Abstract

In the effective index method, which models photonic crystal fibers by means of a step-index waveguide analogy, two critical parameters are the effective index of the cladding and the effective core radius. It has been shown that the use of an effective core radius as a function of the relative air hole diameter, or also of the relative wavelength, can improve the accuracy of this method. We show, by comparison with a rigorous finite-difference frequency-domain method, that the reported improved fully vectorial effective index methods have commonly adopted a radius of the equivalent circular unit cell, which does not give the best accurate effective cladding index as compared with the use of an equivalent circular unit cell having the same area as the hexagonal unit cell. Furthermore, by defining both the radius of the equivalent circular unit cell and the effective core radius as a function of the relative air hole diameter, and the relative wavelength, we believe that the fully vectorial effective index method can be further enhanced in terms of accuracy of both the effective cladding index and the modal index.

© 2008 Optical Society of America

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References

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2007

2006

M. Szpulak, W. Urbanczyk, E. Serebryannikov, A. Zheltikov, A. Hochman, Y. Leviatan, R. Kotynski, and K. Panajotov, "Comparison of different methods for rigorous modeling of photonic crystal fibers," Opt. Express 14, 5699-5714 (2006).
[CrossRef] [PubMed]

Y. Li, C. Wang, Y. Chen, M. Hu, B. Liu, and L. Chai, "Solution of the fundamental space-filling mode of photonic crystal fibers: numerical method versus analytical approaches," Appl. Phys. B 85, 597-601 (2006).
[CrossRef]

K. N. Park, T. Erdogan, and K. S. Lee, "Cladding mode coupling in long-period gratings formed in photonic crystal fibers," Opt. Commun. 266, 541-545 (2006).
[CrossRef]

Y. Li, C. Wang, N. Zhang, C. Wang, and Q. Xing, "Analysis and design of terahertz photonic crystal fibers by an effective-index method," Appl. Opt. 45, 8462-8465 (2006).
[CrossRef] [PubMed]

Y.-Z. Xu, X.-M. Ren, X. Zhang, and Y.-Q. Huang, "A fully vectorial effective index method for accurate dispersion calculation of photonic crystal fibres," Chin. Phys. Lett. 23, 2476-2479 (2006).
[CrossRef]

P. St. J. Russell, "Photonic-crystal fibers," J. Lightwave Technol. 24, 4729-4749 (2006).
[CrossRef]

2005

R. K. Sinha and A. D. Varshney, "Dispersion properties of photonic crystal fiber: comparison by scalar and fully vectorial effective index methods," Opt. Quantum Electron. 37, 711-722 (2005).
[CrossRef]

M. Koshiba and K. Saitoh, "Simple evaluation of confinement losses in holey fibers," Opt. Commun. 253, 95-98 (2005).
[CrossRef]

K. N. Park and K. S. Lee, "Improved effective-index method for analysis of photonic crystal fibers," Opt. Lett. 30, 958-960 (2005).
[CrossRef] [PubMed]

2004

M. Koshiba and K. Saitoh, "Applicability of classical optical fiber theories to holey fibers," Opt. Lett. 29, 1739-1741 (2004).
[CrossRef] [PubMed]

Y. Li, C. Wang, and M. Hu, "A fully vectorial effective index method for photonic crystal fibers: application to dispersion calculation," Opt. Commun. 238, 29-33 (2004).
[CrossRef]

2003

R. K. Sinha and S. K. Varshney, "Dispersion properties of photonic crystal fibers," Microwave Opt. Technol. Lett. 37, 129-132 (2003).
[CrossRef]

M. D. Nielson, N. A. Mortensen, J. R. Folkenberg, and A. Bjarklev, "Mode-field radius of photonic crystal fibers expressed by the V parameter," Opt. Lett. 28, 2309-2311 (2003).
[CrossRef]

J. C. Knight, "Photonic crystal fibres," Nature 424, 847-851 (2003).
[CrossRef] [PubMed]

2002

2001

2000

M. Midrio, M. P. Singh, and C. G. Someda, "The space filling mode of holey fibers: an analytical vectorial solution," J. Lightwave Technol. 18, 1031-1037 (2000).
[CrossRef]

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

1998

1997

Appl. Opt.

Appl. Phys. B

Y. Li, C. Wang, Y. Chen, M. Hu, B. Liu, and L. Chai, "Solution of the fundamental space-filling mode of photonic crystal fibers: numerical method versus analytical approaches," Appl. Phys. B 85, 597-601 (2006).
[CrossRef]

Chin. Phys. Lett.

Y.-Z. Xu, X.-M. Ren, X. Zhang, and Y.-Q. Huang, "A fully vectorial effective index method for accurate dispersion calculation of photonic crystal fibres," Chin. Phys. Lett. 23, 2476-2479 (2006).
[CrossRef]

IEE Proc.

T. Sørensen, J. Broeng, A. Bjarklev, T. P. Hansen, E. Knudsen, S. E. B. Libori, H. R. Simonsen, and J. R. Jensen, "Spectral macro-bending loss considerations for photonic crystal fibres," IEE Proc. : Optoelectron. 149, 206-210 (2002).

J. Lightwave Technol.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Microwave Opt. Technol. Lett.

R. K. Sinha and S. K. Varshney, "Dispersion properties of photonic crystal fibers," Microwave Opt. Technol. Lett. 37, 129-132 (2003).
[CrossRef]

Nature

J. C. Knight, "Photonic crystal fibres," Nature 424, 847-851 (2003).
[CrossRef] [PubMed]

Opt. Commun.

Y. Li, C. Wang, and M. Hu, "A fully vectorial effective index method for photonic crystal fibers: application to dispersion calculation," Opt. Commun. 238, 29-33 (2004).
[CrossRef]

M. Koshiba and K. Saitoh, "Simple evaluation of confinement losses in holey fibers," Opt. Commun. 253, 95-98 (2005).
[CrossRef]

K. N. Park, T. Erdogan, and K. S. Lee, "Cladding mode coupling in long-period gratings formed in photonic crystal fibers," Opt. Commun. 266, 541-545 (2006).
[CrossRef]

Opt. Express

Opt. Fiber Technol.

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

Opt. Laser Technol.

Y. Li, C. Wang, Z. Wang, M. Hu, and L. Chai, "Analytical solution of the fundamental space filling mode of photonic crystal fibers," Opt. Laser Technol. 39, 322-326 (2007).
[CrossRef]

Opt. Lett.

Opt. Quantum Electron.

R. K. Sinha and A. D. Varshney, "Dispersion properties of photonic crystal fiber: comparison by scalar and fully vectorial effective index methods," Opt. Quantum Electron. 37, 711-722 (2005).
[CrossRef]

Other

A. Bjarklev, J. Broeng, and A. S. Bjarklev, Photonic Crystal Fibres (Kluwer, 2003).
[CrossRef]

T. A. Birks, D. Mogilevtsev, J. C. Knight, P. St. J. Russell, J. Broeng, P. J. Roberts, J. A. West, D. C. Allan, and J. C. Fajardo, "The analogy between photonic crystal fibres and step index fibres," in Technical Digest of the Optical Fiber Communication Conference, 1999, and the International Conference on Integrated Optics and Optical Fiber Communication, OFC/IOOC '99 (IEEE, 1999), pp. 114-116.
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

(Color online) Schematic of the principle of the EIM. An air-silica PCF with air holes of pitch Λ and diameter d is approximated as a SIF with an effective cladding index n clad and an effective core radius ρ. The rectangular box B1–B4 is used in numerical methods to compute n clad .

Fig. 2
Fig. 2

In the analytical solution of the n clad , the hexagonal unit cell is approximated as a circular unit cell with a radius of R. Two values of R are considered: R 1 = Λ / 2 (the dashed curve) and R 2 = Λ [ 3 / ( 2 π ) ] 1 / 2 (the dotted–dashed curve), which ensures that the circular cell has the same area as the hexagonal cell.

Fig. 3
Fig. 3

(Color online) (a) Values of n clad obtained by FDFD and the VEIM, and (b) Δ n clad = | n clad FDFD n clad VEIM | . Numbers are for values of d / Λ ranging from 0.3 to 0.8, the wavelength range is from 0.4 to 2.0   μm , and R 1 and R 2 are the two values of R used in the VEIM.

Fig. 4
Fig. 4

(Color online) Δ n mode = | n mode FDFD n mode VEIM | for relative air hole diameters d / Λ of (a) 0.3, (b) 0.4, (c) 0.5, (d) 0.6, (e) 0.7, and (f) 0.8 over the wavelength range of 0.4 2.0   μm . Values of 0.577 and 0.625 stand for ρ / Λ = 3 / 3 [16] and ρ / Λ = 0.625 [20], respectively. The other three values of ρ / Λ are taken from Xu et al. [22], Park and Lee [21], and Zhao et al. [24]. I, II, and III refer to the three values of n clad used in the VEIM: (I) n clad by FDFD, (II) n clad by the VEIM using R 2 , and (III) n clad by the VEIM using R 1 .

Fig. 5
Fig. 5

(Color online) Δ n clad = | n clad FDFD n clad VEIM | when fitted values of R are used in the VEIM.

Fig. 6
Fig. 6

(Color online) Δ n mode = | n mode FDFD n mode VEIM | when fitted values of R and ρ are used in the VEIM.

Fig. 7
Fig. 7

(Color online) Core radii ρ / Λ fitted from Eq. (4) and those obtained by the VEIM by using numerical n clad and n mode by FDFD. Numbers are for values of d / Λ ranging from 0.3 to 0.8.

Fig. 8
Fig. 8

(Color online) Field distribution of the y-polarized fundamental mode at 1.5   μm for Λ = 2.3   μm and (a) d / Λ = 0.3 and (b) d / Λ = 0.6 .

Tables (4)

Tables Icon

Table 1 Comparison of n clad at 1.5 μm Obtained by VEIM, PWM, and FDFD for Different Values of d ∕Λ a

Tables Icon

Table 2 Comparison of n mode at 1.5 μm Obtained by MM, FDFD, and VEIM with Different Combinations of Effective Core Radius and Effective Cladding Index to Give the Best Accurate Results Shown in Fig. 4 a

Tables Icon

Table 3 Values of Coefficients m i , j

Tables Icon

Table 4 Values of Coefficients n i , j

Equations (4)

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( P 1 ( γ a ) γ a P 1 ( γ a ) + I 1 ( κ a ) κ a I 1 ( κ a ) ) ( n silica 2 P 1 ( γ a ) γ a P 1 ( γ a ) + n air 2 I 1 ( κ a ) κ a I 1 ( κ a ) ) = [ ( 1 γ a ) 2 + ( 1 κ a ) 2 ] 2 ( β clad k ) 2 ,
( J 1 ( u ρ ) u ρ J 1 ( u ρ ) + K 1 ( w ρ ) w ρ K 1 ( w ρ ) ) ( n silica 2 J 1 ( u ρ ) u ρ J 1 ( u ρ ) + n c l a d 2 K 1 ( w ρ ) w ρ K 1 ( w ρ ) ) = [ ( 1 u ρ ) 2 + ( 1 w ρ ) 2 ] 2 ( β core k ) 2 ,
R Λ = i = 0 j = 0 5 m i j ( λ Λ ) i ( d Λ ) i .
ρ Λ = i = 0 j = 0 5 n i j ( λ Λ ) i ( d Λ ) j .

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