Reconfigurable vortex beam generator based on the Fourier transformation principle
Aiping Liu, Chang-Ling Zou, Xifeng Ren, Wen He, Mengze Wu, Guangcan Guo, Qin Wang
aa r X i v : . [ phy s i c s . op ti c s ] A p r Reconfigurable vortex beam generator based on the Fourier transformationprinciple
Aiping Liu, Chang-Ling Zou, a) Xifeng Ren, Wen He, Mengze Wu, Guangcan Guo, and Qin Wang Affiliation: Institute of quantum information and technology, Nanjing University of Posts and Telecommunications,Nanjing 210003, China, and Key Lab of Broadband Wireless Communication and Sensor Network Technology,Nanjing University of Posts and Telecommunications, Ministry of Education, Nanjing 210003,China Affiliation: Key Laboratory of Quantum Information, University of Science and Technology of China, CAS,Hefei, 230026, China, and Synergetic Innovation Center of Quantum Information and Quantum Physics,University of Science and Technology of China, Hefei, 230026, China
A method to generate the optical vortex beam with arbitrary superposition of different orders of orbitalangular momentum (OAM) on a photonic chip is proposed. The distributed Fourier holographic gratingsare proposed to convert the propagating wave in waveguides to a vortex beam in the free space, and thecomponents of different OAMs can be controlled by the amplitude and phases of on-chip incident light basedon the principle of Fourier transformation. As an example, we studied a typical device composed of nineFourier holographic gratings on fan-shaped waveguides. By scalar diffraction calculation, the OAM of theoptical beam from the reconfigurable vortex beam generator can be controlled on-demand from − nd to 2 nd by adjusting the phase of input light fields, which is demonstrated numerically with the fidelity of generatedoptical vortex beam above 0.69. The working bandwidth of the Fourier holographic grating is about 80 nmwith a fidelity above 0.6. Our work provides an feasible method to manipulate the vortex beam or detectarbitrary superposition of OAMs, which can be used in integrated photonics structures for optical trapping,signal processing, and imaging. I. INTRODUCTION
Orbital angular momentum (OAM) of a single photonhas inherent infinite and orthogonal dimension for encod-ing quantum information, which is proposed to be a fas-cinating area of research since 1992. Because of the un-bounded dimensions of the OAM, it is possible to encodea single photon by OAM in a high dimensional space.
The potential application of OAM in quantum attracta lot of interest. Efficient OAM generation,
OAM en-tanglement storage, OAM data transmission and OAMdetection have been realized recently.As the real-time manipulation of the OAM com-ponents of a vortex beam is necessary for variousapplications, it is desirable for a re-configurable vor-tex beam generator. Among various experiment plat-forms for vortex beam generation, the photonic chip ap-proach is of great interest because of its high stability andscalability. In order to be compatible with presentphotonic integrated circuit fabrication technology, meth-ods of generating OAM based on the light scattering bygratings in the cavities can be converted to free spacebeam with non-zero OAM.
Alternatively, the holo-graphic method has been proposed, however, the order ofOAM obtained by holography is determined by the geom-etry of the holographic grating, and is not reconfigurablewhen for a given holography. It is proposed that byputting two holography grating on two separate waveg-uides, the generated beam profile can be controlled by ad-justing the input lights in two waveguides, which points a) Electronic mail: [email protected] out a promising approach for reconfigurable vortex beamgeneration. Recently, the waveguide-based holographygrating has been demonstrated, representing the firststep towards reconfigurable vortex beam generation anddetection.In this work, a reconfigurable vortex beam generator(RVBG) on a photonic chip is proposed, which is basedon the Fourier holographic gratings connected to an ar-ray of waveguides. Arbitrary superposition of OAM in anoptical vortex beam can be generated by controlling thephase and amplitude of lights in the waveguides, or be de-tected as a reversal process. As an example, we show theprocedure to construct the RVBG by nine Fourier holo-graphic gratings on nine fan-shaped waveguides ranged ina disk. By numerical simulation, we demonstrated thatthe vortex beam with OAM from − nd to 2 nd can beselectively generated with high fidelity and large wave-length bandwidth. Besides of the vortex beam with sin-gle OAM, vortex beams with a superposition of differentOAMs can also obtained by the proposed device with ap-propriated incident light phases and amplitudes in waveg-uides. II. PRINCIPLE
As shown in Fig. 1(a), the proposed RVBG is com-posed of nine fan-shaped waveguides ranged in a disk,where the holographic gratings are fabricated on the topof the waveguides near the center of the disk. In thisRVBG, the propagating waves in the fan-shaped waveg-uides (in plane) will be scattered by the holographic grat-ing, and the scattered light form the vortex beam in freespace (in vertical direction) together. What is more, thegenerated beam is reconfigurable that the componentsof different OAMs can be manipulated by controlling theamplitude and phase of the input lights, because the vor-tex beam generated by the holographic grating is basedon the principle of Fourier transformation.The basic principle of the Fourier holographic grat-ing can be explained as following: the holographic grat-ings on each fan-shaped waveguide can generate a su-perposition of OAM beams. For (2 N + 1) -fan-shapedwaveguide components, based on the Fourier principle,the j th ( j ∈ [0 , N ]) waveguide can generate a beam as asuperposition of OAMs E b,j ( r, φ ) = A j N X l = − N F ( l, r ) e ilφ e ij ( l + N ) π N +1 , (1)where ( r, φ ) are the cylindrical coordinator, A j is theamplitude of the input light in j th waveguide, F ( l, r ) isthe radial field distribution of vortex beam with OAM of l th order ( l ∈ [ − N, N ]). Therefore, when the input lightsto the waveguides are with equal intensity but with aphase gradient of n ∆(∆ = π N +1 ), i.e. A j = Ae − ijn ∆ , (2)then the generated beam should be E ( r, φ ) = X j E b,j ( r, φ ) = A · F ( n, r ) e inφ . (3)It means that only the n th OAM can be generated asthe constructive interference between the scattered lightfrom all gratings, while the component of other OAMare suppressed due to the destructive interference. Inreversal, if l th OAM beam shine on the fans, then theinput light can be converted to all waveguide modes, butwith a phase gradient of l ∆.By controlling the amplitude and phases of the inputlight, we can generate the arbitrary superposition of theOAM beams. If the input photon in the superposition ofdifferent paths be represented by a vector | Φ i in = { A j } T ,the output of RVBG is | Φ i out = U F | Φ i in , (4)with U F is the unitary matrix for Fourier transformation, | Φ i out is defined on the basis of OAMs.If we prepare a multimode interferometer (MMI) for inverse Fourier transformation on the same chip andguide all the output of the waveguides to the MMI asshown in Fig. 1(b), then | Φ i out = U F U + F | Φ i in = | Φ i in , (5)we can utilize the Fourier transformation principle againto convert the superposition of OAMs inputs to super-position of light outputs in different paths, i.e. the in-put free-space photon with l th OAM be converted to light output in l th port. In reverse, the input light from l th waveguide can be converted to vortex beam of l th OAM. Generally, the photonic integrated circuits allowsthe real-time control of the phase, combining with the ar-bitrary unitary evolution in linear optics, reconfigurablewaveguide mode converter can be realized.
There-fore, arbitrary conversion between the waveguide pathsto the OAMs can be realized if the reconfigurable modeconverter U R = U † F U with U be the demanded arbitraryunitary, i.e. | Φ i out = U F U R | Φ i in = U | Φ i in . (6)Reversely, we have | Φ i in = U † | Φ i out , the vortex beaminput to the RVBG can be converted to the arbitrarysuperposition of output in the waveguide, could be usedfor detecting and analyzing the OAM components of thevortex beam.Here, we also provide the detailed procedure to calcu-late the fan-shaped waveguide grating. For the waveg-uide h j ( j = 0 , ... h j on the center ofthe round, while the other eight waveguides are shieldedfrom the target vortex beam by an opaque as shown inFig. 1(c). At the same time, a guiding wave propagatesin the waveguide from outside to inside of the fan, whichwill interfere with the combination of vortex beam. Theguiding wave in h j can be given as A j = A ∗ e − ik j r , where k j is the wave vector. The combination of vortex beamto form the holographic grating on the j th waveguide isa superposition of the target optical vortex beams withdifferent orders and phase, which has the electric fieldform as E b,j ( r, φ ) = A P l = − F ( l, r ) e ilφ × e ijl ∆ . For theRVBG, the phase difference of different vortex compo-nents changes according to the Fourier transformation,which guarantees that the output of nine holographicgratings are orthogonal to each other. The interference ofthe waves from two directions lead to string with blackand bright patterns on the surface of the waveguides.The interference pattern on each waveguide is formedseparately. The Fourier holographic grating is gener-ated by dividing the interference region into many pix-els (60 nm × nm ) and setting a gray value of G ( x, y )for each pixel. According to the phase difference δθ between the target vortex beam and the guiding wave,the simple binary function is applied, i.e., G ( x, y ) = 0 if δθ < . π , or else G ( x, y ) = 1. The inset of Fig. 1(c)gives the binary gray image of the Fourier holographicgrating. III. RESULTS
To verify the proposed Fourier holographic gratingscheme, we performed numerical simulation based on apractical experimental model. The Fourier holographicgrating is composed of Si N waveguide on a silica sub-strate, with the refractive index of Si N and silica as2.035 and 1.45 at the wavelength of 670 nm. The field (a) (c) (1)(2) (3) (11)(10) z rφx y (b) electriclight
FIG. 1. (color online). The schematic illustration of theFourier holographic grating. (a) The vortex beam generatedby nine incident wave couple to the fan-shaped holographicgratings on a chip. (b) The generation of vortex beam withan arbitrary superposition of OAMs by combining holographicgratings and arbitrary unitary mode converters. (c) The pre-cedure for preparating the Fourier holographic gratings. Thecombination of target vortex beam and the guiding wave inter-fere to form the Fourier holographic grating on the h j waveg-uide with the other waveguides shielded from the combinationof target vortex beam. The inset of (11) is the binary grayimage of the Fourier holographic grating. spatial distribution of generated beam is numerically cal-culated by the scalar diffraction theory.In the following, we do not consider the waveguidemode converter, just simply assume we can have ar-bitrary input to the waveguide. Due to the principleof holography, when the RVBG is constructed by ninegratings ( N = 4 , ∆ = 2 π/ A j e − iθ j , where θ j is the phase lightincident on the guiding wave. According to the principleof holography, the electric field of the generated light is E b ( r, φ ) = P j =0 E b,j ( r, φ ), which can be estimated as E b ( r, φ ) ∝ X l = − F ( l, r ) e ilφ ( X j =0 A j e ij ( l +4)∆ ) , (7)leads to a superposition of OAMs. The scattered lightstogether constitute the vortex beam with appropriate θ j ,as shown in Fig. 2, in which the result is obtained onthe plane with a distance of 10 λ from the up surface ofthe waveguide. The waist of the target vortex beam isposited on the top surface of the waveguide with a diam-eter of 1 µ m. When θ j = 2∆, an optical vortex beamwith − th OAM is obtained, which has a donut-shapedamplitude distribution [Fig. 2(a2)] and a helical phase of − π with a singular in the center [Fig. 2(a4)]. When θ j is changed to be θ j = 3∆, the order l of the optical vortexbeam turns into -1, in which there is amplitude distribu-tion deviated from the donut-shape [Fig. 2(b2)] and a he-lical phase of − π for the phase distribution [Fig. 2(b4)].A Gauss beam is obtained with θ j = 4∆ as shown inFig.2(c2) and 2(c4), where the donut-shaped amplitude l = l =- l =- l = l = π φ FIG. 2. (color online). The amplitude and phase distributionsof the optical vortex beams: (a1)-(e1) and (a2)-(e2) are theamplitude distributions for the target and generated opticalvortex beams, respectively. (a3)-(e3) and (a4)-(e4) are thephase distributions for the target and generated optical vortexbeams, respectively. The columns from left to right are theresults of the optical vortex beams with l = − , − , , , and2, respectively. distribution and the helical phase disappear. When θ j isset to be 5∆, the generated optical vortex beam with 1 st OAM is obtained as shown in Fig.2(d2) and Fig.2(d4), inwhich there is an opposite helical phase of 2 π comparedto the case of − st OAM. For θ j = 6∆, an optical vortexbeam with 2 nd OAM is generated, which has a big donut-shaped amplitude distribution and a helical phase of 4 π as shown in Fig.2(e2) and 2(e4), respectively. The rela-tion between θ j and the order l of the generated OAMis given in Tab.1. With the Fourier holographic grating,the order l of the generated OAM can be manipulatedfreely by the phase difference θ j of wave incident on thewaveguides.To qualify the quality of the generated vortex beam,the fidelity is introduced as F = | R E ∗ ( x ) E t ( x ) dx | R | E ( x ) | dx R | E t ( x ) | dx , (8)where E ( x ) and E t ( x ) are the amplitudes of the gen-erated and target optical vortex beams, respectively. Be-cause of the finite number of imaging pixels, the differ-ence exists between the obtained beam and the targetoptical vortex beam, which makes the fidelity F < /
10 of the max-imum is selected to calculate the F . When the order l of OAM being manipulated between − | l | = 1 and 2, the generated vortex beam has rela-tive high fidelities of about 0.82 and 0.91, respectively.While | l | = 0, the generated vortex beam has a relativelylow fidelity, which is attributed to the blank core of thefan-shaped holographic grating. TABLE I. The order and the fidelity of the generated OAMfor different incident phase difference ∆ φ .∆ φ π/ π/ π/ π/ π/ l -2 -1 0 1 2 F With the Fourier holographic grating, the OAM com-ponents of generated optical vortex beam can be reconfig-ured from − nd to 2 nd order freely. Further, we studiedthe property of Fourier holographic grating working ondifferent wavelengths. The Fourier holographic grating isobtained with the interference of two waves at λ = 670nm in free space, and the light incident on the waveguideto generate the optical vortex beam is changed. Fig-ure 3(a) gives the fidelity of the generated OAM as afunction of the incident wavelength for the five differentorders. The highest fidelity is obtained near 670 nm asdesigned, and the fidelity reduces when the wavelengthof the light incident on the waveguide is far away from670 nm. From Fig. 3(a), it is shown that the Fourierholographic grating has a working bandwidth of 80 nm(from 640 nm to 720 nm) with the fidelities above 0.6 forall of the five orders. So the designed Fourier holographicgrating is practical for the information processing tasks.Besides of the vortex beam with single OAM order, theFourier holographic grating also can generate the beam ofarbitrary superposition of OAMs. When the light inputon the waveguides is a superposition of waves with addi-tional phase shift φ j to the j ∆, which can be representas A ′ wj = A ∗ e − ij ∆ − iφ j , (9)an optical beam consists of a superposition of OAMscan be obtained. For example, when φ = 8 π/ φ = 10 π/
9, a superposition of 0 and 1 st OAM is ob-tained as shown in Fig. 3(b). Figure 3(c) is the cor-responding target vortex beam with the intensity distri-bution on the left column and phase distribution on theright column. The fidelity of the generated vortex beamwith a superposition of 0 and 1 st OAM is about 0.5234.Setting φ = 10 π/ φ = 12 π/
9, a superposition of1 st and 2 nd OAM is obtained with a fidelity of 0.7792 asshown in Fig. 3(d). The superposition states of OAMwith arbitrary orders can be generated by changing thewave phase incident on the waveguides.The generation of vortex beam with arbitrary superpo-sition of OAMs broadens the application of OAM, sincequantum superposition plays a key role in quantum com-munication. The OAM also provide larger Hilbert spacefor encoding information by a single photon, could boostthe rate of key generation. Combine with the advantagesof the photonic integrated circuits, the ultra-fast recon-figurability and scalability allow the stable and compact . . .
600 640 680 720 760Wavelength(nm) F i d e lit y ( a . u . ) (a) (b)(c)(d)(e) FIG. 3. (color online) (a) The fidelity of the generated vortexbeam as a function of the working wavelength. Black (square),red (circular), blue (triangle), green (double cross) and yellow(star) lines are the fidelities of the vortex beam with the OAMbe l = − , − , , , and 2, respectively.(b) and (c) are thegenerated and target vortex beam with a superposition of l = 0 and 1. (d) and (e) are the generated and target vortexbeam with superposition of 1 st and 2 nd OAM. The left andright columns are the intensity and phase distributions of theelectric field, respectively. photonic chip for high-speed signal emitting, receivingand processing.
IV. CONCLUSION
A Fourier holographic grating is proposed to gener-ate the optical vortex beam with reconfigurable arbi-trary superposition of orbit angular momentum. Theswitching between the OAM orders of − , − , , , and 2is demonstrated numerically by controlling the phase ofincident light based on the Fourier transformation prin-ciple. This Fourier holographic grating can work with afidelity above 0.69 for all of the five orders and a work-ing bandwidth of 80 nm. A reconfigurable vortex beamgenerator makes the information processing based on theOAM encoding more feasible for practical applications,since our device posses the advantages as reconfigurable,stable, compact and scalable. V. ACKNOWLEDGEMENTS
This work was supported by the National Key R& D program (Grant No.2016YFA0301300), the Na-tional Natural Science Foundation of China (GrantNo. 11504183, 61505195, 61590932, 11774333, 61475197,11774180), the Anhui Initiative in Quantum InformationTechnologies (No. AHY130300), the Scientific ResearchFoundation of Nanjing University of Posts and Telecom-munications (NY214142), and the Fundamental ResearchFunds for the Central Universities. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerd-man, Orbital angular momentum of light and the transformationof Laguerre-Gaussian laser modes. Phys. Rev. A 45, 8185 (1992). A. Mair, A. Vaziri, G. Weihs, A. Zeilinger, Entanglement of theorbital angular momentum states of photons. Nature 412, 313(2001). X.-F. Ren, G.-P. Guo, B. Yu, J. Li, G.-C. Guo, The orbital an-gular momentum of down-converted photons. J. Opt. B: Quant.Semiclass. Opt. 6, 243 ( 2004). S. Yu, Potentials and challenges of using orbital angular momen-tum communications in optical interconnects. Opt. Express 23,3075 (2015). D. P. O’Dwyer, C. F. Phelan, Y. P. Rakovich, P. R. Eastham,J. G. Lunney, J. F. Donegan, Generation of continuously tun-able fractional optical orbital angular momentum using internalconical diffraction. Opt. Express 18, 16480 (2010). B. M. Heffernan, R. D. Niederriter, M. E. Siemens and J. T.Gopinath, Tunable higher-order orbital angular momentum usingpolarization-maintaining fiber. Opt. Lett. 42, 2683(2017). D.-S. Ding, W. Zhang, Z.-Y. Zhou, S.Shi, G.-Y. Xiang, X.-S.Wang, Y.-K.Jiang, B.-S. Shi and G.-C. Guo, Quantum Storageof Orbital Angular Momentum Entanglement in an Atomic En-semble. Phys. Rev. Lett. 114, 050502 (2015). J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, et al., Terabit free-space data transmission employing orbital angular momentummultiplexing. Nat. Photon. 6, 488( 2012). A.-P. Liu, X. Xiong, X.-F. Ren, Y.-J. Cai, et al., Detecting or-bital angular momentum through division-of-amplitude interfer-ence with a circular plasmonic lens. Sci. Rep. 3, 2402( 2013). R. Ionicioiu, Sorting quantum systems efficiently. Sci. Rep. 6,25356(2016). F.-X. Wang, W. Chen, Z.-Q. Yin, S. Wang, et al., Scalableorbital-angular-momentum sorting without destroying photonstates. Phys. Rev. A 94, 033847 (2016). X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, et al., Inte-grated Compact Optical Vortex Beam Emitters. Science 338, 363(2012). A. Liu, G. Rui, X. Ren, Q. Zhan, et al., Encoding photonicangular momentum information onto surface plasmon polaritonswith plasmonic lens. Opt. Express 20, 24151 ( 2012). H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Opticalparticle trapping with higher-order doughnut beams producedusing high efficiency computer generated holograms, J. Mod.Opt. 42, 217 (1995). C. T. Nadovich, W. D. Jemison, D. J. Kosciolek, and D. T.Crouse, Focused apodized forked grating coupler. Opt. Express25, 26861(2017). N. Zhou, S. Zheng, X. Cao, S. Gao, et al., Design, Fabrication andDemonstration of Ultra-Broadband Orbital Angular Momentum(OAM) Modes Emitter and Synthesizer on Silicon Platform, inOptical Fiber Communication Conference 1, p. Th2A.9. 2 (2018). C. J. Brooks, A. P. Knights, and P. E. Jessop, Vertically-integrated multimode interferometer coupler for 3D photonic cir-cuits in SOI. Opt. Express 19, 2916 (2011). M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, Exper-imental realization of any discrete unitary operator. Phys. Rev.Lett. 73, 58 (1994). J. Carolan, C. Harrold, C. Sparrow, et al., Universal linear optics.Science 349, 711 (2015). F. Flamini, L. Magrini, A. S Rab, N. Spagnolo, et al., Thermally-reconfigurable quantum photonic circuits at telecom wavelengthby femtosecond laser micromachining. Light Sci. Appl. 4, e354(2015). W. R. Clements, P. C. Humphreys, B. J. Metcalf, W. S. Koltham-mer, and I. A. Walmsley, Optimal design for universal multiportinterferometers. Optica 3, 1460 (2016). A. Liu, C.-L. Zou, X. Ren, Q. Wang, G.-C. Guo, On-chip gen-eration and control of the vortex beam. Appl. Phys. Lett. 108,181103 (2016). Y.-H. Chen, L. Huang, L. Gan, Z.-Y. Li, Wavefront shaping ofinfrared light through a subwavelength hole. Light Sci. Appl. 1,e26 (2012).24