Study On The Propagation Properties Of Pearcey Beams

The Pearcey beam (PB)-based on the Pearcey function, has infinite energy. It dis-plays an inherent self-focusing and self healing properties on propagation. Investigating the properties of PB can serve to apply it to microscopy, optical trapping and microma-nipulation.Firstly, we will introduce the caustic, and show that the Pearcey function discribes diffraction near a cusp caustic. Then we calculate the Pearcey integral to get the complex amplitude of PB. The Fourier transformation of Pearcey function gives a 6 line centrated on a parabola, modulated by a frequency domain coordinates biquadratic dependent phase factor. We also calculate the Poynting vector and angular momentum. The result reveals that the transverse Poynting vector is much smaller than that along the transport direction.Furthermore, we derived an analytical expression for the optical field of PB diffract-ing in free space. The expression reveals the PB retains its form as a Pearcey function when transport distance evolves, although the scaling in transverse section is different, accompanied by the Pearcey pattern translating linearly in the section. By numerical sim-ulation, we present the intensity pattern of the PB in different tranverse section. The PB will self-focus at a specific distance and focus to an infinite bright line. With the prop-agation distance furthur increasing, the PB inverted mirroring the behavior before. In addition, we calculate the Poynting vector of different transverse section, and uncover that the energy flow leads to these phenomena. Next we derived the expression for the PB partially blocked by a Gaussian absorption obstacle analytically. We find that the prop-erties of the PB have nothing to do with the obstacle, whereas the transmission distance long enough, the influence of the obstacle will gradually disappear, namely the PB has self-healing property.Specifically, we calculate the diffraction pattern of the PB blocked by a Gaussian absorption obstacle numerically. Results show that the larger waist radius of Gassian ob-stacle or the closer distance to the main lobe, the self-healing process of the PB is slower. That is, it needs longer propagation distance to recover the intensity of the PB, as well as the the intensity recovery on the main lobe and on the point of the obstacle. When the PB partially obstructed, a highlight spot appears during propagation, of which the intensity is larger than that without an obstacle. The intensity of the highlight spot is larger with larger obstacle or the obstacle closer to the main lobe. Finally, we calculate the Poynting vector, and find that the self-healing of the PB is caused by the convergence of energy from the side to the point of obstacle.