## Abstract

We present a unique comparison of ridge-type directional couplers (DC) and multi-mode interferometers (MMI) in terms of their transformational relationship. The two devices are intimately related as the MMI is derived from the DC. We show for the first time the continuous evolution from the two-mode coupling characteristic of DC to the multi-mode mixing and interference characteristic of MMI, as the DC is structurally transformed into the MMI. We also show that DC can be designed to have the MMI features of compactness and polarization-insensitivity, two traits that reflect their shared lineage. However, the design of DC requires careful control of a large set of design parameters, while the MMI design is more robust and involves fewer design variables.

©2004 Optical Society of America

## 1. Introduction

Directional couplers (DC) [1] and multi-mode interference (MMI) couplers are two of the most important devices for optical signal routing and processing. In particular, since its introduction in 1995 by Soldano and Penning [2], MMI has gained widespread usage in power coupling, splitting, switching and coarse wavelength-division multiplexing, because of its compactness, polarization insensitivity and fabrication tolerance relative to directional couplers. Traditionally, these devices have been made in rib or buried waveguides which are weakly guiding. A performance comparison for DC and MMI based on the rib waveguides has been given in [3]. In recent years, with increasing focus on photonic integration, strongly guiding ridge waveguides are increasingly being used as building blocks for optical devices [4,5]. So far, however, no comparable analysis has been performed for DC and MMI based on *ridge* waveguides.

MMI, to zeroth order, may be considered as a fused directional coupler where the gap has been filled. A legitimate question, then, is: When and how does a DC become an MMI? This question is answered in this paper with a rigorous, constructive, and bottom-up approach starting with the ridge waveguide. Specifically, we show, for the first time, an evolutionary lineage between the DC characteristics and the MMI characteristics as one is structurally transformed to the other. It will be shown that understanding this transformational relationship will offer insight into the design of compact MMI and the comparative properties of DC and MMI.

The directional coupler considered consists of two parallel, symmetric waveguides [1]. When light is launched into the access waveguide port, it excites equally the *even* and *odd* supermodes, which then propagate with different velocities given by the propagation constants, β_{e} and β_{o}, respectively. After the two supermodes are *π* out of phase, the light is completely transferred to the adjacent waveguide. The distance in which complete transfer occurs is defined as the *coupling length* and is given by:

where *n*
_{e} and *n*
_{o} are the effective indices of the even and odd supermodes. As a passive coupler, *L*
_{c} is the most important parameter as it determines the size of the device, hence for compactness Lc should be as small as possible. Moreover, because the input optical signal could have a random polarization, ideally the coupling length should be the same for both TE and TM if polarization insensitive operation is important.

The MMI is based on the self-imaging principle arising from multimode interference [2]. For a multimode waveguide supporting *m* lateral modes, the output field at a distance *z*=*L* is given in the 2-D approximation by] $\psi (y,L)={\displaystyle \sum _{\upsilon =0}^{m-1}}{c}_{\upsilon}{\phi}_{\upsilon}(y)\mathrm{exp}\left[j\frac{\upsilon \left(\upsilon +2\right)\pi}{3{L}_{\pi}}L\right]$, where *c*
_{ν} is the excitation coefficient for mode ν, *φ*_{υ}
is the *ν*th eigenmode, and *L*_{π}
is the *beat length*, defined as

*n*
_{0} and *n*
_{1} and are the modal effective indices of the fundamental and first-order modes, *n*
_{r} is the effective index of the slab waveguide, *λ*
_{o} is the operating wavelength, and *W*_{eq}
is the equivalent MMI width, which for the present case is essentially the same as the physical width of the ridge slab waveguide. As a cross coupler, the crossover output is the mirror image of the input about the plane of symmetry (*y*=0). There are two self-imaging mechanisms. Self-imaging in which the output is independent of the modal excitation (c_{ν}) is called *general interference*; in this case, cross coupling occurs at *L*
_{c}=*p*(3*L*_{π}
), where *p* is an odd integer. In some MMI selective excitation in which c_{ν}=0 for ν=2, 5, 8, … is possible by launching a symmetric input field *ψ* (*y*,0) at *y*=±*W*_{eq}
/6. This is called *restricted interference* and in this case, the mirror images of the input field occur at *L*
_{c}=*pL*_{π}
. The advantage of using ridge waveguide, as compared to rib waveguide, is that *L*_{π}
will be smaller, hence a MMI based on ridge waveguide will be more compact. An MMI is also relatively polarization insensitive.

The large lateral index contrast in a ridge waveguide enables it to bend and meander with very small radii, but also makes it more difficult to construct single-mode waveguides which are required by many devices, including the ring resonator [5], the directional coupler, and the Mach-Zehnder interferometer [7]. Laterally a single-mode waveguide is typically about 0.6 µm or less in width. Aside from the stringent fabrication requirement, this small width could incur other problems—high propagation loss per unit length [8], low fiber coupling efficiency, and high birefringence. Therefore, in this study we will use ridge waveguide with a wider width which, although multi-mode, has the advantages of low loss, low birefringence, and good fabrication tolerance. We have shown elsewhere [9] that, by using a judicious combination of critical width and critical ridge height, it is possible to design waveguides that are effectively both polarization-independent and “single-mode”, with reasonably low loss for the fundamental modes, and with the critical width around 1 to 2 µm which is much more robust and easy to fabricate using standard photolithography. In the following sections we consider DC and MMI based on a ridge waveguide with the critical width of 1.5 µm. All simulations are based on the 3D finite-difference beam propagation method and finite-difference eigenmode solutions ^{1}[6].

## 2. Polarization-independent directional couplers

The coupling length, *L*
_{c}, of a directional coupler depends on many factors. Most of all it depends exponentially on the gap separation; for the ridge-type DC of interest here, a small coupling length would require a miniscule (or zero) gap. For later comparison with MMI, we fix the gap width at *g*=0.5 µm, and vary other design parameters that affect *L*
_{c}. First, a shallow etch in the region between the waveguides could be used so that the evanescent coupling here resembles that of a rib waveguide, while a deeper etch on the outer edge is retained to provide strong overall confinement, as shown in Fig. 1(a). In addition to the gap etch depth (*d*_{g}
), the coupling length is also very sensitive to the waveguide width (*w*), the core thickness (*d*_{c}
) and the core-cladding index step, since these parameters all affect the optical confinement and hence the evanescent field strength. Reducing the waveguide width or core thickness (within limits), for example, can effectively reduce the coupling length, as the light mode tends to expand horizontally thereby inducing stronger coupling. We find that the smaller the core thickness or the core index, the smaller the coupling length will be. Moreover, the TE and TM coupling lengths can be made identical by varying the gap etch depth or the waveguide width. The following results are based on an InP/InGaAsP/InP waveguide structure where the upper cladding thickness is 1.5 µm, the cladding index is 3.17, the background index is 1.5, and the operation wavelength is fixed at 1.55 µm. The computed fundamental mode of the input port waveguide is used as the input mode excitation.

Figure 1(b) and (c) show the coupling lengths for TE and TM as a function of the etch depth, *d*_{g}
, and the waveguide width, *w*, respectively. Note that the coupling length is equal at some critical width. Below the critical width the TM mode has a higher effective index or optical confinement, and hence a larger coupling length, while above the critical width the situation is reversed. Similarly, when the gap etch depth is varied from 2 µm (bottom of the core layer) to 1.7 µm (top of the core layer), it can be seen that the TE coupling length is initially larger, and then becomes smaller than that of TM. This is because the TE polarization is more sensitive to the dielectric perturbation in the horizontal direction, and so the TE coupling length decreases at a faster rate than TM as the etch depth is reduced. Therefore, the TE and TM coupling lengths will intersect at some point. At this critical etch depth the directional coupler is said to be *polarization independent*, since the power coupling efficiency will be the same for both TE and TM.

In summary, for ridge waveguide-based directional couplers, there exist various specific combinations of etch depth, core thickness, core index and waveguide width, for which the coupling lengths are both reasonably small and polarization-independent. The polarization independence is unique to ridge-type DC and is not found in rib waveguides. This unique feature is exhibited also in the single-mode, sub-micron regime, even though the waveguides here are highly birefringent [4]. However, the fabrication tolerance for these directional couplers is tight since the coupling length is sensitive to so many inter-dependent parameters, including a core index that is usually not known exactly.

## 3. Transformation from DC to MMI

In this section we address the question of how the DC characteristics might evolve into the MMI characteristics if the gap depth is continuously reduced above the core layer. We investigate the change in mode coupling behavior in a directional coupler as the structure is transformed through the steps (a) to (e) depicted in Fig. 2. The waveguide width is fixed at *w*=1.5 µm, and their center-to-center separation is *D*=2 µm. The resulting structure in Fig. 2(d) is an MMI with the smallest possible width given by *W*
_{mmi}=(*D*+*w*). For convenience we refer to this general-interference MMI as Type I, since it may be considered as the direct descendent from the directional coupler of Fig. 2(a). The *restricted interference* MMI can be formed by expanding the MMI width to *W*
_{mmi}=3*D*, as shown in Fig. 2(e). This MMI may be considered as a derivative of Type I, so for brevity it is referred to as Type II.

The evolution in the transfer function with reducing gap depth is shown in Fig. 3. Plotted are the normalized powers for the TE polarization in the bar (solid) and cross (dotted) waveguides, given by the overlap integrals between the calculated field at the current z position and the input mode field over the waveguide cross-sections. Note that polarization independence in the DC is not preserved as the etch depth is changed. Initially, it is clear that the device is in the DC mode. As the gap depth is reduced, there are increasing ripples in the transfer curves for both TE and TM arising from the dielectric perturbation in the gap, and there is a continuous reduction in the coupling length to about 100 µm. Eventually, as the light sees a much bigger waveguide the power distribution is no longer determined by the coupling between the supermodes, but rather shaped by the multimode interference inside the much bigger waveguide. When the etch depth is 1.1 µm (or 0.4 µm above the top of the core layer), the device characteristic is already MMI-like, with the coupling length still at about 100 µm. When the etch depth is 0.6 µm, the 3-dB coupling length is also well defined, and the MMI characteristic is fully developed even though there is still a sizable etched gap.

Another view of the transition between DC and MMI is presented by the change in the modal effective indices as the gap depth is reduced, as shown in Fig. 4. These effective indices are obtained directly using the FD mode solver that computes the symmetrical and anti-symmetrical modes of the coupled structures. Initially, with very deep gap, the coupling between the two waveguides is weak and the two eigen-indices are nearly equal. When the etch depth is 0.1 µm above the bottom of the core layer, the coupling increases rapidly and the supermode indices start to separate. These supermodes evolve continuously to become the fundamental (*m*=0) and first-order (*m*=1) modes of the Type I MMI. When the etch depth is 1 µm above the bottom of the core (or 0.5 µm above the top of the core), *n*
_{0}=3.243494 and *n*
_{1}=3.221202, giving the smallest *L*_{π}
≈35 µm according to Eq. (2). Since the functional length of MMI (whether as cross coupler or 3-dB splitter) scales with *L*_{π}
, we can see why MMI is always more compact than the DC from which it evolves. For Type I MMI, cross coupling will occur first at 3*L*_{π}
≈105 µm. When Type I is transformed into Type II by increasing the MMI width, it is seen that *n*
_{0} and *n*
_{1} then approach each other, meaning that *L*_{π}
is increasing, such that *L*_{π}
=102 µm when *W*
_{mmi}=3D. For Type II MMI, the smallest coupling length is *L*_{π}
. Hence, both types of MMI have about the same coupling length *L*_{c}
even though they have very different widths.

This result is consistent with Eq. (2), from which it follows that the ratio of the coupling lengths for Type I and Type II MMI is given by ${L}_{c}^{I}$
/${L}_{c}^{\mathit{\text{II}}}$
=3(*D*+*w*)^{2}/(3*D*)^{2}≈1 for *w*=1.5 µm and *D*=2.0 µm. This prediction is also borne out by BPM simulations of the MMI’s. The BPM results are summarized in Table 1. For Type I MMI, cross coupling first occurs at *L*
_{c}=110 µm for TE and 101 µm for TM, corresponding to the 3*L*_{π}
values. As the MMI width is increased the corresponding *L*
_{c} increases. Near *W*
_{mmi}=6 µm where restricted interference dominates, cross coupling starts to appear at a smaller value, i.e., *L*
_{c}=104 µm for TE and 99 µm for TM, corresponding to *L*_{π}
instead of 3*L*_{π}
.. The corresponding 3-dB coupler lengths *L*
_{3dB} are (3*L*_{π}
)/2=55 µm (for TE) for Type I, and *L*_{π}
/2=52 µm (TE) for Type II. Remarkably, these values are in good agreement with those evident from Fig. 3(e) and (f), indicating that they are shaped early in the transition from DC to MMI.

In summary, we have shown, for the first time, the continuous evolution from the two-mode coupling characteristic of DC to the multi-mode mixing and interference characteristic of MMI, as the DC is structurally transformed into the MMI. Not only is the resulting MMI more compact than the original DC, the same compact size can be given by two different types of MMI based on the two different self-imaging mechanisms. Furthermore, we have also shown that DC can be designed to have the MMI features of compactness and polarization-insensitivity, two traits that reflect their shared lineage.

## Footnotes

^{1} | RSoft Inc., BeamPROP Version 5.5, and Apollo Photonics APSS Version 2.1. Simulations are typically run in 3D semi-vectorial mode with Transparant Boundary Condition (TBC), scheme parameter of 0.5, Pade order (1,0), and mesh size of 0.02 µm for the x and y (transverse) directions. |

## References and links

**1. **R. A. Forber and E. Marom, “Symmetric directional coupler switches,” IEEE J. Quantum Electron **QE-22**, 911 (1986). [CrossRef]

**2. **Lucas B. Soldano and Erik C. M. Pennings, “Optical Multi-mode Interference Devices based on Self-Imaging: Principles and Applications,” J. Lightwave Technology **13**, 615 (1995). [CrossRef]

**3. **M. Rajarajan, B. M. A. Rahman, and K. T. V. Grattan, “A Rigorous Comparison of the Performance of Directional Couplers with Multimode Interference Devices,” J. Lightwave Technology **17**, 243 (1999). [CrossRef]

**4. **Yong Ma, Seoijin Park, Liwei Wang, and Seng Tiong Ho, “Ultracompact Multimode Interference 3-dB Coupler with Strong Lateral Confinement by Deep Dry Etching,” IEEE Photon. Technol. Lett. **12**, 492 (2000). [CrossRef]

**5. **M. K. Chin, C. Youtsey, W. Zhao, T. Pierson, Z. Ren, S. L. Wu, L. Wang, Y. G. Zhou, and S. T. Ho, “GaAs microcavity channel-dropping filter based on a race-track resonator,” IEEE Photon. Technol. Lett. **11**, 1620 (1999). [CrossRef]

**6. **W. P. Huang, C. Xu, S. Chi, and S. K. Chaudhuri, “The finite-difference vector beam propagation method: analysis and assessment,” J. Lightwave Technology **10**, 295 (1992). [CrossRef]

**7. **N. Yoshimoto, Y. Shibata, S. Oku, S. Kondo, and Y. Noguchi, “Design and demonstration of polarization-insensitive Mach-Zehnder switch using a lattice-matched InGaAlAs/InAlAs MQW and deep-etched high-mesa waveguide structure,” J. Lightwave Technol. **17**, 1662 (1999). [CrossRef]

**8. **Zhixi Bian, B. Liu, and A. Shakouri, “InP-based passive ring-resonator-coupled lasers,” IEEE J. Quantum Electron. **39**, 859 (2003). [CrossRef]

**9. **M. K. Chin, C. W. Lee, S. Y. Lee, and S. Darmawan, “High-index contrast waveguides and devices,” Opt. Commun. (to be published).