Waveguide Coupler I Waveguide Coupler waveguide to other waveguides
Transcription
Waveguide Coupler I Waveguide Coupler waveguide to other waveguides
Waveguide Coupler I Class: Integrated Photonic Devices Time: Fri. 8:00am ~ 11:00am. Classroom: 資電206 Lecturer: Prof. 李明昌(Ming-Chang Lee) Ming-Chang Lee, Integrated Photonic Devices Waveguide Coupler n1 > n0 Waveguide 1 n0 n1 n0 Waveguide 2 n1 n0 • How to switch the power from one waveguide to other waveguides Ming-Chang Lee, Integrated Photonic Devices 1 Two Types of Directional Couplers I. Planar Waveguide Coupler (Out-of-Plane) (Cross Section) II. Dual-Channel Waveguide Coupler (In-Plane) Ming-Chang Lee, Integrated Photonic Devices Coupled Mode Theory for Directional Coupler monochromatic waveguide modes E ( x, y, z, t ) = AL( z )ψ ( x, y ) exp(− jωt ) Axial (Longitudinal) Part Transverse Part Time harmonic L( z ) is a complex amplitude including phase term exp( j β z ) ψ ( x, y ) represents the field distribution of a single waveguide Ming-Chang Lee, Integrated Photonic Devices 2 Coupled Mode Theory for Directional Coupler N1(x,y) N2(x,y) We define the eigen modes in each optical waveguide satisfying Maxwell’s equations ∇ × E p = jωµ0 H p 2 ∇ × H p = − jωε 0 N p E p (p = 1,2) Np(x,y): refractive index The electromagnetic fields of the coupled waveguide is summation of two eigen modes E = A1 ( z )E1 + A2 ( z )E 2 H = A1 ( z )H1 + A2 ( z )H 2 A1 and A2 is the amplitude of optical fields Ming-Chang Lee, Integrated Photonic Devices Coupled Mode Theory for Directional Coupler The summed fields should also satisfy Maxwell’s equation ∇ × E = jωµ0 H 2 ∇ × H = − jωε 0 N E The vector formula (1) N(x,y) denote the entire refractive index of coupled waveguide ∇ × ( AE) = A∇ × E + ∇A × E = A∇ × E + dA uz × E dz (the amplitude only varies at z-direction) (2) Perturbation! Combine (1) and (2) (u z × E1 ) (u z × H1 ) dA1 dA + (u z × E2 ) 2 = 0 dz dz (3) dA1 dA + jωε 0 ( N 2 − N12 ) A1 E1 + (u z × H 2 ) 2 + jωε 0 ( N 2 − N 2 2 ) A2 E2 = 0 (4) dz dz Ming-Chang Lee, Integrated Photonic Devices 3 Coupled Mode Theory for Directional Coupler ∞ ∫ ∫ ∞ −∞ −∞ ∞ ∫ ∫ E1* ⋅ (4) − H1* ⋅ (3) dxdy = 0 (5) E2* ⋅ (4) − H 2* ⋅ (3) dxdy = 0 (6) ∞ −∞ −∞ From Eq. (5) ∞ ∞ * * dA1 dA2 ∫−∞ ∫−∞ u z ⋅ (E1 × H 2 + E 2 × H1 )dxdy + ⋅ ∞ ∞ * * dz dz u ⋅ (E1 × H1 + E1 × H1 )dxdy ∫ ∫ ωε ∫ ∫ ( N − N )E ⋅ E dxdy + jA ∫ ∫ u ⋅ (E × H + E × H )dxdy ωε ∫ ∫ ( N − N )E ⋅ E dxdy =0 + jA ∫ ∫ u ⋅ (E × H + E × H )dxdy z −∞ −∞ ∞ ∞ 1 ∞ 0 −∞ −∞ ∞ z −∞ −∞ ∞ 0 ∞ 2 ∞ 2 * 1 ∞ 1 z * 1 * 1 2 2 * 1 1 1 2 −∞ −∞ −∞ −∞ * 1 2 1 1 2 * 1 1 (7) Ming-Chang Lee, Integrated Photonic Devices Coupled Mode Theory for Directional Coupler From Eq. (6) ∞ ∞ * * ∫ ∫ u ⋅ (E × H + E × H )dxdy ∫ ∫ u ⋅ (E × H + E × H )dxdy ωε ∫ ∫ ( N − N )E ⋅ E dxdy + jA ∫ ∫ u ⋅ (E × H + E × H )dxdy ωε ∫ ∫ ( N − N )E ⋅ E dxdy =0 + jA ( ) ⋅ × + × dxdy u E H E H ∫ ∫ dA2 dA1 + ⋅ dz dz −∞ −∞ ∞ ∞ z 2 −∞ −∞ z 2 ∞ 2 ∞ 2 ∞ 1 ∞ ∞ −∞ −∞ 2 2 2 2 * * 2 2 2 2 * * 2 z ∞ 0 1 * 0 −∞ −∞ ∞ −∞ −∞ 1 ∞ 2 2 2 * 2 2 1 −∞ −∞ 2 1 * z 2 * 2 2 (8) 2 Here we separate the transverse and axial dependencies of electromagnetic fields E p = E p exp( j β p z ) H p = H p exp( j β p z ) (p = 1,2) Ming-Chang Lee, Integrated Photonic Devices 4 Coupled Mode Theory for Directional Coupler dA1 dA + c12 2 exp[ j ( β 2 − β1 ) z ] + j χ1 A1 + jκ12 A2 exp[ j ( β 2 − β1 ) z ] = 0 dz dz (9) (from (7)) dA2 dA + c21 1 exp[− j ( β 2 − β1 ) z ] + j χ 2 A2 + jκ 21 A1 exp[− j ( β 2 − β1 ) z ] = 0 (10) (from (8)) dz dz where κ pq = ωε 0 ∫ ∞ ∞ ∫ ∫ ∞ c pq ∞ −∞ −∞ ∞ ∞ −∞ −∞ χp = ∞ ∞ ( N 2 − N q 2 )E p* ⋅ Eq dxdy u z ⋅ (E p* × H q + Eq × H p* )dxdy u z ⋅ (E p* × H p + E p × H p* )dxdy ωε 0 ∫ ∫ ∫ ∫ u z ⋅ (E p* × H p + E p × H p* )dxdy −∞ −∞ ∫ ∫ = ∫ ∫ ∞ −∞ −∞ ∞ −∞ −∞ ∞ ∫ ∞ −∞ −∞ ( q perturb p) ( q perturb p) ( N 2 − N p 2 )E p* ⋅ E p dxdy u z ⋅ (E p* × H p + E p × H p* )dxdy Ming-Chang Lee, Integrated Photonic Devices Coupled Mode Theory for Directional Coupler N1 ( x, y ) N 2 ( x, y ) Waveguide I κ12 • Waveguide II c12 χ1 χ is much smaller than κ Ming-Chang Lee, Integrated Photonic Devices 5 Coupled Mode Theory for Directional Coupler (a). Consider χp χ p is real number χ p = χ p* (b). Consider c pq Recall 1 Pp = ∫ ∫ Re(E p × H p* ) ⋅ u z dxdy −∞ −∞ 2 ∞ ∞ ∫ ∫ ∞ −∞ −∞ ∞ (E p* × H p + E p × H p* ) ⋅ u z dxdy = 4 Pp = 4 ) ( ( * 1 E exp(− jωt ) + E exp( jωt ) 2 * 1 H = H exp(− jωt ) + H exp( jωt ) 2 E= ) If Ep, Hp is the eigen mode ∞ c pq ∫ ∫ = ∫ ∫ ∞ −∞ −∞ ∞ ∞ −∞ −∞ u z ⋅ (E p* × H q + Eq × H p* )dxdy c21 = c12* u z ⋅ (E p × H p + E p × H p )dxdy * * Ming-Chang Lee, Integrated Photonic Devices Coupled Mode Theory for Directional Coupler (c). Consider κ pq Suppose there is no coupling loss, the total power should be constant P=∫ ∞ ∫ ∞ −∞ −∞ * 1 Re(E × H ) ⋅ u z dxdy 2 and dP =0 dz jA1* A2 (κ 21* − κ12 − 2δ c12 ) exp(− j 2δ z ) − jA1 A2* (κ 21 − κ12* − 2δ c12* ) exp( j 2δ z ) = 0 where δ= ( β 2 − β1 ) 2 for every z κ 21 = κ12* + 2δ c12* • The coupling coefficients are complicate conjugated when – Two modes are in the same waveguide or – Two waveguides are phase matching δ= β 2 − β1 2 =0 Ming-Chang Lee, Integrated Photonic Devices 6 Coupled Mode Theory for Directional Coupler General expression of coupled mode theory dA1 = jκ a A2 exp( j 2δ z ) + jγ a A1 dz [(9) − (10) × c12 exp( j 2δ z ) = 0] dA2 = jκ b A1 exp(− j 2δ z ) + jγ b A2 dz [(10) − (9) × c21 exp(− j 2δ z ) = 0] κa = where κb = κ12 − c12 χ 2 1 − c12 γa = 2 and κ 21 − c12* χ1 1 − c12 γb = 2 κ 21c12 − χ1 1 − c12 2 κ12 c12* − χ 2 1 − c12 2 Ming-Chang Lee, Integrated Photonic Devices Coupled Mode Theory for Directional Coupler In most cases, we suppose two waveguides are sufficient separated χ p ,q ∼ 0 Then and C pq 1 κ12 = κ 21 and γ a = γ b = 0 The general coupled mode theory is simplified by dA1 = jκ12 A2 exp[ j ( β 2 − β1 ) z ] dz dA2 = jκ 21 A1 exp[− j ( β 2 − β1 ) z ] dz Ming-Chang Lee, Integrated Photonic Devices 7 Coupled Mode Theory for Directional Coupler The coupling between mode is given by the coupled mode equations for the amplitudes of the two modes (guide 1 and guide 2) d A1 ( z ) = j β1 A1 ( z ) + jκ12 A2 ( z ) dz d A2 ( z ) dz = j β 2 A2 ( z ) + jκ 21 A1 ( z ) A1 = A1 exp( j β1 z ) A2 = A2 exp( j β 2 z ) β1 : the propagation constant of guide 1 β 2 : the propagation constant of guide 2 κ12 : the coupling coefficient from 2 to 1 κ 21 : the coupling coefficient from 1 to 2 In general, κ12 and κ21 are not equal. We suppose the coupling coefficients are approximated and real κ 21 = κ12 = κ Ming-Chang Lee, Integrated Photonic Devices Solutions of Directional Coupler δ κ A1 ( z ) = cos( gz ) + j sin( gz ) A1 (0) − j sin( gz ) A2 (0) exp( j ( β1 + δ ) z ) g g κ δ A2 ( z ) = − j sin( gz ) A1 (0) + cos( gz ) + j sin( gz ) A2 (0) exp( j ( β 2 − δ ) z ) g g Where g ≡ κ + δ 2 2 2 and δ ≡ β 2 − β1 2 If the initial condition A1(0) = 1 and A2(0) = 0, the solution is δ A1 ( z ) = cos( gz ) − j sin( gz ) exp( j ( β1 + δ ) z ) g A2 ( z ) = j κ g sin( gz ) exp( j ( β 2 − δ ) z ) • The propagation constant of each eigen mode becomes the average of two individual propagation constants in the coupling region Ming-Chang Lee, Integrated Photonic Devices 8 Transferred Power Phase Mismatching Phase Matching Transferred Power Transferred Power P1 P2 Normalized Distance (gz) δ =0 P1 P2 Normalized Distance (gz) δ = 2κ 2 * 2 2 sin ( gz ) P1 ( z ) = A1 ( z ) ⋅ A1 ( z ) = cos ( gz ) + δ g2 β − β1 2 2 2 and δ ≡ 2 2 Where g ≡ κ + δ P ( z ) = A ( z ) ⋅ A ( z )* = κ sin 2 ( gz ) 2 2 2 2 g2 • With a phase difference δ, the power transfer is incomplete Ming-Chang Lee, Integrated Photonic Devices Directional Coupler with Phase Matching Suppose the two waveguides are identical β1 = β 2 = β and Real κ 21 = κ12 = κ The coupled mode equation becomes d A1 ( z ) = j β A1 ( z ) + jκ A2 ( z ) dz d A2 ( z ) dz = j β A2 ( z ) + jκ A1 ( z ) If the initial condition A1(0) = 1 and A2(0) = 0, the solution is A1 ( z ) = cos(κ z ) exp( j β z ) A2 ( z ) = j sin(κ z ) exp( j β z ) Ming-Chang Lee, Integrated Photonic Devices 9 Directional Coupler with Phase Matching and Loss Intensity If we consider the loss α α A1 ( z ) = cos(κ z ) exp( j β z ) exp(− 2 z ) A2 ( z ) = j sin(κ z ) exp( j β z ) exp(− α z ) 2 P1(z) P1 ( z ) = A1 ( z ) A1* ( z ) = cos 2 (κ z ) exp(−α z ) * 2 P2 ( z ) = A2 ( z ) A2 ( z ) = sin (κ z ) exp( −α z ) P2(z) Propagation Distance The optical power is completely transferred as propagation length (L) L= (2m + 1)π , m = 0,1,2 … 2κ Ming-Chang Lee, Integrated Photonic Devices What is the coupling coefficient? Suppose two identical slab waveguides n2 E1 w n1 s n2 E2 n1 x n2 h,γ ∞ κ= ωε 0 ∫ (n12 − n0 2 )E1* ⋅ E2 dx −∞ ∞ ∫ uˆ −∞ z ⋅ (E1* × H1 + E1 × H1* )dx 2h 2γ exp(−γ s ) κ= β dTE (h 2 + γ 2 ) κ≈ k0n1 γ h2 β z 2β 2 + γ 2 − h2 exp(−γ s ) k2 2 β dTM (h + γ ) 2 2 ⋅ • The coupling coefficient is an exponential function of gap s. Ming-Chang Lee, Integrated Photonic Devices 10 Transfer Matrix Expression Suppose a lossless coupler Ai1 Ao1 L Waveguide 1 Waveguide 2 Ao 2 Ai 2 recall 2 Ao1 1 − c = Ao 2 jc A1 ( z ) = cos(κ z ) exp( j β z ) A2 ( z ) = j sin(κ z ) exp( j β z ) A1(0)=1, A2(0)=0 2 where c = sin (κ L) A i1 1 − c Ai 2 jc 2 c 2 : power coupling ratio Varying L affects the power coupling ratio Ming-Chang Lee, Integrated Photonic Devices Experimental Measurement 3µm 3µm GaAs 3µm L = 2.1 mm (100% coupler) L = 1 mm (3dB coupler) Ming-Chang Lee, Integrated Photonic Devices 11 Summary of Waveguide Coupling When you design a waveguide coupler, you have to • Consider the coupling coefficient (κ) • Consider the phase matching (∆β) • Consider the coupling length (L) 2 2 2 sin ( gz ) exp(−α z ) P1 ( z ) = cos ( gz ) exp(−α z ) + δ g2 2 P ( z ) = κ sin 2 ( gz ) exp(−α z ) 2 g2 g2 ≡ κ 2 +δ 2 Ming-Chang Lee, Integrated Photonic Devices Supermodes Actually, the power oscillating between two waveguides can be thought as the beating between two supermodes. For example, suppose the two waveguides are identical, A1 ( z ) = A0 cos(κ z ) exp( j β z ) A2 ( z ) = A0 j sin(κ z ) exp( j β z ) if A1 (0) = A0 A2 (0) = 0 E ( z ) = A1 ( z )ψ 1 ( x, y ) + A2 ( z )ψ 2 ( x, y ) = A0 [ψ 1 ( x, y ) cos(κ z ) exp( j β z ) + jψ 2 ( x, y )sin(κ z ) exp( j β z )] 1 1 = A0 ψ 1 [ exp( jκ z ) + exp(− jκ z )] exp( j β z ) + ψ 2 [ exp( jκ z ) − exp(− jκ z ) ] exp( j β z ) 2 2 1 1 = A0 (ψ 1 −ψ 2 ) exp [ j ( β − κ ) z ] + (ψ 1 +ψ 2 ) exp [ j ( β + κ ) z ] 2 2 Ea ( z ) Eb ( z ) Ming-Chang Lee, Integrated Photonic Devices 12 Supermodes 1 1 E ( z ) = A0 (ψ 1 −ψ 2 ) exp [ j ( β − κ ) z ] + (ψ 1 +ψ 2 ) exp [ j ( β + κ ) z ] 2 2 = A0ψ a exp( j β a z ) + A0ψ b exp( j βb z ) = A0 Ea ( z ) + A0 Eb ( z ) Where βb ψb 1 ψ a = 2 (ψ 1 −ψ 2 ) ψ = 1 (ψ +ψ ) 2 b 2 1 ψ1 Waveguide II β a = β − κ βb = β + κ βa ψa ψ2 Waveguide I and ψ1 Waveguide I (Symmetric) ψ2 Waveguide II (Antisymmetric) Ming-Chang Lee, Integrated Photonic Devices Supermodes βb βa + ψb ψa ( βb − β a ) L = 2mπ , m = 0,1, 2,... βa βb ψb + ψa ( βb − β a ) L = 2(m + 1)π , m = 0,1, 2,... Lπ = π π = βb − β a 2κ Ming-Chang Lee, Integrated Photonic Devices 13 Polarization Dependent Because the coupling coefficients of TE modes and TM modes are different, we can design a polarization splitter. TE Suppose two identical waveguides TE+TM ls ls = Nπ κTE (2 N ± 1)π = 2κTM TM where N :integer and κ TE κ TM are coupling coefficient for TE and TM Ming-Chang Lee, Integrated Photonic Devices Symmetric Curved Waveguide Coupling Symmetric Curved Waveguide Parallel Waveguide A1(z) A1(s) Waveguide 1 Waveguide 1 z A2(z) s s Waveguide 2 Waveguide 2 A2(s) d A1 ( z ) = j β1 A1 ( z )dz + jκ12 A2 ( z )dz d A2 ( z ) = j β 2 A2 ( z )dz + jκ 21 A1 ( z )dz d A1 ( s) = j β1 A1 ( s)ds + jκ12 ( s ) A2 ( s)ds d A2 ( s ) = j β 2 A2 ( s )ds + jκ 21 ( s) A1 ( s)ds • Line coordinate vs. curve coordinate • The amplitude increment can be modelled in point-to-point correspondence Ming-Chang Lee, Integrated Photonic Devices 14 Asymmetric Curved Waveguide Coupling Waveguide 2 Y Waveguide 2 Y Tangent θ1= 900 R l θ2= 900 X Waveguide 1 θ1 θ2 Y=0 D X=0 X: Axial coordinate for waveguide 2 Y: Axial coordinate for waveguide 1 X =Y l Waveguide 1 X X Y If 2 R + D = tan( 2 R ) then θ1 = θ 2 • If we can define a function to link X,Y curve coordinates such that θ1 = θ2, the asymmetric curved waveguide coupling can be treated as a symmetric curved waveguide coupling. Ming-Chang Lee, Integrated Photonic Devices Asymmetric Curved Waveguide Coupling The asymmetric curved waveguide coupling can be expressed by coupled mode theory. d A1(x) = j β1 ( x) A1 ( x)dx + jκ 2 ( y ) A2 ( y )dy d A2 (y) = j β 2 ( y ) A2 ( y )dy + jκ1 ( x) A1 ( x)dx where X Y = tan( ) 2R + D 2R β1 ( x) and 1/ 2 dy κ1 = κ 0 ⋅ cos(π − 2θ ) ⋅ dx 1/ 2 dx = ⋅ − ⋅ κ κ cos( π 2 θ ) 0 2 dy l Coupling coefficient due to gap The angle of β1 ( x), β 2 ( y) π − 2θ β 2 ( y) Ming-Chang Lee, Integrated Photonic Devices 15 Asymmetric Curved Waveguide Coupling Consider the coupler is lossless; that is, the coupling power is conservative κ1dx = κ 2 dy The coupled mode theory becomes d A1(x) = j β1 ( x) A1 ( x) + jκ1 ( x) A2 ( y ) dx d A2 (y) = j β ( y ) A2 ( y ) + jκ ( y ) A1 ( x) 2 2 dy d A1(x) = j β1 ( x) A1 ( x)dx + jκ1 ( x) A2 ( y )dx d A2 (y) = j β 2 ( y ) A2 ( y )dy + jκ 2 ( y ) A1 ( x)dy A2 ( y0 ) A2 ( y ) A1 ( x) A1 ( x0 ) A1 ( x) C11 ( x, y ) C12 ( x, y ) A1 ( x0 ) A ( y ) = C ( x, y ) C ( x, y ) ⋅ A ( y ) 22 2 21 2 0 2 2 2 2 C11 ( x, y ) + C21 ( x, y ) = C12 ( x, y ) + C22 ( x, y ) = 1 C11 ( x, y ) ⋅ C12 ( x, y )* + C21 ( x, y ) ⋅ C22 ( x, y )* = 0 Due to power conservation Ming-Chang Lee, Integrated Photonic Devices Example of Asymmetric Curved Waveguide Coupling (Vertical Coupling) Ring R w: width Waveguide d: thickness s: gap (i) (ii) Waveguide microdisk (i) Tangent Position (cross section) (ii) Deviated Position (cross section) Ming-Chang Lee, Integrated Photonic Devices 16 Example of Asymmetric Curved Waveguide Coupling (Vertical Coupling) Coupling Coefficient (m-1) For a silicon curved waveguide with 20-um radius of curvature and 0.8 um width, coupling coefficient (k0) is analyzed as follow: Gap: 0.2 µm Gap: 0.4 µm Gap: 0.6 µm Coupling Position (µm) The effective coupling length is only from -5µm to 5µm Ming-Chang Lee, Integrated Photonic Devices Example of Asymmetric Curved Waveguide Coupling (Vertical Coupling) Because the coupling coefficient is dependent on the gap spacing, the power coupling ratio is also a function of gap spacing. Gap Spacing R: 20 µm A2 ( y0 ) Transfer back to the original waveguide A1 ( x) A1 ( x0 ) Power Coupling Ratio A2 ( y ) Straight Waveguide Gap Spacing (µm) Curved Waveguide Ming-Chang Lee, Integrated Photonic Devices 17 Coupler Fabrication Gold Ming-Chang Lee, Integrated Photonic Devices 18