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Historically, infinity was long considered a vague concept—boundless, endless, larger than the largest—without any quantifiable mathematical foundation. This view changed in the 1800s through the pioneering work of Georg Cantor, who showed that infinite sets follow their own seemingly paradoxical mathematical rules. In 1924, David Hilbert highlighted the strangeness of infinity through a thought experiment now referred to as the Hilbert Hotel paradox, or simply Hilbert's Hotel. The paradox describes a "fully" occupied imaginary hotel having an infinite number of single-occupancy rooms. The manager can always find a room for new guests by simply shifting current guests to the next highest room, leaving the first room vacant. The investigation of wavefield singularities has uncovered the existence of a direct optical analogy to Hilbert's thought experiment. Since then, efforts have been made to investigate the properties of Hilbert's Hotel by controlling the dynamics of phase singularities in "fractional" order optical vortex beams. Here, we have taken such proposals to the next level and experimentally demonstrated Hilbert's Hotel using both phase and polarization singularities of optical fields. Using a multi-ramped spiral-phase-plate and a supercontinuum source, we generated and controlled fractional order vortex beams for the practical implementation of Hilbert's Hotel in scalar and vector vortex beams. Using a multi-ramped spiral-phase-plate, we show the possibility for complicated transitions of the generalized Hilbert's Hotel. The generic experimental scheme illustrates the usefulness of structured beams in visualizing unusual mathematical concepts and also for fractional vector beams driven by fundamental and applied research. [ABSTRACT FROM AUTHOR]

Nonlinear frequency conversion processes depend on the polarization state of the interacting beams. On the other hand, vector vortex beams have space-variant polarization in the transverse beam plane. In light of these two points, is it possible to do nonlinear frequency conversion of the vector vortex beam in single-pass geometry and retain the characteristics of the beam? To address this question, here, we report an experimental scheme for single-pass second harmonic generation (SHG) of vector vortex beams. Using an ultrafast Ti:Sapphire laser of pulse width ∼17 fs and a set of spiral phase plates in a polarization based Mach–Zehnder interferometer (MZI), we have generated vector vortex beams of order as high as lp = 12 at an average power of 860 mW. Using two contiguous bismuth borate crystals with the optic axis orthogonal to each other, we have frequency-doubled the near-IR vector vortex beam into visible vector vortex beam with order as high as lsh = 24. The maximum output power of the vector vortex beam of order, lsh = 2, is measured be as high as 20.5 mW at a single-pass SHG efficiency of 2.4%. Controlling the delay in MZI, we have preserved the vector vortex nature of both the pump and frequency-doubled beams. Measurement on the mode purity confirms the generation of high quality vector vortex beams at pump and SHG wavelengths. This generic scheme can be used to generate vector vortex beams across the electromagnetic spectrum in all time scales, continuous-wave to ultrafast. [ABSTRACT FROM AUTHOR]

We report on a simple experimental scheme to generate and control the orbital angular momentum (OAM) spectrum of the asymmetric vortex beams in a nonlinear frequency conversion process. Using a spiral phase plate (SPP) and adjusting the transverse shift of the SPP with respect to the incident Gaussian beam axis, we have transformed the symmetric (intensity distribution) optical vortex of order l into an asymmetric vortex beam of measured broad spectrum of OAM modes of orders l, l − 1, l − 2, …, 0 (Gaussian mode). While the position of the SPP determines the distribution of the OAM modes, we have also observed that the modal distribution of the vortex beam changes with the shift of the SPP of all orders and finally results in a Gaussian beam (l =0). Using single-pass frequency doubling of the asymmetric vortices, we have transferred the pump OAM spectra, l, l − 1, l − 2, …, 0, into the broad spectra of higher order OAM modes, 2l, 2l − 1, 2l − 2, …, 0 at green wavelength, owing to OAM conservation in nonlinear processes. We also observed an increase in single-pass conversion efficiency with the increase in asymmetry of the pump vortices producing a higher power vortex beam of mixed OAM modes at a new wavelength than that of the pure OAM mode. [ABSTRACT FROM AUTHOR]

We report on linear and nonlinear generation of ultrafast hollow Gaussian beams (HGBs). Using only two spiral phase plates (SPPs) having phase variation corresponding to vortex orders, l=1 and 2, and an experimental scheme, we have generated high power, ultrafast HGBs of orders up to 3 at 1064 nm. Based on single-pass, frequency doubling of the HGBs in a 5mm long, MgO doped, periodically poled LiNbO3 (MgO:PPLN) crystal, we have produced HGBs of average output power in excess of 250 mW at 532 nm and order as high as 6. Experimentally, we verified that the frequency doubled HGBs have orders twice those of the pump HGBs. Like the Gaussian beams, the HGBs of all orders have an optimum focusing condition for the highest conversion efficiency. On the contrary to previous reports, we observed that the propagation of the vortex beam of order, l, through a SPP corresponding to the vortex order of, -l, results in HGBs of the same order, jlj. [ABSTRACT FROM AUTHOR]

We report on a simple and compact experimental scheme to generate high-power, ultrafast, higher-order vortex-array beams. Simply by using a dielectric microlens-array (MLA) and a plano-convex lens, we have generated array-beams carrying the spatial property of the input beam. Considering the MLA as a 2D sinusoidal phase-grating, we have numerically calculated the intensity pattern of the array-beams in close agreement with the experimental results. Using vortex beams of order as high as l = 6, we have generated vortex array-beam with individual vortices of orders up to l = 6. We have also theoretically derived the parameters controlling the intensity pattern, size, and the array-pitch and verified experimentally. The single-pass frequency-doubling of vortex-array at 1064 nm in a 1.2 mm long BiBO crystal produced green vortex-array of order, lsh = 12, twice the order of pump beam. Using lenses of different focal lengths, we have observed the vortex-arrays of all orders to follow a focusing dependent conversion similar to the Gaussian beam. The maximum power of the green vortex-array is measured to be 138 mW at a single-pass efficiency as high as ~3.65%. This generic experimental scheme can be used to generate the array beams of desired spatial intensity profile across a wide wavelength range by simply changing the spatial profile of the input beam. [ABSTRACT FROM AUTHOR]