1. Introduction

Optical beams carrying orbital angular momentum (OAM), also called optical vortices, containing helical phase fronts and azimuthal component of the wave vector, have attracted much interest for the significant properties [1–3]. As the OAM lights carry angular momentum of lħ per photon, it has been predicted to be an efficient route to release the emergency requirement in optical communication systems by using OAM lights [4–6].Terabit data transmission has been realized by multiplexing four OAM lights with different topological charges for free-space communication [7]. Besides, due to the disparate energy distribution of circular symmetric doughnut-shape, OAM lights can be developed for many applications in optical microscopy [8], micromanipulation [9], optical trapping [10], and quantum information [11].

Though OAM lights play great roles in the above domains, stable generation and propagation of OAM beam is a long-standing challenge. Much effort has been paid and various designs have been introduced to generate OAM lights using silicon wires [12, 13], annular grating [14]. Helically twisted fibers have also been proposed to excite optical vortex [15]. Furthermore, various optical antennas with chiral assignments have been proposed to generate optical vortex [16, 17]. However, most of the proposed applications radiate light with fixed topological charges.

We here demonstrate the realization of a phased optical antenna array to manipulate the topological charge of the radiated OAM light. The interference of the OAM radiated from each element of the optical antenna array is theoretically calculated and observed. Different radiated field distributions can be realized by varying the input OAM in each antenna element. Properly manipulating the input phase in each element, OAM light with the index from −2 to 4 can be obtained from the superposition of low-index OAM lights radiated from the antenna elements.

2. Design principle

The main concept of this optical antenna array is schematically shown in Fig. 1(a). Eight cylinder waveguides are homogeneously arranged on a circle, and each waveguide can solely work as an optical antenna. By manipulating the radiation field of each optical antenna, the antenna array can actively radiate OAM lights with different topological charges.

 

Fig. 1 Geometry of the optical antenna for OAM modulation. (a) Schematic of the optical antenna array for OAM generation. (b) Side view and (c) front view of the effect geometry of each optical antenna.

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Figure 1(b) depicts one optical antenna in the array. It is composited of a silver tube and an inside core of air. The radius of the tube is designed as R = 0.7 μm, and the thickness of the tube is t = 0.2 μm. The whole height of the optical antenna is h = 2 μm. The working frequency of this optical antenna is designed in the infrared range with center wavelength of 1550 nm. As in the terahertz and optical frequency range, the silver cannot be taken as perfect electric conductor (PEC) any more, and its dispersion must be considered. Therefore, silver with dispersion parameters extracted from the Drude model is used in the simulation. As shown in Fig. 1(b), the excitation light is incident onto the bottom of the antenna, and reradiated from the top end of the antenna.

As each waveguide is circularly symmetrical, lights with circular symmetrical electric field distributions can be coupled into the antennas and propagates inside the antenna [18]. Thus circularly, radially or azimuthally polarized lights can stably propagate in this tube-shaped optical antenna. Consequently, the circularly polarized light and OAM lights are adopted as the feeding light for this antenna array.

2.1 Light propagating in single antenna

The electric field carrying OAM has a helical phase front of exp(ilφ), where the angle φ is the azimuthal angle and the integer l stands for the topology charge of the OAM. The angular momentum contained in each photon equals to lћ. With a certain radius of the tube, the topology charge l of the OAM that can propagate in the tube is limited. The relationship between the radius of the tube and the amount l is written by the formula$l=2\pi (R-t/2)/{\lambda }_{eff}$, where λ_eff is the effect wavelength of the light. It can be calculated that the maximum integer l is 2 that can propagate in our antenna with the designed radius and thickness at the wavelength of 1550nm. Consequently, we choose the circularly polarized light and OAM lights with l = 1 as the illuminating source for each antenna element to manipulate OAM light with tunable topological charges. And the circularly polarized light is considered as the special case of OAM light with topological charge l = 0.

The software we use to simulate the phased array is the commercial electromagnetic simulation software of CST 2014. In the simulation process, the boundary conditions are set to be open (add space), and the minimum and maximum mesh sizes are 0.014 μm and 0.1 μm, respectively.

A field monitor is set in the transverse plane with 1 μm away from the output port of the waveguide to check the radiated electric field. When the OAM lights propagate through each optical antenna element, the radiated electric fields are simulated and depicted in Fig. 2. We mainly focus on three constituents of the radiated field, namely the instant electric field, the average power intensity and the phase distribution. When the light carrying OAM with charge l = 0 is fed into a single antenna, the output electric field elements at 1550 nm are presented in Figs. 2(a)-2(c). The instant x-component of the electric field presents a circular dot and a concentric ring with opposite amplitude to the inner dot, as shown in Fig. 2(a). In this case, the average power pattern as a solid dot is observed in Fig. 2(b). Furthermore, the output phase distribution of the incident circularly polarized light is analogous to the instant electric field, as concentric rings are observed in Fig. 2(c). However, when the incident light contains OAM with l = 1, the outgoing electric field presents disparate distributions at the same wavelength. First, a pair of spiral arms rotating clockwise can be observed in Fig. 2(d), indicating that the E_x component of the electric field has a helical phase front, which is confirmed by the phase distribution in Fig. 2(f). The power of the radiation field for OAM light with topological charge l = 1 presents a doughnut shape in the transverse plane, which is the direct symbol of light carrying OAM. The above electric field distributions validate that the OAM lights with l = 0 and 1 can stably transmit through each element of the antenna array.

 

Fig. 2 Electric field of the radiated circularly polarized light from the single waveguide at 1550nm. (a)-(c) are respectively the instant electric field of Ex, the average energy intensity, and the phase distribution of the transmitted circularly polarized light in single antenna element. (d), (e) and (f) depict the corresponding elements of the radiated OAM light with topological charge l = 1.

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The transmission of the waveguide is also analyzed for different OAM modes, and the simulated results are shown in Fig. 3. We can see that the transmission of l = 0 and l = 1 modes are relatively high in the range from 1350 nm to 1600 nm, and the value of the transmission are 0.985 and 0.935 at 1550 nm, respectively. It can be also observed that the transmission of l = 2 is close to zero in the same range, indicating that this OAM mode is forbidden in the waveguide with radius of 0.7 μm in this frequency band.

 

Fig. 3 Transmission spectra of each waveguide when propagating OAM lights with different topological charges.

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2.2 Principle of OAM index manipulating

After analyzing the transmission property of the antenna element, the radiation character of the antenna array composed by the homogeneously circularly arranged antennas is then calculated. It is clear that the interfered field is related to the amount and the phase of the propagating OAM light in each optical antenna. Particular distributions can be predicted by manipulating the phase difference between the neighboring optical antenna elements.

When optical antennas with quantity of m equally displaced in a circular ring are simultaneously excited by the light carrying OAM of charge l, the cascaded electric field can be described as follows using the phased array antenna theory:

In formula (1), E₀ is the amplitude and the integer l stands for the topological charge of the OAM light radiated by each antenna. θ is the phase shift between the neighboring optical antenna elements in clockwise direction. It can be clearly seen that when θ is endued with a certain value (integer times of 2π/m), the radiated electric field around the normal axis of the antenna array would be the OAM light with tunable index of:

Furthermore, it can be predicted that the radiated field contains two components of angular momentum. One is the interfered field from the antenna elements with the in-phase OAM, and the field is identical to the incident field in each antenna. The second component is the OAM caused by the cascade field of different phase determined by θ. When θ is minus and integer times of 2π/m, the superposed OAM has the same rotation direction with the incident OAM, thus increases the amount of the OAM of the radiated field. On the contrary, when the value of θ is larger than zero and smaller than π, the cascaded OAM has opposite rotation to the incident OAM in each antenna. And it would result in the decrease of the topological charge of the radiated OAM, and even vary the rotation of the radiated OAM if θ is large enough.

2.3 Optical antenna array with circularly polarized incidence

We then simulate the interference of an antenna array composed of eight identical tube-shaped optical antennas based on formula (1). The antenna array is schematically shown in Figs. 4(a) and 4(b). In the front view, the centers of the eight antennas are homogeneously placed in a circle with the radius of 2.7 μm. We then manipulate the radiation phase of each antenna to tune the topological charges of the OAM light radiated by the whole antenna array. In simulation, eight waveguides ports are utilized to feed the optical antennas with OAM lights carrying certain phase differences of θ.

 

Fig. 4 Simulated results of the phased array when propagating OAM lights with certain phase differences at 1550nm. (a)The schematic of the circle optical antenna array with eight antenna elements. (b) Front view of the antenna array, displaced uniformly in a circle with radius of R = 2.7 μm. (c)-(i) The phase distribution of the interfered electric field at the aperture of each antenna with phase difference of −135°, −90°, −45°, 0°, 45°, 90°, 135° between neighboring antenna elements in clockwise direction. (c₁)-(i₁) The instant x-component electric field in near field range with phase difference of −135°, −90°, −45°, 0°, 45°, 90°, 135° between neighboring antenna elements in clockwise direction. (c₂)-(i₂) The intensity of the radiation field in with the corresponding phase difference above each figure.

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Firstly, the antenna elements are separately incident with the circularly polarized light. We set the incident light of each optical antenna to have a phase difference of θ between the neighbor elements in the clockwise direction, and therefore the outgoing light also has the phase difference of θ as the optical antennas are identical. When the value of θ is 0 degree, the eight antenna elements have the same propagation phase. In the transverse plane with a distance of 4.5 um (3λ) to the top end of the antenna array, the x-component of the interfered field presents a series of concentric ring-shaped pattern, as shown in Fig. 4(f1), which is actually an azimuthally polarized field. And the power distribution in the same plane depicts a solid dot with a bright center, where the energy reaches its maximum as shown in Fig. 4(f2). The concentric ring-shaped phase distributions depicted in Fig. 4(f) still indicate that the azimuthally polarized light is realized.

Next, OAM lights with tunable topological charges can be obtained when the value of θ changes. The value of θ is set from −3π/4 to 3π/4 with a step of π/4 for the antenna array to validate the manipulation characterize of the OAM light. When θ equals to π/4, a helical phase distribution and a clockwise rotating field can be observed, as shown in Figs. 4(g) and 4(g1), respectively. The field indicates that an interfered light carrying OAM with topological charge of l = −1 is realized. The average power distribution in this case is a doughnut-shaped ring, as seen in Fig. 4(g2). Subsequently, when θ increases to π/2, a turbine-shaped electric field with two pairs of spiral arms is obtained, as depicted in Fig. 4(h1). It indicates that the topological charge of the OAM light decreases to be −2 when θ increases to π/2. Furthermore, light carrying OAM of charge −3 is also realized when the value of θ increases to 3π/4, and the interfered electric fields are illustrated in Figs. 4(i) - 4(i2). When the phase difference between the neighboring antennas decreases to be minus, for example, -π/4, -π/2, and −3π/4, OAM with the same absolute topological charges but opposite rotation direction can be realized as the positive value of θ. The phase distributions and electric fields in the case of θ equals to -π/4, -π/2, and −3π/4 are respectively shown in the columns in Figs. 4(c)-4(e). In these cases, similar electric field distributions can be observed to the above results. However, the rotation directions of the interfered OAM for negative θ are anticlockwise, which is opposite to the case of positive θ. Above results demonstrate the feasibility of manipulating the topological charges of the interfered OAM by tuning the phase difference between the radiated radial polarized lights. The topological charge of the radiated OAM light varies from −3 to 3 when the gradient-phased circularly polarized light is incident into the antenna array.

It is notable that the power distributions of the interfered fields are well shaped as annular rings when |θ| is less than 3π/4. However, when θ increases to be ± 3π/4, the power intensity patterns show discontinuousness with radial branches though the energy intensities in the center are zero. It is not difficult to understand that the discontinuousness of the power pattern comes from the discontinuous phase distribution of the antenna array, which is not obvious when |θ| is small.

2.4 Optical antenna array with OAM light incidence

Subsequently, we alter the incident lights to be OAM light with topological charge of l = 1. The phase difference θ between the neighboring antennas is still varied from -3π/4 to 3π/4 with a step of π/4. The radiated field distributions are depicted in Fig. 5. When phase difference θ equals to -3π/4, the interfered light has four pairs of clockwise rotating arms (see Fig. 5(a)), indicating that an OAM light with topological charge of 4 is obtained. When the value of θ gradually increases to 3π/4, the topological charge of the radiated light accordingly varies from 4 to −2. Expectably, the radiated azimuthally polarized light appears when θ is π/4.

 

Fig. 5 Interfered electric field distribution of the optical antenna array when fed with OAM lights of l = 1 at 1550nm. (a)-(g) The instant E_x distribution in the case of θ equals to -3π/4, -π/2, -π/4,, 0,π/4, π/2, and 3π/4, respectively. (a₁)-(g₁) The average power distribution for the interfered field with corresponding θ above each figure.

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The above results validate the manipulation capability of this antenna array for lights carrying OAM. It can be concluded that the manipulation range of the topological charges can be greatly enlarged by employing lights carrying OAM with larger topological charge as the feeding source. As to experiment, the phase difference between the neighboring antenna elements can be achieved using the theory proposed in [19, 20]. According to the SPP phase delay theory in metallic nano-slits, the phase shift can be tunable by varying the width of the radius of the antennas, which will be carried out in our following research.

3. Conclusion

We have theoretically demonstrated an optical phased antenna array to generate OAM beams with tunable topological charges by using lights carrying OAM with l = 0 or 1 as the feeding source. When the phase difference between the neighboring antennas varies with a step of π/4, the superposed light beams present tunable OAM states. This approach provides a new route to realize high index OAM generation, which would have potential application in high integrated on-chip optical communication systems and optical detecting domain.