Optical Waveguide Theory

1 Motto Optical Waveguide Theory Manfred Hammer . Theoretical Electrical Engineering Paderborn University, Paderborn, Germany MMET'08, Mathematical Methods in Electromagnetic Theory Odesa, Ukraine, June 29 July 2, 2008. Paderborn University Summer Semester 2023.. Theoretical Electrical Engineering, Paderborn University Phone: +49(0)5251/60-3560. Warburger Stra e 100, 33098 Paderborn, Germany E-mail: 1 2. Maxwell equations Course overview SI, in matter, time domain, differential form: Optical Waveguide Theory A Photonics / integrated optics; Theory , motto; phenomena, introductory examples. D = f , E(r, t): electric field, B Brush up on mathematical tools. E = B , D(r, t): (di-)electric displacement, C Maxwell equations, different formulations, interfaces, energy and power flow. B = 0, B(r, t): magnetic induction (field, flux density), D Classes of simulation tasks: scattering problems, mode analysis, resonance problems.

2 H(r, t): magnetic field (..), E Normal modes of dielectric Optical waveguides, mode interference. H = Jf + D , f (r, t): density of free charges, F Examples for dielectric Optical waveguides. Jf (r, t): density of free currents, G Waveguide discontinuities & circuits, scattering matrices, reciprocal circuits. D = 0 E + P, P(r, t): polarization, H Bent Optical waveguides; whispering gallery resonances; circular microresonators. M(r, t): magnetization, I Coupled mode Theory , perturbation Theory . B = 0 (H + M). Hybrid analytical / numerical coupled mode Theory . 0 : free space permittivity, J A touch of photonic crystals; a touch of plasmonics . (+ constitutive relations) 0 : free space permeability. Oblique semi-guided waves: 2-D integrated optics. Summary, concluding remarks. Valid for more than a century, firm basis for further considerations. 3 4. Formalities Optical waveguides: phenomena, examples Beam propagation in free space Guided light propagation Organization of the course: Waveguide end facet Lectures ( 14 ) Crossing of two waveguides Homework (7 ) Modes of 1-D multilayer slab waveguides Tutorials, Exercises (13 ) Modes of 2-D channel waveguides Exam Circular step-index Optical fibers Evanescent coupling between waveguides Bent waveguides Related textbooks (examples): Circular microring-resonator C.

3 Vassallo, Optical Waveguide Concepts, Elsevier, Amsterdam (1991), Microdisk resonator K. Okamoto, Fundamentals of Optical Waveguides, Academic Press, San Diego, USA (2000), CROW. R. Ma rz, Integrated Optics: Design and Modeling, Artech House, Norwood, USA (1995), Snyder, Love, Optical Waveguide Theory , Chapman and Hall, London, UK (1983);. Waveguide corner & general introductory texts on classical electrodynamics. Photonic crystal Waveguide Exciting TET ! 5 6. Optical Waveguide Theory Course overview Optical Waveguide Theory Task: solve A Photonics / integrated optics; Theory , motto; phenomena, introductory examples. E = B , D = f , D = 0 E + P, B Brush up on mathematical tools. H = Jf + D , B = 0, B = 0 (H + M), (& ..). C Maxwell equations, different formulations, interfaces, energy and power flow. D Classes of simulation tasks: scattering problems, mode analysis, resonance problems.

4 E Normal modes of dielectric Optical waveguides, mode interference. F Examples for dielectric Optical waveguides. In this course: G Waveguide discontinuities & circuits, scattering matrices, reciprocal circuits. specialization to problems relevant for integrated optics, H Bent Optical waveguides; whispering gallery resonances; circular microresonators. theoretical basis for the mostly numerical solution, I Coupled mode Theory , perturbation Theory . Hybrid analytical / numerical coupled mode Theory . approximate concepts, J A touch of photonic crystals; a touch of plasmonics . examples. Oblique semi-guided waves: 2-D integrated optics. Summary, concluding remarks. 7 2. Vector calculus, keywords Vector calculus, keywords Ingredients: (here: Cartesian coordinates) Ingredients: (here: Cartesian coordinates). x x Del, nabla: = y . Space and time coordinates: r = y (x, y, z), t. z.

5 Z x . Ax Gradient: grad = = y . Scalar and vector fields: (r, t), A(r, t), A = Ay . z . Az Inner product: A B = Ax Bx + Ay By + Az Bz . Divergence: divA = A = x Ax + y Ay + z Az .. Ay Bz Az By y Az z Ay Vector product: A B = Az Bx Ax Bz . Curl: curlA = rotA = A = z Ax x Az . Ax By Ay Bx x Ay y Ax . Time derivatives: , t , , t . Laplacian: = = 2 , t Ax = x2 + y2 + x2 , A = Ay . Az 3 4. Dirac delta Fourier transform, 1-D. A linear functional 1-D: A function f (x) C of one variable: Z Z . that extracts the value of a function at one point: 1 ikx 1. f (x) = f (k) e dk, f (k) = f (x) e ikx dx. Z 2 2 . b f (x0 ), if a < x0 < b, 1-D: f (x) (x x0 ) dx = . a 0 otherwise; Arbitrary: positioning of factors 1/ 2 , signs of exponents. (x x0 ) = 0, if x 6= x0 . f^. 1 + f2 = f 1 + f 2 . Z f (x) = f ( x) f (k) = f ( k). f (r0 ), if r0 V, 3-D: f (r) (r r0 ) dV =. V 0 otherwise; f (x) = f ( x) f (k) = f ( k).

6 (r r0 ) = 0, if r 6= r0 . f R f ( k) = f (k). Z . 1. Implications: manifold. (x) = eikx dk. 2 . 5 6. Fourier transform Directionally constant systems A linear PDE in two unknowns 3-D: A field (r): (A xx + B yy + C xy + D x + E y + F) (x, y) = 0, Z Z. 1 1. (r) = 3 (k) eik r d3 k, (k) = 3. (r) e ik r d3 r. coefficients A(x, y), .. , F(x, y). 2 2 . If the system is constant in x, x A = .. = x F = 0 , 4-D: A field (r, t): Z. ZZ write as (x, y) = (k, y) eikx dk . 1. (r, t) = 4. (k, ) ei(k r t) d3 k d , Z. 2 . ZZ B yy + (E + ikC) y + (F + ikD k2 A) (k, y) eikx dk = 0, 1. (k, ) = 4 (r, t) e i(k r t) d3 r dt.. 2 B yy + (E + ikC) y + (F + ikD k2 A) (k, y) = 0, (for all k), .. a set of DEs in one unknown. 7 8. Directionally constant systems General solution of the wave equation . 1 2. A linear PDE in two unknowns 2 2 (r, t) = 0, (r, 0) = 0 (r), t (r, 0) = 0 (r), c t (A xx + B yy + C xy + D x + E y + F) (x, y) = 0, ZZ.

7 1. coefficients A(x, y), .. , F(x, y). & (r, t) = (k, ) ei(k r t) d d3 k, (2 )2.. If the system is constant in x, x A = .. = x F = 0 , 2 2. k + 2 (k, ) = 0, c use an ansatz (x, y) = (y) eikx . (k, ) = af (k) ( k ) + ab (k) ( + k ), k = c |k|, 2.. B yy + (E + ikC) y + (F + ikD k A) (y) = 0, Z . 1 i(k r k t) + a (k) ei(k r + k t) d3 k, (r, t) = a f (k) e b .. a DE in one unknown, with parameter k. (2 )2. (& boundary conditions, ..). (r, 0) = 0 (r), t (r, 0) = 0 (r) .. af (k), ab (k). 8 9. A touch of variational calculus A touch of variational calculus Example: Functional: L : U R, C, Z . U = {u : [0, ] R | u(0) = u( ) = 0}, u L(u), ( x u)2 dx L : U R, L(u) = 0Z . a map from a space U of functions to real / complex numbers.. u2 dx 0. d ( .. ). Stationary functional: L(u + s v) = 0 for all v, ds s=0. u satisfies DE & , L stationary at u, the variation of L at u vanishes for arbitrary directions v.

8 D x2 u = u, = L(u), L(u + s v) = 0 v. ds s=0 u(0) = u( ) = 0. Restriction of a functional: .. to a parametrized family of functions; Restrict L, L(a) = L(u|a ). extremization with respect to these parameters, Approximate solution approximations of stationary points of the functional. L stationary at a, a L = 0. of DE / eigenproblem. 10 11. Course overview ..? Optical Waveguide Theory A Photonics / integrated optics; Theory , motto; phenomena, introductory examples. This concerns time harmonic fields .. with angular frequency .. , B Brush up on mathematical tools. for vacuum wavenumber .. , speed of light .. , and wavelength .. C Maxwell equations, different formulations, interfaces, energy and power flow. D Classes of simulation tasks: scattering problems, mode analysis, resonance problems. The problem is governed by the Maxwell curl equations in the E Normal modes of dielectric Optical waveguides, mode interference.

9 Frequency domain for the electric field .. and magnetic field .. , for F Examples for dielectric Optical waveguides. (lossless) uncharged dielectric, nonmagnetic linear (isotropic) media G Waveguide discontinuities & circuits, scattering matrices, reciprocal circuits. with (piecewise constant) relative permittivity .. : H Bent Optical waveguides; whispering gallery resonances; circular microresonators.. (.) . I Coupled mode Theory , perturbation Theory . Hybrid analytical / numerical coupled mode Theory . J A touch of photonic crystals; a touch of plasmonics . [ M. Hammer, A. Hildebrandt, J. Fo rstner, Journal of Lightwave Technology 34(3), 997 (2016) ]. Oblique semi-guided waves: 2-D integrated optics. Summary, concluding remarks. 2 3. Maxwell equations, Fourier transform Maxwell equations, frequency domain D = f , E = i B , B = 0, H = J f + i D . D = f , E = B , B = 0, H = Jf + D F(r, t) R F (r, ) = (F (r, )).

10 Z Z r r 1 i t 1. & F(r, t) = F (r, ) e d , F (r, ) = F(r, t) e i t dt . 2 2 at frequency 0 : F (r, ) = F (r) ( 0 ) + F (r) ( + 0 ). 2 2. 1n o F (r) ei 0 t + F (r) e i 0 t , . F(r, t) =. 2 n o E(r, t), D(r, t), B(r, t), H(r, t), f (r, t), Jf (r, t) F(r, t) = Re F (r) ei 0 t , E (r, ), D (r, ), B (r, ), H (r, ), f (r, ), J f (r, ), 1. F(r, t) = F (r) ei 0 t + . 2. D = f , E = i B , B = 0, H = J f + i D . E (r), D (r), B (r), H (r), f (r), J f (r), exp(i 0 t), (Caution: arbitrary choice of exp( i t) !). D = f , E = i 0 B , B = 0, H = J f +i 0 D . Caution: Decorations , , 0 are ususally omitted; context determines interpretation of symbols. 4 5. Polarization Magnetization P : density of electric dipole moment (bound charges). M : density of magnetic dipole moments (bound currents). As m V 1 A m2 Vs D = 0 E + P , [D ] = [P ] = 3. , [E ] = , H = B M , [H ] = [M] = , [B ] = T = 2 , m m 0 m 3 m.