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Mie scattering efficiency of a large spherical particle embedded in an absorbing medium
Journal of Quantitative Spectroscopy & Radiative Transfer 70 (2001) 709–714 www.elsevier.com/locate/jqsrt
Mie scattering e(ciency of a large spherical particle embedded in an absorbing medium I. Wayan Sudiartaa ; ∗ , Petr Chyleka; b a
Atmospheric Science Program, Department of Physics, Dalhousie University, Halifax, Nova Scotia, Canada B3H 3J5 b Department of Physics, New Mexico State University, Las Cruces, NM 88003, USA
Abstract The Mie scattering calculations are usually performed for non-absorbing spherical particles embedded in a non-absorbing medium. We consider a case of an absorbing sphere placed in an absorbing medium. We 2nd, by numerical calculation for large size parameter of the order of 104 , that the scattering e(ciency of a spherical particle in an absorbing medium approaches the re4ectance of a plane surface at perpendicular incidence. ? 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction The Mie scattering e(ciency, Qsca , is given by Qsca (x; m) =
∞
2 (2n + 1)(|an |2 + |bn |2 ); x2
(1)
n=1
where x is size parameter and an and bn are the scattered 2eld expansion coe(cients [1,2]. For the case of a spherical particle embedded in a non-absorbing medium, Herman [3] suggested the asymptotic limit of the scattering e(ciency to be lim Qsca = 1 + R(0◦);
x→∞
∗
Corresponding author. Tel.: +1-902-494-5191; fax: +1-902-494-5191. E-mail address: [email protected] (I.W. Sudiarta). 0022-4073/01/$ - see front matter ? 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 2 - 4 0 7 3 ( 0 1 ) 0 0 0 3 9 - 5
(2)
710
I.W. Sudiarta, P. Chylek / Journal of Quantitative Spectroscopy & Radiative Transfer 70 (2001) 709–714
Fig. 1. Scattering e(ciency of a sphere with index of refraction 1.33 +0.1i embedded in a non-absorbing host medium (mmed = 1). The crosses show computed Qsca . The limiting value, given by Eq. (3), is 1.069299. ◦
where R(0 ) = |(m − 1)=(m + 1)|2 is a re4ectance of a plane surface at normal incidence and m is a complex relative refractive index of a sphere. This limit seems to be intuitively plausible if we consider that the sphere becomes a planar surface at an in2nitely large radius. Thus, the Qsca is expected to be equal to one (the diFraction part) plus the re4ection of a planar surface. The same limiting value was deduced by Chylek [4,5] by considering an asymptotic limit of the Eq. (1). Acquista et al. [6] argued that the Chylek’s derivation may not be correct, since some terms were neglected without a proper justi2cation. Van de Hulst [2] and later Deirmendjian [7] suggested an alternative expression for an asymptotic form of a scattering e(ciency of a large absorbing sphere in the form =2 lim Qsca = 1 + (|r1 |2 + |r2 |2 ) cos( )sin( ) d ; (3) x→∞
0
where r1 and r2 are the Fresnel re4ection coe(cients [1,2] for radiation incident at the angle . The 2rst term in Eq. (3) represents the diFraction and the second term the radiation re4ected at the front surface of a sphere. Bohren and Herman [8] provided numerical calculations for a sphere of the size parameter up to the order of 104 to show that the numerical results, using the Mie scattering e(ciency (1), seem to approach the limit given by Eq. (3). Our calculation of the Qsca extended to the size parameter of the order of 105 supports this conclusion (Fig. 1). To verify this numerical result, we have calculated the absorption e(ciency (Fig. 2) by integrating the internal 2elds and obtained the scattering e(ciency from the relation lim Qsca = 2 − lim Qabs :
x→∞
x→∞
(4)
Comparison of results obtained from Eqs. (3) and (4) is provided in Table 1. 2. Scattering efficiency of a spherical particle embedded in an absorbing medium In some applications we may have particles embedded in an absorbing medium. As an example we may consider small particles submerged in an absorbing liquid, or atmospheric aerosols
I.W. Sudiarta, P. Chylek / Journal of Quantitative Spectroscopy & Radiative Transfer 70 (2001) 709–714
711
Fig. 2. Absorption e(ciency of a spherical particle with the refractive index m1 = 1:33 + 0:1i and m2 = 1:75 + 0:58i embedded in a non-absorbing host medium (mmed = 1). The crosses and circles show computed Qabs for refractive indices m1 and m2 , respectively.
Table 1 Calculated large radius limit of Qabs and the right hand side of Eq. (3) M 1.33 1.75 1.55 1.29
+ + + +
0.1i 0.58i 0.155i 0.0472i
Qabs
Qsca = 2 − Qabs
Qsca from Eq. (3)
0.9307 0.8310 0.8964 0.9396
1.0693 1.1690 1.1036 1.0604
1.069299 1.169033 1.103645 1.060395
at the wavelength where the atmosphere itself has a non-negligible absorption (for example at infrared wavelengths due to absorption by atmospheric gases). The scattering formalism for spherical particles embedded in an absorbing medium was considered previously by several investigators [9 –12]. The scattering e(ciency of a spherical particle embedded in an absorbing medium can be written in the form [12,13] 1 Qsca (x; m) = 2(exp(2xI )=2xI + (1 − exp(2xI ))=(2xI )2 ) ∞ 2 ∗ 2 ∗ 2 ∗ Re k (2n + 1)(−i|an | n (x)n (x) + i|bn | n (x)n (x)) ; (5) |x|2 Re(k) n=1
where x = 2 rmmed = is a complex size parameter (r is the radius of sphere, mmed the refractive index of host medium, the wavelength in vacuum), k is the wave number, xI is an imaginary part of the size parameter, n (x) and n (x) are the Ricatti–Bessel functions and their derivatives
712
I.W. Sudiarta, P. Chylek / Journal of Quantitative Spectroscopy & Radiative Transfer 70 (2001) 709–714
Fig. 3. Qsca for a sphere with refractive index m1 = 1:33 + 0:1i and m2 = 3:0 + 0:1i, embedded in an absorbing medium with refractive index mmed = 1 + 10−4 i. R1 (0) = 0:021861 and R2 (0) = 0:250468 is the re4ectance of a planar interface at normal incidence, for refractive indices m1 and m2 , respectively.
with respect to the argument, Re means a real part of and asterisk denotes a complex conjugate expression. The above formulation is identical to that of Lebedev et al. [12], except that the term in the squared brackets—representing the decaying incident 2eld—was added [9,13]. 3. Numerical results The scattering e(ciency of a spherical particle embedded in an absorbing medium (calculated from the Eq. (5)) for the real part of the size parameter up to the order of 104 is shown in Fig. 3. It is apparent that the scattering e(ciency approaches the value given by the re4ection of a 4at surface at perpendicular incidence, suggesting that we have lim Qsca = R(0◦):
x→∞
(6)
Comparing this expression with those for a particle in a non-absorbing medium, we see that: (a) the diFraction term (equal to one in the case of non-absorbing medium) is not present, and (b) the term representing the re4ection by a sphere in Eq. (3) is replaced by a re4ection by a 4at surface at perpendicular incidence. The calculations for diFerent refractive indices of the absorbing sphere (Fig. 4) suggest an agreement with the asymptotic limit given by Eq. (5). The convergence of Qsca to its asymptotic limit is much faster in an absorbing medium than in non-absorbing one. 4. Conclusion and discussion In the large sphere limit the Mie scattering e(ciency for a spherical particle embedded in a non-absorbing medium approaches the limit given by Eq. (3), representing the sum of diFraction contribution and the geometric optics limit of the scattering by a sphere. In the case
I.W. Sudiarta, P. Chylek / Journal of Quantitative Spectroscopy & Radiative Transfer 70 (2001) 709–714
713
Fig. 4. Scattering e(ciency for diFerent refractive indices of sphere with real part of the size parameter =5000; the imaginary part of a refractive index of a sphere is 0.1 and the refractive index of the host medium is mmed =1+0:001i. R(0) is the planar re4ectance at normal incidence. The Integration of R means the integral on the right hand side of Eq. (3).
of a spherical particle embedded in an absorbing medium, the scattering e(ciency approaches the Fresnel re4ection coe(cient for a plane surface at perpendicular incidence. There seems to be no “diFraction” term (equal to one in the case of non-absorbing medium) and the scattering by a sphere (the second term of the right hand side of Eq. (3)) is replaced by a scattering by a plane surface at perpendicular incidence (the second term on the right hand side of Eq. (2)). We conjecture that the radiation that would reach the sphere tangentially in a non-absorbing medium is absorbed by an absorbing medium, in the case of a large sphere. This missing (absorbed) tangential radiation leads to the removal of the diFraction contribution and to the modi2cation of the re4ectance of a sphere (the second term in Eq. (3)) into the re4ectance by a plane surface at perpendicular incidence (the second term in Eq. (2)). Acknowledgements The reported research was supported in part by the Natural Sciences and Engineering Research Council of Canada and by Atmospheric Science section of the U.S. National Science Foundation. The authors thank to Gorden Videen for helpful discussions. References [1] Bohren CF, HuFman DR. Absorption and scattering of light by small particles. New York: Wiley; 1983. [2] Van de Hulst, HC. Light scattering by small particles. New York: Wiley; 1957. p. 225 – 6. [3] Herman M. Infra-red absorption, scattering, and total attenuation cross-sections for water spheres. Quart J R Meteorol Soc 1962;88:143–50. [4] Chylek P. Large-sphere limits of the Mie-scattering functions. J Opt Soc Am 1973;63(6):699–706. [5] Chylek P. Asymptotic limits of the Mie-scattering characteristics. J Opt Am 1975;65(11):1316–8. [6] Acquista C, Cohen A, Cooney JA, Wimp J. Asymptotic behavior of the e(ciencies in Mie scattering. J Opt Soc Am 1980;70(8):1023–5. [7] Bohren CF, Herman BM. Asymptotic scattering e(ciency of a large sphere. J Opt Soc Am 1979;69(11): 1615–6.
714
I.W. Sudiarta, P. Chylek / Journal of Quantitative Spectroscopy & Radiative Transfer 70 (2001) 709–714
[8] Deirmendjian, D. Electromagnetic scattering on spherical polydispersions. New York: Elsevier; 1969. p. 34 –7. [9] Mundy WC, Roux JA, Smith AM. Mie scattering by sphere in an absorbing medium. J Opt Soc Am 1974;64(12):1593–7. [10] Chylek P. Light scattering by small particles in an absorbing medium. J Opt Soc Am 1977;67(4)561–3. [11] Quinten M, Rostalki J. Lorenz-Mie theory for sphere immersed in an absorbing host medium. J Part Part System Charact 1996;13:89–96. [12] Lebedev N, Gartz M, Kreibig U, Stenzel O. Optical extinction by spherical particles in an absorbing medium: application to composite absorbing 2lms. Eur Phys J D 1999;6:365–73. [13] Sudiarta IW, Chylek P. Mie scattering formalism for spherical particles embedded in an absorbing medium. J Opt Soc Am A 2001; in press.
Mie scattering e(ciency of a large spherical particle embedded in an absorbing medium I. Wayan Sudiartaa ; ∗ , Petr Chyleka; b a
Atmospheric Science Program, Department of Physics, Dalhousie University, Halifax, Nova Scotia, Canada B3H 3J5 b Department of Physics, New Mexico State University, Las Cruces, NM 88003, USA
Abstract The Mie scattering calculations are usually performed for non-absorbing spherical particles embedded in a non-absorbing medium. We consider a case of an absorbing sphere placed in an absorbing medium. We 2nd, by numerical calculation for large size parameter of the order of 104 , that the scattering e(ciency of a spherical particle in an absorbing medium approaches the re4ectance of a plane surface at perpendicular incidence. ? 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction The Mie scattering e(ciency, Qsca , is given by Qsca (x; m) =
∞
2 (2n + 1)(|an |2 + |bn |2 ); x2
(1)
n=1
where x is size parameter and an and bn are the scattered 2eld expansion coe(cients [1,2]. For the case of a spherical particle embedded in a non-absorbing medium, Herman [3] suggested the asymptotic limit of the scattering e(ciency to be lim Qsca = 1 + R(0◦);
x→∞
∗
Corresponding author. Tel.: +1-902-494-5191; fax: +1-902-494-5191. E-mail address: [email protected] (I.W. Sudiarta). 0022-4073/01/$ - see front matter ? 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 2 - 4 0 7 3 ( 0 1 ) 0 0 0 3 9 - 5
(2)
710
I.W. Sudiarta, P. Chylek / Journal of Quantitative Spectroscopy & Radiative Transfer 70 (2001) 709–714
Fig. 1. Scattering e(ciency of a sphere with index of refraction 1.33 +0.1i embedded in a non-absorbing host medium (mmed = 1). The crosses show computed Qsca . The limiting value, given by Eq. (3), is 1.069299. ◦
where R(0 ) = |(m − 1)=(m + 1)|2 is a re4ectance of a plane surface at normal incidence and m is a complex relative refractive index of a sphere. This limit seems to be intuitively plausible if we consider that the sphere becomes a planar surface at an in2nitely large radius. Thus, the Qsca is expected to be equal to one (the diFraction part) plus the re4ection of a planar surface. The same limiting value was deduced by Chylek [4,5] by considering an asymptotic limit of the Eq. (1). Acquista et al. [6] argued that the Chylek’s derivation may not be correct, since some terms were neglected without a proper justi2cation. Van de Hulst [2] and later Deirmendjian [7] suggested an alternative expression for an asymptotic form of a scattering e(ciency of a large absorbing sphere in the form =2 lim Qsca = 1 + (|r1 |2 + |r2 |2 ) cos( )sin( ) d ; (3) x→∞
0
where r1 and r2 are the Fresnel re4ection coe(cients [1,2] for radiation incident at the angle . The 2rst term in Eq. (3) represents the diFraction and the second term the radiation re4ected at the front surface of a sphere. Bohren and Herman [8] provided numerical calculations for a sphere of the size parameter up to the order of 104 to show that the numerical results, using the Mie scattering e(ciency (1), seem to approach the limit given by Eq. (3). Our calculation of the Qsca extended to the size parameter of the order of 105 supports this conclusion (Fig. 1). To verify this numerical result, we have calculated the absorption e(ciency (Fig. 2) by integrating the internal 2elds and obtained the scattering e(ciency from the relation lim Qsca = 2 − lim Qabs :
x→∞
x→∞
(4)
Comparison of results obtained from Eqs. (3) and (4) is provided in Table 1. 2. Scattering efficiency of a spherical particle embedded in an absorbing medium In some applications we may have particles embedded in an absorbing medium. As an example we may consider small particles submerged in an absorbing liquid, or atmospheric aerosols
I.W. Sudiarta, P. Chylek / Journal of Quantitative Spectroscopy & Radiative Transfer 70 (2001) 709–714
711
Fig. 2. Absorption e(ciency of a spherical particle with the refractive index m1 = 1:33 + 0:1i and m2 = 1:75 + 0:58i embedded in a non-absorbing host medium (mmed = 1). The crosses and circles show computed Qabs for refractive indices m1 and m2 , respectively.
Table 1 Calculated large radius limit of Qabs and the right hand side of Eq. (3) M 1.33 1.75 1.55 1.29
+ + + +
0.1i 0.58i 0.155i 0.0472i
Qabs
Qsca = 2 − Qabs
Qsca from Eq. (3)
0.9307 0.8310 0.8964 0.9396
1.0693 1.1690 1.1036 1.0604
1.069299 1.169033 1.103645 1.060395
at the wavelength where the atmosphere itself has a non-negligible absorption (for example at infrared wavelengths due to absorption by atmospheric gases). The scattering formalism for spherical particles embedded in an absorbing medium was considered previously by several investigators [9 –12]. The scattering e(ciency of a spherical particle embedded in an absorbing medium can be written in the form [12,13] 1 Qsca (x; m) = 2(exp(2xI )=2xI + (1 − exp(2xI ))=(2xI )2 ) ∞ 2 ∗ 2 ∗ 2 ∗ Re k (2n + 1)(−i|an | n (x)n (x) + i|bn | n (x)n (x)) ; (5) |x|2 Re(k) n=1
where x = 2 rmmed = is a complex size parameter (r is the radius of sphere, mmed the refractive index of host medium, the wavelength in vacuum), k is the wave number, xI is an imaginary part of the size parameter, n (x) and n (x) are the Ricatti–Bessel functions and their derivatives
712
I.W. Sudiarta, P. Chylek / Journal of Quantitative Spectroscopy & Radiative Transfer 70 (2001) 709–714
Fig. 3. Qsca for a sphere with refractive index m1 = 1:33 + 0:1i and m2 = 3:0 + 0:1i, embedded in an absorbing medium with refractive index mmed = 1 + 10−4 i. R1 (0) = 0:021861 and R2 (0) = 0:250468 is the re4ectance of a planar interface at normal incidence, for refractive indices m1 and m2 , respectively.
with respect to the argument, Re means a real part of and asterisk denotes a complex conjugate expression. The above formulation is identical to that of Lebedev et al. [12], except that the term in the squared brackets—representing the decaying incident 2eld—was added [9,13]. 3. Numerical results The scattering e(ciency of a spherical particle embedded in an absorbing medium (calculated from the Eq. (5)) for the real part of the size parameter up to the order of 104 is shown in Fig. 3. It is apparent that the scattering e(ciency approaches the value given by the re4ection of a 4at surface at perpendicular incidence, suggesting that we have lim Qsca = R(0◦):
x→∞
(6)
Comparing this expression with those for a particle in a non-absorbing medium, we see that: (a) the diFraction term (equal to one in the case of non-absorbing medium) is not present, and (b) the term representing the re4ection by a sphere in Eq. (3) is replaced by a re4ection by a 4at surface at perpendicular incidence. The calculations for diFerent refractive indices of the absorbing sphere (Fig. 4) suggest an agreement with the asymptotic limit given by Eq. (5). The convergence of Qsca to its asymptotic limit is much faster in an absorbing medium than in non-absorbing one. 4. Conclusion and discussion In the large sphere limit the Mie scattering e(ciency for a spherical particle embedded in a non-absorbing medium approaches the limit given by Eq. (3), representing the sum of diFraction contribution and the geometric optics limit of the scattering by a sphere. In the case
I.W. Sudiarta, P. Chylek / Journal of Quantitative Spectroscopy & Radiative Transfer 70 (2001) 709–714
713
Fig. 4. Scattering e(ciency for diFerent refractive indices of sphere with real part of the size parameter =5000; the imaginary part of a refractive index of a sphere is 0.1 and the refractive index of the host medium is mmed =1+0:001i. R(0) is the planar re4ectance at normal incidence. The Integration of R means the integral on the right hand side of Eq. (3).
of a spherical particle embedded in an absorbing medium, the scattering e(ciency approaches the Fresnel re4ection coe(cient for a plane surface at perpendicular incidence. There seems to be no “diFraction” term (equal to one in the case of non-absorbing medium) and the scattering by a sphere (the second term of the right hand side of Eq. (3)) is replaced by a scattering by a plane surface at perpendicular incidence (the second term on the right hand side of Eq. (2)). We conjecture that the radiation that would reach the sphere tangentially in a non-absorbing medium is absorbed by an absorbing medium, in the case of a large sphere. This missing (absorbed) tangential radiation leads to the removal of the diFraction contribution and to the modi2cation of the re4ectance of a sphere (the second term in Eq. (3)) into the re4ectance by a plane surface at perpendicular incidence (the second term in Eq. (2)). Acknowledgements The reported research was supported in part by the Natural Sciences and Engineering Research Council of Canada and by Atmospheric Science section of the U.S. National Science Foundation. The authors thank to Gorden Videen for helpful discussions. References [1] Bohren CF, HuFman DR. Absorption and scattering of light by small particles. New York: Wiley; 1983. [2] Van de Hulst, HC. Light scattering by small particles. New York: Wiley; 1957. p. 225 – 6. [3] Herman M. Infra-red absorption, scattering, and total attenuation cross-sections for water spheres. Quart J R Meteorol Soc 1962;88:143–50. [4] Chylek P. Large-sphere limits of the Mie-scattering functions. J Opt Soc Am 1973;63(6):699–706. [5] Chylek P. Asymptotic limits of the Mie-scattering characteristics. J Opt Am 1975;65(11):1316–8. [6] Acquista C, Cohen A, Cooney JA, Wimp J. Asymptotic behavior of the e(ciencies in Mie scattering. J Opt Soc Am 1980;70(8):1023–5. [7] Bohren CF, Herman BM. Asymptotic scattering e(ciency of a large sphere. J Opt Soc Am 1979;69(11): 1615–6.
714
I.W. Sudiarta, P. Chylek / Journal of Quantitative Spectroscopy & Radiative Transfer 70 (2001) 709–714
[8] Deirmendjian, D. Electromagnetic scattering on spherical polydispersions. New York: Elsevier; 1969. p. 34 –7. [9] Mundy WC, Roux JA, Smith AM. Mie scattering by sphere in an absorbing medium. J Opt Soc Am 1974;64(12):1593–7. [10] Chylek P. Light scattering by small particles in an absorbing medium. J Opt Soc Am 1977;67(4)561–3. [11] Quinten M, Rostalki J. Lorenz-Mie theory for sphere immersed in an absorbing host medium. J Part Part System Charact 1996;13:89–96. [12] Lebedev N, Gartz M, Kreibig U, Stenzel O. Optical extinction by spherical particles in an absorbing medium: application to composite absorbing 2lms. Eur Phys J D 1999;6:365–73. [13] Sudiarta IW, Chylek P. Mie scattering formalism for spherical particles embedded in an absorbing medium. J Opt Soc Am A 2001; in press.