Expression for the reflectance of randomly rough surfacesderived with the Fresnel approximation
Ivan Ohlidal
In this theoretical paper the formulas expressing the reflectance of randomly rough surfaces are derivedwithin the framework of the scalar theory of diffraction. The Fresnel approximation has been used in themathematical procedure since geometrical conditions of some real reflectometers do not correspond to theconditions of the Fraunhofer diffraction of light. The formulas found are generally different from the for-mulas derived within the framework of the Fraunhofer approximation. By means of the numerical analysisof these formulas the differences between both the Fresnel and the Fraunhofer approximations are shown.It is also found that the formulas corresponding to the Fresnel approximation give the same results as theformulas derived by means of the Fraunhofer approximation if certain geometrical conditions are valid.These geometrical conditions are determined in this paper. The formulas corresponding to thin films withrandomly rough boundaries are also introduced.
I. Introduction
Optical methods are often used- for studies of theproperties of randomly rough surfaces and thin filmswith randomly rough boundaries. The method mostoften used in practice is that where the reflectancemeasured at normal incidence is considered.1-6 It ispossible to show that the reflectance of the above-mentioned systems is determined by their optical andstatistical properties and the geometrical conditions ofthe experimental arrangements.1-6 The reflectance isoften calculated within the framework of the scalartheory of diffraction. The authors dealing with thederivation of the reflectance by means of the scalartheory have used the Fraunhofer approximation so far(see, e.g., Refs. 1-4). In this paper the formulas ex-pressing the reflectance of randomly rough surfaces andthin films with random boundaries will also be derivedwith the scalar theory. However, it will be assumed thatthe geometrical conditions of the experimental ar-rangement correspond to the Fresnel approximation.New physical facts follow from the formulas derivedunder this assumption. The derived formulas will also
The author is with Purkyne University, Faculty of Science, De-partment of Solid State Physics, Kotlarska 2, 611 37 Brno, Czecho-slovakia.
be discussed through numerical analysis. With thisnumerical analysis we shall show that sometimes it isimpossible to use the simpler formulas found earlierwithin the framework of the Fraunhofer approximationfor an interpretation of experimental results. More-over, formulas expressing the reflectance of thin filmswith randomly rough boundaries will be introduced.
11. Theory
First we deal with a randomly rough surface sepa-rating two media that are homogeneous and isotropicfrom the optical point of view. The medium containingincident and reflected waves is always nonabsorbing.A physical model of the randomly rough surface is de-termined by the following assumptions:
(1) The surface is locally smooth (see, e.g., Ref. 4).(2) Slopes of irregularities are relatively small (i.e.,
corresponding angles are of the order of 1 orsmaller).
(3) Heights of irregularities are smaller than thewavelength of incident light.
(4) The rough surface is generated by a stationaryrandom process (x,y).
The geometrical conditions corresponding to theexperimental arrangement are as follows:
(a) A plane monochromatic wave falls perpendicu-larly on the mean plane of the rough surface.
(b) The dimensions of the irradiated surface aremuch greater than the value of the wavelength. Theirradiated area in the mean plane of the surface is anoblong with dimensions 2X and 2Y.
Fig. 1. Geometrical representation of the experimental conditions:k1 and/or k2 is the wave vector of incident and/or scattered wave, 02
and 03 are the angles characterizing the geometry of light scattering,D is the area of the detector (a circle), M is the mean plane of therandomly rough surface, ao is the acceptance angle of the detector,Ro is the distance between M and D, ro is the distance between a pointlying in M and a point lying in D, and 2X and/or 2Y is the linear di-
mensions of the irradiated area.
(c) The value of the half-acceptance angle of thedetector (see Fig. 1) fulfills the following relation: o_ 0.1 rad (the area of the detector is formed by acircle).
(d) Ro > 2X,2Y, where Ro is the distance between themean plane of the surface and the area of the light de-tector (see Fig. 1) holds. This assumption will bespecified in detail below.
In practice we must consider a near-normal incidenceof light. However, this fact has no influence on thefollowing calculations and considerations.
The electric field (P) at point P (see Fig. 1) is givenby the Helmholtz-Kirchhoff integral7
4 7rJis an an)
where E1 and/or d&B/On is the local electric field and/orits derivative according to the local normal n (see Fig.2), S is the irradiated area of the rough surface, and / isdefined by the equation
i = exp(ik2r)/r, (2)
where k = k2 = I k2 = ki = 27rno/X, X is the wave-length, and r is the distance between points P and Q (seeFig. 2). Below we will assume that no = 1.
With assumption (1) we can use the Kirchhoffboundary conditions, i.e.,
R1 = (1 + R)Ao exp(-ik ¢)
d -i(1 - fl)knAo exp(-ik ),an
(3)
where AO is the amplitude of the incident wave, n, is thecoordinate of n in the direction of the z axis, and R is theFresnel coefficient of a corresponding smooth surfaceat the normal incidence. This apparently fulfills
r = ro - Ro0/ro,
where ro = [R' + (x - xO)2 + (y - yo)2]'/ 2, and x0, yo,and Ro are coordinates of point P, -~(x,y).
The Fraunhofer approximation is used in followingcalculations if we limit the Taylor series of function roto linear terms with x and y. This approximation isjustified when the following inequality is fulfilled:
X 2 + y 2
2XR <<1 (4)
It is apparent that Inequality (4) cannot be fulfilled inthe reflectometers, the geometrical conditions of whichhave been stated through assumptions (1)-(4) (there isusually no optical element between the sample and thedetector, and X,Y 10-3to 10-2 m,Ro 1 m). Fromthe general point of view this fact requires the use of theFresnel approximation in the mathematical formulationof the problem. It means that we have to write
ro = Ro + 1 [(x - Xo)2 + (y - yo)21.2Ro
(5)
Before the following calculations, it is suitable to specifythe geometrical conditions in the real reflectometermore precisely. In practice the following inequalitiesare usually fulfilled:
X 2 Y2). ' A >> 1,[Xo' XR<
[(xt - XO)2 + (y - yo) 2Ima/R2 « 1,
(6)
(7)
where max denotes a maximum quantity written inbrackets. Now it is evident that we can write
(P) = - RfAoexp(ikRo) fX S exp(-ik)
X exp i k [(x -Xo) 2 + (y-yo)2I dydx. (8)
The light flux measured with the detector is propor-tional to the value (A(P)E* (P)) (the symbol ( ) de-notes the statistical mean value of the correspondingrandom quantity). It holds that8
z
D
x(x,y)
Fig. 2. Geometrical representation of a randomly rough surface:t(x,y) is a random function describing the surface mathematically,n is the local normal of the surface, r is the distance between a pointlying in D and a point lying in the surface, no and/or n is the refractive
In Eq. (10) the first and/or second term correspondsthe coherent and/or incoherent flux.
Let us deal with coherent flux F at first.applying the method of stationary phase,9 we get
(2(P)) =fAoexp(ikRo)9(k) forPe A,
(2(P) ) = for P A,
(9) X(k 1 - k) = J exp[-i2k(z - i)]w(zzi;T)dzdzl,
(10) where w(z,z,;-) is the 2-D distribution of the probabilityto density of random variables t and A. Let us introduce
the substitutionBy
(11)
where A denotes the irradiated oblong in the meanplane (the diffraction of light on oblong A is ne-glected),
S(k) = |' exp(-i2kz)w(z)dz, (12)
and w(z) is the 1-D distribution of the probabilitydensity of the random height variable. Let us assumethat the area of the irradiated oblong is a part of thedetector area. Then
F = ko4XYAojf~j29i(k)j2, (13)
where ko is a coefficient of the proportionality. De-fining the coherent reflectance as R = Fc/F 0 (F0 is theincident light flux given by the equation Fo =k04XYA2), we obtain
R = RoI1(k)12, (14)
where R = [(n- 1)2 + 2]/[(n + 1)2 +)k 2], i = n + ik,and n and/or k is the index of refraction and/or ab-sorption of the medium bounded with a randomly roughsurface. The relative coherent reflectance is given as
Pc = Rc/RYo = IX(k) 12. (15)
If we assume that function w (z) is in the form of thenormal distribution, i.e.,
W(Z) = exp(-z 2/2a 2), (16)(22r 112,
we obtain
X(k) = exp(-8r 2 02 /X 2 ), (17)
where o- is the standard deviation of random function. Thus it is evident that the equations for the coherent
reflectance of the randomly rough surface derived at thevalidity of the introduced geometrical assumptions areidentical with those expressing this quantity in the caseof the Fraunhofer approximation." 34 It is also evidentthat the coherent reflectance is not dependent on thegeometrical factors describing the experimental con-ditions.
Incoherent light flux Fi is connected with the quan-tity D[B(P)]. It holds that
D[A(P)J = x 4' f 4'f 3 [X(k1 - k) - 9(k)I2I
X exp[ik(ro - roj)]dydyidxdx1, (18)
where
= (QoA k2 )/(47r2R2), j M t(xj,yj), rol = r(xj,y),
X -Xi rcos~o, Y - Yi= Sin. (19)
Then it can be written as
ro-rol = [(X - xo)r cos + (y - o)T sinfp]/Ro.
Let the function w(z,zI;r) be given as follows:
W(Z,ZI;T) = W(Z)6(Z - z)C(O)+ W(Z)W(Z)[1 - C(T)]
(20)
(21)
where (z - z1) is the Dirac function, w(z) and/or w(z1)is given by Eq. (16), and C(-) is the correlation coeffi-cient of the surface expressed as
C(T) = exp(-r 2 /T 2 ), (22)
where T is the correlation length. After performing theintegration over s° and T, we obtain
D[2(P)] = rxT2 [ - 1S(k)12] fX Y
X exp j-2 [(x-x 0)2 + (y_ yo)2 ] dydx, (23)
4R1
where 1:(k) is given by Eq. (17). It is apparent that wecan write
lim F = ko4XYRoA2(1 - 1X(k)12).TA-~~~~~~~ (24)
We also get the same result for Fi in the case of theFraunhofer approximation.' 0 This means that theincoherent reflectance Ri = Fi/F 0 (the relative one is pi= Ri/Ro) of the randomly rough surface with enormousvalues of parameter T is practically identical in both theFresnel and Fraunhofer approximations. Moreover,the values of Ri are independent of the geometricalconditions of an experimental arrangement for thesegreat values of T. In this paper we shall deal with therandomly rough surfaces characterized by relativelysmall or moderate values of T(T/X < 10). Namely,these surfaces are more interesting from the practicalpoint of view. Let
xo = RO sin0 2 cOs0 3 , Yo = RO sin0 2 sin0 3 . (29)
Owing to assumption (3), we can write sinO2 02.Then
parameters a0, X/Ro, and Y/Ro express the geometricalconditions of the experimental arrangement. Depen-dence Ri or pi on X/RO and Y/Ro represents the new fact
(27) which is important in the interpretation of experimentalresults obtained with the considered reflectometer. Inthe general case total reflectance RT of the randomlyrough surface measured with a reflectometer is givenas
RT = RC + R. (34)
We can limit ourselves to the linear terms owing to xand y in the series of Eq. (27) if the values of parametersbil/a and b2 /a are sufficiently small. Then we have
r4(A4 X2 + Y 2=R= O3170 X) R a 1 - exp[-(4i-7r/X)21I
X exp [ ( a)1 (35)
By means of Eq. (35) it is possible to estimate conditionsfor which equality Ri = RiF is fulfilled. Namely, we canput R = RiF within the framework of the typical ex-perimental accuracy (ART/IRO = 0.01) when the fol-lowing inequality is valid:
ARIo 0.01 (36)
where AR is given by Eq. (35). Moreover, it is evidentthat equality R = RiF is fulfilled for any value of awhen
Fi =k R0 Jo f D[(P)]02d3d02-
After performing the integration of Eq. (30) we get
R. = Rol1 - exp[-(4ro/X)2] 1 - exp [-( A ao)|
n n-k k rT nkyk+ 91 EE R(n~k,p,q) 0.nn Xn
n=2 k=O q=O p=O L1?
I [ (1-rT %121 (n-p-q)/2 1 (IrT 2,dX 1-exp a- A °f° | tE
whereFX A 11L Xwhere
K,(n-kn an)[(n - k - q)!!(k - p)!!p!!q!!(n - k + 1)(k + 1)]
We can rewrite Eq. (31) in the following way:
Ri = RiF + AR,
where
RiF = olo{1 - exp[-(47ra/X) 2] I - [- (_rT x 21
30)4 4X2 + y 2
A 2 o'3X10 4(37)
Conclusions following from Relations (36) and (37) arevery important from the practical point of view, sincethey enable us to find the experimental conditions (i.e.,the values of X/Ro, Y/RO, and ao) justifying the utili-zation of the simple Formula (33) at the interpretationof Ri measured by means of the considered reflecto-meters. It is evident that Relations (36) and (37) can
(31) be easily fulfilled for the relative small values of T (i.e.,T/X < 1) or ai (i.e., /X < 0.05).
Now let us deal with the normal (Gaussian) surfaceoccurring more frequently in practice. This surface isdescribed by an n-dimensional normal distribution ofthe probability density11 from the statistical point ofview, so that after a similar mathematical procedure as
32) before, we get the following equations:
(33)
is the incoherent reflectance of a studied rough surfacecorresponding to the Fraunhofer approximation.3 4
Quantity AR represents the difference between theFraunhofer and Fresnel approximations. It is evidentthat the series expressing AR is quickly convergent.Equation (31) shows that in the case of the Fresnel ap-proximation the reflectance (Ri or pi) of the randomlyrough surface is dependent on the parameters o/X, T/X,ao, X/RO, and Y/Ro. The parameters u/X and T/Xcharacterize statistical properties of this surface, and
i n n-k kRi=RiF+ Roexp[-(47r/X) 2 ] 7' L' E' E
rn=1 n=2 k=O q=O p=O
X K1(n,k,p,q)m!
X(x r n Xn-kYk 1 - exp [-(- ao)2]LX-A/m) I I- /m(n-p-q)/2 1 7rT _2,uJ
The principal conclusions following from Eq. (38) areidentical to those corresponding to Eq. (31).
The theoretical results introduced can be generalizedto thin films with randomly rough boundaries. Herewe shall deal with two models of the system formed bya nonabsorbing substrate covered by a nonabsorbingidentical and/or general thin film. The boundaries ofthe two films fulfill the same assumptions as the simplerough surface studied in the preceding part of thispaper. Both the boundaries of the identical thin film(ITF) are identical from the geometrical and statisticalpoints of view, whereas the boundaries correspondingto the general thin film (GTF) are mutually indepen-dent from the same points of view (for details see Refs.4 and 12). It means that the ITF and/or GTF is locallyrepresented by a plane-parallel and/or wedge-shapedthin film. From this fact it follows that the formulasfor RC and/or Ri corresponding to the ITF are mathe-matically identical to Eq. (14) and/or Eqs. (31) and (38).The difference is only in the expression of flo. Namely,in both equations describing the ITF quantity No mustbe replaced by the formula for the reflectance of thecorresponding thin film with ideally smooth boundaries(see, e.g., Ref. 13). The situation is more complicatedwhere the GTF is concerned. Local field A within theirradiated surface of the upper boundary is determinedby the following equation (see, e.g., Ref. 4):
= {1 + r + (1-rl) (-1)mrlIr~m+m=O
X exp[i(m + 1)(47r/X)njdLI} exp(-ikt 1 ), (40)
where dL = d + t1 - 2, d is the mean thickness of theGTF (the distance between the mean planes of both theboundaries), ¢, and/or {2 is a random function de-scribing the upper and/or lower boundary, nj is the re-fractive index of the thin film, and r and/or r2 is theFresnel coefficient of the upper and/or lower boundaryof the corresponding system with the smooth bounda-ries at the normal incidence. Let us assume thatw(z1,z2;i-) is given by Eq. (21). After applying the samemathematical procedure as before, we obtain
1.00 I 7I >'-
0 20
0 0.05 alo 0.15 0.20
Fig. 3. Relative incoherent reflectance as a function of a/A. Curve1, Fraunhofer approximation; curves 2,3,4, Fresnel approximation(2: L/Ro = 0.01; 3: LIRo = 0.02; 4: LIRo = 0.033). In all cases itis valid that ao = 0.04 rad; 1 < T/X < 20. For example, or = 0.03 m
andT= 3mand0.15um < X 3 m.
AmI = (- I)-n 1 , m = (m + 1)-nid,X X
Bq4 r [(q+ 1)n-1], Mq-(q+1) 4vq mj,X
j(s)l = exp(-s 2 yj'/2), j= 1,2, s =Bq, Mq,
pi (o1 ,T) is given by Eq. (31) (- and T must be replacedby a, and T1),
uj,Tj and/or 2,T2 correspond to the upper and/or lowerboundary, and M is determined by an experimentalerror of Ri. If the rough boundaries are formed by thenormal (Gaussian) surfaces, quantities pi(a,,Tj), pi(TI),and pi(T 2 ), and pi(T') must be expressed by means ofEq. (38). In the same way as before we can prove thatwith the chosen geometrical conditions of the experi-mental arrangement the formula for R of GTF isidentical to that derived within the framework of theFraunhofer approximation. That formula is introducedin Ref. 4.
Ri= rlpi(ai,Tl) + E rr2 (+l)(1-r )2 (Bm)12[1- 2(M )12]
In this section the numerical analysis of Eq. (31) willbe performed. Then this section will be devoted to theanalysis of incoherent reflectance Ri of the randomlyrough surface because coherent reflectance R, is thesame for both the Fresnel and Fraunhofer approxima-tions. We will deal with the spectral dependences ofpi, Ri, and/or RT,PT(RT/RO) since they are very im-portant ones from the experimental point of view. Itmeans that values of parameters a and T will be fixedin all presented dependences of pi, Ri, and/or PTRT.We will be interested in the surfaces characterized by10X > T > X. Namely, the surfaces with T -X can alsobe considered as the locally smooth surfaces within theframework of the usual experimental accuracy provedby our experimental results1'- 6 and the results of otherauthors (see, e.g., Ref. 5). It means that in the spectraldependences upper limit Xmax of X values is given byT/Xmax - 1. For simplicity we shall assume that X =Y = L. In Fig. 3 the dependence of pi on the quantityi/X is introduced. Curve 1 corresponds to the Fraun-
hofer approximation [Eq. (33)], and curves 2, 3, and 4represent the Fresnel approximation for three differentvalues of parameter L/RO (the values of C, T and a0 arefixed). It is evident that the value of the difference Ap= PiF - Pi strongly increases with increasing L/RO valuefor oi/X ' 0.1 in the considered interval of oi/X values.Figure 3 also shows that for the chosen values of T andao we can make a considerable mistake if we neglect thedifference between the formula derived within theFresnel approximation and the formula derived withinthe Fraunhofer one for c/X > 0.1 and L/RO > 0.02. Itis also apparent that we can decrease the differencebetween both approximations in any way if we appro-priately decrease the value of L/RO. In Fig. 4 spectraldependences of total relative reflectance PT = PC + Piare given. These dependences correspond to the de-pendences of pi introduced in Fig. 3. In Fig. 4 spectraldependence PC is also given, so that it is possible to
1.00 ------
0.80 "..-3
0.60
0.40
0.20
Fig. 4. Relative total reflectance PT as a function of a/X. Curves1-4 correspond to curves 1-4 in Fig. 3. Curve 5 denotes the relative
coherent reflectance Pc corresponding to curves 1-4.
1.00 I
0.80 3b
0.60
0.40 / 'l
0.20-
0 0.05 010 0.15 0.20
Fig. 5. Relative incoherent reflectance as a function of a/X. Curvesla, 2a, 3a, and/or lb, 2b, 3b correspond to the Fresnel and/orFraunhofer approximation (1 - ao = 0.01 rad; 2 - ao = 0.04 rad; 3 -ao = 0.06 rad). In all cases it is valid that LIRo = 0.02; 1 < T/X < 20.For example, a = 0.03 Am and T = 3 gm and 0.15 Am • < 3 Am.
1.00 1 00 ' ~3a ' - -- -- ''
3b
0.80
0.60-
0.40 -' lb
0.20 ,l
0 0.05 0.10 015 0r/ 0.20
Fig. 6. Relative incoherent reflectance as a function of a/X. Curvesla, 2a, 3a and/or 1lb, 2b, 3b correspond to the Fresnel and/or Fraun-hofer approximation (1 - 1 S T/X S 6,7, 2 - 1 S T/X • 20, 3-1 T/X 40). In all cases it is valid that ao = 0.04 rad; L/Ro = 0.02. Forexample, a = 0.03 Mm; 1 - T = 1 um; 2- T = 3 Am; 3 - T = 6 m; and0.15 Mm • X Xma, where 1 - Xmal = 1 am, 2 - mac = 3 m, 3 -max
= 6 Mm.
compare contributions of the two components PC andpi to the total relative reflectance PT in the spectralinterval considered. Figure 5 shows the dependence piand PiF on oi/X for three different values of acceptanceangle a 0 (the values of oAT and L/RO are fixed). Figure6 gives pi and PiF as functions of c/X for three differentvalues of correlation length T (the values of a, a0 andL/RO are fixed). Both Figs. 5 and 6 show that the in-fluence of parameters a 0 and T on the values of Ap ismore complicated in comparison with the influence ofparameter L/RO. In this way, we can see that depen-dences Ap = fl(ao) and Ap = f2 (T) have a maximum fora certain value of o-/X, and the absolute value of this
Fig. 7. Relative incoherent reflectance as a function of X. Curvesl a, 2a and/or l b, 2b correspond to the Fresnel and/or Fraunhoferapproximation. 1 - = 0.03 Am, 2 - a = 0.05 ,m. In all cases it is
valid that T = 3 Am; aO = 0.04 rad; L/Ro = 0.02.
maximum depends on the values of the other parame-ters. Figure 7 gives pi and PiF as functions of X for twovalues of a. It is apparent that the values of Ap mustincrease with the increasing value of a. The foregoingfigures show that PiF > pi for all values of c/X, T/X, ao,and L/RO. An illustration of spectral dependences of'RT and Ri is given in Fig. 8. In this numerical examplewe consider the randomly rough surface of silicon.Optical constants were taken according to Ref. 17. Letus note that we get the same qualitative results if weperform the numerical analysis of Eq. (38).
From the foregoing numerical analysis, these con-clusions are formed: There is a difference betweenincoherent and/or total reflectance of the randomlyrough surfaces derived within the Fresnel approxima-tion and the same quantity derived within the Fraun-hofer one for certain values of parameters C/X, T/X, ao,and L/RO (and/or X/Ro, Y/R0). By means of a suitablechoice of the geometrical conditions of the reflectome-ter, it is possible to decrease or remove (within experi-mental accuracy) this difference. Further, it can beseen that within the usual experimental accuracy thereare no differences between the Fresnel and Fraunhoferapproximations for most real geometrical conditions ifthe values of c/X and T/X are sufficiently small (i.e., C/X< 0.05 and T/X - 1). For example, it means that in thevisible spectral region the reflectance of most polishedoptical surfaces measured by means of real reflectom-eters can be correctly explained using the Fraunhoferapproximation. However, a substantial differencebetween these approximations can arise for surfaceswith higher values of a-. Examples of such surfaces areintroduced in Figs. 7 and 8. In practice we encounteredsurfaces of this kind in our optical studies of rough sil-icon surfaces prepared by means of the special methodof grinding.4 Similar rough surfaces also originateduring the anodic oxidation of smooth silicon sur-faces,121 4 thermal oxidation of GaAs,15"6 etc.
IV. Conclusion
In this theoretical paper the formulas expressing thetotal reflectance of the randomly rough surfaces havebeen derived within the scalar theory of diffraction andthe Fresnel approximation. It has been shown that theformula describing the coherent reflectance is alwaysidentical for both the Fraunhofer and the Fresnel ap-proximations. The formulas found for the incoherentreflectance obtained within the Fresnel approximationare generally very different from the corresponding onesderived within the Fraunhofer approximation. In thecase of the Fresnel approximation the reflectance of therandomly rough surfaces generally depends even onparameters X/Ro and Y/Ro (X, Y are the linear di-mensions of the irradiated surface, and RO is the dis-tance between the surface and detector), which is themain physical difference in comparison to the Fraun-hofer approximation. Discussing the derived formulaswe have shown that it is not necessary to distinguishbetween the Fresnel and Fraunhofer approximationsas for the incoherent and/or total reflectance of the veryslightly rough surfaces. Further, the introduced nu-merical analysis has proved that substantial dis-crepancies can exist between both approximations forcertain values of parameters /X, T/X, a0 , X/RO, andY/Ro. However, by means of the appropriate choice ofexperimental conditions (i.e., a0, X/RO, and Y/R) wecan decrease or remove these discrepancies.
The formula expressing the incoherent reflectanceof thin films with randomly rough boundaries has beenintroduced as well.
The theoretical results presented will be used in anexperimental study of some rough metallic and semi-conductor surfaces and even in systems formed by asubstrate covered by a thin film with randomly roughboundaries.
The author wishes to thank V. Zavadil for preparingthe program for the numerical calculations.
0.30
0.20
0.10
00.4 0.5 0.6 0.7 0.8 0.9 A UM 0
Fig. 8. Spectral dependences of absolute total reflectance RT andabsolute incoherent reflectance Ri for the randomly rough surface
of silicon. a = 0.03 m; T = 3 m; o = 0.04 rad; L/Ro = 0.02.
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November
3-5 Spectroscopy in Support of Atmospheric Measure-ments, OSA Topical Mtg., Sarasota, Fla. OSA, 1816Jefferson Pl. N. W., Washington, D.C. 20036
3-5 Application of Accelerators in Research & Industry, 6thConf., N. Tex. State U. J. L. Duggan, Phys. Dept., N.Tex. State U., Denton, Tex. 76203
3-7 Microscopical Identification of Asbestos Course, ChicagoN. Daerr, McCrone Res. Inst., 2508 S. Michigan Ave.,Chicago, Ill. 60616
10-14 Fusion Methods (Hotstage Microscopy) Course, ChicagoN. Daerr, McCrone Res. Inst., 2508 S. Michigan Ave.,Chicago, Ill. 60616
10-14 Scanning Electron Microscopy Course, Chicago N.Daerr, McCrone Res. Inst., 2508 S. Michigan Ave.,Chicago, Ill. 60616
10-14 APS Div. of Plasma Physics, San Diego W. W. Havens,Jr., 335 E. 45 St., New York, N.Y. 10017
11-14 Magnetism and Magnetic Materials, 26th Ann. Conf.,Dallas I. S. Jacobs, GE CR&D, P.O. Box 8, Schenec-tady, N.Y. 12301
15 OSA New England Sec., Itek Corp., Lexington B. A.Horowitz, 76 Judith Rd., Newton Ctr., Mass. 02159
16-20 Electronic Imaging, Int. Conf., Wash., D.C. R. Wood,SPSE, Central P.O. Box 28327, Wash., D.C. 20005
17-23 Inter. Exhibition on Laser-Light-Sound and Holographyin Arts, Budapest Dir. of the App. Biophysics Lab.,Tech. Univ. of Budapest, Krusper u. 2-4., H-1111Budapest, Hungary
18-21 Symp. Optika 80, Budapest V. Vaddsz, Anker koz 1,1061, Budapest, Hunlry
