## Abstract

One of the important properties of an optical wavefront is the transverse field distribution of the beam, that is, the distribution of its amplitude and phase as well as its polarization state. This field distribution determines the propagation behavior of the beam, and its angular momentum. The ability to generate an arbitrary complex scalar optical wavefront accurately is essential in modern optical applications. Although the ability to generate scalar beams is useful, in an increasing number of cases it is desirable to create arbitrary vectorial beams. In general, vectorial beams are defined as beams having space-variant (transversely nonuniform) polarization state. Polarization is a fundamental property of electromagnetic fields. Accordingly, the state of polarization of light has substantial influence in most optical experiments and in the theoretical models developed to interpret them. In optics the polarization state of a light field can significantly affect the propagation of many fully or partially coherent paraxial light fields. However, its influence can never be ignored in nonparaxial conditions when the field propagates in free space. Significant polarization effects can occur during interaction with material interfaces such as gratings, or arrangements of nanoparticles. Comprehensive background information on polarization optics can be found in textbooks (see, for example, Collett [2003] and Brosseau [1998]). Recent years have witnessed a growing interest, theoretically as well as experimentally, in space-variant polarization-state manipulation that can be exploited in a variety of applications. These include polarization encoding of data (Javidi and Nomura [2000]), neural networks and optical computing (Davidson, Friesem and Hasman [1992a]), optical encryption (Mogensen and Glückstad [2000]), tight focusing (Quabis, Dorn, Eberler, Glöckl and Leuchs [2000]), imaging polarimetry (Nordin, Meier, Deguzman and Jones [1999]), material processing (Niziev and Nesterov [1999]), and atom trapping and optical tweezers (Liu, Cline and He [1999]). The study of polarization manipulation has grown into a new branch of modern physical optics known as polarization singularities (see, for example, Nye [1999] and Soskin and Vasnetsov [2001]). In a scalar field, such singularities appear at points or lines where the phase or the amplitude of the wave is undefined or changes abruptly. One class of such dislocations is formed by vortices, which are spiral phase ramps about a singularity. Vortices are characterized by a topological charge l = frac(1, 2) π ∮ ∇ φ d s, where φ is the phase of the beam and l is an integer. Until recently, research had focused mainly on field dislocations in scalar waves. However, if we allow for the polarization to be space-varying, disclinations can arise (see, for example, Nye [1983], Dennis [2002], and Freund, Mokhun, Soskin, Angelsky and Mokhun [2002]). Disclinations are points or lines of singularity in the pattern or direction of a transverse field. An example is the center of a beam with radial or azimuthal polarization. Different techniques for obtaining space-variant polarization manipulation by use of nonuniform anisotropic polarization elements have been reported in the literature. In general, a polarization optical element is any optical element that can modify the state of polarization of a light beam, such as a polarizer, retarder, rotator, or depolarizer. Space-variant polarization elements can be implemented as space-variant computer-generated sub-wavelength dielectric or metal gratings (Hasman, Bomzon, Niv, Biener and Kleiner [2002]), polarization-sensitive materials such as azobenzene-containing materials (Todorov, Nikolova and Tomova [1984]), and liquid-crystal devices (Davis, McNamara, Cottrell and Sonehara [2000]). For the most general case, the transmission, retardation and optical-axis orientation of such elements depend on the location across the face of the element. In order to analyze the beam emerging from a space-variant polarization element, we must resort to Jones and Mueller polarization-transfer matrix methods. The Jones calculus assumes completely polarized light and coherent addition of waves, whereas the Mueller calculus assumes partially polarized incoherent addition of waves. The space-dependent transfer matrices of Jones and Mueller can be calculated by expressing the local behavior of the element as a polarizer and retarder, while the local orientation of the optical axis can be obtained by applying the rotation matrix (see, for example, Collett [2003]). Moreover, Gori, Santarsiero, Vicalvi, Borghi and Guattari [1998] introduced an important method for analyzing partially coherent sources with space-varying partial polarization utilizing the beam coherence-polarization matrix (BCP). This approach can be viewed as an approximate form of Wolf's general tensorial theory of coherence (Wolf [1954]). Sometimes the local transmission and retardation of the polarization element cannot be determined in a straightforward manner, for instance when using gratings for which the period is close to or smaller than the incident beam's wavelength. In this case, a direct solution of Maxwell's equations is required. This can usually be accomplished by using numerical approaches, such as rigorous coupled wave analysis (RCWA; see Moharam and Gaylord [1986] and Lalanne and Morris [1996]), or utilizing finite-difference time-domain methods (FDTD) for analyzing nonuniform polarization optics (see, for example, Mirotznik, Prather, Mait, Beck, Shi and Gao [2000] and Jiang and Nordin [2000]). Complex vectorial fields can be produced either by utilizing polarization elements (see, for example, Bomzon, Biener, Kleiner and Hasman [2002b]) or by using interferometric techniques involving two orthogonally polarized beams (see, for example, Tidwell, Ford and Kimura [1990]). A coherent summation, inside the laser resonator, of two orthogonally polarized TEM_{01} modes was demonstrated by Oron, Blit, Davidson, Friesem, Bomzon and Hasman [2000]. Several designing approaches for obtaining vectorial fields having space-varying polarization distribution have been presented (see, for example, Niv, Biener, Kleiner and Hasman [2004] and Tervo, Kettunen, Honkanen and Turunen [2003]). In a remarkable paper first published in 1956, Pancharatnam [1956] (reprinted in Pancharatnam [1975]) considered the phase of a beam of light whose polarization state is modified. Pancharatnam showed that a cyclic change in the state of polarization of the light is accompanied by a phase shift determined by the geometry of the cycle as represented on the Poincaré sphere (Brosseau [1998]). Therefore, space-variant polarization-state manipulations are accompanied by a phase modification that results from the Pancharatnam-Berry phase (Berry [1987], Bomzon, Kleiner and Hasman [2001d]). In order to investigate the propagation behavior of complex vectorial fields as well as the angular momentum (Allen, Padgett and Babiker [1999]), it is necessary to consider the resulting geometrical phase distribution. The calculation of the space-variant Pancharatnam phase is based on the rule proposed by Pancharatnam [1956] for comparing the phases of two light beams in different states of polarization as the argument of the vectorial projection between the two polarization states. The propagation of paraxial vector fields has been extensively studied theoretically. Several vectorial treatments have been presented in both coherent (see, for example, Gori [2001]) and partially coherent light fields (see, for example, James [1994] and Seshadri [1999]). The simplest approach to studying arbitrarily polarized beams is to decompose the representative field vector at any point of a section into orthogonal linearly or circularly polarized parts. Free-propagation problems can then be performed as the analysis of the propagation of a pair of scalar waves (Goodman [1996]). On the other hand, for specific beams with axial-symmetric polarization distribution, significant results have been obtained by representing the field at a typical point as a superposition of radial and azimuthal components. Jordan and Hall [1994] showed that the propagation process of an azimuthally polarized Bessel-Gauss beam can be analyzed by means of a single one-dimensional propagation integral with a suitable kernel. A general vectorial decomposition of electromagnetic fields with application to propagation-invariant and rotating fields was presented by Pääkkönen, Tervo, Vahimaa, Turunen and Gori [2002]. Vectorial Talbot self-imaging and vectorial nondiffracting beams have been investigated for a wave field with periodic variations of the polarization state (e.g., Mishra [1991]), and have been demonstrated experimentally (Arrizón, Tepichin, Ortiz-Gutierrez and Lohmann [1996]). Moreover, Gori, Santarsiero, Borghi and Piquero [2000] introduced an important method for analyzing the propagation of partially coherent beams with space-varying partial polarization by extension of the van Cittert-Zernike theorem using the beam coherence polarization matrix (see, for example, Mandel and Wolf [1995]). In the nonparaxial case, e.g., the propagation of the beam emerging from a lens with high numerical aperture (NA), one must resort to a vectorial formulation that takes into account polarization effects and nonuniformity of the amplitude over the wavefront. A mathematically tractable representation for dealing with polarization was developed by Debye [1909], and a representation for handling apodization was addressed by Hopkins [1943]. ...

Original language | English |
---|---|

Title of host publication | Progress in Optics |

Publisher | Elsevier |

Pages | 215-289 |

Number of pages | 75 |

ISBN (Print) | 0444515984, 9780444515988 |

DOIs | |

State | Published - 1 Jan 2005 |

Externally published | Yes |

### Publication series

Name | Progress in Optics |
---|---|

Volume | 47 |

ISSN (Print) | 0079-6638 |

## ASJC Scopus subject areas

- Electronic, Optical and Magnetic Materials
- Surfaces and Interfaces