Expand this Topic clickable element to expand a topic
Skip to content
Optica Publishing Group
Journal Home About Issues in Progress Current Issue All Issues Feature Issues

Nonlinearity enhancement in finite coupled-resonator slow-light waveguides

Yan Chen and Steve Blair

Open Access Open Access

Abstract

In this paper, we derive the exact dispersion relation of one dimensional periodic coupled-resonator optical waveguides of finite length, from which the reduced group velocity of light is obtained. We show that the group index strongly depends on the number of cavities in the system, especially for operation at the center frequency. The nonlinear phase sensitivity shows an enhancement proportional to the square of the group index (or light slowing ratio). Aperiodic coupled ring-resonator optical waveguides with optimized linear properties are then synthesized to give an almost ideal nonlinear phase shift response. For a given application and bandwidth requirement, the nonlinear sensitivity can be increased by either decreasing resonator length or by using higher-order structures. The impact of optical loss, including linear and two-photon absorption is discussed in post-analysis.

©2004 Optical Society of America

Full Article  |  PDF Article
More Like This
Slow light enhanced optical nonlinearity in a silicon photonic crystal coupled-resonator optical waveguide

Nobuyuki Matsuda, Takumi Kato, Ken-ichi Harada, Hiroki Takesue, Eiichi Kuramochi, Hideaki Taniyama, and Masaya Notomi
Opt. Express 19(21) 19861-19874 (2011)

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

V. Govindan and S. Blair
Opt. Express 15(7) 3922-3930 (2007)

Optical characterization of coupled resonator slow-light rib waveguides

Jeremy Goeckeritz and Steve Blair
Opt. Express 18(17) 18190-18199 (2010)

View More...

Cited By

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

Alert me when this article is cited.


Figures (12)
Fig. 1.
Fig. 1. Coupled resonator waveguides: (a) coupled Fabry-Perot type cavities in thin film and (b) coupled ring-resonators, where the shaded portions in dotted-boxes represent the unit cells.

Download Full Size | PDF

Fig. 2.
Fig. 2. The filter responses (a) transmission, (b) phase and (c) group delay of periodic coupled ring-resonator waveguides with central frequency νm and miniband width Δν=33.33 GHz. The solid line represents the response of an 11th order filter and dotted line for a 10th order filter. An effective refractive index of 1.5 and central wavelength of 500 nm are used.

Download Full Size | PDF

Fig. 3.
Fig. 3. Dispersion relations (a), normalized group velocity (b) and group index (c) of infinite, 10th, and 11th order periodic coupled ring-resonator waveguides, and the underlying material, respectively. ω 0=2πνm , and X=kd,βd and kNd for the material, infinite waveguide and finite waveguide respectively. The insets show details in the central band.

Download Full Size | PDF

Fig. 4.
Fig. 4. Nonlinear response produced by a periodic 11-cavity coupled ring-resonator waveguide where optical bistability occurs at high intensity input. There’s also a small region where the two bistable cycles overlap, producing a region where there are three stable states.

Download Full Size | PDF

Fig. 5.
Fig. 5. Nonlinear phase sensitivity of periodic coupled ring-resonator waveguides: (a) varying with coupling strength (The numbers shown beside the line are the coupling coefficients) but constant length; (b) varying with ring length but constant coupling.

Download Full Size | PDF

Fig. 6.
Fig. 6. Ideal linear filter response for nonlinear phase shift

Download Full Size | PDF

Fig. 7.
Fig. 7. Linear filter responses (a) transmission, (b) phase and (c) group delay of optimized aperiodic coupled ring-resonator waveguides having 4 THz FSR, ~100 GHz bandwidth, and finesse of 40. The arrow indicates filter order increasing from 3 to 11.

Download Full Size | PDF

Fig. 8.
Fig. 8. (a) Group delay vs. filter order for optimized aperiodic coupled ring-resonator waveguides. For N≥3, group delay is proportional to filter order. (b) Physical length of coupled ring-resonator waveguides and underlying material of equal group delay vs. filter order.

Download Full Size | PDF

Fig. 9.
Fig. 9. Nonlinear response of optimized aperiodic coupled ring-resonator waveguides. The arrow indicates filter order increasing from 3 to 11.

Download Full Size | PDF

Fig. 10.
Fig. 10. (a) Nonlinear sensitivity at low intensity as a function of filter order, for the optimized aperiodic coupled ring-resonator waveguide. (b) Nonlinear sensitivity of a 7th order optimized aperiodic coupled ring-resonator waveguide vs. FSR, given a bandwidth ~100 GHz.

Download Full Size | PDF

Fig. 11.
Fig. 11. Nonlinear sensitivity of optimized aperiodic coupled ring-resonator waveguides having constant bandwidth but increased filter order, studied at the normalized intensity to produce: (a) π/4 phase shift and (b) π phase shift, respectively.

Download Full Size | PDF

Fig. 12.
Fig. 12. Nonlinear responses of a 9th order optimized aperiodic coupled ring-resonator waveguide: (a) at different frequencies; (b) to optical loss, where the linear attenuation is α=1 cm-1 [19]; the normalized two photon absorption (TPA) coefficients are K=0,0.03,0.1, corresponding to zero, moderate and large TPA, respectively. The TPA figure of merit [20] is given as T=8πK, where T<1 is desired.

Download Full Size | PDF

Tables (1)

Tables Icon

Table 1. Optimized coupling coefficients t of aperiodic coupled ring-resonator waveguides (N=3 to 11)

View Table

Equations (25)

Equations on this page are rendered with MathJax. Learn more.

[ A n + A n − ] = M ̂ [ A n + 1 + A n + 1 − ]
M ̂ = [ 1 ⁄ t 1 r 1 * ⁄ t 1 * r 1 ⁄ t 1 1 ⁄ t 1 * ]
M ̂ = j t [ − e jkd r − r e − jkd ]
t 1 = j t e − j k d
cos ( β d ) = Re { 1 t 1 } = sin ( k d ) t
v g = c n t 2 − sin 2 ( k d ) cos ( k d ) ,
n g = c v g = n cos ( k d ) t 2 − sin 2 ( k d ) .
Δ ω = 2 π Δ ν = 4 FSR sin − 1 ( t ) ,
M ̂ N = [ 1 ⁄ t N r N * ⁄ t N * r N ⁄ t N 1 ⁄ t N * ]
t N = T N e j ϕ N ,
ϕ N = k N L N .
M ̂ N = M ̂ sin ( β L N ) sin ( βd ) − I ̂ sin ( N β d ) sin ( β d ) ,
1 t N = 1 t 1 sin ( β L N ) sin ( βd ) − sin ( N β d ) sin ( β d ) .
t N = j t sin ( β d ) e jkd sin ( β L N ) − j t sin ( N β d ) .
tan ( k N L N ) = cos ( k d ) t tan ( β L N ) sin ( β d ) ,
d k N d ω = n g N c = 1 L N d d ω [ tan − 1 ( RHS ) ]
n g N = n g t t 2 sin 2 ( β d ) cos 2 ( β L N ) + cos 2 ( k d ) sin 2 ( β L N ) [ cos ( k d ) sin ( β d ) −
− n d sin ( 2 β L N ) 2 L N ( sin ( k d ) sin ( β d ) n g + cos ( k d ) cos ( β d ) n ) ]
g . d . = n g N L N ⁄ c .
n g N = n t 2 cos 2 ( β L N ) + sin 2 ( β L N ) ,
n g N ( odd ) = n t 2 ,
n g N ( even ) = n ,
d Δ Φ d ( n 2 I in ) ∝ n g N 2 = ( n t 2 ) 2 ,
d Δ Φ d ( n 2 I in ) ∝ L
FOM = L m L w { d Δ Φ ⁄ d I } w { d Δ Φ ⁄ d I } m ,
Select as filters
Select Topics Cancel
Top
       
© Copyright 2024 | Optica Publishing Group. All rights reserved, including rights for text and data mining and training of artificial technologies or similar technologies.
Privacy | Terms of Use