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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 13 — Jul. 1, 2013
  • pp: 15268–15279
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A SNAP coupled microresonator delay line

M. Sumetsky  »View Author Affiliations


Optics Express, Vol. 21, Issue 13, pp. 15268-15279 (2013)
http://dx.doi.org/10.1364/OE.21.015268


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Abstract

A delay line fabricated of a chain of SNAP (Surface Nanoscale Axial Photonics) coupled microresonators is demonstrated. In contrast to resonant delay lines demonstrated to date, the slow light in this structure is enhanced by the 2R (Rotation + Reflection) effect realized due to the 3D propagation of light along the surface of a SNAP fiber. Here, the delay line coupled to a single input/output waveguide (i.e., operating in the reflection mode) is considered. Depending on the coupling parameters and loss, the delay time in this device is either proportional to the density of resonances averaged over the pulse spectrum or tends to zero. The delay line is fabricated of 20 coupled microresonators with the total length of 1.2 mm and footprint area of 0.05 mm2. It exhibits the record low insertion loss (< 3 dB), small speed of light (<c/250), and large (>1 ns) delay time along the 0.1 nm bandwidth achieved for the miniature microresonator delay lines. The feasibility of significant improvement of the SNAP delay line characteristics (larger delay time and bandwidth, smaller losses and dimensions, and anti-reflecting apodization) is discussed.

© 2013 OSA

1. Introduction

A general method to reduce the effective speed of light in a solid optical material is based on modulation of its refractive index [1

1. J. B. Khurgin, “Slow light in various media: a tutorial,” Adv. Opt. Photon. 2(3), 287–318 (2010). [CrossRef]

]. Modulation introduces multiple turns and, on average, sets light to propagate slower. A common structure exhibiting the slow light of this kind is a chain of coupled microresonators [2

2. F. Morichetti, C. Ferrari, A. Canciamilla, and A. Melloni, “The first decade of coupled resonator optical waveguides: bringing slow light to applications,” Laser Photonics Rev. 6(1), 74–96 (2012). [CrossRef]

,3

3. M. Notomi, “Strong light confinement with periodicity,” Proc. IEEE 99(10), 1768–1779 (2011). [CrossRef]

]. The performance of this structure obeys the general limitation of linear photonic systems: the time delay within the predetermined frequency bandwidth has the fundamental upper limit per resonance [4

4. G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron. 37(4), 525–532 (2001). [CrossRef]

]. To increase the delay within the desired bandwidth, it is necessary to increase the density of resonances and, hence, to boost the structure dimensions. For example, to keep the propagation bandwidth and increase the delay time, the number of elements in a chain of coupled microresonators needs to be increased proportionally.

The dimensions of a delay line with large number of coupled microresonators can still be very small for photonic crystal circuits where the individual microresonator (MR) diameter is as small as a few microns [3

3. M. Notomi, “Strong light confinement with periodicity,” Proc. IEEE 99(10), 1768–1779 (2011). [CrossRef]

,5

5. M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nat. Photonics 2(12), 741–747 (2008). [CrossRef]

] and for the ring microresonator photonic circuits where the MR diameter is as small as a few tens of microns [2

2. F. Morichetti, C. Ferrari, A. Canciamilla, and A. Melloni, “The first decade of coupled resonator optical waveguides: bringing slow light to applications,” Laser Photonics Rev. 6(1), 74–96 (2012). [CrossRef]

,6

6. F. N. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1(1), 65–71 (2007). [CrossRef]

]. A significant progress in the improvement of the fabrication precision and reduction of the propagation loss of these structures has been achieved. For example, the standard deviation of the perimeter of silicon MRs fabricated in [6

6. F. N. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1(1), 65–71 (2007). [CrossRef]

] is ~12 nm corresponding to the 0.4 nm standard deviation in their resonance spectra. Similar precision is achieved in the fabrication of photonic crystal MR chains [5

5. M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nat. Photonics 2(12), 741–747 (2008). [CrossRef]

]. However, the enormously high precision and small attenuation of light achieved are still insufficient for the practical realization of these delay lines [7

7. M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, “Statistics of light transport in 235-ring silicon coupled-resonator optical waveguides,” Opt. Express 18(25), 26505–26516 (2010). [CrossRef]

].

As compared to the lithographic fabrication technology, the recently introduced SNAP (Surface Nanoscale Axial Photonics) platform [8

8. M. Sumetsky, “Localization of light in an optical fiber with nanoscale radius variation,” in CLEO/Europe and EQEC 2011 Conference Digest, postdeadline paper PDA_8.

14

14. M. Sumetsky and Y. Dulashko, “SNAP: Fabrication of long coupled microresonator chains with sub-angstrom precision,” Opt. Express 20(25), 27896–27901 (2012). [CrossRef]

] allows fabricating photonic circuits that surpass silicon photonics by two orders of magnitude in both the fabrication accuracy and attenuation. An example of a SNAP device is illustrated in Fig. 1
Fig. 1 Illustration of a SNAP device.
. In this device, light coupled from a microfiber (micron-diameter waist of a biconical fiber taper connected to the light source and detector) experiences the resonant propagation along the SNAP fiber. The direction of propagation is primarily transverse to the fiber axis such that the axial speed of light is very small. Due to the self-interference in the process of rotation along the SNAP fiber surface, light exhibits complex resonant transmission spectra determined by the nanoscale variation of the SNAP fiber radius [8

8. M. Sumetsky, “Localization of light in an optical fiber with nanoscale radius variation,” in CLEO/Europe and EQEC 2011 Conference Digest, postdeadline paper PDA_8.

, 9

9. M. Sumetsky and J. M. Fini, “Surface nanoscale axial photonics,” Opt. Express 19(27), 26470–26485 (2011). [CrossRef]

, 13

13. M. Sumetsky, “Theory of SNAP devices: basic equations and comparison with the experiment,” Opt. Express 20(20), 22537–22554 (2012), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-20-20-22537. [CrossRef]

].

In contrast to the regular photonic waveguides, the axial propagation of light along the SNAP fiber is initially slow because of its transverse rotation along the fiber surface. The supplementary introduction of a periodic modulation of the fiber radius (a MR chain) allows one to slow down the effective axial speed further by multiple reflections along the axial direction. Therefore, the axial speed of light in a SNAP coupled MR chain is reduced due to the 2R (Rotation + Reflection) effect. This effect is made possible due to the 3D propagation of light along the fiber surface. The SNAP coupled MR structure is a special case of the vertically coupled resonator (VCR) structure introduced and investigated in [15

15. A. B. Matsko, A. A. Savchenkov, and L. Maleki, “Vertically coupled whispering-gallery-mode resonator waveguide,” Opt. Lett. 30(22), 3066–3068 (2005). [CrossRef]

, 16

16. M. Sumetsky, “Vertically-stacked multi-ring resonator,” Opt. Express 13(17), 6354–6375 (2005). [CrossRef]

]. As opposed to the previously demonstrated SCISSOR and CROW delay lines [2

2. F. Morichetti, C. Ferrari, A. Canciamilla, and A. Melloni, “The first decade of coupled resonator optical waveguides: bringing slow light to applications,” Laser Photonics Rev. 6(1), 74–96 (2012). [CrossRef]

,17

17. 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(4), 722–731 (2002). [CrossRef]

,18

18. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24(11), 711–713 (1999).

], which are basically 2D structures, realization of VCR are possible only in 3D.

Section 2 of this paper theoretically considers the general properties of the group delay of a SNAP device coupled to a single waveguide (microfiber). It is shown that, depending on the sign of the introduced parameter, called the group delay identifier (GDI), the average group delay is either proportional to the density of resonances or equals zero. In Section 3, the delay line consisting of 20 coupled SNAP MRs is fabricated and characterized. In Section 4, it is shown that, for the coupling parameters introduced, the average group delay of this device vanishes in agreement with the negative sign of its GDI. Next, the coupling parameters are tuned to approach positive GDI. As the result, the record large for miniature coupled resonator chains average group delay exceeding 1 ns with the record low insertion loss < 3 dB is demonstrated. The experimental results are in a good agreement with theory. In Section 5, the measured transmission amplitude data is used for calculation of the pulse propagation delay time, which is in a good agreement with average delay time and theory. Section 6 summarizes and discusses the results obtained.

2. Average group delay in a resonance structure “in reflection”

The group delay in a photonic structure is determined through the derivative of the phase of the transmission amplitude S with respect to the frequency ν (or wavelength λ=c/ν) as [4

4. G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron. 37(4), 525–532 (2001). [CrossRef]

]
τ=12πIm(1SSν)=λ22πcIm(1SSλ)
(1)
where c is the speed of light in vacuum and n is the material refractive index. For example, in the vicinity of a well defined resonance ν=ν0 of an all-pass MR coupled to a single waveguide, the transmission amplitude can be approximately expressed through the attenuation and coupling parameters, σa and σc(see e.g., [19

19. V. S. Ilchenko and A. B. Matsko, “Optical resonators with whispering-gallery modes - Part II: Applications,” IEEE J. Sel. Top. Quantum Electron. 12(1), 15–32 (2006). [CrossRef]

]):
S(ν)=νν0i(σaσc)νν0i(σa+σc).
(2)
Then the group delay
τ(ν)=2σc[(νν0)2σa2+σc2][(νν0)2+σa2+σc2]24σa2σc2.
(3)
From here, the delay-bandwidth product (DBP)
τ(ν)dν={1ifσa<σc,0ifσa>σc.
(4)
Thus, for a relatively small loss, σa<σc, the group delay τ(ν) found from Eq. (3) is always positive and the DBP approaches its maximum value 1. However, for a larger loss, σa>σc, the group delay can be both positive and negative, while the DBP is zero (Fig. 2
Fig. 2 Group delay as a function of frequency in the vicinity of an all-pass resonance calculated from Eq. (3) for σc = 2σa corresponding to DBP = 1 (upper blue curve) and for σc = 0.5σa corresponding to DBP = 0 (lower red curve).
).

The single-resonance propagation through the all-pass MR is a simplest case of resonant propagation “in reflection” since light enters and exits the resonator through the same waveguide. The effect similar to that described by Eq. (4) holds for more complex structures with many resonances. In the case of our concern, the transmission amplitude of a SNAP device coupled to a microfiber (Fig. 1) is [13

13. M. Sumetsky, “Theory of SNAP devices: basic equations and comparison with the experiment,” Opt. Express 20(20), 22537–22554 (2012), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-20-20-22537. [CrossRef]

]
S(λ,z1)=S0i|C|2G0(λ,z1,z1)1+DG0(λ,z1,z1)
(5)
where S0 is the out-of-resonance transmission amplitude, C and D are the microfiber/SNAP fiber coupling parameters, and G0(λ,z1,z2) is the Green’s function of the Schrödinger equation
Ψzz+(E(λ)V(z))Ψ=0,E(λ)=κλλresiγλres,V(z)=κΔreff(z)r0,κ=2(2πnλres)2,
(6)
which describes the propagation of light along the SNAP fiber with radius r0 refractive index n, and effective radius variation (ERV) Δreff(z) near the resonance wavelength λ=λres [8

8. M. Sumetsky, “Localization of light in an optical fiber with nanoscale radius variation,” in CLEO/Europe and EQEC 2011 Conference Digest, postdeadline paper PDA_8.

, 9

9. M. Sumetsky and J. M. Fini, “Surface nanoscale axial photonics,” Opt. Express 19(27), 26470–26485 (2011). [CrossRef]

, 13

13. M. Sumetsky, “Theory of SNAP devices: basic equations and comparison with the experiment,” Opt. Express 20(20), 22537–22554 (2012), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-20-20-22537. [CrossRef]

]. Parameter γ in Eq. (6) determines attenuation of light. From Eqs. (1) and (3), the DBP along the spectral interval which contains N resonances ν1,ν2,...,νN is
ν1<ν<νN+1τ(ν)dν=n=1Nνnνn+1dνGν[1[G+S0(S0Di|C|2)1]1(G+D1)]=NdG[1[G+S0(S0Di|C|2)1]1(G+D1)]={NifΛ>00ifΛ<0
(7)
where
Λ=|C|2|S0|2Re(S0)ImD
(8)
Parameter Λ defined by Eq. (8) is referred to as a group delay identifier (GDI), since the sign of this parameter determines whether the average group delay is proportional to the density of resonances or equals zero. Since the integral 2πντ(ν)dν determines the phase of the transmission amplitude, it is reasonable to suggest that the delay experienced by a pulse propagating along the resonant SNAP device shown in Fig. 1 can be either proportional to the density of resonances for Λ>0 or negligible for Λ<0. Introducing the density of states ρ(ν) defined by the equation νρ(ν)dν=N(ν), we get from Eq. (7) the Krein-Friedel-Loyd formula (see e.g [20

20. 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).

].), τ(ν)=ρ(ν)which is valid, though, for Λ>0 only.

The SNAP resonant delay line is determined by the shape of the potential well V(z) in Eq. (6), or, equivalently, by the ERV Δreff(z). Its design consists of seeking for the ERV, which produces the desired transmission power and delay time spectrum, using Eqs. (1), (5), and (6). One way to increase the density of states and, thus, the average delay time in a quantum well is to increase its length. Alternatively, the density of states and the delay time can be increased by modulation of the potential in a quantum well. The latter, though, can be achieved at the expense of the width of the transmission band. While the investigation of a quantum well delay line of a general shape is of a special interest, here we consider the simplest case of a quantum well with a periodically modulated potential.

3. Fabrication of the coupled resonator chain

A critical factor limiting the design of miniature delay lines demonstrated to date [2

2. F. Morichetti, C. Ferrari, A. Canciamilla, and A. Melloni, “The first decade of coupled resonator optical waveguides: bringing slow light to applications,” Laser Photonics Rev. 6(1), 74–96 (2012). [CrossRef]

,3

3. M. Notomi, “Strong light confinement with periodicity,” Proc. IEEE 99(10), 1768–1779 (2011). [CrossRef]

,5

5. M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nat. Photonics 2(12), 741–747 (2008). [CrossRef]

7

7. M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, “Statistics of light transport in 235-ring silicon coupled-resonator optical waveguides,” Opt. Express 18(25), 26505–26516 (2010). [CrossRef]

] was the fabrication precision, typically in excess of several nm [6

6. F. N. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1(1), 65–71 (2007). [CrossRef]

,7

7. M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, “Statistics of light transport in 235-ring silicon coupled-resonator optical waveguides,” Opt. Express 18(25), 26505–26516 (2010). [CrossRef]

]. Since the fabrication precision in SNAP can be two orders of magnitude better [14

14. M. Sumetsky and Y. Dulashko, “SNAP: Fabrication of long coupled microresonator chains with sub-angstrom precision,” Opt. Express 20(25), 27896–27901 (2012). [CrossRef]

] and attenuation of light two orders of magnitude smaller [11

11. M. Sumetsky, K. Abedin, D. J. DiGiovanni, Y. Dulashko, J. M. Fini, X. Liu, and E. M. Monberg, “Coupled high Q-factor surface nanoscale axial photonics (SNAP) microresonators,” Opt. Lett. 37(6), 990–992 (2012). [CrossRef]

,21

21. M. Pöllinger, D. O’Shea, F. Warken, and A. Rauschenbeutel, “Ultrahigh-Q tunable whispering-gallery-mode microresonator,” Phys. Rev. Lett. 103(5), 053901 (2009). [CrossRef]

], it is possible to realize miniature delay lines with much larger delay times and much smaller losses. The design of a SNAP delay line presented below, though, takes into account the resolution of the measurement device used (Luna Optical Vector Analyzer (OVA), 1.3 pm spectral resolution), which limits the detected group delay time by the value much smaller than 6 ns. This limitation can be released with a better spectral resolution or with a pulse propagation measurement.

The MR chain was designed [13

13. M. Sumetsky, “Theory of SNAP devices: basic equations and comparison with the experiment,” Opt. Express 20(20), 22537–22554 (2012), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-20-20-22537. [CrossRef]

] to have a transmission bandwidth ~0.1 nm (12.5 GHz). To reconcile with the measurement resolution, the number of resonators was set to 20, such that the average group delay of 20 resonances distributed along the 12.5 GHz bandwidth was expected to be ~20/12.5 GHz ~1-2 ns, i.e., the value that can be resolved with the Luna OVA resolution at 1550 nm equal to1.264 pm = 0.158 GHz = 1/(6.31ns).

Fabrication of the MR chain followed the iterative approach described in [14

14. M. Sumetsky and Y. Dulashko, “SNAP: Fabrication of long coupled microresonator chains with sub-angstrom precision,” Opt. Express 20(25), 27896–27901 (2012). [CrossRef]

]. In brief, the MRs were introduced by periodic local annealing of the silica fiber with radius r0=19μm by a focused CO2 laser beam. Annealing of the fiber led to a release of the tension, which was frozen-in in the process of fiber drawing. Consequently, this led to variation of the ERV by the nanometer-scale value determined by the laser beam power [10

10. M. Sumetsky, D. J. DiGiovanni, Y. Dulashko, J. M. Fini, X. Liu, E. M. Monberg, and T. F. Taunay, “Surface nanoscale axial photonics: robust fabrication of high-quality-factor microresonators,” Opt. Lett. 36(24), 4824–4826 (2011). [CrossRef]

]. The axial interval between the laser exposures (i.e., between MRs) was set to 60 μm. After characterization of the MR chain described below, it was corrected iteratively to arrive at the MR effective radii equal to each other with the accuracy of 1Å.

The SNAP fiber was characterized using the microfiber scan method [22

22. T. A. Birks, J. C. Knight, and T. E. Dimmick, “High-resolution measurement of the fiber diameter variations using whispering gallery modes and no optical alignment,” IEEE Photon. Technol. Lett. 12(2), 182–183 (2000). [CrossRef]

, 23

23. M. Sumetsky and Y. Dulashko, “Radius variation of optical fibers with angstrom accuracy,” Opt. Lett. 35(23), 4006–4008 (2010). [CrossRef]

]. To this end, an adiabatic biconical fiber taper with a single-mode micrometer diameter waist (microfiber) was fabricated. The taper ends were connected to the Luna OVA as illustrated in Fig. 1. The taper was translated along the SNAP fiber contacting with the period of 2 μm. The transmission amplitudes measured at contact points were recorded. The amplitudes exhibited resonance behavior with the free spectral range ΔλFSR=λres2/(2πnr0)14nm corresponding to the fiber radius r0=19μm, wavelength λres1550nm, and refractive index of silica n=1.45. In the case of contact to the microfiber region with the thinnest diameter, the microfiber/SNAP fiber coupling was strong and the transmission amplitude exhibited significant loss of several dB due to the increased coupling to non-resonant modes. To perform the SNAP fiber characterization [13

13. M. Sumetsky, “Theory of SNAP devices: basic equations and comparison with the experiment,” Opt. Express 20(20), 22537–22554 (2012), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-20-20-22537. [CrossRef]

,14

14. M. Sumetsky and Y. Dulashko, “SNAP: Fabrication of long coupled microresonator chains with sub-angstrom precision,” Opt. Express 20(25), 27896–27901 (2012). [CrossRef]

], the taper was translated along its axis and a contact point at the taper region with a larger diameter (roughly, 3-4 μm) was chosen. At this point, the resonant spectrum was well-pronounced while the off-resonance losses were negligible. Importantly, in the experiment of this paper the taper waist was intentionally made thinner than that required for the SNAP fiber characterization to allow for the possibility of tuning the coupling parameters performed in Section 4.

The surface plot of the transmission amplitude |S(λ,z)| in the resonance region measured as a function of wavelength, λ, and coordinate along the SNAP fiber, z, is shown in Fig. 3(a)
Fig. 3 Experimental characterization (a) and theoretical modeling (b) of the resonant transmission amplitude of the fabricated 20 coupled MR chain. The surface plots of experimental data are obtained with 2 μm resolution along the fiber axis and 1.24 pm wavelength resolution. The theoretical modeling is performed with the same spatial resolution and 0.62 pm spectral resolution. Black line is the calculated ERV.
. Figure 3(b) shows the surface plot of the transmission amplitude and ERV (black curve) determined theoretically following [13

13. M. Sumetsky, “Theory of SNAP devices: basic equations and comparison with the experiment,” Opt. Express 20(20), 22537–22554 (2012), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-20-20-22537. [CrossRef]

]. It is seen that the magnitude of the introduced periodic ERV was ~4 nm. The surface plot in Fig. 3(b) corresponds to negligible attenuation γ and the following parameters in Eq. (5):

S0=0.950.19i,|C|2=0.01μm1,D=0.03+0.03iμm1
(9)

4. Characterization of the SNAP delay line

The experimentally measured surface plot of the group delay for the device considered in Section 3 is shown in Fig. 4(a)
Fig. 4 (a) – Experimentally measured surface plot of the group delay of the device considered in Section 3. (b) Comparison of the experimental and theoretical group delay spectra along the blue dashed line in Fig. 4(a). (c) – Spectra shown in Fig. 4(b) averaged over 6 pm. (d) – The transmission amplitude spectra corresponding to the group delay spectra in Fig. 4(b) found from data depicted in Fig. 3.
. From Eqs. (9) and (8), the GDI of this device is negative, Λ=0.02μm−1, which suggests that its average group delay tends to zero. This is confirmed by comparison of a sample experimental spectrum along the dashed line in Fig. 4(a) shown in Fig. 4(b) with this spectrum averaged over 6.3 pm depicted in Fig. 4(c) (blue curves). In Figs. 4(b) and 4(c), the corresponding theoretical spectra calculated with 2 times better resolution are also shown (red curves). For completeness, Fig. 4(d) shows the corresponding experimental and theoretical spectra of the transmission amplitude (blue and red curves, respectively).

Since the parameters of this device correspond to the zero average group delay, it does not possess the required pulse delay functionality. Physically, the major part of the ingoing light pulse reflects from the microfiber/SNAP fiber coupling region and propagates along the microfiber directly rather than delays in the SNAP fiber.

The non-zero average delay is realized by tuning the coupling parameters to arrive at the condition of positive GDI. To this end, the group delay plots are measured at different contact points along the microfiber, which correspond to different microfiber diameter and, hence, to different parameters S0, C, and D in Eq. (5). As an example, Fig. 5
Fig. 5 Experimental characterization (a) and theoretical modeling (b) of the resonant transmission amplitude of the fabricated MR chain with coupling parameters defined by Eq. (10). As in Fig. 3, the experimental data is obtained with 2 μm resolution along the fiber axis and 1.24 pm wavelength resolution, while the theoretical modeling is performed with the same spatial resolution and 0.62 pm spectral resolution.
shows the experimental (a) and theoretical (b) surface plots of the transmission amplitude corresponding to the contact point at a microfiber of around 2 μm and the device with parameters
γ=0.6pm,S0=0.850.1i,|C|2=0.042μm1,D=0.021+0.024iμm1.
(10)
For these parameters, the GDI is positive, Λ=0.025μm−1, and, therefore, the group delay distribution shown in Fig. 6(a)
Fig. 6 (a) – Experimentally measured surface plot of the group delay of the MR chain with modified coupling parameters described in Section 4. (b) – Comparison of the experimental and theoretical group delay spectra along the blue dashed line in Fig. 6(a). Green bold curve – the spectrum of the pulse considered in Section 5. (c) – Spectra shown in Fig. 6(b) averaged over 6 pm. (d) – Spectra shown in Fig. 6(b) averaged over 0.03 nm. (e) – The transmission amplitude spectra corresponding to the group delay spectra in Fig. 6(b) found from data depicted in Fig. 5.
is also positive. Obviously, the performance of the fabricated delay line depends on the position of the microfiber coupled to the MR chain. Here it is assumed that the microfiber is situated close to the left hand side of the chain at the position determined by the blue dashed line in Fig. 6(a). The experimentally measured and theoretically calculated spectra at this position are shown in Fig. 6(b) (blue and red curves, respectively). The DBP calculated from these spectra is 17.2 (experimental) and 17.5 (theoretical). The deviation of these numbers from N=20 (the number of resonances in the transmission band equal to the number of MRs) predicted by Eq. (7) is due to insufficient spectral resolution of the Luna OSA near the transmission band edges. The group delay oscillations (ripple) in Fig. 6(b) are primarily caused by reflections from the chain edges. It is suggested that these oscillations can be removed by apodization (impedance matching) of the MR chain [24

24. M. Sumetsky and B. Eggleton, “Modeling and optimization of complex photonic resonant cavity circuits,” Opt. Express 11(4), 381–391 (2003). [CrossRef]

,25

25. A. Melloni, F. Morichetti, C. Ferrari, and M. Martinelli, “Continuously tunable 1 byte delay in coupled-resonator optical waveguides,” Opt. Lett. 33(20), 2389–2391 (2008). [CrossRef]

]. Figure 6(b) is followed by the same spectra averaged over 6 pm (Fig. 6(c)) and 0.03 nm (Fig. 6(d)). The averaged spectra in Fig. 6(c) and 6(d) show that the expected delay of this device should be close to 1 ns, which is confirmed in Section 5 with the pulse propagation modeling. Figure 6(e) shows the transmission amplitude spectrum corresponding to Fig. 6(b), while this spectrum averaged over 6 pm and 0.03 nm is shown in Fig. 6(f), and 6(g), respectively. Ideally, in the absence of losses, the transmission amplitude of the all pass device equals unity. In our case, the experimental transmission amplitude spectrum in Fig. 6(e) approaches the values close to unity over the whole transmission bandwidth. This suggests that the loss of this device is primarily happens at the microfiber/SNAP fiber contact area and is not expected to change significantly with increasing the number of MRs N in the chain. From Fig. 6(f) and 6(g), the average insertion loss of this device is 3 dB only.

5. Pulse propagation

The averaged delay time spectra depicted in Figs. 6(c) and 6(d) suggest that the pulse launched into the fabricated device will experience the delay time close to 1 ns. In this Section, this fact is confirmed by pulse propagation modeling. The output pulse spectrum Eout(ν) is determined from the input pulse spectrum, Ein(ν), from Eout(ν)=S(ν)Ein(ν), where S(ν) is the transmission amplitude. The relations between the input and output pulses in the time and frequency domains are

E¯in(t)=Ein(ν)exp(2iπνt)dνE¯out(t)=Eout(ν)exp(2iπνt)dν=S(ν)Ein(ν)exp(2iπνt)dν
(11)

In calculations, the input pulse Ein(ν) had the Gaussian profile with 4 GHz (0.032 nm) FWHM (Fig. 6(b), green solid curve). Figure 7(a)
Fig. 7 (a) – Pulse propagation modeling using the experimental transmission amplitude spectrum depicted in Figs. 6(b) and 6(e). Solid curve – the input pulse, dashed curve – the output pulses. The directly transmitted pulse and the pulse, which was transmitted after one reflection from the coupling region back into the MR chain, are circled. (b) – Modeling of the pulse propagation for the theoretical transmission amplitude shown in Figs. 6(b) and 6(e) (green dashed curve) and for the model of the transmission amplitude defined by Eq. (12).
shows this pulse in the time domain (green solid curve) and the corresponding time dependence of the output field, which was calculated from Eq. (11) using the experimental transmission amplitude depicted in Figs. 6(b) and 6(e). From Fig. 7(a), the largest center peak of the output field is delayed by 1.09 ns, which is in a good agreement with the average delay time found from Figs. 6(c) and 6(d). Similar good agreement is found for the theoretical transmission amplitude (dashed curve in Fig. 7(b)), which results in 1.12 ns time delay. It is seen from Figs. 7(a) and 7(b) that, due to the absence of apodization of the MR chain [24

24. M. Sumetsky and B. Eggleton, “Modeling and optimization of complex photonic resonant cavity circuits,” Opt. Express 11(4), 381–391 (2003). [CrossRef]

,25

25. A. Melloni, F. Morichetti, C. Ferrari, and M. Martinelli, “Continuously tunable 1 byte delay in coupled-resonator optical waveguides,” Opt. Lett. 33(20), 2389–2391 (2008). [CrossRef]

], a significant part of the input pulse is transmitted through the microfiber directly without delay and, also, experiences the reflection near the microfiber/SNAP fiber on the way out of the SNAP fiber and, therefore, exhibits the delay two times larger than the center pulse. Suppression of these pulses (circled in Fig. 7(a)) by modification of the chain structure is not considered here.

The results shown in Fig. 7 allow the interpretation with simple analytical modeling. If the input pulse spectrum is localized near frequency ν0 inside the transmission band of N coupled MRs then the propagation time is estimated as τ0=N/Δνb, where Δνb is the bandwidth. In the neighborhood of ν0, the transmission amplitude is approximately periodic with the period 2π/τ0=2πΔνb/N (see e.g., Fig. 6(b)). For the simplest periodic model of the transmission amplitude with harmonic group delay ripple, which also assumes positive GDI, Λ>0, and the absence of losses,
S(ν)=exp[2iπτ0(νν0)+iζcos(2πτ0(νν0))],
(12)
where parameter ζ determines the amplitude of the group delay ripple. The time delay and ripple amplitude parameters in this equation, τ0=1.12ns and ζ=0.6, were chosen to fit the output field in Fig. 7(b). The corresponding output field E¯out(mod)(t) (Fig. 7(b), yellow dotted curve) is in a reasonable agreement with the output field E¯out(t) obtained with the experimental transmission amplitude and its theoretical fit.

6. Discussion and summary

To the best of the author’s knowledge, the SNAP device demonstrated above possesses the largest delay time, smallest insertion loss, and smallest effective speed of light as compared to the previously reported miniature delay lines [2

2. F. Morichetti, C. Ferrari, A. Canciamilla, and A. Melloni, “The first decade of coupled resonator optical waveguides: bringing slow light to applications,” Laser Photonics Rev. 6(1), 74–96 (2012). [CrossRef]

,3

3. M. Notomi, “Strong light confinement with periodicity,” Proc. IEEE 99(10), 1768–1779 (2011). [CrossRef]

,5

5. M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nat. Photonics 2(12), 741–747 (2008). [CrossRef]

,6

6. F. N. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1(1), 65–71 (2007). [CrossRef]

,26

26. A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments and ultimate limits,” J. Opt. 12(10), 104008 (2010). [CrossRef]

]. Though the goal of the present paper was to demonstrate a miniature delay line based on microscopic resonators, Table 1

Table 1. Comparison of characteristics of the state of the art resonance delay lines having micron-scale [5,6,26] and millimeter-scale [25,27] elements with the SNAP delay line demonstrated in this paper.

table-icon
View This Table
compares characteristics of the fabricated device with characteristics of the state of the art delay lines consisting both of micron-scale [5

5. M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nat. Photonics 2(12), 741–747 (2008). [CrossRef]

,6

6. F. N. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1(1), 65–71 (2007). [CrossRef]

,26

26. A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments and ultimate limits,” J. Opt. 12(10), 104008 (2010). [CrossRef]

] and millimeter-scale [25

25. A. Melloni, F. Morichetti, C. Ferrari, and M. Martinelli, “Continuously tunable 1 byte delay in coupled-resonator optical waveguides,” Opt. Lett. 33(20), 2389–2391 (2008). [CrossRef]

,27

27. N. K. Fontaine, J. Yang, Z. Pan, S. Chu, W. Chen, B. E. Little, and S. J. B. Yoo, “Continuously Tunable Optical Buffering at 40 Gb/s for Optical Packet Switching Networks,” J. Lightwave Technol. 26(23), 3776–3783 (2008). [CrossRef]

] MRs. This Table classifies delay lines by their configuration (SCISSOR [17

17. 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(4), 722–731 (2002). [CrossRef]

], CROW [18

18. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24(11), 711–713 (1999).

], reflecting CROW [2

2. F. Morichetti, C. Ferrari, A. Canciamilla, and A. Melloni, “The first decade of coupled resonator optical waveguides: bringing slow light to applications,” Laser Photonics Rev. 6(1), 74–96 (2012). [CrossRef]

], and VCR [15

15. A. B. Matsko, A. A. Savchenkov, and L. Maleki, “Vertically coupled whispering-gallery-mode resonator waveguide,” Opt. Lett. 30(22), 3066–3068 (2005). [CrossRef]

,16

16. M. Sumetsky, “Vertically-stacked multi-ring resonator,” Opt. Express 13(17), 6354–6375 (2005). [CrossRef]

]), material, MR diameter, number of MRs, MR chain length, the achieved delay, effective speed of light, bandwidth, and insertion loss. The effective speed of light was calculated as the ratio of the chain length over the delay time. Then, due to the double propagation of the chain length, a reflecting CROW can possess a two times smaller effective speed of light than the same chain in transmission. The delay line fabricated in [6

6. F. N. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1(1), 65–71 (2007). [CrossRef]

] possessed ~1 nm transmission bandwidth. However, due to fabrication errors (the standard deviation of MR circumference ~12 nm leading to the spectral standard deviation ~0.4 nm), the resonant transmission spectrum of the SCISSOR chain with the best performance demonstrated in [6

6. F. N. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1(1), 65–71 (2007). [CrossRef]

] exhibited strong oscillations in excess of 5 dB with the period of ~0.1 nm and the insertion loss of 22 dB (Table 1, line 1). Similar fabrication errors characterize the precision of MRs in other resonance delay lines (both with microscopic and millimeter-scale elements) demonstrated to date. For this reason, the photonic crystal MR chain of Ref [5

5. M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nat. Photonics 2(12), 741–747 (2008). [CrossRef]

]. (Table 1, line 2) exhibited similar transmission power oscillations. To fix the fabrication errors and demonstrate tunable delay lines, the authors of Refs [25

25. A. Melloni, F. Morichetti, C. Ferrari, and M. Martinelli, “Continuously tunable 1 byte delay in coupled-resonator optical waveguides,” Opt. Lett. 33(20), 2389–2391 (2008). [CrossRef]

27

27. N. K. Fontaine, J. Yang, Z. Pan, S. Chu, W. Chen, B. E. Little, and S. J. B. Yoo, “Continuously Tunable Optical Buffering at 40 Gb/s for Optical Packet Switching Networks,” J. Lightwave Technol. 26(23), 3776–3783 (2008). [CrossRef]

]. applied thermal tuning. The latter allowed fabrication of ~0.1 nm bandwidth devices consisting of up to 8 millimeter-scale resonators both in SCISSOR and reflected CROW configurations [25

25. A. Melloni, F. Morichetti, C. Ferrari, and M. Martinelli, “Continuously tunable 1 byte delay in coupled-resonator optical waveguides,” Opt. Lett. 33(20), 2389–2391 (2008). [CrossRef]

,27

27. N. K. Fontaine, J. Yang, Z. Pan, S. Chu, W. Chen, B. E. Little, and S. J. B. Yoo, “Continuously Tunable Optical Buffering at 40 Gb/s for Optical Packet Switching Networks,” J. Lightwave Technol. 26(23), 3776–3783 (2008). [CrossRef]

] (Table 1, lines 3 and 4). Tuning has been also demonstrated for CROWs with microscopic ring resonators (Table 1, line 5). Unfortunately, correction of fabrication errors by thermal tuning is problematic for CROWs with large number of resonators since, for large N, it is practically impossible to perform the spatial recognition of the defects from the device spectrum. To solve the problem, it is necessary to develop a method of local characterization of defects similar to that depicted in the surface plots of Figs. 3-6. Such characterization is unfeasible for the modern planar photonic fabrication technologies. In contrast, due to the much higher fabrication accuracy and lower loss, it was possible to fabricate the SNAP delay line having 0.1 nm transmission band and achieve a good agreement with theoretical modeling of the exactly uniform MR chain without thermal tuning. The fabricated device exhibits the slowest speed of light c/250, the largest (among the delay lines with microscopic MRs) delay time of 1 ns, and the smallest insertion loss (Table 1, line 6). The footprint of this delay line is 0.04 × 1.2 = 0.05 mm2, which is less than the footprint 0.09 mm2 of the smallest SOI MR device [6

6. F. N. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1(1), 65–71 (2007). [CrossRef]

] that achieved though a smaller delay of 0.5 ns. It is believed that the major loss of the SNAP delay line described above ~3 dB has the non-resonant nature and is primarily caused by scattering in the region of coupling with the microfiber, while the internal loss of the MR chain is relatively small. Therefore, it is suggested that the SNAP delay lines with larger numbers of MRs will possess similar low insertion loss.

Appendix: Note added in proof

Recently, a new type of slow light miniature delay line with a breakthrough performance has been proposed and demonstrated [28

28. M. Sumetsky, “Dispersionless impedance-matched low-loss optical bottle resonator slow light delay line,” arXiv:1305.6591, http://arxiv.org/abs/1305.6591.

]. This delay line consists of a single 3 mm long and 0.12 mm2 SNAP bottle resonator with a semi-parabolic nanoscale radius variation rather than a series of coupled microresonators considered above. It exhibits the record dispersionless 2.58 ns (3 bytes) delay of 100 ps pulses with 0.44 dB/ns intrinsic loss and 1.2 dB/ns full loss.

Acknowledgment

The author is grateful to Y. Dulashko for assisting in the experiments and D. J. DiGiovanni for useful discussions.

References

1.

J. B. Khurgin, “Slow light in various media: a tutorial,” Adv. Opt. Photon. 2(3), 287–318 (2010). [CrossRef]

2.

F. Morichetti, C. Ferrari, A. Canciamilla, and A. Melloni, “The first decade of coupled resonator optical waveguides: bringing slow light to applications,” Laser Photonics Rev. 6(1), 74–96 (2012). [CrossRef]

3.

M. Notomi, “Strong light confinement with periodicity,” Proc. IEEE 99(10), 1768–1779 (2011). [CrossRef]

4.

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron. 37(4), 525–532 (2001). [CrossRef]

5.

M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nat. Photonics 2(12), 741–747 (2008). [CrossRef]

6.

F. N. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1(1), 65–71 (2007). [CrossRef]

7.

M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, “Statistics of light transport in 235-ring silicon coupled-resonator optical waveguides,” Opt. Express 18(25), 26505–26516 (2010). [CrossRef]

8.

M. Sumetsky, “Localization of light in an optical fiber with nanoscale radius variation,” in CLEO/Europe and EQEC 2011 Conference Digest, postdeadline paper PDA_8.

9.

M. Sumetsky and J. M. Fini, “Surface nanoscale axial photonics,” Opt. Express 19(27), 26470–26485 (2011). [CrossRef]

10.

M. Sumetsky, D. J. DiGiovanni, Y. Dulashko, J. M. Fini, X. Liu, E. M. Monberg, and T. F. Taunay, “Surface nanoscale axial photonics: robust fabrication of high-quality-factor microresonators,” Opt. Lett. 36(24), 4824–4826 (2011). [CrossRef]

11.

M. Sumetsky, K. Abedin, D. J. DiGiovanni, Y. Dulashko, J. M. Fini, X. Liu, and E. M. Monberg, “Coupled high Q-factor surface nanoscale axial photonics (SNAP) microresonators,” Opt. Lett. 37(6), 990–992 (2012). [CrossRef]

12.

M. Sumetsky, D. J. DiGiovanni, Y. Dulashko, X. Liu, E. M. Monberg, and T. F. Taunay, “Photo-induced SNAP: fabrication, trimming, and tuning of microresonator chains,” Opt. Express 20(10), 10684–10691 (2012), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-20-10-10684. [CrossRef]

13.

M. Sumetsky, “Theory of SNAP devices: basic equations and comparison with the experiment,” Opt. Express 20(20), 22537–22554 (2012), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-20-20-22537. [CrossRef]

14.

M. Sumetsky and Y. Dulashko, “SNAP: Fabrication of long coupled microresonator chains with sub-angstrom precision,” Opt. Express 20(25), 27896–27901 (2012). [CrossRef]

15.

A. B. Matsko, A. A. Savchenkov, and L. Maleki, “Vertically coupled whispering-gallery-mode resonator waveguide,” Opt. Lett. 30(22), 3066–3068 (2005). [CrossRef]

16.

M. Sumetsky, “Vertically-stacked multi-ring resonator,” Opt. Express 13(17), 6354–6375 (2005). [CrossRef]

17.

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(4), 722–731 (2002). [CrossRef]

18.

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24(11), 711–713 (1999).

19.

V. S. Ilchenko and A. B. Matsko, “Optical resonators with whispering-gallery modes - Part II: Applications,” IEEE J. Sel. Top. Quantum Electron. 12(1), 15–32 (2006). [CrossRef]

20.

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).

21.

M. Pöllinger, D. O’Shea, F. Warken, and A. Rauschenbeutel, “Ultrahigh-Q tunable whispering-gallery-mode microresonator,” Phys. Rev. Lett. 103(5), 053901 (2009). [CrossRef]

22.

T. A. Birks, J. C. Knight, and T. E. Dimmick, “High-resolution measurement of the fiber diameter variations using whispering gallery modes and no optical alignment,” IEEE Photon. Technol. Lett. 12(2), 182–183 (2000). [CrossRef]

23.

M. Sumetsky and Y. Dulashko, “Radius variation of optical fibers with angstrom accuracy,” Opt. Lett. 35(23), 4006–4008 (2010). [CrossRef]

24.

M. Sumetsky and B. Eggleton, “Modeling and optimization of complex photonic resonant cavity circuits,” Opt. Express 11(4), 381–391 (2003). [CrossRef]

25.

A. Melloni, F. Morichetti, C. Ferrari, and M. Martinelli, “Continuously tunable 1 byte delay in coupled-resonator optical waveguides,” Opt. Lett. 33(20), 2389–2391 (2008). [CrossRef]

26.

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments and ultimate limits,” J. Opt. 12(10), 104008 (2010). [CrossRef]

27.

N. K. Fontaine, J. Yang, Z. Pan, S. Chu, W. Chen, B. E. Little, and S. J. B. Yoo, “Continuously Tunable Optical Buffering at 40 Gb/s for Optical Packet Switching Networks,” J. Lightwave Technol. 26(23), 3776–3783 (2008). [CrossRef]

28.

M. Sumetsky, “Dispersionless impedance-matched low-loss optical bottle resonator slow light delay line,” arXiv:1305.6591, http://arxiv.org/abs/1305.6591.

OCIS Codes
(060.2340) Fiber optics and optical communications : Fiber optics components
(230.3990) Optical devices : Micro-optical devices
(140.3945) Lasers and laser optics : Microcavities

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: March 4, 2013
Revised Manuscript: May 29, 2013
Manuscript Accepted: June 5, 2013
Published: June 19, 2013

Citation
M. Sumetsky, "A SNAP coupled microresonator delay line," Opt. Express 21, 15268-15279 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-13-15268


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References

  1. J. B. Khurgin, “Slow light in various media: a tutorial,” Adv. Opt. Photon.2(3), 287–318 (2010). [CrossRef]
  2. F. Morichetti, C. Ferrari, A. Canciamilla, and A. Melloni, “The first decade of coupled resonator optical waveguides: bringing slow light to applications,” Laser Photonics Rev.6(1), 74–96 (2012). [CrossRef]
  3. M. Notomi, “Strong light confinement with periodicity,” Proc. IEEE99(10), 1768–1779 (2011). [CrossRef]
  4. G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron.37(4), 525–532 (2001). [CrossRef]
  5. M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nat. Photonics2(12), 741–747 (2008). [CrossRef]
  6. F. N. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics1(1), 65–71 (2007). [CrossRef]
  7. M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, “Statistics of light transport in 235-ring silicon coupled-resonator optical waveguides,” Opt. Express18(25), 26505–26516 (2010). [CrossRef]
  8. M. Sumetsky, “Localization of light in an optical fiber with nanoscale radius variation,” in CLEO/Europe and EQEC 2011 Conference Digest, postdeadline paper PDA_8.
  9. M. Sumetsky and J. M. Fini, “Surface nanoscale axial photonics,” Opt. Express19(27), 26470–26485 (2011). [CrossRef]
  10. M. Sumetsky, D. J. DiGiovanni, Y. Dulashko, J. M. Fini, X. Liu, E. M. Monberg, and T. F. Taunay, “Surface nanoscale axial photonics: robust fabrication of high-quality-factor microresonators,” Opt. Lett.36(24), 4824–4826 (2011). [CrossRef]
  11. M. Sumetsky, K. Abedin, D. J. DiGiovanni, Y. Dulashko, J. M. Fini, X. Liu, and E. M. Monberg, “Coupled high Q-factor surface nanoscale axial photonics (SNAP) microresonators,” Opt. Lett.37(6), 990–992 (2012). [CrossRef]
  12. M. Sumetsky, D. J. DiGiovanni, Y. Dulashko, X. Liu, E. M. Monberg, and T. F. Taunay, “Photo-induced SNAP: fabrication, trimming, and tuning of microresonator chains,” Opt. Express20(10), 10684–10691 (2012), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-20-10-10684 . [CrossRef]
  13. M. Sumetsky, “Theory of SNAP devices: basic equations and comparison with the experiment,” Opt. Express20(20), 22537–22554 (2012), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-20-20-22537 . [CrossRef]
  14. M. Sumetsky and Y. Dulashko, “SNAP: Fabrication of long coupled microresonator chains with sub-angstrom precision,” Opt. Express20(25), 27896–27901 (2012). [CrossRef]
  15. A. B. Matsko, A. A. Savchenkov, and L. Maleki, “Vertically coupled whispering-gallery-mode resonator waveguide,” Opt. Lett.30(22), 3066–3068 (2005). [CrossRef]
  16. M. Sumetsky, “Vertically-stacked multi-ring resonator,” Opt. Express13(17), 6354–6375 (2005). [CrossRef]
  17. 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. B19(4), 722–731 (2002). [CrossRef]
  18. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett.24(11), 711–713 (1999).
  19. V. S. Ilchenko and A. B. Matsko, “Optical resonators with whispering-gallery modes - Part II: Applications,” IEEE J. Sel. Top. Quantum Electron.12(1), 15–32 (2006). [CrossRef]
  20. 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).
  21. M. Pöllinger, D. O’Shea, F. Warken, and A. Rauschenbeutel, “Ultrahigh-Q tunable whispering-gallery-mode microresonator,” Phys. Rev. Lett.103(5), 053901 (2009). [CrossRef]
  22. T. A. Birks, J. C. Knight, and T. E. Dimmick, “High-resolution measurement of the fiber diameter variations using whispering gallery modes and no optical alignment,” IEEE Photon. Technol. Lett.12(2), 182–183 (2000). [CrossRef]
  23. M. Sumetsky and Y. Dulashko, “Radius variation of optical fibers with angstrom accuracy,” Opt. Lett.35(23), 4006–4008 (2010). [CrossRef]
  24. M. Sumetsky and B. Eggleton, “Modeling and optimization of complex photonic resonant cavity circuits,” Opt. Express11(4), 381–391 (2003). [CrossRef]
  25. A. Melloni, F. Morichetti, C. Ferrari, and M. Martinelli, “Continuously tunable 1 byte delay in coupled-resonator optical waveguides,” Opt. Lett.33(20), 2389–2391 (2008). [CrossRef]
  26. A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments and ultimate limits,” J. Opt.12(10), 104008 (2010). [CrossRef]
  27. N. K. Fontaine, J. Yang, Z. Pan, S. Chu, W. Chen, B. E. Little, and S. J. B. Yoo, “Continuously Tunable Optical Buffering at 40 Gb/s for Optical Packet Switching Networks,” J. Lightwave Technol.26(23), 3776–3783 (2008). [CrossRef]
  28. M. Sumetsky, “Dispersionless impedance-matched low-loss optical bottle resonator slow light delay line,” arXiv:1305.6591, http://arxiv.org/abs/1305.6591 .

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