Abstract

A Green’s-function method is employed to provide a rigorous analysis to the propagation and coupling phenomena in holey fibers. The analysis is carried out for an arbitrary grid of circular air holes of the fiber guide, while the electromagnetic field is taken to be a vector quantity. Application of the Green’s-function concept leads to a coupled system of equations incorporating as unknowns the field expansion coefficients to cylindrical wave functions within the air holes. The propagation constants of the guided waves are computed accurately by determining the singular points of the corresponding system’s matrix. Field distribution and dispersion properties of guided modes as well as coupling phenomena between parallel-running holey fibers are investigated, and numerical results are presented.

© 2004 Optical Society of America

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References

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  3. T. M. Monro, D. J. Richardson, P. J. Bennett, “Holey optical fibers: an efficient modal model,” J. Lightwave Technol. 17, 1093–1102 (1999).
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  4. M. Midrio, M. P. Singh, C. G. Someda, “The space filling mode of holey fibers: an analytical vectorial solution,” J. Lightwave Technol. 18, 1031–1037 (2000).
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  10. D. S. Jones, Theory of Electromagnetism (Pergamon, Oxford, UK, 1964).
  11. C. N. Capsalis, N. K. Uzunoglu, “Coupled wave propagation in closely spaced dielectric rod waveguides,” Int. J. Infrared Millim. Waves 7, 813–831 (1986).
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  12. N. K. Uzunoglu, “Scattering from inhomogeneities inside a fiber waveguide,” J. Opt. Soc. Am. 71, 259–273 (1981).
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  14. M. Abramowitz, I. E. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).
  15. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
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2003 (2)

2002 (1)

2001 (1)

2000 (3)

1999 (2)

1997 (1)

1996 (1)

Y. Kugawa, Y. Sun, Z. Mahmood, “Regular boundary integral formulation for the analysis of open dielectric/optical waveguides,” IEEE Trans. Microwave Theory Tech. 44, 1441–1450 (1996).
[CrossRef]

1986 (1)

C. N. Capsalis, N. K. Uzunoglu, “Coupled wave propagation in closely spaced dielectric rod waveguides,” Int. J. Infrared Millim. Waves 7, 813–831 (1986).
[CrossRef]

1981 (1)

1980 (1)

A. D. Yaghjian, “Electric dyadic Green’s function in the source region,” Proc. IEEE 68, 248–263 (1980).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. E. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

Andres, M. V.

Andres, P.

Bennett, P. J.

Birks, T. A.

Botten, L. C.

Capsalis, C. N.

C. N. Capsalis, N. K. Uzunoglu, “Coupled wave propagation in closely spaced dielectric rod waveguides,” Int. J. Infrared Millim. Waves 7, 813–831 (1986).
[CrossRef]

Ferrando, A.

Guan, N.

Habu, S.

Himeno, K.

Jones, D. S.

D. S. Jones, Theory of Electromagnetism (Pergamon, Oxford, UK, 1964).

Knight, J. C.

Kugawa, Y.

Y. Kugawa, Y. Sun, Z. Mahmood, “Regular boundary integral formulation for the analysis of open dielectric/optical waveguides,” IEEE Trans. Microwave Theory Tech. 44, 1441–1450 (1996).
[CrossRef]

Kuhlmey, B. T.

Lu, Tao.

Mahmood, Z.

Y. Kugawa, Y. Sun, Z. Mahmood, “Regular boundary integral formulation for the analysis of open dielectric/optical waveguides,” IEEE Trans. Microwave Theory Tech. 44, 1441–1450 (1996).
[CrossRef]

Martijn de Sterke, C.

Maystre, D.

McPhedran, R. C.

Midrio, M.

Miret, J. J.

Monro, T. M.

Renversez, G.

Richardson, D. J.

Silvestre, E.

Singh, M. P.

Someda, C. G.

St. Russell, P.

Steel, M. J.

Stegun, I. E.

M. Abramowitz, I. E. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

Sterke, C. M.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

Sun, Y.

Y. Kugawa, Y. Sun, Z. Mahmood, “Regular boundary integral formulation for the analysis of open dielectric/optical waveguides,” IEEE Trans. Microwave Theory Tech. 44, 1441–1450 (1996).
[CrossRef]

Takenaga, K.

Uzunoglu, N. K.

C. N. Capsalis, N. K. Uzunoglu, “Coupled wave propagation in closely spaced dielectric rod waveguides,” Int. J. Infrared Millim. Waves 7, 813–831 (1986).
[CrossRef]

N. K. Uzunoglu, “Scattering from inhomogeneities inside a fiber waveguide,” J. Opt. Soc. Am. 71, 259–273 (1981).
[CrossRef]

Wada, A.

White, T. P.

Yaghjian, A. D.

A. D. Yaghjian, “Electric dyadic Green’s function in the source region,” Proc. IEEE 68, 248–263 (1980).
[CrossRef]

Yevick, D.

IEEE Trans. Microwave Theory Tech. (1)

Y. Kugawa, Y. Sun, Z. Mahmood, “Regular boundary integral formulation for the analysis of open dielectric/optical waveguides,” IEEE Trans. Microwave Theory Tech. 44, 1441–1450 (1996).
[CrossRef]

Int. J. Infrared Millim. Waves (1)

C. N. Capsalis, N. K. Uzunoglu, “Coupled wave propagation in closely spaced dielectric rod waveguides,” Int. J. Infrared Millim. Waves 7, 813–831 (1986).
[CrossRef]

J. Lightwave Technol. (5)

J. Opt. Soc. Am. (1)

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

Opt. Lett. (4)

Proc. IEEE (1)

A. D. Yaghjian, “Electric dyadic Green’s function in the source region,” Proc. IEEE 68, 248–263 (1980).
[CrossRef]

Other (3)

M. Abramowitz, I. E. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

D. S. Jones, Theory of Electromagnetism (Pergamon, Oxford, UK, 1964).

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Figures (13)

Fig. 1
Fig. 1

Holey fiber geometry: infinite medium with refractive index n 1 having a finite number of circular-cross-section air tubes of infinite length parallel to each other along the z axis. Each air-hole tube is defined in terms of its center position ( x j ,   y j ) and its radius a j .

Fig. 2
Fig. 2

Geometry of the triangular air-hole lattice. All air holes have the same radius a, and Λ is the distance between neighboring air holes (pitch). The absence of a hole in the center forms the lightguiding core of the holey fiber.

Fig. 3
Fig. 3

The four possible field distributions allowed to propagate in a holey fiber with two perpendicular symmetry planes: (a) (b) for the plane y = 0   ( x = 0 ) being an electric wall and the plane x = 0   ( y = 0 ) being a magnetic wall, (c) (d) for the planes x = 0 and y = 0 being electric (magnetic) walls. The corresponding symmetries of the transversal components E x , E y , H x , H y follow from Eqs. (20) and (21).

Fig. 4
Fig. 4

Normalized intensity profiles of the electric field components for the fundamental mode of a holey fiber with a / Λ = 0.49 and k 0 Λ = 5.0 for the case of the plane y = 0 being an electric wall: (a) E x (maximum level -21.2 dB), (b) E y (maximum level 0 dB), (c) E z (maximum level -10.1 dB). The contour lines are spaced by 2 dB.

Fig. 5
Fig. 5

Normalized intensity profiles of the electric field components for the fundamental mode of a holey fiber with a / Λ = 0.49 and k 0 Λ = 5.0 for the case of the plane x = 0 being an electric wall: (a) E x (maximum level 0 dB), (b) E y (maximum level -20 dB), (c) E z (maximum level -9.3 dB). The contour lines are spaced by 2 dB.

Fig. 6
Fig. 6

Normalized intensity profiles of the electric field components of a holey fiber with a / Λ = 0.49 and k 0 Λ = 5.0 for the case of the planes x = 0 and y = 0 being electric walls: (a) E x (maximum level -1.2 dB), (b) E y (maximum level 0 dB), (c) E z (maximum level -3 dB). The contour lines are spaced by 2 dB.

Fig. 7
Fig. 7

Normalized intensity profiles of the electric field components of a holey fiber with a / Λ = 0.49 and k 0 Λ = 5.0 for the case of the planes x = 0 and y = 0 being magnetic walls: (a) E x (maximum level 0 dB), (b) E y (maximum level -0.5 dB), (c) E z (maximum level -3.9 dB). The contour lines are spaced by 2 dB.

Fig. 8
Fig. 8

Normalized intensity profiles of the electric field components for the fundamental mode of a holey fiber with a / Λ = 0.3 and k 0 Λ = 10.0 for the case of the plane y = 0 being an electric wall: (a) E x (maximum level -24 dB), (b) E y (maximum level 0 dB), (c) E z (maximum level -17.1 dB). The contour lines are spaced by 2 dB.

Fig. 9
Fig. 9

Normalized intensity profiles of the electric field components for the fundamental mode of a holey fiber with a / Λ = 0.3 and k 0 Λ = 10.0 for the case of the plane x = 0 being an electric wall: (a) E x (maximum level 0 dB), (b) E y (maximum level -25.1 dB), (c) E z (maximum level -16.1 dB). The contour lines are spaced by 2 dB.

Fig. 10
Fig. 10

Computed geometrical dispersion in the wavelength interval 1.2–1.8 μm for triangular lattice holey fibers possessing a pitch Λ = 1.2   μ m and different ratios a / Λ . H is the number of air holes. The solid curve corresponds to H = 18 , a / Λ = 0.49 , the dashed curve to H = 36 , a / Λ = 0.3 , the dotted curve to H = 36 , a / Λ = 0.2 , and the dotted-dashed curve to H = 90 , a / Λ = 0.1 .

Fig. 11
Fig. 11

Computed dispersion in the wavelength interval 1.2–1.8 μm for triangular lattice holey fibers possessing a pitch Λ = 2.3   μ m and air-hole radius a = 0.5   μ m . H is the number of air holes. The solid curve corresponds to H = 6 (one layer), the dashed curve to H = 18 (two layers), and the dotted curve to H = 36 (three layers).

Fig. 12
Fig. 12

Normalized intensity profiles of the y electrical component for the even and odd nearly y-polarized modes of two coupled holey fiber cores with a / Λ = 0.3 and k 0 Λ = 5.0 : (a) even mode possesses an electric wall at the y = 0 plane, (b) odd mode possesses electric walls at the planes x = 0 and y = 0 . The contour lines are spaced by 2 dB.

Fig. 13
Fig. 13

Normalized intensity profiles of the x electrical component for the even and odd nearly x-polarized modes of two coupled holey fiber cores with a / Λ = 0.3 and k 0 Λ = 5.0 : (a) even mode possesses an electric wall at the x = 0 plane, (b) odd mode possesses magnetic walls at the planes x = 0 and y = 0 . The contour lines are spaced by 2 dB.

Tables (1)

Tables Icon

Table 1 Normalized Propagation Constant Convergence Pattern for the Case of Fig. 4

Equations (53)

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E ( r ) = e ( x ,   y ) exp ( - j β z ) ,
H ( r ) = h ( x ,   y ) exp ( - j β z ) ,
E ( r ) = k 0 2 ( 1 - n 1 2 ) j = 1 H V j ( n 0 ) G1 ( r ,   r )     E ( r ) d V ,
G1 ( r ,   r ) = ( I + k 1 - 2 )   exp ( - jk 1 | r - r | ) 4 π | r - r | ,
G1 ( r ,   r ) = - ρ ˆ ρ ˆ δ ( r - r ) / k 1 2 - j 8 π - + d k m = - + ( - 1 ) m a 1 2 M m , k ( 1 ) ( r ,   k 1 ) M - m , - k ( 3 ) ( r ,   k 1 ) + N m , k ( 1 ) ( r ,   k 1 ) N - m , - k ( 3 ) ( r ,   k 1 ) , ρ < ρ M m , k ( 3 ) ( r ,   k 1 ) M - m , - k ( 1 ) ( r ,   k 1 ) + N m , k ( 3 ) ( r ,   k 1 ) N - m , - k ( 1 ) ( r ,   k 1 ) , ρ > ρ ,
M m , k ( i ) ( r ,   k 1 ) = ρ ˆ   jm ρ   C m ( i ) ( a 1 ρ ) - ϕ ˆ   C m ( i ) ( a 1 ρ ) ρ exp ( jm ϕ ) exp ( jkz ) ,
N m , k ( i ) ( r ,   k 1 ) = 1 k 1 ρ ˆ jk   C m ( i ) ( a 1 ρ ) ρ - ϕ ˆ   mk ρ   C m ( i ) ( a 1 ρ ) + z ˆ a 1 2 C m ( i ) ( a 1 ρ ) exp ( jm ϕ ) exp ( jkz )
E ( r j ) = n = - + [ a n j M n , - β ( 1 ) ( r j ,   k 0 ) + b n j N n , - β ( 1 ) ( r j ,   k 0 ) ] , ρ j < a j ,
× M m , k ( i ) ( r ,   k j ) = k j N m , k ( i ) ( r ,   k j ) ,
× N m , k ( i ) ( r ,   k j ) = k j M m , k ( i ) ( r ,   k j ) .
- + d z   exp [ - j ( k + β ) z ] = 2 π δ ( β + k ) ,
0 2 π d ϕ   exp [ j ( n - m ) ϕ ] = 2 π δ mn = 2 π , m = n 0 , m n ,
M n , - β ( 3 ) ( r j ,   k 1 ) N n , - β ( 3 ) ( r j ,   k 1 ) = λ = - + H λ - n ( 2 ) ( a 1 ρ ij ) exp [ - j ( λ - n ) ϕ ij ]× M λ , - β ( 1 ) ( r i ,   k 1 ) N λ , - β ( 1 ) ( r i ,   k 1 ) .
E ( r i ) k 0 2 - k 1 2 = E ( r i ) k 0 2 - k 1 2 - π j 2 a 1 2 n = - + { M n , - β ( 1 ) ( r i ,   k 1 )× [ a n i u 1 n ( 3 ) ( a i ) + b n i u 2 n ( 3 ) ( a i ) ] + N n , - β ( 1 ) ( r i ,   k 1 )× [ a n i u 3 n ( 3 ) ( a i ) + b n i u 4 n ( 3 ) ( a i ) ] } - π j 2 a 1 2 j i n = - + M n , - β ( 1 ) ( r i ,   k 1 )× m = - + H n - m ( 2 ) ( a 1 ρ ij ) exp [ - j ( n - m ) ϕ ij ]× [ a m j u 1 m ( 1 ) ( a j ) + b m j u 2 m ( 1 ) ( a j ) ] + N n , - β ( 1 ) ( r i ,   k 1 )× m = - + H n - m ( 2 ) ( a 1 ρ ij ) exp [ - j ( n - m ) ϕ ij ]× [ a m j u 3 m ( 1 ) ( a j ) + b m j u 4 m ( 1 ) ( a j ) ] ,
n = - + [ c n i M n , - β ( 1 ) ( r i ,   k 1 ) + d n i N n , - β ( 1 ) ( r i ,   k 1 ) ] = 0 ,
ρ i < a i ,
a n i u 1 n ( 3 ) ( a i ) + b n i u 2 n ( 3 ) ( a i ) + j i m = - + H n - m ( 2 ) ( a 1 ρ ij )
× exp [ - j ( n - m ) ϕ ij ]
× [ a m j u 1 m ( 1 ) ( a j ) + b m j u 2 m ( 1 ) ( a j ) ] = 0 ,
a n i u 3 n ( 3 ) ( a i ) + b n i u 4 n ( 3 ) ( a i ) + j i m = - + H n - m ( 2 ) ( a 1 ρ ij )
× exp [ - j ( n - m ) ϕ ij ] [ a m j u 3 m ( 1 ) ( a j )
+ b m j u 4 m ( 1 ) ( a j ) ] = 0 , n Z , i = 1 ,   2 , , H .
e i = [ a - N i ,   a - N + 1 i , , a N i ,   b - N i ,   b - N + 1 i , , b N i ] ,
B     [ e 1 e 2 e H ] T = 0 ,
B ij = K ij 0 0 K ij     U 1 j ( 1 ) U 2 j ( 1 ) U 3 j ( 1 ) U 4 j ( 1 )
B ii = U 1 i ( 3 ) U 2 i ( 3 ) U 3 i ( 3 ) U 4 i ( 3 ) .
K ij = { k μ , ν ij } = { H μ - ν ( 2 ) ( a 1 ρ ij ) exp [ - j ( μ - ν ) ϕ ij ] } ,
1 μ , ν 2 N + 1 ,
U λ i ( 1 , 3 ) = diag [ u λ , - N ( 1 , 3 ) ( a i ) ,
u λ , - N + 1 ( 1 , 3 ) ( a i ) , , u λ , N ( 1 , 3 ) ( a i ) ] , λ = 1 ,   2 ,   3 ,   4 ,
E ( r ) = π j 2 k 1 2 - k 0 2 k 1 2 - β 2 j = 1 H n = - N + N { M n , - β ( 3 ) ( r i ,   k 1 ) [ a n i u 1 n ( 1 ) ( a i ) + b n i u 2 n ( 1 ) ( a i ) ] + N n , - β ( 3 ) ( r i ,   k 1 ) [ a n i u 3 n ( 1 ) ( a i ) + b n i u 4 n ( 1 ) ( a i ) ] } ,
E t = 1 β 2 - k 0 2 n 2   ( j β t E z - j ω μ 0 z ˆ × t H z ) ,
H t = 1 β 2 - k 0 2 n 2   ( j β t H z + j ω 0 n 2 z ˆ × t E z ) ,
E z ( even / odd ) function of y b n j = ± ( - 1 ) n b - n j ,
E z ( even / odd ) function of x b n j = ± b - n j ,
1 < β / k 0 < n 1 .
D g = - λ c   d 2 n eff d λ 2 ( ps   nm - 1   km - 1 ) ,
n ^ S × E ( r S ) , n ^ S × S × E ( r S ) , n ^ S × G0 ( r S , r ) ,
n ^ S × S × G0 ( r S , r )
V [ × × E ( r )     G1 ( r ,   r ) - E ( r ) ×
× G1 ( r ,   r ) ] d V = I S ,
I S = S [ n ^ S × E ( r S )     S × G1 ( r S , r ) + n ^ S × S × E ( r S ) G1 ( r S , r ) ] d S ,
- E ( r ) + k 0 2 ( 1 - n 1 2 ) j V j ( n 0 ) E ( r )
  G0 ( r ,   r ) d V = I b ,
G1 ( r ,   r ) ( I - z ˆ z ˆ )   exp ( - jk 1 | z | ) 4 π | z | exp ( ± jk 1 z ˆ     r ) ,
u 1 n ( i ) ( z ) = ( - 1 ) n z m - n , β ( i ) ( ρ j ,   k 1 )     m n , - β ( 1 ) ( ρ j ,   k 0 ) ρ j d ρ j = a 1 a 0 z a 1 2 - a 0 2   [ a 1 C n ( i ) ( a 1 z ) J n ( a 0 z ) - a 0 C n ( i ) ( a 1 z ) J n ( a 0 z ) ] ,
u 2 n ( i ) ( z ) = ( - 1 ) n z m - n , β ( i ) ( ρ j ,   k 1 )     n n , - β ( 1 ) ( ρ j ,   k 0 ) ρ j d ρ j = - n β k 0   C n ( i ) ( a 1 z ) J n ( a 0 z ) ,
u 3 n ( i ) ( z ) = ( - 1 ) n z n - n , β ( i ) ( ρ j ,   k 1 )     m n , - β ( 1 ) ( ρ j ,   k 0 ) ρ j d ρ j = - n β k 1   C n ( i ) ( a 1 z ) J n ( a 0 z ) ,
u 4 n ( i ) ( z ) = ( - 1 ) n z n - n , β ( i ) ( ρ j ,   k 1 )     n n , - β ( 1 ) ( ρ j ,   k 0 ) ρ j d ρ j = a 1 a 0 z k 0 k 1 ( a 1 2 - a 0 2 )   [ a 1 k 0 2 C n ( i ) ( a 1 z ) J n ( a 0 z ) - a 0 k 1 2 C n ( i ) ( a 1 z ) J n ( a 0 z ) ] ,
a j = ( k j 2 - β 2 ) 1 / 2 ,
M m , k ( i ) ( r ,   k j ) = m m , k ( i ) ( ρ ,   k j ) exp ( jm ϕ ) exp ( jkz ) ,
N m , k ( i ) ( r ,   k j ) = n m , k ( i ) ( ρ ,   k j ) exp ( jm ϕ ) exp ( jkz ) ,
i = 1 ,   3 , j = 0 ,   1 ,

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