Geometrical optics treatment of circular lightguides:general wall material

Bertram S. Frost and Philip M. Gourlay

A previously developed theory for the transmission of radiation in over moded hollow circular waveguides isextended to cover walls of arbitrary material. The extension relies on approximations to the Fresnelreflection coefficients near grazing angles of incidence for a material characterized by its complex refractiveindex.

1. Introduction

A geometrical optics theory for the propagation oflight (IR or visible) along hollow straight waveguides ofcircular cross section was presented in a previous pa-per,1 to be referenced here as FGHS. The theoryaddressed the situation where a beam of radiation,typically produced by a laser, was focused to a waistand directed into one end of the waveguide. The waistwas centered on the waveguide axis, although the beamcould be directed at a finite angle to the axis. Thetheory gave expressions for the irradiance distribution,across the guide at a distance z along it for the totaltransmission coefficient and for the transmission coef-ficient for that component of the radiation whichwould pass through a polarizer at the exit end.

The waveguide wall materials covered by the FGHStheory were either a dielectric with real refractive in-dex or what could be termed a classical metal, to bedefined more precisely. At the high frequencies forwhich the theory is intended (and imposed by thecondition that the waveguide radius a >> the free spacewavelength ), the refractive indices of many materi-als, metals in particular, fall outside these specialcases. The purpose of this paper is to extend theoriginal theory to more general types of wall material.

II. Approximation to the Reflection Coefficients

In FGHS, an essential technique in the generation ofclosed form expressions for various features of the

transmitted radiation was the use of exponential ap-proximations for the reflection coefficients. To regainthese for a wall material of general complex refractiveindex v = r- ivi, we start wth the Fresnel2 expressionsfor the amplitude reflection coefficients for electricfield, respectively, parallel or perpendicular to theplane of incidence:

= 2-CsO2 - 2 sinolv2- cos20 + 2 sing

_4p2-COS2-sin

= - -cos 2o - sino-COS2( + sing

Here 0 is the complement of the usual angle of inci-dence; i.e., is the angle between the direction ofpropagation of the incident radiation and the reflect-ing surface. The 'transverse directions t and t inwhich E is resolved are such that t, to and the direc-tion of propagation are a right-handed set for bothincident and reflected radiation.

From a consideration of the power series expansion

j2-COS20 = /2 - 1j (1 + ski 0 v2-cos2 ~ ~ ~ V -1;, Ai_ + sin24+

2 v 2 -1

it can be seen that the replacement of vV2 - cos2 by1 is valid in conditions which make

I -iJ» 1/2 sin 2

. (1)

The authors are with University of Queensland, Physics Depart-ment, St. Lucia, 4067 Queensland, Australia.

Received 3 June 1987.0003-6935/87/235112-06$02.00/0.© 1987 Optical Society of America.

If the refractive index is large in magnitude, as is thecase with metals, this condition holds for all angles .For other materials we shall demand that v be suffi-ciently large to enforce Eq. (1) over the range of small kof interest to the present application. It could beremarked that a dielectric guide with v so close to 1 as

5112 APPLIED OPTICS / Vol. 26, No. 23 / 1 December 1987

to prevent this condition would hardly find a usefulapplication. With Eq. (1) satisfied we have the follow-ing approximations for the reflection coefficients:

1-all sink h v2

alsinOal - sino whe e al

From these one finds

ll 1 - 2 Rel sinp + iI 2 sin20

laill 1 + 2 Rea1 sino + |a0-l2 sin 2

which has a minimum value

1 + co(argu 1)a1 m 1 + cos(argull)

when'k = 0km, the Brewster angle, given in this approxi-mation by

sinm =-

Also, with 6 = arga, one finds

2 Imll sin5I all 1 2 sin

2_-1

with the additional restriction that all passes from thethird to the fourth quadrant as 0 increases through 'm.When = ckmn, 11 = -

1/2r. It should be noted that

power series expansions in sino for the above I all I ortankll are not particularly useful because the Brewsterangle may lie well within the range of small 0 of inter-est.

In the case of a1 , an expansion to first order in thesmall quantity sino/a I is suffiient to cover the cases ofinterest. One finds

2 Rea, :tI 1 - | sing,

2 Imaltano 1 2 sino,

with the additional restriction that a I stays in thesecond quadrant.

The exponential approximations to the reflectioncoefficients, on which the FGHS theory depends, maynow be stated in terms of the variable t = tank asfollows. (Note that Ima is negative or zero)

Ia11 I = exp(-3 11 4)

= exp(-031 1m/ )

for t < tm'

for > t,

= -=-7 + -ll ) for < m'

M- lt - ~. -elmt/t for > ems

I (2)

} (3)

( = 1/I ,1

= 2 Reap

El = -2 Imall.

For a we use

laJ = exp(-Ol ),

a , =7r -e C,

(4)

(5)

where

2 Real

2 Imale I= 1- 12

It should be noted that the approximations for I aIII and6II for > (m cover the region just beyond the Brewsterangle and are not intended to hold when t > 1, i.e., nearnormal incidence.

The special case of the classical metal, assumed inthe original FGHS theory3 is regained by setting

all v = exp(-ir/4),q

where the quantity q, presumed <<1, is given in termsof the resistivity p of the metal by q = 0.183Jp7X. Thelimited experimental data4 that exist on the refractiveindices of metals in the IR or visible region indicatethat this expression is far from reliable. It is worth apassing remark that metals typically have I >> 1,which is sufficient to retain all v, and hence I3 0lhM2.

The above expressions also return to the special caseof a dielectric with real refractive index v, as covered inFGHS. In such cases the Brewster angle Om is typical-ly outside the range of small 'k of interest.

11. Irradiance and Transmission Coefficients

For the most part, the methods and qualitative con-clusions of the FGHS paper carry over unaltered tothis more general case, and the reader is referred to theearlier paper for definitions of quantities involved be-low. The modification consists of replacing FGHSSec. IIB. by the above section and making consequen-tial changes to expressions for metal waveguides. For''metal" one should now read "any material with aBrewster angle inside the range of 0 of interest". Forsuch materials, FGHS Eq. (38) is altered to

= -exp(- 01t ' Pw°t,(°)4,

so that FGHS, Eqs. (42) and (43) become

J= P(Om) exp(-- 13a2)4dt

+ exp(-Z 0l 13) JP(°t)(d, (6)

1 December 1987 / Vol. 26, No. 23 / APPLIED OPTICS 5113

where

= P(O,4) exp(-Z /3t2>d4. (7)( a

If zia > l/,Bflu, the second term in J11becomes negli-gible, and the upper limit of integration in the firstterm can be extended to a. Then J11and J 1 have thesame form. If z is further increased, one reaches theasymptotic region, for which only the axial (Q = 0)value of P is sensed within the integrals with respect tot. Since ,B > p3i, the asymptotic region is again de-fined in terms of the effective to of an arbitrary sourcedistribution by

z> 2

a flto (8)

= 1l 1 + 11-1-Allt) exp -llm) if Mu, > ems (16b)

T 21 (1 - exp(-11 )], (17)

where, in terms of the (tangent of) the truncation angle

1 = &,s ' -1 .f a (18)

For the remaining source distribution considered inFGHS, that of the misdirected Gaussian, the simpleexpression (11) for irradiance no longer applies, and werevert to the general expression

I(z,r,O) = Il + I, (19)

(9)

and the asymptotic transmission coefficient remainsas given by FGHS Eq. (51),

T(z)- aI1 + I )ztg flii 13

Though previous expressions for fil and Al in terms ofq have been abandoned in general, it is still typical ofmetals that All is so much larger than # 1 that the firstterm in Eq (9) is insignificant, and the discussion inFGHS Sec. II.D remains valid.

The calculations of irradiance and transmission co-efficient over the whole (or, at least, preasymptotic)range of z and for specific source distributions, asoutlined in FGHS Sec. II.E, are readily extended to themore general wall material. The transmission coeffi-cient is in all cases a sum of p = , components,

T(z) = Tll(z) + T,(z). (10)

In the special case of a source distribution independentof 0 (i.e., directed along the axis), the irradiance is givenas

I(z,r,O) 1 fl (0) T11 + f () T,W(z) Irar T (11)

For the axially directed Gaussian source, one findsT1 = 1 [1 + ( - 1) exp(-2l 1 l/t2)], (12)

T = 1 (13)1e

where

1p = 1 + /2to Z 0 a (14)

There will usually be a range of z, namely,

max 1 21 < z < 2 (15)#12 fl 1 J2 a t ;201

over which T11 0 and T 1/2, i.e., over which T staysnearly constant at 0.5.

For the truncated uniform source distribution, alsoaxially directed, one finds

Tll = 1 (1 - exp(-111))

where

(20)I, -f(°) JP(Z)

The integrals Jp, Eqs. (6) and (7) evaluate toJ1 = (2r)W(O) exp(-2t /t1)

X (ll-'I -/2 exp(-Kl2)(exp(2K 0 F0 ) + exp(-2K 0F0 ))

+ /2./ Fll exp(F1)[erf(K, - F ll) - erf(KI + Fll)]

+ F exp(F2 erf(Fl)I

+ exp(-K2)I1/2[exp(2K0F0) + exp(-2K0F0 )]

+ '/2V7rFo exp(K + F0)[erfc(K - F) - erfc(K + F)JI),

(21)

J, = (2ir)'W(O) exp(-2t /tg)

X 1- [1 + rF exp(4,) erf(F,)].

In these expressions, with p = , ,

Kp(z) = - s(to

Fp(M,) = T_ tO( cos0°),

(22)

while K0 , Fo are, respectively, K11 and Fjj for z = 0, i.e.,for ll = 1. The quantities p are those given in Eq. (14).

Through these expressions, Eq. (19) gives the irradi-ance distribution at any z. In a further calculation oftotal power in the guide,

W(z) = rdr r dOl,o Jo (23)

the integral with respect to r is trivial, but the integralwith respect to 0 would need to be done numerically ingeneral. In the special case that K11 > 1, J11 approxi-mates to a form resembling JI, and FGHS Eq. (63) isregained. This happens either for sufficiently large zor when rim > t/\I- so that the source distribution fitscomfortably inside the Brewster angle. In that casethe analytic form FGHS Eq. (64) for the transmissioncoefficient may be regained, namely, T(z), is the sum ofthe two terms

5114 APPLIED OPTICS / Vol. 26, No. 23 / 1 December 1987

if M, < Ems, (16a)

Tp(z) = exp(GP- 2t2/t2) I1 + H (a ) [1 - 1 - exp(-Gp)]}

(24)

where

2t2GpW = 2to l

HI,(0,) = -H 1(0 1) = fil(01) - f(0 1) = 2h(01 ).

Both for present and future use, it is convenient todefine the function h(O) by

h(0) = I2 0A 2- I AA2] cos20

+ I A )IA) cos(bx - by) sin20. (25)

IV. Depolarization of the Transmitted Radiation

The corresponding Sec. I.F of FGHS produced ex-pressions for the transmitted irradiance and power forthat component of the radiation which is plane polar-ized in the azimuthal plane 0 = x, i.e., such as would beobserved to pass through a polarizer placed over theguide exit with its polarization axis at angle x withrespect to the x-z plane. We shall generalize thoseexpressions, not only for the more general wall materi-al but also to account for an arbitrary (uniform) polar-ization of the input beam (as opposed to the planepolarization considered in FGHS).

The initial beam polarization is specified by com-plex amplitudes Ax, Ay of field components in the x-zand y-z planes, respectively, where I Ax 12 + IAy 12 = 1.Consider a particular n-times reflected ray, passingthrough the point Q(r,6,z) in the guide. This ray leftthe source with azimuthal coordinates 6' = 0 or 6 + ir, asgiven by FGHS Eq. (29). At that stage the field com-ponents parallel and perpendicular to the plane ofincidence had amplitudes All and AI, respectively,where

All = Ax cos0' + Ay sin0',

A =-A, sin'+ Ay cosW'.

At Q, then, the field components Ell and E 1 differ withrespect to factors Apan. The field in the x plane, i.e.,the plane-polarized component we seek, satisfies Ex AI, an cos(6 - ) - A a sin(6 - x). The contributionof this ray to the irradiance at Q due to x plane-polarized radiation is proportional to IEXl 2, where

IExI2 I JAIII IaiI2n cos2(0 - X) +|A,'la 'I2n sin 2(0 - x)

-2 Re[AlAlaani.xn sin(O - x) cos(0 - x).

The quantities IAIII2, IA 112, and AIIA±* occurring hereturn out to be independent of the choice between thetwo possible values of 6'.

Specifically, on writing A., = I Aj exp(ib,,) and Ay I Aj exp(iby), we find

IAI2= f 1(0) = IA)12 cos

20 + IA)

2sin20

+ 21AjI IA) cos(6b - by) sinG cosO, (26)

IA±12 fl(0) - IA . 2 sir 2 0 + IAy 2 cos20

- 21A I AA cos(bx - by) sin0 cosO,

A IAI fa(0) = IA.1 I A) cos(x - by)(cos20 - sin20)

- (|AXI 2- A)') sinG coso

+iIAjI IAA sin(x -by).

(27)

(28)

At this stage we take note of the approximations tothe complex reflection coefficients suggested in Sec. II.Then, on collecting the dependence on the astigmaticray geometry and the intensity distribution of thesource and converting the sum over contributions ofvarious rays to an integration with respect to A, as inFGHS, we find the irradiance of the x plane-polarizedcomponent of radiation to be

I,(z,r,0) =- J 1l(0,z)fII(0) cos(0 - x)

+ J1(O,z)f(0) sin2(0 - X)

-2 Re(Ja(0,z)fa(O)) sin(O - x) cos(o - x)I. (29)

Here Jll and JI are given by Eqs. (6) and (7), while

= d4P(0,) exp[-! (1311 + f 1 - Ell + E±

+ ex{-/2 - + - EII{)tm)}

X J td#P(0,() exl{-'/ 2 i a (_ + Eiit)t]

(30)

When the source distribution is independent of 0,the integrals with respect to r and over the guide crosssection may be executed to yield a transmission coeffi-cient (i.e., fraction of the total source power) for xplane-polarized radiation:

TX(Z) = /211 + (1 + T h(x) (31)

where T is given by Eq. (10) and either Eqs. (12)-(14)or Eqs. (16)-(18), the function h is defined by Eq. (25),and where Ta results from the evaluation of Ja asfollows. For the axially directed Gaussian source,

Ta(z) = Re{1- [1 - exp(-2 /t )I

+ 1 exp(-L + Q2) [exp(-(Ka + Q)2

)2S oa

- CFQ erfc(Ka + Q)]), (32)

where

1 December 1987 / Vol. 26, No. 23 / APPLIED OPTICS 5115

I/ Z (fl, _ iE')�2X exi- 2 a I

a/ (11 + - Ell +E)

L = 1/2 [f32 +i(2- llm){m], (3

to

izto(7r + M

4a 21a

For the uniform source distribution truncated at ,

Ta(z) = Re{ ' [1 - exp(-la)]}I21a I

if Mu <em, or

Re [1 - exp(-1a.,/")]1

+ , exp(-L + Q2) [exp(-(K + Q)2)

- exp(-(J/j + Q)2)

+ 7Q erf(K + Q) - EFQ erf(gla + Q)]}, (3t

if a> 'em, where on this occasion

la .z(+ iI+2±)

+ -(Fa-Q) erfc(Ka - Fa + Q)j

+ /2 exp[(Ya + Q)2 ]exp[-(ga + Fa + Q)2]

- JW(Fa + Q) erfc(Ka + F + Q) )

where, in addition to Eq. (33), the quantities involvedare

Fa tcOs(O

3)

1' = /g2 Z ( i

while K, Fa are similarly defined in terms of la, and K0,F0 are gained by setting la = 1 (i.e., z = 0). With theseexpressions for Jp (p = I, I, a), Eq. (29) gives theirradiance over the whole range of z for which thegeometrical optics theory is valid [see FGHS Eq. (35)].However, a further integration over the waveguidecross section, to get the total transmitted power of xplane-polarized radiation, would require a numericalintegration with respect to in general. The exceptionto this occurs when IK0 > 1, so that the second term inEq. (30) becomes negligible and the upper limit ofintegration in the first term may be extended to .The physical conditions to be satisfied here are eitherthat the source distribution fits comfortably inside theBrewster angle or that z is sufficiently large to ensureelimination of Ell radiation traveling in directions out-side the Brewster angle, namely, za > 2/fll2 in theusual case (for metals) that ll is >>Al. The conditionI Kai > 1 is slightly stronger than, hence guarantees, K11> 1, which was used in deriving Eq. (24). The tech-nique used there was to integrate with respect to before performing the (specialized) integrals with re-spect to , and if this is done for Eq. (29), one findsTx(z) = - exp(-2t'2/tO)

(ll - exp(GjI)[Hj(X) + H2 (0,,X)F2 (Gll) + H(01,X)F3(G1 )]

+ 11' exp(G,)[Hj(x) - H2(01,x)F2(G±) + H3(01,x)F3(G,

a [1llim + 2 Elltm)tmJ, (35)

K = rIam

iztu(r + ElI2)

4aFor the misdirected Gaussian source, Eq. (31) is no

longer appropriate, and we return to Eq. (29) for theirradiance. Jll and J are given by Eqs. (21) and (22),and further use of the techniques used there gives

Ja = (27r)Y'W(O) exp(-2t /t1)

{la-' [1 - 1/2 exp(-K2) [exp(2KOFO) + exp(-2KoF0)]

+ /2 JFa exp(Fa)[erf(Ka - Fa) - erf(Ka + Fa)]

+ CrFa exp(Fa) erf(Fa)]

+ I'-' exp(-L)

X (/2 exp[(Fa -Q) 2]Iexp[-(K - + Q)2]

-2 ReI1-' exp(Ga)[-h(x) + 2ijAXA) sin(3, - by)

X sin(20, - 2X)F2 (Ga)

+ H3(0j,x)F3(Ga)II),

where h is defined by Eq. (25), andH = 1 + h(x),

H2 = cos(20, -2X) + 2h(01),

H3 = h(20 -X),

F2(G) = 1 1 - exp(-G)

GF,(G) 1 2[2 + exp(-G)] 6[1 - exp(-G)]

GG2

t2G,(Z) 2

t2l(p = 11, , a).

This extends (and corrects5) FGHS Eq. (73) and thelast paragraph of FGHS Sec. II.F to take account ofboth the general wall material and an arbitrary sourcepolarization.

5116 APPLIED OPTICS / Vol. 26, No. 23 / 1 December 1987

I = + 1 t2 Za - (O , - if , ),4 0 a

V. Concluding Remarks

The fact that the refractive indices of metals differsignificantly from the classical expression at IR andvisible frequencies is one that needs to be taken intoaccount in applications of the FGHS theory. Thepresent extension not only allows this but also coversthose dielectrics which have significant absorption.For the purpose of designing lightguides, it would seemthat materials would be better classified as a smallBrewster angle (i.e., within the intensity distributionof a typical input beam) or as a large Brewster angle (inwhich case expressions for I, T, and T, are somewhatsimplified), and a distinction between metal and di-electric is irrelevant. The difficulty we reported earli-er,1 of reconciling the theory with the experimentalresults of Ohlmann et al., can now be attributed to awrong presumption about the refractive indices of themetals involved.

The work of J. P. Crenn warrants highlighting as thetheoretical approach which bears the closest relation-ship to our treatment. The differences betweenCrenn's approach6 and ours with respect to dielectricguides was discussed in FGHS. Crenn has extendedhis work to metallic guides,7 but the essential differ-ence remains, that his effective averaging of the I andI reflection coefficients forestalls any consideration

of polarization effects. Comparison of either theorywith experiment in the case of metals is hinderedsomewhat by a lack of data on complex refractiveindices.References

1. B. S. Frost, P. M. Gourlay, N. R. Heckenberg, and S. T. Shanahan"Geometrical Optics Treatment of Circular Lightguides," Appl.Opt. 24, 4414 (1985).

2. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, NewYork, 1941).

3. Expressions for 61 given at the beginning of Sec. II.B of Ref. 1contain an error in sign. There is also a typographical error (/) in611 for a metal. Fortunately these errors do not affect any finalexpressions given in Ref. 1.

4. H. E. Bennett and J. M. Bennett "Validity of the Drude Theoryfor Silver, Gold and Aluminum in the Infrared," in Proceedings,International Colloqium on Optical Properties and ElectronicStructure of Metals and Alloys, Paris, 1965 (North-Holland,Amsterdam, 1966), pp. 175-88.

5. The correction to FGHS Eq. (73) consists of replacing 4 by 2 in thefourth line and replacing 1/4 by 1/2 in the definition of H whichfollowed.

6. J. P. Crenn, "Optical Theory of Gaussian Beam TransmissionThrough a Hollow Circular Waveguide," Appl. Opt. 21, 4533(1982).

7. J. P. Crenn, "Gaussian Beam Transmission through CircularWaveguide with Conducting Wall Material," Appl. Opt. 24, 3648(1985).

Meetings Calendar continued from page 5103

1988May

31-2 June OSA Photon Correlation Techniques & Applica-tions Top. Mtg., Wash., DC OSA Mtgs. Dept., 1816Jefferson Pl., N.W., Wash., DC 20036

20-23 10th Symp. on Thermophysical Properties, Gaithers-burg A. Cezairliyan, Rm. 124 Hazards Bldg., NBS,Gaithersburg, MD 20899

20-24 14th Int. Laser Radar Conf., San Candido L. Stefan-utti, IROE, CNR, Via Panciatichi, 64-50127 Firenze,Italy

24-29 5th Conf. of the Australian Optical Soc., Sydney AOSDept. of Theoretical Phys., U. Sydney, 2006, Austra-

June lia

4-8 Medical China '88, Hangzhou G. Kallman, KallmanAssocs., 5 Maple Court, Ridgewood, NJ 07450

5-9 1988 Computer Vision & Pattern Recognition Mtg., AnnArbor IEEE, E. 42nd St., New York, NY 10017

6-10 Infrared Technology & Applications course, LondonSira Ltd., South Hill, Chislehurst, Kent BR7 5EH,England

7-9 Optical Storage for Large Systems Mtg., New YorkTOC, P.O. Box 14817, San Francisco, CA 94114

14-19 China Optoelectronics Laser '88, Beijing CahnersExpo. Group, P.O. Box 70007, Wash., DC 20088

15-17 Spatial Light Modulators & Applications Mtg., LakeTahoe OSA Mtgs. Dept., 1816 Jefferson Pl., N.W.,Wash., DC 20036

17-5 July 1988 Summer Inst., History & Philosophy of Sci., BerlinW. Woodward, Dept. of Psych., Conant Hall, U. NewHampshire, Durham, NH 03824

26-29 Photochemistry for Imaging Symp., MinneapolisSPSE, 7003 Kilworth Lane, Springfield, VA 22151

26-30 Engineering & Industrial Sensing for Advanced Manu-facturing Technologies Mtg., Dearborn SPIE, P.O.Box 10, Bellingham, WA 98227

July

11-15 History of Science Soc. & the British Soc. for the Historyof Science Joint Mtg., Manchester AIP, 335 E. 45thSt., New York, NY 10017

12-15 Ultrafast Phenomena Top. Mtg., Mt. Hei SecretariatUltrafast Phenomena, Opto, Ltd., 5-206 Maeno-choHeights, 6-10 Maeno-cho, Itabashiku, Tokyo 174, Ja-pan

18-22 Int. Quantum Electronics Conf., Tokyo IQEC '88 Sec-retariat, OITDA, 20th Mori Bldg. 7-4, Nishi-Shimba-shi 2-chome, Minato-ku, Tokyo 105, Japan

continued on page 5129

1 December 1987 / Vol. 26, No. 23 / APPLIED OPTICS 5117