Table I
Spherical Targets of Pure Acrylic and Acrylic–Water Composite Used in Experimental Measurements
Target | Radius of the sphere (cm) | Water in acrylic by volume (%) | Mode radius rm of spherical water inclusions (cm) |
---|
311 | 3.1731 | 0 | — |
314 | 3.1744 | 1.6 | 0.02 |
315 | 3.1731 | 2.7 | 0.02 |
Table II
Phase Function Measurements of Homogeneous Acrylic Sphere and Acrylic-Water Composite Spheres for Linearly Polarized Radiation of Wavelength λ = 3.1835 cm with Electric Vector E in the Scattering Plane
| Phase function I22 of a spherical target |
---|
|
|
---|
Scattering angle (deg) | | 314 | 315 |
---|
311 | 98.4% Acrylic | 97.3% Acrylic |
---|
100% Acrylic | 1.6% Water | 2.7% Water |
---|
5 | 456.90 | 577.33 | 583.61 |
10 | 278.67 | 398.08 | 419.21 |
15 | 149.24 | 258.83 | 285.44 |
20 | 48.97 | 125.98 | 143.59 |
25 | 24.21 | 67.03 | 70.42 |
30 | 17.68 | 24.61 | 19.52 |
35 | 15.16 | 9.03 | 3.73 |
40 | 18.12 | 6.23 | 2.97 |
45 | 22.84 | 10.64 | 7.77 |
50 | 21.95 | 13.44 | 10.88 |
55 | 25.90 | 19.10 | 13.51 |
60 | 13.10 | 12.52 | 8.13 |
65 | 11.71 | 8.80 | 4.63 |
70 | 16.89 | 9.43 | 5.77 |
75 | 29.12 | 17.94 | 11.29 |
80 | 21.17 | 15.21 | 10.10 |
85 | 8.47 | 7.66 | 5.14 |
90 | 3.24 | 3.01 | 3.04 |
95 | 6.48 | 3.86 | 3.86 |
100 | 16.68 | 11.09 | 9.77 |
105 | 19.06 | 14.74 | 12.88 |
110 | 9.96 | 9.02 | 8.62 |
115 | 2.36 | 3.03 | 3.10 |
120 | 4.23 | 0.92 | 1.74 |
125 | 4.53 | 2.60 | 2.13 |
130 | 3.91 | 3.76 | 3.13 |
135 | 1.59 | 3.88 | 4.06 |
140 | 1.37 | 1.24 | 1.98 |
145 | 8.73 | 2.47 | 2.32 |
150 | 16.46 | 4.59 | 3.26 |
155 | 16.50 | 4.44 | 2.48 |
160 | 11.01 | 1.68 | 1.22 |
165 | 9.60 | 3.05 | 1.24 |
170 | 29.98 | 17.88 | 7.83 |
Note: The cross-polarized component of scattered radiation was found to be zero for all three spheres.
Table III
Results of a Waveguide Method of Determination of the Refractive Index of Pure Acrylic and of the Effective Refractive Index m7 of the Acrylic–Water Composite with the Mode Radius of Spherical Water Inclusions rm = 0.02 cm
Material | Real part of refractive index | Imaginary part of refractive index |
---|
100% Acrylic | 1.686 | −0.007 |
98% Acrylic | 1.718 | −0.011 |
1.6% Water | | |
97.3% Acrylic | 1.750 | −0.013 |
2.7% Water | | |
Table IV
Measured Normalized Extinction Cross Section Qext and Normalized Scattering Cross-Section Qsc Obtained by Numerical Integration of the Measured Phase Function and Calculated Normalized Absorption Cross-Section Qabs of Investigated Targets at the Wavelength λ = 3.1835 cm
Target | Qext | Qsc | Qabs |
---|
311 Pure acrylic | 2.24 ± 0.05 | 2.08 ± 0.05 | 0.16 ± 0.10 |
314 1.6% Water | 2.41 ± 0.05 | 2.06 ± 0.05 | 0.35 ± 0.10 |
315 2.7% Water | 2.48 ± 0.05 | 2.02 ± 0.05 | 0.46 ± 0.10 |
Note: Radiation is linearly polarized with electric vector
E in the scattering plane.
Table V
Effective Refractive Indices of Acrylic–Water Composite Material Calculated Using the Eight Different Effective Medium Approximation Described in the Text
| 1.6% Water | 2.7% Water |
---|
|
|
|
---|
Effective refractive index given by | Re(m) | Im(m) | Re(m) | Im(m) |
---|
Eq. (1) m1 Volume average of refractive indices | 1.782 | −0.045 | 1.848 | −0.073 |
Eq. (2) m2 Volume average of dielectric constants | 1.916 | −0.165 | 2.065 | −0.254 |
Eq. (3) m3 Bruggeman mixing rule | 1.723 | −0.009 | 1.750 | −0.011 |
Eq. (4) m4 Maxwell Garnett mixing rule | 1.722 | −0.009 | 1.747 | −0.011 |
Eq. (5) m5 Maxwell Garnett inverted | 1.847 | −0.116 | 1.953 | −0.183 |
Eq. (11) m6 Chylek-Srivastava rule with modified gamma size distribution | 1.724 | −0.011 | 1.752 | −0.013 |
Measured, m7 Waveguide method | 1.718 | −0.011 | 1.750 | −0.013 |
Note: Refractive index of water at λ = 3.1835 cm is taken to be
mA = 7.70–2.48
i and of acrylic to be
m0 = 1.686–0.007
i.
Table VI
Reduced χ2 Values Characterizing the Goodness of the Fit Between the Measured Phase Function and Phase Function Calculated Using Considered Effective Medium Approximations
| Reduced χ2 value for |
---|
|
|
---|
| 1.6% Water | 2.7% Water |
---|
Eq. (1) m1 Volume average of refractive indices | 6.01 | 4.21 |
Eq. (2) m2 Volume average of dielectric constants | 11.45 | 8.45 |
Eq. (3) m3 Bruggeman mixing rule | 0.80 | 2.01 |
Eq. (4) m4 Maxwell Garnett mixing rule | 0.81 | 2.03 |
Eq. (5) m5 Maxwell Garnett inverted | 10.66 | 8.27 |
Eq. (6)–(10), m6 Modified gamma size distribution | 0.62 | 1.41 |
Waveguide measurement, m7 | 0.63 | 1.41 |
Note: Approximations leading to reduced
χ2 ≤ 2 are considered to be acceptable, while those with
χ2 > 2 are considered nonacceptable. The
χ2 value for a homogeneous acrylic sphere is found to be 0.55.
Table VII
Normalized Extinction, Scattering, and Absorption Cross Section of Acrylic–Water Composite Spheres at λ = 3.1835 cm Calculated from the Mie Theory Using Effective Refractive Indices from Table V
| Sphere 314 | Sphere 315 |
---|
| 1.6% Water | 2.7% Water |
---|
|
|
|
---|
Effective refractive index given by | Qext | Qsc | Qabs | Qext | Qsc | Qabs |
---|
Eq. (1) m1 Volume average of refractive indices | 2.74 | 1.81 | 0.93 | 2.82 | 1.78 | 1.03 |
Eq. (2) m2 Volume average of dielectric constants | 2.62 | 1.37 | 1.25 | 2.51 | 1.26 | 1.25 |
Eq. (3) m3 Bruggeman mixing rule | 2.47 | 2.14 | 0.33 | 2.49 | 2.12 | 0.37 |
Eq. (4) m4 Maxwell Garnett mixing rule | 2.46 | 2.14 | 0.32 | 2.49 | 2.13 | 0.36 |
Eq. (5) m5 Maxwell Garnett inverted | 2.72 | 1.52 | 1.20 | 2.57 | 1.32 | 1.25 |
Eq. (6)–(10), m6 Chylek-Srivastava iteration with modified gamma size distribution | 2.46 | 2.11 | 0.35 | 2.51 | 2.08 | 0.43 |
Waveguide measurement, m7 | 2.46 | 2.09 | 0.37 | 2.50 | 2.09 | 0.41 |
Experimental results | 2.41 | 2.06 | 0.35 | 2.48 | 2.02 | 0.46 |
| ±0.05 | ±0.05 | ±0.10 | ±0.05 | ±0.05 | ±0.10 |
Note: Radiation is linearly poolarized with electric vector
E in the scattering plane.