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We consider spatial shaping of partially coherent fields in two types of optical systems: a 2F Fourier-transforming system with the beam shaping element in the input plane and a 4F imaging system with the element in the intermediate Fourier plane. Different representations of the spatially partially coherent field in terms of fully coherent fields are examined to permit reduction of the dimensionality of the propagation integrals. The standard Mercer-type coherent-mode representation of the incident cross-spectral density (CSD) function is compared to expansions of CSD in either spatially or angularly shifted elementary field modes, all sharing the same spatial profile. In Fourier-transforming systems, the angular elementary-field representation proves computationally superior, while in imaging systems the spatially shifted elementary-field expansion is the best choice. Considering the Fourier-plane element as a generalized pupil, the latter leads to the concept's generalized amplitude associated with the elementary field and to a generalized transfer function of the system. These concepts reduce to the standard point spread function and the optical transfer function in the limit of spatial incoherence at the object plane. Examples of the effects of partial coherence in spatial beam shaping are given.

A representation of partially spatially coherent and partially polarized stationary electromagnetic fields is given in terms of mutually uncorrelated, transversely shifted, fully coherent, and polarized elementary electric-field modes. This representation allows one to propagate non-paraxial partially coherent vector fields using techniques for spatially fully coherent fields, which are numerically far more efficient than methods for propagating correlation functions. A procedure is given to determine the elementary modes from the radiant intensity and the far-zone polarization properties of the entire field. The method is applied to quasi-homogeneous fields with rotationally symmetric cos(n) theta radiant intensity distributions (theta being the diffraction angle with respect to the optical axis and n being an integer). This is an adequate model for fields emitted by, e.g., many light-emitting diodes.

An electromagnetic method based on rigorous diffraction theory of gratings is applied to the analysis of fields in semiconductor laser cavities. The method is based on the Fourier modal method; it is fully rigorous for infinitely periodic resonators and highly accurate for single resonators when absorbing boundary conditions are applied. Fundamental-mode intracavity and near-field distributions are evaluated for some selected geometries, and resonance frequencies are predicted.

A coherent-mode representation for spatially and spectrally partially coherent pulses is derived both in the space-frequency domain and in the space-time domain. It is shown that both the cross-spectral density and the mutual coherence function of partially coherent pulses can be expressed as a sum of spatially and spectrally and temporally completely coherent modes. The concept of the effective degree of coherence for nonstationary fields is introduced. As an application of the theory, the propagation of Gaussian Schell-model pulsed beams in the space-frequency domain is considered and their coherent-mode representation is presented.

A theory of open laser resonators is formulated within the framework of the electromagnetic coherence theory. It is shown that if only one Fox-Li mode contributes to the field at a given frequency, then the field at that frequency is necessarily completely coherent in view of the space-frequency counterpart of the recently introduced degree of coherence of electromagnetic fields [Opt. Express 11, 1137 (2003)]. It is also shown that the relation between the number of Fox-Li modes and the new degree of coherence is analogous to the relation established in the scalar theory of laser resonator modes. Difficulties that arise with the formerly introduced visibility-based definition of the electromagnetic degree of coherence are briefly discussed.

We construct the coherent-mode representation for fluctuating, statistically stationary electromagnetic fields. The modes are shown to be spatially fully coherent in the sense of a recently introduced spectral degree of electromagnetic coherence. We also prove that the electric cross-spectral density tensor can be rigorously expressed as a correlation tensor averaged over an appropriate ensemble of strictly monochromatic vectorial wave functions. The formalism is demonstrated for partially polarized, partially coherent Gaussian Schell-model beams, but the theory applies to arbitrary random electromagnetic fields and can find applications in radiation and propagation and in inverse problems.

The modal theory for spectrally partially coherent nonstationary plane waves is introduced. The theory is first developed in the space-frequency domain and then extended to the space-time domain. Propagation properties of the coherent modes are analyzed. The concept of the overall degree of coherence is extended to the domain of nonstationary fields, and it is shown that the overall degree of coherence of partially coherent plane-wave pulses is the same in the space-frequency and space-time domains. The theory is applied to the recently introduced concept of spectrally Gaussian Schell-model plane-wave pulses.

Conditions for exactly self-imaging nonparaxial fields that are periodic also in the transverse direction are introduced. The theory is first derived by assuming full coherence and then extended into the domain of partial coherence. Different types of solutions are discussed, and some illustrations of the existence of solutions and intensity distributions of the fields are presented.

Partially coherent fields with the electric field parallel to the azimuthal coordinate are analyzed by use of the exact angular spectrum representation. The known results for fully coherent fields are used to find the permitted forms of azimuthally polarized, partially coherent fields. The derived result is then used to show that this class of fields is severely restricted because the azimuthal polarization state is particularly sensitive to the correlation properties of the electric-field components. Two examples of azimuthally polarized fields are briefly examined. The first is a class of nondiffracting fields that retain the polarization state upon propagation, whereas the second is an example in which the azimuthal polarization is broken because the cross-spectral density function is not of the permitted form.

Improved diffraction efficiencies can be obtained in the paraxial domain of diffractive optics by considering light explicitly as an electromagnetic rather than a scalar field because of the extra freedoms provided by the state of polarization. For example, diffractive beam splitters with 100% efficiency are made possible by means of space-variant subwavelength-carrier surface-relief elements. Some aspects of the general design theory of polarization-modulating elements for vector fields, including design freedoms and constraints, are presented. Upper bounds of diffraction efficiency are derived and compared with those for the scalar case. Iterative design algorithms are developed. Several design examples with different constraints are presented, and the effects of replacing continuous-fringe structures by pixel structures containing locally linear gratings are evaluated.