Lecture 7: Optical waveguides Petr Kužel Types of guiding structures: • Planar waveguides (integrated optics) • Fibers (communications) Theory: • Rays and field approach • Various shapes and index profiles Attenuation and dispersion Coupling of light into the waveguide Homogeneous planar waveguide x x z n n n y c s f n α Series of total internal reflections (angle of incidence f): α > ns > nc sin f n f n f Transverse resonance condition nc α f E nf 1 E2 ns = ( ω − − α + α ) E1 E10 exp i t ikn f ( x cos f z sin f ) = ( ω − α + α ) E2 E20 exp i t ikn f (x cos f z sin f ) δ ()= i c = ()= E1 x h e E2 x h Phase change δ ()= i s = ()= E2 x 0 e E1 x 0 during total reflection Waveguide equation Continuity of E1 and E2 lead to: (− α + δ + δ )= exp 2ikn f hcos f i c i s 1 Waveguide dispersion equation: α = δ + δ + π 2kn f hcos f c s 2 m Solution numeric or graphic kh increase → number of modes increase symmetric waveguide → at least one guided mode non-symmetric waveguide, small kh → no guided mode Graphic solution Graphic solution of the waveguide equation 40 α 2 k nf h cos f 10π 30 8π TM3 20 π Phase TE3 6 TM2 TE2 TM1 4π 10 TE1 TM0 2π TE0 0 α α 0 30 c s 60 90 α f Longitudinal propagation nc α E nf f 1 E2 ns Longitudinal propagation constant: ω β = kn sin α = n sin α (ω) f f c f f Effective refractive index: β N = = n sin α k f f Waveguide dispersion curves: β(ω) or N(ω) Waveguide dispersion Waveguide dispersion N(ω) N = nf Guided modes TE0-3 TM0-3 N 0 2 1 3 N = ns ω Waveguide dispersion Waveguide dispersion β(ω) Guided modes TE0-3 TM0-3 β = n ω/c β f 3 2 β ω 1 = ns /c 0 ω Wave approach to waveguides Linear combinations of the basis functions should satisfy the continuity conditions at the interfaces • One-dimensional planar waveguide: harmonic plane waves • Rectangular waveguide: analytic solution does not exist • Circular waveguide: Bessel functions • Circular waveguide with parabolic n-profile: Gaussian beams Planar waveguide E H + − E k k 0 0 α α 0 0 z n0 H E Ei0 r0 E 01 y E (tangential) Et1 s1 x n1 α 1 E H − + k k (tangential) E 1 1 i1 Er1 E12 n2 E H E − + k2 k2 H Attenuation in waveguides Mechanisms: • Intrinsic – Residual absorption (SiO2, impurities: OH ) Rayleigh scattering (proportional to ω4) •Extrinsic • Large inhomogeneities (fabrication of the fiber) • geometrical irregularities (curvature, surface defects…) • Losses at the fiber input and output (Fresnel reflection, aperture...) λ = µ SiO2 fibers: 0.2 dB/km with 1.55 m I α[dB] =10 log 1 I2 Attenuation in waveguides Dispersion Broadening of pulse during the propagation (limits the rate of the information transfer) 1. Modal dispersion (tens of ns per km) ∆τ n − n 1 = Nmax − Nmin ≈ f s Single mode fibers, gradient index profile L c c c 2. Material dispersion (hundreds of ps per km) ∆τ λ d 2n 2 = − f ∆λ µ L c dλ2 It vanishes for SiO2 at 1.27 m ω ∆τ λ 2 3. Waveguide dispersion (N depends on ) 3 = − d N ∆λ L c dλ2 (tens of ps per km).