Nonlinear-optical vacuum ultraviolet generation at maximum ... - IAP

15 August 2002

Optics Communications 209 (2002) 335–347 www.elsevier.com/locate/optcom

Nonlinear-optical vacuum ultraviolet generation at maximum atomic coherence controlled by a laser-induced Stark chirp of two-photon resonance S.A. Myslivets a, A.K. Popov a,b,*, T. Halfmann c, J.P. Marangos d, Thomas F. George b a

b

Institute of Physics of Russian Academy of Sciences, 660036 Krasnoyarsk, Russia Office of the Chancellor/Departments of Chemistry and Physics and Astronomy, University of Wisconsin-Stevens Point, Stevens Point, WI 54481, USA c Fachbereich Physik der Universit€at Kaiserslautern, 67653 Kaiserslautern, Germany d Blackett Laboratory, Imperial College of Science, Medicine and Technology, Prince Consort Road, London SW7 2BW, UK Received 14 March 2002; received in revised form 19 June 2002; accepted 27 June 2002

Abstract A novel scheme is analyzed for efficient generation of vacuum ultraviolet radiation through four-wave mixing processes assisted by the Stark chirp of two photon resonance. In this three-laser technique, a delayed-pulse of strong off-resonant infrared radiation sweeps the laser-induced Stark shift of a two-photon transition and facilitates maximum two-photon coherence induced by the second ultraviolet laser. A judiciously delayed third pulse beats with this coherence and generates short-wavelength radiation. A numerical simulation of transient processes, which ensure maximum coherence, including those leading to Stark-chirped rapid adiabatic passage, is presented. Potential high power and energy conversion efficiency of the weak long-wavelength radiation to the VUV range is shown for a differencefrequency scheme based on parametric amplification of this signal and strong four-wave coupling at maximum atomic coherence. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 42.50.Hz; 42.50.Ct; 42.50.Gy Keywords: Coherent quantum control; Rapid adiabatic passage; Four-wave mixing; Vacuum ultraviolet generation

1. Introduction *

Corresponding author. E-mail addresses: [email protected] (A.K. Popov), [email protected] (T. Halfmann), [email protected] (J.P. Marangos), [email protected] (T.F. George). URLs: http://www.kirensky.ru/popov, http://www.quantumcontrol.de, http://www.lsr.ph.ic.ac.uk/, http://www.uwsp.edu/ admin/chancell/tgeorge.

Frequency conversion techniques, based upon the nonlinear optical response of a gaseous medium with coherent radiation, provide a wellestablished powerful tool for the generation of tunable coherent short-wavelength vacuum ultra-

0030-4018/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 2 ) 0 1 7 3 1 - 5

336

S.A. Myslivets et al. / Optics Communications 209 (2002) 335–347

violet (VUV) radiation [1]. These light sources are of considerable interest for laser-lithography, high resolution microscopy and spectroscopy. However, these sources produce limited powers due to the relatively small nonlinear susceptibilities available in an atomic or molecular gas when the applied and generated fields are not in resonance with transitions. In resonance, severe problems arise due to the detrimental effects of strong absorption and undesirable dispersion for the fundamental and generated VUV radiation. Conversion efficiencies, obtained with these techniques, are therefore rather small, typically in the range of 106 –104 (a review is given in [2]). The resulting few nanojoule VUV pulses are thus of low peak power (1–100 W) which precludes their utility in a wide range of nonlinear optical and spectroscopic applications. These barriers to high conversion efficiency may be overcome by manipulating coherently driven media through nonlinear interference effects at quantum transitions (see, e.g., the reviews in [3–5]) in ways to minimize resonant absorption and maximize nonlinear susceptibilities. A dressing laser is used to induce strong coherent couplings between one of the states, responsible for absorption and another excited state. The opportunities to manipulate transparency with quantum interference were identified more than three decades ago [6–9], including a detailed analysis of favorable conditions for completely vanished absorption and amplification without population inversion [10,11] and the experimental realization of these effects in neon discharge [12]. Destructive interference at quantum transitions has been also extensively studied in the context of electromagnetically induced transparency (EIT) [13,14] and coherent population trapping [15], which may result in cancellation of absorption for some of the coupled fields and in the improvement of phase matching. By the appropriate choice of detuning and intensities, the nonlinear polarization responsible for the conversion process may be aided by the constructive interference and exceed the polarization component responsible for the absorption (furthermore labeled as linear). When optimized in such a way, VUV radiation generated in a frequency conversion process has been pro-

duced with conversion efficiencies up to 102 [16– 18]. While EIT may lead to increased conversion efficiencies, the use of large coupling fields to produce transparency in a Doppler-broadened medium is in conflict with the requirements for maximizing the nonlinear susceptibility. In effect, despite constructive interference, the large Autler– Townes splitting results in much reduced nonlinear susceptibilities compared to resonant values. Furthermore, when tunability is required, resonances with the generated field cannot be used to provide an enhancement in the nonlinear optical response of the medium. To achieve the goal of high conversion efficiency and tunable generation, a closely related approach based on population trapped (darkstate) atoms has been utilized [19,20]. In a simple two-level atom, the polarization leading to nonlinear mixing depends upon the coherence between the ground and excited states. This property reaches a maximum of 1/2 for an equal amplitude coherent superposition of ground and excited states. Therefore, in such a medium comprised of a high density of dark-state atoms, a large nonlinear polarization can be induced while absorption for the driving resonant fields is decreased. The system may then be regarded as an oscillator driven with maximum amplitude. If another electromagnetic field is introduced, the beat frequency produced by interaction with the oscillator may be generated with high conversion efficiency, which in principle can approach unity over a length shorter than the coherence length. In this system the nonlinear polarization responsible for generation may become as large as the terms responsible for absorption and dispersion. Thus, nonlinear interference processes in media with maximum coherence can substantially improve frequency conversion between the coupled fields. Frequency-mixing processes at maximum coherence were studied theoretically [21,22] as well as demonstrated experimentally [19,20,23,24] by a number of workers considering both adiabatic laser pulses and steady-state conditions. In the latter case, the lowest relaxation rate and therefore maximum coherence are achievable at the Raman transitions associated with the ground electronic state. In the up-conversion experiments, maximum


coherence has been established in a lambda-type coupling scheme, similar to that employed for coherent anti-Stokes Raman scattering (CARS), which uses a pulse sequence closely related to stimulated Raman scattering involving adiabatic passage (STIRAP) [25]. Population is driven from a ground state j0i to an excited state j2i in a Raman-type process, induced by a pump laser (frequency x1 ) and a Stokes laser (frequency x2 ) and mediated by an intermediate state j1i. Provided the laser frequencies are tuned close to the twophoton resonance, the laser intensities are sufficient to guarantee adiabatic population evolution and the pulse delay is chosen appropriately, maximum coherence can be established between the ground and excited states. In an optically dense medium, the counter-intuitive pulse sequence conditions can be automatically created through the propagation of two pulses close to Raman resonance to form a pair of matched pulses [24]. For maximum coherence on the (forbidden) 0–2 transition (frequency x20 ), the interaction with a further field x3 will result in sum- or differencefrequency mixing due to the interaction between the atomic oscillator and the optical field leading to fields at the frequencies x ¼ x3 x20 . The field at the frequency x3 may be either off [20,23] or on resonance [24], but in either case near unity conversion efficiency for the third field to be achieved. Thus, this is a very promising technique for frequency conversion into the UV and VUV. However, because in the lambda-type coupling scheme the atomic frequency x20 is limited in magnitude for any feasible atomic system (e.g., is only 10 650 cm1 in the Pb system as explored in the Harris group), the generated radiation cannot reach far into the vacuum-ultraviolet spectral region beyond 185 nm [24]. Multiphoton excitations cannot be used in order to reach higher-lying states, because laser-induced Stark shifts, which are intrinsic to multiphoton transitions, perturb the adiabatic population dynamics and prohibit the preparation of maximum coherence [26–28]. To overcome these difficulties, we propose a new scheme for the generation of maximum coherence, based upon the recently developed Starkchirped rapid adiabatic passage technique (SCRAP) [25,29]. The technique relies on a rapid

337

adiabatic passage process (RAP), driving the population from the ground state to an excited state by any convenient single- or multiphoton process, induced by a pump laser (frequency x1 ). The process is closely related to RAP, mediated by a frequency chirp in the pump laser [30,31]. In SCRAP the chirp in the pump laser frequency is replaced by a Stark shift of the excited state, induced and controlled by another strong laser pulse (frequency xSt ), which is delayed with respect to the pump laser. SCRAP can be used to generate a non-persistent maximum coherence during the interaction with the laser pulses or a steady maximum coherence afterwards [32]. Provided the dynamic Stark shifts induced by the laser with frequency xSt are larger than the shifts induced by the pump radiation field (which usually is the case for a strong Stark shifting laser), any multiphoton transition may be used for the pump transition. In contrast to the lambda-type coupling scheme as discussed above, it is therefore possible to create coherence between a highly excited state and the ground state, e.g., by a two-photon pump transition. If ultraviolet radiation is used for the pump laser, coherence for states with energies up to 10 eV may be efficiently created. When another weak (probe) to-be-converted radiation field (frequency x2 ) is introduced during the interaction or with an appropriate delay, the maximum coherence stimulates efficient generation of beat frequencies at x3;4 ¼ 2x1 x2 . If a twophoton pump transition induced by ultraviolet radiation and laser frequencies x2 in the visible or infrared spectral region is used, even the differencefrequency-mixing process x3 ¼ 2x1 x2 provides vacuum-ultraviolet radiation far below 150 nm, while the sum-frequency-mixing process x4 ¼ 2x1 þ x2 yields wavelengths much shorter than 100 nm. Additionally, because the convertible laser field does not rely on resonances, the generated radiation is tunable without any restrictions. The technique can also be applied to inhomogeneously broadened media, e.g., gas cells, if the two-photon Rabi frequency and/or laser-induced Stark shift are larger than the resonance Doppler-width in the medium. Therefore, frequency conversion processes based upon SCRAP-induced maximum coherence provide a powerful tool for the highly

338


efficient generation of broadly tunable vacuumultraviolet radiation. This paper studies features of four-wave mixing at maximum coherence produced through SCRAP. The major dependencies are illustrated by numerical simulation with the aid of a model relevant to the experiments which are under way.

2. Principal equations Higher-order nonlinearities and considerably smaller particle number densities are characteristic of nonlinear optics of free atoms and molecules as compared to that in solid states. Therefore, oneand multiphoton resonances or quasi-resonances are used, wherever possible, in order to enhance nonlinear-optical responses in gases and metal vapors. We consider a four-wave mixing process x3;4 ¼ 2x1 x2 controlled by a strong auxiliary infrared off-resonant field ESt . A typical scheme of such coupling and of the corresponding quasiresonant levels, which give the dominant contribution to nonlinear response and to an ac-Stark shift, are depicted in Fig. 1(a). This scheme also enables us to evaluate the magnitudes of various material parameters (polarizabilities), which are characteristic for atoms such as mercury and noble gases and are used in our further numerical simulations. Level m is assumed to give the major contribution to two-photon coherence at the transition gn, while level f determines the dynamic Stark shift of the two-photon resonance, which is controlled with the field ESt (In the general case, other fields may contribute as well to the dynamic

Fig. 1. Energy levels and coupled fields.

Stark shift.) In order to enhance the nonlinear polarization, the frequency x2 is chosen such that the generated sum- or difference-frequency is tuned in the vicinity of level l. This level is located above n for the sum-frequency process and below for the difference-frequency generation. In specific cases, level m may play the role of level l and also give a major contribution to the sum- or difference-frequency nonlinear response. Then the scheme reduces to a four-level or even three-level scheme (see Fig. 1(b)), if level m determines the ac-Stark shift of the two-photon resonance as well. The magnitude and evolution of atomic oscillations at 2x1 (two-photon coherence at the transition gn) in time and space are the most important factors which determine the nonlinear optical polarization P ðx3;4 ) created through scattering of the field E2 on these oscillations. The offdiagonal elements of the density matrix are a quantitative measure of the coherence, while the diagonal elements represent the populations of the levels. Consequently, we employ density-matrix equations in further direct analysis of the evolution of these most important physical variables of the problem under consideration. In the general case, a sum over all intermediate levels m,l and f must be taken (see, e.g., [2] and references therein). All the fields are assumed pulsed, where the durations of the pulses are much shorter than the shortest relaxation time in the quantum system. Through a time-dependent laser-induced shift of the two-photon resonance gn, which may be treated as a dynamic shift of level n, the field ESt allows the control of two-photon excitation by the field E1 . By this process, half of the population of the ground level g can be moved to the upper state n by the end of the pulse E1 , which further ensures persistent maximum amplitude of the induced oscillations at 2x1 until relaxation process are developed. Alternatively, transient maximum coherence at the two-photon transition gn is achieved during the rapid adiabatic passage when the entire population of the ground state g is transferred to the upper state n by the end of the pulse E1 . Furthermore, with a judicious delay of the pulse E2 , one can achieve maximum nonlinear polarization at the frequency x3;4 and, therefore, high conversion efficiency of the visible radiation


to the VUV range. Density matrix elements, which are proportional to product of the probability amplitudes of quantum states, describe directly coherence and populations of the corresponding levels. 2.1. Equations for interacting electromagnetic fields

339

Hereafter we omit the prime on z and t, assuming a transformation to a new coordinate system. The polarization P can be calculated with the aid of the density matrix qij as X P¼N qij dji ; ð5Þ i;j

The propagation of optical waves in a nonconducting medium is described by the wave equation r2 E

1 o2 E 4p o2 P ¼ 2 ; c2 ot2 c ot2

ð1Þ

where P is the electric polarization, and we assume the nonlinear medium to be isotropic and all the waves to be identically polarized and to propagate along the z-axis: Ej ðz; tÞ ¼ Re fEj ðz; tÞ exp½iðxj t kj zÞ g; Pj ðz; tÞ ¼ Re fPj ðz; tÞ exp½iðxj t kj0 zÞ g:

ð2Þ

Here kj is the modulus of a wave vector at frequency xj ; kj0 is the wave vector of nonlinear polarization at the same frequency, and Ej ðz; tÞ and Pj ðz; tÞ are slowly varying envelopes. Assuming a uniform field distribution over the cross-section and taking into account Eq. (2) and k x=c, one can write Eq. (1) in the approximation of slowly varying amplitudes as oE 1 oE 2ik ð3Þ þ ¼ 4pk 2 P ; oz c ot where P ¼ P L þ P ðFWMÞ eiDkz , and Dk is the phase mismatch of the interacting waves. We restrict ourselves to the major process under consideration and assume phase matching to be fulfilled so that Dk ¼ 0: In general, Dk ¼ 0 will not be the case; however, by choice of an appropriate buffer gas of opposite dispersion, it is well known that this phase matching condition can be fulfilled. Optimum frequency mixing will occur for values of Dk, other than Dk ¼ 0 and these can be maintained in practice by careful control of the sample pressure or path length. In the local time coordinate system with z0 ¼ z and t0 ¼ t z=c, we can express Eq. (3) in the form 0

oE=oz ¼ 2p ikP :

ð4Þ

where N is the number density of the resonance atoms, dji are electric-dipole transition matrix elements, and qij are density matrix components oscillating at the corresponding frequencies. The problem then reduces to calculating the off-diagonal elements of the density matrix. 2.2. Density matrix of the coherently driven quasiresonant five-level system First, we will consider quasi-resonant interactions depicted in Fig. 1(a) taking into account that the generalization through the sum over all intermediate off-resonant levels is straightforward and will be done in the final formulas below (see, e.g., Section 2.3). We assume that the durations of all laser pulses, as well as of the entire process under consideration, are much shorter than all relaxation times in the system, and initially only the lower level g is populated and that the populations of levels m, l and f are negligibly small during the entire process. This allows us to neglect such terms as qml ; qlf , etc. Besides relaxation, we also neglect photoionization. Then in the rotation-wave representation, where oscillations with the atomic frequencies are subtracted, the density-matrix equations can be written as q_ mn ¼ i½Vmn qn þ Vmg qgn =h; q_ ln ¼ i½Vln qn þ Vlg qgn =h; q_ nf ¼ iVnf qn =h; q_ gl ¼ i½qgn Vnl þ qg Vgl =h; q_ gf ¼ iqgn Vnf =h;

q_ gm ¼ i½qgn Vnm þ Vgm qg =h;

q_ gn ¼ i½Vgm qmn qgm Vmn þ Vgl qln qgl Vln qgf Vfn =h; q_ n ¼ 2Im ½Vnm qmn þ Vnl qln þ Vnf qfn =h; qg ¼ 1 qn ;

ð6Þ

340


where q_ ij ¼ dqij =dt; Vij ðtÞ ¼ hgk ðtÞ expfiXij tg (for energy-levels Ej > Ei ), Vji ¼ Vij ; qji ¼ qij . Here gk ðtÞ ¼ dij Ek ðtÞ=2 h ðk ¼ 1; 2; 3; StÞ are slowly varying Rabi frequencies corresponding to the couplings depicted in Fig. 1 (e.g., g1 ðtÞ ¼ dgm E1 ðtÞ=2 h), and dij are electric-dipole transition momenta. Eq. (6) are written under assumption of two-photon quasi-resonance jXgn j ¼ j2x1 xgn j xij and that the conditions for one-photon detunings, jXij j ¼ jxk xij j jgk j s1 i ;

ð7Þ

are fulfilled, where si is the minimum pulse length. We look for the solution of the coupled equations in the form qij ðtÞ ¼ rij ðtÞ expfiXij tg; Vmn ðtÞ= h ¼ ag1 ðtÞ expfiXmn tg;

present the generalized two-photon Rabi frequency and the laser-induced (Stark) shift of the two-photon resonance. 2.3. Generalized two-level scheme In the more general case of a multilevel system, a sum over all intermediate states m must be taken. The corresponding level scheme and radiative coupling are shown in Fig. 2. Taking into account that Xmn Xgm ; Xln Xgl and Xgf Xnf in the vicinity of a two-photon resonance, we introduce the coefficients (polarization tensors) ci and gi depending on specific atoms and on contributing quantum transitions X c1 ¼ dgm dmn =2h2 Xgm ;

a ¼ dmn =dgm ;

m

X

where rij ðtÞ are slowly varying envelopes. Then using (7) and j_rij j jXij rij j for all the off-diagonal elements except rgn , we can rewrite Eqs. (6) as

c2 ¼

rmn ¼ ½ag1 rn þ g1 rgn =Xmn ; rln ¼ ½g2 rn þ g3 rgn =Xln ;

g1g ¼

rnf ¼ gSt rn =Xnf ; rgl ¼ ½rgn g2 þ rg g3 =Xgl ; rgf ¼ rgn gSt =Xgf ; rgm ¼ ½a g1 rgn þ g1 rg =Xgm ; r_ gn ¼ i½g1 rmn ag1 rgm þ g3 rln

g1n ¼

g2 rgl gSt rgf Xgn rgn ; r_ n ¼ 2Im½a g1 rmn þ g2 rln þ gSt rfn ; rg ¼ 1 rn :

dgl dnl =2h2 Xgl ;

l

X

2

jdgm j =2h2 Xgm ;

m

X

jdnm j2 =2h2 Xnm ;

m

g2 ¼

X

ð12Þ

jdnl j2 =2h2 Xnl ;

l

ð8Þ Here Xnf ¼ xSt xnf ; Xgl ¼ 2x1 x2 xgl ; Xgf ¼ 2x1 þ xSt xgf , etc. After simple algebra, Eqs. (8) reduce to 2 g2 g3 g1 r_ n ¼ 2 Im a þ rgn ; Xmn Xln ag12 g2 g3 þ ; r_ gn ¼ i ðXSt Xgn Þrgn þ ggn rn þ Xgm Xgl

g3 ¼

X

jdgl j2 =2h2 Xgl ;

l

gs ¼

X

jdnf j2 =2h2 Xnf ;

f

ð9Þ where the quantities 2 ag1 g2 g3 ag12 g2 g3 ggn ¼ þ Xmn Xln Xgm Xgl

ð10Þ

and 2

XSt ¼

2

2

2

2

jg3 j jgSt j jg1 j a2 jg1 j jg2 j þ þ þ þ Xln Xgf Xmn Xgm Xgl

ð11Þ Fig. 2. Generalized two-level scheme and radiative couplings.


r_ n ¼ Im ½ðr1 þ r2 Þrgn ¼ Im ½ðx2 1 =x10 þ x2 x3 =x20 Þrgn ;

and the quantities r1 ¼ c1 E12 ;

r 2 ¼ c 2 E2 E3 ; 2

s1 ¼ ðg1g þ g1n ÞjE1 j =2;

s ¼ gs jESt j2 =2; 2

s2 ¼ g2 jE2 j =2;

2

s3 ¼ g3 jE3 j =2: ð13Þ

Then Eqs. (9) take the form r_ n ¼ Im ½ðr1 þ r2 Þrgn ; r_ gn ¼ iðXSt Xgn Þrgn iðr1 þ r2 Þðrn 1=2Þ; ð14Þ where the laser-induced shift of the two-photon resonance is described as XSt ¼ s þ s1 þ s2 þ s3 :

ð15Þ

Eqs. (4) for the pulse envelopes can be written as c dE3 =dZ ¼ i 2 E2 rgn þ E3 ð1 rn Þ ; g3 k2 c2 E3 rgn þ E2 rn ; dE2 =dZ ¼ i k 3 g3 k1 c 2 1 E1 rgn dE1 =dZ ¼ i k3 g3 g1g g1n þ ð1 rn Þ þ r n E1 ; g3 g3 ð16Þ hk3 N g3 . where Z ¼ z=L; L1 ¼ 4p We introduce the new variables x1 ¼ c1 E10 E1 ; x2 ¼ c2 E20 E2 and x3 ¼ c2 E20 E3 , where Ei0 ; x10 ¼ 2 2 c1 E10 ; x20 ¼ c2 E20 and x30 ¼ c2 E20 E30 are the corresponding amplitudes at the entrance to the media. Then Eqs. (16) take the form dx3 c E20 ¼ i 2 x2 rgn þ x3 ð1 rn Þ ; dZ g3 E20 dx2 k2 c2 E20 g2 ¼ i x r x r þ 2 n ; 3 gn dZ k3 g3 E20 g3 ð17Þ dx1 k1 c E10 ¼ i 2 1 x1 rgn dZ k3 g3 E10 g1g g þ ð1 rn Þ þ 1n rn x1 : g3 g3 Taking into account that, r1 ¼ x21 =x10 ;

341

r2 ¼ x2 x3 =x20 ;

ð18Þ

the equations for the generalized two-level system can be presented in the form

r_ gn ¼ iðXSt Xgn Þrgn iðx21 =x10 þ x2 x3 =x20 Þ ðrn 1=2Þ; XSt ¼ s þ

g1g g1n jx1 j2 g2 jx2 j2 g3 jx3 j2 þ : 2c1 x10 2c2 x20 2c2 x20 ð19Þ

The change of the photon numbers along the ðiÞ medium at any instant of time Wph (for i ¼ 2; 3) scaled to the photon number of the field E2 at the entrance to the medium at the same instance, and the relative change of the photon number inteðiÞ grated over time eph is expressed as 2

2

x2 jxi ðt; zÞj jxi ðt; z ¼ 0Þj ; xi jx2 ðt; z ¼ 0Þj2 R R 2 2 x2 jxi ðt; zÞj dt jxi ðt; z ¼ 0Þj dt ðiÞ : eph ðzÞ ¼ R 2 xi jx2 ðt; z ¼ 0Þj dt ðiÞ

Wph ðt; zÞ ¼

ð20Þ 3. Distribution of populations, coherence and conversion efficiency along the medium In this section, we analyze the behavior of the coupled fields along the medium. In further analysis, we assume that the Rabi frequencies of all the fields at the entrance to the medium are given by the Gaussian expressions h i 2 xj ðz ¼ 0Þ ¼ xj0 exp ðt Dsj Þ =2s2j eiuj0 ; ð21Þ h i sðtÞ ¼ s0 exp ðt DsSt Þ2 =s2St : ð22Þ Here s0 is the amplitude of the Stark shift, which is assumed constant along the medium, 2sj is the width of the jth pulse (at the level 1=e of the power maximum), Dsj is the time delay of the jth pulse relative to t ¼ 0, and uj0 is the initial phase of the jth field at the entrance to the nonlinear medium. In the case where dynamic self-shift of the resonance is negligible ðg1g ¼ g1n Þ, the principal contribution to XSt is determined by ESt , such that the entire Stark shift is described by Eq. (22). We choose the intensity and frequency detuning of the Stark field such that the Stark shift XSt can compensate for the two-photon detuning Xgn . In order

342


to provide a two-photon resonance, the requirement js0 j > jXgn j must be fulfilled. Then the resonance condition is met at two instants of time: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t1;2 ¼ DsSt sSt lnðs0 =Xgn Þ: The key idea of SCRAP is to maximize the transition probability at t1 and to minimize it at t2 , which is possible if the evolution is adiabatic at t1 and diabatic at t2 . This leads to the requirement [29] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1

x10

; ð23Þ ln Xgn s 2 Xgn where s is a characteristic pulse lengths. The length of all pulses is assumed to be approximately equal. The SCRAP process is most easily understood in terms of the dressed states, i.e., the eigenstates of the atomic system including the interaction with the radiation fields. The dressed states in a twolevel system, consisting of the bare states jgi and jni driven with the two-photon Rabi frequency ggn ðtÞ by a pump laser pulse, read jaþ i ¼ cos hjgi sin hjni; ja i ¼ sin hjgi þ cos hjni;

ð24Þ

where the mixing angle is defined by the equation 2

2 1=2

tan½hðtÞ ¼ ½1 þ Xgn ðtÞ =ggn ðtÞ Xgn ðtÞ=ggn ðtÞ:

ð25Þ

By analyzing the dressed states one easily obtains the conditions for complete population transfer from the ground state jgi to the excited state jni via one of the dressed states. Consider e.g. state jaþ i. Population can be prepared in this dressed state at the beginning of the interaction, if cos h ¼ 1, thus the detuning Xgn is very large and positive. If at the end of the interaction sin h ¼ 1, thus the detuning Xgn is very large and negative, all the population is transferred to the excited state jni. These conditions refer to a frequency chirp in the detuning. The chirp can be implemented either by sweeping the laser frequency across the atomic resonance (RAP) or by modifying the resonance by externally controlled Stark shifts (SCRAP). In the case of SCRAP the Stark shift is induced by a second, strong radiation field which is suitably

delayed with respect to the pump laser. The Stark shift driving the population transfer process occurs either in the rising or the decaying wing of the Stark shifting laser [25,29]. Besides efficient coherent population transfer the SCRAP process can be used to provide maximum coherence during the transfer process, when the populations in the ground and excited states become equal. The coherence generated in this way is not permanent, but evolves as the full transfer process is being completed (see Fig. 3(a)). Even this non-persistent coherence may be used for efficient frequency conversion during the SCRAP process. Another way to generate a maximum coherence utilizes the SCRAP technique to provide a permanent coherent superposition of the ground and excited states [32] with equal populations (see Fig. 3(b)), which can be used afterwards in the frequency conversion processes. In the following, we will discuss the differences in these two approaches with respect to the robustness of the preparation in extended media, the efficiency of the frequency conversion processes, and the pulse shape of the generated radiation. In a multilevel system with appreciable difference of principal quantum numbers of the ground and excited states, two-photon excitation may substantially contribute to the dynamic shift of the resonance. This makes transient effects, and consequently the choice of the parameters ensuring maximum coherence and rapid adiabatic passage, more complicated.

Fig. 3. Dynamics of population rn (dashed) and coherence jrgn j (solid) at the entrance to the medium controlled by the dynamic shift of the two-photon resonance. S ¼ 75; R1 ¼ 15; X2 ¼ 105 ; X3 ¼ 0: (a) (SCRAP) sSt =s1 ¼ 1:6, dsSt ¼ 2; d ¼ 35, (b) (HSCRAP) sSt =s1 ¼ 1, dsSt ¼ 1:33, d ¼ 3:15:


In this section, we apply our theory for simulating the features of two-photon resonant FWM controlled by SCRAP with the aid of the following numerical model: k1 ¼ 268:8 nm, kSt ¼ 1064 nm, k2 ¼ 532 nm, s1 ¼ 3 ns. Then k3 ¼ 179:8 nm, k4 ¼ 107:3 nm (here ki is the wavelength of radiation with frequency xi ). The other parameters in Eq. (17) are taken as c1 =g3 ¼ 0:032, c2 =g3 ¼ 0:345, g2 =g3 ¼ 0:119, g1g =g3 ¼ 0:092, g1n =g3 ¼ 0:011: This model is appropriate, e.g., for quantum transitions in Hg. In further numerical simulations, we will reduce all parameters by the magnitude of the fundamental pulse width and by its inverse magnitude, where appropriate, by introducing the parameters T ¼ t=s1 , ds ¼ Ds=s1 , dsSt ¼ DsSt =s1 , d ¼ Xgn s1 ; S ¼ s0 s1 , Xj ¼ xj0 s1 . The latter determines the amplitude of the two-photon Rabi frequency at the entrance to the medium as R1 ¼ r10 s1 ¼ X1 : As outlined above, in order to achieve transfer efficiency close to unity, one must fulfill the adiabatic condition discussed in [29,32]. Through a proper choice of the two-photon Rabi frequency R1 , Stark shift S, static detuning d and pulse delay dsSt , one can ensure various values and evolution in time of the populations and coherence at the entrance to the medium. Figs. 3(a) and (b) demonstrate possible dynamics leading either to maximum population transfer (SCRAP) (Fig. 3(a)) and to transient maximum coherence or to equal populations (HSCRAP) (Fig. 3(b)) and to permanent maximum coherence (H stands for ‘‘half’’). The generated radiation depends strongly on the delay at which the pulse E2 to be converted is

343

applied. For the cases depicted in Fig. 3, one can choose either a relatively short interval, when the populations of the initial and final levels g and n become equal and the coherence amplitude reaches its maximum or a ‘‘plateau’’ with almost constant amplitude of the two- photon coherence jrgn j in time. Furthermore, in our simulations, we have assumed that the delay of the pulse E2 is set as ds2 ¼ 5, so that this pulse overlaps the plateau at the time dependence of the coherence jrgn j at the entrance to the medium (Fig. 3(b)), or ds2 ¼ 0, so that the maximum of the convertible pulse coincides with the maximum of the time dependence of the coherence (Fig. 3(a)). Figs. 4(a)–(d) computed for the differencefrequency processes show that the shape of the generated and convertible pulses may vary substantially along the medium, subject to the time delay ds at the entrance to the medium. It is seen that if the pulse E2 overlaps the plateau of the time dependence of jrgn j, there is no significant transformation of the pulse shape along the medium (Figs. 4(c) and (d)). On the contrary, if the maximum of the pulse E2 coincides with the instant when the populations of the levels rg and rn become equal at the entrance to the medium, the change of pulse shapes is most significant (Figs. 4(a) and (b)). In the case of Figs. 4(c) and (d) all parts of the pulse E2 are converted homogeneously. Alternatively, in the case of Figs. 4(a) and (b), the wings of the pulse E2 are not converted and enhanced, since the pulse of coherence jrgn j is substantially shorter than that of E2 . This leads to the appearance of sharp peaks in the pulse shape of both the generated and convertible radiations.

Fig. 4. Shapes of the generated jx3 j2 =jx002 j2 (a), (c) and convertible jx2 j2 =jx002 j2 (b), (d) pulses at the medium lengths corresponding to the conversion maxima. (a), (b): SCRAP, Z ¼ 55; (c), (d): HSCRAP, Z ¼ 4:8. All other parameters are the same as in the corresponding plots of Fig. 3.

344


Fig. 5. Change of the pulse power Wph (a) and energy eph (b)–(d) (see Eq. (20)) for the generated E3;4 (solid) and convertible E2 (dashed) fields, scaled to the corresponding values for the convertible field E2 at the entrance of the medium, vs. the medium length, (a)–(c): down-conversion process x3 ¼ 2x1 x2 (d): up-conversion process x4 ¼ 2x1 þ x2 . (The solid and dashed curves on plots (a)–(c) coincide.) (a), (b): SCRAP, (a): T ¼ 0:65, which corresponds to the conversion maximum (see Figs. 4(a) and (b)); (c), (d): HSCRAP. All other parameters are the same as in the corresponding plots of Fig. 3.

Fig. 5 shows the evolution of the number of the emitted photons along the medium which is the same for both generated and convertible fields in difference frequency FWM and opposite in sign for the sum-frequency process. Plot (a) ðWph Þ displays the power at the instance corresponding to the maximum in Fig. 4(a) (and (b)). Plots (b)–(d) ðeph Þ display the evolution of the pulse energy of the emitted radiations (see Eq. (20)) along the medium. Both are scaled to the corresponding photon numbers for the convertible field E2 at the entrance of the medium. Plots (c) and (d) are computed for the case where the pulse E2 is delayed so that it overlaps the plateau at the time dependence of the coherence jrgn j (HSCRAP) at the entrance to the medium. The simulation shows that there is no difference between the plots for energy and power in this case. On the contrary, in the case where the second pulse overlaps the timedependent part of the coherence (SCRAP), the difference between energy and power becomes significant (Fig. 5(a) and (b)) because the pulse shapes experience change along the medium (Fig. 6). In the SCRAP regime, the conversion efficiency for the total photon number per pulse approaches 60%, while for the HSCRAP regime it is about 10%. The corresponding amplification factor for convertible radiation for the difference-frequency process in the SCRAP regime is about 1.6. Because the energy of VUV photons may several times exceed that of the convertible field (which is about three times for our model), energy conversion from the convertible to the generated field may exceed

Fig. 6. Evolution the pulse shape Wph ðT Þ of the generated radiation (the same as for the convertible radiation) along the medium, with all parameters the same as in Figs. 5(a) and (b) (SCRAP).

100% (180% in our model). Due to amplification, the corresponding photon conversion efficiency from long-wavelength to VUV radiation may exceed 100% too. As seen from Fig. 5(a), the amplification factor of the convertible radiation for some instants is about 14 and corresponding power conversion efficiency makes up about 40. The figures show that the dynamics of the fields along the medium is strongly determined by the coherent nonlinear coupling, i.e., by the change of both the polarization amplitude and phase along the medium. The difference in evolution of the field


E2 in plots (c) and (d) in Fig. 5 is due to the fact that E2 is depleted during the course of the summixing process and, by contrast, enhanced in the difference-frequency-mixing process. The simulation reveals a substantial difference in the evolution, depending on such coupling parameters at the entrance to the medium like the static detuning and delays between pulses, which control the evolution of the coherence in time and along the medium. Since the pulse shape of the driving field E1 varies along the medium, consequently the populations rg , rn , as well as the coherence rgn , vary along the medium too. Therefore, the system being optimally prepared at the entrance (Fig. 3) may evolve to less than optimum along the medium (Fig. 7). One of the major effects that control FWM coupling and energy transfer from fundamental to generated radiation along the medium is the evo-

345

Fig. 7. Population rn and coherence jrgn j versus time controlled by the dynamic shift of the two-photon resonance. Solid: SCRAP, Z ¼ 80; dashed: HSCRAP, Z ¼ 40. All other parameters are the same as in the corresponding plots of Fig. 3.

lution of the phase mismatch between the nonlinear polarization and generated field, which accounts for the nonlinear phase mismatch. The latter depends on the dynamics of the level populations, which is different for SCRAP and

Fig. 8. Change of the phase mismatch between nonlinear polarization and generated radiation h along the medium (in units of p). SCRAP – the same parameters and instant as in Fig. 5(a); HSCRAP – the instant of time corresponds to the maximum of the pulse E2 . The other parameters are the same as in corresponding plots of Fig. 5.

(a)

(b)

(c)

Fig. 9. Dynamics of population and coherence at the entrance to the medium (a) and corresponding energy conversion vs. medium lengths (b) produced through p-coupling with the Gaussian pulse of the fundamental radiation E1 tuned to the static two-photon resonance, R1 ¼ 1:772; and with the Stark-field off.

346


HSCRAP regimes. Such differences in evolution of the phase are displayed in Figs. 8(a)–(d). The plots display correlation of the peaks in dh=dZ with characteristic oscillation of conversion along the medium for these two processes. Fig. 9 is computed for the fundamental pulse energy, which corresponds to full population transfer by the end of the fundamental pulse in the absence of the control Stark radiation. As seen from the comparison with Fig. 5(b), this less robust regime provides about 10 times less conversion efficiency.

4. Conclusions We have shown and discussed the feasibility of enhanced frequency conversion efficiency in a medium prepared by full or half Stark-chirped rapid adiabatic passage. While the persistent maximum coherence initially being prepared at the entrance of the medium through the half-SCRAP (HSCRAP) process may disappear by the end of the medium, the transient maximum coherence may be provided all along the interaction length. Therefore, in the latter case, the generated field may grow over a longer distance, and the power conversion efficiency may become larger as well. On the other hand, the related distortion in the pulse shape may result in considerable differences between the power and pulse energy conversion efficiencies. Although the optimizations for different instants are different, it is shown that, subject to multi-parameter optimization, the number of generated VUV photons at maximum coherence may become compared to those for the convertible long-wavelength radiation at the medium entrance due to their amplification inherent to a differencefrequency scheme and effective FWM coupling at maximum atomic coherence. It must be recognized that the evolution of the coherence will be influenced by the decay of excited state amplitudes through accompanying (and inevitable) photoionization processes and by dephasing due to impacts with the resulting photoelectrons. Photoionization can be controlled by a careful choice of laser intensities, and the choice of appropriate atomic systems and electron impact

dephasing can be minimized by working with low atom densities while maintaining a sufficient sample length to ensure good conversion.

Acknowledgements This work was supported by the European Union Research Training Network, contract number HPRN-CT-1999-00129, by the Deutsche Forschungsgemein schaft and by the German– Israeli Foundation (1-644-118.5/1999). AKP and SAM acknowledge support by EU INTAS (project 99-00019) and Russian Foundation for Basic Research (project 02-02-16325a). The authors are grateful to K. Bergmann for valuable discussions and encouragement over the course of this work. We would like to thank T. Rickes and E. Korsunsky, University of Kaiserslautern, for valuable discussion on the paper, as well as L. P. Yatsenko, Ukrainian Academy of Sciences, Kiev for comments on [32], prior to publication.

References [1] R. Wallenstein, Laser Optoelektr. 3 (1982) 29. [2] V.G. Arkhipkin, A.K. Popov, Soviet Physics: Uspekhi 30 (1987) 952 [Translated from Uspekhi Fizicheskikh Nauk 153 (1987) 423]. This is a shortened version of the book in Russian by V. G. Arkhipkin and A. K. Popov, Nonlinear Light Conversion in Gases (Siberian Branch of Nauka Press, Novosibirsk, 1987), 142 pages. [3] A.K. Popov, S.G. Rautian, in: A.V. Andreev, O.Kocharovskaya, P. Mandel (Eds.), Coherent Phenomena and Amplification without Inversion, SPIE Proceedings, vol. 2798, 1996, p. 49. [4] A.K. Popov, Bull. Russ. Acad. Sci. (Physics) 60 (1996) 927 [Translated by Allerton Press, NY, from Izvestiya RAN, serya Fizika 60 (1996) 99] (http://xxx.lanl.gov/abs/quantph/0005108). [5] A.K. Popov, SPIE Proc. 3485 (1998) 252. [6] M.S. Feld, A. Javan, Phys. Rev. 177 (1969) 540. [7] Th. Hansch, R. Keil, A. Schabert, Ch. Schmeltzer, P. Toschek, Z. Phys. 226 (1969) 293. [8] Th. Hansch, P. Toschek, Z. Phys. 236 (1970) 213. [9] T.Ya. Popova, A.K. Popov, S.G. Rautian, R.I. Sokolovskii, Zh. Eksp. Teor. Fiz. 57 (1969) 850 [JETP 30 (1970) 466] (http://xxx.lanl.gov/abs/quant-ph/0005094). [10] T.Ya. Popova, A.K. Popov, Zhurn. Prikl. Spektrosk. 12 (1970) 989 [Transl. in Engl.: J. Appl. Spectr. 12 (1970) 734] (http://xxx.lanl.gov/abs/quant-ph/0005047).

S.A. Myslivets et al. / Optics Communications 209 (2002) 335–347 [11] T.Ya. Popova, A.K. Popov, Izv. Vysh. Uchebn. Zaved. Fizika 11 (1970) 38 [Transl. in Engl.: Soviet Phys. J. 13 (1970) 1435] (http://xxx.lanl.gov/abs/quant-ph/0005049). [12] I.M. Beterov, Cand. Sci. Dissertation, Institute of Semiconductor Physics, Novosibirsk, December 1970. [13] S.E. Harris, J.E. Field, A. Imamoglu, Phys. Rev. Lett. 64 (1990) 1107. [14] J.P. Marangos, J. Mod. Opt. 45 (1998) 471. [15] E. Arimondo, Prog. Opt. 35 (1996) 259. [16] G.Z. Zhang, K. Hakuta, B.P. Stoicheff, Phys. Rev. Lett. 71 (1993) 3099. [17] C. Dorman, I. Kucukkara, J.P. Marangos, Phys. Rev. A 61 (2000) 013802. [18] C. Dorman, I. Kucukkara, J.P. Marangos, Opt. Commun. 180 (2000) 263. [19] P. Hemmer, D. Katz, J. Donoghue, M. Cronin-Golomb, M. Shahriar, P. Kumar, Opt. Lett. 20 (1995) 982. [20] M. Jain, H. Xia, G.Y. Yin, A.J. Merriam, S.E. Harris, Phys. Rev. Lett. 77 (1996) 4326. [21] S.E. Harris, G.Y. Yin, M. Jain, H. Xia, A.J. Merriam, Philos. Trans. Roy. Soc. London (Math. Phys. Eng. Series) 355 (1997) 2291.

347

[22] M.D. Lukin, P.R. Hemmer, M. L€ offler, M.O. Scully, Phys. Rev. Lett. 81 (1998) 2675. [23] K. Hakuta, M. Suzuki, M. Katsuragawa, J.Z. Li, Phys. Rev. Lett. 79 (1997) 209. [24] A. Merriam, S.J. Sharpe, H. Xia, D. Manuszak, G.Y. Yin, S.E. Harris, Opt. Lett. 24 (1999) 625. [25] N. Vitanov, T. Halfmann, B.W. Shore, K. Bergmann, Annu. Rev. Phys. Chem. 52 (2001) 763. [26] L.P. Yatsenko, S. Guerin, T. Halfmann, K. B€ ohmer, B.W. Shore, K. Bergmann, Phys. Rev. A 58 (1998) 4683. [27] S. Guerin, L.P. Yatsenko, T. Halfmann, B.W. Shore, K. Bergmann, Phys. Rev. A 58 (1998) 4691. [28] K. B€ ohmer, T. Halfmann, L.P. Yatsenko, B.W. Shore, K. Bergmann, Phys. Rev. A 64 (2001) 02340. [29] T. Rickes, L.P. Yatsenko, S. Steuerwald, T. Halfmann, B.W. Shore, N.V. Vitanov, K. Bergmann, J. Chem. Phys. 113 (2000) 534. [30] M.M.T. Loy, Phys. Rev. Lett. 36 (1976) 1454. [31] M.M.T. Loy, Phys. Rev. Lett. 41 (1978) 473. [32] L.P. Yatsenko, N.V. Vitanov, B.W. Shore, T. Rickes, K. Bergmann, Opt. Commun. 204 (2002) 413.