## Uniform coil optical resonator and waveguide: transmission spectrum, eigenmodes, and dispersion relation

Optics Express, Vol. 13, Issue 11, pp. 4331-4340 (2005)

http://dx.doi.org/10.1364/OPEX.13.004331

Acrobat PDF (1013 KB)

### Abstract

The coil optical resonator (COR) is an optical microfiber coil tightly wound on an optical rod. The resonant behavior of this all-pass device is determined by evanescent coupling between the turns of the microfiber. This paper investigates the uniform COR with *N* turns. Its transmission characteristics are surprisingly different from those of the known types of resonators and of photonic crystal structures. It is found that for certain discrete sequences of propagation constant and interturn coupling, the light is completely trapped by the resonator. For *N* →∞, the COR spectrum experiences a fractal collapse to the points corresponding to the second order zero of the group velocity. For a relatively small coupling between turns, the COR waveguide behavior resembles that of a SCISSOR (side-coupled integrated spaced sequence of resonators), while for larger coupling it resembles that of a CROW (coupled resonator optical waveguide).

© 2005 Optical Society of America

## 1. Introduction

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]

3. S. G. Leon-Saval, T. A. Birks, W. J. Wadsworth, and P. St.J. Russell, “Supercontinuum generation in submicron fibre waveguides,” Opt. Express , **12**, 2864–2869 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-13-2864. [CrossRef] [PubMed]

2. G. Brambilla, V. Finazzi, and D. J. Richardson, “Ultra-low-loss optical fiber nanotapers,” Opt. Express , **12**, 2258–2263 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2258. [CrossRef] [PubMed]

3. S. G. Leon-Saval, T. A. Birks, W. J. Wadsworth, and P. St.J. Russell, “Supercontinuum generation in submicron fibre waveguides,” Opt. Express , **12**, 2864–2869 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-13-2864. [CrossRef] [PubMed]

^{6}[4

4. M. Sumetsky, “Optical fiber microcoil resonator,” Opt. Express , **12**, 2303–2316 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2303. [CrossRef] [PubMed]

5. K. J. Vahala, “Optical microcavities,” Nature , **424**, 839–846 (2003). [CrossRef] [PubMed]

*Q*whispering gallery mode microcavities, i.e.

*Q*~10

^{10}[5

5. K. J. Vahala, “Optical microcavities,” Nature , **424**, 839–846 (2003). [CrossRef] [PubMed]

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]

2. G. Brambilla, V. Finazzi, and D. J. Richardson, “Ultra-low-loss optical fiber nanotapers,” Opt. Express , **12**, 2258–2263 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2258. [CrossRef] [PubMed]

3. S. G. Leon-Saval, T. A. Birks, W. J. Wadsworth, and P. St.J. Russell, “Supercontinuum generation in submicron fibre waveguides,” Opt. Express , **12**, 2864–2869 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-13-2864. [CrossRef] [PubMed]

6. M. Sumetsky, Y. Dulashko, J. M. Fini, A. Hale, and D. J. DiGiovanni, “Demonstration of the microfiber loop optical resonator,” Optical Fiber Communication Conference, Postdeadline papers, Paper PDP10, Anaheim (2005), http://www.ofcnfoec.org/materials/PDP10.pdf.

7. M. Sumetsky, Y. Dulashko, J. M. Fini, and A. Hale, “Optical microfiber loop resonator,” Appl. Phys. Lett. **86**, 161108 (2005). [CrossRef]

4. M. Sumetsky, “Optical fiber microcoil resonator,” Opt. Express , **12**, 2303–2316 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2303. [CrossRef] [PubMed]

8. C. K. Madsen, S. Chandrasekhar, E. J. Laskowski, M. A. Cappuzzo, J. Bailey, E. Chen, L. T. Gomez, A. Griffin, R. Long, M. Rasras, A. Wong-Foy, L. W. Stulz, J. Weld, and Y. Low, “An integrated tunable chromatic dispersion compensator for 40 Gb/s NRZ and CSRZ,” Optical Fiber Communication Conference, Postdeadline papers, Paper FD9, Anaheim (2002).

9. G. Bourdon, G. Alibert, A. Beguin, B. Bellman, and E. Guiot, “Ultralow Loss Ring Resonators Using 3.5% Index-Contrast Ge-Doped Silica Waveguides,” IEEE Photon. Technol. Lett. **15**, 709–711 (2003). [CrossRef]

10. B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very High-Order Microring Resonator Filters for WDM Applications,” IEEE Photon. Technol. Lett , **16**, 2263–2265 (2004). [CrossRef]

11. J. Niehusmann, A. Vörckel, P. H. Bolivar, T. Wahlbrink, W. Henschel, and H. Kurz, “Ultrahigh-quality-factor silicon-on-insulator microring resonator,” Opt. Lett. **29**, 2861–2863 (2004). [CrossRef]

12. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. , **24**, 711–713 (1999). [CrossRef]

13. J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, “Designing coupled-resonator optical waveguide delay lines,” J. Opt. Soc. Am. B **21**, 1665–1672 (2004). [CrossRef]

14. J. E. Heebner, R. W. Boyd, and Q-H. Park, “SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides,” J. Opt.Soc. Am. B **19**, 722–731 (2002). [CrossRef]

*N*turns. It is shown that the spectral and transmission properties of the uniform COR are unexpectedly different from those of the known types of resonators and photonic crystal structures. The analytical expression for the transmission amplitude of a uniform COR is derived in section 2. In section 3, the transmission time delay of a COR is described as a function of dimensionless propagation constant,

*B*, and interturn coupling parameter,

*K*. It is shown that, for certain discrete sequences of these parameters, the light is completely trapped by the resonator. For

*N*→∞, the transmission time delay profile in the plane (

*B, K*) experiences a fractal collapse to the points with coordinates

*K*=½ and

*B*=

*π*(2

*n*-½), where

*n*is an integer. In section 4, the dispersion relation and transmission properties of a long COR are considered. It is shown that the COR with many turns (

*N*→∞) has no stop bands. It is also shown that the coupling parameter

*K*=½ corresponds to the crossover between two regimes of propagation: with (

*K*>½) and without (

*K*<½) zeroing of the group velocity. At the crossover point

*K*=½, the group velocity has a second order zero (i.e. the group velocity becomes zero simultaneously with the inversed group velocity dispersion). At

*K*<½, the behavior of a COR waveguide resembles that of a SCISSOR, whereas at

*K*>½, it resembles the behavior of a CROW.

## 2. Solution of coupled wave and continuity equations for a uniform COR

*A*(

*s*) exp(

*iβs*)

*F*(

*x, y*) exp(

*iωt*), where s is the coordinate along the microfiber,

*x*and

*y*are the transversal coordinates,

*t*is the time,

*β*is the propagation constant along the microfiber, and

*ω*is the frequency. It is convenient to define the amplitude of the field at a turn

*m*as

*A*

_{m}(

*s*) and to consider s as the common coordinate along turns, so that 0<

*s*<

*S*, where

*S*is the length of the turn. The propagation of light along the coil is described by the coupled wave equations [4

4. M. Sumetsky, “Optical fiber microcoil resonator,” Opt. Express , **12**, 2303–2316 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2303. [CrossRef] [PubMed]

*κ*is the coupling coefficient between adjacent turns. In Eq. (1),

*A*

_{1}(0) represents the input wave at the beginning of the COR and

*AN*(

*S*) represents the output wave at the end of the COR. In addition to Eq. (1), the following continuity condition should be taken into account:

*n*

_{eff}is the effective refractive index of the microfiber and

*c*is the speed of light. For the lossless COR considered below, |

*T*|=1. Eq. (3) shows that the optical properties of COR depend on three dimensionless parameters: the number of turns,

*N*, the dimensionless propagation constant,

*B*=

*βS*, and the coupling parameter,

*K*=

*κS*. The parameters

*B*and

*K*, which define the eigenmodes of the COR can be determined from the condition

*t*

_{d}=∞.

## 3. Time delay resonances and spectral behavior of a COR

*N*=2) is physically similar to the transmission spectra of the loop and ring resonators [4

**12**, 2303–2316 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2303. [CrossRef] [PubMed]

*κS*=(

*π*/2)(2

*m*-1), and (2) constructive interference between the electromagnetic field at the adjacent turns,

*βS*=(

*π*/2)(4

*n*-2

*m*+1), where

*m*and

*n*are integers. If these conditions are met, the light in the input and output waveguides of the COR is completely separated from the light in its coil-shaped section. During the roundtrip, the light transfers from one turn to another 2

*m*-1 times and fully returns to the original turn with the same phase. In Ref. [4

**12**, 2303–2316 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2303. [CrossRef] [PubMed]

*N*>2, more than two turns are coupling simultaneously leading to complex interplay of propagation along the length of the microfiber and propagation through interturn coupling. Nevertheless, under certain conditions, the COR with

*N*>2 possesses eigenmodes. Figure 2 shows the surface relief of the time delay

*t*

_{d}(

*B*,

*K*) for the CORs with different numbers of turns,

*N*. Because

*t*

_{d}(

*B*+2

*π,K*)=

*t*

_{d}(

*B*,

*K*), only one full period of

*t*

_{d}(

*B*,

*K*), 2

*πn*<

*B*<2

*π*(

*n*+1), where

*n*is an integer, is shown. From these plots, the spectrum of the COR with the fixed coupling parameter,

*K*, is determined along a vertical line with

*K*=const, as illustrated in Fig. 3. Figure 3(a) shows the time delay dependences on

*B*of the COR with

*N*=4 for

*K*=0.5, 1.5, 2.5, and 3.5. Fig. 3(b) and (c) compare these dependencies with the characteristic time delay spectrum of a CROW with 4 turns and a SCISSOR. The spectra of an

*N*-ring CROW and a SCISSOR have, respectively, the characteristic

*N*peaks and a single peak in the interval of periodicity. However, the shape of the COR spectrum qualitatively depends on the value of

*K*. The COR time delay spectral dependencies shown in Fig. 3(a) can be understood by comparing these dependences with the corresponding cross-sections of the

*N*=4 plot in Fig. 2.

*K*, only for

*N*=2 and

*N*=3. The periodicity for

*N*=2 follows from the discussion in the beginning of this section, whereas for

*N*=3 an intuitive explanation of the periodicity has not been found. For

*N*≥4, the dependence on coupling parameter,

*K*, is no longer periodic. In all plots, the dark (infinitely small) points breaking the light lines correspond to the eigenvalues of COR. For each

*N*the eigenvalues ues (

*K*

^{(N)}

_{g}) can be described by a pair of integers

**g**=(

*g*

_{1},

*g*

_{2}), where

*g*

_{1}counts the eigenvalues along the

*B*-axis and

*g*

_{2}- along the

*K*-axis. For any number of turns,

*N*, the COR with fixed coupling parameter,

*K*, has no more than two eigenvalues. This means that the first number of an eigenmode can have only two values,

*g*

_{1}=1,2. However, for the fixed

*K*, the number of resonances in the time delay may be as high as

*N*. The second number of an eigenmode,

*g*

_{2}, grows with

*K*from 1 to infinity. The analytical behavior of the all-pass transmission amplitude in the neighborhood of an eigenvalue,

*)*, is described as

*T(B,K)*≈(

*B*-

*(K))*/(

*B*-

*Γ*

^{(N)}*

_{g}

*(K)*), where

*(K)*=

*)*+

*d*K g ( N ) )and

*(N)*_{g}(K-^{2}*T(B,K)*and the causality arguments, it follows that the imaginary component of

*(K)*does not contain a term linear in

*K*-

*)*=0. Consequently, the singularities of the time delay,

*t*

_{d}(

*B,K*), in the neighborhood of the COR eigenvalues have the form:

*B*and

*K*. In particular, the condition

*N*, which is shown in the lower rows of plots in Fig. 2. All features of the spectrum including the eigenvalues which appear for some values of

*N*in the lower row, do not disappear for larger

*N*but rather move along the straight axial lines towards the point of spectral collapse, (

*B*

_{c}

*,K*

_{c})=(

*π*(2

*n*-1/2), 1/2), (

*n*is an integer). The size of these features shrinks proportionally to the distance from the collapse point. Figure 2 indicates the fractal behavior of spectrum for

*N*→∞, which, in contrast to the spectrum behavior of other types of periodic structures for

*N*→∞ [12

12. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. , **24**, 711–713 (1999). [CrossRef]

13. J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, “Designing coupled-resonator optical waveguide delay lines,” J. Opt. Soc. Am. B **21**, 1665–1672 (2004). [CrossRef]

*B*

_{c}

*,K*

_{c}). Consequently, for

*N*→∞ any series of the eigenvalues, (

*)*, with fixed g tends to (

*B*

_{c}

*,K*

_{c}).

*N*≫1, the scaling law of the collapse can be determined as follows. Note that the number of identical features in the lower row in Fig. 2 along the

*K*-axis is proportional to

*N*. At the same time, the length of an individual feature,

*B*

_{c}

*,K*

_{c}), grows proportional to its number q:

*C*

^{(N)}

*q*. Here

*C*

^{(N)}is the characteristic size of the smallest feature near the collapse point. Therefore, the total length of identical features in Fig. 2 is proportional to

*C*

^{(N)}

*q*~

*C*

^{(N)}

*N*

^{2}. On the other hand, Fig. 2 also suggests that

*C*

^{(N)}

*q*~1. The latter yields the scaling law for characteristic size of features in the neighborhood of the collapse point as

*C*

^{(N)}

*~N*

^{-2}. In particular, all eigenvalues approach the collapse point 2

*K*

_{c}=½ according to the law

*-Kc~N*

^{-2}. Fig. 4 numerically confirms the

*N*

^{-2}law of collapse. It shows the behavior of time delay in the neighborhood of a collapse point rescaled with the scaling factor of

*C/N*

^{2}for

*N*=5,10,15,20,25, and 30. For better visualization, the upper left points of the V-shaped features are connected with the lines. The

*N*

^{-2}law is manifested by the fact that, for large

*N*, these lines become horizontal. Table 1 shows the magnitudes of the smallest eigenvalue coupling parameters,

*N*. The

*N*

^{-2}law is specified for this series as

*g*

_{1}=1,2 and

*g*

_{2}=1, 2, 3, 4, 5 for the COR with

*N*=1,2,3,4,10 is shown in Fig. 5. For each

*N*, the plot of eigenmodes is places next to the corresponding distribution of the time delay (compare with Fig. 2). The arrows in Fig. 5 connect the eigenmodes with the corresponding eigenvalues. Interestingly, the spatial distribution of the eigenmodes is smooth and has no correlation with the length of the turn,

*S*. The eigenvalues having equal values of coupling parameter,

*K*, correspond to the eigenmodes with similar spatial dependences.

## 3. Dispersion relation for infinite COR

*N*→∞, the first and last equations in Eq. (1) can be ignored. Then Eq. (1) has a partial solution

*s*)=exp[2

*iκ*cos(

*ξS*)

*s*±

*iξSm*], where the integer

*m*is a number of a turn and

*ξ*is an effective propagation constant. The continuity condition, Eq. (2), applied to this solution leads to the dispersion relation:

*ω=cβ/n*

_{eff}is the frequency of electromagnetic field. Eq. (7) indicates that the coil optical waveguide does not have stop bands. From this equation, three qualitatively different situations occur depending on the value of coupling parameter,

*K*. If

*K*<½, the dispersion relation is monotonic. This case is illustrated in Fig. 6(a1) and is qualitatively similar to the dispersion relation for a SCISSOR, Fig. 6(c) [14

14. J. E. Heebner, R. W. Boyd, and Q-H. Park, “SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides,” J. Opt.Soc. Am. B **19**, 722–731 (2002). [CrossRef]

*K*=½. The corresponding dispersion relation is shown in Fig. 6(a2). In this case, the function

*ω(ξ)*has inflection points at

*n*is an integer. In the vicinity of these points,

*ω*(

*ξ*)≈

*ω*(

*ξ*

_{n})+

*α*(

*ξ-ξ*

_{n})

^{3}and the group velocity is zeroing simultaneously with the inverse group velocity dispersion. Similar situation has been recently investigated in photonic crystal waveguides [15

15. A. Figotin and I. Vitebskiy, “Electromagnetic unidirectionality in magnetic photonic crystals,” Phys. Rev. B **67**, 165210 (2003). [CrossRef]

*K*>½, the dispersion relation has the minima and maxima points corresponding to zero group velocity. Propagation of light in the neighborhood of these points is similar to propagation near the band edges of photonics crystals and CROWs. This dispersion relation is illustrated in Fig. 6(b) [12

12. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. , **24**, 711–713 (1999). [CrossRef]

13. J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, “Designing coupled-resonator optical waveguide delay lines,” J. Opt. Soc. Am. B **21**, 1665–1672 (2004). [CrossRef]

## 4. Summary

*B,K*). For

*N*→∞, the spectral characteristics of the COR experience the fractal collapse in the plane (

*B,K*) to the points (

*π*(2

*n*-½),½), where

*n*is an integer. The dispersion relation of an infinitely long COR shows that the group velocity has the second order zero in the collapse points. Depending on the value of the coupling parameter, the COR may behave similarly to the CROW (

*K*>½) or SCISSOR (

*K*<½).

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]

2. G. Brambilla, V. Finazzi, and D. J. Richardson, “Ultra-low-loss optical fiber nanotapers,” Opt. Express , **12**, 2258–2263 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2258. [CrossRef] [PubMed]

**12**, 2864–2869 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-13-2864. [CrossRef] [PubMed]

6. M. Sumetsky, Y. Dulashko, J. M. Fini, A. Hale, and D. J. DiGiovanni, “Demonstration of the microfiber loop optical resonator,” Optical Fiber Communication Conference, Postdeadline papers, Paper PDP10, Anaheim (2005), http://www.ofcnfoec.org/materials/PDP10.pdf.

7. M. Sumetsky, Y. Dulashko, J. M. Fini, and A. Hale, “Optical microfiber loop resonator,” Appl. Phys. Lett. **86**, 161108 (2005). [CrossRef]

**12**, 2303–2316 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2303. [CrossRef] [PubMed]

## References and links

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 , |

2. | G. Brambilla, V. Finazzi, and D. J. Richardson, “Ultra-low-loss optical fiber nanotapers,” Opt. Express , |

3. | S. G. Leon-Saval, T. A. Birks, W. J. Wadsworth, and P. St.J. Russell, “Supercontinuum generation in submicron fibre waveguides,” Opt. Express , |

4. | M. Sumetsky, “Optical fiber microcoil resonator,” Opt. Express , |

5. | K. J. Vahala, “Optical microcavities,” Nature , |

6. | M. Sumetsky, Y. Dulashko, J. M. Fini, A. Hale, and D. J. DiGiovanni, “Demonstration of the microfiber loop optical resonator,” Optical Fiber Communication Conference, Postdeadline papers, Paper PDP10, Anaheim (2005), http://www.ofcnfoec.org/materials/PDP10.pdf. |

7. | M. Sumetsky, Y. Dulashko, J. M. Fini, and A. Hale, “Optical microfiber loop resonator,” Appl. Phys. Lett. |

8. | C. K. Madsen, S. Chandrasekhar, E. J. Laskowski, M. A. Cappuzzo, J. Bailey, E. Chen, L. T. Gomez, A. Griffin, R. Long, M. Rasras, A. Wong-Foy, L. W. Stulz, J. Weld, and Y. Low, “An integrated tunable chromatic dispersion compensator for 40 Gb/s NRZ and CSRZ,” Optical Fiber Communication Conference, Postdeadline papers, Paper FD9, Anaheim (2002). |

9. | G. Bourdon, G. Alibert, A. Beguin, B. Bellman, and E. Guiot, “Ultralow Loss Ring Resonators Using 3.5% Index-Contrast Ge-Doped Silica Waveguides,” IEEE Photon. Technol. Lett. |

10. | B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very High-Order Microring Resonator Filters for WDM Applications,” IEEE Photon. Technol. Lett , |

11. | J. Niehusmann, A. Vörckel, P. H. Bolivar, T. Wahlbrink, W. Henschel, and H. Kurz, “Ultrahigh-quality-factor silicon-on-insulator microring resonator,” Opt. Lett. |

12. | A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. , |

13. | J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, “Designing coupled-resonator optical waveguide delay lines,” J. Opt. Soc. Am. B |

14. | J. E. Heebner, R. W. Boyd, and Q-H. Park, “SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides,” J. Opt.Soc. Am. B |

15. | A. Figotin and I. Vitebskiy, “Electromagnetic unidirectionality in magnetic photonic crystals,” Phys. Rev. B |

**OCIS Codes**

(060.2340) Fiber optics and optical communications : Fiber optics components

(190.0190) Nonlinear optics : Nonlinear optics

(230.5750) Optical devices : Resonators

(230.7370) Optical devices : Waveguides

**ToC Category:**

Research Papers

**History**

Original Manuscript: March 30, 2005

Revised Manuscript: May 18, 2005

Published: May 30, 2005

**Citation**

M. Sumetsky, "Uniform coil optical resonator and waveguide: transmission spectrum, eigenmodes, and dispersion relation," Opt. Express **13**, 4331-4340 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-11-4331

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### References

- 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]
- G. Brambilla, V. Finazzi, and D. J. Richardson, �??Ultra-low-loss optical fiber nanotapers,�?? Opt. Express, 12, 2258- 2263 (2004) , <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2258">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2258</a> [CrossRef] [PubMed]
- S. G. Leon-Saval, T. A. Birks, W. J. Wadsworth and P. St.J. Russell, �??Supercontinuum generation in submicron fibre waveguides,�?? Opt. Express, 12, 2864-2869 (2004), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-13-2864">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-13-2864</a>. [CrossRef] [PubMed]
- M. Sumetsky, �??Optical fiber microcoil resonator,�?? Opt. Express, 12, 2303-2316 (2004), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2303">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2303</a>. [CrossRef] [PubMed]
- K. J. Vahala, �??Optical microcavities,�?? Nature, 424, 839-846 (2003). [CrossRef] [PubMed]
- M. Sumetsky, Y. Dulashko, J. M. Fini, A. Hale, and D. J. DiGiovanni, �??Demonstration of the microfiber loop optical resonator,�?? Optical Fiber Communication Conference, Postdeadline papers, Paper PDP10, Anaheim (2005), <a href= "http://www.ofcnfoec.org/materials/PDP10.pdf">http://www.ofcnfoec.org/materials/PDP10.pdf</a>.
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