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Modal approach to optical forces between waveguides as derived by transformation optics formalism.

Optical force between two lossless waveguides has been described by two approaches. One approach is the explicit description of the force by the Maxwell stress tensor. Another approach is to describe the modal force in terms of the derivative of the eigenmode frequency with respect to the distance variation. Here, we analytically prove the equivalence of these two approaches for lossless waveguides having arbitrary cross sections through the use of transformation optics formalism. Our derivation is applicable to both pressure and shear forces. We also show that these two approaches are not equivalent in the presence of loss.