We perform a quantum-orbit analysis for the dependence of high-order-harmonic yield on the driving field ellipticity and the polarization properties of the generated high harmonics. The electron trajectories responsible for the emission of particular harmonics are identified. It is found that, in elliptically polarized driving field, the electrons have ellipticity-dependent initial velocities, which lead to the decrease of the ionization rate. Thus the harmonic yield steeply decreases with laser ellipticity. Besides, we show that the polarization properties of the harmonics are related to the complex momenta of the electron. The physical origin of the harmonic ellipticity is interpreted as the consequence of quantum-mechanical uncertainty of the electron momentum. Our results are verified with the experimental results as well as the numerical solutions of the time dependent Schrödinger equation from the literature.
© 2013 Optical Society of America
High-order harmonic generation (HHG) from atoms and molecules exposed to intense laser fields has been intensively investigated in the past two decades [1, 2]. As a highly nonlinear phenomenon, it provides versatile source of applications such as generating coherent attosecond pulses in the extreme ultraviolet (XUV) regime as well as monitoring and controlling electron dynamics with attosecond and Ångstörm resolutions [4–7]. There is a general agreement that the physical origin of HHG process can be understood within the classical three-step model (CTM) [8, 9]. First, a electron tunnels into the continuum through the potential barrier formed by the Coulomb potential and the laser field. Then it is accelerated in the laser field treated as a free particle. Finally, it may recombine with the parent ion and a high energy XUV photon is emitted. The quantum-mechanical version of CTM based on strong-field approximation (SFA), known as Lewenstein model, is able to give a quantitative treatment of HHG [10, 11]. With the saddle-point approximation applied to SFA , the harmonic amplitude can be expressed as a coherent sum of a few complex electron trajectories (see  for details). These complex trajectories, involving complex times and momenta, can be interpreted using the concept of Feynman’s path integrals . We refer to this model as quantum-orbit model (QO).
HHG driven by elliptically polarized laser pulse attracts more attention motivated by developing new techniques for generating isolated attosecond pulse [15–18] and studying basic physical processes such as tunnel ionization in strong laser field . The ellipticity of the laser field provides an additional control parameter for laser-matter interactions and introduces some new features in strong field laser-matter interaction process [20,21]. It is known that if the laser pulse is elliptically polarized, elliptically polarized high harmonics would be obtained . However, CTM can not explain HHG for elliptical polarization, because the electrons following classical trajectories can not recombine with the parent ion. Some properties of the generated elliptically polarized high harmonics, e.g., the rotation angle of the harmonics, can be explained by CTM as the angle between the direction of the electron momentum at the instant of the recombination and the major polarization axis of the laser field. But the harmonic ellipticity and the dependence of the harmonic yield on the driving field ellipticity can’t be explained using CTM. Lewenstein model based on SFA can provide a theoretical treatment for the harmonic ellipticity and the dependence of the harmonic yield on the driving field ellipticity [23, 24]. However, a qualitative explanation for the physical origin is missed. This question is essential for researchers to gain a better understanding of HHG process and further optimize techniques to generate circularly polarized attosecond pulses  or isolated linearly polarized attosecond pulses .
Recently, a semiclassical model was introduced to qualitatively and quantitatively explain the dependence of the high-order-harmonic yield on driving laser ellipticity . According to CTM, electrons released from the atom accelerate in the strong oscillating electric field. Only those electrons which can return to the parent ion core are responsible for HHG. However, in an elliptically polarized field, the transverse component of the driving field will cause transverse displacements of the electron trajectories, preventing the electrons from returning to the ion core and thereby diminishing the generation of high harmonics. Quantum features of the ionization process have been added by assuming that the electron has an initial transverse velocity at the exit of the tunnel. And then the transverse displacement caused by the laser field can be compensated. The dependence of the harmonic yield on the laser ellipticity is related to different ionization rates of the electrons with different initial velocities [27, 28]. Based on this assumption, authors in Ref.  derived an analytical equation for the dependence of the harmonic yield on laser ellipticity. However, their analytical results show a visible deviation with the measured data and the numerical results the time dependent Schrödinger equation (TDSE). The semiclassical electron trajectories were also employed in Ref. [29, 30], to study the polarization properties of the harmonics. In their work, the authors used the semiclassical electron trajectories to describe the motion of the electron wave packet along the major axis of the laser field whereas the motion in the transverse direction is considered full quantum mechanically. They derived an analytical formula for harmonic ellipticity in Ref. [30, 31] and the analytical results show good agreement with numerical solutions of TDSE. Besides, they propose the explanation of the origin of the harmonic ellipticity in terms of the quantum-mechanical spreading of the electronic wave packet after ionization. Within the semiclassical model, the influence of the symmetry of the atomic ground state on harmonic ellipticity is also discussed in Ref. . Below we will show that using the QO the full process can be treated full quantum mechanically and good agreement of our results with TDSE solutions will be also achieved and a consistent explanation of the origin of the harmonic ellipticity can be given.
In this paper, we perform a systematical quantum-orbit description of HHG with intense elliptically polarized laser field. The dependence of the harmonic yield on the laser ellipticity and polarization properties of the high harmonics are investigated. Quantum electron trajectories responsible for the generation of harmonics are identified. We show that for quantum trajectories contributing to the generation of harmonics, the electron has an nonzero initial velocity at the tunnel exit which facilitate the recombination of the electron with the parent ion. This initial velocity naturally emerges from the quantum mechanical phenomenon of tunneling and we do not require a priori assumption. The initial velocity changes the ionization probability of the electron trajectory thus has a significant influence on the harmonic yield. This physical picture has been well documented in the literatures [26–28]. In this work, a simple formula is given to calculate the ellipticity dependence of the harmonic yield. Our calculation is based on the same physical picture, but we applied less additional simplifications and provided more accurate results. Another point of our results is about the polarization properties of HHG. We show that two components of the electron velocity, along and perpendicular to the major polarization axis of the lase field, dictate the vectorial properties of the HHG and are mapped into the HHG polarization state. Since we deal with quantum trajectories, momenta of the electrons are complex. The non-zero phase difference leads the harmonics to be ellipticity polarized. A qualitative understanding of the physical origin is offered as well.
This paper is organized as follows. In Sec. 2 we briefly discuss QO used here to describe HHG in elliptically polarized field. Quantum trajectories responsible for the generation of particular harmonics are identified. The results of QO are utilized to calculate the dependence of the harmonic yield on the laser ellipticity. In Sec. 3 we derive formulas to calculate HHG ellipticity of different harmonic orders, using complex electron momentum at the instant of recombination obtained by QO. A physical interpretation is provided. Finally, we conclude in Sec. 4.
2. Dependence of harmonic yield on laser ellipticity
We will begin with the standard analysis relied on SFA . The expression of the induced dipole moment at the frequency Ω can be written as (atomic units are used throughout):19], so we neglect this part below. The phase S(p, t, t′) is the quasi-classical action of the form:
The quantum orbits are the trajectories along which the phase of the multidimensional integral Eq. (1) is stationary. They are obtained by finding the stationary points of the action, i.e., by differentiating the action with respect to the integration variables p, t′ and t, yielding correspondingly:Equations (3)–(5) have a transparent physical interpretation. Equation (3) expresses the return condition for the electron. Equations (4) and (5) represent the energy conservation at the moment of ionization and recombination, respectively.
We consider the HHG process in a monochromatic elliptically polarized laser field with electric vector:Eq. (7) into the saddle-point Eqs. (3)–(5) and introducing ps = ps(ti, tr) from Eq. (3) into Eqs. (4) and (5), we obtain a system of two equations, which can be solved numerically for the complex variables ti and tr. In Fig. 1, we present examples of the solutions. Real and imaginary parts of the ionization and recombination times are plotted in Figs. 1(a) and 1(b), respectively. As one can see, there are two quantum orbits with different travel times within one optical cycle, which are well known as short and long trajectories. It is also known that when taking propagation effects into account, long trajectories have an undesirable phase-matching conditions and thus only short trajectories contribute to the harmonics [33–35]. Below we will concentrate on the shot trajectories only.
To gain an intuitive picture about the quantum electron orbit, we give a visualized presentation by plotting the real parts of the trajectories. The position R(t) occupied by the electron is given by [15, 36]Eq. (8) is complex and satisfies the conditions R(tr) = R(ti) = 0. The real part of R(t) is displayed in Fig. 2, with elliptically polarized fields of different wavelengths. Some interesting features can be observed. First, the starting points of the trajectories are not at the origin (the position of the parent ion) but the vicinity of the origin. Mathematically speaking, since ti is a complex variable, R(Re(ti)) ≠ R(ti). Physically, this is due to the fact that the electron must tunnel out and it can’t appear at the origin. Second, The return positions are almost at the origin. Note that we do not add this constraint a priori. It is a natural property of the quantum orbit calculation. This is very different from classical trajectories in elliptically polarized fields. As we pointed out above, using CTM, the electrons can not return to the ion core, for the reason that the transverse component of the driving field would cause transverse displacements of the electron trajectories. Third, we can deduce from the second feature that the electron has non-zero initial velocity at the tunnel exit. The initial velocity compensates the transverse displacement caused by electric field. The same conclusion can also be drawn from the saddle-point Eqs. (3)–(5) mathematically. There is critical difference between the saddle-point equations in linearly and elliptically polarized field [37, 38]. For the former, owing to Eq. (3), ps is along the laser field. Thus, the initial velocity ps + A(ti) of the electron is purely imaginary considering Eq. (4). However, for elliptical polarization, the initial velocity has a none-zero real part. The imaginary part emerges from tunneling, and the real part compensates the transverse displacement and forces the electron to return to the origin. The initial velocities are displayed in Fig. 3. For non-zero ellipticity, the real part of the two velocity components, along and perpendicular the major polarization axis, are not zero, and both of them increase with the laser elipticity. Below we will show that this phenomenon would greatly influence the harmonic yield.
Next, we calculate the dependence of the harmonic yield on laser ellipticity. For a particular harmonic order, we calculate the saddle-points (ps, ti, tr) with laser ellipticity varying from 0 to 1. Inserting the saddle-points into Eq. (2), the stationary points of the action S(ps, ti, tr) are obtained. They are all complex too. The modulation of the harmonic yield Γ as a function of the laser ellipticity is given by [39, 40]Fig. 4 by solid curves. In panels (a) and (b), the target atom is neon. The intensity of the laser pulse used in panel (a) is 5.4 × 1014W/cm2 with the wavelength of 810nm. The 27th harmonic is selected. Laser intensity in panel (b) is 5.4×1014W/cm2 with the wavelength of 405nm. The 11th harmonic is selected. In panels (c) and (d), the target atom is helium. The intensity of the laser pulse used in panel (c) is 7.7×1014W/cm2 with the wavelength of 810nm. The 19th harmonic is selected. Laser intensity in panel (d) is 7.7×1014W/cm2 with the wavelength of 405nm. The 11th harmonic is selected. As one can see, when increasing the ellipticity, the harmonic yields steeply decrease. The dependence of the harmonic yields on laser ellipticity are approximately Gaussian. For laser ellipticity lager than 0.5, there are almost no harmonics generated. Another characteristic we can find from Fig. 4 is that the threshold ellipticity (the fundamental ellipticity for which the harmonic yield is two times lower than for the linearly polarized fundamental field) is smaller for long wavelength (810nm) than short wavelength (405nm). As a result, we find that the HHG sensitivity to the driving laser ellipticity increases with the drive wavelength. The theoretical results are compared with a recent experiment , the results of which are also shown in Fig. 4 by filled hexagram (left panels) and filled diamond (right panels). The prediction of QO matches very well with the measured ellipticity-dependent high-order-harmonic yields. Instead, results of the semiclassical model used in Ref.  show general trends but a obvious deviation with the experimental measurements. Our results show that by treating the HHG process fully quantum mechanically, the QO analysis can give a more realistic description about the ellipticity-dependent high-order-harmonic yield.
Although the prediction of Eq. (9) shows good agreement with the experiment, but it does not offer a clear physical interpretation. Next, we will derive a simple formula to calculate the ellipticity dependence of the harmonic yield with transparent physical meaning. As we have pointed out, the electron has a non-zero initial velocity. In elliptically polarized field, the initial velocity is complex. The real part of the initial velocity compensate the transverse displacement cased by the laser field, so it can be viewed as the initial transverse velocity v⊥. In tunneling theory, at the tunnel exit the electron wave-packet has a simple Gaussian dependence on v⊥ derived from Eq. (9)Eq. (10), the modulation of the harmonic yield can be expressed as: Eq. (11) are also displayed in Fig. 4 by the dashed curves, which also show good agreement with the experiment data. In contrast to Eq. (9), the expression provides a transparent interpretation about the ellipticity dependence of the harmonic yield, together with Fig. 3. For lager ellipticity, the electron has a lager initial velocity. As a result, the Gaussian dependence of the harmonic yield on initial velocity causes the yield to decrease. This interpretation is similar with the Semiclassical model. However, our quantum description gives more realistic initial momentum as well as the tunnel time, thus can provide the correct results in consistent with the experimental data. Besides, small deviations of the prediction of Eq. (11) appear for the relatively larger ellipticity. Results of Eq. (11) show a slightly higher yield in comparison with numerical results of Eq. (9) and measured data. The reason is that only the ionization step is included in Eq. (11). But the probability of recombination can also affect the harmonic yield. According to detailed-banance principle, recombination can be viewed as the inverse process of ionization. The recombination probability is related with the imaginary part of tr in view of Eq. (10). However, as shown in Fig. 1, Im(tr) is close to zero, which is much smaller than Im(ti). Thus the influence of recombination can be neglected. We can conclude that the simple formula given by Eq. (11) is a reasonable approximation evaluating the modulation of the harmonic yield.
3. Harmonic polarization properties in elliptically polarized laser field
We now discuss the polarization properties of HHG in elliptically polarized field. A unified theory of HHG for an elliptically polarized driving field based on SFA provides theoretical treatment for polarization properties of harmonics . However, it doesn’t gives a clear explanation of the physical origin of the non-zero harmonic ellipticity. Very recently, Strelkov et al. propose the explanation of the origin of the harmonic ellipticity with their analytical model . They show that this ellipticity originates from quantum-mechanical uncertainty of the electron motion. Their analytical model is based on semiclassical electron trajectories and includes tunneling coherently . Below, we will analyze the harmonic ellipticity fully quantum mechanically based on QO and a coincident conclusion can be made.
The harmonic ellipticity ζ is calculated by 27], i.e. Eq. (14) into Eq. (13), we find Fig. 5. It is shown that the imaginary part of v‖ is negligible, but the imaginary part of v⊥ is comparable with the real part. Thus the phase difference is not 0 or π. In comparison, CTM would give real valued v‖ and v⊥. This is why harmonic ellipticity can not be explained by CTM.
Next, we calculate harmonic ellipticity using Eq. (15), shown in Fig. 6. One can see that the QO result provides very good agreement with the numerical result for almost all harmonics. Small deviations appear in low harmonic orders. This may due to that we don’t consider the influence of Coulomb potential. The Coulomb effects can be added coherently using strong-field eikonal approximation . It is shown in Ref.  that the Coulomb modification for low harmonic orders is stronger than that for high harmonic orders. The effect of the Coulomb potential on high energy electrons is much weaker. Thereby for higher harmonics orders, our results agree quite well with the numerical results.
We now discuss the physical origin of the ellipticity. As often the case in quantum mechanics, only the observable should be real-valued quantity. In HHG process, the observable is the emitted light, with real valued photo energy. Other characteristics, including the drift momenta, the ionization and recombination times do not in principle have to be real valued. The QO calculation, including the quantum mechanical phenomenon of tunneling, predicts complex valued momenta, ionization and recombination times. Their imaginary parts are substantial, originating from tunneling. Viewing Eq. (10), it is nothing else but the energy-time uncertainty relationship. It indicates that the uncertainty of the electron momentum at the tunnel exit is related with the time which the electron spent in the potential barrier, i.e., the imaginary part of ionization time. During the process of the electron accelerated in the laser field, the momentum is not measured. Accordingly, the uncertainty of the momentum is also exist at the instant of recombination. The uncertainty of the momentum is mapped into the polarization state of the harmonics and cause the harmonics to be elliptically polarized. Therefore, we can make a conclusion that the physical origin of harmonic ellipticity is the uncertainty of the electron momentum. Our conclusion based on quantum-orbits analysis and strong field approximation is in consistent with Ref. .
In summary, we have performed a systematical quantum-orbit analysis of high harmonic generation in intense elliptically polarized laser field. Both the dependence of the harmonic yield on laser ellipticity and the harmonic polarization properties have been investigated. For the former, we have shown that when increasing the laser ellipticity, the initial velocity of the electron increases, causing the reduce of the ionization rate, thereby diminish the generation of harmonics. A physically transparent formula has been presented to calculated the ellipticity dependence of harmonic yield and the results show very good agreement with a recently done experiment. For the latter, we have found that for a spherically symmetrical atom ground state, the parallel and perpendicular components of the electron’s return momentum are mapped on the polarization state of the harmonics. Because the two two orthogonal components are complex quantity, the phase difference of them are nonzero, leading the harmonics to be elliptically polarized. Our calculations match well with the numerical solutions of TDSE. Furthermore, we have provided an explanation of the physical origin of the harmonic ellipticity. The harmonic ellipticity originates from quantum uncertainty of the electron momentum of the relevant quantum electron trajectory. Our work can help researchers gain more insight into HHG process in elliptically polarized laser field and develop new techniques to generated elliptically polarized attosecond pulse.
This work was supported by the NNSF of China under grants 11234004 and 60925021, the 973 Program of China under grant 2011CB808103, and the Doctoral fund of Ministry of Education of China under grant 20100142110047.
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