Orbital Angular Momentum Spectrum Measurement of Ultra-broadband Optical Vortices Based on Field Reconstruction

During the last decade, an optical vortex (OV) has widely drawn attention since it refers to a unique fact that the spatial phase of an electromagnetic wave changes linearly, at its cross-section, with the azimuthal angle φ around the center point, named singularity point where there is no field distribution [1]. This special property of phase profile is described by a factor of exp(ilφ), where l, called topological charge, can be any integer value and lħ indicates the orbital angular momentum (OAM) per photon, which provides a new degree of freedom in phase controlling. These characters of OVs have attracted enormous attention owing to its various applications in different fields such as optical tweezers and quantum communication. Since preceding research mainly concentrated on the unique field distribution of OVs, for many cases, they have simply used temporally-continuous OVs so far. In contrast, our group has recently demonstrated the generation of fewcycle ultrashort OV pulses for applications in spectroscopy and intense field [2]. The topological charge l of the generated OV pulses, however, has been investigated simply by counting pronged fork-dislocation lines in interferograms so far, and thus its purity and dispersion relation have not been quantitatively evaluated. For applications such as quantum information processing or nonlinear spectroscopy by OV pulses, measurement of OAM spectrum and evaluation of frequency-resolved OAM dispersion is essentially crucial. The commonly utilized method with computer-generated holograms (CHGs) on spatial light modulator (SLM) to obtain OAM spectrum for continuous OVs [3], however, is not suitable for ultra-broadband OV pulses because it inevitably causes angular dispersion by the diffraction effect of CHGs. On the other hand, although there has been a method determining topological charges of polychromatic OVs [4], it can only determine the dominant part of l and is consequently not capable of frequency-resolved measurement of OAM spectrum and detection of OAM dispersion of ultra-broadband OV pluses. In the present paper, we demonstrate a frequency-resolved measurement of OAM spectrum of ultra-broadband OV pulses. Our new method overcomes the problem of angular dispersion by using SLM and enables quantitative evaluation of topological charge distribution and its dispersion. The procedure of our measurement is shown in two principal steps. The first one is based on reconstruction of the field information of the OV pulses to be measured. After selecting certain wavelength out of the whole spectral region of the OV pulses and a tilted (quasi-) plane wave, they interfere with each other and form the interferogram, where the field information of the OV pulses at this wavelength is conserved. By Takeda’s method [5], the field distribution of the OV pulses can be reconstructed from this interferogram. Next, thanks to the Fourier-relationship between azimuthal angle φ and topological charge l, we are able to attain the OAM spectrum amplitude as a function of l and radial coordinate r by projecting the reconstructed field (in spatial coordinates: r and φ) onto OAM domain. Finally, the OAM power spectrum at this selected wavelength of the pulses is obtained by integrating total energy in each charge l, with respect to r. We demonstrate our method in resolving OAM spectra of both pure and mixed-type OV pulses at different wavelengths. Next, by acquiring frequency-resolved OAM power spectrum in a comparably large wavelength range, the OAM dispersion relation can be obtained as a function of charge l and wavelength. It is worth mentioning that, in particular, our method is capable of determining relative phase (including the positive/negative sign) between topological-chargeresolved electric-field amplitudes.