Optical Angular Momentum Beam Generation Using Coherent Beam Combination

1. Introduction

Coherent beam combination (CBC) technology, including filled aperture CBC (FA-CBC) and tiled aperture CBC (TA-CBC) [1,2,3,4,5,6,7,8,9,10,11], has been developed in the last two decades, mainly for applications in laser weapons [4,12] and high energy physics [6,10]. FA-CBC offers high efficiency and robustness but it is limited by the number of emitters (not higher than 12 beams was demonstrated for high average power system [13]). In contrast, TA-CBC offers direct scalability beyond hundreds of combined beams, but has low practical efficiency [5,10,11], and difficulty scaling to high average powers. Both technologies require effective adaptive optics subsystems for outdoor applications, such as laser weapons and free-space optical communications (FSOC), over long propagation distances in horizontal directions [8,9,14,15] and/or in Earth-satellite FSOC links [16,17,18].

Since the 1990s, optical angular momentum (OAM) beams have been examined theoretically and experimentally, attracting [19,20,21,22,23] growing interest in material processing, optical tweezing and FSOC [16,18,23,24,25,26,27]. The benefits of OAM technology include propagation invariance as ‘diffraction-free’ solutions of paraxial wave equation, enhanced resilience to atmospheric turbulence, and orthogonality, enabling potential increase in FSOC channel bandwidth thanks to spatial multiplexling. However, its scalability in power and practical realization in laboratory and outdoor applications are problematic for a single aperture output.

The purpose of this study was to merge both technologies, that is, to search for the possibilities of CBC technology for generation of OAM beams (OAM-CBC). Preliminary results were presented in [28], where a segmented vortex wavefront was constructed in the near field owing to a special helix-like stepped phase shift in the hexagonal lattice. Here, we develop this approach for special type of tilted and phased CBC (T&P-CBC) concept, enabling the effective construction of segmented vortex wavefronts in the near field. Section 2 describes the semi-analytical model of CBC propagation and the OAM-CBC concept. The main results, analysis, and discussion are presented in Section 3, and conclusions are drawn in .

2. Tilted and Phased CBC for Optical Angular Momentum generation

Here, we give only a brief description of the analysis method. The full mathematical model is described in detail in [30,31]. To start, we provide the main characteristics of CBC hexagonal architecture (see Figure 1). The center and chief ray directions of each (l, n)-th emitter/beam of the CBC are defined by the geometry of hexagonal lattice; N_cr is the number of crowns; N_lat = 3N_cr (N_cr +1)+1 is the number of emitters; a is the half period of the lattice; r_a is the sub aperture radius; f.f. = r_a/a is the filling factor, f is the focal length. In the classical CBC configuration, each chief ray is directed to the common focal point F, with a spherical segmented wavefront of CBC lattice of radius f. Diffraction profiles of CBC in the far field are characterized by the Airy radius r_Airy= 0.61λ/NA_lat and Fresnel range Z_Fresnel=λ/NA_lat², where NA_lat= (N_cr +1/2)2a/f_. is the numerical aperture of lattice and λ is the wavelength.

We apply a truncated Gaussian beam as a ‘prototype’optical field. To simplify calculations, we used the once determined and calculated approximate solution of a truncated Gaussian beam, for given beam radius w₁ at the aperture and truncation level ε_tr = exp(-2(r_a/w₁)²) (see [32] for details), which is valid over a wide range of propagation distances in the vicinity of focal point F. Note that the ratio of Rayleigh range of an individual beam to the caustics length defined by Fresnel range is proportional to N_cr². Typically, for long-distance propagation, the Rayleigh range is comparable to f (‘collimation case’), whereas the Fresnel range is N_lat times shorter. We calculated the coherent sum of all input beams for specific distribution of phases and tilts for a chosen plane near focal point F.

Evidently, various T&P-CBC realizations exist in the (1+3)×N_lat dimensional space of control parameters. Exploration of this subject is far beyond the scope of this study. Here, we focus on the OAM-CBC case. We define a specific segmented vortex wavefront with a step-like helix phase and tilt vector [θ_x , θ_y ] in the near field as follows:

The OAM-CBC amplitude $B_{m_{\mathrm{OAM}}}$ with given $m_{\mathrm{OAM}}$ - optical angular momentum number can be described as follows:

$$B_{m_{\mathrm{OAM}}}={\displaystyle \sum _{l,n}{A}_{l,n}\exp \left[ik{\Phi }_{l,n}\right]}\mathrm{where}{\Phi }_{l,n}={\displaystyle \frac{{r_{l,n}}^2}{f}}+m_{\mathrm{OAM}}{\displaystyle \frac{{\phi }_{l,n}}{k}}$$

(1)

Here, l and n are the indices of the sub-aperture, A_l,n is the amplitude of (l,n) sub-aperture, r_l,n and ϕ_l,n are the radius to the lattice center and azimuth angle of each sub-aperture, f is the focal length, k is the wavenumber.

Next, we calculate vortex tilt vector [θ_x , θ_y] by gradient $\nabla {\Phi }_{l,n}$as follows:

$$\nabla {\Phi }_{l,n}=\left[{\theta }_{x,l,n},{\theta }_{y,l,n}\right]=\left[\begin{array}{cc}{\displaystyle \frac{r_{l,n}\cos {\phi }_{l,n}}{f}}+m_{\mathrm{OAM}}{\displaystyle \frac{\sin {\phi }_{l,n}}{k{r}_{l,n}}},& {\displaystyle \frac{r_{l,n}\sin {\phi }_{l,n}}{f}}-m_{\mathrm{OAM}}{\displaystyle \frac{\cos {\phi }_{l,n}}{k{r}_{l,n}}}\end{array}\right]$$

(2)

Note, that for $m_{\mathrm{OAM}}$≠ 0 the geometrical-optics caustics of OAM-CBC vortex segmented wavefront has a coiling non-linear helix shape with an inner hole (lower right part of Figure 1) corresponding to the evolving hexagonal helix of the CBC in the near field (upper left part in Figure 1). The coiling helix for m_OAM =0 transforms into a distinct focal point F for classical spherical CBC case. This phenomenon is a direct geometrical-optics interpretation of vanishing irradiance at r = 0 and annular amplitude distribution in caustics for vortex OAM beam.

3. Results

We tested the proposed OAM-CBC concept for a large hexagonal lattice consisted of N_cr= 6 outer crowns (N_lat=127 emitters), assuming very dense packing (f.f.=0.99). To neglect the peculiarities of diffraction, we assumed low truncation losses ε_tr=0.05. Thus, an “almost” Gaussian beam is emitted by an individual emitter with a very low power content at higher diffraction orders in the caustics region. The hexagonal architecture of the CBC reflects the six-folded symmetry of 1-st diffraction orders (see Figure 2).

The basic compliance between the geometrical-optics caustics (Figure 1) and wave-optics demonstrate a dominant annular profile in the zero-order diffraction area (Figure 2, Figure 3, and Figure 4). In Figure 2 the irradiance maps for $m_{\mathrm{OAM}}$ > 0 (Figure 2b-d) exhibit the expected properties of the conjunction of CBC and OAM beams. The zero-diffraction order has a nearly annular shape and 1-st diffraction orders resemble its replicas.

The typical dependencies of the far field profiles and power in bucket (PIB) curves for OAM-CBC are shown in Figure 3 and Figure 4.

With an increase in $m_{\mathrm{OAM}}$, the amplitude profiles worsen with flattening and dissipation of power density out of the low diffraction lobes (Figure 3 and Figure 4). Moreover, above $m_{\mathrm{OAM}}$ >5, the 2D maps resemble the aberrated and scattered partly coherent beams with the minimum at the axis. These features are particularly highlighted for $m_{\mathrm{OAM}}$=6 (Figure 2d), for which six-fold symmetries of the vortex and hexagonal lattice overlap.

To examine the orthogonality of a set of OAM-CBC amplitude profiles in the far field and the feasibility in FSOC, we have calculated correlation coefficients γ_m,k defined as follows:

$${\gamma }_{m,k}={\displaystyle \frac{\langle B_m{B_k}^{*}\rangle }{\sqrt{\langle B_m{B_m}^{*}\rangle \langle B_k{B_k}^{*}\rangle }}}$$

(3)

where $\langle B_m{B_k}^{*}\rangle$ denotes the discrete 2D summation of $B_m{B_k}^{*}$ over finite area in the far field; (15 x 15 r_Airy in our case).

The results of γ_m,k calculations for a trail of 9 configurations with $m_{\mathrm{OAM}}$ϵ(0,8) are collected in Table 1.

Nonperfect orthogonality is caused by the characteristics of OAM-CBC concept (discrete 2D array of hexagonal symmetry), numerical errors, approximate solution of truncated Gaussian beam, and discrete summation over limited area.

The values of γ_m,k were in range [0.0048, 0.1]. For a narrow range of $m_{\mathrm{OAM}}$ ϵ (0,5) average correlation coefficient γ_m,k = 0.7%, whereas for a $m_{\mathrm{OAM}}$ ϵ (0,8) it increases to 1.5%. We suppose that a correlation coefficient of 1%, corresponding to the cross-channel crosstalk, is acceptable for practical applications in FSOC. Thus, we can conclude that the T&P-CBC sufficiently satisfies the main conditions and requiremnets of OAM generation.

4. Conclusions

Numerical experiments demonstrated the feasibility of merging of OAM and CBC technologies. For a densely packed CBC hexagonal array, the vortex wavefront was emulated with a step-like evolving helix of phase and tilts vectors in the near field.

i/ The irradiance maps for $m_{\mathrm{OAM}}$ > 0 exhibit the expected properties of combination of CBC and OAM beams. The zero-diffraction order has a nearly annular shape and 1-st diffraction orders resemble its replicas. However, the differences between classical OAM beams and OAM-CBC are substantial owing to their hexagonal lattice properties.

ii/ The imperfect orthogonality of OAM-CBC set is attributed to the specfic characteristics of OAM-CBC concept (discrete 2D array of hexagonal symmetry), numerical errors, and discrete summation over a limited area. For the $m_{\mathrm{OAM}}$ range of (0,5) the average correlation coefficient, corresponding to cross-channel crosstalk, is 0.7%, which is acceptable for practical applications in FSOC.

We conclude that the proposed here T&P-CBC concept fulfills the primary conditions and requirements for OAM generation. In further research we intend to examine the rules of OAM-CBC algebra and the impact of lattice architecture parameters and imprefections.