1. Introduction

An optical beam with orbital angular momentum (OAM), also called an optical vortex or phase singularity, is a laser beam carrying a helical phase front. That is, the phase structure rotates as the beam propagates. The order, $\ell$, of the OAM is defined by the number of 2$\pi$ phase shifts in the beam’s azimuthal direction. OAM beams have attracted significant interest for their application in fields ranging from optical communications to quantum optics [1–6].

OAM beams with different orders are orthogonal to each other, making them useful for robust high bandwidth optical communications because multiple beams can be multiplexed with low crosstalk [7]. By multiplexing four OAM modes, Gibson et al. achieved terabit per second data transmission over a free-space link [4]. OAM beams also have applications in micromanipulation and optical tweezers [8,9], optical microscopy [10], and quantum information [11].

Vortex electro-magnetic waves have been used in radio-frequency applications, such as radar, for many years [12], but effective and efficient generation of optical OAM beams remains a challenge. Technologies including holograms [4], annular gratings [13], twisted fibers [14], and chiral optical antennas [15] have been demonstrated previously. Recently, groups in China [16] and Russia [17] have proposed using optical phased arrays (OPAs) to generate OAM beams.

OPAs comprise multiple spatially separate emitters with stable relative phases. The beams from the individual emitters interfere in the far-field to form a coherent and contiguous optical wavefront that is the Fourier transform the emitter plane. By controlling the phase of each emitter in the array it is possible to control the shape of the wavefront. By constructing an OPA with an appropriate arrangement of emitters and applying an increasing phase offset in the azimuthal (angle around beam propagation axis) direction it is possible to generate OAM beams in the far field.

OPAs have a number of advantages for OAM beam generation. Most previous methods radiate fixed topological charges (fixed OAM order) while an OPA can change its OAM order, and do so rapidly. OPAs enable fast conversion of one beam type to another, and will be able to rapidly encode information in data transmission applications. With OPAs, fast phase shifts of MHz or GHz rates are possible at each subaperture [18–20].

In this Letter, we demonstrate, for the first time to our knowledge, generation and radiation of OAM beams from an OPA. The phase of each emitter in the OPA is actively measured and stabilized using digital controllers, enabling individual control of each emitter and change of the OAM order. The architecture can be readily scaled to hundreds of emitters with each subaperture able to phase shift at GHz rates.

2. Setup and control architecture

We have previously demonstrated an OPA with 7 emitters in a hexagonal arrangement [19–21]. This OPA uses digital controllers implemented on board a field-programmable gate array (FPGA) to sense and control the relative phase of each emitter, enabling precise control of the interfered wavefront. By switching off the central emitter and controlling the phases of the outer ring of emitters, we are able to generate beams with OAM.

The OPA presented here (Fig. 1) uses electro-optic modulators (EOMs) to individually control the phase of each element in the array. The standard telecommunications wavelength of 1550 nm is used because of the ready availability of modulators and detectors at this frequency. Light from a 1550 nm fiber laser is split into seven channels. Six of these channels are guided through EOMs which are then connected to an optical head assembly which launches the beams into free space. The optical head comprises a hexagonal arrangement of cleaved polarization-maintaining fibers with a separation of 250 $\mu$m between neighbouring fibers. Detailed information about the optical head can be found in [22] and [23]. A microlens array placed 0.5 mm from the optical head is used to increase the fill-factor of the array. The seventh channel from the laser is frequency shifted using an acousto-optic modulator (AOM) to serve as a local oscillator reference. The local oscillator is launched into free space using a collimator and is interfered with the six channels from the optical head at a free space beam splitter. One output of the beam splitter is observed by an infra-red beam profiler, while the heterodyne beat signals between the array channels and local oscillator are detected with a photodetector at the other output.

 

Fig. 1. Schematic of the OPA and digital control system. FPGA, field-programmable gate array; ADC, analogue-to-digital converter; DAC, digital-to-analogue converter; PRBS, pseudo-random bit sequence; EOM, electro-optic phase modulator; AOM, acousto-optics modulator; $f_{h}$, heterodyne frequency.

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All phase sensing and control is done using digital signal processing aboard an FPGA, which has the effect of reducing the complexity of the optical setup by transferring it to the digital domain. To distinguish and measure the phase of each individual emitter we employ a technique called digitally enhanced heterodyne interferometry (DEHI) [24]. DEHI uses pseudo-random bit sequences (PRBS) applied to each emitter channel to discriminate multiple inteferometric signals at a single photodetector without sacrificing the sensitivity of conventional interferometry [25]. The FPGA used for this experiment supports both digital and analogue outputs. Six digital outputs are used to generate 1023-bit M-sequences, while the phase feedback and control signals are output from six analogue outputs. The radio-frequency PRBS and DC phase control signals are combined using bias tees and fed to the EOMs.

The photodetector at the output of the array is connected to an analogue-to-digital converter (ADC) interfaced with the FPGA meaning all six heterodyne beat signals with encoded PRBS are available to the FPGA for processing. In the FPGA, the heterodyne beat notes corresponding to the individual channels are recovered by separately demodulating the PRBS codes by correlating them with delay-matched versions corresponding to the desired channel. The six demodulated heterodyne signals are sent to separate digitally implemented phase-locked loops (PLLs) [26] which measure the phase of each emitter. One emitter is designated as the reference and the phase of each of the other channels is calculated relative to this reference. The result provides five error signals that are fed to separate proportional-integral (PI) controllers that produce the feedback signals for each emitter channel which are output to the bias-tees, and subsequent EOMs, via digital-to-analogue converters (DACs). The feedback bandwidth of the control system is 800 Hz and is dominated by the time required to average over the full 1023 bits of the PRBS. The feedback bandwidth can be increased by averaging over a shorter or partial PRBS at the expense of degraded signal-to-noise (SNR) ratio in the PLL. The optimum trade-off between bandwidth and SNR needs to be tailored to the intended application of the OPA and also depends on the PRBS modulation frequency and ADC sampling rate [25].

3. Simulations

For an array of N emitters spaced equidistantly around the perimeter of a circle emitting radiation at the same frequency, the electric field, $E$, for a point in the far field can be described using phased array antenna theory where the field of the synthesized beam is represented as the sum of fields of the N subapertures of the array:

OAM of order $\ell$ is generated by applying a phase shift of $2\pi \ell /N$ between successive emitters around the circle ($\phi _n = 2\pi \ell (n-1)/N$). The angular momentum of each photon is equal to $\ell \hbar$. The number of emitters in the outer ring of the OPA limits the available $\ell$ numbers to the range $\ell \leq N-1$. With six emitters in the outer ring it is possible to achieve OAM up to order 5 since an OAM of order 6 will mean a 2$\pi$ phase difference between neighboring emitters, equivalent to zero order topological charge. However, it should be noted that, with this 6-emitter setup, an OAM of order 5 is equivalent to an OAM of order −1. Another way to consider this is that a topological charge of 5 in the clockwise direction around the beam axis is the same as a topological charge of −1 in the anti-clockwise direction.

The far-field interference of the emitters was simulated in MATLAB by calculating and summing the complex Gaussian amplitudes at a distance sufficiently far from the emitter array to ensure good overlap of the individual beams. Beam profiles were simulated for OAM of order $\ell$ = 0, 1, 2, 3, 4 and 5. The simulated beam profiles are shown in Fig. 2(a).

 

Fig. 2. Simulations (a) and measurements (b) of the beam profiles for OAM beams with orders $\ell$ = 0, 1, 2, 3, 4 and 5.

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4. Results and discussion

The phase setpoints of the OPA emitters were set to the necessary phase offsets required to generate OAM beams of order $\ell$ = 0, 1, 2, 3, 4 and 5. The resulting beams were viewed on the beam profiler and the results are shown in Fig. 2(b).

The beam profiles show that all OAM beams with $\ell$>0 exhibit the classic central null indicating a phase singularity. It can be seen that the pairs $\ell$ = 1 and 5 and $\ell$ = 2 and 4 have very similar intensity patterns, indicating that they are of equal magnitude in order in opposite directions of rotation with a symmetry about $\ell$ = 3. The simulated beam profiles show distinct intensity patterns outside the central lobe that allow these clockwise/anti-clockwise pairs to be differentiated visually, however, the results from the beam profiler are not sufficiently clear to display these differences.

The controller achieved emitter phase actuation speeds up to 12 kHz, limited by the bias tees used to combine the phase actuation signal with the radio-frequency PRBS codes onto the EOMs. However, we have previously shown with 2 and 3 element OPAs that this control architecture is able to achieve phase change speeds in excess of 1 MHz using suitable high speed DACs [19,20].

Due to the low fill-factor and number of emitters, only around 9-10% of the optical energy emitted by the OPA was coupled into the desired OAM mode. Increasing the fill-factor and number of emitters will increase the efficiency of the OAM generation. As shown in [17], an OPA based on the one presented here with 36 emitters can achieve 90-95% of the emitted mode energy coupled into the desired topological charge in the far field.

The fidelity of the OAM beam is limited by the amplitude and phase noise of the emitters. The amplitudes of the individual emitters were initially matched to within 1% by using polarization controllers prior to the EOMs to rotate the polarization angle of the light so that a partial attenuation was achieved through the highly polarizing EOMs, but the amplitudes were not continuously measured or controlled. Due to temperature changes in the laboratory and relaxation in the polarization controllers, after 24 hours the amplitudes of the emitters could be found to differ by up to 10%. This amplitude mismatch does not effect the beam pointing (the directional propagation angle of the main lobe) but does effect the interference fringe contrast and depth of the central null. The fringe visibility was measured to be approximately 95% with the measurement precision being limited by the noise of the beam profiler. This indicates very good coherence between emitters in the far field, and that the OAM beam is robust to significant differences in emitter power. Amplitude errors can be reduced in future by using fixed optical attenuators.

The phase error between emitters impacts the coherence and contrast of the OAM beam as well as the stability of the beam pointing. To measure the phase error between emitters the beam profiler was replaced with an out-of-loop photodetector connected to a second ADC. The signal from this ADC was fed to separately implemented digital PLLs on the FPGA which measured the relative phase of the emitters. Figure 3 shows a typical phase time series over 60 seconds, (a), and amplitude spectral denisty, (b), of the relative phase between the reference emitter and another OPA channel. These data were used to calculate the root mean square (RMS) relative phase error. The typical RMS phase error was measured to be $\lambda$/833 cycles over a bandwidth of 6 kHz, where $\lambda$ is the wavelength of the laser light. This corresponds to a combination efficiency of over 99.9% [27]. For this OPA with $\lambda$ = 1550 nm and emitters separated by 250 nm, this stability corresponds to an RMS angular pointing precision of 7.4 $\mu$rad. The total unambiguous steering range of the OPA depends on the separation between the outermost emitters and the emitted wavelength, and was 3.1 mrad for this setup.

 

Fig. 3. Phase time series (a) and amplitude spectral density (b) measurements of the typical relative phase between emitters. Blue, emitters phase locked; red, emitters free-running.

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The coherence and contrast of the interfered beam are dependent on the depth of the PRBS modulation as well as the relative emitter phase error. The PRBS codes are necessary to distinguish and measure the phase of each emitter, but reduce the coherence of the beam by spreading the carrier signal into broadband noise. The minimum modulation depth must be used to maintain maximum coherence of the OAM beam, but provide sufficient SNR for precise phase measurement. A modulation depth of $\pi$/8 was used for these tests which provided robust phase locking while preserving 97% of the fringe interference efficiency. Lower modulation depths can be used at the expense of requiring longer averaging in the PLL filter to maintain sufficient SNR.

5. Conclusion

We have demonstrated, for the first time to our knowledge, the generation and control of OAM beams with variable topological order using a digitally controlled OPA. The digital controller provides excellent control and phase stabilization of the generated beam and is scalable to hundreds of emitters [25] and GHz frequency phase modulation [19,20].

OAM beams generated using this method are compatible with a large variety of OAM receivers and have potential applications in optical communications, micromanipulation, microscopy, and quantum information. The interference side lobes can be reduced by selecting a more appropriate microlens array to improve the fill-factor of the array. Work is currently underway to build an OPA of this design with more emitters, and to reduce the size of the apparatus to produce a chip-scale integrated OPA. Expanding the hexagonal layout of the optical head used here will enable emitted energy to more efficiently couple into the desired OAM mode, and the beam to more closely approximate a true Laguerre-Gaussian. Alternatively, the same number of emitters can be laid out in a single large ring, increasing the number of available OAM modes at the expense of fill-factor and efficiency. Although DEHI was used here to phase-up the array, other methods such as LOCSET and stochastic-parallel gradient descent algorithms are also suitable depending on the desired application. A chip-scale OPA may be able to operate without closed-loop control due to the inherent mechanical stability of the small form-factor.