Optical vortex beams carrying orbital angular momentum (OAM) are widely investigated for their unique performance in recent years. They can be used to extend the capacity of optical communications system due to the orthogonality of different channels. In the receiver side of a multiplexing optical vortices system, verifying the OAM spectrum is of great importance. A new kind of diffraction element called Dammann vortex grating can distribute energies among different diffraction orders equally. Based on this unique characteristic, we reported a new algorithm to analyze the spot of each diffraction order. The OAM spectrum in the receiver side can then be obtained. In the experiment, the OAM spectrum measurement of at most six-channel multiplexing optical vortices is realized. The experimental results illustrate that the OAM spectrum gained by this approach is highly consistent with the theoretical value.
© 2016 Optical Society of America
Optical vortices carrying orbital angular momentum (OAM) are attracting more and more attention for their unique characteristics. Allen et al. have proved that the complex amplitude of an optical vortex comprises an azimuthal phase term exp (ilφ), where φ is the azimuthal angle and l is the topological charge . Each photon in the optical vortex has an OAM lħ (ħ is Planck’s constant divided by 2π). In recent years, optical vortices have widely applied in lots of fields, such as optical tweezers , generating vector beams [3–5 ], quantum information process , optical trapping and guiding of cold atoms [7,8 ] and so on.
Optical vortices have the potential in extending the capacity of optical communication system [9–15,17 ]. Owing to the orthogonality of optical vortices , the mode-division multiplexing can be introduced in traditional optical communications. And the mode-division multiplexing can be realized simultaneously with other traditional multiplexing domains such as wavelength multiplexing and polarization multiplexing. Hence the optical transmission capacity could be improved by multiple orders of magnitude. Huang et al. have realized the 100 Tbit/s free-space data transmission by employing three-dimensional multiplexing of polarization, wavelength and OAM .
In the receiver side of a multiplexing optical vortices transmission system, it is necessary to verify which OAM modes they contain and the OAM spectrum. Researchers have done a lot in this aspect. The cascaded Mach-Zehnder interferometer with Dove prisms can separate OAM modes according to the parity of their topological charge [18–20 ]. A method called OAM modes sorter is reported in recent years [21–23 ]. It is made up of a transforming and a phase-correcting optical element, which can focus each input OAM mode to a different lateral position. By measuring the intensity of each OAM mode through a stop and a power meter, the OAM spectrum of the multiplexing optical vortices can be obtained.
In this paper, inspired by the equal intensity distribution of Dammann vortex grating [24,25 ], we experimentally demonstrate a new method that using a binary Dammann vortex grating reported in reference  and the gray-scale algorithm we report to evaluate OAM spectrum in the receiver side of optical vortices communication system. We first generate multiplexing optical vortices consisting of single OAM modes with different topological charge and different intensity proportion. Then the gray-scale algorithm we design is utilized to analyze the gray image of the far-field diffraction pattern when multiplexing optical vortices propagate through Dammann vortex gratings. The output of the gray-scale algorithm is the measured OAM spectrum. In the experiment, the OAM spectrum of at most six-channel multiplexing is obtained. The experimental results fit well with the theoretical value. The availability, limitations, dynamic range and other parameters of this approach are discussed in this paper too.
2. Basic principle
Supposing that a single OAM mode with topological charge l is a part of a beam of multiplexing optical vortices. When the multiplexing beams propagate through a spiral phase plate (SPP) with order –l, there will emerge a bright spot at the center of beams. The bright spot’s power is that of the l-th order OAM mode in the multiplexing beams. After measuring each OAM mode’s power by using different SPP, the OAM spectrum of the multiplexing optical vortices can be verified. To make the measurement more convenient, we introduce a Dammann vortex grating and the gray-scale algorithm we design in the experiment. The adding of Dammann vortex grating allows us to analyze multiple spots simultaneously. And the gray-scale algorithm we design can calculate the OAM spectrum automatically.
The concept of the OAM spectrum measurement of multiplexing optical vortices is shown in Fig. 1 . A beam of multiplexing optical vortices is incident into a binary Dammann vortex grating. The far-field diffraction pattern is observed by a CCD camera. The gray-scale algorithm is used to analyze the received pattern, which are the main and the most important part in this technique.
Figure 2 illustrates the flow chart of the gray-scale algorithm. The first step of the algorithm is to determine which single OAM mode is existed in the multiple optical vortices. In other words, this step is to find where the bright spot emerged in the center of the beams in the far-field diffraction array. We can set a stationary gray value. If the gray we measured in the center of the beam, is lower than the stationary value, the single OAM mode with the topological charge shown in this location can be considered to be absent. In our experiment, the stationary gray value is set as 25 (the input image is a 256 order gray image). The second step is only for each single OAM mode that has been proved presented in the multiple beams, which is to calculate the sum of each pixel’s gray value of the center bright spot. It is easy to understand that the sum of every pixel’s gray value of the bright spot received by the camera is proportional to the real intensity of the spot when the received power is lower than the camera’s threshold. So if we can get the sum of gray value of each bright spot, the intensity proportion of single OAM modes in different order can be obtained. The last step is to compute the gray value’s proportion of all the single OAM modes in present. Then the OAM spectrum measurement is completed.
3. The experimental setup
The experimental setup for the measurement of OAM spectrum of multiplexing optical vortices is sketched in Fig. 3 .
The fundamental Gaussian mode whose wavelength is 1550 nm is emitted from a laser diode (LD) and is coupled into a single mode fiber. Then it is collimated and transformed into horizontal linearly polarized Gaussian beam by a polarized beam splitter (PBS). The reason why adding a PBS here is because the liquid-crystal spatial light modulator (SLM) we use can realize pure-phase modulate only for horizontal linearly polarized beams. The beam expander (BE), which is consist of a concave lens with focal length −50 mm and a convex lens with focal length 100 mm, is used to realize the double expansion of the OAM beam. The SLM has a resolution of 1920 × 1080 pixels and the pixel pitch of 8.0 μm (Holoeye). We upload the designed hologram of SPP on SLM1 to generate multiplexing optical vortices with specific intensity proportion of single OAM mode. SLM2 is used to upload the hologram of binary Dammann vortex grating, with a grating constant of 0.32 mm. The convex lens can realize the optical field transformation. The infrared CCD camera with the spectrum range of 900 nm~1700 nm is placed at the image focal plane of the convex lens and is used to observe the far-field diffraction pattern diffracted by the Dammann vortex grating.
4. Measuring OAM spectrum
As we introduced previously, we use Dammann vortex grating here is because of its unique characteristic of equal intensity distribution in different diffraction order. However, in experiments some other factors, for instance, the resolution of SLM can’t satisfy the giant phase jump of Dammann vortex grating, may cause the unequal intensity distribution. And it will lead to the inaccuracy of the measurement. Hence, we first need to measure the intensity distribution of different diffraction order, and then use it to compensate the measured OAM spectrum.
We upload a pure black screen on SLM1, to obtain the far-field diffraction patterns of Dammann vortex grating when Gaussian beams propagate through it, as shown in Fig. 4(a) . Figure 4(a) also illustrates the location corresponding relationship between the diffraction order of Dammann vortex grating and the OAM states it can detect. We figure out the sum of every pixel of each diffraction order’s spot. After the normalized calculation, the intensity spectrum of each diffraction order can be obtained, which is displayed in Fig. 4(b).
We first generate a beam of multiplexing optical vortices consisting of 4th order and −4th order single OAM modes with intensity proportion of 1:2. The SPP we upload on SLM1 and the observed multiplexing optical vortices is shown in Figs. 5(a) and 5(b) , respectively. Figure 5(c) illustrates the RGB and the gray image of the patterns received by the CCD camera after uploading Dammann vortex grating on SLM2. One important thing to notice here is that the image processing is only for the gray image of the diffraction pattern. We present the RGB image here is to show the diffraction pattern more clearly. In the image processing of Fig. 5(c) through gray-scale algorithm, the collection area of the central bright spot is a circle with diameter of 30 pixels, as shown in Fig. 5(d). The basic principle of the diameter’s selection is that the selection area should include the entire center bright spot exactly, and shouldn’t contain any other peripheral spots. Of course, the diameter’s selection can be changed with the change of optical path, camera’s resolution and any other factor. But as for a fixed measurement system, this diameter should be stationary and must obey the basic selection principle as described previously. The output OAM spectrum is shown in Fig. 5(e), where the blue, green and yellow bar denote the practically measured result without the compensation of intensity spectrum, with the compensation and the theoretical value.
We also generate three-channel, four-channel and six-channel multiplexing optical vortices by uploading different SPPs on SLM1. Then utilizing the gray-scale algorithm to compute the OAM spectrum. The diameter selection of the collection area in the algorithm is the same as the situation of two-channel beams, for we didn’t adjust any element of the optical path. The measurement results are displayed in Fig. 6 .
From Fig. 5 and Fig. 6 we can find that our method can realize the measurement of OAM spectrum in the receiver side of multiplexing optical vortices communications. The measurement results of two-channel, three-channel, four-channel and six-channel fit well with the theoretical spectra. The relative error of six-channel measurement is a little lager compared with the other conditions in our experiment. The reason can be understood as follows. The difference value of the adjacent diffraction order’s topological charge of Dammann vortex grating we use in the experiment is 1. With more and more channels getting filled, the mode discrimination became more and more difficult because of the narrow mode spatial separation. And it will contribute to the reduction of the measuring accuracy of each OAM mode’s power (the sum of each pixel’s gray value).
We make a simulation to illustrate the relationship between the measuring accuracy and the mode spacing. We use a Laguerre-Gaussian mode as an optical vortex in the simulation. Figures 7(a)–7(e) are five different multiplexing optical vortices. They both consist of two OAM channels with equal intensity. One is LG01 mode, the others are LG02 mode, LG03 mode, LG04 mode, LG05 mode and LG06 mode, respectively. The mode spacing of beams shown in Figs. 7(a)–7(e) is 1, 2, 3, 4 and 5. When they propagate through a −1st order SPP, the far-field diffraction pattern is shown in Figs. 7(f)–7(J). Based on the gray-scale algorithm we design and the selection principle of collecting area of the central spot, we calculate the intensity proportion between single LG01 mode and the multiplexing optical vortices. The line chart of the calculation result is displayed in Fig. 7(k), which has shown the relative measurement error decreases with the increasing of channel spacing. The relative error is 0.39% when channel spacing is 5. The larger the difference of adjacent channels’ topological charge is, the more accurate the measuring result will be.
Figure 7 also means that the approach we demonstrate is invalid for multiplexing optical vortices with smaller mode spacing. However, it has little influence on the application in multiplexing optical vortices communications. In the OAM multiplexing optical communication system, the mode spacing is usually larger to reduce the crosstalk between different channels and decrease the bit error rate (BER) . In addition, the large mode spacing is also conducive to the de-multiplexing of OAM beams .
There may be one doubt that how many channel multiplexing can our approach measure the OAM spectrum exactly. In the experiment, we accomplish the OAM spectrum measurement of at most six-channel multiplexing beams. As for the Dammann vortex grating we use in the experiment, the OAM spectrum measurement of more than six channel multiplexing is inaccurate. But if we use some other kinds of Dammann vortex grating under special design, such as the grating given in reference  and , the number of detectable OAM channel can be extended. We can also design the Dammann grating according to the OAM composition of the beams received by the receiver side. For instance, if the received beams consist of −30th, −25th, −20th, −10th, −5th, + 5th, + 10th, + 20th, + 25th and + 30th single OAM beams, we can design a 3 × 5 binary Dammann vortex grating. And the difference value of adjacent diffraction orders’ topological charge is set as 5, as shown in Fig. 8(a) . The far-field diffraction pattern when Gaussian beams propagate through it is displayed in Fig. 8(b). Figure 8(c) illustrates the values of the topological charges at different diffraction orders. In this case, our technique can realize the OAM spectrum measurement of at most fifteen-channel multiplexing beams.
Our approach is utilizing the gray value of the spots to measure the OAM spectrum. So the scattered light and the thermal noise of the CCD camera may have influence on the measurement results. An aperture stop and the NDF in the optical path could filter out the scattered light outside. Reducing the grating constant can contribute to the larger distance between adjacent diffraction orders, which will eliminate scattered light from adjoining orders. The thermal noise of the camera is inevitable. When the camera is working for a long time, the measurement accuracy will decrease. At that time, cooling will be helpful to reduce the effects result from the camera’s thermal noise.
In this paper, we have demonstrated a new method that using the gray-scale algorithm and Dammann vortex grating to accomplish the measurement of OAM spectrum. The gray-scale algorithm we design can be used to obtain the intensity proportion of different OAM modes without using power meter. In our experiment, the OAM spectra of at most six-channel multiplexing beams are obtained. But it is not the biggest detection range of our method. By replacing the grating in our experiment by a specially designed grating, the detectable number of OAM channels can be extended. The operability and the limitations of our approach are also discussed.
The biggest advantage of our method is the simple operation in practice. The most complex part of this technique, the gray-scale algorithm, can be done entirely by the image processing program. And the processing program can be developed entirely by MATLAB and some other software. What we should do is just to get the far-field diffraction patterns of Dammann vortex grating. Then input the gray-scale diffraction pattern in the algorithm. The output is the measured OAM spectrum.
We acknowledge the support of National Basic Research Program of China (973 Program) under contract of No. 2014CB340002 and No. 2014CB340004.
References and links
1. L. Allen, M. W. Beijersbergen, R. J. 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]
4. S. Fu, C. Gao, Y. Shi, K. Dai, L. Zhong, and S. Zhang, “Generating polarization vortices by using helical beams and a Twyman Green interferometer,” Opt. Lett. 40(8), 1775–1778 (2015). [CrossRef] [PubMed]
6. J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum Correlations in Optical Angle-Orbital Angular Momentum Variables,” Science 329(5992), 662–665 (2010). [CrossRef] [PubMed]
7. T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997). [CrossRef]
8. X. Xu, K. Kim, W. Jhe, and N. Kwon, “Efficient optical guiding of trapped cold atoms by a hollow laser beam,” Phys. Rev. A 63(6), 063401 (2001). [CrossRef]
9. A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
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. M. J. Willner, H. Huang, N. Ahmed, G. Xie, Y. Ren, Y. Yan, M. P. J. Lavery, M. J. Padgett, M. Tur, and A. E. Willner, “Reconfigurable orbital angular momentum and polarization manipulation of 100 Gbit/s QPSK data channels,” Opt. Lett. 38(24), 5240–5243 (2013). [CrossRef] [PubMed]
16. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011). [CrossRef]
17. H. Huang, G. Xie, Y. Yan, N. Ahmed, Y. Ren, Y. Yue, D. Rogawski, M. J. Willner, B. I. Erkmen, K. M. Birnbaum, S. J. Dolinar, M. P. J. Lavery, M. J. Padgett, M. Tur, and A. E. Willner, “100 Tbit/s free-space data link enabled by three-dimensional multiplexing of orbital angular momentum, polarization, and wavelength,” Opt. Lett. 39(2), 197–200 (2014). [CrossRef] [PubMed]
19. X. Qi and C. Gao, “Experimental study of detecting orbital angular momentum states of spiral phase beams,” Wuli Xuebao 60(1), 014208 (2011).
20. C. Gao, X. Qi, Y. Liu, J. Xin, and L. Wang, “Sorting and detecting orbital angular momentum states by using a Dove prism embedded Mach-Zehnder interferometer and amplitude gratings,” Opt. Commun. 284(1), 48–51 (2011). [CrossRef]
21. G. C. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105(15), 153601 (2010). [CrossRef] [PubMed]
22. M. P. J. Lavery, D. J. Robertson, G. C. G. Berkhout, G. D. Love, M. J. Padgett, and J. Courtial, “Refractive elements for the measurement of the orbital angular momentum of a single photon,” Opt. Express 20(3), 2110–2115 (2012). [CrossRef] [PubMed]
23. M. P. J. Lavery, G. C. G. Berkbout, J. Courtial, and M. J. Padgett, “Measurement of the light orbital angular momentum spectrum using an optical geometric transformation,” J. Opt. 13(6), 064006 (2011). [CrossRef]
26. A. J. Willner, Y. Ren, G. Xie, Z. Zhao, Y. Cao, L. Li, N. Ahmed, Z. Wang, Y. Yan, P. Liao, C. Liu, M. Mirhosseini, R. W. Boyd, M. Tur, and A. E. Willner, “Experimental demonstration of 20 Gbit/s data encoding and 2 ns channel hopping using orbital angular momentum modes,” Opt. Lett. 40(24), 5810–5813 (2015). [CrossRef] [PubMed]
27. S. Fu, T. Wang, S. Zhang, and C. Gao, “Integrating 5 × 5 Dammann gratings to detect orbital angular momentum states of beams with the range of −24 to +24,” Appl. Opt. 55(7), 1514–1517 (2016). [CrossRef]