A SNAP coupled microresonator delay line

M. Sumetsky

doi:10.1364/OE.21.015268

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]. 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 [2,3]. 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]. 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 [3,5] and for the ring microresonator photonic circuits where the MR diameter is as small as a few tens of microns [2,6]. 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] 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]. However, the enormously high precision and small attenuation of light achieved are still insufficient for the practical realization of these delay lines [7].

As compared to the lithographic fabrication technology, the recently introduced SNAP (Surface Nanoscale Axial Photonics) platform [8–14] 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. 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, 9, 13].

Fig. 1 Illustration of a SNAP device.

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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, 16]. As opposed to the previously demonstrated SCISSOR and CROW delay lines [2,17,18], 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]

τ = - \frac{1}{2 π} Im (\frac{1}{S} \frac{\partial S}{\partial ν}) = \frac{λ^{2}}{2 π c} Im (\frac{1}{S} \frac{\partial S}{\partial λ})

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 = ν_{0}

of an all-pass MR coupled to a single waveguide, the transmission amplitude can be approximately expressed through the attenuation and coupling parameters,

σ_{a}

and

σ_{c}

(see e.g., [19]):

S (ν) = \frac{ν - ν_{0} - i (σ_{a} - σ_{c})}{ν - ν_{0} - i (σ_{a} + σ_{c})} .

Then the group delay

τ (ν) = \frac{2 σ_{c} [{(ν - ν_{0})}^{2} - σ_{a}^{2} + σ_{c}^{2}]}{{[{(ν - ν_{0})}^{2} + σ_{a}^{2} + σ_{c}^{2}]}^{2} - 4 σ_{a}^{2} σ_{c}^{2}} .

From here, the delay-bandwidth product (DBP)

\int_{- \infty}^{\infty} τ (ν) d ν = {\begin{matrix} 1 & if σ_{a} < σ_{c}, \\ 0 & if σ_{a} > σ_{c} . \end{matrix}

Thus, for a relatively small loss,

σ_{a} < σ_{c}

, 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,

σ_{a} > σ_{c}

, 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 σ_c = 2σ_a corresponding to DBP = 1 (upper blue curve) and for σ_c = 0.5σ_a corresponding to DBP = 0 (lower red curve).

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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]

S (λ, z_{1}) = S_{0} - \frac{i | C |^{2} G_{0} (λ, z_{1}, z_{1})}{1 + D G_{0} (λ, z_{1}, z_{1})}

where

S_{0}

is the out-of-resonance transmission amplitude, C and D are the microfiber/SNAP fiber coupling parameters, and

G_{0} (λ, z_{1}, z_{2})

is the Green’s function of the Schrödinger equation

Ψ_{z z} + (E (λ) - V (z)) Ψ = 0, E (λ) = κ \frac{λ - λ_{r e s} - i γ}{λ_{r e s}}, V (z) = κ \frac{Δ r_{e f f} (z)}{r_{0}}, κ = - 2 {(\frac{2 π n}{λ_{r e s}})}^{2},

which describes the propagation of light along the SNAP fiber with radius

r_{0}

refractive index

n

, and effective radius variation (ERV)

Δ r_{e f f} (z)

near the resonance wavelength

λ = λ_{r e s}

[8, 9, 13]. Parameter

γ

in Eq. (6) determines attenuation of light. From Eqs. (1) and (3), the DBP along the spectral interval which contains

N

resonances

ν_{1}, ν_{2}, ..., ν_{N}

is

\begin{array}{l} \int_{ν_{1} < ν < ν_{N + 1}} τ (ν) d ν = \sum_{n = 1}^{N} \int_{ν_{n}}^{ν_{n + 1}} d ν \frac{\partial G}{\partial ν} \cdot [\frac{1}{[G + S_{0} {(S_{0} D - i | C |^{2})}^{- 1}]} - \frac{1}{(G + D^{- 1})}] \\ = N \int_{- \infty}^{\infty} d G [\frac{1}{[G + S_{0} {(S_{0} D - i | C |^{2})}^{- 1}]} - \frac{1}{(G + D^{- 1})}] = {\begin{matrix} N & if Λ > 0 \\ 0 & if Λ < 0 \end{matrix} \end{array}

where

Λ = \frac{| C |^{2}}{| S_{0} |^{2}} Re (S_{0}) - Im D

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

2 π \int_{}^{ν} τ (ν) d ν

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

\int_{}^{ν} ρ (ν) d ν = N (ν)

, we get from Eq. (7) the Krein-Friedel-Loyd formula (see e.g [20].),

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 $Δ r_{e f f} (z)$ . 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 [2,3,5–7] was the fabrication precision, typically in excess of several nm [6,7]. Since the fabrication precision in SNAP can be two orders of magnitude better [14] and attenuation of light two orders of magnitude smaller [11,21], 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] 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]. In brief, the MRs were introduced by periodic local annealing of the silica fiber with radius $r_{0} = 19$ μm 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]. 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, 23]. 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 $Δ λ_{F S R} = λ_{r e s}^{2} / (2 π n r_{0}) \approx 14$ nm corresponding to the fiber radius $r_{0} = 19$ μm, wavelength $λ_{r e s} \approx 1550$ nm, 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 [13,14], 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). Figure 3(b) shows the surface plot of the transmission amplitude and ERV (black curve) determined theoretically following [13]. 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):

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.

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S_{0} = 0.95 - 0.19 i, | C |^{2} = 0.01 μ m^{- 1}, D = 0.03 + 0.03 i μ m^{- 1}

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). From Eqs. (9) and (8), the GDI of this device is negative, $Λ = - 0.02$ μm⁻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).

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.

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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 $S_{0}$ , C, and D in Eq. (5). As an example, Fig. 5 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 = 0.6 pm, S_{0} = 0.85 - 0.1 i, | C |^{2} = 0.042 μ m^{- 1}, D = 0.021 + 0.024 i μ m^{- 1} .

For these parameters, the GDI is positive,

Λ = 0.025

μm⁻¹, and, therefore, the group delay distribution shown in Fig. 6(a) 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 [24,25]. 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.

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.

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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.

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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 $E_{o u t} (ν)$ is determined from the input pulse spectrum, $E_{i n} (ν)$ , from $E_{o u t} (ν) = S (ν) E_{i n} (ν)$ , where $S (ν)$ is the transmission amplitude. The relations between the input and output pulses in the time and frequency domains are

\begin{array}{l} {\bar{E}}_{i n} (t) = \int E_{i n} (ν) \exp (2 i π ν t) d ν \\ {\bar{E}}_{o u t} (t) = \int E_{o u t} (ν) \exp (2 i π ν t) d ν = \int S (ν) E_{i n} (ν) \exp (2 i π ν t) d ν \end{array}

In calculations, the input pulse $E_{i n} (ν)$ had the Gaussian profile with 4 GHz (0.032 nm) FWHM (Fig. 6(b), green solid curve). Figure 7(a) 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 [24,25], 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.

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).

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The results shown in Fig. 7 allow the interpretation with simple analytical modeling. If the input pulse spectrum is localized near frequency $ν_{0}$ inside the transmission band of $N$ coupled MRs then the propagation time is estimated as $τ_{0} = N / Δ ν_{b}$ , where $Δ ν_{b}$ is the bandwidth. In the neighborhood of $ν_{0}$ , the transmission amplitude is approximately periodic with the period $2 π / τ_{0} = 2 π Δ ν_{b} / N$ (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 [- 2 i π τ_{0} (ν - ν_{0}) + i ζ \cos (2 π τ_{0} (ν - ν_{0}))],

where parameter

ζ

determines the amplitude of the group delay ripple. The time delay and ripple amplitude parameters in this equation,

τ_{0} = 1.12

ns and

ζ = 0.6

, were chosen to fit the output field in Fig. 7(b). The corresponding output field

{\bar{E}}_{o u t}^{(m o d)} (t)

(Fig. 7(b), yellow dotted curve) is in a reasonable agreement with the output field

{\bar{E}}_{o u t} (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 [2,3,5,6,26]. Though the goal of the present paper was to demonstrate a miniature delay line based on microscopic resonators, Table 1 compares characteristics of the fabricated device with characteristics of the state of the art delay lines consisting both of micron-scale [5,6,26] and millimeter-scale [25,27] MRs. This Table classifies delay lines by their configuration (SCISSOR [17], CROW [18], reflecting CROW [2], and VCR [15,16]), 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] 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] 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]. (Table 1, line 2) exhibited similar transmission power oscillations. To fix the fabrication errors and demonstrate tunable delay lines, the authors of Refs [25–27]. 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 [25,27] (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] 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.

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.

View Table

The transmission amplitude of the demonstrated SNAP delay line exhibits oscillations in the transmission band, which causes reflection of pulses at the MR chain edge adjacent to the microfiber. To remove these oscillations and suppress the reflection, the MR chain should be appropriately apodized [24, 25]. Generally, the apodization (impedance matching) problem is not reduced to the simple problem of optimization in the tight-binding approximation [24], since, for relatively wide transmission bands, the inter-resonator coupling is not constant and strongly depends on wavelength (frequency). To design the MR chain, parameters of the transmission amplitude in Eq. (5) need to be adjusted in addition to the introduction of the optimized ERV. To this end, a special design of the coupler (which can be a microfiber used in this paper, as well as planar waveguide, fiber with an angled facet, and prism) may be needed. Increasing the bandwidth of the delay line from 0.1 nm to 1 nm is feasible. This can be accomplished, for example, by fabrication of the MR chain in the oversaturated regime, which enables creation of smaller MRs with larger individual eigenvalue spacing [11]. The major limitation of the detected delay time is related to the resolution of the Luna OVA used in the experiments. For this reason, the effective speed of light in the first narrowest (fundamental) transmission band (Figs. 5 and 6(a)) was not measured. It is supposed to be $c / 1500$ corresponding to irresolvable 6 ns of time delay. Thus, SNAP devices exhibiting much longer delay time, smaller effective speed of light, similar or smaller losses, and wider transmission band can be realized.

While the delay line considered is a stand-along device, combination of SNAP devices with planar photonic circuits is feasible. Figure 8 illustrates SNAP fibers coupled to a planar circuit through waveguides. For example, combination of SNAP fibers with planar fast (all-optical) switches adds tunability and allows to create miniature optical buffers. Thermal tuning of SNAP devices can also be accomplished by positioning SNAP fibers next to miniature heaters.

Fig. 8 The cartoon illustrating combination of SNAP delay lines with planar photonic circuits.

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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]. 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.

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A SNAP coupled microresonator delay line

Abstract

1. Introduction

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

3. Fabrication of the coupled resonator chain

4. Characterization of the SNAP delay line

5. Pulse propagation

6. Discussion and summary

Appendix: Note added in proof

Acknowledgment

References

Cited By

Figures (8)

Tables (1)

Equations (12)

Optics Express