Propagation of electromagnetic helmholtz gauss beams through paraxial optical systems and spatial correlation vortices

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(1)INSTITUTO TECNOLÓGICO Y DE ESTUDIOS SUPERIORES DE MONTERREY CAMPUS MONTERREY GRADUATE PROGRAMS IN MECHATRONICS AND INFORMATION TECHNOLOGIES. DOCTOR OF PHILOSOPHY INFORMATION TECHNOLOGIES AND COMMUNICATIONS MAJOR IN OPTICAL SCIENCES Propagation of electromagnetic Helmholtz-Gauss beams through paraxial optical systems and spatial correlation vortices by Raúl Ignacio Hernández Aranda. MAY 2008.

(2) Propagation of electromagnetic Helmholtz-Gauss beams through paraxial optical systems and spatial correlation vortices A Dissertation Presented by Raúl Ignacio Hernández Aranda Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Information Technologies and Communications Major: Optical Sciences. Thesis Advisor: Dr. Julio C. Gutiérrez Vega, ITESM Campus Monterrey Thesis Committee: Dr. Grover A. Swartzlander, Jr., The University of Arizona Dr. Rodolfo Rodrı́guez y Masegosa, ITESM Campus Monterrey Dr. Hugo Alarcón Opazo, ITESM Campus Monterrey Dr. Carlos M. Hinojosa Espinosa, ITESM Campus Monterrey. Optics Center Instituto Tecnológico y de Estudios Superiores de Monterrey Campus Monterrey May 2008.

(3) Instituto Tecnológico y de Estudios Superiores de Monterrey Campus Monterrey Graduate Program in Mechatronics and Information Technologies The committee members hereby recommend the dissertation presented by Raúl Ignacio Hernández Aranda to be accepted as a partial fulfillment of the requirements for the Degree of Doctor of Philosophy in Information Technologies and Communications, Major in Optical Sciences.. Dr. Julio C. Gutiérrez Vega Dissertation Advisor. Dr. Grover A. Swartzlander, Jr.. Dr. Rodolfo Rodrı́guez y Masegosa. Committee Member. Committee Member. Dr. Hugo Alarcón Opazo. Dr. Carlos M. Hinojosa Espinosa. Committee Member. Committee Member. Dr. Graciano Dieck Assad Director of Graduate Programs in Mechatronics and Information Technologies. i.

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(5) Declaration. I hereby declare that I composed this dissertation, and that the work presented here is original, describing my own research, except where indicated. No part of this thesis has been submitted for any other degree. A number of original results where found in collaboration with my supervisors, Prof. Julio C. Gutiérrez Vega, and Prof. Grover A. Swartzlander, Jr.. Any views expressed in this thesis are those of the author, and do not represent in any way those of Instituto Tecnológico y de Estudios Superiores de Monterrey.. Raúl Ignacio Hernández Aranda Monterrey, N.L., México May 2008. iii.

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(7) Abstract The propagation of electromagnetic beams through a large variety of optical systems has received much attention during the last two decades. Similar attention has been paid to singular points arising on fields that represent optical beams, also known as optical vortices. This thesis studies the propagation of electromagnetic Helmholtz-Gauss beams through optical systems that can be represented by an ABCD matrix. Closed form expressions for the field distributions are derived, showing that the propagation of these beams can be described by the paraxial transformation of two parameters. Three special cases of ABCD systems are considered, namely a free-space segment, a quadratic index medium, and an unapertured thin lens. A complete characterization of these beams is made by studying their polarization properties and angular spectrum distribution. An important phenomenon known as the focal-shift is studied for the case of vector Mathieu-Gauss beams, and the existence of this shift is confirmed theoretically by means of two methods. All studies previously mentioned are performed assuming highly coherent sources. Partially coherent optical systems are considered as well, but we now look at singularities of the wave function describing the optical beam, which are commonly known as optical vortices. It is shown that partial coherence requires a beam to exhibit Rankine vortex characteristics, in analogy to vortices in fluids. We suggest a method to study the coherence properties of a beam by looking at its cross correlation function. Potential applications of this method are proposed, for instance we introduce a “vortex stellar interferometer” to determine the angular extent of distant objects such as stars, and the measurement of orbital angular momentum content in an optical beam is briefly discussed.. v.

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(9) Acknowledgements First, I would like to express my sincere gratitude to my dissertation advisor, Prof. Julio Gutiérrez, for his guidance and help to bring this work to completion, and for his continued support and friendship through all these years. It is a blessing to work under his guidance. I would also like to thank Prof. Grover Swartzlander for introducing me to the field of singular optics, for guiding me and making me feel so welcomed during my stay at the College of Optical Sciences, in Tucson. I also thank him for the many hours in the lab, and for the very fruitful lunchtime discussions. It has been a privilege to collaborate with him and his group. Thanks indeed to Erin, Jenn, and Rukiah, for making the lab more enjoyable. I am also grateful to the members of my Dissertation Committee, Dr. Rodolfo Rodrı́guez y Masegosa, Dr. Hugo Alarcón Opazo, and Dr. Carlos Hinojosa Espinosa, for taking the time to review my manuscript and helping me improve it, and also for the good food on Thursdays. Many thanks to all my friends at Tec, especially to “La Pecera” past and present members: Raco, Carlos, Servando, Bandrés, Ponce, Jezzini, Peter, Josué, Rod, Polo, and Dorilián for their incredible tolerance to my jokes, for listening to my nonsensical conversations, and for adding so much fun to this journey. I would also like to thank all my friends in Tucson for taking good care of me, and for their direct and indirect contributions to the conclusion of this thesis. A great deal of thanks to my brother and sister in Christ, Brett Major and Andrea Norcross, for all the fun, the good music, and better food. Massive thanks to my family in Christ, in Tucson: Jon Heine, Tom, Maggie, Marten, Lu, Tee Wei, Pick Chung, Keqian, Nick, Lisa, Bill and Donna Wilson, Clay and Bonnie Williams, Dede, Akong, Jason, John and Sharon Lew, Jeong-Yeol and Seunghee, Peter, Ji-Young, Raymond, Rolphin, Karen, Cathy, Mercy, Catherine, Makiko, Andrew, Christopher, Meryanne, Ivy, Dai Xin, and all ISF people I am missing, for their everyday prayers, the Friday night fellowship, the food, the foosball games, and mainly for loving me as they do. Quiero agradecer infinitamente a toda mi familia, principalmente a papá y mamá. A papá porque su recuerdo ha sido una de mis mayores motivaciones, y porque su amor y cuidado siguen presentes en nuestras vidas. A mamá, por todo su amor, por saber sonreir siempre y alegrar mis dı́as, por sus incontables horas de sueño en el sillón esperando mi llegada, por sus consejos y su interés en mi trabajo, y por dejarme seguir siendo niño. Gracias a mi hermano Luis por todo su apoyo, por sus palabras de aliento, y por ser mi mejor amigo. Gracias a mi hermana Anel, por sus juegos y travesuras, y por todo su cariño. Gracias también a Lore por ser mi segunda hermana. Y especialmente, gracias a Luisito, por su incansable curiosidad y su inagotable ternurna, por la alegrı́a que ha traı́do a nuestras vidas durante estos tres años. Gracias también a mi hermano Julio Ernesto por su motivación y amistad en todo momento. Agradezco también a mi hermano José Guadalupe, por su confianza y contribuciones para ponerle sabor al caldo..

(10) Gracias también a toda la Familia González, y en especial a papá Ray y mamá Fina, por todo su cariño y apoyo, y por recibirme siempre con una sonrisa. Finalmente, quiero agradecer a Marcela por su amor incondicional, por acompañarme en este recorrido y estar siempre a mi lado aun en la distancia, por su paciencia, comprensión, motivación, y cuidados, por compartir conmigo un proyecto de vida, y ser al mismo tiempo novia, esposa, y mi mejor amiga. Gracias Bebé. Above all, I would like to thank God for all the blessings in my life. For the life we have in Christ, for his justice and love. And because if you acknowledge Him in all your ways, he will make your paths straight. (Pr. 3:6). RAÚL IGNACIO HERNÁNDEZ ARANDA. Instituto Tecnológico y de Estudios Superiores de Monterrey May 2008.

(11) To Marcela.

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(13) Contents Committee Declaration. i. Declaration. iii. Abstract. v. List of Figures. xvii. 1 General Overview. I. 1. 1.1. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1. 1.2. Thesis structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2. 1.3. Main Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2. Electromagnetic Helmholtz-Gauss beams. 2 Wave equation for electromagnetic fields. 5 7. 2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7. 2.2. Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 9. 2.3. Wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3.1. Helmholtz Equation . . . . . . . . . . . . . . . . . . . . . . . . . 10. 2.3.2. Paraxial Wave Equation (PWE) . . . . . . . . . . . . . . . . . . . 11. 3 Generalized vector Helmholtz-Gauss beams 3.1. 13. Definition of generalized vector Helmholtz-Gauss beams . . . . . . . . . . 14 3.1.1. TM polarized gVHzG beams . . . . . . . . . . . . . . . . . . . . . 15 xi.

(14) 3.2. 3.3. 3.1.2. TE polarized gVHzG beams . . . . . . . . . . . . . . . . . . . . . 16. 3.1.3. Classification of the generalized vector HzG beams . . . . . . . . 17. 3.1.4. Propagation through ABCD systems . . . . . . . . . . . . . . . . 19. 3.1.5. Poynting vector of the generalized vector HzG beams . . . . . . . 20. 3.1.6. Propagation of the vector angular spectrum . . . . . . . . . . . . 21. 3.1.7. Remarks on the coordinates systems and polarization basis . . . . 22. Physical discussion of the propagation properties. . . . . . . . . . . . . . 24. 3.2.1. Free space propagation . . . . . . . . . . . . . . . . . . . . . . . . 24. 3.2.2. Propagation through a GRIN medium . . . . . . . . . . . . . . . 27. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32. Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4 Focal shift in vector Mathieu-Gauss beams 4.1. 4.2. 4.3. II. 35. Vector Mathieu-Gauss beams . . . . . . . . . . . . . . . . . . . . . . . . 36 4.1.1. General description of a vector Mathieu-Gauss beam . . . . . . . 36. 4.1.2. Focusing of vector Mathieu-Gauss beams . . . . . . . . . . . . . . 38. 4.1.3. Focal shift in vector Mathieu-Gauss beams . . . . . . . . . . . . . 39. Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.2.1. Focal shift of vMG beams Vs. Polarization . . . . . . . . . . . . . 44. 4.2.2. Focal shift Vs. Gaussian-Fresnel number . . . . . . . . . . . . . . 44. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45. Spatial coherence and optical vortices. 5 Optical Vortices. 47 49. 5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49. 5.2. Mathematical definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 50. 6 The Optical Rankine Vortex (ORV) 6.1. 53. Geometrical optics analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 53.

(15) 6.2. Wave-optics analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55. 6.3. Anomalous circulation of light . . . . . . . . . . . . . . . . . . . . . . . . 59. 6.4. Potential applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61. 6.5. 6.4.1. Vortex stellar interferometer . . . . . . . . . . . . . . . . . . . . . 62. 6.4.2. Orbital angular momentum (OAM) content measurements . . . . 63. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63. 7 General conclusions 7.1. 65. Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66. Bibliography. 67.

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(17) List of Figures 2.1. Different models for the description of optical phenomena. . . . . . . . .. 3.1. ABCD system for the propagation of gVHzG beams. . . . . . . . . . . . 14. 3.2. Physical picture of the decomposition of a gVHzG beam propagating in free space, in terms of fundamental vector Gaussian beams whose mean propagation axes lie on the surface of a double cone. . . . . . . . . . . . . 25. 3.3. Propagation of the transverse intensity distribution and electric field for circularly polarized gVHzG beams, constructed with a finite superposition of vector Gaussian beams. (Left column) Initial profile, (Right column) Fourier transformed profile. The parameter data for the propagations are included within the text. . . . . . . . . . . . . . . . . . . . . 29. 3.4. Propagation of the transverse intensity distribution and electric field for generalized vector Bessel-cosine-Gauss, Mathieu-Gauss, and parabolicGauss beams. (Left column) Initial profile, (Right column) Fourier transformed profile. The parameter data for the propagations are included within the text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31. 4.1. Focusing system under consideration. The lens is assumed to be thin and unapertured, f defines the lens focal distance, i.e. geometrical focus, z0 the actual focus, and ∆z the focal shift. . . . . . . . . . . . . . . . . . . 38. 4.2. Transverse intensity profiles of a TM type focused vMG beam with seed function W = Wme (ξ, η, ε), ε = 5 and beam orders (a) m = 0, (b) m = 1, (c) m = 2, and (d) m = 7. Plots in the top row correspond to initial profiles at plane z = 0, bottom row images show the intensity profile near the focal region, at plane z/f = 0.95. . . . . . . . . . . . . . . . . . . . . 41 xv. 8.

(18) 4.3. Transverse intensity profiles of a TM type focused vMG beam with seed function W = Wme (ξ, η, ε) + iWmo (ξ, η, ε), ε = 5 and beam orders (a) m = 4, and (b) m = 7. Plots in (c) and (d) correspond to a right-circularly polarized vMG beam with seed function W = Wme (ξ, η, ε), ε = 5, and beam orders m = 0 and m = 7 respectively. Plots in the top row show initial profiles at plane z = 0, bottom row images show the intensity profile near the focal region, at plane z/f = 0.95. . . . . . . . . . . . . . 42. 4.4. Beam waist plotted against the normalized propagation distance z/f , for ε = [0, 5, 25] and mode orders m = 0 through 7. (a) Calculation of beam waist using the encircled energy criteria and (b) using the mean squared radius of the intensity distribution. The minimum beam waist is achieved just before the geometric focus, i.e. z/f = 1, indicating a focal shift ∆z towards the lens. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43. 4.5. Relative focal shift versus Gaussian-Fresnel number Nw of the beam for ε = 5 and beam orders m = [0, 1, 7]. A change in Nw was achieved by changing the Gaussian waist w1 from 600 µm to 2000 µm. All simulation parameters are the same as in Fig. 4.2. Notice how narrower beams present larger shifts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45. 5.1. Amplitude profiles of (a) a large-core vortex and (b) small-core vortex embedded in a Gaussian background beam of waist w0 . (c) Phase profile of the two optical vortices. The topological charge is m = 1, a negative value of m represents a variation of the phase from 0 to 2πm, measuring in the clockwise direction. The line plots are one dimensional amplitude profiles, that show the vortex core size in comparison to the waist of the background beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52. 6.1. Laboratory system for the generation of a partially coherent optical vortex. An extended incoherent source illuminates the vortex lens at a distance d. The projection of the light source through the center of the vortex lens demarks a boundary of radius a′ between solid-body and fluid-like circulation of light. The vortex charge density inside this region is uniform. 54. 6.2. Relationship between vectors r, ρj , and sj defined at the plane of the screen, (x, y). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55. 6.3. (a) Intensity I(r, θ), (b) phase, and (c) cross-correlation χ(r, θ) profiles for a single member of an ensemble. . . . . . . . . . . . . . . . . . . . . . 56.

(19) 6.4. (a) Scatter plot showing the random locations of the composite vortices, the circle represents the size of the projected source on the screen, namely a′ . (b) Distribution of vortex positions for an ensemble of 300 members (N’=300). (c) Relative standard deviation σ of histograms for different values of a′ . The small values of |Rpd − σ|, squared plot in (c), suggest an equivalence between Rpd and σ. The continuous line is the least-squares fit: Rpd /w ≈ σ/w = a′ /(a′ + w). . . . . . . . . . . . . . . . . . . . . . . . 57. 6.5. Ensemble average (a) intensity and (b) cross-correlation profiles of a partially coherent vortex beam. The ensemble is constructed from N ′ = 300 members. Note the presence of the circular ring dislocation in the cross correlation, which is characteristic for a beam carrying an optical vortex.. 58. 6.6. Solid lines: Ensemble average topological charge hM (r)i as a function of the distance r from the optical axis, note that the distance is normalized in terms of the dislocation radius Rpd . Dashed lines: Normalized ensemble average azimuthal wave vector hkθ (r)iRpd . Black (gray) lines correspond to uniform (Gaussian) background beams. The circulation 2πhM (r)i is anomalous for the case of a Gaussian background beam. . . . . . . . . . 61. 6.7. Configuration of a Wavefront Folding Interferometer (WFI) for crosscorrelation measurements. Lenses L1 and L2 image the aperture A1 onto the vortex lens V L with unity magnification. L3 and L4 image the plane at a distance d′ , from the vortex lens, onto the camera after passing through the WFI. D1 and D2 are dove prisms, and F is a filter used to eliminate temporal coherence effects. . . . . . . . . . . . . . . . . . . . . 62. 6.8. Cross correlation profiles of partially coherent vortices composed by N ′ = 1000 ensemble members carrying individual vortices of topological charges m0 = 1 through 6. The number of ring dislocations in the cross correlation is equal to the value of the topological charge. . . . . . . . . . . . . . . . 63.

(20) Chapter 1 General Overview 1.1. Motivation. A vast majority of physical phenomena can be described in terms of waves, for instance light, sound, ocean waves, earthquakes, electric signals, thermal radiation, and many others. Among them the study of light phenomena stands out, and has captivated our minds since ancient times. This thesis is a study of light phenomena, aiming at improving our understanding of how light propagates, diffracts, and interferes. It is clear that the advancement of optical sciences has contributed enormously to the development of technologies and systems that now play an important, and almost indispensable role in our every day life. Good examples are laser systems for all types of applications, optical fibers in telecommunications, liquid crystals displays, security systems, and measuring devices for metrology and medical systems. Regarding the subjects of these thesis, a good understanding of vector solutions to the wave equation is essential for the development of laser cutting technologies, particle and atom trapping, as well as for radar, antennas, and fiber optics technologies. Similarly, optical vortices play a key role in applications of optical micromanipulation, particle rotation, information encoding, filtering, and quantum information systems. Therefore, we embraced this thesis project in the hope that our findings will bring new perspectives in the subjects of electromagnetic solutions of the wave equation, and singularities in optical wave fields, also known as optical vortices. 1.

(21) 2. CHAPTER 1. GENERAL OVERVIEW. 1.2. Thesis structure. This thesis is divided into two parts. Part I presents the results regarding the description of generalized vector Helmholtz-Gauss beams and their propagation through ABCD optical systems. Chapter 2 provides an introduction to the work done by different authors on the study of solutions to the wave equation. Next we present the theoretical framework required to describe the propagation of electromagnetic fields in free space, which is provided by Maxwell’s equations. Chapter 3 gives an additional account of different vector solutions to Maxwell’s equations. The theoretical description of the generalized vHzG (gVHzG) beams is provided here, and the propagation of gvHzG beams through ABCD systems is studied. Two particular cases are considered, namely the propagation through a GRIN medium, and the propagation through a focusing system. Chapter 4 addresses the phenomena of focal shift for these beams, where we describe the shift by means of two different approaches. Each chapter concludes by summarizing the most important results. In Part II we study optical vortices in spatial partially coherent light. A review and introduction to the field of optical vortices are given in Chapter 5. In Chapter 6 we examine in detail the properties of an optical vortex in a spatially incoherent beam, and show that it is analogous to Rankine vortices that appear in fluids. The analysis is based on a geometrical and wave optics description. Important applications of our results are outlined. An experimental setup for the study of partially coherent vortices is also presented. The chapter closes with some concluding remarks. Chapter 7 provides the general conclusions of this thesis, and describes some of the future work that can be done on the basis of this research.. 1.3. Main Contributions. This dissertation addresses several problems in optical sciences, mainly related to finding general vectorial solutions of Maxwell’s wave equations, and to improving our understanding of optical vortices in partially coherent systems. The main contributions of this thesis and its collaborators are the following: • The finding of generalized solutions to the vector paraxial wave equation, and closed form expressions for their propagation through paraxial optical systems, which are independent of the coordinate system chosen for the description of the electromagnetic field..

(22) 1.3. MAIN CONTRIBUTIONS. 3. • Characterization of the focal shift phenomena in this type of generalized solutions, by means of two different methods. • Introduction of the optical Rankine vortex in partially coherent light, and its description as a general state of optical vortices occurring in nature. • A method for measuring the angular extent of spatially unresolved distant objects. • A method for the determining the orbital angular momentum content of a light beam. As a result of this research, a number of articles have been published, or are in preparation for publication in scientific journals, and as conference proceedings. The following is a list of the related works.. Journal papers • R. I. Hernandez-Aranda, R. Celechovský, and G. A. Swartzlander, Jr., “Experimental propagation of spatial correlation vortices,” in preparation. • R. I. Hernandez-Aranda and J. C. Gutiérrez-Vega, “Focal shift in vector MathieuGauss beams,” Opt. Express 16 (8), 5838–5848 (2008). • G. A. Swartzlander, Jr., and R. I. Hernandez-Aranda, “Optical Rankine vortex and anomalous circulation of light,” Phys. Rev. Lett. 99, 163901 (2007). • R. I. Hernandez-Aranda, J. C. Gutiérrez-Vega, M. Guizar-Sicairos, and M. A. Bandres, “Propagation of generalized vector Helmholtz-Gauss beams through paraxial optical systems,” Opt. Express 14 (20), 8974–8988 (2006). Conference proceedings • R. I. Hernandez-Aranda, J. C. Gutiérrez-Vega, “Focal shift in vector MathieuGauss beams,” Proc. SPIE 6422, 64221K (2007). • R. I. Hernandez-Aranda, J. C. Gutiérrez-Vega, M. Guizar-Sicairos, and M. A. Bandres, “Propagation of focused vector Helmholtz-Gauss beams,” Frontiers in Optics, OSA Technical Digest Series (Optical Society of America, 2006), paper JWD2..

(23) 4. CHAPTER 1. GENERAL OVERVIEW • R. I. Hernandez-Aranda, J. C. Gutiérrez-Vega, and M. A. Bandres, “Propagation dynamics of vector Mathieu-Gauss beams,” Proc. SPIE 6290, 629011 (2006)..

(24) Part I Electromagnetic Helmholtz-Gauss beams. 5.

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(26) Chapter 2 Wave equation for electromagnetic fields 2.1. Introduction. Different approaches can be adopted for the description of optical phenomena. In general they are considered as four different models, shown in Fig. 2.1. The main differences between each of the models consist on the simplifications made, and consequently on the accuracy of description that one can obtain for a specific problem. Geometrical optics is usually regarded as the simplest model, where the optical field is represented as a bundle of rays propagating along straight lines in free space, and which reflect and refract at smooth surfaces following Snell’s law. Although this model is extensively used in the design of macroscopic optical elements, it does not consider the effects of diffraction and interference, which are a consequence of the wave nature of light. In the wave optics model, light is described by a scalar wave function satisfying the wave equation, but in addition to reflection and refraction phenomena, diffraction and interference effects are now taken into account. However, when one is interested in studying the interaction with structures at the wavelength scale, the behavior of light at the boundary regions of an optical element, or when the polarization of light plays an important role, the wave optics model is inadequate. Therefore, when the vectorial nature of light becomes relevant the electromagnetic optics model is required to account for the effects of polarization. In this model, light is regarded as an electromagnetic wave, where the wave function has now a vectorial nature, and is described by a pair of three dimensional vector fields which obey Maxwell’s equations. The most complex model, and complete, for describing optical phenomena is provided by quantum optics, where the fields are now considered to be quantized. The 7.

(27) 8. CHAPTER 2. WAVE EQUATION FOR ELECTROMAGNETIC FIELDS. Figure 2.1: Different models for the description of optical phenomena.. quantum optical description is required when one is dealing with systems consisting of a few photons, or for describing the microscopic interaction between light and matter. Although this model provides the most complete description for optical phenomena, the calculations usually become very complicated, and sometimes unnecessary. In this thesis we will not make use of the quantum optics model, and will restrict ourselves to work within the framework of the remaining three models previously described. The spatial characteristics of light beams propagating in free space are most commonly described by solutions of the scalar Helmholtz equation. In recent years special solutions to the Helmholtz equation, usually referred to as ‘nondiffracting’ [1, 2], have received much attention, these possess interesting properties such as their suppressed diffraction divergence during propagation, the self-healing (or self-reconstruction) ability to reshape their intensity profiles after an obstacle, and in some cases the transportation of orbital angular momentum (OAM) [3, 4]. Good examples of these solutions are Bessel beams [1], Mathieu beams [5], and parabolic beams [6]. However, the physical realization of these type of solutions is unfeasible, since the generation of an ideal nondiffracting beam will require infinite energy. Approximate solutions have been proposed to overcome this limitation, mainly by means of apodization of the beams [7, 8]. A general type of approximate solutions to nondiffracting beams was presented by Gutiérrez-Vega and Bandres, who named them Helmholtz-Gauss beams [9], which constitute exact solutions to the scalar paraxial wave equation (PWE). It was in a later work [10], where the same authors introduced two new families of electromagnetic beams which are solutions to the vector paraxial wave equation (vPWE), namely vector Helmholtz-Gauss (vHzG) and vector Laplace-Gauss beams (vLpG), which are constructed from scalar solutions to the Helmholtz and Laplace equations respectively. However, the propagation of these beams was not completely characterized, especially their polarization properties. Therefore, a general study of solutions of this type became one of the main projects of this thesis..

(28) 2.2. MAXWELL’S EQUATIONS. 2.2. 9. Maxwell’s Equations. Our starting point for the electromagnetic analysis of vector beams are the Maxwell’s Equations, from which the wave equation and its special cases are derived. This section briefly comments on the fundamental mathematical and physical tools required for the analysis proposed here. At every ordinary point in space an electromagnetic field is subject to Maxwell’s equations, written in their differential form these are. ∇·D = ρ. (2.1). ∇·B = 0. (2.2). ∇×E = −. (2.3). ∂B ∂t ∂D ∇×H = J+ ∂t. (2.4). where E stands for the electric field, D = ǫE + P is the electric displacement, B is the magnetic field, and H = B/µ − M the magnetic field strength. The constants ǫ and µ are respectively the permittivity and permeability of the medium, while the P and M vectors are the polarization and magnetization vectors of the material. The vector J represents the current density and ρ is the density of electric charge. The relations between E and D, and B and H are commonly known as constitutive relations. For a linear and isotropic medium, these relations describe the properties of the material, taking the form D = ǫE and H = B/µ. The simplest form of equations (2.1-2.4) is obtained for a linear and homogeneous medium, in the absence of charge sources, where they reduce to. ∇ · (ǫE) = 0. ∇ · (µH) = 0. ∂H ∂t ∂E ∇×H = ǫ ∂t ∇ × E = −µ. (2.5a) (2.5b) (2.5c) (2.5d). It is possible to decouple the equations relating the electric and magnetic fields, and obtain an equation in terms of only one of these fields, indeed this procedure leads us to the derivation of the wave equation..

(29) 10. 2.3. CHAPTER 2. WAVE EQUATION FOR ELECTROMAGNETIC FIELDS. Wave equation. Under the assumption that the medium in which the field propagates is linear and homogeneous, i.e. the permittivity and permeability are spatially invariant, the wave equation can be easily derived from Eqs. (2.5a), (2.5c), and (2.5d) as follows . ∂H ∇ × ∇ × E = ∇(∇ · E) − ∇ E = ∇ × −µ ∂t ∂ = −∇2 E = −µ (∇ × H) ∂t 2 ∂ E ∇2 E = µǫ 2 , ∂t 2. . and the final form of the wave equation for the electric field is ∇2 E − µǫ. ∂2E = 0, ∂t2. (2.6). where µǫ = 1/v 2 , and v is the speed of light in the medium. Once the electric field is known, the expression for the magnetic field can be calculated by using Eq. (2.5c), however it is also possible to obtain a similar expression to Eq. (2.6) for the magnetic field by repeating the same procedure outlined above.. 2.3.1. Helmholtz Equation. Let us assume a solution to Eq. (2.6) of the form E(r, t) = U(r) exp(−iωt),. (2.7). where r = (x, y, z), and the temporal dependence of the field is given by the harmonic function exp(−iωt). Inserting Eq. (2.7) in Eq. (2.6) and performing the temporal differentiation, we obtain ω2 2 ∇ U(r) + 2 U(r) exp(−iωt) = 0, (2.8) v since the temporal dependence is already known, this equation can be reduced to ∇2 U(r) + k 2 U(r) = 0,. (2.9). which only depends on the spatial coordinates. Eq. (2.9) is commonly known as the vector Helmholtz equation, where k = 2π/λ = ω/v is known as the wave number..

(30) 2.3. WAVE EQUATION. 11. Finding an analytical solution to Eq. (2.9) is not an easy task, however there are different ways to deal with this problem. For instance, the simplest approach is to assume that there is no coupling between the field components, and apply the argument that if both E and H satisfy the wave equation, then all their components should satisfy an equivalent scalar equation, or which is equivalent [∇2 +k 2 ](Ej , Hj ) = 0, where j = (x, y, z) indicates the field component in each coordinate. The scalar Helmholtz equation is separable in 11 coordinate systems [11, 12, 13], from these, only four can be considered as cylindrical systems in the sense that they include a single coordinate along the z axis, and are naturally appealing to search for solutions that represent propagating fields [14]. It has also been shown in different works that solving the scalar version of Eq. (2.8) yields nondiffracting beams [1, 5, 6]. We will make extensive use of this type of solutions in the following sections.. 2.3.2. Paraxial Wave Equation (PWE). Solutions of the Helmholtz equation in cylindrical systems can be decomposed into transverse and longitudinal components, and are found to represent beams propagating in the z direction. Let us assume a function of the form U(r) = Ψ(r) exp(ikz z),. (2.10). where kz is calculated from the relationship k 2 = kt2 + kz2 , which gives the magnitude of the wave number k in terms of the transverse and longitudinal wave numbers, respectively. Substituting Eq. (2.10) into Eq. (2.9) and simplifying yields ∂2 ∂ 2 2 2 ∇t + 2 + 2ikz + (k − kz ) Ψ(r) = 0. (2.11) ∂z ∂z If the field U travels along directions which do not deviate significantly from the optical axis z, then kz ≃ k, and Eq. (2.11) now reads as ∂ ∂2 2 Ψ(r) = 0. (2.12) ∇t + 2 + 2ikz ∂z ∂z A further assumption is made by requiring the field Ψ to be a slowly varying function of z, which means ∂2Ψ ∂Ψ , ≪ kz 2 ∂z ∂z and Eq. (2.12) is reduced to a simpler form, namely ∂ 2 Ψ(r) = 0. ∇t + 2ikz ∂z. (2.13).

(31) 12. CHAPTER 2. WAVE EQUATION FOR ELECTROMAGNETIC FIELDS. which is known as the vector paraxial wave equation (vPWE). The solutions introduced by Bandres and Gutiérrez-Vega in [10] are indeed localized vector beam solutions to Eq. 2.13. The properties of these solutions will be discussed in detail throughout the first part of this document, since they constitute a special case of the more general beams discussed in this dissertation..

(32) Chapter 3 Generalized vector Helmholtz-Gauss beams The problem of finding vector beam solutions of Maxwell equations has been studied by several researchers [15]–[22]. In particular Stratton presents an excellent discussion in his book [11], Bouchal et al. [2] and, Volke-Sepulveda and Ley-Koo [22] have studied solutions in the context of propagation invariant optical fields (PIOFs), however due to their infinite energy requirement only approximate realizations of these type of solutions are physically possible. In this direction, the existence of the vector Helmholtz-Gauss (vHzG) beams, which constitute a general family of localized electromagnetic beams, was demonstrated theoretically for propagation in free space in [10]. Special cases of the vHzG beams are the transverse electric (TE) and transverse magnetic (TM) Gaussian vector beams [16], nondiffracting vector Bessel beams [17], azimuthally polarized Bessel– Gauss beams [18, 19], Mathieu-Gauss and Parabolic-Gauss beam modes in cylindrical waveguides and cavities [23, 24], and scalar Helmholtz-Gauss beams [9, 25]. In this chapter, we introduce a useful generalized form of the vHzG beams, to which we will refer as generalized vHzG (gVHzG) beams. The paraxial propagation of the gVHzG beams is studied, not only in free space, but also through more general types of paraxial optical systems characterized by complex ABCD matrices, including lenses, Gaussian apertures, cascaded paraxial systems, and systems having quadratic amplitude as well as phase variations about the axis. By following a coordinate-free approach, rather than proposing solutions in a particular coordinate system, it was possible to derive an elegant and closed-form expression for the electromagnetic field, the vector angular spectrum, and the Poynting vector at the output plane of the ABCD system. It is found that the gVHzG beams are a class of vector fields which exhibit the property of form-invariance under paraxial optical transformations. The formulation described in 13.

(33) 14. CHAPTER 3. GENERALIZED VECTOR HELMHOLTZ-GAUSS BEAMS. this thesis can be useful in applications where the polarization of the fields is of major concern [20, 21, 26, 27, 28].. 3.1. Definition of generalized vector Helmholtz-Gauss beams. Consider an electromagnetic paraxial beam with time dependence exp (−iωt) travelling in the z direction (unit vector b z) through an ABCD axisymmetric optical system, with input and output planes located at z = z1 and z = z2 , as shown in Fig. 3.1. The system is characterized by an ABCD matrix with, in general, complex elements A, B, C, and D that satisfy the unimodular relation AD − BC = 1.. Figure 3.1: ABCD system for the propagation of gVHzG beams. Let us redefine the position vector R and wave vector K as R = (r, z) and K = (k, kz ), where now r = (x, y) = (r, θ) and k = (kx , ky ) = (k, φ). (3.1). are the positions at the transverse planes of the configuration and frequency spaces, respectively. Additionally, we will denote a general vector field which is a solution to Eq. (2.13) as z, F = f + fz b. (3.2). where f = (fx , fy ) and fz represent the transverse and longitudinal parts of the field. The transverse nabla operators in the configuration and frequency spaces are denoted bx ∂/∂kx + k by ∂/∂ky , respectively. e =k b∂/∂x + y b∂/∂y and ∇ by ∇ = x Throughout this thesis, and unless stated otherwise, the suffixes “1” and “2” denote the evaluation of the respective physical quantity (or operator) at the input (z = z1 ) and output (z = z2 ) transverse planes of the ABCD system..

(34) 3.1. DEFINITION OF GENERALIZED VECTOR HELMHOLTZ-GAUSS BEAMS15 It is known from [10] that the transverse part of a field with the form given in Eq. (3.2), can be expressed as the product of a function U(r) and a Gaussian modulating envelope. Following the analysis in [11] it was also found that U satisfies the 2D vector Helmholtz equation, for instance it represents two independent solutions corresponding to the TM and TE polarizations, which are expressed as U(1) = ∇W (r) and U(2) = −ẑ × U(1) ,. (3.3). where U(1) is associated with the TM polarization, and U(2) with the TE.. 3.1.1. TM polarized gVHzG beams. We begin the analysis by defining the transverse electric e(r) and magnetic h(r) vectors of a first-class (TM) gVHzG beam at the input plane of the ABCD system as r ǫ iKr12 b z × e1 (r1 ), (3.4) ∇1 W (r1 ; κ1 ), h1 (r1 ) = e1 (r1 ) = exp 2q1 µ. where K = |K| = ω(µǫ)1/2 is the wave number.. The electric vector field in Eqs. (3.4) results from the product of two functions, each depending on one parameter. First, the Gaussian modulation is characterized by a complex beam parameter q1 = q1R + iq1I , where the superscripts R and I denote the real and imaginary parts of a complex quantity, respectively. For simplicity in notation, when dealing with the ABCD system, the parameter q1 will be used. The physical meaning of q1 is contained in the known relation 1 1 2 = +i , q1 R1 Kw12. (3.5). where R1 is the radius of curvature of the phase fronts, and w1 is the 1/e amplitude spot size of the Gaussian modulation. By assuming a complex value for q1 we are allowing for the possibility that the Gaussian apodization has an initial converging (q1R < 0) or diverging (q1R > 0) spherical wavefront. Additionally, the condition q1I < 0 must be fulfilled in order to satisfy the physical requirement that the field amplitude vanishes as r becomes arbitrarily large. The vectorial nature of the transverse electric field in Eq. (3.4) is provided by the gradient ∇1 W (r1 ; κ1 ). The scalar function W (r1 ; κ1 ) is a solution of the two-dimensional Helmholtz equation [∇21 + κ21 ] W = 0, and physically describes the transverse shape of an ideal scalar nondiffracting beam [2, 5, 6]. Since the vector field e1 (r1 ) in Eq. (3.4) is directly determined from the scalar function W (r1 ; κ1 ), we will refer to the latter as a seed.

(35) 16. CHAPTER 3. GENERALIZED VECTOR HELMHOLTZ-GAUSS BEAMS. function. Formally, the function W can be expanded in terms of purely monochromatic plane waves as [29]. W (r1 ; κ1 ) =. Z. π. g(φ) exp [iκ1 (x1 cos φ + y1 sin φ)] dφ,. (3.6). −π. where κ1 and g(φ) are the transverse wave number and the angular spectrum of the ideal scalar nondiffracting beam, respectively. Since g(φ) is arbitrary, an infinite number of profiles can be obtained, see section 3.1.7 for important special cases. It should be mentioned that the independence of g(φ) from the radial coordinate in K-space comes from the fact that we are dealing with a purely monochromatic nondiffracting beam. The angular spectrum of a polychromatic nondiffracting beam would indeed posses a dependence on the transverse spatial frequency components, this problem has been studied by different authors [30, 31] regarding nondiffracting X waves. The fields in Eq. (3.4) are purely transverse, and correspond to the zeroth-order electric and magnetic fields of the perturbative series expansion of Maxwell’s equations provided by Lax et al. [15]. However, the next-order correction in the Lax expansion also shows that small longitudinal field components must be present, and they can be obtained from the transverse components through. ez,1 hz,1. 3.1.2. 2 iKr12 iκ1 r1 i W + ∇1 W · , = ∇1 · e1 = − exp K 2q1 K q1 r i iKr12 ǫ r1 = ∇1 · h1 = − exp (b z × ∇1 W ) · . K µ 2q1 q1. (3.7a) (3.7b). TE polarized gVHzG beams. All the calculations and formulae presented in this work have been made for a TM type solution, throughout this thesis we only deal with the explicit expressions for the TM beams. However, expressions for the second-class (TE) beams can be readily obtained from Eqs. (3.4) by applying the duality property, i.e. replacing E with (µ/ǫ)1/2 H and (µ/ǫ)1/2 H with −E, namely (T E) e1 (r1 ). = − exp. . iKr12 2q1. . [b z × ∇1 W ] ,. (T E) h1 (r1 ). =. r. ǫ exp µ. . iKr12 2q1. . ∇1 W. (3.8).

(36) 3.1. DEFINITION OF GENERALIZED VECTOR HELMHOLTZ-GAUSS BEAMS17 The corresponding longitudinal components are obtained by applying the divergence operator as shown in Eqs. (3.7), iKr12 r1 i (T E) (b z × ∇1 W ) · , (3.9a) ez,1 = ∇1 · e1 = exp K 2q1 q1 2 r ǫ r1 iKr12 iκ1 i (T E) exp W + ∇1 W · . (3.9b) hz,1 = ∇1 · h1 = − K µ 2q1 K q1. 3.1.3. Classification of the generalized vector HzG beams. In the theory of nondiffracting beams, the parameter κ1 in Eq. (3.6) is customarily assumed to be real and positive [2, 5, 6]. For the sake of generality, we will let κ1 = I κR 1 + iκ1 be arbitrarily complex allowing the possibility of having three kinds of beams: • Ordinary VHzG (oVHzG) beams correspond to purely real κ1 = κR 1 for which the seed function W (r1 ; κ1 ) is a two-dimensional purely oscillatory (or standing wave) function. The physical meaning of κR 1 is clear, as it governs the oscillatory behavior of the function W in the transverse direction. Far from the z axis, the spatial period of the field oscillations tends monotonically to 2π/κR 1 . The special R 2 case when κ1 > 0 and q1 = −iKw1 /2 is purely imaginary leads to the ordinary vector solutions discussed in Ref. [10], where w1 is the standard 1/e amplitude spot size of the Gaussian apodization. • Modified VHzG (mVHzG) beams correspond to purely imaginary κ1 = iκI1 . In this case W r1 , κI1 his an evanescent seed function which satisfies the modified i 2 Helmholtz equation ∇21 − κI1 W = 0. Two concrete examples of seed functions of the modified kind are provided by the cosh-Gaussian beams [32] and the modified Bessel beams [33, 34]. • Generalized VHzG (gVHzG) beams correspond to the general case when κ1 is complex. As we will see, the gVHzG can be interpreted as intermediate solutions between oVHzG and mVHzG beams. A word of caveat is important, as it was mentioned the fields in Eq. (3.4) are purely transverse, however in Eq. (3.2) we assumed a solution composed of a transverse and a longitudinal part, therefore a longitudinal component has to be present. This component is indeed given by Eq. (3.7), however in order that the fields in Eq. (3.4) satisfy the paraxial approximation, it is needed that K ≫ 1/w1 , i.e. the Gaussian spot width is.

(37) 18. CHAPTER 3. GENERALIZED VECTOR HELMHOLTZ-GAUSS BEAMS. many wavelengths wide, and additionally that K ≫ |κ1 |, i.e. the spatial transverse beam oscillations must be many wavelengths wide. Two limiting cases of the gVHzG beams are of particular interest: (a) pure vector nondiffracting beams and (b) the generalized vector Laplace-Gauss beams. A brief discussion of each is presented next. Pure vector nondiffracting beams Pure vector nondiffracting beams are obtained when q1I → ∞, under this condition the longitudinal components for the H(T M ) and E(T E) fields are negligible, and the fields become purely TM and TE polarized, respectively. In this case, the expressions for the TM solution reduce to (T M ). E(T M ) = e1. (T M ). +b zez,1. = ∇1 W (r1 ; κ1 ) − b z. iκ21 W (r1 ; κ1 ), K. (3.10a). (T M ). H(T M ) = h1 , r ǫ b z × ∇1 W (r1 ; κ1 ), = µ. and for the TE case. (3.10b). (T E). E(T E) = e1 (T E). H. = b z × ∇1 W (r1 ; κ1 ), =. (T E) h1. (T E) +b zhz,1 ,. r ǫ iκ21 = ∇1 W (r1 ; κ1 ) − b z W (r1 ; κ1 ) , µ K. (3.11a). (3.11b). which are indeed the expressions for vector nondiffracting beams, for instance when these are expressed in circular coordinates they describe the vector Bessel beams introduced by Bouchal and Olivik [17]. Vector Mathieu and vector parabolic nondiffracting beams are obtained when Eqs. (3.10) and (3.11) are expressed in elliptic and parabolic coordinates respectively [22]. Generalized vector Laplace-Gauss (gVLpG) beams The second limiting case of the gVHzG is obtained when κ1 = 0, which leads to the generalized vector Laplace-Gauss beams, for which the seed function W (r1 , κ1 ) → W̄ (r1 ) now only depends on the transverse coordinates (x, y), and is a solution of the 2D Laplace.

(38) 3.1. DEFINITION OF GENERALIZED VECTOR HELMHOLTZ-GAUSS BEAMS19 equation ∇21 W̄ (r1 ) = 0 [10]. From Eq. (3.3) we observe that there are also two types of solutions satisfying the Laplace equation, their corresponding expressions are obtained from Eqs. (3.4), (3.8), and by a simplification of Eqs. (3.7) and (3.9), namely. (1). e1 (r1 ) = G(r1 , q1 )∇1 W̄ (r1 ),. r1 (1) ez,1 (r1 ) = G(r1 , q1 )∇1 W̄ (r1 ) · , q1 r ǫ (1) h1 (r1 ) = G(r1 , q1 )[b z × ∇1 W̄ (r1 )], µ r ǫ r1 (1) G(r1 , q1 )[b z × ∇1 W̄ (r1 )] · , hz,1 (r1 ) = − µ q1. (3.12a) (3.12b) (3.12c) (3.12d). for the first type solution, and for the second type they are. (2) z × ∇1 W̄ (r1 ) , e1 (r1 ) = −G(r1 , q1 ) b r1 (2) z × ∇1 W̄ (r1 ) · , ez,1 = G(r1 , q1 ) b q1 r ǫ (2) h1 (r1 ) = G(r1 , q1 )∇1 W̄ (r1 ), µ r r1 ǫ (2) G(r1 , q1 )∇1 W̄ (r1 ) · . hz,1 = − µ q1. (3.13a) (3.13b) (3.13c) (3.13d). Just like in the general case of the gVHzG beams, the gVLpG beams can also be expressed in different coordinate systems, however to the knowledge of the author, this beams have not been studied in detail, and indeed they constitute an area of opportunity for future work, as they are not discussed in this thesis.. 3.1.4. Propagation through ABCD systems. Paraxial propagation of the electric vector field e1 (r1 ) through an ABCD system can be performed by solving the Huygens diffraction integral [35, 36] ZZ ∞ 2 K exp (iKL0 ) iK 2 2 e2 (r2 ) = Ar1 − 2r1 ·r2 + Dr2 d r1 , (3.14) e1 (r1 ) exp i2πB 2B −∞. where e2 (r2 ) is the output transverse electric field, and L0 is the optical path length from the input to the output plane of the ABCD system measured along the optical axis, i.e. L0 = z2 − z1 . The vector integral in Eq. (3.14) can be treated as a pair of.

(39) 20. CHAPTER 3. GENERALIZED VECTOR HELMHOLTZ-GAUSS BEAMS. independent scalar integrals by decomposing the transverse vector e1 (r1 ) into orthogonal linearly polarized parts. After substituting each Cartesian component of Eq. (3.4) and the expansion given in Eq. (3.6) into Eq. (3.14), the integration can be performed by applying the changes of variables xj = uj cos φ − vj sin φ, and yj = uj sin φ + vj cos φ, for j = 1, 2. Upon returning to the original variables and regrouping the Cartesian components into a vector function, we obtain (see the detailed derivation in the appendix A) κ1 κ2 B κ1 exp −i G(r2 , q2 )∇2 W (r2 ; κ2 ), (3.15a) e2 (r2 ) = κ2 2K 1/2 ǫ b h2 (r2 ) = z × e2 (r2 ), (3.15b) µ. where. exp (iKL0 ) exp G(r2 , q2 ) = A + B/q1. . iKr22 2q2. . ,. (3.16). is the output field of a scalar Gaussian beam with input parameter q1 , travelling axially through the ABCD system. The transformation laws for the parameters q1 and κ1 from the plane z1 to z2 are q2 =. Aq1 + B , Cq1 + D. κ2 =. κ1 q1 . Aq1 + B. (3.17). Equations (3.15) constitute one of the main results of this thesis, since they provide a closed form expression for the propagation of an arbitrary gVHzG through an axisymmetric ABCD optical system, with real or complex matrix elements. The presence of the fundamental Gaussian beam G(r2 , q2 ) in Eq. (3.15a) provides the confinement mechanism, which ensures the transverse intensity distribution vanishes at large values of r, and that the beam is square integrable. As expected, Eqs. (3.15) reduce to Eq. (3.4) when the ABCD matrix becomes the identity matrix. Similarly to the method used for finding the electric and magnetic fields along the propagation coordinate at the input plane, Eqs. (3.7) and (3.9), the longitudinal components of the electric ez,2 and magnetic hz,2 fields, at the output plane, can be readily obtained by applying the operator [(i/K) ∇2 ·] over the corresponding transverse components e2 (r2 ) and h2 (r2 ), respectively.. 3.1.5. Poynting vector of the generalized vector HzG beams. The time-averaged Poynting vector is given by hSi = Re (E × H∗ ) /2. It can be decomposed as hSi = hsz i b z + hsi , where hsz i is the longitudinal part which determines the.

(40) 3.1. DEFINITION OF GENERALIZED VECTOR HELMHOLTZ-GAUSS BEAMS21 flow of energy in the direction of propagation z, and hsi is the transverse part which determines the flow of energy perpendicular to this direction. The Poynting vector can be calculated at the input (j = 1) and output (j = 2) planes, the general expression is given by 1 1 hSj i = Re(E × H∗ ) = 2 2. 1/2 n µ h io ǫ z − Re ez,j e∗j + |ej |2 b h∗z,j hj , µ ǫ. (3.18). where the transverse and longitudinal components for the electric and magnetic fields have been already defined for both TM and TE type solutions. Using the corresponding expressions for the electric and magnetic fields of the TM type solution, Eqs. (3.4) and (3.7), the Poynting vector in Eq. (3.18) reduces to 1 hSj i = 2. 1/2 2 iκj ǫ rj 2 2 ∇j W ∗ + z + Re |fj | |∇j W | b W + ∇j W · µ K qj ∗ rj b z × ∇j W · (b z × ∇j W ) , qj. (3.19). where f1 = exp (iKr12 /2q1 ) , and f2 = (κ1 /κ2 ) exp (−iκ1 κ2 B/2K) G(r2 , q2 ). For a pure nondiffracting beam, i.e. |qjI | → ∞, the Poynting vector reads as 1 hSj i = 2. 2 1/2 iκj ǫ 2 ∗ W ∇j W z + Re |∇j W | b , µ K. (3.20). which is indeed the expression obtained in Ref. [22] for ideal nondiffracting beams. From Eq. (3.19) we note that the energy flux density along the longitudinal direction is proportional to the squared magnitude of the transverse electric field vector. Since, in general, W and qj are complex quantities, the beam exhibits a transverse flow of energy whose radial part is a manifestation of diffraction. For a paraxial beam, it is expected that the longitudinal part of the energy flux be much more significant than the transverse part, in fact a simple comparison of the orders of magnitude in Eq. (3.19) reveals that the longitudinal flow is at least Kw1 times the transverse one. Finally, for a lossless medium, the light power carried by the beam along the longitudinal direction remains constant for any z plane.. 3.1.6. Propagation of the vector angular spectrum. The electric field e(r) of the gVHzG beam at either the input or output planes of the ABCD system admits a plane wave expansion of the form.

(41) 22. CHAPTER 3. GENERALIZED VECTOR HELMHOLTZ-GAUSS BEAMS 1 e(r) = 2π. ZZ. ∞. −∞. e e(k) exp (ik · r) d2 k,. (3.21). where e e (k) is the vector angular spectrum, whose functional form is obtained by Fourier inversion ZZ ∞ 1 e e(r) exp (−ik · r) d2 r. (3.22) e(k) = 2π −∞. After inserting Eqs. (3.4) and (3.15a) into Eq. (3.22), and solving the integrals, we obtain the vector angular spectra at the input and output planes of the ABCD system, namely q1 κ21 q1 k12 e q1 κ1 e e1 (k1 ) = i exp −i exp −i ∇1 W k1 ; , (3.23a) 2K 2K K 2 κ1 K q2 κ22 + κ1 κ2 B e Bκ 2 e 2 W k2 ; e e2 (k2 ) = exp −i , (3.23b) G(k2 , q2 )∇ κ2 q2 2K Kκ1. bx ∂/∂kx + k by ∂/∂ky is the transverse nabla operator in the K space, and e =k where ∇ q2 k22 iq2 exp (iKL0 ) e exp −i (3.24) G(k2 , q2 ) = K A + B/q1 2K is the Fourier transform of the Gaussian function G(r2 , q2 ) in Eq. (3.16).. 3.1.7. Remarks on the coordinates systems and polarization basis. The electromagnetic fields in Eqs. (3.4), (3.15) and (3.23) are completely general in the sense that they do not depend on a particular coordinate system. The vector beam solutions are constructed starting from scalar solutions of the 2D Helmholtz equation, which can be formally expanded in terms of plane waves according to Eq. (3.6). Although this integral expansion constitute a general integral solution, it is important to note that the 2D Helmholtz equation can also be solved in several orthogonal coordinate systems using the separation of variables method [9, 11, 12]. This fact leads to have complete and orthogonal families of eigenfunctions of the 2D Helmholtz equation. Of particular interest are the families of eigenfunctions of the 2D Helmholtz equation in Cartesian, polar, elliptic, and parabolic coordinates [9]. For Cartesian coordinates (x, y), generalized vector Gaussian beams can be constructed from superpositions of fundamen→ tal plane waves of the form W = exp(i− κ 1 ·r1 ) = exp [iκ1 (x cos φ + y sin φ)], see for example the cosine-Gauss beams studied in Ref. [9]. The case of the polar coordinates (r, θ) corresponds to eigenfunctions W = Jm (κr) exp(imθ) for which gVHzG beams reduce to.

(42) 3.1. DEFINITION OF GENERALIZED VECTOR HELMHOLTZ-GAUSS BEAMS23 the mth-order generalized vector Bessel-Gauss beams [18]. For elliptic coordinates (ξ, η), generalized vector even Mathieu-Gauss beams of mth-order and ellipticity parameter ε can be constructed from the eigenfunctions W = Jem (ξ, ε)cem (η, ε), where Je(·) and ce(·) are the radial and angular even Mathieu functions of mth-order and parameter ε, respectively [5]. For parabolic coordinates (u, v), generalized vector even parabolic-Gauss √ √ beams can be constructed from the eigenfunctions W = Pe u 2κ; p Pe v 2κ; −p , where Pe(·) is a parabolic cylinder function of parameter p and even parity [6]. On the other hand, the gradient operator in Eqs. (3.4) and (3.15a) can also be expressed in several coordinate systems [11, 12], with the consequence that several polarization basis may be used to decompose the field vector e2 (r2 ) at any point r2 into two orthogonal polarized transverse parts. For instance, in polar coordinates, the transverse vector fields can be split into radial and azimuthal polarized components, or in elliptic coordinates, into elliptic and hyperbolic polarized components. Explicit expressions for these polarization basis, associated to particular coordinate systems, can be found elsewhere [11, 12]. It is important to emphasize that a general vector beam solution is found by the superposition of TM and TE vector modes, i.e. E = αE(T M ) + βE(T E) ,. (3.25). where α and β are arbitrary constants. By combining the different polarization basis with the different seed functions W (r2 ; κ2 ), a large variety of beam profiles with specific polarization states could be constructed through superposition. Circular polarization basis √ b ± = (b Consider the basis of circular polarizations u x ± ib y) / 2. It is easy to verify that b± the gradient of W in Eqs. (3.4) and (3.15a) can be expressed in the basis of vectors u as ∂W ∂W 1 ∂W ∂W 1 + b +√ b−. ∇W = √ −i +i u u (3.26) ∂y ∂y 2 ∂x 2 ∂x Now, gVHzG beams with pure left (+) and right (−) circularly polarized beams can be constructed from Eq. (3.26) using the superposition e± = e(T M ) ± ie(T E) at either the input (j = 1) or output (j = 2) planes of the ABCD system, we have explicitly √ ∂W ∂W ± b±, u (3.27) ∓i ej (rj ) = 2fj ∂xj ∂yj. where f1 = exp (iKr12 /2q1 ) , and f2 = (κ1 /κ2 ) exp (−iκ1 κ2 B/2K) G(r2 , q2 ). A similar approach can be applied for the vector angular spectra in Eq. (3.23) to derive the.

(43) 24. CHAPTER 3. GENERALIZED VECTOR HELMHOLTZ-GAUSS BEAMS. corresponding circular polarization states. Finally, from Eqs. (3.4) and (3.8), it is clear that the polarization of the transverse electric field of the TM and TE beams is defined entirely by the operations ∇W and b z × ∇W, respectively. Both vector fields Ψ(1) = ∇W and Ψ(2) = b z × ∇W constitute two independent vector solutions of the 2D vector Helmholtz equation ∇2 Ψ + κ2 Ψ = 0 [11, 12]. Now, if we set the function W = Wm to be the m-th eigensolution belonging to a countable set of complete and orthogonal solutions of the scalar Helmholtz equation, then, because of the linearity and the one-to-one mapping of the gradient operator, the properties of linear independence, orthogonality, and completeness exhibited by the family of scalar solutions Wm are transferred to the corresponding families of vector fields Ψ(1) and Ψ(2) . Additionally, the transverse fields of the TM and TE beams are orthogonal, even when their seed functions Wm are equal. In this sense, the gVHzG beams exhibit similar polarization properties as the ideal vector nondiffracting beams [17, 22, 2] and waveguides with constant cross-section [23, 24, 11].. 3.2. Physical discussion of the propagation properties. In Section 3.1 we have demonstrated that localized vector beam solutions of the Maxwell equations can be propagated through an ABCD optical system in a closed and coordinatefree form. Particular attention was focused on the propagation of the vector beams and their vector angular spectra. Several considerations for the coordinate system and polarization basis were also discussed. Now, we turn our attention to study the propagation properties of gVHzG beams. To this end we consider two special cases, namely free space propagation and propagation through a GRIN medium.. 3.2.1. Free space propagation. First, we consider the free space propagation along a distance L = z2 − z1 . The input and output fields are given by Eqs. (3.4) and (3.15a), with A = 1, B = L, C = 0, and D = 1. From Eqs. (3.17) the propagation parameters are transformed as q2 = q1 + L,. κ2 =. κ1 q1 , q1 + L. (3.28). where we note that the product q2 κ2 = q1 κ1 remains constant under free-space propagation..

(44) 3.2. PHYSICAL DISCUSSION OF THE PROPAGATION PROPERTIES. 25. Figure 3.2: Physical picture of the decomposition of a gVHzG beam propagating in free space, in terms of fundamental vector Gaussian beams whose mean propagation axes lie on the surface of a double cone. In a similar fashion to the scalar HzG beams [9], in order to gain a basic understanding of the properties of the gVHzG beams propagating in free space, one may consider that a gVHzG beam is formed as a superposition of fundamental vector Gaussian beams (see Fig. 1), whose mean propagation axes lie on the surface of a double cone, and with amplitudes modulated angularly by the function g(φ). This physical picture is evident after replacing Eqs. (3.28) into Eq. (3.15a), and observing that the TM polarized gVHzG beams in vacuum can be rewritten as Z π e2 (r2 ) = g(φ)g2 (r2 ; φ)dφ, (3.29) −π. where − → κ21 L exp (iKL) iKr22 κ 1 · r2 − → g2 (r2 ; φ) = i exp −i κ 1, exp exp i 2K ζ ζ 2q1 ζ ζ. (3.30). → with ζ ≡ 1 + L/q1 and − κ 1 = κ1 (cos φb x + sin φb y). Equation (3.30) represents the free space propagation along a distance L of a tilted Gaussian beam with input parameter → q1 , whose mean wave vector has a projection − κ 1 over the transverse plane [9], and whose → polarization vector points in direction of the vector − κ 1. The generatrix of the double cone, shown in Fig. 3.2, corresponds to the linear propagation of the centroid of the individual Gaussian beams, and from Eqs. (3.15a) it is found to be I I R |q1 |2 κI1 κR 1 q1 + κ1 q1 rgen (z) = + (z − z1 ) . (3.31) Kq1I Kq1I In Fig. 3.2 we identify three important transverse planes: • The initial plane at z = z1 ..

(45) 26. CHAPTER 3. GENERALIZED VECTOR HELMHOLTZ-GAUSS BEAMS • The waist plane (z = zwaist ) corresponds to the plane where the width of the elementary Gaussian beams is minimum, i.e. where the radial factor exp (iKr22 /2q2 ) becomes a real Gaussian envelope. Using this condition, from Eqs. (3.28) we get zwaist = z1 − q1R . At the waist plane the parameter q becomes purely imaginary qwaist = iq1I , whereas the parameter κ reduces to κwaist = κ1 1 − iq1R /q1I . From the general expression of the Poynting vector in Eq. (3.19), we note that if W and κ are set to be purely real, then at the waist plane the energy flow becomes purely longitudinal. • The vertex plane (z = zvertex ) corresponds to the plane where the main propagation axes of the constituent Gaussian beams intersect. As shown in Fig. 3.2, the pseudo-nondiffracting region delimits the zone where significant interference of the constituent vector Gaussian beams occurs, and where the transverse beam profile exhibits a standing-wave behavior. The evaluation of the condition rgen = 0 in Eq. (3.31) yields zvertex = z1 −. |q1 |2 κI1 . I R I κR 1 q1 + κ1 q1. (3.32). Note that at the vertex plane the parameter κ becomes purely real, κvertex = I R I κR 1 + κ1 q1 /q1 , with the consequence that at this plane the beam profile belongs to the oVHzG kind with qvertex = κ1 q1 /κvertex . At the vertex plane the extent of the pseudo-nondiffracting region is maximum, and the 1/e amplitude Gaussian spot size can be calculated with wvertex = [K Im (1/qvertex ) /2]−1/2 .. In general, the initial, waist, and vertex planes are located at as shown in Fig. 3.2. The ordinary VHzG beams studied in special case when z1 = 0, q1R = 0, and κI1 = 0, for which the z = 0, and the cone generatrix reduces to the expected rgen =. different axial positions, Ref. [10] constitute the three planes coincide at κR 1 /K z.. R On the other side, the mVHzG beams occur when κR 1 = 0; if we additionally set q1 = 0, then rgen = r0 = q1I κI1 /K becomes a constant, and therefore the mVHzG beams may be viewed as a superposition of vector Gaussian beams, whose axes are parallel to the z axis and lie on the surface of a circular cylinder of radius r0 .. Finally, we should remark the fact that the transverse fields of the gVHzG beams propagating in free space satisfy the paraxial wave equation [∇21 + i2K∂/∂z] {e1 , h1 } = 0, and correspond to the zeroth-order purely transverse electric and magnetic fields of the perturbative series expansion of the Maxwell equations provided by Lax et al. [15]..

(46) 3.2. PHYSICAL DISCUSSION OF THE PROPAGATION PROPERTIES. 3.2.2. 27. Propagation through a GRIN medium. Let us now consider the propagation of the gVHzG beams through a graded refractiveindex (GRIN) medium with quadratic index variation n(r) = n0 (1−r2 /2a2 ). The ABCD transfer matrix from plane z1 to plane z2 = z1 + L is given by A B cos(L/a) a sin(L/a) = . (3.33) C D − sin(L/a)/a cos(L/a) For a general input vector field of the form (3.4), the propagated vector field at a distance z2 is described by Eq. (3.15a). Substitution of the matrix elements in Eq. (3.33) into Eqs. (3.17) yields the parameter transformations: q2 = a. q1 cos(L/a) + a sin(L/a) , −q1 sin(L/a) + a cos(L/a). κ2 =. κ1 q1 , q1 cos(L/a) + a sin(L/a). (3.34). where we note that, under propagation, the parameters q and κ vary periodically with a longitudinal period 2πa, therefore, the initial field distribution reproduces itself after a distance 2πa. In order to show the role played by the gVHzG beams as intermediate solutions between oVHzG and mVHzG beams, let us assume that the input field at z1 = 0 belongs to the oVHzG kind (i.e. κI1 = 0), with an initial real Gaussian apodization of width w1 (i.e. q1 = iq1I = −iKw12 /2). For a propagation distance L = LF = πa/2, the ABCD matrix Eq. (3.33) reduces to [0, a; −1/a, 0], which is indeed identical to the matrix transformation from the first to the second focal plane of a converging thin lens of focal length a, i.e. a Fourier transformer. At the Fourier plane L = LF , we see from Eqs. I (3.34) that the two parameters q2 = ia2 /q1I and κ2 = iκR 1 q1 /a become purely imaginary. It is now evident that if an oVHzG profile is Fourier transformed, a mVHzG profile will be obtained, and vice versa. The intermediate profiles belong to the gVHzG kind where, for the particular case of the GRIN medium, the transition between both types of beams is characterized by the continuous transformations given in Eqs. (3.34). The special case when the parameter a of the GRIN medium is equal to the Rayleigh distance of the initial Gaussian apodization (i.e. zR = Kw12 /2) is of particular interest. From Eqs. (3.34), we see that the Gaussian width q2 = q1 = −ia remains constant under propagation, and also that the wave number κ2 = κ1 exp (−iL/a) rotates at a I constant rate over the complex plane κR as the beam propagates through the 2 , κ2 GRIN medium. In order to make a distinction of the case when the beam propagates through the GRIN medium with a = Kw12 /2, we will refer to it as having a balanced propagation, and non-balanced otherwise..

(47) 28. CHAPTER 3. GENERALIZED VECTOR HELMHOLTZ-GAUSS BEAMS. Numerical examples We now provide numerical examples to show the propagation of gVHzG beams. Figs. √ 3.3 and 3.4 show the propagation through a GRIN medium, with a = 1/ 2π m, of the transverse intensity distribution and orientation of the electric field for several circularly polarized gVHzG beams with κ1 = 30 mm−1 . The input fields are given by Eq. (3.27) with j = 1 for K = 2π/λ, and λ = 632.8 nm. The graphs were generated by calculating the field at planes (z1 = 0) and (z2 = LF ), which correspond to the initial and Fourier planes, respectively. An animation of the propagating fields showing more details, can be found online in Ref. [60], where the videos were generated by computing the field distribution at 200 transverse planes, evenly spaced, from the input (z1 = 0) to the output (z2 = 4LF = 2πa) planes. Eq. (3.27), with j = 2, was used to generate the left and right circularly polarized fields as the case may be. The fields shown in Fig. 3.3 correspond to a seed function W (r1 ) given by the superposition of N plane waves of the general form W (r1 ) =. N X n=1. An exp [iκ1 r1 cos (θ1 − φn )] ,. (3.35). where An are complex amplitudes. For Fig. 3.3(a) we have chosen a left circularly √ polarized oVHzG beam in a balanced condition (q1 = −ia = −i/ 2π) with N = 3, An = {1, 1, 1}, and φn = {90◦ , −30◦ , −150◦ }. In this case, the width of the constituent Gaussian beams remains constant under propagation because the beams are balanced. Following the established convention, at a given z plane, the transverse components of the fields rotate anti-clockwise for left-handed circular polarization as time increases. The field at the plane z2 = LF is shown Fig. 3.3(b), where we note that for the selected amplitudes An = 1 the beam polarization becomes purely radial. To show the nonbalanced condition of Fig. 3.3(c), we propagated the same gVHzG but now setting √ q1 = 0.4 − i0.8/ 2π and keeping all remaining parameters unchanged. For this case, the width of the constituent Gaussian beams change under propagation, and reach a minimum at the plane where q2 in Eqs. (3.34) becomes purely imaginary (∼ 1.22LF ) In Fig. 3.3(e) we show a right circularly polarized balanced oVHzG beam constructed with N = 8 constituent Gaussian beams. By means of the amplitudes and phases of the coefficients An it is possible to adjust the polarization state of the resulting beam. In this case we set An = i such that the electric field at the plane z2 = LF now becomes purely azimuthal, as shown in Fig. 3.3(f). In Fig. 3.3(g) we set An = exp (−iπn/4) such that the electric field vectors at each point on the plane z2 = LF become parallel, as shown in Fig. 3.3(h)..

(48) 3.2. PHYSICAL DISCUSSION OF THE PROPAGATION PROPERTIES. 29. Figure 3.3: Propagation of the transverse intensity distribution and electric field for circularly polarized gVHzG beams, constructed with a finite superposition of vector Gaussian beams. (Left column) Initial profile, (Right column) Fourier transformed profile. The parameter data for the propagations are included within the text..

(49) 30. CHAPTER 3. GENERALIZED VECTOR HELMHOLTZ-GAUSS BEAMS. Figures 3.4(a) to 3.4(d) depict the vector propagation of ordinary vector Bessel-Gauss (VBG) beams with even parity. The first propagation shown in Fig. 3.4(a) corresponds to a balanced input VBG beam of the form in Eq. (3.27), with seed function W (r1 ) = iJ3 (κ1 r1 ) cos (3θ1 ) . The presence of the imaginary factor i in the seed function causes that, at the plane z2 = LF , the VBG beam be azimuthally polarized and belongs to the modified VHzG kind. The second propagation shown in Fig. 3.4(c) corresponds to a non-balanced VBG beam, with W (r1 ) = J3 (κ1 r1 ) cos (3θ1 ) and complex input √ parameter q1 = 0.4 − i0.8/ 2π. The propagation of a fourth-order helical vector Mathieu-Gauss (VMG) beam is shown in Figs. 3.4(e) and 3.4(f). The seed function is given by the superposition of even and odd Mathieu beams [5, 9], namely W (r1 ) = Je4 (ξ, 3)ce4 (η, 3) + iJo4 (ξ, 3)se4 (η, 3), where (ξ, η) are the elliptic coordinates defined as x = h cosh ξ cos η and y = h sinh ξ sin η, with h being the semifocal distance. In this case the vector beam is balanced, but the initial field belongs to the gVHzG kind with κ1 = 30 + i15 mm−1 . The input field is given by Eq. (3.27), where the Cartesian partial derivatives are expressed in elliptic coordinates as follows ∂ 1 ∂ ∂ sinh ξ cos η = − cosh ξ sin η , ∂x ∂ξ ∂η h cosh2 ξ − cos2 η ∂ 1 ∂ ∂ cosh ξ sin η = + sinh ξ cos η . ∂y ∂ξ ∂η h cosh2 ξ − cos2 η. (3.36a) (3.36b). As the beam propagates, the parameters q and κ vary according to Eq. (3.34). For this value of κ1 , the typical elliptic annular intensity pattern of the ordinary helical VMG beams occurs approximately at z ≃ 0.28LF , while the expected circular annular pattern of the modified helical VMG beams occurs at z ≃ 1.28LF . Finally, in Fig. 3.4(g) we show the propagation of a travelling vector Parabolic-Gauss (VPG) beam with TM polarization. The electric field is given directly by Eq. (3.15a), with a seed function given by the superposition of even and odd Parabolic nondiffracting beams [6, 9], namely √ √ √ √ W (r1 , κ) = Pe u 2κ; p Pe v 2κ; −p + iPo u 2κ; p Po v 2κ; −p , with parabolicity parameter p = 2. The Cartesian derivatives are expressed in the.

(50) 3.2. PHYSICAL DISCUSSION OF THE PROPAGATION PROPERTIES. 31. Figure 3.4: Propagation of the transverse intensity distribution and electric field for generalized vector Bessel-cosine-Gauss, Mathieu-Gauss, and parabolic-Gauss beams. (Left column) Initial profile, (Right column) Fourier transformed profile. The parameter data for the propagations are included within the text..