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

Maksim SkorobogatiyGénie Physique

École Polytechnique de Montréal (Université de Montréal)

S. Jacobs, S.G. Johnson and Yoel FinkOmniGuide Communications & MIT

Presented at Photonics Europe, SPIE 2004

2Coupled Mode Theory and perturbation formulations for high-index contrast waveguides

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 index-contrast waveguides with shifting dielectric boundaries generally fail as these methods do not correctly incorporate field discontinuities on the dielectric interfaces.

Direction of propagation

y

x

z

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.

3 Method of perturbation matching

thPosition of the n perturbed dielectric interface:

for every and (0,2 )

x= ( , , )

y= ( , , )

z= ( , , )

n

n

n

s

x s

y s

z s

n

Unperturbed fiber profile

yx

Perturbed fiber profile

•Dielectric profile of an unperturbed fiber o(,,s) can be mapped onto a perturbed dielectric profile (x,y,z) via a coordinate transformation x(,,s), y(,,s), z(,,s).

•Transforming Maxwell’s equation from Cartesian (x,y,z) onto curvilinear (,,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.

Coupled Mode Theory

- modal expansion coefficients, - original propagation constants

ˆ ˆ ˆ

ˆ ˆ, Hermitian

o

o

C

CiB BC HC

s

B H

(x,y,z)o(,,s) mapping

F(,,s) F((x,y,z),(x,y,z),s(x,y,z))

4 Method of perturbation matching, applications

a)

b)

c)

TR

Static PMD due to profile distortions

Scattering due to stochastic profile variations

d)

Modal Reshaping by tapering and scattering (Δm=0)

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

5

Rs=6.05a Rf=3.05a

L

n=3.0

n=1.0

High index-contrast fiber tapers

Transmission properties of a high index-contrast non-adiabatic taper. Independent check with CAMFR.

th

f s

s

f s

s

Position of the n inter-layer

boundary:

R Rx= Cos( ) (1+ ( ))

R

R Ry= Sin( ) (1+ ( ))

R

z=s

n

n

z

L

z

L

Convergence of scattering coefficients ~ 1/N2.5

When N>10 errors are less than 1%

6 High index-contrast fiber Bragggratings

3.05a

L

n=3.0

n=1.0

w

Transmission properties of a high index-contrast Bragg grating. Independent check with CAMFR.

thPosition of the n inter-layer

boundary:

2x= Cos( ) (1+ sin( ))

2y= Sin( ) (1+ sin( ))

z=s

n

n

z

z

Convergence of scattering coefficients ~ 1/N1.5

When N>2 errors are less than 1%

7 OmniGuide hollow core Bragg fiber

Very high dispersion

Low dispersion

Zero dispersion

[2/a]

[2c

/a]

HE11

8 PMD of dispersion compensating Bragg fibers

11 11

11

( ) | 1, | |1, |2HE HE

HE

HPMD

y

x

thPosition of the n inter-layer

boundary:

x= Cos( ) (1+ ( ))

y= Sin( ) (1- ( ))

z=s

n n

n n

f

f

"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

9

ps/n

m/k

m

Find Dispersion

Find PMD

Adjust Bragg mirror layer thicknesses to:

• Favour large negative

dispersion at 1.55m

• Decrease PMD

Iterative design of low PMD dispersion compensating Bragg fibers

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

Using the guided and evanescent modes of an unperturbed PxTal

waveguide to predict eigen modes or scattering coefficients for a perturbed PxTal waveguide

Uniform unperturbed waveguide

Uniform perturbed waveguide (eigen problem)

Nonuniform perturbed waveguide

(scattering problem)

20.25

0.2 ; 0.3

3.37

0.5

core reflector

cyl

c

ar a r a

n in air

a m

11 Defining coordinate mapping in 2D

( ) ( )x zx x f x f z

y y

z z

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

13 Back scattering of the fundamental mode

14 Transmission through long tapers

15 Scattering losses due to stochastic variations in the waveguide walls

16 Scattering losses due to stochastic variations in the waveguide walls

17 Negating imperfections by local manipulations of the refractive index