NAIS  - Next-Generation Active Integrated-Optic Subsystems

Workpackage 3: Design

Properties of a strongly bent dielectric waveguide

Calculated by Milan Hubálek, IREE AS CR

In the framework of Workpackage 3 of the NAIS project, Manfred Hammer presented here the results of the rigorous analysis of the following bent waveguide problem:

Geometry of the problem:

     The structure is parametrized in terms of
  • ns, nf, nc: the refractive indices in the interior of the bend, in the core region, and of the cladding material,
  • R: the bend radius, measured from the origin to the core center,
  • d: the core thickness,
  • omega = k c = 2 pi c/lambda: The angular frequency of the light, specified in terms of the vacuum speed of light c, vacuum wavenumber k, and vacuum wavelength lambda.

Data for the simulations on this page (see [1]):
ns = 1.6,   nf  = 1.7,   nc = 1.6,   d = 1.0 µm,   lambda = 1.3 µm.

[1] C. Vassallo, Optical waveguide concepts, Elsevier, Amsterdam, 1991

We present results obtained with the  help of the OlympIOs C2V software package (bend mode solver) for bend radii R from 3 to 200 µm. The radius of curvature is measured from the origin to the centre of the waveguide core. Only TE polarized modes are analyzed here.

Below you find

Effective refractive indices as a function of the radius of curvature

Radius R {µm)

Re {Neff}

Im {Neff}
3 1.78756

1.2482×10–1
4 1.75106

8.941×10–1
5 1.73006

6.775×10–2
6 1.71644

5.326×10–2
7 1.706984

4.305×10–2
8 1.700096

3.5496×10–2
9 1.694852

2.9705×10–2
10 1.690717

2.5163×10–2
15 1.678698

1.23627×10–2
20 1.672873

6.802×10–3
25 1.669439

3.958×10–3
30 1.667194

2.371×10–3
35 1.665635

1.441×10–3
40 1.664506

8.82×10–4
45 1.663671

5.40×10–4
50 1.663037

3.32×10–4
60 1.662188

1.21×10–4
70 1.661669

4.44×10–5
80 1.661339

1.59×10–5
90 1.661117

5.66×10–6
100 1.660963

1.98×10–6
120 1.660768

2.44×10–7
140 1.660654

2:93×10–8
160 1.660580

3.54×10–9
180 1.660530

4.28×10–10
200 1.660494

5.29×10–11

 

   

Re{Neff} as a function of the radius R

Approximate expressions:

Left: 3 µm < R < 10 µm;     Neff  = 1.659187095 –  0.3015221124e–R + 0.2664257194 R–1 + 0.4913766319 R–2

Right: 3 µm < R < 200 µm; Neff  = 1.662814325 + 0.2185996756 e–R + 0.5173445544 R–1 – 0.1628990731 R–2 – 0.07003386733R– 0.5

Field and equivalent index distributions

The next set of pictures shows the distributions of absolute values of the electric field vector (Ey) of the eigenmodes (left) and the equivalent refractive index profile of the bent waveguide (right). The core of the waveguide structure is symbolized by the two vertical lines. The origin of the coordinate system  is moved to the centre of the core. The equivalent index profile obtained by the conformal mapping technique is given by the expression
nR = n exp[(x – R)/R]  after [2].
 [2] M. Heilblum, J.H. Harris: "Analysis of curved optical waveguide by conformal transformation", IEEE J. Quantum Electron. QE-11, pp. 75-83, 1975, corrected in QE-12, p. 313, 1976

Green - the original step-index refractive index profile n, violet - the equivalent profile nR, blue - the effective refractive index  Neff

 

r = 3 µm: Very strongly leaky mode of the "whispering gallery" type

 

r = 5 µm

                                                                                               

 

r = 10 µm:  Mode leakage starts to be of a "tunnelling type" at the outer boundary while still of a "whispering gallery type" at the inner boundary.

 

 r = 20 µm

 

 r = 30 µm: The mode starts to be of a standard tunelling leaky mode, reflecting also from the inner boundary  

 

 r = 50 µm: Very weakly lossy (leaky) waveguide

 

r = 80 µm

 

r = 100 µm: Very low-loss bent waveguide

 

r = 140 µm

 

r = 200 µm

Comparison of field distributions for various radii of curvature

red – r = 3 µm, green – 10 µm, blue – 25 µm, yellow – 100 µm; brown – 200 µm

The page was created on 13 July 2002 by Jiri Ctyroky