Two-dimensional tunable orbital angular momentum generation using

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Letter

Vol. 42, No. 23 / December 1 2017 / Optics Letters

Two-dimensional tunable orbital angular momentum generation using a vortex fiber YOUCHAO JIANG,1,2 GUOBIN REN,1,2,* YA SHEN,1,2 WEI JIAN,1,2 AND SHUISHENG JIAN1,2

YAO XU,1,2 WENXING JIN,1,2 YUE WU,1,2

1

Key Laboratory 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 29 August 2017; revised 31 October 2017; accepted 3 November 2017; posted 6 November 2017 (Doc. ID 305813); published 1 December 2017

We demonstrate the two-dimensional tunable orbital angular momentum (OAM) generation in a ring-core (vortex) fiber. The LP11 mode generated by an all fiber fused coupler is coupled into a vortex fiber. Because the vector modes of the LP11 mode group in the vortex fiber are no longer degenerate, the mode status will change between linearly polarized modes (LPMs) and complex OAM modes periodically during propagation. The generated OAM can be tuned smoothly by filtering the mixed mode with different polarization directions or changing the wavelength at a certain polarization directions. The two-dimensional tuning of OAM from l
−1 to l
1 is experimentally demonstrated in an all fiber OAM generator. © 2017 Optical Society of America OCIS codes: (060.2310) Fiber optics; (060.2400) Fiber properties; (060.2280) Fiber design and fabrication; (050.4865) Optical vortices. https://doi.org/10.1364/OL.42.005014

An optical vortex carrying orbital angular momentum (OAM) is characterized by a helical phase front of exp
il ϕ, in which ϕ refers to the azimuth angle, and l is the topological charge number [1]. As one kind of structured light beam, optical vortices have recently attracted plenty of attention because of their intriguing properties and widespread applications, such as optical tweezers [2], higher dimensional optical communication [3], microscopy [4], and optical sensors [5]. In recent years, there is an increasing interest in generating and manipulating optical vortices based on optical fibers due to the inherent advantages of fibers, such as delivery and compactness [6–10]. As an important aspect of manipulating OAM beams, continuously tunable OAM is desirable for applications such as particle manipulation [11,12] and optical communications [13]. Several mechanisms have been presented to tune the OAM for different applications, such as changing the relative phase between the cos and sin components [14,15]; changing the amplitudes of the cos and sin components [16]; filtering a mixed 0146-9592/17/235014-04 Journal © 2017 Optical Society of America

mode, which contains a couple of OAM modes of opposite chirality with different polarization directions [17]. All these methods are achieved by controlling one parameter, such as phase difference, amplitude, or polarization direction, thus, the generated OAM is only tunable in one dimension. Meanwhile, these experiments require complex devices and operation to control the phase between the components, or align the non-circular-symmetrical linearly polarized modes (LPMs) to the non-circular-symmetrical polarization-maintaining fiber (PMF) end, or generate different vector modes (VMs) in few mode fibers (FMFs). In 2016, Niederriter et al. introduced that OAM can be tuned by changing the wavelength and varying coherence simultaneously [18]. But, the coherence just controls the amplitude of OAM, and the OAM tuning from l  −1 to l  1 cannot be achieved by only varying the coherence. In this Letter, we present a method to generate twodimensional continuously tunable OAM by breaking the degeneracy in the LP11 mode group with a ring-core (vortex) odd fiber. The LPeven 11 ∕LP11 mode evolution in a vortex fiber is inodd vestigated, and the conversion from LPeven 11 ∕LP11 to the mixed mode, which contains a couple of OAM modes with opposite chirality, is demonstrated. The OAM can be smoothly tuned by filtering the mixed mode with different polarization directions or changing the wavelength at a certain polarization direction. This also allows us to achieve tunable OAM at any wavelength. The two-dimensional tunable OAM will have a wide range of applications, such as OAM mode division multiplexing, sensing, imaging, and particle manipulation. The vortex fiber plays a pivotal role in the OAM generation and manipulation. In vortex fibers, the index separation of two adjacent VMs reach an order of 1 × 10−4 , which is large enough to break the degeneracy within the LPM group; in other words, the VMs are no longer degenerate in vortex fibers [19]. Thus, when a LPM is injected into a vortex fiber, the output is not always LPMs because of the variable relative phases between the VMs. Generally, the guided VMs of the FMFs in the same group have approximately the same effective index. Take this reason into account, the relative phase differences between the VMs

Vol. 42, No. 23 / December 1 2017 / Optics Letters

Letter are a constant zero as they propagate in the FMFs. Hence, the modes observed at the fiber output are linearly combined VMs with no phase deviations, leading to the generation of LPMs. However, the VMs in the same LPM group are no longer degenerate in vortex fibers. The relative phase
θ between the two adjacent VMs can be calculated from θ  Δβ × z, where Δβ is the propagation constant difference, and z represents the propagation distance. It is easy to know that the θ changes from 0 to 2π continually and periodically as the VMs propagate in vortex fibers. Since the high-order LPMs are four-fold degenerate, the results of combining two adjacent VMs, which are in the same LPM group with varying relative phase, can be described as 8 9 H E e2;m  exp
iθTM0;m > > > > < = H E e2;m  exp
iθTE0;m M 1;m
θ 
l  1; H E o2;m  exp
iθTM0;m > > > > : ; H E o2;m  exp
iθTE0;m (1a)

M l ;m
θ 

8 9 H E el1;m  exp
iθEH el−1;m > > > > < HEe =  exp
iθEH o l 1;m H E ol1;m > > : H E ol1;m

 

l −1;m exp
iθEH el−1;m exp
iθEH ol−1;m

> > ;

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 La  l ;m

Lbl;m   H E e2;m iTM0;m ;
l  1  TE0;m  iH E o2;m   H E el1;m  iEH el−1;m ;
l > 1 or EH ol−1;m  iH E ol1;m   pffiffiffi xˆ cos
l ϕ  iˆy sin
l ϕ  2 exp
iπ∕4F l ;m
r ;
l ≥ 1; xˆ sin
l ϕ iˆy cos
l ϕ (3) where Lal;m and Lbl;m represent two different complex mode states. Since the transverse electric field distributions of LPMs can be described by cosine or sine functions of the azimuth angle, Eq. (3) can be rewritten as  a   s;y  pffiffiffi Ll ;m
LPc;x l ;m  iLPl ;m  ;  2 exp
iπ∕4F l ;m
r × c;y Lbl;m
LPs;x l ;m  iLPl ;m  (4)


l > 1;

(1b) where M l ;m
θ represents a different mode status, l refers to the azimuthal index, m refers to the radial index, and superscripts e and o refer to the even and odd modes. Because TM0;m and TE0;m are circular-symmetrical, the H E o2;m and H E e2;m modes are actually equivalent for the results of the combination when l  1. It is obvious that M l ;m
θ is a LPM when θ is 0 or π. According to the odevity of VMs, the equations in Eq. (1) can be divided into two groups: the two VMs have opposite odevity, such as the second and third equations of Eqs. (1a) and (1b); and the two VMs have a same odevity, such as the first and fourth equations in Eqs. (1a) and (1b). For the second and third equations of Eqs. (1a) and (1b), it has been investigated that combining two different VMs with a π∕2 phase shift can generate tunable OAM [17]. When θ  π∕2, we can rewrite the second and forth equations of Eqs. (1a) and (1b) as ( a )   C l ;m H E e2;m i TE0;m ;
l  1  TM0;m iH E o2;m C bl;m   H E el1;m iEH ol−1;m or ;
L > 1 EH el−1;m  iH E ol1;m   xˆ exp
il ϕ iˆy exp
il ϕ
l ≥ 1;  F l ;m
r × xˆ exp
il ϕ iˆy exp
il ϕ

where the superscripts c (s) denotes the cosine (sine) function, and x (y) denotes polarized direction. In previous studies, tunable OAM can also be achieved by the combination of two even (odd) LPMs, which have orthogonal polarization directions with a π∕2 phase shift. Then, the continually tunable OAM can be achieved by filtering Lal;m and Lbl;m with different polarization directions, as described in [16]. In a word, as LPMs propagate in vortex fibers, M l ;m
θ (mode status) changes between LPMs and complex OAM modes periodically. Although LPMs are four-fold degenerate, the tunable OAM can be achieved from filtering the complex OAM mode despite of the LPM type. A home-made vortex fiber is fabricated by a conventional standard modified chemical vapor deposition (MCVD) process. The refractive index profile of the vortex fiber is shown in Fig. 1. The core radius and fiber radius are about 4.7 μm and 60 μm, respectively. The refractive index difference is about 0.022 between the inner layer core and cladding, and it is 0.039 between the outer layer core and cladding. The effective indices of second-order VMs are shown in the top right corner of Fig. 1. The values of index differences between two adjacent VMs are more than 1 × 10−4 , which is large enough for the sake of breaking degeneracy within the LP11 mode group [19].

(2) C al;m

C bl;m

and represent two different complex mode where states (states a and b), respectively, F l ;m represents the solution of the Bessel equation, and ϕ is the azimuthal coordinate. The continually tunable OAM can be achieved by filtering C al;m and C bl;m with different polarization directions as described in [17]. For the first and fourth equations of Eqs. (1a) and (1b), when θ is π∕2, the two equations can be rewritten as

Fig. 1. Fiber refractive index profile of the vortex fiber. The inset at the top left corner is an image of the fiber cross section, and the inset at the top right corner is the effective index of second-order VMs.

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Fig. 2. Phase difference (θ) between H E e21 and TE01 modes versus the propagation length when the LPodd 11 mode propagates in the vortex fiber. The insets show the evolution of intensity patterns. The arrows represent the polarization direction. odd It is known that the LPeven 11 (LP11 ) mode is a combination of TM01 and HE21 (TE01 and HE21 ) modes. Therefore, when the odd LPeven 11 ∕LP11 mode propagates in the vortex fiber, the phase difference between two corresponding VMs changes from 0 to 2π continually and periodically. Taking the LPeven 11 mode as an example, Fig. 2 shows that the phase difference (θ) between H E e21 and TE01 modes changes as the propagation distance increases. When θ is 0 or π, the results are LP11 modes; when θ is π∕2 or 3π∕2, the results are complex OAM modes, which are introduced in Ref. [17]. Since the propagation constant (β) decreases as the wavelength increases, the phase difference between two adjacent VMs can be controlled by changing the wavelength, which is easier to achieve than changing the fiber length in the experiment. The schematic of the experiment setup is shown in Fig. 3. A home-made mode selective coupler (MSC) is used as a mode converter from the LP01 mode to the LP11 mode [20,21]. Then, the generated LP11 mode is directly coupled into the vortex fiber. After propagating 60 cm in the vortex fiber, the output beam is collimated using a 20 times objective lens. The variation of the output can be observed by a charge coupled device (CCD). When the donuts are obtained, a polarizer is placed after the objective lens as a filer. Different LP11 modes can be generated by tuning the wavelength and polarization controller, and the mode type can be distinguished by a polarizer. The evolutions of the LPeven 11 and LPodd modes as the wavelength changes from 1531 nm to 11 1544 nm with a step length of 1 nm are shown in Fig. 4.

Fig. 3. Schematic of the experiment setup. PC, polarization controller; SMF, single mode fiber; TMF, two mode fiber; MSC, mode selective coupler; VF, vortex fiber; Obj., objective; BS, beam splitter; HWP, half-wave plate; Col., collimator; Pol., polarizer (used when capturing Figs. 5 and 6).

Letter

odd Fig. 4. Evolutions of the LPeven 11 and LP11 modes as the wavelength changes from 1531 nm to 1544 nm with a step length of 1 nm.

The phase difference between the TM01 and HE21 (from LPeven 11 ) modes changes π from Figs. 4(a4)–4(a11), so the wavelength range is about 14 nm for a circle. Similarly, the wavelength range is about 10 nm for the HE21 and TE01 modes. It should be noted that the patterns in Fig. 4 are captured without a polarizer in the experiment setup. The combination of two different VMs introduced in Ref. [17] is precisely a mode state of the evolutions of the LP11 modes. When the donuts, which contain a couple of OAM modes of opposite chirality, appear during the evolution, such as Figs. 4(a4), 4(a11), 4(b1), 4(b6), and 4(b11), we fix the wavelength and add a polarizer after the objective in the experiment setup. The results of filtering the donuts, which are odd chosen from the evolutions of the LPeven 11 and LP11 modes, are shown in Fig. 5. Figures 5(a) and 5(b) show two donuts without filtering, Figs. 5(a1)–5(a4) and 5(b1)–5(b4) show intensity patterns with different filtering directions, Figs. 5(a5)– 5(a8) and 5(b5)–5(b8) are the corresponding interference patterns. Take the results of filtering the donut that derives from the LPeven 11 mode as an example. It is obvious that Figs. 5(a2) and 5(a4) are two LPeven 11 modes with orthogonal polarization directions. It can also be confirmed from Figs. 5(a5) and 5(a7) that Figs. 5(a1) and 5(a3) are two linearly polarized OAM modes with opposite chirality, and their polarization directions are

Fig. 5. Results of filtering the complex OAM modes. The donut in odd Fig. 4(a)/Fig. 4(b) derives from the evolution of the LPeven 11 ∕LP11 mode, the arrows represent the polarization directions.

Letter

Fig. 6. Evolution of the linearly polarized OAM mode, which originates from Fig. 5(a1), as the wavelength changes from 1531 nm to 1544 nm with a step of 1 nm. (a1)–(a14) The intensity patterns, (b1)–(b14) the corresponding interference patterns. Inten., Intensity patterns; Inter., Interference patterns; the arrows represent the polarization directions.

orthogonal. Their OAM mode purities are about 95% and 90%, respectively, which are calculated by using the modified ring method [7,22]. Hence, when we continually rotate the polarizer, the OAM can be continually tuned from OAM (l  −1) to OAM (l  1). A mode selective switcher among the LPeven 11 and OAM (l  1) modes is also achieved just by rotating the polarizer at four specific polarization directions. Similarly, the tunable OAM and mode selective switcher among the LPodd 11 and OAM (l  1) modes can be obtained by filtering the donut that derives from the LPodd 11 mode. The calculated OAM mode purities are about 90% and 93% in Figs. 5(b1) and 5(b3), respectively. The OAM mode purity depends on the purity of the input LP11 mode because the OAM generation is based on the evolution of the LP11 mode in vortex fibers. The purity of LP11 modes that are generated by the MSC can reach up to nearly 100%. If we hold the direction of the polarizer when a linearly polarized OAM mode is obtained, such as Figs. 5(a1), 5(a3), 5(b1), and 5(b3), the OAM can also be continually tuned by changing the wavelength, which displays an additional dimension of tuning the OAM. The tuning process of OAM, which originates from the LPeven 11 mode [Fig. 5(a1)], is shown in Fig. 6 when we change the wavelength from 1531 nm to 1544 nm with a step of 1 nm. Figures 6(a4)– 6(a11) show the OAM continually changing from l  −1 to l  1, Figs. 6(b4)–6(b11) are the corresponding interference patterns. The mechanism is that the OAM oscillates between l  −1 to l  1 periodically, as a sin (cos) function of phase difference when the phase difference between the cos and sin components changes from 0 to 2π periodically [14,15]. Compared with Figs. 4(a4) and 4(a11), Figs. 6(a4) and 6(a11) are two linearly polarized OAM modes; meanwhile, Figs. 4(a4) and 4(a11) are two complex OAM modes. Hence, tunable OAM can be achieved from two dimensions. One dimension is to filter the complex OAM modes, as Fig. 5 shows; the other is to change the wavelength (phase difference between two components of the linearly polarized OAM modes), as Fig. 6 shows. The mode state is insensitive to temperature, but it is sensitive to bending and twisting of the fiber, because fiber

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bending will affect the phase difference between the two VMs. During our experiment, the fiber is kept in a nearly straight line. However, we can also use this effect of bending to control the phase difference between the VMs and get the desired mode state at any wavelength with any fiber length. odd In conclusion, the evolutions of the LPeven 11 ∕LP11 modes propagating in a vortex fiber are theoretically and experimentally investigated, the mode status changes periodically between LPMs and complex OAM modes during propagation. The continually tunable OAM and mode selective switcher among odd the LPeven 11 ∕LP11 and OAM (l  1) modes are achieved by filtering the complex OAM modes. The OAM can also be continually tuned by changing the wavelength (phase difference) when the polarizer is kept in a certain direction. The twodimensional tunable OAM can be achieved at any wavelength with any fiber length. A simple all fiber tunable OAM generator and mode selective switcher has a potential use in a wide range of applications, such as OAM mode division multiplexing, sensing, imaging, and particle manipulation. Funding. National Natural Science Foundation of China (NSFC) (61178008, 61275092). 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. N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, Science 340, 1545 (2013). 4. S. Fürhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, Opt. Lett. 30, 1953 (2005). 5. N. Cvijetic, G. Milione, E. Ip, and T. Wang, Sci. Rep. 5, 15422 (2015). 6. S. Ramachandran, P. Kristensen, and M. F. Yan, Opt. Lett. 34, 2525 (2009). 7. N. Bozinovic, S. Golowich, P. Kristensen, and S. Ramachandran, Opt. Lett. 37, 2451 (2012). 8. C. Brunet, P. Vaity, Y. Messaddeq, S. LaRochelle, and L. A. Rusch, Opt. Express 22, 26117 (2014). 9. S. Li, Q. Mo, X. Hu, C. Du, and J. Wang, Opt. Lett. 40, 4376 (2015). 10. P. Gregg, P. Kristensen, and S. Ramachandran, Optica 2, 267 (2015). ˇ 11. T. Cižmár, H. I. C. Dalgarno, P. C. Ashok, F. J. Gunn-Moore, and K. Dholakia, Appl. Phys. Lett. 98, 081114 (2011). 12. C. H. J. Schmitz, K. Uhrig, J. P. Spatz, and J. E. Curtis, Opt. Express 14, 6604 (2006). 13. P. Gregg, P. Kristensen, A. Rubano, S. Golowich, L. Marrucci, and S. Ramachandran, in Conference on Lasers and Electro-Optics (Optical Society of America, 2016), paper JTh4C.7. 14. R. D. Niederriter, M. E. Siemens, and J. T. Gopinath, Opt. Lett. 41, 3213 (2016). 15. X. Zeng, Y. Li, Q. Mo, W. Li, Y. Tian, Z. Liu, and J. Wu, IEEE Photon. J. 8, 1 (2016). 16. Y. Jiang, G. Ren, H. Li, M. Tang, W. Jin, W. Jian, and S. Jian, IEEE Photon. Technol. Lett. 29, 901 (2017). 17. Y. Jiang, G. Ren, Y. Lian, B. Zhu, W. Jin, and S. Jian, Opt. Lett. 41, 3535 (2016). 18. R. D. Niederriter, M. E. Siemens, and J. T. Gopinath, Opt. Lett. 41, 5736 (2016). 19. S. Ramachandran and P. Kristensen, Nanophotonics 2, 455 (2013). 20. A. Witkowska, S. G. Leon-Saval, A. Pham, and T. A. Birks, Opt. Lett. 33, 306 (2008). 21. R. Ismaeel, T. Lee, B. Oduro, Y. Jung, and G. Brambilla, Opt. Express 22, 11610 (2014). 22. Y. Jiang, G. Ren, H. Li, M. Tang, Y. Liu, Y. Wu, W. Jian, and S. Jian, Appl. Opt. 56, 1990 (2017).