Encoding and decoding of orbital angular momentum for wireless optical interconnects on chip

Beams carried orbital angular momentum (OAM) are proposed for wireless optical interconnects on chip and a full scheme of encoding and decoding of OAM at single frequency is demonstrated with numerical simulation. With proposed structure, beams with OAM order of −3 to 4 are generated and four orders of them (0 to 3) are used to encode and decode data so that the increased data density of two folds is achieved. According to such results, we believe that if OAM is utilized as an additional dimension in wireless optical interconnects, the data density can be significantly increased since the adopted orders of OAM could be infinite in principle. Moreover, such improvement could be easily applied to the existing architecture without any more complex technology. ©2012 Optical Society of America OCIS codes: (130.3120) Integrated optics devices; (130.4110) Modulators; (060.4510) Optical communications. References and links 1. L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. 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Introduction
Since the pioneering work of Allen et.al, the optical orbital angular momentum (OAM) has stimulated much interest [1][2][3]. Owing to unique characteristics of spiral flow of electromagnetic energy and helical wave-front [4,5], OAM could be applied on optical tweezers and spanner [6][7][8], light microscopy [9,10], quantum and wireless communications [11][12][13], etc. Specifically, the OAM is very attractive for data transmission due to the nature of unlimited number of potential states so that OAM beams can carry an infinite amount of data in principle [11,14]. It has been demonstrated that the wireless communications with OAM multiplexing can work in both optical and radio frequency domains with higher data rate and capacity [15][16][17][18].
As the development of CMOS technology, it becomes urgent to increase the transmission data rate in computer systems [19]. Among the proposed solutions, optical interconnect on chip is considered as the most promising one [20][21][22]. In 2010, A. Alu et al. proposed that wireless optical interconnects could be employed to obtain higher performances for communication on chip [23]. However, in the reported work of the wireless optical interconnects, only plane or spherical wave is considered. As mentioned above, if OAM could be employed for such systems, the transmission data rate could be dramatically increased and as a result, the power consumption per bit could be dramatically reduced.
In this work, aiming to exploit a new dimension for wireless optical interconnects, optical OAM is proposed for wireless optical links within layers or chips so that the data density can be increased. As the first step, encoding and decoding OAM on optical light with integrated structure on silicon chip are demonstrated with numerical simulation. The optical OAM is generated by a ring cavity with one input waveguide, in which the OAM order only depends on the mode order of ring. Therefore, optical OAM can be encoded by electro-optic tuning of the mode order of ring cavity at a certain wavelength with high speed. Moreover, the encoded OAM beam can be easily decoded using gratings and Mach-Zehnder interferometers. Based on such a coding approach with multi-nary symbol, the data density can be significantly increased by taking advantage of an additional dimension of OAM beams. Fig. 1. The proposed structure with one ring, one bus waveguide and eight download units and each download unit is constituted by an arc waveguide and a grating. Two strips around the ring represent p-doped and n-doped regions for electrical tuning of the ring. The inset shows the profile of the output grating. Figure 1 is the schematic of the proposed structure for producing and encoding optical OAM on chip. There are one ring cavity, one bus waveguide, and eight download units (each including an arc waveguide and a grating). The eight download units are equally coupled to the ring and equidistantly distributed around the circle, i.e. eight-fold rotationally symmetrical location. If the central frequency of input light, which is injected from the input-port of bus, deviates from the resonance of ring cavity, the light would pass through the thru-port of bus waveguide. On the contrary, if input light satisfies the resonance condition of ring cavity, it would be coupled into the ring and propagates along the ring as whispering-gallery-mode (WGM) and downloaded by each of the eight arc waveguides. Finally, the lights propagating in all arc waveguides would be combined and transformed into free space light (or travels in homogenous and isotropic medium) at the end of gratings. Since the phase shift of propagating light at each downloaded unit is successively varied, the phase of output light beam superposed from gratings would be azimuthally dependent. In other words, an OAM beam with azimuthal phase dependence of exp( ) ilϕ could be generated, which is similar to the principle reported in Ref [24,25].

Generating beams with OAM on chip
Specifically for the resonant light, the circumference of the ring ( L ) and propagation constant of light ( β ) satisfy the resonant relation: where n is the WGM order of light in the ring. The phase shift of a circle is 2nπ along the ring so that the phase difference between each download unit is ( ) 2 /8 n π ⋅ and would be carried by light beam in free space. Here, considering the periodicity of phase shift in optical interference, the n can be rewritten as where N is the number of download units, p is a positive integer, and m is expressed as modulo N with an integer value from 0 to 1 N − . Here, N is considered as 8 and then m can be taken as an integer from 0 to 7. Thus the effective phase difference ( eff φ Δ ) between any two adjacent download units is ( ) For further understanding of the produced OAM beam, three dimensional finite difference time domain (3D-FDTD) method is employed to calculate the near and far electromagnetic field distributions of the structure shown in Fig. 1. Such structure is assumed to be fabricated on silicon-on-insulator (SOI) substrate with top cladding silicon dioxide layer. The thickness and refractive indices of silicon/silicon dioxide layers are 340-nm/3-μm and 3.45/1.46, respectively. The radiuses of the ring and all arcs are 16 μm and 5 μm, respectively. To achieve the critical coupling state of ring cavity for obtaining maximum intensity of the WGM light, the distance between the bus and ring as well as those between arcs and ring should be carefully designed. Here, the optimized distances (edge to edge) are calculated as 20 nm and 60 nm, respectively. The width of all waveguides (ring, bus, and arcs) is 500 nm while that of grating is gradually increased to 2 μm within the length of 7 μm. All gratings are designed as the height of 100 nm, eight cycles with pitch of 500 nm and duty-cycle of 50%. For 0 m = the near field distribution is a series of concentric circles (Fig. 2(b1)) while the phase difference is nearly zero along each circle in the far field phase distribution ( Fig. 2(b2)) so that it corresponds to 0 l = .

Encoding and decoding data with OAM beam on the single frequency carrier
In order to convey information with OAM states, it is required to modulate different orders of OAM on optical carrier at a certain wavelength. With the proposed structure, different OAM orders could be encoded by varying the WGM order of ring cavity following Eq. (3). Furthermore, the decoding can be done by receiving elements such as optical gratings or nano-antennas, which relies on detecting phase information of beams with OAM at different azimuthal distributions as Ref [26][27][28]. In principle, N types of OAM of beams can be produced by N arcs and could be distinguished by N receiving elements. However, due to the beam spreading of different order of OAM, only half or less of the total number of types can be clearly and easily distinguished [26]. So only quaternary encoding and decoding ( l = 0, 1, 2, and 3) are considered although there are eight download units. As shown in Fig.  3, OAM is encoded on light beam at layer1 and decoded by superposition of lights from four receiving elements at layer2 which are rotationally symmetric on a circle. , , , f f f f ) among received lights and exported to two ports in binary format.
According to Eq. (1), the WGM order at a certain frequency could be adjusted by changing β , which could be easily achieved by varying the refractive index of Si ( Si n ). As demonstrated for silicon optical intensity modulator, the Si n could be varied by carrier injection or depletion with p-n junctions with very high speed [29,30]. As shown in Fig. 1, we assume that there is a proper p-n junction around the ring cavity for varying Si n , which are similar to the structure of silicon ring modulator [29]. Then via properly adjusting the refractive index, the WGM order at a certain optical frequency could be tuned. As an example, the WGM order at frequency of 194.68 THz is 368 ( 0 m = ) while it could be tuned to 369 ( 1 m = ) or 367 ( 7 m = ) with index variation ( Si n Δ ) of 0.0116 or −0.0116 and the OAM order is tuned from 0 l = to 1 or −1. The optical transmission spectra at the thru port of bus waveguide with varied Si n Δ are shown in Fig. 4(a) and the corresponding near field patterns are shown in Figs. 4(b-d). Similarly, higher order of OAM also can be obtained by increasing is determined by the applied voltage on p-n junction, the quantized voltage signal can be encoded to different order of OAM with our proposed structure. Specifically for structure shown as layer1 in Fig. 3, the input binary signals are transformed to quantized quaternary voltage signals in electric domain and applied on p-n junctions so that four different orders of OAM beams are consequently generated. Comparing to encoding data using OAM at different wavelengths, such approach is more efficient in bandwidth utilization and can increase the data rate due to multi-nary coding assisted by the additional dimension of OAM. Decoding signals from beams with OAM are achieved by two steps: free-space light is received and transformed to multi-guided lights using gratings, and then signals are recovered by superposition of the multi-guided lights. Here, we assume that the input binary signals of gratings with the same structure parameters as the gratings in the layer 1. Thus, the encoded free-space light with OAM is received and transferred to four-guided lights by four gratings. Although the phase shift of received light of each grating is time-varied, there is still a stable phase difference of 2 /4 l π between any two adjacent gratings so that the binary signals could be recovered by coherent superposition among the four-guided lights, which could be achieved by Mach-Zehnder interferometers (MZIs) or arrayed-waveguide gratings [31]. Since each OAM state represents a quaternary number (i.e. two binary numbers), the recovered signals will be exported to two binary outputs at each OAM state using these four-guided lights. The procedure of recovering signals is introduced with more details in Appendix-A. Here, with weighting factor depicted by Eq. (5) in Appendix-A, the relationships between outputs of two ports and OAM order versus Si n Δ is shown in Fig. 4(f). It could be seen that OAM order of 0~3 are encoded by varying Si n and the signals are successfully recovered and exported to binary optical signals at port1 and port2. It should be mentioned that the free carrier absorption of electrical tuning is not considered in our simulation. The required variation of Si n Δ is as large as 0.035 so that free carrier absorption introduced by electrical injection should be taken into account. For a practical device, one possible solution is to adopt a ring with much larger radius so that the requirement of Si n Δ could be small enough. With such quaternary coding, the transmitted data rate on chip could be increased two folds (deduced from 2 log 4 for quaternary signal) and the power consumption per bit could be reduced by half in succession. Obviously, if higher multi-ary coding is employed, data rate could be dramatically increased as 2 log N folds for N -nary signal, and more details can be found in Appendix-B. Here, only a simple case of quaternary coding is considered in order to demonstrate the operating principle more clearly. Actually, there is no physical limit to achieve higher order OAM coding with our proposed structure. Furthermore, it should be mentioned that the proposed structure could be easily applied on the existing optical interconnects architecture and what should be done is replacing the intensity modulator with the encoder and adding a decoder before the photodetectors.
It is worth pointing out that such proposed structure is not only potential for wireless optical interconnects, moreover, but also can serve as OAM generator or OAM modulator for free-space optical communications, quantum communications, and optical manipulations. Among most of these applications, if the tuning speed is not very critical, thermal tuning can also be adopted to switch from several to dozens of OAM states since silicon refractive index could be varied within a rather wide range with thermo-optic effect.

Summary
A full solution of adopting optical OAM for wireless optical interconnects on chip is proposed, which includes generation of OAM beams and encoding/decoding data on OAM. With the help of numerical simulation, we found that taking full advantage of the OAM correctly can increase the data density substantially in the wireless optical interconnections on chip. Furthermore, the fabrication of the proposed structure is compatible with CMOS technology and the existing optical interconnects architecture. And we believe that the integrated OAM generator or modulator also can be applied for free-space optical communications, quantum communications, and optical manipulations with distinguished feature.
phase shifter before the superposition using Mach-Zehnder interferometers or arrayedwaveguide gratings. Such process for decoding and derivation of weighting factors can be easily extended to more orders of OAM of beams. The data processing procedures for normal binary transmission system and our proposed system are depicted in Fig. 6(a) and 6(b), respectively. For such two systems, the symbol rate ( BN R ) are supposed to be the same since BN R is only determined by speed of modulation. For a normal system, there are an intensity modulator (such as integrated microring or Mach-Zehnder modulator) and a photodetector. The input binary signals are directly encoded, thus the bit rate ( b R ) is equal to the symbol rate ( BN R ). For our proposed one, the input binary signals are firstly transformed to quaternary signals in electric domain and then are encoded. Thus, b R is equal to ( ) 2 log 4 * BN R due to modulation of quaternary signals, namely the transmission date rate is increased two folds. Similarly, for coding with N -nary signals, data rate could be increased 2 log N folds.