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Selective control of vibrational modes with sequential femtosecond-scale laser pulses
Chemical Physics Letters 515 (2011) 137–140
Contents lists available at SciVerse ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Selective control of vibrational modes with sequential femtosecond-scale laser pulses Chen-Wei Jiang a,b, Xiang Zhou b,c, Rui-Hua Xie a,d, Fu-Li Li a, Roland E. Allen b,⇑ a
MOE Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter, and Department of Applied Physics, Xi’an Jiaotong University, Xi’an 710049, China Department of Physics, Texas A&M University, College Station, TX 77843, USA c Department of Physics, Wuhan University, Wuhan 430072, China d Department of Physics, Hubei University, Wuhan 430062, China b
a r t i c l e
i n f o
Article history: Received 1 April 2011 In final form 1 September 2011 Available online 7 September 2011
a b s t r a c t For nonresonant femtosecond-scale laser pulses, we showed earlier that the optimum FWHM pulse duration is 0.42 T if one wishes to achieve maximum relative excitation of a specific vibrational mode with period T. Here we show that much greater enhancement of a specific mode can be achieved with a sequence of laser pulses which have both this optimized duration and optimized delays between pulses. One can thus enhance a selected set of modes that are particularly useful in identifying or characterizing a chemical system, to lift them out of the background of less informative modes. Ó 2011 Elsevier B.V. All rights reserved.
There is widespread interest in vibrational control using ultrafast laser pulses with a variety of techniques [1–8]. Here, as in an earlier paper [9], we consider the specific case of nonresonant femtosecond-scale laser pulses, with fluences of 0:1 kJ=m2 (or intensities of 1011 W= cm2 Þ, applied to an arbitrary chemical system. Similar pulses are used in impulsive stimulated Raman scattering and related techniques [10–24]. In Ref. [9] we found that the optimum FWHM pulse duration is 0.42 T (for an approximately Gaussian profile) if one wishes to achieve maximum relative excitation of a specific vibrational mode with period T. In the present Letter we now consider a sequence of pulses with this optimized duration. We employ both the simple model defined by Eqs. (3) and (11) of Ref. [9] and our density-functional-based SERID technique, which is defined in the present context by Eqs. (8)-(11) of Ref. [9]. In the SERID simulations for C60 , we again find that seven of the ten Raman-active modes are appreciably excited: modes Ag ð1Þ; Ag ð2Þ, Hg ð1Þ; Hg ð4Þ; Hg ð5Þ; Hg ð6Þ and Hg ð8Þ, with periods of 61.5, 20.6, 125, 39.6, 27.0, 23.0, and 19.0 femtoseconds, respectively. The relative magnitudes for the vibrational excitations depend on the pulse parameters, of course, and Figure 1 shows results for a simulation with a 26 fs pulse at 1500 nm. As in Ref. [9], we characterize the strength of the vibrational response of a specific mode by its maximum kinetic energy K max . In the figures k we show the numerical Fourier transform of the total kinetic energy, which has a peak at 2 xk with a strength proportional to the response of a normal mode with angular frequency xk . For this reason, the horizontal scale in Figure 1 and subsequent figures ⇑ Corresponding author. Fax: +1 979 845 2590. E-mail address: [email protected] (R.E. Allen). 0009-2614/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2011.09.008
shows 1/2 the frequency actually used in performing the Fourier transform. For the pulse of Figure 1, three modes are appreciably excited in the range of frequencies shown (up to 1000 cm1 Þ: Hg ð1Þ; Ag ð1Þ, and Hg ð4Þ. In the remainder of the Letter we will focus on these modes, and the criteria for achieving relative enhancement of either one mode or a set of modes. It will be seen below that the principal results are so simple and robust that they should be valid for other modes and more general chemical systems. We can start by again confirming the prediction of Ref. [9] that the optimum pulse duration s is 0.42 T. Figure 2 shows the results of a SERID simulation in which s is chosen to be 52.5 fs, which is predicted to yield maximum relative enhancement of the Hg ð1Þ mode. It can be seen that this mode, which is already naturally dominant for pulse durations in this general range, is now totally dominant, with almost no contribution from the other modes. Now let us consider the effect of a sequence of pulses. First, using the simple model, we performed numerical calculations for the three modes visible in Figure 1, with a sequence of two laser pulses having the same duration and wavelength as in Figure 2, and separated by a delay t1 between their onset times, as depicted in Figure 3. Then we performed the much more complex SERID simulations, whose results are displayed in Figures 5–7 and discussed below. One can characterize the energy deposited in a mode by the ratio of the maximum kinetic energy after the pulse sequence is completed to the value it has after a single pulse. When each set of points is plotted as a function of the variable delay t 1 , called Dt in Figure 4, one finds that it is perfectly fitted by a single cosine function:
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C.-W. Jiang et al. / Chemical Physics Letters 515 (2011) 137–140
Figure 1. Fourier transform of total kinetic energy of C60 after a 26 fs (FWHM) 1500 nm laser pulse with a fluence of 0:055 kJ=m2 . The 26 fs pulses represented in the following figures have this same wavelength and fluence. Here, and in the following figures, the frequency given on the horizontal scale is 1/2 the frequency actually used in performing the numerical Fourier transform, for the reason given in the text. The results in this and the following figures, except for Figure 4, were obtained in density-functional-based SERID simulations. Three vibrational modes are appreciably excited in the range of frequencies shown, below 1000 cm1 .
Figure 4. Ratios of maximum kinetic energy after two laser pulses to that after one pulse, in numerical simulation with the simple model defined by Eqs. (3) and (11) of Ref. [9]. Each set of points can be fitted with a single cosine function of the time delay Dt, which is given in femtoseconds.
max 2 K max two =K one ¼ 2 cosð2pt 1 =T k Þ þ 2 ¼ 4 cos ðpt 1 =T k Þ
ð1Þ
where T k is the period of the mode labeled by k. The simple model thus implies that the optimum delay time is simply equal to the period of the mode being selected. This is not an immediately obvious result, since the numerical simulations of the present Letter use the full vector potential A(t) in Eq. (11) of Ref. [9], rather than the approximate form for the electric field in Eq. (2) of that paper, and a simple analytical solution does not
Figure 2. Fourier transform of total kinetic energy of C60 after a 52.5 fs (FWHM) 1500 nm laser pulse with a fluence of 0:113 kJ=m2 . The 52.5 fs pulses represented in the following figures have this same wavelength and fluence. The Hg ð1Þ mode is overwhelmingly dominant. As the inset shows, the Ag ð1Þ and Hg ð4Þ modes are observed, but only very weakly.
Figure 3. Definition of the pulse delay times t1 and t2 . Here s represents the fullwidth-at-half-maximum (FWHM) pulse duration.
Figure 5. Fourier transform of the total kinetic energy of C60 after two 52.5 fs (FWHM) pulses with different time delays. (a) A delay of 125 fs enhances the Hg ð1Þ vibrational response by a factor of 4. (b) A delay of 187.5 fs almost totally suppresses this mode.
C.-W. Jiang et al. / Chemical Physics Letters 515 (2011) 137–140
139
Figure 6. Fourier transform of total kinetic energy of C60 after two 26 fs (FWHM) pulses separated by various time delays. In (a), the originally small response of the Ag ð1Þ mode is now totally dominant. In (b), the Hg ð1Þ mode is again dominant. In (c), the Hg ð1Þ mode is almost completely suppressed, with the combined Ag ð1Þ and Hg ð4Þ modes enhanced.
Figure 7. Fourier transform of total kinetic energy of C60 after four 26 fs (FWHM) pulses separated by various time delays (solid curves). The dashed curves are for one laser pulse, as also shown in Figure 6. In (a), with t 1 ¼ 187:5 fs and t2 ¼ 277:0 fs, the Hg ð4Þ mode is totally dominant. In (b), with t 1 ¼ 62:5 fs and t2 ¼ 186:0 fs , the Ag ð1Þ mode is dominant. In (c), with t1 ¼ 92:3 fs and t2 ¼ 217:8 fs, Hg ð1Þ regains dominance.
appear to exist. In addition, the simple model predicts that a given mode can be completely quenched if the delay time is chosen to be e.g. 3/2 the vibrational period. In other words, this model implies that one can use the simplest imaginable picture for constructive or destructive interference of successive laser pulses. An important question, of course, is the validity of the model, which totally neglects many important effects (and simply parameterizes others). To address this issue, as mentioned above, we performed independent SERID simulations for sequences of pulses applied to C60 . As in Ref. [9], and our earlier studies of photoinduced processes in molecules [25,26] and materials [27], these simulations employ, in a nonorthogonal basis, (i) the timedependent Schrödinger equation for the electron dynamics, (ii) Ehrenfest’s theorem for the nuclear dynamics, (iii) the timedependent Peierls substitution for the coupling of the laser pulse to the electrons, and (iv) the density-functional-based approach of Frauenheim and co-workers [28,29]. A complete description of the computational details can be found in Refs. [25–27]. In order to test the validity of Eq. (1), we performed simulations with the time delay t1 chosen to be equal to first the period of the
Hg ð1Þ mode and then 3/2 its period. The results, shown in Figure 5, are in remarkably good agreement with the prediction of Eq. (1) for these two limiting cases, with a fourfold increase in the energy deposited in the Hg ð1Þ mode for the first case and nearly complete suppression in the second case. Recall again that these results are obtained in detailed simulations with many interacting electronic and nuclear degrees of freedom, and that they include anharmonic effects in the vibrations, nonlinear effects in the response to the applied field, and no simplifying assumptions about the electronic polarizability tensor. The most interesting issue, of course, is whether one can achieve a relative enhancement of the weaker modes by optimizing both the pulse duration and the delay between pulses, and this was explored in calculations like those represented in Figure 6, with a sequence of 2 pulses, and Figure 7, where 4 pulses were used. In Figure 6a, both the pulse duration s and the pulse delay t1 were chosen to optimize the response of the Ag ð1Þ mode, which as a result becomes totally dominant, in contrast to the results of Figure 2.
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C.-W. Jiang et al. / Chemical Physics Letters 515 (2011) 137–140
What might appear to be a relatively modest change in the delay t 1 , with exactly the same pulse duration s, produces the dramatic change in response that can be seen in Figure 6b, where the Hg ð1Þ mode reassumes dominance. But with an even longer delay of 188 fs, the Hg ð1Þ mode is essentially totally suppressed, and both of the other modes make their appearance. One expects that a longer train of pulses can yield even greater selectivity, and this is verified by the results of Figure 7: With 4 pulses, one can select any of the three modes under consideration to be almost totally dominant with a proper choice of the pulse delays t 1 and t2 (as defined in Figure 3) as well as the duration. The very strong dependence of the vibrational excitations on the pulse duration and delay times is dramatically apparent when one compares Figure 7 with Figure 2. In summary, we have demonstrated that the relative response of a specific vibrational mode can be enormously enhanced by optimizing the delay times, as well as the individual pulse duration, in a sequence of nonresonant femtosecond-scale laser pulses applied to a general chemical system. Our conclusions are based on numerical calculations with a simple and general model and, more convincingly, on density-functional-based simulations for the specific example of C60 . The response in the present simulations is not as strong as it would be with resonant pulses and higher fluences, but the qualitative predictions should nevertheless be experimentally observable and potentially useful.
References
Acknowledgements
[23]
This work was supported by the Robert A. Welch Foundation (Grant A-0929), China Scholarship Council, Fundamental Research Funds for the Central Universities, National Natural Science Foundation of China (Grant 11074199), and Special Prophase Project of the National Basic Research Program of China (Grant 2011CB311807). C J. would like to thank Zhibin Lin and Meng Gao for helpful discussions, and the Texas A&M University Supercomputing Facility for the use of its parallel supercomputing resources.
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Contents lists available at SciVerse ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Selective control of vibrational modes with sequential femtosecond-scale laser pulses Chen-Wei Jiang a,b, Xiang Zhou b,c, Rui-Hua Xie a,d, Fu-Li Li a, Roland E. Allen b,⇑ a
MOE Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter, and Department of Applied Physics, Xi’an Jiaotong University, Xi’an 710049, China Department of Physics, Texas A&M University, College Station, TX 77843, USA c Department of Physics, Wuhan University, Wuhan 430072, China d Department of Physics, Hubei University, Wuhan 430062, China b
a r t i c l e
i n f o
Article history: Received 1 April 2011 In final form 1 September 2011 Available online 7 September 2011
a b s t r a c t For nonresonant femtosecond-scale laser pulses, we showed earlier that the optimum FWHM pulse duration is 0.42 T if one wishes to achieve maximum relative excitation of a specific vibrational mode with period T. Here we show that much greater enhancement of a specific mode can be achieved with a sequence of laser pulses which have both this optimized duration and optimized delays between pulses. One can thus enhance a selected set of modes that are particularly useful in identifying or characterizing a chemical system, to lift them out of the background of less informative modes. Ó 2011 Elsevier B.V. All rights reserved.
There is widespread interest in vibrational control using ultrafast laser pulses with a variety of techniques [1–8]. Here, as in an earlier paper [9], we consider the specific case of nonresonant femtosecond-scale laser pulses, with fluences of 0:1 kJ=m2 (or intensities of 1011 W= cm2 Þ, applied to an arbitrary chemical system. Similar pulses are used in impulsive stimulated Raman scattering and related techniques [10–24]. In Ref. [9] we found that the optimum FWHM pulse duration is 0.42 T (for an approximately Gaussian profile) if one wishes to achieve maximum relative excitation of a specific vibrational mode with period T. In the present Letter we now consider a sequence of pulses with this optimized duration. We employ both the simple model defined by Eqs. (3) and (11) of Ref. [9] and our density-functional-based SERID technique, which is defined in the present context by Eqs. (8)-(11) of Ref. [9]. In the SERID simulations for C60 , we again find that seven of the ten Raman-active modes are appreciably excited: modes Ag ð1Þ; Ag ð2Þ, Hg ð1Þ; Hg ð4Þ; Hg ð5Þ; Hg ð6Þ and Hg ð8Þ, with periods of 61.5, 20.6, 125, 39.6, 27.0, 23.0, and 19.0 femtoseconds, respectively. The relative magnitudes for the vibrational excitations depend on the pulse parameters, of course, and Figure 1 shows results for a simulation with a 26 fs pulse at 1500 nm. As in Ref. [9], we characterize the strength of the vibrational response of a specific mode by its maximum kinetic energy K max . In the figures k we show the numerical Fourier transform of the total kinetic energy, which has a peak at 2 xk with a strength proportional to the response of a normal mode with angular frequency xk . For this reason, the horizontal scale in Figure 1 and subsequent figures ⇑ Corresponding author. Fax: +1 979 845 2590. E-mail address: [email protected] (R.E. Allen). 0009-2614/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2011.09.008
shows 1/2 the frequency actually used in performing the Fourier transform. For the pulse of Figure 1, three modes are appreciably excited in the range of frequencies shown (up to 1000 cm1 Þ: Hg ð1Þ; Ag ð1Þ, and Hg ð4Þ. In the remainder of the Letter we will focus on these modes, and the criteria for achieving relative enhancement of either one mode or a set of modes. It will be seen below that the principal results are so simple and robust that they should be valid for other modes and more general chemical systems. We can start by again confirming the prediction of Ref. [9] that the optimum pulse duration s is 0.42 T. Figure 2 shows the results of a SERID simulation in which s is chosen to be 52.5 fs, which is predicted to yield maximum relative enhancement of the Hg ð1Þ mode. It can be seen that this mode, which is already naturally dominant for pulse durations in this general range, is now totally dominant, with almost no contribution from the other modes. Now let us consider the effect of a sequence of pulses. First, using the simple model, we performed numerical calculations for the three modes visible in Figure 1, with a sequence of two laser pulses having the same duration and wavelength as in Figure 2, and separated by a delay t1 between their onset times, as depicted in Figure 3. Then we performed the much more complex SERID simulations, whose results are displayed in Figures 5–7 and discussed below. One can characterize the energy deposited in a mode by the ratio of the maximum kinetic energy after the pulse sequence is completed to the value it has after a single pulse. When each set of points is plotted as a function of the variable delay t 1 , called Dt in Figure 4, one finds that it is perfectly fitted by a single cosine function:
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C.-W. Jiang et al. / Chemical Physics Letters 515 (2011) 137–140
Figure 1. Fourier transform of total kinetic energy of C60 after a 26 fs (FWHM) 1500 nm laser pulse with a fluence of 0:055 kJ=m2 . The 26 fs pulses represented in the following figures have this same wavelength and fluence. Here, and in the following figures, the frequency given on the horizontal scale is 1/2 the frequency actually used in performing the numerical Fourier transform, for the reason given in the text. The results in this and the following figures, except for Figure 4, were obtained in density-functional-based SERID simulations. Three vibrational modes are appreciably excited in the range of frequencies shown, below 1000 cm1 .
Figure 4. Ratios of maximum kinetic energy after two laser pulses to that after one pulse, in numerical simulation with the simple model defined by Eqs. (3) and (11) of Ref. [9]. Each set of points can be fitted with a single cosine function of the time delay Dt, which is given in femtoseconds.
max 2 K max two =K one ¼ 2 cosð2pt 1 =T k Þ þ 2 ¼ 4 cos ðpt 1 =T k Þ
ð1Þ
where T k is the period of the mode labeled by k. The simple model thus implies that the optimum delay time is simply equal to the period of the mode being selected. This is not an immediately obvious result, since the numerical simulations of the present Letter use the full vector potential A(t) in Eq. (11) of Ref. [9], rather than the approximate form for the electric field in Eq. (2) of that paper, and a simple analytical solution does not
Figure 2. Fourier transform of total kinetic energy of C60 after a 52.5 fs (FWHM) 1500 nm laser pulse with a fluence of 0:113 kJ=m2 . The 52.5 fs pulses represented in the following figures have this same wavelength and fluence. The Hg ð1Þ mode is overwhelmingly dominant. As the inset shows, the Ag ð1Þ and Hg ð4Þ modes are observed, but only very weakly.
Figure 3. Definition of the pulse delay times t1 and t2 . Here s represents the fullwidth-at-half-maximum (FWHM) pulse duration.
Figure 5. Fourier transform of the total kinetic energy of C60 after two 52.5 fs (FWHM) pulses with different time delays. (a) A delay of 125 fs enhances the Hg ð1Þ vibrational response by a factor of 4. (b) A delay of 187.5 fs almost totally suppresses this mode.
C.-W. Jiang et al. / Chemical Physics Letters 515 (2011) 137–140
139
Figure 6. Fourier transform of total kinetic energy of C60 after two 26 fs (FWHM) pulses separated by various time delays. In (a), the originally small response of the Ag ð1Þ mode is now totally dominant. In (b), the Hg ð1Þ mode is again dominant. In (c), the Hg ð1Þ mode is almost completely suppressed, with the combined Ag ð1Þ and Hg ð4Þ modes enhanced.
Figure 7. Fourier transform of total kinetic energy of C60 after four 26 fs (FWHM) pulses separated by various time delays (solid curves). The dashed curves are for one laser pulse, as also shown in Figure 6. In (a), with t 1 ¼ 187:5 fs and t2 ¼ 277:0 fs, the Hg ð4Þ mode is totally dominant. In (b), with t 1 ¼ 62:5 fs and t2 ¼ 186:0 fs , the Ag ð1Þ mode is dominant. In (c), with t1 ¼ 92:3 fs and t2 ¼ 217:8 fs, Hg ð1Þ regains dominance.
appear to exist. In addition, the simple model predicts that a given mode can be completely quenched if the delay time is chosen to be e.g. 3/2 the vibrational period. In other words, this model implies that one can use the simplest imaginable picture for constructive or destructive interference of successive laser pulses. An important question, of course, is the validity of the model, which totally neglects many important effects (and simply parameterizes others). To address this issue, as mentioned above, we performed independent SERID simulations for sequences of pulses applied to C60 . As in Ref. [9], and our earlier studies of photoinduced processes in molecules [25,26] and materials [27], these simulations employ, in a nonorthogonal basis, (i) the timedependent Schrödinger equation for the electron dynamics, (ii) Ehrenfest’s theorem for the nuclear dynamics, (iii) the timedependent Peierls substitution for the coupling of the laser pulse to the electrons, and (iv) the density-functional-based approach of Frauenheim and co-workers [28,29]. A complete description of the computational details can be found in Refs. [25–27]. In order to test the validity of Eq. (1), we performed simulations with the time delay t1 chosen to be equal to first the period of the
Hg ð1Þ mode and then 3/2 its period. The results, shown in Figure 5, are in remarkably good agreement with the prediction of Eq. (1) for these two limiting cases, with a fourfold increase in the energy deposited in the Hg ð1Þ mode for the first case and nearly complete suppression in the second case. Recall again that these results are obtained in detailed simulations with many interacting electronic and nuclear degrees of freedom, and that they include anharmonic effects in the vibrations, nonlinear effects in the response to the applied field, and no simplifying assumptions about the electronic polarizability tensor. The most interesting issue, of course, is whether one can achieve a relative enhancement of the weaker modes by optimizing both the pulse duration and the delay between pulses, and this was explored in calculations like those represented in Figure 6, with a sequence of 2 pulses, and Figure 7, where 4 pulses were used. In Figure 6a, both the pulse duration s and the pulse delay t1 were chosen to optimize the response of the Ag ð1Þ mode, which as a result becomes totally dominant, in contrast to the results of Figure 2.
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C.-W. Jiang et al. / Chemical Physics Letters 515 (2011) 137–140
What might appear to be a relatively modest change in the delay t 1 , with exactly the same pulse duration s, produces the dramatic change in response that can be seen in Figure 6b, where the Hg ð1Þ mode reassumes dominance. But with an even longer delay of 188 fs, the Hg ð1Þ mode is essentially totally suppressed, and both of the other modes make their appearance. One expects that a longer train of pulses can yield even greater selectivity, and this is verified by the results of Figure 7: With 4 pulses, one can select any of the three modes under consideration to be almost totally dominant with a proper choice of the pulse delays t 1 and t2 (as defined in Figure 3) as well as the duration. The very strong dependence of the vibrational excitations on the pulse duration and delay times is dramatically apparent when one compares Figure 7 with Figure 2. In summary, we have demonstrated that the relative response of a specific vibrational mode can be enormously enhanced by optimizing the delay times, as well as the individual pulse duration, in a sequence of nonresonant femtosecond-scale laser pulses applied to a general chemical system. Our conclusions are based on numerical calculations with a simple and general model and, more convincingly, on density-functional-based simulations for the specific example of C60 . The response in the present simulations is not as strong as it would be with resonant pulses and higher fluences, but the qualitative predictions should nevertheless be experimentally observable and potentially useful.
References
Acknowledgements
[23]
This work was supported by the Robert A. Welch Foundation (Grant A-0929), China Scholarship Council, Fundamental Research Funds for the Central Universities, National Natural Science Foundation of China (Grant 11074199), and Special Prophase Project of the National Basic Research Program of China (Grant 2011CB311807). C J. would like to thank Zhibin Lin and Meng Gao for helpful discussions, and the Texas A&M University Supercomputing Facility for the use of its parallel supercomputing resources.
[24] [25] [26] [27]
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