DOI:
10.1039/C6RA16259B
(Paper)
RSC Adv., 2016,
6, 100295-100299
Selective excitation and control of the molecular orientation by a phase shaped laser pulse
Received
23rd June 2016
, Accepted 17th October 2016
First published on 17th October 2016
Abstract
We demonstrate that the selective excitation and further control of molecular orientation can be realized by a dual-color shaped laser pulse with V-style spectral phase modulation. Due to the instructive or destructive interference between the odd and even rotational wave-packet contributions, selective excitation of the molecular orientation is realized in the half rotational periods by the V-style phase modulated femtosecond laser pulse, and the positive and negative molecular orientations can be freely switched by varying the carrier-envelope phase of the shaped laser pulse. Moreover, by varying the modulation position of the V-style phase-shaped femtosecond laser pulse, the maximum degrees of the positive and negative molecular orientation in the half rotational periods can be continuously manipulated. Our scheme can provide a new method for the manipulation of the molecular orientation induced by the phase shaped laser pulse, and also has significant applications in the control of the molecular rotational wave-packet.
1. Introduction
An anisotropic quantum system, such as aligned molecules, has attracted both physicists' and chemists' interest because of its widespread applications in chemical reaction dynamics,1–3 high-order harmonic generation,4,5 ultrafast molecular imaging,6 and attosecond science.7 Comparing with the molecular alignment, which suffers from an averaging effect over the direction of the molecular axis, the molecular orientation with a “head-versus-tail” order is faced with greater challenges, but has more important applications in various related areas. So far, several different techniques have been utilized to realize the molecular orientation, such as the strong dc field (i.e., the so called “brute force” method),8 the weak dc field combined with an intense laser field,9–11 or the asymmetric intense laser field.12–17 However, the Stark effect induced by the strong dc electric field may influence the experimental results, and the presence of the dc field will also limit the further applications of the oriented molecules. Therefore, the field-free molecular orientation induced by a short asymmetric laser pulse has achieved more and more attentions, such as the dual-color or multi-color laser pulse,12–15 or the half-cycle laser pulse.16,17
For those who dedicated on the research of the molecular alignment and orientation, an important issue is how to control the evolution of the molecular rotational wave-packet. With the advent of the ultrafast pulse shaping technique, the shaped laser pulse with the spectral phase modulation has shown to be one of the effective methods to manipulate the aligned or oriented molecules. Various phase modulation schemes, including the phase step modulation, the cubic phase modulation, and the closed feedback phase optimization, have been applied to the study of the molecular alignment or orientation control.18–25 Recently, we demonstrated that, by the V-style spectral phase modulation, the degree and temporal structure of the molecular alignment can be effectively manipulated.26 In this paper, we further show that the molecular orientation in half rotational periods can be selectively excited and further controlled by the V-style spectral phase modulation, and the selective excitation can be attributed to the instructive or destructive interference between the odd and even rotational wave-packet contributions. Furthermore, by varying the carrier-envelope phase of the shaped laser pulse, the positive and negative molecular orientations can be freely switched. Finally, we show that the degree of the molecular orientation in the half rotational periods can be continuously manipulated by scanning the modulation position of the V-style spectral phase modulation.
2. Theoretical model
We consider that a linear molecule is subjected to a linearly polarized dual-color laser pulse with the fundamental-wave (FW) and its second-harmonic (SH) fields, and the dual-color laser field can be written as |
E(t) = EFW(t)cos(ω0t) + ESH(t)cos(2ω0t + ϕCEP)
| (1) |
where EFW(SH)(t) = E0
exp((−2
ln
2)t2/τ02) is the pulse envelope with the Gaussian distribution, E0 is the laser field amplitude, ω0 is the laser central frequency, τ0 is the pulse duration, and ϕCEP is the carrier-envelope phase (CEP) of the laser pulse. For ϕCEP = 0 or π, the overall electric field of the dual-color pulse is totally reverse. Based on the rigid rotor model, the time-dependent Schrödinger equation of the interaction between the linear molecule and dual-color laser field can be approximated as |
 | (2) |
with the effective Hamiltonian |
 | (3) |
where B and μ0 are respectively the rotational constant and the permanent dipole moment of the molecule, J is the angular momentum, θ is the angle between the molecular axis and the direction of the dual-color laser field, α∥ and α⊥ are respectively the polarizability components parallel and perpendicular to the molecular axis, and β∥ and β⊥ are respectively the hyperpolarizability components parallel and perpendicular to the molecular axis. The degree of the molecular orientation is usually given by the expectation value of cos
θ (i.e., 〈cos
θ〉). When the thermal equilibrium of the molecular ensemble is considered, the expectation value of cos
θ should be averaged over the Boltzmann distribution, and written as |
 | (4) |
where gJ is the spin degeneracy factor, Q is the rotational partition function, k is the Boltzmann constant, and T is the molecular rotational temperature.
In our theoretical simulation, the time-dependent Schrödinger equation in eqn (2) is numerically solved by the split-operator method.27,28 The CO molecule is used as the example, and the molecular parameters are set as B = 1.93 cm−1, μ0 = 0.112 D, α∥ = 2.294 Å3, α⊥ = 1.77 Å3, β∥ = 2.748 × 109 Å5, β⊥ = 4.994 × 108 Å5.29,30 Thus, the rotational period of the CO molecule can be calculated as Trot = 1/(2Bc) ≈ 8.64 ps, where c is the velocity of the light in vacuum. The central frequency of the fundamental wave field is ω0 = 12
500 cm−1, corresponding to the central wavelength of 800 nm. The pulse duration of the transform-limited laser pulse is τ0 = 50 fs, and all the peak intensities of the unshaped and shaped laser pulses are I = 1 × 1013 W cm−2. The molecular rotational temperature is set to be T = 30 K.
3. Results and discussion
We first show the time revolved molecular orientation induced by the transform-limited femtosecond dual-color laser pulse, as depicted in Fig. 1, together with the odd (green dashed line) and even (blue dotted line) rotational wave-packet contributions. It can be seen that the odd and even rotational wave-packet contributions show the same evolution behavior around the full rotational periods, while exhibit the inverse evolution behavior around the half rotational periods. Since the molecular orientation results from the interference between the odd and even rotational wave-packet contributions, it will experience the destructive interference around the half rotational periods while instructive interference around the full rotational periods. Therefore, the result is that only the total contributions for the molecular orientation can be found around the full rotational periods. However, if the symmetrical contributions between the odd and even rotational wave-packets around the half rotational periods can be broken, it can be suggested that the selective excitation and further control of the molecular orientation can be realized.31,32
 |
| Fig. 1 Time revolved molecular orientation (red solid line) induced by a transform-limited femtosecond dual-color laser pulse, together with its odd (green dashed line) and even (blue dotted line) rotational wave-packet contributions. | |
Here, we utilize a V-style phase modulated dual-color laser pulse to realize the selective excitation and further control of the molecular orientation. The details of the V-style phase modulation has been described elsewhere.26,33 Briefly, the shaped laser field by the V-style phase modulation in frequency domain EV(ω) can be written as EV(ω) = E(ω) × exp[iΦ(ω)], where E(ω) is the Fourier transform of the transform-limited laser field E(t), as shown in eqn (1), and Φ(ω) is the V-style phase modulation function defined by Φ(ω) = τ|ω − ω0 − δω|, here τ and δω represent the modulation depth and the modulation position, respectively. Thus, the shaped laser field in time domain EV(t) can be given by the reverse Fourier transform of EV(ω). Fig. 2(a) presents the schematic diagram of the V-style spectral phase modulation in frequency domain, and the amplitude profile of the 50 fs transform-limited laser pulse in time domain is shown in Fig. 2(b), together with the corresponding shaped dual-color laser pulse with different modulation parameters (Fig. 2(c)–(e)). One can see from Fig. 2(c)–(e) that, two time-delayed sub-pulses with controllable relative intensity ratio can be formed by the V-style phase modulation, where the modulation depth τ determines the time separation Δt of the two sub-pulses with Δt = 2τ, and the modulation position δω determines the relative intensity ratio between the two sub-pulses. Therefore, the shaped laser pulse with the V-style spectral phase modulation is similar to the simplest extension of a single pulse to a train of two pulses with controllable relative intensity ratio.34–36
 |
| Fig. 2 (a) Schematic diagram of the 50 fs dual-color laser spectrum (green solid line) and the V-style spectral phase modulation Φ(ω) = τ|ω − ω0 − δω| with τ = 2 ps and δω = 0 cm−1 (blue dashed line). (b)–(e) The amplitude profile of the unshaped dual-color laser pulse (b) and the shaped laser pulse by the V-style spectral phase modulation with τ = 2 ps and δω = 0 cm−1 (c), together with τ = 4 ps and δω = −20 (d) and 20 cm−1 (e). | |
By employing the shaped laser pulse formed by the V-style spectral phase modulation, we can realize the selective excitation and control of the rotational wave-packet in the molecular orientation. Fig. 3 shows the time revolved molecular orientation (red solid lines) induced by the V-style spectral phase shaped femtosecond laser pulse for carrier-envelope phase ϕCEP = 0 (a) and π (b), together with the odd (green dashed lines) and even (blue dotted lines) rotational wave-packet contributions. Here, the modulation depth is set as τ = 2.16 ps, corresponding to the time separation of the two sub-pulses Δt = 2τ = 4.32 ps, and the modulation position is δω = 0 cm−1. This indicates that two sub-pulses with equal intensity and the time separation of half rotational period of the molecule can be formed by the V-style spectral phase modulation. One can see from Fig. 3 that, due to the precise excitation of the second sub-pulse at the position of the half rotational period, the symmetrical contributions of the odd and even rotational wave-packets are broken, and selective excitation of the molecular orientation at these positions is realized. Furthermore, by observing Fig. 3(a) and (b), one can see that this selective excitation of the positive and negative molecular orientation can be freely switched by simply changing the carrier-envelope phase ϕCEP from 0 to π.
 |
| Fig. 3 Time revolved molecular orientation (red solid lines) induced by the V-style spectral phase shaped femtosecond laser pulse with the modulation depth τ = 2.16 ps and the modulation position δω = 0 cm−1 for carrier-envelope phase ϕCEP = 0 (a) and π (b), together with their odd (green dashed lines) and even (blue dotted lines) rotational wave-packet contributions. | |
In order to explore the physical origin why the selective excitation of the molecular orientation in the half rotational periods can take place by the V-style spectral phase modulation, we further investigate the population in each rotational state, which can be obtained by the Fourier transform of the molecular orientation signal, and the results are presented in Fig. 4. As can be seen, both the odd and even rotational states are populated under the excitation of the transform-limited laser pulse (red squares). However, only the odd rotational states are populated under the case of the V-style phase modulation for both the carrier-envelope phase ϕCEP = 0 (green circles) and π (blue triangles). This can be illustrated by considering both the pattern of the field amplitude and the contributions of the odd and the even rotational wave-packets. For ϕCEP = 0, the pattern of the field amplitude is shown in Fig. 2(c). The odd and even rotational wave-packet contributions by the first sub-pulse of the V-style phase modulation, which is the same as that shown in Fig. 1, are then excited by the second sub-pulse at the position of the half rotational periods. Considering the slope of the curves of the odd (i.e., positive) and even (i.e., negative) rotational wave-packet contributions,37 the instructive interference for the odd rotational wave-packet contributions while the destructive interference for the even rotational wave-packet contributions will occur due to the excitation of the second sub-pulse. Therefore, the result is that only the odd rotational states are populated. Similarly, for ϕCEP = π, both the field pattern of the two sub-pulses shown in Fig. 2(c) and the contributions of the odd and even rotational wave-packets presented in Fig. 1 are totally reverse, and therefore the same result can be obtained, i.e., only the odd rotational states are populated.
 |
| Fig. 4 Fourier transform of the molecular orientation signal induced by the transform-limited laser pulse (red squares) and the V-style spectral phase shaped femtosecond laser pulse with τ = 2.16 ps and δω = 0 cm−1 for ϕCEP = 0 (green circles) and π (blue triangles). | |
Finally, we demonstrate the maximum degrees of the positive (red squares) and negative (blue circles) molecular orientation in the half rotational periods by varying the modulation position δω with the modulation depth τ = 2.16 ps, and the result is shown in Fig. 5. One can see that the degree of the molecular orientation at the half rotational periods can be continuously controlled by varying the modulation position. For δω = 0 cm−1, corresponding to a shaped laser pulse with two sub-pulses of equal intensity, the maximal value of the degree of the molecular orientation is observed. With the increase of the absolute value of the modulation position δω, the maximal degree of the molecular orientation decreases and gradually approaches to zero, which means that the selective excitation of the molecular orientation will disappear. The reason is that the larger absolute value of the modulation position δω will progressively diminish one of the two shaped sub-pulses, and it is back to the case of the excitation with single laser pulse. Therefore, the selective excitation of the molecular orientation can not only be switched by varying the carrier-envelope phase ϕCEP but also be continuously controlled by scanning the modulation position δω.
 |
| Fig. 5 The maximum degrees of the positive (red squares) and negative (blue circles) molecular orientation in the half rotational period by varying the modulation position δω with the modulation depth τ = 2.16 ps. | |
It should be pointed out that, although the selective excitation of the molecular orientation in this work is obtained at the relatively lower molecular rotational temperature T = 30 K, the effect is still applicable for higher temperatures (e.g., room temperature), except for a much lower degree of the molecular orientation. In fact, it is a common method to obtain the higher (or lower) degree of the molecular orientation by reducing (or increasing) the molecular rotational temperature.
4. Conclusions
In summary, we have shown that the selective excitation of the molecular orientation is realized and further controlled by a shaped laser pulse with the V-style spectral phase modulation. By precisely control the modulation depth, the selective excitation of the molecular orientation at half rotational periods is obtained. The positive and negative molecular orientations can be freely switched by changing the carrier-envelope phase, and the degree can also be continuously manipulated by scanning the modulation position. The physical origin of the selective excitation by the V-style phase modulation can be well illustrated by considering the instructive and destructive interferences between the odd and even rotational wave-packet contributions. We believe that these results provide a valuable method to manipulate the molecular orientation degree and direction, and are also expected to be significant for the applications in various related areas.
Acknowledgements
This work was partly supported by National Natural Science Fund (Grants No. 51132004 and 11474096), the Special Fund for Theoretical Physics Research Program of the National Natural Science Foundation (Grant No. 11547223), and Shanghai Municipal Science and Technology Commission (Grant No. 14JC1401500).
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