{"id":69502,"date":"2022-03-01T10:56:51","date_gmt":"2022-03-01T14:56:51","guid":{"rendered":"https:\/\/seniordesign.digitalscholar.rochester.edu\/spring2022\/uncategorized\/cai-thesis\/"},"modified":"2024-03-04T16:40:54","modified_gmt":"2024-03-04T20:40:54","slug":"cai-thesis","status":"publish","type":"post","link":"https:\/\/www.hajim.rochester.edu\/senior-design-day\/cai-thesis\/","title":{"rendered":"Full Poincar\u00e9 Fields Composed of a Discrete Number of Plane Waves"},"content":{"rendered":"\n
Author<\/h2>\n\n\n\n
Lingyi Cai<\/p>\n\n\n\n
Mentor<\/h2>\n\n\n\n
Miguel Alonso<\/p>\n\n\n\n
Abstract<\/h2>\n\n\n\n
In recent years, full Poincar\u00e9 (FP) fields have attracted an increasing amount of interest. Conventionally, the FP fields are composed of beams with Gaussian modes and Laguerre-Gaussian modes. In this thesis, we numerically and analytically analyzed the superposition of multiple paraxial plane waves and discovered the pattern of such superpositions composing FP fields. MATLAB scripts are used to simulate both the transverse plane distribution and the polarization states in the Poincar\u00e9 sphere. We found that three waves are sufficient to compose a FP field, and from the FP configurations, we proposed a hypothesis of the requirements to compose FP fields with only three plane waves.<\/p>\n\n\n\n
Background<\/h2>\n\n\n\n
The polarization of electromagnetic waves is defined as the statistical description of the geometric behavior of the oscillations of the field vector. In isotropic media, the oscillation of the field is always perpendicular to the direction of the propagation of the wave. When multiple coherent waves are propagating in directions with a slight difference, the longitudinal polarization component will be ignorable and the polarization distribution over the transverse plane will be nonuniform.<\/p>\n\n\n\n
Poincar\u00e9 sphere<\/strong><\/h3>\n\n\n\n
In 1852, George Gabriel Stokes introduced the Stokes parameters which describe the polarization with the total intensity I, the fractional degree of polarization p, and the shape parameters of the polarization ellipse. Divide the parameters by the intensity gives the normalized Stokes vector s<\/strong>2D<\/sub>=(s1<\/sub>,s2<\/sub>,s3<\/sub>). When s1<\/sub>, s2<\/sub>, and s3<\/sub> are used as a basis of coordinate, every polarization state represented by the normalized Stokes parameters will correspond to a single point interior of a unit sphere centered at the origin, which is called the Poincar\u00e9 sphere.<\/p>\n\n\n\n
\n
\nFigure 1.(a) Polarization ellipse showing the shape parameters. (b) Visualization of a Poincar\u00e9 sphere with several polarization states labeled.<\/figcaption><\/figure>\n<\/div><\/div>\n<\/div><\/div>\n\n\n\n
Full Poincar\u00e9 Field<\/strong><\/h3>\n\n\n\n
Full Poincar\u00e9 fields are fields with nonuniform polarization distribution which spans the entire surface when projected onto the Poincar\u00e9 sphere. Amber M. Beckley, et al. (2010) first introduced that the beams can be constructed from a coaxial superposition of a fundamental Gaussian mode and a spiral-phase Laguerre-Gauss mode having orthogonal polarizations. Our goal in this thesis is to use plane waves instead to compose Full Poincar\u00e9 fields.<\/p>\n\n\n\nFigure 2. Evolution of the distribution of polarization ellipses for FP beams where\u00a0\u00ea_1\u00a0and\u00a0\u00ea_2\u00a0are, respectively, (a) RHC and LHC, (b) LHC and RHC, and (c) vertical and horizontal. In all movies, the radial distance is scaled by\u00a0w(z) and the angle between the\u00a0x-axis and the purple line represents\u00a0\u03d5_\u03be (z).<\/figcaption><\/figure>\n\n\n\n
Results<\/h2>\n\n\n\nFigure 3. (a) Polarization distribution over the transverse plane (b) The polarization states mapped onto Poincar\u00e9 sphere. (c) The plot of polarization states projected onto s1<\/sub>-s2<\/sub> plane. (d) The plot of polarization states is projected onto s2<\/sub>-s3<\/sub> plane. (e) The plot of polarization states projected onto s1<\/sub>-s3<\/sub> plane.<\/figcaption><\/figure>\n\n\n\nFigure 4. (a) Polarization distribution over the transverse plane (b) The polarization states mapped onto Poincar\u00e9 sphere. (c) The plot of polarization states projected onto s1<\/sub>-s2<\/sub> plane. (d) The plot of polarization states is projected onto s2<\/sub>-s3<\/sub> plane. (e) The plot of polarization states projected onto s1<\/sub>-s3<\/sub> plane.<\/figcaption><\/figure>\n\n\n\n
Hypothesis<\/h2>\n\n\n\n
Requirement for Composing FP Fields with Three Plane Waves<\/strong>:<\/p>\n\n\n\n
For three fully polarized plane waves E<\/strong>1<\/sub>,E<\/strong>2<\/sub>,E<\/strong>3<\/sub>, exist E<\/strong>1<\/sub>, such that: <\/p>\n\n\n\n
(1)k<\/strong>2<\/sub>+k<\/strong>3<\/sub>=mk<\/strong>1<\/sub>, where m\u2208R; <\/p>\n\n\n\n
In this thesis, we numerically and analytically analyzed the superposition of multiple paraxial plane waves and discovered the pattern of such superpositions composing FP fields.<\/p>\n","protected":false},"author":6242,"featured_media":39,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_coblocks_attr":"","_coblocks_dimensions":"","_coblocks_responsive_height":"","_coblocks_accordion_ie_support":"","_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"categories":[136,2976],"tags":[],"coauthors":[8612],"class_list":["post-69502","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-opt-ope-archive","category-optical-archive"],"acf":[],"yoast_head":"\n
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