Polarization and losses of whispering-gallery waves alongtwisted trajectories

M. E. MarhicDepartment of Electrical Engineering and Computer Science, Northwestern University, Evanston, Illinois 60201

(Received 10 October 1978)

A geometrical optics treatment shows that whispering-gallery waves following twisted trajectoriescan be in eigenstates of polarization if oa = (radius of curvature X torsion) is a constant. The Jonesvectors of these eigenpolarizations are calculated, along with their propagation constants and powerattenuation constants. The curve representing the evolution of an arbitrary input polarization makesa constant angle with the circles passing through the points corresponding to the eigenpolarizations,on the Poincar6 sphere or in any one of the equivalent complex-plane representations. Experimentscarried out with a cylindrical glass tube give results in good agreement with theoretical conclusions.The theory predicts that the smallest attenuation constant for three-dimensional trajectories ininfrared and far-infrared guides should be about (1 + cr2 times larger than that for TE waves alongtwo-dimensional trajectories of the same curvature, providing a practical criterion for the design ofguides exhibiting twisted trajectories.

1. INTRODUCTION

Low-loss propagation of infrared light in metallic wave-guides operating on the whispering-gallery principles 2 reliesupon the fact that the attenuation of TE waves at grazingincidence is small. In the guides of Refs. 1 and 2 the normalto the guiding surface is always in a fixed plane, or nearly so,so that if a TE wave is launched in that plane at the input ofthe guide, the electric field remains parallel to the surface ofthe guide at any subsequent location. Consequently, thelosses along these two-dimensional trajectories are easy tocalculate, and, for the preferred TE excitation, depend onlyupon the limiting form of the TE losses per reflection neargrazing incidence.1' 2

With the three-dimensional, or twisted, ray trajectoriesproposed recently,3 however, the situation is more complicatedas one cannot a priori expect the polarization state to remainthe same at all locations along the ray, and even if polarizationeigenstates do exist, they are unlikely to be either TE or TM.The reason for this complexity is that as the rays bounce atgrazing incidence along the interface, they experience a con-tinuous rotation of the plane of incidence due to the torsionof the three-dimensional curve, as well as different losses forthe TE and TM components, both factors contributing to theprogressive modification of the polarization along the raytrajectory.

In this paper we study the conditions necessary for the ex-istence of polarization eigenstates for whispering-gallery wavesalong twisted trajectories and calculate the propagationconstants for these eigenstates. We then study the evolutionof an arbitrary polarization state. Finally, we specialize theanalysis to a number of particular cases and describe experi-ments designed to test the theoretical predictions.

II. POLARIZATION EIGENSTATES

Consider a light ray Gos guided by reflection along the in-terface z between two media of respective indices of refractionn1 and n2, as shown in Fig. 1. As the angle of incidence ismade to approach 90°, i.e., as the angle 4 is made to approachzero, the distance between consecutive reflection points suchas Pm and Pmni vanishes and the jagged ray trajectory Go

becomes the smooth curve Go, which is a geodesic of 2. Inwhat follows, 4 should be understood to be a small quantity,even though it is exaggerated in the figures for the sake ofclarity, so that all statements made and equations written areaccurate to a high degree of accuracy and become exact for4 = 0.

At any point P along Go one defines three mutually or-thogonal unit vectors I, j, and 6 (tangent, principal normral,and binormal, respectively), forming what is called the movingtrihedron. A property of geodesics is that their principalnormal coincides with the normal to the surface at any pointP. or that their rectifying plane (spanned by F and 6) coincideswith the tangent plane to the surface at P. Light propagationalong Go can thus be viewed as a series of reflections by theplanes tangent to z at Pm, Pm+i, etc., or by the rectifyingplanes of Go at the same points.

To study the propagation of the light ray from a point justbefore Pm to a point just before Pm+i, we can examine pro-jections of the three-dimensional geometry onto planes of thetrihedron at the point P located half way between Pm andPm+ 1 on Go. Figures 2(a) and 2(b) show the orthogonal pro-jections onto the normal plane (spanned by f and 6) and theosculating plane (spanned by F and j); for the sake of sim-plicity, the same names are used for the projections as for theoriginal entities.

The plane of incidence of light ray Go at Pm is spanned byP-5m and PmP1 ,,+t, while at Pm+i it is spanned by iPm+i andPmPm+i. Hence, the rotation of the plane of incidence ex-perienced by Go, between the two reflections is measured bythe angle e of the projections of Pm and m+ I onto the planeperpendicular to PmPm+,, or the normal plane of Co at P [Fig.2(a)]. It can be shown that

e = T ds, (1)

where r is the torsion of Go at P, and ds is the element oflength along G0 between Pm and Pm+ (Tr, and therefore E, ispositive where the trajectory is right handed). From Fig. 2b,we have

ds = 2po, (2)

1218 J. Opt. Soc. Am., Vol. 69, No. 9, September 1979 0030-3941/79/091218-09$00.50 © 1979 Optical Society of America 1218

where the eigenvalue r is a complex number to be determined.Equation (6) means that both TE and TM components arealtered by exactly the same amount, so that the ellipse de-scribing the polarization is left unchanged in shape from onereflection to the next. Equation (6) can be rewritten as

I rIIcose - r rIsinE l(E ' = (0l\-rll sine rI cose- r)E , E \0/

(7)

n 2

FIG. 1. Geometry for the study of whispering-gallery waves; , 5, t) isthemoving trihedron at P.

wherep = !PCI = IPmCI = IPm+ICI istheradiusofcurvatureof Go at P.

We assume that the electromagnetic field is monochromaticso that it is in a well-defined state of elliptical polarization ateach point in space. For such a field the time dependence canbe dispensed with and one need only study complex phasorsof projections of the field along arbitrary axes in order to fullycharacterize the polarization. A natural approach here is touse only the two components Ell and E 1 which are, respec-tively, obtained by projecting the electric field phasor E ontothe plane of incidence and its normal at the reflection pointtowards which the light is propagating. Thus the field beforePm, measured in this standard fashion, is given by the Jonesvector,

E = ~E(3)(E

The sign convention is such that the components of the elec-tric field are positive if they point in the direction of i and 6,respectively.

Upon reflection at Pm, the TM and TE componehts aremodified by their respective amplitude reflection coefficientsrll and r1 . Measuring this reflected field with respect to theplane of incidence at Pm, we obtain the intermediate expres-sion

Em = (no )E (4)

(where the now unnecessary subscript m has been dropped),which can only be satisfied if r is a solution of the character-istic equation,

rw1 cose

-rII sine

-r rI sine

r I cose - r= 0, (8)

the two roots of which are given by

rd. = 2-1 cosE Jr, + riq + [(rj - rlI) 2 - 4r rl! tan2k]112j. (9)

Since we are dealing with nearly grazing incidence, the ex-pressions for r 1 and rnj can be expanded to first order in 4,yielding'

r~l = 1 - 2,n(-n 2 -)-1/2

and

r, = 1 - 20(n 2 -)-/2

where n = n 2/nl. Consequently,

r I + rll = 2 - 20(n2 + 1)(n 2 -)-1/2

and

rI- rj = 24(n 2-1)1/2.

Go

(10)

(11)

(12)

(13)

Ic

(a)

A

Im.i

To obtain a vector comparable to Em, however, we mustmeasure the field with respect to the plane of incidence atPm+1 towards which the light is propagating. The plane ofincidence at Pm+ 1 being obtained from that at Pm by a rota-tion of angle e, we have the proper expression for the fieldarriving at Pm+, 5

Em+i ( cose sinE\ ( rjj cose r 1 sineEl

\-sinE cos5E ~ -rj1 sie r1 cos;

The resulting matrix governs the evolution of the polar-ization and the amplitude of the field from one reflection tothe next, and can be used to study them along the entire lightpath by successive multiplications, and the method couldconceivably be applied to trajectories of arbitrary shapes.From now on, however, we specialize the analysis to paths forwhich polarization eigenstates can be found, as this rendersthe calculations fairly simple. We say that the field is in apolarization eigenstate in the neighborhood of P if

Em+t = rEm, (6)

1219 J. Opt. Soc. Am., Vol. 69, No. 9, September 1979

P t

m t(b)

Ab

FIG. 2. (a) Projection of Fig. 1 onto the (j5, i) plane. (b) Projection of Fig.1 onto the (j, b) plane. The angle e is taken to be positive if A rotatesclockwise when P goes from Pm to Pnr,1.

M. E. Marhic 1219

G. j

Also to first order in 0, we have

E = Tds = 2TpO = 2rqo,

where we have introduced the important quantity

0r = Tp.

We also have

sine tane c 2cro,

and

speak about eigenpolarizations and to study the propagationof an arbitrary input polarization by decomposing it upon the

(14) eigenpolarizations. In the rest of the paper we assume thatwe are in this situation, and to further simplify the discussion,we also assume that a >> 0 (right-handed trajectories); minor

(15) modifications of what follows are required for left-handedhelixes. Under those circumstances, we can now study howthe amplitude and phase of the eigenvectors Es vary with sin the limit of grazing incidence. For small 0 we can rewriter± as

(22)cose 1. r4 =1+ i13d ds,

Substituting all these expressions into Eq. (9), we find that

r+ = 1-0[(n2 + 1)(n2 - 1)-1/2 i (n2 - 1 -4a2)12

(16)

The eigenpolarizations corresponding to these values of rare then determined from either one of the two equationsobtained from Eq. (7). The Jones vectors are determined onlywithin arbitrary multiplicative constants, and can take on anumber of equivalent forms, such as

{EIIj r I sinel=E (rd - r11 cosE

or (17)

(El) = (ri cosE -r,)

~E, r 11sine

where one must substitute the values of rd, given by Eq. (16).The Jones eigenvectors so obtained are

{E~ll1 2of[l - 24(n2 - )-1/2] 1

kE, 4= \(n2

- 1)1/2 F (n2- 1 - 4a 2)1/2

(18a)

(EII = 2(n2 - 1)1/2 i (n2 - I - 4a2)1/2 1(El6 \ 2af[l- 2¢n2(0- l)-1/] J

As we let 0 approach zero, we see that the eigenpolarizationsreach well defined limits, functions only of cr and n. Thus,for curves with a constant a,6 the polarization state of thewhispering-gallery waves, referred to the moving trihedron,can remain constant at any point along the trajectory, pro-vided it is initially in either one of the polarization eigenstates.It is convenient to write the Jones vectors of the latter in eitherone of the equivalent forms

where, from Eqs. (2) and (16), the propagation constant d,is given by

13d = i(2p)Y1'(n 2 + 1)(n 2 - 1)-1/2 i (n2 - 1 - 4a2)1/21.(23)

Deleting the now unnecessary subscript m, we can also re-write Eq. (6) as

E+ + dE± = r±E± = (1 + i/ds)E±,

which can be integrated, with the result

Es = E+o exp (i J' ir ds),

(24)

(25)

where E+o is the initial value of Ed at s = 0. Letting 3 + = Fir+ ifl., we see that i+, and 13

dr govern the evolution of theamplitudes and phases, respectively, of the eigenpolariza-tions.

When a single eigenpolarization is excited, the phase termis of no concern, and the power carried by the wave variesas

Pa = P-o exp (-2 f ds)

= P4, exp (- a ds. (26)

where P+O is the initial value of the power, and where we haveintroduced the power attenuation constant

= p- Re[(n2 + 1)(n 2 - 1)-1/2 i (n 2 - 1 - 4U2 )1 /2 ] (27)

III. EVOLUTION OF AN ARBITRARYPOLARIZATION STATE

where

(E 1, ) (A IF B)

A = (n2-1)12/2a

and

B = (n2-1 -4 4a2

)1/2/2a = (A

2- 1)1/2.

In Sec. II we have calculated the power attenuation con-stants for the eigenpolarizations, but have not analyzed what

(19) happens in the case of an arbitrary input polarization. Ofparticular importance is the question of how fast such a po-larization is going to decay into the least attenuated eigenstate,because the answer will determine the minimum length ofguide required to reach steady-state propagation with thateigenpolarization. Also of interest is the study of what hap-

(20) pens when there is no least-attenuated eigenstate (a+ = a-),in which case an arbitrary input polarization keeps repeatingitself periodically and no steady-state propagation is ever

(21) reached.

Equations (19) show that the ellipses representing the ei-genpolarizations are symmetric with respect to P + 6 (or P -6). Clearly, it is only when ai = const that it is conceivable to

1220 J. Opt. Soc. Am., Vol. 69, No. 9, September 1979

When the input polarization is not an eigenstate, one candecompose it upon the two eigenpolarizations, i.e., find thetwo eigenstate phasors E+o and E- 0 such that

M. E. Marhic 1220

(28) which leads to

By the linearity of Maxwell's equations, this linear combi-nation remains the same at all subsequent values of s, sothat

E(s) = E+0 exp (i 0 f3+ ds) + E-0 exp (i g f-ds).

(29)

Equation (29) enables one to study the evolution of an ar-bitrary input field, in terms of both the power carried and thestate of polarization. The variation of the power carried bythe wave, proportional to E.E*, can be calculated from Eq.(29). It has, in general, a complicated expression since itcontains an interference term coming from the beat betweenthe eigenpolarizations, unless the latter happen to be or-thogonal, i.e., satisfy the relation

E+ -EL = 0.

dw =s.

w2-2Aw + 1 (35)

The left-hand side of Eq. (35) can be integrated since, asassumed earlier, a is not a function of s; the right-hand sidecan be integrated even if r is not a constant. Denoting theinitial value of w at s = 0 by wo,we obtain,

S W dw Ad0 w2Aw = 1 JoTrds = T(s)

wo W- 2Aw + 1 = J(36)

where T(s) is the integrated torsion of the three-dimensionaltrajectory. Letting

w- 2Aw + 1= (w - w+)(w -w-,

where

(30) wi= A ± B,

Rather than studying the power carried by the wave for anarbitrary input polarization, we will concentrate on theanalysis of the evolution of the state of polarization along theguide, which will already give us important information aboutthe situation. In doing so we will learn much that will be ofuse in the selection of an experimental situation adequate fora convincing verification of the theory as a whole. The stateof polarization can, for example, be conveniently studied bymeans of the single complex function

w(s) = E 1 .E, (31)

The expression for w could be obtained from Eq. (29), but weprefer to derive it in a different way, which will directly leadto interesting forms. We obtain the differential equation forw(s) by taking the ratio of the two equations derived from Eq.(5), namely,

wm+i = EBl i) _ r11 cose (E1/Ew)m + rj sinem Eljm+1 -ri sine (E1/E)m + r 1 cose

= r cose Wm + r1 sine-r 11 sine wm + r 1 cosE

and expandingnotation we let

and

where

(32)

both sides to first order. To simplify the

ril = 1 -p -1p'g ds

r = 1- p-lpIds,

ktti = n 2 (n 2 -)-1/2

AI = (n 2- )-1/2.

and

(33)

(34)

Letting wm+ 1 = wm + dw, and deleting the now unnecessarysubscript m, we obtain

w + dw =(1-p-1A11dsw+Tds-rwds+ 1-p-1,' ds

(w -p- pllwds + rds)(1 + rwds + pt lMds)

- w - p-tLtiwds + rds + TW2 ds + p-'M±wds,

1221 J. Opt. Soc. Am., Vol. 69, No. 9, September 1979

-i 0

-i

(37)

(38)

we obtain

W- W+ = W-W+ e(w+-w)T(s) = WO W+ e 2BT(s)

w-w- wo-w- wO-w-

(39)

An alternate form can be obtained by solving for w, with theresult

w = A - Bcoth BT(s)-coth-'[(wo-A)/B]$. (40)

The evolution of the polarization as a function of s can thenbe conveniently studied with the help of either Eq. (39) or (40),and visualized in the complex w plane, on the Poincar6 sphere,or on a stereographic projection of the latter onto any one ofits tangent planes.7 In the w representation, it can readily beshown that the locus of w for given w+, w-, and w0 is a portionof a curve making a constant angle 8 = (argument of B) withthe arcs of circle going from w+ to w- (see Appendix A).These curves may take on a variety of shapes depending upon6, and a few representative examples are shown in Fig. 3.From Eq. (37) we see that

w+w- = 1, (41)

Im (w)

- 72 <8<02T<<8-.

Re (w)

FIG. 3. w-plane representation of the possible loci of w(s) as a functionof a.

M. E. Marhic 1221

E(O) = Eio,+ E-0.

so that wv- must lie inside the unit circle and w+ outside (see

Appendix B), or both must lie on the unit circle. If wo =w,

then w = w d for all s, as one would expect from the definitionof the eigenpolarizations. However, if wo /- w e, w generallyspirals away from one of the eigenpolarizations (which maybe said to be unstable) and toward the other one (stable), nomatter what wo is; since --7r/2 < 6 < 7r/2 (see Appendix B), w -is stable. For 6 = 0 the curve reduces to an arc of circle joiningw+ to w-; for 6 = i7r/2, both eigenpolarizations have the sameattenuation constant and, as a result, w never reaches eitherw+ or w-, but keeps traveling at a constant rate around thecircle of Apollonius such that8

Iw-w+I/Iw-W-I = Iwo-W+j/IWO-W-I.

A quantity of interest is the scale length St over which aninitial polarization w0 approaches the stable eigenpolarization.Assuming for simplicity that r is a constant, we see from Eq.(39) that st must be such that the magnitude of the expo-nential term varies by a substantial amount, and we thus de-

fine St by the condition I Re(2Brst) I = 1. More explicitly,

St = pI Re(n 2 - 1 - 4U2 )1/21 -1 (42)

If wo is close to the stable eigenpolarization, w will reach thelatter in a fraction of st, while if wo is near the unstable one,

it will take many st to reach steady state9 ; for wo in an inter-mediate position, the transient regime will extend over adistance of order st.

Finally, let us see under what circumstances the eigenpo-larizations can be orthogonal. In terms of the Jones vectors,orthogonality is described by Eq. (30), which is equivalentto

w+w* + 1 = 0. (43)

-1 (w')

Im (w)

Re (w)

(w'+)

-i

FIG. 4. W'-plane representation of the locus of w'(s) for a = 0.

By stereographic projection of the Poincar6 sphere onto itstangent plane at the point representing linear polarizationinclined at 450 (or appropriate bilinear transformation of w),we obtain the w' representation wherein w A = +1, as shownin Fig. 4.7 Any initial polarization w0 evolves toward w?- withthe scale length

st = pI ReRn 2- 1)1/2]1-1. (47)

B. a -/- 0We are now dealing with twisted trajectories.

Together with Eq. (41), the latter condition leads to w+ +we = 0. This means that in order for the eigenpolarizationsto be orthogonal w+ must be purely imaginary, i.e., the axesof the corresponding ellipses must be parallel and perpen-dicular to the interface.' 0

IV. PARTICULAR CASES

As stated before, we assume that a- is a positive constant,although r and p may vary with respect to s.

A. c=OExcluding the trivial cases p = 0, these curves are two-

dimensional and the results of our analysis reduce to those

already established for such trajectories." 2 The limiting formof Eq. (38) (see Appendix B) shows that

w- = 0

and (44)

i.e., the eigenstates are orthogonal linearly polarized TE andTM waves. Their respective attenuation constants are givenby

a(- = 2p-1 Re[(n 2 - 1)-1/2] = aTE (45)

and

1. n real (and positive)For n = 1 there is no interface to speak of, and this partic-

ular value separates two ranges of n with different charac-teristics.

a. O<n<1A representative example of this category is that of light

guided in glass along a glass-air interface (ni 1.5, n2 1).From Eqs. (20) and (21) we see that both A and B are purelyimaginary. Letting A = iAi, where Ai is real, we then haveB = i (A? + 1)1/2, so that

wv = i[Ai + (A? + 1)1/21, (48)

which implies that the eigenpolarizations are orthogonal. Infact, it can be shown that 0 < n < 1 is a necessary and suffi-cient condition for orthogonality of the eigenpolarizations.Equation (27) yields ao, = 0, as expected for total internalreflection. Since there are no losses, we have a situation wherew± is never reached unless wo = w+; this is shown in Fig. 5,where the w representation has been used.

b. n>1This would, for example, correspond to the case of light

guided in air along an air-glass interface (n1 1, n2 1.5).A is always real and we denote it by Ar, but there are twodistinct regimes separated by the condition Ar = 1, or a- = a,

where

a+ = 2p-1 Re[n2 (n2 - 1)-1/2] = aTM. (46) c = 2'-(n 2- 1)1/2.

M. E. Marhic 12221222 J. Opt. Soc. Am., Vol. 69, No. 9, September 1979

(49)

w+ = X0,

Im (wi

AI >> 1.

Re (w

1

This may, for example, be the case for infrared or far-in-frared radiation propagating inside metallic guides; underthose circumstances I n J may be of the order of 100, and therewill, therefore, be a wide range of values of a for which Eq. (53)will be satisfied. When that condition holds, we can use A-1as a small parameter to expand w± and a+. Keeping the firstsignificant terms, we find that

w+J 21A1 >> 1, jw-j 1/21A1 << 1,

o+ 2p-'Re[n 2 (n 2- 1)-/ 2 ] = aTM,

andFIG. 5. w-plane representation of the locus of w(s) for a- = 0 and 0 < n< 1. Numerical values are n = (2)-1f2, a = 1.

(i). 0 < a < a,. Here B = (A? -1)/2 is real, and Eq. (38)shows that w± is real as well; i.e., both eigenpolarizations arelinear, making an angle of magnitude Itan-'[Ar + (Ar - 1)1/211with respect to the plane of incidence and the tangent plane,respectively, i.e., generally not orthogonal. For a = 0 we re-cover the orthogonal TE and TM states, and for a = a, theeigenpolarizations become identical, aligned with 6 + P. Theattenuation constants are finite and in general different: fora- = 0 they yield aTE and aTM which can be very different ifn is large, but for a = a, the two eigenpolarizations are at-tenuated by exactly the same amount. The polarization wproceeds towards w- along an arc of circle. Equation (42)becomes

St = pI2(o2 - a2

)1

/2, (50)

which shows that it takes a longer distance to reach w- as aapproaches a,.

(ii) a > Ge. Now, B = i(1 - A2)1/2 is purely imaginary.Since A is real,

(51)

is not purely imaginary and the eigenpolarizations are ellip-tical, with opposite handedness, but not orthogonal. Theattenuation constants are given by

(52)

which shows that both eigenpolarizations are attenuated bythe same amount; furthermore, the common attenuationconstant depends only on p and not on a itself. As a -> a, A,

- 0 and wa - ±i; i.e., both eigenstates become circularlypolarized and orthogonal. Even for finite a, B is purelyimaginary (6 = iw/2) and w keeps rotating around one of theeigenpolarizations without ever reaching it.

Figure 6 summarizes some of the results concerning the casewhere n is real. The same w' representation as in Fig. 4 isused. The two segments of the real and imginary axes limited

1223 J. Opt. Soc. Am., Vol. 69, No. 9, September 1979

a_ - (1 + a2 )2pr1Re[(n 2 - )-1/2] = (1 + a 2 )aTE. (56)

This shows that the (+) eigenpolarization is almost TM, andits attenuation constant is very nearly that of TM waves (alonga two-dimensional trajectory with the same p). The (-) ei-genpolarization is almost TE, however, its attenuation con-stant is about (1 + a2 ) times higher than that of the low-lossTE waves, provided, of course, a is not so large as to violateEq. (53). This latter conclusion is important since it showsthat the minimum losses of metallic waveguides for the in-frared and far-infrared regions of the spectrum will increasedrastically as a approaches and exceeds unity (for a fixed p),and this fact will have to be taken into account when designingwaveguides of this type.

Im (WI

WI WI.

CrTO < C =R- C R e(w')

-1 w.1 0 W' I

W-2

FIG. 6. w/-plane representation of the loci of WI and w'(s) for 0 • a <0¾

M. E. Marhic 1223

by the unit circle are the locus of wi for all values of a. Thecircular trajectories of w' resulting from a arbitrary w0 areshown for a, < or and a2 > ac, with corresponding eigenpo-larizations w1, and w%2.

2. n largeWe are now restricting our attention to the situation where

n (generally complex) and a are such that

-i

(53)

(54)

(55)

w_ = A, + i(1 - A?1/2

a+ = pr(n 2+ 1)(n 2

-)-1/2

X = 27rb

FIG. 7. Right-handed circular helix.

V. EXPERIMENTS

In order to verify the predictions of the preceding theoryin an arbitrary situation, it is necessary to have an accurateknowledge of n, p, and r along the trajectory of interest.Furthermore, in most situations one must be able to preciselycharacterize the input and output states of polarization byellipsometry, to verify that w either evolves according to ex-pectations for wo #d w±, or remains equal to w± if wo = w+.Also, measurements of at can in general be made easily onlyif the polarization is known to remain in either one of theeigenstates along the whole trajectory.

There is a particular case, however, where several featurespredicted by the theory can be verified without complicatedellipsometry. This is the situation studied in Sec. B l b (ii)where n > 1 and a > cc. We saw that under those circum-stances the two eigenpolarizations have the same attenuationconstant, and that for very large a they become circularlypolarized and orthogonal. When the eigenpolarizations areorthogonal and have the same attenuation constant, it followsfrom Eq. (29) that any input polarization will decay with thesame attenuation constant as well. Furthermore, if conditionsare such that the relative phase change between two eigen-polarizations is a multiple of 27r, then the output polarizationwill be identical to the input polarization.

We verified these predictions along a trajectory in the shapeof a circular helix, obtained by directing a He-Ne laser beamtangentially onto the inner surface of a soda-lime glass tube(n2 1.52), as shown in Fig. 7. By an appropriate choice ofthe coordinate system, the parametric equations of the circularhelix can be written in the form

x = acost, y = asint, z = bt, (57)

where a (1.664 cm) is the radius of the cylinder, 27rb the pitchof the helix, and t the angle indicated on Fig. 7. Circular he-lixes are the simplest three-dimensional curves, having con-stant radius of curvature and torsion given by

p = (a2 + b2 )/a

1224 J. Opt. Soc. Am., Vol. 69, No. 9, September 1979

and

T = b/(a2 + VA),

respectively, so that

a = b/a.

The length s along the trajectory is related to t by

s = (a2 + b 2 )'/ 2 t.

(58)

(59)

(60)

In the experiments, we adjusted the input beam to have t= 27r at the exit of the cylinder. Under those circumstances,it can be shown that the transmittance T(27r) of the guide isgiven by

T(27r)27 = x n2 + 1= exp (1 + 1

2/47

2a

2)1/2 (n 2 - 1)1/2 (61)

Strictly speaking, this expression is valid only for the ei-genpolarizations, but, as discussed earlier, should apply wellto any input polarization if a» >> a.

The eigenpolarizations being circular for a >> ac and havingthe same attenuation coefficient, it follows that if linearlypolarized light is injected into the guide, it will emerge linearlypolarized as well, although possibly with a different orienta-tion determined by the phase shift between the eigenpolari-zations. This phase shift is given by

A = (+r - )S, (62)

which for a >> a, reduces to A\ 2t; for the trajectory of Fig.7, A 47r, so that the input and output polarizations shouldbe practically identical.

A large value of a was obtained by using a long glass tube.For 1 = 122 cm we had a = 11.67, compared to a, = 0.57, en-suring that the approximating considerations on polarizationand losses should apply very well. We first checked theconclusion that the input and output polarizations should beidentical. Rotating a linear polarizer in the input beam, wefound it necessary to rotate a second linear polarizer in theoutput beam by the same amount in order to extinguish thelight, in good qualitative agreement with the theoreticalprediction.

The power transmittance was measured for both TE andTM input polarizations and was found to be 18.4% and 17.9%,respectively, to be compared with a theoretical value T(27r)= 21%. Considering unknown sources of error, such as scat-tering from the surface along the trajectory, the agreementbetween theoretical and experimental values is quite good.The agreement between the values for the two different po-larizations, presumably affected to a similar extent by scat-tering, is even better, and indicates that the losses are indeedindependent of polarization for a» >> a, being equal, in par-ticular, to the losses for the eigenpolarizations.

Attempts were made to check the theoretical predictions,especially Eq. (56), for CO2 laser radiation (10.6-,gm wave-length) propagating along metallic guides, but it was founddifficult to fabricate guides with accurately known values ofn and having a shape appropriate to compensate for the strongdiffraction of the infrared light. A narrow flat metallic stripof well-known n was twisted into helical shapes, but was thenfound unable to contain the laterally spreading beam. On theother hand, appropriately confining helical guides were made

M. E. Marhic 1224

out of commercially available copper tubing,3 but a coatingon the surface reduced the index so much as to make Eq. (56)inapplicable for interesting values of a, say a Ž 1; flatteningthe surface to measure n by ellipsometry modified it so muchas to render the measurements meaningless.

VI. CONCLUSION

We have shown that eigenstates of polarization exist forwhispering-gallery waves following general helixes along in-terfaces and have calculated the complex propagation andpower attenuation constants associated with these eigenpo-larizations. Simple experiments carried out with a cylindricalair-glass interface verified a number of theoretical conclusionswith good quantitative agreement, rendering it likely that allpredictions based on this theory should be valid, providedaccurately known values of n, p, and r be used.

Of particular interest is the conclusion that the smallestattenuation constant for twisted trajectories in metallicwaveguides for the infrared and far-infrared exceeds the op-timum attenuation constant aTE of the corresponding two-dimensional trajectories by the factor (1 + a2

). Quantitativeverification of this prediction will have to await the develop-ment of metallic guides with high-quality surfaces and crosssections appropriate to confine beam spread, but, if borne out,it should constitute an important design criterion for flexiblelaser delivery systems in the infrared and far-infrared regionsof the electromagnetic spectrum.

APPENDIX A. LOCUS OF THE POINTREPRESENTING THE POLARIZATION

Consider first the w representation. The locus of w(s) issuch that

dw= T(W -W+)(W -W-) (63)

ds

or

W-W+ We-W+-= e2Bs(s). (39)w-w- wo-w-

The circle passing through w, w+, and w-, as seen in Fig. 8,can be represented by the equation

wc.-W+ W -W- - e 2 T(sc), (64)

wc-.w_ w-w-

with s, = 0 for w0 = w, which leads to the differential equa-tion

dwc w -w+ (dw, 2 ( 2~d w e2T() + (wC - w_)2Te2r(sc)ds, w - we-

(65)

For wc w this expression reduces to

dwc I Z= (w - w+)(w - w-), (66)dsc 1o w+ - w-

so that

dw /dwu¾ - w- B.ds s/dsl 2

(67)

dwHdscW

w_ W+s

FIG. 8. Trajectory wis) and circles wc(s.)

Since (dwlds) 1 and (dwc/ds)c I o represent the directionsof the tangents to the locus of w (s) and the circle passingthrough w, w+, and w-, respectively, we see that the formermakes an angle ( with the latter equal to the argument of thecomplex number B. This holds for any location of w(s) and,therefore, its trajectory has the property of intersecting all thearcs of circle joining w+ to w - which it encounters under thesame angle 5; furthermore, the same holds true for all trajec-tories obtained by varying the initial polarization wo.

Consider now any other complex representation w" of thepolarization obtained from w by an appropriate bilineartransformation. Such a transformation preserves circles aswell as angles and, therefore, a trajectory of wT (s) will alsointersect the arcs of circles joining w- to w+ with the sameangle 6.

This invariance can be viewed as a consequence of the factthat the equivalent complex representations of polarizationare obtained by stereographic projections of the Poincar6sphere onto its tangent planes. The inverse stereographicprojection from the w plane being conformal and preservingcircles as well, it follows that the angular relationship betweenthe trajectory of the point representing the polarization andthe circles passing through the eigenpolarization holds trueon the Poincar6 sphere itself, and by stereographic projectiononto any tangent plane, in any wa' representation as well.

If the eigenpolarizations are orthogonal, i.e., diametricallyopposed on the Poincar6 sphere, then the loci of the polar-ization on that sphere are loxodromes of the sphere (the ei-genpolarizations being viewed as the poles). Even if the ei-genpolarizations are not orthogonal, the w"(s) trajectories canalways be transformed into equiangular spirals by bilineartransformations which send either w + or w?- to infinity; theyare, therefore, plane loxodromes.

APPENDIX B. SIGN CONVENTIONS FORSQUARE ROOTS

In the text we have made certain assumptions about thesigns associated with the quantities (n2 - 1)1/2 and (n2 - I -4a

2)

11 2 which we will now justify. It is clear that the signchosen for (n2

- 1)1/2 should be such as to lead to attenuationand not to gain for the TE and TM waves for a = 0, and thisleads to the condition,

Re[(n 2- 1)1/2] > 0 (68)

The uncertainty on the sign of (n2 - - 4

2)1/

2 is less im-

portant, as both signs must be considered on the same footing.

1225 J. Opt. Soc. Am., Vol. 69, No. 9, September 1979 M. E. Marhic 1225

It is natural, however, to adopt the sign convention whichreduces to the previous one for aF = 0, i.e., to require that

Rej(n 2 - 1 - 4a2)1/

21 > 0. (69)

These sign conventions have been adhered to in the deri-vations of Eqs. (44)-(46), (55), and (56), and in the discussionsof the various complex-plane representations.

For Re[(n 2 - 1)1/2] = 0 or Re[(n2

- 1 - 4a2

)1/2] = 0, the sign

convention for the remaining imaginary parts is arbitrary, andone can in fact arrive at either choice by continuity from caseswhere the real parts do not vanish. Since the labeling in thiscase is of no consequence, any convenient choice can bemade.

'E. Garmire, T. McMahon, and M. Bass, "Low-loss optical trans-migsion through bent hollow metal waveguides," Appl. Phys. Lett.31, 92-94 (1977).

2H. Krammer, "Light waves guided by a single curved metallic sur-face," Appl. Opt. 17, 316-319 (1978).

3M. E. Marhic, L. I. Kwan, and M. Epstein, "Optical surface wavesalong a toroidal metallic guide," Appl. Phys. Lett. 33, 609-611(1078); "invariant properties of helical-circular metallic wave-guides," Appl. Phys. Lett; 33, 874-876 (1978).

4 E. Kreyszig, Introduction to Differential Geometry and RiemannianGeometry (University of Toronto, Toronto, 1968), p. 32.

5If ni is complex, an appropriate attenuation term should be intro-duced in what follows. We omit this term, together with a phaseterm, since they have no bearing on the eigenpolarizations, and ifneeded can easily be taken into account in the expressions of thepropagation and attenuation constants.

6Three-dimensional curves with a = const make a constant angle y= tan-kr with a fixed direction, and are called general helixes; Ref.4, p. 46.

7 R. M. A. Azzam and N. M. Bashara, Ellipsometry and PolarizedLight (North-Holland, Amsterdam, 1977), Chap. 1.

5H. S. M. Coxeter, Introduction to Geometry (Wiley, New York,1969), p. 88.

9Here, "to reach" should be understood in a physical rather than amathematical sense, meaning to come so close to the eigenpolari-zation as to be practically indistinguishable from it.

'°Another possibility is w = 0, w, which can be viewed as a limitingform of the general statement, since the point at infinity in the wplane belongs, in particular, to the imaginary axis.

Modes of dielectric waveguides of arbitrary cross sectionalshape

L. Eyges and P. GianinoRome Air Development Center, Deputy for Electronic Technology, Hanscom AFB, Massachusetts 01731

P. WintersteinerARCON Corporation, 260 Bear Hill Road, Waltham, Massachusetts 02154

(Received 22 January 1979)A technique is presented for determining the modal propagation properties of a homogeneous

cylindrical dielectric waveguide of arbitrary cross sectional shape and index n 1 embedded in amedium of index n2. Both the weakly guiding case in which n 1z n 2 and the general case of arbi-trary index difference are discussed theoretically. In both cases the approach is to derive integralrepresentations for appropriate components of E and B. These satisfy the appropriate Helmholtzequations inside and outside the guide and also guarantee that the boundary conditions are satisfied.On expansion of the components in certain sets of basis functions, the representations become a setof linear equations. The vanishing of the determinant of this set yields the propagation constants ofthe various mqdes. Numerical results are given for weakly guiding fibers of various shapes. Amongthese are rectangles and ellipses, which make comparisons with previous work possible.

INThODUCTIONWith the increased interest in fiber optic or integrated

optic communication techniques has come a need to know inincreasing detail the propagation properties of cylindricaldielectric waveguides.1- 3 In fact, the literature has so bur-geoned that it would be excessive to try to quote all the indi-viduail papers that discuss the subject. Happily, there are bynow several excellent books and review articles that make thisunnecessary.4' 5

The problem of electromagnetic propagation down a di-electric cylinder is closely related to the problem of scatteringof electromagnetic waves from the same cylinder. It is thennot surprising that methods used for the scattering problem6

have also been applied to the problem of propagation. Thus,point matching2 and ray tracing7 techniques have been ap-

plied. Despite much work, however, there is no one generalanalytic method that applies to single guides of arbitrarycross-sectional shape and to the coupling between two or moresuch guides.

In this paper we present such a method for single guides;its extension to coupled guides is in preparation. The methodis largely analytic although finally machine computation isrequired. It not only provides new techniques for irregularlyshaped guides, but also has the minor advantage of providingneat derivations of the standard formulas for single circularguides. The present method is an extension of recent newtechniques for solving the problem of scattering by irregularlyshaped dielectric bodies, and in the static limit, for solving theproblem of an irregular dielectric or permeable body in anexternal field.6' 8' 9 The general idea here is the same as in

1226 J. Opt. Soc. Am., Vol. 69, No. 9, September 1979 0030-3941/79/091226-10$00.50 i�� 1979 Optical Society of America 1226