## Abstract

We report an approach to the increase of signal channels in free-space optical communication based on composed optical vortices (OVs). In the encoding process, conventional algorithm employed for the generation of collinearly superimposed OVs is combined with a genetic algorithm to achieve high-volume OV multiplexing. At the receiver end, a novel Dammann vortex grating is used to analyze the multihelix beams with a large number of OVs. We experimentally demonstrate a digitized system which is capable of transmitting and receiving 16 OV channels simultaneously. This system is expected to be compatible with a high-speed OV multiplexing technique, with potentials to extremely high-volume information density in OV communication.

©2011 Optical Society of America

## 1. Introduction

The light beam with helical wave front of $\mathrm{exp}(il\varphi )$, known as optical vortex (OV), has been studied for decades [1], where *ϕ* is the azimuthal angle and *l* is referred to as topological charge. In 1992, Allen *et al* discovered that such a beam carries orbital angular momentum (OAM) of $l\hslash $ per photon [2]. Investigation and application of OAM has since become an area of great interest, ranging from optical trapping to detection in astrophysics [3,4].

One promising prospect offered by OAM is in optical communication. It has been suggested that the infinite OAM eigenstates can enable a single photon to carry unlimited amount of information [5,6]. At macroscopic level, as pointed out in [7], OV beams can serve as carriers of information in free-space optical (FSO) communication with great security advantages. A system of this kind is based on the creation of helical states of different topological charges, thus can be named as charge variable free-space optical (CFSO) communication. Since it is possible to transfer several OVs simultaneously, a multichannel CFSO system can be achieved. In such a system a set of helical states are chosen as a basis, each one matches an independent signal channel, whose existence is provided as one bit “1”, absence as “0”. Different combination of the OVs thus represents different binary information.

In [8], Bouchal *et al* presented a seminal method to realize this kind of system by generating pseudo non-diffraction beams which can transfer up to four OVs simultaneously. The system accordingly possesses four signal channels, which are insufficient compared to other existing FSO methods. Another method based on spatially separated OVs might be free of this problem. In [9,10], Khonina *et al* employed a single diffractive optical element (DOE) to generate multiple beams simultaneously, each one of which had a combination of two OVs and was examined by another analyzing DOE. The communication system based on this method has been proposed in [11], where 20 beams were generated simultaneously by a DOE, each one of which contained 2 or 3 OVs and was examined by another 24-order analyzing DOE. Obviously, this system has a large number of signal channels and lower requirement on carrier spatial frequencies [12], as the diffraction orders of the analyzing DOE are circumferentially distributed. However since these 20 beams are spatially separated and have to be analyzed independently, such a system needs a cumbersome decoding end, which is undesirable.

In [13], Lin *et al* put forward another CFSO system by using a phase-only element to generate a multihelix beam composed of collinearly superimposed OVs. Although this system’s experimental setup is simpler, like the one proposed by Bouchal [8], only 4 signal channels are available. Further increment was constrained by difficulties at both encoding and decoding terminals.

In this paper, based on Lin’s method, an advanced CFSO system with a large number of signal channels is presented. In Sections 2.1 and 3.1, we first briefly review the conventional approaches employed to encode and decode multiple helical states and analyze their constraints. In Sections 2.2 and 3.2 we propose respectively an improved algorithm and a novel grating to surmount these difficulties. Combination of these two methods results in a broadband CFSO system, for which we achieved 30 signal channels in simulation and 16 in experiment. Finally, the probability of integrating our system with another high-speed modulation CFSO system is discussed.

## 2. Generation of a multihelix beam

#### 2.1 Review of Lin’s algorithm

Theoretically, to modulate a laser beam to contain N collinearly superimposed OVs of charge $\left\{{l}_{m}\right\}$ with weight$\left\{{A}_{m}{}^{2}\right\}$, implementation of the transmittance function

is required. Mostly Eq. (1) appears in a complex form, including both amplitude and phase modulations. For simplicity, it is preferable to reduce the amplitude-phase function to a phase-only approximation with tolerable loss in diffraction efficiency and reconstruction accuracy.Such kind of problem—transforming complex distribution into phase-only distribution is common, for which several iterative and non-iterative methods have been developed [14]. For the generation of multihelix beam, Lin *et al* proposed a method which can be deemed as analogous to the Adaptive-Additive algorithm [15], a well-known iterative method for the phase-only DOE synthesis [16]. Although we suppose the multihelix beam can also be generated by non-iterative ones, the specific approach to do so remains as the topic of future research.

In Lin’s method, rather than calculating the complex values of pixels, the algorithm deals with the weights of helical states. In the iteration process the phase pattern is defined by the weight coefficients$\left\{{B}_{m}^{s}\right\}$ in

*β*used to vary the modulus of $\left\{\left|{B}_{m}^{s}\right|\right\}$ in each step. Furthermore as soon as more than 8 collinear states are required, the stagnation effect becomes significant, mainly because the initial set of parameters $\left\{{B}_{m}^{0}\right\}$ are set equal to $\left\{\left|{A}_{m}\right|\right\}$, which lead to immature convergence in most cases. For example, for the target beam containing 18 equal-energy OVs whose spectrum is shown in Fig. 1(a) , the algorithm stagnates at 6th iteration, and the resultant DOE completely fails to reach the goal, as shown in Fig. 1(b). Therefore how to find the optimal set of parameters $\left\{\right\{{B}_{m}^{0}\},\beta \}$ for different target beams constitutes a problem for high-volume OV multiplexing.

#### 2.2 GA improved algorithm

An inherently parallel search technique—genetic algorithm (GA) can be combined with Lin’s algorithm to address this difficulty. GA has long been known for its capability of solving various optimization problems through mimicking the natural evolutionary process [17,18]. Since there is not a clear principle for the choice of$\left\{\right\{{B}_{m}^{0}\},\beta \}$, GA’s parallel search provides a reasonable approach to find the global optimum. Using the example of 18 OVs to illustrate, when the optimal $\left\{\right\{{B}_{m}^{0}\},\beta \}$ found by GA is brought into Lin’s algorithm, the iteration does not stagnate until 157th steps and the resultant DOE generates an almost identical spectrum with great uniformity and nearly 95.6% diffraction efficiency, as shown in Fig. 1(c).

In the initiation of the process, a group of N strings of binary digits are randomly generated, which are called the population of chromosomes. Each string is regarded as an individual (trial solution) corresponding to one set of parameters$\left\{\right\{{B}_{m}^{0}\},\beta \}$, where $\left\{{B}_{m}^{0}\right\}$ vary within the interval between 0 and the largest$\left|{A}_{m}\right|$, and *β* within [0.05, 0.1]. Second, each individual is converted to the real$\left\{\right\{{B}_{m}^{0}\},\beta \}$ which subsequently undergoes Lin’s algorithm. The whole population is then ranked according to the individuals’ fitness defined by the corresponding reconstruction accuracy value. Third, the GA repeated the typical selection, single-node crossover and mutation operators successively until the next generation of N ‘offspring’ are generated, which replace the current population. (For more detail about these operators, please see [17,18]). The GA then goes back to the second step and begins the next generation. The same procedure is iterated until the best reconstruction accuracy value of each generation does not vary significantly. The last generation’s most fitted individual corresponds to the optimal$\left\{\right\{{B}_{m}^{0}\},\beta \}$. We find for the multihelix beam composed of less than 30 helical states, a population of 40 individuals generally suffices to find the optimum, which on average can be achieved at around the 30th generation. Choice on the parameters applied to the operators is based on [19]. After the optimal$\left\{\right\{{B}_{m}^{0}\},\beta \}$is found, the set of parameters $\left\{\right\{{B}_{m}^{t}\},\beta \}$is automatically achieved, by which the phase pattern is calculated, here t denotes the step when Lin’s algorithm terminates. The phase pattern is subsequently encoded onto a spatial light modulator (SLM), which facilitates easy encoding of required multihelix beam and dynamic modulation.

## 3. Analyzing the multihelix beam

#### 3.1 Conventional vortex grating and its constraints

To analyze the multihelix beam, a method which can simultaneously identify each distinct OV is required, and large-volume detection is preferable. Notwithstanding there exist a number of measurement techniques, some of them are specified for single state [20,21], while others are complicated to implement [22,23]. To facilitate multiple signal channels, the most straightforward approach, to our knowledge is the diffractive grating which includes amplitude and phase forms [24]. Because of the amplitude grating’s relatively low diffraction efficiency, we will focus on phase grating exclusively in this paper.

To begin with, the transmittance function of 1-D vortex phase grating with period *d* and embedded charge *l* can be written by a Fourier series as [25]

*p*will possess partial plane wave (i.e. the state of charge 0). In far field, this is characterized by a central bright spot which can serve as the criteria for detection.

However, the coefficient${A}_{n}$of the conventional phase grating dwindles rapidly as order$\left|n\right|$increases. This poses a problem that the energy of central bright spots at higher orders are extremely low, which seriously constrains the number of signal channels available in a CFSO system. To resolve this difficulty, Gibson *et al* proposed a kind of diffraction grating calculated by GS algorithm which can produce 8 equal-energy OVs [26]. In spite of its high diffraction efficiency, this kind of grating is in continuous phase form, accordingly difficult to implement on a phase plate which in some occasions is indispensible. The analyzing DOE presented by Khonina *et al*, which we mentioned in the introduction part may be a better option for our purpose. In [9,10], binary DOEs capable of determining 8 and 24 OVs were employed respectively to analyze mulithelix beams composed of 2 OVs. The working principle of this kind of analyzer was illustrated in [12].

In this paper, we put forward an alternative approach—binary Dammann vortex grating which can achieve a much wider detection range with even less fabrication complexity.

#### 3.2 Dammann vortex grating as the analyzer

The Dammann grating is well known for its uniform energy distribution among designated diffraction orders [27,28]. Each period of the grating is divided into segments with 0 and pi phase shift by a set of transition points which are calculated for maximum diffraction efficiency among these equal-energy orders [29]. Integrating this concept with a spiral phase pattern $\mathrm{exp}(il\varphi )$ results in the binary-phase Dammann vortex grating. This can be done by using the same set of transition points to binarize each period of the product of a spiral phase pattern and a linear diffraction grating [30]. Although the Fourier expansion of the resultant transmittance function yields exactly the same form as Eq. (3), unlike the conventional vortex grating, the parameters ${A}_{n}$ are equal for those selected orders. Figure 2 shows this difference.

Although the way that Dammann vortex grating works as an analyzer has no difference from the way that conventional vortex grating works, the intensity of the bright spots at the designated orders is irrelevant to${A}_{n}{}^{2}$, which makes it possible to detect those OVs matching the high diffraction orders. To substantiate this point, we use the grating of Fig. 2 to measure the multihelix beam containing OVs of charge { + 4, + 8,-6}. Figure 3(b) shows that the bright spots at {-2, −4, + 3} orders of a Dammann vortex grating are equal; while for the conventional vortex grating, as shown in Fig. 3(a), central spots at −2 order is much brighter than the one at + 3 order, and the spot at −4 order is practically impossible to distinguish.

When Dammann vortex grating is extended to two dimensions, detecting range is greatly expanded. Now each order $({n}_{x},{n}_{y})$ is characterized by an equal-energy OV of charge${n}_{x}{l}_{x}+{n}_{y}{l}_{y}$, (${l}_{x}$and ${l}_{y}$are the horizontal and vertical intrinsic charges respectively). To verify its detecting ability, we use a $5\times 7$ Dammann vortex grating as the analyzer for a CFSO system with 30 signal channels. Figure 4 shows the OVs at different orders of the grating. Detecting results are depicted in Fig. 5 and Fig. 6 . Here to simplify the system, we use the intensity values of the central pixels at different orders to determine the component of the multihelix beam. Note that we do not consider those orders matching the OVs which are not chosen as channels, because the values at these orders have no effect in decoding the transferred information. In this case, we have already transferred two pieces of 30-bit information, the first one being {1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1} and the second one being {1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 0 1}.

For comparison, we also utilize a conventional phase grating to measure the same multihelix beams. Now due to the grating’s uneven power distribution, the central bright spots at high orders are very weak. Figure 7 shows that when some states are absent, changes of the corresponding central pixels are indistinct (see the values at the orders with charge −29 and −6), which in practice would lead to high bit error rate.

It may have been noticed that the in Fig. 6 the intensity values of the central pixels are different in spite of using Dammann vortex grating and the even power of incoming OVs. However, this phenomenon does not falsify our previous argument. To accurately measure the energy of a bright spot, it is necessary to integrate over the area which the peak produced by the plane wave envelops, whereas the wave front of a multihelix beam after passing a diffraction grating is complicated. Each peak caused by plane wave is surrounded by the intricate interference patterns of other OVs. Since the central pixel approximation is satisfactory, this wave front analysis, although possible [31], would only complicate a digital communication system.

## 4. Experiment

We first generate the multihelix beam using a spatially filtered 532nm laser beam from a Quantum Ventus 532 laser system. The beam is collimated and impinged onto a Hamamatsu X8267-11 reflect type parallel aligned liquid crystal spatial light modulator (PAL-LCSLM). The pixel size of the SLM is 26µm and the working area is 4cm^{2}. The SLM is operating in pure phase mode, and the gray scale levels of input images will be translated into the phase levels of liquid crystals.

To demonstrate the digital communication principle, we choose 16 topological charges $\{\pm 1,\pm 2,\pm 3,\pm 4,\pm 5,\pm 6,\pm 7,\pm 8\}$as a signal basis. Two multihelix beams are generated. One contains all 16 states, and the other one has {-1, −2, −6, + 5, + 7} states absent, representing two 16-bit signals. Their phase patterns are shown in Fig. 8 . The reflected multihelix beams are then propagated 1 meter and analyzed by a fabricated Dammann vortex grating, with horizontal and vertical intrinsic charges of + 1 and + 5 respectively. The Dammann vortex grating is capable of detecting 24 OVs of topological charges ranging from −12 to + 12, more than enough for the generated 16 signal channels. We fabricated the Dammann vortex grating on SU-8 photoresist using conventional UV lithography technique. The refractive index of SU-8 photoresist at 532nm is 1.59. According to this value, the calculated π-phase thickness of the photoresist is 451nm, and the typical measured thickness is 447nm. The microscopic images of the fabricated grating are shown in Fig. 9 .

After passing through the grating, the beam is focused onto a CCD camera by a converging lens with 300mm focal length. Figure 10 shows the intensity profiles captured by the CCD camera and Fig. 11 depicts the intensity values of central pixels at the corresponding diffraction orders. Although the intensity profile is too blurred to directly identify the central bright spots, the pixels’ intensity values clearly indicate the change in the power spectrum. In Fig. 11(b), it is noted that at the orders with topological charges {-5, + 1}, there is some unwanted noise, which can be mistaken as signal. This problem can be easily solved by setting an arbitrarily chosen threshold value according to various practical systems.

For further increasing the number of signal channels two technical matters should be noticed. First, in the grating fabrication process, our equipment can approximately reach 20µm accuracy, which is relatively low. This hinders us from producing the grating that generates a larger array of equal-energy OVs as detecting channels. Thus, in order to realize a wider band in CFSO, higher fabrication accuracy is needed. Second, as the number of signal channels increases, OVs of higher topological charges will be used. Since the radius of OV scales with *l* (*l*is the topological charge), the optical system with a larger aperture is necessary. With these problems surmounted we estimate a CFSO system with 50 channels should be possible in practice.

## 5. Discussion

It has been noticed that the slow refresh rate of SLM impedes high-rate transmission of information in CFSO. Recently, Gao *et al* proposed an experimental setup to address this problem [32], which involves a Michelson interferometer with two Porro prisms to generate multihelix beam. Although this system’s transmission speed is independent on the refresh rate of SLM, its information density is limited, as only two OVs are included in the generated multihelix beam.

Another approach to circumvent the refresh rate problem is proposed by Celechovsky and Bouchal [33]. In this system the multihelix beam is generated by illuminating a static DOE with an array of spatially separated laser diodes, each one of which corresponds to an OV in the multihelix beam. High speed modulation of signals can be realized by switching the array of laser diodes which work at a refresh rate several orders higher than that of SLM. The DOE is calculated by Lin’s method, but the phase pattern is given by

We find combining this method with those of us can result in the CFSO system of both wideband and high transmission rate. First, as the difference between Eq. (4) and Eq. (2) is some constant part in the iterative process, applying the GA improved algorithm to calculate the DOE produced by Eq. (4) is straightforward, which can significantly increase the number of collinear OVs. Although the calculation time will be longer as Eq. (4) has lower symmetry than Eq. (2), this should not pose a problem since only one running is required. Second, the grating utilized by Celechovsky is capable of detecting four states in parallel, with a CFSO system using a large number of collinear OVs, the Dammann vortex grating will become a preferable option as analyzer.

However, we anticipate there will be some adjusting difficulty when more laser diodes are used to implement signal channels. As this difficulty’s character is only technical rather than physical, the CFSO system with extremely large information density should be realizable in practice.

## 6. Conclusion

In this work, we have presented an FSO system capable of large-volume OV multiplexing and de-multiplexing. Phase masks calculated by a GA improved algorithm are used to generate signal channels containing collinearly superimposed OVs. At the receiver end, Dammann vortex gratings are employed as analyzer. Despite it has a larger number of signal channels than previously achieved, this advanced system does not need additional experimental setup. Also in the discussion section, we have shown that integrating a high-speed modulation method with our system is possible. Although many practical problems, such as the various distortions pointed out in [11], must be addressed first, we believe this technique can lead to extremely high data density.

Furthermore, as pointed out in [7], the FSO system based on composed OVs should be compatible with other FSO techniques such as wavelength division multiplexing, thus has the prospect of dramatically increasing the FSO communication channels.

## Acknowledgement

This work was partially supported by the Ministry of Science and Technology of China under Grant no. 2009DFA52300 for China-Singapore collaborations and the National Research Foundation of Singapore under Grant No. NRF-G-CRP 2007-01. The fabrication part was supported by the Ministry of Science and Technology of China under Grant no 2010CB327702. ZXW would like to thank the “100 Projects” of Creative Research of Nankai University of China. XCY would like to thank the National Natural Science Foundation of China under Grant No. 10974101 for studies of optical vortices.

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