As the number of transistors on a many-core processor increases, Rent’s Rule [1] implies that the total number of wires of all lengths will increase according to a power law distribution, requiring an ever-increasing number of layers of global interconnect on the die. However, metal interconnects are becoming the performance bottleneck for modern integrated circuits (ICs) with deep sub-micron feature size and multi-core architecture. Various problems associated with the metal interconnect, such as power dissipation, signal latency, limited bandwidth and electric mitigation, become more and more serious[2, 3] as the feature size decreases and number of cores increases. Building on-chip, massively parallel photonic interconnect network composed of dielectric waveguides is believed to be an effective solution to satisfy the ever-increasing demands for the interconnect capability[3, 4]. By relying on optical waveguides for all global communication, every wire longer than a few millimeters could be replaced by a waveguide capable of carrying many communication channels at different frequencies [3]. Various components and devices that are required in such a interconnect network, such as on-chip lasers[5, 6], filters and modulators[7, 8, 9, 10, 11, 12], and photo detectors[13, 14], were invented in recent years. One of the problems faced by the on-chip optical interconnect is the crossing of two waveguides. Having two interconnect lines that crosses each other is a very desirable feature of an interconnect network. For metal interconnects, the crossing is realized by routing one interconnect line above or below the other as a “bridge”, through another interconnect layer or through the poly-silicon layer. Such a “bridge” structure cannot however be realized on sort length scales for two optical waveguides because of the drastic bending losses. Two dielectric waveguides can cross directly , as we see in Fig. 1(a), however, an optical signal propagating through such crossing will typically experience scattering, both into radiation modes and to guided modes, generating a back-reflected wave and crosstalk. Although the power loss due to scattering into radiation modes may be compensated by increasing the power carried by the signal, the increased background power level may corrupt other signals in the network, especially when the cross is close to a photodetector. Further, back-reflection and crosstalk are more serious problems since they degrade the integrity of the signal and create errors in the photonic network. While these errors could in principle be mitigated by some error correction scheme, this would create some significant overhead and reduce the interconnect bandwidth. The ability to efficiently intersect waveguides with little or no signal deterioration would greatly increase architectural design flexibility, offering the promise that the global interconnect could be encapsulated in a single optical layer on a bonded die. Previous works tackled this problem by expanding the waveguide around the cross[15, 16] or by varying the angle of the cross[17]. In this paper, we describe an effective and easy-to-realize method to reduce the power loss at the crossing of two on-chip dielectric waveguides without changing the lateral shape of the cross.

We first study the propagation of a guided mode through a crossing of two waveguides. The three dimensional (3D) model used in the simulation is shown in Fig. 1(a). The simulations were performed with the finite-difference time-domain method using MEEP, a freely available software package with subpixel smoothing for increased accuracy[18, 19]. Uniform grids of 12.5nm are used with time step of 2.1 × 10^-17 second selected by the program by default. Perfectly matched layers (PMLs) are placed at each side of the model to absorb the out-going propagation. The two waveguides that cross each other are identical, and are made of Si (n = 3.6) buried in a host medium of SiO₂ (n = 1.4). The cross section of the waveguide has a rectangular shape of size a×b where a = 250nm, b = 500nm. To launch the incident guided mode, a sinusoidal current source whose frequency corresponds to that of the 1.55μm free space wavelength is placed inside the incident waveguide far away from the crossing (not shown in Fig. 1(a)). The electric and magnetic field at different time steps on plane i, ii and iii (refer to Fig. 1(a)) are then recorded. The record begins after the transient signals at each plane have disappeared, and the magnitude and phase of the signals for the 1.55μm wavelength component can then be extracted from the time domain data. For plane i and ii, the electric and magnetic field of the 1.55μm wavelength component actually contain both the guided mode and the radiating mode components of the waveguide, and the electric field can be expanded into the eigenmodes of the waveguide as

here E _i,ii is the phasor of the electric field at 1.55μm wavelength achieved above, on plane i or ii. E _g is the guided eigenmode, and E _r is the radiating eigenmodes related to the propagating constant k _r (can be real or imaginary), of the waveguide. k _t is the transverse wavenumber related to the eigenvalue k _r. The integral should go through all possible k_r and k _t, and x is the local coordinates on the plane i or ii. Notice the waveguide is designed to support only one guided mode. We are interested in the magnitude of the guided mode propagating along the exiting waveguide and the crossing waveguide, i.e. a_i, and a_ii in Eq. (1). This can be achieved using the orthogonality property of the waveguide eigenmodes[20]. Cross product both sides by H ^* _g(x), the complex conjugate of the magnetic field of the guided mode, and integrate over the whole plane, we have

because the integral from the radiating modes give zero due to the orthogonality property [20]. P_g is twice the power carried in the guided mode and is defined as P_g = ∫_x E _g × H ^* _gdx. The integral should go over the whole plane to the infinity. However, since the guided mode decays evanescently as it goes away from the waveguide, the integral over a finite size, but large enough plane is a good approximation, which is what we do here. The field distribution of the guided mode E _g, H _g is achieved from numerical simulations with finite element method (FEM) using a commercially available software package COMSOL^TM, and the mode shape is shown in Fig. 2(a) together with the 3D plot of the waveguide. From Eq. (2) we can calculate the magnitude of the guided mode (thus the power carried in it) that propagates along the exiting and crossing waveguide.

The field recorded on plane iii at different time steps contains both the incident and the reflected signals, while we are only interested in the reflection due to the existence of the crossing. To get rid of the incident signal, the former simulation is repeated with the crossing waveguide removed, but all the other conditions remained the same. The electric and magnetic field at the same position as plane iii is then recorded at the same time points. This is the “incident field” and can be subtracted from the former-recorded field on plane iii to achieve the reflected part. Such a reflected field still contains both the radiating and the guided mode components, and the guided mode part can then be calculated following a similar procedure as described above. Finally, the simulation for the incident waveguide only also gives the magnitude of the incident guided mode, which can be used for normalization purpose. The power radiated into the surrounding medium is achieved by subtracting the power carried in the exiting guided mode and the two crossing guided modes from that carried in the incident guided mode.

 

Fig. 1. The first row gives the geometries of the crossings: (a), the original crossing; (b), the modified crossing with one dielectric block on the top and the other at the bottom; (c), the modified crossing with only one block on the top. Each crossing can be described as two interfaces, and the middle row depict one of the interfaces corresponding to each crossing: (a′), an interface between an incident waveguide and a slab; (b′), an interface between an incident waveguide and a slab with one bar on the top and the other at the bottom; (c′), similar to b′, but with only one bar on the top. When the right hand side of each interface is cut at a distance b (the width of the waveguide) and mounted with an exiting waveguide, they give the original or the modified crossing(s). The bottom row gives the instantaneous H_z distribution when a guided mode in the waveguide hits the interface, for the corresponding situations shown in the second row. The field distributions shown here are on the plane that is parallel to the x̂ – ŷ plane and cut the waveguides in half.

Download Full Size | PDF

The simulation results for the waveguide crossing clearly show its influence to the interconnect. In fact, only 75.5% of the power carried in the incident guided mode goes through the crossing and remain as the guided mode in the exiting waveguide. Each of the the guided mode in the crossing waveguide propagating in opposite directions carries 3.9% of the power , and 1.1% is reflected back as guided mode propagating to the opposite direction in the incident waveguide. The rest of it, about 15.6% of the incident power is radiated and distributed into the open space. Thus it is clear that it takes not many crosses before the signal reaches an power level that is not acceptable. Further more, the crosstalk, although looks relatively small (3.9%) compared to the incident signal, deserves great attentions considering the fact that the signals in the different parts of an optical interconnect network are designed to be of different strength[3]. Malfunction of the circuit may be trigered by such unwanted crosstalk signals.

An intuitive understanding of the problem is to treat the crossing waveguide as a slice of width b cut from a slab waveguide of thickness a that is infinite in the x̂ – ŷ plane. Such a slab waveguide supports guided modes that may propagate in any direction in the x̂ – ŷ plane. The mode is confined in the ẑ direction and is uniform along the direction perpendicular to the propagation, which is shown in Fig. 2(b) together with the 3D plot of the slab. Considering an interface between such a dielectric slab and the dielectric waveguide, as we shown in Fig. 1(a′). When the guided mode carried in the incident waveguide with mode shape shown in Fig. 2(a) hits the interface, the guided modes in the dielectric slab are to be excited. Because the mode shapes of the respective guided modes in the two parts (the dielectric slab and the dielectric waveguide) are obviously different, guided modes propagating to all directions in the x̂ – ŷ plane in the dielectric slab are excited. Fig. 1(a″) gives the simulated field distribution of such a situation. The plot gives the instantaneous field distribution of H_z, at the plane parallel to x̂ – ŷ plane and cut the waveguide and the slab in half. As we see from the field plot, the original, well confined guided mode in the waveguide begins to diverge as the wave enters and propagates down the slab. Radiating modes at both sides of the interface are also excited to balance the boundary conditions. Now suppose the slab is cut at a distance of b from the original interface with an exiting waveguide mounted, which in turn gives the original situation of two waveguides crossing each other, as we see in Fig. 1(a). The already-diverged wave is then to excite the guided mode in the exiting waveguide. Again, due to the obvious mode shape mismatch, reflections in forms of guided modes of the dielectric slab are excited, together with radiating modes at both sides. The guided modes in the dielectric slab that are bouncing back and fourth in between the two interfaces become the crosstalk (i.e. the guided modes in the crossing waveguide) propagating along the crossing waveguide to both directions.

 

Fig. 2. The structures considered in the problem and the shape of the guided mode they support, respectively. (a), a waveguide of 250nm × 500nm cross section. (b), an infinite slab of 250nm thick. (c), an infinite slab of 250nm thick, with a bar of 50nm×500nm on the top and at the bottom. (d), similar to (c) but with only one bar on the top. The mode shape plots give the magnitude of the Poynting vector along the propagation direction of the mode. All the structures are made of Si (n = 3.6) buried in a host medium of SiO₂ (n = 1.4)

Download Full Size | PDF

From the former analysis, it is obvious that the mode shape mismatch between the guided modes of the waveguide and the dielectric slab contributes mostly to the signal deterioration at a waveguide crossing. Thus a heuristic solution is to provide a better mode match at the interfaces. To accomplish this, we add a block above and below the crossing region, as we see in Fig. 1(b). We use the same material (Si) as the waveguides for the dielectric blocks in our study, but in general they can be made of other dielectrics if only the refractive index is larger than that of the host medium. The interface between the incident waveguide and the crossing waveguide now becomes the one shown in Fig. 1(b′), where the right hand side of the interface is a dielectric slab with dielectric bars sitting on each side. The bars are of the same width of the incident waveguide. Such a slab-with-bars structure also supports a guided mode that propagates in between the two bars, being confined in both the ŷ and ẑ directions. The guided mode shape for a dielectric slab of thickness 250nm with dielectric bars of 500nm wide and 50nm thick is shown in Fig. 2(c) together with a 3D plot of the structure. Such a guided mode can match to the incident guided mode of the original waveguide much better than the guided mode in a dielectric slab. The wave propagation through an interface between the incident waveguide and the slab-with-bars is also simulated and shown in Fig. 1(b″). Similar to Fig. 1(a″), here the instantaneous H_z distribution at the plane in the middle of the waveguide and the slab is shown. As we can see from the plot, the guided mode in the slab-with-bars region is indeed excited, so that the mode shape of the original waveguide mode is much better reserved. Again, if we cut the structure at a distance of b = 500nm from the current interface with the exiting waveguide mounted, we get the modified crossing shown in Fig. 1(b). The guided mode in the slab-with-bars region can then better coupled to the guided mode in the exiting waveguide. In this means, the blocks added above and below the crossing as shown in Fig. 1(b) may help the signal go through the crossing.

The numerical simulation of the modified crossing (Fig. 1(b)) verifies the idea we presented. The simulation process is similar to that of the original crossing (Fig. 1(a)) that we discussed before. The two blocks above and below the crossing are identical, and situations of different block thicknesses are studied. The power carried in the guided modes in the exiting waveguide, the crossing waveguide (to either direction), the incident waveguide but propagates backward (the reflection), and the power that is radiated away from the waveguide (all normalized to the power carried in the incident guided mode) are shown in Fig. 3(a) and 3(b) as solid lines with solid marks. Zero-thickness-block corresponds to the case of the original crossing. As the thickness of the blocks increase, it is clear that the crosstalk, the radiation and the reflection decreases obviously. In fact, the crosstalk power in each direction of the crossing waveguide decreases monolicitly from about 3.9% to only 0.19% as the thickness of the blocks increases from 0 (the original crossing) to about 175nm. At the same time, the power portion of the reflected guided mode also decreases greatly , from 1.15% to about only 0.09% which is achieved for blocks of 150nm thick. The obvious decrease in the reflection and crosstalk should greatly reduce the possibility of circuit malfunction. The useful portion of the power, i.e. the one that transmits the crossing and stay in the guided mode, increases from 75.5% to 89% when the block is 175nm thick, and the total radiation into the surrounding medium is reduced to about 10% from 15.6%. Thus, by adding two blocks to the crossing, the influence from the crossing to the interconnect network is greatly reduced. As the thickness of the blocks further increases, the situation begins to deterioriate: crosstalk, reflection and radiation begin to increase while the transmission begin to decrease. For the current design of waveguides, an optimized design is achieved for blocks of 150nm-175nm thick. In Fig. 4 we give the instantaneous H_z distribution at the plane parallel to x̂ – ŷ and cut the waveguides in half, for the original crossing (Fig. 4(a)) and the modified crossing with two blocks of 175nm thickness (Fig. 4(b)). The two plots are of the same color scale. Notice that the crosstalk is obviously launched for the original crossing, while for the modified crossing, the crosstalk is much weaker.

The two blocks below and above the crossing can be fabricated before and after the formation of the waveguides, requiring two extra lithography. This is in general compatible with the current planar IC process with which the on-chip optical interconnect network is fabricated. Rather than using two blocks placed symmetrically above and below the crossing, using only one block placed at either side (refer to Fig. 1(c)) may also provide reasonable improvement. In fact, a dielectric slab with a single dielectric bar placed on top forms a wedge waveguide. Such a structure also supports a guided mode confined in both the ŷ and the ẑ direction (the mode shape shown in Fig. 2(d)), and can match to the incident guided mode of the dielectric waveguide at the interface. The geometry of such an interface and the simulation result for a guided mode of the waveguide propagating through it are shown in Fig. 1(c′) and 1(c″). A guided mode to the right of the interface is also launched, although the confinement is not as good as the situation shown in Fig. 1(b″). The advantage of using only one dielectric block is that the fabrication process is simplified. Of course, as we see in the mode shape plot, the mode is not symmetric along ẑ axis, on the contrary to the mode shape of the original waveguide. Thus the effectiveness is expected to be not as good as that of the two-block case. Nevertheless, as we see in Fig. 3, the existence of a single block indeed helps reduce the crosstalk and reflection while increasing the transmission for moderate block thicknesses up to about 75nm. However, the radiation loss is not effectively reduced by the single block. As the the thickness of the block further increases, the mode shape shown in Fig. 2(d) becomes further asymmetric, thus is further mismatched with the symmetric mode of the waveguide shown in Fig. 2(a). Because of this, the crosstalk and the reflectance begins to increase as the thickness of the block increases. Thus, a trade off in the performance is required when single-block design is used which gives simplified process.

 

Fig. 3. The simulation results for the modified crossing with two blocks placed above and below symmetrically (identified as Symmetric in the plots), and the modified crossing with only one block (identified as Asymmetric in the plots). Crosstalk and reflection shown in (a), and the transmission and radiation shown in (b), are given as the percentage of the power in the incident guided mode.

Download Full Size | PDF

 

Fig. 4. The instantaneous H_z distribution for (a) the original crossing, and (b) the modified crossing with two blocks of optimized thickness 175nm, on thex̂ – ŷ plane that goes through the middle of the waveguides. Refer to Fig. 1(a) and Fig. 1(b).

Download Full Size | PDF

In conclusion, we successfully demonstrated, by numerical simulations, an effective method to reduce the signal deterioration caused by the crossing of two dielectric waveguides in a on-chip optical interconnect network. By slightly modify the geometry of the crossing, the crosstalk, reflection and radiation caused by the crossing can be greatly reduced. The method is compatible with the planar IC process. We expect this method to enhance greatly the performance of the optical interconnect, which is a critical part for the next generation multi-core IC chips. Since the current method does not change the lateral geometry of the cross, it will not influence the placement of the optical network. This is important especially when the waveguides are relatively closely placed, such as in a parallel array. The experimental demonstration of this idea is currently under study.