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A Multimode Inte

S. K. Raghuwanshi Department of Electronics Engine

Indian School of Mines Dhanbad, India

[email protected]

Abstract—The finite difference beam propagatis one of the most powerful technique to invnonlinear lightwave propagation phenomena waveguides such as curvilinear directional coand combining waveguides, S-shaped bendtapered waveguides. BPM is also quite importaof ultra short light pulse propagation in optinumerous algorithms for BPM have beesuccessfully applied to a very wide class ofwaveguide structures. In this paper, we hMultimode interference (MMI) star coPropagation Method (BPM) having S-bend wcompared the MMI star coupler with the simphas been found that power distribution in all fwaveguide is almost identical, which reflects th

Keywords- MMI star coupler; Linear wwaveguide; Beam Propagation Method.

I. INTRODUCTION Curvilinear directional couplers, branchin

waveguides, S-shaped bend waveguidewaveguides are indispensable components iintegrated optical circuits. In practical dirhowever, light coupling in the S-shaped regions in the front and rear of parallel wavtaken into account so as to evaluate characteristics precisely [1]. For axially varthe FEM stationary mode analysis should be the paraxial wave approximation and Galerk[3]. A lot of the waveguides used in actual devices, however, have non-uniform structurtapers, and crosses in the propagation directipaper, we discuss the transparent boundary cbeam propagation methods (BPMs) that havfor the analysis of such non-uniform structurdescribe the advantages of MMI star coupleMMI coupler. Section (II) explains aboutransparent boundary condition (TBC). Sectabout the results and discussion.

II. THEORY This paper describes the how to crea

coupler. The MMI star coupler further devMMI coupler [10]. It consists of one input wide MMI coupler, and four S-bend output w

erference Star Coupler Waveguide

eering Santos

Department of EleIndian Sch

Dhanbsantoshrus

ting method (BPM) vestigate linear and

in axially varying ouplers, branching

d waveguides, and ant for the analysis ical fibers. To date en proposed and f dielectric optical have analyzed the oupler by Beam aveguide. We have

ple MMI coupler. It four output S-bend

he desired result.

waveguide; S-bend

ng and combining es, and tapered in constructing an rectional couplers,

bend waveguide veguides should be

the propagation rying waveguides, modified by using

kin’s method [2] -optical waveguide res such as bends, ion [4] -[6]. In this condition (TBC) of ve been developed res [7] -[8]. Also, er over the simple ut the theory of tion (III), explains

ate an MMI star velops the simple thin waveguide, a waveguides. Here,

thin waveguide are the single m

propagates whether thick wavemultimode, where 8 even and 7An infinitely wide area would there would thus be no reflectisuch an area cannot be assumelimited analysis window has tradiated waves are reflected at core area, where they interact winteraction disturbs the propagathe calculation accuracy. To rewindows, we need some artifithis paper, we discuss the condition (TBC) that is norma[17].

Figure 1. Layout o

The TBC was developed efficiently suppress the reflectito implement into computer prbe applied to 2D problems herapplied to 3D problems. As window contains nodes at nodes at and analysis window. Following Hincorporate the influences of th

into the nodes at and

A. Left-hand Boundary Consider the left-hand boun

the influence of the hypothetianalysis window) into the nod

with S-Bend

sh Kumar ectronics Engineering hool of Mines, bad, India [email protected]

and S-bend waveguide mode, where only one even mode

guide behaves as 7 odd mode propagates [9]-[10]. have no analysis boundary and

ions at the boundaries. Because ed in actual design, however, a to be used. In actual structures, the boundaries and return to the

with the propagating fields. This ating fields and greatly degrades educe the reflections on analysis icial boundary conditions so, In

simple transparent boundary ally used in the FD-BPM [11] -

of MMI star coupler

by Hadley [5] as a way to ons at boundaries, and it is easy

rograms. Although the TBC will re, it can easily be extended and shown in Fig. 2, the analysis

to . The hypothetical are assumed to be outside the

Hadley’s line of thinking [5], we he nodes at and

.

ndary in Fig. 2. We incorporate ical node at (outside the de at (inside the analysis


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window). The wave function of the left-traveling wave with the x-directed wave number is expressed as

We denote the x-coordinates and the fields of the nodes at as and as and we assume that

The fields and are inside the analysis window, and is a hypothetical field whose influence should be

incorporated into the field inside the analysis window.

Figure 2. Nodes and are outside the analysis area.

Dividing eq. (4) by eq. (3) and eq. (3) by eq. (2), we get,

where, . Substituting the ratio of to ,

which is equal to , into eq. (6), we get

Equation (8) can also be derived by substituting the x-directed wave number,

which is obtained from eq. (5), into

which is obtained from eq. (6). It should be noted that since the wave travels leftward, the real part of the x-directed wave number , , should be negative. When it is positive, which implies reflection at the left-hand boundary, the sign should be changed from plus to minus.

B. Right-hand Boundary Consider the right-hand boundary in Fig. 2. We incorporate

the influence of the hypothetical node at (outside the analysis window) into the node at (inside the analysis window). The wave function of the right-traveling wave with the x-directed wave number is expressed as

We denote the -coordinates and the fields of the nodes as and ,

and we assume that

where, and are the fields inside the analysis window and is hypothetical field whose influence should be incorporated into the field inside the analysis window. Dividing eq. (13) by (12) and eq. (14) by (13), we get,

where, . Substituting the ratio of to ,

which is equal to , into eq. (16), we get

Equation (18) can also be derived by substituting the x-directed wave number,

which is obtained from eq. (15), into

Similar to what we saw in the case of the left-hand boundary, since the wave travels rightward, the real part of the x-directed wave number should be negative. When it is positive, which implies reflection occurs at the right-hand boundary, the sign should be changed from plus to minus.

III. RESULTS AND DISCUSSIONS In this section, we use the Beam propagation method based

on the above formulation in section (II) to simulate the pulse propagation through MMI starcoupler as shown in Fig. 1. Here guide refractive index is , cladding refractive index , and operating wavelength of light

are simulation parameters taken for the MMI star coupler. Here, the width and length of smaller section are

, and respectively and the width and length of wide section are , and respectively. Since there are four output channels in the MMI star coupler, it is necessary to find where the simple MMI coupler produces four intensity maxima. As seen from the simulation that, this happens where the second waveguide in the MMI coupler is about long. In this simulation, we are using the TE polarization and Transparent Boundary Condition (TBC). Figure 3 shows the plot of region of interest vs. length of wide waveguide. It is apparent from this figure that the centers of the two points are at


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approximately on the axis. To focon the centers of the points, change the lenwaveguide as . Figure 4 sregion of interest vs. revised length of Analogically to the procedures above, find for the upper and outermost intensity maxithat it’s approximately ,

from top to bottom of respectively. The output waveguide must bconnects the start point of the second wintensity maxima. Then we have taken the fowaveguide on top of the second waveguide

and at a different angle accordingintensity maximum. Figure 5 shows the 2-Dfour S-bend output waveguide. Figure 6 showoptical field propagation, output field pro

, output field profile at fixingof optical field propagation for the case

respectively. Figur Vs. width . It is apparent

Fig. 7 that electric field are maxima in the twapparent that due to wide angle, intensitFigure 8 shows the 3-D-plot of refractive

. Figure 9 shows the powewith the fundamental mode of four S-bend wcan see from this figure, the resulting pathsThis means that the power distribution inwaveguides is divided into two parts accordiinclination and this output is almost identicthe desired result.

Figure 3. Plot of region of interest Vs. length of w

Figure 4. Plot of region of interest Vs. revised length

cus more precisely ngth of the second shows the plot of wide waveguide. a vertical position ma. It is apparent

, and wide waveguide be on a line that waveguide to the our S-bend output e from to g to the outermost D simulation with ws the 2-D Plot of file at fixing

and 3-D plot of (a)-(d)

re 7 shows the t from Fig. 6 and

wo groups. It is also ty are decreasing. e index profile at er overlap integral waveguide. As we s are very similar. n all four output ing to the angle of cal, which reflects

wide waveguide

of wide waveguide

Figure 5. 2-D simulation with

Figure 6. 2-D Plot of optical field propa, output field profile at fixing

propagation for the case of (a) -(d)

Figure 7.

Figure 8. 3-D-plot of refractive

h four S-bend output waveguide

agation, output field profile at fixingg and 3-D plot of optical field

respectively

Vs. Width

e index profile at


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Figure 9. Power overlap Integral with fundamental mode

IV. CONCLUSION We have analyzed the MMI Star coupler having S-bend

output waveguide. The power loss found to be more for wide angle S-bend output waveguide. The loss occurs in the output waveguide with less angle and wide angle are and respectively. This is due to splitting of light into many parts and it may no longer remains guided. The main advantages of star coupler are that light is guided at the output and there is a less substantial loss of the MMI star coupler compare to the existing simple MMI coupler. The power distribution over the length are nonuniform due to the odd/even mode distribution.

REFERENCES [1] J. J. Wu, “A multimode interference coupler with exponentially tapered

waveguide,” Progress In Electromagnetics Research C, vol. 1, pp. 113–122, 2008.

[2] Lucas B. Soldano and Erik C. M. Pennings, “Optical Multi-Mode Interference Devices Based on Self-Imaging: Principles and Applications,” Journal of lightwave technology, vol. 13, no. 4, april 1995.

[3] M. D. Feit and J. A. Freck, Jr., “ Light propagation in graded-index optical fibers,” Appl. Opt., vol. 17, pp. 3990-3998, 1978.

[4] J. Yamauchi, J. Shibayama, and H. Nakano, “Beam Propagation method using Pade approximant operators,” Trans. IEICE Jpn., vol. J77-C-I, pp. 490-494, 1994.

[5] G.. R. Hadley, “Wide-angle beam propagation using Pade approximant operators,” Opt. Lett., vol. 17, pp. 1426-1428, 1992.

[6] Hongzhen Wei, Jinzhong Yu, Zhongli Liu, Xiaofeng Zhang, Wei Shi, and Changshui Fang, “Fabrication of 4×4 tapered MMI coupler with large cross section,” IEEE Photonics technology letters, vol. 13, no. 5, May 2001.

[7] G. R. Hadley, “Transparent boundary condition for beam propagation method,” Opt. Lett., vol. 16, pp. 624-626, 1992.

[8] G.. R. Hadley, “ A multistep method for wide angle beam propagation,” Integrated Photon. Res., vol. 17, pp. ITu 15-1, pp. 387-391, 1993.

[9] S. K. Raghuwanshi, “Comparative study of asymmetric versus symmetric planar slab dielectric optical waveguides,” Indian Journal of Physics, vol. 84, no. 7, pp. 831-846, 2010.

[10] Santosh Kumar, S. K. Raghuwanshi, “Analysis of step discontinuities in rectangular MMI coupler by beam propagation method,” to be presented to IEEE International Conference on Optical Engineering 2012, 26-28 July, 2012, VTU, Belgaum, India.

[11] N. Talebi and M. Shahabadi, “Application of generalized multipole technique to the analysis of discontinuities in substrate integrated waveguides,” Progress In Electromagnetics Research, vol. 69, pp. 227–235, 2007.

[12] S. K. Raghuwanshi, Santosh Kumar, “Step discontinuity analysis in an asymmetric single mode thin planar slab taper optical waveguide” i-manager’s Journal on Electronics Engineering (JELE), vol. 2, no. 2, pp. 43-48, Dec 2011-Feb 2012.

[13] N. S. Lagali, M. R. Paiam, R. I. MacDonald, K. Worhoff, and A. Driessen, “ Analysis of generalized Mach-Zehnder interferometers for variable ratio power splitting and optimized switching,” J. Lightwave Technol., vol. 17, pp. 2542-2550, Dec. 1999.

[14] M. Koshiba and Y. Tsuji, “A wide-angle finite element beam propagation method,” IEEE Photon. Technol. Lett., vol.8, pp. 1208-1206, 1996.

[15] P. L. Liu and B.J. Li, “Study of form birefringence in waveguide devices using the semivectorial beam propagation method,” IEEE Photon. Technol. Lett., vol. 3, pp. 913-915, 1991.

[16] Y. Chung and N. Dagli, “Assessment of finite difference beam propagation,” IEEE J. Quantum Electron, vol. 26, pp. 1335-1339, 1990.

[17] L. Thylen, “The beam propagation method: An analysis of its applicability,” Opt. Quantum Electron, vol. 15, pp. 433-439, 1983.

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