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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References

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Cavity-Enhanced Spectroscopies (1)

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

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. Lett. (3)

Optics Express (1)

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

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 (3)

M. J. Adams, An Introduction to Optical Waveguides (Wiley, New York, 1981)

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

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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