Abstract

In contrast to recent reports of localization-impaired transport in long slow-light waveguides, we demonstrate light transport in silicon coupled-resonator optical waveguides (CROWs) consisting of up to 235 coupled microrings without localization over frequency bands that are several hundred gigahertz wide. Furthermore, from the unique statistical signatures provided by time-domain propagation delay measurements, we demonstrate the spectrally correlated nature of light propagation in CROWs.

©2010 Optical Society of America

1. Introduction

Coupled resonator optical waveguides (CROWs) are long sequences of micro-resonators fabricated on a chip that guide light from one end of the chain to the other by nearest-neighbor coupling. These devices have been proposed for applications in slow light, temperature insensitive high-order optical filters, dispersion compensation, etc [17]. A single CROW waveguide can readily achieve multiple bands of slow and filtered light in the same physical structure, unlike band-edge photonic crystal slow light waveguides [810]. This characteristic is particularly useful for on-chip wavelength division multiplexing, temperature-insensitive optical filters, and nonlinear optics, in which several wavelengths of light are often used simultaneously. However, practical applications require relatively long CROWs, consisting of hundreds of coupled resonators [7,1113], while maintaining good performance.

In spite of great theoretical interest in this novel form of waveguiding, the practical applications requirement that several hundreds of resonators must be chained together in a relatively disorder-free manner has constituted a technological challenge [11,12]. Disorder in CROWs has thus far been a severe practical problem, since in a multi-resonator ensemble, the resonance frequencies of the constituent resonators must be precisely aligned. Active resonance tuning mechanisms, e.g., thermal heaters placed over each ring [14,15], are impractical for ensembles consisting of hundreds or thousands of rings. Moreover, although the benefit of microrings and CROWs increases for higher quality factors, such resonators are also harder to align. In fact, coupled-resonator and photonic crystal waveguides that are about a hundred lattice periods in length have shown disorder-induced localization of light [16,17], which though fundamentally interesting and potentially useful for some applications [8], is generally considered problematic for most device applications.

Indeed, there have been few reports of long CROWs, in each case with transmission characteristics that show considerable ripple [7,12,13]. Moreover, the recently-demonstrated disorder-induced localization of slow light in CROWs and photonic crystal waveguides indicates serious challenges to making long structures, since all the eigenmodes of a one-dimensional waveguide are localized in theory for any value of disorder [18,19]. Disorder-tolerance can be achieved by lowering the finesse, F, of the CROW by designing for a larger value of the coupling coefficient κ, since F = π/[2sin−1(κ)], and compensating for the reduced overall group delay (τg α N/κ) by increasing the number of coupled resonators, N. Scaling up the number of resonators within a single CROW waveguide, while maintaining suitable “ballistic” [20,21] propagation characteristics, poses a significant technical challenge, as failure rates of such series-coupled structures may be thought to scale as a power law with the number of unit-cells in the exponent.

Here we show that silicon waveguide with up to 235 coupled microring resonators can be realized with very good transmission characteristics. CROW devices as shown in Figs. 1(a) -1(b) were fabricated at the IBM Microelectronics Research Laboratory using a CMOS compatible optical lithography process on silicon-on-insulator (SOI) wafers. The coupling lengths are shorter, and the resonators much more compact, in SOI CROWs than low-index contrast (e.g., polymer [4,22] or silicon oxynitride [6]) CROWs. Compared to the delay line structures in Ref. 4 and Ref. 6, our structure occupies 0.01% and 0.00001%, respectively, of the area per unit cell. However, in high index contrast materials such as SOI, issues of disorder are more important, since scattering losses from sidewall roughness can be significant and small errors in lithography can cause significant variations in inter-resonator coupling [2325]. Nevertheless, as Fig. 1(c) shows, there was relatively little degradation of transmission characteristics in going from 35 to 235 microrings, when comparing similar data in previous reports [7,12,13].

 figure: Fig. 1

Fig. 1 Coupled-resonator optical waveguides (CROWs). a, Silicon microring CROWs consisting of 35, 65, 95, 135 and 235 coupled microrings were fabricated using a CMOS compatible process on 200 mm silicon-on-insulator wafers and cleaved into 4 mm long chips, as shown in this optical microscope image. b, Scanning electron micrographs of selected regions, including the input/output waveguides and microring resonators formed from single-mode silicon wire waveguides (width = 0.5 μm) in the racetrack configuration (bending radius = 6.5 μm). c, Transmission (insertion loss) spectrum, i.e., the drop port response normalized to the input port (black: 35 ring structure; gray: 235 ring structure, + 7 dB net external amplification using an L-band EDFA).

Download Full Size | PPT Slide | PDF

Here, we present unambiguous evidence of non-localized slow light propagation over bands that are several hundred gigahertz wide, using high-resolution statistical measurements of transmission and infrared camera modal imaging. Furthermore, from the unique statistical signatures provided by novel time-domain propagation delay measurements, we demonstrate the correlated nature of light transport in coupled-resonator waveguides. The well-behaved light transmission characteristics of these structures shows that devices and systems consisting of hundreds of rings can play a significant role in the development of the advanced on-chip optical interconnects expected to enable computer systems with performance at the Exaflops level [26].

2. Experimental methods

CROWs were fabricated at the IBM Microelectronics Research Laboratory on silicon-on-insulator wafers. The waveguides were single-mode, with transverse dimensions approximately 0.50 μm x 0.20 μm, which is sufficient to strip the transverse-magnetic polarization of light after a few bends. The inter-resonator coupling gaps vary from 150 nm (apodized input section, which consists of 3 unit cells each at the input and output sections) to 270 nm (the remaining section of the waveguide), achieving the over-coupled regime of waveguide-ring coupling. The inter-ring coupling coefficient over the central portion of the CROW varies from |κ| ≅ 0.80 to 0.88 over the range of wavelengths in Fig. 1, and can be precisely calculated for each transmission band from the measured spectral bandwidth [27].

Issues of disorder are important in lithographically-fabricated waveguides since scattering losses from sidewall roughness are proportional to the variance of the line-edge roughness (typical values are 1 nm2 to 10 nm2). Here, a double thermal oxidation step was carried out on the wafer during the fabrication process, each step growing 20 nm of oxide on the surface of the silicon waveguide. The oxide grown after the first step was etched away by a buffered HF dip before the second layer was grown. This roughness-reduction process was used and characterized in an earlier study of 100-ring silicon CROWs, which estimated a line-edge roughness of 1.1 nm and a correlation length of 60 nm [12].

Each segment of the waveguides shown in Fig. 1 consists of a microring resonator. While all of the constituent oscillators should match identically in resonant frequency and inter-resonator coupling, in practice it is known that these ideal conditions are spoiled by unavoidable disorder in fabrication [12], which results in phase decoherence during transport [28]. Disorder manifests as a high-frequency “ripple” in the spectrally-resolved measurements of transmission; e.g., see Fig. 1(c).

Optical coupling into and out of the waveguides was achieved using tapered and lensed polarization-maintaining fibers. Transmission and group delay were measured using an optical vector network analyzer (OVA 5000 from Luna Technologies), based on the principle of swept-wavelength interferometry, with 1.4 pm spectral resolution. A mode-hop free rapidly-tunable laser source was used along with Mach-Zehnder interferometers, polarization controllers and photodiodes to measure the four elements of the Jones matrix of the device under test (two for each polarization). A discrete Fourier transform of the raw data, followed by time-domain windowing, obtained the time-domain impulse response of the device under test. The instrument measured up to 6 ns of group delay, which was determined by the window of the time-domain filter. The procedures used to calculate group delay from the measured transfer function, and its calibration against a known standard (e.g., acetylene gas cell) are described in [29].

The transverse-electric (TE) polarization of light was used in the reported measurements. The fiber-to-fiber insertion loss was −17 dB. This number includes the loss in coupling from the input fiber and output fiber to the silicon waveguides (estimated to be −8 dB/coupler from measurements on waveguides of various lengths), and residual losses from waveguide bends, radiation, etc. To overcome the coupling losses and maintain a high signal to noise ratio at the detector, the output of the OVA (average power 200 μW) was amplified (before the chip) by an L-band EDFA, which was used in the saturation regime (constant current mode, output power + 18 dBm), followed by a programmable passive attenuator to reduce the power level incident on the silicon chip to a sufficiently low level to avoid nonlinearities. No amplification was performed after the chip. In this way, it was ensured that the amplifier noise contribution remained constant, regardless of the spectral variations in the CROW response. Calibration measurements showed that the EDFA and attenuator combination added only 0.14 ps RMS noise to the measured propagation delay data.

Propagation delay measured data were corrected for the shorter “input” and “drop” straight waveguide sections in longer structures. As shown in Fig. 1(a), chips were prepared for measurement by cleaving between parallel planes, and the input and output waveguides for the 235-ring CROW were cumulatively 2.85 mm shorter compared to the input and output waveguides of the 35-ring CROW. Based on the measured group index in straight waveguides (4.25 over the range of wavelengths covered by the tunable laser), this length difference translated to 40.5 ps difference in the measured propagation time. Similar relative differences were calculated for the 65, 95 and 135 ring CROWs to be 6.1 ps, 12.2 ps, and 20.3 ps, respectively.

3. Measurements

3.1 Transmission (insertion loss) spectra

Figure 2 shows a typical set of transmission spectra from a group of 5 CROWs consisting of 35, 65, 95, 135, and 235 microrings on a single chip, showing well-aligned passbands over a wide range of wavelengths. The width of the passbands increased slightly with wavelength for all the chips measured because of the dispersion of the silicon waveguides [27]. However, the average level of transmission in the passband, as well as high on-off extinction, remained approximately the same for each CROW, thus showing the high level of ring-to-ring uniformity achieved in these CROW structures. Based on transfer-matrix simulations of transmission through disordered CROWs which yield similar traces to the measurements, we infer that the standard deviation of the microring optical path length is around 10 nm, and the standard deviation in the coupling coefficient is less than 0.045.

 figure: Fig. 2

Fig. 2 Transmission spectra. Transmission (insertion loss) spectra were measured with resolution 1.4 pm, for 35, 65, 95, 135, and 235 ring waveguides. Shown here are the averages of 128 measured traces. Relative to the 35-ring CROW, measurements for 65, 95, 135 and 235 ring CROWs were amplified by 7, 4, 3, and 7 dB, respectively in order to boost the power level detected at the photoreceiver.

Download Full Size | PPT Slide | PDF

Linear fits to the spectrally-integrated power transmitted through the CROWs of different length indicated an average loss per ring of −0.08 dB/ring. On nearby test sites, the waveguide propagation loss was obtained by measuring transmission through four waveguides (without resonators) of different lengths, but with the same number of bends, and was measured to be −3.9 dB/cm. The insertion loss of the 35 ring CROW by itself was only about −3 dB, inferred from the loss per ring measurement.

As the transverse-magnetic (TM) mode has a larger bending loss and propagation loss close to the cutoff, it is stripped off after propagation through a few bends. Therefore, the effect of polarization conversion on the spectral response in rings [30,31] can be neglected. In view of the high polarization selectivity of the CROWs (TE-polarized light is transmitted but not TM) and the polarization selectivity of the fiber patchcords, the in-built instrumental averaging of the Luna OVA 5000 [29] over the four components of the Jones matrix ultimately reports only on the desired TE polarization.

3.2 Group delay spectra

Figure 3 shows the group delay spectra for the CROWs over the same range of wavelengths as in Fig. 2. The slowing factor, c/v g, where v g is the group velocity, varies from 16.0 to 14.6 over the measured bands. The band-edge and stop-band regions, clearly indicated by the significantly increased group delay ripple at those wavelengths, are excluded from the statistical analysis presented here. Within the central region of the passband (approximately two-thirds of the edge-to-edge span), spectrally-dependent properties such as the density of states and the localization length should be approximately constant [16] and therefore, the variation of measured group delay with wavelength can be used as a statistical variable.

 figure: Fig. 3

Fig. 3 Propagation (group) delay spectra. Group delay spectra were measured with resolution 1.4 pm, for 35, 65, 95, 135, and 235 ring waveguides, over the same range of wavelengths as in Fig. 2. Spectral regions of large variation in delay correspond exactly to the stopbands of the intensity spectrum shown in Fig. 2.

Download Full Size | PPT Slide | PDF

Group delay ripple (GDR) here is defined as the difference between the measured group delay τ(λ) and τrms, the root-mean-squared group delay over the central portion of the passband. GDR is an important parameter in the study of devices for optical communications, since spectral components of GDR that are on the order of the signal bandwidth affect the signal spectrum by imposing an average chromatic dispersion, which can be compensated at the receiver. Therefore, in the study of fiber Bragg gratings, the average dispersion over the (narrow) band of interest is usually subtracted out from the phase variations, and the residual phase variation is used as the noise statistic which determines performance degradation [32]. The averaging bandwidth depends on the spectral width of the pulses used in data transmission. For example, 40 Gbps modulated optical data streams will average over ripple with spectral components less than 100 pm, effectively sensing them as a constant group delay [33].

Here, we use ripple statistics not to investigate the performance degradation of data transmission but the nature of light propagation in the CROWs. The band-to-band variation in average group delay was less than 0.1 ps/ring over a wavelength variation of 20 nm. A linear slope imposed on the spectral variation of the measured group delay by the EDFA of −1.81 ps/nm was subtracted from all the data. Calibration measurements showed that the EDFA added only 0.14 ps RMS noise to the measured propagation delay data, which is negligible in the context of approximately 30-200 ps of group delay ripple that characterizes the CROWs.

4. Discussion

In a regime where disorder may play a significant role, it is very important to realize that measurement of only the average transmission properties of a long waveguide constitutes an incomplete story of transport. This is because: (a) absorption reduces the average transmission of light in a similar way as does localization (exponentially with length) [3436], and (b) the average propagation time through the waveguide scales similarly in both the localized and non-localized regimes (linearly with length) [37,38]. A distinction between the localized and non-localized transport regimes can be obtained only through an analysis of the statistical properties of the transmission intensity and propagation time and further, by directly imaging the light propagating through the structure.

4.1 IR imaging

Eigenmodes of light propagation in a 235-ring waveguide were imaged using an infrared camera diagnostic method developed for multi-ring waveguides [39]. Selected wavelengths in the stop-band, band-edge and at locations throughout the passband, as indicated in Fig. 4(a) , were imaged under cw excitation from a narrow-linewidth tunable laser, using a microscope fitted with an infrared camera. Several image fields were stitched together laterally, but no data correction was made for the absorptive decay of intensity with length, thus showing clearly the low intrinsic loss of the propagating modes, quantified to be −0.08 dB/ring, see Fig. 5(a) . Since the chip was imaged through a semi-transparent and scattering polymeric cladding layer, the camera did not resolve individual microrings, but nevertheless clearly showed the general trends of propagation. Although modes at only a few selected wavelengths are shown in Fig. 4(b) for clarity, the results for other wavelengths were very similar, except at two sharp disorder-induced notches in the passband. Based on Fig. 4(b), it is clear that light can be transmitted throughout the entire length of a 235-ring CROW without localization.

 figure: Fig. 4

Fig. 4 Infrared images of CROW waveguide modes. a, Transmission (insertion loss) spectrum for a single passband of a 235 ring CROW, with measurements at selected wavelengths labeled (i)-(v). b, Intensity profiles of the eigenmodes at the wavelengths (ii)-(iv) measured with an infrared camera technique [39] show that non-localized excitations (extended throughout the entire waveguide length) were observed throughout the passband, in contrast with out-of-band (i) and band-edge (v) wavelengths. No correction was made in these images for the absorptive decay of intensity with length of the propagating modes [see Fig. 5(a)].

Download Full Size | PPT Slide | PDF

 figure: Fig. 5

Fig. 5 Transmission (intensity) measurements. a, The mid-band average of the transmitted intensity (in dB), measured without amplification, decreased linearly with length (−0.08 dB/resonator), except for one anomalous waveguide on the measured chip (65 rings) as discussed in the text. The errorbars represent the standard deviation, i.e., ripple, in the measured intensity over the flat portion of the band. b, The probability distribution function (PDFs) of the normalized intensity transmission (I^I/I) for the all the CROWs show agreement with the Rayleigh distribution, shown by the dashed lines, indicating non-localized transport through the waveguide. In contrast, the localized regime would show considerably different (long-tailed) statistics [8,34,40].

Download Full Size | PPT Slide | PDF

4.2 Transmission measurements

High-resolution spectral measurements (Figs. 2 and 3) reveal strong, well-resolved transmission passbands for CROWs composed of 35 to 235 coupled microrings. The high-frequency transmission “ripple” bears the signature of random disorder in the fabrication of CROWs, and can be seen to increase with the number of unit cells. Both disorder and loss contribute to a bandwidth narrowing with increasing length of only 0.15% per ring, i.e., 100 additional rings reduced the bandwidth by 15% compared to the bandwidth of the 35 ring structure. A wide passband exceeding 380 GHz (in each band) was therefore maintained even in the 235 ring structure, sufficient for many high-speed optical signal applications. As shown in Fig. 5(a), a linear fit to the spectrally-integrated power transmitted at mid-band for CROWs of different length indicated an average loss per ring of only −0.08 dB/ring, i.e., the insertion loss of a 35 ring CROW by itself was less than −3 dB. The 65 ring CROW may have had a defective input coupler or damaged cleaved facet, which lowered the overall transmission through the device, but did not affect the transmission statistics. As shown in Fig. 5(b), the measured probability distribution of the normalized intensity agreed with the Rayleigh distribution [dashed line, P(I^)=exp(I^)] which indicates transport occurred in the non-localized regime even in the longest CROW. This negative-exponential statistical signature of non-localized propagation is significantly distinct from the long-tailed log-normal distributions reported for diffusive and localized light [8,34,40].

4.3 Group delay measurements

Further evidence of non-localized transport was obtained from measurements of the transmission delay, which comprises the summed contributions from each of the rings encountered by photons traveling from input to output ports, and is therefore the sum of a large number of random variables. The average delay value <τ> (units of picoseconds, ps), shown in Fig. 6(a) , was obtained by the root-mean-squared average of the measured group delay data over the central one-half region of the transmission band. As expected [4,12], the average delay increased linearly with the number of resonators, N, with a slope <τ>/N = 0.73 ps/ring. This linear scaling was confirmed by additional measurements on more than 800 CROW bands measured over 16 different chips. However, as mentioned before, the average propagation time through the waveguide does not by itself provide conclusive evidence that transmission occurs in the non-localized regime; the average delay is indeed expected to scale linearly with length in both the localized and non-localized regimes [35]. Therefore, we examined the distributions of the normalized time delay of propagation (τ^τ/<τ>, where τ represents the raw group delay data, and the denominator <τ> is linearly proportional to N as discussed in the earlier paragraph), which are plotted in Fig. 4b, using a logarithmic scale on the vertical axis for clarity. As shown by the dashed lines in Fig. 6(b), the delay time distributions were well described by Gaussian statistics, characteristic of the ballistic, i.e., non-localized propagation regime only [37]. In contrast, diffusive or localized propagation would result in much wider (polynomial) tails to the distribution, as has been previously demonstrated for disordered microwave waveguides [36,41].

 figure: Fig. 6

Fig. 6 Propagation (time) delay measurements. a, The measured propagation delay averaged over the middle of a transmission band <τ> (units of picoseconds) increased linearly with length (L = 35, 65, …, 235 rings). The errorbars represent the standard deviation, i.e., ripple, in the measured delay over the flat portion of the band. b, The probability distributions of normalized delay τ^τ/<τ> were peaked at unity (i.e., τ = <τ>). The dashed line is a Gaussian (normal distribution) fit to the data, which indicates ballistic propagation statistics and the absence of localization, previously estimated to be a severe constraint on achieving >100 resonator lengths of CROWs. In fact, the self-averaging properties of longer chains of resonators yielded better fits to normal statistics than for the shorter waveguides, where finite-size effects caused an asymmetric lineshape in the tails of the distributions.

Download Full Size | PPT Slide | PDF

4.4 Statistical properties of group delay measurements

In fact, the self-averaging properties of longer chains of resonators yielded better fits to normal statistics (i.e. Figure 6(b), panels 3-5) than for the shorter waveguides (i.e. Figure 6(b), panels 1-2), where finite-size effects caused an asymmetric lineshape [42]. As shown in Fig. 7(a) , in the case of the three longest waveguides with 95, 135, and 235 coupled microrings, the normalized delay distributions converged to a single-parameter distribution [37] whose width describes the average level of group delay ripple per ring, equal to 0.19 ps/ring. This value can potentially be reduced with further improvements in fabrication, or by post-fabrication trimming [43].

 figure: Fig. 7

Fig. 7 Scaling of delay statistics with CROW length. a, The measured probability distribution functions (PDFs) of the normalized group delay τ^τ/<τ> are shown, using a logarithmic scale on the vertical axis for clarity, for the waveguides labeled (3)-(5) in Fig. 5(a). With increasing length, the distributions converged to a single-parameter Gaussian distribution, shown by the dashed black line. b, The variance of the measured delay (ps2) increased with the square of the number of resonators (N), as shown by the dashed fit, var τ = 0.0346 N2 + (12.9 ps)2 where the second term was the typically measured group delay ripple of the measurement apparatus. This scaling behavior was different from that of conventional waveguides or cascaded fiber Bragg gratings, and as discussed in the text, demonstrated that the individual resonator excitations are mutually correlated.

Download Full Size | PPT Slide | PDF

Moreover, since Fig. 7(a) shows that the width of the distribution of normalized delay was a constant, i.e., independent of N, it follows that the variance of the delay itself, var(τ), scaled as N2, as shown by the dotted line in Fig. 7(b). This prediction was in excellent agreement with var(τ) extracted directly from the measured data, shown by the squares in Fig. 7(b), and moreover agrees with numerical simulations (see Section 4.5). This behavior was in contrast to what is expected from conventional photonic waveguides, or a sequence of cascaded fiber Bragg gratings, in which cases var(τ) scales linearly with N [28,33,44].

To explain this behavior, we recall that according to statistical theory, the variance of the mean of an ensemble of uncorrelated random variables of sample size N decreases as N−1, which is commonly called the law of large numbers [45]. For an ensemble of correlated random variables, however, theory dictates that the variance of the sample mean (i.e., average delay per ring) reaches a constant value, independent of N, and is equal to the degree of correlation [45], so that the total delay variance of an N-ring waveguide thus scales as N2. This latter case is indeed the behavior shown in Fig. 7(b), and can be explained by the physical nature of light propagation in a CROW, in which coherent oscillations of all N coupled resonators constitute each of the propagating eigenmodes. The delay values measured across the passband are therefore shown to be correlated random variables, with a sample space increasing linearly with the length of the waveguide. Therefore, Fig. 7 provides experimental evidence of the mutually-correlated physical mechanism by which light propagates in a CROW. Understanding the statistical scaling behavior with length is particularly relevant for phase-sensitive applications e.g., in coherent optics, power combining, waveguide quantum light circuits and slow light.

Moreover, the fact that only nearest neighbors are directly coupled fulfils the criterion for applicability of the generalized Central Limit Theorem [46] and thus, the distribution of total delay will tend to a Gaussian probability distribution as N becomes large, exactly as shown by Fig. 7(a). The Gaussian fit is characteristic of only the ballistic propagation regime [37]; the statistics in the near-localization (diffusive) regime would show distributions with wider polynomially-decaying tails [47], becoming even wider in the localized regime, in fact, with exponential divergence of the higher moments [37]. Furthermore, localization would destroy any long-range phase correlation across the 235-ring length of the structure. Therefore, increasing the length of a waveguide structure wherein transport occurs in the localized regime would decrease the number of resonators that are mutually coupled [16,48], leading to an entirely different scaling behavior (exponentially growing with N) from Fig. 7(b) [37].

Transmission through CROWs was simulated using the transfer matrix method [22,49]. Numerical values of the coupling coefficients were chosen to match the experimentally measured bandwidths, free spectral ranges, and delays, as shown in Figs. 2 and 3. Disorder was introduced in both the coupling coefficients and in the eigenfrequencies of individual resonators, using independent and uncorrelated Gaussian distributions. The statistics of Monte-Carlo simulations of transmission were used to obtain Fig. 8 .

 figure: Fig. 8

Fig. 8 Simulation results of group delay ripple in disordered CROWs. The yellow dots are obtained from simulations of the calculated complex transmission coefficient over the passband, in the same way as done for the analysis of measured data. The dashed line shows that a quadratic function can be fit through these points. The inset shows the probability density distribution of the variance of the group delay, from Monte Carlo simulations at a particular value of N, the number of resonators.

Download Full Size | PPT Slide | PDF

Figure 8 confirms the experimentally-measured scaling trend shown in Fig. 7(b), i.e., that the variance of the propagation delay scales quadratically with the number of resonators in the CROW. The vertical axis of Fig. 8 (simulation) covers a larger range of values than in Fig. 7(b) (measurements), showing that the quadratic scaling of the variance of τ with N remains valid, in well-apodized CROWs, even to significantly higher values of group delay ripple.

5. Conclusion

In order to realize the advanced on-chip optical interconnects expected to enable computer systems with performance at the Exaflops level, it is foreseeable that ensembles of hundreds, or even thousands, of resonators will be required. The band-edge localization regime may eventually impact the transport characteristics as the length of the CROW waveguides is increased, since the passband width narrows by approximately 0.15% per ring (see Fig. 2) from the effects of disorder and loss. However, using the data obtained here, we can estimate that ballistic transport will persist until the number of rings exceeds approximately 650, until which point the undesirable and unpredictable effects of localization will not impede system performance. While further reduction of disorder may be required in order to provide an additional performance margin, our results indicate that coupled-resonator optical waveguides can be designed with sufficient control over the effects of disorder to yield large optical component ensembles suitable for large-scale high-performance silicon photonics.

Acknowledgements

The authors would like to acknowledge the staff at the IBM Microelectronics Research Laboratory, where the devices were fabricated. This work was supported in part by the National Science Foundation under grants ECCS-0642603 and ECCS-0925399, the California Institute for Telecommunications and Information Technology at the University of California, San Diego (UCSD), and a UCSD-IBM Joint Study Agreement.

References and links

1. 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). [CrossRef]  

2. J. E. Heebner, P. Chak, S. Pereira, J. E. Sipe, and R. W. Boyd, “Distributed and localized feedback in microresonator sequences for linear and nonlinear optics,” J. Opt. Soc. Am. B 21(10), 1818–1832 (2004). [CrossRef]  

3. P. Sanchis, J. Marti, W. Bogaerts, P. Dumon, D. Van Thourhout, and R. Baets, “Experimental results on adiabatic coupling into SOI photonic crystal coupled-cavity waveguides,” IEEE Photon. Technol. Lett. 17(6), 1199–1201 (2005). [CrossRef]  

4. J. K. S. Poon, L. Zhu, G. A. DeRose, and A. Yariv, “Transmission and group delay of microring coupled-resonator optical waveguides,” Opt. Lett. 31(4), 456–458 (2006). [CrossRef]   [PubMed]  

5. Y. A. Vlasov, W. M. J. Green, and F. Xia, “High-throughput silicon nanophotonic wavelength-insensitive switch for on-chip optical networks,” Nat. Photonics 2(4), 242–246 (2008). [CrossRef]  

6. 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]   [PubMed]  

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

8. L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, and P. Lodahl, “Cavity quantum electrodynamics with Anderson-localized modes,” Science 327(5971), 1352–1355 (2010). [CrossRef]   [PubMed]  

9. S. Mazoyer, P. Lalanne, J. C. Rodier, J. P. Hugonin, M. Spasenović, L. Kuipers, D. M. Beggs, and T. F. Krauss, “Statistical fluctuations of transmission in slow light photonic-crystal waveguides,” Opt. Express 18(14), 14654–14663 (2010). [CrossRef]   [PubMed]  

10. T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2(8), 465–473 (2008). [CrossRef]  

11. J. B. Khurgin, “Optical buffers based on slow light in electromagnetically induced transparent media and coupled resonator structures: comparative analysis,” J. Opt. Soc. Am. B 22(5), 1062–1074 (2005). [CrossRef]  

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

13. X. Luo and A. W. Poon, “Many-element coupled-resonator optical waveguides using gapless-coupled microdisk resonators,” Opt. Express 17(26), 23617–23628 (2009). [CrossRef]  

14. H. Shen, M. H. Khan, L. Fan, L. Zhao, Y. Xuan, J. Ouyang, L. T. Varghese, and M. Qi, “Eight-channel reconfigurable microring filters with tunable frequency, extinction ratio and bandwidth,” Opt. Express 18(17), 18067–18076 (2010). [CrossRef]   [PubMed]  

15. D. Dai, L. Yang, and S. He, “Ultrasmall Thermally Tunable Microring Resonator With a Submicrometer Heater on Si Nanowires,” J. Lightwave Technol. 26(6), 704–709 (2008). [CrossRef]  

16. S. Mookherjea, J. S. Park, S. H. Yang, and P. R. Bandaru, “Localization in silicon nanophotonic slow-light waveguides,” Nat. Photonics 2(2), 90–93 (2008). [CrossRef]  

17. J. Topolancik, B. Ilic, and F. Vollmer, “Experimental observation of strong photon localization in disordered photonic crystal waveguides,” Phys. Rev. Lett. 99(25), 253901 (2007). [CrossRef]  

18. D. C. Herbert and R. Jones, “Localized States in Disordered Systems,” J Phys. C 4, 1145 (1971).

19. D. J. Thouless, “Relation between Density of States and Range of Localization for One Dimensional Random Systems,” J Phys. C 5, 77 (1972).

20. D. K. Ferry, Semiconductor transport (Taylor & Francis, London, New York, 2000), pp. x, 368 p.

21. E. Akkermans, and G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University Press, Cambridge, 2007), pp. xviii, 588 p.

22. G. T. Paloczi, Y. Y. Huang, A. Yariv, and S. Mookherjea, “Polymeric Mach-Zehnder interferometer using serially coupled microring resonators,” Opt. Express 11(21), 2666–2671 (2003). [CrossRef]   [PubMed]  

23. F. Grillot, L. Vivien, S. Laval, D. Pascal, and E. Cassan, “Size influence on the propagation loss induced by sidewall roughness in ultrasmall SOI waveguides,” IEEE Photon. Technol. Lett. 16(7), 1661–1663 (2004). [CrossRef]  

24. T. Barwicz and H. A. Haus, “Three-dimensional analysis of scattering losses due to sidewall roughness, in microphotonic waveguides,” J. Lightwave Technol. 23(9), 2719–2732 (2005). [CrossRef]  

25. C. Ferrari, F. Morichetti, and A. Melloni, “Disorder in coupled-resonator optical waveguides,” J. Opt. Soc. Am. B 26(4), 858–866 (2009). [CrossRef]  

26. DARPA IPTO, “Exascale Computing Study: Technology Challenges in Achieving Exascale Systems,” (2008).

27. M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, D. K. Gifford, and S. Mookherjea, “Waveguide dispersion effects in silicon-on-insulator coupled-resonator optical waveguides,” Opt. Lett. 35(18), 3030–3032 (2010). [CrossRef]   [PubMed]  

28. R. Adar, C. H. Henry, M. A. Milbrodt, and R. C. Kistler, “Phase coherence of optical waveguides,” J. Lightwave Technol. 12(4), 603–606 (1994). [CrossRef]  

29. D. K. Gifford, B. J. Soller, M. S. Wolfe, and M. E. Froggatt, “Optical vector network analyzer for single-scan measurements of loss, group delay, and polarization mode dispersion,” Appl. Opt. 44(34), 7282–7286 (2005). [CrossRef]   [PubMed]  

30. A. Melloni, F. Morichetti, and M. Martinelli, “Polarization conversion in ring resonator phase shifters,” Opt. Lett. 29(23), 2785–2787 (2004). [CrossRef]   [PubMed]  

31. F. Morichetti, A. Melloni, and M. Martinelli, “Effects of polarization rotation in optical ring-resonator-based devices,” J. Lightwave Technol. 24(1), 573–585 (2006). [CrossRef]  

32. M. H. Eiselt, C. B. Clausen, and R. W. Tkach, “Performance characterization of components with group delay fluctuations,” IEEE Photon. Technol. Lett. 15(8), 1076–1078 (2003). [CrossRef]  

33. H. Chotard, Y. Painchaud, A. Mailloux, M. Morin, F. Trepanier, and M. Guy, “Group delay ripple of cascaded Bragg grating gain flattening filters,” IEEE Photon. Technol. Lett. 14(8), 1130–1132 (2002). [CrossRef]  

34. E. Kogan and M. Kaveh, “Random-matrix-theory approach to the intensity distributions of waves propagating in a random medium,” Phys. Rev. B Condens. Matter 52(6), R3813–R3815 (1995). [CrossRef]   [PubMed]  

35. P. Markoš and C. M. Soukoulis, “Intensity distribution of scalar waves propagating in random media,” Phys. Rev. B 71(5), 054201 (2005). [CrossRef]  

36. J. A. Sánchez-Gil, V. Freilikher, I. Yurkevich, and A. A. Maradudin, “Coexistence of Ballistic Transport, Diffusion, and Localization in Surface Disordered Waveguides,” Phys. Rev. Lett. 80(5), 948–951 (1998). [CrossRef]  

37. C. Texier and A. Comtet, “Universality of the Wigner time delay distribution for one-dimensional random potentials,” Phys. Rev. Lett. 82(21), 4220–4223 (1999). [CrossRef]  

38. A. Z. Genack, P. Sebbah, M. Stoytchev, and B. A. van Tiggelen, “Statistics of Wave Dynamics in Random Media,” Phys. Rev. Lett. 82(4), 715–718 (1999). [CrossRef]  

39. M. L. Cooper, G. Gupta, J. S. Park, M. A. Schneider, I. B. Divliansky, and S. Mookherjea, “Quantitative infrared imaging of silicon-on-insulator microring resonators,” Opt. Lett. 35(5), 784–786 (2010). [CrossRef]   [PubMed]  

40. M. Stoytchev and A. Z. Genack, “Observations of non-Rayleigh statistics in the approach to photon localization,” Opt. Lett. 24(4), 262–264 (1999). [CrossRef]  

41. A. A. Chabanov and A. Z. Genack, “Statistics of dynamics of localized waves,” Phys. Rev. Lett. 87(23), 233903 (2001). [CrossRef]   [PubMed]  

42. J. K. S. Poon, L. Zhu, G. A. DeRose, and A. Yariv, “Polymer Microring Coupled-Resonator Optical Waveguides,” J. Lightwave Technol. 24(4), 1843–1849 (2006). [CrossRef]  

43. C. K. Madsen and J. H. Zhao, “Postfabrication optimization of an autoregressive planar waveguide lattice filter,” Appl. Opt. 36(3), 642–647 (1997). [CrossRef]   [PubMed]  

44. T. Goh, S. Suzuki, and A. Sugita, “Estimation of waveguide phase error in silica-based waveguides,” J. Lightwave Technol. 15(11), 2107–2113 (1997). [CrossRef]  

45. W. B. Davenport, and W. L. Root, An Introduction to the Theory of Random Signals and Noise, Lincoln Laboratory Publications (McGraw-Hill, New York, 1958), p. 393.

46. B. Rosén, “On the central limit theorem for sums of dependent random variables,” Probab. Theory Relat. Fields 7, 48–82 (1967).

47. B. A. van Tiggelen, P. Sebbah, M. Stoytchev, and A. Z. Genack, “Delay-time statistics for diffuse waves,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 59(6), 7166–7172 (1999). [CrossRef]  

48. Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100(1), 013906 (2008). [CrossRef]   [PubMed]  

49. J. K. S. Poon, J. Scheuer, S. Mookherjea, G. T. Paloczi, Y. Y. Huang, and A. Yariv, “Matrix analysis of microring coupled-resonator optical waveguides,” Opt. Express 12(1), 90–103 (2004). [CrossRef]   [PubMed]  

References

  • View by:
  • |
  • |
  • |

  1. 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).
    [Crossref]
  2. J. E. Heebner, P. Chak, S. Pereira, J. E. Sipe, and R. W. Boyd, “Distributed and localized feedback in microresonator sequences for linear and nonlinear optics,” J. Opt. Soc. Am. B 21(10), 1818–1832 (2004).
    [Crossref]
  3. P. Sanchis, J. Marti, W. Bogaerts, P. Dumon, D. Van Thourhout, and R. Baets, “Experimental results on adiabatic coupling into SOI photonic crystal coupled-cavity waveguides,” IEEE Photon. Technol. Lett. 17(6), 1199–1201 (2005).
    [Crossref]
  4. J. K. S. Poon, L. Zhu, G. A. DeRose, and A. Yariv, “Transmission and group delay of microring coupled-resonator optical waveguides,” Opt. Lett. 31(4), 456–458 (2006).
    [Crossref] [PubMed]
  5. Y. A. Vlasov, W. M. J. Green, and F. Xia, “High-throughput silicon nanophotonic wavelength-insensitive switch for on-chip optical networks,” Nat. Photonics 2(4), 242–246 (2008).
    [Crossref]
  6. 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] [PubMed]
  7. M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nat. Photonics 2(12), 741–747 (2008).
    [Crossref]
  8. L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, and P. Lodahl, “Cavity quantum electrodynamics with Anderson-localized modes,” Science 327(5971), 1352–1355 (2010).
    [Crossref] [PubMed]
  9. S. Mazoyer, P. Lalanne, J. C. Rodier, J. P. Hugonin, M. Spasenović, L. Kuipers, D. M. Beggs, and T. F. Krauss, “Statistical fluctuations of transmission in slow light photonic-crystal waveguides,” Opt. Express 18(14), 14654–14663 (2010).
    [Crossref] [PubMed]
  10. T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2(8), 465–473 (2008).
    [Crossref]
  11. J. B. Khurgin, “Optical buffers based on slow light in electromagnetically induced transparent media and coupled resonator structures: comparative analysis,” J. Opt. Soc. Am. B 22(5), 1062–1074 (2005).
    [Crossref]
  12. F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1(1), 65–71 (2007).
    [Crossref]
  13. X. Luo and A. W. Poon, “Many-element coupled-resonator optical waveguides using gapless-coupled microdisk resonators,” Opt. Express 17(26), 23617–23628 (2009).
    [Crossref]
  14. H. Shen, M. H. Khan, L. Fan, L. Zhao, Y. Xuan, J. Ouyang, L. T. Varghese, and M. Qi, “Eight-channel reconfigurable microring filters with tunable frequency, extinction ratio and bandwidth,” Opt. Express 18(17), 18067–18076 (2010).
    [Crossref] [PubMed]
  15. D. Dai, L. Yang, and S. He, “Ultrasmall Thermally Tunable Microring Resonator With a Submicrometer Heater on Si Nanowires,” J. Lightwave Technol. 26(6), 704–709 (2008).
    [Crossref]
  16. S. Mookherjea, J. S. Park, S. H. Yang, and P. R. Bandaru, “Localization in silicon nanophotonic slow-light waveguides,” Nat. Photonics 2(2), 90–93 (2008).
    [Crossref]
  17. J. Topolancik, B. Ilic, and F. Vollmer, “Experimental observation of strong photon localization in disordered photonic crystal waveguides,” Phys. Rev. Lett. 99(25), 253901 (2007).
    [Crossref]
  18. D. C. Herbert and R. Jones, “Localized States in Disordered Systems,” J Phys. C 4, 1145 (1971).
  19. D. J. Thouless, “Relation between Density of States and Range of Localization for One Dimensional Random Systems,” J Phys. C 5, 77 (1972).
  20. D. K. Ferry, Semiconductor transport (Taylor & Francis, London, New York, 2000), pp. x, 368 p.
  21. E. Akkermans, and G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University Press, Cambridge, 2007), pp. xviii, 588 p.
  22. G. T. Paloczi, Y. Y. Huang, A. Yariv, and S. Mookherjea, “Polymeric Mach-Zehnder interferometer using serially coupled microring resonators,” Opt. Express 11(21), 2666–2671 (2003).
    [Crossref] [PubMed]
  23. F. Grillot, L. Vivien, S. Laval, D. Pascal, and E. Cassan, “Size influence on the propagation loss induced by sidewall roughness in ultrasmall SOI waveguides,” IEEE Photon. Technol. Lett. 16(7), 1661–1663 (2004).
    [Crossref]
  24. T. Barwicz and H. A. Haus, “Three-dimensional analysis of scattering losses due to sidewall roughness, in microphotonic waveguides,” J. Lightwave Technol. 23(9), 2719–2732 (2005).
    [Crossref]
  25. C. Ferrari, F. Morichetti, and A. Melloni, “Disorder in coupled-resonator optical waveguides,” J. Opt. Soc. Am. B 26(4), 858–866 (2009).
    [Crossref]
  26. DARPA IPTO, “Exascale Computing Study: Technology Challenges in Achieving Exascale Systems,” (2008).
  27. M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, D. K. Gifford, and S. Mookherjea, “Waveguide dispersion effects in silicon-on-insulator coupled-resonator optical waveguides,” Opt. Lett. 35(18), 3030–3032 (2010).
    [Crossref] [PubMed]
  28. R. Adar, C. H. Henry, M. A. Milbrodt, and R. C. Kistler, “Phase coherence of optical waveguides,” J. Lightwave Technol. 12(4), 603–606 (1994).
    [Crossref]
  29. D. K. Gifford, B. J. Soller, M. S. Wolfe, and M. E. Froggatt, “Optical vector network analyzer for single-scan measurements of loss, group delay, and polarization mode dispersion,” Appl. Opt. 44(34), 7282–7286 (2005).
    [Crossref] [PubMed]
  30. A. Melloni, F. Morichetti, and M. Martinelli, “Polarization conversion in ring resonator phase shifters,” Opt. Lett. 29(23), 2785–2787 (2004).
    [Crossref] [PubMed]
  31. F. Morichetti, A. Melloni, and M. Martinelli, “Effects of polarization rotation in optical ring-resonator-based devices,” J. Lightwave Technol. 24(1), 573–585 (2006).
    [Crossref]
  32. M. H. Eiselt, C. B. Clausen, and R. W. Tkach, “Performance characterization of components with group delay fluctuations,” IEEE Photon. Technol. Lett. 15(8), 1076–1078 (2003).
    [Crossref]
  33. H. Chotard, Y. Painchaud, A. Mailloux, M. Morin, F. Trepanier, and M. Guy, “Group delay ripple of cascaded Bragg grating gain flattening filters,” IEEE Photon. Technol. Lett. 14(8), 1130–1132 (2002).
    [Crossref]
  34. E. Kogan and M. Kaveh, “Random-matrix-theory approach to the intensity distributions of waves propagating in a random medium,” Phys. Rev. B Condens. Matter 52(6), R3813–R3815 (1995).
    [Crossref] [PubMed]
  35. P. Markoš and C. M. Soukoulis, “Intensity distribution of scalar waves propagating in random media,” Phys. Rev. B 71(5), 054201 (2005).
    [Crossref]
  36. J. A. Sánchez-Gil, V. Freilikher, I. Yurkevich, and A. A. Maradudin, “Coexistence of Ballistic Transport, Diffusion, and Localization in Surface Disordered Waveguides,” Phys. Rev. Lett. 80(5), 948–951 (1998).
    [Crossref]
  37. C. Texier and A. Comtet, “Universality of the Wigner time delay distribution for one-dimensional random potentials,” Phys. Rev. Lett. 82(21), 4220–4223 (1999).
    [Crossref]
  38. A. Z. Genack, P. Sebbah, M. Stoytchev, and B. A. van Tiggelen, “Statistics of Wave Dynamics in Random Media,” Phys. Rev. Lett. 82(4), 715–718 (1999).
    [Crossref]
  39. M. L. Cooper, G. Gupta, J. S. Park, M. A. Schneider, I. B. Divliansky, and S. Mookherjea, “Quantitative infrared imaging of silicon-on-insulator microring resonators,” Opt. Lett. 35(5), 784–786 (2010).
    [Crossref] [PubMed]
  40. M. Stoytchev and A. Z. Genack, “Observations of non-Rayleigh statistics in the approach to photon localization,” Opt. Lett. 24(4), 262–264 (1999).
    [Crossref]
  41. A. A. Chabanov and A. Z. Genack, “Statistics of dynamics of localized waves,” Phys. Rev. Lett. 87(23), 233903 (2001).
    [Crossref] [PubMed]
  42. J. K. S. Poon, L. Zhu, G. A. DeRose, and A. Yariv, “Polymer Microring Coupled-Resonator Optical Waveguides,” J. Lightwave Technol. 24(4), 1843–1849 (2006).
    [Crossref]
  43. C. K. Madsen and J. H. Zhao, “Postfabrication optimization of an autoregressive planar waveguide lattice filter,” Appl. Opt. 36(3), 642–647 (1997).
    [Crossref] [PubMed]
  44. T. Goh, S. Suzuki, and A. Sugita, “Estimation of waveguide phase error in silica-based waveguides,” J. Lightwave Technol. 15(11), 2107–2113 (1997).
    [Crossref]
  45. W. B. Davenport, and W. L. Root, An Introduction to the Theory of Random Signals and Noise, Lincoln Laboratory Publications (McGraw-Hill, New York, 1958), p. 393.
  46. B. Rosén, “On the central limit theorem for sums of dependent random variables,” Probab. Theory Relat. Fields 7, 48–82 (1967).
  47. B. A. van Tiggelen, P. Sebbah, M. Stoytchev, and A. Z. Genack, “Delay-time statistics for diffuse waves,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 59(6), 7166–7172 (1999).
    [Crossref]
  48. Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
    [Crossref] [PubMed]
  49. J. K. S. Poon, J. Scheuer, S. Mookherjea, G. T. Paloczi, Y. Y. Huang, and A. Yariv, “Matrix analysis of microring coupled-resonator optical waveguides,” Opt. Express 12(1), 90–103 (2004).
    [Crossref] [PubMed]

2010 (5)

2009 (2)

2008 (7)

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[Crossref] [PubMed]

Y. A. Vlasov, W. M. J. Green, and F. Xia, “High-throughput silicon nanophotonic wavelength-insensitive switch for on-chip optical networks,” Nat. Photonics 2(4), 242–246 (2008).
[Crossref]

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] [PubMed]

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

D. Dai, L. Yang, and S. He, “Ultrasmall Thermally Tunable Microring Resonator With a Submicrometer Heater on Si Nanowires,” J. Lightwave Technol. 26(6), 704–709 (2008).
[Crossref]

S. Mookherjea, J. S. Park, S. H. Yang, and P. R. Bandaru, “Localization in silicon nanophotonic slow-light waveguides,” Nat. Photonics 2(2), 90–93 (2008).
[Crossref]

T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2(8), 465–473 (2008).
[Crossref]

2007 (2)

J. Topolancik, B. Ilic, and F. Vollmer, “Experimental observation of strong photon localization in disordered photonic crystal waveguides,” Phys. Rev. Lett. 99(25), 253901 (2007).
[Crossref]

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

2006 (3)

2005 (5)

2004 (4)

2003 (2)

M. H. Eiselt, C. B. Clausen, and R. W. Tkach, “Performance characterization of components with group delay fluctuations,” IEEE Photon. Technol. Lett. 15(8), 1076–1078 (2003).
[Crossref]

G. T. Paloczi, Y. Y. Huang, A. Yariv, and S. Mookherjea, “Polymeric Mach-Zehnder interferometer using serially coupled microring resonators,” Opt. Express 11(21), 2666–2671 (2003).
[Crossref] [PubMed]

2002 (1)

H. Chotard, Y. Painchaud, A. Mailloux, M. Morin, F. Trepanier, and M. Guy, “Group delay ripple of cascaded Bragg grating gain flattening filters,” IEEE Photon. Technol. Lett. 14(8), 1130–1132 (2002).
[Crossref]

2001 (1)

A. A. Chabanov and A. Z. Genack, “Statistics of dynamics of localized waves,” Phys. Rev. Lett. 87(23), 233903 (2001).
[Crossref] [PubMed]

1999 (5)

B. A. van Tiggelen, P. Sebbah, M. Stoytchev, and A. Z. Genack, “Delay-time statistics for diffuse waves,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 59(6), 7166–7172 (1999).
[Crossref]

M. Stoytchev and A. Z. Genack, “Observations of non-Rayleigh statistics in the approach to photon localization,” Opt. Lett. 24(4), 262–264 (1999).
[Crossref]

C. Texier and A. Comtet, “Universality of the Wigner time delay distribution for one-dimensional random potentials,” Phys. Rev. Lett. 82(21), 4220–4223 (1999).
[Crossref]

A. Z. Genack, P. Sebbah, M. Stoytchev, and B. A. van Tiggelen, “Statistics of Wave Dynamics in Random Media,” Phys. Rev. Lett. 82(4), 715–718 (1999).
[Crossref]

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).
[Crossref]

1998 (1)

J. A. Sánchez-Gil, V. Freilikher, I. Yurkevich, and A. A. Maradudin, “Coexistence of Ballistic Transport, Diffusion, and Localization in Surface Disordered Waveguides,” Phys. Rev. Lett. 80(5), 948–951 (1998).
[Crossref]

1997 (2)

C. K. Madsen and J. H. Zhao, “Postfabrication optimization of an autoregressive planar waveguide lattice filter,” Appl. Opt. 36(3), 642–647 (1997).
[Crossref] [PubMed]

T. Goh, S. Suzuki, and A. Sugita, “Estimation of waveguide phase error in silica-based waveguides,” J. Lightwave Technol. 15(11), 2107–2113 (1997).
[Crossref]

1995 (1)

E. Kogan and M. Kaveh, “Random-matrix-theory approach to the intensity distributions of waves propagating in a random medium,” Phys. Rev. B Condens. Matter 52(6), R3813–R3815 (1995).
[Crossref] [PubMed]

1994 (1)

R. Adar, C. H. Henry, M. A. Milbrodt, and R. C. Kistler, “Phase coherence of optical waveguides,” J. Lightwave Technol. 12(4), 603–606 (1994).
[Crossref]

1967 (1)

B. Rosén, “On the central limit theorem for sums of dependent random variables,” Probab. Theory Relat. Fields 7, 48–82 (1967).

Adar, R.

R. Adar, C. H. Henry, M. A. Milbrodt, and R. C. Kistler, “Phase coherence of optical waveguides,” J. Lightwave Technol. 12(4), 603–606 (1994).
[Crossref]

Assefa, S.

Avidan, A.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[Crossref] [PubMed]

Baba, T.

T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2(8), 465–473 (2008).
[Crossref]

Baets, R.

P. Sanchis, J. Marti, W. Bogaerts, P. Dumon, D. Van Thourhout, and R. Baets, “Experimental results on adiabatic coupling into SOI photonic crystal coupled-cavity waveguides,” IEEE Photon. Technol. Lett. 17(6), 1199–1201 (2005).
[Crossref]

Bandaru, P. R.

S. Mookherjea, J. S. Park, S. H. Yang, and P. R. Bandaru, “Localization in silicon nanophotonic slow-light waveguides,” Nat. Photonics 2(2), 90–93 (2008).
[Crossref]

Barwicz, T.

Beggs, D. M.

Bogaerts, W.

P. Sanchis, J. Marti, W. Bogaerts, P. Dumon, D. Van Thourhout, and R. Baets, “Experimental results on adiabatic coupling into SOI photonic crystal coupled-cavity waveguides,” IEEE Photon. Technol. Lett. 17(6), 1199–1201 (2005).
[Crossref]

Boyd, R. W.

Cassan, E.

F. Grillot, L. Vivien, S. Laval, D. Pascal, and E. Cassan, “Size influence on the propagation loss induced by sidewall roughness in ultrasmall SOI waveguides,” IEEE Photon. Technol. Lett. 16(7), 1661–1663 (2004).
[Crossref]

Chabanov, A. A.

A. A. Chabanov and A. Z. Genack, “Statistics of dynamics of localized waves,” Phys. Rev. Lett. 87(23), 233903 (2001).
[Crossref] [PubMed]

Chak, P.

Chotard, H.

H. Chotard, Y. Painchaud, A. Mailloux, M. Morin, F. Trepanier, and M. Guy, “Group delay ripple of cascaded Bragg grating gain flattening filters,” IEEE Photon. Technol. Lett. 14(8), 1130–1132 (2002).
[Crossref]

Christodoulides, D. N.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[Crossref] [PubMed]

Clausen, C. B.

M. H. Eiselt, C. B. Clausen, and R. W. Tkach, “Performance characterization of components with group delay fluctuations,” IEEE Photon. Technol. Lett. 15(8), 1076–1078 (2003).
[Crossref]

Comtet, A.

C. Texier and A. Comtet, “Universality of the Wigner time delay distribution for one-dimensional random potentials,” Phys. Rev. Lett. 82(21), 4220–4223 (1999).
[Crossref]

Cooper, M. L.

Dai, D.

DeRose, G. A.

Divliansky, I. B.

Dumon, P.

P. Sanchis, J. Marti, W. Bogaerts, P. Dumon, D. Van Thourhout, and R. Baets, “Experimental results on adiabatic coupling into SOI photonic crystal coupled-cavity waveguides,” IEEE Photon. Technol. Lett. 17(6), 1199–1201 (2005).
[Crossref]

Eiselt, M. H.

M. H. Eiselt, C. B. Clausen, and R. W. Tkach, “Performance characterization of components with group delay fluctuations,” IEEE Photon. Technol. Lett. 15(8), 1076–1078 (2003).
[Crossref]

Fan, L.

Ferrari, C.

Freilikher, V.

J. A. Sánchez-Gil, V. Freilikher, I. Yurkevich, and A. A. Maradudin, “Coexistence of Ballistic Transport, Diffusion, and Localization in Surface Disordered Waveguides,” Phys. Rev. Lett. 80(5), 948–951 (1998).
[Crossref]

Froggatt, M. E.

Garcia, P. D.

L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, and P. Lodahl, “Cavity quantum electrodynamics with Anderson-localized modes,” Science 327(5971), 1352–1355 (2010).
[Crossref] [PubMed]

Genack, A. Z.

A. A. Chabanov and A. Z. Genack, “Statistics of dynamics of localized waves,” Phys. Rev. Lett. 87(23), 233903 (2001).
[Crossref] [PubMed]

A. Z. Genack, P. Sebbah, M. Stoytchev, and B. A. van Tiggelen, “Statistics of Wave Dynamics in Random Media,” Phys. Rev. Lett. 82(4), 715–718 (1999).
[Crossref]

B. A. van Tiggelen, P. Sebbah, M. Stoytchev, and A. Z. Genack, “Delay-time statistics for diffuse waves,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 59(6), 7166–7172 (1999).
[Crossref]

M. Stoytchev and A. Z. Genack, “Observations of non-Rayleigh statistics in the approach to photon localization,” Opt. Lett. 24(4), 262–264 (1999).
[Crossref]

Gifford, D. K.

Goh, T.

T. Goh, S. Suzuki, and A. Sugita, “Estimation of waveguide phase error in silica-based waveguides,” J. Lightwave Technol. 15(11), 2107–2113 (1997).
[Crossref]

Green, W. M. J.

M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, D. K. Gifford, and S. Mookherjea, “Waveguide dispersion effects in silicon-on-insulator coupled-resonator optical waveguides,” Opt. Lett. 35(18), 3030–3032 (2010).
[Crossref] [PubMed]

Y. A. Vlasov, W. M. J. Green, and F. Xia, “High-throughput silicon nanophotonic wavelength-insensitive switch for on-chip optical networks,” Nat. Photonics 2(4), 242–246 (2008).
[Crossref]

Grillot, F.

F. Grillot, L. Vivien, S. Laval, D. Pascal, and E. Cassan, “Size influence on the propagation loss induced by sidewall roughness in ultrasmall SOI waveguides,” IEEE Photon. Technol. Lett. 16(7), 1661–1663 (2004).
[Crossref]

Gupta, G.

Guy, M.

H. Chotard, Y. Painchaud, A. Mailloux, M. Morin, F. Trepanier, and M. Guy, “Group delay ripple of cascaded Bragg grating gain flattening filters,” IEEE Photon. Technol. Lett. 14(8), 1130–1132 (2002).
[Crossref]

Haus, H. A.

He, S.

Heebner, J. E.

Henry, C. H.

R. Adar, C. H. Henry, M. A. Milbrodt, and R. C. Kistler, “Phase coherence of optical waveguides,” J. Lightwave Technol. 12(4), 603–606 (1994).
[Crossref]

Huang, Y. Y.

Hugonin, J. P.

Ilic, B.

J. Topolancik, B. Ilic, and F. Vollmer, “Experimental observation of strong photon localization in disordered photonic crystal waveguides,” Phys. Rev. Lett. 99(25), 253901 (2007).
[Crossref]

Kaveh, M.

E. Kogan and M. Kaveh, “Random-matrix-theory approach to the intensity distributions of waves propagating in a random medium,” Phys. Rev. B Condens. Matter 52(6), R3813–R3815 (1995).
[Crossref] [PubMed]

Khan, M. H.

Khurgin, J. B.

Kistler, R. C.

R. Adar, C. H. Henry, M. A. Milbrodt, and R. C. Kistler, “Phase coherence of optical waveguides,” J. Lightwave Technol. 12(4), 603–606 (1994).
[Crossref]

Kogan, E.

E. Kogan and M. Kaveh, “Random-matrix-theory approach to the intensity distributions of waves propagating in a random medium,” Phys. Rev. B Condens. Matter 52(6), R3813–R3815 (1995).
[Crossref] [PubMed]

Krauss, T. F.

Kuipers, L.

Kuramochi, E.

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

Lahini, Y.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[Crossref] [PubMed]

Lalanne, P.

Laval, S.

F. Grillot, L. Vivien, S. Laval, D. Pascal, and E. Cassan, “Size influence on the propagation loss induced by sidewall roughness in ultrasmall SOI waveguides,” IEEE Photon. Technol. Lett. 16(7), 1661–1663 (2004).
[Crossref]

Lee, R. K.

Lodahl, P.

L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, and P. Lodahl, “Cavity quantum electrodynamics with Anderson-localized modes,” Science 327(5971), 1352–1355 (2010).
[Crossref] [PubMed]

Luo, X.

Madsen, C. K.

Mailloux, A.

H. Chotard, Y. Painchaud, A. Mailloux, M. Morin, F. Trepanier, and M. Guy, “Group delay ripple of cascaded Bragg grating gain flattening filters,” IEEE Photon. Technol. Lett. 14(8), 1130–1132 (2002).
[Crossref]

Maradudin, A. A.

J. A. Sánchez-Gil, V. Freilikher, I. Yurkevich, and A. A. Maradudin, “Coexistence of Ballistic Transport, Diffusion, and Localization in Surface Disordered Waveguides,” Phys. Rev. Lett. 80(5), 948–951 (1998).
[Crossref]

Markoš, P.

P. Markoš and C. M. Soukoulis, “Intensity distribution of scalar waves propagating in random media,” Phys. Rev. B 71(5), 054201 (2005).
[Crossref]

Marti, J.

P. Sanchis, J. Marti, W. Bogaerts, P. Dumon, D. Van Thourhout, and R. Baets, “Experimental results on adiabatic coupling into SOI photonic crystal coupled-cavity waveguides,” IEEE Photon. Technol. Lett. 17(6), 1199–1201 (2005).
[Crossref]

Martinelli, M.

Mazoyer, S.

Melloni, A.

Milbrodt, M. A.

R. Adar, C. H. Henry, M. A. Milbrodt, and R. C. Kistler, “Phase coherence of optical waveguides,” J. Lightwave Technol. 12(4), 603–606 (1994).
[Crossref]

Mookherjea, S.

Morandotti, R.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[Crossref] [PubMed]

Morichetti, F.

Morin, M.

H. Chotard, Y. Painchaud, A. Mailloux, M. Morin, F. Trepanier, and M. Guy, “Group delay ripple of cascaded Bragg grating gain flattening filters,” IEEE Photon. Technol. Lett. 14(8), 1130–1132 (2002).
[Crossref]

Notomi, M.

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

Ouyang, J.

Painchaud, Y.

H. Chotard, Y. Painchaud, A. Mailloux, M. Morin, F. Trepanier, and M. Guy, “Group delay ripple of cascaded Bragg grating gain flattening filters,” IEEE Photon. Technol. Lett. 14(8), 1130–1132 (2002).
[Crossref]

Paloczi, G. T.

Park, J. S.

M. L. Cooper, G. Gupta, J. S. Park, M. A. Schneider, I. B. Divliansky, and S. Mookherjea, “Quantitative infrared imaging of silicon-on-insulator microring resonators,” Opt. Lett. 35(5), 784–786 (2010).
[Crossref] [PubMed]

S. Mookherjea, J. S. Park, S. H. Yang, and P. R. Bandaru, “Localization in silicon nanophotonic slow-light waveguides,” Nat. Photonics 2(2), 90–93 (2008).
[Crossref]

Pascal, D.

F. Grillot, L. Vivien, S. Laval, D. Pascal, and E. Cassan, “Size influence on the propagation loss induced by sidewall roughness in ultrasmall SOI waveguides,” IEEE Photon. Technol. Lett. 16(7), 1661–1663 (2004).
[Crossref]

Pereira, S.

Poon, A. W.

Poon, J. K. S.

Pozzi, F.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[Crossref] [PubMed]

Qi, M.

Rodier, J. C.

Rosén, B.

B. Rosén, “On the central limit theorem for sums of dependent random variables,” Probab. Theory Relat. Fields 7, 48–82 (1967).

Sánchez-Gil, J. A.

J. A. Sánchez-Gil, V. Freilikher, I. Yurkevich, and A. A. Maradudin, “Coexistence of Ballistic Transport, Diffusion, and Localization in Surface Disordered Waveguides,” Phys. Rev. Lett. 80(5), 948–951 (1998).
[Crossref]

Sanchis, P.

P. Sanchis, J. Marti, W. Bogaerts, P. Dumon, D. Van Thourhout, and R. Baets, “Experimental results on adiabatic coupling into SOI photonic crystal coupled-cavity waveguides,” IEEE Photon. Technol. Lett. 17(6), 1199–1201 (2005).
[Crossref]

Sapienza, L.

L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, and P. Lodahl, “Cavity quantum electrodynamics with Anderson-localized modes,” Science 327(5971), 1352–1355 (2010).
[Crossref] [PubMed]

Scherer, A.

Scheuer, J.

Schneider, M. A.

Sebbah, P.

A. Z. Genack, P. Sebbah, M. Stoytchev, and B. A. van Tiggelen, “Statistics of Wave Dynamics in Random Media,” Phys. Rev. Lett. 82(4), 715–718 (1999).
[Crossref]

B. A. van Tiggelen, P. Sebbah, M. Stoytchev, and A. Z. Genack, “Delay-time statistics for diffuse waves,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 59(6), 7166–7172 (1999).
[Crossref]

Sekaric, L.

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

Shen, H.

Silberberg, Y.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[Crossref] [PubMed]

Sipe, J. E.

Smolka, S.

L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, and P. Lodahl, “Cavity quantum electrodynamics with Anderson-localized modes,” Science 327(5971), 1352–1355 (2010).
[Crossref] [PubMed]

Soller, B. J.

Sorel, M.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[Crossref] [PubMed]

Soukoulis, C. M.

P. Markoš and C. M. Soukoulis, “Intensity distribution of scalar waves propagating in random media,” Phys. Rev. B 71(5), 054201 (2005).
[Crossref]

Spasenovic, M.

Stobbe, S.

L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, and P. Lodahl, “Cavity quantum electrodynamics with Anderson-localized modes,” Science 327(5971), 1352–1355 (2010).
[Crossref] [PubMed]

Stoytchev, M.

A. Z. Genack, P. Sebbah, M. Stoytchev, and B. A. van Tiggelen, “Statistics of Wave Dynamics in Random Media,” Phys. Rev. Lett. 82(4), 715–718 (1999).
[Crossref]

M. Stoytchev and A. Z. Genack, “Observations of non-Rayleigh statistics in the approach to photon localization,” Opt. Lett. 24(4), 262–264 (1999).
[Crossref]

B. A. van Tiggelen, P. Sebbah, M. Stoytchev, and A. Z. Genack, “Delay-time statistics for diffuse waves,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 59(6), 7166–7172 (1999).
[Crossref]

Sugita, A.

T. Goh, S. Suzuki, and A. Sugita, “Estimation of waveguide phase error in silica-based waveguides,” J. Lightwave Technol. 15(11), 2107–2113 (1997).
[Crossref]

Suzuki, S.

T. Goh, S. Suzuki, and A. Sugita, “Estimation of waveguide phase error in silica-based waveguides,” J. Lightwave Technol. 15(11), 2107–2113 (1997).
[Crossref]

Tanabe, T.

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

Texier, C.

C. Texier and A. Comtet, “Universality of the Wigner time delay distribution for one-dimensional random potentials,” Phys. Rev. Lett. 82(21), 4220–4223 (1999).
[Crossref]

Thyrrestrup, H.

L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, and P. Lodahl, “Cavity quantum electrodynamics with Anderson-localized modes,” Science 327(5971), 1352–1355 (2010).
[Crossref] [PubMed]

Tkach, R. W.

M. H. Eiselt, C. B. Clausen, and R. W. Tkach, “Performance characterization of components with group delay fluctuations,” IEEE Photon. Technol. Lett. 15(8), 1076–1078 (2003).
[Crossref]

Topolancik, J.

J. Topolancik, B. Ilic, and F. Vollmer, “Experimental observation of strong photon localization in disordered photonic crystal waveguides,” Phys. Rev. Lett. 99(25), 253901 (2007).
[Crossref]

Trepanier, F.

H. Chotard, Y. Painchaud, A. Mailloux, M. Morin, F. Trepanier, and M. Guy, “Group delay ripple of cascaded Bragg grating gain flattening filters,” IEEE Photon. Technol. Lett. 14(8), 1130–1132 (2002).
[Crossref]

Van Thourhout, D.

P. Sanchis, J. Marti, W. Bogaerts, P. Dumon, D. Van Thourhout, and R. Baets, “Experimental results on adiabatic coupling into SOI photonic crystal coupled-cavity waveguides,” IEEE Photon. Technol. Lett. 17(6), 1199–1201 (2005).
[Crossref]

van Tiggelen, B. A.

A. Z. Genack, P. Sebbah, M. Stoytchev, and B. A. van Tiggelen, “Statistics of Wave Dynamics in Random Media,” Phys. Rev. Lett. 82(4), 715–718 (1999).
[Crossref]

B. A. van Tiggelen, P. Sebbah, M. Stoytchev, and A. Z. Genack, “Delay-time statistics for diffuse waves,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 59(6), 7166–7172 (1999).
[Crossref]

Varghese, L. T.

Vivien, L.

F. Grillot, L. Vivien, S. Laval, D. Pascal, and E. Cassan, “Size influence on the propagation loss induced by sidewall roughness in ultrasmall SOI waveguides,” IEEE Photon. Technol. Lett. 16(7), 1661–1663 (2004).
[Crossref]

Vlasov, Y.

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

Vlasov, Y. A.

Y. A. Vlasov, W. M. J. Green, and F. Xia, “High-throughput silicon nanophotonic wavelength-insensitive switch for on-chip optical networks,” Nat. Photonics 2(4), 242–246 (2008).
[Crossref]

Vollmer, F.

J. Topolancik, B. Ilic, and F. Vollmer, “Experimental observation of strong photon localization in disordered photonic crystal waveguides,” Phys. Rev. Lett. 99(25), 253901 (2007).
[Crossref]

Wolfe, M. S.

Xia, F.

M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, D. K. Gifford, and S. Mookherjea, “Waveguide dispersion effects in silicon-on-insulator coupled-resonator optical waveguides,” Opt. Lett. 35(18), 3030–3032 (2010).
[Crossref] [PubMed]

Y. A. Vlasov, W. M. J. Green, and F. Xia, “High-throughput silicon nanophotonic wavelength-insensitive switch for on-chip optical networks,” Nat. Photonics 2(4), 242–246 (2008).
[Crossref]

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

Xu, Y.

Xuan, Y.

Yang, L.

Yang, S. H.

S. Mookherjea, J. S. Park, S. H. Yang, and P. R. Bandaru, “Localization in silicon nanophotonic slow-light waveguides,” Nat. Photonics 2(2), 90–93 (2008).
[Crossref]

Yariv, A.

Yurkevich, I.

J. A. Sánchez-Gil, V. Freilikher, I. Yurkevich, and A. A. Maradudin, “Coexistence of Ballistic Transport, Diffusion, and Localization in Surface Disordered Waveguides,” Phys. Rev. Lett. 80(5), 948–951 (1998).
[Crossref]

Zhao, J. H.

Zhao, L.

Zhu, L.

Appl. Opt. (2)

IEEE Photon. Technol. Lett. (4)

M. H. Eiselt, C. B. Clausen, and R. W. Tkach, “Performance characterization of components with group delay fluctuations,” IEEE Photon. Technol. Lett. 15(8), 1076–1078 (2003).
[Crossref]

H. Chotard, Y. Painchaud, A. Mailloux, M. Morin, F. Trepanier, and M. Guy, “Group delay ripple of cascaded Bragg grating gain flattening filters,” IEEE Photon. Technol. Lett. 14(8), 1130–1132 (2002).
[Crossref]

F. Grillot, L. Vivien, S. Laval, D. Pascal, and E. Cassan, “Size influence on the propagation loss induced by sidewall roughness in ultrasmall SOI waveguides,” IEEE Photon. Technol. Lett. 16(7), 1661–1663 (2004).
[Crossref]

P. Sanchis, J. Marti, W. Bogaerts, P. Dumon, D. Van Thourhout, and R. Baets, “Experimental results on adiabatic coupling into SOI photonic crystal coupled-cavity waveguides,” IEEE Photon. Technol. Lett. 17(6), 1199–1201 (2005).
[Crossref]

J. Lightwave Technol. (6)

J. Opt. Soc. Am. B (3)

Nat. Photonics (5)

Y. A. Vlasov, W. M. J. Green, and F. Xia, “High-throughput silicon nanophotonic wavelength-insensitive switch for on-chip optical networks,” Nat. Photonics 2(4), 242–246 (2008).
[Crossref]

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

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

S. Mookherjea, J. S. Park, S. H. Yang, and P. R. Bandaru, “Localization in silicon nanophotonic slow-light waveguides,” Nat. Photonics 2(2), 90–93 (2008).
[Crossref]

T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2(8), 465–473 (2008).
[Crossref]

Opt. Express (5)

Opt. Lett. (7)

Phys. Rev. B (1)

P. Markoš and C. M. Soukoulis, “Intensity distribution of scalar waves propagating in random media,” Phys. Rev. B 71(5), 054201 (2005).
[Crossref]

Phys. Rev. B Condens. Matter (1)

E. Kogan and M. Kaveh, “Random-matrix-theory approach to the intensity distributions of waves propagating in a random medium,” Phys. Rev. B Condens. Matter 52(6), R3813–R3815 (1995).
[Crossref] [PubMed]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

B. A. van Tiggelen, P. Sebbah, M. Stoytchev, and A. Z. Genack, “Delay-time statistics for diffuse waves,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 59(6), 7166–7172 (1999).
[Crossref]

Phys. Rev. Lett. (6)

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[Crossref] [PubMed]

A. A. Chabanov and A. Z. Genack, “Statistics of dynamics of localized waves,” Phys. Rev. Lett. 87(23), 233903 (2001).
[Crossref] [PubMed]

J. A. Sánchez-Gil, V. Freilikher, I. Yurkevich, and A. A. Maradudin, “Coexistence of Ballistic Transport, Diffusion, and Localization in Surface Disordered Waveguides,” Phys. Rev. Lett. 80(5), 948–951 (1998).
[Crossref]

C. Texier and A. Comtet, “Universality of the Wigner time delay distribution for one-dimensional random potentials,” Phys. Rev. Lett. 82(21), 4220–4223 (1999).
[Crossref]

A. Z. Genack, P. Sebbah, M. Stoytchev, and B. A. van Tiggelen, “Statistics of Wave Dynamics in Random Media,” Phys. Rev. Lett. 82(4), 715–718 (1999).
[Crossref]

J. Topolancik, B. Ilic, and F. Vollmer, “Experimental observation of strong photon localization in disordered photonic crystal waveguides,” Phys. Rev. Lett. 99(25), 253901 (2007).
[Crossref]

Probab. Theory Relat. Fields (1)

B. Rosén, “On the central limit theorem for sums of dependent random variables,” Probab. Theory Relat. Fields 7, 48–82 (1967).

Science (1)

L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, and P. Lodahl, “Cavity quantum electrodynamics with Anderson-localized modes,” Science 327(5971), 1352–1355 (2010).
[Crossref] [PubMed]

Other (6)

D. C. Herbert and R. Jones, “Localized States in Disordered Systems,” J Phys. C 4, 1145 (1971).

D. J. Thouless, “Relation between Density of States and Range of Localization for One Dimensional Random Systems,” J Phys. C 5, 77 (1972).

D. K. Ferry, Semiconductor transport (Taylor & Francis, London, New York, 2000), pp. x, 368 p.

E. Akkermans, and G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University Press, Cambridge, 2007), pp. xviii, 588 p.

DARPA IPTO, “Exascale Computing Study: Technology Challenges in Achieving Exascale Systems,” (2008).

W. B. Davenport, and W. L. Root, An Introduction to the Theory of Random Signals and Noise, Lincoln Laboratory Publications (McGraw-Hill, New York, 1958), p. 393.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1 Coupled-resonator optical waveguides (CROWs). a, Silicon microring CROWs consisting of 35, 65, 95, 135 and 235 coupled microrings were fabricated using a CMOS compatible process on 200 mm silicon-on-insulator wafers and cleaved into 4 mm long chips, as shown in this optical microscope image. b, Scanning electron micrographs of selected regions, including the input/output waveguides and microring resonators formed from single-mode silicon wire waveguides (width = 0.5 μm) in the racetrack configuration (bending radius = 6.5 μm). c, Transmission (insertion loss) spectrum, i.e., the drop port response normalized to the input port (black: 35 ring structure; gray: 235 ring structure, + 7 dB net external amplification using an L-band EDFA).
Fig. 2
Fig. 2 Transmission spectra. Transmission (insertion loss) spectra were measured with resolution 1.4 pm, for 35, 65, 95, 135, and 235 ring waveguides. Shown here are the averages of 128 measured traces. Relative to the 35-ring CROW, measurements for 65, 95, 135 and 235 ring CROWs were amplified by 7, 4, 3, and 7 dB, respectively in order to boost the power level detected at the photoreceiver.
Fig. 3
Fig. 3 Propagation (group) delay spectra. Group delay spectra were measured with resolution 1.4 pm, for 35, 65, 95, 135, and 235 ring waveguides, over the same range of wavelengths as in Fig. 2. Spectral regions of large variation in delay correspond exactly to the stopbands of the intensity spectrum shown in Fig. 2.
Fig. 4
Fig. 4 Infrared images of CROW waveguide modes. a, Transmission (insertion loss) spectrum for a single passband of a 235 ring CROW, with measurements at selected wavelengths labeled (i)-(v). b, Intensity profiles of the eigenmodes at the wavelengths (ii)-(iv) measured with an infrared camera technique [39] show that non-localized excitations (extended throughout the entire waveguide length) were observed throughout the passband, in contrast with out-of-band (i) and band-edge (v) wavelengths. No correction was made in these images for the absorptive decay of intensity with length of the propagating modes [see Fig. 5(a)].
Fig. 5
Fig. 5 Transmission (intensity) measurements. a, The mid-band average of the transmitted intensity (in dB), measured without amplification, decreased linearly with length (−0.08 dB/resonator), except for one anomalous waveguide on the measured chip (65 rings) as discussed in the text. The errorbars represent the standard deviation, i.e., ripple, in the measured intensity over the flat portion of the band. b, The probability distribution function (PDFs) of the normalized intensity transmission ( I ^ I / I ) for the all the CROWs show agreement with the Rayleigh distribution, shown by the dashed lines, indicating non-localized transport through the waveguide. In contrast, the localized regime would show considerably different (long-tailed) statistics [8,34,40].
Fig. 6
Fig. 6 Propagation (time) delay measurements. a, The measured propagation delay averaged over the middle of a transmission band <τ> (units of picoseconds) increased linearly with length (L = 35, 65, …, 235 rings). The errorbars represent the standard deviation, i.e., ripple, in the measured delay over the flat portion of the band. b, The probability distributions of normalized delay τ ^ τ/<τ> were peaked at unity (i.e., τ = <τ>). The dashed line is a Gaussian (normal distribution) fit to the data, which indicates ballistic propagation statistics and the absence of localization, previously estimated to be a severe constraint on achieving >100 resonator lengths of CROWs. In fact, the self-averaging properties of longer chains of resonators yielded better fits to normal statistics than for the shorter waveguides, where finite-size effects caused an asymmetric lineshape in the tails of the distributions.
Fig. 7
Fig. 7 Scaling of delay statistics with CROW length. a, The measured probability distribution functions (PDFs) of the normalized group delay τ ^ τ/<τ> are shown, using a logarithmic scale on the vertical axis for clarity, for the waveguides labeled (3)-(5) in Fig. 5(a). With increasing length, the distributions converged to a single-parameter Gaussian distribution, shown by the dashed black line. b, The variance of the measured delay (ps2) increased with the square of the number of resonators (N), as shown by the dashed fit, var τ = 0.0346 N2 + (12.9 ps)2 where the second term was the typically measured group delay ripple of the measurement apparatus. This scaling behavior was different from that of conventional waveguides or cascaded fiber Bragg gratings, and as discussed in the text, demonstrated that the individual resonator excitations are mutually correlated.
Fig. 8
Fig. 8 Simulation results of group delay ripple in disordered CROWs. The yellow dots are obtained from simulations of the calculated complex transmission coefficient over the passband, in the same way as done for the analysis of measured data. The dashed line shows that a quadratic function can be fit through these points. The inset shows the probability density distribution of the variance of the group delay, from Monte Carlo simulations at a particular value of N, the number of resonators.

Metrics