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Selective enhancement of single-order and two-order harmonics from He atom via two-color and three-color laser fields
Chemical Physics 527 (2019) 110497
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Selective enhancement of single-order and two-order harmonics from He atom via two-color and three-color laser fields Yi Lia, Liqiang Fenga,b, Yan Qiaob,c,
T
⁎
a
Laboratory of Molecular Reaction Dynamics, Liaoning University of Technology, Jinzhou 121001, China State Key Laboratory of Molecular Reaction Dynamics, Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Dalian 116023, China c Department of Pathophysiology, Basic Medical College of Zhengzhou University, Zhengzhou 450001, China b
ARTICLE INFO
ABSTRACT
Keywords: High-order harmonic generation Selective enhancement of single-order harmonics Selective enhancement of two-order harmonics Laser waveform control Two-color and three-color combined fields
An effective method to achieve the selective enhancement of single-order and two-order harmonics from He atom has been proposed through laser waveform control of two-color and three-color laser fields. Firstly, with the optimization of ω-2ω two-color laser waveform, not only the wavelength tunable single-order harmonic (i.e. from 182th order to 328th order) can be enhanced by 17 times compared with the neighbor harmonics, but also the enhancement of two-order harmonics can also be obtained. Theoretical analyses show that the enhancement of the specific harmonics is coming from the folded structure on the short quantum path of the harmonic emission peak. Detail analyses show that the above folded structure happens in the reversed sub-peak region of the half-cycle laser waveform, where the deceleration and further acceleration of the free electron can be found in this structure. Moreover, this structure is dependent on the pulse duration of the controlling pulse. However, it is not very sensitive to the wavelength of the controlling pulse. Further, by properly adding a 3ω pulse, not only the efficiency of spectral continuum can be enhanced by 300 times compared with that from the two-color field, but also the generation of single-order harmonic with the intensity enhancement of 20 times can be found. Furthermore, the enhancement of the harmonic spectrum can also be achieved when using the 4ω or 6ω controlling pulses.
1. Introduction High-order harmonic generation (HHG) as one of the most important nonlinear optical phenomena in strong field physics has been investigated over 30 years due to its important applications in (a) the generations of single attosecond pulses (SAPs) [1–8] by filtering the spectral continuum or (b) the selective generation of single-order harmonic from the spectral continuum [9–11]. Currently, the HHG can be obtained from atoms, molecules and solids [12–15]. However, to obtain the useful light sources in practice, the HHG from atoms and molecules are the common schemes, which can be explained by the semi-classical three-step model (TSM) with the processes of ‘ionization, acceleration and recombination’ [16]. On the basis of TSM, (a) the harmonic cutoff is proportional to the laser intensity and the square of the laser wavelength. (b) The harmonic yield is sensitive to the ionization probability and the occupancy of the ground state. Thus, to produce the intense SAPs in XUV and X-ray regions, many attractive schemes have been proposed, such as, the waveform control in HHG by using (1) the multi-color laser field [17–22]; (2) the
⁎
frequency-chirp modulations [23–25]; (3) the polarization gating technology [26–28] and (4) the inhomogeneous effect of the laser field [29–34]. The improvement of harmonic yield by using (5) the high Rydberg initial state in atomic system [35,36]; (6) the high vibrating initial state in molecular system [37] and (7) the resonance enhancement ionization scheme [38,39]. As discussed before, the generation of single-order harmonic is another important application in HHG [9–11,40–43]. Generally, the enhancement of the specific harmonic order can be achieved by using the intra-atomic phase matching scheme via the single-color, two-color, and even three-color fields [9–11,40–43]. However, the spectral region of the enhanced single-order harmonic is narrow and the enhanced ratio between the selective harmonic and its neighbors is only by a factor of 4 times. Recently, with the development of optical parametric chirped-pulse amplification technology, it is possible to generate any optical waveform via the multi-color combined fields [44–51]. Thus, in this paper, through the waveform control of the two-color and three-color laser fields, we found the optimal waveforms to achieve the enhancement of
Corresponding author. E-mail address: [email protected] (Y. Qiao).
https://doi.org/10.1016/j.chemphys.2019.110497 Received 24 March 2019; Received in revised form 12 May 2019; Accepted 15 August 2019 Available online 16 August 2019 0301-0104/ © 2019 Elsevier B.V. All rights reserved.
Chemical Physics 527 (2019) 110497
Y. Li, et al.
single-order and even two-color harmonics. The spectral region of the selective single-order harmonic is from 182th order to 328th order. Moreover, the enhanced ratios between the selective harmonics and the neighbors are from 17 times to 20 times. Theoretical analyses show that the enhancement of the specific harmonics is coming from the folded structure on the short quantum path of the harmonic emission process, which is attributed to the deceleration and further acceleration of the free electron in the reversed sub-peak region of the half-cycle laser waveform. 2. Theoretical part The model is chosen to be He atom in this paper. In detail, He atom driven by the laser field can be described by the three dimensional timedependent Schrödinger equation [52–60],
(r , t ) = H (t ) (r , t ) = t
i
1 2
2
+ V (r ) + z·E (t )
(r , t ),
Fig. 1. The effect of b1 on the HHG spectra driven by two-color laser field. The two-color field is combined by 20 fs-1600 nm and 10 fs-800 nm with I1 = I2 = 1.0 × 1014W/cm2.
(1)
3. Results and discussion
where, V (r ) = 1.535 r is the soft Coulomb potential of He. E (t ) = E (t ) z is the linearly polarized laser field, which can be expressed as, 2 j]
Ej exp[ 4 ln(2)(t )2
E (t ) =
cos( j t + bj t 2),
3.1. Selective enhancement of single-order and two-color harmonics via two-color laser field In this part, we try to find the specific waveforms to achieve the enhancement of single-order and even two-order harmonics. Here, the waveform of the laser field is mainly controlled by two chirp parameters (b1, b2). Fig. 1 shows the effect of b1 on the HHG spectra driven by a twocolor laser field. The two-color field is combined by 20 fs-1600 nm and 10 fs-800 nm with I1 = I2 = 1.0 × 1014W/cm2. For the chirp-free pulse case (b1 = b2 = 0), a spectral continuum from 50th order to 130th order can be found. Moreover, the intensity of spectral region from 55th order to 70th order is enhanced by 6 times compared with the neighbors. As the negative chirp of 1600 nm pulse added, not only the harmonic cutoff is extended, but also the enhanced spectral region is extended. Especially for the cases of b1 = −0.00007, b2 = 0 and b1 = −0.00006, b2 = 0, the selective enhancement of the single 291th order and 328th order harmonics can be obtained, respectively. The enhanced ratios between the above two selective harmonics and their neighbors are 17 times. To better understand the harmonic emission process and the reason behind the enhancement of single-order harmonic, we present the timeprofiles of laser fields and the time-frequency analyses of HHG for some specific chirp modulations, as shown in Fig. 2. In detail, for the case of b1 = b2 = 0 [Fig. 2(a)], there are lots of harmonic radiation peaks (HRPs) during the laser-He atom driven time, which are marked as Pi. As can be seen, although the largest emitted photon energy comes from P1, its intensity is very weak and it can be ignored compared with the others. On the contrary, the intensity of P2 is very strong and the cutoff of P2 is around 130th order, which means the signal of spectral continuum mainly comes from P2. Generally, on the basis of TSM, the HRP is contributed by two harmonic emission paths. In detail, when the electron is ionized around A point, the free electron with the later ionization time will first recombines with its parent ion around B point, thus leading to the short quantum path of P2; while, the free electron with the former ionization time will later recombines with its parent ion around C point, which leads to the long quantum path of P2. However, there is a reversed sub-peak from t = 0.25 T to t = 0.75 T in this waveform (from t = 0 to t = 1.0 T). T is the optical cycle of 1600 nm pulse. Thus, the free electron will experience the deceleration and further acceleration in this sub-peak region, which leads to a folded structure (open white circles) on the short quantum path of P2. For the chirp-free pulse case, this folded region covers from 55th order harmonic to 70th order harmonic. Thus, the intensity of this spectral region is higher than
(2)
j =1 3
where, Ej, ωj, τj and bj are the laser amplitudes, the laser frequencies, the full width at half maximum (FWHM) and the linear chirp parameters of the three-color field. It should be illustrated that the units of the chirp parameters bj are rad/s2. However, for convenience, the units of chirp parameters are omitted in the following discussion. (r , t ) is the time-dependent wave function, which can be expressed as, lmax = 50
(r , t ) = l=0
1 (r , t ) Yl0 ( ). r l
(3)
Here, l (r , t ) and Yl0 ( ) are the radial and spherical harmonics functions. Put Eq. (3) into the TDSE, we can obtain a set of coupled partial differential equations, i
t
l
1 2 l (l + 1) + V (r ) + 2 r2 2r 2
(r , t ) =
l
(r , t ) + zE (t )[cl
l + 1 (r ,
t ) + cl
1
l 1 (r ,
t )].
(4) (l + 1)2 (2l + 3)(2l + 1)
Here cl =
are the coupling constants related to Clebsch-
Gordan coefficients. In the present calculations, the radius and the number of partial waves are chosen to be r = 800 a.u. and lmax = 50, respectively. The grid spacing of radius is chosen to be dr = 0.1 a.u.. Further, the timedependent wave function can be advanced using the standard secondorder split-operator method with dt = 0.05 a.u. [61]. Through Fourier transforming the time-dependent dipole acceleration a(t), the HHG spectrum can be expressed as,
S( ) =
1 2
a (t ) e
i t dt
2
,
(5)
(r , t )|[H (t ), [H (t ), z ]]| (r , t ) . where, a (t ) = The time-frequency analyses (or called the time-profile) of the HHG can be expressed as [62],
A (t , ) =
a (t )
where W (x ) =
1
W ( (t
eixe
x2 2 2
t )) dt ,
(6)
is the Morlet wavelet with ξ = 36 a.u.. 2
Chemical Physics 527 (2019) 110497
Y. Li, et al.
Fig. 2. The laser profiles and the time-frequency analyses of HHG for the cases of (a) b1 = b2 = 0; (b) b1 = -0.00008, b2 = 0; (c) b1 = −0.00007, b2 = 0 and (d) b1 = −0.00006, b2 = 0.
the neighbors. For the case of b1 = −0.00008, b2 = 0 [Fig. 2(b)], as discussed before, when the electron is ionized around A point, the recombination times for the short and long quantum paths are around B and C points, respectively. Compared with the result shown in Fig. 2(a), not only the laser amplitude round t = 0.75 T is enhanced, but also the accelerated time of the free electron is increased. Thus, the free electron will receive more energy during its accelerated process, which is responsible for the extension of P2 and the harmonic cutoff. However, there are two reversed sub-peaks during the harmonic emission process (i.e. from t = 0.25 T to t = 0.75 T and from t = 0.75 T to t = 1.25 T). As a result, two folded structures on the short quantum path of P2 can be found. For the second folded structure (open white circles), the folded region is from 209th order harmonic to 217th order harmonic, which is responsible for the selective enhancement of these harmonics on the HHG spectrum. For the cases of b1 = −0.00007, b2 = 0 [Fig. 2(c)] and b1 = −0.00006, b2 = 0 [Fig. 2(d)], we see that before the recombination process between the free electron and its parent ion (around B point), there are also two reversed sub-peaks (i.e. from t = 0.25 T to t = 0.75 T and from t = 0.75 T to t = 1.25 T). Thus, two folded structures can also be found on the short quantum path of P2. However, for the second folded structure (open white circles), the folded regions are only cover 291th order harmonic (for b1 = −0.00007, b2 = 0 case) and 328th order harmonic (for b1 = −0.00006, b2 = 0 case), respectively, thus leading to the selective enhancement of the single-order harmonic on the HHG spectrum. We see that by properly choosing the chirp of the fundamental field, the enhancement of single-order harmonic can be obtained. Whether the single-order harmonic enhancement can be further controlled? With that in mind, we investigate the effect of b2 on the enhancement of single-order harmonic, as shown in Fig. 3. The chirp parameter of 1600 nm pulse is chosen to be b1 = −0.00006. Clearly, as b2 negatively increases, the enhancement of single-order harmonic is disappeared, as shown in Fig. 3(a). As b2 positively increases from 0 to 0.00008, the wavelength tunable single-order harmonics from 182th order to 328th order can be selectively enhanced by almost 17 times in comparison with their neighbors, as shown in Fig. 3(b). As b2 further increases (i.e.
0.00009 ≤ b2 ≤ 0.00015), the enhancement of two-order harmonics with the intensity improvement of nearly 17 times can be found, as shown in Fig. 3(c). To understand the mechanism behind the order-change of the enhanced single-order harmonic or the enhancement of two-order harmonics, in Fig. 4, we present the time-profiles of laser fields and the time-frequency analyses of HHG for some specific chirp modulations described in Fig. 3. Through analyzing the results shown in Fig. 2, we know that the enhancement of single-order harmonic is attributed to the second folded structure on the HRP. Now, for the case of b1 = −0.00006, b2 = −0.00004 [Fig. 4(a)], we see that as the negative chirp of 800 nm pulse introduced, the second reversed sub-peak is disappeared. Thus, only the first folded structure with the lower-order harmonics can be found on the short quantum path of P2. This is the reason behind the decrease and even disappearance of the single-order harmonic enhancement in the higher-order harmonic region. For the case of b1 = −0.00006, b2 = 0.00006 [Fig. 4(b)], there are three subpeaks from t = 0 to t = 1.75 T. However, before the recombination time between the free electron and its parent ion (around B point), there are still two sub-peaks in this region, which leads to two folded structures on P2. For the second folded structure (open white circles), the folded harmonic order is around 205th order, which is responsible for the enhancement of this harmonic on the HHG spectrum. For the case of b1 = −0.00006, b2 = 0.00011 [Fig. 4(c)], before the recombination time for the short quantum path of P2 (around B point), there are three sub-peaks in this region, which leads to three folded structures on P2. For the second and third folded structures (open white circles), the folded regions cover 151th order harmonic and 355th order harmonic, respectively, which is the reason behind the enhancement of two-order harmonics on the HHG spectrum. Of course, as b2 changes, the subpeaks in the half laser waveform (from t = 0 to t = 1.75 T) will be slightly changed, which leads to the different folded regions on the short quantum path of P2 and is responsible for the selective enhancement of the wavelength tunable single-order and two-order harmonics on the HHG spectrum. Through the above analyses, we see that by properly controlling the 3
Chemical Physics 527 (2019) 110497
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Fig. 3. The effect of b2 on the HHG spectra driven by two-color laser field. The chirp parameter of 1600 nm pulse is chosen to be b1 = −0.00006.
Firstly, the pulse duration effect of 800 nm pulse on the enhancement of single-order harmonic has been shown and discussed in Figs. 5 and 6. Here the pulse duration means the total pulse duration of the laser field. As can be seen, in Fig. 5, for the longer pulse duration (i.e. 20 T800 ∼ 12 T800, where T800 is the optical cycle of 800 nm pulse), a broader spectral region can be enhanced and the bandwidth of the enhanced spectral region is reduced as pulse duration decreased. When the pulse duration of 800 nm pulse is chosen from 11 T800 to 8 T800, the enhancement of the nearly single-order harmonic can be found. While, as pulse duration of 800 nm pulse further decreases, the enhancement of single-order harmonic is disappeared. To clearly see the pulse duration effect on the single-order harmonic enhancement, the laser profiles and the corresponding time-frequency analyses of HHG for different 800 nm pulse durations have been shown in Fig. 6. In detail, for the cases of longer pulse durations (i.e. 20 T800 and 13 T800), the laser amplitudes around t = 1.25 T are large enough and the second reversed sub-peaks from t = 0.75 T to t = 1.25 T are very obvious. As a result, the deceleration of the free electron in this sub-peak region is very obvious, which leads to a broader folded region on the short quantum path of P2, as shown in Fig. 6(a) and (b). This is the reason behind the enhancement of the broader spectral region on the HHG spectrum. For the cases of middle pulse durations [i.e. 11 T800, 10 T800 (corresponding to 10 fs-800 nm pulse), 9 T800 and 8 T800], the laser amplitudes around t = 1.25 T are decreased compared with those from the longer pulse durations. Thus, the deceleration of the free electron will be reduced in comparison with those from the longer pulse durations, which leads to the reduction of the folded region on the short quantum path of P2, as shown in Fig. 6(c)–(e). Particularly, for the cases of 11 T800, 9 T800 and 8 T800, the folded regions are around the single 328th order, 312th order and 311th order harmonics, respectively, which are responsible for the enhancement of these single-order harmonics on the HHG spectra. For the case of shorter pulse duration (i.e. 7 T800), the second reversed sub-peak from t = 0.75 T to t = 1.25 T is almost disappeared. Therefore, there is no folded region in the largerorder harmonic region [see Fig. 6(f)], which is responsible for the disappearance of the single-order harmonic enhancement. Secondly, the wavelength effect of the controlling pulse on the enhancement of single-order harmonic has been shown and discussed in Figs. 7 and 8. As can be seen in Fig. 7, when using the 10 fs-400 nm [Fig. 7(a)] or the 10 fs-1064 nm [Fig. 7(b)] controlling pulses, the
Fig. 4. The laser profiles and the time-frequency analyses of HHG for the cases of (a) b1 = −0.00006, b2 = −0.00004; (b) b1 = −0.00006, b2 = 0.00006 and (c) b1 = −0.00006, b2 = 0.00011.
laser waveform via the two-color chirped pulse, the wavelength tunable single-order harmonics (or two-order harmonics) can be selectively enhanced. Considering the practical applications, the generation of single-order harmonic is more useful. Thus, in the following discussion, we will focus on the laser parameters effects on the single-order harmonic enhancement. 4
Chemical Physics 527 (2019) 110497
Y. Li, et al.
Fig. 5. The effect of 800 nm pulse duration on the enhancement of single-order harmonic.
Fig. 6. The laser profiles and the time-frequency analyses of HHG for the controlling pulse durations of (a) 20 T800; (b) 13 T800; (c) 11 T800; (d) 9 T800; (e) 8 T800 and (f) 7 T800.
enhancement of single-order and even two-order harmonics can also be found. That is to say the enhancement of single-order harmonic is not very sensitive to the wavelength of the controlling pulse. However, the enhanced ratios of the single-order harmonics from adding 400 nm and adding 1064 nm pulses are lower than those from adding 800 nm controlling pulse. To clearly see the controlling wavelength effect on the single-order harmonic enhancement, the laser profiles and the corresponding time-frequency analyses of HHG for the cases of adding 400 nm and 1064 nm controlling pulses have been shown in Fig. 8. In detail, for the case of adding 400 nm controlling pulse with b1 = −0.00004, b2 = 0 [Fig. 8(a)], the number of sub-peaks from t = 0 to t = 1.25 T are increased. Thus, the harmonic emission process in this region becomes much more complicated. For instance, due to the multisub-peaks structure, when the electron is ionized around A point, it will experience many times ‘acceleration-deceleration-further acceleration’
before it recombines with its parent ion around B point. Thus, there are three folded regions on the HRP. The first folded region covers from 80th order harmonic to 110th order harmonic, thus leading to the little enhancement of this spectral region on the HHG spectrum. The second and the third folded regions are around 165th order harmonic, which leads to the enhancement of this single-order harmonic on the HHG spectrum. For the case of adding 1064 nm controlling pulse with b1 = −0.00005, b2 = 0 [Fig. 8(b)], after the electron ionized around A point and before the free electron recombined with its parent ion around B point, there is only one reversed sub-peak during this half laser waveform region. Therefore, only one folded structure with the folded region around 119th order harmonic can be found on P2, which is responsible for the enhancement of this single-order harmonic on the HHG spectrum.
5
Chemical Physics 527 (2019) 110497
Y. Li, et al.
Fig. 7. The wavelength effect of the second pulse on the enhancement of single-order harmonic (a) λ2 = 400 nm and (b) λ2 = 1064 nm. The laser intensities are unchanged.
color field will be much more useful than that from two-color field. Furthermore, by properly adding 400 nm (4ω pulse) or 267 nm controlling pulses (6ω pulse), both the improvement of total HHG yield and the enhancement of single-order harmonic can be obtained, as shown in Fig. 9(b). However, the signal to noise ratio of the enhanced singleorder harmonic from adding 400 nm or 267 nm pulses are decreased compared with that from adding 532 nm pulse. To understand the improvement of total HHG yield and the singleorder harmonic in three-color field, in Figs. 10 and 11, we present the laser profiles and the time–frequency analyses of HHG for the cases of adding 532 nm, 400 nm and 267 nm controlling pulses. Clearly, with the introduction of the third pulse, the laser amplitudes around t = 0 for all cases are increased compared with that from two-color field. Thus, the ionization probability around A point can be increased, which is responsible for the intensity enhancement of total HHG yield. In detail, for the cases of adding 532 nm with b3 = ± 0.00004, the laser amplitudes around t = 1.0 T are large enough. Thus, the second reversed sub-peaks from t = 0.6 T to t = 1.0 T are very obvious. Consequently, the deceleration and further acceleration of the free electron can be found in this sub-peak structure, which leads to the folded region on P2, as shown in Fig. 10(a) and (c). This is the reason behind the enhancement of single-order harmonic shown in Fig. 9(a). While, when b3 is chosen to be zero, the laser amplitude around t = 1.0 T is decreased compared with those from two-color field or three-color field with b3 = ± 0.00004. Thus, the deceleration process of the free electron and the corresponding folded region on P2 can not be found, as shown in Fig. 10(b), which is responsible for the disappearance of the single-order harmonic enhancement. For the cases of adding 400 nm pulse with b3 = −0.00001 [Fig. 11(a)] and 267 nm pulse with b3 = −0.00005 [Fig. 11(b)], the number of reversed sub-peaks from t = 0 to t = 1.25 T will be increased. Thus, the harmonic emission process in this region will be much more complicated, as similarly discussed in Fig. 8(a). However, the folded structure around the singleorder harmonic region can also be found on P2, thus leading to the enhancement of single-order harmonic on the HHG spectrum. However, due to the increased laser amplitudes around D point, the intensity of P1 is also enhanced compared with that from adding 532 nm pulse case. As a consequence, the intensity and the interference structure of the spectral region from 50th order to 200th order can also be enhanced. Thus, the signal to noise ratio of the selective single-order harmonic from adding 400 nm or 267 nm pulses will be decreased compared with that from adding 532 nm pulse.
Fig. 8. The laser profiles and the time-frequency analyses of HHG for the cases of adding (a) 10 fs-400 nm pulse with b1 = −0.00004, b2 = 0 and (b) 10 fs1064 nm pulse with b1 = −0.00005, b2 = 0.
3.2. Selective generation of single-order harmonics via three-color laser field As we know that the lower conversion efficiency of HHG is a shortcoming for its applications. Thus, in this part, we try to improve the efficiency of HHG spectrum by adding a third controlling pulse. The first two-color field is also chosen to be 20 fs-1600 nm and 10 fs-800 nm with I1 = I2 = 1.0 × 1014W/cm2. The chirp parameters of these two pulses are chosen to be b1 = −0.00006 and b2 = 0.00006. Firstly, the third controlling pulse is chosen to be 6.67 fs-532 nm with I3 = 1.0 × 1014W/cm2 and the effect of b3 on the enhancement of single-order harmonic driven by this three-color field has been shown in Fig. 9(a). It is found that with the introduction of third controlling pulse, the intensity of total HHG spectrum can be enhanced by 300 times compared with that from two-color field. Moreover, by properly choosing the chirp parameter of the third pulse, the selective enhancement of single-order harmonic with the intensity enhancement of 20 times can be found (i.e. b3 = ± 0.00004). Due to the improvement of total HHG yield, the generation of single-order harmonic from three6
Chemical Physics 527 (2019) 110497
Y. Li, et al.
Fig. 9. (a) The effect of b3 on the enhancement of single-order harmonic driven by three-color field. The first two-color field is also chosen to be 20 fs-1600 nm and 10 fs-800 nm with I1 = I2 = 1.0 × 1014W/cm2. The chirp parameters of these two pulses are chosen to be b1 = −0.00006 and b2 = 0.00006. The third controlling pulse is 6.67 fs-532 nm with I3 = 1.0 × 1014W/cm2. (b) The wavelength effect of the third pulse on the enhancement of single-order harmonic. The pulse intensity and pulse duration of the third pulse are unchanged.
Fig. 11. The laser profiles and the time-frequency analyses of HHG for the cases of adding (a) the third 6.67 fs-400 nm pulse with b3 = −0.00001 and (b) the third 6.67 fs-267 nm pulse with b3 = −0.00005.
enhancement of single-order and even two-color harmonics with the intensity enhancement of 17 times can be obtained. Moreover, the spectral region of the selective single-order harmonic is from 182th order to 328th order. Theoretical analyses show that the enhancement of the specific harmonics is coming from the folded structure on the HRP, which is attributed to the deceleration and further acceleration of the free electron in the reversed sub-peak region of the half-cycle laser waveform. Further, with the introduction of the third controlling pulse, the efficiency of HHG spectrum can be enhanced by almost 300 times compared with that from two-color field. In addition, the selective generation of the single-order harmonic with the intensity enhancement of 20 times can also be found. The present paper proposes an attractive method to obtain the high-intensity single-order harmonic, which is helpful for the attosecond science community. It should be illustrated that the final HHG yield is determined not only by the single-atom response, but also by the macroscopic response [63–65]. However, the present investigation only considers the singleatom response of HHG, which is a shortcoming of this paper. Thus, both
Fig. 10. The laser profiles and the time-frequency analyses of HHG for the cases of adding the third 532 nm controlling pulses with (a) b3 = −0.00004; (b) b3 = 0.0 and (c) b3 = 0.00004.
4. Conclusion In conclusion, we theoretically investigate the laser waveform control in the enhancement of single-order and even two-order harmonics via two-color and three-color combined fields. It is found that with the optimization of two-color laser waveform, the selective 7
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considering the single-atom and macroscopic response on HHG is necessary in further investigations.
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8
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Selective enhancement of single-order and two-order harmonics from He atom via two-color and three-color laser fields Yi Lia, Liqiang Fenga,b, Yan Qiaob,c,
T
⁎
a
Laboratory of Molecular Reaction Dynamics, Liaoning University of Technology, Jinzhou 121001, China State Key Laboratory of Molecular Reaction Dynamics, Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Dalian 116023, China c Department of Pathophysiology, Basic Medical College of Zhengzhou University, Zhengzhou 450001, China b
ARTICLE INFO
ABSTRACT
Keywords: High-order harmonic generation Selective enhancement of single-order harmonics Selective enhancement of two-order harmonics Laser waveform control Two-color and three-color combined fields
An effective method to achieve the selective enhancement of single-order and two-order harmonics from He atom has been proposed through laser waveform control of two-color and three-color laser fields. Firstly, with the optimization of ω-2ω two-color laser waveform, not only the wavelength tunable single-order harmonic (i.e. from 182th order to 328th order) can be enhanced by 17 times compared with the neighbor harmonics, but also the enhancement of two-order harmonics can also be obtained. Theoretical analyses show that the enhancement of the specific harmonics is coming from the folded structure on the short quantum path of the harmonic emission peak. Detail analyses show that the above folded structure happens in the reversed sub-peak region of the half-cycle laser waveform, where the deceleration and further acceleration of the free electron can be found in this structure. Moreover, this structure is dependent on the pulse duration of the controlling pulse. However, it is not very sensitive to the wavelength of the controlling pulse. Further, by properly adding a 3ω pulse, not only the efficiency of spectral continuum can be enhanced by 300 times compared with that from the two-color field, but also the generation of single-order harmonic with the intensity enhancement of 20 times can be found. Furthermore, the enhancement of the harmonic spectrum can also be achieved when using the 4ω or 6ω controlling pulses.
1. Introduction High-order harmonic generation (HHG) as one of the most important nonlinear optical phenomena in strong field physics has been investigated over 30 years due to its important applications in (a) the generations of single attosecond pulses (SAPs) [1–8] by filtering the spectral continuum or (b) the selective generation of single-order harmonic from the spectral continuum [9–11]. Currently, the HHG can be obtained from atoms, molecules and solids [12–15]. However, to obtain the useful light sources in practice, the HHG from atoms and molecules are the common schemes, which can be explained by the semi-classical three-step model (TSM) with the processes of ‘ionization, acceleration and recombination’ [16]. On the basis of TSM, (a) the harmonic cutoff is proportional to the laser intensity and the square of the laser wavelength. (b) The harmonic yield is sensitive to the ionization probability and the occupancy of the ground state. Thus, to produce the intense SAPs in XUV and X-ray regions, many attractive schemes have been proposed, such as, the waveform control in HHG by using (1) the multi-color laser field [17–22]; (2) the
⁎
frequency-chirp modulations [23–25]; (3) the polarization gating technology [26–28] and (4) the inhomogeneous effect of the laser field [29–34]. The improvement of harmonic yield by using (5) the high Rydberg initial state in atomic system [35,36]; (6) the high vibrating initial state in molecular system [37] and (7) the resonance enhancement ionization scheme [38,39]. As discussed before, the generation of single-order harmonic is another important application in HHG [9–11,40–43]. Generally, the enhancement of the specific harmonic order can be achieved by using the intra-atomic phase matching scheme via the single-color, two-color, and even three-color fields [9–11,40–43]. However, the spectral region of the enhanced single-order harmonic is narrow and the enhanced ratio between the selective harmonic and its neighbors is only by a factor of 4 times. Recently, with the development of optical parametric chirped-pulse amplification technology, it is possible to generate any optical waveform via the multi-color combined fields [44–51]. Thus, in this paper, through the waveform control of the two-color and three-color laser fields, we found the optimal waveforms to achieve the enhancement of
Corresponding author. E-mail address: [email protected] (Y. Qiao).
https://doi.org/10.1016/j.chemphys.2019.110497 Received 24 March 2019; Received in revised form 12 May 2019; Accepted 15 August 2019 Available online 16 August 2019 0301-0104/ © 2019 Elsevier B.V. All rights reserved.
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single-order and even two-color harmonics. The spectral region of the selective single-order harmonic is from 182th order to 328th order. Moreover, the enhanced ratios between the selective harmonics and the neighbors are from 17 times to 20 times. Theoretical analyses show that the enhancement of the specific harmonics is coming from the folded structure on the short quantum path of the harmonic emission process, which is attributed to the deceleration and further acceleration of the free electron in the reversed sub-peak region of the half-cycle laser waveform. 2. Theoretical part The model is chosen to be He atom in this paper. In detail, He atom driven by the laser field can be described by the three dimensional timedependent Schrödinger equation [52–60],
(r , t ) = H (t ) (r , t ) = t
i
1 2
2
+ V (r ) + z·E (t )
(r , t ),
Fig. 1. The effect of b1 on the HHG spectra driven by two-color laser field. The two-color field is combined by 20 fs-1600 nm and 10 fs-800 nm with I1 = I2 = 1.0 × 1014W/cm2.
(1)
3. Results and discussion
where, V (r ) = 1.535 r is the soft Coulomb potential of He. E (t ) = E (t ) z is the linearly polarized laser field, which can be expressed as, 2 j]
Ej exp[ 4 ln(2)(t )2
E (t ) =
cos( j t + bj t 2),
3.1. Selective enhancement of single-order and two-color harmonics via two-color laser field In this part, we try to find the specific waveforms to achieve the enhancement of single-order and even two-order harmonics. Here, the waveform of the laser field is mainly controlled by two chirp parameters (b1, b2). Fig. 1 shows the effect of b1 on the HHG spectra driven by a twocolor laser field. The two-color field is combined by 20 fs-1600 nm and 10 fs-800 nm with I1 = I2 = 1.0 × 1014W/cm2. For the chirp-free pulse case (b1 = b2 = 0), a spectral continuum from 50th order to 130th order can be found. Moreover, the intensity of spectral region from 55th order to 70th order is enhanced by 6 times compared with the neighbors. As the negative chirp of 1600 nm pulse added, not only the harmonic cutoff is extended, but also the enhanced spectral region is extended. Especially for the cases of b1 = −0.00007, b2 = 0 and b1 = −0.00006, b2 = 0, the selective enhancement of the single 291th order and 328th order harmonics can be obtained, respectively. The enhanced ratios between the above two selective harmonics and their neighbors are 17 times. To better understand the harmonic emission process and the reason behind the enhancement of single-order harmonic, we present the timeprofiles of laser fields and the time-frequency analyses of HHG for some specific chirp modulations, as shown in Fig. 2. In detail, for the case of b1 = b2 = 0 [Fig. 2(a)], there are lots of harmonic radiation peaks (HRPs) during the laser-He atom driven time, which are marked as Pi. As can be seen, although the largest emitted photon energy comes from P1, its intensity is very weak and it can be ignored compared with the others. On the contrary, the intensity of P2 is very strong and the cutoff of P2 is around 130th order, which means the signal of spectral continuum mainly comes from P2. Generally, on the basis of TSM, the HRP is contributed by two harmonic emission paths. In detail, when the electron is ionized around A point, the free electron with the later ionization time will first recombines with its parent ion around B point, thus leading to the short quantum path of P2; while, the free electron with the former ionization time will later recombines with its parent ion around C point, which leads to the long quantum path of P2. However, there is a reversed sub-peak from t = 0.25 T to t = 0.75 T in this waveform (from t = 0 to t = 1.0 T). T is the optical cycle of 1600 nm pulse. Thus, the free electron will experience the deceleration and further acceleration in this sub-peak region, which leads to a folded structure (open white circles) on the short quantum path of P2. For the chirp-free pulse case, this folded region covers from 55th order harmonic to 70th order harmonic. Thus, the intensity of this spectral region is higher than
(2)
j =1 3
where, Ej, ωj, τj and bj are the laser amplitudes, the laser frequencies, the full width at half maximum (FWHM) and the linear chirp parameters of the three-color field. It should be illustrated that the units of the chirp parameters bj are rad/s2. However, for convenience, the units of chirp parameters are omitted in the following discussion. (r , t ) is the time-dependent wave function, which can be expressed as, lmax = 50
(r , t ) = l=0
1 (r , t ) Yl0 ( ). r l
(3)
Here, l (r , t ) and Yl0 ( ) are the radial and spherical harmonics functions. Put Eq. (3) into the TDSE, we can obtain a set of coupled partial differential equations, i
t
l
1 2 l (l + 1) + V (r ) + 2 r2 2r 2
(r , t ) =
l
(r , t ) + zE (t )[cl
l + 1 (r ,
t ) + cl
1
l 1 (r ,
t )].
(4) (l + 1)2 (2l + 3)(2l + 1)
Here cl =
are the coupling constants related to Clebsch-
Gordan coefficients. In the present calculations, the radius and the number of partial waves are chosen to be r = 800 a.u. and lmax = 50, respectively. The grid spacing of radius is chosen to be dr = 0.1 a.u.. Further, the timedependent wave function can be advanced using the standard secondorder split-operator method with dt = 0.05 a.u. [61]. Through Fourier transforming the time-dependent dipole acceleration a(t), the HHG spectrum can be expressed as,
S( ) =
1 2
a (t ) e
i t dt
2
,
(5)
(r , t )|[H (t ), [H (t ), z ]]| (r , t ) . where, a (t ) = The time-frequency analyses (or called the time-profile) of the HHG can be expressed as [62],
A (t , ) =
a (t )
where W (x ) =
1
W ( (t
eixe
x2 2 2
t )) dt ,
(6)
is the Morlet wavelet with ξ = 36 a.u.. 2
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Fig. 2. The laser profiles and the time-frequency analyses of HHG for the cases of (a) b1 = b2 = 0; (b) b1 = -0.00008, b2 = 0; (c) b1 = −0.00007, b2 = 0 and (d) b1 = −0.00006, b2 = 0.
the neighbors. For the case of b1 = −0.00008, b2 = 0 [Fig. 2(b)], as discussed before, when the electron is ionized around A point, the recombination times for the short and long quantum paths are around B and C points, respectively. Compared with the result shown in Fig. 2(a), not only the laser amplitude round t = 0.75 T is enhanced, but also the accelerated time of the free electron is increased. Thus, the free electron will receive more energy during its accelerated process, which is responsible for the extension of P2 and the harmonic cutoff. However, there are two reversed sub-peaks during the harmonic emission process (i.e. from t = 0.25 T to t = 0.75 T and from t = 0.75 T to t = 1.25 T). As a result, two folded structures on the short quantum path of P2 can be found. For the second folded structure (open white circles), the folded region is from 209th order harmonic to 217th order harmonic, which is responsible for the selective enhancement of these harmonics on the HHG spectrum. For the cases of b1 = −0.00007, b2 = 0 [Fig. 2(c)] and b1 = −0.00006, b2 = 0 [Fig. 2(d)], we see that before the recombination process between the free electron and its parent ion (around B point), there are also two reversed sub-peaks (i.e. from t = 0.25 T to t = 0.75 T and from t = 0.75 T to t = 1.25 T). Thus, two folded structures can also be found on the short quantum path of P2. However, for the second folded structure (open white circles), the folded regions are only cover 291th order harmonic (for b1 = −0.00007, b2 = 0 case) and 328th order harmonic (for b1 = −0.00006, b2 = 0 case), respectively, thus leading to the selective enhancement of the single-order harmonic on the HHG spectrum. We see that by properly choosing the chirp of the fundamental field, the enhancement of single-order harmonic can be obtained. Whether the single-order harmonic enhancement can be further controlled? With that in mind, we investigate the effect of b2 on the enhancement of single-order harmonic, as shown in Fig. 3. The chirp parameter of 1600 nm pulse is chosen to be b1 = −0.00006. Clearly, as b2 negatively increases, the enhancement of single-order harmonic is disappeared, as shown in Fig. 3(a). As b2 positively increases from 0 to 0.00008, the wavelength tunable single-order harmonics from 182th order to 328th order can be selectively enhanced by almost 17 times in comparison with their neighbors, as shown in Fig. 3(b). As b2 further increases (i.e.
0.00009 ≤ b2 ≤ 0.00015), the enhancement of two-order harmonics with the intensity improvement of nearly 17 times can be found, as shown in Fig. 3(c). To understand the mechanism behind the order-change of the enhanced single-order harmonic or the enhancement of two-order harmonics, in Fig. 4, we present the time-profiles of laser fields and the time-frequency analyses of HHG for some specific chirp modulations described in Fig. 3. Through analyzing the results shown in Fig. 2, we know that the enhancement of single-order harmonic is attributed to the second folded structure on the HRP. Now, for the case of b1 = −0.00006, b2 = −0.00004 [Fig. 4(a)], we see that as the negative chirp of 800 nm pulse introduced, the second reversed sub-peak is disappeared. Thus, only the first folded structure with the lower-order harmonics can be found on the short quantum path of P2. This is the reason behind the decrease and even disappearance of the single-order harmonic enhancement in the higher-order harmonic region. For the case of b1 = −0.00006, b2 = 0.00006 [Fig. 4(b)], there are three subpeaks from t = 0 to t = 1.75 T. However, before the recombination time between the free electron and its parent ion (around B point), there are still two sub-peaks in this region, which leads to two folded structures on P2. For the second folded structure (open white circles), the folded harmonic order is around 205th order, which is responsible for the enhancement of this harmonic on the HHG spectrum. For the case of b1 = −0.00006, b2 = 0.00011 [Fig. 4(c)], before the recombination time for the short quantum path of P2 (around B point), there are three sub-peaks in this region, which leads to three folded structures on P2. For the second and third folded structures (open white circles), the folded regions cover 151th order harmonic and 355th order harmonic, respectively, which is the reason behind the enhancement of two-order harmonics on the HHG spectrum. Of course, as b2 changes, the subpeaks in the half laser waveform (from t = 0 to t = 1.75 T) will be slightly changed, which leads to the different folded regions on the short quantum path of P2 and is responsible for the selective enhancement of the wavelength tunable single-order and two-order harmonics on the HHG spectrum. Through the above analyses, we see that by properly controlling the 3
Chemical Physics 527 (2019) 110497
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Fig. 3. The effect of b2 on the HHG spectra driven by two-color laser field. The chirp parameter of 1600 nm pulse is chosen to be b1 = −0.00006.
Firstly, the pulse duration effect of 800 nm pulse on the enhancement of single-order harmonic has been shown and discussed in Figs. 5 and 6. Here the pulse duration means the total pulse duration of the laser field. As can be seen, in Fig. 5, for the longer pulse duration (i.e. 20 T800 ∼ 12 T800, where T800 is the optical cycle of 800 nm pulse), a broader spectral region can be enhanced and the bandwidth of the enhanced spectral region is reduced as pulse duration decreased. When the pulse duration of 800 nm pulse is chosen from 11 T800 to 8 T800, the enhancement of the nearly single-order harmonic can be found. While, as pulse duration of 800 nm pulse further decreases, the enhancement of single-order harmonic is disappeared. To clearly see the pulse duration effect on the single-order harmonic enhancement, the laser profiles and the corresponding time-frequency analyses of HHG for different 800 nm pulse durations have been shown in Fig. 6. In detail, for the cases of longer pulse durations (i.e. 20 T800 and 13 T800), the laser amplitudes around t = 1.25 T are large enough and the second reversed sub-peaks from t = 0.75 T to t = 1.25 T are very obvious. As a result, the deceleration of the free electron in this sub-peak region is very obvious, which leads to a broader folded region on the short quantum path of P2, as shown in Fig. 6(a) and (b). This is the reason behind the enhancement of the broader spectral region on the HHG spectrum. For the cases of middle pulse durations [i.e. 11 T800, 10 T800 (corresponding to 10 fs-800 nm pulse), 9 T800 and 8 T800], the laser amplitudes around t = 1.25 T are decreased compared with those from the longer pulse durations. Thus, the deceleration of the free electron will be reduced in comparison with those from the longer pulse durations, which leads to the reduction of the folded region on the short quantum path of P2, as shown in Fig. 6(c)–(e). Particularly, for the cases of 11 T800, 9 T800 and 8 T800, the folded regions are around the single 328th order, 312th order and 311th order harmonics, respectively, which are responsible for the enhancement of these single-order harmonics on the HHG spectra. For the case of shorter pulse duration (i.e. 7 T800), the second reversed sub-peak from t = 0.75 T to t = 1.25 T is almost disappeared. Therefore, there is no folded region in the largerorder harmonic region [see Fig. 6(f)], which is responsible for the disappearance of the single-order harmonic enhancement. Secondly, the wavelength effect of the controlling pulse on the enhancement of single-order harmonic has been shown and discussed in Figs. 7 and 8. As can be seen in Fig. 7, when using the 10 fs-400 nm [Fig. 7(a)] or the 10 fs-1064 nm [Fig. 7(b)] controlling pulses, the
Fig. 4. The laser profiles and the time-frequency analyses of HHG for the cases of (a) b1 = −0.00006, b2 = −0.00004; (b) b1 = −0.00006, b2 = 0.00006 and (c) b1 = −0.00006, b2 = 0.00011.
laser waveform via the two-color chirped pulse, the wavelength tunable single-order harmonics (or two-order harmonics) can be selectively enhanced. Considering the practical applications, the generation of single-order harmonic is more useful. Thus, in the following discussion, we will focus on the laser parameters effects on the single-order harmonic enhancement. 4
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Fig. 5. The effect of 800 nm pulse duration on the enhancement of single-order harmonic.
Fig. 6. The laser profiles and the time-frequency analyses of HHG for the controlling pulse durations of (a) 20 T800; (b) 13 T800; (c) 11 T800; (d) 9 T800; (e) 8 T800 and (f) 7 T800.
enhancement of single-order and even two-order harmonics can also be found. That is to say the enhancement of single-order harmonic is not very sensitive to the wavelength of the controlling pulse. However, the enhanced ratios of the single-order harmonics from adding 400 nm and adding 1064 nm pulses are lower than those from adding 800 nm controlling pulse. To clearly see the controlling wavelength effect on the single-order harmonic enhancement, the laser profiles and the corresponding time-frequency analyses of HHG for the cases of adding 400 nm and 1064 nm controlling pulses have been shown in Fig. 8. In detail, for the case of adding 400 nm controlling pulse with b1 = −0.00004, b2 = 0 [Fig. 8(a)], the number of sub-peaks from t = 0 to t = 1.25 T are increased. Thus, the harmonic emission process in this region becomes much more complicated. For instance, due to the multisub-peaks structure, when the electron is ionized around A point, it will experience many times ‘acceleration-deceleration-further acceleration’
before it recombines with its parent ion around B point. Thus, there are three folded regions on the HRP. The first folded region covers from 80th order harmonic to 110th order harmonic, thus leading to the little enhancement of this spectral region on the HHG spectrum. The second and the third folded regions are around 165th order harmonic, which leads to the enhancement of this single-order harmonic on the HHG spectrum. For the case of adding 1064 nm controlling pulse with b1 = −0.00005, b2 = 0 [Fig. 8(b)], after the electron ionized around A point and before the free electron recombined with its parent ion around B point, there is only one reversed sub-peak during this half laser waveform region. Therefore, only one folded structure with the folded region around 119th order harmonic can be found on P2, which is responsible for the enhancement of this single-order harmonic on the HHG spectrum.
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Fig. 7. The wavelength effect of the second pulse on the enhancement of single-order harmonic (a) λ2 = 400 nm and (b) λ2 = 1064 nm. The laser intensities are unchanged.
color field will be much more useful than that from two-color field. Furthermore, by properly adding 400 nm (4ω pulse) or 267 nm controlling pulses (6ω pulse), both the improvement of total HHG yield and the enhancement of single-order harmonic can be obtained, as shown in Fig. 9(b). However, the signal to noise ratio of the enhanced singleorder harmonic from adding 400 nm or 267 nm pulses are decreased compared with that from adding 532 nm pulse. To understand the improvement of total HHG yield and the singleorder harmonic in three-color field, in Figs. 10 and 11, we present the laser profiles and the time–frequency analyses of HHG for the cases of adding 532 nm, 400 nm and 267 nm controlling pulses. Clearly, with the introduction of the third pulse, the laser amplitudes around t = 0 for all cases are increased compared with that from two-color field. Thus, the ionization probability around A point can be increased, which is responsible for the intensity enhancement of total HHG yield. In detail, for the cases of adding 532 nm with b3 = ± 0.00004, the laser amplitudes around t = 1.0 T are large enough. Thus, the second reversed sub-peaks from t = 0.6 T to t = 1.0 T are very obvious. Consequently, the deceleration and further acceleration of the free electron can be found in this sub-peak structure, which leads to the folded region on P2, as shown in Fig. 10(a) and (c). This is the reason behind the enhancement of single-order harmonic shown in Fig. 9(a). While, when b3 is chosen to be zero, the laser amplitude around t = 1.0 T is decreased compared with those from two-color field or three-color field with b3 = ± 0.00004. Thus, the deceleration process of the free electron and the corresponding folded region on P2 can not be found, as shown in Fig. 10(b), which is responsible for the disappearance of the single-order harmonic enhancement. For the cases of adding 400 nm pulse with b3 = −0.00001 [Fig. 11(a)] and 267 nm pulse with b3 = −0.00005 [Fig. 11(b)], the number of reversed sub-peaks from t = 0 to t = 1.25 T will be increased. Thus, the harmonic emission process in this region will be much more complicated, as similarly discussed in Fig. 8(a). However, the folded structure around the singleorder harmonic region can also be found on P2, thus leading to the enhancement of single-order harmonic on the HHG spectrum. However, due to the increased laser amplitudes around D point, the intensity of P1 is also enhanced compared with that from adding 532 nm pulse case. As a consequence, the intensity and the interference structure of the spectral region from 50th order to 200th order can also be enhanced. Thus, the signal to noise ratio of the selective single-order harmonic from adding 400 nm or 267 nm pulses will be decreased compared with that from adding 532 nm pulse.
Fig. 8. The laser profiles and the time-frequency analyses of HHG for the cases of adding (a) 10 fs-400 nm pulse with b1 = −0.00004, b2 = 0 and (b) 10 fs1064 nm pulse with b1 = −0.00005, b2 = 0.
3.2. Selective generation of single-order harmonics via three-color laser field As we know that the lower conversion efficiency of HHG is a shortcoming for its applications. Thus, in this part, we try to improve the efficiency of HHG spectrum by adding a third controlling pulse. The first two-color field is also chosen to be 20 fs-1600 nm and 10 fs-800 nm with I1 = I2 = 1.0 × 1014W/cm2. The chirp parameters of these two pulses are chosen to be b1 = −0.00006 and b2 = 0.00006. Firstly, the third controlling pulse is chosen to be 6.67 fs-532 nm with I3 = 1.0 × 1014W/cm2 and the effect of b3 on the enhancement of single-order harmonic driven by this three-color field has been shown in Fig. 9(a). It is found that with the introduction of third controlling pulse, the intensity of total HHG spectrum can be enhanced by 300 times compared with that from two-color field. Moreover, by properly choosing the chirp parameter of the third pulse, the selective enhancement of single-order harmonic with the intensity enhancement of 20 times can be found (i.e. b3 = ± 0.00004). Due to the improvement of total HHG yield, the generation of single-order harmonic from three6
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Fig. 9. (a) The effect of b3 on the enhancement of single-order harmonic driven by three-color field. The first two-color field is also chosen to be 20 fs-1600 nm and 10 fs-800 nm with I1 = I2 = 1.0 × 1014W/cm2. The chirp parameters of these two pulses are chosen to be b1 = −0.00006 and b2 = 0.00006. The third controlling pulse is 6.67 fs-532 nm with I3 = 1.0 × 1014W/cm2. (b) The wavelength effect of the third pulse on the enhancement of single-order harmonic. The pulse intensity and pulse duration of the third pulse are unchanged.
Fig. 11. The laser profiles and the time-frequency analyses of HHG for the cases of adding (a) the third 6.67 fs-400 nm pulse with b3 = −0.00001 and (b) the third 6.67 fs-267 nm pulse with b3 = −0.00005.
enhancement of single-order and even two-color harmonics with the intensity enhancement of 17 times can be obtained. Moreover, the spectral region of the selective single-order harmonic is from 182th order to 328th order. Theoretical analyses show that the enhancement of the specific harmonics is coming from the folded structure on the HRP, which is attributed to the deceleration and further acceleration of the free electron in the reversed sub-peak region of the half-cycle laser waveform. Further, with the introduction of the third controlling pulse, the efficiency of HHG spectrum can be enhanced by almost 300 times compared with that from two-color field. In addition, the selective generation of the single-order harmonic with the intensity enhancement of 20 times can also be found. The present paper proposes an attractive method to obtain the high-intensity single-order harmonic, which is helpful for the attosecond science community. It should be illustrated that the final HHG yield is determined not only by the single-atom response, but also by the macroscopic response [63–65]. However, the present investigation only considers the singleatom response of HHG, which is a shortcoming of this paper. Thus, both
Fig. 10. The laser profiles and the time-frequency analyses of HHG for the cases of adding the third 532 nm controlling pulses with (a) b3 = −0.00004; (b) b3 = 0.0 and (c) b3 = 0.00004.
4. Conclusion In conclusion, we theoretically investigate the laser waveform control in the enhancement of single-order and even two-order harmonics via two-color and three-color combined fields. It is found that with the optimization of two-color laser waveform, the selective 7
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considering the single-atom and macroscopic response on HHG is necessary in further investigations.
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