Significantly enhanced conversion efficiency of high-order harmonic generation by introducing chirped laser pulses into scheme of spatially inhomogeneous field

Jia-Xin Han; Jun Wang; Yue Qiao; Ai-Hua Liu; Fu-Ming Guo; Yu-Jun Yang

doi:10.1364/OE.27.008768

1. Introduction

In recent years, high-order harmonic generation (HHG) has been one of the most important phenomena from interactions between strong lasers and matters [1,2]. At present, the low efficiency of HHG restricts its practical applications [3]. Many people have been working to improve the efficiency of HHG [4–7]. The waveform of incident laser pulse is closely related to the generation of high harmonics, optimized waveform of chirped laser pulses can greatly enhance the yield of HHG [8]. There are many kinds of techniques to generate isolated attosecond pulses (IAPS), such as the polarization gating technique [9], and two-color or multicolor fields scheme [10,11], few-cycle laser pulses scheme [12,13] and so on [14–21]. However, the pump pulse intensities used in these mentioned schemes are often greater than 10¹⁴Wcm⁻² and the spatial distribution of these pump laser pulses are generally homogeneous. Most recently, employing a spatially inhomogeneous field, which was generated by a modest femtosecond laser pulse irradiating on designed nanostructures, has become an efficient method for HHG. Compared with the homogeneous field, the spatially inhomogeneous field shows spatial gradient, thus the ionized electrons recombined with the parent nucleus will gain higher energy, thus the pump laser intensity can be lowered [22]. For example, Kim et al. obtained the 17th-order of harmonic enhanced by three orders of magnitude by using driving pulse (10¹¹Wcm⁻²) irradiating around the bow tie shaped nanostructure [23]. By optimizing the distance between bow-tie shaped gold nanostructure. Xue et al. obtained a broadband super continuum and a 35-as IAP [24]. Feng et al. added a terahertz field into the driving laser pulse, and the efficiency of harmonics was increased by two orders of magnitude and produced five IAPS from 60 as to 22 as [25]. The same year, by properly adding a half-cycle pulse into the inhomogeneous field, a 279 eV continuum in the HHG spectrum develops, then by directly superposing this continuum, an IAP with the FWHM of 25 as can be generated [26]. Feng et al. theoretically obtained a series of sub-30-as pulses utilizing the polarization gating two-color inhomogeneous field, by introducing ultraviolet pulses, the efficiency of harmonics can be improved by two orders of magnitude [9]. Yuan et al. theoretically obtained a sub-80-as pulse by using two-color spatially inhomogeneous field [27].

The physical mechanism behind HHG has been described with the so-called three-step model, which is the same for inhomogeneous fields [28]: in the first step, the electron tunnels through the coulomb potential barrier formed by the strong laser field, then the ionized electron oscillates in the field, and finally, the electron recombines with the parent ion and emits harmonic photons. The photon energy equals to the sum of ionization potential of the atom or molecule and the kinetic energy gained by the recombining electron in the laser field [29,30].

The generation of harmonics and the synthesis of IAPS in inhomogeneous field have been studied extensively [31–40]. However, using chirped laser pulses to observe their effects on the HHG and synthesis of IAPS are seldomly investigated. In this paper, we theoretically investigated the harmonic generation from a hydrogen atom in the presence of spatially inhomogeneous fields through the interaction between a few-cycle chirped laser pulses and a nano-tip structure, and significantly enhanced efficiency of high harmonic generation is demonstrated. By systematically varying the chirp parameters and adjusting the distance between atoms and nano-tips, enhancement of the ionization rate can be achieved, leading to enhancements both in harmonic yield by up to three orders of magnitude and in the harmonic cutoff energy. The effects of different types of chirps on harmonics are also compared.

The structure of this paper is as follows: section two presents the theoretical model and numerical method to simulate HHG driven by chirped laser field. The results and discussion are presented in section three. The conclusion of the paper is given in section four. Atomic units are used throughout in this paper unless specified otherwise.

2. Method

We theoretically investigated the HHG process from a hydrogen atom in a chirp spatially inhomogeneous field by solving the one-dimensional (1D) TDSE by using the split operator in length-gauge.

i \frac{\partial ψ (x, t)}{\partial t} = [- \frac{1}{2} \frac{\partial^{2}}{\partial x^{2}} + V (x, R) + V_{E} (x, t)] ψ (x, t) .

V (x, R) = - Z / \sqrt{{(x - R)}^{2} + a} .

R is the position of atom. Where

V (x, R)

is the soft Coulomb potential of the hydrogen atom:the parameters are Z = 0.5612 and a = 0.367, it represents that the energies of ground and first excited state are −0.5 and −0.125, respectively. Here

V_{E} (x, t)

is the potential of the laser field. For an inhomogeneous spatial field, the interaction energy is

V_{E} (x, t) = \int_{0}^{x} E (x^{'}, t) d x^{'} .

Inhomogeneous electric field can be expressed as the product of the spatial function part and the temporal function part:

E (x, t) = E_{s} (x) E_{t} (t) .

Temporal function

E_{t} (t)

can be expressed as

E_{t} (t) = E_{0} f (t) \sin [ω t + φ + δ (t)] .

Where

E_{0}

and

ω

are the electric field peak amplitude and the frequency of the laser pulse, respectively;

φ =0

is the carrier-envelope phase of the driving field, and

f (t) = \exp [- 4 \ln 2 {(t / σ)}^{2}]

is the Gaussian temporal envelope.

δ (t) =- β tanh(\frac{t - t_{0}}{κ}),

it's a hyperbolic tangent function that we added to change the shape of the electric field [41]. The form of the chirp is controlled by adjusting the three parameters:

β

(to adjust the frequency sweeping range), and a positive chirp can be obtained when the chirp parameter β is adopted with a minus number,

κ

= 200 a.u. (for adjusting the slope of the chirping function), and

t_{0}

is set at the middle of the sweep. When

β

= 0,

δ (t)

is 0, and the laser pulse is chirp-free. Spatial function

E_{x} (x)

is expressed as

E_{s} (x) = \frac{ε \cdot x + μ}{x + η} .

It is in close proximity to the real spatial field distribution in experiments (When this pulse interacts with the nano-tip nanostructure, a spatially inhomogeneous field will be generated due to the resonance effect). Spatial factors

ε

,

μ

, and

η

are used to change the degree of spatial inhomogeneity. In addition, we can change the inhomogeneous of the spatial field by changing the thickness, angle, height of the nano-tip. In the experiment, an input chirp laser of 3800-nm carrier wavelength is focused on a metallic nanostructure consisting of nano-tip-shaped gold elements, meanwhile, injection with a hydrogen atom jet. When exposed to the femtosecond pulse, free electrons are confined within one apex of the nano-tip element, which makes the pulse capable of producing a strong field enhancement around the vertices and enables the generation of high harmonics. Also one can also adjust the shape of the nano-tip structure to change the spatial heterogeneity in experiments. The solution to Eq. (1) can be expressed as

ψ (x, t + Δ t) = e^{- i Δ t (T + V)} \cdot ψ (x, t),

where

T = - \frac{1}{2} \frac{\partial^{2}}{\partial x^{2}},

V = V (x) + \int_{0}^{x} E (x^{'}, t) d x^{'},

T for kinetic energy and V for potential energy. Equation (5) is solved by means of the splitting-operator fast-Fourier transform technique. The initial wave packet is constructed by the imaginary time-propagation method. And the wave function at any time can be expressed as:

ψ (x, t + Δ t) = e^{- i Δ t / 2 (T + V)} e^{- i Δ t V} e^{- i Δ t T / 2} \cdot ψ (x, t) + ο (Δ t^{3}) .

Moreover, a mask function with the form cos^1/8 was used to avoid the wave packet near the boundaries contribute to the HHG. Then time-dependent induced dipole moment can be given by Ehrenfest theorem

a (t) = 〈 ψ (x, t) | - \frac{d V (x, R)}{d x} - E (x, t) | ψ (x, t) 〉 .

The HHG power spectrum can be determined by Fourier transforming the time-dependent dipole acceleration, which is given by

p (ω) = {| \frac{1}{\sqrt{2 π}} \int_{0}^{t^{'}} a (t) \exp (- i ω t) d t |}^{2} .

To further explain the physical mechanism of HHG, a time-frequency distributions of the induced dipole moment can be performed by means of wavelet transform:

A_{ω} (t_{0,} ω_{0}) = \int_{t_{i}}^{t_{f}} a (t) w_{t_{0}, ω} (t) d t,

w_{t_{0}, ω} (t)

is the wavelet transform kernel, which can be

w_{t_{0}, ω} (t) = \sqrt{ω} W [ω (t - t_{0})],

where the kernel of the wavelet is

W (x) = \frac{1}{\sqrt{τ}} e^{i x} e^{- x^{2} / 2 τ^{2}} .

The intensities of IAPs are given as follows:

I_{I A P_{s}} (t) = | \sum_{q} (\int_{- \infty}^{+ \infty} a (t) e^{- i q ω t} d t) e^{i q ω t} |^{2},

q is the order of harmonics.

3. Results and discussion

In our calculation, we first study the harmonic spectra of hydrogen atom irradiated by a chirped laser pulse with inhomogeneous spatial distribution. The intensity of the chirped laser pulse is I₀ = $3 \times 10^{12} {W/cm}^{2},$ which is lower than the damage threshold of nanomaterials. Duration and wavelength of the pulse are one cycle and 3800nm, respectively; As for parameters, $E_{s} (x) = \frac{ε \cdot x + μ}{x + η},$ $ε$ = 12.02, $μ$ = 0.8001, $η$ = 340.7 are adopted. At the beginning, the nano-tip was set at the position of −20nm in the one-dimensional space, and the mask function was located −19nm away from the nano-tip. By changing the distance L between the atom and the nano-tip in the spatial inhomogeneous field and the chirp parameters of the incident laser pulse, we can make variation of the intensity and bandwidth of IAPS. Figure 1(a) shows the ratio of intensity and bandwidth of IAPS at different positions varies with the chirp parameter β (From 0 to −0.6 with −0.1 step length); Fig. 1(b) shows the intensity of IAPS varies with chirp parameters when the distance between the atom and the nano-tip changes;Fig. 1(c) shows the time bandwidth of IAPS in different distances of the atom and the nano-tip varies with the chirp parameters. From Fig. 1(b) and Fig. 1(c), we can see that (with the distance L fixed) as the chirp parameter β decreases, the intensity of the IAPS increases and the attosecond pulse bandwidth narrows. This is consistent with the variation trend of Fig. 1(a). In other words, we can achieve the optimal IAPS by adjusting the ratio of intensity and bandwidth. It can be found that the closer the atom is to the nano-tip, the larger negative chirp parameter β is adapted, the higher the intensity of the IAPS can be obtained. A detailed analysis to explain this phenomenon will be shown later in this paper.

Fig. 1 (a)-(c) Ratio of the intensity to bandwidth of IAPS, intensity and bandwidth of IAPS obtain under the different chirp parameters and distances between the hydrogen atom and the nano-tip.

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The continuum spectrum generated when the short laser pulse drives the system, as it’s shown in Fig. 2. As we all know, harmonics generated twice in every single period. However, there exist many periods for longer pulses, clear harmonic peaks will appear. But for a short pulse, in inhomogeneous case, we can only see the harmonics emitted in half a period, and the interference during the periods will disappear and harmonics emitted in half a period will form a continuum spectrum. Figure 2 describes the enhancement of the HHG yield. By adjusting the position of atom and altering the laser chirp parameter β, we found a clear clue. First of all, we present that the effects of the distance between the atom and the nano-tip on the HHG spectra. The spectra showing five kinds of colors are generated by the chirp-free driving laser field, and the distances L between the atom and the nano-tip are 12nm (A), 11nm (B), 10nm (C), 9nm (D), 8nm (E), respectively. In Fig. 2(a), as closer to the nano-tip, the efficiency of harmonics is improved; considering the inhomogeneous character of field, the closer the atom gets to the tip, the released electron will acquire more energy from the inhomogeneous electric field. Secondly, we investigated the HHG spectra under the influence of a chirped laser pulse with decreasing β (From 0 to −0.6 with −0.1 step length) values as shown in Fig. 2(b). It can be seen from Fig. 2(b) that a boost of the HHG energy cutoff arises as the absolute value of the chirp parameters increase. At the same time, there is a more obvious phenomenon. With the introduction of the chirp pulse compared with the harmonics generated by chirp free pulse, the harmonic efficiency becomes higher and higher, and when the chirp parameter β = −0.6, the efficiency of harmonics can no longer be improved. From the above, the chirp effect cannot only broaden HHG platform but also improve the conversion efficiency of high harmonics in the same spatial distribution. The reason of harmonic cutoff expansion and efficiency improvement can be explained by observing the peaks of the inhomogeneous electric field and ionization probability in Figs. 3(e)-3(f) in the following parts.

Fig. 2 HHG spectra obtained under different distances L in (a) and different chirp parameters β in (b).

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Fig. 3 Time-frequency profile of the HHG power spectra of hydrogen atoms driven in (a) β = 0, L = 8nm; (b) β = −0.6, L = 8nm; (c) β = −0.6, L = 12nm. (d) Coupling field in time and space for $φ =0 .$ Black dot curves, red dash curves and green solid curve in (e) and (f) are electric field and ionization probability corresponding to (a), (b) and (c).

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Systematical analyses of the HHG efficiency enhancements are shown in Fig. 3. In Figs. 3(a)-3(c), we show the quantum time-frequency profile of HHG power spectra of hydrogen atom. The distribution of incident laser electric field in time and space is shown in Fig. 3(d). Figure 3(e) and Fig. 3(f) represent the electric field and ionization probability corresponding to time frequency, respectively. In Fig. 3(a) and Fig. 3(b), the atoms are all 8nm away from the nano-tip, the difference is, chirp parameter β = 0 in Fig. 3(a) and β = −0.6 in Fig. 3(b).The dependence of harmonic order on the ionization time is also displayed in Fig. 3(f). Black dot curves in Figs. 3(e) and 3(f) correspond to Fig. 3(a). For comparison, red dash curves in Figs. 3(e) and 3(f) correspond to Fig. 3(b). We can see that the higher harmonic efficiency is under the chirp results from Fig. 3(a) and Fig. 3(b). It can be clearly explained from the Fig. 3(f). In the previous paper and the work of others [28,34], the classical and quantum can be a good agreement. And the instants of the ionized electrons’ releasing and returning their parent ions can be easily seek through the classical trajectory simulation. As for β = −0.6, the electron ionized around 0.8T₀ (the red arrow) (T₀ corresponds to the optical period), while as β = 0, the electron ionized around 0.6T₀ (the black arrow). Obviously, ionization rate with chirp is higher than that without chirp at ionization time contributing to harmonics in Fig. 3(f), this is why the harmonic efficiency is improved in Fig. 3(b). In addition, the probability of ionizing electrons returning to the parent nucleus is increased and the recombination time is more concentrated [28]. Similarly, as for the green solid curve in Fig. 3(f), the atom is 12nm away from the nano-tip, we can clearly see that the electron ionized around 0.6T₀ (the green arrow) in Fig. 3(f), which corresponds to the time-frequency analyses in Fig. 3(c) and the green solid curve in Fig. 3(e). It can be seen from this that the harmonic efficiency is increased due to the ionization probability is significantly increased at the position close to the nano-tip compared with that in Fig. 3(b). The harmonic order is calculated by $q ω_{0} = I_{p} + E_{k i n}$ , where $I_{p}$ is the ionization energy of the ground state for hydrogen atom, E_kin is the kinetic energy obtained from the laser field when the electron returns to the parent ion. As it is shown in Fig. 3(e), due to the red curve shows a stronger electric field at ionization time than the green curve or the black curve, the electrons get more kinetic energy E_kin when they accelerate in the laser field, so the harmonic order in Fig. 3(b) is extended.

In the above discussion, we show the enhancement of HHGs when the chirp parameter β is adopted with a minus number. If β is plus or chirp form is changed, what will happen? Next, we will compare the effect on HHG when the laser chirp from is changed. As it is shown in Fig. 4 (the distance L is the same as Fig. 3(b)), Fig. 4(a) shows the chirp form we have used, and Fig. 4(c) shows the chirp form $δ (t) = α {(t / ν)}^{2}$ ( $ν$ = 200 a.u.) for comparison. Different from $δ (t) =- β tanh(\frac{t - t_{0}}{κ}),$ $δ (t) = α {(t / ν)}^{2}$ shows a linear chirp pulse in Fig. 4(c). In Fig. 4(a), the green line represents the laser electric field with chirp parameter β = −0.6, it can be seen from Fig. 4(b) that the harmonics efficiency of the green line is significantly improved and increased by nearly three orders of magnitude than that from chirp-free pulse (black dash dot line). Compared with chirp-free pulse, the red line (chirp parameter β = 0.6) shows lower efficiency. The reason of efficiency improvement has been explained by observing the instantaneous electric field intensity and ionization probability in Figs. 3(e)-3(f).

Fig. 4 (a) and (c) display the laser fields in time domain with different chirp forms. (b) and (d) show the HHG spectra of hydrogen atom obtained with chirp pulses parameters $β$ and $α$ , respectively.

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Similarly, in Fig. 4(c), the green solid line represents the laser electric field with negative chirp. We can see from Fig. 4(d) that the negative chirp corresponds to higher harmonic efficiency, which is increased by almost two orders of magnitude. This phenomenon also can be clearly explained by observing the instantaneous electric field intensity and ionization probability, as negative chirp pulses is more likely to ionize than positive chirp pulses. By comparison, we can get the conclusion that chirp effect has an impact on the efficiency of high order harmonics, and differentchirp forms show certain differences in improving the efficiency of HHG.

4. Conclusion

In our work, we investigated the influence of chirp effects on harmonics in the scheme of spatially inhomogeneous field. Contributions of two field features are taken into account: temporal chirp and spatial inhomogeneity. In our pre-work [28], it can be found that in the case of the inhomogeneous electric field in space, the instantaneous of ionized electrons coming back to the core is highly concentrated. In the case of spatial inhomogeneous field, the harmonic spectrum will be greatly expanded and the attosecond pulse will be shorter. On this basis, the influence of time domain chirp effect on harmonic efficiency is further discussed. When we fixed the position of the atom in this field and changed the chirp parameters, it is found that the harmonic efficiency can be adjusted. When β = −0.6, the harmonics efficiency is significantly increased by nearly three orders of magnitude. At the same time, we can synthesize higher intensity IAPS by selecting proper atom positions and laser chirp parameters. By now, we have studied the effect of harmonics at different positions in spatial inhomogeneous fields. In order to compare with experiment, the propagation of harmonics at different positions satisfies Maxwell's equation. Since the phase information of HHG is different, we also need to coherent superposition the harmonics at different positions, this will be our future work.

Funding

National Natural Science Foundation of China (NSFC) (11604119, 11627807, 91850114).

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Significantly enhanced conversion efficiency of high-order harmonic generation by introducing chirped laser pulses into scheme of spatially inhomogeneous field

Abstract

1. Introduction

2. Method

3. Results and discussion

4. Conclusion

Funding

References

Cited By

Figures (4)

Equations (7)

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