Orbital beam laser unity free download

Introduction

In the last decades, the great attention is paid to laser beams with orbital angular momentum (OAM) [1&#x;3] due to their particular properties, which have found a lot of applications [4&#x;8]. The beams of this kind are referred to as optical vortices or vortex beams due to the presence of the transverse circulation component of the Pointing vector [1]. In particular, the feasibility of using optical vortices for information coding and transmission is intensely studied [7, 8]. The generation of vortex beams has become a new field of the new optical science &#x; singular optics [9]. However, the development of relevant technologies requires creation of fast devices for generation of vortex beams with a tunable topological charge or OAM. Generation methods based on spatial light modulators (SLM) [10&#x;12] are insufficiently fast and, as a rule, inefficient in conversion of one beam type into another. Their application allows beam OAM to be tuned with a frequency no higher than few kilohertzs. As a result, the development of new high-speed methods and devices for generation of vortex laser beams comes to the forefront. In some tasks, they should operate under conditions of high radiation intensity. In our opinion, the method of formation of vortex optical beams with the changeable orbital angular momentum based on an array of coherent fiber radiators is most adequate to the formulated problem. It implements the approach, whose idea occurred to us upon publication of Lachinova and Vorontsov [13]. The approach is based on the control over the phase of individual radiating subapertures. These subapertures are arranged hexagonally and make up a cluster (array) to provide for the phase progression of 2mÀ while circling around the center of the synthesized beam. The main advantage of this approach is the possibility of fast (with a frequency higher than 10⁹ Hz) phase shift at subapertures, which provides for a change of OAM.

This review analyzes the possibility of creating a system for generation of laser beams, in particular, vortex laser beams, with the spatial structure controllable in real time based on the Coherent Beam Combining principles [13, 14]. First, some peculiarities of formation of vortex beams synthesized from the different number of sub-beams are determined [15&#x;19]. Then, the corresponding experimental setup is schematically described, along with its main operating principles and experimental results obtained with it [20, 21]. Finally, the results of numerical simulation of the propagation of synthesized beams in the turbulent atmosphere are given [15&#x;17, 22].

Generation of Vortex Beam Based on Coherent Combining of Fields From Elements of the Fiber Cluster. Numerical Experiment

The complex amplitude of the field of a synthesized vortex beam can be presented in the following form

where N_a is the number of sub-beams (subapertures) in the array, and are coordinates of the centers of sub-beams arranged in a circle or a hexagon, a_sub is the sub-aperture radius. It should be noted that the central sub-beam has the coordinates x_c = 0, y_c = 0. In addition, if the synthesized beam is formed from several sub-beam rings (N_a= 18, 36, 60, and so on), internal rings may be absent.

We take the Laguerre&#x;Gaussian beam

as a reference for the comparison with the synthesized beam given by Equations (1&#x;4). Here, and ¸ = arctan(y/x) are polar coordinates, a is the beam radius, m is the radial mode index, and l is the value of the topological charge.

The amplitude and phase distributions for the field of this beam with the topological charge l = 3 are shown in Figure 1. It can be seen that the number of subapertures making up the synthesized vortex beam analogous to the fundamental Laguerre&#x;Gaussian beam (5) determines the radius of an individual subaperture. In addition, it is obvious that the number of subapertures also determines the maximal possible value of the topological charge of the vortex beam, which can be achieved in this way.

Figure 1. (A) Amplitude and (B) phase distributions of the fields of Laguerre &#x;Gaussian (Equation 5) and synthesized vortex (Equations 1&#x;4) beams with a topological charge l = 3 [17].

Figure 2 shows the amplitude and phase distribution of the field of the synthesized vortex beam with the topological charge l = 3 upon propagation in the free space to the distance equal to a half diffraction length (k₀ = 2À/» is the wave number, » is wavelength) corresponding to the traditional Laguerre&#x;Gaussian beam of the radius a. It can be seen that during the propagation, the amplitude and phase of vortex beams synthesized from 18 or 36 subapertures behave similarly to those of the Laguerre&#x;Gaussian beam &#x; the ring amplitude distribution and the screw phase distribution with the progression of 2lÀ while circling around the beam center are observed. However, as the beam synthesized of six apertures propagates, no vortex components appears, and only the interference pattern from the interaction between subaperture radiation is seen. This is explained by the fact that no less than 3l radiation sources are necessary to generate a field with the nonzero orbital angular momentum l. In this case, for formation of a vortex beam with the topological charge l = 3, the minimal number of subapertures arranged in a circle should be nine.

Figure 2. (A) Amplitude and (B) phase distributions of the fields of the Laguerre &#x;Gaussian and synthesized vortex beams shown in Figure 1 during the beam propagation in a free space to a distance z = k₀a² [17].

In the study of the influence of the synthesized beam structure (number of sub-beam rings) on the formation of an optical vortex, it was found that the radiation from subapertures lying in the inner ring of the fiber cluster weakly affects the formation of the orbital angular momentum of the synthesized vortex beam. In this case, the maximal topological charge l_max is determined by the number of subapertures in the outer ring () and is independent of the number of inner rings (l_max=/3). It should be also noted that for the topological charge close to the maximal value l_max, the smoothed amplitude and phase distributions of the field form much more quickly (at shorter distances) if there is only one ring (no inner rings). The main property of field (1&#x;4) manifests itself at the propagation in a free space. It consists in the fact that the intensity and phase distributions of the synthesized and ordinary Laguerre&#x;Gaussian beams are widely different in the near zone (at short distances z   z_d) and close in the far zone (at distances comparable with the Rayleigh diffraction length z_d = k₀a² and longer) [16, 17]. Let us consider the free propagation of the field with initial distribution (1&#x;4) by representing it as an expansion in azimuthal modes

and calculate the energy fractions corresponding to the modes of different order n

where,

We have succeeded in determining [17] that as the synthesized beam propagates at the initial part of the distance in the near-axial zone (whose size is comparable with the initial size of the beam), the energy transfers between modes corresponding to different values of n. In the course of the propagation, the energy of the mode n, whose order is equal to the given topological charge l, increases and saturates to the level close to % of the total energy. Thus, as the synthesized vortex beam propagates to the distance exceeding several tenth fractions of the Rayleigh diffraction length, the beam properties in the near-axial zone approach the properties of the Laguerre&#x;Gaussian beam (5).

We have studied the orbital angular momentum of the synthesized beam (1), as well as the structure of its wavefront [18, 19]. For this purpose, we have calculated the transverse component of the Pointing vector, which can be represented in the following form in the paraxial approximation [1]

where I(r, z) and Æ(r, z) are the intensity and phase of the field E(x, y, z). With allowance for Equation (10), we can write the equation for the normalized (specific) density of the orbital angular momentum [1]

where is the power of radiation, n_z is the unit vector in the direction of the radiation propagation axis.

The orbital angular momentum of the laser beam at the distance z upon normalization to the beam power [1] is calculated by as follows:

The calculation of the orbital angular momentum by Equation (12) for the initial field specified by Equations (1&#x;4) gives zero values for any values of the parameters l and N_a both in the initial plane and at any distance from it.

It should be reminded that as the number of subapertures in the fiber cluster increases, the transverse phase distribution in the central part of the beam in the course of its propagation becomes closer to the screw structure characteristic of the Laguerre&#x;Gaussian vortex beam. It was assumed in the calculations that all the radiating subapertures form nested hexagonal rings. This structure in the initial plane is shown in Figure 3.

Figure 3. Geometry of subaperture arrangement (A), intensity (B), and phase (C) distribution for the synthesized beam of 36 subapertures in the plane; l = 1 [18].

In this configuration of 36 subapertures, according to Equations (1&#x;4), the subaperture phase Æ_sub takes 24 different values. Every three phase values at the subapertures lying at six rays outgoing from the center are equal.

We assume that the propagation of laser beams in the atmosphere is described quite accurately by the parabolic equation for the complex amplitude of the field:

where n₁(x, y, z) are variations of the refractive index (in this section n₁(x, y, z) = 0). Our computational schemes employ the method for numerical solution of Equation (13) based on splitting by physical factors [23] implemented in the parallel code [24].

To study the dependence of the received OAM value on the radius of the receiving aperture, the integration over the unlimited plane in Equation (12) was replaced with the integration over a circle with the radius a_t and the center at the beam axis

where

Figure 4 shows the results of calculation of the intensity, phase, and OAM density distributions for the beam formed of 36 sub-beams at the distance , where , is the radius of the synthesized beam as determined in Figure 3A.

Figure 4. Intensity (A), phase (B), and OAM density (Equation 11) (C) distributions of the synthesized vortex beam at the end of the distance z/z_d0 = , simulation domain of L/a₀ = 80, × computational grid. Squares show the zones containing peripheral screw phase dislocations at the vertices of the regular hexagon [18].

The OAM density (Equation 11) distribution is shown in Figure 4C. The light shade in this figure corresponds to the positive density, while the dark one is for the negative density values. It can be seen that the OAM density is nonzero in the central part, where the density is positive. In addition, there are six zones located at the hexagon vertices at the beam periphery, within which the OAM density is nonzero. The OAM density at these zones takes both positive and negative values. Thus, in the interference field of the synthesized beam, the zone of nonzero OAM density is multiply connected and alternating-sign in contrast to the simply connected OAM density of the Laguerre&#x;Gaussian beam [25].

The received power and L_z(a_t) as functions of the radius a_t are shown in Figure 5. One can see that for the receiving aperture with the radius a_t/a₀ &#x; 3 the OAM value is close to unity, while for the receiving aperture with the radius a_t/a₀  6 the OAM value is close to zero.

Figure 5. Energy and OAM as functions of the radius of the receiving aperture. The parameters of the calculations are the same as in Figure 4 [18].

The behavior of the OAM curve in Figure 5 demonstrates how the presence of positive and negative values of the OAM density described by Equation (11) affects the OAM value within the limited aperture (14). It can be seen that L_z(a_t) tends to zero at a_t &#x; . Thus, the OAM value of the synthesized beam fully intercepted by the receiving aperture is equal to zero at any point of the path. This result reflects the principle of OAM conservation.

The study of the phase gradient circulation demonstrates [18, 19] that this characteristic also reflects the features of hexagonal arrangement of subapertures. In the near-axial part of the beam, there is a limited zone, within which the integral of the OAM density is equal to unity. The circulation of the phase gradient over the perimeter of this zone is equal to 2À.

Thus, in Dudorov et al. [15] and Aksenov et al. [16&#x;19] we have studied theoretically the influence of the number of radiating apertures N_a, and their arrangement on the phase and intensity distribution of the synthesized beam in the far wave field for different given values of OAM. It has been shown that OAM of the beam synthesized by the proposed method is equal to zero at the complete interception by the receiving aperture. However, the aperture limitation of the beam in the receiving area allows us to separate the central part of the synthesized beam and to assign the non-zero OAM to it. In this case, the central ring of the vortex beam carries about 50% of the power emitted by the synthesized aperture at N_a = 6, l = 1 and 70% of the power at N_a = 18, l = 1.

Vortex Beam Generation Based on Coherent Combining of Fields of the Fiber Cluster. Experimental Setup and Laboratory Studies

Vortex beams with l = 1 and l =2 were obtained experimentally for the first time by us with an array of six coherent Gaussian sub-beams arranged in a circle by setting a fixed phase shift between neighboring sub-beams [20]. In Aksenov et al. [21], the experimental setup was significantly modernized, and new experimental data were obtained. Further on, we follow the paper Aksenov et al. [21].

Consider the experiment and, in the first turn, the experimental setup in detail (Figure 6).

Figure 6. Diagram of the experimental setup: narrowband laser (1), optical fiber amplifier (2), fiber splitter (3), phase modulators (4), fiber collimators (5), long-focus lens (6), beam splitting plate (7), lenslet (8), beam profiler (9), Shack&#x;Hartmann sensor (10), computer (11), pinhole (12), broadband photodetector (13), optimizing multichannel SPGD processor (14), control computer (15), oscilloscope (16), spiral phase plate (17). Arrangement of subapertures for coherent beam combining: 1) in the experiment on synthesis of a beam with maximal intensity at the axis, 2) in the experiment on synthesis of a vortex beam [21].

In our experiment, the linearly polarized radiation of narrow-band (&#x;f = 3 MHz) semiconductor laser 1 with the central wavelength » = 1, nm and the output power up to mW was amplified by fiber amplifier 2 with the tunable amplification factor from 0 to 33 dBm. The amplified radiation was split into eight channels (power of every channel up to 15 mW) with fiber splitters 3. Six or seven working channels were used in the experiment. Every channel was connected by an optical fiber with one of seven integral LiNbO₃ phase modulators 4 with the modulation frequency up to MHz with the controllable phase shift in the range from 0 to  6À in response to the applied voltage. Seven fiber collimators 5 (F = mm) arranged hexagonally and forming a cluster of sub-beams [(1) in Figure 6] were installed at the output.

Each collimator formed a sub-beam with the Gaussian intensity distribution and the diameter d_sub = 22 mm. The optical axes of all the sub-beams were aligned in parallel to each other thus forming an aperture with the diameter D = 90 mm. The set of sub-beams was focused by lens 6 with the focal length F = mm. Ophir-Spiricon SPU beam profiler 9 with 1X10 micro-objective 8 was set in the focal plane of lens 6. The recorded intensity distribution of the synthesized beam was displayed at the monitor of computer To provide for the phase control of the synthesized beam, beam-splitting plate 7 was set in the radiation propagation channel. Plate 7 directed a part of the beam to Thorlabs PDA10CF-EC photodetector 13 equipped with pinhole 12. Photodetector 13 recorded the intensity of the interference maximum formed as a result of sub-beam superposition in the plane of pinhole 12. The signal from photodetector 13 was processed by multichannel processor 14 operating according to the stochastic parallel gradient descent (SPGD) algorithm [26]. The multichannel SPGD processor [27] with the clock frequency up to kHz ( kHz in the experiment) generated control signals for phase modulators 4 to maintain the maximal level of the signal at photodetector 13. Control parameters were set with computer 15. According to the SPGD algorithm, the maximum of the signal recorded by photodetector 13 corresponded to the in-phase state of all the sub-beams. Thus, the feedback loop providing for the phase synchronization of radiation of the constructed fiber array was formed.

To synthesize the vortex beam, it is first necessary to perform the initial synchronization of sub-beam phases, since the system is unstable due to thermal and acoustic fluctuations in the system elements including optical fiber [28].

The phase synchronization of the sub-beams forming the cluster was performed with the SPGD algorithm. As the SPGD was turned on, the in-phase state of the channels was achieved for the time shorter than 10 ms. Then the SPGD controller was turned off, and the signal drop was recorded at photodetector 13. The typical oscillogram of this process is shown in Figure 7. The time for one reading is s. The time for signal drop down to the 1/ level upon averaging over 10 realizations is s. The signal drop time is caused by random phase progressions in fiber channels under experimental conditions and characterizes the particular system. During this time, the phase state of the system can be considered as frozen and sufficient for the phase modulation. Once the state of phase synchronization was achieved for all the six beams, the SPGD controller applied the voltages corresponding to the necessary phase shift to the phase modulators [20, 21]. The intensity distribution recorded by beam profiler 9 is shown in Figure 8.

Figure 7. Oscillogram of the signal at photodetector 13 with the SPGD controller turned off (top oscillogram). Dynamics of voltage at phase modulators (bottom oscillogram). Once SPGD is turned off, all voltages at the phase modulators are zero. The blue vertical line in the top oscillogram is for the time of SPGD turning off, the red horizontal line is for the signal decrease 2 times [20].

Figure 8. Experimental intensity distribution of the vortex beam l = 1 (left); result of numerical simulation [21] (right).