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Abstract
Efficient coupling between integrated optical waveguides and optical fibers is essential to the success of silicon photonics. While many solutions exist, perfectly vertical grating couplers that scatter light out of a waveguide in the direction normal to the waveguide’s top surface are an ideal candidate due to their potential to reduce packaging complexity. Designing such couplers with high efficiencies, however, has proven difficult. In this paper, we use inverse electromagnetic design techniques to optimize a high efficiency two-layer perfectly vertical silicon grating coupler. Our base design achieves a chip-to-fiber coupling efficiency of 99.2% (−0.035 dB) at 1550 nm. Using this base design as a starting point, we run subsequent constrained optimizations to realize vertical couplers with coupling efficiencies over 96% and back reflections of less than −40 dB which can be fabricated using 65 nm-resolution lithography. These results demonstrate a new path forward for designing fabrication-tolerant ultra high efficiency grating couplers.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Optical couplers are an essential component in silicon photonic systems. Grating couplers in particular present a number of advantages over alternative coupling methods [1] by providing a flexible means of interfacing high-index-contrast integrated optical devices with the outside world. In general, a grating coupler consists of a high index slab with either partially- or fully-etched periodic corrugations which cause light to scatter out of the slab at a desired angle. Most challenging are grating couplers that couple light in the direction normal to the slab’s top surface (“perfectly vertically”), as they are more susceptible to strong back reflections which results in lower coupling efficiencies and system feedback problems. Compared to grating couplers which scatter light at a small angle, designing perfectly vertical grating couplers is inherently more difficult [2–7].
In general, grating couplers are designed to couple light into a single mode fiber. In order to do so, the grating must be chirped (i.e., the grating parameters must be varied gradually along the length of the grating) such that the intensity of scattered light matches the slowly-varying Gaussian-like mode of the optical fiber. Fulfilling this requirement necessitates that we independently optimize each period of the grating. In doing so, it is essential that we have enough degrees of freedom in each period of the grating in order to control the amplitude and phase of the different modes to which an input wave can couple. If we consider an input guided mode propagating through a period of the grating as depicted in the left half of Fig. 1 (the right half depicts the case of a free-space wave incident on the grating which can be handled in a similar manner), the relevant modes of the system are the transmitted guided mode within the grating, the reflected guided mode, and the downward scattered wave. We must control the amplitude and phase of these three modes, and hence there are six variables in total. It is important to note that by properly manipulating the amplitude and phase of the transmitted wave and by suppressing the total downward scattered wave and reflected mode of the grating, we automatically obtain the desired upward-scattered mode as a result.
Fig. 1 Diagrams of the possible modes that an input (blue dashed arrow) can couple to in a grating coupler. The left diagram depicts the chip-to-fiber coupling case while the right depicts the fiber-to-chip coupling case. In both situations, the input mode can couple to four modes of the grating. By controlling the amplitude and phase of three of these modes (represented by red arrows), the input is automatically coupled to the output mode (blue solid arrow) in the desired way. This tells us that at least six degrees of freedom are required to independently control all of the variables (amplitudes and phases) of the grating coupler.
In the case of a typical partially-etched grating coupler, the available degrees of freedom are the local period, local duty factor, grating film thickness, and etch depth. It would appear that there are not enough degrees of freedom (of which there are four) to independently control the six amplitudes and phases that are present in the grating coupler. This becomes evident when we attempt to use the available degrees of freedom. First, we can select the film thickness and etch depth to control the amplitude and phase of the downward scattered wave in order to reduce scattering in the downward direction [8, 9]. Next, we choose the duty factor which controls the amplitude of the transmitted wave (and hence controls the intensity distribution of the scatter wave–a process which we call mode matching [10]). Finally we choose the local period of the grating in order to set the desired phase of the transmitted wave and therefore the angle of the scattered beam. This leaves us with no additional degrees of freedom with which to independently control the backward reflected wave. Often, this problem is partially handled by choosing to scatter light at a small angle from normal which inherently results in a reduced back reflection [10]. Nonetheless, without additional degrees of freedom, our control over all six of these variables is limited, which ultimately places limits on the maximum efficiency of the grating coupler.
A promising source of the needed degrees of freedom can be found by adding a second layer to the grating coupler, as shown in Fig. 2. So-called “two-layer grating couplers”, which have recently received attention due to their inherent high directionality (defined as the fraction of scattered power which is coupled into upwards-propagating modes) [11–13] are formed by stacking two partially- or fully-etched grating couplers on top of one another. When designed properly, these two-layer gratings act as a phased array of scatterers which couple light out of the waveguide with a high directionality and is depicted in the left half of Fig. 2. A guided wave propagating from left to right along the grating will encounter grooves in the top and bottom surfaces of the grating which couple some of the propagating power into free-space modes. The grating achieves high directionality if three conditions are satisfied. First, the horizontal separation between grooves in the top and bottom layer is chosen such that the guided mode accumulates a phase shift of π/2 as it propagates from a groove in the bottom layer to its adjacent top layer groove. Second, the thicknesses of the top and bottom layers and etch depths of the grating are chosen such that a wave scattered out of a bottom groove acquires an additional π/2 phase shift relative to a wave scattered out of a top groove. Third, the scattering strength of the two layers should be equal. The resulting asymmetry will result in scattered waves which interfere constructively in the upwards direction and destructively in the downwards direction. In addition to enabling a directionality of nearly 100%, the relative shift between the top and bottom layers produces two sets of reflected guided waves within the grating which are π out of phase as depicted in the right half of Fig. 2. These waves destructively interfere, resulting in an exceptionally low back reflection, even for perfectly vertical coupling.
Fig. 2 Description of the operation of two-layer grating couplers. The two-layer grating resembles two fully-etched grating couplers which have been stacked one on top of the other with the top layer shifted forward by a small amount relative to the bottom layer. When each grating layer is one quarter of a wavelength thick and the spacing between grooves in each layer is a quarter wavelength, then constructive interference in the upwards direction and destructive interference in the downwards direction can be achieved. Furthermore, under these conditions, waves reflected back into the input of the grating destructively interfere leading to an inherently low back reflection.
Based on these results, it is apparent that the particular structure shown in Fig. 2 adds additional degrees of freedom which control both the amplitude and phase of the reflected and downward-scattered waves. This leaves us with the amplitude and phase of the transmitted wave (and hence the angle and intensity of the scatter field), which we can control using the period and duty factor of the grating. At least conceptually, this new structure provides enough degrees of freedom to independently control all of the variables of the grating coupler. In practice, it is difficult to achieve exact cancellation of the reflected and downward scatter waves, especially when chirping the grating to achieve the desired mode matching. We handle this by applying numerical optimization techniques which can simultaneously chirp the grating and fine-tune the parameters in order to achieve the desired wave cancellation. This fine-tuning is achieved by allowing (1) the duty factors of the top and bottom layer to be adjusted independently of one another and (2) the relative spacing between grooves in the two layers to be adjusted. Because of the favorable asymmetries of the structure depicted in Fig. 2, we find that adjusting these parameters is sufficient for designing grating couplers whose efficiencies approach 100%. These numerical optimization techniques, which we describe in the next section, have enabled us to design these two-layer grating coupler structures with record breaking coupling efficiencies.
2. Grating coupler optimization
The introduction of a second grating layer inherently increases the complexity of the grating coupler and hence the difficulty associated with designing an efficient structure. In order to handle this complexity, we have employed gradient-based optimization using the adjoint method [14–16] which is capable of rapidly optimizing electromagnetic structures with many degrees of freedom. In conjunction with the adjoint method, we take advantage of boundary smoothing techniques which are particularly well suited to grating couplers whose shapes are constrained [17].
Successful optimization of a grating coupler is strongly contingent on three important choices: (1) the figure of merit that properly describes its efficiency, (2) the parameterization of the geometry, and (3) the initial choice for the geometry. In general, we design grating couplers with the intention of coupling an input waveguide mode to a desired output waveguide (e.g. an optical fiber) or free-space mode. As is such, the most appropriate figure of merit is the mode overlap between the fields scattered from the grating and a desired field profile. This mode overlap describes the fraction of power in an electromagnetic field which can couple into a desired mode, which in the case of grating couplers is most commonly the approximately Gaussian mode profile of an optical fiber. Assuming that no light is reflected back towards the grating, the mode overlap expression used to calculate the grating coupler efficiency can be simplified to [17]
where E is the incident electric field, Hm is the magnetic fields of the desired mode profile, and Psrc and Pm are the total input source power to the system and the power carried in the desired mode, respectively. The integral is computed over a plane large enough to encompass the incident and desired fields. For our purpose, Em and Hm correspond to the electric and magnetic field of a Gaussian beam with a 10.4 μm mode field diameter.The optimization process consists of improving the grating coupler efficiency given by Eq. (1) by varying designable parameters which define the structure. In order to couple to a Gaussian output, the intensity of light scattered from the grating should grow and then fall gradually over the length of the grating. This suggests that the scattering strength of the grating should also gradually increase and decrease along the grating’s length. The evolution of the grating parameters that accomplishes this is given by the grating chirp function. It is advantageous to choose a parameterization of the structure which we can guarantee will generate a well behaved grating chirp. We have found it convenient to represent this mathematically as a Fourier expansion over the length of the grating (this is not to be confused with the periodicity of the grating teeth). To understand this parameterization, consider Fig. 3. Each tooth and immediately preceding gap of the grating is assigned an (increasing) integer index. The local period, labeled Λt and Λb (where the t refers to the top layer and the b refers to the bottom layer) in Fig. 3, are then defined as a function of this index. In our optimizations, the functional form of the period is expressed as a Fourier series expansion given by
where n is the grating index, M is the total number of Fourier terms, and N is the number of periods in the grating. In this expression, the coefficients an and bn are the design parameters of the grating; by modifying their values, an arbitrary grating chirp can be realized (assuming enough Fourier terms are present). In order to control how rapidly the dimensions of the grating can change along the grating’s length, the number of Fourier terms can be reduced. In practice, we have found five sine and five cosine Fourier components to be sufficient for representing the chirp functions. The duty factor (defined as the fraction of a period of the grating which is unetched), meanwhile, is expressed in an identical fashion.Fig. 3 The grating coupler is parameterized in terms of local “periods” and duty factors which evolve along the length of the grating. The distance between the onset of one gap and the next gap of the grating is the period Λ while the duty factor describes the fraction of this period that is unetched. Each period of the grating is assigned an index n and the way in which the period and duty factor evolve along the grating is defined using a smooth chirp function of this index n.
A final requirement for the optimization of grating couplers is the selection of a “good” starting structure. The choice of a good starting point is particularly important for solving non-convex problems using convex optimization techniques which make no guarantees about global optimality of the solution. This is further complicated by the wave nature of electromagnetics which tends to introduce many local optima. In the case of grating couplers, however, we can take advantage of our understanding of the physical principles of the device in order to select a starting structure that will enable local optimization to find a highly efficient structure. As we noted previously, the strength with which we scatter light must gradually increase from zero along the length of the grating in order to produce the desired Gaussian mode profile. Since we have constrained ourselves to a single layer thickness and etch depth, the primary way to accomplish this is with grating slots that are very small at the beginning of the grating and slowly increase over the length of the grating. Based on this intuition, it makes sense to choose a starting structure with small slots (or high duty factor). Given this initial duty factor, the period should be chosen to generate a beam that propagates at the correct angle in order to ensure that our starting design is as close as possible to a desirable solution. In the case of the two-layer grating with a large duty factor, the period is given by (see Appendix for derivation)
where λ is the free-space wavelength, D is the duty factor, ne1, ne2, and ne3 are the effective indices of a wave propagating through the different sections of the grating (depicted in Fig. 10), n0 is the cladding index, θ is the angle relative to normal that the generated beam makes, and m is the desired diffraction order (which is generally equal to 1).With a starting geometry, grating parameterization, and an appropriate figure of merit selected, we may optimize our structure with relative confidence that a reasonable solution will be found.
3. Optimization results
For the purpose of these optimizations, we use a custom finite difference frequency domain (FDFD) solver to simulate Maxwell’s equations. Easy access to the internals of the simulation software simplifies the application of our optimization methods. Because 1D grating couplers are significantly wider than they are thick, they are effectively modeled in two dimensions.
The starting structure for the optimization is shown in Fig. 4 and consists of a uniform two-layer silicon grating clad both on top and underneath with silicon dioxide. The grating structure is excited by the fundamental TE mode of the waveguide shown at the left edge of Fig. 4 at a wavelength of 1550 nm. In the following optimizations, we only consider this single wavelength (although the same method could be applied to a broadband figure of merit if desired). Given this wavelength, we choose the layer thicknesses to be 110 nm to produce the desired π/2 phase shift between the top and bottom layer, resulting in a total film thickness of 220 nm. Based on our previous reasoning, we choose a starting duty factor of 80%, a grating period of 586 nm, and a shift between the top and bottom layers of 160 nm to ensure that the starting efficiency is reasonably high, which in this case was 48%. Finally, a total of 26 grating periods are used, which we found sufficient for scattering all of the light out of the grating.
Fig. 4 Plot of the initial starting geometry used in the optimization of a two layer SiO2-clad silicon grating coupler. A uniform grating with a duty factor of 80% and a period of 586 nm is chosen for both the top and bottom layers. This choice of period and duty factor results in a nearly vertical beam with a high directionality.
This initial structure is modified using our gradient-based optimization methods in order to maximize the efficiency with which the grating couples light into a 10.4 μm mode field diameter Gaussian beam corresponding to the approximate mode of a single mode fiber which is situated 2 μm above the grating and oriented normally to its top surface. This efficiency is evaluated using Eq. (1) where Hm is the magnetic field of the desired Gaussian fiber mode. It is worth noting that while the modes of real single mode fibers may not be exactly Gaussian, our method can be used for any desired field profile (i.e. with non-Gaussian Hm) as long as that field is known.
The optimization process is executed until the figure of merit changes by less than 10−5, which takes 67 iterations of the Broyden-Fletcher-Goldfarb-Shanno (BFGS) minimization algorithm (requiring 166 simulations in total and about two hours of computation time on a single computer core). The corresponding optimized structure and simulated electric field is displayed in Fig. 5. As desired, light is coupled vertically and downwards coupling is almost entirely suppressed. The wavefronts, furthermore, are very flat and well-behaved; the quality of this generated beam is reflected by a record chip-to-fiber coupling efficiency of 99.2% (−0.035 dB). This coupling efficiency is significantly higher than previously reported values [13] and does not rely on coupling to narrow mode field diameter fibers or steep emission angles as in previous related work [12].
Fig. 5 Optimization results for our perfectly vertical two layer grating coupler. The real part of Ez, which has been overlayed with an outline of the optimized refractive index, shows perfectly vertical emission with an extremely high directionality and flat wavefronts. This optimized structure is very well mode-matched to the mode of a 10 μm mode field diameter single mode fiber located 2 μm above the grating surface which is reflected by a chip-to-fiber efficiency of 99.2% (−0.035 dB).
The mode matching capabilities of the optimized grating are more quantitatively demonstrated by the slices of Ez shown in Fig. 6. As is readily apparent, the magnitude of the simulated electric field very closely matches the desired Gaussian field profile, with the exception of some weak rippling. We attribute this rippling to the inherently discrete rectangular etches of the grating. This has a minimal impact on the overall efficiency, which is demonstrated by the calculated mode match of over 99%. As with the electric field amplitude, the simulated phase is very flat over the majority of the beam’s width, deviating only near the edges of the beam. This deviation does not lead to an appreciable decrease in efficiency since the field amplitude is nearly zero far away from the beam’s center where the phase begins to fluctuate. In addition to mode matching, the back reflection of waveguide mode incident on the grating coupler is suppressed by −43 dB for our optimized design.
Fig. 6 Plots comparing the simulated electric field amplitude (top) and phase (bottom) of the optimal grating coupler design to the desired amplitude and phase. The visible deviation in the simulated phase is inconsequential as it occurs only when the field amplitude is very small.
The unprecedented efficiency of this optimized grating is a direct consequence of the chirping of the grating period and duty factor visible in the insets of Fig. 5 which is plotted as a function of grating index (position along the grating) in Fig. 7. The optimal chirp function for the duty factor (upon which the scattering strength of the grating depends most strongly) of the top layer roughly matches the behavior of the theoretical scattering parameter for an ideal grating coupler matched to a Gaussian field profile [18]. The chirp function of the bottom layer’s duty factor, meanwhile, deviates from this ideal behavior, highlighting the strength of our optimization methods: its ability to design structures that would otherwise be difficult or impossible to design “by hand.”
Fig. 7 Plot of the chirp functions for the optimized two-layer grating. The blue curves show the the period as a function of the position (index) along the grating. The red curves, meanwhile, show the duty factor along the grating. In both sets of curves, the solid trace corresponds to the top layer while the dotted curve corresponds to the bottom layer of the grating.
Figure 7 also highlights the primary shortcoming of these optimization results: the optimal design contains duty factors that approach 100%. This means that the optimal structure contains features that are as small as a few nanometers wide and hence cannot be fabricated using available deep UV lithography. Intuitively, if we are willing to sacrifice some amount of efficiency, we should be able to constrain the design such that the optimal design can be fabricated. We might try to accomplish this by simply eliminate features below a minimum size after the optimization is complete or by placing lower bounds on grating dimensions, making it impossible for small features to ever develop during the design process [19–21]. Unfortunately, these brute force approaches will likely leave us with a non-optimal design. Instead, it is desirable to introduce this minimum feature size constraint directly into the optimization and use our unconstrained design as a starting point.
In order to accomplish this, we modify our original figure of merit such that the efficiency of the grating is penalized when small features form. The new figure of merit is given by
where F(E, H, p) is our new constrained figure of merit, p is the set of design variables (i.e. grating dimensions), η(E, H) is the mode match efficiency discussed previously, and fpenalty(p) is a function of the design variables which penalizes the efficiency when the feature sizes of the grating are too small. For this penalty function, we use a smooth approximation of a rect function which is positive when a feature in the grating is greater than zero and smaller than the specified minimum feature size and tends towards zero otherwise. This function is computed for each gap width in the grating and summed up before subtracting from the efficiency. Because this penalization process operates on each individual gap and tooth in the grating, we no longer use a Fourier series parameterization in subsequent optimizations, but instead opt for parameterization in which each gap and tooth dimension is a separate independent design variable. Since we begin with an optimized design with a smooth chirp function, there is less danger of falling into a non-robust local optimum. Finally, in order to control the influence of this penalty function, we adjust its maximum value and the steepness of its edges.For the purpose of exploring the impact of minimum feature size on grating coupler efficiency, we ran a series of constrained optimizations for a specific set of minimum feature sizes. Beginning with the previous optimized structure shown in Fig. 5, we introduce a penalty function which is very weakly weighted and then gradually increase its strength in a series of separate optimizations. In total we perform three optimizations per minimum feature size, strictly enforcing the minimum feature size before running the third optimization.
The result of this process, which we performed for a set of minimum feature sizes between 0 nm and 180 nm, is shown in Fig. 8. Using constrained optimization, we are able to maintain exceptionally high efficiencies out to more practical feature sizes. Of particular interest is the 65 nm constraint which, due to the maturity of the 65 nm CMOS platform, shows future promise for silicon photonics [22]. For a minimum feature size of 65 nm, we have achieved an optimized efficiency of 96.9% (−0.137 dB). Furthermore, better than −0.5 dB is achievable out to a minimum feature size of over 130 nm, roughly corresponding to lithography technologies that have already been used in commercial nanophotonic settings [1]. As the minimum feature size is further increased, the efficiency of the optimized design begins to fall off quickly. This is a direct consequence of our inability to match to the gradually changing tails of the Gaussian mode. These results are very promising for both current and future applictions, especially considering that such structures have already been fabricated using an existing CMOS process [12] which is capable of resolving the feature sizes we have considered.
Fig. 8 Plot of the coupling efficiency for perfectly vertical grating couplers optimized with a minimum feature size constraint. Using the “ideal” optimized result as a starting point, additional constrained optimizations are performed in order to design gratings that can be fabricated with lithography that has a limited resolution.
In addition to maintaining a high efficiency at the design wavelength, our feature-size-constrained designs also maintain a reasonable bandwidth. Figure 9 shows the coupling efficiency plotted as a function of wavelength for the “ideal” unconstrained design and the 65 nm design. In both cases, the 1 dB bandwidth is about 24 nm. This is largely due to the number of periods in the grating: in general, the more periods a grating has, the narrower its bandwidth will be. The number of grating periods used was kept the same, independent of the minimum feature size. If necessary for a given application, this bandwidth could be increased by coupling into a fiber with a smaller mode field diameter which would allow us to reduce the number of periods within the grating. In addition to the number of grating periods, the insertion loss bandwidth is also reduced some as a result of the narrower wavelength response of the back reflection plotted in the bottom of Fig. 9.
Fig. 9 Plots of the coupling efficiency (top) and back reflection (bottom) as a function of wavelength for the ideal optimized result (blue dashed line) and the 65 nm constrained minimum feature size result (red solid line). Both cases achieve a high peak coupling efficiency at 1550 nm as well as a modest 1 dB bandwidth of 24 nm. The reflection in both cases is exceptionally low, reaching below −40 dB at 1550 nm.
Fig. 10 Diagrams of a two-layer grating with large duty factor (left) and smaller duty factor (right). In both diagrams, d denotes layer offset and Λ denotes grating period (which for the sake of simplicity are assumed to be equal in both layers). The lines marked ne represent the effective index in each section of the grating. Although depicted here as being approximately equal, the thicknesses of the two layers need not be equal, and thus ne2 and ne3 will not necessarily be the same.
Depending on the application, the reflective properties of the grating may be very important. In the case of both our unconstrained and constrained designs, the back reflection into the input waveguide is exceptionally low at the design wavelength. At 1550 nm, the unconstrained and constrained designs achieve reflections of −43 dB and −41 dB, respectively. Although not explicitly part of the figure of merit used during optimization, reducing reflection is important to improving the overall insertion loss of the device. It is thus not unexpected that the optimal solution has a very low reflectivity. It is worth noting that while the reflection is so low at the design wavelength, it increases very quickly as the wavelength deviates from 1550 nm, as indicated by the −20 dB bandwidth of only 8 nm. In situations in which low reflection is required, the useable bandwidth of the device may be significantly smaller than the 1 dB insertion loss bandwidth. This is in part due to our choice to emit vertically, which likely increases the sensitivity of the back reflection to variations in wavelength. If a more broadband back reflection and coupling efficiency is required, gratings designed to couple light at a small angle could be used. Designing such gratings is a minor modification to the process we have presented here.
In our previous discussion, we considered only the chip-to-fiber coupler case. Depending on the application, the fiber-to-chip coupling case may be equally important. Unfortunately, it is not immediately apparent that the two cases should act in a reciprocal way. In order to get a sense of how well our designs perform in he fiber-to-chip coupling case, we have performed additional simulations of the structure at the design wavelength of 1550 nm. In these simulations, the grating is excited by a 10.4 μm mode field diameter Gaussian beam and the coupling efficiency into the grating coupler waveguide as well as the back reflection into the input fiber are calculated. In the case of our unconstrained design, we found that the coupling efficiency was identical to the chip-to-fiber case shown previously while the back reflection into the fiber was slightly reduced to −44 dB. In the case of our 65 nm constrained design, the coupling efficiency was once again the same as the chip-to-fiber case shown preivously, while the back reflection was increased to −37 dB (which is still uniquely low). Despite this small increase in back reflection, it is important to note that the design was optimized for the chip-to-fiber coupling case. We expect that re-optimizing for the fiber-to-chip case will result in lower reflections.
4. Conclusion
Using improved gradient-based shape optimization methods, we have designed two-layer silicon grating couplers which couple 1550 nm light perfectly vertically into single mode fibers with over 99% efficiency. Applying constrained optimization techniques, we have demonstrated that exceptionally high efficiencies can be maintained even when enforcing minimum feature sizes. Of particular interest, we have achieved chip-to-fiber coupling efficiencies in excess of 96.9% (−0.137 dB) that can be fabricated using 65 nm lithography, a result which to our knowledge is the highest reported.
In our optimizations, we considered only 1D grating couplers which is generally sufficient for the purpose of designing single polarization grating couplers. Equally important is the polarization-splitting grating coupler which splits mixed-polarization light from an optical fiber into two output waveguides [23]. Designing these polarization-splitting grating couplers requires full 3D simulations and are thus not directly compatible with the work we have presented here. Fortunately, our optimization methods are sufficiently general that they can be applied to full three dimensional problems, and we hope to tackle this challenge in future work. Such challenges are a perfect opportunity for us to make use of inverse electromagnetic design. Between the rapid growth of silicon photonic processing and the power of these optimization tools, truly efficient vertical optical coupling is well on its way to becoming a reality.
Appendix A: Two-layer grating equations
In the design of gratings, it is important to understand how the grating parameters (effective index of the guided waves, period, and duty factor) affect the angle of the beam generated by the grating. Knowing this relationship is essential from the standpoint of choosing an initial geometry for an optimization as well as understanding the chirp function that results from an optimization.
To derive a relationship relating the physical grating parameters to the angle of the generated beam, we begin with the grating equation which is true of any periodic grating coupler:
In this expression, kg is the wavenumber of a guided wave propagating along the grating, Λ is the grating period, m is the diffraction order, k0 is the wavenumber of the cladding material, and θ is the angle that the generated beam makes relative to the grating normal. For the sake of simplicity, kg is often approximated as being equal to the effective wavenumber of the mode propagating in the unetched waveguide at the input to the grating, however for even modest etch depths, this is not accurate. If we instead rewrite (5) in terms of the phase that a wave accrues over one period of the grating, we need not make any approximations for kg. Multiplying each side of Eq. (5) by Λ and rearranging terms slightly yields, where φg = kgΛ is the phase a guided wave in the grating must accrue over one period in order to generate a free-space wave propagating at an angle θ to normal. This expression defines the relationship between the physical parameters of the grating and the angle of the generated beam.It is desirable to use Eq. (6) in order to derive expressions for the period of two-layer grating with a specified duty factor D and desired coupling angle θ. To do so, there are three different grating configurations that must be considered independently. Fig. 10 depicts the two configurations that we care about the most. In the left diagram, the grating has a large duty factor (i.e. small gaps), which results in sets of fully separated gaps. In this case, a wave propagating through the grating will travel through regions with up to three different effective refractive indices, which will determine the phase acquired by the wave along one period of the grating. In the diagram on the right, the grating has a smaller duty factor which results in overlapping gap regions. In these overlapping gap regions, the effective index is equal to that of the cladding index. A third possible configuration occurs when the duty factor becomes very small resulting in a grating which resembles an inverted version of the top diagram in Fig. 10. This is not a desirable regime of operation, however, and will not be considered.
In the case of a large duty factor, the phase acquired by a guided wave over one period of the grating is given by
where k1, k2, and k3 are the wavenumbers corresponding to the effective indices marked in Fig. 10 and D is the duty factor of the grating. Substituting this result into (6) and solving for Λ yields the grating period needed to generate a beam at the desired angleA second important quantity is the horizontal shift between the bottom and top layers of the grating, labeled d in Fig. 10. As discussed previously in this manuscript, the guided wave must accrue a phase of π/2 between the gap in the bottom layer and the gap in the top layer in order to achieve asymmetric emission from the grating. Equating the phase that the wave picks up between the beginning of the first gap and the second gap to π/2 and solving for d yields the shift between the layers in the case of a large duty factor:
Note that these two results are only valid when the gaps of the top and bottom grating do not overlap, that is when d ≥ (1 − DF)Λ. Substituting for Λ in this expression yields a definition for “large duty factor:”
For a 220 nm thick silicon grating clad in SiO2 with roughly equal layer thicknesses, a “large” duty factor is over 74%.For duty factors which are below this value, we must use a different set of expressions corresponding to the case depicted in the bottom of Fig. 10. The derivation of the period relevant to this configuration can be found in the same manner as was used for Eq. (8) and has as a result
where we have implicitly used the fact that the shift between layer needs to be Together, the expressions derived in this section provide us with the tools we need to design the most basic two-layer grating.Funding
Office of Naval Research (ONR) (N00014-14-1-0505 and N00014-16-1-2237).
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