Battling imperfections in high indexcontrast systems from Bragg

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Battling imperfections in high index-contrast systems – from Bragg fibers to planar photonic crystals

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 Omni. Guide Communications & MIT Some slides are courtesy of Prof. Steven Johnson

1 All Imperfections are Small for systems that work x z y • Material

1 All Imperfections are Small for systems that work x z y • 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: small De ~ Dx / Rbend • Roughness: small De or boundary shift Hitomichi Takano et al. , Appl. Phys. Let. 84, 2226 2004 Weak effects, long distances: hard to compute directly — use perturbation theory

2 Perturbation Theory for Hermitian eigenproblems given eigenvectors/values: …find change & for small Solution:

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

Perturbation Theory for electromagnetism (no shifting material boundries) 3 ecore+De Dielectric boundaries do not

Perturbation Theory for electromagnetism (no shifting material boundries) 3 ecore+De Dielectric boundaries do not move …e. g. absorption gives imaginary Dw = decay!

4 Losses due to material absorption Material absorption: small perturbation Im(e) EH 11 Large

4 Losses due to material absorption Material absorption: small perturbation Im(e) EH 11 Large differential loss TE 01 strongly suppresses cladding absorption TE 01 (like ohmic loss, for metal) l (mm)

Perturbation formulation for high-index contrast waveguides and shifting material boundaries 5 Standard perturbation formulation

Perturbation formulation for high-index contrast waveguides and shifting material boundaries 5 Standard perturbation formulation and coupled mode theory in 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. b+ Elliptical deformation lifts degeneracy Degenerate b- 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

6 Method of perturbation matching Unperturbed fiber profile eo(r, q, s) rn q Perturbed

6 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.

7 Method of perturbation matching, applications Static PMD due to profile distortions b) Scattering

7 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

8 High index-contrast fiber tapers n=1. 0 Rs=6. 05 a n=3. 0 L Transmission

8 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 9 n=1. 0 w n=3. 05 a Convergence of

High index-contrast fiber Bragg gratings 9 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 B. Temelkuran et al. , Nature 420, 650

Omni. Guide hollow core Bragg fiber B. Temelkuran et al. , Nature 420, 650 (2002) [2 pc/a] 10 HE 11 Zero dispersion Very high dispersion Low dispersion [2 p/a]

PMD of dispersion compensating Bragg fibers 11 r q y x "Analysis of general

PMD of dispersion compensating Bragg fibers 11 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 12 Optimization by varying layer

Iterative design of low PMD dispersion compensating Bragg fibers 12 Optimization by varying layer thicknesses h 3 ps/nm/km h 1 h 2 Find Dispersion Find PMD Adjust Bragg mirror layer thicknesses to: • Favour large negative dispersion at 1. 55 mm • Decrease PMD

13 Method of perturbation matching in application to the planar photonic crystal waveguides Uniform

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

14 Perturbation matched CMT Perfect PC Scattering region Perfect PC 1 T R "Modelling

14 Perturbation matched CMT Perfect PC Scattering region Perfect PC 1 T R "Modelling the impact of imperfections in high index-contrast photonic waveguides. ", M. Skorobogatiy, Opt. Express 10, 1227 (2002), PRE (2003)

15 Eigen modes of a perfect PC

15 Eigen modes of a perfect PC

16 Perturbation matched CMT Mapping a perfect PC onto a perturbed one Perturbation matched

16 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

17 Perturbation matched CMT Mapping a perturbed PC onto a perfect one Mapping system

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

18 Defining coordinate mapping in 2 D

18 Defining coordinate mapping in 2 D

19 Finding the new modes of the uniformly perturbed photonic crystal waveguides

19 Finding the new modes of the uniformly perturbed photonic crystal waveguides

20 Back scattering of the fundamental mode

20 Back scattering of the fundamental mode

21 Transmission through long tapers

21 Transmission through long tapers

22 Scattering losses due to stochastic variations in the waveguide walls Hitomichi Takano et

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

23 Scattering losses due to stochastic variations in the waveguide walls

23 Scattering losses due to stochastic variations in the waveguide walls

24 Negating imperfections by local manipulations of the refractive index

24 Negating imperfections by local manipulations of the refractive index

25 Statistical analysis of imperfections from the images of 2 D photonic crystals. Maksim

25 Statistical analysis of imperfections from the images of 2 D 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

26 Image Analysis By using object recognition and image processing techniques, one can find

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

27 Characterization of individual features

27 Characterization of individual features

28 Fractal nature of the imperfections Self-similar profile of roughness Hurst exponent H=0. 43

28 Fractal nature of the imperfections Self-similar profile of roughness Hurst exponent H=0. 43 Correlation length l=35 nm Standard deviation and mean do not characterize roughness uniquely … But fractal dimension and correlation length do. Roughness

29 Deviation of an underlying lattice from perfect

29 Deviation of an underlying lattice from perfect

30 Hurst exponent Roughness of a hole wall in a planar PC

30 Hurst exponent Roughness of a hole wall in a planar PC

31 Hurst exponent and structure function Fractal behavior is lost for length scales >

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

32 Distribution of parameters characterizing individual features

32 Distribution of parameters characterizing individual features