An intermediate approximation, in which the modelled area is made closest to the actual area of the geometry was also considered and is illustrated in figure 4.8c. In this method, boundary cells are included in the geometry selectively according to the distance of the cell-centre to the actual boundary of the
0.1 1 10 100 1000 10000 100000 1e+06 1e+07
100 1000 10000 100000 1e+06 1e+07 1e+08 1e+09
Q Error %
Theoretical Q factor WGM (x,1)
WGM (x,2) WGM (x,3)
Figure 4.11: The percentage error of resonant Q factor of the E wave modes supported by an infinite cylinder (n = 2.82, r = 1.35µm) mod- elled by CST Microwave Studio with a multi mesh with space step size varying between 0.04µm and 0.004µm. 1st radial order modes are deonted by WGM(x,1), 2nd order modes by WGM(x,2) and 3rd order modes by WGM(x,3).
geometry until the modelled geometry fulfils the aforementioned criterion. For a circle of radius r at r0 = (x0, y0, z0), the distance to a cell-centre at
rc = (xc, yc, zc) from the boundary will be |rc− r0| − r. A downside which
soon becomes obvious is that geometries that are meant to be symmetrical might become assymetric due to the non uniform operation of selecting some boundary cells. Further it is unsuitable for situations illustrated by figure 4.1b, where it might produce a staggered edge. However, this approximation is now persued for circular geometries. The spectrum of the total energy inside the infinite cylinder defined previously is presented in figure 4.12 for three step sizes 0.05µm, 0.025µm and 0.0125µm using the same area technique with a computational domain of 10µm × 10µm truncated by TLM matched boundaries.
0.01 0.1 1 160 180 200 220 240 Magnitude (normalised) Frequency(THz) Dl = 0.05 DL = 0.025 DL = 0.0125 data
Figure 4.12: The energy spectrum of an infinite cylinder excited by a line source inside the resonator of E type waves for same area discreti- sation along with the analytical result for step sizes of 0.05µm, 0.025µm, and 0.0125µm.
the same area results tend to converge to the analytical resonant frequencies. The amplitude of the baseline seems to be slightly higher for λ/40 (0.0125µm) step size indicating that the power loss is generally smaller. There is a signifi- cant error in resonant frequency predicted in λ/10 (0.05µm) step size towards the higher frequencies. This effect can be related to mesh dispersion, previ- ously observed even when using inside and outside discretisations at the same step size. However, the curves seem to converge towards the analytical res- onant frequencies when the step size is decreased. Therefore, the same area technique accurately represents the domain so that the resonant frequencies converges with decreasing step size.
Now that each resonant frequency is resonably accurately modelled as ob- served from the energy spectrum, the complex resonances (resonant frequency and Q factor) are extracted using the modified difference Prony method. The
complex resonances are extracted from all the time signals of duration of 4000f s at each measurable space point within the resonator, individually. The median of the complex resonances over the resonator is then calculated for each resonance and are presented in table 4.5. The error percentages in resonant frequency and Q factor are given in figures4.13and4.14respectively. Table 4.5: Resonant frequencies and Q factors extracted from time signals of duration of 4000fs for an infinite cylinder (n = 2.82, r = 1.35µm) modelled with same area technique for step sizes of 0.05µm, 0.025µm, 0.0125µm.
∆l = 50nm ∆l = 25nm ∆l = 12.5nm
WGM Res. Q factor Res. Q factor Res. Q factor
order Freq. Freq. Freq.
(THz) (THz) (THz) WGM(6,2) 150.954 156 151.815 153 151.908 156 WGM(9,1) 151.048 3572 151.696 24641 151.843 53497 WGM(10,1) 165.004 1381 165.834 38395 165.987 146327 WGM(7,2) 166.668 342 167.607 331 167.807 326 WGM(11,1) 178.799 2092 179.868 17549 180.122 402087 WGM(8,2) 181.804 835 183.136 764 183.500 751 WGM(12,1) 192.341 3575 193.764 25952 194.126 646188 WGM(9,2) 197.094 2109 198.571 2157 198.922 2135 WGM(13,1) 205.783 21613 207.610 153406 208.004 129674 WGM(7,3) 207.495 106 209.291 98 209.444 101.409 WGM(10,2) 211.873 2977 213.728 6407 214.194 5247 WGM(14,1) 218.660 16066 221.319 363019 221.850 107892 WGM(8,3) 223.039 185 225.406 173 WGM(11,2) 226.366 3073 228.629 10588 229.168 16314 WGM(15,1) 232.660 7124 235.019 239323 235.601 965641 WGM(9,3) 238.761 351 241.458 301 242.104 291 WGM(12,2) 240.400 832 243.372 5182 244.07 43075 WGM(16,1) 246.249 5435 248.767 40255 249.312 303433
0.01 0.1 1 10 150 160 170 180 190 200 210 220 230 240 250 Frequency Error % Frequency(THz) WGM (x,1) DL=0.05um WGM (x,2) DL=0.05um WGM (x,3) DL=0.05um WGM (x,1) DL=0.025um WGM (x,2) DL=0.025um WGM (x,3) DL=0.025um WGM (x,1) DL=0.0125um WGM (x,2) DL=0.0125um WGM (x,3) DL=0.0125um
Figure 4.13: The percentage error of resonant frequencies of the an infinite cylinder (n = 2.82, r = 1.35µm) modelled by same area method with space steps of 0.05µm, 0.025µm and 0.0125µm.
for a space step of 0.0125µm. The resonant frequency has a clear dependency on the space step and the error is decreased almost in proportion to the space step. The error in E waves observed using the finest mesh (0.0125µm) has some resonances (at ≈ 165T Hz and 210T Hz) with slightly deviated error values from the general trend, which may have resulted from the extraction method’s error in resolving close resonances.
The error in Q factor increases with the analytical Q factor over around 1000 according to figure 4.14. The error decreases by almost one order of magnitude when the space step is halved. The error is less than 10% for Q’s of 2000 for 0.05µm, 8000 for 0.025µm and 60000 for 0.0125µm step sizes. The Q factors tend to decrease with the surface roughness [11] with practical resonators. In this instance, the stair step approximation presented by the mesh is the source of an effective surface roughness in the structure simulated. However, the errors obtained for the resonant frequency and Q factor are
0.1 1 10 100 1000 10000 100000 1e+06 1e+07 1e+08
10 100 1000 10000 100000 1e+06 1e+07 1e+08 1e+09
Q Error % Theoretical Q factor WGM (x,1) DL=0.05um WGM (x,2) DL=0.05um WGM (x,3) DL=0.05um WGM (x,1) DL=0.025um WGM (x,2) DL=0.025um WGM (x,3) DL=0.025um WGM (x,1) DL=0.0125um WGM (x,2) DL=0.0125um WGM (x,3) DL=0.0125um
Figure 4.14: The percentage error of resonant Q factors of the an infinite cylinder (n = 2.82, r = 1.35µm) modelled by same area method with space steps of 0.05µm, 0.025µm and 0.0125µm.
higher than those observed using multi-mesh technique. Yet, the results are more orderly, and can be predicted to produce slightly better results than the multi-mesh approach when the space step is halved further (λmin/80). But,
the computation time complexity would in this case be 8 times that of the finest mesh used in this study.
In conclusion so far, the same area method provides an improvement over the inside and outside discretisations considering the resonant frequency. The simulated resonant frequency converge to the analytical values as the step size decreases. However, the Q factors are underestimated due to surface roughness still present in the same area discretisation technique. Hence, other discretisation techniques are considered next to solve the factor of the surface roughness and its detrimental effects on simulated Q factor.