We expect an improvement in any of these aspects, when the parasitic grating in-plane scattering is reduced. The comparison with regard to in-plane scattering suppression can be directly translated also to SOI 2D GCs, since in the SOI designs only the local periods need to be changed in accordance to the SOI refractive index.

<figure><img src="image_60.png" /><figcaption>Figure 5.5: A comparison between uniform and segmented a-Si:H 2D grating couplers (2D GCs) for C-band in a single-port simulation. The relevant parameters are the target- and cross-polarization per single waveguide port. Three designs are compared - a uniform 2D GC as a reference, a segmented 2D GC with 4 sections and a segmented 2D GC with 6 sections. The curves represent (a) the normalized out-coupled power, (b) the mode field overlap and (c) the coupling efficiency of a 2D GC.</figcaption></figure>

We begin with results from the single-port simulations. Figure 5.5 shows a comparison between the three designs in terms of target- and cross-polarization, whereat out-coupled power, mode field overlap and coupling efficiency are analyzed. It is evident that the segmentation in any of the two variants reduces the cross-polarization's power and improves at the same time the target-polarization's out-coupling efficiency. In segmented designs, the mode field overlap has only a minor improvement by about 5%, since it is not the target parameter for optimization. The results show an unusual effect of the segmentation as a type of an apodization technique - in 2D GCs it can be used for enhancing the out-coupled power by reducing the parasitic in-plane scattering and the related cross-polarization. Accordingly the coupling efficiency is improved by 0.6-0.7 dB at 1550 nm, depending on the particular design (-4.3/-4.4 dB for the 6/4-segment design vs. -5 dB for the reference, see
Fig. 5.5 (c)). The consequences of the reduced cross-polarization can be further inspected in Fig. 5.6, which shows the polarization split ratio (per single waveguide) and the polarizations' angular relation and crosstalk between the signals from waveguide Ports 1 and 2. The zero-crossings of the crosstalk are indicated additionally, when they occur at points, which are not included in the simulation. The cross-polarization reduction is expressed in an increased polarization split ratio. The angular relation is corrected towards the orthogonal state $Δψ = 90°$ particularly for wavelengths larger than the central wavelength of 1550 nm. Accordingly, the polarization crosstalk becomes lower in this wavelength range. The difference between the segmented structures with 4 and 6 sections is minimal, which indicates that even more segments may be necessary. If we permit a minimal difference between the local diameters of 20 nm, the case with 6 segments represents the maximal achievable improvement using this technique. Next, we continue with a simultaneous

<figure><img src="image_61.png" /><figcaption>Figure 5.6: A comparison between uniform and segmented a-Si:H 2D grating couplers (2D GCs) for C-band in a single-port simulation. The relevant parameters are the polarization split ratio per single waveguide port and the polarizations' angular relation and crosstalk between signals from both waveguides. Three designs are compared - a uniform 2D GC as a reference, a segmented 2D GC with 4 sections and a segmented 2D GC with 6 sections. The curves represent (a) the polarization split ratio, (b) the polarizations' angular relation, (c) the polarizations' crosstalk.</figcaption></figure>

dual-port excitation to investigate the coupling of the even- and odd-polarization, which result from in-phase vs. anti-phase superposition of the target- and cross-polarization originating from different waveguides. Figure 5.7 shows the x-component (y equivalent) of the even- and odd-polarization in terms of normalized out-coupled power (Fig. 5.7 (a)), mode field overlap (Fig. 5.7 (b)) and coupling efficiency (Fig. 5.7 (c)). The PDL resulting from the different coupling spectra of the even- and odd-polarization is shown in Fig. 5.7 (d). Also in this case the zero-crossings are indicated additionally, when they occur at points, which are not included in the simulation. The PDL results from the gap between the out-coupled powers of both polarizations (Fig. 5.7 (a)) and from the wavelength-shifted mode field overlap maximum (Fig. 5.7 (b)). With an increasing number of segments, we achieve especially a correction of the overlaps' spectral shift. Thus, for wavelengths > 1550 nm, the difference between the coupling efficiency of the even- and odd-polarization is decreased. This corresponds to a PDL improvement in this wavelength range. If we permit a maximal PDL of 0.5 dB, the achieved correction is still insufficient.
<figure><img src="image_62.png" /><figcaption>Figure 5.7: A comparison between uniform and segmented a-Si:H 2D grating couplers (2D GCs) for C-band in a dual-port simulation. The relevant parameters are the even- and odd-polarization, resulting from in- or anti-phase superposition of the signals from both GC waveguides. Three designs are compared - a uniform 2D GC as a reference, a segmented 2D GC with 4 sections and a segmented 2D GC with 6 sections. The curves represent (a) the normalized out-coupled power, (b) the mode field overlap and (c) the total coupling efficiency and (d) the polarization-dependent loss (PDL) of a 2D GC.</figcaption></figure>

Further suppression of the in-plane scattering is necessary.

For completeness, the same comparison between a reference, uniform 2D GC and a segmented 2D GC with 4 sections will be made in the O-band as well. Compared to the device with 4 segments, a design with 6 segments shows a minor improvement, which does not justify the enhanced fabrication effort. For that reason, the results for the 6-segment device will be omitted.

<table><caption>Table 5.3: Geometric parameters of a 2° sheared, segmented O-band a-Si:H 2D GCs of Type II with 4 sections. 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 etch depth is larger than that of the reference uniform 2D GC: 120 nm vs. 100 nm. The i-th segment has a local period $Λ_i$ and a local diffracting elements' diameter $w_{Λ_i}$. The coupling angles are $φ_{out} = 45°, θ_{out} = 9°$.</caption><tr><td>Number of segments</td><td>4</td></tr><tr><td>Segment 1: Λ<sub>1</sub>, w<sub>Λ1</sub></td><td>453 nm, 180 nm</td></tr><tr><td>Segment 2: Λ<sub>2</sub>, w<sub>Λ2</sub></td><td>461 nm, 220 nm</td></tr><tr><td>Segment 3: Λ<sub>3</sub>, w<sub>Λ3</sub></td><td>469 nm, 260 nm</td></tr><tr><td>Segment 4: Λ<sub>4</sub>, w<sub>Λ4</sub></td><td>473 nm, 280 nm</td></tr></table>
The geometric parameters of the segmented O-band 2D GC are given in Tab. 5.3. A larger etch depth of 120 nm than the reference design's etch depth of 100 nm is considered to compensate for the small diffraction elements in the first two segments. The etch depth of 120 nm is applied for all segments.

<figure><img src="image_63.png" /><figcaption>Figure 5.8: A comparison between uniform and segmented a-Si:H 2D grating couplers (2D GCs) for O-band in a single-port simulation. The relevant parameters are the target- and cross-polarization per single waveguide port. Two designs are compared - a uniform 2D GC as a reference and a segmented 2D GC with 4 sections. The curves represent (a) the normalized out-coupled power, (b) the mode field overlap and (c) the coupling efficiency of a 2D GC.</figcaption></figure>

<figure><img src="image_64.png" /><figcaption>Figure 5.9: A comparison between uniform and segmented a-Si:H 2D grating couplers (2D GCs) for O-band in a single-port simulation. The relevant parameters are the polarization split ratio per single waveguide port and the polarizations' angular relation and crosstalk between signals from both waveguides. Two designs are compared - a uniform 2D GC as a reference and a segmented 2D GC with 4 sections. The curves represent (a) the polarization split ratio, (b) the polarizations' angular relation, (c) the polarizations' crosstalk.</figcaption></figure>
Figure 5.8 shows the target- and cross-polarization of the uniform and the segmented design, in a similar way to C-band (single-port simulations). Accordingly, Fig. 5.9 shows the polarization split ratio per single waveguide and the polarizations' angular relation and crosstalk for the signals from both GC waveguides. The even- and odd-polarizations and their PDL are presented in Fig. 5.10 (dual-port simultaneous excitation).

The segmented design has a slightly shifted spectrum from the reference one, but this plays a minor role for their comparison. Also in the O-band, the same trends shown in C-band can be observed. In segmented 2D GCs, the maximal coupling efficiency is slightly enhanced by about 0.7 dB (-3.2 dB at 1310 nm vs. -3.9 dB at 1300 nm for the reference). The polarizations' split ratio, the angular relation and the corresponding polarization crosstalk are improved for wavelengths > 1310 nm. Regarding the even- and odd-polarizations and their PDL, better characteristics are achieved in the same wavelength range, while performance at wavelengths < 1310 nm is sacrificed. Although O-band 2D GCs outperform their C-band counterparts, the achievement of sufficiently low PDL levels is challenging for these devices as well. In our recent publication [22], segmented 2D GCs in C- and O-band

<figure><img src="image_65.png" /><figcaption>Figure 5.10: A comparison between uniform and segmented a-Si:H 2D grating couplers (2D GCs) for O-band in a dual-port simulation. The relevant parameters are the even- and odd-polarization, resulting from in- or anti-phase superposition of the signals from both GC waveguides. Two designs are compared - a uniform 2D GC as a reference and a segmented 2D GC with 4 sections. The curves represent (a) the normalized out-coupled power, (b) the mode field overlap and (c) the coupling efficiency and (d) the polarization-dependent loss (PDL) of a 2D GC.</figcaption></figure>

based on SOI have been investigated as a coupling interface in coherent transmission systems. The designs account for the reduction of OSNR penalties, caused by the limitations of polarizations' non-orthogonality and crosstalk. The combination by the segmented 2D GCs (especially in O-band) with an appropriate DSP improves significantly the system-level
performance. By contrast, conventionally designed 2D GCs are related to large penalties even when advanced DSP is used. A strategy for the further design improvement of 2D GCs, which could erase the necessity of a dedicated DSP, will be presented in the following.

### 5.2.2 Two-Dimensional Grating Couplers with Elongated and Individually Oriented Perturbing Elements

Undoubtedly, the out-standing work of Luxtera towards a 2D GC optimization delivered the most remarkable results. Excellent performance with a low insertion loss and a low PDL could be demonstrated, on both 200 mm and 300 mm platforms [23–30]. Luxtera’s team was among the first who implemented successfully a non-zero waveguide-to-grating shear angle, improving with this the coupling efficiency. However, due to their commercial development activities, no explanation of the design with such an angle has been provided. In fact, an explicit information about a non-perpendicular angle between the 2D GC’s waveguides can be found in Luxtera’s patents only, e.g. Refs. [35–37]. In the present work, the systematic design of our sheared 2D GCs have been described in Chap. 4.

Furthermore, to optimize the PDL Luxtera proposed the adoption of a special clover-like *scatterer’s shape* for the 2D GCs’ perturbing elements. The latter inspired many groups to make investigations on the importance of the scatterer’s shape, see e.g. Refs. [31–33], but also in this case, no explanation about the physical processes determining the choice of such a shape has been given. It is possible that Luxtera pursued similar purposes as we do in the present work, namely - to reduce the scattering in undesired directions and preserve the diffraction efficiency towards the SMF (Luxtera assigns the latter often as a scattering as well). Possibly, the choice of the name - *scatterer’s shape* - is a hint about the actual effect that has been manipulated, which is the *in-plane* scattering and not the out-of-plane diffraction.

Here, an approach for an in-plane scattering optimization is proposed, which can be used as an alternative to the scatterer’s shape of Luxtera. Some of the descriptions and analyses below have been previously published in Ref. [38]. The basic idea behind the optimization method is the following. A starting point is the hypothesis from our previous work [39] that the in-plane scattering’s strength is very large, when perturbing elements with identical scattering pattern are periodically arranged. The possibility to use objects with different sizes and periodicity helps reducing the in-plane scattering to some extend, but not sufficiently to reach a low PDL (cf. 5.2.1). Alternatively, we still may use perturbing elements of identical size, shape and periodicity, but in the same time we have to ensure that neighboring objects have abruptly different local scattering patterns. To reach that goal, we need first elongated elements such as ellipses or ovals, whereat the exact choice of a shape is not determining. Second, two neighboring objects may be rotated by 90° to each other. A grating comprising such objects has been assigned as a zig-zag tilted ovals array [38]. The sudden change of the orientations of adjacent objects allows for the cancellation of a forwards-scattered local wave by a backwards-scattered one. The exact perturbing elements’ size is thus determining for the degree of cancellation of the forwards- and backwards-scattered fields.
<figure><img src="image_66.png" /><figcaption>Figure 5.11: A schematic of an optimized 2D grating coupler (2D GC), comprising elongated perturbing elements with different orientations. The major axis of a perturbing element is denoted as $w_l$, the minor axis is denoted as $w_s$.</figcaption></figure>

Figure 5.11 illustrates an exemplary geometry of a 2D GC, which may be optimized for a low in-plane scattering in this manner. Its most distinctive property is the fact that the scattering's cancellation is not reached by elements of a complex shape, but by simple objects, differing in their abruptly changing orientations and local scattering profiles. The simple shape of the perturbing elements is the most significant advantage of this optimization approach, compared to the Luxtera's one. Thus, the 2D GCs can be realized with a low-resolution DUV lithography such as the 248 nm DUV used in our fabrication platform. Furthermore, no optical proximity correction (OPC) is necessary, but only an appropriate shape biasing. By contrast, Luxtera's scatterers require OPC in combination with a 193 nm DUV lithography. In view of the expenses' gap between 248 nm and 193 nm DUV, the 2D GCs proposed in this work can be realized with the lowest DUV fabrication cost. At this point, other alternatives to Luxtera's design should be acknowledged. 2D GCs with a low PDL, comprising uni-directional elongated scatterers, have been reported in Ref. [40]. A post-simulation of the given geometry (performed for comparison purposes in this work) suggests that the low PDL results partially at the cost of a low out-coupled power. Furthermore, although the feature sizes should be compatible with a 193 nm or 248 nm DUV lithography, the reported test device has been fabricated by an electron beam lithography. It is difficult to predict, whether the given PDL values will remain the same, when a DUV lithography is used. In another recent work Ref. [20], apodized 2D GC with rectangular scatterers has been proposed for a low PDL. Since the designed geometry differs strongly from the fabricated one, it is hard to predict, whether the obtained results are in a close relation to the design method. In a most recent work Ref. [41], an optimization approach, including perturbing elements with different orientations has been proposed. However, the basic principle of the reported device differs from those presented here. A gradual orientations' change is used to achieve and improved mode field overlap - a technique, which may be rather associated with the apodization/chirping method. Due to the larger degrees of freedom used there, it can be concluded that the success of the reported design is achieved by the sufficient number of apodized segments (cf. the discussion from 5.2.1 in this chapter). Lastly, the resulting feature sizes are significantly smaller than the proposed ones in this work and experimental validation of the suggested design is missing.
A final remark will be made with regard to the experimentally reported PDL values. There are only few works, in which the repeatability of the determined PDL has been investigated statistically. Reference [31] provides a wafer map of the distribution of the PDL within the 1 dB bandwidth for a variety of 2D GCs. Unfortunately, the test structures were fabricated by an electron beam lithography, which is not given in large-scale manufacturing platforms. In Refs. [20, 32, 33, 40] very low PDL values are reported, but again without any discussion of the measurement uncertainty and the statistical range. In fact, Ref. [33] shows results from 3 chips, which is, however, not sufficient for a statistical evidence. It is a well-known problem that parameters taking very low values are challenging for a characterization with a good confidence. In this context, objective statistical studies, considering measurement uncertainties, are still not widely available. In this work, statistical results will be provided to evaluate the applicability of the proposed designs in the target 0.25 µm BiCMOS platform.

<figure><img src="image_67.png" /><figcaption>Figure 5.12: A comparison between a reference and an optimized design of a C-band 2D grating coupler (2D GC). (a) The coupling efficiency of a target- and cross-polarization per single waveguide, (b) the corresponding polarization split ratio, (c) the polarizations' angular relation $\Delta\psi$ between signals from both GC arms, (d) the corresponding polarization crosstalk. The optimized design shows a significant improvement in terms of polarization splitting.</figcaption></figure>

In the following, the exact optimization procedure for the proposed grating is briefly outlined. As already indicated - the achievement of a sufficient suppression of in-plane scattered fields requires an exact sizing of the perturbing elements. The optimization procedure involves the simultaneous adaption of several parameters:

* The perturbing elements' minor and major axis, denoted as $w_s$ and $w_l$. Typically, when the minor axis is too short, the overall out-coupled power decreases. This leads to a lower cross-polarization and thus a low PDL. However, in the same time the coupling
efficiency drops as well. Therefore, models with a low PDL in combination with a low coupling efficiency are not of interest in this work. As a lower limit for the coupling efficiency, at least -4.5 dB for both polarizations has been set. The PDL should be < 0.5 dB within the 2D GC's 1 dB-bandwidth.

* The grating period. Each simulation run, in which new perturbing elements' shape is investigated, requires in the same time the adaption of the grating period, so that a desired angle $\theta_{out}$ results at the 2D GC's symmetry plane.

* The etch depth. Generally, the elongated perturbing elements, designed with the restrictions of a 248 nm DUV lithography, have a lower perturbation strength than their circular counterparts. For that reason, together with an appropriate perturbing elements' dimensions, we look for a proper etch depth that guarantees for the sufficient out-coupled power.

### 5.2.2.1 Linear Two-Dimensional Grating Couplers

The optimization procedure is first applied to linear 2D GCs. In the following, the best performing devices are presented. An analysis of tolerances regarding deviations from the target perturbing elements' shape is carried out as well.

**Designs for C-Band** The investigation of the optimization technique begins in C-band, where both SOI and a-Si:H material platforms are considered.

**SOI** All models below are designed for the coupling angles $\varphi_{out} = 45^\circ, \theta_{out} = 8^\circ$ at a wavelength of 1550 nm. To demonstrate the success of the applied optimization approach, a reference and an optimized SOI 2D GC design are compared. The reference design is already known from 4.2.2.3, Chap. 4. We begin with sheared 2D GCs of Type II. Later, it is shown exemplary that 2D GCs of Type I cannot be optimized, using an equivalent geometry. Generally, the sufficient scattering suppression in Type I gratings appears very challenging. The reference and the optimal design have the following geometries:

* Reference: a shear angle $\alpha = 2^\circ$ (Type II), a grating period $\Lambda = 622$ nm, an etch depth $d = 120$ nm, circular perturbing elements with a diameter $w_\Lambda = 440$ nm.

* Proposed optimized design: a shear angle $\alpha = 2^\circ$ (Type II), a grating period $\Lambda = 594$ nm, an etch depth $d = 140$ nm, elongated perturbing elements with a minor axis $w_s = 230$ nm and a major axis $w_l = 320$ nm (cf. Fig. 5.11).

As in previous analyses, single-port simulations are used to determine the fiber coupling efficiency of a target- and cross-polarization as well as their split ratio, when a given 2D GC arm is used as a source. The polarizations' angular relation and crosstalk, between the signals from both 2D GC arms are determined as well. A dual-port simultaneous simulation is used to evaluate the fiber coupling efficiency of an even- and odd-polarization as well as the PDL resulting from the difference between their coupling efficiencies.
<figure><img src="image_68.png" /><figcaption>Figure 5.13: Even vs. odd polarizations (x-component) of a reference sheared 2D grating coupler (2D GC) of Type II for C-Band. (a) The out-coupled power, (b) the mode field overlap, (c) the coupling efficiency and (d) the polarization-dependent loss (PDL).</figcaption></figure>

Figure 5.12 shows a comparison between the reference and the optimized design in terms of cross-polarization suppression (Fig. 5.12 (a)), split ratio (Fig. 5.12 (b)), polarizations' non-orthogonality (Fig. 5.12 (c)) and polarization crosstalk (Fig. 5.12 (d)). In the optimized design, the cross-polarization's maximum is reduced by 16 dB, compared to the reference. This is directly reflected in the polarizations' split ratio - while in the reference design the split ratio does not exceed 20 dB, the optimized design has a split ratio > 23 dB in the range 1500 nm-1580 nm. Finally, the polarizations' angular relation $\Delta\psi$ in a wavelength range covering the 1 dB-bandwidth of the optimized design shows a remarkable improvement in terms of orthogonality with a maximally 3.5° deviation from the ideal case of 90°. The orthogonality uniformity is significantly better. The corresponding polarization crosstalk remains below -24 dB within the considered bandwidth.

It can be expected that the improved polarization splitting performance will be directly translated to the PDL levels (even- vs. odd-polarization) in receiver-side 2D GCs. Figure 5.13 shows (a) the out-coupled power, (b) the mode field overlap and (c) the coupling efficiency of the even- and odd-polarization (x-component) and (d) the PDL of the reference design. For comparison, Fig. 5.14 (a)-(d) shows the same parameters for the optimized design. Thanks to the cross-polarization suppression in the optimized design, the gap between the out-coupled power of the even- and odd-polarization is significantly decreased, as can be seen in Fig. 5.13 (a) and Fig. 5.14 (a). The mode field overlap spectra approach the same central wavelength (cf. Fig. 5.13 (b) and Fig. 5.14 (b)). Overall, this leads to a remarkable improvement of the coupling efficiency spectra of both polarizations, which are now centered at the same wavelength with only a minor spectrum difference (Fig. 5.14 (c)). The PDL becomes thus significantly lower. The maximal coupling efficiency of the optimized
design is -4.1 dB at 1550 nm with a 34 nm 1 dB-bandwidth - from 1534 nm-1568 nm. In this range, the PDL reaches maximally 0.55 dB, which is slightly higher than the target value. In structures with conically etched perturbing elements, the PDL might be lower due to the decreased perturbation strength.

<figure><img src="image_69.png" /><figcaption>Figure 5.14: Even vs. odd polarizations (x-component) of an optimized sheared 2D grating coupler (2D GC) of Type II for C-Band. (a) The out-coupled power, (b) the mode field overlap, (c) the coupling efficiency and (d) the polarization-dependent loss (PDL).</figcaption></figure>

It becomes evident that there is an optimization chain: waveguide *split ratio* → *orthogonality* and *polarization crosstalk* between signals from both GC arms → *PDL* between signals from both GC arms combined with different phase relations. This can be well explained: with an improving split ratio, the signal of one of the waveguides coupled to the fiber approaches the x-polarization state and the signal from the other one - the y-polarization state. Accordingly, the polarizations' orthogonality comes even closer to 90° and the polarization crosstalk becomes lower. Thus, an arbitrary combination of orthogonal signals without crosstalk causes no PDL. From Figs. 5.12 (b)-(d) and 5.14 (d), we can estimate a required value for each of the examined parameter to reach a PDL of 0.5 dB (via interpolation and evaluation at 1565 nm, where the PDL is exactly 0.5 dB).

We obtain the following limits:

*   Minimal split ratio 25 dB → maximal deviation from the orthogonality state 3°, i.e. maximal polarization crosstalk of -26 dB → maximal PDL of 0.5 dB.

Because the parameters are closely connected with each other, the analyses within the next paragraphs will be mostly focused on the PDL. Out of its behavior, we can directly
conclude in what value range the other parameters are. In some cases, where the simulation duration is too long, the single-port simulations will be considered instead.

Next, it will be illustrated that sheared 2D GCs of Type I are more challenging with respect to in-plane scattering suppression. We use the design parameters of the previously designed 2D GC of Type II and change only the way of defining the shear angle, i.e. the grating area is now rhombus-shaped. Figure 5.15 shows (a) the coupling efficiency of the target- and cross-polarization, (b) the coupling efficiency of the even- and odd-polarization. It is evident that for the same perturbing elements' size and periodicity, the cross-polarization is significantly higher. For that reason, the even- and odd- polarization are still shifted to each other. Within the permitted geometric range for a compatibility with a 248 nm DUV lithography, no appropriate perturbing elements' geometry could be found for 2D GCs of Type I, which fulfills the criteria in terms of coupling efficiency and PDL.

<figure><img src="image_70.png" /><figcaption>Figure 5.15: The performance of a sheared 2D grating coupler (2D GC) of Type I with the same perturbing elements' geometry and periodicity as the optimized 2D GC of Type II. (a) The coupling efficiency of the target- and cross-polarization (single-port simulation), (b) the coupling efficiency of the even- and odd-polarization (dual-port simulation).</figcaption></figure>

Lastly, the optimal 2D GC design is investigated, regarding its robustness against deviations of the perturbing elements' shape. Only the coupling efficiency of the even- and odd-polarization and the PDL are presented. We consider the following cases:

* $\Delta w_s = \Delta w_I = +10$ nm, i.e. both axes are 10 nm larger $\rightarrow w_s = 240$ nm, $w_I = 330$ nm

* $\Delta w_s = \Delta w_I = -10$ nm, i.e. both axes are 10 nm smaller $\rightarrow w_s = 220$ nm, $w_I = 310$ nm

* $\Delta w_s = \text{const.}, w_I = +10$ nm, i.e. the minor axis is unchanged, the major axis is 10 nm larger $\rightarrow w_s = 230$ nm, $w_I = 330$ nm

* $\Delta w_s = \text{const.}, w_I = -10$ nm, i.e. the minor axis is unchanged, the major axis is 10 nm smaller $\rightarrow w_s = 230$ nm, $w_I = 310$ nm

* $\Delta w_s = +10$ nm, $w_I = \text{const.}$, i.e. the minor axis is 10 nm larger, the major axis is unchanged $\rightarrow w_s = 240$ nm, $w_I = 320$ nm

* $\Delta w_s = -10$ nm, $w_I = \text{const.}$, i.e. the minor axis is 10 nm smaller, the major axis is unchanged $\rightarrow w_s = 220$ nm, $w_I = 320$ nm
<figure><img src="image_71.png" /><figcaption>Figure 5.16: A performance variation of the optimized 2D grating coupler (2D GC) in C-band, when the perturbing elements' size varies: $\Delta w_s = \Delta w_I = +10\,\text{nm} \rightarrow w_s = 240\,\text{nm}, w_I = 330\,\text{nm}$ or $\Delta w_s = \Delta w_I = -10\,\text{nm} \rightarrow w_s = 220\,\text{nm}, w_I = 310\,\text{nm}$. The compared parameters are (a)-(b) the coupling efficiency of the even- and odd-polarization, (c)-(d) the corresponding polarization-dependent loss (PDL).</figcaption></figure>

<figure><img src="image_72.png" /><figcaption>Figure 5.17: A performance variation of the optimized 2D grating coupler (2D GC) in C-band, when the perturbing elements' size varies: $\Delta w_s = \text{const.}, \Delta w_I = +10\,\text{nm} \rightarrow w_s = 230\,\text{nm}, w_I = 330\,\text{nm}$ or $\Delta w_s = \text{const.}, \Delta w_I = -10\,\text{nm} \rightarrow w_s = 230\,\text{nm}, w_I = 310\,\text{nm}$. The compared parameters are (a)-(b) the coupling efficiency of the even- and odd-polarization, (c)-(d) the corresponding polarization-dependent loss (PDL).</figcaption></figure>
Figure 5.16 shows the trade-offs, which occur in the first two cases: $\Delta w_s = \Delta w_I = \pm 10$ nm. When the perturbing elements' size is larger, the PDL is affected, becoming maximally 0.65 dB within the 1 dB-bandwidth. The coupling efficiency remains unchanged, the spectral shift is small. When the perturbing elements' shape is smaller than expected, the PDL remains below 0.5 dB within the wavelength range of interest. In fact, there is even a small improvement at 1568 nm of around 0.15 dB. This is at the cost of a coupling efficiency decrease of about 0.3 dB. The wavelength shift is again not significant.

As can be seen in Figs. 5.17, 5.18, similar results can be observed for the both cases $\Delta w_s = \pm 10$ nm or $\Delta w_I = \pm 10$ nm. A larger axis leads to a PDL deterioration with maximal values of about 0.6-0.65 dB and is thus more critical than a deviation towards smaller dimensions, for which the PDL remains < 0.5 dB. The larger PDL is related to a slightly better maximal coupling efficiency of -4 dB, while in the opposite case, the efficiency decreases to about -4.3 dB. In summary, deviations of the perturbing elements' size within a 20 nm range are well tolerable within the 2D GC's 1 dB-bandwidth in C-band. No significant wavelength shift occurs. Basically, an increase of a given dimension leads to a PDL deterioration and a coupling efficiency improvement and vice versa. The worst-case PDL becomes 0.65 dB. Conically etched perturbing elements in real structures may have better tolerances, due to the lower perturbation strength.

<figure><img src="image_73.png" /><figcaption>Figure 5.18: A performance variation of the optimized 2D grating coupler (2D GC) in C-band, when the perturbing elements' size varies: $\Delta w_s = +10$ nm, $\Delta w_I = \text{const.}$ $\rightarrow w_s = 240$ nm, $w_I = 320$ nm or $\Delta w_s = -10$ nm, $\Delta w_I = \text{const.}$ $\rightarrow w_s = 220$ nm, $w_I = 320$ nm. The compared parameters are (a)-(b) the coupling efficiency of the even- and odd-polarization, (c)-(d) the corresponding polarization-dependent loss (PDL).</figcaption></figure>

**a-Si:H** The advances, which are achievable in SOI structures, can be directly translated also to the a-Si:H based structures, discussed in Sect. 5.1. Here, an improved a-Si:H design with a Metal3 back-reflector is briefly presented. Since the devices are applicable at the
transmitter-side, only the relevant aspects there are discussed, that is, the coupling effi-
ciency, the split ratio and the polarizations' angular relationship $\Delta\psi$ with the corresponding
polarization crosstalk. An optimized design may have the following geometry: a shear angle
$\alpha = 2^\circ$ (Type II), a grating period $\Lambda = 582$ nm, stretched perturbing elements with a minor
axis $w_s = 210$ nm and a major axis $w_l = 330$ nm, an etch depth $d = 140$ nm. The design
angles are $\varphi_{out} = 45^\circ$, $\theta_{out} = 9^\circ$ at 1550 nm. Figure 5.19 shows the coupling efficiency of
the target- and cross-polarization and the corresponding split ratio (per single waveguide).
The polarizations' angular difference and the related polarization crosstalk between signals
from both GC arms is given as well. Compared to the design with a back-reflector from
Sect. 5.1 we observe a 1 dB improvement of the maximal coupling efficiency, which is
here -2.4 dB at 1550 nm. A split ratio better than 23 dB is reachable in a 55 nm range from
1500 nm-1555 nm. The orthogonality deviation within the bandwidth 1530 nm-1570 nm is
maximally 4°, corresponding to a polarization crosstalk below -23.5 dB. The parameters'
improvement is comparable to the SOI case. The example shows that the optimization
method works well also for 2D GC with a comparatively high coupling efficiency.

<figure><img src="image_74.png" /><figcaption>Figure 5.19: An a-Si:H 2D grating coupler (2D GC) for C-band with a Metal3 back-reflector - an optimized design with individually oriented elongated perturbing elements. (a) The coupling efficiency of the target- and cross-polarizations. (b) The corresponding split ratio (per single waveguide). (c) The polarization' angular difference $\Delta\psi$. (d) The corresponding polarization crosstalk (between signals from both GC arms).</figcaption></figure>

**A Brief Outlook - Possible Modifications** The proposed way to define an array of stretched perturbing elements with alternating orientations is not the only one thinkable possibility to design 2D GCs with a low in-plane scattering. More degrees of freedom may be included to achieve an even better performance. A possible extension will be shown here, using the a-Si:H design from the previous paragraph as a starting point. Although
the example here considers a-Si:H as a material, the same configuration may be used in SOI as well. Because the potential modifications have been investigated at a later point, no experimental structures were available at the moment of the dissertation's preparation. The example given in this paragraph may be considered as an outlook for the further design optimization. The design modification is illustrated in Fig. 5.20.

<figure><img src="image_75.png" /><figcaption>Figure 5.20: A schematic of an optimized 2D grating coupler (2D GC) design, comprising stretched perturbing elements, combined with circular perturbing elements.</figcaption></figure>

Along with the stretched objects, circular perturbing elements may be included as well. This adds an additional local scattering pattern and increases the degrees of freedom for an in-plane scattering cancellation. The exemplary design has the following dimensions: a shear angle $\alpha = 2^\circ$, a grating period $\Lambda = 582$ nm, stretched perturbing elements with a minor axis $w_s = 210$ nm and a major axis $w_l = 330$ nm, circular perturbing elements with a diameter $w_{circ} = 265$ nm and an etch depth $d = 140$ nm. The only modification, compared to the previously developed a-Si:H design is the inclusion of the circular perturbing elements. Their diameter is calculated such that the circle has the same area as the ellipse/oval. If we assume an ellipse:

$$w_{circ} = \sqrt{w_s \cdot w_l} \qquad (5.1)$$

If we consider an oval, the surface is calculated from the two half-circles with a diameter $w_s$ plus a small connecting rectangle with sidewalls lengths $w_s$ and $w_l - w_s$, that is:

$$w_{circ} = \sqrt{w_s^2 + \frac{4}{\pi} w_s \cdot (w_l - w_s)} \qquad (5.2)$$

The calculated $w_{circ}$ differs little in dependence on the chosen formula. Figure 5.21 shows the target- and cross-polarization's coupling spectra, the corresponding split ratio together with the polarizations' angular difference and crosstalk. The figure compares the previously shown optimized design as a reference and the modified optimized design. The latter is slightly wavelength-shifted, the maximal coupling efficiency remains unaltered. The cross-polarization is further reduced (by 2-3 dB), compared to the initially optimized design. For wavelengths > 1540 nm a constant split ratio improvement of about 2.5 dB can be achieved. The polarization' orthogonality is further improved with a maximal deviation of only 2.5°. The polarization crosstalk is below -27.5 dB.
<figure><img src="image_76.png" /><figcaption>Figure 5.21: A comparison of optimized a-Si:H 2D grating couplers (2D GCs) for C-band with a Metal3 back-reflector - an optimized design with stretched perturbing elements (PE) vs. an optimized design combining stretched and circular perturbing elements (PE). (a) The coupling efficiency of the target- and cross-polarization. (b) The corresponding split ratio (per single waveguide). (c) The polarization' angular difference $\Delta\psi$. (d) The corresponding polarization crosstalk (between signals from both GC arms).</figcaption></figure>

## Designs for O-Band
Here, the same analyses are repeated in O-band.

### SOI
We begin with the comparison between a reference and an optimized sheared 2D GC of Type II in SOI. The structures are developed for the coupling angles $\phi_{out} = 45^{\circ}$, $\theta_{out} = 8^{\circ}$ at a wavelength of 1310 nm and have the following geometries:

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

* Proposed optimized design: a shear angle $\alpha = 2^{\circ}$, a grating period $\Lambda = 480$ nm, an etch depth $d = 140$ nm, elongated perturbing elements with a minor axis $w_s = 180$ nm and a major axis $w_l = 260$ nm.

Figure 5.22 compares the target- and cross-polarization coupling efficiency and the corresponding split ratio of both designs (single-port simulation). The reference O-band structure has a better split ratio than its C-band counterpart. Further improvement can be achieved by the proposed new design. For wavelengths around the maximal transmission, the split ratio's enhancement is more than 10 dB. The split ratio is > 23 dB in the range 1275 nm - 1330 nm. The optimized design is also responsible for the improvement of the uniformity of $\Delta\psi$. In a 40 nm bandwidth around the maximum, the deviation from the orthogonal state is maximally 3.2°, corresponding to a maximal polarization crosstalk of -25 dB.
<figure><img src="image_77.png" /><figcaption>Figure 5.22: A comparison between a reference and an optimized design of a O-band 2D grating coupler (2D GC). (a) The coupling efficiency of a target- and cross-polarization per single waveguide, (b) the corresponding polarization split ratio, (c) the polarizations' angular relation $\Delta\psi$ between signals from both GC arms, (d) the corresponding polarization crosstalk. The optimized design shows a significant improvement in terms of polarization splitting.</figcaption></figure>

In the next step, the even and odd polarization's coupling efficiency and PDL are determined from dual-port simultaneous simulations. Figures 5.23 and 5.24 can be used for the comparison between the reference and the proposed 2D GC. Again, the proposed design with a lower cross-polarization makes it possible that the even- and odd-polarization's spectra come closer to each other: the out-coupled power's gap is closing and the overlap's wavelength shift vanishes. The optimized design reaches a coupling efficiency of -3.3 dB at 1310 nm, the 1 dB-bandwidth is 21 nm from 1303 nm to 1324 nm. Regarding the PDL, values lower than 0.5 dB can be obtained for wavelengths > 1280 nm.

For completeness, the 2D GC's robustness against perturbing elements' size deviations is analyzed for the O-band design as well. The considered cases cover the same range as in C-band with the following length combinations:

* $\Delta w_s = \Delta w_I = +10$ nm, i.e. both axes are 10 nm larger $\rightarrow w_s = 190$ nm, $w_I = 270$ nm

* $\Delta w_s = \Delta w_I = -10$ nm, i.e. both axes are 10 nm smaller $\rightarrow w_s = 170$ nm, $w_I = 250$ nm

* $\Delta w_s = \text{const.}, w_I = +10$ nm, i.e. the minor axis is unchanged, the major axis is 10 nm larger $\rightarrow w_s = 180$ nm, $w_I = 270$ nm

* $\Delta w_s = \text{const.}, w_I = -10$ nm, i.e. the minor axis is unchanged, the major axis is 10 nm smaller $\rightarrow w_s = 180$ nm, $w_I = 250$ nm
* $\Delta w_s = +10\,\text{nm}, w_I = \text{const.}$, i.e. the minor axis is 10 nm larger, the major axis is unchanged $\rightarrow w_s = 190\,\text{nm}, w_I = 260\,\text{nm}$

* $\Delta w_s = -10\,\text{nm}, w_I = \text{const.}$, i.e. the minor axis is 10 nm smaller, the major axis is unchanged $\rightarrow w_s = 170\,\text{nm}, w_I = 260\,\text{nm}$

<figure><img src="image_78.png" /><figcaption>Figure 5.23: Even vs. odd polarizations (x-component) of a reference sheared 2D grating coupler (2D GC) of Type II for O-Band. (a) The out-coupled power, (b) the mode field overlap, (c) the coupling efficiency and (d) the polarization-dependent loss (PDL).</figcaption></figure>

<figure><img src="image_79.png" /><figcaption>Figure 5.24: Even vs. odd polarizations (x-component) of an optimized sheared 2D grating coupler (2D GC) of Type II for O-Band. (a) The out-coupled power, (b) the mode field overlap, (c) the coupling efficiency and (d) the polarization-dependent loss (PDL).</figcaption></figure>
<figure><img src="image_80.png" /><figcaption>Figure 5.25: A performance variation of the optimized 2D grating coupler (2D GC) in O-band, when the perturbing elements' size varies: $\Delta w_s = \Delta w_I = +10\,\text{nm} \rightarrow w_s = 190\,\text{nm}, w_I = 270\,\text{nm}$ or $\Delta w_s = \Delta w_I = -10\,\text{nm} \rightarrow w_s = 170\,\text{nm}, w_I = 250\,\text{nm}$. The compared parameters are (a)-(b) the coupling efficiency of the even- and odd-polarization, (c)-(d) the corresponding polarization-dependent loss (PDL).</figcaption></figure>

<figure><img src="image_81.png" /><figcaption>Figure 5.26: A performance variation of the optimized 2D grating coupler (2D GC) in O-band, when the perturbing elements' size varies: $\Delta w_s = \text{const.}, \Delta w_I = +10\,\text{nm} \rightarrow w_s = 180\,\text{nm}, w_I = 270\,\text{nm}$ or $\Delta w_s = \text{const.}, \Delta w_I = -10\,\text{nm} \rightarrow w_s = 180\,\text{nm}, w_I = 250\,\text{nm}$. The compared parameters are (a)-(b) the coupling efficiency of the even- and odd-polarization, (c)-(d) the corresponding polarization-dependent loss (PDL).</figcaption></figure>

Figures 5.25, 5.26 and 5.27 show the simulation results for the considered cases. The results confirm the observations in C-band. Again, the spectrum shift is small. A shrunk perturbing element in either direction is not an issue, with the PDL remaining < 0.5 dB and
a maximal coupling efficiency of -3.4 dB. As previously observed, the larger perturbing elements are responsible for a PDL enhancement with a maximum of 0.7 dB when both $\Delta w_s = \Delta w_I = +10$ nm, and an upper limit of 0.6 dB in the other cases. The PDL deterioration comes with an improved coupling efficiency of -3.1 dB. Generally, independent of the given band, during fabrication it should be ensured that the perturbing elements' size does not get too large or too circular. An appropriate shape biasing may be advantageous.

<figure><img src="image_82.png" /><figcaption>Figure 5.27: A performance variation of the optimized 2D grating coupler (2D GC) in O-band, when the perturbing elements' size varies: $\Delta w_s = +10$ nm, $\Delta w_I = \text{const.}$ $\rightarrow w_s = 190$ nm, $w_I = 260$ nm or $\Delta w_s = -10$ nm, $\Delta w_I = \text{const.}$ $\rightarrow w_s = 170$ nm, $w_I = 260$ nm. The compared parameters are (a)-(b) the coupling efficiency of the even- and odd-polarization, (c)-(d) the corresponding polarization-dependent loss (PDL).</figcaption></figure>

**a-Si:H** For completeness, the proposed design will be translated to an a-Si:H 2D GC with a Metal3 back-reflector. Again, the transmitter-side relevant parameters will be shown, i.e. the coupling efficiency, the split ratio, the polarizations' angular relationship $\Delta\psi$ and the polarization crosstalk. For a-Si:H, the only parameter that is changed is the grating period, which is now 470 nm. The complete design geometry includes: a shear angle $\alpha = 2^\circ$, a grating period $\Lambda = 470$ nm, stretched perturbing elements with a minor axis $w_s = 180$ nm and a major axis $w_I = 260$ nm, an etch depth $d = 140$ nm. The design angles are $\phi_{out} = 45^\circ$, $\theta_{out} = 8^\circ$ at 1320 nm, with a small deviation from the target wavelength of 1310 nm. An extended, optimized design with a combination of stretched and circular perturbing elements has the same geometric parameters, further including the circular elements' diameter $w_{circ} = 216$ nm.

Figure 5.28 shows the coupling efficiency of the target- and cross-polarization, the corresponding split ratio (per single-waveguide), the polarization angles' difference and the polarization crosstalk (between both GC waveguides) for both optimized cases. Compared to the reference design with Metal3 from Sect. 5.1, a small coupling efficiency improvement is achieved - from -2.8 dB to -2.5 dB. As expected, the most significant advance is in terms
of polarization splitting: the improved design with elongated elements has a split ratio
> 23 dB from 1290 nm to 1330 nm. The extended optimized design, combining elongated
and circular elements, has an even better split ratio with values > 23 dB from 1285 nm to
1353 nm. In a 40 nm bandwidth around the central wavelength, the design with elliptical
elements has a variation from $\Delta\psi = 88^\circ$ to $\Delta\psi = 91^\circ$ and the combined design both elon-
gated and circular elements - from $\Delta\psi = 88^\circ$ to $\Delta\psi = 92^\circ$. In both cases the polarization
crosstalk lower than -27.5 dB. Although no further improvement could be achieved by
the design with combined elements' shape in this aspect, both structures have a decent
polarizations' orthogonality and a low polarization crosstalk.

<figure><img src="image_83.png" /><figcaption>Figure 5.28: A comparison of optimized a-Si:H 2D grating couplers (2D GCs) for O-band with a Metal3 back-reflector - an optimized design with stretched perturbing elements (PE) vs. an optimized design combining stretched and circular perturbing elements (PE). (a) The coupling efficiency of the target- and cross-polarization. (b) The corresponding split ratio (per single waveguide). (c) The polarization' angular difference $\Delta\psi$. (d) The corresponding polarization crosstalk (between signals from both GC arms).</figcaption></figure>

5.2.2.2 Focusing Two-Dimensional Grating Couplers

After an approach for an improved polarization handling has been introduced and ap-
propriate designs for linear 2D GCs in C- and O-band have been found, the last aspect
that remains open for an investigation is the potential size reduction of the proposed 2D
GCs. Here, 2D FGCs are analyzed as a low-footprint alternative to linear 2D GCs. Particu-
larly, we look for polarization-optimized designs, using the same approach with elongated
perturbing elements with an individual orientation (zig-zag tilted ovals). The simulation
procedure is as given in 2.1.3.1, Chap. 2. Because the present analyses have been carried
out at the final stage of this work, the following paragraphs will be used as an outlook and
will indicate some points that need to be addressed in future works.
Before looking for an appropriate diffracting elements' design, the determination of their locations will be briefly explained. As a construction method, design1 in combination with tilted waveguides from Ref. [34] was chosen. To obtain the required positions, the intersections of two arc arrays I and II have to be found. First, each arc array I or II can be defined by the parametrized positions $(x_{n,I}, y_{n,I})$ and $(x_{n,II}, y_{n,II})$, with $n = 1, 2, \dots, N$, $N$ the number of arcs within an array. The positions of the grating perturbing elements result from the intersections of these two arc arrays.

<figure><img src="image_84.png" /><figcaption>Figure 5.29: A description of the modeling procedure of 2D focusing grating couplers (2D FGCs). The grating perturbing elements' positions are found by the intersection of two arc arrays I, II. The arc array I consists of the parametrized arcs $(x_{1,I}, y_{1,I})$, $(x_{2,I}, y_{2,I})$, \dots, $(x_{n,I}, y_{n,I})$, \dots, $(x_{N,I}, y_{N,I})$. The arc array II can be defined analogously.</figcaption></figure>

The following parametrization can be used for the arc arrays I and II, defined in the angular range $[-\gamma, \gamma]$ (Fig. 5.29):

$$
\begin{align}
x_{n,I} &= x_{\text{offset},I} + (x_{0,I} + (n-1) \cdot \Lambda) \cdot \cos t_1, & t_I \in [-\gamma, \gamma] \nonumber \\
y_{n,I} &= y_{\text{offset},I} + (y_{0,I} + (n-1) \cdot \Lambda) \cdot \sin t_1, & t_I \in [-\gamma, \gamma] \nonumber \\
x_{n,II} &= x_{\text{offset},II} + (x_{0,II} + (n-1) \cdot \Lambda) \cdot \cos t_2, & t_{II} = t_I + 90^\circ \tag{5.3} \\
y_{n,II} &= y_{\text{offset},II} + (y_{0,II} + (n-1) \cdot \Lambda) \cdot \sin t_2, & t_{II} = t_I + 90^\circ \nonumber \\
n &= 1, 2, \dots, N, \nonumber
\end{align}
$$

where $\Lambda$ is the period in an arc array. The offset points $(x_{\text{offset},I}, y_{\text{offset},I})$, $(x_{\text{offset},II}, y_{\text{offset},II})$ are
chosen in such a way that the arc arrays cross each other at their central region. The points $(x_{0,I}, y_{0,I})$, $(x_{0,II}, y_{0,II})$ give the position, where the first arc of a given array I and II is defined. This determines the length of the focusing taper, which transforms a waveguide with an initial width $w_{start}$ to the final $w_{end}$ directly at the grating (Fig. 5.29). Exemplary arcs $(x_{1,I}, y_{1,I})$, $(x_{1,II}, y_{1,II})$ and $(x_{2,I}, y_{2,I})$, $(x_{2,II}, y_{2,II})$ and their intersections can be seen in Fig. 5.29. The parameters depicted for the arc array I can be defined analogously for the arc array II. For the following analyses, we choose the values:

$$ 
\begin{aligned}
x_{0,I} &= y_{0,II} = 20 \mu\text{m}, & y &= 24^\circ \\
x_{\text{offset},I} &= x_{0,I} + \frac{\Lambda \cdot N}{2} \\
y_{\text{offset},I} &= 0 \\
x_{\text{offset},II} &= 0 \\
y_{\text{offset},II} &= y_{0,II} + \frac{\Lambda \cdot N}{2}
\end{aligned} 
\qquad (5.4) 
$$

After the discrete mathematical definition of the arc arrays, their intersections have to be found. This is enabled by the MATLAB function intersections.m, which is an open-source code, available from Ref. [42].

<figure><img src="image_85.png" /><figcaption>Figure 5.30: 2D focusing grating couplers (2D FGCs) for C-band: a comparison between a reference design with circular perturbing elements and a proposed optimized design with elongated perturbing elements. The plots show a target- vs. cross-polarization (single-port simulation). (a) The normalized out-coupled power, (b) the mode field overlap, (c) the coupling efficiency, (d) the corresponding polarizations' split ratio.</figcaption></figure>

In the following, reference and proposed optimized designs for C- and O-band will be presented. Because of the large simulation models and computational time, only the
excitation from one of both 2D GC arms is considered. The evaluated parameters are: target- and cross-polarization and the polarizations' split ratio (per single waveguide). If a large split ratio is reached, a good performance in terms of orthogonality between the signals of both GC arms and a low PDL between arbitrary incident polarizations can be expected. We begin with SOI designs for C-band with the following geometries:

* Reference: a shear angle $\alpha = 2^\circ$, a grating period $\Lambda = 588$ nm, an etch depth $d = 140$ nm, circular perturbing elements with a diameter $w_\Lambda = 330$ nm.

* Proposed optimized design: a shear angle $\alpha = 2^\circ$, a grating period $\Lambda = 594$ nm, an etch depth $d = 140$ nm, elongated perturbing elements with a minor axis $w_s = 210$ nm and a major axis $w_l = 510$ nm.

The coupling angle is $\theta_{out} = 6^\circ$ at the symmetry plane. Similarly, a reference and an optimized design for O-band can be defined:

* Reference: a shear angle $\alpha = 3^\circ$, a grating period $\Lambda = 487$ nm, an etch depth $d = 140$ nm, circular perturbing elements with a diameter $w_\Lambda = 280$ nm, a coupling angle $\theta_{out} = 8^\circ$ at the symmetry plane.

* Proposed optimized design: a shear angle $\alpha = 3^\circ$, a grating period $\Lambda = 490$ nm, an etch depth $d = 140$ nm, elongated perturbing elements with a minor axis $w_s = 180$ nm and a major axis $w_l = 440$ nm, a coupling angle $\theta_{out} = 10^\circ$ at the symmetry plane.

<figure><img src="image_86.png" /><figcaption>Figure 5.31: 2D focusing grating couplers (2D FGCs) for O-band: a comparison between a reference design with circular perturbing elements and a proposed optimized design with elongated perturbing elements. The plots show a target- vs. cross-polarization (single-port simulation). (a) The normalized out-coupled power, (b) the mode field overlap, (c) the coupling efficiency, (d) the corresponding polarizations' split ratio.</figcaption></figure>
Figures 5.30 and 5.31 compare the reference and the optimized design for C-band and O-band with respect to the parameters: (a) normalized out-coupled power, (b) mode field overlap, (c) coupling efficiency and (d) polarizations' split ratio in a single-port simulation. In both cases, a clear advantage of the optimized design can be observed - both the out-coupled power and the mode field overlap of the cross-polarization are significantly suppressed compared to the reference design. In C-band, this results in a split ratio better than 23 dB within the bandwidth 1535 nm - 1580 nm. In O-band, the corresponding wavelength range is from 1305 nm - 1345 nm. Further optimization would be advantageous for O-band, however, the current dimensions are at the limit of the design rules for the target 0.25 µm BiCMOS technology. The adoption of a modified grating, comprising both stretched and circular elements (cf. 5.2.2.1), may be a better alternative for O-band 2D FGCs.

<figure><img src="image_87.png" /><figcaption>Figure 5.32: A comparison of 2D focusing grating couplers (2D FGCs) for C-band with different waveguide heights $h_{Si}$: $h_{Si} = 220$ nm vs. $h_{Si} = 300$ nm. Both designs comprise elongated perturbing elements with the same dimensions. The plots show a target- vs. cross-polarization (single-port simulation). (a) The normalized out-coupled power, (b) the mode field overlap, (c) the coupling efficiency, (d) the corresponding polarizations' split ratio.</figcaption></figure>

There are some basic differences in comparison to the linear 2D GCs. First, the perturbing elements' shape for 2D FGCs must be further stretched. While the linear 2D GCs show a good suppression of the cross-polarization's out-coupled power in combination with a moderate mode field overlap efficiency, the 2D FGCs reach an improvement mainly by a strong deterioration of the mode field overlap. This has the disadvantage of excess loss, due to the persisting power out-coupling into the undesired polarization. Looking at the maximally reachable coupling efficiency in C-band, we observe about 1.5 dB deterioration compared to linear 2D GCs: the maximum coupling efficiency of the optimized 2D FGC
design is -5.7 dB (cf. -4.1 dB for optimized linear 2D GCs). Similarly, in O-band we obtain
-5.5 dB, which is more than 2 dB worse compared to its linear 2D GC alternative with a
-3.3 dB efficiency. Apparently, the large gap in terms of efficiency cannot be caused only by
the cross-polarized power. An inspection of both models showed that the low coupling
efficiency is related to a large power in the target-polarization, which is not coupled out
by the grating and keeps propagating within the structure. Typically, this issue is caused
either by a too shallow etch depth or by an inappropriate BOX thickness.

Particularly, the computationally smaller C-band model is investigated regarding these
two parameters. A variation of the etch depth and the BOX values shows that the present
coupling efficiency is the maximally reachable one. Regarding this matter, the following
explanation can be given: for a given design wavelength, there is an optimal combination
of a waveguide height, an etch depth (both related to the effective refractive index), a
BOX thickness and a coupling angle. Reaching a local maximum for a given set of fixed
parameters does not necessarily mean that the global maximum is found. The behavior of
our 2D FGCs suggests that the waveguide height is potentially not optimal for the coupling
efficiency. To verify this hypothesis, a modified C-band 2D FGC is simulated with the geo-
metric parameters: a waveguide height $h_{Si} = 300$ nm, a shear angle $\alpha = 3^\circ$, a grating period
$\Lambda = 575$ nm, an etch depth $d = 200$ nm, elongated perturbing elements with a minor axis
$w_s = 210$ nm and a major axis $w_l = 510$ nm. While the perturbing elements' size is kept
constant, the remaining parameters are adapted for a new, exemplary waveguide height
and a coupling angle of $\theta_{out} = 10^\circ$ at the symmetry plane.

Figure 5.32 shows a comparison of the C-band designs for both waveguide heights. In-
deed, a better power out-coupling is reached by the 300 nm 2D FGC, leading to a coupling
efficiency improvement of about 0.7 dB. Apparently, a slightly modified design of the per-
turbing elements' shape is necessary to reduce further the cross-polarization and reach
an additional coupling efficiency improvement. In O-band, an enhancement of the power
out-coupling in a dedicated design could be observed as well, however, at the cost of a
mode field overlap deterioration. The latter may be related to a non-optimal perturbing
elements' shape and/or periodicity.

The present analysis showed that the zig-zag-tilted ovals technique can be applied success-
fully also to low-footprint 2D FGCs. The lower coupling efficiency of the initially developed
designs is not related to the perturbing elements' optimization, since reference 2D FGCs
with circular diffracting elements suffer from the same problem. The coupling efficiency
limitation is rather a combination of the impact of several factors, including the waveguide
geometry. The concerns on the 2D FGCs' optimization in terms of efficiency remain open
for a potential investigation in future works. The present work suggested the transition
to other waveguide thicknesses, making the 2D FGCs potentially more suitable for the
integration in the BEOL of this platform, in which the used a-Si:H waveguides offer more
design flexibility.
### 5.2.3 Experimental Validation

In this subsection, experimental results for SOI designs, comprising elongated zig-zag tilted perturbing elements, are presented. A focus is set on the measurement of the PDL, which is expressed as the difference between the coupling efficiencies (in dB) of the polarization with a maximal transmission and the polarization with a minimal transmission. This corresponds to the PDL definition used in simulations. Low values for this parameter are an indication that the polarizations' split ratio is sufficiently large and the polarizations' non-orthogonality and crosstalk are low.

#### 5.2.3.1 Results in C-band

In the first fabrication iteration, the producibility of the proposed elongated perturbing elements has been investigated, starting with the linear 2D GCs for C-band. The target geometry includes a waveguide-grating shear angle of $\alpha = 2^\circ$, a grating period $\Lambda = 594$ nm, an etch depth $d = 140$ nm, elongated perturbing elements with a minor axis $w_s = 230$ nm and a major axis $w_l = 320$ nm (as proposed in 5.2.2.1). It has been found that an appropriate shape biasing is necessary, i.e. the diffracting elements must be laid out with larger and/or stretcher dimensions. Both elliptical/oval and rectangular layout shapes may be used to obtain a desired geometry. Figure 5.33 shows a scanning electron microscope (SEM) picture after the reactive ion etch step. The determination of the exact dimensions of the gratings' elements out of the SEM photograph is not achievable with a large accuracy, because of the limited resolution. Deviations of at least $\pm 10$ nm are possible. A roughly estimated range of the dimensions is: 210-230 nm for the minor axis and 300-320 nm for the major axis. Thus, it is possible that the fabricated features are slightly smaller than desired. This can be expressed in a lower coupling efficiency compared to simulations (cf. the analysis of tolerances in 5.2.2.1).

<figure><img src="image_88.png" /><figcaption>Figure 5.33: An optimized 2D grating coupler (2D GC) for C-band comprising a zig-zag-tilted ovals array: (a) a schematic representation, (b) an exemplary array after the reactive ion etch step. (The figure is published in Ref. [38]).</figcaption></figure>
The 2D GC shown in Fig. 5.33 is the first design that has been investigated thoroughly in experiments. Because of the back-to-back test structure in the layout, the parameter that can be best examined in such a measurement is the PDL, out of which we can get an idea of the polarization handling properties of a 2D GC. The results below have been initially reported in our work, Ref. [38]. To characterize the proposed design, back-to-back test structures, comprising two 2D GCs, connected by linear tapers and waveguides, are fabricated in our 0.25 µm photonic BiCMOS technology on 200 mm wafers with a partially processed BEOL stack. The stack height has no importance for the PDL investigation. A setup, including a tunable laser Agilent 81960A (1505 nm - 1625 nm), followed by a programmable polarization controller Agilent 8169A, is employed. Cleaved SMFs support the light in- and out-coupling on a chip. The signal is detected by a power meter Agilent 81634B. For the accurate power normalization, a slim photodiode S132Ce is placed in front of both SMF facets to measure the optical power loss in both off-chip paths, using a power meter PM100D by Thorlabs. Due to the short Si waveguides, no waveguide loss is considered. To prepare a full wafer statistic a semi-automated 300 mm wafer probe system by Formfactor is used. The measurements are performed at a fixed height. Different polarization states are scanned: for a fixed polarization state, a wavelength sweep is carried out. The first and the last wavelength sweeps during the polarization scan are done for the same polarization state, and are used to control the coupling stability in the end of the polarization sweep. The PDL is determined in the following way. First, the wafer mean maximum transmission wavelength is found and a 20 nm bandwidth around this wavelength is chosen. In this wavelength range, the maximal PDL is determined using 9 polarizations. The number of polarizations is sufficient to find the maximal coupling efficiency difference within the given bandwidth and has been determined in preliminary experiments. Finally, the PDL is averaged over the 61 chips on the wafer.

Figure 5.34 (a) shows exemplary measured coupling spectra of the 9 different polarizations with a zoom in the considered bandwidth: the wafer-averaged central wavelength is 1555 nm with a mean coupling efficiency of -4.7 dB (the standard deviation $\sigma_{CE}$ is 0.2 dB). The lower mean coupling efficiency may be related to smaller perturbing elements. The 20 nm bandwidth considered for the PDL evaluation is from 1545 nm to 1565 nm. Figure 5.34 (b), (c), (d) shows a wafer map, a histogram and a cumulative distribution function (CDF) of the maximal PDL. Because no index-matching gel can be used on this setup, there is an uncertainty in the PDL determination, caused by Fabry-Perot ripples. To reduce their impact, the measured curves are smoothed by a local regression fitting algorithm. The wafer maximal root-mean-square error, representing the measurement uncertainty, is $\pm 0.1$ dB. The wafer-averaged maximal PDL in the investigated wavelength range is 0.5 dB with a standard deviation $\sigma_{PDL}$ of 0.18 dB. Furthermore, 67% of the wafer chips have a maximal PDL < 0.55 dB (Fig. 5.34 (d)). Although further optimization of the shape biasing is desirable to obtain an even better wafer uniformity, the present results confirm the suitability of the proposed C-band design for a fabrication in a 0.25 µm photonic BiCMOS technology. Furthermore, the present work shows for the first time statistical results for the PDL of optimized 2D GCs, fabricated with a 248 nm DUV lithography.
<figure><img src="image_89.png" /><figcaption>Figure 5.34: Experimental results for an optimized C-band 2D grating coupler (2D GC) comprising a zig-zag-tilted ovals array. (a) Exemplary measured coupling spectra of different polarizations on a single chip. A zoom of the evaluation bandwidth from 1545 nm to 1565 nm is shown. (b) A wafer map, (c) a histogram and (d) a cumulative distribution function (CDF) of the maximal polarization-dependent loss (PDL) within the considered bandwidth. (A similar figure is published in Ref. [38]).</figcaption></figure>

### 5.2.3.2 Results in O-band

Next, results from first experiments for linear O-band 2D GCs will be presented. Due to the limited availability of the automated wafer probe system, the present analyses have been performed on dice measurements. The chosen 9 chips originate from wafer locations, which are used for process control measurements. For that reason, statistical range can be evaluated also in this case. Another missing component at the time of the measurements was a programmable polarization controller for O-band. No polarization sweep could be performed. Thus, the PDL determination is based on the minimization or maximization of the optical power transmission.

In the following, we compare designs with circular and elongated perturbing elements. The former design is used as a reference and has the parameters: a shear angle $α = 2°$, a grating period $Λ = 480$ nm, an etch depth $d = 140$ nm, circular perturbing elements with a diameter $w_Λ = 280$ nm. The target design with elongated diffracting elements comprises a shear angle $α = 2°$, a grating period $Λ = 480$ nm, an etch depth $d = 140$ nm, elongated
perturbing elements with a minor axis $w_s = 180$ nm and a major axis $w_l = 260$ nm (as proposed in 5.2.2.1). Several biased shapes have been tested in the layout. Here, the best performing variant will be presented.

The measurement setup includes a lightwave measurement system by Agilent (8164B) with a tunable laser Agilent 81600b (sweep range 1280-1340 nm) and an optical power meter Agilent 81634B. The laser is followed by a manual polarization controller. Cleaved single mode fibers are used for in- and out-coupling. The loss in the off-chip paths is determined using the calibrated photodiode Thorlabs SM05PD5A, which is placed in front of the facet of each fiber.

<table><caption>Table 5.4: An experimental comparison of a reference O-band 2D grating coupler (2D GC) with circular perturbing elements and a proposed O-band 2D GC with zig-zag-tilted ovals. The considered parameters are the average coupling efficiency $\sigma_{CE}$ with a standard deviation $\sigma_{CE}$ and the average maximal PDL in 20 nm bandwidth with a standard deviation $\sigma_{PDL}$.</caption><thead><tr><td>Design</td><td>Bandwidth</td><td>CE ±σ<sub>CE</sub></td><td>PDL ±σ<sub>PDL</sub></td></tr></thead><tbody><tr><td>Reference</td><td>1305-1325 nm</td><td>-4.5 dB ± 0.4 dB @1315 nm</td><td>2 dB ± 1.2 dB</td></tr><tr><td>Proposed</td><td>1300-1320 nm</td><td>-4.8 dB ± 0.5 dB @ 1310 nm</td><td>1.1 dB ± 0.5 dB</td></tr></tbody></table>

The back-to-back test structures are fabricated analogously to their C-band equivalents and comprise two 2D GCs, connected by linear tapers and waveguides. Also in this case, the devices are fabricated in our 0.25 µm photonic BiCMOS technology on 200 mm wafers with a partially processed BEOL stack. It should be noted that the 2D GCs' etch depth is this time 130 nm, instead of the target 140 nm. For that reason, a higher coupling loss is expected.

<figure><img src="image_90.png" /><figcaption>Figure 5.35: An experimental comparison of (a) a reference O-band 2D grating coupler (2D GC) with circular perturbing elements and (b) an optimized O-band 2D GC with zig-zag-tilted ovals.</figcaption></figure>

Because of the lack of a programmable polarization controller, we use a 1D-1D GC back-to-back test structure to find the polarization state, which corresponds to the Si TE<sub>00</sub> mode on the chip. Due to the test structures' orientation, this polarization state will be equally split into both 2D GC arms. The choice of this polarization state is necessary to ensure a
balanced coupling to the 2D GC. With the adjusted polarization, the coupling to the 2D-2D GC configuration is optimized. After the best coupling position is found, the polarization is changed so that a maximal or minimal power transmission results. The signal is controlled at different wavelengths to make sure that the largest difference is found. The procedure is repeated for all considered test structures on 9 chips.

Figure 5.35 shows exemplary coupling spectra of the reference design with circular diffracting elements and the investigated design with elongated diffracting elements. The curves are measured on the same chip. In addition, Tab. 5.4 compares both structures in terms of mean coupling efficiency and PDL, including their standard deviations. The PDL is again evaluated in a 20 nm bandwidth around the central wavelength of each structure. For the reference design, the bandwidth is 1305-1325 nm. For the zig-zag tilted ovals design, the bandwidth is 1300-1320 nm. While the device with elongated perturbing elements shows a slight deterioration in terms of efficiency, the average PDL and its variation are fairly improved. In spite of the achieved progress, the investigated zig-zag-tilted ovals design does not reach the performance benchmarks of its C-band equivalent.

The inspection of SEM photographs shows that the desired aspect ratio is not achieved for the optimized design, i.e. the ovals are still too circular (Fig. 5.36 (a)). Alternative test structures have also the problem that their ovals are either too circular or with too large dimensions. For that reason, further shape biases are investigated next. Our observations show that the smaller the desired minor axis *w*<sub>s</sub>, the stretcher the major axis *w*<sub>l</sub> must be laid out. Thus, the first-generation test ovals were laid out with a too low bias. To optimize the layout design procedure, new test structures for O-band 2D GCs are presently under development, which examine variants in a more suitable range. An SEM of an improved design is shown in Fig. 5.36 (b).

<figure><img src="image_91.png" /><figcaption>Figure 5.36: A comparison of SEM photographs of O-band zig-zag-tilted ovals 2D grating couplers (2D GCs). (a) A presently tested device with non-optimal shape biasing. (b) A device with an improved shape biasing for stretcher perturbing elements.</figcaption></figure>
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# 6 Conclusions and Outlook

High-index contrast 2D GCs are fascinating devices: although simple in their composition, they tend to interact with light in an unusual way. Understanding the behavior of 2D GCs is not as trivial as it initially seems. The present thesis has the ambition to provide the reader with a systematic and comprehensive description of all important physical properties of 2D GCs. The author hopes that the summary in this work may be a helpful guideline for interested researchers, who would like to develop own designs for a particular fabrication platform.

The present dissertation was based on 3 milestones, namely: 1) The development of reliable methods for the analysis of 2D GCs. 2) The thorough investigation of the interplay between different physical effects in 2D GCs. 3) The optimization of 2D GCs considering the requirements of a 0.25 µm photonic BiCMOS technology. The progress within this work went from the bottom to the top through these three steps.

The first aspect of the methods' advancement was an essential basis to make proper conclusions in the subsequent analyses. Hereby, it was particularly important to get acquainted with the trade-offs between numerical accuracy and computational cost. Moreover, dedicated evaluation techniques and mathematical formulations were necessary to guarantee for the appropriate post-processing of simulation results. The establishment of a reliable chain of methods for the proper modeling, simulation and post-simulation analysis was the key to determine 2D GCs' design metrics reliably. On the other hand, the accurate experimental handling of the sensitive coupling devices was not less critical. The interplay between theoretical and experimental analysis was the best means to understand the complex behavior of 2D GCs as a next step.

In this second part of the work, the central objective of all analyses became the *polarization*. As a first important advance, a diffraction condition for two basic orthogonal polarizations was formulated. For a given wavelength and effective refractive index, the combination of a non-zero waveguide-to-grating shear angle and a grating period were responsible for the variation of the resulting coupling angle at the symmetry plane. It was shown that a larger coupling angle at the symmetry plane requires an increasing of both the shear angle and the grating period. The mathematical formulation of these dependences made the development of coupling structures for different optical bands, material compositions, and
coupling angles possible. Keeping the polarization as a central objective, the next examined aspect was the polarization splitting capability of 2D GCs. Three performance-related parameters were introduced. First, it was discovered that a 2D GC converts partially a given target-polarization state into its orthogonal counterpart, assigned as a cross-polarization. The ratio between the target- and cross-polarization - the split ratio - was the first investigated parameter. The results showed that the cross-polarization scales with the grating perturbation strength and contributes not only to the split ratio's deterioration, but is also a limiting factor for the coupling efficiency. In addition, the excitation of a cross-polarization means that signals originating from both GC arms are no more fiber coupled to purely x- and y-polarized fields respectively, but have a mixed polarization composition. It was shown that these modified polarizations are generally not orthogonal and can be associated with a polarization crosstalk. This was the second analyzed parameter, for which a large wavelength dependence was found. Although orthogonal signals could be observed at the 2D GC's central wavelength, all other wavelengths were affected by the polarizations' non-orthogonality and crosstalk, defining strict limitations of the usable 2D GC bandwidth. The last contribution of the cross-polarization could be observed in receiver-side 2D GCs, where a given target signal (channel 1) can be superposed with the cross-polarized component from the other communication channel 2 with a different phase relation. Depending on the particular superposition, a large PDL results, which was the third important parameter for this analysis. In the end, a dependence chain could be defined between split ratio, polarizations' non-orthogonality and crosstalk, and PDL. A good split ratio means that polarizations from both GC arms approach the desired x- and y-polarization states after fiber coupling. Accordingly, the signals from the two GC arms become orthogonal to each other and the polarization crosstalk vanishes. Thus, any combination of orthogonal polarizations without crosstalk is coupled with the same efficiency, leading thus to a low PDL. Therefore, a 2D GC has ideally a high split ratio, a low polarization crosstalk and a low PDL.

Apparently, to obtain these parameters as desired, the cross-polarization needs to be eliminated. It became crucial to discover the physical origin of the cross-polarization. The most important contribution of this work was the identification of the grating in-plane scattering as the physical effect behind the cross-polarization and the consequential issues. The in-plane scattering results from the finite dimensions of the perturbing elements with respect to the optical waveguide mode. In summary, the analysis of physical effects in 2D GCs showed that diffraction and in-plane scattering take place in parallel and are not necessarily interrelated. The former physical effect is the desired one, the latter has to be engineered in such a way that the 2D GC handles properly any polarizations' combination. Only the proper in-plane scattering control is a guarantee for an efficient 2D GC with good polarization splitting capabilities in terms of a low polarizations' non-orthogonality and crosstalk, and a low PDL.

After the systematization of these fundamental dependences, the final objective in this work became the exploration of a method to reach the goals given above, namely, to realize an efficient 2D GC with excellent polarization handling properties. While it is not possible
to eliminate the in-plane scattering completely, an appropriate local scattering design can be applied to reduce the total scattering strength and reach thus a low cross-polarization. Two different approaches were proposed and analyzed, considering two optical bands - C- and O-band - and two material platforms - SOI and a-Si:H in the platform's BEOL. Both optimization approaches have the basic idea that large total in-plane scattering may be avoided, when the grating's perturbing elements have different local scattering profiles. As a first technique, the segmented 2D GCs were developed. Different local scattering patterns were achieved by using circular perturbing elements with different sizes and periodicity, grouped into several segments. The numerical analyses showed an improvement in terms of enhanced polarizations' split ratio, reduced polarizations' non-orthogonality and crosstalk, and improved coupling efficiency. However, the achieved advancement was not sufficient to obtain a low PDL. For that reason, a second optimization approach had to be developed, which can be considered as the highlight of the present work. The method relies on a grating, comprising zig-zag tilted elongated perturbing elements. In such an array, adjacent perturbing elements have abruptly different local scattering pattern, due to their varying orientations. An appropriate design of the minor and major axis of the diffracting elements ensures the local cancellation of forwards- and backwards scattered waves of adjacent objects. In an extended variant of this method, both elongated and circular elements can be used, achieving even better scattering suppression. The most important properties of the proposed design is its simplicity - basic geometric shapes can be used with dimensions appropriate for a 248 nm DUV lithography. In addition, no optical proximity correction, but only a suitable shape biasing is necessary. The feasibility of the optimization technique was demonstrated by wafer-level statistics, investigating the PDL. In fact, this work provided for the first time statistical results for the PDL of optimized 2D GCs fabricated with a 248 nm DUV lithography. Satisfactory results were obtained for C-band designs, without further shape biasing optimization. The measured wafer-averaged maximal PDL in a 20 nm was 0.5 dB. The results in O-band showed that further optimization of the shape biasing is necessary. Possibilities to improve the O-band layout design procedure were analyzed. Presently, optimized O-band devices are being prepared for fabrication. Finally, numerical analyses of 2D GCs with optimized zig-zag tilted elongated perturbing elements and a metallic back-reflector showed that such devices may reach a coupling efficiency between -2.5 dB and -2 dB, while keeping the excellent polarization handling properties unaltered.

As an outlook, the on-chip footprint reduction via 2D FGCs was briefly investigated. Although the polarization splitting performance could be improved in a similar way with the zig-zag tilted ovals technique, the focusing designs required an additional optimization in terms of coupling efficiency. Further work is necessary in the future to investigate this matter. The transition to other waveguide geometries may be necessary. In addition, the extended optimization technique, involving stretched and circular perturbing elements, may be more advantageous for 2D FGCs.

The most of the analyses showed that O-band 2D GCs developed under the boundaries of the given 0.25 µm BiCMOS technology have better characteristics both in terms of
coupling efficiency and polarization handling, compared to their C-band equivalents. The consideration of a metallic back-reflector promises for the achievement of a performance, comparable to the best reported devices so far, with the decisive advantage of a reduced complexity. This observation is particularly interesting with regard to the potential deployment of O-band coherent DCIs, willing to combine the best of the IM-DD and the coherent detection worlds. Because the proposed 2D GCs can be realized with a relatively low cost and effort, the optimization technique may contribute to the cost reduction of coherent transceivers, which is necessary for them to penetrate in the data center domain. The proposed design technique is presently undergoing a patent registration procedure, which could make it commercially available.

As a final remark, the present work demonstrates that the good understanding of the physical background of any optical component is crucial for its appropriate optimization. Although the behavior of high-index contrast photonic devices is often not trivial to comprehend, it is worth investing efforts in finding an appropriate model of their physical properties, instead of relying on blind searching methods. The devoted time to understand single devices guarantees in return a more relaxed and reliable design of full systems.
Danksagung

An dieser Stelle möchte ich meinen ganz herzlichen Dank an alle richten, die den Erfolg
dieser Doktorarbeit möglich gemacht haben.

Ich danke Herrn Prof. Klaus Petermann dafür, dass er mich als letzte Doktorandin vor
seiner Pensionierung aufgenommen hat. Es war für mich eine besondere Ehre, von einer
der lebenden Legenden der optischen Kommunikationstechnik betreut werden zu dürfen.
Vor allem bin ich Prof. Petermann sehr dankbar, immer die richtigen Fragen gestellt zu
haben, sodass ich Wichtiges und Unwichtiges voneinander trennen konnte. Die Heraus-
forderungen von Herrn Petermann haben mich dazu motiviert, nach Antworten zu suchen,
die nicht sofort und nicht leicht zu finden waren. Ohne seine Motivation wäre ich nicht
zu den Einsichten gekommen, die letztlich den eigentlichen wissenschaftlichen Kern der
Doktorarbeit darstellen.

Ein ganz spezieller Dank geht an Herrn Prof. Lars Zimmermann für seine ausgeprägte
Unterstützung in diesen Jahren. Dabei meine ich nicht nur fachlichen Beratungen und die
zahlreichen wissenschaftlichen Diskussionen. Prof. Zimmermann hat mich motiviert, nicht
allein im bequemen Bereich der Feldtheorie zu verbleiben, sondern mich ebenso intensiv
mit experimentellen Untersuchungen auseinander zu setzen. Während der Laborarbeit
konnte ich die echten Probleme viel besser verstehen. Es ist für mich daher auch sehr
wichtig geworden, eine praktisch relevante Lösung zu liefern. Dank Prof. Zimmermann
hat sich die Balance zwischen Theorie und Praxis in dieser Arbeit deutlich verbessert. Des
Weiteren bin ich Prof. Zimmermann für seinen moralischen Beistand dankbar, den ich in
manch kritischer Situation besonders geschätzt habe.

Einen weiteren Dank geht an Herrn Prof. Heino Henke, der sich auch nach seiner Pen-
sionierung bemüht, eine interessante Vorlesung für seine Studenten anzubieten. Die dort
behandelten elektromagnetischen Fragestellungen haben mich zum ersten Mal auf die
Idee gebracht, was der mögliche Kern aller Problemen im Rahmen meiner Arbeit sein
könnte.
Bei Herrn Prof. Rolf Schuhmann möchte ich mich für die fachmännische Hilfe bei der Einarbeitung in die Simulationssoftware CST bedanken. So konnte ich eine gute Übereinstimmung zwischen theoretische und praktische Ergebnisse erreichen.

Einen ganz herzlichen Dank möchte ich an alle Kollegen von der TU Berlin und vom IHP richten. Mit Pascal Seiler hatte ich eine angenehme und fruchtbare Zusammenarbeit, da sich unsere Dissertationen erfreulich gegenseitig ergänzten. Unsere gemeinsamen Diskussionen waren in vielen Situationen hilfreich, um voran zu kommen. Besonders dankbar bin ich für die enorme Hilfe bei Laborarbeiten, die Spitze davon das nicht-triviale kohärente Übertragungsexperiment in der TU-Berlin-Umgebung. Weiterhin waren die System-Simulationen von Pascal Seiler eine wertvolle Ergänzung meiner Analysen. Ein Dank geht an Karsten Voigt, der mir sowohl durch Simulationen verschiedener Hilfsstrukturen für meine Layouts als auch durch seine praktische Erfahrung im Labor viel geholfen hat. Ebenso danke ich Georg Winzer, der eine zuverlässige Hilfestellung bei jedem Layout-Problem liefern konnte und viele Layout-Zellen für mich elegant programmiert hat. Ich bedanke mich auch für die Unterstützung des IHP Technologie-Teams, insbesondere bei Christian Mai, der bei der technologischen Realisierung vieler spezieller Designwünsche sein Bestes gab und damit die Erzielung vieler Messergebnisse erst ermöglichte. Ich bin weiterhin dankbar für die wirklich angenehme Zusammenarbeit mit Jochen Kreissl, von dem ich auch etwas über andere Gitter gelernt habe. Vielen Dank an Anna Peczek und Alexandra Kroh, die mich mit verschiedenen Messungen oder Hilfe im Labor am IHP unterstützt haben. Für die schönen Bilder während und nach meiner Verteidigung danke ich Andrzej Gajda.

Abschließend möchte ich mich bei den restlichen Mitgliedern des Promotionsausschusses – Herrn Prof. Manfred Berroth, Herrn Prof. Wim Bogaerts und Herrn Prof. Ronald Freund – für Ihre Zeit bedanken. Während dieses Prüfungsverfahrens habe ich viel gelernt.

Meine tiefste Dankbarkeit und Liebe gehört aber meiner Familie in Bulgarien, ohne die nichts von dem, was ich erreicht habe, möglich gewesen wäre. Eine solche Familie ist eine Gottesgabe und ein großes Glück, das ich jedem wünsche.