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Optics Express

Limitations on nonlinear pulse propagation in coupled-resonator slow-light waveguides

Vishnupriya Govindan, Steve Blair

Published Jan 1, 2007• 5 citations
Publisher

Abstract

Under the constraint of fixed pulse distortion, we study the bandwidth-delay product and nonlinear phase shift performance of coupled-resonator slow-light waveguides with designs that produce maximally-flat transmission and maximally-flat group delay responses. Even though improvement in bandwidth-delay product can be obtained with increasing number of resonators, the nonlinear response fails to improve beyond a certain number of resonators due to the increased filter bandwidth necessary to maintain a fixed pulse distortion. However, as expected, the nonlinear response improves with resonator finesse and degrades with resonator loss.

DOI
10.1364/oe.15.003922

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Under the constraint of fixed pulse distortion, we study the bandwidth-delay product and nonlinear phase shift performance of coupled-resonator slow-light waveguides with designs that produce maximally-flat transmission and maximally-flat group delay responses. Even though improvement in bandwidth-delay product can be obtained with increasing number of resonators, the nonlinear response fails to improve beyond a certain number of resonators due to the increased filter bandwidth necessary to maintain a fixed pulse distortion. However, as expected, the nonlinear response improves with resonator finesse and degrades with resonator loss.
One of the key issues when dealing with slow light is the inevitable effect of dispersion, the result of which is to broaden a pulse upon propagation. Moreover, the dispersion tends to increase when the group velocity, the speed at which the pulse propagates, decreases. To see this, we note that if we write the group velocity vg ≡ c/ng, where ng is the group index, then we find that ng = n+ωdn/dω. Since for the materials we are interested in, the refractive index does not vary strongly in magnitude, and slow light is obtained when the derivative dn/dω is large and positive. However, as the variations in the magnitude of n itself are limited, the large derivative can only be maintained over a narrow bandwidth ∝ (dn/dω)−1. Therefore, a large delay implies a narrow bandwidth, and hence a limited pulse width. Here we discuss how to avoid this limit by using nonlinear effects in the form of gap solitons. Though this argument is quite simple, it can be made rigorous using the Kramers-Kronig relations. In this way it can be shown that the achievable delay in a linear, two-port device is subject to a limitation of the product of the delay and the bandwidth (Lenz et al., 2001).
All resonant systems throughout science and engineering, independent of their physical implementation have a bandwidth that is inversely related to the decay time. A similar relation exists in all slow-light systems, where the group index (and therefore the delay for a given footprint) is inversely related to the bandwidth. Therefore, resonant or slow-light systems can either store a broad signal for a short time, or a narrow signal for a long time, but cannot achieve large delay for broad bandwidth signals.Here we discuss our recent work on non-reciprocal optical systems that are not constrained by the delay-bandwidth limit. We show that large, broadband optical delay is not a pipe dream and is achievable with current optical technology. We discuss the underlying physics of delay and bandwidth in non-reciprocal optical systems and present an experimental implementation, based on a figure-9 cavity. We demonstrate a delay-bandwidth product 30 times above the seemingly fundamental time-bandwidth limit of traditional systems. Furthermore, we show that the optical pulse can be released after an arbitrary number of round trips, providing the control and tunability lacking from conventional spiral waveguide or fibre loop delay lines.
Slow light with a markedly low group velocity is a promising solution for optical buering and advanced time-domain optical signal processing.It is also anticipated to enhance linear and nonlinear eects and so miniaturize functional photonic devices because slow light compresses optical energy in space.Photonic crystal waveguide devices generate on-chip slow light at room temperature with a wide bandwidth and low dispersion suitable for short pulse transmission.This paper rst explains the delay-bandwidth product, fractional delay, and tunability as crucial criteria for buering capacity of slow light devices.Then the paper describes experimental observations of slow light pulse, exhibiting their record high values.It also demonstrates the nonlinear enhancement based on slow light pulse transmission.
We study analytically pulse distortion in linear slow light systems, and provide some useful limits on these devices. Additionally, we also show that the contributions of phase and amplitude broadening can be de-coupled and quantified. It is observed that phase broadening is generally smaller than amplitude broadening in conventional slow light media (lorentzian gain profile) except for very large fractional delays, where it becomes larger. Upon these expressions, we may envisage new strategies to minimize the distortion in the delaying of pulses, depending on the specific application and the required fractional delay. To overcome the residual distortion, we show that nonlinear systems can lead to a resharpening of the pulses and a re-generation of the filtered frequencies.
Slow light has been proposed as a potential solution to all-optically tunable delay line. However, a slow-light element may degrade data quality in an optical communication system while decreasing group velocity of optical pulses. In this paper, pattern dependence of signal distortion is identified as a main reason for data degradation, which is caused by narrow-band amplitude and phase responses of the slow-light elements. We define figure of merit involving pulse delay and data degradation to optimize slow-light devices. It is shown that the pattern dependence can be reduced by detuning slow-light devices away from the signal carrier frequency, which allows using narrow-band slow-light techniques to increase normalized delay up to 0.8, withQQQimprovement of 2 dB.