# Gradient Force of optical near field

Source: Principles of Nano-Optics, for Lukas Novotny and Bert Hecht.

The equations above represent the electric field in the second medium when a light hit a surface and the condition of TIR (total internal reflection) is satisfied. Actually this is what called Evanescent field. The point is if I want to calculate the gradient force on a particle in this field, F(gradient)=0.5a*gradient|E|^2, where "a" is the Polarizability and "E" is the electric field. How does the gradient force hold in y- direction? If I make the derivative of the field in y direction it will vanish according to the equation above because there is no variable y that exist in the equation in y axis, on another hand how I read in some articles that they say the gradient force is exist in y direction but with no explanation or derivation!

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. This is kind of an ideal case because any beam of electric field has finite span in space, otherwise it will have infinite energy! If your electric field has finite span, for example a cylindrical beam with a Gaussian envelope, then your electric field has $$y$$ dependence and you get a gradient force along $$y$$ direction. I did not find the article you read but I guess they assumed finite span in space.