J. Elschner, A. Rathsfeld, G. Schmidt Optimisation of diffraction gratings with DIPOG

Transcription

1 W eierstraß-institut für Angew andte Analysis und Stochastik Matheon MF 1 Workshop Optimisation Software J Elschner, A Rathsfeld, G Schmidt Optimisation of diffraction gratings with DIPOG Mohrenstr 39, Berlin rathsfeld@wias-berlinde June 1, 2005

2 Outline Diffractive Optical Elements Finite Element Simulation Inverse Problems Optimization of Optical Gratings Examples Summary Matheon MF 1 Workshop June 1, (38)

3 Outline Diffractive Optical Elements Finite Element Simulation Inverse Problems Optimization of Optical Gratings Examples Summary Matheon MF 1 Workshop June 1, (38)

4 Outline Diffractive Optical Elements Finite Element Simulation Inverse Problems Optimization of Optical Gratings Examples Summary Matheon MF 1 Workshop June 1, (38)

5 Outline Diffractive Optical Elements Finite Element Simulation Inverse Problems Optimization of Optical Gratings Examples Summary Matheon MF 1 Workshop June 1, (38)

6 Outline Diffractive Optical Elements Finite Element Simulation Inverse Problems Optimization of Optical Gratings Examples Summary Matheon MF 1 Workshop June 1, (38)

7 Outline Diffractive Optical Elements Finite Element Simulation Inverse Problems Optimization of Optical Gratings Examples Summary Matheon MF 1 Workshop June 1, (38)

8 Outline Diffractive Optical Elements Finite Element Simulation Inverse Problems Optimization of Optical Gratings Examples Summary Matheon MF 1 Workshop June 1, (38)

9 Diffractive Optical Elements incoming field reflected modes i i (E,H ) photoresist chrome silicon transmitted modes periodical grating (details of surface geometry in order of wavelength) Matheon MF 1 Workshop June 1, (38)

10 Diffractive Optical Elements glass 174% 171% 177% 171% 174% beam splitter Matheon MF 1 Workshop June 1, (38)

11 Diffractive Optical Elements non-periodic grating, manufactured at BIFO Matheon MF 1 Workshop June 1, (38)

12 Diffractive Optical Elements Microoptical and Other Diffractive Elements: realize old and new optical functions of optical devices eg by diffraction at surfaces and interfaces: forming and splitting of laser beams, diffraction and absorption smallest size resp smallest details manufactured by means of semi-conductor industry, thin layer technology (photo resist exposition, try and wet etching, ion beam etching, vapor deposition) more pictures from B Kleemann (Carl Zeiss Oberkochen): Matheon MF 1 Workshop June 1, (38)

13 Diffractive Optical Elements Einsatz des Laser-Gitters (Schema) in Littrow-Anwendung Faltspiegel Laserkammer Gitter 5x10 LNP 5000 MM Matheon MF 1 Workshop June 1, (38)

14 Diffractive Optical Elements Diffraktive Linse in einem Projektionsobjektiv Objektiv mit Diffraktiver Linse (DL) λ = 248nm+/-05pm ΝΑ = 07 Feld = 26x8mm 2 β = -025 Material: Quarz CHL = 02µm/pm Bandbreite ~07pm Objektiv mit DL: CHL= 01µm/pm Bandbreite~14pm f DL = 1250mm d min ~ 3µm Matheon MF 1 Workshop June 1, (38)

15 Diffractive Optical Elements Applications: Microscopy Spectroscopy Interferometry Correction of image abberations Wave forming elements (CGH) Used in: Objectives for photography, microscopes, projectors Microelectronic circuits Solar technique Optical image processing Laser technique (data storage) Document security Matheon MF 1 Workshop June 1, (38)

16 Finite Element Simulation Simplest example: TM polarization (magnetic field orthogonal to cross section plane) classical diffraction (incoming wave in cross section plane) Incident wave Reflected modes Upper line Γ + FEM domain Ω Lower line Γ Transmitted modes Matheon MF 1 Workshop June 1, (38)

17 Finite Element Simulation Maxwell s equations problem reduces to scalar Helmholtz equation for transversal component v of amplitude of time harmonic magnetic field vector v(x 1, x 2 ) exp( iωt) v(x 1, x 2 ) + k 2 v(x 1, x 2 ) = 0, k := ω µε where: ε electric permitivity µ magnetic permeability ω circular frequency of incoming light k wavenumber Matheon MF 1 Workshop June 1, (38)

18 Finite Element Simulation Maxwell s equations problem reduces to scalar Helmholtz equation for transversal component v of amplitude of time harmonic magnetic field vector v(x 1, x 2 ) exp( iωt) v(x 1, x 2 ) + k 2 v(x 1, x 2 ) = 0, k := ω µε where: ε electric permitivity µ magnetic permeability ω circular frequency of incoming light k wavenumber Helmholtz equation fulfilled in domains with constant (contin) wavenumber k transmission condition through interfaces between different materials (continuous function, jump of derivative) Matheon MF 1 Workshop June 1, (38)

19 Finite Element Simulation Floquet s theorem: Incoming plane wave of the form exp(iαx 1 iβx 2 ) leads to a quasi-periodic solution v(x 1 + d, x 2 ) = v(x 1, x 2 ) exp(iαd) where: d period of grating α := k + sin θ, θ angle of incidence β := k + cos θ k + wavenumber of cover material (air) Matheon MF 1 Workshop June 1, (38)

20 Finite Element Simulation Floquet s theorem: Incoming plane wave of the form exp(iαx 1 iβx 2 ) leads to a quasi-periodic solution v(x 1 + d, x 2 ) = v(x 1, x 2 ) exp(iαd) where: d period of grating α := k + sin θ, θ angle of incidence β := k + cos θ k + wavenumber of cover material (air) v(x 1, x 2 ) exp( iαx 1 ) = v(x 1, x 2 ) = c j (x 2 ) exp(ij 2π d x 1), j= j= A + j exp(iα jx 1 + iβ j + x 2) + exp(iαx 1 iβx 2 ) x 2 > x max Matheon MF 1 Workshop June 1, (38)

21 Finite Element Simulation v(x 1, x 2 ) = j= A j exp(iα jx 1 iβ j x 2), α j := k + sin(θ) + 2π d j, β± j := x 2 < x min [k ± ] 2 [α j ] 2 Rayleigh series with Rayleigh coefficients A ± j Efficiency e ± j of jth mode: rate of energy radiated into direction of mode e ± j := (β ± j /β+ 0 ) A ± j 2 boundary condition for Helmholtz equation: left and right boundary line: quasi-periodicity upper and lower boundary line: derivative of solution in x 2 direction equals x 2 derivative of Rayleigh expansion Matheon MF 1 Workshop June 1, (38)

22 Finite Element Simulation Ω 1 k2{ + i(α, 0)}u { + i(α, 0)}ϕ + 1 (T (k ) 2 Γ α u)ϕ = 1 (k + ) 2 Ω uϕ + 1 (k + ) 2 Γ + (2iβe iβx 2 )ϕ, Γ + (T + α u)ϕ ϕ H 1 per(ω) a(u, ϕ) = F(ϕ), ϕ H 1 per(ω) Matheon MF 1 Workshop June 1, (38)

23 Finite Element Simulation Triangulation (Shevchuk): Matheon MF 1 Workshop June 1, (38)

24 Finite Element Simulation red: Air green: resist blue: SiO 2 Wavel: 633 nm Polariz: TM Angle inc: 40 Period: 17 µm Matheon MF 1 Workshop June 1, (38)

25 Finite Element Simulation Tranor= -2 Tranor= -1 Tranor= 0 Tranor= red: Air green: resist blue: SiO 2 Wavel: 633 nm Polariz: TM Angle inc: 40 eff Period: 17 µm inc_angle theta Matheon MF 1 Workshop June 1, (38)

26 Finite Element Simulation Real_Part Matheon MF 1 Workshop Imaginary_Part June 1, (38)

27 Finite Element Simulation DIPOG: Program package for numerical solution of direct and inverse problems for optical gratings Matheon MF 1 Workshop June 1, (38)

28 Finite Element Simulation DIPOG: Program package for numerical solution of direct and inverse problems for optical gratings FEM based on program package PDELIB of our institute Matheon MF 1 Workshop June 1, (38)

29 Finite Element Simulation DIPOG: Program package for numerical solution of direct and inverse problems for optical gratings FEM based on program package PDELIB of our institute Rigorous treatment of outgoing wave condition by coupling with boundary elements Treatment of large wavelength (ie: many wavelengths λ over period d) by generalized FEM Generalized FEM due to Babuška/Ihlenburg/Paik/Sauter over uniform rectangular grids special combination of local Helmholtz solution as trial functions (cf Partition of Unity Method by Babuška/Melenk or ultra-weak approach by Cessenat/Despres) Optimization Matheon MF 1 Workshop June 1, (38)

30 Finite Element Simulation TMT -1, FEM TMT -1, GFEM TET -2, FEM TET -2, GFEM TER -1, FEM TER -1, GFEM TMR 0, FEM TMR 0, GFEM Efficiency (energy percentage) in dependence of DOF in one dim simple binary grating discretized by uniform rectangular partition, comparison of conventional FEM with generalized GFEM Matheon MF 1 Workshop June 1, (38)

31 Inverse Problems (Reconstruction of gratings) Application: quality check Matheon MF 1 Workshop June 1, (38)

32 Inverse Problems (Reconstruction of gratings) Application: quality check given: measured farfield data = energy ratio and phase shifts (ie Rayleigh coefficients) of propagating reflected and transmitted modes under different angles of incidence resp under different wavelengths Matheon MF 1 Workshop June 1, (38)

33 Inverse Problems (Reconstruction of gratings) Application: quality check given: measured farfield data = energy ratio and phase shifts (ie Rayleigh coefficients) of propagating reflected and transmitted modes under different angles of incidence resp under different wavelengths sought: Grating (represented eg by profile curve or by space dependent function of refractive index) which realizes the prescribed farfield data Matheon MF 1 Workshop June 1, (38)

34 Inverse Problems (Reconstruction of gratings) Application: quality check given: measured farfield data = energy ratio and phase shifts (ie Rayleigh coefficients) of propagating reflected and transmitted modes under different angles of incidence resp under different wavelengths sought: Grating (represented eg by profile curve or by space dependent function of refractive index) which realizes the prescribed farfield data challenging mathematical problem: severely ill posed inverse problem theoretical investigations: Uniqueness, Stability of Solution, Convergence of numerical algorithms Matheon MF 1 Workshop June 1, (38)

35 Inverse Problems F(k, u 1,, u L ; γ) := c 1 c 2 L B 1 [B(k, θ l )u l w l ] 2 + L 2 (Ω) l=1 L F (θ l ) u l A meas (θ l ) 2 l 2 l=1 + γ { c 3 k 2 2 H 1/2 per (Ω) + c 4 L l=1 u l 2 H 1 per(ω) } arguments of objfunctional: Wavenumber function k and field functions u l, l = 1,, L for different angles of incidence Regularization parameter: γ precondhelmholtz equ: B 1 [B(k, θ l )u l w l ] Difference of farfield data: F (θ l )u l A meas (θ l ) with measured data A meas (θ l ) and data F (θ l )u l corresponding to u l Matheon MF 1 Workshop June 1, (38)

36 Inverse Problems F(k, u 1,, u L ; γ) min k 2 Hper 1/2 (Ω), u l Hper(Ω), 1 l = 1,, L Discretization by FEM finite dimensional problem Method of conjugate gradients for optimization (resp SQP method for a modified objectfunction) Matheon MF 1 Workshop June 1, (38)

37 Inverse Problems F(k, u 1,, u L ; γ) min k 2 Hper 1/2 (Ω), u l Hper(Ω), 1 l = 1,, L Discretization by FEM finite dimensional problem Method of conjugate gradients for optimization (resp SQP method for a modified objectfunction) Example: Number of angles of incidence: L = 25 Degrof Freedom of FEM: L times Degrof Freedom for k: 400 Matheon MF 1 Workshop June 1, (38)

38 Inverse Problems RefrInd (380) Real P x prescribed function k over cross section of grating y Matheon MF 1 Workshop June 1, (38)

39 Inverse Problems RefrInd (380) RefrInd (1480) Real P Real P x y x y prescribed function k over cross section of grating and reconstructed function k Matheon MF 1 Workshop June 1, (38)

40 Optimization of Optical Gratings (Optimal design) Inverse problems with a restricted number of geometry parameters (difficult ill posed problem turns into simple well posed problem) Design problem: design grating to realize a desired farfield pattern Matheon MF 1 Workshop June 1, (38)

41 Optimization of Optical Gratings (Optimal design) Inverse problems with a restricted number of geometry parameters (difficult ill posed problem turns into simple well posed problem) Design problem: design grating to realize a desired farfield pattern Objective functional: Φ ( e ± j (λ m, θ n ), A ± j (λ m, θ n ) ) inf Φ ( e ± j (λ m, θ n ), A ± j (λ m, θ n ) ) eg = e ± j (λ m, θ n ) e ± j,desired (λ m, θ n ) 2 λ m,θ n Matheon MF 1 Workshop June 1, (38)

42 Optimization of Optical Gratings Optimization parameters: Widths, heights, and position of rectangles in (multi-layered) binary grating Coordinates of polygonal profile (interface of two material regions) Refractive indices of material Matheon MF 1 Workshop June 1, (38)

43 Optimization of Optical Gratings Optimization parameters: Widths, heights, and position of rectangles in (multi-layered) binary grating Coordinates of polygonal profile (interface of two material regions) Refractive indices of material Mathematical properties of problem: objective functional: non-linear and smooth (FEM discretization even discont) domain: lower dimensional box constraints: eventually many non-linear smooth functionals computation of function and gradient: time consuming Matheon MF 1 Workshop June 1, (38)

44 Optimization of Optical Gratings Optimization parameters: Widths, heights, and position of rectangles in (multi-layered) binary grating Coordinates of polygonal profile (interface of two material regions) Refractive indices of material Mathematical properties of problem: objective functional: non-linear and smooth (FEM discretization even discont) domain: lower dimensional box constraints: eventually many non-linear smooth functionals computation of function and gradient: time consuming Optimization methods: global method (Simulated annealing) gradient based methods (Interior point method, Method of conjugate gradients, Augmented Lagrangian method) gradients: integral representation including solution of dual problem or representation by solution of original variational equ with new right-hand side discretization of gradient via FEM Matheon MF 1 Workshop June 1, (38)

45 Optimization of Optical Gratings a (ũ, ϕ) = F u,χ (ϕ), ϕ H 1 per(ω) F u,χ (ϕ) := 1 k 2 Ω [ ] {k 2 χ uϕ + x χ y ( x u + iαu) y ϕ + y u( x ϕ + iαϕ) [ + y χ x x u y ϕ + y u x ϕ ] [ x χ x y u y ϕ x u x ϕ + α 2 uϕ ] ]} y χ y [( x u + iαu) ( x ϕ + iαϕ) y u x ϕ A ± j (u) p = A ± j (ũ) where p is a coordinate of a corner point c at the polygonal interface and where χ is some cut off function of corner point c (piecewise linear at interface, one at c, zero over Γ ± ) Matheon MF 1 Workshop June 1, (38)

46 Optimization of Optical Gratings simple polygonal grating: 3 rd and 4 th component of gradient (conical illumination) Φ = 0004(e tr ) (e tr ) (e re ) (p tr ) (p tr ) (p re ) 2 Matheon MF 1 Workshop June 1, (38)

47 Optimization of Optical Gratings Objfunct(colisols)+Grads(arrs), Lev th param rd param Matheon MF 1 Workshop June 1, (38)

48 Examples Simple example wavelength: λ=625 nm period: 0 µm x 15 µm bounds for y-coordinates: -065 µm y 065 µm cover material: Air refractive index of substrate: n=145 grating: polygonal profile grating with four corners direction of illumination: D = (sin θ cos φ, cos θ, sin θ sin φ), θ = 48, φ = 10 TE polarization: incident electric field wavevector and normal of grating plane reflefficiencies and phase shifts: TE polarized, ie projections onto D (0, 1, 0) number of finite elements (DOF) at level 3: Matheon MF 1 Workshop June 1, (38)

49 Examples Simple example wavelength: λ=625 nm period: 0 µm x 15 µm bounds for y-coordinates: -065 µm y 065 µm cover material: Air refractive index of substrate: n=145 grating: polygonal profile grating with four corners direction of illumination: D = (sin θ cos φ, cos θ, sin θ sin φ), θ = 48, φ = 10 TE polarization: incident electric field wavevector and normal of grating plane reflefficiencies and phase shifts: TE polarized, ie projections onto D (0, 1, 0) number of finite elements (DOF) at level 3: Φ = e re e re p re p re Matheon MF 1 Workshop June 1, (38)

50 Examples Φ = 0 for high level simulation and for grating (which is to be reconstructed?): Matheon MF 1 Workshop June 1, (38)

51 Examples Simulated annealing cooling factor: 095 size of neighbourhood: 003 cooling steps: 150 restarts: 100 computing time: 375 h value for solution: Φ = Matheon MF 1 Workshop June 1, (38)

52 Examples Simulated annealing cooling factor: 095 size of neighbourhood: 003 cooling steps: 150 restarts: 100 computing time: 375 h value for solution: Φ = Conjugate gradients initial solution: (03,01), (06,02), (09,02), (12,01) iterations: 138 number of computed gradients: 385 computing time: 5 min value for solution: Φ = < 3972 Matheon MF 1 Workshop June 1, (38)

53 Examples Conjugate gradients initial solution: solution of simulated annealing iterations: 23 number of computed gradients: 89 value for solution: Φ = < Matheon MF 1 Workshop June 1, (38)

54 Examples Conjugate gradients initial solution: solution of simulated annealing iterations: 23 number of computed gradients: 89 value for solution: Φ = < Conjugate gradients initial solution: exact solof higher level iterations: 8 number of computed gradients: 51 value for solution: Φ = < Matheon MF 1 Workshop June 1, (38)

55 Examples Conjugate gradients initial solution: solution of simulated annealing iterations: 23 number of computed gradients: 89 value for solution: Φ = < Conjugate gradients initial solution: exact solof higher level iterations: 8 number of computed gradients: 51 value for solution: Φ = < Conjugate gradients initial solution: exact sol+o(005) iterations: 23 number of computed gradients: 89 value for solution: Φ = < solution not recovered! Matheon MF 1 Workshop June 1, (38)

56 Examples Conjugate gradients level: 2 initial solution: exact sol+o(005) value for solution after 10 iterations: Φ = < level: 3 initial solution: solution of level 2 value for solution after 10 iterations: Φ = < level: 4 initial solution: solution of level 3 value for solution after 10 iterations: Φ = < level: 5 initial solution: solution of level 4 value for solution after 10 iterations: Φ = < value for solution after 100 iterations: Φ = < Matheon MF 1 Workshop June 1, (38)

57 Examples Al Al Al Al Quarz Quarz Initial and optimal solution for polarization grating Matheon MF 1 Workshop June 1, (38)

58 Examples 100 TE, Reflexion TM, Reflexion TE, Transmission TM, Transmission 100 TE, Reflexion TM, Reflexion TE, Transmission TM, Transmission Polarization grating: TE light reflected, TM light transmitted Matheon MF 1 Workshop June 1, (38)

59 Examples 100 effic curve optim result Efficiency Wavelength Optimization of 20 layers for maximization of transmitted energy Matheon MF 1 Workshop June 1, (38)

60 Examples 100 effic curve optim result Efficiency Wavelength Optimization of additional binary grating over the 20 layers Matheon MF 1 Workshop June 1, (38)

61 Summary Optical gratings can be simulated by standard FEM Matheon MF 1 Workshop June 1, (38)

62 Summary Optical gratings can be simulated by standard FEM Advantage: This is a rigorous method Matheon MF 1 Workshop June 1, (38)

63 Summary Optical gratings can be simulated by standard FEM Advantage: This is a rigorous method Complex geometries can be treated Matheon MF 1 Workshop June 1, (38)

64 Summary Optical gratings can be simulated by standard FEM Advantage: This is a rigorous method Complex geometries can be treated Relatively large wavenumbers can be treated by generalized FEMs Matheon MF 1 Workshop June 1, (38)

65 Summary Optical gratings can be simulated by standard FEM Advantage: This is a rigorous method Complex geometries can be treated Relatively large wavenumbers can be treated by generalized FEMs Algorithms useful for reconstruction problems and design of gratings The corresponding DIPOG routines are still under development Matheon MF 1 Workshop June 1, (38)

66 Summary Optical gratings can be simulated by standard FEM Advantage: This is a rigorous method Complex geometries can be treated Relatively large wavenumbers can be treated by generalized FEMs Algorithms useful for reconstruction problems and design of gratings The corresponding DIPOG routines are still under development Thank you for your attention Matheon MF 1 Workshop June 1, (38)