Multiphoton Processes

processes of interaction of electromagnetic radiation with matter that are accompanied by the absorption or emission (or both) of several electromagnetic quanta (photons) in an elementary event.

The main difficulty in observing multiphoton processes is their extraordinarily low probability as compared to single-photon processes. Before the appearance of lasers, only two-photon processes, such as resonance fluorescence, Rayleigh scattering, Brillouin scattering, and Roman scattering, were observed in the optical range during scattering of light. During resonance fluorescence (Figure 1,a) an atom or molecule simultaneously absorbs one photon of exciting radiation ђω₁ and emits one photon ђω₂ of the same energy in an elementary event. In the process the scattering ion returns to the same energy level S₁. In an elementary event of Brillouin and Raman scattering, as a result of the absorption and emission of photons the scattering particle is at an energy level that satisfies the law of conservation of energy for the two-photon process as a whole: the increase in the particle’s energy £₂ – &₁ is equal to the difference between the energies of the absorbed and the emitted photons ђω₁ – ђω₂ (Figure l,b). After the appearance of lasers, it became possible to observe processes of multiphoton excitation, in which several photons of exciting radiation are absorbed simultaneously in an elementary event (Figure 1, c). Thus, during two-photon excitation an atom or molecule simultaneously absorbs two photons ђω₁ and ђω₂ and is in an excited state with energy ε₂ = ε₁ + (ђω₁ + ђω₂)

Figure 1. Quantum transitions for two-photon processes: (a) resonance fluorescence, (b) Raman scattering and Brillouin scattering, (c) two-photon excitation

The concept of multiphoton processes was developed in quantum field theory to describe the interaction of radiation with matter. The interaction is described in terms of elementary single-photon absorption and emission events. An elementary event involving simultaneous participation of p photons corresponds to the p-approximation of perturbation theory; the p-photon transition may be considered as a transition that passes in p stages through p – 1 intermediate states of the system: first one photon is absorbed (or emitted) and the system passes from state ε0 to state ε1, then a second photon is absorbed (or emitted) and the system passes into state ε2, and so on. Finally, as a result of p elementary single-photon events, the system passes into the final state, εp.

In the case of multiphoton processes involving the absorption or forced emission of p photons of identical frequency ω the probability of the transition is proportional to the number of photons of the frequency to the p-th power—that is, to the radiation intensity to the p-th power.

The probability of multiphoton processes in which p photons participate differs from the probability of multiphoton processes involving (p – 1) photons by a factor of the order of (E_el / /E_at)², where E_el is the intensity of the electric field of the radiation and E_at is the average intensity of the intra-atomic electric field (∼ 10⁹ volts per cm) in the optical band for non-resonant permitted dipole electric transitions. For all nonlaser sources of radiation, E_el ≪ E_at, and the probability of a transition decreases sharply with an increase in the number of photons. In the case of laser sources such high densities of radiation power (10¹⁵ W/cm²) have been reached that E_el/E_at ∼ 1, and the probability of multiphoton processes involving a large number of photons becomes comparable to probabilities of single-photon transitions.

The selection rules for multiphoton processes differ from the selection rules for single-photon processes. In systems with a center of symmetry, dipole electric transitions involving an even number of photons are permitted only between states of identical parity, and transitions involving an odd number of photons are permitted between states with different parity. One of the most fundamental applications of multiphoton processes—multiphoton spectroscopy—is based on the selection rules for multiphoton processes. Measurement of multiphoton absorption spectra makes it possible to study by optical methods the energy states whose excitation from the ground state is prohibited in single-photon processes.

In contrast to single-photon processes, the law of conservation of energy can be satisfied in multiphoton processes when the resultant transition of an atom from a lower to a higher energy state occurs not only with absorption but also with emission of individual photons. Therefore, multiphoton processes underlie the methods used to convert the radiation frequency of lasers and to develop frequency-tunable laser radiation sources (such as harmonic generators, combination-frequency generators, and parametric light generators). The development of frequency-tunable high-power optical radiation sources also is possible on the basis of multiphoton processes.