Fiber coupler for generating orbitalangular momentum modes
Yan Yan,1,* Jian Wang,1 Lin Zhang,1 Jeng-Yuan Yang,1 Irfan M. Fazal,1 Nisar Ahmed,1 Bishara Shamee,1
Alan E. Willner,1 Kevin Birnbaum,2 and Sam Dolinar21Department of Electrical Engineering, University of Southern California, Los Angeles, California 90089, USA
2Jet Propulsion Lab, 4800 Oak Grove Drive, Pasadena, California 91109, USA*Corresponding author: [email protected]
Received July 7, 2011; revised September 20, 2011; accepted September 27, 2011;posted September 28, 2011 (Doc. ID 150671); published October 31, 2011
An optical beam that carries orbital angular momentum(OAM) is characterized as having a helical phase wave-front of expð�inφÞ [1]. An OAM beam with a topologicalcharge number of n has a 2nπ azimuthal phase change ofits wavefront. The OAM modes with different chargenumber n are spatially orthogonal to each other. Eachþ or − charge is spatially orthogonal to all other charges.OAM has wide applications in atom trapping, opticaltweezers, and photon entanglement [2]. Recently, OAMhas gained much interest for increasing the transmissioncapacity in optical communication systems given by theability to carry independent data streams on orthogonalmodes [3].A key challenge has been the generation of OAM
modes from conventional Gaussian modes. This has ty-pically been accomplished by using discrete spatial lightmodulators [3], which are generally bulky and expensive.A laudable goal would be to use compact fiber technol-ogy for efficiently generating OAM modes in an inte-grated photonic component. Reports have shown thegeneration of a single OAMmode in a conventional multi-mode fiber using stress or acoustic waves [4,5].In this Letter, we propose and show by simulation a
new (to our best knowledge) approach to generate OAMmodes in a fiber coupler consisting of a central ring andfour external cores. By designing the size of the externalcores and controlling the polarization state and phase ofthe four input lights, one can selectively generate OAMmodes with odd charge numbers of n in the central ring.The purity of the OAM modes and generation efficiencycan exceed 99% using <2mm long fiber. This fiber cou-pler could serve as an integrated OAM light source andcould also be used as the transmitter and receiver in aspatial-mode multiplexing and demultiplexing system.Figure 1 shows the structure of the proposed fiber cou-
pler. The glass materials are (i) background, Schott LLF1with nL ¼ 1:53; (ii) ring, SF6 with nH1 ¼ 1:76; and (iii) ex-ternal cores, SF4 with nH2 ¼ 1:71 (at 1550 nm). Fabrica-tion of this kind of high-contrast index fiber has beendemonstrated [6]. The ring size is fixed with an innerradius of 5 μm and an outer radius of 6:1 μm. The size
of the external cores is designed to excite the OAMmodewith different charge numbers of n.
The OAM modes in a single ring waveguide are theHEn;m and EHn;m modes with their electric fields Ez,Eφ, and Er having the expression of Er;φ;zðrÞ expð�inφÞ[5,7]. The purpose of using the ring structure and high-contrast index is to remove the Neff degeneracy of theHE and EH modes [8]. Our approach to generate theOAM mode is schematically illustrated in Fig. 1. Fourbeams are launched into the four external cores to excitethe fundamental mode HE1;1. When the effective refrac-tive indices (Neff ) of theHE1;1 mode in the external coresand HEn;1 mode in the ring are matched, the lights in theexternal cores would couple into the ring to excite theHEn;1 modes. Figure 2 shows theNeff of theHE-polarizedmodes in the ring (red horizontal lines) andHE1;1 mode inthe external cores as a function of the core radius (bluecurve) at λ ¼ 1550 nm, which is calculated by finite ele-ment analysis in the COMSOL software package. By chan-ging the core size to match the Neff of modes in the ringand cores, one can generate OAM modes with differentcharge numbers of n. The conditions for the input lightsinclude the following: (i) the input lights are ellipticallypolarized; the top and bottom input lights have the samepolarization with ellipticity jExj=jEyj ¼ ε; (ii) the left andright inputs have the same polarization with ellipticityjExj=jEyj ¼ 1=ε; and (iii) the phase difference ΔΦ
Fig. 1. (Color online) Structure of the fiber coupler and thephase and polarization state of the input lights. The horizontal(red) electric field components have a phase difference of �90°relative to the vertical (blue) components.
between the x- and y-polarized E fields is�90°. WhenΔΦis 90° (−90°), it will generate OAM modes where n is po-sitive (negative). When the lights are coupled from the ex-ternal cores into the ring, the input y-polarized E fieldmainly contributes to the even mode HEe
m;1 while the in-put x-polarized E field mainly contributes to the oddmodes HEo
m;1. Finally, an OAM mode HEOAMm;1 ¼ HEe
m;1 þiHEo
m;1 is obtained in the ring. For example, Fig. 3(a)shows theNeff of symmetric and asymmetric modes com-posed of HE3;1 mode in the ring and HE1;1 mode in thefour cores where the insets show the power distributionand E field of the even and odd modes. Figure 3(b) showsthe group velocity of the symmetric and asymmetricmodes with the external core radius r ¼ 2 μm. The groupvelocity mismatch between the symmetric and asym-metric modes Δβ1 ¼ βasym − βsym ¼ 1=vg;asym − 1=vg;symis zero at λ ¼ 1550 nm and does not exceed 0:1 pm=mmover 1540–1560 nm. This is small when the coupling lengthis about severalmillimeters. Figures 1–3 illustrate the con-cept and physics of using an optical fiber coupler withmultiple coherent input lights to generate OAM modesat one output.We use the beam propagation method in the RSoft
BPM software package to investigate the performance ofthe generated OAM modes. Figure 4 shows the intensity(top) and phase (bottom) of the azimuthal component ofthe electric field of the input and the output generatedOAM modes with different charge numbers of n. Thephase increases clockwise for charge number n > 0,while the phase decreases clockwise for n < 0. Theparameters used in the computation are (i) the ring sizewith the inner radius of 5:0 μm, the outer radius of 6:1 μm,and the radius of external cores for generating OAM
modes with n ¼ �1, �3, �5, �7, and �9 are 2.2, 2.0,1.6, 1.3, and 1:0 μm, respectively; (ii) the input polariza-tion state of ε is 0, 0.3, 0.4, 0.6, and 0.8, respectively; and(iii) the offsets between the cores and ring is chosen tohave the coupling length of around 1mm, which can beread out in Fig. 6(b). The wavelength is λ ¼ 1550 nm.
To estimate the purity of the generated OAM modes,we calculate the OAM power weight, which is defined asCi ¼ ∬ Fðx; yÞψ�
i ðx; yÞdxdy andP jCij2 ¼ 1 [9], where
Fðx; yÞ is the electric field of the generated OAMmode and ψ iðx; yÞ is the electric field of the OAM eigen-mode HEi;1 or EHi;1 in the central ring. jCij2 is the OAMpower weight with n ¼ i, which is also known as theOAM spectra [10]. Figure 5 shows the OAM weight spec-tra (in the unit of dB) of each OAM mode in Fig. 4. Thetop row is theHE-polarized OAMweight spectra, and thebottom row is the EH-polarized OAMweight spectra. Thedesired OAMweight of the generated beam can be>0:99.The cross talk among the different charges can be lowerthan −15 dB. This proves that the proposed approachcould generate OAM modes with high purity.
We investigate the dependence of the OAM modegeneration on the external core offset, input light polar-ization state, and wavelength. Figure 6 shows the depen-dence of the coupling length on the offset. The offset isdefined as the distance between the external core and thering as shown in Fig. 1. Figure 6(a) shows the opticalpower (normalized to input power) in the external coreas a function of the propagation length when the OAMmode n ¼ 1 is generated. The extinction ratio is definedas 10×log10½ðP0 − PcÞ=Pc�, where P0 is the input power inone external core and Pc is the power left in the core atthe coupling length. Figure 6(b) shows the couplinglengths of n ¼ 1, 3, 5, 7, and 9 as the functions of the off-set between the core and the ring. The coupling length isaround 1mm when the offset is around 0:6−0:8 μm.
The performance of the OAM mode generation can beoptimized by controlling the polarization state of the in-put light. Figure 7 shows the extinction ratio and OAMweight as a function of the polarization state with ellip-ticity ε ¼ jExj=jEyj. For each charge number of n, there isan optimized polarization state εopt of the input light toachieve the maximum extinction ratio. As the chargenumber n increases from 1 to 9, εopt also increases from0 to 0.8. This can be explained as follows: when n is asmall number, the azimuthal electric field Eφ of theHEn;1 mode dominates over the radial field Er. As n in-creases, jErj=jEφj of mode HEn;1 increases. As ε in-creases, the input lights provide more radial electricfield components. When ε is approximately matched withthe ratio jErj=jEφj of the HEn;1 mode, one could obtainthe maximum extinction ratio. The OAM weight is not
Fig. 2. (Color online) Effective refractive index of the mode inthe cores and the ring. Blue curve, effective refractive index ofthe fundamental mode in the external cores as a function of thecore radius. Red horizontal lines, effective refractive index ofthe HEn;1 mode in the central ring.
Fig. 3. (Color online) (a)Neff of the symmetric and asymmetric(even and odd) modes composed of HE3;1 in the ring and HE1;1in the cores. White arrows illustrate the electric field vectordirection. (b) Group velocity difference of asymmetric andsymmetric modes Δβ1¼ βasym−βsym¼ 1=vg;asym−1=vg;asym.
Fig. 4. (Color online) Intensity and phase of the azimuthalcomponents of the generated OAM modes with odd chargenumbers of þ1, −3, þ5, −7, and þ9 in the central ring,respectively.
influenced significantly by ε. It can remain >0:9 in arelatively wide range of ε for each n.Figure 8 shows the dependence of the OAM generation
on wavelength by considering the waveguide dispersion.
The 10 dB extinction ratio bandwidth for ten modes are10–17 nm, and the 95% OAM weight bandwidth is10–40 nm. The input light for each charge number n is setto be the optimized polarization according to results inFig. 7. The performance of higher order OAM modes isvery sensitive to the wavelength variation, because thehigher order OAM modes in the ring and the fundamentalmodes in the cores have much larger waveguide disper-sion. As a result, the input light would excite other OAMmodes in the vicinity of the desired OAM modes andtherefore reduce the weight and extinction ratio of thedesired OAM mode.
In conclusion, we propose a new approach to generateOAM modes in a fiber-based coupler, which could beextended to spatial multiplexing and demultiplexing sys-tems. One may generate several OAM modes by placingmore sets of external cores and inputs around the ring.Also, this fiber coupler could work in a reciprocal way todemultiplex OAM modes in the ring by coupling themback to the external cores. Further work would alsoinclude the efficient generation of four coherent inputsbefore the coupler and the fabrication tolerance of thefiber coupler.
We acknowledge the support of the Defense AdvancedResearch Projects Agency (DARPA) under the InPhoprogram. We are thankful for the fruitful discussion withProf. Moshe Tur at Tel Aviv University, Israel.
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Fig. 5. (Color online) Each column shows the OAM weight spectra of the generated OAM modes in decibels with different chargenumbers of n. The top is the spectra of the HE modes, and the bottom is the spectra of the EH modes. The OAM weight determinesthe purity of the desired generated OAM mode. The purity of each OAM mode is >0:99, and the cross talk among these modes islower than −15dB.
Fig. 6. (Color online) Dependence of OAM generation perfor-mance on the offset of external cores. (a) Light power in theexternal core as a function of the propagation distance for threeoffsets when n ¼ 1. (b) Coupling length as a function of the off-set (n ¼ 1–9, odd numbers).
Fig. 7. (Color online) Dependence of OAM generation on theinput polarization state. For some charge number with appro-priate polarization, the weight could exceed 99% and the extinc-tion ratio could exceed 20dB (i.e., the generation efficiency is>0:99).
Fig. 8. (Color online) Dependence of OAM generation onthe input wavelength. The higher order OAM mode is moresensitive to the wavelength change because of the larger wave-guide dispersion.
