Table 1
Comparison between Direct and Enhanced Convergence: First-Order Efficiency for a Sinusoidal Grating in TE Polarization, h/d = 0.3
| Direct | Enhanced |
---|
|
|
|
---|
n | e1 | ∊ | e1 | ∊ |
---|
10 | 0.5034545 × 10−2 | 1.5 × 10−1 | 0.1735547 × 10−1 | 8.3 × 10−2 |
15 | 0.1073396 × 10−1 | 5.3 × 10−2 | 0.1204894 × 10−1 | 1.3 × 10−5 |
20 | 0.1045474 × 10−1 | 4.0 × 10−3 | 0.1157684 × 10−1 | 5.1 × 10−4 |
25 | 0.1140532 × 10−1 | 3.2 × 10−3 | 0.1163189 × 10−1 | 3.1 × 10−5 |
30 | 0.1161950 × 10−1 | 2.2 × 10−4 | 0.1163870 × 10−1 | −8.4 × 10−6 |
35 | 0.1165000 × 10−1 | 1.1 × 10−5 | 0.1163798 × 10−1 | −7.3 × 10−7 |
40 | 0.1162548 × 10−1 | −3.2 × 10−5 | 0.1163793 × 10−1 | −4.4 × 10−8 |
45 | 0.1163059 × 10−1 | 5.1 × 10−5 | 0.1163793 × 10−1 | 2.2 × 10−8 |
50 | 0.1164204 × 10−1 | 1.2 × 10−5 | 0.1163793 × 10−1 | 2.0 × 10−9 |
55 | 0.1164090 × 10−1 | −1.7 × 10−5 | 0.1163793 × 10−1 | −3.7 × 10−11 |
60 | 0.1163616 × 10−1 | −4.0 × 10−6 | 0.1163793 × 10−1 | −4.5 × 10−11 |
Table 2
Comparison between Direct and Enhanced Convergence: First-Order Efficiency for a Sinusoidal Grating in TE Polarization, h/d = 0.4
| Direct | Enhanced |
---|
|
|
|
---|
n | e1 | ∊ | e1 | ∊ |
---|
10 | 0.3897061 | −7.2 × 10−1 | 0.4018525 | −4.5 × 10−1 |
15 | 0.4713435 × 101 | −1.4 × 101 | 0.1649737 | −7.3 × 10−2 |
20 | 0.3635994 × 101 | −1.5 × 101 | 0.1568326 | 3.1 × 10−2 |
25 | 0.3848452 × 101 | −1.2 × 101 | 0.1687898 | 4.8 × 10−3 |
30 | 0.3357765 | −5.5 | 0.1698172 | 3.0 × 10−4 |
35 | 0.1276039 × 101 | −2.4 × 101 | 0.1701259 | −4.0 × 10−4 |
40 | 0.4143993 × 102 | −2.2 × 102 | 0.1700261 | −1.3 × 10−4 |
45 | 0.1665033 × 103 | −1.6 × 103 | 0.1699795 | 1.3 × 10−5 |
50 | 0.1406855 × 104 | −8.2 × 103 | 0.1699760 | 1.1 × 10−5 |
55 | 0.7075471 × 104 | −6.9 × 104 | 0.1699781 | 2.5 × 10−6 |
60 | 0.7847034 × 105 | −4.1 × 105 | 0.1699792 | −4.8 × 10−7 |
Table 3
Efficiencies for f(x) = h/2 cos(2πx/d) in TE Polarization with A = 9, σ = 0.13, and 60 Terms of the Taylor Expansion
h/d | e0 | e1 | e2 | ∊ |
---|
0.05 | 0.786 | 0.105 | 0.002 | −1.6 × 10−15 |
0.10 | 0.337 | 0.310 | 0.021 | −4.2 × 10−15 |
0.15 | 0.030 | 0.403 | 0.082 | −1.2 × 10−14 |
0.20 | 0.051 | 0.299 | 0.176 | −3.7 × 10−14 |
0.25 | 0.266 | 0.110 | 0.257 | −1.2 × 10−13 |
0.30 | 0.423 | 0.012 | 0.277 | −4.5 × 10−11 |
0.35 | 0.415 | 0.062 | 0.230 | −9.5 × 10−9 |
0.40 | 0.337 | 0.170 | 0.162 | −4.8 × 10−7 |
0.45 | 0.317 | 0.211 | 0.131 | −3.4 × 10−6 |
0.50 | 0.355 | 0.161 | 0.161 | 2.1 × 10−4 |
Table 4
Efficiencies for f(x) = h/2 cos(2πx/d) in TM Polarization with A = 9, σ = 0.13, and 60 Terms of the Taylor Expansion
h/d | e0 | e1 | e2 | ∊ |
---|
0.05 | 0.743 | 0.125 | 0.004 | 1.3 × 10−14 |
0.10 | 0.263 | 0.319 | 0.050 | 3.0 × 10−14 |
0.15 | 0.012 | 0.313 | 0.180 | 3.3 × 10−15 |
0.20 | 0.031 | 0.126 | 0.359 | −5.0 × 10−14 |
0.25 | 0.050 | 0.001 | 0.474 | 7.7 × 10−10 |
0.30 | 0.000 | 0.071 | 0.429 | 1.1 × 10−6 |
0.35 | 0.194 | 0.164 | 0.239 | 2.6 × 10−4 |
Table 5
Efficiencies for f(x) = h/2g(2πx/d) in TE Polarization with A = 9, σ = 0.20, F = 10, and 30 Terms of the Taylor Expansion
h/d | e0 | e1 | e2 | ∊ |
---|
0.05 | 0.856 | 0.071 | 0.001 | −3.8 × 10−15 |
0.10 | 0.515 | 0.230 | 0.012 | −1.2 × 10−14 |
0.15 | 0.180 | 0.358 | 0.052 | −2.1 × 10−11 |
0.20 | 0.012 | 0.366 | 0.128 | −1.1 × 10−8 |
0.25 | 0.029 | 0.261 | 0.224 | −6.6 × 10−7 |
0.30 | 0.136 | 0.125 | 0.307 | −9.9 × 10−6 |
0.35 | 0.230 | 0.043 | 0.342 | −5.6 × 10−5 |
0.40 | 0.287 | 0.041 | 0.315 | −3.4 × 10−4 |
0.45 | 0.363 | 0.078 | 0.242 | −3.7 × 10−3 |
Table 6
Efficiencies for f(x) = h/2g(2πx/d) in TM Polarization with A = 9, σ = 0.20, F = 10, and 40 Terms of the Taylor Expansion
h/d | e0 | e1 | e2 | ∊ |
---|
0.05 | 0.826 | 0.085 | 0.003 | 6.4 × 10−15 |
0.10 | 0.447 | 0.241 | 0.035 | 1.8 × 10−14 |
0.15 | 0.134 | 0.296 | 0.137 | −5.6 × 10−12 |
0.20 | 0.011 | 0.195 | 0.299 | −1.2 × 10−8 |
0.25 | 0.001 | 0.049 | 0.451 | −1.5 × 10−6 |
0.30 | 0.001 | 0.002 | 0.497 | 6.7 × 10−5 |
0.35 | 0.071 | 0.068 | 0.394 | 5.1 × 10−3 |
Table 7
Efficiencies in Quadruple Precision Arithmetic for f(x) = h/2 cos(2πx/d) in TE Polarization with A = 9, σ = 0.13, and 100 Terms of the Taylor Expansion
h/d | e0 | e1 | e2 | ∊ |
---|
0.05 | 0.786186 | 0.105346 | 0.001561 | −6.7 × 10−16 |
0.10 | 0.337470 | 0.310013 | 0.021252 | −6.7 × 10−16 |
0.15 | 0.029851 | 0.403106 | 0.081969 | −4.4 × 10−16 |
0.20 | 0.050728 | 0.298916 | 0.175721 | −2.2 × 10−16 |
0.25 | 0.266100 | 0.110307 | 0.256643 | 1.1 × 10−16 |
0.30 | 0.422850 | 0.011638 | 0.276937 | 1.2 × 10−15 |
0.35 | 0.415025 | 0.062114 | 0.230374 | 1.6 × 10−14 |
0.40 | 0.336781 | 0.169979 | 0.161630 | 3.2 × 10−12 |
0.45 | 0.316566 | 0.211003 | 0.130714 | −6.9 × 10−9 |
0.50 | 0.354656 | 0.161227 | 0.161446 | −1.8 × 10−6 |
0.55 | 0.355306 | 0.100680 | 0.221733 | −1.3 × 10−4 |
0.60 | 0.291421 | 0.100459 | 0.255925 | −4.2 × 10−3 |