Transcript Modelling and Simulation of Passive Optical Devices

Modelling and Simulation
of Passive Optical Devices
João Geraldo P. T. dos Reis and Henrique J. A. da Silva
Introduction
Integrated Optics is a field of increasing interest and there has been considerable progress in the design and development of high performance devices based on optical waveguiding structures, particularly those built by the
diffusion of Titanium strips in a Lithium Niobate substrate, due to the extremely good electrooptical properties of the latter. This has called for an accurate knowledge of the components characteristics, essential for a correct
device design and simulation. However, the dielectric structures usually employed in this kind of devices forbid the use of traditional analytical methods. Several numerical methods have been proposed. Among these, the
finite differences and the beam propagation methods are the most popular for their accuracy and efficiency. These algorithms were combined with several others to develop modelling and simulation tools with the Matlab
software package, making possible to accurately predict the dynamic and static behaviour of several passive optical devices, such as directional couplers and Mach-Zehnder modulators, from their individual models and to
simulate the complete structure of an optical integrated circuit. To obtain a general basic device model, which reflects reality as much as possible, it is necessary to understand not only the diffusion of Titanium in Lithium
Niobate, employed in the construction of diffused optical waveguides, but also their propagation characteristics and the electrooptic control of optical waves.
Refractive Index Profile
Travelling-Wave Electrode Analysis
The diffusion of a Ti strip in a LiNbO3 substrate produces a distribution that is strongly dependent on the
diffusion process characteristics (temperature, duration, strip width and thickness) and is characterised by
a combination of error and exponential functions
The control of an electrooptic device is done through metallic electrodes that are deposited on the substrate
surface and produce an electric field that induces a variation in the waveguides’ refractive index. The
analysis is based on the assumption that a TEM wave propagates through the electrodes and this has been
shown to be an excellent approximation up to frequencies of several dozens of GHz. The calculation of the
external field distribution and the transmission line characteristics are done using a conformal mapping
approach. The propagation of the modulating wave in the electrodes is completely characterised and the
interaction between the optical and modulating fields is readily determined.

C0
erf  w2  x  DS   erf  w2  x  DS exp   y DP 2
C x , y  
2

The induced refractive index change is dependent on the wavelength and polarisation of the incoming
light beam, due to the anisotropic nature of LiNbO3 and is related to the Titanium concentration by

s










no e x , y  no e    o e H o e  C x , y
Electrooptic Control Analysis
oe
Extraordinary Refractive Index
The electrooptic effect is a mechanism through which it is possible to change the refractive index of a
device’s waveguides and induce a phase variation in the propagating beam. Nevertheless, the optical and
modulating fields propagate at different velocities, and the beam will experience different phase variations
as it propagates through the device. The overall effect is determined by
16
6
14
5
12
10
Dp [m ]
D s [ m ]
4
3
0.01
8
0.008
6
0.006
4
0.004
2
0.002
1
2
tu
re
1.5
ra
-5
pe
1
0
te m
2:50
9 00
3:00
2:30
2:40
2 :1 0
2:20
1 :50
2 :0 0
1:30
d u ra ti on
1:40
1:10
9 75
1:20
0:50
1:00
0 :3 0
0 :4 0
0:10
tu
ra
m
pe
0 :20
re
1 05 0
te
2:50
90 0
3:00
2:30
2:40
2:1 0
2 :2 0
1:5 0
1 :3 0
2:00
d ur a tio n
1:40
1:10
975
1:20
0:50
1 :0 0
0 :3 0
0 :4 0
0:10
10 50
0 :2 0
0
2
0
1 125
0
2.5
12 00
1 20 0
11 25
0















 L 

  t0   sin tan1 
  t0    o e L  1
t0  
e sin L  tan1 

2
  
  
2  2   





 2
m0o
3
0.5
5
y [um]
0
x [um]
DS and DP variation with t [hours] and T [ºC]
This expression can be directly applied to calculate the response of a phase or interferometric modulator.
For directional coupler based intensity modulators, some changes in the analysis are required and the
complete response is computed by segmenting the device into infinitesimal directional couplers.
Extraordinary index variation profile
Transverse Field Profile
After defining the dielectric structure of the device, its optical
characteristics must be determined in order to simulate the
propagation of light through its waveguides. However, since the
complex shape of the index profile does not lend itself to an
analytical analysis, it was necessary to develop numerical tools
that can be used with any type of profile.
Non-Uniform Discretisation Result
Simulations and Results
2.2
The results were obtained for
interferometric and directional
coupler
based
intensity
modulators. The devices were
designed to have the same
characteristic impedance (25)
and the same active length,
equal to the coupling length of
the directional coupler modulator (5.63mm).
2
1.8
1.6
1.4
1.2
1
2
1
The transverse dielectric domain is divided by a non-uniform
grid into small cells in which the electric field is defined. To
increase the accuracy, the density of cells is higher in the areas
where the index change is more significant. Substituting the
partial derivatives in the wave equation by five-point finite
differences and applying the adequate boundary conditions
between the cells, the full partitioning procedure produces a
matrix problem of finding the eigen values and functions of
6
4
0
2
-1
0
-2
-4
-2
-6
y [um]
x [um]
Field Transverse Profile
1
Copper electrodes, with 2 m
thickness, were positioned in a
complementary
coplanar
arrangement, producing a field
overlap of =0.911 and
=0.552 for the interferometric
and directional coupler cases.
The corresponding calculated
velocity mismatch factors were
=0.429 and =0.414.
0.7
0.6
0.5
0.4
0.3
0
2
1
6
4
0
2
-1
which is solved by conventional matrix algorithms to obtain the
propagation constants, te, and mode profiles, Ete, of the optical
waveguide.
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
20
20
30
10
0
-2
-4
-2
-6
y [um]
x [um]
Non-uniform grid and corresponding
fundamental optical mode
Optical Propagation Analysis
10
-20
y [um]
1
1
0.9
0.6
0.8
0.4
0.7
0.2
0.6
0
0.5
-0.2
0.4
-0.4
0.3
-0.6
0.2
-0.8
0.1
-1
0
20
20
30
10
0
-10
-20
y [um]
6
0.5
-20
y [um]
x [um]
Optical mode profiles for the
Directional Coupler Modulator
-50
x [um]
Optical mode profiles for the
Interferometric Modulator
Relative Integrated Phase Shift Response
Relative Integrated Phase Shift Response
0
0
-5
-5
-10
-10
20.Log(ß/ßo)
0
10
-15
-15
-20
-20
-25
-25
15
4
0
20
20
3
15
10
2
5
50
0
-5
1
40
30
20
10
0
[um]
-10
-20
-30
-40
-30 -3
10
-50
f3dB = 7.1 GHz
10
-2
10
-1
10
0
10
1
10
-30 -3
10
2
f3dB = 8.5 GHz
10
-2
10
-1
[GHz]
10
0
10
1
10
2
[GHz]
[mm]
-10
0
-20
Overlaped Modulating and Optical Field Profiles
Modulation Depth Frequency Response
Modulation Depth Frequency Response
-20
Field Distribution
0
0
-15
-5
-10
-5
-5
0
10.Log(m(f))
-10
10.Log(m(f))
um
y
-20
-30
5
-10
-15
5
1
4
-20
10
3.5
0.5
3
-15
15
0
2
20
40
30
1.5
30
1
0.5
-20
-30
[um]
-40
[mm]
0
Normalised field amplitude distributions for a dir.
coupler and a Mach-Zehnder interferometer
-25
f3dB = 23.3 GHz
2.5
-10
x
0
-10
-10
7
0
F
0
10
5
10
z
50
10
20
0
20.Log(ß/ßo)
[um]
1
20
1
x [um]
Transverse Optical Mode Profile
0.8
-5
[um]
where the operator H models the half-step propagation through
a homogeneous medium and N introduces a correction due to
the index variation. Using some properties of the Fourier
Transform,
0
-50
-10
N
F exp jk nx, y z F exp jk z 2E k ,k , z 
-20
y [um]
x [um]
-15
H

-20
-30
Transverse Optical Mode Profile
-20
z 

E  x , y , z  z   exp j 
 k B   exp j z  


 2
 
z 

 exp j 
 k B   E  x , y , z 
 2
 
EF k x , k y , z  z   exp jk z z 2 
0
-10
-10
Field Distribution
-15
H
0
0
-10
Overlaped Modulating and Optical Field Profiles
After the characterisation of each optical device by a three
dimensional refractive index distribution, and the determination
of the transverse propagating modes, the response of a
waveguiding device to an arbitrary optical excitation can be
found by applying the beam propagation method. This method,
derived from the wave equation and based on the Fast Fourier
Transform algorithm, takes the optical input and propagates it
step by step throughout the device. The step propagation is
based on the expression
50
10
20
0
0.8
0.1
M te Ete    Ete 
1
0.9
0.2
2
te
Transverse Optical Mode Profile
Transverse Optical Mode Profile
20
10
0
um
-10
-20
-30
Overlap profile between the
optical and modulating fields for
both devices
-20 -3
10
10
-2
10
-1
10
0
10
1
10
[GHz]
Responses for the integrated phase
shift and modulation depth of the
interferometric modulator
2
-30 -3
10
f3dB = 19.0 GHz
10
-2
10
-1
10
0
10
1
[GHz]
Responses for the integrated phase
shift and modulation depth of the
Dir. Coupler based modulator
10
2