JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Analysis of Irregular Reflectors

JAMES N. HANSON

Cleveland State University, Mathematics Department, Cleveland, Ohio 44115 andTRW Inc., Equipment Laboratories, Cleveland, Ohio 44117

(Received 24 August 1965)

The flux distribution resulting from the light from an emitting surface being scattered by reflection froman irregular mirror surface is formulated. The particular case of the limb-darkened sun, a parabolic mirror,and a plane focal surface are analyzed using an example of geometrical scattering that occurs for thin metaloptical surfaces possessing a show-through due to honeycomb backing. A matrix algorithm for computingthe flux distribution is developed.

INDEX HEADINGS: Reflectors; Mirrors; Scattering.

INTRODUCTION

FOR space exploration and for many new astro-nomical requirements, there is need for large

lightweight optics of varying degrees of accuracy, inparticular, lightweight metal mirrors for scientificsatellite experiments and for space solar-power systems.Because of the missions and because of the requirementof minimum weight, mirrors will exhibit imperfectionsseldom necessary to consider in the design of traditionaloptics. For example, a mirror to be used in a spacesolar-power system will probably be quite large, saythirty feet in diameter, and will require much lessoptical precision than that demanded for astronomicalinstruments. Hence, the reflecting characteristics of itssurface will show a large and complicated scattering ofthe reflected light. This scattering bears very stronglyon the solar flux reflected through a given aperture and,therefore, on the efficiency of the system. Furthermore,mirrors used in space will probably be subjected todeformations caused by thermal gradients and forcesdue to rotation. It is not the object of this paper toanalyze a mirror's performance when subjected torotation or thermal gradients, but to develop a suffi-ciently general mathematical model of the mirror opticswhich can facilitate evaluation of the optical effects ofmechanical and thermal deformations.

Only geometrical scattering of the reflected light dueto small deviations from the intended geometry of thereflecting surface are considered. This mode of scatter-ing is exhibited by many mirrors intended for space useas energy collectors whether they are to be used forpower systems or for flux-measuring systems.

Such an analysis is important in determining theefficiency of a space solar-power system or an astro-nomical light collector. This analysis permits the deter-mination of the flux distribution in the focal surface interms of an integral equation containing the emittancefunction, the reflector geometry, and the surfacedeformation function.

FORMULATION OF PROBLEM

Let ui,v1 be parametric coordinates in the emittingsurface Si; let u2,v2 be parametric coordinates on thereflecting surface S2, and let U3,V3 be parametric coordi-nates on the focal surface 83. Let qPi(ui,vi,aiIi) be theradiance of SI and let 02(u1,v1,u 2,V2,u3,v3) be the scatter-ing function for S2. a1 is the angle from the normal of SIat (ul,v1 ) of a vector, r1, originating at (ul,vl) and ,31 isthe corresponding azimuthal angle measured withrespect to directions it and v1. 44 and 02 are, in general,functions of wavelength. The energy (erg sec- sr-')leaving an area dA 1 on Si is

'P1 (ui,vi,ai,0i)dA 1,

and the energy (erg sec') intercepted by dA2 on 82 is

0I(ui,v3,aI,ffl)dA I(dA2/r2) COSta,

where 02 is the angle ri makes with the normal to S2at dA2. Therefore, the fraction of the flux deviatedfrom dA 2 in the direction of dA 3 at (U 3 ,v 3 ) on S3 is givenby

'P2 (u1 ,V1,U2,V 2,U 3,V 3)01 (uiviaiSi)dA 1 (dA 2 /r2 ) cos52

and the energy (erg sect sr-') intercepted by dA3 onS3 is

dE(u1,V1,u 2 ,v 2 ,u3 ,v 3 ) = 'P' 24A 1 (adA 2 /r2)

XCOSa 2 (dA 3/r 2) COSo 3,

where r2 is the vector from dAA2 to dAA3, and 03 is theangle r2 makes with the normal to S3 at dA 3. Thus theresulting flux, F= OE/0A 3 through dA 3 is

(82/aA 10A 2)(aE/0A 3)=44' 2 cosa2 cosa3(rlr 2) 2 . (1)

Rewriting the area elements in terms of their respectivesurface parameters and then integrating A1 over Siand A2 over S3 gives the total flux received at (u 3 ,V3 ),'

'W. Graustein, Differential Geometry (The MacmillanCompany, Inc., New York, 1931), Chap. 4.

741

VOLUME 56, NUMBER 6 JUNE 1966

JAMES N. HANSON

#/

-LIMB OF "

DISKI

TRIG. 1. Radiance (solar) matrix.

where

F(3'-3) =/ 0102 -5li5Y-2F ) / )/1 1t2 f2i(

XDdiZdvZr2- 2D2du2 dv2 COSa2 COSa3, (2)

dA I = D1 (ui,v1)du1dv8

flux on the sphere is the image of 4. This obvious resultfulfills the requirement that 4' is the pseudoradianceresulting from 01 and 02. A formula for 4i may be ob-tained by considering the incremental flux, AF, resultingfrom A£4 of the mirror surface. From Eq. (2),

tiAF= litat I 1k12 COScx2

X COSaa3 (dA i/ri 2 ) (dii 2/r22 ). (4)

In most cases of practical interest, e.g. the sun, stellarfields, planetary bodies, the aurora, etc., the opticalfield of view is sufficiently small so that L82 for any AA2is very nearly a constant. Hence, the above limit yields

AF Flim 01'026),,

A,'2-0 r272 cosa 2AA 2 1(5)

where

(6)'k=½' cosci, dw-=rr 2 cosaldil 1.

Equation (5) then gives

A~=-= todlaw St

(7)

dii 2 = D2 (12,v2)du2dV2 . (3)

FORMULATION OF THE PSEUDORADIANCE

The evaluation of Eq. (2) is the subject of this paper.This evaluation can be simplified if the scattering by S2can be mathematically separated from the geometryof S2. This separation is accomplished by the formula-tion of a pseudoemitlance. The pseudoemittance AP is anemittance from surface Si replacing 4l, such that if 4'were reflected by a perfectly smooth surface elementat S2 it would produce the same flux distribution on S3as does the actual ki when reflected and scattered bythe actual surface element with scattering function 02.Hence, the scattering by S2 is absorbed in the formula-tion of the pseudoradiance function 4'. In the case ofthe solar disk, VI is called the pseudosun. The introduc-tion of a pseudoradiance function greatly simplifies theintegrand in Eq. (2) by replacing qb102 by 4'=4'(u:,)!i.e., by a function of it, and v1 only. There is a corre-sponding change of the limits of integration, since 4' isnonzero over a larger area on SI than was 01; forexample, the limits defined by the radius of the solardisk are replaced by limits defined by the radius of thepseudosun.

It is convenient to assume that the radiant energyemanates from the conical projection of Si onto a sphereabout At of S2 and that the pseudoradiance is theresulting flux intercepted by this same sphere. If thepseudoradiance 4,, is referred back onto the projectionof Si and if 4A 2 is perfectly reflecting, then the resulting

Usually 102 is given as the distribution of flux about thespecularly reflected ray r2. If i2 does not depend on 12

then the pseudoradiance may be used throughout themirror surface provided, also, that 02 is independentof t2 and v2. In most applications this will be the case.In particular, for long-focus mirrors, the light is nearlynormally incident on the mirror surface. The radiancefunction for the solar disk is radial and, if 'k2 variesradially about the direction r2, then the radiance func-tion of the pseudosun is

12 2 r

+(P)-,f 10=0 | " TP cosO)IrdrdO, (8)

where r, R, and p are arcs as viewed from A2. R is theradius of the solar disk and r and 0 are the radial andazimuthal coordinates on the solar disk. The point(r,6) = (0,0) is the center of the solar disk. p is the radialcoordinate of the pseudosun.

MATRIX ALGORITHM FOR COMPUTING THEPSEUDORADIANCE (PSEUDOSUN)

In nearly all cases the integration in Eq. (8) cannotbe performed explicitly nor can standard methods ofnumerical integration be easily employed. Problems ofinterest are the sun, moon, planets, aurora, etc. Theseobjects usually subtend sufficiently small angles asviewed from the mirror so that the projected region onthe sphere may be regarded as planar. With this in

742 Vol. 56

ANALYSIS OF IMPRECISE REFLECTORS

mind, the matrix algorithm for the solar disk and aradial scattering function are developed.

For ease in representation, the subscripts 1 and 2 areomitted from 4) and only the prime is used for distinc-tion. Let the projected solar disk be partitioned by aCartesian grid of equal-area elements as shown inFig. 1. Let the radiance in one particular square by 4'and let a square array containing at least all the non-zero radiances be called the radiance matrix (solarmatrix). Then the squares about the image of thissquare contains amounts A# =r'sAco of the incidentflux q'. The scattering matrix, Fig. 2, is formed byscattering a beam of unit flux incident on AA 2 andrecording the fraction of the incident flux in each of thegrid squares of the projected radiance grid. The twomatrices are made to be square and dimensionally equalby appending a sufficient number of zero columns androws. If instead of unit radiance the radiance O' wereused, then the image matrix, Fig. 3, of elements q5)oAcwwould result. Hence, the scattering matrix when appliedto the radiance of every square in the radiance matrixgives an image matrix centered about each image squareof the radiance matrix. The sum of all the incrementalfluxes falling in a given square from all image matricesapproximates the total flux for points within thatsquare. The distribution of these total fluxes numeri-cally defines the pseudosun matrix. Since thepseudosun's radiance depends on p, it is only necessaryto compute the total fluxes for squares lying on a radiusof the imaged sun.

To derive inductively an algorithm for the radianceof the pseudosun, consider a 3X3 partitioning of thesolar and scattering matrices. Let A, B, and C, respec-tively, denote the solar, scattering, and image matrices,such that

whereA=(FS), B= (<ale), C= (4Jo), (9)

(~j =0ij'AW. (10)

If the A and B matrices are 3X3, then the C matrix is5 X 5 and the radiance along a diameter of the pseudosunis given, from limb to limb, by 4'13, 4'23, 433, 434, 35- Inmatrix notation Vi/3 may be represented as the productof a row vector in 4) and a column vector in O1,

{3i _(i+1-a)l3 T L¾)iE1 in terl-m)2 of m 2 ; i= 1B, 5. (b1)

{-2,i> 3- il-)1-(3

Equation (11), in terms of matrices A and B. becomes

(12)

where A is is the row vector formed by the ith row of A,AT is the transpose of A, and B is the matrix resulting

FIG. 2. Scattering matrix.

from interchanging the ith column(n+1-i) column. We now define twoand 63 such that

of B with thenew matrices a

A 33 oQ2\ 'B 3

3 032a=10 022) 0 =k023 022)

where O/ is the iXj zero matrix. Therefore, Eq. (12)becomes

a =

lpi3= f,, , i<3

63 (i+1-a)*Lata*1T. (13)

If this analysis is extended to the general case of nXn,A and B matrices, where n is odd to insure that A and Bare centrosymmetric, then a and 63 are (2n-1)X (2n- 1) matrices and the general expression for finifor the case of the solar disk and a radial scatteringfunction is

ki =E is(il-) r ha*2T; i = 1 *** 0 (14)a=i

Thus, i/i,,, is the pseudosun radiance wherei= 1 is at the limb and i=n is at the center. By thesymmetry of a and 63 and by some matrix identities2

3(i+1-)a)*-[a*"]T= 63 (i+-aI) aT]*a

= ((63 T) (ill-a) a

={ ((63 a) T] T) (i+1-a) a

= {[a6t3T]T} (i±l,,),a

= f(ta]T) (i-I-1-a)a=E (B]T)ail)

(15)

which is simply the a(i+± -a) element of the product

2 L. Mirsky, Introduction to Linear Algebra (Oxford Press,London, 1955), Chap. 3.

June 1966 743

a =�i3

Vli3= F_ _F3(,+,_.)EA .. ]T,.=J1, i<3

i -2,0; 3

JAMES N. HANSON

FIG. 3. Image matrix.

matrix a(6d. Therefore, Eq. (14) becomes

n4/'i=Zi (,63)a(i+ I..a); i= lx * *, it.

aFo

From the definitions of (l and 653,

a0_3=( 1fl_)( 1

I

(20)

0 n-1\

On_1n 1

where n is the number of nonzero ibin and tm is the radiusof the circle of scattering.

Equation (20) is exact when the partitionings of thesolar and scattering matrices exactly divide theirdiameters into an odd number of divisions, otherwise asmall error occurs that decreases with finer partitioning.

THE SCATTERING FUNCTION

The scattering function for the case of geometricalscattering is derived from the shape for the individualsurface irregularities assumed to be perfectly smooth

(16) reflecting surfaces. If a bundle of parallel rays strikesthe reflector surface perpendicularly, then the reflectedflux distribution through a distant surface determinesthe scattering function. Hence, the scattering functionmay be determined by computing what fraction of thetotal reflected flux is reflected into the r2 direction persteradian,

/AB 0 nn-1

A-n n- ) (17)

This equation shows that only the AB partition of a63need be considered in the computation of Vlin. Therefore,from Eq. (16) the expression for fi, in its simplestform is

nCin=EA (AB).(ji_.-); i= 1 ***Xa (18)

a=1

where B=3 since B is centrosymmetric. Examinationof Eq. (18) indicates that thin is the sum of the elementsof the diagonals of the AB matrix as shown.

02AW =F(U3 ,v3 )/f Irdo, (21)

where 1, is the reflected flux.The shape of the imperfect surface may be repre-

sented by a double Fourier series, however, in thepresent analysis a very simple representation is used.Only the first term of a Fourier series is used, so thatthe imperfection is a surface of revolution with thefollowing central cross section,

z= (b/2) cos(2irx/X), -(X/4)<x< (X/4), (22)

where b/2 is the amplitude or depth of the individualsurface imperfection and X/2 is the corresponding semi-

(AB),,1

'Pin = E, (A B).(2.)i

2

4b2n = E (AB)a(3a)

3

)/3n = E- (A B) .(4 a)'i

(A B) 12

(A B) 21 (A B) 22

(A B) 31 (A B)32

(A B)1 3

(A B)23 ...(A B) 33 . . .

01

U

T.

WI

1I0

2I0

l x10o

0 R 2R

FIG. 4. Pseudosun for various imiR.

T1

I I

II II I -IIl l l l

I

04I 06�w

744 Vol. 5 6

In terms of the trace of a matrix (2), tin can be writtenas

)/i.= TrEQi] (19)where C, is the matrix obtained by replacing everyelement jk of C=AB by the j(n+1-i) element, i.e.,a 900 counterclockwise rotation, of the square sub-matrix formed by the first i rows and columns of C.

Since it has been defined to be an odd integer, i maybe related to the continuous variable p by the proportion

p= (t.+R)i1(n-1),

.,

ANALYSIS OF IMPRECISE REFLECTORS

wavelength or width. This is very nearly the case of theshow-through of a honeycomb backed surface. If a rayfalling perpendicularly on a reflector of a central crosssection with a surface roughness characterized byEq. (22) is reflected onto the plane z=r2 and if thedimensions of the imperfection are small compared tothe distance to the intercepting plane, then the deflec-tion of the incident ray in this plane is

t= 2r2(dz/dx) = br2[- sin (27rx/X)].

I000

100

(23)

L-et Po be the total power (erg sec-') incident on theimperfection and let P (x) be the power intercepted in acircle of radius x about (0,0) in the (x,y) plane. There-fore, assuming the incident bundle to be homogeneous,we find that P(x) is given by the following proportion:

P (x) =[X2/ (X/4)2]P 0.

F(f/g)

10

(24)

Equations (23) and (24) when applied to the identity,

dP/dt= (dP/dx)/(dt/dx)

give the power scattered as

dP 8P, x 8 sin-1 (1/tm)=- - (25)di r2br2 cos(2rx/X) T2 tmEl-t(/tmj.)2'

where i is the radial distance in the z= r2 plane of thepoint of intersection with the reflected ray and wheretm is defined by

tm= 27rr2(b/X). (26)

The scattered flux is given by

dP dP 4 sin-l (t/lm)-=-- - Po. (27)dA 3 27rtdt 7r3 it El - (/t.)2]1

Hence, the scatter function for a sinusoidal surface is

4 sin-' (1/tm)(28)

7r3 it. El- (t/j.)2]2

From Eq. (28), it is seen that b and X always appearas the ratio b/X and, hence, the scattered flux from asurface having a roughness characterized by Eq. (22)is scattered independently of the absolute size of b andX provided they are both small in comparison to r2.

A NUMERICAL EXAMPLE

Examples of the variation of the pseudosun of Eq.(28) with tm/R are shown in Fig. (4) where the wave-length-integrated law of solar limb darkening has beenused as the radiance function.3 The integral

rtm+RfrI I '(p)pdpd6_________ J p=o J e=o

3 G. Kuiper, Editor, The Sun (University of Chicago Press,Chicago, 1953) ,p. 99.

0 .5 1.0

FIG. 5. Focal-plane flux distribution measured in relative units.

must be the same for any tm/R and be equal to the solarconstance, S= 1.33X106 (erg sec-'-cm-2). Figure (5)shows the flux distribution F=F( () in the focal planewhere t is the distance from the paraboloid axis. Theintegral

f FdA 3J 3

must be constant for all values of tmIR and y and mustbe equal to the solar flux intercepted by the mirror,S7r[2f tan (-y/2)]2 . -y is the rim angle of the paraboloid.The maximum deviation, in the focal plane of thereflected rays from the pseudosun is taken as unity ofthe t axis, where,

1-cos-y} sinp secy(max-_f 1- |-

1 + cos/cos ('+p)'(29)

The calculations for Fig. (5) have a truncation errorwhich decreases with finer partitionings of the emitter,mirror, and focal plane. For the pseudosun n= 20 issufficient to yield convergence in the first four significantfigures, while a similar accuracy for the focal-plane fluxdistribution requires that the mirror surface be parti-tioned into at least 1000 incremental equal areas. InFig. (5) the solid, dashed, dot-dashed, dotted, and solidlines, respectively, are for tm/R= 0,1,2,3,4 and the num-bers 1 through 5 indicate rim angles of 100, 200, 300,400, and 500, respectively.

745June 1966