Kelsey Ulmer, Junshan Lin, and David P. Nicholls, "Monte Carlo–transformed field expansion method for simulating electromagnetic wave scattering by multilayered random media," J. Opt. Soc. Am. A 39, 1513-1523 (2022)
We present an efficient numerical method for simulating the scattering
of electromagnetic fields by a multilayered medium with random
interfaces. The elements of this algorithm, the Monte
Carlo–transformed field expansion method, are (i) an interfacial
problem formulation in terms of impedance-impedance operators, (ii)
simulation by a high-order perturbation of surfaces approach (the
transformed field expansions method), and (iii) efficient computation
of the wave field for each random sample by forward and backward
substitutions. Our perturbative formulation permits us to solve a
sequence of linear problems featuring an operator that is deterministic, and its LU decomposition matrices
can be reused, leading to significant savings in computational effort.
With an extensive set of numerical examples, we demonstrate not only
the robust and high-order accuracy of our scheme for small to moderate
interface deformations, but also how Padé summation can be used to
address large deviations.
No data were generated or analyzed in the presented research.
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Absolute Value of the Mean of the Energy Defect as the Lateral () and Vertical () Discretizations Were
Refineda
1.0501e-03
1.0814e-03
1.0802e-03
1.0827e-03
3.8155e-05
1.1569e-06
1.1161e-06
1.1277e-06
3.9156e-05
5.3976e-10
5.5434e-10
5.5156e-10
3.9349e-05
1.1695e-11
2.3820e-13
2.4384e-13
Correlation length, perturbation size, and number of Taylor
orders fixed at ${l_c} =
1$, $\varepsilon
= 0.1$, and $N =
20$, respectively.
Table 2.
Standard Deviation of the Energy Defect as the Lateral () and Vertical () Discretizations Were
Refineda
1.8407e-06
1.8379e-06
1.8221e-06
1.8365e-06
1.4026e-08
8.6896e-10
8.6678e-10
8.6775e-10
1.2821e-08
1.2812e-08
1.2818e-08
Correlation length, perturbation size, and number of Taylor
orders fixed at ${l_c} =
1$, $\varepsilon
= 0.1$, and $N =
20$, respectively.
Table 3.
Absolute Value of the Mean of the Energy Defect as the Number of Taylor
Orders Was Varied for Assorted
Values of the Perturbation Size a
0.01
2.5860e-07
8.9270e-11
5.8876e-12
5.8876e-12
0.05
1.8107e-04
1.8469e-06
4.0214e-09
3.9343e-09
0.1
2.8511e-03
1.0254e-04
1.3407e-07
6.2869e-08
0.2
4.6765e-02
3.2213e-03
2.5526e-05
4.6869e-06
Correlation length and lateral/vertical discretization
fixed at ${l_c} =
1$ and ${N_x} =
{N_z}={ 2^5}$, respectively.
Table 4.
Standard Deviation of the Energy Defect as the Number of Taylor
Orders Was Varied for Assorted
Values of the Perturbation Size a
0.01
1.5934e-11
0.05
2.8624e-07
1.0078e-10
1.4301e-16
1.3036e-16
0.1
2.9170e-05
1.2568e-07
2.7749e-12
1.3858e-14
0.2
5.1089e-03
1.1718e-04
3.9580e-07
7.0365e-08
Correlation length and lateral/vertical discretization
fixed at ${l_c} =
1$ and ${N_x} =
{N_z}={ 2^5}$, respectively.
Table 5.
Absolute Value of the Mean of the Energy Defect as the Number of Taylor
Orders Was Varied for Assorted
Values of the Perturbation Size Calculated via Taylor
Summationa
0.1
7.8669e-08
9.5712e-09
9.6114e-09
9.6099e-09
0.2
2.4983e-05
5.4705e-06
1.3548e-08
1.4908e-07
0.4
4.0993e-01
1.6469e03
2.3238e11
1.5042e28
Correlation length and lateral/vertical discretization
fixed at ${l_c} =
1$ and ${N_x} =
{N_z}={ 2^6}$, respectively.
Table 6.
Absolute Value of the Mean of the Energy Defect as the Number of Taylor
Orders Was Varied for Assorted
Values of the Perturbation Size Calculated via Padé
Summationa
0.1
6.9990e-09
9.5992e-09
9.6114e-09
9.6099e-09
0.2
2.5644e-06
1.3940e-07
1.3943e-07
1.3987e-07
0.4
1.2791e-03
2.3498e-06
1.7707e-06
1.7775e-06
0.6
2.5179e-02
1.8212e-04
6.7339e-06
6.5450e-06
0.8
1.2679e-01
8.4429e-03
1.6977e-05
2.0144e-05
1
3.6152e-01
1.1361e-01
2.6508e-03
1.5363e-03
1.1
9.3667e-01
9.3128e-01
1.0762e-02
6.0850e-03
1.2
1.6302
0.9515
0.0654
3.2271e-02
Correlation length and lateral/vertical discretization
fixed at ${l_c} =
1$ and ${N_x} =
{N_z}={ 2^6}$, respectively.
Tables (6)
Table 1.
Absolute Value of the Mean of the Energy Defect as the Lateral () and Vertical () Discretizations Were
Refineda
1.0501e-03
1.0814e-03
1.0802e-03
1.0827e-03
3.8155e-05
1.1569e-06
1.1161e-06
1.1277e-06
3.9156e-05
5.3976e-10
5.5434e-10
5.5156e-10
3.9349e-05
1.1695e-11
2.3820e-13
2.4384e-13
Correlation length, perturbation size, and number of Taylor
orders fixed at ${l_c} =
1$, $\varepsilon
= 0.1$, and $N =
20$, respectively.
Table 2.
Standard Deviation of the Energy Defect as the Lateral () and Vertical () Discretizations Were
Refineda
1.8407e-06
1.8379e-06
1.8221e-06
1.8365e-06
1.4026e-08
8.6896e-10
8.6678e-10
8.6775e-10
1.2821e-08
1.2812e-08
1.2818e-08
Correlation length, perturbation size, and number of Taylor
orders fixed at ${l_c} =
1$, $\varepsilon
= 0.1$, and $N =
20$, respectively.
Table 3.
Absolute Value of the Mean of the Energy Defect as the Number of Taylor
Orders Was Varied for Assorted
Values of the Perturbation Size a
0.01
2.5860e-07
8.9270e-11
5.8876e-12
5.8876e-12
0.05
1.8107e-04
1.8469e-06
4.0214e-09
3.9343e-09
0.1
2.8511e-03
1.0254e-04
1.3407e-07
6.2869e-08
0.2
4.6765e-02
3.2213e-03
2.5526e-05
4.6869e-06
Correlation length and lateral/vertical discretization
fixed at ${l_c} =
1$ and ${N_x} =
{N_z}={ 2^5}$, respectively.
Table 4.
Standard Deviation of the Energy Defect as the Number of Taylor
Orders Was Varied for Assorted
Values of the Perturbation Size a
0.01
1.5934e-11
0.05
2.8624e-07
1.0078e-10
1.4301e-16
1.3036e-16
0.1
2.9170e-05
1.2568e-07
2.7749e-12
1.3858e-14
0.2
5.1089e-03
1.1718e-04
3.9580e-07
7.0365e-08
Correlation length and lateral/vertical discretization
fixed at ${l_c} =
1$ and ${N_x} =
{N_z}={ 2^5}$, respectively.
Table 5.
Absolute Value of the Mean of the Energy Defect as the Number of Taylor
Orders Was Varied for Assorted
Values of the Perturbation Size Calculated via Taylor
Summationa
0.1
7.8669e-08
9.5712e-09
9.6114e-09
9.6099e-09
0.2
2.4983e-05
5.4705e-06
1.3548e-08
1.4908e-07
0.4
4.0993e-01
1.6469e03
2.3238e11
1.5042e28
Correlation length and lateral/vertical discretization
fixed at ${l_c} =
1$ and ${N_x} =
{N_z}={ 2^6}$, respectively.
Table 6.
Absolute Value of the Mean of the Energy Defect as the Number of Taylor
Orders Was Varied for Assorted
Values of the Perturbation Size Calculated via Padé
Summationa
0.1
6.9990e-09
9.5992e-09
9.6114e-09
9.6099e-09
0.2
2.5644e-06
1.3940e-07
1.3943e-07
1.3987e-07
0.4
1.2791e-03
2.3498e-06
1.7707e-06
1.7775e-06
0.6
2.5179e-02
1.8212e-04
6.7339e-06
6.5450e-06
0.8
1.2679e-01
8.4429e-03
1.6977e-05
2.0144e-05
1
3.6152e-01
1.1361e-01
2.6508e-03
1.5363e-03
1.1
9.3667e-01
9.3128e-01
1.0762e-02
6.0850e-03
1.2
1.6302
0.9515
0.0654
3.2271e-02
Correlation length and lateral/vertical discretization
fixed at ${l_c} =
1$ and ${N_x} =
{N_z}={ 2^6}$, respectively.
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