Abstract

In theory, no more than 80% of the light diffracted by a circular aperture can be directly coupled into one mode of an ordinary optical fiber. To improve coupling efficiency, and illustrate an inverse method for designing optical fibers with a desired mode shape, we have made an optical fiber that guides an approximate Airy pattern as one of its modes. The fiber’s attenuation was 11.0 dB/km at a 1550 nm wavelength, the match between the fiber’s mode and the ideal infinite Airy pattern was 93.7%, and the far field resembled a top-hat beam. The guidance mechanism has strong similarities to photonic bandgap guidance.

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References

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2013 (3)

2006 (1)

2004 (1)

2003 (1)

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borelli, D. C. Allan, and K. W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424, 657–659 (2003).
[Crossref]

2002 (1)

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature 420, 650–653 (2002).
[Crossref]

2000 (1)

F. Brechet, P. Roy, J. Marcou, and D. Pagnoux, “Singlemode propagation into depressed-core-index photonic-bandgap fibre designed for zero-dispersion propagation at short wavelengths,” Electron. Lett. 36, 514–515 (2000).
[Crossref]

1988 (1)

1978 (2)

P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. 68, 1196–1201 (1978).
[Crossref]

D. Küppers, J. Koenings, and H. Wilson, “Deposition of fluorine-doped silica layers from a SiCl4/SiF4/O2 gas mixture by the plasma-CVD method,” J. Electrochem. Soc. 125, 1298–1302 (1978).
[Crossref]

1969 (1)

V. N. Melekhin and A. B. Manenkov, “Dielectric tube as a low-loss waveguide,” Sov. Phys. Tech. Phys. 13, 1698–1699 (1969).

Allan, D. C.

J. A. West, C. M. Smith, N. F. Borelli, D. C. Allan, and K. W. Koch, “Surface modes in air-core photonic band-gap fibers,” Opt. Express 12, 1485–1496 (2004).

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borelli, D. C. Allan, and K. W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424, 657–659 (2003).
[Crossref]

Beach, R. J.

Benoit, G.

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature 420, 650–653 (2002).
[Crossref]

Bigot, L.

Borelli, N. F.

J. A. West, C. M. Smith, N. F. Borelli, D. C. Allan, and K. W. Koch, “Surface modes in air-core photonic band-gap fibers,” Opt. Express 12, 1485–1496 (2004).

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borelli, D. C. Allan, and K. W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424, 657–659 (2003).
[Crossref]

Bouwmans, G.

Brechet, F.

F. Brechet, P. Roy, J. Marcou, and D. Pagnoux, “Singlemode propagation into depressed-core-index photonic-bandgap fibre designed for zero-dispersion propagation at short wavelengths,” Electron. Lett. 36, 514–515 (2000).
[Crossref]

Buck, J. A.

J. A. Buck, Fundamentals of Fiber Optics (Wiley, 2004).

Calvet, P.

Caplen, J. E.

J. E. Caplen, J.-P. Desandro, and J. Koh, “Two step etching process for an optical fiber preform,” International patent application WO2003052173 A1 (June26, 2003).

Coulombier, Q.

Dawson, J. W.

Delplace, K.

Desandro, J.-P.

J. E. Caplen, J.-P. Desandro, and J. Koh, “Two step etching process for an optical fiber preform,” International patent application WO2003052173 A1 (June26, 2003).

Dong, L.

Douay, M.

Dunn, C.

Fink, Y.

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature 420, 650–653 (2002).
[Crossref]

Gallagher, M. T.

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borelli, D. C. Allan, and K. W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424, 657–659 (2003).
[Crossref]

Ghatak, A.

A. Ghatak and K. Thyagarajan, Introduction to Fiber Optics (Cambridge University, 1998).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Gu, G.

Guenther, R.

R. Guenther, Modern Optics (Wiley, 1990).

Hart, S. D.

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature 420, 650–653 (2002).
[Crossref]

Hawkins, T. W.

Heebner, J. E.

Hudonnot, E.

Joannopoulos, J. D.

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature 420, 650–653 (2002).
[Crossref]

Jones, M.

Kalichevsky-Dong, M. T.

Koch, K. W.

J. A. West, C. M. Smith, N. F. Borelli, D. C. Allan, and K. W. Koch, “Surface modes in air-core photonic band-gap fibers,” Opt. Express 12, 1485–1496 (2004).

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borelli, D. C. Allan, and K. W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424, 657–659 (2003).
[Crossref]

Koenings, J.

D. Küppers, J. Koenings, and H. Wilson, “Deposition of fluorine-doped silica layers from a SiCl4/SiF4/O2 gas mixture by the plasma-CVD method,” J. Electrochem. Soc. 125, 1298–1302 (1978).
[Crossref]

Koh, J.

J. E. Caplen, J.-P. Desandro, and J. Koh, “Two step etching process for an optical fiber preform,” International patent application WO2003052173 A1 (June26, 2003).

Kong, F.

Küppers, D.

D. Küppers, J. Koenings, and H. Wilson, “Deposition of fluorine-doped silica layers from a SiCl4/SiF4/O2 gas mixture by the plasma-CVD method,” J. Electrochem. Soc. 125, 1298–1302 (1978).
[Crossref]

Love, J. D.

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

Manenkov, A. B.

V. N. Melekhin and A. B. Manenkov, “Dielectric tube as a low-loss waveguide,” Sov. Phys. Tech. Phys. 13, 1698–1699 (1969).

Marcou, J.

F. Brechet, P. Roy, J. Marcou, and D. Pagnoux, “Singlemode propagation into depressed-core-index photonic-bandgap fibre designed for zero-dispersion propagation at short wavelengths,” Electron. Lett. 36, 514–515 (2000).
[Crossref]

Marom, E.

Melekhin, V. N.

V. N. Melekhin and A. B. Manenkov, “Dielectric tube as a low-loss waveguide,” Sov. Phys. Tech. Phys. 13, 1698–1699 (1969).

Messerly, M. J.

Müller, D.

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borelli, D. C. Allan, and K. W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424, 657–659 (2003).
[Crossref]

Mussot, A.

Pagnoux, D.

F. Brechet, P. Roy, J. Marcou, and D. Pagnoux, “Singlemode propagation into depressed-core-index photonic-bandgap fibre designed for zero-dispersion propagation at short wavelengths,” Electron. Lett. 36, 514–515 (2000).
[Crossref]

Parsons, J.

Pax, P. H.

Quiquempois, Y.

Roddier, F.

Roy, P.

F. Brechet, P. Roy, J. Marcou, and D. Pagnoux, “Singlemode propagation into depressed-core-index photonic-bandgap fibre designed for zero-dispersion propagation at short wavelengths,” Electron. Lett. 36, 514–515 (2000).
[Crossref]

Samson, B.

Senior, J. M.

J. M. Senior, Optical Fiber Communications (Prentice-Hall, 1985).

Shaklan, S.

Smith, C. M.

J. A. West, C. M. Smith, N. F. Borelli, D. C. Allan, and K. W. Koch, “Surface modes in air-core photonic band-gap fibers,” Opt. Express 12, 1485–1496 (2004).

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borelli, D. C. Allan, and K. W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424, 657–659 (2003).
[Crossref]

Snyder, A. W.

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

Temelkuran, B.

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature 420, 650–653 (2002).
[Crossref]

Thyagarajan, K.

A. Ghatak and K. Thyagarajan, Introduction to Fiber Optics (Cambridge University, 1998).

Toyoshima, M.

Valentin, C.

Venkataraman, N.

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borelli, D. C. Allan, and K. W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424, 657–659 (2003).
[Crossref]

Wei, K.

West, J. A.

J. A. West, C. M. Smith, N. F. Borelli, D. C. Allan, and K. W. Koch, “Surface modes in air-core photonic band-gap fibers,” Opt. Express 12, 1485–1496 (2004).

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borelli, D. C. Allan, and K. W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424, 657–659 (2003).
[Crossref]

Wilson, H.

D. Küppers, J. Koenings, and H. Wilson, “Deposition of fluorine-doped silica layers from a SiCl4/SiF4/O2 gas mixture by the plasma-CVD method,” J. Electrochem. Soc. 125, 1298–1302 (1978).
[Crossref]

Yariv, A.

P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. 68, 1196–1201 (1978).
[Crossref]

A. Yariv, Optical Electronics in Modern Communications (Oxford University, 1997).

Yeh, P.

Appl. Opt. (1)

Electron. Lett. (1)

F. Brechet, P. Roy, J. Marcou, and D. Pagnoux, “Singlemode propagation into depressed-core-index photonic-bandgap fibre designed for zero-dispersion propagation at short wavelengths,” Electron. Lett. 36, 514–515 (2000).
[Crossref]

J. Electrochem. Soc. (1)

D. Küppers, J. Koenings, and H. Wilson, “Deposition of fluorine-doped silica layers from a SiCl4/SiF4/O2 gas mixture by the plasma-CVD method,” J. Electrochem. Soc. 125, 1298–1302 (1978).
[Crossref]

J. Opt. Soc. Am. (1)

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

Nature (2)

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature 420, 650–653 (2002).
[Crossref]

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borelli, D. C. Allan, and K. W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424, 657–659 (2003).
[Crossref]

Opt. Express (4)

Sov. Phys. Tech. Phys. (1)

V. N. Melekhin and A. B. Manenkov, “Dielectric tube as a low-loss waveguide,” Sov. Phys. Tech. Phys. 13, 1698–1699 (1969).

Other (10)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

R. Guenther, Modern Optics (Wiley, 1990).

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

J. M. Senior, Optical Fiber Communications (Prentice-Hall, 1985).

A. Ghatak and K. Thyagarajan, Introduction to Fiber Optics (Cambridge University, 1998).

J. A. Buck, Fundamentals of Fiber Optics (Wiley, 2004).

http://doi.org/10.15125/BATH-00173 .

J. E. Caplen, J.-P. Desandro, and J. Koh, “Two step etching process for an optical fiber preform,” International patent application WO2003052173 A1 (June26, 2003).

COMSOL Multiphysics, https://www.comsol.com/ .

A. Yariv, Optical Electronics in Modern Communications (Oxford University, 1997).

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Figures (12)

Fig. 1.
Fig. 1. Field ϕ0(R) of the ideal Airy pattern, with the 2D intensity pattern (center saturated) in the inset. The radial coordinate R is normalized so that the first dark ring lies at R1=1. The Airy disk is lobe N=0, the first bright ring is lobe N=1, etc. Here and elsewhere we include negative values of R for visual effect; the absolute value of R is taken.
Fig. 2.
Fig. 2. Normalized index distribution ν(R) calculated using Eq. (3) for (a) the exact Airy pattern of Eq. (5) and (b) the Airy mode of Eq. (8) and Table 1. The refractive index in the “Airy cladding” R>1 is notably high compared to that in the Airy disk R<1.
Fig. 3.
Fig. 3. (a) First four basis functions fN,i(R) in lobe N=1 and (b) field distributions of the ideal Airy pattern (solid red curve) and our Airy mode (dashed black curve). The inset magnifies lobe 1.
Fig. 4.
Fig. 4. (a) Normalized index νc of the uniform cladding versus truncation radius Rc for the N=3 lobe and (b) field distributions ψ(R) for Airy modes (red curves) truncated at three different Rc (dashed lines), together with the untruncated Airy mode (gray curve). The curves are labeled with the corresponding νc, which can be compared to the vertical scale of Fig. 2(b).
Fig. 5.
Fig. 5. (a) Field distributions of the truncated Airy mode (red) and the ideal Airy pattern it approximates (blue). The middle curve of Fig. 4(b) enlarges the truncation region. (b) Normalized index distributions of fibers guiding the truncated (red) and untruncated (blue) Airy modes.
Fig. 6.
Fig. 6. Field distributions of selected LPlm modes of the truncated Airy fiber, together with ν(R) of Fig. 5(b) for scale. The fundamental mode (LP01) has no zeros, and the Airy mode (LP04) has three annular zeros.
Fig. 7.
Fig. 7. (a) Airy mode at design wavelengths of λ0 (red), 0.8λ0 (black), and 1.2λ0 (blue). Inset: the overlap Ω with the ideal Airy pattern ϕ0(R) versus wavelength λ/λ0. The dashed lines mark the wavelengths in the main plot. (b) The normalized effective index ν of the fiber’s l=0 modes versus frequency λ0/λ. The red curve is the Airy mode (LP04). Insets: mode fields on either side of the first anticrossing, where the most Airy-like mode changes identity.
Fig. 8.
Fig. 8. (a) Physical refractive index profiles, relative to undoped silica, of the final Airy fiber design (red) and the measured PCVD preform (blue). The radial coordinate r is normalized to a, the radius of the Airy disk, and negative r is a copy of positive r for visual effect. (b) Same as (a) but for just the center of the preform, including a trial preform (black) made without burn-off etching; contrast the vertical scale with (a). Inset: a schematic diagram of the PCVD process.
Fig. 9.
Fig. 9. (a) Optical micrograph of one end of 3  cm of the Airy fiber, illuminated by the microscope’s white light source at the other end, and (b) a cross-sectional intensity plot derived from (a) along a row of pixels through the center.
Fig. 10.
Fig. 10. (a) Measured near-field image of the output of the fiber when trying to excite the Airy mode. The minimum intensity in the first dark ring is 0.1% of the peak intensity; it still looks dark even though the center of the image is saturated. Inset: an unsaturated image of the central bright disk, to the same scale. (b) The mode field ψ as deduced from the measurement, assuming that the field changes sign in successive rings. Red and blue represent positive and negative phases, respectively, while white means the plot is saturated.
Fig. 11.
Fig. 11. Measured near-field images when the fiber was sharply bent (1  cm diameter) to couple light to other modes. These can be contrasted with the Airy mode in Fig. 10(a).
Fig. 12.
Fig. 12. Far-field intensity distributions (a) calculated for an ideal Airy pattern, (b) calculated for an Airy pattern truncated to the first three rings, and (c) measured for the Airy mode of our fiber. The 1D plots below are horizontal slices through the centers of the 2D plots above.

Tables (2)

Tables Icon

Table 1. Constants P, C, and A in Eqs. (7) and (8) in Each Lobe N=0 to 3, for Our M=3 Approximation to the Airy Pattern

Tables Icon

Table 2. Parameters Defining Our Truncation of the Airy Mode by a Uniform Cladding, Using Eqs. (10) and (11) and the Untruncated Field of Eq. (8) and Table 1

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

T2ψ+(k2n2β2)ψ=0,
n2(r,θ)=T2ψ(r,θ)k2ψ(r,θ)+neff2,
ν(R)={d2ψdR2+1RdψdR}ψ,
ν(R)=a2k2(n2neff2)
ϕ0(R)=J1(j11R)j11R,
d2fdR2+1RdfdR+P2f=0.
f(R)=J0(PR)+CY0(PR),
ψN(R)=i=1MAN,ifN,i(R),
Ω={Ψ1Ψ2dA}2{Ψ12dA}{Ψ22dA},
ψc(R)=DK0(WR),
1ψ(Rc)dψdR|R=Rc=WK1(WRc)K0(WRc),
n2(r)=n02+ν(r/a)ν0a2k2.

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