# 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 *E _{z}* and

*H*coexistence can they exist, and when

_{z}*E*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,

_{z}*LP*

_{mn}= EH_{m}+_{1n}HE_{m-1},

_{n}, where

*EH*and

_{m}+_{1n}*HE*are two solutions to Maxwell equations under the same boundary conditions [11].

_{m-1},_{n }In order to facilitate the discussion of mode polarization, we transform the polar coordinates into Cartesian coordinates via following equations:

We solved for HE_{31}, EH_{11} mode using:

where 6_{0} 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 LP_{21} = HE_{31} + EH_{11}, we can derive the formula of LP_{21} 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 6_{0} = 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 *в _{0}* = 0° and 90°.