nonadiabatic molecular alignment of linear molecules...

1 3

DOI 10.1007/s00340-016-6562-1Appl. Phys. B (2016) 122:288

Nonadiabatic molecular alignment of linear molecules probed by strong‑field ionization yields of photoelectrons

G. Kaya1,2 · N. Kaya2,3 · J. Strohaber4 · N. A. Hart1 · A. A. Kolomenskii1 · H. A. Schuessler1

Received: 14 May 2016 / Accepted: 4 November 2016 © Springer-Verlag Berlin Heidelberg 2016

anisotropic polarizability of the molecule, which tends to align with its axis of highest polarizability along the laser polarization direction [1]. If the laser pulse is short, i.e., its duration is less than the characteristic molecular rota-tional period, the laser pulse cannot align the molecule adiabatically, and the resulting alignment process is called nonadiabatic or field-free alignment [2, 3]. In nonadiabatic alignment, the short laser pulse creates in each molecule a rotational wave packet, i.e., a coherent superposition of the rotational states, which will dephase and rephase in time after the laser pulse has ended. After the impulsive interac-tion of molecules with the laser pulse, the absence of the aligning field prevents a distortion of the molecular rota-tional dynamics caused by the field, thus giving this advan-tage for the dynamics observation compared to adiabatic alignment [4–6]. Since the pioneering works on the theory [7, 8] and the first experimental demonstration of nonadi-abatic molecular alignment [9], this phenomenon contin-ues attracting the interest of researchers (see [10, 11]). A variety of applications of the molecular alignment was pro-posed, such as controlling chemical reactions [12], selec-tively controlled alignment of isotopes [13], imaging of molecular structures [14], nanolithography with molecular beams [15], pulse compression [16], and quantum informa-tion processing [17]. Femtosecond laser technology allows aligning molecules and observing the molecular dynam-ics by using a pump–probe technique [18]. Alignment ini-tiated by a pump pulse creates a dynamically anisotropic medium evolving in time, and hence, concurrent variations of the interaction with such a medium of a probe pulse can be observed by ionization [19–21], fragmentation [22–26], high harmonic generation [14, 26–29] or by spatial and spectral modulation of the probe pulse [30–35].

It should be noted that rotational coherence spectroscopy (RCS) [36], which is based on quantum beats that arise

Abstract The dynamics of rotational wave packets of laser-aligned linear molecules were studied with femtosec-ond laser-driven strong-field ionization (SFI). The dynam-ics were observed as a function of the delay between a femtosecond probe pulse and a linearly polarized aligning pump pulse. The induced nonadiabatic molecular align-ment was directly monitored by the total SFI yield. The measured revival signatures were compared to the calcu-lated degree of molecular alignment taking into account the effects of electronic structure and symmetry of the molecules. By fitting the calculated alignment parameter to the measured experimental data, we also determined the molecular rotational constants of N2, CO, O2, and C2H2 gas molecules.

1 Introduction

When a molecule is exposed to a strong laser pulse, the interaction creates an induced dipole moment due to the

This article is part of the topical collection “Enlightening the World with the Laser” - Honoring T. W. Hänsch guest edited by Tilman Esslinger, Nathalie Picqué, and Thomas Udem.

* H. A. Schuessler [email protected]

1 Department of Physics, Texas A&M University, College Station, TX 77843, USA

2 Science Program, Texas A&M University at Qatar, Doha 23874, Qatar

3 Department of Physics, Giresun University, Giresun 28200, Turkey

4 Department of Physics, Florida A&M University, Tallahassee, FL 32307, USA

G. Kaya et al.

1 3

288 of 12

from the coherent excitation of rotational levels, offered for the spectroscopy of the rotational states a rich variety of detection schemes, including a pump–probe photoioni-zation approach. Even earlier [37], transient birefringence resulting from the time evolution of a coherent superposi-tion of laser-induced rotational states was used for their observation. To measure the degree of alignment, a weak field polarization technique that employs Raman scattering was proposed [38]. The depolarization of the probe pulse induced by the transient birefringence due to field-free alignment was also employed for monitoring rotational dynamics [39]. In another approach, the coherences in the molecular rotational dynamics initiated by the nonresonant dynamic Stark effect and by stimulated Raman scattering were detected via refractive index modulation, resulting in the Stokes and anti-Stokes sidebands of the scattered probe pulse [40].

By using the combined effect of the laser field and a static electric extraction field, the orientation effect of quantum-state-selected polar molecules was achieved and was assessed by the probe-induced angular distribution of ions in Coulomb explosion [41] or of photoelectrons due to tunneling ionization [42]. A nonadiabatic field-free orienta-tion of a heteronuclear diatomic molecule by femtosecond two-color laser fields was reported in Ref. [43]. The proce-dure to quantitatively extract the degree of alignment from measured strong-field ionization (SFI) yields was devel-oped in [44]. It uses above threshold ionization (ATI) pho-toelectron spectroscopy and photoelectron–photofragment coincidences [45, 46]. The isotopic rotational constants and fragmentation channels were determined using correlated measurement of rotational and mass or photoelectron spec-tra by combining Fourier transform rotational coherence spectroscopy with resonance-enhanced multiphoton ioniza-tion [47].

The degree of alignment can be improved by reducing the temperature of the gas [7, 48, 49] and also by selecting single states with state separators. A combination of an elec-trostatic field, nonresonant femtosecond laser excitation, and the preparation of state-selected molecules using a hexapole was implemented [50] to improve the degree of alignment up to ⟨cos2θ⟩ = 0.82. The degree of the alignment can be fur-ther improved by using supersonic beams of polar molecules that are deflected using inhomogeneous electric fields. The quantum-state selectivity of the deflection is used to spatially separate molecules according to their quantum state [51], achieving a very high degree of alignment ⟨cos2θ⟩ = 0.972. With a Stark-selected beam of ground-state OCS molecules in an approach that uses a quantum-state-dependent deflec-tion in an inhomogeneous electrostatic dipole field, also high alignment degrees were reached, ⟨cos2θ⟩ = 0.84 [52]. A combination of strong inhomogeneous static electric field and the ac field from the laser pulse are used for separation

of polar molecules in a supersonic beam according to their quantum states to demonstrate an unprecedented degree of laser-induced one-dimensional alignment ⟨cos2θ⟩ = 0.97 and strong orientation of state-selected iodobenzene molecules with an estimated temperature of 1.0 K. [41].

The molecular ionization rate is often maximal, when the maximum of the initial electron density distribu-tion corresponds to the laser polarization direction of the applied field. However, not all diatomic molecules have their HOMO, which donates electrons most readily, aligned along the molecular axis. For example, for an π electron configuration, it is preferentially lined up in the direction perpendicular to the molecular axis with no electron den-sity along the internuclear axis. This is unlike the σ elec-tron configuration, for which the maxima of the electron density are preferentially aligned in the direction parallel to the molecular (internuclear) axis due to the shared electron density distribution directly between the bonding atoms. Therefore, depending on the configuration of the highest occupied molecular orbital (HOMO), qualitatively differ-ent alignment signatures can be expected. The contribu-tions to total ionization yield not only from HOMO but also from energetically lower orbitals such as HOMO-1 and HOMO-2 were investigated in high harmonic generation studies [53, 54]. The techniques to resolve SFI channels for polyatomic molecules were developed in Refs. [45, 46].

Our goal is to investigate under the same experimental conditions the nonadiabatic alignment process initiated with femtosecond pump pulses of linear molecules N2, CO, O2, and C2H2 by detecting SFI yields of photoelec-trons produced by the probe pulses. Previously, the nona-diabatic alignment in CO [55] and C2H2 [56] was studied with a pump–probe scheme by detecting produced ions. The choice of the molecules in our study was determined by their variety of configurations of HOMOs. The meas-ured molecular alignment is compared to the calculated time dependence of the molecular alignment parameter ⟨⟨

cos2 θ⟩⟩

, and the effects of the electronic structure and symmetry of molecules on the signatures of alignment revivals are investigated.

2 Experimental setup

The pump–probe setup is depicted in Fig. 1. A mode-locked Ti/sapphire laser oscillator provides ~20 fs pulses at a repe-tition rate of 80 MHz. These pulses are seeded into a regen-erative laser amplifier, which outputs ~50 fs laser pulses at a repetition rate of 1 kHz with a central wavelength of 800 nm. Since shorter pulses have a higher peak intensity for a given pulse energy, temporal compression of the laser pulses in the focus was achieved by maximizing the inte-grated SFI yield, while adjusting the grating compressor

Nonadiabatic molecular alignment of linear molecules probed by strong-field ionization yields…

1 3

of 12 288

in the laser amplifier. The maximum pulse energy from the amplified laser system was ~1 mJ. The linearly polarized incident light was split by a beam splitter into two beams with about equal intensities to serve as pump and probe pulses. A half-wave plate and a polarizer were placed into the pump arm assuring proper linear polarization and for adjusting the intensity of the pump pulse.

In the experiments, different gases (N2, CO, O2, or C2H2 from Matheson TRIGAS, with purities of 99.9995, 99.9, 99.997, and 99.6%, respectively) were admitted to the chamber through an adjustable leak valve. The pressure in the vacuum chamber for the various gases studied was adjusted to 6 × 10−6 mbar, which was about three orders of magnitude higher than the background pressure. The photoelectron signal from aligned molecules was meas-ured as a function of the time delay between the linearly polarized pump and probe laser pulses. This time delay was precisely adjusted using an optical delay line with a trans-lational stage controlled by a stepping motor (GTS150, ESP300, Newport), which provided a high temporal resolu-tion of ~0.67 fs. Another beam splitter was used to recom-bine the two beams, and the laser beams were focused in a vacuum chamber by a 20-cm achromatic lens. The pump pulse created a rotational wave packet, while the delayed probe pulse ionized molecules in the focus producing elec-trons. During the experiment, the pump and probe beams were horizontally polarized along the axis connecting the centers of the left and right microchannel plate (MCP) detectors. The ionized electrons were ejected by the prob-ing laser field toward the MCP detectors. These electrons traveled within a μ-metal TOF tube in a magnetic field-free region. The signals from the MCP detectors were ampli-fied by a preamplifier (ZKL-2 Mini-Circuits), and a FAST ComTech MCS6 multiple-event time digitizer with 100-ps time resolution was used for data acquisition. Laser pulses were detected behind one of the mirrors on the probe arm by a photodiode, and this signal was used to trigger the data

acquisition software. The total number of counts, depend-ing on the pump–probe delay, was registered as the output signal. We note that this setup was also used for ATI experi-ments; in particular, the intensity measurements were refer-enced to the well-defined kink in the ATI spectra of Xe at two times the ponderomotive energy, which was measured with the same setup [57]. The pulse duration was measured using a frequency-resolved optical gating autocorrelator (Grenouille 8–20, Swamp Optics).

When using linearly polarized pump pulses, the photo-electron momentum distribution showed a sharp maximum in the direction of the field polarization [58], and direct electrons have a narrower distribution than re-scattered electrons [59]. In ATI spectra, the total number of the direct electrons (typically with energies up to 2Up, where Up is the ponderomotive energy of an electron in the oscillating field) is significantly larger than that of the re-scattered electrons in the plateau region, which have larger ener-gies, namely from 2Up to10Up [60, 61]. The solid angle of our MCP detector is relatively small, and therefore mostly direct electrons emitted close to the probe field polarization direction are detected.

3 Theory

The induced dipole potential of a linear mole-cule in a linearly polarized field along the Z-axis is VL(t) = −1

/

2[

�α cos2 θ + α⊥]

E2(t) [62–64]. The interaction of a linearly polarized pump pulse with linear molecules can be described by the effective Hamiltonian H = H0 − 1

/

2(

�α cos2 θ + α⊥)

E2(t), where H0 = hBcJ2 is the field-free Hamiltonian with the rotational molecu-lar constant B and the angular momentum operator J, θ is the angle between the molecular axis and the field polari-zation of the aligning pulse, E(t) is the time-dependent electric field of the aligning pulse, and Δα = α‖ − α⊥ is

Fig. 1 Experimental setup. BS beam splitters, TS translational stage, WP wave plate, M flat mirrors, L achromatic focusing lens, PM power meter, MCP microchannel plates. The dots depict electrons flying toward the MCP detectors

G. Kaya et al.

1 3

288 of 12

the polarizability difference between the components par-allel α‖ and perpendicular α⊥ to the molecular axis. Cen-trifugal distortion effects are not taken into account in our calculations.

The degree of molecular alignment is characterized by the quantity

⟨⟨

cos2 θ⟩⟩

which is calculated by a dou-ble averaging procedure: First, the Schrödinger equa-tion, i� ∂

∂tΦJM(t) = HΦJM(t), is solved with a Runge–

Kutta procedure (typically taking into account terms up to Jmax = 25–30) for each initial molecular rotational state

∣

∣ΦJ0M0(t)⟩

=∑

JM dJ0M0JM (t) exp

(

−iEJ t/

�)

|JM� in terms of a series in free-field rotor functions |JM� with eigenenergies EJ = hBcJ(J + 1) [10, 65, 66]. Then, the degree of molecular alignment at time t is found know-ing population amplitudes dJ0M0

JM (t) of the rotational states, as

⟨

cos2 θ⟩

J0M0(t) =

⟨

ΦJ0M0(t)∣

∣ cos2 θ∣

∣ΦJ0M0(t)⟩

[3]. The evolution of the rotational wave packet in time can be described as

∣

∣ΦJ0M0(τ )

⟩

=∑

J ,M

dJ0M0

JM|JM�

exp [−iπτJ(J + 1)] [67], where we have written the time-dependent phase term explicitly, and the time τ is given in units of Trev = 1

/

(2Bc) with a rotational constant B. Thus, for the time dependence of the observable of interest, it follows

where ⟨

J ′M ′∣

∣ cos2 θ |JM� ∝ δ{J ,J ′J ,J ′±2

}δM,M ′ [3] and

ϕJ ,J ′ = exp(−iπτ[

J ′(J ′ + 1)− J(J + 1)]

). For the terms with the same angular momentum quantum number,J = J ′, the time-dependent phase disappears, but for the terms with J = J ′ ± 2, the time dependence is given by the phase factors ϕJ ,J±2 = exp [−iπτ(4J + 6)]. At integer values τ = 0, 1, 2, … which correspond to full revivals, the phase term ϕJ ,J±2 = 1 for all the J quantum num-bers. At half revivals τ = 1

/

2, 3/

2, . . ., the phase terms are π shifted for all of the J’s, resulting in the phase term ϕJ ,J±2 = −1, and thus overall distribution has the same magnitude, but the opposite sign compared to full reviv-als. At quarter revivals, the phase term ϕJ ,J±2 depends on the even or odd parity of J-state: At τ = 1

/

4, 5/

4, . . ., the phase term ϕJ ,J±2 = i for even J-states and ϕJ ,J±2 = −i for odd J-states. At τ = 3

/

4, 7/

4, . . ., the phase term ϕJ ,J±2 = −i for even J-states and ϕJ ,J±2 = i for odd J-states. The even and odd states contribute to the signal; however, their contributions are shifted by π, and there-fore they interfere destructively. Indeed, the expression for the time dependence of Eq. (1) at quarter revivals has a phase shift of π

/

2 with regard to the signal observed at half and full revivals.

(1)

⟨

cos2 θ

⟩

J0M0

(τ ) =⟨

ΦJ0M0(τ )

∣

∣ cos2 θ

∣

∣ΦJ0M0(τ )

⟩

=∑

J ′,M ′

∑

J ,M

d∗J0M0

J ′M ′ dJ0M0

JM

⟨

J′,M

′∣

∣ cos2 θ |J ,M�ϕJ ,J ′ ,

To understand the observed revival signatures, we con-sidered the molecular ionization rate. Depending on the configuration of the HOMO, qualitatively different align-ment signatures are expected based on the angular depend-ence of the SFI yield in the interaction of the aligned molecules with the probe pulse. To take into account the possible deviation of the direction of the maximal SFI yield relative to the molecular axis, we add an additional term � to the phase: ϕ′

J ,J±2 = exp [−i(πτ(4J + 6)+�)], where � is determined by the configuration of the HOMO.

Before the laser interaction, the gas ensemble is assumed to be in thermal equilibrium having a Boltzmann distribution with temperature T . In the quantum mechani-cal approach, an ensemble is described by a statistical mixture of states |J0M0�, having angular momenta J0 = 0, 1, 2,… and their projections on the laser polarization axis M0 = −J0, −(J0 − 1),… 0,… (J0 − 1), J0 with the prob-abilities of these states following the Boltzmann distribu-tion PJ0 ∼ gJ0(2J0 + 1) exp

(

−EJ0

/

kT)

, where EJ0 is the rotational energy of |J0M0� state and k is the Boltzmann constant. The (2J0 + 1) term accounts for the degeneracy within a given J0 state. In the case of a molecule containing two identical nuclei, gJ0 is the relative weight between odd and even J-states, which comes from the nuclear spin sta-tistics as an additional factor. For gerade symmetry states, bosonic nuclei have gJ0 = (2I + 1)(I + 1), if J is even and gJ0 = (2I + 1)I, if J is odd, and fermionic nuclei have gJ0 = (2I + 1)I, if J is even and gJ0 = (2I + 1)(I + 1), if J is odd; I is the nuclear spin of identical particles exchanged during rotation [68, 69]. For ungerade electronic states, the ratio of odd to even states is reversed.

Finally, the degree of alignment of the ensemble at tem-perature T can be found by averaging the alignment degree of a single initial state over the Boltzmann distribution, including nuclear spin statistical factors, as [70]

where ⟨

cos2 θ⟩

J0M0(t) is the degree of alignment for a sin-

gle initial state.To illustrate the dependence of the degree of alignment

on temperature, we calculated the N2 temporal alignment signatures for a set of temperatures T = 50, 100, 200, and 300 K (Fig. 2a). The summation of the contributions of dif-ferent rotational states up to Jmax = 30 is shown in Fig. 2b.

We note that the duration of the used laser pulse of 50 fs is much shorter than revival periods for all investigated molecules. Consequently, the details of the pulse shape

(2)

⟨⟨

cos2 θ

⟩⟩

(t)

=

∑

J0

∑J0M0=−J0

gJ0⟨

cos2 θ⟩

J0M0(t) exp

(

−hBcJ0(J0 + 1)/

kT)

∑

J0

∑J0M0=−J0

gJ0 exp(

−hBcJ0(J0 + 1)/

kT)

,


1 3

of 12 288

only slightly influence the alignment, and therefore the ini-tial action of the pump pulse can be considered as a kick of transferred angular momentum proportional to �α and the laser fluence [71, 72]. The correspondence between quan-tum and classical quantities in the molecular alignment is discussed in Ref. [73, 74].

The presented treatment is valid only for linear mol-ecules and a linearly polarized laser field that aligns only the molecular axis leaving the rotation about this axis unrestricted. For molecules similar to an asymmetric top, the rotation about the molecular axis is not isotropic, and it was shown [75] that an intense, elliptically polar-ized, nonresonant laser field can simultaneously force all three axes of a molecule to align along given axes fixed in space. An enhanced degree of field-free three-dimen-sional alignment was also obtained using two orthogo-nally polarized, time-separated laser pulses [76]. Field-free molecular revivals of 3D alignment of asymmetric top molecules using elliptically polarized laser pulses was also demonstrated [39] and is most efficient at low temperatures.

4 Results and discussion

In this section, we present experimental results for N2, CO, O2, and C2H2 gases followed by the interpretation of the observed revival signal for each gas. For this interpretation, we consider the total wave function of a linear molecule, which according to the Born–Oppenheimer approxima-tion can be factorized, Ψtot = ψel × ψvib × ψrot × ψns, as a product of the electronic wave function ψel, the vibrational wave function ψvib, the rotational wave function ψrot, and

the nuclear spin wave function ψns. We present observed temporal signatures of the alignment and compare them to the calculated time evolution of the alignment parameter ⟨⟨

cos2 θ⟩⟩

(t), which depends on the laser parameters and the molecular properties.

Figure 3a shows the calculated alignment parameter ⟨⟨

cos2 θ⟩⟩

at temperature T = 300 K with our laser param-eters and the properties of N2 molecules with the rotational constant B = 1.9896 cm−1 [77, 78]; Fig. 3b depicts the measured signal of the SFI yield for N2 molecules aligned by a linearly polarized pump pulse. The SFI yields are smaller at Trev/4 and 3Trev/4 and larger at Trev/2 and Trev. This can be explained in the following way. 14N2 is a boson, and therefore, the total wave function Ψtot is symmetric [79]. At room temperatures, the electronic ψel (3

∑+g ), and

the vibrational ψvib wave functions of 14N correspond to the ground state, and both are symmetric [79]. Therefore, to understand the ψrot behavior, we should consider the sym-metric and antisymmetric forms of the nuclear spin wave function ψns of 14N2. For Ψtot to be symmetric, both ψrot and ψns must be symmetric or antisymmetric. For 14N, the nuclear spin is I = 1 and the nuclear spin of a N2 molecule takes only values Itot = 0, 1, or 2. Consequently, N2 exists in even (Itot = 0, 2) and odd (Itot = 1) forms. A state degen-eracy with Itot is 2 Itot +1, so the even-N2 and odd-N2 sta-tistical weights are 6 and 3, respectively. Due to the relative abundance of the even-N2 versus odd-N2, 2:1 for even/odd states, the two opposed Trev/4 revival signals do not cancel completely; however, the signal has a reduced amplitude compared to the Trev and Trev/2 revivals, as is observed in the experiment (Fig. 3b). Consequently, our measured SFI yield in Fig. 3b varies in phase with the time dependence of the molecular alignment parameter in Fig. 3a. In Fig. 3b,

Fig. 2 a Dependence of the degree of alignment on the gas temperature calculated for N2. Jmax is the maximum number of rotational states taken into account in the calculation. b Summation of the contributions of different rotational states for different Jmax for N2 at T = 300 K

G. Kaya et al.

1 3

288 of 12

we also presented the corresponding fit curve based on Eq. (2) for the alignment parameter

⟨⟨

cos2 θ⟩⟩

. Thus, we can conclude that � = 0 for the N2 measurements.

As described, the molecular ionization rate is there-fore maximal when molecules are aligned along the laser polarization direction and the configuration of the HOMO is parallel to the molecular axis, because such configuration makes ejection of an electron easier. This agrees with the σg HOMO symmetry of the neutral nitrogen molecule [80], where the molecular ionization rate is maximal, when the molecules are aligned along the probe polarization direc-tion. As seen in the inset of Fig. 3a, N2 has its maximum electron density along the internuclear (molecular) axis in its σg HOMO configuration. By fitting the curve within one full revival period and taking the same laser parame-ters as experimental ones, we also found the characteristic rotational constant and the corresponding revival time for nitrogen gas. We note that no offset for the time axis was introduced. The theoretical values of the rotational constant

Fig. 3 a Calculated time dependence of the molecular alignment parameter induced by the pump pulse in N2; the configuration of the HOMO is also shown; b the time evolution of the measured SFI sig-

nal for N2 molecules and the fit curve. The pump and probe pulses were polarized collinearly and had peak intensities of 2 × 1013 and 7.8 × 1013 W/cm2, respectively

Fig. 4 Frequency spectrum amplitudes of the measured time-depend-ent SFI yield for N2 shown in Fig. 3b. The numbers on spectral peaks, which correspond to the allowed transitions, show the frequencies in terms of 4J + 6. The series (6, 14, 22, 30, 38, 46, 54, …) Bc and (10, 18, 26, 34, 42, 50, 58, …) Bc come from even and odd values of J’s, respectively


1 3

of 12 288

1.9896 cm−1 [77, 78] and the revival time 8.383 ps agree well with our experimental values of the rotational constant B = 1.999(+0.01)cm−1 and the corresponding revival time 8.342 ps; the number in parentheses here and below shows the deviation of the experimental value of B from the theo-retical one.

In Fig. 4, we show the corresponding frequency spec-trum amplitudes of the measured time-dependent SFI yield for N2 obtained by a Fourier transform (FT) analysis. The spectrum reveals two sequences of allowed Raman tran-sitions obtained from the matrix elements of

⟨⟨

cos2 θ⟩⟩

, which are calculated as (EJ+2 − EJ)/h = (4J + 6)Bc with the selection rule ΔJ = ±2. Namely, these sequences are as follows: (6, 14, 22, 30, 38, …)Bc for even J’s and (10, 18, 26, 34, 42,…) Bc for odd J’s. Recall that the nuclear spin of the nitrogen nucleus is 1, and both even and odd rotational states are permitted for the nitrogen molecule [13]. The relative ratio of the even and odd J-states is 2∶1, which fol-lows from the nuclear spin statistics and gives an additional factor of ~2 for even J’s in the spectrum [25, 81, 82].

Figure 5a shows the calculated alignment parameter ⟨⟨

cos2 θ⟩⟩

at temperature T = 300 K with our laser param-eters and the properties of CO molecules with the rotational constant B = 1.9226 cm−1 [77]. The CO molecule has two nonidentical nuclei. Because of this lack of symmetry, even and odd states in an ensemble of CO molecules are equally populated, and the contributions that come from the even and odd states completely cancel each other at quar-ter revivals [69, 83]. As a result, no quarter revivals appear in the calculated time dependence of the molecular align-ment parameter

⟨⟨

cos2 θ⟩⟩

and in the measured SFI sig-nal, and only half and full revivals are observed, as shown in Fig. 5b. The calculated time dependence of molecular alignment in Fig. 5a changes in phase with the measured SFI yield, so there is no need for any additional term aris-ing from the phase term in the calculation of the alignment parameter, thus � = 0. This is similar to N2, since CO also has σ-type HOMO [80] and no nodal plane along the internuclear axis. While the theoretical rotational constant for CO is 1.9226 cm−1 [77] and revival time 8.675 ps, we

Fig. 5 a Calculated time dependence of the molecular alignment parameter for CO; the configuration of the HOMO is also shown; b the time evolution of the measured SFI signal for CO and the fit curve

are depicted. Collinearly polarized pump and probe pulses were used with peak intensities of 2 × 1013 and 7.9 × 1013 W/cm2, respectively

G. Kaya et al.

1 3

288 of 12

evaluated the characteristic rotational constant from the fit to our experimental data as B = 1.939(+0.016)cm−1, and the corresponding periodic revival time is 8.601 ps.

In Fig. 6, we presented the frequency spectrum ampli-tudes of the time-dependent molecular alignment signal for CO, which is expected to have peaks at frequencies (4J + 6)Bc for all J’s, giving the sequence (6, 10, 14, 18, 22, 26, 30, 34, 38, 42, …)Bc.

As an example of an πg type HOMO configuration [80], we investigated the O2 molecule, and Fig. 7a, b shows the time dependences of the molecular alignment parameter ⟨⟨

cos2 θ⟩⟩

calculated at a temperature of T = 300 K for our laser parameters and O2 molecule with a rotational constant B = 1.4376 cm−1 [77, 84] as well as the meas-ured temporal alignment profile for O2 gas. Oxygen 16O is bosonic, so the total wave function Ψtot is symmetric. The ground state of the electronic wave function, ψel, of 16O is (3

∑−g ) and antisymmetric, while the vibrational wave

function, ψvib, is symmetric. Then, for the ground state Ψtot to be symmetric, ψrot and ψns must be symmetric and antisymmetric, respectively, or vice versa. For an O2 mol-ecule, the nuclear spin Itot = 0, and hence only odd rota-tional J-states are populated [85]; consequently, the rela-tive abundance of molecules with even and odd J’s is 0:1. Therefore, only a wave packet with odd J exists, and strong

Fig. 6 Frequency spectrum amplitudes of the measured time-depend-ent SFI yield for CO shown in Fig. 5b. The number on each spectral peak shows the frequency from the series (6, 10, 14, 18, 22, 26, 30, 34, 38, 42, …) Bc

Fig. 7 a Calculated time dependence of molecular alignment param-eter for O2, following the action of the pump pulse; the configura-tion of the HOMO is also shown. b The time evolution of the meas-

ured SFI signal for O2 and the fit curve. Linearly polarized pump and probe pulses were used with peak intensities of 2 × 1013 and 8.1 × 1013 W/cm2, respectively


1 3

of 12 288

effects of alignment and antialignment are observed at the quarter and the three-quarter revivals of O2 with the ampli-tudes comparable to the full and half revivals, as shown in

Fig. 7a. However, when the graph of Fig. 7a and experi-mentally measured dependence in Fig. 7b are compared, one can see that the shapes of the calculated and experi-mental dependences are different, which has the following explanation. The HOMO of O2 is of πg type and has nodes of the electron densities [86] in the direction of and also perpendicular to the molecular axis, as shown in the inset of Fig. 7a. Because of the exact twofold symmetry of this molecular orbital, the SFI signal from O2 maximizes near 45°, and the SFI signal is minimal when the molecular axis is exactly parallel or perpendicular to the laser polariza-tion. Indeed, the fitting function calculated with the value � = π/2 gives a good agreement with the experiment (Fig. 7b). The theoretical value of the rotational constant B = 1.4376 cm−1 [77, 84] and the corresponding periodic revival time 11.601 ps compare favorably with our experi-mental values found by fitting B = 1.461(+0.023) cm−1 and the revival period 11.415 ps.

In Fig. 8, we show the corresponding frequency spec-trum amplitudes of the time dependence of the measured SFI yield for O2 obtained by the Fourier transform (FT). This figure exhibits the experimental spectrum, showing

Fig. 8 Frequency spectrum amplitudes of the measured time-depend-ent SFI yield for O2, shown in Fig. 7b. The number on each spectral peak shows the frequency in terms of 4J + 6 from the series (10, 18, 26, 34, 42, 50, 58, …) Bc for odd J

Fig. 9 a Calculated for C2H2 time dependence of the molecular alignment parameter; the configuration of the HOMO is also shown. b The time evolution of the measured SFI signal for C2H2 and the fit

curve. We used linearly polarized pump and probe pulses with peak intensities of 3 × 1013 and 7.6 × 1013 W/cm2, respectively

G. Kaya et al.

1 3

288 of 12

series (10, 18, 26, 34, 42,…)Bc, since only odd J levels are permitted for O2 due to the nuclear spin statistics.

Figure 9a shows the alignment parameter ⟨⟨

cos2 θ⟩⟩

calculated at temperature T = 300 K with our laser param-eters for C2H2 molecules with the rotational constant B = 1.1766 cm−1 [87] and the measured time trace of the SFI yield with variations due to alignment of C2H2 mol-ecules. Acetylene C2H2 has both even and odd J-states, which are populated with the ratio 1:3 [85]. As a result of this fact, even and odd wave packets partially cancel each other, and some alignment and antialignment are observed at quarter revivals as shown in Fig. 9a, b. The experimental results for C2H2 molecules at quarter revival, half revival, three-quarter revival, and full revival are in good agreement with the calculated time dependence of molecular align-ment

⟨⟨

cos2 θ⟩⟩

parameter, but with the inverted polarity. This can be understood from the configuration of HOMO for acetylene, which is dominated by a πu orbital [88]. Molecular orbitals of πu type have higher electron density above/below the internuclear axis, with a node along the internuclear axis, as shown in the inset of Fig. 9a. Conse-quently, the temporal structure of C2H2 revivals in Fig. 9b has inverted polarity as compared to the calculated molecu-lar alignment parameter

⟨⟨

cos2 θ⟩⟩

in Fig. 9a. With � = π , the temporal signature of the molecular alignment param-eter

⟨⟨

cos2 θ⟩⟩

is inverted, as shown in Fig. 9b, and the cal-culated function gives a shape similar to the experimental one. The theoretical value of the molecular constant for C2H2, B = 1.1766 cm−1 [87], and the respective revival time 14.175 ps match well the characteristic rotational

constant B = 1.180(+0.003) cm−1 and the revival period 14.133 ps following from the fitting procedure in our data.

The frequency spectrum of the time-dependent SFI yield for C2H2 is shown in Fig. 10. The number on each spectral peak shows the frequency in terms of 4J + 6 which corre-sponds to allowed Raman transitions, given by the series (6, 14, 22, 30, 38, 46, 54, …)Bc for even J and by the series (10, 18, 26, 34, 42, 50, 58, …)Bc for odd J. The relative ratio of the amplitudes of the even and odd spectral components is close to 1:3, following from the nuclear spin statistics.

We note that although the qualitative interpretation of our results can be provided based only on considerations involving HOMO configurations, in more complex cases it can be necessary to take into account ionization occurring from several orbitals using a description involving Dyson orbitals, which represent the overlap between an initial wave function for the neutral system and the electron wave function of the corresponding ionized system [54, 89].

5 Conclusions

The revival signatures of the linear molecules N2, CO, O2, and C2H2 produced by linearly polarized femtosecond pump pulses have been studied under nonadiabatic align-ment conditions and measured by the detection of the SFI photoelectron yield in a pump–probe setup with a variable delay. For N2 we observed strong alignment effects at mul-tiples of Trev/2 and reduced amplitude signals (not com-pletely suppressed) at odd multiples of Trev/4 due to the ratio of even/odd states 2:1 for the nuclear spin statistics factors of N2. For CO, containing nonidentical nuclei, there is no additional factor arising from the nuclear spin statis-tics. Thus, the revivals at odd multiples of Trev/4 are com-pletely canceled, whereas the revivals at multiples of Trev/2 remain. Because the HOMO configuration donates electrons most readily, the HOMO symmetry of molecules is impor-tant for interpretation of the observed alignment signals. Therefore, the symmetry effect of the HOMO electron den-sity distribution on the shape of the temporal dependence of the SFI yields of photoelectron measured with aligned molecules was discussed. N2 and CO have their maximum electron densities along the internuclear (molecular) axis due to σg and σ HOMOs, respectively. The molecular ion-ization rate is maximal when molecules are aligned along the laser polarization direction, because such configuration makes ejection of an electron easier. Consequently, in case of N2 and CO, our measured SFI yields were modulated in phase with the time dependency of the molecular alignment parameter. In the case of O2, since only odd J-states are rel-evant, we observed strong alignment signals at Trev/4 and 3Trev/4 revivals as well as at multiples of Trev/2. For O2 mol-ecules, the ionization rate has minima when the molecular

Fig. 10 Frequency spectrum amplitudes of the measured time-dependent SFI yield for C2H2, shown in Fig. 9b. The numbers on the spectral peaks show frequencies in terms of 4J + 6, which corre-spond to allowed Raman transitions from the series (6, 14, 22, 30, 38, 46, 54, …)Bc for even J and the series (10, 18, 26, 34, 42, 50, 58, …) Bc for odd J. The relative ratio of the magnitudes of the spectral har-monics corresponding to even and odd J correlates with the nuclear spin statistics ratio 1∶3


1 3

of 12 288

axis is parallel or perpendicular to the laser polarization due to the πg HOMO configuration. The measured SFI yield is temporally modulated with a (−π/2) phase shift with respect to the time dependence of the molecular alignment param-eter, which agreed with the experimental data. For the C2H2, the nuclear spin statistics ratio of even and odd states is 1:3, and the strong alignment effects were observed at multiples of Trev/2, and the reduced amplitude alignment signals (not completely suppressed) were at odd multiples of Trev/4. Due to the πu HOMO symmetry of C2H2, which has higher elec-tron density above/below the internuclear axis with a node along the direction of this axis, our experimental SFI yields for C2H2 have a polarity-inverted dependence compared to the molecular alignment parameter [a (−π) shift]. By taking into account these additional phase shifts, a good agreement between calculated and measured alignment signatures was obtained. In addition, the spectra of revivals reveal peaks corresponding to sequences of transitions between even and odd rotational states with amplitude ratios in agreement with the population ratio following from the nuclear spin statistics. Thus, we have demonstrated that the implemented alignment monitoring by measuring SFI photoelectron yields presents a viable and efficient way of studying rota-tional molecular dynamics. By fitting the molecular align-ment parameter

⟨⟨

cos2 θ⟩⟩

, modified by taking into account discussed additional phase shifts to the experimental data, we determined the characteristic rotational constants and corresponding revival periods, which are in good agreement with the theoretical values found in the literature.

5.1 Personal recollections of Hans Schuessler dedicated to Ted Haensch

As a friendly colleague, I was privileged to watch your many contributions to fundamental atomic spectroscopy. Early on, lasers were considered a passing fad, since they were not widely tuneable. In Heidelberg, your He–Ne laser tubes in Schmelzer’s Applied Physics laboratory were long and fragile, while the small hollow cathode light sources in Kopfermann’s Physikalischen Institut, when combined with radio frequency techniques, were well suited for hyperfine spectroscopy. Several years later, matters changed dramati-cally. You had gone to Stanford and I to Seattle. I guess we both were enjoying the stimulating atomic physics atmos-phere at these West Coast universities, where a starting sci-entist could work independently and realize his dreams. On one of my visits to Stanford, you picked me up in a Kar-mann Ghia sports car, keeping with this famous Volkswagen brand. During this visit, I realized the playfulness in your research by watching your wavemeter, where the mov-ing arm was on a rail track. This playful treat continued to be reflected in many of your subsequent works. I was also impressed by your widely applicable nitrogen pumped dye

laser, so much that on my return to Seattle I wanted to build such a setup for spectroscopy with ion traps, but no such luck, Hans Dehmelt insisted for me to wait until I have my own laboratory at Texas A&M University. Decades full of wonderful experiments have passed, and still today thanks to your work and help, I use now frequency comb lasers for sensing and precision spectroscopy, and the wonderful jour-ney continues.

Acknowledgements This work was supported by the Robert A. Welch Foundation Grant No. A1546, and the Qatar Foundation under the Grant NPRP 6-465-1-091.

References

1. N. Xu, J. Li, J. Li, Z. Zhang, Q. Fan, in Lasers: Applications in Science and Industry, ed. K. Jakubczak (InTech, 2011)

2. T. Seideman, J. Chem. Phys. 103, 7887 (1995) 3. J. Ortigoso, M. Rodrıguez, M. Gupta, B. Friedrich, J. Chem.

Phys. 110, 3870 (1999) 4. I.S. Averbukh, N.F. Perelman, Phys. Lett. A. 139, 449 (1989) 5. R.W. Robinett, Phys. Rep. 392, 1 (2004) 6. G.H. Lee, H.T. Kim, J.Y. Park, C.H. Nam, T.K. Kim, J.H. Lee,

H. Ihee, J. Korean Phys. Soc. 49, 337 (2006) 7. N.E. Henriksen, Chem. Phys. Lett. 312, 196 (1999) 8. T. Seideman, Phys. Rev. Lett. 83, 4971 (1999) 9. F. Rosca-Pruna, M.J.J. Vrakking, Phys. Rev. Lett. 87, 153902

(2001) 10. H. Stapelfeldt, T. Seideman, Rev. Mod. Phys. 75, 543 (2003) 11. T. Seideman, E. Hamilton, in Advances in Atomic, Molecular,

and Optical Physics, ed. by P.R. Berman, C.C. Lin (Academic Press, New York, 2005), p. 289

12. S. Henrik, Phys. Scr. 2004, 132 (2004) 13. S. Fleischer, I.S. Averbukh, Y. Prior, Phys. Rev. A. 74, 041403

(2006) 14. J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. Pepin, J.C.

Kieffer, P.B. Corkum, D.M. Villeneuve, Nature 432, 867 (2004) 15. R.J. Gordon, L. Zhu, W.A. Schroeder, T. Seideman, J. Appl.

Phys. 94, 669 (2003) 16. R.A. Bartels, T.C. Weinacht, N. Wagner, M. Baertschy, C.H.

Greene, M.M. Murnane, H.C. Kapteyn, Phys. Rev. Lett. 88, 013903 (2001)

17. K.F. Lee, D.M. Villeneuve, P.B. Corkum, E.A. Shapiro, Phys. Rev. Lett. 93, 233601 (2004)

18. A.H. Zewail, J. Phys. Chem. A 104, 5660 (2000) 19. V. Loriot, E. Hertz, B. Lavorel, O. Faucher, J. Phys. B. 41,

015604 (2008) 20. D. Pavicić, K.F. Lee, D.M. Rayner, P.B. Corkum, D.M. Ville-

neuve, Phys. Rev. Lett. 98, 243001 (2007) 21. I.V. Litvinyuk, K.F. Lee, P.W. Dooley, D.M. Rayner, D.M. Vil-

leneuve, P.B. Corkum, Phys. Rev. Lett. 90, 233003 (2003) 22. E. Péronne, M.D. Poulsen, C.Z. Bisgaard, H. Stapelfeldt, T. Sei-

deman, Phys. Rev. Lett. 91, 043003 (2003) 23. M. Kenzo, S. Takayuki, N. Didier, J. Phys. B. 37, 753 (2004) 24. I. Nevo, L. Holmegaard, J.H. Nielsen, J.L. Hansen, H. Sta-

pelfeldt, F. Filsinger, G. Meijer, J. Kupper, PCCP 11, 9912 (2009)

25. P.W. Dooley, I.V. Litvinyuk, K.F. Lee, D.M. Rayner, M. Spanner, D.M. Villeneuve, P.B. Corkum, Phys. Rev. A 68, 023406 (2003)

26. R. Velotta, N. Hay, M.B. Mason, M. Castillejo, J.P. Marangos, Phys. Rev. Lett. 87, 183901 (2001)

27. T. Kanai, S. Minemoto, H. Sakai, Nature 435, 470 (2005)

G. Kaya et al.

1 3

288 of 12

28. C. Vozzi, F. Calegari, E. Benedetti, J.P. Caumes, G. Sansone, S. Stagira, M. Nisoli, R. Torres, E. Heesel, N. Kajumba, J.P. Maran-gos, C. Altucci, R. Velotta, Phys. Rev. Lett. 95, 153902 (2005)

29. C.B. Madsen, A.S. Mouritzen, T.K. Kjeldsen, L.B. Madsen, Phys. Rev. A. 76, 035401 (2007)

30. J.-F. Ripoche, G. Grillon, B. Prade, M. Franco, E. Nibbering, R. Lange, A. Mysyrowicz, Opt. Commun. 135, 310 (1997)

31. S. Varma, Y.H. Chen, J.P. Palastro, A.B. Fallahkair, E.W. Rosenthal, T. Antonsen, H.M. Milchberg, Phys. Rev. A. 86, 023850 (2012)

32. Y. Feng, H. Pan, J. Liu, C. Chen, J. Wu, H. Zeng, Opt. Express 19, 2852 (2011)

33. F. Calegari, C. Vozzi, S. Gasilov, E. Benedetti, G. Sansone, M. Nisoli, S. De Silvestri, S. Stagira, Phys. Rev. Lett. 100, 123006 (2008)

34. J. Wu, H. Cai, Y. Peng, Y. Tong, A. Couairon, H. Zeng, Laser Phys. 19, 1759 (2009)

35. N. Kaya, G. Kaya, M. Sayrac, Y. Boran, S. Anumula, J. Stro-haber, A.A. Kolomenskii, H.A. Schuessler, Opt. Express 24, 2562 (2016)

36. P.M. Felker, J. Phys. Chem. 96, 7844 (1992) 37. J.P. Heritage, T.K. Gustafson, C.H. Lin, Phys. Rev. Lett. 34,

1299 (1975) 38. V. Renard, M. Renard, S. Guérin, Y.T. Pashayan, B. Lavorel, O.

Faucher, H.R. Jauslin, Phys. Rev. Lett. 90, 153601 (2003) 39. A. Rouzée, S. Guérin, O. Faucher, B. Lavorel, Phys. Rev. A 77,

043412 (2008) 40. P.J. Bustard, R. Lausten, B.J. Sussman, Phys. Rev. A 86, 053419

(2012) 41. L. Holmegaard, J.H. Nielsen, I. Nevo, H. Stapelfeldt, F. Fils-

inger, J. Küpper, G. Meijer, Phys. Rev. Lett. 102, 023001 (2009) 42. L. Holmegaard, J.L. Hansen, L. Kalhoj, S. Louise Kragh, H. Sta-

pelfeldt, F. Filsinger, J. Kupper, G. Meijer, D. Dimitrovski, M. Abu-samha, C.P.J. Martiny, L. Bojer Madsen, Nat. Phys. 6, 428 (2010)

43. S. De, I. Znakovskaya, D. Ray, F. Anis, N.G. Johnson, I.A. Bocharova, M. Magrakvelidze, B.D. Esry, C.L. Cocke, I.V. Lit-vinyuk, M.F. Kling, Phys. Rev. Lett. 103, 153002 (2009)

44. J. Mikosch, C.Z. Bisgaard, A.E. Boguslavskiy, I. Wilkinson, A. Stolow, J. Chem. Phys. 139, 024304 (2013)

45. A.E. Boguslavskiy, J. Mikosch, A. Gijsbertsen, M. Spanner, S. Patchkovskii, N. Gador, M.J.J. Vrakking, A. Stolow, Science 335, 1336 (2012)

46. J. Mikosch, A.E. Boguslavskiy, I. Wilkinson, M. Spanner, S. Patchkovskii, A. Stolow, Phys. Rev. Lett. 110, 023004 (2013)

47. C. Schröter, K. Kosma, T. Schultz, Science 333, 1011 (2011) 48. M. Machholm, N.E. Henriksen, Phys. Rev. Lett. 87, 193001

(2001) 49. E. Péronne, M.D. Poulsen, H. Stapelfeldt, C.Z. Bisgaard, E.

Hamilton, T. Seideman, Phys. Rev. A 70, 063410 (2004) 50. O. Ghafur, A. Rouzee, A. Gijsbertsen, W.K. Siu, S. Stolte, M.J.J.

Vrakking, Nat. Phys. 5, 289 (2009) 51. F. Filsinger, J. Küpper, G. Meijer, L. Holmegaard, J.H. Nielsen,

I. Nevo, J.L. Hansen, H. Stapelfeldt, J. Chem. Phys. 131, 064309 (2009)

52. J.H. Nielsen, P. Simesen, C.Z. Bisgaard, H. Stapelfeldt, F. Fils-inger, B. Friedrich, G. Meijer, J. Kupper, Phys. Chem. Chem. Phys. 13, 18971 (2011)

53. B.K. McFarland, J.P. Farrell, P.H. Bucksbaum, M. Gühr, Science 322, 1232 (2008)

54. O. Smirnova, Y. Mairesse, S. Patchkovskii, N. Dudovich, D. Vil-leneuve, P. Corkum, M.Y. Ivanov, Nature 460, 972 (2009)

55. D. Pinkham, R.R. Jones, Phys. Rev. A 72, 023418 (2005) 56. X. Xie, K. Doblhoff-Dier, H. Xu, S. Roither, M.S. Schöffler, D.

Kartashov, S. Erattupuzha, T. Rathje, G.G. Paulus, K. Yaman-ouchi, A. Baltuška, S. Gräfe, M. Kitzler, Phys. Rev. Lett. 112, 163003 (2014)

57. N.A. Hart, J. Strohaber, G. Kaya, N. Kaya, A.A. Kolomenskii, H.A. Schuessler, Phys. Rev. A 89, 053414 (2014)

58. D. Comtois, D. Zeidler, H. Pépin, J.C. Kieffer, D.M. Villeneuve, P.B. Corkum, J. Phys. B At. Mol. Opt. Phys. 38, 1923 (2005)

59. A. Rudenko, K. Zrost, E. Th, A.B. Voitkiv, B. Najjari, V.L.B.D. Jesus, B. Feuerstein, C.D. Schröter, R. Moshammer, J. Ullrich, J. Phys. B At. Mol. Opt. Phys. 38, L191 (2005)

60. P. Colosimo, G. Doumy, C.I. Blaga, J. Wheeler, C. Hauri, F. Catoire, J. Tate, R. Chirla, A.M. March, G.G. Paulus, H.G. Mul-ler, P. Agostini, L.F. DiMauro, Nat. Phys. 4, 386 (2008)

61. G.G. Paulus, W. Nicklich, H. Xu, P. Lambropoulos, H. Walther, Phys. Rev. Lett. 72, 2851 (1994)

62. T. Seideman, J. Chem. Phys. 107, 10420 (1997) 63. J.D. Graybeal, Molecular Spectroscopy (McGraw-Hill, New

York, 1988) 64. S.M. Purcell, P.F. Barker, Phys. Rev. Lett. 103, 153001 (2009) 65. F.H.M. Faisal, A. Abdurrouf, Phys. Rev. Lett. 100, 123005

(2008) 66. P.F. Bernath, P.F. Bernath, Spectra of Atoms and Molecules

(Oxford University Press, New York, 1995) 67. S. Fleischer, Y. Khodorkovsky, E. Gershnabel, Y. Prior, I.S. Aver-

bukh, Isr. J. Chem. 52, 414 (2012) 68. B.H. Bransden, C.J. Joachain, Physics of Atoms and Molecules

(Prentice-Hall, Englewood Cliffs, 2003) 69. D.A. McQuarrie, J.D. Simon, Physical Chemistry: A Molecular

Approach (University Science Books, Sausalito, 1997) 70. H. Zeng, P. Lu, J. Liu, W. Li, in Progress in Ultrafast Intense

Laser Science VIII, ed. by K. Yamanouchi, M. Nisoli, T.W. Hill (Springer, Berlin, 2012), p. 47

71. M. Leibscher, I.S. Averbukh, H. Rabitz, Phys. Rev. Lett. 90, 213001 (2003)

72. J. Floß, I.S. Averbukh, J. Phys. Chem. A 120, 3206 (2016) 73. Y. Khodorkovsky, K. Kitano, H. Hasegawa, Y. Ohshima, I.S.

Averbukh, Phys. Rev. A 83, 023423 (2011) 74. J.-M. Hartmann, C. Boulet, J. Chem. Phys. 136, 184302 (2012) 75. J.J. Larsen, K. Hald, N. Bjerre, H. Stapelfeldt, T. Seideman,

Phys. Rev. Lett. 85, 2470 (2000) 76. J.G. Underwood, B.J. Sussman, A. Stolow, Phys. Rev. Lett. 94,

143002 (2005) 77. W.M.L.D.R. Haynes, CRC Handbook of Chemistry and Physics:

A Ready-Reference Book of Chemical and Physical Data (CRC Press, Boca Raton, 2011)

78. J. Bendtsen, J. Raman Spectrosc. 2, 133 (1974) 79. P. Atkins, J. de Paula, R. Friedman, Quanta, Matter, and Change:

A Molecular Approach to Physical Chemistry (Oxford Univer-sity Press, Oxford, 2009)

80. X.M. Tong, Z.X. Zhao, C.D. Lin, Phys. Rev. A 66, 033402 (2002)

81. F.H.M. Faisal, A. Abdurrouf, K. Miyazaki, G. Miyaji, Phys. Rev. Lett. 98, 143001 (2007)

82. K. Miyazaki, M. Kaku, G. Miyaji, A. Abdurrouf, F.H.M. Faisal, Phys. Rev. Lett. 95, 243903 (2005)

83. R.B. Philip, J. Per, Molecular Symmetry and Spectroscopy, 2nd edn. (NRC Research Press, Ottawa, 2006)

84. W.H. Fletcher, J.S. Rayside, J. Raman Spectrosc. 2, 3 (1974) 85. H. Li, W. Li, Y. Feng, J. Liu, H. Pan, H. Zeng, Phys. Rev. A 85,

052515 (2012) 86. J. Itatani, D. Zeidler, J. Levesque, M. Spanner, D.M. Villeneuve,

P.B. Corkum, Phys. Rev. Lett. 94, 123902 (2005) 87. M. Herman, A. Campargue, M.I. El Idrissi, Auwera J. Vander, J.

Phys. Chem. Ref. Data 32, 921 (2003) 88. R. Torres, N. Kajumba, J.G. Underwood, J.S. Robinson, S.

Baker, J.W.G. Tisch, R. de Nalda, W.A. Bryan, R. Velotta, C. Altucci, I.C.E. Turcu, J.P. Marangos, Phys. Rev. Lett. 98, 203007 (2007)

89. M. Spanner, S. Patchkovskii, Chem. Phys. 414, 10 (2013)

nonadiabatic molecular alignment of linear molecules...

Documents