Poarization in optics has two meaning. One is the reaction of the medium to the electric field. The other is the polarization state of the optical field. Which one you mean?

Hello Ruma -- If the laser propagates along the y axis, then the electric field of the incident beam may have components in the x direction and z direction. If the initial polarization is linear with the field exactly in the x or z direction, then the polarization will remain the same regardless of whether or not you apply a voltage. If the initial polarization features at least some component in both the x and z directions, the polarization will evolve even if you don't apply a voltage in the z direction, but the way the polarization evolves can be changed by applying a voltage. This is because the electro-optic effect allows you to change the index of refraction experienced by one of the polarization components independent from the other. So to answer your question exactly: the polarization of the incident beam will change depending on the initial polarization of the laser and on the difference between the indices of refraction along the x (nx) and z (nz) directions, where nz is subject to the electro-optic effect. A brief internet search for KTP yielded the attached link, though there are many free resources online for learning more about the basics on polarization and electro-optics. Hope this helps, ~Eric

Dear Xiaoguang Zhang,

Thank you for your valuable reply. In my case I have to consider both, so kindly explain me both the cases.

Dear Eric Cunningham,

Thanks for your kind reply. But if we propagate light angularly (θ = 90° and ϕ = 21.3° ) through the media then after applying electric field (along z axis) what will be the polarization rotation/change of the incident beam within the media. Should we consider that light-amplitude for a beam propagating along the y axis requires the beam to be polarized 45 degree relative to the x axis with the electric field along the z (polar axis) direction?

KTP is the bire-axial crystal. I do not know exactly what do its electro-optic coefficients look like. I have to check them, and answer you afterwards.

Hi Ruma --

Generally speaking, it is not useful to have the laser propagating at some angle with respect to the principal axes of the crystal. The reason for this is as follows: if the beam propagates directly along the y axis, the difference between the two different indices of refraction, nx and nz, experienced by the x-polarized and z-polarized components is as large as possible; if instead, for example, the beam propagates at some angle relative to the y-axis within the x-y plane, then one component of the wave is still polarized in the z direction and experiences the index of refraction nz, but the other component of the wave is polarized in some mixture of the x- and y-direction, meaning that the effective refractive index for this component of the wave is some value between nx and ny.

Mathematically-speaking: if propagating along the y-axis, the difference in the refractive indices experienced by the two polarization components of the laser is nz-nx, which is greater than the difference in the refractive indices experienced by the two polarization components when the laser does not propagate along the y-axis (i.e. nz - nx > nz - nx,y,angle).

All of this is important because polarization modulation depends completely on the difference between these refractive indices, and it is advantageous when the difference is large because the modulator does not require as long of a propagation distance/as high of a voltage/etc. in order to function.

To answer your second question: in general, it can indeed be useful to launch the beam in the y-direction with a polarization 45 degrees between the x- and z-axes, but this will depend on what you're hoping to accomplish using the electro-optic modulator (i.e. amplitude, phase, or polarization modulation; see link for general information).

Good luck, ~Eric 

https://www.rp-photonics.com/electro_optic_modulators.html

Dear Xiaoguang Zhang

Please take your time to help me out. Thank you.

Dear Eric

I am going through your answer. I hope that will help me to understand the theory behind the polarization. But I need the value of phase retardation between the slow and fast axis of the crystal. I am also searching and studying to solve the issue. Thank you.

Is there any analytical solution of the polarization retardation, kindly suggest.