LIGHT REFLECTION FROM ROUGH SURFACES

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RSIC - 588

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LIGHT REFLECTION FROM ROUGH SURFACES

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/ , bY V. K. Polyanskiy

17. P. Rvachev I

Optika i Spektroskopiya, 20, No. 4, 701 -708 (1966)

Translated from the Russian

September 1966

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. 15 September 1966 RSIC - 588


bY V. K. Polyanskiy

V. P. Rvachev

Optika i Spektroskopiya, 20, No. 4, 701 -708 (1966)

Translated from the Russian

DISTRIBUTION OF THIS DOCUMENT IS UNLIMITED

Translation Branch

Redstone Scientific Information Center

Research and Development Directorate

U. S. Army Missile Command

Redstone Arsenal, Alabama 35809

. The effect of the scattering of polarized light by a dull dielectric

surface has been studied. The zeroth diffraction approximation proves to be more detailed than does the ray-optics approximation. In such a n approximation, we must repeatedly explain the fact of light depolarization and the dependence of polarization effects upon the scattering conditions and also the displacement of the maximum of the scattering curve for la rger observation angles.

There a r e a t present two methods for studying the interaction of light with a rough interface between two media. consists of mathematically studying the diffraction of plane waves by a rough interface. Using this method, Rayleigh (V. Strutt) ' gave an approximate solution to the problem of plane-wave scattering by a sinusoidal surface for normal incidence. Later on, a method was developed by Andronov and Leontovich' and Antokol'skiy3 ; finally, a r t ic les by Brekhovskikh4 described a solution to the problem of plane- wave scattering by a periodic surface for a rb i t ra ry angles of incidence, and gave the resul ts of specific calculations for a sinusoidal and broken periodic (sawtooth) surface. Another method uses the fact that a rough surface can be represented a s a set of microareas statistically distributed with respect to orientation, each of which reflects according to the laws of ray optics5. These representations a r e a basis f o r a r t ic les by Gershun and Popov6, Ivanov and Toporets7, Mullamaa', Gorodinskiy', and a number of others.

The f i r s t ( s t r ic te r )

We should note that these two methods for solving a particular problem coincide a t a very small number of points and a r e , in essence, developed independently. Of particular importance i s an art icle by Isakovich" which is devoted t o the problem of wave scattering by a totally reflecting statistically-rough surface. This i s the main point of an ar t ic le which is quite familiar to us which notes the tendency of two methods of investigating the optical propert ies of dull surfaces to converge.

The purpose of this ar t ic le i s t o study the scattering properties of dull sur faces in polarized light and to explain some observed effects using light-diffraction representations in addition to those of ray optics. Here , we will res t r ic t ourselves to considering the effects a t the interface of isotropic t ransparent media.

1. The experimental device intended for measuring brightness is similar to the one described by Gershun and Popov' but has in addition

1

an interference light f i l t e r , polar izer , and analyzer. The F E U - 19 photomultiplier i s used to receive radiation; the signal generated by the photomultiplier i s sent to the reference unit UF-206 . The optical systems of the illuminator and receiver make it possible to ass ign aperture angles of 2 " to 12' f o r the illuminating and recorded beams; this corresponds to the minimum solid angle w = 1 - s te r . In order to provide appropriate light f luxes, a high-power heater tube (K-30, 17 v, 170 w) was used a s the light source, and was enclosed by a jacket cooled by circulating water. An interference light filter X = 550 nm was used

the principal planes of the polar izer and analyzer did not exceed 20'; the e r r o r in reading the angles of incidence o r observation was 10 ' while the sums of these angles were 2' . the stand of a goniometric c i rc le . device.

, t o monochromatize the light. The e r r o r in reading the orientations of

The specimens were placed on Figure 1 gives the principle of this

4 6 7

1 -

2 -

3 - 4 - 5 -

2. being a

Ape r tu r e Diaphragm of Illuminator Interference Light Filter of Receiver (550 nm) 8 - Photomultiplier Polar izer 9 - Reference Device Object (Total Length of Optical Analyze r

6 - Field Diaphragm 7 - Aperture Diaphragm

System 2. 3 m)

Figure 1. Optical Diagram of Device.

Experimental Basis. We shal l consider a rough surface a s set of statistically distributed mic roa reas . We shall assume

the incident light beam is parallel] monochromatic, and l inearly polarized. azimuthal angle 9 between the direct ion of oscil lation of the e lec t r ic

The state of polarization of this beam will be descr ibed by the

2

I . vector and the plane of incidence. coinciding with the plane of incidence; the angles of incidence p and observation a a re read off relative to the normal to the surface. The index of refraction n of the second medium relative to the f i r s t is assumed to be r ea l and greater than unity. approximation we shall assume that rays reflected f r o m microareas oriented uniquely will propagate in every given direction. The normals of the microareas under consideration lie in the plane of incidence. The orientation of each microarea is determined by the angle y between the normal to the surface and the normal to the microarea.

Observations will be made in a plane

In accordance with the ray-optics

In the ray-optics approximation the effect of a set of identically oriented microareas is equivalent to the effect of one a r e a having a corresponding magnitude and orientation. mic roa reas are, in general, ' not coherent; the total intensity of the beam reflected in a given direction is determined as a s u m of par t ia l intensities.

Rays reflected by different

If we assume that the reflected beam consists of a se t of singly- reflected rays (i. e. , multiple reflections a r e not considered), then, as in the case of a polished surface, the reflected beam is plane- polarized but the plane of polarization is turned so that the azimuthal angle cp' between the plane of oscillation of the electr ic vector and the

I I plane of observation is determined by the relationship

where r (angle of refraction) is determined by the Sne11 law s i n ( a t P ) / 2 = n sin r. In order to obtain (1) we must expand the electric vector in the incident beam into s- and p-components , use the Fresne l formulas fo r each of them, and find the ratio of these components in the reflected beam. We must set a = P f o r a polished surface.

If we let cp = 4 5 " in the incident beam, then the s- and p-components of the electr ic vector in the incident beam a r e the same; in this sense, the incident plane-polarized beam i s equivalent to a beam of ordinary (unpolarized) light. Measurements using different types of analyzers corresponding to the t ransmission of electric vectors with azimuths 0, ~ / 2 , cp', cp' f ~ / 2 , all other conditions being equal, make it possible to obtain information a s to the changes both ordinary beams and polarized beams undergo upon reflection. ings No and NT/2 (in view of the above-stipulated inherent equivalence

Thus, f r o m the first pair of read-

of the incident plane-polarized f lux with azimuth 0 = 45" owing to the fact that the p- and s-components of the electr ic vector a r e equivalent) we obtain

No - NTIh

No 4- NT/2

P =

for the degree of polarization of ordinary light when reflected f rom the surface under consideration. ized the reading No' * T I / 2 is zero. ized after reflection, NT1 * TI/2 # 0. be computed f rom the relationship

If the reflected beam is completely polar- Since the beam i s partially depolar-

The degree of depolarization will

The quantities P and A character ize the changes in beams of dif- fe ren t types (ordinary and polarized) although the incident beams a r e plane-polarized in both cases . difficult to obtain the relationship

Under the above assumptions, it is not

cos2 (7 a + P t r) - cos2 (7 a . 4 - P - r)

cos 2 ( a + P t r) t cos2 ( ! L E - r)

1 - t g z c p I

(2) - P = - 1 + tg2cp'

for the degree of polarization.

This assumption must be satisfied in the same manner a s in the assumption that the outer component of the reflected f l u x maintains the state of polarization of the incident flux". In this ca se A = 0.

3 . Experimental Results. The selection of the object for experimental study was determined by the requirements that the inner component be absent,and that there be no ellipticity in the reflected beam. In other words, the object mus t only have an outer component and possess a r e a l index of refraction. glass with refractive index n = 1.515 was chosen (determined according to the Brewster angle) where one surface was polished and the other dulled mechanically (electrocorundum No. 240). By experimentally studying the s t ructure and curve of br ightness of the beam reflected by the polished surface of the specimen, we established that the imposed requirements a r e satisfied within the limits df experimental e r r o r .

F o r this purpose , a plate of black

4

. This agrees with Gorodinskiy's data'. the polished surface the light becomes completely polarized and the orientation of the electr ic vector corresponds to expressior_ (1). tionship (2 ) i s a l so well satisfied,

In a beam m i r r o r reflected by

Rela-

Figures 2 and 3 , which give typical experimental curves , charac- te r ize the resul ts of investigating the polarization character is t ics of beams reflected f rom a dull surface.

P

1.0

0.8

0.6

0.4

0.2

n4 20 0 30 40 50 60 70 7 o + P

1 - p = 30" 2 - p = 4 5 " 3 - p = 60"

4 - Polished Glass (Theoretical and Experimental)

Figure 2. Degree of Polarization of a Light Beam Reflected from a Dull Surface.

The resul ts given i n Figure 2 a r e a l so found to agree with Gorodinskiy' s results ' obtained in experiments using ordinary light. Our curves permit us to conclude that light polarization, upon reflection f r o m a dull surface, depends considerably upon the angle of incidence p. When p decreases , the degree of polarization has a smaller maximum and the maximum i s displaced in the direction of lower values for the half-sums of the angles of incidence and observation (a t p)/2. When (a 9 p) /2 i s fixed and l e s s than o r about equal to 45", polarization increases with a decrease in p; i f (a t p) /2 2 45", polarization dec reases when /3 decreases .

The resul ts given in Figure 3 characterize depolarization caused by reflection f rom a dull surface. Here, the polarization propert ies as a function of the angle of incidence prove to be much more clear ly

A

0.1 1 0 20 40 60 80'0

1 - p = 3 0 " 3 - p = 6 0 " 2 - p '45"

Figure 3 . Depolarization of a Light Beam Upon Reflection f r o m a Dull Surface.

defined, low but increases rapidly when the angle of observation increases . F o r large angles of incidence the polarization is great and decreases rapidly when the angle of observation increases . When p = 4 5 " the depolarization was almost constant for our experiment when the angle of observation changed.

Fo r small angles of incidence the polarization is relatively

A s was the case f o r reflection f r o m a dull surface, the electr ic- vector orientation predominating in a beam reflected f rom a dull su r - face is determined so accurately f rom Equation (1) that the measured angles a r e no different f rom those computed. very accurate for beams reflected f r o m a dull surface yielded some- what exaggerated resu l t s for this dull surface; the g rea t e r the surface roughness and the smaller the angle of incidence, the grea te r the devi- ation of the resul ts f rom those calculated. The assumption a s to the state of polarization remaining constant upon reflection is only weakly satisfied and the magnitude of the depolarized f l u x can prove grea te r than the polarized one.

Equation (2) which i s

4. Discussion of Experimental Results. The idea of representing the diffusion-reflected beam a s a set of paral le l beams mi r ro r - r e f l ec t ed f rom microareas makes it possible to obtain satisfactory resu l t s when the polarization plane i s rotated fo r reflection of a plane-polarized beam; i t a lso makes it possible, at least qualitatively, to f o r m an opinion a s to the distribution function of the mic roa reas with respec t to direction'*!

6

The presence of a depolarized component in the beam and i t s low (in comparison to that computed using the Fresne l formulas) polarization upon reflection f r o m a dull surface is usually associated with the p re s - ence of rays which a r e reflected more than once. nation is not sufficient since, preceeding only f rom the ray-optics representations, we a r e often unable t o explain such experimental facts as the deflection in the intensity maximum f rom the direction of m i r r o r reflection toward l a rge r angles of observation, the relatively high depolarization component in the reflected beam, and also a n w b e r of other effects. depolarized fluxes a r e roughly equivalent, we must assume on the bas i s of the concept of multiple reflection that the number of times multiple reflection occurs is a t least 26-23 Limes greater than the number of t imes single reflection occurs. (For every reflection f rom the glass surface, which possesses , a s is known, a reflection coefficient of the order of 4-570 and for equal angles of incidence and reflection (6 50") the beam must attenuate by a factor of 20-25. ) However, such anassump- tion contradicts Gershun' s data6 which deals with the distribution function of the direction of the microareas.

However, this expla-

Thus, in o rde r to explain the fact that the polarizedand

The effects which a r e observed when light is reflected f rom a dull surface can be explained in greater detail i f we use diffraction representations (even when we r e s t r i c t ourselves to the zeroth diffraction approximation).

Let us consider the interaction of a plane wave and an aperiodic uniform reflecting lattice. In other words, fo r the sake of simplicity, we shall assume that the dull surface consists of a set of plane bands with paral le l adjacent edges. The bands a r e of different widths I and a r e tilted differently. The inclination of the bands i s given by the angle y. The bands under consideration are similar to the microareas ; qualitatively, such a simplification for the object does not affect the resul ts (however, it i s necessary to introduce a correct ion (the form factor) for quantitative calculations). Figure 4, which shows one such band, a lso gives the notation which will be necessary below.

The intensity distribution in the diffraction pattern is determined by Upon using ordinary methods to compute the path the path difference 6.

difference, we obtain, in our case,

p - Angle of Incidence a - Angle of Observation

a 0 - Direction of Principal Maximum Path Difference 6 = AD-AC

Figure 4. Calculation of Reflection Allowing for Diffraction Effects.

The path difference 6 = 0 for the direction of the principal (zeroth) It is c l ea r f rom Figure 4 that this occurs when diffraction maximum.

a 0 = p t 2y which is the same a s the mir ror - re f lec t ion condition.

Relationship ( 3 ) shows that the direction of the zeroth diffraction maxima, created by each individual mic roa rea (principal pa r t of the curve) is determined only by their orientation, i. e . , the direction distribution f (y ) . The direction of propagation of the maxima of the f i r s t and higher o rde r s is determined not only by the direction distribution function but also by the s ize distribution function.

In contrast to the ray-optics approximation the principal diffraction maximum proves to be blurred. diffraction broadening of the reflected mic roa rea of the pencil of r ays allowing for the fact that the direction to the first minimum i s de te r - mined by the condition 6 = X we a r r ive at the resul t

Using relationship ( 3 ) to es t imate the

A 21 ’ A a Z -

where broadening increases with an inc rease in t h e angle of incidence p and the angle of inclination of t h e m i c r o a r e a y.

In the case of oblique incidence the diffract ion pat tern proves to be asymmetric. maximum(bet ter termed principal beam) a l so proves to be asymmetr ic .

In par t icular , the intensity distribution in the principle

8

. In fact , let us determine the directions a0 t 2+ and a0 - 25 for which the path difference is the same in absolute value and opposite in sign. A s i s not difficult to see, the corresponding conditions can be writ ten on the bas i s of ( 3 ) in the fo rm

We can easily ver i fy that the last equality is satisfied by nonzero values of + and 5 only when

Since the intensity in a given direction is determined (for ordinary light) by the path difference, we can conclude on the bas i s of relation- ships (5) and (6) that within the l imits of each diffraction maximum when the angle of observation increases the intensity first increases rapidly ( f rom ze ro to some maximum) and then drops more slowly to zero. If the maximum under consideration is represented in the f o r m of a fringe on the band pattern, the fringe proves to be asymmetr ic - - turned in the direction of increasing angles of observation read off f r o m the normal. (Note in passing that this statement a lso r e fe r s to the fo rm of the spectral line found in diffraction spectral devices for oblique incidence. ) Asymmetricity in the principal beams leads to asymmetr ic i ty in the curve. In fact, if two principal beams (two principal maxima) caused by diffraction f r o m different mic roa reas partially intersect , then assuming these beams a r e mutually incoherent, the total intensity is calculated a s the sum of the intensities in each beam:

In such a case, the position of the resulting maximum i s determined by the condition (Figure 5):

, d d - 11 (a) = - - I 2 (a) . da da

By extending such a consideration to an a rb i t r a ry number of beams we a r e led to the conclusion that the maximum of reflection must be shifted i n the direction of l a rge r angles of observation for oblique illu- mination of a dull glass. that such a shift in the maximum invariably leads to e r r o r in deter- mining the distribution function of the microareas by famil iar methods 7 ’ 8

This was also observed experimentally. Note

n Y

I

Figure 5. Illustration of the Shift in Maximum for Addition of Asymmetric Pulses.

If the incident beam is plane-polarized, it is natural to assume that a beam reflected f rom one microarea is polarized and characterized by the azimuth cp' in Equation (1).

If rays reflected f rom microareas oriented uniquely propagate in a direction determined by the angle of observation a and i r radiat ionand observation a r e performed using strictly paral le l beams, then, assuming there i s no multiple reflection, the reflected beam becomes plane- polarized with varying azimuth.

The presence of multiple reflection involves some depolarization; however, the intensity contributed by multiple reflection can barely exceed 3-5% of the total flux.

The depolarization resulting from the finite dimensions of the angular aper ture of the receiver system and illuminator system is negligibly small under our conditions since ( see Section 1) the aperture angles lie within the limits of 12'-15' (the d iameters of the diaphragms of the collimator and receiver a r e 0. 8 mm and the focal lengths of the objectives a r e 300 rnm).

In fact, the principal beams result ing f r o m diffraction on the micro- a r e a s with different orientations a r e propagated in the direction of observation. These beams a r e incoherent and a r e polarized in different planes. (For normal incidence on the mic roa rea , the reflected beam retains the azimuth of the incident beam (45") ; when the angle of incidence i s very close to the Brewster angle the azimuth is 90" ; f o r further increase, the azimuthal angle reaches 135 O . )

10

When the dimensions of the microareas a r e comparable with the wavelength of light we may find a good deal of beam overlapping. Thus, i f Z cos (p + a) /2 z 51 then in accordance with ( 3 ) , the principal beams from the microareas propagate in the direction of observation and the normal to the microarea lies within the limits of 12". beam overlapping is , in general, equivalent to an increase in the aper - tu re of the observer system; therefore, it i s an unnecessary precaution to attempt to reduce the aperture angle of the la t ter to l e s s than 1". Overlapping of the principal beams occurring as a resul t of diffraction on the microareas with different orientation and therefore characterized by different azimuths must lead to a partial depolarization of the resul- tant beam.

The effect of

This effect has a lso been observed experimentally.

Beam overlapping must be reduced when the effective dimensions of the microareas a r e increased, i. e. , when the f i r s t factor in expres- sion (3),Z cos (a + p) /2 , is la rger o r when the angle of incidence p is smaller. When there is little beam overlapping (diffraction maxima f r o m different microareas) there i s little depolarizatios. increases depolarization must increase. This effect has been observed experimentally and is shown in Figure 3 ( for small a). the angle of incidence and a decrease i n Z resul ts in conditions where the dull surface begins to introduce a m i r r o r c ~ m p o n e n t ~ ' ~ ' ~ . case depolarization decreases with an increase in p. The appearance of this m i r r o r effect can be explained by the features of the curves in Figure 3 for l a rge r values of p and a. effects compensate for one another; i n our case this occurs whenp =45".

When p

An increase in

In this

Under cer ta in conditions, these

I A reduction in the dimensions of the surface microstructure (pro- cessing with finer abrasives) i s accompanied by the appearance of par t ia l coherence which leads to a reduction in surface depolarization

described by Gorodinskiy' explain this, l and, in the long run, to m i r r o r (coherent) reflection. The laws

Thus, we can say that depolarization, upon reflection, is due not only to the presence of multiple reflection and finite aper ture of the observer device but a lso (mainly) to diffraction by the surface micro- a reas .

i i

0 LITERATURE CITED

1. V. Strutt (Rayleigh), Teoriya zvuka (Theory of Sound), Pub. by AS USSR, Vol. 11, Moscow, 1954.

2. A. A. Andronov, M. A. Leontovich, ZhRFKhO, 38, 4851 1926. - 3. M. L. Antokol'skiy, DAN SSSR (Reports of the Acad. Sci . , USSR), - 62, 203, 1948.

4. L. M. Brekhovskikh, ZhETF (Journal of Experimental and I Theoretical Physics) , 23, - 275, 289, 1952.

5. P. Bougner, Opticheskiy t raktat o gradats i i sveta (Optical T rea t i s e on Light Gradation), Pub. by AS USSR, Moscow, 1950.

6. A. A. Gershun, 0. I. Popov, Transactions of the GO1 (State Optical Institute) - 24, No. 143, 3, 1955.

7. A. P. Ivanov, A. S. Toporets, ZhTF (Journal of Technical Physics) , - 26, 623, 631, 1956.

I 8. Yu. Mullamaa, Issled. PO fizike atmosf. (Study of Atmospheric Physics) , AS ESSR, 2, 5, 1962.

I 9. G. M. Gorodinskiy, Opt. i spektr. (Optics and Spectroscopy), - 16, ~ 112, 1964.

10. I. A. Isakovich, ZhETF, 23, 305, 1952. - I 11. A. S . Toporets, ZhETF, 20, 390, 1950.

12. A. S. Toporets, Opt. i spektr., 16, 102, 1964.

I 13. G. S. Landsberg, Optika (Optics), GTI, Moscow, 1957.

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1 Technical Information

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INTERNAL

Headquarters U. S. Army Missile Command Redstone Arsenal, Alabama A": AMSMI-D

AMS1-E. Mr. Lowers MI-XS; ~ r . tarter AMSMI-Y M I - R , Mr. &Daniel AMSMI-RAP MI-RBLLI USACDC-Lno AMSMI-RB, Mr. Croxton AMSMI-RBT

National Aeronautics and Space Administration Marshall Space Flight Center Huntsville, Alabama ATTN: MS-T, Mr. Wiggins

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UNCLASSIFIED ~n

Security Classification - DOCUMENT CONTROL DATA - RLD

(Security cla..ilicatim of title. body ol abmtmet and indexin# mnotation mumt be entend when the overall report ia claamified) IG I A ING A TIYITY (c WOt I O 2.. REPORT 5 E C U R I T V C L A 5 5 I F I C A T I O N

'XeJslone Scientific fXot%ation Center Unclassified Research and Develo ment Directorate U.S. Arm Missile C?ommand Redstone irsenal, Alabama 35809


2 b . GROUP

N /A REPORT T I T L E

Optika i Spektroskopiya, 20, No. 4, 701-708 (1966)

Translated from the Russian DESCRIPTIVE NOTES (Typ. of nport and inclumive date.)

AUTHOR(S) (Lael n m e . H N ~ name. initial)

Polyanskiy, V. K. and Rvachev, V. F.

. R E P 0 RT DATE 7.. TOTAL NO. O F PAGES 76. NO. O F REF5

15 September 1966 16 13

N /A a. CONTRACT OR O R A N T NO. 9.. ORIOINATOR'5 REPORT NUMBCRfs)

RSIC - 5 88 b PROJECT NO.

ab. OTHER R PORT NO(S) (Any othotnwnbmrm that may ba aem1mM mi. Import?

e. N /A

d. AD 0. A V A I L ABILITY/LIMITATION NOTICES

Distribution of this document i s unlimited.

1. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY

None Same as No. 1 I

3. ABSTRACT

The effect of the scattering of polarized light by a dull dielectric surface has been studied. detailed than does the ray-optics approximation. In such an approximation, we must repeatedly explain the fact of l ight depolarization and the dependence of polarization effects upon the scattering conditions and also the displacement of the maximum of the scattering curve for larger observation angles.

The zeroth diffraction approximation proves to be more

17 Security Classification

UNC LASSIFLED Security Classification

14. KEY WORDS

Scattering properties Polarized light Dielectric surface Dull surfaces Light diffraction

INSTIUJC?

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The assignment of l inks. rules . and weights i s

18 UNC LASSIFED ___ Security Classification