Abstract

We discuss a transform technique for analyzing the wave vector content of microstructured optical fiber (MOF) modes, which is computationally efficient and gives good physical insight into the nature of the mode. In particular, if the mode undergoes a transition from a bound state to an extended state, this is evident in the spreading-out of its transform. The method has been implemented in the multipole formulation for finding MOF modes, but are capable of adaptation to other formulations.

© 2004 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  3. B. T. Kuhlmey, R. C. McPhedran, and C. M. de Sterke, “Modal cutoff in microstructured optical fibers,” Opt. Lett.,  27, 1684–1686 (2002).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  6. T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. M. de Sterke, and L. C. Botten, “Multipole method for microstructured optical fibers I: formulation,” J. Opt. Soc. Am. B19, 2322–2330 (2002), and “Erratum,” J. Opt. Soc. Am. B20, 1581 (2003).
    [CrossRef]
  7. B. T. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. M. de Sterke, and R. C. McPhedran, “Multipole formulation for microstructured optical fibers II: implementation and results,” J. Opt. Soc. Am. B 19, 2331–2340 (2002).
    [CrossRef]
  8. http://www.physics.usyd.edu.au/cudos/mofsoftware/
  9. T. A. Birks, J. C. Knight, and St. J. Russel, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22961–963 (1997).
    [CrossRef] [PubMed]
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  11. G. Allaire, C. Conca, and M. Vanninathan, “The Bloch Transform and applications,” 29th Congress of Numerical Analysis, ESAIM: Proceedings 3, 65–84 (1998), http://www.edpsciences.org/articlesproc/Vol.3/conca/conca.htm
  12. P. R. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides-I: Summary of results,” IEEE Trans. Microwave Theory Tech. MTT-23, 421–429 (1975).
    [CrossRef]

2003 (2)

P. St J. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003).
[CrossRef] [PubMed]

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

2002 (4)

1997 (1)

1975 (1)

P. R. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides-I: Summary of results,” IEEE Trans. Microwave Theory Tech. MTT-23, 421–429 (1975).
[CrossRef]

Allaire, G.

G. Allaire, C. Conca, and M. Vanninathan, “The Bloch Transform and applications,” 29th Congress of Numerical Analysis, ESAIM: Proceedings 3, 65–84 (1998), http://www.edpsciences.org/articlesproc/Vol.3/conca/conca.htm

Allan, D. C.

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

Birks, T. A.

Borrelli, N. F.

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

Botten, L. C.

B. T. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. M. de Sterke, and R. C. McPhedran, “Multipole formulation for microstructured optical fibers II: implementation and results,” J. Opt. Soc. Am. B 19, 2331–2340 (2002).
[CrossRef]

T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. M. de Sterke, and L. C. Botten, “Multipole method for microstructured optical fibers I: formulation,” J. Opt. Soc. Am. B19, 2322–2330 (2002), and “Erratum,” J. Opt. Soc. Am. B20, 1581 (2003).
[CrossRef]

Conca, C.

G. Allaire, C. Conca, and M. Vanninathan, “The Bloch Transform and applications,” 29th Congress of Numerical Analysis, ESAIM: Proceedings 3, 65–84 (1998), http://www.edpsciences.org/articlesproc/Vol.3/conca/conca.htm

de Sterke, C. M.

Gallagher, M. T.

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

Knight, J. C.

Koch, K. W.

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

Kuhlmey, B. T.

Maystre, D.

McIsaac, P. R.

P. R. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides-I: Summary of results,” IEEE Trans. Microwave Theory Tech. MTT-23, 421–429 (1975).
[CrossRef]

McPhedran, R. C.

Mortensen, N. A.

Muller, D.

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

Reed, M.

M. Reed and B. Simon, Methods of Modern Mathematical Physics IV: Analysis of Operators (Academic, New York, 1978).

Renversez, G.

Robinson, P. A.

Russel, St. J.

Russell, P. St J.

P. St J. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003).
[CrossRef] [PubMed]

Simon, B.

M. Reed and B. Simon, Methods of Modern Mathematical Physics IV: Analysis of Operators (Academic, New York, 1978).

Smith, C. M.

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

Vanninathan, M.

G. Allaire, C. Conca, and M. Vanninathan, “The Bloch Transform and applications,” 29th Congress of Numerical Analysis, ESAIM: Proceedings 3, 65–84 (1998), http://www.edpsciences.org/articlesproc/Vol.3/conca/conca.htm

Venkataraman, N.

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

West, J. A.

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

White, T. P.

B. T. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. M. de Sterke, and R. C. McPhedran, “Multipole formulation for microstructured optical fibers II: implementation and results,” J. Opt. Soc. Am. B 19, 2331–2340 (2002).
[CrossRef]

T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. M. de Sterke, and L. C. Botten, “Multipole method for microstructured optical fibers I: formulation,” J. Opt. Soc. Am. B19, 2322–2330 (2002), and “Erratum,” J. Opt. Soc. Am. B20, 1581 (2003).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

P. R. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides-I: Summary of results,” IEEE Trans. Microwave Theory Tech. MTT-23, 421–429 (1975).
[CrossRef]

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

Nature (1)

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

Opt. Express (2)

Opt. Lett. (2)

Science (1)

P. St J. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003).
[CrossRef] [PubMed]

Other (4)

T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. M. de Sterke, and L. C. Botten, “Multipole method for microstructured optical fibers I: formulation,” J. Opt. Soc. Am. B19, 2322–2330 (2002), and “Erratum,” J. Opt. Soc. Am. B20, 1581 (2003).
[CrossRef]

http://www.physics.usyd.edu.au/cudos/mofsoftware/

M. Reed and B. Simon, Methods of Modern Mathematical Physics IV: Analysis of Operators (Academic, New York, 1978).

G. Allaire, C. Conca, and M. Vanninathan, “The Bloch Transform and applications,” 29th Congress of Numerical Analysis, ESAIM: Proceedings 3, 65–84 (1998), http://www.edpsciences.org/articlesproc/Vol.3/conca/conca.htm

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Figures (2)

Fig. 1.
Fig. 1.

Field maps and total Bloch transform of a mode consisting essentially of a superposition of 6 Bloch waves. Note that the fields are depicted in the direct space (r-space), whereas the Bloch transform is in the reciprocal space (k-space): the white hexagon on the Bloch transform map depicts the edges of the first Brillouin zone. Here Λ=2.3 µm, λ=1.55 µm, d/Λ=0.15, and n silica=1.44402036.

Fig. 2.
Fig. 2.

Fundamental mode of two MOFs with different pitch, but with same d/Λ=0.3 and N r=8. The field distribution changes considerably between the two values of the pitch, but the Bloch transform remains a single peak centered on the origin. For all figures λ=1.55 µm and n silica=1.44402036.

Equations (14)

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𝓑 n ( k ) = l = 1 N i exp ( i k · c l ) B n ( c l ) .
V ( r ) = m = 1 N B exp ( i k B m · r ) v k B m ( r ) ,
B n ( c l ) = m = 1 N B B ̂ n m exp ( i k B m · c l ) ,
c 𝓛 , G · c 2 π ,
𝓑 T ( k ) = n 1 sup k 2 ( 𝓑 n ( k ) ) 𝓑 n ( k ) .
𝓑 n ( k B m ) = l = 1 N i j = 1 N B B ̂ n j exp ( i ( k B m k B j ) · c l ) ,
= N i B ̂ n m + j = 1 , j m N B B ̂ n j l = 1 N i exp ( i ( k B m k B j ) · c l ) .
G 𝓛 * , j m , k B m k B j + G N i 1 2 Λ 4 .
𝓑 n ( k B m ) N i B ̂ n m .
B ̂ n m ( Σ j = 1 N B B ̂ n j 2 ) 1 2 .
l = 1 N i B n ( c l ) 2 = 1 𝓐 FBZ FBZ 𝓑 n ( k ) 2 d k N i i = 1 N B B ̂ n i 2 ,
𝓑 n ( k ) = B ̂ n sin ( N 1 ( k k b ) · u 1 Λ 2 ) sin ( ( k k b ) · u 1 Λ 2 ) sin ( N 2 ( k k b ) · u 2 Λ 2 ) sin ( ( k k b ) · u 2 Λ 2 ) ,
f ( x ) = { sin ( a x ) sin ( x ) if x m π , m a if x = m π , m .
δ k m 2 1.91 N m Λ .

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