## Abstract

We present a compact and efficient design for slanted grating couplers (SLGC’s) to vertically connect fibers and planar waveguides without intermediate optics. The proposed SLGC employs a strong index modulated slanted grating. With the help of a genetic algorithm-based rigorous design tool, a 20µm-long SLGC with 80.1% input coupling efficiency has been optimized. A rigorous mode analysis reveals that the phase-matching condition and Bragg condition are satisfied simultaneously with respect to the fundamental leaky mode supported by the optimized SLGC.

© 2004 Optical Society of America

## 1. Introduction

Both photonic crystal [1–2] and high index contrast waveguide [3–4] photonic devices are being actively investigated for use in dense planar lightwave circuits (PLCs). To fully realize highly integrated PLCs, an efficient optical connection interface between single mode fibers or fiber arrays and high-index-contrast waveguides is required. However, coupling light into small waveguides is challenging due to the mode size mismatch between the fiber and the waveguide. The core size of single-mode fiber is typically 4–9µm while the core size of high-index-contrast waveguides is usually less than 1 to 2µm. This results in prohibitive insertion loss for simple pigtail coupling.

A common approach reported in the literature is to use tapered couplers [5–8]. In traditional adiabatic waveguide tapers, the small size waveguide is gradually tapered to a size comparable to the incident fiber core to improve the coupling between the small waveguide and the fiber. In order to avoid radiation loss, the tapering section can be quite long (on the order of one millimeter or more). On the other hand, inversely tapered waveguides (with a core size of a few hundred nanometers tapered to a small tip of tens of nanometers) can shorten the tapering length to the order of tens of micrometers [8]. An alternative taper technique is to fabricate a second waveguide layer on top of the small size high index contrast waveguide [7]. The top waveguide has a low index contrast and is designed to be single mode with a large core size comparable to the fiber core size. Efficient light coupling first takes place between the incident fiber and this auxiliary waveguide. Then the light is routed from one layer to the other by means of horizontal and/or vertical tapers of the waveguides in both layers. The above approaches share a number of challenges. These include (1) difficulty in achieving vertical tapering; (2) generally complicated processes to fabricate and integrate the tapered structure with other devices; and (3) input and output coupling can take place only on the edge of a PLC chip, which limits the number of optical I/O’s.

An alternative approach is to use grating couplers, which have been extensively studied both theoretically and experimentally [9–24]. In grating couplers, coupling takes place out of the plane of the waveguide, which permits optical I/O’s to populate any area of a PLC rather than being confined to just the edge of a PLC chip. This approach allows denser integration and PLC testing before the wafer is diced into separate chips. A particularly attractive possibility is to have the incident and exiting light at normal incidence to the waveguide layer. One drawback, however, is traditional grating couplers usually work in the weak coupling regime, in which the grating introduces a small perturbation to the underlying waveguide. They therefore require long coupling length (>100µm) in order to maximize the coupling efficiency.

To shorten the coupling length of the grating couplers, Taillaert, et al. [25] recently proposed the use of strong index modulated gratings for normal incidence coupling between single mode fibers and compact planar waveguides. Because of operation in the strong coupling regime, they realized a short (10µm long) grating coupler design with 74% theoretical coupling efficiency between single-mode fiber and a 240nm thick GaAs-AIO_{x} single mode waveguide at a wavelength of 1550nm. The measured performance of a fabricated device was 19%. Their approach employed a symmetric, rectangular shaped grating for surface-normal operation. To break the grating symmetry and realize unidirectional coupling, an extra first-order grating reflector and a multilayer Bragg-mirror like bottom reflector were introduced, which complicates the device.

In this paper, we propose a slanted grating coupler (SLGC) in which high coupling efficiency can be achieved with use of a single grating layer that operates in the strong coupling regime [26]. We were originally motivated to exam such structures by the development of a new etching technique that readily achieves slanted etches [27]. As shown in Fig. 1, our SLGC utilizes a slanted grating to couple light from a normally oriented fiber into a waveguide without any intermediate optics. There are three advantages of our approach. First, employment of an asymmetric slanted parallelogramic grating profile breaks the symmetry problem associated with the surface normal coupling between fiber and the planar waveguide. This slanted grating can therefore unidirectionally couple light into the desired waveguide direction and suppress coupling in the opposite direction. Therefore both the first-order grating reflector and the bottom reflector in Taillaert’s design can be eliminated. Second, recent studies [28–31] reveal that parallelogramic shaped gratings are excellent for coupler applications because they generally have a larger radiation factor and higher radiation directionality compared to other grating profiles. Finally, our SLGCs also utilize strong index modulation gratings to further strength the coupling effect and keep the length of the coupler short. The strong index modulation is introduced by means of a high index material for the grating ridges and a grating thickness on the order of a wavelength. All these features enable us to realize efficient and compact SLGCs designs.

However, the utilization of a strong index modulated parallelogramic grating in our SLGCs places new difficulties in their simulation and design. There has been little work reported in the literature on strong index modulated grating couplers, especially their theory. In this paper, we will investigate the rigorous design of SLGCs with a powerful parallel rigorous design tool recently developed in our group [33]. In particular, we also present a brief theory to understand and explain our optimized SLGCs. Through a rigorous mode analysis, we find that the leaky modes supported by SLGCs are much more complicated than and completely independent from the unperturbed output waveguide mode. The phase-matching condition and the Bragg condition of our SLGCs are both satisfied with respect to the fundamental leaky mode of the SLGCs. Therefore these physical principles of our SLGC are different from those of conventional weak grating couplers, where the phase-matching and Bragg condition are usually assumed to be realized with respect to the unperturbed output waveguide mode. We believe that this theory can also be applied to other strong grating coupler designs.

The rest of the paper is organized as follows. In Section 2 the structure of our proposed SLGC is first described. Then we introduce the rigorous simulation method and the micro-genetic algorithm (µGA) global optimization method we adopted for design of SLGCs. Section 3 presents and discusses two µGA optimized SLGC designs, namely uniform and non-uniform SLGCs. In Section 4, a detailed physical analysis of the SLGCs presented in Section 3 is carried out. The leaky mode characteristics of these SLGCs are solved with a rigorous mode solver. Then the phase-matching condition and the Bragg condition of SLGCs are analyzed. The paper is summarized in Section 5.

## 2. SLGC simulation and design method

Since our SLGC operates in the strong coupling regime, the effect of the grating cannot be treated as a small perturbation to the waveguide as in conventional approaches to analyzing gratings that operate in the weak coupling regime. On the other hand, rigorous electromagnetic grating theories such as rigorous coupled wave analysis (RCWA) [34] cannot be applied because of the finite extent of the grating coupler. Taillaert et al. [25] employed an eigenmode expansion method in the simulation of their device. Unfortunately, this method will be inefficient in computation if applied to our case because of the slanted shape of the grating. Owing to the need to divide each grating unit cell into a number of rectangular sub-cells to approximate the slant of the grating, we have therefore applied the finite difference time domain (FDTD) method [35] to rigorously analyze and design the SLGCs. In this paper, we address 2D simulation and design of the SLGCs.

Figure 2 is a 2D cross-section view of the SLGC shown in Fig. 1 and it is also the schematic geometry of the SLGCs simulated by 2D FDTD. The underlying 2D slab waveguide is a 1.0µm thick single mode waveguide with core and cladding refractive indices of 1.5073 and 1.46 respectively. The slanted grating is positioned on top of the waveguide core and embedded in the upper cladding of the waveguide. Silicon nitride (n=2.0 at a wavelength λ_{0}=1.55µm) is used as the ridge material for the grating. The high index contrast between the grating and waveguide materials together with a thick grating layer strongly disturbs the underlying waveguide mode in the coupling region. Because of this strong coupling mechanism, efficient coupling can occur with a short grating. The fiber we simulated is a single mode fiber with a core size of 8.3µm and core and cladding refractive indices of 1.470 and 1.4647, respectively. This fiber is simulated as a 2D slab waveguide. Berenger perfect matched layer (PML) boundary conditions [36] are used to terminate and minimize the simulation region. The whole structure fits in an overall FDTD simulation area of 40µm×8µm. A square Yee cell of 36nm is used in both the x and y directions. We found that 7000 time steps are sufficient for the FDTD simulation to reach steady state. Note that only TE polarization (electric field out of the plane) is considered in this paper.

As shown in Fig. 2, the fundamental mode of the fiber waveguide is sourced at the top of the FDTD simulation region and propagates toward the grating coupler and is eventually coupled into the waveguide traveling to the right. Four FDTD field power monitors are defined in Fig. 2 for monitoring the optical efficiencies of reflection (*η _{R}*) back toward the input space, transmission (

*η*) into the lower cladding, and left and right coupling efficiencies (

_{T}*η*and

_{LCE}*η*) to the slab waveguide. These parameters are calculated from FDTD simulation results by:

_{RCE}where *P _{j}* and

*P*are the total power detected on each monitor and

_{RCE}*P*is the power in the incident fiber mode. The

_{i}*MOI*term in Eq. (1) is the mode overlap integral between the actual FDTD field distribution detected on a monitor and the analytical mode profile of the waveguide at the specific location of the monitor, which indicates how much of the power detected on the monitor will actually be guided by the waveguide.

Parameters considered to maximize *η _{RCE}* of the SLGCs include grating period, Λ, in the x direction; the grating depth along the slanted direction, t; the fill factor (grating ridge width/period), f; the slant angle, θs; and the relative lateral position between the fiber and the slanted grating, d. To quickly and efficiently explore these parameters, we apply a parallel rigorous design tool recently developed by our group [33]. This design tools employs a parallel small population size genetic algorithm, called micro-GA (µGA) [37] as the global optimization method and a 2D FDTD method for rigorous electromagnetic computation. GAs [38] are patterned after natural evolutionary processes, e.g., survival of the fittest. Because of its small population size (usually 5), our µGA is more computationally efficient than the conventional GA (CGA) [39]. It is straightforward to apply µGA to SLGC design by simply encoding all of the SLGC parameters as float variables in µGA. However, it is critical to define a proper fitness (or objective) function for the success of the µGA optimization. The fitness function of SLGC optimization is defined as:

where *c* is a positive scaling coefficient. The purpose of µGA optimization is to minimize *f* and therefore maximize *η _{RCE}*. Usually it takes approximately 100 µGA generations to converge to an optimal SLGC design.

## 3. Simulation results

#### 3.1 Uniform SLGC

Let us first consider a simple SLGC, which we will call a uniform SLGC that has a fixed fill factor for all of the grating ridges. For µGA optimization, the grating period, grating depth, fill factor, the slant angle, and the relative fiber/waveguide lateral position, are set as independent variables. Note that we change the period of the slanted grating during the optimization. As discussed later in Section 4, this is required to meet the complicated phase-matching condition of the SLGC. With only five variables involved in the optimization, µGA converges very fast and gives us several designs with similar performance. Table 1 shows the variable ranges of µGA and the final optimized values for one of the designs.

The geometry of this particular uniform SLGC design is shown in Fig. 3(a). Figure 3(b) shows a corresponding image plot of the magnitude squared of the time averaged electric field from 2D FDTD simulation. *η _{RCE}*,

*η*,

_{LCE}*η*and

_{R}*η*are 66.8%, 0.69%, 6.63%, and 18.48% respectively. We can see several salient features of SLGCs from this design. First, with a grating period of 1.0263µm, the slanted grating spans less than 20µm. It is well known that it is essentially impossible for traditional weak index-modulated grating couplers to achieve high coupling efficiency within such a short coupling length. Second, we notice that the slanted grating of SLGC greatly suppress the left coupled light. The left coupling efficiency (

_{T}*η*) is only 0.69%, which demonstrates the excellent unidirectional coupling capability of the SLGC. It is also interesting to note that the power coupled toward the right without considering the mode overlap integral (i.e., just the power ratio term in

_{LCE}*Eq. (1)*) is 74.2%. This means that 7.4% of the incident power (or 10% of the coupled power) radiates away from the waveguide along the propagation direction due to a mismatch between the monitored field and the waveguide mode. This loss originates from the mode mismatch between the coupled leaky mode and the output waveguide mode, and boundary scattering at the right edge of the slanted grating.

#### 3.2 Non-uniform SLGC

In order to further improve the performance of SLGC, non-uniform fill factor SLGC designs are considered. For this case, the fill factor of every grating ridge in the SLGC is independently changed. This idea is motivated by prism coupling, in which it is possible to realize 100% coupling efficiency by varying the gap between the bottom side of the prism and the slab waveguide, which changes the local coupling coefficient [32]. For weak grating couplers, both varying the grating depth [11] and the fill factor [24] of individual grating ridges have been studied to improve their performance. For fabrication simplicity, here we investigate the non-uniform fill factor approach.

In µGA optimization of non-uniform SLGC, the fill factors of all 18 grating ridges are varied independently in the range of 10% to 90%. The ranges for all other variables are the same as those in the previous uniform SLGC case. Therefore 22 variables are optimized and approximately 400–500 generations are required for µGA to converge to a reasonable design. Figure 4(a) shows the geometry of an optimized non-uniform SLGC. Figure 4(b) is an image plot of the magnitude squared time average electric field. The optimized SLGC parameters are: Λ=1.013µm, *t*=1.6266µm, θ_{s}=35.940 and *d*=10.115µm. These values are very close to those of uniform SLGC except the lateral position d of the incident fiber. This SLGC has a right coupling efficiency (*η _{RCE}*) of 80.1%, which is 13% greater than the uniform SLGC design. The improvement mainly comes from the decrease of the transmitted light (

*η*) to the substrate, which is 10.04%. Efficiency of

_{T}*η*and

_{R}*η*also decrease to 4.02% and 0.19% respectively.

_{LCE}Figure 5 illustrates the µGA optimized fill factor as a function of the ridge position in the x direction. It is clear that the fill factor of the grating ridges gradually decreases along the coupling direction. The fill factors of the four right most grating ridges are essentially at the lower allowable limit used in the µGA optimization. However, the average fill factor of all of the grating ridges is 23.48%, which is almost the same as that of the uniform case. Notice that the incident fiber is also pushed to the left compared to the uniform SLGC design. The locally varied fill factor profile causes redistribution of the field inside the coupling region to minimize the reflected, transmitted and scattered light, which in turn results in better performance.

Further 2D FDTD simulation reveals that this non-uniform SLGC design has a relaxed lateral fiber alignment tolerance and a relatively broad spectral response, which is shown in Fig. 6 and Fig. 7, respectively. These features may be attractive for many photonic systems.

## 4. Physical analysis and discussion

While our rigorous numerical design tool, µGA-2D FDTD, is well suited for the design of SLGCs, it does not give intuitive insight into the principles of SLGC operation due to the built-in random process of µGA and the pure numerical nature of FDTD. To physically analyze the µGA optimized SLGC designs, it is crucial to find the leaky mode characteristics of the SLGCs. This is complicated by the fact that the presence of the grating strongly affects the waveguide mode, which is not the case for gratings that operate in the weak coupling regime.

We therefore use a rigorous grating theory, the Rigorous Coupled-Wave Analysis or RCWA method, to solve the so-called homogeneous problem [40] for SLGCs for the complex propagation constants *γ _{m}*=

*β*+

_{m}*iα*(

_{m}*m*is the mode index) of the leaky modes supported by the SLGC. The real part of the propagation constant,

*β*, is responsible for the phase-matching condition of SLGCs as will be discussed later and

_{m}*α*is the radiation factor of the leaky modes, which determines the coupling length of the grating coupler. We will skip the detail and tedious mathematical formulations of the RCWA procedure and refer the interested reader to the literature, such as Ref. [41]. Also, the well-known Muller root searching algorithm [42] is used to find the zeros of the determinant of the RCWA characteristic matrix. To apply RCWA, the proposed finite length SLGCs have to be infinitely extended in the x direction, as illustrated in Fig. 8. Also, the SLGCs must be divided into multiple sub-layers in the y direction in order to model the sloped ridge of the SLGCs with RCWA. Because we designed the SLGC as an input coupler for coupling light to the right in the waveguide, only the forward propagating leaky modes with positive

_{m}*β*are responsible for this coupling and will therefore be solved.

_{m}First we examine the leaky modes of the uniform SLGC design discussed in section 3.1. The 1.6263µm deep slanted grating layer is divided into 100 sub-layers in the RCWA mode analysis. At λ_{0}=1.55µm two lowest order leaky modes are found for this SLGC, a fundamental mode with *γ _{0}*=

*β*+

_{0}*iα*=6.122037+i0.319629 and a higher order mode with

_{0}*γ*=

_{1}*β*+

_{1}*iα*=6.02677+i0.0464558. We can define the effective index of a leaky mode as:

_{1}where *k*
_{0} is the wave propagation constant in vacuum. With this definition, the effective indices of the fundamental and higher order mode are 1.51025 and 1.48674. On the other hand, a simple mode analysis of the single mode 2D slab waveguide reveals that the effective index of its fundamental mode is 1.476.

Equipped with accurate mode effective indices, we can now investigate the phase-matching condition of strong coupling SLGC. In general, the phase matching of the grating can be expressed with the well-known grating equation as:

in which *k _{ix}* is the

*x*component of the incident k vector, and Λ is the grating period in the x direction, and

*q*is the diffraction order of the slanted grating. For surface normal incidence,

*k*is zero. For the µGA optimized value Λ=1.02633µm, it is easy to see that the phase-matching condition of the SLGC can be satisfied with respect to the fundamental leaky mode when

_{ix}*q*=1. In other words, the +1 diffraction order of the grating provides the required phase matching between the incident fiber mode and the fundamental leaky mode of the SLGC. Note that this phase-matching mechanism is very different from that of conventional weak coupling grating couplers. In weak grating couplers, even though the +1 diffraction order is also responsible for the phase matching, the phase matching is always assumed to be realized for the unperturbed waveguide mode, and the period of the coupler can be analytically calculated. However, analytical determination of the grating period of our SLGC from the phase-matching condition is impossible because the fundamental leaky mode of SLGCs is a function of all of grating parameters, including the grating period itself. Consequently, realization of the phase-matching condition must be part of the SLGCs’ design process. In this sense, the design and optimization of SLGCs is far more complicated than weak grating couplers.

The phase-matching condition of the SLGC suggests that its fundamental leaky mode plays a central role. We further notice that the radiation factor of the fundamental leaky mode is about one order of magnitude greater than that of the higher order mode. The radiation factor can be related to Bragg diffraction. A k-vector diagram of the SLGC is constructed to confirm this.

The k-vector diagram for this uniform SLGC is shown in Fig. 9. Note that all k vectors in the figure are normalized by *k*
_{0} for convenience. As a zeroth order approximation, the slanted grating layer of the SLGC can be treated as a homogeneous layer with an average index defined as the volume average between the two materials forming the grating [28]:

The solid circle with a radius of 1.6004 denotes this average index for the grating layer. *K⃗ _{inc}* is the normally incident k-vector and the dotted slanted line refers to the orientation of the slanted grating ridges relative to the k

_{y}axis, which is 34.98° in this case. The two dotted vertical lines L

_{1}and L

_{2}at

*k*=1.476 and 1.51025 correspond to the effective indices of the waveguide mode and the fundamental leaky mode, respectively. If we draw the grating vector

_{x}*K⃗*perpendicular to the orientation of the slanted ridges, the diffracted k-vector,

_{G}*k⃗*, which is equal to vector sum of (

_{final}*K⃗*+

_{inc}*K⃗*), terminates on L

_{G}_{2}because phase matching condition is satisfied. If Bragg condition is obeyed, the diffracted k-vector,

*k⃗*, will terminate on the circle. As seen in Fig. 9, this is the case. It is clear from this k-vector diagram that the Bragg diffraction condition is simultaneously satisfied for the +1 diffraction order of the slanted grating while phase matching between the incident fiber mode and the fundamental leaky mode of the SLGC is also satisfied. Bragg diffraction suppresses other diffraction orders and therefore enforces unidirectional coupling of the SLGC. For surface-normal operation, the occurrence of Bragg diffraction is the main advantage of an asymmetric parallelogramic grating shape compared to a symmetric grating. Note that finding a SLGC design that simultaneously satisfies phase matching and the Bragg condition for a grating that operates in the strong coupling regime strongly demonstrates the powerful optimization capability of our µGA-2D FDTD design tool.

_{final}It is interesting that the direction of *k⃗ _{final}* in Fig. 9 is tilted at about 20° with respect to the

*k*axis. This implies that the phase front in the SLGC coupling region for the structure of Fig. 3 should also be tilted about 20° with respect to the x-axis. The tilted phase front can be clearly seen from the phase distribution of this SLGC calculated by 2D-FDTD, which is shown in Fig. 10. The tilted wavefront is quite flat when the light is coupled in by SLGC and is gradually rippled in the area around the right end interface of the slanted grating. The tilted wave front and the mismatch between the effective indices of the waveguide mode and the fundamental leaky mode of SLGC (the distance between L

_{x}_{1}and L

_{2}as shown in Fig. 9) will induce scattering loss at the boundary between the slanted grating region and the undisturbed waveguide region. In general this scattering loss is more severe than that of grating couplers that operate in the weak coupling regime where the mode mismatch is assumed to be negligible.

The above RCWA mode analysis procedure explains the behavior of the µGA- optimized uniform SLGC. Nevertheless, it cannot be directly applied to the analysis of the non-uniform SLGC discussed in Section 3.2 because of the varying fill factor of the grating ridges. In order to apply a similar the mode analysis, the non-uniform SLGC is first approximated as a uniform SLGC with a fill factor of 23.48%, which is the average of the fill factors of all of the grating ridges. For this simplified SLGC, two leaky modes can be found with RCWA mode analysis: a fundamental mode with propagation constant *γ _{0}*=

*β*+

_{0}*iα*=6.201752+0.3148340i (

_{0}*n*=1.52991) and a higher order mode with

_{eff}*γ*=

_{1}*β*+

_{1}*iα*=6.029924+0.040366i (

_{1}*n*=1.4875). By following a similar procedure, it is straightforward to show that both the phase-matching and Bragg conditions are satisfied for the fundamental leaky mode. The k-vector diagram and the phase distribution of the 2D FDTD simulation are shown in Figs. 11(a) and 11(b), respectively. Although these figures are very similar to those for the uniform SLGC case, we notice that the phase ripple near the right hand termination of the grating in Fig. 11(b) is much less than in Fig. 10, which means that the transition from the coupling grating to the output waveguide is smoother and hence the scattering loss is lessened.

_{eff}Finally, there is one more point worth mentioning. In both SLGC designs discussed above, we found that they can support a higher order leaky mode in addition to the main fundamental leaky mode. It seems that SLGCs generally support multiple leaky modes because of the introduction of a high index material as the grating ridge for strongly coupled SLGCs. It is well known that alternative leaky channels are not desirable in traditional weak grating couplers. However, as demonstrated by these µGA-optimized SLGCs, if properly designed, the higher order mode will be greatly suppressed and therefore its presence will not appreciably affect the performance of SLGCs.

## 5. Conclusions

In this paper, we proposed the utilization of strong index-modulated slanted grating couplers (SLGCs) as a potential coupler technology for surface-normal coupling between fibers and waveguides for dense PLCs. The major advantage of our SLGCs is that they can realize high efficiency unidirectional coupling for surface-normal operation in a very short coupling length, i.e., on the order of the width of a fiber mode. With the help of a powerful µGA-2D FDTD design tool, high efficiency SLGCs, both uniform and non-uniform, have been designed.

Rigorous mode analysis shows that the phase-matching mechanism of SLGCs is different from the tradition grating couplers with weak index modulation. Both the phase-matching and Bragg conditions are satisfied with respect to the fundamental leaky mode of the SLGCs instead of the output waveguide mode. 2D FDTD simulation also shows that SLGCs have a large tolerance for the lateral fiber alignment and a broadband response. Such grating couplers, taking advantage of planar processing, can offer the potential to surmount the difficulties typically associated with coupling from fibers oriented normally to a waveguide surface.

An important next step in evaluating the method is to extend 2D results present in this paper to a 3D analysis. Further research also needs to be done in applying similar concept to other waveguide materials, for example SOI waveguides. Currently, we are investigating the fabrication feasibility of SLGCs.

## Acknowledgments

This work was supported by Defense Advanced Research Projects Agency grant N66001-01-1-8938 and National Science Foundation grant EPS-0091853.

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