Reflectivity of corrugated surfaces

Physica IV, no 1 1

December 1937

REFLECTIVITY

OF

CORRUGATED

SURFACES

b y L. S. O R N S T E I N a n d Miss A. VAN D E R B U R G Communication from the Physical Institute of the University of Utrecht Summary

The scattering of a p~ane surface, due to small irregularities is calculated assuming, that the surface is composed of small planes, fortuitously orientated. The probability and the scattering are given in tables. The reflection of a c o r r u g a t e d surface can be described with the help of two methods. T h e first m e t h o d is indicated b y R a y 1 e i g h *), who used a F o u r i e r analysis of the surface. This m e t h o d m u s t be used when the irregularities of the surface are small in comparison with the w a v e l e n g t h of the incident light. The second m e t h o d considers the surface as being built of small planes, which reflect the light i n d e p e n d e n t of each other. This m e t h o d can only be used when the irregularities are large in comparison with the wavelength. Using this m e t h o d , the i n t e n s i t y of the reflected light in different directions is calculated, m a k i n g an a s s u m p t i o n on the distribution of the e l e m e n t a r y planes over the possible positions. Suppose, t h a t originally the normals of all e l e m e n t a r y planes are parallel to one direction. Accidental influences cause deviations from this position. The p r o b a b i l i t y of a small deviation is supposed to be i n d e p e n d e n t of the actual deviation of the normal. W h e n the normals are considered as points on a sphere, we are concerned in the p r o b l e m of the diffusion of points on a hemisphere, originating f r o m the top. The diffusion on a sphere satisfies the following differential equation : at

--

s i n 0 31} s i n l } ~ g .

,

*) R a y 1e i g h, The Theory of Sound II, p. 89. --

Physica IX'

1 181

--

74*

1182

L. S. ORNSTEIN

AND

MISS

A. VAN

DER

BURG

where W (9, t) is the probability per unit of solid angle. W is no function of 1~ in this case because of the s y m m e t r y r o u n d the polar axes. B y the substitution ~ = cos ~ the diff. eq. is changed into :

of which the solution is: 0<3

W = Z A,, e-":" ~l)t p , (~).

P,, (~) = Legendre function.

0

The coefficients A,, dan be found with the help of the d e v e l o p m e n t of W for t = 0. W 0z, 0) is a a-function in 0 and 1

f w (~, o) d~ = i. --1

"We get : co

W=J-Z

e -'':'§

(2n

§

P,, (~.).

0

F r o m this formula one can calculate the mean value of ~: ~

C- 2 t .

If we substitute the value obtained in the formula, we get :

W=~r(;)

2

(2~+l)P.(~)

~:cos,~,

0

where thus W is given as a function of ~ and the mean vs quantity. The solution for a hemisphere is: W(~) = W(a)+

of this

W(=--~)

W e get for this solution: co

w(~?=

z

n(n+l)

(~) ~

(2n+l)P,,(~)

0, 2, 4 9 , .

In this formula the total n u m b e r of points is 2r~ : .rr~2

f 2:~ ~in 4} W (4}) dl} = 2~. 0

F o r the values 0.95, 0.9, 0.8, 0.7, 0.6, 0.5 and 0.4 of the p a r a m e t e r the distribution W (9) is caiculated. Table I. Fig. 1.

REFLECTIVITY

OF CORRUGATED

1 183

SURFACES

TABLE I

ix=cos

O'

1.00

1o 508 , 8~ 11 ~ 15 ~ 18~ 21~

0.90 0.88 0.85 0.80

25~ 28~ 31~ 36~

0.75 0.70

41~ 45~ 49~

0.60 0.55 0.50 0.45 0.40 0,35 0.30 0.25 0.20 0.15

0.10 0.05 0.00

w (0)

0 o

0.9998 3.9990 0.996 0.990 0.980 0.966 0.95 0.93

0.65

F:0.95

8

3~

53~ 56~ 60 ~ 63~ 66~ 69~ 72~

75~ 78~ 81~ 84~ 87~ 90 ~

g=0.9

g=0.8

~ - = 0.7

9.66

4.652

2.976

9.32

4.576 4.~66

w (~)

w (8)

19.65 19.89 19.14 18.27 16.20 13.34 10.14 7.42 5.00 2.77

4.99 3.74

3.~3 3.427 3.001

1.00

3.09 2.30

2,743 2.396

0.36

1.40

0.04

0.85 0.50 0.30

1.905 1.509 1.190 0.934

0.00

871 6 o4

0.18 0.10 0.06 0.03 0.02

0.730 0.567 0.439 0.338 0.259

0.01 0.01 0.00

0.197 0.150 0.114

8

0.088 0.069

7

0.056 0.049 0.046

w (0)

g=0.6

w (8)

g:O.5

W (&)

~=0.4

tu (0)

2.136

1.634

2.~7

2.120

1.626

2,823

2.060

1.898

2.609

1.952

1. ~ 8

1.274

2.~77

1. ~ 3

1.449

1.229

1.985

1.628 1.484 1.352 1.231 1.121 1.021 0.931

1.364 1.285 1.212 1.143 1.080 1,021

1.187 1.147 1.110 1.075 1.042 1,012 0.985 0.960 0.937 0.917 0.899

1,727 1,498

1.297 1.121 0.966 0.831 0.714 0.614 0.527 0.454 0.393 0.344 0.304 0.274 0.252 0.240 0.236

0.849 0.777

0.713 0.657 0.610 0.570

0.537 0.512 0.494 0.484 0.480

90.968 0.919 0.876

0.837 0.802 0.773 0,747

1.32l

0.727

0.883 0.870 0.859

0.711

0.851

0.700 0.693 0.691

0.845 0.842 0.840

Considering one square cm of the surface, the distribution of the normals is supposed to be given by: a W (cos l}), this is the number of elementary planes with normals in the direction I}, per unit of solid angle. The condition, that the projection of all planes on the surface i~ equal to unity, determines the constant a: 9/2 1 f aW(cosl~) . s cos 8 . 2 = sin 0 d0 1 as W2 0 2~. f W (cos 8) sin l~ da 0

s = area of one elementary plane. as depends on g and must be calculated for each case (see table III) We now will give the scattering of light, when a beam of light, parallel to the normal falls on the surface and also treat the case, that the incident beam forms an angle of 45 ~ with the normal. What is the intensity of the reflected light in different directions, supposed, that the planes reflect all the incident light ? In answering this question, secondary reflections will be neglected.

1184 90:

L. S. ORNSTEIN AND MISS A. VAN DER BURG

.~._a~w(~)

70 /12 = 0,9S 60

SO

~\

t0

30

s=~

~

~

- - ~

4O

0

10

20

30

~0

50

60

70

80

90 ~

,~60

Fig. 1. D i s t r i b u t i o n of W (b) for d i f f e r e n t v a l u e s of 5- ( I n t e n s i t y of t h e r e f l e c t e d l i g h t a t a n a n g l e of 2 b w i t h t h e n o r m a l . A n g l e of i n c i d e n c e 0~

I n the case of an incident b e a m , p e r p e n d i c u l a r to the surface the p a r t of the light, falling on planes with n o r m a l s b e t w e e n 3 a n d 3 + d3 is : a W (3) . ~ cos 3 . 2 = sin 3 d3. This is reflected in the solid angle. 2~ sin 2 3 . 2 d 3 . So the i n t e n s i t y per unit of solid angle in the direction 21~ is : 0,(5"

REFLECTIVITY

OF CORRUGATED

SURFACES

1185

This formula holds only, when no light is intercepted. For this interception two reasons exist : Firstly, it is possible, that the light reflected by one plane is intercepted by another. Secondly, when the angle of incidence is not zero the incident light may already be intercepted. This effect becomes important, when a considerable part of the elementary planes hold an oblique position. The interception of reflected light will always be very small. At an angle of incidence of 45 ~ the interception of incident light will only be small, when the majority of the planes show angles with the surface, which are smaller than 45 ~ This is the case when ~ > 0.5.

Intensity o/ the reflected light in di//erent directions /or di/ferent values o/~ Angle o~ incidence 0 ~ I (28) = ~era - W (8).

(Table II).

T A B L E II

20

~=0.95 I (20-)

•ii(20.)

~=0.8 I (20.)

--=0.7 ~/(20.)

-=o.6~ I (20.)

~=o.s 1 (20.)

~=0.4

0.815 0,813 0.794 0.758 0.672 0.554 0.421

0.425

0.2279

0.1667

0. I345

0.1136

0.0978

0.410

0,2242

0.1650

0.1336

0.1130

0.352

0,2090

0.1581

0.1298

0.1109

0.308

0.266

0.1230

0.1069

0.0943

0.220

0,1834 0,1679 0.1470 0.1344 0.1174 0.0933 0.0739

0.1461

0.207 0.115

0.1275

0.1123

0.1007

0.0909

0.1116 0.0967 0.0839

0.1026

0.0948

0.0878

0.0935 0.0852

0.0831 0.0842

0.0821

0o

2~ 6o 10~ 16~ 22~ 30 ~ 36~ 43~ 51~ 56~ 63~ 73~ 82~

,

= 0" 9

0.041 0.015

0.165 0.136 0.101 0.062

(0.006)

0.037

I (20")

0.0849

T A B L E III ~a

0.95 0.90 0.80 0.70 0.60 0.50 0.40 0.00

0.0415 0.044 0.049 0.056

0.063 0.069 ~ 0.074 0.079 s

The values of ,a/4 in table III are determined graphically and are used for the calculation of table II. Physica I V

75

1186

L. S. ORNSTEIN AND MISS A. VAN DER BURG

Angle o[ incidence 45 ~ TL

Ds

I I

I 0 Fig. 2.

Again the intensity in a direction B (fig. 2) is given by the formula: ~a

I (=, ~) ----~- W (I~), where ~, ~3are the co~3rdinates of the direction t3 in relation to the axes 0C (direction of reflection for horizontal elementary planes). is the angle between the normal on the surface and the normal on elementary planes, which reflect the light in the direction B.

Calculation o/~/or di//erent directions, P till H (table IV). See fig. 3. C

A

A

~

t

\

l)z i . "

IN

Fig. 3.

REFLECTIVITY

OF

CORRUGATEDSURFACES

~3

7,o ,xT=o,gs so

~o '~/ Y

. ~: 0,e

I ,.~ ==0.S ~'''~ t.o

090~

~ '= o.o

~

~4

\~-~--~

0

~ S bO

~

90 ~

Fig. 4. Distribution Of the intensity in the plane of incidence. Angle of incidence 45%

1187

1188

L. S. O R N S T E I N A N D MISS A. VAI~I D E R B U R G

9.o

3 8,o

7,O

~_-0,gS

S,0 . . . .

so

~p

2.o

90 ~

't,5 ~

0 0(4

~5 ~ .

90*

o("

F i g . 5. D i s t r i b u t i o n of the intensity in a plane perpendicular t o t h e i n c i d e n t b e a m . A n g l e o f i n c i d e n c e 45 ~.

REFLECTIVITY

1189

OF CORRUGATED SURFACES

TABLE IV

Direction of incidence

0r

Direction of

P

0o

At

5~

Aa B1 B~

~o

B3

!~o

A~

Ct C2 C3 C4

C5

~ 0o

0o 180 ~ 90 ~ 0o 180 ~ 90 ~ 0o 180 ~ 90 ~ 45 ~ 135 ~

b

incidence

2~ ' 2030 , 3032 , 6~ 6~ 8~ ' 12~ ' 12~ ' 17~ ' 14~ ' 16~

D,

0o

Dt

~ 45 ~

Ds D, D~ E~ Ea

Ft

61"3o" 9c;~

Fz G

11#30'

H

135 ~

~ 0o

180 ~ 90 ~ 45 ~ 135 ~ 0o 90 ~ 0o 90 ~ 0o 0o

,8" 22~ , 22030 , 31 ~ 27~ 31~ 33045 , 46016 , 45 ~ 60 ~ 56015 , 67030 '

Intensity o~ the reflected light in the direction P till H. Angle o~ incidence 45 ~ (table V). TABLE V

Direction of

~=0.95

incidence

I (~)

P A~ A, Aa B, B_o

0.815 0.801 0.801 0.788 0.735 0.735 0.658 0.517 0.517 0.328 0.448 0.375 0.185 0.185 0.046 0.095 0.046 0.028 0.017 0.017 0.000 0,000 0.000

B3 Ct C2 C3 C, C5 D~ D., Da D, D~ E, E3 Ft F,~ G H

~/=0.9

(~) •/=0.8

~=0.7

~=0.6

~'=0.5

I (~)

~=0.4 I (~)

0.425 0.422 0.922 0.418 0.404 0.404 0.384 0.341 0.341 0.274 0.319 0.292 0.207 0.207 0.105 0.150 0.105 0.084 0.020 0.025 0.003 0.005 0.001

0.228 0.227 0.227 0.226 0.223 0.223 0.218 0.206 0.206 0.186 0.199 0.192 0.164 0.164 0.119 0.141 0.119 0.108 0.056 0.061 0.022 0.029 0.012

0.167 0.166 0.166 0.166 0.164 0.164 0.162 0.157 0.157 0.147 0.154 0.150 0.136 0.136 0.112 0.124 0.112 0.106 0.071 0.074 0.040 0.047 0.027

0.134 0,134 0.134 0.134 0.134 0.134 0,132 0.129 0,129 0,123 0.127 0.125 0.117 0. I17 0.103

0.114 0.114 0.114 0.113 0.113 0.113 0.112 0.111 0.111 0.107 0.109 0.108 0.104 0.104 0.095 0.099 0.095 0.092 0 078 0.080 0.064 0.068 0.057

0.098 0.098 0.098 0.098 0.097 0.097 0.097 0.096 0.096 0.095 0.095 0.095 0.092 0.092 0.088 0.090 0.088 0.087 0.079 0.080 0.071 0.073 0.068

s (~)

1 (a~)

0.112 0.103 0.099 0.976 0.079 0.054 0.059 0.044

=: 0,0 gives an uniform distribution of the normals over the sphere: W (0) = 1. Whatever the direction of the incident beam is, the intensity in all directions is the same (provided there is no interception of light). Received October 13th, 1937.