The exact expression is easier to derive by yourself.

See chapters 11 to 14, particularly tables 14-1 and table 14-6 of:  A. W. Snyder & J. D. Love "Optical Waveguide Theory" (1983) http://www.springer.com/gb/book/9780412099502

Also: A. W. Snyder, "Asymptotic Expressions for Eigenfunctions and Eigenvalues of a Dielectric or Optical Waveguide", IEEE Trans. Microwave Theory & Technique, vol. MIT-17, no 12, (1969), pp 1380-1388.

Background discussion and potentially useful, but less comprehensive information in responses here: https://www.researchgate.net/post/Difference_between_TE-TM_modes_and_horizontal-vertical_polarization

Are you planning to launch the TM02 mode in isolation?  The HE22 and TE02 modes have very similar propagation constants.  Expect significant mode coupling during propagation in a real world fibre.  Exciting only the TM02 mode at launch may be tricky too.

thanks