Wigner Distribution Moments for Beam Characterization

Optical beam characterization is an important task for different applications such as imaging, metrology, light-matter interaction, optical communication, etc. An optical beam can encode information in its temporal-frequency spectrum, polarization, spatial structure, and statistical properties. Successful exploitation of the encoding capabilities of light requires the synthesis of beams with specific characteristics and monitoring of their parameters during beam propagation. This Chapter is focused on the characterization of the spatial structure of paraxial quasi-monochromatic scalar beams. The description of such beams by their mutual intensity (MI) is presented in Section 2. In Section 3 the transformation of the MI of beam during its propagation through first-order optical systems, often called ABCD systems, is studied. Note that the first-order optical systems are completely described by their ray transformation matrix. Along this Chapter we will use this matrix formalism which significantly simplifies the solutions of many discussed problems. The Wigner distribution (WD) provides an alternative way for beam characterization exploring the concept of phase space. Its definition and transformation under beam propagation is considered in Section 4. Neither the MI nor the WD can be measured directly, and their reconstruction is a cumbersome task. A more accessible characterization of beams via the moments of the WD is presented in Section 5. These moments are grouped in orders such that the lower the order the more global beam characteristics it represents. The physical meaning and properties of firstand second-order moments are discussed in Section 6. In Section 7 the Poincare sphere is introduced for classification and comparison of optical beams.