Battling imperfections in high index-contrast systems

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

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All Imperfections are Small for systems that work
Material absorption small imaginary De
Nonlinearity small De E2
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
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
ecoreDe
e.g. absorption gives imaginary Dw decay!
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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 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.
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. Soljacic, S.A.
Jacobs and Y. Fink, J. Opt. Soc. Am. B, vol. 19,
p. 2867, 2002
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Method of perturbation matching
e(x,y,z)
eo(r,q,s)
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 Maxwells equation from Cartesian
  (x,y,z) onto curvilinear (r,q,s), coordinate
  system brings back an unperturbed dielectric
  profile, while adding additional terms to
  Maxwells equations due to unusual space
  curvature. These terms are small when
  perturbation is small, allowing for correct
  perturbative expansions.
• Rewriting Maxwells 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))
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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 (?m0)
T
d)
R
Inter-Modal Conversion (?m?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
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High index-contrast fiber tapers
n1.0
Convergence of scattering coefficients
1/N2.5 When Ngt10 errors are less than 1
Rs6.05a
Rf3.05a
n3.0
L
Transmission properties of a high index-contrast
non-adiabatic taper. Independent check with
CAMFR.
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High index-contrast fiber Bragggratings
n1.0
3.05a
Convergence of scattering coefficients
1/N1.5 When Ngt2 errors are less than 1
w
n3.0
L
Transmission properties of a high index-contrast
Bragg grating. Independent check with CAMFR.
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OmniGuide hollow core Bragg fiber
B. Temelkuran et al., Nature 420, 650 (2002)
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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
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

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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)
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Perturbation matched CMT
1
T
R
"Modelling the impact of imperfections in high
index-contrast photonic waveguides.", M.
Skorobogatiy, Opt. Express 10, 1227 (2002), PRE
(2003)
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Eigen modes of a perfect PC
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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
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Perturbation matched CMT
Mapping a perturbed PC onto a perfect one
Mapping system Hamiltonian onto the one of a
perfect PC curvature corrections
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Defining coordinate mapping in 2D
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Finding the new modes of the uniformly perturbed
photonic crystal waveguides
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Back scattering of the fundamental mode
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Transmission through long tapers
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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
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Negating imperfections by local manipulations of
the refractive index
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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.phy
s.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
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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
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Characterization of individual features
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Fractal nature of the imperfections
Self-similar profile of roughness
Hurst exponent H0.43 Correlation length l35nm
Standard deviation and mean do not characterize
roughness uniquely
Roughness
But fractal dimension and correlation length do.
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Deviation of an underlying lattice from perfect
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Hurst exponent
Roughness of a hole wall in a planar PC
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Hurst exponent and structure function
Fractal behavior is lost for length scales gt 100nm
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Distribution of parameters characterizing
individual features