Perturbative numerical modeling of microstructure fibers

John M. Fini

doi:10.1364/OPEX.12.004535

Optics Express
Vol. 12,
Issue 19,
pp. 4535-4545
(2004)
•https://doi.org/10.1364/OPEX.12.004535

Perturbative numerical modeling of microstructure fibers

John M. Fini

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John M. Fini, "Perturbative numerical modeling of microstructure fibers," Opt. Express 12, 4535-4545 (2004)

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Abstract

Modeling of microstructure fibers often involves severe computational bottlenecks, in particular when a design space with many degrees of freedom must be analyzed. Perturbative versions of numerical mode-solvers can substantially reduce the computational burden of problems involving automated optimization or irregularity analysis, where perturbations arise naturally. A basic theory is presented for perturbative multipole and boundary element methods, and the speed and accuracy of the methods are demonstrated. The specific optimization results in an elliptical-hole birefringent fiber design, with substantially higher birefringence than the intuitive unoptimized design.

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Fig. 1. For many realistic fibers, irregularity in hole geometry is small, but large enough to seriously impact optical properties such as birefringence. A perturbative approach is then natural.

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Fig. 2. Standard multipole and boundary-element methods require many large-matrix operations for each geometry, since there are many effective index values in the search. A perturbative approach needs very few large-matrix operations for each perturbed geometry.

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Fig. 3. Test of perturbative method for random hole irregularities. The fiber has three rings of cladding holes with spacing 2 microns, diameter 1 micron, and index 1. The wavelength was 1630 nm and the substrate index was set to 1.45. The geometric perturbations consisted of independently displacing six holes (σ_x =σ_y = .02Λ), and placing the remaining 30 holes to preserve sixfold rotational symmetry. Dashed lines indicate expected error trends (blue and pink), the unperturbed loss value (red), and ideal agreement between standard and perturbative methods (black).

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Fig. 4. For a birefringent fiber with elliptical holes, four holes of the inner ring were initially misaligned. An automated optimization of birefringence adjusts orientations of these four holes, ultimately aligning them with the orientation of the fixed holes. This is an optimization test with an intuitive optimum design. Modes are calculated at wavelength λ = 1.17Λ.

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Fig. 5. A consistency check confirms that the estimates of birefringence perturbation agree with non-perturbative results at each step in the optimization.

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Fig. 6. Birefringence is plotted for two optimizations where all 18 holes are free to rotate. Both initially x-oriented and y-oriented holes arrive at equivalent fiber geometries with a substantial improvement in birefringence. Again λ = 1.17Λ.

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Fig. 7. Orientation angles for the six inner holes are plotted versus optimization step. All holes are initially oriented to ϕ_j = 90° and ϕ_j = 0 for the left and right optimizations respectively. Both optimizations are converging to equivalent geometries, rotated 120 degrees from each other. The optimal perturbations at each step approximately maintain point-reflection symmetry about the origin (from which “hidden” curves can be inferred).

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Equations (21)

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M (n_{eff}, λ) v = 0 .

M (n) = M (n_{eff} = n; λ, p)

M (n) = M_{0} (n) + δ M_{1} (n) + δ^{2} M_{2} (n) + \dots

M_{1} (n) = \frac{\partial M}{\partial δ} \approx \frac{M (n; λ, p_{0} + ε p_{1}) - M (n; λ, p_{0} - ε p_{1})}{2 ε} .

v = v_{0} + δ v_{1} + δ^{2} v_{2} + \dots

n = n_{0} + δ n_{1} + δ^{2} n_{2} + \dots

[M_{0} (n_{0} + δ n_{1} + \dots) + M_{1} (n_{0} + \dots) + \dots] [v_{0} + δ v_{1} + \dots] = 0,

δ n_{1} = \frac{- u_{0}^{†} δ M_{1} (n_{0}) v_{0}}{u_{0}^{†} M_{0}^{'} (n_{0}) v_{0}}

M_{0} (n_{0}) v_{0} = 0 .

M_{0} (n_{0}) δ v_{1} + [δ n_{1} M_{0}^{'} (n_{0}) + δ M_{1} (n_{0})] v_{0} = 0,

u_{0}^{†} [δ n_{1} M_{0}^{'} (n_{0}) + δ M_{1} (n_{0})] v_{0} = 0 .

M_{0}^{p} M_{0} (n_{0}) = I - {\hat{v}}_{0} {\hat{v}}_{0}^{†} .

δ v_{1} = {- M}_{0}^{p} [δ n_{1} M_{0}^{'} (n_{0}) + δ M_{1} (n_{0})] v_{0} .

[M_{0} (n_{0})] [δ^{2} v_{2}] + [δ n_{1} M_{0}^{'} (n_{0}) + δ M_{1} (n_{0})] [δ v_{1}] + [M_{2}] [v_{0}] = 0

M_{2} = (δ^{2} v_{2}) M_{0}^{'} (n_{0}) + \frac{1}{2} {(δ n_{1})}^{2} M_{0}^{″} (n_{0}) + δ M_{1}^{'} (n_{0}) (δ n_{1}) + δ^{2} M_{2} (n_{0})

v_{0} = Bx .

A^{†} [δ n_{1} M_{0}^{'} (n_{0}) + δ M_{1} (n_{0})] Bx = 0,

- A^{†} δ M_{1} (n_{0}) Bx = δ n_{1} A^{†} M_{0}^{'} (n_{0}) Bx,

{[\nabla_{p} n_{Bi}]}_{j} = \frac{\partial n_{slow}}{\partial ϕ_{j}} - \frac{\partial n_{fast}}{\partial ϕ_{j}} .

p^{m + 1} = p^{m} + {[Δ p]}^{m} .

{[Δ p]}^{m} = \frac{δ \nabla_{p} n_{Bi}}{∣ \nabla_{p} n_{Bi} ∣} .

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Equations (21)

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