Geometrical-optics approximation of forward scattering by gradient-index spheres Xiangzhen Li,* Xiang’e Han, Renxian Li, and Huifen Jiang School of Science, Xidian University, Xi’an 710071, China *Corresponding author:
[email protected] Received 31 January 2007; revised 19 April 2007; accepted 19 April 2007; posted 19 April 2007 (Doc. ID 79591); published 9 July 2007
By means of geometrical optics we present an approximation method for acceleration of the computation of the scattering intensity distribution within a forward angular range (0 – 60°) for gradient-index spheres illuminated by a plane wave. The incident angle of reflected light is determined by the scattering angle, thus improving the approximation accuracy. The scattering angle and the optical path length are numerically integrated by a general-purpose integrator. With some special index models, the scattering angle and the optical path length can be expressed by a unique function and the calculation is faster. This method is proved effective for transparent particles with size parameters greater than 50. It fails to give good approximation results at scattering angles whose refractive rays are in the backward direction. For different index models, the geometrical-optics approximation is effective only for forward angles, typically those less than 60° or when the refractive-index difference of a particle is less than a certain value. © 2007 Optical Society of America OCIS codes: 080.0080, 290.4020, 290.5850, 200.0200.
1. Introduction
The gradient-index (GRIN) medium has found many applications in modern industries. For example, continuous refractive-index models are used for a crystalline lens; a waveguide model is considered for human photoreceptors; and GRIN models are widely used for the optical fiber [1]. The atmosphere of the Earth has a refractive index that decreases with height because the density decreases at higher altitudes. To probe the applications of the GRIN medium optically, one needs to calculate the theoretical scattering intensities of different sized particles. Although the rigorous way to do this was given by Lorenz and Mie more than 100 years ago, the calculation of spherical harmonic functions [2] has proved difficult and time-consuming. The larger the particle the more CPU time is needed. Therefore the Fraunhofer diffraction (FD) theory is often introduced into particle analysis as an approximation method. Nevertheless, FD theory is not valid for all situations [3,4]. For example, to measure a particulate system of 0003-6935/07/225241-07$15.00/0 © 2007 Optical Society of America
bimodal distribution with a large size ratio requires the particle size of one fraction to be small. Thus one would have to detect the scattering light pattern within a large angular range. In this case, the FD theory does not work properly because it is invalid for large-angle scattering. On the basis of the Mie theory, contributions of emergent rays of different types, e.g., diffracted, specularly reflected, and refracted rays undergoing internal reflections, can be extracted through the Debye series decomposition. However the scattering calculation by a large particle is not fast enough. Hence it becomes urgent to develop an approximate algorithm for scattering intensity calculation that will be faster than the Lorenz–Mie theory and more accurate than the FD theory. Studies of approximation methods of the scattering light pattern for homogeneous or coated particles [5] have been carried out by several researchers. Schiff [6] developed a high-energy approximation method that describes scatter intensity within the forward direction (0 –10°) for stratified particles on condition that the refractive indices of the core and layer be close to 1 and the dimensionless size parameter be more than 50. Chen [7] found the high-energy approximation effective for refractive indices as large as 2 but with an 1 August 2007 兾 Vol. 46, No. 22 兾 APPLIED OPTICS
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angular validity range that shrinks when the particle size increases. By means of a geometrical-optics approximation (GOA), Glantschnig and Chen [8] derived a formula with which to calculate scattering intensity in the forward angular range for water droplets. Belafhal et al. [9] studied light scattering by absorbing and nonabsorbing spherical particles in the Wentzel– Kramers–Brillouin approximation and developed a new mathematical description of light scattering. Min et al. [10] calculated the optical cross sections of size and shape distributions of spheroids, and Grynko and Shkuratov [11] studied the scattering characteristics of semitransparent particles with various shapes. Xu et al. [12] combined the FD theory with geometrical optics and proposed the whole GOA method to accelerate the computation of light intensity for homogeneous absorbent particles and coated particles. Xu et al. [13] studied the extension of the GOA to on-axis Gaussian beam scattering by a spherical particle. However, to the best of our knowledge, the GOA of forward scattering by GRIN particles has never been researched. Here we develop a more general method of GOA that can be used for the calculation of GRIN spheres. In Section 2 we describe the GOA for light scattering by GRIN spheres. The numerical results are presented in Section 3 and compared with those of the Mie theory and the Debye series. The speed of calculation of the three methods are compared in Section 4. Conclusions are given in Section 5. 2. Geometrical-Optics Approximation of Gradient-Index Spheres
where represents the complement of the incident angle, ⬘ is the complement of the internal refracted angle, and N is the number of light–sphere interactions. The ray trajectories through the particle are straight lines; see solid line in Fig. 1. If the refractive index of the sphere is not uniform, the Descartes law in Eq. (2) is no longer applicable. For a GRIN sphere, the ray trajectories through the particle are curved; see the dash– dot curve in Fig. 1. m0 and m1 are the refractive indices of the center and the surface and R is the radius of the sphere. The refractive index of environment ms ⫽ 1.0. For a sphere with a spherically symmetric refractive index m共r˜兲 immersed in a uniform medium, the relationship between incident and scattering angles for a spherically symmetric particle cannot be described by the Descartes law [Eq. (2)]. In 2005 Vetrano et al. derived geometrically the relationship between incident and scattering angles of a spherically symmetric particle [15]: ⫽ 2 ⫺ 2共N ⫺ 1兲关共r ⫽ R兲 ⫺ 共r ⫽ rm兲兴.
(3)
As shown in Fig. 1, the ray trajectories through the GRIN sphere are determined by 共r兲 and rm, and 共r兲 can be written in a nondimensional form:
共r兲 ⫽ 0 ⫹
冕
r
e
r冑关m共r˜兲r˜兴2 ⫺ e2 rm
dr,
(4)
From a geometrical-optics viewpoint, scattered light amplitude is considered to be the superposition of diffracted, refracted, and reflected fractions [5], that is,
where e ⫽ ms cos共兲 is a constant, 0 equals zero under ordinary conditions, r is a position of one point of a light ray, and r˜ ⫽ r兾R is the nondimensional sphere radius. r˜m is described by
s ⫽ sdiffraction ⫹ srefraction ⫹ sreflection.
2 关m共r˜兲r˜兴2 ⫺ e ⫽ 0.
(1)
In 1637 Descartes [14] described geometrically the relationship of incident angles to scattering angles of a uniform sphere as ⫽ 2 ⫺ 2共N ⫺ 1兲⬘,
(2)
(5)
For a spherically symmetric particle with a radius of R, the dimensionless parameter is set as ␣ ⫽ 2R兾, where is the wavelength of the incident light and rm represents the minimal distance from the light ray to the center of the sphere. Corbin et al. [16] believed that 90% of the scattering energy is pro-
Fig. 1. Comparison of the light path inside a sphere with a uniform refractive index (solid curves) and the light path inside a sphere with a spherically symmetric refractive index m共r˜兲 (dashed curves): (a) m0 ⬎ m1 and (b) m0 ⬍ m1. 5242
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duced by reflection and first refraction and other emerging rays can be neglected. Therefore N can be set to be N ⫽ 1, 2 in this paper. Combined with Eq. (3), the phase shift and the scattering intensity of a spherically symmetric particle can be derived. The dimensionless scattering intensity is defined as [5,17] i1,2 ⫽ ␣2k1,22D,
k1,2 ⫽ 共1 ⫺
兲共⫺1,2兲
2 1,2
N⫺2
s1,2 ⫽ 冑i1,2 exp共j1,2兲.
for N ⫽ 1, for N ⫽ 2, 3, 4 . . . .
(7)
s ⫽ sreflection ⫹ srefraction ⫹ sdiffraction, 僆 关0°, 20°兴, s ⫽ sreflection ⫹ srefraction, 僆 关20°, 60°兴, (13) where sdiffraction ⫽ ␣2J1共␣ sin 兲兾共␣ sin 兲 and J1 is the first-order Bessel function. Suppose that the particle is illuminated by an unpolarized monochromatic beam with a wavelength of and an incident intensity of I0. The scattering intensity I共兲 at distance f can finally be obtained by
D is usually called divergence or gain, a parameter denoting the influence of the shape of a particle on the angular intensity distribution. It can be expressed as D⫽
sin cos , sin ⱍd兾dⱍ
(8)
where 1,2 are Fresnel reflection coefficients defined as 1 ⫽
sin ⫺ m1 sin ⬘ , sin ⫹ m1 sin ⬘
2 ⫽
m1 sin ⫺ sin ⬘ . m1 sin ⫹ sin ⬘
(9)
Phase shifts [17] 1,2 that are due to reflection and correspond to perpendicular and parallel polarized components are expressed by the Fresnel reflection coefficients. Xu et al. [12] used a one-dimensional piecewise cubic spline interpolation to obtain the amplitudes and phases of the rays. However, this method could result in deviation. Here we apply the same scattering angles to calculate the amplitudes and phases of the rays. Hence, the incident angle of the reflected light should be determined by the scattering angle. In terms of the van de Hulst discussion of the phase change that is due to optical length and focal lines, the combined phases reflection1,2 and refraction1,2 in the foregoing cases can finally be obtained: reflection1,2 ⫽ 兾2 ⫹ 2␣ sin共reflection兲 ⫹ 1,2, refraction1,2 ⫽ 3兾2 ⫹ 2␣ sin共兲 ⫺
2L ,
(10)
where reflection ⫽ 兾2 is the incident angle of the reflected light and L is the optical path length inside the sphere of the N ⫽ 1 rays and is described by
冕冑 R
L⫽
rm
2m共r˜兲2r˜
关m共r˜兲r˜兴2 ⫺ e
2
dr.
(11)
(12)
The final amplitude functions can then be obtained [12]:
(6)
where subscript 1 represents perpendicular polarized components in polarization and subscript 2 represents parallel polarized components. k1,2 is introduced to characterize the fraction of the emerging rays in the incident intensity: k1,2 ⫽ 1,2
With the scattering angles and the combined phases, the amplitude functions for the two polarizations can be written as [17]
I共兲 ⫽
2I0 82f 2
共ⱍs1共兲ⱍ2 ⫹ ⱍs2共兲ⱍ2兲.
(14)
Here we define index 2I0兾共82f 2兲 as 1.0 and wavelength of light ⫽ 632.8 nm. 3. Comparison of the Geometrical-Optics Approximation with Other Theories
With two typical index models, the scattering intensity distribution is calculated by the GOA method. Parameters of the Mie theory and the Debye series are the same as the ones for the GOA. Parameter p ⫽ N ⫺ 1 is the mode of refraction. A comparison is made of the results with those obtained by use of the Mie theory and the Debye series. The valid range of the GOA method is researched. A.
Index Model for a Water Droplet
The refractive index is described by m共r˜兲 ⫽ m0 ⫹ 共m1 ⫺ m0兲
ebr˜ ⫺ 1 eb ⫺ 1
.
(15)
This refractive index model is often used for a water droplet with an inhomogeneous temperature and when b is a real number. Here the profile parameter of b is 6. Two kinds of scattering intensity distribution were calculated by the GOA method in the forward direction 共0–60°兲. The results were compared with those obtained with the Mie theory and the Debye series; see Fig. 2. In Fig. 2(a) the refractive indices of the surface and the center are, respectively, m1 ⫽ 1.30 and m0 ⫽ 1.33, and the radius is R ⫽ 10 m. In Fig. 2(b) the refractive indices of the surface and the center are, respectively, m1 ⫽ 1.33 and m0 ⫽ 1.30, and the radius is the same as the one in Fig. 2(a). Figure 2 shows that the scattering intensity distribution by the GOA method coincides approximately with that obtained with the Mie theory and the Debye series for GRIN spheres. In consider1 August 2007 兾 Vol. 46, No. 22 兾 APPLIED OPTICS
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Fig. 2. Comparison of intensity of calculation by other theories and by the GOA method: (a) R ⫽ 10 m, m0 ⫽ 1.33, m1 ⫽ 1.30 and (b) R ⫽ 10 m, m0 ⫽ 1.30, m1 ⫽ 1.33.
ation of the clarity of the figures, the scattering intensity distribution of a particle with a larger radius is not offered by the GOA method. A great number of numerical calculations show that the GOA method fails to work well for particles whose dimensionless parameter ␣ is smaller than 50. To study the valid range of the GOA method, research was conducted on the large index difference between the surface index and the center index. The trajectories of a portion of N ⫽ 2 rays and N ⫽ 3 rays through the sphere with the large index difference are shown in Fig. 3. As indicated in Fig. 3, a portion of the scattering angles of N ⫽ 2 rays are backward angles and a portion of the scattering angles of N ⫽ 3 rays are forward angles. For different index models, the valid range of the GOA method differ. As indicated in Fig. 4(a), for the index difference ⌬m ⫽ |m0 ⫺ m1| ⬎ 3, the scattering intensity distribution by the GOA method differs from that obtained with the Mie theory and the Debye series for GRIN spheres at 30–50°, and the scattering intensities by the GOA method and the Debye series are smaller than that obtained with the Mie theory at 35–60°. As shown in Fig. 4(b), a large portion of the scattering angles of N ⫽ 2 rays fall between 60° and 180° and a small portion of the scattering angles of N ⫽ 3 rays are between 30° and 60°. Therefore we do know that the contribution of N ⫽ 2 rays decreases, and the contribution of N ⫽ 3 rays should be taken into account. As indicated in Figs. 4(c) and 4(d), with the
N ⫽ 3 rays taken into account, the scattering intensity distribution by the Debye series coincides approximately with that of the Mie theory and the scattering intensities by the GOA method and the Debye series approximate those of the Mie theory. However, the results by GOA failed to give good scattering intensity distribution. The comparison is illustrated in Figs. 4(c) and 4(d), which shows that Xu et al. [13] and Hovenac and Lock [18] made qualitative analyses. Within the framework of GO, the surface wave rays, tunneling rays, and complex rays are not taken into account. When the index difference is too large, more attention should be paid to the surface wave because of its more important contribution to scattering. The surface wave plays an important role with scattering angles around 90° [17]. As studied by Xu [17], the surface wave has a greater influence on smaller particles than on larger ones. However the surface wave effect is also concerned with the index difference for a GRIN sphere. As shown in Figs. 2 and 4, the surface wave proportionally increases in area when the size of the particle increases. Although we tried to combine numerically the surface wave with the geometric rays, so far we have not been successful. In this case, the intensity distribution calculated by the GOA method under this condition may have remarkable approximation errors. B.
Index Model for Gradient-Index Fiber
The refractive index is represented by
冉
m共r˜兲 ⫽ m0 1 ⫺
m02 ⫺ m12 m02
冊
1兾2
r˜2
.
(16)
This refractive index model is often used for the optical fiber, and this is the one we studied as the refractive-index model for a GRIN sphere. Equation (4) can be rewritten as
冋冑
共R兲 ⫽ 1兾2 sin⫺1 Fig. 3. Path of light rays through a GRIN particle with a large index difference. 5244
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m02 ⫺ 2e2 m0 ⫺ 4共m0 ⫺ m1 兲e 4
2
2
2
册
⫹ . 4 (17)
Fig. 4. Valid range of the GOA method for a particle with an index model expressed by Eq. (15): R ⫽ 15 m, m0 ⫽ 4.1, m1 ⫽ 1.1.
The optical path length L is derived from Eq. (11):
再
m02
冋
2m12 ⫺ m02
2
册冎
L⫽
R 2
L⫽
Rm02 R 2m12 ⫺ m02 冑 m12 ⫺ e2 ⫹ ln ⫹ 冑m12 ⫺ e2 ⫺ ln 2 2 2 2 2 冑 冑 4 m1 ⫺ m0 2 m1 ⫺ m0
冑m12 ⫺ e2 ⫺
2冑m02 ⫺ m12
sin⫺1
再冋
冑m04 ⫺ 4共m02 ⫺ m12兲e2
⫺
册 冋
m0 ⬎ m1,
,
冑m02 ⫺ 4共m02 ⫺ m12兲e2 2冑m12 ⫺ m02
册冎
,
m0 ⬍ m1.
(18)
Fig. 5. Comparison of intensity of calculation by other theories and by the GOA method: (a) R ⫽ 10 m, m0 ⫽ 1.49, m1 ⫽ 1.47 and (b) R ⫽ 10 m, m0 ⫽ 1.47, m1 ⫽ 1.49. 1 August 2007 兾 Vol. 46, No. 22 兾 APPLIED OPTICS
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Table 1. Comparison of Speed of Calculation
Radius of Particle R (m) 10 100 500 800 1000
Mie Theory (ms)
GOA (ms) [Index Model Eq. (15)]
GOA (ms) [Index Model Eq. (16)]
Debye Series (ms)
600 5800 19490 61310 78400
70 690 3460 7230 8820
2 12 55 80 110
79 746 3816 26145 31240
By use of Eqs. (17) and (18), two kinds of scattering intensity distribution can be calculated by the GOA method in the forward direction 共0–60°兲. The results are compared with those obtained with the Mie theory and the Debye series. In Fig. 5(a) the refractive indices of the surface and the center are, respectively, m1 ⫽ 1.47 and m0 ⫽ 1.49, and the radius is R ⫽ 10 m. In Fig. 5(b) the refractive indices of the surface and the center are, respectively, m1 ⫽ 1.49 and m0 ⫽ 1.47, and the radius is the same as the one in Fig. 5(a). Also shown in Fig. 5 is that the scattering intensity distribution by the GOA method coincides approximately with that obtained with the Mie theory and the Debye series for transparent particles. Compared with the case in Subsection 3.A, the restrictions on the use of the GOA are similar. The valid range of this index model by the GOA method is researched. A great deal of calculation yields the valid range. When the index difference is larger than 1.0, the scattering intensity distribution intensities calculated by the GOA under this condition can have remarkable approximation errors. 4. Comparlson of Speed of Calculation
When the radius varied from 10 to 1000 m, numerical calculations were performed of the water droplet refractive-index model with refractive indices of the center and the surface, respectively, of 1.33 and 1.30 and the GRIN fiber refractive-index model with refractive indices of the center and the surface, respectively, of 1.49 and 1.47 to test the calculation efficiency of the three methods on an Intel Pentium 3.0 GHz PC with 1.0 Gbyte random access memory. As illustrated in Table 1, the calculation time of the rigorous Mie theory increased rapidly with the increased radius. For the index model shown in Eq. (16), 共R兲 can be expressed by a unique function of Eq. (17) and L can be expressed by Eqs. (18). The calculation time of the GOA theory varied from 2 to 110 ms. With the index model expressed by Eq. (15), 共R兲 and L can only be numerically integrated by a general-purpose integrator that uses a 21-point Gauss–Kronrod rule and the calculation time of the case (70 – 8820 ms) is much more than that of the index model from Eq. (16). 5. Conclusions
We have treated light scattering as the superposition of diffraction, reflection, and refraction. An approxi5246
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mation method or the GOA method has been developed to calculate the scattered light pattern within the forward direction (0 – 60°) for GRIN particles. With two typical continuous refractive-index models, the calculation time of the GOA theory is compared with that obtained from the Mie theory and the Debye series and the CPU time is greatly decreased. When the index difference is out of the valid range, the surface wave plays an important role and the contribution of N ⫽ 3 rays should be taken into account. However, it is difficult to derive the incident angle of N ⫽ 2 rays from the scattering angle of N ⫽ 3 rays and to combine numerically the surface wave with geometric rays. Neither can it work well for particles with a dimensionless parameter ␣ of less than 50.
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