Solution to LP2i Mode
From Maxwell equations, we can obtain natural modes, TE and TH modes. However, the TE and TH modes only exist when m = 0; when m is not zero, TE mode and TH mode cannot exist. Only in the form of Ez and Hz coexistence can they exist, and when Ez accounts for a large proportion they are called EH modes, otherwise are named HE modes. Linear polarized mode is constituted by the degeneration of the EH mode and the HE mode, LPmn = EHm+1nHEm-1,n, where EHm+1n and HEm-1,n are two solutions to Maxwell equations under the same boundary conditions .
In order to facilitate the discussion of mode polarization, we transform the polar coordinates into Cartesian coordinates via following equations:
We solved for HE31, EH11 mode using:
where 60 can take either 0° or 90°, standing for even or odd mode respectively, indicating that both are mutually orthogonal. In this way, by the expression LP21 = HE31 + EH11, we can derive the formula of LP21 mode as follows:
In Eq. 3.5, it can be obtained that the transverse component along the X-direction is zero, i.e., the light field is only in the Y-direction of polarization, and contains 60 = 0° and 90° two modes. Similarly, LP21 mode can be derived from the expression LP21 = HE31 - EH11:
In this condition, the transverse component in Y direction is zero, that is to say, only in the X direction exists the polarized light field, and comprising of two modes with в0 = 0° and 90°.