Show by direct calculation, using Mueller matrices, that a...

Show by direct calculation, using Mueller matrices, that a unitirradiance beam of natural light passing through a vertical linear polarizer is converted into a vertical $\mathscr{P}$ -state. Determine its relative irradiance and degree of polarization.

Key Concepts

Mueller Matrices

Mueller matrices are 4x4 matrices used to describe how the polarization state of light, represented by the Stokes vector, is transformed when it interacts with optical elements. They provide a systematic way to calculate the output Stokes parameters after the light passes through devices such as polarizers, retarders, and depolarizers.

Stokes Parameters

Stokes parameters constitute a set of values that fully describe the polarization state of a light beam. They include total intensity and the differences in intensity between orthogonal polarization components, allowing for the quantitative assessment of polarization properties such as the degree and orientation of the polarization.

Linear Polarization

Linear polarization refers to the state where the electric field oscillates in a single plane. When unpolarized or naturally polarized light passes through a linear polarizer oriented in a specific direction (vertical in this case), the polarizer selects the component of the light that oscillates in that plane, converting the beam into a linearly polarized state.

Degree of Polarization

The degree of polarization quantifies how much of the light's intensity is contained in the polarized component, ranging from 0 (completely unpolarized) to 1 (fully polarized). After passing through a linear polarizer, unpolarized light typically emerges fully polarized, indicating a unit degree of polarization.

Relative Irradiance

Relative irradiance relates to the intensity of the light transmitted through the polarizer compared to the incident light intensity. For a completely unpolarized beam of unit irradiance passing through an ideal linear polarizer, the transmitted beam has an irradiance of one-half the incoming intensity due to the polarizer's projection effect on the light’s polarization components.

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