Maximum relative excitation of a specific vibrational mode
via optimum laser pulse duration
For molecules and materials responding to femtosecond-scale optical laser pulses, we predict maximum relative excitation of a Raman-active vibrational mode with period when the pulse has an FWHM duration . This result follows from a general analytical model, and is precisely confirmed by detailed density-functional-based dynamical simulations for C and a carbon nanotube, which include anharmonicity, nonlinearity, no assumptions about the polarizability tensor, and no averaging over rapid oscillations within the pulse. The mode specificity is, of course, best at low temperature and for pulses that are electronically off-resonance, and the energy deposited in any mode is proportional to the fourth power of the electric field.
For a quarter century there has been considerable interest in optimizing the vibrational response of molecules and materials to ultrafast laser pulses Nelson-1985-CPL (); Nelson-1985-JCP (); Nelson-1990 (); Dresselhaus-1992 (); Shank-1995 (); banin-1994 (); smith-1996 (); Nazarkin-1998 (); Nazarkin-1999 (); merlin-1999 (); Kapteyn-2003 (); Mathies-2004 (); Corkum-2004 (); Torralva2001 (); Zhang (). This problem is directly relevant to the broader issue of coherent control in physical, chemical Rabitz-2000 (); Rabitz-Kapteyn-2001 (); Murnane-2002 (); Gerber-control-1998 (), and biological Tsen-2007 (); Tsen-Sankey-2007 (); Sankey-2008 (); Sankey-2009 () systems.
Here we consider excitation via impulsive stimulated Raman scattering and related techniques using femtosecond-scale optical pulses. We find that the optimum full-width-at-half-maximum (FWHM) pulse duration for exciting a specific vibrational mode with angular frequency is given by . Our prediction results from a general analytical model, and is precisely confirmed by completely independent density-functional-based simulations for C Dexheimer1993 (); Hohmann1994 (); Boyle2005 (); Larrmann2007 (); Bhardwaj2003 () and a small carbon nanotube Dresselhaus-2008 (); Liu2002 (); Hulman2004 (). Unlike the model, these simulations include anharmonic effects in the vibrations, nonlinear effects in the response to the applied field, and no simplifying assumptions about the electronic polarizability tensor.
Our general model consists of the following:
(1) The electric field has the form
with and off resonance. Since the oscillations of will average out to over a period that is short compared to the response time of the vibrating nuclei, we will actually replace the square of Eq. (1) by the envelope function
with . This form has the following nice features Graves (): (i) The duration is finite and need not be truncated. (ii) The FWHM duration is exactly half the full duration . (iii) A plot reveals that it closely resembles a Gaussian. (iv) The slope is zero at beginning and end.
(2) The initial conditions are imposed on the normal-mode coordinates . This approximation is valid for the expectation value below Eq. (4) at low temperature, or an ensemble average at higher temperature, in the linearized Eq. (3).
where the anharmonic and damping terms (in the normal-mode coordinates ), nonlinear terms (in the electric field ), and off-diagonal terms (in the polarizability tensor ) have been neglected, with the electric dipole moment given by , , , and the radiation field taken to be linearly polarized in the -direction. Equation (3) follows from e.g. the Heisenberg equations of motion for the quantized Hamiltonian
with and , when the term involving vanishes or is neglected (e.g. in taking rapid oscillations of the electric field to average out to zero, an approximation which also causes the lowest-order nonlinear term to vanish in Eq. (4)).
The total energy in vibrational mode is therefore
is, of course, equivalent to the maximum kinetic energy .
The maximum response is then given by the extremum of the function in brackets, which occurs at , or .
We have tested this prediction by performing independent supercomputer simulations for C and a small carbon nanotube, using the density-functional-based approach of Frauenheim and co-workers Porezag1995 (); Seifert1996 (), together with semiclassical electron-radiation-ion dynamics (SERID), which is defined by the following equations allen-2008 ():
(1) Time-dependent Schrödinger equation in a nonorthogonal basis:
With atoms and a minimal basis set, the matrices are . A time step of 50 attoseconds was used. The simulation time after completion of the pulse was 2000 fs for C and 1000 fs for the nanotube.
(2) Ehrenfest’s theorem (in a nonorthogonal basis):
where is any nuclear coordinate. The Hamiltonian matrix , overlap matrix , and effective ion-ion repulsion were determined by the methods and results of Refs. Porezag1995 () and Seifert1996 () and later work by this group.
(3) Coupling of electrons to the radiation field through the time-dependent Peierls substitution
where . is the vector potential, which in the present simulations was taken to have the form
with , so that , although Eq. (11) was actually used. The polarization vector was taken to lie along the -axis, with the -axis pointing down the axis of a nanotube.
Within the present DFT-based model, the energy gap for electronic excitations is eV for C and eV for a nanotube with a periodicity length of 5 unit cells. (This model nanotube, with only atoms, has a substantial gap because small wavenumbers are not allowed. An infinitely long (3,3) nanotube would be metallic Liu2002 ().) The laser pulse photon energy was chosen to be eV for C and eV for the nanotube, and is thus off-resonance in both cases.
For C, seven of the ten Raman-active modes were appreciably excited: modes , , , , , and , with periods of fs, fs, fs, fs, fs, fs, and fs respectively. For the nanotube only an mode with a period of fs was observed, and this is consistent with the experimental difficulty of observing the radial breathing mode in the Raman spectrum Hulman2004 ().
As in Ref. Zhang (), it is natural to characterize the strength of the vibrational response of a specific mode by its maximum kinetic energy . In the harmonic approximation, after completion of the laser pulse, the velocity is proportional to , so is proportional to . A numerical Fourier transform of the total kinetic energy therefore shows a peak at , with a strength proportional to the response of the normal mode with angular frequency . Figure 1 shows our results for the Fourier transform of the total kinetic energy for C following a fs, eV, V/nm pulse.
Figure 2 shows that the detailed simulations agree with Eq. (7) regarding the dependence of the maximum kinetic energy of mode on the field amplitude (for the range of amplitudes considered here) when the pulse duration is fixed: .
When is fixed, on the other hand, is predicted to be proportional to the square of the function involving in Eq. (7), and is maximal when . Figure 3 shows, for example, that the mode in C dominates for fs, and the mode for fs, in agreement with the analytical model of Eqs. (2)-(3), and in qualitative agreement with other simulations Zhang () and experiment Dexheimer1993 (); Hohmann1994 (); Bhardwaj2003 (); Boyle2005 (); Larrmann2007 ().
In Fig. 3, the agreement between the predictions of the analytical model (curves) and the completely independent DFT-based simulations (points) is truly remarkable. The only adjustable parameter for each curve is the effective polarizability parameter .
There is similarly remarkable agreement for the carbon nanotube, as can be seen in Fig. 4. In this case, two photon energies were used: eV and eV, which are respectively below and above the excitation gap of eV for this model nanotube, as discussed above. Figure 4 shows the results for the maximum kinetic energy of the one prominent mode observed in this case. (Recall that the polarization vector for the pulse was chosen to point across the nanotube, so that only radial modes should be directly excited, and this is a severe constraint for the small (3,3) nanotube, as is also observed in experiment Hulman2004 ().) The panels in Fig. 4 correspond to: (a) eV, fs; (b) eV, V/nm; (c) eV, fs; (d) eV, V/nm. It is interesting that the analytical model still works in this case even for above-the-gap excitation, but we find less satisfactory agreement for C, and certainly do not expect the model of Eq. (3) to be generally valid when there is a substantial population of electrons in excited states at the end of the pulse. In this case other effects will ordinarily be of dominant importance Dresselhaus-1992 (); Torralva2001 ().
As mentioned above, the electronic response in the DFT-based simulations can be characterized by an effective polarizability parameter if one fits Eq. (7) to the results of the simulations. If the model leading to Eq. (7) is in fact consistent with the detailed simulations, then the values obtained when (a) varying the electric field amplitude at constant (FWHM) pulse duration and (b) varying at constant should also be reasonably consistent. Table 1 shows the results for all of the prominent Raman-active modes. The two procedures do lead to generally consistent values, with the differences for the , , and modes presumably arising from their short periods, so that the approximation of averaging out the oscillations in Eq. (2) is not valid. However, the prediction regarding optimum pulse duration works even for these modes, as can be seen in Fig. 3.
|carbon nanotube||E æ|
|C æ||A æ|
|nanotube, above gap||E æ|
It has long been suspected that there might be an optimum pulse duration. For example, in Ref. Zhang () the pulse envelope was assumed to have a Gaussian form: (with a slight change in notation). Simulations were then performed with a model that essentially lies between the analytical model of Eqs. (2)-(3) and our much more extensive density-functional-based simulations, defined by Eqs. (8)-(11). It was found that the optimum value of was about . The corresponding FWHM duration then has an optimum value . In both our general model and DFT-based simulations, on the other hand, the optimal FWHM pulse duration is found to be given by . Both results seem to be consistent with the existing experiments Dexheimer1993 (); Hohmann1994 (); Bhardwaj2003 (); Boyle2005 (); Larrmann2007 (), and both are larger than the naïve prediction of . In our result, the reason appears to be the phase lag apparent in Eq. (5).
It should be emphasized that our result of is for the maximum response of this mode when the intensity of the laser pulse is kept fixed while the duration is varied. On the other hand, if the fluence is instead held fixed (while is varied), Eqs. (5) and (7) are changed to
is the energy in the pulse. If is held constant, the maximum in Eq. (13) is achieved as :
Then gives . However, Eq. (15) holds for all modes, so there is no relative enhancement of any preferred mode.
It should also be emphasized that the success of the simple analytical model of Eq. (3) results from the fact that the additional effects described below Eq. (3) are not very important for the pulse intensities, time durations, and systems considered here. The various effects omitted in the model will lead to richer (or messier) behavior in regimes where nonlinear effects, anharmonicity, damping, etc. become comparable in importance to the lowest-order effects included in the model. In future studies with density-functional-based simulations based on Eqs. (8)-(11) it may be interesting to sort out the influence of the higher-order effects.
Finally, as emphasized below Eq. (1), the present results are for off resonance, in contrast to those of e.g. Refs. Torralva2001 () and smith-1996 (). In particular, Smith and Cina considered pulses whose central frequencies are near resonance with an electronic transition, in a model with two electronic levels and one vibrational degree of freedom. They found that the momentum increment (for the nuclei) falls off less rapidly with increasing offset (from resonance) than does the population loss (from the electronic ground state), and they presented a rather complex strategy for optimizing the various parameters for these “preresonant” pulses. The work of this group and others is thus complementary to that of the present paper.
In summary, we find remarkable agreement between the general model of Eqs. (2)-(3) and the detailed DFT-based simulations based on Eqs. (8)-(11) for C and a carbon nanotube. At fixed pulse intensity, both of these approaches predict maximum excitation of a Raman-active vibrational mode with period when the pulse has an FWHM duration .
Acknowledgements.This work was supported by the Robert A. Welch Foundation (Grant A-0929) and the China Scholarship Council, and we wish to thank the Texas A&M University Supercomputing Facility for the use of its parallel computing resources.
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