An Electrostatic Catastrophe Machine as an Attosecond Pulse Generator

Transcription

1 Optics ad Phtics Jural, 014, 4, Published Olie December 014 i SciRes. A Electrstatic Catastrphe Machie as a Attsecd Pulse Geeratr Adrey Giti Max Br Istitute fr Nliear Optics ad Shrt Pulse Spectrscpy, Berli, Germay agiti@mbi-berli.de Received 11 Octber 014; revised 8 Nvember 014; accepted 1 December 014 Cpyright 014 by authr ad Scietific Research Publishig Ic. This wrk is licesed uder the Creative Cmms Attributi Iteratial Licese (CC BY). Abstract The geerati f a attsecd pulse i the ultravilet rage is described i the terms f the catastrphe thery. A simple criteri f tuelig is prpsed. The criteri allws cstructig the quasiclassical mdel f the geeratr f attsecd laser pulses based the iteracti f a electric field f extremely pwerful femtsecd pulse with the valece electr i the ptetial well f the gas atm. Keywrds Ultrafast Optics, Catastrphe Thery, Bhr Mdel f the Atm 1. Itrducti Sice the advet f the laser i 1960, there has bee a sustaied iterest i the quest f geeratig laser pulses f the shrtest durati ad f the maximum pwer. A pulse is the packet f mchrmatic waves ad the cetral frequecy f the packet is the s-called carrier frequecy f the pulse. Thus, there is a fudametal physical limit f durati f a pulse. It is the perid f its carrier frequecy. The pulse whse durati is f the rder f the perid f its carrier frequecy is called the ultrashrt pulse. I the visible rage f the electr-magetic spectrum, the ultra shrt laser pulse ca have femtsecd duratis (1 fs = s). Such laser pulses ca be directly prduced by mder mde-lcked lasers [1] []. By usig the techique f chirped pulse amplificati [3]-[8] the pwer f the femtsecd pulse ca be brught up t the Petawatt level (1 PW = W). Fcusig with parablic mirrrs [9] allws gettig the itesity f this pulse at the target abut 10 W/cm, which crrespds t the electric field with the stregth well abve the iteratmic electric field (abut several vlts per agstrm, 10 9 V/cm). The s-called attsecd (1 as = s) pulse ca be created ly i the EUV regis f the spectrum. Hw- Hw t cite this paper: Giti, A. (014) A Electrstatic Catastrphe Machie as a Attsecd Pulse Geeratr. Optics ad Phtics Jural, 4,

2 ever, i these spectral regis the mde-lckig methd ad the chirped pulse amplificati methd are lger applicable. Frtuately, there are the s-called catastrphe machies which trasfrm smth chages f the iput sigal it a quick chage f their states [10]-[14]. As a example, the gravitatial catastrphe machie iveted by T. Pst [10] [11] ca be csidered. I this machie the ceter f gravity is represeted as a small heavy ball i a gravitatial ptetial well. The ball takes a psiti that gives a lcal miimum f its ptetial eergy. Let the iitial ptetial well have a sigle miimum, but slwly chagig uder a exteral ifluece f this ptetial well. I this case a secd lcal ptetial miimum appears ear the first e. This way the secd lcal ptetial miimum gradually ges dw ad the first miimum ges up. I the mmet the first miimum disappears, the ball jumps t the secd ptetial miimum. This jump is called a catastrphe. The heavy ball i the gravitatial ptetial well ca be replaced by a electr i the electrstatic ptetial well ad the exteral ifluece by a electric field f a femtsecd laser pulse. I the same way we ca create a electrstatic catastrphe machie i which the electr jumps frm e lcal miimum with high eergy t ather e with lwer eergy. If the differece f the eergy levels is f the rder f tes f electr vlt, the electr jump is accmpaied by emissi f attsecd electrmagetic pulse i the ultravilet rage f spectrum. The aim f the article is t explai the wrk f the hypthetical electrstatic geeratr f attsecd pulses frm the pit f view f the catastrphe thery ad classical mechaics, ad t use the btaied ccepts fr a quatum descripti f the real (quatum) electrstatic geeratr f attsecd pulses.. The Hypthetical Attsecd Electrmagetic Pulses Geeratr Let s csider a classical particle with a elemetary charge i a ptetial well V ( x ), where V is a electric ptetial (i vlts) at the pit x. Assume that the shape f the ptetial well is described by a biquadratic equati 4 V( xa) = x Ax, (1) ; where A is a parameter. This ptetial well has a mirrr symmetry. If A < 0, the the well has a miimum at pit x = 0, ad if A > 0, the it has miima at tw pits 1, x = with the value f ( ) A V x ; A = A 4 1, x = ± with values ( ) ad a maximum at the pit 3,4 0 V ; 0 x3,4 A =. Thus, i case A > 0, the ptetial well has a W-shaped prfile with tw miima at pits x 1 и x, separated by a ptetial barrier with the height f V (Figure 1(a)): ( ) ( ) 3,4 1, V V x ; A V x ; A = A 4. () Sice the distace betwee the miima (i.e. the width f the ptetial barrier) x x x = A (3) 1 ( ) is prprtial t the square rt f the parameter A, the height f the ptetial barrier with the distace betwee the miima x by the frmula ( ) 4 4 V is assciated V = A = x. (4) If we add the ptetial f the electric field f the laser pulse with the maximum stregth E (i the frm f a liear fucti Ex ) t the iitial W -shaped ptetial fucti, the resultig fucti takes the frm (Figure 1(b)) 4 ( ;, ) ( ; ) V xae V xa+ E x= x Á x + E x (5) ad its ptetial miima will be redistributed. The state f the system (a classical particle with elemetary charge i a ptetial well) is described by the iteral variable x ad the ctrl variables A ad E : V( xae ;, ). We assume that the evluti f the system is quasistatic r adiabatic. Whe the ctrl variables A ad E have fixed values, the system settles it V xae ;, : a equilibrium state where the iteral variable x miimizes (lcally) the fucti ( ) 338

3 where (a) Figure 1. The iitial electrstatic ptetial well V ( ) uifrm electric field E : (a) E = 0, (b) E > 0. (b) x (a) ad its defrmati by a exteral U( xae ;, ) = 0, (6) ( ) 3 Cmbiig (6) ad (7) e gets the equati f equilibrium states U x; A, E = dv dx = 4x Ax + E. (7) 3 4x Ax E 0 + =. (8) xae which is the xae -space it tw regis. I the re-. I the regi lcated geerally belw M we Equati (8) gives a surface M i the three-dimesial space with crdiates (,, ) s-called catastrphes surface (Figure ). The surface M divides (,, ) gi lcated geerally higher tha M we have U( xae ;, ) > 0 have U( xae ;, ) < 0. Let the x -axis be vertical t the plae f ctrl variables (, ) fud by drawig a vertical lie passig thrugh the pit (, ) surface М at the pits (,, ) fld, the umber f slutis based the parameters ( AE, ) may be equal t 1, r 3 (Figure ). Miima f the fucti V ( x ) are achieved at thse values f x fr which the fucti ( ) AE. The slutis f Equati (8) ca be AE f the ctrl plae. The lie itersects the xae, where x is the desired sluti. As the catastrphes surface М has a U x chages its sig frm mius t plus. Therefre, the case whe Equati (5) has e sluti crrespds t a miimum f the ptetial well. Whe it has three slutis, the middle sluti crrespds t the maximum. Whe it has tw slutis, e sluti is t the miimum r the maximum, but the secd sluti is the miimum. At the pits f the duble sluti the vertical lie tuches the surface M, s if e lks frm the tp at the surface M, e ca see a visible path B alg which the surface beds (Figure ). This path B i the ( AE, ) -plae is called the discrimiat set f U. This set separates the pits (, ) sluti t U( t ) = 0 ( utside B ) frm thse givig three slutis ( iside B ) (Figure 3). Thus, if the ctrl variables (, ) AE f the ctrl plae givig e AE vary, the catastrphe ca happe: the charged particle ca jump suddely frm the high lcal ptetial miimum t the lwer e (This jump is accmpaied by the emissi f electrmagetic radiati). I this case the mmet f the catastrphe is determied by the priciple f maximum delay [13]: the state f the system is determied by the lcal miimum util it exists. U x = has a The discrimiat set B ca be fud usig the cditi that at these pits the equati ( ) 0 duble rt, i.e. U( x) = U ( x) = 0. Excludig the variable x frm the crrespdig equatis we btai B : 3 4x Ax E 0 + = (9) 1x A 0 =, (10) 3 8A = 7E. (11) The geerati f ultrashrt pulses is a prcess whe A is cstat ad E is a fucti f time t. Let 3 A = cst > 0, the the discrimiat set B degeerates it tw pits E+ = + ( 3) ( A 3) (pit i 339

4 Figure. The cusp catastrphe. 3 Figure 3) ad E ( 3) ( A 3) Figure 3. The discrimiat set the ctrl plae. = (pit i Figure 3). If E E <, the there is ly the right ptetial lcal miimum (pit 3 i Figure 3), if E < E < E+, the there are tw ptetial lcal miima (pits 1, 0, 1 i Figures 3), ad if E > E +, there is ly the left ptetial lcal miimum (pit 3 i Figure 3). Let a femtsecd laser prduce a ultra shrt laser pulse i the frm f tw big scillatis ad the amplitude f the first (psitive) scillati is bigger tha E + ad the amplitude f the secd (egative) scillati is smaller tha E (Figure 4). O the attsecd time scale, the femptsecd pulse ca be csidered quasistatic. If the quasistatic laser pulse falls the system the charged particle i the secd ptetial well, we have a geerati f ultrashrt pulses which are described by a fur-strke cycle (Figure 5). Strke 1 The leadig edge f the psitive scillati raises the charged particle i the secd ptetial miimum util this ptetial well disappears. Durig this time iterval, the particle reserves the eergy E+ = x E. Strke Accrdig t the priciple f maximum delay, i the mmet whe the psitive amplitude f the leadig edge equals E + the secd ptetial well disappears ad a catastrphe ccurs: the charged particle jumps frm the high secd miimum x E + t the lw first miimum x E + ad radiates a attsecd pulse with a carrier frequecy f ω = x E+ e ħ. Strke 3 The leadig edge f the egative scillati raises the charged particle i the first ptetial miimum util the ptetial well disappears. Durig this time iterval the particle reserves the eergy E = x E. 340

5 Figure 4. The femptsecd laser pulse. Figure 5. The fur-strke cycle f the hypthetical attsecd geeratr. Strke 4 Accrdig t the priciple f maximum delay, i the mmet whe the egative amplitude f the leadig edge equals E the first ptetial well disappears ad a catastrphe ccurs: the charged particle jumps frm the high first miimum x E t the lw secd miimum x E ad radiates a attsecd pulse with a carrier frequecy f ω = x E e ħ. At the ed f the furth strke the system returs t its iitial state ad the cycle ca be repeated. Thus, if this catastrphe machie culd exist i ature, it wuld be a perfect attsecd pulse geeratr. 3. The Real Attsecd Electrmagetic Pulses Geeratr There is a real geerati f attsecd pulses i which a electric field f a fcused extremely pwerful femtsecd pulse iteracts with a valece electr i the ptetial well f the ble gas atm [15]-[17]. Nte that the wrk f a real geeratr f attsecd pulses ca be explaied by usig the ccepts f the hypthetical geeratr f attsecd pulses ad the s-called semi classical apprximati f quatum mechaics. Accrdig t de Brglie, electrs have wave prperties. A electr is described by a wave fucti. The wave fucti has a wave legth λ. I semi classical Bhr mdel f the atm (1913), valece electrs rtate i circular statiary rbits arud the atm ucleus [18]. The statiary rbit satisfies the stadig wave cditi: the whle umber f the electr wavelegths λ must fit alg the circumferece f the rbit [18]: πr = λ, (1) where is a iteger, r is the radius f the -rbit. I this case the eergy level f the electr i the - rbit (the s-called iizati eergy [19]) is 341

6 Z V = 13, 6, (13) where Z is the effective uclear charge [0]. I ctrast t the classical particle with a elemetary charge, a electr des t lie at the bttm f the electrstatic ptetial well, but lies at the -th eergy level V. Let the atm be illumiated by a fcused femtsecd pwerful laser pulse. If the stregth E f the electric field f the laser pulse is clse the stregth f the Culmb field f the atm ucleus, the resultig ptetial well fr the valece electr becmes a superpsiti f the Culmb ptetial well ad the liear fucti E x (i vlts) [1]: Ze V ( x) = k Ex, (14) x V x, Equati (14), has the ptetial barrier with the height V max (see Figure 6). The width f the barrier is determied by the distace betwee the turig pits x 1 ad x, where the ptetial V ( x ) is equal t the basic eergy level V : where e is the charge f the electr, k is Culmb s cstat. Nte that the resultig ptetial ( ) The quadratic equati with respect t x gives tw slutis: e k E( t) x = V. (15) x E x Vx k is the left turig pit (a particle frm the regi I falls it the regi II), + e= 0 (16) + V V 4e ke x1 = (17a) E x + V + V 4e ke = (17b) E is the right turig pit (a particle frm the regi II falls it the regi III). Fr further calculatis it is ecessary t chse a simple criteri f the barrier width at which the electr tuels thrugh the barrier. The cditi f the statiary rbit, Equati (1), ad the cditi fr tuelig (the width f the barrier is cmparable t the wavelegth f the electr λ ) allw us t frmulate a simple criteri: the electr tuels thrugh the barrier if the barrier width equals the diameter f the statiary rbit divided by (pits ad i Figure 7). Nte that this criteri ca be writte i the arithmetic frm as Figure 6. The resultig ptetial V( x ). Value V 1 = 13.9 crrespds t the basic eergy level f the hydrge atm Z = 1, = 1. ( ) 34

7 Figure 7. The tuelig curve the ctrl plae. x + x 1 V ( x x ) =. (18) 16 = ke e. (19) The geerati f ultrashrt pulses is a prcess whe V is cstat ad E is a fucti f time t, as we ca see i Figure 7. Let V = cst > 0, the the tuelig curve T degeerates it tw pits: = V (pit i Figure 7) ad E E = V (pit i Figure 7). If 16ke 16ke E < E < E, the tuelig is t impssible (pits 1, 0, 1 i Figure 7). If E = E, the the left ptetial barrier ca be tueled by the electr (pit i Figure 7) ad if E = E +, the right ptetial barrier ca be tueled by the electr (pit i Figure 7). Let a femtsecd laser prduce a ultrashrt laser pulse i the frm f tw scillatis where the amplitude f the first (psitive) scillati is bigger tha E + ad the amplitude f the secd (egative) scillati is smaller tha E (Figure 4). O a attsecd time scale, the femtsecd pulse ca be csidered as quasistatic. Accrdig t Keldysh [] [3], the prcess f tuelig iizati f the valece electr is quasistatic t, if the carrier frequecy f the laser pulse ω is sigificatly less tha the frequecy f a electr ω t tuelig thrugh the ptetial barrier e E ω << ωt, mv where e, m ad V are the charge, mass ad eergy f the electr, ad E is the maximum amplitude f the laser pulse. If a quasistatic laser pulse falls a quasistatic system electr i the ptetial well f the atm ucleus, we have the geerati f ultrashrt pulses which is described by a six-strke cycle (Figure 8). Strke 1 Whe the electric stregth E i the psitive scillati icreases frm 0 t E +, the width f the right ptetial barrier reduces. Whe the electric stregth E equals E +, the electr tuels thrugh the right ptetial barrier. Strke Whe the electric stregth E reduces frm E + t 0, the electr reserves the eergy Z V = 13, 6. Strke 3 Whe the electric stregth E equals 0, the ptetial barrier disappears ad, accrdig t the priciple f maximum delay, a catastrphe ccurs: the electr jumps frm the zer eergy level t the basic eergy level ad radiates a attsecd pulse with a carrier frequecy f ω = e ħ. Strke 4 Whe the electric stregth E i the egative scillati reduces frm 0 t E, the width f the V 343

8 Figure 8. The six-strke cycle f the real attsecd geeratr. right ptetial barrier reduces t. Whe the electric stregth E equals E, the electr tuels thrugh the left ptetial barrier. Strke 5 Whe the electric stregth E icreases frm E t 0, the electr reserves the eergy Z V = 13, 6 Strke 6 Whe the electric stregth E equals 0, the ptetial barrier disappears ad, accrdig t the priciple f maximum delay, a catastrphe ccurs: the electr jumps frm the zer eergy level t the basic eergy level ad radiates a attsecd pulse with a carrier frequecy ω = V e ħ. At the ed f the sixth strke the system returs t its iitial state ad the cycle ca be repeated. I the table f the elemets there is a peridic tred fr iizati eergy [19]: each perid begis at a miimum fr the alkali metals, ad eds at a maximum fr the ble gases. S, t geerate attsecd pulses the hydrge r the ble gases are used. As the H iizati eergy [19] is ev, He 4.58 ev, Ne 1.56 ev, Ar ev, Kr ev, Xe 1.13 ev, Hg ev, R ev, the crrespdig radiati refers t the sft EUV regi f the spectrum. The durati f the ultra shrt pulse i a pht eergy rage f 10 ev t 5 ev cat be less tha 100 as. I the catastrphe thery the priciple f maximum delay is widely used [13]. I this article we have used this priciple t. Hwever, it des t allw takig it accut the kietic eergy f a electr scillatig i a exteral laser field the s-called pder mtive eergy. T prduce the hard EUV-rays r eve X-rays pder mtive eergy must be take it accut. I this case, the priciple f maximum decelerati shuld be replaced by a differet, mre suitable priciple. 4. Cclusi The trasfrmati frm the iput femtsecd pulse i the visible spectrum t the utput attsecd pulse i the ultravilet spectrum is a trasfrmati f a smth chagig iput sigal t a quickly chagig utput sigal, s it is a field f iterest f the catastrphe thery. We prpse a criteri fr tuelig (18) ad a quasiclassical mdel f the trasfrmati f femtsecd laser pulses it attsecd pulses described as a electrstatic catastrphe machie. 344

9 Refereces [1] Yariv, A. (1965) Iteral Mdulati i Multimde Laser Oscillatrs. Jural f Applied Physics, 36, [] Giti, A.V. (013) Applicati f the Samplig ad Replicati Operatrs t Describe Mde-Lcked Radiati. Optics ad Phtics Jural, 3, [3] Stricklad, D. ad Muru, G. (1985) Cmpressi f Amplified Chirped Optical Pulses. Optics Cmmuicatis, 56, [4] Giti, A.V. (008) Gemetrical Methd fr Calculatig the Grup Velcity Dispersi f Stretcher Takig it Accut the Ifluece f Optical System Parameters. Quatum Electrics, 38, [5] Giti, A.V. (01) Tautchrism Priciple ad Gratig Dispersive Delay Lies. Applied Optics, 51, [6] Giti, A.V. (010) Zer-Distace Pulse Frt as a Grup Delay Characteristic f the Tw-Gratig Cmpressr. Optics Cmmuicati, 83, [7] Giti, A.V. (01) Usig the Ufldig Methd fr Dispersi Calculatis f Reflective Gratig Delay Lies i the Chirped Pulses Amplifier. Optics Cmmuicati, 85, [8] Giti, A.V. (013) Zer-Distace Pulse Frts f a Cmpressr, a Stretcher, ad the Optical System f the Stretcher. Optics Cmmuicati, 95, [9] Giti, A.V. (009) Optimal Tw-Mirrr System fr Laser Radiati Fcusig. Quatum Electrics, 39, [10] Pst, T. ad Stewart, I. (1978) Catastrphe Thery ad Its Applicatis. Pitma Publishig, Ltd., Ld. [11] Bruce, J.W. ad Gebli, P.G. (1993) Curves ad Sigularities. Cambridge Uiversity Press, Cambridge. [1] Zeema, E.C. (1976) Catastrphe Thery. Scietific America, 34, [13] Gilmre, R. (1981) Catastrphe Thery fr Scietists ad Egieers. Wiley, New Yrk. [14] Brcker, T. ad Lader, L. (1975) Differetiable Germs ad Catastrphes. Uiversitat Regesburg Fachbereich Mathematik. Cambridge Uiversitz Press, Cambridge. [15] Crkum, P.B. (1993) Plasma Perspective Strg-Field Multipht Iizati. Physical Review Letters, 71, [16] Hellerer, T. (01) Attsciece Ges OPCPA. New Laser Develpmets i High Harmic Geerati. Optik & Phtik, 7, [17] Uiberacker, M., Uphues, T., Schultze, M., Verhef, A.J., Yakvlev, M.F., Klig, V., Rauscheberger, J., Kabachik, N.M., Schrцder, H., Lezius, M., Kmpa, K.L., Muller, H.-G., Vrakkig, M.J.J., Hedel, S., Kleieberg, U., Heizma, U., Drescher, M. ad Krausz, F. (007) Attsecd Real-Time Observati f Electr Tuellig i Atms. Nature, 446, [18] Bhr Mdel. Frm Wikipedia, the Free Ecyclpedia. [19] Iizati Eergy. Frm Wikipedia, the Free Ecyclpedia. [0] Effective Nuclear Charge. Frm Wikipedia, the Free Ecyclpedia. [1] Pastr, A.A. ad Serdbitsev, P.Y. (007) Methds f Geerati f Attsecd Pulses f Laser Radiati ad Electr Beams Attsecd Pulses i Itese Field f Femtsecd Durati. [] Keldysh, L.V. (1965) Iizati i the Field f a Strg Electrmagetic Wave. Jural f Experimetal ad Theretical Physics, 0, [3] Ppv, V.S. (004) Tuel ad Multipht Iizati f Atms ad Is i a Strg Laser Field (Keldysh Thery). Physics-Uspekhi, 47,

10