I have never worked with fiber-based SPR, but I assume the concepts are the same.  Honestly, I've seen worse dips than this that people have used for measurements, myself included.  For two dips that far apart (~50nm), my first guess would be multiple refractive indexes surrounding your fiber along it's length.  I don't know your precise setup.  Is a portion of your fiber coated with or in contact with plastic?  Could part of the fiber be dirty or have oil on it?  Differences in the optical fiber or metal coating could also be the cause, but those seems less likely. 

My experience is with prism based SPR and it involved determining the thickness of cellulose deposited on the gold/glass slide while being exposed to various enzymes. Since SPR is sensitive to, among other things, thickness of materials in the vicinity of resonance SPR curves showing two resonance angles were encountered when cellulose depositions of two average thicknesses were present in the region of interest. That is to say we were observing SPR in a region where on average cellulose fibers had deposited in mounds of say either 5-10 nm or 30-40 nm (which is shown in the sketch I attached); this gave us an idea of how "rough" the region was.

I'm unable to say with certainty if this is what you are encountering but from the look of the resonance curve it seems like there may be some non uniformity in the layers adjacent to the SPR sensing region of the fiber. 

 And the above mentioned sketch ... 

i got this with polarizer but may be different in thickness of MMf that gives 2 dips....thanks