Selective excitation and control of the molecular orientation by a phase shaped laser pulse

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) = E_FW(t)cos(ω₀t) + E_SH(t)cos(2ω₀t + ϕ_CEP) (1)

where E_FW(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 ϕ_CEP is the carrier-envelope phase (CEP) of the laser pulse. For ϕ_CEP = 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 aswith the effective Hamiltonianwhere 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 aswhere g_J 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 ų, α_⊥ = 1.77 ų, β_∥ = 2.748 × 10⁹ Å⁵, β_⊥ = 4.994 × 10⁸ Å⁵.^29,30 Thus, the rotational period of the CO molecule can be calculated as T_rot = 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 E_V(ω) can be written as E_V(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 E_V(t) can be given by the reverse Fourier transform of E_V(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 ϕ_CEP = 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 ϕ_CEP 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 ϕ_CEP = 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 ϕ_CEP = 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 ϕ_CEP = 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 ϕ_CEP = 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 ϕ_CEP = 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 ϕ_CEP 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).

References

1. H. Stapelfeldt and T. Seideman, Colloquium: Aligning molecules with strong laser pulses, Rev. Mod. Phys., 2003, 75, 543.
2. T. Suzuki, S. Minemoto, T. Kanai and H. Sakai, Optimal Control of Multiphoton Ionization Processes in Aligned I₂ Molecules with Time-Dependent Polarization Pulses, Phys. Rev. Lett., 2004, 92, 133005.
3. I. V. Litvinyuk, K. F. Lee, P. W. Dooley, D. M. Rayner, D. M. Villeneuve and P. B. Corkum, Alignment-Dependent Strong Field Ionization of Molecules, Phys. Rev. Lett., 2003, 90, 233003.
4. T. Kanai, S. Minemoto and H. Sakai, Quantum interference during high-order harmonic generation from aligned molecules, Nature, 2005, 435, 470.
5. T. Kanai, S. Minemoto and H. Sakai, Ellipticity Dependence of High-Order Harmonic Generation from Aligned Molecules, Phys. Rev. Lett., 2007, 98, 053002.
6. M. Lein, Molecular imaging using recolliding electrons, J. Phys. B: At., Mol. Opt. Phys., 2007, 40, R135.
7. P. Lan, P. Lu, W. Cao, Y. Li and X. Wang, Carrier-envelope phase-stabilized attosecond pulses from asymmetric molecules, Phys. Rev. A, 2007, 76, 021801.
8. B. Friedrich and D. Herschbach, Spatial orientation of molecules in strong electric fields and evidence for pendular states, Nature, 1991, 353, 412.
9. H. Sakai, S. Minemoto, H. Nanjo, H. Tanji and T. Suzuki, Controlling the Orientation of Polar Molecules with Combined Electrostatic and Pulsed, Nonresonant Laser Fields, Phys. Rev. Lett., 2003, 90, 083001.
10. A. Goban, S. Minemoto and H. Sakai, Laser-Field-Free Molecular Orientation, Phys. Rev. Lett., 2008, 101, 013001.
11. Y. Sugawara, A. Goban, S. Minemoto and H. Sakai, Laser-field-free molecular orientation with combined electrostatic and rapidly-turned-off laser fields, Phys. Rev. A, 2008, 77, 031403.
12. S. De, et al., Field-Free Orientation of CO Molecules by Femtosecond Two-Color Laser Fields, Phys. Rev. Lett., 2009, 103, 153002.
13. J. Wu and H. Zeng, Field-free molecular orientation control by two ultrashort dual-color laser pulses, Phys. Rev. A, 2010, 81, 053401.
14. M. Muramatsu, M. Hita, S. Minemoto and H. Sakai, Field-free molecular orientation by an intense nonresonant two-color laser field with a slow turn on and rapid turn off, Phys. Rev. A, 2009, 79, 011403.
15. S. Zhang, J. Shi, H. Zhang, T. Jia, Z. Wang and Z. Sun, Field-free molecular orientation by a multicolor laser field, Phys. Rev. A, 2011, 83, 023416.
16. M. Machholm and N. E. Henriksen, Field-Free Orientation of Molecules, Phys. Rev. Lett., 2001, 87, 193001.
17. A. Matos-Abiague and J. Berakdar, Sustainable orientation of polar molecules induced by half-cycle pulses, Phys. Rev. A, 2003, 68, 063411.
18. M. Spanner, E. A. Shapiro and M. Ivanov, Coherent control of rotational wave-packet dynamics via fractional revivals, Phys. Rev. Lett., 2004, 92, 093001.
19. M. Renard, E. Hertz, S. Guérin, H. R. Jauslin, B. Lavorel and O. Faucher, Control of field-free molecular alignment by phase-shaped laser pulses, Phys. Rev. A, 2005, 72, 025401.
20. C. Horn, M. Wollenhaupt, M. Krug and T. Baumert, Adaptive control of molecular alignment, Phys. Rev. A, 2006, 73, 031401.
21. A. Rouzée, E. Hertz, B. Lavorel and O. Faucher, Towards the adaptive optimization of field-free molecular alignment, J. Phys. B: At., Mol. Opt. Phys., 2008, 41, 074002.
22. O. Ghafur, A. Rouzée, A. Gijsbertsen, W. K. Siu, S. Stolte and M. J. J. Vrakking, Impulsive orientation and alignment of quantum-state-selected NO molecules, Nat. Phys., 2009, 5, 289.
23. T. Suzuki, Y. Sugawara, S. Minemoto and H. Sakai, Optimal control of nonadiabatic alignment of rotationally cold N₂ molecules with the feedback of degree of alignment, Phys. Rev. Lett., 2008, 100, 033603.
24. S. Zhang, C. Lu, T. Jia, Z. Sun and J. Qiu, Field-free molecular alignment control by phase-shaped femtosecond laser pulse, J. Chem. Phys., 2011, 135, 224308.
25. S. Zhang, C. Lu, J. Shi, T. Jia, Z. Wang and Z. Sun, Field-free molecular alignment by shaping femtosecond laser pulse with cubic phase modulation, Phys. Rev. A, 2011, 84, 013408.
26. S. Xu, Y. Yao, C. Lu, J. Ding, T. Jia, S. Zhang and Z. Sun, Manipulating field-free molecular alignment by V-shaped femtosecond laser pulses, Phys. Rev. A, 2014, 89, 053420.
27. M. D. Feit, J. A. Fleck Jr and A. Steiger, Solution of the Schrödinger equation by a spectral method, J. Comput. Phys., 1982, 47, 412.
28. A. D. Bandrauk and H. Shen, Exponential split operator methods for solving coupled time-dependent Schrödinger equations, J. Chem. Phys., 1993, 99, 1185.
29. K. A. Peterson and T. H. Dunning Jr, The CO molecule: the role of basis set and correlation treatment in the calculation of molecular properties, J. Mol. Struct.: THEOCHEM, 1997, 400, 93.
30. M. Pecul, Density functional and coupled cluster calculations of dynamic hyperpolarizabilities and their geometry derivatives, Chem. Phys. Lett., 2005, 404, 217.
31. S. Fleischer, I. Sh. Averbukh and Y. Prior, Selective alignment of molecular spin isomers, Phys. Rev. Lett., 2007, 99, 093002.
32. M. Renard, E. Hertz, B. Lavorel and O. Faucher, Controlling ground-state rotational dynamics of molecules by shaped femtosecond laser pulses, Phys. Rev. A, 2004, 69, 043401.
33. R. Stoian, M. Wollenhaupt, T. Baumert and I. V. Hertel, Temporal pulse tailoring in laser manufacturing technologies, in Laser Precision Microfabrication, Springer, Berlin, 2010, pp. 121–144.
34. S. Zhang, C. Lu, T. Jia, Z. Wang and Z. Sun, Field-free molecular orientation enhanced by two dual-color laser subpulses, J. Chem. Phys., 2011, 135, 034301.
35. M. Leibscher, I. S. Averbukh and H. Rabitz, Molecular alignment by trains of short laser pulses, Phys. Rev. Lett., 2003, 90, 213001.
36. C. Z. Bisgaard, M. D. Poulsen, E. Peronne, S. S. Viftrup and H. Stapelfeldt, Observation of Enhanced Field-Free Molecular Alignment by Two Laser Pulses, Phys. Rev. Lett., 2004, 92, 173004.
37. Y. Li, P. Liu, S. Zhao, Z. Zeng, R. Li and Z. Xu, Active control of the molecular rotational wave packet using two laser pulses, Chem. Phys. Lett., 2009, 475, 183.

---