Theory of Optical Waveguide

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1 Theor of Optical Waveguide Class: Integrated Photonic Devices Time: Fri. 8:am ~ :am. Classroom: 資電 6 Lecturer: Prof. 李明昌 (Ming-Chang Lee Reflection and Refraction at an Interface (TE n kˆi H i E i θ θ E r H r kˆr Transverse Electric (TE or s wave Medium n Medium H t E t θ kˆt Snell s law: n sin( θ n sin( θ Refraction Coefficient: r TE Er ncosθ ncosθ E n cosθ + n cosθ i i

2 Reflection and Refraction at an Interface (TM kˆi kˆr E i E r Transverse Electric (TM or p wave n H i θ θ H r Medium n E t Medium H t θ kˆt Snell s law: n sin( θ n sin( θ Refraction Coefficient: r TM Er ncosθ ncosθ E n cosθ + n cosθ i Eternal Reflection of TE Wave (Low to High

3 Eternal Reflection of TM Wave (Low to High n Brewster Angle: θb tan n Internal Reflection of TE Wave (High to Low Snell s law: n sin( θ n sin( θ Critical Angle: θ c n n sin ( For θ > θ c ϕ sin θ sin θc tan cos θ

4 Internal Reflection of TM Wave (High to Low Critical Angle: θ c n n sin ( For θ > θ c ϕ sin θ sin θc tan cos sin θ θc Summar of Reflection at Interface The phase changes either b π or b for eternal reflection. The phase has abrupt change across Brewster angle for TM wave. The Brewster angle is smaller than critical angle. The phase changes continuousl for total internal reflection

5 Slab Waveguide (D Waveguide Ra Optics Phase front Cladding n 3 Core n Xa X X-a n 3 n n Cladding n n Wave Optics Standing Wave Propagating Wave Derivation of Basic Equations Cladding n 3 a Core n Substrate n General Waveguide Mode Epression: ~ E E(, ep[ j( β ωt] ~ H H (, ep[ j( β ωt] Transverse Component Longitudinal Component F. Law A. Law E jβe jωµ H E + jβe jωµ H E E jωµ H H jβh jωεn E H + jβh jωεn E H H jωεn E

6 Polariation of Optical Fields in Slab (Planar Waveguide (TE (TM E H E,H is independent of -ais Transverse Electric (TE: E, H E H Transverse Magnetic (TM: H, E H E For slab waveguide, the optical modes can alwas be decomposed into TE and TM TE-Polaried Optical Modes in Slab Waveguide E, E and E is -direction independent; that is, E The epression of optical wave: waveguide E ( r, t E( rep( jωt E( r, t E (, ep( jωt The optical wave propagates in -direction E( r, t E (, ep( jωt E( r, t E ( ep( jβ ep( jωt E ( ep( jβ jωt E( E( Propagating Wave

7 TE-Polaried Optical Modes in Slab Waveguide E( r + n( ω k E( r E( r, t E ( ep( jβ jωt E + ( nk β E H H β E ωµ j ωµ de d ( n could be in core, substrate or cladding What is the solution of (? What is the β? k n 3 k n k n Suppose n >n >n 3 cladding inde substrate inde core inde n 3 cladding core n n substrate Leaked Leaked Guided Guided Not eist!

8 TE-Polaried Optical Guided Modes in Slab Waveguide E where Acos( ha φep[ γ 3( a] Acos( h φ Acos( ha + φep[ γ ( + a] h k n β γ β k n γ β k n 3 3 ( > a ( a a ( < a Xa X X-a n 3 n n ep ep cos Boundar Condition : E is continuous at a and -a Trivial! Boundar Condition : H is continuous at a and -a j de H should be continuous at the boundar ωµ d TE-Polaried Optical Modes in Slab Waveguide γ 3Acos( ha φep[ γ 3( a] de h Asin( h φ d γ Acos( ha + φ ep[ γ ( + a] de d a de d + a and h sin( ha + φ γ cos( ha + φ h sin( ha φ γ 3 cos( ha φ de d a ( > a ( a a ( < a de d + a Xa X X-a w tan( u + φ u ' where w tan( u φ u n 3 n n u ha w γ a ' w γ 3a mπ u + tan mπ φ + tan w ( + tan u w ( tan u ' w ( u ' w ( u m,,,...

9 Smmetric and Asmmetric Smmetric Asmmetric ( tan ( tan ( tan ( tan ' ' u w u w m u w u w m u π φ π + ( tan π φ π m u w m u ' 3 w w γ γ γ ' 3 w w γ γ a -a a -a Electric Fields of High Order Mode ( ( ( ] ( ep[ cos( cos( ] ( ep[ cos( 3 a a a a a h a A h A a h a A E < > + + γ φ φ γ φ The waveguide is smmetrical. m: even cos(h m: odd sin(h

10 Mode Number 5 m m m m 3 For eample, how man solutions to satisf the eigenvalue equation for a smmetric waveguide (n n 3? mπ w tan( u + u 5 m 4 m 5 m h u/a 6 ( Cut-Off Condition γ 3 h γ k n 3 k n k n β ' β For a guided mode, Guided Range kn < β < kn < h < k ( n n We define V a k ( n n u ha< a k ( n n V High-order modes have large u: um ( + > um (

11 Cut-Off Condition What is the highest mode can be supported in the waveguide? Recall ' mπ w( m w( m ( tan ( tan ( um + + um ( um ( Suppose the highest mode No. is M, then u(m is approimatel given b um ( V (close to cut-off Therefore, um ( a k ( n n ' wm ( a k ( n n3 wm ( π ( n n M 3 ( + tan ( um ( n n Cut-Off Condition As we know, u(m has to be less than V; that is, π ( n n + < V M 3 ( tan ( um ( n n Therefore, for a waveguide, the accommodated mode no (M. M TE V tan n n3 π π n n int For smmetric waveguides, there is alwas a TE mode (m eisting.

12 TM-Polaried Optical Modes in Slab Waveguide Homework: E + ( nk β E β H E ωµ µ H Reciprocal ε n E d H + ( nk β H d E β ωε n H H j ωµ de d E j ωε n dh d. What is the cut-off condition for TM mode?. For a silicon slab with.5-um thickness (n3.4, air-cladding on the two sides, what are the β s of TM mode and TE mode (m for wavelength from 5 nm to 56 nm? Optical Power (Background S E H Ponting vector S ( E H H E E H nˆ Volume Adv surface A nds volume Volume ( H E E H dv surface ( E H nds Volume E H ( ε E + µ H dv t t surface ( E H nds

13 Optical Power (Background Volume E H ( ε E + µ H dv t t surface ( E H nds nˆ ε ( E t t W e ε ( H t t W h volume Electric Stored Energ t Volume Magnetic Stored Energ ( W + W dv S nds S represents the power flow densit e h surface Optical Power (Background The intensit of the power densit can be calculated through the time-average of the ponting vector over one wave ccle: T S E Hdt T For monochromatic wave ( ep( ω * ep( ( ep( ω * ep( E E j t + E j t H H j t + H j t Re( * S E H * P S uˆ Re( E H ˆ sds usds s s

14 Optical Power of Slab Waveguide For TE mode, the onl non-ero component is E. And because wave is propagate in -direction, we onl consider E and H. Therefore, the power of TE mode is Longitude (Propagation * P S uˆ Re( E H ˆ sds usds s s Re( * β TE E H ωµ P dd E dd Recall E E E + * ( Optical Power of Slab Waveguide Xa X X-a n 3 n n Clad Core Sub For slab waveguide (TE polariation, Suppose the field amplitude is A Longitude (Propagation β PTE d E d ωµ P Core P P Sub Clad βaa P PCore + PSub + Pclad + + ' ωµ w w ' ωµ w w βaa cos ( u + φ ωµ w βaa cos ( u φ ωµ w' βaa sin ( u φ sin ( u φ

15 Effective Waveguide Thickness dte a+ + γ γ 3 n n n 3 a / γ / γ 3 samller γ represents the optical field penetrates farther Mode Confinement dte a+ + γ γ 3 d / d / Γ TE E ( d E ( d n n n 3 Γ TM d / ( d / ( ( n H d n H d a Γ TE ( a + + d γ + h / γ γ + h / γ TE 3 3 Γ TE ( a d + γ q + h / γ + γ q + h / γ TM / γ / γ 3 dtm β β where q + a+ + γ q γ q 3 3 k k β β q + 3 k k3

16 Mode Confinement Low order modes have better mode confinement than high order modes TE modes have better mode confinement than TM modes Approimated Solutions of Waveguide Modes h + β k n v,,, ν ν r where π k λ and h ν ( ν + π W eν Effective width Suppose the waveguide is well-confined σ λ n c e M + ( r c π nr W W n n β kn r σ for TM, for TE Smmetric Waveguide (Fundametial Mode

17 Eample Refractive inde of SiO :.45 Wavelength:.3 µm Refractive inde of Si 3 N 4 :. 3 µm. How man TE modes? Refractive inde of SiO :.45 π π 3 V a ( nr nc ( λ.3 M TE V tan n n V 6 π π n n π int π int int for smmetric waveguide 7 TE modes Eample Refractive inde of SiO :.45 Wavelength:.3 µm Refractive inde of Si 3 N 4 :. 3 µm Refractive inde of SiO :.45. What is the propagation constant β of the fundamental mode (TE? σ λ n c e M + ( r c π nr W W n n W e ( ( µ m π π π k nr h m λ W W n n π e M + ( r c ( + π π h W 3.3 β 9.694( µ e (TE actual value: (9.6664

18 Rectangular Waveguide (D Waveguide Diffused Waveguide Channel Waveguide Rib Waveguide Inde Profile of A Channel Waveguide n 3 n 5 n 4 Thickness d n a n Width n > n, n 3, n 4, n 5

19 Mode Epression Magnetic field intensit: H (, X( Y( ep( jh+ φ ep( jh+ ψ ep[ γ ( + a] ep[ γ ( d] cos( h+ φ ep[ γ ( d] ep[ γ ( a] ep[ γ ( d] Y d ep[ γ ( + a] cos( h+ ψ cos( h+ φ cos( h+ ψ ep[ γ ( a] cos( h+ ψ Y -d ep[ γ ( + a] ep[ γ ( + d] cos( h+ φ ep[ γ ( + d] ep[ γ ( a] ep[ γ ( + d] X -a X a TE Polariation (TE-Like H, E and H is predominant Width: a d H H + + ( nk β H d H ωµ H E H + β ωε β H E ωεn j H E ωεn j H H β n This modelling was proposed b Marcatili E Thickness: d Suppose width > thickness

20 TM Polariation (TM-like H, E and H is predominant d H d H + + ( nk β H H ωµ H E H β ωε β H E ωεn j H E ωεn j H H β n Width: a E Thickness: d Suppose width > thickness TE-Polaried Optical Modes in Channel Waveguide (Marcatili s Method For simplicit, we assume the rectangular waveguide is smmetrical cladding H Acos( h φcos( h ψ Acos( ha φ ep[ γ ( a]cos( h ψ Acos( h φep[ γ ( d]cos( hd ψ ( ( (3 n n ( ( n (3 n Where n h h + k n β ( γ h + k n β ( h + γ + k n β (3 And π φ ( p p,,. π TE pq ψ ( q q,,. from instead of E

21 TE-Polaried Optical Modes in Channel Waveguide Boundar Condition : E is continuous at a, -a H n E Boundar Condition : H is continuous at d, -d E Given a p and q H H π ha ( p + tan π hd ( q + tan And n γ ( n h γ ( h β k n ( h + h γ γ k ( n n h k ( n n h B.C. B.C. (TM eigenvalue eq. in slab waveguide (TE eigenvalue eq. in slab waveguide Five unknowns: β, h, h, γ, γ Mode Profile X( p H(, X( Y( p Y( q q p p q q

22 Mode Profile (Consider Polariation Electric field TE TM TE TM TE TM Rib Waveguide n h n n d a It is difficult to be analed b Marcatili method!

23 Effective Inde Methods h n n d n a We suppose the polariation is in -direction (TE. The wave equation for the TE mode is H H + + [ k n (, β ] H Effective Inde Methods Suppose the electromagnetic fields can be epressed with the separation of variables H H + + [ k n (, β ] H H (, X( Y( Divided b XY d X d Y + + [ k n (, β ] X d Y d d Y + [ k n (, kneff ( ] Y d d X + [ k neff (, β ] X d N eff (: effective inde distribution

24 Effective Inde Methods d Y + [ k n (, k neff ( ] Y d d X + [ k neff (, β ] X d d Y + [ k n (, k neff ( ] Y d d X + [ k neff (, β ] X d Procedure of solving problems Divide the waveguide into several sections (horiontall or verticall. Consider either Y component or X component onl. Calculate the propagation constant in each section. β Calculate the effective indices n eff k Consider the other component based on the effective indices D rectangular waveguide problem Several slab waveguide problems Effective Inde Methods n n d n a n d n d+h n n + + d n n n eff ( n eff ( n eff (3 n n eff ( n eff ( n n n eff (3 a

25 Comparison of Effective Inde Method and Marcatili s Method Normalied Inde β kn b kn kn a EI. d MT. kn β kn For same waveguide, the propagation constant calculated from effective inde method is larger than that of Marcatili s Method. The accurate propagation constant is between these two values.

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)

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) 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