Analytical solution to the optical transfer function of a scattering medium with large particles

Analytical solution to the optical transfer functionof a scattering medium with large particles

Eleanor P. Zege and Alexander A. Kokhanovsky

A new analytical expression for the optical transfer function of multiple-scattering media such as clouds,mists, and dust aerosols is given in terms of their microphysical characteristics. The geometrical opticsapproximation is used to find local optical parameters of a scattering medium, including the simpleapproximation of the phase function, which is the key to the solution of the problem consideredhere. The optical transfer function is taken within a small-angle approximation of the radiative transfertheory. A comparison with Monte Carlo data shows a fairly satisfactory accuracy of our analyticformulas.

Key words: Radiative transfer theory, small-angle approximation optical transfer function, geometri-cal optics.

1. Introduction

It is the optical transfer function (OTF) that is thebasic characteristic responsible for image transferthrough a scattering medium (e.g., atmospheric aero-sols, clouds, and fogs).1 4 The OTF of the scatteringmedium depends on the medium's microstructureand thickness. When the effect of the microstruc-ture on image-transfer characteristics is studied, theMie calculations are used to obtain single-scatteringcharacteristics, and then the Monte Carlo simulationsprovide the OTF's.4 5 It is a direct and reliabletechnique, but how cumbersome and time consumingit is! Moreover, it is not exactly what we need fordesign estimations.

The phase functions of real scattering media (aero-sols, clouds, mists, sea water, etc.) are highly ex-tended, and the scattering is predominantly in theforward direction. That is why the small-angle ap-proximation (SAA)1"2 can be used to rescue the situa-tion, and it provides an elegant OTF solution for thisparticular case. But the strongly extended phasefunctions are applicable only to scattering by largeparticles.6-10 In addition the Mie calculations be-come more cumbersome the larger the particles.The geometrical optics approximation' 10 is an ad-equate tool to use for the study of large particles.

The authors are with the Institute of Physics, Prospect F.Scoryna 70, Minsk 220602, Belarus.

Received 19 April 1993; revision received 1 January 1994.0003-6935/94/276547-08$06.00/0.© 1994 Optical Society of America.

In this paper we consider imaging through a scatter-ing medium that is a polydispersion of large sphericalparticles where the diffraction parameter p = ka >> 1,k = 2,rr/X, X is the wavelength, and a is the particleradius. Let f(a) be the particle-size distribution(PSD), and let m = n - ix be the complex refractiveindex of the particle. The optical parameters of thescattering medium are the extinction E and scatteringa coefficients and the phase function P(O) ( is thescattering angle). Geometrical optics solutions for a,a, and P(0) are known.7-10

Essentially we have simplified the known geometri-cal optics solution for P(0) (see Appendix A).Combining the obtained formula with the SAA solu-tion for the OTF provides the desired analyticalsolution, which directly relates the OTF of the me-dium to its particle characteristics, m and f(a). Theresults of the combined Mie and Monte Carlo compu-tations5 were used to test our simple result, and ahigh accuracy for fog and clouds has been obtained.

The solution obtained provides an ample scope forthe study of the effect of the microstructure on thetransfer characteristics of clouds and fog.

2. Theory

For scattering media with a moderate optical thick-ness 7 and an extended phase function, the OTF S(w*)can be estimated with the SAA1'11:

S(o*) = eXp[-T + q(W*)l], (1)

20 September 1994 / Vol. 33, No. 27 / APPLIED OPTICS 6547

1r = El, (w*) = o a(w*y)dy,

I1 du~~w~y) 2 o,(6)J0(o*y0)0d0.

2

of Eq. (7) can be used (the error will be < 2%):

(2)|1 - -(1 - 2 X

11,(w* a) = 4bŽ1

Here w* = v1 is the angular dimensionless frequency,v is the spatial frequency, I is the observation depth,Jo(AO) is the zero-order Bessel function, and u(O) =uP(O). The OTF accuracy provided by Eq. (1) wasdiscussed earlier,5 and we too shall touch on thispoint in Section 4.

We consider only large particles, so we can use thereported geometrical optics result7 -' 0 for the extinc-tion coefficient E = 2N7rM2 , where M2 = fo a2 f(a)da,and N is the particle number density. The simpleformula for the function u(O) [Eqs. (Al) and (A10)] isderived in Appendix A. Taking into account Eq.(A10) we can rewrite the expression for ,q(*) as

3

Q(W*) = N 7Taf* )ai=l

rli(W*) = N ra2f(a)q(w*, a)da,

Substitution of Eqs. (8) and (9) into Eq. (4) gives

11W°) = 5 ra3f(a)da + N rra2f(a)

28x 1 - -1,rra - 12a2 da,

112 = TrNM22a2[l + (*/) 2 ]1/2

- T(O)NiTM2* (r/)1/2erf[w*/2( )1/2],4w*

(3)

(10)

(11)

(12)

where

(4) M.= f aif(a)da, Mj* = 5 aje-cf(a)da.( T (13)

11i()*, a) = 2 dy OJO(*y)qj(0)d0,

where

() The function 1 (w*) can be represented in the moreconvenient form of

71(*) = NTr(M2 - M2) - 28(Ml - Ml)

qj(O) = 4J,2 (Op)/02 ,

q3(0) = T(0)exp(- p0 2 - c),

q2(0) = exp(-aO), 63 4 (l a

6 3 J(6)

and where c = 4Xp, and T(0), a, and f3 depend only onn (see Appendix A). Substituting Eq. (6) in Eq. (5),we obtain

2 2(2 + b2)(1 -Tl1((S -a) arccos b- Seb

4x U+(1 - b) + -

2(W*, a) =

(7)

1

2ot2 [1 + (*/a) 2 ]1/ 2

113(w , a) = T(0)ec (8)

Here, erf(x) = 2/(Qr)1/2 lox exp(-x 2 )dx is the probabilityintegral, and U+(x) is the Heaviside unit function[U+(x) = 1 at x 2 0 and U+(x) = 0 at x < 0], b = /a,and = w*/2k. The following simple approximation

(14)

where Mj = f, aif(a)da. As follows from Eq. (14) ato* - 0(b - 0), the value of

'9 = N'rM2 (1 - */WP21), P21 = kM2 /M1, (15)

while at a* 0(8 o)

,ql = 8kNM3/3w*. (16)

From Eqs. (11) and (12) we obtain at o* -- c

N7rM2

112 = 2ux(o*

NwrM2*T(O) (7/p)1/2113 4w (17)

and at w* - 0

N7rM 2112 = 2 1 -

2 l(*212 I -o I l 3 rrNM2*T(O) [ -

11 = 4F -~ 12y (*)2j. (18)

6548 APPLIED OPTICS / Vol. 33, No. 27 / 20 September 1994

b < 1

(9)

As seen from Eqs. (16) and (17), at a* >> 1, theratios 12/1 and 13/11 are independent of a* and areof the order of the small value P32-1 = (ka32)-1 anda32 = M3/M2 . For instance, for water droplets in thevisible spectrum (X = 0, M2* = M2 ) with regard to thedata of Table 1 and formulas (16) and (17), we have112/1l = 0. 2 /P32 and 113/11 = 3 .7 /P32. So, the largerthe scatterers, the more decisive is the contribution ofdiffraction into 1(W*) at high frequencies. For typi-cal water clouds in the visible spectrum with P32 60,we obtain1 2/'9 1 = 0.003 and1 3 /11 = 0.06.

On the other hand, for nonabsorbing particles atthe low frequencies (o* << 1) the contributions of thegeometrical optics terms and diffraction are compar-able [see Eqs. (15) and (18)]. For strongly absorbingparticles, when M2* -> 0 and 13()*) -> 0 the contribu-tion of the diffraction component is decisive at allfrequencies (ato * << land 2/91 < 1/2x 2).

The analytical expressions can be obtained forthe integrals of Mj and Mj in formulas (11), (12), and(14) for many typical PSD. Some of them are givenin Table 1.

So, formulas (1)-(3), (11), (12), and (14) give us thedesirable solution and directly relate the OTF to theparticle characteristics f(a) and m. Finally, fromEqs. (3), (11), (12), and (14) we have 11(w*) [see Eq. (1)]in the form

= - Z2 - v2 (1 - Z1)

83 ~~4a32

+ - a1 2 (1 - Z1) + r Z3

1 T(0)y

2+2[1 + (w*/a)2]l/2 +

X (/p)1/2erf[w*/2(3)1/2]}1TNM 2, (19)

where (see Table 1)

Z = MJ/Mj, Y = M 2*/M 2. (20)

The first five terms in Eq. (19) originate from theFraunhofer diffraction, and the sixth and seventhterms are due to the beams reflected from the surfaceand transmitted through the particle, respectively.

Note, that the Mj* function in Eq. (20) can be givenas

M* = k 1t Mk+j,k =O k

(21)

where a = 4rx/X. At ot << 1, from Eq. (21) we have

M2* = M2 (1 - 2aCa32), (22)

where a32 = M 3/M 2 is an effective radius of thepolydispersion. As would be expected, at * x0from Eqs. (1) and (19), it follows that

S(-) = exp(-T) (23)

i.e., the OTF is determined only by the unscatteredlight.

At (o* = 0 a new and useful expression for thetransmission coefficient can be obtained from Eqs. (1)and (19).

3. Transmission Coefficient under Normal Incidence

Really, the OTF at w = 0 gives the transmissioncoefficient of a layer' illuminated by a normallyincident wide parallel beam t, [Lo = 1) (o is thecosine of the angle between the incidence directionand the normal to the layer boundary), i.e.,t(T, = 1) = S(w* = 0). From Eqs. (1) and (19) at

Table 1. PSD Moments M} and Values of Z = Mj/M for the Most Frequently Used PSD'sa

N f(a) Mj Z

Gamma distributions ( R+1a exp(-a/ao) (ao\i 1(p. + 1 + j) P(L + 1 + j, A),

kaol r(L + 1) 1 i [(p. + 1) =o)*

A= 2

po = kao

Log-normal distributionc exp(-0.5c-2 In2 a/am) . U221 + erf(x)

a 1(2'nj'/2 (am)J expy-j-2 2

au(2,rr)~~~ ~~~~~~~~~ 2 2d l I ~2* 7

2p

x~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

aHere, P(n, y) is the incomplete gamma function, F(x) is the gamma function, and erf(x) is the probability integral.12

bHere, ao is the mode radius, and p. is the half-width parameter.cHere, ao is the median radius and (r is the rms of the particle radius logarithm.


aij = MiAMj

(* = we have

[1 1 - M2 * T(0)1t(r, p.0 = 1) = expj-T[2 - M 813 Jj (24)

where the parameters a and 13 depend only on the realpart of the refractive index (at X " n). Here =2NTrM21, which follows from the geometrical opticsapproximation. It is a new expression for the trans-mission coefficient that relates it directly to the PSDmoments and the value of m.

The following formula obtained through the SAA isknown':

t(T, Ro = 1) = exp[ -r(1 - A*)], (25)

where A* = AF(0o) is some efficient single-scatteringalbedo, A = u/E is the true single-scattering albedo,and F(0o) is a part of the light flux singly scattered inthe on-axis region 0 < 00. Strictly speaking, the A*value is not specified here: It depends on the choiceof the value of 00, which differs in different works,"ranging from Tr/6 to Tr/2. Using formula (24), onecan not only avoid this uncertainty, one can alsorelate the A* value directly to the particle characteris-tics (n, X, and PSD moments) by

ances made for the data of Table 1, (o) is of the form

1(03*) = S{1 + 83P(. + 4,A) -P(p + 3,A)

2A

Tr( + 2) [ -P(P + 2, A)]A3

6+rr( + )(p + 2) [1 - P(R, A)

1 T(0)y

2a2 [1 + (*/a) 2 ]1/2 4w*

(29)

where E = Nmrra02(. + 1)(p + 2)/ p 2, A = p.w*/2po,y = (1 + 4Xpo/ p)-(,f+3), and P(n, y), is the incompletegamma function.' 2

The proposed solution gives an ample scope for thestudy of the effect of the medium's microstructure onimage degradation during image transfer throughfogs and clouds. Formula (29) for the gamma PSD isespecially simple when p. is an integer, because in thiscase the incomplete gamma function P(n, y) iS 2

(26)

As follows from Eq. (24), in the two limiting cases(X = 0 and Xp >' 1), the transmission coefficient doesnot depend on X, and for nonabsorbing media and wehave

P(n, y) = 1 - exp(-y) Y!j=0

(30)

Figure 1 compares the OTF estimated throughEqs. (1) and (29) with the results of Monte Carlocalculations5 combined with the Mie formulas toprovide the single-scattering characteristics. Let uscall such an algorithm the Monte Carlo-Mie method.

A* = 1 + I + T(0)2 4a

2 813(27) 1 .U

0.9While at strong absorption when M2* -* 0, we obtain

1 1A* = + - (28)

where the term 1/4x 2 is small (see Table 2 inAppendix A). As follows from Eqs. (27) and (28) forwater droplets in the visible spectrum (m = 1.33)A* = 0.94, and for the dust aerosol in the visible(m = 1.53 - 0.008i and kXao >> 1) A* = 0.54.

The accuracy obtained with Eq. (24) is high.Comparison with different computation results hasshown that at < 7 and a32 1OX errors of thetransparency estimations do not exceed 7%. Someillustrations will be given in Section 4.

0.8

CO) 0.7

0.6

0.5

0.4

I - of --- ----------------

. . .. .I ,. . . . I . . . . . . . I . . . . . . . . .i I . . . . . . . . . I

0 20 40 60 80 100

4. Numerical Results and Comparison with Monte CarloSimulations

Consider a scattering medium with a gamma PSD6

(see Table 1). As follows from Eq. (19), with allow-

Fig. 1. OTF calculated at T = 1, = 0.7 pm, n = 1.33, K = 0, andp. = 6 based on the proposed Eqs. (1) and (29) for ao = 4 pAm (solidcurve), a = 8 m (long-dashed curve), and a = 50 pim (short-

dashled curve), and based on Monte Carlo calculations for a = 4 (asterisks).


X (,7/p)1/2erf(w*/2CP) I

1 1 T(O)M2*A* = 1 + 2+ �� .2 2 at 4PM2

Here data are presented for a model6 cloud C1, whichis the gamma PSD with p. = 6 and a = 4 um. Asseen from Fig. 1, at 7 = 1 both methods give practi-cally the same results for all (o*. Figure 1 also showsthe results of the calculations for polydispersion withlarge particles at a0 = 8 plm and a0 = 50 pim.

As seen from both Eq. (1) and Fig. 1, the functionD(03*) = -ln[S(03*)/T] for nonabsorbing scatteringmedia at small o* values does not practically dependon the mode radius ao. It originates for two reasons:First, at X = 0 the geometrical optics phase function,12(03*), and 1(03*) do not depend on a0 . Second, at03* -. 0, the diffraction componentX 1(X*)/E is propor-tional to the small value P21-1 [see Eq. (15)]. On thecontrary, at large *, dependence of the functionD(w*) on the mode radius is essential [see Eq. (16) andFig. 1]. The parameter a32 of the PSD can beobtained through the OTF measurement.'3" 4

Analysis of the data reveals that the OTF dependenceon p at a fixed value of a32 = ao(l + 3/p) is ratherpoor,'3 and the a32 value is the most influentialparameter of the microstructure relating to the OTFestimations.

Figure 2 gives the T dependence of F = -ln[S(w*, T)]

for a few 0* values. One can see that the values ofthe transmission coefficient, which is S(03* = 0, T),

agree with the results obtained with the MonteCarlo-Mie method for all T < 7. Our simple ap-proach ensures good accuracy, especially for low(03* 0 0) and high (* 2 50) angular frequencies.The errors of our formula, which grow as 7 increases,originate from the use of the SAA.

Figure 3 illustrates the OTF dependence on the

5

4

(f3 ,U)

2 ,,

,. . . . . . . . . .. . . . . . . . . .1 ,, , _

C0 ~~2 4 6 8

Fig.2. Plot of the cloud Cl model of -ln S(o*) versus T at =0.45plm based on Eqs. (1) and (29) for a* = 0 (solid curve), ok = 3(long-dashed curve), a* = 12 (medium-dashed curve), and w = 50(short-dashed curve), and on Monte Carlo calculations for the samevalues of o (asterisks).

1.0

0.9

0.8

!) 0.7

0.6

0.5

-

20... . 40... . 60... . 80... . 100.. .. I'0 20 40 60- 80 100

Fig. 3. OTF calculated with the proposed Eqs. (1) and (29) at'r =

1, A = 0.7 pLm, n = 1.33, p. = 6, and ao = 4 pAm, for x = 10-2 (solidcurve), K = 10-3 (short-dashed curve), and K = 10-4 (long-dashed

curve).

microstructure parameters and the particle absorp-tion. As seen from the figure, the OTF of theabsorbing and scattering media shows a pronounceddependence on absorption at low frequencies.

We have demonstrated only the data for wateraerosols. Image transfer through soil aerosols isalso an important problem. Figure 4 gives the OTFfor dust aerosols with m = 1.53-0.008i and p. = 2 atdifferent mode radii of the PSD. Naturally, the OTF

0.62

0.58

)

0.54

. . . . . . ...... I I I I I.. . . . . . . I . . . . . . . I . . . 1

20 40 60 80 100

W*

Fig. 4. OTF calculated with the proposed Eqs. (1) and (29) at'r =1, A = 0.7 pum, n = 1.53, K = 0.008, and p. = 2 for ao = 30 pum (solidcurve), ao = 10 .m (short-dashed curve), and ao = 5 pum (long-

dashed curve).


. . . . . . .

At

I , I

I

I �I

I �I ,

II

I,I

II

value for dust aerosols is lower than that for waterbecause of the strong light absorption in particles.

5. Conclusion

We have managed to directly relate the OTF to theparticle-size distribution parameters and to the com-plex refractive index of the particles. To do so, wecombined the special geometrical optics solution foroptical parameters of particles and the small-anglesolution for the OTF. The accuracy of the derivedformulas appears to be high enough: the error of theOTF estimation is no more than 7% at a32 > 10 upto = 7 for nonabsorbing particles, and it provideshigh accuracy at any r for absorbing and scatteringmedia. This approach is indispensable in the studyof the effect of the microstructure on imaging throughaerosols, clouds, and fogs and in the consideration ofthe OTF changes in the processes of condensation,coagulation, or evaporation of drops that are bothnatural and artificially stimulated.

The use of our OTF formula can save much time incomparison with conventional techniques such as theMie calculations for optical particle parameters or theMonte Carlo calculations that provide OTF.

Relation (16) is the key to the inverse problem, andthe parameter a32 of the PSD can easily be obtainedthrough OTF measurements.13,'4 Such a methodcan be considered as a generalization of the small-angle method15 of particle sizing that takes intoaccount the multiple-scattering effect.

reflection inside the particle, with

Il(a, 0) - p2Jl2 (po)02

I2a -p2R(0)4

_ p2T(O)(A3)

Here J,(p0) is the first-order Bessel function, c =4Xp and

_N 1j2(1 - q2)1/2 - (n2 - q2)1/2]2

R(0) = 2j1 N2(1 - q2 )1/2 + (n2- q2)1/2J

( 2n 4 (nq - 1)3(n - q)3 (1 + q4 )

\n2 - 1 2q5(1 + n2 - 2nq)2

n - q(1 + n2 - 2nq)1/2

N, = 1, N2 = n2.

(A4)

(A5)

0q = cos -

(A6)

For any particle in the small-angle region, which isof interest in OTF investigations, and for all anglesfor absorbing particles, the terms in Eq. (A2) withj >3 can be neglected.7

To simplify this result, we suggest using the follow-ing approximations:

Appendix A

Here we give a convenient analytical approximationfor the phase function P(0) of polydispersions of largespherical particles

P(0) = 2u I(a, O)f(a)da, f (a)da= 1. (A)k'rNf~od.(l

R(0) = exp(-a0),

2n - 1A(O) = 1 - (n- 1)

where

1 2n 4

(n - 2n + 1,

Here N is the number concentration, k = 2r/X, isthe wavelength, c is the scattering coefficient, I(a, 0)is the dimensionless Mie intensity, and a is theparticle radius.

It is known that the Mie calculation are the morecumbersome for particles with larger diffraction pa-rameters p = ka. The geometrical optics approxima-tion for I(a, 0) gives

I(a, 0) = E Ij(a, 0).j=l

(A2)

Here I(a, 0) is the radiance of the Fraunhoferdiffraction, I 2(a, 0) and I 3(a, 0) are, respectively, theradiance of the light reflected by the particle surfaceand that transmitted by the particle without any

(A8)

and the values of a and 1 depend only on the real partn of the complex refractive index (see Table 2). Theerror of approximation (A7) does not exceed 15% at0 < 350 andn = 1.33-1.6.

Thus, instead of Eqs. (A1)-(A6) we have

I(a, 0) = T I qj,IX cj 3

(A9)

Table 2. Values of a and , as Functions of the Real Part of theRefractive Index n

n a 13

1.33 3.0 4.71.4 2.7 3.61.53 2.4 2.41.6 2.3 2.0


T(0) = T(O)exp(- p02

- c),

(A7)

4J, 2 (op)qL= 02 q2 = exp(-a0),

q3 = T()exp(- p02 - c),

and

r(0) = N7M 2

x [D(0) + exp(-ao) + yT()exp(-1302 - c)],

(A10)

where

M2= a 2 f(a)da,

Table 3. The Angular Scattering Coefficient a(e) (km-) for the ModelCloud Cl Provided by Mie Calculations6 and Eq; (AlO) with Relative

Percent Discrepancies A

a(0)

0 (deg) Mie Theory Eq. (A10) A (%)

0 2.81 x 10-4 2.53 x 10-4 9.91 1.98 x 10-4 1.77 x 10-4 10.72 7.56 x 10-3 7.05 x 10-3 6.83 2.22 x 10-3 2.16 x 10-3 2.44 8.18 x 10-2 8.06 x 10-2 1.45 4.48 x 10-2 4.38 x 10-2 2.26 3.08 x 10-2 3.01 x 10-2 2.37 2.36 x 10-2 2.33 x 10-2 1.58 1.93 x 10-2 1.93 x 10-2 <19 1.66 x 10-2 1.67 x 10-2 <1

10 1.47 x 10-2 1.49 x 10-2 -1.415 9.78x 10-' 1.03 x 10-2 -5.325 5.22 x 10-' 5.42 x 10-' -3.735 2.74 x 10-' 2.32 x 10-' 15

M2* = a2 exp(-c)f(a)da, (All)sorbing spheres when co = 2irNM2, we have

U(0) = uP(0).

For the gamma particle-size distribution we have

= 2( 1)( 2+

1

(1 + 4xpo/[L)-+3po = kao,

Po (-n)nr(2n + 3)r(2n + 5 + pL)Dp = . 2

RL n=O r(lI1+ 3)r (n + 2)r(n + 3)n!

(oPo

2p,

P(O) = (1 + 3/p.)(1 + 4/p)po2 /2. (A17)

.12) Thus, at p - 0, P(O) = p 2/2, as would be ex-pected.7'8 Let us consider the known model6 of cloudC1, which is the gamma distribution of water dropswith p. = 6 and aO = 4 pm. For cloud C1, from Eq.(A17) at X = 0.7 pLm we have P(O) = 1611. The Miecalculation 6 gives P(O) = 1680, and the discrepancy isapproximately 4%. The accuracy of the simple phasefunction formula Eq. (A15) can be judged with Table3, where the data of the Mie calculation and of the Eq.(A10) approximation are compared for the cloud Cl

L13) model at = 0.7 pm. The discrepancy reaches avalue of 15% only at 0 = 35° and is smaller for valuesof 0 < 35°. Naturally, the error of the proposed

82n phase function approximation decreases with increas-ing particle radii and absorption.

(A14)

To obtain the last formula we use the knownexpansion' 2 given by

J, 2(Z) = (2 Anz2n2n=OA- (- I)-F(2n + 3)_ A1.\

n- 4nF2(n + 2)r(n + 3)n! \-_1

From Eq. (A10) at 0 = 0 we have

P(O) = rNM 2(po/p.)2 (p. + 4)(p. + 3)/u,(A16)

where the geometrical optics components are ne-glected. As follows from Eq. (A16), for large nonab-

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Analytical solution to the optical transfer function of a scattering medium with large particles