Transfer function simulation of all-air-gap filters based on - nusod

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nstitute oficrostructure Technologies andnalyticsIMA

Technological ElectronicsUniversity of Kassel

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Transfer function simulation of all-air-gap filters based on eigenmodes

Friedhard Römer, Matthias Streiff, Cornelia Prott, Sören Irmer, Andreas Witzig, Bernd Witzigmann, and Hartmut Hillmer

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Overview

• All-air-gap filters• Implementation and properties• Optimization goals

• Simulation• Relation of eigenmodes and filter transfer functions• Mode coupling efficiency with external stimulus• Finite Difference Stationary Harmonic (FDSH) vs. eigenmode

• Results• Performance impact of deflected membranes • Cavity length influence

• Conclusion and Outlook

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Filter ImplementationTunable all-air-gap filter with 3λ/4 InP- λ/4 air DBR´s

SEM images WLI characterizationinstable curvature stable curvature

Tuning is achieved by electrostatic actuation of the inner membranes

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Filter Geometry and Properties

~850nm20µmStructure 3~820nm40µmStructure 2~905nm20µmStructure 1

dcavD

Stimulation by direct fiber coupling(single mode fiber: dcore = 8.3µm, ∆=0.36%)

Cleaved fiber

<5µm

Filter surface

Filter geometry

Transmission is measured by a large area detector below the sample

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Optimization goals

Insertion Loss• lower signal to noise ratio• less Amplifier stages

Linewidth (FWHM)• higher channel density• lower signal to noise ratio

Mode spectrum• lower crosstalk• lower signal to noise ratio

T: filter transmission, λ: wavelength

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Quasi 3D body of revolution model

Coordinate system

Separation of variable φ Solving eigenmode equation:

• Boundary reflections are suppressed by PMLs• Only vertical modes are substantial for filter function• Eigenvalue k2: real{k} = wave number, photon lifetime τph = 1/|2c imag{k}|

lateral mode vertical mode

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Evaluating the eigenmode

Cavity loss: energy irradiationUpper mirror (1): P1

Intrinsic and diffraction: Pml

Lower mirror (2): P2

Cavity energy: Wcav

FP transmission function

Cavity time constants andEquiv. FP mirror reflectance

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Coupling efficiency

Stimulus

Field distribution

Extraction

Compound transfer function

Coupling coefficients: κµ

Horizontal sectionoverlap integral

Typical source fields have spatially linear polarization: only modes with ν=1 are excited.Orthogonality of eigenmodes in the horizontal section required.

Computed overlap of (vertical) modes is approx. 10-5.

Filter modes

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Eigenmode vs. Stationary Harmonic Anlaysis

Eigenmode analysis providesgood accuracy for real world applicationsanalytical transfer functions (R/T)low computation time

• real wavenumber k, independent variable• boundary source term constitutes stimulus• frequency stepping

boundary source

Stationary harmonic

Eigenmode

• solving for complex wavenumber kresults in eigenvalues

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Filter 1 (20µm): spectral portrait

× no deflection

× 16nm deflection

× -8nm deflection

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Filter 1 (20µm): transfer functions

Deflection has low influence on the mode quality, coupling efficiency is more affected.

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Filter 1 (20µm): characterization and simulation

straight membranes

Influence of the suspensions disturbs rotational symmetry

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Filter 2 (40µm): spectral portrait

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Filter 2 (40µm): transfer functions

Deflection has low influence on the mode loss, but strong influence on the coupling efficiency

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Filter 2 (40µm): characterization and simulation

56 nm deflection, stable cavity

Better agreement due to lower influence of the suspensionsDifference in transmission is due to the substrate absorption

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Cavity length impact

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Conclusion

Analysis of passive devices by eigenmodes• yields analytical expression of the transfer functions• good orthogonality of eigenfunctions in a horizontal section (≈10-5)• about 25 faster than Stationary Harmonic Analysis• good agreement with characterization if suspension influence is low

Next steps:• radial displacement of source• full 3D geometry with suspensions to better fit the experiment

Acknowledgements:• W. Fichtner (Integrated Systems Laboratory, Swiss Federal Institute of Technology Zurich)• A. Tarraf, I. Kommallein, D. Gutermuth (Technolgical Electronics, University of Kassel)• Funding by the German DFG, BMBF under contract numbers Hi763/3-1 and 01BC150, and the Nanofond Hessen is gratefully acknowledged

Transfer function simulation of all-air-gap filters based on - nusod

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