Tunable orbital angular momentum generation in ... - OSA Publishing

Letter

Vol. 41, No. 15 / August 1 2016 / Optics Letters

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Tunable orbital angular momentum generation in optical fibers YOUCHAO JIANG,1,2 GUOBIN REN,1,2,* YUDONG LIAN,1,2 BOFENG ZHU,1,2 WENXING JIN,1,2 SHUISHENG JIAN1,2

AND

1

Key Lab of All Optical Network & Advanced Telecommunication Network of EMC, Beijing Jiaotong University, Beijing 100044, China Institute of Lightwave Technology, Beijing Jiaotong University, Beijing 100044, China *Corresponding author: [email protected]

2

Received 30 May 2016; revised 2 July 2016; accepted 4 July 2016; posted 7 July 2016 (Doc. ID 267186); published 25 July 2016

We present a method in this Letter to generate optical vortices with tunable orbital angular momentum (OAM) in optical fibers. The tunable OAM optical vortex is produced odd by combining different vector modes HEeven 2;m (HE2;m ) even and TE0;m (TM0;m ) when l 1 or combining HEl 1;m odd even (HEodd l 1;m ) and EHl −1;m (EHl −1;m ) when l > 1 with a π∕2 phase shift. The vortex can be regarded as a result of overlapping two orthogonal optical vortex beams of equal helicity but opposite chirality with a π∕2 phase shift. We have experimentally demonstrated the smooth variation of OAM from l −1 to l 1 by adjusting a polarizer at the output end of the fiber. © 2016 Optical Society of America OCIS codes: (050.4865) Optical vortices; (060.2310) Fiber optics; (060.2340) Fiber optics components; (060.2330) Fiber optics communications; (060.4230) Multiplexing. http://dx.doi.org/10.1364/OL.41.003535

An optical vortex carrying orbital angular momentum (OAM) is characterized by a helical phase front of expil ϕ, in which ϕ refers to the azimuth angle and l is the topological charge number [1]. Because of their ability to carry OAM, these optical vortices have wide applications in optical tweezers [2], atom manipulation [3], higher dimensional optical communication [4,5], microscopy [6], and so forth. Recently, OAM has attracted great attention for increasing transmission capacity and spectral efficiency in optical communication [7–9]. There have been many attempts to generate and manipulate OAM beams, including cylindrical lens, mode converters, q-plates, spiral phase plates, spatial light modulators (SLM), metamaterials-based phase plates, and silicon integrated devices [1,4,10–13]. Among them, the most common method for vortex beam generation is to use SLM. Meanwhile, there is an increasing interest in the generation of OAM beams with optical fibers because of the inherent advantages of fibers, such as remote delivery and compactness [14–21]. A fiber must support high order modes (HOMs) for carrying OAM. So far, the methods to generate OAM in fiber can be classified into two types according to the source modes. For the first type, the OAM is generated by combing two linearly polarized modes, where the generated 0146-9592/16/153535-04 Journal © 2016 Optical Society of America

vortex has no spin angular momentum (SAM) [14,20]; for the second type, the OAM is generated by combining two vector modes, where the generated vortex has SAM characterized by 1 or −1 [15,18]. Specifically, for the second type, the generation of OAM is based on the linear combination of two degenerate orthogonal components of the same vector mode. Since the even and odd variants of both HE and EH modes are degenerate modes in circularly symmetrical fibers, the OAM modes are also eigenmodes of the propagation constant β. Hence, the OAM modes can stably propagate in fiber [18]. Control of the OAM in a beam is important in many applications. The OAM in an optical vortex beam can be adjusted by the wavefronts’ helicity and the photon flux [22]. However, since the radius of an optical vortex scales with its helicity [23], applications that require controlled geometry or photon density will be limited to fixed OAM. The tunable OAM optical vortex produced by the overlap of two collinear optical vortex beams of equal helicity but opposite chirality could overcome this problem because it maintains a constant geometry and total intensity during tuning. Nevertheless, with the method of OAM generation in fibers so far, the OAM has only one kind of chirality after generation no matter whether the SAM is zero or not. We find two optical vortex beams of equal helicity but opposite chirality can be achieved by combining two different vector modes in fiber. In this Letter, we present a method to generate optical vortices with tunable OAM in optical fibers based on the combination of different vector modes in the same nearly degenerate group. The tunable OAM optical vortex is produced by comodd bining the different vector modes HEeven 2;m (HE2;m ) and TE0;m even (TM0;m ) when l 1 or combining HEl 1;m (HEodd l 1;m ) and even (EH ) when l > 1 with a π∕2 phase shift. Then EHodd l −1;m l −1;m the vortex can be regarded as a result of overlapping two orthogonal optical vortex beams of equal helicity but opposite chirality with a π∕2 phase shift, where l refers to the azimuthal index and m refers to the radial index. The OAM can be smoothly tuned just through a polarizer at the output end of fiber. Moreover, the controllable mode switching among −l , 0, and l can also be achieved during the tuning. The high order vector modes in optical fiber could be classified as HEe;o , EHe;o , TE, and TM modes, where e and o refer

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to the even and odd modes, respectively. In the Cartesian coordinate system, with the weakly guiding approximation, these HOMs have the following transverse electric field distributions [24]: HEel1;m xˆ cosl ϕ − yˆ sinl ϕ l ≥ 1; F l ;m r HEol1;m xˆ sinl ϕ yˆ cosl ϕ EHel−1;m xˆ cosl ϕ yˆ sinl ϕ l > 1; F l ;m r EHol−1;m xˆ sinl ϕ − yˆ cosl ϕ TM0;m xˆ cosϕ yˆ sinϕ F 1;m r l 1; TE0;m xˆ sinϕ − yˆ cosϕ

Fig. 1. Phase of HEe3;1 , iEHo1;1 and HEe3;1 iEHo1;1 . The first line and second line are the xˆ and yˆ polarization directions, respectively.

(1) where F l ;m represents solution of the Bessel equation [the radial field distribution of the corresponding scalar mode (LP)], and ϕ is the azimuthal coordinate. In previous investigations, the OAM is generated by the following linear combination of the two degenerate orthogonal components of the same vector mode [15,18,19,21]: V l ;m HEel1;m iHEol1;m EHel−1;m iEHol−1;m V l ;m σˆ expil ϕ; l ≥ 1 ; (2) F l ;m r σˆ expil ϕ; l > 1 where the superscripts and subscripts l denote the SAM and OAM; σˆ xˆ iˆy represents left- or right-handed circular polarization, respectively, which is also denoted by SAM with S 1. A scalar field of a classic optical vortex is V F l ;m r expil ϕ; therefore, we can also regard Eq. (2) as two OAM modes with the same l in the xˆ and yˆ polarization directions, and there is a π∕2 phase shift between them. Because the two degenerate orthogonal components of HE or EH modes have the same propagation constant β, the OAM modes are stable and can propagate in fiber. But it is not necessary that the generation and propagation of OAM must be in the same fiber. Now, we just focus on the generation of OAM modes. By observing Eq. (1), the combination of different vector modes with the same l (modes in the same group) results in the following equation: Ca e HEl 1;m iEHol−1;m l ;m EHel−1;m iHEol1;m C bl;m xˆ expil ϕiˆy expil ϕ l > 1; F l ;m r xˆ expil ϕiˆy expil ϕ (3) C al;m

C bl;m

and represent two different complex OAM where states (states a and b, respectively). From Eq. (1) we know that TE and TM modes have the same expressions as those of EHo and EHe modes. Then the following equation can be obtained from Eq. (3): a C 1;m HEe2;m iTE0;m C b1;m TM0;m iHEo2;m xˆ expiϕ iˆy expiϕ : (4) F 1;m r xˆ expiϕ iˆy expiϕ

Specifically, Eq. (4) is a special circumstance (l 1) of Eq. (3). The same as for Eq. (2), we regard Eq. (3) as two OAM modes with l of opposite signs in the xˆ and yˆ polarization directions, and there is a π∕2 phase shift between them. Hence, C al;m and C bl;m are overlaps of two collinear optical vortex beams of equal helicity but opposite chirality in fibers. For example, as shown in Fig. 1, the phases of HEe3;1 iEHo1;1 in the xˆ and yˆ polarization directions have equal helicity but opposite chirality. The interference vortex V int of two classic optical vortices can be described as follows [22]: V int F l ;m r a expil ϕ φa b exp−il ϕ − φb

Br; ϕ expiφϕ;

(5)

with constant, positive, and real amplitudes a, b and phase offsets φa and φb . The Br; ϕ and φϕ are the amplitude and phase distribution, respectively. The mixing amplitudes a and b determine the modulation amplitude, c a − b∕a b. The phase profile of an interfered vortex is φϕ arctan c tanl ϕ α φa φb ∕2;

(6)

where α φa − φb ∕2l . The local helicity of the interference is ∂φ cl ; ∂ϕ cos2 l ϕ α c 2 sin2 l ϕ α

and the average OAM can be smoothly varied by changing the modulation parameter c. In order to get an OAM-tunable optical vortex, we set a polarizer at the output end of the fiber and let C al;m and C bl;m pass through it; then the output derived from Eq. (5) can be described as PCa l ;m

PCbl;m F l ;m r

cospexpil ϕ sinpexpil ϕ π∕2 cospexpil ϕ sinpexpil ϕ − π∕2

0 ≤ p ≤ π∕2;

;

(7a)


Letter

PC al;m PC bl;m

F l ;m r

−cosp expil ϕ π sinp expil ϕ π∕2 ; −cosp expil ϕ π sinp expil ϕ − π∕2

where PCal;m and PCbl;m denote the output of C al;m and C bl;m passing through the polarizer, and p is the angle between the direction of the polarizer and the positive direction of the x-axis. The experimental setup for the generation and detection of an OAM-tunable optical vortex is sketched in Fig. 2. The output beam from a tunable laser is divided into two paths by an optical coupler with a proportion of 1:1. One path is used to generate OAM beams, and the other path is used as a reference beam to interfere with the generated OAM beams. Two adjustable attenuators are inserted in both paths to ensure their power in the same order of magnitude at the end to obtain an optimal interference. A mechanical long-period grating (LPG) is applied on a two-mode fiber (TMF) fabricated by our lab as a mode converter. In order to mitigate the effects of higher order modes before the LPG, a mode stripper realized by the tight bending of TMF is used to ensure a pure fundamental mode (1st OM) launching. In addition, a fiber Bragg grating (FBG) written in the TMF is used to mitigate the effects of 1st OM after the LPG. The TMF output is collimated using a 20× objective lens. A polarizer after the 20× objective lens is used to adjust the p derived from Eq. (7). The polarization of reference beam can be adjusted by the half-wave plate. The TMF can support six vector modes or two LP modes, including the first order HEe;o 1;1 (LP0;1 ) and the second order , TM , TE (LP ) HEe;o 0;1 0;1 1;1 modes. The calculated effective 2;1 refractive indices of the 1st OM and second order modes (2nd OMs) of the TMF as a function of wavelength (λ) are depicted in Fig. 3. In order to realize mode coupling between the 1st OM and 2nd OMs, the period (Λ) of the LPG should be equal to the modal beat length, namely, Λ λ∕n1 − n2 , where n1 and n2 are the effective indices of the two order modes. The calculated period of the LPG is also shown in Fig. 3. The effective period can be changed by varying the angle of the grating relative to the fiber [20]. Because the effective mode index differences among HEe;o 2;1 , TM0;1 , and TE0;1 are very small, this is neglected when calculating the period of LPG. It is known from Fig. 3 that the effective mode indices from high to low are TE0;1 , HEe;o 2;1 , and TM0;1 . Hence, around

Fig. 2. Experimental setup for the generation and detection of tunable OAM beams. PC, polarization controller; MS, mode stripper; LPG, long-period grating; FBG, fiber brag grating; Rot., rotator; MPS, metal parallel slab; Obj., objective; Pol., polarizer; BS, beam splitter; HWP, half-wave plate; Col., collimator.

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π∕2 ≤ p ≤ π;

(7b)

the calculated period at a certain wavelength, the longer period is advantageous to the coupling from HE1;1 to TE0;1 and HEe;o 2;1 , while the shorter period is advantageous to the coupling from HE1;1 to HEe;o 2;1 and TM0;1 . The transmission spectra of the FBG written in the TMF is shown in Fig. 4. The rightmost dip corresponds to the coupling of the 1st OM from the forward to backward direction, the leftmost dip corresponds to the coupling of the 2nd OMs from the forward to backward direction, and the middle dip represents the coupling from forward 1st OM to backward 2nd OMs. Note the rightmost dip at wavelength 1554.64 nm; at this wavelength, the 1st OM is reflected while the 2nd OMs can pass through. Hence, the wavelength is set as 1554.64 nm during the experiment. By the integrated application of varying pressure applied to the mechanical LPG and metal parallel slab, adjusting the input polarization and rotating the fiber, the vortex beams with tunable OAM can be obtained. Then the TMF output is collimated, and the beam intensity is imaged using a CCD camera. The topological charge number can be identified through the interference pattern.

Fig. 3. Calculated effective indices of the 1st OM and 2nd OMs. Calculated period of the LPG for mode coupling between 1st OM and 2nd OMs. Top inset: image of fiber cross section. Bottom inset: refractive indices of TE01 , HE21 , and TM01 modes.

Fig. 4. Transmission spectra of FBG written in the TMF.

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Letter group. We have experimentally demonstrated the optical vortices with tunable OAM in a TMF, and the results also validated the proposed method. The average OAM can be smoothly varied by adjusting the polarizer at the output end of the TMF. The optical vortices with tunable OAM have huge advantages in many application areas over traditional optical vortices, such as atomic manipulation and micromanipulation. In addition, the mode switching among l −1; 0; 1 is achieved just through adjusting the polarizer in our experimental setup; this is helpful to the OAM mode-division multiplexing in high-capacity transmission systems.

Fig. 5. Experimental results and corresponding simulations of intensity and interference patterns. Sim. Inten., simulation intensity; Exp. Inter., experimental intensity; Sim. Inter., simulation interference pattern; Exp. Inter., experimental interference pattern.

Because the electric field distributions of TE and TM modes are circularly symmetrical, we can always treat the HE2;m iTE0;m as HEe2;m iTE0;m by selecting a proper coordinate system, the same as TM0;m iHE2;m . The experimental results of HE2;1 iTE0;1 and TM0;1 iHE2;1 are shown in Figs. 5(a) and 5(b), respectively. As shown in Figs. 5(a) and 5(b), the first row is the simulation intensity profiles of different polarizations, and the second row is the corresponding experiment results; the third and fourth rows are interferences patterns of simulation and experiment, respectively, with different polarizations. It can be confirmed that the tunable OAM modes are generated by HE2;m and TE0;m modes or by HE2;m and TM0;m modes through comparing (a6) and (b6) or comparing (a8) and (b8). Then the phase relationship between HEe2;m and TE0;m modes can be confirmed by comparing (a13) and (a15), similar to confirmation of TM0;m and HE2;m modes by comparing (b13) and (b15). Based on experimental OAM model images, we evaluate mode purity using the modified ring method derived from [18]. The OAM mode purity is about 88.5% and 84.1% for (a5) and (a7), and 81.6% and 83.1% for (b5) and (b7), respectively. The OAM mode purity could be further improved by accurately controlling the excitation of HOMs in optical fiber, including mode selection, amplitude, and phase control. Note that (a13) and (a15) are the OAM modes with opposite sign and perpendicular polarization directions; the results are agreement with the theoretical analysis. (a14) and (a16) have the same interference patterns as the LP11 mode; this means that l 0 and the experimental setup can be used as a mode switcher between LP and OAM modes. The variation from (a13) to (a16) is smooth during our experiment, and this can also be obtained from the analysis of Eq. (7). The generation of higher order optical vortices with tunable OAM can be achieved by employing a few-mode fiber which supports more HOMs. The potential technical challenges are to excite the specific higher order modes and the phase control. With our method, the two vector modes in the same nearly degenerate group correspond to a LP mode. The higher order LP modes can be achieved easily by the LPG, or phase plates [25]. Then the tunable OAM vortices could be generated by manipulating the phase difference of the two vector modes. In summary, we presented a method to generate optical vortices with tunable OAM in optical fibers based on the combination of different vector modes in the same nearly degenerate

Funding. National Natural Science Foundation of China (NSFC) (61178008, 61275092). Acknowledgment. We would like to thank Yunlong Bai and Liangying Wu for writing the FBG. REFERENCES 1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992). 2. J. E. Curtis, B. A. Koss, and D. G. Grier, Opt. Commun. 207, 169 (2002). 3. J. W. R. Tabosa and D. V. Petrov, Phys. Rev. Lett. 83, 4967 (1999). 4. G. Gibson, J. Courtial, and M. J. Padgett, Opt. Express 12, 5448 (2004). 5. A. Vaziri, G. Weihs, and A. Zeilinger, Phys. Rev. Lett. 89, 240401 (2002). 6. S. Fürhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, Opt. Lett. 30, 1953 (2005). 7. N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, Science 340, 1545 (2013). 8. S. Ramachandran, P. Gregg, P. Kristensen, and S. E. Golowich, Opt. Express 23, 3721 (2015). 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, Adv. Opt. Photon. 7, 66 (2015). 10. L. Marrucci, C. Manzo, and D. Paparo, Phys. Rev. Lett. 96, 163905 (2006). 11. A. M. Yao and M. J. Padgett, Adv. Opt. Photon. 3, 161 (2011). 12. P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, Appl. Phys. Lett. 100, 013101 (2012). 13. E. Karimi, S. A. Schulz, I. De Leon, H. Qassim, J. Upham, and R. W. Boyd, Light Sci. Appl. 3, e167 (2014). 14. D. McGloin, N. B. Simpson, and M. J. Padgett, Appl. Opt. 37, 469 (1998). 15. P. Z. Dashti, F. Alhassen, and H. P. Lee, Phys. Rev. Lett. 96, 043604 (2006). 16. S. Ramachandran, P. Kristensen, and M. F. Yan, Opt. Lett. 34, 2525 (2009). 17. Y. Yan, J. Wang, L. Zhang, J.-Y. Yang, A. E. Willner, I. M. Fazal, K. Birnbaum, N. Ahmed, and S. Dolinar, Opt. Lett. 36, 4269 (2011). 18. N. Bozinovic, S. Golowich, P. Kristensen, and S. Ramachandran, Opt. Lett. 37, 2451 (2012). 19. C. Brunet, P. Vaity, Y. Messaddeq, S. LaRochelle, and L. A. Rusch, Opt. Express 22, 26117 (2014). 20. S. Li, Q. Mo, X. Hu, C. Du, and J. Wang, Opt. Lett. 40, 4376 (2015). 21. P. Gregg, P. Kristensen, and S. Ramachandran, Optica 2, 267 (2015). 22. C. H. J. Schmitz, K. Uhrig, J. P. Spatz, and J. E. Curtis, Opt. Express 14, 6604 (2006). 23. J. E. Curtis and D. G. Grier, Phys. Rev. Lett. 90, 133901 (2003). 24. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983). 25. M. M. Ali, Y. Jung, K.-S. Lim, M. R. Islam, S.-U. Alam, D. J. Richardson, and H. Ahmad, IEEE Photon. Technol. Lett. 27, 1713 (2015).