Abstract

We propose and analyze a general approach for coupling a free space uniformly polarized beam to a desired hollow waveguide mode, thus enabling a single mode operation. The required spatial polarization state manipulation is achieved by use of inhomogeneous anisotropic subwavelength structures. Demonstration is obtained by coupling a linearly polarized CO2 laser beam at a wavelength of 10.6μm to the TE01, TM01, EH11, EH21, and EH31 modes of a 300μm diameter dielectric-coated hollow metallic waveguide. Full polarization and intensity analysis of the beam at the waveguide’s inlet and outlet ports indicates a high coupling efficiency to a single waveguide mode. Finally, shaping the waveguide mode to a nearly diffraction limited linearly polarized beam and to a radially polarized vectorial vortex are also demonstrated.

© 2007 Optical Society of America

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References

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    [CrossRef]
  6. S. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. Engeness, M. Soljacic, S. Jacobs, J. Joannopoulos, and Y. Fink, "Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers," Opt. Express 9, 748-779 (2001).
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    [CrossRef] [PubMed]
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  10. O. Shapira, K. Kuriki, N. D. Orf, A. F. Abouraddy, G. Benoit, J. F. Viens, A. Rodriguez, M. Ibanescu, J. D. Joannopoulos, Y. Fink, and M. M. Brewster, "Surface-emitting fiber lasers," Opt. Express 14, 3929-3935 (2006).
    [CrossRef] [PubMed]
  11. I. Bassett and A. Argyros, "Elimination of polarization degeneracy in round waveguides," Opt. Express 10, 1342-1346 (2002).
    [PubMed]
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  13. M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, and E. A. Cornell, "Laser-guided atoms in hollow-core optical fibers," Phys. Rev. Lett. 75, 3253-3256 (1995).
    [CrossRef] [PubMed]
  14. C. D. Poole, J. M. Wiesenfeld, D. J. DiGiovanni, and A. M. Vengsarkar, "Optical fiber-based dispersion compensation using higher order modes near cutoff," J. Lightwave Technol. 12, 1746-1758 (1994).
    [CrossRef]
  15. T. Engeness, M. Ibanescu, S. Johnson, O. Weisberg, M. Skorobogatiy, S. Jacobs, and Y. Fink, "Dispersion tailoring and compensation by modal interactions in OmniGuide fibers," Opt. Express 11, 1175-1196 (2003).
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  17. E. Hasman, N. Davidson, and A. A. Friesem, "Heterostructure multilevel binary optics," Opt. Lett. 16, 1460-1462 (1991),
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  18. E. Hasman and A. A. Friesem, "Analytic optimization for holographic optical elements," J. Opt. Soc. Am. A 6, 62-72 (1989),
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    [CrossRef] [PubMed]
  21. R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
    [CrossRef]
  22. M. E. Marhic, and E. Garmire, "Low-order TE0q operation of a CO2 laser for transmission through circular metallic waveguides," Appl. Phys. Lett. 38, 743-745 (1981).
    [CrossRef]
  23. E.  Hasman, G.  Biener, A.  Niv, and V.  Kleiner, "Space-variant polarization manipulation," in Progress in Optics, E. Wolf ed. (Elsevier, Netherlands, Amsterdam, 2005) Vol. 47.
  24. A. Niv, G. Biener, V. Kleiner, and E. Hasman, "Propagation-invariant vectorial Bessel beams obtained by use of quantized Pancharatnam-Berry phase optical elements," Opt. Lett. 29, 238-240 (2004),
    [CrossRef] [PubMed]
  25. A. Niv, G. Biener, V. Kleiner, and E. Hasman, "Rotating vectorial vortices produced by space-variant subwavelength gratings," Opt. Lett. 30, 2933-2935 (2005).
    [CrossRef] [PubMed]
  26. W. S. Mohammed, A. Mehta, M. Pitchumani, and E. G. Johnson, "Selective excitation of the TE01 mode in hollow-glass waveguide using a subwavelength grating," Photon. Technol. Lett. 17, 1441 (2005).
    [CrossRef]
  27. Y. Yirmiyahu, A. Niv, G. Biener, V. Kleiner, and E. Hasman, "Vectorial vortex mode transformation for a hollow waveguide using Pancharatnam-Berry phase optical elements," Opt. Lett. 31, 3252-3254 (2006)
    [CrossRef] [PubMed]
  28. A. Niv, G. Biener, V. Kleiner, and E. Hasman, "Spiral phase elements obtained by use of discrete space-variant subwavelength gratings," Opt. Commun. 251, 306-314 (2005).
    [CrossRef]
  29. R. George and J.  A. Harrington, "Infrared transmissive, hollow plastic waveguides with inner Ag-AgI coatings," Appl. Opt. 44, 6449-6455 (2005).
    [CrossRef] [PubMed]
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    [CrossRef]

2006 (5)

2005 (6)

2004 (1)

2003 (1)

2002 (2)

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, Y. Fink, "Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission," Nature 420, 650 (2002),
[CrossRef] [PubMed]

I. Bassett and A. Argyros, "Elimination of polarization degeneracy in round waveguides," Opt. Express 10, 1342-1346 (2002).
[PubMed]

2001 (1)

2000 (1)

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
[CrossRef]

1999 (1)

V. G. Niziev and A. V. Nesterov, "Influence of beam polarization on laser cutting efficiency," J. Phys. D 32, 1455-1461 (1999).
[CrossRef]

1995 (1)

M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, and E. A. Cornell, "Laser-guided atoms in hollow-core optical fibers," Phys. Rev. Lett. 75, 3253-3256 (1995).
[CrossRef] [PubMed]

1994 (1)

C. D. Poole, J. M. Wiesenfeld, D. J. DiGiovanni, and A. M. Vengsarkar, "Optical fiber-based dispersion compensation using higher order modes near cutoff," J. Lightwave Technol. 12, 1746-1758 (1994).
[CrossRef]

1991 (1)

1989 (1)

1984 (1)

M. Miyagi and S. Kawakami, "Design theory of dielectric-coated circular metallic waveguides for infrared transmission," J. Lightwave Technol. 2, 116-126 (1984).
[CrossRef]

1981 (1)

M. E. Marhic, and E. Garmire, "Low-order TE0q operation of a CO2 laser for transmission through circular metallic waveguides," Appl. Phys. Lett. 38, 743-745 (1981).
[CrossRef]

1964 (1)

E.J. Marcatili and R. A. Schmeltzer, "Hollow Metallic and Dielectric Waveguides for Long Distance Optical Transmission of Lasers," Bell. Syst. Tech. J. 43, 1783-1809 (1964).

1961 (1)

1897 (1)

Lord Rayleigh, "On the passage of electric waves through tubes, or the vibrations of dielectric cylinders," Phil. Mag. 43, 125-132 (1897).

Abouraddy, A. F.

Anderson, D. Z.

M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, and E. A. Cornell, "Laser-guided atoms in hollow-core optical fibers," Phys. Rev. Lett. 75, 3253-3256 (1995).
[CrossRef] [PubMed]

Argyros, A.

Bassett, I.

Benoit, G.

O. Shapira, K. Kuriki, N. D. Orf, A. F. Abouraddy, G. Benoit, J. F. Viens, A. Rodriguez, M. Ibanescu, J. D. Joannopoulos, Y. Fink, and M. M. Brewster, "Surface-emitting fiber lasers," Opt. Express 14, 3929-3935 (2006).
[CrossRef] [PubMed]

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, Y. Fink, "Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission," Nature 420, 650 (2002),
[CrossRef] [PubMed]

Biener, G.

Blit, S.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
[CrossRef]

Bomzon, Z.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
[CrossRef]

Brewster, M. M.

Chen, H.

Chuang, S. L.

Codemard, C.

J. S. Kim, C. Codemard, J. Nilsson, and J. K. Sahu, "Erbium-ytterbium co-doped hollow optical fibre laser," Electron Lett. 42, 515 (2006).
[CrossRef]

Cornell, E. A.

M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, and E. A. Cornell, "Laser-guided atoms in hollow-core optical fibers," Phys. Rev. Lett. 75, 3253-3256 (1995).
[CrossRef] [PubMed]

Cueva, G. R.

Dai, M.

Davidson, N.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
[CrossRef]

E. Hasman, N. Davidson, and A. A. Friesem, "Heterostructure multilevel binary optics," Opt. Lett. 16, 1460-1462 (1991),
[CrossRef] [PubMed]

Davis, J. A.

DiGiovanni, D. J.

C. D. Poole, J. M. Wiesenfeld, D. J. DiGiovanni, and A. M. Vengsarkar, "Optical fiber-based dispersion compensation using higher order modes near cutoff," J. Lightwave Technol. 12, 1746-1758 (1994).
[CrossRef]

Engeness, T.

Evans, G. H.

Fink, Y.

Friesem, A. A.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
[CrossRef]

E. Hasman, N. Davidson, and A. A. Friesem, "Heterostructure multilevel binary optics," Opt. Lett. 16, 1460-1462 (1991),
[CrossRef] [PubMed]

E. Hasman and A. A. Friesem, "Analytic optimization for holographic optical elements," J. Opt. Soc. Am. A 6, 62-72 (1989),
[CrossRef]

Garmire, E.

M. E. Marhic, and E. Garmire, "Low-order TE0q operation of a CO2 laser for transmission through circular metallic waveguides," Appl. Phys. Lett. 38, 743-745 (1981).
[CrossRef]

George, R.

Harrington, J.  A.

Hart, S. D.

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, Y. Fink, "Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission," Nature 420, 650 (2002),
[CrossRef] [PubMed]

Hasman, E.

Ibanescu, M.

Jacobs, S.

Joannopoulos, J.

Joannopoulos, J. D.

O. Shapira, K. Kuriki, N. D. Orf, A. F. Abouraddy, G. Benoit, J. F. Viens, A. Rodriguez, M. Ibanescu, J. D. Joannopoulos, Y. Fink, and M. M. Brewster, "Surface-emitting fiber lasers," Opt. Express 14, 3929-3935 (2006).
[CrossRef] [PubMed]

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, Y. Fink, "Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission," Nature 420, 650 (2002),
[CrossRef] [PubMed]

Johnson, E. G.

W. S. Mohammed, A. Mehta, M. Pitchumani, and E. G. Johnson, "Selective excitation of the TE01 mode in hollow-glass waveguide using a subwavelength grating," Photon. Technol. Lett. 17, 1441 (2005).
[CrossRef]

Johnson, S.

Kawakami, S.

M. Miyagi and S. Kawakami, "Design theory of dielectric-coated circular metallic waveguides for infrared transmission," J. Lightwave Technol. 2, 116-126 (1984).
[CrossRef]

Kim, J. S.

J. S. Kim, C. Codemard, J. Nilsson, and J. K. Sahu, "Erbium-ytterbium co-doped hollow optical fibre laser," Electron Lett. 42, 515 (2006).
[CrossRef]

Kleiner, V.

Kuriki, K.

Marcatili, E.J.

E.J. Marcatili and R. A. Schmeltzer, "Hollow Metallic and Dielectric Waveguides for Long Distance Optical Transmission of Lasers," Bell. Syst. Tech. J. 43, 1783-1809 (1964).

Marhic, M. E.

M. E. Marhic, and E. Garmire, "Low-order TE0q operation of a CO2 laser for transmission through circular metallic waveguides," Appl. Phys. Lett. 38, 743-745 (1981).
[CrossRef]

Mehta, A.

W. S. Mohammed, A. Mehta, M. Pitchumani, and E. G. Johnson, "Selective excitation of the TE01 mode in hollow-glass waveguide using a subwavelength grating," Photon. Technol. Lett. 17, 1441 (2005).
[CrossRef]

Minin, S.

Miyagi, M.

M. Miyagi and S. Kawakami, "Design theory of dielectric-coated circular metallic waveguides for infrared transmission," J. Lightwave Technol. 2, 116-126 (1984).
[CrossRef]

Mohammed, W. S.

W. S. Mohammed, A. Mehta, M. Pitchumani, and E. G. Johnson, "Selective excitation of the TE01 mode in hollow-glass waveguide using a subwavelength grating," Photon. Technol. Lett. 17, 1441 (2005).
[CrossRef]

Montgomery, D.

M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, and E. A. Cornell, "Laser-guided atoms in hollow-core optical fibers," Phys. Rev. Lett. 75, 3253-3256 (1995).
[CrossRef] [PubMed]

Moreno, I.

Nesterov, A. V.

V. G. Niziev and A. V. Nesterov, "Influence of beam polarization on laser cutting efficiency," J. Phys. D 32, 1455-1461 (1999).
[CrossRef]

Nilsson, J.

J. S. Kim, C. Codemard, J. Nilsson, and J. K. Sahu, "Erbium-ytterbium co-doped hollow optical fibre laser," Electron Lett. 42, 515 (2006).
[CrossRef]

Niv, A.

Niziev, V. G.

V. G. Niziev and A. V. Nesterov, "Influence of beam polarization on laser cutting efficiency," J. Phys. D 32, 1455-1461 (1999).
[CrossRef]

Orf, N. D.

Oron, R.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
[CrossRef]

Pitchumani, M.

W. S. Mohammed, A. Mehta, M. Pitchumani, and E. G. Johnson, "Selective excitation of the TE01 mode in hollow-glass waveguide using a subwavelength grating," Photon. Technol. Lett. 17, 1441 (2005).
[CrossRef]

Poole, C. D.

C. D. Poole, J. M. Wiesenfeld, D. J. DiGiovanni, and A. M. Vengsarkar, "Optical fiber-based dispersion compensation using higher order modes near cutoff," J. Lightwave Technol. 12, 1746-1758 (1994).
[CrossRef]

Renn, M. J.

M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, and E. A. Cornell, "Laser-guided atoms in hollow-core optical fibers," Phys. Rev. Lett. 75, 3253-3256 (1995).
[CrossRef] [PubMed]

Rodriguez, A.

Sahu, J. K.

J. S. Kim, C. Codemard, J. Nilsson, and J. K. Sahu, "Erbium-ytterbium co-doped hollow optical fibre laser," Electron Lett. 42, 515 (2006).
[CrossRef]

Schmeltzer, R. A.

E.J. Marcatili and R. A. Schmeltzer, "Hollow Metallic and Dielectric Waveguides for Long Distance Optical Transmission of Lasers," Bell. Syst. Tech. J. 43, 1783-1809 (1964).

Shapira, O.

Skorobogatiy, M.

Snitzer, E.

Soljacic, M.

Temelkuran, B.

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, Y. Fink, "Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission," Nature 420, 650 (2002),
[CrossRef] [PubMed]

Vdovin, O.

M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, and E. A. Cornell, "Laser-guided atoms in hollow-core optical fibers," Phys. Rev. Lett. 75, 3253-3256 (1995).
[CrossRef] [PubMed]

Vengsarkar, A. M.

C. D. Poole, J. M. Wiesenfeld, D. J. DiGiovanni, and A. M. Vengsarkar, "Optical fiber-based dispersion compensation using higher order modes near cutoff," J. Lightwave Technol. 12, 1746-1758 (1994).
[CrossRef]

Viens, J. F.

Wang, Z.

Webb, K. J.

Weisberg, O.

Wieman, C. E.

M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, and E. A. Cornell, "Laser-guided atoms in hollow-core optical fibers," Phys. Rev. Lett. 75, 3253-3256 (1995).
[CrossRef] [PubMed]

Wiesenfeld, J. M.

C. D. Poole, J. M. Wiesenfeld, D. J. DiGiovanni, and A. M. Vengsarkar, "Optical fiber-based dispersion compensation using higher order modes near cutoff," J. Lightwave Technol. 12, 1746-1758 (1994).
[CrossRef]

Yang, M.

Yin, J.

Yirmiyahu, Y.

Appl. Opt. (2)

Appl. Phys. Lett. (2)

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
[CrossRef]

M. E. Marhic, and E. Garmire, "Low-order TE0q operation of a CO2 laser for transmission through circular metallic waveguides," Appl. Phys. Lett. 38, 743-745 (1981).
[CrossRef]

Bell. Syst. Tech. J. (1)

E.J. Marcatili and R. A. Schmeltzer, "Hollow Metallic and Dielectric Waveguides for Long Distance Optical Transmission of Lasers," Bell. Syst. Tech. J. 43, 1783-1809 (1964).

Electron Lett. (1)

J. S. Kim, C. Codemard, J. Nilsson, and J. K. Sahu, "Erbium-ytterbium co-doped hollow optical fibre laser," Electron Lett. 42, 515 (2006).
[CrossRef]

J. Lightwave Technol. (2)

M. Miyagi and S. Kawakami, "Design theory of dielectric-coated circular metallic waveguides for infrared transmission," J. Lightwave Technol. 2, 116-126 (1984).
[CrossRef]

C. D. Poole, J. M. Wiesenfeld, D. J. DiGiovanni, and A. M. Vengsarkar, "Optical fiber-based dispersion compensation using higher order modes near cutoff," J. Lightwave Technol. 12, 1746-1758 (1994).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

J. Phys. D (1)

V. G. Niziev and A. V. Nesterov, "Influence of beam polarization on laser cutting efficiency," J. Phys. D 32, 1455-1461 (1999).
[CrossRef]

Nature (1)

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, Y. Fink, "Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission," Nature 420, 650 (2002),
[CrossRef] [PubMed]

Opt. Commun. (1)

A. Niv, G. Biener, V. Kleiner, and E. Hasman, "Spiral phase elements obtained by use of discrete space-variant subwavelength gratings," Opt. Commun. 251, 306-314 (2005).
[CrossRef]

Opt. Express (6)

Opt. Lett. (5)

Phil. Mag. (1)

Lord Rayleigh, "On the passage of electric waves through tubes, or the vibrations of dielectric cylinders," Phil. Mag. 43, 125-132 (1897).

Photon. Technol. Lett. (1)

W. S. Mohammed, A. Mehta, M. Pitchumani, and E. G. Johnson, "Selective excitation of the TE01 mode in hollow-glass waveguide using a subwavelength grating," Photon. Technol. Lett. 17, 1441 (2005).
[CrossRef]

Phys. Rev. Lett. (1)

M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, and E. A. Cornell, "Laser-guided atoms in hollow-core optical fibers," Phys. Rev. Lett. 75, 3253-3256 (1995).
[CrossRef] [PubMed]

Other (3)

J. A. Harrington, Infrared Fiber Optics and their Applications, (SPIE Press, 2004).
[CrossRef]

E.  Hasman, G.  Biener, A.  Niv, and V.  Kleiner, "Space-variant polarization manipulation," in Progress in Optics, E. Wolf ed. (Elsevier, Netherlands, Amsterdam, 2005) Vol. 47.

S. C. Tidwell, G. H. Kim, and W. D. Kimura, "Efficient radially polarized laser beam generation with a double interferometer," Appl. Opt.  32, 5222- (1993).
[CrossRef] [PubMed]

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Figures (8)

Fig. 1.
Fig. 1.

(color online) Schematic representation of the experimental apparatus. Insets illustrate false color intensity distributions and polarization orientation (black arrows) of the beam at different locations along the experimental apparatus.

Fig. 2.
Fig. 2.

Electric field lines of metallic hollow waveguide modes.

Fig. 3.
Fig. 3.

(color online) Coupling coefficients for the TE 0m (left) and the EH 1m (right) sets of modes.

Fig. 4.
Fig. 4.

Scanning Electron Microscope (SEM) images of PBOEs for coupling a linearly polarized 10.6μm wavelength beam to hollow waveguide modes of different azimuthal order n.

Fig. 5.
Fig. 5.

(color online) Measured intensity and polarization for coupling to the TE 01 mode at the waveguide’s inlet (top) and outlet (bottom) ports. The first column depicts false color intensity representation with dashed line indicating the waveguide’s inner circumference. The second column depicts false color representation of the intensity after a linear polarizer whose orientation is given by the yellow arrows. The third column shows the measured polarization ellipse’s orientation, with bar length indicating intensity. The fourth column shows the measured (dots) and predicted (solid lines) intensity cross sections.

Fig. 6.
Fig. 6.

(color online) Measured intensity and polarization for coupling to the TM 01 mode at the waveguide’s inlet (top) and outlet (bottom) ports. The first to fourth columns depict the intensity (dashed line indicates the waveguide inner circumference), intensity after a polarizer (arrows indicate polarizer orientation), measured polarization orientation, and intensity cross-section (dots-measured, solid line-predicted), respectively.

Fig. 7.
Fig. 7.

(color online) Measured intensity and polarization for coupling the high order modes at the waveguide’s inlet and outlet ports. The first to forth columns depict the intensity (dashed line indicates the waveguide’s inner circumference), intensity after a polarizer (arrows indicate polarizer orientation), measured polarization orientation, and intensity cross-section (dots-measured, solid line-predicted), respectively.

Fig. 8.
Fig. 8.

(color online) Measured intensity and polarization for the inverse coupling and transformation of the TE 01 mode to: (a) a linearly polarized beam by use of PBOE with n=0 (red line shows the focus of a Gaussian beam having a similar width.), (b) a radially polarized beam by use of PBOE with n=1. (c) Azimuthally polarized beam obtained without a second PBOE. The first to fourth columns depicts the intensity, intensity after a polarizer (arrows indicate polarizer orientation), measured polarization orientation, and intensity cross-section (dots-measured, solid line-predicted), respectively.

Equations (8)

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E φ , 0 m = J 1 ( u 0 m r a ) φ ,
E r , 0 m = J 1 ( u 0 m r a ) r .
E nm = J n 1 ( u nm r a ) [ ± cos ( n φ + φ 0 ) r sin ( n φ + φ 0 ) φ ] ,
E out , n = cos ( 2 θ φ ) r + sin ( 2 θ φ ) φ .
θ = 1 n 2 φ + φ 0 2 ,
E f , n = A n ( r ) E out , n ,
A n ( r ) = iλf 2 π r 2 0 2 π R 0 λf r x J 1 n ( x ) dx .
η nm = E f , n E nm 2 E f , n E f , n E nm E nm .

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