Abstract

An analytic expression is obtained for the confinement loss of model anti-resonant fibres consisting of concentric regions of air and glass. Hankel functions in the regions surrounding the air core are approximated by their asymptotic form; apart from this, results are correct to leading order in the small parameter 1/(k0rc), where rc is the core radius and k0 the free space wavenumber. The results extend and generalise previous solutions for propagation in a hollow glass tube and a thin-walled capillary. Comparison with exact numerical calculations shows that the analytic expression provides an accurate description of the loss, including its dependence on the mode, the core radius and the widths of the surrounding glass and air regions. The relevance of the results to the recent generation of hollow-core, anti-resonant photonic crystal fibres is discussed.

© 2017 Optical Society of America

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    [Crossref] [PubMed]
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2017 (2)

B. Debord, A. Amsanpally, M. Chafer, A. Baz, M. Maurel, J.M. Blondy, E. Hugonnot, F. Scol, L. Vincetti, F. Gérôme, and F. Benabid, “Ultralow transmission loss in inhibited-coupling guiding hollow fibres,” Optica 4(2), 209–217 (2017).
[Crossref]

L.D. van Putten, E.N. Fokoua, S.A. Mousavi, W. Belardi, S. Chaudhuri, J.V. Badding, and F. Poletti, “Exploring the effect of the core boundary curvature in hollow antiresonant fibers,” IEEE Photon. Technol. Lett. 29(2), 263–266 (2017).
[Crossref]

2016 (5)

2015 (1)

2014 (2)

2013 (4)

2012 (1)

2011 (1)

2010 (1)

2009 (1)

2007 (1)

2002 (2)

2001 (1)

1993 (1)

J.-L. Archambault, R.J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Lightwave Technol. 11(3), 416–423 (1993).
[Crossref]

1980 (1)

M. Miyagi and S. Nishida, “Transmission characteristics of dielectric tube leaky waveguide,” IEEE Trans. Microwave Theory Tech. 28(6), 536–541 (1980).
[Crossref]

1978 (1)

1964 (1)

E.A.J. Marcatili and R.A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).
[Crossref]

Abeeluck, A.K.

Abokhamis, M.S.

Abu Hassan, M.R.

Ahmed, G.

Alharbi, M.

Amsanpally, A.

Archambault, J.-L.

J.-L. Archambault, R.J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Lightwave Technol. 11(3), 416–423 (1993).
[Crossref]

Baddela, N.K.

Badding, J.V.

L.D. van Putten, E.N. Fokoua, S.A. Mousavi, W. Belardi, S. Chaudhuri, J.V. Badding, and F. Poletti, “Exploring the effect of the core boundary curvature in hollow antiresonant fibers,” IEEE Photon. Technol. Lett. 29(2), 263–266 (2017).
[Crossref]

Baskiotis, C.

Baz, A.

Beaudou, B.

Belardi, W.

L.D. van Putten, E.N. Fokoua, S.A. Mousavi, W. Belardi, S. Chaudhuri, J.V. Badding, and F. Poletti, “Exploring the effect of the core boundary curvature in hollow antiresonant fibers,” IEEE Photon. Technol. Lett. 29(2), 263–266 (2017).
[Crossref]

W. Belardi and J.C. Knight, “Effect of core boundary curvature on the confinement losses of hollow antiresonant fibers,” Opt. Express 21(19), 21912–21917 (2013).
[Crossref] [PubMed]

Benabid, F.

Bierlich, J.

Black, R.J.

J.-L. Archambault, R.J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Lightwave Technol. 11(3), 416–423 (1993).
[Crossref]

Blondy, J.M.

Bradley, T.

Bures, J.

J.-L. Archambault, R.J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Lightwave Technol. 11(3), 416–423 (1993).
[Crossref]

Burger, S.

Chafer, M.

Chaudhuri, S.

L.D. van Putten, E.N. Fokoua, S.A. Mousavi, W. Belardi, S. Chaudhuri, J.V. Badding, and F. Poletti, “Exploring the effect of the core boundary curvature in hollow antiresonant fibers,” IEEE Photon. Technol. Lett. 29(2), 263–266 (2017).
[Crossref]

Debord, B.

Douay, M.

Edavalath, N.N.

Eggleton, B.J.

Engeness, T.D.

Février, S.

Fink, Y.

Fokoua, E.N.

L.D. van Putten, E.N. Fokoua, S.A. Mousavi, W. Belardi, S. Chaudhuri, J.V. Badding, and F. Poletti, “Exploring the effect of the core boundary curvature in hollow antiresonant fibers,” IEEE Photon. Technol. Lett. 29(2), 263–266 (2017).
[Crossref]

Fourcade-Dutin, C.

Frosch, T.

Frosz, M.H.

Gérôme, F.

Günendi, M.C.

Hartung, A.

Hayes, J.R.

J.R. Hayes, F. Poletti, M.S. Abokhamis, N.V. Wheeler, N.K. Baddela, and D.J. Richardson, “Anti-resonant hexagram hollow core fibers,” Opt. Express 23(2), 1289–1299 (2015).
[Crossref] [PubMed]

F. Poletti, J.R. Hayes, and D.J. Richardson, “Optimising the performances of hollow antiresonant fibres,” in Proc. European Conference on Optical Communication (ECOC)2011, paper Mo.2.LeCervin.2.

Headley, C.

Hu, J.

Hugonnot, E.

Ibanescu, M.

Jacobs, S.A.

Joannopoulos, J.D.

Johnson, S.G.

Knight, J.C.

Kobelke, J.

Lacroix, S.

J.-L. Archambault, R.J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Lightwave Technol. 11(3), 416–423 (1993).
[Crossref]

Lee, R.K.

Litchinitser, N.M.

Marcatili, E.A.J.

E.A.J. Marcatili and R.A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).
[Crossref]

Margulis, W.

Marom, E.

Maurel, M.

Ménard, J.-M.

Menyuk, C.R.

Miyagi, M.

M. Miyagi and S. Nishida, “Transmission characteristics of dielectric tube leaky waveguide,” IEEE Trans. Microwave Theory Tech. 28(6), 536–541 (1980).
[Crossref]

Mousavi, S.A.

L.D. van Putten, E.N. Fokoua, S.A. Mousavi, W. Belardi, S. Chaudhuri, J.V. Badding, and F. Poletti, “Exploring the effect of the core boundary curvature in hollow antiresonant fibers,” IEEE Photon. Technol. Lett. 29(2), 263–266 (2017).
[Crossref]

Nishida, S.

M. Miyagi and S. Nishida, “Transmission characteristics of dielectric tube leaky waveguide,” IEEE Trans. Microwave Theory Tech. 28(6), 536–541 (1980).
[Crossref]

Ouyang, G.X.

Pearce, G.J.

Poletti, F.

L.D. van Putten, E.N. Fokoua, S.A. Mousavi, W. Belardi, S. Chaudhuri, J.V. Badding, and F. Poletti, “Exploring the effect of the core boundary curvature in hollow antiresonant fibers,” IEEE Photon. Technol. Lett. 29(2), 263–266 (2017).
[Crossref]

J.R. Hayes, F. Poletti, M.S. Abokhamis, N.V. Wheeler, N.K. Baddela, and D.J. Richardson, “Anti-resonant hexagram hollow core fibers,” Opt. Express 23(2), 1289–1299 (2015).
[Crossref] [PubMed]

F. Poletti, “Nested antiresonant nodeless hollow core fiber,” Opt. Express 22(20), 23807–23828 (2014).
[Crossref] [PubMed]

F. Poletti, J.R. Hayes, and D.J. Richardson, “Optimising the performances of hollow antiresonant fibres,” in Proc. European Conference on Optical Communication (ECOC)2011, paper Mo.2.LeCervin.2.

Popp, J.

Poulton, C.G.

Quiquempois, Y

Richardson, D.J.

J.R. Hayes, F. Poletti, M.S. Abokhamis, N.V. Wheeler, N.K. Baddela, and D.J. Richardson, “Anti-resonant hexagram hollow core fibers,” Opt. Express 23(2), 1289–1299 (2015).
[Crossref] [PubMed]

F. Poletti, J.R. Hayes, and D.J. Richardson, “Optimising the performances of hollow antiresonant fibres,” in Proc. European Conference on Optical Communication (ECOC)2011, paper Mo.2.LeCervin.2.

Rugeland, P.

Russell, P.St.J

Russell, P.St.J.

Sakai, J.

Schmeltzer, R.A.

E.A.J. Marcatili and R.A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).
[Crossref]

Schmidt, M.A.

Schwuchow, A.

Scol, F.

Sillard, P.

Skorobogatiy, M.

Soljacic, M.

Sterner, C.

Uebel, P.

van Putten, L.D.

L.D. van Putten, E.N. Fokoua, S.A. Mousavi, W. Belardi, S. Chaudhuri, J.V. Badding, and F. Poletti, “Exploring the effect of the core boundary curvature in hollow antiresonant fibers,” IEEE Photon. Technol. Lett. 29(2), 263–266 (2017).
[Crossref]

Viale, P.

Vincetti, L.

Wadsworth, W.J.

Wang, Y.Y.

Weisberg, O.

Wheeler, N.V.

Wiederhecker, G.S.

Wondraczek, K.

Xu, M.

Xu, Y.

Yariv, A.

Yeh, P.

Yu, F.

Adv. Opt. Photon. (1)

Bell Syst. Tech. J. (1)

E.A.J. Marcatili and R.A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).
[Crossref]

IEEE J. Sel. Topics Quantum Electron. (1)

F. Yu and J.C. Knight, “Negative curvature hollow-core optical fiber,” IEEE J. Sel. Topics Quantum Electron. 22(2), 4400610 (2016).
[Crossref]

IEEE Photon. Technol. Lett. (1)

L.D. van Putten, E.N. Fokoua, S.A. Mousavi, W. Belardi, S. Chaudhuri, J.V. Badding, and F. Poletti, “Exploring the effect of the core boundary curvature in hollow antiresonant fibers,” IEEE Photon. Technol. Lett. 29(2), 263–266 (2017).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

M. Miyagi and S. Nishida, “Transmission characteristics of dielectric tube leaky waveguide,” IEEE Trans. Microwave Theory Tech. 28(6), 536–541 (1980).
[Crossref]

J. Lightwave Technol. (2)

J.-L. Archambault, R.J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Lightwave Technol. 11(3), 416–423 (1993).
[Crossref]

Y. Xu, G.X. Ouyang, R.K. Lee, and A. Yariv, “Asymptotic matrix theory of Bragg fibres,” J. Lightwave Technol. 20(3), 428–440 (2002).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (2)

Opt. Express (12)

P. Rugeland, C. Sterner, and W. Margulis, “Visible light guidance in silica capillaries by antiresonant reflection,” Opt. Express 21(24), 29217–29222 (2013).
[Crossref]

A. Hartung, J. Kobelke, A. Schwuchow, K. Wondraczek, J. Bierlich, J. Popp, T. Frosch, and M.A. Schmidt, “Double antiresonant hollow core fiber – guidance in the deep ultraviolet by modified tunneling leaky modes,” Opt. Express 22(16), 19131–19140 (2014).
[Crossref] [PubMed]

J.R. Hayes, F. Poletti, M.S. Abokhamis, N.V. Wheeler, N.K. Baddela, and D.J. Richardson, “Anti-resonant hexagram hollow core fibers,” Opt. Express 23(2), 1289–1299 (2015).
[Crossref] [PubMed]

F. Yu, M. Xu, and J.C. Knight, “Experimental study of low-loss single-mode performance in anti-resonant hollow-core fibers,” Opt. Express 24(12), 12969–12975 (2016).
[Crossref] [PubMed]

L. Vincetti, “Empirical formulas for calculating loss in hollow core tube lattice fibers,” Opt. Express 24(10), 10313–10325 (2016).
[Crossref] [PubMed]

S.G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T.D. Engeness, M. Soljacic, S.A. Jacobs, J.D. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express 9(13) 748–779 (2001).
[Crossref] [PubMed]

W. Belardi and J.C. Knight, “Effect of core boundary curvature on the confinement losses of hollow antiresonant fibers,” Opt. Express 21(19), 21912–21917 (2013).
[Crossref] [PubMed]

B. Debord, M. Alharbi, T. Bradley, C. Fourcade-Dutin, Y.Y. Wang, L. Vincetti, F. Gérôme, and F. Benabid, “Hypocycloid-shaped hollow-core photonic crystal fiber Part I: Arc curvature effect on confinement loss,” Opt. Express 21(23), 28597–28608 (2013).
[Crossref]

G.J. Pearce, G.S. Wiederhecker, C.G. Poulton, S. Burger, and P.St.J Russell, “Models for guidance in kagome-structured hollow-core photonic crystal fibres,” Opt. Express 15(20), 12680–12685 (2007).
[Crossref] [PubMed]

S. Février, B. Beaudou, and P. Viale, “Understanding origin of loss in large pitch hollow-core photonic crystal fibers and their design simplification,” Opt. Express 18(5), 5142–5150 (2010).
[Crossref] [PubMed]

F. Yu, W.J. Wadsworth, and J.C. Knight, “Low loss silica hollow core fibers for 3–4 μm spectral region,” Opt. Express 20(10), 11153–11158 (2012).
[Crossref] [PubMed]

F. Poletti, “Nested antiresonant nodeless hollow core fiber,” Opt. Express 22(20), 23807–23828 (2014).
[Crossref] [PubMed]

Opt. Lett. (2)

Optica (2)

Other (1)

F. Poletti, J.R. Hayes, and D.J. Richardson, “Optimising the performances of hollow antiresonant fibres,” in Proc. European Conference on Optical Communication (ECOC)2011, paper Mo.2.LeCervin.2.

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Figures (3)

Fig. 1
Fig. 1

Schematic diagram of glass-air multi-layered structures. Air regions are white, glass regions are shaded, and black lines represent the boundaries between layers. N refers to the number of complete anti-resonant layers in each structure: N = 0 represents a hollow glass tube; N = 1 a thin-walled capillary; N = 2 a jacketed capillary; N = 3 two nested capillaries; N = 4 as for N = 3 but with an outer glass jacket. In each case the outermost region extends to infinity.

Fig. 2
Fig. 2

Contour plots of the confinement loss of HE11 and TE01 modes, incurred in propagation through λ0, for N = 2 as a function of the width of the glass and air layers, for rc0 = 15 and = 2.25. Loss is scaled by dividing by (λ0/rc)5, as in Table 1. Contour levels are at 1, 2, 4, 8, 16, 32 in all plots, with the innermost contour at a scaled loss of 1. x and y axes are scaled relative to the anti-resonant thickness; so 1 corresponds to ϕi = π/2. Scaled loss at (1, 1) is given in Table 1. ‘exact’ corresponds to numerical calculations and ‘approx’ to results from Eq. (1).

Fig. 3
Fig. 3

Confinement loss incurred in propagation through λ0 of HE11 and TE01 modes for N = 4 as a function of the width of the glass and air layers, for rc0 = 15 and = 2.25. Loss is scaled by dividing by (λ0/rc)7. x axis is scaled relative to the anti-resonant thickness, as in Fig. 2. For each curve, three layers are held at the anti-resonant value given by ϕi = π/2, and the width of the other is varied. Approx curves are from Eq. (1); other curves are from numerical calculations with variation of the width of the given layer.

Tables (2)

Tables Icon

Table 1 Confinement loss of low order modes incurred in propagation through a distance of λ0 (in dB and divided by (λ0/rc)N+3) as a function of number of layers N. x0 is the zero of the relevant Bessel function, given by Eqs. (41), (42) or (43). All calculations are carried out for structures with ϕi = π/2 for all i. Approx values refer to Eq. (1); other results are for full numerical calculations at the given value of rc0. Error refers to the difference between approximate and numerical results for rc0 = 15. = 2.25 in all cases.

Tables Icon

Table 2 Numerical values of confinement loss, scaled as in Table 1, calculated for rc0 = 15 and = 2.25. The ‘shift’ values are the % variation of layer thicknesses, relative to those given by ϕi = π/2, that minimise loss in each case, and ‘minimum’ is the optimised, scaled loss. Values of the % shift are listed from the centre outwards, with successive glass and air layers labelled by g1, a2, g3, a4. For comparison, scaled loss values with ϕi = π/2 for all i from Table 1 are also given.

Equations (102)

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8.686 ( x 0 2 π ) N + 2 ( 1 ( 1 ) ) N + 1 ( λ 0 r c ) N + 3 i = 1 N 1 sin 2 ϕ i ( TE modes )
8.686 ( x 0 2 π ) N + 2 ( ( 1 ) ) N + 1 ( λ 0 r c ) N + 3 i = 1 N 1 sin 2 ϕ i ( TM modes )
8.686 ( x 0 2 π ) N + 2 1 2 N + 1 + 1 ( 1 ) ( N + 2 ) / 2 ( λ 0 r c ) N + 3 i = 1 N 1 sin 2 ϕ i ( HE / EH modes ) .
ϕ i ( g ) = 2 π ( 1 ) w i ( g ) λ 0 or ϕ i ( a ) = x 0 w i ( a ) r c
w ( g ) r c = 1 4 ( 1 ) λ 0 r c and w ( a ) r c = π 2 x 0
E z ( r , θ ) = ( a H n ( 1 ) ( k r ) + b H n ( 2 ) ( k r ) ) cos ( n θ + ϕ )
H ˜ z ( r , θ ) = ( c H n ( 1 ) ( k r ) + d H n ( 2 ) ( k r ) ) cos ( n θ + ψ )
k = ( k 0 2 β 2 ) 1 / 2
E r ( r , θ ) = i k 2 [ β E z r + k 0 r H ˜ z θ ]
E θ ( r , θ ) = i k 2 [ β r E z θ k 0 H ˜ z r ]
H ˜ r ( r , θ ) = i k 2 [ β H ˜ z r k 0 r E z θ ]
H ˜ θ ( r , θ ) = i k 2 [ β r H ˜ z θ k 0 E z r ] .
M ( k 2 , R ) ( a 2 b 2 c 2 d 2 ) = M ( k 1 , R ) ( a 1 b 1 c 1 d 1 )
M ( k , R ) = ( H n ( 1 ) ( k R ) k 0 k H n ( 1 ) ( k R ) H n ( 2 ) ( k R ) k 0 k H n ( 2 ) ( k R ) 0 n β k 2 R H n ( 1 ) ( k R ) 0 n β k 2 R H n ( 2 ) ( k R ) 0 n β k 2 R H n ( 1 ) ( k R ) 0 n β k 2 R H n ( 2 ) ( k R ) H n ( 1 ) ( k R ) k 0 k H n ( 1 ) ( k R ) H n ( 2 ) ( k R ) k 0 k H n ( 2 ) ( k R ) )
E z ( r , θ ) = ( a H n ( 1 ) ( k r ) + b H n ( 2 ) ( k r ) ) cos ( n θ )
H ˜ z ( r , θ ) = ( c H n ( 1 ) ( k r ) + d H n ( 2 ) ( k r ) ) sin ( n θ ) .
E z ( r , θ ) = ( a H n ( 1 ) ( k r ) + b H n ( 2 ) ( k r ) ) sin ( n θ )
H ˜ z ( r , θ ) = ( c H n ( 1 ) ( k r ) + d H n ( 2 ) ( k r ) ) cos ( n θ )
( a f b f c f d f ) = M 1 ( k f , r N ) M ( k N , r N ) M 1 ( k N , r N 1 ) M ( k 1 , r 1 ) M 1 ( k 1 , r c ) M ( k c , r c ) ( a c b c c c d c )
a c = b c and c c = d c
b f = 0 and d f = 0 .
x = k c r c = ( k 0 2 β 2 ) 1 / 2 r c and D = k 0 r c = 2 π r c λ 0 .
β = k 0 ( 1 x 2 D 2 ) 1 / 2
k = k 0 ( ( 1 ) + x 2 D 2 ) 1 / 2 , k R = D R r c ( ( 1 ) + x 2 D 2 ) 1 / 2 , n β k 2 R = n D r c R ( 1 x 2 / D 2 ) 1 / 2 ( 1 ) + x 2 / D 2 .
k = k 0 x D , k R = x R r c , n β k 2 R = D n r c R ( 1 x 2 / D 2 ) 1 / 2 x 2 .
M g = ( H n ( 1 ) ( k R ) ( 1 ) H n ( 1 ) ( k R ) H n ( 2 ) ( k R ) ( 1 ) H n ( 2 ) ( k R ) 0 n D r c R 1 1 H n ( 1 ) ( k R ) 0 n D r c R 1 1 H n ( 2 ) ( k R ) 0 n D r c R 1 1 H n ( 1 ) ( k R ) 0 n D r c R 1 1 H n ( 2 ) ( k R ) H n ( 1 ) ( k R ) 1 ( 1 ) H n ( 1 ) ( k R ) H n ( 2 ) ( k R ) 1 ( 1 ) H n ( 2 ) ( k R ) )
M g 1 = i π r c D R r c ( 1 ) × ( 1 ( 1 ) H n ( 2 ) ( k R ) 1 H n ( 2 ) ( k R ) n D r c R 1 ( 1 ) H n ( 2 ) ( k R ) 0 1 ( 1 ) H n ( 1 ) ( k R ) 1 H n ( 1 ) ( k R ) n D r c R 1 ( 1 ) H n ( 1 ) ( k R ) 0 n D r c R 1 ( 1 ) H n ( 2 ) ( k R ) 0 1 ( 1 ) H n ( 2 ) ( k R ) H n ( 2 ) ( k R ) n D r c R 1 ( 1 ) H n ( 1 ) ( k R ) 0 1 ( 1 ) H n ( 1 ) ( k R ) H n ( 1 ) ( k R ) )
k R = D R r c ( 1 ) 1 / 2
M a = ( H n ( 1 ) ( k R ) D x H n ( 1 ) ( k R ) H n ( 2 ) ( k R ) D x H n ( 2 ) ( k R ) 0 D n x 2 r c R H n ( 1 ) ( k R ) 0 D n x 2 r c R H n ( 2 ) ( k R ) 0 D n x 2 r c R H n ( 1 ) ( k R ) 0 D n x 2 r c R H n ( 2 ) ( k R ) H n ( 1 ) ( k R ) D x H n ( 1 ) ( k R ) H n ( 2 ) ( k R ) D x H n ( 2 ) ( k R ) )
M a 1 = i π 4 x 2 D R r c ( D x H n ( 2 ) ( k R ) H n ( 2 ) ( k R ) D n x 2 r c R H n ( 2 ) ( k R ) 0 D x H n ( 1 ) ( k R ) H n ( 1 ) ( k R ) D n x 2 r c R H n ( 1 ) ( k R ) 0 D n x 2 r c R H n ( 2 ) ( k R ) 0 D x H n ( 2 ) ( k R ) H n ( 2 ) ( k R ) D n x 2 r c R H n ( 1 ) ( k R ) 0 D x H n ( 1 ) ( k R ) H n ( 1 ) ( k R ) )
k R = x R r c .
T ( k i , r i , r i 1 ) = M ( k i , r i ) × M 1 ( k i , r i 1 ) ,
( a f b f c f d f ) = M 1 ( k f , r N ) T ( k N , r N , r N 1 ) T ( k 2 , r 2 , r 1 ) T ( k 1 , r 1 , r c ) M ( k c , r c ) ( a c b c c c d c ) .
H n ( 1 ) ( z ) ( 2 π z ) 1 / 2 exp [ i ( z n π 2 π 4 ) ]
H n ( 2 ) ( z ) ( 2 π z ) 1 / 2 exp [ i ( z n π 2 π 4 ) ]
T g R 1 R 2 × ( cos ϕ ( g ) ( 1 ) sin ϕ ( g ) n D r c R 1 1 ( 1 ) sin ϕ ( g ) 0 ( 1 ) t 1 sin ϕ ( g ) cos ϕ ( g ) n D Δ R r c R 1 R 2 1 ( 1 ) cos ϕ ( g ) n D r c R 2 1 ( 1 ) sin ϕ ( g ) n D r c R 1 1 ( 1 ) sin ϕ ( g ) 0 cos ϕ ( g ) ( 1 ) sin ϕ ( g ) n D Δ R r c R 1 R 2 1 ( 1 ) cos ϕ ( g ) n D r c R 1 1 ( 1 ) sin ϕ ( g ) 1 ( 1 ) t 1 sin ϕ ( g ) cos ϕ ( g ) )
t 1 = ( 1 + n 2 D 2 r c R 1 r c R 2 1 ( 1 ) )
ϕ ( g ) = D ( 1 ) Δ R r c
T a = R 1 R 2 ( cos ϕ ( a ) x D sin ϕ ( a ) n x r c R 1 sin ϕ ( a ) 0 D x t 2 sin ϕ ( a ) cos ϕ ( a ) n D x 2 Δ R r c r c R 1 r c R 2 cos ϕ ( a ) n x r c R 2 sin ϕ ( a ) n x r c R 1 sin ϕ ( a ) 0 cos ϕ ( a ) x D sin ϕ ( a ) n D x 2 Δ R r c r c R 1 r c R 2 cos ϕ ( a ) n x r c R 2 sin ϕ ( a ) D x t 2 sin ϕ ( a ) cos ϕ ( a ) )
t 2 = ( 1 + n 2 x 2 r c R 1 r c R 2 )
ϕ ( a ) = x Δ R r c .
( a f b f c f d f ) = M g 1 ( k f , r c ) M a ( k c , r c ) ( a c b c c c d c )
k c r c = x and k f r c = D ( 1 ) 1 / 2 .
( a c b c c c d c ) = 1 2 ( 1 1 p p )
1 2 M a ( k c , r c ) ( 1 1 p p ) = ( J n ( x ) D x J n ( x ) 0 D n x 2 J n ( x ) ) + p ( 0 D n x 2 J n ( x ) J n ( x ) D x J n ( x ) ) v _ 1 + p v _ 2 .
J n ( x ) + p J n ( x ) n x ( 1 x 2 D 2 1 1 ) = i x D ν TM J n ( x )
J n ( x ) n x ( 1 x 2 D 2 1 1 ) + p J n ( x ) = i p x D ν TE J n ( x )
ν TE = 1 ( 1 ) and ν TM = ( 1 ) .
x = x 0 + δ x = x 0 + x 1 D + x 2 D + p = p 0 + δ p = p 0 + p 1 D + p 2 D 2 +
J n ( x 0 ) + p 0 n x 0 J n ( x 0 ) = 0 n x 0 J n ( x 0 ) + p 0 J n ( x 0 ) = 0
J 0 ( x 0 ) = 0 or J 1 ( x 0 ) = 0 .
J n ( x 0 ) + n x 0 J n ( x 0 ) = 0 or J n 1 ( x 0 ) = 0
J n ( x 0 ) n x 0 J n ( x 0 ) = 0 or J n + 1 ( x 0 ) = 0 .
x 1 n x 0 p 1 = i x 0 ν TM x 1 + n x 0 p 1 = i x 0 ν TE
( x ) = ( δ x ) = x 0 ν D
( x ) = ( δ x ) = x 0 D 1 2 ( ν TE + ν TM ) .
( β ) = k 0 x 0 D 2 ( δ x )
( β ) = ( x 0 2 π ) 2 λ 0 2 r c 3 1 ( 1 ) ( TE solutions )
( β ) = ( x 0 2 π ) 2 λ 0 2 r c 3 1 ( 1 ) ( TM solutions )
( β ) = ( x 0 2 π ) 2 λ 0 2 r c 3 1 2 1 ( 1 ) ( HE / EH solutions ) .
( J 0 ( x ) D x J 0 ( x ) )
T g = ( cos ϕ ( g ) 1 ν sin ϕ ( g ) ν sin ϕ ( g ) cos ϕ ( g ) )
T a = ( cos ϕ ( a ) x D sin ϕ ( a ) D x sin ϕ ( a ) cos ϕ ( a ) )
T = ( P Q R S )
( i D x 1 ) ( P ( x ) Q ( x ) R ( x ) S ( x ) ) ( J 0 ( x ) D x J 0 ( x ) ) = 0 ( N odd )
( i ν 1 ) ( P ( x ) Q ( x ) R ( x ) S ( x ) ) ( J 0 ( x ) D x J 0 ( x ) ) = 0 ( N even )
N = 1 ( i D x , 1 ) ( O ( 1 ) O ( 1 ) _ O ( 1 ) O ( 1 ) ) ( i O ( D ) + O ( 1 ) , i O ( D ) _ + O ( 1 ) )
N = 2 ( i ν , 1 ) ( O ( 1 ) O ( 1 ) O ( 1 ) O ( D ) _ ) ( i O ( 1 ) + O ( D ) , i O ( 1 ) + O ( D ) _ )
N = 3 ( i D x , 1 ) ( O ( D ) O ( D ) _ O ( D ) O ( D ) ) ( i O ( D 2 ) + O ( D ) , i O ( D 2 ) _ + O ( D ) )
N = 4 ( i ν , 1 ) ( O ( D ) O ( D ) O ( D 2 ) O ( D 2 ) _ ) ( i O ( D ) + O ( D 2 ) , i O ( D ) + O ( D 2 ) _ )
N = 5 ( i D x , 1 ) ( O ( D 2 ) O ( D 2 ) _ O ( D 2 ) O ( D 2 ) ) ( i O ( D 3 ) + O ( D ) , i O ( D 3 ) _ + O ( D 2 ) ) .
D 2 Q J 0 + D P J 0 x = i ( D S J 0 x + R J 0 x 2 )
J 0 ( x ˜ 0 ) = 1 D P J 0 x Q | x ˜ 0 .
( D 2 Q J 0 + D P J 0 x ) | x ˜ 0 δ x + i ( D S J 0 x + R J 0 x 2 ) | x ˜ 0 δ x = i ( D S J 0 x + R J 0 x 2 ) | x ˜ 0 + O ( δ x ) 2 .
i ( D S J 0 x + R J 0 x 2 ) | x ˜ 0 = i ( Q R P S ) J 0 x 2 Q | x ˜ 0 = i J 0 x 2 Q | x ˜ 0
δ x = i ( x ˜ 0 D ) 2 J 0 ( Q J 0 ) | x ˜ 0 .
( δ x ) = ( x 0 D ) 2 1 Q 2 ( x 0 ) .
D S J 0 + R J 0 x = i ν ( D Q J 0 + P J 0 x )
( δ x ) = x 0 ν D 1 S 2 ( x 0 ) .
N = 1 ( δ x ) = ( x 0 ν D ) 2 1 sin 2 ϕ 1 ( g )
N = 2 ( δ x ) = ( x 0 ν D ) 3 1 sin 2 ϕ 1 ( g ) sin 2 ϕ 2 ( a )
N = 3 ( δ x ) = ( x 0 ν D ) 4 1 sin 2 ϕ 1 ( g ) sin 2 ϕ 2 ( a ) sin 2 ϕ 3 ( g )
N = 4 ( δ x ) = ( x 0 ν D ) 5 1 sin 2 ϕ 1 ( g ) sin 2 ϕ 2 ( a ) sin 2 ϕ 3 ( a ) sin 2 ϕ 4 ( a )
( w _ 1 ( x ) + i w _ 2 ( x ) ) ( x ) ( v _ 1 ( x ) + p v _ 2 ( x ) ) = 0
( w _ 3 + i w _ 4 ( x ) ) ( x ) ( v _ 1 ( x ) + p v _ 2 ( x ) ) = 0
w _ 1 = ( 0 1 n D r c r N 1 1 0 ) w _ 2 = ( ν TM 0 0 0 ) w _ 3 = ( n D r c r N 1 1 0 0 1 ) w _ 4 = ( 0 0 ν TE 0 )
w _ 1 = ( 0 1 D n x 2 r c r N 0 ) w _ 2 = ( D x 0 0 0 ) w _ 3 = ( D n x 2 r c r N 0 0 1 ) w _ 4 = ( 0 0 D x 0 ) .
( q 21 q 42 q 22 q 41 ) ( q 11 q 32 q 12 q 31 ) = i ( q 11 q 42 q 12 q 41 + q 21 q 32 q 22 q 31 )
q i j ( x ) = w _ i ( x ) ( x ) v _ j ( x ) .
q 11 ( x ˜ 0 ) q 32 ( x ˜ 0 ) q 12 ( x ˜ 0 ) q 31 ( x ˜ 0 ) = 0
δ x = i ( q 11 q 42 q 12 q 41 + q 21 q 32 q 22 q 31 ) | x ˜ 0 ( q 11 q 32 q 12 q 31 ) | x ˜ 0 + ( q 21 q 42 q 22 q 41 ) | x ˜ 0 ( q 11 q 32 q 12 q 31 ) | x ˜ 0 .
q 21 ( x ˜ 0 ) q 42 ( x ˜ 0 ) q 22 ( x ˜ 0 ) q 41 ( x ˜ 0 ) = 0
δ x = i ( q 11 q 42 q 12 q 41 + q 21 q 32 q 22 q 31 ) | x ˜ 0 ( q 21 q 42 q 22 q 41 ) | x ˜ 0 + ( q 11 q 32 q 12 q 31 ) | x ˜ 0 ( q 21 q 42 q 22 q 41 ) | x ˜ 0 .
N = 1 ( δ x ) = ( x 0 D ) 2 1 2 ( ν TE 2 + ν TM 2 ) 1 sin 2 ϕ 1 ( g )
N = 2 ( δ x ) = ( x 0 D ) 3 1 2 ( ν TE 2 + ν TM 2 ) 1 sin 2 ϕ 1 ( g ) sin 2 ϕ 2 ( a ) C 2
N = 3 ( δ x ) = ( x 0 D ) 4 1 2 ( ν TE 4 + ν TM 4 ) 1 sin 2 ϕ 1 ( g ) + sin 2 ϕ 2 ( a ) sin 2 ϕ 3 ( g ) C 3
N = 4 ( δ x ) = ( x 0 D ) 5 1 2 ( ν TE 5 + ν TM 5 ) 1 sin 2 ϕ 1 ( g ) + sin 2 ϕ 2 ( a ) sin 2 ϕ 3 ( g ) sin 2 ϕ 4 ( a ) C 4
C 2 = 1 f 1 2 ( 1 f 2 2 ) ( 1 ( 1 ) 3 + 1 f 2 ( 2 f 2 ) )
C 3 = 1 f 1 2 ( 1 f 2 2 ) 2 ( 1 2 4 + 1 f 2 ( 2 + 1 f 2 ) )
C 4 = 1 f 1 2 f 3 2 ( 1 f 2 2 ) 2 ( 1 f 4 2 ) 2 ( 1 + F ( , f 2 , f 4 ) )
f 1 = 1 + n 2 r c 2 x 0 2 r 1 r 2 , f 2 = ± 1 f 1 n r c 2 x 0 r 1 r 2 r 2 r 1 r c cos ϕ 2 ( a ) sin ϕ 2 ( a ) , f 3 = 1 + n 2 r c 2 x 0 2 r 3 r 4 , f 4 = ± 1 f 3 n r c 2 x 0 r 3 r 4 r 4 r 3 r c cos ϕ 4 ( a ) sin ϕ 4 ( a ) ,
F = + 1 5 + 1 ( ( 2 + 1 ) ( 2 f 2 ( f 4 + 1 ) + f 4 2 ) + 2 ( f 2 2 ( f 4 1 ) 2 2 f 2 ( f 4 2 1 ) f 4 ( f 4 + 2 ) ) ) .