Reflective Optics for Obtaining Prescribed IrradiativeDistributions from Collimated Sources

J. H. McDermit and T. E. Horton

The differential equations are derived for reflective optical surfaces in systems that give prescribed spa-tial irradiation from collimated sources. The equations are derived for axisymmetric configurations as-suming specular reflective surfaces. The formulation considers general source radiant emittance distri-butions, general receiver shapes, and receiver irradiation distributions. The formulation of the differen-tial equations is effected through the vector formulation of ray-trace equations for the systems, formula-tion of differential radiant energy balance equations, and appropriate differentiation of the ray-traceequations. The differential equations are derived for two configurations.

Introduction

The laser has a variety of power applications, e.g.,medical applications, machining, spectroscopy, andphotography.L The power output of the laser can beused more efficiently and more effectively for some ofthese applications if prescribed spatial irradiativedistributions can be obtained. In some instancesthis can be accomplished through the use of refrac-tive optical systems. However, for some applica-tions, especially those employing high energy lasers,it is well known that refractive optics must be dis-carded in favor of reflective optics (due to the prob-lem of heating, break down of the refractive materi-als, etc.).

This paper treats the problem of designing an axi-symmetric reflective optical system that converts acollimated source of nonuniform intensity distribu-tion (similar to a laser source) into a prescribed spa-tial intensity distribution on a receiving surface.Both the collimated source and the receiver are as-sumed to be axisymmetric and to have prescribedradial intensity distributions. The reflective surfac-es are assumed to be perfectly specular. Their de-sign is achieved through the analytical rendering ofthe concepts of geometrical optics in conjunctionwith conservation of energy considerations to yielddifferential equations for the surfaces.

Physically, the problem of designing to achieve adesired radiative distribution has two aspects: (1)achieving the targeting of rays on the desired sur-

J. H. McDermit is with Lockheed Missiles & Space Company,Inc., Huntsville Research & Engineering Center, Huntsville, Ala-bama 35807; T. E. Horton is with the Department of MechanicalEngineering, University of Mississippi.

Received 5 March 1973.

face; (2) achieving the correct ordering of rays toachieve the desired distribution of energy. The firstaspect is achieved mathematically by the ray-traceequation. Achieving the second results in the differ-ential energy balances.

The analysis that follows consists of developingray-trace equations through application of the law ofreflection, developing differential energy balance rela-tions, and combining a differentiated form of the ray-trace equations with the differential energy balancerelations to obtain the differential equations of thereflective surfaces. The development of the ray-trace equations is expedited by a vectorial formula-tion of the law of reflection presented in Ref. 2.This formulation introduces the gradient of the re-flecting surface function as a vector normal to thesurface. With this approach the ray-trace equationsare written as a system of partial differential equa-tions in terms of the reflective surfaces.

Differentiation of the ray-trace equations yieldsexpressions relating differential changes of sourceand receiver coordinates. Numerical differentiationof the ray-trace equations has been used in the de-sign of optical systems for some time. However, inthis paper analytical differentiation of ray-traceequations, such as that presented in Refs. 3 and 4, isused. Differential energy balances are the result ofequating the energy transferred from a differential ofthe source surface to a differential of the receivingsurface. The differential energy balances are thencoupled with the differentiated ray-trace equationsand the ray-trace equations to yield the differentialequations of the reflective surfaces.

This procedure for developing the specular surfacedifferential equations may be applied to a variety ofsituations; however, for illustration, the procedure isdeveloped only for two general geometrical configu-

1444 APPLIED OPTICS / Vol. 13, No. 6 / June 1974

[(VI/IVn) + (Vn+1/IVn+1j)]-VSn = 0, (3)

(V. X V+) VS. = 0,

Vn

Fig. 1. Specular reflection.

rations. The first configuration contains a singlespecular surface, and the second configuration con-tains two reflective surfaces.

Illustrative solutions of the differential equationsare given for the single reflection case when thesource is assumed to have a cosine distribution andthe receiving surface, which is the inside of a frus-tum, is to be uniformly irradiated. The illustrationof the two reflection cases is that of a parabolicsource distribution which is to irradiate uniformly anexternal cylindrical surface.

In the following the complete development of thedifferential equations of the single reflective case isfirst given. This is followed by a more abbreviatedbut nevertheless complete development of the differ-ential equation for the two reflection case. The re-mainder is devoted to presentation of the two illus-trative example solutions of the design problem.

General Ray-Trace Formulation

The first aspect of the formulation is to obtain theray-trace expressions that satisfy the law of reflec-tion on each reflective surface. The law of reflectionrequires the angle between an incident ray and thesurface normal to be equal to the angle between thereflected ray and the surface normal and that the in-cident ray, reflected ray, and the surface normal areall coplanar. Consider the general reflection depict-ed in Fig. 1. For the nth surface in a system, thelaw of reflection requires csO = cs~n' which is sat-isfied providing

[(V/IVnl) + (Vn+1/IVn++1 )]*N, = 0.

(4)

are seen to be partial differential equations in termsof the surface functions. A set of equations of thistype for each reflective surface combination consti-tutes the ray-trace equations for the system.

All the analyses in this work are for axisymmetricconfigurations; therefore, the coplanar requirementgiven by Eq. (4) will be automatically satisfied.

The coordinates and parameters used in the fol-lowing analyses are defined either in the text or bythe figures. The subscripts used are defined as fol-lows: (1) source coordinate, parameter, or radiantemittance distribution; (2) first reflective surfacecoordinate, function, or parameter; (3) second reflec-tive surface (where applicable) coordinate, function,or parameter; (4) receiver coordinate, parameter, orirradiation distribution.

Analysis-One Reflective Surface

Ray Trace

Consider the general, axisymmetric nonuniformcollimated source and geometric arrangement givenin Fig. 2, where R2, Z 2 are general coordinates of thereflective surface, R4, Z4 are general coordinates ofthe receiver surface, and K24 is the axial displace-ment of the specular surface relative to the receiversurface as indicated. The law of reflection appliedto this configuration requires

[(V12 /1V121) + (V2 4/1V2 41)]-N2 = 0, (5)

where because of the collimated nature of rays andthe geometry chosen,

V12 /IV1 21 = k. (6)

In terms of the coordinates the vector V24 connect-ing the reflecting and receiving surfaces becomes

K4 {C}

R

14 RECEIVER

_Z

J4 {A}(1)

The law of reflection also requires that V,, Vn+L,and Nn are coplanar. This is satisfied providing

[V X V.+j]-N, = 0. (2)

The dependence of Eqs. (1) and (2) upon the func-tional form of the reflective surface is found by re-calling that the gradient of a surface function is avector normal to the surface, i.e.,

VS, = Nn.

Thus the general reflection expressions

REFLECTIVESURFACE

K2 4 10}

Fig. 2. Geometric arrangement for a collimated source-one re-flective surface. (Note: nondimensional complements of geo-

metric parameters are in braces.)

June 1974 / Vol. 13, No. 6 / APPLIED OPTICS 1445

Nn

Vn+1

Sn(Cl, (2, 3)

V 24 = j(R 4 - R2) - k(K2 4 + Z 2 - Z4)

so that the quantity required in Eq. (5) becomes

V2 4 j(R4 - R2) - k(K, + Z2 - Z4)IV241 [(R 4 - R2)

2 + (K24 + Z 2 - Z 4 )2]1/2 (7)

Letting R2 be the independent variable, the axisym-metric surface can be denoted functionally as

Z2 = f(R2 )

Thus, the surface function can be written as

S2(R2, Z2) = Z2- f(R2) = 0-

Now recalling the relationship between surface nor-mals and the gradient of the surface

N2 =-VS2,

the surface normal can be expressed as

N2 =-k + f', (8)

wheref= df/dR2.

Substituting the contributions (6), (7), and (8) into(5), one obtains the conditions for satisfying the lawof reflection [Eq. (5)].

+ j(R4- R2)- k(K24 + Z2 - Z4)k + [(R4 - R2)2 + (K24 + Z 2 Z4)2]1/2

(-k + jf') = 0,

which can be simplified to

(R4 - R2)A1 - (f')2]/2f' - (K24 + f - Z4) = 0- (9)

Equation (9) is the ray-trace equation for the config-uration shown in Fig. 2. This differential equationrelates the source, reflector, and receiver coordinatesfor any ray emitted from the source and striking thereceiver in one reflection.

Differential Ray-Trace

The above relation places constraints on the spec-ular surface function that assure that source raysstrike the receiver; however, to achieve a prescribedintensity distribution a particular ordering of raysstriking the receiver must be achieved. Therefore,consider two source rays on a common plane thatcontains the optical or Z axis and that are a radialdisplacement dRL apart. These two rays will be re-flected at positions that are separated by dR 2 anddZ2 , where dZ2 = f'dR 2 , and will be received at posi-tions that are separated by dR4 and dZ4, where dR4= R 4' dZ 4. Since the ray-trace equations relate thesurface coordinates of the source, receiver, and re-flector, the differential of this equation will relate thedifferentials of these surface coordinates. Thus, re-calling that for this collimated source case that theradial coordinates of the reflective surface and the

source are taken as identical, differentiation of theray-trace Eq. (9) with respect to R 2 yields

(R4 - R2)2f'f" + (f')2 - 1][R'4(dZ4/dR 2) - 1]= 2f'[(dZ4/dR2) - f'] - 2(K24 + f - Z4)f". (10)

Noting the similarity of certain terms in this expres-sion and those in the ray-trace equation,

K24 + f - Z4 = (R4 - R2)[ - (f)2]/2f'l,

and substituting this into Eq. (10) and rearranging,the following expression relating differential changesin ray patterns on one surface to differential changeson the other is obtained:

f" 1 jdZ4 , 1- (f/)2

f' R4 -R 2 dR2 LR4 1 + (f')2

1 + (f)2] - (11)

This differential expression, which relates the or-dering of ray patterns satisfying the ray-trace equa-tions, is termed appropriately the differential ray-trace equation.

Energy Balance

An additional constraint on the ordering of raysemitted from the source and striking the receiver isthat energy must be conserved. To satisfy the con-servation of energy constraint for a perfectly specularreflection, the energy emitted by a differential ele-ment of source and irradiating a differential elementof the receiver must be equal. Let the receiver spa-tial irradiation be prescribed by Q4(R4,Z4) and thesource radiant emittance distribution be denoted byQ,(R9. Thus for a perfect system, balancing theradiant energy emitted by a differential ring on thesource and received by a differential ring on the re-ceiver yields the following:

2rR 2dR 2Q,(R 2) = 2rR4[(dR 4)2 + (dZ 4 )2]112 Q4(R 4, Z,

which can be written

dZ 4 Q,(R 2) R 2 1

dR4 Q4(R 4,Z 4) R4 [(R'4)2 + 1]1'2

whereR'4 = dR4/dZ4.

Differential Equation of the Reflective Surface

(12)

Combining the differential energy balance of Eq.(12) with the differentiated ray trace of Eq. (11), thefollowing is obtained:

f" _ 1 | Q1(R2) R 2 1

f' R 4 -R 2 Q 4 (R 4 ,Z 4 ) R 4 [(R' + 11/2~ [~4 1-(f')2 2/121' - 1 . (13)[R4 1 + (f)2 + 1 + (f')2

Solutions to this differential equation satisfy both

1446 APPLIED OPTICS / Vol. 13, No. 6 / June 1974

the energy balance and ray-trace conditions and thusrepresent general solutions to the problem underconsideration.

Introducing the dimensionless variables

z = //J2 , s = R2/J2,r = R4/J2 , v = Z4/J2,

(14)

and nondimensionalizing the geometric parametersin the following manner,

A = J4/J2 C = K 4 /J2 P = K 24 /J2 (15)

The total radiant energy from the source is given by

nJ2

= 27r f Q(R2)R2dR2.

Thus the radiative functions can be nondimensional-ized as

Ei(s) = J22Q1(R 2)/4,

E 4(r, v) = J 22Q4(R 4, Z4 )/J?.

(16)

Upon applying Eqs. (14), (15), and (16) to Eqs. (13)and (9), the differential equation and ray-trace equa-tion can be expressed in dimensionless form as

'_/ 1_ I El(s) s 1z' r -s E4(r, v) r [(r')2 + 11/2

X[ i + )2 2 '1 (Z2- 1} (17)+ Z) 1 + ()2J 4

to that of the previous section; therefore, explana-tions will be less detailed.

Ray TraceConsider the arrangement in Fig. 3. The law of

reflection for the two reflective surfaces requires

[(V12/V 12 1) + (V23/V 231)]- N 2 = 0,

[(V/IV231) + (V34/V 341)A N 3 = 0.

(20)

(21)

The ray vectors are

V 12 = k, (22)

V23= j(R3 - R2) + k(K23 + Z 3 - Z2), (23)

V34 = j(R4 - R3) + k(K24 + Z4 - K, - Z3). (24)

Letting Z2 and Z 3 be the independent variables ofthe first and second reflective surfaces, respectively,the surfaces will be denoted functionally as

R2 = f2(Z2,R3= f 3(Z 3)

and the surface functions will be written as

S 2(R 2, Z2) = R 2 - f 2 (Z2 ) = 0,

S3(R3, Z3) = R3 - 3 (Z 3 ) = 0.

Thus the surface normals are

N2 = VS2 = -kf2',

N3 = -VS3 = -i + kf3',(25)

(26)

(r - s) [1 (Z)2] (P + z - ) 0

for no ray crossing between the reflective surface andthe receiver, the boundary conditions are

when Let thefollows:

z(O) = 0, (r,v) = (A,0)

andz'(0) = -P/A + [(P/A) 2 + 11/2. (19)

The second boundary condition is obtained from theray-trace eqution.

In order to realize a solution to Eq. (17), in addi-tion to the boundary conditions, one must have thefollowing ingredients; E1 (s), the source radiant emit-tance distribution specified as a function of the inde-pendent variable s; E 4(rv), the receiver irradiationdistribution must be specified as a function of thereceiver coordinates; and the receiver profile must bespecified by the coordinates and slope.

Analysis-Two Reflective Surfaces

The arrangement in Fig. 2 would not be desirablefor receivers that intersect the axis of symmetry;therefore, the arrangement shown in Fig. 3 with tworeflective surfaces will be analyzed, and the differen-tial equations of the surfaces will be derived.

The procedure for this development is very similar

f2' = df2/dZ2,f3'= df3/dZ3.

various groups of terms be represented as

R 23 = R 3 - R 2;

Z23 = K23 + Z 3 - Z- ;

R 3 4 = R 4 - R3;

Z34 = K2 4 + Z 4 - K 23 - Z3 -

I - K24 {P3 IK41_1K4(Cl

Fig. 3. Geometric arrangement for a collimated source-two re-flective surfaces. (Note: nondimensional complements of geo-

metric parameters are in braces.)

June 1974 / Vol. 13, No. 6 / APPLIED OPTICS 1447

(18) where

Substituting the components (22) through (26) into(20) and (21) and performing the indicated opera-tions (20) and (21) can be simplified to the followingforms:

f2' = R 2 3 /[Z 2 3 + (Z232 + R 232)1/2]; (27)

R23(Z 34 + R342)"2 + R34(Z 2 + R2 (28)

Z23(Z342 + R3 42 )1 2 + Z 3 4 (Z232 + R232)1/2-

Equations (27) and (28) are the ray-trace equationsfor this system. The first equation relates thesource, first reflective surface, and second reflectivesurface coordinates (noting that R1 = R2). The sec-ond equation relates the first reflective surface, sec-ond reflective surface, and receiver coordinates. Bycombining the two equations, an expression relatingthe terminal coordinates can be obtained. This canbe done by eliminating R2 3 and Z 23 from Eq. (28) bythe use of Eq. (27). The results are

4f2'f3'[1 - (2')2] - f2'(1- f3')Z342

+ [1 - (/2 ') 2 ]t[1 - (2/)2][1 - ()2]

+ 4/ 3'1R 342 - 2f3'[1 + (f21)2]Z34R34 = 0. (29)

The above form of the ray-trace equations, in addi-tion to showing clearly the relationship between raycoordinates, indicates an obvious condition for whichthe system of coupled differential Eq. (27) and (28)can be simplified, i.e., f2' = 1. This condition alsorequires Z2 3 = 0 for Eq. (27) to be satisfied, whichimplies K2 3 0 and Z3 -Z2 R2 RL. There-fore, only designs in which the first reflecting surfaceis a right circular cone with a half-angle of 45° willbe considered in this analysis. This assumption re-duces Eq. (29) to the form:

(K 2 4 + 4 - Z 3)R/f3) 2- 1]/2f3'1 = R 4 - f. (30)

Equation (30) relates the coordinates of the source,the two reflective surfaces, and the receiver for anyray emitted from the source.

Energy Balance

Performing a differential energy balance betweenthe source and receiver, one obtains

27RdRQ,(Rl) = 2irR4[(dR4)2 + (dZ4 )2]"2Q4(R4, Z4),

which can be written as (recalling R 1 - R2 =Z2

Z3)

dZ4 = Q,(Z 3) Z3 1dZ3 Q4 (R 4 , Z4 ) R 4 [(R4 ')2 + 1]1/2'

whereR,' = dR4/dZ4.

Differential Ray Trace

Differentiating Eq. (30) with respect to Z3

4 12f3'R4' [(f3)2 - 1]1 - [(f3)2 + 1]dZ3

+ 2f3"[(R 4 - f3) - f 3'(K24 + Z4 - Z3)] = 0. (32)

Differential Equation of the Reflective SurfaceThe ray-trace Eq. (30) can be solved for R 4 - ,

and this can be substituted into the differential Eq.(32). Upon combining this differential form of theray-trace equation with the differential energy bal-ance of Eq. (31), the following is obtained:

f " 1 J Q,(R,) Z3

/3' K 24 + Z 4 - Z3 (Q4 (R 4, Z4 ) R [(R 4')2 + 1]1"2

x f[1+ 1)2 1 + ('Y] - (2)

Nondimensionalizing the variables and parametersin the following manner,

z = Z3/J2 s = 3/J3 r = R4/J2

v = Z4/J2 A = J31J2 P= K24/J2B = J 4/J2 D = J4/J2 C = K4/J2

E,(z) = J 2Q,(R,)/'t E4(r, v) = J 22Q4(R 4, Z4)/4,

where b is the total radiant energy.Introducing Eqs. (34) into Eqs. (33) and (30), the

nondimensional form of the differential equation. andray-trace equation are, respectively,

s" 1 v El(z) z 1sI P + v -z UE(r' v) r [(rt)2 + 1]1/2

X 2As'r' + 1- (As') 2 1}1 + (As') 2 1 + (As') 2 J

F(As')2- 1

(35)

(36)

For no ray crossing between reflective surfaces andno ray crossing between the reflective surface andthe receiver, the boundary conditions are; when s(0)= A, (r, v) = (D, 0)

s() A { P + [( p ) +] "} (37)

Equation (35) is the differential equation of the re-flective surface, and solutions to this equation can berealized by use of the boundary conditions (37) forspecified source emittance and receiver irradiationdistributions and a prescribed receiver profile.

Example Results-One Reflective Surface

(3¢) The differential Eq. (17) of the reflective surfacehas been solved along with the ray-trace Eq. (18)subject to the boundary conditions for the followingvalues of the geometric parameters:

1448 APPLIED OPTICS / Vol. 13, No. 6 / June 1974.

E4 (r,v) =0.04674

A=3

(S) COS 7rS

Fig. 4. Conditions for the illustrative solution to the differentialEq. (17).

A=3 C= 1 P =2for the following receiver profile which is a frustumof a cone with a half-angle of 450

r = v + A

for the following source radiant emittance distribu-tion

g = cot-'z(1).

The illustrative example presented above was ar-bitrarily chosen to illustrate a solution of the differ-ential equation. In Ref. 5 the differential Eq. (17) issolved for the case of a Gaussian radiant emittancedistribution uniformly irradiating the interior of aright circular cylinder. The solutions are presentedfor various values of the geometric parameters.

Example Results-Two Reflective Surfaces

The differential Eq. (35) has been solved alongwith the ray-trace Eq. (36) subject to the boundaryconditions [Eq. (37)] for the following conditions.

Geometric parameters:

A = 3 B = 0.5 C = 2 P = 3,

Receiver profile (a cylinder of length 2 and radius0.5):

r = 0.5.

Source radiant emittance distribution:

El(z) - 1-Z 2.

Receiver irradiation distribution:

-2.0El(s) = cos(rs/2)

and for a uniform receiver irradiation distributionthat is the total source radiant emittance divided bythe receiver area, i.e.,

-1.5

N0-

- -10N

E4(r, v) = 2rf E,(s)sds/2rf r[(r')2 + 1]"/2dv

= 0.04674.

Note that the total radiant energy was not speci-fied. By looking at the differential Eq. (17), it canbe seen that the solution depends on the ratio of thesource radiant emittance and the receiver irradiationdistributions and is thus independent of the total ra-diant energy.

The conditions outlined above are depicted in Fig.

-0.5

0

Fig. 5. Reflective surface variation for a collimated source-onereflection.

Upon solving the differential Eq. (17) for the spec-ified conditions, it was found that the reflective sur-face was nearly a cone. Thus, rather than plottingthe actual surface, the variation of the surface from acone that passes through the end points of the sur-face was plotted. This variation (Az) can be ex-pressed as

Az(s) = z(s) - z(1)s. (38)

The reflective surface profile, as represented by Eq.(38), is plotted in Fig. 5. Before the results in Fig. 5can be fully interpreted z(1) must be known. How-ever, it is easier to visualize the situation in terms ofthe half-angle of the cone that passes through the re-flective surface end points. Figure 2 shows that thehalf-angle can be expressed as follows:

Fig. 6. Conditions for the illustrative solution to the differentialEq. (35).

June 1974 / Vol. 13, No. 6 / APPLIED OPTICS 1449

0 0.2 0.4 0.6 0.6 1.0

-0 = tan-' A[s(1) - s(0)].

-0.8

o -0.6

< -0.4

-0.2

00 0.2 0.4 0.6 0.8 1.0

Fig. 7. Reflective surface variation for a collimated source-tworeflections.

27rf El(z)zdzE4(r, v) = (C - ) = 0.25(C - v).

2 7 r f r[(r')2 + 1]"2dv

Thus a cylinder is being irradiated in a nonuni-form manner as specified by E 4(rv). (Note that0.25 would be the uniform irradiation value.) Theabove conditions are depicted in Fig. 6.

As in the case of one reflective surface, the solu-tion of the differential equation revealed that the re-flective surface was nearly a frustum of a cone.Again the results are presented as the variationfrom a frustum of a cone passing through the endpoints of the reflective surface. This variation canbe expressed as

As(z) = s(z) - s(0) - z[s(1) - s(0)1. (39)

The reflective surface profile, as represented by Eq.(39), is plotted in Fig. 7. Again, with this type ofrepresentation, s(1) or the cone frustum half-anglemust be given to specify completely the surface. Forthis configuration

As was the case in the previous section, the illus-trative example presented above was arbitrarily cho-sen to illustrate a solution of the differential equa-tion. In Ref. 5 the differential equation is solved fora Gaussian radiant emittance distribution and inRef. 6 for a parabolic radiant emittance distribution.These solutions are for the uniform irradiation of flatcircular receivers. The solutions are presented forvarious values of the geometric parameters.

Conclusions

The differential equations of specular xisymme-tric optical surfaces have been derived for arbitrarysource radiant emittance distributions, arbitrary re-ceiver irradiative distribution, and a general receiversurface. The equations are derived for two geomet-ric configurations, and illustrative solutions are pre-sented for each configuration.

The ideas presented here could also be used to de-velop the differential equations of aspherical lensesthat are to perform the same function as the reflec-tive surfaces in this paper. The formulation wouldbe identical to that presented here except the ray-trace equations would be formulated from the law ofrefraction rather than the law of reflection.

J. H. McDermit was supported by an NDEA Fel-lowship during a portion of this work.

References1. L. Levi, Applied Optics (Wiley, New York, 1968), p. 322.2. J. A. Plamondon and T. E. Horton, Int. J. Heat Mass Trans.

10, 665 (1966).3. H. P. Bhasin, "Radiant Distribution and Transmission Char-

acteristics for Specular Conical Configurations," Masters The-sis, University of Mississippi (June 1968).

4. D. P. Feder, J. Opt. Soc. Am. 58, 1494 (1968).5. J. H. McDermit, "Curved Reflective Surfaces for Obtaining

Prescribed Irradiation Distributions," Ph.D. Dissertation, Uni-versity of Mississippi (August 1972).

6. T. E. Horton and J. H. McDermit, J. Heat Trans., TransASME, C94 (4), 453 (1972).

1450 APPLIED OPTICS / Vol. 13, No. 6 / June 1974

- 10

-