## Abstract

We show that it is possible to obtain 2×2 waveguide couplers with new power splitting ratios for cross coupling of 7%, 64%, 80% and 93% by cascading two short MMI sections. These couplers have simple geometry and low loss. They offer valuable new possibilities for designing waveguide power taps, high-Q ring resonators, ladder-structure optical filters, and loop-mirror partial reflectors.

©2007 Optical Society of America

## 1. Introduction

The 3-dB multi-mode interference (MMI) coupler, in either the 1×2 or the 2×2 form, has been extensively used as power splitter/combiners in Mach-Zehnder interferometers, ring resonators, and other optical waveguide circuits owing to their relative compactness [1]. Couplers for unequal splitting of power are also needed for other photonic-integration applications such as power taps, high-Q ring resonators [2–4], ladder-structure optical filters [5], and loop-mirror partial reflectors [6–8]. For 2×2 couplers, the conventional constant-width MMI design is known to be capable of providing only seven different cross-coupling ratios, i.e. 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 [9, 10]. For low-loss power taps and high-Q ring resonators, a K value that is even lower than 0.15 would be very useful. For loop-mirror partial reflectors [6–8], a K value that is even closer to 0.5 than 0.72 would also be highly desirable. Although it is possible to obtain arbitrary splitting ratios by using a bent MMI coupler [11,12], for typical 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 and the very high angular sensitivity of the K value. We show in this letter that it is possible to realize new K values of 0.07, 0.64, 0.80, and 0.93 by cascading two short MMI sections. The devices have low loss and do not involve angled or curved geometries.

## 2. Building-block MMI’s

The MMI sections to be discussed in this letter are all 2×2 waveguide couplers. A schematic diagram of a generic 2×2 MMI coupler is shown in Fig. 1(a). For this group of MMI sections, we set the input and 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 all MMI sections. In order to minimize the size of the couplers, *s* is generally chosen to be as short as the lithographic process to be used 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, 9]

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

*W*is formed,

_{e}*λ*

_{0}is the vacuum wavelength, and

*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 the Beam
Propagation Method (BPM) to obtain more realistic assessments. In these studies, the
epitaxial slab waveguide on InP substrate is assumed to have a 1.79-μm thick (mostly InAlAs) upper cladding, a 0.45-μm thick core layer containing 3 quantum wells with the absorption edge at 1.48-μm, and a 0.23-μm thick lower cladding (also mostly InAlAs). 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 substrate to keep the number of confined
modes large. The effective index of the TE fundamental slab mode is 3.2147 at λ_{0}=1.55 μm. In order to match the low-order modes in 2-D simulation to those in 3-D simulation, the equivalent effective index of the lateral cladding is set to 2.3. The effective width, *w*, of the access waveguides is assumed to be 2.2 μm and their separation is set to be *s* = 5 μm.

Four relatively short MMI sections with K ≥ 0.5 have been considered as the building blocks of cascaded MMI’s. For MMI-A (K = 0.5), we have We=s+w=7.2μm=1.44s. For given values of *s* and *w*, this is the minimum practical effective width for a low loss MMI [1]. The device length *L _{A}* is (3/2)

*L*

_{πA}=3.1104

*As*

^{2}=215 μm. The actual length in μm is obtained not by Eq. (1) but by a 3-D BPM simulation tool [13]. For MMI-B (K=0.5,

*r*= 3), we have

*W*= 3

_{e}*s*= 15 μm and

*L*= (1/2)

_{B}*L*

_{πB}=4.5

*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*

_{πC}=3

*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*

_{πD}=3.75

*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 basic MMI’s are deduced from Bachmann et al. [9]. 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, the total insertion losses of these four basic MMI’s lye in the range of 4 to 5%.

_{U}## 3. Cascaded 2×2 MMI couplers

Since all four basic MMI sections are 2×2 couplers with the same *s* value, they can be simply aligned and cascaded. In this paper, the MMI sections are actually butt jointed without any interconnecting waveguides. When MMI-U and MMI-V are cascaded, the transfer matrix of the cascaded device can be obtained by the following matrix multiplication.

where HUV_{X±} and HUV_{B±} 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

We have examined the results of all possible combinations using Eq. (2) and verified their consistency with the results of 3-D BPM simulations. The resultant K-values are: KAB=0, KAC=0.15 (0.146), KAD=0.07 (0.075), KBC=0.85 (0.854), KBD=0.93 (0.925), KCD=0.64 (0.641), and KDD=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 by using a constant-width MMI [9, 10]. The optical field maps of these MMI’s obtained by 2-D BPM simulation are shown in Fig. 2. Additional information on these four cases are given below.

#### KAD=0.07

The total length of this cascaded device is [(3/2)*L*
_{πA}+(3/5)*L*
_{πD}]=[3.1104*As*
^{2}+3.75*As*
^{2}] =474 μm. The MPA and matrix multiplication give |HAD_{X}|^{2}=(0.075) and |HAD_{B}|^{2}=(0.925). The total insertion loss is 5% according to the 3-D simulation.

#### KBD=0.93

This cascaded device is complementary to the 7% A+D device. The reason behind this complementary relation is that the transfer functions of MMI-B are complementary in phase changes to those of MMI-A as is evident in Table I. The total length is [(1/2)*L*
_{πB}+(3/5)*L*
_{πD}]=[4.5*As*
^{2}+3.75*As*
^{2}]=570 μm. The total insertion loss is 6%.

#### KCD=0.64

The total length is [(3/4)*L*
_{πC}+(3/5)*L*
_{πD}]=[3*As*
^{2}+3.75*As*
^{2}]=466 μm. The MPA and matrix multiplication give predictions of |HCD_{X}|^{2}=(0.643) and |HCD_{B}|^{2}=(0.357). The total insertion loss is 5%.

#### KDD=0.80

In this case, two MMI-D’s are cascaded with the double images shifting successively in the same direction. The total length is (6/5)*L*
_{πD}=7.5*As*
^{2}=518 μm. The MPA and matrix multiplication give predictions of |HDD_{X}|^{2}=(0.80) and |HDD_{B}|^{2}=(0.20). The total insertion loss is 6%. If the second MMI-D section is flipped over horizontally, the result is a constant-width MMI-2D of length (6/5)*L*
_{πD} with K2D=0.72 (0.724).

Since the cascaded devices are butt jointed without interconnecting waveguides, the total insertion losses of the cascaded devices are much less than the sum of the total insertion losses of the constituent MMI sections. For instance, the insertion loss of the straight-edged MMI-2D is essentially the same as that of the shorter MMI-D, and the insertion loss of the step-edged MMI-DD is only 1% higher than that of MMI-2D. For all four devices, cascading increases the total insertion loss by less than 2% over those of the single sections alone. This means that, at the junction plane, the optical field is so well focused around the double images that hardly any optical field is intercepted by the abrupt change of width at the junction. At the junction plane, the near absence of optical field away from the focused images also makes the devices relatively insensitive to process induced distortions of the steps, provided that the location of the junction plane does not deviate from the actual image plane by much more than the equivalent Rayleigh length of the images.

## 4. Wavelength sensitivity of cascaded couplers

When the input wavelength deviates from the design wavelength of 1.55 μm, the total transmittance decreases because *L*
_{πU} and *L*
_{πV} are inversely proportional to the wavelength and the device length no longer matches the correct sum of their specific multiples. The results of 3-D simulation using another simulation tool based on modal expansion [14] are plotted in
Fig. 3(a). This simulation tool tracks the wavelength dependence of the mode profiles exactly. The calculated values of the total transmittance are, however, about 2% lower than those obtained by the other simulation tool [13]. This is partly because only the part of the output optical field that matches exactly the fundamental mode in the output waveguide is taken as the output. Otherwise, the differences are not significant. The bandwidth at 85% total transmittance (i.e. -1 dB from the maximum) is approximately inversely proportional to the device length. This is a general property of all MMI’s. Each of the MMI images mimics the Gaussian-like input field and has the same effective Rayleigh length. The bandwidth is proportional to the fractional change of the wavelength that moves the image by an effective Rayleigh length. Each of the MMI images moves in proportion to the device length multiplied by the fractional change of the wavelength. Additional simulations show that there is no discernible difference between the 1-dB bandwidths of MMI-DD and MMI-2D. These are two MMI’s of the same width and length, one with stepped edges and the other with perfectly straight edges. The result shows that the steps on the edges do not cause noticeable additional loss when the wavelength deviates from the design value.

The dependences of the K values on the wavelength deviation are plotted in Fig. 3(b). Within the 1-dB bandwidth of each of the four devices, the variation of the K value is found to be less than 0.02.

## 5. Applications

The K values of 7% and 93% are obviously very useful when one wishes to tap off only a small amount of optical power from the waveguide. The K values of 93% and 80% would also be useful in the construction of ladder-structure optical filters [5]. The value of 7% is very attractive for constructing a high-Q ring resonator. The filtering property of a symmetric single-ring resonator is very similar to that of a symmetric Fabry-Perot resonator. The coupling factor K is equivalent to 1-R in the Fabry-Perot resonator where R is the mirror reflectance. The finesse of a lossless symmetric ring resonator is, therefore, given by

For previously available K values of 0.15, and 0.5, the (lossless) finesses are respectively 19.3, and 4.44. By using the available new K value of 0.07, one can obtain a much higher (lossless) finesse of 43.3.

A partially reflecting loop mirror can be constructed by using a 2×2 coupler [6–8]. The access waveguides at the far end would be connected together through a waveguide loop. At the near end, one access waveguide would serve as the input and the other as the output. For a lossless loop mirror, the input reflectance is given by

For K=|H_{X±}|^{2}=0.5, 0.64, 0.72 (and 0.28), 0.80, 0.85 (and 0.15), and 0.93 (and 0.07), we get respectively R=1, 0.92, 0.81, 0.64, 0.51, and 0.27. The transmittance to the output port is 1-R. The newly available K value of 64% makes it possible to obtain a high reflectance of 92% with 8% output coupling.

## 6. Conclusions

In summary, we have obtained new power splitting ratios of K=0.07, 0.64, 0.80, and 0.93 by cascading two short MMI sections. These couplers have low insertion loss and simple geometries requiring no angled or curved patterns. They provide new opportunities to improve the performance of waveguide power taps, high-Q ring resonators, ladder-structure optical filters, and loop-mirror partial reflectors.

## Acknowledgements

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

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