If we have C major, the diatonic chords are:
C, Dm, Em, F, G, Am, Bdim, C
If we turn each of these chords to diatonic 13th chord versions of themselves:
Cmaj13, Dmin13, Emin13, Fmaj13, G13, Amin13, Bdim13
The notes of these chords are now equivalent to the modes of each starting root.
C Ionian, D Dorian, E Phrygian, F Lydian, G Mixolydian, A Aeolian, B Locrian.
Here's a picture of what I mean from one of my other questions. So I was wondering if this is a known relationship in harmonic theory and are people using it. For example, using extended chords to modulate to modes and vice versa.