Abstract

We consider the general problem of free-space beam shaping for coupling in and out of higher order modes (HOMs) in optical fibers with high purity and low loss. We compare the performance of two simple phase structures – binary phase plates (BPPs) and axicons – for converting Gaussian beams to HOMs and vice versa. Both axicons and BPPs allow for excitation of modes with high purity (>15 dB parasitic mode suppression), or conversion of HOMs to near-Gaussian beams (M2 < 1.25). Axicon coupling in single-clad fibers allows for lower loss (0.85 ± 0.1 dB) conversion than BPPs (1.7 ± 0.1 dB); but BPPs are compatible with any fiber design, and allow for rapid switching between modes. The experiments detailed here use all commercial components and fibers, allowing for a simple means to investigate the unique properties of multi-mode fibers.

© 2015 Optical Society of America

Full Article  |  PDF Article

Corrections

Jeff Demas, Lars Rishøj, and Siddharth Ramachandran, "Free-space beam shaping for precise control and conversion of modes in optical fiber: errata," Opt. Express 23, 33587-33587 (2015)
https://www.osapublishing.org/oe/abstract.cfm?uri=oe-23-26-33587

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References

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2015 (3)

2014 (1)

2013 (1)

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[Crossref] [PubMed]

2012 (5)

2011 (3)

2009 (1)

2007 (4)

2006 (1)

2005 (2)

2003 (1)

S. Ramachandran, J. W. Nicholson, S. Ghalmi, and M. F. Yan, “Measurement of multipath interference in the coherent crosstalk regime,” IEEE Photonics Technol. Lett. 15(8), 1171–1173 (2003).
[Crossref]

2002 (1)

1993 (1)

1987 (1)

1974 (1)

R. H. Stolen, J. E. Bjorkholm, and A. Ashkin, “Phase‐matched three‐wave mixing in silica fiber optical waveguides,” Appl. Phys. Lett. 24(7), 308–310 (1974).
[Crossref]

Armstrong, J. P.

Ashkin, A.

R. H. Stolen, J. E. Bjorkholm, and A. Ashkin, “Phase‐matched three‐wave mixing in silica fiber optical waveguides,” Appl. Phys. Lett. 24(7), 308–310 (1974).
[Crossref]

Barankov, R. A.

Betzig, E.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2011).
[Crossref] [PubMed]

Birks, T. A.

Bjorkholm, J. E.

R. H. Stolen, J. E. Bjorkholm, and A. Ashkin, “Phase‐matched three‐wave mixing in silica fiber optical waveguides,” Appl. Phys. Lett. 24(7), 308–310 (1974).
[Crossref]

Bolle, C.

Boyd, R. W.

Bozinovic, N.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[Crossref] [PubMed]

Burrows, E. C.

Carpenter, J.

Charraut, D.

Chen, Y.

Dally, A.

Davidson, M. W.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2011).
[Crossref] [PubMed]

Dawson, J. W.

Demas, J.

DeSantolo, A.

J. W. Nicholson, J. M. Fini, A. DeSantolo, P. S. Westbrook, R. S. Windeler, T. Kremp, C. Headley, and D. J. DiGiovanni, “A higher-order mode fiber amplifier with an axicon for output mode conversion,” Proc. SPIE 9344, 93441V (2015).

DeSantolo, A. M.

Dholakia, K.

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
[Crossref]

DiGiovanni, D. J.

J. W. Nicholson, J. M. Fini, A. DeSantolo, P. S. Westbrook, R. S. Windeler, T. Kremp, C. Headley, and D. J. DiGiovanni, “A higher-order mode fiber amplifier with an axicon for output mode conversion,” Proc. SPIE 9344, 93441V (2015).

J. W. Nicholson, J. M. Fini, A. M. DeSantolo, X. Liu, K. Feder, P. S. Westbrook, V. R. Supradeepa, E. Monberg, F. DiMarcello, R. Ortiz, C. Headley, and D. J. DiGiovanni, “Scaling the effective area of higher-order-mode erbium-doped fiber amplifiers,” Opt. Express 20(22), 24575–24584 (2012).
[Crossref] [PubMed]

DiMarcello, F.

Dimarcello, F. V.

Drachenberg, D. R.

Durnin, J.

Esmaeelpour, M.

Essiambre, R.

Feder, K.

Fini, J. M.

J. W. Nicholson, J. M. Fini, A. DeSantolo, P. S. Westbrook, R. S. Windeler, T. Kremp, C. Headley, and D. J. DiGiovanni, “A higher-order mode fiber amplifier with an axicon for output mode conversion,” Proc. SPIE 9344, 93441V (2015).

J. W. Nicholson, J. M. Fini, A. M. DeSantolo, X. Liu, K. Feder, P. S. Westbrook, V. R. Supradeepa, E. Monberg, F. DiMarcello, R. Ortiz, C. Headley, and D. J. DiGiovanni, “Scaling the effective area of higher-order-mode erbium-doped fiber amplifiers,” Opt. Express 20(22), 24575–24584 (2012).
[Crossref] [PubMed]

Galbraith, C. G.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2011).
[Crossref] [PubMed]

Galbraith, J. A.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2011).
[Crossref] [PubMed]

Gao, L.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2011).
[Crossref] [PubMed]

Ghalmi, S.

S. Ramachandran, J. W. Nicholson, S. Ghalmi, M. F. Yan, P. Wisk, E. Monberg, and F. V. Dimarcello, “Light propagation with ultralarge modal areas in optical fibers,” Opt. Lett. 31(12), 1797–1799 (2006).
[Crossref] [PubMed]

S. Ramachandran, J. W. Nicholson, S. Ghalmi, and M. F. Yan, “Measurement of multipath interference in the coherent crosstalk regime,” IEEE Photonics Technol. Lett. 15(8), 1171–1173 (2003).
[Crossref]

Gnauck, A. H.

Goto, M.

Gregg, P.

Grosjean, T.

Headley, C.

J. W. Nicholson, J. M. Fini, A. DeSantolo, P. S. Westbrook, R. S. Windeler, T. Kremp, C. Headley, and D. J. DiGiovanni, “A higher-order mode fiber amplifier with an axicon for output mode conversion,” Proc. SPIE 9344, 93441V (2015).

J. W. Nicholson, J. M. Fini, A. M. DeSantolo, X. Liu, K. Feder, P. S. Westbrook, V. R. Supradeepa, E. Monberg, F. DiMarcello, R. Ortiz, C. Headley, and D. J. DiGiovanni, “Scaling the effective area of higher-order-mode erbium-doped fiber amplifiers,” Opt. Express 20(22), 24575–24584 (2012).
[Crossref] [PubMed]

Heebner, J. E.

Huang, H.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[Crossref] [PubMed]

Ibrahim, I. A.

Isenhower, L.

Johnson, E. G.

Karimi, E.

Kremp, T.

J. W. Nicholson, J. M. Fini, A. DeSantolo, P. S. Westbrook, R. S. Windeler, T. Kremp, C. Headley, and D. J. DiGiovanni, “A higher-order mode fiber amplifier with an axicon for output mode conversion,” Proc. SPIE 9344, 93441V (2015).

Kristensen, P.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[Crossref] [PubMed]

Lai, K.

Leon-Saval, S. G.

Leuchs, G.

Lindlein, N.

Lingle, R.

Liu, X.

Marrucci, L.

McCurdy, A. H.

McGloin, D.

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
[Crossref]

Mehta, A.

Milkie, D. E.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2011).
[Crossref] [PubMed]

Mirhosseini, M.

Mohammed, W. S.

Monberg, E.

Mumtaz, S.

Nicholson, J. W.

J. W. Nicholson, J. M. Fini, A. DeSantolo, P. S. Westbrook, R. S. Windeler, T. Kremp, C. Headley, and D. J. DiGiovanni, “A higher-order mode fiber amplifier with an axicon for output mode conversion,” Proc. SPIE 9344, 93441V (2015).

J. W. Nicholson, J. M. Fini, A. M. DeSantolo, X. Liu, K. Feder, P. S. Westbrook, V. R. Supradeepa, E. Monberg, F. DiMarcello, R. Ortiz, C. Headley, and D. J. DiGiovanni, “Scaling the effective area of higher-order-mode erbium-doped fiber amplifiers,” Opt. Express 20(22), 24575–24584 (2012).
[Crossref] [PubMed]

S. Ramachandran, J. W. Nicholson, S. Ghalmi, M. F. Yan, P. Wisk, E. Monberg, and F. V. Dimarcello, “Light propagation with ultralarge modal areas in optical fibers,” Opt. Lett. 31(12), 1797–1799 (2006).
[Crossref] [PubMed]

S. Ramachandran, J. W. Nicholson, S. Ghalmi, and M. F. Yan, “Measurement of multipath interference in the coherent crosstalk regime,” IEEE Photonics Technol. Lett. 15(8), 1171–1173 (2003).
[Crossref]

Ortiz, R.

Pax, P. H.

Peckham, D. W.

Piquerey, V.

Planchon, T. A.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2011).
[Crossref] [PubMed]

Ramachandran, S.

J. Demas, P. Steinvurzel, B. Tai, L. Rishøj, Y. Chen, and S. Ramachandran, “Intermodal nonlinear mixing with Bessel beams in optical fiber,” Optica 2(1), 14–17 (2015).
[Crossref]

P. Gregg, M. Mirhosseini, A. Rubano, L. Marrucci, E. Karimi, R. W. Boyd, and S. Ramachandran, “Q-plates as higher order polarization controllers for orbital angular momentum modes of fiber,” Opt. Lett. 40(8), 1729–1732 (2015).
[Crossref] [PubMed]

J. Demas and S. Ramachandran, “Sub-second mode measurement of fibers using C2 imaging,” Opt. Express 22(19), 23043–23056 (2014).
[Crossref] [PubMed]

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[Crossref] [PubMed]

Y. Chen, L. Yan, L. Rishøj, P. Steinvurzel, and S. Ramachandran, “Dynamically tunable optical bottles from an optical fiber,” Opt. Lett. 37(16), 3327–3329 (2012).
[Crossref] [PubMed]

P. Steinvurzel, K. Tantiwanichapan, M. Goto, and S. Ramachandran, “Fiber-based Bessel beams with controllable diffraction-resistant distance,” Opt. Lett. 36(23), 4671–4673 (2011).
[Crossref] [PubMed]

D. N. Schimpf, R. A. Barankov, and S. Ramachandran, “Cross-correlated (C2) imaging of fiber and waveguide modes,” Opt. Express 19(14), 13008–13019 (2011).
[Crossref] [PubMed]

N. Lindlein, G. Leuchs, and S. Ramachandran, “Achieving Gaussian outputs from large-mode-area higher-order-mode fibers,” Appl. Opt. 46(22), 5147–5157 (2007).
[Crossref] [PubMed]

S. Ramachandran, J. W. Nicholson, S. Ghalmi, M. F. Yan, P. Wisk, E. Monberg, and F. V. Dimarcello, “Light propagation with ultralarge modal areas in optical fibers,” Opt. Lett. 31(12), 1797–1799 (2006).
[Crossref] [PubMed]

S. Ramachandran, “Dispersion-tailored few-mode fibers: a versatile platform for in-fiber photonic devices,” J. Lightwave Technol. 23(11), 3426–3443 (2005).
[Crossref]

S. Ramachandran, J. W. Nicholson, S. Ghalmi, and M. F. Yan, “Measurement of multipath interference in the coherent crosstalk regime,” IEEE Photonics Technol. Lett. 15(8), 1171–1173 (2003).
[Crossref]

S. Ramachandran, Z. Wang, and M. Yan, “Bandwidth control of long-period grating-based mode converters in few-mode fibers,” Opt. Lett. 27(9), 698–700 (2002).
[Crossref] [PubMed]

Randel, S.

Ren, Y.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[Crossref] [PubMed]

Rishøj, L.

Rubano, A.

Ryf, R.

Saffman, M.

Saleh, S. S.

Sandoz, P.

Schimpf, D. N.

Siegman, A. E.

Sierra, A.

Sridharan, A. K.

Steinvurzel, P.

Stolen, R. H.

R. H. Stolen, J. E. Bjorkholm, and A. Ashkin, “Phase‐matched three‐wave mixing in silica fiber optical waveguides,” Appl. Phys. Lett. 24(7), 308–310 (1974).
[Crossref]

Suarez, M. A.

Supradeepa, V. R.

Tai, B.

Tantiwanichapan, K.

Thomsen, B. C.

Tur, M.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[Crossref] [PubMed]

Wadsworth, W. J.

Wang, Z.

Westbrook, P. S.

J. W. Nicholson, J. M. Fini, A. DeSantolo, P. S. Westbrook, R. S. Windeler, T. Kremp, C. Headley, and D. J. DiGiovanni, “A higher-order mode fiber amplifier with an axicon for output mode conversion,” Proc. SPIE 9344, 93441V (2015).

J. W. Nicholson, J. M. Fini, A. M. DeSantolo, X. Liu, K. Feder, P. S. Westbrook, V. R. Supradeepa, E. Monberg, F. DiMarcello, R. Ortiz, C. Headley, and D. J. DiGiovanni, “Scaling the effective area of higher-order-mode erbium-doped fiber amplifiers,” Opt. Express 20(22), 24575–24584 (2012).
[Crossref] [PubMed]

Wilkinson, T. D.

Williams, W.

Willner, A. E.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[Crossref] [PubMed]

Windeler, R. S.

J. W. Nicholson, J. M. Fini, A. DeSantolo, P. S. Westbrook, R. S. Windeler, T. Kremp, C. Headley, and D. J. DiGiovanni, “A higher-order mode fiber amplifier with an axicon for output mode conversion,” Proc. SPIE 9344, 93441V (2015).

Winzer, P. J.

Wisk, P.

Witkowska, A.

Yan, L.

Yan, M.

Yan, M. F.

S. Ramachandran, J. W. Nicholson, S. Ghalmi, M. F. Yan, P. Wisk, E. Monberg, and F. V. Dimarcello, “Light propagation with ultralarge modal areas in optical fibers,” Opt. Lett. 31(12), 1797–1799 (2006).
[Crossref] [PubMed]

S. Ramachandran, J. W. Nicholson, S. Ghalmi, and M. F. Yan, “Measurement of multipath interference in the coherent crosstalk regime,” IEEE Photonics Technol. Lett. 15(8), 1171–1173 (2003).
[Crossref]

Yilmaz, Y. O.

Yue, Y.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[Crossref] [PubMed]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

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Contemp. Phys. (1)

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IEEE Photonics Technol. Lett. (1)

S. Ramachandran, J. W. Nicholson, S. Ghalmi, and M. F. Yan, “Measurement of multipath interference in the coherent crosstalk regime,” IEEE Photonics Technol. Lett. 15(8), 1171–1173 (2003).
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Nat. Methods (1)

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2011).
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Opt. Express (4)

Opt. Lett. (9)

P. Steinvurzel, K. Tantiwanichapan, M. Goto, and S. Ramachandran, “Fiber-based Bessel beams with controllable diffraction-resistant distance,” Opt. Lett. 36(23), 4671–4673 (2011).
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S. Ramachandran, J. W. Nicholson, S. Ghalmi, M. F. Yan, P. Wisk, E. Monberg, and F. V. Dimarcello, “Light propagation with ultralarge modal areas in optical fibers,” Opt. Lett. 31(12), 1797–1799 (2006).
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Optica (1)

Proc. SPIE (1)

J. W. Nicholson, J. M. Fini, A. DeSantolo, P. S. Westbrook, R. S. Windeler, T. Kremp, C. Headley, and D. J. DiGiovanni, “A higher-order mode fiber amplifier with an axicon for output mode conversion,” Proc. SPIE 9344, 93441V (2015).

Science (1)

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
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A. E. Siegman, “How to (maybe) measure laser beam quality,” in DPSS (Diode Pumped Solid State) Lasers: Applications and Issues, M. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics (Optical Society of America, 1998), paper MQ1.

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Figures (10)

Fig. 1
Fig. 1 (a) Binary phase plate (BPP) input coupling setup; SLM phase profile includes a BPP designed for LP0,9 (diameter = 3.8 mm) and a lens (f = 465 mm) as shown on the left-hand side of the figure; the mode in the far field of the SLM is shown inset (contrast and brightness enhanced for visibility); (b) Axicon input coupling setup; the setup upstream of the SLM is identical to (a); SLM phase profile is an axicon with angle α = 0.787° as shown on the left-hand side of the figure; Bessel-Gauss beam in the focal plane of the axicon and the beam in the focal plane of lens f2 are shown inset (contrast and brightness enhanced for visibility); (c) Facet image of the fiber under test (FUT); (d) Refractive index profile of the FUT; Modal purity is characterized via frequency-domain C2 imaging (fC2).
Fig. 2
Fig. 2 Input coupling simulations; (a) Comparison of the simulated electric fields for a Gaussian beam converted by a BPP and the target LP0,9 mode; (b) MPI as a function of mode order for the mode converted by BPP; (c) Comparison of the simulated electric fields for a Gaussian beam converted by an axicon and the target LP0,9 mode; (d) Simulated MPI as a function of mode order for the mode converted by axicon; (e) Simulated overlap between the converted and target modes as a function of normalized Gaussian width for binary (red) and axicon (blue) conversion; (f) Incoherent sum of simulated MPI to all parasitic modes as a function of normalized Gaussian width for binary (red) and axicon (blue) conversion; Dashed red and blue lines in (e) and (f) are optimal widths which correspond to those used for the simulations in (a), (b), (c) and (d).
Fig. 3
Fig. 3 (a) Sample fC2 trace for BPP excitation of LP0,9; (b) LP0,9 mode image for BPP excitation, contrast and brightness enhanced for visibility. (c) Sample fC2 trace for axicon excitation of LP0,9; (d) LP0,9 mode image for axicon excitation, contrast and brightness enhanced for visibility.
Fig. 4
Fig. 4 Parasitic mode suppression as measured by fC2 imaging for various changes to the system alignment; Mode images for different conditions inset, contrast and brightness enhanced for visibility; (a) Suppression as a function of BPP size; (b) Suppression as a function of FUT offset with respect to the coupling lens for BPP excitation; (c) Suppression as a function of axicon angle; (d) Suppression as a function of FUT offset for axicon excitation.
Fig. 5
Fig. 5 Simulated Gaussian-like intensity profiles of a converted LP0,9 mode; (a) BPP conversion on a linear scale; (b) BPP conversion on a log scale; (c) Axicon conversion on a linear scale; (d) Axicon conversion on a log scale. The “qualitatively Gaussian point” (QGP), as defined in the text, is marked in each figure, and a zoomed-in plot of the QGP region is inset in each top right corner.
Fig. 6
Fig. 6 (a) Schematic of the M2 measurement setup; (b) Experimentally stitched mode image of the converted mode from the BPP output coupling setup; (c) Image of the converted mode with increased gamma contrast (Γ = 1/5); (d) Experimental intensity line cut of in the x direction as a function of z; (e) Wx2(z) measurement with parabolic fit corresponding to Mx2 = 8.05; (f) Experimental intensity line cut of in the y direction as a function of z; (g) Wy2(z) measurement with parabolic fit corresponding to My2 = 9.65.
Fig. 7
Fig. 7 M2 as a function of power transmitted through the iris for (a) BPP output coupling; (b) Axicon output coupling; Red and blue lines correspond to simulations, and markers indicate experimental measurements; Mode images with increased gamma-contrast (Γ = 1/5) for various iris diameters are insets.
Fig. 8
Fig. 8 Calculated loss at the QGP with respect to the optimal condition, and M2 as a function of various changes in system alignment; Inset: stitched mode images for different conditions (log scale image for visibility) (a) Relative loss and M2 as a function of BPP diameter; (b) Relative loss and M2 as a function of BPP offset; (c) Relative loss and M2 as a function of axicon angle; (d) Relative loss and M2 as a function of axicon offset.
Fig. 9
Fig. 9 Parasitic mode suppression as a function of mode order measured by fC2 for (a) BPP input coupling, (b) axicon input coupling; simulations delineated with solid line, measurements are marked with crosses, example mode images (brightness and contrast enhanced for visibility) inset; (c) Simulated loss at the QGP as a function of mode order for axicon and BPP output coupling, measurements from section 4 shown with markers.
Fig. 10
Fig. 10 (a) Schematic for a double clad fiber, core V number is 2.1, Δn is variable; (b) Simulated parasitic mode suppression for input coupling as a function of Δn, measurements from section 3 shown with markers; (c) Simulated loss at the QGP as a function of Δn, measurements from section 4 shown with markers.

Equations (4)

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α = J 0 ( m ) k 0 ( n 1 ) R c M
η 0 , m = | r d r d ϕ ψ i * ( r , ϕ ) ψ 0 , m ( r , ϕ ) | 2 r d r d ϕ | ψ i ( r , ϕ ) | 2 r d r d ϕ | ψ m ( r , ϕ ) | 2
W 2 ( z ) = W 0 2 + M 4 ( λ π W 0 2 ) 2 ( z z 0 ) 2
M 2 = π W n f W f f λ z

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