Generation, Transmission, Detection, and Application of Vortex Beams

Preface
Introduction
Contents
About the Author
1 Introduction
	1.1 Optical Vortices
	1.2 Orbital Angular Momentum
		1.2.1 Background and Meaning
		1.2.2 Principle of OAM Multiplexing Technology
		1.2.3 OAM Multiplexing Communication System Model
	1.3 Vortex-Beam Generation
		1.3.1 Space-Generation Methods
		1.3.2 Fiber-Generation Methods
		1.3.3 Comparison of Vortex-Beam Generation Methods
	1.4 Transmission Characteristics of Vortex Beams
		1.4.1 Atmospheric-Turbulence Effect
		1.4.2 Research Methods of Beam-Propagation Characteristics
		1.4.3 Transmission Characteristics of Vortex Beams
	1.5 Transmission Characteristics of Vortex Beams
		1.5.1 Traditional Adaptive-Optics Correction Technology
		1.5.2 AO Correction Without a Wavefront Sensor
		1.5.3 Vortex Beam Phase Distortion Correction
	1.6 Separation and Detection of Vortex Beams
		1.6.1 Fork Grating
		1.6.2 Interference Characteristics
		1.6.3 Diffraction Characteristics
		1.6.4 Reconstructed Wavefront
	References
2 Vortex-Beam Spatial-Generation Method
	2.1 Basic Principle of Vortex Beams
	2.2 Types of Vortex Beams
		2.2.1 Laguerre–Gaussian Beams
		2.2.2 Bessel Beams
		2.2.3 Hermite–Gaussian Beams
	2.3 Vortex-Beam Generation Methods
		2.3.1 Computer-Generated Hologram Method
		2.3.2 Mode-Conversion Method
		2.3.3 Spiral-Phase-Plate Method
		2.3.4 Spatial Light Modulator Method
		2.3.5 Optical-Waveguide Device-Conversion Method
	2.4 Higher-Order Radial LG Beams
	2.5 Generation of Fractional Vortex Beams
		2.5.1 Principle of LG-Beam Preparation by the Holographic Method
		2.5.2 Experimental Study on the Orbital Angular Momentum of Fractional Laguerre–Gaussian Beams
	References
3 Vortex-Beam Generation Using the Optical-Fiber Method
	3.1 Introduction
	3.2 Optical-Fiber Mode Theory
		3.2.1 Wave Equation
		3.2.2 Vector Modes in Optical Fiber
		3.2.3 Guide-Mode Cutoff and Distance Cutoff
		3.2.4 Scalar Modulus Under a Weakly-Conducting Approximation
		3.2.5 Analysis of the Principle of Using Optical Fiber to Generate Vortex Light
	3.3 Analysis of the Influencing Factors of a Vortex Light Generated by Optical Fiber
		3.3.1 Influence of the Incident Wavelength on the Vortex Light
		3.3.2 Influence of the Refractive-Index Difference Between the Inside and Outside of the Optical Fiber on the Vortex Light
		3.3.3 Influence of the Fiber-Core Radius on the Vortex Light
		3.3.4 Effect of the Incident Angle on the Excitation Efficiency of Vortex Light
		3.3.5 Effect of Off-Axis Incident Fiber on Vortex Light
	3.4 Experiments Using Few-Mode Fibers to Generate Vortex Light
		3.4.1 Principle of Using Few-Mode Fibers to Generate Vortex Light
		3.4.2 Analysis of the Excitation Efficiency of Vortex Light
		3.4.3 Experimental Research
		3.4.4 Phase Verification
	3.5 Changing the Fiber Structure to Produce Vortex Light
		3.5.1 Structural Design
		3.5.2 Influence of the Low-Refractive-Index Layer on OAM Mode
	References
4 Superposition Characteristics of High-Order Radial Laguerre–Gaussian Beams
	4.1 Introduction
	4.2 Influence of the Radial Index on the Superposition State of High-Order Radial LG Beams
		4.2.1 Interference and Superposition of LG Beams with the Same Topological Charge
		4.2.2 Interference and Superposition of LG Beams with the Same Radial Index
		4.2.3 Interference Superposition of LG Beams with Arbitrary Radial Indices and Topological Charges
	4.3 Influence of the Transmission Distance on the Superposition State of High-Order Radial LG Beams
	4.4 Influence of the Beam-Waist Radius on the Superposition State of High-Order Radial LG Beams
	4.5 Effect of Off-Axis Parameters on the Superposition State of High-Order Radial LG Beams
	4.6 Experiment on a Superimposed High-Order Radial LG Beam
		4.6.1 Experimental Device
		4.6.2 Hologram Production
		4.6.3 Analysis of Results
	References
5 Transmission Characteristics of Vortex Beams
	5.1 Introduction
	5.2 Transmission of LG Beams in Atmospheric Turbulence
		5.2.1 Theoretical Analysis
		5.2.2 Propagation Characteristics of an LG Beam Passing Through a Slanted Channel of Atmospheric Turbulence
	5.3 Transmission of Bessel–Gaussian Beams in Space
		5.3.1 Theory of BG-Beam Propagation in Turbulence
		5.3.2 Characteristics of a BG Beam Passing Through an Atmospheric-Turbulence Channel
	5.4 Research on the Stability of Orbital Angular Momentum in the Slanted Propagation of a Vortex Beam
		5.4.1 Comparison of the Light-Intensity Distribution of Vortex Beams
		5.4.2 Comparison of the Harmonic Components of a Vortex Beam
	References
6 Adaptive-Optics Correction Technology
	6.1 Introduction
	6.2 Basic Adaptive-Optics Principles
		6.2.1 Adaptive-Optics Correction Technology
		6.2.2 Shack–Hartmann Algorithm
		6.2.3 Phase-Recovery Algorithm
		6.2.4 Stochastic Parallel Gradient-Descent Algorithm
		6.2.5 Phase-Difference Algorithm
	6.3 Wavefront Correction of an OAM Beam After Passing Through Atmospheric Turbulence
		6.3.1 Phase-Recovery Algorithm
		6.3.2 Stochastic Parallel Gradient-Descent Algorithm
		6.3.3 Phase-Difference Algorithm
	6.4 Experimental Research
		6.4.1 Phase-Recovery Algorithm
		6.4.2 Stochastic Parallel Gradient-Descent Algorithm
		6.4.3 Phase-Difference Algorithm
	References
7 Crosstalk Analysis of an OAM-Multiplexing System Under Atmospheric Turbulence
	7.1 Introduction
	7.2 Propagation Theory of OAM Beams in Atmospheric Turbulence
		7.2.1 Multiphase-Screen Transmission Method
		7.2.2 Random Phase-Screen Generation
		7.2.3 Crosstalk of an OAM-Multiplexed Beam in Atmospheric Turbulence
	7.3 Analysis of the Intensity Phase of an OAM-Multiplexed Beam in Atmospheric Turbulence
		7.3.1 Formation of an OAM-Multiplexed Beam
		7.3.2 Influence of Different Transmission Conditions on the Light Intensity and Phase
	7.4 Spiral-Spectrum Characteristics of OAM-Multiplexed Beams Under Atmospheric Turbulence
		7.4.1 OAM-Multiplexed Beam Spiral-Spectrum Theory
		7.4.2 Spiral-Spectrum Analysis Under Different Transmission Conditions
	7.5 Analysis of the Bit Error Rate of an OAM-Multiplexed Beam Under Atmospheric Turbulence
		7.5.1 Bit-Error-Rate Theory of OAM-Multiplexed Beams
		7.5.2 Analysis of the Bit Error Rate Under Different Transmission Conditions
	7.6 Experiment on the Influence of Atmospheric Turbulence on an OAM-Multiplexed Beam
		7.6.1 Principle of the Experiment
		7.6.2 Analysis of Results
	References
8 Properties of a Superimposed Vortex Beam
	8.1 Introduction
	8.2 Fabrication of the Vortex-Beam Superposition State Using a Grating Method
		8.2.1 Theoretical Analysis
		8.2.2 Grating Superposition
	8.3 Double OAM Beam Prepared Using the Phase-Superposition Method
		8.3.1 Theoretical Analysis
		8.3.2 Characteristic Analysis of Superimposed Vortex Beams with Different Topological Charges
	8.4 Vortex-Beam Superposition Interference Experiment
		8.4.1 Experimental Design
		8.4.2 Experiment of the Grating-Superposition Method
		8.4.3 Result Analysis of the Grating-Superposition Method
		8.4.4 Experiment on the Phase-Superposition Method
		8.4.5 Results and Analysis of the Phase-Superposition Method
	References
9 Vortex-Beam Detection
	9.1 Introduction
	9.2 Separate Detection of OAM States Using the Coordinate-Transformation Method
		9.2.1 Theoretical Foundations
		9.2.2 Superimposed Light-Field Distribution for Different Topological-Charge Numbers
		9.2.3 OAM-State Multiplexing System Based on the Coordinate-Transformation Method
	9.3 Using a Grating to Detect the Angular Momentum of a Vortex-Light Orbit
		9.3.1 Transmission Function of a Grating and Its Representation
		9.3.2 Vortex Light Field and Its Diffraction
		9.3.3 Phase Correction and the Fan-Out Technique
		9.3.4 Periodic-Gradient Grating
	9.4 Using Interferometry to Detect the Vortex-Light Phase
		9.4.1 Vortex Self-Interference Detection Method
	9.5 Measuring the Vortex Optical Phase Using Diffraction Methods
		9.5.1 Triangle Diffraction Method
		9.5.2 Square-Hole Diffraction Method
		9.5.3 Single-Slit-Diffraction Detection Method
		9.5.4 Circular-Hole-Diffraction Detection Method
	9.6 Summary
	References
10 Diffraction Characteristics of a Vortex Beam Passing Through an Optical System
	10.1 Diffraction Model of a Vortex Beam Passing Through a Mak-Cass Antenna
		10.1.1 Mak-Cass Antenna Structure
		10.1.2 Mak-Cass-Antenna Diffraction Model
	10.2 Analysis of the Diffraction Characteristics of a Vortex Beam Passing Through a Mak-Cass Antenna Optical System
		10.2.1 Diffracted Light-Field Model
		10.2.2 Diffraction Spots and Phase Distribution
		10.2.3 Spiral-Spectrum Distribution
		10.2.4 Transmission Efficiency of the Mak-Cass Antenna
	10.3 Analysis of the Diffraction Characteristics of a Vortex Beam Passing Through an Aperture Diaphragm
		10.3.1 Theoretical Model of Aperture-Diaphragm Diffraction
		10.3.2 Theoretical Diffraction Analysis of a Vortex Beam Passing Through a Diaphragm
		10.3.3 Analysis of the Experimental Diffraction Pattern of a Vortex Beam Passing Through an Aperture
		10.3.4 Aperture-Diaphragm Detection-Effect Comparison
	10.4 Summary
	References
11 Propagation Characteristics of a Partially Coherent Vortex-Beam Array in Atmospheric Turbulence
	11.1 Beam-Array Overview
	11.2 Intensity Distribution of a Radial Partially Coherent Vortex-Beam Array in Atmospheric Turbulence
		11.2.1 Mathematical Model of a Radial Partially Coherent Vortex-Beam Array
		11.2.2 Cross-Spectral Density Function on an Observation Plane
		11.2.3 Expression of the Light Intensity on the Observation Plane
		11.2.4 Effect of Light-Source Parameters on the Light-Intensity Characteristics in Non-Kolmogorov Turbulence
		11.2.5 Influence Analysis of Radial-Array Parameters
		11.2.6 Influence Analysis of Single Partially Coherent Vortex-Beam Parameters
	11.3 Influence of Non-Kolmogorov Turbulence Parameters on Light-Intensity Characteristics
		11.3.1 Impact Analysis of Non-Kolmogorov Turbulence Intensity
		11.3.2 Non-Kolmogorov Turbulence Internal- and External-Scale Influence Analysis
	11.4 Summary
	References
12 Propagation Characteristics of Scalar Partially Coherent Vortex Beams in Atmospheric Turbulence
	12.1 Basic Theory of Laguerre–Gaussian–Schell-Mode Vortex Beams
		12.1.1 Laguerre–Gaussian–Schell Beams
		12.1.2 Laguerre–Gaussian–Schell Vortex-Beam Model
		12.1.3 Propagation Theory of a Laguerre–Gaussian–Schell Vortex Beam in Atmospheric Turbulence
	12.2 Phase-Singularity Evolution of Far-Field Laguerre–Gaussian–Schell Vortex Beams
		12.2.1 Relationship Between a Phase Singularity and the Topological Charge
		12.2.2 Effect of the Transmission Distance on the Phase-Singularity Evolution
		12.2.3 Effect of the Correlation Length on the Phase-Singularity Evolution
	12.3 Intensity Distribution of a Laguerre–Gaussian–Schell Vortex Beam in Atmospheric Turbulence
		12.3.1 Effect of the Atmospheric-Turbulence Intensity on the Light-Intensity Distribution
		12.3.2 Influence of the Internal and External Atmospheric-Turbulence Scales on the Light-Intensity Distribution
	12.4 Beam Propagation of a Laguerre–Gaussian–Schell Vortex Beam in Atmospheric Turbulence
		12.4.1 Analysis of Variation of a Beam Spread with Light-Source Parameters
		12.4.2 Analysis of the Beam Spread with Atmospheric-Turbulence Intensity
	References
13 Propagation Characteristics of Partially Coherent Vector Vortex Beams in Atmospheric Turbulence
	13.1 Polarization Theory of Partially Coherent Vector Beams
	13.2 Cross-Spectral Density Matrix of Partially Coherent Vector Vortex Beams in Atmospheric Turbulence
		13.2.1 Intensity and Degree of Polarization
		13.2.2 Polarization-Direction Angle
	13.3 Polarization-Distribution Degree of Partially Coherent Vector Vortex Beams in Atmospheric Turbulence
		13.3.1 Influence of the Light-Source Parameters on the Degree of Polarization
		13.3.2 Influence of Atmospheric Turbulence on the Polarization Degree
		13.3.3 Variation of the Polarization Degree with the Transmission Distance
	13.4 Polarization-Direction-Angle Distribution of Partially Coherent Vector Vortex Beams in Atmospheric Turbulence
		13.4.1 Influence of Atmospheric Turbulence on the Polarization-Direction Angle
		13.4.2 Influence of the Transmission Distance on the Polarization-Direction Angle
		13.4.3 Polarization-Direction Angle Detection of the Topological Charge
		13.4.4 Polarization-Direction-Angle Model of a Far-Field Diffracted Light Field
		13.4.5 Results of Using the Polarization-Direction Angle to Detect the Topological Charge
		13.4.6 Analysis of the Influence of the Light-Source Parameters on the Detection Effect
	13.5 Summary
	References
14 Vortex-Beam Information Exchange
	14.1 Flexibility of an OAM Vortex-Beam Topological Charge
		14.1.1 Conversion of a Single OAM Beam
		14.1.2 Conversion of an OAM-Multiplexed Beam
	14.2 Principle of OAM Vortex-Beam Channel Reconstruction
		14.2.1 OAM-Beam Information Exchange
		14.2.2 OAM-Beam Mode Switch
	14.3 Demultiplexing an OAM-Multiplexed Vortex Beam
	14.4 Experimental Research on the Channel Reconstruction of OAM Vortex Beams
		14.4.1 Experimental Research on OAM Information Exchange
		14.4.2 Experimental Research on Exchanging Two Information Beams Among Three OAM-Multiplexed Beams
		14.4.3 Experimental Research on the Mode Conversion of One Beam in Multiple OAM-Multiplexed Beams
		14.4.4 Experimental Research on Exchanging Two OAM Multiplexed Beams with the Same Mode and Different Information
		14.4.5 Experimental Research on Deleting/Adding a Beam Pattern in Multiple OAM-Multiplexed Beams
	References