Transmission Characteristics of SpecularlyReflecting Light Pipes Uniformly Irradiatedby Obliquely Inclined RaysW. Richard Powell

The light-pipe efficiency or fraction of the incident energy transmitted from one end to the other is ex-pressed as a function of the wall reflectivity p and -geometrical parameters. Light pipes with both circu-lar and rectangular cross sections are considered. The analysis includes skew rays and is mathematical-ly exact within the limits of geometrical optics and a constant specular wall reflectivity. The distribu-tion along the light pipe of the average intensity incident upon the walls is also derived.

1. Introduction

This study is concerned primarily with the calcu-lation of the efficiency of hollow light pipes withspecularly reflecting walls of circular or rectangularcross sections. Such a light pipe is preferred over asolid dielectric light pipe if (1) no dielectric materialis available with low absorption at the wavelength ofinterest; or (2) high power levels are employed thatwould cause excessive heating of the dielectric; or (3)the angle of inclination of the rays exceeds that forwhich total internal reflection is possible.

The transmission of ir radiation through lightpipes with circular' and rectangular2 cross sectionhas been analyzed but skew rays were not included,and certain mathematical approximations of limitedvalidity were used. The present study includes theskew rays and is mathematically exact for both cir-cular and rectangular light pipe cross sections.

I1. Analysis

The wavelength of the radiation is assumed to besmall compared to the dimensions of the light pipeso that geometrical optics applies. The entranceend of the light pipe is assumed to be uniformly irra-diated by rays incident at an angle 0 with respect toend surface normal. The specular wall reflectivity pis assumed to be constant and equal to 1 - a, wherea is the absorption coefficient.

The analytical procedure to be used involves asubdivision of the entrance end surface into zonessuch that all the rays passing through a particularzone suffer exactly the same number of reflections on

the walls before leaving the light pipe. Then thetransmission efficiency of the light pipe or the frac-tion of the incident radiation passing through it is

t = ZA&'1 (1)

where Ai is the fraction of the total entrance area oc-cupied by the zone for which rays experience precise-ly i reflections. This method will first be illustratedfor the rectangular light pipe because the summationin Eq. (1) includes only four terms. An infinitesummation is required for a light pipe with a circularcross section.

A. Rectangular Light Pipe

Consider a rectangular light pipe of length L andsides of width a and b standing upright on the firstquadrant of a horizontal xy plane with one edgecoincident with the z axis as shown in Fig. 1. Con-sider an incident ray that just grazes the entrancepoint (0, 0, L) and continues on to strike the xyplane at the point (xo, yo, 0). Then

x = -L tanG sink,

y, = L tanO cosO,

(2a)

(2b)

where 0 is the angle the projection in the xy plane ofthis ray makes with the y axis. If we temporarilyimagine that the walls of the light pipe are transpar-ent and that the entrance end is opaque, a rectangu-lar shadow with corners at (xo, yo), [(xo + a), yo],[(xo + a), (o + b)], and [xo, (yo + b)] would exist inthe xy plane. This shadow can contain only onepoint (-ma, nb) with m and n both nonnegative inte-gers, and these integers are given by

The author is with the Applied Physics Laboratory, Johns Hop-kins University, Silver Spring, Maryland 20910.

Received 6 August 1973.

m = integral part of [(L a) tanG sin~k,

n = integral part of [1 + (L/a) tanG coskl.

952 APPLIED OPTICS / Vol. 13, No. 4 / April 1974

(3a)

Al' (L tan + b-n b) /b (7a)

y,- , -I- /

/ -I -X * -' / --

~'1- _/_

SAO~nb, 0)

(a, b, )

- If I _ -(-ma, 0.0 ) ( , ; a, 0, O)

Fig. 1. Rectangular light pipe geometry defining ray angles 0and 4) and the four shadow region quadrants I, II, m, and IV il-lustrated for the case m = 5 and n = 2. Dotted construction linesrepresent the intersection in the xy plane of imaginary mirror

image surfaces parallel to the walls of the light pipe.

All the rays passing through that portion of theentrance surface of the light pipe associated withthat section of the shadow labeled I in Fig. 1 are re-flected m times by the sides of width b and n timesby the sides of width a. Thus these rays are trans-ferred through the light pipe with an efficiency

El= PbmPa , (4a)

where subscripts for possibly different wall reflectivi-ties are retained to illustrate one obvious extensionto a more general case. Likewise for the regions la-beled II, III, and IV in Fig. 1 we have

El, = Pb'm+l)p n (4b)

E, = Pb(P+l)P :, (4c)

Alv' = (nb - L tanG)/b. (7b)

Thust(L, b, 0, pa) = Pan Al' + peon -1Alv', (8)

where, as before, n is given by Eq. (3b). Equation(8) can be applied twice, once for each polarizationcomponent with different reflectivities, p1 and p,,,and as observed in Ref. 2 the degree of polarizationcan be enhanced, as the transmission efficiency differsfor the two components.

B. Circular Light Pipe

Prior studies3 of the ray geometry in specularly re-flecting circular cylinders with length L and diame-ter D have shown that it is possible to subdivide theentrance end surface of the cylinder into zones suchthat all rays incident upon the end surface at angle 0from the normal and passing through the ith zonestrike the walls of the cylinder precisely i times.The fraction of the entrance surface occupied by theith zone iS4

Ai = X(i + 1) - 2X(i) + X(i - 1),

whereX(i) = (2/r)(i sin'-l i - Ass),

= (L/D) tanG,

andfj = [1 - (#i)2]/2!,

or if (f/i) > 1, then,4' = 0,

(9)

(10)

(11)

(12a)

(12b)

provided that the number of wall reflections i is notthe minimum possible number io the integral part ofthe geometrical parameter A.

For the zone with only io reflections,

Ai, = X(i0 + 1).

The fraction of the total entrance area associatedwith region I is

A = (ma + a - L tnO sinq)(L tanG coso + b -nb)/Iab,(5a)

and likewise for regions II, III, and IV we have

A11 = (L tanG sin - ma) (L tanG cosq + b - nb)/ab, (5b)

AI, = (L tnO sin - ma) (nb - L tanG cosk)/ab, (5c)

AIv = (ma + a - L tanG sink) (nb - L tanG coso)/ab. (5d)

Thus the efficiency of the rectangular light pipe is

t(L,a, b, 0, , Pa, Pb) = AE 1 + AIIE1 + A El, + AEjv.(6)

Consider the special case with no skew rays andthe incident radiation parallel to the sides of widthb. Then 0 = 0, and as a consequence m = AI = AI,,= 0.

The fraction of the energy remaining after i reflec-tions is p, and as in Eq. (1)

t = >Aipi.io

(14)

The error e associated with truncating the infinitesum in Eq. (14) so as to omit terms with more thanim reflections is less than

e < p(im +) Ai = plim+l) -IAi('m+ 1) 10 )

(15)

and both factors approach zero as m is increased.Results for various wall reflectivities are illustratedin Fig. 2.

C. Wall Flux Distribution

Even for the circular light pipe, the local intensityincident upon the walls is not uniform around thetube unless the incident flux is axially symmetric (allX equally present). However, for most rectangular

April 1974 / Vol. 13, No. 4 / APPLIED OPTICS 953

E =v Pb Pa (4d) (13)

Z

0.8

I-.4z

0 .0.2

0.0.85

0 2 4 6 8 10 12 14 16 18 2;

GEOMETRICAL PARAMETER, p = (L/D) tan 0

Fig. 2. Fraction of energy incident at angle 0 from the axistransmitted through a circular tube of length L and diameter D

vs the geometrical parameter A for various wall reflectivities.

and all circular light pipes, the. local intensity inci-dent upon the walls tends to become more uniformlydistributed around the light pipe as the radiationpasses through the tube. This effect is quite rapid..in circular tubes. The wall absorption per unitlength of tube Q can be easily calculated once thetransmission efficiency is known.

Consider the transmission efficiency as a functionof length alone, i.e., Eqs. (6) or (14) with all otherparameters held constant. At L = 0, t = 1, and as Lincreases t decreases monotonically, approachingzero for large L if p #d 1 and 0 #d 0 as illustrated inFig. 2. Let energy enter a light pipe of length Lo atthe rate IoAo cosG, where Io is the incident beam in-tensity and Ao is the entrance area. If the length isincreased by AL, the change in the fraction emergingAt will be negative, and the rate at which energy isdissipated on the light pipe is increased by AtIoAocosO. Since that section of the light pipe from theentrance to Lo is unaffected by the additional incre-ment of length, all this increased dissipation mustoccur in.the increment of length AL. Thus the wallflux absorbed per unit length is

Q = -(at/aL)Aj 0 cos, (16a)

or,Q = A, cosO Y b pi,

whereb = -(aA't/ dL). (16c)

Thus the magnitude of the slope of the curves in Fig.2 is the relative distribution of the absorbed wall fluxin circular tubes. For the circular tube, the averagelocal wall absorption per unit area is

q = Q/7rD, (17a)

andq = Q/2ab (17b)

for the light pipe with rectangular cross sections.The average intensity incident upon the wall is

I = q/c = q/(l -p). (18)

111. Summary

The procedure for accurately calculating the trans-mission efficiency and average wall intensity distri-bution for both circular and rectangular cross sectionlight pipes has been derived and illustrated for en-trance areas uniformly irradiated with radiationfrom a particular direction. These results can be in-tegrated over the full 27r sr with a weighting coeffi-cient characteristic of the incident intensity distribu-tion to obtain the transmission characteristics for abeam with finite angular size.

References1. R. C. Ohlmann, P. L. Richards, and M. Tinkham, J. Opt. Soc.

Am. 48, 531 (1958).2. T. 0. Poehler and R. Turner, Appl. Opt. 9, 971 (1970).3. S. H. Lin and E. M. Sparrow, Appl. Opt. 4, 277 (1965).4. Equations (9) through (12) are closely related to, but differ

from, Eqs. (12) through (14) of Ref. 3.

PLASTRACTS, a monthly abstracting service covering all major U.S.plastics publications is being published monthly by Syllabus,Incorporated. The reference guide covers Materials, Processing Equipment, Properties Testing, and Applications. For a freesample copy, please contact: B. Levy, Syllabus, Inc., 10-64 JacksonAvenue, Long Island City, N.Y. 11101.

954 APPLIED OPTICS / Vol. 13, No. 4 / April 1974