Instituto Tecnol´ogico y de Estudios Superiores de Monterrey Campus Monterrey Diffraction of Plane Waves by Finite-Radius Uniform-Phase Disks and Spiral Phase Plates by Jos´eHip´olito Garc´ıa Gracia Thesis Presented to the Program of Graduate Studies in Mechatronics and Information Technologies in partial fulfillment of the requirements for the degree of Master of Science in Electronic Systems May, 2008 c Jos´eHip´olito Garc´ıa Gracia, 2008 Instituto Tecnol´ogico y de Estudios Superiores de Monterrey Campus Monterrey Division of Mechatronics and Information Technologies Program of Graduate Studies The members of the committee recommend that the present thesis by Jos´eHip´olito Garc´ıa Gracia be accepted in partial fulfillment of the requirements for the academic degree of Master of Science, in: Electronic Systems Thesis Committee: Dr. Julio C´esar Guti´errez Vega Thesis Advisor Dr. Carlos M. Hinojosa Espinosa Dr. Rodolfo Rodr´ıguez y Masegosa Thesis Reader Thesis Reader Dr. Graciano Dieck Assad Chair of Graduate Studies May, 2008 A mis pap´as Reconocimientos Hay muchas personas a quienes agradecer por su apoyo para permitir hacer este trabajo posible. En primer lugar quisiera brindarle mi m´as profundo agradecimiento al Dr. Julio C´esar Guti´errez Vega, por aceptar ser mi asesor, por sus ense˜nanzas y su comprensi´on durante estos dos a˜nos. Adem´as quiero agradecer a mis sinodales, el Dr. Carlos Manuel Hinojosa Espinosa y el Dr. Rodolfo Rodr´ıguez y Masegosa, por tomarse el tiempo de leer este documento y por sus valiosos comentarios. Uno nunca puede olvidarse de la familia, porque sin ella no ser´ıamos nada. Gracias a mis padres, por su apoyo incondicional en todas mis empresas y por siempre cuidar de mi bienestar y de mi progreso, todo el tiempo alent´andome a ser mejor cada d´ıa. A mi hermana Aline, gracias por siempre estar al pendiente de m´ı. Muchas gracias a todos los miembros de la pecera: a los que est´an, Ra´ul, Servando y Dorili´an, por toda su ayuda para la realizaci´on de este trabajo; a los que ya se fueron, Rodrigo y Josu´e, por hacer ameno el inicio de mi estancia en el Centro de Optica.´ Finalmente, muchas gracias a Mayela, por brindarme su amor y su apoyo incondi- cional, a pesar de los momentos en que este trabajo me imped´ıa estar a su lado, porque siempre est´acuid´andome y preocup´andose por mi. Gracias por ser parte de mi vida, te amo con todo mi ser. Jose´ Hipolito´ Garc´ıa Gracia Instituto Tecnol´ogico y de Estudios Superiores de Monterrey Mayo 2008 v Diffraction of Plane Waves by Finite-Radius Uniform-Phase Disks and Spiral Phase Plates Jos´eHip´olito Garc´ıa Gracia, M.Sc. Instituto Tecnol´ogico y de Estudios Superiores de Monterrey, 2008 Thesis Advisor: Dr. Julio C´esar Guti´errez Vega Scalar Fresnel and Fraunhofer diffraction by phase objects has been widely studied in recent years. However, to the best of my knowledge, no analysis of diffraction by finite-radius phase objects with variable transmittances has been done. In this work, the Fresnel diffraction of plane waves by two different kinds of circular phase objects is studied. First, a finite-radius disk with uniform phase shift is analyzed, including the effects that varying the disk and outer-plane transmittances has on the diffraction pattern. Comparisons with the exact results for a circular aperture, presented by Dubra and Ferrari, and with Fourier propagation are done for the sake of validation. Next, the Fresnel diffraction of plane waves by spiral phase plates of finite radius is considered. The integer-step spiral phase plate is studied as a particular case to study the vortex distribution and behavior within the diffraction pattern. Finally, the fractional-step spiral phase plate is considered as a more general case, and the behavior of the optical vortices in the diffracted field is analyzed. Contents Reconocimientos v Abstract vi Chapter 1 Introduction 1 1.1 Diffraction by circular and phase objects . ... 2 1.2 Structure .................................. 4 Chapter 2 Theoretical background 6 2.1 The Huygens-Fresnel integral in cylindrical coordinates . ........ 6 2.2 Opticalvortices ............................... 8 Chapter 3 The uniform-phase disk 10 3.1 Non-unitary transmittances and Babinet’s Principle . ....... 14 Chapter 4 The finite-radius spiral phase plate 20 4.1 Integer-stepspiralphaseplate . 20 4.1.1 Opticalvortexdistribution. 24 4.2 Fractional-stepspiralphaseplate . 29 4.2.1 Optical vortex distribution for α = n + 1/2............ 32 6 4.2.2 Optical vortex distribution for α = n + 1/2............ 34 Chapter 5 Conclusions 40 Appendix A Partitioned Gaussian quadrature method 43 Appendix B Fourier space propagation 45 Bibliography 47 Vita 50 vii Chapter 1 Introduction Diffraction effects have been widely studied for centuries, since the first accurate report of such a phenomenon was made by Grimaldi and was posthumously published in 1665. However, the real evolution of diffraction theory began in 1678, when Christian Huygens expressed that each point on the wavefront of a light disturbance acted as a source for secondary spherical waves, and that the resulting pattern a certain distance from the original is formed by the envelope of these secondary spherical waves. The growth of the diffraction theory was stalled for more than a century due to the fact that Isaac Newton supported the corpuscular theory of light. Despite this long period of silence, in the 1800’s the wave theory of light and the study of diffraction phenomena had several important breakthroughs. In 1804, Thomas Young introduced the concept of interference, proving its validity with his famous double slit experiment. In 1818, Augustin Fresnel merged the ideas of Huygens and Young in his memoirs, by adding the concept of interference to Huygens’ secondary spherical wavelets. This corrected version of Huygens’s idea is currently known as the Huygens-Fresnel principle [1, 2]. Ever since Young carried out his double slit interference experiment, it has become one of the fundamental arrays used in diffraction experiments. More recent diffraction analysis include variations of the slit experiment [3, 4, 5], circular apertures and obsta- cles [6, 7], among others. In recent years, analysis of diffraction patterns generated by phase objects, as denoted by Sussman [8], have been gaining increasing relevance. This is because phase objects are able to produce structures of phase singularities known as optical vortices. The purpose of this work is to explore the Fresnel diffraction of a plane wave by three different kinds of phase objects, a uniform-phase disk, an finite-radius interger- step spiral phase plate, and finishing with the more general case of a finite-radius fractional-step spiral phase plate. This research partially extends the work done by M. V. Berry on the diffraction of a plane wave by integer and fractional step spiral phase plates [9] by considering a finite-radius spiral phase plate, thus incorporating the effects of the border diffracted wave. It also extends the analysis of the finite-radius integer-step spiral phase plate made by Kotlyar et al. [10, 11], by considering the 1 interference of the portion of the plane wave diffracted by the spiral phase plate with the rest of the plane wave which goes unaffected by the plate. It is worth noting that the Huygens-Fresnel principle used throughout this work belongs to what is known as the scalar paraxial diffraction theory, which focuses on obtaining a complex amplitude function as an approximation to the complete elec- tromagnetic field which should be calculated. For a complete and rigorous solution to the diffraction of light by screens and obstacles, one must resort to the vectorial non-paraxial diffraction theory, which is beyond the scope of this work. 1.1 Diffraction by circular and phase objects In this chapter an overview is given on some of the research done so far regarding light diffraction by circular and phase objects, since this work is based on combinations of both cases. Diffraction by circular objects has been a topic of research interest since Young carried out his slit experiment, and it keeps generating research articles today. Although Fresnel diffraction is quite an old subject, its quantitative study was severely limited because solving the diffraction integrals is quite a challenge. In 1984, J. E. Harvey and J. L. Forgham revised the spot of Arago, more commonly known as Poisson’s spot, as a tool to detect whether a light wavefront has aberratioins and how bad they are [12]. They began with a brief description of the Fresnel diffraction pattern of a circular aperture. Then, Harvey and Forgham studied diffraction by an opaque obstacle, starting from an annular aperture and increasing the outer radius to infinity, for both plane wave and Gaussian illumination. Finally, they observed modifications in Arago’s spot caused by introducing aberrations into the incident light beam. In 1985, D. S. Burch published his results on the analysis of Fresnel diffraction by a circular aperture, in which he studied both on-axis and off-axis intensities [13]. Although he was able to produce analytical formulae for the on-axis intensity, Burch was forced to use an “auxiliary function” approximation to the Fresnel integrals required to compute the off-axis diffraction pattern. In 1987, R. E. English Jr. and N. George also studied the on-axis intensity of Fresnel diffraction generated by a circular aperture for plane wave illumination, but they used an alternative formulation of the diffraction integral which included a partial derivative with respect to the propagation coordinate, and used this result to extend their analysis to tapered light [14]. They also did a brief comment on the behavior of the axial intensity of the Fresnel diffraction pattern by a circular obstacle.