Abstract

The optical microfiber coil resonator with self-coupling turns is suggested and investigated theoretically. This type of a microresonator has a three-dimensional geometry and complements the well-known Fabry-Perot (one-dimensional geometry, standing wave) and ring (two-dimensional geometry, traveling wave) types of microresonators. The coupled wave equations for the light propagation along the adiabatically bent coiled microfiber are derived. The particular cases of a microcoil having two and three turns are considered. The effect of microfiber radius variation on the value of Q-factor of resonances is studied.

© 2004 Optical Society of America

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  1. J. C. Knight, G. Cheung, F. Jacques, and T. A. Birks, “Phase-matched excitation of whispering-gallery-mode resonances by a fiber taper,” Opt. Lett. 22, 1129–1131 (1997).
    [CrossRef] [PubMed]
  2. V. I. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Dispersion compensation in whispering-gallery modes,” J. Opt. Soc. Am. A 20, 157 (2003).
    [CrossRef]
  3. S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a Fiber-Taper-Coupled Microresonator System for Application to Cavity Quantum Electrodynamics,” Phys. Rev. Lett. 91, 04902 (2003).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  10. A. Ghatak and K. Thyagarajan, Introduction to fiber optics (Cambridge University Press, 1998).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  14. L. Tong, J. Lou, and E. Mazur, “Single-mode guiding properties of subwavelength-diameter silica and silicon wire waveguides,” Opt. Express,  12, 1025–1035 (2004).
    [CrossRef] [PubMed]
  15. M. Ibsen, M. K. Durkin, R. Feced, M. J. Cole, M. N. Zervas, and R. I. Laming, “Dispersion compensating fibre Bragg gratings,” Proceedings of SPIE,  4532, 540–551 (2001).
    [CrossRef]
  16. M. V. Berry, “Anticipations of the geometric phase,” Physics Today 43, 34–40 (1990).
    [CrossRef]
  17. W.W. Lui, C.-L. Xu, T. Hirono, K. Yokoyama, and W.-P. Huang, “Full-vectorial wave propagation in semiconductor optical bending waveguides and equivalent straight waveguide approximations,” J. Lightwave Technol. 16, 910–914 (1998).
    [CrossRef]
  18. M. J. Adams, An introduction to optical waveguides (John Wiley and sons, New York, 1981).

2004 (1)

2003 (3)

V. I. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Dispersion compensation in whispering-gallery modes,” J. Opt. Soc. Am. A 20, 157 (2003).
[CrossRef]

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a Fiber-Taper-Coupled Microresonator System for Application to Cavity Quantum Electrodynamics,” Phys. Rev. Lett. 91, 04902 (2003).
[CrossRef]

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature,  426, 816–819 (2003).
[CrossRef] [PubMed]

2001 (2)

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE Journ. Quant.Electron. 37, 525–532 (2001).
[CrossRef]

M. Ibsen, M. K. Durkin, R. Feced, M. J. Cole, M. N. Zervas, and R. I. Laming, “Dispersion compensating fibre Bragg gratings,” Proceedings of SPIE,  4532, 540–551 (2001).
[CrossRef]

2000 (1)

1998 (1)

1997 (2)

J. C. Knight, G. Cheung, F. Jacques, and T. A. Birks, “Phase-matched excitation of whispering-gallery-mode resonances by a fiber taper,” Opt. Lett. 22, 1129–1131 (1997).
[CrossRef] [PubMed]

B.E. Little, S.T. Chu, H.A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[CrossRef]

1990 (1)

M. V. Berry, “Anticipations of the geometric phase,” Physics Today 43, 34–40 (1990).
[CrossRef]

1983 (1)

S. C. Rashleigh, “Origin and control of polarization effects in single-mode fibers,” J. Lightwave Technol.,  1, 312–331 (1983).
[CrossRef]

1982 (1)

1972 (1)

Adams, M. J.

M. J. Adams, An introduction to optical waveguides (John Wiley and sons, New York, 1981).

Ashcom, J. B.

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature,  426, 816–819 (2003).
[CrossRef] [PubMed]

Babich, V. M.

V. M. Babich and V. S. Buldyrev, Short-Wavelength Diffraction Theory (Springer-Verlag, Berlin, 1991).
[CrossRef]

Berry, M. V.

M. V. Berry, “Anticipations of the geometric phase,” Physics Today 43, 34–40 (1990).
[CrossRef]

Birks, T. A.

Buldyrev, V. S.

V. M. Babich and V. S. Buldyrev, Short-Wavelength Diffraction Theory (Springer-Verlag, Berlin, 1991).
[CrossRef]

Cheung, G.

Chodorow, M.

Chu, S.T.

B.E. Little, S.T. Chu, H.A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[CrossRef]

Cole, M. J.

M. Ibsen, M. K. Durkin, R. Feced, M. J. Cole, M. N. Zervas, and R. I. Laming, “Dispersion compensating fibre Bragg gratings,” Proceedings of SPIE,  4532, 540–551 (2001).
[CrossRef]

Durkin, M. K.

M. Ibsen, M. K. Durkin, R. Feced, M. J. Cole, M. N. Zervas, and R. I. Laming, “Dispersion compensating fibre Bragg gratings,” Proceedings of SPIE,  4532, 540–551 (2001).
[CrossRef]

Eggleton, B. J.

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE Journ. Quant.Electron. 37, 525–532 (2001).
[CrossRef]

Feced, R.

M. Ibsen, M. K. Durkin, R. Feced, M. J. Cole, M. N. Zervas, and R. I. Laming, “Dispersion compensating fibre Bragg gratings,” Proceedings of SPIE,  4532, 540–551 (2001).
[CrossRef]

Foresi, J.

B.E. Little, S.T. Chu, H.A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[CrossRef]

Gattass, R. R.

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature,  426, 816–819 (2003).
[CrossRef] [PubMed]

Ghatak, A.

A. Ghatak and K. Thyagarajan, Introduction to fiber optics (Cambridge University Press, 1998).

Haus, H.A.

B.E. Little, S.T. Chu, H.A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[CrossRef]

He, S.

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature,  426, 816–819 (2003).
[CrossRef] [PubMed]

Hirono, T.

Huang, W.-P.

Ibsen, M.

M. Ibsen, M. K. Durkin, R. Feced, M. J. Cole, M. N. Zervas, and R. I. Laming, “Dispersion compensating fibre Bragg gratings,” Proceedings of SPIE,  4532, 540–551 (2001).
[CrossRef]

Ilchenko, V. I.

Jacques, F.

Kippenberg, T. J.

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a Fiber-Taper-Coupled Microresonator System for Application to Cavity Quantum Electrodynamics,” Phys. Rev. Lett. 91, 04902 (2003).
[CrossRef]

Knight, J. C.

Laine, J.-P.

B.E. Little, S.T. Chu, H.A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[CrossRef]

Laming, R. I.

M. Ibsen, M. K. Durkin, R. Feced, M. J. Cole, M. N. Zervas, and R. I. Laming, “Dispersion compensating fibre Bragg gratings,” Proceedings of SPIE,  4532, 540–551 (2001).
[CrossRef]

Lenz, G.

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE Journ. Quant.Electron. 37, 525–532 (2001).
[CrossRef]

Little, B.E.

B.E. Little, S.T. Chu, H.A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[CrossRef]

Lou, J.

L. Tong, J. Lou, and E. Mazur, “Single-mode guiding properties of subwavelength-diameter silica and silicon wire waveguides,” Opt. Express,  12, 1025–1035 (2004).
[CrossRef] [PubMed]

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature,  426, 816–819 (2003).
[CrossRef] [PubMed]

Lui, W.W.

Madsen, C. K.

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE Journ. Quant.Electron. 37, 525–532 (2001).
[CrossRef]

Maleki, L.

Matsko, A. B.

Maxwell, I.

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature,  426, 816–819 (2003).
[CrossRef] [PubMed]

Mazur, E.

L. Tong, J. Lou, and E. Mazur, “Single-mode guiding properties of subwavelength-diameter silica and silicon wire waveguides,” Opt. Express,  12, 1025–1035 (2004).
[CrossRef] [PubMed]

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature,  426, 816–819 (2003).
[CrossRef] [PubMed]

NÖckel, J. U.

J. U. NÖckel, “2-d Microcavities: Theory and Experiments,” in Cavity-Enhanced Spectroscopies,R. D. van Zee and J. P. Looney, eds. (Academic Press, San Diego, 2002).

Painter, O. J.

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a Fiber-Taper-Coupled Microresonator System for Application to Cavity Quantum Electrodynamics,” Phys. Rev. Lett. 91, 04902 (2003).
[CrossRef]

Rashleigh, S. C.

S. C. Rashleigh, “Origin and control of polarization effects in single-mode fibers,” J. Lightwave Technol.,  1, 312–331 (1983).
[CrossRef]

Russell, P. St. J.

Savchenkov, A. A.

Shaw, H.J.

Shen, M.

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature,  426, 816–819 (2003).
[CrossRef] [PubMed]

Slusher, R. E.

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE Journ. Quant.Electron. 37, 525–532 (2001).
[CrossRef]

Snyder, A.W.

Spillane, S. M.

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a Fiber-Taper-Coupled Microresonator System for Application to Cavity Quantum Electrodynamics,” Phys. Rev. Lett. 91, 04902 (2003).
[CrossRef]

Stokes, L.F.

Thyagarajan, K.

A. Ghatak and K. Thyagarajan, Introduction to fiber optics (Cambridge University Press, 1998).

Tong, L.

L. Tong, J. Lou, and E. Mazur, “Single-mode guiding properties of subwavelength-diameter silica and silicon wire waveguides,” Opt. Express,  12, 1025–1035 (2004).
[CrossRef] [PubMed]

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature,  426, 816–819 (2003).
[CrossRef] [PubMed]

Vahala, K. J.

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a Fiber-Taper-Coupled Microresonator System for Application to Cavity Quantum Electrodynamics,” Phys. Rev. Lett. 91, 04902 (2003).
[CrossRef]

Wadsworth, W. J.

Xu, C.-L.

Yokoyama, K.

Zervas, M. N.

M. Ibsen, M. K. Durkin, R. Feced, M. J. Cole, M. N. Zervas, and R. I. Laming, “Dispersion compensating fibre Bragg gratings,” Proceedings of SPIE,  4532, 540–551 (2001).
[CrossRef]

IEEE Journ. Quant.Electron. (1)

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE Journ. Quant.Electron. 37, 525–532 (2001).
[CrossRef]

J. Lightwave Technol. (3)

S. C. Rashleigh, “Origin and control of polarization effects in single-mode fibers,” J. Lightwave Technol.,  1, 312–331 (1983).
[CrossRef]

W.W. Lui, C.-L. Xu, T. Hirono, K. Yokoyama, and W.-P. Huang, “Full-vectorial wave propagation in semiconductor optical bending waveguides and equivalent straight waveguide approximations,” J. Lightwave Technol. 16, 910–914 (1998).
[CrossRef]

B.E. Little, S.T. Chu, H.A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Nature (1)

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature,  426, 816–819 (2003).
[CrossRef] [PubMed]

Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. Lett. (1)

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a Fiber-Taper-Coupled Microresonator System for Application to Cavity Quantum Electrodynamics,” Phys. Rev. Lett. 91, 04902 (2003).
[CrossRef]

Physics Today (1)

M. V. Berry, “Anticipations of the geometric phase,” Physics Today 43, 34–40 (1990).
[CrossRef]

Proceedings of SPIE (1)

M. Ibsen, M. K. Durkin, R. Feced, M. J. Cole, M. N. Zervas, and R. I. Laming, “Dispersion compensating fibre Bragg gratings,” Proceedings of SPIE,  4532, 540–551 (2001).
[CrossRef]

Other (4)

M. J. Adams, An introduction to optical waveguides (John Wiley and sons, New York, 1981).

V. M. Babich and V. S. Buldyrev, Short-Wavelength Diffraction Theory (Springer-Verlag, Berlin, 1991).
[CrossRef]

J. U. NÖckel, “2-d Microcavities: Theory and Experiments,” in Cavity-Enhanced Spectroscopies,R. D. van Zee and J. P. Looney, eds. (Academic Press, San Diego, 2002).

A. Ghatak and K. Thyagarajan, Introduction to fiber optics (Cambridge University Press, 1998).

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Figures (4)

Fig. 1.
Fig. 1.

Microcoils having a - two turns, b - three turns, c - multiple turns and non-uniform pitch.

Fig. 2.
Fig. 2.

Microcoil with two coupling turns. The turns are assumed to be parallel in the region of coupling. (x,y,s) is the common for two turns local coordinate system, n(s) is their normal and b(s) is their binormal at point s.

Fig.3.
Fig.3.

Q-factors of the two-turn microcoil as a function of separation of the coupling parameter from the resonant value. The fiber is uniform and propagation losses are ignored.

Fig. 4.
Fig. 4.

Group delay dependencies on the inverse propagation constant, 2πnf /β 0, for the value of refractive index of the fiber nf = 1.46; a - a microcoil with two turns; b - a microcoil with three turns.

Equations (56)

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E ( q 1 , q 2 , s ) = F 0 ( q 1 , q 2 , s ) exp ( i s β ( s ) ds )
β ( s ) = β 0 + δβ ( s ) , δβ ( s ) < < β 0 .
E t ( x , y , s ) = A 1 ( s ) exp ( i s 1 s β 1 ( s ) ) F 0 ( x , y ) + A 2 ( s ) exp ( i s 1 s β 2 ( s ) ) F 0 ( x , y p ( s ) )
s ( A 1 ( s ) A 2 ( s ) ) = i ( 0 χ 12 χ 21 0 ) ( A 1 ( s ) A 1 ( s ) ) , χ pq ( s ) = κ ( s ) exp ( 2 i s 1 s ( β p ( s ) β q ( s ) ) ds )
A 2 ( s 1 ) = A ( s 1 + S ) exp ( i s 1 s 1 + S β 1 ( s ) ds )
d ds ( A 1 A 2 A m A M 1 A M ) = i ( 0 χ 12 ( s ) 0 0 0 0 χ 21 ( s ) 0 χ 23 ( s ) 0 0 0 0 χ 32 ( s ) 0 0 0 0 0 0 0 0 χ M 1 , M 2 ( s ) 0 0 0 0 χ M 2 , M 1 ( s ) 0 χ M 1 , M ( s ) 0 0 0 0 χ M , M 1 ( s ) 0 ) ( A 1 A 2 A m A M 1 A M ) ,
χ pq ( s ) = κ pq ( s ) exp ( 2 i s 1 s ( β p ( s ) β q ( s ) ) ds ) .
A m + 1 ( s 1 ) = A m ( s 1 + S ) exp [ i s 1 s 1 + S β m ( s ) ds ] , m = 1,2 M 1 .
Δβ ( s ) = 1 2 ( β 1 ( s ) β 2 ( s ) )
Δβ ( s ) < < κ ( s )
T = exp ( i s 1 s 1 + S β 1 ( s ) ds ) + Ξ 1 + exp ( i s 1 s 1 + S β 1 ( s ) ds ) Ξ * exp ( i s 1 s 1 + S β 2 ( s ) ds ) ,
Ξ = 1 2 [ ( 1 ε 1 + ε 2 ) e iK ( 1 + ε 1 * + ε 2 * ) e iK ] ,
t d = n f c d ln ( T ) d β 0 ,
K = K m = ( 2 m 1 ) π 2 ,
β 0 S = β 0 n S = 2 n π + π 2 + Im ε 1 s 1 s 1 + S ds δ β 1 ( s ) ,
Δ t d mn 2 n f S c ( K K m ) 2 + ( Re ε 1 ) 2 ( β 0 β 0 n ) 2 S 2 + 1 4 [ ( K K m ) 2 + ( Re ε 1 ) 2 ] 2 .
Q = βS ( K K m ) 2 + ( Re ε 1 ) 2 .
Re ε 1 = Δ β 0 κ 0 .
Δ β 0 β 0 ~ Δ r r .
T ( β ) = Q 2 ( β ) Q 2 * ( β ) ,
Q 2 ( β ) = e iβS 2 1 / 2 i sin ( 2 1 / 2 K ) e iβS sin 2 ( 2 1 / 2 K )
K = K m ( 1 ) = ( 2 m 1 ) π / 2 1 / 2 , βS = β n S = ,
K = K ( 2 ) = 2 1 / 2 [ ε arcsin ( 3 1 / 2 ) + m π ] , βS = β S = ( 2 n + ε 2 ) π , ε = ± 1 ,
2 E + ( ( E , ln ( n 2 ) ) ) + ( 2 πn λ ) 2 E = 0 ,
r ( q 1 , q 2 , s ) = r ( 0,0 , s ) + q 1 e 1 + q 2 e 2 ,
e 1 = n cos ( Λ ( s ) ) b sin ( Λ ( s ) ) ,
e 2 = n sin ( Λ ( s ) ) b cos ( Λ ( s ) ) ,
Λ ( s ) = s τ ( s ) ds
n ( q 1 , q 2 , s ) = n 0 ( q 1 , q 2 ) + Δn ( q 1 , q 2 , s ) ,
Δ = 1 G ( q 1 , q 2 , s ) [ s 1 G ( q 1 , q 2 , s ) s + j = 1 2 q j G ( q 1 , q 2 , s ) q j ]
= ( 1 G ( q 1 , q 2 , s ) s , q 1 , q 2 )
G ( q 1 , q 2 , s ) = 1 ρ ( s ) ( q 1 cos Λ ( s ) q 2 sin Λ ( s ) )
E ( q 1 , q 2 , s ) = F 0 ( q 1 , q 2 , s ) exp ( i s β ( s ) ds ) .
Λ 0 ( s ) = s s + S ds τ ( s )
ρ ( s ) = 4 π 2 R p 2 ( s ) + 4 π 2 R 2 , τ ( s ) = 2 πp ( s ) p 2 ( s ) + 4 π 2 R 2
ρ ( s ) = 1 R , τ ( s ) = p ( s ) 2 π R 2 .
e 1 ( 0 ) = n ,
e 2 ( 0 ) = b
G ( q 1 , q 2 , s ) = 1 q 1 R
cos ( Λ ( s ) ) = 1 ,
sin ( Λ ( s ) ) = 1 2 π R 2 0 s p ( s ) ds .
κ ( s ) = ( n f 2 n e 2 ) β { 2 π 2 λ 2 ∫∫ x 2 + y 2 < r 2 dxdy [ F 0 x ( x , y ) ( F 0 x ( x , y p ( s ) ) + F 0 y ( x , y ) ( F 0 y ( x , y p ( s ) ) ]
+ 1 2 n f 2 x 2 + y 2 = r 2 dl r ( x F 0 x ( x , y ) + y F 0 y ( x , y ) ) ( F 0 x ( x , y p ( s ) ) x + F 0 y ( x , y p ( s ) ) y ] } ,
x 2 + y 2 < dxdy ( F 0 x ( x , y ) F 0 x ( x , y ) + F 0 y ( x , y ) F 0 y ( x , y ) ) = 1 .
Δβ ( s ) = 1 2 ( β 1 ( s ) β 2 ( s ) )
A 1 ( s 1 ) = A 01
A 2 ( s 1 ) = A 02
A 1 ( s 1 + S ) = 1 2 [ ( A 01 + A 02 ) ( 1 + ε 1 + ε 2 ) e iK + ( A 01 A 02 ) ( 1 ε 1 * + ε 2 * ) e iK ] ,
A 2 ( s 1 + S ) = 1 2 [ ( A 01 + A 02 ) ( 1 ε 1 + ε 2 ) e iK ( A 01 A 02 ) ( 1 + ε 1 * + ε 2 * ) e iK ]
K = s 1 s 1 + S ds κ ( s ) ,
ε 1 = i s 1 s 1 + S dsΔβ ( s ) [ exp ( 2 i s s 1 + S ds′ κ ( s′ ) ) 1 ] ,
ε 2 = 2 i s c 1 s c 2 dsκ ( s ) ( s 1 s ds Δβ ( s′ ) ) [ s 1 s ds Δβ ( s′ ) exp ( 2 i s′ s ds′′ κ ( s′′ ) ) ] .
ε 2 + ε 2 * = ε 1 ε 1 *
T = A 2 ( s 1 + S ) A 1 ( s 1 ) exp ( i s 1 s 1 + S ds β 2 ( s ) )
A 1 ( s 1 + S ) exp ( i s 1 s 1 + S ds β 1 ( s ) ) = A 2 ( s 1 )
β 1 ( s 1 + S ) = β 2 ( s 1 )

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