EE 529 Semiconductor Optoelectronics – Optical Waveguides EE529 Semiconductor Optoelectronics Optical Waveguides 1. Modes in planar waveguides 2. Ray-optics approach 3. EM-wave approach 4. Modes in channel waveguides “Photonic Devices,” Jia-Ming Liu, Chapter 2 “Theory of Optical Waveguides,” by H. Kogelnik, in “Guided-wave Optoelectronics,” T. Tamir, ed., Chapter 2, Springer Verlag Lih Y. Lin
EE 529 Semiconductor Optoelectronics Why study waveguides for integrated – Optical Waveguides optoelectronics? Cleaved reflecting surface Semiconductor laser Photonic integrated circuits W L Stripe electrode Oxide insulator p -GaAs (Contacting layer) p -Al x Ga 1-x As (Confining layer) p -GaAs (Active layer) n -Al x Ga 1-x As (Confining layer) 2 1 3 Substrate n -GaAs (Substrate) Current paths Substrate Electrode Cleaved reflecting surface Elliptical laser Light: Science and Applications (2012) Active region where J > J th . beam (Emission region) Modulator Directional coupler W av eg u id es In V ( t ) A B L o E le ctro de V ( t ) B C O u t E lec trod e B In D A A W av eg u id e L iN bO F ib ers 3 L iN bO E O Sub str ate 3 2 Lih Y. Lin
EE 529 Semiconductor Optoelectronics – Optical Waveguides E-M Field in a Planar Waveguide Warm-up question: What kind of structure can be a waveguide? Assuming a monochromatic wave propagating in z-direction: ω = − β ω = j t j z j t ( , ) ( ) ( , ) E r t E r e E x y e e ∇ + = 2 2 2 ( ) ( ) ( ) 0 E r k n r E r ∂ 2 + − β = 2 2 2 ( , ) ( ) ( , ) 0 E x y k n E x y Region I: 1 ∂ 2 x ∂ 2 + − β = 2 2 2 ( , ) ( ) ( , ) 0 E x y k n E x y Region II: 2 ∂ 2 x ∂ 2 + − β = 2 2 2 ( , ) ( ) ( , ) 0 E x y k n E x y Region III: 3 ∂ 2 x 3 Lih Y. Lin
EE 529 Semiconductor Optoelectronics – Optical Waveguides Modes in a Planar Waveguide − β 2 2 2 ( ) k n Modal solutions are sinusoidal or exponential, depending on the sign of i Boundary conditions: ∂ ( , ) E x y ( , ) and E x y must be continuous at the interface between layers. ∂ x > > n n n Assuming , let’s draw possible waveguide modes: 2 3 1 kn 1 kn 3 kn 2 0 x β n 1 n 2 n 3 (The technique you learned from solving optical waveguide modes can be applied to the design of many photonic components.) 4 Lih Y. Lin
EE 529 Semiconductor Optoelectronics – Optical Waveguides Guided Modes in a Planar Waveguide Examples of guided modes in a symmetrical waveguide m: Mode order Q: How to define the mode order? Q: Can we obtain an infinite number of solutions to β with continuous values? 5 Lih Y. Lin
EE 529 Semiconductor Optoelectronics Experimental Observation of – Optical Waveguides Waveguide Modes Q1: How to choose the laser wavelength? Q2: How to create different modes? Q3: How to tell which side is air, which side is the substrate? 6 Lih Y. Lin
EE 529 Semiconductor Optoelectronics – Optical Waveguides Do things in simple ways first. → Geometrical optics. 7 Lih Y. Lin
EE 529 Semiconductor Optoelectronics Ray Patterns in the – Optical Waveguides Three-Layer Planar Waveguide Remember that only discrete values of β are allowed. How to solve for allowable β ? Step 1: Determine the relation between β and the angle of the optical ray. Different modes have different angles. ∝ + γ sin( ) E hx In the guided region, β + h = 2 2 2 2 k n 2 Overall propagation constant → Propagation constant in x h − θ = 1 tan For the m-th mode, m β m → Propagation constant in z Lower-order mode has smaller θ m and larger β m. 8 Lih Y. Lin
EE 529 Semiconductor Optoelectronics – Optical Waveguides Ray Patterns for Different Modes β φ 2 n − − φ = ≤ 1 1 1 sin sin 2 kn n 2 2 n n n − φ ≥ ≤ φ ≤ 1 3 1 3 sin sin 2 2 n n n 2 2 2 kn 1 kn 3 kn 2 0 Higher-order β Lower-order 9 Lih Y. Lin
EE 529 Semiconductor Optoelectronics – Optical Waveguides Reflection at a Dielectric Interface Step 2: Determine phase changes at the interfaces. = = , E rE E tE 3 1 2 1 θ − θ cos cos n n = 1 1 2 2 = 1 + For TE wave: r t r TE θ + θ TE TE cos cos n n 1 1 2 2 θ − θ cos cos n n n = = + 2 1 1 2 1 ( 1 ) For TM wave: r t r TM TM TM θ + θ n cos cos n n 2 2 1 1 2 = ϕ = ϕ | | exp( ), | | exp( ) r r j r r j TE TE TE TM TM TM → Phase change accompanies reflection. Ref: Pedrotti 3 , “Introduction to Optics,” Sec. 23.1-23.3 10 Lih Y. Lin
EE 529 Semiconductor Optoelectronics – Optical Waveguides Phase Change on Total Internal Reflection 2 θ − 2 θ θ − θ − θ θ − φ 2 2 2 2 2 2 2 2 φ 2 sin sin sin sin sin n n sin n n n = = 1 c 1 1 2 = 1 = c 1 1 2 TE TM 1 tan tan θ θ θ θ θ 2 2 2 cos cos 2 cos n cos sin n n 1 1 1 1 1 1 2 c 2 11 Lih Y. Lin
EE 529 Semiconductor Optoelectronics – Optical Waveguides Dispersion Equation Step 3: Define transverse resonance condition. Transverse resonance condition: θ − φ − φ = π 2 cos 2 2 2 kn h m m : mode number f c s θ cos kn f h : phase shift for the transverse passage through the film φ = φ 2 ( ) : phase shift due to total internal reflection from film/cover interface , TM c TE φ = φ 2 ( ) : phase shift due to total internal reflection from film/substrate interface , TM s TE Dispersion equation ( β vs. ω ): θ − φ − φ = π cos kn h m → Solve for θ . f c s β < < ≡ = θ n N n sin Effective guide index N n s f f k 12 Lih Y. Lin
EE 529 Semiconductor Optoelectronics Graphical Solution – Optical Waveguides of the Dispersion Equation Symmetrical waveguide, φ s = φ c Asymmetrical waveguide, φ s ≠ φ c For a symmetrical waveguide, there is always a solution (no cutoff) for fundamental mode (m = 0). Increasing h (and/or decreasing λ ) will support more modes. 13 Lih Y. Lin
EE 529 Semiconductor Optoelectronics – Optical Waveguides Typical β – ω diagram Cut-off kn s kn f Higher-order β Lower-order 14 Lih Y. Lin
EE 529 Semiconductor Optoelectronics – Optical Waveguides Numerical Solution for Dispersion Equation (I) Define: Normalized frequency and film thickness ≡ − 2 2 V kh n n f s Normalized guide index − 2 2 N n ≡ s b − 2 2 n n f s b = 0 at cut-ooff (N = n s ), and approaches 1 as N → n f . Measure for the asymmetry 4 − − 2 2 2 2 n n n n n ≡ ≡ f s c s c for TE, for TM a a − − 2 2 4 2 2 n n n n n f s c f s a = 0 for perfect symmetry (n s = n c ), and a approaches infinity for strong asymmetry (n s ≠ n c , n s ~ n f ). 15 Lih Y. Lin
EE 529 Semiconductor Optoelectronics – Optical Waveguides Numerical Solution for Dispersion Equation (II) For TE modes, dispersion relation + b b a θ − φ − φ = π → − − − = π + + 1 1 cos kn h m 1 tan tan V b m f c s − − 1 1 b b ν = m : Mode number (Normalized) cut-off frequency: − = 1 tan V a 0 = + π V V m 0 m # of guided modes allowed: 2 h = − 2 2 m n n λ f s <Example> AlGaAs/GaAs/AlGaAs double heterostructure, n = 3.55/3.6/3.55. Determine a waveguide thickness supporting 0 th , 0 th and 1 st order modes for λ = 1.55 µ m.. 16 Lih Y. Lin
EE 529 Semiconductor Optoelectronics – Optical Waveguides The Goos-Hänchen Shift ϕ d = z s β s d − = − θ 2 2 1 / 2 ( ) tan kz N n For TE modes s s − − θ 2 2 1 / 2 ( ) tan N n = s kz For TM modes s 2 2 N N + − 1 2 2 n n s f The lateral ray shift indicates a penetration depth: z = tan s x s θ 17 Lih Y. Lin
EE 529 Semiconductor Optoelectronics – Optical Waveguides Effective Waveguide Thickness Effective thickness: = + + h h x x eff s c Normalized effective thickness: ≡ − 2 2 H kh n n eff f s For TE modes: 1 1 = + + H V + b b a Minimum H → Maximum confinement Effective waveguide thickness cannot be zero, even for symmetrical waveguide (a = 0). Example: Sputtered glass, n s = 1.515, n f = 1.62, n c = 1, a = 3.9. Determine the minimum effective waveguide thickness. 18 Lih Y. Lin
EE 529 Semiconductor Optoelectronics – Optical Waveguides Ray-optic approach can solve for the effective index, but this is not good enough. Why? 19 Lih Y. Lin
EE 529 Semiconductor Optoelectronics – Optical Waveguides Guided E-M Wave in a Planar Waveguide Define: κ = − β = − γ 2 2 2 2 2 n k c c c κ = − β 2 2 2 2 n k f f κ = − β = − γ 2 2 2 2 2 n k s s s ∂ ∂ 2 2 + − β = → − γ = 2 2 2 2 ( , ) ( ) ( , ) 0 0 E x y n k E x y E E Cover: ∂ c ∂ c 2 2 x x ∂ ∂ 2 2 + − β = → + κ = 2 2 2 2 ( , ) ( ) ( , ) 0 0 E x y n k E x y E E Film: f f ∂ ∂ 2 2 x x ∂ ∂ 2 2 + − β = → − γ = 2 2 2 2 ( , ) ( ) ( , ) 0 0 E x y n k E x y E E Substrate: s s ∂ ∂ 2 2 x x 20 Lih Y. Lin
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