nentially decaying function, the later used as an approximation of the field diffracted by a uniform grating.

The mode field overlap is characterized by an overlap integral. The latter will be derived here, by considering the orthogonality relations for the SMF's eigenmodes.

<figure><img src="image_25.png" /><figcaption>Figure 2.19: A schematic with the definitions used for the derivation of an overlap integral between the upwards diffracted grating fields $\vec{E}_i, \vec{H}_i$, related to the power $P_{up} = P_i$ and the single mode fiber (SMF) fundamental mode. The SMF is placed at the local coordinates $(u, v, w = \text{const.})$, which are defined by the angles $\varphi_F, \theta_F$. The fiber's effective area is assigned as $A_F$. Arbitrary SMF eigenmodes $\vec{E}_{l,m}, \vec{H}_{k,n}$, $(l, m, k, n \in \mathbb{Z})$, weighted by transmission $a_{l,m}, a_{k,n}$ and back-propagation coefficients $b_{l,m}, b_{k,n}$ are shown.</figcaption></figure>

Assume the 2D GC's radiated fields as incident fields at the fiber facet with an effective area $A_F$ and the position $(u, v, w = \text{const.})$ (Fig. 2.19). In the following, all given fields are complex.

## Definitions

* Tangential incident fields: $\vec{E}_i, \vec{H}_i$

* Fiber eigenmodes:

$$ 
\begin{aligned}
\vec{E}_F &= \sum_{l,m} (a_{l,m} + b_{l,m}) \vec{E}_{l,m}, & l, m \in \mathbb{Z} \\
\vec{H}_F &= \sum_{k,n} (a_{k,n} - b_{k,n}) \vec{H}_{k,n}, & k, n \in \mathbb{Z}
\end{aligned}
\qquad (2.40) 
$$

Here, the coefficients $a$ represent forwards-propagating waves and determine the transmission through the fiber facet with an effective area $A_F$. The coefficients $b$ account for
backwards-propagating waves. Particularly, we are interested in the calculation of the coefficients $a$. An orthogonality relation between the fiber's eigenmodes is given as:

$$ \int_{A_F} \vec{E}_{l,m} \times \vec{H}_{k,n}^* d\vec{A} = - \int_{A_F} \vec{H}_{l,m} \times \vec{E}_{k,n}^* d\vec{A} = \delta_{lk}\delta_{mn} = \begin{cases} 1, & \text{if } l=k \text{ and } m=n \\ 0, & \text{otherwise} \end{cases} \quad (2.41) $$

where $\delta$ represents the Kronecker's delta.

### Continuity Condition at $w = \text{const.}$

The unknown coefficients $a$ and $b$ can be determined by using the continuity of the tangential fields at the fiber facet $w = \text{const}$.

$$ \begin{array}{l} \text{(i.)} \qquad \vec{E}_i = \vec{E}_F \\ \qquad \vec{E}_i = \sum_{l,m} (a_{l,m} + b_{l,m}) \vec{E}_{l,m} \end{array} \quad (2.42) $$

To make use of the orthogonality relations, we make on both sides of the equation the cross product with $\vec{H}_{k,n}^*$ and integrate over the area $A_F$.

$$ \begin{align} \int_{A_F} \vec{E}_i \times \vec{H}_{k,n}^* d\vec{A} &= \sum_{l,m} \int_{A_F} (a_{l,m} + b_{l,m}) \vec{E}_{l,m} \times \vec{H}_{k,n}^* d\vec{A} \nonumber \\ &= \int_{l=k,m=n} \int_{A_F} (a_{l,m} + b_{l,m}) \vec{E}_{l,m} \times \vec{H}_{l,m}^* d\vec{A} \nonumber \\ \Rightarrow a_{l,m} + b_{l,m} &= \frac{\int_{A_F} \vec{E}_i \times \vec{H}_{l,m}^* d\vec{A}}{\int_{A_F} \vec{E}_{l,m} \times \vec{H}_{l,m}^* d\vec{A}} \tag{2.43} \end{align} $$

Analogously,

$$ \begin{array}{l} \text{(ii.)} \qquad \vec{H}_i = \vec{H}_F \\ \qquad \vec{H}_i = \sum_{k,n} (a_{k,n} - b_{k,n}) \vec{H}_{k,n} \end{array} \quad (2.44) $$

This time, we make on both sides of the equation the cross product with $-\vec{E}_{l,m}^*$ and again integrate over the area $A_F$.

$$ \begin{align} - \int_{A_F} \vec{H}_i \times \vec{E}_{l,m}^* d\vec{A} &= - \sum_{k,n} \int_{A_F} (a_{k,n} - b_{k,n}) \vec{H}_{k,n} \times \vec{E}_{l,m}^* d\vec{A} \nonumber \\ &= - \int_{l=k,m=n} \int_{A_F} (a_{l,m} - b_{l,m}) \vec{H}_{l,m} \times \vec{E}_{l,m}^* d\vec{A} \nonumber \\ \Rightarrow a_{l,m} - b_{l,m} &= \frac{\int_{A_F} \vec{E}_{l,m}^* \times \vec{H}_i d\vec{A}}{\int_{A_F} \vec{E}_{l,m}^* \times \vec{H}_{l,m} d\vec{A}} = \frac{\int_{A_F} \vec{E}_{l,m}^* \times \vec{H}_i d\vec{A}}{\int_{A_F} \vec{E}_{l,m}^* \times \vec{H}_{l,m}^* d\vec{A}} \tag{2.45} \end{align} $$
Finally, the unknown coefficients $a$ can be obtained.

(i.) + (ii.)

$$a_{l,m} = \frac{1}{2} \left( \int_{A_F} \vec{E}_i \times \vec{H}_{l,m}^* d\vec{A} + \int_{A_F} \vec{E}_{l,m}^* \times \vec{H}_i d\vec{A} \right) \frac{1}{\int_{A_F} \vec{E}_{l,m} \times \vec{H}_{l,m}^* d\vec{A}} \quad (2.46)$$

The coefficients $b$ can be calculated analogously ((i.) - (ii.)). Since we have no coupling to back-propagating waves, we obtain in any case $b_{l,m} = 0$. The estimation of $b$ can be done as a verification of the intermediate steps and as a proof for the transparent boundaries.

## Overlap

The overlap can be defined as the quotient of the power transmitted into the fiber $P_{l,m}^+$ and the incident grating field power. With $P_{up} \equiv P_i$ at the fiber facet:

$$\begin{aligned} \eta_F &= \frac{P_{l,m}^+}{P_i} \\ \eta_F &= \frac{\Re \int_{A_F} a_{l,m} \vec{E}_{l,m} \times \vec{H}_{l,m}^* d\vec{A}}{\Re \int_{A_F} \vec{E}_i \times \vec{H}_i^* d\vec{A}} = |a_{l,m}|^2 \frac{\Re \int_{A_F} \vec{E}_{l,m} \times \vec{H}_{l,m}^* d\vec{A}}{\Re \int_{A_F} \vec{E}_i \times \vec{H}_i^* d\vec{A}} \end{aligned} \quad (2.47)$$

Now, we substitute $a_{l,m}$ from (2.46) and assume $\int_{A_F} \vec{E}_{l,m} \times \vec{H}_{l,m}^* d\vec{A}$ to be real. We obtain:

$$\eta_F = \frac{1}{4} \left| \int_{A_F} \vec{E}_i \times \vec{H}_{l,m}^* d\vec{A} + \int_{A_F} \vec{E}_{l,m}^* \times \vec{H}_i d\vec{A} \right|^2 \frac{1}{\Re \int_{A_F} \vec{E}_{l,m} \times \vec{H}_{l,m}^* d\vec{A} \cdot \Re \int_{A_F} \vec{E}_i \times \vec{H}_i^* d\vec{A}} \quad (2.48)$$

## Coupling Efficiency

The overlap integral (2.48) still does not take into account the limited out-coupled power efficiency, defined as the portion of the input waveguide power $P_{WG}$ that is directed by the grating into the proper direction $P_{up} \equiv P_i$. We can assign this parameter as an input-output grating efficiency:

$$\eta_{io} = \frac{P_i}{P_{WG}} = \frac{\Re \int_{A_F} \vec{E}_i \times \vec{H}_i^* d\vec{A}}{\Re \int_{A_{WG}} \vec{E}_{WG} \times \vec{H}_{WG}^* d\vec{A}} \quad (2.49)$$

with $A_F$ the fiber effective area, $A_{WG}$ the waveguide effective area, $\vec{E}_{WG}$, $\vec{H}_{WG}$ the Si waveguide fields.

The final expression for the coupling efficiency is:

$$\begin{aligned} \eta &= \eta_F \cdot \eta_{i,o} \\ \eta &= \frac{1}{4} \left| \int_{A_F} \vec{E}_i \times \vec{H}_{l,m}^* d\vec{A} + \int_{A_F} \vec{E}_{l,m}^* \times \vec{H}_i d\vec{A} \right|^2 \frac{1}{\Re \int_{A_{WG}} \vec{E}_{WG} \times \vec{H}_{WG}^* d\vec{A} \cdot \Re \int_{A_F} \vec{E}_{l,m} \times \vec{H}_{l,m}^* d\vec{A}} \end{aligned} \quad (2.50)$$
**SMF Reference Mode** In (2.50), the SMF's LP<sub>01,u</sub>-mode can be substituted as the mode of interest defined at (u,v,w = const.). The field distribution is commonly approximated as a Gaussian beam:

$$ 
\begin{aligned}
E_u(u, v) &= E_0 \exp\left(\frac{(u-u_0)^2}{w_0^2}\right) \exp\left(\frac{(v-v_0)^2}{w_0^2}\right) \\
H_v(u, v) &= \frac{E_0}{Z_F} \exp\left(\frac{(u-u_0)^2}{w_0^2}\right) \exp\left(\frac{(v-v_0)^2}{w_0^2}\right) \quad Z_F = \frac{Z_0}{n_{\text{SMF}}} = \frac{376.73}{n_{\text{SMF}}} \Omega
\end{aligned}
$$

with $n_{\text{SMF}} \approx 1.447$ [39] and the mode field diameter $2w_0 = 10.4$ µm at 1550 nm or $2w_0 = 9.2$ µm at 1310 nm, according to the Corning's SMF-28 data sheet [40]. The central position of the Gaussian beam is given by $(u_0, v_0)$ and its field amplitudes are normalized, so that:

$$ \iint_{A_F} E_u H_v^* du dv = 1. $$

The LP<sub>01,v</sub>-mode can be defined in an analogous way. In this work, the Gaussian beam's position is always set in such a way that the fiber-grating mode field overlap is maximized. The position is also balanced between both arms of the 2D GC.

<figure><img src="image_26.png" /><figcaption>Figure 2.20: A schematic for the illustration of two parameters, related to the polarization splitting in 2D grating couplers (2D GCs). (a) Waveguides (WG) 1 and 2 are excited separately, delivering a certain field distribution at a tilted plane with coordinates (u,v,w = const.), related to the angles ($\phi_F = 45^\circ, \theta_F$). Out of each field distribution, a set of Stokes parameters related to a polarization state $P_1$ or $P_2$ results. (b) Exemplary polarization states $P_1$ and $P_2$ on the Poincaré sphere. (c) Ideally, $P_1$ and $P_2$ are composed of one component (x- or y). The polarization split ratio SR is a ratio between the coupling efficiency CE of the desired component to its orthogonally converted counterpart. The split ratio refers to a single waveguide. The angular relationship between $P_1$ and $P_2$ on the Poincaré sphere shows, whether the signals from WG 1 and 2 are orthogonal to each other. Non-orthogonal relation results in polarization crosstalk between both waveguides. (Adapted from Ref. [38] under a CC BY 4.0 license.)</figcaption></figure>
### 2.1.4.3 Polarizations’ Splitting and Orthogonality

Along with the coupling performance, another important task for a 2D GC is to act as a polarization splitter/combiner. Particularly, it is of large importance, whether the 2D GC is able to split/combine polarizations orthogonally. In the following, two parameters used for the characterization of the polarization splitting/combining behavior of 2D GCs will be distinguished. The content in the following paragraphs has been partially published in our previous work in Ref. [38].

Assume the configuration in Fig. 2.20 (a), where the 2D GC is placed in the Cartesian (x,y,z)-plane. The input fields from either waveguide 1 or 2 are radiated towards a tilted plane, which is rotated by ($\phi_F = 45^\circ, \theta_F$) with respect to the grating’s coordinates and corresponds to the position where a SMF would be placed ($u, v, w = \text{const.}$). First, waveguide 1 is excited with the fundamental TE mode of a silicon waveguide. The radiated field evaluated at the tilted reference plane can be evaluated in terms of coupling efficiency. In addition a set of Stokes parameters ($S_0, S_1, S_2, S_3$) and angles ($2\psi_1, 2\chi_1$) on the Poincaré sphere (Fig. 2.20 (b)) can be determined (polarization state $P_1$). The bold font used here denotes a Stokes vector. Next, the procedure is repeated after the excitation of waveguide 2, resulting in a polarization $P_2$. Exemplary polarization states $P_1$ and $P_2$ are marked in Fig. 2.20 (b). According to the waveguides’ definition, we could expect that excitation of waveguide 1 will result in an x-polarized field in the SMF and waveguide 2 will deliver a y-polarized field. For the corresponding waveguides, these polarizations are assigned as target-polarizations. Any power conversion, which leads to the excitation of the orthogonal counterpart of the target-polarization is assigned as a cross-polarization. With other words:

* waveguide 1 - x: target-polarization, y: cross-polarization;

* waveguide 2 - y: target-polarization, x: cross-polarization.

From the perspective of a single waveguide, we can define the polarization split ratio $SR$ as the ratio between the power coupling efficiencies $CE$ of the target- and cross-polarization:

$$SR = \frac{CE \text{ target-polarization}}{CE \text{ cross-polarization}} \Rightarrow SR_1 = \frac{CE_x}{CE_y} \quad SR_2 = \frac{CE_y}{CE_x} \quad (2.51)$$

Due to symmetry, the split ratios are equal $SR_1 = SR_2$. This parameter gives us a good intuitive impression about the polarization splitting quality of a 2D GC. In the ideal case, when each waveguide excites only the target-polarization, the split ratio is infinite and perfect polarization separation is guaranteed. Otherwise, signals coded in different polarizations will be erroneously superposed, e.g. the target x-polarization from waveguide 1 and the cross-x-polarization originating from waveguide 2 (and in opposite). To estimate how critical this superposition is, another parameter needs to be introduced. The latter accounts for waveguides 1 and 2 not alone, as in the definition of split ratio, but characterizes the interaction between the fields of both waveguides with their x- and y-components. The important statement about this characterization is whether the signals originating from waveguides 1 or 2 are orthogonal to each other or not. The investigation of this aspect is important, because orthogonal signals can be separated from each other by applying
orthogonal transformations - similarly to the polarization demultiplexing of a signal rotated arbitrarily during its propagation in a SMF. By contrast, non-orthogonal signals cannot be separated perfectly from each other by this means. A polarization crosstalk will be given in such a case, which is the second important parameter of interest here. The essential difference between both used parameters is that the split ratio gives a polarization conversion considering a single waveguide, while the polarization crosstalk refers to the relationship between the signals from both waveguides (illustration in Fig. 2.20 (c)).

<figure><img src="image_27.png" /><figcaption>Figure 2.21: An exemplary representation of two non-orthogonal polarization states on the Poincaré sphere (in a 2D cross-section). (a) A rotation of the Stokes vector $P_1$ by 180° should result in $P_2$. In the non-orthogonal case, the polarization states $\hat{P}_1, \hat{P}_2$ differ by the angle $2\Delta\psi$. (b) A Thales triangle for the determination of a polarization crosstalk. (Adapted from Ref. [38] under a CC BY 4.0 license.)</figcaption></figure>

In the following, the determination of the polarization crosstalk will be explained in more detail. All given fields below are complex. The orthogonality relation of two polarizations can be determined from the Stokes vectors $P_1$ and $P_2$ with their corresponding angles on the Poincaré sphere ($2\psi_1, 2\chi_1$) and ($2\psi_2, 2\chi_2$) (Fig. 2.20 (a), (b)). The orthogonality condition is:

$$ \begin{gathered} |\psi_1 - \psi_2| = 90^{\circ} \\ \chi_1 = -\chi_2. \end{gathered} \qquad (2.52) $$

First, the calculation of the Stokes parameters will be shown. The latter is based on the procedure used for their practical measurement [41]. We use the definitions:

$$ \begin{aligned} S_0 &= I(0^{\circ}) + I(90^{\circ}) \\ S_1 &= I(0^{\circ}) - I(90^{\circ}) \\ S_2 &= I(45^{\circ}) - I(135^{\circ}) \\ S_3 &= I_{RHC} - I_{LHC}, \end{aligned} \qquad (2.53) $$

where $I$ is the field intensity of the polarization states, given in the parentheses. The latter are linear ($0^{\circ}, 90^{\circ}, 45^{\circ}, 135^{\circ}$) and right-handed or left-handed circular (RHC, LHC). In our case, the intensity is calculated as the square of the magnitude of a given electric field distribution, integrated over the evaluation plane. Since the fields are normalized, the parameter $S_0$, giving to the total field intensity, is equal to 1. A normalized Stokes vector can be represented in such a case as a set of ($S_1, S_2, S_3$). The transformation through
polarization plates is described mathematically with the help of their Jones matrices [42].

$$S_0 = l(0^\circ) + l(90^\circ) = \iint |E_u|^2 + |E_v|^2 dudv = 1 \quad \text{normalization} \qquad (2.54)$$

$$S_1 = l(0^\circ) - l(90^\circ) = \iint |E_u|^2 - |E_v|^2 dudv. \qquad (2.55)$$

Transformations representing polarizing plates:

$$\text{linear polarization } \pm 45^\circ : \begin{bmatrix} \hat{E}_u \\ \hat{E}_v \end{bmatrix} = \frac{1}{2} \begin{bmatrix} 1 & \pm 1 \\ \pm 1 & 1 \end{bmatrix} \cdot \begin{bmatrix} E_u \\ E_v \end{bmatrix} \qquad (2.56)$$

$$\text{right/left hand circular polarization : } \begin{bmatrix} \hat{\hat{E}}_u \\ \hat{\hat{E}}_v \end{bmatrix} = \frac{1}{2} \begin{bmatrix} 1 & \pm j \\ \mp j & 1 \end{bmatrix} \cdot \begin{bmatrix} \hat{E}_u \\ \hat{E}_v \end{bmatrix}. \qquad (2.57)$$

Note the linear polarizations' equivalence $-45^\circ \equiv 135^\circ$. Passing the fields through the polarizing plates yields:

$$S_2 = l(45^\circ) - l(135^\circ) = \iint |\hat{E}_{u,\text{pol}45}|^2 + |\hat{E}_{v,\text{pol}45}|^2 dudv - \iint |\hat{E}_{u,\text{pol}135}|^2 + |\hat{E}_{v,\text{pol}135}|^2 dudv \qquad (2.58)$$

$$S_3 = l_{\text{RHC}} - l_{\text{LHC}} = \iint |\hat{\hat{E}}_{u,\text{pol}45,\text{RHC}}|^2 + |\hat{\hat{E}}_{v,\text{pol}45,\text{RHC}}|^2 dudv - \iint |\hat{\hat{E}}_{u,\text{pol}135,\text{RHC}}|^2 + |\hat{\hat{E}}_{v,\text{pol}135,\text{RHC}}|^2 dudv. \qquad (2.59)$$

The degree of polarization (DoP) $p$ and angles on the Poincaré sphere can be calculated from the Stokes parameters as given in Ref. [43], whereat the DoP here is $p = 1$:

$$p = \frac{\sqrt{S_1^2 + S_2^2 + S_3^2}}{S_0} = 1$$

$$S_1 = p \cos 2\chi \cos 2\psi = \cos 2\chi \cos 2\psi$$

$$S_2 = p \cos 2\chi \sin 2\psi = \cos 2\chi \sin 2\psi$$

$$S_3 = p \sin 2\chi = \sin 2\chi$$

$$\Rightarrow 2\psi = \arctan \frac{S_2}{S_1}, \quad 2\chi = \arcsin S_3. \qquad (2.60)$$

After the determination of the angular relation between the signals from the two GC arms, the corresponding polarization crosstalk can be calculated in the next step. For a simplicity, we set $2\chi = 0$. In Fig. 2.21 (a), a 2D cross-section of the Poincaré sphere is shown. Here, the exemplary polarization vectors $\mathbf{P}_1$ and $\mathbf{P}_2$ are such, that the angular difference is $2\Delta\psi < 180^\circ$. For orientation, the normalized Stokes vectors of a x-polarization (1, 0, 0) and a y-polarization (-1, 0, 0) are given. For a system with two orthogonal Stokes vectors, we should be able to cancel one of the two polarizations by rotating it by $180^\circ$. If the assumption of orthogonality is wrong, an error occurs. To determine this error, we rotate e.g. the Stokes vector $\mathbf{P}_1$ by $180^\circ$, obtaining $\hat{\mathbf{P}}_1$.
The polarization states $\hat{P}_1$ and $\hat{P}_2$ are not identical and differ by the angle:

$$2\Delta\tilde{\psi} = 180^{\circ} - (2\psi_2 - 2\psi_1) = 180^{\circ} - 2\Delta\psi. \qquad (2.61)$$

Now, we consider the Thales right-angled triangle in gray, spanned by $\hat{P}_1, \hat{P}_2, \hat{P}_2$, where $\hat{P}_2$ is the orthogonal counterpart of $\hat{P}_2$ (Fig. 2.21 (b)). In the case of orthogonality, the polarizations $\hat{P}_1$ and $\hat{P}_2$ would be coincident and the depicted triangle would not exist. The latter indicates otherwise different paths between the points $\hat{P}_2, \hat{P}_2$ and $\hat{P}_1, \hat{P}_2$. To account for this deviation, the line segment $2 \cdot m$ can be used, where $m$ is a fraction of the total line length 2 (with $p = 1$) between the points $\hat{P}_2$ and $\hat{P}_2$. The parameter $m$ is defined in the interval $m \in [0, 1]$ and its value gives the polarization crosstalk in terms of power. To calculate $m$ as a function of the polarization angles' difference $\Delta\psi$, we apply the geometric mean theorem for the hypotenuse and the altitude $h$:

$$h^2 = [1 + (1 - 2m)] \cdot 2m. \qquad (2.62)$$

Out of the smaller highlighted triangle, we obtain:

$$h = \sin(2\Delta\tilde{\psi}) \Rightarrow h^2 = \sin^2(2\Delta\tilde{\psi}). \qquad (2.63)$$

By setting (2.62) and (2.63) to be equal, a quadratic equation for $m$ results in the end. With the boundary condition that $m$ should vanish in the orthogonal case $\Delta\psi = 90^{\circ}$, i.e. $\Delta\tilde{\psi} = 0^{\circ}$, the final solution is:

$$m = \frac{1}{2} (1 - \cos(2\Delta\tilde{\psi})) = \sin^2(\Delta\tilde{\psi}). \qquad (2.64)$$

The polarization crosstalk in dB can be calculated as $m[dB] = 10 \lg m$.

## 2.2 Measurement Methodology

Although the experimental characterization of 2D GCs requires a relatively simple means, several important objectives should be considered to reach confidence in the measurement results. This includes:

* Proper power normalization in loss measurements. Any loss contributions off- and on-chip must be determined as precisely as possible. Any overestimation of the additional loss terms may lead to rather optimistic coupling efficiency results. It is mandatory to describe carefully the means for power normalization, when measurement results are presented.

* Reliable approach for relative measurements. Here, the focus is on the accurate determination of the polarization split ratio. An appropriate test structure is required for a coupling-efficiency-independent measurement. Robustness against mechanical instability is desirable as well.

* Wafer statistics. When we develop a component such as an integrated 2D GC, which is intended for a large-scale fabrication, the repeatability of its performance is crucial.
It is not correct to give the best chip on a wafer as a benchmark. A given GC's parameter should be characterized by a mean value and a standard deviation. A full, automated wafer-scale measurement is best suited for that purpose. If this should not be attainable, at least 9 chips at fixed wafer positions should be measured. This number of chips is typical for process control measurements.

In the following subsections, these points will be discussed in further detail.

<figure><img src="image_28.png" /><figcaption>Figure 2.22: An exemplary setup for the characterization of passive on-chip components (not to scale). A tunable laser source in the C- or O-band is used for a signal generation. A manual or programmable polarization controller is used for polarization management. Two single-mode fibers (SMFs) are used for in- and out-coupling. Their position is controlled by mechanical positioners. The signal is received by a power meter and can be further processed on a computer. A photonic chip contains a device under test (DUT), which comprises in the simplest case back-to-back structures with 1D or 2D grating couplers (GCs) at each end. The coupling is further supported by an external light source and a microscope camera, which may introduce limitations in the SMF positioning range. Such factors need to be considered, when laying out a test structure.</figcaption></figure>

### 2.2.1 Basic Setup

In the beginning of our discussion, a basic measurement setup will be shown schematically (Fig. 2.22). With small modifications, each passive device measurement involves these components. A tunable laser source is used for a signal generation in the C- or O-band. A polarization controller is placed prior to chip-coupling, either to ensure the proper polarization state on chip, or to generate different polarization states or arbitrarily polarized light via scrambling. In the first case, a manual polarization controller is sufficient. In the second one, programmable polarization controllers come into question, which have typically a higher insertion loss. The in- and out-coupling is reached via cleaved standard SMFs, which are adjusted by mechanical positioners with three axes (six axes also possible). The signal is finally received by a photodetector and programming tools can be used for
the depiction of a measurement on a computer. The devices under test (DUT) include in the simplest case 2D (or 1D) GCs at both ends, connected by tapers and waveguides. Depending on the investigated problem, the basic configuration can be extended by other components. Aside from mechanical positioners, an external light source and a microscope camera are used as helping tools for coupling. Although these components have no direct impact on the measurement, their placement may introduce restrictions regarding the adjustable coupling angles and the positioning range of the SMFs. These factors need to be taken into account, when laying out a test structure.

### 2.2.2 Power Normalization in Loss Measurements

The power normalization in coupling loss measurements needs to take into account off- and on-chip losses. Off-chip losses are mostly caused by fiber attenuation, connector losses and losses from other components, e.g. a programmable polarization controller. Furthermore, the cleaved fiber facet may suffer from impurity or mechanical damages, due to erroneous cleaving or undesired fiber contacts with the chip surface. The best way to take all off-chip losses into account and to eliminate some of them is the calibration with a photodetector directly in front of the facets of the in- and out-coupling fibers. The calibration takes place at both sides, so that for the receiver path, the fiber at the photodetector must be switched over to the laser source. Apparently, it must be ensured that all connectors and fiber facets are sufficiently clean, so that no loss and a corresponding error in the calibration occur, when the fibers are switched back. In this work, a calibrated photodiode Thorlabs SM05PD5A has been used to determine the off-chip loss in most of the measurements. The measured photocurrent at the in- or the out-coupling fiber facet, is transferred to an optical power, by using the provided responsivity data. Ideally, calibrations at several wavelengths are made to cover a certain band, due to the responsivity's wavelength dependence. Alternatively, only the band central wavelength can be used. This causes a wavelength-dependent error of the determined input power of maximally ±0.1 dB, according to the photodiode data sheet for C- and O-band. In some measurements, a slim photodiode S132Ce is used for calibration purposes, combined with a power meter PM100D by Thorlabs. In this case, the optical power instead of the photocurrent is directly measured.

The second aspect is the determination of the on-chip loss. Typically, short deeply-etched rib waveguides and bends with sufficiently large radii are used to minimize the impact of their loss on the measurement. For longer waveguides, the cut-back [44] or the optical frequency domain reflectometry [45] methods are most commonly used for their loss determination. These techniques distinguish themselves for their simplicity, reliability and robustness, which make them suitable for automated wafer-scale measurements [46]. In the photonic BiCMOS platform relevant for this work, the loss for deeply-etched rib waveguides is typically 3 dB/cm. Another important component on the chip is the taper, which scales a waveguide from the grating width of about 12 µm down to a single-mode waveguide with a 400-500 nm width. Typically, simulations predict a nearly lossless behavior for a sufficiently long taper. However, the specific of our sheared 2D GCs with a non-zero
waveguide-to-grating angle, makes it possible that a mode, coupled by the grating, enters the taper under an angle different from expected. The deviation can result from both material/geometric variations or from fiber misalignment. The angled mode coupling could generally cause higher-order modes. In addition, in rib waveguides, modes in the slab may also occur, leading to spectrum oscillations. The latter could make the determination of the maximum coupling value difficult. The uncertainty due to such artifacts may reach ±0.5 dB. Finally, Fabry-Perot oscillations in the measured spectrum should be mentioned. The latter result in back-to-back configurations, due to back-reflections at the grating and also due to reflections at the SMF's facet. Typical spectrum variation of ±0.1 dB is given. In summary, there are different power loss and oscillation sources, which need to be quantified for an exact determination of the 2D GC's coupling loss. The accumulated uncertainty makes it difficult to guarantee for a loss accuracy better than ±0.5 dB.

<figure><img src="image_29.png" /><figcaption>Figure 2.23: (a) A schematic of a device for the measurement of a polarization split ratio. (b) An incident polarization for a split ratio $SR = 1$, leading to an extinction ratio (ER) of infinity. (c) An incident polarization for a split ratio $SR \to \infty$, leading to an ER of 1 and vanishing resonances. (d) A camera picture of an exemplary fabricated device with assigned components. Other abbreviations: GC: grating coupler, MMI: multi-mode interferometer. (Adapted from Ref. [47], under a CC BY 4.0 license.)</figcaption></figure>

### 2.2.3 A Measurement Technique for the Polarization Split Ratio

An approach for the reliable characterization of the 2D GC's polarization splitting performance has been initially reported in our work, Ref. [47]. Here, the description of the test structure design follows Ref. [47]. The characterization of the polarization split ratio is enabled by the integrated device, schematically depicted in Fig. 2.23 (a). It comprises a 2D GC under study, connected with two Si waveguides via tapers. For linear 2D GCs, adiabatic tapers with a length $\ge 250$ µm can be used. The Si waveguides are typically rib waveguides with the same etch depth as the 2D GC (deeply etched rib waveguides). The core width is 500 nm and the slab width 2 µm in C-band. The tapers and the waveguides
are not depicted in the schematic, but can be seen in the camera picture of an exemplary fabricated device in Fig. 2.23 (d). Furthermore, the test structure comprises a low-loss delay line of 200 µm length on the lower arm, a MMI and two 1D FGCs as outputs. The MMI's split ratio is wavelength dependent, but the variation is small.

The device has the following working principle - the 2D GC splits an incident wave of an arbitrary polarization into the two waveguide arms. If we denote these complex amplitudes as $\mathbf{a}_1$ and $\mathbf{b}_1$, the ratio of their absolute squares corresponds to the 2D GC's split ratio $SR$, i.e. to the power ratio between the polarizations in the two waveguides (Fig. 2.23 (a)). The amplitudes are normalized to the 2D GC's input power

$$SR = \frac{|\mathbf{a}_1|^2}{|\mathbf{b}_1|^2}, \quad |\mathbf{a}_1|^2 + |\mathbf{b}_1|^2 = 1. \qquad (2.65)$$

After the delay line and the MMI, the two signal paths interfere constructively or destructively at the outputs, depending on the wavelength and the corresponding phase delay $\Delta\varphi$ in the delay line. The ratio between the maximal and minimal signal transmission is known as an extinction ratio (ER) and can be measured after out-coupling with a 1D FGC.

From the ER we are able to estimate the 2D GC's split ratio by using a simple matrix model [48]. The relation between the complex input waves $\mathbf{a}_1$ and $\mathbf{b}_1$ and output waves $\mathbf{a}_2$ and $\mathbf{b}_2$ can be mathematically written as a multiplication of a matrix $T_{\text{MMI}}$ describing the MMI transfer properties and a matrix $T_{\Delta\varphi}$ describing the delay line:

$$\begin{bmatrix} \mathbf{a}_2 \\ \mathbf{b}_2 \end{bmatrix} = T_{\text{MMI}} \cdot T_{\Delta\varphi} \begin{bmatrix} \mathbf{a}_1 \\ \mathbf{b}_1 \end{bmatrix} \quad \text{with} \qquad (2.66)$$

$$T_{\text{MMI}} = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & j \\ j & 1 \end{bmatrix}, \quad T_{\Delta\varphi} = \begin{bmatrix} e^{i\Delta\varphi} & 0 \\ 0 & 1 \end{bmatrix}. \qquad (2.67)$$

If we take e.g. the first output, the power measured there is proportional to the absolute square of the wave amplitude $\mathbf{a}_2$

$$P_{\text{out}} \sim |\mathbf{a}_2|^2 = \frac{1}{2} (|\mathbf{a}_1|^2 + |\mathbf{b}_1|^2 + 2|\mathbf{a}_1||\mathbf{b}_1| \sin\Delta\varphi). \qquad (2.68)$$

The maximum and the minimum of $P_{\text{out}}$ can be obtained by maximizing or minimizing $\sin\Delta\varphi$. The maximum-to-minimum ratio is the ER. If we further replace $\mathbf{a}_1$ and $\mathbf{b}_1$ through their split ratio $SR$, we obtain

$$ER = \left( \frac{1 + \sqrt{SR}}{1 - \sqrt{SR}} \right)^2. \qquad (2.69)$$

Measured ERs can be easily translated into split ratios. The working principle is further illustrated by two examples, which do not consider the GC's filter spectrum. In Fig. 2.23 (b), the incident polarization is such that the signal is equally split between the two waveguides. The split ratio is therefore 1 and from (2.69) we obtain an ER of infinity. If we consider another case in Fig. 2.23 (c), where the incident polarization is such that all the power is coupled in one of the waveguides, the split ratio is this time $SR \to \infty$ and the ER is 1, which means that minimal and maximal levels are the same and no resonances result. In our
experiment, the situation in Fig. 2.23 (c) is pursued: we look for a polarization, which will be coupled into one of the 2D GC arms. Ideally, smooth transmission without resonances should result in this case. Out of the minimal measured ER, we can estimate the best achievable split ratio of the considered structure, according to (2.69).

To be able to measure the split ratio according to the given definition, we attempt to reproduce the scenarios depicted in Fig. 2.23 (b), (c) during all measurements. In the first case, the incident polarization is split equally between both 2D GC arms. Such a polarization state can be found by using an auxiliary structure, comprising two 1D FGCs connected to each other in a back-to-back configuration. The 1D FGCs can be aligned on the chip in such a way that maximal coupling efficiency results for a polarization state oriented as in Fig. 2.23 (b). After the desired polarization state is found, a measurement on the 2D GC with a delay line MMI can be done. Due to the equal splitting, very large ERs will be observed. In the next step, this polarization state can be rotated by $2\psi = 90^\circ$ on the Poincaré sphere, using a programmable polarization controller. This corresponds to the scenario in Fig. 2.23 (c). The corresponding ER can be then measured as well and used for the determination of the split ratio SR according to the definition used here. Although this method may have a limited precision, it still gives us the opportunity to observe trends, which is the main purpose of this investigation.

The principle of our measurement using a delay line interferometer may first look rather complicated, but in manual measurements, it has two obvious advantages. First, the signals at the two MMI outputs are only phase shifted. Thus, it is sufficient to measure only one of the outputs, which reduces the time for the measurement and/or the setup complexity. Second, if we measure on two outputs, we need to guarantee that they are equally well coupled, which is practically very difficult in manual measurements. In this experiment, the in- and out-coupling efficiency has no impact on the determined split ratios. The mechanical coupling stability during the long wavelength sweep is also less crucial.

### 2.2.4 Wafer Statistics

**Objectives** Among the largest strengths of 2D GCs compared to lateral spot-size converters is the possibility for an automated wafer-scale characterization. This is especially important for a high-yield production, which makes 2D GCs currently a more mature solution for EPICs. Simultaneously, higher standards for the performance of 2D GCs are given: it is not important that a single device reaches a good behavior, even when it is a record-high one. Rather, it is decisive that most of the devices show a similarly good operation.

Several 2D GC's design parameters can vary on a wafer. Here, they will be categorized, depending on their particular impact:

1. Parameters with an impact on the grating's *effective refractive index* and the maximum transmission *wavelength*. Such parameters include:

- The waveguide height;

- The size and the etch depth of the 2D GC's diffracting elements;
- The thickness of BEOL layers, covering the waveguide and the grating.

2. Parameters with an impact on the 2D GC's *out-coupled power* such as:

- The BOX thickness;

- The thickness of BEOL layers, independent on their distance to the grating.

Typically, we target at a fixed wavelength for a given application. The quality of any 2D GC's parameter can be characterized in terms of a mean value and a $3\sigma$-interval ($\sigma$ the standard deviation). Often, a standard Gaussian statistical wafer distribution can be assumed for the majority of optical parameters. The $3\sigma$-interval defines a span, where 99 % of the values of a certain parameter are expected. The robustness of GCs against variations is often given by their high-index contrast nature and cannot be influenced significantly. Mostly, the choice of an appropriate oxide thickness below the grating or the choice of an appropriate target wavelength can be helpful, provided that such flexibility is given. Otherwise, an optimization of the fabrication process itself is the only way towards improved performance stability. A weighting between complexity and reachable best operation is necessary at this point. In this work, all fabricated 2D GCs have been characterized statistically. At TU Berlin, a manual wafer-probe system has been mostly used for that purpose, which is the main limiting factor for the number of chips, which could be measured on a wafer. The minimal number of chips is 9 (typical for process control measurements), which can be placed as illustrated in Fig. 2.24. The exact positions depend on the total number of chips on the wafer.

<figure><img src="image_30.png" /><figcaption>Figure 2.24: An exemplary representation of 9 chips on a wafer, at which 2D grating couplers (2D GCs) can be characterized.</figcaption></figure>

**Fully-Automated Wafer-Scale Measurements** Although automated wafer-scale measurements are well-established in electronics, there are only very few vendors that are specialized in automated optical measurement equipment. The reason is that optical measurements are significantly more sensitive to alignment, coupling distance adjustment, environmental changes etc. Thus, the fiber alignment and positioning system is among the most challenging components in a fully automated wafer test system. In contrast to
electronic tests, where contact pads are available, the photonics coupling takes place
at a certain distance and under a non-zero coupling angle. The nearly vertical coupling
requires a precise control over the height and an appropriate calibration. In addition,
precise alignment and mechanical stability are required. Verification and calibration are
not trivial and are connected with certain variations [49, 50]. Exemplary automated wafer
probe systems are provided by Formfactor [49] and Keysight [50]. Both vendors invested
effort in high-precision software kits to fulfill the requirement of equal coupling conditions
over the whole wafer. Dedicated calibration tools are necessary to satisfy the stringent
requirements of optical wafer-scale characterization. In addition, it is of large importance
for volume production that high throughput is ensured. Thus, it is desirable that all electric,
optic and electro-optic test are performed in one step [49, 50].

Furthermore, short- and long-term repeatability of the measured results is another issue. A
short-term interval covers several hours, while long-term periods may extend over several
months. It has been shown that coupling loss variation in short-term test remains below
0.1 dB [51]. By contrast, long-term reproducibility is affected by more factors such as
contamination of the fiber facets, mechanical instability, temperature changes, polarization
shift due to fiber movement or other mechanical defects in connectors and other connect-
ing fiber patches. In the same work, the long term repeatability of measured coupling loss
of less than 0.8 dB has been determined [51]. Automated wafer characterization at IHP
Microelectronics is enabled by a fully-automated system by Formfactor. Similar measure-
ment principles and performance in terms of repeatability apply for this test station. Some
of the experimental results in the present work are obtained on this setup.

**Summary** Every measurement, even performed with the best state-of-the-art equipment, contains inevitably a level of uncertainty. For any reported experimental results, a careful description of all setup specifics is mandatory, including the aspects of calibration, stability and repeatability. The range of possible deviations needs to be determined.
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# 3 Fabrication Platform

In early years, Si photonics was a niche research field and the experimental work did not consider constraints given by a certain foundry process. Presently, several Si photonics platforms with a variable device portfolio are available, e.g. Refs. [1-4]. The 2D GCs designed and presented in this thesis consider the boundary conditions for an integration in a 0.25 µm photonic BiCMOS technology. In this chapter, the basic differences between a pure CMOS and a BiCMOS process will be explained. The advantages and specifics of the SiGe:C heterojunction bipolar transistor (HBT) will be highlighted. Afterwards, the combination of SOI photonics with the electronic process in bulk Si for the realization of a photonic BiCMOS platform will be briefly reviewed. An alternative approach for the integration of photonics and electronics both on bulk Si wafers, in a 3D integration manner, will be outlined in the end of this chapter.

## 3.1 Bipolar CMOS

This section gives a brief introduction to the idea of the bipolar CMOS technology and discusses in further detail the realization of high-speed SiGe:C HBTs, which is the most distinctive electronic component in the technology platform relevant for this work.

### 3.1.1 CMOS vs. Bipolar Technology

The following paragraphs are based on Refs. [5, 6]. The information here is intended to give a brief overview, but not to go into technological details on the BiCMOS integration. For a more detailed information, the reader may refer to the publications given in the bibliography.

The motivation for a bipolar CMOS technology is to combine the advantages of metal oxide semiconductor (MOS) and bipolar transistors on a single platform. The MOS transistors are known for their low power consumption and ease of design, however, they have limitations in drive capability and speed. By contrast, bipolar transistors allow a larger current drive and a better high-frequency performance [5].
Figure 3.1 gives a schematic comparison of the cross-sections of two classical components - the n-type metal oxide semiconductor field effect transistor (MOSFET) transistor and the (planar) npn bipolar junction transistor (BJT). Apparently, in the MOSFET, the charge carrier flow and the electric current are horizontally directed. By contrast, the npn transition in the BJT is vertical, which implies a vertical charge carrier flow. This fundamental difference is related to different design concerns, which are reflected in the fabrication technology of both transistor types. The combination of these two technologies - the bipolar CMOS (BiCMOS) - is hence related to compromises in the performance of these devices.

<figure><img src="image_31.png" /><figcaption>Figure 3.1: A schematic cross-section of (a) a n-MOSFET (adapted from Ref. [6], Chap. 2) and (b) a npn-BJT (adapted from Ref. [5], Chap. 1). The components differ by the direction of the charge carrier and electric current flow. The co-fabrication of both transistor types is related to performance trade-offs.</figcaption></figure>

Among the most critical trade-offs is the n-well formation, which is necessary for both the p-MOS and the npn-BJT. For the p-MOS, the n-well optimization is performed with respect to the threshold voltage, punchthrough voltage, source and drain capacitance and body effect. On the other hand, for the BJT's collector formation, the n-well doping and thickness are related to a compromise between high transit frequency and large breakdown voltage. The choice of an appropriate well profile is also technology dependent (n-well, p-well or twin-well) [6].

The efforts to combine properly CMOS with bipolar technology with a minimal number of additional processing steps have been shifted with the time towards the pursuit to reduce the total fabrication time. Current BiCMOS processes attempt rather to add bipolar transistor modules with a minimal disturbance of the standard CMOS process. Additional optimization steps can be added depending on the target application [5].

Often, a comparison between CMOS and BiCMOS has been made. In the first place, it should be noted that CMOS scales significantly faster than BiCMOS. The integration of bipolar modules gets even more challenging with decreasing CMOS nodes. However, there are certain advantages of BiCMOS, which still makes the technology competitive [7]:

* Reliable high-speed performance. The bipolar transistors come with a higher break-
down voltage at the same speed than most of the CMOS transistors.

* Large transconductance and low frequency noise.

* Better BiCMOS passive devices, due to the optimized BEOL, usually not available in CMOS.

With these characteristics, the BiCMOS technologies target at applications for even increasing frequencies and very demanding requirements with respect to power and reliability. Exemplary application fields are the high-speed, high-data-rate communication systems or smart mobility systems [7].

### 3.1.2 SiGe:C Heterojunction Bipolar Transistor

The IHP Microelectronics' fabrication platform related to this work offers high-speed electronics enabled by SiGe:C HBTs based on a 200 mm technology. As a most decisive figure-of-merit, the product of transit frequency and collector-emitter breakdown voltage is used: $f_T \times BV_{CEO}$. Reaching simultaneously a high $f_T$ and a high $BV_{CEO}$ is a challenging task. The HBTs available in this platform are a result from a long development process and designs for different target applications are available. The choice of a SiGe doped base instead of a pure Si base has several advantages. First, the base bandgap in SiGe is reduced, compared to Si. Thus, the barrier height for the emitter-to-base electron flow is smaller, leading to a larger collector current. With this, larger current gain can be reached with SiGe HBTs compared to Si BJTs. Moreover, the SiGe base offers more design flexibility. When a moderate current gain is chosen, the base doping can be optimized for reaching higher $f_T$ and/or higher maximal oscillation frequency $f_{max}$. To maximize both parameters, the boron profile in the base must be as thin and highly doped as possible [5]. For the realization of such a base, the boron diffusion during layer growth and high-temperature anneals must be minimized. A possibility of the reduction of the boron diffusion is the additional carbon doping of the base. This approach is used for the HBTs in this technology as well [8]. Another important aspect is the collector design, responsible for the combination of $f_T \times BV_{CEO}$. The IHP's collectors are formed without subcollectors and deep trenches. The low collector resistance is achieved by high-dose ion implantation after shallow trench formation [9, 10]. The best in class $f_T/f_{max}$ performance has been consistently demonstrated in 0.13 µm CMOS environment ($f_T/f_{max}$ of 505 GHz / 720 GHz) [11]. The latest generation of 0.25 µm HBTs at IHP show $f_T/f_{max}$ values of up to 220 GHz / 290 GHz [12].

## 3.2 Monolithic Photonic BiCMOS

IHP's SiGe:C BiCMOS is a prominent but not the only example of a high-performance 200 mm technology. Most important characteristics of these technologies is the $f_T$ scaling in vertical transistor structures, thus not necessarily advancing lithography beyond 248 nm DUV capabilities, corresponding to a 0.13 µm CMOS. The photonic BiCMOS platform of IHP has been developed from previous electronic BiCMOS technology families featuring a
0.25 µm CMOS and a 5-metal BEOL. The BiCMOS technologies based on a 0.13 µm CMOS have a 7-metal BEOL. Both technology families rely on a 248 nm DUV lithography. For small and intermediate fabrication volumes 200 mm technologies have proven advantageous in terms of cost. However, the 248 nm DUV lithography is not an optimum for nano-waveguide Si photonic structures. Due to the improved resolution at a 193 nm wavelength, finer features can be fabricated. So far, best performing 2D GCs were fabricated using 193 nm DUV lithography [13]. The question whether comparable 2D GCs may be realized with a 248 nm DUV is addressed by this thesis.

As discussed in the previous section, the co-integration of CMOS and bipolar technology is not a trivial task. Even more demanding is the ambition to go for monolithic EPICs. Along with the strict conditions for the realization of fast and reliable electronics, additional requirements for the highly sensitive photonic circuits result.

<figure><img src="image_32.png" /><figcaption>Figure 3.2: An illustration of the photonic modules’ integration in a 0.25 µm electronic BiCMOS flow. Abbreviations: (Bi)CMOS: (bipolar) complementary metal oxide semiconductor, HBT: heterojunction bipolar transistor, BEOL: backend of line, SOI: silicon on insulator, WG: waveguide. (Representation after Ref. [12].)</figcaption></figure>

Currently, there are only few EPIC platforms worldwide. On the other hand, monolithic EPIC platforms have several clear advantages. Among the most important ones is the possible realization of the shortest interconnects between photonic and electronic devices. By contrast, hybrid platforms require wire-bonding or flip chip techniques for interfacing between PICs and EICs, which is inevitably related to additional parasitic resistances, capacitances and inductances with an impact on the whole device performance [12, 14–16]. In addition, a technology with a common BEOL stack for photonics and electronics is
favorable compared to dual backend approaches with expensive photonic-electronic co-
packaging [14].

The main challenge for the monolithic photonic-electronic integration is the frontend of
line (FEOL) with different substrate requirements. On the one hand, high-performance
BiCMOS devices are typically realized in bulk Si, while photonic devices require SOI wafers.
The translation of BiCMOS on SOI is limited by the deep collector and the high thermal
resistivity of the buried oxide, which is a significant issue for the heat dissipation of high-
speed circuits [14]. To overcome these problems, a mixed-substrate technology - the local
SOI approach [17] - has been introduced. First, an SOI substrate with a 2 µm thick buried
oxide and a 220 nm SOI layer, intended for photonic applications, is taken. No BiCMOS
requirements are considered at that point. Afterwards, the SOI layer and the buried oxide
are removed locally by a sequence of plasma and wet etch steps. These locally etched areas
are intended for the BiCMOS electronics and are re-filled in the next step by a selective Si
epitaxy. Chemical mechanical polishing (CMP) is used for the planarization. This approach
leads to the final monolithic integration of photonics and BiCMOS electronics, achieving
an unaltered BiCMOS performance in comparison to the pure electronic technology [14].
Figure 3.2 illustrates the integration of photonic modules within the 0.25 µm BiCMOS
according to Ref. [12].

<figure><img src="image_33.png" /></figure>

Figure 3.3: A schematic cross-section of a photonic BiCMOS platform. In the frontend of line (FEOL) high-speed heterojunction bipolar transistors (HBTs) in bulk Si are integrated in parallel with silicon on insulator (SOI) photonic devices, such as waveguides, coupling interfaces, multimode interferometers (MMIs), Ge photodetectors etc. Both electronics and photonics share the same backend of line (BEOL) stack, consisting of multiple SiO₂ layers with different thicknesses and material constants. The BEOL offers 5 metals for the realization of the shortest interconnects between photonics and electronics. (Adapted from Ref. [18] under a CC BY 4.0 license.)
Figure 3.3 shows an exemplary cross-section of the IHP’s monolithic photonic-electronic platform. In the FEOL, SOI photonics, including coupling devices, waveguides, tapers, directional couplers, MMIs, Mach-Zehnder modulators and Ge photodetectors are depicted on the left-hand side. The photonic devices are realized in parallel with high-speed SiGe:C HBTs in bulk Si (depicted on the right-hand side). Both photonics and electronics share a common BEOL layer stack, comprising multiple SiO₂ and Si₃N₄ with different thicknesses and refractive indices. The specific geometric and material properties of the covering layers are taken into account in all 2D GC designs. It should be noted that this has no impact on the general observations and design rules, which have been derived in the scope of this thesis. Generally, the 2D GC designs would require only small modifications to be adopted to other fabrication platforms. Finally, the platform offers three thin (Metal1, Metal2, Metal3) and two thick metals (TopMetal1, TopMetal2) for the realization of the shortest interconnects between photonics and electronics.

## 3.3 BiCMOS With BEOL Photonic Layers

The monolithic photonic BiCMOS platform presented in the previous section shows the established fabrication routine for photonic-electronic integration. In the recent years, an alternative approach has been investigated, which is based on the principles of the 3D photonic integration. Some of our preliminary ideas and analyses on that concept have been published in Ref. [18] and will be given in this section. 3D silicon photonics is based on the idea of the extension of the number of photonic layers, thus improving the integration density and the design flexibility of photonic components [19]. Several materials come into question for that purpose: hydrogenated amorphous silicon (a-Si:H) [20], silicon nitride (Si₃N₄) [21, 22] or aluminum nitride (AlN) [23]. From the perspective of GC enabled devices, a high-index contrast material is preferable. Among the listed materials, a-Si:H offers the highest refractive index contrast, and has been proved suitable for inter- or intra-chip vertical grating coupling [24]. The initial disadvantage of the high absorption loss of a-Si at telecom wavelengths could be overcome by a hydrogen passivation [25]. For these reasons, a-Si:H has been considered a promising material for 3D photonic transmitters or transceivers as a part of the IHP’s photonic BiCMOS platform.

The basic motivation for a 3D photonic integration is the following. After years of research in silicon photonics, it is now well-known that the silicon modulator and the modulator drivers are the bottleneck for the realization of high-speed all-silicon transmitters [26]. Due to the fundamental limitations of all-silicon phase shifters, silicon Mach-Zehnder modulators typically show opto-electrical bandwidth < 50 GHz, large modulation loss and $V_π$ and considerable power consumption [27–30]. For these reasons, many groups started considering the accomplishment of silicon hybrid transmitters, in which the modulator is made of material with distinct second-order nonlinearities (Pockels effect). Modulators based on various materials have been reported, e.g. nonlinear polymers [31], lead zirconate titanate [32], barium titanate [33] and lithium niobate [34–37]. Moreover, attempts towards the co-integration on a full-flow silicon platform have been undertaken [33, 36]. On the
one hand, polymers suffer from temperature and long-term stability issues. Polymers are in general not FEOL compatible. On the other hand, all inorganic Pockels materials listed here are not compatible with common CMOS processes and require a post fabrication bonding step. From the perspective of a foundry process flow, bonding of a Pockels material in the FEOL would require significant modifications of the FEOL fabrication routine, adding processing time and complexity. In comparison, bonding after the BEOL has been processed is less challenging and both IBM [33] and Sandia [36] demonstrated their modulators in the BEOL. If we consider keeping the modulator there, an extension with another photonic layer at that level is necessary.

<figure><img src="image_34.png" /><figcaption>Figure 3.4: A schematic representation of a 3D photonic BiCMOS platform with electronic and photonic components in the frontend of line (FEOL) extended by a photonic amorphous Si:H (a-Si:H) layer at the TopMetal1 level. Grating couplers (GCs) can be used as in- and out-coupling interfaces, which may or may not use Metal3 as a back-reflector. A 2D GC is needed for a polarization-multiplexed system. A Pockels material can be bonded for the realization of high-speed modulators and transmitters in the backend of line (BEOL). The realization on both silicon-on-insulator (SOI) and bulk Si wafers is possible. (Adapted from Ref. [18] under a CC BY 4.0 license.)</figcaption></figure>

Figure 3.4 shows an exemplary cross-section of the concept of a 3D photonic BiCMOS platform with an additional a-Si:H photonic layer in the BEOL at the TopMetal1 level. The combination of the a-Si:H layer with a Pockels material of choice could enable the realization of high-speed transmitters, which meet the requirements for future generation DCIs. The separation of the metal levels, where Pockels materials and a-Si:H are integrated, requires a thick enough interlayer stack. The interlayer dielectric stacks above the thin metal layers (Metal1, Metal2, Metal3) have a thickness of around 900 nm, which is not sufficient. For that reason, the integration of a-Si:H after processing TopMetal1 is favorable, since the
interlayer dielectric stack between TopMetal1 and TopMetal2 with a thickness of around 3 µm fulfills best the requirements.

In the photonic a-Si:H layer, typical components such as waveguides, directional couplers, MMIs etc. can be realized in the same way as their SOI counterparts. Grating couplers can be used as in- and out-coupling interfaces, whereat the input interface can be substituted in future by a hybrid integrated laser source, following the principles outlined e.g. in Refs. [38–40]. For polarization-multiplexed systems, 2D GCs are necessary as an output. Regarding the efficiency of GCs, a low thickness variation of silicon dioxide (SiO₂) is very decisive. There are several process points in the BEOL, which cause variation in the SiO₂ cladding. Before the fabrication of each metal layer, a CMP is done to create suitable process conditions. This means that we have four CMP modules (PreMetal1, PreMetal2, PreMetal3, PreTopMetal1), which will increase the thickness variation. If we consider the whole BEOL with the planarized TopMetal1 topography, a typical thickness variation of the SiO₂ cladding of around 330 nm results. In this context, the adoption of Metal3 as a mirror below the GCs is not only to increase the coupling efficiency, but – more important – to reduce the influence of the SiO₂ thickness below the grating. For a SiO₂ thickness between a-Si:H and Metal3 of around 3 µm, a thickness variation of around 210 nm can be expected.

A deviation in that range is acceptable for the 2D GCs, which will be shown in Chap. 5.
A typical transmitter configuration requires integrated photodetectors for setting the operation point of the modulator. In the case of a BEOL integrated modulator, an interlayer coupling scheme such as in Ref. [23] will be needed to access a frontend photodiode. The interlayer coupling is not shown in Fig. 3.4, being still under development. The possibility for non-SOI photodetector has been investigated on our BiCMOS platform as well [41]. In this concept, a Si₃N₄ photonic layer has been used in the FEOL. This outlines the potential possibility to realize complete transceivers on bulk Si wafers. As a very premature concept, an in-depth analysis is still forthcoming. Nevertheless, the analysis of 2D GCs in this context is of interest for this thesis, since reachable maximal coupling efficiency limits can be explored.
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# 4 Investigation of Fundamental Physical Effects in Two-Dimensional Grating Couplers

Among the most significant objectives of this work is the systematic explanation of the physical properties of 2D GCs. This includes:

1) The description of the *diffraction* in two directions. This is possible by the derivation of a 2D diffraction condition, in which all material and geometric 2D GC's parameters are put into a clear relation with the two angles of diffraction $\varphi_{out}, \theta_{out}$. The reliability of the derived formulas can be shown in simulations and experiments.

2) The demonstration of the importance of *in-plane scattering* effects. The latter have been underestimated in 2D GCs until now. This work demonstrates the fundamental limitations due to in-plane scattering and proposes new means to overcome in-plane-scattering-related issues.

In this chapter, the basic description of these effects will be summarized. Methods for the 2D GCs optimization by an appropriate diffraction and scattering management will be reviewed in Chap. 5.

## 4.1 Diffraction

This section is dedicated to the derivation of a 2D diffraction condition, which links the geometric and material 2D GC's properties to two angles of diffraction - an azimuth angle $\varphi_{out}$ and an off-plane angle $\theta_{out}$ (Fig. 4.1). Out of this condition, general design rules for our sheared 2D GCs are derived. Their feasibility is verified in numerical simulations.
<figure><img src="image_35.png" /><figcaption>Figure 4.1: An illustration of a sheared 2D grating coupler (2D GC) with a rhombus-shaped grating area, placed in the plane (x,y,z = 0). An incident wave vector $\vec{k}$ is transformed to a vector $\vec{k}_{in}$ due to the angled interface between waveguide and grating. The vector $\vec{k}_{in}$ is defined by the effective refractive index of the 2D GC $n_{eff,GC}$ and the angle of incidence $\varphi_{in}$. The 2D GC's wave vector $\vec{K}$ is defined by the period $\Lambda$ and the shear angle $\alpha$. The grating's wave vector $\vec{K}$ transforms the input vector $\vec{k}_{in}$ into the wave vector $\vec{k}_{out}$, which is related to the diffraction angles ($\varphi_{out}, \theta_{out}$).</figcaption></figure>

### 4.1.1 Derivation of a 2D Diffraction Condition

A part of the following analyses has been previously published in Ref. [1]. The starting point is the consideration of the wave vectors of the field in the grating plane and the field radiated outside of the grating. The continuity of the tangential fields at the grating's interface leads to a phase condition, which gives an information about the grating geometry that ensures light deflection under desired radiation angles. We focus here entirely on the coupling to a given target SMF polarization. For that reason, only the target diffraction order will be considered.

For all calculations, an output 2D GC is examined. The grating area is assumed rhombus-shaped as illustrated in Fig. 4.1. A Si waveguide mode propagating in the grating is referred as an input field. A field radiated by the grating is denoted as an output field. All derivations are made for the x-polarized waveguide mode. For a y-polarization all calculations may be performed analogously. Note that the equations for the x-polarization can be converted to equations for the y-polarization by taking into account, that the y-polarization is principally the same field, rotated by 90°.

#### 4.1.1.1 Phase Condition

We start with the formulation of the phase condition of a sheared 2D GC with a rhombus-shaped area and perpendicular feeding waveguides. A phase condition for a sheared 2D GC with tilted waveguides can be found in Ref. [2].
**Input Field** To implement the 2D nature of a 2D GC with respect to a single input arm, the grating and the input field from this arm must be oblique to each other. A modified input field vector with two components and/or a sheared grating wave vector allows for a 2D diffraction pattern. We consider a waveguide with an effective refractive index $n_{eff,WG}$. When this waveguide is periodically perturbed to form a grating, a lower effective refractive index $n_{eff,GC}$ within the grating results. Thus, for a wave propagating from the waveguide into the grating (and vice versa), a Fresnel reflection and refraction can be expected (Fig. 4.1). In sheared 2D GCs, the transition from the waveguide to the grating takes place at an oblique angle. Depending on the 2D GC's type, we may have two scenarios. When the grating area is rhombus shaped, an angled interface between waveguide and grating is given. When the grating is square-shaped, but the feeding waveguide is angled, the incident field propagates in an oblique direction with respect to the interface. In any case, a purely x-polarized waveguide mode that propagates in y-direction will change its propagation direction after a transition into the grating. Thus, the following wave vector transformation takes place:

$$ \begin{gathered} \text{x-pol. waveguide mode: } \vec{k}_{WG} = k_0 n_{eff,WG} \vec{e}_y \\ \text{grating mode: } \vec{k}_{GC} = k_0 n_{eff,GC} (\cos \varphi_{in} \vec{e}_x + \sin \varphi_{in} \vec{e}_y) =: \vec{k}_{in} \end{gathered} \quad (4.1) $$

with $k_0 = \frac{2\pi}{\lambda_0}$, $\lambda_0$ the free-space wavelength, $n_{eff,WG}$, $n_{eff,GC}$ the effective refractive indices of the waveguide and grating mode respectively, $\varphi_{in}$ the angle of transmittance from the waveguide into the grating, which can be initially estimated using the Snell's law. The mode that propagates within the grating is further referred as an input field.

**Grating Wave Vector** The wave vector of a rhombus-shaped grating can be defined as:

$$ \vec{K} = \frac{2\pi}{\Lambda} \sin \alpha \cdot \vec{e}_x + \frac{2\pi}{\Lambda} \cos \alpha \cdot \vec{e}_y \quad (4.2) $$

with $\Lambda$ the grating period and $\alpha$ the shear angle.

If we consider the path difference between two diffracting elements

$$ \Delta \vec{r} = \Lambda (\sin \alpha \cdot \vec{e}_x + \cos \alpha \cdot \vec{e}_y) \quad (4.3) $$

we obtain

$$ \vec{K} \cdot \Delta \vec{r} = \frac{2\pi}{\Lambda} \Lambda \sin^2 \alpha + \frac{2\pi}{\Lambda} \Lambda \cos^2 \alpha = 2\pi. \quad (4.4) $$

This short calculation check confirms that the condition for constructive interference between two partial diffracted waves is fulfilled. Due to the non-zero shear angle, the grating vector $\vec{K}$ can be used for the manipulation of the components of $\vec{k}_{in}$ in two angular directions.

**Continuity Condition** To obtain the phase condition for a sheared 2D GC, the continuity of the tangential fields at the interface between the grating and the outer space must be
considered. For the field phases it means that the wave vectors must be continuous for all x,y.

### Input Wave Vector

$$ \vec{k}_{in} = k_0 n_{eff,GC} (\cos \varphi_{in} \vec{e}_x + \sin \varphi_{in} \vec{e}_y) = k_0 (\tilde{n}_{eff,GC} \vec{e}_x + \tilde{n}_{eff,GC} \vec{e}_y) \text{ with} \quad (4.5) $$

$$ \tilde{n}_{eff,GC} = n_{eff,GC} \cdot \sin \varphi_{in} \quad (4.6) $$

$$ \tilde{n}_{eff,GC} = n_{eff,GC} \cdot \cos \varphi_{in}. \quad (4.7) $$

### Grating Wave Vector

$$ \vec{K} = \frac{2\pi}{\Lambda} \sin \alpha \cdot \vec{e}_x + \frac{2\pi}{\Lambda} \cos \alpha \cdot \vec{e}_y. \quad (4.8) $$

### Output Wave Vector

$$ \vec{k}_{out} = k_0 n_1 \sin \theta_{out} (\cos \varphi_{out} \vec{e}_x + \sin \varphi_{out} \vec{e}_y) \quad (4.9) $$

where $\varphi_{out}$ is the angle in the horizontal plane $z = \text{const.}$ and $\theta_{out}$ is the angle with respect to the vertical. Normally, the propagation angle $\theta_{out}$ in the free space is of interest, the outer space has a refractive index $n_1 = 1$ in this case. When we substitute the cladding's refractive index $n_{cladd}$, we obtain the angle of propagation within the cladding $\theta_{out,cladd}$. At the cladding-air interface, the Snell's law holds:

$$ n_{cladd} \sin \theta_{out,cladd} = n_1 \sin \theta_{out} \quad (4.10) $$

### Continuity at $z = 0$

$$ \vec{k}_{in} - \vec{K} = \vec{k}_{out} \quad (4.11) $$

#### x-component:

$$ k_0 \tilde{n}_{eff,GC} - \frac{2\pi}{\Lambda} \sin \alpha = k_0 n_1 \cos \varphi_{out} \sin \theta_{out} \\ \tilde{n}_{eff,GC} - \frac{\lambda}{\Lambda} \sin \alpha = n_1 \cos \varphi_{out} \sin \theta_{out}, \quad (4.12) $$

#### y-component:

$$ k_0 \tilde{n}_{eff,GC} - \frac{2\pi}{\Lambda} \cos \alpha = k_0 n_1 \sin \varphi_{out} \sin \theta_{out} \\ \tilde{n}_{eff,GC} - \frac{\lambda}{\Lambda} \cos \alpha = n_1 \sin \varphi_{out} \sin \theta_{out}, \quad (4.13) $$

with the definitions of $\tilde{n}_{eff,GC}$, $\tilde{n}_{eff,GC}$ given in (4.6), (4.7). For the y-polarization the same equation system results, if we let $\varphi_{in} \to 90^\circ - \varphi_{in}$, $\varphi_{out} \to 90^\circ - \varphi_{out}$, $\alpha \to 90^\circ - \alpha$ under the condition, that all named angles are mathematically positive.
### 4.1.1.2 Sheared 2D GC Design Parameters

For given 2D GC geometric parameters - a period $A$ and a shear angle $\alpha$ - the radiation pattern may be determined from (4.12) and (4.13).

$$ (4.13) : (4.12) \Rightarrow \varphi_{out} = \arctan \left( \frac{\tilde{n}_{eff,GC} - \frac{\lambda}{A} \cos \alpha}{\tilde{n}_{eff,GC} - \frac{\lambda}{A} \sin \alpha} \right) \quad (4.14) $$

$$ \text{From (4.13)} \Rightarrow \theta_{out} = \arcsin \left( \frac{1}{n_1 \sin \varphi_{out}} \left( \tilde{n}_{eff,GC} - \frac{\lambda}{A} \cos \alpha \right) \right). \quad (4.15) $$

Usually, we are interested in finding design parameters of a 2D GC for a given desired combination ($\varphi_{out} = 45^{\circ}$, $\theta_{out}$), which will be equivalent to the fiber position $\varphi_{out} = \varphi_{F}$, $\theta_{out} = \theta_{F}$. To do so, we can use again (4.12) and (4.13), this time with the shear angle $\alpha$ and the period $A$ as unknown parameters.

To linearize the equation system, following parameters may be introduced:

$$ \begin{aligned} \frac{\cos \alpha}{A} &=: A \\ \frac{\sin \alpha}{A} &=: B. \end{aligned} \quad (4.16) $$

Thus, the linear system of equations that have to be solved with respect to $A$ and $B$ is:

$$ \begin{bmatrix} \lambda & 0 \\ 0 & \lambda \end{bmatrix} \cdot \begin{bmatrix} A \\ B \end{bmatrix} = \begin{bmatrix} \tilde{n}_{eff,GC} - n_1 \sin \varphi_{out} \sin \theta \\ \tilde{n}_{eff,GC} - n_1 \cos \varphi_{out} \sin \theta \end{bmatrix}. \quad (4.17) $$

Finally, the 2D GC's design parameters $\alpha$ and $A$ are obtained:

$$ \begin{aligned} \alpha &= \arctan \frac{B}{A} \\ A &= \frac{1}{B} \sin \alpha = \frac{1}{A} \cos \alpha. \end{aligned} \quad (4.18) $$

It should be noted that the waveguide-to-grating shear angle, initially proposed by Taillaert [2], has been reported often in the literature, e.g. Refs. [3–7], however, without a detailed explanation, what the exact design procedure is and whether the latter describes well strongly perturbed 2D GCs with a shear angle. Strongly perturbed sheared gratings require a full 3D numerical analysis [2]. Therefore, the derived relationships in this work (4.12), (4.13), (4.18) will be verified numerically in the next subsection.

### 4.1.2 Numerical Verification of the 2D Diffraction Condition

From the relations (4.14)-(4.18), we can see that the combination of a shear angle $\alpha$ and a grating period $A$ results in a combination of the coupling angles ($\varphi_{out}$, $\theta_{out}$). The dependence is used to design different 2D GCs for various coupling angles $\theta_{out}$, combined with a constant $\varphi_{out} = 45^{\circ}$. Numerical simulations are carried out for exemplary cases combining different ($\alpha,A$) and a non-sheared 2D GC is used as a reference.
As already mentioned in Chap. 2, there are basically two ways to realize a shear angle in practice. A possibility is to etch a rhombus-shaped grating array and keep the waveguides perpendicular - a design, which is assigned here as a sheared 2D GC of Type I. The second option is to keep the grating rectangular, but tilt instead the waveguides with respect to the grating - a configuration assigned as a 2D GC of Type II. Figure 4.2 illustrates both types of 2D GCs, which have been investigated in this work. In the following, several exemplary designs of sheared 2D GCs Type I and II for C-band will be analyzed.

<figure><img src="image_36.png" /><figcaption>Figure 4.2: Two types of sheared 2D grating couplers (2D GCs) investigated in this work. Type I - rhombus-shaped grating area and perpendicular waveguides. Type II - square-shaped grating area and angled waveguides.</figcaption></figure>

With regard to the results that will be shown in the remaining part of this work, the following specifications need to be made. As noted in Chap. 2, all results obtained by simulations are evaluated at the SMF-plane with coordinates (u, v, w = const.) calculated by (2.33), which are tilted with respect to the Cartesian (x, y, z), according to the fiber angles $\phi_F = \phi_{out}$, $\theta_F = \theta_{out}$. An on-chip input polarization $E_x$ is thus translated to the polarization $E_u$, respectively $E_y$ to $E_v$. To avoid confusion, the following equivalence will be set: $E_x = E_u$, $E_y = E_v$. Although the polarizations $E_u, E_v$ are calculated, they will be further denoted as $E_x$ and $E_y$.

<table><caption>Table 4.1: Numerically estimated coupling angles $\phi_{out}, \theta_{out}$ at a 1550 nm wavelength for designs with varying shear angle $\alpha$ and grating period $\Lambda$. Both 2D grating coupler's (2D GC's) polarizations are excited simultaneously. The designs are intended for coupling to a Si rib waveguide with a 220 nm height and slab etch depth equal to the grating etch depth $d$. The buried oxide thickness is 2 µm.</caption><thead><tr><th>Type and<br>shear angle α</th><th>Period λ</th><th>perturbing elements'<br>diameter w<sub>λ</sub> and etch depth d</th><th>x-pol.<br>φ<sub>out</sub>, θ<sub>out</sub></th><th>y-pol.<br>φ<sub>out</sub>, θ<sub>out</sub></th></tr></thead><tbody><tr><td>0°</td><td>615 nm</td><td>420 nm and 120 nm</td><td>90°, 8°</td><td>0°, 8°</td></tr><tr><td>Type I, 2°</td><td>620 nm</td><td>420 nm and 120 nm</td><td>45°, 8°</td><td>45°, 8°</td></tr><tr><td>Type II, 2°</td><td>612 nm</td><td>420 nm and 120 nm</td><td>50°, 8°</td><td>40°, 8°</td></tr><tr><td>Type I 3°</td><td>633 nm</td><td>425 nm and 120 nm</td><td>45°, 12°</td><td>45°, 12°</td></tr><tr><td>Type II 3°</td><td>624 nm</td><td>425 nm and 120 nm</td><td>49°, 12°</td><td>41°, 12°</td></tr><tr><td>Type I, 4°</td><td>643 nm</td><td>430 nm and 120 nm</td><td>45°, 15°</td><td>45°, 15°</td></tr><tr><td>Type II, 4°</td><td>637 nm</td><td>430 nm and 120 nm</td><td>50°, 16°</td><td>40°, 16°</td></tr></tbody></table>
In contrast to 1D GCs, the perturbation strength of a single diffraction element in 2D GCs is lower. For that reason, 2D GCs require a larger etch depth, compared to the "standard" 70 nm. An etch depth of 120 nm is found appropriate for 2D GCs. The diameter of the diffraction elements should be sufficiently large, but its exact value is not determining. It is important that the design rules for a 248 nm DUV lithography are not violated. The distance between two diffracting elements should not be less than 180 nm, which defines the limit for the maximal diffracting elements' size. Table 4.1 summarizes the details of the exemplary designs used here to demonstrate the principle of the 2D diffraction condition. The angles are obtained via spatial FFT (cf. 2.1.4.1) after both input 2D GC's polarizations are excited simultaneously. The evaluation wavelength is 1550 nm. The geometric properties of the perturbing elements are nearly the same - the small diameter variation ensures a constant duty cycle of about 0.67-0.68 (depending on the type), where the duty cycle is defined as the ratio of the diffracting element's diameter to the period. As stated in (4.12), (4.13), (4.18), different combinations of shear angle and grating period result in different combinations of the coupling angles ($\varphi_{out}$, $\theta_{out}$).

<figure><img src="image_37.png" /><figcaption>Figure 4.3: A comparison of (a) the normalized out-coupled power, (b) the mode field overlap and (c) the coupling efficiency of 2D grating couplers (2D GCs) without and with a shear angle $\alpha$ (a grating of Type I). The fiber position is given by the angles $\varphi_F = 45^\circ$, $\theta_F = 8^\circ$.</figcaption></figure>

In the case of zero shear angle, it is evident that no rotation in $\varphi$-direction takes place. The propagation angle $\theta_{out}$ is the same for both polarizations, but the angle $\varphi_{out}$ differs by 90°. It can be further observed that for a desired maximum transmission wavelength, the coupling angle $\theta_{out}$ at a fixed $\varphi_{out} = 45^\circ$ can be increased, by increasing simultaneously the shear angle and the grating period. Although some of the designs have a slight deviation $\Delta\varphi_{out} = \pm 5^\circ$, this is not critical and corresponds to a very small wavelength mismatch. More significant changes in terms of efficiency occur, when $\Delta\varphi_{out} > \pm 10^\circ$, as will be shown
exemplary later in this section. The results in Tab. 4.1 confirm that there is not a single "proper" shear angle, but the latter can be adapted according to the desired coupling angle $\theta_{out}$. The differences between gratings of Type I and II scale with an increasing shear angle and can be explained by different effective refractive indices of both types. The latter are caused by the different shape and material distribution of the grating's area.

Next, the designs in Tab. 4.1 are examined with regard to their coupling efficiency. In Fig. 4.3 two designs without and with a shear angle (Type I) are compared first. The evaluation plane is oriented along the symmetry axis between the 2D GC's arms $\phi_F = 45^\circ$. The coupling angle in both cases is $\theta_{out} = \theta_F = 8^\circ$. Figure 4.3 (a) shows the out-coupled power normalized with respect to the waveguide input power and Fig. 4.3 (b) the mode field overlap of both designs. The out-coupled power differs more profound for wavelengths larger 1560 nm. More remarkably, the maximal mode field overlap is significantly deteriorated in the non-sheared case, due to the missing rotation towards the symmetry plane and the corresponding different optimal coupling positions for both polarizations. This results in about 3 dB lower coupling efficiency at 1550 nm, as can be seen in Fig. 4.3 (c). The shear angle is crucial for the improvement of the 2D GC's mode field overlap and the overall coupling efficiency.

<figure><img src="image_38.png" /><figcaption>Figure 4.4: A comparison of (a) the normalized out-coupled power, (b) the mode field overlap and (c) the coupling efficiency of 2D grating couplers (2D GCs) with different shear angles $\alpha = 2^\circ, 3^\circ, 4^\circ$, corresponding to different coupling angles $\theta_{out} = 8^\circ, 12^\circ, 15^\circ$ at $\phi_{out} = 45^\circ$. The 2D GCs are of Type I - with a rhombus-shaped grating area and perpendicular waveguides.</figcaption></figure>

Figure 4.4 shows the normalized out-coupled power, the mode field overlap and the coupling efficiency of the considered 2D GC designs of Type I. Analogously, Fig. 4.5 shows
the same parameters for 2D GCs of Type II. In both cases, the mode field overlap does not depend significantly on the chosen combination of a shear angle and a grating period. Once again, it is shown that different shear angles $\alpha$ adjust different coupling angles $\theta_{out}$ at the 2D GC's symmetry plane. It should be noted that the maxima of some of the overlap curves are slightly larger than the theoretical limit of 80% [2]. The latter value results from an estimation, assuming an exponential profile of the grating field distribution. Deviations of the actual field profile from the assumed function may cause small deviations in the range of ±5%.

<figure><img src="image_39.png" /><figcaption>Figure 4.5: A comparison of (a) the normalized out-coupled power, (b) the mode field overlap and (c) the total coupling efficiency of a 2D GCs with different shear angles $\alpha = 2°, 3°, 4°$, corresponding to different coupling angles $\theta_{out} = 8°, 12°, 16°$ at $\varphi_{out} = 45°$. The 2D GCs are of Type II - with a square-shaped grating area and tilted waveguides.</figcaption></figure>

Regarding the out-coupled power efficiency, differences can be observed both between structures of different types (because of the effective refractive indices $n_{eff,GC}$ variation) and with different shear angles (because of the changed out-coupling angles $\theta_{out}$). Both parameters - $n_{eff,GC}$ and $\theta_{out}$ - have an impact on the BOX thickness dependence of the out-coupled power at a given wavelength. The wavelength-dependence of the out-coupled power may be responsible for an asymmetric spectrum of a 2D GC, as e.g. in the case of the structure with $\alpha = 2°$, Type I in Fig. 4.4 (c). Another effect may be a (quasi-)shifted maximum as in the case of the structures with $\alpha = 3°, 4°$, Type II (Fig. 4.5 (c)). The spectrum shift in this situation is not caused by a coupling angle mismatch that would be reflected in the mode field overlap (cf. Fig. 4.5 (b)). Appropriate designs have such $n_{eff,GC}$ and coupling angle that the power distribution is nearly constant within the desired wavelength range. With certain fixed design specifications, this is not always achievable.
As could be shown, the role of the shear angle is to "shift" the desired coupling angle $\theta_{out}$ towards the symmetry plane $\varphi_{out} = 45^{\circ}$ for both polarizations. With this, coupling angles $\theta_{out}$ for wavelengths different from the desired one are coupled out at $\varphi_{out} \neq 45^{\circ}$. An example for the coupling-angles-wavelength combinations obtained by a spatial FFT is given for the structure with a shear angle $\alpha = 3^{\circ}$ of Type I in Fig. 4.6. For this design, a change of $\Delta\theta_{out} = \pm 3^{\circ}$ is combined with $\Delta\varphi_{out} \approx \pm 10^{\circ}$ (for the x-polarization on example). This means that unlike for 1D GCs, there is only a limited possibility to shift the grating spectrum's maximum by adapting the SMF's tilt angle $\theta_{F}$. The reason is that a deviating $\theta_{F}$ would not be optimally coupled at $\varphi_{F} = 45^{\circ}$, because the grating fixes the required combination of angles for each wavelength. The mismatch $\varphi_{out} \neq \varphi_{F} = 45^{\circ}$ is responsible for a mode field overlap deterioration. Figure 4.6 shows further the mode field overlap when the fiber is tilted by the default coupling angle $\theta_{F} = 12^{\circ}$ and by two other angles with $\Delta\theta_{F} = \pm 3^{\circ}$. The evaluation plane is always fixed at $\varphi_{F} = 45^{\circ}$. The overlap penalty due to the symmetry plane deviation is about 8%.

<figure><img src="image_40.png" /><figcaption>Figure 4.6: A mode field overlap of a 2D grating coupler (2D GC) with a shear angle $\alpha = 3^{\circ}$ of Type I and a design coupling angle $\theta_{out} = 12^{\circ}$ at $\varphi_{out} = 45^{\circ}$. A comparison between the case of optimal fiber tilt angle $\theta_{F} = \theta_{out}$ and two other coupling cases with $\theta_{F} = \theta_{out} \pm 3^{\circ}$ at $\varphi_{F} = 45^{\circ}$ is done. The table indicates the expected ($\varphi_{out}, \theta_{out}$) combinations for each wavelength, which are fixed by the 2D GC. The combined deviation from both ($\varphi_{out}, \theta_{out}$) determines the wavelength shift. The mismatch between $\varphi_{out} \neq \varphi_{F} = 45^{\circ}$ is responsible for a mode field overlap deterioration.</figcaption></figure>

## 4.2 In-Plane Scattering

In this section, the importance of the grating in-plane scattering is demonstrated in theory and experiment. As shown in the previous section, the out-coupled fields of 2D GCs are always associated with a constructive interference and can be described appropriately by the formulation of a 2D diffraction condition using wave vectors. By contrast, the fields deflected within the grating plane are not necessarily related to a coherent field
superposition and their description using wave vectors may not be sufficiently general. For
that reason, field deflection within the grating plane, assigned here as in-plane scattering,
is mathematically handled in a more rigorous way, using analytical methods. The in-plane
scattering's existence is proven mathematically in a simplified configuration. Although only
qualitative, this analytical study is able to give us a good feeling about the origin of parasitic
in-plane field components in 2D GCs. Next, the effect of the cross-polarization, i.e. the
orthogonal conversion of a given polarization within the grating plane and its subsequent
fiber coupling, is thoroughly investigated as the most significant consequence of in-plane
scattering. It is shown that the cross-polarization is related to coupling and polarization
splitting issues. For the experimental evidence of the in-plane scattering, higher-order
modes excitation in different 2D GC designs is finally studied.

4.2.1 Analytical Investigation of In-Plane Scattering

The following analytical study of the in-plane scattering in 2D GCs has been initially published in Ref. [8]. A simple analytical method is used to describe the scattering by an array of dielectric cylinders with similar properties as the diffracting elements of a 2D GC. Only a limited number of cylinders can be taken into account. The field problem is simplified in several aspects, so that the results can be interpreted only qualitatively. The limitations of the analytical description are summarized in the end of the subsection.

Si photonic GCs are typically considered as diffractive structures, for which only the diffrac-
tion condition is of interest. In 1D GCs, the grating perturbing element is defined completely
along the waveguide's cross-section and is continuous with respect to the wave front. For
this reason, diffraction is the dominant process in 1D GCs. However, in 2D GCs, the perturb-
ing elements are discrete and have a finite dimension with respect to the waveguide and
its wave front. For that reason, an incident wave will not only be diffracted according to the
grating's diffraction condition, but also scattered in the grating plane. The scattering being
initially a random process [9] can receive a more systematic nature, when the scattering
elements are periodic along the propagation direction of the incident wave. Such situation
is given in 2D GCs, therefore, scattering needs to receive as much attention as diffraction.
Here, it is shown that scattering is an issue, which can be responsible for many undesired
effects in 2D GCs such as polarization conversion (i.e. cross-polarization), polarization
crosstalk, PDL and higher-order modes excitation in Si waveguides. These effects are not
predicted by the 2D GC's diffraction condition. The latter states that the polarization of a
wave will be preserved and only the propagation direction will be changed according to
the chosen grating period (no cross-polarization). The diffraction condition can be applied
to higher-order modes, however, their refractive index is significantly smaller and requires
for the same design coupling angle a significantly larger period than the 2D GCs designed
for the fundamental TE₀₀ have. For that reason, lateral higher-order modes coupled to
2D-GC-interfaced Si components cannot result from diffraction.

Scattering of a plane wave by a dielectric or conductive cylinder has been a well-known
electromagnetic problem for many years. Currently, various solutions of special cases are
present in the literature. Good summaries of classical solutions were available already in
the 70s [10, 11]. Simplified analyses consider the cylinder infinite in its length. In addition, the material properties include perfect metals or lossless dielectrics. In more advanced calculations, finite conductivity, dielectric losses (see e.g. Ref. [12] and the references therein), anisotropy [13] or finite length [14] are taken into account. For a certain group of problems, e.g. in the optical domain, the plane wave as an incident wave is substituted by a Gaussian beam [15]. Another interesting aspect is the scattering not only by a single cylinder, but also by a periodic array [12, 15].

The grating area of a silicon photonic 2D GC combines many special conditions in its scattering nature. It consists of a dielectric cylinder array, which is periodic in two directions. The cylinders have a finite length and a size comparable to the wavelength. The material is dielectric with a wavelength-dependent refractive index. Moreover, in our case the cylinders are radially stratified, due to the BiCMOS BEOL filling layers. Obviously, the scattering taking place in 2D GCs is far more complex to describe than is the diffraction. A simple analytical formulation, which clearly states how a scattered field is distributed, in what direction it propagates, and what polarization components it has, is not available. The aim in this work is not to derive an absolutely exact analytical description of the scattering in 2D GCs, but to deliver a mathematical proof, based on a simplified configuration, that scattering can be the reason for a polarization conversion in 2D GCs. The understanding of this fact is almost intuitive, since a plane wave scattered by a circular cylinder results in a cylindrical wave. A previous analysis of the scattering properties of 2D photonic crystals [16] is used as a basis to illustrate that in 2D GCs scattering takes place in parallel to diffraction. For the next analysis, the following simplifications are assumed. The incident wave is a plane wave instead of a Gaussian beam. The grating's perturbing elements are considered as infinitely long dielectric cylinders. These assumptions would be eligible, if a 2D GC is fully etched, the waveguide mode is very well-confined and the grating area is smaller than the incident wave front. For the scattering's description, the relations from Ref. [16] are adapted to the particular problem. A scattered field by a single cylinder and by a cylinder array is represented. Since the derivations are well-explained in Ref. [16], only the significant basic steps are outlined.

<figure><img src="image_41.png" /><figcaption>Figure 4.7: A schematic representation of two scattering problems: (a) scattering by a dielectric cylinder, (b) scattering by a dielectric cylinder array. (Adapted from Ref. [8] under a CC BY 4.0 license.)</figcaption></figure>
For an incident P-polarized wave, the electric field is parallel to the (x,y)-plane, the magnetic field is parallel to the cylinder axis ($\vec{H} = H_z \vec{e}_z$) and is used for the determination of the scattered field. We consider complex fields. The assumed time-dependence in [16] is $\exp(-j\omega t)$. The latter is omitted in the fields' representation.

In general, a non-zero angle of incidence $\phi_i$ can be assumed (Fig. 4.7), which can be related to a waveguide-grating shear angle. The propagation direction of the incident field is depicted by the vector $\vec{n}_i$ with the subscript i for "incident". Here, we are particularly interested in the case $\phi_i = 180^\circ$ (a propagation in x-direction, according to the definitions in Fig. 4.7). Such a propagation is given in perfect vertically coupled gratings, for which no polarization conversion and orthogonal polarization separation is expected [17]. Assuming a propagation direction in $\vec{e}_x$ and a y-polarized incident field, we expect no x-polarized component of the total field.

To obtain the magnetic field in the exterior, the sum of the incident field and the scattered field at the cylinder or the cylinder array is calculated. Once the superposition is calculated for $H_z$, the electric field components result from the Ampere-Maxwell's equation. In Fig. 4.7 the considered cases are illustrated - Fig. 4.7 (a) shows the simple case of a single cylinder, while in Fig. 4.7 (b) we have the extended case of a cylinder array. We assign the refractive indices $n_{1,2}$ in the regions ① and ②, and a free-space wavelength $\lambda_0$.

All calculations presented in the following are adapted from Ref. [16]. We begin with a single cylinder (Fig. 4.7 (a)), which will be later extended to a cylinder array (Fig. 4.7 (b)). In the first half-space ①, the incident wave and the scattered wave have a wave number $k_1$ and the wave impedance $Z_1$, depending on the material refractive index $n_1$. In our case, $n_1$ is the Si waveguide's refractive index. In the cylinder with a radius $a$ (half-space ②), we have the refractive index $n_2$, for which we assume $\text{SiO}_2$, with the corresponding wave number $k_2$ and wave impedance $Z_2$. The listed parameters are defined as:

$$k_1 = \frac{2\pi}{\lambda_0} n_1 \quad Z_1 = \sqrt{\frac{\mu_0}{\varepsilon_0 n_1^2}} \quad k_2 = \frac{2\pi}{\lambda_0} n_2 \quad Z_2 = \sqrt{\frac{\mu_0}{\varepsilon_0 n_2^2}} \quad (4.19)$$

The first step is to expand the incident plane wave in a sum of cylindrical waves (Jacobi-Anger identity). For both cases (single cylinder or cylinder array), the expansion in Bessel-functions $J_m(k_1\rho)$ with the coefficients $p_0$ is the same and can be written in matrix form, as a vector product. Here, [.] represents a column vector with elements defined by the index $m$.

$$H_z = H_0 e^{-jk_1(\cos\varphi_1 x + \sin\varphi_1 y)} = H_0 e^{-jk_1\rho\cos(\varphi-\varphi_1)} = H_0 \psi^j(\rho, \varphi), \quad (4.20)$$

$$\psi^j(\rho, \varphi) = \Phi_j^T \cdot p_0 \quad \text{with} \quad \Phi_j = [J_m(k_1 \rho) e^{im\varphi}], \quad p_0 = [(-j)^m e^{im\varphi_1}], \quad m = 0, \pm 1, \pm 2, \dots \quad (4.21)$$

The scattered field of a single cylinder can be expressed as a sum of Hankel functions of first kind $H_m^{(1)}(k_1\rho)$, weighted by unknown scattering coefficients $a_0^s$.

$$\psi^s = \Phi_H^T \cdot a_0^s \quad \text{with} \quad \Phi_H = [H_m^{(1)}(k_1 \rho) e^{im\varphi}]. \quad (4.22)$$

The scattering coefficients are related to the incident wave coefficients by the T-Matrix $T$ [16, 18], which results from the continuity condition at the cylinder's surface.
For a P-polarized wave the T-Matrix is given as:

$$T = [\tau_m \delta_{mn}], \quad \text{with} \quad \tau_m = -\frac{Z_2 J_m(k_1 a) J'_m(k_2 a) - Z_1 J_m(k_2 a) J''_m(k_1 a)}{Z_2 J'_m(k_2 a) H_m^{(1)}(k_1 a) - Z_1 J_m(k_2 a) H_m^{(1)'}(k_1 a)} \quad (4.23)$$

$$a_0^s = T p_0 \quad (4.24)$$

with $\delta_{mn}$ the Kronecker's delta. The parameters $J'_m, H_m^{(1)'}$ represent the first spatial derivative of the Bessel and Hankel functions. Now, we extend the solution for a single cylinder to a $h$-periodic cylinder array along the $y$-axis (Fig. 4.7 (b)), using the Floquet principle. The scattered field is now modified as:

$$\psi^s(x, y) = \sum_{l=-\infty}^{\infty} e^{ik_1 \cos \varphi_l l h} \phi_{H,l}^T a_0^s \quad \text{with} \quad (4.25)$$

$$\Phi_{H,l} = [H_m^{(1)}(k_1 \rho_l) e^{im\varphi_l}], \quad \rho_l = \sqrt{x^2 + (y-lh)^2}, \quad \sin \varphi_l = \frac{y-lh}{\rho_l} \quad (4.26)$$

The T-Matrix $T$ will be substituted by the aggregate T-matrix $\bar{T}$ with the following steps:

$$L = [L_{mn}], \quad L_{mn} = \sum_{l=1}^{\infty} H_{n-m}^{(1)}(k_1 l h) [e^{-jk_1 \cos \varphi_l l h} + (-1)^{n-m} e^{jk_1 \cos \varphi_l l h}] \quad (4.27)$$

$$\bar{T} = (I - TL)^{-1} T \quad (4.28)$$

$$a_0^s = \bar{T} p_0, \quad (4.29)$$

where $I$ is the identity matrix.

<figure><img src="image_42.png" /><figcaption>Figure 4.8: An analytically calculated electric field distribution for the scattering problem of a dielectric cylinder array, placed along the $y$-axis. The incident wave is a $y$-polarized plane wave, propagating in the $x$-direction. (Adapted from Ref. [8] under a CC BY 4.0 license.)</figcaption></figure>
The final magnetic field in the exterior is a superposition of the incident and the scattered field components. The electric field can be obtained by the Ampere-Maxwell's equation.

$$ \psi(x,y) = \Phi_j^T p_0 + \sum_{l=-\infty}^{\infty} e^{ik_1 \cos\phi_l l h} \Phi_{H,l}^T a_0^s \quad (4.30) $$

$$ \vec{H} = H_0 \psi(x,y) \vec{e}_z \quad (4.31) $$

$$ \vec{E} = \frac{1}{j\omega\epsilon_0 n_1^2} \text{rot } \vec{H} \quad (4.32) $$

In summary, the following scattered fields result for each of the two cases:

$$ H_z = H_0 \psi \quad \text{P-polarized wave} $$

$$ \psi = \psi^i + \psi^s \quad i \text{- incident, s} \text{-scattered} $$

$$ \psi^i = \Phi_j^T p_0, \quad \text{with} \quad \Phi_j = [J_m(k_1 p) e^{im\phi}], \quad p_0 = [(-j)^m e^{im\phi}], \quad m = 0, \pm 1, \pm 2, \dots $$

Single cylinder

$$ \psi^s = \Phi_H^T \cdot a_0^s $$

$$ \Phi_H = [H_m^{(1)}(k_1 p) e^{im\phi}] $$

$$ a_0^s = T p_0 $$

Cylinder array

$$ \psi^s = \sum_{l=-\infty}^{\infty} e^{ik_1 \cos\phi_l l h} \Phi_{H,l}^T a_0^s $$

$$ \Phi_{H,l} = [H_m^{(1)}(k_1 p) e^{im\phi_l}], $$

$$ \rho_l = \sqrt{x^2 + (y-lh)^2}, \quad \sin\phi_l = \frac{y-lh}{\rho_l} $$

$$ a_0^s = \bar{T} p_0 \quad (4.33) $$

Independent of the specific case, the finally resulting $H_z$ field is a sum of cylindrical waves with both x- and y-dependence. When calculating the curl of this field, we will obtain not only the initial polarization $E_y$, but also an $E_x$-polarized part, which is an indication for the origin of cross-polarization in the 2D GCs. Figure 4.8 shows the magnitude of the calculated $E_x$ and $E_y$ field components for a case, which considers the 2D GC's material and geometric properties. The cylinders' material is SiO$_2$, while the exterior's material is Si. The matrix operations are carried out in MATLAB with a Bessel/Hankel functions order $m = 0, \pm 1, \dots, \pm 19$ and a wavelength of 1550 nm. The discrete spatial locations have a resolution of 15 points per wavelength. The calculated field distribution in Fig. 4.8 results from a y-polarized incident field, which passes through the cylinder array in $\vec{e}_x$-direction. The plotted fields are scaled equally, so we can clearly see that the initial $E_y$-polarization predominates. The scattering at the cylinder array appears to spoil its modal purity, but the effect is small for this short 1D array. The cross-polarization $E_x$ looks rather stochastically distributed without a well-defined propagation direction. This can be explained by the small number of scattering elements and by the missing second periodicity in propagation direction, which could enhance the $E_x$ strength and directivity.

This example shows that an array of cylindrically shaped elements leads to the partial conversion of a given incident polarization into its orthogonal counterpart. This happens even when the incident wave propagates perpendicularly with respect to the given array.
Since 2D GCs are basically an arrangement of such cylindrical elements, the orthogonal polarization conversion there (i.e. the cross-polarization) is inevitable. The present analytical model is limited in its capability to describe cross-polarization, as its complexity grows significantly, when we go for a 2D array of scattering objects. In this case, the scattered fields of the first row should be used as a new incident fields on the next row etc. For even higher accuracy, the assumed plane wave must be substituted by a Gaussian beam. In such a case, the Gaussian beam must be expanded in plane waves and the scattering problem must be solved for each partial plane wave. Obviously, the complexity of the analytical model becomes easily comparable with the one of a numerical model and looses the manageability typical for analytical representations. Nevertheless, the simplified analytical formulation is valuable as a qualitative mathematical proof of the origin of cross-polarized fields in 2D GCs. In the following subsections, the consequences of the in-plane scattering will be presented. Their impact on chip or system level will be analyzed in simulations and experiments.

### 4.2.2 Cross-Polarization

As already indicated, the conversion of a given target-polarization into its orthogonal counterpart - an effect assigned as a cross-polarization - is a phenomenon in 2D GCs, which can be explained by scattering processes in the grating plane. There are several important consequences from the cross-polarization, particularly:

1. Limited coupling efficiency. The cross-polarization scales with the grating's perturbation strength, since larger objects enhance the scattering. This puts limitations to the 2D GC's out-coupled power.

2. Limited split ratio. Especially in polarization multiplexed systems, the 2D GCs' purpose is twofold. They act not only as a coupling interface, but also as a polarization splitter/combiner. The presence of cross-polarization inevitably limits the splitting performance of a 2D GC. The latter two issues have been discussed in detail in the scope of this work and have been initially published in Ref. [19].

3. Polarization-dependent loss (PDL). In receiver-side 2D GCs random incident polarizations are given. Depending on their orientation, the grating splits them into x- and y-polarized components with a different power ratio and/or phase relation. This means that a given target-polarization will be superposed with the cross-polarization from the other signal path with a different phase and amplitude relation. This leads to a large PDL. The problem is relevant also for systems without polarization multiplexing. An investigation of this issue has been published in Ref. [20].

4. Polarizations' non-orthogonality and crosstalk. Consider two signals coded into two different channels 1, 2, which are orthogonally polarized to each other. The presence of cross-polarization rotates the polarization states 1, 2 on the Poincaré sphere, so that the latter may become non-orthogonal to each other. A polarization crosstalk results as a consequence.
5. System-level penalties. The presence of a cross-polarized part in a given signal leads to an additional optical signal-to-noise ratio (OSNR) penalty. The latter may be compensated in the case of orthogonal signals, but becomes critical for non-orthogonal signals. The latter two issues were investigated in the previous works [19, 21, 22] with a focus on dual-polarization quadrature amplitude modulation (QAM) systems.

### 4.2.2.1 Split Ratio and Coupling Efficiency Limitations

The basic investigation of the cross-polarization and, particularly, its impact on the 2D GC's split ratio and coupling efficiency are summarized here, following the previously published work in Ref. [19]. The analysis is performed numerically and experimentally and considers the two types of sheared 2D GCs (cf. Fig. 4.2). The impact of different shear angles and etch depths are studied in more detail.

<figure><img src="image_43.png" /><figcaption>Figure 4.9: A comparison of the target- and cross-polarization of different sheared 2D grating couplers (2D GCs). Two types of gratings are considered - Type I with a rhombus-shaped grating area and Type II with angled waveguides. (a) A shear angle $α = 2°$, an etch depth $d = 120$ nm, (b) a shear angle $α = 2°$, an etch depth $d = 90$ nm, (c) a shear angle $α = 3°$, an etch depth $d = 120$ nm. The evaluation plane is tilted, according to the fiber angles $φ_F = 45°$, $θ_F = 8°$. (Adapted from Ref. [19] under a CC BY 4.0 license.)</figcaption></figure>

**Numerical Analysis** For the numerical analysis of cross-polarization related effects, out-coupling 2D GCs for the C-band are considered. Only one of the waveguides is excited, in our case corresponding to the y-polarization. The radiated fields are evaluated at a tilted plane ($φ_F = 45°$, $θ_F$), corresponding to the SMF position. We calculate the power, which is
properly coupled into the y-polarization in the SMF. The portion of power coupled into the x-polarization is then referred as a cross-polarization. The difference between the calculated y- and x-polarization powers in dB at a wavelength of interest corresponds to the 2D GC's split ratio. A large split ratio indicates that the 2D GC is a good polarization beam splitter/combiner (PBS/C). Because in practical measurements it is difficult to determine exactly the maximal transmission wavelength, a mean split ratio in a wavelength range of 15 nm around the y-polarization's maximum is evaluated. In this interval, the coupling efficiency changes by no more than 0.3 dB, which is typically the accuracy limit in transmission measurements. For symmetry reasons, the obtained split ratios are the same, also when a single-port simulation of the other 2D GC arm are carried out. The designs for the investigation of cross-polarization effects are such that maximally large diffracting elements are given. Their diameter and the grating period take into account the design rules of the 248 nm DUV lithography used for the fabrication. The designs are chosen on the one hand to compare structures of Type I and II with different shear angles and on the other hand - structures with different etch depths and fixed remaining geometric parameters. Following geometries and coupling angles are designed, assuming a coupling to/from a Si rib waveguide with a 220 nm height and slab etch depth equal to the grating etch depth.

a) A shear angle $\alpha = 2°$ of Type I or II, a grating period $\Lambda = 622$ nm, circular perturbing elements with a diameter $w_\Lambda = 440$ nm, an etch depth $d = 120$ nm, coupling angles $\varphi_{out} = 45°, \theta_{out} = 8°$ at 1550 nm.

b) A shear angle $\alpha = 2°$ of Type I or II, a grating period $\Lambda = 622$ nm, circular perturbing elements with a diameter $w_\Lambda = 440$ nm, an etch depth $d = 90$ nm, coupling angles $\varphi_{out} = 45°, \theta_{out} = 8°$ at 1590 nm.

c) A shear angle $\alpha = 3°$ of Type I or II, a grating period $\Lambda = 636$ nm, circular perturbing elements with a diameter $w_\Lambda = 450$ nm, an etch depth $d = 120$ nm, coupling angles $\varphi_{out} = 45°, \theta_{out} = 12°$ at 1550 nm.

<table><caption>Table 4.2: Numerically estimated split ratios *SR* in a 15 nm wavelength range near the y-polarization's transmission maximum for different sheared 2D grating couplers (2D GCs). The coupling angle is 8° at the symmetry plane.</caption><thead><tr><th>Shear angle α</th><th>Etch depth d [nm]</th><th>Wavelength [nm]</th><th>2D GC Type</th><th>Split ratio SR [dB]</th></tr></thead><tbody><tr><td rowspan="2">2°</td><td rowspan="2">120</td><td rowspan="2">1530-1545</td><td>I</td><td>9.5</td></tr><tr><td>II</td><td>11.5</td></tr><tr><td rowspan="2">2°</td><td rowspan="2">90</td><td rowspan="2">1585-1600</td><td>I</td><td>14.2</td></tr><tr><td>II</td><td>16.5</td></tr><tr><td rowspan="2">3°</td><td rowspan="2">120</td><td rowspan="2">1570-1585</td><td>I</td><td>7.4</td></tr><tr><td>II</td><td>9.8</td></tr></tbody></table>
The first two designs a) and b) are used to investigate the impact of the etch depth. Because all other parameters are kept unaltered, the maximum transmission wavelength is shifted to 1590 nm for the decreased etch depth of 90 nm. The designs with equal etch depth a) and c) can be compared regarding the impact of the shear angle. For these two designs, different radiation angles $\theta_{out} = 8^{\circ}, 12^{\circ}$ result. To ensure the same coupling conditions e.g. in terms of vertical distance to the grating, the evaluation angle is fixed at $\theta_{F} = 8^{\circ}$. The fixed fiber coupling angle is more convenient for comparison of experimental results as well. With $\theta_{F} = 8^{\circ}$ the design with a shear angle $\alpha = 3^{\circ}$ has a maximum shifted near 1580 nm. Figure 4.9 shows the simulated out-coupling spectra for the two types of sheared gratings, comparing the three different cases. In any of them, we see that the two types of gratings Type I and Type II differ slightly from each other. Another common characteristic in all cases is that the cross-polarization (x-polarized) has a central wavelength, which is shifted to larger wavelengths compared to the y-polarization. The shift is larger in the case of the gratings of Type II. Figure 4.9 (a) and (b) compares the geometries with different etch depths, the remaining parameters fixed. Figure 4.9 (a) and (c) compares models comprising different shear angles. For each structure, a mean split ratio in a 15 nm interval around the maximal transmission wavelength is given in Tab. 4.2. Because of the larger cross-polarization wavelength shift for the Type II gratings, their split ratios appear better than for the Type I grating. Comparing the structures with shear angles $2^{\circ}$ and $3^{\circ}$, we see that the latter shows around 1-2 dB worse split ratio. This shear angle is combined with a different grating period, so that the worse behavior may be caused not by the shear angle alone, but by the combination of shear angle and period.

<figure><img src="image_44.png" /><figcaption>Figure 4.10: An electric field distribution and a power loss in percent caused by fields, propagating further in a 2D grating coupler (2D GC). The shear angle is $\alpha = 2^{\circ}$. The input mode is purely y-polarized. After propagating through the 2D GC, a portion of its power is radiated under $\varphi_{out} = 45^{\circ}, \theta_{out} = 8^{\circ}$ and coupled into a SMF. Power losses caused by fields that propagate further within the grating are shown. The propagating mode has two polarization states. For an etch depth $d = 90$ nm, the power remains mainly within the initial y-polarization (24%) and 10% are converted into the cross-polarization. For an etch depth $d = 120$ nm the power loss is mainly caused by the cross-polarization conversion (24%). (Adapted from Ref. [19] under a CC BY 4.0 license.)</figcaption></figure>
A comparison between the structures with different etch depths in Tab. 4.2 shows clearly that a larger etch depth, corresponding to a stronger grating perturbation strength, leads to about 4-5 dB worse split ratio. For a good splitting functionality, shallowly etched 2D GCs should be preferred. However, unlike the shear angle, the etch depth has a strong impact on the grating out-coupled power and coupling efficiency. If the perturbation strength is too weak, a large amount of power remains guided and is not radiated by the grating. With an increasing etch depth, more cross-polarization occurs and again - further propagating (this time) cross-polarized fields remain in the grating and limit the out-coupled power efficiency. This trade-off is illustrated by the numerical example in Fig. 4.10 for the designs a) and b) of Type I. On the left hand side, we have an input mode, which is purely y-polarized. During the mode's propagation through the 2D GC, a portion of its power is coupled out towards a SMF. Power losses are caused by a radiation towards the substrate (not represented) and by the power further guided in the structure. In Fig. 4.10, we clearly see that in both cases the further propagating mode has an y-polarized part (in the waveguide on the right hand side) and an x-polarized part (in the upper waveguide). For the etch depth *d* = 90 nm the power remains mainly within the initial y-polarization (24%) and 10% are converted into the cross-polarization. On the other hand, for an etch depth *d* = 120 nm the power loss is mainly caused by the cross-polarization conversion (24%) and overall we do not increase the amount of the radiated power. The cross-polarization conversion increases with an increasing grating perturbation strength - compared to 1D GCs, this is an additional loss mechanism limiting the maximal achievable coupling efficiency.

<figure><img src="image_45.png" /><figcaption>Figure 4.11: Exemplary measured interferometric curves for the characterization of different sheared 2D grating couplers (2D GCs) in terms of split ratio. Two types of gratings are considered - Type I with a rhombus-shaped grating area and Type II with angled waveguides. (a) A shear angle $\alpha$ = 2°, an etch depth $d$ = 120 nm, (b) a shear angle $\alpha$ = 2°, an etch depth $d$ = 90 nm, (c) a shear angle $\alpha$ = 3°, an etch depth $d$ = 120 nm. (Adapted from Ref. [19] under a CC BY 4.0 license.)</figcaption></figure>
**Experimental Measurement of the Polarizations' Split Ratio**
The experimental verification of the numerically predicted polarizations' split ratios is carried out by manual wafer measurements, using the test structure and method described in 2.2.3, Chap. 2. We compare structures of Type I and II as well structures with different shear angles $α = 2°$ and $α = 3°$ on the same wafer. For the comparison of the structures with different etch depths $d = 90$ nm and $d = 120$ nm, two separate wafers are available. To account for fabrication variations on these wafers, we determine an averaged split ratio over 10 chips. A laser source Agilent 81960 with a wavelength range from 1505 nm to 1625 nm is used, followed by a programmable polarization controller Agilent 8169A and two standard SMFs for the in- and out-coupling together with their alignment equipment. The signal is finally detected by a power sensor Agilent 81634B. Back-to-back reference structures with 1D-1D FGCs and 2D-2D GCs are used for the determination of the proper coupling angles at both sides. They are chosen such that the 2D GCs at the input and the 1D FGCs at the output have a maximal coupling centered at the same wavelength. The 1D-1D FGC back-to-back structure is additionally used for polarization adjustment, as described in 2.2.3, Chap. 2. The 1D FGCs have a period of 610 nm, a perturbing elements' width of 315 nm and an etch depth of 70 nm. The wavelength sweep is performed in 1 pm increment for the MMI-based measurement, to make the ER determination as accurate as possible. The ER is first averaged in the given 15 nm bandwidth. Its value is then converted to a split ratio $SR$ according to (2.69) in 2.2.3, Chap. 2. The split ratios determined on different chips are finally averaged to account for the wafer deviations (a mean value $\overline{SR}$ and a standard deviation $σ$).

Figure 4.11 shows exemplary interferometric curves for each of the considered 2D GC designs. The polarization is such, that the incident field is coupled into one of both 2D GC arms. The depicted wavelength range covers the 15 nm evaluation bandwidth around the spectral maximum. The vertical axes are scaled equally. The transmission includes the coupling losses at the input and the output. It has been observed that split ratios above 20 dB, which correspond to ERs of below 1.5 dB become difficult to measure, because other effects like Fabry-Perot resonances and noise start to predominate. For that reason, a larger distance between the SMF and the 2D GC has been chosen for the measurements of the structures with a 90 nm etch depth. Parasitic effects could be minimized at the cost of a higher insertion loss (Fig. 4.11 (b)). Since a relative measurement is carried out, the coupling loss is not critical for the ER estimation.

In Tab. 4.3 the bandwidth- and wafer-averaged split ratio $\overline{SR} \pm \sigma$ for each of the considered cases are summarized. All structures, which are directly compared to each other, show a similar standard deviation. The differences between devices of Type I and Type II vary from one device geometry to another and we cannot generally state that one of the types has an advantageous splitting behavior. The lack of systematic difference can be caused by variations of the perturbing elements' size. In addition, SOI height deviations have also an impact, as they change the grating perturbation strength. These fabrication variations can make it difficult to observe differences between the gratings with shear angles $α = 2°, 3°$ as well. Being difficult to distinguish from other fabrication variations, the contrast between
<table><caption>Table 4.3: Experimentally estimated mean split ratios $\overline{SR}$ in a 15 nm wavelength range near the maximal transmission for different sheared 2D grating couplers (2D GCs). The values are averaged over 10 chips with a standard deviation $\sigma$. The coupling angle is 8°.</caption><thead><tr><th>Shear angle α</th><th>Etch depth d [nm]</th><th>Wavelength [nm]</th><th>2D GC Type</th><th>SR ± σ [dB]</th></tr></thead><tbody><tr><td rowspan="2">2°</td><td rowspan="2">120</td><td rowspan="2">1527-1542</td><td>I</td><td>16.4 ± 1.4</td></tr><tr><td>II</td><td>17.7 ± 1.6</td></tr><tr><td rowspan="2">2°</td><td rowspan="2">90</td><td rowspan="2">1579-1594</td><td>I</td><td>19.7 ± 1.1</td></tr><tr><td>II</td><td>19.1 ± 1.7</td></tr><tr><td rowspan="2">3°</td><td rowspan="2">120</td><td rowspan="2">1542-1557</td><td>I</td><td>15.3 ± 1.0</td></tr><tr><td>II</td><td>17.7 ± 0.9</td></tr></tbody></table>

the 2D GCs of different types or with different shear angles appears to be of minor importance. Compared to simulations, the difference between the split ratios for $d = 120$ nm and $d = 90$ nm is lower - between 1 dB and 3 dB and better pronounced for the 2D GCs of Type I. The deviations from the simulation can result for the following reason. In real structures, the perturbing elements are not cylindrically, but conically etched. This results in a lower perturbation strength than assumed in simulations, so that we generally obtain better split ratios than expected. Moreover, deeply etched features are more affected by that problem than shallowly etched ones. This means that the perturbation strength does not scale with the etch depth in the same way, as assumed in the simulations. Because the perturbation difference between 2D GCs with $d = 90$ nm and $d = 120$ nm is lower than assumed, their split ratios differ less from each other. In addition, effects related to non-orthogonality may have also an impact on the maximally reachable split ratio. Nevertheless, the distinction between structures with different etch depths is still observable, which shows that the grating's perturbation strength has the most pronounced impact on the 2D GC's splitting performance.

**System-Level Behavior** The following paragraph is a kind contribution by Pascal M. Seiler, who performed and evaluated system-level simulations and experiments. In these analyses, the impact of the limited split ratio is examined in a coherent system that does not take this issue into account (no dedicated DSP applied). As a criterion, the OSNR penalty at a bit error ratio (BER) of $10^{-3}$ is used. The penalty is defined as the difference between the OSNR levels in dB, which are necessary to reach the target BER. In simulations, penalties for different dual-polarization (DP) QAM formats are determined (results originally published in Ref. [19]). Experimentally, 16-QAM single-polarization (SP) and DP transmission are compared (results originally published in Ref. [21]).

**Simulation Results for Different QAM Formats (by Pascal M. Seiler)** In an integrated DP coherent detection receiver, the 2D GC's limited polarization split ratio, can be expressed as a superposition between two data streams with the same source. The system behavior in such a case is comparable to the in-band crosstalk, discussed in long haul transmission systems [23]. We perform an exemplary simulation, where the split ratio is varied and
the OSNR penalty for reaching the forward error correction (FEC) limit of BER = $10^{-3}$ is determined. The penalty results from the comparison of the same polarization in SP and DP operation. A 28 Gbaud transmission using a laser with a linewidth of 100 kHz is assumed (local oscillator effects are neglected). Polarization mode dispersion is not taken into account.

Figure 4.12 shows the OSNR penalty over the 2D GC split ratio for different QAM formats. If we permit a 1 dB penalty (corresponding to the dashed line in Fig. 4.12), we can estimate the minimal split ratio required for the desired QAM format. For quadrature phase shift keying (QPSK), 8-PSK, 16-QAM, 32-QAM and 64-QAM, the required split ratios are 16.5 dB, 21.5 dB, 23.5 dB, 26.5 dB and 29 dB, respectively. The 2D GC designs discussed in the present section reach mostly split ratios better than 16.5 dB, which make them suitable for DP QPSK without the necessity of dedicated DSP compensation methods. However, for higher order modulation formats, the limited split ratio becomes more critical. The integration of 2D GCs in such systems may be possible only with appropriate signal post-processing. The latter may increase the receiver power consumption and complexity. Therefore, the suppression of cross-polarization effects in 2D GCs is desirable for the realization of cost-efficient integrated higher order DP QAM systems.

<figure><img src="image_46.png" /><figcaption>Figure 4.12: An exemplary simulation of the optical signal-to-noise ratio (OSNR) penalty for reaching the forward error correction (FEC) limit of a bit error ratio BER = $10^{-3}$, plotted over different 2D grating coupler (2D GC) split ratios. The considered DP QAM formats are QPSK, 8-PSK, 16-QAM, 32-QAM and 64-QAM. Further abbreviations: DP: dual-polarization, QAM: quadrature amplitude modulation, (Q)PSK: (quadrature) phase shift keying. (Adapted from Ref. [19], used under a CC BY 4.0 license. Author: Pascal M. Seiler.)</figcaption></figure>

### A 16-QAM Experiment (by Pascal M. Seiler)
To verify the predicted performance deterioration, the polarization splitting/combining limitations of 2D GCs are investigated in a 16-QAM system experiment. The 2D GCs are considered at the transmitter side, but due to reciprocity, the same effects are present at the receiver as well. An exemplary 2D GC design is chosen, which has been analyzed with respect to the split ratio in this section (design c) of Type II on p. 105). To study the impact of the chosen 2D GC, a simple on-chip
structure is designed, which comprises two 1D FGCs as inputs, combined at the two arms of a 2D GC. The 1D FGCs have a period of 610 nm, perturbing elements' width of 315 nm and an etch depth of 70 nm. An exemplary configuration is shown in Fig. 4.13 (a), the shown structures do not correspond to the measured ones. The test structure is fabricated in a standard full photonic BiCMOS flow.

<figure><img src="image_47.png" /><figcaption>Figure 4.13: (a) A test structure (photonic integrated circuit, PIC) used to investigate the polarization combining performance of a 2D grating coupler (2D GC). For a single-polarization (SP) operation, a signal at only one of the inputs is coupled - the 2D GC acts as a coupling interface. For a dual-polarization operation, both inputs are active - the 2D GC acts as a coupler and polarization combiner. (b) A schematic representation of the experimental setup used for a 16-QAM transmission experiment. The inset shows 160k bits of a recovered 16-QAM at 10 GBd for a single polarization. Further abbreviations: QAM: quadrature amplitude modulation, ECL: external cavity laser, DP: dual-polarization, IQM: in-phase quadrature modulator, DAC: digital-to-analog converter, PBS: polarization beam splitter, VOA: variable optical attenuator, PC: polarization controller, POW: power meter, OBPF: optical bandpass filter, OSA: optical spectrum analyzer, Rx: receiver, RTO: real-time oscilloscope. (Author: Pascal M. Seiler, re-print with permission. The figure is published in Ref. [21].)</figcaption></figure>

An intradyne measurement setup is deployed to study the polarization combining performance of a 2D GC (Fig. 4.13 (b)). A commercial IQ transmitter (ID photonics OMFT, 1540.30 nm) is used to generate a DP 16-QAM. The modulator is driven by a DAC (Fujitsu OOLA DK 2A), which supplies four 10 GBd random patterns with a peak-to-peak voltage of approximately 290 mV. The 10 GBd is limited by the available equipment and has no impact on the cross-polarization. The -0.5 dBm (DP) output power are then amplified with an EDFA (IPG EAD-1K-C) to +20 dBm, before split into the respective orthogonal polarizations (denoted as X and Y) using a PBS. The 1D GCs, used as chip interfaces, also act as polarization filters, removing the residual unwanted polarizations after the PBS. Remaining power imbalances of the modulator and PBS are compensated using variable optical
attenuators (VOAs) and a calibrated photodiode at the fiber facet at the chip input. Using the VOAs, one of the polarizations can be blocked of to have either a SP, or DP signal at the 2D GC. The coupling is closely monitored using a power meter and a 20 dB splitter. The SP/DP signal is then filtered using a 1 nm filter to remove amplified spontaneous emission (ASE) noise and subsequently noise loaded using an EDFA (Fiber Labs AMP-FL8011-CB). The input power at the coherent receiver (Finisar CPRV1222A) for the SP and DP signal are -3 dBm and -0.2 dBm, respectively. The local oscillator, LO (Agilent 81960A, 1540.05 nm) is set to +14 dBm. Only one polarization is detected at the coherent receiver, since the impact of cross-polarization on a single polarization is sufficient for this investigation. The detected polarization state is not changed between the SP and DP measurements. The polarization states within the setup are closely controlled using polarization controllers. For this purpose, only the coherent receiver's Y output is connected to the real-time oscilloscope (Tektronix DPO71304SX, 50 GSa/s, 13 GHz bandwidth). The Y output has per specification a polarization extinction ratio of 35.5 dB at 1550 nm.

An offline processing is done with the Tektronix OM1106 optical modulation analyzer software. The DSP includes a frequency offset compensation and a least-mean square filter based on a radius-directed algorithm. The OSNR is varied between 21 dB and 33 dB per polarization and the respective BER are measured by switching between SP and DP using the VOAs. The determined BERs for SP and DP signal are shown in Fig. 4.14. The DP signal is overall showing a penalty in comparison to the SP signal. At a BER of $10^{-3}$, an OSNR penalty of approximately 2 dB can be found. According to Fig. 4.12, the corresponding split ratio is about 16 dB. This value is in the expected range for a structure with this geometry (cf. Tab. 4.3 for a Type II design with a shear angle of 3° and an etch depth of $d = 120$ nm).

<figure><img src="image_48.png" /><figcaption>Figure 4.14: Bit error ratio vs. optical signal-to-noise ratio (BER vs. OSNR) for a transmission with a single active polarization (single-polarization, SP) and two active polarizations (dual-polarization, DP). The 2D grating coupler (2D GC) is included at the transmitter-side. The curves are evaluated for one of the receiver polarizations. The OSNR penalty at BER = $10^{-3}$ in the DP case is approximately 2 dB. (Author Pascal M. Seiler, re-print with permission. The figure is published in Ref. [21].)</figcaption></figure>

#### 4.2.2.2 Polarizations' Non-Orthogonality

Some of the results and discussions shown in the following paragraph have been published in Ref. [22]. The penalties presented in the previous paragraph are related to the splitting
performance limitations of 2D GCs. It can be expected that these negative effects can be compensated by appropriate DSP routines. Since a 2D GC converts partially a x-polarization into a y-polarization and vice versa, the final effect can be compared to the polarization rotation, which takes place in an optical fiber. In coherent receivers, the compensation of this rotation is a standard procedure. However, the recovery of the original x- and y-signals assumes that they are combined/rotated orthogonally. In systems, where the orthogonality is not preserved (e.g. a fiber transmission with PDL), power and OSNR imbalance between the two polarizations can occur [24]. This motivated the investigation of the splitting/combining performance of 2D GCs in terms of orthogonality. A numerical analysis can be carried out, following the procedure in 2.1.4.3, Chap. 2.

In this paragraph, two exemplary 2D GCs for C- and O-band are compared with regard to the polarizations' non-orthogonality. The design geometries for a coupling angle $\theta_{out} = 8^\circ$ at the symmetry plane are:

* C-band: a shear angle $\alpha = 2^\circ$, a grating period $\Lambda = 622$ nm, circular perturbing elements with a diameter $w_\Lambda = 440$ nm, an etch depth $d = 120$ nm.

* O-band: a shear angle $\alpha = 2^\circ$, a grating period $\Lambda = 480$ nm, circular perturbing elements with a diameter $w_\Lambda = 280$ nm, etch depth $d = 120$ nm.

Both designs are of Type II and are intended for coupling to a Si rib waveguide with a 220 nm height and slab etch depth equal to the grating etch depth. For the simulations, a single-port excitation of a given target-polarization is done separately for both input waveguides 1 and 2. The evaluation plane is fixed at the SMF coordinates with $\phi_F = \phi_{out} = 45^\circ$ and $\theta_F = \theta_{out} = 8^\circ$. At this plane, we calculate first the coupling efficiency of the target- and cross-polarization from each waveguide. Due to symmetry, results for one of both waveguides will be shown. In addition, the Stokes parameters of the fields from waveguide 1 and 2 are calculated as well. Out of them, the angles on the Poincaré sphere are determined, whereat the azimuth angles' difference $\Delta\psi = \psi_1 - \psi_2$ is of interest ($\chi_1, \chi_2$ negligible). Finally, a polarization crosstalk is determined out of this difference, using (2.64) from 2.1.4.3 in Chap. 2.

In the following we investigate the designs in a 40 nm bandwidth around their maximum transmission. The design for C-band is centered at 1540 nm, the considered bandwidth is 1520-1560 nm. The design for O-band is centered at 1310 nm, the considered bandwidth is 1290-1330 nm. Figure 4.15 compares multiple parameters, relevant for the coupling and splitting performance of both devices. The coupling efficiencies of both designs (target- and cross-polarization) can be seen in Fig. 4.15 (a) for C-band and (b) for O-band. Apparently, the O-band design outperforms its C-band counterpart in terms of efficiency, with a target-polarization's maximal coupling efficiency of -3.3 dB at 1310 nm vs. -5 dB at 1550 nm. In parallel, the C-band 2D GC shows higher levels of cross-polarization, which directly reflects on the polarization split ratio (Fig. 4.15 (c) vs. (d)). The worse splitting behavior in C-band is further reflected in the orthogonality relation $\Delta\psi$ between waveguides 1 and 2. Within the considered bandwidth, the variation in C-band is in the range $77^\circ$ to $132^\circ$ (Fig. 4.15 (e)), compared to $84^\circ$ to $102^\circ$ in O-band (Fig. 4.15 (f)).
<figure><img src="image_49.png" /><figcaption>Figure 4.15: A comparison of 2D grating couplers (2D GCs) for C- and O-band. (a) Target-and cross-polarization spectra in C-band. (b) Target- and cross-polarization spectra in O-band. (c) The polarization split ratio in C-band. (d) The polarization split ratio in O-band. (e) The polarization angles' difference in C-band. (f) The polarization angles' difference in O-band. (g) The polarization crosstalk in C-band. (h) The polarization crosstalk in O-band. (Adapted from Ref. [22] used under a CC BY 4.0 license.)</figcaption></figure>

This is finally reflected into the polarization crosstalk, which reaches up to -3.5 dB at 1560 nm, whereas the maximum in O-band is -13.5 dB at 1330 nm (Fig. 4.15 (g), (h)). It should be noted that the overall better performance of the O-band design is rather related to the boundaries of our fabrication platform, which are more advantageous for the O-band
2D GCs (also experimentally observed, cf. [25]). Interestingly, independent of the optical band and the achieved split ratio, both 2D GCs combine the signals orthogonally at their central wavelengths. Accordingly, wavelengths other than the design wavelength show non-orthogonality and crosstalk. Although this behavior looks like a good fortune, it becomes obvious that non-optimized 2D GCs can be used in a very narrow wavelength range. This is particularly disadvantageous, when robustness e.g. against laser emission fluctuations is required. The latter is mandatory for systems using uncooled lasers, which are economically reasonable for data centers. Since 2D GCs are candidates for photonic sub-systems in DCIs, where stringent requirements in terms of power consumption and cost are given, the improvement of the 2D GCs' polarization splitting/combining performance is essential. In this work, substantial efforts were made to optimize 2D GCs in this aspect, with results following later in Chap. 5.

### 4.2.2.3 Polarization-Dependent Loss

With respect to undesired effects caused by scattering and cross-polarization, we focus here particularly on receiver-side 2D GCs, which are exposed to randomly polarized light. Here, it will be illustrated that the cross-polarization and the related polarization-splitting issues can be associated with a PDL at the receiver. This PDL is expressed in a different coupling efficiency, depending on the incident polarization.

<figure><img src="image_50.png" /><figcaption>Figure 4.16: An illustration of different incident polarizations at a receiver-side 2D grating couplers (2D GCs). The different orientations are responsible for a different signal splitting manner between both 2D GC arms. (a) and (b) An ideal coupling of a purely y- or x-polarized incident wave. (c) An incident polarization for an in-phase signal splitting, assigned as an even-polarization. (d) An incident polarization for an anti-phase signal splitting, assigned as an odd-polarization.</figcaption></figure>

For the subsequent analysis, we shall make parallels between single- and dual-port simulations. Out of the former, we can determine the coupling efficiency of a given target-polarization, as well as its orthogonally converted component - the cross-polarization.
Afterwards, we can continue with a dual-port excitation, i.e. the simultaneous activation of Ports 1 and 2. There, the final x-polarization, for instance, will be a superposition of the target-polarization from Port 2 and the cross-polarization from Port 1. To account for the polarization diversity at the receiver, different amplitude/phase relations between the signals from the two ports have to be defined. Particularly, we are interested in the scenarios depicted in Fig. 4.16 (c) and (d). The polarization in Fig. 4.16 (c) is related to signals in the two GC arms, which are in-phase, e.g. +x- and +y-polarized. The incident polarization in Fig. 4.16 (d) corresponds to port signals in anti-phase, e.g. -x- and +y-polarized. To distinguish between these two cases, the polarization, leading to an in-phase splitting, is called an *even*-polarization. Accordingly, the polarization for an anti-phase splitting is assigned as the *odd*-polarization. It should be noted that these polarization states can be met in the literature under different names, such as S- and P-polarization or 45°- and 135°-polarization [26–29]. In accordance to the assignment of the even- and odd- incident polarization, the most important circumstance for any difference in their behavior is the way, how the target-polarization and cross-polarization are superposed. In the even case, the target- and cross-polarization are in-phase, while in the odd case, they are in anti-phase. Respectively, any other polarization orientations result in different superpositions of the target- and cross-polarization. The variable scenarios for the superposition of a target- and cross-polarization may be expressed in a PDL. The largest PDL can be expected between the even- and the odd-combination, because the cross-polarized signal is maximized in these cases.

<figure><img src="image_51.png" /><figcaption>Figure 4.17: Coupling efficiencies of a target- and cross-polarization in parallel with the x-component of an even- and odd-polarization for 2D grating couplers (2D GCs) with the same geometry and of different types. (a), (c) Type I; (b), (d) Type II. Both 2D GCs reach a relatively high levels of cross-polarization. In the same time, their even- and odd-polarization are strongly wavelength-shifted from each other, leading to a large polarization-dependent loss (PDL). It is hypothesized that the cross-polarization is responsible for the PDL.</figcaption></figure>
To illustrate this behavior, we consider now two exemplary 2D GC designs of different types. A single-port simulation is used to show the coupling spectrum of the target- and cross-polarization (scenarios in Fig. 4.16 (a) and (b) equivalent). Afterwards, the same structure is simulated in a dual-port simulation (simultaneous excitation), whereat the according phase conditions are set to reproduce the even or odd case (Fig. 4.16 (c) and (d)). The considered structures operate in C-band and differ only by their type and have a waveguide-to-grating shear angle of 2° (Type I or II), a grating period of 622 nm and circular diffracting elements with a diameter of 440 nm and an etch depth of 120 nm (cf. the structures in 4.2.2.1). Compared to previous simulations, the full BEOL has been omitted here, in order to save simulation time and hard disk space. Due to the evaluation at a lower distance to the grating, the coupling efficiency is better than with a full BEOL stack. Since the coupling efficiency is not a subject of investigation here, the deviation is acceptable.

Figure 4.17 (a) and (b) shows the single-port simulation of both designs, indicating the coupling efficiency of the target- and cross-polarization. In parallel, Fig. 4.17 (c) and (d) shows the x-component of the even- and odd-polarization, determined from the dual-port simulation. Due to symmetry, the y-component behaves in the same way and is not shown. Large cross-polarization and large PDL can be observed simultaneously. The PDL is a result from the wavelength-shifted spectra of the even- and odd-polarizations. It is not predefined, whether the even or the odd polarization will be shifted in a given direction, as can be seen from the comparison between 2D GCs of Type I and Type II. The PDL is among the most critical problems in 2D GCs. Based on these observations, in this thesis, it is hypothesized that the large difference between the even- and odd-polarizations is entirely caused by the cross-polarization and the grating’s in-plane scattering as its physical origin. All developed optimization methods (cf. Sect. 5.2, Chap. 5) are based on this assumption. It is worth mention that 2D GCs with a low coupling efficiency will inherently have also a poor cross-polarization out-coupling. For that reason, low-efficiency 2D GCs achieve “automatically” a low PDL as well. The challenge for the optimization methods is thus not only to reduce the scattering/cross-polarization and improve the PDL, but in the same time to deliver a reasonable coupling efficiency.

### 4.2.3 Higher-Order Modes

To find an experimental evidence for the grating’s in-plane scattering, the polarization and mode coupling in 2D GC interfaced Si waveguides is investigated here. With regard to polarization splitting, it is expected that 2D GCs with a vertical fiber coupling $\theta_F = 0°$ are able to separate x- from y-polarized signals perfectly [17]. If we consider Fig. 4.18, this would mean that an incident y-polarization will propagate in waveguides (WG) 1 and 3, and the x-polarization will be coupled to WG 2 and 4. However, this assumption is correct only, when no in-plane scattering is present. In this subsection, it will be shown that even zero-angle coupled 2D GCs suffer from cross-polarization and limited polarization splitting capabilities. Furthermore, it will be shown that 2D GCs designed for different coupling angles change only the modal composition of the undesired coupled fields. Even when higher-order modes can be efficiently filtered in single-mode waveguides, their excitation
may be a further limiting factor for the coupling efficiency.

For the subsequent analysis, the following notations will be made. Considering Fig. 4.18, we assume an incident polarization, such that all power should be coupled into WG 1 (an example with $\theta_F \neq 0°$). Our target-polarization is thus the y-polarized $TE_{00}$ mode (short: $TE_{00,y}$), which results from diffraction at the 2D GC perturbing elements matching the fiber angle $\theta_{out} = \theta_F$. In addition, we investigate, whether other higher-order y-polarized modes are coupled, particularly $TE_{10,y}$. The latter cannot result from out-of-plane diffraction, because its effective refractive index does not fulfill the 2D GC's diffraction condition for the given angle $\theta_F$. The presence of $TE_{10,y}$ will be the first indication of in-plane scattering

<figure><img src="image_52.png" /><figcaption>Figure 4.18: A schematic representation of a 2D grating coupler (2D GC) with a Gaussian beam excitation from top. Several field components may be excited in a Si waveguide due to grating in-plane scattering. The grating plane is defined in the Cartesian $(x,y,z)$ coordinates, the Gaussian beam is tilted under the fiber angles ($\varphi_F = 45°, \theta_F$). The arrows indicate the polarization of the excitation source (in yellow), which is chosen such that all the power should be completely coupled into the $TE_{00}$ mode of the Si waveguide WG 1 by diffraction (target-polarization: $TE_{00,y}$). The example illustrates the case $\theta_F \neq 0°$. The excitation of higher-order modes, e.g. $TE_{10,y}$ can be attributed to in-plane scattering. The power coupled in the other waveguide WG 2 (marked in blue) is assigned as a cross-polarization, which can be also composed of $TE_{00,x}$, $TE_{10,x}$ etc. For the given incident field, the presence of these components is an indication for in-plane scattering as well. (Adapted from Ref. [8] under a CC BY 4.0 license.)</figcaption></figure>

processes in 2D GCs. Similarly, the cross-polarization (here the x-polarization), coupled to waveguide WG 2 will be analyzed. Also in this case, two modes are considered: $TE_{00,x}$ and $TE_{10,x}$. For the given incident field, the presence of the $TE_{00,x}$ and $TE_{10,x}$ components in the Si waveguides can be also attributed to grating in-plane scattering (cf. 4.2.1). Table 4.4 summarizes the notations used in this subsection. The excitation of undesired polarizations and modes is investigated experimentally, considering 3 different 2D GC designs in C-band.
<table><caption>Table 4.4: A summary of the notations, used to assign a desired target-polarization with a TE₀₀ modal component, resulting from diffraction and in-plane-scattering-related polarizations/modes.</caption><thead><tr><td>Full assignment</td><td>Short assignment</td><td>Desired?</td><td>Physical process</td></tr></thead><tbody><tr><td>target-polarization y in mode TE₀₀</td><td>TE₀₀,y</td><td>yes</td><td>diffraction</td></tr><tr><td>polarization y in mode TE₁₀</td><td>TE₁₀,y</td><td>no</td><td>in-plane scattering</td></tr><tr><td>cross-polarization x in mode TE₀₀</td><td>TE₀₀,x</td><td>no</td><td>in-plane scattering</td></tr><tr><td>cross-polarization x in mode TE₁₀</td><td>TE₁₀,x</td><td>no</td><td>in-plane scattering</td></tr></tbody></table>

As a figure-of-merit, the ER between the target TE₀₀,y and the remaining polarizations/-modes is used, which is determined on different wafer sites. More detailed results and discussion can be found in the original publication, Ref. [8].

**Experimental Approach and Setup** We begin with the description of the experimental method, used for the higher-order mode's characterization.

<figure><img src="image_53.png" /><figcaption>Figure 4.19: (a) A camera picture of an exemplary device used for the experimental determination of the polarization state and the modal composition of fields, excited by a 2D grating coupler (2D GC) in Si components. (b) A detailed picture of a 2D GC under test, the input 2D GC varies according to the three considered designs. (c) A schematic of the components following the upper and the lower arms of a 2D GC. For the experiments, only the upper part is relevant, which contains a TE₁₀ → TE₀₀ mode converter and 1D GCs for out-coupling. Output 1 (O1) delivers the portion of power of the TE₀₀ mode. Output 2 (O2) delivers the portion of power of the TE₁₀ mode. (d) A detailed picture of the output sections. (Adapted from Ref. [8] used under a CC BY 4.0 license.)</figcaption></figure>
**Test Structures** All test structures are fabricated in a photonic BiCMOS short flow, in which the BEOL has not been completely processed. This is eligible for the current investigation, since the BEOL has no impact on the higher-order mode excitation by the 2D GCs and its omission reduces the fabrication time. The minimum feature size is defined by the 248 nm DUV lithography we used. Figure 4.19 (a) shows an exemplary device picture, taken with a microscope camera. The relevant device parts are indicated by the dashed ellipse in yellow. Figure 4.19 (b) shows a more detailed picture of a 2D GC under test. Figure 4.19 (c) illustrates schematically the structures in the upper and the lower arms of the 2D GC. Only the upper arm is relevant for the experiments. There, we apply after down-tapering a TE₁₀ → TE₀₀ mode converter, which can extract a TE₁₀ mode propagating in the waveguide towards Output 1 (O1) and convert it to the fundamental TE₀₀ mode propagating into the waveguide towards Output 2 (O2). Therefore, for the polarization in question, O1 gives us information about the power coupled into the fundamental TE₀₀ mode, while O2 shows the share of TE₁₀. At each output, identical 1D FGCs are used (a period 610 nm, perturbing elements with a width of 315 nm, an etch depth 70 nm, a design coupling angle 8° at 1550 nm). Figure 4.19 (d) shows a detailed camera picture of the output sections.

<table><caption>Table 4.5: 2D grating coupler (2D GC) designs used for the investigation of their polarizations' and modal on-chip excitation.</caption><thead><tr><td>Abbr.</td><td>Coupling angle</td><td>Shear angle</td><td>Grating period [nm]</td><td>Perturbing elements diameter [nm]</td><td>Etch depth [nm]</td></tr></thead><tbody><tr><td>M1</td><td>0°</td><td>0°</td><td>585</td><td>360</td><td>120</td></tr><tr><td>M3</td><td>8°</td><td>2°, Type I</td><td>622</td><td>440</td><td>120</td></tr><tr><td>M4</td><td>8°</td><td>2°, Type II</td><td>622</td><td>440</td><td>120</td></tr></tbody></table>

Three different 2D GCs are compared during the measurements. All of them are intended for coupling to a Si strip waveguide with a 220 nm height. The first 2D GC is designed for a zero angle of incidence $\theta_F$ (cf. Fig. 4.18) and has no waveguide-to-grating shear angle. The other two GCs are designed for a 8° coupling angle and differ by the realization of their waveguide-to-grating shear angle (Type I vs. Type II). The geometrical details are summarized in Tab. 4.5 listed with their abbreviations that will be used to refer to the structures. The TE₁₀ → TE₀₀ mode converter consists of multiple adiabatic taper sections, which are designed as in Ref. [30]. Its characterization is done by Karsten Voigt in Fimmwave/PhotonDesign. The conversion efficiency of TE₁₀ to TE₀₀ (O2) is better than 98.6% (-0.06 dB) for the whole C-band. The portion of TE₁₀ power further propagating towards O1 is less than 1‰ (-30 dB). The fundamental TE₀₀ arrives at O1 with an efficiency better than 99.6% (-0.02 dB) and less than 4.5‰ (-23.5 dB) can couple down and propagate towards O2.

**Measurement Setup and Procedure** Manual wafer measurements are performed on 9 chips. Our setup consists of a tunable laser source Agilent 81940A, followed by a
manual polarization controller. Standard SMFs are used for the in- and out-coupling. The measured signal is detected by a power meter Agilent 81634B. The measurement steps are as follows:

* Position and polarization at the 2D GC adjusted for a maximal out-coupling at O1:
  → measure the signal at O1, corresponding to the target TE<sub>00,y</sub>,
  → measure the signal at O2, corresponding to TE<sub>10,y</sub>.

* Position kept constant and polarization at the 2D GC adjusted for a minimal out-coupling at O1:
  → measure the signal at O1, corresponding to the cross-polarized TE<sub>00,x</sub>,
  → measure the signal at O2, corresponding to the cross-polarized TE<sub>10,x</sub>.

<figure><img src="image_54.png" /><figcaption>Figure 4.20: Exemplary measured coupling spectra of the modes TE<sub>00,y</sub> (target-polarization), TE<sub>10,y</sub>, TE<sub>00,x</sub>, TE<sub>10,x</sub> for different 2D grating couplers (2D GCs). The considered structures are (a) M1: 2D GC with a vertical coupling and without a shear angle; (b) M3: 2D GC with a 10° coupling angle and a 2° shear angle (Type I); (c) M4: 2D GC with a 10° coupling angle and a 2° shear angle (Type II). (Adapted from Ref. [8] under a CC BY 4.0 license.)</figcaption></figure>

By setting the polarization for a maximal or minimal transmission, we have to perform the measurement only at the upper arm of the 2D GC. For each structure, we compare the ER between the target TE<sub>00,y</sub> and the remaining polarizations/modes near the maximum transmission wavelength of the target TE<sub>00,y</sub>. For the statistical measurements, we intentionally choose a larger coupling distance, in order to exclude the distance variation as a factor and to move easily from one chip to another on the wafer. The insertion loss
(in- and out-coupling) is between 4 dB and 5 dB higher than for the case of an optimal coupling height. Similarly to the measurements in 4.2.2.1 (see Tab. 4.3), we determine the ERs by first averaging in a 15 nm range around the wavelength of maximal transmission. The obtained value is then averaged over the 9 chips on the wafer.

<figure><img src="image_55.png" /><figcaption>Figure 4.21: A coupling spectrum of the output 1D focusing grating coupler (1D FGC), measured at 10°. (Adapted from Ref. [8] used under a CC BY 4.0 license.)</figcaption></figure>

**Experimental Results and Discussion** First experimental results indicated that the 1D FGC and the 2D GCs M3 and M4 require a coupling angle of 10° instead of 8° (see e.g. simulations in Fig. 4.9 (a)). The deviation is most likely caused by variations of the perturbing elements' size or etch depth. The coupling angle of M1 is 0°. Figure 4.20 shows exemplary measured coupling spectra of the considered polarizations/modes for the investigated designs (for a maximized coupling). An exemplary spectrum of the output 1D FGC is shown in Fig. 4.21. In all plots in Fig. 4.20 it is evident that 2D GCs excite cross-polarized modes TE<sub>00,x</sub>, TE<sub>10,x</sub>, as well as higher-order TE<sub>10,y</sub>, independent of their particular geometry. The statistical analysis helps us to examine the differences between the considered designs more precisely.

<table><caption>Table 4.6: Mean extinction ratios (ERs) ±σ between the target TE<sub>00,y</sub> and the other considered polarizations/modes averaged over 9 chips. Different 2D grating coupler (2D GC) designs are compared. The analyzed structures are M1: 2D GC with a vertical coupling and without a shear angle; M3: 2D GC with a 10° coupling angle and a 2° shear angle (Type I); M4: 2D GC with a 10° coupling angle and a 2° shear angle (Type II).</caption><thead><tr><th>TE<sub>00,y</sub> vs.</th><th>M1: mean ER ±σ [dB]</th><th>M3: mean ER ±σ [dB]</th><th>M4: mean ER ±σ [dB]</th></tr></thead><tbody><tr><td>TE<sub>10,y</sub></td><td>10.6 ±1.7</td><td>12.7 ±2.1</td><td>11.4 ±2.2</td></tr><tr><td>TE<sub>00,x</sub></td><td>27 ±2.1</td><td>23.8 ±2.7</td><td>25.3 ±3.7</td></tr><tr><td>TE<sub>10,x</sub></td><td>16.8 ±0.9</td><td>16.8 ±1.1</td><td>17.0 ±1.4</td></tr></tbody></table>

The results for designs M1, M3, M4 are summarized in Tab. 4.6. The mean extinction ratio of the target TE<sub>00,y</sub> vs. TE<sub>10,y</sub>, TE<sub>00,x</sub> and TE<sub>10,x</sub> is first wavelength-averaged around the maximal
transmission wavelength of each structures. The wavelength ranges are 1545-1560 nm
for M1 and 1552-1567 nm for M3 and M4. Afterwards, the values are wafer averaged
with the corresponding standard deviation σ. Comparing the results in Tab. 4.6, we can
basically observe that designs with a non-zero coupling angle reduce the parasitic coupling
to the TE₁₀,γ mode compared to the vertically coupled structure M1. In the same time,
the cross-polarized TE₀₀,ₓ becomes larger in M3 and M4. The different shear angle type is
responsible for the variation between both structures. With regard to TE₁₀,ₓ, all 3 designs
behave similarly in average. Since higher-order modes will be filtered by single-mode
waveguides, the lower coupling to TE₁₀,γ means that structures M3 and M4 with a tilted
coupling angle improve their coupling efficiency, compared to the zero-angle coupled M1.
This happens at the cost of a deteriorated polarization splitting, as the TE₀₀,ₓ coupling
efficiency increases.

In the original publication Ref. [8] the role of the coupling position is discussed in addition.
Penalties with regard to imbalanced coupling are analyzed in terms of coupling efficiency,
bandwidth and PDL. As this investigation does not change the revealed trends and conclu-
sions here, interested reader may refer to Ref. [8] for further information on that matter.
As most important observation, and contrary to prior expectations, it becomes evident
that vertically coupled devices induce cross-polarization as well (TE₀₀,ₓ and TE₁₀,ₓ). Thus,
out-of-plane diffraction and in-plane scattering appear to be parallel existing physical
processes in all 2D GCs.
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# 5 Design Optimization and Characterization

In this chapter, the 2D GC's design development with respect to coupling efficiency and polarization handling is outlined. First, possibilities to enhance the out-coupled power are investigated, regardless of polarization-related issues. Afterwards, methods to reduce in-plane scattering and cross-polarization related problems are discussed. In the end, it is shown that 2D GCs with a satisfying coupling efficiency, a low PDL and an outstanding polarization splitting performance can be realized without a complication of the standard 0.25 µm photonic BiCMOS process.

## 5.1 Improvement of the Out-Coupled Power Efficiency

There are two widely spread methods to enhance the out-coupled power efficiency of a GC, which have been initially developed for 1D GCs [1]:

1) Via the adoption of an enhanced Si waveguide thickness [2–8].

2) Via the application of a back-reflector below the grating area [9–16].

The idea behind the first concept is to improve the ratio between upwards and downwards diffracted light, that is, to reduce the power of the downwards diffracted wave in favor of the upwards diffracted one. This should be achieved by enhancing the thickness of the Si waveguide in the vicinity of the grating. The abrupt height change, is related to an increased power, diffracted in upwards direction. Typically, we start with a standard 220 nm Si waveguide height, which is locally enhanced by an epitaxial growth in the area, where the grating should be defined. Although there has been a successful demonstration of an enhanced 1D GC on our platform [6], there are issues with regard to the integration compatibility. For that reason, this optimization approach is not considered in this work. The second way to increase the out-coupled power efficiency is to try to re-gain the downwards diffracted light. To do so, a sufficiently good metal or Bragg mirror is necessary. However, this is not the only prerequisite. A proper phase match between the upwards
diffracted wave and the back-reflected wave must be ensured. The latter is typically fulfilled for a proper oxide thickness between the grating and the mirror [1]. The main disadvantage of this optimization method is the complicated fabrication procedure. For the formation of a dielectric Bragg mirror, a dedicated wafer fabrication is required [10, 11]. Alternatively, non-standard double-SOI substrates [14] may be considered. On the other hand, the application of a metal mirror often requires non-CMOS compatible materials and bonding techniques [9, 12, 13, 15, 16]. It should be noted that in the recent years, an alternative approach for reducing the downwards diffraction has been proposed - the dual-layer grating, designed using the adjoint optimization method [17]. The proposed device combines two gratings, the second one intended for the elimination of the downwards propagating light. By additionally applying apodization as a method for an improved mode field overlap, a nearly unity coupling efficiency for 1D GCs with perfectly vertical coupling angle is predicted. However, in the case of 2D GCs, the in-plane scattering, which takes place in multiple directions, needs to be considered as a degree of freedom. This may result in an overall very complex design with a non-trivial fabrication, even when the minimum feature size for a 0.25 µm photonic BiCMOS is guaranteed. Moreover, the elimination of the downwards diffracted light is achievable in a very limited bandwidth, which could exclude e.g. the possibility to use 2D GCs in WDM systems. For that reason, such an optimization procedure is not considered in this work.

<table><caption>Table 5.1: Sheared a-Si:H 2D grating couplers (2D GCs) of Type II, designed for C- and O-band. The designs are intended for an a-Si:H rib waveguide with a 220 nm height and a slab etch depth equal to the grating etch depth.</caption><thead><tr><th>Band, refractive index</th><th>shear angle & grating period</th><th>perturbing elements' diameter & etch depth</th><th>coupling angles φ<sub>out</sub> = 45°, θ<sub>out</sub></th></tr></thead><tbody><tr><td>C-band, n<sub>aSi</sub> = 3.57</td><td>2° & 593 nm</td><td>400 nm & 100 nm</td><td>9°</td></tr><tr><td>O-band, n<sub>aSi</sub> = 3.66</td><td>2° & 458 nm</td><td>280 nm & 100 nm</td><td>9°</td></tr></tbody></table>

Here, an alternative possibility to form a 2D GC with a back-reflector has been investigated. Some of the analyses below have been originally published in Ref. [18]. The concept, already presented in Sect. 3.3 of Chap. 3, involves the realization of 2D GCs in an a-Si:H photonic layer in the BiCMOS BEOL, whereat bulk Si substrate is considered (Fig. 3.4, p. 82). Doing so, the metals in the BEOL can be used as back-reflectors to enhance the out-coupled power of the 2D GCs. The TopMetal1 level has been chosen as an appropriate position for the definition of the a-Si:H layer. In contrast to FEOL 2D GCs, there are no specific BEOL filling layers at TopMetal1 and only around 3 µm SiO₂ covers the 2D GCs. One of the three thin metal levels below TopMetal1 (Metal1, Metal2 or Metal3) is a candidate to form a back-reflector. Among them, Metal3 allows for an appropriate SiO₂ thickness separation of 3.06 µm between grating and mirror. The exact value is given by the BEOL specific layer thicknesses in our BiCMOS platform. Moreover, the choice of Metal3 guarantees for the lowest SiO₂ thickness variation, which is determining for the wafer-scale stability of the 2D GC's performance. Table 5.1 shows sheared 2D GCs of Type II, designed for C- and
O-Band. A light coupling to an a-Si:H rib waveguide with a 220 nm height and a slab etch depth equal to the grating etch depth is considered.

To find an appropriate SiO₂ thickness separation between the grating and an underlying reflecting material, a parameter sweep is carried out. The refractive index of SiO₂ is about 1.45. Due to the long simulation times, the C-band design is considered only. The latter requires less number of grid cells, typically ~ 70 million vs. > 100 million in O-band. In addition, Si with a constant refractive index is assumed as a substrate material ($n_{Si} = 3.47$) and only the upwards diffracted normalized out-coupled power is detected (no mode field overlap evaluation). Figure 5.1 shows the dependence of the out-coupled power on the underlying SiO₂ thickness, out of which the separation to Metal3 of 3.06 µm has been determined as appropriate.

<figure><img src="image_56.png" /><figcaption>Figure 5.1: An a-Si:H 2D grating coupler (2D GC) design – dependence of the normalized out-coupled power on the SiO₂ thickness separation from the Si substrate (mode field overlap not considered). The wavelength is 1550 nm. (Adapted from Ref. [18] used under a CC BY 4.0 license.)</figcaption></figure>

Next, the C- and O-band 2D GCs are simulated for two cases. 1) Bulk Si below the grating. In the absence of Metal3, around 8.37 µm thick SiO₂ layer separates the grating from the underlying bulk Si substrate. The thickness results as a sum of the layer thicknesses defined in the BiCMOS BEOL and its exact value may vary, depending on the exact position of the a-Si:H layer. 2) Metal3 below the grating with a 3.06 µm thick SiO₂ separation. The conductivity of Metal3 is calculated from the measured mean value of its sheet resistance, which is 55 mΩ. Figure 5.2 shows the out-coupled power, the mode field overlap and the coupling efficiency of the C-band 2D GC. Figure 5.3 shows the corresponding parameters of the O-band 2D GC. For symmetry reasons, only one of both 2D GC waveguides is excited (single-port simulation), with a polarization assigned as the target-polarization (here, the y-polarization). The corresponding cross-polarization (here, the x-polarization) is given as well. For both C- and O-band, the mode field overlap changes only little, depending on the back-reflector below the grating. The small differences between the bulk Si and the Metal3 case are most possibly due to the different distances between the reflector and the grating. As expected, the out-coupled power is increased by the metal mirror, however, not only for the target-polarization, but also for the cross-polarization. Overall, the split ratio at the
maximum transmission wavelength remains unchanged in C-band (about 10 dB) and is minimally improved in the O-band (by about 1.5 dB at 1300 nm: 14.5 dB vs. 16 dB). Finally, the coupling efficiency in the C-band design with a metal mirror improves by about 1.6 dB - from -5 dB to -3.4 dB at 1550 nm. For the O-band design, an increase of 1.1 dB is achieved - from -3.9 dB to -2.8 dB at 1300 nm. Obviously, the coupling efficiency increase is not as large as compared to reported 2D GCs with gold back-reflectors [15, 16]. This is due to the lower reflectivity of Metal3 (AlCu alloy), compared to gold. Moreover, power losses, caused by the conversion into cross-polarization cannot be compensated by a back-reflector. A final comment will be given regarding the O-band coupling efficiency, which is about 0.5-1 dB better than in C-band - a phenomenon, observed in SOI 2D GCs as well [21]. This may be given by the lower in-plane scattering strength of the perturbing elements in O-band. A different a-Si:H waveguide thickness may be potentially more appropriate for C-band 2D GC. The current 220 nm have been chosen to allow for a direct comparison between a-Si:H and SOI waveguides and GCs. The waveguide thickness may be adapted in the future, depending on the desired application.

<figure><img src="image_57.png" /><figcaption>Figure 5.2: An a-Si:H 2D grating coupler (2D GC) for C-band at the TopMetal1 level. Two different back-reflectors are compared: bulk Si at a distance of 8.37 µm and Metal3 at a distance of 3.06 µm. The curves represent (a) the normalized out-coupled power, (b) the mode field overlap and (c) the total coupling efficiency of a 2D GC. (Adapted from Ref. [18] under a CC BY 4.0 license.)</figcaption></figure>

A question that remains open for an experimental investigation is the impact of Metal3 imperfections on the wafer-scale performance of 2D GCs. Particularly, metal thickness variation and surface roughness need to be analyzed in more detail. It should be noted that the performance improvement of 2D GCs, achieved by an additional a-Si:H layer in the BEOL and by the adoption of a BEOL metal as a back-reflector, is applicable only at the transmitter side. The BEOL and a-Si:H temperature requirements are not compatible with the fabrication of Ge photodetectors, so that a receiver-side 2D GC still needs to be realized on SOI.
<figure><img src="image_58.png" /><figcaption>Figure 5.3: An a-Si:H 2D grating coupler (2D GC) for O-band at the TopMetal1 level. Two different back-reflectors are compared: bulk Si at a distance of 8.37 µm and Metal3 at a distance of 3.06 µm. The curves represent (a) the normalized out-coupled power, (b) the mode field overlap and (c) the coupling efficiency of a 2D GC. (Adapted from Ref. [18] under a CC BY 4.0 license.)</figcaption></figure>

## 5.2 Optimized Polarization Handling

Apparently, the most significant problems of 2D GCs result from an undesired in-plane scattering. This section is devoted to the analysis of approaches for its minimization, which is related to an improved polarization splitting behavior and a reduced PDL. Along with moderately successful methods, a new design concept is presented, which is a simple, working solution for the elimination of scattering-related issues in 2D GCs. In the first two subsections 5.2.1 and 5.2.2, the optimization methods are presented and numerically supported. Experimental validation is given in the last subsection 5.2.3.

### 5.2.1 Segmented Two-Dimensional Grating Couplers

The strong in-plane scattering strength in 2D GCs is most likely caused by the constructive superposition of the local scattered fields of periodic objects with identical geometry. Since it has been observed that the scattering strength is lower for shallowly etched objects, the first idea for in-plane scattering reduction was to combine shallowly and deeply etched arrays, e.g. in a dual etch configuration [19]. In gratings with such a geometry, the first several periods are shallowly etched for a lower in-plane scattering, followed by deeply etched perturbing elements for an enhanced power out-coupling. However, such 2D GCs showed no improvement, because once the light reached the deeply etched region, strong scattering and cross-polarization resulted rapidly after a small number of periods. Therefore, going away from the unitary nature of the scattering array requires another
means. A possible alternative includes the design of gratings, which have objects with different sizes and a varying distance between them. This leads us to the old concept of the apodization, combined with period chirping. In apodized gratings, the diffracting elements have gradually increasing sizes from the beginning of the array. To ensure a diffraction in a single, clearly defined direction, the periodicity is adapted, depending on the objects' size and the corresponding local effective refractive index. Having a grating with objects of different sizes and with different periodicity between them, a certain reduction of the in-plane scattering could be expected. It should be noted that in this work apodization has not been investigated with regard to the improvement of the mode field overlap between the grating and the fiber mode, which is the usual motivation to apply this optimization method. In Ref. [20] an "inversely" apodized grating has been shown: the obtained results reveal the potential of apodization for an improved polarization handling.

<figure><img src="image_59.png" /><figcaption>Figure 5.4: A segmented a-Si:H 2D grating coupler (2D GC) for C-band, comprising (a) 4 or (b) 6 segments. The i-th segment has a local period $Λ_i$ and a local diffracting elements' diameter $w_{Λ_i}$.</figcaption></figure>

The apodization of 2D GCs can be realized in two ways: either by using diffracting elements of different sizes in every single row or by defining segments with a given number of rows, which gather objects with the same geometry. The latter special type of apodization is assigned here as a segmentation. With $N$ objects in a given direction, we would need in the first case $N$ different diameters for each diffracting element. The exact adjustment of a very small difference between the diameters can be very challenging for lithography techniques with a limited resolution. For example, if two neighboring diffracting elements in a grating have a diameter difference of < 10 nm, this difference may not be transferred to the fabricated structure at all. For that reason, the segmented design is the preferred method in this work.

Due to the significantly simpler modeling, the segmentation as a method to reduce in-plane scattering and cross-polarization has been investigated for the case of a-Si:H BEOL integrated 2D GCs (cf. Fig. 3.4, p. 82). Particularly, the modeling is significantly easier in the absence of the BEOL grating filling layers. Nevertheless, the same observations can be translated to SOI integrated 2D GCs. To ensure similar conditions as for SOI 2D GCs, the
investigated designs have a Si substrate below the grating and no metal mirrors have been modeled. Thus, the pure contribution of the segmentation can be observed. To make the designs comparable with the a-Si:H 2D GCs presented in the previous section, the SiO₂ thickness separation to the Si substrate is kept 8.37 µm. Results from simulations in the C-band will be presented first.

<table><caption>Table 5.2: Geometric parameters of sheared, segmented C-band a-Si:H 2D grating couplers (2D GCs) of Type II with a different number of segments. The designs are intended for the coupling to an a-Si:H rib waveguide with a 220 nm height and a slab etch depth equal to the grating etch depth. The i-th segment has a local period $Λ_i$ and a local diffracting elements' diameter $w_{Λ_i}$. For both given structures, the shear angle is 2°, the etch depth is 100 nm and the coupling angles are $φ_{out} = 45°, θ_{out} = 9°$.</caption><thead><tr><th>Number of segments</th><th>4</th><th>6</th></tr></thead><tbody><tr><td>Segment 1: Λ<sub>1</sub>, w<sub>Λ1</sub></td><td>573 nm, 300 nm</td><td>573 nm, 300 nm</td></tr><tr><td>Segment 2: Λ<sub>2</sub>, w<sub>Λ2</sub></td><td>581 nm, 340 nm</td><td>577 nm, 320 nm</td></tr><tr><td>Segment 3: Λ<sub>3</sub>, w<sub>Λ3</sub></td><td>589 nm, 380 nm</td><td>581 nm, 340 nm</td></tr><tr><td>Segment 4: Λ<sub>4</sub>, w<sub>Λ4</sub></td><td>593 nm, 400 nm</td><td>585 nm, 360 nm</td></tr><tr><td>Segment 5: Λ<sub>5</sub>, w<sub>Λ5</sub></td><td>—</td><td>589 nm, 380 nm</td></tr><tr><td>Segment 6: Λ<sub>6</sub>, w<sub>Λ6</sub></td><td>—</td><td>593 nm, 400 nm</td></tr></tbody></table>

In the following, three designs are compared - the uniform C-band a-Si:H 2D GC from the previous section and two segmented designs, comprising either 4 or 6 segments. Apart from the modified gratings, all material and geometric details are as given in the previous Sect. 5.1, cf. Tab. 5.1. Figure 5.4 illustrates the geometry of the segmented gratings, in which a fixed number of sections are defined. In each of them, the diffracting elements' diameter $w_Λ$ and the periodicity $Λ$ vary. As a general rule of thumb, a change of $Δw_Λ = ±20$ nm or a change of the etch depth $Δd = ±10$ nm, results in an effective refractive index change of $Δn_{eff,GC} = ±0.02$. Therefore, to keep the coupling angle constant and gradually increase the diffracting elements' diameter, e.g. with a $Δw_Λ = 20$ nm increment, the local period must be adapted by $ΔΛ = 4$ nm. Table 5.2 summarizes the geometric specifics of the segmented 2D GCs.

We compare the structures regarding the following parameters:

* Single-port simulations (Port 1 and 2 excited separately): the coupling efficiency of a given target-polarization, the cross-polarization and the polarizations' split ratio per single waveguide; the polarizations' angular relation $Δψ$ (i.e. the non-orthogonality) and the polarization crosstalk between waveguides with Ports 1 and 2. The latter two parameters are evaluated in a 40 nm bandwidth around the maximal transmission wavelength.

* Dual-port simulations (Port 1 and 2 excited simultaneously): the coupling efficiency of the even- and odd polarizations, defined as in Fig. 4.16 (c) and (d) on p. 115; PDL, i.e. the absolute difference between the coupling efficiency of these two polarizations (in dB).