5. Bragg diffraction gratings
5.4. Optimization procedure
5.4.2. Enhancement of the modal overlap
5. Bragg diffraction gratings 93 order to reach higher efficiencies, the upward reflection has to be decreased by adjusting the coupling strength along the grating.
5. Bragg diffraction gratings 94 first rib-groove pair in the proximity of the waveguide by sweeping the corresponding FF from 0.01 to 0.99 in steps of 0.02 for a set of periods from 520 nm to 660 nm in 20 nm steps and saving the dimensions for the best-calculated efficiency. The procedure is applied for all subsequent pairs and is defined as a generation. This routine is repeat-ed for the whole grating again and again until the result converges. The number of itera-tions is defined as the product of the number of generaitera-tions and rib-groove pairs. The second step of the algorithm serves to refine the obtained dimensions since only a pre-defined set of periods is used to ensure a short simulation time. Therefore, each element is optimized separately starting from a length variation of ±10 nm down to ±1 nm. Here a generation is defined when all elements are investigated within a single optimization cycle. This routine is also repeated for the whole grating until no higher coupling effi-ciency is obtained. The number of iterations during this step corresponds to the product of the number of generations and elements [117].
Figure 5.8: (a) Simplified algorithm flowchart for the optimization of the individual grating elements. (b) Typical coupling efficiency progression.
1 10 100 1000
-1.0 -0.8 -0.6 -0.4 -0.2 0.0
Coupling efficiency [dB]
Number of iterations Sweep FF and Λfor
each rib-groove pair
Yes
No
Adjust each rib and groove length
Yes
No Periodic
structure Start
End efficiency?Max.
efficiency?Max.
(a)
(b)
5. Bragg diffraction gratings 95 The optimization algorithm is implemented in MATLAB, and the 2D simulations are carried out by the open source software CAMFR based on the eigenmode expansion method. The advantage of this method in comparison to the FDTD tool previously used is that the simulation domain is not discretized into very fine grids but divided in re-gions of the same refractive index [118]. Hence, the computational effort is lessened and the optimization time is dramatically reduced. However, the final structure is simu-lated based on RSoft FullWAVE since it delivers a more accurate result.
It should also be noted that the field excitation in CAMFR occurs in the waveguide and not the fiber. Thus, the coupling efficiency is determined from the opposite direction as the ratio of the output power in the fiber to the waveguide input power. Owing to the reciprocity of the configuration, both directions have to deliver the same results [106].
Figure 5.8(b) represents the typical efficiency progression of the optimized structure as a function of the total iteration steps. The first iterations are responsible for the fast en-hancement since it suffices to adapt the FF of the grating region in the proximity of the waveguide to decrease the coupling strength and approach the targeted Gaussian profile.
While this step improves the efficiency by ~0.4 dB, the following refinement step yields
~0.1 dB enhancement.
In order to emphasize the origin of the coupling efficiency improvement, the diffracted field profiles of the starting periodic structure and the finally obtained aperiodic grating are compared to each other. Both profiles and the fiber Gaussian function are depicted in Figure 5.9. It is clear that the similarity of the scattered field at the aperiodic grating is much better than for the case of a periodic structure. Hence, the modal overlap be-tween both profiles is larger, and the coupling efficiency is improved to more than –0.4 dB.
The dimensions of the optimized nonuniform structure, called GC1, are shown in Table 5.1 where the lengths of the individual grooves and ribs are labeled gi and bi respective-ly. In order to obtain a more accurate result, the grating coupler is simulated using RSoft FullWAVE. Figure 5.10 depicts the corresponding electric field distribution at the target wavelength and the spectral efficiency between 1500 nm and 1600 nm.
5. Bragg diffraction gratings 96
Figure 5.9: Fiber and diffracted field profiles of (a) the initial periodic structure and (b) the obtained aperiodic grating after the optimization proce-dure.
Table 5.1: Dimensions of the optimized nonuniform grating GC1 in nm. g1
corresponds to the length of the nearest groove to the output waveguide.
g1 b1 g2 b2 g3 b3 g4 b4 g5 b5 g6 b6 g7 b7 g8 b8 g9
42 515 81 550 67 471 434 87 516 133 478 149 218 427 156 409 269 b9 g10 b10 g11 b11 g12 b12 g13 b13 g14 b14 g15 b15 g16 b16 g17 b17
338 268 318 303 313 277 308 288 312 289 313 291 312 276 319 291 315 g18 b18 g19 b19 g20 b20 g21 b21 g22 b22 g23 b23 g24 b24 g25 b25 g26
282 316 296 296 299 320 282 304 274 331 274 303 298 324 289 283 302
Figure 5.10: (a) Electric field distribution of the obtained nonuniform grat-ing at 1550 nm. (b) Simulated spectral efficiency and upward reflection.
-20 -15 -10 -5 0 5
0.0 0.2 0.4 0.6 0.8
1.0 Diffracted field Fiber profile
Normalized intensity
z [µm]
-20 -15 -10 -5 0 5
0.0 0.2 0.4 0.6 0.8
1.0 Diffracted field Fiber profile
Normalized intensity
z [µm]
(a) (b)
1500 1520 1540 1560 1580 1600
-14 -12 -10 -8 -6 -4 -2 0
Rup, ηGC [dB]
Wavelength [nm]
Rup ηGC
Max. Min.
(a) Al
Si SiO2 SiO2
(b)
5. Bragg diffraction gratings 97 The electric field distribution shows that the power is less coupled at the waveguide edge due to the large FF, which corresponds to a narrow groove length, and the maxi-mum coupling is shifted toward the middle of the grating. The improved modal overlap produces a very high coupling efficiency, which reaches –0.26 dB at 1550 nm, a 1 dB bandwidth larger than 43 nm, and a 3 dB bandwidth of 75 nm. In addition, the upward reflection is reduced to less than –13 dB. Thus, using a metal mirror and apodizing the grating based on the adapted algorithm significantly enhances the coupling efficiency while still retaining a large bandwidth.
The dimensions of the grating in Table 5.1 show that shaping the grating field profile to a Gaussian-like function necessitates very narrow grooves in the proximity of the output waveguide with lengths smaller than 100 nm in addition to some ribs with similar criti-cal dimensions. g1 and b4, with values of 42 nm and 87 nm respectively, may be espe-cially technologically challenging. As these structures are aimed to be concretely fabri-cated, it is meaningful to carry out the optimization with more relaxed dimensions and compare the results with the ideal case. For this purpose, the algorithm is run again with different prerequisites. The second grating, called GC2, has to exhibit a minimum rib and groove length of 100 nm and 60 nm respectively to be able to be realized by means of electron beam lithography. The third structure, meanwhile, called GC3, has to be lim-ited to nearly 120 nm for both parameters so that it can be fabricated using deep UV lithography for example.
The obtained structures are verified using RSoft FullWAVE, and the corresponding spectral efficiencies are shown in Figure 5.11. The critical dimensions of GC2 are re-laxed to gmin = 60 nm and bmin = 115 nm with a coupling efficiency of –0.33 dB at 1550 nm, whereas GC3 with gmin = 110 nm and bmin = 115 nm still exhibits a high effi-ciency of –0.41 dB. The 1 dB bandwidth of all three structures is around 43 nm. Hence, it can be seen that relaxing the minimal lengths to values in the range of 100 nm does not degrade the performance dramatically, and the theoretically achievable coupling efficiency is still better than –0.5 dB. These dimensions are technologically manageable, and the resulting gratings can be fabricated using the aforementioned procedures. In this thesis, both structures GC2 and GC3 are realized based on the electron beam lithogra-phy, and the fabrication process is described in chapter 5.5.
5. Bragg diffraction gratings 98
1530 1540 1550 1560 1570
-1.0 -0.8 -0.6 -0.4 -0.2 0.0
GC3 GC2
Coupling efficiency [dB]
Wavelength [nm]
GC1
Figure 5.11: Simulated spectral efficiency of the three designed gratings with variable critical dimensions.
Finally, after optimizing the grating profile in the z-direction, the modal overlap has also to be maximized along the lateral direction. In fact, ηO is composed of two terms with ηO = ηO,z·ηO,y, where ηO,y is assumed to be 1 in 2D simulations. Thus, in order to obtain a high total efficiency using concrete 3D structures, the optimal grating width wGC has to be determined. This is done by simulating the fundamental waveguide mode profile for different widths and calculating the lateral overlap with the fiber mode. The result is illustrated in Figure 5.12 and shows that the grating width has to range between 11 µm and 16 µm in order to reach an overlap higher than 90%. Values outside this interval yield a modal mismatch and decrease the total coupling efficiency [117].
0 10 20 30
0.0 0.2 0.4 0.6 0.8 1.0
Lateral modal overlap ηO,y
Grating width [µm]
Figure 5.12: Calculated modal overlap along the lateral direction versus the grating width.
5. Bragg diffraction gratings 99