Blueshift of high-order harmonic generation in crystalline silicon subjected to intense femtosecond near-infrared laser pulse
aa r X i v : . [ phy s i c s . a t o m - ph ] J u l Blueshift of high-order harmonic generation incrystalline silicon sub jected to intense femtosecondnear-infrared laser pulse
Boyan Obreshkov , Tzveta Apostolova , Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences,Tsarigradsko chausse 72, 1784 Sofia, Bulgaria Institute for Advanced Physical Studies, New Bulgarian University, 1618 Sofia, BulgariaE-mail:
Abstract.
We present the generation of high order harmonics in crystalline silicon subjected to intensenear-infrared 30fs laser pulse. The harmonic spectrum extends from the near infrared to theextreme ultraviolet spectral region. Depending on the pulsed laser intensity, we distinguishtwo regimes of harmonic generation: (i) perturbative regime: electron-hole pairs born duringeach half-cycle of the laser pulse via multiphoton and tunnel transitions are accelerated inthe laser electric field and gain kinetic energy; the electron-hole pairs then recombine in theground state by emitting a single high-energy photon. The resultant high harmonic spectrumconsists of sharp peaks at odd harmonic orders. (ii) non-perturbative regime: the intensity ofthe harmonics increases, their spectral width broadens and the position of harmonics shifts toshorter wavelengths. The blueshift of high harmonics in silicon are independent on the harmonicorder which may be helpful in the design of continuously tunable XUV sources.
1. Introduction
High harmonic generation (HHG) is a nonlinear optical process which occurs when a target(atomic gas, plasma, amorphous or crystalline solid) is irradiated by an intense laser beam.Subsequently the photoexcited state emits radiation in the form of high harmonics of the drivinglaser frequency. HHG in atomic gases is usually interpreted within a semi-classical three-stepmodel [1, 2] including photoionization, acceleration of the freed electron by the laser electricfield and subsequent radiative re-combination of the electron with its parent ion resulting inemission of single high-energy photon. A typical HHG spectrum in the gas-phase consists ofodd order harmonics with a plateau region and sharp cutoff at ¯ hω max = I p + 3 . U p , where I p is the atomic ionization potential and U p = e E / mω L is the ponderomotive energy; here e isthe electronic charge, m is the electron mass, E is the peak electric field strength and ω L is thedriving laser frequency.More recently HHG has been observed in bulk ZnO crystals subjected to intense mid-infraredlaser pulses [3]. The measured HHG spectrum displays features similar to the gas phase withhigh harmonic orders extending into the UV region. In contrast to gas phase harmonics, thecutoff energy for solid-state HHG was found to scale linearly with the peak field strength. Alsothe measured yield of high harmonics differs substantially from the prediction of perturbativeonlinear optics. These observations suggest that strong-field driven electron dynamics in solidsis qualitatively different from the prediction of the semi-classical three-step model applicable tothe gas phase. Subsequently, HHG was demonstrated in wide-band gap solids [4, 5, 6, 7] and ingraphene [8]. On this basis solid-state HHG has promising potential in development of compactultrafast and coherent XUV sources (cf. Ref. [9]).In this paper we present numerical results of HHG in bulk silicon driven by linearly polarized30fs laser pulse of 800 nm wavelength: Sec. 2 presents the time evolution of the electric field ofthe transmitted pulse and the characteristics of HHG spectra in crystalline silicon as a functionof the peak laser intensity, Sec. 3 includes our main conclusion.
2. Numerical results and discussion
Details of the theoretical approach and methodology can be found in Refs. [10, 11, 12]. In brief,we solve numerically the time-dependent Schr¨odinger equation for valence electrons subjectedto a linearly polarized intense ultrashort laser pulse with near-infrared wavelength 800 nm. Thetime-dependent electric field of the transmitted pulse is presented as E = E ext + E ind ; the appliedlaser electric field is parameterized by a temporary Gaussian function E ext ( t ) = e e − ln(4)( t − t ) /τ L F cos ω L t (1)where e is the laser polarization vector, ω L is the laser oscillation frequency (corresponding tophoton energy ¯ hω L = 1 .
55 eV), τ L = 30 fs is the pulse length, t specifies the position of thepulse peak, F = I / is the electric field strength and I is the peak laser intensity. The inducedelectric field E ind = − π P is a result of the polarization of the solid, here the macroscopicpolarization P ( t ) = R t dt ′ J ( t ′ ) is expressed in terms of the transient photo-current J = tr [ ρ v ],where ρ is the one-electron density matrix and v is the velocity operator.The static band structure of silicon is obtained from empirical pseudo-potential method. Thedirect bandgap energy (3.2 eV) associated with the Γ ′ → Γ interband transition specifies thethreshold for electron-hole pair excitation. To calculate the transient photocurrent J , we samplethe Brillouin zone by a Monte Carlo method using 5000 quasi-randomly k -points generated fromthree-dimensional Sobol sequence in a cube of edge length 4 π/a , where a = 5.43 ˚A is the bulklattice constant of Si; 4 valence and 16 initially unoccupied conduction bands are included in theexpansion of the time-dependent wave-functions over static Bloch orbitals. The wave-functionsof valence electrons were propagated forward in time for small equidistant time steps of δt =0.7 attoseconds. The spectrum of high harmonic generation inside the bulk is obtained fromthe Fourier transformation of the component of the photocurrent onto the laser polarizationdirection I ( ω ) = (cid:12)(cid:12)(cid:12)(cid:12)Z dte iωt J ( t ) · e (cid:12)(cid:12)(cid:12)(cid:12) (2)The time evolution of the electric field of the transmitted laser pulse in bulk silicon is shownin Fig.1. For the relatively low laser intensity, I = 3 × W/cm in Fig.1a, the pulse exhibitstemporary Gaussian profile with field strength E = E ext /
12 and ǫ ≈
12 is the static bulkdielectric constant of Si. The associated HHG spectrum shown in Fig. 2a consists of clean oddorder harmonic peaks, i.e. close to zero crossing of the oscillating electric field electron-holepairs recombine and XUV photons are emitted during each half-cycle. Though photoionizationinvolves perturbative three-photon transition across the direct bandgap in the low-intensityregime, non-perturbative effects in HHG are exhibited as the spectral intensities of the 5th to9th harmonics are of comparable magnitude and comprise a plateau region. This effect can betraced to the accelerated motion of charge carriers in their respective bands. For the increasedpeak intensity, shown in Fig.1b,c the peak of the transmitted pulse undergoes a progressive timedelay relative to the peak of the applied laser. For I = 7 × W/cm , Fig.1c the trailingdge of the pulse becomes steeper as a result of increasing population in the conduction band.Both the harmonic yield and the cutoff energy for HHG increase moderately with the increaseof intensity, cf. Fig. 2b,c.For the highest peak intensity shown in Fig.1d, I = 9 × W/cm , the non-linear responseof electrons becomes prominent: the temporal profile of the pulse is strongly distorted, the peakintensity is pushed to the back of the pulse where a steep edge is formed and frequency up-chirpis exhibited. This regime corresponds to high level of electronic excitation: each Si atom absorbs1.2 eV energy (i.e. each atom absorbs nearly one laser photon). The harmonic peaks, shown inFig.2d, superimpose onto a continuous background, the spectral width of individual harmonicsbroadens and their central wavelength undergoes a blue shift that covers the spacing betweenadjacent orders. Noticeably the position of the fundamental harmonic is unchanged and theshift of the position of harmonic peaks δω q ≈ . ω L is independent on the harmonic order q , i.e.harmonics are mainly emitted at the back of the pulse when the driving frequency is up-chirped.
3. Conclusion
In summary we presented calculation of HHG spectrum in bulk silicon induced by intense 30fsnear-infrared laser pulse. The result describes HHG in a wide range of peak laser intensities. Inthe low intensity regime I ≤ × W/cm , the temporal profile of the transmitted pulse isweakly distorted by the laser-matter interaction and exhibits relatively high degree of temporalcoherence. In this regime, HHG spectrum in silicon consists of clean odd-order harmonics withplateau region and sharp cutoff. The temporal coherence of the transmitted pulse deteriorateswith the increased laser intensity I ≥ × W/cm , the peak of the pulse undergoes progressivetime delay relative to the applied laser and becomes subject to self-steepening. In this regime,the up-chirp in the back part of the pulse results in blueshift of high-order harmonic generationin silicon. Acknowledgment
The material is based upon work supported by the Air Force Office of Scientific Research underaward number FA9550-19-1-7003. Support from the Bulgarian National Science Fund undercontracts No. 08-17 (B.O) and No. KP-06-KOST (T.A) is acknowledged.
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Nat. Photon , pp. 291-296.[10] Apostolova T, Obreshkov B 2018 Diam. Relat. Mater. , pp. 165-172[11] Apsotolova T, Obreshkov B 2018 Opt. Quant. Electron :408[12] Apsotolova T, Obreshkov B, Gnilitskyi I 2020 Appl. Surf. Sci , In press, doi: 10.1016/j.apsusc.2020.146087 igure 1. (a-c) Time evolution of the pulsed electric field (in V/˚A) transmitted in bulk silicon.The laser intensity at the pulse peak is 3 × W/cm in (a), 5 × W/cm in (b), 7 × W/cm in (c) and 9 × W/cm in (d). The laser is linearly polarized along the [001]direction, the laser wavelength is 800 nm and the pulse duration is 30 fs. (cid:1) (cid:2) (cid:3) (cid:4) (cid:2) (cid:5) (cid:6) (cid:3) (cid:7) (cid:8)(cid:9) (cid:10)(cid:11) (cid:8)(cid:9) (cid:10)(cid:12) (cid:8)(cid:9) (cid:10)(cid:13) (cid:8)(cid:9) (cid:10)(cid:14) (cid:8)(cid:9) (cid:10)(cid:15) (cid:8)(cid:9) (cid:10)(cid:8) (cid:8)(cid:9) (cid:9) (cid:16)(cid:17)(cid:18)(cid:19)(cid:20)(cid:2)(cid:6)(cid:21)(cid:22)(cid:20)(cid:18)(cid:23)(cid:4)(cid:18) (cid:9) (cid:15) (cid:13) (cid:11) (cid:24) (cid:8)(cid:9) (cid:8)(cid:15) (cid:8)(cid:13) (cid:25)(cid:17)(cid:26) (cid:1) (cid:2) (cid:3) (cid:4) (cid:2) (cid:5) (cid:6) (cid:3) (cid:7) (cid:8)(cid:9) (cid:10)(cid:11) (cid:8)(cid:9) (cid:10)(cid:12) (cid:8)(cid:9) (cid:10)(cid:13) (cid:8)(cid:9) (cid:10)(cid:14) (cid:8)(cid:9) (cid:10)(cid:15) (cid:8)(cid:9) (cid:10)(cid:8) (cid:8)(cid:9) (cid:9) (cid:16)(cid:17)(cid:18)(cid:19)(cid:20)(cid:2)(cid:6)(cid:21)(cid:22)(cid:20)(cid:18)(cid:23)(cid:4)(cid:18) (cid:9) (cid:15) (cid:13) (cid:11) (cid:24) (cid:8)(cid:9) (cid:8)(cid:15) (cid:8)(cid:13) (cid:8)(cid:11) (cid:8)(cid:24) (cid:25)(cid:27)(cid:26) (cid:1) (cid:2) (cid:3) (cid:4) (cid:2) (cid:5) (cid:6) (cid:3) (cid:7) (cid:8)(cid:9) (cid:10)(cid:11) (cid:8)(cid:9) (cid:10)(cid:12) (cid:8)(cid:9) (cid:10)(cid:13) (cid:8)(cid:9) (cid:10)(cid:14) (cid:8)(cid:9) (cid:10)(cid:15) (cid:8)(cid:9) (cid:10)(cid:8) (cid:8)(cid:9) (cid:9) (cid:16)(cid:17)(cid:18)(cid:19)(cid:20)(cid:2)(cid:6)(cid:21)(cid:22)(cid:20)(cid:18)(cid:23)(cid:4)(cid:18) (cid:9) (cid:15) (cid:13) (cid:11) (cid:24) (cid:8)(cid:9) (cid:8)(cid:15) (cid:8)(cid:13) (cid:8)(cid:11) (cid:8)(cid:24) (cid:15)(cid:9) (cid:15)(cid:15) (cid:15)(cid:13) (cid:15)(cid:11) (cid:15)(cid:24) (cid:14)(cid:9) (cid:25)(cid:23)(cid:26) (cid:1) (cid:2) (cid:3) (cid:4) (cid:2) (cid:5) (cid:6) (cid:3) (cid:7) (cid:8)(cid:9) (cid:10)(cid:11) (cid:8)(cid:9) (cid:10)(cid:12) (cid:8)(cid:9) (cid:10)(cid:13) (cid:8)(cid:9) (cid:10)(cid:14) (cid:8)(cid:9) (cid:10)(cid:15) (cid:8)(cid:9) (cid:10)(cid:8) (cid:8)(cid:9) (cid:9) (cid:16)(cid:17)(cid:18)(cid:19)(cid:20)(cid:2)(cid:6)(cid:21)(cid:22)(cid:20)(cid:18)(cid:23)(cid:4)(cid:18) (cid:9) (cid:15) (cid:13) (cid:11) (cid:24) (cid:8)(cid:9) (cid:8)(cid:15) (cid:8)(cid:13) (cid:8)(cid:11) (cid:8)(cid:24) (cid:15)(cid:9) (cid:15)(cid:15) (cid:15)(cid:13) (cid:25)(cid:21)(cid:26) Figure 2.
HHG spectra is bulk silicon as a function of the peak laser intensity (a) I = 3 × W/cm , (b) I = 5 × W/cm , (c) I = 7 × W/cm and (d) I = 9 × W/cm2