Near-field scattering by passive and active small sphericalparticles
Ehud Tirosh and Ariel Cohen
Department of Atmospheric Sciences, Hebrew University of Jerusalem, Jerusalem, Israel
Received February 1, 1988; accepted November 22, 1988
Near-field scattering functions of small spheres are discussed and presented as a function of the distance from thescatterer. Although the total scattered energy defined by the angular integral of the scattering function does notdepend on the distance from the particle, the scattering at any given angle in the near field does. In particular,apparent negative scattered energy flux values along the radius vector in given angular ranges are emphasized. Theappearance of negative values was reported previously; however, they are shown here to reach any high normalizedvalue, relative to the total scattering, depending mainly on the scatterer size, the distance from the scatterer, and thecomplex index of refraction. The negative values that were treated previously merely as numerical results areexplained in terms of the Ricatti-Bessel-function asymptotic expansion. It is also shown that, excluding thescattering region near 90 deg, the physical significance of these high negative values in the near field is usuallynegligible relative to the overall field when the coherent interference with the incident wave is taken into account.In the far field the negative values disappear.
INTRODUCTION
The near-field scattering functions play an important role inthe determination of the minimal distances between parti-cles required for using single-scattering values and multiple-scattering calculations. For example, only when the dis-tances between spherical particles can be regarded as the farfield in terms of the asymptotic behavior of the Ricatti-Bessel functions can the scattered field be treated as a spher-ical wave depending on R'1.
When the behavior of the near field is examined, a series ofnew phenomena can be observed (see, for example, Ref. 1).Kamiuto2' 3 reported the existence of negative scattered in-tensities in a certain range of scattering angles in the nearfield along the radial direction, although the integral over allscattering angles along the radial direction is independent ofthe distance from the particle.
This result means that the energy flow of the scatteringwave alone can be directed toward the particle, within thementioned scattering-angle ranges.
Below we investigate the effect of the size parameter onthe normalized negative range of values for active and pas-sive particles (particles with negative and positive imaginaryparts of the refractive index). Since the near scattered fieldof a very small particle (a = 2ra/X << 1) always interfereswith the incident field, it is shown that the energy flow hasno physical significance in most angular ranges. It is impor-tant to note that disregarding the interference with the inci-dent field results in other apparent paradoxes, as reported,for example, by Kerker et al.
4 The scattering functions foractive emitting spheres predict that the negative extinctionof such particles, meaning the emission of the active particle,creates a parallel beam. The emitted beam is a result of theconstructive interference between the incident and the scat-tered fields as explained previously by Cohen. 5 It is sug-gested also that, in the case of small passive (or active)
particles, the negative (or positive) values of the scatteredenergy cannot be treated independently without accountingfor the presence of the incident field.
For large particles this effect disappears.
THEORY
The overall energy flux in the region of a scattering particlecan be expressed in terms of the well-known Poynting vec-tor 6
(S) = 1/2 Re(E X H).
This vector is given by
(S) = (S) + (S5 ) + (S'),
where
(Si) = /2 Re[E(') X R(i)],
(S,) = /2 Re[E(s) X (s)],
(S') = 1/2 Re[E(') X H(s) + E(s) X H(i)]
(1)
(2)
(3)
(4)
(5)
and where i and s represent the incident and the scatteredfields, respectively.
For a nonabsorbing medium the integral of the incidentradiation [Eq. (3)] over all direction vanishes. The integralsof Eqs. (4) and (5) over all scattering directions yield thescattering and extinction cross sections, respectively.
In far-field directional scattering measurements, the scat-tered radiation can be well separated from the incident radi-ation, and therefore only Eq. (4) is taken into account.
This cannot be true for near-field measurements, andboth Eq. (3) and Eq. (5) must be taken into account as well.
The normalized phase functions for a randomly polarizedincident radiation are defined by
where A and r are the Ricatti-Bessel functions.P8(0) is normalized according to
1/2 fPs(J)dW = 1,
where A = cos(0).
(16)
(17)
ASYMPTOTIC RELATIONS
By substitution of the asymptotic values of the Ricatti-Bessel function for large arguments into Eq. (7), the well-
(8) known far-field expressions for P8 (0) are obtained.The asymptotic relations are
lim &11W(z) = (i)+leiZ,Z_aa
(9)
(11)
(12)
(13)
(14)
for which a is the dimensionless size of the particle, a = 27ra/X, where a is the particle radius and X is the wavelength inthe medium; n is the dimensionless distance from the parti-cle center, n = r/a, where r is the distance; ti') is the Ricatti-Bessel function of the second kind, which is defined byt$')(z) = zh(1)(z), where h(1)(z) is the spherical Hankel func-tion of the first kind; p}1) is the associated Legendre func-tion; and mB1 and eBI are the magnetic and electric Miecoefficients, respectively, given by
and hence the far-field P8 (0) values are bounded.There is a special interest in the asymptotic behavior of
the Ricatti-Bessel function for small values of the argument.This approximation is useful for the analysis of near-fieldscattering from small particles.
The asymptotic expressions are7
* l) ~~(an) 1+1 .1 3 5(21 + 1)lim 01(an) = -
an-0 ~ 1 3 .5 ... (21+ 1) (a) 1
li V)(a7= (I + I)°fB +il-- 3 5(21 + 1)aij-0 1 3 -5--.(21 + 1) or)l
(19)
Since an is real, these expressions generally tend to have asmall real value, but they have an infinite imaginary valuewhen an is small.
Since eB, and mB can have in practice any complex value,the importance of the imaginary parts of Eqs. (19) is clear.
The asymptotic relations for eBI and mBl for a - 0 are 6
a21+1 neB il
a21+3
l1(1 + 1)(21 + 1)[1 .3. 5...(21 +1)]2 (n 1), (20)
where n is the complex refractive index.It should be emphasized that, although approximations
(20) are valid for complex values of n, erroneous results mayoccur if they are used for real values of n.
The reason is that in relations (20) either the real part orthe imaginary part is identically zero for real n, in contradic-
(15)
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(c)Fig. 1. Normalized phase function P8 (O) for (a) a = 0.001,q = 1; (b)a = 0.001, 7 = 10; (c) a = 0.001, t7 = 100. Symbols: 0, n = 2.5; X, n= 2.5 ± 0.75i.
(c)Fig. 2. Normalized phase function P(O) for (a) a = 0.01, = 1; (b) a= 0.01, i = 10; (c) a = 0.01, n >> 1 (far field). Symbols: 0, n = 2.5;X, n = 2.5 0.75i.
526 J. Opt. Soc. Am. A/Vol. 6, No. 4/April 1989
tion to the exact expression. This nonzero part is the domi-nant one in P3 (0) calculations and in absorption cross-sec-tion calculations and therefore cannot be neglected.
The asymptotic expression for W is
W= - 4 6l n 2 - 1 23 2 + 2 (a << 1). (21)
A CDC Cyber 180/855 computer was used, and the calcula-tions were made at single precision, which had an accuracy of13 digits.
RESULTS
Near-Field Scattered RadiationThe near-field scattering was calculated for a wide spectrumof parameters by using the methods described in the preced-ing section. Representative results are shown in Figs. 1-4.Refractive-index values were n = 2.5 (scattering particle),2.5 + 0.75i (absorbing particle), and 2.5 - 0.75i (active parti-cle). The size parameter a was between 0.001 and 1.
20-
10
LLJ QI0-
-20-
-20- i. . . . . . .i . . . . .
-IH
- I
H
In
100THETA
(a)
0 50 100 150THETA
(a)
THETA
(b)
1 .
I1.
I1.0
uIJIa
0 50 100 150 200THETA
(b)
Fig. 3. Normalized phase function P 8(O) for (a) a = 0.1,?) = 1; (b) a= 0.1, - = 10. Symbols: 0 n = 2.5; X, n = 2.5 + 0.75i;A, n = 2.5 -0.75i.
3.
2
0 50 100 150 200THETA
(c)
Fig. 4. Normalized phase function P(0) for (a) a = 1, i = 1; (b) a =1, 71 = 10; (c) a = 1, ? >> 1 (far field). Symbols: 0, n = 2.5; X n = 2.5+ 0.75i; A, n = 2.5 - 0.75i.
200
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Table 1. Leading Terms for Different n for a Small Particlea
Value for the Following nTerm 1 10 100
P1(1)Q1(l) -0.492553 X 10-18 -0.492552 X 10-18 -0.492552 X 10-18PA(1)Q2 (1) -0.126216 X 10-12 -0.126204 X 10-16 -0.124967 X 10-20
-P 2 (1)Q1 (1) -0.126217 X 0-18 -0.126229 X 10-20 -0.127479 X 10-22Pj(1)Q 1(2) -0.129871 X 10-12 -0.129867 X 10-16 -0.129445 X 10-20P 1(1)Q2 (2) 0.175911 X 10-16 0.175903 X 10-21 0.175278 X 10-26P1(1)Q1(3) -0.863838 X 0-18 -0.863788 X 10-23 -0.844755 X 10-28P1(2)Qj(1) 0.259743 X 10 -12 0.259755 X 10-16 0.261034 X 10-20P1 (2)Q2 (1) -0.342682 X 10-16 -0.342665 X 10-21 -0.340544 X 10-26P1(3)Qj(1) 0.259152 X 10-17 0.259168 X 10-22 0.262880 X 10-27Rj(1)Sj(1) -0.492553 X 0-18 -0.492552 X 10-18 -0.492552 X 10-18R1(1)S2 (1) -0.126216 X 10-12 -0.126204 X 10-16 -0.124967 X 10-20
-R 2 (1)S1 (1) -0.126217 X 10-18 -0.126229 X 10-20 -0.127479 X 10-22R 1(1)S1 (2) -0.129871 X 10-12 -0.129867 X 10-16 -0.129445 X 10-20Rl(l)S 2 (2) 0.175911 X 10-16 0.175903 X 10-21 0.175278 X 10-26R 1 (1)S1 (3) -0.863838 X 0-18 -0.863788 X 10-23 -0.844755 X 10-28R1(2)Si(l) 0.259743 X 10-12 0.259755 X 10-16 0.261034 X 10-20R1 (2)S2 (1) -0.342682 X 10-16 -0.342665 X 10-21 -0.340544 X 10-26Rj(3)Sj(1) 0.259152 X 10-17 0.259168 X 10-22 0.262880 X 10-27
P.(0) -0.727903 X 10-14 .250987 X 10-18 0.984997 X 10-18
a = 0.001; n = 2.5 + 0.75i; 0 = 0.
As can be seen from Figs. 1-4, one can get any large valuefor IP8(0)I in the near field, provided that a is small.
In the region where 1P8(0)I >> 1, all curves have a similarshape, and P8(0) a-2n-4. This fact is directly related tothe large values of the imaginary parts of r and ' when theirarguments become small. This make terms higher than thefirst the dominant terms in the expansion of Eq. (7).
In order to verify this fact and to obtain some quantitativeestimations of the relative contribution of each term, wecalculated the different terms in Eq. (7) separately.
For the sake of convenience Eq. (7) is written as
P =(O) = Re i (Pl - P2) X 3 (Ql - Q2)
+ (R - R') X 3 (31-32)J/J- (22)
For the definitions of P1, P2, etc., see the analogous Eqs.(11)-(14).
Examples of the relative contributions of the differentterms to P8(0) are given for a small particle (a = 0.001) inTable 1 and for a medium particle (a = 1) in Table 2. Thedifferent terms can be identified according to Eq. (22).
Some points related to these tables should be emphasized:
(1) The first term, which is the Rayleigh dipole term, isindependent of ?, as can be proved analytically. It can beseen that in the far field it becomes the dominant term for,the small particle, where it exhibits Rayleigh scattering. Inthe near-field region, other terms become dominant for thesmall particle, and they are some orders of magnitude biggerthan the dipole term.
(2) For bigger particles a is not small even when 7 = 1,and all the terms are weakly dependent on ; hence the near-field scattering does not vary as much from the far-field
Table 2. Leading Terms for Different X for a MediumParticlea
-P2(1)Q(1) -0.238571 -0.120478-P 2 (1)Q2 (1) -0.524887 X 10-1 -0.524887 X 10-1
Pj(1)Q 1 (2) -0.147092 -0.382894 X 10-1Pl (1)Q2 (2) 0.244051 X 10-1 -0.669113 X 10-2
-P 2 (1)QI(2) -0.713175 X 10-1 -0.721501 X 10-2-P 2 (1)Q2 (2) 0.137822 X 10-1 -0.226691 X 10-2
P1 (2)Qj(1) 0.228661 -0.375754 X 10-1Pj(2)Q 2 (1) -0.533082 X 10-1 -0.695554 X 10-2
-P 2 (2)Ql(l) -0.398561 X 10-2 -0.697134 X 10-2-P 2 (2)Q2 (1) -0.887881 X 10-2 -0.230996 10-2
P1(2)Q1(2) -0.378827 X 10-2 -0.378827 X 10-2P1 (2)Q2 (2) -0.176873 X 10-1 -0.504627 X 10-3
-P 2 (2)Q1 (2) -0.676280 X 10-2 -0.536290 X 10-3
-P 2(2)Q2(2) -0.115539 X 10-3 -0.115539 X 10-3Rj(1)Sj(1) -0.439372 -0.439372RI(1)S2(1) -0.119285 -0.118104
-R 2(1)SI(l) -0.238571 -0.120478-R2 (1)S2 (1) -0.524887 X 10-1 -0.524887 X 10-1
Rj(1)S 1 (2) -0.147092 -0.382894 X 10-1Rl(l)S 2 (2) 0.244051 X 10-1 -0.669113 X 10-2
-R2(1)S,(2) -0.713175 X 10-1 -0.721501 X 10-2-R2(1)S2(2) 0.137822 X 10-1 -0.226691 X 10-2
Rj(2)S 1 (1) 0.228661 -0.375754 X 10-1R1(2)S2 (1) -0.533082 X 10-1 -0.695554 X 10-2
-R 2 (2)S(1) -0.398561 X 10-2 -0.697134 X 10-2-R2(2)S2(1) -0.887881 X 10-2 -0.230996 X 10-2
Rl(2)S 1 (2) -0.378827 X 10-2 -0.378827 X 10-2R1(2)S2 (2) -0.176873 X 10-1 -0.504627 X 10-3
-R2(2)S1 (2) -0.676280 X 10-2 -0.536290 X 10-3-R 2 (2)S 2 (2) -0.115539 X 10-3 -0.115539 X 10-3
P'(0) 0.158751 X 101 0.169557 X 101
a = 1; n = 2.5 + 0.75i; 0 =0.
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528 J. Opt. Soc. Am. A/Vol. 6, No. 4/April 1989
scattering as it would for small particles. This fact is appar- X1012
ent from Figs. 4(a) and 4(b). 4
Total and Scattered FieldsIn Figs. 5-7 the normalized incident field Pi [Eq. (7)] and the 2 -total normalized field PTOT = P, + Pi + P' are given for a =0.001, 0.1, 1.0 for some values of?).
The vector Pi represents the electromagnetic radiation in L 0the absence of a particle, whereas PTOT represents the dis- tturbed radiation in the presence of the particle. The physi- Ccal interpretations of Figs. 5-7 are clear:
-2-
(1) At the near field, as is mentioned above, only thetotal radiation has a real physical meaning. Comparison ofthe data of Figs. 1-4 with those of Figs. 5-7 shows that the I4total radiation at the vicinity of the small particle exceeds 0 50 100 150 200
the scattered radiation by many orders of magnitude at THETAscattering angles that are not in the region of 90 deg. The (a)difference between the total radiation and the incident radi-ation gives an indication of the amount of disturbance exhib- X1013ited by the particle. The scattered field is, of course, negligi- 2
ble, and no paradoxical inward flux really occurs. It isinteresting that in the case of a = 1 Ps is of the same order ofmagnitude as PTOT and that PTOT is larger than P in theforward direction, which means that constructive interfer- Ience occurs.
(2) Going farther from the particle, PTOT approaches the <value of Pi. This represents, of course, the fact that, far I° 0enough from the particle, its influence on the original inci- IH -dence field is negligible. a_
For larger particles, the influence on the original incidencefield is also larger, and therefore the distance at which PTOTand Pi coincide becomes larger too, as can be seen from Fig.7(c). However, far enough from the particle, P can beseparated from Pi (in particular in the backward direction, -2where in fact P and Pi are of opposite signs), although the 0 50 100 150 200
former is much smaller than the latter. In this case P, is the THETAusual far-field scattered radiation. (b)
The above facts can also be demonstrated well with some M14of Kamiuto's original data [Figs. 8(a)-8(c)]. 4
Scattering at 90 DegreesFrom Figs. 1-4 it can be seen that the scattering approaches 2-a minimum at a scattering angle near 90 deg. This fact isexplained readily by the observation that in this directionthe interaction between the incident radiation and the scat- U otered radiation is minimal. H
In order to see the detailed behavior of the scattering near90 deg, we calculated its value for a = 0.001 and for differentvalues of 7j. The real part of n had a constant value of 2.5, -2-
and its imaginary value was varied between -5 and 5. InFigs. 9(a)-9(c) the calculated results are shown.
The main results are as follows:0 50 200 50 200
(1) For a real Rayleigh particle, P8 (90 deg) = 0.75 (the THETAfar-field value) independently of . (c)
(2) In contrast a complex particle has this value only in Fig. 5. PTOT and P as functions of 0 for (a) n = 2.5, a = 0.001, 1 =;
the far field. In the near field there is an inward flux for an (b) n = 2.5, a = 0.001, = 2; (c) n = 2.5, a = 0.001, = 10. Symbols:absorbing particle and an outward flux for an active particle. 0, PTOT; X, Pi.
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(c)Fig. 6. PTOT and Pi as functions of 0 for (a) n = 2.5, a = 0.1, q = 1;(b) n = 2.5, a = 0.1, n = 2; (c) n = 2.5, a = 0.1, 7 = 10. Symbols: 0,PTOT; X, Pi.
THETA
(c)
Fig. 7. PTOT, Pi, and P8as functions of 0 for (a) n = 2.5, a = 1, q = 1;(b)n=2.5,a=1, = 2;(c)n=2.5,a=1,q7=10. Symbols: 0,PTOT; X, Pi; A, Ps.
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530 J. Opt. Soc. Am. A/Vol. 6, No. 4/April 1989 E. Tirosh and A. Cohen
200-
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0L -
0 0)
- 100 -
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THETA
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150 200 -i6
CLU
00)
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I |I 1 I I-4 -2 0
NI
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I I I I I |2 4 6
NI
(b)
6-
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000-
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200
Fig. 8. (a) Normalized phase function P(O) for n = 1.342 + 0.01iand a = 0.001. Symbols: 0, = 1; X, = 2. (b), (c) PTOT and Pi asfunctions of 0 for n = 1.342 + 0.01i, a = 0.25 and (b) = 1 or (c) t = 2.Symbols: 0, PTOT; X, Pi.
-6I I I I I 1
-4 -2 0
NI
(c)
I I I I I2 4 6
Fig. 9. (a) P8(0) as a function of the imaginary part of the refractiveindex (NI). The real part is 2.5, and a = 0.001. Symbols: 0,?) = 1;X,7= 1.5. (b) Detail of (a). Symbols: , = 1; -.-- , j = 1.5.(c) Same as (a) but for different values of : 0,) = 2; X, ? = 10.
6'
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Vol. 6, No. 4/April 1989/J. Opt. Soc. Am. A 531
The scattering intensity at 90 deg is approximately pro-portional to the imaginary part of the refractive index. Thechange in the 90-deg component with is due to the bendingof the field lines of the scattered Poynting vector that makesa contribution to the 90-deg flux from points at directionsother than 90°. This is a consequence of the tangentialcomponent of the scattered Poynting vector, which existsonly in the near-field region.
The same bending exists for the other near-field terms,namely, Pi + P'.
An enlightening graphic presentation of these field linesfor an absorbing particle was given by Bohren and Huff-man.1
As we can see from Fig. 9, these results do not hold for realsmall particles,, for which, because of the symmetry proper-ties of a Rayleigh particle, the 900 scattering does not de-pend on .
SUMMARY
In scattering measurements, the separation between the in-cident and the scattered fields can be done accurately only ata large distance from each scatterer. Multiple-scatteringcalculations involve several scattering centers, which scatterthe incident plane wave, as well as the nonnegligible contri-butions of the light scattered by all other scattering centers.
For each multiple-scattering calculation it is important toinvestigate the asymptotic behavior of the scattering func-tions in order to distinguish between the far-field values andthe near-field effects. Here we have shown that not only dothe scattering values differ in the near field from the Mie far-field values but their physical interpretation must be han-dled carefully and in particular must account for the pres-ence of the incident field.
Hence the concept of the near field introduces changes inthe radial values and in the tangential values of the Poyntingvector and also changes in the physical width of the requiredplane-wave assumption that determines the space in whichthe incident and the scattering waves are inseparable.
The near-field values merely emphasize the necessity ofconsidering the interference of the two fields in all anglesand not only in the forward angle ( = 0), as is required bythe optical theorem. However, the exact determination ofthe required normalized minimal distances among particlesof various sizes for which far-field results can be used isbeyond the scope of this presentation.
E. Tirosh is also with EL-OF, Electro-Optics IndustriesLtd., P.O. Box 1165, Rehovot 76 110, Israel.
REFERENCES
1. C. F. Bohren and D. R. Huffman, Absorption and Scattering ofLight by Small Particles (Wiley, New York, 1983), par. 12.1.8.
2. K. Kamiuto, "Near-field scattering by a large spherical particleembedded in nonabsorbing medium," J. Opt. Soc. Am. 73, 1819-1822 (1983).
3. K. Kamiuto, "Near field scattering by a small spherical particleembedded in nonabsorbing medium," J. Opt. Soc. Am. A 1, 840-844 (1984).
4. M. Kerker, D. S. Wang, H. Chew, and D. D. Cooke, "Does Lo-renz-Mie scattering theory for active particles lead to a para-dox?" Appl. Opt. 19, 1231-1232 (1980).
5. A. Cohen, "Extinction of active and passive particles," Appl. Opt.19, 2655-2656 (1980).
6. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford,1975).
7. M. Abramowitz and I. A. Stegun, Handbook of MathematicalFunctions (Dover, New York, 1972).
