1. Introduction

A one-dimensional (1D) periodic coupled cavity array or coupled resonator optical waveguide (CROW) [1–3] has enabled dispersion-managed slow-light with a group index (n_g) of 30 to 170 [4–6], where n_g = c/v_g (c: the speed of light in a vacuum, v_g: the group velocity) is the slowing down factor of slow-light. Propagation loss is the most serious issue in photonic on-chip slow-light waveguides [3,7], and the total waveguide length longer than 1 mm was rarely reported [8–10]. In CROWs the use of high Q cavities greatly reduced the loss and greatly increased the number of the cavities to be coupled [5,6,11]. There is a fundamental trade-off relationship between n_g and the BW in a slow-light waveguide [3,7] and in a CROW n_g and BW basically follow a simple sinusoidal dispersion [1,2,6] (See Appendix 1). With our previous mode-gap-nanocavity-based CROW we realized a fairly high n_g (~40) and a wide BW (3.5 nm) by setting the cavity spacing L at 5a (a: lattice constant.) [5] However, setting L at a small value (5a) caused the dispersion curve to deviate significantly from an ideal sinusoidal curve [5,11] due to the relatively weak confinement of the nanocavity, and a further reduction in L to access a lower n_g regime was difficult in terms of mode-gap design [12]. Thanks to the recent demonstrations of the validity of a low n_g (5-10) dispersion-managed waveguide for certain device applications [8,13], the CROW is also promising for such wideband and moderate (10-40) slow-light applications [14] in addition to the ultra-slow-light applications.

In this study, we focus on the use of a multiple-hole-tuned L3 nanocavity with a Q of ~10⁶ and a mode volume of less than (λ/n)³ [15,16] (λ: cavity wavelength, n: refractive index at the cavity centre) instead of mode-gap nanocavity [12]. Compared with a mode-gap nanocavity [5,6], an L3 nanocavity has a smaller footprint, stronger cavity confinement (or a smaller mode volume V), and wider mode separation between the 0th and 1st cavity modes (when the centre line defect is terminated [17]). The last feature is important if we are to realize a single-mode wideband CROW mode that does not overlap with the other CROW modes, which is difficult for ring-waveguide-based CROWs [4,9,10]. Unlike the mode-gap-nanocavity-based CROW [5,6,18,19], a multiple-hole-tuned L3 nanocavity that requires many holes to enhance Q (See Fig. 1(a) where the footprint is shown as a yellow box) is incompatible with a CROW arrayed for Γ-K orientation (θ = 0^◦). Instead an L3 nanocavity can be easily coupled to another L3 nanocavity placed off-axis (θ>0^◦) out of the Γ-K orientation [20,21]. We employed this slanted coupled cavity model, developed it into a large-scale L3 CROW, and described a preliminary feasibility demonstration [22]. In this letter we report extensive and detailed experimental studies of large scale CROWs, with up to 1,000 coupled cavities, realized by employing a slanted array of L3 nanocavities. Every L3 nanocavity had a footprint of several micrometres and an intrinsic Q factor of 6 × 10⁵ thanks to multiple-hole tuning [15], which enabled a single-mode slow-light BW of over 30 nm in 50 coupled cavities and of over 20 nm in 1,000 coupled cavities. The latter had a total waveguide length (L_CROW) of over 1 mm, which is even difficult to achieve with line-defect-based dispersion-managed photonic crystal waveguides [7].

 

Fig. 1 Designs and analysis of L3-based slanted CROW up to N = 5. (a) Cavity design and detailed cavity array layout. The lattice constant (a) and hole radius (r) were respectively 420 and 100 nm in the simulation and 408 and 95 nm in the experiment. The thickness (t) and refractive index (n) of the Si slab were 200 nm and 3.46, respectively. The hole shifts annotated A, B, and C to enhance Q [16] were 0.055a, 0.296a, and 0.148a, respectively. The L3 footprint is displayed as a broken yellow line. L3s (Total number: N) were periodically arrayed with a spacing of L on the green line slanted by θ from a Γ-K orientation along which L3 point defects were placed. L_v, the spacing of the rows on which the L3 cavities were placed, was set at 3 (rows.) (b) Index profile (the inset shows the mode profile of a single L3 nanocavity) and (c) the 1st (λ = 1594 nm), (d) the 2nd (λ = 1585 nm), (e) the 3rd (λ = 1575 nm), and (f) the 4th (λ = 1567 nm) CROW modes (N = 4, θ = 30°) obtained by FDTD simulations. (g) A scanning electron microscope image of a slanted CROW (N = 5, θ = 30°) sample. The locations of L3s are shown by overlaid yellow ovals. (h) Theoretical κ as a function of L for fixed θ (solid lines). The broken green line shows κ for L_v = 3 corresponding to the experiment. n_g is the theoretical group index at the CROW band centre. The two data plots indicated by surrounding circles were simulated using the conventional L3 design [24]. (i) Experimental transmission spectrum with five discrete CROW modes (N = 5, θ = 30°). (j) Sinusoidal curve fitting of the CROW modes shown in (i). (k) Experimentally evaluated κ compared with theoretical κ.

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2. Basic design and studies on short slanted L3 CROWs

Figure 1(a) shows the elementary unit design of our L3 CROW with a slant angle θ that we studied here. The dimensionless coupling constant κ [6] (See also Appendix 1) of CROWs excluding ring-cavity-based configurations [23] can be controlled simply by changing the cavity spacing L. Meanwhile, here we fixed the row spacing between adjacent L3 nanocavities (L_v) to be 3 rows, and changed L by changing θ. Unlike other cavities the L3 nanocavity has a strongly anisotropic spatial cavity mode profile as shown in Fig. 1(d), hence we anticipated a new class of κ control achieved by controlling the array orientation θ beyond simply setting θ = 30° [20,21]. As the first step, we analysed an L3 CROW consisting of 4 cavities (N = 4) by using a finite-difference time-domain (FDTD) simulation as shown in Figs. 1(b)-1(g). As expected from the CROW mode profile obtained with the FDTD simulation [Fig. 1(d)], at a θ of around 30° and L_v = 2, κ was enhanced by one order of magnitude from the value of 0.002 obtained in a modegap-nanocavity-based CROW at L = 5a [6] (a: lattice constant of photonic crystal). The slanted array (θ>>0°) allows the conventional L3 design [24] to make the spacing closer to L_v = 2. Although the multiple-hole tuning employed here limits the minimum L_v to 3, the slant array allowed us to obtain an L smaller than 3a (θ>60°), which is smaller than 8a for the straight array (θ = 0°). In this study, we fixed L_v at 3, and wefocused on a κ range of 0.002-0.01, which has not been explored for a 1D chain-like CROW. More complex designs combining a zigzag array and hole-tuning [25] was recently proposed with the aim of achieving a high normalized delay-bandwidth product (NDBP), which is a benchmark of a flat dispersion bandwidth [7,26,27], beyond the sinusoidal dispersion of the 1D chain-like CROW in the tight binding model [1,2,6]. However, as shown later in Section 4 and Appendix 1, a simple chain-like CROW has a fairly high NDBP. Furthermore, unlike the line-defect-based dispersion-managed waveguides and modegap-cavity-based CROWs which are bound to a Γ-K orientation (θ = 0°), the slanted CROW studied here provides a dispersion-managed waveguide out of Γ-K orientation. Hence the simple chain-like design is suitable as the first step study of the large-scale L3 CROW. Next, we fabricated L3 CROWs consisting of 5 cavities (N = 5) with θ values of = 19°, 30°, and 46° and measured their transmittance. CROW mode was formed due to the high Q fundamental cavity mode. The single cavity Q factor (theoretically 2 × 10⁶) was experimentally determined to be ~6 × 10⁵ around 1,550 nm [16] by measuring N = 1 CROW (reference single cavity). Since the cavity to input/output waveguide coupling was designed to be weak, 5 discrete coupled cavity modes were observed as expected (Fig. 1(i)). The Q factor of the coupled cavity modes in Fig. 1(i) was ~6 × 10⁴, which is much lower than 6 × 10⁵ in the single cavity due to the adjacent L3 cavities working as disorder and coupling loss to the input/output waveguides. We compared the numerical and experimental κ values and their θ dependence. In both the simulation and the experiment, the discrete CROW modes were well fitted by a sinusoidal dispersion curve as shown in Fig. 1(j) and showed good agreement as seen in Fig. 1(k), which presents the potential for controlling the dispersion (κ) by θ and enhancing the BW_CROW up to ~30 nm (at n_g = 12, κ = 0.01).

3. Experimental studies on large-scale slanted L3 CROWs

Encouraged by these results as the first step, we increased the number of L3 cavities N to 50, 200, and 1,000 [22] for several values of θ as shown in Fig. 2 (see Appendix 3 for details.) We believed that ~10-times higher Q of L3 cavity employed here would significantly reduce the propagation loss in CROWs, thus allowing long (N>>10) CROWs to be formed. For these CROW samples, we minimized the spacing between the CROW and the input/output W1 waveguides to obtain a continuous CROW spectrum and minimize insertion loss. We successfully fabricated the samples with a set at 408 nm as shown in Fig. 2 with the fabrication accuracy that we recently reported [15]. Note that for all the θ values studied here, L_CROW, namely the waveguide length (L_WG) of the CROW, exceeded 1 mm at N = 1,000 as shown in Table 1. The transmission spectra obtained at all N and θ values are compared in Fig. 3(d). For the longest L_CROW of 3.3 mm (N = 1,000, θ = 19°), the transmission in the CROW band was detected, although a wide transmission band could not be obtained. More importantly a fairly wide CROW band was obtained even at N = 1,000 in the 30° to 60° θ range. A wider spectrum was obtained at a smaller N, reflecting the fact that because of the sinusoidal dispersion, the propagation loss enhanced by n_g [3,14] became serious as wavelength was closer to the bandedge. At N = 50, the total bandwidth BW_CROW in Fig. 3(d) (See Appendix 4 for details) ranged from 6.7 nm to 32 nm as summarized in Table 1 depending on θ, κ and n_g. The maximum BW of 32 nm (frequency of 4.0 THz and κ of 0.0105) is 5.3 times larger than the value of 6 nm in the previous nanocavity-based CROW [6] and comparable to line-defect-based dispersion-managed photonic crystal waveguides with an n_g value exceeding 10 [26,27]. Very recently another utilization of coupling of L3-based cavities up to N = 800 and bandwidth of ~18 nm as designed in [25] was reported, which focused on higher n_g (~40) and its totallength was ~0.5 mm [28]. As shown in Fig. 3(b), the CROW modes formed by the 0th mode and the 1st mode were clearly separated thanks to the wide cavity mode spacing (~40 nm) [16]. Therefore the mode discussed here was always single-mode. The relationships between θ, κ, n_g, BW, and NDBP will be analysed in detail later in Fig. 5. In Figs. 3(b) and (d), there were Fabry-Perot fringes on the wide-band CROW spectra with an amplitude of ~10 dB as with previous photonic crystal CROWs [5,6]. (See Appendix 3 for detail.) In Fig. 3(c) we evaluated the propagation loss (α) by the cut-back method (see Appendix 4) since the long L_CROW exceeding 1 mm allowed evaluation of the loss at an order of dB/mm. As shown in Table 1, α evaluated at the centre (~1,550 nm) of the CROW band was ~10 dB/mm or less for a wide θ range, which is surprisingly as low as that of the 1,000 coupled L3 nanocavities. Table 1 suggests that α was proportional to n_g rather than to n_g² around the center of the CROW band, indicating the minor impact of backscattering [29]. There is a certain fluctuation among the cavities (See Appendix 3) but the impact on the characteristics of the CROW seems minimal. This is likely because BW_CROW in this study is much wider than the fluctuation unlike the ultra-narrow-band and ultra-slow-light CROWs [6].

 

Fig. 2 Experimental large-scale CROW samples. (a) Schematic of the design of on-chip waveguide with large-scale CROW samples. All samples have an 8 μm-wide WG part and a 16 μm-long conversion (8 μm to 0.5 μm) part. The reference WG has a 1.85 mm-long nanowire part (width: 500 nm). The CROW sample has two coupling W1 waveguide parts (total length: ~30 μm) in addition to the CROW part (length: L_CROW). (b)-(c) Optical microscope images of the top view of CROW samples with θ = 30°. (b) N = 50 and 200 and (c) N = 1,000. (d)-(e) SEM images of slanted L3 CROWs. (d) is wide area image of the sample with θ = 30°. The inset is expanded image of a L3 nanocavity. (e) Expanded images for all fabricated array orientations (θ).

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Table 1. Specifications and characteristics of large-scale slanted L3 CROWs evaluated by transmission measurement

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Fig. 3 (a) Spectrum of single L3 nanocavity. The blue line is a Lorentzian fitting. FWHM: full width at half maximum. (b) A wide-range transmission spectrum of a CROW for θ = 37° and N = 50. (c) Loss plots obtained by comparing the transmission of N = 50, 200, and 1,000 samples (cut back method.) (d) Transmission spectra of CROWs with N = 50 (blue lines), N = 200 (green lines) and N = 1,000 (red lines) at various θ values. The grey lines are the spectra of reference Si waveguides.

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To demonstrate that these CROWs actually acted as slow-light waveguides, we performed time-of-flight (TOF) measurements [6] by injecting a short pulse into the CROWs. Thanks to the high transmittance, we were able to collect TOF data from CROWs over a millimetre in length formed by 1,000 cavities, which is still challenging even with line-defect-based high NDBP photonic crystal waveguides [9,14,26,27]. As described in the Appendix 5, we injected a 17 ps (~0.21 nm in wavelength domain) pulse from one port of a CROW and detected the pulse from another port after it had travelled through the sample. We scanned the tunable laser wavelengths from 1,535 to 1,570 nm in 0.1 nm steps and recorded the time response of the output power for N = 1,000 CROWs as shown in Figs. 4(a) and 4(b). Reference waveguide (total length: 5.6 mm, including a 1.85 mm-long and 500 nm-wide photonic-wire part) exhibited a constant delay over the entire scanned wavelength range as shown by the magenta line. The delay mapping plots (raw data) as reported in Figs. 4(a) and 4(b) were consistent with the expected sinusoidal dispersion. Figure 4(c) compares time-dependent waveforms obtained after the pulses had passed the CROW (N = 1,000, θ = 37°). There was a clear large pulse delay (several times the pulse width) at bandwidth above 21 nm. The waveforms of pulses after they had passed through the 1.8 mm-long CROW were not seriously broadened or distorted compared to the waveforms of pulses after propagation through the reference waveguide, revealing an almost flat dispersion. Since we have demonstrated the successful transmission of pulse trains modulated around 10 GHz in CROWs with similar transmission spectra [5,6], the L3 CROWs presented here are expected to work as a delay line and/or a buffer for these fast modulation signals. Next, we performed a correction to take account of the delay in the reference waveguide and the on-chip input/output waveguides as shown in Fig. 2(a) and obtained the net flight (delay) time in the CROW as plotted in Figs. 4(d)-4(g) with corresponding n_g (See Appendix 5 for details of the correction). A fairly high n_g of 13~40 and a wide effective BW_CROW of 12~22 nm were obtained. The values of BW_NDBP, where the n_g variation is less than ± 10% [7], were (16.0, 16.2, 15.3, 10.5) nm, and the centre values of n_g inside BW_NDBP (n_g(C)) were (14.3, 14.5, 17.6, 25.3) at θ = (30°, 37°, 46°, and 60°), respectively. The TOF-determined n_g values agree with those in Table 1 evaluated from the BW_CROW of a shorter CROW (N = 50) and are in the useful slow-light range despite the large BW. From the BW and n_g described above, fairly high NDBP values from 0.147 to 0.174 were obtained as plotted in Fig. 5(a). The TOF-determined BW_NDBP values were equal to or higher than 0.5 × (BW_CROW at N = 50) and therefore agreed well with the theoretical value of 0.575 × BW_CROW, shown as the red broken line, for the sinusoidal dispersion curve. The deviation of the experimental NDBP (red dots) from the theoretical value (red broken line) in Fig. 5(a) was fairly small at θ> = 46° where BW_NDBP was limited by dispersion (n_g variation was within ± 10%). At θ = 60° [Fig. 4(g)] the long wavelength edge of BW_NDBP was limited not by dispersion but by high propagation loss, which further increased the deviation. Figures 3 and 4 suggest that the light propagation in the CROWs plotted in Fig. 4(d)-4(g) were free from disorder-induced localisation [5,9] at least in the NDBP band where dispersion was sinusoidal.

 

Fig. 4 Results of time-of-flight experiment for slanted L3 CROWs with N = 1,000, obtained by 17-ps pulse propagation. (a)-(b) Time dependent output waveform spectrum after pulses had passed through the CROW (N = 1,000). (a) θ = 37°and (b) 60°. Vertical magenta lines show the pulse delay in the reference waveguide corrected by the delay in the waveguides out of the CROW. (c) Extracted time-dependent waveforms after pulses had passed through the CROW (N = 1,000, θ = 37°). Pulse wavelengths are displayed on the corresponding waveforms. The waveform obtained from the reference waveguide is corrected as the length of the waveguides are identical to that of the CROW sample since every sample having different CROW design has different waveguide design as shown in Fig. 2(a). Y-scale (intensity) was normalized in every waveform. (See Appendix 5 for detail). (d)-(g), Flight time (left) and n_g (right) as a function of wavelength obtained for θ values of (d) 30°, (e) 37°, (f) 46°, and (g) 60° after correction. The blue lines show the results of a ten-point moving average. The grey area overlaid on the plot shows BW_NDBP in which n_g changed within ± 10%. n_g values (n_g(C)) for NDBP and DBP evaluation were determined at the centre value of the ± 10% n_g range.

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Fig. 5 (a) Experimental DBP, NDBP, and L_CROW values plotted as a function of L in large-scale CROWs. DBP was evaluated from L_CROW/c × {(n_g(C))-(n_g of vacuum)} × (BW_DBP in ω). The experimental NDBP was evaluated from (n_g(C)) × (BW_NDBP in ω)/(ω at the centre of CROW band). The broken red line shows the theoretical NDBP ( = 0.639c/($\pi$ω₀L), where ω₀/2π = 193 THz, a = 408 nm) for sinusoidal dispersion curve. (T) were obtained in this study for N = 1,000 CROWs and (P) were obtained in L = 5a and N = 400 in [5] and L = 7a and N = 150 in [6] previously. (b) κ, NDBP and θ plots as a function of L in short CROWs. The experimental values in this study (T) were measured in CROWs with N = 50 and evaluated from BW_CROW shown in Table 1. The experimental NDBP values in the previous study (P) were according to the κ values reported in [6].

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4. Delay bandwidth product in various CROWs

Figure 5 plots the Γ-K oriented (θ = 0°) mode-gap-cavity-based CROWs investigated in previous studies (P) [5,6] in addition to the slanted (θ = 19°~79°) L3 CROWs investigated in this study (T). In Fig. 5(b), NDBP values obtained from the experiment agreed surprisingly well with the theoretical NDBP of 0.575 BW_CROW for the sinusoidal curve (see Appendix 1), flexibly control n_g and/or BW even if NDBP is constant. Such flexible κ control suggests that coupled photonic crystal nanocavities basically follow the tight binding model. A problem was that the NDBP measured in a short-period (L = 5a) large-scale mode-gap-cavity-based CROW was significantly lower than that of the sinusoidal curve as shown in Fig. 5(a) because of the structural problem [11]. On the other hand, in a slanted L3 nanocavity array the NDBP is essentially determined by sinusoidal dispersion curve although it may be limited by the propagation loss when n_g is high. Another important feature of CROW is that the NDBP depends on L but not on κ (See Appendix 1). Therefore, by changing κ we can utilized in a slanted L3 array by combining the anisotropic cavity mode distribution and slant angle (θ) control (Fig. 5(b) shows only a case of L_v = 3) in contrast to the mode-gap-cavity-based CROW where only linear control of κ by changing L is allowed. Here we chose a κ value of 0.002 to 0.01, which mainly enhances BW_CROW and (1/n_g) several times. As a result, we realized an ultra-wideband and moderate-n_g CROW band.

Finally, we evaluated the delay bandwidth product (DBP) [3] as a benchmark of a long slow-light waveguide. Here DBP was determined by n_g(c), BW_DBP, and L_CROW as described in the caption of Fig. 5. Except the (P) at L = 7a where BW_DBP was BW_CROW, BW_DBP was the bandwidth in which TOF signal was detected (see Figs. 4(d)-4(f)). As plotted in Fig. 5(a), unprecedentedly high DBP values exceeding 200 were achieved in the 30° to 46° θ range and the highest value was 220 at θ = 37° thanks to L_CROW exceeding 1 mm in this study. Previously, a very high NDBP value of around 0.3 was demonstrated experimentally in PhC waveguides [26,27,30], but an L_WG value of much shorter than 1 mm kept DBP much lower than 100. Meanwhile, an L_CROW value longer than that obtained in this study (3.35 mm) was realized by using ring waveguide cavities [9], but the DBP was only about 70 (for a single sinusoidal band) due to the narrow BW_CROW. With regards to dispersion-engineered slow-light waveguides, which usually have a highly complicated structure [9], fabricating a waveguide longer than 1 mm while overcoming loss is a serious challenge. A 4-mm-long dispersion-managed slow-light waveguide has been reported [8], but it lacks a wide flat dispersion band with an n_g higher than 10. Therefore, the realization of a DBP of over 200 in a purely passive way is a landmark achievement with respect to all on-chip slow-light technologies. Note that a W1 waveguide [31], which is a standard waveguide in a photonic crystal often useful for slow-light applications [32,33], has a flat dispersion band with n_g ~5 and its high n_g band is highly dispersive, hence the L3 CROW presented here is superior to a W1 waveguide in terms of providing a wide flat dispersion.

5. Conclusion

Throughout the paper we have demonstrated utilization of the large-scale coupled L3 nanocavities with slanted coupling, that can provide long wideband dispersion-managed slow-light waveguide for PhC orientations slanted from Γ-K. The ultrahigh-Q L3 nanocavity based CROW provides both a powerful dispersion-managed slow-light waveguide and a flexible on-chip waveguide alternative to W1-based photonic crystal waveguides. The successful formation of large-scale coupled nanocavities free from a common long line-defect [5,6] encourages the study of novel 2D coupled cavities [21,34]. We are now developing nanocavities with a smaller footprint and a higher Q from the L3 nanocavity employed in this study. This approach will make the CROW a more flexible platform on which to form a wideband dispersion-managed waveguide on a photonic crystal chip.

Appendix 1 Tight binding model and theoretical BW, n_g, DBP, and NDBP in 1D-chain-like CROW

In a simple tight-binding model highly applicable to a dense periodic nanocavity array, the dispersion is given by ω=ω₀ (1+ κ cos(k_nL)) (where ω/2π is frequency, ω₀ is ω of an uncoupled cavity, k_n is the wavenumber of the n-th CROW mode, L is the cavity spacing, and κ is the coupling coefficient) [1,2] and BW is 2κω₀. n_g at ω₀ is c/(2$\pi$ω₀κL) and its dispersion is proportional to (sin(k_n L))⁻¹. n_g inside the NDBP band, where n_g varies within ±10% including ω₀, is between n_g (ω₀) and 1.222 n_g (ω₀), and n_g at the centre is 1.111 n_g (ω₀). BW_NDBP (Δω) is 0.575×2κω₀. The delay in the CROW (over light propagation in a vacuum) is L_CROW (n_g −1)/c and 0.555L_CROW/($\pi$ω₀κL) at n_g >>1. DBP is 0.555L_CROW/($\pi$ω₀κL)×2κω₀=1.111 L_CROW/($\pi$L). NDBP is n_g (centre)×Δω/ω₀= 0.639c/($\pi$ω₀L). Above mentioned relations are illustrated in Fig. 6.

 

Fig. 6 Tight binding model in 1D chain-like CROW.

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Appendix 2 FDTD simulation

For the simulation, we used our own FDTD code as in previous studies [1,6,12,15,16]. For the simulation of coupled cavities, a mesh size of 30 nm was applied and a was 420 nm. Q, the cavity wavelength, and the mode volume of the fundamental mode of a single L3 were 1.9×10⁶, 1,578 nm, and 0.8 (λ/n)³, respectively. Because of the weak coupling to outer W1 waveguides, the CROW modes split into separate peaks in the spectrum and every CROW mode was selectively excited by the narrow band source and evaluated as shown in Figs. 1(c)-1(f). κ was evaluated by sinusoidal curve fitting according to [2]. Due to the lack of wavelength resolution in FDTD (which may be overcome by greatly increasing the calculation time), we could not evaluate the CROW mode at a κ of ~0.001 or less where the mode spacing was ~ 1 nm or less.

Appendix 3 Device preparation

The samples were prepared by using electron beam lithography according to [15]. The minimum step of the position setting was 0.125 nm and the wavelength fluctuation of the cavity mode was σ=0.36 nm (σ: standard deviation) within the length of 1 mm. Wavelength accuracy is not guaranteed when the in-wafer position spacing is larger than 1 mm. The spectrum of the fundamental L3 cavity mode with a full width at half maximum (FWHM) linewidth of 2.6 pm (~300 MHz) is shown in Fig. 3(a) corresponding to a Q factor of 6×10⁵. The actual on-chip layouts of large-scale CROWs and external on-chip Si waveguides are presented in Figs. 2(a)-2(c). Detailed SEM images of CROWs with different θ values are shown in Fig. 2(d). At θ=30°~46°, a CROW with N=1,000 and others (N=50, 200) CROWs with other N values were patterned on the same wafer at the same time but placed on separate chips with different lengths. The chip length for the former and the latter were 5.6 and 2.8 mm, respectively. Other CROW samples were patterned on the same chip with the same total chip length of 5.6 mm. Almost all the external Si waveguides had a large width of 8 μm to simplify the air-bridge-formation process by wet etching with buffered hydrogen fluoride acid. After etching the sacrificial layer, a 1500-nm-thick air undercladding layer was formed between the Si photonic crystal membrane and the bottom 1,500-nm-thick SiO₂ layer.

Appendix 4 Spectrum and loss measurement in large-scale CROWs

The spectra shown in Fig. 3 were collected by tunable laser source and high-sensitivity photodiode [6]. In the experimental analysis we assumed that dispersion of the CROW follows tight binding model as shown in Appendix I. At N=50 we regarded BW_CROW as full transmission band width shown in Fig. 3(d) and neglected the BW narrowing due to the propagation loss which gave undervaluation of BW. Since we did not use spot-size converters or anti-reflection coating on the on-chip Si waveguide edge, the waveguide edge acted as a mirror with a reflectivity of ~30%, which was one of the origins of visible Fabry-Perot fringes in the measured transmission spectra. The reflection or impedance mismatch at the joint of the CROW and the on-chip PhC waveguide may be another origin of the fringe. The coupling loss between the external optical fibre and on-chip waveguide was ~10 dB. To collect a transmission spectrum, we used the same high-resolution tuneable laser and high-sensitivity optical power meter used in previous work [6,15]. Fig. 3(b) shows the wide range scan of the transmission spectrum of a CROW (N=50, θ=37°), where we can see the clear separation of the CROW mode (1,543~1,575 nm) formed by the fundamental cavity mode and another CROW mode (1,501~1,522 nm) formed by the first high-order cavity mode. The details of the latter will be presented elsewhere. We performed a rough waveguide propagation loss (α) evaluation using the cut-back method. We ignored the loss of the 8-μm-wide Si waveguide (less than 1 dB/cm) and the coupling error at the waveguide edges (within a few decibels). One problem was that we had to separate CROWs with N=1,000 from other shorter CROWs (spacing>>millimetre) because of the large footprint of the former as shown in Figs. 2(a)-2(c). As a result, there was an in-wafer size fluctuation that caused a wavelength shift of a few nanometres with the former. But thanks to the wide BW_CROW, we could neglect the wavelength deviation among the CROWs and perform the α evaluation as shown in Fig. 2(c). We averaged the spectrum range, 1,555 nm±0.5 nm (for θ=79°, it was 1,557 nm±0.5 nm), over about 20 fringes, and have plotted it in Fig. 3(c). The error bars show the intensity of the Fabry-Perot interference and cavity-wavelength-fluctuation-induced fringes. α was evaluated from the line fitting slope and its values are shown in Table 1.

Appendix 5 Time-of-flight dispersion measurement

Our experimental setup for TOF measurement is shown in Fig. 7. We generated a short pulse (< 5 ps, limited by the laser source linewidth) by using a tuneable CW laser and a fast LN modulator. We injected the pulse from one of the ports of the on-chip waveguide, and detected it from another port after it had travelled through the sample. The output signal was amplified by an EDFA and recorded with an optical sampling oscilloscope after passing through a 65 GHz optical bandpass filter. Unfortunately, most of the on-chip Si waveguides outside of the CROW had total lengths of several millimetres and wide cross-sections (200 nm × 8,000 nm), which increased the pulse width to 17 ps (~0.21 nm) even in the reference waveguide as shown in Fig. 4(c). This caused serious coupling loss in the external fibre. However, if we inject a pulse directly from an edge of a PhC waveguide [5,6], such pulse broadening in the external waveguide can be greatly reduced. Otherwise, we can improve both the coupling loss and pulse broadening by installing a mode converter [35] at the edge of the waveguide. The latter can also be improved by greatly reducing the Si waveguide width.

 

Fig. 7 Schematic setup and apparatus for TOF single-pulse propagation measurement. VOA: Variable fiber optical attenuator. Pol: optical polarizer. EDFA: Erbium-doped fiber amplifier. BPF: Bandpass filter. PPG: Pulse pattern generator. See [15] for details.

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To extract TOF data from the wavelength-delay mapping data, the flight time outside the CROW should be corrected. This was accomplished by comparing the TOF data of the sample including the CROW with the TOF data of the reference waveguide.

For the comparison, the different waveguide configurations were corrected as follows. As shown in Fig. 3(a), every sample had similar small-width converters whose n_g can be regarded as that of a wide waveguide (n_g=3.7). The reference waveguide had a 1.85 mm-long nanowire waveguide (n_g=4.2) and a 3.75 mm-long wide waveguide (n_g=3.7). (We measured the fabrication error for a nominal chip length of 5.6 mm.) Two coupling W1 waveguides with a total length of ~30 μm were assumed to have an n_g of 6.0. The total length of the wide waveguide in a CROW sample was about (5.6 mm)-(30 μm)-L_CROW×cos θ. The delay of the pulse in the reference waveguide shown in Figs. 4(a) and 4(b) was corrected by assuming that the delay in a wide waveguide had a total waveguide length identical to that of the corresponding CROW sample. The delays in all the samples were determined by peak fitting every waveform. The difference in the pulse peak of the CROW and the magenta line represent the delay in the CROW coupled to a wide waveguide (n_g=3.7) with an identical length. The flight time in the CROW plotted in Figs. 4(d)-4(g) was obtained by adding the flight time in the equivalent n_g=3.7 waveguide to the relative delay in the CROW.

Funding

Core Research for Evolutional Science and Technology (CREST) (JPMJCR15N4).

Acknowledgment

We thank T. Tamamura and H. Onji for their support in fabricating the devices, H. Onji for his support in measuring the devices, D. H. Smith for helpful discussions, and H. Gotoh and T. Sogawa for their continuous encouragement.

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