Abstract—This paper presents a strategy for the robust design

of an optoelectronic communication system consisting of

non-linearly tapered waveguide and a photodetector. The

Inductive Design Exploration Method (IDEM) is implemented to

reduce the potential uncertainties existing when simulation

models are combined in a design process. This method assists us

in looking for robust solutions rather than an optimal solution

for this problem. A robust ranged set of design solutions could

perform better under uncertainty than optimal single solutions

which may deteriorate significantly when uncertainties make

conditions or assumptions change. In this paper, the emphasis is

on the method.

I. INTRODUCTION

he design of a tandem non-linearly tapered waveguide and

photodetector offers enhanced means for signal

propagation and dispersion control, especially for specialized

applications operating at specific wavelengths and low

receiver intensities. Typical optical propagation and

dispersion control components cause losses and significantly

decrease the signal-to-noise ratio of the signal.

In designing such complex systems, uncertainty is one of

the most important factors [1]. In addition to the uncertainty

caused by the changes of environmental conditions and

control factors [2, 3], the designers have to face several

potential uncertainties. An important source of uncertainty is

caused by simplifing assumptions of the model. This kind of

uncertainty, unfortunately, is common in the design of

complex systems because it is often impossible to create an

accurate, yet efficient simulation model for use in the early

stage of design. Another challenging source of uncertainty

arises when simulations are combined in a design process. The

uncertainties of each sub-system are propagated through a

Manuscript received April 9, 2008.

Chenjie Wang is a graduate student with the System Realization

Laboratory of The George W. Woodruff School of Mechanical Engineering,

Georgia Institute of Technology, Savannah, GA 31407, USA (e-mail:

cwang44@mail.gatech.edu)

Vivek Krishnamurthy is a graduate student with the School of Electrical

and Computer Engineering, Georgia Tech (e-mail: vivek@ece.gatech.edu)

Benjamin Klein is Assistant Professor with the School of Electrical and

Computer Engineering, Georgia Tech (e-mail: ben.klein@gtsav.gatech.edu)

P.D. Yoder is Assoicate Professor with the School of Electrical and

Computer Engineering, Georgia Tech(e-mail: doug.yoder@gtsav.gatech.edu)

Janet K. Allen is Professor with the System Realization Laboratory of The

George W. Woodruff School of Mechanical Engineering, Georgia Institute of

Technology, Savannah, GA, 31407, USA., (corresponding author, phone: (912) 963-6901, fax: (912) 966-7910, e-mail: janet.allen@me.gatech.edu)

model chain and the final solution performance estimation

may have a large degree of uncertainty. Such uncertainty is

also called “propagated uncertainty” [4].

To handle these uncertainties, robust design using the

Inductive Design Exploration Method (IDEM) [4] is

implemented to look for satisficing solutions instead of the

optimal ones. The main strategy in IDEM includes: (i)

identifying feasible ranged sets of design space instead of

single point design solution, and (ii) systematically

compromising between target achievements and providing for

potential uncertainties [5].

Using the Inductive Design Exploration Method, IDEM,

our goal is to design a non-linearly tapered waveguide and a

photodetector whose design solutions are feasible under

different kinds of uncertainties. Therefore, in Section 2 we

introduce this design problem and the theoretical foundations

of robust design using IDEM are introduced in Section 3. In

Section 4, we describe details of modeling the design

problems and associated decisions as well as present the result

obtained. Results are discussed in the Section 5.

II. NON-LINEARLY TAPERED WAVEGUIDE AND

PHOTODETECTOR

Optical communication systems use light as the

transmission medium. Generally speaking, an optical

communication system consists of a transmitter, which

encodes a message into an optical signal, a channel, which

carries the signal to its destination, and a receiver, which

reproduces the message from the received optical signal. The

heart of this receiver is a highly sensitive photodetector. In the

present work, we seek to improve upon photodetector

responsivity through the introduction of a passive integrated

tapered optical coupler, without penalty to 3dB bandwidth.

The communication system is shown in the Fig. 1.

A. Non-linearly Tapered Waveguide

A waveguide is a structure that guides light or other types of

waves. It is a passive or quasi-passive optical component

monolithically integrated with the active components: sources,

detectors, modulators and electronic devices. Guided wave

components are used to route optical signals on chips and also

required for the functions of directional coupling, filtering and

modulation. Lithium Niobate (LiNbO3) and similar dielectric

materials having large electro-optic coefficients are very

suitable for these kinds of applications [6].

Robust Design of Nonlinearly Tapered Waveguide and Photodetector

Chenjie Wang, Vivek Krishnamurthy, Benjamin Klein, Member, IEEE, P.D. Yoder, Senior Member,

IEEE, and Janet K. Allen

T

Proceedings of the 2008 IEEE Systems and InformationEngineering Design Symposium, University of Virginia,Charlottesville, VA, USA, April 25, 2008

FAMOpt.4

1-4244-2366-8/©2008 IEEE 123

Fig. 1. Optoelectronic Communication System Model

Fig. 2. Non-linearly tapered Waveguide Geometry

In this paper, the goal of the waveguide is to provide good

optical coupling. Fig. 2 indicates the geometry of the

non-linearly tapered coupler, where,

2b1 is the width at the broad end

2b2 is the width at the narrow end

L is the length of tapered coupler

au defines the taper profile.

L is set to 3.5µm and au is set 0.7. Therefore, in this

problem, there are only two design variables, i.e., b1 and b2.

Considering the imperfections observed in tapered

waveguides which is unfortunately impossible to predict in the

early stages of design, we treat the imperfection of the coupler

as noise. In our work, the uncertainty from it is +/- 10% of the

boundary of the control factors. The design variable range is

shown in Table I. TABLE I.

COUPLER DESIGN VARIABLES RANGE

Lower Bound Upper Bound

b1 0.2µm 40µm

b2 0.2µm 8µm

B. Photodetector

The photodetector converts the optical signal to an

electrical signal. A substantial reverse bias applied to the

photodetector during operation. Photons entering the

photodetector’s active region may be absorbed, leading to the

production of electron-hole pairs, whose motion through the

active region under the influence of a strong electric field

couples electrical current to an external circuit. Maximizing

the photodetector’s quantum efficiency is critical for both

long haul telecommunications applications, in which signal

attenuation over optical fibers can be substantial, as well as for

low photon flux applications. In this paper, the design

problem focuses on increasing the transmission of the passive

tapered waveguide and the bandwidth of the photodetector

itself

The geometry of the photodetector is presented in Fig. 3.

The bias is applied across at the P and N 'terminals' of the

semiconductor structure. The top layer of the photodetector

mesa is P-doped having extra holes, the middle layer is the

absorption layer having a narrow electronic band-gap and the

bottom layer is N-doped, having extra electrons.

Fig. 3. Photo Detector Geometry

In Fig. 3, W is the mesa width of the substrate. h is the height

of the intrinsic layer, and Lp is the length of the photodiode.

For an electron-hole pair created in the intrinsic region due

to photon absorption, there is a transit time associated with the

collection of these charge carriers, which depends on the

position of photogeneration, Eq. 1, and Eq. 2 are the

maximum transit time for electrons and holes, respectively.

These transit times can be used to define the expectation value

of the photodetector’s electrical response to an optical

impulse, Eq. 3.

/n nsh Vτ = (1)

/p ps

h Vτ = (2)

( ) 1 ( ) 1 ( )n p

n n p p

q t q tI t U t U tτ τ

τ τ τ τ

= − − + − −

( 0)t > (3)

q is the fundamental unit of charge Coulombs, ns

V is an

average electron saturated drift velocity of 1e5m/s, ps

V is an

hole saturated drift velocity of 5e4m/s, and U is the

Heaviside function, Eq. 4

( ) 0 ( 0)

1 ( 0)

U t t

t

= <

≥

(4)

The frequency response of photodetector is given by Eq 5.

We seek the 3dB frequency which is the bandwidth of the

photodiode.

( )

1L j

IF

j R C

ω

ω=

+

(5)

In Eq.5, ( )I ω is the Fourier transform of Eq 3, L

R is the

load resistance of 50 Ω in this paper; and j

C is junction

capacitance, Eq 6. E is dielectric permittivity 9 × 8.85e-12

F/m.

j

WLpC

h= Ε (6)

One figure of merit for a photodetector is its efficiency at

converting photons to electrical current.. This is known as its

1-4244-2366-8/©2008 IEEE 124

quantum efficiency (QE), which is the probability that a

photon incident on the photodetector is actually absorbed. A

good approximation for a waveguide photodiode’s quantum

efficiency is given in Eq. 7.

1Lpin out

in

P PQE e

P

α− Γ−

= = − (7)

Γ is the optical confinement factor, which is a measure of

how much of the power associated with a optical signal is

actually within the absorbing region of the photodetector,

which can be calculated by the waveguide model solving

Maxwell’s equations. α is the optical absorption coefficient

– a bulk material parameter. Here, we assume its value is 10e4

cm-1

.Because the photodetector is integrated in tandem with

the passive tapered optical waveguide, the system quantum

efficiency (Eff) is the product of this QE and transmission of

waveguide. Eff should be higher than 30% to make the system

work reliably. Again, in order to save the computational cost,

h is set 0.2 um in the design problem for this paper. The design

variable range is shown in Table II. TABLE II

PHOTODETECTOR DESIGN VARIABLES RANGE

Lower Bound Upper Bound

Lp 5µm 500µm

W 4µm 16µm TABLE III

CDSP FOR WAVEGUIDE AND PHOTODETECTOR PROBLEM

Given

• Length of the tapered waveguide L=3.5µm

• the taper profile variable au =0.7

• optical absorption coefficient α =10e4cm-1

• load resistance L

R =50 Ω

• dielectric permittivity E=9× 8.85e-12 F/m

• height of the intrinsic layer h=0.2µm

• T1=f(b1, b2);

• bandwidth=f(Lp, W)

• Eff=(T1, L)

• Propagated Uncertainty +-10% in system

performances

• Bandwidth target: G1= 120GHz (the larger the

better)

• System quantum efficiency objective: G2=1

Find

• b1, b2, Lp, W

Satisfy

Constraints

• b1>b2

• system quantum efficiency Eff>0.3

• bandwidth >35 GHz

Bounds

• 0.2µm <b1<1.5µm, 0.2µm <b2<1µm

• 50µm <Lp<500µm, 5µm <W<15µm

• di-*di

+0, di

-,di

+=0

Goals

Maximize bandwidth and Eff under propagated

uncertainty

• d1-=1- A1(x)/ G1

• d2-=1- A2(x) /G2

Minimize

• Z=W1 d1-+W2 d2

-

This design problem contains two conflicting design

objectives. The higher bandwidth is better, while high band-

width may reduce the quantum efficiency of the photodetector.

In this design problem, we look for a robust solution to the

conflicting design problems.

The compromise Decision Support Problem (cDSP) [8,9] is

implemented to explain the whole design problem in this

paper. It is used to determine the values of the design variables,

which satisfy a set of constraints and bounds while achieving

as closely as possible a set of conflicting goals. The cDSP for

waveguide and photodetector design problem is presented in

Table III. The overall objectives of this design problem are to

maximize the bandwidth and system quantum efficiency with

robustness under uncertainty. If the design variables deflect

within the solution range, the performances of system can still

keep stable in feasible range.

III. INDUCTIVE DESIGN EXPLORATION METHOD (IDEM)

In this section, we explain the procedure to design the

coupler and photo detector using the Inductive Design

Exploration Method (IDEM) [5].

IDEM can be used in very complex product design chains.

The problem in this paper is still a preliminary and simple

design. The flow of information for this design problem is

shown in Fig. 4.

Photonic Crystal

Waveguide Model

b1

b2

Lp

W

Bandwidth

System EfficiencyQE Model

Transmission 1

Bandwidth

Model

Lp

Fig. 4. Information map for the coupler and photo detector design

problem

A. Overview of Solution Procedure

The overall procedure for IDEM is illustrated in Fig. 5.

The procedure includes the following steps:

Step 1: It is necessary to define the rough design space (x

space in Fig. 5), the interdependent space (y space in

Fig. 5), and the performance space (z spaces in Fig. 5).

Discrete points are generated in each of these spaces.

Step 2: The discrete points which are generated are

evaluated based on the mapping models (models f and

g in Fig. 5) and the evaluated data sets which are

composed of a discrete input point and output range

are stored in a database.

1-4244-2366-8/©2008 IEEE 125

Step 3: Feasible regions in y and x spaces are

sequentially identified, along with a given final

performance range in z space. We call this step

Inductive Discrete Constraints Evaluation (IDCE).

Although the example given here is simple, the procedure is

available for finding robust ranged sets of specifications in all

types (sequential, parallel, and hierarchical) of sub-system

network [5].

f

gCauses/Effects (Deductive)

x

y

z

f

gCauses/Effects (Deductive)

f

gCauses/Effects (Deductive)

x

y

z

X space

Y space

Z space

STEP 1

X space

Y space

Z space

X space

Y space

Z space

STEP 1

X space

Y space

Z space

STEP 2

X space

Y space

Z space

X space

Y space

Z space

STEP 2

Goals/means (Inductive)

Given performancerequirements

X space

Y space

Z space

STEP 3

Identified feasibledesign space

Goals/means (Inductive)

Given performancerequirements

X space

Y space

Z space

STEP 3

Goals/means (Inductive)

Given performancerequirements

X space

Y space

Z space

STEP 3

Identified feasibledesign space

g

f

Fig. 5 Solution search procedure for multi-level robust design [5]

B. Inductive Discrete Constraints Evaluation (IDCE)

In this step, designers sequentially search for feasible

ranges in the spaces of interdependent variables and design

variables, based on the data generated in the last step. The

IDCE process includes the following three steps:

Step (a): find satisfactory points in an input space with

given constraints (feasible ranges) in an output space

based on Hyper Dimensional Error Margin Indices

(HD_EMIs)

Step (b): obtain contours for the borders of the feasible

regions in an input space, creating border points

between discrete satisfying points and points that are

not satisfying, and

Step (c): sequentially repeat steps (a) and (b) to find

feasible regions at the lower levels using the borders of

feasible regions obtained in the previous step (b) as the

constraint bound in the output space [5].

Hyper-Dimensional Error Margin Indices:

The first step of the IDCE is to check if the mapping of each

discrete point from the input space to the output space stays

within the satisfying output range. HD_EMIs is used for this

feasibility check. If more than half of the nearest neighbor

points obtained from the mapping model are feasible points,

then we assume that the mean is in the feasible range. When

the mean vector of an output range is not in the feasible range,

then the HD_EMIs of all outputs are [5]:

_ 1 allHD EMI = − (8)

When the mean vector is inside the feasible range, we identify

the value of the HD_EMI in each output direction [6]:

( )

( )

-_

-i

HD EMI Min

=

j i

i

j i

mean B u

mean B u

g

g

(9)

In Eq. 6, an HD_EMI in the ith

direction is the minimum

HD_EMI among all HD_EMIs that are calculated using

discrete points on a constraint boundary (B), their projected

points on output boundary in the ith

direction (Bi), and the

mean of the output range (mean). Details of the HD_EMI

calculation are clarified in Fig. 6. Cleary, with HD_EMIi

increasing, the output range moves farther away from the

constraint boundary in the ith

direction.

jB

Feasible region

1

2

mean

Output range

Constraint boundary

given or obtained

from previous DCE

task

Output

boundary

Input space

Output space

Output space

i

jB

( )- j imean B ug

( )- i

j imean B ug

Fig. 6 HD_EMI calculation in a direction [5]

C. Waveguide and Photodetector Design using IDEM

The IDEM for designing waveguide and photodetector is

shown in Fig. 7. The resolutions for discrete points are fixed

as 0.3µm, 0.4µm, 25µm and 0.25µm 2% for b1, b2, Lp, W, and

T1, respectively. These resolutions are selected considering

the computational cost. First, we search the entire feasible

ranges in property space (T1, Lp and W) with given

performance requirements. The required HD-EMIs (HD-EMI2

and HD-EMI3) for mapping models (QE models and

bandwidth model) are set as greater than or equal to unity,

which means all quantified uncertainty must be satisfied.

Among the obtained feasible space for T1, Lp and W, we select

the value of W that has the largest feasible space for the rest of

the properties (T1 and Lp), because we want to maintain the

feasible region as large as possible until the end of the design

process to achieve robustness. The design process is

implemented in MATLAB.

When HD_EMI2 and HD_EMI3 are given as unity, the

largest feasible range in y space is achieved at W=5.75µm,

which is considered as a satisficing solution which makes the

system performance robust to the potential uncertainties. The

feasible space of the Lp and T1 is shown in Fig. 8. Satisfactory

discrete points (circular points) and boundary points

(diamond points) between the feasible and infeasible space are

shown in the figure. From this figure, we can determine Lp in

term of cDSP of HD_EMI and then obtain the T1 performance

range for designing the waveguide in the next step.

1-4244-2366-8/©2008 IEEE 126

Fig. 8 Achieved feasible design space in Lp and T1

Since the achieved feasible design space can be further

reduced, we further selected the HD_EMIs with various

design scenarios. A cDSP is coupled with the IDCE process

described above in order to find the best HD_EMI as

described in Table IV. The goal of selecting HD_EMI is to

find the satisficing design values of Lp and W, which

maximizes design performances, and obtain the value of T1,

which is the performance feasible range of waveguide. TABLE IV

THE CDSP FOR SEEARCHING THE BEST HD_EMI

Given

• HD_EMI target, i =10

• Eff=[0.35, 1] the larger, the better

• bandwidth=[35, ∞] (GHz) the larger, the better

Find

• HD_EMI i, Lp, W

Satisfy

Constraints

• G(HD_EMI i, Lp, W, Eff, bandwidth) = (Lp,

W): the IDCE procedure to obtain two design

variables

• Num(L,W)1: it should be a design range

• HD_ EMI i1, where i=2,3: entire output range

can satisfy the constraints

Goals

Maximize the HD_EMI in output space to both

make the system more robust in more uncertain

design environment and maximize the

performance

• HD_ EMIi / HD_EMI target, i +di- =1

• di-*di

+0

• di-,di

+=0

Minimize

• Objective function Z

Scenario 1: Z= di- where i=2;

Scenario 2: Z= di- where i=3;

Scenario 3: Z= ∑ wdi- where i=2,3, w=0.5

In this paper, we formulate three different scenarios of

photodetector design part.

Scenario 1: Find the satisficing design specifications for

maximizing the bandwidth.

Scenario 2: Find the satisficing design specifications for

maximizing the quantum efficiency.

Scenario 3: Find the robust design specifications for both

performances.

Three scenarios lead to different length of photodetector

design value and the performance range of T1 may be different.

b1and b2 in the waveguide design model should be designed

to realize these T1 performances.

All the goals of three scenarios in cDSP are to maximize the

HD_EMI. In IDEM, the larger HD_EMI is, the farther the

performances keep away from the feasible boundaries. In this

design problem, since the feasible boundaries are lower

bounds of the performances, largest HD_EMI also means the

maximized system performances, i.e., largest bandwidth and

Eff.

IV. DISCUSSION OF RESULTS

The results of the design exploration are shown in Table V. TABLE. V.

DESIGN RESULTS WITH THREE SCENARIOS

Scenarios 1 2 3

Lp (µm) 225 475 275

W (µm) 5.75 5.75 5.75

b1 (µm) 0.5

b2 (µm) 0.2

Required

HD_EMIs

1

2

3

1.70

3.52

6.02

1.70

5.61

2.03

1.70

5.05

5.35

Bandwidt

h

(GHz)

Min

Mean

Max

67.8

75.3

82.9

33.9

37.7

41.4

56.5

62.8

69.1

Eff Min

Mean

Max

0.573

0.637

0.710

0.729

0.810

0.891

0.625

0.695

0.764

In Scenario 1, we minimize the deviation of HD_EMI3 from

the target for HD_EMI3. This is for the purpose of maximizing

the bandwidth. Therefore, the HD_EMI3 (6.02) and mean of

bandwidth (75.3GHz) are both largest among three scenarios.

In Scenario 2, the deviation of HD_EMI2 is minimized from

the target. This design process found a solution that

maximizes the quantum efficiency of the system. It is not

strange that HD_EMI2 (5.61) and mean of Eff (0.810) are

largest. In Scenario 3, minimizing an objective function that is

equally weighted summation of the deviations from the targets

Inductive Discrete Constraints Evaluation process

Feasible region

Infeasible region

X=(b1, b2) Y=(T1) Z=(bandwidth, E)

Solution Range

of b1, b2

Required

HD-EMI1

Solution

Range of Lp,

W

Required

HD-EMI2

HD-EMI3

Compromise Decision Support Problem

For Selecting the best HD-EMI

Fig. 7 IDEM for designing waveguide and photodetector system

1-4244-2366-8/©2008 IEEE 127

of HD_EMI2 and HD_EMI3 design specifications are

indentified. The design process is to obtain robust solutions

from two conflicting design problems. In this scenario, the

efforts to maximize the two design objectives are balanced.

The mean of bandwidth is smaller than Scenario 1 which is to

maximize bandwidth but larger than Scenario 2; the mean of

Eff is smaller than Scenario 2 which is to maximize Eff, but

larger than Scenario 1.

In all scenarios, the HD_EMIs is large enough to accept.

The larger HD_EMI is, the farther the design performance is

away from the boundary. Since potential uncertainties which

are not expected by the designers may make originally optimal

design solutions totally infeasible, it is reasonable to keep the

solution far away from the boundary, especially for these

models not highly nonlinear.

Although the solutions provided listed in this paper seems

single point solutions, all the points around the solutions are

acceptable. In other word, even though the actual variables are

a little different from the solutions designed, the final

performance will not be influenced.

The design problem in this paper is a simplified distributed

design problem, in which one design performance of one

model is another’s input design variable. In the further study

of more complex waveguide and photodetector system with

more interconnections, IDEM will show more advantages of

handling with uncertainty propagation.

There are also some limitations of design solutions in this

paper. First, in order to save the computational cost, some

design variables are set as constant variables. Although it does

not influence the feasibility of the design performance, the

design solutions are still affected. Considering this paper is the

starting point of further research, we still accept the results.

Secondly, the design solution has a main limitation caused by

IDEM, i.e., discretization of a design space. Although the

exact border boundary generation algorithm is introduced to

improve the performance of exploring a design space, the

discretization errors are still unavoidable while checking the

feasibility of a mean performance based on discretized

feasible and infeasible points [1]. The finer resolution may

reduce the error, but it also increases the computational cost.

Thirdly, IDEM only considers the feasible performance, but

not stable performance as some other robust design methods

[10].

V. CONCLUDING REMARKS

In this paper, we have focused on designing a

communication system consisting of tapered waveguide and

photodetector. IDEM is implemented for handling propagated

uncertainties existing in the distributed collaborative design

environment, such as the design problem in this paper. The

final design solution is robust to potential uncertainties with

high HD_EMI which ensures the performance far away from

the feasibility boundary. IDEM is also shown as useful to be

implemented in this design problem.

Future work will focus on more complex waveguide

structures with more design variables and interconnections

with photodetector. Moreover, future study will focus on how

to reduce the errors and limitations caused by IDEM and how

to improve it to support collaborative and distributed design

environment better.

ACKNOWLEDGMENT

The cost of computer time has been underwritten by The

Systems Realization Laboratory, The George W. Woodruff

School of Mechanical Engineering, Georgia Tech.

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