Lecture 7: Optical

Petr Kužel Types of guiding structures: • Planar waveguides (integrated ) • Fibers (communications) Theory: • Rays and field approach • Various shapes and index profiles Attenuation and Coupling of into the 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 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 : ω β = kn sin α = n sin α (ω) f f c f f Effective : 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)