How do liquid crystals rotate the plane of polarized light (electric field of light) which is used in LCD displays? Currently, I am studying liquid crystal displays. I have studied that liquid crystals are used in LCD screens for controlling and rotating the plane of vibration of the incoming light. I didn't get the exact physics phenomenon behind it. Please elaborate more on it in a scientific way on the physics behind the behaviour of liquid crystals.

 A: The key physics is that liquid crystals shown in your diagram are birefringent material. This means that the polarization (orientation) of the light changes the effective index it sees.
In the case you are looking at OFF (a) the light is essentially guided by the helical/twisting nematic liquid crystals. The polarization follows the orientation of the liquid crystal if the helical pitch is much larger than the wavelength. The construction is usually done by applying rubbing on two substrates and then having them twisted at 90 degrees. The boundary conditions require that the liquid crystals line up with rubbed "grooves" along the plate so that when no electric field is applied the liquid crystals twist when going from one plate to another. For a mathematical treatment of the polarization "guiding" you can read Section 2.4.1.1 of the textbook Fundamentals of Liquid Crystals, 2nd Ed by Yang and Wu. I believe the first edition also has it at the same section. The Wikipedia article on it doesn't really describe the physics, but it has a nicer picture.
In the ON (b) state, a large voltage is applied so that all the liquid crystal molecules/directors line up. In this case, no polarization "guiding" is occurring so that the linear polarized light hits a perpendicular polarizer and is blocked.
If you are interested in the physics of how the liquid crystals are oriented and interplay of the elastic energies and electric fields, you can look up the Frank-Oseen Energy. For even more detail, you can look up the work by Pierre-Gilles de Gennes whose work in liquid crystals go him a Nobel Prize. To be honest, liquid crystals can be cumbersome and hard, so how much you want to dig will depend on what you need to do with it.
A: The liquid crystal “state” can be thought of as being between the solid (molecules fixed in both position and orientation) and liquid (molecules with both random position and orientation) state.
In the liquid crystal “state” molecules can have random position but there is some degree of order as to their orientation.  Not all liquid crystal molecules point in the same direction but over time there is an average non-zero (zero in the liquid state) direction in which the molecules point, and this direction is called the director.
As might be expected from what has been written the molecules tend to be long tread-like (nematic), helical (chiral nematic) or arranged in planes and the direction order breaks down above a certain temperature above which the material exhibits a liquid phase.
A liquid crystal is an anisotropic material and so what happens to light as it passes through depends on the direction of travel and polarisation of the light relative to the director.
If the molecules which make up a liquid crystal have a permanent or induced dipole moment, then applying an external electric field will change the alignment of the director.
The speed of propagation of light through a liquid crystal depends on the orientation of the plane of polarisation of the light relative to the director, one when the plane of polarisation is parallel to the director and one when it is perpendicular to the director, thus a liquid crystal is birefringent and possesses two refractive indices.
This means that linearly polarised light entering a nematic liquid crystal will emerge elliptically polarised or possibly linearly polarised because the differential speed results in a change of phase between light with a component plane polarised along the director and light with a component plane polarised perpendicular to the director.
With helical molecules (chiral nematic) and the direction of the light along the helical axis, right and left polarised light will travel at different speeds. Thus, if linearly polarised light, which can be thought of as the sum of left and right circularly polarised light, enters the right and left components travel at different speeds and when they emerge from the crystal their sum results in a plane polarised wave which has been rotated relative to the incident plane polarised wave.
