Novel adiabatic coupler for III-V nano-ridge laser grown on a Si photonics platform

While III-V lasers epitaxially grown on silicon have been demonstrated, an efficient approach for coupling them with a silicon photonics platform is still missing. In this paper, we present a novel design of an adiabatic coupler for interfacing nanometer-scale III-V lasers grown on SOI with other silicon photonics components. The starting point is a directional coupler, which achieves 100% coupling efficiency from the III-V lasing mode to the Si waveguide TE-like ground mode. To improve the robustness and manufacturability of the coupler, a linear-tapered adiabatic coupler is designed, which is less sensitive to variations and still reaches a coupling efficiency of around 98%. Nevertheless, it has a relatively large footprint and exhibits some undesired residual coupling to TM-like modes. To improve this, a more advanced adiabatic coupler whose geometry is varied along its propagation length is designed and manages to reach ∼100% coupling and decoupling within a length of 200 μm. The proposed couplers are designed for the particular case of III-V nano-ridge lasers monolithically grown using aspect-ratio-trapping (ART) together with nano-ridge engineering (NRE) but are believed to be compatible with other epitaxial III-V/Si integration platforms recently proposed. In this way, the presented coupler is expected to pave the way to integrating III-V lasers monolithically grown on SOI wafers with other photonics components, one step closer towards a fully functional silicon photonics platform. © 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement


Introduction
Leveraging the well-established fabrication technology developed originally by the electronics industry, the domain of silicon photonics has seen an exponential growth over the last decade. Various passive Si optical components including highly confined waveguides (WG) [1], grating couplers [2], directional/tapered couplers [3] and arrayed waveguide gratings (AWGs) [4] as well as excellent modulators [5] and photodetectors [6] have been demonstrated. For a recent review see [7].
However, given the fact that Si has an indirect bandgap, which inhibits efficient light emission, there is a lack of high-performance optical sources directly integrated on the silicon photonics platform. Direct-bandgap III-V semiconductors, known to be efficient light emitters, could in principle fill this gap. This has motivated extensive research towards efficient approaches allowing the integration of III-V semiconductor materials on silicon, including heterogeneous integration [8,9] and hybrid integration [10]. Several groups have now also developed novel epitaxial processes providing high quality III-V materials directly grown on silicon substrates, demonstrating optically or electrically pumped lasing [11][12][13][14][15][16]. An efficient scheme for interfacing these devices with standard silicon photonics devices has not been presented yet. The preferred coupling scenario obviously depends on the approach chosen for the hetero-epitaxy process. Some groups use a thick buffer layer to reduce the III-V defect density in the active layers [13,17,18]. In that case the lasing mode is typically separated from the Si layer by several In the following we will first introduce the technology platform considered and the overall design methodology. Section 3 then presents the design of a directional coupler, schematically shown in Fig. 2(a), to illustrate that efficient optical coupling between the nano-ridge laser and a silicon waveguide is feasible. However, as the directional coupler is very sensitive to process variations, section 4 discusses the design of a more robust linearly tapered adiabatic coupler ( Fig. 2(b)) with better tolerance to fabrication and wavelength variations. Finally, section 5 describes a more advanced adiabatic coupler (shown in Fig. 2(c)) whose geometry along its propagation direction is optimized to further improve the coupling efficiency and reduce the coupler length. design methodology. Section 3 then presents the design of a directional coupler, schematically shown in Fig. 2(a), to illustrate that efficient optical coupling between the nano-ridge laser and a silicon waveguide is feasible. However, as the directional coupler is very sensitive to process variations, section 4 discusses the design of a more robust linearly tapered adiabatic coupler ( Fig. 2(b)) with better tolerance to fabrication and wavelength variations. Finally, section 5 describes a more advanced adiabatic coupler (shown in Fig. 2(c)) whose geometry along its propagation direction is optimized to further improve the coupling efficiency and reduce the coupler length.

Integration process
The proposed design approach considers the case of GaAs based nano-ridges grown on Si using the aspect ratio trapping (ART) and nano-ridge engineering (NRE) techniques described in [24,25]. A TEM image of a cross-section of such a nano-ridge grown on a standard Si substrate is shown in Fig. 1(a). Thanks to ART, defects are efficiently trapped during the growth inside narrow SiO 2 trenches . Afterwards, NRE, through controlling the growth rate on different crystal planes, allows to form box-shaped GaAs nano-ridges above the defective trenches. In principle this nano-ridge can also be grown on an SOI substrate, which would allow for the straightforward integration with Si waveguides on the same wafer. Figure 3 shows a possible process flow to fabricate such a coupler. Before III-V growth, starting from an SOI wafer, the Si ridge, which will later define the trench for III-V epitaxy, and the Si waveguide are defined are defined and planarized with SiO 2 through a standard shallow trench isolation (STI) process ( Fig. 3(b)). Next the epitaxial growth of the nano-ridges can be carried out using metal-organic vapor phase epitaxy (MOVPE), the same way as we did for the structure shown in Fig. 1(a). The high-aspect-ratio trench for growing the GaAs nano-ridge can be defined using a tetramethylammonium hydroxide (TMAH) wet-etch step, resulting in exposed {111} facets at the bottom as shown in Fig. 3(c), which help suppressing anti-phase domains. The waveguide region should be protected during this step. The nano-ridge growth starts with a thin GaAs nucleation layer deposited in the trenches followed by GaAs growth inside and outside the trench (Fig. 3(d)). Afterwards, three compressively strained InGaAs QWs with about 20% indium are embedded in the fully relaxed GaAs nano-ridge, shown in Fig. 3(e). A passivation layer of InGaP covering the full ridge is deposited to suppress carrier losses through surface recombination ( Fig. 3(f)). After nano-ridge growth,the whole structure can be planarized with SiO 2 . Fig. 4(a) shows the resulting schematic cross-section of the proposed coupler. QWs and passivation layer are not included in this sketch. In line with our previous experiments, in the remainder of the paper the thickness of the silicon layer H Si is chosen to be 350 nm, and it is etched 300 nm resulting in a 50 nm remaining Si layer

Integration process
The proposed design approach considers the case of GaAs based nano-ridges grown on Si using the aspect ratio trapping (ART) and nano-ridge engineering (NRE) techniques described in [24,25]. A TEM image of a cross-section of such a nano-ridge grown on a standard Si substrate is shown in Fig. 1(a). Thanks to ART, defects are efficiently trapped during the growth inside narrow SiO 2 trenches . Afterwards, NRE, through controlling the growth rate on different crystal planes, allows to form box-shaped GaAs nano-ridges above the defective trenches. In principle this nano-ridge can also be grown on an SOI substrate, which would allow for the straightforward integration with Si waveguides on the same wafer. Figure 3 shows a possible process flow to fabricate such a coupler. Before III-V growth, starting from an SOI wafer, the Si ridge, which will later define the trench for III-V epitaxy, and the Si waveguide are defined are defined and planarized with SiO 2 through a standard shallow trench isolation (STI) process ( Fig. 3(b)). Next the epitaxial growth of the nano-ridges can be carried out using metal-organic vapor phase epitaxy (MOVPE), the same way as we did for the structure shown in Fig. 1(a). The high-aspect-ratio trench for growing the GaAs nano-ridge can be defined using a tetramethylammonium hydroxide (TMAH) wet-etch step, resulting in exposed {111} facets at the bottom as shown in Fig. 3(c), which help suppressing anti-phase domains. The waveguide region should be protected during this step. The nano-ridge growth starts with a thin GaAs nucleation layer deposited in the trenches followed by GaAs growth inside and outside the trench (Fig. 3(d)). Afterwards, three compressively strained InGaAs QWs with about 20% indium are embedded in the fully relaxed GaAs nano-ridge, shown in Fig. 3(e). A passivation layer of InGaP covering the full ridge is deposited to suppress carrier losses through surface recombination ( Fig. 3(f)). After nano-ridge growth,the whole structure can be planarized with SiO 2 . Figure 4(a) shows the resulting schematic cross-section of the proposed coupler. QWs and passivation layer are not included in this sketch. In line with our previous experiments, in the remainder of the paper the thickness of the silicon layer H Si is chosen to be 350 nm, and it is etched 300 nm resulting in a 50 nm remaining Si layer next to the waveguides. next to the waveguides.

Design methodology and simulation tools
As discussed in [20] the nano-ridge laser operates in the lowest order TE-like mode of the device. Therefore the coupler design should maximize the coupling efficiency η C from this mode to the TE-like fundamental mode of the Si WG. Given the aforementioned fabrication process, the height of the Si waveguide is fixed (H Si = 350 nm) and only the geometrical parameters in plane can be manipulated. Hence the free variables include the width of the Si WG W Si , the spacing W Gap defined as the distance between the right edge of the GaAs trench and the left edge of the Si WG and the length of the coupler L. The effective refractive index for the TE-like ground mode of this nano-ridge is calculated to be n T E I I I -V = 2.64 at a wavelength of 1310 nm. All the simulations were carried out using commercial software (Lumercial MODE Solution [26]). The Finite Difference Eigenmode (FDE) solver is used to calculate the eigenmodes of the individual waveguides and within the coupler while the Bidirectional EigenMode Expansion (EME) solver is used to simulate the light propagation through the proposed couplers. We monitor the TE-like and TM-like ground modes for the standalone WGs and the lowest-order symmetric and anti-symmetric TE-like and TM-like supermodes at both input and output sides of the couplers but include 20 optical modes inside the simulation region for a good accuracy. The simulation results of the optimized linearly tapered coupler conducted by EME is confirmed by (Finite-Difference Time-Domain) FDTD simulation. Except where otherwise mentioned, all simulations were carried out for a wavelength of 1310 nm.

Directional coupler
In the case of the directional coupler, W Si and W Gap are kept constant along the propagation direction (z). The remaining free variables are hence the spacing W Gap between both waveguides and the width W Si of the Si waveguide. To facilitate full coupling from the TE-like lasing mode Φ T E I I I-V of the III-V nano-ridge, shown in the inset of Fig. 4(b), to the TE-like mode Φ T E Si of the Si WG, the two modes have to be phase-matched. In practice this means the width of the Si WG has to be chosen such that the effective refractive index n T E Si of its TE-like ground mode is equal to that of the TE-like ground mode of the III-V nano-ridge, n T E I I I -V = 2.64. From Fig. 4(b), which shows the effective indices of the first two eigenmodes of the Si WG as function of its width, we can derive this phase matching happens for W Si = W match = 295 nm (for a wavelength of 1310 nm). Figure 4(c), which shows a top view of the propagating field for W Gap = 240 nm, proves the optical power is indeed fully exchanged between both WGs in a periodic fashion. The coupling length L C for this configuration is 45 µm. To obtain full field exchange in a directional coupler, the phase-matching condition has to be rigorously met and the coupler length has to be exactly equal to L c , imposing stringent requirements to the fabrication process. The L c is highly wavelength dependent, as shown in Fig. 4(d), decreasing from 63 µm to 40 µm in the wavelength range 1260 nm to 1360 nm. It is also found to be extremely sensitive to variations of Si WG width ∆W Si and of gap ∆W Gap , with This proves the directional coupler is highly sensitive to variations.

Design methodology and simulation tools
As discussed in [20] the nano-ridge laser operates in the lowest order TE-like mode of the device. Therefore the coupler design should maximize the coupling efficiency η C from this mode to the TE-like fundamental mode of the Si WG. Given the aforementioned fabrication process, the height of the Si waveguide is fixed (H Si = 350 nm) and only the geometrical parameters in plane can be manipulated. Hence the free variables include the width of the Si WG W Si , the spacing W Gap defined as the distance between the right edge of the GaAs trench and the left edge of the Si WG and the length of the coupler L. The effective refractive index for the TE-like ground mode of this nano-ridge is calculated to be n TE III-V = 2.64 at a wavelength of 1310 nm. All the simulations were carried out using commercial software (Lumercial MODE Solution [26]). The Finite Difference Eigenmode (FDE) solver is used to calculate the eigenmodes of the individual waveguides and within the coupler while the Bidirectional EigenMode Expansion (EME) solver is used to simulate the light propagation through the proposed couplers. We monitor the TE-like and TM-like ground modes for the standalone WGs and the lowest-order symmetric and anti-symmetric TE-like and TM-like supermodes at both input and output sides of the couplers but include 20 optical modes inside the simulation region for a good accuracy. The simulation results of the optimized linearly tapered coupler conducted by EME is confirmed by (Finite-Difference Time-Domain) FDTD simulation. Except where otherwise mentioned, all simulations were carried out for a wavelength of 1310 nm.

Directional coupler
In the case of the directional coupler, W Si and W Gap are kept constant along the propagation direction (z). The remaining free variables are hence the spacing W Gap between both waveguides and the width W Si of the Si waveguide. To facilitate full coupling from the TE-like lasing mode Φ TE III-V of the III-V nano-ridge, shown in the inset of Fig. 4(b), to the TE-like mode Φ TE Si of the Si WG, the two modes have to be phase-matched. In practice this means the width of the Si WG has to be chosen such that the effective refractive index n TE Si of its TE-like ground mode is equal to that of the TE-like ground mode of the III-V nano-ridge, n TE III-V = 2.64. From Fig. 4(b), which shows the effective indices of the first two eigenmodes of the Si WG as function of its width, we can derive this phase matching happens for W Si = W match = 295 nm (for a wavelength of 1310 nm). Figure 4(c), which shows a top view of the propagating field for W Gap = 240 nm, proves the optical power is indeed fully exchanged between both WGs in a periodic fashion. The coupling length L C for this configuration is 45 µm.
To obtain full field exchange in a directional coupler, the phase-matching condition has to be rigorously met and the coupler length has to be exactly equal to L c , imposing stringent requirements to the fabrication process. The L c is highly wavelength dependent, as shown in Fig. 4(d), decreasing from 63 µm to 40 µm in the wavelength range 1260 nm to 1360 nm. It is also found to be extremely sensitive to variations of Si WG width ∆W Si and of gap ∆W Gap , with This proves the directional coupler is highly sensitive to variations.

Proposed configuration
Given that the directional coupler exhibits a low tolerance to fabrication and temperature variations, a more robust and variation-tolerant linearly tapered adiabatic coupler is proposed. Figure 2(b) shows the proposed design. The left edge of the Si waveguide is kept parallel with the III-V nano-ridge such that W Gap remains z-invariant. The right edge on the other hand is linearly varying along the coupler. The full coupler can be divided in three segments. In the first segment with length L 1 the Si waveguide starts with a sharp tip and ends with a width of W 1 . It has a This segment is included in the coupler to reduce perturbations introduced by the start of the Si coupler. In the second and main segment the width of the silicon waveguide varies from W 1 and expands to W 2 (W 2 >W 1 ) over a length L 2 , with a slope W 2 −W 1 L 2 . The variables W 1 , W 2 and L 2 are the main parameters to be optimized for the coupler. The last segment consists of a straight WG with constant width W 2 to guide the coupled light away from the amplifier section, towards devices further away on the photonic IC.
The top view of the coupler and the respective TE-like optical modes at different cross-sections are shown in Fig. 5: Φ TE III-V is the TE-like ground mode in the standalone III-V nano-ridge, Ψ TE +1 is the symmetric TE-like supermode at the position of W 1 , Ψ TE +2 is the symmetric TE-like supermode at the position of W 2 and finally Φ TE Si is the TE-like ground mode in the standalone Si WG. For easier recognition of the different modes, the symbol Φ is used to represent the modes of the standalone waveguides while Ψ is used for denoting the supermodes in the coupling region. The TM-like optical modes are supposed to be highly suppressed in the amplifier section itself but given the asymmetric structure could be excited if the coupler is not well designed. They are denoted as Φ TM III-V and Φ TM Si (in the standalone WGs) and Ψ TM +1 and Ψ TM +2 for the TM-like supermodes at the position W 1 and W 2 . Figure 2(b) shows the proposed design. The left edge of the Si waveguide is kept parallel with the III-V nano-ridge such that W Gap remains z-invariant. The right edge on the other hand is linearly varying along the coupler. The full coupler can be divided in three segments. In the first segment with length L 1 the Si waveguide starts with a sharp tip and ends with a width of W 1 . It has a slope W 1 L 1 . This segment is included in the coupler to reduce perturbations introduced by the start of the Si coupler. In the second and main segment the width of the silicon waveguide varies from W 1 and expands to W 2 (W 2 > W 1 ) over a length L 2 , with a slope W 2 −W 1 L 2 . The variables W 1 , W 2 and L 2 are the main parameters to be optimized for the coupler. The last segment consists of a straight WG with constant width W 2 to guide the coupled light away from the amplifier section, towards devices further away on the photonic IC.
The top view of the coupler and the respective TE-like optical modes at different cross-sections are shown in Fig. 5: Φ T E I I I-V is the TE-like ground mode in the standalone III-V nano-ridge, Ψ T E +1 is the symmetric TE-like supermode at the position of W 1 , Ψ T E +2 is the symmetric TE-like supermode at the position of W 2 and finally Φ T E Si is the TE-like ground mode in the standalone Si WG. For easier recognition of the different modes, the symbol Φ is used to represent the modes of the standalone waveguides while Ψ is used for denoting the supermodes in the coupling region. The TM-like optical modes are supposed to be highly suppressed in the amplifier section itself but given the asymmetric structure could be excited if the coupler is not well designed. They are denoted as Φ T M I I I -V and Φ T M Si (in the standalone WGs) and Ψ T M +1 and Ψ T M +2 for the TM-like supermodes at the position W 1 and W 2 .

Optimisation of coupler width
The coupling from the III-V lasing mode Φ T E I I I -V to the Si WG mode Φ T E Si can be decomposed into three individual coupling processes. The initial optical mode Φ T E I I I -V in the active III-V nano-ridge is converted to Ψ T E +1 at the start of the Si WG. The transition from Ψ T E +1 to Ψ T E +2 happens in the taper region. The last step is the conversion from Ψ T E +2 to Φ T E Si , after which the optical field is guided further in the standalone Si WG towards following optical components. The total coupling efficiency is the product of the efficiency of each individual mode coupling process.
The widths W 1 and W 2 are optimized first. The coupling between two optical modes is approximated by taking the square of their overlap integral. Figure 6

Optimisation of coupler width
The coupling from the III-V lasing mode Φ TE III-V to the Si WG mode Φ TE Si can be decomposed into three individual coupling processes. The initial optical mode Φ TE III-V in the active III-V nano-ridge is converted to Ψ TE +1 at the start of the Si WG. The transition from Ψ TE +1 to Ψ TE +2 happens in the taper region. The last step is the conversion from Ψ TE +2 to Φ TE Si , after which the optical field is guided further in the standalone Si WG towards following optical components. The total coupling efficiency is the product of the efficiency of each individual mode coupling process.
The widths W 1 and W 2 are optimized first. The coupling between two optical modes is approximated by taking the square of their overlap integral. Figure 6(    In addition, coupling to the TM-like mode should be suppressed as much as possible. Figure 6 To avoid the risk of coupling to TM-like modes, the tapered coupler should include point A but exclude B, C and D, namely W B <W 1 <W A <W 2 <W C <W D . Combining this set of conditions with the ones derived above we find 235 nm<W 1 <275 nm and 315 nm<W 2 <325 nm for W Gap = 240 nm from Fig. 6. Obviously the last condition cannot be fulfilled, which means that for the narrower gaps no configuration can be defined, which at the same time assures a high coupling efficiency and low coupling to unwanted modes. This can also be seen from Table 1, which gives the allowable values for W 1 and W 2 for a wider range of values of W Gap and shows that for this linearly tapered adiabatic coupler design the gap between the III-V nano-ridge and the silicon waveguide W Gap has to be chosen larger than 220 nm. As we will discuss in the next section this in turns limits how compact (short) we can make the coupler.

Optimisation of coupler length
If we follow the above derived design criteria for W 1 and W 2 a high coupling efficiency should in principle be possible, at least if the lengths L 1 and L 2 are chosen sufficiently long to allow for an adiabatic transition from Ψ TE +1 to Ψ TE +2 . As discussed above, the first coupler segment with length L 1 is included in the coupler to avoid scattering at the point where the Si waveguide is introduced. Simulations showed that less than 0.3% loss is introduced as soon as L 1 >5 µm. In the remainder of this paper, we therefore take L 1 = 10 µm.   Fig. 5 (b), the strong coupling to Ψ T M −2 is indeed expected. Fig. 6 (b) shows the evolution of the normalized field in the III-V and Si WGs respectively, when L 2 = 150 nm. The ripples in the second half of the taper are attributed to the interference between the modes Ψ T E +2 and Ψ T M −2 . The related power exchange can be observed also clearly in Fig. 6 (c), which depicts the 2-D field distribution along the coupler. If the gap is increased to W Gap = 240 nm, the maximum coupling happens at L 2 = 310 µm with η C ∼ 98% (Fig. 6 (d)). In this case, the remaining ∼ 2% of the optical power is partly coupled to supermode Ψ T E −2 centered in III-V nano-ridge and partly coupled to the TM-like supermode Ψ T M +2 centered in the Si WG. Fig. 6 (e) and (f) show how the field exchanges along the coupler when L 2 = 310 µm. Some ripples are still present, due to interference between the supermodes Ψ T E +2 and Ψ T E −2 , but they gradually decrease in amplitude along the propagation axis as the two modes get decoupled. Hence, the chosen parameters W Gap = 240 nm, W 1 = 275 nm, W 2 = 315 nm result in good coupling efficiency and an overall decent design, albeit at the cost of larger footprint, related to the reduced coupling strength between both waveguides for this wider gap. As we now also clearly showed simply reducing the gap is not a good solution, in section 5 we will propose a more advanced design which provides a better compromise between decoupling the optical modes at the entrance and exit of the coupler on the one hand and increasing the coupling strength in the center part.

Tolerance to Fabrication Variations
In our process, the exact size of the III-V nano-ridge is determined by the epitaxy process. A change in growth temperature, growth time or mask loading might cause slight deviations in the nano-ridge dimensions. Fig. 7 (a) and (b) plot the coupling efficiency η C and the effective refractive index n T E I I I −V as function of a deviation in height ∆H or width ∆W of the nano-ridge dimensions, for a coupler with W Gap = 240 nm, W 1 = 275 nm, W 2 = 315 nm, L 1 = 10 µm, L 2 = 310 µm. Fig. 7 (a) shows that 30 nm variation in III-V nano-ridge height causes the coupling efficiency drop from 98% to 84%. Fig. 7 (b) shows that 15 nm variation in width is acceptable with η C decreasing by 16% from 98% to 82%. But for width variations above The second segment with length L 2 , should convert supermode Ψ TE +1 to Ψ TE +2 adiabatically. Therefore, we swept L 2 to optimise the coupling efficiency while keeping a compact footprint. To illustrate that a too narrow separation W Gap between both waveguides will indeed introduce coupling to unwanted TM-like modes, both the simulation results for W Gap = 150 nm and When W Gap = 150 nm the coupling shows an optimum η C ∼ 80% for L 2 = 150 µm. The remaining light is mostly coupled to the TM-like mode Ψ TM −2 centered in the III-V nano-ridge. As L 2 further increases, more severe coupling between Ψ TE +2 and Ψ TM −2 can be seen. With W 2 close to the phase matching

Advanced adiabatic coupler
Compared with the narrow-band and variation-intolerant directional coupler, the linearly tapered coupler already exhibits a broadband and more fabrication tolerant performance. Nevertheless, with a length of 310 µm its footprint is large compared to the length of a typical III-V integrated laser or amplifier (100 − 600 µm) and in particular with respect to our earlier demonstrated nano-ridge laser, which was only ∼ 100µm long [20]. In addition insufficient decoupling at the point D for the modes Φ TE Si and Φ TM III-V as shown in Fig. 6(b), the strong coupling to Ψ TM −2 is indeed expected. Figure 7(b) shows the evolution of the normalized field in the III-V and Si WGs respectively, when L 2 = 150 nm. The ripples in the second half of the taper are attributed to the interference between the modes Ψ TE +2 and Ψ TM −2 . The related power exchange can be observed also clearly in Fig. 7(c), which depicts the 2-D field distribution along the coupler. If the gap is increased to W Gap = 240 nm, the maximum coupling happens at L 2 = 310 µm with η C ∼ 98% (Fig. 7(d)). In this case, the remaining ∼ 2% of the optical power is partly coupled to supermode Ψ TE −2 centered in III-V nano-ridge and partly coupled to the TM-like supermode Ψ TM +2 centered in the Si WG. Figures 7(e) and 7(f) show how the field exchanges along the coupler when L 2 = 310 µm. Some ripples are still present, due to interference between the supermodes Ψ TE +2 and Ψ TE −2 , but they gradually decrease in amplitude along the propagation axis as the two modes get decoupled. Hence, the chosen parameters W Gap = 240 nm, W 1 = 275 nm, W 2 = 315 nm result in good coupling efficiency and an overall decent design, albeit at the cost of larger footprint, related to the reduced coupling strength between both waveguides for this wider gap. As we now also clearly showed simply reducing the gap is not a good solution, in section 5 we will propose a more advanced design which provides a better compromise between decoupling the optical modes at the entrance and exit of the coupler on the one hand and increasing the coupling strength in the center part.

Advanced adiabatic coupler
Compared with the narrow-band and variation-intolerant directional coupler, the linearly tapered coupler already exhibits a broadband and more fabrication tolerant performance. Nevertheless, with a length of 310 µm its footprint is large compared to the length of a typical III-V integrated laser or amplifier (100 − 600 µm) and in particular with respect to our earlier demonstrated nano-ridge laser, which was only ∼ 100µm long [20]. In addition insufficient decoupling at the beginning and end of the linearly tapered coupler decreases the maximum coupling efficiency and introduces extra coupling to unwanted modes, which could impact the laser performance. Therefore, a more compact adiabatic coupler design with more efficient coupling and decoupling is desired.
To improve the design, a semi-analytical approach as described in [27] is adopted. The 3D schematic view of the coupler is shown in Fig. 2(c) while the top view can be found in Fig. 8(a). Now both the width W Si (z) of the Si WG and the spacing W Gap (z) between the III-V trench and the Si WG are allowed to vary along the propagation direction. Their values are derived from optimised functions κ(z) and ∆β(z) for the coupling coefficient and the dephasing between the lasing mode Φ TE III-V and the Si WG mode Φ TE Si : κ(z) = κ max · sinθ(z) (1) ∆β(z) = −∆β max · cosθ(z) (2) with κ max the maximum coupling coefficient, at the device center z = L 2 , and ∆β max the maximum dephasing, at the beginning z = 0 and the end z = L of the device. θ(z) is a monotonically increasing function of z, with θ(0) = 0 and θ(L) = π. The coupling coefficient κ is calculated as permittivity function describing the waveguide profile and U i is the TE-like optical field of the corresponding WG.
It is found that the most efficient taper is obtained if θ(z) varies as a Blackman function [28]. We let The values for the Si WG effective index n T E Si (z) and the normalized coupling coefficient κ(z) κ ma x optimized according to this procedure are plotted in Fig. 8(b). Figure 8(c) shows the according geometrical parameters W Gap and W Si . They are obtained from the known κ max , ∆β max and θ(z) using an FDE mode solver. The staircase-like pattern in the curve for W Si (z) in Fig. 8(c) originates from the simulation resolution of 1 nm.   The maximum coupling constant κ max is calculated by choosing W Si ( L 2 ) = W match = 295 nm (the phase matching condition) and W Gap ( L 2 ) = 150 nm to realize a high coupling strength. Although above we found that the choice W Gap = 150 nm introduces high coupling to unwanted modes in the linearly tapered coupler, we will show this is not the case for the optimised adiabatic coupler discussed here. ∆β(z) is given by: The maximum dephasing ∆β max occurs at the positions z = 0 and z = L, where we choose W Gap (0) = W Gap (L) = 850 nm, W Si (0) = 280 nm and W Si (L) = 310 nm to assure complete decoupling between both WGs. It is found that the most efficient taper is obtained if θ(z) varies as a Blackman function [28]. We let The values for the Si WG effective index n TE Si (z) and the normalized coupling coefficient κ(z) κ max optimized according to this procedure are plotted in Fig. 8(b). Figure 8(c) shows the according geometrical parameters W Gap and W Si . They are obtained from the known κ max , ∆β max and θ(z) using an FDE mode solver. The staircase-like pattern in the curve for W Si (z) in Fig. 8(c) originates from the simulation resolution of 1 nm. Figure 9(a), shows how the coupling efficiency towards the different supermodes varies as function of L. The green curve shows how the coupling efficiency η C towards Ψ TE +2 firstly rapidly increases with L, and then approaches unity for L = 150 µm and beyond while the undesired coupling towards the TM-like modes Ψ TM +1 and Ψ TM +2 remains low. Figures 9(b) and 9(c) show how the optical power is exchanged between both waveguides for the optimized coupler with L = 200 µm. These results confirm the optimized adiabatic coupler allows for a high coupling efficiency and improves the coupling and decoupling processes along the taper, without spurious coupling to undesired modes.

Tolerance to fabrication variations
As the III-V nano-ridges are developed for lasing in the O-band, the couplers are expected to have high coupling efficiency over the 1260 − 1360 nm. Further, in our fabrication process, the exact size of the III-V nano-ridge is determined by the epitaxy process. A change in growth temperature, growth time or mask loading might cause slight deviations in the nano-ridge dimensions. Therefore, the tolerance to wavelength and fabrication variations of the three coupler implementations is discussed in this section. Figure 10 shows how the coupling efficiency of the three couplers varies versus wavelength. For the directional coupler, even though 100% coupling efficiency is achieved at ∼ 1310 nm, η c rapidly drops away from this central wavelength, by 37% at 1260 nm. To the contrary, the variations in coupling efficiency is only ∼ 4% and ∼ 2% for the linearly tapered coupler and the advanced adiabatic coupler respectively over a 100 nm wide wavelength range, proving their broadband character. This follows from the fact that, although n TE III-V varies relatively strongly with wavelength (shown in Fig. 10), the deviation from the phase matching condition remains small as also the effective index of the silicon waveguide decreases with wavelength.   Figure 11(b) shows that 10 nm variation in width is acceptable with η C decreasing by ∼ 20%. But for width variations above 30 nm the coupling efficiency drastically drops. The different impact from the nano-ridge height and width variations on the coupling efficiency can be understood by the related changes in the effective index of the nano-ridge. n TE III-V is more strongly influenced by width variations than by height variations. These simulations indicate that the linearly tapered coupler and the advance adiabatic coupler are strongly improved in terms of fabrication tolerances with respect to the directional coupler. Although the tolerance of the advanced adiabatic coupler is slightly worse than that of the longer linearly tapered adiabatic coupler, it allows the freedom to optimise either towards compact size or towards high processing tolerance, dependent on the application. Fig. 11. Coupling efficiency as a function of (a) nano-ridge height variation, (b) width variation.

Conclusion
We proposed a novel adiabatic coupler working in the O-band for coupling light from a III-V nano-ridge laser to a Si waveguide. The coupler is compatible with our earlier presented process for the monolithic epitaxy of GaAs based nano-ridges on silicon using the aspect ratio trapping combined with nano-ridge engineering. We showed that a directional coupler design can reach a coupling efficiency of 100% but is very sensitive to process and wavelength variations. To improve this, a more robust linearly tapered adiabatic coupler is presented, which is more tolerant to process variations and can still achieve a 98% coupling efficiency. However, this comes at the cost of an increased footprint and some residual coupling to unwanted modes. To prevent this undesired coupling from influencing the laser operation, a more advanced adiabatic couple, which better decouples the modes at the beginning and end of the coupler was then proposed. This device exhibits a coupling efficiency >99% at 1310 nm and >97% over the full O-band, for a coupler length of 200 µm, without coupling to undesired modes. Ultimately the objective is to allow for electrical injection of these devices. The necessary metal contacts should thereby be implemented without introducing excessive loss, e.g. through separating them from the nano-ridges by a narrow fin [29]. In such a case, the optical modes of the nano-ridge should hardly be affected and the design procedure outlined above would remain applicable. Finally it is important to note that the design procedure proposed here can directly be transferred to other III-V nano-lasers epitaxially grown on silicon recently proposed in literature [11,15,16,19].
The presented adiabatic coupler is expected to connect III-V devices with the whole on-chip Si photonics circuit. Besides, it helps explore the potential of the epitaxial III-V to make more complex and functional devices with the help of external Si circuits to provide feedback or/and form a cavity.