Abstract

This paper presents an expression for the eigenvalue equation and fields in a Bragg fiber by calculating the phase change that is based on geometrical optics. The quarter-wave stack condition makes it possible to consider that the Bragg fiber has approximately no cladding from the electromagnetic point of view, despite the fact that the Bragg fiber has a periodic cladding. As a result, its eigenvalue equation can be represented in terms of the zeros of Bessel functions and only the core parameters, for a specific case. The eigenvalue equations for HE and EH modes in the Bragg fiber have a formal equivalence to those for TE and TM modes, respectively, in the circular metallic waveguide. Results obtained are in agreement, under a specific limit, with those derived by an asymptotic expansion method.

© 2006 Optical Society of America

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References

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2006 (1)

2005 (2)

J. Sakai and P. Nouchi, "Propagation properties of Bragg fiber analyzed by a Hankel function formalism," Opt. Commun. 249, 153-163 (2005).
[CrossRef]

J. Sakai, "Hybrid modes in a Bragg fiber: general properties and formulas under the quarter-wave stack condition," J. Opt. Soc. Am. B 22, 2319-2330 (2005).
[CrossRef]

2004 (2)

2003 (3)

2002 (2)

2001 (1)

1999 (1)

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, "Single-mode photonic bandgap guidance of light in air," Science 285, 1537-1539 (1999).
[CrossRef] [PubMed]

1998 (1)

1996 (1)

1978 (1)

1976 (1)

1975 (1)

P. R. McIsaac, "Symmetry-induced modal characteristics of uniform waveguides - 1: Summary of results," IEEE Trans. Microwave Theory Tech. MTT-23, 421-4429 (1975).
[CrossRef]

1960 (1)

J. B. Keller and S. I. Rubinow, "Asymptotic solution of eigenvalue problems," Ann. Phys. (N.Y.) 9, 24-75 (1960).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, 1972), Chap. 9.

Allan, D. C.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, "Single-mode photonic bandgap guidance of light in air," Science 285, 1537-1539 (1999).
[CrossRef] [PubMed]

Andrés, P.

Atkin, D. M.

Birks, T. A.

Bjarklev, A.

Botten, L. G.

Broeng, J.

Chang, H.

Cregan, R. F.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, "Single-mode photonic bandgap guidance of light in air," Science 285, 1537-1539 (1999).
[CrossRef] [PubMed]

de Sandro, J. P.

Deyerl, H.

Ferrando, A.

Guo, S.

Guobin, R.

Hansen, T. P.

Huang, Y.

Jakobsen, C.

Jensen, J. B.

Keller, J. B.

J. B. Keller and S. I. Rubinow, "Asymptotic solution of eigenvalue problems," Ann. Phys. (N.Y.) 9, 24-75 (1960).
[CrossRef]

Knight, J. C.

Kuhlmey, B. T.

Kurokawa, K.

K. Tajima, J. Zhou, K. Kurokawa, and K. Nakajima, "Low water peak photonic crystal fibers," in Proceedings of the European Conference on Optical Communication (ECOC, 2003), paper Th4.1.6.

Lee, R. K.

Love, J. D.

Mangan, B. J.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, "Single-mode photonic bandgap guidance of light in air," Science 285, 1537-1539 (1999).
[CrossRef] [PubMed]

Marom, E.

Maystre, D.

McIsaac, P. R.

P. R. McIsaac, "Symmetry-induced modal characteristics of uniform waveguides - 1: Summary of results," IEEE Trans. Microwave Theory Tech. MTT-23, 421-4429 (1975).
[CrossRef]

McPhedran, R. C.

Miret, J. J.

Monsoriu, J. A.

Mortensen, N. A.

Nakajima, K.

K. Tajima, J. Zhou, K. Kurokawa, and K. Nakajima, "Low water peak photonic crystal fibers," in Proceedings of the European Conference on Optical Communication (ECOC, 2003), paper Th4.1.6.

Nouchi, P.

J. Sakai and P. Nouchi, "Propagation properties of Bragg fiber analyzed by a Hankel function formalism," Opt. Commun. 249, 153-163 (2005).
[CrossRef]

Ouyang, G.

Ouyang, G. X.

Renversez, G.

Rizzi, P. A.

P. A. Rizzi, Microwave Engineering: Passive Circuits (Prentice Hall, 1988), Chap. 5.

Roberts, P. J.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, "Single-mode photonic bandgap guidance of light in air," Science 285, 1537-1539 (1999).
[CrossRef] [PubMed]

Rubinow, S. I.

J. B. Keller and S. I. Rubinow, "Asymptotic solution of eigenvalue problems," Ann. Phys. (N.Y.) 9, 24-75 (1960).
[CrossRef]

Russell, P.

P. Russell, "Photonic crystal fibers," Science 299, 358-362 (2003).
[CrossRef] [PubMed]

Russell, P. St. J.

Sakai, J.

Sasaki, J.

Shuqin, L.

Silvestre, E.

Simonsen, H.

Snyder, A. W.

Sørensen, T.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, 1972), Chap. 9.

Sterke, G. M.

Tajima, K.

K. Tajima, J. Zhou, K. Kurokawa, and K. Nakajima, "Low water peak photonic crystal fibers," in Proceedings of the European Conference on Optical Communication (ECOC, 2003), paper Th4.1.6.

Terrel, M.

Vienne, G.

Weijun, L.

White, T. P.

Xu, Y.

Yariv, A.

Yeh, P.

Yu, C.

Zhi, W.

Zhou, J.

K. Tajima, J. Zhou, K. Kurokawa, and K. Nakajima, "Low water peak photonic crystal fibers," in Proceedings of the European Conference on Optical Communication (ECOC, 2003), paper Th4.1.6.

Ann. Phys. (N.Y.) (1)

J. B. Keller and S. I. Rubinow, "Asymptotic solution of eigenvalue problems," Ann. Phys. (N.Y.) 9, 24-75 (1960).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Microwave Theory Tech. (1)

P. R. McIsaac, "Symmetry-induced modal characteristics of uniform waveguides - 1: Summary of results," IEEE Trans. Microwave Theory Tech. MTT-23, 421-4429 (1975).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. (1)

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

J. Opt. Soc. Am. B (3)

Opt. Commun. (1)

J. Sakai and P. Nouchi, "Propagation properties of Bragg fiber analyzed by a Hankel function formalism," Opt. Commun. 249, 153-163 (2005).
[CrossRef]

Opt. Express (5)

Opt. Lett. (1)

Science (2)

P. Russell, "Photonic crystal fibers," Science 299, 358-362 (2003).
[CrossRef] [PubMed]

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, "Single-mode photonic bandgap guidance of light in air," Science 285, 1537-1539 (1999).
[CrossRef] [PubMed]

Other (3)

K. Tajima, J. Zhou, K. Kurokawa, and K. Nakajima, "Low water peak photonic crystal fibers," in Proceedings of the European Conference on Optical Communication (ECOC, 2003), paper Th4.1.6.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, 1972), Chap. 9.

P. A. Rizzi, Microwave Engineering: Passive Circuits (Prentice Hall, 1988), Chap. 5.

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Figures (4)

Fig. 1
Fig. 1

Schematic structure of the Bragg fiber. r c , core radius; n c , refractive index of core; n a and n b , indices of layer with thickness a and b, respectively; Λ = a + b , period in the cladding; r ca , caustic radius.

Fig. 2
Fig. 2

View of ray travel from point P to point Q existing on the core-cladding boundary. At the middle point of PQ the ray touches the caustic S with radius r ca . The core radius is r c , and its index is n c . The ray makes angles ϕ r c , ϕ θ c , and ϕ z c with respect to the r, θ, and z axes, respectively, at the core-cladding boundary. The Q P and Q S are projections of point Q onto a plane perpendicular to the z axis and including points P and S, respectively. Translations of PQ along the θ and z directions are indicated by δ θ c and δ z c . ϕ r , ca , ϕ θ , ca , and ϕ z , ca denote the ray angles at the caustic.

Fig. 3
Fig. 3

View of ray propagating from point Q on the core-cladding boundary to point T on the outer side of cladding layer “a” with thickness a and index n a . The ray makes angles ϕ r a , ϕ θ a , and ϕ z a with respect to the r, θ, and z axes, respectively, at the inner interface of each layer. The T Q is the projection of the point T onto a plane perpendicular to the z axis and including the point Q. Translations of QT along the θ and z directions are indicated by δ θ a and δ z a . The core radius is denoted by r c .

Fig. 4
Fig. 4

Top and side views of two ray paths traveling from the caustic to point D ( r , θ , z ) to constitute the core field. One ray tangent to the caustic surface leaves a point B on the caustic and directly reaches point D. The other ray leaves the caustic at another point A and arrives at the same point D after reflecting at a point C on the core-cladding boundary. Points H A , H B , and H 0 are situated on the cross line between the caustic surface and θ = 0 . Arcs AH A and BH B are perpendicular to the z axis. Angles AOH A and BOH B are rational translations of the direct and reflected rays, respectively. H 0 is on the z = 0 plane.

Equations (54)

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PQ ¯ = 2 r c cos ϕ r c sin 2 ϕ z c .
δ z c = PQ ¯ cos ϕ z c = 2 r c cos ϕ r c cos ϕ z c sin 2 ϕ z c .
δ θ c = sin 1 ( 2 cos ϕ r c cos ϕ θ c sin 2 ϕ z c ) .
cos ϕ θ , ca = sin ϕ z c
r ca = r c [ 1 ( cos ϕ r c sin ϕ z c ) 2 ] 1 2 ,
r c cos ϕ r c sin ϕ z c = ( r c 2 r ca 2 ) 1 2 .
cos 1 ( cos ϕ θ c sin ϕ z c ) = 1 2 sin 1 ( 2 cos ϕ r c cos ϕ θ c sin 2 ϕ z c ) = cos 1 ( r ca r c )
sin 1 ( 2 cos ϕ r c cos ϕ θ c sin 2 ϕ z c ) = 2 cos 1 ( cos ϕ θ c sin ϕ z c ) .
sin 2 ϕ z a QT ¯ 2 + 2 r c cos ϕ r a QT ¯ 2 r c a = 0 .
QT ¯ a cos ϕ r a .
δ z a = QT ¯ cos ϕ z a = a cos ϕ z a cos ϕ r a .
δ θ a = sin 1 ( a cos ϕ θ a r c cos ϕ r a ) a cos ϕ θ a r c cos ϕ r a .
β = n i k 0 cos ϕ z i
κ i = [ ( n i k 0 ) 2 β 2 ] 1 2
n c k 0 sin ϕ z c = n c k 0 ( 1 cos 2 ϕ z c ) 1 2 = κ c .
ν = n c k 0 r c cos ϕ θ c ( ν = 0 , 1 , 2 , ) .
ν = n c k 0 r ca cos ϕ θ , ca = κ c r ca
n c k 0 r c cos ϕ r c = [ ( κ c r c ) 2 ν 2 ] 1 2
r ca = ν κ c
ϕ θ i π 2 ( i = a , b ) ,
n i k 0 cos ϕ r i κ i ( i = a , b )
P a n a k 0 QT ¯ n a k 0 cos ϕ z a           δ z a n a k 0 cos ϕ θ a      
r c δ θ a n a k 0 a cos ϕ r a = κ a a
n c k 0 PQ ¯ n c k 0 cos ϕ z c     δ z c n c k 0 cos ϕ θ c       r c δ θ c + Φ add = 2 π μ ,
Φ add = Φ ca + Φ R ,
Φ R = { π the S component in all cases and the P component with large incident angle 0 the P component with small incident angle .
2 n c k 0 r c [ cos ϕ r c cos ϕ θ c cos 1 ( cos ϕ θ c sin ϕ z c ) ] + Φ ca + Φ R = 2 π μ
2 { [ ( κ c r c ) 2 ν 2 ] 1 2 ν cos 1 ( ν κ c r c ) } + Φ ca + Φ R = 2 π μ .
2 [ κ c ( r c 2 r ca 2 ) 1 2 ν cos 1 ( r ca r c ) ] + Φ ca + Φ R = 2 π μ ,
cos X = sin ( π 2 X ) π 2 X = ν κ c r c .
2 κ c r c ν ( π ν κ c r c ) + Φ ca + Φ R 2 π μ
2 κ c r c ν π + Φ ca + Φ R 2 π μ .
κ c r c = π ( μ + ν 2 1 4 ) j ν , μ .
κ c r c = π ( μ + ν 2 3 4 ) j ν , μ .
( + k 2 ) ψ = 0 ,
ψ ( r , θ , z ) = j A j ( r , θ , z ) exp [ i k S j ( r , θ , z ) ]
BD ¯ = ( r 2 r ca 2 ) 1 2 sin ϕ z c .
AC ¯ + CD ¯ = PQ ¯ BD ¯ = [ 2 ( r c 2 r ca 2 ) 1 2 ( r 2 r ca 2 ) 1 2 ] 1 sin ϕ z c
δ z BD ¯ = ( r 2 r ca 2 ) 1 2 cot ϕ z c .
δ z AC ¯ + CD ¯ = [ 2 ( r c 2 r ca 2 ) 1 2 ( r 2 r ca 2 ) 1 2 ] cot ϕ z c .
arc BH B = r ca [ θ cos 1 ( r ca r ) ] .
arc AH A = r ca [ θ + cos 1 ( r ca r ) 2 cos 1 ( r ca r c ) ] .
H B H 0 ¯ = z BD ¯ cos ϕ z c = z ( r 2 r ca 2 ) 1 2 cot ϕ z c
H A H 0 ¯ = z ( AC ¯ + CD ¯ ) cos ϕ z c = z [ 2 ( r c 2 r ca 2 ) 1 2 ( r 2 r ca 2 ) 1 2 ] cot ϕ z c
P 1 ( r , θ , z ) = n c k 0 BD ¯ + n c k 0 cos ϕ θ , ca     arc BH B + n c k 0 cos ϕ z , ca     H B H 0 ¯
= β z + ν [ θ cos 1 ( r ca r ) ] + κ c r 2 r ca 2 ,
P 2 ( r , θ , z ) = n c k 0 ( AC ¯ + CD ¯ ) + n c k 0 cos ϕ θ , ca       arc AH A + n c k 0 cos ϕ z , ca       H A H 0 ¯ = β z + ν [ θ + cos 1 ( r ca r ) 2 cos 1 ( r ca r c ) ] + κ c [ 2 ( r c 2 r ca 2 ) 1 2 ( r 2 r ca 2 ) 1 2 ]
P 2 ( r , θ , z ) = β z + ν [ θ + cos 1 ( r ca r ) ] κ c ( r 2 r ca 2 ) 1 2 + 2 π μ Φ ca Φ R .
A ( r , θ , z ) = A 0 s 1 2 .
A 1 ( r , θ , z ) = A 0 κ c ( r 2 r ca 2 ) 1 4 .
A 2 ( r , θ , z ) = exp ( i Φ R ) A 0 κ c ( r 2 r ca 2 ) 1 4 .
ψ ( r , θ , z ) = 2 A 0 κ c ( r 2 r ca 2 ) 1 4 cos [ κ c ( r 2 r ca 2 ) 1 2 ν cos 1 ( r ca r ) π μ + Φ ca 2 ] × exp [ i ( β z + ν θ ) ] exp [ i ( π μ Φ ca 2 ) ] .
ψ ( r , θ , z ) ( 1 ) μ 1 κ c r cos [ κ c r ( 2 ν + 1 ) π 4 ]
J ν ( κ c r ) ( 2 π κ c r ) 1 2 cos [ κ c r ( 2 ν + 1 ) π 4 ] .

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