difference between TE11 and TE10 mode and also tm01 mode

how to explain TE11, TE10, TM01 mode?

Please can anyone explain the m and n variables?

also explain the half wavelength variation of the field..

No need to re-invent the wheel - any basic electromagnetic text will have this. There's a nice free online one here -- see chapter 9.

It depends on your structure.i mean if it's a rectangular or circular waveguide. m and n are the integers that define the modal and its associate eigen value...