The excitation vibration of the Raman active molecule was assumed as a stimulated simple harmonic oscillator when excited by the injected light field[19], which can be described as follows:
where ωn is the natural vibration frequency of the harmonic oscillator, m is the reduced nuclear mass, and γ is the damping constant, as follows γ~exp[-(tc-tw)/(ta·T2)], and in which the parameters tc, tw, ta, and T2 represent the interval of adjacent pulse within the burst, pulse duration, molecule oscillation duration, and dephasing time, respectively. q(t) represents the deviation of the internuclear distance from its equilibrium value q0, and F(t) denotes the force that acts on the molecule vibrational mode. The response of the molecule vibrational mode to a unit impulse excitation started at t=τ can be expressed as follows[20]:
where ωL and ωS are the pump field frequency and Stokes field frequency, respectively. ζ and ωd denote the damping ratio and the frequency of damped oscillator respectively, and given by the following:
Furthermore, the response of the vibrational mode to the driven force F(t) of arbitrary excitation can be represented by the following integral:
Substituting Eq.(2) into the Eq.(4), we arrive at the following equation:
Equation (5) is called the Duhamel integral, which represents the molecule oscillation driven by the applied excitation force. Assuming that the optical excitation obeys a Gaussian type, the applied optical excitation signal of a multi-pulse burst can be described as follows:
where t0 is the peak moment of a sub-pulse excitation signal, tp represents the temporal duration of a sub-pulse, n is the number of sub-pulses contained in a pulse-burst group, A denotes the optical amplitude, and r is the spot radius. The driven force exerted on the molecular oscillation by the applied optical field can be given by the following[19]:
in which ε0 is the permittivity of free space, α is the optical polarizability of the molecule, E(t) is the applied optical field, and the angular brackets denote a time average over an optical period. Taking the KGW crystal as an example, the response of the molecule vibrational mode to the multi-pulse burst excitation regime can be simulated by inserting Eqs (6) and (7) into the Duhamel integral, and the normalized analysis results are shown in Fig.1, in which the temporal duration of a sub-pulse was 20 ps.
When the optical field interacts with a Raman active molecule, the molecule will initiate a stimulated vibration in response to the optical excitation signal, thereby transferring from the ground state to an excited state. As can be seen from the simulated results of Fig.1, the stimulated vibration is a kind of damped oscillator that lasts for a certain period of time until it is annihilated, and this period is named here as the oscillation duration. In particular, the simulated vibration of the single-pulse excitation signal, shown in Fig.1 (a), is the damped free oscillator, where the duration is referred to as the free oscillation duration and the frequency is referred to as the nature frequency. The impact of the multi-pulse burst excitation signal on the molecule stimulated vibration is mainly reflected in two aspects, as following: on the one hand, the multi-pulse burst excitation regime can drive the response oscillation multiple times to maintain a high amplitude state, which we consider as the oscillation becoming more active; and on the other hand, the multi-pulse burst excitation regime can effectively prolong the oscillation duration. By comparing the figures of 1 (a), 1 (b), and 1 (c), the results imply that the larger the number of sub-pulses in a pulse-burst group becomes, the more significant the effects are. The multi-pulse excitation can enhance the weakened molecular oscillation caused by the damping effect, prompting it to return to the initial oscillation at nature frequency multiple times. The positive influences can enhance the interaction between the optical field and the vibrational mode, which would improve the Raman gain. The Raman gain coefficient can be described as follows[21]:
in which gs is the Raman gain factor with the unit of cm/MW, Ng is the number density of molecules, σr is the scattering cross section, λS is the Stokes wavelength, h is the Planck constant, and $\delta {\bar \omega _{\rm{s}}}$ denotes the average spectral width of the stimulated scattering, n is the refractive index, c and ε0 are the light speed and dielectric constant in a vacuum, respectively. Taking the 768 cm–1 vibrational mode of KGW as an example, we simulated the normalized average gain within an optical excitation period of single, dual, and three sub-pulses, respectively, as shown in Fig.2.
It can be seen from Figure 2 that the three sub-pulses excitation scheme shows the largest Raman gain which is approximately three times that of the single pulse. This is the result of multiple oscillations with a high amplitude, which means that the Raman molecule would become more active under multi-pulse excitations and appear as an increase in Raman gain. But, if the temporal distance of the adjacent sub-pulses is longer than the free oscillation duration (Δt1 > Δt2) as shown in Fig.1(d), the multi-pulse burst regime does not have an enhancement effect on the stimulated vibration, which is the same as the excitation influence of the single pulse regime. Therefore, it is important to measure the free oscillation duration in order to set an appropriate temporal interval of adjacent sub-pulses for multi-pulse burst pumping scheme, which will be discussed in the next section.