Coherent control of high-order harmonic generation by phase jump pulses

Yang Xiang; Yueping Niu; Hongmei Feng; Yihong Qi; Shangqing Gong

doi:10.1364/OE.20.019289

1. Introduction

In recent years, high-order harmonic generation (HHG) has become one of the hot topics of super intense laser physics. Its wide applications on the generation of attosecond (as) pulses make it attract great interests around the world [1, 2]. The physical origin of the HHG process can be understood by the well-known three-step model (TSM) proposed by Corkum [3]. Firstly a free electron is born by tunneling through the potential barrier formed by the electric field and the Coulomb potential, then oscillates in the laser field and recombines with the parent ion at last. During the recombination, a photon is emitted. Usually, the HHG spectrum contains only discrete odd harmonics since this process occurs in each half cycle of the laser field. In order to generate isolated attosecond pulse (IAP), a broad continuum spectrum is needed. Thanks to the development of the laser technology, ultra-short and ultra-strong laser pulses can be generated in experiment and the evolution of the electric field of the pulse can be well controlled [4, 5], which provide powerful tools for the control of IAP. Till now, many control ways has been proposed for the generation of IAP, such as polarization gate [6–8], two-color (or multi-color) [9–12], chirp [13] and static electric field control [14, 15], and so on.

In this paper, we will explore another way to control the HHG, i.e., the phase jump, which has been widely used in the control of light-atom interactions. For example, Torosov et al. demonstrated that complete population inversion in two-state model can be controlled by the jump phase [16], Qian et al. found that the breakdown of dipole blockade may occur when atoms are driven by a phase-jump pulse [17], and recently Jha et al. found that the population transfer can be significantly enhanced by suitable phase jump pulses [18]. As described in our previous work, the asymmetry of laser field leads to the extension of the HHG spectrum cutoff [19]. If proper phase jump parameters are chosen, the asymmetry of the laser field will be enlarged and the cutoff of the HHG spectrum will extend dramatically. Moreover, a broad continuum spectrum can be expected which benefits to the generation of isolated attosecond pulse. The rest of this paper is organized as follows. The principle and method are described in Sec. 2. Then the numerical results and analysis are presented in Sec. 3. At last, the conclusions are given in Sec. 4.

2. Principle and method

In our calculations, the target atom is the helium atom. The interaction between the helium atom and the laser field can be described by the following three-dimensional (3D) time-dependent Schrödinger equation (TDSE) with a single active electron [20] and dipole approximation [The atom units (a.u.) are used in all equations in this paper, unless otherwise mentioned]:

i \frac{\partial ψ (r, t)}{\partial t} = [- \frac{1}{2} \nabla^{2} + V_{c} (r) - r \cdot E (t)] ψ (r, t),

where,

r

is the position vector,

V_{c} (r)

is the effective Coulomb potential, and

E (t) = E (t) \hat{z}

is the electric field of the laser pulse with polarization direction along z axis. For helium atom, the effective Coulomb potential is expressed as

V_{c} (r) = - \frac{1}{r} [1 + (1 + 27 r / 16) e^{- 27 r / 8}]

with

r = | r |

[21] and the ionization potential

I_{p}

is 0.904a.u.. Equation (1) can be solved effectively by using a partial-wave decomposition of the wave function method, together with the Peaceman-Rachford scheme [22]. When the time- dependent wave function

ψ (r, t)

is obtained, the mean acceleration can be calculated by means of the Ehrenfest’s theorem [23]:

d_{A} (t) \equiv 〈 \ddot{z} (t) 〉 = - 〈 ψ (r, t) | \frac{\partial V (r)}{\partial z} - E (t) | ψ (r, t) 〉,

The HHG power spectrum

P (ω)

can be obtained by taking the Fourier transforms of

d_{A} (t)

:

P (ω) = {| \int d_{A} (t) e^{- j ω t} d t |}^{2} .

3. Numerical results and analysis

Usually, a laser pulse can be expressed as:

E (t) = F f (t) \cos (ω t + ϕ_{i}),

where,

F

,

f (t)

,

ω

and

ϕ_{i}

are the amplitude, envelope, frequency and carrier-envelope phase of the laser pulse, respectively. As mentioned in [18], the electric field of a pulse with a phase jump can be expressed as:

E (t) = {\begin{cases} F f (t) \cos (ω t + ϕ_{i}) if {t<t}_{0}, \\ F f (t) \cos (ω t + ϕ_{i} + ϕ_{j}) if t \geq t_{0}, \end{cases}

where,

ϕ_{j}

is the jump phase introduced into the electric field at

t = t_{0}

. The phase jump laser field in Eq. (5) can be realized by femtosecond pulse shaping technology, which is described in [16]. In this paper,

F

is 0.12 a.u. (corresponding to the intensity of

5.0 \times 10^{14}

W/cm²) and

ω

is 0.057 a.u. (corresponding to the wavelength of 800nm). The envelope is

f (t) = e^{- 2 \ln 2 {(t / τ)}^{2}}

with the duration

τ

. In order to obtain a larger asymmetry of the electric filed, we consider the cases that the phase jump occurs at the zero points near the pulse center with a jump phase of

π

. Figure 1 displays the HHG spectra driven by the laser pulses with

τ = 10

fs and

ϕ_{i} = 0

for jump time at

t_{0} = - 0.75 T

,

- 0.25 T

,

0.25 T

, and

0.75 T

, where

T

is the period of the laser field. As a reference, the HHG spectrum without phase jump is also shown. From this figure, we can see that the jump time

t_{0}

has significant effects on the HHG spectrum. For

t_{0} = - 0.75 T

, the cutoff of the HHG spectrum is about at 280th-order harmonic which is much larger than the well-known value

I_{p} + 3.17 U_{p}

(about 77 harmonics,

U_{p} = F^{2} / 4 ω^{2}

is the ponderomotive energy of the electron) and a very broad continuum spectrum appears. For

t_{0} = - 0.25 T

, the HHG cutoff extends to about 300 harmonics and the intensity of the continuum spectrum is higher than that for

t_{0} = - 0.75 T

, except that near the cutoff. For

t_{0} = 0.25 T

, the intensity of continuum spectrum increases further and the cutoff has not apparent variation. While for

t_{0} = 0.75 T

, the intensity of the continuum spectrum is a little higher than that for

t_{0} = 0.25 T

, while the cutoff decreases to about 270 harmonics.

Fig. 1 The HHG spectra driven by the pulses with different jump times.

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In order to explore the underlying mechanism of the HHG spectrum variation with the jump time, we show in Fig. 2 the return kinetic energy (RKE) of the electron vs. born time and return time for different jump times, as well as the electric field. The RKE can be obtained by solving Newton equation. From Fig. 2(a), we can see that due to the enlargement of the asymmetry of the laser field by phase jump, the maximum RKE (MRKE) reaches about 13.8 $U_{p}$ (corresponding to the cutoff of 280 harmonics). In Fig. 2(b), the MRKE increases to $14.8 U_{p}$ (corresponding to the cutoff of 299 harmonics), due to that the asymmetry of the laser field is enlarged compared with that in Fig. 2(a). In Fig. 2(c), the MRKE is about $15.0 U_{p}$ (corresponding to the cutoff of 303harmonics). This value is comparable with the MRKE in Fig. 2 (b), since the asymmetry of the electric field has little changes compared with that in Fig. 2(b). In Fig. 2(d), due to the decrease of the asymmetry of the laser field, the MRKE reduces to $13.5 U_{p}$ (corresponding to the cutoff of 274 harmonics). Comparing the classical results with the cutoffs of the HHG spectra in Fig. 1, it is easy to find that they agree well.

Fig. 2 The RKE of the electron as a function of born and return time and the electric fields for different jump times. (a) $t_{0} = - 0.75 T$ ; (b) $t_{0} = - 0.25 T$ ; (c) $t_{0} = 0.25 T$ ; (d) $t_{0} = 0.75 T$ .

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Now we begin to discuss the variation of the intensity of the continuum spectra with jump time. According to the TSM, the continuum spectrum comes from the electrons with RKE between the MRKE and the second MRKE, due to their few recollision times with the parent ion [24]. From Fig. 2(a), we can see that the electrons with RKE between $3.17 U_{p}$ and $13.8 U_{p}$ contribute to the continuum spectrum, which are mainly born at $t_{A_{i}}$ [the time corresponding to peak $A_{i}$ ( $i = 1, 2, 3, 4$ ), the same below]. In Fig. 2(b), the electrons with RKE between $3.0 U_{p}$ and $14.8 U_{p}$ mainly born at $t_{B_{i}}$ ( $i = 1, 2, 3, 4$ ) and return at $t_{B}$ , which contribute to the generation of continuum spectrum. According to the ADK theory [25], the tunneling ionization rate of the electron increases nonlinearly with the electric field intensity, i.e., the stronger the electric field, the larger the birth rate of free electrons and the higher the HHG spectrum generated by them. Apparently, the electric field $| E (t_{B_{i}}) |$ is stronger than $| E (t_{A_{i}}) |$ ( $i = 1, 2, 3$ . Though $| E (t_{A_{4}}) |$ is a little higher than $| E (t_{B_{4}}) |$ , both of them are very low and the electrons born at these peaks have not significant contributions to the HHG spectrum), so the intensity of the continuum spectrum for $t_{0} = - 0.25 T$ is higher than that for $t_{0} = - 0.75 T$ . For the case of $t_{0} = 0.25 T$ , as shown in Fig. 2(c), the continuum spectrum mainly comes from the electrons born at $t_{C_{i}}$ ( $i = 1, 2, 3, 4, 5$ ) and $| E (t_{C_{i}}) |$ is stronger than $| E (t_{B_{i}}) |$ ( $i = 1, 2, 3, 4$ ). Therefore, the intensity of continuum spectrum for $t_{0} = 0.25 T$ enhances compared with that for $t_{0} = - 0.25 T$ . As is displayed in Fig. 2(d), the electrons which contribute to the generation of continuum spectrum are mainly born at $t_{D_{i}}$ ( $i = 1, 2, 3, 4, 5$ ). Compare Figs. 2(c) and 2(d), one can find $| E (t_{D_{i}}) | > | E (t_{C_{i}}) |$ ( $i = 1, 2, 3, 4, 5$ ), so the continuum spectrum for $t_{0} = 0.75$ has a higher intensity.

As mentioned in the introduction, the generation of IAP requires continuum spectrum. On the other hand, the coherence of the continuum spectrum has a great influence on the temporal profile of the generated IAP. So it is necessary to discuss the coherence of the continuum spectra in Fig. 1. Usually, there are two dominant quantum paths with different burst times contribute to each harmonic, i.e., the long path and the short path. The long path means the electron is born at an earlier time but returns at a later time while the short path is just the contrary. The different burst times of the two paths may led to decoherence of the harmonics [26]. In the map of RKE vs. born time, the positive and negative slope sections are corresponding to the long and short paths respectively. While in the map of RKE vs. return time, the positive and negative slope sections are corresponding to the short and long paths respectively, as marked in Fig. 2 (a). From Fig. 2(a), we can see that only short paths exist for the electrons with RKE between $3.17 U_{p}$ and $8.44 U_{p}$ . This means that the photons generated by these electrons burst with highly coherence, which is beneficial to the generation of IAP. Selecting the harmonics between the 120th and 160th order (These harmonics come from the electrons with RKE from $5.4 U_{p}$ to $7.48 U_{p}$ . For comparison, we only consider the same harmonic orders in the following discussions), an isolated 76as pulse can be obtained, as show in Fig. 3(a) . For $t_{0} = - 0.25 T$ , though both short and long paths exist for the selected continuum spectrum, a regular IAP can be obtained with duration about 65as, as Fig. 3(b) displays. As can be seen from Fig. 2(b), for the electrons with RKE between $5.4 U_{p}$ and $7.48 U_{p}$ , there are four short and two long paths. Moreover, $| E (t_{_{B_{i}}}^{l}) |$ is much weaker than $| E (t_{B_{i}}^{s}) |$ ( $i = 1, 2 .$ $t_{B_{i}}^{l}$ and $t_{B_{i}}^{s}$ represent the born times of the long path and the short path of $B_{i}$ in the selected RKE range, the same below respectively). Therefore, the birth rates of the electrons through long paths are much lower than those through short paths. So the harmonics come mainly from the short paths and an IAP can be obtained. For the same reason, an isolated 77as pulse can be generated for $t_{0} = 0.25 T$ , as shown in Fig. 3(c). However, as can be seen from Fig. 2(d), $| E (t_{D_{i}}^{l}) |$ enhances significantly compared with $| E (t_{B_{i}}^{l}) |$ ( $i = 1, 2$ ). Thus, the long paths may have obvious effects on the HHG spectrum. So in Fig. 3(d), an apparent tail appears in the generated attosecond pulse. From the analysis above, considering the intensity and the duration of the generated IAP, we can see that the optimal jump times are at $t_{0} = - 0.25 T$ and $t_{0} = 0.25 T$ , i.e., the nearest two zero points to the pulse center.

Fig. 3 The generated IAPs from the spectra of Fig. 1 driven by different pulses. (a) $t_{0} = - 0.75 T$ ; (b) $t_{0} = - 0.25 T$ ; (c) $t_{0} = 0.25 T$ ; (d) $t_{0} = 0.75 T$ . The harmonics used are from the 120th to 160th order.

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To verify the classical analysis above, we perform the time-frequency analysis of the HHG spectra driven by the phase jump laser pulses with different jump times using the Morlet wavelet [27], as shown in Fig. 4 . In this figure, the peaks A’, B’, C’ and D’ are corresponding to the peaks A, B, C and D in Fig. 2, respectively. From this figure, we can clearly see that the intensity increases from peak A’ to D’. For peaks A’, B’, and C’, the intensity of the long path is much weaker than those of the short path, so the continuum spectra are highly coherent. While for peak D’, the intensity of the long path increases apparently compared with the other three peaks and are comparable with the short path, which leads to the phase mismatch of the continuum spectrum. These mean that the classical analysis agrees very well with the quantum theory.

Fig. 4 The wavelet time-frequency profiles of the HHG spectra driven by the laser pulses with different phase jump times. (a) $t_{0} = - 0.75 T$ ; (b) $t_{0} = - 0.25 T$ ; (c) $t_{0} = 0.25 T$ ; (d) $t_{0} = 0.75 T$ .

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For a further investigation, we consider the IAP generation for different CEPs. Figure 5 shows the IAPs driven by phase jump laser pulses with different $ϕ_{i}$ . The jump time $t_{0} = (- 0.5 π - ϕ_{i}) / ω$ is the nearest zero point to the pulse center. As can be seen from Fig. 5, the duration of the IAP varies in a small range (65as~77as), when the CEP $ϕ_{i}$ varies from $- π$ to 0. This means that for the pulse with any CEP, an isolated sub-100 as pulse can be obtained by controlling the $π$ -phase jump at the nearest zero point to the pulse center, which confirms the conclusion we have drawn before.

Fig. 5 The generated IAP from the HHG spectrum driven by phase jump laser pulses with different $ϕ_{i}$ . The harmonic orders used are from 120 to 160.

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4. Conclusions

In conclusion, we investigated the HHG spectrum driven by the laser pulses with a phase jump in details. Due to the enlargement of the electric field caused by the $π$ -phase jump, the cutoff of the HHG spectrum extends dramatically. The intensity and the coherence of the continuum spectrum can be controlled by adjusting the jump time. If the jump time is at the zero point which is the nearest to the pulse center, a continuum spectrum with highly coherence and a high intensity can be obtained, from which a regular IAP with duration less than 100as can be generated without any phase compensation.

Acknowledgments

This work was supported by the National Nature Science Foundation of China (Grant Nos. 11074263, 60978013, and 60921004), the high-performance grid computing platform of and the Doctor Fund of Henan Polytechnic University (Grant No. B2011-076).

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Coherent control of high-order harmonic generation by phase jump pulses

Abstract

1. Introduction

2. Principle and method

3. Numerical results and analysis

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (5)

Equations (5)

Optics Express