## Abstract

We show that it is possible to obtain 2×2 waveguide couplers with arbitrary power splitting ratios by interconnecting a pair of unequal-width waveguides as the phase-tuning section into the middle of two short MMI sections. These couplers have simple geometry and low loss. They offer valuable new possibilities for designing waveguide-based photonic integrated circuits.

©2008 Optical Society of America

## 1. Introduction

Multi-mode interference (MMI) couplers of unequal power splitting are attractive for photonic integration circuits (PICs) applications such as power taps, high-Q ring resonators [1–3], ladder-structure optical filters [4], and loop-mirror partial reflectors [5]. A coupler with freely chosen power splitting ratio is also valuable in Mach-Zehnder interferometer (MZI) when losses and gain are asymmetrically distributed between both different-length arms [6]. However, the conventional constant-width MMI design for 2×2 couplers is known to be capable of providing only seven different cross-coupling ratios, i.e. power splitting ratio K’s (defined as the cross output power divided by the sum of bar and cross output powers), of 1, 0.85, 0.72, 0.5, 0.28, 0.15, and 0 [7, 8]. Therefore, a reasonable way for obtaining arbitrary power splitting ratio is getting more significant. In principle, the way to obtain free choice of K-value is to introduce a phase-difference into the devices [9]. Although, electrical bias and current-injection have been investigated to induce a small variation of refractive index in semiconductor optical amplifier (SOA)-based waveguides [10]. In practical, it is not only difficult to partially control refractive index changes without electric-isolation but also most of the strong refractive index changes are usually accompanied by considerable absorption and waveguide loss. Several other approaches have been investigated to obtain arbitrary power splitting ratios by using a bent MMI coupler [11], a butterfly-like MMI with a long cladding-gap [12] or taper-width [6], and an etching-depth-controlled conventional 2×2 MMI [13]. Considering the mask resolution and process variability, the final K-value obtained after fabrication is more susceptible to error due to the very small bend angle involved with the very high angular sensitivity of the K-value and unacceptable process deviations in waveguide width, depth and gap. In our previous report [14], we have shown that it is possible to realize new K values of 0.07, 0.64, 0.80, and 0.93 by immediately cascading two short MMI sections. In this paper, we further realize freely chosen power splitting ratio by adding a pair of unequal-width waveguides as a phase-shifter between two short MMI sections. The devices have low loss and do not involve angled or curved geometries.

## 2. Design of 2×2 couplers with arbitrary power splitting ratio

The idea to achieve a coupler with freely chosen power splitting ratio is based on two short MMI sections interconnected by a phase-shifter in the middle as illustrated in the schematic diagram of Fig. 1(a). Different to a phase-shifter formed by a pair of unequal-length waveguides as usually used in MZI, this phase-shifter is of the same length L_{PS} but of unequal waveguide widths (*w*+δ_{a} and *w*+δ_{b}). Waveguides “*a*” and “*b*” are located at the same (bar)- and opposite (cross)-side to the input waveguide, respectively. The MMI sections to be discussed in this paper are all 2×2 waveguide couplers. For the MMI sections, we set the input/output waveguide pairs of all the devices to have the same center-to-center separation *s*. This distance *s* will serve as the common unit of length scaling for the MMI sections. In order to minimize the size of the couplers, *s* is generally chosen to be as short as the lithographic process would permit. The beat length (*L _{π}*) between the two lowest guided modes of the same polarization in the MMI waveguide is given according to the standard mode propagation analysis (MPA) by [1, 7]

where *n _{s}* is the effective refractive index of the slab waveguide from which the MMI waveguide of effective width

*W*is formed,

_{e}*λ*is the vacuum wavelength, and

_{0}*r*≡

*W*/

_{e}*s*. Equation (1) holds for both TE and TM polarizations [1]. In addition to analyses by MPA, we also carry out numerical simulations by beam propagation method (BPM) to obtain more realistic assessments. In this studies, the epitaxial slab waveguide on n

^{+}-InP substrate is assumed to have a 1.79 µm-thick InAlAs upper p-type cladding, a 0.14 µm-thick core layer containing triple quantum wells with the absorption edge at wavelength =1.48 µm, and a 0.14 µm-thick InAlAs lower n-type cladding. The basic layer structure in the waveguide ridge is shown schematically in Fig. 1(b). The side walls of the MMI section are assumed to be etched down to the InP substrate to keep the number of confined modes large, to reduce phase error of the confined modes, and to suppress polarization sensitivity. The

*n*of the TE fundamental slab mode is 3.2147 at λ

_{s}_{0}=1.55 µm (the base wavelength in this study). In order to match the low-order modes in 2-D simulation to those in 3-D simulation, the equivalent effective refractive index of the lateral cladding is set to 2.3. Considering the resolution and limitation of our photolithographic process, the width of the access waveguides

*w*is assumed to be 2.2 µm for only existing fundamental mode inside the input/output access waveguides and their separation is set to be

*s*=5 µm. The following four relatively short MMI sections with K≥0.5 have been considered as the building blocks.

For MMI-A (K=0.5, *r*=1.44), we have *W _{e}*=

*s*+

*w*=1.44s=7.2 µm. For the given values of

*s*and

*w*, this is the minimum practical effective width for a low loss MMI [1]. The device length

*L*is (3/2)

_{A}*L*=3.1104

_{πA}*As*

^{2}=215 µm according to MPA. For MMI-B (K=0.5,

*r*=3), we have

*W*=3

_{e}*s*=15 µm and

*L*=(1/2)

_{B}*L*=4.5

_{πB}*As*

^{2}=311 µm. The access waveguides for MMI-B are located exactly at ±

*W*/6 from the centerline of the multimode waveguide. For MMI-C (K=0.85,

_{e}*r*=2), we have

*W*=2

_{e}*s*=10 µm and

*L*=(3/4)

_{C}*L*=3

_{πC}*As*

^{2}=207 µm. The access waveguides are located exactly at ±

*W*/4 from the centerline of the MMI-C waveguide. For MMI-D (K=0.72,

_{e}*r*=2.5), we have

*W*=2.5

_{e}*s*=12.5 µm,

*L*=(3/5)

_{D}*L*=3.75

_{πD}*As*

^{2}=259 µm. The geometric centers of the access-waveguide pairs at two ends are offset in opposite directions by ±0.25

*s*from the center of the MMI-D waveguide.

The cross- and bar-transfer functions of these four basic MMIs are deduced from Bachmann et al. [7] and summarized in Table I. In this table, *β _{U}* is the phase constant for the fundamental mode in the MMI section U, and

*L*is the length of that MMI section. The ± signs in the subscripts are significant only for MMI-D in which the access waveguides are not symmetrically placed. The asymmetry leads to two cross optical paths of unequal lengths. According to the 3-D BPM simulations [15], the total insertion losses of these four basic MMIs are in the range of 4 to 5%. Since all four basic MMI sections are 2×2 couplers with the same

_{U}*s*value, they can be simply aligned and cascaded. Using 3-D EigenMode Expansion (EME) method [16] to check the mode coupling efficiency for those interconnecting junctions as indicated in Fig. 1(a) by circular dot-line, the results show that only about 1% mode-mismatch loss is presented when one phase-shifter waveguide width, either “a” or “b”, is extended/reduced to 2.4/2.0 µm from 2.2 µm. Thus the phase-shifter waveguides can be considered as lossless, and their transfer functions are expressed as

where *β _{a}* and

*β*are the propagation constant of fundamental mode for waveguide “

_{b}*a*” and “

*b*”, respectively; and L

_{PS}is the length of phase-shifting section. The transfer matrix of this cascaded device can be obtained by the following matrix multiplication.

where *HX*
^{U(PS)V}
_{±} and *HB*
^{U(PS)V}
_{±} are respectively the transfer functions to the cross output port and the bar output port for the cascaded device. The K-value is given by

**When L _{PS}**=

**0**, the MMI sections are actually butt-jointed without any interconnecting waveguides. We have examined the results of all possible combinations using Eq. (3) and verified their consistency with the results of 3-D BPM simulations. The resultant K-values are: K

^{AB}=0, K

^{AC}=0.15 (0.146), K

^{AD}=0.07 (0.075), K

^{BC}=0.85 (0.854), K

^{BD}=0.93 (0.925), K

^{CD}=0.64 (0.643), and K

^{DD}=0.80 (0.80). The alternative values given between parentheses are the MPA values. Among these, 0.07, 0.93, 0.64, and 0.80 are new K values that are so far not known to be possible obtained by using a constant-width MMI [7, 8]. Their detailed design concept, geometry and simulated 2-D field map combining with wavelength sensitivity have been reported in [14]. Using this concept, another shortened coupler of K=0.28 was also obtained by a half of MMI-D (MMI-hD) cascading another MMI of

*W*=1.94

_{e}*s*(MMI-E); it had been presented with a detailed discussion in [17]. In order to show the benefit from our cascaded MMIs in comparison with conventional constant-width ones, a summarized map is shown in Fig. 2 to illustrate their necessary device length for each obtainable K-value by the unit length of

*As*according to MPA.

^{2}**When L _{PS}≠0** and the waveguide width in the phase-tuning section is the same as those of input/output access waveguides,

*i.e.*δ

_{a}=δ

_{b}=0, seven 2×2 cascaded MMI devices to be applied as the starting-point to achieve power splitter with freely chosen K-values are shown by simulated 2-D field maps in Fig. 3. According to 3-D BPM simulation, the resultant Kvalues of these seven cases are still the same as those of butt-jointed cascaded ones. The total insertion loss is 4% (0.17 dB) for A+A, 8% (0.36 dB) for A+B, A+C, and A+D, and 12% (0.55 dB) for B+D, C+D, and D+D.

For asymmetric cases (**L _{PS}≠0 and δ_{a}≠δ_{b}**), continuous variations of L

_{PS}following with a series of consequent phase differences can induce a predictable change in K-values. In order to realize the shortest power splitter for each given K-value, we firstly analyze the MMI devices by using Eq. (3) and summarize the calculated results in Fig. 4 by solid-line to give a whole view about K-values vs. the total length of devices. Fig. 4(a) is for the cases of δ

_{a}=0.1/0.2 µm and δ

_{b}=0, while Fig. 4(b) is for the cases of δ

_{a}=0 and δ

_{b}=0.1/0.2 µm. The horizontal dash lines help us to catch the needed device length for each starting point simultaneously corresponding to those summarized in Fig. 2. Moreover, the 2-D EME simulation results, marked by open circles (δ

_{a,b}=0.1 µm) and triangles (δ

_{a,b}=0.2 µm), are also consistent with those calculation results from MPA and matrix multiplication of Eq. (3).

In fact, each case can be a full-range power splitter with a sinusoidal change in K-value when the L_{PS} is linearly extended to a π-phase shifting at L_{PS}=621 and 328 µm for δ_{a(b)}/δ_{b(a)}=0.1/0 and 0.2/0 µm, respectively. For clarity, each coupler shown Fig. 4 is in range of its best application region. In the case of *β _{a}*>

*β*as demonstrated in Fig. 4(a), the K-value for the A+D configuration starts decreasing and then increasing as the total length increases. On the other hand, a reverse trend for the K-value is observed in the B+D and C+D configurations. However, in case of

_{b}*β*<

_{a}*β*as demonstrated in Fig. 4(b), all K-values can be fine adjusted in either up or down without any saturated point before reaching a π-phase shifting. This result is caused by the asymmetric cross-state transfer function of MMI-D as indicated in Table I. In brief, the case of

_{b}*β*<

_{a}*β*offers us a chance to obtain relative shorter MMI-based couplers with K-values in those three special continuously tunable range of K=0.07-0.15 (A+D), 0.93-0.8 (B+D), and 0.64-0.5 (C+D).

_{b}## 4. Conclusions

By introducing a pair of unequal-widths waveguides as a phase shifter into the middle of cascaded MMI sections, 2×2 waveguide couplers with arbitrary power splitting ratio are demonstrated. These compact MMI-based devices use only rectangular geometry without any bent, curved, and tapered waveguides. They offer valuable possibilities for designing waveguide-based photonic integrated circuits.

## Acknowledgement

This work was supported by the Ministry of Education, Taiwan under the Aim for the Top University Plan.

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