Firstly, in the case of the longitudinal MOKE, the reading of the x components of the magnetization vector is first done. The x component as applied in the first order matrix understanding of the longitudinal MOKE is that of the applied magnetic field directionality. The applied magnetic field will be in the plane of incidence, and will be parallel to the surface of the sample. Now once again in this setup, the Kerr angle is measured. It can be identified that the Kerr angle is proportional to the magnetization vector M. The magnetization vector component is actually projected into field direction. For obtaining the complete magnetization vector at this point, the MOKE measurement is performed with the samples and with the external magnetic field, at an observed rotation of around 90 degree. Now an interesting aspect here is that of the orientation of sample and the field constant nature in plane incidence.
Distinction between the angles cos and sin are seen to exist because of the orientation sample alignment with respect to the applied field alignment. The below diagram shows how the sample rotation and the angle for M vector that would appear in the case of the longitudinal MOKE. For measuring for vector MOKE it will be necessary to rotate both the applied magnetic field and the sample by at least a degree of 90 degrees or so. This will ensure that magnetization component scattering is controlled on the bottom place. my component is in the scattering plane or the bottom panel.
A distinction is observed here. The applied field orientation is necessary for the vector MOKE technique. Here the in-plane components of magnetization vector are both calculated. It is identified that the normal MOKE calculations as understood in Kerr rotation would not be applicable here. Both components combined cannot however be applied to the entire magnetization vector because of how there is an unknown proportionality. This unknown proportionality exists between magnetization and Kerr rotation. It can also be an intensity change as well.