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~el~4t,) UTTA for filrthe'r dissemination with NAVWEPS REPORT 7736Sout li', tj . d thoe imposod by security NOTS TP 2695

regulatij.i3 COPY 7 -

RELATION BETWEEN SURFACE ROUGHNESSAND SPECULAR REFLECTANCE

AT NORMAL INCIDENCE

H. E. Bennettand

J. 0. Porteus

Research Department

ABSTRACT. Expressions relating the roughness of a plane surface to its

specular reflectance at normal incidence are presented and are verified

experimentally. The expres:-ions are valid for the case when the root

mean square surface roughness is small compared to the wavelength of

light. If light of a sufficiently long wavelength is used, the decrease

in measured soecular reflectance due to surface roughness is a functiononly of the root mean square height of the surface irregularities. Long-

wavelength specular reflectance measurements thus provide a simple and

sensitive method for accurate measurement of surface finish. This methodis particularly useful for surface finishes too fine to be measured

accurately by conventional tracing instruments. Surface roughness must

also be considered in precise optical measurements. For example, a non-

negligible systematic error in specular reflectance measurements will be

made even if the root mean square surface roughness is less than 0.01

wavelength. The roughness of even optically polished surfaces may thus

be important for measurements in the visible and ultraviolet regions of

the spectrum.

XEROX I.t' )

4 U.S. NAVAL ORDNANCE TEST STATION

China Lake, CaliforniaCE ~19 May 1961

U.S. NAVAL ORDNANCE TEST -STATION

AN ACTIVITY OF THE BUREAU OF NAVAL WEAPONS

W. W. HOLLISTER, CAPT., USN WM. B. McLEAN, PHD.Commanider Technical Director

FOREWORD

This paper reports a new, high-precision, reflection method ufmeasuring surface roughness. It provides for the replacement of thecrude surface measuring devices, such as the profilometers found inmaclIne shops. This capability has been needed to study the mechanismof polishing metal surfaces which has been an art to dqte. Use ofhigher temperatures and bearing speeds demands the transfer of metalpolishing from an art to a technological basis.

The work was supported by Exploratory and Foundational Researchfunds, Task Assignment WEPTASK No. R36OFRI06/RO1101001.

The text of this report is a reprint of the article of the sametitle published in the Journal of the Optical Society of America,Volume 51, Number 2 (February 1961), pp. 123-129. 0

0@

CHAS. E. WARINGHead,, Research Department

@0

Released under theauthority of:

WM. B. McLEANTechnical Director

NOTS Technical Publication 2695 0

NAVWEPS Report 77,,6

Publishet by ........ .................. . Research DepartmentReprinted from ......... .Journal of the Optical Society of America

. aVol. 51, No. 2 (February 1961)Collation ..... ............... .. Cover, 4 leaves, abstract cardsFirst printing ........ .................. .. 150 numbered copiesSecurity classification .......... ......... UNCLASSIFIED

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0 0

ii0

Reprinted from JUoxAL Oi ir E Oi'rm-. Socii.ry o," A.Ni EICA, Vol. 51, No. 2, 123-129, February, 1961 iPrinted In U. S. A.

Relation Between Surface Roughness and Specular Reflectanceat Normal Incidence

H. E. BENNETT AND J. 0. PORTEUS.ichelson Laboratory, China Lake, California

(Received July 11, 1960)

Expressions relating the roughness of a plane surface to its specular reflectance at normal incidence arepresented and are verified experimentally. The expressions are valid for the case when the root mean squaresurface roughness is small compared to the wavelength of light. If light of a sufficiently long wavelengthis used, the decrease in measured specular reflectance due to surface roughness is a function only of the rootmean square height of the surface irregularities. Long-wavelength specular reflectance measurements thus

*provide a simple and sensitive method for accurate measurement of surface finish. This method is particu-larly useful for'surface finishes too fine to be measured accurately by" conventional tracing instruments.Surface roughness must also be considered in precise optical measurements. For example, a non-negligiblesystematic error in specular reflectance measurements will be made even if the root mean square surfaceroughness i less than 0.01. wavelength. The roughness of even optically polished surfaces may thus beimportant for measurements in the visible and ultraviolet regions of the spectrum.

INTRODUCTIONi slope, so that reflectan'ce measurements have been of

T HE retlectance of a surface is a sensitive function only limited fusefulness as a method of determining

of its rotighness. However, an adequate and surface roughness.

experimentally verifled theory relating these properties The present investigation suggests that if somewhatha. been lacking. This paper describes such a theory loger wavelengths are used for such surfaces, theaiol its verifici ion for the case of normal incidence. characteistics of.the surface other than roughness

Several experimental inve.sigations of the relation become unimportant and specular reflectance measure-

between the roughnesS of machined metal surfaces or nients at nearly normal incidence provide a simple

ground glass surfaces and the specutlar or difftse and precise *method of determining the root mean

rellectance'have been reported. Light in the visible 0tuare roughness o[ a plane strface. This method can"

region was used and in most cases the reflectance Vas best be applied tg surfaces with a root mean square

measured at oblique incideirce, since at these IWave- roughness of less than 50 uiii. The roughness of surfaceslengths the .surfatre irregutlarities are comparable in in this range is of considerable practical importance,magnitude to the wavelength and the amount of light and current methods of measurement are not completelywhich is specularly rellected at normal incidence is satisfactory. The reflectance method is easily applied,quite small. Under these circumstances the rellectance does not cligturb the surface, and is particularly usefuldepends not only on thesurface toughness but also onl for surfaces not amenable to other techniques.dterspe s ol the stirface oghes btit sal o Little attention has been given to the -effects ofother aspects of the surface, e.g., the root mean square surface roughness on optical measurements when the

ILord Rayleigh, Nature 64, 385 (1901). surface irregularities are very small relative to the2 F. Jentzsch, Z. tech. Physik 7, 310 (1926). wavelength antd diffraction effects predominate. This3 H. Hasunuma and J. Nara, J. iwhys. Soc. Japan 11, 69 (1956). situation is rather surprisig in view of the extremely

W. E. K. Middleton and G. Wysze:ki, J. Opt. Soc. Am. 47,1020 (1957). large number of measurements of the specular reflect-

'R. S. Hunter, J. Opt. Soc. Am. 36, 178 (1946). ance, transmittance, or polarization of various materials5J. Guild, J. Sci. instr. 17, 178 (19,0).E. A. Ollard, J. Electrodepositor's Tech. Soc. 24, 1 (1949). which have been made. Although the reflectance is

Oj. Hailing, J. Sci. Instr, 31, 318 (1954). more sensitive to surface roughness than is the trans-123

o 0

124 1-. E. BENNETT AN]) J. 0. PORTEUS Vol. 51

miittanlce, 2 -9 a significant systematic error will exist inl Here r,1(0)(10 refers to tilc fraction of the reflected lightprecise optical measurements of either quantity if tile which is scattered into an an&~ bctwc~n 0 and 0+dOsurfaces involved are not suffliciently smooth. Thc at anl anigle 0 from the noninal to the surfacc. If m isdegree of smoothness required is much morc critical thc root mean square slope of the profile of the surface,than has beeni prcviously upse.the autocovariance length a is, as is shown in ,the

Appendix,THEORY a (3

Expressions for the relation between reflectance and ThrfriterflcI n o teomais esudroot mean squiare rotdghness may be Obtained from thle Thrfrif thhelcac ntenrasmauestatistical treatment of the retlection of electromagnetic wiha0ntuetlacpac nl 0 o ihradiation from a rough surface derivred by )avies.' atnral incidence the contribution from diffuseAlthloughaw this theoryc was "developed in connection reflectance iswith the scattering of radar waves from rough water ri0d=R-oX(O.~~ 4Surfaces, it is equally valid in the optical region. Thle A)2W (4surface is represented by a statis;tical mure havingthe following properties: (1) T1he root mean Squareroughlness a-, detined as thle roo t Mean Squlare dleviation Note that this contribution to thle measured refled'tanceof the surface from the mneai surface level, is small decreases, very rapidly witha increasing wavelength.compared with thie wavelength X. (2) Trhe surface is For sitdlicienlvl long wavelengths the ds/fstse reflectanceperfectly conducting and hence would have a specular mayv therefore be neglected. Thle m'leasured reflectance isreflectance of unity, if it were perfectly sinoof h. (3) then essentially specCLlar and is given by Eq. (1). ItThe distribution of heights of the surface irregularities depoenids only on a- and is not affected by the root meanis Gaussian about the mecan. (4) The autocovariance0 l s(Iurre slope of the surface.function of th , surface irregyularities is also Gauissian 'Pu summiarize, the complete exp~ressioin for thewith standard deviation a. The surface has the statis- measured reflectance R istical properties of stlat ionarity and ergodicity w~ith25rrespect to position along the surface. R =R, exp[- (4iro-)i/Xil ±R.- - (aiX)" (AO0)'. (5 )

Iia surface is illuminatedl with a parallel beami of "

monochromatic light, the reilectance may be divided*into* two comnponen ts, one of which arises fiom specular If reflectance measurements are madie at sufficient',.reflection andl the other from dliffuse reflection .or long wavelenrtlw, a- canl he. calculated directly from 0

scattering. D~avies' expressioiu fur the specular coin- . 1mau ~rfetnesic q 5 eue"tponenlt for a perfect conductor redt'ice.s; for tlfe case 'Eq. (1). At shorter xfavelenigtlis, however, the retli-

of oral inidnc ~oex[-(.-bra) 2 X\2j Sic(e no atnce near thei pbrmnal will be a function of both th~.,material is perfectly conducting, in orkler to aipply surface -runghness andI tile* root mean square slope oiDavies' theory to anl q~tual metal su,1rface it is nicexsarY thle surface irregularities. Byv measuring the.reflectankc

to odiy tis xprssiR sighly o gve or he at two wavelengths, one of which is long -enoughspecular reflectance at normal incidlence that the effect oif Ole slope maty be neglected, it should

be -possible to determine both thle surface r~oughness*R, = Ro exp[ - (47r )2 V\], (1)t an~d thle root mean square slope of the surface

.where R. is the specular reflectance of thle rough surface -irregularities. .* .andI Ro that of a perfectly smooth surface ofthe same When the wavelength is long enough so that thleA'iaterial. Thle 'angular depeildence of othe dliffuIsely diffuse reflectance nlay. be nieglected, Eq. (5) may bereflected light can also be obtained from Davies'wrtetheory. If Jo is included as before, for ligljt at nbrnl. to,RoR &(~a)~'~l]1"5 . (incdence the expression reduces to ThuP, if RoeR is'plotted onl seinilog paper~ vs 1/X2, a

rd(0)d0=Rj21(a, 1X)2(o_/X)(CoO. + lyf'sin0o straight line through the origin with a slope iich is_______Xexp[- (7ra 5ino0)2jX2]do. (2) d"irectly proportional to U2~ is obtained. It is convenient

to u4~ this equation to calculate the value of the root'R. WV. Wood, Plusical Optics (The Macmillan Comnpany, New mean sqluare roughness from the experimental values

Yo~k, 1934), 3rrJ cd., p.41.H0I. Davies, T'roc. fnst. Ele. Engrs. 101, 209 (1054). of thie reflectance at normal incidence. Approximate

"1Davies refers to this func~lon as the autocorrelatinn function. values Of the rouguhness miay be obtained inl this wvayH-owever, wve shall use the term autocovariancc functioif as a even if the contribution fromt diffuse reflectance is notmore appropriate name when the function in question is notnormalized. For a discussion of die properties of such functio,; negligible.see j. H-. Laning, Jr., and R. 1-. B~affin, Random P'roccsses inl The surface rouighness may also be obtained fromA itloiatic Control (McGraw-I-ill Book Company, Inc., New York,1956); and R. B3. Blackman and J. wV. ruikey,'The )Icasurcmenl measurements, of thle total diffuse reflectance using anof Power Spectra (Dover Pu blications, ie'v York, 1958). integrating sphere. If RlRo is near unity, Eq. (5) may

FebruaryI961R. ELATION B3,TWE EN SURFACE ROUGHNESS AND REFLECTANCE 125

be expanded to give finish. Roughnesses of the samples were 21, 8, and 32,iin. root mean square as measured with a profilometer.o-=X(Ro-R)V4rRul. (7) Some lapping was necessary for the 21 -in. sample.

Replacing RP-R in Eq. (7) with R,, the total diffuse The plate-glass disks were ground using grindingreflectance, yields particles of 5-22 y average particle si'e. The ground

disks and a plane plate-glass reference disk were! ]?=Ru(-trr)'X'. (8) aluminized in one evaporation, and the reflectance atIf the ratio R,1u'R1 is measured at a fixed wavelength, essentially normal incidence was measured as a functionI the surfaceioug is ire atla d waveleonth to of wavelength in the 2- 2 2-,u infrared region using the

th ispropCeoiaitly a dovord t reflectometer reported previously."3 For 'highly reflect-piricallv h ihrporto s ed e r- ing samples an accurac" in measured reflectanccovaluespircally by Engelhard,'- wh.Jo u, Ced it to inake a cor-rection1 for the sur~ace roughness of gauge blocks, of about 0.1,,( can be" obtained with this instrument.i Since a solid angle of only 0.03 sr about the direction

The increasingly, important part played hy thesurface roughness as the wavelength becomes longer o sn each of the two relections from the sample, onlymay be easily understood from a physical point of n

eaiylight reflected ams clryi eoddIview. Consider a nominally plane surface made up of dalost specularly is recorded.many small facets randomly oriented in various RESULTSdirections. If the dimensions of the facets are largecompared with the wavelength of light, the reflectance The theoretical wavelength dependence of theof a surface in a given direction is determined entirely decrease in reflectance caused by surface roughness isby geometrical optics and is a function only of the in good agieement with experiment. A typical relativeinclinations of the facet,,. j\s the wavelength becomes reflectance vs wavelength curve is shown in Fig. 1.longer, diffraction effects become important, and the The circles represent experimental values. The solidreflectance is a functioo of both the illclination and line was computed on the assumption that only thethe size of the facets. A, the wavelengfh l)ecomes ,s.till specularly reflected light need be considered. Thelonger, so that the dimensions of the facets become relletance is then only a function of u- and not of m.ver small by comparion, the reflectance of the urface The solid line fits the experimental points quite vellwill be determined almost entirely by diffraction above 0.9I reflectance, but begins to decrease tooeffects. The surface roughness will thel'be the only rapidly for lower rellectances" indicating that hereimportant parameter. th'e is an appreciable contribution from diffuse

Although it is assumed in.l)avi.,' theory that the reflectance near the normal. The dotted line, whichsurface is perfectl , iotropic, thi., condlition is y no was computed assuming a contribution from bothmean, nece.sary. 'requently in practice, particularly, specular and diffuse rellectance, fits the experimentalwith mach'ned.surfaces, there i-,.a preferred direction data to about 0.75 rejlectanq. At lower reflectancestechnically referred to as the lay of the surface. .in the relluirement that o-t <X is violated, and the theorythe case of such an anisotrolpiL surface, the colept ()f would not be expected tojhold. These curves demon-a mean sli4are slolpe must be extended. Suplpose for strate that if a- is to be deterynined from Eq. (1) withoutexample that the autocovariance function for such'asurface is . Gaussian, but with two different auto- 100covariance lengths a and b corresponding to two 90ortliogonal directions x and v along the surface. It can 'be shown by a derivation similar to that for the <' Go -isotropic surface that the (uantity in. in Eq. (5) is /jeplaced by mno,, where na and no, are the root mean U.o osquare slopes measufed in the x and y, directions, W, .orespectively. o

EXPERIMENTAL W

To test the theory presented above, a series of flat .40ol2-in.-diam disks of various roughnesses were prepared.The disks were overcoated with an opaque, evaporated 2 6 10 4 iS 22aluminum filn, and the reflectance was measured as a WAVELENGTH A (A)

function of wavelength. Both steel and plate-glass tic.. 1. Relative reflectance of a fineh" ground glass surface asdisks were used. The steeLdisks were made of AISI a function of wavelength. Circles indicate experimental points.'lihe dottcd cuteVe was calculated frome Eq. (5) and the solidtype 01 tool steel hardened to Rockwell 58 .6(, and curve from Eq. (1). Both curves coincide above R/Ro=0.90.a fine feed surface grinder was used to obtain tile - m0) 13 H. E. Bennett and \V. F,. Koehler, J. Opt. So. Am. 50: 112 E. Engelhard, Natl. Bur. Standards ('irc. No. 581, 1 (1957). (1960).

04

*1

a

126 11. E. BENNETT ANI) J. 0, PORt, US Vol.51

danger of a significant systematic error, the wavelength TABLE II. Roughnesses of ground steel surfaces.must be Wulient__l___y - -=-- ....

must be sufficiently long that the diffuse reflectance - - ----- ___

near the normal is negligible. It is interesting to o-Profilometer r Optical a Optical a Optical/

observe, however, that even at a relative reflcctance (microinches) (microinches) (microns) 0profilometer

of 0.40 tile value of o- computed from tile exprintal 32 51.0 1.30 1.6values of the retlectance and neglecting tb contribution 8 15.8 0.40 2.0

f r mth 2.5,i 6.2 0.16 2.5from tie diffuse reflectance is in error by less than a,factor of 2.

fThe root mean square surface roughness values for

ground glass surfaces obtained by this retlectance An important advatage of the relectance methodmethod are recorded in Table I and are in good 'Nwhen smoother surfaces are to be measured is mostagreement with results obtained using other methods easily explained by comparison with the profilometer.of measurennt. Tn cases where the relative specular The protilometer soylus is of a fixed size, and hencereflectance was l es wthan ere( at the longest wae the percent uncertainty in the measurements in-reflectanc asa less suraac 0.coX) stoother.ngeweerve-length measured, a small correction was applied to cirees as the surface becomes smoother. However in

the observed retlectance for tile diffuse reflectance near the reflectance method, a shorter wavelength is used

the normal in accordance with Eq. (5). In making this for smoother surfaces so that the percent uncer-correction the value of , for the rougher ground glass tainty remains the same. Thus, when using the reflect-surfaces was assumed to me e tul to that measured fr ance method, the accuracy with which the rough-the smoothei" surfaces. Although the irregularities on ness of a rather smooth surface can be measured is thetie S110hr ufcs lhuhtl reuaii soi Same a,; that for a much rougher one.a ground glass surface are so closely spaced and soirregularin shape that the usual techniqus of rough- The precision of ineasurement of surface roughness

which may be obtained using the rellectance method i,ness; meas"Urement fail,. Pire.t oi "Ihas reported a. value 'of about 1 wavelength of visible light for tie average good. Since the square of o- appears in the exponent in

peak to valley depth of.a inel: ground glass surface. Eq. (5), the reflectance is sensitive to a small changeThis tigure is in good agreement with the results in -. Even if a crude reflectometer is used and therepoited in the last column of Tale .. uncertainty in the measured rellectance is - -l , the

The root mnean square surface roughness values for uncertainty in the value of a- i, :5n at a relativetile ground steel surfaces obtained lv this rellectance of Q.Q0. For a 2-pin, root mean squartmethod are recorded in Table II along with the roughness surface, this uncertijinty is .-0.1 ,in. Withcorretsponding proildmeter values and the ratio the best reflectometer in our laboratory, tbis un-

t(optical I o-(protilormeter). Although the surface rough- certainty can be reduced by an order of i.agnitude.The reflectance method is free from other dis,-ness of the samples change, by an order.of magnitude, , a

thle roughness valtues obtained by those two methods advantages of previously used methods of measuringare une ,~strLaice riiness. For ^ample, most convenitionlad

are generally eompatible. However, as the sudace . oa dhiaon e ti st wich

becQmes smoother, the di.,crepancy between the two methols emethods increases until for tile smoothest surface the leaves a deep' scr~l ch on metal surfaces. Such .insfru-

roughness obtained from the optical measurements is ments are insensitive to roughnesses of less than a few

2 times that obtained using the proiloneter. This microinches, even if the surface is composed of very

result would be.expected if the tip of the protilometer "4road, shallow grooves. If.the surface is irregular with

stylus did not bottom in the grooves on the smoother deep microscratches in which the diamond point can-

surfaces. Since the diameter of the tracing stylus is not bottom, these tracing instruments ciauiot be usedI at all Une r oIM) ,yin., or about two order of magnitude larger than at all. pnder restricted conditions an linterferometric

the rouglness to be measured, failure of the tip to method an be used," ' but determining root meanbottom is easily understood. * square roughness in this way is difficult and -time

consuming Comnpared to the reflectance method.

T.xiu.E J. RnLghnesses of ground glass surfaces. if a small and relatively inexpensive spectrometer_were fitted with sodium chlorle and cesium bromide

Average . prisms, it would be possible to cover tie 1-40-y wave-particle length range with sufficient resolution for surface

size roughness measurement. If a smooth, flat sample ofGrade ('omprsition (microns) (microns) the same material were used as a standard, the data

302 emery powder 22 1.0 necessary for calculating the root mean square rough-W6 garnet powder 12 0.7 ness from Eq. (1) could be obtained by simply turning303-j emery powder I11 0.6\Vl gamet powder 5 0.2 the wavelength control until tie reflectance of the rough

305 emery powder 5 0.15 surface is 90% of that of the *tandard. If the wave-

16 NV. F, Koehler and W. C. White, J. Opt. Soc. Am. 45, 1011J,. W. Preston, Trans. Opt. Soc. (London) 23, 141 (1922). (1955).

USt0

Februiary 1961 RELATI ON B ETlWE EN SUTRFACE. RoUIT HN ESS AND R EFLECTANCE 127

length drum were calibrated inl values of the root rlian F. I sd Ouortz Fuse Id Pyrox P1o. Fsquare roughness, no calculationi would be necessary, 2.7A U9.65A A I

and the roughn11eSS could be read directly from the Idrum ting Although11 the surface., to be m~asured III Iwould have to be carefully cleaned., this mncasurig 1,0

procedlure wvou ld be rapid, versa t ile, anrd niond(est ructLiVe. h ~ itIt could be performed by unskilled personnel anid W 00would not be subject to operator error.,

32500 Z 0

OPTICALLY POLISHED SURFACESI

Sinice vrery smnall Surface roughnesses may be of bO 20010201110 5

importance inl optical measurements, it is important rms SURFACE ROUGHNESS, (A)to consider whether thle theory tits the experimenitaldata for veryr siall values of r _X. Thle dlerivation is riot l'. 3. E~rror causcd byN Surface roughness in reflectance

valid ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ i whnrX<,adide udrteecru- IrsurnGents as a function of wavelength. Dotted lines indicatevali whe o- <<Iaiidiiided wder heseCir~n-t he surface rou~ghoesses of ty-pical optically polished glass surficessaces HuIyagens' principle, en which thle thieory- is calCUlated from the experim'ental data of Koehler and White.

based, would be expected to fail. Huwev~er, the changein relative reflectance with wavelength calcu~la ted Ani example of the importance which the surfacetrom Eq. (5) tits the experimental data for the roug(hnIess of .optically polished surfaces may have

smohetgrud-lssad telsrfcs tth when mieasuring the intrinsic specular relectance oflongaest wavelengths which couldl he measured with the iteasin hviblan.utvoetrgnsfte

present eq uipment. Tyvpical. results, for a g'rot nd-glass spcrm sshw inig. ntslgueherornsurface are shown-i in lig. -2. The solidl line represents, th ) e~ue spcua retca ,ecue ufc

the alclatd dcrese n relecane cascdh a rollghness is plotted vs wavelength.* Since a log-logsurface roughniess so small that the ratiO of a X\ wa5*po suete acltdvle o -wihgv

onlyo isou used, 1.e Notelte that the caclae which meaeuredonlyabot 001.Not tht te clcuatel ald llel"LIVIparticular error inl reflectance lie onl straight lines when

reflectatnces show *no tenclency to (liverpe ,ven for the0P plotted as a funictioni of wavelen~gth. Since opticall ~

smalle~t valuie of a X, whrNhNpcla elcac polished glass surfaces are amongy the smoothestof he rourlsuracewason ~ ~% essthn tiit ofitsurfaces available, the roughnesscs *of some* typical

plane surface. Since the theoretical expression al)- otclyplse lse resonfrcmaio.proaches the correct lilnit as thle surface hecoilics upoe.atontheruheso'olse ls vr

caused 1w Surface roughIlness beoe) eoilem their data indicated that the irregularities onl a polishedprolpdle that the calculated Curve holds for e, eissmaller glass surface wyere not symmiletrical about an a~eragrevalue;; of a X thanl were obtainied exp~erimentally. 'planle, they used] anl wsnsvmtrical distribution

function. Also, thecy report only the dlistributioni of the

maxima of the irregularities rather thari that of all

with the theory presented here, however, it is necessary0 ~to use a Gauissian distriutioni fundiion. A5 triangular

Z 3.04 shape was assuImed for the irregularities. The numbers

presenlterl here for thle roughnlesscs of various polishedgla. s surfaces are thus ondy approximate, but Idoindicate the rangaeuof errors which onle *71ighlt expect to

La:2.0- 0bfind inl reflectance mecasuremnents At shorter wave-

W lengths using even optically polished surfaces. ForZ 0 example? if df materhd ~having the surface roughlness of

ow (.0- 00)1 3 Jlint glas werui used, an error of about 1%1ix would be expect ed iif reflectance measurements made

inl thle visible. This error is anl order of magnitudegrcater thanl the accuracy with which thp reflectance

0.

0 .005 .010 .015 1 canl be measured."3 It would be expected to be nearlyrms SURFACE ROUGHNESS 'I WAVELENGTH X two orders of magnitudle or 100 times greater thanl the

y10.~~~ ~ ~ (. Ero ae nrfecac nesrmnt Iel ufc racy of measyremnert inl the ultraviolet.rougliness-is neglected. Circles represent experimental points. ____

Thle solid line- was calculated from Eqt. (1). W. F. Kochler-, J. Opt. Soc Am. 413, 743 (1953).

0

128 HI. E. I3ENN13,TT AND) J. 0. IPORT"EUS Vol. 51

CONCLUSION We then consider

Tile expressions relating the reflectance at normalincidence to the surface roughness give~a yavelcngthi Jim (1/X)(1/Y) I'C(0zy/,1x)'dxdy 6dlependence wvhich is verified by experiment. In acici- f,- w fJ,,tion, from measurement of the reflectance a. value ofthe root mean square surface roughness canl be obtained r)_ (y)X [ &a,)2X Xwhich is in good agreement with the results using other JmW 1X(1/I) J_ y()(z8)r'x

techniques. Since the surface roughness canl be cleter- X)

mined with precision regardless of the root mean square +2z(Oz/Ox)rx(x) (drx/dx)Slope Of thle Surface irregularities if the measurements,±z(x/)'id.().are made at sufficiently 0long wavelengths, a possible +2d~~X2d'd.(0,application to measurement of surface finish is sug- We now letgrestqd. The thieory may also be applied to opticallypolished surfaces, since exlperimenltal results show that rx (x) = (X/X± ) [(x/ ) + (X/2 ) +Q1;it holds even for surfaces which havea aroot mnean sqluare - (X/2) - < x< - (,Y/2)roughlness of less than one hundredth of the wavelength = (-X/X±+ xi IK (,Y/2) (1employed. -A non-negligible system~atic error in reflect- =(YX/+ t (x/ + (X/ 2 + I1]ance meiasurements made in the visible and ultraviolet (/)< X2+may result from surface roughness even if good(/2<<(/)+optically polished surfaces are employed, where is aniarbitrarily small constant, and

ACKNOWLEDGMENTS (,y)=l ylI <(11/2)

The authors, would like to thank Dr. jean Af.lBennett, =0; li> (Y;'2).Dr. W. F. Koehler, 'ancl Dr. D. E. Ziln-i'er, all of.Michelson Laboratory, for helpful cliscussions. This causes; all terms except the first inl the square

bracket of Eq. (10) to vanish outside of a finite reg-ion,APPENDIX ' andI henrce give a vanishing contribution to the integnil

inl the limit. Furthermorg, the righlt-iand side of EilInl this appendix we develop the relationship betwen 011 eoe qiaettotergthn ie

in, the root mnean square slope, and a, lie autocovariance -Zlengt~i, of the Surface. Let z~~)rep~resent the surface'. Eq. (9). Combining t hese results yields

height as a function of p-osition along thle Surface.,Following I'avie, 1" we approximpate the actual surface ',m2= Jlm (j/X)(,/Y) F(&-Xi-f)dxdy. (1-3)by a svrfacb of infinite extent with the samie statistical JfI .Icharacter throughou01.t. The mean square slobe in'2measured in anl arbitrary clireCtiov, whiich we take as Application of thle twio-dIirensionail'Parseval relationtjie x dire~tion, is then cfefined by 'to Eq. (13), which is now ifl proper' formi, yields

( '\2 ,12 r~rOx-' 2

lirn~~l/X)( a/' O/X)2dj.rdy, (9) 1)2= Jim ( 1 X(/)U i ~ I(~)dd'-X1 ~2~2 x~-,.. ~ J., 0 8x (14)

where we assume that z2 and (az,'Ox)2 are noninlinite wee[zycxfiv eoe tetociesoaat all points onl thle surface. Thle relationship between ii e rnfr f(z,~a) oo'antecneto

involves; the Fourier transform of the autocovariance grahdc of Eq. (14) in terms of Ezxl-]f directly.' Thisfunction. In order to establish this relationship, oOC myedn ypata ifrnitineteeutomay apply the two-dimensional Parseval relation'7 to dlefining the inverse F~ourier transform:0Eq. (9). However in order to do this tilhe equation lltatfirst be expressed in anl appropriate form.' - 0

To convert Eq. (9) to the proper form for application Z 10y (rv) =1 ( Zy-f(1V ~, -+~dd.(5of the Parseval relation, we introduce the new function O xjj_~(5zxrY(xvy)=z(x,y)rxv(x)rYyv). H-ere rxy and rl- are func-tions which may be used to control the behiavior 6f We thus infer thatzxy(x,y) in the region X/2< jxl < -, F/2< jyj <ci-.

. . JBochner and K. ('landrasekharari, Fourier Trans wills r-]( ,v) 27rh(Ezxy], (11,z) (6(Princeton University Press, Princeton, New Jersey, 19409), 1). 67. Lax If

0V

February1961 RELATION BETWEEN SURFACE ROUGHNESS AND REFLECTANCE 129

6and finally that Using the above definition of rx and ry causes this,w - lim (1/X)(1/W) f 4r2 u2 [z:'],(u,v)f 2dudv expression to reduce to

X_ -- A (s,) = lir (1/X) (/Y)

=4r2.Jo(Af(,,,v)} (17)X/2 Y12

where Af(u,v) denotes the Fourier transform of the z(x,y)z(s-x, 1-y)dxdy, (19)autocovariance function of z, and mo-0 denotes the second -X./2 -Y/2

6moment with respect to.nu and the zeroth moment which is the usual definition of the autocovariancewith respect to v of :4 ,(uiv).

To show that the expression function.All that remains, to obtain the relationship between

A, 6v) = lima (1/X)(1/Y) zx1]2(nu,v)P ,hY and a2 is to substitute the Fourier transform ofx --,W the Gaussian autocovariance function

A (s'j) = U2 exp[- (s2'+ 1)/adQis in accordance with the customary definition of theautocovariance function is perfectly straightforward. into Eq. (17). One thus obtainsTaking the inverse Fourier transform of both sidesyields ,,2 47r 2 Jf f i 2[o 27ra2 exp -r2 a2(u2+v 2)jdudv (20)

A(s,,)= lim (l/a)(1 ) f f [0(x,y)rx(x)ry(y)]

or

X[z(s-x, 1-y)rx(s-x)r (I-y)]dxdy. (18) ,P 2= 2 (/a)2. (21)

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