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If I undertsand your scenario correctly the two beams overlapping at the detector would be 1) a beam that has travelled through the length of the fibre, and back; and 2) a beam that hasn't travelled through the fibre at all. There's a couple of problems with this: gyroscope fibres are often very long (kilometres), but the coherence length of lasers is not ...


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There is no $y$ dependence in your expression for the electric field, and hence there is no gradient force along the $y$-direction you are asking in this case. The expression of the electric field you gave is for a plane wave whose wave vector is in the xz plane. Any plane wave dose not change along the perpendicular direction of its propagation direction. ...


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