Linear and Triangle Order of NX3 Optical Directional Couplers: Variation Coupling Coefficient

Dedi Irawan1, Saktioto1,2, Jalil Ali 1 1Advanced Photonics Science Institute, Faculty of Science, Universiti Teknologi Malaysia

Johor, Malaysia. dedi.dawan@yahoo.com 2Physics Dept, Math and Sciences Faculty, University of Riau, Pekanbaru, Indonesia

Abstract

Fabrication of single-mode fiber coupler has widely expands. However, directional fiber coupler

geometry always affects the power propagation either two or three ports. This paper describes

power launching by NX3 single mode fiber, examined using a matrix transfer for linear and

triangle order, calculated from the eigenvalue and the eigenvector. This eigenvalue is referred to

the coupling coefficient, where it can be expressed as an effective power transmitted to another

fiber. The ratio of coupling coefficient between adjacent axial fibers varies, which can be

achieved by adjusting separation of fiber and refractive index of core and cladding. A calculation

has been shown in 3D that both power transmission and phase are affected by not only the

geometry order, but also the variation of coupling coefficient, assuming the propagation

constants, cross section, and separation of coupling length fiber axis are held constant. This

calculation can be applied for any sources of wavelength and junction.

Keywords: Optical directional coupler, power transfer, coupling coefficient, matrix transfer.

1. Introduction

The great utilization of single mode fiber coupler has led to the extensive research and

development. The development and fabrication of single-mode fiber coupler has been widely

expanding for both tunable filter and optical waveguide switch in telecommunication and device

sensor. Optical directional couplers are commonly used for several guided wave devices and for

optical switching networks [1]. Directional couplers may refer as devices that used to combine

and split optical signals. Significant interaction between three coupled waveguides will occur and

thus lead to a significant power exchange between the waveguides [2]. For waveguides that are in

parallel condition and propagating with the same phase velocities of guided modes causes the

losses and the fraction of power or total power to be transferred from one waveguide to the other

waveguides [3]. The power transfer is investigated by constructing a matrix transfer model. Then,

Photonic Fiber and Crystal Devices: Advances in Materials and Innovationsin Device Applications IV, edited by Shizhuo Yin, Ruyan Guo, Proc. of SPIE Vol. 7781

77810J · © 2010 SPIE · CCC code: 0277-786X/10/$18 · doi: 10.1117/12.862573

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the coupling coefficient varied as a function of position. Otherwise, propagation constants, and

cross section of fibers are hold to be constant. The input power is only at one input port source

and focusing on the power splitting phenomenon as the function of phase variation along the

coupling length axis [4]. Based on the placement, linear and triangle order of NX3 fiber couplers

can be used as power splitters and routers respectively. N represents power input ports and 3 is

for output ports. For the extent of application purpose, it can be developed as optical switch. An

optical switching device is built by fabricating coupler on electro-optic material and applying a

voltage into the active region [5].

2. Optical Directional Couplers; Matrix transfer model

The physical properties of the waveguides are explained using matrix methods to describe the

optical directional couplers. Modes for a coupled structure consist of two or many waveguides

[6]. Consider the fibers coupler consists of three waveguides that has three input ports and three

output ports. 1 2 3(0), (0) and (0)A A A are the amplitudes of input power while

1 2 3( ), ( ) and ( )A z A z A z are the amplitudes of output power of waveguide 1, W1, waveguide 2, W2,

and waveguide 3 W3 respectively, as shown in Figure 1.

Figure 1: (a) Three cylindrical 3X3 directional coupler, (b) coupling is linear order placement, (c) coupling in triangle order placement.

The modes of these three isolated waveguides can be formalized as follows [7].

11 1 12 2

( ) ( ) ( )dA z j A z j A zdz

β κ= − − (1a)

(b) Input power

WG WG2 WG3 � WG2

WG1

WG3

(c) Input power

�

(a)

WG2

WG3

WG1

Input power

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22 2 21 1 23 3

( ) ( ) ( ) ( )dA z j A z j A z j A zdz

β κ κ= − − − (1b)

33 3 32 2

( ) ( ) ( )dA z j A z j A zdz

β κ= − − (1c)

Where ( )A z represents the coefficient of mode on z. The value of 12 13 21 2 23 31 32, , , , , ,κ κ κ κ κ κ κ are

the coupling constants between waveguide 1, waveguide 2 and waveguide 3, and 1 2 3, ,β β β are

their propagation constants. The coupled mode equation for the 3X3 coupler can be modified in

term of matrix [8].

1 1 12 13 1

2 21 2 23 2

3 31 32 3 3

( ) ( )( ) ( )( ) ( )

A z k k A zd A z j k k k A zdz

A z k k A z

β

β

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥= −⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

(2)

Consider a directional coupler has three identical and equally spaced

waveguides, 1 2 3β β β β= = = , and 21 23,κ κ κ= = . Since the interaction between the three

waveguides occurs from the waveguide 1 to the nearest waveguide, waveguide 2, thus the

coupling coefficient of waveguide 1 to waveguide 3, while 13 31and κ κ can be neglected. Based

on eigenvectors of matrix 3X3, [ ]V can be constructed as,

[ ]1 1 2

1 2 2 02

1 1 2

V

⎡ ⎤−⎢ ⎥

= ⎢ ⎥⎢ ⎥

− −⎢ ⎥⎣ ⎦

(3)

and by defining[ ] [ ] [ ]1U V A−= , Equation 1 can be transformed to matrix differential equation:

1 1

2 2

3 3

2 0 0( ) ( )( ) 0 2 0 ( )( ) 0 0 ( )

kU z U zd U z j k U zdz

U z U z

β

ββ

⎡ ⎤+⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥= − −⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎢ ⎥⎣ ⎦

(4)

Substituting Equation 1 to [ ] [ ][ ]A V U= will yield matrix transfer for 3X3 optical directional

coupler as follows.

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( )( )( )

( ) ( )

( ) ( )

( )( )( )

1 1

2 2

3 3

1 1cos 2 1 sin 2 cos 2 12 22 0

sin 2 cos 2 sin 2 02 2 0

1 1cos 2 1 sin 2 cos 2 12 22

j z

jz z zA z A

j jA z e z z z AA z A

jz z z

β

κ κ κ

κ κ κ

κ κ κ

−

⎡ ⎤+ − −⎢ ⎥⎢ ⎥⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥= − −⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎢ ⎥− − +⎢ ⎥⎣ ⎦

(5)

The waveguides are assumed having constant cross section, otherwise coupling changes with the

position along the coupling length and due to the separation of fiber axis and refractive indices of

core and cladding [9].

3. Result and Discussion

Since power is fed in one waveguide, the power transfers between three waveguides can be

determined from matrix transform which is given by Equation (5) in two conditions. The first

condition, power is launched in to waveguide 1 or waveguide 3 as a linear order. By

substituting ( )1 2 3(0) 1, and (0) (0) 0A A A= = = in Equation (5) as shown in Figure 4 will yield as

follow.

( ) ( )11 cos 2 12

j zA z z e βκ −= − (6a)

( ) ( )21 sin 22

j zA z j z e βκ −= − (6b)

( ) ( )31 cos 2 12

j zA z z e βκ −= + (6c)

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Figure 4. Power transfer between the three waveguides with variable coupling coefficient

The second condition, power input is fed in the center of the three waveguides as a triangle order.

Similarly by substituting ( )2 1 3(0) 1, and (0) (0) 0A A A= = = in Equation (5) will yield as follow:

( ) ( ) ( )1 31 sin 22

j zA z A z j z e βκ −= = − (7a)

( ) ( )2 cos 2 j zA z z e βκ −= (7b)

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Figure 5. Power transfer between the three waveguides with variable coupling coefficient

The input power is transferred completely in to the two others waveguides both for matrix of

(100) and (010) when the power of initial waveguide is minimum, then it can be used as a

splitters or routers. Increasing the coupling coefficient from 1 10.1 mm to 1.5 mm− − gives

significant changes in phase of propagation.

4. Conclusions

Calculation of power splitting in NX3 single mode fiber coupling using matrix transform have

shown that the power launching in one waveguide transferred into the others. In NX3 single mode

fiber coupler, power splitting depends on order placement of waveguide position. Increasing of

coupling coefficient will change phase of power transferred in to another waveguides. The higher

coupling coefficient, the shorter coupling length. When power input is fed into linear order, it will

be transferred gradually to two other waveguides and launched completely when waveguide 1

and 2 are minimums. If power input is fed into triangle order, the input power will be divided and

distributed into two other waveguide with the same values.

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5. Acknowledgement

We would like to thank the Institute of Advanced Photonic Science, Faculty of Science,

Universiti Teknologi Malaysia (UTM), and Physics Dept. University of Riau for generous

support in this research.

6. References

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[5] H. Kogelnik and R. V. Schmidt, “Switched directional couplers with alternating βΔ ,”

IEEE Journal of Quantum Electronics, VOL. QE-12, no. 7, 396-401 (July 1976).

[6] T. Wongcharoen, B.M.A Rahman, K.T.V. Grattan, “Finite Element Analysis of Optical

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[8] Chin-Lin Chen, “Foundations For Guided-Wave Optics, Perdue University, West

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[9] Saktioto, Jalil Ali, Mohamed Fadhali, “Theoretical and empirical comparison of

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