IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 29, NO. 11, JUNE 1, 2017
901
Tunable Orbital Angular Momentum Generation Based on Two Orthogonal LP Modes in Optical Fibers Youchao Jiang, Guobin Ren, Haisu Li, Min Tang, Wenxing Jin, Wei Jian, and Shuisheng Jian Abstract— We present a method to generate tunable orbital angular momentum (OAM) based on linearly polarized (LP) modes in optical fibers. The tunable OAM is produced by combining two even (odd) LP modes with orthogonal polarization directions. The tunability of OAM is realized by controlling the power proportion of the two LP modes through a polarizer. From another perspective, the generated mixed vortex can also be regarded as a result of overlapping two orthogonal optical vortices of equal helicity but opposite chirality, then the tunable OAM can be achieved through filtering the mixed mode using a polarizer. Index Terms— Optical vortices, optical fibers, fiber optics.
I. I NTRODUCTION
A
N OPTICAL vortex carrying orbital angular momentum (OAM) is characterized by a helical phase front of exp(±ilφ), in which φ refers to the azimuth angle and l refers to the topological charge number [1]. Because of the helical phase structure, the OAM beam cancels out at the axis, resulting in a phase singularity at which the phase is indeterminate and the amplitude is zero. The optical vortices have wide applications such as optical tweezers [2], higher dimensional optical communication [3]–[5] and microscopy [6]. In recent years, there is an increasing interest in fiberbased OAM mode generation due to the inherent advantages of fibers, such as delivery and compactness [7]–[14]. As an important aspect of manipulating OAM beams, tunability of the OAM is indispensable in many applications such as optical tweezers [15]. Generally, methods for fiber-based OAM mode generation can be classified into two types according to the source modes. For one type, the OAM modes can be generated by combining two linearly polarized (LP) modes with π/2 phase shift, where the generated vortex has no spin angular momentum (SAM) [7], [13]. For the other type, the OAM modes can be generated by combining two vector modes with π/2 phase Manuscript received October 8, 2016; revised February 5, 2017; accepted April 9, 2017. Date of publication April 12, 2017; date of current version May 3, 2017. This work was supported by the National Natural Science Foundation of China under Grant 61178008 and Grant 61275092. (Corresponding author: Guobin Ren.) The authors are with the Key Laboratory of All Optical Network & Advanced Telecommunication Network of EMC, Institute of Lightwave Technology, Beijing Jiaotong University, Beijing 100044, China (e-mail:
[email protected];
[email protected];
[email protected];
[email protected];
[email protected];
[email protected];
[email protected].). Color versions of one or more of the figures in this letter are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LPT.2017.2693680
shift, where the generated vortex has SAM characterized by +1 or -1 [8], [11]. These methods are all consist of two aspects; one is the phase shift, the other one is the mode type (LP modes or vector modes). Therefore, the achievement of tunable OAM can be conducted from the two aspects. For the first aspect, by combining an even and an odd LP modes owning the same polarization directions with continually tunable phase shift, the continually tunable OAM can be obtained [16]. It should be noted that the tunable OAM can also be obtained by combining an even and an odd vector modes (the same vector modes with different odevity) with continually tunable phase shift, although there have been no reports about it. For the second aspect, a mixed OAM mode, which can be regarded as a result of overlapping two orthogonal optical vortices of equal helicity but opposite chirality, is obtained by combining two near-degenerate vector modes in the same LP mode group with a ±π/2 phase shift. Then, the OAM can be smoothly tuned by filtering the mixed mode with different polarization directions [17]. In this letter, we present a method which is realized through combining two even (odd) LP modes owning orthogonal polarization directions with a ±π/2 phase shift to achieve the tunability of OAM. The generated OAM can be turned by controlling the power proportion of the two LP modes through a polarizer. From another perspective, the generated mixed vortex can also be regarded as a result of overlapping two orthogonal optical vortices of equal helicity but opposite chirality, the tunability of OAM can be obtained through filtering the mixed mode using a polarizer. II. T HEORY In the Cartesian coordinate system, with the weakly guiding approximation, the transverse electric field distributions of high order LP modes can be described by cosine or sine functions of the azimuth angle. Considering x- and y-polarization directions, the electric field distributions of LP modes that be described as four expressions: ⎫ ⎧ ⎧ c,x ⎫ LPl,m ⎪ x cos (lφ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ ⎨ LPs,x ⎬ x sin (lφ) l,m , (l ≥ 1) (1) =Fl,m (r ) c,y LPl,m ⎪ y cos (lφ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ ⎩ s,y ⎭ y sin (lφ) LPl,m where x and y represent the polarization directions, subscripts l and m denote transverse and radial indices, superscripts c (s) denotes cosine (sine) function and x denotes polarized direction, Fl,m represents solution of
1041-1135 © 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
the the (y) the
902
IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 29, NO. 11, JUNE 1, 2017
Fig. 1. Modal patterns of LP1,1 mode. (a) even mode in x-direction, (b) odd mode in x-direction, (c) odd mode in y-direction, (d) even mode in y-direction.
Bessel equation. According to the electric field distributions and polarization directions, LP modes can be classified as even and odd modes in both x- and y- polarization directions. For example, Fig. 1 presents the four different L P 1,1 mode s,y patterns, which are even modes of (a) LPc,x 1,1 , (d) LP1,1 and c,y s,x odd modes of (b) LP1,1 , (c) LP1,1 . The electric field of a classical LP OAM mode in optical fibers can be described as V±l,m = Fl,m (r ) exp (±ilφ) = Fl,m (r ) (cos (lφ) ±i sin (lφ)), (l≥1).
Fig. 2. (a) The Cartesian coordinate system, d1 and d2 mean the polarizer directions when the value of θ is π/4 and 3π/4. (b) The Absolute value of cosine and sine components as a function of θ . At point d1 , the two components have the same absolute value and same signs, while at point d2 , the two components have the same absolute value but opposite signs.
(2)
According to Eq. (1), the LP OAM modes in x and y polarization directions can be obtained from the follow equations:
x V±l,m x cos (lφ) ± i x sin (lφ) = Fl,m (r ) . (3) yV±l,m y cos (lφ) ± i y sin (lφ) Eq. (3) reflects the relation between LP modes and LP OAM modes. The LP OAM mode can be obtained by combining an even and an odd LP modes owning the same polarization directions with a ±π/2 phase shift. From Eq. (1) and (2), combining two even (odd) LP modes which have orthogonal polarization directions with a ±π/2 phase shift, one can obtain
a M±l,m x cos (lφ) ±i y sin (lφ) = F (4) (r ) l,m b y cos (lφ) ±i x sin (lφ) M±l,m a b and M±l,m represent two different mixed states where M±l,m (state a and b), respectively. In order to achieve the tunability a b and M±l,m . of OAM, a polarizer is used to filter the M±l,m As a result, the output derived from Eq. (4) is expressed as
P M a±l,m b P M±l,m
cos (θ ) cos (lφ) ±i si n(θ ) sin (lφ) = p Fl,m (r ) , (5) sin (θ ) cos (lφ) ±i cos(θ ) sin (lφ)
where P M a±l,m and P M b±l,m denote the output of filtered a b and M±l,m , respectively; θ is the angle between the M±l,m direction of polarizer and the positive direction of x-axis as shown in Fig. 2(a), ranging between 0 and π; p is the unit vector with the direction of polarizer. From Eq. (2), we know the LP OAM mode consists of a real cosine component and an imaginary sine component. In Eq. (5), with the θ changing from 0 to π, the absolute value of the two components vary as Fig. 2(b) shows. In Fig. 2(b), the two components have the same absolute value and same signs at point d1 (θ = π/4), and they have the same absolute value but opposite signs at point d2 ( θ = 3π/4). According to Eq. (5), at these two points,
Fig. 3. (a) The transverse electric field intensity and phase distributions of s,y LPc,x 2,1 and iLP2,1 modes, respectively. (b) The schematic of tunable OAM mode generation.
we get the OAM modes with opposite helicity as expressed in Eq. (6) and Eq. (7), √
Pd1 M a±l,m 2 exp (±ilφ) , (6) = p F × (r ) l,m b exp (±ilφ) Pd1 M±l,m 2 √
Pd2 M a±l,m 2 −ex p (∓ilφ) , (7) = p Fl,m (r ) × b Pd2 M±l,m 2 exp (∓ilφ) where the subscripts d1 and d2 mean the values of θ locate at point d1 and d2 in Fig. 2(b), respectively. s,y For example, LPc,x 2,1 and LP2,1 are two odd modes with orthogonal polarization directions as shown in the upper panel of Fig. 3(a). The transverse phase distribution of s,y LPc,x 2,1 and i LP2,1 are shown in second row of Fig. 3(a) s,y when a π/2 phase shift is added to LP2,1 . By combining s,y LPc,x 2,1 and i LP2,1 , the mixed mode is obtained. Then the tunability of OAM is achieved by controlling the power proportion of the two LP modes through a polarizer. The process is shown in Fig. 3(b) schematically and the results are shown in Fig. 4. The double-headed arrows in first row of Fig. 4 represent different polarization directions (different value of θ ) ranging from 0 to π with an increment of π/8. The transvers electric field intensity in different polarization directions are shown in second row and the corresponding transverse phase distributions are shown in third row of Fig. 4. The two LP modes that constitute the mixed mode are show in Fig. 4(e1) and (e5). Because their polarization directions are orthogonal, so, they can be separated by the polarizer.
JIANG et al.: TUNABLE OAM GENERATION BASED ON TWO ORTHOGONAL LP MODES IN OPTICAL FIBERS
direction of d1 ), √
2 P M a±l,m Fl,m (r ) = p× b P M ±l,m 2
cos(θ ) exp (±ilφ) − sin(θ )ex p (∓ilφ) . × cos(θ ) exp (±ilφ) + sin(θ )ex p (∓ilφ)
903
(10)
Eq. (4) and Eq. (9) are equivalent, same as to Eq. (5) and Eq. (10). In previous investigations, the interference vortex Vint of LP OAM modes can be described as [15]:
Vint = Fl,m (r ) a exp (i (lφ + ϕa )) + b exp (−i (lφ − ϕb )) = B (r, φ) exp (i ϕ (φ)) , (11) with constant, positive and real amplitudes a, b and phase offsets ϕa and ϕb . The B (r, φ) and ϕ (φ) are 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 Fig. 4. The simulation result of filtering the mixed mode in different polarization directions. The first row of the two panels: polarizer directions (the value of θ ); the second row of the two panels: mode patterns; the third row of the two panels: transverse phase distributions.
Fig. 4(e3) and (e7) represent the two LP OAM modes corresponding to point d1 and d2 in Fig. 2(b), respectively. As Fig. 4(p3) and (p7) show, their transverse phases vary from small to large anticlockwise and clockwise, respectively. Hence, the two LP OAM modes have orthogonal polarization directions and their helicity is opposite. In this method, when the mixed mode is filtered in x- and y- polarization directions, the outputs maintain LP modes; when the mixed mode is filtered in d1 - and d2 - polarization directions, the outputs are LP OAM modes. The electric field of the mixed mode is expressed by Eq. (4) which is based on the LP modes with x- and y- polarization directions. For analyzing the tunable OAM conveniently, we re-express the electric field based on LP OAM mode basis with d1 - and d2 - polarization directions. According to Eq. (4), and following relations cos (lφ) = 0.5{exp (ilφ) + exp (−ilφ)},
(8a)
sin (lφ) = −0.5i {exp (ilφ) − exp (−ilφ)}, (8b) 1 x + y), (8c) d 1 = √ ( 2 1 d 2 = √ (− x + y), (8d) 2 where the d 1 and d 2 are the unit vectors in d1 - and d2 - polara b ization directions as Fig. 2(a) shows, the M±l,m and M±l,m can be re-expressed as:
a √ 2 M±l,m d 1 exp (±ilφ)− d 2 ex p (∓ilφ) Fl,m (r ) . = b M±l,m d1 exp (±ilφ)+ d 2 ex p (∓ilφ) 2 (9) The new expressions of P M a±l,m and P M b±l,m can be obtained with θ = θ −π/4 (the angle between the polarizer and positive
ϕ (φ) = arctan [c tan (l (φ + α))] + (ϕa + ϕb ) /2,
(12)
where α = (ϕa − ϕb ) /(2l). The local helicity of the interference is ∂ϕ/∂φ = cl/(cos 2 [l (φ +α)] + c2 sin2 [l (φ +α)]), (13) the average OAM can be smoothly varied by changing the modulation parameter c. According to Eq. (11), the Eq. (10) can be re-expressed again as, √
P M a±l,m 2 Fl,m (r ) = p× b P M ±l,m 2
cos θ exp (±ilφ) + sin(θ )ex p (∓i (lφ + π)) , × cos(θ ) exp (±ilφ) + sin(θ )ex p (∓ilφ) (0 ≤ θ ≤ π/2), (14a) ⎧ ⎫ √ ⎨ ⎬ 2 a P M ±l,m = Fl,m (r ) p× ⎩ ⎭ 2 P M b±l,m
−cos(θ ) exp(±i (lφ + π))+sin(θ )ex p(∓i (lφ + π)) × , −cos(θ ) exp(±i (lφ + π))+sin(θ )ex p(∓ilφ) (π/2 ≤ θ ≤ π). (14b) III. R EALIZATION A PPROACHES LP mode excitation using phase plates have been widely used in recent mode division multiplexing transmission [18], the excitation of LP mode is convenient. Considering the realization of tunable LP OAM based on two LP modes with the same polarization directions in few-mode polarizationmaintaining optical fibers (FM-PMFs) [16], [19], the realization of our method is possible. The LP mode order that can be achieved determines the practical maximum value of the OAM topological charge. In addition, the high order vector modes in fibers are H E e,o , E H e,o , TE and TM modes. In the Cartesian coordinate system, with weakly guiding approximation, these six high
904
IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 29, NO. 11, JUNE 1, 2017
order modes have the following transverse electric field distributions [20]:
e H E l+1,m x cos (lφ)− y sin (lφ) , (l ≥ 1), = Fl,m (r ) o x sin (lφ)+ y cos (lφ) H E l+1,m (15)
e E H l−1,m x cos (lφ)+ y sin (lφ) = Fl,m (r ) , (l > 1), E Ho x sin (lφ)− y cos (lφ)
l−1,m x cos (φ) + y sin (φ) T M 0,m , (l = 1). = F1,m (r ) x sin (φ) − y cos (φ) T E 0,m Eq. (15) means that a vector mode can be regarded as a combination of two LP modes. By observing Eq. (4) and Eq. (15), we find that the x- and y- components in the two equations are same and the difference is just the ±π/2 phase shift between the two components. It is known that two LP modes (one even and one odd with the same polarization directions) will convert to a LP OAM mode after propagating a suitable distance in FM-PMFs because of the birefringence as presented in [19]. Therefore, it indicates that by injecting a vector mode into a FM-PMF with right angle, between the symmetry axis of mode pattern and the PMF cross-sectional axis, the mixed mode expressed as Eq. (4) will be obtained after propagating a suitable distance. The proposed method is an important supplement to the methods system of fiber-based OAM generation. Considering the convenient LP mode excitation using phase plates, the method will be easier to implement compared with [17]. Moreover, for tuning the OAM, the experiment setup needs to be adjusted incessantly to obtain different phase shift between two LP modes in [16], whereas we just need to turn the direction of polarizer which is set at the output end of the fiber. Hence, tuning the OAM is more convenient and stable with the method. IV. C ONCLUSION In summary, we presented a method to generate tunable OAM based on combination of two even (odd) LP modes owning orthogonal polarization directions with a ±π/2 phase shift. The OAM can be turned by controlling the power proportion of the two LP modes through a polarizer. From another perspective, the generated mixed vortex can be regarded as a result of overlapping two orthogonal optical vortices of equal helicity but opposite chirality, then the tunability of OAM can be achieved just through filtering the mixed mode using a polarizer. Two possible approaches for realizing the method have been presented. One is to inject two even (odd) LP modes with orthogonal polarization directions to a FM-PMF, the other one is to inject one vector mode to a FM-PMF. The method theoretically provides a new different mechanism to obtain
tunable OAM which has potential uses in optical tweezers, fiber communications, imaging. R EFERENCES [1] L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of LaguerreGaussian laser modes,” Phys. Rev. A, Gen. Phys., vol. 45, no. 11, pp. 8185–8189, 1992. [2] J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun., vol. 207, pp. 169–175, Jun. 2002. [3] G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, and V. Pas’ko, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Exp., vol. 12, no. 22, pp. 5448–5456, 2004. [4] A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, threedimensional entanglement for quantum communication,” Phys. Rev. Lett., vol. 89, no. 24, p. 240401, 2002. [5] H. Li et al., “Broadband orbital angular momentum transmission using a hollow-core photonic bandgap fiber,” Opt. Lett., vol. 41, no. 15, pp. 3591–3594, 2016. [6] S. Fürhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Spiral interferometry,” Opt. Lett., vol. 30, no. 15, pp. 1953–1955, Aug. 2005. [7] D. McGloin, N. B. Simpson, and M. J. Padgett, “Transfer of orbital angular momentum from a stressed fiber-optic waveguide to a light beam,” Appl. Opt., vol. 37, no. 3, pp. 469–472, 1998. [8] P. Z. Dashti, F. Alhassen, and H. P. Lee, “Observation of orbital angular momentum transfer between acoustic and optical vortices in optical fiber,” Phys. Rev. Lett., vol. 96, no. 4, p. 043604, Feb. 2006. [9] S. Ramachandran, P. Kristensen, and M. F. Yan, “Generation and propagation of radially polarized beams in optical fiber,” Opt. Lett., vol. 34, no. 16, pp. 2525–2527, 2009. [10] Y. Yan et al., “Fiber coupler for generating orbital angular momentum modes,” Opt. Lett., vol. 36, no. 21, pp. 4269–4271, 2011. [11] N. Bozinovic, S. Golowich, P. Kristensen, and S. Ramachandran, “Control of orbital angular momentum of light with optical fibers,” Opt. Lett., vol. 37, no. 13, pp. 2451–2453, 2012. [12] C. Brunet, P. Vaity, Y. Messaddeq, S. LaRochelle, and L. A. Rusch, “Design, fabrication and validation of an OAM fiber supporting 36 states,” Opt. Exp., vol. 22, no. 21, pp. 26117–26127, 2014. [13] S. H. Li, Q. Mo, X. Hu, C. Du, and J. Wang, “Controllable all-fiber orbital angular momentum mode converter,” Opt. Lett., vol. 40, no. 18, pp. 4376–4379, Sep. 2015. [14] P. Gregg, P. Kristensen, and S. Ramachandran, “Conservation of orbital angular momentum in air-core optical fibers,” Optica, vol. 2, no. 3, pp. 267–270, 2015. [15] C. H. J. Schmitz, K. Uhrig, J. P. Spatz, and J. E. Curtis, “Tuning the orbital angular momentum in optical vortex beams,” Opt. Exp., vol. 14, no. 15, pp. 6604–6612, 2006. [16] R. D. Niederriter, M. E. Siemens, and J. T. Gopinath, “Continuously tunable orbital angular momentum generation using a polarizationmaintaining fiber,” Opt. Lett., vol. 41, no. 14, p. 3213, 2016. [17] Y. Jiang, G. Ren, Y. Lian, B. Zhu, W. Jin, and S. Jian, “Tunable orbital angular momentum generation in optical fibers,” Opt. Lett., vol. 41, no. 15, p. 3535, 2016. [18] M. M. Ali et al., “Characterization of mode coupling in few-mode FBG with selective mode excitation,” IEEE Photon. Technol. Lett., vol. 27, no. 16, pp. 1713–1716, Aug. 15, 2015. [19] X. Zeng et al., “Experimental investigation of LP11 mode to OAM conversion in few mode-polarization maintaining fiber and the usage for all fiber OAM generator,” IEEE Photon. J., vol. 8, no. 4, pp. 1–7, Aug. 2016. [20] A. W. Snyder and J. D. Love, Optical Waveguide Theory. London, U.K.: Chapman & Hall, 1983.
