## Abstract

Based on the conventional iterative algorithm, we present a pattern search assisted iterative (PSI) algorithm to simultaneously generate multiple orbital angular momentum (OAM) modes using a single phase-only element. The PSI algorithm shows a favorable operation performance for generating 100 randomly spaced OAM modes and 50 evenly spaced OAM modes with high diffraction efficiency (>93%), low relative root-mean-square error (R-RMSE) and low standard deviation. Moreover, we can also manipulate the relative power distribution of the generated OAM modes simply by setting the initial weight coefficients in the PSI algorithm.

© 2015 Optical Society of America

## 1. Introduction

The light beam carrying helical wavefront, known as orbital angular momentum (OAM) beam or vortex beam, has been studied for decades. It was shown by Allen in 1992 that the helically phased beams comprising an azimuthal phase term exp(*ilφ*), possess an OAM of *lℏ* per photon, where *l* is referred to topological charge and *φ* is azimuthal angle [1–3
]. OAM is an intrinsic property of various types of helically phased beams, ranging from electron beams to radio waves. Recently, OAM-carrying beams have seen wide applications in different areas, such as optical manipulation, optical trapping, optical tweezers, optical vortex knots, imaging and quantum information processing [4–9
].

Due to the intrinsic spatial orthogonality of OAM modes with different topological charge numbers, one promising prospect offered by OAM is in optical communications [10]. For this reason, a series of research works on OAM modes multiplexing and multicasting in optical communications have been reported recently for increasing the transmission capacity [11–13 ]. In the conventional OAM communication experiments, one element (e.g. spiral phase plate or spatial light modulator) only provides one OAM mode [14], which is not scalable. The cost and complexity of an OAM multiplexing system will rapidly grow with the increase of the multiplexed OAM modes and resultant required multiple elements. In order to increase the capacity of optical communications in a scalable and cost-effective way, it is highly desirable to develop a simple method for generating a large number of OAM modes with a single element. Moreover, simultaneous generation of multiple OAM modes using a single element from a single input Gaussian beam is also an important basic function in an OAM multicasting system.

In 2004, Gibson demonstrated the transfer of information encoded as OAM modes of a light beam using computer-generated hologram (CGH) [10]. Although the CGH method is versatile, the relatively low diffraction efficiency of CGH may limit its applications for generating a large number of OAM modes. Phase-only elements are more desirable in most instances with higher efficiency and readier implementation. Some iterative algorithms, such as the well-known Gerchberg-Saxton (GS) algorithm and the adaptive-additive (AA) algorithm [15, 16
], have been developed to optimize phase-only elements. In 2005, Lin *et al* put forward an iterative algorithm for the generation of multiple collinear OAM modes with a phase-only element [17–19
]. 4 OAM modes have been successfully generated. Further improvement was made by Wang *et al* in 2011 [20]. They demonstrated a free-space optical communication system which was capable of transmitting and receiving 16 OAM modes in experiment. Recently, Yan *et al* experimentally demonstrated multicasting 5 and 7 OAM modes from a single input OAM mode by using the specially designed sliced phase pattern [13]. However, for generating a large number (>50) of OAM modes to greatly increase the communication capacity, those methods may not achieve the target with favorable performance. The efficiency will become relatively low with the increase of the OAM modes. In this scenario, a laudable goal would be to develop a method for generating a large number of OAM modes with high efficiency by a single element.

In this paper, a pattern search assisted iterative (PSI) algorithm for the simultaneous generation of multiple OAM modes is presented. We achieve 100 randomly spaced OAM modes and 50 evenly spaced OAM modes generated through a single phase-only element in simulation. Furthermore, we can optionally manipulate the weight (i.e. relative power distribution) of each OAM mode during the simultaneous generation of multiple OAM modes using the PSI algorithm.

## 2. Pattern search assisted iterative (PSI) algorithm

Usually, a single OAM mode can be generated by an element with a transmission function of exp(*ilφ*). For the simultaneous generation of *n* OAM modes $\mu $, the mathematical description of the desired transmission function can be expressed as

Since the weight coefficients $\text{{}{A}_{{l}_{m}}\text{}}$ is settled at first, the parameter R-RMSE is determined by $\text{{}{C}_{{l}_{m}}\text{}}$ or $\text{{}{B}_{{l}_{m}}\text{}}$. Then it becomes a simple minimization problem. We need to find the suitable $\text{{}{B}_{{l}_{m}}\text{}}$ for minimizing R-RMSE. To solve the problem, Lin *et al* proposed an iterative algorithm, which is a spontaneous optimization algorithm [17]. It is a highly effective method for generating multiple OAM modes with a single phase-only element. However, when we use this method to generate more than 10 OAM modes, the performance of the algorithm gets worse, mainly because the initial set of parameters $\text{{}{B}_{{l}_{m}}^{0}\text{}}$ are set equal to $\text{{}{A}_{{l}_{m}}\text{}}$. The unsuitable choice of $\text{{}{B}_{{l}_{m}}^{0}\text{}}$ will lead to immature convergence of the iterative algorithm.

For the suitable choice of $\text{{}{B}_{{l}_{m}}^{0}\text{}}$, we introduce the pattern search optimization algorithm. Pattern search is a numerical optimization method, which does not require calculating the gradient of the problem to be optimized [21, 22 ]. Hence pattern search can be used to solve the functions that are not continuous or differentiable. So it can provide a reasonable approach to getting the initial $\text{{}{B}_{{l}_{m}}^{0}\text{}}$. The following provides specific steps, as shown in Fig. 1(a) .

Firstly, in the initiation of the process, we set ${\Delta}_{0}>0$, $\left\{{b}_{{l}_{m}}^{0}\right\}=\left\{{A}_{{l}_{m}}\right\}$ and the iteration counter $k=0$. Here, we use $\left\{{b}_{{l}_{m}}^{k}\right\}$ instead of $\text{{}{B}_{{l}_{m}}\text{}}$ in the pattern search process. According to Eqs. (2)-(4) , we can get the $R\text{-}RMSE({b}_{{l}_{m}}^{k})$, which is a function of $\left\{{b}_{{l}_{m}}^{k}\right\}$ and $R\text{-}RMSE$.

Secondly, we perform the search and possibly poll steps until an improved $\left\{{b}_{{l}_{m}}^{k+1}\right\}$ with lowest $R\text{-}RMSE$is found.

Thirdly, if $R\text{-}RMSE({b}_{{l}_{m}}^{k+1})<R\text{-}RMSE({b}_{{l}_{m}}^{k})$, then update ${\Delta}_{k+1}\ge {\Delta}_{k}$. Otherwise, $R\text{-}RMSE({b}_{{l}_{m}}^{k+1})\ge R\text{-}RMSE({b}_{{l}_{m}}^{k})$, set $\left\{{b}_{{l}_{m}}^{k+1}\right\}=\left\{{b}_{{l}_{m}}^{k}\right\}$ and update ${\Delta}_{k+1}<{\Delta}_{k}$.

Finally, increase $k$ to $k+1$, and go back to the second step. Perform this loop until ${\Delta}_{k}$cannot be smaller, which means the step length of the pattern search process is small enough to achieve the optimal parameter $\text{{}{B}_{{l}_{m}}^{0}\text{}}$. Then output the $\left\{{b}_{{l}_{m}}^{k}\right\}$, and the pattern search process is ended.

After that, the initial parameters $\text{{}{B}_{{l}_{m}}^{0}\text{}}$are achieved, which is equated to the output of the PS process$\left\{{b}_{{l}_{m}}^{k}\right\}$. We can also use the Patternsearch function in Matrix Laboratory (Matlab) to do the optimization. After we get $\text{{}{B}_{{l}_{m}}^{0}\text{}}$, we can perform the iterative algorithm, as shown in Fig. 1(b), and get the phase-only approximate function $g(\phi )$. Thus we can design the phase pattern to generate the multiple OAM modes through$g(\phi )$.

## 3. Results and discussions

By using the PSI algorithm described above, we simultaneously generate multiple randomly spaced and evenly spaced OAM modes. We calculate the diffraction efficiency, R-RMSE and standard deviation to evaluate the performance of PSI algorithm. We also compare the performance of PSI algorithm with the previous algorithms. Moreover, by setting the initial weight coefficients $\text{{}{A}_{{l}_{m}}\text{}}$ in PSI algorithm, we optionally manipulate the power of each generated OAM mode.

#### 3.1 Simultaneous generation of multiple randomly spaced OAM modes

Here, we show the simulation results of randomly spaced OAM modes by using PSI algorithm. And we also compare it with the Lin’s algorithm and the genetic algorithm (GA) improved algorithm.

Firstly, we get 20 randomly spaced equal-power OAM modes by each algorithm as shown in Fig. 2
. The target spectrum is shown in Fig. 2(a). And in Fig. 2(b), we get the spectrum through Lin’s algorithm. As we can see, the Lin’s algorithm fails to get the target, mainly because the initial set of parameters $\text{{}{B}_{{l}_{m}}^{0}\text{}}$ are set equal to $\text{{}{A}_{{l}_{m}}\text{}}$. To improve the algorithm, Wang *et al* introduced the genetic algorithm [20]. The spectrum by GA improved algorithm is shown in Fig. 2(c). The efficiency is improved by the GA improved algorithm, but the relative power of each OAM mode has great difference. Figure 2(d) shows the spectrum by PSI algorithm with both high efficiency and average relative power. So, the achieved spectrum by PSI algorithm has the most similar power distribution to the target.

In order to quantitatively assess the performance of each algorithm, we can compare the diffraction efficiency, the relative root-mean-square error (R-RMSE), and the standard deviation of each algorithm. The diffraction efficiency is defined as the ratio of the power falling into the desired OAM modes to the total power, which has been normalized to unity. R-RMSE is achieved by Eq. (4), which indicates the power difference between the simulation spectrum and the target spectrum. The standard deviation $\sigma $ is normally defined as

The performance of three different algorithms is summarized in Table 1 . As we can see, the diffraction efficiency of the GA improved algorithm and PSI algorithm are both higher than 90%, showing relatively small loss. Moreover, the R-RMSE of PSI algorithm is lower than the other two algorithms. Especially, the standard deviation of PSI algorithm is almost one-order magnitude lower than the others, which leads to the power equalization of each OAM mode. According to the above analyses, the PS improved algorithm shows better performance than the other two algorithms.

We study the convergence of the PSI algorithm. The iterative process in the PSI algorithm is convergent. Shown in Fig. 3 are the convergence curves of R-RMSE. Figure 3(a) shows the R-RMSE convergence curve for generating 20 randomly spaced equal-power OAM modes. One can easily see from the curve that the R-RMSE is convergent after 65 iterations. The R-RMSE convergence curve for generating 50 randomly spaced equal-power OAM modes is depicted in Fig. 3(b). One can also clearly see that the R-RMSE is convergent after 91 iterations.

Remarkably, the complexity and computational speed are another two important parameters to evaluate the efficiency of an algorithm. The complexity of the PSI algorithm is almost the same as the GA improved algorithm. The two algorithms both need to search for suitable initial parameters $\text{{}{B}_{{l}_{m}}^{0}\text{}}$ first, and then perform the iterative algorithm. Lin’s algorithm sets the initial set of parameters $\text{{}{B}_{{l}_{m}}^{0}\text{}}$ equal to $\left\{{A}_{{l}_{m}}\right\}$, and then do the iterative algorithm.

To compare the computational speed of these algorithms, we perform them on the same computer with 3.4 GHz Intel Core i3-2130 CPU and 16 GBytes RAM. For generating 20 randomly spaced equal-power OAM modes, the average computing time of the Lin’s algorithm is 67.87 s, the PSI algorithm costs 93.53 s, and the GA improved algorithm takes 96.43 s. The computing time of all the three algorithms is less than 100 s, while the PSI algorithm shows improved operation performance.

Moreover, in order to evaluate the algorithm under a more realistic scenario, we study the performance of the phase patterns loaded onto the practical spatial light modulator (SLM). Here, we take commercially available Holoeye PLUTO phase only SLM as an example. The resolution of the SLM is 1920x1080 pixels with 256 gray levels covering 0-2π phase modulation. By using the GA improved algorithm and PSI algorithm, we get realistic SLM phase patterns for generating 20 randomly spaced equal-power OAM modes. The phase patterns loaded onto SLM are 1080x1080 pixels with 256 gray levels, which are shown in Fig. 4(a) and 4(b) . We then compare the performance of these two realistic SLM phase patterns for generating OAM modes. The simulation results of the OAM spectra are depicted in Fig. 4(c) and 4(d). The OAM spectra of the original phase patterns (i.e. no consideration of realistic SLM) using the GA improved algorithm and PSI algorithm are also shown for reference. Figure 4(c) shows the OAM spectra of the original phase pattern (blue one) and realistic SLM phase pattern (red one) using the GA improved algorithm. The OAM spectra of the original phase pattern (blue one) and realistic SLM phase pattern using the PSI algorithm are depicted in Fig. 4(d). By comparing the simulation results shown in Fig. 4, one can see that the performance of realistic SLM phase pattern is slightly degraded compared with the original phase pattern. One can also find that the PSI algorithm shows better performance than the GA improved algorithm even using the realistic SLM phase pattern.

Furthermore, we also calculate the diffraction efficiency, R-RMSE, and standard deviation of the two realistic SLM phase patterns using the GA improved algorithm and PSI algorithm and compare them with the original phase patterns. The simulation results are listed in Table 2 . One can clearly see that the realistic SLM phase pattern using the PSI algorithm still shows favorable performance for generating multiple OAM modes with high diffraction efficiency, low R-RMSE, and low standard deviation.

To further show the performance of the algorithm for a large number superposition of multiple OAM modes, we use the GA improved algorithm and PSI algorithm to get 50 and 100 OAM modes, respectively. The simulation results are shown in Fig. 5 . And the corresponding performance is summarized in Table 3 . As shown in the figure and table, the PSI algorithm can also get better performance with almost the same diffraction efficiency with the GA improved algorithm and lower standard deviation. Figure 5(e) and 5(f) are the phase patterns for generating the 100 randomly spaced OAM modes by GA improved algorithm and PSI algorithm, respectively.

#### 3.2 Simultaneous generation of multiple evenly spaced OAM modes

In most OAM communication experiments, evenly spaced OAM modes are always employed. So, it is necessary to evaluate the performance of the algorithm for generating evenly spaced OAM modes. In this session, the simulation results of evenly spaced OAM modes with space 1, 3, 5 and 10 are shown in Fig. 6 .

From Fig. 6, one can clearly see that the PSI algorithm shows a better performance in each case. The quantitative analyses are listed in Table 4 . The two algorithms can both achieve high diffraction efficiency above 95%. The R-RMSE and standard deviation of PSI algorithm are much lower than that of GA improved algorithm. As a result, the achieved spectrum by PSI algorithm tends to be closer to the target.

#### 3.3 Manipulating the weight of each OAM mode

In a multiplexing communication system, multiple channels are multiplexed together to increase the communication capacity. In a multicasting communication system, one input channel is multicasted to multiple channels for multi-users. It is noted that the loss of different channels might be different for many reasons. Consequently, in practice we usually need to flexibly manipulate the power of each OAM mode to compensate the channel-varying loss of different OAM modes. By changing the target weight coefficients as desired, we can use PSI algorithm to get the corresponding OAM spectrum. The simulation spectrums for manipulating the weight of 10 OAM modes are shown in Fig. 7 .

As seen from Fig. 7, both GA improved algorithm and PSI algorithm can control the power of different OAM modes. To assess the performance of power manipulation, we can also calculate the diffraction efficiency and R-RMSE. But the standard deviation cannot be used in this case, because the OAM modes have different powers. The performance of the algorithms is summarized in Table 5 . The R-RMSE is calculated with the power ratio in Table 5, which can actually illustrate the ability for manipulating different power ratio.

Table 5 shows that the PSI algorithm has higher diffraction efficiency and lower R-RMSE, which means it has a better performance than the GA improved algorithm. We also evaluate the two algorithms for manipulating 50 OAM modes. The obtained results are shown in Fig. 8 . It is found the PSI algorithm still shows a better performance with a high diffraction efficiency 97.87% and a low R-RMSE 1.86%.

As a consequence, by employing a single phase-only element such as spatial light modulator (SLM), we can use the PSI algorithm to generate a large number of OAM modes with favorable performance in OAM communication systems. The scheme is scalable and cost-effective. Moreover, we can also easily manipulate the relative power distribution of different OAM modes, which can enhance the flexibility of OAM communication systems.

## 4. Conclusion

In summary, we have presented the PSI algorithm to simultaneously generate multiple OAM modes with a single phase-only element. The PSI algorithm shows favorable performance with high diffraction efficiency, low R-RMSE and low standard deviation for generating randomly spaced or evenly spaced multiple OAM modes. Additionally, we can also flexibly manipulate the relative power distribution of the generated multiple OAM modes simply by setting the initial weight coefficients in the PSI algorithm. The obtained simulation results indicate that the PSI algorithm might find useful applications in OAM communication systems which require a large number of OAM modes.

## Acknowledgments

This work was supported by the National Basic Research Program of China (973 Program) under grant 2014CB340004, the National Natural Science Foundation of China (NSFC) under grants 11274131, 11574001 and 61222502, the Program for New Century Excellent Talents in University (NCET-11-0182), the Wuhan Science and Technology Plan Project under grant 2014070404010201, the Fundamental Research Funds for the Central Universities (HUST) under grants 2012YQ008 and 2013ZZGH003, and the seed project of Wuhan National Laboratory for Optoelectronics (WNLO). The authors thank Shuhui Li and Jun Liu for helpful discussions.

## References and links

**1. **L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A **45**(11), 8185–8189 (1992). [CrossRef] [PubMed]

**2. **A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics **3**(2), 161–204 (2011). [CrossRef]

**3. **S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photonics Rev. **2**(4), 299–313 (2008). [CrossRef]

**4. **K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photonics **5**(6), 335–342 (2011). [CrossRef]

**5. **L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science **292**(5518), 912–914 (2001). [CrossRef] [PubMed]

**6. **M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics **5**(6), 343–348 (2011). [CrossRef]

**7. **M. R. Dennis, R. P. King, B. Jack, K. O’Holleran, and M. J. Padgett, “Isolated optical vortex knots,” Nat. Phys. **6**(2), 118–121 (2010). [CrossRef]

**8. **S. Bernet, A. Jesacher, S. Fürhapter, C. Maurer, and M. Ritsch-Marte, “Quantitative imaging of complex samples by spiral phase contrast microscopy,” Opt. Express **14**(9), 3792–3805 (2006). [CrossRef] [PubMed]

**9. **A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature **412**(6844), 313–316 (2001). [CrossRef] [PubMed]

**10. **G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express **12**(22), 5448–5456 (2004). [CrossRef] [PubMed]

**11. **J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics **6**(7), 488–496 (2012). [CrossRef]

**12. **N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science **340**(6140), 1545–1548 (2013). [CrossRef] [PubMed]

**13. **Y. Yan, Y. Yue, H. Huang, Y. Ren, N. Ahmed, M. Tur, S. Dolinar, and A. Willner, “Multicasting in a spatial division multiplexing system based on optical orbital angular momentum,” Opt. Lett. **38**(19), 3930–3933 (2013). [CrossRef] [PubMed]

**14. **M. W. Beijersbergen, R. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. **112**(5–6), 321–327 (1994). [CrossRef]

**15. **R. W. Gerchberg and W. O. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik (Stuttg.) **34**, 275–284 (1971).

**16. **V. Soifer, V. Kotlyar, and L. Doskolovich, *Iterative Methods for Diffractive Optical Elements Computation* (Taylor & Francis, 1997).

**17. **J. Lin, X. C. Yuan, S. H. Tao, and R. E. Burge, “Collinear superposition of multiple helical beams generated by a single azimuthally modulated phase-only element,” Opt. Lett. **30**(24), 3266–3268 (2005). [CrossRef] [PubMed]

**18. **J. Lin, X. Yuan, S. H. Tao, and R. E. Burge, “Synthesis of multiple collinear helical modes generated by a phase-only element,” J. Opt. Soc. Am. A **23**(5), 1214–1218 (2006). [CrossRef] [PubMed]

**19. **J. Lin, X. C. Yuan, S. H. Tao, and R. E. Burge, “Multiplexing free-space optical signals using superimposed collinear orbital angular momentum states,” Appl. Opt. **46**(21), 4680–4685 (2007). [CrossRef] [PubMed]

**20. **Z. Wang, N. Zhang, and X. C. Yuan, “High-volume optical vortex multiplexing and de-multiplexing for free-space optical communication,” Opt. Express **19**(2), 482–492 (2011). [CrossRef] [PubMed]

**21. **V. Torczon, “On the convergence of pattern search algorithms,” SIAM J. Optim. **7**(1), 1–25 (1997). [CrossRef]

**22. **C. Audet and J. E. Dennis Jr., “Analysis of generalized pattern searches,” SIAM J. Optim. **13**(3), 889–903 (2002). [CrossRef]