speckle field of curved, rotating surfaces of gaussian roughness illuminated by a laser light spot

Speckle field of curved, rotating surfaces of Gaussian roughness illuminatedby a laser light spot*

Joachim C. Erdmann and Robert I. GellertBoeing Commercial Airplane Company, Seattle, Washington 98124

(Received 5 May 1976)

The first- and second-order space-time correlation functions of the optical field and the intensity, respectively,are calculated for the case of a rotating symmetrical curved surface with Gaussian roughness, scattering laserlight into the far-field zone from a small illuminated spot. The incident beam is either collimated or spherical.The diameter of the illuminated area is small compared with the radii of curvature. The mean surface isapproximated by a torus with two principal radii of curvature. The sphere, the cone, and the flat surface arespecial cases. Optical phase variations linear and quadratic in the surface coordinates are included in thecalculation. The surface is assumed to be very rough. It is found that the correlation function of the intensitydepends on the radius of the incident beam, the local surface velocity, and the radii of curvature in theilluminated region. Curvature decreases the decay time of the correlation functions. This is confirmed byexperiments carried out with the flat disk, the sphere, and the cone. The temporal cross-correlation functionof the outputs of two detectors in the speckle field obtained with an incident spherical wave is very sensitiveto normal displacements of the surface.

I. INTRODUCTION

The purpose of this investigation is to study the effectof rotation and shape of bodies with rotational symmetryon the speckle field of laser light scattered by such ob-jects. Examples are spheres, cylinders, and cones.The incident light beam illuminates only a small part ofthe surface area so that at constant rotation rate thelocal tangential velocity can be regarded as constant

1194 J. Opt. Soc. Am., Vol. 66, No. 11, November 1976

over the illuminated area. In this respect, our ap-proach is based on differential geometry and differsfrom "global" analyses of entire objects under illumina-tion. 1-5

In particular, we first calculate the space-time cor-relation function G1 of the scattered field by extendingBeckmann's application of scalar diffraction theory toa very rough surface. 6 The correlation function G2 of

Copyright i 1976 by the Optical Society of America 1194

z

FIG. 1. Surface coordinate system. The x-y plane is a tan-gential plane.

the intensity is then found from the relation 7 9

G 2(T) = I G (0) 1 2 +I G,(r) I 2 , (1)

which holds whenever the scattered field has a Gaussianprobability density function. Here T denotes the lagtime variable, but the relation is also valid for generalspace-time correlations.

After the calculation is carried out, the analytical re-sults are compared with experimental information ob-tained with the sphere, the cone, and the flat plate.Reasonable agreement is found for all configurationsthat have been investigated.

II. STATEMENT OF PROBLEM AND SPECIALASSUMPTIONS

In Fig. 1 we consider the rough surface of a solidbody with rotational symmetry, rotating about the sym-metry axis v at constant rate n. A collimated laserbeam with wave vector k1 is incident from an arbitrarydirection (with the exception of grazing incidence). Theincident electric field is arbitrarily polarized. Itsscalar complex amplitude A1 has a Gaussian beam pro-file of effective radius ro, illuminating a small ellip-tical spot on the surface. The beam axis interceptsthe surface at a point P0 which is chosen as the originof a Cartesian coordinate system x, y, z. The z axispoints in the direction of the outer normal of the "mean"surface which is obtained after averaging over theroughness.

The z axis and the incident beam axis form a planewhich is the plane of incidence on the mean surface.It also contains the x axis in the direction that rendersk1, positive. The x and y axes together form the tangen-


tial plane on the mean surface at the origin. The tan-gential velocity u in the illuminated area in general hascomponents uX and u, in the tangential plane.

Given the configuration just described, the analyticalproblem consists of calculating the space-time correla-tion function G1 of the scattered field with complex am-plitude A2 and the correlation function G2 of the intensityIA2 12 in the far-field zone.

To facilitate the mathematical treatment, the fol-lowing conditions are assumed to be fulfilled: (i) Thesurface properties are uniform everywhere; (ii) thesurface is isotropically rough in two dimensions withnormally distributed roughness D of zero mean, vari-ance u2, and correlation distance X; (iii) both a and Xare of similar magnitude as the optical wavelength X orlarger; and (iv) the beam radius ro is large comparedwith the correlation distance X and the wavelength X,but small compared with the radii of principal curva-ture at any point of the surface.

These conditions limit application of the results tovery rough surfaces that can be approximated by thenormal distribution. The restrictions imposed on thebeam radius will be used to treat the surface velocityas a constant. Furthernmore, the dependence of theoptical phase on the surface coordinates can then belimited to linear and quadratic terms as in the Fresnelapproximation. A sufficiently large beam radius, onthe other hand, is required for proper sampling of thesurface statistics at any time.

III. SCATTERED FIELD

We now define the scattering angles a,, %2, cy3 at anarbitrary point P on the mean surface, following Beck-mann's convention. 10 At the origin these angles willbe further subscripted by 0 as shown in Fig. 1.

a, is the angle of incidence of the beam with wavevector ki on the mean surface and is defined by

cos01 = - (n -kl)/k, (2)

where k = 27T/X and n is the unit vector of the outer nor-mal through the point P of the mean surface. Further-more,

cosa 2 = (n k2 )/k, (3)

where k2 is the wave vector of the scattered field. a3is then the angle between the planes formed by n, k,and n, k2 , so that

cosaY3 = (nxkl) * (nxk 2)/(sincxl sinag2) (4)

which can be defined only when ac l 0, 2: i 0. At theorigin, for example, !30 is the angle measured counter-clockwise in the x-y plane between the positive x axisand the projection of k2 into the x-y plane.

We also define the scattering vector6

v = k, - k 2 = k[(sina 1 o - sinac20 COS C3 0 )e,

- sina 2 0 sina 30 e,- (cosc110 + CoS a 20)e ]X (5)

where e,, e., e, are unit vectors in the x, y, z di-rections, respectively. In a far-field analysis, v is aconstant vector.

J. C. Erdmann and R. I. Gellert 1195

On the rough surface, the true local scattering anglesE1, E2, E3 are in general different from a,, a, a3 .Let s be the true local normal unit vector which can beexpressed by the slopes D,= 8/ax and ty= a/ay of theroughness C(x, y):

s = (- e,- ey+ e,)(1 + + 2)-1 /2. (6)

If in Eqs. (2)-(4), n is replaced by s, one obtains thedefinitions of the local scattering angles El, E2 , E3 onthe rough surface.

From the viewpoint of geometrical optics one canask which roughness slopes Ax and by contribute most tothe field observed at far distance for a particular scat-tering vector v. The required condition is the reflec-tion condition

kl-k2=v=-slkl-k2l I (7)

This means that the scattering vector must be anti-parallel to the local normal of the reflecting roughnesselements with suitable orientation. The solution ofEq. (7) is

Zx =-VX /vZ, by=-vy/vz 2 (8)

where v,, vy, v, are the components of v in Eq. (5).This approximation of the scattering slopes will be use-ful later [see Eq. (41)].

Following Beckman, 11 we can write for the scatterfield at the far distance Ro from the surface after ad,tion to a Gaussian beam

A 2 = ike2 rikR0 CO JJ P(x, y)F(¢,, by)

xeiv.rdxdy,

Here the Gaussian pupil function

P(x, y) = exp[- (X2 cosa 0 +y2)/r 0]

FIG. 2. Generation of the surface by revolution of the merid-ian (shown as upper part of the contour by a solid line) aboutthe rotational axis v. Also shown are the radii a, b, c, andthe angle /3 (the angle between the local surface normal at P0and the rotational axis v is 17r-P).

edLpta-

In order to characterize the optical phase v .r in Eq.(9), we shall first describe the mean surface of revolu-tion. In Fig. 2 the solid upper part of the contourrepresents a meridian which generates the surface byrotation about the axis v in the same plane, Also shown

(9) is the origin P0 of the coordinate system x, y, z as inFig. 1, The tangential velocity u at P0 is directedvertically out of the drawing plane in Fig. 2, and wedefine y as the angle between u and the positive x axis.

tau)

suppresses the contributions from surface elementsoutside the illuminated area so that the integration canbe extended from - oo to + -. Furthermore, the functionF is defined by"l

F=Fa Ux+Fc Y-Fb, (11)

with

Fa = (1 - R) sinal + (1 + R) sina 2 cosa 3 X (12)

Fb = (1 + R) cosa, - (1 - R) cosa, (13)

Fc= (1 +R) sina 2 sina3 . (14)

F is a function of the scattering angles a,, a2 , a3 andof the reflection coefficient R which in turn depends onthe polarization state of the incident beam and on thetrue local scattering angles El, E2, E3. On a two-di-mensionally rough surface the Fresnel relations forthe reflection coefficient R are of little help, becausethe true local polarization state of the incident field isin general unknown, even if it is defined with respectto the x-y plane. For this reason we shall assume thatF of Eq. (11) is a random variable that depends on Axand gy, and we shall resort to a statistical analysis inSec. IV.


In the vicinity of the origin P0 the surface can berepresented by a torus which is obtained if we replacethe meridian of the surface by the osculating circle ofradius b. As shown in Fig. 2, its center has a distancea from the axis of rotation. A surface point P can thenbe characterized by two parameters q and AP. The pa-rameter q is the angle between the two planes of themeridians through PO and P, respectively (the rota-tional angle out of plane in Fig. 2). The parameter AO3measures the angle between the normal through thepoint P0 and the normal through a point P' that lies onthe meridian through P0 and transfers into P by rotationthrough the angle q (in-plane in Fig. 2).

When q and Avi are sufficiently small so that

sinq q and sinAP3=Aji, (15)

the parameter form of the torus can be calculated as' 2

p = [cq cosy+ (ba3+ 2cq2 sing) siny]e.

- [cq siny - (bA3+ 2cq2 sing) cosy] ey

- I[b(Ap) 2+ cq2 cosg] eg.. (16)

A general point on the illuminated part of the surfaceis then given by


\k z

X-y plane

I

r = p + Dee, (17)

with components x, y, z. The aperture coordinatesx, y are related to q and AO3 by the transformation

q= (xcosy- y siny)/c, Ai3= (x siny+ y cosy)/b, (18)

where b and c are the two radii of principal curvatureat the point P0 .

In the range where Eqs. (15) are valid, the velocitycan be regarded as constant. From Eqs. (15) and (18)we then obtain the restrictions for the incident beamradius rA/b2 < 1 and r ? «c<< 1 which will give an errorof less than 1% in p given by Eq. (16) if put into the form

ro 0<0. 1 min(b, c) . (19)

For given aperture coordinates x- and y-body rota-tations change the optical phase only through changes ofthe z contribution v. z, caused by the fluctuations of theroughness C. Rotations about the v axis leave the meansurface representation, Eq. (16), invariant. Hence,during a time interval t, t + T the z coordinate changesfrom

z (t) = p± + t (x, y) (20)

to

Z(t+oT)=P2+ (X-UrT, y-U,7-) . (21)

Here the argument of g has been changed through aGalilei transformation in the x-y plane. Similarly,the slopes 9x and t, of the roughness function changeduring the time interval t, t+ T from

txfy(X' A) (22a)

to

Cx, y(X-UX T. Y-UY T), (22b)

Returning to Eq. (9), we substitute Eqs. (17) and (20)and obtain for the optical phase

v~r=b(x, y)+v 2D, (23)

where c =v . p represents the phase modulation by themean curved surface and ve T the phase modulation in-troduced by the roughness. Since we approximate thesurface part of interest by a torus, 4' can be writtenas the quadratic form

b =Blx+B2x2+B3y+B4y2+B5xy . (24)

surface velocity, and the curvature parameters of thesurface. The following surfaces are special cases ofthe torus: (a) the sphere (a = 0). (b) the cone (b - o,( 0 0), and (c) the circular cylinder (b o-, j = 0). Thecase of a flat surface moving in its own plane can bederived by putting B2 = B 4 = B5 = 0.

IV. CORRELATION FUNCTION GI

We want to calculate

G,(Av, 7T)= (A2 (Y, t)A2* (v+ v, t + T)v(21

which is the space-time correlation function of thescattered field, differing from the mutual coherencefunction of the analytic signal'3 by a factor exp(- iwT),where w is the laser frequency. Substituting Eqs. (9)and (23) into Eq. (26) we obtain

k2 CO2a

G, = R jjjjP(x1, Yl)P(X 2 , Y2)

xexpfi[1(x,, lY) -'I(x 2, Y2)]}

x (F(expi,, t, )F(- V 2 , t2)

x expli (Ve1 Cl - V.2 C2)] ) dxl dy, dX2 dY2 (2'

Corresponding to Eqs. (20) to (22), g2, Ux2, and ,y2

have to be evaluated at the point x2 - U, T, Y2 - U Tr, ifAl, Dxl, and g,, are taken at the point xl, yl. Throughthis transformation in the x-y plane the right side ofthe last equation becomes a function of T. To calculatethe ensemble average as an expectation value, we re-quire the joint probability density function of the sixnormal random variables Cl 2, D, Ax , Lty, ly2-Because , Ax, and by are statistically independent, thedensity factorizes into three joint densities for thevariable pairs (pI, g2), (Cxl, x), ( yl Y2). Each den-sity is determined by the matrix of second moments ofthe variables. 14 Furthermore, if the second momentof the roughness is given by

(gl g2= ,C(xlY ,, X2 -ux_ , y 2 -uY vr) (28)

C being the autocorrelation coefficient of the roughness,we can apply the lemma15

(H' (f )H' (f+ g)) =-d 2 (H(f)H(f +g))/dg2' (29)

and obtain

Substituting Eqs. (5) and (16) and applying the transfor-mation Eq. (18) in the last expression, we obtain aftera comparison of coefficients

BI=v,

B2 = (V. u/u) cos 2 y sinO/2c

-v,(sin 2y/2b+ cos2 y cos3/2c),

B3=v,, .(25)

B 4 = (v . u/u) sin2y sing/2c

- vZ(cos2 y/2b + sin2y cos3/2c),

B5 = (v . u/u) sinf3/c - vea siny cosy/bc .

The coefficients B, to B5 are completely determined bythe scattering vector, the direction of the rotational


/rr \,2)= - a2 A2Iand

(Lyl Cy2 = - C

Ax = X2 - X- UT

1~Y=Y2-yi-UY T .

Separating the D-dependent part of the term in angularbrackets in Eq. (27), we then find by standard calcula-tion the characteristic function

x(v,,, - v22) = (exp[i(v., 1 -V.2 V 2)2 )

= exp(- a2 Av 2/2)e-g"-c) (32)

where we have written Av, = vz2- v,1 and

g =,C2 V,2 s (33)

For ordinary rough surfaces the rms roughness u is

J. C. Erdmann and R. 1. Gellert 1197

of the order of the wavelength X or larger, and thereholds g>> 1 (in fact, g> 100). Substantial contributionsto the integral Eq. (27) can then come only from thoseaperture elements for which Con 1. We assume thatC has the form

C=exp[- (AiX, + /)/X 2] , (34

where X is the correlation distance of the roughness.Ax and A, have been given in Eqs. (30) and (31). Thefunction

p = e-g(l-C)

with C given by Eq. (34) can be expanded about thesaddle point coordinates 6 Ax= A= 0 or

4 =U.T, T ?s=UYT

in the plane formed by

= X 2 -x 1 , X 7=Y2-Y1 -

With

q2=glx

one obtains for the first two terms

(35)

(36)

(Q i*7\

which will be approximated by16

p = exp[- q 2(A+ A2)] (40)

For very large values of g this approximation is alwaysvery good.

For UT> r0 the surface elements for which C 1,are outside the illuminated area, and contributions toGI do not arise because of the pupil function P(x, y),Eq. (10). This means that G, must decay on a timescale comparable with the transit time of a surfaceelement through the illuminated area. The choice ofro then sets the scale for the decay time of GI, to-gether with u.

To calculate the average

(F, F2) = (F(Mx, yI)F(4 2, C,2))

in Fq. (27), we follow an approach already applied byHagfors to the backscatter case. 2 In essence, we as-sume that most of the surface slopes of importancewith a particular scattering vector have values closeto the mean values T. and Ty defined by Eqs. (8). Thismeans that F, F2 has a maximum for 4,l = 6,? = ;x, Xyl

= Cy2 = .Expanding F. F2 about 4 and 'y and substitut-ing C= 1 in the second moments, Eqs. (30) and (31),we find that all terms but the zero-order term [F(Zx,T,)]2 are very small and can be neglected. In the cal-culation of the average we can replace a,, C2, (X3 bythe center values olO, C!0 C!30, respectively. Sub-stituting then Eqs. (8) into Eq. (11), we find

(F, F2) (F') = 4(R2)(1 + CoscYosCoSa 0

- sinalo sinCY20 COSa30 )'/

(coso'10 + COS0120)2 . (41)

Here (R2) is the mean square of the reflection coeffi-cient. For the backscatter case (a2O =- aloe, a3o =0)


one obtains the cos'2Q 0 dependence found by Hagfors. IFurthermore, if we replace in Eq. (41) (R2) by unity,we regain the angular dependence found by Beckmannfor perfect conductivity. 17

Substituting Eqs. (30)-(32), (35), (40), and (41) intoEq. (27) and setting

A =k cosal,((F2)) 1j2 /(2T2rrR,) ,

we can write

Gl(Av, T) = A2e-(aAvz)2/2

x f11 P(xl, YI)P(X2, Y2)

(42)

x exp {-q [ (X1-X2 I ,u T) + (YI -Y2 + UT)']

+ i[4 (xI, Yl) - ) (x2, Y2)] }dxI dy, dX2 dY2 -kv 1) (43)

Here the phase c is given by Eqs. (24) and (25). Carry-ing out the integration, we obtain a product of many ex-

(38) ponential factors of which only those with argumentsproportional to q , q , and q2'uv, can noticeably de-viate from unity. Retainingonlythesignificantfactors,

(39) one finds

G,(V, T) = 7r-rAo2/(Ii h, cCOSCao)S(AVX Av5y Av U)

X D,(Avx , Avy)TI(T)T2(T)Dt (T)

with

S = exp {- [V-X /2h2 + F52 /2h2

+ 2t(AVx2 /COS2ablo + AVy2) + (UAV,)2]}

D± = expr(4 /o ÷ [ tx(F.1)2]+ 2},

+ AvY(,, B 4 + -B5)/2h2

= exp[- (QMU2+ Q2u2 + Q2 U U)7r2]

T2=exp[(Ux~u, U U)y)T]X

Dt = exp[i(FXU±+ UyUy)T

x exp[i(AB2u2 + AB 4uy + 5-ABSUXUY)T2] 1

h2= 2q2+ Cos 2lo /rg,

h2 =2q2+ l/r2

Q2 = cos2c0!0 /2r'± + 2r /(2 cos'cY 0)

Q2 1/2r2 + B42rO

2 B 4 Bsro,

U', = ro[AuxT2 /(2 cos'aj 0)+ AV, 4 ],

an 2[dVxb /(4 COS2C~lo) + 2 AVy B4]

and

ABi = BJ2 -Bil I

(44)

(45)

(46)

(47)

(48)

(49)

(50)

(51)

(52)

(53)

(54)

(55)

(56)

(57)

(58)

with B, given by Eqs. (25). Furthermore, U,= 2(v

+ ±V2), etc.

The spatial part S of G, does not depend on the shapeof the surface. It depends on the tangential compo-nents of the scattering vector and on the roughness pa-


rameters in correspondence with Beckmann's treatise. 18

The aperiodic temporal part T, is a Gaussian deter-mined by the rotational velocity, the beam radius, andthe curvature radii of the surface. The factor T2 isthe temporal cross correlation, significant only whenthe surface is curved. It shifts the peak of the correla-tion function T = T1T., either to positive or negativevalues of T, depending on the sign of U * U.

The periodic factor D, results from the Doppler shiftassociated with the surface rotation. The periodic fac-tor D8 is observable only for curved surfaces when thedetector or the laser is moved.

V. CORRELATION FUNCTION G2

From Eq. (1) with Eq. (44) there follows for the cor-relation function of the intensity:

G2 (AV, 7) = K 2 [S2 (0) + S 2 (Av)T 12(T)T 2(T)], (59)

where

K= r2 r2A 2 /(hjh 2 cosU10) , (60)

and where S, T1 , T2 are given by Eqs. (45), (47), (48),respectively. There holds T1 (0)T2(0) =1. A specialcase of Eq. (59) follows for Av =0:

G2 (0, 7) =K 2 S2 (O)[1 + T 2(T)] . (61)

Here we can rewrite T 2 in the form

T 2 =exp(- 12 T2) (62)

where

its 2 4 (Q2 U2 + Q2y U2y + Q2Y U"U,) .(63)

Here .o. can be regarded as the rms "speckle frequency"which is the rms frequency of the stochastic signal inthe output of a square-law detector at the observationpoint in the far field, observable when the surface isrotating. This view is substantiated by Fourier trans-formation of T 2: The result is the power spectraldensity of the frequency X

f(w,) = exp(- w2/2w2)/[Wc (27r) /2] (64)

which is the normal density with variance w.

Referring to Eqs. (52)-(54), we note that the mean-square speckle frequency originates in two differenttypes of contributions, those crr 2 and those cr r2. Theformer stem from the transit of scatterers through theilluminated area, while the latter are the result ofcurvature (coefficients B2 , B4, B5 ). The relative mag-nitude of the different terms depends on the incidentbeam radius and the curvature radii. As one would ex-pect, an increase of the light spot radius or a decreaseof a curvature radius enhances the curvature contribu-tion to S.

VI. ILLUMINATION WITH AN UNCOLLIMATED BEAM

If the incident beam is not collimated, the scatteringvector varies over the illuminated part of the surface.Let v0 be the scattering vector at P0 , the center of il-lumination. At all other points the scattering vector vdiffers from vo by a vector that can be expanded in


terms of the coordinates x and y. It is then easilyfound that Eq. (23) for the optical phase has to bechanged into

v . r = i (x, y) + k(x2 cos2 oCZ0 + y2)/d, + DV,

+ higher-order terms, (65)

where all higher-order terms are small and can beneglected. ds is the curvature radius of the sphericalwave front. Its value should be taken with the positivesign, if the center of curvature (focal point) falls infront of the illuminated surface, otherwise with thenegative sign. The additional terms xx2 and y2 can beabsorbed in the coefficients B2 and B4 of Eqs. (25) sothat one should use instead of B2 and B 4

B = B2 + k cos2 110 /d 8 ,

B4= B4 + k/d, .

(66)

Other changes can occur in the constant K of Eq. (60)which, however, do not affect the r- and Av-dependenceof the correlation functions G, and G2. Curvature of thesurface and curvature of the wave front can be treatedby the same formalism, at least as far as the opticalphase is affected.

VII. EXPERIMENTS AND DISCUSSION

The foregoing analysis provides a basis to comparewith experimental results the various forms of thecorrelation function G2 of the intensity, observable inthe far field of a rotating surface that is illuminated bya small laser light spot. The basic shape of the cor-relation function is Gaussian, and its decay time de-pends on the surface velocity, the radius of the incidentbeam, and the curvature of the surface as well as of theincident wave front. The details also depend on whetherone records the autocorrelation function of a single de-tector output or the cross-correlation function of twodetector outputs at different scattering angles. All de-tails are contained in the temporal function T = T1 T2 ,given by Eqs. (47) and (48).

To check the validity of these equations, the simplescattering apparatus shown in Fig. 3 has been used.Not shown in Fig. 3 is the detector (ITT-FW130 photo-multiplier with 0. 25 mm aperture), located at a dis-tance of approximately 1 m. Several aluminum bodieswith the shape of a sphere (as shown in Fig. 3), a cone,and a flat disk were carefully machined and sandeduniformly with a fine grit. A 4-mW He-Ne laser beamwas slightly expanded and then focused through lensesof large F number on the surface, resulting in spotsizes ranging from 1OX to 150X. The spot sizes weredetermined by calibrating measurements with the flatdisk (see below) and were found to be in reasonableagreement with the values derived from the diffractionformulas (ro = 1. 22XFo, where F0 denotes the F number;for a more accurate determination of the spot size,see Ref. 19).

A rotation rate of n =0. 15 revolutions per secondwas maintained uniformly by a precision electric motor(Globe). The detector output was analyzed by a Saicor43A Correlation & Probability Analyzer.


to the table top. Thus, the surface velocity u in thecoordinate system of Fig. 1 has only a y component.One then finds from Eqs. (61) to (63) and from Eqs. (53)and (25),

G2 /(K2 S2 (Q)) g2 =1+exp(- 2wS2 ) , (67)

where the mean-square speckle frequency is now

-Os U=" = 2(1/r+ B4r0)uy, (68)

with

B4=-vcos!3/2c . (69)

We also specify specular scattering with o = 20 =Oso that

v.= 27rV7/X . (70)

The following surfaces have been investigated:

1i Rotatingflat disk. Light spot on the flat surfaceat a distance R from the center of rotation:

W*=8vrn2 R 2 /r2 . (71)

-I. Rotating sphere of radius b. The rotational axisforms the angle 6 with the surface normal through thecenter of illumination:

2= 8rOn2 sin2 6(b2 /r2 + 412r2r /X2) (72)

III. Rotating cone with subtending angle 2Eo. Light

FIG. 3. Scattering apparatus showing a sphere intercepted by spot at an axial distance D from the tip:a thin laser beam. A speckle pattern is shown. The detector o2= 8en 2(X/r0 )2[(D/X)2 tan 2E0 + 27r2(ro/X)4 cos2

0] . (73)is located to the left outside the field of view. The horizontalorientation of the rotational axis can be varied. The simplest scattering case occurs with the rotat-

In an initial test the conditions were establishedwhichresult in an exponential probability density function ofthe scattered intensity. This is the proper density pro-vided the optical field has a Gaussian density (thedensity of E 2 can be calculated from the density of Eby transformation of the variable 2 0 ) . To be sure on thispoint, a parallel investigation of the statistical func-tions of the optical field was also carried out (Gaussiandistribution and Rayleigh distribution of the envelope),and the exponential density of the intensity has alwaysbeen found, when the average speckle size was muchlarger than the detector area.

The roughness pattern of the surface was checked bytaking electron micrographs of aluminum pieces withsurfaces prepared and finished by the same specifica-tions. One example is shown in Fig. 4. The rough-ness (see the scale in Fig. 4) was chosen just largeenough to result in a practically uniform speckle dis-tribution in the vicinity of the specular scattering di--rection where the measurements of C2 were carried out.A typical speckle pattern is included in Fig. 3 and mayperhaps be visible in the reproduction.

We now discuss the light-scattering experimentscarried out with the rotating flat plate, the sphere, andthe cone.

In all cases the rotational axis and the detector alsolie in the plane of incidence which in Fig. 3 is parallel FIG. 4. Electron micrograph of a sanded aluminum surface.

1200 J. Opt. Soc. Am., Vol. 66, No. I1, November 1976 J. C. Erdmann and R. I. Gellert 1200

TABLE I. Measurements with a rotating flat disk (see alsoFigs. 5 and 6). The values of the local velocity in the illumi-nated region measured directly are compared with those ob-tained from the correlation displays and the spot size.

T scale Radius R Vdiret VowelNo. of trace (ms/cm) (mm) (mm/s) (mm/s)

1 2 5 4.81 4.412 0.8 10 9.61 10.003 0.4 15 14.42 14.364 0. 2 20 19.23 19.685 0.2 25 24.03 24.616 0.2 30 28.84 28.73

on the other hand, the surface velocity is known, Eq.(68) can serve to find the light spot radius ro. It iseasy to compare the value of the local velocity mea-sured directly with the value calculated from the cor-relation display, once ro is known. The result is sum-marized in Table I. The error is of the order of a fewpercent and stems mainly from some uncertainty of thespot size and from slight distortions of the correlationdisplay.

In the case of the rotating sphere the moving speckleexhibits predominantly a rapid lateral movement whichwill be further discussed together with the results ofcross correlation. The speckle frequency defined by

FIG. 5. Sequence of autocorrelation displays (Nos. 1-6 fromtop to bottom) obtained by light scattering from the plane sur-face of a rotating disk. The width of each display depends onthe transit time of a scatterer across the illuminated area.The lag time scales for each trace and the radial distance ofthe light spot from the center of rotation can be found in TableI. The correlator has 400 lag time bins for each trace shown.

ing flat disk. Inspecting the moving speckle one findsthat it consists mainly of size and shape fluctuationscomparable to a stationary "boiling" motion. The widthof the autocorrelation function G2 in this case is entire-ly determined by the transit time of scatterers passingthrough the illuminated area. This is also found ex-perimentally. A sequence of correlation displays isreproduced in Fig. 5. All of these displays exhibit thebasic Gaussian dependence on the lag time which istypical for the function T2 [see Eqs. (47) and (62)].This is shown further in Fig. 6 where logj0T2 is plottedversus the square of the lag time in units of the correla-tor bin number. The resulting straight lines havenegative slopes from which the rms speckle frequencyw, can be found and then either the rotation rate n fromEq. (71) or the surface velocity from Eq. (68). If,

1201 J. Opt. Soc. Am., Vol. 66, No. I 1, November 1976

1.0

0.9

0.8

0.7

0.6

0.5

0.4

T12

0.3

0.2

(aINT NO.)2

FIG. 6. Semilogarithmic plot of the correlation displays 4, 5,and 6 of Fig. 5 vs the square of the lag time in units of thecorrelator bin number shows the straight-line dependence in-dicating Gaussian dependence.


16

14

12

10

15 30 45 60 75 90

8 (DEGREE)

FIG. 7. Mean speckle frequency co multiplied by the factorK = ro/X of a light-scattering rotating sphere (radius 3.75 cm)plotted vs the orientation angle 6 of the rotational axis for dif-ferent spot radii ro. X= 0. 6328 p .

Eq. (72), multiplied by ro /X = K, has been plotted in Fig.7 versus the axis orientation angle 6 for different spotsizes. The basic sinusoidal dependence on 6, expressedby Eq. (72), is well established. Of particular interest,however, is the influence of the curvature of the sur-face on the speckle frequency. It can be concludedfrom Eq. (72) that the product WK must be independentof ro if the second term on the right-hand side is insignif-icant. In fact, the plots in Fig. 7 referring to smallvalues of K tend to fall together, while considerablespread occurs for larger values of K. A closer investi-gation shows that the plotted values are well consistentwith a sphere radius b = 3. 7 cm which is the actualvalue.

The cone as a surface of varying curvature radius isparticularly interesting for the investigation of curva-ture effects on the correlation functions. Here we havebeen using a cone with subtending angle 2Eo= 30°. Fora set of values of D (axial distance of the light spot fromthe tip) the correlation function T,(T) has been recorded,and from it the speckle frequency w3 defined by Eq. (73)has been calculated. The product WSK is plotted versusD in Fig. 8 for different spot sizes. Again, for smallvalues of the spot size the curvature contribution to 2is small so that the associated plots fall closely to-gether. For larger values of the spot size, on the otherhand, the curvature contribution becomes importantand, close to the tip of the cone, becomes predominant.From Eq. (73) there follows that the curvature contri-bution is independent of D, while the transit termvanishes at the tip (some caution is required, because


the spot radius becomes of similar magnitude as thecurvature radius at the tip so that our analysis mayno longer hold). All of these special features can alsobe seen in Fig. 8.

More information on the peculiarities of the specklemotion can be obtained by looking at the cross-correla-tion function of the outputs of two detectors at two dif-ferent positions in the speckle field (Av * 0). The tem-poral part of the correlation function G2 is then Y2= TIT2 [see Eqs. (61), (47), and (48)]. For a flat sur-face and with a collimated incident beam there holdsT2 = 1, and T, is independent of Av. Thus, we do notexpect temporal cross correlation for flat surfacesand collimated beams. In fact, one observes mostlyuncorrelated "boiling" motion of the speckle. Thissituation changes as soon as the incident wave front ismade spherical or the flat disk is replaced by a curvedsurface. The uncorrelated motion then gives way to avery rapid sweep motion in the speckle field that de-serves further discussion.

Lining up the two detectors parallel to the specklesweep direction (at right angle to the axis of rotation),one can now expect strong cross correlation between thetwo outputs. The factor of importance is T2, given byEq. (48). This factor introduces a shift of the correla-tion peak on the T scale by

A= U. * U/(Q2U2 + Qyu2 + Q2 (74)

which can be either positive or negative. Let B be the

0 130

0 50

1 42

8 25

& 23

0 8

V 17

0 20 40 60 80 100 120 140

D (n-n)

FIG. 8. Mean speckle frequency w, multiplied by robA of alight-scattering rotating cone (subtending angle 2E0 = 301) plottedvs the axial distance D of the light spot from the tip for differ-ent spot radii ro.


O/

I--,"

I

allI

Z

4

3

-1.5 -1.0 -0.5

LAG TIME T IMS]

0.5

FIG. 9. Cross-correlation functions of the outputs of two de-tectors placed in the moving speckle scattered by the planesurface of a rotating disk. Illumination with anf/3 single lens.The top trace refers to the focused condition (focal point on thesurface). The remaining traces are obtained by defocusing insteps of 0.025 mm. Distance between detectors: B =6 mm.

distance between the two detectors. We can then de-fine a "speckle velocity" v, by

v, = B/ATr, (75)

although only for the regime where AT * 0.

As an example we consider a rotating flat disk and aspherical incident wave front. The curvature contribu-tion to T then depends on B' and BV given by Eqs. (66)in which we must put B2 = B4 = 0. If the incident beam isfocused directly on the surface, the wave-front curva-ture vanishes, 21 and U and -AT become zero. We arethen left with the uncorrelated boiling motion. Veryslight defocusing, however, returns the fast specklesweep immediately. All of these effects can be easilyobserved. In particular, the sense of the sweep mo-tion reverses as one passes the focal point from theoutside of the surface to its inside. In Fig. 9 we showa set of cross-correlation displays obtained in the man-ner just described, although only one sign of the sur-face distance d from the focal point is represented.A completely similar set skewed towards the oppositedirection of the lag time scale can be obtained with theopposite sign of d.

VIII. CONCLUSIONS

The salient properties of moving speckle in the far-field zone of a rotating curved surface scattering laser


ant.IV I- V I

100

125

225 _


light from a small illuminated spot can be analyzedthrough the space-time correlation function G2 of thescattered light intensity. The calculation of G2 has beenpresented under assumptions of which Gaussian surfaceroughness is the most important one. Furthermore,the incident beam diameter must be small comparedwith the radii of curvature and also sufficiently smallto ascertain almost constant velocity in the illuminatedarea. No restrictive assumptions about the reflectioncoefficient of the surface are necessary, if a statisticaldescription is sufficient. The surface has rotationalsymmetry and is approximated by a torus with twoprincipal radii of curvature. Furthermore, only thoseoptical phase differences are included which are linearand quadratic in the surface coordinates as in theFresnel approximation.

It is found that the correlation function G2 is a well-suited observable for the study of speckle effects as-sociated with rotating curved surfaces. It depends onthe radius of the incident beam, the surface velocity,and the curvature of the surface or the incident wavefront. Curvature, expressed by the inverse of the twoprincipal radii of the torus, increases the speckle fre-quency or decreases the decay time of the correlationfunction. This effect has been measured in the case ofthe sphere and the cone, and reasonable agreement ofthe experimental results with the analytical findings hasbeen obtained. Furthermore, a study of temporalcross correlation in the moving speckle field has beencarried out with two detectors, using a flat disk asscattering surface and a spherical incident wave front.Again, reasonable agreement has been found withanalytical expressions.

*Presented at the 1976 Asilomar Conference on Speckle Phe-nomena in Optics, Microwaves, and Acoustics, sponsored bythe Optical Society of America, Pacific Grove, Calif., Feb.24-26, 1976.

IV. A. Fock, "Electromagnetic Diffraction and PropagationProblems (Pergamon, Oxford, 1965).

2T. Hagfors, "Relations Between Rough Surfaces and TheirScattering Properties as Applied to Radar Astronomy, "s inRadar Astronomy, edited by J. V. Evans and T. Hagfors(McGraw-Hill, New York, 1968), p. 187.

3N. George, "Wavelength Sensitivity of Speckle from an Ex-tended Object, " 1976 Asilomar Conference on Speckle Phe-nomena in Optics, Microwaves, and Acoustics, sponsored bythe Optical Society of America, Pacific Grove, Calif., Feb.24-26, 1976.

4D. L. Fried, "Statistics of Scattering from a ModeratelyRough Surface, " 1976 Asilomar Conference on Speckle Phe-nomena in Optics, Microwaves, and Acoustics, sponsored bythe Optical Society of America, Pacific Grove, Calif., Feb-24-26, 1976.

5J. E. Pearson, Effects of Speckle in Adaptive Optical Sys-tems, 1976 Asilomar Conference on Speckle Phenomena inOptics, Microwaves, and Acoustics, sponsored by the Op-tical Society of America, Pacific Grove, Calif., Feb. 24-26,1976.

6P. Beckmann andA. Spizzichino, The Scattering of Electromag-netic Wavesfrom Rough Surfaces (MacMillan, New York, 1963).7R. J. Glauber, "Coherent and Incoherent States of the Radia-

tion Field, " Phys. Rev. 131, 2766-2788 (1963).8B. Crosignani, P. Di Porto, and M. Bertolotti, Statistical

Properties of Scattered Light (Academic, New York, 1975).

9J. W. Goodman, "Statistical Properties of Laser Speckle Pat-terns, " in Laser Speckle and Related Phenomena, edited byJ. C. Dainty (Springer, Berlin, 1975).

t°Reference 6, p. 18."Reference 6, p. 26, Eq. (1).12I. S. Sokolnikoff, TensorAnalysis (Wiley, New York, 1951).' 3M. Born and E. Wolf, Principles of Optics, 3rd ed. (Perga-

mon, New York, 1965), p. 499.14J. S. Bendat, Principles and Applications of Random NoiseTheory (Wiley, New York, 1958).

15 Reference 14, p. 129.'GE. Madelung, Die Mathenmatischen Hilfsmittel des Physikers

(Springer, Berlin, 1957), p. 80.'7Reference 6, p. 27.81Reference 6, p. 88, Eq. (59).

19R. Pitlak, "Laser Spot Size for Single Element Lenses,Electro-Optical Systems Design, Sept. 1975, p. 30-31.

20W. Mendenhall and R. L. Schaefer, Mathematical Statisticswith Applications (Duxbury, North Scituate, Mass., 1973).

2 1 Reference 13, p. 446-

Copyright © 1976 by the Optical Society of America 12041204 J. Opt. Soc. Am., Vol. 66, No. I 1, November 1976

speckle field of curved, rotating surfaces of gaussian roughness illuminated by a laser light spot

Documents