x → = { x 1 , x 2 , z } {\displaystyle ~{\vec {x}}=\{x_{1},x_{2},z\}~} is the spatial coordinate vector, with z giving the direction of motion, e → {\displaystyle ~{\vec {e}}~} is the polarization vector of the pulse, k {\displaystyle ~k~} is the wave number in the z direction, a ^ {\displaystyle ~{\hat {a}}~} is the annihilation operator of the photon in a specific mode M ( x → ) {\displaystyle ~M({\vec {x}})~} [clarification needed].