Abstract

It was recently reported that a photonic crystal fiber (PCF) with no structural core guides light if a permanent chiral twist is introduced by spinning the fiber preform during the draw. The intriguing guidance mechanism behind this novel effect has many remarkable features; for example, it intrinsically supports circularly polarized helical Bloch modes (HBMs) that carry multiple optical vortices, making twisted PCFs of interest in fields such as optical micro-manipulation, imaging, quantum optics, and optical communications. Here we report for the first time, to the best of our knowledge, that a twisted coreless PCF supports not just one but a family of guided HBMs, each member of which has a unique transverse field distribution and harmonic spectrum. By making detailed interferometric measurements of the near-field phase and amplitude distributions of HBMs, and expanding them as a series of Bessel beams, we are able to extract the amplitude of each azimuthal and radial HBM harmonic. Good agreement is found with the numerical solutions of Maxwell’s equations. The results shed light on the properties of this curious new optical phenomenon.

© 2019 Optical Society of America

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References

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2019 (1)

2018 (3)

2017 (3)

2016 (1)

R. Beravat, G. K. L. Wong, M. H. Frosz, X. M. Xi, and P. St.J. Russell, Sci. Adv. 2, e1601421 (2016).
[Crossref]

2015 (2)

Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, Nat. Photonics 9, 796 (2015).
[Crossref]

R. Liu, D. B. Philips, F. Li, D. L. Andrews, and M. J. Padgett, J. Opt. 17, 45608 (2015).
[Crossref]

2013 (1)

C. Schulze, A. Dudley, D. Flamm, M. Duparré, and A. Forbes, New J. Phys. 15, 073025 (2013).
[Crossref]

2011 (1)

M. Boguslawski, P. Rose, and C. Denz, Phys. Rev. A 84, 013832 (2011).
[Crossref]

2010 (1)

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, Nat. Photonics 4, 611 (2010).
[Crossref]

2009 (1)

Q. Wang, O. Ronneberger, and H. Burkhardt, IEEE Trans. Pattern Anal. Mach. Intell. 31, 1715 (2009).
[Crossref]

Ahmed, G.

Andrews, D. L.

R. Liu, D. B. Philips, F. Li, D. L. Andrews, and M. J. Padgett, J. Opt. 17, 45608 (2015).
[Crossref]

Beravat, R.

P. Roth, Y. Chen, M. C. Günendi, R. Beravat, N. N. Edavalath, M. H. Frosz, G. Ahmed, G. K. L. Wong, and P. St.J. Russell, Optica 5, 1315 (2018).
[Crossref]

P. St.J. Russell, R. Beravat, and G. K. L. Wong, Philos. Trans. R. Soc. London A 375, 20150440 (2017).
[Crossref]

R. Beravat, G. K. L. Wong, M. H. Frosz, X. M. Xi, and P. St.J. Russell, Sci. Adv. 2, e1601421 (2016).
[Crossref]

Bliokh, Y.

Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, Nat. Photonics 9, 796 (2015).
[Crossref]

Boguslawski, M.

M. Boguslawski, P. Rose, and C. Denz, Phys. Rev. A 84, 013832 (2011).
[Crossref]

Bonaccorso, F.

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, Nat. Photonics 4, 611 (2010).
[Crossref]

Burkhardt, H.

Q. Wang, O. Ronneberger, and H. Burkhardt, IEEE Trans. Pattern Anal. Mach. Intell. 31, 1715 (2009).
[Crossref]

Cardano, F.

Chen, Y.

Cheng, C.

Z. Li, H. Liu, X. Zhang, Y. Zhang, R. Zhang, S. Xu, Y. Tang, X. Wang, J. Zhang, L. Ma, and C. Cheng, Opt. Exp. 26, 28228 (2018).
[Crossref]

D’Amelio, R.

D’Errico, A.

Denz, C.

M. Boguslawski, P. Rose, and C. Denz, Phys. Rev. A 84, 013832 (2011).
[Crossref]

Dudley, A.

C. Schulze, A. Dudley, D. Flamm, M. Duparré, and A. Forbes, New J. Phys. 15, 073025 (2013).
[Crossref]

Duparré, M.

C. Schulze, A. Dudley, D. Flamm, M. Duparré, and A. Forbes, New J. Phys. 15, 073025 (2013).
[Crossref]

Edavalath, N. N.

Ferrari, A. C.

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, Nat. Photonics 4, 611 (2010).
[Crossref]

Flamm, D.

C. Schulze, A. Dudley, D. Flamm, M. Duparré, and A. Forbes, New J. Phys. 15, 073025 (2013).
[Crossref]

Forbes, A.

C. Schulze, A. Dudley, D. Flamm, M. Duparré, and A. Forbes, New J. Phys. 15, 073025 (2013).
[Crossref]

Frosz, M. H.

Günendi, M. C.

Hasan, T.

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, Nat. Photonics 4, 611 (2010).
[Crossref]

Jiang, X.

Joly, N. Y.

Li, F.

R. Liu, D. B. Philips, F. Li, D. L. Andrews, and M. J. Padgett, J. Opt. 17, 45608 (2015).
[Crossref]

Li, Z.

Z. Li, H. Liu, X. Zhang, Y. Zhang, R. Zhang, S. Xu, Y. Tang, X. Wang, J. Zhang, L. Ma, and C. Cheng, Opt. Exp. 26, 28228 (2018).
[Crossref]

Liu, H.

Z. Li, H. Liu, X. Zhang, Y. Zhang, R. Zhang, S. Xu, Y. Tang, X. Wang, J. Zhang, L. Ma, and C. Cheng, Opt. Exp. 26, 28228 (2018).
[Crossref]

Liu, R.

R. Liu, D. B. Philips, F. Li, D. L. Andrews, and M. J. Padgett, J. Opt. 17, 45608 (2015).
[Crossref]

Ma, L.

Z. Li, H. Liu, X. Zhang, Y. Zhang, R. Zhang, S. Xu, Y. Tang, X. Wang, J. Zhang, L. Ma, and C. Cheng, Opt. Exp. 26, 28228 (2018).
[Crossref]

Marrucci, L.

Napiorkowski, M.

Nori, F.

Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, Nat. Photonics 9, 796 (2015).
[Crossref]

Padgett, M. J.

M. J. Padgett, Opt. Express 25, 11265 (2017).
[Crossref]

R. Liu, D. B. Philips, F. Li, D. L. Andrews, and M. J. Padgett, J. Opt. 17, 45608 (2015).
[Crossref]

Philips, D. B.

R. Liu, D. B. Philips, F. Li, D. L. Andrews, and M. J. Padgett, J. Opt. 17, 45608 (2015).
[Crossref]

Piccirillo, B.

Rodríguez-Fortuño, F. J.

Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, Nat. Photonics 9, 796 (2015).
[Crossref]

Ronneberger, O.

Q. Wang, O. Ronneberger, and H. Burkhardt, IEEE Trans. Pattern Anal. Mach. Intell. 31, 1715 (2009).
[Crossref]

Rose, P.

M. Boguslawski, P. Rose, and C. Denz, Phys. Rev. A 84, 013832 (2011).
[Crossref]

Roth, P.

Russell, P. St.J.

R. Sopalla, G. K. L. Wong, N. Y. Joly, M. H. Frosz, X. Jiang, G. Ahmed, and P. St.J. Russell, Opt. Lett. 44, 3964 (2019).
[Crossref]

P. Roth, Y. Chen, M. C. Günendi, R. Beravat, N. N. Edavalath, M. H. Frosz, G. Ahmed, G. K. L. Wong, and P. St.J. Russell, Optica 5, 1315 (2018).
[Crossref]

P. St.J. Russell, R. Beravat, and G. K. L. Wong, Philos. Trans. R. Soc. London A 375, 20150440 (2017).
[Crossref]

R. Beravat, G. K. L. Wong, M. H. Frosz, X. M. Xi, and P. St.J. Russell, Sci. Adv. 2, e1601421 (2016).
[Crossref]

Schulze, C.

C. Schulze, A. Dudley, D. Flamm, M. Duparré, and A. Forbes, New J. Phys. 15, 073025 (2013).
[Crossref]

Sopalla, R.

Sun, Z.

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, Nat. Photonics 4, 611 (2010).
[Crossref]

Tang, Y.

Z. Li, H. Liu, X. Zhang, Y. Zhang, R. Zhang, S. Xu, Y. Tang, X. Wang, J. Zhang, L. Ma, and C. Cheng, Opt. Exp. 26, 28228 (2018).
[Crossref]

Urbanczyk, W.

Wang, Q.

Q. Wang, O. Ronneberger, and H. Burkhardt, IEEE Trans. Pattern Anal. Mach. Intell. 31, 1715 (2009).
[Crossref]

Wang, X.

Z. Li, H. Liu, X. Zhang, Y. Zhang, R. Zhang, S. Xu, Y. Tang, X. Wang, J. Zhang, L. Ma, and C. Cheng, Opt. Exp. 26, 28228 (2018).
[Crossref]

Wong, G. K. L.

R. Sopalla, G. K. L. Wong, N. Y. Joly, M. H. Frosz, X. Jiang, G. Ahmed, and P. St.J. Russell, Opt. Lett. 44, 3964 (2019).
[Crossref]

P. Roth, Y. Chen, M. C. Günendi, R. Beravat, N. N. Edavalath, M. H. Frosz, G. Ahmed, G. K. L. Wong, and P. St.J. Russell, Optica 5, 1315 (2018).
[Crossref]

P. St.J. Russell, R. Beravat, and G. K. L. Wong, Philos. Trans. R. Soc. London A 375, 20150440 (2017).
[Crossref]

R. Beravat, G. K. L. Wong, M. H. Frosz, X. M. Xi, and P. St.J. Russell, Sci. Adv. 2, e1601421 (2016).
[Crossref]

Xi, X. M.

R. Beravat, G. K. L. Wong, M. H. Frosz, X. M. Xi, and P. St.J. Russell, Sci. Adv. 2, e1601421 (2016).
[Crossref]

Xu, S.

Z. Li, H. Liu, X. Zhang, Y. Zhang, R. Zhang, S. Xu, Y. Tang, X. Wang, J. Zhang, L. Ma, and C. Cheng, Opt. Exp. 26, 28228 (2018).
[Crossref]

Zayats, A. V.

Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, Nat. Photonics 9, 796 (2015).
[Crossref]

Zhang, J.

Z. Li, H. Liu, X. Zhang, Y. Zhang, R. Zhang, S. Xu, Y. Tang, X. Wang, J. Zhang, L. Ma, and C. Cheng, Opt. Exp. 26, 28228 (2018).
[Crossref]

Zhang, R.

Z. Li, H. Liu, X. Zhang, Y. Zhang, R. Zhang, S. Xu, Y. Tang, X. Wang, J. Zhang, L. Ma, and C. Cheng, Opt. Exp. 26, 28228 (2018).
[Crossref]

Zhang, X.

Z. Li, H. Liu, X. Zhang, Y. Zhang, R. Zhang, S. Xu, Y. Tang, X. Wang, J. Zhang, L. Ma, and C. Cheng, Opt. Exp. 26, 28228 (2018).
[Crossref]

Zhang, Y.

Z. Li, H. Liu, X. Zhang, Y. Zhang, R. Zhang, S. Xu, Y. Tang, X. Wang, J. Zhang, L. Ma, and C. Cheng, Opt. Exp. 26, 28228 (2018).
[Crossref]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

Q. Wang, O. Ronneberger, and H. Burkhardt, IEEE Trans. Pattern Anal. Mach. Intell. 31, 1715 (2009).
[Crossref]

J. Opt. (1)

R. Liu, D. B. Philips, F. Li, D. L. Andrews, and M. J. Padgett, J. Opt. 17, 45608 (2015).
[Crossref]

Nat. Photonics (2)

Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, Nat. Photonics 9, 796 (2015).
[Crossref]

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, Nat. Photonics 4, 611 (2010).
[Crossref]

New J. Phys. (1)

C. Schulze, A. Dudley, D. Flamm, M. Duparré, and A. Forbes, New J. Phys. 15, 073025 (2013).
[Crossref]

Opt. Exp. (1)

Z. Li, H. Liu, X. Zhang, Y. Zhang, R. Zhang, S. Xu, Y. Tang, X. Wang, J. Zhang, L. Ma, and C. Cheng, Opt. Exp. 26, 28228 (2018).
[Crossref]

Opt. Express (1)

Opt. Lett. (2)

Optica (2)

Philos. Trans. R. Soc. London A (1)

P. St.J. Russell, R. Beravat, and G. K. L. Wong, Philos. Trans. R. Soc. London A 375, 20150440 (2017).
[Crossref]

Phys. Rev. A (1)

M. Boguslawski, P. Rose, and C. Denz, Phys. Rev. A 84, 013832 (2011).
[Crossref]

Sci. Adv. (1)

R. Beravat, G. K. L. Wong, M. H. Frosz, X. M. Xi, and P. St.J. Russell, Sci. Adv. 2, e1601421 (2016).
[Crossref]

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Figures (6)

Fig. 1.
Fig. 1. Overview of the experimental setup. Light from the reference and sample arms is superimposed to produce an interference pattern on a CCD camera, while the phase of the reference beam is swept from 0 to 2 π . Top right, pattern written onto the SLM; lower left, SEM of the PCF microstructure.
Fig. 2.
Fig. 2. Measured and simulated near-field (a) intensity and (b) phase distributions for LCP ( s = + 1 ) versions of Mode 1 (twist period of 5 mm) and Modes 2 and 3 (twist period of 3.6 mm). The distributions for RCP modes are very similar (not shown). The simulations were based on perfectly six-fold rotationally symmetric structures with a channel diameter of 2.1 μm and spacing of 5.8 μm (average values taken from SEMs). The measured effective mode radii for Modes 1 to 3 are 12.5, 12.11, and 11.63 μm, respectively.
Fig. 3.
Fig. 3. (a) Comparison of reconstructed experimental field intensity distributions for LCP Mode 2, including (left) all spectral components in the range 40 A + 40 and (right) only those values of A corresponding to m = A 0 + 6 m . (b) Same as (a) but for RCP Mode 2.
Fig. 4.
Fig. 4. Harmonic spectra for LCP and RCP versions of Modes 1, 2, and 3, extracted from the measured field distributions using Eq. (3) and summed over radial order p . The numbered peaks correspond to the values of m expected for each mode. (The weak background peaks are caused by small errors in six-fold symmetry in the fiber structure.) Note that the OAM order of each harmonic is m s .
Fig. 5.
Fig. 5. Mode 2 amplitudes a p m as a function of p and A , calculated using Eq. (4), from FEM simulations in a perfect PCF with a twist period of 3.6 mm.
Fig. 6.
Fig. 6. (a) PBGs (white) and passbands (red) of the twisted coreless PCFs, plotted as a function of the radius. The twist periods are 5 (upper) and 3.6 mm (lower). The band structure rises to the higher values of effective index n eff in proportion to the square of the radius ρ . The effective indices of Modes 1, 2, and 3 (calculated by FEM) are marked with horizontal dashed lines. PBGs are encountered at the radii marked by red circles, creating the conditions for bound modes to form. (b) Contrast-enhanced field patterns for Mode 2 at different values of radius in the two main crystallographic directions, illustrating how the microstructure of the Bloch field changes as the edge of the mode is approached. The distances between the unit cells in the two directions are a M and a K .

Equations (4)

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E HBM = P ( ρ , ϕ α z ) e i ( A 0 ϕ + β z ) , P ( ρ , θ ) = P ( ρ , θ + 2 m π /N ) ,
E HBM ( ρ , ϕ ) = p = 1 m = a p m J m ( u p m ρ R ) e i m ϕ
E HBM = p = 1 m = a p m ( x ^ + i s y ^ ) 2 J m ( u p m ρ R ) e i ( m s ) ϕ ,
a p m = A E HBM J m ( u p m ρ R ) e i ( m s ) ϕ ρ d ϕ d ρ π J m + 1 2 ( u p m ) .

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