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 $(n_{1}{}^{2}+2)/3$ multiplied by the Mie macroscopic field, where ${n}_{1}$ 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.

Vadim A. Markel J. Opt. Soc. Am. A 33(7) 1244-1256 (2016)

References

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Values of the Rms Fractional Difference ${\chi}_{s}$ Used in Comparing ${S}_{1},\hspace{0.5em}{S}_{2},\hspace{0.5em}{S}_{3},$ and ${S}_{4}$ Values Based on DDA versus Mie Solutions for a Sphere at Various Orientation Anglesa

Electric Field Amplitudes at DDA Dipole Sites Compared with $({{n}_{1}}^{2}+2)/3$ 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)

$|{E}_{x}|$

$|{E}_{y}|$

$|{E}_{z}|$

|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 $\mathit{ka}=1.5,$ with incident wavelength $\mathrm{\lambda}=0.6328\mathit{\mu}\mathrm{m}$ and refractive index ${n}_{1}=1.30,$ 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 $({n}_{1}^{2}+2)/3$ 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)

$|{E}_{x}|$

$|{E}_{y}|$

$|{E}_{z}|$

|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 $\mathit{ka}=1.5,$ with incident wavelength $\mathrm{\lambda}=0.6328\mathit{\mu}\mathrm{m}$ and refractive index ${n}_{1}=1.80,$ 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 ${n}_{2}=1.1$

Rms Fractional Difference χ for ${n}_{2}=1.3$

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 ${\mathit{ka}}_{1}=0.75,\hspace{0.5em}{\mathit{ka}}_{2}=1.50,$ with indices of refraction ${n}_{1}=1.30$ for the inner core and ${n}_{2}$ 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

${\mathit{ka}}_{1}$

${\mathit{ka}}_{2}$

N

Rms Fractional Difference χ for ${n}_{2}=1.1$

Rms Fractional Difference χ for ${n}_{2}=1.3$

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 ${n}_{1}=1.30$ for the inner core and ${n}_{2}=1.10$ or ${n}_{2}=1.30$ 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 ${\chi}_{s}$ Used in Comparing ${S}_{1},\hspace{0.5em}{S}_{2},\hspace{0.5em}{S}_{3},$ and ${S}_{4}$ Values Based on DDA versus Mie Solutions for a Sphere at Various Orientation Anglesa

Electric Field Amplitudes at DDA Dipole Sites Compared with $({{n}_{1}}^{2}+2)/3$ 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)

$|{E}_{x}|$

$|{E}_{y}|$

$|{E}_{z}|$

|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 $\mathit{ka}=1.5,$ with incident wavelength $\mathrm{\lambda}=0.6328\mathit{\mu}\mathrm{m}$ and refractive index ${n}_{1}=1.30,$ 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 $({n}_{1}^{2}+2)/3$ 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)

$|{E}_{x}|$

$|{E}_{y}|$

$|{E}_{z}|$

|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 $\mathit{ka}=1.5,$ with incident wavelength $\mathrm{\lambda}=0.6328\mathit{\mu}\mathrm{m}$ and refractive index ${n}_{1}=1.80,$ 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 ${n}_{2}=1.1$

Rms Fractional Difference χ for ${n}_{2}=1.3$

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 ${\mathit{ka}}_{1}=0.75,\hspace{0.5em}{\mathit{ka}}_{2}=1.50,$ with indices of refraction ${n}_{1}=1.30$ for the inner core and ${n}_{2}$ 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

${\mathit{ka}}_{1}$

${\mathit{ka}}_{2}$

N

Rms Fractional Difference χ for ${n}_{2}=1.1$

Rms Fractional Difference χ for ${n}_{2}=1.3$

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 ${n}_{1}=1.30$ for the inner core and ${n}_{2}=1.10$ or ${n}_{2}=1.30$ 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.