Molecular photoelectron holography with circularly polarized laser pulses Weifeng Yang, Zhihao Sheng, Xingpan Feng, Miaoli Wu, Zhangjin Chen, and Xiaohong Song* Department of Physics, College of Science, Shantou University, Shantou, Guangdong 515063, China * [email protected]
Abstract: We investigate the photoelectron momentum distribution of molecular-ion H 2+ driven by ultrashort intense circularly polarized laser pulses. Both numerical solutions of the time-dependent Schrödinger equation (TDSE) and a quasiclassical model indicate that the photoelectron holography (PH) with circularly polarized pulses can occur in molecule. It is demonstrated that the interference between the direct electron wave and rescattered electron wave from one core to its neighboring core induces the PH. Moreover, the results of the TDSE predict that there is a tilt angle between the interference pattern of the PH and the direction perpendicular to the molecular axis. Furthermore, the tilt angle is sensitively dependent on the wavelength of the driven circularly polarized pulse, which is confirmed by the quasiclassical calculations. The PH induced by circularly polarized laser pulses provides a tool to resolve the electron dynamics and explore the spatial information of molecular structures. ©2014 Optical Society of America OCIS codes: (190.7110) Ultrafast nonlinear optics; (320.7150) Ultrafast spectroscopy.
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#201832 - $15.00 USD Received 22 Nov 2013; revised 13 Jan 2014; accepted 13 Jan 2014; published 29 Jan 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.002519 | OPTICS EXPRESS 2519
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1. Introduction To image electron motion and the structures in atoms and molecules is one of the most important aims in ultrafast physics and attosecond science [1–11]. Driven by a strong laser field, a liberated electron is accelerated and redirected to the parent ion within a very short duration [12]. Electron-ion recollision leads to a number of typical processes, such as highorder harmonic generation, above-threshold ionization, and nonsequential double ionization. Electron motion in these processes encodes detailed structural and dynamical information of the atoms or molecules [4–11]. Recently, a novel interference structure of photoelectron holography (PH) has been observed in experiment where metastable Xenon atoms are ionized by intense 7-micrometer pulses from a free-electron laser [13]. The PH records underlying electron dynamics on a sub-cycle time scale, enabling photoelectron spectroscopy with a time resolution almost two orders of magnitude higher than the duration of the ionization pulse [13]. In traditional optical holography [14], laser light is divided into two beams: one beam illuminates a target and then imprints on a recording medium. This beam is the signal wave. The other beam which is known as the reference wave is shone directly onto the recording medium. The interference between a signal and a reference wave is recorded to reconstruct the information of the target. Since the laser-driven electron motion is fully coherent, the concept of holography can be extended to strong field ionization. In strong field ionization, the recollision electron can be used as the signal wave and the ion itself is the target, while direct photoelectron which does not interact with the ion is the reference wave. The signal and reference waves with the same final momentum interfere with each other and record the PH encoding temporal and spatial information about both the ion and the recollision electron [13,15,16]. A simulation by solving the time-dependent Schrödinger equation (TDSE) has well reproduced the experimental results of the Xenon atom [16]. Further investigation shows that both short and long rescattered electron trajectories in molecule can be imaged by PH with attosecond temporal resolution [17]. It is indicated that PH induced by intense near-infrared laser pulse can occur not only in the tunneling regime but also in the multiphoton regime [18]. In [19], Y. Huismans et al. reported the scaling laws of PH interference patterns with the laser pulse duration, wavelength, and intensity. So far, investigations about PH were based on the physical picture of electron-ion recollision driven by a linearly polarized laser pulse. When diatomic molecules are driven by
#201832 - $15.00 USD Received 22 Nov 2013; revised 13 Jan 2014; accepted 13 Jan 2014; published 29 Jan 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.002519 | OPTICS EXPRESS 2520
a linearly polarized laser pulse, the interference patterns from different rescattering cores will overlap [17]. To our knowledge, the PH with a circularly polarized laser pulse has not been reported yet. In this work, we explore the PH of molecular-ion H 2+ driven by intense circularly polarized laser pulses. When the molecule is driven by a circularly polarized laser pulse, the liberated electron cannot collide with its parent ion. Consequently, the physical picture of the molecular PH with circularly polarized laser pulses is much clearer. It is found that the interference between the direct photoelectron and rescattered photoelectron from one core to its neighboring core induces the molecular PH with circularly polarized laser pulses. This is demonstrated by numerical solutions of the TDSE and confirmed by simulations with a quasiclassical model. Moreover, both the results of the TDSE and the quasiclassical calculation indicate that there is a tilt angle between the interference pattern of the PH and the direction perpendicular to the molecular axis, and the tilt angle becomes smaller with the increasing of the wavelength of the driven circularly polarized laser. This paper is organized as follows. The numerical methods are briefly introduced in Sec. II. The characteristics of the PH and the physical mechanism of the PH are discussed in Sec III. We summarize our results and conclude in Sec. IV. 2. Theory We have carried on the numerical calculation by solving the two-dimensional (2D) TDSE. i
p2 ∂ Ψ (r , t ) = + p ⋅ A ( t ) + V (r ) Ψ (r , t ) ∂t 2
(1)
Here, V(r) is the soft-Coulomb potential of H 2+ ion. In our computation
V ( x, y | R ) =
−Z ( x + R 2) + y + a 2
2
+
−Z ( x − R 2) 2 + y 2 + a
,
(2)
where R is the internuclear distance, Z is the effective charge (Z = 1), and a is the parameter of soft-Coulomb potential (a = 0.47, the corresponding ionization potential of H 2+ reproduced 1sσ
here is I p g = 0.65 a.u. when R = 10 a.u.). We assume that the molecular axis is coincident with the x axis. The wave function at a given time ti is split as [20–22] Ψ (ti ) = Ψ (ti )[1 − Fs ( rc )] + Ψ (ti ) Fs ( rc ) = Ψ I (ti ) + Ψ II (ti )
(3)
Here, Fs ( rc ) = 1 (1 + e − ( r − rc )/ Δ ) is a split function that separates the whole space wave function Ψ (ti ) into the inner (0 → rc ) wave function Ψ I (ti ) and outer ( rc → rmax ) wave function Ψ II (ti ) . ∆ represents the width of crossover region. We choose rmax = 409.6 a.u., rc = 100 a.u., and Δ = 20 a.u.. The exact time evolution of Ψ (ti ) is evaluated using the Crank-Nicolson method [21–24]. We calculate C ( p, ti ) = Ψ II (ti )
e − i [ p − A ( ti )]⋅r 2 d r, 2π
(4)
then Ψ II is propagated to the final time as
Ψ II ( ∞, ti ) = C (p, ti )
eip⋅r 2 d p, 2π
(5)
#201832 - $15.00 USD Received 22 Nov 2013; revised 13 Jan 2014; accepted 13 Jan 2014; published 29 Jan 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.002519 | OPTICS EXPRESS 2521
∞1 2 with C ( p, ti ) = exp −i [ p − A (t ')] dt ' C ( p, ti ) . The outer region wave function is ti 2 propagated by the above Eq. (4) so that there is no boundary problem any more. The final momentum distribution is obtained as
dP( p) = 2 En dEn
C ( p, t ) . i
(6)
i
Here, En is the electron energy associated with p . The total pulse duration Tp is 13 optical cycles. The vector potential of the pulse is A (t ) = eˆx
Ex0
ω
f (t )sin(ωt ) + eˆ y
E y0
ω
f (t ) cos(ωt )
(7)
where eˆx / y is the polarization direction and ε = E y 0 / E x 0 = 1 ( E x 0 = E0 and E y 0 = ε E0 ) is the
ellipticity of the laser pulse, with a smooth sine squared pulse envelope f (t ) = sin 2 (π t / Tp ) . E0 is 0.134 a.u. corresponding to a peak intensity 6.0 × 1014 W/cm2. This vector potential is to ∂A is zero. ensure the total electric area of the field E (t ) = − ∂t 3. Results and discussions
Figure 1 reports the TDSE simulation of photoelectron momentum distribution (PMD) of H 2+ driven by a circularly polarized laser pulse. Short wavelength reduces the excursion time so that the recollision rate with the neighboring core can be increased. Moreover, it has been demonstrated that the tunneling regime is not a necessary condition for the holographic pattern, which can be observed under the conditions formally attributed to multiphoton regime [18]. The appearance of the holographic pattern at short wavelengths requires higher laser intensities [17, 18]. We found that this criterion also applies to the circularly polarized laser pulse case. Thus, the laser frequency used in our simulation is ω = 0.114 a.u. ( λ = 400 nm) and the peak laser intensity is 6.0 × 1014 W/cm2. To clearly show the interference structure, we first give the PMD spectrum on linear scale in Fig. 1(a). Two kinds of interference patterns with different tilt angles from the direction perpendicular to the molecular axis are observed (denoted by “T” and “P”, respectively). The PMD on logarithmic scale in Fig. 1(b) further show that both of these two interference patterns have a cutoff of Pcutoff = 2.356 = 4 U p (Up = E02/4ω2) corresponding to 8Up in energy. As we know, for a circularly polarized laser pulse, the cutoff energy of the direct photoelectron is 8Up [25]. Hence, the direct photoelectron plays an essential role in the interference patterns “P” and “T”. The interference pattern “T” (marked by the yellow lines in Fig. 1(a)) comes from the two-center interference generated by direct electrons ionized separately from the two cores. It can be seen that the interference fringes are not perpendicular to the molecular axis. Electron emission from the H 2+ ion driven by a circularly polarized laser pulse with wavelength of 800nm was investigated in previous work [26]. Both experimental and theoretical results demonstrated that there is a rotation angle in PMD, which results from a complex laser-driven electron dynamics inside the molecule influencing the instant of ionization and the initial momentum of the electron [26]. Here similar rotation angle is observed in the two-center interference pattern. In Fig. 1(b), the two-center interference fringes are also observed in the higher energy region with much lower intensity. Recently, it has been demonstrated that for long internuclear distances, the liberated electron collides with its neighboring ion can generate the maximum kinetic energy of 32 Up for both linear and circular polarizations [25].The interference pattern in the high energy is due to the collision of electrons with neighboring cores. Unlike the direct electron, the
#201832 - $15.00 USD Received 22 Nov 2013; revised 13 Jan 2014; accepted 13 Jan 2014; published 29 Jan 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.002519 | OPTICS EXPRESS 2522
interference fringes for the recollision electron are perpendicular to the molecular axis. This is because that the instant of ionization and the initial momentum of the electron have little influence on the recollision electron.
Fig. 1. Photoelectron momentum distribution of H 2+ driven by a circularly polarized laser pulse (a) on linear scale and (b) on logarithmic scale. The parameters of laser electric field are peak intensity I 0 = 6.0 × 1014 W/cm2, wavelength λ = 400 nm, and duration Tp = 17.35 fs. The lines are guides to the eye.
Besides the two-center interference, there exists another kind of unexpected interference pattern “P” (marked by white dashed lines in Fig. 1(a)) which has “fork” structures in the second and fourth quadrants. The “fork” structures have been identified as the typical interference structure of the PH with linearly polarized laser pulses in previous experimental and theoretical works [13, 16]. As a result, we infer that the interference pattern “P” in Fig. 1(a) is the PH with circularly polarized laser pulses. The direct photoelectron serves as a reference wave, and the rescattered electron from the neighboring core serves as signal wave. The interference between them leads to the strong-field PH with circularly polarized laser pulses. Note that the interference pattern “P” also has a tilt angle from the direction perpendicular to the molecular axis. However, this tilt angle is in opposite direction to that of the two-center interference. For comparison, we first construct an atomic-like reference with a single scattering center V ( x, y ) = −1.13 x 2 + y 2 + a which provides the same ionization potential of the corresponding molecule with R = 10 a.u [15]. Figures 2(a) and 2(b) show PMD by numerical simulations of the TDSE for the atomic-like reference driven by a linearly and circularly polarized pulse, respectively. As expected, a “fork” structure, i.e., the interference pattern of the PH, can be identified in atomic-like reference driven by a linearly laser pulse whose polarization direction is along y axis (see Fig. 2(a)). It has been demonstrated that this holographic pattern arises from the interference between the reference and rescattered signal electron wave packets that are emitted during the same quarter cycle of the laser field [15, 16]. The maximum momentum of the electron inducing the PH is smaller than that in Fig. 1(b), since the cutoff energy of the direct electron driven by a linearly polarized laser (2Up) is smaller than that driven by a circularly polarized laser (8Up). When a circularly polarized laser is applied to atom, the electron cannot be driven back to its parent ion, which means there is no signal wave. Therefore, the interference structure of the PH disappears in Fig. 2(b). This is quite different with the case shown in Fig. 1. When H 2+ ion is considered (see Fig. 1), the electron can be rescattered by the neighboring ion. Consequently, the signal wave still exists, and so does the interference structure of the PH. It should be noted that no two-center interference structure occurs in atomic-like reference (see Figs. 2(a) and 2(b)). Next, we
#201832 - $15.00 USD Received 22 Nov 2013; revised 13 Jan 2014; accepted 13 Jan 2014; published 29 Jan 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.002519 | OPTICS EXPRESS 2523
consider the molecule driven by a linearly polarized pulse and assume that the molecular axis is along x direction in Fig. 2(c). The polarization of the laser electric field is perpendicular to the molecular axis. It can be seen that a “fork” interference structure of the PH is present along the polarization direction of the laser pulse. Moreover, two-center interference fringes can be observed. The energy of the cutoff is 10.2Up which corresponds to the maximum kinetic energy of the recolliding electron with its parent ion. Due to the nonzero initial velocity, collision with its neighboring ion can also occur which generates the two-center interference structure in the higher energy region. These interference patterns are similar to those in Fig. 1. However, as expected, there is no tilt angle in this case, and the interference of the PH and two-center interference are in same direction, which is different with that in Fig. 1.
+
Fig. 2. Photoelectron momentum distribution of atomic-like reference [(a) and (b)] and of H 2 [(c)]. The driven lasers are a linearly polarized pulse [(a) and (c)], and a circularly polarized laser pulse [(b)]. The parameters of laser electric field are same with those of Fig. 1. The results are plotted on a logarithmic scale.
To further demonstrate that the interference pattern “P” shown in Fig. 1 is indeed a holographic-type structure, we consider a simplified picture based on a quasiclassical laserinduced collision-recollision model [16, 17, 23, 25]. The quasiclassical analysis allows one to identify the physical process leading to certain interference structure in the photoelectron spectrum. When a diatomic molecule is ionized by a circularly polarized laser pulse, the liberated electron cannot revisit its parent ion and can only be rescattered by the neighboring core. If the PH occurs, the signal wave must be the photoelectron rescattered by the neighboring core. In quasiclassical calculations, we focus on the interference between the photoelectron rescattered by its neighboring core and the direct photoelectron. For simplicity, we assume the envelope f (t ) = 1 and consider the continuous-wave field in the quasiclassical calculations. If the signal electron is ionized at a particular phase φ, the laser electric field is E (t ) = eˆx E0 cos(ωt + φ ) + eˆ y E0 sin(ωt + φ ) . The velocity and the position can be calculated by integrating the classical equation of motion. The initial condition of the electron is at rest immediately after ionization. The velocities along x direction and y direction [in atomic units (a.u.)] are t
v x (t , φ ) = − E x (t ')dt ' = − 0
E0
[sin(ωt + φ ) − sin(φ )]
(8)
[cos(ωt + φ ) − cos(φ )],
(9)
ω
and t
v y (t , φ ) = − E y (t ')dt ' = 0
E0
ω
respectively. The electron displacements x and y can be written as
#201832 - $15.00 USD Received 22 Nov 2013; revised 13 Jan 2014; accepted 13 Jan 2014; published 29 Jan 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.002519 | OPTICS EXPRESS 2524
t
x(t , φ) = v x (t ')dt ' 0
=
E0
ω
[cos(ωt + φ) − cos( φ) + ω sin( φ)t ] − x0 2
(10)
and t
y (t , ϕ ) = v y (t ')dt ' 0
=
E0
[sin(ωt + φ) − sin( φ) − ω cos( φ)t ] − y0 , 2
ω
respectively. Here, r0 = x02 + y02 =
(11)
Ip
is the initial position of electron after ionization. As E0 we know, recollision can only occur between the electron and its neighboring ion when driven by a circularly polarized laser. The displacement in quasiclassical calculations must satisfy x(tc , φ) = ± R and y (tc , φ) = 0 , in which tc is the traveling time of the scattered signal electron. We assume that the signal electron is elastically scattered by the neighboring ion at an angle θc. After the recollision, the finial velocities along the x and y direction are t
v xf (t , φ ) = − E x (t ')dt ' tc
=−
E0
ω
[sin(ωt + φ ) − sin(ωtc + φ )] + v(tc , φ ) cos(θ c )
(12)
and t
v yf (t , φ ) = − E y (t ')dt ' tc
=
E0
ω
[cos(ωt + φ ) − cos(ωtc + φ )] + v(tc , φ ) sin(θ c ).
(13)
The vector potential is A (t ) = 0 after the end of the pulse. Thus, the electron momentum along x direction measured at the detector is Pxf =
E0
ω
sin(ωtc + φ ) + v (tc , φ ) cos(θ c ).
(14)
The electron momentum along y direction is Pyf = −
E0
ω
cos(ωtc + φ ) + v(tc , φ ) sin(θ c ).
(15)
The direct electron, i.e., the reference wave, is ionized at a phase φ ' and its traveling time is tr. The phase accumulated between the reference and the scattered signal waves does not change after the recollision of the signal electron. The traveling time tr up to which the phase accumulation of the reference wave packet needs to be calculated is the time interval between the collision time of the signal wave packet and the ionization time for the reference wave packet [16]. Therefore, if the PH and interference occur, the signal and reference electrons should satisfy the interference condition: ωtc + φ = ωt r + φ ' . The velocities along the x and y direction of reference electron are v xf (t r , φ ') = v(tc , φ ) cos(θ c ), v yf (t r , φ ') = v(tc , φ ) sin(θ c ) .
#201832 - $15.00 USD Received 22 Nov 2013; revised 13 Jan 2014; accepted 13 Jan 2014; published 29 Jan 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.002519 | OPTICS EXPRESS 2525
Based on the above discussion, we can write out the phase difference between the signal electron and the reference electron as 2 2 2 2 tc v ( t ', φ ) + v y ( t ', φ ) t r v ( t ', φ ' ) + v y ( t ', φ ' ) (φ − φ ' ) ΔΦ = x dt '− x . (16) dt '− I p 0 0 ω 2 2
The phase difference between the signal electron and the reference electron will lead to holographic-type interference fringes ~cos(ΔΦ).
Fig. 3. Comparison of photoelectron momentum distribution of the TDSE simulations [(a)-(c)] and interference fringes ∼cos(ΔΦ) of the quasiclassical calculations [(d)-(f)]. The wavelengths of the laser are 320 nm [(a) and (d)], 420 nm [(b) and (e)], and 520 nm [(c) and (f)]. In the quasiclassical calculations, only the photoelectron scattered by the neighboring core is considered as the signal wave of the PH. The phase difference v (t ',φ ) + v (t ',φ ) r v (t ',φ ') + v (t ',φ ') (φ − φ') leads to interference fringes ∼cos(ΔΦ). ΔΦ = c dt '− dt '−I 2
t
0
2
x
y
2
2
t
0
2
x
y
2
p
ω
We compare the interference fringes ~cos(ΔΦ) from the quasiclassical calculations with the interference pattern “P” in TDSE simulations for different laser wavelengths in Fig. 3. The TDSE results are shown in upper panel and the results of solving the quasiclassical model are shown in lower panel. The wavelengths of the driven laser are 320 nm (Figs. 3(a) and (d)), 420 nm [Figs. 3(b) and 3(e)], and 520 nm [Figs. 3(c) and 3(f)]. To guide the eye, solid curves are drawn in Figs. 3(a) to 3(c) to show clearly the title angle trend of the holographic-type interference pattern “P” in TDSE simulations. It can be seen that the results predicted by the quasiclassical model are consistent qualitatively with the TDSE results. Firstly, both the TDSE and the quasiclassical model predict a tilt angle between the interference pattern of the PH and the direction perpendicular to the molecular axis. Secondly, both the TDSE and the quasiclassical model show that the tilt angle is dependent on the laser wavelength. With
#201832 - $15.00 USD Received 22 Nov 2013; revised 13 Jan 2014; accepted 13 Jan 2014; published 29 Jan 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.002519 | OPTICS EXPRESS 2526
increasing laser wavelength, the tilt angle becomes smaller. Compared with the TDSE simulations, the calculations of the quasiclassical model seem to overestimate the tilt angle. It is because that only the laser electric field is considered, while other factors, such as the Coulomb potential and the initial velocity distribution of the electron, are neglected in the calculations of the quasiclassical model. Thirdly, both the quasiclassical model and the TDSE results demonstrate that the cutoff of the holographic-type interference fringes in momentum spectrum becomes larger with longer wavelengths. This is because that the cutoff of the holographic-type interference fringes is determined by the direct electron which has a cutoff energy of 8Up driven by circularly polarized laser pulses. With increasing the laser wavelength, the value of the Up becomes larger, which induces larger cutoff in momentum distribution. As the above discussion, the interference structures in the quasiclassical analysis come from the interference between the direct photoelectron (i.e., the reference wave) and the photoelectron rescattered by its neighboring core (i.e., the signal wave). The results of the quasiclassical calculation agree very well with the simulations of the TDSE. Therefore, it can be confirmed that the rescattered photoelectron from one core to its neighboring core serves as the signal wave which interferes with the direct photoelectron, i.e. the reference wave, and generates the molecular PH with circularly polarized laser pulses. 4. Conclusions
In summary, the PH in molecule induced by a circularly polarized laser pulse has been investigated by solving the corresponding 2D TDSE and a quasiclassical recollision model numerically. The quasiclassical model has been demonstrated to agree with the TDSE results. Both of them show that the interference between the direct photoelectron and rescattered photoelectron from one core to its neighboring core generates the PH in molecule. Moreover, both of the calculations of the TDSE and the quasiclassical model predict that there is a tilt angle between the interference pattern of the PH and the direction perpendicular to the molecular axis. Furthermore, the tilt angle is dependent on the wavelength of the driven circularly polarized laser in molecular PH. Compared with the PH with a linearly polarized laser pulse, the PH with a circularly polarized laser pulse has some advantages: first, only the electron scattered by its neighboring ion contributes to the signal wave, the physical picture of PH with a circularly polarized laser pulse is much clearer; second, the recollision with the neighboring can access the detailed spatial information of the molecule, i.e. the internuclear distance; third, the holographic interference has different rotation angle from the two-center interference, so different interference processes can be resolved. As a result, the interference dynamics of the PH with circularly polarized laser pulses proposed in the present work can offer a new way to explore the electronic dynamics and access molecular spatial structures. Acknowledgments
We thank Dr. J. Chen and Dr. Z. N. Zeng for helpful discussions on PH. The work was supported by the National Natural Science Foundation of China (Grant Nos. 11374202, 11274220, and 11274219), Guangdong Natural Science Foundation (Grant No. S2013010016061), Foundation for High-level Talents in Higher Education of Guangdong, STU Scientific Research Foundation for Talents, the National Basic Research Program of China (Grant No. 2010CB923200), and the Open Fund of the State Key Laboratory of High Field Laser Physics (SIOM). W. Yang was also sponsored by the Foundation for the Author of National Excellent Doctoral Dissertation of PR China (Grant No. 201215) and “Yang Fan” talent project of Guangdong Province.
#201832 - $15.00 USD Received 22 Nov 2013; revised 13 Jan 2014; accepted 13 Jan 2014; published 29 Jan 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.002519 | OPTICS EXPRESS 2527
Abstract: We investigate the photoelectron momentum distribution of molecular-ion H 2+ driven by ultrashort intense circularly polarized laser pulses. Both numerical solutions of the time-dependent Schrödinger equation (TDSE) and a quasiclassical model indicate that the photoelectron holography (PH) with circularly polarized pulses can occur in molecule. It is demonstrated that the interference between the direct electron wave and rescattered electron wave from one core to its neighboring core induces the PH. Moreover, the results of the TDSE predict that there is a tilt angle between the interference pattern of the PH and the direction perpendicular to the molecular axis. Furthermore, the tilt angle is sensitively dependent on the wavelength of the driven circularly polarized pulse, which is confirmed by the quasiclassical calculations. The PH induced by circularly polarized laser pulses provides a tool to resolve the electron dynamics and explore the spatial information of molecular structures. ©2014 Optical Society of America OCIS codes: (190.7110) Ultrafast nonlinear optics; (320.7150) Ultrafast spectroscopy.
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#201832 - $15.00 USD Received 22 Nov 2013; revised 13 Jan 2014; accepted 13 Jan 2014; published 29 Jan 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.002519 | OPTICS EXPRESS 2519
16. X. B. Bian, Y. Huismans, O. Smirnova, K. J. Yuan, M. J. J. Vrakking, and A. D. Bandrauk, “Subcycle interference dynamics of time-resolved photoelectron holography with midinfrared laser pulses,” Phys. Rev. A 84(4), 043420 (2011). 17. X. B. Bian and A. D. Bandrauk, “Attosecond Time-Resolved Imaging of Molecular Structure by Photoelectron Holography,” Phys. Rev. Lett. 108(26), 263003 (2012). 18. T. Marchenko, Y. Huismans, K. J. Schafer, and M. J. J. Vrakking, “Criteria for the observation of strong-field photoelectron holography,” Phys. Rev. A 84(5), 053427 (2011). 19. Y. Huismans, A. Gijsbertsen, A. S. Smolkowska, J. H. Jungmann, A. Rouzée, P. S. W. M. Logman, F. Lépine, C. Cauchy, S. Zamith, T. Marchenko, J. M. Bakker, G. Berden, B. Redlich, A. F. G. van der Meer, M. Yu. Ivanov, T. M. Yan, D. Bauer, O. Smirnova, and M. J. J. Vrakking, “Scaling Laws for Photoelectron Holography in the Midinfrared Wavelength Regime,” Phys. Rev. Lett. 109(1), 013002 (2012). 20. Q. Liao, P. Lu, P. Lan, W. Cao, and Y. Li, “Phase dependence of high-order above-threshold ionization in asymmetric molecules,” Phys. Rev. A 77(1), 013408 (2008). 21. S. Chelkowski, C. Foisy, and A. D. Bandrauk, “Electron-nuclear dynamics of multiphoton H2+ dissociative ionization in intense laser fields,” Phys. Rev. A 57(2), 1176–1185 (1998). 22. M. Protopapas, C. H. Keitel, and P. L. Knight, “Atomic physics with super-high intensity lasers,” Rep. Prog. Phys. 60(4), 389–486 (1997). 23. W. F. Yang, X. H. Song, Z. N. Zeng, R. X. Li, and Z. Z. Xu, “Quantum path interferences of electron trajectories in two-center molecules,” Opt. Express 18(3), 2558–2565 (2010). 24. W. F. Yang, X. H. Song, and Z. J. Chen, “Phase-dependent above-barrier ionization of excited-state electrons,” Opt. Express 20(11), 12067–12075 (2012). 25. K. J. Yuan and A. D. Bandrauk, “Angle-dependent molecular above-threshold ionization with ultrashort intense linearly and circularly polarized laser pulses,” Phys. Rev. A 84(1), 013426 (2011). 26. M. Odenweller, N. Takemoto, A. Vredenborg, K. Cole, K. Pahl, J. Titze, L. P. H. Schmidt, T. Jahnke, R. Dörner, and A. Becker, “Strong Field Electron Emission from Fixed in Space H2+ Ions,” Phys. Rev. Lett. 107(14), 143004 (2011).
1. Introduction To image electron motion and the structures in atoms and molecules is one of the most important aims in ultrafast physics and attosecond science [1–11]. Driven by a strong laser field, a liberated electron is accelerated and redirected to the parent ion within a very short duration [12]. Electron-ion recollision leads to a number of typical processes, such as highorder harmonic generation, above-threshold ionization, and nonsequential double ionization. Electron motion in these processes encodes detailed structural and dynamical information of the atoms or molecules [4–11]. Recently, a novel interference structure of photoelectron holography (PH) has been observed in experiment where metastable Xenon atoms are ionized by intense 7-micrometer pulses from a free-electron laser [13]. The PH records underlying electron dynamics on a sub-cycle time scale, enabling photoelectron spectroscopy with a time resolution almost two orders of magnitude higher than the duration of the ionization pulse [13]. In traditional optical holography [14], laser light is divided into two beams: one beam illuminates a target and then imprints on a recording medium. This beam is the signal wave. The other beam which is known as the reference wave is shone directly onto the recording medium. The interference between a signal and a reference wave is recorded to reconstruct the information of the target. Since the laser-driven electron motion is fully coherent, the concept of holography can be extended to strong field ionization. In strong field ionization, the recollision electron can be used as the signal wave and the ion itself is the target, while direct photoelectron which does not interact with the ion is the reference wave. The signal and reference waves with the same final momentum interfere with each other and record the PH encoding temporal and spatial information about both the ion and the recollision electron [13,15,16]. A simulation by solving the time-dependent Schrödinger equation (TDSE) has well reproduced the experimental results of the Xenon atom [16]. Further investigation shows that both short and long rescattered electron trajectories in molecule can be imaged by PH with attosecond temporal resolution [17]. It is indicated that PH induced by intense near-infrared laser pulse can occur not only in the tunneling regime but also in the multiphoton regime [18]. In [19], Y. Huismans et al. reported the scaling laws of PH interference patterns with the laser pulse duration, wavelength, and intensity. So far, investigations about PH were based on the physical picture of electron-ion recollision driven by a linearly polarized laser pulse. When diatomic molecules are driven by
#201832 - $15.00 USD Received 22 Nov 2013; revised 13 Jan 2014; accepted 13 Jan 2014; published 29 Jan 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.002519 | OPTICS EXPRESS 2520
a linearly polarized laser pulse, the interference patterns from different rescattering cores will overlap [17]. To our knowledge, the PH with a circularly polarized laser pulse has not been reported yet. In this work, we explore the PH of molecular-ion H 2+ driven by intense circularly polarized laser pulses. When the molecule is driven by a circularly polarized laser pulse, the liberated electron cannot collide with its parent ion. Consequently, the physical picture of the molecular PH with circularly polarized laser pulses is much clearer. It is found that the interference between the direct photoelectron and rescattered photoelectron from one core to its neighboring core induces the molecular PH with circularly polarized laser pulses. This is demonstrated by numerical solutions of the TDSE and confirmed by simulations with a quasiclassical model. Moreover, both the results of the TDSE and the quasiclassical calculation indicate that there is a tilt angle between the interference pattern of the PH and the direction perpendicular to the molecular axis, and the tilt angle becomes smaller with the increasing of the wavelength of the driven circularly polarized laser. This paper is organized as follows. The numerical methods are briefly introduced in Sec. II. The characteristics of the PH and the physical mechanism of the PH are discussed in Sec III. We summarize our results and conclude in Sec. IV. 2. Theory We have carried on the numerical calculation by solving the two-dimensional (2D) TDSE. i
p2 ∂ Ψ (r , t ) = + p ⋅ A ( t ) + V (r ) Ψ (r , t ) ∂t 2
(1)
Here, V(r) is the soft-Coulomb potential of H 2+ ion. In our computation
V ( x, y | R ) =
−Z ( x + R 2) + y + a 2
2
+
−Z ( x − R 2) 2 + y 2 + a
,
(2)
where R is the internuclear distance, Z is the effective charge (Z = 1), and a is the parameter of soft-Coulomb potential (a = 0.47, the corresponding ionization potential of H 2+ reproduced 1sσ
here is I p g = 0.65 a.u. when R = 10 a.u.). We assume that the molecular axis is coincident with the x axis. The wave function at a given time ti is split as [20–22] Ψ (ti ) = Ψ (ti )[1 − Fs ( rc )] + Ψ (ti ) Fs ( rc ) = Ψ I (ti ) + Ψ II (ti )
(3)
Here, Fs ( rc ) = 1 (1 + e − ( r − rc )/ Δ ) is a split function that separates the whole space wave function Ψ (ti ) into the inner (0 → rc ) wave function Ψ I (ti ) and outer ( rc → rmax ) wave function Ψ II (ti ) . ∆ represents the width of crossover region. We choose rmax = 409.6 a.u., rc = 100 a.u., and Δ = 20 a.u.. The exact time evolution of Ψ (ti ) is evaluated using the Crank-Nicolson method [21–24]. We calculate C ( p, ti ) = Ψ II (ti )
e − i [ p − A ( ti )]⋅r 2 d r, 2π
(4)
then Ψ II is propagated to the final time as
Ψ II ( ∞, ti ) = C (p, ti )
eip⋅r 2 d p, 2π
(5)
#201832 - $15.00 USD Received 22 Nov 2013; revised 13 Jan 2014; accepted 13 Jan 2014; published 29 Jan 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.002519 | OPTICS EXPRESS 2521
∞1 2 with C ( p, ti ) = exp −i [ p − A (t ')] dt ' C ( p, ti ) . The outer region wave function is ti 2 propagated by the above Eq. (4) so that there is no boundary problem any more. The final momentum distribution is obtained as
dP( p) = 2 En dEn
C ( p, t ) . i
(6)
i
Here, En is the electron energy associated with p . The total pulse duration Tp is 13 optical cycles. The vector potential of the pulse is A (t ) = eˆx
Ex0
ω
f (t )sin(ωt ) + eˆ y
E y0
ω
f (t ) cos(ωt )
(7)
where eˆx / y is the polarization direction and ε = E y 0 / E x 0 = 1 ( E x 0 = E0 and E y 0 = ε E0 ) is the
ellipticity of the laser pulse, with a smooth sine squared pulse envelope f (t ) = sin 2 (π t / Tp ) . E0 is 0.134 a.u. corresponding to a peak intensity 6.0 × 1014 W/cm2. This vector potential is to ∂A is zero. ensure the total electric area of the field E (t ) = − ∂t 3. Results and discussions
Figure 1 reports the TDSE simulation of photoelectron momentum distribution (PMD) of H 2+ driven by a circularly polarized laser pulse. Short wavelength reduces the excursion time so that the recollision rate with the neighboring core can be increased. Moreover, it has been demonstrated that the tunneling regime is not a necessary condition for the holographic pattern, which can be observed under the conditions formally attributed to multiphoton regime [18]. The appearance of the holographic pattern at short wavelengths requires higher laser intensities [17, 18]. We found that this criterion also applies to the circularly polarized laser pulse case. Thus, the laser frequency used in our simulation is ω = 0.114 a.u. ( λ = 400 nm) and the peak laser intensity is 6.0 × 1014 W/cm2. To clearly show the interference structure, we first give the PMD spectrum on linear scale in Fig. 1(a). Two kinds of interference patterns with different tilt angles from the direction perpendicular to the molecular axis are observed (denoted by “T” and “P”, respectively). The PMD on logarithmic scale in Fig. 1(b) further show that both of these two interference patterns have a cutoff of Pcutoff = 2.356 = 4 U p (Up = E02/4ω2) corresponding to 8Up in energy. As we know, for a circularly polarized laser pulse, the cutoff energy of the direct photoelectron is 8Up [25]. Hence, the direct photoelectron plays an essential role in the interference patterns “P” and “T”. The interference pattern “T” (marked by the yellow lines in Fig. 1(a)) comes from the two-center interference generated by direct electrons ionized separately from the two cores. It can be seen that the interference fringes are not perpendicular to the molecular axis. Electron emission from the H 2+ ion driven by a circularly polarized laser pulse with wavelength of 800nm was investigated in previous work [26]. Both experimental and theoretical results demonstrated that there is a rotation angle in PMD, which results from a complex laser-driven electron dynamics inside the molecule influencing the instant of ionization and the initial momentum of the electron [26]. Here similar rotation angle is observed in the two-center interference pattern. In Fig. 1(b), the two-center interference fringes are also observed in the higher energy region with much lower intensity. Recently, it has been demonstrated that for long internuclear distances, the liberated electron collides with its neighboring ion can generate the maximum kinetic energy of 32 Up for both linear and circular polarizations [25].The interference pattern in the high energy is due to the collision of electrons with neighboring cores. Unlike the direct electron, the
#201832 - $15.00 USD Received 22 Nov 2013; revised 13 Jan 2014; accepted 13 Jan 2014; published 29 Jan 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.002519 | OPTICS EXPRESS 2522
interference fringes for the recollision electron are perpendicular to the molecular axis. This is because that the instant of ionization and the initial momentum of the electron have little influence on the recollision electron.
Fig. 1. Photoelectron momentum distribution of H 2+ driven by a circularly polarized laser pulse (a) on linear scale and (b) on logarithmic scale. The parameters of laser electric field are peak intensity I 0 = 6.0 × 1014 W/cm2, wavelength λ = 400 nm, and duration Tp = 17.35 fs. The lines are guides to the eye.
Besides the two-center interference, there exists another kind of unexpected interference pattern “P” (marked by white dashed lines in Fig. 1(a)) which has “fork” structures in the second and fourth quadrants. The “fork” structures have been identified as the typical interference structure of the PH with linearly polarized laser pulses in previous experimental and theoretical works [13, 16]. As a result, we infer that the interference pattern “P” in Fig. 1(a) is the PH with circularly polarized laser pulses. The direct photoelectron serves as a reference wave, and the rescattered electron from the neighboring core serves as signal wave. The interference between them leads to the strong-field PH with circularly polarized laser pulses. Note that the interference pattern “P” also has a tilt angle from the direction perpendicular to the molecular axis. However, this tilt angle is in opposite direction to that of the two-center interference. For comparison, we first construct an atomic-like reference with a single scattering center V ( x, y ) = −1.13 x 2 + y 2 + a which provides the same ionization potential of the corresponding molecule with R = 10 a.u [15]. Figures 2(a) and 2(b) show PMD by numerical simulations of the TDSE for the atomic-like reference driven by a linearly and circularly polarized pulse, respectively. As expected, a “fork” structure, i.e., the interference pattern of the PH, can be identified in atomic-like reference driven by a linearly laser pulse whose polarization direction is along y axis (see Fig. 2(a)). It has been demonstrated that this holographic pattern arises from the interference between the reference and rescattered signal electron wave packets that are emitted during the same quarter cycle of the laser field [15, 16]. The maximum momentum of the electron inducing the PH is smaller than that in Fig. 1(b), since the cutoff energy of the direct electron driven by a linearly polarized laser (2Up) is smaller than that driven by a circularly polarized laser (8Up). When a circularly polarized laser is applied to atom, the electron cannot be driven back to its parent ion, which means there is no signal wave. Therefore, the interference structure of the PH disappears in Fig. 2(b). This is quite different with the case shown in Fig. 1. When H 2+ ion is considered (see Fig. 1), the electron can be rescattered by the neighboring ion. Consequently, the signal wave still exists, and so does the interference structure of the PH. It should be noted that no two-center interference structure occurs in atomic-like reference (see Figs. 2(a) and 2(b)). Next, we
#201832 - $15.00 USD Received 22 Nov 2013; revised 13 Jan 2014; accepted 13 Jan 2014; published 29 Jan 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.002519 | OPTICS EXPRESS 2523
consider the molecule driven by a linearly polarized pulse and assume that the molecular axis is along x direction in Fig. 2(c). The polarization of the laser electric field is perpendicular to the molecular axis. It can be seen that a “fork” interference structure of the PH is present along the polarization direction of the laser pulse. Moreover, two-center interference fringes can be observed. The energy of the cutoff is 10.2Up which corresponds to the maximum kinetic energy of the recolliding electron with its parent ion. Due to the nonzero initial velocity, collision with its neighboring ion can also occur which generates the two-center interference structure in the higher energy region. These interference patterns are similar to those in Fig. 1. However, as expected, there is no tilt angle in this case, and the interference of the PH and two-center interference are in same direction, which is different with that in Fig. 1.
+
Fig. 2. Photoelectron momentum distribution of atomic-like reference [(a) and (b)] and of H 2 [(c)]. The driven lasers are a linearly polarized pulse [(a) and (c)], and a circularly polarized laser pulse [(b)]. The parameters of laser electric field are same with those of Fig. 1. The results are plotted on a logarithmic scale.
To further demonstrate that the interference pattern “P” shown in Fig. 1 is indeed a holographic-type structure, we consider a simplified picture based on a quasiclassical laserinduced collision-recollision model [16, 17, 23, 25]. The quasiclassical analysis allows one to identify the physical process leading to certain interference structure in the photoelectron spectrum. When a diatomic molecule is ionized by a circularly polarized laser pulse, the liberated electron cannot revisit its parent ion and can only be rescattered by the neighboring core. If the PH occurs, the signal wave must be the photoelectron rescattered by the neighboring core. In quasiclassical calculations, we focus on the interference between the photoelectron rescattered by its neighboring core and the direct photoelectron. For simplicity, we assume the envelope f (t ) = 1 and consider the continuous-wave field in the quasiclassical calculations. If the signal electron is ionized at a particular phase φ, the laser electric field is E (t ) = eˆx E0 cos(ωt + φ ) + eˆ y E0 sin(ωt + φ ) . The velocity and the position can be calculated by integrating the classical equation of motion. The initial condition of the electron is at rest immediately after ionization. The velocities along x direction and y direction [in atomic units (a.u.)] are t
v x (t , φ ) = − E x (t ')dt ' = − 0
E0
[sin(ωt + φ ) − sin(φ )]
(8)
[cos(ωt + φ ) − cos(φ )],
(9)
ω
and t
v y (t , φ ) = − E y (t ')dt ' = 0
E0
ω
respectively. The electron displacements x and y can be written as
#201832 - $15.00 USD Received 22 Nov 2013; revised 13 Jan 2014; accepted 13 Jan 2014; published 29 Jan 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.002519 | OPTICS EXPRESS 2524
t
x(t , φ) = v x (t ')dt ' 0
=
E0
ω
[cos(ωt + φ) − cos( φ) + ω sin( φ)t ] − x0 2
(10)
and t
y (t , ϕ ) = v y (t ')dt ' 0
=
E0
[sin(ωt + φ) − sin( φ) − ω cos( φ)t ] − y0 , 2
ω
respectively. Here, r0 = x02 + y02 =
(11)
Ip
is the initial position of electron after ionization. As E0 we know, recollision can only occur between the electron and its neighboring ion when driven by a circularly polarized laser. The displacement in quasiclassical calculations must satisfy x(tc , φ) = ± R and y (tc , φ) = 0 , in which tc is the traveling time of the scattered signal electron. We assume that the signal electron is elastically scattered by the neighboring ion at an angle θc. After the recollision, the finial velocities along the x and y direction are t
v xf (t , φ ) = − E x (t ')dt ' tc
=−
E0
ω
[sin(ωt + φ ) − sin(ωtc + φ )] + v(tc , φ ) cos(θ c )
(12)
and t
v yf (t , φ ) = − E y (t ')dt ' tc
=
E0
ω
[cos(ωt + φ ) − cos(ωtc + φ )] + v(tc , φ ) sin(θ c ).
(13)
The vector potential is A (t ) = 0 after the end of the pulse. Thus, the electron momentum along x direction measured at the detector is Pxf =
E0
ω
sin(ωtc + φ ) + v (tc , φ ) cos(θ c ).
(14)
The electron momentum along y direction is Pyf = −
E0
ω
cos(ωtc + φ ) + v(tc , φ ) sin(θ c ).
(15)
The direct electron, i.e., the reference wave, is ionized at a phase φ ' and its traveling time is tr. The phase accumulated between the reference and the scattered signal waves does not change after the recollision of the signal electron. The traveling time tr up to which the phase accumulation of the reference wave packet needs to be calculated is the time interval between the collision time of the signal wave packet and the ionization time for the reference wave packet [16]. Therefore, if the PH and interference occur, the signal and reference electrons should satisfy the interference condition: ωtc + φ = ωt r + φ ' . The velocities along the x and y direction of reference electron are v xf (t r , φ ') = v(tc , φ ) cos(θ c ), v yf (t r , φ ') = v(tc , φ ) sin(θ c ) .
#201832 - $15.00 USD Received 22 Nov 2013; revised 13 Jan 2014; accepted 13 Jan 2014; published 29 Jan 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.002519 | OPTICS EXPRESS 2525
Based on the above discussion, we can write out the phase difference between the signal electron and the reference electron as 2 2 2 2 tc v ( t ', φ ) + v y ( t ', φ ) t r v ( t ', φ ' ) + v y ( t ', φ ' ) (φ − φ ' ) ΔΦ = x dt '− x . (16) dt '− I p 0 0 ω 2 2
The phase difference between the signal electron and the reference electron will lead to holographic-type interference fringes ~cos(ΔΦ).
Fig. 3. Comparison of photoelectron momentum distribution of the TDSE simulations [(a)-(c)] and interference fringes ∼cos(ΔΦ) of the quasiclassical calculations [(d)-(f)]. The wavelengths of the laser are 320 nm [(a) and (d)], 420 nm [(b) and (e)], and 520 nm [(c) and (f)]. In the quasiclassical calculations, only the photoelectron scattered by the neighboring core is considered as the signal wave of the PH. The phase difference v (t ',φ ) + v (t ',φ ) r v (t ',φ ') + v (t ',φ ') (φ − φ') leads to interference fringes ∼cos(ΔΦ). ΔΦ = c dt '− dt '−I 2
t
0
2
x
y
2
2
t
0
2
x
y
2
p
ω
We compare the interference fringes ~cos(ΔΦ) from the quasiclassical calculations with the interference pattern “P” in TDSE simulations for different laser wavelengths in Fig. 3. The TDSE results are shown in upper panel and the results of solving the quasiclassical model are shown in lower panel. The wavelengths of the driven laser are 320 nm (Figs. 3(a) and (d)), 420 nm [Figs. 3(b) and 3(e)], and 520 nm [Figs. 3(c) and 3(f)]. To guide the eye, solid curves are drawn in Figs. 3(a) to 3(c) to show clearly the title angle trend of the holographic-type interference pattern “P” in TDSE simulations. It can be seen that the results predicted by the quasiclassical model are consistent qualitatively with the TDSE results. Firstly, both the TDSE and the quasiclassical model predict a tilt angle between the interference pattern of the PH and the direction perpendicular to the molecular axis. Secondly, both the TDSE and the quasiclassical model show that the tilt angle is dependent on the laser wavelength. With
#201832 - $15.00 USD Received 22 Nov 2013; revised 13 Jan 2014; accepted 13 Jan 2014; published 29 Jan 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.002519 | OPTICS EXPRESS 2526
increasing laser wavelength, the tilt angle becomes smaller. Compared with the TDSE simulations, the calculations of the quasiclassical model seem to overestimate the tilt angle. It is because that only the laser electric field is considered, while other factors, such as the Coulomb potential and the initial velocity distribution of the electron, are neglected in the calculations of the quasiclassical model. Thirdly, both the quasiclassical model and the TDSE results demonstrate that the cutoff of the holographic-type interference fringes in momentum spectrum becomes larger with longer wavelengths. This is because that the cutoff of the holographic-type interference fringes is determined by the direct electron which has a cutoff energy of 8Up driven by circularly polarized laser pulses. With increasing the laser wavelength, the value of the Up becomes larger, which induces larger cutoff in momentum distribution. As the above discussion, the interference structures in the quasiclassical analysis come from the interference between the direct photoelectron (i.e., the reference wave) and the photoelectron rescattered by its neighboring core (i.e., the signal wave). The results of the quasiclassical calculation agree very well with the simulations of the TDSE. Therefore, it can be confirmed that the rescattered photoelectron from one core to its neighboring core serves as the signal wave which interferes with the direct photoelectron, i.e. the reference wave, and generates the molecular PH with circularly polarized laser pulses. 4. Conclusions
In summary, the PH in molecule induced by a circularly polarized laser pulse has been investigated by solving the corresponding 2D TDSE and a quasiclassical recollision model numerically. The quasiclassical model has been demonstrated to agree with the TDSE results. Both of them show that the interference between the direct photoelectron and rescattered photoelectron from one core to its neighboring core generates the PH in molecule. Moreover, both of the calculations of the TDSE and the quasiclassical model predict that there is a tilt angle between the interference pattern of the PH and the direction perpendicular to the molecular axis. Furthermore, the tilt angle is dependent on the wavelength of the driven circularly polarized laser in molecular PH. Compared with the PH with a linearly polarized laser pulse, the PH with a circularly polarized laser pulse has some advantages: first, only the electron scattered by its neighboring ion contributes to the signal wave, the physical picture of PH with a circularly polarized laser pulse is much clearer; second, the recollision with the neighboring can access the detailed spatial information of the molecule, i.e. the internuclear distance; third, the holographic interference has different rotation angle from the two-center interference, so different interference processes can be resolved. As a result, the interference dynamics of the PH with circularly polarized laser pulses proposed in the present work can offer a new way to explore the electronic dynamics and access molecular spatial structures. Acknowledgments
We thank Dr. J. Chen and Dr. Z. N. Zeng for helpful discussions on PH. The work was supported by the National Natural Science Foundation of China (Grant Nos. 11374202, 11274220, and 11274219), Guangdong Natural Science Foundation (Grant No. S2013010016061), Foundation for High-level Talents in Higher Education of Guangdong, STU Scientific Research Foundation for Talents, the National Basic Research Program of China (Grant No. 2010CB923200), and the Open Fund of the State Key Laboratory of High Field Laser Physics (SIOM). W. Yang was also sponsored by the Foundation for the Author of National Excellent Doctoral Dissertation of PR China (Grant No. 201215) and “Yang Fan” talent project of Guangdong Province.
#201832 - $15.00 USD Received 22 Nov 2013; revised 13 Jan 2014; accepted 13 Jan 2014; published 29 Jan 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.002519 | OPTICS EXPRESS 2527
