4e726f20   
   From: siegman@stanford.edu   
      
   Let's recall that the basic reason for using a _confocal_ resonator is   
   that in a confocal resonator the resonance frequencies of all the   
   even-numbered higher-order transverse modes of the resonator line up and   
   overlap exactly on top of the axial-mode resonance frequencies for the   
   lowest-order (TEM00) mode.   
      
   At the same time, the resonance frequencies of all the odd-numbered   
   transverse modes all line up at exactly halfway between these axial   
   modes.   
      
   If the input beam is not perfectly aligned and/or focused and/or   
   mode-matched into the interferometer, all that happens is that you   
   excite a coherent mixture of even (or odd) modes inside the   
   interferometer -- but since these have individual resonances all lined   
   up right on top of each other in frequency, you get a single resonance   
   transmission line.   
      
   The net result is that you do not need to perfectly align  and focus and   
   carefully "transverse mode match" the input beam into the lowest-order   
   TEM00 mode of the resonator (at least for higher-order transverse mode   
   orders up to some point where the paraxial mode approximation begins to   
   fail).   
      
   In brief: transverse mode matching requirements are greatly relaxed.   
      
   There are, however, three problems you have to be concerned about in   
   this:   
      
   1)  Since this overlapping occurs right at the TEM00 axial modes for   
   even-symmetry modes but halfway in between (at the "half-axial modes")   
   for odd-symmetry modes, the effective free spectral range for the   
   interferometer is halved.   
      
   2)  And, you will find out that if you have a reasonably high finesse in   
   your interferometer, and you want to have a substantial relaxation in   
   mode matching requirement,  the confocal condition in the interferometer   
   must be very accurately met.  In practical terms, the effective finesse   
   will deteriorate very rapidly with *very* small changes in length of the   
   interferometer.  (Been there, done that!)   
      
   3)  And, something I'd never really pondered before:   If your input   
   beam profile and alignment is such that it excites a _mixture_ of even   
   and odd modes (pEven power in even modes, pOdd power in odd modes) in   
   the interferometer, then you will get alternate resonance responses,   
   spaced by half the FSR of a planar FP of the same length, with relative   
   amplitudes of pEven and pOdd.   
      
   [LASERS, p. 763, and others.  I need to look back and dig out:  Who was   
   it who first recognized this?]   
      
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