106m–1) k op (

Fig -3:Natureofdependenceofparametricgain coefficient gpara onopticalphononwavenumber kop for threedifferentcases,viz.(i) B0 =0T,(ii) B0 =11T,(iii) B0 = 14.2Tforn0=1020m 3 and E0 =12.5×108Vm 1 .

Fig. 3 shows the nature of dependence of parametric gain coefficient para g on wave number op k for the cases, viz. absence of magnetostatic field ( 0 0 B  T) and presence of magnetostatic field ( 0 11,14.2 B  T) for 20 0 10 n  m 3 and 8 0 12.510 E  Vm 1 ( 0,th E  ).Itcanbeobservedthatinthe absence of magnetostatic field ( 0 0 B  T), para g remains constant for 6210 op k  m 1 and increases quadrically for 6210 op k  m 1. In the presence of magnetostatic field ( 0 11,14.2 B  T), para g increasesquadricallyfortheplotted regimeof op k .Acomparisonamongalltheabovethreecases reveal that the gain coefficient satisfies the inequality condition: 000 14.2110()()() paraBTparaBTparaBT ggg  . Hence we conclude from this figure that the parametric gain coefficient canbeenhancedbyincreasingthewavenumber and simultaneous application of externally applied magnetostaticfield.

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n0 =1024m–3

n0 =1022m–3 n0 =1020m–3

4 6 8 10 12 14 16 18 20 2

MagnetostaticField,T) B0 (

Fig 4:Natureofdependenceofparametricgain coefficient gpara onmagnetostaticfield B0 forthree differentvaluesofdopingconcentration n0 with E0 = 12.5×108Vm 1 .

Fig. 4 shows the nature of dependence of parametric gain coefficient para g onmagnetostaticfield 0B forthreedifferent valuesofdopingconcentration 0n .Itcanbeobservedthatin allthethreecases, para g showsasharppeakataparticular valueof 0B (i.e.14.2Tfor 20 0 10 n  m 3,11Tfor 22 0 10 n  m 3,3Tfor 24 0 10 n  m 3).Thisbehaviourcanbeexplainedas follows: (i) For 20 0 10 n  m 3, the peak at 0 14.2 B  T (corresponding 0 c  )isduetothefactor 22 0 ()0 c  [Eq.(17)].(ii)For 22 0 10 n  m 3,thepeakat 0 11 B  Tisdue tofactor rs  (i.e. 22 cs  )[Eq.(17)].(iii)For 24 0 10 n  m 3 , thepeakat 0 3 B  Tisduetoparameter rop  (i.e. 22 ~ cop  ) [Eq.(17)].Acomparisonamongthethreecasesrevealsthat withincreasing 0n ,thepeakinthevalueof para g becomes morehigherandshiftstowardslowervaluesof 0B .Hence, we conclude from this figure that externally applied magnetostaticfieldplaysanimportantroleinenhancingthe parametric gain coefficient for the onset of optical parametric amplification in III V semiconductors. The increaseindopingconcentrationfurtherenhancesthegain coefficient and shifts the peak towards lower values of magnetostaticfield.

1022 1023

DopingConcentration,m–3)

P a r a m e t r i c G a i n C o e f f i c i e n t , (

105

104

) ( m 1 ) g para 101

103

102

106 B0=14.2T B0=0T

Fig -5:Natureofdependenceofparametricgain coefficient gpara ondopingconcentration n0 forthree differentvaluesofmagnetostaticfield B0 with E0 = 12.5×108Vm 1 10.5 11 11.5 12 12.5 13 13.5 14 14.5 10 PumpAmplitude, ×108Vm–1) E0 (

Fig 6:Natureofdependenceofparametricgain coefficient gpara onpumpamplitude E0 intheabsence(B0 = 0T)andpresenceofmagnetostaticfield(B0 =14.2T)for n0 =1022m 3

Fig. 5 shows the nature of dependence of parametric gain coefficient para g on doping concentration 0n for three

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differentvaluesofmagnetostaticfield 0B .TheresultsofFig. 5supportresultsofFig.4.

Fig. 6 shows the nature of dependence of parametric gain coefficient para g on pump amplitude 00,() th EE  for the cases, viz. absence of magnetostatic field ( 0 0 B  T) and presenceofmagnetostaticfield( 0 14.2 B  T)for 20 0 10 n  m 3. We observed that in both the cases para g increases quadricallywithrespectto E0.Thus,higherpumpfieldyield higherparametricgaincoefficient.

4. CONCLUSIONS

In the present study, a detailed numerical analysis of parametricamplificationofopticalphononsinmagnetized doped III V semiconductors has been undertaken. The hydrodynamic model of semiconductor plasma has been successfullyappliedtostudytheeffectsofexternallyapplied magnetostaticfieldanddopingconcentrationonthreshold pump amplitude and gain coefficient for the onset of parametric process in III V semiconductor crystals duly irradiatedbyslightlyoff resonantnottoohighpowerpulsed laserswithpulsedurationsufficientlylargerthantheoptical phonon lifetime. The threshold pump amplitude can be reducedwhileparametricgaincoefficientcanbeenhanced by increasing the wave number and simultaneous application of externally applied magnetostatic field. The threshold pump amplitude can be reduced by proper selection of magnetostatic field (around resonance conditions). The increase in doping concentration further lower the threshold pump amplitude and shifts the dip towardslowervaluesofmagnetostaticfield.Theparametric gain coefficient can be enhanced by proper selection of magnetostatic field (around resonance conditions). The increaseindopingconcentrationfurtherenhancesthegain coefficient and shifts the peak towards lower values of magnetostatic field. Moreover, higher pump field yield higher parametric gain coefficient. The technological potentiality of a transversely magnetized weakly polar doped semiconductor plasma as the hosts for parametric devices like parametric amplifiers and oscillators are established. In III V semiconductor crystals, parametric amplificationandoscillationintheinfraredregimeappears quite promising under the resonance conditions and replacestheconventionalideaofusinghighpowerpulsed lasers.

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