Refraction by spherical particles in the intermediate scattering region

15 January

1997

OPTICS

COMMUNICATIONS ELSJWIER

Oprics Communications

134 (1997) 233-240

Full length article

Refraction by spherical particles in the intermediate scattering region G.H. Meeten Schlumberger Cambridge Resrurch, High Cross, Madingley Road, Cambridge CB3 OEL. UK

Received 13 May 1996; accepted 28 August 1996

Abstract Mie theory computations of the refraction efficiency for spherical particles Rayleigh, Rayleigh-Gans and Anomalous Diffraction approximations. Attention

are compared with predictions of the is given to the intermediate region of

particle size, where for extinction the Rayleigh-Gans and Anomalous Diffraction approximations have been shown to merge with each other and with Mie computations. In this region the Mie refraction efficiency is shown to have a first maximum which is not given by any approximation. The Rayleigh and Rayleigh-Gans refraction efficiencies are independent of particle size and approximate to the Mie results in the intermediate region, but not at larger particle sizes. Apart from the first maximum, the AD approximation is shown to closely predict the extrema and zeros of the refraction efficiency. In the intermediate region the Mie computations reveal a first maximum in the refraction efficiency which is not modelled by either of the RG or the AD approximations. Consequently a region where the Rayleigh-Gans and Anomalous Diffraction approximations merge with each other and with Mie computations is not found for refraction. The first maximum in refraction efficiency predicted by the Mie theory occurs when the optical wavelength inside a particle is about 20% greater than the particle’s diameter. It is shown to result mainly from the contributions to refraction by the magnetic dipole and electric quadrupole.

1. Introduction

Light falling on an assembly of scattering objects, such as a suspension of colloidal particles, will in general suffer both refraction and extinction. The complex index of refraction n = 11’- in” of a dilute assembly depends on the number of particles per unit volume N, and the complex forward scattering amplitude per scatterer S(O). For a less dilute and often strongly turbid assembly, typical of industrially-important concentrated suspensions, experiments show that n” becomes difficult to measure owing to multiple scattering, but that n’ remains measurable for 0030-4018/97/$17.00

fractions approaching close-packing [l]. For such suspensions n’ appears to depend only on the single-particle optical properties. In contrast with the highly non-linear concentration-dependence shown by n” [l], it depends linearly on volume fraction. Critical angle methods for measuring n’ of suspensions have been described [3-51, and Mohammadi [6] has recently reviewed the use of such measurements for investigating the microstructure of concentrated suspensions. Chj2ek and Li [7] have compared the optical extinctions predicted by two common approximations to the exact Mie theory for spheres; the

volume

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Rayleigh-Gans (RG), and the Anomalous Diffraction (AD). A sphere of radius a is described by the dimensionless size parameter x = 27~a/h, where A is the optical wavelength in the medium external to the sphere. The dielectric property of the sphere is described by m, the refractive index of the sphere relative to that of the external medium. Rayleigh scattering describes the region where the scatterer is small enough relative to the wavelength to be modelled as a point optical dipole, i.e. x -K 1. RG scattering describes a region of arbitrary X, but of small phase shift caused by the scattering particle, i.e. x(m - 1) < 1. For particles large enough for ray optics and diffraction to be applied the AD approximation is valid. The intermediate case as defined by van de Hulst [s] is where the particle size parameter x and the relative refractive index m satisfy x Z+ 1 and x(m - 1) +K 1; here the regions of validity of the RG and AD approximations overlap. For spherical particles Chflek and Li [7] compared the two approximations in this region, and showed the calculated extinctions to converge with each other, and with the exact Mie theory. See e.g. Table 1 of Ref. [7], and Table 14 of Ref. [8]. The scattering efficiency Q,,, is generally complex, such that its real part Q:,, determines n”, while its imaginary part P& determines n’ [8,9]. Here we investigate refraction in the intermediate region and compare the Rayleigh, RG and AD approximations with exact Mie theory.

3m,v n = n’ - in” = m, + -2i x3

ik3

S(O)

’

(2)

where v is the volume fraction of scatterers. A complex refractive index implies a complex extinction efficiency Q,,, = Q:,, + i Q:,, . The conventionally-defined extinction efficiency, here denoted by Q&t'is as given by van de Hulst [8];

An analogous refraction efficiency PAf = ;s”(O)

(4)

was defined [9] such that Pie, = Q’L,,. The real part of the refractive index increment is % -8

p:,, x’

(5)

and its imaginary part is

Forward-scattered light from an assembly of particles is wholly coherent with the incident light. Interference will occur between the transmitted and the forward-scattered light. The resultant far-field amplitude will generally differ from that of the incident light. The amplitude and the complex refractive index n will generally depend on the particle number density N and the optical properties of each scattering particle. Van de Hulst [8] gave 2rrm,N n = n’ - in” = m, + -

233-240

where $0) = S’(0) + is’(O) is the relative complex forward-scattering amplitude per scatterer, m, is the refractive index of the external medium, and i = m. The imaginary part S(O) determines the phase of the scattered light relative to the incident light, and depending on its sign, the suspension’s real refractive index n’ can be smaller or larger than that of the external medium. The transmitted light intensity is determined by S’(O), the intensity transmittance over a distance z being exp< - 4nn”z./&,) for light of free-space wavelength A,. For spherical particles of dimensionless size X, Eq. (1) becomes [9]

dn’ -= dv 2. Extinction and refraction

134 (1997)

WL

d n” du-8

3m, Q',,, ---

x’

(6)

Conservation of energy applied to extinction shows [8] that Q&, = Q:,, + Q:,,. The absorption efficiency Q’,, describes the conversion of incident light to heat inside the scatterer. The scattering efficiency Q:,, describes the removal of incident light via scattering. A particle whose permittivity is real at the frequency of the incident light will have pa,, = 0. A similar separation of Pi._., into P&, + Pibs has been discussed [9], where P,& = 0 for a particle whose permittivity is real. In the following we consider

G.H. Meeten /Optics

Communications

only non-absorbing scatterers in a non-absorbing external medium such that m is real, and P& = PiC,.

134 (1997) 233-240

to an assumption of the RG approximation. Thus Eq. (8) in the limit of m --) 1 gives the RG refraction approximation P’sea = t(m-

3. Models for Pica

235

1).

x

3.1. Rayleigh approximation 3.3. Anomalous difSraction approximation

Rayleigh scattering describes the region x -SK1 where the scatterer is small relative to the wavelength and so can be modelled by a point optical dipole. For a sphere the expansion of S(O) in powers of x as given by van de Hulst [S] gives S(0)

= ix 3( $$+)

+ Y(

$J,

(7)

Here the forward-scattered amplitude results from interference between light transmitted through the scattering object and undiffracted light transmitted through the external medium. For a spherical particle van de Hulst [8] writes s(o) =x2(;

+ exPCipip)

+ 1 -exP(-id), P2

and Eq. (4) gives P’ -=sea X

4p

m2-

i m2+2

(10)

1 1’

(8)

3.2. Rayleigh-Gans approximation The scattering particle is modelled as an assembly of dipole scatterers, which interact only in the far field. The scattering particle is assumed not to affect the phase and amplitude of the incident wave. The moment of each dipole, induced by the incident wave field, is unmodified by the other dipoles which comprise the particle. Wavelets in the far field which originate from the dipoles in a given scattering particle will generally possess different phases. Interference between the wavelets in the far field causes the angle-variation of the scattered amplitude, and a particle size dependence of the extinction efficiency. At zero scattering angle the wavelets from all the dipoles of a RG scattering particle will be in phase in the far field, regardless of their position in the particle. Thus the far-field interference will be wholly constructive, as if all dipoles were gathered to a point. On this basis it was asserted [9] that Eq. (8) would also represent RG scattering. This is erroneous. The Rayleigh approximation giving Eq. (8) can be derived from the electrostatic dipole moment induced in a dielectric sphere by a uniform external electric field. The field inside the sphere is uniform but 3/(m2 + 2) times the external field [lo], contrary

where p = 2 x( m - 1) is the relative phase lag of a diametral ray. Eqs. (4) and (10) give [9] P’sea -=X

4 X

sin p

_-_ ( P2

cos p P

I.

(‘1)

3.4. Me scattering Mie theory gives an exact description of scattering from an isotropic sphere, the accuracy being limited only by the computation. Van de Hulst [S] gives S(O)=+

i II=1

(2n+l)(a,+b,),

(12)

where a, and b, are the Mie coefficients, which are generally complex. Thus Q:,, and P& may be obtained via Eqs. (3) and (4) respectively, by computing the real and imaginary parts of a, and b,. Well-known methods [11] were used to sum the terms in Eq. (12) to a sufficient precision, which was checked as follows. For x < 0.5, PsLaand Q:,, from Eq. (12) compared closely with analytical expressions given by Wiscombe [l 11. (We obtained an analytical expression for PIC, by taking the imaginary part of Wiscombe’s expression.) For x > 0.5 the computed Q:,, was compared with data from the tabulations of Wickramasinghe [ 121. In all cases excellent agreement was obtained. For x > 0.5 no

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134 (1997) 233-240

numerical data for P& was found. Imprecision of the computed data is un-noticeable on the scale of the figures shown herein.

4. Results 4.1. Comparison of approximate and exact theories

Figs. 1, 2, 3, and 4, show P&/x for the various models, plotted as a function of x, for m = 1.1, 1.3, 1.5, and 1.7, in that order. As x and m increase, short-period modulations of the long-period Mie PiJx data become apparent. Similar modulations for extinction have been described [8] as bumps (small x) or ripple (large x). Both are discussed later. The AD approximation (11) is shown to agree (Figs. 1 to 4) with the broad features of the Mie theory results if short-period modulations are neglected. It is particularly good at predicting the zeros and extrema of PL.Jx, except for the first maximum, which we discuss in Section 4.4. The AD approximation (11) to P&/x predicts the first zero (at x0) for a diametral phase shift, relative to the medium outside the scatterer, of pO= 2(m - 1)x, = 4.493 rad. Fig. 5 compares the Mie and AD predictions by plotting 2 x0( m - 1) from the

0

10

20

30

X

Fig. 2. f$, /X versus x for m = 1.3.Mie theory; continuous line. AD approximation; short dash. RG approximation; long dash. Rayleigh approximation; dot-dash.

Mie theory, and p. from the AD approximation, against m. The Mie theory value of 2x,(m - 1) is seen to agree closely with p. for small m, and to increasingly deviate from p. as m increases from 1. The fluctuations each side of p. for increasing m 1.5

0.3 1

_____._____._._._._.~.~.~.~.-.~.-.-.-.-.-.-.-.-.-.-.-.-.

0.2

5

0.’

6

0

-0.5

-O.l0

50

I

I

I

I

0

5

10

15

0

X

Fig. 1. P& /x versus x for m = 1.1. Mie theory; continuous line. AD approximation; short dash. RG approximation; long dash. Rayleigh approximation; dot-dash.

Fig. 3. Pk, /x versus x for m = 1S. Mie theory; continuous line. AD approximation; short dash. RG approximation; long dash. Rayleigh approximation; dot-dash.

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7

_.___._.__-._._._ ._._._._._._._._._._._._._._._._._._._._.. 6.5 0

: : :

0 k

o-

0

Ai

: : '$

0

_~a___“.._..__.___.__________~________

TE E

: :

0.5-

0

6

0

5.5

5 5

0

10

15

1

1.2

Fig. 4. Pk, /x versus x for m = 1.7. Mie theory; continuous line. AD approximation; short dash. RG approximation; long dash. Rayleigh approximation; dot-dash.

originate from bumps or ripple in PiJx. However, only for m greater than 1.7 does the deviation exceed 2%. The AD approximation (11) predicts the first minimum (at .K& for a diametral phase shift of Pain = 5.764 rad. Fig. 6 compares the Mie and AD

4.6

0

4.6

& x

___.

B_________‘o________.________o___..

0

3 3

4.4

0

0

3 4.2

4

1.6

2

Fig. 6. 2x,&m - 1) versus relative refractive index m. Mie theory; squares. AD approximation ( pmin= 5.764 rad); short dash.

predictions of the first minimum by plotting 2(m l)xmin from the Mie theory, and pmin from the AD approximation, versus m. The same kind of behaviour is shown as for the first zero, discussed above. The agreement between the approximate and exact Mie predictions for both cases is surprisingly good, apart from deviations caused by bumps or ripple. 4.2. Intermediate region

5

2

1.6

m

X

_

1.4

I1

I

I

I

I

1.2

1.4

1.6

1.6

m

Fig. 5. 2x,&m - 1) versus relative refractive index m. Mie theory; squares. AD approximation ( p,, = 4.493 rad); short dash.

For x in the region of 1, Figs. 1, 2, 3, and 4 show that there is little to choose between the approximations compared with the exact Mie theory. An intermediate region, where the AD and RG approximations for refraction overlap and merge with the Mie prediction, does not occur. Overlap of the AD and RG approximations to P&/x occurs only for x --, 0, and both approximations merge with the Mie predictions at x + 0 only when m + 1. Neither the AD nor the RG approximation predict the first maximum of Ps’Jx at xmaX(Fig. 7). Figs. 5 and 6 show that both x0 and x,,,~”from Mie theory are closely proportional to l/(m - 1). This supports the AD model of extinction or refraction as interference between the undiffracted light, and zero-order diffracted light transmitted through the particle.

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G.H. Meeten/Optics

1

I 1.2

I 1.4

I 1.6

1 1.6

Communications 134 (1997) 233-240

I 2

m

Fig. 7. mx,,, Mie theory.

versus relative refractive index

m

computed from

Figs. 1 to 4 show that (&/x),,, from Mie theory is close to the x-independent value of Pk,/x = $(m - 1) obtained from the RG approximation. Fig. 8 compares the Mie theory and RG approximation by plotting (P~Jx>,,J(m - 1) from Mie theory, and 8/3 from the RG approximation, both

l-

2.7!j[1‘__________________________________ k "

versus m. It shows that the Mie and the RG results merge as m + 1, as expected. Even for x(m - 1) 5: 1, which is well outside the expected RG validity obtained from Mie theory is range, (PL&L shown to be only a few % greater than ;(rn - 1). The Mie theory is seen to agree exactly with the Rayleigh approximation for x + 0, and closely with the RG approximation for x = x,,. For a given m it is curious that the RG approximation becomes worse rather than better when x(m - 1) + 0, as x decreases from xmaX to 0. The reason for this behaviour is the RG approximation’s requirement that the electric field inside and outside the particle be the same, compared with the Rayleigh approximation (x -X 1) in which the electric field in a spherical scattering object is 3/(m2 + 2) times the field in the external medium [ 101. Thus if m > 1 the Rayleigh approximation necessitates a reduction of the net optical polarization compared with the RG approximation. 4.3. Ripple in refraction Ripple in extinction has been extensively discussed and modelled. Its origins are reviewed and further analysed by GuimarZes and Nussenzveig [13]. Similar attention to ripple in refraction has been largely absent. Van de Hulst [8] modelled ripple as interference between surface waves and forward-diffracted waves, predicting an extinction ripple period Ax of about 0.8. Chylek [ 141 analysed the x-dependence of the Mie coefficients a,, and b,, showing that the ripple in extinction arose from sharp and narrow resonant peaks in the real parts of these coefficients, giving

2.!j-

Ax= arctan(~)/~.

F

(13)

5

2.25

2-F 1

I

I

I

I

1.2

1.4

1.6

1.6

m

Fig. 8. (P/x),,,~~ /(m - 1) versus relative refractive index m. Mie theory; continuous line. RG approximation; (P/X),,, /(m - 1) = 8/3; dash.

GuimarZes and Nussenzveig [ 131 found Chflek’s result (13) to be an approximation to more precise expressions which agreed closely with extinction computations. No expressions appear to exist for the refraction ripple period, but it was shown by Chilek [ 131 that resonance in the imaginary part of a Mie coefficient occurred for the same x as resonance in the real part. The imaginary part crossed zero with a maximum negative gradient where the real part was a maximum. Thus for small x, where relatively few

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134 (1997) 233-240

239

terms are required for the sum in Eq. (121, the ripple period is expected to be similar for both extinction and refraction. Eq. (13) predicts A x(m) = 0.83(1.3), 0.75(1.5) and 0.69(1.7), which compare well with 0X5,0.76, and 0.70 (in the same order) from computations shown in Figs. 2, 3 and 4. 4.4. Origin of first maximum The first maximum of Pl&/x at x,, in Mie calculations of refraction was noted without explanation by Zimm and Dandliker [15], Chou and Kerker [16], and Champion et al. [9]. It is predicted by neither the AD nor the RG approximation. Thus it originates neither from interference between rays inside and outside the scatterer (AD), nor from farfield interference of light scattered by non-interacting dipoles within the scatterer (RG). We note from Figs. 5 and 6 that both x0 and xmin are approximately proportional to l/cm - 11, but from Fig. 7 that x,,, is approximately proportional to

2

0

3

X

Fig. 9. Contributions to P,, /x for m = 1S. Mie theory, full thick line. a, tern, full thin line. 6, term, short-dash line. a2 term, long-dash line. b, term, dot-dash line.

1/tn.

Examination of the PlJx versus x curves (Figs. 2, 3, and 4) in the region of x between the first maximum and the first ripple maximum shows a weak modulation superimposed on the more slowlyvarying curve. This modulation has been noted for extinction [8], where it was described as bumps, and attributed to the x-dependence of Re(a,) and Re(b,) when n is small and only a few terms dominate the Mie sum. Fig. 9 is a plot of P&/x versus x, and also the first two terms of the imaginary parts sum of Eq. (121, i.e. (4n + 2) Im(a,)/x3 and (4n + 2) Im(b,>/x3, for n = 1 and n = 2. With increasing x the electric dipole term a, decreases monotonically, but the maxima in the magnetic dipole and quadrupole terms b, and b, can be seen to cause the bumps in P&/x where x is about 1.6, and 2.6, respectively. The electric quadrupole term varies more gently with x than the terms in b, and b,, and so contributes less to the bumps. Fig. 9 shows the main cause of the first maximum in P&/x to be the combined effects of the magnetic dipole term b, and the electric quadrupole term u2. It was concluded by van de Hulst [s] that the first bump in the extinction curve at x = 2 (also for m = 1.5) was caused only by the magnetic dipole term.

On the basis of its good general agreement with the Mie theory, it is pertinent to ask whether the AD approximation might be modified to account for the first maximum in Pl&/x. Owing to diffraction, a ray of width ph cannot propagate as a ray over distances larger than p*A [81. Thus for a spherical particle, the ray basis of the AD approximation or spherical particles will be invalid for x < T, exactly the region of the first maximum, and the AD approximation cannot be modified to account for the first maximum. In the RG approximation, the scattering particle is modelled as an assembly of dipoles which interact only uia interference between wavelets scattered from them in the far field. For refraction we have shown this approximation to neither approximate to the Mie theory, nor to account for the first maximum. The coupled dipole model (CDM) incorporates near-field interaction between the dipoles and may be regarded as an extension of the RG approximation sensu stricto. Buitenhuis et al. [17] have shown the CDM to model very closely the scattered radiance calculated from Mie theory, and on this basis an application of the CDM to refraction would be of interest, but it is outside the scope of this paper.

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5. Summary

Exact Mie calculations of the refraction efficiency Pica for spheres have been compared with predictions from various approximations_ For the particle size parameter x of about 2 or less, the Rayleigh, Rayleigh-Gans and Anomalous Diffraction approximations give a refraction efficiency only moderately close to the exact Mie results. None of the approximations predict the maximum of the refraction efficiency in the region of x = 2, which is explained by the close proximity of maxima in the magnetic dipole and electric quadrupole terms of the Mie theory. For x > 2 the AD approximation was shown to give a fair approximation to the Mie results, particularly to predict the extrema (except the first) and zeros of Pica/x. Unlike extinction, we do not find for refraction an intermediate region of x where the Rayleigh-Gans and Anomalous Diffraction approximations merge with each other and the Mie calculations.

Acknowledgements

This work was carried out within the DTI Colloid Technology Project.

References

111A.

Killey and G.H. Meeten, J. Chem. Sot. Faraday Trans. 2, 77 (1981) 587. 121K. Alexander, A. Killey, G.H. Meeten and M. Senior, J. Chem. Sot. Faraday Trans. 2, 77 (1981) 361. [31 G.H. Meeten and A.N. North, Meas. Sci. Technol. 2 (1991) 441. [41 G.H. Meeten and A.N. North, Meas. Sci. Technol. 6 (1995) 214. bl J.E. Geake, C.S. Mill and MS. Mohammadi, Meas. Sci. Technol. 5 (1994) 53 1. b1 M. Mohammadi, Adv. Colloid Interface Sci. 62 (1995) 17. t71 P. Chjlek and J. Li, Optics Comm. 117 (1995) 389. [81 H.C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981). [91 J.V. Champion, G.H. Meeten and M. Senior, J. Colloid Interface Sci. 72 (1979) 47 1. [lOI C.J.F. Bottcher, Theory of Electric Polarisation (Elsevier, Amsterdam, 1952). 1111 W.J. Wiscombe, Appl. Optics 19 (1980) 1505. Light Scattering Functions for Small iI21 N.C. Wickramasinghe, Particles (Adam Hilger, London, 1973). iI31 L.G. GuimarZes and H.M. Nussenzveig, J. Modem Optics 41 (1994) 625. [141 P. Chylek, J. Opt. Sot. Am. 66 (1976) 285. [ISI B.H. Zimm and W.B. Dandliker, J. Phys. Chem. 58 (1954) 644. [I61 A. Chou and M. Kerker, J. Phys. Chem. 60 (1956) 562. [I71 I. Buitenhuis, J.K.G. Dhont and H.N.W. Lekkerkerker, J. Colloid Interface Sci. 162 (1994) 19.