Generation of Sources of Light with Well Defined Orbital Angular Momentum

In this work, a technique to produce spatial electromagnetic modes with definite orbital angular momentum is presented. The method is based in the construction of binary diffractive gratings generated by computer. In the classical regime the gratings produce the well known Laguerre-Gaussian modes distributions when illuminated by a plane wave. In the quantum regime the grating is placed in the signal path of a spontaneous parametric down conversion layout and the diffraction pattern, observed in the coincidence count rate, shows that the single photons are projected onto spatial states consistent with a Laguerre-Gaussian modes distribution.


Introduction
Quantum engineering is a branch of science that emerged recently. Its objective is to understand how to improve the manipulation and control of physical systems for the development of new technologies approaching the quantum regime. Of the infinity of processes that occur at the quantum level, the interaction of radiation with matter is of fundamental importance. This is why the interest in understanding the properties of radiation and how to apply them in handling quantum systems has been growing in recent years. One of such properties is the ability to transmit energy and momentum to material systems at a microscopic scale. The energy and linear momentum of radiation are well studied concepts. However, a complete analysis of the optical orbital angular momentum and its potential applications is still under development. The concept itself has gained much interest since its discovery in 1992 [1][2][3] due to the fact that beams with well defined orbital angular momentum can be easily produced in practice by means of different methods [4][5][6][7][8]. At the same time, the extraordinary perspectives of application of their properties in the manipulation and trapping of matter, quantum information and photonics were immediately evident [9][10][11][12]. Experimental studies involving electromagnetic orbital angular momentum in nonlinear optics began shortly in the generation of frequency doubling [13,14], where the phase matching conditions for helical beams proven to be the same as for plane waves. A similar phenomenon occurs in more general cases of frequency mixing [15,16]. Since then, too much interest has been paid to the study of the conservation of the orbital angular momentum, spatial correlation and entanglement, of single photons produced by spontaneous parametric down conversion [17][18][19][20][21][22][23][24]. Studies of these phenomena intends to give a better understanding of the transfer of angular momentum between matter and light at the quantum level. The purpose of this work is to present a procedure to generate sources of light in definite orbital angular momentum states in the classical as well as in the quantum regime. In section 2, the Laguerre-Gaussian modes are considered, from the theoretical point of view, as solutions of the paraxial Helmholtz equation with definite orbital angular momentum. In section 3, the procedure used to generate Laguerre-Gaussian modes distributions is presented. Section 4 contains the experimental setup used to generate single photons in definite spatial states consistent with a Laguerre-Gaussian modes distribution. A spontaneous parametric down conversion (SPDC) layout is used as the source of single photons. The article closes with some conclusions.

Orbital angular momentum of light
The fundamental Gaussian beams of light are characterized by their plane wavefronts and Gaussian transverse intensity distributions (see Figure 1). For those modes the linear momentum density 0 E × B points along the direction of propagation, meaning that there is no component of the angular momentum in that direction. For higher-order modes, as e.g. Laguerre-Gaussian beams, the points of constant phase are delayed with respect to each other and the wavefronts are no longer plane surfaces [25]. The transverse intensity distributions in those cases have also complex structures (see Figure 2). These spatial modes are predicted by the Maxwell wave description of light as solutions of the wave equation in the paraxial approximation [26]. In particular, the cylindrically symmetric solutions known as the Laguerre-Gaussian modes have the form [27] where E 0 is a constant vector, is an integer number, z R is the Rayleigh range, ω(z) is the transverse width of the beam (assuming that the waist is placed at z = 0), R(z) is the radius of is the associated Laguerre polynomial [27]. These modes form a complete set of orthogonal solutions to the paraxial Helmholtz equation. The phase θ is responsible for the helical shape of the wavefronts (see Figure 2) and is directly related to the orbital angular momentum of the beam.
Considering the local angular momentum density of the electromagnetic field [28] and using the expression (1), it is possible to show that the ratio of orbital angular momentum to energy in the direction of propagation is /ω [29], suggesting that the Laguerre-Gaussian modes posses definite orbital angular momentum proportional to . In this way, the integer number determines the number of sheets in the helical surface of the wavefront and the amount of orbital angular momentum of each axial mode of the electromagnetic field.

Production of light with orbital angular momentum
Different techniques have been developed in order to generate structured beams in practice. For example, it is possible to transform Hermite-Gaussian beams into Laguerre-Gaussian ones using a set of two cylindrical lenses called mode converter [1,7,8]. The converting process is based on the decomposition of a Laguerre-Gaussian and a diagonally oriented Hermite-Gaussian modes into Hermite-Gaussian ones. By changing the separation of the set of lenses it is possible to revert the helicity of the resulting beam. However, the most common technique in the production of helical wavefronts with any value of orbital angular momentum is the use of diffractive gratings generated by computer [4][5][6]. In this case the interference pattern of a plane wave and the mode to be generated is recorded in a photographic film. The resulting grid has an −fold dislocation (a fork) at the center of the pattern. When a plane wave illuminates this pattern, the diffracted beam acquire the desired distributions of phase and intensity. Spatial light modulators (SLMs) have also been used to generate beams of a wide range of phase and intensity distributions [10,23]. They consist in pixelated liquid crystal devices instead of the photographic film, in which different holographic patterns can be displayed.
In this work the Laguerre-Gaussian beams were produced by means of binary diffraction gratings. In order to make the design of these holograms, the interference process of a plane wave and a Laguerre-Gaussian beam of the same frequency and amplitude is considered. The total amplitude of the electric field at a point P in the far field is (see Figure 3) where the first term is the plane wave and the second one the Laguerre-Gaussian mode. Note the additional term in the phase of this field corresponding to the phase angle that characterizes the helical beams. The total irradiance of the electromagnetic field at point P is given by  Assuming that r 1 and r 2 are much greater than the separation a between the sources one obtains ( Figure 3) Next, it is assumed that the boundaries between the clear and dark fringes of the interference pattern are found whenever the argument of the periodic term of (5) crosses integer multiples of π, i.e.
where (r, θ) are the polar coordinates, s is the number of dislocations in the grid, m is an integer number and Λ is the period of the grid at large distances from the dislocation [6,9]. By assigning different values to these parameters one may generate diverse grids that produce particular spacial light distributions. In Figure 4 we present two diffractive gratings constructed with s = 2 (a) and s = 6 (b). As they were illuminated by a plane wave, the well known distributions of Laguerre-Gaussian modes were observed. In order to evaluate the vorticity of these modes we use the self-interference method [1], which consists in superposing two modes of the same charge and opposite handedness. Then, an interference pattern with N = 2 regions of constructive interference can be observed. In Figure 5 we show the interference pattern of the Laguerre-Gaussian mode distribution generated by a grating with charge s = 2 superposed with itself but rotated πrad. We can observe a central mode of order 0, a second order mode composed by a distribution of four maxima, a third order one with eight maxima and so on. This means that the corresponding Laguerre-Gaussian modes are associated to = 0, ±2, ±4, . . . Similar experiments with different spatial distributions allow to conclude that Laguerre-Gaussian modes associated to = sn, n = 0, 1, 2, . . . can be generated by means of grids with s−fold dislocations.

Spatial correlation of light with orbital angular momentum
In this section the experimental setup used to test the generation of light with definite orbital angular momentum in the quantum regime is presented. A SPDC layout is used as a source of single photons. In our setup the pump beam is generated by a violet diode laser, of wavelength λ = 405nm and a bandwidth of 0.78nm, with horizontal polarization. A type I BBO crystal cut at 30 • with respect to the optical axis and a semi-cone angle of 5 • is placed to 13cm from the laser output as shown in Figure 6. A beam with a spot of transversal diameter 1.5mm and power of 100mW is incident on the BBO crystal. Two light collectors are placed at a distance of 80cm from the crystal, on the signal and idler arms of the SPDC setup. The collectors consist on coupling lenses connected to optical fiber to send the down converted light to the detection system. For the measurement of count rates, two avalanche photodiodes (APD), with a detection efficiency for 810nm single photons of about 60%, are used. When the APDs detect a photon, a 3.5V TTL pulse of 15ns is triggered. Then the coincidence system detects either individual or coincidence counts. A diffractive grating to produce Laguerre-Gaussian modes with = 0, ±2, ±4, . . . (see Figure  4(a)) is set on the signal arm of the SPDC source at a distance of 27cm from the BBO crystal. An He-Ne laser of wavelength 633nm is used to track the course of signal photons in order to align the grid. The first collector is then placed at the position of maximum counts of idler photons while the second collector make an horizontal scanning on the signal arm searching for coincidences. The aim of each idler photon is to localize in both, time and position, the corresponding signal photon. In Figure 7 it is shown the diffraction pattern observed in coincidence count rate. Observe that this distribution shows that the signal photons are projected onto spacial states consistent with the three lowest orders of a Laguerre-Gaussian modes distribution, suggesting that they posses definite orbital angular momentum.

Concluding remarks
A technique for the generation of Laguerre-Gaussian modes with different values of angular momentum was established. This technique consist in the design of diverse types of diffractive gratings by means of the interference pattern of a plane wave and different Laguerre-Gaussian modes. The angular momentum of the resulting modes was determined by observing the interference pattern of two modes with opposite helicity. The designed holograms shown to produce helical modes with orbital angular momentum proportional to the number of dislocations in the grids. A spontaneous parametric down conversion source was used to produce single photons and the coincidence count rate diffraction pattern, when a fork grid is placed in the signal path was observed. The experiment allowed to conclude that the parametric down converted photons were projected onto spacial states consistent a Laguerre-Gaussian modes distribution.