Power splitting ratio couplers based on MMI structures with high bandwidth and large tolerance using silicon waveguides

Power splitting ratio couplers based on MMI structures with high

bandwidth and large tolerance using silicon waveguides

Cao-Dung Truong a, Trung-Thanh Le b,*a Hanoi University of Science and Technology, No. 1 Dai Co Viet, Hanoi, Viet Nam

b Hanoi University of Natural Resources and Environment, No. 41A, K1 Road, Phu Dien, Hanoi, Viet Nam

Received 24 November 2012; received in revised form 16 January 2013; accepted 18 January 2013

Available online 26 January 2013

Abstract

We show that it is possible to obtain 2 � 2 couplers based on multimode interference (MMI) structures with nineteen new power-

splitting ratios by cascading three or four MMI couplers. The other aim of this study is to use silicon waveguides, that are compatible

with the existing CMOS (Complementary Metal-Oxide-Semiconductor) fabrication technology, for designing the proposed

devices. The proposed MMI couplers with new power splitting ratios have simple geometries and low losses. These MMI

couplers can offer valuable new possibilities for designing MMI waveguide-based photonic integrated circuits such as all-optical

interconnects, microring resonators, clock distribution, Mach Zehnder Interferometer based on MMI couplers and other all-optical

processing applications. The transfer matrix method (TMM) and modified effective index method (MEIM) along with the support of

the 3D Beam Propagation Method (3D BPM) are used to optimize the proposed devices.

# 2013 Elsevier B.V. All rights reserved.

Keywords: All-optical processing; Integrated optics; Multimode interference (MMI) couplers; Silicon on insulator (SOI); CMOS technology

www.elsevier.com/locate/photonics

Available online at www.sciencedirect.com

Photonics and Nanostructures – Fundamentals and Applications 11 (2013) 217–225

1. Introduction

Multimode interference (MMI) couplers of unequal

power splitting ratios are attractive for photonic

integrated circuit (PICs) applications such as power

taps, high-Q ring resonators [1,2], ladder-structure

optical filters [3] and loop mirror partial reflectors. A

coupler with freely chosen power splitting ratio is

particularly valuable in Mach–Zehnder interferometer

(MZI) structure when loss and gain are asymmetrically

distributed between both different-length arms [4]. In

the literature, it is showed that a conventional MMI

* Corresponding author. Tel.: +84 985 848 193.

E-mail addresses: [email protected],

[email protected] (T.-T. Le).

1569-4410/$ – see front matter # 2013 Elsevier B.V. All rights reserved.

http://dx.doi.org/10.1016/j.photonics.2013.01.002

coupler with a rectangular shape provides only seven

fixed different power splitting ratios [5]. Therefore,

finding a reasonable way for obtaining a variety of

power splitting ratios is particularly important to realize

all-optical signal processing devices based on MMI

couplers [6].

In principle, the way to achieve free choice of power

splitting ratios is to introduce a phase difference at some

special position within the MMI device or between two

couplers in the MZI. The introduction of such a phase

shift will lead to new phase relations between the self-

images at the output plane. Depending on the material

system used for fabricating the MMI couplers, several

approaches have been proposed to obtain arbitrary

power splitting ratios. One of the most commons ways

to ‘‘tune’’ the coupling coefficient of a coupler is to use

an MZI structure [7–9]. The tuning of the refractive

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C.-D. Truong, T.-T. Le / Photonics and Nanostructures – Fundamentals and Applications 11 (2013) 217–225218

index using the carrier related plasma effect has been

performed directly within the MMI region. The same

methods can be applied to the devices designed on other

materials but using the thermo-optic effect [10] or

electro-optic [11] to change the refractive index at

special shapes such as butterfly-like MMI, exponential

tapered MMI and angled MMIs to produce a free

selection of the coupling coefficients. The fourth

approach is to use an etching-depth-controlled conven-

tional 2 � 2 MMI [12].

In this paper, we propose a new approach for

achieving MMI couplers with new power splitting ratios

by interconnecting three or four MMI sections together.

The proposed devices can provide nineteen new power

splitting ratios. In addition, in this study silicon

waveguides are used for designing the proposed

devices. An analytical analysis is used to design the

device. Finally, the transfer matrix method (TMM) and

modified effective index method (MEIM) along with

the support of the 3D Beam Propagation Method (3D

BPM) are used to optimize the proposed devices.

2. General theory

Fig. 1 shows the structure of a single MMI coupler,

where a1 and a2 are the complex amplitudes at two

input ports and b1, b2 are complex amplitudes at two

output ports. The 2 � 2 MMI coupler has a width of

WMMI and a length of LMMI. The width of access

waveguides at input and output ports is assumed to be

Wa. We set s as the distance or separation between two

parallel access waveguides. In order to minimize the

size of the device, separation s is chosen to be small as

possible.

The idea to achieve an MMI coupler with variable

power splitting ratios is to cascade three to four 2 � 2

MMI couplers together. The 2 � 2 MMI couplers have

Fig. 1. Structure of a basic 2 � 2 coupler: (a) inplane view

the same separation s. For an MMI coupler, the beat

length is the most important parameter of an MMI

structure. The beat length Ln between the two lowest

guided modes is given by [1]:

Lp ¼4nsW

2e

3l0

(1)

where l0 is the operating wavelength, ns and nc are the

core refractive index and cladding refractive index,

respectively and We is the effective width of the

MMI coupler that can be expressed by:

We ¼ WMMI þp

l0

ns

nc

� �2s

n2s � n2

c

� ��1=2(2)

where s = 0 for transverse electric (TE) mode and s = 1

for transverse magnetic (TM) mode.

It is well-known that the complex optical field

amplitudes at output ports and input ports of the MMI

coupler are expressed by [7]:

b ¼ Ma (3)

where b = [b1, b2]T, a = [a1, a2]T and M is the transfer

matrix of the 2 � 2 MMI coupler.

In general, the width of the MMI coupler can be

written by WMMI = rs, where r is a constant factor

depending on the multimode interference mechanism

[13,14]. For a single MMI coupler with the rectangular

shape, there have been four coupling coefficients K

(cross-power coupling ratio) corresponding to general

interference and restricted interference mechanisms as

follows [15,16]:

Case A (MMI-A) (K = 0.5, r = 1.44): The transfer

matrix of the MMI coupler in this case is given by:

MA ¼1ffiffiffi2p e� jp=4 e jp=4

e jp=4 e� jp=4

� �(4)

of a 2 � 2 MMI coupler and (b) cross-sectional view.

C.-D. Truong, T.-T. Le / Photonics and Nanostructures – Fundamentals and Applications 11 (2013) 217–225 219

Case B (MMI-B) (K = 0.5, r = 3): The access

waveguides for MMI-B is located exactly at positions

�WMMI/6 from center-line of the multimode wave-

guide. The transfer matrix of the MMI coupler in this

case is given by:

MB ¼1ffiffiffi2p e jp=4 e� jp=4

e� jp=4 e jp=4

� �(5)

Case C (MMI-C) (K = 0.85, r = 2): The access

waveguides in this case is located exactly at positions

�WMMI/4 from the centerline of the multimode

waveguide. The transfer matrix of the MMI coupler

in this case is given by:

MC ¼1ffiffiffi2p

cosð� 3p

8Þe j3p=8 cosð�p

8Þe� jp=8

cosð�p

8Þe� jp=8 cosð� 3p

8Þe j3p=8

0B@

1CA

(6)

Case D (MMI-D) (K = 0.72, r = 2.5): The locations

of access waveguides in this case is at �0.25 s from the

center of the MMI region. The transfer matrix of the

MMI coupler is given by:

MD ¼2ffiffiffi5p

cosð� 3p

10Þe j3p=10 cosð� p

10Þe� j3p=10

cosð� p

10Þe� j3p=10 cosð� 3p

10Þe j3p=10

0B@

1CA(7)

Since four basic MMI sections MMI-A, MMI-B,

MMI-C and MMI-D have the same separation s, they

can be aligned and cascaded to achieve new MMI

structures. In this study, the MMI sections are actually

butt jointed without any interconnecting waveguides.

When MMI-A, MMI-B, MMI-C and MMI-D sections

Fig. 2. Schematic diagram of 2 � 2 MMI coupler based on cascaded 2 � 2

sections.

are cascaded as shown in Fig. 2(a) and (b), the overall

transfer matrix of the cascaded device can be obtained

by:

M ¼ M3M2M1 (8)

M ¼ M4M3M2M1 (9)

where Mi (i = 1, 2, 3, 4) are the transfer matrices of four

MMI sections MMI-A, MMI-B, MMI-C or MMI-D. By

changing properly the positions of the four type MMI

sections A, B, C and D in the configurations of AAD,

ADB, ADCD, BDAD, BDCD, CAD, CDA, CDBD,

CDC, DBD, DCA, DCB, DCD, DDA, DDAD, DDB,

DDBD, DDC, DDCD (Fig. 2), it is possible to achieve

nineteen new power splitting ratios as follows: 0.02,

0.04, 0.084, 0.11, 0.12, 0.16, 0.276, 0.37, 0.38, 0.46,

0.49, 0.54, 0.55, 0.65, 0.73, 0.734, 0.88, 0.91, 0.96.

3. Simulation results and discussion

It is well-known that the three dimensional finite-

difference time-domain (3D-FDTD) method is a

general method to solve Maxwell’s partial differential

equations numerically in the time-domain. Simulation

results for devices based on the silicon waveguide using

3D-FDTD method can achieve a very high accuracy.

However, due to the limitation of computer resource and

memory requirement it is difficult to apply the 3D-

FDTD method to the modeling of large devices on

silicon waveguides. Meanwhile, the BPM was shown to

be a quite suitable method that has sufficient accuracy

for simulating devices based on SOI channel wave-

guides [17,18]. In this paper, we use the BPM along

with the support of the modified effective index method

(MEIM) for designing proposed structure [13]. This

MMI sections: (a) three 2 � 2 MMI sections and (b) four 2 � 2 MMI


modified effective index method uses beat length Lp as

the invariant. The aim of the MEIM is to find a matching

value of the cladding index for the effective index

method that forces the beat length Lp in the equivalent

2D model (Lp(2D)) to be equal to the beat length in an

accurate 3D model (Lp(3D)). By varying the value of

the cladding refractive index in the 2D model, it is

possible to find an effective cladding index that

produces the same beat length as the accurate 3D

model. The 3D-BPM [19] can be used to compute the

beat length of the 3D waveguide to high accuracy. The

advantage of using the MEIM is that a full 3D solution is

only required once for establishing the matching

cladding index. Then the IEIM can be used to compute

the propagating fields in the MMI structure quickly,

with little further computational effort. Using this

modified effective index method, we found that the

refractive index of the TE fundamental mode is 2.82 at

l0 = 1550 nm in the 2D simulation. In order to match

the low-order modes in the 2D simulation, the

equivalent effective refractive index of the lateral

cladding is calculated to be 2.19 [13].

The waveguide cross-section used in our design is

shown in Fig. 1(b). The core thickness is hco = 220 nm

Fig. 3. BPM simulation results of four basic MMI couplers (a) power coupl

lengths.

and the width of the access waveguides is Wa = 500 nm.

In addition, widening the access waveguides improves

the performance of devices. By using the 3D-BPM, it is

found that the access waveguides can be widened via a

taper having a length of Ltp = 5 mm in order to achieve

the lowest loss.

In order to minimize the size of the device, the

separation s is chosen small as possible. This means that

the width of the MMI coupler is large enough to limit

crosstalk between two adjacent parallel waveguides. In

our design, we found that the minimal separation is

chosen to be s = 2.5 mm. First, an analytical method [1]

is used to find the length of the MMI-A, MMI-B, MMI-

C and MMI-D. Then the BPM is used to find optimal

lengths of these MMI couplers.

For the case MMI-A, the width of the MMI-A

coupler is found to be AA = 3.6 mm and the MMI length

LA is found to be LA = (3/2)LpA = 54.72 mm. For the

MMI-B, the width and length of the MMI-B are

WB = 7.5 mm and LB = (1/2)LpB = 73.38 mm, respec-

tively. For the MMI-C, we have the width and length of

WC = 5 mm and LC = (3/4)LpC = 50.68 mm and for the

MMI-D, we have WD = 6.25 mm, MMI length LD = (3/

5)LpD = 62.03 mm. The BPM simulation results are

ing ratios at different lengths and (b) BPM simulations at the optimal


Fig. 4. BPM simulation results for some cascaded MMIs in order to achieve new power coupling ratios.

shown in Fig. 3. Fig. 3(a) shows the power splitting ratio

of the MMI-A, MMI-B, MMI-C and MMI-D at

different lengths of the MMI couplers. The simulation

results show that the optimal lengths of MMI-A, MMI-

B, MMI-C and MMI-D are calculated to be 54.5, 74,

51.5 and 62 mm, respectively. Fig. 3(b) shows the field

propagation within the MMI-A, MMI-B, MMI-C and

MMI-C couplers at the optimal lengths. The insertion

loss at the optimal lengths in four cases calculated to be

�0.4 dB by using the BPM.

By cascading four type MMI sections MMI-A,

MMI-B, MMI-C and MMI-D, it is possible to achieve

nineteen new values of the power coupling ratios k.

Without loss of generality, in this study, we have carried

out the BPM simulations for ten cascaded cases,

including CDA, DDA, AAD, ADB, DDC, DCA, DCB,

BDCD, DDBD, and CDAD. The BPM simulations for

these cases are shown in Fig. 4. Since the cascaded

devices are butt jointed without connecting wave-

guides, the total insertion losses of the cascaded MMI

couplers are much less than the sum of the total

insertion losses of the MMI sections. The simulation

results show that the insertion loss for three cascaded

MMI couplers is about �0.42 dB and for four cascaded

MMI coupler is about �0.83 dB at the central

wavelength of 1550 nm.

Now we investigate the wavelength sensitivity of the

proposed devices. When the wavelength deviates from

the operating wavelength of 1550 nm, the total

transmittance decreases because the beat lengths of

the four-type MMI couplers are inversely proportional

to the wavelength. Without loss of generality, the power

coupling ratios k and total transmittance at different

wavelengths for five cascaded cases AAD, ADB, CDA,

CDAD and DDC are shown in Fig. 5. The bandwidth at

85% total transmittance (i.e. �1 dB from the maximum)

is approximately inversely proportional to the device

length. The dependences of the k values on the

wavelength deviation are plotted in Fig. 5(a). Within

the 1 dB bandwidth of each of the the cascaded MMI

devices, the variation of the k value is found to be less

than 0.02.

Next, fabrication tolerances of the proposed devices

are investigated. Without loss of generality, we only

investigate two types of cascaded MMI structures

including three MMI couplers cascaded AAD and four

MMI couplers cascaded CDAD. Then we change the

length of one MMI coupler in the cascaded MMI


Fig. 5. (a) The power coupling ratios and (b) total normalized output power of the some cascaded MMI structures at different wavelengths.

structure to obtain the output power. Fig. 6 shows the

BPM simulation results of the output powers of the two

MMI structures at different MMI lengths. Here we

change the length of the type MMI-D coupler. It is

Fig. 6. The power coupling ratio of the three and four cascaded

obvious from the simulations that fabrication tolerances

of both MMI cascaded couplers are very large. The

fabrication tolerance of the MMI lengths for three

cascaded MMI structure is �167 nm for a power

MMI coupler with different lengths of the MMI-D coupler.


Fig. 7. The power coupling ratio of the three and four cascaded MMI coupler with different widths of the MMI-D coupler.

coupling ratio variation of 0.002. The fabrication

tolerance for four cascaded MMI structures is

�167 nm for a power coupling ratio variation of less

than 0.01. These fabrication tolerances are easily

achieved by using the e-beam or 193 nm deep UV

lithography [20,21]. Simularly, the fabrication tolerance

of the MMI cascaded structure is also very large. Fig. 7

shows the output powers of three cascaded MMI and

four cascaded MMI couplers at different width of the

MMI-D coupler.

The locations of the input and output access

waveguides are important to the performance of the

devices. By using the BPM, the normalized output

powers for the three and four cascaded MMI couplers

at different positions of the access waveguide

are shown in Fig. 8. The BPM simulations show

Fig. 8. The power coupling ratio of the three and four cascaded M

that the power coupling ratio is unchanged within a

change of �20 nm in the position of the access

waveguide.

For some practical applications such as MMI based

ring resonators, MMI based MZI structure, the phases

of the output signals are particularly important.

Therefore, study of the phase errors due to fabrication

tolerance is necessary. Fig. 9 shows the output phases

of the output signals for three cascaded MMI and four

cascaded MMI couplers at different lengths, widths

and positions of the access waveguides. The BPM

simulation results show that within a change of

�20 nm in the length or width, the change in phase is

only about 0.28. A fabrication tolerance of �20 nm

can be obtained by using the existing CMOS

fabrication technology.

MI coupler at different positions of the access waveguide.


Fig. 9. The phases of the output signals at different lengths, widths and positions of the access waveguides.

4. Conclusion

In summary, a new method for achieving MMI

structures with new power splitting ratios have been

presented in this study. Nineteen new power splitting

ratios can be achieved by cascading three or four short

MMI sections. These cascaded MMI couplers have low

insertion losses and simple geometries requiring no

angled or bent waveguides. The design of the proposed

devices has been verified and optimized using analytical

and numerical simulation methods. The cascaded MMI

structures can be useful for all-optical computing

systems and photonic integrated circuits such as power

taps, high-Q ring resonators, clock distribution circuits,

ladder-structure optical filters, and loop-mirror partial

reflectors.

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Power splitting ratio couplers based on MMI structures with high bandwidth and large tolerance using silicon waveguides