Stephen D. Druger and Burt V. Bronk, "Internal and scattered electric fields in the discrete dipole approximation," J. Opt. Soc. Am. B 16, 2239-2246 (1999)
The calculated scattering matrix elements and interior electric fields for a dielectric sphere based on the discrete dipole approximation (DDA) are compared with the exact Mie solution for homogeneous and composite spheres. For homogeneous spheres the macroscopic average field produced at each DDA dipole site by the incident field combined with the field from all DDA sites is found to be approximated by the factor multiplied by the Mie macroscopic field, where is the refractive index. This holds to surprising accuracy, considering the finite wavelength and the small number of dipoles used in the DDA approximation. The approximate relation is most accurate near the center of the sphere and least accurate at the interface. The relation also holds for electric fields within composite spheres, with poorer agreement near each interface, where the refractive index changes. The dependence of this relation on parameters of the model is examined.
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Values of the Rms Fractional Difference Used in Comparing and Values Based on DDA versus Mie Solutions for a Sphere at Various Orientation Anglesa
No. of Dipoles
619
1791
5449
16,279
619
1791
5449
16,279
619
1791
5449
16,279
619
1791
5449
16,279
Table 2
Electric Field Amplitudes at DDA Dipole Sites Compared with Multiplied By the Mie Solution for the Electric Field at the Corresponding Point in a Homogeneous Sphere: 47,937 Dipolesa
Method
Radial Distance (μm)
Sample Location (x, y, z)
|E|
Mie
0
(0, 0, 0)
1.269
0.00
0.00
1.269
DDA
1.270
0.00
0.00
1.270
Mie
0.01641
(-0.0134, -0.0067, -0.0067)
1.2454
0.0006
0.0354
1.2459
DDA
1.2465
0.0006
0.0351
1.2470
Mie
0.03350
(-0.03350, 0, 0)
1.2565
0.0000
0.0871
1.2595
DDA
1.2577
0.0000
0.8064
1.2607
Mie
0.05014
(-0.0402, -0.0268, -0.0134)
1.2009
0.0069
0.1039
1.2054
DDA
1.2019
0.0069
0.1031
1.2064
Mie
0.06701
(-0.0536, -0.0402, 0)
1.2135
0.0139
0.1337
1.2209
DDA
1.2147
0.0138
0.1327
1.2220
Mie
0.08396
(-0.0804, -0.0201, -0.0134)
1.1576
0.0098
0.1965
1.1742
DDA
1.1589
0.0098
0.1950
1.1753
Mie
0.10074
(-0.0804, -0.0603, -0.0067)
1.1324
0.0287
0.1881
1.1482
DDA
1.1335
0.0286
0.1867
1.1492
Mie
0.11741
(-0.1139, -0.0201, -0.0201)
1.0814
0.0126
0.2574
1.1117
DDA
1.0827
0.0124
0.2553
1.1125
Mie
0.13435
(-0.1340, -0.0067, -0.0067)
1.0727
0.0048
0.2841
1.1097
DDA
1.0774
0.0054
0.2802
1.1132
Mie
0.15088
(-0.1273, -0.0737, -0.0335)
0.9791
0.0449
0.2594
1.0139
DDA
1.0752
0.1273
0.2287
1.1066
The solutions are compared for a dielectric sphere of size with incident wavelength and refractive index represented in the DDA by 47,937 dipoles. Fields E are relative to unit magnitude for the incident field.
Table 3
Electric Field Amplitudes at DDA Dipole Sites Compared with Multiplied by the Mie Solution for the Electric Field at the Corresponding Point in a Homogeneous Sphere: 15,967 Dipolesa
Method
Radial Distance (μ)
Sample Location (x, y, z)
|E|
Mie
0
(0, 0, 0)
1.8809
0.0
0.0
1.8809
DDA
1.8809
0.0
0.0
1.8809
Mie
0.03348
(-0.0193, -0.01933, -0.01933)
1.7830
0.0137
0.4207
1.8320
DDA
1.7654
0.0136
0.4109
1.8126
Mie
0.06696
(-0.0387, -0.0387, -0.0387)
1.7437
0.0482
0.7380
1.8941
DDA
1.7152
0.0479
0.7232
1.8620
Mie
0.10090
(-0.07731, -0.05799, -0.0290)
1.4232
0.1290
1.2418
1.8932
DDA
1.4071
0.1287
1.2132
1.8624
Mie
0.13426
(-0.1063, -0.0580, -0.0580)
1.3667
0.1342
1.2339
1.8462
DDA
1.3633
0.1237
1.1908
1.8143
Mie
0.15096
(-0.1160, -0.0966, 0)
0.8852
0.2526
1.2370
1.5419
DDA
1.1466
0.4504
0.6689
1.4018
The solutions are compared for a dielectric sphere of size with incident wavelength and refractive index represented in the DDA by 15,967 dipoles. Fields E are relative to unit magnitude for the incident field.
Table 4
Accuracy of the Relation between DDA Local Fields with Mie Macroscopic Fields as a Function of Dipole Number for Homogeneous and Inhomogeneous Spheresa
Dipole Number
Rms Fractional Difference χ for
Rms Fractional Difference χ for
x Component
(y, z)
(x, y, z)
x Component
(y, z)
(x, y, z)
305
0.0261
0.3773
0.0363
0.0441
0.3294
0.0590
2969
0.0158
0.2247
0.0235
0.0228
0.2144
0.0350
6031
0.0144
0.2112
0.0187
0.0203
0.1859
0.0360
11,981
0.0125
0.1974
0.0187
0.0186
0.1720
0.0283
23,871
0.0099
0.1721
0.0157
0.0161
0.1520
0.0248
47,937
0.0091
0.1526
0.0147
0.0143
0.1357
0.0211
The calculations were for a layered sphere with with indices of refraction for the inner core and for the outer layer as indicated. Shown are the overall values of χ for all three components (x, y, z) in Eq. (4) and for the x and the (y, z) components considered alone.
Table 5
Accuracy of the Relation between DDA Local Fields with Mie Macroscopic Fields as a Function of Particle Size at Constant Dipole Numbera
N
Rms Fractional Difference χ for
Rms Fractional Difference χ for
x Component
(y, z)
(x, y, z)
x Component
(y, z)
(x, y, z)
0.75
1.50
23,871
0.0099
0.1721
0.0157
0.0161
0.1520
0.0248
0.35
0.70
23,871
0.0098
0.2533
0.0156
0.0167
0.4397
0.0260
0.20
0.40
23,871
0.0098
0.2756
0.0156
0.0169
0.7246
0.0262
0.10
0.20
23,871
0.0098
0.2843
0.0155
0.0161
0.1520
0.0248
0.010
0.020
23,871
0.0098
0.2874
0.0155
0.0169
1.4094
0.0263
0.325
0.75
2969
0.0117
0.3671
0.0190
0.0235
0.5468
0.0361
The calculations were for a layered sphere with indices of refraction for the inner core and or as indicated for the outer layer. Shown are the overall values of χ for all three components (x, y, z) in Eq. (4) and for the x and the (y, z) components considered alone.
Tables (5)
Table 1
Values of the Rms Fractional Difference Used in Comparing and Values Based on DDA versus Mie Solutions for a Sphere at Various Orientation Anglesa
No. of Dipoles
619
1791
5449
16,279
619
1791
5449
16,279
619
1791
5449
16,279
619
1791
5449
16,279
Table 2
Electric Field Amplitudes at DDA Dipole Sites Compared with Multiplied By the Mie Solution for the Electric Field at the Corresponding Point in a Homogeneous Sphere: 47,937 Dipolesa
Method
Radial Distance (μm)
Sample Location (x, y, z)
|E|
Mie
0
(0, 0, 0)
1.269
0.00
0.00
1.269
DDA
1.270
0.00
0.00
1.270
Mie
0.01641
(-0.0134, -0.0067, -0.0067)
1.2454
0.0006
0.0354
1.2459
DDA
1.2465
0.0006
0.0351
1.2470
Mie
0.03350
(-0.03350, 0, 0)
1.2565
0.0000
0.0871
1.2595
DDA
1.2577
0.0000
0.8064
1.2607
Mie
0.05014
(-0.0402, -0.0268, -0.0134)
1.2009
0.0069
0.1039
1.2054
DDA
1.2019
0.0069
0.1031
1.2064
Mie
0.06701
(-0.0536, -0.0402, 0)
1.2135
0.0139
0.1337
1.2209
DDA
1.2147
0.0138
0.1327
1.2220
Mie
0.08396
(-0.0804, -0.0201, -0.0134)
1.1576
0.0098
0.1965
1.1742
DDA
1.1589
0.0098
0.1950
1.1753
Mie
0.10074
(-0.0804, -0.0603, -0.0067)
1.1324
0.0287
0.1881
1.1482
DDA
1.1335
0.0286
0.1867
1.1492
Mie
0.11741
(-0.1139, -0.0201, -0.0201)
1.0814
0.0126
0.2574
1.1117
DDA
1.0827
0.0124
0.2553
1.1125
Mie
0.13435
(-0.1340, -0.0067, -0.0067)
1.0727
0.0048
0.2841
1.1097
DDA
1.0774
0.0054
0.2802
1.1132
Mie
0.15088
(-0.1273, -0.0737, -0.0335)
0.9791
0.0449
0.2594
1.0139
DDA
1.0752
0.1273
0.2287
1.1066
The solutions are compared for a dielectric sphere of size with incident wavelength and refractive index represented in the DDA by 47,937 dipoles. Fields E are relative to unit magnitude for the incident field.
Table 3
Electric Field Amplitudes at DDA Dipole Sites Compared with Multiplied by the Mie Solution for the Electric Field at the Corresponding Point in a Homogeneous Sphere: 15,967 Dipolesa
Method
Radial Distance (μ)
Sample Location (x, y, z)
|E|
Mie
0
(0, 0, 0)
1.8809
0.0
0.0
1.8809
DDA
1.8809
0.0
0.0
1.8809
Mie
0.03348
(-0.0193, -0.01933, -0.01933)
1.7830
0.0137
0.4207
1.8320
DDA
1.7654
0.0136
0.4109
1.8126
Mie
0.06696
(-0.0387, -0.0387, -0.0387)
1.7437
0.0482
0.7380
1.8941
DDA
1.7152
0.0479
0.7232
1.8620
Mie
0.10090
(-0.07731, -0.05799, -0.0290)
1.4232
0.1290
1.2418
1.8932
DDA
1.4071
0.1287
1.2132
1.8624
Mie
0.13426
(-0.1063, -0.0580, -0.0580)
1.3667
0.1342
1.2339
1.8462
DDA
1.3633
0.1237
1.1908
1.8143
Mie
0.15096
(-0.1160, -0.0966, 0)
0.8852
0.2526
1.2370
1.5419
DDA
1.1466
0.4504
0.6689
1.4018
The solutions are compared for a dielectric sphere of size with incident wavelength and refractive index represented in the DDA by 15,967 dipoles. Fields E are relative to unit magnitude for the incident field.
Table 4
Accuracy of the Relation between DDA Local Fields with Mie Macroscopic Fields as a Function of Dipole Number for Homogeneous and Inhomogeneous Spheresa
Dipole Number
Rms Fractional Difference χ for
Rms Fractional Difference χ for
x Component
(y, z)
(x, y, z)
x Component
(y, z)
(x, y, z)
305
0.0261
0.3773
0.0363
0.0441
0.3294
0.0590
2969
0.0158
0.2247
0.0235
0.0228
0.2144
0.0350
6031
0.0144
0.2112
0.0187
0.0203
0.1859
0.0360
11,981
0.0125
0.1974
0.0187
0.0186
0.1720
0.0283
23,871
0.0099
0.1721
0.0157
0.0161
0.1520
0.0248
47,937
0.0091
0.1526
0.0147
0.0143
0.1357
0.0211
The calculations were for a layered sphere with with indices of refraction for the inner core and for the outer layer as indicated. Shown are the overall values of χ for all three components (x, y, z) in Eq. (4) and for the x and the (y, z) components considered alone.
Table 5
Accuracy of the Relation between DDA Local Fields with Mie Macroscopic Fields as a Function of Particle Size at Constant Dipole Numbera
N
Rms Fractional Difference χ for
Rms Fractional Difference χ for
x Component
(y, z)
(x, y, z)
x Component
(y, z)
(x, y, z)
0.75
1.50
23,871
0.0099
0.1721
0.0157
0.0161
0.1520
0.0248
0.35
0.70
23,871
0.0098
0.2533
0.0156
0.0167
0.4397
0.0260
0.20
0.40
23,871
0.0098
0.2756
0.0156
0.0169
0.7246
0.0262
0.10
0.20
23,871
0.0098
0.2843
0.0155
0.0161
0.1520
0.0248
0.010
0.020
23,871
0.0098
0.2874
0.0155
0.0169
1.4094
0.0263
0.325
0.75
2969
0.0117
0.3671
0.0190
0.0235
0.5468
0.0361
The calculations were for a layered sphere with indices of refraction for the inner core and or as indicated for the outer layer. Shown are the overall values of χ for all three components (x, y, z) in Eq. (4) and for the x and the (y, z) components considered alone.