E_x = A_xexp(i(kz - ωt)), E_y = A_yexp(i(kz - ωt + φ)).

We can write the components as a column vector, which is called a Jones vector.

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We may factor out the common dependence on z and t and just write 

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The most general Jones vector of a polarized beam propagating along the z-axis is given by this equation.  The intensity of the beam is proportional to A_x² + A_y².

The Jones vector for horizontally polarized light is

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The Jones vector for vertically polarized light is

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The normalized Jones vector for light polarized at 45^o,  where A_x = A_y = A is

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(Normalization means that |E|² = 1.)

The normalized Jones vector for right-hand circularly polarized light is

 .

For the normalized vectors we have

If a polarized beam with field vector E is incident on a polarization-changing medium such as a polarizer or a wave plate, then the result is a beam in another polarization state given by E' with E'_x = m₁₁E_x + m₁₂E_y,  E'_y = m₂₁E_x + m₂₂E_y.  We write

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The 2 by 2 transformation matrix is called the Jones matrix. 

The table below lists the Jones matrices for common optical elements. 

If we require the Jones matrix for an optical element which has been rotated through an angle θ with respect to the direction given in the table above, we must multiply the above matrix by the usual matrices for rotation.

M(θ) = R(θ) M R(-θ), where

To find the Jones matrix for a sequence of polarization transformations, for example a linear polarizer followed by a quarter wave plate, we simply multiply the individual Jones matrices together in the correct order.  If an incident beam of light with field vector E passes through a sequence of four polarizing elements, M₁ followed by M₂, M₃ and M₄, then the resultant field vector E' is given by

E' = M₄ M₃ M₂ M₁ E.

The eigenvectors of the Jones matrix M of an optical device correspond to the polarization states which propagate through the optical device represented by M unchanged.  The beam enters and emerges in the same polarization state.  The eigenvalues, which in general are complex numbers, tell us about changes in amplitude and phase produced by the optical device.

Link: The Jones Calculus

AI Study Tip:

Example prompts:  (a good review of the above material)

• I have a Jones vector
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  Walk me through the steps to identify the state of polarization.  Is it linear, circular, or elliptical?  If circular, is it left circularly polarized (LCP) or right circularly polarized (RCP) based on the standard optics convention.
• Write the normalized Jones vector for a linearly polarized wave oriented at an angle 30^o from the x-axis.  Then show how to represent a state that has a phase lag of π/2 in the y-component relative to the x-component.
• Explain the effect of a quarter-wave plate on linearly polarized light.  If the incident light is polarized at 45^o to the fast axis of the quarter-wave plate, what is the resulting Jones vector?  Why does this specific orientation produce circular polarization?
• Compare a half-wave plate and a quarter-wave plate.  How does a half-wave plate change the handedness'of circularly polarized light, and how does it rotate the plane of linearly polarized light?