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  • Vol. 37, Iss. 13 — Jul. 1, 2012
  • pp: 2451–2453
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Control of orbital angular momentum of light with optical fibers

Nenad Bozinovic, Steven Golowich, Poul Kristensen, and Siddharth Ramachandran  »View Author Affiliations


Optics Letters, Vol. 37, Issue 13, pp. 2451-2453 (2012)
http://dx.doi.org/10.1364/OL.37.002451


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Abstract

We present a fiber-based method for generating vortex beams with a tunable value of orbital angular momentum from 1 to +1 per photon. We propose a new (to our knowledge) method to determine the modal content of the fiber and demonstrate high purity of the desired vortex state (97% after 20 m, even after bends and twists). This method has immediate utility for the multitude of applications in science and technology that exploit vortex light states.

© 2012 Optical Society of America

Because of their ability to carry orbital angular momentum (OAM) [1

1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992). [CrossRef]

], vortex beams have generated considerable interest in the recent past, finding applications in the areas of optical tweezers [2

2. J. E. Curtis, B. A. Koss, and D. G. Grier, Opt. Commun. 207, 169 (2002). [CrossRef]

,3

3. M. K. Kreysing, T. Kießling, A. Fritsch, C. Dietrich, J. R. Guck, and J. A. Käs, Opt. Express 16, 16984 (2008). [CrossRef]

], higher dimensional classical and quantum communications [4

4. G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, Opt. Express 12, 5448 (2004). [CrossRef]

,5

5. A. Vaziri, G. Weihs, and A. Zeilinger, Phys. Rev. Lett. 89, 240401 (2002). [CrossRef]

], atom manipulation [6

6. J. W. R. Tabosa and D. V. Petrov, Phys. Rev. Lett. 83, 4967 (1999). [CrossRef]

], and microscopy [7

7. S. W. Hell and J. Wichmann, Opt. Lett. 19, 780 (1994). [CrossRef]

].

The most common methods for vortex beam generation use spatial light modulators (SLM). A fiber-based generation technique, on the other hand, would yield advantages commonly derived from fibers, such as remote delivery and compactness. In addition, the prospect of exploiting fiber nonlinear optical properties [8

8. S. Ramachandran, C. Smith, P. Kristensen, and P. Balling, Opt. Express 18, 23212 (2010). [CrossRef]

] would enable temporal and spectral control of vortex beams.

In order to carry OAM, a fiber must support higher-order modes (HOMs). Specifically, it can be shown that the linear combination of two HE21 HOMs with a ±π/2 phase shift between them will result in OAM states [9

9. A. Snyder and J. D. Love, Optical Waveguide Theory (Springer, 1983).

,10

10. P. Z. Dashti, F. Alhassen, and H. P. Lee, Phys. Rev. Lett. 96, 043604 (2006). [CrossRef]

]. Systems that induce stress in a multimode fiber [11

11. D. McGloin, N. B. Simpson, and M. J. Padgett, Appl. Opt. 37, 469 (1998). [CrossRef]

] or utilize acoustic long-period gratings [10

10. P. Z. Dashti, F. Alhassen, and H. P. Lee, Phys. Rev. Lett. 96, 043604 (2006). [CrossRef]

] to achieve this π/2 phase shift have been demonstrated. However, mode coupling in fibers destabilizes the OAM states, leading to multipath interference (MPI) [12

12. S. Ramachandran, J. Nicholson, S. Ghalmi, and M. Yan, IEEE Photon. Technol. Lett. 15, 1171 (2003). [CrossRef]

,13

13. R. Olshansky, Appl. Opt. 14, 935 (1975). [CrossRef]

]. In most multimode fibers, the TE and TM modes always coexist with the desired HE21 modes and will couple with them to produce linearly polarized (LP) states at the output [14

14. S. Ramachandran, P. Kristensen, and M. F. Yan, Opt. Lett. 34, 2525 (2009). [CrossRef]

]. Not being true eigenmodes of the fiber, the LP modes cannot carry OAM, and to the best of our knowledge, OAM states have been demonstrated only in short (30cm), straight fibers.

In this Letter, we show that the OAM states can be created with 97% purity in a 20 m long fiber, even in the presence of bends and twists. To achieve this, we designed and fabricated a so-called vortex fiber that lifts the degeneracy among the higher-order LP11 family of modes, thereby minimizing the coupling to the TE and TM modes [14

14. S. Ramachandran, P. Kristensen, and M. F. Yan, Opt. Lett. 34, 2525 (2009). [CrossRef]

]. Figure 1 shows the experimental setup. Using standard single mode fiber (SMF), a 50 nm wide 1550 nm LED and a narrowband CW tunable laser (Agilent 8168F) were multiplexed into a 20 m long vortex fiber. Thereafter, using a microbend grating (40 mm length, 475 μm period) [15

15. J. N. Blake, B. Y. Kim, H. E. Engan, and H. J. Shaw, Opt. Lett. 12, 281 (1987). [CrossRef]

], with only the LED source turned on, we obtained 18 dB of mode conversion from the input fundamental mode to the desired HE21odd mode [see the transmission spectrum in Fig. 1(b) obtained using the optical spectrum analyzer (OSA)]. Next, we switched the source to the laser, tuned to the resonant mode-conversion wavelength (1527 nm). The vortex fiber was then cleaved approximately 20 m after the output of the microbend grating and imaged onto a camera (VDS, NIR-300, InGaAs).

Fig. 1. (a) Experimental setup, (b) grating resonance spectrum used to deduce HE21odd mode-conversion level, (c) camera image showing l=1 OAM, s=1 SAM state.

In order to determine the purity of the vortex states thus obtained, we have developed a new method that analyzes fiber output projections onto left circular (LC) and right circular (RC) polarization states. In addition, to observe the phase of the beam, we interfered a vertical (V) polarization projection with the reference beam. Using a combination of nonpolarizing beam splitters (NPBSs), quarter wave plate (QWP) and polarizing beam displacing prisms (PBDPs), we devised a setup capable of recording these projections in one camera shot [horizontal (H) projection was also recorded]. Previously, it was shown that a linear combination of two l=±1 OAM modes will have a total OAM of topological charge that lies in 1l1. [16

16. C. H. J. Schmitz, K. Uhrig, J. P. Spatz, and J. E. Curtis, Opt. Express 14, 6604 (2006). [CrossRef]

]. By adjusting paddles on a commercial polarization controller mounted on our vortex fiber [PolCon2 of Fig. 1(a)], we were able to tune the output OAM state from l=1 to l=1 [Fig. 1(c) shows the output of the l=1 OAM and s=1 spin angular momentum (SAM) state].

In general, light at the fiber output can contain contributions from the six vector modes: HE11x,y, HE21even,odd, TM01, and TE01 [9

9. A. Snyder and J. D. Love, Optical Waveguide Theory (Springer, 1983).

]. To analyze the purity of the OAM states, it is more convenient to introduce a so-called vortex basis set:
V11+(r,θ)=def(HE11x+iHE11y)/2=(x^+iy^)F01/2,V11(r,θ)=def(HE11xiHE11y)/2=(x^iy^)F01/2,V21+(r,θ)=def(HE21e+iHE21o)/2=eiθ(x^+iy^)F11/2,V21(r,θ)=def(HE21eiHE21o)/2=eiθ(x^iy^)F11/2,VT+(r,θ)=def(TM01iTE01)/2=eiθ(x^+iy^)F11/2,VT(r,θ)=def(TM01+iTE01)/2=eiθ(x^iy^)F11/2,
where we have used the fact that, for this fiber, the vector modes may be accurately expressed as linear combinations of LP modes through the weak guidance approximation, and Flm(r) denotes the radial wave functions of the LPlm modes [9

9. A. Snyder and J. D. Love, Optical Waveguide Theory (Springer, 1983).

]. Note that in this representation, the V21+, modes correspond to the desired OAM states in the fiber. The total electric field can be expressed as
E(r,θ)=l=[11,21,T]s=[+,]γlsVls(r,θ),
(1)
where γls is the mode field complex amplitude of the vortex basis vectors. We denote mode power contributions as
MPIls=def10log10(|γls|2/Ptot),
(2)
where Ptot=ls|γls|2. To experimentally measure the mode amplitudes, we consider the intensity of the LC polarization projection:
|P+E(r,θ)|2=|γ11+F01(r)+γ21+eiθF11(r)+γT+eiθF11(r)|2.
The key approximation in our analysis is that the |γ11+||γT+| term may be neglected, which is valid when most of the power is confined to the HE21 modes, as is the case in the experiments presented below. In addition, for simplicity we confine attention to the points on the radius r0, for which F01(r0)F11(r0) (we observe that r0 conveniently corresponds to the radius of the LC and RC projection ring). The LC azimuthal intensity at r0 now simplifies to
|P+E(r0,θ)|2DC+Δ1cos(θ+ϕ21,11)+Δ2cos(2θ+ϕ21,T),
(3)
where ϕij indicates the phase difference between the two corresponding modes, and we define
DC=def|γ11+|2+|γ21+|2+|γT+|2,
(4)
Δ1=def2|γ11+||γ21+|,
(5)
Δ2=def2|γ21+||γT+|.
(6)
By taking the Fourier series of |P+E(r0,θ)|2, the coefficients DC, Δ1, and Δ2 can be determined; the mode powers (|γls|2) can then be obtained by solving Eqs. (4)–(6). Figure 2(a) shows an example of a measurement of |P+E(r0,θ)|2—the azimuthal intensity variation of the image that occurs due to interference between the V21+, V11+, and VT+ modes. Figure 2(b) shows the example of the Fourier series analysis, and Fig. 2(c) illustrates the powers of the extracted modes. An equivalent procedure was also repeated for the RC projection to calculate amplitudes of the negative helicity modes.

Fig. 2. (a) Azimuthal intensity profile of LC projection for radius r0; (b) Fourier series coefficients for the profile in (a); (c) extracted modal power contributions.

The high purity of the vortex states implies that only two modes—the HE21even,odd pair—are dominantly present in the fiber. Here, we would like to emphasize the analogy of the two HE21 modes with the two linearly polarized fundamental modes—HE11x,y. As with the two fundamental modes, we first note that since the HE21 modes are degenerate, they can easily couple to each other under controlled perturbations. Second, just as a linear combination of two fundamental modes with ±π/2 phase shift will create a circularly polarized state, the same linear combination of the two HE21 modes results in an OAM state. This analogy has also been elegantly represented by a higher-order Poincaré sphere by other authors studying OAM [17

17. M. J. Padgett and J. Courtial, Opt. Lett. 24, 430 (1999). [CrossRef]

,18

18. G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, Phys. Rev. Lett. 107, 053601 (2011). [CrossRef]

]. A general linear combination of the two l=±1 OAM modes will have a topological charge with noninteger total OAM [16

16. C. H. J. Schmitz, K. Uhrig, J. P. Spatz, and J. E. Curtis, Opt. Express 14, 6604 (2006). [CrossRef]

].

Fig. 3. (a) Mode powers as the PolCon2 was adjusted to obtain the desired superposition of the OAM states, (b) observed camera images at points A–C.

We would like to thank P.E. Steinvurzel and P. Gregg for helpful discussions. This work was funded by Defense Advanced Research Projects Agency (DARPA) grant no. HR0011-11-1-0004.

References

1.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992). [CrossRef]

2.

J. E. Curtis, B. A. Koss, and D. G. Grier, Opt. Commun. 207, 169 (2002). [CrossRef]

3.

M. K. Kreysing, T. Kießling, A. Fritsch, C. Dietrich, J. R. Guck, and J. A. Käs, Opt. Express 16, 16984 (2008). [CrossRef]

4.

G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, Opt. Express 12, 5448 (2004). [CrossRef]

5.

A. Vaziri, G. Weihs, and A. Zeilinger, Phys. Rev. Lett. 89, 240401 (2002). [CrossRef]

6.

J. W. R. Tabosa and D. V. Petrov, Phys. Rev. Lett. 83, 4967 (1999). [CrossRef]

7.

S. W. Hell and J. Wichmann, Opt. Lett. 19, 780 (1994). [CrossRef]

8.

S. Ramachandran, C. Smith, P. Kristensen, and P. Balling, Opt. Express 18, 23212 (2010). [CrossRef]

9.

A. Snyder and J. D. Love, Optical Waveguide Theory (Springer, 1983).

10.

P. Z. Dashti, F. Alhassen, and H. P. Lee, Phys. Rev. Lett. 96, 043604 (2006). [CrossRef]

11.

D. McGloin, N. B. Simpson, and M. J. Padgett, Appl. Opt. 37, 469 (1998). [CrossRef]

12.

S. Ramachandran, J. Nicholson, S. Ghalmi, and M. Yan, IEEE Photon. Technol. Lett. 15, 1171 (2003). [CrossRef]

13.

R. Olshansky, Appl. Opt. 14, 935 (1975). [CrossRef]

14.

S. Ramachandran, P. Kristensen, and M. F. Yan, Opt. Lett. 34, 2525 (2009). [CrossRef]

15.

J. N. Blake, B. Y. Kim, H. E. Engan, and H. J. Shaw, Opt. Lett. 12, 281 (1987). [CrossRef]

16.

C. H. J. Schmitz, K. Uhrig, J. P. Spatz, and J. E. Curtis, Opt. Express 14, 6604 (2006). [CrossRef]

17.

M. J. Padgett and J. Courtial, Opt. Lett. 24, 430 (1999). [CrossRef]

18.

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, Phys. Rev. Lett. 107, 053601 (2011). [CrossRef]

OCIS Codes
(060.2340) Fiber optics and optical communications : Fiber optics components
(060.4230) Fiber optics and optical communications : Multiplexing
(140.3300) Lasers and laser optics : Laser beam shaping
(050.4865) Diffraction and gratings : Optical vortices
(260.6042) Physical optics : Singular optics

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: March 8, 2012
Manuscript Accepted: April 20, 2012
Published: June 18, 2012

Citation
Nenad Bozinovic, Steven Golowich, Poul Kristensen, and Siddharth Ramachandran, "Control of orbital angular momentum of light with optical fibers," Opt. Lett. 37, 2451-2453 (2012)
http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-37-13-2451


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References

  1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992). [CrossRef]
  2. J. E. Curtis, B. A. Koss, and D. G. Grier, Opt. Commun. 207, 169 (2002). [CrossRef]
  3. M. K. Kreysing, T. Kießling, A. Fritsch, C. Dietrich, J. R. Guck, and J. A. Käs, Opt. Express 16, 16984 (2008). [CrossRef]
  4. G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, Opt. Express 12, 5448 (2004). [CrossRef]
  5. A. Vaziri, G. Weihs, and A. Zeilinger, Phys. Rev. Lett. 89, 240401 (2002). [CrossRef]
  6. J. W. R. Tabosa, and D. V. Petrov, Phys. Rev. Lett. 83, 4967 (1999). [CrossRef]
  7. S. W. Hell, and J. Wichmann, Opt. Lett. 19, 780 (1994). [CrossRef]
  8. S. Ramachandran, C. Smith, P. Kristensen, and P. Balling, Opt. Express 18, 23212 (2010). [CrossRef]
  9. A. Snyder, and J. D. Love, Optical Waveguide Theory (Springer, 1983).
  10. P. Z. Dashti, F. Alhassen, and H. P. Lee, Phys. Rev. Lett. 96, 043604 (2006). [CrossRef]
  11. D. McGloin, N. B. Simpson, and M. J. Padgett, Appl. Opt. 37, 469 (1998). [CrossRef]
  12. S. Ramachandran, J. Nicholson, S. Ghalmi, and M. Yan, IEEE Photon. Technol. Lett. 15, 1171 (2003). [CrossRef]
  13. R. Olshansky, Appl. Opt. 14, 935 (1975). [CrossRef]
  14. S. Ramachandran, P. Kristensen, and M. F. Yan, Opt. Lett. 34, 2525 (2009). [CrossRef]
  15. J. N. Blake, B. Y. Kim, H. E. Engan, and H. J. Shaw, Opt. Lett. 12, 281 (1987). [CrossRef]
  16. C. H. J. Schmitz, K. Uhrig, J. P. Spatz, and J. E. Curtis, Opt. Express 14, 6604 (2006). [CrossRef]
  17. M. J. Padgett, and J. Courtial, Opt. Lett. 24, 430 (1999). [CrossRef]
  18. G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, Phys. Rev. Lett. 107, 053601 (2011). [CrossRef]

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