The half Kapitza-Dirac effect of molecule in an intense standing-wave laser field is studied with a focus on the influence of the molecular orbital symmetry and the molecular alignment on the photo-electron angular distributions (PADs). In standing-wave laser fields, the PADs split along the scattering angle due to the momentum change of electron with photons when it escapes from the laser fields. The structures and the symmetry of PADs are severely affected by the molecular orbital symmetry and the molecular alignment. For molecule in ground state (σg), the PADs are severely changed by the molecular alignment only when the photoelectron kinetic energy is sufficiently high. For molecule in the first excited state (σμ), the molecular alignment distinctively changes the PADs, irrelevant to the kinetic energy of photoelectrons. When the molecules are aligned either parallel with or perpendicular to the laser polarization, the PADs are symmetric about an axis. In other cases, the PADs do not show any symmetry. These results indicate that the molecular alignment can be used to control the splitting in the half Kapitza-Dirac effect.
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In 1988, Bucksbaum et al.  showed that the photoelectrons from atoms in an intense standing-wave laser field could be deflected into two separate peaks. They termed this phenomenon as Kapitza-Dirac (KD) effect, a prediction proposed by Kapitza and Dirac back in 1933 that a free electron beam would be split into two coherent electron beams when the free electron beam passes through a standing-wave light . The following theoretical treatment, however, showed that this was actually a half-process of the KD effect, since the electrons were produced from ionization of atoms in the laser fields, not originally free electrons . In 2001, Freimund et al.  demonstrated the diffraction of an electronic beam into separate, coherent beams in a standing-wave laser field. This was the first observation for the full process of KD effect originally proposed. Since it survived some conceptual and theoretical debates on the diffraction of matter waves , the KD effect excited great interest in various matter waves, and the half KD effect of atoms were observed from 1980’s [6–9].
The splitting pattern of electron beams in the KD effect is determined by three factors: the incident angle of laser, the kinetic energy of electrons and the intensity of the standing-wave laser. So one can control the splitting of electron beams by varying the three factors. Since the momentum change of electron is just several photon momenta in full KD effect, the splitting effect is weak. In order to obtain highly splitting electron beams, one should use the half KD effect as performed by Bucksbaum et al., where the momentum change of electron reaches as much as several hundreds to thousands of photon momenta. For the half KD effect of atoms, one controls the splitting of the ejected electrons only by changing the laser field, such as the intensities of two counter-propagation laser beams. Changing the laser intensity, however, may lead to many unexpected effects due to nonlinear dependence of the half KD splitting on the laser intensity. So one should make a deliberated scheme to control the splitting of electron beams. The usage of molecules as an electron source can solve this difficulty, since the interaction of molecules with intense laser field is affected by more parameters, such as the molecular orbital symmetry and the molecular alignment. In the KD effect of free electron beams or atom, the electron emission direction is not easy to control. In the half KD effect of molecules, however, the electron emission direction can be controlled by the molecular alignment, which benefits the application of KD effect .
Many new features appear in photoionization of molecules in intense laser fields, such as the dependence of the ionization rate and photoelectron angular distributions (PADs) on the molecular orbital symmetry and the molecular alignment with respect to the laser polarization [10–16]. It is reasonable to wonder that the molecular orbital symmetry and the molecular alignment significantly affect the PADs in the molecular half KD effect. However, the way that the molecular orbital symmetry and the molecular alignment affect the splitting PADs in the molecular half KD effect is an open topic. Thank to the progress in controlling the molecular alignment [17–19], it is much easier to control the molecular alignment by a short laser pulse. And the proposed scheme is feasible in experiments.
The half KD effect of atoms in intense laser fields is featured by a splitting of the PADs along the scattering angle, which is caused by the momentum change of photoelectron with photons when it escapes from the laser fields. But what if it is molecule? To answer this question, in this paper, using a nonperturbative scattering theory for above-threshold ionization (ATI) developed by Guo, Åberg and Crasemann (GAC) , we study the half KD effect of molecule with a focus on the influence of the molecular orbital symmetry and the molecular alignment on the half KD effect. The GAC theory, as well as the related treatments [21–23], had received notable success in explaining the intense laser field phenomena, such as the half KD effect performed by Bucksbaum et al. , and the KD diffraction of free electrons observed by Freimund et al. .
Among all molecules, the molecule is of special importance. This molecule contains only one electron, so the multi-electron effect during photoionization is avoided. The initial state of molecules can be obtained analytically by the linear combination of atomic orbitals (LCAO) , then the complex computations to obtain the molecular orbitals is also avoided. With a purpose to show how the molecular alignments and the molecular orbital symmetry to affect the half KD splitting of molecules, we take in its ground state, which is a bonding electronic state σg, and in its first excited state, which is an antibonding electronic state σu, as the initial state of moelcule and calculate the PADs from ionized by a standing-wave laser field. Experimentally, compared with in σg state, molecule in σu state is difficult to prepare for its short survival time. The half KD splitting of molecules with antibonding electronic state, however, offers an important reference to that of molecules with bonding electronic states, and the predicted features can be observed for the other molecules with antibonding electronic states, such as O2, NO and CO2 molecules. So for theoretical completeness, we also study the KD splitting of molecule in its σu electronic state.
This paper is arranged as follows. In Sec. 2 we introduce the formula used in this paper. The results are presented and discussed in Sec. 3, and the conclusions are drawn in Sec. 4.
2. Theoretical treatment
According to GAC theory, the photoionization process of molecules and atoms in a strong laser field contains two steps: (1) the initial bound electron absorbs multiple photons and then is excited into a Volkov state; (2) the electron in the Volkov state escapes from the laser field to become a free photoelectron that is detected experimentally. In each transition step, the energy and the momentum are conserved. The Volkov states are not the final states, but the intermediate states . To theoretically study the photoionization processes of atoms and molecules in an intense laser field, the strong-field approximation (SFA) is widely used. In this treatment, the Coulomb interaction of the parent ion to the ionized electron can be neglected compared with the stronger laser electric strength. Within SFA, the photoelectron in the intermediate state is treated as a quantum-field Volkov state given by Eq. (1) is defined as  Eq. (1) differ from that for traveling-wave mainly in the momentum phase factor, k(Na1 – Na2), which have no classical-field correspondence and possess a unique advantage. The momentum phase factor allows an arbitrarily large momentum transfer between two laser modes of the standing-wave field, by which the large splitting in the angular distributions of photoelectrons can be formed in the KD effect.Eq. (5) is obtained by the minimum coupling principle as Eq. (5), Φ (R,r) is the initial wave function of molecule with the internuclear distance R = |R| denoting the distance between two atomic cores of the molecule. According to LCAO theory , the initial wave function of molecule can be exactly expressed by an infinite series or obtained by four atomic orbitals , in which the two lower energy states contribute the most to the molecular state. In this paper, the initial wave function for molecule is chosen as the linear combination of the ground states of two hydrogen atoms centered at and , respectively, and can be expressed as Fig. 1(a) and the orientation of R is defined by the polar angle θR and azimuthal φR as shown in Fig. 1(a), where z axis is along the laser propagation direction. In this paper we use a linearly polarized laser field and choose x axis along the laser polarization direction. In Eq. (7), the plus and the minus signs on the right hand side correspond to the ground state in bonding molecular orbital symmetry σg and the first excited state in antibonding molecular orbital symmetry σμ, respectively; S(R) = ∫ drΦ1(r1)Φ2(r2)* is the atomic orbital overlap integral. For photoelectrons with a fixed kinetic energy, the differential ionization rate is Fig. 1(b); j is the number of absorbed photons in the first step, s is the number of photons returning to the background laser field in the second step, and n = j – s is the number of net photons absorbed during ionization; Eb is the electron binding energy of molecule. The arguments of the generalized phased Bessel functions in Eq. (8) can be written as Eq. (8) is performed over all the possible combinations of j and s for a fixed n.
The factor F in Eq. (8) is defined as3] Eq. (10), Φ (R,P±) is the Fourier transform of the initial wave function Φ (R,r) of molecules. For molecule in bonding or antibonding orbital symmetry, the Fourier transform takes the following form 11, 24]. The factor corresponds to the molecule in a bonding orbital symmetry, and to that in an antibonding orbital symmetry. One advantage of the present simplified wave function is its analytical simplicity, by which one can easily analyze the physical process. Numerical results show that the present wave function only changes the absolute value of the ionization rate compared with the ionization rate that calculated using the wave function given in Ref. . The differences between the PADs calculated using both the wave functions are less, since the PAD discloses the relative variation of the ionization rate. Such a treatment has been used to explain the physical mechanism of the ionization suppression of molecules [10–12]. Another advantage is that the interference factor described by the sine or cosine function stems from the geometric symmetry of homonuclear diatomic molecules. It provides a common factor in the molecular wave function. So we expect the present treatment holds for other homonuclear diatomic molecules or complex but symmetric distributed molecules . For those molecules, the KD splitting will show some new features other than the features predicted in this paper and needs further detailed study.
The interference factor reflects the principal influence of the molecular orbital symmetry. According to the interference theory , the interference pattern varies with the photoelectron energy and the internuclear distance of molecules. Mathematically, the interference effect can be seen from the inner product of the momentum of photoelectrons with the internuclear vector of molecules. Thus, it depends on the value of PR and the relative direction of the two vectors. In our calculations, the internuclear distance is chosen as the stablest one, which keeps to the value of PR less than π for photoelectrons with energy lower than 100 ω, approximately. For bonding orbital symmetry, the modulation to the PADs is not severe for lower-energy photo-electrons due to the constructive quantum interference effect. For antibonding orbital symmetry, the interference factor generally acts as destructive and its modulation to the PADs is distinct and sensitive to the molecular alignment.
3. Numerical results and discussion
According to Eq. (8), we calculate the PADs by varying the scattering angle θf from 0° to 180° and the azimuthal angle ϕf from −90° to 90° at a given molecular alignment. In all calculations, the laser beam is chosen as linearly polarized with wavelength of 800 nm. The peak intensity of the laser beam varies from 1013 W/cm2 to 2×1014 W/cm2, and the Keldysh parameter is larger than unity for molecule at R=2.0 a.u. in this laser intensity range. Since the response of electrons to the laser field is faster than molecular oscillation and rotation times , the nuclei of the molecules are treated as fixed in space. In this paper, the molecule is in a state with R = 2.0 a.u.. The alignment of the internuclear vector is described by two angles, θR and φR, in the spherical coordinates with the laser propagation vector as z direction. When the molecules are aligned along the laser propagation, θR takes 0 and φR is arbitrary. When the molecules are aligned in the laser polarization plane, θR takes 0.5π and φR=0.
3.1. Angular splitting of PADs under different laser intensities
Photoionization of atoms and molecules in intense standing-wave laser fields is featured by the large angle splitting along the scattering angles in PADs. The photon exchange, when the electrons in Volkov states escape from the laser fields, causes the angular splitting. The electrons absorb photons from one propagating laser mode and immediately move mainly along the laser polarization, accordingly the electrons have a momentum P in the same direction; and then emit photons into the other propagating laser mode when they leave the laser field to become free electrons. The large momentum change of a photoelectron is in the laser propagation direction. Due to the large momentum change which is mostly perpendicular to the original momentum P, the photoelectrons leave the laser field in directions departing away from the laser polarization plane with large angles. As a result the splitting occurs in the PADs. According to the geometry relation between the original momentum P and the final momentum of photoelectron Pf, we obtain the half splitting angle α as 
Our calculations show that the angular splitting is enlarged as the laser intensity increases before the total ponderomotive parameter 2up reaches the next integer. The PADs of the 2nd and the 10th ATI peak for two laser intensities depicted in Fig. 2 clearly show these features. The splitting is along the scattering angle θf, and the splitting PADs vary notably with the azimuthal angle ϕf. In order to show the influence of the used wave functions for the same initial state, in Fig. 2(d) we also compare the normalized PADs calculated using the wave function given by Eq. (13) and that given in Ref. . The similarity between two PADs approves the present treatment. For linear polarization, the photoelectrons generally are emitted along the laser polarization, and the ionization yield decreases oscillatory when the photoelectrons emit away from the laser polarization. This dependence leads the PADs to vary with these two angles. In order to show the overall variation of the PADs, we will next resort to the 3D plots of PADs, and depict 2D plots for detailed comparison when necessary.
Our calculations show a multiple splitting. When the two-laser-beam ponderomotive parameter 2up reaches the next integer, the splitting angle becomes zero, and the splitting vanishes. As the value of 2up further increases, a new central splitting appears. The previous splitting peaks, however, remain as a pair of side peaks. And the multiple splitting appears when the total ponderomotive parameter exceeds an integer. The multiple splitting tests the integer property of the ponderomotive parameter (for more detail, see Ref. ). So far, there have been no clear experimental indications for the integer property of ponderomotive parameter which was predicted from nonperturbative quantum electrodynamics [3, 20, 22]. We hope this new effect can be observed in the future with further improved experimental techniques.
The angular splitting is also affected by the kinetic energy of photoelectrons. The central splitting becomes narrow as the kinetic energy of photoelectrons increases, which can be seen when the plots (a) and (b) in Fig. 2 are compared. This implies a smaller splitting angle for higher-energy photoelectrons. Meanwhile, besides the highest peaks in the PADs, more smaller peaks appear as the photoelectrons kinetic energy increases. These smaller peaks manifest the maximum value of the Bessel function which describes the transition amplitude of the electron from the initial bound state to the Volkov state just like the jets of the PADs in single mode case .
The change of laser intensity leads to multiple splitting effects. So, in order to easily control the splitting in the half KD effect, it is natural to seek for the additional degrees of freedom of molecules. In traveling laser fields, the molecular orbital symmetry evidently changes photoionization of molecules has been found. Firstly, the overall ionization rate may be modulated by the interference factor. It has been found in single-mode case that the ionization rate of molecules may be much less than that of atoms with a like binding energy . This is the so-called ionization suppression of molecules due to the destructive interference [11, 12]. Secondly, the interference factor modulates the shape of PADs and the modulation becomes more notable for higher-energy photoelectrons. When the momentum of photoelectrons is small, the modulation on PADs is notable only for molecule in the antibonding case due to the small value of PR. When the momentum of photoelectrons is large, the value of PR becomes large, and the modulation becomes notable for molecules with both bonding and antibonding shells. Thirdly, the molecular alignment fundamentally changes the symmetry property of PADs. For example, the molecular PADs have only inversion symmetry in general. While, they become four-fold symmetric when the molecules are aligned either parallel with or perpendicular to the laser polarization [16, 26]. Fourthly, the molecular orbital symmetry determines the role of the molecular alignment. For the bonding orbital symmetry, the calculated PADs notably vary with the molecular alignment only when the photoelectrons’ kinetic energy is sufficiently high. But for the antibonding orbital symmetry, the calculated PADs always vary significantly with the molecular alignment. It is expected that the molecular orbital symmetry and the molecular alignment greatly change the PADs in standing-wave laser fields, which helps to control the splitting of the KD effect. In the following subsection, we focus on the PADs in various molecular alignments to show how the splitting is controlled by the molecular alignment.
3.2. Influence of molecular alignment on PADs
Our calculations show that the splitting PADs are greatly changed by the molecular alignment. The orbital symmetry, however, plays a key role in the molecular alignment changing the PADs.
For molecule in a bonding state σg, the molecular alignment affects the PADs only when the photoelectron kinetic energy is sufficiently high. For lower-energy photoelectrons, the PADs depend less on the molecular alignment. The PADs are almost the same for various molecular alignments for the photoelectrons with kinetic energy lower than 15ω. The splitting PADs show two series of broad peaks symmetrically located beside θf = 90°. Since the interference factor is a cosine function of P·R and the internuclear distance R is small, the value of the interference factor varies in a small range from 1, thus the PADs vary less with the molecular alignment for lower-energy photoelectrons. As the kinetic energy of photoelectrons increases, the influence of the molecular alignment becomes notable, and the splitting in PADs become narrower and narrower. In Fig. 3, we show 3D plots of the PADs for the molecules aligned along (Figs. 3(a) and 3(c)) and perpendicular to (Figs. 3(b) and 3(d)) the laser polarization in the polarization plane. From this figure we see that the PADs of the 10th ATI peak (Figs. 3(a) and 3(b)) are almost the same while those of the 20th peak (Figs. 3(c) and 3(d)) show notable differences for two different molecular alignments.
For molecule in an antibonding state σμ, the molecular alignment becomes a key factor that affects the PADs. The PADs severely vary with the molecular alignment, no matter what the kinetic energy of photoelectrons is. Fig. 4 depicts 3D plots of the PADs for the molecules aligned along (Figs. 4(a) and 4(c)) and perpendicular to (Figs. 4(b) and 4(d)) the laser polarization in the polarization plane. Considerable differences appear between the PADs for different molecular alignments. One notable difference lies in the ionization rate along ϕf = 0°. As the molecules align along the laser polarization, in each side of the splitting PADs, there is a broad knap around a sharp peak with the top located at ϕf = 0°, while for the molecules aligned perpendicular to the laser polarization, the broad knaps are split and there appears a broad valley around ϕf = 0°, and the sharp peak mentioned above is suppressed into two small tubers located around ϕf = 0°. Another difference is the appearance of new side peaks for the molecules aligned perpendicular to the laser polarization. The dramatic variation of PADs with the molecular alignment is due to the small value of PR in the sine function. As the kinetic energy of photoelectrons increases, the PADs show many more new features and still dramatically vary with the molecular alignment, as shown in Figs. 4(c) and 4(d) for the 10th ATI peak. For molecules aligned along the laser polarization, the PADs also show many broad peaks. While for the molecules aligned perpendicular to the laser polarization, the PADs exhibit many sharp peaks, symmetrically located at the two sides of the plane determined by ϕf = 0° and θf = 90°. There still exists a sharp valley around ϕf = 0° due to the destructive interference. The valley becomes more and more narrow when the value of PR tends to π.
In Fig. 5 we depict two PADs for the molecule prepared in antibonding state σμ and aligned along the laser propagation. The PADs show several pairs of broad peaks on the two sides of θf = 90° located different θf angles. The broad valley is the splitting due to the half KD effect. As the kinetic energy of photoelectrons increases, the peaks become steeper and steeper, while the valley becomes narrower and narrower. For molecule prepared in bonding states σg, the PADs do not show much difference between the molecules aligned in the polarization plane and those aligned in the laser propagation direction, or between different ATI peaks, if the photoelectron kinetic energy is not high enough. Although we do not have detailed discussions on these bonding cases here.
The above discussions are for the molecules aligned along different special directions with respect to the propagation and the polarization of the laser beams. For molecules aligned along arbitrary directions, the PADs show many distinct features, among which the most notable one is the asymmetry in the splitting PADs. The asymmetry of the PAD manifests the unequal ionization rates about the axes θf = 90° and ϕf = 0°. For molecule prepared in a bonding states σg, the asymmetry of the PAD becomes obvious only when the photoelectron kinetic energy is high enough. While for molecule prepared in an antibonding states σμ, the asymmetry of the PAD is generally expected. In Fig. 6 we depict the PADs of the second ATI peak for two arbitrarily aligned molecule prepared in an antibonding state σμ. The plot (a) is the 3D PAD for the molecules aligned in the polarization plane and the molecular axis departs from the polarization axis by an angle of , i.e. and . For such a molecular alignment, the PAD is still symmetric about θf = 90°, but shows distinctive asymmetry along ϕf, and the minima in the main peaks appear around ϕf = 45°. The PADs are highly non-symmetric, as show in Fig. 6 (b) for and . In such a case, the molecules are aligned neither in the laser polarization plane nor along the laser beam. The peaks in the left side are obviously higher than those in right side, which means that photoelectrons are emitted in a small rang of one side of the half KD splitting. This can be used to control the motion of electron beam in KD effect. In order to qualitatively show the variation, the 2D PADs for the molecular alignments corresponding to (a) and (b) are depicted in (c) and (d). From these figures, we see the PADs are still symmetric about the central split along θf for the molecular alignment , but are asymmetric for molecules aligned neither parallel with nor perpendicular to the laser polarization.
The half KD effect observed by Bucksbaum et al. in atoms shined by standing-wave laser exists in molecules. In standing-wave laser fields, the PADs of molecules split along the laser beam, which is caused by the photon exchange between the two counter propagating modes of the standing-wave. The splitting angle varies with the laser intensity and the kinetic energy of photoelectrons. We draw conclusions: 1) The change of laser intensity results in many effects, among which the most attracting one is the multiple splitting when the ponderomotive parameter 2up exceeds an integer. It is difficult to control the splitting electron beams by changing the laser intensity. 2) The shapes of PADs, as well as the splitting, are greatly affected by the molecular orbital symmetry and the molecular alignment. The molecular orbital symmetry determines the role of the molecular alignment in changing the PADs, and the degree of the molecular alignment affecting the PADs also varies with the kinetic energy of photoelectrons. For molecule in the bonding state, the PADs do not change much with the molecular alignment unless the photoelectron kinetic energy is sufficiently high. For molecule in the antibonding state, the molecular alignment distinctively changes the PADs, no matter what the energy of photoelectrons is. 3) For molecule in the antibonding state, the PADs show two-fold splitting: One is along the laser propagation and features the half KD effect; the other is along the molecular axis and is caused by the destructive interference, which also reflects the initial electron distribution of molecules. 4) The PADs do not show any symmetry if the molecules are aligned neither parallel with nor perpendicular to the laser polarization. This feature indicates that the molecular alignment can be used to control the beam splitting in the KD effect.
This work is supported by the Chinese National Natural Science Foundation under Grant No. 10774153, 61078080, 11174304, 11104167 and the 973 Program of China under Grant Nos. 2010CB923203, and 2011CB808103, and Excellent Middle-Aged and Youth Scientist Award Foundation of Shandong Province (No. BS2011SF021).
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