## Abstract

We propose an ultra-broadband multimode interference (MMI) coupler with a wavelength range exceeding the O, E, S, C, L and U optical communication bands. For the first time, the dispersion property of the MMI section is engineered using a subwavelength grating structure to mitigate wavelength dependence of the device. We present a 2 × 2 MMI design with a bandwidth of 450nm, an almost fivefold enhancement compared to conventional designs, maintaining insertion loss, power imbalance and MMI phase deviation below 1dB, 0.6dB and 3°, respectively. The design is performed using an in-house tool based on the 2D Fourier Eigenmode Expansion Method (F-EEM) and verified with a 3D Finite Difference Time Domain (FDTD) simulator.

© 2013 OSA

## 1. Introduction

Multimode interference (MMI) couplers are important devices in optical integrated circuits (OIC). MMIs present several advantages compared to other waveguide coupling structures such as directional couplers, including smaller size, broader operational bandwidth and relaxed fabrication tolerances [1]. For that reason, MMIs are often used in Mach-Zehnder interferometers (MZIs) [2], splitters and combiners with uniform [3] or arbitrary [4] power splitting ratios, 90° hybrids for optical coherent receivers [5, 6] and polarization beam splitters [7]. Several studies related to size reduction [8] or performance improvement [5, 6, 9] of MMIs have been published. In order to improve tolerances to dimensional variations, it is advantegeous to maximize the ratio between the widths of access ports and the multimode region [10]. While this approach also yields some improvement of bandwidth, it does not provide control over the wavelength response of the device. In fact, broading the MMI bandwidth has received comparatively little attention in literature, with bandwidth typically limited to about 100nm for 2 × 2 MMIs for an insertion loss penalty of 1dB.

In this paper we propose, for the first time, a design procedure for virtually wavelength independent MMI couplers. We demonstrate a 2 × 2 MMI design with a bandwidth of 450nm exceeding the wavelength range of O, E, S, C, L and U bands (1260nm – 1675nm). Furthermore, the device length is about one half compared to a conventional 2 × 2 MMI. This unprecedented performance is achieved by simultaneously engineering the dispersion of multiple guided modes in the multimode region of the coupler, which is composed of a specifically designed subwavelength grating.

Subwavelength gratings (SWGs) are periodic or aperiodic structures in which diffraction effects are supressed by using a grating pitch (Λ) substantially smaller than the wavelength (Λ ≪ *λ*). SWGs have been employed in planar waveguides for fibre-chip surface grating couplers [11], microphotonic waveguides [12], lenses [13], waveguides crossings [14], fibre-chip edge couplers [15, 16], wavelength multiplexers [16], and ultra-fast optical switches [17]. Recently, their application to dispersion engineering in integrated optics was proposed [18] and applied to design of colorless directional couplers [19].

Our SWG-MMI structure schematics is shown in Fig. 1. The device comprises a subwavelength grating multimode section and tapered transitions between conventional interconnecting waveguides and SWG access ports. The tapered transitions have a double purpose: they widen the waveguide mode laterally, and match the effective index of the silicon-wire waveguides to the significantly lower effective index of the SWG region. The number of periods are *P _{MMI}* and

*P*in the multimode waveguide and the tapered access ports, respectively.

_{T}The device was designed using our in-house 2D simulation tool based on the Fourier Eigen-mode Expansion Method (F-EEM), specifically optimized for Floquet-Bloch mode calculations [20]. The nominal silicon waveguide geometry that we use is shown in Fig. 2(a), with a core thickness *h* = 260nm and refractive indexes *n _{Si}* = 3.476 (core),

*n*

_{SiO2}= 1.444 (substrate) and

*n*

_{SU–8}= 1.58 (superstrate) at

*λ*= 1.55

*μ*m. We have applied the Effective Index Method (EIM) to obtain a simplified 2D structure for TE (in-plane principal electric field component,

*E*) polarization. As illustrated in Fig. 2(a) we model the waveguide as a slab of effective index

_{x}*n*in the central guiding region, surrounded by SU-8. Our EIM model includes both geometrical and material dispersion. Calculations were verified with 3D FDTD simulations.

_{f}This paper is organized as follows. In section 2, we briefly discuss the working principle and the geometrical considerations for a conventional MMI coupler and the main factors limiting the bandwidth of MMI devices. In section 3 we propose a design procedure exploiting subwavelength grating waveguide dispersion engineering to substantially broaden MMI bandwidth. Simulation results, including insertion loss, power imbalance and MMI phase deviation between outputs are discussed in section 4. Finally, in section 5 we draw conclusions.

## 2. MMI couplers: working principle and bandwidth limitations

MMI couplers are based on the self-imaging principle [1], where *N*-fold images (copies) of the field at the input plane are formed at the output plane after propagation in a multimode waveguide of width *W _{MMI}*. Figure 2(b) shows the geometry of a conventional 2 × 2 MMI coupler. A fundamental mode in one of the input ports excites a set of modes

*ϕ*in the multimode section (

_{m}*m*= 0,...,

*M*– 1), with mode propagation constants

*β*= 2

_{m}*πn*

_{eff,m}/

*λ*, where

*n*

_{eff,m}is the effective index of the

*m*-th mode. For perfect (error-free) imaging, it is required that propagation constants follow a parabolic relation [1]:

*L*=

_{π}*π*/(

*β*

_{0}−

*β*

_{1}) is the beat length between the two lowest order modes.

*L*can be expressed, in terms of its wavelength dependence, as

_{π}*n*is the equivalent index of the multimode region resulting from the effective index method [1].

_{f}As an example, let us consider a 2 × 2 MMI coupler based on paired interference, as it generally yields shorter and more tolerant devices [10], so that the length of the multimode region is *L*_{MMI} = *L _{π}*/2. In a paired MMI, access waveguides are located at

*x*= ±

*W*

_{MMI}/3 and separated sufficiently for negligible coupling (< −40dB). This condition is fulfilled with a center-to-center separation

*S*= 2

*μ*m, which yields an MMI section width

*W*

_{MMI}= 6

*μ*m [see Fig. 2(b)].

Imaging quality of an MMI coupler primarily depends on two parameters. First, the propagation distance (device length, *L*_{MMI}) where images are formed is proportional to *L _{π}*, which is intrinsically wavelength dependent (Eq. 2). This length is set at the center wavelength (

*λ*

_{0}), e.g.

*L*

_{MMI}=

*L*(

_{π}*λ*

_{0})/2 in the example above. Second, deviations from the optimal parabolic relation Δ

*β*between the mode propagation constants (Eq. 1) increase for higher order modes [21]. To partially compensate this effect, the access ports width (

_{m}*W*) is generally designed to excite only a limited number of lower order modes [22].

_{a}Let us briefly inspect the wavelength dependence of *L _{π}* and Δ

*β*. Since optimal MMI,

_{m}*L*

_{MMI}, length is proportional to the beat length

*L*, which is wavelength dependent, the optimal length varies with wavelength. That is, images will be formed at different positions along propagation direction

_{π}*z*for different wavelengths. Wavelength dependence of the beat length is the main source of the MMI bandwidth limitation. This is shown in Fig. 2(c), where insertion loss is plotted as a function of device length for a conventional MMI coupler with multimode region width

*W*

_{MMI}= 6

*μ*m and access port width

*W*= 1.2

_{a}*μ*m. Insertion loss is defined here as IL = −10log(|

*s*

_{31}|

^{2}+ |

*s*

_{41}|

^{2}), where

*s*

_{31}and

*s*

_{41}are the transmission parameters from port 1 to port 3 and 4, respectively [see Fig. 2(b)]. Minimal loss is obtained at different device lengths for different wavelengths. Interestingly, since the insertion loss minima have similar values (≤ 0.25dB), wavelength independent behavior is expected if

*L*dependence with wavelength is mitigated by juducious design, as will be shown in the next section. In addition to reducing

_{π}*L*wavelength dependence, the parabolic relation between mode propagation constants (Eq. 1) needs to be fulfilled over the operational bandwidth to assure good MMI performance. From Eq. (1), it is observed that wavelength dependence of Δ

_{π}*β*is inversely proportional to

_{m}*L*(

_{π}*λ*). Consequently, provided that the beat length is designed substantially wavelength independent and access port width

*W*is wide enough so as to excite only limited number of the lower order modes that fulfill Eq. (1), the parabolic phase relation will hold and high quality imaging is expected for a broad wavelength range. These considerations are the basis of our ultra-wideband MMI device design.

_{a}## 3. Bandwidth widening of MMI couplers using dispersion engineering with subwavelength gratings

We propose a new type of MMI couplers in which the multimode section comprises a subwavelength grating structure (Fig. 1) to effectively flatten the wavelength dependence of *L _{π}*. In our SWG-MMI design, the calculations are carried out using Floquet-Bloch modes of the periodic structure. Note that since the SWG is non-diffractive, the field evolution of the Floquet-Bloch modes strongly resembles that of conventional waveguide modes, so that the behaviour of the MMI can still be described by the conventional waveguide mode expressions given by Eqs. (1) and (2).

The subwavelength grating in the multimode region is designed so that the beat length of its two lowest order Floquet-Bloch modes is approximately constant with wavelength. The SWG geometry provides two degrees of freedom to control waveguide dispersion: grating pitch (Λ) and duty cycle (DC = *a*/Λ). The range of values for pitch and duty cycle are limited by technological and practical considerations. Specifically, we constrain our geometry to a duty cycle of 50%, since this choice maximizes the minimum feature size for a given pitch. Furthermore, the device needs to be designed to operate below the Bragg condition over the entire bandwidth to avoid bandgap opening, so that Λ < *λ _{min}*/(2

*n*

_{eff}), where

*n*

_{eff}is the effective index of the fundamental Floquet-Bloch mode of the periodic structure at the minimum operational wavelength

*λ*. From elementary effective permitivity theory [23] for the electric field parallel to the grating lines, it follows: ${n}_{\text{eff}}\simeq {\left({n}_{Si}^{2}\cdot \text{DC}+{n}_{SU-8}^{2}\cdot \left(1-\text{DC}\right)\right)}^{1/2}\simeq 2.72$. This yields an upper bound for the pitch of Λ ≃ 230nm at the wavelength

_{min}*λ*= 1.25

_{min}*μ*m. We then calculate the beat length,

*L*, as a function of pitch and wavelength. The results are shown in the contour map of Fig. 3(a). It should be pointed out that such a systematic calculation can be performed efficiently in a comparatively short time using our in-house simulation tool [20]. Albeit larger values of pitch are desirable to ease device fabrication, this implies a rapidly varying coupling length as the Bragg resonance is approached [white region in Fig. 3(a)]. On the other hand, pitch values below 170nm results in narrower

_{π}*L*bandwidths and indeed also smaller minimum feature sizes. Therefore, from Fig. 3(a) an optimal region is identified near Λ = 190nm. As shown in Fig. 3(b),

_{π}*L*wavelength dependence is mitigated for the SWG-MMI with a sub-wavelength grating pitch of Λ = 190nm and a duty cycle of 50%. The wavelength dependence of the beat length of a conventional MMI device with same width

_{π}*W*

_{MMI}= 6

*μ*m is shown for comparison. These results were confirmed with 3D simulations carried out with MEEP [24].

For Λ = 190nm, the wavelength averaged beat length is *L _{π}* = 46.5

*μ*m, so that the number of periods for a paired 2 × 2 MMI is given by

*P*

_{MMI}= (

*L*/2)/Λ, i.e.

_{π}*P*

_{MMI}= 122 periods. This results in a length of the multimode section 23.18

*μ*m, which almost coincides with one half of the wavelength-averaged beat length. Note that a conventional MMI of the same width is about twice as long (

*L*

_{MMI}= 48.2

*μ*m). The shortening of the SWG-based device is attributed to the lower equivalent index

*n*in the multimode region [see Eq. (3)] compared to a conventional MMI device.

_{f}The deviation from the ideal parabolic relation between the propagation constants [Eq. (1)] can be estimated with the *modal phase error* defined as MPE* _{m}* = |Δ

*β*− Δ

_{m,real}*β*| ·

_{m}*L*

_{MMI}. The MPE should be below 45° for high quality imaging [25]. Figure 3(c) shows that the MPE for the first five modes of the MMI is well below this threshold in a 500nm wavelength range. The width of access waveguides

*W*is designed to ensure that 90% of the input power is carried by these first five modes, resulting in

_{a}*W*= 1.2

_{a}*μ*m. This requires an adiabatic transition between the interconnecting silicon wire waveguide of width

*W*= 0.45

_{g}*μ*m, and the SWG access waveguide of width

*W*= 1.2

_{a}*μ*m. Note that this transition must furthermore adapt the effective index of the wire waveguide to the effective index of the SWG region, to prevent Fresnel reflections at the MMIs input and output planes. We use a taper geometry similar to that proposed in [16], shown schematically in Fig. 1, where the waveguide core between the SWG segments is gradually reduced to provide the effective index matching. The pitch of the taper is identical to the subwavelength grating pitch used in the multimode region (Λ = 190nm). Losses were found to be negligible for a taper length

*L*= 10

_{T}*μ*m, with

*P*= 53 subwavelength periods [26].

_{T}## 4. Evaluation of MMI broadband performance

We evaluate the bandwidth performance of the wavelength independent 2 × 2 MMI, in terms of insertion loss, IL = −10log(|*s*_{31}|^{2} + |*s*_{41}|^{2}), power imbalance, PI = |10log(|*s*_{31}|^{2}/|*s*_{41}|^{2})|, and MMI phase deviation from the nominal 90° phase difference between the MMI outputs, PD = |∠ (*s*_{31}/*s*_{41})| − 90°, where ∠ denotes the phase of a complex number. Insertion loss of the SWG-MMI design is compared in Fig. 4(a) with a conventional MMI coupler with similar dimensions: *W*_{MMI} = 6*μ*m, *W _{a}* = 1.2

*μ*m,

*S*= 2

*μ*m,

*L*= 10

_{T}*μ*m and

*L*

_{MMI}= 48.2

*μ*m at

*λ*= 1.47

*μ*m. Power imbalance and MMI phase deviations are shown in Fig. 4(b) and Fig. 4(c), respectively. The SWG-MMI shows a significant bandwidth enhancement in all these parameters: insertion loss and power imbalance are less than 1dB and 0.6dB, respectively, and the MMI phase deviation is less than 3° within a wavelength range of 450nm. This corresponds to an almost fivefold bandwidth enhancement compared to conventional MMI devices. These results have been verified with RSoft FullWAVE [27], a 3D FDTD simulator. As in the 2D case, material dispersion was included in the 3D calculations. Note that due to the inherent approximations of the 2D model device dimensions must be re-optimized in the 3D case. This was done by keeping the 50% duty-cycle in the SWG, and iteratively adjusting the pitch of the SWG and the length of the MMI region around the values obtained with the 2D model. According to this 3D FDTD design, the SWG pitch is Λ = 198nm, and the number of periods of the MMI region and the access tapers are

*P*

_{MMI}= 94 and

*P*= 51, respectively. It is noticed from the results shown in Fig. 4(a–c) that our 2D design approach accurately describes the performance of the device, and gives a good initial point for the final design with reduced computational effort.

_{T}Simulated field propagation (electric field conponent *E _{x}*) in the SWG-MMI device is shown in Fig. 4(d), for wavelengths

*λ*= 1.26

*μ*m (O band),

*λ*= 1.47

*μ*m (S band) and

*λ*= 1.675

*μ*m (U band). As expected, in all cases the field pattern propagated along the MMI is almost invariant. Thus, images are formed properly at approximately the same

*z*-position, confirming that the wavelength dependence of the beat length,

*L*, has been minimized. The SWG-based MMI thus covers most of the bands presently used in optical communications (O, E, S, C, L and U) in the wavelength range from 1260nm to 1675nm.

_{π}## 5. Conclusions

With proper device design, MMI bandwidth is solely limited by the wavelength dependence of the beat length *L _{π}*. We have shown that by engineering the dispersion properties of a multimode region using subwavelength gratings, MMI wavelength dependence can be mitigated. Our subwavelength grating dispersion engineered MMI device exhibits substantially wavelength independent beat length and consequently an ultra-broadband response, covering most of the bands used for communications in the wavelength range of 1260nm–1675nm. The device has calculated insertion loss, power imbalance and MMI phase deviations of less than 1dB, 0.6dB and 3° in a 450nm bandwidth, that is an almost fivefold bandwidth enhancement compared to conventional MMI designs. Furthermore, the length of the device is reduced to one half compared to a conventional design. The design has been performed using our in-house 2D simulation tool and verified with a commercial 3D FDTD software. The proposed method of planar waveguide dispersion engineering with subwavelength gratings opens promising prospects for designing new types of MMI couplers and other broadband integrated optical devices.

## Acknowledgments

This work was supported by the Spanish Ministerio de Ciencia (project TEC2009-10152), the Andalusian Regional Ministry of Science, Innovation and Business (project P07-TIC-02946), and the European Mirthe project ( FP7-2010-257980).

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