The dependence of the population of the target state, and the excitation cross section and rate, on pulse duration Q, laser intensity I, and a molecular matrix element A is discussed for two-photon molecular excitation. Perturbative and rotating wave approximation (RWA) expressions for the observables are obtained; the latter are used to discuss the validity of the former. For example, the perturbative cross section and rate increasingly underestimate the RWA results as I increases for given A and Q. Two- and ten-level model dipolar molecules are employed for illustrative purposes. The results are relevant for understanding two-photon excitation processes and their enhancement and include discussions of the roles the permanent dipole- and virtual-state mechanisms in such processes and of the validity of using intensity-independent cross sections to gauge the strength of such excitations.
The dependence of the population of the target state, and the excitation cross section and rate, on pulse duration Q, laser intensity I, and a molecular matrix element A is discussed for two-photon molecular excitation. Perturbative and rotating wave approximation (RWA) expressions for the observables are obtained; the latter are used to discuss the validity of the former. For example, the perturbative cross section and rate increasingly underestimate the RWA results as I increases for given A and Q. Two- and ten-level model dipolar molecules are employed for illustrative purposes. The results are relevant for understanding two-photon excitation processes and their enhancement and include discussions of the roles the permanent dipole- and virtual-state mechanisms in such processes and of the validity of using intensity-independent cross sections to gauge the strength of such excitations.
The dynamics and the cross sections associated with the excitation of a ground-state dipolar molecule to an excited state, by the simultaneous absorption of two photons, are discussed using the rotating wave approximation and time-dependent perturbation theory. The excitations of ten- and two-energy level representations of model dipolar molecules are used for illustrative purposes. The temporal development of the final state of the excitation process and the two-photon excitation cross sections are discussed as functions of laser intensity, laser–molecule interaction time and the two types of excitation mechanism (which are available for dipolar molecules, relative to the one mechanism for a non-polar molecule). The results are used to discuss the validity of perturbation theory treatments of the problem and various strategies for the enhancement of the intrinsically weak two-photon excitation process.
One of the purposes of this paper is to develop an analytical many-level generalized rotating wave approximation (GRWA), including the effects of permanent dipoles, for the excitation of many-level molecules through the simultaneous absorption of two photons. Included are expressions for the two-photon laser–molecule coupling C, and its two components Cd and Cv corresponding to the direct permanent dipole and the virtual state excitation mechanisms, respectively, and related observables such as the time-dependent populations of the initial and final states of the excitation process and resonance profiles. This GRWA treatment also includes an energy shift parameter e, which causes shifts in the position of the resonance energy as the laser intensity increases. The effects of permanent dipoles are very different in Cv and e versus Cd. These effects have been discussed previously for Cd using analytic two-level RWA approaches. The analytical results for Cv and e obtained here are new as is their use in discussing the influence of permanent dipoles in the parts of the two-photon excitation process involving virtual states. In the absence of permanent dipoles Cd is zero whereas Cv and e are not; they equal the corresponding perturbation theory results. The GRWA and perturbative results are related by Bessel function damping functions which, for dipolar molecules, damp out the divergence of the perturbative results as the laser intensity increases. Illustrative examples are given for a two-photon excitation involving a model chromophore which has a significant virtual state contribution through both the laser–molecule coupling and the energy shift parameter.
A model two-level dipolar molecule, and the rotating wave approximation and perturbation theory, are used to investigate the optimization and the laser intensity dependence of the two-photon excitation rate via the direct permanent dipole mechanism. The rate is proportional to the square of the laser intensity I only for small intensities and times when perturbation theory is applicable. An improvement on perturbation theory is provided by a small time RWA result for the rate which is not proportional to I2; rather it is proportional to the square of an effective intensity Ieff. For each laser intensity the optimum RWA excitation rate as a function of time, for low intensities, is proportional to I, not I2, and for high intensities it is proportional to Ieff. For a given two-photon transition the laser-molecule coupling optimizes for an intensity Imax which, for example, leads to a maximum possible excitation rate as a function of time. The validity of the RWA results of this paper, and the importance of including the effects of virtual excited states, are also discussed briefly.
A rotating-wave approximation (RWA) is developed to describe the interaction of a two-level system, which has permanent dipole moments, with two pulsed lasers. The RWA expressions for the time-dependent populations of the molecular states are applied to model laser-molecule interactions and tested by comparison with exact results. The results are used to discuss the pulsed-laser phase control of molecular excitation through the interplay of competing one- and two-photon resonances involving the effects of a nonzero difference d between the permanent dipoles of the two states involved in the transition; the competition vanishes if d=0.
The absolute laser phase dependence of the time-dependent populations of the molecular states, including the steady-state (long time) populations of the states, associated with the interaction of a molecule with a pulsed laser is investigated using illustrative two-level examples. One-photon transitions, including the effects of permanent dipoles, are discussed as a function of the pulse duration, intensity, and (absolute) laser phase, for selected laser frequencies. The effects of laser phase can be large, depending on the values of the pulse duration for a given frequency and intensity. The effects of permanent dipoles, relative to no permanent dipoles, are significant for large laser field strengths ε0. When the laser-molecule coupling parameter b=μ12ε0/E21⩾0.2, where μ12 and E21 are the transition dipole and energy difference between the ground and excited states, respectively, the dynamics of the pulse-molecule interaction are (strongly) phase dependent, independent of pulse duration, whereas the corresponding steady-state populations of the molecular states may or may not be phase-dependent depending on the pulse duration. Analytical rotating wave approximations for pulsed laser-molecule interactions are useful for interpreting the dynamics and the steady-state results as a function of field strength and pulse duration, including the effects of permanent dipole moments. The results reported in this paper are based on molecular parameters associated with an S0→S1 electronic transition in a dipolar molecule. However, they are presented in reduced form and therefore can be scaled to other regions of the electromagnetic spectrum. Short, intense pulses at or beyond the limits of current laser technology will often be required for the types of absolute laser phase effects of this paper to be appreciable for electronic excitations. The discussion, in the UV-VIS, also suffers from the use of a two-level model and from the requirement of field intensities that can be beyond the Keldysh limit. For other spectral regions, these absolute laser phase effects will be much more readily applicable.
The spectral and dynamic effects associated with a non-zero difference, d, between the permanent dipole moments of two molecular states involved in a transition, are investigated for a variety of electromagnetic field-molecule interactions. Both analytical and exact numerical techniques are used to solve the time-dependent Schrodinger equation and to obtain the temporal populations of the molecular states and the associated absorption spectra, for molecular interactions with a Gaussian pulsed, or one or two continuous wave (CW), laser(s).;Analytical rotating wave approximation results are derived for a single CW laser interacting with a two-level d {dollar}\\ne{dollar} 0 molecule with an excited state lifetime, and for a two-level d {dollar}\\ne{dollar} 0 molecule interacting with two CW lasers. A symmetry-adapted Riemann product integral method for exact calculations of pulsed laser-molecule interactions is developed and applied to specific problems. The Floquet technique for the treatment of periodic Hamiltonians is extended to accommodate the interaction of a molecule with two CW lasers.;The one-field RWA results are used to predict and interpret the effects of permanent dipoles on the temporal behaviour of the molecular states for a decaying two-level system interacting with a CW laser, and for a non-decaying two-level system interacting with a pulsed laser. It is shown that the excited state lifetime and the pulse duration can act as internal and external probes, respectively, of the temporal effects of permanent dipoles, relative to the often-studied d = 0 problem. The two-field analytical RWA solutions are used to help discuss and predict the effects of permanent dipole moments in two-laser multi-photon transitions. Several examples of multi-photon, two-colour, two-level resonance profiles, calculated in the RWA and by using exact Floquet techniques, are used to demonstrate how particular multi-photon transitions can be optimized, and to help quantify the conditions for the validity of the two-colour RWA. The effects of neighbouring energy levels on the transition of interest, for both probe and pump-probe laser-molecule interactions, are investigated using a series of three- to seven-level model molecular systems.