Identification Quantum Number Of Orbital Angular Momentum And Radial Modes Of Laguerre-Gaussian Beams

It is well known that photons carry both spin and orbital angular momentum.The spin angular moment is associated with circular polarization,which has only two values of σh(σ=±1).Therefore the polarization of a photon can be described by a point in a two-dimensional state space.In contrast to spin angular momentum,orbital angular momentum(lh)is associated with the transverse structure of a photon and can offer an unbounded Hilbert space due to the unlimited values of l∈ {0,±1,±2,…}.A photon can carry an orbital angular momentum of lh and an azimuthal phase exp(ilφ).A light beam with orbital angular momentum is called the vortex beam.Orbital angular momentum of photos characterizes both high dimensions and optics vortices,and shows a great of potential applications in quantum and classical fields,and attracts a lot of attention,including high-capacity optical communications,quantum computation,quantum cryptography,optical microscopy and micromanipulation.The most common form of helical phase structure is Laguerre-Gaussian modes.LaguerreGaussian modes are solutions of the paraxial wave equation in a cylindrical coordinate and characterized by two modes indices,the radial mode index p ∈{0,1,2,…} and the azimuthal mode index l∈ {0,±1,±2,…}.The Laguerre-Gaussian modes constitute a complete basis set for representing the transverse structure of a paraxial photon field in free space.Now interest in Laguerre-Gaussian mode has spread to many fields,such as quantum key distribution and high-dimensional entanglement.Thus the need to develop sorters of Laguerre-Gaussian modes is becoming increasing important.This thesis mainly focuses on the identification quantum number of orbital angular momentum and radial modes of Laguerre-Gaussian beams.First,a brief introduction is made to describe the theoretical background of Laguerre-Gaussian modes.Second,we give a review about methods of detecting the orbital angular momentum of Laguerre-Gaussian modes in optics systems.Third,We proposed several methods to measure Laguerre-Gaussian modes according to the orbital angular momentum quantum number l and the radial quantum number p.The main contents and innovations of this thesis are summarized as follows:1.Based on the interference and diffraction,we propose three methods to detect the orbital angular momentum of a vortex beam.1)When a vortex beam with a spiral phase structure passes through dynamic angular double slits,the interference pattern changes alternatively between destructive and constructive at the angular bisector of the angular double slits.This change is due to their phase difference.Based on this property,we experimentally demonstrate a simple method to precisely and efficiently determine the orbital angular momentum of vortex beams.Furthermore,this scheme allows for the simultaneous determination of the modulus and the sign of the orbital angular momentum quantum number l of vortex beams.The work is detailed in the second chapter of the thesis.2)We study the diffraction pattern in the far field when a vortex beam passes through an arc slit and demonstrate experimentally that a light spot of the diffraction pattern has a displacement which is linear to the orbital angular momentum of the incident vortex beam.Based on this property,this method is capable of measuring both modulus and sign of the orbital angular momentum quantum number l of the vortex beam.Furthermore,this scheme allows identifying multiple orbital angular momentum states simultaneously.The work is detailed in the second chapter of the thesis.3)We present an interferometer method in which a Sagnac interferometer with a Dove prism is placed on each arm to separate the different orbital angular momentum of photons into different output ports,namely,orbital angular momentum sorters.We demonstrate experimentally the feasibility of orbital angular momentum sorter by dividing the photons into five different orbital angular momentum states.Using the cascade interferometers,we also sort the superposition state successfully.Experimental results are in good agreement with the theoretical predictions.Compared with other methods,this method is more stable and can be used at the single-photon level.Furthermore,this method can also be used to couple orbital angular momentum modes with spatial modes,a very important method for manipulating orbital angular momentum states.The work is detailed in the second chapter of the thesis.2.First,we present a theoretical model to sufficiently investigate the optical rotational Doppler effect based on modal expansion method.We find that the frequency shift content is only determined by the surface of spinning object and the reduced Doppler shift is linear to the difference of mode index between input and output orbital angular momentum light,and linear to the rotating speed of spinning object as well.An experiment is carried out to verify the theoretical model.We explicitly suggest that the spatial spiral phase distribution of spinning object determines the frequency content.Based on this theoretical model,we propose an orbital angular momentum complex spectrum analyzer that enables simultaneous measurements of the power and phase distributions of orbital angular momentum modes by employing the rotational Doppler effect.The original orbital angular momentum mode distribution is mapped to an electrical spectrum of beat signals using a photodetector.The power and phase distributions of superimposed orbital angular momentum beams are successfully retrieved by analyzing the electrical spectrum.We also extend the measurement technique to other spatial modes,such as linear polarization modes.These results represent a new landmark in spatial mode analysis.The work is detailed in the third chapter of the thesis.3.We propose and demonstrate a radial mode sorter based on the fractional Fourier transform to efficiently decompose the Laguerre-Gaussian mode according to its radial index p.We experimentally characterize the performance of our implementation by separating individual radial modes as well as superposition states.The reported scheme can,in principle,achieve unit efficiency and thus can be suitable for applications that involve quantum states of light.Then we propose and experimentally demonstrate a scheme to accomplish complete LaguerreGaussian mode sorting,which consists of a novel,robust radial mode sorter that can be used to couple radial modes to polarizations,an l-dependent phase shifter and an orbital angular momentum mode sorter.The l-dependent phase shifter is a building block to connect the radial mode sorter and the subsequent orbital angular momentum mode sorter.The work is detailed in the fourth chapter of the thesis.