Directionality and Spatial Coherence

Measurement of spatial coherence of light [Invited]

The most frequently used experimental techniques for measuring the spatial coherence properties of classical light fields in the space-frequency and space-time domains are reviewed and compared, with some attention to polarization effects. In addition to Young's classical two-pinhole experiment and several of its variations, we discuss methods that allow the determination of spatial coherence at higher data acquisition rates and also permit the characterization of lower-intensity light fields. These advantages are offered, in particular, by interferometric schemes that employ only beam splitters and reflective elements, and thereby also facilitate spatial coherence measurements of broadband fields.

Paraxial propagation of a class of Bessel-correlated fields

The propagation of a novel class of paraxial spatially partially coherent beams exhibiting Bessel-type correlations is studied in free space and in paraxial optical systems. We show that, under certain conditions, such beams can have functionally identical forms of the absolute value of the complex degree of spatial coherence not only at the source plane and in the far zone, but also at all finite propagation distances. Under these conditions the degree of spatial coherence properties of the field is a shape-invariant quantity, but the spatial intensity distribution is only approximately shape-invariant. The main properties of this class of model beams are demonstrated experimentally by passing a coherent Gaussian beam through a rotating wedge and measuring the coherence of the ensuing beams with a Young-type interferometer realized with a digital micromirror device.

Bessel-correlated supercontinuum fields

We examine the spatial coherence properties of supercontinuum fields generated by illuminating rotating bulk media with intense pulsed beams. Theoretical models are presented, which indicate the possibility of generating a class of Bessel-correlated fields (in time-averaged sense) using tilted plane-parallel glass plates and wedges as media for generation of supercontinuum radiation. In special cases, the ensuing fields have a strictly identical functional form in the spatial and angular domains. Some of the main results are verified experimentally by measuring the spatial coherence properties of bulk-generated supercontinuum fields using a wavefront-folding interferometer.

Elementary-field analysis of partially coherent beam shaping

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.

Spatial coherence of broad-area laser diodes

We model the spatial coherence of broad-area laser diodes (BALDs) by representing the mutual intensity as superpositions of individually fully coherent but mutually uncorrelated fields. Consideration of spectroscopic modal structure measurements and intensity-based mode recovery shows that the standard Mercer-type coherent-mode expansion can lead to unsatisfactory results for real BALDs. However, we show that a so-called shifted elementary-field method provides a sufficiently accurate tool for spatial coherence and propagation modeling even if the modal structure of the BALD is severely distorted.

Numerical modeling of spatial coherence using the elementary function method

The elementary function method is an approximate method for propagation calculations in spatially, partially coherent light in two dimensions. In this paper, we present the numerical application of this method to a 248 nm UV excimer laser source. We present experimental results of the measurement of the degree of spatial coherence and the beam profile of this source. The elementary function method is then applied to the real beam data and used to simulate the effects of imaging an opaque edge with a source of varying degrees of spatial coherence. The effect of spatial coherence on beam homogenization is also presented.

Shifted-elementary-mode representation for partially coherent vectorial fields

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.

Genuine cross-spectral densities and pseudo-modal expansions

A necessary and sufficient non-negative definiteness condition for the cross-spectral density (CSD) is provided. It is also shown that any genuine CSD can be expanded in terms of the so-called pseudo-modes of the source, understood as coherent contributions, not orthogonal to one another, that, superposed in an uncorrelated way, give rise to the CSD. Their evaluation is analyzed by means of an illustrative example.

Devising genuine spatial correlation functions

The choice of the mathematical form of spatial correlation functions for optical fields is restricted by the constraint of nonnegative definiteness. We discuss a sufficient condition for ensuring the satisfaction of such a constraint.

Propagation of spatially partially coherent emission from a vertical-cavity surface-emitting laser

Recently we observed a strong reduction of spatial coherence of the emission of large-aperture vertical-cavity surface-emitting lasers when they are driven by microsecond electrical pulses [Opt. Express 13, 9337 (2005)]. We study the influence of this partial spatial coherence on the propagation characteristics. The spatial decoherence manifests itself in the formation of a Gaussian far-field intensity distribution. We measure the transverse pulse profile from near to far field and see that the shape-invariant far-field regime starts after 140 microm in the pulsed regime as opposed to several centimeters in continuous wave operation. This value matches quantitatively calculations made with a novel partially coherent propagation model without any free fitting parameters.