Angular Momentum in
Colliding Airy-vortex
beams
by
Carmelo Guadalupe Rosales Guzm´
an
A dissertation submitted in partial fulfillment of the
requirements for the degree of
MASTER OF SCIENCE IN OPTICS
at the
Instituto Nacional de Astrof´ısica, ´
Optica y
Electr´
onica
September 2010
Tonantzintla, Puebla, M´exico
Advisor:
Dr. Rub´
en Ramos Garc´ıa, INAOE
Prof. Kishan Dholakia, St Andrews University
c
INAOE 2010
The author hereby grants to INAOE permission to
reproduce and to distribute copies of this thesis
The fact that light carries orbital angular momentum (OAM) associated to singularities em-bedded in the phase function of optical fields was recently discovered by Allen. A couple of years later, Courtial demonstrated that light beams might also posses OAM without the pres-ence of a singularity. Over the past years, OAM has been the subject of intense experimental and theoretical study, including its conservation. Laguerre-Gaussian and Bessel are among the most well known beams with phase singularities carrying OAM. Fairly recently, Dholakiaet al. generated, by inserting a phase singularity into the main lobe of an Airy beam, a novel kind of beam carrying OAM. As was expected, the vortex accelerates in a direction perpendicular to the beam axis of propagation similar to an Airy beam. Motivated by this peculiarity, we experimentally generated two Airy-vortex beams with opposite acceleration in order to study the OAM during the collision. To our knowledge, there are no reports on the analysis of OAM in collisions of beams carrying OAM. Two experiments were carried out, the first one involves collisions of two beams carrying the same amount of OAM. In the second one, we collide beams carrying opposite OAM. Analysis of the phase function from several planes perpendicular to the propagation axis lead us to conclude that after the collision of two beams having the same OAM, the phase singularity does not disappear. Whereas in the case where the beams have opposite OAM, the phase singularity annihilate during the collision. These experiences suggest a conservation of the OAM associated with phase singularities. Analysis of the phase was done by means of a new technique proposed by Mazilu et al., which allows to detect and track the singularities embedded in the phase function.
This thesis would not have been possible without the financial support of CONACyT. It is also a pleasure to thank those who supported me along this journey, first to my supervisor Rub´en Ramos-Garc´ıa who assisted me wisely in many ways, as well as my co-supervisor Kishan Dholakia for giving me the opportunity to work in his laboratory under his supervision. I would also like to show my gratitude to the Instituto Nacional de Astrof´ısica ´Optica y Electr´onica (INAOE) where I found an encouraging place to carried out my studies as well as the department of physics and astronomy at St. Andrews University where I did most of the experimental work. My gratitudes extend to Maichael Mazilu and J¨org Baumgartl who transmitted me a lot of their knowledge and experience and for their help in the lab. I am heartily thankful to my family as well as Valeria who have always encouraged me to succeed. Lastly, I offer my regards to all of those who supported me in any respect during the completion of the project.
1 Introduction 1
References . . . 5
2 Optical phase singularities 7 2.1 Wave Dislocations . . . 8
2.2 Generation of optical vortices . . . 9
2.2.1 Hermite-Gaussian and Laguerre-Gaussian modes . . . 9
2.2.2 Direct generation from a laser cavity . . . 11
2.2.3 Generation using cylindrical lenses mode converter . . . 12
2.2.4 Generation using Spiral Phase Plates . . . 13
2.2.5 Generation using diffractive optical elements . . . 15
2.2.6 Generation using spatial light modulators . . . 16
2.3 Phase singularities and orbital angular momentum of light . . . 17
2.4 Topological charge of optical vortices . . . 19
2.5 Phase cuts . . . 21
References . . . 23
3 The Airy-vortex beam 27 3.1 The Airy Wave Packet . . . 28
3.2 The Finite Energy Airy Beam . . . 30
3.3 Airy-Vortex Beam . . . 33
3.4 Phase cuts in the Airy-vortex phase function . . . 35
3.4.1 Rotating phase cuts in the phase function . . . 36
References . . . 38
4 The Experiment 39
4.1 Setup Description . . . 39
4.2 Reconstruction of the electromagnetic field . . . 42
4.2.1 Implementation . . . 44
References . . . 46
5 Results 47 5.1 Intensity images directly taken with CCD camera . . . 48
5.2 Electric field reconstruction . . . 49
5.2.1 Laguerre-Gaussian beams . . . 49
5.2.2 Airy-vortex beams . . . 50
5.3 Phase cuts in reconstructed electric fields . . . 51
5.3.1 Phase cuts in LG beams . . . 51
5.4 Phase cuts in colliding Airy-vortex beams . . . 52
5.4.1 Airy-vortex beams carrying equal charges . . . 53
5.4.2 Airy-vortex beams carrying opposite charges . . . 58
6 Conclusions 63 References . . . 65
Appendices 69 A Topics on Airy beams 69 A.1 Finite energy Airy beam deduction . . . 69
A.2 Total power contained in the finite energy Airy beam . . . 71
A.3 Fourier transform of the finite energy Airy beam . . . 71
B Matlab scripts 73 B.1 Matlab generating phase mask script . . . 73
2.1 The dislocation is perpendicular to the vortex line . . . 8
2.2 The wave dislocation is parallel to the vortex line . . . 9
2.3 Intensity plot of Hermite-Gaussian modes . . . 10
2.4 Intensity plots of Laguerre-Gaussian modes . . . 11
2.5 An LGnm output of a conventional laser can be transformed by means of a cylindrical lens mode converter into an LG beam carrying an vortex along its propagation axis . . . 12
2.6 Spiral phase plate, and its phase distribution . . . 13
2.7 Wave fronts of a Gaussian beam before and after passing through a SPP. . . 14
2.8 Variable phase plate . . . 15
2.9 Holograms to create beams with optical vortices . . . 15
2.10 Second-harmonic generation. . . 18
2.11 In the immediate vicinity of vortex, azimuthal phase term occur in helicoidal front of the optical field . . . 19
2.12 Phase singularities in 2-D are represented by points, whereas in 3-D are repre-sented by lines. . . 20
2.13 Plot of an analytic multivalued function. Branch points are enclosed by circle. Branch cuts are usually lines between pairs of these branch points . . . 21
2.14 The phase function of an Airy-vortex beam shows maximums along branch cuts. Plots of branch cuts are called phase cuts. . . 22
3.1 Square modulus of an Airy function Ai(x). For negative numbers it decays oscillating whereas for positive it decays very fast . . . 28
3.2 Acceleration in infinite and finite energy Airy beams . . . 32
3.3 Intensity profile of 2D (a) diffraction-free infinite and (b) finite energy Airy
beams, along with the generating phase mask. . . 32
3.4 A mask with a vortex in its center (a) is generated in order to produce an Airy-vortex beam from a broad Gaussian. . . 33
3.5 The Mathematical expression for an Airy-vortex beam is computed by propa-gating the generalized Huygens-Fressnel integral of the field ˆuAV . . . 34
3.6 The intensity profile of an Airy vortex beam (b) along with its phase profile (a) is shown forz = 0 and `= 1. . . 35
3.7 Airy vortex beam. (a) Amplitud , (b) Phase and (c) phase cuts. . . 36
3.8 By adding a varying phase ϕr to a phase function we can observe rotation of branch cuts and phase cuts around branch points. From (a) to (l)ϕr is increasing in steps of π/11. We should focusing our attention in the branch cut emerging from the main vortex we can observe its rotation in the clockwise direction. . . . 37
4.1 Experimental setup. . . 40
4.2 SLM was addressed with a Matlab script allowing us to control several parameter of the beam. . . 41
4.3 The five-points method to compute the gradient of a function . . . 43
4.4 Picture of the experimental setup. . . 45
5.1 Intensity pattern of beams . . . 48
5.2 Collision of two Airy-vortex beams with topological charges +1 and -1 . . . 49
5.3 Plots of one and two Laguerre-Gaussian beams . . . 50
5.4 Plots of theoretical an experimental intensity and phase of an Airy-vortex beam 50 5.5 Comparison of Theoretical and experimental plots of intensity and phase of two colliding Airy-vortex beams . . . 51
5.6 Phase cuts in LG beams evidence discontinuities in the phase function. . . 52
5.7 Phase and phase cuts in a LG beam of charge two as the phase increases in steps of π/12. . . 53
5.8 Rotating phase cuts for Airy-vortex beams . . . 54
5.9 Rotation of phase cuts makes eviden the presence of the two principal vortices in the collision of two Airy-vortex beams . . . 55
5.10 Collision of two Airy-vortex beams carrying equal topological charge. Branch points translate at the same time that phase cuts rotates around them . . . 56
5.11 Branch cuts in the collision of two Airy-vortex beams carrying equal charge. We can observe the branch point before and after collision, this is, the topological charge of the system remains constant. . . 57 5.12 Rotation of phase cuts in Airy vortex beams carrying opposite charge −1 for the
first beam (left) and +1 for the second (right). . . 58 5.13 Colliding Airy beams of opposite charge . . . 59 5.14 During the collision of two Airy-vortex beams both vortices cancel each other. . 60
Introduction
Singularities are the locus of points where mathematical quantities become infinite, or change abruptly. In the field of optics, we can find singularities in the intensity, in the phase, or in the polarization of electromagnetic waves.
Optical phase singularities were experienced at first indirectly through defects in interfer-ence patterns and observed only as dark spots. Though these singularities can be created by random processes such as light scattering, the singular structures carried by well directed laser beams are of particular importance. Main attention has been focused on the phase singularities embedded in Gaussian beams, and interesting effects such as attraction and annihilation of a pair of vortices of opposite topological charges has been predicted and verified [1, 2]. In recent times they have shown to be able to exert mechanical forces capable to move and rotate small objects [3, 4, 5]. These forces are exerted by the light field surrounding the singularity and their characteristics are determined by the helical nature of the wavefronts associated with the singularity and a radial increase in irradiance.
The foundations of wave singularities, as Michael Berry has pointed out [6], laid in three astonishing papers published as early as the 1830’s. Two by William Whewell from 1833 and 1836 in which he reported phase singularities of tide waves. And a third one from 1838 by George Biddell Airy, in which he recognized that the rainbow is a particular example of a singularity, a line where light rays are focused (often referred as a caustic). At these points, ray optics predicts infinite brightness.
In 1947 Goos and Hanchen[7] discovered that, when a finite-width optical beam hits a di-electric interface under total internal reflection, the center of the reflected beam undergoes a spatial shift and as result a spatial interference structure arises. In 1950 Wolter found
ically that this interference phase pattern clearly showed singularities in the phase function[8]. He speculated this phenomenon was quite general in optical fields.
In 1951 Braunbek realized that three plane waves are sufficient to produce phase singularities [9]. He also observed that in singularities, the phase is not well defined and, at the same time, the amplitude becomes zero, moreover, the energy flows in closed “stream lines” .
Perhaps the most important study from this early period is the direct computation and plots by Braunbek and Laukien of Sommerfeld’s exact solution of a plane wave diffracted by an infinite half-plane[10]. The intensity plot revealed the existence of points where the amplitude becomes zero, whereas a plot of the phase function makes evident the existence of phase singularities embedded in it. Berry interpret this phase singularities as originating from the interference of three waves: the incident wave, the reflected wave, and the edge wave from the diffracting line at the end of the half-plane without which, the interference pattern on the illuminated side would be a simple standing wave[11].
The essential role of phase singularities started to be recognized only after the seminal publication by Nye and Berry in 1974 [12]. They introduced the concept of wave dislocations and observed several similarities between phase singularities and dislocations in crystal lattices. It was until 1981 that Berry recognized phase singularities as the most remarkable features of wave fronts [13]. In the same year Baranova et al. observed phase dislocations in speckle fields[14]. A year, later they show the existence of phase singularities in the form of screw wave-front dislocations [15] and established their important features in a speckle field.
The terms “phase dislocations”, “phase singularity lines”, “optical vortices” (introduced by Cullet et al. [16]) and “screw wave front dislocations” are quite often used as synony-mous. However, as Dennis et al. pointed out these expressions are not equivalent but rather complementary[17]. An optical vortex has a complicated structure with a dark core (zero-amplitude axis) around which phase circulates forming a spiral. A phase dislocation is the locus of the zero amplitude and it can be a line or even a surface. A screw wave-front dislo-cation is an helicoidal structure that appears around the dislodislo-cation line. The first reported ideas of artificial introduction of phase singularities into a smooth wave-front beam, belongs to Bryngdahl[18] and Bryngdahl and Lee [19].
Phase singularities are now recognized as important features in all waves. In light, they are optical vortices and the wave intensity is zero, in contrast to caustics where the intensity is (geometrically) infinite. In acoustics, the singularities are threads of silence; ; in superfluids, quantized vortices; and in superconductors, quantized lines of magnetic flux.this
The recognition that light may also have angular momentum associated to the phase struc-ture of light was carried out by Allen et al[20] in 1992 . In their works they established that any beam with amplitude distributionu(r, φ, z) =u0(r, z)ei`φ, carries angular momentum about the beam axis. Such angular momentum could be separated into orbital and spin components. The former determined solely by the azimuthal phase dependence. They not only established a theoretical framework for OAM but also proposed an optical system to transform Hermite-Gaussian (HG) into Laguerre-Hermite-Gaussian (LG) modes and vice versa (both arriving as solution of the paraxial wave equation in Cartesian and cylindrical coordinates respectively). Moreover, they proposed an experiment to measure the mechanical torque induced by the transfer of OAM associated with such a transformation. In 1993 Beijersbergen et al.[21] also presented a design of a mode converter, consisting of two cylindrical lenses to transform HG modes into LG. The prediction that beams with an azimuthal dependence of ei`φ would have an OAM of L = `
~ per photon, where` can take any integer value was also established in these works. For a given value of`, the beam has `intertwined helical phase fronts possessing a singularity on the beam axis. Therefore the cross sectional intensity pattern of all such beams has a doughnut shape.
An eigenfunction description of the OAM was proposed by van Enk and Nienhuis[22]. Allen et al. proposed a matrix formulation for the propagation of beams with OAM [23]. An equiva-lent geometric transformation for spin and OAM was developed by Allen and Beijersbergen[24]. Also an equivalent description of the Poincar´e-sphere for polarization was carried out out by Padgett and Courtial [25] and by Soares et al. [26] with the aim of representing HG and LG beams.
In an independent way and in the context of quantum mechanics, Berry and Balaz [27] proposed in 1979 a new type of solution to the 1-D Schr¨odinger for a free particle, in the form of “Airy wave packets”. These solutions were showed to posses amazing features like its ten-dency to freely accelerate in free space, as well as, its non diffracting property. Motivated by this work, Besieris et all [28] investigated for similar solutions to the 3-D Schr¨odinger equa-tion. However, it was until 2007 when Siviloglou and Christodoulides, taking advantage of the similarity between the paraxial and Schr¨odinger wave equations, derived a new solution to the former [29]. Further more, they were also able to generate them experimentally, through the use of spatial light modulators by encoding a cubic phase front on an incident Gaussian beam[30]. They show that this novel kind of beams with self-healing properties, represent a new class of non-diffracting beams that do not propagate in a straight line but exhibit a constant transversal acceleration. Using these qualities, a group from St Andrews University led by Kishan
Dho-lakia, was able to transport particles along parabolic trajectories [31], clearing optically entire regions of microparticles. Using the same generating method as in Airy beams, Dholakia and coworkers were able to generate theoretically and experimentally Airy-like beams (Airy-vortex), that feature a main vortex in the maximum intensity lobe of the beam [32]. They were also able to show that the vortex travels along with the principal lobe, describing a parabolic trajectory in a plane perpendicular to the propagation axis. Motivated by this idea and in collaboration with the same group, we performed an experimental set up to produce collisions between two Airy-vortex beams OAM, in the search of a conservation of OAM. The method we used is based in the reconstruction and analysis of the phase function for several planes perpendicular to the propagation axis. This method employs a novel technique developed by the same group that allow us to reconstruct the electromagnetic field in planes perpendicular to the propagation axis.
The chapters in this disertation are organized as follows. Chapter 2 is entirely devoted to phase singularities. We go first through the phase singularities viewed as wave dislocations (section 2.1). A general overview of some existing methods to generate optical vortices will be given in section 2.2. We follow with the concept of the topological charge of an optical vortex (section 2.4) and its relation with OAM (section 2.3). In the last section we introduce the concept of “phase cuts” which is a novel technique through which we can reconstruct the phase function of an electromagnetic field, to perform a further analysis of the phase singularities embedded in it. Chapter 3 is dedicated to Airy vortex-beam. We start with the Airy wave packets that emerged as solutions of the 1-D Scrh¨odinger equation (section 3.1),the antecedent of Airy and Airy-vortex beams. In the last section of this chapter we explain how to apply the phase cuts method to an Airy-vortex beam. The experimental setup is described in chapter 4 as well as its implementation in the acquisition of data. Some of our main results are presented in chapter 5. And final chapter 6 is devoted to some conclusions related to our work.
References
[1] G. Indebetouw. Optical vortices and their propagation. J. Mod Opt., 40(1):73–87, 1993.
[2] I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov. Optics of light beams with screw dislocations. Opt. Commun., 103(5-6):422–428, 1993.
[3] Takahiro Kuga, Yoshio Torii, Noritsugu Shiokawa, Takuya Hirano, Yukiko Shimizu, and Hiroyuki Sasada. Novel optical trap of atoms with a doughnut beam. Phys. Rev. Lett, 78(25):4713–4716, 1997.
[4] David Grier. A revolution in optical manipulation. Nature, 424(6950):810–816, 2003.
[5] M. Babiker and D. L. Andrews. Optical manipulation of atoms and molecules using structured light. African Physical Review, 1(1):18–33, 2007.
[6] M. V. Berry. Making waves in physics. three wave singularities from the miraculous 1830s.Nature, 403(6765):21, 2000.
[7] F. Goos and H. H¨anchen. Ein neuer und fundamentaler versuch zur totalreflexion. Ann. Phys, 436(7-8):333–346, 1947.
[8] H. Wolter. Zur frage des lichtweges bei totalreflexion. Z. Naturforsch. Teil A, 5:276–283, 1950. [9] W. Braunwbek. Zur darstellung von wellenfeldern. Z. Naturforsch, Teil A, 6:12–15, 1951. [10] W. Braunbek and G. Laukien. Einzelheiten zur halbebenen-beugung. Optik, 9:174–179, 1952. [11] M. V. Berry. Exuberant interference: rainbows, tides, edges, (de)coherence. Phil. Trans. R. Soc.
Lond. A, 360(1794):1023–1037, 2002.
[12] J. F. Nye and M. V. Berry. Dislocations in wave trains. Proc. R. Soc. Lond., 336(1605):165–190, 1974.
[13] Michael Berry. Physics of defects, chapter 7: Singularities in waves and rays. North-Holland Publishing Company, 1981.
[14] N. B. Baranova, B. Y. Zel’dovich, A. V. Mamaev, N. F. Pilipetskii, and V. V. Shkunov. Sov. Phys. JETP Lett, 33:195, 1981.
[15] N. B. Baranova, A. V. Mamaev, N. F. Pilipetskii, and V. V. Shkunov. Sov. Phys. JETP Lett, 56:983, 1982.
[16] P. Coullet, L. Gill, and F. Rocca. Optical vortices. Opt. Commun., 73(5):403–408, 1989.
[17] M. R. Dennis, K. O’Holleran, and M. J. Padgett. Progres in opticss, volume 53, chapter 5: Singular Optics: optical vortices and polarization singularities. Elsevier, 2009.
[18] O. Bryngdahl. Image formation using self-imaging techniques. J. Opt. Soc. Am, 63(4):416–419, 1973.
[19] O. Bryngdahl and W. Lee. Shearing interferometry in polar coordinates. J. Opt. Soc. Am, 64(12):1606–1615, 1974.
[20] L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman. Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes. Phys. Rev. A, 45(11):8185–8189, 1992.
[21] M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman. Astigmatic laser mode converters and transfer of orbital angular momentum.Opics Communications, 96(1-3):123– 132, 1993.
[22] S. J. van Enk and G. Nienhuis. Eigenfunction description of laser beams and orbital angular momentum of light. Opt. Commun., 94(1-3):147–158, 1992.
[23] L. Allen, J. Courtial, and M. J. Padgett. Matrix formulation for the propagation of light beams with orbital and spin angular momenta. Phys. Rev. E, 60(6):7497–7503, 1999.
[24] L. Allen and M. J. Padgett. Equivalent geometric transformations for spin and orbital angular momentum of light. J. of Modern Optics, 54(4):487–491, 2007.
[25] M. J. Padgett and J. Courtial. Poincar´e-sphere equivalent for light beams containing orbital angular momentum. Opt. Lett., 24(7):430–432, 1999.
[26] W. C. Soares, D. P. Caetano, and J. M. Hickmann. Hermite-bessel beams and the geometrical rep-resentation of nondiffracting beams with orbital angular momentum. Opt. Express, 14(11):4577– 4582, 2006.
[27] M. V. Berry and N. L. Balazs. Nonspreading wave packets. Am. J. Phys, 47(3):264–267, 1979. [28] I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkosky. Nondispersive accelerating wave packets.
Am. J. Phys, 62(6):519–521, 1994.
[29] G. A. Siviloglou and D. N. Christodoulides. Accelerating finite energy airy beams. Opt. Lett, 32(8):979–981, 2007.
[30] G. A. Siviloglou, J. Broky, A. Dogariu, , and D. N. Christodoulides. Observation of accelerating airy beams. Phys. Rev. Lett, 99(21):213901, 2007.
[31] J. Baumgartl, M. Mazilu, and K. Dholakia. Optically mediated particle clearing using airy wavepackets. Nature Photonics, 2(11):675–678, 2008.
[32] M. Mazilu, J. Baumgartl, T. Cizmar, and K. Dholakia. Accelerating vortices in airy beams. Proc. SPIE, 7430, 2009.
Optical phase singularities
Contents
2.1 Wave Dislocations . . . 8
2.2 Generation of optical vortices . . . 9
2.2.1 Hermite-Gaussian and Laguerre-Gaussian modes . . . 9
2.2.2 Direct generation from a laser cavity . . . 11
2.2.3 Generation using cylindrical lenses mode converter . . . 12
2.2.4 Generation using Spiral Phase Plates . . . 13
2.2.5 Generation using diffractive optical elements . . . 15
2.2.6 Generation using spatial light modulators . . . 16
2.3 Phase singularities and orbital angular momentum of light . . . . 17
2.4 Topological charge of optical vortices . . . 19
2.5 Phase cuts . . . 21
References . . . 23 When three or more waves interfere, light intensity vanishes at some points. At this places the phase become undefined (singular), and in general, all 2π phase values occur around such points, leading to a circulation of the optical energy. This points have receive different names, such as: phase singularities, nodal points, wave dislocations and optical vortices. Phase sin-gularities can be also seen as topological objects embedded in wave-front surfaces, possessing topological charges, which are attributed to an helicoidal spatial structure of the wave front
around a phase singularity. This structure is similar to a crystal lattice defect, and therefore was at first known as wave-front screw dislocation. Section 2.1 is devoted to this topic. Some of the methods encountered in optics to generate phase singularities are briefly reviewed in section 2.2. The treatment of phase singularities as topological charges will be discussed in section 2.4. In section 2.3 we will delve in the related to OAM associated to phase singularities. Finally in section 2.5 we introduce the concept of branch points and branch cuts, we will also describe a novel method for the analysis of optical vortices. In particular this method was applied to analyze the orbital angular momentum in Airy-vortex beams.
2.1
Wave Dislocations
Optical phase singularities are also known as “wave dislocations” due to several similarities observed in dislocations in crystal lattices [1]. Thess similitudes can be observed if we plot the wave function
ψ = (x+iy)ei~k·~r (2.1)
which is a plane wave ei~k·~r multiplied by a function (x+iy) which represents a vortex along
the linex =y = 0 embedded in a plane wave. The example when~k is parallel to the y−axis
is showed in fig. 2.1a, where the vortex line (along the z−axis) is perpendicular to ~k. The similitude with a crystal edge dislocation can be observed in Fig. 2.1b.
(a) Edge wave dislocation (b) Crystal edge
disloca-tion. Taken from the Im-press Education website [2].
The case of vortex line parallel to is showed in 2.2a. This type of dislocation has helical wavefronts, whose handedness depends on whether k·z > 0 (left-handed helicoid) or k·z <
0 (right-handed helicoid). This wavefront dislocations are related to screw dislocation 2.2b. Mixed edge-screw dislocations occur when the direction of propagation of the wave is neither perpendicular nor parallel to~k, and here, as in general, the wavefronts around the singularity are helicoids.
(a) Screw wave dislocation (b) Crystal screw
disloca-tion. Taken from the Im-press Education website [3].
Figure 2.2: The wave dislocation is parallel to the vortex line
The so called Burgers’ vector of the crystal dislocations was thus equated to the wave vector of the carrier wave. The optical current always circulates around the singularities, regardless of its nature (edge or screw), although the streamlines of current away from the vortex line display interesting spiral features.
2.2
Generation of optical vortices
2.2.1
Hermite-Gaussian and Laguerre-Gaussian modes
In this section we introduce expressions for Hermite-Gaussian (HG) and Laguerre-Gaussian (LG) modes, which will turn out to be essential for an understandings of the direct generation of vortices in a laser cavity as well as the generation through laser modes converters.
HG and LG modes arises as solutions of the 3-D paraxial wave equation in Cartesian and cylindrical coordinates, respectively. The former gives a family of solutions expressed in terms
of Hermite polynomials multiplied by a Gaussian envelope.
uHnm = C
HG nm
1 +ζ2e
−(x2+y2) w2(ζ) e
−ik(x2+y2)z
2(z2+(z/ζ)2)ei(n+m+1) arctan(ζ)H
n
x√2
w(ζ) !
Hm
y√2
w(ζ) !
(2.2)
These Hermite-Gauss solutions (or modes) are categorized by their indices n,m and “order”
N =n+m and given by the expression[4] with
ζ = zλ
πw2 0
, w(ζ) = wo(1 +ζ2)1/2 (2.3)
where w0 the beam waist, Hn the nth Hermite polynomial and CnmHG a normalization
con-stant. Intensity profile for some of the first modes are shown in figure 2.3, HG00 coincides with the Gaussian beam and is usually the output of an ordinary laser.
(a)HG00 (b)HG01 (c)HG02 (d)HG10
(e)HG11 (f)HG12 (g)HG20 (h)HG21
Figure 2.3: Intensity plot of Hermite-Gaussian modes
The solutions in cylindrical coordinates give rise to a family of modes, expressed as the product of a Laguerre polynomial of radial indexpand axial index`(LG`
p), a Gaussian envelope
radial index `. The electric field of Laguerre-Gaussian mode, is given by[5, 6]
LGp` =
e−i`φ
ω(ζ)`+1ρ
`L` m
ρ2
ω2
0(1 +iζ)
eikz−
ρ2
ω02(1+iζ)−i(`+2p+1) arctan(ζ) (2.4)
L`p are the associated Laguerre polynomials, ζ, w0, w(ζ) as defined in equation2.3. LG modes with p= 0 are single annular rings with a well defined azimuthal phase given by`. Forp >0, the modes are multiringed with p+ 1 radial nodes. In figure 2.4 we have plotted the intensity for some of the first modes. As we can observeLG00 is the same as HG00.
(a)LG00 (b)LG01 (c)LG02 (d)LG10
(e) LG11 (f)LG12 (g) LG20 (h)LG21
Figure 2.4: Intensity plots of Laguerre-Gaussian modes
2.2.2
Direct generation from a laser cavity
Perhaps the first optical system able to generate an optical beam containing a single vortex was the one developed by Tamm and Weiss[7]. They demonstrated that residual astigmatism of a laser could be controlled such thatHG10and HG01modes were frequency degenerate, and coherently interfered to give the hybrid mode T EM10∗ , (a superposition of HG10 and HG01). This mode has a transverse intensity pattern with the appearance of a ring donut and can also be expressed as an incoherent sum of the LG modesLG10 and LG−10. These are characterized by the fact that their phases rotates around the vortex in opposite directions. Although these two modes have identical intensity distributions, when passed through a pair of cylindrical
lenses they are transformed into a HG10 and HG01 modes respectively, providing thus a way to identify the sign of the vortex in the original mode. Tam and Weiss used this information to control the coupling of a small fraction of the T EM10∗ amplitude back into the laser, in such a way the output could be stabilized or switched between LG10 and LG−10 modes. Similar laser systems that emit LG beams directly have been developed including Amiel et al.[8] and Okida and Omatsu[9]. The latter demonstrated direct production of high power LG mode and suggested the possibility of the generation of high power optical vortexes in the visible and ultra-violet regimes.
2.2.3
Generation using cylindrical lenses mode converter
Following the work of Tamm and Weiss on low-order modes, Beijersbergen et al. demonstrated that exist an external-cavity way to obtain pure LG modes of any order [10]. Cylindrical lens mode-converter works by direct analogy with the waveplate for polarization [11, 12]. A mode converter consisting of two cylindrical lenses of focal lengthf separated byf /√2 can transform suitably oriented Hermite-Gaussian modes into simply related Laguerre-Gaussian modes with
` = ± | m−n | and p = min(m, n), acting as a π/2-converter in analogy with the quarter-waveplate (fig. 2.5).
Figure 2.5: AnLGnmoutput of a conventional laser can be transformed by means of a cylindrical
lens mode converter into an LG beam carrying an vortex along its propagation axis
When the cylindrical lenses are separated by 2f then they act as a π-converter, and it only rotates the Hermite-Gaussian mode. The combination of π/2 and π-converters provides considerable freedom in manipulating the mode and its orbital angular momentum. The analogy between transverse modes and polarization states extends to a description of modes and their transformation with an equivalent to the Poincar´e sphere[13].
The operating principle of mode converters is based on the fact that both HG and LG modes form complete sets of solutions to the paraxial wave equation. Hence any arbitrary paraxial distribution can be described as a superposition of HG or LG terms with the appropriate weighting and phase factors. Therefore, it follows that a LG mode can be described as a superposition of various HG modes and vice versa.
In transforming HG into LG modes with cylindrical lenses there are two main problems: firstly, this method requires the generation of a high-order HG beam as an input; secondly, any imperfection in the cylindrical lenses shape or a misalignment in the arrangement leads to a residual astigmatism in the resulting LG mode. This is manifested mainly in two things: the intensity pattern is deformed, losing its circular profile and; upon propagation, higher index vortex splits into multiple vortices [14].
2.2.4
Generation using Spiral Phase Plates
Another approach to generate beams with phase singularities was provided by Woerdman and coworkers[15, 16] through the use of Spiral Phase Plates (SPP). A SPP is an optical element constructed from a piece of disk-shaped transparent material with homogeneous refractive index
n and variable heighth. This height increases linearly with the azimuthal angle θ, resembling a spiral staircase (fig. 2.6a) and is given by
h=hs
θ
2π +h0, (2.5)
where hs is the step height and h0 is the base height of the device (fig. 2.6b).
(a) Spiral Phase plate. Taken from skulls in the stars webpage[17]
(b) Phase step (c) Phase distribution
Figure 2.6: Spiral phase plate, and its phase distribution
The spiraling thickness variation imposes an azimuthal retardation on the optical field (the thicker the plate, the greater the phase shift), creating the helicoidal phase distribution of an
optical vortex (fig. 2.6c). The azimuth-dependent optical phase delay is given by
φ(ϕ, λ) = 2π
λ
(n−n0)hsθ
2π +nh0
, (2.6)
where n0 is the refractive index of the surrounding medium.
If the height of the step corresponds to a phase difference of 2π, a SPP inserted in the waist of a Gaussian beam will imprint an azimuthal phase profile ofeiϕ, generating the optical vortex along the beam axis (fig. 2.7).
Figure 2.7: Wave fronts of a Gaussian beam before and after passing through a SPP.
If the step height of a SPP is not an integer multiple of 2π, in addition to the on-axis optical vortex, a radial phase discontinuity is created. Closer inspection reveals that this radial line has an intricate vortex structure. For half-integer step heights, there is a chain of vortices with alternating sign on propagation [18]. These vortex points are simply the intersection of vortex lines with the viewing plane. Such vortex line can be observed by inspecting successive planes [19]. SPP with half-integer step heights have been deliberately fabricated for experiments in quantum optics [15].
In order to be effective, the phase plate must be smooth and accurately shaped to a frac-tion of the wavelength. Several groups have employed precise micro-machining techniques to manufacture them[20]. However, even if it is successfully produced, it is only applicable to a single wavelength of light and will produce a specific topological charge.
A more versatile way to create optical vortices is by means of an adjustable spiral phase plate [21]. They are created by twisting a piece of line-cracked Plexiglas in such a way one tab of the phase plate is directly perpendicular to the incident light, and the other tab is bent at some angleθ away from the other (Fig. 2.8). Because of the azimuthally varying tilt around the center of the phase plate, a laser directed to this point will acquire a phase singularity. These phase plates can be used with multiple wavelengths and are able to produce vortices within a
wide range of topological charges.
Figure 2.8: Variable phase plate
2.2.5
Generation using diffractive optical elements
The use of diffractive optical components to transform spatially coherent, flat phase beams into beams containing optical vortices began to be explored by a couple of groups in the early 1990s. Soskin et al. discovered in 1990, that when a diffraction grating is modified to include an edge dislocation at its center, in the form of a fork. To create this diffracting grating, the interference pattern between a plane wave and the beam one desires to produce is recorded as a hologram on photographic ¨ıˇnAlm. The resulting grating has a “pitchfork” dislocation with a,
`-value imposed corresponding to the difference between the number of lines above and below
the dislocation. Holograms designed to create a beam with ` = 1 and `= 2 are shown in Fig. 2.9a and 2.9b respectively.
(a)`= 1 (b)`= 2
Figure 2.9: Holograms to create beams with optical vortices
Once developed, the ¨ıˇnAlm can be illuminated by a plane wave to produce a ¨ıˇn, Arst-order, diffracted beam that has both the intensity and phase of the desired beam[22, 23]. This forked
design, which has become synonymous of the generation of optical vortices, can be implemented either as an amplitude or phase grating. In theory, a phase June grating can diffract all of the incident energy into the first diffraction order, but in practice this is not achievable and a fraction of the energy ends up always in other orders. Since each order is diffracted through different angles, we can simply use spatial filters to select the firts order. The big advantage of this approach lies in the current availability of high-quality spatial light modulators (SLMs) which will be discussed in next section.
2.2.6
Generation using spatial light modulators
SLMs are computer-controlled pixellated liquid crystal devices that has replaced the holographic ¨ıˇnAlm. The interference pattern need not be experimentally derived anymore, but can be, simply computed-calculated and displayed on the device. Holograms calculated in this way allow a simple laser beam to be converted into any beam with an exotic phase and amplitude structure. A further advantage is that the pattern can be changed many times per second, so the transformed beam can be adjusted to meet the experimental requirements. This is the easiest way to generate beams carrying phase singularities. Also the spatial variations of phase can be transfered upon reflexion or transmission to the incoming beam. When working by reflection they features efficiencies that typically exceeds 50%.
In many cases manufacturing process of SLMs tended to produce some residual astigmatism in the resulting beam; however, it is possible to use an algorithm to calculate a correction hologram that can be added to any hologram design [24]. This powerful algorithm originally developed for crystallography by Gerchberg and Saxton [25], is now used widely in the design of computer-generated holograms. This is an iterative method that relies on the ability to perform high-speed Fourier transforms to calculate, given any hologram, the intensity and phase distribution in the far-field, and inverse transforms to invert the distribution back to the hologram plane to give the phase of the required hologram (typically it converges within a few iterations). This algorithm can also be modified to generate LG and other beams [26].
The phase information of the diffracted beam is encoded in the spatial irregularity of the phase pattern provided by the diffraction grating. Furthermore, the depth of the phase mod-ulation can be used to locally control the intensity of the diffracted light. This was first used to reduce the on-axis intensity surrounding an optical vortex, so that the near field intensity pattern was a closer approximation to an LG mode [27]. The technique was fully exploited in the precise generation of specific superpositions of LG modes to create vortex loops, links,
and knots [28]. In recent years, spatial light modulators have been also used in applications as diverse as optical tweezers[29] and adaptive optics[30].
2.3
Phase singularities and orbital angular momentum
of light
It has been know since 1873 with the work of Maxwell, that light carries energy, momentum and angular momentum. However the small force obtained with the available light sources, along with some other effects such as radiometric, thermophoretic and photophoretic in low pressure gases, made direct measurements difficult. The existence of optical angular momentum was plainly stated by Poynting who used a mechanical analogy to suggest a connection with circular polarization[31]. This reasoning was endorsed independently by Holbourn[32] and by Beth[33] in 1936. In particular, Beth observed the rotation of a birefringent plate in a circularly polarized plane wave. Recently, in an experimental work, rotation of microscopic birefringent material was observed due to the transfer of angular momentum carried by circularly-polarized light of a Gaussian beam[34]. The hypothesis that the polarization state was the principal vehicle for the transport of optical angular momentum was modified in 1992 by Allenet al.[10]. They realized that optical singularities in Laguerre-Gaussian (LG) laser modes carry angular momentum associated with their optical vortices. This fact made clear that along with the polarization-related angular momentum, there was an additional contribution associated with spatial phase variation within the field. They also proposed an experiment to observe transfer of the OAM associated with its optical vortex to a macroscopic object. In the experiment a pair of cylindrical lenses suspended on a torsion fiber would reverse the helicity of a LG beam and suffer a reaction torque. However, this torque is very small for reasonable power beams and it has turned out to be easier measuring such effects on a microscopic scale, in the context of optical trapping experiments. More evidence for vortex angular momentum comes from experiments where LG beams have been used in second harmonic generation (Fig. 2.10). Second harmonic generation can be seen as the annihilations of two photons of low frequency and the generation of one photon with double frequency, that is twice the energy. To ensure efficient second harmonic generation it is also important to ensure a conservation of linear momentum. More over, also exist a conservation of angular momentum, which implies that in second harmonic generation, two photons with orbital angular momentum `~ per photon should combine to form a single photon with an orbital angular momentum of 2`~ [35, 36].
Figure 2.10: Second-harmonic generation.
Orbital angular momentum and optical vortices are often incorrectly used as synonymous though the former may exist without the presence of the second, as we explain now. An optical vortex is a locus in space around which the optical phase advances or retards by a multiple of 2π. At the vortex center, the phase is singular and the intensity becomes zero, therefore the vortex itself carries neither linear momentum nor angular momentum. By contrast, it is only in the immediate vicinity of the vortex, that the azimuthal phase term occur in helicoidal phase fronts of the optical field (Fig.2.11). Furthermore, in 1997 Courtial et al. [37] demonstrated than an elliptical Gaussian beam focused by a cylindrical lens can posses orbital angular momentum without the presence of an optical vortex, since an optical beam can have an azimuthal phase gradient without a nearby phase singularity. A further discussion lies beyond the scope of this work, however most studies of orbital angular momentum have involved beams with optical vortex lines along the beam axis.
The physical origin of the orbital angular momentum is associated with the rotation of the momentum density. The electric and magnetic fields at any point lie in the plane tangent to the helical phase front. This means that the local momentum density, 0E~ ×B~, is normal to the phase front, hence the momentum density itself follows a helical path along the beam. For
` = 1 the phase fronts have the form of a simple screw thread, whereas for ` = 2 the phase fronts form a double helix.
(a)
Figure 2.11: In the immediate vicinity of vortex, azimuthal phase term occur in helicoidal front of the optical field
ratio of angular momentum flux to energy flux for a LG mode of order ` can be written as
J cP =
l+σz
ω (2.7)
withσz =±1 for left/right-handed circularly polarized light,σz = 0 for linearly polarized light,
and P is the linear momentum density.
This can be interpreted as each photon of energy ~ω carrying σz quanta of “spin” angular
momentum associated with the polarization state and` quanta of “orbital” angular momentum associated with the spatial distribution of the field, each quantum having magnitude~.
Optical phase singularities as morphological objects rips of wave front remain the same under perturbations. For instance, addition of a small coherent background does not destroys a vortex, but only shifts spatially its position to another place where the field amplitude has a zero value. For vortices with multiple charge this operation will split an initially m-charged vortex into m single-charge vortices. Intuition suggests that the total topological charge would be conserved in a beam propagating in free space.
2.4
Topological charge of optical vortices
In two dimensions phase singularities occur at points, while in three dimensions they occur at lines. This can be explained as follows: in two dimensions a zero of the complex field must be a place where the real imaginary parts become zero the imaginary becomes zero, both occur along lines (see fig.2.12a), these two sets of contour lines intersect at points, the nodal points of the complex field. In three dimensions the real part as well as the imaginary becomes zero
along surfaces, hence they intersect at curved lines (fig.2.12b).
(a) 1-D Phase singularities (b) 2-D Phase singularities
Figure 2.12: Phase singularities in 2-D are represented by points, whereas in 3-D are represented by lines.
Optical vortices embedded in a host light beam behave much as charged particles. They may rotate around the beam axis, repel or attract each other, and annihilate in collision. For the next definition, lets represent the complex wave function as, ψ(~r, t) =%(~r, t)eiϕ(~r,t), in terms of its real modulus%(~r, t) (giving the intensityI =%2) and phaseϕ(~r, t). The dislocation strength or vortex topological charge s is defined by the circulation of the phase gradient around the singularity[38],
s= 1 2π
I
dr· ∇ψd~r (2.8)
Since the phase changes by a multiple of 2π the result is an integer. The sign of s gives the sign of the singularity, positive if the phase increases in a right-handed sense and negative if it increases in the opposite direction. Under appropriate conditions, it also measures an orbital angular momentum of the vortex associated with the helical wave-front structure.
To compute the topological charge value of an optical vortex we need information of the wave front or the phase distribution. Several techniques have been developed to compute indirectly the topological charge. A commonly used technique is to interfere the measured wave front with its mirror image [12, 39] or a uniform wave front, in which case the interferograms reveal the topological charge value of the measured optical vortex. Some other interferometric methods have also been proposed, for example, Leachet al. proposed a method based on a Mach-Zehnder interferometer with a Dove prism placed in each arm[40].
2.5
Phase cuts
In the mathematical field of complex analysis branch cuts are curves in the complex plane across which an analytic multivalued function is discontinuous. They are also known as cut lines, slits, or branch lines and can exist with ends possibly open, closed, or half-open. Branch cuts are usually, but not always, taken between a pair of branch points. A branch point is a locus in which a multivalued function is discontinuous when going around an arbitrarily small circuit around it.
More precisely, lets consider a small circular path around a point z0 represented by the equation z = z0 + reiθ, with r > 0 a small constant and θ varying in a counterclockwise direction about the pointz0 . If we have a function ω =f(z) such thatω =f(z0+reiθ) takes on different values as θ increases from 0 to 2π, then the pointz0 is called a branch point of the function and the different values of ω are called branches of the function. A line which starts at a branch points is called a branch cut or branch line (see fig 2.13).
Figure 2.13: Plot of an analytic multivalued function. Branch points are enclosed by circle. Branch cuts are usually lines between pairs of these branch points
Extrapolating the above definition to the phase singularities we can see that branch cuts correspond to discontinuities in the phase function and branch points corresponds to optical vortices. For this reason, line-discontinuities in the wavefront are referred to as phase cuts. A formal demonstration of the existence of branch points in the phase function goes beyond the scope of this work. The complete formalism can be found in an article by Fried and Jeffrey [41].
Detection of branch cuts (and therefore branch points) embedded in the phase function can be done with the help of the gradient operator. An exampled of a phase function is presented in fig.2.14and corresponds to an Airy-vortex beam (which will be introduced in chapter 3). In fact this is the case in which we are interested. As we can observe, branch cuts corresponds to
maximums in the phase function. Plots of branch cuts can be done with the help of Matlab. The program used is presented in appendix B.2. The original version was provided by M. Mazilu though minimal changes were done for this final version. The branch cuts plot of an Airy-vortex beam are shown in fig. 2.14b. From now on we will refer to these plots as phase cuts, a name given by Mazilu.
(a) Phase of an Airy-vortex beam (b) Phase cuts of the phase of an Airy-vortex beam
Figure 2.14: The phase function of an Airy-vortex beam shows maximums along branch cuts. Plots of branch cuts are called phase cuts.
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The Airy-vortex beam
Contents
3.1 The Airy Wave Packet . . . 28 3.2 The Finite Energy Airy Beam . . . 30 3.3 Airy-Vortex Beam . . . 33 3.4 Phase cuts in the Airy-vortex phase function . . . 35
3.4.1 Rotating phase cuts in the phase function . . . 36
References . . . 38 A new type of solution to the 1-D Schr¨odinger equation in terms of Airy functions was proposed by Berry and Balaz in 1979 [1] and it was shown to possess amazing features. Perhaps the most relevant, is its tendency to freely accelerate in free space. We consider of high relevance this discovering and hence we will dedicate the section 3.1 to talk about it . The similitude between 1-D Schr¨odinger equation and the paraxial wave equation in optics motivated the investigation of a similar solution to this equation. As result the Airy beam was generated featuring surprising properties similar to the quantum case. A complete discussion is presented in section 3.2. More recently, an interesting proposal by Dholakia’s group lead to the generation of Airy beams carrying orbital angular momentum (OAM). This OAM was given by adding an optical vortex to the main lobe of an Airy beam, they were able to show that this optical vortex also experiences a transverse acceleration in the same way Airy beams does. Section 3.3 is dedicated to its mathematical description and generation. Finally, in section 3.4, we describe the application of phase cuts to track the position of the optical vortex embedded in the Airy beam.
3.1
The Airy Wave Packet
In the context of quantum mechanics, Landau and Lifshitz found a solution for the Schr¨odinger equation of a particle in a homogeneous external field in terms of Airy functions [2]. Later, in 1979, Berry and Balaz proposed a new type of solution to the 1-D Schr¨odinger equation for a free particle with mass m [1]
− ~
2
2m ∂2ψ
∂x2 =i~
∂ψ
∂t, (3.1)
also in terms of Airy functions. Such nonstationary solution was
ψ(x, t) =Ai
B
~2/3
x− B
3t2 4m2
e(iB3t/2m~)[x−(B3t2/6m2)], (3.2)
which evolves according to Schr¨odinger equation (3.1) from the wave packet
ψ(x, t= 0) =Ai(Bx/~)2/3 (3.3)
with B an arbitrary constant and Ai(x) representing the Airy function whose square modulus is sketched in Fig. 3.1.
Figure 3.1: Square modulus of an Airy function Ai(x). For negative numbers it decays oscil-lating whereas for positive it decays very fast
This novel kind of solution (eq. 3.2) seems to contradict some of the statements of quan-tum mechanics. First, this wave packet has the ability to freely accelerate in free space.This apparently contradicting Ehrenfest’s theorem which states that the center of mass of a packet should move with constant speed in the absence of any external potential[3]. Berry and Balaz explained this apparent paradox by saying that the Airy packet does not have a well defined center of mass. This happens because the Airy function is not square integrable and thus can-not represent the probability density for a single particle. By using an analogy with ray theory,
they assigned a particle to each plane wave and argued that the Airy packet corresponds to an ensemble of an infinite number of particles. The straight trajectories of these particles in a space-time diagram are enveloped by a parabolic caustic.
The acceleration term can be obtained by setting the argument of the Airy function in Eq. (3.2) to a constant and differentiating it with respect to time [4] to obtain v and from it the acceleration term
v = B 3t
2m2 from which a=
B3
2m2. (3.4)
Another remarkable feature is the tendency to propagate without spreading as can be seen when taking the|ψ|2 in equation (3.2). This feature seems to go against the dispersion nature in the Schr¨odinger equation that suggests that all wave packets in free space should change their form as they propagate. A physical explanation to this apparently paradox was provided by Greenberger [5], based on the equivalence principle. It states that all forces disappear in a free-falling system. By using a generalized Galilean transformation he was able to change the forced Schr¨odinger equation
i~∂tχ(x, t) + ~
2
2m∂
2
xχ(x, t) +
B3x
2mχ(x, t) = 0 (3.5)
to a force-free equation
i~∂tψ(x0, t) + ~
2
2m∂
2
x0ψ(x0, t) = 0 (3.6)
by transforming the wave function in (3.5) from an initial gravitating frame to a free falling frame with a transformation relation given by
χ(x, t) =ψ(x0, t)e−i(m~)[ξ(t)x0−Rdtξ˙(t)/2], (3.7)
where ˙ξ(t) is the time derivative of the function ξ(t) and
md
2ξ(t)
dt2 =−
B3
2m. (3.8)
From equation (3.7) he realized that the wave functionψ(x0, t) is precisely the Airy packet in (3.2). He concluded that this Airy packet does not spread out because it actually represents a free non-relativistic particle falling in a constant gravitational field and the transformation to
the free falling system only introduces a phase factor. A detailed explanation is also given by Besieris et al.[6].
It was also emphasized by Berry and Balaz using a purely classical analysis of trajectories, that the Airy packet is the only nontrivial solution (apart from a plane wave) that propa-gates without changing its form. However, it was until 1996 that a formal derivation of this nonspreading wave packet was given by Unnikrishnan and Rau [7]. In the same article they provided with a quantum mechanical proof to the assertion that this is the only solution hav-ing this properly. This probe was derived from first principles ushav-ing a decomposition of the quantum mechanical evolution operator of a free particle.
3.2
The Finite Energy Airy Beam
In 1993 Besieris et al.[6] motivated by the work of Berry and Balaz on the 1-D Schr¨odinger equation investigated a class of nonspreading solutions to the 3-D Schr¨odinger equation
i~∂tψ(~r, t) + ~
2
2m∇
2
ψ(~r, t) = 0, (3.9)
involving accelerating Airy-type solutions and the 3-D scalar wave equation
1
c2∂ 2
t − ∇
2
ψ(~r, t) = 0 (3.10)
By using
ψ(~r, t) =G(ρ, ζ)eiβη (3.11)
with ζ =z−ct, η=z+ctand β an arbitrary parameter, they were able to transform the 3-D scalar wave equation to the 2-D Schr¨odinger equation
i4β∂ζG(ρ, ζ) +∇2TG(ρ, ζ) = 0, (3.12)
where∇2
T is the transverse Laplacian. The Airy packet solution derived was given by
Having in mind this antecedent they finally arrived to an Airy packet solution to the 3-D scalar wave equation given by
ψ(~r, t) =Ai[2β(x−3βζ2/4)]Ai[2β(y−3βζ2/4)]ei2β2ζ(x+y−2βζ2/3)eiβη (3.14)
They emphasized the fact that in modeling particles with nondispersive wave packets it is important their localization and the finiteness of their total energy content. However they shown that if the localization of particles is given, it is possible to overcome the infinite energy problem by using the energy of the central portions of the non-dispersive wave packets and an appropriate choice of parameters, These packets can be rendered with very large amplitudes around their centers compared to their tails.
In the context of optics new features of the Airy packets were demonstrated for the first time in 1994 by Nassaret. al.[4]. In 2007 taking advantage of the similitude between the Schr¨odinger equation and the paraxial equation of diffraction
∇2
Tφ−i2k
∂φ
∂z = 0 (3.15)
where ∇2
T = ∂2/∂x2 +∂2/∂y2, Siviloglou and Christodoulides[8]were able to provide with a
new kind of solution to the paraxial wave equation. Starting with the 1-D paraxial equation of diffraction
i∂φ ∂ξ +
1 2
∂2φ
∂s2 = 0 (3.16)
and the initial condition φ(s, ξ) = Ai(s)eas they obtained an expression for the finite energy Airy beam (Apendix A.1)
φ(s, ξ) =Ai(s−(ξ/2)2+iaξ)eas−aξ2/2−iξ3/12+ia2ξ/2+isξ/2
(3.17)
where s = x/x0 is a dimensionless transverse coordinate, x0 is an arbitrary transverse scale,
ξ =z/kx2
0 is a normalized propagation distance, k= 2πn/λ is the wave number of the optical wave, n is the refraction index, λ is the wavelength of light and a is an arbitrary positive constant. The later ensures confinement of the infinite Airy tail, which is essential for a physical realization of such a beam. The total power contained in this finite energy Airy wave packet is computed in Appendix A.2. This finite energy Airy beam is no longer diffraction-free, however, it has been shown that its intensity profile remains almost invariant for up to 50 cm. One of the main features of Airy beams is its ability to freely accelerate in a direction perpendicular
to the axis of propagation. This acceleration dynamics can be clearly seen in Fig. 3.2, where the beam’s parabolic trajectory becomes evident for infinite (Fig. 3.2a) and finite energy Airy beams (Fig. 3.2b).
(a) Infinite energy (a=0) (b) Finite energy (a=0.04)
Figure 3.2: Acceleration in infinite and finite energy Airy beams
It is straight forward to extend this solution to two dimensions. The initial field envelope is given by
φ(x, y, z= 0) =Ai
x x0
Ai
y y0
eωx1+ y
ω2. (3.18)
The intensity profile of such a 2-D Airy beam forx0 =y0 = 1 and ω1 =ω2 = 1 is shown in Fig. 3.3a for a diffraction free Airy beam and in Fig. 3.3b for a finite energy Airy beam .
(a) a=0 (b) a=0.04 (c) Phase mask
Figure 3.3: Intensity profile of 2D (a) diffraction-free infinite and (b) finite energy Airy beams, along with the generating phase mask.
Even though the theory for the physical realization of these novel beams was already estab-lished, it was not clear yet how they could be generated experimentally. A couple of months after the theoretical prediction, Christodoulides et al.[9] reported the first observation of the
finite energy Airy beams. The clue was in the Fourier transform of equation (3.17) (Apendix A.3)
Φ0(k) =e−ak
2
ei3(k
3−3a2k−ia3)
(3.19)
From this equation, is some how clear, the Airy beam could be generated from a broad Gaussian beam (given bye−ak2) through a Fourier transformation provided that a cubic phase is imposed (which is evident from the termeik3/3). Figure 3.3c shows the 2-D cubic phase mask generated to produce the Airy beam.
3.3
Airy-Vortex Beam
Recently, Dholakia et al., developed a novel technique to produce Airy beams carrying orbital angular momentum [10]. This “Airy-vortex” beam is characterized by a dark spot in the center of the maximum intensity lobe. Paraphrasing their own words, “there are two ways in which these beams can be generated. One is to illuminate with a Laguerre-Gaussian beam an SLM encoding a cubic phase. Second is to include in the cubic phase mask the phase singularity corresponding to a Laguerre-Gaussian (Fig. 3.4a) beam and use a broad Gaussian beam to illuminate the SLM ” (Fig.3.4).
(a) Mask (b) Airy-vortex beam
Figure 3.4: A mask with a vortex in its center (a) is generated in order to produce an Airy-vortex beam from a broad Gaussian.
A detailed mathematical treatment for Airy-vortex beams can be found in [10] as well as an experimental demonstration that the vortex embedded in the Airy beam also exhibits a transversal acceleration.
We now describe briefly the method followed by Dholakia and coworkers in the deduction of a general expression for Airy-vortex beam. Starting out from the 2-D equation for the field
profile in the SLM plane, given by
ˆ
uAV(kx, ky, z =zSLM) = (kx+iky)`e−a0x
2
0(k2x+k2y)e3i(x 3
0k3x−3a20x0kx−ia30)e3i(x30ky3−3a02x0ky−ia30) (3.20)
where subscriptAV stands for Airy-vortex and `gives the vortex order. The amplitude co-efficient containing the vorticity was transformed to a polynomial expression using the binomial expansion
(x+iy)` = `
X
n=0
`!
n!(`−n)!x
n(iy)(`−n) (3.21)
The above equation is introduced in 3.20. Then the generalized Huygens-Fresnel integral for the field is computing. Finally a propagation through a distance f, a Fourier lens of focal lengthf, again a distance f and lastly a distance z is performed to obtain an expresion for the Airy-vortex beam(Fig.3.5).
Figure 3.5: The Mathematical expression for an Airy-vortex beam is computed by propagating the generalized Huygens-Fressnel integral of the field ˆuAV
To perform mathematically this procedure, use of the ABCD matrix required
A B
C D
!
= 1 f+z
0 1
!
1 0
−1/f 1 !
1 f
0 1 !
, (3.22)
one can arrived to the final expression for an Airy-vortex beam which is given by
uV A(x, y, z) =
ik 2πB ` X n=0 `!
n!(`−n)!(i)
`−nP
n(`−n)exp
2a3 0 3 −i
kD
2B(x
2+y2)
. (3.23)
The Pij functions are determined by using the recurrence relationships
Pij =
∂P(i−1)j
∂cx
and Pji =
∂Pi(j−1)
∂cy
