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Multi-objective and constrained design of fibre Bragg gratings using Evolutionary Algorithms

Steven Manos and Leon Poladian

Open Access Open Access

Abstract

We apply a multi-objective evolutionary algorithm to a grating problem where only very specific features of the transmission spectrum are specified during the optimisation process. The design problem analysed here relates to the passive extraction of 10 GHz clock signals from a 10 Gbps OTDM RZ encoded data stream. Four spectral features of interest such as bandwidth and passband quality are explicitly defined. Using a real-encoded evolutionary algorithm along with an elitist multi-objective selection method, we arrive at a group of solutions which each satisfy the objectives to various degrees in the presence of manufacturing and other design constraints.

©2005 Optical Society of America

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Figures (7)
Fig. 1.
Fig. 1. Overview of the FBG design parameters and there contribution to various features of the profile.

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Fig. 2.
Fig. 2. Outline of the various spectral traits extracted using a peak-finding algorithm and used to evaluate the objectives of a design.

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Fig. 3.
Fig. 3. Scatterplots of FBG solutions in design parameter space. Light grey dots indicate the initial population, and black dots indicate the final non-dominated set after 1000 generations.

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Fig. 4.
Fig. 4. Visualisation of the four dimensional non-dominated set in three dimensions after dimensionality reduction. The axis of the FBG spectra and profiles have been removed for clarity but are in the range λ → [1549.5,1550.5] nm, T → [-90,0] dB for the spectra and z → [-1,1] cm, q 0 → [0,10]cm-1 for the FBG profiles. The red lines represent the phase profile in the range ϕ → [0, π] . The two clusters labeled A and B were extracted using a hierarchical clustering algorithm. The designs labeled I, II, III, IV correspond to the corner designs shown in greater detail in Fig. 5.

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Fig. 5.
Fig. 5. Transmission spectra and FBG q and phase ϕ profiles of the four designs corresponding to the corner designs on the non-dominated set (Fig. 4). Designs I and IV are most optimal with respect to bandwidth, but design I simultaneously exhibits the worst (widest) peak separation. Design II is the best with respect to peak separation, but worst in terms of the transmission. Design III simultaneously exhibits optimal peak full-width half-maximum and transmission objectives, but average bandwidth and peak separation objectives.

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Fig. 6.
Fig. 6. Instead of looking at non-linear reduction from 4 dimensions to 3 dimensions as in Fig. 4, we can further reduce the non-dominated set to 2 dimensions. Values of the design variables L,z 0,α,n can then be shown as height above the x-y plane, giving us an indication of how the design variables change as you move around the designs which form the non-dominated set. The plots for q 0, ϕ are not shown since they are relatively constant across the population with average values of q 0ϕ = 9.997cm-1 and ϕ ¯ =π.

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Fig. 7.
Fig. 7. Plots for the transmission, bandwidth, peak separation and peak FWHM objective values across the 2-dimensional representation of the non-dominated set. The tradeoff and harmony between spectral characteristics is obvious here, for example, minimal FWHM results in non-optimal BW (conflicting relationship) but on the other hand results in optimal transmission (harmonious relationship).

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Tables (3)

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Table 1. Outline of the parameter bounds

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Table 2. Minimum and maximum values of the FBG design parameters in the non-dominated set.

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Table 3. Minimum and maximum values of the FBG spectral characteristics and objectives in the non-dominated set.

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Equations (10)

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

F = ∑ λ i w i ∣ R ( λ i ) − R ̄ ( λ i ) ∣ p
f i ( a ) ≤ f i ( b ) , i = 1 , … , M and ∃i ∈ ( 1 , … , M ) , f i ( a ) < f i ( b )
q ( z ) = q 0 cos α ( π 2 ∣ 2 z L ∣ n ) e iϕ ( z )
+ iu ′ z δ + δu z δ + q ( z ) e + iϕ ( z ) v z δ = 0
+ iv ′ z δ + δv z δ + q ( z ) e - iϕ ( z ) u z δ = 0
δ = 2 π n ̄ ( 1 λ − 1 λ B )
T κ = [ cosh ( qh ) sinh ( qh ) e + iϕ sinh ( qh ) e − iϕ cosh ( qh ) ]
T δ = [ e + iδh 0 0 e − iδh ]
[ u ( − L 2 ) v ( − L 2 ) ] = T κ T δ … T κ T δ [ u ( L 2 ) v ( L 2 ) ]
T dB ( λ ) = 20 log 10 ∣ 1 u ( − L 2 ) ∣
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