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Any polarization state can be represented as a superposition of two mutually orthogonal polarization states  and , or R and L.

On the other hand, another vector representation called Stokes vectors can represent partially polarized and unpolarized lights.

In the Jones vector (7), the maximum amplitudes Ax and Ay are real quantities. The presence of the exponent with imaginary arguments causes Ex and Ey to be complex quantities.

Since the Jones vector is a column matrix of rank 2, any pair of orthogonal Jones vectors can be used as a basis of the mathematical space spanned by all the Jones vectors.

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To facilitate the treatment of complicated polarization problems at the amplitude level, in 1941, R. Clark Jones developed a matrix calculus for treating these problems, commonly called the Jones matrix calculus.

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where A is a constant vector representing the amplitude, ω is the angular frequency, k is the wave vector (wave number), and r is the position in space.

The description of the polarizing behavior of the optical field in terms of amplitudes was one of the first great success of the wave theory of light. The solution of the wave equation in terms of transverse components leads to elliptically polarized light and its degenerate linear and circular forms.

Before we proceed to find the Jones vectors for various states of polarized light, we discuss the normalization of the Jones vector; it is customary to express the Jones vector in normalized form.

Without loss of generality, we consider the time evolution of the electric field vector at the origin z = 0. Thus (3) becomes:

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As a general rule, the most approximate choice of matrix method is to use the Jones calculus for amplitude superposition problems and the Mueller formalism for intensity superposition problems.

For the purpose of describing various representations of the polarization states, we consider propagation along the z axis. Since light wave is a transverse wave, the electric field vector must lie in the xy plane (perpendicular to z the propagation axis).

Stockes vectors can be used for the superposition of incoherent intensities. Similarly, we can superpose coherent amplitudes, that is, Jones vectors.

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In particular, we can resolve the basic linear polarization states  and  into two circular polarization states R and L and vice versa.

The Jones calculus involves complex quantities contained in 2x1 column matrices (the Jones vector) and 2x2 matrices (the Jones matrices).

The Jones vector specifies the polarization state of the wave uniquely. It contains complete information about the amplitudes and the phases of the electric field vector's x and y components.

Circular polarization states are seen to consist of linear oscillations along the x and y directions with equal amplitude , but with a phase difference of .

Similarly, a linear polarization state can be seen as a superposition of two oppositely sensed circular polarization states.

The polarization state of a light beam can be described in terms of the amplitudes(Ax, Ay) and the phase angles (δx,δy) of the x and y components of the electric field vector.

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On the basis of the amplitude results, many results could be understood (e.g., Young's interference experiment, circularly polarized light). However, even using the amplitude formulation, numerous problems become difficult to treat, such as the propagation of the field through several polarizing components.

When a light beam propagates in an isotropic and homogeneous medium, the beam can be represented by its electric field E(r,t), which can be written:

From the normalization condition (14) we see that Ax2 = 1. Thus, suppressing eiδx because it is unimodular, the normalized Jones vector for linearly horizontally polarized light is written: