Abstract

A novel full vector model based on the supercell lattice method is presented to analyze photonic crystal fiber (PCF). From symmetry analysis waveguide modes, we classify PCF modes into nondegenerate or degenerate pairs according to the minimum waveguide sectors and their appropriate boundary conditions. We describe how the modes of the PCF can be labeled by step-index fiber analogs, with the exception of modes that have the same symmetry as the PCF. When the doublet of the degenerate pairs both have the same symmetry as the PCF, they will be split into two nondegenerate modes.

© 2003 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |

  1. J. C. Night, P. St. Russell, �??New way to guide light,�?? Science 296, 276-277 (2002).
    [CrossRef]
  2. T. A. Birks, J. C. Knight, P. St. J. Russell, �??Endlessly single-mode photonic crystal fiber,�?? Opt. Lett. 22, 961-963 (1997).
    [CrossRef] [PubMed]
  3. A. Ferrando, E. Silvestre, �??Nearly zero ultraflattened dispersion in photonic crystal fibers,�?? Opt. Lett. 25, 790-792 (2000).
    [CrossRef]
  4. AL. Gaeta, �??Nonlinear propagation and continuum generation in microstructured optical fibers,�?? Opt. Lett. 27, 924-926 (2002).
    [CrossRef]
  5. T. A. Birks, D. Mogilevtsev, J. C. Knight, P. St. J. Russell, J. Broeng, P. J. Roberts, J. A. West, D. C. Allan, J. C. Fajardo, �??The analogy between photonic crystal fibres and step index fibres,�?? in Optical Fiber Communication Conference, OSA (Optical Society of America, Washington, D.C., 1998), FG4, 114-116.
  6. J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic crystals: molding the flow of light, (New York : Princeton University Press, 1995).
  7. Shangping Guo, Sacharia Albin, �??Simple plane wave implementation for photonic crystal calculations,�?? Opt. Express 11, 167-175 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-2-167">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-2-167</a>
    [CrossRef] [PubMed]
  8. T. M. Monro, D. J. Richardson, N. G. R. Broderick, P. J. Bennett, �??Modeling large air fraction holey optical fibers,�?? J. Lightwave Technol. 18, 50-56 (2000).
    [CrossRef]
  9. D. Mogilevtsev, T. A. Birks, P. St. Russell, �??Localized function method for modeling defect mode in 2-d photonic crystal,�?? J. Lightwave Technol. 17, 2078-2081 (1999).
    [CrossRef]
  10. M. Koshiba, �??Full vector analysis of photonic crystal fibers using the finite elelment method,�?? IEICE Electron, E85-C, 4, 881-888 (2002).
  11. Fabrizio Fogli, Luca Saccomandi, Paolo Bassi, Gaetano Bellanca, Stefano Trillo, �??Full vectorial BPM modeling of Index-Guiding Photonic Crystal Fibers and Couplers,�?? Opt. Express 10, 54-59 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-1-54">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-1-54</a>
    [CrossRef] [PubMed]
  12. T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, L. C. Botten, �??Multipole method for microstructured optical fibers. I. Formulation,�?? J. Opt. Soc. Am. B 19 (10): 2322-2330 OCT (2002).
    [CrossRef]
  13. Boris T. Kuhlmey, Thomas P. White, Gilles Renversez, Daniel Maystre, Lindsay C. Botten, C. Martijn de Sterke, Ross C. McPhedran, �??Multipole method for microstructured optical fibers. II. Implementation and results,�?? J. Opt. Soc. Am. B 19 (10): 2331-2340 OCT (2002).
    [CrossRef]
  14. F. G. Omenetto, A. J. Taylor, M. D. Moores, J. Arriaga, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, �??Simultaneous generation of spectrally distinct third harmonics in a photonic crystal fiber,�?? Opt. Lett. 26, 1158 (2001).
    [CrossRef]
  15. Anatoly Efimov, Antoinette J. Taylor, Fiorenzo G. Omenetto, Jonathan C. Knight, William J. Wadsworth, Philip St. J. Russell, �??Nonlinear generation of very high-order UV modes in microstructured fibers,�?? Opt. Express 11, 910-918 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-8-910">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-8-910</a>
    [CrossRef] [PubMed]
  16. F. Brechet, J. Marcou, D. Pagnoux, P.Roy, �??Complete analysis of the characteristics of propagation into photonic crystal fibers, by the finite element method,�?? Opt. Fibre Technol. 6, 181-191 (2000).
    [CrossRef]
  17. Ferrando A, Silvestre E, Miret JJ, Juan Jose Miret, Pedro Andres, Miguel V. Andres, �??Vector description of higher-order modes in photonic crystal fibers,�?? J. Opt. Soc. Am. B 17, 1333-1340 (2000).
    [CrossRef]
  18. M. J. Steel, T. P. White, C. M. Sterke, R. C. McPhedran, L. C. Botten, �??Symmetry and degeneracy in microstructured optical fibers,�?? Opt. Lett. 26, 488-490 (2001).
    [CrossRef]
  19. M. J. Steel, T. P. White, C. M. Sterke, R. C. McPhedran, L. C. Botten, �??Symmetry and birefringence in microstructured optical fibers,�?? OFC 2001, 3, WDD88-1 -WDD88-3 (2001).
  20. A. W. Snyder, Optical waveguide theory, (New York: Chapman and Hall, 1983).
  21. Wang Zhi, Ren Guobin, Lou Shuqin, Jian Shuisheng, �??Supercell lattice method for the photonic crystal fibers,�?? Opt. Express 11, 980-991 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX- 11-9-980">http://www.opticsexpress.org/abstract.cfm?URI=OPEX- 11-9-980</a>
    [CrossRef] [PubMed]
  22. Wang Zhi, Ren Guobin, Lou Shuqin, Jian Shuisheng, �??A Novel Supercell Overlapping Method for Different Photonic Crystal Fibers,�?? submitted to J. Lightwave Technol.
  23. I. S. Gradshtein, and I. M. Ryzhik, Tables of integrals, series and products, (New York: Academic, 1994).
  24. Isidoro Kimel, Luis R. Elias, �??Relations between Hermite and Laguerre Gaussian modes,�?? IEEE J. Quant. Electron. 29, 2562-2567 (1993).
    [CrossRef]
  25. 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]

IEEE J. Quant. Electron.

Isidoro Kimel, Luis R. Elias, �??Relations between Hermite and Laguerre Gaussian modes,�?? IEEE J. Quant. Electron. 29, 2562-2567 (1993).
[CrossRef]

IEEE, Trans. Microwave Theory Tech.

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]

IEICE Electron

M. Koshiba, �??Full vector analysis of photonic crystal fibers using the finite elelment method,�?? IEICE Electron, E85-C, 4, 881-888 (2002).

J. Lightwave Technol.

J. Opt. Soc. Am. B

Opt. Express

Opt. Fibre Technol.

F. Brechet, J. Marcou, D. Pagnoux, P.Roy, �??Complete analysis of the characteristics of propagation into photonic crystal fibers, by the finite element method,�?? Opt. Fibre Technol. 6, 181-191 (2000).
[CrossRef]

Opt. Lett.

Science

J. C. Night, P. St. Russell, �??New way to guide light,�?? Science 296, 276-277 (2002).
[CrossRef]

Other

T. A. Birks, D. Mogilevtsev, J. C. Knight, P. St. J. Russell, J. Broeng, P. J. Roberts, J. A. West, D. C. Allan, J. C. Fajardo, �??The analogy between photonic crystal fibres and step index fibres,�?? in Optical Fiber Communication Conference, OSA (Optical Society of America, Washington, D.C., 1998), FG4, 114-116.

J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic crystals: molding the flow of light, (New York : Princeton University Press, 1995).

M. J. Steel, T. P. White, C. M. Sterke, R. C. McPhedran, L. C. Botten, �??Symmetry and birefringence in microstructured optical fibers,�?? OFC 2001, 3, WDD88-1 -WDD88-3 (2001).

A. W. Snyder, Optical waveguide theory, (New York: Chapman and Hall, 1983).

Wang Zhi, Ren Guobin, Lou Shuqin, Jian Shuisheng, �??A Novel Supercell Overlapping Method for Different Photonic Crystal Fibers,�?? submitted to J. Lightwave Technol.

I. S. Gradshtein, and I. M. Ryzhik, Tables of integrals, series and products, (New York: Academic, 1994).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (12)

Fig. 1.
Fig. 1.

Reconstruction of the dielectric structure of a triangular lattice TIR-PCF with the parameters Λ=2.3µm, d/Λ=0.6, P 1=50, P 2=500. (a) Three-dimensional (3-D) refractive-index reconstruction. (b) Cross section along the y=0 axis of index reconstruction.

Fig. 2.
Fig. 2.

Minimum sectors for waveguides with C symmetry. The modes of waveguides are classified into eight classes (p=1,….8). Solid lines indicate short-circuit boundary conditions, dashed lines indicate open-circuit boundary conditions.

Fig. 3.
Fig. 3.

Mode index of the first 24 modes in TIR-PCF with structural parameters Λ=2.3µm, d/Λ=0.8 at λ=633nm.

Fig. 4.
Fig. 4.

Total modal intensity distribution of HE11 mode (a) and 2-D electric vector distributions of polarization doublet modes (b), (c) with parameters Λ=2.3µm, d/Λ=0.8, at wavelength λ=633nm.

Fig. 5.
Fig. 5.

Electric vector distributions for modes 3, 4, 5, 6 corresponding to (a) TE01, (b), (c) HE21, and (d) TM01 mode of step-index fiber.

Fig. 6.
Fig. 6.

Intensity distribution of the combination of modes 4 and 5 (HE21).

Fig. 7.
Fig. 7.

Electric vector distributions of mode 7–14.

Fig. 8
Fig. 8

Intensity distributions of nondegenerate mode HE311, degenerate mode EH11, HE12 and EH21.

Fig. 9.
Fig. 9.

Modal intensity distribution of EH311 mode (a) and 2-D electric vector distributions of EH311 and EH312 modes (b), (c) with parameters Λ=2.3µm, d/Λ=0.9, at wavelength λ=0.633µm.

Fig. 10.
Fig. 10.

Minimum sectors for waveguides with C symmetry. The waveguides modes are divided into eight classes (p=1,….6). Solid lines indicate short-circuit boundary conditions; dashed lines indicate open-circuit boundary conditions.

Fig. 11.
Fig. 11.

Electric vector distributions for modes 3–6 of the square lattice PCF.

Fig. 12.
Fig. 12.

intensity distributions of nondegenerate modes HE211 and TE01 of a square lattice PCF.

Tables (2)

Tables Icon

Table 1. Mode-index (neff ), mode class (p), degeneracy, computation error (Δn) and label for first 14 modes

Tables Icon

Table 2. Mode index (neff ), mode class (p), degeneracy, computation error (Δn), and label for first eight modes

Equations (16)

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

E j ( x , y , z ) = [ e j t ( x , y ) + e j z ( x , y ) ] e j ( β j z ω t ) ,
( t 2 β j 2 + k 2 n 2 ) e x = x ( e x In n 2 x + e y In n 2 y ) ,
( t 2 β j 2 + k 2 n 2 ) e y = y ( e x In n 2 x + e y In n 2 y ) ,
e x ( x , y ) = a , b = 0 F 1 ε a b x ψ a ( x ) ψ b ( y ) ,
e y ( x , y ) = a , b = 0 F 1 ε a b y ψ a ( x ) ψ b ( y ) ,
ψ i ( s ) = 2 i 2 π 1 4 ( i ) ! ω exp ( s 2 2 ω 2 ) H i ( s ω ) ,
n 2 ( x , y ) = a , b = 0 P 1 1 P 1 a b cos 2 π a x l 1 x cos 2 π b y l 1 y + a , b = 0 P 2 1 P 2 a b cos 2 π a x l 2 x cos 2 π b y l 2 y ,
In n 2 ( x , y ) = a , b = 0 P 1 1 P 1 a b In cos 2 π a x l 1 x cos 2 π b y l 1 y + a , b = 0 P 2 1 P 2 ab ln cos 2 π a x l 2 x cos 2 π b y l 2 y ,
L [ e x e y ] = [ I abcd ( 1 ) + k 2 I abcd ( 2 ) + I abcd ( 3 ) x I abcd ( 4 ) x I abcd ( 4 ) y I abcd ( 1 ) + k 2 I abcd ( 2 ) + I abcd ( 3 ) y ] [ e x e y ] = β j 2 [ e x e y ] ,
I abcd ( 1 ) = + ψ a ( x ) ψ b ( y ) t 2 [ ψ c ( x ) ψ d ( y ) ] dx dy ,
I abcd ( 2 ) = + n 2 ψ a ( x ) ψ b ( y ) ψ c ( x ) ψ d ( y ) dx dy ,
I abcd ( 3 ) x = + ψ a ( x ) ψ b ( y ) x [ ψ c ( x ) ψ d ( y ) In n 2 x ] dx dy ,
I abcd ( 3 ) y = + ψ a ( x ) ψ b ( y ) y [ ψ c ( x ) ψ d ( y ) In n 2 y ] dx dy ,
I abcd ( 4 ) x = + ψ a ( x ) ψ b ( y ) x [ ψ c ( x ) ψ d ( y ) In n 2 y ] dx dy ,
I abcd ( 4 ) y = + ψ a ( x ) ψ b ( y ) y [ ψ c ( x ) ψ d ( y ) In n 2 x ] dx dy .
e t = i k 2 n 2 β 2 { β t e z ( μ 0 ε 0 ) 1 2 k z ̂ × t h z } .

Metrics