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What help us determine the polarization of electromagnetic wave . Does perpendicular electric and magnetic field determine it or does the direction of propagation ?

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The polarization of an electromagnetic wave follows the direction of the electric field. For example, if the electric component is oscillating along the x-axis and the magnetic field is oscillating in the y-axis, the polarization will be along the x-axis.

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You can uniquely define the polarisation of a plane wave from any of the following:

  1. The electric field vector as a function of time $\vec{E}(t)$ and the magnetic field (or induction) $\vec{H}(t)$ (or $\vec{B}(t)$;

  2. The wavevector $\vec{k}$ and two scalar functions of time, the latter being the transverse components (in the plane at right angles to $\vec{k}$) of either the electric or magnetic field (or magnetic induction);

In the case of a nearly monochromatic wave, the vector functions of time in 1. and the two scalars in 2. can be reduced to complex scalars, which define the amplitude and phase of sinusoidally varying quantities. The alternative 2. together with an implicit knowledge of the wavevector is what we are using when we represent a write down a pure polarisation state as a $2\times 1$ vector $\psi$ of complex scalars, called the Jones vector. The scalars define magnitude and phase of the two transverse components of $\vec{E}$ (or $\vec{H}$, $\vec{B}$, as appropriate).

If only the relative phase of the two complex scalars is important, we can represent a pure polarisation state by an implicit definition of the wavevector and three real scalars: the Stokes parameters $s_j = \psi^\dagger \sigma_j \psi$, where the $\sigma_j$ are the Pauli spin matrices.

A partially polarised state is most readily thought of in quantum terms: we consider a general partially-polarised state to be a classical probabilistic mixture of pure polarisation states, defined by the $2\times2$, Hermitian complex density matrix (as well as an implicit definition of the wavevector direction). An equivalent characteristation is through the Mueller calculus, as discussed in my answer here. The classical description, in terms of random processes, is much fiddlier, messier and subtler than the quantum, and takes a full chapter in Born and Wolf, Principles of Optics" to describe (Emil Wolf was one of the pioneers in the rigorous description of partially polarised and partially coherent light).

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    $\begingroup$ Extra Note: In plasma physics, the polarization is defined by the orientation and variation of the electric (or magnetic) field with respect to the background, quasi-static magnetic field. It's similar to astronomy where the wave vector is the axis of orientation but we use the quasi-static magnetic field. $\endgroup$ – honeste_vivere Mar 15 '16 at 17:38

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