Extreme Ultraviolet Sources Generation by Using the Two-Color Multi-Cycle Weak Inhomogeneous Field

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1 Commun. Theor. Phys. 63 (2015) Vol. 63, No. 1, January 1, 2015 Extreme Ultraviolet Sources Generation by Using the Two-Color Multi-Cycle Weak Inhomogeneous Field FENG Li-Qiang ( ) 1,2,3, and LI Wen-Liang ( ) 2,3 1 College of Science, Liaoning University of Technology, Jinzhou , China 2 State Key Laboratory of Molecular Reaction Dynamics, Dalian Institute of Chemical Physics Chinese Academy of Sciences, Dalian , China 3 Xinjiang Institute of Engineering, Urumqi , China (Received December 23, 2013; revised manuscript received May 4, 2014) Abstract An efficient method for attosecond extreme ultraviolet source generation under the two-color multi-cycle weak pulse has been theoretically presented by using the concept of the plasmonic field enhancement in the vicinity of metallic nanostructures. The results show that by properly choosing the inhomogeneity of the two-color multi-cycle (20 fs) weak pulse (10 13 W/cm 2 ), not only the harmonic cutoff has been extended, resulting in a broadband XUV continuum, but also the single short quantum path has been selected to contribute to the harmonic. As a result, two isolated XUV pulses with durations of 68as and 66as can be obtained. PACS numbers: Ky, Rm, Re Key words: high-order harmonic generation, attosecond pulse, multi-cycle pulse, plasmonic field enhancement 1 Introduction The coherent XUV sources are high demand due to its important ingredients for physical, material and chemical science [1 3] etc. High-order harmonic generation (HHG) as one of the most attractive method for producing isolated XUV radiation has been widely investigated in the past two decades. [4 9] The physical picture of HHG can be well explained by the classical three-step model: [10] ionization, acceleration, and recombination. During the recombination, a maximum harmonic cutoff with E max = I p U p can be obtained, where I p is the ionization potential and U p = I/4ω 2 is the ponderomotive energy of the free electron in the laser field. Fig. 1 Schematic illustration of linear electric field enhancement and harmonic emission using a nanostructure of bow-tie elements. In the past several years, the studies of HHG mostly focus on supercontinuum extension, due to a broader supercontinuum will support the shorter attosecond pulses generation, for instance, using the few-cycle driving laser, [11] the polarization gating technique, [12] the multi-color field control, [13] the chirp control, [14] and the multi-cycle driving pulse [15 16] etc. Recently, an alternative technique, named plasmonic nanostructures assisted harmonic enhancement has attracted a lot of interest. [17 20] In this scheme, it is not necessary to utilize extra cavities to amplify the input pulse power, and the local electric fields can be enhanced by more than 20 db. [21 22] The underlying mechanism of the plasmonic field enhancement can be described by the two-step model, as shown in Fig. 1: first, when a low intensity input pulse couples to the nanostructure plasmon mode. Due to the collective oscillation effect of free charges, the negative charges are redistributed around one apex and the positive charges around the other one, resulting in a large resonant enhancement of the local field. Secondly, by injecting rare gases into this enhanced field, the HHG can be produced or extended. However, due to surface plasmon resonance, the enhanced laser is spatial dependence, which is called spatially inhomogeneous field. For instance, (i) from the experimental side, Kim et al. [17] showed that by using this plasmonic nanostructures assisted scheme, the laser field has been enhanced by Supported by the Scientific Research Fund of Liaoning University of Technology of China under Grant No. X and the Scientific Research Fund of Liaoning Provincial Education Department under Grant No. L Corresponding author, lqfeng lngy@126.com c 2015 Chinese Physical Society and IOP Publishing Ltd

2 No. 1 Communications in Theoretical Physics 87 three orders of magnitude compared with the input pulse, and the XUV pulses from the 7th (114 nm) to the 17th (47 nm) can be obtained. However, the outcome of Kim et al. has been put under controversy more recently, [18 19] i.e. whether the harmonic emission is in fact the coherent (HHG) or only an incoherent atomic line emission. (ii) From the theoretical side, Luo et al. [23] obtained a 27as XUV pulse by using the 10 fs/2000 nm inhomogeneous field with I = W/cm 2. Yavuz et al. [24] obtained a 130as pulse by using the 4 fs/800 nm inhomogeneous field with I = W/cm 2. However, the above schemes strongly rely on the few-cycle pulse control, which is also a technical difficulty for experimental realization. Thus, in this paper, a promising approach based on the combination of the two-color multi-cycle weak pulse and the plasmonic field harmonic enhancement scheme has been theoretically investigated. It shows that by properly choosing the inhomogeneity of the two-color multi-cycle weak field, a smooth XUV continuum with a 124 ev bandwidth can be obtained, further resulting in two isolated sub-70as XUV pulses. 2 Theoretical Method The interaction of He atom with an inhomogeneous laser field can be investigated by solving the timedependent Schrödinger equation (TDSE) in the length gauge [25 26] ϕ(x, t) i = H(t)ϕ(x, t), t ϕ(x, t) [ i = 1 2 ] t 2 x 2 + V (x) + xe(x, t) ϕ(x, t), (1) where V (x) = 1.0/ x is the soft Coulomb potential of the He atom. Due to the spatial dependence of the laser field, the synthesized laser field can be expressed as, E(x, t) = f 0 (t)e 0 (1 + g(x)) cos(ω 0 t) + f 1 (t)e 1 (1 + g(x)) cos(ω 1 t). (2) Here E i, and ω i (i = 0, 1) are the amplitude and the frequency of the 800 nm fundamental pulse and the 1200 nm controlling pulse. g(x) represents the functional form of the inhomogeneous electric field and it can be written as a power series of the form g(x) = N i=1 β ix i. The coefficients β i determine the inhomogeneities of the fields, which can be obtained by fitting the actual electric field that results from an FE simulation considering the real geometry of different nanostructures. [27] For the plasmonic present enhanced fields, to a first approximation one can employ g(x) = βx, and this is a good approximation as far as the strong field physics is concerned since the excursion of the electron is very small. [24,28 29] Besides, the computationally demanding task for the higher nonlinearity enhancement is beyond the scope of this paper. The envelope function can be expressed as, f i (t) = exp[ 4 ln(2)t 2 /τ 2 i ], (3) where τ i (i = 0, 1) is the pulse durations of the two pulses. Equation (1) can be solved by using the standard second-order split-operator method. [30 32] The generated harmonics can be obtained by Fourier transforming the time-dependent dipole acceleration a(t), S(ω 0 ) exp( iω 0 t)a(t)dt 2, (4) where a(t) can be expressed by the Ehrenfest theorem, [33] a(t) = d2 x dt 2 = ϕ(x, t) [H(t), [H(t), x]] ϕ(x, t) = ϕ(x, t) (V (x) + xe(x, t)) ϕ(x, t) V (x) (xe(x, t)) = ϕ(x, t) + ϕ(x, t). (5) x x Finally, the XUV pulses can be obtained by harmonic superposing as follows, I(t) = a q e iqω0t 2, (6) where a q = a(t) e iqω0t dt. 3 Results and Discussion q Fig. 2 (a) HHG spectra driven by the two-color homogeneous field (β = 0.0) and inhomogeneous field with β = 0.002, 0.004, and The laser parameters are chosen to be 20 fs/800 nm, I 0 = W/cm 2 and 20 fs/1200 nm, I 1 = W/cm 2. (b) HHG spectra driven by the above two-color inhomogeneous field with β = and the single-color 20 fs/800 nm and 5 fs/800 nm homogeneous fields with I = W/cm 2.

3 88 Communications in Theoretical Physics Figure 2(a) shows the HHG spectra driven by twocolor multi-cycle weak field with different inhomogeneous parameters. The laser field is chosen to be 20 fs/800 nm, I0 = W/cm2 and 20 fs/1200 nm, I1 = W/cm2. For comparison, the harmonic spectrum of the two-color homogeneous field case (β = 0.0) has also been shown. Clearly, with the increasing of the inhomogeneous parameter β, the harmonic cutoff has been remarkably enhanced. However, for the larger inhomogeneous parameter (i.e. β = 0.005), although the harmonic cutoff has a remarkable extension, the large modulation on the harmonic cutoff is failed to the generation of the isolated attosecond pulse. Thus, through our calculations, β = (which corresponds to an inhomogeneous region of nm) is the optimal inhomogeneous parameter for the harmonic extension and a smooth 124 ev bandwidth has been obtained. According to the three step model,[10] it is possible to approximately extract the enhanced pulse intensity using Vol. 63 a given harmonic cutoff, namely 4ω 2 (Emax Ip ). (7) 3.17 For example, for the optimal inhomogeneous parameter case (β = 0.004), the maximum harmonic cutoff is Emax = 140ω0, and if the ω in Eq. (7) is chosen to be single ω0 (800 nm), then the enhanced pulse intensity equals to I 0 = W/cm2, which is one order of magnitude higher than the two-color input field. Now, to provide the advantages for using the plasmon field enhancement scheme, in Fig. 2(b), we present the harmonic spectra of the above two-color field with β = case and the single 800 nm/20 fs (800 nm/5 fs), I 0 = W/cm2 with β = 0.0 cases. Clearly, the harmonic cutoff of the two-color weak inhomogeneous field case is as large as the other two single-color intense homogeneous field cases. However, due to the inhomogeneous effect, the harmonic modulation on the plateau region of the inhomogeneous field case is much smaller than that of the two homogeneous field cases, which is beneficial to isolated XUV source generation. I= Fig. 3 The time-frequency distributions of the HHG spectra for the cases of (a) the above two-color homogeneous field; (b) the above inhomogeneous field with β = 0.004; (c) the single-color 20 fs/800 nm homogeneous field with I 0 = W/cm2 ; (d) the single-color 5 fs/800 nm homogeneous field with I 0 = W/cm2. emission process, in Fig. 3, we present the time-frequency wavelet transformation of the dipole acceleration a(t),[34] Z A(t, ω0 ) = a(t0 ) ω0 W (ω0 (t0 t))dt0, (8) distributions of harmonic spectra, obtained by using the where W (ω0 (t0 t)) is the Morlet wavelet with the formula To better understand the harmonic extension and

4 No. 1 Communications in Theoretical Physics 89 of W (ξ) = ( 1 α ) e iξ e ξ2 /2α 2, (9) the isolated XUV sources. Moreover, the multi-cycle weak pulses are much better for experimental realization. and α is chosen to be 30 in our calculations. Clearly, for the two-color homogeneous field case (Fig. 3(a)), there are three main energy peaks in the amplitude region for the present field and the maximum value of the highest peak is very close to that of the second highest peak, thus resulting in a shorter supercontinuum near the cutoff, which can be clearly observed in Fig. 2(a) solid black line. Moreover, each peak receives two similar contributions, named the short and the long quantum paths, [35 36] which is responsible for the larger harmonic modulation. However, with the introduction of the inhomogeneous parameter, as shown in Fig. 3(b) for the case of two-color inhomogeneous field with β = 0.004, not only the maximum value of the highest peak has been remarkably extended as illuminated in Fig. 2(a), but also the single short quantum path has been well selected to contribute to the harmonic, which is responsible for the small modulation on the harmonic plateau and will be favorite to support an isolated XUV pulse generation. Figures 3(c) and 3(d) show the harmonic time-frequency for the cases of the single 20 fs/800 nm and 5 fs/800 nm homogeneous fields with I = W/cm 2. We see that although the maximum harmonic cutoff is as large as the above optimal harmonic extension case (two-color with β = 0.004), the shorter supercontinuum in the cutoff region (for the 20 fs/800 nm case) and the same contributions from the two quantum paths (for the 5 fs/800 nm case) are both unbeneficial to isolated attosecond source generation. Thus, we see that the plasmon field enhancement scheme provides much more advantages (multi-cycle pulses with low intensities) on the harmonic emission compared with using the homogeneous field. Figure 4 shows the temporal profiles of XUV pulses. In particular, by respectively superposing the optimal harmonics (two-color multi-cycle weak pulse with β = 0.004) from the 60th to the 100th orders and from the 100th to the 140th orders, two isolated attosecond XUV pulses with durations of 68as and 66as can be directly obtained, as shown in Fig. 4(a). For comparison, the generations of the XUV pulses from the few-cycle intense pulse case (single 5 fs/800 nm with I = W/cm 2 ) has also been shown. Particularly, by superposing its harmonics from the 105th to the 140th orders, two attosecond pulse trains with durations of 86as can be obtained, as shown in Fig. 4(b). However, from a practical viewpoint, an isolated XUV pulse is more favorable for the timeresolved dynamics measurement. Thus, the plasmon field enhancement scheme combined with the two-color multicycle weak pulse is more favorable for the generations of Fig. 4 (Color online) Temporal profiles of the XUV pulses. The superposed harmonics are chosen (a) from the 60th to the 100th orders (solid black line) and from the 100th to the 140th orders (solid red line) of the above two-color inhomogeneous field with β = case; (b) from the 105th to the 140th orders of the single-color 5 fs/800 nm homogeneous field with I = W/cm 2 case. 4 Conclusion In conclusion, we have theoretically investigated the harmonic extension and the attosecond XUV pulse generation by using the two-color multi-cycle weak field. The results show that by using the plasmonic nanostructures assisted harmonic enhancement, the harmonic cutoff has been remarkably enhanced under the two-color multi-cycle weak inhomogeneous field, resulting in a 124 ev XUV supercontinuum with single short quantum path contribution. Further, by properly superposing the harmonics, two isolated XUV pulses with durations of 68as and 66as have been obtained. It notes that the synthesized twocolor multi-cycle weak field we have chosen here can be easy available in many laboratories, that is, 20 fs/800 nm, I 0 = W/cm 2 combined with 20 fs/1200 nm, I 1 = W/cm 2. Thus, we hope that the above scheme can be experimentally realized in future and has implications for the real applications of isolated attosecond XUV pulses. Acknowledgments The authors thank Professor Keli Han for providing us the computational code used in the present work.

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