Alignment structures of rotational wavepacket created by two strong femtosecond laser pulses

Hongyan Jiang; Chengyin Wu; He Zhang; Hongbing Jiang; Hong Yang; Qihuang Gong

doi:10.1364/OE.18.008990

1. Introduction

When molecules are irradiated by strong femtosecond laser pulses, nonadiabatic rotational excitation brings molecules to the states with higher angular momentum while keeping the azimuthal quantum number unchanged. A rotational wavepacket is therefore generated. After the laser is over, the evolution of the wavepacket leads to the transient alignment with the molecular axis along the laser polarization direction [1]. The alignment degree of molecules is characterized by the average of $< \cos^{2} θ >$ with θ being the angle between the laser polarization direction and the molecular axis. Many experimental techniques have been developed to measure the alignment degree [2–4]. Meanwhile, field-free aligned molecules have attracted widespread interest for application [5–9]. Because highly aligned molecules are required for practical application, many efforts have been made to improve the alignment degree. Previous results showed that the alignment degree could be improved by decreasing the rotational temperature of molecules [10] or by increasing the laser intensity [11]. However, molecules will be ionized if the laser intensity is higher than the threshold intensity of ionization. In order to circumvent the maximum intensity limit for a single laser pulse, multiple-pulse method is also proposed to improve the alignment degree of molecules [12–15]. In the multiple-pulse control scheme, one or several additional laser pulses are applied at selected times after the first aligning laser pulse. The peak alignment can be increased to a high level without destroying molecules. Recently, optimization of field-free molecular alignment was achieved both theoretically and experimentally by using phase- and amplitude-shaped femtosecond laser pulses [16–18].

For two-pulse alignment of molecules, it has been demonstrated that the alignment degree is influenced by the time delay between the two laser pulses. The alignment degree of the coherent rotational wavepacket created by the first aligning laser pulse can be enhanced or annihilated by a second laser pulse [19–26]. In our recent papers, we studied the population distribution for the rotational wavepacket created by two laser pulses. The results show that the populations of molecular rotational states can be enhanced or suppressed for a specific rotational state [25,26]. The oscillatory dependence of the rotational population on the time delay is also observed by Meijer et al. for NO molecules irradiated by two femtosecond laser pulses [27]. In our previous studies, we observed the enhancement, annihilation and split of the alignment structures depending on the time delay. However, this observation was not quantitatively explained [24–26]. In this paper, we systemically study the alignment structures of molecular rotational wavepacket created by two strong femtosecond laser pulses. By separately calculating the self-coupling term and the cross-coupling term, the observed alignment structures are quantitatively explained for the rotational wavepacket created by two strong femtosecond laser pulses

2. Theory

Fig. 1 shows the generation process of molecular rotational wavepacket created by linear molecules and two strong femtosecond laser pulses. A molecule initially in a rotational state $| J_{0}, M_{0} 〉$ is irradiated by a strong femtosecond laser pulse. The laser-molecule interaction generates a rotational wavepacket via a series of Raman processes. The wavepacket can be expanded as:

Ψ_{1} (t) = \sum_{J_{i}} P_{J_{0}, J_{i}} \exp (- i E_{J_{i}} t) | J_{i} 〉,

where

E_{J_{i}}

and

P_{J_{0}, J_{i}}

are respectively the energy eigenvalue and the coefficient of rotational state

| J_{i} 〉

. By numerically solving the time-dependant Schrödinger equation for the linear molecule irradiated by strong femtosecond laser pulses,

P_{J_{0}, J_{i}}

is obtained at the end of the laser pulse. The formulas are written in atomic units. In the expression, we omit the quantum number M₀ because the cylindrical symmetry of the interaction keeps the azimuthal quantum number unchanged. By a kind of phase matching between different rotational states, the evolution of the wavepacket results in transient molecular alignment.

Fig. 1 Rotational wavepacket created by linear molecules and two strong femtosecond laser pulses.

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During the evolution of the wavepacket, a second laser pulse is applied. Each J_i component of the wavepacket $Ψ_{1} (t)$ produces a sub-wavepacket $Φ_{J_{i}} (t)$ , which can be written as:

Φ_{J_{i}} (t) = P_{J_{0}, J_{i}} \exp (- i E_{J_{i}} Δ t) \sum_{J} P_{J_{i}, J} \exp [- i E_{J} (t - Δ t)] | J 〉,

where

Δ t

is the delay time between the two laser pulses. Coherent superpositions of these sub-wavepackets generate the final wavepacket

Ψ_{2} (t)

, which can be written as:

Ψ_{2} (t) = \sum_{J_{i}} Φ_{J_{i}} (t) = \sum_{J_{i}} P_{J_{0}, J_{i}} \exp (- i E_{J_{i}} Δ t) \sum_{J} P_{J_{i}, J} \exp [- i E_{J} (t - Δ t)] | J 〉 .

The phase factor $\exp (- i E_{J_{i}} Δ t)$ in Eq. (2) and (3) denotes the phase that the J_i component gains in the freely evolution period between the two laser pulses. Our previous studies demonstrate that quantum interferences among these sub-wavepackets enable the population to be enhanced or suppressed for a specific rotational state in the final wavepacket by controlling the phase factor [25,26]. In the following, we consider the effect of the factor $\exp (- i E_{J_{i}} Δ t)$ on the alignment structures.

For the wavepacket created by two laser pulses with a time delay $Δ t$ , the alignment $< \cos^{2} θ >$ can be written as:

\begin{array}{l} 〈 \cos^{2} θ 〉 (t) = \sum_{J_{0}, M_{0}} g_{J_{0}, M_{0}} < \cos^{2} θ > {(t)}_{J_{0}, M_{0}} = \sum_{J_{0}, M_{0}} g_{J_{0}, M_{0}} 〈 Ψ_{2} (t) | \cos^{2} θ | Ψ_{2} (t) 〉 \\ = \sum_{J_{0}, M_{0}} g_{J_{0}, M_{0}} \sum_{J_{i}} 〈 Φ_{J_{i}} (t) | \cos^{2} θ | Φ_{J_{i}} (t) 〉 + \sum_{J_{0}, M_{0}} g_{J_{0}, M_{0}} \sum_{J_{i}} \sum_{J_{i}^{'} \neq J_{i}} 〈 Φ_{J_{i}} (t) | \cos^{2} θ | Φ_{J_{i}^{'}} (t) 〉, \end{array}

where

g_{J_{0}, M_{0}}

is the Boltzmann averaging factor of different initial state

| J_{0}, M_{0} 〉

. This expression demonstrates that, for the rotational wavepacket created by two laser pulses, the alignment structures are composed of the self-coupling term

\sum_{J_{0}, M_{0}} g_{J_{0}, M_{0}} \sum_{J_{i}} 〈 Φ_{J_{i}} (t) | \cos^{2} θ | Φ_{J_{i}} (t) 〉

and the cross-coupling term

\sum_{J_{0}, M_{0}} g_{J_{0}, M_{0}} \sum_{J_{i}} \sum_{J_{i}^{'} \neq J_{i}} 〈 Φ_{J_{i}} (t) | \cos^{2} θ | Φ_{J_{i}^{'}} (t) 〉

. Substituting Eq. (2) into the self-coupling term, we find that the phase factor

\exp (- i E_{J_{i}} Δ t)

is totally cancelled. The self-coupling term can therefore be expanded into a superposition of a series of

A_{J} \cos [ω_{J} (t - Δ t) + φ_{J}]

with

ω_{J}

the Raman frequency. The expansion means that the self-coupling term produces one series of structures. These structures are called basal alignment structures. They locate at

n \cdot T_{r} / 2 + Δ t

with T_r the molecular rotational period. Unlike the self-coupling term, the phase factor

\exp (- i E_{J_{i}} Δ t)

cannot be cancelled in the cross-coupling term. Additional phase factor

\exp [- i (E_{J_{i}^{'}} - E_{J_{i}}) Δ t]

is left. This phase factor has a modulation effect on the alignment structures. Depending on the sign of the additional phase, the cross-coupling term can be expanded into a series of

A_{J} \cos [ω_{J} t + φ_{J}]

and a series of

A_{J} \cos [ω_{J} (t - 2 Δ t) + φ_{J}]

. Based on these expansions, we know that the cross-coupling term generate two series of structures. These structures are called modulation structures and locate at

t = n \cdot T_{r} / 2

and

n \cdot T_{r} / 2 + 2 Δ t

, respectively. For different time delays between the two laser pulses, the superposition of the basal alignment structures and the modulation structures generates different alignment structures, which have been observed in the experiment.

3. Experiment

The experimental setup and detection technique have been described in our previous paper [4]. A Ti:sapphire chirped-pulse amplifier (TSA-10, Spectra-Physics Inc., USA) delivers laser pulses with a central wavelength of 800 nm and a pulse duration of 110 fs at a repetition rate of 10 Hz. The laser pulse is split into two strong pump pulses and one weak probe pulse. The relative timings are precisely adjusted by two computer-controlled translational stages. The two linearly polarized pump pulses have the same polarization and completely overlap in space. The probe pulse is slightly elliptically polarized. The pump beam and the probe beam are focused with a 15 cm focal length lens into a 20 cm long gas cell at a small angle. The pure heterodyne alignment signal is obtained by subtracting two heterodyne signals measured respectively by a left-handed and right-handed elliptically polarized probe laser. The signal is proportional to $(〈 \cos^{2} θ 〉 - 1 / 3)$ and directly reproduces the alignment structure of the rotational wavepacket.

Fig. 2 shows the alignment signal for the rotational wavepacket of N₂O created by a single or double laser pulses. The initial N₂O molecules are at room temperature. Nonadiabatic rotational excitation generates a coherent rotational wavepacket after the irradiation of one strong femtosecond laser pulse. The laser intensity is estimated to be 6.0 × 10¹² W/cm² through measuring its pulse duration and focusing size. Fig. 2(a) shows the alignment structure for the rotational wavepacket of N₂O created by a single laser pulse. The classical rotational period T_r of N₂O is 39.9 ps. At the half or full rotational period, all the components of the wavepacket evolve in phase and the wavepacket exhibits transient alignment or antialignment. The alignment structure fully revives every rotational period. Fig. 2(b)-2(e) show the alignment structures for the rotational wavepacket of N₂O created by two laser pulses with different time delays. These observations demonstrate that the alignment structure created by the first aligning laser pulse can be annihilated, enhanced or split by the second laser pulse depending on the time delay between the two laser pulses. In the following section, we will analyze these different alignment structures using the model we propose in the theory section.

Fig. 2 Experimentally measured alignment signal of N₂O created by (a) a single laser pulse and (b)-(e) double laser pulses with different time delays. The arrow marks the time the second laser pulse is applied.

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4. Results and discussion

Fig. 3(a) shows the experimentally measured alignment structures for the rotational wavepacket of N₂O created by two laser pulses with ∆t = 8.21 ps. There are three series of alignment structures. One locates at $n \cdot T_{r} / 2 + Δ t$ . The other two locate respectively at $n \cdot T_{r} / 2$ and $n \cdot T_{r} / 2 + 2 Δ t$ . Fig. 3(b) exhibits the basal alignment structures described by the self-coupling term. These structures located at $n \cdot T_{r} / 2 + Δ t$ . Fig. 3(c) exhibits the modulation structures described by the cross-coupling term. There are two series of structures, respectively located at $n \cdot T_{r} / 2$ and $n \cdot T_{r} / 2 + 2 Δ t$ . The coherent superposition of the self-coupling term and the cross-coupling term is shown with the red line in Fig. 3(a). It reproduces the three series of alignment structures observed in the experiment for the rotational wavepacket of N₂O created by two laser pulses with ∆t = 8.21 ps.Fig. 4(a) shows the experimentally measured alignment structures for the rotational wavepacket of N₂O created by two laser pulses with ∆t = 9.99 ps. There are only two series of alignment structures when the time delay is approximately a quarter of a rotational period. One locates at $n \cdot T_{r} / 2 + Δ t$ . The other locates at $n \cdot T_{r} / 2$ . Fig. 4(b) exhibits the basal alignment structures described by the self-coupling term. The structures locate at $n \cdot T_{r} / 2 + Δ t$ and their shapes are similar to those shown in Fig. 3(b). Fig. 4(c) exhibits the modulation structures described by the cross-coupling term. Different from the two series of modulation structures shown in Fig. 3(c), there is only one series of modulation structures when the time delay is around a quarter rotational period. These structures locate at $n \cdot T_{r} / 2$ . The superposition of the self-coupling term and the cross-coupling term also reproduces the two series of alignment structures observed in the experiment for the rotational wavepacket of N₂O created by two laser pulses with ∆t = 9.99 ps.The calculations above show that the self-coupling term generates one series of basal alignment structures, whose shapes have no relationship with the time delay between the two laser pulses. However, the modulation structures produced by the cross-coupling term sensitively depend on the time delay. Two series of modulation structures are generated for a general time delay. The coherent superposition of the basal alignment structures and the modulation structures generates three series of alignment structures. However, when the time delay is around a quarter rotational period, the two series of modulation structures produced by the cross-coupling term will overlap in time and merge into one series of structures. With this time delay, the coherent superposition of the self-coupling term and the cross-coupling term generates two series of alignment structures. These calculations are consistent with our experimental observations.

Fig. 3 (a) Evolution of rotational wavepacket of N₂O created by two laser pulse with ∆t = 8.21 ps. Black line represents the measured alignment signal, red line represents the theoretical superposition of the self-coupling term and the cross-coupling term, (b) basal alignment structures described by the self-coupling term and (c) modulation structures described by the cross-coupling term.

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Fig. 4 The same as Fig. 3 but ∆t = 9.99 ps.

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Fig. 5(a) shows the experimentally measured alignment structures for the rotational wavepacket of N₂O created by two laser pulses with ∆t = 19.93 ps. The alignment structures are annihilated when the time delay is around a half rotational period. Fig. 5(b) and 5(c) show the basal alignment structures and the modulation structures described by the self-coupling term and the cross-coupling term, respectively. Based on these calculations, we know that the basal alignment structures and the modulation structures have opposite phases in addition to the temporal overlap. The superposition cancels each other out and leads to the annihilation of the alignment structures.

Fig. 5 The same as Fig. 3 but ∆t = 19.93 ps.

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Fig. 6(a) shows the experimentally measured alignment structures for the rotational wavepacket of N₂O created by two laser pulses with ∆t = 39.82 ps. There are only one series of alignment structures when the time delay is approximately a full rotational period. Moreover, the alignment structures are enhanced after the irradiation of the two laser pulses. Fig. 6(b) and 6(c) exhibit the basal alignment structures and the modulation structures described by the self-coupling term and the cross-coupling term, respectively. These calculations show that the basal alignment structures and the modulation structures are in phase in addition to the temporal overlap. Their superposition therefore enhances the alignment structures and is consistent with the experimental observation.

Fig. 6 The same as Fig. 3 but ∆t = 39.82 ps.

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5. Conclusion

We numerically solve the time-dependant Schrödinger equation for the linear molecule irradiated by two strong femtosecond laser pulses. The alignment structures of the rotational wavepacket are composed of self-coupling term and cross-coupling term. The former generates basal alignment structures and the latter modulation structures. Their superposition generates exotic alignment structures depending on the time delay. Theses calculations reproduce our experimental observations for the rotational wavepacket of N₂O created by two strong femtosecond laser pulse. This study quantitatively explains the annihilation, enhancement and split of rotational wavepacket observed in previous reports.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under grant Nos. 10974005, 10634020, and 10821062 and the National Basic Research Program of China under grant No. 2006CB921601.

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Alignment structures of rotational wavepacket created by two strong femtosecond laser pulses

Abstract

1. Introduction

2. Theory

3. Experiment

4. Results and discussion

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (6)

Equations (4)

Optics Express