G. Kajtár, P. Markoš

Analysis of 2D periodic structures using RCWA

Department of Physics Institute of Nuclear and Physical Engineering Faculty of Electrical Engineering and Information Technology Slovak University of Technology

gergely.kajtar@stuba.sk Ilkovičova 3

Bratislava 812 19 Slovak Republic

mailto:gergely.kajtar@stuba.sk

OUTLINE

RCWA in a nutshell

Analysis of 2D lamellar grating

Comparison of computational methods

Results

Conclusion

RCWA IN A NUTSHELL

Goal: to calculate diffraction efficiency of a

periodic structure for a given diffraction order

< Diffraction efficiency of sth order

RCWA IN A NUTSHELL

Rigorous vector analysis for electromagnetic wave

propagation through periodic medium.

Maxwell equations:

Helmholtz's wave equations for electric and

magnetic fields:

RCWA IN A NUTSHELL

Fourier decomposition of quantities:

Electric/magnetic field intensities: F = (Ex, Ey, Hx, Hy)

Relative permittivity

Floquet theorem:

RCWA IN A NUTSHELL

Eigenproblem

Solutions:

Above the grating

Inside the grating

Below the grating

Q – eigenvectors of C

L – eigennumbers of C

RCWA IN A NUTSHELL

Boundary conditions – tangential vectors of

electric and magnetic intensities are conserved –

are used to formulate algebraic problem for

unknown coefficients r and t.

Calculate magnetic field intensities by electric

field intensities:

RCWA IN A NUTSHELL

Transfer matrix approach

RCWA IN A NUTSHELL

ENHANCEMENTS

Lee‘s inverse rules to improve convergence

Pendry‘s renormalization to avoid exponentially

growing amplitudes of evanescent waves

𝜖 𝜖 −1 𝜖−1 𝜖

NUMERICAL SIMULATION PROGRAM

Input parameters

Incident wave properties – wavelength, angle of incidence,

polarisation angle, conical angle

Grating properties – spatial periods, permittivity,

thickness, grating shape, etc..

Number of Fourier modes (truncation order)

Output parameters

Diffraction angles

Diffraction efficiencies (transmittance and reflectance)

ANALYSIS OF 2D PERIODIC GRATING

Spatial periods: 400 nm

Filling factors: 0.25

Relative permittivity of rods: 4

Relative permittivity of gaps: 1

Rel. perm. of sub- & superstrate: 1

Thickness: 100 nm

Normal incidence

Vector E paralell to x

COMPARISON OF COMPUTATIONAL

METHODS

Thanks to J. Šoltýs, International Laser Centre, Bratislava for RSoft data. .

RCWA 7 works with size of matrix 450x450. Transfer matrix 24 works with size of matrix 1152x1152.

RCWA is faster.

COMPARISON OF COMPUTATIONAL

METHODS

Thanks to J. Šoltýs, International Laser Centre, Bratislava for RSoft data.

RESULTS

DIFFERENT REL. PERMITTIVITIES

Number of Fourier modes: 6

RESULTS

DIFFERENT GRATING THICKNESS

Number of Fourier modes: 6

RESULTS

DIFFERENT FILLING FACTORS

RESULTS – INVERSE GRATING

Air holes in grating with rel. perm. 4.

Guided resonances + Fabry-Perot interference

CONCLUSION

RCWA rigorous vector method for analysis of

propagation of EM waves through periodic

structures

Fast, effective compared to transfer matrix method

Subwavelength

Evanescent waves, near – field

Diffraction efficiencies

Various periodic structures (e.g. photonic crystal slab)

Various materials of grating (metal, dielectric,...)

This work was supported by the Slovak Research and Development Agency under the contract No. APVV-0108-11 and STU Project for Young Scientists NUMOPE.