UNIVERSITY OF THE WITWATERSRAND MASTERS DISSERTATION Skyrmions and vectorial wavefunctions Author: Pedro ORNELAS Supervisors: Prof. Andrew FORBES A dissertation submitted in fulfillment of the requirements for the degree of Master of Science in the Structured Light Laboratory School of Physics April 29, 2024 http://www.wits.ac.za https://za.linkedin.com/in/keshaan-singh-a7a6ab203 http://www.structured-light.org https://structured-light.org/ https://www.wits.ac.za/physics/ iii Declaration of Authorship I, Pedro ORNELAS, declare that this thesis titled, “Skyrmions and vectorial wave- functions” and the work presented in it are my own. I confirm that: • This work was done wholly or mainly while in candidature for a research de- gree at this University. • Where any part of this thesis has previously been submitted for a degree or any other qualification at this University or any other institution, this has been clearly stated. • Where I have consulted the published work of others, this is always clearly attributed. • Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely my own work. • I have acknowledged all main sources of help. • Where the thesis is based on work done by myself jointly with others, I have made clear exactly what was done by others and what I have contributed my- self. Signed: Date: 2024/04/29 v vii UNIVERSITY OF THE WITWATERSRAND Abstract Master of Science Skyrmions and vectorial wavefunctions by Pedro ORNELAS The study and generation of robust structured light stand as compelling areas of fo- cus in the exploration of future classical and quantum photonic technologies. While the appeal of structuring light in all its degrees of freedom (DOFs) is undeniable, achieving the generation of intricate light resilient to noise from multiple sources, such as faulty detectors, stray white light, and atmospheric turbulence, is impera- tive for its practical integration into forthcoming technologies. Recently, there has been a lot of interest in generating states of light with identi- fiable topological features which are robust to local deformations thus providing such states with a possible mechanism for noise rejection. Topological structures known as optical skyrmions have garnered a lot of interest in the optics commu- nity of late as their magnetic counterparts have shown great promise as potential low-power information carriers. It has been shown that skyrmionic structures may be realised in classical free-space optical beams where their spatial and polarization DOFs are appropriately combined and manipulated to generate what are known as vector beams. Furthermore with the emergence of quantum structured light allow- ing for the manipulation of an individual photon’s DOFs, such topological structures may also be utilized as a resource for photonic based quantum technologies. In this dissertation we investigate the generation of classical optical skyrmions through the use of Bessel-Gaussian optical modes possessing interesting propagation dy- namics which mimic magnetic systems under the application of a magnetic field. Furthermore, we extend the study of optical skyrmions to the quantum realm by generating and characterizing the topology of the quantum analogue to classical vector beams: hybrid entangled states where the spatial DOF of one photon is en- tangled with the polarization DOF of another. In this case the skyrmionic topology emerges as a shared property of both photons and can be identified through inves- tigating their mutual correlations. We postulate a novel topological characterization of entangled states with the corollary that smooth deformations of these states do not change their topology and thus do not change how they are characterized. We show that the topology remains intact even when entanglement is fragile and fur- ther discuss how a typical mechanism for entanglement decay can be characterized as a smooth deformation. Lastly, we investigate the topological resilience of hy- brid entangled states in the presence of isotropic noise usually attributed to external sources. We demonstrate the invariance of the topology of these states to varying levels of isotropic noise and discuss the associated mechanism for this invariance. HTTP://WWW.WITS.AC.ZA ix Acknowledgement I would like to acknowledge and thank the following people, without whom this degree would not have been possible. To my supervisor, Andrew Forbes. In the brief period during which I’ve had the privilege of being your student, I’ve acquired a wealth of knowledge. Your unwa- vering work ethic and passion for groundbreaking science have undeniably trans- formed you into an inspiration and a joy to work with. Beyond the realm of academia, you’ve succeeded in cultivating an exceptional work environment, assembling a team of warm and enthusiastic individuals who have become more than just col- leagues—they’ve become family. I am genuinely excited about the prospect of con- tinuing our work together and furthering my learning journey under your guidance. To my mentors, Keshaan Singh, Isaac Nape and Robert De Mello Koch. Each of you has played a pivotal role in shaping my journey toward success, providing in- valuable advice, and imparting essential theoretical and experimental knowledge. A heartfelt thank you to Keshaan, my first mentor, for guiding my initiation into the scientific realm. Prof. De Mello Koch, your unwavering willingness to dedicate time to address my myriad of questions and embrace even my most naive ideas is truly appreciated. And to Isaac, a remarkable friend and mentor during my transition into quantum optics, your support has been immeasurable. You serve as a genuine inspiration for aspiring young scientists, and I aspire to emulate your success. To my friends, Cade Peters, Declan Mahony, Leerin Perumal and Reuben Neate. To Reuben, I extend my heartfelt gratitude for your friendship spanning the last 15 years. Your influence not only helped to propel me into a career in science but has also played a crucial role in my personal growth. Declan, your unwavering support throughout my academic journey has been truly indispensable. Your compassion has made a lasting impact on me. Cade and Leerin, I feel incredibly fortunate to have shared this academic journey with you. From the late nights to early morn- ings, navigating the highs and lows, I couldn’t have asked for better companions on this journey. To my lab and colleagues, Neelan Gounden, Light Mkumbuza, Mwezi Koni, Chané Moodley, Bereneice Sephton, Wagner Tavares Buono, Michael De Oliveira. Each one of you brings joy into my every day, making it an absolute pleasure to step into the office. I am grateful for the continuous support and the wonderful times we have shared and will share in the future. Thank you To the my family, Armando Ornelas, Ana Paula Ornelas, Victor Ornelas, Marta Or- nelas. Expressing my deepest gratitude to my parents, Armando and Ana Paula, whose unwavering support has been the bedrock of all my achievements. Your ef- forts in creating a loving home have made me feel constantly loved and valued. A special thanks to my older siblings, Marta and Victor, for their steadfast support in x every endeavor—both financially and emotionally. Your own remarkable achieve- ments, both socially and academically, have served as inspirations in my pursuit of further studies. Your enduring presence has been a consistent source of strength, and I am truly grateful to have you as pillars of support that I can always count on. Last but not least, I would like to thank the University of the Witwatersrand, the joint Council of Scientific and Industrial Research (CSIR) and Department of Science and Technology (DST)-Interbursary Support (IBS) financial assistance. xi Contents Declaration of Authorship iii Abstract vii Acknowledgement ix 1 Introduction 1 1.1 Structured light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.1 Polarization and spin angular momentum of light . . . . . . . . 2 1.1.2 Spatial distribution and orbital angular momentum of light . . 3 Free-space mode families . . . . . . . . . . . . . . . . . . . . . . 5 1.1.3 Scalar vs vector beams . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 Skyrmions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.1 Skyrmion mapping . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.2 Realizations of optical skyrmions . . . . . . . . . . . . . . . . . . 14 1.2.3 Constructing optical skyrmions with vector beams . . . . . . . 15 1.3 Quantum structured light and vectorial wavefunctions . . . . . . . . . 18 1.3.1 Structuring single photons . . . . . . . . . . . . . . . . . . . . . 19 1.3.2 Entanglement of photons . . . . . . . . . . . . . . . . . . . . . . 20 High-dimensional entangled states . . . . . . . . . . . . . . . . . 21 Hybrid entanglement and vectorial wavefunctions . . . . . . . 22 1.3.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2 Experimental Techniques 25 2.1 Structured light techniques . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.1.1 Amplitude and phase modulation . . . . . . . . . . . . . . . . . 25 2.1.2 Polarization control . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.1.3 Vector beam generation . . . . . . . . . . . . . . . . . . . . . . . 31 Digital devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Geometric optics . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.1.4 Vector beam characterization . . . . . . . . . . . . . . . . . . . . 35 2.1.5 Vector beam setup . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.2 Quantum structured light techniques . . . . . . . . . . . . . . . . . . . . 40 2.2.1 Entanglement through SPDC . . . . . . . . . . . . . . . . . . . . 40 Spatial mode entanglement from SPDC . . . . . . . . . . . . . . 42 Optimising the spiral bandwidth . . . . . . . . . . . . . . . . . . 44 OAM detection of single photons . . . . . . . . . . . . . . . . . . 45 2.2.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Spatial-to-Polarization Conversion . . . . . . . . . . . . . . . . . 47 2.2.3 Tools for quantum state characterization . . . . . . . . . . . . . 49 Quantum State Tomography . . . . . . . . . . . . . . . . . . . . 49 Concurrence, fidelity and purity . . . . . . . . . . . . . . . . . . 52 2.2.4 Quantum stokes . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 xii 2.2.5 Bell measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3 Synthetic spin dynamics of Bessel Skyrmions 57 3.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.2 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4 Non-local skyrmions as topologically resilient quantum entangled states of light 67 4.1 Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.2 Creating and detecting quantum skyrmions . . . . . . . . . . . . . . . . 70 4.3 Topology and entanglement . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.4 Topological resilience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.5 Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 81 5 Topological rejection of noise by non-local quantum skyrmions 83 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.3 Experimental results and discussion . . . . . . . . . . . . . . . . . . . . 87 5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 6 Conclusion 91 A Stereographic Projection 93 B Topological resilience against quantum Gouy phase 95 C Additional Data Sets for "Non-local skyrmions as topologically resilient quantum entangled states of light" 99 C.1 Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Bibliography 103 xiii List of Figures 1.1 Polarization DOF of light . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Intensity and phase (insets) of Hermite-Gaussian modes . . . . . . . . 6 1.3 Intensity and phase (insets) of Laguerre-Gaussian modes . . . . . . . . 7 1.4 Intensity and phase (insets) of Bessel-Gaussian . . . . . . . . . . . . . . 8 1.5 Vector Beams vs scalar beams . . . . . . . . . . . . . . . . . . . . . . . . 10 1.6 Creation of VBs through the superpositions of orthogonally polarized LG modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.7 Skyrmion mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.8 Skyrmions of varying texture and topology . . . . . . . . . . . . . . . . 14 1.9 Skyrmionic mapping with LG modes . . . . . . . . . . . . . . . . . . . . 16 1.10 Diverse skyrmionic topologies using VBs . . . . . . . . . . . . . . . . . 18 1.11 Structuring single photons’ in spatial and polarization DOFs . . . . . . 19 2.1 Working illustration of the SLM display . . . . . . . . . . . . . . . . . . 26 2.2 Illustration of the different modulation techniques . . . . . . . . . . . . 28 2.3 Experimental intensity images of scalar modes . . . . . . . . . . . . . . 29 2.4 Polarization optics used for complete control over the polarization state of incident light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.5 Concept for generation of arbitrary vector modes using SLMs . . . . . 32 2.6 The concept of geometric phase . . . . . . . . . . . . . . . . . . . . . . . 33 2.7 Vector beam (VB) generation using geometric optics . . . . . . . . . . . 35 2.8 Setup for the generation and detection of vector beams . . . . . . . . . 38 2.9 Experimental results for VB generation . . . . . . . . . . . . . . . . . . 39 2.10 Spontaneous parametric down-conversion . . . . . . . . . . . . . . . . 41 2.11 Types of nonlinear crystals with different phase-matching conditions . 42 2.12 OAM detection scheme for high dimensional states . . . . . . . . . . . 46 2.13 Measurement outcomes for simultaneous projective measurements on OAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.14 Experiment for generation of Hybrid entangled state . . . . . . . . . . 48 2.15 Conceptual idea of tomography with a 3D object . . . . . . . . . . . . . 50 2.16 QST and density matrix reconstruction . . . . . . . . . . . . . . . . . . . 52 2.17 Experimental Quantum Stokes parameters . . . . . . . . . . . . . . . . 55 2.18 Experimental Bell measurement curves . . . . . . . . . . . . . . . . . . 56 3.1 Construction of BG skyrmions and observation of their propagation dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.2 Experimental diagram for BGℓ based optical skyrmions . . . . . . . . . 63 3.3 Plot showing the experimentally measured Skyrme number . . . . . . 65 3.4 Path traced by Stokes vector during propagation . . . . . . . . . . . . . 66 4.1 Non-local quantum skyrmions . . . . . . . . . . . . . . . . . . . . . . . 69 4.2 Experimental quantum skyrmion . . . . . . . . . . . . . . . . . . . . . . 70 4.3 Traversing the quantum skyrmionic landscape . . . . . . . . . . . . . . 73 xiv 4.4 Topology of quantum entangled states . . . . . . . . . . . . . . . . . . . 74 4.5 Quantum topological invariance to entanglement decay . . . . . . . . . 76 4.6 The origin of topologically protected wavefunctions . . . . . . . . . . . 78 4.7 Entanglement Decay as smooth deformation . . . . . . . . . . . . . . . 80 4.8 Entanglement Decay as smooth deformation . . . . . . . . . . . . . . . 80 5.1 Conceptual image of non-local skyrmion in the presence of isotropic noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.2 Skyrmionic mappings in the presence of isotropic noise . . . . . . . . . 88 5.3 Measured Skyrmion number in the presence of isotropic noise . . . . . 89 A.1 Stereographic projection of entangled state onto state space . . . . . . . 94 B.1 Relative phase as a smooth deformation . . . . . . . . . . . . . . . . . . 97 C.1 Reconstructed Density Matrices . . . . . . . . . . . . . . . . . . . . . . . 100 C.2 Reconstructed Quantum Stokes . . . . . . . . . . . . . . . . . . . . . . . 101 C.3 Reconstructed state topologies . . . . . . . . . . . . . . . . . . . . . . . 102 xv List of Tables 1.1 Projective measurement outcomes for separable and non-separable states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1 A summary of useful polarization transformations . . . . . . . . . . . . 30 C.1 Table of experimental data for generated non-local skyrmions . . . . . 99 xvii List of Publications Journal Papers 1. Singh, K., Ornelas, P., Dudley, A., and Forbes, A. "Synthetic spin dynam- ics with Bessel-Gaussian optical skyrmions". Optics Express 31, 15289–15300 (2023). 2. Nape, I., Sephton, B., Ornelas, P., Moodley, C. and Forbes, A. Quantum struc- tured light in high dimensions. APL Photonics 8 (2023). 3. Peters, C., Ornelas, P., Nape, I. and Forbes, A. Spatially resolving classical and quantum entanglement with structured photons. Physical Review A 108, 053502 (2023). 4. Gounden, N., Epstein, J., Ornelas, P., Beck, G., Nape, I. and Forbes, A., "Pop- per’s conjecture with angular slits and twisted light." Scientific Reports, 13(1), p.21701 (2023). 5. Ornelas, P., Nape, I., de Mello Koch, R. and Forbes, A. Non-local skyrmions as topologically resilient quantum entangled states of light. Nature Photonics, 1–9 (2024). International Conferences 1. Ornelas, P., de Mello Koch, R., Forbes, A., "Embedding the skyrmionic topol- ogy into the stokes field”, 3rd International Conference on Light and Light- Based Technologies, Talk, 2022. 2. Ornelas, P., Nape, I. and Forbes, A., "Quantum entangled skyrmions and their quantum topology". In Complex Light and Optical Forces XVII (p. PC124360J). SPIE, Talk, 2023. 3. Ornelas, P., Singh, K. and Forbes, A., "Evolution of Bessel skyrmions in free- space". In Complex Light and Optical Forces XVII (p. PC1243605). SPIE, Talk, 2023. 4. Ornelas, P., Nape, I., de Mello Koch, R., and Forbes, A., "Revealing the Skyrmionic Topology of Entanglement”. International Symposium on Plasmonics and Nanopho- tonics, Talk, 2023. National Conferences 1. Ornelas, P., de Mello Koch, R., Forbes, A., "Links and Twists within the Stokes Field". 66th Annual Conference of the South African Institute of Physics, Talk, 2022. 2. Ornelas, P., Buono, W., Forbes, A., "Modal Description of Optical Elements". 66th Annual Conference of the South African Institute of Physics, Talk, 2022. xviii 3. Ornelas, P., Nape, I., de Oliveira, A., Souto Ribeiro, P., Forbes, A., "Multi- channel, turbulence resistant Quantum Key Distribution". 66th Annual Con- ference of the South African Institute of Physics, Poster, 2022. 4. Ornelas, P., Nape, I., de Mello Koch, R., and Forbes, A., "Beyond doughnuts and mugs, tying photons together". 67th Annual Conference of the South African Institute of Physics, Talk, 2023. 5. Ornelas, P., Nape, I., de Mello Koch, R., and Forbes, A., "Non-local topological rejection of noise". 67th Annual Conference of the South African Institute of Physics, Talk, 2023, 6. Ornelas, P., Singh, K. and Forbes, A., "Emulating magnetic skyrmions with light". 67th Annual Conference of the South African Institute of Physics, Talk, 2023. 1 Chapter 1 Introduction Structured light being the manipulation of light in all its degrees of freedom (DOFs) [1], is an emerging field that promises many technological and scientific advance- ments such as providing higher speed optical communications [2–6], enabling the control and trapping of particles through the use of optical tweezers [7, 8], high reso- lution microscopy and imaging [9–11], topography measurements [12, 13], detection of motion for optical metrology [14, 15] and many more. Beyond classical sources of light, structured light has also been employed to advance the utility of single pho- tons to produce complex, high dimensional quantum states [16–23] suitable for use in future quantum technologies such as quantum cryptography [24–30], quantum communication [31–34], quantum computing [35, 36] and quantum metrology [37, 38] to name just a few. Beyond the significant utility offered by way of increased dimensionality and com- plexity, structured light also allows for the creation of robust states of light allowing for their feasible implementation in noisy environments. Approaches to develop these robust states range from increasing their tolerance to noise, which as an exam- ple can be done through an increase in dimensionality [39, 40], or by engineering and identifying state properties which are not altered by the presence of noise in the sys- tem [41–44]. The latter scheme is typically desired as it allows for reliable informa- tion encoding into invariant state properties. Topology is one such property which is invariant to local perturbations and has been investigated in optics by way of em- bedding topological structures such as loops, links and knots into different DOFs of free-space optical beams [45–48]. Recent observations of their resilience against local phase abberations and system misalignments [49] have prompted further in- vestigations into practical encoding schemes using their associated topological in- variants [50–52]. Other more recent topological candidates known as skyrmions [53] have gathered much interest in the field of optics [54–58] since their magnetic counterparts have shown great promise as potential low-power information carri- ers [59, 60]. The combination of polarization and spatial DOFs of free-space optical beams used to generate vector beams, optical beams with a spatially varying po- larization, have been shown to exhibit skyrmionic topology [57]. The plethora of tools available to generate and reliably characterize vector beams [61–66] has lead to a flurry of research over the past few years into creating different types of optical skyrmions with arbitrary order [57, 67] and vectorial textures [68, 69] and inves- tigating their resilience to perturbations [70, 71]. Furthermore, it is comparatively simpler to achieve higher-order optical skyrmionic topologies than their lower di- mensional counterparts [50, 52] thus providing a possible advantage when using the skyrmionic topology as a high capacity information carrier. However, investi- gations into optical skyrmionic topology is still very much in its infancy with open questions around the robustness of the skyrmionic topology as well as how these 2 Chapter 1. Introduction structures may manifest in quantum structured light. In this chapter we consider the manipulation of light’s spatial and polarization DOFs for the generation of arbitrary vectorial fields and study the accompanying skyrmionic topology embedded within. This is followed by an overview of entanglement be- tween quantum structured photons and the realization of hybrid entangled states, where the polarization of one photon is entangled with the spatial DOF of another. 1.1 Structured light 1.1.1 Polarization and spin angular momentum of light It is well-known that light is an electromagnetic wave, where electric and magnetic fields oscillate perpendicular to one another [72] (as presented in Fig. 1.1 (a)), leading to a property known as polarization - the direction in which the electric field (by convention) oscillates. In the paraxial regime the electric field oscillation in the plane perpendicular to propagation is considered. With this in mind, a general description for the polarization state of a light beam can be given by [73] |P⟩ = β1 |e1⟩+ β2 |e2⟩ , (1.1) a 2-dimensional vector decomposed into the orthogonal basis vectors, |e1⟩ , |e2⟩ and where β1,2 are complex numbers, whose relative amplitudes and phases control the polarization in the transverse plane. It is important to note that we have adopted the quantum bra-ket notation [74] where a vector is denoted by a ket, |·⟩, and its complex conjugate by a bra, ⟨·|. If we assume that |e1⟩ , |e2⟩ point along the x and y axes of the plane perpendicular to propagation then modifying the relative ampli- tude changes the orientation of the electric field oscillation, an example of which is shown in Fig. 1.1 (b) where the vertical oscillation has been modified to horizontal through a change in amplitude of the orthogonal components. Furthermore, rel- ative phase between the polarization components induces a delay between them such that an elliptical polarization oscillation is achieved, an example of which is shown in Fig. 1.1 (b) where an induced phase shift of π 2 between the components yields a circular oscillation of the beams polarization. Importantly, it was discov- ered that photons whose polarization state is circular carry spin angular momentum (SAM) [75, 76] of ±h̄ per photon, where the sign is dependent on the handedness of the polarization. The states associated with SAM are the left-circular, |L⟩, and right- circular, |R⟩, polarization states. If we consider expressing the polarization in terms of these spin states, normalize the resultant state and ignore global phase terms, we can rewrite Eq. 1.1 as follows |P⟩ = cos ( θ 2 ) |R⟩+ eiα sin ( θ 2 ) |L⟩ , (1.2) where θ ∈ [0, π] and α ∈ [0, 2π] are parameters controlling the relative ampli- tude and phase between the two orthogonal components, respectively. Therefore, a sphere is a natural choice for the space of all possible transverse polarization states. Similar to the Bloch sphere for spin states in quantum mechanics [74], polarization states are found on the surface of the Poincaré sphere [73], depicted by Fig. 1.1 (c). 1.1. Structured light 3 It is convention to write Eq. 1.2 in the right-, |R⟩, and left- , |L⟩, circular basis as this follows the Bloch sphere convention of writing the spin states at the poles. However any choice of orthogonal bases is valid and in this work in addition to the circu- lar basis, {|R⟩ , |L⟩}, we will also use the horizontal and vertical basis, {|H⟩ , |V⟩}, diagonal and anti-diagonal basis, {|D⟩ , |A⟩}, and the general orthogonal arbitrary elliptical basis, {|P+⟩ , |P−⟩}, described by Eq. 1.2. Since polarization states are so readily described using simple vectors, it is conve- nient to describe them and operations on them using one of two formalisms, the Jones formalism [77], and the Stokes formalism [78]. The Jones formalism describes a polarization state as a 2-dimensional complex vector, |P⟩ = [ cos ( θ 2 ) eiα sin ( θ 2 )]T, which lives on a 2D Hilbert space [74]. Under this formalism unitary transforma- tions can be written as simple 2×2 matrices acting on the state (discussed in more detail in section 2.1.2). Conventionally, the polarization basis states are written as the following Jones vectors |H⟩ = [ 1 0 ]T, |V⟩ = [ 0 1 ]T, |D⟩ = 1√ 2 [ 1 1 ]T, |A⟩ = 1√ 2 [ 1 −1 ]T, |R⟩ = 1√ 2 [ 1 i ]T and |L⟩ = 1√ 2 [ 1 −i ]T. Whilst compact, the Jones formalism does not yield outcomes which are measured directly as it contains imag- inary components. This is where the Stokes formalism is useful as it builds a real- valued 3-dimensional vector living on S2 (a 3D sphere of unit radius) describing the polarization state. The Hopf map [79] maps a state described by the Jones formalism to the Stokes formalism as follows (β1, β2) → S⃗ = (2Re{β1β2}, 2Im{β1β2}, |β1|2 − |β2|2}) = (S1, S2, S3). Each component of the Stokes vector which describes a state can be extracted from a series of intensity measurements thus making the Stokes formalism useful when conducting experiments. It is important to note that by con- vention the Stokes vector has 4 components, with the first component, S0, being the total intensity of the field. If the field is considered to be normalized and fully polar- ized (S2 0 = S2 1 + S2 2 + S2 3) then this component can be ignored as its information can be inferred from the other components. Matrices which transform Stokes vectors are known as Mueller matrices [78] however are not of particular interest in this work. Polarization states can be represented either by the Stokes vector in the paraxial plane, as indicated in Fig. 1.1 (d) on the left, or by its corresponding polarization ellipse where the ellipse parameters can be directly extracted from the angles which give a position on the surface of the Poincaré sphere. 1.1.2 Spatial distribution and orbital angular momentum of light We will consider the spatial modes that are solutions to the Helmholtz equation in free-space [80, 81] ( ∇2 + k2)U(x, y, z) = 0, (1.3) as appropriate smooth functions of the spatial coordinates x, y, z ∈ R. Here, ∇2 = ∂2 ∂x2 + ∂2 ∂y2 + ∂2 ∂z2 is the Laplacian operator in cartesian coordinates, k is the magnitude of the wave vector and U(x, y, z) is a complex scalar field. It should be noted that the Helmholtz equation arises from considering fields that satisfy the wave equa- tion, ( ∇2 − 1 c2 ∂2 ∂t2 ) F(x, y, z, t) = 0, and assuming that F(x, y, z, t) can be written in a separable manner, F(x, y, z, t) = U(x, y, z)T(t) [80]. Assuming that the field can be written as U(x, y, z) = u(x, y, z)e−ikz where e−ikz is a plane wave which points 4 Chapter 1. Introduction FIGURE 1.1: a) An electromagnetic wave consisting of an electric and magnetic field which are orthogonal to one another. The direction of the electric field oscillation gives the po- larization of the field. b) Examples of (top) vertically, (middle) horizontally and circularly polarized light. c) The Poincaré sphere with right- and left-circular polarization states repre- sented at the poles. d) Representations of polarization states, using a (left) Stokes vector in 3D and (right) a polarization ellipse. along the direction of propagation of the field, we have that Eq. 1.3 after appropriate application of the product rule and simplification, becomes ∇2 ⊥u(x, y, z)− 2ik ∂u(x, y, z) ∂z + ∂2u(x, y, z) ∂z2 = 0, (1.4) where ∇⊥ = ∂2 ∂x2 + ∂2 ∂y2 . Since we are interested in paraxial fields, we are free to perform the paraxial approximation which assumes that the divergence of the resultant spatial modes with propagation is minor [82], i.e., u(x, y, z) does not vary significantly with change in z, which allows for the removal of the third term in Eq. 1.4 since ∣∣∣ ∂2u(x,y,z) ∂z2 ∣∣∣ ≪ ∣∣∣ ∂u(x,y,z) ∂z ∣∣∣. The result of this is the paraxial, helmholtz equation in free-space ∇2 ⊥u(x, y, z)− 2ik ∂u(x, y, z) ∂z = 0, (1.5) which when solved under different symmetries yields diverse sets of mode fam- ilies that form complete orthonormal basis sets for complex scalar fields defined on R2. The completeness of these basis sets allow for a description of any arbitrary complex scalar field as linear combinations of basis elements [83]. Furthermore, the orthornormality of these basis sets allow for their unique detection, i.e., elements from these basis modes can be distinguished from one another [83, 84]. Another fea- ture that all of these mode families share is that they are propagation invariant up to 1.1. Structured light 5 a scaling factor which implies that they retain features such as orthornormality upon propagation [83, 85]. These features make such modes highly useful and practical to generate and detect in an experimental setting [84]. Free-space mode families The first mode family that will be considered is the Hermite-Gaussian (HG) modes which have cartesian symmetry and are derived when solving the Helmholtz equa- tion in cartesian coordinates. They are defined as follows [81, 86] u(x, y, z) ≡ HGm n (x, y, z) = w0 w(z) √ 2(1−n−m) πn!m! Hm (√ 2 x w(z) ) Hn (√ 2 y w(z) ) exp ( − r2 w2(z) ) × exp ( −i [ kr2 2R(z) − (n + m + 1)ϕG(z) + kz ]) , (1.6) where w0 is the beam radius at z = 0, k = 2π λ , λ is the wavelength, r = √ x2 + y2, (1.7) w(z) = w0 √ 1 + ( z zR )2 , (1.8) is the beam radius after propagation, R(z) = z [ 1 + ( zR z )2 ] , (1.9) is the radius of curvature of the beam, ϕG = arctan ( z zR ) , (1.10) is the mode dependent phase gathered by the beam on propagation, known as gouy phase [87, 88], zR = πw2 0 λ , (1.11) is the Rayleigh range of the beam [89] and Hm,n are the Hermite polynomials of the m-th, n-th order with m, n ∈ Z+ [90]. The Hermite polynomials are one- dimensional functions with m, n nodes along the x- and y-axes, respectively. In Fig. 1.2 the intensity and phase (placed as insets) of HG modes with varying indices are shown. HGs are identified with a clear disjointed intensity lobe structure where lobes of intensity are distributed along the x and y-axes. Clearly there are m, n null intensity "lines" across the x- and y-axes, respectively. The phase of each mode is a binary field, seen as a "checkerboard" pattern in Fig. 1.2, where the phase oscillates 6 Chapter 1. Introduction between 0 and π with the switch between phase values occurring at the position of the intensity nulls. FIGURE 1.2: Intensity and phase (insets) of Hermite-Gaussian modes with varying m and n indices. Another two mode families of interest are the Laguerre-Gaussian (LG) and Bessel- Gaussian (BG) modes which are cylindrically symmetric and are derived when solv- ing the Helmholtz equation in polar coordinates [91]. Whilst LG modes are derived using the paraxial approximation of the Helmholtz equation, BG modes are the finite energy analogue to Bessel beams which are solutions to the full Helmholtz equation [92]. The LG modes are defined by [81] u(x, y, z) ≡ LGℓ p(r, ϕ) = w0 w(z) √ 2p! π(p + |ℓ|)! ( √ 2r w(z) )|ℓ| L|ℓ| p ( 2r2 w2(z) ) exp ( − r2 w2(z) ) × exp ( −i [ k r2 2R(z) − (2p + |ℓ|+ 1)ϕG(z) + kz + ℓϕ ]) , (1.12) where ℓ ∈ Z and p ∈ Z+ are the azimuthal and radial indices, respectively and L|ℓ| p are the associated Laguerre polynomials [93]. In Fig. 1.3 the intensity and phase (placed as insets) of LG modes with different radial and azimuthal orders are 1.1. Structured light 7 shown. The radial order, p, sets the number of radial nulls in the mode, or alter- natively, the number of concentric intensity rings in the mode is set by p + 1. The effect of changing p is also evident in the inset phase of each mode where the phase structure presents some discontinuity lines at the radii where one finds the intensity nulls. Changing the azimuthal index results in a larger intensity null at the centre of the mode. More significant is that the phase of a mode possessing an aziumthal index of ℓ performs ℓ full 2π rotations about the azimuth. For ℓ < 0, the rotation of the phase from 0 to 2π occurs in the clockwise direction while for ℓ > 0 it occurs in the anti-clockwise direction. In such beams with non-zero azimuthal index one also observes regions of undefined phase at the centre of these beams, i.e., an optical singularity [94, 95]. These optical singularities are often defined by their topological charge, the number 2π rotations performed by the phase as well as the rotation di- rection. FIGURE 1.3: Intensity and phase (insets) of Laguerre-Gaussian modes with varying ℓ and p indices. BG modes have a similar structure to LG modes, also possessing an azimuthal in- dex, ℓ, except that their radial behaviour is set by a continuous parameter, kr =√ k2 − k2 x − k2 y, the radial wavevector. BG beams are defined by [81] 8 Chapter 1. Introduction u(x, y, z) ≡ BGℓ kr (r, ϕ, z) = Jℓ(krr) exp ( − r2 w2(z) ) exp (i [ℓϕ − kz]) , (1.13) where Jℓ is the ℓ-th order bessel function of the first kind [96]. BG modes are of par- ticular interest due to their non-diffracting nature [92], ability to "self-heal" around obstacles [97, 98], and interesting propagation dynamics [92, 99]. In Fig. 1.4 the in- tensity and phase (placed as insets) of BG modes with varying azimuthal index and radial wave vector are shown. It is clear that the structure of BG modes is almost identical to that of LG modes with a high radial order, except that the energy dis- tribution spread between the central ring and concentric rings heavily favour the central ring, which is not the case for LG modes. FIGURE 1.4: Intensity and phase (insets) of Bessel-Gaussian modes with varying ℓ index and magnitude of radial kr vector. An aspect of these cylindrically symmetric modes which have made them popular in the community, is their ability to carry orbital angular momentum (OAM), as a result of the number of "twists" in their phase structure [100]. The number of full 2π rotations performed by the phase function, gives the number of helical twists the wavefront of the light performs within a single wavelength. Much like polariza- tion attributing single photons with SAM, the spatial distribution of an individual 1.1. Structured light 9 photon’s wavefunction can endow it with an OAM of ℓh̄ per photon. This OAM property will be revisited in the context of individual photons described by quan- tum states in section 1.3. In this work, we are particularly interested in the cylindrically symmetric BG and LG modes, with particular focus paid to LG modes in chapters 2, 4 and 5 and BG modes in chapter 3. Apart from their ability to carry OAM, the phase structure embedded in these modes allow for every possible relative phase between two orthogonal modes to be achieved in the transverse plane, a feature that is essential when constructing skyrmionic topologies, which will be discussed in section 1.2. 1.1.3 Scalar vs vector beams In the previous sections the manipulation of polarization and spatial DOFs were dis- cussed separately however these two DOFs can be manipulated independently. This allows for the generation of more complex beams, known as vector beams (VBs), where the polarization varies across the transverse plane of the beam [101, 102]. One can formulate a general description of such VBs through a straightforward generalization of Eq. 1.2 |Ψ⟩ = cos ( θ(x, y) 2 ) |R⟩+ eiα(x,y) sin ( θ(x, y) 2 ) |L⟩ , (1.14) where now the parameters which control the polarization state, θ and α are made to be functions of position. This concept is illustrated in Fig. 1.5 (a) where each block corresponds to a unique position in the plane perpendicular to the direction of propagation of a beam, and is associated its own polarization. In the case where the polarization is uniform across the beam profile, θ(x, y) is a constant, we have a scalar beam (SB). Mathematically these two types of beams can be distinguished by a property known as separability [103]. SBs are beams whose spatial and polariza- tion DOFs are completely separable, while for VBs these DOFs are non-separable. As discussed previously, the choice of polarization basis is arbitrary, one is free to choose any pair of spatially dependent functions to control the spatial dependence of polarization across the beam profile. Due to their orthonormality it is convenient to express Eqn. 1.17 in terms of previously defined spatial modes |Ψ⟩ = u1(x, y) |R⟩+ u2(x, y) |L⟩ . (1.15) As an example if we consider the case shown in Fig. 1.5 (b) (top row) where u1(x, y) = u2(x, y) = LG1 0 , the spatial components can be factorized out of the ex- pression, thus making Eqn. 1.15 separable. However if we instead consider the case shown in Fig. 1.5 (b) (bottom row) where u1(x, y) = LG1 0 and u2(x, y) = LG−1 0 , the spatial components cannot be factorized out of the expression thus making Eqn. 1.15 non-separable. The degree of non-separability of these beams also referred to as the vector quality factor (VQF) can be precisely quantified using a metric known as concurrence used to characterize the degree of entanglement between quantum subsystems [103–105] VQF = 2 √ ⟨u1|u1⟩ ⟨u2|u2⟩ − | ⟨u1|u2⟩ |2 (1.16) 10 Chapter 1. Introduction where |ui⟩ = ∫ |x, y⟩ ui dxdy is the state description of the spatial modes with |x, y⟩ identifying a particular position state and 〈 ui ∣∣uj 〉 = ∫ |x, y⟩ u∗ i uj dxdy is the inner product of spatial mode states with the indices i, j ∈ 1, 2. It is important to note that the description given above is specific to pure/coherent states and a more general description used to quantify the degree of entanglement in quantum states is given in section 2.2.3. Given that we have chosen our spatial modes to be orthonor- mal, this implies that 〈 ui ∣∣uj 〉 = 1 2 δij where δij is the Kronecker delta. For the case of the separable SB and non-separable VB discussed it is easy to verify that VQF = 0 and VQF = 1, respectively. FIGURE 1.5: a) Inhomogeneous polarization structure achieved through combination of spa- tial and polarization DOFs. b) Difference between scalar and vector modes, where the po- larization is homogeneous in the former and inhomogeneous in the latter. To see how we can construct a plethora of diverse VBs, we may consider writing Eqn. 1.15 in terms of LG modes |Ψ⟩ = LGℓ1 p1 |R⟩+ LGℓ2 p2 |L⟩ . (1.17) Since LG modes are cylindrically symmetric they can be rewritten as (for simplicity we set p = 0 and z = 0) LGℓ 0(r, ϕ) = e − r2 w2 0 √ 1 π|ℓ|! (√ 2r w0 )|ℓ| exp (iℓϕ) , (1.18) which indicates that the LG modes can be written in the separable form LGℓ 0(r, ϕ) = A(r)eiℓϕ, where the amplitude of the mode is dependent on the radial coordinate and the phase depends on the azimuthal coordinate. An example of this with ℓ1 = 0 and ℓ2 = 1 is depicted graphically in Fig. 1.6 (a). From this it is clear to see that the polarization changes in both the radial and azimuthal directions with the rela- tive amplitudes and phases between orthogonal polarization components changing in the radial and azimuthal directions, respectively. This simple construction allows for an intuitive approach to create a plethora of diverse VBs, two examples of which are shown in Fig. 1.6 (b). Fig. 1.6 (b) depicts the polarization structure in two ways. The first representation being the Stokes vector plot which depicts a VB as a 3 di- mensional vector field over the transverse plane. Here every position is associated 1.1. Structured light 11 with a vector which points along the Poincaré sphere thus specifying a particular polarization state. From the previously discussed Hopf map, the Stokes vector field, S⃗, can be computed using Eq. 1.18 to give S⃗ =  2Re{LGℓ1 0 LGℓ2∗ 0 } 2Im{LGℓ1 0 LGℓ2∗ 0 }∣∣∣LGℓ1 0 ∣∣∣2 − ∣∣∣LGℓ2 0 ∣∣∣2  . (1.19) The second representation, the polarization ellipse plot, shows the actual oscilla- tion of the electric field at each point in the transverse plane. FIGURE 1.6: a) VB creation with LG modes, where their cylindrical symmetry allows for clear decomposition into amplitude and phase components. b) VBs formed from LGs with {ℓ1, p1, ℓ2, p2} = {0, 0, 1, 0} (left) and {2, 0,−2, 0} (right). The VBs are represented using a Stokes vector plot and polarization ellipse plot. The study of VBs as well as the development of their generation and detection toolkit [101] has motivated several different applications such as the manipulation of the polarization and beam shape of tightly focused beams [106–108] allowing for optical control of nanoparticles [109], increased trapping efficiency in optical trap- ping experiments [7, 110–112], improved resolution for optical microscopy [113–115] and increased information capacity for optical communications [2, 116] to name just a few. Most recently it was shown that the degree of non-separability of VBs is invariant to unitary transformations induced by chiral liquids, optical fibres, atmo- spheric turbulence and complex optical aberrations even though the underlying spa- tial modes are heavily distorted [42], thus qualifying the degree of non-separability as a viable resource for high capacity optical communication through highly aber- rated systems [43, 44]. Beyond these advances VBs have also recently been used for the generation of skyrmionic topologies [57] which will be discussed in the proceed- ing section. 12 Chapter 1. Introduction 1.2 Skyrmions In the early 1960s Tony Skyrme proposed a non-linear meson field theory to de- scribe sub-atomic particles as excitations of a single fundamental field, the pion [117–120]. To accomplish this he used the inherent mathematical structure of ba- sic pion theory to postulate topologically non-trivial pion field configurations, now called skyrmions. A skyrmion is a topologically stable field configuration, character- ized by an integer topological invariant, the skyrmion number. By definition, these skyrmions cannot be smoothly deformed by any means to attain a new field config- uration with a different topology or equivalently, skyrmion number. The skyrmion mapping was originally conceptualized as a map between 3-spheres (spheres that live in R4) where the skyrmion number indicates the number of times the first sphere wraps the second. However, this notion has since been extended to allow for mapping between spheres of arbitrary dimensions, i.e., maps from n- spheres to n-spheres. Furthermore the generality of this definition and its associated topological conserved quantity has allowed the notion to be extended beyond its ini- tial intent [121, 122]. In particular, skyrmions have been instrumental in advances in magnetism and spintronics [123–130], nuclear physics [131, 132] and have recently featured as optical realizations [54, 57, 69, 133, 134]. 1.2.1 Skyrmion mapping In this work, of particular interest is the skyrmion topology formed by the map- ping from the real sphere (a sphere living in real space), S2, to a parameter sphere, S2. Since one can stereographically project a sphere onto a plane (See appendix A), we can consider mappings from the real plane to the parameter sphere where the skyrmion number defines the number of times the parameter sphere is wrapped given a full traversal of the real plane. To realize such a mapping we may consider a simple vector field mapping the real plane onto a unit sphere, n⃗(r, ϕ) = cos(α(ϕ)) sin(β(r)) sin(α(ϕ)) sin(β(r)) cos(β(r))  , (1.20) where (r, ϕ) are coordinates on R2, α(ϕ) = mϕ + γ with γ ∈ [0, 2π) and m ∈ Z and β(r) is an arbitrary continuous function of r. For simplicity, we choose β(r) to be a monotonically increasing or decreasing function of r and β(r) ∈ [0, π] (It must be noted that such a description rules out topological structures such as skyrmioniums [135] and target skyrmions [136] where the z-component of n⃗(r, ϕ) would complete multiple oscillations between 1 and -1 with distance from the origin, however these structures are not of interest in the current work). An example of such a mapping is shown in Fig 1.7 where the transverse plane is mapped onto a parameter sphere. To quantify the skyrmion number, we consider that it computes the number of times R2 wraps S2, therefore a computation over the solid angle of a unit sphere computes the skyrmion number [53, 121] 1.2. Skyrmions 13 FIGURE 1.7: Mapping of transverse real plane, onto parameter space. Having a non-trivial skyrmion topology means that all possible states are achieved in the transverse plane at least once. N = 1 4π ∫∫ R2 n⃗ · [ ∂⃗n ∂x × ∂⃗n ∂y ] dxdy = 1 4π ∫∫ R2 n⃗ · [ ∂⃗n ∂r × ∂⃗n ∂ϕ ] drdϕ (1.21) where the multiplication of a constant 1 4π ensures that the above computation com- putes only N wrappings and not N × surface Area of unit sphere. To calculate the skyrmion number of the vector field defined in Eq. 1.20 we first calculate [ ∂⃗n ∂r × ∂⃗n ∂ϕ ] = α′(ϕ)β′(r) sin (β(r))  cos (α(ϕ)) sin (β(r)) sin (α(ϕ)) sin (β(r)) cos2 (α(ϕ)) cos (β(r)) + sin2 (α(ϕ)) cos (β(r))  (1.22) where f ′(·) indicates the derivative of the single-variable function, f (·), with respect to its variable. Then 14 Chapter 1. Introduction n⃗ · [ ∂⃗n ∂x × ∂⃗n ∂y ] = α′(ϕ)β′(r) sin (β(r)) [cos2 (α(ϕ)) sin2 (β(r)) + sin2 (α(ϕ)) sin2 (β(r)) + cos2 (β(r)) cos2 (α(ϕ)) + sin2 (α(ϕ)) cos2 (β(r))], (1.23) which after using the simple trigonometric identity, sin2 x + cos2 x = 1, reduces to n⃗ · [ ∂⃗n ∂x × ∂⃗n ∂y ] = α ′ (ϕ)β ′ (r) sin (β(r)) . (1.24) Finally, substituting this result into Eq. 1.21 and integrating over all of space yields [69] N = 1 4π α(ϕ)] ϕ=2π ϕ=0 cos β(r)]r→∞ r=0 = qm (1.25) where q = 1 2 [cos β(r)]r→∞ r=0 and m = 1 2π [α(ϕ)] ϕ=2π ϕ=0 [69]. Clearly, the skyrmion topology is set by the parameters m and q known as the vorticity and polarity, re- spectively. Furthermore, the parameter γ, known as the helicity, has no bearing on the topology of the vector field but simply controls its texture [69, 121]. Fig 1.8 de- picts the different skyrmionic structures that are achievable through the tuning of these three parameters. Topologically equivalent Néel- (see Fig 1.8 (a)) and Bloch- type (See Fig 1.8 (b)) skyrmions are realised when N = 1 and γ = 0, π 2 , respectively. Furthermore, N = −1 and |N| > 1 correspond to anti-skyrmions (See Fig 1.8 (c)) and higher-order skyrmions (See Fig 1.8 (d)), respectively. FIGURE 1.8: Skyrmions of different textures such as a) Néel- and b) Bloch-type as well as different topologies such as c) anti-skyrmion and d) higher-order skyrmions are all achiev- able through the manipulation of the vorticity, polarity and helicity DOFs. 1.2.2 Realizations of optical skyrmions Optical skyrmions realized from linear optical fields was first observed by Tsesses. et. al. in 2018 in the evanescent electric field formed by the interference of surface plasmon polariton (SPP) waves within a hexagonal resonator [133]. Here, n⃗ = E⃗ where E⃗ is the electric field formed by the superposition of standing SPP waves within the resonator. These optical configurations have been observed to form peri- odic lattices of Néel-type skyrmions with more general topological textures still yet 1.2. Skyrmions 15 to be observed [121]. Beyond the electric field, it was observed that the excitation of a waveguided mode with a beam carrying OAM generates spin fields with individual isolated skyrmionic configurations [54] or lattice structures [55]. Optical skyrmions have also been realized in structured materials through the exploitation of multilay- ered chiral structures [56]. Further, by coupling space and time to form what are known as supertoroidal light pulses, skyrmionic topology was observed with vary- ing textures at different cross-sections within these pulses [58]. Furthermore, whilst the construction suggested by Eq. 1.20 focused on mappings from real-space to a parameter sphere, it is also possible to replace real space with some alternative such as momentum space [137, 138]. In this work we are concerned with the generation of skyrmionic structures embed- ded within VBs [57] where the spatially varying polarization structure is described by the Stokes vector field, S⃗(x, y) (now playing the role of n⃗), and can then be as- cribed a skyrmion number based on the number of times the transverse plane of the beam wraps the Poincaré sphere (the chosen parameter space). The large struc- tured light toolkit available for the manipulation of spatial and polarization DOFs has allowed for the realisation of a vast array of skyrmionic topologies with rel- ative ease when compared to other optical realisations. Constructing VBs using propagation invariant modes such as LG modes ensures that the VB is topologi- cally stable, i.e., if no external perturbations are present the topology of the VB re- mains intact during propagation [57, 121]. Furthermore, the polarity, vorticity and helicity can all be controlled via an appropriate choice of spatial modes and rela- tive phases between the orthogonal polarization components thus allowing for the generation of Néel- and Bloch-type skyrmions as well as anti-skyrmions and ar- bitrary higher-order skyrmions [57, 69]. Skyrmionium and bimeron structures are achievable through a change in the choice of spatial mode basis [69] and polariza- tion basis [68], respectively. Beyond 2D skyrmions, 3D skyrmionic structures called skyrmionic hopfions which realise mappings from the 3D real space to the 4D hyper- sphere have also been achieved by tracking lines of constant polarization in propa- gating VBs [139–141]. 1.2.3 Constructing optical skyrmions with vector beams Suppose we construct the following VB (where we have set p = 0) |Ψ⟩ = LGℓ1 |R⟩+ LGℓ2 |L⟩ . (1.26) If we consider the simplified version of the LG modes given in Eq. 1.18 then we can identify certain boundary behaviours of the polarization vector field created by Eq. 1.27. We consider the vector field at three distinct regions, r → 0, r → ∞, and r → r0 > 0 with ϕ ∈ [0, 2π) depicted in Fig. 1.9 (a) as the red, green and blue regions, respectively. If |ℓ1| = |ℓ2|, then the amplitude structure of the LG modes are identi- cal, thus the polarization state is independent of the radial coordinate since the rel- ative amplitude between orthogonal polarization components is always 1. Suppose then that we select LG modes such that |ℓ1| ̸= |ℓ2| and without loss of generality ℓ2 > ℓ1. In the regions, r → 0 and r → ∞ the dominating behaviour is set by the function (√ 2r w0 )ℓ thus at r → 0 we will only observe right circular polarization and at r → ∞ we will only observe left circular polarization. In between these two regions we have that the relative amplitude of the two LG modes will change. A simple 16 Chapter 1. Introduction example is shown in Fig. 1.9 (b) where ℓ1 = 0 and ℓ2 = 1. Clearly the beam with ℓ1 = 0 will dominate in the central region and the beam with ℓ2 = 1 will dominate "far-away" from the centre. Finally, observing the region r → r0 > 0 with ϕ ∈ [0, 2π) we see that the relative amplitudes between the orthogonal polarization components is constant, however the relative phase changes with azimuthal position. Fig. 1.9 (c) depicts the case where ℓ1 = 0 and ℓ2 = 2. The relative phase between the orthogonal components is given by ei(ℓ2−ℓ1)ϕ, therefore one complete revolution in the azimuthal direction yields ℓ2 − ℓ1 complete relative phase cycles between the two orthogonal polarization components. To summarize, LG beams are an appropriate basis to con- struct skyrmionic topologies as they allow for all possible relative amplitudes and phases between orthogonal polarization states within the transverse plane. FIGURE 1.9: a) Simplest construction of a skyrmionic mapping, modelled after a stereo- graphic map from an infinite real spatial plane, onto a spherical parameter space. b) Radial cross-section of amplitudes for two LG modes, with ℓ = 0, 1. c) Azimuthal cross-section of relative phase function for two LG modes, with ℓ = 0, 2. We now consider modifying Eq. 1.19 so that it takes the form of Eq. 1.20. This allows for a simple extraction of an analytic expression for the skyrmion number in terms of parameters of the chosen LG modes. To do this, it is first necessary to locally normal- ize Eq. 1.27. Since only the relative amplitudes between the orthogonal polarization states matter (not the amplitudes of individual orthogonal polarization states) it is convenient to normalize the state as follows |Ψ⟩ = |R⟩+ v(r) |L⟩ 1 + v(r) , (1.27) where v(r) = f (r)ei∆ℓϕ, f (r) = ∣∣∣ LGℓ2 LGℓ1 ∣∣∣ and ∆ℓ = ℓ2 − ℓ1. Modifying the Stokes vector in Eq. 1.19 with Eq. 1.27 yields 1.2. Skyrmions 17 S⃗ = g( f (r)) cos (∆ℓϕ)} g( f (r)) sin (∆ℓϕ)} h( f (r))  , (1.28) where g( f (r)) = 2 f (r) 1+ f (r)2 and h( f (r)) = 1− f 2(r) 1+ f 2(r) . We note that g( f (r)) ∈ [0, 1] and h( f (r)) ∈ [−1, 1] for f (r), r ∈ [0, ∞). Furthermore, since we require a mapping from the real plane to the unit sphere, we require that S⃗ · S⃗ = 1 =⇒ g2( f (r)) + h2( f (r)) = 1. Therefore, we can set g( f (r)) = sin(Θ(r)) and h( f (r)) = cos(Θ(r)) where Θ(r) ∈ [0, π]. It must be noted that the explicit form of Θ(r) depends solely on the ratio of LG modes given by f (r) which is set by their ℓ-indices. Since the Stokes vector now has the form of Eq. 1.20 we can extract the skyrmion number N = ∆ℓm, (1.29) where m = 1 2 [cos Θ(r)]r→∞ r=0 . Further, by evaluating f (r) at r = 0 and r → ∞ for different cases of relative ℓ-index between the two LG modes we find that [57, 142] m =  0, |ℓ1| = |ℓ2| 1, |ℓ1| > |ℓ2| −1, |ℓ1| < |ℓ2|. Here, ∆ℓ plays the role of vorticity with |∆ℓ| setting the absolute order of the skyrmion topology and m plays the role of polarity as before. Furthermore, by including a rel- ative phase of γ between the two orthogonal polarization components in Eq. 1.27 we can control the helicity of the skyrmion as well. By manipulating these parame- ters it is possible to generate a plethora of skyrmionic modes with varying skyrmion number (Fig. 1.10 (a)) and vectorial textures (Fig. 1.10 (b)). 18 Chapter 1. Introduction FIGURE 1.10: Skyrmion topologies generated by different LG modes. a) (column) Topolog- ically equivalent skyrmions generated by different LG mode superpositions. Insets give the ℓ value for the right and left circular components respectively. (row) Different topological structures are also generated through the change in relative topological charges between the two LG modes. b) Topologically equivalent structures with different vectorial textures are generated through an induced relative phase between the orthogonal components. 1.3 Quantum structured light and vectorial wavefunctions The particle-wave duality of light naturally suggests that while light can be quan- tized as photons, individual photons still possess wave-like features present if one studies their probability amplitude associated with properties of their wavefunction distributed in time and space [143]. Since we are particularly interested in the spa- tial and polarization DOFs we can consider the decomposition of a photon into said DOFs. Fig. 1.11 depicts the decomposition of a photon state into polarization (a) and spatial (b) DOFs which can be written as follows [143] |Ψ⟩ = ∫∫ Φspatial(r) |r⟩ |P⟩ d3r, (1.30) using the continuous spatial |r = (x, y, z)⟩ coordinate basis, with corresponding prob- ability amplitudes, Φspatial(r), that constitute the photon wavepacket, while |P⟩ de- notes the polarization state of the photon. Here we have once again decided to use Dirac notation when referencing a state which lives in a d-dimensional Hilbert space, Hd, where the dimensionality is dictated by the DOF in question and further the shorthand notation |·⟩ |·⟩ ... is used in place of writing down the tensor product between states, |·⟩ ⊗ |·⟩ ⊗ ... . Written in the above form, it is clear that applying 1.3. Quantum structured light and vectorial wavefunctions 19 the classical structured light toolbox to the quantum realm involves structuring the particle- (angular momentum) and wave- (spatially dependent complex probability amplitude) like properties of photons. FIGURE 1.11: A decomposition of a photon into a) polarization and b) spatial DOFs. The polarization of a photon is connected to its inherent SAM and the spatial distribution of a photon’s wavefunction can be associated to its OAM 1.3.1 Structuring single photons The polarization degree of freedom was initially the best candidate for photon infor- mation processing due to its ease of control with conventional linear optical elements and was used to demonstrate numerous fundamental tests of quantum mechanics (Bell inequality violations [144] and quantum erasers [145]) and the initial demon- strations of quantum communication and cryptography (quantum key distribution [146], teleportation [147] and superdense coding [148]). However, since the basis for the polarization DOF only consists of two elements, a photon is restricted to a 2D state. Instead we may consider extending the dimensionality of the photon’s state by manipulating its OAM as this can take on any integer value, thus is in principle infi- nite dimensional. We discussed earlier that the phase of spatially structured beams with non-zero OAM have an azimuthal dependence. Similarly, a photon possessing OAM can be described by a wavefunction whose distribution is characterized by the azimuthal phase dependent factor exp(iℓϕ) [100]. Optical modes with this charac- teristic are eigenstates of the OAM operator L̂z = ih̄ ∂ ∂ϕ . In cylindrical coordinates r = (r, ϕ, z), these modes take the form [143] Φ(r) = R(r) exp(iℓϕ), (1.31) 20 Chapter 1. Introduction where R(r) is an enveloping function that determines the radial amplitude profile of the photon. The form of Eq. 1.18 matches that of Eq. 1.31 thus the LG mode family forms a natural basis to describe OAM states, |ℓ⟩. Considering the case where the radial order, p = 0, we can write the OAM states in position basis |ℓ⟩ = ∫ R2 LGℓ(r) |r⟩ d2r. (1.32) It is clear that OAM states described in the position basis reveal the full content of the spatial wavefunction of the photon. The probability amplitude of the photon in the transverse plane is given by ∣∣∣LGℓ ∣∣∣, while the characteristic azimuthal phase is given by arg(LGℓ). These two pieces of information play an important role in describing the photon statistics as well as the evolution of the wavefunction due to propagation. A photon may be described in the OAM basis as follows |ψ⟩ = ∑ ℓ cℓ |ℓ⟩ , (1.33) where the complex coefficients cℓ can be tailored for a given application. For exam- ple, in quantum key distribution, such superpositions can be optimised to achieve unparalleled security [24, 29, 149, 150], transmit quantum states through fiber [151] or to tailor novel projective measurements for state characterisation [39, 152–154]. 1.3.2 Entanglement of photons In the classical regime, non-separability was engineered between internal degrees of freedom of a system, i.e, the non-separability between the spatial and polariza- tion DOFs of a single beam. However, in the quantum regime, non-separability also known as entanglement is achieved through two or more subsystems or photons that are spatially separated [155]. These subsystems cannot be described indepen- dently from one another, for example, for two photons entangled in position, one cannot extract the information of a single photon’s position without affecting the position of the other photon. Suppose we consider a simple case of two spatially separated photons (photon A and photon B), each occupying a 2-level polarization system described by the ba- sis {|R⟩ , |L⟩}. Each photon’s state can be described by the superposition given in Eq. 1.2. The composite system can be described as a tensor product between the two subsystems |ψ⟩sep AB = |ψ⟩A |ψ⟩B = a1 |R⟩A |R⟩B + a2 |R⟩A |L⟩B + a3 |L⟩A |R⟩B + a4 |L⟩A |L⟩B , (1.34) where the complex coefficients ai control the relative amplitudes and phases be- tween each of the states. The state |ψ⟩sep AB is a separable tensor product (|ψ⟩sep AB = |ψ⟩A |ψ⟩B) of the two subsystems. If a1 = a2 = a3 = a4 then the state is not entangled and said to be fully separable. Furthermore, this composite system is fully described through the use of four basis vectors thus the state lives on a 4 dimensional hilbert space, H4 AB. Whilst this composite system is not entangled, we are free to select a 2D state that lives on a subspace, H2 AB of H4 AB which is entangled. Suppose instead that we selected the state 1.3. Quantum structured light and vectorial wavefunctions 21 Projection Separable Non-separable |R⟩A ⟨R|A |R⟩A (|R⟩B + |L⟩B) |R⟩A |L⟩B |L⟩A ⟨L|A |L⟩A (|R⟩B + |L⟩B) |L⟩A |R⟩B TABLE 1.1: Projective measurement outcomes (normalization factors have been omitted) for the separable state described in Eq. 1.34 and non-separable state described in Eq. 1.35 |ψ⟩ent AB = 1√ 2 (|R⟩A |L⟩B + |L⟩A |R⟩B) , (1.35) which is obtained from Eq. 1.34 by setting a1 = a4 = 0 and a2 = a3 = 1√ 2 It is clearly not possible to factor the state of either photon (|ψ⟩ent AB ̸= |ψ⟩A |ψ⟩B) therefore the state is non-separable, thus entangled. To illustrate this idea of separability and non- separability further, we consider the outcomes of both scenarios when we project photon A onto the basis states and observe the corresponding outcome on photon B. For simplicity we have assumed the case where a1 = a2 = a3 = a4 in Eq. 1.34. The measurement outcomes have been summarized in table 1.1. In the case of the separable state, regardless of the projection done on photon A, photon B always collapses into the same superposition of spin states. This demonstrates the fact that the state of photon B is independent of the state of photon A, and therefore they are not entangled. For the non-separable state, we observe the opposite behaviour. A projection of photon A onto the state |R⟩ yields a collapse of photon B’s state into |L⟩ and vice versa. Therefore, photon A and B cannot be considered independent of one another, and are thus entangled. This relationship between the photons is always true despite any distance of separation between the photons. This prompted Einstein questioning the reality [156], calling this "spooky action at a distance" and has since been termed quantum entanglement [157]. The example given above serves to highlight the differences between entangled and non-entangled systems. In the proceeding section we will consider a more general definition for entanglement in the context of a 2-photon system. High-dimensional entangled states To generalise the description of entangled photons beyond d = 2, we need to add more linearly independent terms to Eq. (1.35) such that the state remains non-separable. However, as discussed before, the SAM DOF of photons can only form a basis con- taining 2 orthogonal components, thus this DOF is inherently limited to 2 dimen- sions. Instead, to create high dimensional entangled states we consider the infinite dimensional OAM DOF. It is important to note that any infinite dimensional DOF can be used in the discussion that follows and further that the DOFs used for each photon need not be the same. The composite system of two photons, whose states are each described by superpositions of OAM states can be written as |ψ⟩AB = ∑ ℓA,ℓB=−∞,−∞ cℓA ℓB |ℓA⟩A |ℓB⟩B , (1.36) where cℓA ℓB are the complex coefficients which define state. As mentioned before, such a state is not entangled if it is factorizable as follows 22 Chapter 1. Introduction |ψ⟩AB = |ψA⟩A |ψB⟩B , (1.37) where |ψA⟩A = ∑∞ ℓA=−∞ cℓA |ℓA⟩A and |ψB⟩B = ∑∞ ℓB=−∞ cℓB |ℓB⟩B. If we consider the restriction applied on the coefficients of the state then we observe that for a sep- arable state cℓA ℓB = cℓA cℓB . However, if cℓA ℓB ̸= cℓA cℓB then |ψ⟩AB ̸= |ψA⟩A |ψB⟩B , (1.38) and so by definition the state is non-separable and thus entangled. This provides a more general definition of entanglement for a 2-photon system. Hybrid entanglement and vectorial wavefunctions In section 1.1.3 we discussed free-space VBs where the polarization was spatially varying across the transverse profile of the beam. This behaviour originates from the non-separability between the spatial and polarization DOFs of the optical field. In fact these VBs are analogous to specific hybrid entangled states [22, 158] where the spatial DOF of one photon is entangled with the polarization DOF of another photon. Such a state can be written in the form |Ψ⟩AB = 1√ 2 (|ℓ1⟩A |R⟩B + |ℓ2⟩A |L⟩B) (1.39) where the OAM of photon A is entangled with the SAM of photon B. Expressing the OAM DOF of photon A in position basis, |Ψ⟩AB = 1√ 2 ∫ R2 |rA⟩ (LGℓ1(rA) |R⟩B + LGℓ2(rA) |L⟩B) drA, (1.40) the correlations between position, r, of photon A and polarization state of photon B become evident. Here the joint wavefunction of the entangled photons is not a simple complex scalar field, but is rather a complex vector field given by ψ⃗AB(rA) = LGℓ1(rA) |R⟩B + LGℓ2(rA) |L⟩B . (1.41) Clearly, Eq. 1.41 has the same form as the VB defined in Eq. 1.17. This implies that beyond engineering correlations between OAM and SAM, it is possible to pre- cisely engineer vectorial wavefunctions describing the entanglement between two photons. Exploiting the similarities between VBs and hybrid states has proven to be immensely fruitful. Several techniques used to generate and detect classical VBs have been de- ployed for the rapid realization and characterization of arbitrary hybrid states useful for higher dimensional quantum communication and cryptography [30, 116]. One study revealed that it is possible to characterize a quantum channel by studying its effects on a VB passed through it, so long as the channel only perturbs a single DOF 1.3. Quantum structured light and vectorial wavefunctions 23 [158]. Further, a quantum inspired analysis of VBs revealed the invariance of their degree of non-separability to certain noisy channels [42]. This inspired the use of classical non-separability as a resource for communication over noisy channels [43, 44]. In this work, we further explore the connections between VBs and hybrid states, namely the skyrmionic properties they share and demonstrate the robustness of this new DOF against typical sources of noise. 1.3.3 Outline Our work is structured as follows: In chapter 2 we will discuss classical and quan- tum structured light techniques. Classical techniques will be discussed to generate and control the spatial and polarization DOFs of light using liquid crystal technol- ogy and linear polarization optics. Following this, the generation of arbitrary vector modes using dynamic and geometric phase devices will be demonstrated. Finally, Stokes polarimetry will be employed as a technique for the characterization of vector beams. The classical techniques will then be extended into the quantum realm. The generation of high dimensional entangled states through spontaneous parametric down-conversion as well as the detection of OAM states, will be discussed. These high-dimensional OAM-OAM entanglement states form an ideal platform to con- struct arbitrary 2D hybrid entangled states. In the experimental setup required to generate these arbitrary hybrid states we will pay particular attention to the spatial- to-polarization conversion system allowing for the conversion of initial OAM entan- gled states to hybrid entangled states. To characterize such states we will discuss 2D quantum state tomography, a measurement technique capable of fully charac- terizing a 2D state as well as several other useful parameters (such as concurrence, fidelity and purity) that can be calculated once the state is reconstructed. The quan- tum analogue to Stokes polarimetry will also be developed. In Chapter 3 we will demonstrate the generation of optical skyrmions through the use of Bessel Gaussian modes. Observing the transformations of their polarization structures on propagation reveals a beating between skyrmion types whilst leaving their topology unaltered. Furthermore, we observe the procession of the Stokes vec- tors with propagation, analogous to the procession of magnetic spins in the presence of a magnetic field. Lastly, through the use of a frozen wave construction, it will be shown that the evolution of the central on-axis polarization state can be precisely tuned during propagation. In Chapter 4 we generate various 2D hybrid entangled states where the polariza- tion of one photon is entangled to the OAM of another. We then present a novel result, that is the generation of non-local skyrmionic entangled states. To this end, we will discuss the characterization of quantum entangled states according to their topology, along with the implications that come along with it, such as characteriz- ing any transformation that does not alter the topology as a smooth deformation or coordinate transformation. Lastly, resilience of the non-local topological features of the entangled states against entanglement decaying noise will be demonstrated and further the noise will be precisely associated to a coordinate transformation. Lastly, in Chapter 5 we will build upon the ideas of chapter 4, where we then con- sider the effects of isotropic noise on the non-local topology of entangled states. We will demonstrate through theory and experiment that isotropic noise does not alter the topology of these states except when the maximally mixed state limit is reached. 24 Chapter 1. Introduction Furthermore, we will describe the isotropic noise transformation, as a simple resiz- ing of the state space, thus not altering the topology as expected. 25 Chapter 2 Experimental Techniques In this chapter, we focus on the experimental techniques to control the polarization, amplitude and phase at every position within a classical optical field. We explore the use of static geometric optics as well as digital Spatial Light Modulators (SLMs) for the creation of vectorial beams. Furthermore we discuss Stokes polarimetry, an essential tool for the analysis of vectorial structures. We then consider the quantum analogue to the classical set of tools, by considering the generation and detection of entangled biphoton states. Complete control over 2 separate degrees of freedom (high dimensional spatial DOF and 2D polarization degree of freedom), allows for the digital creation and detection of a plethora of vectorial biphoton wavefunctions as well as different non-local topologies. 2.1 Structured light techniques 2.1.1 Amplitude and phase modulation The goal of spatial amplitude and phase modulation of light, can be simply de- scribed as finding and performing the transformation Ui(x, y) T−→ U f (x, y) =⇒ U f (x, y) = TUi(x, y) where Ui, f (x, y) are input (output) spatially dependent com- plex scalar fields, respectively with T(x, y) being the transfer function required to engineer the desired final output field. Expressing the Ui(x, y) and U f (x, y) as sep- arable products of their amplitude and phase components we see that the transfer function is simply given by T(x, y) = A f (x, y) Ai(x, y) ei(ϕ f (x,y)−ϕi(x,y)), (2.1) where Ai( f )(x, y) = |Ui( f )(x, y)| and ei(ϕi( f )(x,y) = arg(Ui( f )(x, y)). Clearly, to im- plement such transformations generally, we either require devices which can mod- ulate the amplitude and phase simultaneously, or if we only have access to either amplitude or phase control, we can make use of Fourier modulation techniques to acquire the desired output fields [91]. In this chapter, we will focus on the use of phase-only devices to implement the desired modulation of an incoming optical field, Ui(x, y). The first device we will discuss is the Liquid Crystal Spatial Light Modulator (which we will call an SLM), which can be digitally controlled to gener- ate any desired mode. The second device is a static geometric phase device known as the q-plate, which will be discussed in more depth in a proceeding section under the context of generating vector modes. 26 Chapter 2. Experimental Techniques The SLM functions as a Liquid Crystal (LC) display and allows the user to dy- namically tune the orientation of the LCs within every pixel of the screen. The ori- entation of these LCs is set by an applied voltage across the LCs within every pixel. Due to the birefringence of the LCs, controlling their orientation equips the user with a tool to locally tune the refractive index of the SLM screen thus inducing a spatially dependent phase delay of the incoming light [159]. It is important to note that these devices are birefringent, therefore only light of a particular polarization will experi- ence this phase delay. It is standard to orientate the SLM screen so that it modulates horizontally polarized light. A simplified structure of the SLM screen and function is depicted in Fig.2.1. FIGURE 2.1: A working illustration of the SLM display. A zooming into the SLM screen, shows that the LC molecules are sandwiched between a glass substrate, transparent film and alignment film on the one side, and another alignment film, mirror and silicon substrate on the other side. Controlling the applied voltage across each pixel allows for dynamic control over the orientation of the LC molecules, and thus the local refractive index, which in turn induces a phase delay on incident light. The SLM can be controlled via the use of gray-scale images, where the local gray-scale value sets the applied voltage at that pixel, and thus the induced phase shift. SLM technology has matured to a point where a wide range of devices with specific properties are currently available on the market. For this work, the Holoeye Pluto SLM series has been used, and is considered an all-rounded device with a pixel size of 8 µm and a resolution of 1920×1080 pixels. These SLMs are capable of an 8-bit encoding depth, that is a 0 to 2π phase step modulation broken up into 256 levels. The voltage can be discretely controlled via the use of specially designed gray-scale images where the discrete phase shift levels from 0 to 2π are encoded into the colour gradient from black (0) to white (2π). These gray-scale images are known as digital holograms, and can be directly computed for any desired field modulation. The two modulation techniques we will focus on are phase (with and without grating) and complex amplitude modulation. As mentioned before the SLM is a phase-only device, which means that any modulation of incoming light is achieved through the use of a phase holograms. In general, one can define the hologram that 2.1. Structured light techniques 27 encodes Φ∗(r) as [143, 160] H(Φ(r)) = f (Φ∗(r))× h (arg(Φ∗(r)) , (2.2) where f (·) is the relation chosen to describe the grating depth capable of mod- ulating for the correct amplitude in the image plane, while h(·) encodes the phase, arg(Φ∗(r)). By setting f (·) = 1, and h = mod [arg(Φ∗(r)), 2π], we perform a phase modula- tion that allows for the construction of the desired mode. This technique is known as phase modulation and works by constructing the desired phase of the final field at the SLM plane, and through propagation, the amplitude of the field is distributed to construct the desired field amplitude. An example of this modulation technique is shown in Fig. 2.2 a), where the phase hologram modulates the input field into the desired output field (Ideal) after propagation. The simplicity of this technique does not come without cost however, as there is typically some component of the incident light that is left unmodulated due to the inherent inefficiencies of the SLM. This unmodulated light follows the same optical path as the modulated light thus resulting in the final field being some superposition of the unmodulated (Ui) and the desired output field (U f ), shown as the superposition "Ui + U f " in Fig. 2.2 a) with an exaggerated contribution from the unmodulated light. To avoid this issue, we can make use of a grating which will separate out the modulated and unmodulated light into spatially disjoint zeroth and first diffraction orders as shown Fig. 2.2 b). We achieve this by setting h = mod [ arg(Φ∗(r)) + Λgrating, 2π ] , where the grating is controlled by the term Λgrating. However, phase modulation still has shortcomings, the amplitude and phase attained after propagation is not yet ideal, rather there is some contribution from undesired modes albeit the phase can usually be corrected by a second phase modulation at the focus of a lens [161]. For a single-step modulation technique, which produces the desired output field at the plane of the SLM we can employ complex amplitude modulation, where both the phase and amplitude information of the output field is encoded into the phase hologram. The amplitude information can be directly encoded into the function, f (Φ(r)), which is proportional to the field amplitude. An example of this is shown in Fig. 2.2 c) where the SLM has been encoded with the desired field amplitude through control of the grating depth. Explained simply, this technique takes in an input field, usually a planar wave with flat phase and amplitude across the field, and the desired output field amplitude is "carved" out of the input field, whilst the desired phase is simultaneously embedded into the field profile. Due to the existing grating, there is no distortion of the output field due to unmodulated light and furthermore, the SLM plane can be imaged to any plane using a 4f system of lenses. his specific method of complex amplitude modulation is rather efficient, however it may come at the cost of some purity since the field amplitude does not map linearly with conversion ef- ficiency. Several alternative methods of complex amplitude modulation exist [162– 164] where efficiency and modal purity are some of the key parameters discussed when comparing them.[165] One scheme that can produce modes with high pu- rity is when the amplitude function, f (·) is encoded as[163], f (Φ(r)) = J−1 1 (Φ(r)). Here, J−1 1 (·) is the Bessel function of the first kind. The phase function is altered as h = sin ( arg(Φ∗(r)) + Λgrating ) . Some experimentally generated spatial modes 28 Chapter 2. Experimental Techniques using complex amplitude modulation with this mentioned scheme, are shown in Fig. 2.3, and are in excellent empirical agreement with the theoretical modes shown. FIGURE 2.2: Illustration of the different modulation techniques capable of producing high fidelity spatial modes with a phase-only device. a) Phase modulation works by encoding the desired phase into the input field and observing the desired field after propagation. Ideally, all of the light is modulated by this phase, however, realistically, there is a portion of the light that is left unmodulated (Ui) and which interferes with the desired field (U f ). b) Adding a grating to the phase modulation technique, avoids interference issues between the modulated and unmodulated light by separating them out in space. c) Complex amplitude modulation accomplishes a single-step modulation where the desired field is generated at the plane of the SLM and can be imaged directly from the SLM plane Using a 4f imaging system 2.1. Structured light techniques 29 FIGURE 2.3: (Top) Experimental intensity images of scalar modes produced by displaying a digital hologram on a phase-only SLM using complex amplitude modulation. For empirical comparison, the (Bottom) simulated modes are also shown. 2.1.2 Polarization control As discussed in section 1.1.1, we can describe a particular polarization state using Eq.1.1 or equivalently using the Jones formalism |P⟩ = [ cos ( θ 2 ) eiα sin ( θ 2 )] . (2.3) For the purposes of transforming and analysing polarization states, we only re- quire three types of optics: linear polarizers (LPs), half-wave plates (HWPs) quarter- wave plates (QWPs). LPs are used to perform projective measurements onto a chosen linear state, and their action can be described using the Jones formalism as follows [77] LP(θ) = [ cos2(ϕ) cos (ϕ) sin (ϕ) cos (ϕ) sin (ϕ) sin2(ϕ) ] , (2.4) where ϕ is the angle between the optical axis and the horizontal. We can also describe the action of the LP as performing an overlap between an initial state |P⟩ and a chosen linear state |ϕ⟩ = cos(ϕ) |H⟩+ sin(ϕ) |V⟩, such that the final state is given by |P′⟩ = ⟨ϕ|P⟩|ϕ⟩. HWPs and QWPs induce a π and π 2 phase delay, respectively, between orthogo- nal polarization components and in general their action can be described using the Jones formalism as follows HWP(θ) = e−i π 2 [ cos2(ϕ)− sin2(ϕ) 2 cos (ϕ) sin (ϕ) 2 cos (ϕ) sin (ϕ) sin2(ϕ)− cos2(ϕ) ] , (2.5) QWP(θ) = e−i π 4 [ cos2(ϕ) + i sin2(ϕ) (1 − i)) cos (ϕ) sin (ϕ) (1 − i)) cos (ϕ) sin (ϕ) sin2(ϕ) + i cos2(ϕ) ] . (2.6) 30 Chapter 2. Experimental Techniques The phase delay incurred by a state passing through a HWP ensures that it only changes the states orientation, i.e., it maps linear states to linear states (example shown in Fig. 2.4 a) where the HWP rotates the state from |H⟩ to |V⟩) whilst the phase delay induced by a QWP changes the states ellipticity, i.e. it maps linear states to circular states and vice versa (example shown in Fig. 2.4 b) where the QWP maps the input state |H⟩ to |L⟩). The actions of these polarization optics are summarized in Fig. 2.4. With these two optics it is possible to attain every possible polarization state on the Poincaré sphere, in fact it has been shown that the use of two QWPs and one HWP (in any configuration), is sufficient to traverse the entirety of the Poincaré sphere[166]. The transformations between the six most common polarization states, |H⟩ , |V⟩ , |D⟩ , |A⟩ , |R⟩ , |L⟩, is summarized in Table 2.1, along with the orientation angles of QWPs and HWPs required to achieve said states. FIGURE 2.4: Polarization optics used for complete control over the polarization state of inci- dent light. a) Half-wave plate (HWP) commonly used to rotate polarization states from one linear polarization state to another, and b) a Quarter-wave plate (QWP) commonly used to transform between linear and circular polarization states. For horizontally polarized inci- dent light passing through a HWP and QWP orientated with their fast-axes 45◦ with respect to the x-axis, the output light has its polarization changed to vertical and left-circular, re- spectively. wave plate QWP HWP ϕ 0◦ 45◦ 90◦ 0◦ 22.5◦ 45◦ 90◦ |H⟩ |H⟩ |L⟩ |H⟩ |H⟩ |D⟩ |V⟩ |H⟩ |V⟩ |V⟩ |R⟩ |V⟩ |V⟩ |A⟩ |H⟩ |V⟩ |D⟩ |R⟩ |D⟩ |L⟩ |A⟩ |H⟩ |D⟩ |A⟩ |A⟩ |L⟩ |A⟩ |R⟩ |D⟩ |V⟩ |A⟩ |D⟩ |R⟩ |A⟩ |H⟩ |D⟩ |L⟩ |L⟩ |L⟩ |L⟩ |L⟩ |D⟩ |V⟩ |A⟩ |R⟩ |R⟩ |R⟩ |R⟩ TABLE 2.1: A summary of useful polarization transformations. Incident light of a particular polarization (far-left column) is transformed by a QWP and/or HWP, orientated with an angle ϕ with respect to the horizontal axis, into another polarization state. 2.1. Structured light techniques 31 2.1.3 Vector beam generation Possessing the tools for full spatial and polarization control, we can now consider how to combine these tools in order to achieve the generation of vectorial beams, where the polarization structure of these beams is spatially dependent. Digital devices In section 2.1.1, the use of an SLM device to impart a spatially dependent dynamic phase on incident light was discussed. The dynamically controlled phase modula- tion was achieved through control over the orientation of birefringent LCs within each pixel of the device which would then grant the user control over the refractive index, thus inducing a phase shift at each pixel. This allows for complete control over the phase and amplitude at each point in space. However, due to the birefrin- gent nature of this device, only incident light that is horizontally polarized (the exact polarization that gets modulated also depends on properties of the device itself as well as the orientation of the device with respect to the incident light, however, with- out loss of generality, we can configure the optical setup such that the SLM is setup to modulate horizontally polarized light) will be modulated. Therefore, in order to generate vector beams, we require an optical setup which is capable of modulating two orthogonally polarized beams, and then recombining them once again (assum- ing that the two orthogonally polarized beams are not spatially separted when they are modulated.) to form the desired superposition. Fig. 2.5 depicts the idea behind vector beam generation using SLMs. Two horizontally polarized beams are inci- dent on SLMs. The fields U1 and U2 are encoded onto the SLMs with appropriately chosen holograms, and then are embedded onto the individual beams. After this modulation, one of the beams is sent through HWP, which rotates its polarization to vertical polarization before the two beams are superimposed using a beam splitter (BS). After this process, we attain the vector mode |Ψ⟩ = U1 |H⟩+ U2 |V⟩. If one de- sires to change the polarization basis, to some arbitrary basis {|P+⟩ , |P−⟩}, then we just need to pass the beam through an appropriate set of polarization optics (POs) in order to generate the final vector mode |Ψ⟩ = U1 |P+⟩+ U2 |P−⟩. Geometric optics Another approach to generate vector beams is the use of geometric optics to impart different phases onto orthogonal polarization states, simultaneously. Instead of imparting a spatially dependent dynamic phase, as is done with an SLM, geometric optics offer an alternative approach by using the ’memory’ of locally ap- plied transformations. Here, by altering parameters adiabatically in a closed loop fashion, a geometric phase [167] may be induced equal to half the solid area bounded by the transformations in the parameter space. Applied to light, one may use a change in polarization to induce such a relative phase in the electric field, as illus- trated in Fig. 2.6 (a) and is otherwise known as Pancharatnam-Berry phase. [168– 170] To illustrate this phenomenon, we first consider the path that an incident beam’s polarization traces out on the surface of the Poincaré sphere as it passes through a series of polarization optics. In Fig. 2.6 a) diagonally polarized light passes through 32 Chapter 2. Experimental Techniques FIGURE 2.5: Concept for generation of arbitrary vector modes using SLMs. Two horizon- tally polarized beams are incident on SLMs which modulate them independently. Before recombining the beams with a beam splitter (BS), one is passed through a HWP to rotate it to the orthogonal polarization. A vector beam is then generated on the output port of the BS, and the final desired vector beam is generated after passing through an appropriate set of polarization optics (PO). a HWP (A), QWP (B) and then one last QWP (C), such that its polarization trans- formation can be summarized as follows |D⟩ A−→ |H⟩ B−→ |R⟩ C−→ eiϕG |D⟩ or using shorthand |D⟩ CBA−−→ eiϕG |D⟩, where ϕG is the geometric phase acquired due to the closed loop evolution of the state. To find this geometric phase, we consider the path traced out on the surface of the Poincaré sphere as depicted in Fig. 2.6 b). The trans- formations, A,B and C, form a closed path bounding a solid area Ω which means that the accrued geometric phase is given by ϕG = eiΩ/2. Suppose then that instead of applying each transformation consecutively in time, we apply them spatially, that is that at every point across the transverse plane, the incident light sees a waveplate with a different orientation, τ(r, θ), where r, θ are radial and azimuthal coordinates. As such a change in polarization occurs at points across the transverse spatial plane of the incident beam, an associated spatially-varying geometric phase is also gener- ated. This is illustrated in Fig. 2.6 c), showing how geometric phase may be gathered in the transformation from right-circularly polarized light to left-circularly polarized light when passing through four arbitrary points (denoted τ0(r, ϕ) to τ3(r, ϕ)) on an optic (c) which has effective HWP elements with different orientations at each trans- verse coordinate. With the polarization change occurring spatially, the ‘closed path’ is formed by the difference in path between the other elements, resulting in the geo- metric phase being both relative and spatially varying[171, 172]. One may therefore engineer the relative optical axis orientations to generate any number of variable ge- ometric phase acquisitions that may be used to manipulate the spatial mode of the overall beam. 2.1. Structured light techniques 33 FIGURE 2.6: The concept of geometric phase. (a) Illustration of a diagonally polarized state traversing a series of waveplates that take the state through a closed loop transformation which is (b) depicted on the Poincaré sphere. (c) Depiction of an element with spatially varying optical indices of refraction which may be considered as a series of HWPs with varying optical axes. (d) Illustration of associated paths taken by different rays of a light beam traversing an element with four arbitrary HWPs with varying fast axis rotational an- gles. A q-plate (QP) [173] is one such device which does this by patterning the optical axis orientation, such that the relative phase changes azimuthally and thus creates an azimuthal phase across the entire beam [173]. This patterning of the optical axis is described by α(r, ϕ) = qϕ + αo. (2.7) Here the QP is taken to be in the xy-plane, αo is the angle formed by the optical axis from the x-axis, q is a constant defining the number of times the optical axis rotates in a path as it traverses once around the plate center, also known as the QP charge and αo is the permanent offset of the optical axis from the element’s x-axis. A discontinuity occurring at r = 0 when q ̸= 0 is evident due to the nature of the azimuthal coordinate, ϕ, being undefined at this point. 34 Chapter 2. Experimental Techniques A simple description of an ideal QP (also referred to as being ’tuned’) action can is given in terms of its Jones matrix[174]: QP = i [ cos (2α(r, ϕ)) sin (2α(r, ϕ)) sin (2α(r, ϕ)) − cos (2α(r, ϕ)) ] . (2.8) It must be noted that Eq. 2.8 refers to an ideal QP, however QPs typically carry some inherent inefficiency even for their designed wavelength, which expectantly degrades the modal purity of the output mode, as the unmodulated light travels collinear to the modulated light, not unlike phase modulation using the SLM. As- suming that the inefficiency is minor, Eq. 2.8 perfectly captures the QP action. One can further simplify the QP action by identifying the following set of simple selection rules [174]: QP |ℓ, R⟩ → |l + 2q, L⟩ , (2.9a) QP |ℓ, L⟩ → |l − 2q, R⟩ , (2.9b) where we have used the shorthand notation, |ℓ, P+⟩ to denote a state of light with OAM ℓ and polarization P+. While useful, the QP is limited by the physical principles governing geometric phase only. As shown in Eq. 2.9 the operation of the QP yields conjugate OAM contributions to orthogonal polarizations. An alterna- tive to QPs uses the properties of meta-surfaces [175, 176] instead of liquid crystals. Here, the desired phases are achieved by spatially engineering the characteristics of sub-wavelength structures, known as meta-atoms. This is usually achieved through altering the orientation and dimensions of the meta-atom structures. Meta-surfaces can subsequently replicate the action of the QP [177–180], but can then be extended to have more control, enabling the controlled pairing of both geometric and propa- gation phase such that arbitrary polarization states can be mapped to arbitrary in- crements of OAM [181, 182], or even, superpositions thereof [179, 183]. For instance, one may engineer the metasurface such that a generalised conversion of OAM is possible for arbitrary linear polarization bases states,[181] |P+⟩ and |P−⟩, i.e., the action of such a device would yield the following selection rules JP ∣∣ℓ, P+ 〉 → ∣∣ℓ+ m, (P+)∗ 〉 , (2.10a) JP ∣∣ℓ, P−〉→ ∣∣ℓ+ n, (P−)∗ 〉 , (2.10b) with n, m ∈ Z. These devices thus allow one to arbitrarily map between polarizations and orbital angular momentum, allowing one control of the full angular momentum (J) domain and, as such, have been denoted J-Plates. The capabilities of these devices go beyond single-step spatial modulation. Rather, by selecting an appropriate polarization state for the incident light (a superposition of the polarization basis states on which the device acts independently), we attain that the orthogonal components of said state are modulated independently with dif- ferent spatial modes, thus producing a vector mode in a single step. For example, 2.1. Structured light techniques 35 we consider the case shown in Fig. 2.7, where horizontally polarized light with ℓ = 0 is incident on a QP with charge 1 2 . The QP transformation according to the selection rules given in equation 2.9 yields the following output state QP |0, H⟩ = 1√ 2 QP (|0, R⟩+ |0, L⟩) (2.11) = 1√ 2 (|1, L⟩+ |−1, R⟩) (2.12) where |1⟩ = eiϕ and |−1⟩ = e−iqϕ. As before, a set of POs allows for the mod- ulation of the basis polarization states of the vector beam. Finally, a lens is then used to observe the far-field of the QP (this is necessary as the QP performs phase modulation) where the desired vector beam is found |Ψ⟩ = LG1 |P+⟩+ LG−1 |P−⟩. FIGURE 2.7: Vector beam (VB) generation using geometric optics. Horizontally polarized light is incident on a q-plate which independently modulates the right- and left- circular components of the field. Polarization optics (POs) are used to shift to the polarization basis of choice, |P+⟩ , |P−⟩ desired polarization basis and a lens generates the desired final VB at its focal plane. 2.1.4 Vector beam characterization Following the creation of VBs it is necessary to characterize their spatially dependent polarization and deduce its VQF. Since VBs naturally couple the polarization and 36 Chapter 2. Experimental Techniques spatial degrees of freedom of light it becomes necessary to measure both degrees of freedom simultaneously. To do this we employ a technique known as Stokes polarimetry [63] which reconstructs the stokes field through a series of spatially- resolved intensity measurements by projecting different polarization states onto a CCD. The stokes field, S⃗, as previously discussed, is a vector field with 4 compo- nents, such that S⃗ = [ S0 S1 S2 S3 ]T, where Si = Si(x, y) are scalar fields known as the stokes parameters. With this in mind, characterizing a VB surmounts to mea- suring the Stokes parameters which enables the reconstruction of the stokes field of the VB. Each stokes parameter can be extracted from two intensity measurements according to S0 = IH + IV (2.13a) S1 = IH − IV (2.13b) S2 = ID − IA (2.13c) S3 = IR − IL, (2.13d) where IH, IV , ID, IA, IR, IL are spatially resolved intensity measurements obtained when projecting the VB into horizontal, vertical, diagonal, anti-diagonal, right-circular and left-circular states, respectively. If the degree of polarization (DOP) of the beam is unity (or close to unity), then the required number of measurements can be re- duced to 4, a process typically known as reduced Stokes polarimetry [63]. In this process, we can still extract the desired stokes parameters by measuring IH, IV , ID, IR and computing the stokes parameters using S0 = IH + IV (2.14a) S1 = 2IH − S0 (2.14b) S2 = 2ID − S0 (2.14c) S3 = 2IR − S0, (2.14d) Having extracted the Stokes parameters, we are then left with characterizing the quality of the VB. This is done by calculating its VQF from the extracted Stokes parameters. To do so we start by modifying the Stokes parameters above as follows [105, 184] S0 = I ( |u1|2 + |u2|2 ) , (2.15a) S1 = I ( |u1|2 − |u2|2 ) (2.15b) S2 = 2I (|u1||u2| cos(arg(|u1|)− arg(|u2|))) (2.15c) S3 = 2I (|u1||u2| sin(arg(|u1|)− arg(|u2|))) , (2.15d) where I = IH + IV . We can then compute the terms within Eq. 1.16 as follows ⟨u1|u2⟩ = ∫ |u1||u2|i(arg(|u1|)−arg(|u2|)dxdy) = 1 2s0 (s2 − is3) , (2.16) 2.1. Structured light techniques 37 ⟨u1|u1⟩ ⟨u2|u2⟩ = ∫ |u1|2dxdy ∫ |u2|dxdy = 1 4 [ 1 − ( s1 s0 )2 ] , (2.17) where si = ∫ Sidxdy. Then substituting Eq. 2.16 and Eq. 2.17 into Eq. 1.16 we obtain VQF = √ 1 − s2 1 + s2 2 + s2 3 s2 0 . (2.18) We note that SBs have a VQF = 0 denoting the maximal separability between po- larization and spatial DOFs and VBs VQF = 1 denoting the maximal non-separability between polarization and spatial DOFs. 2.1.5 Vector beam setup Combining all the techniques discussed up to this point, allows for the full genera- tion, detection and analy