Maksim Skorobogatiy Génie Physique École Polytechnique de Montréal

1. Battling imperfections in high index-contrast systems – from Bragg fibers to planar photonic crystals Maksim Skorobogatiy Génie Physique École Polytechnique de Montréal S. Jacobs, S.G. Johnson and Yoel Fink OmniGuide Communications & MIT Some slides are courtesy of Prof. Steven Johnson

2. x z y All Imperfections are Small for systems that work • Material absorption:small imaginary De • Nonlinearity:small De ~ |E|2 • Acircularity (birefringence):small e boundary shift • Variations in waveguide size:small e boundary shift • Bends:smallDe ~ Dx / Rbend • Roughness:smallDe or boundary shift Hitomichi Takano et al., Appl. Phys. Let. 84, 2226 2004 Weak effects, long distances: hard to compute directly — use perturbation theory

3. Perturbation Theoryfor Hermitian eigenproblems given eigenvectors/values: …find change & for small Solution: expand as power series in & (first order is usually enough)

4. Perturbation Theory for electromagnetism (no shifting material boundries) Dielectric boundaries do not move ecore ecore+De …e.g. absorption gives imaginary Dw = decay!

5. Losses due to material absorption Material absorption: small perturbation Im(e) EH11 Large differential loss TE01 strongly suppresses cladding absorption (like ohmic loss, for metal) TE01 l (mm)

6. Perturbation formulation for high-index contrast waveguides and shifting material boundaries Standard perturbation formulation and coupled mode theoryin a problem of high index-contrast waveguides with shifting dielectric boundaries generally fail as these methods do not correctly incorporate field discontinuities on the dielectric interfaces. Elliptical deformation lifts degeneracy b+ b- Degenerate bo of unperturbed fiber "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. Weisberg, T.D. Engeness, M. Solja¡ci´c, S.A. Jacobs and Y. Fink, J. Opt. Soc. Am. B, vol. 19, p. 2867, 2002

7. Perturbed fiber profile Unperturbed fiber profile rn y x q Method of perturbation matching eo(r,q,s) e(x,y,z) mapping • 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). • 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. F(r,q,s) F(r(x,y,z),q(x,y,z),s(x,y,z))

8. 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) T d) R 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

9. High index-contrast fiber tapers n=1.0 Convergence of scattering coefficients ~ 1/N2.5 When N>10 errors are less than 1% Rs=6.05a Rf=3.05a n=3.0 L Transmission properties of a high index-contrast non-adiabatic taper. Independent check with CAMFR.

10. High index-contrast fiber Bragggratings n=1.0 Convergence of scattering coefficients ~ 1/N1.5 When N>2 errors are less than 1% 3.05a w n=3.0 L Transmission properties of a high index-contrast Bragg grating. Independent check with CAMFR.

11. HE11 [2pc/a] Zero dispersion Very high dispersion Low dispersion [2p/a] OmniGuide hollow core Bragg fiber B. Temelkuran et al., Nature 420, 650 (2002)

12. r y q x PMD of dispersion compensating Bragg fibers "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

13. h3 h2 h1 ps/nm/km Iterative design of low PMD dispersion compensating Bragg fibers Optimization by varying layer thicknesses • Find Dispersion • Find PMD • Adjust Bragg mirror layer thicknesses to: • Favour large negative • dispersion at 1.55mm • Decrease PMD

14. Method of perturbation matching in application to the planar photonic crystal waveguides Uniform perturbed waveguide (eigen problem) Uniform unperturbed waveguide GOAL: Using eigen modes of an unperturbed 2D photonic crystal waveguide to predict eigen modes or scattering coefficients associated with propagation in a perturbed photonic crystal waveguide Nonuniform perturbed waveguide (scattering problem)

15. Scattering region Perfect PC Perfect PC Perturbation matched CMT 1 T R "Modelling the impact of imperfections in high index-contrast photonic waveguides.", M. Skorobogatiy, Opt. Express10, 1227 (2002), PRE (2003)

16. Eigen modes of a perfect PC

17. Perturbation matched CMT Mapping a perfect PC onto a perturbed one Perturbation matched expansion basis Regions of field discontinuities are matched with positions of perturbed dielectric interfaces

18. Perturbation matched CMT Mapping a perturbed PC onto a perfect one Mapping system Hamiltonian onto the one of a perfect PC + curvature corrections

19. Defining coordinate mapping in 2D

20. Finding the new modes of the uniformly perturbed photonic crystal waveguides

21. Back scattering of the fundamental mode

22. Transmission through long tapers

23. Scattering losses due to stochastic variations in the waveguide walls Hitomichi Takano et al., Appl. Phys. Let. 84, 2226 2004

24. Scattering losses due to stochastic variations in the waveguide walls

25. Negating imperfections by local manipulations of the refractive index

26. Statistical analysis of imperfections from the images of 2D photonic crystals. Maksim Skorobogatiy – Canada Research Chair, and Guillaume Bégin Génie Physique, École Polytechnique de Montréal Canada www.photonics.phys.polymtl.ca Opt. Express, vol. 13, pp. 2487-2502 (2005) Images used in the paper for statistical analysis are courtesy of A. Talneau, CNRS, Lab Photon & Nanostruct, France

27. Image Analysis By using object recognition and image processing techniques, one can find and analyze the constituent features of an image Once the defects are found and analyzed, one can predict degradation in the performance of a photonic crystal

28. Characterization of individual features

29. Fractal nature of the imperfections Self-similar profile of roughness Hurst exponent H=0.43 Correlation lengthl=35nm Standard deviation and mean do not characterize roughness uniquely … Roughness But fractal dimension and correlation length do.

30. Deviation of an underlying lattice from perfect

31. Hurst exponent Roughness of a hole wall in a planar PC

32. Hurst exponent and structure function Fractal behavior is lost for length scales > 100nm

33. Distribution of parameters characterizing individual features