(3) Here, the factor of -i comes from (-i) in the third order of perturbative expansion. Here, t 1 and t 2 are the time of interaction between the system and the second and third laser pulses, respectively, whereas t 4 is the time at which the system produces the non-linear response signal. Further, R can be decomposed into 8 different Liouville pathways, each representing a separate Feynman diagram and giving rise to a non-linear response signal. Each of the response signals is also accompanied by a phase factor that makes it possible to have an independent spatial direction of detection for each diagram.