extinction and scattering by soft spheres

Extinction and scattering by soft spheres

A. Y. Perelman

On the basis of an approximate relation, the Mie series are replaced by new ones, which can be summed exactlywith the aid of various modified and generalized forms of the addition theorem for cylindrical functions. Thesums obtained simultaneously simplify the initial expressions for the scattering characteristics and preservetheir analytical nature. The conventional approximations for the amplitude functions and the efficiencyfactors rigorously follow from the new approaches if the optical parameters are properly restricted. Theacceptable domains of these approaches contain the long wavelength region and are extensive enough to studythe various (including inverse) scattering problems for the real disperse systems, in which the particles aresuspended in a medium with similar optical properties.

1. IntroductionLight scattering by small spherical particles with

arbitrary optical properties exposed to a plane wavecan be described in terms of the known series which, infact, are linear or quadratic combinations of the scat-tering coefficients.1-3 The problem with which we areconcerned is deriving relatively simple approximateexpressions for the sums of these series to obtain someinsight into extinction and absorption by sphericalparticles and to solve the inverse scattering problemsin optics. No such approach suitable for all values ofthe optical parameters is likely to exist. Therefore,the approaches are to be constructed in certain condi-tions with respect to the scattering coefficients.

We discuss scattering from soft particles whose rela-tive refractive index, by definition, is close to unity.Obviously, this qualitative definition cannot serve asthe mathematical basis for the desired approaches toderive, and it is replaced by the one expressed analyti-cally. The new definition4 of a soft particle is based onthe relevant approximate representation of the scat-tering coefficients. This approximation appears to bequite natural since it allows the series to be summed forall field characteristics, and, at the same time, theapproaches obtained in such a way reflect the essentialfeatures of the phenomenon of scattering. So, theseapproaches turn into a number of conventional ap-proximations for field vectors and cross sections giventhat the same diffraction conditions have been im-

The author is with Forest Academy, Leningrad 194018, U.S.S.R.Received 10 February 1989.0003-6935/91/040475-10$05.00/0.© 1991 Optical Society of America.

posed.45 It is worthwhile that the new approachesallow the Mie scattering characteristics to be repre-sented with reasonable accuracy if the values of therefractive index and the size parameter belong to theacceptable domain, which is sure to cover the longwavelength region. The most significant result is thefact that the acceptable domain proved to be extensiveenough to study some real scattering problems. Itshould be added that the integral representations ob-tained for the Mie efficiency factors are useful to solveinverse problems in ocean optics.

In Sec. II a way to derive new small angle approxima-tions of Mie amplitudes is given. The leading term ofthese approximations is found and it is proved to con-vert into the Rayleigh and Rayleigh-Gans approxima-tions of Mie amplitudes if the parameters of scatteringare properly restricted. None of the above approxi-mations is demonstrated to satisfy the optical theo-rem. As shown in Sec. III, the new approximations canbe modified for the optical theorem to be fulfilled. Itshould be noted that, in fact, the independent scatter-ing angle term of the modified approximations wasderived earlier in Ref. 5.

In Secs. IV and V the approximation of the Mieefficiency factor for extinction is decomposed into twoparts. The long wavelength approximations of theseparts and the corresponding (for scattering and ab-sorption) Mie efficiency factors are verified to coin-cide. The approximations obtained of Mie efficien-cies are shown to be suitable for solving inverseproblems of particle sizing. A typical inversion formu-la is presented in Sec. VI, its acceptable domain isdescribed in Sec. IV on the basis of the data taken fromRef. 5. The results of this paper and some otherworks4-6 on the same subject are briefly discussed andreviewed in Secs. VII and VIII.

Let us list the basic notations and definitions: a =

1 February 1991 / Vol. 30, No. 4 / APPLIED OPTICS 475

radius of a sphere, X = wavelength in a vacuum, 0 =scattering angle, ml and m2 = refractive indices insideand outside the sphere:{m = mJm2 = m'-im"(m2 > 0), ju = cosO,

x 27rm 2aX1, y =mx.

Further, we put akn = scattering coefficients, 7rn(,4) andTn(A) = angle-dependent functions, SkG(L) = amplitudefunctions (amplitudes), Ak(Az) = small angle approxi-mations of the amplitudes, and QSC, Qa, and Q = effi-ciency factors for scattering, absorption, and extinc-tion, respectively. It is known that'

Q = QSC + Qa, (1)

Q = 2x-2E (2n + 1) Re(aln + a2 ), (2)

n=1

QSC = 2x-2 (2n + 1)(la 112 + la2n ),

n=1

QSC = x-2 J [ISd(A) 2 + IS9(M)12] sindO

in which the amplitudes

S(1 U) = (2n + l)in 1[a 1 nx(y) + a2nTn(A)],

n=1

2(4) = (2n + 1)F1h[a 1 nTn(A) + a2 n7rn(,)],

n=l

i.e., the linear combinations of akn, are used.elsewhere we assume that

k = 1,2 and = n(n+ 1). (6)

Within the far field zone, the spherical components ofthe scattered electric and magnetic field vectors andthe intensity of the scattered light are represented interms of the amplitudes. For a nonabsorbing particle,the relative refractive index m is real, Q = QSC, and

Reakn = akn12 (m = Rem). (7)

The well-known optical theorem 2

Q = 4x-2 ReS(1), S(1) = Sk(l) (8)

establishes the connection between factor Q and theelectric and magnetic fields outside a sphere. Equa-tion (8) can easily be verified by means of Eqs. (2), (5),and (18).

In the Mie problem (ml and M2 are constant) wehave

akn=-.2 akk(mx) + ihk+2n (9)

hkn-ihknhk+2n

hkn + hk+2n

where

tn = X4'n(y)'n(X) - A(Y)'n(X)1

h2n = AYz(y)(X) - 4(Y)n(X),

h3n = xnP(y)x(x) -Yn(Y)XnW1

(12)h4n = Y4"n(Y)x(x) -Xn(Y)Xn(X)

The Riccati-Bessel functions Vtn(x) and Xn(X) satisfythe following differential equation:

nz(x) = X2[Z"(X) + z(x)], (13)

and their power expansions (x - 0),1 1 1 n+ + 0(X.+ 5),{ 1'n(x) = (2n + 1)!! Xn -2(2n + 3)!! x

Xn(x) (2n - 1)!!x-n + O(X-n+2),

allow us to establish the asymptotic behavior

h1 n = O(X2n+2), h2n = 0(X2n+4)

(14)

h3 nh 4 n = O(x) (x - 0)

(3) of the quantities defined by Eqs. (11) and (12).The Rayleigh approximation SR(,) (ImI is not large, x

<< 1) and the Rayleigh-Gans approximation SkG(M)(m(4) - 1 << 1,xlm - << 1) of the Mie amplitudes are of the

forml 2

SjR(p) = i(m 2 + 2)(m 2-l)x3ski, (15)

k = 2i(m -)u 2,Pi(u)X

3!Lk 1, u = 2x sinO/2. (16)

(5) It is easy to see that the above approximations do notsatisfy the optical theorem. In fact, for a nonabsorb-ing particle the amplitudes given by Eqs. (15) and (16)are equal to pure imaginary numbers, whereas factor Q

Here and is for certain equal to a positive number.

II. Small Angle Approximation of AmplitudesTo determine the amplitudes defined by Eq. (5)

within the small angle region of Ol1 << 1, we first find theexpansions of the polynomials 7rn(At) and Tn(A) in pow-ers of 1- u. The angle-dependent functions 7n(A) andTn(A) can be written in the form

n(A) Pn(A), _rn() = !PA(U) - Wrn(L) (17)

where Pn(,) is the Legendre polynomial. Whence,with the aid of the truncated hypergeometric series

Pn(= F(nn + 1,1, 1 -)

we get

7rn() = + (2 ) me+(f) (1 - )"2 _ (2m + 2)!!m!

(18)

= f - V.M) n (-1Y'M(n h r~f)(Irn(A) =n n(A) + Z (2m)!!m! ()

wheremn-I

em(z) = (Z - it). (19)n=O

(10)

The various small scattering angle approximations(11) Ak(x) for Mie amplitudes Sk(G) can be derived on the

basis of Eqs. (18) and (19). Let us construct this

476 APPLIED OPTICS / Vol. 30, No. 4 / 1 February 1991

approximation which is accurate to Q(04). We there-fore have{n(A) = 2- ( ) (21),

2 8

'r()=n (3h - 2) ( 2 8

(20)

and by virtue of Eqs. (5) and (20) the followingrepresentation of the amplitudes holds:

Ak(Y) = H1 1 + H2 1 - (1 - A)Hk + 0(°4) (0 0), (21)

Hk = 0.25[(2k - 1)H12 + (-2k + 5)H22] - 0.5(H1l + H21), (22)

with

Hkr = >3 (n + 0.5)nrlakn.n=1

(23)

For Eq. (21) to simplify, we introduce the concept ofthe soft (Im - 1)1 << 1) particle approximation (the S-approximation) in the Mie problem. Namely, takinginto account the identity t"(x)Xn(x) - 4n(X)xn(X) = 1and bearing in mind Eqs. (9)-(12) and (14), we define(see Refs. 4 and 5):

(1) The simplest form of the S-approximationwhich corresponds to akn given by Eq. (9), with hkn +ihk+2n - ixjml1/2, that is,

akn -lhkn, 1 = i-'mh1

/2 . (24)

(2) The (main form of the) S-approximation, whichcorresponds to akn given by Eq. (10), with

hkn + h+2n x21mI, (25)

that is,

aknf x Im(hkn ihknhk+2n). (26)

Hence, the condition Im -11 << 1 in the S-approxi-mation is used as follows: the approximate equalitiesx Im1/2x mx are applied in the denominators ofthe scattering coefficients [written by Eq. (9) or by Eq.(10)].

The sums of the series defined by Eq. (23) can beexactly calculated in the approximation due to Eq.(24). We get(aF aF= ( , dH xFr)

Hj,1 _x I,H,1= X- I,Hir a' x ay/ O x ay/) (27)

where

Fr= > (n + 0.5)flr-l Xn(X)Pn(Y)-

n=1

By means of Eqs. (13) and (Al), we are able to sum theabove series for every natural number r. In particular,we find

{F, = 0.5[vpo(z) - A(x)1P(y)]

XF2 = V1(z),

with z = y - x and v = xyz- 1, and therefore, Eq. (27)yields

(H11 + H2 1 = 1(x + y)V' 1 (Z),

H12 = l(x + y)[v2

t'2 (z) + 2vi 1(z)],

2 = I(X + y)V242 (Z).

Then, using Eq. (22), we obtain{H = i(m + 1)m2 1m1 12x4z-2 t02(z)

H2 = i(m + 1)m 21ml-/ 2

x4

[z-2 42 (z) + m-1X-2Z1' 1 (z)]

(28)

and finally arrive at the following S-approximation ofMie amplitudes:

Ak(j,) = i(m + )mjmj-1/2x

2[ykz-lp(z) - (1 -)MX2Z-2 (Z)]

z = (m - 1)x, (29)

which, as 0 - 0, remains accurate to Q(04).Let us now prove that Eq. (29) turns into Eqs. (15)

and (16) provided the parameters m, x, and 0 are prop-erly restricted. For, if x << 1, we must keep only thefirst term in Eq. (29), because of relations (z - 0):

z-16l(Z) = z/3 + 0(z3), z-

21' 2 (z) = z/15 + O(Z3). (30)

Hence, we get

Ah() = 3imlml 2(m2- l)x3y-1 (x << 1). (31)

As for a soft particle the approximate equality m(m2 +2) 31mMl/2 may be used, from Eqs. (15) and (31) weconclude that Ak(A) = SR(,g) if x << 1 and Im - 11 << 1.In turn, if xlm - 11 << 1, corresponding to Eq. (30), wemust keep both terms in Eq. (29), and it will have theform ( - 0)

Ak(A) = 3 1im1mjh

1/2(m

2- 1)X3[Ak1 - 5-mX

2(j _ s)] + 0(04),

(xlm - 1) << 1). (32)

Making use of the expansion

_C=M n ( 1) x2n (1 -

U 2 > n!(2n + 3)!!u = 2x sinO/2,

we can rewrite Eq. (16) as (6 - 0)

SRG(A) = 2 i(m - 1)X3k-1- (1-)] + 0(04). (33)

By virtue of Eqs. (32) and (33), we obtain Ak(gu)SkG(g) if xm11 << 1 and Im-11 << 1, since in Eq. (32)we may put m = 1 except for the factor m - 1.

Hence, the S-approximation of Mie amplitudes con-verts into the Rayleigh approximation (naturally, un-der supposition on Im- 1 << 1) and the Rayleigh-Gansapproximation [for small values of 0, accurate to 0(04)]if, respectively, the same restriction is imposed on thescattering conditions.

It should be mentioned that we cannot apply Eq. (8)to the approximation Sk(l) Ak(l) with Ak(l) givenby Eq. (29). The factor Q must be calculated, e.g., inthe manner described in Ref. 1. However, the moreexact approximation of the factor Q is derived other-wise (see Sec. IV).


Ill. Amplitudes and the Optical TheoremWe have already demonstrated the optical theorem

is surely violated for some familiar approximationsused in the Mie problem. To improve the situation,let us first introduce the concept of the sufficient S-approximation in which, by definition, as x - 0, thequantities Reakn and akn12 for a nonabsorbing spherehave the same order with respect to x so that Eq. (7)holds at least for the leading terms in x of its left andright members.

For example, the S-approximation corresponding toEq. (26) is sufficient. Indeed, assuming that x - 0 andm = Rem, from Eqs. (14) and (26) we find that

akn = O(X 4n+4k-2) + i0(x2n+2k-1). (34)

Whence aknl2, Reakn = O(X4n+4k-2) and on the basis ofEqs. (25) and (26), the leading terms of iaknl12 and Reakncoincide and are equal to x-2m-lhkn. Moreover, tak-ing into account the relations 7ri(p) = 1, iri(y) = ,Eqs. (5) and (34), we get Ak(/t) = k l0(x 6 )+ ik-lO(x3), and, with the aid of Eqs. (2) and (3), thefollowing connection between the sufficiency of the S-approximation and the optical theorem can be estab-lished: the quantities Q, QSC, and 4x-2 ReA(1) are ofthe order of O(X4) and their leading terms are equal to6x-2 Re(all + a2l). On the other hand, the S-approxi-mation corresponding to Eq. (24) is not sufficient.Indeed, in this case the leading term of akn12 equalsx 2 m'hkn, whereas Reakn = 0. It means that Iaknl2and Reakn are essentially different.

Mathematically, the difference between the S-ap-proximation and its simplest form is that the formerdeals only with the function An(x) [see Eqs. (11) and(24)] while the latter also involves the function xn(x)[see Eqs. (11), (12) and (26)]. The main analyticalfeatures of the amplitudes in the long wavelength re-gion are governed by the function Xn(x), because of itssingularity at point x = 0.

The amplitudes given by Eq. (29) were derived onthe basis of Eq. (24). Next, as was shown in the pre-ceding section, Eqs. (15) and (16) follow directly fromEq. (29). It means that the Mie amplitudes approxi-mated due to Rayleigh and Rayleigh-Gans are of thesimplest form of the S-approximation as well.

For the amplitudes to be correctly computed bymeans of Eq. (21) in the most extensive domain of theoptical parameters m = m - im" and x, certain supple-mentary conditions must be satisfied. So, within thenear forward scattering region the coincident leadingterms A(1) = H11 + H21 of both amplitudes in Eq. (21)need to be evaluated in terms of the scattering coeffi-cients approximated by Eq. (26). For this reason, inaccordance with Eqs. (5), (18), and (26), the series

A(1) = X 21mI- 1 (n + 0.5)n=1

X [hin + h2n - i(hlnh3 + h2 nh4 n)]

is to be summed. This sum can be found exactly on thebasis of Eqs. (11), (12), and (A5)-(A9). After somereductions, we come to the expression

A(1) = (2 + 1)2 + w(mp)- (-m, - R)81mw I 2m II

where

(35)

co(m,z) = [a(m) + a0 (m)z-2 ]ei(z) + ia 1(m)e1(z) + a2(m)e 2 (z),

a(m) = (m2- 1)

2(m

2 + 1), ao(m) =-2(m2- 1)

2(m - 1)2,

aI(m) = (m + 1) 2 (m4-2m3 -2m 2 - 2m + 1), (3

a2(m) =-ao(m)- al(m),

eiz) =z 1- exp(-it) exp(-iz)ei(z) = t dt, e1(z) = ______

t z

1 - exp(-i2e2 (Z) = Z2

p = 2(m-1)x, R = 2(m + 1)x.

36)

(37)

The above result of summation holds for all values ofx > 0 and m = m'-im" (m' > 0, m" > 0). It followsfrom the fact proved in Appendix A that the series inEqs. (A5)-(A9) are convergent for any positive value ofa and any complex value of 3. Equations (35)-(37) arederived in a different way in Ref. 5.

On the basis of Eq. (21) we conclude that, accurate to0(04), the Mie amplitudes Sk(AL) given by Eq. (5) can beapproximated by the S-approximation amplitudesAk(pu). Thus, we finally obtain

Sk(A) Ak(A) = A(1) - (1 - li)Hh, y = cosO, (38)

where the leading term A(1) has to be computed fromEqs. (35)-(37), and, as for the quantities Hk, theircalculation is admissible in the simplest form of the S-approximation, i.e., by Eq. (28). It should be notedthat the quantities Hk can also be found in terms of themain form of the S-approximation, however, the corre-sponding correction for the near forward scatteringdirections is of little importance.

IV. S-Approximation for the Extinction Efficiency Factorand Its Accuracy

The reasoning in Sec. III justifies the application ofthe optical theorem to the amplitudes found in thesufficient S-approximation. Hence, by virtue of Eqs.(8) and (35), the S-approximation of the Mie extinc-tion efficiency factor for any absorbing and nonabsorb-ing spherical particles may be expressed as

(39)Q = 1 Re[(m2 + 1)2 + (m,p) -W(-m,-R)]

where the quantities defined by Eqs. (36) and (37) havebeen utilized. It means that the approximation ofMie's efficiency factor for extinction found in Ref. 5satisfies the optical theorem. Let us recall some re-sults obtained in Ref. 5 and carry out the investigationof function Q from another point of view.

Taking into account Eq. (37) and passing, as m - 1and p is fixed arbitrarily, to the limit in Eq. (36), wefind co(-1,-R) = 0, (1,p) = 16[-ie1(p) + e2(p)], and,in turn, Eq. (39) yields limQs = Re[2 - 4iel(p) +4e2(p)]. In other words, if m tends to unity when x(m- 1) is equal to any fixed value, we have


z)

Table 1. Values of p(m)

m 1.00-1.06 1.08 1.10 1.12 1.14 1.16 1.18 1.20 1.22

p(m) 128 60 31 25 23 18 15 14

limQS = QH,

where

QH = 4 ReK(ip),

(40)

1 exp(-w) exp(-w) - 1KMw =-+ ~ + (41)

2 w w2

Table II. Values of pi

j 1 2 3 4

Pi 4.1 10.8 17.2 23.5

is the general form of the well-known approximation ofanomalous diffraction.' It means that the van deHulst approximation can be regarded as the leadingterm of function Qs provided that the above restric-tions with respect to parameters m and x are fulfilled.Analogously, function Qs has been proved5 to coincidewith the Rayleigh aproximation QR and the Rayleigh-Gans approximation QRG of the Mie extinction effi-ciency factors before if parameters m and x are proper-ly restricted, that is,

Qs=QRifX<<1, Im-i<<1,

QS=QRGifXIm-1I1<<, Im - 11<<i.

It is worth mentioning that Mie factor Q and its S-approximation Qs are the even functions of parameterm, whereas this property is violated for factors QR,

QRG, and QH. The assertion obviously results from theanalytical representations of the factors mentioned.

Then, we introduce the formal decompositionw(m,z) = w+(m,z) + iw-(m,z) which is uniquely definedby means of the equalities co+(m - z) = w+(m,z) andco-(m,-z) = -- (m,z). Corresponding with this con-struction of w(m,z) we can write

Qs = Qs + Qw,

whereI Q= Re[(M2 + 1)2 + W (mp)

(43),W+(m,z) = [a(m) + ao(m)z-2 ci(z) + a,(m)s(z) + a2 (m)c(z),{ cu(mp) + wi(mR)]|s =R~ 41mlm ']'

(44)(m,z) = [a(m) + a0 (m)z- 2]si(z) + al(m)a(z) + a2(m)Y(z).

Here, the functions

ci(z) = J tl(l - cost)dt, si(z) = J t-1 sintdt,o o

s(z) = z' sinz, a(z) = z1 cosz, (45)

c(z) = z 2

(1 - cosz), y(z) = z-1 sinz,

and the notations from Eqs. (36) and (37) are used.When decomposing Eq. (39) by means of Eqs. (42)-

(44), we distinguished between the unit imaginarynumber i included in functions w(m,p) and w(-m,-R)directly and due to the parameter m = m' - im", sincethe equations cw+(m,z) = Rew(m,z) and w-(m,z) =

Imcw(m,z) are obviously satisfied if and only if m =Rem. In turn, to obtain the S-approximation for fac-tors QSC and Qa, Eqs. (43) and (44) were continuedanalytically from the set m > 0 into the domain m = m'-im" (m'>0, m">0).

Functions Q+ and Q- appear to be the relevant ap-proximations of Mie factors QSC and Qa, respectively,as shown in Sec. V on the basis of modified representa-tions of functions QS.

The region of the real applications of the S-approxi-mations can be characterized by the results of calcula-tions5 for the nonconservative scattering summarizedbelow.

An acceptable domain exists,

(46)

in which Mie factor Q is represented by its S-approxi-mation Qs with the error that does not exceed 5%except, possibly, at points on the interval 0 < x < 2,where this error can attain 5-25% in the indispensablecondition that the absolute error IQs - Q is of theorder of 10-n, with n 2 2. For m > 1, the behavior ofthe upper bound x(m) of the acceptable domain ispresented in Table I (taken from Ref. 5) in terms of the

(42) phase shift

p(m) = 2(m - 1)x(m). (47)

To illustrate the accuracy of the S-approximationQs, let us recall that Mie factor Q is known 7 to havesuccessive maxima xj(m) > 0, at which the phase shiftsp = pj almost do not depend on m. The first fournumbers of

pj = 2(m-1)xj(m), O <P <P2 < ... (48)

(computed in Ref. 1) are listed in Table II.The data of Tables I and II shows that Mie factor Q

can be fairly well approximated by its S-approxima-tion Qs for the range of parameters m and x permittinga discussion of some real problems (see Sec. VII). Themode of the S-approximation Q Qs, 0 < x <x(m) (m = Rem), makes its use possible in the solu-tion of inverse scattering problems. Such a typicalproblem is presented in Sec. VI.

V. S-Approximation for Scattering and AbsorptionEfficiency Factors

To achieve the physical meaning of functions QS letus regroup the terms in Eqs. (43) and (44). After somecalculations we obtain


O<X<X(M), M=1+M (M>-1),

QS = Re t (mp) t (-m,R)

(49)t+(m,z) = 4a(m)ci(z) + ao(m)Hi(z) - aj(m)H(z),

{Q - Im ~ (m,p) + ~(-m,R)IIS 161mlm (50)

-7(m,z) = 4a(m)si(z) + ao(m)hi(z) - al(m)h(z),

where

H(z) = 2 - 4s(z) + 4c(z), h(z) =-4a(z) + 4y(z), (51)

Hi(z) = J tH(zt)dt, hi(z) = J th(zt)dt.

Here, we have used the relations{Hi(z) = 1 - 4c(z) + 4z-2ci(z),

hi(z) =-4y(z) + 4z-2si(z),

which are easily verified.Point x = 0 is a removable singularity of functions

Q'. For this reason, functions QS can be expandedinto Taylor series in powers of x, and we establish theasymptotic behavior (x - 0):

(QS = X4 Reus(m)(m2- 1)2] + x

6 Reu 2 (m)(m2- 1)2] +

(52)Q = x Im[v1'(m)(m 2

- 1)] + X3Im[v2(m)(m

2- 1)2] +...,

where

QH = QH + QH,QHC= X

2UH(m) +...

Q{ = xV1H(m) + X V2H(m) +...,

(56)

where{ ulH(m) = 2(m' - 1)2(1 + tan2

03),

vlH(m) = - (m'- 1) tanfl, V2 H(M) = -4(m'- 1)2 tan2#,

3

with tanf = m -(m 1)-i. It is plain that Eq. (56)essentially differs from Eq. (54).

Comparing the asymptotic behavior of functionsQSC and Q+, as well as that of the functions Qa and Q-and taking into consideration Eqs. (1), (42), and (48),we see that functions QS may be regarded as the S-approximations for corresponding Mie factors.Namely, the S-approximations of the form QSC Q+SQa nQ, Q-, Q+ + Q- are valid within their accept-able domains, which, in any case, include the parame-ters m and x satisfying the conditions I' - im" - 11 <<L1 and O < x << 1. Here we do not dwell on thisquestion.

On the basis of Eqs. (B1) and (B2), we can concludethat Eqs. (42), (49), and (50) actually deliver the S-approximation integral representations of Mie factorsQ, QSC, and Qa, respectively. These representationsare appropriate for the Mellin transforms to find.This is used in the next section.

U(u i) = 8 2 U2 MvlS(m) = - in

2(i, + u2),

l '( ) _ 4 M2(M2 + 2),

) = - 8 m2(m

2+ 1),

1351m1 VI. Particle Size Distribution from the S-Approximation

Density function (particle size distribution) p(a) canbe evaluated by inversion of the integral equation

v2,(m) = 2 m2(m2 + 10).2251m1

The long wavelength approximations of factors QSCand Qa are known1' 2 to have the form (x - 0):

Qsc = x4 Re[u1 (m ( 2

-1)21 + x6

Re[u2(M)(m2

- 1)2] + ... ,

(54)IQQ = x Im[v1(m)(m 2 - 1)] + x3 Im[v2 (m) (M

2- 1)21 + .. ..

where

{u1(i) = 3(M2

+ 2)2'

v1 (m) = - 2 4in +2 '

16(=M2

- 2)U2 (in = 5(M2 + 2)3

v2 (i) = -4(m 4 + 27m 2 + 38)

15(m2

+ 2)2(2m

2+ 3)-

From Eqs. (53) and (55) we get

ul,(1) = uJ(1) = 8/27, U2,(1) = U2 (1) = -16/135,

v1 8(1) = vj(1) = -4/3, v2,(1) = v2(1) = -88/225,

i.e., for an arbitrary sphere, functions Q+ and Q- affordsatisfactory asymptotes for the long wavelength ap-proximations of Mie factors QSC and Qa, respectively.

On the other hand, as shown in Ref. 1, the longwavelength approximation of the van de Hulst factorQH is of the form (x - 0);

T(m,X) = J Qf(a)da, f(a) = ra2p(a), (57)

where T(m,X) is the spectral extinction coefficient(turbidity) and kernel Q is the exact or some approxi-mate expression for the extinction efficiency factor.

Equation (57) is an example of the ill-posed inverseproblem, therefore it must be solved by some of theregularization methods.- 10 As is usual with manyproblems of this type, Eq. (57) can be regularized withthe help of the numerical solution, e.g., by the mostwidely used method described in Ref.11. On the otherhand, analytical inversion is well worth using providedthat the inversion formula is available. This formulamust certainly be regularized, too, but in this casethere are specific methods of regularization based onthe fact that any finite-dimensional operator is bound-ed.12 To secure the optimal utilization of the experi-mental information, the method of regularizationmust be chosen properly. Naturally, this conditionhas to be satisfied for all methods of solving the ill-posed problem.

Many inverse optical problems deal with integralequations of the first kind with a product type kernel.The various integral transformations (due to Fourier,Laplace, Hankel, Mellin) are convenient for theseproblems to solve. A number of analytical solutions of


Eq. (57) in the anomalous diffraction approximation ofvan de Hulst have been published in Refs. 13-21.

We have shown that, within the acceptable domain,kernel Q of Eq. (57) can be replaced by its S-approxi-mation with uniformly small error. In this section, bymeans of the Mellin convolution method, an outline forthe solution of Eq. (57) in the S-approximation of itskernel Q will be discussed. To apply the method, wemust represent the functions Q in the form relevantto compute their Mellin transforms. The sought forrepresentations are given by Eqs. (49) and (50), be-cause all summands of the functions tj(m,z) have thesame strip of convergence -2 < Rep < 0, and, conse-quently, the property of linearity of the Mellin trans-formation can be used.

For simplicity we limit our discussion to a nonab-sorbing particle. We therefore have Q = Qs, Q- = 0,and, by virtue of Eq. (42), kernel Q in Eq. (57) is to betaken as

= 1+(mp)2+(-mR) (m = Rem). (58)

Using the integral representations of the summands offunction V+(m,z) and the Mellin transforms of thesesummands [see Eqs. (49), (Bi), (B6), and (B7)], we findthat, for any positive number t, the relation

M#+(m,tx) = 4t-Prc(p) (m) (p )2 p-l2] (59)

holds in the strip -2 < Rep < 0. The functions on theright-hand side of Eq. (59) are defined by Eqs. (36) and(B5). It is possible now to use the Mellin convolutiontheorem and the inversion formula for the Mellintransformation to solve Eq. (57). Due to Eqs. (58),(59), and (B6), this method of solution yields

f(a) = . J G(a,p)L(m,p)dp (-2 < c < 0), (60)

where

g(m,v) = 16m 2T(m,X), v = 47r-1,

G(a,p) = J g(mv)(av)-Pdv,

L(mp) = (27r)-'(1 -p2)A(mp),

(61)A(m,p) = (p + 1)rj(p)[l(mp) - 1(-_,p)]-1,

1(m,p) = Im - 1P- 1[a(m)(p + 1)2

+ a(m)(p - 1) - aj(m)(p 2-1)].

Here, the notations of Eqs. (36) and (B5) are utilized.To evaluate the contour integral in Eq. (60), we split itsinner integral G(ap) into two expressions as follows:

jG(a,p) G7 (ap) + J [cO(m) + c2 (m)v-2](av)-Pdv,

|G (a, p)!- E -g(m j) (aj) -PA j.

0 <Vj<7

Here, the nodes of the quadrature formula vj, the pa-rameter r, and the coefficients co(m) and c2(m) are to

be determined from the experimental data with re-spect to the turbidity T(m,X). After some manipula-tion we find the solution of Eq. (57) in the form of

2irf(a) = g(m,vPj)k(aj)Avj + cO(m)T40(ar)O<l<T

+ C2 (m)T-0 2 (aT), (62)

where/ 1 c+i-

W = 27ri Ej ip -

1 c+i-

OWx = .1 (p +27r( [-ij

1 c+i-

t2(X) 2ri Le i- (P

with the kernel A(mp) given by Eq. (61) and -2 < c <0. It is easy to show that, in the region Rep < 0 (at leastif Im -11 << 1), the equation l(mp) = l(-m,p) has theonly zero p = -1, which is the removable singularity ofthe integrands in Eq. (63). Therefore, the set of singu-lar points of analytic functions A(mp) and Fc(p) coin-cide provided that Rep < c. On the basis of Jordan'slemma applied to the region Rep < c with -2 < c < 0,the contour integrals in Eq. (63) can be computed bythe residue theorem. With the help of Eq. (B9) we getexpansions of the form

x) - (4n 2- 1)a(m,n)x2 n,

n=l

(x) = (2n -1)a(mn)x2 ,

n=1

q52 (X) = (2n + 1)a(m,n)x2 -,n=1

in which a(m,n) = (-)n(2n - 1)[l(m,-2n) -I(-m,-2n)] 1-/(2n)! These power series are obviouslyconvergent throughout the entire complex plane x.

It may also be noted that, after passing the limit, asthe relative refractive index m tends to unity and thephase shift p is fixed, the above result of inversion ofEq. (57) with kernel Q = Qs is proved to coincide withthat found in Ref. 14 in the case of the inversion of thisintegral equation with kernel Q = QH. The assertion isbased on equalities al(1) = -16 and a(+1) = ao(Gl) =a 1(-1) = 0. The equalities follow from Eq. (36).

Vll. Use in Ocean OpticsAll the scattering problems dealing with optical pa-

rameters m and x that belong in the acceptable domainof the S-approximation are within the scope of realapplications of this approximation. Let us consider,e.g., the S-approximation Qs of Mie factor Q. If ab-sorption is negligible, we can use the results of Sec. IV;from which it follows that, if refractive index m2 out-side particles is known, the upper bound of the relativerefractive index m = m11 is the only quantity to beestimated from the data of the measurements for the


(63)

1)A(m,p)x-Pdp,

1)A(m,p)x-Pdp,

1)A(m,p)x-Pdp,

acceptable domain to describe. In fact, the corre-sponding value of the size parameter xu = x(mj) maybe numerically evaluated [see Table I and Eq. (47)],and since the function x = x(m) decreases with increas-ing m-values, the rectangle 1 < m • m,, 0 < x(m) < xuin the Cartesian coordinates m and x is definitely in-cluded in the acceptable domain of the S-approxima-tion Qs. As a practical example, let us describe theacceptable domains of S-approximation Q for the par-ticles which are of importance in ocean optics. We usethe experimental data taken from Ref. 3.

Subscripts B and T refer to biogenous and terrige-nous groups of particles, respectively, which form thefundamental systems presented in seawater. On thesupposition that (outside the absorption bands) therefractive index of pure seawater is equal to m2 = 1.34,the upper bounds of the relative refractive indices areequal to mB = 1.02-1.04 and mT = 1.14-1.16. Makinguse of the data given in Table I and Eq. (47), we getXB(1.0 4 ) = - and XT(1.1 6 ) = 72. It means that, fromthe viewpoint of the S-approximation method, for abiogenous group of particles, the spectral measure-ments may be carried out without restriction; on theother hand, the lowest wavelength X = XT of the spec-tral interval for a terrigenous group of particles exists.The wavelength XT is to be found from the inequality27rm2aTXC

1< XT(1.1

6). It yields X > XT 0.29 ,um,

because the radii of the particles under considerationdo not exceed aT _ 2.5 ,gin.

Hence, S-approximation Qs of Mie factor Q is cor-rect within the acceptable domain that completelycovers the ranges of parameters m and x, which reallyoccur in the main components (suspended particles,dissolved organic matter) of oceanic water, at least inthe wavelength region of X > 0.29 m outside theabsorption bands. It means that S-approximation Qscan be utilized when solving some (including inverse)scattering problems in ocean optics.

Vill. ConclusionsThe various forms of the S-approximation method

applications are based on a number of relations, andonly one of them is not exact. Such an approximaterelation given by Eq. (26) [or by Eq. (24)] reflects theintrinsic meaning of the Mie scattering problem inview of its analytical nature. So, for each fixed valueof the parameter m _ 1 the absolute error las - akniappears to be uniformly small with respect to the ac-ceptable values of the parameter x, every natural num-ber n, and k = 1,2; here, as is the notation for thequantity on the right-hand side of Eq. (26). In thelong run, this fact and the symmetric dependence ofthe quantities defined in Eqs. (1)-(5) on the scatteringcoefficients account for the above proved properties ofthe S-approximations for different optical characteris-tics.

We have shown:(1) The Mie amplitudes are better represented by

their S-approximations than by the Rayleigh approxi-mation (for Im -11 << 1) and the Rayleigh-Gans ap-proximation (for 101 << 1).

(2) The Mie amplitudes can be replaced by their S-approximations for the ranges of parameters m and x,which are much wider than those for the Rayleigh-Gans approximation.

(3) In the long wavelength region (x << 1), the as-ymptotic behavior of all the Mie efficiencies is certain-ly reproduced from their corresponding S-approxima-tions.

(4) Within the acceptable domain [Eq. (46)] the Miefactor Q = QSC is restored more accurately and uni-formly by its S-approximation Qs = QS from Eq. (43)than by the approximation anomalous diffraction QH= QSC from Eq. (41). This is the most considerableimprovement on the interval 0.95 < m S 1.07 (for allvalues of x > 0). It should also be observed that, as x- 0, both the Mie factor QSC and its S-approximationQ' are of the same order of O(x4), whereas the factorQHC is of the order of O(x 2) [see Eqs. (52), (54), and(56)]. On the other hand, for large values of parameterm, the relative error

(m'X) = QS Q [x >x(m), m =Rem]

increases appreciably as x increases [e.g., A(1.24,a-) =17%] so that the factor Qs is actually useless in theshort wavelength region (x >> 1) even if m > 1.20-1.24.However, the comparatively wide range of acceptableparameters m and x allows us to solve many real prob-lems in the S-approximation of factor Q.

In addition, the following results were obtained ear-lier:

(5) The S-approximation Qs improves the Ray-leigh-Gans approximation of Mie factor Q for conser-vative scattering.5

(6) The S-approximation for Mie amplitudes at 0 =7r (backscattering) was evaluated in Ref. 6.

Thus, we are able to conclude that the scatteringcharacteristics can be replaced by their S-approxima-tions provided optical parameters m and x are in theacceptable domain. So, in the case of the extinctionefficiency factor for a nonabsorbing spherical particle,this domain is described by Eq. (46) and the data arepresented in Table I. In practical use, it is importantthat the S-aproximation proves to be in fairly goodagreement with Mie factor Q for frequently encoun-tered disperse systems in which the particles are sus-pended in a medium with similar optical properties.

Appendix A. Series Used to Derive the S-ApproximationThe symbol definitions introduced by Eq. (45) are

used and the following notations: p = 2(# - a), R =2(o + a), ij = (x? + x - 2Xj)1/2.

The addition theorem for the Bessel functions canbe written as

{ I (2n + 1)Pn(=i)Pn(-i)Pn(A) XiXjS(Wij)'n=O

E' (2n + 16(Xi)Xn(Xj)Pn0A) = XiXjU(Wij)s , l j

n=O


(A2)

From this we have4 5

i (n + 0.5) 1 n _XrI sm12 s 34 d

(n + ° 5) II 4n(Xr) Xn(x 4) 1 1 sinw1 2 COSW34 dj.

n=O r=1 x X4 4 -1 W1 2W3 4

With the aid of Eqs. (A3) and (A4) we obtain

7~~~~~~Z (n + 0.5) n ( n/3(a) = a1(4 + 2Rp)[ci(R) - ci(p)]n=0 12803

+ R2

_ p2

+ (R2 + 4Rp + p2)[s(R) - s(p)]

+ (3R2 + 4Rp - p2)c(R) - (-R

2+ 4Rp + 3p

2)c(p)j,

N (n + = 1 4[ci(R) - ci(p)]n n ~~128n=O

+ (R2 - 2)[s(R) + s(p)]

+ (3R2+ p

2)c(R) - (R2 + 3p

2)C(p),

(n + 0.5) n n(Y = 1a a {(4 + 2Rp)[si(R) + si(p)]n'~~~~O ~~128/3

+ (R2 + 4Rp + p

2)[a(R) + a(p)]

+ (3R2 + 4Rp -p2)y(R) + (-R

2+

4Rp + 3p

2)Y(p)j,

Appendix B. Relations Used in the Inverse ProblemThe integral representations

(A3) H(z) = 2z J (1- t2) sin(zt)dt,

(A4) Hi(z) = 2z J dt J u2

(1 - t2) sin(ztu)du,

fo 0

J Jeiz o dt io t~u

(A5)

(A6)

h(z) = 2z J (1 - t2) cos(zt)dt,

hi(z) = 2z J dt J u2

(1 - t2) cos(ztu)dul

1 11 si(z) =z J dt Jo cos(ztu)du

can be checked by direct calculationEqs. (45) and (51).

The Mellin transform

Mf(x) = Jf(x)xP-ldx

(B1)

(B2)

on the basis of

(B3)

of the function f(x) = x sinx yields

M(x sinx) = pfc(p),

(A7)

rz(n + 0.5)V,2(/,p)i/4(a)x'(a) = 1(4-2Rp)[si(R) + si(p)]

+ (R2 -4Rp + p

2)&(R) + a(p)]

+ (3R2 -4

Rp - p2)y(R) + (-R2 - 4Rp + 3p

2)y(p)1, (A8)

Z (n + 0.5)V'n(#)Vn(0Kn(C1)Xn(a) + tn(a))x'(a)I

n=O

= 64 14[si(R) + si(p)] + (R2 -2)[a(R) - a(p)]

+ (3R2 + p2)'y(R) + (R2 + 3p

2)Y(p)1. (A9)

To establish the domains of convergence of the seriesgiven by Eqs. (A3)-(A5), we use the formulas22

J,(v/coshy) (2 rv tanh) -1/2 exp[-v(y - tanhy)],

Yp(vcoshy) -(0.5rv tanhy)yY/ 2 exp[v(-y - tanhy)],(A10)

(B4)

where

r,(p) = r(p) cos rP.2 (B5)

From Eqs. (Bi) and (B4) we derive the Mellin trans-forms as follows:

4r,(p) _______

MH(x) = ' 2 MHi(x) -

(p-2)~~~r~p

Mci(x)= C . (B6)p

The transforms given by Eqs. (B4) and (B6) are obvi-ously valid in the strip

-2 < Rep < 0.

Note that

2rc(p)rc(1 - p) = r

and (n = 0,1,2,. . .):

(B7)

(B8)

describing, as v becomes infinite, the asymptototic be-havior of the cylindrical functions. As Eq. (A10) holdsfor any positive value of y, we conclude that the seriesin Eqs. (A3) and (A4) converge for all non-negative and(if, e.g., X3 = X4) positive numbers xr, respectively.Analogously, the domains of convergence of the seriesin Eqs. (A5)-(A9) include at least the entire quadranta, , > 0. The analytic continuation of the expansionsin Eqs. (A5)-(A9) makes it possible to use those forevery positive number a and all complex values of fi.It should be noted that if this addition theorem isapplied in the form of Eq. (A2) such extension fails.

lim (p + 2n)rc(p) = (), lim (p + 2n + 1)rc(p) = o.p--2n (2n). p--2n-1

(B9)

References1. H. C. van de Hulst, Light Scattering by Small Particles (Wiley,

New York, 1957).2. C. F. Bohren and D. R. Huffman, Absorption and Scattering of

Light by Small Particles (Wiley, New York, 1983).3. K. S. Shifrin, Introduction to Ocean Optics (Gidrometeoizdat,

Leningrad, 1983).4. A. Y. Perelman, "An Application of Mie's Series to Soft Parti-

cles," Pageoph 116,1077-1088 (1978).


5. A. Y. Perelman, "Extinction Efficiency Factor for SuspendedOceanic Particles," Izv. Akad Nauk S.S.S.R. Fiz. Atmos. Okeana22, 242-250 (1986).

6. A. Y. Perelman, "The Scattering of Light by a TranslucentSphere Described in Soft-Particle Approximation," Dokl. AkadNauk S.S.S.R. 281, 51-54 (1985).

7. K. S. Shifrin, Scattering of Light in a Turbid Medium (Goste-chizdat, Moscow, 1951), Chap. 7.

8. A. N. Tichonov and V. Y. Arsenin, Solution of Ill-Posed Prob-lems (Winston-Wiley, New York, 1977).

9. S. Twomey, Introduction to the Mathematics of Inversion inRemote Sensing and Indirect Measurements (American Else-vier, New York, 1977).

10. D. L. Phillips, "A Technique for the Numerical Solution ofCertain Integral Equations of the First Kind," J. Assoc. Comput.Mach. 9, 84-97 (1962).

11. S. Twomey, "On the Numerical Solution of Fredholm IntegralEquations of the First Kind by the Inversion of the LinearSystem Produced by Quadrature," J. Assoc. Comput. Mach. 10,97-108 (1963).

12. A. Y. Perelman, "Solution of Integral Equations of the FirstKind with Kernel Depending on the Product," Sov. J. Vychisl.Math. Math. Phys. 7, 94-112 (1967).

13. K. S. Shifrin and V. F. Raskin, "Spectral Transparency andInverse Problem of the Scattering Theory," Opt. Spectrosc. 11,268-271 (1961).

14. K. S. Shifrin and A. Y. Perelman, "Determination of the ParticleSpectrum of a Dispersed System from Data of Its Transparen-cy," Opt. Spectrosc. 15, 533-542 (1963).

15. K. S. Shifrin and A. Y. Perelman, "Calculation of Particle Distri-bution by the Data on Spectral Transparency," Pageoph 58,208-220 (1964).

16. M. A. Box and B. H. J. McKellar, "Further Relations BetweenAnalytic Inversion Formulas for Multispectral Extinction Da-ta," Appl. Opt. 20, 3829-3831 (1981).

17. G. Viera and M. A. Box, "Information Content Analysis ofAerosol Remote-Sensing Experiments Using an Analytic Eigen-function Theory: Anomalous Diffraction Approximation,"Appl. Opt. 24, 4525-4533 (1985).

18. C. B. Smith, "Inversion of the Anomalous Diffraction Approxi-mation for Variable Complex Index of Refraction Near Unity,"Appl. Opt. 21, 3363-3366 (1982).

19. A. L. Fymat, "Remote Monitoring of Environmental ParticulatePollution: a Problem in Inversion of First-Kind Integral Equa-tions," Appl. Math. Comput. 1, 131-185 (1975).

20. M. Bertero, C. D. Mol, and E. R. Pike, "Particle Size Distribu-tion from Spectral Turbidity: a Singular-System Analysis,"Inverse Problems 2, 247-258 (1986).

21. J. D. Klett, "Anomalous Diffraction Model for Inversion ofMultispectral Extinction Data Including Absorbing Effects,"Appl. Opt. 23, 4499-4508 (1984).

22. M. Abramowitz and I. A. Stegun, Eds., Handbook of Mathemat-ical Functions (Dover, New York, 1965), Chapt. 9.


extinction and scattering by soft spheres

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