## Abstract

A grating coupler with fully etched slots is optimized for fiber coupling into SOI slab waveguides. Such coupler can be produced in one lithography step together with other SOI components. Theoretical maximal coupling efficiency of 49% is demonstrated with a 3dB bandwidth of 35nm. Strong reflection from the fully etched grating was avoided through an antireflection interface. Constructive interference is used to decrease radiation into the substrate and the filling factor is optimized for optimal power coupling into the fiber mode. It was also demonstrated, that the chirped grating approach is inapplicable for fully etched gratings.

©2009 Optical Society of America

## 1. Introduction

The trend towards high speed and large bandwidth in telecommunication drives the development of ever smaller components and away from electrical towards optical devices. Over decades the technology in silicon microelectronics has grown and enabled to inexpensively fabricate those devices on silicon wafers. Since silicon is transparent at wavelengths typically used in telecommunications (1310 nm and 1550 nm), the silicon-on-insulator (SOI) technology is suitable for photonics and compatible to complementary metal oxide semiconductor (CMOS) technology [1,2]. Hence, optical and electrical components can be fabricated on a single chip. Moreover, the high refractive index of silicon results in smaller wavelengths and also strong light confinement within the silicon waveguide (refractive index ${n}_{si}=3.45$) which is situated above an SiO_{2} buried oxide layer (BOX) with much lower refractive index (${n}_{Si{O}_{2}}=1.45$).

One of the challenges in nanophotonics and subject of current research is the coupling into and out of these nanoscale devices directly from lasers or from optical fibers which are magnitudes larger in size. A technique that is straight forward and widely used is butt coupling, where the light beam is focused into the silicon waveguide at the edge of a chip. The constraint that light can only be coupled into the nanostructure at the edges, and the high demand on the alignment of the system to focus the beam directly on the facet of such a small waveguide are disadvantageous here. Another technique is the taper coupler for compact mode conversion [3,4]. Apart from the fact that coupling is also done into or out of a cleaved edge of a chip, exact cleaving alignment is necessary [3] or additional waveguide deposition [4] is required. A third solution is the grating coupler where periodic perturbations of the waveguide’s refractive index lead to scattering of the propagating light out of the waveguide. The big advantage of such a grating coupler is that one is not limited to couple light at the edges. This enables testing devices anywhere on a chip. Several designs of grating couplers have been published [5–8], where simulated coupling efficiencies of up to 92% and bandwidths in the order of a few tens of nanometers were achieved.

The grating coupling efficiency is dominated by mainly two factors. The light is scattered not only into the air, but a part is also lost to the bottom into the substrate. In addition to that, to achieve high efficiency the light scattered to the top should have a field distribution that matches the fiber mode. To overcome the first issue, grating couplers with gold bottom mirrors [7] or multilayer bottom reflectors [5,6] were suggested to avoid radiation into the substrate. However, the process to fabricate such structures is not compatible to CMOS processing technology. Therefore, Roelkens et al. have recently suggested [9,10] and demonstrated [11] structures where the Bloch modes that propagate in the grating were modified such that a better directionality is achieved. They have done this by deposition of an additional silicon-layer before etching. This makes the process better compatible to CMOS processing technology. Better directionality can also be achieved through slanted gratings in the cladding material [12]. Grating couplers that lead to a better modematch with the fiber mode have also been designed. Nonuniform perturbation of the refractive index is required [6]. Even better mode match can be achieved in long nonuniform gratings coupled through free space optics [13].

All grating couplers proposed so far are achieved by shallow or slanted etching of slots into the waveguide or cladding. Such gratings have the advantage that coupling from the waveguide into the grating occurs with little reflection and the etched slots radiate efficiently. However, a significant drawback, for instance, of such shallow etched gratings is the additional lithography step. In this paper the design of a grating-fiber coupler with fully etched slots is proposed. All structures on the wafer can be fabricated in just one step because the second lithography step for shallow etching is not necessary. This grating would offer a fast and economic experimental solution for the coupling problem. Due to the fully etched slots, the periodic perturbation of the index contrast is very strong, which makes coupling into the grating from the slab waveguides more difficult to achieve. We have therefore developed an anti-reflection interface section that enhances coupling in order to reach a coupling efficiency comparable to shallow etched gratings.

## 2. Design approach

#### 2.1 Geometrical structure

Figure 1
depicts the grating to fiber coupler and its geometrical parameters. The height of the silicon slab waveguide is $h=220\text{\hspace{0.17em} nm}$. It is surrounded by air on top of it and by a BOX layer with thickness *d*. A silicon substrate below the BOX caters for mechanical stability. The period of the grating is *a*, *w* is the air slot width ($w/a$ is the filling factor). *N* is the number of slots in the grating and defines the overall coupler length $L=N\cdot a$. The grating’s lateral mode width is chosen to fit the fiber mode width, what is achieved with waveguide lateral width ${w}_{g}=12\text{\hspace{0.17em} \mu m}$ [6]. For coupling reasons, the widths of the first and the last slot are given by ${w}_{c}=w/2$ as discussed later. *ϕ* is the angle of the fiber towards the grating normal and, thus, the angle of the beam that is incident on or scattered from the grating.

Full wave simulations using CST Microwave Studio [14], based on the finite integration technique (FIT) [15,16], are carried out to evaluate the field radiated from the grating. The simulation volume corresponds to the one depicted in Fig. 1. The two dimensional (2D) simulations presented here assume the waveguide extension in x direction to be infinite. In 3D case the coupling efficiency is determined by the overlap integral of the scattered field and fiber mode profiles. This overlap integral can be split into two orthogonal contributions. The contribution from integrating along x-direction can be derived from a single 3D calculation. For coupling into a single mode fiber with a mode field diameter of 10µm this calculation yields an optimum waveguide width ${w}_{g}=12\text{\hspace{0.17em} \mu m}$ and a coupling efficiency contribution of 97%. The contribution from the direction orthogonal to x is obtained from the 2D calculation as described further.

We approach the design of the grating coupler as if we consider coupling from the grating to the fiber. That is, in the simulation we excite the slab waveguide with the fundamental TE-mode that is coupled into the grating, whose slots should radiate a field into free space that is coupled into the fiber. Due to reciprocity, the results would be the same as if a wave is coupled from the fiber to the grating. The design approach is explained in more detail in the following sections.

#### 2.2 Scattering angle

Each slot of the grating approximately radiates a spherical wave. All the single waves superpose and shape the radiated field. For a specific wavelength *λ* the distance between two radiating slots *a* determines towards which direction the single spherical waves interfere constructively and, thus, along which angle *ϕ* the phase front propagates. If *ϕ* is too large, due to the mechanical constraints, the fiber core will be relatively far away from the grating and the beam divergence will limit the coupling efficiency. If, on the other hand, *ϕ* is too small, the beam coming from the fiber would be strongly reflected from the grating back into the fiber or vice versa. For our coupler it is chosen $\varphi =-10\xb0$ at the operating free space wavelength ${\lambda}_{0}=1550\text{\hspace{0.17em} nm}$ or in terms of frequency ${f}_{0}=c/{\lambda}_{0}=193.4\text{\hspace{0.17em} THz}$. The positive angle is defined in the clock wise direction. The negative angle indicates that the scattered wave propagates in the opposite direction to the excitation mode. To achieve *ϕ*, the corresponding *a*, which is constant relative to the guided wavelength, can be determined. Hence, *a* will vary with the slot width *w* because the effective refractive index of the grating is a function of *w*.

#### 2.3 Coupling efficiency

Three factors determine the overall efficiency of the coupler. We are designing the coupler by optimizing those factors step by step.

First of all, there is the radiation efficiency ${\eta}_{1}$ of the grating which indicates how much of the energy in the slab waveguide is scattered by the slots. It is dependent on two factors. The better the slab mode can be coupled into the grating, and the less energy is transmitted through the grating, the more energy is being scattered. Thus, we define

where *R* is the reflection coefficient of the grating back into the slab waveguide and *T* is the transmission coefficient through the grating. That part which is neither reflected nor transmitted is scattered by the slots.

A second efficiency ${\eta}_{2}$ indicates what part of this scattered energy is radiated into the direction of the fiber. Initially the grating radiates about 50% of the energy into the air, whereas the other 50% are radiated downward into the BOX. At the lower BOX-silicon-interface, a part of the power is transmitted into the substrate and a part of it is reflected. The reflected part propagates upward again through the BOX toward the air and interferes with the wave that is radiated upward by the grating directly. The interference should be constructive to maximize the scattered power radiated upward into the air. This leads to the definition

which is the ratio of the power radiated to the air and the power scattered by the grating. Since *d* defines the distance that the wave propagates in the BOX, ${\eta}_{2}$ can be maximized by finding the optimum BOX thickness ${d}_{opt}$ for constructive interference of the two partial waves at the waveguide-air-interface [7,17,18].

We now have defined what amount of the total power is scattered in the direction to the fiber by ${\eta}_{1}$ and ${\eta}_{2}$. In order to obtain the total coupling efficiency, we need to know how much energy is coupled into the fundamental mode of the fiber. This partial efficiency is ${\eta}_{3}$, which is given by the following equations:

The enumerator of ${\eta}_{3a}$ is the overlap integral of the E-field distributions squared of the scattered wave and the fundamental fiber mode along the coordinate z', which is parallel to the fiber input interface. This is normalized by the powers of the two waves. The scattered electric field was evaluated close to the grating. This is justified as the fiber core will be positioned much closer than the Rayleigh length of the scattered beam. ${\eta}_{3b}$ is the same efficiency as ${\eta}_{3a}$ but evaluated for an integration direction perpendicular to the yz-plane. It depends on the lateral width of the grating, which was determined from a 3D simulation as ${w}_{g}=12\text{\mu m}$, in accordance with Ref [6]. ${\eta}_{3c}$ in the Eq. (6) is the Fresnel transmission of a normally incident electromagnetic wave, where ${\beta}_{1}$ and ${\beta}_{2}$ are the phase constants of the wave in air and in the fiber medium, respectively. Hence, ${\eta}_{3}$ is higher if the scattered field resembles the fiber mode more closely.

The overall coupling efficiency of the grating-fiber-configuration is given by the product of the three efficiencies defined.

The presented coupling efficiency is valid for the ideal structure with a perfectly aligned optical system. A disorder introduced by the manufacturing process may lead to additional losses. However, for the typical deviations of 5 nm [19] the disorder induced losses are expected to be low taking into account the very short length of the grating.

2.4 Chirped grating

For a uniform grating, where all slots have the same width, the power in the grating decreases exponentially by $P(z)={P}_{0}\cdot \mathrm{exp}(-2\gamma z)$. Thus the scattered power has also an exponentially decaying spatial distribution. This results in an inherent mismatch of the radiated field and the fiber mode, such that the maximum achievable ${\eta}_{3a}$ is about 80% for a uniform grating [6]. *γ* is the loss factor of the grating per length and is constant over *z* due to constant *w*. A more Gaussian like profile for the radiated field can be achieved if the loss factor is designed to be *z*-dependent. Thus, *w* should vary with *z* such that the power that is lost in the grating per length matches the fiber mode. Hence,

with

From this follows the *z* dependent loss factor to achieve a Gaussian profile for the radiated field

Since *a* is a function of *w*, the correct chirp of the grating can be determined with the help of the *a*-dependence of *γ* by

where ${a}_{i}$ is the corresponding grating period to the *i*th slot, with ${a}_{0}=0$. The width of the *i*th slot can then be determined, since *a* and *w* are directly related.

## 3. Simulation results and discussion

#### 3.1 Radiation efficiency

As described in the previous section, the design is being approached by optimizing the three defined partial efficiencies individually. Coupling from the slab mode into the scattered wave is the first challenge. Due to the fully etched slots the waveguide is strongly modified and perturbation theory is not applicable which approximates the z component of the scattered wave vector as a propagation constant of the unperturbed waveguide shifted by $2\pi /a$. Thus, scattering of Bloch modes should be considered as presented in Fig. 2 , which have very different dispersion in comparison to the guided modes of the unperturbed waveguide. Due to the fully etched slots and strong refractive index contrast between air and silicon, the Bloch modes dispersion curves are flat, in particular close to the band edge where we have to operate the grating in order to achieve $\varphi =10\xb0$ (see Fig. 2(a)). With a slab mode the grating Bloch mode above and below the second photonic band gap (PBG) can be excited. As we excite modes with positive group velocity, the mode below band gap scatters light at negative angles and the mode above the band gap scatters light at positive angles.

There is strong reflection close to the band gap for modes above and below band gap, as shown in Fig. 2(b) with a grey line. That is, the group velocity of the Bloch modes is relatively small and coupling is more difficult to achieve than for gratings with shallow etched slots. However, coupling from a slab mode into a Bloch mode can be optimized for a narrow range of frequencies if the coupling interface to the grating coupler is properly adjusted. We choose to couple from the silicon slab directly into the *xy* symmetry plane of the 1D-Bragg stack. That is why at the input there is effectively half of the slot present (see Fig. 1). In Fig. 2(b) the reflected intensity is plotted with a black line for a grating with such an antireflection interface at the input and output. We can observe resonant coupling into the mode below the band edge and strong reflection above the band edge. Therefore, we define the radiation angle to be $\varphi =-10\xb0$.

The resonant coupling to the mode below the band gap can be explained through Bloch mode consideration. The Bloch modes can be described as a sum of forward and backward propagating plane waves in every layer of the Stack. The 1D transfer matrix simulations show that in the mode below the band gap these forward and backward propagating waves interfere *in anti-phase* at the center planes of dielectric and air layers. At the same time the coupling from silicon into air leads to a strong reflection at the interface, which is *in phase* to the incident field. Thus the backward propagating plane wave of the Bloch mode and the reflection at the silicon-air interface can compensate each other when their amplitudes become equal. On the other hand in the mode above the band gap the forward and backward propagating waves interfere *in phase* at the symmetry planes. To compensate the reflection the coupling from air into silicon should be considered, which is not possible in a slab waveguide configuration.

Having defined the direction of radiation and given the parameters in section 2.1, everything is known to find the function $a(w)$. It is depicted in Fig. 3 . Thus increasing slot width should be compensated by the increasing lattice constant to achieve the same scattering angle.

We have then investigated the dependence of ${\eta}_{1}$ on *w*. Therefore, different uniform gratings with $N=20$ have been simulated and Fig. 4
shows the behavior of ${\eta}_{1}$ over *w* at the operating wavelength ${\lambda}_{0}$. It is observed, that the radiation efficiency is larger for gratings with wider slots and it saturates at about $w>400\text{\hspace{0.17em} nm}$. The transmission through a grating falls since a wider slot is a stronger radiator and for the same number *N* of slots, more power is radiated and less is transmitted through the grating. The result also confirms that coupling from the slab waveguide into the grating is not degraded for stronger gratings.

#### 3.2 BOX width optimization

Now, the BOX width is determined that maximizes the power radiated to the top and minimizes the power radiated into the substrate. The ${\eta}_{2}$ dependence on *d* for a fixed radiation angle *ϕ* is depicted in Fig. 5
. The maximum achievable value for ${\eta}_{2}$ with our structure is ${\eta}_{2,\mathrm{max}}=73\%$ and its minimum value is ${\eta}_{2,\mathrm{min}}=36\%$. It is a periodic function of BOX thickness with a periodicity that is consistent with the expected value $\Delta d=({\lambda}_{0}\mathrm{cos}\varphi )/(2{n}_{BOX})=0.53\text{\hspace{0.17em} \mu m}$.

The standard BOX width of 2 µm would be suboptimal for the grating coupler. The next optimized BOX widths are approximately 2.25 µm and 2.75 µm. We chose ${d}_{opt}=2.775\text{\hspace{0.17em} \mu m}$ for our simulations.

#### 3.3 Mode matching in uniform gratings

Having obtained the results so far, we know how we can optimize the grating such that maximum power is scattered to the fiber. Now, ${\eta}_{3a}$ is calculated for different uniform gratings with $N=20$, ${d}_{opt}$ and $\varphi =-10\xb0$. The result is plotted in Fig. 6(a)
. The slot width *w* does not have a very strong effect on the mode match. The scattering profile is a section of an exponentially decaying function on length of $L=20a$. It can be shown that the coupling integral does not vary much until the decay length becomes much smaller than the grating length what is observed at slot widths of more than 400nm. This is due to the fact that for wider slots, the radiated power is more and more concentrated in a region that is smaller than the width of the Gaussian beam of the fiber.

Both, ${\eta}_{3a}(w)$ and ${\eta}_{1}(w)$ are functions of *w*. That is, an optimum slot width ${w}_{opt}$ can be found where the product ${\eta}_{1}(w)\cdot {\eta}_{3}(w)$ is maximized, this function is shown in Fig. 6(b). ${\eta}_{2}$, ${\eta}_{3b}$ and ${\eta}_{3c}$ stay unaffected, they are independent of *w*. Due to the flat maximum we chose for our coupler the smallest slot width ${w}_{opt}=350nm$ at the maximum, because a smaller slot width results in wider bandwidth due to the steeper dispersion curve. The overall coupling efficiency at ${\lambda}_{0}$ is maximized to $\eta =\eta 1\cdot \eta 2\cdot \eta 3=49\%$. The efficiencies over wavelength are visualized in Fig. 7
.

The coupling efficiency is maximal at the wavelength 1.55 µm and decreasing at shorter and longer wavelengths. ${\eta}_{3}$ is the factor that mainly limits the bandwidth. Due to the flat dispersion curve the radiation angle is relatively strongly dependent on the wavelength and a slightly different radiation angle would lead to strong mismatch of the radiated field and the fiber mode. All data of the designed grating is given in the following table:

As we can see from Fig. 7, ${\eta}_{1}$ is relatively high. ${\eta}_{2}$ is already maximized for this structure and cannot be optimized further. Figure 8 visualizes the radiated field of the uniform grating along its phasefront and the field distribution of the fundamental fiber mode. We can see that the two fields have a certain expected mismatch.

#### 3.4 Chirped grating

There is room to increase ${\eta}_{3}$ theoretically by a chirped grating as described in section 2.4. Figure 9(a) depicts the loss factor $\gamma (z)$, determined by Eq. (9), to achieve a Gaussian distribution of the radiated field at 97% of the scattered power. Through the function $\gamma (w)$, which is depicted in Fig. 9(b), and Eq. (11) the discrete slot widths can be computed.

Simulations have shown that chirped gratings with fully etched slots and the restriction that it is coupled directly to the fiber are not more effective than their uniform counterparts. One reason is the strong discretization of the gradual function. We can see in Fig. 10(a) that ${\eta}_{1}$ is quite low in comparison to uniform grating presented in Fig. 7 and shows resonances at frequencies close to the band edge. This indicates that coupling into these chirped gratings is not as efficient as for uniform gratings with antireflection interface. The gradual change in the slot width is not accepted by the wave as an adiabatic coupling. Reflections lead to distortion of the scattered profile and the overlap integral results only in ${\eta}_{3a}\approx 80\%$ which is no essential improvement.

## 4. Conclusion

We have presented the design of a grating-to-fiber coupler that can be fabricated with fully etched slots. Resonant coupling into the dielectric mode of the grating is achieved by choosing an antireflection interface of the photonic crystal and operating below the second bandgap with negative scattering angle. A total maximum coupling efficiency of $\eta =49\%$ is theoretically achieved, which is comparable to the efficiency of shallow etch gratings. The 3dB bandwidth of 35nm is approximately two times smaller in comparison to shallow etched grating, which is unavoidable due to the flat mode close to the band edge of the fully etched structures. However, this does not pose a particular problem as the grating coupler is intended to be designed specifically for a device application at a fixed wavelength and narrow bandwidth.

Additional loss occurs mainly to radiation into the substrate and mismatch of the radiated field and the Gaussian beam of the fiber. The radiation in the substrate was reduced by optimization of the BOX width. The mode mismatch loss we have tried to improve with a nonuniform slot width along the grating. However, even though we have a gradually changing taper at the input, the coupling into a chirped grating is more difficult than into uniform gating with antireflection interface and thus the fiber coupling efficiency is not improved in comparison to a uniform grating.

## Acknowledgements

This research is supported by the German Research Foundation (DFG). The authors acknowledge the support from CST, Darmstadt, Germany with their Microwave Studio software.

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