Abstract

Beams with polarization singularities have attracted immense recent attention in a wide array of scientific and technological disciplines. We demonstrate a class of optical fibers in which these beams can be generated and propagated over long lengths with unprecedented stability, even in the presence of strong bend perturbations. This opens the door to exploiting nonlinear fiber optics to manipulate such beams. This fiber also possesses the intriguingly counterintuitive property of being polarization maintaining despite being strictly cylindrically symmetric, a prospect hitherto considered infeasible with optical fibers.

© 2009 Optical Society of America

Cylindrical vector beams (CVBs), of which radially polarized beams are a subclass, are characterized by a nonuniform spatial distribution of their polarization vector (see top row of Fig. 1 ). This property results in several interesting ramifications, such as (a) focal spot sizes smaller than the diffraction limit [1], (b) the feasibility of electron acceleration [2], (c) optical tweezers with minimal scattering forces [3], (d) a focused pattern resembling emissions from an atomic dipole [4], and (e) higher efficiency for laser-machining of metals [5].

The preferred means to generate CVBs is to use a mode-selective element to excite them, since they are eigen-solutions of cylindrical resonators. In free-space resonators this has been achieved by adding intracavity gratings [6] or conical prisms [7]. Variants of this technique have been demonstrated with optical fibers [8, 9], which are also cylindrically symmetric resonators. The lowest-loss means to obtain CVBs with high modal purity from fibers have involved inducing the required mode transformations in the fiber itself, either with fiber gratings [10] or waveguide transitions [11].

All these intra- and extra-cavity CVB generation techniques face a debilitating mode-instability problem. The first higher-order mode group in a cylindrically symmetric resonator comprises the almost-degenerate azimuthally [Fig. 1a, TE01 in guided waves] and radially [Fig. 1d, TM01 in guided waves] polarized beams. In a fiber, this mode group additionally comprises two strictly degenerate mixed states [Figs. 1b, 1c, HE21]. Thus, even the slightest perturbations to the cylindrical symmetry of a resonator (free-space or fiber) induces coupling between them, resulting in the formation of the more familiar (undesired) Hermite–Gaussian-like beam—the so-called LP11 mode in a fiber [see colored-line pairs in Figs. 1e, 1f, 1g, 1h illustrating different orientations of the LP11 mode generated by different linear combinations of the CVBs of the top row in Fig. 1]. This means that while previously demonstrated CVB generation techniques in fibers may produce such a mode immediately after the mode converters, to our knowledge a fiber-optic means of delivering them has been infeasible.

Here we describe a class of fibers that addresses this fundamental instability problem by lifting the near-degeneracy of the TM01 (radially polarized), TE01 (azimuthally polarized), and HE21 modes. Our experiments reveal that this fiber maintains either the azimuthally or radially polarized mode with exceptional purity (>99.8%), after propagation over fiber lengths greater than 20m, even in the presence of extreme bends and twists (of radii of curvature as small as 5mm). Our demonstration also points to another intriguing phenomenon previously not observed in fibers—the ability to maintain the polarization state of a signal even though the fiber is strictly cylindrically symmetric. This is because CVBs, in contrast to Gaussian beams, naturally maintain their polarization state during propagation.

Achieving mode stability in a fiber requires separation of the propagation constants (β=2πneffλ, where λ=free-space wavelength, and neff=effective refractive index of the mode) of the different CVBs. The neff of a mode is closely related to its intensity pattern. It follows that solutions within a mode group with similar intensity profiles (such as those depicted in Fig. 1) would have substantially similar neff. A full vectorial numerical solver yields the propagation constants of the HE01, HE21, and TM01 modes of a fiber, but substantial analytical insight into this modal degeneracy problem can be gained by means of a first-order perturbative analysis [12]. First, we calculate their scalar propagation constants (identical in the scalar approximation), following which the real propagation constants are obtained through a vector correction given by

δβTE01=0,δβTM01=2(δ1+δ2),δβHE21=(δ2δ1),
where
δ1=Δnmax2a2ncoβrE(r)E(r)r(Δn(r)Δnmax)rdr,
δ2=Δnmax2a2ncoβE2(r)(Δn(r)Δnmax)rdr,
where r is the radial coordinate, E(r) is the electric field for the scalar mode, a is the size of the waveguiding core, Δn(r) is its refractive index profile relative to the index of the infinite cladding, and Δnmax is the maximum value that this function takes (typically, the index of the core). Equation (1) provides interesting physical intuition into the relationship between the field profile of a mode and its propagation constant. Phase shifts at index discontinuities (index steps) critically depend on an incident wave’s polarization orientation, and the propagation constant of a mode represents phase accumulation. Thus the search for a solution that substantially separates the propagation constants of the TE01, TM01, and HE21 modes boils down to a search for a waveguide that yields high fields [E(r)] and field gradients [E(r)r] at index steps. Thus we conclude that a waveguide whose profile mirrors that of the mode itself, i.e., an annular waveguide resembling an antiguide, would be more suitable for maximizing Δnmax while also maximizing field gradients at index steps.

Figure 2a shows the measured refractive index profile of the fiber we fabricated to test this concept. Also shown in this plot is the simulated mode profile for the LP11 mode guided by this fiber. Note that this fiber possesses the annular high-index rings that we expect to help with mode stability. However, it also has a step-index central core, as do conventional fibers. This core does not detract from the design philosophy discussed above, but instead it allows for the fundamental mode to be Gaussian shaped, enabling low-loss coupling, either from free-space lasers or conventional single-mode fibers (SMFs). Figure 2b shows the neff for the TM01, HE21, and TE01 modes in this fiber, measured by recording grating resonance wavelengths for a variety of grating periods (the grating coupling schematic and measurement technique are described in the following paragraph). The neff for the desired TM01 mode (radially polarized mode) is separated by at least 1.8×104 from any other guided mode of this fiber. For conventional fibers, the three curves would be indistinguishable in the scale of this plot, with neff differences of the order of 105.

Figure 3a shows the experimental setup used to generate CVBs in this fiber. The fiber input is spliced to the SMF, so that the light entering our fiber is in the conventional fundamental mode. Thereafter, a fiber grating, comprising periodic microbends, couples the fundamental mode to the desired vector mode. Figure 3b shows the mode conversion spectra when the grating period Λ, is 800μm—the measured conversion efficiency exceeds 99.8% at the respective resonance wavelengths λres, for each vector mode. Repeating this measurement for different grating periods yields the neff of the three vector modes as a function of wavelength, as plotted in Fig. 2b. The HE21 resonance can be excited with any state of polarization (SOP) of light entering the microbend grating, but the TE01 or TM01 modes are excited only when the input SOP is parallel or perpendicular, respectively, to the plane of microbend perturbation. Thus we can use the input SOP as well as the grating period to choose the mode we excite, as well as the wavelength at which we do so.

Figure 3c shows experimentally recorded near-field images of the fiber output when the grating is tuned either to the radially polarized (TM01), or the azimuthally polarized (TE01) resonance. All images were obtained after mode propagation over fiber lengths exceeding 20  m. Both beams have annular shapes and appear to be remarkably pure and stable—a condition that is maintained as we perturb the fiber (with bends and twists of radii as small as 5mm). With the polarizer in the beam path, only the projections of the mode that are aligned with polarizer are transmitted, leading to the double-humped intensity profile that rotates in opposite directions as the polarizer is rotated. The insertion loss, measured with a power meter, is less than 5% (0.2dB), and intensity measurements along the azimuth of the annular intensity profiles revealed variations of less than 1% (0.05dB).

Figure 4a illustrates the setup used to quantify the polarization and modal purity of these modes. The distinction from the setup of Fig. 3a is that the CVB fiber has a microbend grating at both the input as well as the output. Since gratings are reciprocal devices, light enters the CVB fiber; gets converted to the desired CVB by the first grating; and, after propagation over 20  m in this mode, gets reconverted to a conventional, Gaussian-like, fundamental mode of an SMF by the second grating. This facilitates using conventional fiber optic measurement equipment to study the stability of the CVBs in the fiber, since the input and the output are conventional beams. We record the SOP of this setup on a Poincare sphere [Fig. 4b] with a commercial polarization analyzer while perturbing the fiber with multiple twists and bends with radii of curvature down to 1cm. The top part of Fig. 4b shows a reference measurement on a conventional SMF—as expected, perturbations on an SMF result in an output that traverses virtually every point on the surface of the Poincare sphere. In contrast, a similar measurement with our fiber shows only small changes in the SOP, as represented by the limited “smearing” of the blue trace on the bottom sphere in Fig. 4b. The polarization extinction ratio, measured by recording the minimum and maximum transmitted power with a rotating polarizer at the SMF output of the bottom setup of Fig. 4a, is 28.7dB, corresponding to a modal purity level of 99.8%.

In summary, we have demonstrated the first, to the best of our knowledge, fiber in which cylindrical vector modes can be generated and maintained (propagated over greater than tens of meters) with exceptional modal purity. Since fibers are well known for their ability to offer nonlinear and dispersive tailoring of light, this demonstration opens the door to studying, and exploiting, nonlinear phenomena with such vector beams. Our results also reveal a new means of obtaining polarization-maintaining operation from fibers, even when the fibers are strictly cylindrically symmetric (in index or geometry), since vector beams, unlike the conventional fundamental mode of fibers, are naturally polarization preserving.

 figure: Fig. 1

Fig. 1 Intensity patterns (white is maximum intensity) of the first higher-order mode group in a fiber. Arrows show the orientation of the electric field in each beam. The top row shows cylindrical vector modes that are the exact vector solutions in a fiber, while the bottom row shows the resultant, unstable LP11 modes commonly obtained at a fiber output. Specific linear combinations of pairs of top row of modes, resulting in the variety of LP11 modes obtained at a fiber output, are shown by colored lines.

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 figure: Fig. 2

Fig. 2 (a) Measured refractive index profile (relative to silica index) for fabricated fiber, and corresponding LP11 mode intensity profile. (b) Effective index for the three vector components of the scalar LP11 mode for fiber shown in (a). neff of radially polarized (TM01) mode separated by 1.8×104 from other modes.

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 figure: Fig. 3

Fig. 3 (a) Experimental setup: the ring-design fiber is spliced to the SMF (bottom branch) for spectral measurements shown in (b), or cleaved and imaged on camera (top branch) for measurements shown in (c). (b) Measured grating resonance spectra for coupling from fundamental LP01 mode to desired antisymmetric mode—efficiency >99.8%. (c) Experimentally recorded near-field images for the radially polarized (TM01) mode (top) and azimuthally polarized (TE01) mode (bottom). Clean annular intensity profile for both. Image rotation with polarizer in beam path consistent with expected polarization orientation for the two modes, confirming the polarization state of the two beams.

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 figure: Fig. 4

Fig. 4 (a) Setup to compare polarization-maintaining characteristics of the CVB fiber and conventional SMF. Input and output gratings on the CVB fiber ensure that light entering or exiting the setup is conventionally polarized (i.e., spatially uniform, as in Gaussian beams), thus facilitating measurements with conventional fiber-optic test sets (such as polarization analyzers). (b) Poincare sphere representation of output SOP, to measure polarization-state variations as both fibers are perturbed. Red traces represent evolution of SOP on the front surface of the sphere (as viewed in the figure), while blue traces show SOP states on back surface of the sphere.

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1. R. Dorn, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003). [CrossRef]   [PubMed]  

2. Y. I. Salamin, Phys. Rev. A 73, 043402 (2006). [CrossRef]  

3. Q. Zhan, Opt. Express 12, 3377 (2004). [CrossRef]   [PubMed]  

4. L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, Phys. Rev. Lett. 86, 5251 (2001). [CrossRef]   [PubMed]  

5. A. V. Nesterov and V. G. Niziev, J. Phys. D 33, 1817 (2000). [CrossRef]  

6. M. A. Ahmed, J. Schulz, A. Voss, O. Parriaux, J.-C. Pommier, and T. Graf, Opt. Lett. 32, 1824 (2007). [CrossRef]   [PubMed]  

7. Y. Kozawa and S. Sato, Opt. Lett. 30, 3063 (2005). [CrossRef]   [PubMed]  

8. J.-L. Li, K.-I. Ueda, M. Musha, A. Shirakawa, and L.-X. Zhong, Opt. Lett. 31, 2969 (2006). [CrossRef]   [PubMed]  

9. G. Volpe and D. Petrov, Opt. Commun. 237, 89 (2004). [CrossRef]  

10. P. Z. Dashti, F. Alhassen, and H. P. Lee, Phys. Rev. Lett. 96, 043064 (2006). [CrossRef]  

11. A. Witkowska, S. G. L-Saval, A. Pham, and T. A. Birks, Opt. Lett. 33, 306 (2008). [CrossRef]   [PubMed]  

12. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).

References

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  1. R. Dorn, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003).
    [Crossref] [PubMed]
  2. Y. I. Salamin, Phys. Rev. A 73, 043402 (2006).
    [Crossref]
  3. Q. Zhan, Opt. Express 12, 3377 (2004).
    [Crossref] [PubMed]
  4. L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, Phys. Rev. Lett. 86, 5251 (2001).
    [Crossref] [PubMed]
  5. A. V. Nesterov and V. G. Niziev, J. Phys. D 33, 1817 (2000).
    [Crossref]
  6. M. A. Ahmed, J. Schulz, A. Voss, O. Parriaux, J.-C. Pommier, and T. Graf, Opt. Lett. 32, 1824 (2007).
    [Crossref] [PubMed]
  7. Y. Kozawa and S. Sato, Opt. Lett. 30, 3063 (2005).
    [Crossref] [PubMed]
  8. J.-L. Li, K.-I. Ueda, M. Musha, and A. Shirakawa, and L.-X. Zhong, Opt. Lett. 31, 2969 (2006).
    [Crossref] [PubMed]
  9. G. Volpe and D. Petrov, Opt. Commun. 237, 89 (2004).
    [Crossref]
  10. P. Z. Dashti, F. Alhassen, and H. P. Lee, Phys. Rev. Lett. 96, 043064 (2006).
    [Crossref]
  11. A. Witkowska, S. G. L-Saval, A. Pham, and T. A. Birks, Opt. Lett. 33, 306 (2008).
    [Crossref] [PubMed]
  12. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).

2008 (1)

2007 (1)

2006 (3)

J.-L. Li, K.-I. Ueda, M. Musha, and A. Shirakawa, and L.-X. Zhong, Opt. Lett. 31, 2969 (2006).
[Crossref] [PubMed]

Y. I. Salamin, Phys. Rev. A 73, 043402 (2006).
[Crossref]

P. Z. Dashti, F. Alhassen, and H. P. Lee, Phys. Rev. Lett. 96, 043064 (2006).
[Crossref]

2005 (1)

2004 (2)

G. Volpe and D. Petrov, Opt. Commun. 237, 89 (2004).
[Crossref]

Q. Zhan, Opt. Express 12, 3377 (2004).
[Crossref] [PubMed]

2003 (1)

R. Dorn, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

2001 (1)

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, Phys. Rev. Lett. 86, 5251 (2001).
[Crossref] [PubMed]

2000 (1)

A. V. Nesterov and V. G. Niziev, J. Phys. D 33, 1817 (2000).
[Crossref]

1983 (1)

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).

Ahmed, M. A.

Alhassen, F.

P. Z. Dashti, F. Alhassen, and H. P. Lee, Phys. Rev. Lett. 96, 043064 (2006).
[Crossref]

Beversluis, M. R.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, Phys. Rev. Lett. 86, 5251 (2001).
[Crossref] [PubMed]

Birks, T. A.

Brown, T. G.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, Phys. Rev. Lett. 86, 5251 (2001).
[Crossref] [PubMed]

Dashti, P. Z.

P. Z. Dashti, F. Alhassen, and H. P. Lee, Phys. Rev. Lett. 96, 043064 (2006).
[Crossref]

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

Graf, T.

Kozawa, Y.

Lee, H. P.

P. Z. Dashti, F. Alhassen, and H. P. Lee, Phys. Rev. Lett. 96, 043064 (2006).
[Crossref]

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

Li, J.-L.

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).

L-Saval, S. G.

Musha, M.

Nesterov, A. V.

A. V. Nesterov and V. G. Niziev, J. Phys. D 33, 1817 (2000).
[Crossref]

Niziev, V. G.

A. V. Nesterov and V. G. Niziev, J. Phys. D 33, 1817 (2000).
[Crossref]

Novotny, L.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, Phys. Rev. Lett. 86, 5251 (2001).
[Crossref] [PubMed]

Parriaux, O.

Petrov, D.

G. Volpe and D. Petrov, Opt. Commun. 237, 89 (2004).
[Crossref]

Pham, A.

Pommier, J.-C.

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

Salamin, Y. I.

Y. I. Salamin, Phys. Rev. A 73, 043402 (2006).
[Crossref]

Sato, S.

Schulz, J.

Shirakawa, A.

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).

Ueda, K.-I.

Volpe, G.

G. Volpe and D. Petrov, Opt. Commun. 237, 89 (2004).
[Crossref]

Voss, A.

Witkowska, A.

Youngworth, K. S.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, Phys. Rev. Lett. 86, 5251 (2001).
[Crossref] [PubMed]

Zhan, Q.

Zhong, L.-X.

J. Phys. D (1)

A. V. Nesterov and V. G. Niziev, J. Phys. D 33, 1817 (2000).
[Crossref]

Opt. Commun. (1)

G. Volpe and D. Petrov, Opt. Commun. 237, 89 (2004).
[Crossref]

Opt. Express (1)

Opt. Lett. (4)

Phys. Rev. A (1)

Y. I. Salamin, Phys. Rev. A 73, 043402 (2006).
[Crossref]

Phys. Rev. Lett. (3)

R. Dorn, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, Phys. Rev. Lett. 86, 5251 (2001).
[Crossref] [PubMed]

P. Z. Dashti, F. Alhassen, and H. P. Lee, Phys. Rev. Lett. 96, 043064 (2006).
[Crossref]

Other (1)

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).

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Figures (4)

Fig. 1
Fig. 1 Intensity patterns (white is maximum intensity) of the first higher-order mode group in a fiber. Arrows show the orientation of the electric field in each beam. The top row shows cylindrical vector modes that are the exact vector solutions in a fiber, while the bottom row shows the resultant, unstable LP11 modes commonly obtained at a fiber output. Specific linear combinations of pairs of top row of modes, resulting in the variety of LP11 modes obtained at a fiber output, are shown by colored lines.
Fig. 2
Fig. 2 (a) Measured refractive index profile (relative to silica index) for fabricated fiber, and corresponding LP 11 mode intensity profile. (b) Effective index for the three vector components of the scalar LP 11 mode for fiber shown in (a). n eff of radially polarized ( TM 01 ) mode separated by 1.8 × 10 4 from other modes.
Fig. 3
Fig. 3 (a) Experimental setup: the ring-design fiber is spliced to the SMF (bottom branch) for spectral measurements shown in (b), or cleaved and imaged on camera (top branch) for measurements shown in (c). (b) Measured grating resonance spectra for coupling from fundamental LP 01 mode to desired antisymmetric mode—efficiency > 99.8 % . (c) Experimentally recorded near-field images for the radially polarized ( TM 01 ) mode (top) and azimuthally polarized ( TE 01 ) mode (bottom). Clean annular intensity profile for both. Image rotation with polarizer in beam path consistent with expected polarization orientation for the two modes, confirming the polarization state of the two beams.
Fig. 4
Fig. 4 (a) Setup to compare polarization-maintaining characteristics of the CVB fiber and conventional SMF. Input and output gratings on the CVB fiber ensure that light entering or exiting the setup is conventionally polarized (i.e., spatially uniform, as in Gaussian beams), thus facilitating measurements with conventional fiber-optic test sets (such as polarization analyzers). (b) Poincare sphere representation of output SOP, to measure polarization-state variations as both fibers are perturbed. Red traces represent evolution of SOP on the front surface of the sphere (as viewed in the figure), while blue traces show SOP states on back surface of the sphere.

Equations (4)

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δ β TE 01 = 0 , δ β TM 01 = 2 ( δ 1 + δ 2 ) , δ β HE 21 = ( δ 2 δ 1 ) ,
where
δ 1 = Δ n max 2 a 2 n co β r E ( r ) E ( r ) r ( Δ n ( r ) Δ n max ) r d r ,
δ 2 = Δ n max 2 a 2 n co β E 2 ( r ) ( Δ n ( r ) Δ n max ) r d r ,

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