Reconfigurable beams with arbitrary polarization and shape distributions at a given plane

Methods for generating beams with arbitrary polarization based on the use of liquid crystal displays have recently attracted interest from a wide range of sources. In this paper we present a technique for generating beams with arbitrary polarization and shape distributions at a given plane using a Mach-Zehnder setup. The transverse components of the incident beam are processed independently by means of spatial light modulators placed in each path of the interferometer. The modulators display computer generated holograms designed to dynamically encode any amplitude value and polarization state for each point of the wavefront in a given plane. The steps required to design such beams are described in detail. Several beams performing different polarization and intensity landscapes have been experimentally implemented. The results obtained demonstrate the capability of the proposed technique to tailor the amplitude and polarization of the beam simultaneously.

Dielectric anomaly and magnetic response of epitaxial orthorhombic YMnO3 thin films

The structure, magnetic response, and dielectric response of the grown epitaxial thin films of the orthorhombic phase of YMnO3 oxide on Nb:SrTiO3 (001) substrates have been measured. We have found that a substrate-induced strain produces an in-plane compression of the YMnO3 unit cell. The magnetization versus temperature curves display a significant zero-field cooling (ZFC)-field cooling hysteresis below the Nel temperature (TN 45 K). The dielectric constant increases gradually (up to 26%) below the TN and mimics the ZFC magnetization curve. We argue that these effects could be a manifestation of magnetoelectric coupling in YMnO3 thin films and that the magnetic structure of YMnO3 can be controlled by substrate selection and/or growth conditions.

Tunable epitaxial growth of magnetoresistive La2/3Sr1/3MnO3 thin films

We report on the growth of epitaxial La2/3Sr1/3MnO3 thin films on buffered Si(001) substrates. We show that a suitable choice of the buffer heterostructure allows one to obtain epitaxial (00h), (0hh), and (hhh) manganite thin films. The magnetotransport properties are investigated and we have found that the low-field magnetoresistance is directly related to the width of the normal-to-plane rocking curves, irrespective of the film orientation. The magnetic anisotropy of these films has also been determined.

Scattering of quantum particles by gravitational plane waves

We consider the coupling of quantum massless and massive scalar particles with exact gravitational plane waves. The cross section for scattering of the quantum particles by the waves is shown to coincide with the classical cross section for scattering of geodesics. The expectation value of the scalar field stress tensor between scattering states diverges at the points where classical test particles focus after colliding with the wave. This indicates that back-reaction effects cannot be ignored for plane waves propagating in the presence of quantum particles and that classical singularities are likely to develop.

Particle creation in a colliding plane wave spacetime: Wave packet quantization

We use wave packet mode quantization to compute the creation of massless scalar quantum particles in a colliding plane wave spacetime. The background spacetime represents the collision of two gravitational shock waves followed by trailing gravitational radiation which focus into a Killing-Cauchy horizon. The use of wave packet modes simplifies the problem of mode propagation through the different spacetime regions which was previously studied with the use of monochromatic modes. It is found that the number of particles created in a given wave packet mode has a thermal spectrum with a temperature which is inversely proportional to the focusing time of the plane waves and which depends on the mode trajectory.

Algebraic decay of velocity fluctuations in a confined fluid

Computer simulations of a colloidal particle suspended in a fluid confined by rigid walls show that, at long times, the velocity correlation function decays with a negative algebraic tail. The exponent depends on the confining geometry, rather than the spatial dimensionality. We can account for the tail by using a simple mode-coupling theory which exploits the fact that the sound wave generated by a moving particle becomes diffusive.

Magnetic field scaling of relaxation curves in small particle systems

We study the effects of the magnetic field on the relaxation of the magnetization of smallmonodomain noninteracting particles with random orientations and distribution of anisotropyconstants. Starting from a master equation, we build up an expression for the time dependence of themagnetization which takes into account thermal activation only over barriers separating energyminima, which, in our model, can be computed exactly from analytical expressions. Numericalcalculations of the relaxation curves for different distribution widths, and under different magneticfields H and temperatures T, have been performed. We show how a T ln(t/t0) scaling of the curves,at different T and for a given H, can be carried out after proper normalization of the data to theequilibrium magnetization. The resulting master curves are shown to be closely related to what wecall effective energy barrier distributions, which, in our model, can be computed exactly fromanalytical expressions. The concept of effective distribution serves us as a basis for finding a scalingvariable to scale relaxation curves at different H and a given T, thus showing that the fielddependence of energy barriers can be also extracted from relaxation measurements.

Rational periodic sequences for the Lyness recurrence

Consider the celebrated Lyness recurrence $x_{n+2}=(a+x_{n+1})/x_{n}$ with $a\in\Q$. First we prove that there exist initial conditions and values of $a$ for which it generates periodic sequences of rational numbers with prime periods $1,2,3,5,6,7,8,9,10$ or $12$ and that these are the only periods that rational sequences $\{x_n\}_n$ can have. It is known that if we restrict our attention to positive rational values of $a$ and positive rational initial conditions the only possible periods are $1,5$ and $9$. Moreover 1-periodic and 5-periodic sequences are easily obtained. We prove that for infinitely many positive values of $a,$ positive 9-period rational sequences occur. This last result is our main contribution and answers an open question left in previous works of Bastien \& Rogalski and Zeeman. We also prove that the level sets of the invariant associated to the Lyness map is a two-parameter family of elliptic curves that is a universal family of the elliptic curves with a point of order $n, n\ge5,$ including $n$ infinity. This fact implies that the Lyness map is a universal normal form for most birrational maps on elliptic curves.

On the bicanonical map of irregular varieties

From the point of view of uniform bounds for the birationality of pluricanonical maps, irregular varieties of general type and maximal Albanese dimension behave similarly to curves. In fact Chen-Hacon showed that, at least when their holomorphic Euler characteristic is positive, the tricanonical map of such varieties is always birational. In this paper we study the bicanonical map. We consider the natural subclass of varieties of maximal Albanese dimension formed by primitive varieties of Albanese general type. We prove that the only such varieties with non-birational bicanonical map are the natural higher-dimensional generalization to this context of curves of genus $2$: varieties birationally equivalent to the theta-divisor of an indecomposable principally polarized abelian variety. The proof is based on the (generalized) Fourier-Mukai transform.

Algebraic decay of velocity fluctuations near a wall

Computer simulations of the dynamics of a colloidal particle suspended in a fluid confined by an interface show that the asymptotic decay of the velocity correlation functions is algebraic. The exponents of the long-time tails depend on the direction of motion of the particle relative to the surface, as well as on the specific nature of the boundary conditions. In particular, we find that for the angular velocity correlation function, the decay in the presence of a slip surface is faster than the one corresponding to a stick one. An intuitive picture is introduced to explain the various long-time tails, and the simulations are compared with theoretical expressions where available.

Optimized back-focal-plane interferometry directly measures forces of optically trapped particles

Back-focal-plane interferometry is used to measure displacements of optically trapped samples with very high spatial and temporal resolution. However, the technique is closely related to a method that measures the rate of change in light momentum. It has long been known that displacements of the interference pattern at the back focal plane may be used to track the optical force directly, provided that a considerable fraction of the light is effectively monitored. Nonetheless, the practical application of this idea has been limited to counter-propagating, low-aperture beams where the accurate momentum measurements are possible. Here, we experimentally show that the connection can be extended to single-beam optical traps. In particular, we show that, in a gradient trap, the calibration product κ·β (where κ is the trap stiffness and 1/β is the position sensitivity) corresponds to the factor that converts detector signals into momentum changes; this factor is uniquely determined by three construction features of the detection instrument and does not depend, therefore, on the specific conditions of the experiment. Then, we find that force measurements obtained from back-focal-plane displacements are in practice not restricted to a linear relationship with position and hence they can be extended outside that regime. Finally, and more importantly, we show that these properties are still recognizable even when the system is not fully optimized for light collection. These results should enable a more general use of back-focal-plane interferometry whenever the ultimate goal is the measurement of the forces exerted by an optical trap.

Optimized back-focal-plane interferometry directly measures forces of optically trapped particles

Back-focal-plane interferometry is used to measure displacements of optically trapped samples with very high spatial and temporal resolution. However, the technique is closely related to a method that measures the rate of change in light momentum. It has long been known that displacements of the interference pattern at the back focal plane may be used to track the optical force directly, provided that a considerable fraction of the light is effectively monitored. Nonetheless, the practical application of this idea has been limited to counter-propagating, low-aperture beams where the accurate momentum measurements are possible. Here, we experimentally show that the connection can be extended to single-beam optical traps. In particular, we show that, in a gradient trap, the calibration product κ·β (where κ is the trap stiffness and 1/β is the position sensitivity) corresponds to the factor that converts detector signals into momentum changes; this factor is uniquely determined by three construction features of the detection instrument and does not depend, therefore, on the specific conditions of the experiment. Then, we find that force measurements obtained from back-focal-plane displacements are in practice not restricted to a linear relationship with position and hence they can be extended outside that regime. Finally, and more importantly, we show that these properties are still recognizable even when the system is not fully optimized for light collection. These results should enable a more general use of back-focal-plane interferometry whenever the ultimate goal is the measurement of the forces exerted by an optical trap.

Holographic optical tweezers combined with back-focal-plane displacement detection

A major problem with holographic optical tweezers (HOTs) is their incompatibility with laser-based position detection methods, such as back-focal-plane interferometry (BFPI). The alternatives generally used with HOTs, like high-speed video tracking, do not offer the same spatial and temporal bandwidths. This has limited the use of this technique in precise quantitative experiments. In this paper, we present an optical trap design that combines digital holography and back-focal-plane displacement detection. We show that, with a particularly simple setup, it is possible to generate a set of multiple holographic traps and an additional static non-holographic trap with orthogonal polarizations and that they can be, therefore, easily separated for measuring positions and forces with the high positional and temporal resolutions of laser-based detection. We prove that measurements from both polarizations contain less than 1% crosstalk and that traps in our setup are harmonic within the typical range. We further tested the instrument in a DNA stretching experiment and we discuss an interesting property of this configuration: the small drift of the differential signal between traps.

Pasquale del Pezzo, Duke of Caianello, Neapolitan mathematician

This article is dedicated to a reconstruction of some events and achievements, both personal and scientific, in the life of the Neapolitan mathematician Pasquale del Pezzo, Duke of Caianello.