Waveguide Couler I Class: Integrated Photonic Devices Time: Fri. 8:am ~ 11:am. Classroom: 資電 6 Lecturer: Prof. 李明昌 (Ming-Chang Lee) Waveguide Couler n 1 > n n Waveguide 1 n 1 n Waveguide n 1 n How to switch the ower from one waveguide to other waveguides
Two Tyes of Directional Coulers I. Planar Waveguide Couler (Out-of-Plane) (Cross Section) II. Dual-Channel Waveguide Couler (In-Plane) Couled Mode Theory for Directional Couler monochromatic waveguide modes Exyt (,,, ) = AL ( ) ψ ( xy, )ex( jωt) Axial (Longitudinal) Part Transverse Part Time harmonic L ( ) is a comlex amlitude including hase term ex( jβ ) ψ ( x, y) reresents the field distribution of a single waveguide
Couled Mode Theory for Directional Couler N 1 (x,y) N (x,y) We define the eigen modes in each otical waveguide satisfying Maxwell s equations E = jωµ H H = jωεn E ( = 1,) N (x,y): refractive index The electromagnetic fields of the couled waveguide is summation of two eigen modes E = A 1( ) E1+ A( ) E A 1 and A is the amlitude of H = A 1( ) H1+ A( ) H otical fields Couled Mode Theory for Directional Couler The summed fields should also satisfy Maxwell s equation The vector formula E = jωµ H H jωε = N E (1) da ( AE) = A E+ A E= A E+ u E d N(x,y) denote the entire refractive index of couled waveguide () (the amlitude only varies at -direction) Perturbation! Combine (1) and () 1 (u da da E ) (u E ) d + d = 1 (3) da da 1 (u H 1) + jωε ( N N1 ) A1E 1+ (u H ) + jωε ( N N ) AE = d d (4)
Couled Mode Theory for Directional Couler E 1 (4) H1 (3) dxdy = E (4) H (3) dxdy = (5) (6) From Eq. (5) da 1 1 1 da u ( E H E H ) dxdy + + d d u ( E 1 H1+ E1 H1 ) dxdy + ja + ja ( N N ) E 1 E1dxdy 1 1 ωε u ( E 1 H1+ E1 H1 ) dxdy ωε ( N N ) E1 Edxdy = u ( E 1 H1+ E1 H1 ) dxdy (7) Couled Mode Theory for Directional Couler From Eq. (6) da 1 1 da1 u ( E H E H ) dxdy + + d d u ( E H + E H ) dxdy + ja + ja ( N N ) E Edxdy ωε u ( E H + E H ) dxdy ωε ( N N1 ) E E1dxdy = u ( E H + E H ) dxdy 1 (8) Here we searate the transverse and axial deendencies of electromagnetic fields E = Eex( jβ ) H = Hex( jβ ) ( = 1,)
Couled Mode Theory for Directional Couler da1 da + c1 ex[ j( β β1) ] + jχ1a1 + jκ1 A ex[ j( β β1) ] = d d (9) (from (7)) da da1 + c1 ex[ j( β β1) ] + jχ A + jκ1a1 ex[ j( β β1) ] = (1) (from (8)) d d where κ c q q = = χ = ωε q q u ( E H + E H ) ( N N ) E E dxdy u E H E H ( q + q ) u E H E H ( + ) ωε u ( E H + E H ) dxdy dxdy dxdy ( N N ) E E dxdy dxdy ( q erturb ) ( q erturb ) Couled Mode Theory for Directional Couler N ( x, y) N ( x, y) 1 Waveguide I Waveguide II χ κ1 c 1 1 χ is much smaller than κ
Couled Mode Theory for Directional Couler (a). Consider χ χ χ = χ is real number (b). Consider c q 1 Re( = ) E H u P dxdy ( E H + E H ) udxdy = 4P = 4 Recall ( Eex( ω ) E ex( ω )) ( Hex( ω ) H ex( ω )) 1 E = j t + j t 1 H = j t + j t If E, H is the eigen mode c q = u E H E H ( q + q ) u E H + E H ( ) dxdy dxdy c = c 1 1 Couled Mode Theory for Directional Couler (c). Consider κ q Suose there is no couling loss, the total ower should be constant P = 1 Re( ) dp d = E H udxdy and ja A ( κ κ δc )ex( j δ) ja A ( κ κ δc )ex( j δ) = 1 1 1 1 1 1 1 1 ( β β1) where δ = κ = κ + δc 1 1 1 The couling coefficients are comlicate conjugated when Two modes are in the same waveguide or Two waveguides are hase matching for every β β δ = = 1
Couled Mode Theory for Directional Couler General exression of couled mode theory da1 = jκ A ex( j δ) + jγ A d a a 1 da = jκ A ex( j δ) + jγ A d b 1 b [(9) (1) c ex( j δ ) = ] 1 [(1) (9) c ex( j δ ) = ] 1 where κ c χ κ = a b 1 1 1 c1 κ c κ = χ 1 1 1 1 c1 and κ c γ = a κ c γ = b χ 1 1 1 1 c1 χ 1 1 1 c1 Couled Mode Theory for Directional Couler In most cases, we suose two waveguides are sufficient searated χ and C 1 q, q Then κ = κ 1 1 and γ = γ = a b The general couled mode theory is simlified by da1 = jκ1 A ex[ j( β β1) ] d da = jκ1a1 ex[ j( β β1) ] d
Couled Mode Theory for Directional Couler The couling between mode is given by the couled mode equations for the amlitudes of the two modes (guide 1 and guide ) da 1( ) = jβ 1A1( ) + jκ1a( ) d da ( ) = jβ A( ) + jκ1a1( ) d A 1 = A1ex( jβ1) A = A ex( jβ ) β1 : the roagation constant of guide 1 β : the roagation constant of guide κ1 : the couling coefficient from to 1 κ1 : the couling coefficient from 1 to In general, κ 1 and κ 1 are not equal. We suose the couling coefficients are aroximated and real κ1 = κ1 = κ Solutions of Directional Couler δ κ A1 ( ) = cos( g) + j sin( g) A1() j sin( g) A() ex( j( 1 ) ) g β + δ g κ A δ ( ) = j sin( g) A1() + cos( g) + j sin( g) A() ex( j( β δ) ) g g Where g κ + δ and β β δ 1 If the initial condition A 1 () = 1 and A () =, the solution is δ A1 ( ) = cos( g) j sin( g) ex( j( 1 ) ) g β + δ κ A( ) = j sin( g)ex( j( β δ) ) g The roagation constant of each eigen mode becomes the average of two individual roagation constants in the couling region
Transferred Power Phase Matching Phase Mismatching Transferred Power P P 1 Transferred Power P P 1 Normalied Distance (g) Normalied Distance (g) δ = δ = κ sin ( g) P1( ) = A1( ) A1( ) = cos ( g) + δ g β β1 κ Where g κ + δ and δ P( ) = A( ) A( ) = sin ( g) g With a hase difference δ, the ower transfer is incomlete Directional Couler with Phase Matching Suose the two waveguides are identical and β1 = β = β κ1 = κ1 = κ Real The couled mode equation becomes da 1( ) = jβ A 1( ) + jκa( ) d da ( ) = jβ A ( ) + jκa1( ) d If the initial condition A 1 () = 1 and A () =, the solution is A1 ( ) = cos( κ)ex( jβ) A( ) = jsin( κ)ex( jβ)
Directional Couler with Phase Matching and Loss If we consider the loss α Intensity P 1 () α A1( ) = cos( κ)ex( jβ)ex( ) α A( ) = jsin( κ)ex( jβ)ex( ) P () Proagation Distance P1( ) = A1( ) A1 ( ) = cos ( κ)ex( α) P( ) = A( ) A ( ) = sin ( κ)ex( α) The otical ower is comletely transferred as roagation length (L) (m + 1) π L =, m =,1, κ What is the couling coefficient? Suose two identical slab waveguides E 1 n w s E n 1 n κ = ωε ( 1 ) E1 E uˆ ( E H + E H ) dx n n dx 1 1 1 1 κ = n 1 n h,γ h γ ex( γ s) ( βd ) TE h + γ x β k n 1 γh β + γ h κ ( ex( ) ) γs βdtm h + γ k The couling coefficient is an exonential function of ga s.
Transfer Matrix Exression Suose a lossless couler A i1 L A o1 Waveguide 1 Waveguide A i recall A1 ( ) = cos( κ)ex( jβ) A( ) = jsin( κ)ex( jβ) A 1 ()=1, A ()= where Ao 1 1 c jc Ai 1 = A o jc 1 c Ai c= sin ( κ L) A o c : ower couling ratio Varying L affects the ower couling ratio Exerimental Measurement 3µm 3µm GaAs 3µm L =.1 mm (1% couler) L = 1 mm (3dB couler)
Summary of Waveguide Couling When you design a waveguide couler, you have to Consider the couling coefficient (κ) Consider the hase matching ( β) Consider the couling length (L) sin ( g) P1 ( ) = cos ( g)ex( α) + δ ex( α) g κ P ( ) = sin ( g)ex( α ) g g κ + δ Suermodes Actually, the ower oscillating between two waveguides can be thought as the beating between two suermodes. For examle, suose the two waveguides are identical, A1 ( ) = A cos( κ)ex( jβ) A( ) = A jsin( κ)ex( jβ) if A1 () = A A() = E ( ) = A 1( ) ψ ( xy, ) + A( ) ψ ( xy, ) 1 [ ψ (, )cos( κ )ex( β ) ψ (, )sin( κ )ex( β )] = A x y j + j x y j 1 1 1 = A ψ1[ ex( jκ) + ex( jκ) ] ex( jβ) + ψ[ ex( jκ) ex( jκ) ] ex( jβ) 1 1 = A ( ψ1 ψ)ex [ j( β κ) ] + ( ψ1+ ψ)ex [ j( β + κ) ] E ( ) a E ( ) b
Suermodes 1 1 E ( ) = A ( ψ1 ψ)ex [ j( β κ) ] + ( ψ1+ ψ)ex [ j( β + κ) ] = Aψ ex( jβ ) + Aψ ex( jβ ) = A E ( ) + A E ( ) a a b b a b Where βb 1 ψa = ψ1 ψ 1 ψb = ψ1+ ψ ( ) ( ) and βa = β κ βb = β + κ βa ψ b ψ1 ψ ψ a ψ ψ1 Waveguide I (Symmetric) Waveguide II Waveguide I Waveguide II (Antisymmetric) Suermodes β b + β a ψ b ψ a ( β β b a) L= mπ, m=,1,,... β b + β a ψ b ψ a ( β β ) L= b a ( m+ 1) π, m=,1,,... π π L π = = β β κ b a
Polariation Deendent Because the couling coefficients of TE modes and TM modes are different, we can design a olariation slitter. Suose two identical waveguides TE TE+TM Nπ (N ± 1) π ls = = κ κ TE l s TM TM where N κ κ :integer and TE TM are couling coefficient for TE and TM Symmetric Curved Waveguide Couling Parallel Waveguide Symmetric Curved Waveguide A 1 () Waveguide 1 A 1 (s) Waveguide 1 s A () Waveguide da1( ) = jβ1a1( ) d+ jκ1a( ) d da( ) = jβa( ) d+ jκ1a1( ) d s Waveguide A (s) d A1() s = jβ1a1() s ds + jκ1() s A() s ds da() s = jβa() s ds+ jκ1() s A1() s ds Line coordinate vs. curve coordinate The amlitude increment can be modelled in oint-to-oint corresondence
Asymmetric Curved Waveguide Couling Waveguide θ 1 =9 θ = 9 Y X X l = Y Waveguide Waveguide 1 X: Axial coordinate for waveguide Y: Axial coordinate for waveguide 1 D If Y R Y= X= X X Y = tan( ) R + D R θ 1 θ l Tangent Waveguide 1 then θ1 = θ If we can define a function to link X,Y curve coordinates such that θ 1 = θ, the asymmetric curved waveguide couling can be treated as a symmetric curved waveguide couling. Asymmetric Curved Waveguide Couling The asymmetric curved waveguide couling can be exressed by couled mode theory. da(x) 1 = jβ1( x) A1( x) dx+ j κ( y) A( y) dy da (y) = jβ( y) A( y) dy+ j κ1( x) A1( x) dx where X Y = tan( ) R + D R and dy κ1 = κ cos( π θ) dx dx κ = κ cos( π θ) dy 1/ 1/ β ( x) 1 π θ β ( y) Couling coefficient l due to ga The angle of β ( x), β ( y) 1
Asymmetric Curved Waveguide Couling Consider the couler is lossless; that is, the couling ower is conservative κ dx = κ dy 1 The couled mode theory becomes da(x) 1 da(x) 1 = jβ1( x) A1( x) dx+ j = jβ 1( xa ) 1( x) + j κ1( xa ) ( y) κ1( x) A( y) dx dx da(y) = jβ( ya ) ( ydy ) + j κ( ya ) 1( xdy ) da (y) = jβ ( ya ) ( y) + j κ( ya ) 1( x) dy A ( y ) A1( x) A ( y) A ( x) 1 A1( x) C11( x, y) C1( x, y) A1( x) A( y) = C1( x, y) C( x, y) A( y) 11 1 1 C( xy, ) + C ( xy, ) = C ( xy, ) + C ( xy, ) = 1 C ( x, y) C ( x, y) + C ( x, y) C ( x, y) = 11 1 1 Due to ower conservation Examle of Asymmetric Curved Waveguide Couling (Vertical Couling) Ring R w: width Waveguide d: thickness s: ga (i) (ii) Waveguide microdisk (i) Tangent Position (cross section) (ii) Deviated Position (cross section)
Examle of Asymmetric Curved Waveguide Couling (Vertical Couling) For a silicon curved waveguide with -um radius of curvature and.8 um width, couling coefficient (k ) is analyed as follow: Couling Coefficient (m -1 ) Ga:.4 µm Ga:. µm Ga:.6 µm Couling Position (µm) The effective couling length is only from -5µm to 5µm Examle of Asymmetric Curved Waveguide Couling (Vertical Couling) Because the couling coefficient is deendent on the ga sacing, the ower couling ratio is also a function of ga sacing. A ( y) A ( x) 1 Transfer back to the original waveguide R: µm Ga Sacing Power Couling Ratio A ( y ) A1( x) Straight Waveguide Ga Sacing (µm) Curved Waveguide
Couler Fabrication Gold