Based on a multimode interference (MMI) coupler in slot waveguide structures, an ultracompact wavelength demultiplexer operating at 1.30 and 1.55μm wavelengths is proposed and designed by using a full-vector mapped Galerkin mode solver and a modified three-dimensional full-vector beam propagation method. The tapered waveguide structures are applied to connect the input/output channels and the MMI section for reducing excess loss. The modal characteristics of the slot waveguides are analyzed and the evolution of the injected field in whole device are demonstrated. The results show that a MMI section of 119.8μm in length, which is only 27.5% length of that of the MMI coupler by using conventional rib waveguides, is achieved with the contrasts of 26.03 and 28.14dB at wavelengths 1.30 and 1.55μm, respectively, and the insertion losses are below 0.2dB at both wavelengths.
©2007 Optical Society of America
The wavelength demultiplexers play very important roles in optical transmission systems using wavelength division multiplexing (WDM) technology, which can increase the number of channels and the information capacity of optical fibers . For optical telecommunications, such as fiber to the home (FTTH) system, much interest has been given to demultiplexing 1.30 and 1.55μm wavelengths.
Several devices have been proposed to perform this function, such as Y-branch devices , Mach-Zehnder interferometers , and multimode interference (MMI) couplers –. Among them, the MMI-based devices are good candidates due to their excellent characteristics, including ease fabrication, large optical bandwidth, and compact size . Several groups have reported wavelength demultiplexers for the operation at 1.30 and 1.55μm wavelengths based on MMI couplers –. However, the length of these devices is still relatively large. While for future photonic integrated circuits (PICs), compact even ultracompact photonic devices are needed. More recently, the progress in silicon photonics enable us to realize the ultracompact photonic devices in high-index contrast material system, most prominently in silicon-on-insulator (SOI) –. Apart from silicon photonic-wires and photonic crystal waveguides, the slot waveguides proposed recently – haven attracted much attention. In a slot waveguide structure, the guided light is strongly confined in a narrow low-index gap (slot region) between two high-index photonic wires. This enables the introduction of new compact photonic devices where the characteristics of active optical materials can be efficiently exploited for modulation, switching, and other applications. So far, various kinds of photonic devices by using slot waveguides haven been proposed or fabricated, including micro-ring resonators , optical modulators , electrically pumped light emitting devices , directional couplers , optical splitters , all-optical logic gates , and optical sensors . Especially, Fujisawa and Koshiba  have proposed an ultrasmall optical splitter based on MMI by using slot waveguides. To the best of our knowledge, however, there has no attempt so far to design a wavelength demultiplexer for 1.30 and 1.55μm operation based on a MMI coupler by using slot waveguides.
In this paper, a 1.30/1.55μm wavelength demultiplexer based on a MMI coupler by using slot waveguides is proposed. The tapered waveguide structures are applied to connect the input/output channels and the MMI section for reducing the excess loss. Since the guided modes in a slot waveguide are true guided modes confined by total internal reflections, thus there are no confinement losses. Therefore, this slot waveguide structure can improve the quality of the self-imaging in MMI section as pointed in Ref.  and lead to the present demultiplexer with ultracompact size and low loss. Moreover, the present device can be monolithically integrated with the other devices proposed earlier due to the same material system. A full-vector mapped Galerkin method (FV-MGM) mode solver  and a modified three-dimensional full-vector beam propagation method (3D-FV-BPM) based on E field  are applied to optimize the parameters and assess the performance of the present device. The field patterns and the effective indices of the guided-modes for the slot waveguides are obtained, and the evolution of the injected field along the propagation direction in whole device are demonstrated.
2. Computational methods
The FV-MGM mode solver based on E field as described in Ref.  is applied to analyze the guided modes for the slot waveguides. In a FV-MGM, the whole space is mapped into a unit square in the absence of the boundary truncation via a tangent-type variable transformation. The transverse electric field components are expanded as a complete of orthogonal, sinusoidal basis functions with finite multinomial and resulting in a matrix eigenvalue equation which is solved by the MATLAB subroutines.
The three-dimensional full-vectorial beam propagation method based on E field is used to simulate optical propagation in the present device, and the basic equation is given as below 
where Ex and Ey are the transverse electric components, k 0 is the wavenumber in free space, n=n(x, y, z) is the refractive index in guiding medium, and n̄ is the reference index. It is noted that Axxx and Axyy are x-dependent operators, Ayxx and Ayxx are y-dependent operators, and Axy and Ayx are cross-coupling terms (CCTs).
A modified alternating direction implicit (ADI) algorithm  is applied to discretize Eq. (1) along the longitudinal direction. Here we introduce the algorithm in brief as follows. A propagation step is split into two substeps. At the first substep, the field propagates in the absence of the CCTs, and then it is evaluated and double used at the second substep. The order of the two substeps is reversed for each transverse electric field component so that the CCTs are always expressed in an implicit form. Therefore, the operator inversion for the CCTs is avoided by using operator splitting technique, and the calculation is efficient since the resulting matrix is tridiagonal. The finite difference (FD) scheme for operators Axxx, Axyy, Ayxx and Ayxx along the transverse directions are the same as those used in Ref. . Since the accuracy is severely restricted by the FD scheme for the CCTs which are indispensable in a 3D-FV-BPM, we construct a six-point FD scheme for approximating the CCTs. The derivation is similar as that described in Ref. , and the results are given as below
Two more adjacent points corresponding to electric fields are added on the RHS in Eq. (3). Therefore, the present formula is a six-point FD scheme and is independent of specific structures of waveguide since they are expressed in explicit forms. So, the procedure is simple and versatile. For boundaries of the computational window, the transparent boundary condition (TBC)  is used in this algorithm in order to eliminate the non-physical reflection.
3. Operating principles
The operating principle of the MMI demultiplexer is based on the self-imaging effect , in which the guided-modes of a multimode waveguide are excited and interfere constructively to produce single or multiple images of an input field launched by usually single-mode optical waveguides at the one end of the structure at periodic intervals along direction propagation. Under the restricted interference mechanism, a direct or mirrored image of the injected field is formed at the coupler length of L MMI=pL π if p is an even or odd integer, respectively, in which L π is the so-called beat length and is defined as below 
where β 0 and β 1 are the propagation constants for the two lowest order modes and can be calculated by using mentioned-above FV-MGM mode solver. To realize the restricted self-imaging effect, the input/out waveguide of a 2×2 MMI coupler should be placed at a lateral offset of , which respect to the center of the MMI region, so that every third mode will not be excited. As a result, the beat length is three times shorter than that of the general interference mechanism.
The proposed wavelength demultiplexer is schematically shown in Fig. 1(a). The device consists of three parts: an input channel, a MMI coupler, and two S-bent output channels, in which one is a bar-coupler and another is a cross-coupler. To make the device as short as possible, the MMI coupler under the restricted interference mechanism is introduced into the present demultiplexer. When two wavelengths of λ 1=1.30μm and λ 2=1.55μm are injected in the input waveguide, they can be separated if the length L MMI of the MMI coupler satisfies the following relation
where p is a positive integer, q is an odd integer, and Lλπ is the beat length at wavelength λ. As a result, a direct image of the injected field is formed for one wavelength, which corresponds to the bar-state, and a mirrored image is formed for another wavelength, which corresponds to the cross-state.
4. Numerical results
The slot waveguide, whose cross section is illustrated in Fig. 1(b), is utilized in the proposed wavelength demultiplexer. The waveguides parameters are as follows: the refractive index of high index region, slot region, and cladding are taken as nf=3.50/3.48 (Si), ns=1.60/1.58, and nc=1.48/1.46 (SiO2) at 1.30/1.55μm wavelengths, respectively. wf is the width of the slot waveguide, hf and hs are the height of the photonic wire and slot region, respectively.
By using our proposed FV-MGM mode solver, it is found that the slot waveguide with wf=250nm, hf=200nm, and hs=100nm only supports the fundamental mode. So, the width is fixed at wf=250nm for both input and output channels. The field patterns of the quasi-TM fundamental mode at wavelength λ=1.55μm are presented in Fig. 2, and the effective indices are 1.9978 and 1.7423 at 1.30/1.55μm wavelengths, respectively. Because of the high index contrast at horizontal interfaces, the normal electric field (major electric component, Ey, under present study) undergoes a large discontinuity, which results in a field enhancement in the low-index region (slot region) with a ratio of . These guided modes are true guided modes confined by total internal reflections, thus there are no confinement losses. Therefore, this slot waveguide structure can improve the quality of the self-imaging as pointed in Ref.  and lead to the present demultiplexer with ultracompact size and low loss.
We also use the FV-MGM mode solver to calculate the beat lengths of the MMI coupler at both λ 1=1.30μm and λ 2=1.55μm in Quasi-TM mode. Figure 3(a) shows the beat lengths L π and their ratio χ = L λ1 π/L λ2 π as the MMI width W MMI increases. From this figure, it is noted that the beat lengths monotonously increase with the increment of W MMI for both wavelengths, and similar behavior is also observed for their ratio. This is because that difference of the propagation constants for the two lowest order modes gradually becomes small as the width of the MMI section increases. Figure 3(b) shows that the beat lengths and their ratio as a function of hs. It is seen that the beat lengths almost linearly increase with increment of hs, but their ratio decreases with the growth of hs. When selecting W MMI=3.0μm and hs=100nm, we obtain a beat length ratio of 1.197≈1.20, in which the beat lengths for λ 1=1.30μm and λ 2=1.55μm are 23.90μm and 19.96μm, respectively, from Fig. 3. If p=5 and q=1 in Eq. (2) are selected, following relation is achieved
Therefore, the MMI region acts as a cross-coupler and bar-coupler at 1.30 and 1.55μm wavelengths, respectively. So the two wavelengths can be separated. By using local guided-modes analysis in Ref. , the length of MMI section by using conventional rib waveguides in SiON/SiO2 material with the width of 7.3μm is 435.5μm. The length of the present MMI coupler is only 27.5% that of the conventional counterpart.
The performance of the proposed wavelength demultiplexer is assessed by the contrast and insertion loss defined as
where P 1 and P 2 are the intensities in the cross and bar output waveguides at λ 1=1.30μm, respectively, or the intensities in the bar and cross output waveguides at λ 2=1.55μm, respectively, and Pi are the intensity in the input waveguides. The following simulations are performed by using the modified 3D-FV-BPM as described in section 2 in quasi-TM modes. Because of the large refractive index difference and the ultrasmall size of the slot region, there is a considerable reflection at the junctions between the MMI section and the out/input waveguides. Therefore, some linear tapers between the input/output waveguides and the MMI section are introduced into the present device as shown in Fig. 1(a), which reduces the excess loss and further improve the quality of self-imaging. The width of the taper varies from 250 to 500nm in a length of 10.0μm, which ensures that this taper is adiabatic. The gap width between the two S-bend output waveguides at the end of the device is chosen as D=3μm with the bent length of Ls=40μm so that the bending-loss and the cross-coupling effect between the two output waveguides are negligible.
The contrast and insertion loss as functions of the width W MMI are shown in Figs. 4(a) and Fig. 4(b), respectively. It is seen that the performance of the demultiplexer is very sensitivity to the structure parameter W MMI. So, the width of MMI width should be accurately controlled since its fabrication tolerance is in nanometer scale. Figures 4(c) and 4(d) give the contrast and insert loss as functions of the length L MMI. Compared with the structure parameter W MMI, the performance is relatively low sensitivity to the structure parameter L MMI whose fabrication tolerance can be up to micrometer scale. For W MMI=3.0μm and L MMI=119.8μm, the contrasts are 26.03 and 28.14dB at λ=1.30μm and λ=1.55μm, respectively, and the insertion losses are below 0.2dB at both wavelengths. So the demultiplexer achieves the low loss characteristics. The evolution of the input field along the propagation distance in the center of the slot region is illustrated in Fig. 5. It is noted that the cross-sate at λ=1.30μm and bar-state at λ=1.55μm are realized.
An ultracompact 1.30/1.55μm wavelength demultiplexer based on a MMI coupler by using slot waveguides is proposed and designed. A short MMI section of 119.8μm in length, which is only 27.5% length of that of the MMI coupler by using conventional rib waveguides, is achieved with the contrasts of 26.03 and 28.14dB in quasi-TM mode at wavelengths 1.30 and 1.55 μm, respectively, and the insertion losses are below 0.2dB at both wavelengths. Although only the demultiplexing configuration is considered in this paper, the proposed device can also operate as multiplexer with the direction of lightwave propagation reversed. Moreover, in principle, further optimizing the refractive index, height, and width of slot region, a compact polarization-independent demultiplexer can be realized, and such considerations will be performed in our future work.
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