We demonstrate all-optical division and multiplication of the state order $\u2113$ for data-carrying orbital angular momentum (OAM) beams. We use linear optical transformations between log-polar and Cartesian coordinates to: (i) divide the OAM state order to convert the OAM order from 2$\u2113$ to $\u2113$ ($\u2113$ = −5, −4, …, +4, +5), and (ii) multiply the OAM state order from $\u2113$ to 2$\u2113$. We analyze the OAM mode purity and the bit-error-rate performance of a classical two-mode OAM multiplexed link for the case of division and multiplication of the OAM state order. The experimental mode purity for halving and doubling OAM state order can reach around 87% and 40%, respectively. We further study the dependence of the OAM mode purity on the displacement of SLMs in simulation. The obtained results show that the transformation for doubling the OAM state order is more sensitive to the increase of the displacement than that for halving the OAM state order. The link bit error rates are below the forward error correction threshold of 3.8 × 10^{−3} for both channels.

Orbital angular momentum (OAM) has recently gained attention in communication systems in which multiple independent OAM beams are spatially multiplexed to increase system capacity.^{1,2} In 1992, an electromagnetic wave with a helical wavefront $ exp ( i \u2113 \theta ) , $ where $\u2113$is the OAM state order, *θ* the azimuthal angle, and *ℏ* the reduced Plank constant, was found to carry an OAM of $\u2113\u210f$ per photon.^{3–5} The beam’s wavefront twists along the propagation axis, producing a central intensity null (i.e., phase singularity) and an annular ring shape. OAM beams with different state orders are orthogonal to each other, which enables efficient OAM-based mode-division multiplexing in the optical^{1,2,6,7} and radio^{8–10} domains. We note that OAM-based quantum information processing^{11–14} has also been used to increase the usable alphabet for quantum systems.^{15}

For certain applications, it might be valuable to manipulate the OAM states of light, namely, translating one OAM state to another state. This function could be used for reconfigurable systems, such as switching and routing applications.^{16} Moreover, modifying the channel spacing for OAM systems could potentially reduce OAM channel crosstalk^{17} (by increasing the state spacing) or enhance mode and system efficiency (by reducing the state spacing).

Several elements have been proposed to add or subtract (i.e., shift) a fixed-order number onto the OAM states of light, including spiral phase plates,^{18,19} spatial light modulator (SLM),^{20} integrated microring resonators,^{21} metametrials,^{22,23} and metasurfaces.^{24,25} In addition to shifting the OAM state order $\u2113$, it might be valuable to perform the division and multiplication of the OAM state order; note that frequency dividers^{26} and multipliers^{27,28} are fairly useful for signal processing systems in which the frequency domain is being manipulated.

Previously, OAM state multiplication combined with frequency multiplication has been achieved using nonlinear harmonic generation.^{29,30} Another demonstration of OAM state multiplication was achieved using linear optical coordinate transformations.^{17} However, there have been few reports of OAM state division, nor has there been data transmission on these beams that are undergoing either division or multiplication.

In this paper, we demonstrate all-optical division and multiplication for data-carrying OAM beams, based on two-step linear optical coordinate transformations.^{31} We experimentally realize the bidirectional transformation between states $ | \u2113 \u27e9 $ and $ | 2 \u2113 \u27e9 $ with $\u2113$ varying from −5 to +5. The experimental mode purity for halving and doubling the OAM state order can reach around 87% and 40%, respectively. We analyze the dependence of the OAM mode purity on the displacement of SLMs in simulation. When the displacement increases from $ 0 \u2009 \mu m \u2009 to \u2009 40 \u2009 \mu m , $ the mode purity of OAM +4 decreases from 89% to 81% for halving OAM +8 and from 98% to 18% for doubling OAM +2. It shows the transformation for doubling the OAM state order is more sensitive to the increase of the displacement than that for halving the OAM state order. We also measure the bit-error-rate (BER) performance of a classical 100-Gbit/s two-mode OAM multiplexed link after division and multiplication of the OAM state order and achieve BERs <3.8 × 10^{−3} for both channels.

Division and multiplication of the OAM state order can be achieved through optical coordinate transformations in two steps using SLMs, as shown in Fig. 1(a). Phase mask 1 and 3 are used to achieve the coordinate transformations; phase mask 2 and 4 are used to correct the phase of the transformed beams. For division of the OAM state order by *n*: (i) unwrap the “donut shape” with a helical phase to a rectangular shape with a linear phase through an optical log-polar to Cartesian coordinate transformation^{17,32} followed by a phase correction pattern; (ii) select $ 1 / n $ of the rectangular shape and wrap it to a donut shape. This second step can be performed through a Cartesian to log-polar coordinate transformation followed by a phase correction pattern. We block a part of the incoming beam, resulting in an $ ( n \u2212 1 ) / n $ power loss in the second step. One possible choice with less power loss is dividing the rectangular shape into *n* parts, wrapping each part to a donut shape, and combining them to one beam. Multiplication of the OAM state order by *n* can be also performed through a two-step coordinate transformation: (i) unwrap the donut shape to *n* copies of a rectangular shape; (ii) wrap each copy of the rectangular shape to an arc shape with angles of $ 2 \pi / n $ and combine all arc shapes to a donut shape. Phase correction patterns are required in both steps. Figure 1(b) shows the intensity and phase profiles in each step for halving and doubling the OAM state order. The helical phase profiles are converted from $ exp ( i \theta ) $ to $ exp ( i 2 \theta ) , $ or inversely, which demonstrates the achievement of a bidirectional transformation between OAM states $ | + 1 \u27e9 $ and $ | + 2 \u27e9 $.

The optical log-polar to Cartesian coordinate transformation, based on two spatially designed phase masks, can map a log-polar coordinate $ ( r , \theta ) $ to a Cartesian coordinate $ ( x , y ) $ with $ x = a 1 \theta $ and $ y = \u2212 a 1 ln ( r / b 1 ) , $ where $ a 1 $ and $ b 1 $ are scaling constants.^{32,33} These phase masks can be described as the phase delay^{32}

where *f* is the lens’ focal length and *λ* is the wavelength of light. The last terms of $ r 2 / 2 f $ and $ ( x 2 + y 2 ) / 2 f $ are the lens functions. The first phase mask unwraps the donut shape to a rectangular shape, and the second one corrects the phase of the output beams after this transformation. We can make *n* copies of the rectangular shape during this transformation by changing the phase delay of the first phase mask as a sum of multiple $ \varphi 1 i ( r , \theta ) $^{34} with different parameters $ a 1 i $ and $ b 1 i $, described by

This phase mask can achieve multiple log-polar to Cartesian transformations simultaneously, resulting in *n* copies of rectangular shapes at different positions with $ x = a 1 i \theta $ and $ y = \u2212 a 1 i ln ( r / b 1 i ) $ on the Fourier plane $ ( x , y ) $. Using Eq. (1b), we can carefully design the second phase mask to achieve different phase corrections for the corresponding copies of the beam.

In the following transformation, we use the coordinate transformation in reverse to map the Cartesian coordinate $ ( x , y ) $ back to the log-polar coordinate $ ( r \u2032 , \phi ) $ with $ \phi = x / a 2 i $ and $ r \u2032 = b 2 i exp ( \u2212 y / a 2 i ) , $ where $ a 2 i $ and $ b 2 i $ are scaling constants of the inverse coordinate transformation. Therefore, we have

Let $ a 2 i = n a 1 i , $ we can get $ r \u2032 = b 2 i ( r / b 1 i ) 1 / n $ and $ \phi = \theta / n $. Assume the incoming beam has an OAM state of $ | \u2113 \u27e9 , $ namely a helical phase front of $ exp ( i \u2113 \theta ) $. After the two-step transformations, the output beam will have a phase front of $ exp ( i n \u2113 \theta n ) = exp ( i n \u2113 \phi ) , $ indicating that the OAM state has been transformed from $ | \u2113 \u27e9 $ to $ | n \u2113 \u27e9 $.

Similar to the OAM multiplication, we can use the same setup to transform the OAM state $ | n \u2113 \u27e9 $ to $ | \u2113 \u27e9 $. We do not copy the rectangular shape in the first step. In the second step, we select $ 1 / n $ of the rectangular shape and wrap it to a donut shape. By setting $ a 2 i = a 1 i / n $, we have $ r \u2032 = b 2 i ( r / b 1 i ) n $ and $ \phi = n \theta $. Therefore, the phase front of $ exp ( i n \u2113 \theta ) $ will be converted into $ exp ( i \u2113 n \theta ) = exp ( i \u2113 \phi ) , $ achieving the transformation from $ | n \u2113 \u27e9 $ to $ | \u2113 \u27e9 $.

Figure 2 shows the experimental setup for division and multiplication of the OAM state order. A 1550 nm laser is divided into two paths of fibers with different delays and then projected onto two SLMs (SLM-1 and SLM-2). These SLMs transform Gaussian beams into OAM beams, and a beam splitter (BS) combines the two OAM beams. The beam is then sent to SLM-3 with phase patterns described as Eqs. (1a) and (2), which performs the log-polar to Cartesian transformation on the focal plane at a distance f = 360 mm from SLM-3. SLM-4 is designed to perform the phase correction and an inverse Cartesian to log-polar transformation simultaneously. Using Eq. (1b), we can achieve such phase pattern by simply adding the phase-correction pattern of the first step and the phase delay of the inverse coordinate transformation’s first phase mask. SLM-5 is used to correct the phase pattern and demultiplex the OAM beams. Both SLM-4 and SLM-5 are placed at a distance f = 360 mm from the prior SLM. At the receiver, the beam is split into two paths. We use a camera to measure the intensity profiles of the output beam, and on the other path, we use a pair of lens to demagnify the output beam and a filter to block the undesired diffraction orders of the light. The beam is then coupled to a single mode fiber for detection.

We first perform division of the OAM state order. As shown in Fig. 3, we measure and simulate the intensity profiles of the OAM states and the corresponding interference patterns with a normal Gaussian beam after halving the OAM states from $ | 2 \u2113 \u27e9 $ to $ | \u2113 \u27e9 $. The input OAM beams have an OAM state order varying from −10 to +10. The donut-shape intensity profiles and the corresponding twisting interference patterns with a normal Gaussian beam demonstrate that the output beams have an OAM state order varying from −5 to +5, as expected. Figure 4 shows the measured mode purity, characterized by the power distribution among OAM modes, of the output beams when the input OAM beams have an OAM order of 0, +2, +4, +6, +8, and +10; mode purity can be defined as the percentage of power that is located in the desired mode. The output beams have the highest power in the OAM order of 0, +1, +2, +3, +4, and +5. We also find that the mode purity decreases from 87% to 53% when the input OAM state order varies from 0 to +10. A possible explanation is that when the helical phase front of the input OAM beam twists faster, the phase front itself introduces higher distortion on the coordinate transformations.

We now turn to realize multiplication of the OAM state order. In order to double the OAM states from $ | \u2113 \u27e9 $ to $ | 2 \u2113 \u27e9 , $ we use the same setup of division and change the phase patterns of SLM-3, SLM-4, and SLM-5 to the desired phase delay described as Eqs. (1a)–(2). The input OAM beams now have an OAM state order varying from −5 to +5. Figure 5 shows the measured and simulated intensity profiles of the OAM beams and the corresponding interference patterns with a normal Gaussian beam. The output beams still keep the donut shapes, and the interference patterns indicate the helical phase fronts of the OAM beams with an order from −10 to +10. Figures 6(a)–6(f) show the measured mode purity of the output beams when input beams have an OAM order of 0, +1, −2, +3, −4, and +5, respectively. The output beams clearly have the highest power with the OAM order of 0, +2, −4, +6, −8, and +10. Figure 6(d) and the inset show a comparison of the mode purity with/without doubling the OAM state order of +3, indicating that the mode purity decreases from 85% to 35% after OAM doubling. Meanwhile, the mode purity decreases from 40% to 28% when the absolute value of input OAM state order increases from 0 to +5. The mode purity is worse than that of the case for OAM division. The degradation of the mode purity might be explained by: (i) displacement of the SLMs, (ii) the diffraction effect when copying the beams, and (iii) an imperfect combination of two semi-annular shapes to a donut shape.

We further investigate the OAM mode purity through evaluating the displacement of SLMs in simulation. In an ideal case, the SLMs would be perfectly aligned, namely, all SLMs would be coaxial. However, in a practical experimental setup, there might be displacements among different SLMs, as shown in the inset of Fig. 7(b). The presence of a displacement causes phase distortions on the coordinate transformations, resulting in a degradation of the OAM mode purity. Figures 7(a) and 7(b) show the simulated power distribution among different OAM states when there exists a displacement (Dt) between SLM-3 and SLM-4. We halve OAM +8 or double OAM +2 to obtain OAM +4 at the receiver. When the displacement increases, the power leaked to other states increases whereas the power on OAM +4 decreases. For halving the OAM order from +8 to +4, the crosstalk between OAM +4 and the neighboring state increases from −14 dB to −3 dB when Dt increases from $ 0 \u2009 to \u2009 80 \u2009 \mu m $. For doubling the OAM order from +2 to +4, the crosstalk between OAM +4 and the neighboring state is −47 dB when Dt = 0, while the power of the neighboring state becomes greater than that of OAM +4 when $ Dt = 12 \u2009 \mu m $. Figures 8(a)–8(d) show the mode purity of the OAM beams after halving OAM +8 and doubling OAM +2 when $ Dt = 0 \u2009 \mu m $ and $ Dt = 40 \u2009 \mu m $, respectively. The mode purity of OAM +4 decreases from 89% to 81% for halving OAM +8 and from 98% to 18% for doubling OAM +2. The obtained results indicate that the mode purity of the output OAM beams for a doubling OAM system is more sensitive to the increase of Dt than that for a halving OAM system. These results may explain why the experimental mode purity for OAM multiplication is much worse than that for OAM division.

We measure the BER performances of a two-mode OAM multiplexed link for the case of division and multiplication of the OAM state order. We multiplex two OAM channels, and each channel carries a 50 Gbit/s Quadrature Phase-Shift Keying (QPSK) signal. First, we turn one channel on and keep the other off. We transmit a normal Gaussian beam though one channel and measure the BER of the single channel. We then turn both channels on and transform the OAM state order to $\u2113$/2 or 2$\u2113$ and measure the BERs of both channels with halving/doubling the OAM order. Last, we turn both channels on while erasing the phase patterns for coordinate transformations on SLM-3, SLM-4, and SLM-5. We can get the BERs of both channels without halving/doubling the OAM order. Figure 9(a) shows the BERs for both channels with OAM order $\u2113$ = −6 and $\u2113$ = −2 with/without halving the OAM state order. Comparing the BERs of the case with/without halving the OAM order, we can see that there is a BER performance penalty. This may be because the decreased state spacing introduces more crosstalk between these two channels. Figure 9(b) shows the measured BER performance of two OAM channels with orders $\u2113$ = −3 and $\u2113$ = −1 with/without doubling the OAM state order. The BERs of both OAM channels can reach levels lower than the forward error correction (FEC) limit of 3.8 × 10^{−3},^{35} demonstrating that we achieve doubling the OAM order without much penalty on BER performance. The reason is that doubling the OAM order deteriorates the mode purity but creates wider state spacing, namely, increasing the OAM channel spacing from 2 to 4. It decreases the interaction between neighboring states, resulting in less crosstalk.

Optical devices for division and multiplication of the OAM order can be described as an operator ** H**, achieving a transformation $ \bm{H} | \u2113 \u27e9 = | n \u2113 \u27e9 , $ where $ n = 1,2,3 , \u2026 $ for the multiplication or $ n = 1 , 1 2 , 1 3 , \u2026 $ for the division. However, our devices act as an operator $ \bm{H} \u2032 $, transforming the initial state to $ \bm{H} \u2032 | \u2113 \u27e9 = \u2211 i = \u2212 \u221e + \u221e b i | i $, where $ b i $ has the maximum value when $ i = n \u2113 \u27e9 $ (i.e., the desired state) and $ b i $ may not be zero when $ i \u2260 n \u2113 $ (i.e., undesired state introducing crosstalk). This is because imperfect coordinate mapping results in a reduction in mode purity. Minimizing the displacement of SLMs and enhancing the SLMs’ transmission efficiency might be helpful to improve the mode purity and achieve an operator $ \bm{H} \u2032 \u2248 \bm{H} . $ OAM state division and multiplication could potentially bring some benefits to classical information processing. For example, OAM state multiplication could be used to create wider state spacing, which might help reduce the interactions between neighboring states, resulting in less crosstalk among OAM multiplexing channels. Moreover, OAM state division could be used to create dense state spacing, which might enhance mode efficiency. This might also be useful in OAM-based quantum systems.

We acknowledge the generous support of Vannevar Bush Faculty Fellowship from ASD(R&E) and ONR, NxGen Partners, Air Force Office of Scientific Research No. FA9550-16-C-0008 and NSF No. ECCS-1509965.

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