Abstract

We numerically study confinement loss in photonic crystal fiber (PCF) tapers and compare our results with previously published experimental data. Agreement between theory and experiment requires taking into account hole shrinkage during the tapering process, which we measure by using a noninvasive technique. We show that losses are fully explained within the adiabatic approximation and that they are closely linked to the existence of a fundamental core-mode cutoff. This cutoff is equivalent to the core-mode cutoff in depressed-cladding fibers, so that losses in PCF tapers can be obtained semiquantitatively from an equivalent depressed-cladding fiber model. Finally, we discuss the definition of adiabaticity in this open boundary problem.

© 2006 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  7. Y. K. Lizé, E. C. Mägi, V. G. Taeed, J. A. Bolger, P. Steinvurzel, and B. J. Eggleton, "Microstructured optical fiber photonic wires with subwavelength core diameter," Opt. Express 12, 3209-3217 (2004).
    [CrossRef] [PubMed]
  8. D. J. Moss, Y. Miao, V. Weed, E. C. Mägi, and B. J. Eggleton, "Coupling to high-index waveguides via tapered microstructured optical fiber," Electron. Lett. 41, 951-953 (2005).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  28. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 9th ed. (Dover, 1965).

2005 (8)

D. J. Moss, Y. Miao, V. Weed, E. C. Mägi, and B. J. Eggleton, "Coupling to high-index waveguides via tapered microstructured optical fiber," Electron. Lett. 41, 951-953 (2005).
[CrossRef]

H. C. Nguyen, B. T. Kuhlmey, E. C. Mägi, M. J. Steel, P. Domachuk, C. L. Smith, and B. J. Eggleton, "Tapered photonic crystal fibres: properties, characterisation and applications," Appl. Phys. B 81, 377-387 (2005).
[CrossRef]

M. A. Foster, J. M. Dudley, B. Kibler, Q. Cao, D. Lee, R. Trebino, and A. L. Gaeta, "Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: experiment and simulation," Appl. Phys. B 81, 363-367 (2005).
[CrossRef]

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

S. Wilcox, L. C. Botten, C. M. de Sterke, B. T. Kuhlmey, R. C. McPhedran, D. P. Fussell, and S. Tomljenovic-Hanic, "Long wavelength behavior of the fundamental mode in microstructured optical fibers," Opt. Express 13, 1978-1984 (2005).
[CrossRef] [PubMed]

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]

H. C. Nguyen, B. T. Kuhlmey, M. J. Steel, C. L. Smith, E. C. Mägi, R. C. McPhedran, and B. J. Eggleton, "Leakage of the fundamental mode in photonic crystal fiber tapers," Opt. Lett. 30, 1123-1125 (2005).
[CrossRef] [PubMed]

G. Renversez, F. Bordas, and B. T. Kuhlmey, "Second mode transition in microstructured optical fibers: determination of the critical geometrical parameter and study of the matrix refractive index and effects of cladding size," Opt. Lett. 30, 1264-1266 (2005).
[CrossRef] [PubMed]

2004 (2)

2003 (1)

V. Finazzi, T. M. Monro, and D. J. Richardson, "The role of confinement loss in highly nonlinear silica holey fibers," IEEE Photon. Technol. Lett. 15, 1246-1248 (2003).
[CrossRef]

2002 (5)

2001 (2)

1997 (1)

1995 (1)

1991 (1)

J. D. Love, W. M. Henry, W. J. Stewart, R. J. Black, S. Lacroix, and F. Gonthier, "Tapered single-mode fibers and devices. 1. Adiabaticity criteria," IEE Proc.-J: Optoelectron. 138, 343-354 (1991).
[CrossRef]

1986 (1)

R. J. Black and R. Bourbonnais, "Core-mode cutoff for finite-cladding light guides," IEE Proc.-J: Optoelectron. 133, 377-384 (1986).
[CrossRef]

1978 (1)

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 9th ed. (Dover, 1965).

Birks, T. A.

T. A. Birks, J. C. Knight, and S. J. Russel, "Endlessly single-mode photonic crystal fiber," Opt. Lett. 22, 961-963 (1997).
[CrossRef] [PubMed]

T. A. Birks, "Reducing losses in photonic crystal fibres," in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference on CD-ROM (Optical Society of America, 2006), p. OFC7.

Black, R. J.

J. D. Love, W. M. Henry, W. J. Stewart, R. J. Black, S. Lacroix, and F. Gonthier, "Tapered single-mode fibers and devices. 1. Adiabaticity criteria," IEE Proc.-J: Optoelectron. 138, 343-354 (1991).
[CrossRef]

R. J. Black and R. Bourbonnais, "Core-mode cutoff for finite-cladding light guides," IEE Proc.-J: Optoelectron. 133, 377-384 (1986).
[CrossRef]

Bolger, J. A.

Bordas, F.

Botten, L. C.

Bourbonnais, R.

R. J. Black and R. Bourbonnais, "Core-mode cutoff for finite-cladding light guides," IEE Proc.-J: Optoelectron. 133, 377-384 (1986).
[CrossRef]

Cao, Q.

M. A. Foster, J. M. Dudley, B. Kibler, Q. Cao, D. Lee, R. Trebino, and A. L. Gaeta, "Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: experiment and simulation," Appl. Phys. B 81, 363-367 (2005).
[CrossRef]

Chandalia, J. K.

de Sterke, C.

de Sterke, C. M.

Domachuk, P.

H. C. Nguyen, B. T. Kuhlmey, E. C. Mägi, M. J. Steel, P. Domachuk, C. L. Smith, and B. J. Eggleton, "Tapered photonic crystal fibres: properties, characterisation and applications," Appl. Phys. B 81, 377-387 (2005).
[CrossRef]

Dudley, J. M.

M. A. Foster, J. M. Dudley, B. Kibler, Q. Cao, D. Lee, R. Trebino, and A. L. Gaeta, "Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: experiment and simulation," Appl. Phys. B 81, 363-367 (2005).
[CrossRef]

Eggleton, B. J.

Finazzi, V.

V. Finazzi, T. M. Monro, and D. J. Richardson, "The role of confinement loss in highly nonlinear silica holey fibers," IEEE Photon. Technol. Lett. 15, 1246-1248 (2003).
[CrossRef]

Foster, M. A.

M. A. Foster, J. M. Dudley, B. Kibler, Q. Cao, D. Lee, R. Trebino, and A. L. Gaeta, "Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: experiment and simulation," Appl. Phys. B 81, 363-367 (2005).
[CrossRef]

M. A. Foster and A. L. Gaeta, "Ultra-low threshold supercontinuum generation in subwavelength waveguides," Opt. Express 12, 3137-3143 (2004).
[CrossRef] [PubMed]

Fussell, D. P.

Gaeta, A. L.

M. A. Foster, J. M. Dudley, B. Kibler, Q. Cao, D. Lee, R. Trebino, and A. L. Gaeta, "Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: experiment and simulation," Appl. Phys. B 81, 363-367 (2005).
[CrossRef]

M. A. Foster and A. L. Gaeta, "Ultra-low threshold supercontinuum generation in subwavelength waveguides," Opt. Express 12, 3137-3143 (2004).
[CrossRef] [PubMed]

Gonthier, F.

J. D. Love, W. M. Henry, W. J. Stewart, R. J. Black, S. Lacroix, and F. Gonthier, "Tapered single-mode fibers and devices. 1. Adiabaticity criteria," IEE Proc.-J: Optoelectron. 138, 343-354 (1991).
[CrossRef]

Henry, W. M.

J. D. Love, W. M. Henry, W. J. Stewart, R. J. Black, S. Lacroix, and F. Gonthier, "Tapered single-mode fibers and devices. 1. Adiabaticity criteria," IEE Proc.-J: Optoelectron. 138, 343-354 (1991).
[CrossRef]

Houdewalter, S. N.

Kibler, B.

M. A. Foster, J. M. Dudley, B. Kibler, Q. Cao, D. Lee, R. Trebino, and A. L. Gaeta, "Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: experiment and simulation," Appl. Phys. B 81, 363-367 (2005).
[CrossRef]

Knight, J. C.

Knox, W. H.

Koshiba, M.

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

Kosinski, S. G.

Kuhlmey, B.

Kuhlmey, B. T.

Lacroix, S.

J. D. Love, W. M. Henry, W. J. Stewart, R. J. Black, S. Lacroix, and F. Gonthier, "Tapered single-mode fibers and devices. 1. Adiabaticity criteria," IEE Proc.-J: Optoelectron. 138, 343-354 (1991).
[CrossRef]

Lee, D.

M. A. Foster, J. M. Dudley, B. Kibler, Q. Cao, D. Lee, R. Trebino, and A. L. Gaeta, "Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: experiment and simulation," Appl. Phys. B 81, 363-367 (2005).
[CrossRef]

Lee, K. S.

Liu, X.

Lizé, Y. K.

Love, J.

A. Snyder and J. Love, Optical Waveguide Theory (Chapman & Hall, 1996).

Love, J. D.

J. D. Love, W. M. Henry, W. J. Stewart, R. J. Black, S. Lacroix, and F. Gonthier, "Tapered single-mode fibers and devices. 1. Adiabaticity criteria," IEE Proc.-J: Optoelectron. 138, 343-354 (1991).
[CrossRef]

Mägi, E. C.

H. C. Nguyen, B. T. Kuhlmey, M. J. Steel, C. L. Smith, E. C. Mägi, R. C. McPhedran, and B. J. Eggleton, "Leakage of the fundamental mode in photonic crystal fiber tapers," Opt. Lett. 30, 1123-1125 (2005).
[CrossRef] [PubMed]

H. C. Nguyen, B. T. Kuhlmey, E. C. Mägi, M. J. Steel, P. Domachuk, C. L. Smith, and B. J. Eggleton, "Tapered photonic crystal fibres: properties, characterisation and applications," Appl. Phys. B 81, 377-387 (2005).
[CrossRef]

D. J. Moss, Y. Miao, V. Weed, E. C. Mägi, and B. J. Eggleton, "Coupling to high-index waveguides via tapered microstructured optical fiber," Electron. Lett. 41, 951-953 (2005).
[CrossRef]

Y. K. Lizé, E. C. Mägi, V. G. Taeed, J. A. Bolger, P. Steinvurzel, and B. J. Eggleton, "Microstructured optical fiber photonic wires with subwavelength core diameter," Opt. Express 12, 3209-3217 (2004).
[CrossRef] [PubMed]

Marom, E.

Maystre, D.

McPhedran, R.

McPhedran, R. C.

Miao, Y.

D. J. Moss, Y. Miao, V. Weed, E. C. Mägi, and B. J. Eggleton, "Coupling to high-index waveguides via tapered microstructured optical fiber," Electron. Lett. 41, 951-953 (2005).
[CrossRef]

Monro, T. M.

V. Finazzi, T. M. Monro, and D. J. Richardson, "The role of confinement loss in highly nonlinear silica holey fibers," IEEE Photon. Technol. Lett. 15, 1246-1248 (2003).
[CrossRef]

Mortensen, N. A.

Moss, D. J.

D. J. Moss, Y. Miao, V. Weed, E. C. Mägi, and B. J. Eggleton, "Coupling to high-index waveguides via tapered microstructured optical fiber," Electron. Lett. 41, 951-953 (2005).
[CrossRef]

Nguyen, H. C.

H. C. Nguyen, B. T. Kuhlmey, M. J. Steel, C. L. Smith, E. C. Mägi, R. C. McPhedran, and B. J. Eggleton, "Leakage of the fundamental mode in photonic crystal fiber tapers," Opt. Lett. 30, 1123-1125 (2005).
[CrossRef] [PubMed]

H. C. Nguyen, B. T. Kuhlmey, E. C. Mägi, M. J. Steel, P. Domachuk, C. L. Smith, and B. J. Eggleton, "Tapered photonic crystal fibres: properties, characterisation and applications," Appl. Phys. B 81, 377-387 (2005).
[CrossRef]

Park, K. N.

Renversez, G.

Richardson, D. J.

V. Finazzi, T. M. Monro, and D. J. Richardson, "The role of confinement loss in highly nonlinear silica holey fibers," IEEE Photon. Technol. Lett. 15, 1246-1248 (2003).
[CrossRef]

Robinson, P.

Russel, S. J.

Saitoh, K.

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

Smith, C. L.

H. C. Nguyen, B. T. Kuhlmey, M. J. Steel, C. L. Smith, E. C. Mägi, R. C. McPhedran, and B. J. Eggleton, "Leakage of the fundamental mode in photonic crystal fiber tapers," Opt. Lett. 30, 1123-1125 (2005).
[CrossRef] [PubMed]

H. C. Nguyen, B. T. Kuhlmey, E. C. Mägi, M. J. Steel, P. Domachuk, C. L. Smith, and B. J. Eggleton, "Tapered photonic crystal fibres: properties, characterisation and applications," Appl. Phys. B 81, 377-387 (2005).
[CrossRef]

Smith, R. E.

Snyder, A.

A. Snyder and J. Love, Optical Waveguide Theory (Chapman & Hall, 1996).

Steel, M.

Steel, M. J.

H. C. Nguyen, B. T. Kuhlmey, E. C. Mägi, M. J. Steel, P. Domachuk, C. L. Smith, and B. J. Eggleton, "Tapered photonic crystal fibres: properties, characterisation and applications," Appl. Phys. B 81, 377-387 (2005).
[CrossRef]

H. C. Nguyen, B. T. Kuhlmey, M. J. Steel, C. L. Smith, E. C. Mägi, R. C. McPhedran, and B. J. Eggleton, "Leakage of the fundamental mode in photonic crystal fiber tapers," Opt. Lett. 30, 1123-1125 (2005).
[CrossRef] [PubMed]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 9th ed. (Dover, 1965).

Steinvurzel, P.

Stewart, W. J.

J. D. Love, W. M. Henry, W. J. Stewart, R. J. Black, S. Lacroix, and F. Gonthier, "Tapered single-mode fibers and devices. 1. Adiabaticity criteria," IEE Proc.-J: Optoelectron. 138, 343-354 (1991).
[CrossRef]

Taeed, V. G.

Tomljenovic-Hanic, S.

Trebino, R.

M. A. Foster, J. M. Dudley, B. Kibler, Q. Cao, D. Lee, R. Trebino, and A. L. Gaeta, "Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: experiment and simulation," Appl. Phys. B 81, 363-367 (2005).
[CrossRef]

Weed, V.

D. J. Moss, Y. Miao, V. Weed, E. C. Mägi, and B. J. Eggleton, "Coupling to high-index waveguides via tapered microstructured optical fiber," Electron. Lett. 41, 951-953 (2005).
[CrossRef]

White, T.

Wilcox, S.

Windeler, R. S.

Xu, C.

Yariv, A.

Yeh, P.

Appl. Phys. B (2)

H. C. Nguyen, B. T. Kuhlmey, E. C. Mägi, M. J. Steel, P. Domachuk, C. L. Smith, and B. J. Eggleton, "Tapered photonic crystal fibres: properties, characterisation and applications," Appl. Phys. B 81, 377-387 (2005).
[CrossRef]

M. A. Foster, J. M. Dudley, B. Kibler, Q. Cao, D. Lee, R. Trebino, and A. L. Gaeta, "Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: experiment and simulation," Appl. Phys. B 81, 363-367 (2005).
[CrossRef]

Electron. Lett. (1)

D. J. Moss, Y. Miao, V. Weed, E. C. Mägi, and B. J. Eggleton, "Coupling to high-index waveguides via tapered microstructured optical fiber," Electron. Lett. 41, 951-953 (2005).
[CrossRef]

IEE Proc.-J: Optoelectron. (2)

R. J. Black and R. Bourbonnais, "Core-mode cutoff for finite-cladding light guides," IEE Proc.-J: Optoelectron. 133, 377-384 (1986).
[CrossRef]

J. D. Love, W. M. Henry, W. J. Stewart, R. J. Black, S. Lacroix, and F. Gonthier, "Tapered single-mode fibers and devices. 1. Adiabaticity criteria," IEE Proc.-J: Optoelectron. 138, 343-354 (1991).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

V. Finazzi, T. M. Monro, and D. J. Richardson, "The role of confinement loss in highly nonlinear silica holey fibers," IEEE Photon. Technol. Lett. 15, 1246-1248 (2003).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (1)

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

Opt. Express (5)

Opt. Lett. (8)

Other (5)

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 9th ed. (Dover, 1965).

BandSOLVE 2.0.0 (Rsoft Design Group, Inc., 2005).

B. T. Kuhlmey, "Theoretical and numerical investigation of the physics of microstructured optical fibres," Ph.D. dissertation (University of Sydney and Université Aix-Marseille III, 2003), http://setis.library.usyd.edu.au/adt/publiclowbarhtml/adt-NU/public/adt-NU20040715.171105/.

T. A. Birks, "Reducing losses in photonic crystal fibres," in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference on CD-ROM (Optical Society of America, 2006), p. OFC7.

A. Snyder and J. Love, Optical Waveguide Theory (Chapman & Hall, 1996).

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Figures (7)

Fig. 1
Fig. 1

(a) Cross section of a PCF. The inner cladding, consisting of N r rings of triangularly arranged holes (diameter d) in a matrix material, surrounds a core consisting of a missing hole at the center of the fiber. The matrix material extends beyond the cladding, forming a region we call outer cladding. The natural external boundary of the fiber separates the outer cladding from the outside medium, or jacket. Typically, the jacket is air or index-matching fluid. The pitch or center-to-center distance between holes is denoted Λ. (b) Schematic “effective” refractive index profile of the PCF: the core and outer cladding have same refractive index, whereas the inner cladding’s effective index is lowered by the presence of holes. The jacket has a refractive index that can be one (PCF in air) or that can be matched to the matrix material (dashed).

Fig. 2
Fig. 2

PCF tapered down along the z direction, with profile ρ ( z ) .

Fig. 3
Fig. 3

(a) Pitch as extracted from ρ measurement (pitch before hole collapse, Λ ρ ) and from transverse probing (pitch after hole collapse Λ tr ). (b) d Λ as a function of the ratio of the pitch before and after collapse Λ tr Λ ρ , for various d Λ values before collapse. (c) Extracted d Λ along the taper: red, mean value; blue, upper and lower edges of the error bars. Thick curves, value obtained after smoothing the ρ measurements and direct transverse measurement Λ tr by fitting to a high-order polynomial. Corresponding error bars are obtained from differentiating Eq. (6) and by using the fitting error as error estimates. Thin light curves, values obtained directly from the measured data, without smoothing. Error bars are calculated by using conservatively estimated measurement errors of ± 1 % for ρ and ± 2 % for the transverse probing technique.

Fig. 4
Fig. 4

Comparison between simulated [(a)–(f)] and measured (g) transmission through a PCF taper. Simulations use the multipole method for the full PCF structure [ N r = 10 , d Λ = 0.4 , [(a)–(c)], and d Λ = 0.43 , [(d)–(f)]. Results for the tapers (a) and (d) are calculated by using the measured taper profile taken from Fig. 3a, using Eq. (3). Curves (b), (c), (e), and (f) for untapered PCFs are for comparison and correspond to the transmission of a segment of PCF having same length as the taper but with constant cross section being either the initial, untapered PCF [(b), (e)] or the cross section of the PCF at the taper’s waist [(c), (f)] for the same parameters as curves (a) and (d), respectively. Note that the low transmission below 0.8 μ m and the increase in transmission above 1.6 μ m in the experimental data are measurements artifacts due to the use of a low-pass filter and second-order diffraction on the spectrometer’s grating.

Fig. 5
Fig. 5

Taper transmission: comparison between results from full multipole simulations of the PCF structure [(a), (b)] and from the simplified depressed-cladding fiber (or w-fiber) model [(c), (d)], assuming d Λ values of 0.4 [(a), (c)] and 0.43 [(b), (d)], respectively. Curves (a) and (b) are taken from Fig. 4. The model used to compute curves (c) and (d) is explained in Section 6, with n m = 1.445 , ρ co = 0.9174 μ m , ρ cl = 12.04 μ m .

Fig. 6
Fig. 6

(a) Local losses as a function of position z in taper in dB/mm, for different wavelengths. This is the function to be integrated in Eq. (3). (b) Contour plot of the V value as a function of wavelength and position ( z ) in the taper. V reaches V c for λ 1.1 , which is a slight overestimate of the actual cutoff wavelength. The V value diagram is insufficient to give a precise indication of losses. Losses and V values obtained from simulations using the depressed-cladding model, with n cl = n fsm , n co = n J = 1.445 , and ρ co = 0.66 Λ = 0.9174 μ m , ρ cl = 3 2 N r Λ = 12.04 μ m before tapering, using the profile of Fig. 3a and d Λ = 0.4 to evaluate n fsm .

Fig. 7
Fig. 7

Adiabaticity criterion: experimental (black, polynomial fit; gray, raw data and numerical derivatives) taper angle as a function of local radius. Theoretical limit angle using k 0 ρ ( n eff n fsm ) 2 π , n eff calculated for a single core ( n co embedded in n fsm , ρ 0 = 0.66 Λ ) at different wavelengths. When the experimental taper angle is below the theoretical limit, the taper is adiabatic. We see that our taper is adiabatic up to 1.5 μ m . Note that because the taper decreases and then increases in diameter nonsymmetrically, the experimental curves have two points for each radius.

Equations (30)

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α = 20 ln ( 10 ) k 0 Im ( n eff ) .
Re ( n eff ) > n fsm .
I ( λ , z ) = I ( λ , 0 ) exp [ 0 z α ( λ , u ) d u ] .
n eff ( λ , z ) = n eff [ λ ρ 0 ρ ( z ) , 0 ] .
A Silica = Λ 2 [ 3 2 π 4 ( d Λ ) 2 ] .
d Λ = 2 Λ 1 π ( 3 2 Λ 2 A silica 0 ) .
V = 2 π ρ co λ n co 2 n cl 2 ,
V c = 2 ln ( ρ cl ρ co ) ,
n fsm ( 1 f ) n co 2 f ,
λ c 2 π ρ co [ ln ( ρ cl ρ co ) 2 f ( n co 2 1 ) ] 1 2 .
Ω < ρ co k 0 n eff n eff 2 π ,
ψ ( r ) = { J 0 ( k c o r ) J 0 ( k c o ρ c o ) , r ρ co A cl J 0 ( k cl r ) J 0 ( k cl ρ co ) + B cl H 0 ( 1 ) ( k cl r ) H 0 ( 1 ) ( k cl ρ co ) , ρ co r ρ cl , B J H 0 ( 1 ) ( k co r ) H 0 ( 1 ) ( k co ρ cl ) , r ρ cl ,
k co = k 0 ( n co 2 n eff 2 ) 1 2 ,
k cl = k 0 ( n cl 2 n eff 2 ) 1 2 ,
U co = ρ co k co ,
U cl = ρ co k cl ,
U J = ρ cl k co = S U co ,
V co = ρ co k 0 ( n co 2 n cl 2 ) 1 2 = ( U co 2 U cl 2 ) 1 2 ,
U cl = 0 ,
U co = V co .
1 = A cl + B cl ,
U co J 1 ( U co ) J 0 ( U co ) = A cl U cl J 1 ( U cl ) J 0 ( U cl ) + B cl U cl H 1 ( 1 ) ( U cl ) H 0 ( 1 ) ( U cl ) .
V co J 1 ( V co ) J 0 ( V co ) = B cl ln ( U cl ) .
B J = A cl J 0 ( S U cl ) J 0 ( U cl ) + B cl H 0 ( 1 ) ( S U cl ) H 0 ( 1 ) ( U cl ) ,
B J U co H 1 ( 1 ) ( S U co ) H 0 ( 1 ) ( S U co ) = A cl U cl J 1 ( S U cl ) J 0 ( U cl ) + B cl U cl H 1 ( 1 ) ( S U cl ) H 0 ( 1 ) ( U cl ) .
B J = A cl + B cl ln ( S U cl ) ln U cl ,
B J V co H 1 ( 1 ) ( S V co ) H 0 ( 1 ) ( S V co ) = B cl 1 S ln U cl .
J 0 ( V co ) J 1 ( V co ) V co ln S = 1 S H 0 ( 1 ) ( S V co ) H 1 ( 1 ) ( S V co ) .
V co J 1 ( V co ) J 0 ( V co ) = 1 ln S .
V co 2 ln S ,

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