We present a theory of high-order harmonic generation by arbitrary polyatomic molecules based on the molecular strong-field approximation (MSFA) in the framework of the S-matrix theory. A polyatomic molecule is modeled by an (N + 1)-particle system, which consists of N heavy atomic (ionic) centers and an electron. We derived various versions (with or without the dressing of the initial and/or final molecular state) of the MSFA. The general expression for the T-matrix element takes a simple form for neutral polyatomic molecules. We show the existence of the interference minima in the harmonic spectrum and explain these minima as a multiple-slit type of interference. This is illustrated by numerical examples for the nitrous oxide (N2O) molecule exposed to strong linearly polarized laser field

Hasović, E.

Milošević, D.B.

Odžak, S.

English

Repositorium für Naturwissenschaften und Technik  (TIB Hannover)

Journal of Physics: Conference SeriesPAPER • OPEN ACCESSHigh-order harmonic generation by polyatomicmoleculesTo cite this article: S Odžak et al 2017 J. Phys.: Conf. Ser. 826 012007 View the article online for updates and enhancements.You may also likeTrilobites, butterflies, and other exoticspecimens of long-range RydbergmoleculesMatthew T Eiles-The scientific career of V S Letokhov (10November 1939–21 March 2009)Victor I Balykin-Polyatomic molecules as quantum sensorsfor fundamental physicsNicholas R Hutzler-This content was downloaded from IP address 194.95.157.29 on 01/06/2023 at 07:44High-order harmonic generation by polyatomicmoleculesS Odžak1, E Hasović1 and D B Milošević1,2,31 Faculty of Science, University of Sarajevo, Zmaja od Bosne 35, 71000 Sarajevo, Bosnia andHerzegovina2 Max-Born Institut, Max-Born Strasse 2a, 12489 Berlin, Germany3 Academy of Sciences and Arts of Bosnia and Herzegovina, Bistrik 7, 71000 Sarajevo, Bosniaand HerzegovinaE-mail: senad.odzak@pmf.unsa.baAbstract. We present a theory of high-order harmonic generation by arbitrary polyatomicmolecules based on the molecular strong-field approximation (MSFA) in the framework of the S-matrix theory. A polyatomic molecule is modeled by an (N + 1)-particle system, which consistsof N heavy atomic (ionic) centers and an electron. We derived various versions (with or withoutthe dressing of the initial and/or final molecular state) of the MSFA. The general expressionfor the T -matrix element takes a simple form for neutral polyatomic molecules. We show theexistence of the interference minima in the harmonic spectrum and explain these minima as amultiple-slit type of interference. This is illustrated by numerical examples for the nitrous oxide(N2O) molecule exposed to strong linearly polarized laser field.1. IntroductionWith the advent of high-power lasers, many interesting nonlinear phenomena have been observedwhen exposing atoms and molecules to strong laser fields (see review articles [1, 2] and referencestherein). High-order harmonic generation (HHG) is a unique highly nonlinear process which hasattracted attention of experimental and theoretical physicists over the past three decades. Asa source of coherent extreme ultraviolet and soft x-ray radiation, this process has importantapplications in time-resolved spectroscopy and attoscience [3, 4, 5, 6, 7]. HHG may be describedby the so-called three-step model [8]. In the first step, the considered molecular target absorbsmore photons from the laser field than is necessary for ionization. Influenced by the laser field,the released electron may return to the parent molecular ion (the second step). In the thirdstep, the electron recombines with the parent ion and one high-energy photon is emitted.HHG carries information about the atomic or molecular targets, like internuclear distanceof diatomic molecules or symmetry and the shape of the highest occupied molecular orbital(HOMO) of simple linear molecules [9, 10]. One powerful method to probe the structure ofdiatomic molecular targets is to analyze characteristic minima in the HHG spectra caused bythe two-center interference of returning electron wave packet. The interference minima are firsttheoretically investigated for the simplest diatomic molecules H +2 and H2 [11, 12, 13]. Theexperimental observation of the minima was first achieved for aligned CO2 molecules [14, 15].While the position of the minima can be determined by a simple interference formula for diatomicmolecules, the case of polyatomic molecules is considerably more complex. In [16, 17] a general25th Annual International Laser Physics Workshop (LPHYS'16)                                                       IOP PublishingIOP Conf. Series: Journal of Physics: Conf. Series 826 (2017) 012007          doi:10.1088/1742-6596/826/1/012007Internation l Conference on R cent Trends in P ysics 2016 (ICRTP2016) I ublishingJournal o Physics: Co ference Series 755 (2016) 011001 i: . / /755/ / 1 1Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distributionof this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.Published under licence by IOP Publishing Ltd 1interference minima condition, which is valid for arbitrary homonuclear diatomic molecules, isderived and tested.The HHG process for polyatomic molecules was analyzed experimentally in [18, 19, 20, 21,22, 23, 24, 25, 26]. From a theoretical point of view, numerical solution of the time-dependentSchrödinger equation is not practical for polyatomic molecules. Therefore, most calculations ofHHG spectra for polyatomic molecules are performed using approximate methods like strong-field approximation [27] or quantitative rescattering theory [24]. In this paper we present themolecular strong-field approximation (MSFA) and apply it to calculate and analyze HHG spectraof nitrous oxide molecule (N2O).2. Theory of molecular HHGThe MSFA theory is based on our previous work [27]. A polyatomic molecule is considered as asystem of N atomic (ionic) centers and an electron which can be freed under the action of thelaser field with the electric field vector E(t) in the molecular HHG process. The Hamiltonian ofthis system, in the dipole approximation and the length gauge, can be written in the formH =N∑k=1p2k2mk+p2e2me+ V (re, r1, . . . , rN )−E(t) ·(ere +N∑k=1ekrk), (1)where pk, rk, mk, and ek, are, respectively, the momenta, coordinates, masses, and charges ofatoms (ions), while pe, re, me, and e are the momentum, coordinate, mass, and charge of theelectron (me = 1 and e = −1 in the atomic units). It is convenient to introduce the Jacobicoordinates and to replace the system of N + 1 coordinates rk (k = 1, . . . , N), re, with thesystem of N relative coordinates Rk, k = 1, . . . , N , and the center-of-mass coordinate Rc.m.. Inthe new coordinates, the Hamiltonian of our system can be written in the formH =P2c.m.2M+N∑k=1[P2k2µk+ Vk(re − rk)]+ VC({R})−E(t) ·(ẽc.m.Rc.m. +N∑k=1ẽkRk), (2)where the laser-free interaction V (re, r1, . . . , rN ) is written as the sum of all electron-kthmolecular center potentials Vk(re − rk) and the potential energy of these centers VC({R}). Theset of (N−1) relative coordinates of atomic (ionic) centers is denoted by {R} ≡ (R1, . . . ,RN−1).The reduced masses are determined by the following expressionsµj =mj+1MjMj+1= λjmj+1 = κj+1Mj , j = 1, . . . , N, (3)where Mj =j∑k=1mk, λj = Mj/Mj+1 and κj = mj/Mj while Pc.m. =∑N+1k=1 pk andẽc.m. =∑N+1k=1 ek are the momentum and charge of the center of mass (we introduced the notationpN+1 = pe and eN+1 = e). The relative momenta and charges are given by (j = 1, . . . , N)Pj = λjpj+1 − κj+1j∑k=1pk, ẽj = λjej+1 − κj+1j∑k=1ek. (4)For a T = 2π/ω–periodic laser field the rate of emission of a harmonic photon having thefrequency ωK = nω+Eei({R})−Eef({R}), wave vector K, and the polarization êK, into a solidangle dΩK̂, is given by [in atomic units (au)] [28]wq′qn =12π(nωc)3 ∣∣∣T q′qn ∣∣∣2 , (5)25th Annual International Laser Physics Workshop (LPHYS'16)                                                       IOP PublishingIOP Conf. Series: Journal of Physics: Conf. Series 826 (2017) 012007          doi:10.1088/1742-6596/826/1/0120072where the T -matrix element isTqq′n =T∫0dtTeinωtdqq′fi (t) = Tqq′n êK =∑ξ=x,y,zT qq′nξ êξ. (6)The time-dependent dipole matrix element dqq′fi (t) can be calculated within the strong-fieldapproximation, fixed-nuclei approximation and solving the integral over the intermediateelectron momentum k by the saddle-point method, as it was done in [27], with the final resultdqq′fi (t) = −i(2πi)3/2 ∞∫0dττ3/2ei[Skst (t′)−Skst (t)+Eeiτ ]×N∑l=1∑aclaηq(ρl, t)〈ψa|ẽNr + ρCl |kst − ẽNA(t)〉 (7)×N∑j=1∑a′cja′η∗q′(ρj , t′)〈kst − ẽNA(t′)|(ẽNr + ρCj ) ·E(t′)|ψa′〉.Integration in the previous equation is performed over the travel time τ = t − t′, while thequasiclassical action is given by Sk(t) =12∫ t−∞ dt′[k − ẽNA(t′)]2 and is calculated at thestationary momentum kst = ẽN∫ tt−τ A(t′)dt′/τ where A(t) = −∫ t−∞E(t′)dt′. The ionization(recombination) matrix element is obtained by expanding the molecular orbitals as a linearcombination of atomic orbitals |ψa〉 with expansion coefficients cja′ (cla). Furthermore we writeρCl = ẽNρl +∑N−1j=1 ẽjlRj , ρk({R}) = λk−1Rk−1 −∑N−1j=k κj+1Rj and we use dressing of theinitial wave function with factors ηd(ρj , t) = exp{ikst ·ρj}, ηu(ρj , t) = exp{i [kst − ẽNA(t)] ·ρj}.The index l on ẽj means that the charge distribution is such that the electron is ionized from(recombined with) the center l. For neutral polyatomic molecules we have ẽN = −1 and ρCl = 0,l = 1, . . . , N , so that the expression for the time-dependent dipole simplifies.3. Numerical resultsNow we are going to demonstrate the application of the presented theory using a simple example.We consider the HHG process of the nitrous oxide (N2O) molecule. Nitrous oxide, commonlyknown as laughing gas, is a colorless, odorless non-flammable gas. The N2O molecule is alinear molecule with N-N (N-O) internuclear distances of 1.128Å(1.184Å) at the equilibriumgeometry. Structure of the N2O molecule belongs to the C∞v point group and its doubledegenerate HOMO, shown in Fig. 1, has the π symmetry. Vertical ionization energy of theN2O molecule is Ip = 12.89 eV.Now, we present the calculated harmonic emission rate of the HHG process in a linearlypolarized laser field, defined as E(t) = E0 sin(ωt)êL and having the intensity 4 × 1014 W/cm2and the wavelength 800 nm. The molecule is oriented along z axis and êL ≡ (θL, ϕL) inspherical coordinates. We fix the laser field in the xz plane (ϕL = 0) and change the polarangle θL ∈ [0◦, 180◦]. In the upper panel of Fig. 2 we show in false colors the logarithm of theharmonic emission rate, Eq. (5), as a function of the angle θL and the harmonic order n.In the remaining panels we show the rates obtained using only the x (middle panel) andonly the z (bottom panel) component of the T -matrix element, Eq. (6). One can see that thecontributions of both components are comparable.The HHG spectra exhibit a pronounced minimum for the angle θL = 90◦. This minimumis caused by the nodal plane of the N2O molecule. Furthermore, there are minima for the25th Annual International Laser Physics Workshop (LPHYS'16)                                                       IOP PublishingIOP Conf. Series: Journal of Physics: Conf. Series 826 (2017) 012007          doi:10.1088/1742-6596/826/1/0120073Figure 1. Coordinate-space wave function for the HOMOs of N2O molecule.angles of θL ≈ 45◦ and θL ≈ 135◦. These type of minima can be related to the interference ofthree contributions of the recombination matrix element, i.e. the interference of the recombiningelectronic wave packets. The three-point interference can be explained by the presented MSFAtheory. It is possible to extract an interference factor [27] which becomes small for some specificvalues of n, θL, and ϕL.4. ConclusionsWe have used a strong-field-approximation-based theory of high-order harmonic generationby polyatomic molecules to calculate the harmonic emission rates of the nitrous oxide N2Omolecules. In our model, a polyatomic molecule is considered as a system of N + 1 particles: Natomic (ionic) centers and one valence electron, emitted by strong-field ionization. In the Born-Oppenheimer and the fixed-nuclei approximation we derived an expression for the harmonicemission T -matrix element. Our theory is formulated in such a way that the strong-fieldapproximation with the laser-field dressed or undressed initial and final states can be treated onthe same footing.We have shown that the harmonic emission rate of the (N2O) molecules exhibits minima forparticular values of the orientation angle θL. The appearance of these minima is related to theposition of the nodal planes of the molecular HOMO as well as to the destructive interferenceof the recombining electronic wave packets. In general, for neutral polyatomic molecules the T -matrix element can be simplified and its recombination part can be extracted as a factor. Thisfactor is responsible for a multicenter interference of the recombining electronic wave packets.AcknowledgmentsWe gratefully acknowledge support by the Federal Ministry of Education and Science, Bosniaand Herzegovina.25th Annual International Laser Physics Workshop (LPHYS'16)                                                       IOP PublishingIOP Conf. Series: Journal of Physics: Conf. Series 826 (2017) 012007          doi:10.1088/1742-6596/826/1/0120074Figure 2. Logarithm of the HHG rate of the N2O molecule presented in false colors as afunction of the orientation angle θL and the harmonic order n. The results are obtained usingboth (top panel), only the x (middle panel), and only the z (bottom panel) component of theT -matrix element. Laser field is linearly polarized having the intensity 4 × 1014 W/cm2 andthe wavelength 800 nm. The initial state is dressed (q′ = d), while the final state is undressed(q = u).25th Annual International Laser Physics Workshop (LPHYS'16)                                                       IOP PublishingIOP Conf. Series: Journal of Physics: Conf. Series 826 (2017) 012007          doi:10.1088/1742-6596/826/1/0120075References[1] Becker W, Grasbon F, Kopold F, Milošević D B, Paulus G G and Walther H 2002 Adv. At. Mol. Opt. Phys.48 35[2] Winterfeldt C, Spielmann C and Gerber G 2008 Rev. Mod. Phys. 80 117[3] Agostini P and DiMauro L F 2004 Rep. Prog. Phys. 67 813[4] Krausz F and Ivanov M 2009 Rev. Mod. Phys. 81 163[5] Gaarde M B, Tate J L and Schafer K J 2008 J. Phys. B 41 132001[6] Midorikawa K, Nabekawa Y and Suda A 2008 Prog. Quant. 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High-order harmonic generation by polyatomic molecules

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High-order harmonic generation by polyatomic molecules

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