Oscar P. Bruno and Michael C. Haslam, "Efficient high-order evaluation of scattering by periodic surfaces: deep gratings, high frequencies, and glancing incidences," J. Opt. Soc. Am. A 26, 658-668 (2009)

We present a superalgebraically convergent integral equation algorithm for evaluation of TE and TM electromagnetic scattering by smooth perfectly conducting periodic surfaces $z=f\left(x\right)$. For grating-diffraction problems in the resonance regime (heights and periods up to a few wavelengths) the proposed algorithm produces solutions with full double-precision accuracy in single-processor computing times of the order of a few seconds. The algorithm can also produce, in reasonable computing times, highly accurate solutions for very challenging problems, such as (a) a problem of diffraction by a grating for which the peak-to-trough distance equals 40 times its period that, in turn, equals 20 times the wavelength; and (b) a high-frequency problem with very small incidence, up to 0.01° from glancing. The algorithm is based on the concurrent use of Floquet and Chebyshev expansions together with certain integration weights that are computed accurately by means of an asymptotic expansion as the number of integration points tends to infinity.

V. I. Tatarskii J. Opt. Soc. Am. A 12(6) 1254-1260 (1995)

References

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Coefficients ${\alpha}_{2\mathcal{l}}$, $\mathcal{l}=0,\dots ,12$, Used in Expansion (45)

$\mathcal{l}$

${\alpha}_{2l}$

1

0

2

3

3

0

4

63

5

1320

6

49,203

7

2,653,560

8

196,707,423

9

19,194,804,720

10

2,385,684,870,723

11

367,985,503,366,800

12

68,980,888,889,771,103

Table 2

Parameters Defining Interpolants of the Function ${\nu}^{*}\left(z\right)$^{
a
}

${z}_{i}$

${z}_{i+1}$

${a}_{0}^{\left(i\right)}$

${a}_{1}^{\left(i\right)}$

${a}_{2}^{\left(i\right)}$

1

25

$1.9349\times {10}^{1}$

5.0731

$-3.4114\times {10}^{-2}$

25

200

$3.8393\times {10}^{1}$

3.5681

$-8.1905\times {10}^{-4}$

200

1000

$6.8173\times {10}^{1}$

3.2753

$-4.8158\times {10}^{-5}$

1000

2000

$9.4061\times {10}^{1}$

3.2126

$-1.0606\times {10}^{-5}$

2000

4000

$1.2388\times {10}^{2}$

3.1820

$-2.6515\times {10}^{-6}$

4000

6000

$1.5245\times {10}^{2}$

3.1684

$-9.9068\times {10}^{-7}$

6000

8000

$1.8265\times {10}^{2}$

3.1591

$-2.7473\times {10}^{-7}$

For $z>8000$ we use the linear relationship ${\nu}^{*}=210+3.1535z$, which leads to slight overestimates of the truncation index $Q=\left[{\nu}^{*}\left(z\right)\right]+1$.

Table 3

Results Provided by Our Method for the TE Test Case Considered in Table 3 of [16]^{
a
}

N

M

$\theta =\pi \u22152$

$\theta =\pi \u22156$

$\mid 1-E\mid $

Error

$\mid 1-E\mid $

Error

4

${2}^{6}$

$1.26\times {10}^{-4}$

$9.56\times {10}^{-4}$

$1.67\times {10}^{-4}$

$6.66\times {10}^{-4}$

6

${2}^{6}$

$1.26\times {10}^{-7}$

$3.14\times {10}^{-7}$

$1.39\times {10}^{-7}$

$1.03\times {10}^{-7}$

8

${2}^{6}$

$8.59\times {10}^{-10}$

$1.45\times {10}^{-10}$

$1.43\times {10}^{-10}$

$1.78\times {10}^{-10}$

10

${2}^{6}$

$1.32\times {10}^{-13}$

$1.28\times {10}^{-13}$

$2.11\times {10}^{-13}$

$1.36\times {10}^{-13}$

Our results for the TM case exhibit nearly identical behavior. Our algorithm’s parameters are $k=1.25\u22150.546$, $d=3$, $a=0.7$, ${M}_{v}=129$ for all the cases considered in this table. The reference solution was computed with the same parameters listed in each row of the table and ${N}^{*}=30$. The execution times required by our code for these tests are very small; for the case $N=10$, for example, the execution time was ${t}_{\mathrm{exe}}=0.43\phantom{\rule{0.3em}{0ex}}\mathrm{s}$.

Table 4

Code Parameters, Computation Times, and Resulting Accuracies for Various Problems in the Classical Resonance Regime, TE Problem (TM Results Are Nearly Identical)^{
a
}

$a\u2215P$

$P\u2215\lambda =1.0$

$P\u2215\lambda =2.0$

$P\u2215\lambda =4.0$

N

M

Error

${t}_{\mathit{exe}}$

N

M

Error

${t}_{\mathit{exe}}$

N

M

Error

${t}_{\mathit{exe}}$

0.5

15

${2}^{7}$

$1.3\times {10}^{-14}$

0.67

20

${2}^{7}$

$1.0\times {10}^{-14}$

0.74

30

${2}^{7}$

$1.0\times {10}^{-14}$

1.02

1.0

25

${2}^{7}$

$3.9\times {10}^{-12}$

0.78

25

${2}^{8}$

$2.2\times {10}^{-13}$

1.32

45

${2}^{8}$

$1.8\times {10}^{-14}$

2.30

1.5

35

${2}^{8}$

$4.4\times {10}^{-14}$

1.82

35

${2}^{8}$

$3.0\times {10}^{-14}$

1.64

55

${2}^{8}$

$2.1\times {10}^{-12}$

2.45

2.0

65

${2}^{8}$

$1.0\times {10}^{-14}$

3.21

65

${2}^{8}$

$1.0\times {10}^{-14}$

2.84

70

${2}^{9}$

$5.2\times {10}^{-13}$

5.96

The code execution time ${t}_{\mathit{exe}}$ is measured in seconds. The reference solution was computed with the same parameters listed in each row of the table and ${N}^{*}=80$. In all examples presented here, the value ${M}_{v}=129$ was used for the evaluation of the Green’s function; see Appendix A.

Table 5

Code Performance for Various Surface Heights a With Parameters and Resulting Accuracy and Energy Balance for Both Cases of TE and TM Scattering from Surface Profile $f\left(x\right)=a\phantom{\rule{0.2em}{0ex}}\mathrm{cos}(2\pi x\u2215P)$ with $\lambda =0.05P$ and $\theta =7\pi \u221518$^{
a
}

$a\u2215P$

N

M

${N}^{*}$

Error (TE)

$\mid 1-E\mid $ (TE)

Error (TM)

$\mid 1-E\mid $ (TM)

${t}_{\mathit{exe}}$

4

550

${2}^{11}$

700

$7.35\times {10}^{-13}$

$5.73\times {10}^{-12}$

$7.58\times {10}^{-13}$

$1.09\times {10}^{-12}$

10

8

1050

${2}^{12}$

1200

$6.14\times {10}^{-12}$

$1.48\times {10}^{-11}$

$7.52\times {10}^{-11}$

$5.19\times {10}^{-11}$

64

12

1600

${2}^{13}$

1700

$1.49\times {10}^{-11}$

$4.00\times {10}^{-12}$

$3.75\times {10}^{-11}$

$8.28\times {10}^{-11}$

236

16

2100

${2}^{13}$

2200

$3.59\times {10}^{-11}$

$8.72\times {10}^{-11}$

$5.74\times {10}^{-11}$

$2.83\times {10}^{-11}$

392

20

2600

${2}^{13}$

2700

$9.80\times {10}^{-11}$

$3.51\times {10}^{-11}$

$2.05\times {10}^{-10}$

$1.55\times {10}^{-10}$

570

In each case there are 40 propagating modes. The code execution time ${t}_{\mathit{exe}}$ is measured in minutes. In all examples presented here we use ${M}_{v}=1025$ to compute the Green’s function; see Appendix A.

Table 6

Code Performance for Grazing Incidence Problem: Incidence Angles θ (Measured in Degrees from Horizontal) with Parameters and Resulting Accuracies for Both Cases of TE and TM Scattering from Surface Profile $f\left(x\right)=a\phantom{\rule{0.2em}{0ex}}\mathrm{cos}(2\pi x\u2215P)$ with $a=P$ and $\lambda =0.01P$^{
a
}

θ (deg)

N

M

${M}_{v}$

Error (TE)

$\mid 1-E\mid $ (TE)

Error (TM)

$\mid 1-E\mid $ (TM)

${t}_{\mathit{exe}}$

1.00

800

${2}^{12}$

${2}^{10}+1$

$3.00\times {10}^{-14}$

$2.84\times {10}^{-14}$

$2.61\times {10}^{-14}$

$2.38\times {10}^{-14}$

50

0.10

800

${2}^{12}$

${2}^{10}+1$

$3.09\times {10}^{-15}$

$2.32\times {10}^{-13}$

$4.80\times {10}^{-14}$

$2.35\times {10}^{-13}$

50

0.01

800

${2}^{12}$

${2}^{12}+1$

$2.47\times {10}^{-13}$

$7.58\times {10}^{-14}$

$1.21\times {10}^{-12}$

$1.45\times {10}^{-12}$

72

In each case there are 200 propagating modes. The code execution time ${t}_{\mathit{exe}}$ is measured in minutes. The reference solution was computed with the same parameters listed in each row of the table and ${N}^{*}=900$. For cases $\theta =1.00\xb0$ and 0.10° we use ${M}_{v}=1025$, while for $\theta =0.01\xb0$ we use ${M}_{v}=4097$; see Appendix A.

Table 7

Code Performance for Various Incidence Wavelengths λ with Parameters and Resulting Accuracies for Both Cases of TE and TM Scattering from Surface Profile $f\left(x\right)=a\phantom{\rule{0.2em}{0ex}}\mathrm{cos}(2\pi x\u2215P)$ with $a=P$ and $\theta =\pi \u22154$^{
a
}

$\lambda \u2215P$

N

M

${N}^{*}$

Error (TE)

$\mid 1-E\mid $ (TE)

Error (TM)

$\mid 1-E\mid $ (TM)

${t}_{\mathrm{exe}}$

0.100

100

${2}^{9}$

200

$1.40\times {10}^{-14}$

$2.22\times {10}^{-16}$

$1.30\times {10}^{-14}$

$1.77\times {10}^{-14}$

0.25

0.050

200

${2}^{10}$

300

$2.40\times {10}^{-14}$

$1.95\times {10}^{-14}$

$2.32\times {10}^{-14}$

$5.22\times {10}^{-14}$

1.3

0.010

750

${2}^{12}$

900

$2.90\times {10}^{-14}$

$7.69\times {10}^{-14}$

$4.85\times {10}^{-14}$

$1.60\times {10}^{-13}$

45

0.005

1500

${2}^{13}$

1600

$6.99\times {10}^{-14}$

$1.43\times {10}^{-13}$

$7.59\times {10}^{-14}$

$5.27\times {10}^{-13}$

221

The number of propagating modes in the above examples are 20, 40, 200, and 400. This is a challenging problem in the resonance (and approaching high frequency) regime. The code execution time ${t}_{\mathrm{exe}}$ is measured in minutes. In all examples presented here we use ${M}_{v}=1025$ to compute the Green’s function; see Appendix A.

Table 8

Convergence Results for a Problem of Scattering by the Composite Surface Depicted in Fig. 2^{
a
}

N

M

Error (TE)

$\mid 1-E\mid $ (TE)

Error (TM)

$\mid 1-E\mid $ (TM)

${t}_{\mathrm{exe}}$

150

${2}^{10}$

$2.14\times {10}^{-4}$

$1.03\times {10}^{-3}$

$1.11\times {10}^{-1}$

$8.67\times {10}^{-1}$

62

175

${2}^{10}$

$3.13\times {10}^{-5}$

$4.94\times {10}^{-5}$

$1.90\times {10}^{-4}$

$2.80\times {10}^{-5}$

74

200

${2}^{10}$

$2.67\times {10}^{-9}$

$3.35\times {10}^{-9}$

$7.13\times {10}^{-9}$

$1.66\times {10}^{-8}$

85

225

${2}^{10}$

$1.30\times {10}^{-14}$

$2.89\times {10}^{-14}$

$2.06\times {10}^{-14}$

$1.42\times {10}^{-14}$

98

Incidence data $\lambda =0.05P$, $\theta =\pi \u22154$; 40 propagating modes. The code execution time ${t}_{\mathit{exe}}$ is measured in seconds. The reference solution was computed using ${N}^{*}=400$. In all examples presented in this table the value ${M}_{v}=257$ was used for the evaluation of the periodic Green’s function; see Appendix A.

Table 9

Comparison of the Spatial Collocation and Spectral Testing Methods, in a TM Problem with Scattering Surface Defined in Eq. (49) with $a=P$, $\lambda =0.1P$, $\theta =\pi \u22154$ (TE Results Are Nearly Identical)^{
a
}

N

Spatial Collocation

Spectral Testing

$\mid 1-E\mid $

Error

$\mid 1-E\mid $

Error

60

$1.67\times {10}^{-1}$

$6.66\times {10}^{-2}$

$1.26\times {10}^{-2}$

$9.56\times {10}^{-3}$

65

$1.39\times {10}^{-2}$

$1.03\times {10}^{-2}$

$1.26\times {10}^{-4}$

$3.14\times {10}^{-3}$

70

$1.43\times {10}^{-4}$

$1.78\times {10}^{-4}$

$8.59\times {10}^{-5}$

$1.45\times {10}^{-4}$

75

$1.80\times {10}^{-6}$

$1.36\times {10}^{-5}$

$1.32\times {10}^{-6}$

$1.28\times {10}^{-6}$

80

$1.65\times {10}^{-8}$

$6.47\times {10}^{-8}$

$2.55\times {10}^{-9}$

$2.35\times {10}^{-9}$

85

$7.18\times {10}^{-11}$

$1.31\times {10}^{-10}$

$9.10\times {10}^{-13}$

$6.75\times {10}^{-13}$

90

$4.00\times {10}^{-14}$

$3.59\times {10}^{-14}$

$2.90\times {10}^{-14}$

$1.60\times {10}^{-14}$

95

$4.00\times {10}^{-14}$

$1.70\times {10}^{-14}$

$1.90\times {10}^{-14}$

$1.20\times {10}^{-14}$

Parameters: $M=513$, ${M}_{v}=257$. The reference solution in each case was obtained by using ${N}^{*}=100$. Similar performance improvements were observed in a wide range of cases, and no cases were found in which spatial collocation resulted in better performance than its spectral counterpart.

Tables (9)

Table 1

Coefficients ${\alpha}_{2\mathcal{l}}$, $\mathcal{l}=0,\dots ,12$, Used in Expansion (45)

$\mathcal{l}$

${\alpha}_{2l}$

1

0

2

3

3

0

4

63

5

1320

6

49,203

7

2,653,560

8

196,707,423

9

19,194,804,720

10

2,385,684,870,723

11

367,985,503,366,800

12

68,980,888,889,771,103

Table 2

Parameters Defining Interpolants of the Function ${\nu}^{*}\left(z\right)$^{
a
}

${z}_{i}$

${z}_{i+1}$

${a}_{0}^{\left(i\right)}$

${a}_{1}^{\left(i\right)}$

${a}_{2}^{\left(i\right)}$

1

25

$1.9349\times {10}^{1}$

5.0731

$-3.4114\times {10}^{-2}$

25

200

$3.8393\times {10}^{1}$

3.5681

$-8.1905\times {10}^{-4}$

200

1000

$6.8173\times {10}^{1}$

3.2753

$-4.8158\times {10}^{-5}$

1000

2000

$9.4061\times {10}^{1}$

3.2126

$-1.0606\times {10}^{-5}$

2000

4000

$1.2388\times {10}^{2}$

3.1820

$-2.6515\times {10}^{-6}$

4000

6000

$1.5245\times {10}^{2}$

3.1684

$-9.9068\times {10}^{-7}$

6000

8000

$1.8265\times {10}^{2}$

3.1591

$-2.7473\times {10}^{-7}$

For $z>8000$ we use the linear relationship ${\nu}^{*}=210+3.1535z$, which leads to slight overestimates of the truncation index $Q=\left[{\nu}^{*}\left(z\right)\right]+1$.

Table 3

Results Provided by Our Method for the TE Test Case Considered in Table 3 of [16]^{
a
}

N

M

$\theta =\pi \u22152$

$\theta =\pi \u22156$

$\mid 1-E\mid $

Error

$\mid 1-E\mid $

Error

4

${2}^{6}$

$1.26\times {10}^{-4}$

$9.56\times {10}^{-4}$

$1.67\times {10}^{-4}$

$6.66\times {10}^{-4}$

6

${2}^{6}$

$1.26\times {10}^{-7}$

$3.14\times {10}^{-7}$

$1.39\times {10}^{-7}$

$1.03\times {10}^{-7}$

8

${2}^{6}$

$8.59\times {10}^{-10}$

$1.45\times {10}^{-10}$

$1.43\times {10}^{-10}$

$1.78\times {10}^{-10}$

10

${2}^{6}$

$1.32\times {10}^{-13}$

$1.28\times {10}^{-13}$

$2.11\times {10}^{-13}$

$1.36\times {10}^{-13}$

Our results for the TM case exhibit nearly identical behavior. Our algorithm’s parameters are $k=1.25\u22150.546$, $d=3$, $a=0.7$, ${M}_{v}=129$ for all the cases considered in this table. The reference solution was computed with the same parameters listed in each row of the table and ${N}^{*}=30$. The execution times required by our code for these tests are very small; for the case $N=10$, for example, the execution time was ${t}_{\mathrm{exe}}=0.43\phantom{\rule{0.3em}{0ex}}\mathrm{s}$.

Table 4

Code Parameters, Computation Times, and Resulting Accuracies for Various Problems in the Classical Resonance Regime, TE Problem (TM Results Are Nearly Identical)^{
a
}

$a\u2215P$

$P\u2215\lambda =1.0$

$P\u2215\lambda =2.0$

$P\u2215\lambda =4.0$

N

M

Error

${t}_{\mathit{exe}}$

N

M

Error

${t}_{\mathit{exe}}$

N

M

Error

${t}_{\mathit{exe}}$

0.5

15

${2}^{7}$

$1.3\times {10}^{-14}$

0.67

20

${2}^{7}$

$1.0\times {10}^{-14}$

0.74

30

${2}^{7}$

$1.0\times {10}^{-14}$

1.02

1.0

25

${2}^{7}$

$3.9\times {10}^{-12}$

0.78

25

${2}^{8}$

$2.2\times {10}^{-13}$

1.32

45

${2}^{8}$

$1.8\times {10}^{-14}$

2.30

1.5

35

${2}^{8}$

$4.4\times {10}^{-14}$

1.82

35

${2}^{8}$

$3.0\times {10}^{-14}$

1.64

55

${2}^{8}$

$2.1\times {10}^{-12}$

2.45

2.0

65

${2}^{8}$

$1.0\times {10}^{-14}$

3.21

65

${2}^{8}$

$1.0\times {10}^{-14}$

2.84

70

${2}^{9}$

$5.2\times {10}^{-13}$

5.96

The code execution time ${t}_{\mathit{exe}}$ is measured in seconds. The reference solution was computed with the same parameters listed in each row of the table and ${N}^{*}=80$. In all examples presented here, the value ${M}_{v}=129$ was used for the evaluation of the Green’s function; see Appendix A.

Table 5

Code Performance for Various Surface Heights a With Parameters and Resulting Accuracy and Energy Balance for Both Cases of TE and TM Scattering from Surface Profile $f\left(x\right)=a\phantom{\rule{0.2em}{0ex}}\mathrm{cos}(2\pi x\u2215P)$ with $\lambda =0.05P$ and $\theta =7\pi \u221518$^{
a
}

$a\u2215P$

N

M

${N}^{*}$

Error (TE)

$\mid 1-E\mid $ (TE)

Error (TM)

$\mid 1-E\mid $ (TM)

${t}_{\mathit{exe}}$

4

550

${2}^{11}$

700

$7.35\times {10}^{-13}$

$5.73\times {10}^{-12}$

$7.58\times {10}^{-13}$

$1.09\times {10}^{-12}$

10

8

1050

${2}^{12}$

1200

$6.14\times {10}^{-12}$

$1.48\times {10}^{-11}$

$7.52\times {10}^{-11}$

$5.19\times {10}^{-11}$

64

12

1600

${2}^{13}$

1700

$1.49\times {10}^{-11}$

$4.00\times {10}^{-12}$

$3.75\times {10}^{-11}$

$8.28\times {10}^{-11}$

236

16

2100

${2}^{13}$

2200

$3.59\times {10}^{-11}$

$8.72\times {10}^{-11}$

$5.74\times {10}^{-11}$

$2.83\times {10}^{-11}$

392

20

2600

${2}^{13}$

2700

$9.80\times {10}^{-11}$

$3.51\times {10}^{-11}$

$2.05\times {10}^{-10}$

$1.55\times {10}^{-10}$

570

In each case there are 40 propagating modes. The code execution time ${t}_{\mathit{exe}}$ is measured in minutes. In all examples presented here we use ${M}_{v}=1025$ to compute the Green’s function; see Appendix A.

Table 6

Code Performance for Grazing Incidence Problem: Incidence Angles θ (Measured in Degrees from Horizontal) with Parameters and Resulting Accuracies for Both Cases of TE and TM Scattering from Surface Profile $f\left(x\right)=a\phantom{\rule{0.2em}{0ex}}\mathrm{cos}(2\pi x\u2215P)$ with $a=P$ and $\lambda =0.01P$^{
a
}

θ (deg)

N

M

${M}_{v}$

Error (TE)

$\mid 1-E\mid $ (TE)

Error (TM)

$\mid 1-E\mid $ (TM)

${t}_{\mathit{exe}}$

1.00

800

${2}^{12}$

${2}^{10}+1$

$3.00\times {10}^{-14}$

$2.84\times {10}^{-14}$

$2.61\times {10}^{-14}$

$2.38\times {10}^{-14}$

50

0.10

800

${2}^{12}$

${2}^{10}+1$

$3.09\times {10}^{-15}$

$2.32\times {10}^{-13}$

$4.80\times {10}^{-14}$

$2.35\times {10}^{-13}$

50

0.01

800

${2}^{12}$

${2}^{12}+1$

$2.47\times {10}^{-13}$

$7.58\times {10}^{-14}$

$1.21\times {10}^{-12}$

$1.45\times {10}^{-12}$

72

In each case there are 200 propagating modes. The code execution time ${t}_{\mathit{exe}}$ is measured in minutes. The reference solution was computed with the same parameters listed in each row of the table and ${N}^{*}=900$. For cases $\theta =1.00\xb0$ and 0.10° we use ${M}_{v}=1025$, while for $\theta =0.01\xb0$ we use ${M}_{v}=4097$; see Appendix A.

Table 7

Code Performance for Various Incidence Wavelengths λ with Parameters and Resulting Accuracies for Both Cases of TE and TM Scattering from Surface Profile $f\left(x\right)=a\phantom{\rule{0.2em}{0ex}}\mathrm{cos}(2\pi x\u2215P)$ with $a=P$ and $\theta =\pi \u22154$^{
a
}

$\lambda \u2215P$

N

M

${N}^{*}$

Error (TE)

$\mid 1-E\mid $ (TE)

Error (TM)

$\mid 1-E\mid $ (TM)

${t}_{\mathrm{exe}}$

0.100

100

${2}^{9}$

200

$1.40\times {10}^{-14}$

$2.22\times {10}^{-16}$

$1.30\times {10}^{-14}$

$1.77\times {10}^{-14}$

0.25

0.050

200

${2}^{10}$

300

$2.40\times {10}^{-14}$

$1.95\times {10}^{-14}$

$2.32\times {10}^{-14}$

$5.22\times {10}^{-14}$

1.3

0.010

750

${2}^{12}$

900

$2.90\times {10}^{-14}$

$7.69\times {10}^{-14}$

$4.85\times {10}^{-14}$

$1.60\times {10}^{-13}$

45

0.005

1500

${2}^{13}$

1600

$6.99\times {10}^{-14}$

$1.43\times {10}^{-13}$

$7.59\times {10}^{-14}$

$5.27\times {10}^{-13}$

221

The number of propagating modes in the above examples are 20, 40, 200, and 400. This is a challenging problem in the resonance (and approaching high frequency) regime. The code execution time ${t}_{\mathrm{exe}}$ is measured in minutes. In all examples presented here we use ${M}_{v}=1025$ to compute the Green’s function; see Appendix A.

Table 8

Convergence Results for a Problem of Scattering by the Composite Surface Depicted in Fig. 2^{
a
}

N

M

Error (TE)

$\mid 1-E\mid $ (TE)

Error (TM)

$\mid 1-E\mid $ (TM)

${t}_{\mathrm{exe}}$

150

${2}^{10}$

$2.14\times {10}^{-4}$

$1.03\times {10}^{-3}$

$1.11\times {10}^{-1}$

$8.67\times {10}^{-1}$

62

175

${2}^{10}$

$3.13\times {10}^{-5}$

$4.94\times {10}^{-5}$

$1.90\times {10}^{-4}$

$2.80\times {10}^{-5}$

74

200

${2}^{10}$

$2.67\times {10}^{-9}$

$3.35\times {10}^{-9}$

$7.13\times {10}^{-9}$

$1.66\times {10}^{-8}$

85

225

${2}^{10}$

$1.30\times {10}^{-14}$

$2.89\times {10}^{-14}$

$2.06\times {10}^{-14}$

$1.42\times {10}^{-14}$

98

Incidence data $\lambda =0.05P$, $\theta =\pi \u22154$; 40 propagating modes. The code execution time ${t}_{\mathit{exe}}$ is measured in seconds. The reference solution was computed using ${N}^{*}=400$. In all examples presented in this table the value ${M}_{v}=257$ was used for the evaluation of the periodic Green’s function; see Appendix A.

Table 9

Comparison of the Spatial Collocation and Spectral Testing Methods, in a TM Problem with Scattering Surface Defined in Eq. (49) with $a=P$, $\lambda =0.1P$, $\theta =\pi \u22154$ (TE Results Are Nearly Identical)^{
a
}

N

Spatial Collocation

Spectral Testing

$\mid 1-E\mid $

Error

$\mid 1-E\mid $

Error

60

$1.67\times {10}^{-1}$

$6.66\times {10}^{-2}$

$1.26\times {10}^{-2}$

$9.56\times {10}^{-3}$

65

$1.39\times {10}^{-2}$

$1.03\times {10}^{-2}$

$1.26\times {10}^{-4}$

$3.14\times {10}^{-3}$

70

$1.43\times {10}^{-4}$

$1.78\times {10}^{-4}$

$8.59\times {10}^{-5}$

$1.45\times {10}^{-4}$

75

$1.80\times {10}^{-6}$

$1.36\times {10}^{-5}$

$1.32\times {10}^{-6}$

$1.28\times {10}^{-6}$

80

$1.65\times {10}^{-8}$

$6.47\times {10}^{-8}$

$2.55\times {10}^{-9}$

$2.35\times {10}^{-9}$

85

$7.18\times {10}^{-11}$

$1.31\times {10}^{-10}$

$9.10\times {10}^{-13}$

$6.75\times {10}^{-13}$

90

$4.00\times {10}^{-14}$

$3.59\times {10}^{-14}$

$2.90\times {10}^{-14}$

$1.60\times {10}^{-14}$

95

$4.00\times {10}^{-14}$

$1.70\times {10}^{-14}$

$1.90\times {10}^{-14}$

$1.20\times {10}^{-14}$

Parameters: $M=513$, ${M}_{v}=257$. The reference solution in each case was obtained by using ${N}^{*}=100$. Similar performance improvements were observed in a wide range of cases, and no cases were found in which spatial collocation resulted in better performance than its spectral counterpart.