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Journals >Photonics Research >Volume 6 >Issue 11 >Page 1008 > Article

Photonics Research
Vol. 6, Issue 11, 1008 (2018)

Accurate extraction of fabricated geometry using optical measurement

Yufei Xing^1、2、*, Jiaxing Dong^1、2, Sarvagya Dwivedi³, Umar Khan^1、2, and Wim Bogaerts^1、2

Author Affiliations

¹Photonics Research Group, Ghent University-IMEC, Ghent, Belgium

²Center of Nano and Biophotonics, Ghent, Belgium

³Electrical and Computer Engineering Department, University of California Santa Barbara, Santa Barbara, California 93106-9560, USA

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DOI: 10.1364/PRJ.6.001008 Cite this Article Set citation alerts

Yufei Xing, Jiaxing Dong, Sarvagya Dwivedi, Umar Khan, Wim Bogaerts. Accurate extraction of fabricated geometry using optical measurement[J]. Photonics Research, 2018, 6(11): 1008 Copy Citation Text

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Abstract

We experimentally demonstrate extraction of silicon waveguide geometry with subnanometer accuracy using optical measurements. Effective and group indices of silicon-on-insulator (SOI) waveguides are extracted from the optical measurements. An accurate model linking the geometry of an SOI waveguide to its effective and group indices is used to extract the linewidths and thicknesses within respective errors of 0.37 and 0.26 nm on a die fabricated by IMEC multiproject wafer services. A detailed analysis of the setting of the bounds for the effective and group indices is presented to get the right extraction with improved accuracy.

w = p_{0} + Σ_{i = 1}^{n} p e_{i} n_{eff}^{i} + Σ_{j = 1}^{n} p g_{j} n_{g}^{j} + Σ_{j = 1, i = 1}^{m}, n p e g_{i j} n_{eff}^{i} n_{g}^{j},

(1)

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t = q_{0} + Σ_{i = 1}^{n} q e_{i} n_{eff}^{i} + Σ_{j = 1}^{n} q g_{j} n_{g}^{j} + Σ_{j = 1, i = 1}^{m}, n q e g_{i j} n_{eff}^{i} n_{g}^{j},

(2)

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m = \frac{n_{eff, 0} Δ L}{λ_{res}},

(3)

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FSR = \frac{λ_{res}^{2}}{n_{g} Δ L},

(4)

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\frac{n_{eff, 0} Δ L}{λ} = m, \frac{(n_{eff, 0} - Δ n_{eff} / 2) Δ L}{λ} > m - 0.5, \frac{(n_{eff, 0} + Δ n_{eff} / 2) Δ L}{λ} < m + 0.5 .

(5)

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m < \frac{n_{eff, 0}}{Δ n_{eff}} .

(5)

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Δ L < \frac{λ}{Δ n_{eff}} .

(6)

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n_{eff} = n_{{eff}_{0}} + \frac{\partial n_{eff}}{\partial w} (w - w_{0}) + \frac{\partial n_{eff}}{\partial t} (t - t_{0}), n_{g} = n_{g_{0}} + \frac{\partial n_{g}}{\partial w} (w - w_{0}) + \frac{\partial n_{g}}{\partial t} (t - t_{0}) .

(8)

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n_{{eff}_{1}} = n_{{eff}_{0}} + \frac{\partial n_{eff}}{\partial w} (w_{1} - w_{0}) + \frac{\partial n_{eff}}{\partial t} (t_{1} - t_{0}),

(7)

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n_{{eff}_{2}} = n_{{eff}_{0}} + \frac{\partial n_{eff}}{\partial w} (w_{2} - w_{0}) + \frac{\partial n_{eff}}{\partial t} (t_{2} - t_{0}),

(8)

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n_{g_{1}} = n_{g_{0}} + \frac{\partial n_{g}}{\partial w} (w_{2} - w_{0}) + \frac{\partial n_{g}}{\partial t} (t_{1} - t_{0}),

(9)

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n_{g_{2}} = n_{g_{0}} + \frac{\partial n_{g}}{\partial w} (w_{1} - w_{0}) + \frac{\partial n_{g}}{\partial t} (t_{2} - t_{0}),

(10)

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Δ n_{eff, rectangle} = n_{{eff}_{2}} - n_{{eff}_{1}} = \frac{\partial n_{eff}}{\partial w} Δ w + \frac{\partial n_{eff}}{\partial t} Δ t,

(11)

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Δ n_{eff, parallelogram} = (- \frac{\frac{\partial n_{eff}}{\partial w} \frac{\partial n_{g}}{\partial t}}{\frac{\partial n_{g}}{\partial w}} + \frac{\partial n_{eff}}{\partial t}) Δ t, when \frac{\partial n_{g}}{\partial t} Δ t < - \frac{\partial n_{g}}{\partial w} Δ w,

(12)

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Δ n_{eff, parallelogram} = (\frac{\partial n_{eff}}{\partial w} - \frac{\frac{\partial n_{g}}{\partial w} \frac{\partial n_{eff}}{\partial t}}{\frac{\partial n_{g}}{\partial t}}) Δ w, when \frac{\partial n_{g}}{\partial t} Δ t > - \frac{\partial n_{g}}{\partial w} Δ w .

(13)

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\frac{Δ n_{eff, rectangle}}{Δ n_{eff, parallelogram}} = a \frac{Δ w}{Δ t} + b,

(16)

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a = \frac{\frac{\partial n_{eff}}{\partial w}}{- \frac{\partial n_{eff}}{\partial w} \frac{\partial n_{g}}{\partial t} / \frac{\partial n_{g}}{\partial w} + \frac{\partial n_{eff}}{\partial t}}, b = \frac{\frac{\partial n_{eff}}{\partial t}}{- \frac{\partial n_{eff}}{\partial w} \frac{\partial n_{g}}{\partial t} / \frac{\partial n_{g}}{\partial w} + \frac{\partial n_{eff}}{\partial t}} .

(17)

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{variation}_{total} = {variation}_{inter-die} + {variation}_{location-dependent} + {variation}_{local} .

(14)

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Δ n_{eff, total} = \frac{\partial n_{eff}}{\partial w} Δ w_{total} + \frac{\partial n_{eff}}{\partial t} Δ t_{total} .

(19)

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Δ n_{eff, local} = (- \frac{\frac{\partial n_{eff}}{\partial w} \frac{\partial n_{g}}{\partial t}}{\frac{\partial n_{g}}{\partial w}} + \frac{\partial n_{eff}}{\partial t}) Δ t_{local} .

(20)

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Δ n_{eff, total} = \frac{\partial n_{eff}}{\partial w} Δ w_{total} + \frac{\partial n_{eff}}{\partial t} Δ t_{total} = 0.002055 \times 40 nm + 0.003916 \times 20 nm = 0.16 .

(21)

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Δ n_{eff, intra-die} = (- \frac{\frac{\partial n_{eff}}{\partial w} \frac{\partial n_{g}}{\partial t}}{\frac{\partial n_{g}}{\partial w}} + \frac{\partial n_{eff}}{\partial t}) Δ t_{intra-die} = 0.0074 \times 2 = 0.0148,

(22)

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{Error}_{w} = 0.31 + 0.06 = 0.37 nm, {Error}_{t} = 0.18 + 0.08 = 0.26 nm .

(23)

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References (21)

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Yufei Xing, Jiaxing Dong, Sarvagya Dwivedi, Umar Khan, Wim Bogaerts. Accurate extraction of fabricated geometry using optical measurement[J]. Photonics Research, 2018, 6(11): 1008

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