Directional Coupler

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INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF OPTICS A: PURE AND APPLIED OPTICS

J. Opt. A: Pure Appl. Opt. 5 (2003) 449–452 PII: S1464-4258(03)62907-7

A simple, fast and accurate method of designing directional couplers byevaluating the phase difference of localsupermodes

Qian Wang and Sailing He

Centre for Optical and Electromagnetic Research, State Key Laboratory of Modern Optical

Instrumentation, Zhejiang University, Yu-Quan, 310027 Hangzhou, People’s Republic of

China,

Joint Laboratory of Optical Communications of Zhejiang University, Yu-Quan,

310027 Hangzhou, People’s Republic of China

and

Division of Electromagnetic Theory, Alfven Laboratory, Royal Institute of Technology,

S-100 44 Stockholm, Sweden

E-mail: [email protected]

Received 30 April 2003, accepted for publication 16 June 2003Published 1 July 2003Online at stacks.iop.org/JOptA/5/449

AbstractThe local supermode formalism and a three-dimensional beam propagationmethod (BPM) are combined in designing directional couplers. The

accumulated phase difference of the two local supermodes in the input oroutput region is evaluated quickly with the assistance of the BPM. Thedesign method is simple, fast and accurate. A comparison of the presentmethod with some other design methods is given. The design results areverified both numerically and experimentally.

Keywords: Directional coupler, beam propagation method, waveguides,integrated optics

1. Introduction

A directional coupler is a fundamental planar lightwave

circuit (PLC) performing the functions of splitting andcombining [1]. Combined with optical delay lines,

such couplers can be used to construct Mach–Zehnder

interferometers (MZIs) [1], interleavers [2] and wavelength

division multiplexers/demultiplexers (WDMs) [3]. An actual

PLC in a directional coupler should consist of the input region,

the central coupling region and the output region (see figure 1)

for connecting to fibres (or fibre arrays) or use in constructing

MZIs or WDM devices. The effective index method can

reduce the computation time greatly through converting the

three-dimensional (3D) model to a two-dimensional model.

However, this method is not accurate in the design of

directional couplers. Therefore, we have to consider the 3D

model in the present paper.

In the design and analysis of directional couplers, the

supermode solution [4,5] has been widely used to calculate the

coupling length of two parallel waveguides. In the supermode

solution the two parallel single-mode waveguides are treatedas a whole, which allows two supermodes (one is symmetric

and the other is asymmetric with respect to the middle plane of

the two identical parallel waveguides). The coupling between

the two waveguides is a result of the interference between

the two supermodes as they propagate along the directional

coupler. When a PLC is used in an optical communication

system, it should be connected to some input and output

fibres. Therefore, a PLC of a directional coupler consists

of the input waveguide region, the central coupling region

and the output waveguide region. The separation between

the two parallel waveguides (in the central coupling region)

increases gradually to e.g. 250 mm in the input and output

waveguide regions used to connect the PLC to fibres or fibre

1464-4258/03/050449+04$30.00 © 2003 IOP Publishing Ltd Printed in the UK 449

http://stacks.iop.org/JOptA/5/449





Q Wang and S He

Input

1

2

3

4

Rin Rout

L

A

A0

B

B0

z axis

x axis

y axis

Central couplingregion

Output

Figure 1. The schematic structure of a directional coupler with aninput region, a central coupling region and an output region.

arrays. The coupling between the two parallel waveguides in

the central region dominatesthe splittingratioof thedirectionalcoupling as compared to the coupling in the input and output

regions. However, if one neglects the coupling in the input

and output regions and uses the coupling length obtained

from the supermode solution (in accordance with the two

parallel waveguides) directly, the actual splitting ratio will be

wrong (as shown below at the end of section 2). Therefore,

the local supermode formalism has to be used in order to

consider the coupling in the input and output regions (which

contain bending parts). In a conventional local supermode

method, each small interval along the directional coupler is

treated as twoparallel waveguides andthe corresponding phase

difference between the two local supermodes is calculated and

accumulated. The relative output powers of the directionalcoupler are then given by P3 = (1 + cos φt)/2 and P4 = (1−cosφt)/2, where φt is the total phase difference between the

two supermodes for the whole directional coupler (including

the input region, the central coupling region and the output

region). In such conventional methods, one has to calculate the

propagation constants of the two supermodes for each small

interval (by using e.g. a finite-difference method [6] for the

cross-sectional profile of the two 3D waveguides) and this is

very time-consuming. Another way to design a directional

coupler (as used in most of the existing commercial software)

is to obtain the length of the central coupling region through

some manual adjustment with a beam propagation method

(BPM) [7–9] around an approximate coupling length (withoutconsidering the couplings in the input and output regions)

calculated from the supermode solution. With this method one

can find accurately the length of the central coupling region,

but one needs to simulate the light propagation with a 3D BPM

repeatedly, particularly in the output region.

2. The design method and evaluation of the phasedifference

In the present paper, we introduce a simple, fast and accurate

design method in which one only needs to calculate the

propagation constants of the two supermodes at one or

two specific positions (instead of at all positions along the

directional coupler as in a conventional local supermode

method). This method combines the local supermode

formalism with the 3D BPM and can give the length of the

central coupling region accurately by simulating the light

propagation in the input and output regions just once (if the

input and output regions have the samestructure, we only need

to simulate the light propagation in the input region with a 3D

BPM once). We take a 3 dB directional coupler as an example

in describing the design procedure, as follows.

2.1. Step 1

Use a 3D BPM to simulate the light propagation in the input

region. The initial field distribution f in at plane 1–2 for

the BPM can be approximated with the eigenmode of one

input waveguide (e.g. waveguide 1). It can also be chosen

as f in = ((s)in +

(a)in )/

√ 2, where

(s)in and

(a)in are the field

profiles of the two supermodes at the input position 1–2. We

then use a 3D BPM to simulate the light propagation in the

input region and obtain the field distribution (denoted as f AA0)

at plane AA0 (the end of the input region).

2.2. Step 2

Evaluate the accumulated phase difference between the two

local supermodes in the input region. On the basis of the local

supermode theory, one has the following formula at the end of

the input region:

f AA0= (s)

c exp

j

AA0

12

β(s)( z) d z

+ (a)c exp

j

AA0

12

β (a)( z) d z

, (1)

where β(s)( z) and β(a)( z) are the propagation constants of

the two supermodes at position z, (s)c and

(a)c are the field

profiles of the two supermodes at position AA0 (the same

anywhere in the central coupling region). From equation (1)

and the orthogonality of the two supermodes, one sees that the

accumulated phases of the two local supermodes in the input

region satisfy

exp

j

AA0

12

β(s)( z) d z

=

f AA0[(s)

c ]∗ d x d y

(s)

c [(s)c ]∗ d x d y

and

exp

j

AA0

12

β(a)( z) d z

=

f AA0[(a)

c ]∗ d x d y

(a)

c [(a)c ]∗ d x d y,

where the asterisk indicates the complex conjugate. Therefore,

the phase difference between the two local supermodes,

φin = AA0

12[β(s)( z) − β(a)( z)] d z, can be calculated using the

following useful formula:

φin = arg

f AA0

[(s)c ]∗ d x d y

(a)c [

(a)c ]∗ d x d y

(s)c [

(s)c ]∗ d x d y f AA0

[(a)c ]∗ d x d y

(2)

450



A simple, fast and accurate method of designing directional couplers

where ‘arg’ denotes the argument (polar angle) of a

complex number. With the above formula we can obtain

the accumulated phase difference between the two local

supermodes in the input region without calculating the

propagation constants of the two local supermodes at everyposition of the input region (certainly with the assistanceof the

3D BPMsimulation (one time) in step 1). If the structure of the

output region differs from that of the input region, we can carryout a similar evaluation, i.e., simulate the light propagationfrom the output to position BB0 and obtain the accumulated

phase difference φout = BB0

34[β(s)( z) − β (a)( z)] d z for the

output region with a formula similar to equation (2).

2.3. Step 3

Calculate the length of the central coupling region from

L = (φt − φin − φout)/(β(s)c − β(a)

c ), (3)

where the total phase difference φt is determined by thedesired

splitting ratio (note that P3

=(1 + cos φt)/2 and P4

=(1−cosφt)/2) andβ(s)c ( z) andβ (a)c ( z) are the propagation constants

of the two supermodes for the two parallel waveguides in the

central coupling region.

The above formulation is of scalar type (like the BPM

used), which is appropriate for the weakly guided buriedsilica (step-index) waveguides considered in the present paper.

In some polarization-sensitive cases, a vectorial (or semi-

vectorial) BPMandsupermodes of vectorform should be used.

However, the above design idea can still be employed.

In order to test the above design method, we choose a3 dB directional coupler (formed by silica-on-silicon buried

waveguides) as a design example. The refractive indicesof the core and cladding are chosen to be 1.454 and 1.445,

respectively. The cross-section of the waveguide is 5 µm ×5 µm and the bending radius for the curved part of the input

and output waveguides is Rin = Rout = 20 000 µm. The

separationdistanceof thetwoparallelwaveguides in the central

coupling region is 7 µm and the separation distance between

thetwoinput(oroutput)waveguidesis45µm. Thewavelength

of the input light is 1.55 µm. The field distribution at plane

AA0 is shown in figure 2; it was obtained from a three-dimensional BPM. From figure 2 onesees that thecoupling has

occurredin theinputregion. Theaccumulated phasedifferencebetween the two local supermodes in the input region is then

obtained from equation (2) as φin = 0.125 31 (=φout). Here

the alternate-direction implicit (ADI) finite-difference BPM is

used in thecalculation. This numerical methodis very efficientand an analysis of its accuracy can be found in e.g. [10].Equation (3) then gives L = 2420 µm for the length

of the central coupling region. If one neglects the coupling

in the input and output regions, the corresponding length

of the central coupling region will be L0 = π/[2k 0(n(s)e −

n(a)e )] = 2880.5 µm, where k 0 is the wavenumber in vacuum;

n(s)e = 1.448392 89 and n

(a)e = 1.44825836 are the effective

refractive indices for the two supermodes in the central

coupling region. This length differs from our design result

by 460.5 µm. Thus, one sees that the coupling in the input andoutput regions cannot be neglected. If we neglect the coupling

in the input andoutput regions in thedesignand use L0 directlyas the length of the central coupling region, the splitting ratio

will become 37.6%:62.4% (instead of 50%:50%).

Figure 2. The field distributionat plane AA0 obtained with a 3DBPM.

20 25 30 35-35 -30 -25 -20

Lateral position(µm)

Present method

Method in Ref. [1]

R e l a t i v e o u t p u t

0.5

0.6

0.7

0.8

0.9

1

1.1

Figure 3. The final output field profile at the central line (along the x -direction) on plane 3–4. Relative intensities below 0.5 are notshown in order to give a clearer comparison.

3. Numerical and experimental verification of thedesign

To verify the above design numerically, we simulate the light

propagation in the whole directional coupler with a 3D BPM.

The final output field profile at the central line (along the

x -direction) on plane 3–4 is shown by the solid curve in

figure 3. If we use the approximate method mentioned in [1]

(cf equations (12) and (13) there), the length of the central

coupling region will be 2503 µm (which is different from our

design result of 2420 µm) and the corresponding output field

profile is shown by the dashed curve in figure 3. From this

figure one sees that the present design method gives a more

accurate length forthecentral couplingregion as theuniformity

of the two output powers is much better.

We have also fabricated a 3 dB directional coupler with

the structural parameters designed with the present method.

The buffer layer, core layer (Ge-doped silica deposited by

451



Q Wang and S He

Table 1. The insertion loss of a fabricated 3 dB directional coupler(fibre-pigtailed) at each output (unit: dB).

Input Output 2 Output 3 Uniformity

1 4.39 4.55 0.164 4.33 4.26 0.07

PECVD) and cladding layer have thicknesses of 20, 5 and

15 µm, respectively. The measured results for the insertion

loss at wavelength 1.55 µm at each output waveguide (with

the input at waveguide 1 or 2) are shown in table 1. In our

experiment, the coupling loss between a fibre and a waveguide

is about 0.6–0.7 dB. Thus, it is reasonable that the measured

insertion loss for each output waveguide is about 1.2–1.4 dB

larger than the calculated insertion loss (since there are two

fibre–waveguide connections in each light path). From this

table one also sees that the uniformity is good.

4. Computational time and accuracy

In the present method we use the BPM to calculate the

accumulated phase difference in the input and output regions.

Therefore, a large number of small incremental steps along the

lengths of the arms are used to simulate the light propagation.

Thecurved waveguides are replaced by a large number of short

step waveguides. The calculated values of ϕin with different

grid sizes are presented in table 2, from which one sees that the

calculation error for the accumulated phase difference is very

small when the grid size z is less than 0.5 µm. These small

phase errors of the directional couplers (for the case where

z 0.5 µm) only cause a very small change in the splitting

ratio of the directional couplers and have almost no effect

on the performance of PLCs (such as MZIs and interleavers)

consisting of directional couplers. On the other hand, the

supermode propagation constants are calculated using a finite-

difference method [7] and thus the lateral mesh grid chosen in

the design can also affect the calculation accuracy for both

the present method and the conventional local supermode

method. Our numerical results have shown that the calculated

propagation constants of the supermodes have a good accuracy

when x 0.1 µm and y 0.1 µm. The computational

times for the present design method and some other methods

are also compared. All the calculations are carried out on a

Pentium-3PC. To obtainan accurate design, we usea mesh grid

with x = y = 0.1 µm and z = 0.5 µm. It takes about

6 minto calculate thephase difference in each small interval by

using a finite-difference method. In our example, the length

(along the z-direction) of the input region is 1232 µm. To

obtain the accumulated phase difference for the input region,

a conventional local supermode method (in which one has to

calculate the propagation constants of the two supermodes for

each small intervalby using the finite-difference method) takes

more than 14000 min (i.e. about ten days). Therefore, this

conventional methodis notpractical foruse in an actualdesign.

With the same mesh grid, a 3D BPM takes 1.16 s for each

propagation step. If one only uses the 3D BPM to adjust the

Table 2. Convergence of ϕin with respect to grid size.

Grid size z (µm) Calculated value of ϕin

1.0 0.125 010.8 0.125 030.5 0.125 310.3 0.125 38

0.1 0.125 38

length of the central coupling region (as done in most of the

existing commercial software), each adjustment takes about

190 min in order to simulate the light propagation along the

directional coupler one time. If N times of adjustment are

neededto findan accurate lengthof thecentral coupling region,

the design procedure will take 190 × N min. However, with

the present design method it takes only about 60 min (since the

3D BPM simulation only needs to be carried out once in the

input region for the present symmetric case).

5. Conclusions

We have introduced a simple, fast and accurate method for

the design of directional couplers. With the assistance of a

3D BPM, the phase difference of the two local supermodes

in the input (or output) region is evaluated quickly and

accurately using formula (2). The method is much faster than

any conventional local supermode method and the manual

adjustment method used in most of the existing commercial

software; also it is more accurate than the approximate method

mentioned in [1]. The present method can also be modified

for the design of asymmetric directional couplers.

Acknowledgment

The support of the Wenner–Gren Foundation is acknowledged.

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Directional Coupler