Modelling of Ridge Waveguide Bends for Sensor Applications

Transcription

1 Modelling of Ridge Waveguide Bends for Sensor Applications Wilfrid Pascher FernUniversität, Hagen, Germany n 1 n2 n 3 R

2 Modelling of Ridge Waveguide Bends for Sensor Applications Wilfrid Pascher FernUniversität, Hagen, Germany n 1 n2 n 3 Radiation losses Evanescent field of a rib waveguide are precisely modelled R by the Method of Lines

3 Why employ the Method of Lines? The MoL is a semianalytic approach

4 Why employ the Method of Lines? The MoL is a semianalytic approach Analytic solution in one coordinate direction (perpendicular to the layers)

5 Why employ the Method of Lines? The MoL is a semianalytic approach Analytic solution in one coordinate direction (perpendicular to the layers) Discretization in the other direction(s) (with Finite Differences) = 3D problem 2D discretization

6 Why employ the Method of Lines? The MoL is a semianalytic approach Analytic solution in one coordinate direction (perpendicular to the layers) Discretization in the other direction(s) (with Finite Differences) = 3D problem 2D discretization For reasons of technology, waveguide structures are multilayered (e.g., planar waveguides) cascaded (e.g., waveguide circuits)

7 Advantages and Disadvantages + precise modeling + low memory and computing time

8 Advantages and Disadvantages + precise modeling + low memory and computing time reduced flexibility = different geometries require new algorithms = extension to hybrid methods

9 Transition to Cylindrical Coordinates n 1 n1 n 2 n 2 R

10 Transition to Cylindrical Coordinates z Propagation ϕ r P o p a a g t on i n 1 n1 n 2 n 2 R

11 Transition to Cylindrical Coordinates z Propagation ϕ r P o p a a g t on i n 1 n1 n 2 n 2 R Propagation z ϕ exp( jβz) exp( jνϕ)

12 Discretization of Straight / Bent Waveguides

13 Discretization of Straight / Bent Waveguides

14 Discretization of Straight / Bent Waveguides Discretization x Analytic Solution r Analytic Solution y z Discretization ψ ψ e h ψ h ψ e

15 Discretization of Straight / Bent Waveguides Discretization x Analytic Solution r Analytic Solution y z Discretization ψ ψ e h ψ h ψ e Discretization x z P x P z

16 Discretization of Straight / Bent Waveguides Discretization x Analytic Solution r Analytic Solution y z Discretization ψ ψ e h ψ h ψ e Discretization x z Analytic solution y r P x P z sin(k y y) J ν ( λr)

17 The Method of Lines (MoL) for circular bends in waveguides 1. Transition cartesian cylindrical (x, y, z) (z, r, ϕ)

18 The Method of Lines (MoL) for circular bends in waveguides 1. Transition cartesian cylindrical (x, y, z) (z, r, ϕ) 2. Discretization of the wave equation 2 z 2 P z (3. Transformation to diagonal form) diag ( λ 2 k )

19 The Method of Lines (MoL) for circular bends in waveguides 1. Transition cartesian cylindrical (x, y, z) (z, r, ϕ) 2. Discretization of the wave equation 2 z 2 P z (3. Transformation to diagonal form) diag ( λ 2 k ) ) 4. Solution of the wave equation J ν ( λk r +...

20 The Method of Lines (MoL) for circular bends in waveguides 1. Transition cartesian cylindrical (x, y, z) (z, r, ϕ) 2. Discretization of the wave equation 2 z 2 P z (3. Transformation to diagonal form) diag ( λ 2 k ) ) 4. Solution of the wave equation J ν ( λk r Field computation (6. Inverse transformation)

21 The Method of Lines (MoL) for circular bends in waveguides 1. Transition cartesian cylindrical (x, y, z) (z, r, ϕ) 2. Discretization of the wave equation 2 z 2 P z (3. Transformation to diagonal form) diag ( λ 2 k ) ) 4. Solution of the wave equation J ν ( λk r Field computation (6. Inverse transformation) 7. Characteristic equation in C radiation loss L Im(n eff )

22 Wave equations in coordinate free form Vector MoL with two potentials Π e, Π h accurate fulfillment of the continuity conditions for all field components

23 Wave equations in coordinate free form Vector MoL with two potentials Π e, Π h accurate fulfillment of the continuity conditions for all field components coordinate free approach: a) Helmholtz equation for Π h { + ε r (z)k 2 0 } Π h = 0

24 Wave equations in coordinate free form Vector MoL with two potentials Π e, Π h accurate fulfillment of the continuity conditions for all field components coordinate free approach: a) Helmholtz equation for Π h { + ε r (z)k 2 0 } Π h = 0 b) Sturm-Liouville differential equation for Π e { + ε r (z)k ε r (z) grad ε r(z) div } Π e = 0

25 Wave equations in cylindrical coordinates Potentials with one component in z direction only Π e,h = k 2 0 exp( jνϕ) ψ e,h a z order proportional to effective model index ν = n eff R using normalized coordinates: e.g. R = k 0 R

26 Wave equations in cylindrical coordinates Potentials with one component in z direction only Π e,h = k 2 0 exp( jνϕ) ψ e,h a z order proportional to effective model index ν = n eff R using normalized coordinates: e.g. R = k 0 R Consideration of the radiation losses n eff, ν complex no artificial increase in the guiding

27 Discretization of the wave equation in the cartesian z direction Partial differential equations in cylindrical coordinates { ( 1 r ) } ν2 r r r r 2 + ε r(z) + 2 z 2 ψ h = 0 { 1 r r ( r ) ν2 r r 2 + ε r(z) + ε r (z) z ( 1 ε r (z) ) } ψ e = 0 z

28 Discretization of the wave equation in the cartesian z direction Partial differential equations in cylindrical coordinates { ( 1 r ) } ν2 r r r r 2 + ε r(z) + 2 z 2 ψ h = 0 { 1 r r ( r ) ν2 r r 2 + ε r(z) + ε r (z) z ( 1 ε r (z) ) } ψ e = 0 z Potentials and dielectric constants continuous discretized ψ e, ψ h Ψ e, Ψ h (column vector) ε r (z) ε e, ε h (diagonal matrix)

29 Discretization of the wave equation in the cartesian z direction Differential operators difference operators ε r (z) z ( 1 ε r (z) 2 z 2 ψ h P zh Ψ h (tridiagonal) ) ψ e Pze ε Ψ e (tridiagonal) z

30 Discretization of the wave equation in the cartesian z direction Differential operators difference operators ε r (z) z ( 1 ε r (z) 2 z 2 ψ h P zh Ψ h (tridiagonal) ) ψ e Pze ε Ψ e (tridiagonal) z Coupled ordinary differential equations 1 r d dr ( r d ) I ν2 dr r 2 I + ε P ε ze e,h P zh } {{ } (tridiagonal) Ψ e,h = 0

31 Transformation to diagonal form with T ε e 1 (tridiagonal) {}}{ (ε e P ε ze) T ε e = λ 2 e = diag ( λ 2 e,k )

32 Transformation to diagonal form with (tridiagonal) {}}{ Te ε 1 (ε e Pze) ε Te ε = λ 2 e = diag ( λ 2 e,k ) Decoupled ordinary differential equations { ( 1 d r d ) } I ν2 r dr dr r 2 I + λ 2 e Ψ e = 0 with the transformed potential Ψ e = T ε e 1 Ψ e

33 Transformation to diagonal form with (tridiagonal) {}}{ Te ε 1 (ε e Pze) ε Te ε = λ 2 e = diag ( λ 2 e,k ) Decoupled ordinary differential equations { ( 1 d r d ) } I ν2 r dr dr r 2 I + λ 2 e Ψ e = 0 with the transformed potential Ψ e = T ε e 1 Ψ e completely analogous for Ψ h = T h 1 Ψ h

34 Solution of the Bessel differential equation 1 r d dr ( r d ) dr + ( λ 2 e,k ν2 r 2 ) Ψ e,k = 0 with ν = n eff R b for one component Ψ e,k of the transformed potential

35 Solution of the Bessel differential equation

36 Solution of the Bessel differential equation Solution in three regions

37 Solution of the Bessel differential equation Solution in three regions 1 finite for r 0 Ψ e,k = A k J ν ( λ e,k r)

38 Solution of the Bessel differential equation Solution in three regions 1 finite for r 0 Ψ e,k = A k J ν ( λ e,k r) 2 Ψ e,k = B k J ν ( λ e,k r) + C k Y ν ( λ e,k r)

39 Solution of the Bessel differential equation Solution in three regions 1 finite for r 0 Ψ e,k = A k J ν ( λ e,k r) 2 Ψ e,k = B k J ν ( λ e,k r) + C k Y ν ( λ e,k r) 3 radiation condition Ψ e,k = D k H (2) ν ( λ e,k r)

40 Radial derivation matrices Γ Transmission line equation A = Γ A with r B B Γ = j λ p ν r ν W B W A q ν

41 Radial derivation matrices Γ Transmission line equation A = Γ A with r B B Γ = j λ p ν r ν W B W A q ν The radial derivation matrix Γ is computed from the cross products and the Wronskian of the Bessel functions p ν = J ν ( r A )Y ν ( r B ) J ν ( r B )Y ν ( r A ) q ν = J ν ( r A )Y ν( r B ) J ν( r B )Y ν ( r A ) r ν = J ν( r A )Y ν ( r B ) J ν ( r B )Y ν( r A ) W A,B = 2 π r A,B with r A,B = j λr A,B

42 Programming of the cylinder functions The problem 1. very high complex order: ν j with small imaginary part

43 Programming of the cylinder functions The problem 1. very high complex order: ν j with small imaginary part 2. order nearly equal to argument: ν λ 1 r for the first eigenvalue

44 Programming of the cylinder functions The problem 1. very high complex order: ν j with small imaginary part 2. order nearly equal to argument: ν λ 1 r for the first eigenvalue The solution 1, 2 Uniform asymptotic series (high argument and order)

45 Programming of the cylinder functions The problem 1. very high complex order: ν j with small imaginary part 2. order nearly equal to argument: ν λ 1 r for the first eigenvalue The solution 1, 2 Uniform asymptotic series (high argument and order) Alternative for the ring regions Multiplication formulas for cross products

46 Uniform asymptotic expansions Bessel functions, e.g. Abramowitz-Stegun (9.3.35) J ν (νy) ( 4ζ 1 y 2 ) 1/4 Ai(ν 2/3 ζ) ν 1/3 i=0 a i (ζ) ν 2i + Ai (ν 2/3 ζ) ν 5/3 i=0 b i (ζ) ν 2i ν with 2 3 ζ3/2 = log y 2 y 1 y 2

47 Uniform asymptotic expansions Bessel functions, e.g. Abramowitz-Stegun (9.3.35) J ν (νy) ν ( 4ζ 1 y 2 ) 1/4 Ai(ν 2/3 ζ) ν 1/3 i=0 a i (ζ) ν 2i + Ai (ν 2/3 ζ) ν 5/3 with 2 3 ζ3/2 = log y 2 y Coefficients a i (ζ), b i (ζ) descending and independent of ν i=0 b i (ζ) ν 2i 1 y 2

48 Uniform asymptotic expansions Bessel functions, e.g. Abramowitz-Stegun (9.3.35) J ν (νy) ν ( 4ζ 1 y 2 ) 1/4 Ai(ν 2/3 ζ) ν 1/3 i=0 a i (ζ) ν 2i + Ai (ν 2/3 ζ) ν 5/3 with 2 3 ζ3/2 = log y 2 y Coefficients a i (ζ), b i (ζ) descending and independent of ν Series terms decrease by 1 ν Series can be truncated already after i = 2 for double precision arithmetic i=0 b i (ζ) ν 2i 1 y 2

49 Uniform asymptotic expansions Bessel functions, e.g. Abramowitz-Stegun (9.3.35) J ν (νy) ν ( 4ζ 1 y 2 ) 1/4 Ai(ν 2/3 ζ) ν 1/3 i=0 a i (ζ) ν 2i + Ai (ν 2/3 ζ) ν 5/3 with 2 3 ζ3/2 = log y 2 y Coefficients a i (ζ), b i (ζ) descending and independent of ν Series terms decrease by 1 ν Series can be truncated already after i = 2 for double precision arithmetic i=0 b i (ζ) ν 2i 1 y 2 Y ν (νy), J ν(νy), Y ν(νy) computed analogously

50 Multiplication theorem for direct calculation of cross products Bessel function of outer radius r B = µ r A C ν ( r B ) = C ν (µ r A ) = µ ν i=0 (δ r A ) i i! C ν i ( r A ) with δ = (µ 2 1)/2 Abramowitz-Stegun (9.1.74)

51 Multiplication theorem for direct calculation of cross products Bessel function of outer radius r B = µ r A C ν ( r B ) = C ν (µ r A ) = µ ν i=0 (δ r A ) i i! C ν i ( r A ) with δ = (µ 2 1)/2 Abramowitz-Stegun (9.1.74) yields series for cross products p ν, q ν p ν = (δ r A ) i µ ν i! q ν i=0 P ν,i ( r A ) Q ν,i ( r A )

52 Multiplication theorem for direct calculation of cross products Bessel function of outer radius r B = µ r A C ν ( r B ) = C ν (µ r A ) = µ ν i=0 (δ r A ) i i! C ν i ( r A ) with δ = (µ 2 1)/2 Abramowitz-Stegun (9.1.74) yields series for cross products p ν, q ν p ν = (δ r A ) i µ ν i! q ν i=0 P ν,i ( r A ) Q ν,i ( r A ) P, Q are computed from a three term recurrence relation Other cross products r ν, s ν are determined from p ν, q ν

53 Multiplication theorem for direct calculation of cross products Bessel function of outer radius r B = µ r A C ν ( r B ) = C ν (µ r A ) = µ ν i=0 (δ r A ) i i! C ν i ( r A ) with δ = (µ 2 1)/2 Abramowitz-Stegun (9.1.74) yields series for cross products p ν, q ν p ν = (δ r A ) i µ ν i! q ν i=0 P ν,i ( r A ) Q ν,i ( r A ) P, Q are computed from a three term recurrence relation Other cross products r ν, s ν are determined from p ν, q ν = Two independent procedures for checking

54 Field computation in every subregion separately Matching on the cylinders r = r A and r = r B

55 Field computation in every subregion separately Matching on the cylinders r = r A and r = r B Transfer of the fields from cylinder r = r A to r = r B H A H B = ỹ1 ỹ 2 ỹ 2 ỹ 1+ Ẽ A ẼB

56 Field computation in every subregion separately Matching on the cylinders r = r A and r = r B Transfer of the fields from cylinder r = r A to r = r B H A H B = ỹ1 ỹ 2 ỹ 2 ỹ 1+ Ẽ A ẼB yields HB = ỸBẼB

57 Field computation in every subregion separately Matching on the cylinders r = r A and r = r B Transfer of the fields from cylinder r = r A to r = r B H A H B = ỹ1 ỹ 2 ỹ 2 ỹ 1+ Ẽ A ẼB yields HB = ỸBẼB with the recurrence Ỹ B = ỹ 2 (ỹ 1 ỸA) 1 ỹ2 ỹ 1+

58 Field computation in every subregion separately Matching on the cylinders r = r A and r = r B Transfer of the fields from cylinder r = r A to r = r B H A H B = ỹ1 ỹ 2 ỹ 2 ỹ 1+ Ẽ A ẼB yields HB = ỸBẼB with the recurrence Ỹ B = ỹ 2 (ỹ 1 ỸA) 1 ỹ2 ỹ 1+ In cartesian coordinates matrices ỹ 1,2 depend only on the layer thickness

59 Field computation in every subregion separately Matching on the cylinders r = r A and r = r B Transfer of the fields from cylinder r = r A to r = r B H A H B = ỹ1 ỹ 2 ỹ 2 ỹ 1+ Ẽ A ẼB yields HB = ỸBẼB with the recurrence Ỹ B = ỹ 2 (ỹ 1 ỸA) 1 ỹ2 ỹ 1+ In cartesian coordinates matrices ỹ 1,2 depend only on the layer thickness now ỹ 1,2 depend on the radii r A, r B now ỹ 1,2 are computed using cylinder functions

60 Determination of the radiation losses Inverse transformation to spatial domain characteristic equation Z(n eff ) H A = 0

61 Determination of the radiation losses Inverse transformation to spatial domain characteristic equation Z(n eff ) H A = 0 Solution det(z(n eff )) = 0 by a search for zeros in the complex plane C

62 Determination of the radiation losses Inverse transformation to spatial domain characteristic equation Z(n eff ) H A = 0 Solution det(z(n eff )) = 0 by a search for zeros in the complex plane C Radiation losses (db/90 ): L = Im(n eff ) R π 10 ln 10

63 Geometry of a Bent Rib Waveguide Sensor gas w t Si Si 3 N 4 Si O 2 n 1 n2 n 3 h 3 h 2 R OIWS108A

64 Geometry of a Bent Rib Waveguide Sensor gas w t Si 3 N 4 Si O 2 n 1 n2 n 3 h 2 h 3 w = 3 5µm n 1 = t = µm n 2 = h 2 = 5µm Si n 3 = 3.5 R OIWS108A

65 Sensitivity Sensitivity in % w [µm] Sensitivity depending on thickness t for w = 3.0 µm

66 Distribution of the Radial Electric Field z [µm] dr [µm] Fundamental mode, w = 3.0 µm

67 Distribution of the Radial Electric Field z [µm] dr [µm] First higher order mode, w = 3.50 µm

68 Distribution of the Radial Electric Field z [µm] dr [µm] First higher order mode, w = 3.13 µm

69 Conclusion: Advantages and Disadvantages The presented model yields most accurate results for propagation constant radiation loss

70 Conclusion: Advantages and Disadvantages The presented model yields most accurate results for propagation constant sensitivity radiation loss evanescent field

71 Conclusion: Advantages and Disadvantages The presented model yields most accurate results for propagation constant sensitivity radiation loss evanescent field Comparison at the TU Delft for a polarization converter: MoL is superior to the Effective Index Method (EIM) and Finite Element Method (FEM)

72 Conclusion: Advantages and Disadvantages The presented model yields most accurate results for propagation constant sensitivity radiation loss evanescent field Comparison at the TU Delft for a polarization converter: MoL is superior to the Effective Index Method (EIM) and Finite Element Method (FEM) Disadvantage only applicable to strictly rotational structures Extension: Combination with other numerical methods