Abstract

Microcoil resonators are a radical new geometry for high Q resonators with unique linear features. In this paper I briefly summarise their linear properties before extending the analysis to nonlinear interactions in microcoil resonators. As expected such nonlinear resonators are bistable and exhibit hysteresis. Finally I discuss possible applications and extensions to such resonators.

© 2008 Optical Society of America

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References

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  1. M. Sumetsky, "Optical fiber microcoil resonator," Optics Express 12,2303 (2004).
    [CrossRef] [PubMed]
  2. M. Sumetsky, "Uniform coil optical resonator and waveguide: transmission spectrum, eigenmodes, and dispersion relation," Optics Express 13,4331 (2005).
    [CrossRef] [PubMed]
  3. M. Sumetsky, "Basic elements for microfiber photonics: Micro/nanofibers and microfibre coil resonators," IEEE J. Light. Tech. 26,21-27 (2008).
    [CrossRef]
  4. F. Xu and G. Brambilla, "Manufacture of 3-D microfiber coil resonators," IEEE Photonics Technology Letters 19,1481-1483 (2007).
    [CrossRef]
  5. F. Xu and G. Brambilla, "Embedding optical microfiber coil resonators in Teflon," Opt. Lett. 32,2164-2166 (2007).
    [CrossRef] [PubMed]
  6. F. Xu, P. Horak, and G. Brambilla, "Optimised design of microcoil resonators," IEEE J. Light. Tech. 25,1561- 1567 (2007).
    [CrossRef]
  7. It should be noted that there are at least two misprints in Sumetsky???s Optics Express paper [2]. Firstly in Eq. 1 there is a factor i missing on the L.H.S. Secondly in the equation for the transmission a factor of exp(i® l) is missing. Both of these misprints do not occur in his original paper.
  8. G. P. Agrawal, Nonlinear Fibre Optics, 3rd ed. (Academic Press, San Diego, 2001).
  9. A. B. Aceves, C. DeAngelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, "Discrete selftrapping, soliton interactions and beam steering in nonlinear waveguide arrays," Phys. Rev. E 53,1172-1189 (1996).
    [CrossRef]
  10. K. Ogusu, "Dynamic Behavior of Reflection Optical Bistability in a Nonlinear Fiber Ring Resonator," IEEE J. Quant. Elect. 32,1537-1543 (1996).
    [CrossRef]
  11. Q. Xu and M. Lipson, "Carrier-induced optical bistability in silicon ring resonators," Opt. Lett. 31,341-343 (2006).
    [CrossRef] [PubMed]
  12. J. Nocedal and S. J. Wright, Numerical Optimisation, 1st ed. (Springer-Verlag, New York, 1999).
    [CrossRef]

2008

M. Sumetsky, "Basic elements for microfiber photonics: Micro/nanofibers and microfibre coil resonators," IEEE J. Light. Tech. 26,21-27 (2008).
[CrossRef]

2007

F. Xu and G. Brambilla, "Manufacture of 3-D microfiber coil resonators," IEEE Photonics Technology Letters 19,1481-1483 (2007).
[CrossRef]

F. Xu, P. Horak, and G. Brambilla, "Optimised design of microcoil resonators," IEEE J. Light. Tech. 25,1561- 1567 (2007).
[CrossRef]

F. Xu and G. Brambilla, "Embedding optical microfiber coil resonators in Teflon," Opt. Lett. 32,2164-2166 (2007).
[CrossRef] [PubMed]

2006

2005

M. Sumetsky, "Uniform coil optical resonator and waveguide: transmission spectrum, eigenmodes, and dispersion relation," Optics Express 13,4331 (2005).
[CrossRef] [PubMed]

2004

M. Sumetsky, "Optical fiber microcoil resonator," Optics Express 12,2303 (2004).
[CrossRef] [PubMed]

1996

A. B. Aceves, C. DeAngelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, "Discrete selftrapping, soliton interactions and beam steering in nonlinear waveguide arrays," Phys. Rev. E 53,1172-1189 (1996).
[CrossRef]

K. Ogusu, "Dynamic Behavior of Reflection Optical Bistability in a Nonlinear Fiber Ring Resonator," IEEE J. Quant. Elect. 32,1537-1543 (1996).
[CrossRef]

Aceves, A. B.

A. B. Aceves, C. DeAngelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, "Discrete selftrapping, soliton interactions and beam steering in nonlinear waveguide arrays," Phys. Rev. E 53,1172-1189 (1996).
[CrossRef]

Brambilla, G.

F. Xu, P. Horak, and G. Brambilla, "Optimised design of microcoil resonators," IEEE J. Light. Tech. 25,1561- 1567 (2007).
[CrossRef]

F. Xu and G. Brambilla, "Manufacture of 3-D microfiber coil resonators," IEEE Photonics Technology Letters 19,1481-1483 (2007).
[CrossRef]

F. Xu and G. Brambilla, "Embedding optical microfiber coil resonators in Teflon," Opt. Lett. 32,2164-2166 (2007).
[CrossRef] [PubMed]

DeAngelis, C.

A. B. Aceves, C. DeAngelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, "Discrete selftrapping, soliton interactions and beam steering in nonlinear waveguide arrays," Phys. Rev. E 53,1172-1189 (1996).
[CrossRef]

Horak, P.

F. Xu, P. Horak, and G. Brambilla, "Optimised design of microcoil resonators," IEEE J. Light. Tech. 25,1561- 1567 (2007).
[CrossRef]

Lederer, F.

A. B. Aceves, C. DeAngelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, "Discrete selftrapping, soliton interactions and beam steering in nonlinear waveguide arrays," Phys. Rev. E 53,1172-1189 (1996).
[CrossRef]

Lipson, M.

Muschall, R.

A. B. Aceves, C. DeAngelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, "Discrete selftrapping, soliton interactions and beam steering in nonlinear waveguide arrays," Phys. Rev. E 53,1172-1189 (1996).
[CrossRef]

Ogusu, K.

K. Ogusu, "Dynamic Behavior of Reflection Optical Bistability in a Nonlinear Fiber Ring Resonator," IEEE J. Quant. Elect. 32,1537-1543 (1996).
[CrossRef]

Peschel, T.

A. B. Aceves, C. DeAngelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, "Discrete selftrapping, soliton interactions and beam steering in nonlinear waveguide arrays," Phys. Rev. E 53,1172-1189 (1996).
[CrossRef]

Sumetsky, M.

M. Sumetsky, "Basic elements for microfiber photonics: Micro/nanofibers and microfibre coil resonators," IEEE J. Light. Tech. 26,21-27 (2008).
[CrossRef]

M. Sumetsky, "Uniform coil optical resonator and waveguide: transmission spectrum, eigenmodes, and dispersion relation," Optics Express 13,4331 (2005).
[CrossRef] [PubMed]

M. Sumetsky, "Optical fiber microcoil resonator," Optics Express 12,2303 (2004).
[CrossRef] [PubMed]

Trillo, S.

A. B. Aceves, C. DeAngelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, "Discrete selftrapping, soliton interactions and beam steering in nonlinear waveguide arrays," Phys. Rev. E 53,1172-1189 (1996).
[CrossRef]

Wabnitz, S.

A. B. Aceves, C. DeAngelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, "Discrete selftrapping, soliton interactions and beam steering in nonlinear waveguide arrays," Phys. Rev. E 53,1172-1189 (1996).
[CrossRef]

Xu, F.

F. Xu, P. Horak, and G. Brambilla, "Optimised design of microcoil resonators," IEEE J. Light. Tech. 25,1561- 1567 (2007).
[CrossRef]

F. Xu and G. Brambilla, "Embedding optical microfiber coil resonators in Teflon," Opt. Lett. 32,2164-2166 (2007).
[CrossRef] [PubMed]

F. Xu and G. Brambilla, "Manufacture of 3-D microfiber coil resonators," IEEE Photonics Technology Letters 19,1481-1483 (2007).
[CrossRef]

Xu, Q.

IEEE J. Light. Tech.

M. Sumetsky, "Basic elements for microfiber photonics: Micro/nanofibers and microfibre coil resonators," IEEE J. Light. Tech. 26,21-27 (2008).
[CrossRef]

F. Xu, P. Horak, and G. Brambilla, "Optimised design of microcoil resonators," IEEE J. Light. Tech. 25,1561- 1567 (2007).
[CrossRef]

IEEE J. Quant. Elect.

K. Ogusu, "Dynamic Behavior of Reflection Optical Bistability in a Nonlinear Fiber Ring Resonator," IEEE J. Quant. Elect. 32,1537-1543 (1996).
[CrossRef]

IEEE Photonics Technology Letters

F. Xu and G. Brambilla, "Manufacture of 3-D microfiber coil resonators," IEEE Photonics Technology Letters 19,1481-1483 (2007).
[CrossRef]

Opt. Lett.

Optics Express

M. Sumetsky, "Optical fiber microcoil resonator," Optics Express 12,2303 (2004).
[CrossRef] [PubMed]

M. Sumetsky, "Uniform coil optical resonator and waveguide: transmission spectrum, eigenmodes, and dispersion relation," Optics Express 13,4331 (2005).
[CrossRef] [PubMed]

Phys. Rev. E

A. B. Aceves, C. DeAngelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, "Discrete selftrapping, soliton interactions and beam steering in nonlinear waveguide arrays," Phys. Rev. E 53,1172-1189 (1996).
[CrossRef]

Other

J. Nocedal and S. J. Wright, Numerical Optimisation, 1st ed. (Springer-Verlag, New York, 1999).
[CrossRef]

It should be noted that there are at least two misprints in Sumetsky???s Optics Express paper [2]. Firstly in Eq. 1 there is a factor i missing on the L.H.S. Secondly in the equation for the transmission a factor of exp(i® l) is missing. Both of these misprints do not occur in his original paper.

G. P. Agrawal, Nonlinear Fibre Optics, 3rd ed. (Academic Press, San Diego, 2001).

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Figures (3)

Fig. 1.
Fig. 1.

(a) Schematic of a microcoil resonator taken from [2]. (b) Transmission spectrum for a lossy OMR with 8 coils. The loss is 0.02dB/mm while the coupling strength was 3mm-1. The green line shows the expected transmission for a straight length of fibre with the same length and loss.

Fig. 2.
Fig. 2.

Normalised group velocity (top row) and dispersion (bottom) for a 8 turn OMR with the same parameters as the one shown in Fig. 1. The figures on the right are expanded versions of the main graphs showing the narrow wavelength region of negative group velocity. Note that off resonance the dispersion is still large (1000’s of picoseconds squared) and increases by several orders of magnitude on resonance. In the graphs the spikes are due to numerical errors at points where the group velocity becomes undefined.

Fig. 3.
Fig. 3.

Nonlinear response of a 3 turn microcoil resonator. Fig. (a) shows the full solution (both stable and unstable branches) as a function of the input power. Fig. (b) shows the output phase while Fig. (c) shows the hysteresis curves for the resonator for a range of wave-lengths. Figyre 3(a) shows the transmission as a function of the normalised input power for a wavelength of λ=1.53022µm. This is on the long wavelength size of the resonance and so one would expect the transmission to decrease as the intensity increases as can be seen. It is also clear that there is a wide range of intensities over which the transmission is no longer single valued and thus one expects the system to exhibit bistability and hysteresis. In this case the maximum contrast between the high and low transmission branches is > 33dB. The limiting factor in the contrast ratio is the fact that as the loss increases the signal intensity drops to the point where the transmission is linear thus reducing the loss. Figure 3(b) shows the output phase as a function of the input intensity and is discussed further in the appendix.

Equations (19)

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i d ds ( A 1 A 2 A 3 A n 1 A n ) = ( 0 κ 0 0 0 0 κ 0 κ 0 0 0 0 κ 0 κ 0 0 0 0 0 κ 0 κ 0 0 0 0 κ 0 ) ( A 1 A 2 A 3 A n 1 A n )
( A 1 ( 0 ) A 2 ( 0 ) A 3 ( 0 ) A n 1 ( 0 ) A n ( 0 ) ) = ( 0 0 0 0 0 e i β l 0 0 0 0 0 e i β l 0 0 0 0 0 0 0 0 0 0 0 e i β l 0 ) ( A 1 ( l ) A 2 ( l ) A 3 ( l ) A n 1 ( l ) A n ( l ) ) + ( A 1 ( 0 ) 0 0 0 0 )
T = A n ( l ) e i β l A 1 ( 0 ) .
A ( 0 ) = B A ( l ) + A in
A ( l ) = K ( l ) A ( 0 )
A ( l ) = [ I K ( l ) B ] 1 K ( l ) A in
A ( 0 ) = [ I B K ( l ) ] 1 A in
A ( s ) = K ( s ) [ I B K ( l ) ] 1 A in
t d = d ϕ d ω
v g = l d ϕ d ω
t ( ω ) 2 = e α l eff ( ω ) 2
d et ( I B K ) = 0 .
d A 1 d s = α A 1 + i γ A 1 ( s ) 2 A 1 ( s ) + i κ A 2 ( s )
d A k d s = α A k + i γ A k ( s ) 2 A j ( s ) + i κ [ A k 1 ( s ) + A k + 1 ( s ) ]
d A n d s = α A n + i γ A n ( s ) 2 A n ( s ) + i κ A n 1 ( s )
A ( 0 ) = B N ( A ) A ( 0 ) + A in
f ( X ) = B N ( X ) X + A in .
A n = B N ( A ) A n 1 + A in
A n = ( k = 0 k = n ( B K ) k ) A in .

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