The idea of this chapter is to find a way of finding robust and simple optimal fields. As described in the introduction to chapter 7 and Ref. [133] this can be done by using local control and changing the functional, parameterizing the fields and using a e.g. gradient scheme or genetic algorithm to do the optimization task. Here it is shown that it can be accomplished directly within the rapidly convergent algorithm. The advantages of doing so are multiple: the convergence is fast and the optimization is unconstrained and global.
The penalty factor α is introduced in the OCT functional to regulate the pulse intensity of the optimized field. It is also suited to reduce the pulse complexity as a rather abstract following discussion will explain.
The central points: dependence of the threshold value for α on the initial guess and the important correlation between high thresholds and robust optimal pulses are illustrated by performing OCT calculations on the potas-sium dimer (described in chapter 6). In all optimizations presented, the shape function was set to s(t) = sin(πt/T) with T = 1.6 ps. Figure 7.3 shows the spectra of pulses optimizing the v00=0 to v00=3 transfer via the first excited state all with a yield above 90%, for different α values. The dependence of the optimized pulse on this penalty factor will be discussed in the following.
First it is reasonable to assume, that multiple pathways exist connect-ing the initial with the target state of the system and moreover that these pathways are not all equivalent. The equivalence statement is central to this argumentation and means that each pathway has a different activa-tion threshold, i.e. there exist pathways that can be excited with rather
Figure 7.3: Optimization results with maximum penalty factor, starting from different initial guess pulses. 1.6 ps at 10 974 cm−1. (b) 1.6 ps at 11 698 cm−1(c) Broadband addition fo 3-fs pulses. (d) Second run with optimized field from (a) as initial guess.
low pulse intensities, such as resonant one-photon processes and other that deserve much higher pulse intensities due to perhaps their non-resonant or multiphoton character. Moreover the equivalence statement means that not all transitions contribute equally much to the yield, i.e. there may be some pathways that are so effective in connecting the initial with the target state that no other pathway has to be excited and perfect control is achieved. Of course each pathway consists of a number of transitions and therefore re-quires that the excitation pulse has frequencies matching these transitions.
It logically follows that a low value for the penalty factor leads to com-plex optimal pulses since the allowed intensity is sufficient to excite many pathways each differing in its transition frequencies consequently leading to broad excitation spectra. On the contrary if the penalty factor is chosen very high the field intensity can be only very modest, and the optimal pulse can only spent a limited amount of energy in building up just the frequen-cies, that excite the very few yield promoting transitions with low activation energy. As a result the optimal field will be very simple and the yield as high as for the low α case. From that one could easily conclude that it is very simple to obtain a realizable and simple pulse, the optimization only has to be performed with highα. But in order to do so the initial guess has to be nearly perfect (i.e. has to have already a high yield) as the maximal value for α depends strongly on the performance of the initial guess. The following argumentation should make this important point clear.
In order to allow for highα values the very few pathways have to be excited having a low activation threshold and high yield. This however deserves
7. Experimentally realizable laser pulses 95 already a tailored pulse with the correct time-frequency ordering, which is normally not known, since this pulse is the goal of the optimization. For this reason it is safe to assume in the following that the initial guess will excite non-optimal transitions, their number and character depending of course on the pulse intensity. In this set a transition with minimum (µ−) and maxi-mum (µ+) dipole moment will exist. Their respective excitation thresholds will be called ˆ²−and ˆ²+, where ˆ²−>ˆ²+. The pulse in the following iteration is given by Eq. (5.10) and therefore its maximum amplitude is∝µ+/α.
Now, if α is substantially larger than one, such that
µ+/α <ˆ²− (7.7)
the new field will not be capable to excite over again the µ− transitions in all the forthcoming iterations. That is in the following iterations the µ−
transition will be eliminated from the pulse’s excitation capabilities until finally no transition is left over. Eq. (5.10) amounts to zero and the zero field results.
On the contrary if α is small enough, such that
µ+/α >ˆ²− (7.8)
no frequencies are eliminated. The algorithm can improve the laser field in the next iterations and Eq. (7.8) will be always fulfilled since improvement of the field means excitation of regions with stronger and stronger transition dipole moments. Of course the transition frequencies of the region excited by the initial guess pulse will remain also in the optimal field ending up in a complex pulse [see Fig. 7.3(a)] of unnecessarily high intensity (1012W/cm2).
The maximum choice of this value therefore depends on the initial guess optimality. In the case of imperfect initial guess the α threshold can be much too low to obtain a simple pulse [see Fig.7.3(a)]. In Fig. 7.3(a) and 7.3(b) the initial guess
ε0(t) = (0.001 a.u.)s(t) cos(ωt) (7.9)
was tried with the center frequency atω= 10 974 cm−1(a) [ω= 11 698 cm−1 in Fig.7.3(b)] which allowed for α = 400 (α = 1100). A comparison of the complex broadband spectrum Fig. 7.3(a) with the two separated frequency bands in Fig. 7.3(b) impressively demonstrates the impact of α. Enormous simplification was attained with a negligible loss of yield. As already pointed out such a good initial is normally unknown. One can accommodate for the lack of knowledge of the optimal frequencies by taking a broadband initial guess, e.g., a few femtosecond cycle pulse. However this provision does not account for the perhaps necessary time-ordering of these optimal frequen-cies. In the considered transfer here, however it already permits the setting
ofα = 1000 [see Fig. 7.3(c)].
A more generally applicable procedure is to perform a first OCT optimiza-tion with a smallαvalue and the use this optimized field, however complex, as initial guess for a second OCT run. In the second run α can be set to unprecedented high values and consequently after a few iteration cycles the intensity of the pulse is so much reduced (109W/cm2) that it can only excite the most robust and strongest transitions. As a result the pulse is very sim-ple and experimentally realizable [see Fig. 7.3(d)]. By this meansα= 2000 could be chosen and the yield was still 94%. When an even higher value for α was chosen, no remarkable further simplification could be obtained (i.e.
α = 3000 gives 90%). The main advantage of this second filtering OCT run is that the dependence of the OCT performance in retrieving robust pulses on the initial guess pulse is completely eliminated. Figures 7.3(c) and 7.3(d) again demonstrate, that simple spectral structure is unequivocally correlated with high α values.
In summary I have shown, that whenever a pulse with a complex time-frequency behavior is optimal it might be necessary to rerun the optimization with this complex pulse as initial guess, allowing for a high penalty factor.
This proposition is based on the fact that most of the time-frequency be-havior only leads to an increase of pulse energy with the consequence of exciting secondary multiphoton or off-resonant pathways. There contribu-tion to the yield being of minor importance. The second highαoptimization extracts from the complex pulse, the time-frequency behavior necessary to excite the most fundamental and important pathway. Looking back it is now clear, that only if the initial guess has the time-frequency distribution necessary to excite the optimal pathway, i.e. the important frequencies must be ordered correctly in time, a highα optimization is possible. Comparing again Figs. 7.3(a),7.3(b),7.3(c) and 7.3(d) the following interpretation can be made: the pulses in Figs. 7.3(b) and 7.3(d) have similar spectra and therefore seem to excite the same pathway, meanwhile a second only slightly different pathway could be isolated Fig. 7.3(c) with merely a different choice of initial guess. Since the frequencies of the robust pathways in Figs. 7.3(b) and 7.3(c) are also inherent in the spectrum Fig. 7.3(a), its complexity can therefore be attributed to the simultaneous excitation of many different pathways.
It would be very valuable if all these possible control mechanisms could be distilled in an isolated fashion, however Figs. 7.3(b) and 7.3(c) show that this is only limitedly possible by choosing different initial guesses. This will be the central topic of the next section.
7. Experimentally realizable laser pulses 97