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We studied a novel family of paraxial laser beams forming an overcomplete yet nonorthogonal set of modes. These modes have a singular phase profile and are eigenfunctions of the photon orbital angular momentum. The intensity profile is characterized by a single brilliant ring with the singularity at its center, where the field amplitude vanishes. The complex amplitude is proportional to the degenerate (confluent) hypergeometric function, and therefore we term such beams hypergeometric gaussian (HyGG) modes. Unlike the recently introduced hypergeometric modes (Opt. Lett. {textbf 32}, 742 (2007)), the HyGG modes carry a finite power and have been generated in this work with a liquid-crystal spatial light modulator. We briefly consider some sub-families of the HyGG modes as the modified Bessel Gaussian modes, the modified exponential Gaussian modes and the modified Laguerre-Gaussian modes.
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We present a novel family of paraxial optical beams having a confluent hypergeometric transverse profile, which we name hypergeometric Gauss modes of type-II (HyGG-II). These modes are eigenmodes of the photon orbital angular momentum and they have the lowest beam divergence at waist of HyGG-II among all known finite power families of paraxial modes. We propose to exploit this feature of HyGG-II modes for generating, after suitable focusing, a light needle having record properties in terms of size and aspect ratio, possibly useful for near-field optics applications.
A special singular limit $omega_1/omega_2to 1$ is considered for the Faddeev modular quantum dilogarithm (hyperbolic gamma function) and corresponding hyperbolic integrals. It brings a new class of hypergeometric identities associated with bilateral sums of Mellin-Barnes type integrals of particular Pochhammer symbol products.
We propose a class of Pade interpolation problems whose solutions are expressible in terms of determinants of hypergeometric series.
General theory of elliptic hypergeometric series and integrals is outlined. Main attention is paid to the examples obeying properties of the classical special functions. In particular, an elliptic analogue of the Gauss hypergeometric function and some of its properties are described. Present review is based on authors habilitation thesis [Spi7] containing a more detailed account of the subject.
We consider the KZ differential equations over $mathbb C$ in the case, when the hypergeometric solutions are one-dimensional integrals. We also consider the same differential equations over a finite field $mathbb F_p$. We study the space of polynomial solutions of these differential equations over $mathbb F_p$, constructed in a previous work by V. Schechtman and the second author. Using Hasse-Witt matrices we identify the space of these polynomial solutions over $mathbb F_p$ with the space dual to a certain subspace of regular differentials on an associated curve. We also relate these polynomial solutions over $mathbb F_p$ and the hypergeometric solutions over $mathbb C$.
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