Presented at Photonics Europe SPIE 2004 Battling imperfections

Presented at Photonics Europe, SPIE 2004 Battling imperfections in high index-contrast systems – from Bragg fibers to planar photonic crystals • Maksim Skorobogatiy • Génie Physique • École Polytechnique de Montréal (Université de Montréal) • S. Jacobs, S. G. Johnson and Yoel

Coupled Mode Theory and perturbation formulations for high-index contrast waveguides 1 Propagation of radiation through a waveguide of generic non-uniform high index-contrast dielectric profile • Standard perturbation formulation and coupled mode theory in a problem of high indexcontrast waveguides with shifting dielectric boundaries generally fail as these methods do not correctly incorporate field discontinuities on the dielectric interfaces. x z Direction of propagation y Other known methods that can solve the problem are: • Method of crossections (expansion into the instantaneous eigen modes). This method requires recalculation of the local eigen modes at each of the different crossesctions along the direction of propagation, and is computationally intensive. • Expansion into the eigen modes of a uniform waveguide with smooth dielectric profile (empty metallic waveguide f. e. ). Convergence of this method with the number N of expansion modes is slow (linear ~1/N). • Traditional FDTD, FETD are surprisingly difficult to use for analysis of small variations as one needs to resolve spatially such variations, and the effect of such variations is only observable after long propagation distances.

2 Method of perturbation matching Unperturbed fiber profile eo(r, q, s) rn q Perturbed fiber profile e(x, y, z) mapping y x • Dielectric profile of an unperturbed fiber eo(r, q, s) can be mapped onto a perturbed dielectric profile e(x, y, z) via a coordinate transformation x(r, q, s), y(r, q, s), z(r, q, s). F(r, q, s) F(r(x, y, z), q(x, y, z), s(x, y, z)) • Transforming Maxwell’s equation from Cartesian (x, y, z) onto curvilinear (r, q, s), coordinate system brings back an unperturbed dielectric profile, while adding additional terms to Maxwell’s equations due to unusual space curvature. These terms are small when perturbation is small, allowing for correct perturbative expansions. • Rewriting Maxwell’s equation in the curvilinear coordinates also defines an exact Coupled Mode Theory in terms of the coupled modes of an original unperturbed system.

3 Method of perturbation matching, applications Static PMD due to profile distortions b) Scattering due to stochastic profile variations a) c) Modal Reshaping by tapering and scattering (Δm=0) d) R T Inter-Modal Conversion (Δm≠ 0) by tapering and scattering "Geometric variations in high index-contrast waveguides, coupled mode theory in curvilinear coordinates", M. Skorobogatiy, S. A. Jacobs, S. G. Johnson, and Y. Fink, Optics Express, vol. 10, pp. 1227 -1243, 2002 "Dielectric profile variations in high-index-contrast waveguides, coupled mode theory, and perturbation expansions", M. Skorobogatiy, Steven G. Johnson, Steven A. Jacobs, and Yoel Fink, Physical Review E, vol. 67, p. 46613, 2003

4 High index-contrast fiber tapers n=1. 0 Rs=6. 05 a n=3. 0 L Transmission properties of a high index-contrast non-adiabatic taper. Independent check with CAMFR. Convergence of scattering coefficients ~ 1/N 2. 5 Rf=3. 05 a When N>10 errors are less than 1%

High index-contrast fiber Bragg gratings 5 n=1. 0 w n=3. 05 a Convergence of scattering coefficients ~ 1/N 1. 5 When N>2 errors are less than 1% L Transmission properties of a high index-contrast Bragg grating. Independent check with CAMFR.

Omni. Guide hollow core Bragg fiber [2 pc/a] 6 HE 11 Zero dispersion Very high dispersion Low dispersion [2 p/a]

PMD of dispersion compensating Bragg fibers 7 r q y x "Analysis of general geometric scaling perturbations in a transmitting waveguide. The fundamental connection between polarization mode dispersion and group-velocity dispersion", M. Skorobogatiy, M. Ibanescu, S. G. Johnson, O. Weiseberg, T. D. Engeness, M. Soljacic, S. A. Jacobs, and Y. Fink, Journal of Optical Society of America B, vol. 19, pp. 2867 -2875, 2002

Iterative design of low PMD dispersion compensating Bragg fibers ps/nm/km 8 Find Dispersion Find PMD Adjust Bragg mirror layer thicknesses to: • Favour large negative dispersion at 1. 55 mm • Decrease PMD

9 Method of perturbation matching in application to the planar photonic crystal waveguides Uniform perturbed waveguide (eigen problem) Uniform unperturbed waveguide Using the guided and evanescent modes of an unperturbed Px. Tal waveguide to predict eigen modes or scattering coefficients for a perturbed Px. Tal waveguide Nonuniform perturbed waveguide (scattering problem)