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
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
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
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
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
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
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!
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,,,...
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
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 6 5 5 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 (
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.
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
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
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
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 3 3 3 / γ / γ 3 dtm β β where q + a+ + γ q γ q 3 3 k k β β q + 3 k k3
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
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 (.45 9.986 λ.3 M TE V 3 9.986 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.3 3 + (.45 3.3( 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.3 3.3 e (TE actual value: (9.6664
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
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
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
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
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!
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
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
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.